import ContractExact
namespace SAct
open ExactRuntime FloatLib.Floats.Formats.BinaryInterchange FloatLib.Floats FloatLib.Numerics Real
abbrev F64:=FloatFormat.binary64
def dyM (n k:Nat):Model F64 :=
  Model.roundDyadic F64 ⟨true,n, -(k:Int)⟩
@[inline] def toF (m:Model F64):Float:=Float.ofModel (Model.toFloatModel m)
def tT:FloatArray :=
  ⟨Array.ofFn (n:=64) fun j=>toF (Model.Transcendentals.CorrectlyRounded.exp (dyM j.val 6))⟩
def tM:FloatArray :=
  ⟨Array.ofFn (n:=64) fun j=>toF (Model.Transcendentals.CorrectlyRounded.expMinus1 (dyM j.val 6))⟩
def tE:FloatArray :=
  ⟨Array.ofFn (n:=88) fun i=>toF (Model.Transcendentals.CorrectlyRounded.exp (dyM i.val 0))⟩
@[inline] def hq (r:Float):Float :=
  ((((Float.ofBits 0xbf56c16c16c16c17*r+Float.ofBits 0x3f81111111111111)*r
    + Float.ofBits 0xbfa5555555555555)*r+Float.ofBits 0x3fc5555555555555)*r
    + Float.ofBits 0xbfe0000000000000)*r+Float.ofBits 0x3ff0000000000000
@[inline] def rnd (Y:UInt64):UInt32 :=
  let ef:=Y>>>52
  let f:=Y&&&0xFFFFFFFFFFFFF
  if 897 ≤ ef&&ef ≤ 1022&&1024 ≤ f&&f < 0xFFFFFFFFFFC00 &&
      2048 < (Y&&&0x1FFFFFFF)-0xFFFFC00 then
    ((Y+0x10000000)>>>29).toUInt32-0xC0000000
  else 0xFFFFFFFF
@[inline] def kOf (ee m:UInt64):UInt64:=if 144-ee < 24 then m>>>(144-ee) else 0
@[inline] def rOf (ee m:UInt64):Float :=
  (if 144-ee < 24 then m&&&((1<<<(144-ee))-1) else m).toInt64.toFloat *
    Float.ofBits ((ee+873)<<<52)
@[inline] def sgF (w:UInt32):UInt32 :=
  let a:=w&&&0x7FFFFFFF
  if a < 0x00800000||0x7F800000 ≤ a then
    (if a==0 then 0x3F000000 else 0xFFFFFFFF)
  else if w < 0x80000000&&0x41900000 ≤ a then 0x3F800000
  else if 0x42AE0000 ≤ a then 0xFFFFFFFF
  else
    let ee:=(a>>>23).toUInt64
    let m:=((a&&&0x7FFFFF)|||0x800000).toUInt64
    let k:=kOf ee m
    let r:=rOf ee m
    let t:=(tE.get! (k>>>6).toNat*tT.get! (k&&&63).toNat) *
      (Float.ofBits 0x3ff0000000000000-r*hq r)
    let d:=Float.ofBits 0x3ff0000000000000+t
    let y:=if w < 0x80000000 then Float.ofBits 0x3ff0000000000000/d else t/d
    rnd y.toBits
@[inline] def thF (w:UInt32):UInt32 :=
  let a:=w&&&0x7FFFFFFF
  if a < 0x00800000||0x7F800000 ≤ a then
    (if a==0 then w else 0xFFFFFFFF)
  else if 0x41180000 ≤ a then (w&&&0x80000000)+0x3F800000
  else
    let ee:=(a>>>23).toUInt64+1
    let m:=((a&&&0x7FFFFF)|||0x800000).toUInt64
    let k:=kOf ee m
    let r:=rOf ee m
    let y:=if k < 64 then
        let nu:=tT.get! k.toNat*(r*hq r)-tM.get! k.toNat
        nu/(Float.ofBits 0x4000000000000000-nu)
      else
        let t:=(tE.get! (k>>>6).toNat*tT.get! (k&&&63).toNat) *
          (Float.ofBits 0x3ff0000000000000-r*hq r)
        (Float.ofBits 0x3ff0000000000000-t)/(Float.ofBits 0x3ff0000000000000+t)
    let o:=rnd y.toBits
    if o==0xFFFFFFFF then o else o+(w&&&0x80000000)
def sg (w:UInt32):UInt32:=let o:=sgF w; if o==0xFFFFFFFF then sigmoidCR w else o
def th (w:UInt32):UInt32:=let o:=thF w; if o==0xFFFFFFFF then tanhCR w else o
noncomputable abbrev one:Float:=Float.ofBits 0x3ff0000000000000
noncomputable abbrev two:Float:=Float.ofBits 0x4000000000000000
noncomputable def rv (x:Float):ℝ:=Model.toReal (Model.ofFloatModel x.toModel)
def fv (x:Float):Prop:=Model.isFinite (Model.ofFloatModel x.toModel)=true
noncomputable def eW (w:UInt32):UInt64:=((w&&&0x7FFFFFFF)>>>23).toUInt64
noncomputable def mW (w:UInt32):UInt64:=((w&&&0x7FFFFFFF&&&0x7FFFFF)|||0x800000).toUInt64
noncomputable def cv (ee m:UInt64):ℝ:=(m.toNat:ℝ)*2^((ee.toNat:ℤ)-150)
noncomputable def eT (k:UInt64) (r:Float):Float :=
  (tE.get! (k>>>6).toNat*tT.get! (k&&&63).toNat)*(one-r*hq r)
noncomputable def thM (ee m:UInt64):Float :=
  let k:=kOf ee m
  let r:=rOf ee m
  if k < 64 then
    let nu:=tT.get! k.toNat*(r*hq r)-tM.get! k.toNat
    nu/(two-nu)
  else (one-eT k r)/(one+eT k r)
noncomputable def thY (w:UInt32):Float:=thM (eW w+1) (mW w)
noncomputable def sgY (w:UInt32):Float :=
  let t:=eT (kOf (eW w) (mW w)) (rOf (eW w) (mW w))
  if w < 0x80000000 then one/(one+t)
  else t/(one+t)
theorem zp (k:ℕ):(2:ℝ)^(-(k:ℤ))=1/2^k:=by rw [zpow_neg,zpow_natCast,one_div]
theorem fv_iff (x:Float):fv x ↔ x.isFinite=true:=by
  rw [fv, ← Model.toDyadic?_isSome_eq_isFinite,Model.toDyadic?_ieee_eq_model _ (by decide),
    Model.toModel_ofFloatModel]
  show _ ↔ x.toModel.unpack.isFinite=true
  cases x.toModel.unpack <;> rfl
theorem pmax:(2:ℝ)^(60:ℕ) ≤ Model.toReal (Model.posMaxFinite F64):=by
  have h:=Model.abs_signed_mul_bpow_le_toReal_posMaxFinite F64 false 1 60 (by decide) (by decide)
  simp only [Bool.false_eq_true,ite_false,Nat.cast_one,one_mul] at h
  change |(2:ℝ)^(60:ℤ)| ≤ _ at h
  rw [abs_of_pos (by positivity)] at h
  exact_mod_cast h
theorem p4:(4:ℝ) ≤ Model.toReal (Model.posMaxFinite F64):=le_trans (by norm_num) pmax
theorem opF {z:Float} {M:Model F64} {v:ℝ} (e:Model.ofFloatModel z.toModel=M)
    (hf:Model.isFinite M=true) (hv:Model.toReal M=Model.roundAt F64 v) :
    fv z∧rv z=Model.roundAt F64 v:=⟨by unfold fv; rw [e]; exact hf,by unfold rv; rw [e]; exact hv⟩
theorem mulF (x y:Float) (hx:fv x) (hy:fv y) (hb : |rv x*rv y| ≤ 4) :
    fv (x*y)∧rv (x*y)=Model.roundAt F64 (rv x*rv y):=by
  have hm : |rv x| * |rv y| ≤ Model.toReal (Model.posMaxFinite F64):=by rw [← abs_mul]; exact hb.trans p4
  exact opF (z:=x*y) (ExecFloat.Binary.toModel_ofFloat_mul_of_isFinite x y ((fv_iff x).1 hx) ((fv_iff y).1 hy))
    (Model.isFinite_mul_of_abs_mul_le_posMaxFinite _ _ (by decide) hx hy hm)
    (Model.toReal_mul_eq_roundAt_of_abs_mul_le_posMaxFinite _ _ (by decide) hx hy hm)
theorem addF (x y:Float) (hx:fv x) (hy:fv y) (h1 : |rv x| ≤ 2) (h2 : |rv y| ≤ 2)
    (_ : |rv x+rv y| ≤ 4):fv (x+y)∧rv (x+y)=Model.roundAt F64 (rv x+rv y):=by
  have hm : |rv x| + |rv y| ≤ Model.toReal (Model.posMaxFinite F64):=le_trans (by linarith) p4
  exact opF (z:=x+y) (ExecFloat.Binary.toModel_ofFloat_add_of_isFinite x y ((fv_iff x).1 hx) ((fv_iff y).1 hy))
    (Model.isFinite_add_of_abs_add_le_posMaxFinite _ _ (by decide) hx hy hm)
    (Model.toReal_add_eq_roundAt_of_abs_add_le_posMaxFinite _ _ (by decide) hx hy hm)
theorem subF (x y:Float) (hx:fv x) (hy:fv y) (h1 : |rv x| ≤ 2) (h2 : |rv y| ≤ 2)
    (_ : |rv x-rv y| ≤ 4):fv (x-y)∧rv (x-y)=Model.roundAt F64 (rv x-rv y):=by
  have hm : |rv x| + |rv y| ≤ Model.toReal (Model.posMaxFinite F64):=le_trans (by linarith) p4
  exact opF (z:=x-y) (ExecFloat.Binary.toModel_ofFloat_sub_of_isFinite x y ((fv_iff x).1 hx) ((fv_iff y).1 hy))
    (Model.isFinite_sub_of_abs_add_le_posMaxFinite _ _ (by decide) hx hy hm)
    (Model.toReal_sub_eq_roundAt_of_abs_add_le_posMaxFinite _ _ (by decide) hx hy hm)
theorem divF (x y:Float) (hx:fv x) (hy:fv y) (h0:rv y≠0) (hb : |rv x/rv y| ≤ 4) :
    fv (x/y)∧rv (x/y)=Model.roundAt F64 (rv x/rv y):=by
  have hz:Model.isZero (Model.ofFloatModel y.toModel)=false:=by
    cases hz:Model.isZero (Model.ofFloatModel y.toModel)
    · rfl
    · exact absurd (Model.toReal_eq_zero_of_isZero _ hz) h0
  exact opF (z:=x/y) (ExecFloat.Binary.toModel_ofFloat_div_of_isFinite x y ((fv_iff x).1 hx) ((fv_iff y).1 hy))
    (Model.isFinite_div_of_abs_div_le_posMaxFinite _ _ (by decide) hx hy hz (hb.trans p4))
    (Model.toReal_div_eq_roundAt_of_abs_div_le_posMaxFinite _ _ (by decide) hx hy hz (hb.trans p4))
theorem inv2le {a b:ℕ} (h:a ≤ b):(1:ℝ)/2^b ≤ 1/2^a :=
  one_div_le_one_div_of_le (by positivity) (pow_le_pow_right₀ (by norm_num) h)
theorem rndA (x:ℝ) :
    |Model.roundAt F64 x-x| ≤ 1/2^53 * |x| + 1/2^256:=by
  have h:=Model.abs_roundAt_sub_le_unitRoundoffAt F64 x
  rw [Model.unitRoundoffAt_eq,Model.underflowErrorAt_eq] at h
  have e1:(2:ℝ)^(-((F64.fracWidth:ℤ)+1))=2^(-((53:ℕ):ℤ)):=by
    congr 1
  have e2:(2:ℝ)^(FloatFormat.minSubnormalExponent F64-1) =
      2^(-((1075:ℕ):ℤ)):=by
    congr 1
  rw [e1,e2,zp,zp] at h
  exact h.trans (add_le_add le_rfl (inv2le (a:=256) (b:=1075) (by norm_num)))
def Rel (a b δ:ℝ):Prop := |a-b| ≤ δ * |b|
def Ap (x:Float) (X d:ℝ):Prop:=fv x∧Rel (rv x) X d
theorem Rel.mono {a b d e:ℝ} (h:Rel a b d) (hde:d ≤ e):Rel a b e :=
  h.trans (mul_le_mul_of_nonneg_right hde (abs_nonneg _))
theorem Rel.le {a b d:ℝ} (h:Rel a b d) : |a| ≤ (1+d) * |b| := by
  have:=abs_sub_abs_le_abs_sub a b; unfold Rel at h; linarith
theorem Rel.ge {a b d:ℝ} (h:Rel a b d):(1-d) * |b| ≤ |a| := by
  have:=abs_sub_abs_le_abs_sub b a; rw [abs_sub_comm] at this; unfold Rel at h; linarith
theorem Rel.zero {a b d:ℝ} (h:Rel a b d) (hb:b=0):a=0:=by
  unfold Rel at h; rw [hb,abs_zero,mul_zero,sub_zero] at h; exact abs_nonpos_iff.1 h
theorem Rel.trans {a b c d e:ℝ} (h1:Rel a b d) (h2:Rel b c e) (hd:0 ≤ d:=by norm_num) :
    Rel a c (d+e+d*e):=by
  have hb:=h2.le
  unfold Rel at *
  calc |a-c| ≤ |a-b| + |b-c| := abs_sub_le a b c
    _ ≤ d * |b| + e * |c| := add_le_add h1 h2
    _ ≤ d*((1+e) * |c|)+e * |c| := by gcongr
    _=(d+e+d*e) * |c| := by ring
theorem Rel.mul {a1 b1 a2 b2 d1 d2:ℝ} (h1:Rel a1 b1 d1) (h2:Rel a2 b2 d2) (hd1:0 ≤ d1:=by norm_num) :
    Rel (a1*a2) (b1*b2) (d1+d2+d1*d2):=by
  have ha2:=h2.le
  unfold Rel at *
  rw [abs_mul b1 b2]
  calc |a1*a2-b1*b2| = |(a1-b1)*a2+b1*(a2-b2)| := by ring_nf
    _ ≤ |a1-b1| * |a2| + |b1| * |a2-b2| := by
        rw [← abs_mul, ← abs_mul]; exact abs_add_le _ _
    _ ≤ (d1 * |b1|)*((1+d2) * |b2|) + |b1| * (d2 * |b2|):=by
        apply add_le_add
        · exact mul_le_mul h1 ha2 (abs_nonneg _) (mul_nonneg hd1 (abs_nonneg _))
        · exact mul_le_mul_of_nonneg_left h2 (abs_nonneg _)
    _=(d1+d2+d1*d2)*(|b1| * |b2|):=by ring
theorem Rel.div {a1 b1 a2 b2 d1 d2:ℝ} (h1:Rel a1 b1 d1) (h2:Rel a2 b2 d2) (hb2:b2≠0)
    (hd1:0 ≤ d1:=by norm_num) (hd2:0 ≤ d2:=by norm_num) (hd2':d2 < 1:=by norm_num) :
    Rel (a1/a2) (b1/b2) ((d1+d2)/(1-d2)):=by
  have hB:0 < |b2| := abs_pos.2 hb2
  have ha2:=h2.ge
  have hA:0 < |a2| := lt_of_lt_of_le (mul_pos (by linarith) hB) ha2
  unfold Rel at *
  rw [div_sub_div _ _ (abs_pos.1 hA) hb2,abs_div,abs_mul,abs_div]
  have hnum : |a1*b2-a2*b1| ≤ (d1+d2) * |b1| * |b2| := by
    calc |a1*b2-a2*b1| = |(a1-b1)*b2-b1*(a2-b2)| := by ring_nf
      _ ≤ |a1-b1| * |b2| + |b1| * |a2-b2| := by rw [← abs_mul, ← abs_mul]; exact abs_sub _ _
      _ ≤ d1 * |b1| * |b2| + |b1| * (d2 * |b2|):=by
          apply add_le_add
          · exact mul_le_mul_of_nonneg_right h1 (abs_nonneg _)
          · exact mul_le_mul_of_nonneg_left h2 (abs_nonneg _)
      _=(d1+d2) * |b1| * |b2| := by ring
  rw [div_le_iff₀ (mul_pos hA hB)]
  have h1d:0 < 1-d2:=by linarith
  calc |a1*b2-a2*b1| ≤ (d1+d2) * |b1| * |b2| := hnum
    _=(d1+d2)/(1-d2)*(|b1| / |b2|)*((1-d2) * |b2| * |b2|):=by
        field_simp
    _ ≤ (d1+d2)/(1-d2)*(|b1| / |b2|)*(|a2| * |b2|):=by
        apply mul_le_mul_of_nonneg_left _ (by positivity)
        exact mul_le_mul_of_nonneg_right ha2 (abs_nonneg _)
theorem Rel.add {a1 b1 a2 b2 d:ℝ} (h1:Rel a1 b1 d) (h2:Rel a2 b2 d) (hb1:0 ≤ b1)
    (hb2:0 ≤ b2):Rel (a1+a2) (b1+b2) d:=by
  unfold Rel at *
  rw [abs_of_nonneg hb1] at h1; rw [abs_of_nonneg hb2] at h2
  rw [abs_of_nonneg (add_nonneg hb1 hb2)]
  calc |a1+a2-(b1+b2)| = |(a1-b1)+(a2-b2)| := by ring_nf
    _ ≤ |a1-b1| + |a2-b2| := abs_add_le _ _
    _ ≤ d*b1+d*b2:=add_le_add h1 h2
    _=d*(b1+b2):=by ring
theorem rnd52 {x:ℝ} (hx:1/2^203 ≤ |x|) :
    Rel (Model.roundAt F64 x) x (1/2^52):=by
  unfold Rel; linarith [rndA x]
theorem rndR {v b d:ℝ} (h:Rel v b d) (hd:d ≤ 1/2) (hb:b=0∨1/2^202 ≤ |b|) :
    Rel (Model.roundAt F64 v) b (d+1/2^51):=by
  rcases hb with hb | hb
  · rw [h.zero hb,Model.roundAt_zero,hb]; unfold Rel; simp
  · have h1:1/2 * |b| ≤ (1-d) * |b| := mul_le_mul_of_nonneg_right (by linarith) (abs_nonneg _)
    exact ((rnd52 (by linarith [h.ge])).trans h).mono (by linarith)
theorem opR {z:Float} {v b d:ℝ}
    (hz : |v| ≤ 4 → fv z∧rv z=Model.roundAt F64 v)
    (h:Rel v b d) (hb:b=0∨1/2^202 ≤ b) (hB:b ≤ 2) (h0:0 ≤ b:=by positivity)
    (hd:d ≤ 1/2:=by norm_num):Ap z b (d+1/2^51):=by
  rw [← abs_of_nonneg h0] at hB
  have h1:(1+d) * |b| ≤ 3/2*2:=mul_le_mul (by linarith) hB (abs_nonneg _) (by norm_num)
  obtain ⟨f,e⟩:=hz (by linarith [h.le])
  exact ⟨f,by rw [e]; exact rndR h hd (by rwa [abs_of_nonneg h0])⟩
theorem sgnW (u:UInt64):(u&&&0x8000000000000000 != 0)=(u>>>63 != 0):=by
  apply Bool.eq_iff_iff.2
  simp only [bne_iff_ne,ne_eq, ← UInt64.toNat_inj]
  rw [UInt64.toNat_and,UInt64.toNat_shiftRight]
  change ¬(UInt64.toNat u)&&&2^(63:ℕ)=0 ↔ ¬(UInt64.toNat u)>>>63=0
  rw [Nat.and_two_pow,Nat.shiftRight_eq_div_pow,Nat.testBit_eq_decide_div_mod_eq]
  have d:(UInt64.toNat u)/2^(63:ℕ) < 2:=Nat.div_lt_of_lt_mul u.toNat_lt
  rw [Nat.mod_eq_of_lt d]
  generalize (UInt64.toNat u)/2^(63:ℕ)=b at d ⊢
  interval_cases b <;> simp
theorem fldW (a:UInt64):Model.ofModelBits (fmt:=F64) a.toBitVec =
    Model.ofFields F64 (a>>>63 != 0) ((a>>>52)&&&2047).toNat
      (a&&&0xFFFFFFFFFFFFF).toNat:=by
  have d:Model.expField (Model.ofModelBits (fmt:=F64) a.toBitVec) =
      ((a>>>52)&&&2047).toNat:=rfl
  have e:Model.fracField (Model.ofModelBits (fmt:=F64) a.toBitVec) =
      (a&&&0xFFFFFFFFFFFFF).toNat:=rfl
  have g:Model.signBit (Model.ofModelBits (fmt:=F64) a.toBitVec) =
      (a>>>63 != 0):=by
    rw [← sgnW]; apply Bool.eq_iff_iff.mpr; simp only [Model.signBit,bne_iff_ne,ne_eq]
    rw [← UInt64.toBitVec_inj]; exact Iff.rfl
  rw [← Model.ofFields_signBit_expField_fracField (Model.ofModelBits (fmt:=F64) a.toBitVec),d,e,g]
theorem valW (a:UInt64) (h0:((a>>>52)&&&2047).toNat≠0)
    (h1:((a>>>52)&&&2047).toNat < 2047) :
    Model.isFinite (Model.ofModelBits (fmt:=F64) a.toBitVec)=true ∧
    Model.toReal (Model.ofModelBits (fmt:=F64) a.toBitVec) =
      (if (a>>>63 != 0) then (-1:ℝ) else 1)*((2^52+(a&&&0xFFFFFFFFFFFFF).toNat:ℕ):ℝ) *
        (2:ℝ)^((((a>>>52)&&&2047).toNat:ℤ)-1075):=by
  rw [fldW]
  have hf:(a&&&0xFFFFFFFFFFFFF).toNat < 2^F64.fracWidth:=by
    rw [UInt64.toNat_and]; exact lt_of_le_of_lt Nat.and_le_right (by decide)
  have he:((a>>>52)&&&2047).toNat < 2^F64.expWidth:=by
    show _ < 2^11; omega
  refine ⟨?_,?_⟩
  · rw [Model.isFinite_ofFields _ _ _ _ he hf]
    show (((a>>>52)&&&2047).toNat != 2047)=true
    simp only [bne_iff_ne,ne_eq]; omega
  · rw [Model.toReal_ofFields_normal _ _ _ _ h0 (by show _ < 2047; exact h1) hf]
    simp only [show F64.fracWidth=52 from rfl,
      show F64.exponentBias=1023 from rfl,Model.pow2_eq_two_pow]
    change _*(2:ℝ)^_=_
    congr 2
    simp only [Int.ofNat_eq_natCast]; ring
theorem ofBitsM (a:UInt64) (hv:¬ (a>>>52)&&&2047=2047) :
    Model.ofFloatModel (Float.ofBits a).toModel=Model.ofModelBits (fmt:=F64) a.toBitVec:=by
  have e1:Model.ofFloatModel (Float.ofBits a).toModel =
      Model.canonicalizeModel (Model.ofModelBits (fmt:=F64) a.toBitVec):=by
    show Model.ofFloatModel (Float.Model.ofBits a)=_
    unfold Float.Model.ofBits; exact Model.ofFloatModel_pack _
  rw [e1]; apply Model.canonicalizeModel_eq_self
  refine ⟨fun hE _=>absurd hE ?_⟩
  intro hE; apply hv
  have:=congrArg BitVec.toNat hE
  have k1:F64.toModel.exponentBits=11:=rfl
  have k2:F64.toModel.mantissaBitsWithoutImplicit=52:=rfl
  simp [Float.Model.UnpackedFloat.unpackExponent,Model.toModelBits,Model.ofModelBits,k1,k2] at this
  change (BitVec.extractLsb' 52 11 (a.toBitVec:BitVec 64)).toNat=2047 at this
  rw [BitVec.extractLsb'_toNat] at this
  apply UInt64.toNat_inj.mp
  simp only [UInt64.toNat_and,UInt64.toNat_shiftRight,UInt64.reduceToNat]
  rw [show (2047:ℕ)=2^(11:ℕ)-1 from rfl,Nat.and_two_pow_sub_one_eq_mod]
  simp only [UInt64.toNat_toBitVec,Nat.shiftRight_eq_div_pow] at this ⊢
  have:=a.toNat_lt; omega
theorem rvB (a:UInt64) (h0:((a>>>52)&&&2047).toNat≠0)
    (h1:((a>>>52)&&&2047).toNat < 2047) :
    fv (Float.ofBits a)∧rv (Float.ofBits a) =
      (if (a>>>63 != 0) then (-1:ℝ) else 1)*((2^52+(a&&&0xFFFFFFFFFFFFF).toNat:ℕ):ℝ) *
        (2:ℝ)^((((a>>>52)&&&2047).toNat:ℤ)-1075):=by
  have hv:¬ (a>>>52)&&&2047=2047:=fun h=>by rw [h] at h1; simp at h1
  have ⟨f,v⟩:=valW a h0 h1
  exact ⟨by unfold fv; rw [ofBitsM a hv]; exact f,by unfold rv; rw [ofBitsM a hv]; exact v⟩
theorem modelW (y:Float):Model.ofFloatModel y.toModel =
    Model.ofModelBits (fmt:=F64) y.toBits.toBitVec:=rfl
theorem rndExact (d:FloatLib.Numerics.Dyadic) (l:d.significand < 2^(53:ℕ)) (g : -1074 ≤ d.exponent)
    (e : |d.toReal| ≤ 4):Model.roundAt F64 d.toReal=d.toReal:=by
  have ⟨h,j⟩:=Model.roundDyadic_of_representable F64 (by decide) d d.significand 0
    (by simp only [pow_zero,mul_one]) l g (e.trans p4)
  rw [← Model.toReal_roundDyadic_eq_roundAt _ (by decide) d h,j]
theorem i64F (i:Int64) (e:(Int.natAbs (Int64.toInt i)) < 2^(53:ℕ)) :
    fv i.toFloat∧rv i.toFloat=(Int64.toInt i):=by
  have g:Model.ofFloatModel i.toFloat.toModel =
      Model.roundDyadic FloatFormat.binary64 (FloatLib.Numerics.Dyadic.ofScaledInt (Int64.toInt i) 0):=by
    have h:=FloatLib.Floats.ExecFloat.Binary.toModel_ofFloat_int64ToFloat i
    rw [FloatLib.Floats.ExecFloat.Binary.toModel_ofFloat] at h
    rw [h, ← Model.ofModel_ofInt_eq_intToFloat FloatFormat.binary64 (by decide)]
    exact Model.ofModel_ofInt_eq_roundDyadic FloatFormat.binary64 (by decide) _
  have d:(FloatLib.Numerics.Dyadic.ofScaledInt (Int64.toInt i) 0).toReal=(Int64.toInt i):=by
    rw [← FloatLib.Numerics.Dyadic.cast_toRat,FloatLib.Numerics.Dyadic.ofScaledInt_toRat]; simp
  have hb : |((Int64.toInt i:ℤ):ℝ)| ≤ (2:ℝ)^(60:ℕ):=by
    have h3:((Int.natAbs (Int64.toInt i):ℕ):ℝ) = |((Int64.toInt i:ℤ):ℝ)| := by
      rw [← Int.cast_natCast,Int.natCast_natAbs,Int.cast_abs]
    have:((Int.natAbs (Int64.toInt i):ℕ):ℝ) ≤ (2:ℝ)^(53:ℕ):=by exact_mod_cast e.le
    rw [← h3]; linarith only [this]
  have ⟨j,l⟩:=Model.roundDyadic_of_representable FloatFormat.binary64 (by decide)
    (FloatLib.Numerics.Dyadic.ofScaledInt (Int64.toInt i) 0) (Int.natAbs (Int64.toInt i)) 0
    (by simp [FloatLib.Numerics.Dyadic.ofScaledInt]) e
    (by simp [FloatLib.Numerics.Dyadic.ofScaledInt]; decide) (by rw [d]; exact hb.trans pmax)
  exact ⟨by rw [fv,g]; exact j,by rw [rv,g,l,d]⟩
theorem i64U (u:UInt64) (h:u.toNat < 2^25):fv u.toInt64.toFloat∧rv u.toInt64.toFloat=u.toNat:=by
  have e:Int64.toInt u.toInt64=u.toNat:=by
    show u.toBitVec.toInt=_
    rw [BitVec.toInt_eq_toNat_cond,UInt64.toNat_toBitVec]
    split <;> omega
  have ⟨f,v⟩:=i64F u.toInt64 (by rw [e]; simp only [Int.natAbs_natCast]; omega)
  exact ⟨f,by rw [v,e]; simp⟩
theorem pw2 (ee:UInt64) (h1:1 ≤ ee.toNat) (h2:ee.toNat ≤ 134) :
    fv (Float.ofBits ((ee+873)<<<52)) ∧
      rv (Float.ofBits ((ee+873)<<<52))=(2:ℝ)^((ee.toNat:ℤ)-150):=by
  have q:((ee+873)<<<52).toNat=(ee.toNat+873)*2^52:=by
    rw [UInt64.toNat_shiftLeft,UInt64.toNat_add]
    simp only [UInt64.reduceToNat,Nat.shiftLeft_eq]
    rw [Nat.mod_eq_of_lt (a:=ee.toNat+873) (by omega)]
    simp only [Nat.reduceMod]; omega
  have f1:(((ee+873)<<<52>>>52)&&&2047).toNat=ee.toNat+873:=by
    rw [UInt64.toNat_and,UInt64.toNat_shiftRight,q]
    simp only [UInt64.reduceToNat,Nat.reduceMod,Nat.shiftRight_eq_div_pow]
    rw [Nat.mul_div_cancel _ (by positivity),show (2047:ℕ)=2^11-1 from rfl,
      Nat.and_two_pow_sub_one_eq_mod]; omega
  have f2:(((ee+873)<<<52)&&&0xFFFFFFFFFFFFF).toNat=0:=by
    rw [UInt64.toNat_and,q]; simp only [UInt64.reduceToNat]
    rw [show (4503599627370495:ℕ)=2^52-1 from rfl,Nat.and_two_pow_sub_one_eq_mod]; simp
  have f3:((ee+873)<<<52>>>63 != 0)=false:=by
    rw [show (ee+873)<<<52>>>63=0 from UInt64.toNat_inj.mp (by
      rw [UInt64.toNat_shiftRight,q]; simp only [UInt64.reduceToNat,Nat.reduceMod,Nat.shiftRight_eq_div_pow]
      exact Nat.div_eq_of_lt (by omega))]; rfl
  have ⟨f,v⟩:=rvB ((ee+873)<<<52) (by rw [f1]; omega) (by rw [f1]; omega)
  refine ⟨f,?_⟩
  rw [v,f1,f2,f3,show ((ee.toNat+873:ℕ):ℤ)-1075=(ee.toNat:ℤ)-150 + -52 by push_cast; ring,
    zpow_add₀ two_ne_zero]
  norm_num; ring
theorem redSpec (ee m:UInt64) (he1:1 ≤ ee.toNat) (he2:ee.toNat ≤ 134) (hm2:m.toNat < 2^24) :
    fv (rOf ee m)∧0 ≤ rv (rOf ee m)∧rv (rOf ee m) ≤ 1/64 ∧
      (rv (rOf ee m)=0∨1/2^150 ≤ rv (rOf ee m)) ∧
      cv ee m=((kOf ee m).toNat:ℝ)/64+rv (rOf ee m):=by
  have hs:(144-ee).toNat=144-ee.toNat:=by
    rw [UInt64.toNat_sub_of_le _ _ (by rw [UInt64.le_iff_toNat_le]; simp; omega)]; simp
  generalize hS:144-ee.toNat=s at hs
  set rI:=(if 144-ee < 24 then m&&&((1<<<(144-ee))-1) else m) with hrI
  have hk:(kOf ee m).toNat=m.toNat/2^s∧rI.toNat=m.toNat%2^s:=by
    unfold kOf; rw [hrI]
    by_cases hc:144-ee < 24
    · have hc':s < 24:=by rw [UInt64.lt_iff_toNat_lt,hs] at hc; simpa using hc
      simp only [hc,ite_true]
      constructor
      · rw [UInt64.toNat_shiftRight,hs,Nat.mod_eq_of_lt (by omega),Nat.shiftRight_eq_div_pow]
      · rw [UInt64.toNat_and,UInt64.toNat_sub_of_le,UInt64.toNat_shiftLeft,hs]
        · simp only [UInt64.reduceToNat,Nat.mod_eq_of_lt (show s < 64 by omega),Nat.shiftLeft_eq,
            one_mul]
          rw [Nat.mod_eq_of_lt (show 2^s < 2^64 from Nat.pow_lt_pow_right (by norm_num) (by omega)),
            Nat.and_two_pow_sub_one_eq_mod]
        · rw [UInt64.le_iff_toNat_le,UInt64.toNat_shiftLeft,hs]
          simp only [UInt64.reduceToNat,Nat.mod_eq_of_lt (show s < 64 by omega),Nat.shiftLeft_eq,
            one_mul]
          rw [Nat.mod_eq_of_lt (show 2^s < 2^64 from Nat.pow_lt_pow_right (by norm_num) (by omega))]
          exact Nat.one_le_two_pow
    · have hc':24 ≤ s:=by
        rw [UInt64.lt_iff_toNat_lt,hs] at hc; simp at hc; omega
      simp only [hc,ite_false]
      have:m.toNat < 2^s:=lt_of_lt_of_le hm2 (Nat.pow_le_pow_right (by norm_num) hc')
      exact ⟨by simp [Nat.div_eq_of_lt this],(Nat.mod_eq_of_lt this).symm⟩
  have hP:(0:ℝ) < 2^((ee.toNat:ℤ)-150):=by positivity
  have e2:(2:ℝ)^s*2^((ee.toNat:ℤ)-150)=1/64:=by
    rw [← zpow_natCast, ← zpow_add₀ two_ne_zero,show ((s:ℕ):ℤ)+((ee.toNat:ℤ)-150) = -6 by omega]
    norm_num
  have hrs:rI.toNat < 2^s:=by rw [hk.2]; exact Nat.mod_lt _ (by positivity)
  have hval:(rI.toNat:ℝ)*2^((ee.toNat:ℤ)-150) ≤ 1/64:=by
    rw [← e2]; exact mul_le_mul_of_nonneg_right (by exact_mod_cast hrs.le) hP.le
  have hr25:rI.toNat < 2^25:=by rw [hk.2]; exact lt_of_le_of_lt (Nat.mod_le _ _) (by omega)
  have ⟨fi,vi⟩:=i64U rI hr25
  have ⟨fp,vp⟩:=pw2 ee he1 he2
  have ⟨fr,vr⟩:=mulF _ _ fi fp (by rw [vi,vp,abs_of_nonneg (by positivity)]; linarith only [hval])
  have hex:rv (rOf ee m)=(rI.toNat:ℝ)*2^((ee.toNat:ℤ)-150):=by
    show rv (rI.toInt64.toFloat*Float.ofBits ((ee+873)<<<52))=_
    rw [vr,vi,vp, ← FloatLib.Numerics.Dyadic.toReal_mk_false]
    exact rndExact _ (by show rI.toNat < 2^53; omega) (by show (-1074:ℤ) ≤ (ee.toNat:ℤ)-150; omega)
      (by rw [FloatLib.Numerics.Dyadic.toReal_mk_false,abs_of_nonneg (by positivity)]; linarith only [hval])
  rw [hex]
  refine ⟨fr,by positivity,hval,?_,?_⟩
  · rcases Nat.eq_zero_or_pos rI.toNat with h0 | h0
    · left; rw [h0]; simp
    · right
      have h2:(1:ℝ)/2^150 ≤ 2^((ee.toNat:ℤ)-150):=by
        rw [← zp]; exact zpow_le_zpow_right₀ (by norm_num) (by omega)
      exact h2.trans (le_mul_of_one_le_left hP.le (by exact_mod_cast h0))
  · unfold cv; rw [hk.1]
    have hm:m.toNat=2^s*(m.toNat/2^s)+rI.toNat:=by rw [hk.2]; exact (Nat.div_add_mod _ _).symm
    rw [show (m.toNat:ℝ)=(2^s*(m.toNat/2^s)+rI.toNat:ℕ) by rw [← hm]]
    push_cast
    rw [add_mul,mul_comm ((2:ℝ)^s),mul_assoc,e2]; ring
theorem gLB (C:ℝ):C*exp (-C) ≤ 1-exp (-C):=by
  have h:=mul_le_mul_of_nonneg_left (add_one_le_exp C) (exp_pos (-C)).le
  rw [mul_add,mul_one, ← exp_add,neg_add_cancel,exp_zero] at h
  linarith
theorem canonR (m:Model F64) :
    Model.toReal (Model.canonicalizeModel m)=Model.toReal m:=by
  rw [Model.toReal_eq_unpackedToReal_toModel (by decide),
    Model.toReal_eq_unpackedToReal_toModel (by decide),Model.toModel_canonicalizeModel]
theorem canonF (m:Model F64) :
    Model.isFinite (Model.canonicalizeModel m)=Model.isFinite m:=by
  rw [← Model.toDyadic?_isSome_eq_isFinite, ← Model.toDyadic?_isSome_eq_isFinite,
    Model.toDyadic?_ieee_eq_model _ (by decide),Model.toDyadic?_ieee_eq_model _ (by decide),
    Model.toModel_canonicalizeModel]
theorem toF_spec (m:Model F64) :
    (fv (toF m) ↔ Model.isFinite m=true)∧rv (toF m)=Model.toReal m:=by
  have e:Model.ofFloatModel (toF m).toModel=Model.canonicalizeModel m :=
    Model.ofFloatModel_toFloatModel m
  exact ⟨by unfold fv; rw [e,canonF],by unfold rv; rw [e,canonR]⟩
theorem dyM_spec (n k:ℕ) (hn:n < 2^53) (hk:k ≤ 1000) :
    Model.isFinite (dyM n k)=true ∧
      Model.toReal (dyM n k) = -((n:ℝ)*2^(-(k:ℤ))):=by
  have hv:(⟨true,n, -(k:ℤ)⟩:FloatLib.Numerics.Dyadic).toReal = -((n:ℝ)*2^(-(k:ℤ))):=by
    rw [Model.Dyadic.toReal_mk]
    change (if true=true then (-1:ℝ) else 1)*(n:ℝ)*(2:ℝ)^(-(k:ℤ))=_
    simp
  have hb : |(⟨true,n, -(k:ℤ)⟩:FloatLib.Numerics.Dyadic).toReal| ≤
      Model.toReal (Model.posMaxFinite F64):=by
    refine le_trans ?_ pmax
    rw [hv,abs_neg,abs_of_nonneg (by positivity)]
    have h1:(n:ℝ) ≤ 2^(53:ℕ):=by exact_mod_cast hn.le
    have h2:(2:ℝ)^(-(k:ℤ)) ≤ 1:=zpow_le_one_of_nonpos₀ (by norm_num) (by omega)
    calc (n:ℝ)*2^(-(k:ℤ)) ≤ 2^(53:ℕ)*1:=by gcongr
      _ ≤ 2^(60:ℕ):=by norm_num
  have hmin:F64.minSubnormalExponent = -1074:=rfl
  have ⟨h,j⟩:=Model.roundDyadic_of_representable F64 (by decide)
    ⟨true,n, -(k:ℤ)⟩ n 0 (by simp) hn (by rw [hmin]; show (-1074:ℤ) ≤ -(k:ℤ); omega) hb
  exact ⟨h,by unfold dyM; rw [j,hv]⟩
theorem expLB {x:ℝ} (hx:x ≤ 100):1/2^200 ≤ exp (-x):=by
  have h1:exp (-100) ≤ exp (-x):=exp_le_exp.2 (by linarith)
  have h2:exp 1 ≤ 3:=by have:=Real.exp_one_lt_d9; linarith
  have h3:exp (100:ℝ) ≤ 3^(100:ℕ):=by
    rw [show (100:ℝ)=((100:ℕ):ℝ)*1 by norm_num,Real.exp_nat_mul]
    exact pow_le_pow_left₀ (exp_pos 1).le h2 100
  have h4:(3:ℝ)^(100:ℕ) ≤ 2^(200:ℕ):=by norm_num
  have h5:exp (-100)=1/exp 100:=by rw [Real.exp_neg,one_div]
  refine le_trans ?_ h1; rw [h5]
  exact one_div_le_one_div_of_le (exp_pos _) (h3.trans h4)
theorem getT (j:ℕ) (hj:j < 64) :
    tT.get! j=toF (Model.Transcendentals.CorrectlyRounded.exp (dyM j 6)):=by
  show (Array.ofFn _)[j]! = _
  rw [getElem!_pos _ _ (by simpa using hj),Array.getElem_ofFn]
theorem getM (j:ℕ) (hj:j < 64) :
    tM.get! j=toF (Model.Transcendentals.CorrectlyRounded.expMinus1 (dyM j 6)):=by
  show (Array.ofFn _)[j]! = _
  rw [getElem!_pos _ _ (by simpa using hj),Array.getElem_ofFn]
theorem getE (i:ℕ) (hi:i < 88) :
    tE.get! i=toF (Model.Transcendentals.CorrectlyRounded.exp (dyM i 0)):=by
  show (Array.ofFn _)[i]! = _
  rw [getElem!_pos _ _ (by simpa using hi),Array.getElem_ofFn]
theorem expTab (n k:ℕ) (hn:n < 2^53) (hk:k ≤ 1000)
    (hx:(n:ℝ)*2^(-(k:ℤ)) ≤ 100) :
    Ap (toF (Model.Transcendentals.CorrectlyRounded.exp (dyM n k)))
        (exp (-((n:ℝ)*2^(-(k:ℤ))))) (1/2^52):=by
  have ⟨hf,hv⟩:=dyM_spec n k hn hk
  have hr:exp (Model.toReal (dyM n k)) <
      Model.Transcendentals.Certified.overflowThreshold F64:=by
    refine lt_of_le_of_lt ?_ (Model.Transcendentals.CorrectlyRounded.one_lt_overflowThreshold (by decide))
    have h0:(0:ℝ) ≤ n*2^(-(k:ℤ)):=by positivity
    rw [hv]; exact exp_le_one_iff.2 (by linarith)
  have h1:=Model.Transcendentals.CorrectlyRounded.toReal_exp_eq_roundAt (by decide) hf hr (budget := {})
  have h2:=Model.Transcendentals.CorrectlyRounded.isFinite_exp (by decide) hf hr (budget := {})
  have ⟨f3,v3⟩:=toF_spec (Model.Transcendentals.CorrectlyRounded.exp (dyM n k))
  refine ⟨f3.2 h2,?_⟩
  rw [v3,h1,hv]
  refine rnd52 ?_
  rw [abs_of_pos (exp_pos _)]
  exact le_trans (inv2le (by norm_num)) (expLB hx)
theorem tT_spec (j:ℕ) (hj:j < 64) :
    Ap (tT.get! j) (exp (-((j:ℝ)/64))) (1/2^52):=by
  have e:(j:ℝ)/64=(j:ℝ)*2^(-((6:ℕ):ℤ)):=by norm_num; ring
  have hj':(j:ℝ) ≤ 64:=by exact_mod_cast hj.le
  rw [getT j hj,e]
  exact expTab j 6 (by omega) (by norm_num) (by rw [← e]; linarith)
theorem tE_spec (i:ℕ) (hi:i < 88) :
    Ap (tE.get! i) (exp (-(i:ℝ))) (1/2^52):=by
  have e:(i:ℝ)=(i:ℝ)*2^(-((0:ℕ):ℤ)):=by norm_num
  have hi':(i:ℝ) ≤ 88:=by exact_mod_cast hi.le
  rw [getE i hi]; conv=>arg 2; rw [e]
  exact expTab i 0 (by omega) (by norm_num) (by rw [← e]; linarith)
theorem tM_spec (j:ℕ) (hj:j < 64) :
    Ap (tM.get! j) (exp (-((j:ℝ)/64))-1) (1/2^52):=by
  have ⟨hf,hv⟩:=dyM_spec j 6 (by omega) (by norm_num)
  have hv':Model.toReal (dyM j 6) = -((j:ℝ)/64):=by rw [hv]; norm_num; ring
  have h0:(0:ℝ) ≤ j:=by positivity
  have hE:exp (-((j:ℝ)/64)) ≤ 1:=exp_le_one_iff.2 (by linarith)
  have hr:exp (Model.toReal (dyM j 6))-1 <
      Model.Transcendentals.Certified.overflowThreshold F64:=by
    refine lt_of_le_of_lt ?_ (Model.Transcendentals.CorrectlyRounded.one_lt_overflowThreshold (by decide))
    rw [hv']; linarith
  have h1:=Model.Transcendentals.CorrectlyRounded.toReal_expMinus1_eq_roundAt (by decide) hf hr (budget := {})
  have h2:=Model.Transcendentals.CorrectlyRounded.isFinite_expMinus1 (by decide) hf hr (budget := {})
  have ⟨f3,v3⟩:=toF_spec (Model.Transcendentals.CorrectlyRounded.expMinus1 (dyM j 6))
  rw [getM j hj]
  refine ⟨f3.2 h2,?_⟩
  rw [v3,h1,hv']
  rcases Nat.eq_zero_or_pos j with h0 | h0
  · subst h0; unfold Rel; simp [Model.roundAt_zero]
  · refine rnd52 ?_
    have hx:(1:ℝ)/64 ≤ (j:ℝ)/64:=by
      have:(1:ℝ) ≤ j:=by exact_mod_cast h0
      linarith
    have h3:(1:ℝ)/3 ≤ exp (-((j:ℝ)/64)):=by
      have:=Real.exp_one_lt_d9
      refine le_trans ?_ (exp_le_exp.2 (show -1 ≤ -((j:ℝ)/64) by
        have:(j:ℝ) ≤ 63:=by exact_mod_cast (by omega:j ≤ 63)
        linarith))
      rw [Real.exp_neg,le_inv_comm₀ (by norm_num) (exp_pos 1)]; linarith
    have:=mul_le_mul hx h3 (by norm_num) (by positivity)
    rw [abs_of_nonpos (by linarith)]
    linarith only [gLB ((j:ℝ)/64),this]
theorem cstE (a:UInt64) (s:Bool) (f e:ℕ) (h:(a>>>63 != 0)=s∧(a&&&0xFFFFFFFFFFFFF).toNat=f ∧
    ((a>>>52)&&&2047).toNat=e∧e≠0∧e < 2047:=by decide):fv (Float.ofBits a) ∧
      rv (Float.ofBits a)=(if s then -1 else 1)*((2^52+f:ℕ):ℝ)*2^((e:ℤ)-1075):=by
  obtain ⟨hs,hf,he,h0,h1⟩:=h; subst hs hf he; exact rvB a h0 h1
theorem cst (a:UInt64) (s:Bool) (f e:ℕ) (q:ℝ) (h:(a>>>63 != 0)=s ∧
    (a&&&0xFFFFFFFFFFFFF).toNat=f∧((a>>>52)&&&2047).toNat=e∧e≠0∧e < 2047:=by decide)
    (hq : |(if s then -1 else 1)*((2^52+f:ℕ):ℝ)*2^((e:ℤ)-1075)-q| ≤ 1/2^52:=by
      norm_num [abs_le]):fv (Float.ofBits a) ∧ |rv (Float.ofBits a)-q| ≤ 1/2^52:=by
  have ⟨f,v⟩:=cstE a s f e h; exact ⟨f,by rw [v]; exact hq⟩
theorem kone:fv one∧rv one=1:=by
  have ⟨f,v⟩:=cstE 0x3ff0000000000000 false 0 1023; exact ⟨f,by rw [v]; norm_num⟩
theorem ktwo:fv two∧rv two=2:=by
  have ⟨f,v⟩:=cstE 0x4000000000000000 false 0 1024; exact ⟨f,by rw [v]; norm_num⟩
theorem hstep {s r c:Float} {S R C:ℝ} (hs:fv s) (hr:fv r) (hc:fv c) (hR:rv r=R)
    (h0:0 ≤ R) (h1:R ≤ 1/64) (he : |rv s-S| ≤ 1/2^50) (hC : |rv c-C| ≤ 1/2^52)
    (hS : |S| ≤ 1:=by norm_num [abs_div]) (hC1 : |C| ≤ 1:=by norm_num [abs_div]) :
    fv (s*r+c) ∧ |rv (s*r+c)-(S*R+C)| ≤ 1/2^50 ∧ |S*R+C| ≤ 1/64 + |C| := by
  have a1 : |rv s| ≤ 2:=by linarith only [abs_sub_abs_le_abs_sub (rv s) S,he,hS]
  have a2 : |rv s*R| ≤ 1/32:=by
    rw [abs_mul,abs_of_nonneg h0]; exact (mul_le_mul a1 h1 h0 (by norm_num)).trans (by norm_num)
  have a3 : |rv s*R-S*R| ≤ 1/2^56:=by
    rw [← sub_mul,abs_mul,abs_of_nonneg h0]; exact (mul_le_mul he h1 h0 (by positivity)).trans (by norm_num)
  obtain ⟨fm,vm⟩:=mulF s r hs hr (by rw [hR]; linarith only [a2])
  rw [hR] at vm
  have a4:=rndA (rv s*R); rw [← vm] at a4
  have a5 : |rv (s*r)| ≤ 1/16:=by
    linarith only [abs_sub_abs_le_abs_sub (rv (s*r)) (rv s*R),a4,a2]
  have a6 : |rv c| ≤ 2:=by linarith only [abs_sub_abs_le_abs_sub (rv c) C,hC,hC1]
  have a8 : |rv (s*r)+rv c| ≤ 3:=by linarith only [abs_add_le (rv (s*r)) (rv c),a5,a6]
  obtain ⟨fa,va⟩:=addF _ c fm hc (by linarith only [a5]) a6 (by linarith only [a8])
  have a7:=rndA (rv (s*r)+rv c); rw [← va] at a7
  have a9 : |S*R| ≤ 1/64:=by
    rw [abs_mul,abs_of_nonneg h0]; exact (mul_le_mul hS h1 h0 (by norm_num)).trans (by norm_num)
  refine ⟨fa,?_,by linarith only [abs_add_le (S*R) C,a9]⟩
  have b1:=abs_le.1 a7; have b2:=abs_le.1 a4; have b3:=abs_le.1 a3; have b4:=abs_le.1 hC
  rw [abs_le]; constructor <;>
    linarith only [b1.1,b1.2,b2.1,b2.2,b3.1,b3.2,b4.1,b4.2,a2,a8,abs_nonneg (rv s*R)]
noncomputable def Q6 (R:ℝ):ℝ :=
  ((((-1/720*R+1/120)*R + -1/24)*R+1/6)*R + -1/2)*R+1
theorem hqSpec (r:Float) (R:ℝ) (hr:fv r) (hR:rv r=R) (h0:0 ≤ R) (h1:R ≤ 1/64) :
    fv (hq r) ∧ |rv (hq r)-Q6 R| ≤ 1/2^50:=by
  obtain ⟨f5,c5⟩:=cst 0xbf56c16c16c16c17 true 1901519842667543 1013 (-1/720)
  obtain ⟨f4,c4⟩:=cst 0x3f81111111111111 false 300239975158033 1016 (1/120)
  obtain ⟨f3,c3⟩:=cst 0xbfa5555555555555 true 1501199875790165 1018 (-1/24)
  obtain ⟨f2,c2⟩:=cst 0x3fc5555555555555 false 1501199875790165 1020 (1/6)
  obtain ⟨f1,c1⟩:=cst 0xbfe0000000000000 true 0 1022 (-1/2)
  obtain ⟨f0,c0⟩:=cst 0x3ff0000000000000 false 0 1023 1
  have ⟨g4,e4,b4⟩:=hstep f5 hr f4 hR h0 h1 (c5.trans (by norm_num)) c4
  have ⟨g3,e3,b3⟩:=hstep g4 hr f3 hR h0 h1 e4 c3 (b4.trans (by norm_num [abs_div]))
  have ⟨g2,e2,b2⟩:=hstep g3 hr f2 hR h0 h1 e3 c2 (b3.trans (by norm_num [abs_div]))
  have ⟨g1,e1,b1⟩:=hstep g2 hr f1 hR h0 h1 e2 c1 (b2.trans (by norm_num [abs_div]))
  obtain ⟨g0,e0, -⟩:=hstep g1 hr f0 hR h0 h1 e1 c0 (b1.trans (by norm_num [abs_div]))
  exact ⟨g0,e0⟩
theorem trunc (R:ℝ) (h0:0 ≤ R) (h1:R ≤ 1/64) :
    |(1-exp (-R))-R*Q6 R| ≤ R/2^48:=by
  have h:=Real.exp_bound (x := -R) (by rw [abs_neg,abs_of_nonneg h0]; linarith) (n:=7)
    (by norm_num)
  have hs : ∑ m ∈ Finset.range 7,(-R)^m/(m.factorial:ℝ)=1-R*Q6 R:=by
    simp only [Finset.sum_range_succ,Finset.sum_range_zero,Nat.factorial,Q6]; push_cast; ring
  rw [hs,abs_neg,abs_of_nonneg h0] at h
  rw [show (1-exp (-R))-R*Q6 R = -(exp (-R)-(1-R*Q6 R)) by ring,abs_neg]
  refine h.trans ?_
  have h6:R^6 ≤ (1/64)^6:=pow_le_pow_left₀ h0 h1 6
  have hc:((Nat.succ 7:ℕ):ℝ)/((Nat.factorial 7:ℕ)*(7:ℕ))=1/4410:=by
    norm_num [Nat.factorial]
  rw [hc]
  calc R^7*(1/4410)=R*(R^6/4410):=by ring
    _ ≤ R*((1/64)^6/4410):=by gcongr
    _ ≤ R*(1/2^48):=mul_le_mul_of_nonneg_left (by norm_num) h0
    _=R/2^48:=by ring
theorem gSpec (r:Float) (R:ℝ) (hr:fv r) (hR:rv r=R) (h0:0 ≤ R) (h1:R ≤ 1/64)
    (hR0:R=0∨1/2^150 ≤ R) :
    Ap (r*hq r) (1-exp (-R)) (1/2^47):=by
  obtain ⟨fq,eq⟩:=hqSpec r R hr hR h0 h1
  have hT:=trunc R h0 h1
  have hE:=add_one_le_exp (-R)
  have hG:R*(63/64) ≤ 1-exp (-R):=by
    have:=mul_le_mul_of_nonneg_left (show 63/64 ≤ exp (-R) by linarith only [hE,h1]) h0
    linarith only [gLB R,this]
  have e1 : |R*rv (hq r)-R*Q6 R| ≤ R*(1/2^50):=by
    rw [← mul_sub,abs_mul,abs_of_nonneg h0]; exact mul_le_mul_of_nonneg_left eq h0
  have hv:Rel (rv r*rv (hq r)) (1-exp (-R)) (3/2^49):=by
    unfold Rel; rw [hR,abs_of_nonneg (show (0:ℝ) ≤ 1-exp (-R) by linarith only [hG,h0])]
    have:=abs_sub_le (R*rv (hq r)) (R*Q6 R) (1-exp (-R))
    rw [abs_sub_comm (R*Q6 R)] at this; linarith only [this,e1,hT,hG,h0]
  have:=opR (mulF r (hq r) hr fq) hv (by
    rcases hR0 with hz | hz
    · left; rw [hz,neg_zero,exp_zero,sub_self]
    · right; linarith only [hG,hz]) (by linarith only [hE,h1]) (by linarith only [hG,h0])
  exact ⟨this.1,this.2.mono (by norm_num)⟩
theorem Rel.neg {a b d:ℝ} (h:Rel a b d):Rel (-a) (-b) d:=by
  unfold Rel at *; rw [abs_neg b,show -a - -b = -(a-b) by ring,abs_neg]; exact h
theorem Rel.b2 {a b d:ℝ} (h:Rel a b d) (hb : |b| ≤ 1) (hd:d ≤ 1:=by norm_num) : |a| ≤ 2:=by
  have:=mul_le_mul (show 1+d ≤ 2 by linarith) hb (abs_nonneg _) (by norm_num); linarith [h.le]
theorem rel1 {d:ℝ} (hd:0 ≤ d):Rel 1 1 d:=by unfold Rel; simpa using hd
theorem thirdE:(1:ℝ)/3 ≤ exp (-1):=by
  have:=Real.exp_one_lt_d9
  rw [Real.exp_neg,le_inv_comm₀ (by norm_num) (exp_pos 1)]; linarith
theorem halfE:exp (-1) ≤ 1/2:=by
  have:=Real.exp_one_gt_d9
  rw [Real.exp_neg,inv_le_comm₀ (exp_pos 1) (by norm_num)]; linarith
theorem kij (k:UInt64):(k>>>6).toNat=k.toNat/64∧(k&&&63).toNat=k.toNat%64:=by
  constructor
  · rw [UInt64.toNat_shiftRight]; simp only [UInt64.reduceToNat,Nat.reduceMod,Nat.shiftRight_eq_div_pow]
  · rw [UInt64.toNat_and]; simp only [UInt64.reduceToNat]
    rw [show (63:ℕ)=2^6-1 from rfl,Nat.and_two_pow_sub_one_eq_mod]
theorem cast64 (k:ℕ):((k/64:ℕ):ℝ)+((k%64:ℕ):ℝ)/64=(k:ℝ)/64:=by
  have:(k:ℝ)=64*((k/64:ℕ):ℝ)+((k%64:ℕ):ℝ):=by exact_mod_cast (Nat.div_add_mod k 64).symm
  rw [this]; ring
theorem erSpec (r:Float) (R:ℝ) (hr:fv r) (hR:rv r=R) (h0:0 ≤ R) (h1:R ≤ 1/64)
    (hR0:R=0∨1/2^150 ≤ R):Ap (one-r*hq r) (exp (-R)) (1/2^50):=by
  obtain ⟨fg,rg⟩:=gSpec r R hr hR h0 h1 hR0
  obtain ⟨f1,v1⟩:=kone
  have hE:=add_one_le_exp (-R)
  have hG0:0 ≤ 1-exp (-R):=by linarith only [gLB R,mul_nonneg h0 (exp_pos (-R)).le]
  have hg:=rg; unfold Rel at hg; rw [abs_of_nonneg hG0] at hg
  have r0:Rel (rv one-rv (r*hq r)) (exp (-R)) (1/2^51):=by
    unfold Rel; rw [v1,abs_of_pos (exp_pos _),show 1-rv (r*hq r)-exp (-R) =
      -(rv (r*hq r)-(1-exp (-R))) by ring,abs_neg]
    linarith only [hg,hE,h1]
  have:=opR (subF _ _ f1 fg (by rw [v1]; norm_num) (rg.b2 (by
    rw [abs_of_nonneg hG0]; linarith only [exp_pos (-R)]))) r0 (Or.inr (by linarith only [hE,h1]))
    (by linarith only [exp_le_one_iff.2 (by linarith only [h0] : -R ≤ 0)])
  exact ⟨this.1,this.2.mono (by norm_num)⟩
theorem expT (k:UInt64) (hk:k.toNat < 5632) (r:Float) (R:ℝ) (hr:fv r) (hR:rv r=R)
    (h0:0 ≤ R) (h1:R ≤ 1/64) (hR0:R=0∨1/2^150 ≤ R) :
    Ap (eT k r) (exp (-((k.toNat:ℝ)/64+R))) (1/2^48):=by
  obtain ⟨hki,hkj⟩:=kij k
  obtain ⟨fE,rE⟩:=tE_spec (k.toNat/64) (by omega)
  obtain ⟨fT,rT⟩:=tT_spec (k.toNat%64) (Nat.mod_lt _ (by norm_num))
  obtain ⟨fer,rer⟩:=erSpec r R hr hR h0 h1 hR0
  have hk':(k.toNat:ℝ)/64 ≤ 88:=by
    have:(k.toNat:ℝ) ≤ 5632:=by exact_mod_cast hk.le
    linarith only [this]
  have hk0:(0:ℝ) ≤ (k.toNat:ℝ)/64:=by positivity
  have eq2:exp (-((k.toNat/64:ℕ):ℝ))*exp (-(((k.toNat%64:ℕ):ℝ)/64)) =
      exp (-((k.toNat:ℝ)/64)):=by rw [← exp_add, ← cast64 k.toNat]; ring_nf
  have eq3:exp (-((k.toNat:ℝ)/64))*exp (-R)=exp (-((k.toNat:ℝ)/64+R)):=by
    rw [← exp_add]; ring_nf
  have p1:=expLB (x:=(k.toNat:ℝ)/64) (by linarith only [hk'])
  have p2:=expLB (x:=(k.toNat:ℝ)/64+R) (by linarith only [hk',h1])
  have q1:exp (-((k.toNat:ℝ)/64)) ≤ 1:=exp_le_one_iff.2 (by linarith only [hk0])
  have q2:exp (-((k.toNat:ℝ)/64+R)) ≤ 1:=exp_le_one_iff.2 (by linarith only [hk0,h0])
  have ⟨fET,rET⟩:=opR (mulF _ _ fE fT) (rE.mul rT) (Or.inr (by rw [eq2]; linarith only [p1]))
    (by rw [eq2]; linarith only [q1])
  rw [eq2] at rET
  have ⟨ft,rt⟩:=opR (mulF _ _ fET fer) (rET.mul rer) (Or.inr (by rw [eq3]; linarith only [p2]))
    (by rw [eq3]; linarith only [q2])
  rw [eq3] at rt
  unfold eT; rw [hki,hkj]
  exact ⟨ft,rt.mono (by norm_num)⟩
theorem expP (j:ℕ) (hj:j < 64) (r:Float) (R:ℝ) (hr:fv r) (hR:rv r=R)
    (h0:0 ≤ R) (h1:R ≤ 1/64) (hR0:R=0∨1/2^150 ≤ R) :
    Ap (tT.get! j*(r*hq r)) (exp (-((j:ℝ)/64))*(1-exp (-R))) (9/2^50):=by
  obtain ⟨fT,rT⟩:=tT_spec j hj
  obtain ⟨fg,rg⟩:=gSpec r R hr hR h0 h1 hR0
  have hj':(j:ℝ) ≤ 63:=by exact_mod_cast (by omega:j ≤ 63)
  have hE1:exp (-((j:ℝ)/64)) ≤ 1:=exp_le_one_iff.2 (by
    linarith only [(by positivity:(0:ℝ) ≤ j)])
  have hE4:1/3 ≤ exp (-((j:ℝ)/64)):=thirdE.trans (exp_le_exp.2 (by linarith only [hj']))
  have hE:=add_one_le_exp (-R)
  have hG:=mul_le_mul_of_nonneg_left (show 63/64 ≤ exp (-R) by linarith only [hE,h1]) h0
  have hG0:0 ≤ 1-exp (-R):=by linarith only [gLB R,hG,h0]
  have:=opR (mulF _ _ fT fg) (rT.mul rg) (by
    rcases hR0 with hz | hz
    · left; rw [hz,neg_zero,exp_zero,sub_self,mul_zero]
    · right; linarith only [mul_le_mul_of_nonneg_right hE4 hG0,gLB R,hG,hz])
    (by linarith only [mul_le_mul_of_nonneg_right hE1 hG0,hE,h1]) (mul_nonneg (exp_pos _).le hG0)
  exact ⟨this.1,this.2.mono (by norm_num)⟩
theorem expU (j:ℕ) (hj:j < 64) (r:Float) (R:ℝ) (hr:fv r) (hR:rv r=R)
    (h0:0 ≤ R) (h1:R ≤ 1/64) (hR0:R=0∨1/2^150 ≤ R) :
    Ap (tT.get! j*(r*hq r)-tM.get! j) (1-exp (-((j:ℝ)/64+R))) (5/2^47):=by
  obtain ⟨fp,rp⟩:=expP j hj r R hr hR h0 h1 hR0
  obtain ⟨fM,rM⟩:=tM_spec j hj
  have hj0:(0:ℝ) ≤ j:=by positivity
  have hj':(j:ℝ) ≤ 63:=by exact_mod_cast (by omega:j ≤ 63)
  have hE1:exp (-((j:ℝ)/64)) ≤ 1:=exp_le_one_iff.2 (by linarith only [hj0])
  have hE:=add_one_le_exp (-R)
  have hG0:0 ≤ 1-exp (-R):=by linarith only [gLB R,mul_nonneg h0 (exp_pos (-R)).le]
  have hEG:0 ≤ exp (-((j:ℝ)/64))*(1-exp (-R)):=mul_nonneg (exp_pos _).le hG0
  have rMn:Rel (-rv (tM.get! j)) (1-exp (-((j:ℝ)/64))) (9/2^50):=by
    have:=rM.neg; rw [neg_sub] at this; exact this.mono (by norm_num)
  have es:=rp.add rMn hEG (by linarith only [hE1])
  rw [show exp (-((j:ℝ)/64))*(1-exp (-R))+(1-exp (-((j:ℝ)/64))) =
    1-exp (-((j:ℝ)/64+R)) by rw [neg_add,exp_add]; ring, ← sub_eq_add_neg] at es
  have hN3:=mul_le_mul_of_nonneg_left (thirdE.trans (exp_le_exp.2 (by linarith only [hj',h1] :
    -1 ≤ -((j:ℝ)/64+R)))) (by positivity:(0:ℝ) ≤ (j:ℝ)/64+R)
  have hN:=gLB ((j:ℝ)/64+R)
  have hP1 : |exp (-((j:ℝ)/64))*(1-exp (-R))| ≤ 1:=by
    rw [abs_of_nonneg hEG]; linarith only [mul_le_mul_of_nonneg_right hE1 hG0,hE,h1]
  have hM1 : |exp (-((j:ℝ)/64))-1| ≤ 1:=by
    rw [abs_of_nonpos (by linarith only [hE1])]; linarith only [exp_pos (-((j:ℝ)/64))]
  have:=opR (subF _ _ fp fM (rp.b2 hP1) (rM.b2 hM1)) es (by
    rcases hR0 with hz | hz
    · rcases Nat.eq_zero_or_pos j with hj1 | hj1
      · left; rw [hz,hj1]; simp
      · right
        have:(1:ℝ) ≤ j:=by exact_mod_cast hj1
        linarith only [hN,hN3,this,h0]
    · right; linarith only [hN,hN3,hz,hj0])
    (by linarith only [exp_pos (-((j:ℝ)/64+R))]) (by linarith only [hN,hN3,hj0,h0])
  exact ⟨this.1,this.2.mono (by norm_num)⟩
theorem dSpec (t:Float) (T d:ℝ) (ht:fv t) (hR:Rel (rv t) T d) (hT0:0 ≤ T) (hT1:T ≤ 1)
    (hd0:0 ≤ d:=by norm_num) (hd:d ≤ 1/4:=by norm_num) :
    Ap (one+t) (1+T) (d+1/2^51):=by
  obtain ⟨f1,v1⟩:=kone
  have ha:Rel (rv one+rv t) (1+T) d:=by rw [v1]; exact (rel1 hd0).add hR (by norm_num) hT0
  exact opR (addF _ _ f1 ht (by rw [v1]; norm_num) (hR.b2 (by rw [abs_of_nonneg hT0]; exact hT1)
    (by linarith only [hd]))) ha (Or.inr (by linarith only [hT0])) (by linarith only [hT1])
    (by linarith only [hT0]) (by linarith only [hd])
theorem dPos {x:Float} {D d:ℝ} (h:Rel (rv x) D d) (hd:d < 1) (hD:0 < D):rv x≠0 :=
  fun e=>by
    have:=h.ge; rw [e,abs_zero,abs_of_pos hD] at this
    linarith only [this,mul_pos (by linarith only [hd]:(0:ℝ) < 1-d) hD]
theorem qSpec {x d:Float} {X D dx dd:ℝ} (hx:fv x) (hd:fv d) (rx:Rel (rv x) X dx)
    (rd:Rel (rv d) D dd) (hD:1 ≤ D) (hD2:D ≤ 2) (hX0:1/2^201 ≤ X) (hX1:X ≤ 2)
    (hdx:0 ≤ dx:=by norm_num) (hdd:0 ≤ dd:=by norm_num) (hdd1:dd ≤ 1/8:=by norm_num)
    (hdx1:dx ≤ 1/8:=by norm_num):Ap (x/d) (X/D) ((dx+dd)/(1-dd)+1/2^51):=by
  have hq:X/D ≤ 2:=by rw [div_le_iff₀ (by linarith only [hD])]; linarith only [hD,hX1]
  have hq0:1/2^202 ≤ X/D:=by rw [le_div_iff₀ (by linarith only [hD])]; linarith only [hD2,hX0]
  exact opR (divF _ _ hx hd (dPos rd (by linarith only [hdd1]) (by linarith only [hD])))
    (rx.div rd (by linarith only [hD]) hdx hdd (by linarith only [hdd1])) (Or.inr hq0) hq
    (le_trans (by positivity) hq0) (by rw [div_le_iff₀ (by linarith only [hdd1])]; linarith only [hdd1,hdx1,hdd])
theorem sgTail (t:Float) (T:ℝ) (ht:fv t) (hR:Rel (rv t) T (1/2^48))
    (hT0:1/2^200 ≤ T) (hT1:T ≤ 1) :
    Ap (one/(one+t)) (1/(1+T)) (1/2^44)∧Ap (t/(one+t)) (T/(1+T)) (1/2^44):=by
  obtain ⟨f1,v1⟩:=kone
  have hT:0 ≤ T:=le_trans (by positivity) hT0
  obtain ⟨fd,rd⟩:=dSpec t T _ ht hR hT hT1
  have h1:Rel (rv one) 1 0:=by rw [v1]; exact rel1 le_rfl
  have a:=qSpec f1 fd h1 rd (by linarith only [hT]) (by linarith only [hT1]) (by norm_num) (by norm_num)
  have b:=qSpec ht fd hR rd (by linarith only [hT]) (by linarith only [hT1])
    (le_trans (inv2le (by norm_num)) hT0) (by linarith only [hT1])
  exact ⟨⟨a.1,a.2.mono (by norm_num)⟩,⟨b.1,b.2.mono (by norm_num)⟩⟩
theorem thTailT (t:Float) (T:ℝ) (ht:fv t) (hR:Rel (rv t) T (1/2^48))
    (hT0:0 < T) (hT1:T ≤ 1/2):Ap ((one-t)/(one+t)) ((1-T)/(1+T)) (1/2^44):=by
  obtain ⟨f1,v1⟩:=kone
  obtain ⟨fd,rd⟩:=dSpec t T _ ht hR hT0.le (by linarith only [hT1])
  have hT:0 < 1-T:=by linarith only [hT1]
  have r0:Rel (rv one-rv t) (1-T) (1/2^48):=by
    unfold Rel at hR ⊢; rw [abs_of_pos hT0] at hR; rw [v1,abs_of_pos hT]
    rw [show 1-rv t-(1-T) = -(rv t-T) by ring,abs_neg]; linarith only [hR,hT1]
  have ⟨fn,rn⟩:=opR (subF _ _ f1 ht (by rw [v1]; norm_num) (hR.b2 (by rw [abs_of_pos hT0]; linarith only [hT1])))
    r0 (Or.inr (by linarith only [hT1])) (by linarith only [hT0]) hT.le
  have:=qSpec fn fd rn rd (by linarith only [hT0]) (by linarith only [hT1]) (by linarith only [hT1])
    (by linarith only [hT0])
  exact ⟨this.1,this.2.mono (by norm_num)⟩
theorem thTailU (nu:Float) (N:ℝ) (hn:fv nu) (hR:Rel (rv nu) N (5/2^47))
    (hN0:1/2^200 ≤ N) (hN1:N ≤ 2/3):Ap (nu/(two-nu)) (N/(2-N)) (1/2^44):=by
  obtain ⟨f2,v2⟩:=ktwo
  have hN:0 < N:=lt_of_lt_of_le (by positivity) hN0
  have hN2:0 < 2-N:=by linarith only [hN1]
  have r0:Rel (rv two-rv nu) (2-N) (5/2^48):=by
    unfold Rel at hR ⊢; rw [abs_of_pos hN] at hR; rw [v2,abs_of_pos hN2]
    rw [show 2-rv nu-(2-N) = -(rv nu-N) by ring,abs_neg]; linarith only [hR,hN1]
  have ⟨fd,rd⟩:=opR (subF _ _ f2 hn (by rw [v2]; norm_num) (hR.b2 (by rw [abs_of_pos hN]; linarith only [hN1])))
    r0 (Or.inr (by linarith only [hN1])) (by linarith only [hN]) hN2.le
  have:=qSpec hn fd hR rd (by linarith only [hN1]) (by linarith only [hN])
    (le_trans (inv2le (by norm_num)) hN0) (by linarith only [hN1])
  exact ⟨this.1,this.2.mono (by norm_num)⟩
theorem thCore (ee m:UInt64) (he1:1 ≤ ee.toNat) (he2:ee.toNat ≤ 134) (hm:m.toNat < 2^24)
    (hc0:1/2^150 ≤ cv ee m) (hc:cv ee m < 19) :
    Ap (thM ee m) ((1-exp (-cv ee m))/(1+exp (-cv ee m))) (1/2^44):=by
  obtain ⟨fr,r0,r1,rz,hid⟩:=redSpec ee m he1 he2 hm
  unfold thM; dsimp only
  split_ifs with hk64
  · have hk':(kOf ee m).toNat < 64:=by rwa [UInt64.lt_iff_toNat_lt] at hk64
    obtain ⟨fn,rn⟩:=expU _ hk' _ _ fr rfl r0 r1 rz
    rw [← hid] at rn
    have h63:((kOf ee m).toNat:ℝ) ≤ 63:=by exact_mod_cast (by omega:(kOf ee m).toNat ≤ 63)
    have hE3:1/3 ≤ exp (-cv ee m):=thirdE.trans (exp_le_exp.2 (by
      linarith only [hid,h63,r1]))
    have hc3:=mul_le_mul_of_nonneg_left hE3 (le_trans (by positivity) hc0)
    have:=thTailU _ _ fn rn (by linarith only [gLB (cv ee m),hc3,hc0]) (by linarith only [hE3])
    rwa [show (2:ℝ)-(1-exp (-cv ee m))=1+exp (-cv ee m) by ring] at this
  · have hk':64 ≤ (kOf ee m).toNat:=by rw [UInt64.lt_iff_toNat_lt] at hk64; simpa using hk64
    have h64:(64:ℝ) ≤ (kOf ee m).toNat:=by exact_mod_cast hk'
    obtain ⟨ft,rt⟩:=expT _ (by exact_mod_cast (by linarith only [hid,h64,r0,hc] :
      ((kOf ee m).toNat:ℝ) < 5632)) _ _ fr rfl r0 r1 rz
    rw [← hid] at rt
    exact thTailT _ _ ft rt (exp_pos _) (le_trans (exp_le_exp.2 (by
      linarith only [hid,h64,r0])) halfE)
theorem rse (f m:ℕ) (hw:f%2^29+1024 < 2^28∨2^28+1024 < f%2^29)
    (hm1:2^52+f ≤ m+1024) (hm2:m ≤ 2^52+f+1024) :
    roundShiftRightEven m 29=2^23+(f+2^28)/2^29:=by
  rw [roundShiftRightEven_def]
  have e1:Nat.shiftRight m 29=m/2^29:=Nat.shiftRight_eq_div_pow m 29
  have e2:Nat.shiftLeft (m/2^29) 29=m/2^29*2^29:=Nat.shiftLeft_eq _ _
  simp only [show (29==0)=false from rfl,Bool.false_eq_true,ite_false,e1,e2,
    show 29-1=28 from rfl]
  split_ifs <;> omega
theorem rndN (Y:UInt64) (h:rnd Y≠0xFFFFFFFF) :
    897 ≤ Y.toNat/2^52∧Y.toNat/2^52 ≤ 1022∧1024 ≤ Y.toNat%2^52 ∧
      Y.toNat%2^52+1024 < 2^52 ∧
      (Y.toNat%2^52%2^29+1024 < 2^28∨2^28+1024 < Y.toNat%2^52%2^29) ∧
      (rnd Y).toNat=(Y.toNat/2^52-896)*2^23+(Y.toNat%2^52+2^28)/2^29 ∧
      ((Y>>>52)&&&2047).toNat=Y.toNat/2^52 ∧
      (Y&&&0xFFFFFFFFFFFFF).toNat=Y.toNat%2^52∧(Y>>>63 != 0)=false:=by
  have hY:=Y.toNat_lt
  have hf:(Y&&&0xFFFFFFFFFFFFF).toNat=Y.toNat%2^52:=by
    rw [UInt64.toNat_and]; simp only [UInt64.reduceToNat]
    rw [show (4503599627370495:ℕ)=2^52-1 from rfl,Nat.and_two_pow_sub_one_eq_mod]
  unfold rnd at h ⊢; dsimp only at h ⊢
  split at h
  · rename_i hc
    rw [if_pos hc]
    simp only [Bool.and_eq_true,decide_eq_true_eq] at hc
    obtain ⟨⟨⟨⟨c1,c2⟩,c3⟩,c4⟩,c5⟩:=hc
    rw [UInt64.le_iff_toNat_le,UInt64.toNat_shiftRight] at c1 c2
    rw [UInt64.le_iff_toNat_le,hf] at c3
    rw [UInt64.lt_iff_toNat_lt,hf] at c4
    rw [UInt64.lt_iff_toNat_lt,UInt64.toNat_sub,UInt64.toNat_and] at c5
    simp only [UInt64.reduceToNat,Nat.reduceMod,Nat.shiftRight_eq_div_pow] at c1 c2 c3 c4 c5
    rw [show (536870911:ℕ)=2^29-1 from rfl,Nat.and_two_pow_sub_one_eq_mod] at c5
    have hm:Y.toNat%2^52%2^29=Y.toNat%2^29:=Nat.mod_mod_of_dvd _ (by norm_num)
    refine ⟨c1,c2,c3,by omega,by rw [hm]; omega,?_,?_,hf,?_⟩
    · rw [UInt32.toNat_sub,UInt64.toNat_toUInt32,UInt64.toNat_shiftRight,UInt64.toNat_add]
      simp only [UInt64.reduceToNat,UInt32.reduceToNat,Nat.reduceMod,Nat.shiftRight_eq_div_pow]
      omega
    · rw [UInt64.toNat_and,UInt64.toNat_shiftRight]
      simp only [UInt64.reduceToNat,Nat.reduceMod,Nat.shiftRight_eq_div_pow]
      rw [show (2047:ℕ)=2^11-1 from rfl,Nat.and_two_pow_sub_one_eq_mod]; omega
    · have:Y>>>63=0:=UInt64.toNat_inj.mp (by
        rw [UInt64.toNat_shiftRight]; simp only [UInt64.reduceToNat,Nat.reduceMod,Nat.shiftRight_eq_div_pow]
        omega)
      rw [this]; rfl
  · exact absurd rfl h
theorem rwN (neg:Bool) (ef f m:ℕ) (h1:897 ≤ ef) (h2:ef ≤ 1022) (h3:1024 ≤ f)
    (h4:f+1024 < 2^52) (hw:f%2^29+1024 < 2^28∨2^28+1024 < f%2^29)
    (hm1:2^52+f ≤ m+1024) (hm2:m ≤ 2^52+f+1024) :
    FastRound.roundW neg m ((ef:ℤ)-1075) =
      (if neg then 0x80000000 else 0)+UInt32.ofNat ((ef-896)*2^23+(f+2^28)/2^29):=by
  have hL:m.log2=52:=by rw [Nat.log2_eq_iff (by omega)]; omega
  unfold FastRound.roundW
  simp only [hL,show 52-23=29 from rfl,rse f m hw hm1 hm2,FastRound.p23,FastRound.p24]
  have hq:(f+2^28)/2^29 ≤ 2^23:=by omega
  rw [if_pos ⟨by norm_num,by omega,by omega,by omega,by omega⟩,add_assoc]
  refine congrArg (fun x=>(if neg=true then 2147483648 else 0)+x) ?_
  generalize (f+2^28)/2^29=q at *
  apply UInt32.toNat_inj.mp
  simp only [UInt32.toNat_add,UInt32.toNat_shiftLeft,Nat.toUInt32,UInt32.toNat_ofNat',UInt32.reduceToNat,
    Nat.reduceMod,Nat.shiftLeft_eq]
  split_ifs <;> omega
theorem encl {y v P:ℝ} {M:ℕ} (hR:Rel y v (1/2^44)) (hv:0 < v) (hP:0 < P)
    (hy:y=M*P) (hM:M < 2^53) :
    ((M:ℝ)-1024)*P ≤ v∧v ≤ ((M:ℝ)+1024)*P:=by
  unfold Rel at hR; rw [abs_of_pos hv,hy] at hR
  have h1:(M:ℝ)*P ≤ 2^53*P:=mul_le_mul_of_nonneg_right (by exact_mod_cast hM.le) hP.le
  have h2:=abs_le.1 hR
  constructor <;> linarith only [h2.1,h2.2,h1,hv,hP]
theorem wordOK (a:UInt32) (h1:2^23 ≤ a.toNat%2^31) (h2:a.toNat%2^31 < 255*2^23) :
    expOf a≠255∧(decodeD a).significand≠0:=by
  have he:(expOf a).toNat=a.toNat/2^23%256:=by
    simp only [expOf,UInt32.toNat_and,UInt32.toNat_shiftRight,UInt32.reduceToNat,Nat.reduceMod,
      Nat.shiftRight_eq_div_pow]
    rw [show (255:ℕ)=2^8-1 from rfl,Nat.and_two_pow_sub_one_eq_mod]
  have ha:=a.toNat_lt
  have hx:a.toNat/2^23%256=a.toNat%2^31/2^23:=by omega
  refine ⟨fun h => ?_,?_⟩
  · have:=congrArg UInt32.toNat h; rw [he] at this; simp at this; omega
  · unfold decodeD
    have hne:¬ expOf a=0:=fun h=>by
      have:=congrArg UInt32.toNat h; rw [he] at this; simp at this; omega
    simp only [hne,ite_false]; omega
theorem fin (y:Float) (v:ℝ) (hv:0 < v) (hR:Rel (rv y) v (1/2^44))
    (ho:rnd y.toBits≠0xFFFFFFFF) (neg:Bool) {z:Model Fmt}
    (hz:Model.toReal z=Model.roundAt Fmt (if neg then -v else v)) :
    toWord z=(if neg then 0x80000000 else 0)+rnd y.toBits:=by
  obtain ⟨h1,h2,h3,h4,hw,hN,hef,hf,hs⟩:=rndN y.toBits ho
  generalize y.toBits.toNat/2^52=ef at *
  generalize y.toBits.toNat%2^52=f at *
  have hyv:rv y=((2^52+f:ℕ):ℝ)*2^((ef:ℤ)-1075):=by
    unfold rv; rw [modelW]
    have ⟨_,v1⟩:=valW y.toBits (by omega) (by omega)
    rw [v1,hef,hf,hs]; simp
  have ⟨lo,hi⟩:=encl hR hv (by positivity) hyv (by omega)
  have hr:rnd y.toBits=UInt32.ofNat ((ef-896)*2^23+(f+2^28)/2^29) :=
    UInt32.toNat_inj.mp (by rw [hN,UInt32.toNat_ofNat']; omega)
  have hW1:=rwN neg ef f (2^52+f-1024) h1 h2 h3 h4 hw (by omega) (by omega)
  have hW2:=rwN neg ef f (2^52+f+1024) h1 h2 h3 h4 hw (by omega) (by omega)
  rw [← hr] at hW1 hW2
  have hok:=wordOK ((if neg then 0x80000000 else 0)+rnd y.toBits)
    (by cases neg <;> simp [UInt32.toNat_add] <;> omega)
    (by cases neg <;> simp [UInt32.toNat_add] <;> omega)
  have c1:(((2^52+f-1024:ℕ):ℤ):ℝ)=((2^52+f:ℕ):ℝ)-1024:=by
    rw [Int.cast_natCast,Nat.cast_sub (by omega)]; push_cast; ring
  have c2:(((2^52+f+1024:ℕ):ℤ):ℝ)=((2^52+f:ℕ):ℝ)+1024:=by
    push_cast; ring
  cases neg
  · rw [FastRound.roundW_signed _ _ _ (by omega)] at hW1 hW2
    simp only [Bool.false_eq_true,ite_false] at hW1 hW2 hz ⊢
    exact FastRound.bracket_spec (by omega) (by omega) hW1 hW2 hok.1 hok.2
      (by rw [c1]; exact lo) (by rw [c2]; exact hi) hz
  · rw [FastRound.roundW_signed _ _ _ (by omega)] at hW1 hW2
    simp only [ite_true] at hW1 hW2 hz ⊢
    refine FastRound.bracket_spec (by omega) (by omega) hW2 hW1 hok.1 hok.2 ?_ ?_ hz
    · rw [Int.cast_neg,c2]; linarith only [hi]
    · rw [Int.cast_neg,c1]; linarith only [lo]
theorem toReal_ofW' (w:UInt32) (hfin:expOf w≠255) :
    Model.toReal (ofW w) =
      (if (decodeD w).negative then -1 else 1)*((decodeD w).significand*(2:ℝ)^(decodeD w).exponent):=by
  rw [Model.toReal_eq,ofW,toDyadic_ofUInt32 _ hfin]
  simp only [FloatLib.Numerics.Dyadic.toReal,FloatLib.Numerics.Dyadic.cast_signedSignificand]
  ring
theorem dec32 (w:UInt32) :
    (eW w).toNat=(expOf w).toNat∧(mW w).toNat=2^23+(fracOf w).toNat ∧
    (w&&&0x7FFFFFFF).toNat=(expOf w).toNat*2^23+(fracOf w).toNat ∧
    (fracOf w).toNat < 2^23 ∧
    (signOf w=true ↔ 0x80000000 ≤ w) ∧
    (w&&&0x80000000).toNat=(if signOf w then 2^31 else 0):=by
  unfold eW mW
  have hw:=w.toNat_lt
  have ha:(w&&&0x7FFFFFFF).toNat=w.toNat%2^31:=by
    rw [UInt32.toNat_and]; simp only [UInt32.reduceToNat]
    rw [show (2147483647:ℕ)=2^31-1 from rfl,Nat.and_two_pow_sub_one_eq_mod]
  have he:(expOf w).toNat=w.toNat/2^23%256:=by
    simp only [expOf,UInt32.toNat_and,UInt32.toNat_shiftRight,UInt32.reduceToNat,Nat.reduceMod,
      Nat.shiftRight_eq_div_pow]
    rw [show (255:ℕ)=2^8-1 from rfl,Nat.and_two_pow_sub_one_eq_mod]
  have hf:(fracOf w).toNat=w.toNat%2^23:=by
    simp only [fracOf,UInt32.toNat_and,UInt32.reduceToNat]
    rw [show (8388607:ℕ)=2^23-1 from rfl,Nat.and_two_pow_sub_one_eq_mod]
  have hs:(w&&&0x80000000).toNat=w.toNat/2^31*2^31:=by
    rw [UInt32.toNat_and]; simp only [UInt32.reduceToNat]
    rw [show (2147483648:ℕ)=2^31 from rfl,Nat.and_two_pow]
    rw [Nat.testBit_eq_decide_div_mod_eq]
    have:w.toNat/2147483648=0∨w.toNat/2147483648=1:=by omega
    rcases this with h | h <;> simp [h]
  refine ⟨?_,?_,?_,?_,?_,?_⟩
  · rw [UInt32.toNat_toUInt64,UInt32.toNat_shiftRight,ha,he]
    simp only [UInt32.reduceToNat,Nat.reduceMod,Nat.shiftRight_eq_div_pow]; omega
  · rw [UInt32.toNat_toUInt64,UInt32.toNat_or,UInt32.toNat_and,ha,hf]
    simp only [UInt32.reduceToNat]
    rw [show (8388607:ℕ)=2^23-1 from rfl,Nat.and_two_pow_sub_one_eq_mod,
      show (8388608:ℕ)=2^23 from rfl,Nat.or_two_pow_eq_add_of_lt (Nat.mod_lt _ (by positivity))]
    omega
  · rw [ha,he,hf]; omega
  · rw [hf]; exact Nat.mod_lt _ (by positivity)
  · unfold signOf
    rw [UInt32.le_iff_toNat_le]
    simp only [bne_iff_ne,ne_eq, ← UInt32.toNat_inj,hs,UInt32.reduceToNat]
    constructor <;> intro h <;> omega
  · unfold signOf
    rw [hs]
    by_cases h:w.toNat/2^31=0
    · have:(w&&&0x80000000)=0:=UInt32.toNat_inj.mp (by rw [hs,h]; rfl)
      simp [this]; omega
    · have h1:w.toNat/2^31=1:=by omega
      have:(w&&&0x80000000)≠0:=fun e=>by
        have:=congrArg UInt32.toNat e; rw [hs,h1] at this; simp at this
      simp [this]; omega
theorem xval (w:UInt32) (h0:expOf w≠0) (h1:expOf w≠255) :
    Model.toReal (ofW w)=(if signOf w then -1 else 1) *
      (((2^23+(fracOf w).toNat:ℕ):ℝ)*(2:ℝ)^(((expOf w).toNat:ℤ)-150)):=by
  rw [toReal_ofW' w h1]
  unfold decodeD
  simp only [h0,ite_false]
noncomputable def C32 (ex fr:ℕ):ℝ:=((2^23+fr:ℕ):ℝ)*(2:ℝ)^((ex:ℤ)-150)
theorem C32_lt (ex fr E0 F0:ℕ) (hfr:fr < 2^23) (hF0:F0 < 2^23)
    (h:ex*2^23+fr < E0*2^23+F0):C32 ex fr < C32 E0 F0:=by
  unfold C32
  rcases Nat.lt_or_ge ex E0 with he | he
  · have h1:((2^23+fr:ℕ):ℝ)*(2:ℝ)^((ex:ℤ)-150) < (2:ℝ)^(24:ℕ)*2^((ex:ℤ)-150):=by
      apply mul_lt_mul_of_pos_right _ (by positivity); exact_mod_cast (by omega:2^23+fr < 2^24)
    have h2:(2:ℝ)^(24:ℕ)*2^((ex:ℤ)-150) ≤ (2:ℝ)^(23:ℕ)*2^((E0:ℤ)-150):=by
      rw [← zpow_natCast, ← zpow_natCast, ← zpow_add₀ two_ne_zero, ← zpow_add₀ two_ne_zero]
      exact zpow_le_zpow_right₀ (by norm_num) (by push_cast; omega)
    have h3:(2:ℝ)^(23:ℕ)*2^((E0:ℤ)-150) ≤ ((2^23+F0:ℕ):ℝ)*2^((E0:ℤ)-150):=by
      apply mul_le_mul_of_nonneg_right _ (by positivity); exact_mod_cast (by omega:2^23 ≤ 2^23+F0)
    linarith
  · have hee:ex=E0:=by omega
    subst hee
    apply mul_lt_mul_of_pos_right _ (by positivity); exact_mod_cast (by omega:2^23+fr < 2^23+F0)
theorem C32_le (ex fr E0 F0:ℕ) (hfr:fr < 2^23) (hF0:F0 < 2^23)
    (h:E0*2^23+F0 ≤ ex*2^23+fr):C32 E0 F0 ≤ C32 ex fr:=by
  rcases (Nat.lt_or_ge (E0*2^23+F0) (ex*2^23+fr)) with h' | h'
  · exact (C32_lt E0 F0 ex fr hF0 hfr h').le
  · have:E0=ex∧F0=fr:=by omega
    rw [this.1,this.2]
theorem e18:exp (-18) ≤ 1/2^25:=by
  have h1:(2.7:ℝ) ≤ exp 1:=by have:=Real.exp_one_gt_d9; linarith
  have h2:(2.7:ℝ)^(18:ℕ) ≤ exp 18:=by
    rw [show (18:ℝ)=((18:ℕ):ℝ)*1 by norm_num,Real.exp_nat_mul]
    exact pow_le_pow_left₀ (by norm_num) h1 18
  rw [Real.exp_neg,inv_eq_one_div]
  exact one_div_le_one_div_of_le (by positivity) (le_trans (by norm_num) h2)
theorem sgS (w:UInt32) (he1:expOf w≠255) :
    Model.toReal (sigmoidModel (ofW w))=Model.roundAt Fmt (realSigmoid (Model.toReal (ofW w))) :=
  (toReal_sigmoidModel (fmt:=Fmt) (by decide) (input:=Model.ofUInt32 w) (by simp [isFinite_ofUInt32,he1])).2
theorem thS (w:UInt32) (he1:expOf w≠255) :
    Model.toReal (tanhModel (ofW w))=Model.roundAt Fmt (tanh (Model.toReal (ofW w))) :=
  (toReal_tanhModel (fmt:=Fmt) (by decide) (input:=Model.ofUInt32 w) (by simp [isFinite_ofUInt32,he1])).2
theorem sgZero (w:UInt32) (h:(w&&&0x7FFFFFFF)=0):sigmoidCR w=0x3F000000:=by
  obtain ⟨_,_,ha,hfr,_,_⟩:=dec32 w
  rw [h] at ha; simp at ha
  have he:expOf w=0:=UInt32.toNat_inj.mp (by simp; omega)
  have hf:fracOf w=0:=UInt32.toNat_inj.mp (by simp; omega)
  have hx:Model.toReal (ofW w)=0:=by
    rw [toReal_ofW' w (by simp [he])]; unfold decodeD; simp [he,hf]
  have hspec:=sgS w (by simp [he])
  rw [hx] at hspec
  change toWord (sigmoidModel (ofW w))=_
  have hv:realSigmoid 0=((2^23:ℤ):ℝ)*2^(-24:ℤ):=by
    unfold realSigmoid; norm_num
  have hw:finishWord (2^23) (-24) false=0x3F000000 :=
    (FastRound.roundW_signed false (2^23) (-24) (by norm_num)).symm.trans (by decide)
  exact FastRound.bracket_spec (by norm_num) (by norm_num) hw hw (by decide) (by decide)
    (le_of_eq hv.symm) (le_of_eq hv) hspec
theorem thZero (w:UInt32) (h:(w&&&0x7FFFFFFF)=0):tanhCR w=w:=by
  have hw:=w.toNat_lt
  have ha:(w&&&0x7FFFFFFF).toNat=w.toNat%2^31:=by
    rw [UInt32.toNat_and]; simp only [UInt32.reduceToNat]
    rw [show (2147483647:ℕ)=2^31-1 from rfl,Nat.and_two_pow_sub_one_eq_mod]
  rw [h] at ha; simp at ha
  have:w=0∨w=0x80000000:=by
    rcases (show w.toNat=0∨w.toNat=2147483648 by omega) with e | e
    · left; exact UInt32.toNat_inj.mp (by rw [e]; rfl)
    · right; exact UInt32.toNat_inj.mp (by rw [e]; rfl)
  rcases this with e | e <;> subst e <;> decide
theorem wv (w:UInt32) (h0:2^23 ≤ (w&&&0x7FFFFFFF).toNat)
    (h1:(w&&&0x7FFFFFFF).toNat < 0x7F800000) :
    expOf w≠255∧Model.toReal (ofW w)=(if signOf w then -1 else 1)*cv (eW w) (mW w) ∧
      cv (eW w) (mW w)=C32 (expOf w).toNat (fracOf w).toNat∧(mW w).toNat < 2^24:=by
  obtain ⟨hee,hm,ha,hfr, -, -⟩:=dec32 w
  have he0:expOf w≠0:=fun e=>by have:=congrArg UInt32.toNat e; simp at this; omega
  have he1:expOf w≠255:=fun e=>by have:=congrArg UInt32.toNat e; simp at this; omega
  have hc:cv (eW w) (mW w)=C32 (expOf w).toNat (fracOf w).toNat:=by unfold cv C32; rw [hee,hm]
  exact ⟨he1,by rw [xval w he0 he1,hc]; rfl,hc,by omega⟩
theorem satW (neg:Bool) (v:ℝ) (h1:1-1/2^25 ≤ v) (h2:v ≤ 1) {z:Model Fmt}
    (hz:Model.toReal z=Model.roundAt Fmt (if neg then -v else v)) :
    toWord z=(if neg then 0x80000000 else 0)+0x3F800000:=by
  have e1:(((2^25-1:ℤ):ℝ)*2^(-25:ℤ))=1-1/2^25:=by norm_num
  have e2:(((2^25:ℤ):ℝ)*2^(-25:ℤ))=1:=by norm_num
  cases neg
  · exact FastRound.bracket_spec (lo:=2^25-1) (hi:=2^25) (e := -25) (v:=v) (by norm_num) (by norm_num)
      ((FastRound.roundW_signed false (2^25-1) _ (by norm_num)).symm.trans (by decide))
      ((FastRound.roundW_signed false (2^25) _ (by norm_num)).symm.trans (by decide)) (by decide)
      (by decide) (by rw [e1]; exact h1) (by rw [e2]; exact h2) (by simpa using hz)
  · exact FastRound.bracket_spec (lo := -(2^25)) (hi := -(2^25-1)) (e := -25) (v := -v) (by norm_num)
      (by norm_num) ((FastRound.roundW_signed true (2^25) _ (by norm_num)).symm.trans (by decide))
      ((FastRound.roundW_signed true (2^25-1) _ (by norm_num)).symm.trans (by decide)) (by decide)
      (by decide) (by rw [Int.cast_neg,neg_mul,e2]; linarith only [h2])
      (by rw [Int.cast_neg,neg_mul,e1]; linarith only [h1]) (by simpa using hz)
theorem sgSat (w:UInt32) (h1:w < 0x80000000) (h2:0x41900000 ≤ w&&&0x7FFFFFFF)
    (h3:w&&&0x7FFFFFFF < 0x7F800000):sigmoidCR w=0x3F800000:=by
  obtain ⟨_,_,ha,hfr,hs, -⟩:=dec32 w
  rw [UInt32.le_iff_toNat_le] at h2; rw [UInt32.lt_iff_toNat_lt] at h3
  simp only [UInt32.reduceToNat] at h2 h3
  obtain ⟨he1,hx,hc, -⟩:=wv w (by omega) h3
  have hneg:signOf w=false:=by
    cases hh:signOf w
    · rfl
    · exact absurd (hs.1 hh) (UInt32.not_le.2 h1)
  have h18:18 ≤ cv (eW w) (mW w):=by
    rw [hc]; exact le_trans (by unfold C32; norm_num) (C32_le _ _ 131 1048576 hfr (by norm_num) (by omega))
  have hspec:=sgS w he1
  rw [hx,hneg] at hspec
  have hE:exp (-cv (eW w) (mW w)) ≤ 1/2^25 :=
    le_trans (exp_le_exp.2 (by linarith only [h18])) e18
  have hp:=exp_pos (-cv (eW w) (mW w))
  have:=satW false (realSigmoid (cv (eW w) (mW w)))
    (by unfold realSigmoid; rw [le_div_iff₀ (by linarith only [hp])]; linarith only [hE,hp])
    (realSigmoid_lt_one _).le (z:=sigmoidModel (ofW w)) (by simpa using hspec)
  change toWord (sigmoidModel (ofW w))=_
  simpa using this
theorem thSat (w:UInt32) (h2:0x41180000 ≤ w&&&0x7FFFFFFF) (h3:w&&&0x7FFFFFFF < 0x7F800000) :
    tanhCR w=(w&&&0x80000000)+0x3F800000:=by
  obtain ⟨_,_,ha,hfr, -, hsb⟩:=dec32 w
  rw [UInt32.le_iff_toNat_le] at h2; rw [UInt32.lt_iff_toNat_lt] at h3
  simp only [UInt32.reduceToNat] at h2 h3
  obtain ⟨he1,hx,hc, -⟩:=wv w (by omega) h3
  have h95:19/2 ≤ cv (eW w) (mW w):=by
    rw [hc]; exact le_trans (by unfold C32; norm_num) (C32_le _ _ 130 1572864 hfr (by norm_num) (by omega))
  have hspec:=thS w he1
  rw [hx] at hspec
  have hE:exp (-2*cv (eW w) (mW w)) ≤ 1/2^26:=by
    have:=mul_le_mul (le_trans (exp_le_exp.2 (by linarith only [h95] :
      -2*cv (eW w) (mW w)+1 ≤ -18)) e18) halfE (exp_pos _).le (by norm_num)
    rw [← exp_add,show -2*cv (eW w) (mW w)+1 + -1 = -2*cv (eW w) (mW w) by ring] at this
    linarith only [this]
  have hp:=exp_pos (-2*cv (eW w) (mW w))
  have hf:=satW (signOf w) (tanh (cv (eW w) (mW w)))
    (by rw [tanh_eq_exp_neg_two,le_div_iff₀ (by linarith only [hp])]; linarith only [hE,hp])
    (Real.tanh_lt_one _).le (z:=tanhModel (ofW w)) (by rw [hspec]; cases signOf w <;> simp [Real.tanh_neg])
  change toWord (tanhModel (ofW w))=_
  rw [hf]; congr 1
  apply UInt32.toNat_inj.mp; rw [hsb]; cases signOf w <;> simp
theorem sgMain (w:UInt32) (h0:2^23 ≤ (w&&&0x7FFFFFFF).toNat)
    (h1:(w&&&0x7FFFFFFF).toNat < 0x42AE0000) (h:rnd (sgY w).toBits≠0xFFFFFFFF) :
    sigmoidCR w=rnd (sgY w).toBits:=by
  obtain ⟨hee,hm,ha,hfr,hs, -⟩:=dec32 w
  obtain ⟨he1,hx,hc,hm2⟩:=wv w h0 (by omega)
  have h87:cv (eW w) (mW w) < 87:=by
    rw [hc]; exact (C32_lt _ _ 133 3014656 hfr (by norm_num) (by omega)).trans_eq (by unfold C32; norm_num)
  obtain ⟨fr,r0,r1,rz,hid⟩:=redSpec (eW w) (mW w) (by omega) (by omega) hm2
  have hk:((kOf (eW w) (mW w)).toNat:ℝ) < 5632:=by linarith
  obtain ⟨ft,rt⟩:=expT _ (by exact_mod_cast hk) _ _ fr rfl r0 r1 rz
  rw [← hid] at rt
  obtain ⟨⟨fy1,ry1⟩,⟨fy2,ry2⟩⟩:=sgTail _ _ ft rt (expLB (by linarith))
    (exp_le_one_iff.2 (by linarith))
  have hspec:=sgS w he1
  change toWord (sigmoidModel (ofW w))=_
  rw [hx] at hspec
  have hpT:=exp_pos (-cv (eW w) (mW w))
  unfold sgY at h ⊢; dsimp only at h ⊢
  by_cases hw:w < 0x80000000
  · have hsg:signOf w=false:=by
      cases hh:signOf w
      · rfl
      · exact absurd (hs.1 hh) (UInt32.not_le.2 hw)
    rw [if_pos hw] at h ⊢
    simp only [hsg,Bool.false_eq_true,ite_false,one_mul] at hspec
    have hv:realSigmoid (cv (eW w) (mW w))=1/(1+exp (-cv (eW w) (mW w))):=rfl
    rw [hv] at hspec
    have:=fin _ _ (by positivity) ry1 h false (z:=sigmoidModel (ofW w)) (by simpa using hspec)
    simpa using this
  · have hsg:signOf w=true:=hs.2 (UInt32.not_lt.1 hw)
    rw [if_neg hw] at h ⊢
    simp only [hsg,ite_true,neg_mul,one_mul] at hspec
    have hv:realSigmoid (-cv (eW w) (mW w)) =
        exp (-cv (eW w) (mW w))/(1+exp (-cv (eW w) (mW w))):=by
      unfold realSigmoid; rw [neg_neg]
      have h1:exp (cv (eW w) (mW w))*exp (-cv (eW w) (mW w))=1:=by
        rw [← exp_add]; simp
      field_simp; linarith
    rw [hv] at hspec
    have:=fin _ _ (by positivity) ry2 h false (z:=sigmoidModel (ofW w)) (by simpa using hspec)
    simpa using this
theorem tanhPos (c:ℝ) (hc:0 < c):0 < tanh c:=by
  rw [tanh_eq_exp_neg_two]
  have:exp (-2*c) < 1:=Real.exp_lt_one_iff.2 (by linarith)
  have hp:=exp_pos (-2*c)
  exact div_pos (by linarith) (by linarith)
theorem thMain (w:UInt32) (h0:2^23 ≤ (w&&&0x7FFFFFFF).toNat)
    (h1:(w&&&0x7FFFFFFF).toNat < 0x41180000) (h:rnd (thY w).toBits≠0xFFFFFFFF) :
    tanhCR w=rnd (thY w).toBits+(w&&&0x80000000):=by
  obtain ⟨hee,hm,ha,hfr, -, hsb⟩:=dec32 w
  obtain ⟨he1,hx,hc,hm2⟩:=wv w h0 (by omega)
  have h95:cv (eW w) (mW w) < 19/2:=by
    rw [hc]; exact (C32_lt _ _ 130 1572864 hfr (by norm_num) (by omega)).trans_eq (by unfold C32; norm_num)
  have hc1:(1:ℝ)/2^150 ≤ cv (eW w) (mW w):=by
    unfold cv
    have h1:(1:ℝ) ≤ (mW w).toNat:=by exact_mod_cast (by omega:1 ≤ (mW w).toNat)
    have h2:(1:ℝ)/2^150 ≤ 2^(((eW w).toNat:ℤ)-150):=by
      rw [← zp]; exact zpow_le_zpow_right₀ (by norm_num) (by omega)
    exact h2.trans (le_mul_of_one_le_left (by positivity) h1)
  have he:(eW w+1).toNat=(eW w).toNat+1:=by rw [UInt64.toNat_add]; simp; omega
  have h2:cv (eW w+1) (mW w)=2*cv (eW w) (mW w):=by
    unfold cv; rw [he]; push_cast
    rw [show ((eW w).toNat:ℤ)+1-150=((eW w).toNat:ℤ)-150+1 by ring,zpow_add₀ two_ne_zero]
    ring
  obtain ⟨fy,ry⟩:=thCore (eW w+1) (mW w) (by omega) (by omega) hm2
    (by rw [h2]; linarith) (by rw [h2]; linarith)
  rw [h2,show (1-exp (-(2*cv (eW w) (mW w))))/(1+exp (-(2*cv (eW w) (mW w)))) =
    tanh (cv (eW w) (mW w)) by rw [tanh_eq_exp_neg_two,neg_mul]] at ry
  have hspec:=thS w he1
  change toWord (tanhModel (ofW w))=_
  rw [hx] at hspec
  have hf:=fin _ _ (tanhPos _ (by linarith)) ry h (signOf w) (z:=tanhModel (ofW w)) (by
    rw [hspec]; cases signOf w <;> simp [Real.tanh_neg])
  rw [hf,add_comm]
  congr 1
  apply UInt32.toNat_inj.mp
  rw [hsb]; cases signOf w <;> simp
theorem sgF_eq (w:UInt32):sgF w =
    if (w&&&0x7FFFFFFF < 0x00800000||0x7F800000 ≤ w&&&0x7FFFFFFF) then
      (if w&&&0x7FFFFFFF==0 then 0x3F000000 else 0xFFFFFFFF)
    else if (w < 0x80000000&&0x41900000 ≤ w&&&0x7FFFFFFF) then 0x3F800000
    else if 0x42AE0000 ≤ w&&&0x7FFFFFFF then 0xFFFFFFFF
    else rnd (sgY w).toBits:=rfl
theorem thF_eq (w:UInt32):thF w =
    if (w&&&0x7FFFFFFF < 0x00800000||0x7F800000 ≤ w&&&0x7FFFFFFF) then
      (if w&&&0x7FFFFFFF==0 then w else 0xFFFFFFFF)
    else if 0x41180000 ≤ w&&&0x7FFFFFFF then (w&&&0x80000000)+0x3F800000
    else if rnd (thY w).toBits==0xFFFFFFFF then rnd (thY w).toBits
    else rnd (thY w).toBits+(w&&&0x80000000):=rfl
theorem sgF_spec (w:UInt32) (h:sgF w≠0xFFFFFFFF):sigmoidCR w=sgF w:=by
  rw [sgF_eq] at h ⊢
  by_cases c1:(w&&&0x7FFFFFFF < 0x00800000||0x7F800000 ≤ w&&&0x7FFFFFFF)=true
  · rw [if_pos c1] at h ⊢
    by_cases c2:(w&&&0x7FFFFFFF==0)=true
    · rw [if_pos c2]; exact sgZero w (by simpa using c2)
    · rw [if_neg c2] at h; exact absurd rfl h
  rw [if_neg c1] at h ⊢
  simp only [Bool.or_eq_true,decide_eq_true_eq,not_or,UInt32.not_lt,UInt32.not_le] at c1
  by_cases c3:(w < 0x80000000&&0x41900000 ≤ w&&&0x7FFFFFFF)=true
  · rw [if_pos c3]; simp only [Bool.and_eq_true,decide_eq_true_eq] at c3; exact sgSat w c3.1 c3.2 c1.2
  rw [if_neg c3] at h ⊢
  by_cases c4:0x42AE0000 ≤ w&&&0x7FFFFFFF
  · rw [if_pos c4] at h; exact absurd rfl h
  rw [if_neg c4] at h ⊢
  rw [UInt32.not_le,UInt32.lt_iff_toNat_lt] at c4; rw [UInt32.le_iff_toNat_le] at c1
  exact sgMain w c1.1 c4 h
theorem thF_spec (w:UInt32) (h:thF w≠0xFFFFFFFF):tanhCR w=thF w:=by
  rw [thF_eq] at h ⊢
  by_cases c1:(w&&&0x7FFFFFFF < 0x00800000||0x7F800000 ≤ w&&&0x7FFFFFFF)=true
  · rw [if_pos c1] at h ⊢
    by_cases c2:(w&&&0x7FFFFFFF==0)=true
    · rw [if_pos c2]; exact thZero w (by simpa using c2)
    · rw [if_neg c2] at h; exact absurd rfl h
  rw [if_neg c1] at h ⊢
  simp only [Bool.or_eq_true,decide_eq_true_eq,not_or,UInt32.not_lt,UInt32.not_le] at c1
  by_cases c3:0x41180000 ≤ w&&&0x7FFFFFFF
  · rw [if_pos c3]; exact thSat w c3 c1.2
  rw [if_neg c3] at h ⊢
  by_cases c4:(rnd (thY w).toBits==0xFFFFFFFF)=true
  · rw [if_pos c4] at h; exact absurd (beq_iff_eq.1 c4) h
  rw [if_neg c4] at h ⊢
  rw [UInt32.not_le,UInt32.lt_iff_toNat_lt] at c3; rw [UInt32.le_iff_toNat_le] at c1
  exact thMain w c1.1 c3 (by simpa using c4)
theorem sg_eq:sg=sigmoidCR:=by
  funext w
  unfold sg
  dsimp only
  by_cases h:sgF w=0xFFFFFFFF
  · simp [h]
  · rw [if_neg (by simpa using h)]; exact (sgF_spec w h).symm
theorem th_eq:th=tanhCR:=by
  funext w
  unfold th
  dsimp only
  by_cases h:thF w=0xFFFFFFFF
  · simp [h]
  · rw [if_neg (by simpa using h)]; exact (thF_spec w h).symm
end SAct
namespace Cv1
open ExactRuntime FloatLib.Floats.Formats.BinaryInterchange FloatLib.Floats
@[inline]def c1Ok(s:ConvShape):Bool:=
 s.w==1&&s.kw==1&&s.strideH==1&&s.strideW==1&&s.padT==0&&s.padL==0&&s.padB==0&&s.padR==0&&s.dilH==1&&s.dilW==1&&0<s.kh&&s.kh<=s.h
def im2c(h kh k ho:Nat)(inp:Array Word):Array Word:=Array.ofFn (n:=ho*k) fun q=>inp[(q.val%k/kh)*h+q.val/k+q.val%k%kh]!
def trn(rows cols:Nat)(Y:Array Word):Array Word:=Array.ofFn (n:=cols*rows) fun q=>Y[(q.val%rows)*cols+q.val/rows]!
theorem c1F(s:ConvShape)(hc:c1Ok s=true):s.w=1∧s.kw=1∧s.strideH=1∧s.strideW=1∧s.padT=0∧s.padL=0∧s.padB=0∧s.padR=0∧s.dilH=1∧s.dilW=1∧0<s.kh∧s.kh<=s.h:=by
 simp only[c1Ok,Bool.and_eq_true,beq_iff_eq,decide_eq_true_eq]at hc
 obtain⟨⟨⟨⟨⟨⟨⟨⟨⟨⟨⟨a1,a2⟩,a3⟩,a4⟩,a5⟩,a6⟩,a7⟩,a8⟩,a9⟩,a10⟩,a11⟩,a12⟩:=hc
 exact⟨a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12⟩
theorem c1HW(s:ConvShape)(hc:c1Ok s=true):s.ho=s.h+1-s.kh∧s.wo=1:=by
 obtain⟨hw,hkw,hsh,hsw,hpt,hpl,hpb,hpr,hdh,hdw,hk0,hkh⟩:=c1F s hc
 constructor
 ·simp only[ConvShape.ho,ConvShape.ekh,hpt,hpb,hdh,hsh,Nat.one_mul,Nat.add_zero,Nat.div_one];omega
 ·simp only[ConvShape.wo,ConvShape.ekw,hpl,hpr,hdw,hsw,hw,hkw,Nat.one_mul,Nat.add_zero,Nat.div_one]
theorem lin_conv1(s:ConvShape)(hc:c1Ok s=true)(hg:s.groups=1)(relu:Bool)(inp wts b:Array Word):
  trn (s.h+1-s.kh) s.cout (linearSpec (s.h+1-s.kh) (s.cin*s.kh) s.cout relu (im2c s.h s.kh (s.cin*s.kh) (s.h+1-s.kh) inp) wts b)
   =conv2dSpec s relu inp wts b:=by
 obtain⟨hw,hkw,hsh,hsw,hpt,hpl,hpb,hpr,hdh,hdw,hk0,hkh⟩:=c1F s hc
 obtain⟨eho,ewo⟩:=c1HW s hc
 have etp:s.taps=s.cin*s.kh:=by simp only[ConvShape.taps,ConvShape.cpg,hg,Nat.div_one,hkw,Nat.mul_one]
 have eog:s.opg=s.cout:=by simp only[ConvShape.opg,hg,Nat.div_one]
 unfold trn linearSpec linearWith conv2dSpec conv2dWith
 apply Array.ext
 · simp only[Array.size_ofFn,eho,ewo,Nat.mul_one]
 intro q h1 h2
 simp only[Array.size_ofFn]at h1
 generalize hH:s.h+1-s.kh=ho at *
 have hq:q/ho<s.cout:=Nat.div_lt_of_lt_mul (by rw[Nat.mul_comm];exact h1)
 have hr:q%ho<ho:=Nat.mod_lt _ (Nat.pos_of_ne_zero (fun z=>by simp[z]at h1))
 have hi:(q%ho)*s.cout+q/ho<ho*s.cout:=by nlinarith
 simp only[Array.getElem_ofFn,eho,ewo,etp,eog,Nat.mul_one,Nat.div_one,Nat.mod_one]
 rw[getElem!_pos _ _ (by simpa using hi),Array.getElem_ofFn]
 have d1:((q%ho)*s.cout+q/ho)/s.cout=q%ho:=by rw[Nat.add_comm,Nat.add_mul_div_right _ _ (Nat.lt_of_le_of_lt (Nat.zero_le _) hq),Nat.div_eq_of_lt hq,Nat.zero_add]
 have d2:((q%ho)*s.cout+q/ho)%s.cout=q/ho:=by rw[Nat.add_comm,Nat.add_mul_mod_self_right,Nat.mod_eq_of_lt hq]
 have g0:q/ho/s.cout=0:=Nat.div_eq_of_lt hq
 simp only[d1,d2,g0]
 congr 2
 funext t
 by_cases ht:t<s.cin*s.kh
 · simp only[ht,ite_true]
   have hx:(q%ho)*(s.cin*s.kh)+t<ho*(s.cin*s.kh):=by nlinarith
   unfold im2c
   rw[getElem!_pos _ _ (by simpa using hx),Array.getElem_ofFn]
   have e1:((q%ho)*(s.cin*s.kh)+t)%(s.cin*s.kh)=t:=by rw[Nat.add_comm,Nat.add_mul_mod_self_right,Nat.mod_eq_of_lt ht]
   have e2:((q%ho)*(s.cin*s.kh)+t)/(s.cin*s.kh)=q%ho:=by rw[Nat.add_comm,Nat.add_mul_div_right _ _ (Nat.lt_of_le_of_lt (Nat.zero_le _) ht),Nat.div_eq_of_lt ht,Nat.zero_add]
   simp only[e1,e2,windowAt,Nat.zero_mul,Nat.zero_add,hdh,hdw,hsh,hsw,hpt,hpl,hw,hkw,Nat.mul_one,Nat.div_one,Nat.mod_one,Nat.sub_zero]
   have hy:q%ho+t%s.kh<s.h:=by have:=Nat.mod_lt t hk0;omega
   simp only[Nat.not_lt_zero,false_or,show ¬(s.h≤q%ho+t%s.kh) from by omega,show ¬(1≤0) from by omega,if_false]
   rw[Nat.add_zero,Nat.add_assoc]
 · simp only[ht,ite_false]
@[inline]def d1Ok(s:ConvShape):Bool:=
 s.w==1&&s.kw==1&&s.strideH==1&&s.strideW==1&&s.padT==0&&s.padL==0&&s.padB==0&&s.padR==0&&s.dilH==1&&s.dilW==1&&0<s.kh&&s.kh<=s.h
def im2d(h kh ho c:Nat)(inp:Array Word):Array Word:=Array.ofFn (n:=ho*kh) fun q=>inp[c*h+q.val/kh+q.val%kh]!
def dwRows(s:ConvShape)(relu:Bool)(inp wts b:Array Word)(lin:Nat->Nat->Nat->Bool->Array Word->Array Word->Array Word->Array Word):Array Word:=
 let ho:=s.h+1-s.kh
 let Ys:=Array.ofFn (n:=s.cin) fun c=>lin ho s.kh 1 relu (im2d s.h s.kh ho c.val inp) (Array.ofFn (n:=s.kh) fun t=>wts[c.val*s.kh+t.val]!) #[b[c.val]!]
 Array.ofFn (n:=s.cin*ho) fun q=>(Ys[q.val/ho]!)[q.val%ho]!
theorem lin_dw1(s:ConvShape)(hc:d1Ok s=true)(relu:Bool)(inp wts b:Array Word):
  dwRows s relu inp wts b linearSpec=depthwiseSpec s relu inp wts b:=by
 have hc':c1Ok s=true:=hc
 obtain⟨hw,hkw,hsh,hsw,hpt,hpl,hpb,hpr,hdh,hdw,hk0,hkh⟩:=c1F s hc'
 have e1:s.depthwise.ho=s.h+1-s.kh:=by
  simp only[ConvShape.ho,ConvShape.ekh,ConvShape.depthwise,hpt,hpb,hdh,hsh,Nat.one_mul,Nat.add_zero,Nat.div_one];omega
 have e2:s.depthwise.wo=1:=by
  simp only[ConvShape.wo,ConvShape.ekw,ConvShape.depthwise,hpl,hpr,hdw,hsw,hw,hkw,Nat.one_mul,Nat.add_zero,Nat.div_one]
 unfold dwRows depthwiseSpec depthwiseWith conv2dWith
 apply Array.ext
 · simp only[Array.size_ofFn];rw[e1,e2];simp[ConvShape.depthwise]
 intro q h1 h2
 simp only[Array.size_ofFn]at h1
 generalize hH:s.h+1-s.kh=ho at *
 have hcin:0<s.cin:=Nat.pos_of_ne_zero (fun z=>by simp[z]at h1)
 have e3:s.depthwise.taps=s.kh:=by simp only[ConvShape.taps,ConvShape.cpg,ConvShape.depthwise,Nat.div_self hcin,Nat.one_mul,hkw,Nat.mul_one]
 have e4:s.depthwise.opg=1:=by simp only[ConvShape.opg,ConvShape.depthwise,Nat.div_self hcin]
 have e5:s.depthwise.cpg=1:=by simp only[ConvShape.cpg,ConvShape.depthwise,Nat.div_self hcin]
 have hq:q/ho<s.cin:=Nat.div_lt_of_lt_mul (by rw[Nat.mul_comm];exact h1)
 have hr:q%ho<ho:=Nat.mod_lt _ (Nat.pos_of_ne_zero (fun z=>by simp[z]at h1))
 simp only[Array.getElem_ofFn,e1,e2,e3,e4,Nat.mul_one,Nat.div_one,Nat.mod_one]
 rw[getElem!_pos (Array.ofFn _) (q/ho) (by simpa using hq),Array.getElem_ofFn]
 unfold linearSpec linearWith
 rw[getElem!_pos (Array.ofFn _) (q%ho) (by simpa using hr),Array.getElem_ofFn]
 simp only[Nat.div_one,Nat.mod_one,Nat.zero_mul,Nat.zero_add]
 congr 2
 · funext t
   by_cases ht:t<s.kh
   · simp only[ht,ite_true]
     unfold im2d
     have hx:(q%ho)*s.kh+t<ho*s.kh:=by nlinarith
     rw[getElem!_pos _ _ (by simpa using hx),Array.getElem_ofFn]
     have a1:((q%ho)*s.kh+t)/s.kh=q%ho:=by rw[Nat.add_comm,Nat.add_mul_div_right _ _ hk0,Nat.div_eq_of_lt ht,Nat.zero_add]
     have a2:((q%ho)*s.kh+t)%s.kh=t:=by rw[Nat.add_comm,Nat.add_mul_mod_self_right,Nat.mod_eq_of_lt ht]
     simp only[a1,a2,windowAt,e5,Nat.mul_one,Nat.div_eq_of_lt ht,Nat.add_zero,Nat.mod_eq_of_lt ht,hdh,hdw,hsh,hsw,hpt,hpl,hw,hkw,ConvShape.depthwise,Nat.div_one,Nat.mod_one,Nat.sub_zero,Nat.zero_mul,Nat.zero_add]
     have hy:q%ho+t<s.h:=by omega
     simp only[Nat.not_lt_zero,false_or,show ¬(s.h≤q%ho+t) from by omega,show ¬(1≤0) from by omega,if_false]
     simp only[ConvShape.cpg,Nat.div_self hcin,Nat.mul_one,Nat.add_assoc]
   · simp only[ht,ite_false]
 · funext t
   by_cases ht:t<s.kh
   · simp only[ht,ite_true]
     rw[getElem!_pos _ _ (by simpa using ht),Array.getElem_ofFn]
   · simp only[ht,ite_false];rfl
end Cv1
namespace Submission
local postfix:max "б"=>USize.toNat
local postfix:max "и"=>Fin.val
open _root_ renaming FloatArray->ἀ,ite_true->ἁ,ite_false->ἂ,rfl->ἅ,UInt64->ἆ,UInt32->ἇ,Nat->Ἀ,Int->Ἁ,Array->Ἂ,USize->Ἄ,Float->Ἅ,Bool->Ἆ
open Bool renaming true->ἃ,false->ἄ
open ExactRuntime renaming Word->Ἃ
open ExactRuntime hiding ConvShape windowAt conv2dWith conv2dSpec depthwiseWith depthwiseSpec avgpoolWith avgpoolSpec maxpoolSpec
open FloatLib.Floats.Formats.BinaryInterchange FloatLib.Floats
structure ConvShape where
  cin:Nat
  h:Nat
  w:Nat
  cout:Nat
  kh:Nat
  kw:Nat
  stride:Nat
  pad:Nat
def ConvShape.ho (s:ConvShape):Nat:=(s.h+2*s.pad-s.kh)/s.stride+1
def ConvShape.wo (s:ConvShape):Nat:=(s.w+2*s.pad-s.kw)/s.stride+1
@[inline] def windowAt (s:ConvShape) (inp:Array Word) (y x t:Nat):Word :=
  let c:=t/(s.kh*s.kw)
  let dy:=t/s.kw%s.kh
  let dx:=t%s.kw
  let yy:=y*s.stride+dy
  let xx:=x*s.stride+dx
  if yy < s.pad∨xx < s.pad∨yy-s.pad ≥ s.h∨xx-s.pad ≥ s.w then 0
  else inp[(c*s.h+(yy-s.pad))*s.w+(xx-s.pad)]!
@[specialize] def conv2dWith (dot:DotFn) (s:ConvShape) (relu:Bool) (inp wts b:Array Word) :
    Array Word :=
  let k:=s.cin*s.kh*s.kw
  Array.ofFn (n:=s.cout*(s.ho*s.wo)) fun q =>
    let o:=q.val/(s.ho*s.wo)
    let y:=q.val/s.wo%s.ho
    let x:=q.val%s.wo
    act relu <| dot (k+1) (fun t=>if t < k then windowAt s inp y x t else oneW)
      (fun t=>if t < k then wts[o*k+t]! else b[o]!)
def conv2dSpec:=conv2dWith dotOf
@[specialize] def depthwiseWith (dot:DotFn) (s:ConvShape) (relu:Bool)
    (inp wts b:Array Word):Array Word :=
  let k:=s.kh*s.kw
  Array.ofFn (n:=s.cin*(s.ho*s.wo)) fun q =>
    let c:=q.val/(s.ho*s.wo)
    let y:=q.val/s.wo%s.ho
    let x:=q.val%s.wo
    act relu <| dot (k+1) (fun t=>if t < k then windowAt s inp y x (c*k+t) else oneW)
      (fun t=>if t < k then wts[c*k+t]! else b[c]!)
def depthwiseSpec:=depthwiseWith dotOf
@[inline] def poolAt (h w k:Nat) (inp:Array Word) (ch y x t:Nat):Word :=
  inp[(ch*h+y*k+t/k)*w+x*k+t%k]!
@[specialize] def avgpoolWith (dot:DotFn) (c h w k:Nat) (inp:Array Word):Array Word :=
  Array.ofFn (n:=c*(h/k*(w/k))) fun q =>
    let ch:=q.val/(h/k*(w/k))
    let y:=q.val/(w/k)%(h/k)
    let x:=q.val%(w/k)
    let s:=dot (k*k) (poolAt h w k inp ch y x) (fun _=>oneW)
    toWord (Model.div (ofW s) (ofW (natW (k*k))))
def avgpoolSpec:=avgpoolWith dotOf
def maxpoolSpec (c h w k:Nat) (inp:Array Word):Array Word :=
  Array.ofFn (n:=c*(h/k*(w/k))) fun q =>
    let ch:=q.val/(h/k*(w/k))
    let y:=q.val/(w/k)%(h/k)
    let x:=q.val%(w/k)
    toWord <| (List.range (k*k)).foldl (fun acc t =>
      Model.maximum acc (ofW (poolAt h w k inp ch y x t))) (ofW (poolAt h w k inp ch y x 0))
open Submission renaming windowAt->ll,ConvShape->lR,poolAt->nI,avgpoolSpec->n9,avgpoolWith->pa
open Nat renaming shiftRight_eq_div_pow->β,mod_eq_of_lt->γ,and_two_pow_sub_one_eq_mod->ζ,mul_le_mul_right->ι,shiftLeft_eq->ο,pow_le_pow_right->Θ,zero_add->Κ,add_mul_mod_self_right->Μ,succ_mul->Ο,add_mul_div_right->Ρ,add_comm->Τ,mod_lt->ϊ,div_eq_of_lt->ώ,mul_comm->Ϗ,mul_lt_mul_of_pos_right->ϐ,add_zero->ϑ,mul_lt_mul''->ϕ,lt_log2_self->ϙ,mul_assoc->Ϛ,div_add_mod->ϛ,and_le_right->ϝ,div_lt_of_lt_mul->Ϟ,zero_le->ϟ,add_mul->Ϡ,toUInt64_eq->Ϣ,two_pow_add_eq_or_of_lt->Ϥ,mul_le_mul->ϧ,div_le_div_right->ϭ,mul_pos->Ϯ,log2_self_le->ϰ,add_sub_cancel_left->ϱ,zero_mul->ϳ,pos_of_ne_zero->ϵ,two_pow_pos->϶,div_lt_iff_lt_mul->ϸ,mul_one->ϻ
open USize renaming lt_iff_toNat_lt->ε,le_size->ς,toNat_zero->Ϧ
open UInt64 renaming toNat_add->η,le_iff_toNat_le->θ,lt_iff_toNat_lt->μ,toNat_shiftRight->ν,toNat_inj->ξ,toNat_sub_of_le->χ,toNat_shiftLeft->ψ,toNat_ofNat_of_lt'->Α,toNat_and->Ε,toNat_sub->Ζ,toNat_zero->ϋ,toNat_mul->ύ,le_trans->ϗ,toNat_toUInt32->Ϫ
open Bool renaming false_eq_true->υ,and_eq_true->Η,eq_iff_iff->ϓ
open UInt32 renaming toNat_inj->Β,lt_iff_toNat_lt->Ι,toNat_and->Ξ,toNat_shiftRight->Υ,toNat_toUInt64->ϩ,toNat_shiftLeft->ϴ
open Array renaming getElem_ofFn->Λ,size_ofFn->Ϝ,size_push->ϣ
open Finset renaming sum_range_succ->Ν,sum_congr->ϔ,mem_range->ϥ
open Model renaming isNaN_eq_false_of_toDyadic?_some->Χ,toReal_roundDyadic_eq_roundAt->ό,toModel_ofFloatModel->ϒ,isFinite_roundDyadic_of_isIEEE_of_abs_toReal_le_posMaxFinite->Ϙ,isInf_eq_false_of_toDyadic?_some->ϡ,abs_signed_mul_bpow_le_toReal_posMaxFinite->ϫ,roundDyadicGeneral_eq_roundDyadic->Ϻ
open FloatLib.Numerics.Dyadic renaming cast_toRat->ϖ,toRat_eq_zero_iff->ϲ
open Int renaming natAbs_mul->Ϩ,abs_eq_natAbs->Ϲ
open Model.Reduction renaming dot_status_eq_zero_of_finite->Ϭ
open Model.Reduction.Internal renaming dotState_exact_toRat->ϯ
open List renaming foldl_cons->Ϸ
open UInt64 renaming reduceToNat->kA,ofNat->lh,toNat->lq,toNat_div->ml,toNat_or->my,toNat_ofNat->mY,toNat_mod->nB,toNat_toBitVec->nR,size->n0,toNat_one->pS open ExactRuntime renaming expOf->kB,Word->kF,idxOf->kG,mantOf->kM,toDyadic_ofUInt32->kV,toWord->kW,signOf->k9,oneW->ld,mantOf_lt_24->lB,decodeD_toRat->lG,ofW->lI,fracOf->lK,Fmt->lU,finishWord->md,finishWord_of_ne->mf,linearSpec->mj,stableNormDomain->mq,dotOf->ms,sqrtRational->mK,expOf_toNat_lt->mO,isFinite_ofUInt32->m6,ofIntDyadic->nb,decodeD->nh,rowMean->no,dallNZ->nC,ofUInt32_bits->pu,act->p8 open FloatLib.Floats.Formats.BinaryInterchange.FloatFormat renaming binary64->kC open USize renaming reduceToNat->kD,size->kX,toNat_add->lJ,toNat_one->mA,ofNat_add->mC,size_eq_two_pow->nT open FloatLib.Floats.Formats.BinaryInterchange.Model renaming roundDyadic->kE,ofFloatModel->kI,ofUInt32->kT,toReal->ls,unpackedToReal->lw,roundedMantissa_shift_quotient->lF,roundDyadic_of_representable->lL,roundAt_mono->lM,toReal_add_eq_roundAt->mH,isFinite->mN,toNatBits_lt_two_pow->mQ,posMaxFinite->mR,modelSignBit->mS,ofNatBits_toNatBits->m3,roundAt->ns,maximum->nt,ofFloatModel_pack->nu,ofFloat32Model->n5,roundRatScaled->n6,add->pb,fma->pd,div->pf,ofModel->pv,roundAt_zero->pD,pow2->p7 open Nat renaming reduceMod->kH,pow_lt_pow_right->lu,mul_lt_mul_of_lt_of_le->ly,add_right_comm->lO,testBit_eq_decide_div_mod_eq->lW,toUSize_eq->lX,div_div_eq_div_mul->lY,le_mul_of_pos_left->lZ,succ_le_of_lt->l7,le_div_iff_mul_le->mc,le_log2->mu,add_mul_div_left->mv,log2_eq_iff->mz,div_eq_of_lt_le->mG,and_one_is_mod->mP,pow_add->m7,log2_lt->m9,one_shiftLeft->na,reducePow->nr,abs_cast->nE,add_one_mul->nO,cast_ofNat->nZ,one_mul->n4,div_le_div_left->pq,div_mul_le_self->pr,add_assoc->pA,cast_one->pJ,cast_sub->pO,sub_zero->pQ open FloatArray renaming emptyWithCapacity->kJ,size->kS,uget->nN,set!->pY open Array renaming ofFn->kK,getElem?_eq_getElem->lC,emptyWithCapacity->lQ,getElem?_eq_none->l5,ext->mi,replicate->mB,getElem?_ofFn->mE,ext_getElem?->mU,size_map->m0,all_eq_true->np,getElem_push_lt->nU,getElem_push_eq->nV,uset_eq_set->pP open Float.Model.UnpackedFloat renaming unpack->kL,mul->lc,accuracyOfFraction->lP,roundWithAccuracy->nk,unpackExponent->nX,round->pp,isFinite->pE,compare->pI,Sign->pK,pack->pT open UInt32 renaming reduceToNat->kN,ofNat->lA,toNat_ofNat'->lE,toNat_add->mh,toNat->mJ,toNat_zero->nj,toNat_max->nw,toNat_sub_of_le->nG,toNat_or->nK,lt_asymm->nP,toNat_sub->pU open FloatLib.Numerics renaming binaryRadix->kO,shiftRightJam->kQ,Dyadic->kU,roundQuotientEven->la,roundShiftRightEven_shiftRightJam->lS open Finset renaming range->kP,sum_range_succ'->lV,sum_sub_distrib->nF,sum_add_distrib->nJ,sum_mul->n8 open FloatLib.Numerics.Dyadic renaming ofScaledInt->kR,signedSignificand->lr,toReal->mb,cast_signedSignificand->nf,ofScaledInt_toRat->nD,add_toRat->pF,mul_toRat->pG,toRat->pR open Float.Model renaming totalExponent->kY,add->pW,mul->pX open FloatLib.Floats.Formats.Flocq renaming bpow->kZ open Float.Model.Format renaming binary64->k0,binary32->li open FloatLib.Numerics.RationalBinary renaming floorLog2->lf,scaleByPowerOfTwo->l9,scaleByPowerOfTwo_ofNat->mw,lessThanPowerOfTwo->mM,atLeastPowerOfTwo->mZ open List renaming range->lj,foldl_append->l6,range_succ->mD,foldl_nil->mT,range'->pV open FloatLib.Numerics.FixedWord renaming roundShiftRightEven->lk open Float.Model.UnpackedFloat.ExtendedMantissa renaming shiftRightOne->lm open BitVec renaming toInt_eq_toNat_cond->lv open Float32 renaming ofBits->lx open Int renaming cast_zero->lH,cast_abs->mI,ofNat_eq_natCast->nY,ModEq->pe,log->ph open Bool renaming false_ne_true->lN,or_eq_true->nM open FloatLib.Floats.Formats.BinaryInterchange.Model.Reduction.Internal renaming dotState->lT,State->me open FloatLib.Floats.ExecFloat.Binary renaming toModel_ofFloat_int64ToFloat->ma,toModel_ofFloat->m5 open FloatLib.Numerics.Radix renaming toReal->mg open FloatLib.Floats.Formats.BinaryInterchange.Model.Reduction.Semantics renaming finiteDot_toList_eq_slice->mL,finiteDotSlice->nQ open MulZeroClass renaming zero_mul->nv open System.Platform renaming numBits->nx open FloatLib.Floats.Formats.BinaryInterchange.Model.Proof renaming div_eq_spec->ny open FloatLib.Floats.Formats.BinaryInterchange.Model.NativeBinary32 renaming ofUInt32_roundRatScaled->nA,toUInt32_ofFields->n7,toUInt32->ps,roundRatScaled->pC open Option renaming some->pj open Min renaming min->po open Decidable renaming not_not->pM open Or renaming inl->pZ open FloatLib.Floats.Formats.BinaryInterchange.Model.Spec renaming div->p0
open _root_ renaming zpow_natCast->AA,getElem!_pos->AB,ite_eq_right->AC,ite_eq_left->AD,mul_le_mul_of_nonneg_right->AE,pow_add->AF,zpow_le_zpow_right₀->AG,mul_lt_mul_of_pos_right->AI,decide_eq_true_eq->AJ,zpow_add₀->AK,two_ne_zero->AL,bne_iff_ne->AM,lt_of_lt_of_le->AN,lt_of_le_of_lt->AO,le_rfl->AP,beq_iff_eq->AQ,dite_eq_left->AT,pow_le_pow_right₀->AU,one_div_pow_le_one_div_pow_of_le->AV,abs_of_pos->AW,lt_irrefl->AX,abs_sub_abs_le_abs_sub->AY,le_trans->AZ,abs_nonneg->Aa,zpow_sub₀->Ac,abs_of_nonneg->Ad,abs_mul->Ae,mul_nonneg->Af,abs_add_le->Ag,zero_add->Ai,ne_eq->Ak,ite_false->Am,getElem!_neg->An,lt_trans->Ao,mul_assoc->Ap,decide_eq_true_iff->As,add_zero->Av,abs_sub_comm->Aw,apply_ite->Ax,beq_self_eq_true->Az,abs_le->A3,pow_succ->A4,abs_zero->A5,decide_eq_true->A6,gt_iff_lt->BC,abs_neg->BE,lt_of_mul_lt_mul_right->BF
open USize renaming reduceToNat->AH,size->Al,toNat_mul->BO,ofNat_mul->BP
open FloatArray renaming emptyWithCapacity->AR
open Array renaming emptyWithCapacity->AS,replicate->Ah,ofFn->Aj,mkEmpty->A0,size_uset->BT,size_set!->BU
open Nat renaming log2->Aq,div_add_mod'->Au,mod_le->Ay,mul_zero->A7,pos_iff_ne_zero->BD,shiftLeft_zero->BH,one_le_two_pow->BI,eq_or_lt_of_le->BJ,add_sub_cancel->BK,add_sub_of_le->BQ
open Model renaming div->Ar,ofFloatModel->BB,toReal_eq->BR
open Or renaming inr->BA
open UInt32 renaming ofBitVec->BG
open Float32 renaming ofBits->BL
open Bool renaming not_eq_true'->BM
open ZMod renaming natCast_mod->BN
open List renaming range'_succ->BS
open Float.Model.Format renaming targetExponent->BV open USize renaming toNat->J0 open Nat renaming toUSize->G0,lt_of_le_of_lt->C0,toUInt64->I0,add_le_add_left->J1,le_add_right->V1,lt_of_lt_of_le->U1,add_lt_add_left->O1,lt_of_sub_eq_succ->N1,add_le_add_right->L1 open Array renaming size->O0 open Nat renaming le_of_lt_succ->J2,le_add_left->K2,le_trans->U2,le_of_ble_eq_true->N2,sub_eq_zero_iff_le->K3,not_lt->W3,add_lt_add_right->X3,add_le_add->Y3,mul_le_mul_left->U3,cast_nonneg->M3,le_of_eq->B3,lt_succ_self->I3,sub_add_cancel->L3,le_mul_of_pos_right->J4,sub_le->K4,sub_add_eq->V4,le_of_not_lt->W4,le_of_lt->X4,lt_succ_of_le->Z4,cast_mul->Y4,sub_lt_left_of_lt_add->G4,le_of_sub_eq_zero->O4,sub_le_sub_right->P4,add_sub_cancel'->M4,lt_of_le_of_ne->N4,lt_irrefl->B4,le_antisymm->C4,cast_pow->I4,lt_trans->K5,cast_add->W5,le_of_succ_le->X5,sub_succ_lt_self->Z5,le_sub_of_add_le->G5,not_lt_zero->U5,lt_add_of_pos_right->P5,cast_le->R5,sub_lt_sub_left->C5,div_one->F5,div_le_self->L5,lt_succ_of_lt->K6,lt_of_lt_of_eq->V6,pair->X6,lt_succ_iff_lt_or_eq->Z6,add_lt_add_of_lt_of_le->P6,ofNat_nonneg->R6,lt_of_succ_lt->B6,strong_induction_on->C6,add_lt_add->E6,ne_of_gt->L6,cast_succ->X7,lt_or_ge->Z7,mul_add_mod->O7,min_le_left->P7,sub_add_comm->R7,mul_lt_mul_of_pos_left->M7,add_lt_add_of_le_of_lt->N7,eq_of_mul_eq_mul_right->B7,le_succ->E7,sub_sub->F7,lt_of_succ_le->I7,zero_lt_of_lt->L7,add_left_cancel->J8,mul_add_div->W8,sub_ne_zero_of_lt->Z8,lt_of_not_le->O8,mod_eq_sub_mod->P8,ne_of_lt->R8,sub_le_of_le_add->M8,add_right_cancel->N8 open Real renaming sqrt->O2 open USize renaming toNat_ofNat_of_lt'->J3,ofNat->C3,toNat_inj->O5,le_iff_toNat_le->U8 open ExactRuntime renaming rationalWord->V3,ConvShape->L4,windowAt->U6,roundRational->C7,sv->L8,PoolShape->Q9,depthwiseSpec->J9 open Array renaming size_replicate->O3 open UInt64 renaming toUInt32->N3,toNat_ofNat'->O6 open Int renaming ofNat_le->U4,cast_natCast->M5,natCast_nonneg->W6,toNat->F6,natAbs->I6,add_le_add_right->T8 open ExactRuntime.ConvShape renaming depthwise->J5,cpg->T6 open Float renaming ofBits->N5,sqrt->W7 open Finset renaming Ico->B5,abs_sum_le_sum_abs->K8,sum_range_zero->X8,sum_nonneg->D8 open Bool renaming and_eq_true_iff->I5,eq_false_iff->F8 open Model renaming modelSignBit->J6,unpackedToReal_finite->N6,ofModelBits->C8 open Float.Model.UnpackedFloat renaming normalize->G6 open Float.Model renaming pack->M6 open Rat renaming cast_mul->J7,cast_ofNat->G7,cast_natCast->T7,cast_zpow->S8 open Int64 renaming toInt->K7 open UInt32 renaming le_iff_toNat_le->V7 open Pinned renaming kernels->U7
local instance:AddLeftMono Real:=inferInstance
local instance:AddRightMono Real:=inferInstance
local instance:AddLeftStrictMono Real:=inferInstance
local instance:PosMulReflectLT Real:=inferInstance
local instance:AddLeftMono Nat:=inferInstance
local instance:AddLeftStrictMono Nat:=inferInstance
local instance:LawfulBEq ἆ:=inferInstance
local instance:LawfulBEq ἇ:=inferInstance
section
open FloatLib.Numerics
lemma cm_h1(j L:Ἀ)(h:j+25<=L):j+1<=L∧23<=L-j∧23<=L:=⟨(J1 ((by decide):1<=25) j).trans h,G5 (by rw[Τ];exact (J1 ((by decide):23<=25) j).trans h),(K2 23 j).trans ((J1 ((by decide):23<=25) j).trans h)⟩
lemma cm_h2(e:Ἁ)(j L:Ἀ)(h:j+25<=L)(c:(L:Ἁ)+e< -126):∃a:Ἀ,e- -149=Int.negSucc a∧e+j- -149=Int.negSucc (a-j)∧a-j+1=a+1-j∧j+2<=a+1:=by
 obtain⟨a,v,y⟩:∃a:Ἀ,e- -149=Int.negSucc a∧j+2<=a:=⟨(F6 (-(e+149)-1)),by omega,by omega⟩
 exact⟨a,v,by (clear * - v y;omega),(R7 ((V1 j 2).trans y)).symm,y.trans (E7 a)⟩
lemma cm(s:Ἆ)(m:Ἀ)(e:Ἁ)(j:Ἀ)(h:j+25<=(Aq m)):
  kE lU ⟨s,kQ m j,e+j⟩=kE lU ⟨s,m,e⟩:=by
 have g:m≠0:=by rintro rfl;simp at h
 have k:=cm_h1 j _ h
 have o:2^(Aq m)<=m:=ϰ g;have u:2^(j+1)<=m:=AZ (Θ (by norm_num) k.1) o;have d:(Aq (kQ m j))=(Aq m)-j:=log2_shiftRightJam m j;have t:kQ m j≠0:=shiftRightJam_ne_zero m j (AZ (Θ (by norm_num) (E7 j)) u);rw[<-Ϻ,<-Ϻ];simp only[Model.roundDyadicGeneral,t,g,AQ,Am,d];clear d o u t g;rw[(by rw[pO ((V1 j 25).trans h)];ring:(((Aq m)-j:Ἀ):Ἁ)+(e+j)=((Aq m):Ἁ)+e),show Fmt.fracWidth=23 from ἅ,show Fmt.minSubnormalExponent=-149 from ἅ,show Fmt.minNormalExponent=-126 from ἅ,if_pos k.2.1,if_pos k.2.2];split_ifs with c1 c2
 rfl;congr 1;obtain⟨a,v,w,x,y⟩:=cm_h2 e j _ h c2;rw[v,w];dsimp only;rw[x];exact lS m j (a+1) y;congr 1;rw[Nat.sub_right_comm _ j];exact lS m j _ (G5 h)
end
lemma ab(n:Ἀ)(f:Ἀ->ἇ)(k:Ἀ)(d:k<n):
  (kK (n:=n) λi=>f (Fin.val i))[k]! =f k:=by rw[AB _ k (by simpa using d),Λ]
namespace Core
noncomputable def tm(a b:Ἃ):Ἁ:=sv a*sv b*2^(idxOf a+idxOf b)
noncomputable def af(n:Ἀ)(fa fb:Ἀ->Ἃ):Ἁ:=∑k∈kP n,tm (fa k) (fb k)
def aG(n:Ἀ)(fa fb:Ἀ->Ἃ):Prop:=∀k,k<n->expOf (fa k)≠255/\expOf (fb k)≠255
lemma aK(n:Ἀ)(d e:Ἀ->kF):
  af (n+1) d e=af n d e+tm (d n) (e n):=by unfold af;rw[Ν]
lemma gm(A B:Ἂ ἇ)(i c:Ἀ):
  dsum A B i c=∑k∈kP c,tm A[i+k]! B[i+k]!:=by
 induction c generalizing i with
 |zero=>simp[dsum]
 |succ c d=>
  rw[dsum,d (i+1),lV,add_comm];exact congrArg₂ _ (ϔ rfl λk _=>by rw[((by exact Nat.succ_add_eq_add_succ i k):i+1+k=i+(k+1))]) rfl
lemma cH(n:Ἀ)(d e:Ἀ->kF):
  dsum (kK (n:=n) λi=>d (Fin.val i)) (kK (n:=n) λi=>e (Fin.val i)) 0 n=
   af n d e:=by rw[gm,af];exact ϔ rfl λk g=>by rw[ϥ]at g;rw[Κ,ab n d k g,ab n e k g]
theorem bI_h1{s a p c d:Ἆ}(h1:s=(false||c))(h2:a=(true&&d)):(s&&(a||(!p&&false)))=(c&&d):=by rw[h1,h2,Bool.false_or,Bool.true_and,Bool.and_false,Bool.or_false]
theorem bI_h2(A B:Ἂ kF)(n:Ἀ)(S:Ἁ)(hS:dsum A B 0 n=S)(hA:A.size=n)(hB:B.size=n)(m:∀k<n,kB A[k]!≠255/\kB B[k]!≠255):dotSpec A B=md S (-298) (decide (n≠0)&&nC A B 0 n):=by
 have hL:(A.map (kT (fmt:=lU))).size=n:=m0.trans hA
 have l:(A.map (kT (fmt:=lU))).size=(B.map (kT (fmt:=lU))).size:=hL.trans (hB.symm.trans m0.symm)
 have x:∀X:Ἂ kF,(∀k (w:k<X.size),kB X[k]≠255)->∀k (j:k<(X.map (kT (fmt:=lU))).size),mN (X.map (kT (fmt:=lU)))[k]=ἃ:=λX h k j=>by rw[Array.getElem_map,m6];exact AM.2 (h k (V6 j m0))
 have e:=x A λk w=>AB A k w▸(m k (hA▸w)).1
 have g:=x B λk w=>AB B k w▸(m k (hB▸w)).2
 have ha:0+(A.map (kT (fmt:=lU))).size<=(A.map (kT (fmt:=lU))).size:=B3 (Κ _)
 have hb:0+(A.map (kT (fmt:=lU))).size<=(B.map (kT (fmt:=lU))).size:=B3 ((Κ _).trans l)
 have j:∀k,0<=k->k<0+(A.map (kT (fmt:=lU))).size->kB A[k]!≠255/\kB B[k]!≠255:=λk _ F=>m k (V6 F ((Κ _).trans hL))
 have d:=(finiteDotSlice_eq A B 0 _ ha hb j).trans ((congrArg (λs=>(dsum A B 0 s:ℚ)*2^(-298:ℤ)) hL).trans (congrArg (λt:Ἁ=>(t:ℚ)*2^(-298:ℤ)) hS))
 have y:=(ϯ (destination:=lU) _ _ l).trans d
 unfold dotSpec
 rw[←Model.Reduction.map_value_dot]
 by_cases v:S=0
 · rw[v,finishWord_zero,Ϭ lU _ _ .nearest l e g ((ϲ _).1 (y.trans (by rw[v,lH,zero_mul])))]
   obtain⟨z,D⟩:=dotStateLoop_flags A B ({}:me lU) 0 _ ha hb j
   simp only[Except.map]
   exact congrArg (λb=>kW (Model.zero lU b)) ((bI_h1 z D).trans (congrArg (λs=>decide (s≠0)&&nC A B 0 s) hL))
 · have hnz:Model.Reduction.Semantics.finiteDot (A.map (kT (fmt:=lU))).toList (B.map (kT (fmt:=lU))).toList≠.ok 0:=by
    rw[Model.Reduction.Semantics.finiteDot_toList_eq_slice _ _ l]
    intro h;injection h with h;rw[d]at h;exact v (Int.cast_eq_zero.1 ((mul_eq_zero.1 h).resolve_right (zpow_ne_zero _ AL)))
   rw[Model.Reduction.dot_status_eq_round_of_finite_nonzero lU _ _ .nearest l e g hnz]
   simp only[Except.map]
   exact(mf _ _ _ _ v y).symm
lemma bI(n:Ἀ)(o p:Ἀ->kF)(u:aG n o p):
  ms n o p=md (af n o p) (-298) (decide (n≠0)&&
   nC (kK (n:=n) λi=>o (Fin.val i)) (kK (n:=n) λi=>p (Fin.val i)) 0 n):=by
 rw[ms];exact bI_h2 _ _ n _ (cH n o p) Ϝ Ϝ λk j=>by rw[ab n o k j,ab n p k j];exact u k j
lemma bT(n:Ἀ)(d e:Ἀ->kF)(g:aG n d e)
  (h:af n d e≠0):
  ms n d e=md (af n d e) (-298) ἄ:=by rw[bI n d e g];simp only[md,h,Am]
@[inline]def f5(w:ἇ):ἆ:=if(w>>>23)&&&255!=0 then 1 else 0
@[inline]def mW(w:ἇ):ἆ:=(UInt32.toUInt64 (w&&&8388607))|||(f5 w<<<23)
@[inline]def iW(w:ἇ):ἆ:=(UInt32.toUInt64 ((w>>>23)&&&255))-f5 w
noncomputable def ae(x:ἆ):Ἁ:=x.toBitVec.toInt
lemma gr(w:ἇ):(mJ (w&&&8388607))<2^(23:Ἀ):=by rw[Ξ];exact AO ϝ (by decide)
lemma an(w:ἇ):(lq (mW w))=kM w:=by
 have g:=gr w;rw[mW,f5,kM,kB,lK];by_cases d:(w>>>23)&&&255=0
 .simp[d]
 .have e:((w>>>23)&&&255!=0)=ἃ:=by simpa using d
  simp only[e,d,ἁ,ἂ];rw[my,ϩ,show(lq ((1:ἆ)<<<23))=2^(23:Ἀ)*1 from ἅ,Nat.or_comm,<-Ϥ g,ϻ,Τ]
lemma am(w:ἇ):(lq (iW w))=kG w:=by
 rw[iW,f5,kG,kB];have g:=mO w;rw[kB]at g;by_cases e:(w>>>23)&&&255=0;simp[e];have d:1<=(mJ ((w>>>23)&&&255)):=BD.2 λh=>e (Β.1 (by simpa using h));simp only[(by simpa using e:((w>>>23)&&&255!=0)=true),e,ἁ,ἂ];rw[χ _ _ (θ.2 (by simpa[ϩ]using d)),ϩ];rfl
lemma ap_h1(n:Ἀ)(e:n<2^(32:Ἀ)):¬(decide (n/2^(31:Ἀ)%2=1)).toNat*2^(31:Ἀ)=0↔2^(31:Ἀ)<=n:=by
 by_cases h:2^(31:Ἀ)<=n
 ·rw[(mG (by (clear * - h;omega)) (by (clear * - e;omega)):n/2^(31:Ἀ)=1)];exact iff_of_true (by decide) h
 rw[ώ (by omega)];exact iff_of_false (by decide) h
lemma ap(w:ἇ):k9 w=ἃ↔2^(31:Ἀ)<=(mJ w):=by
 rw[k9];have d:(w&&&0x80000000).toNat=w.toNat&&&2^(31:Ἀ):=by rw[Ξ];rfl
 rw[Nat.and_two_pow,lW]at d;rw[AM,Ak,<-Β,d];exact ap_h1 _ w.toNat_lt
lemma fy(w:ἇ):(w>>>31!=0)=k9 w:=by
 have d:(mJ (w>>>31))=(mJ w)/2^(31:Ἀ):=by rw[Υ];simp[β]
 have:=w.toNat_lt;rw[ϓ,AM,Ak,<-Β,d,ap];simp
@[inline]def sx(k x:ἆ):ἆ:=bif x>>>k==0 then x else x>>>k
@[inline]def sr(k x r:ἆ):ἆ:=bif x>>>k==0 then r else r+k
@[inline]def ai(x0:ἆ):ἆ:=
 let x1:=sx 32 x0
 let r1:=sr 32 x0 0
 let x2:=sx 16 x1
 let r2:=sr 16 x1 r1
 let x3:=sx 8 x2
 let r3:=sr 8 x2 r2
 let x4:=sx 4 x3
 let r4:=sr 4 x3 r3
 let x5:=sx 2 x4
 let r5:=sr 2 x4 r4
 sr 1 x5 r5
noncomputable def f7(value shift:ἆ):ἆ:=
 let quotient:=value>>>shift
 let remainder:=value&&&((1<<<shift)-1)
 let half:=(1:ἆ)<<<(shift-1)
 if remainder<half then quotient
 else if remainder>half then quotient+1
 else if(quotient&&&1)==0 then quotient
 else quotient+1
@[inline]def de(value shift:ἆ):ἆ:=
 let quotient:=value>>>shift
 let remainder:=value&&&((1<<<shift)-1)
 let half:=(1:ἆ)<<<(shift-1)
 quotient+(if remainder+(quotient&&&1)>half then 1 else 0)
@[inline]def aW(value shift:ἆ):ἆ:=
 if shift==0 then value
 else if shift<64 then de value shift
 else if shift==64 then(if value>0x8000000000000000 then 1 else 0)
 else 0
@[inline]def c9(sign:Ἆ)(e f:ἇ):ἇ:=
 (if sign then(0x80000000:ἇ)else 0)|||((e&&&255)<<<23)|||(f&&&8388607)
@[inline]def d0(n:ἆ):ἇ:=
 (N3 (if n<8388608 then 0 else n-8388608))
@[inline]def az(sign:Ἆ)(product scale:ἆ):ἇ:=
 if product==0 then
  if sign then 0x80000000 else 0
 else
  let leading:=ai product
  let position:=leading+scale
  if position<172 then
   let fraction:=
    if scale<149 then aW product (149-scale)else product<<<(scale-149)
   if fraction==0 then(if sign then 0x80000000 else 0)
   else if fraction==8388608 then c9 sign 1 0
   else c9 sign 0 (N3 fraction)
  else
   let roundedMantissa:=
    if leading<23 then product<<<(23-leading)else aW product (leading-23)
   let carry:=roundedMantissa==16777216
   let normalizedPosition:=if carry then position+1 else position
   if normalizedPosition>425 then(if sign then 0xff800000 else 0x7f800000)
   else
    let normalizedMantissa:=if carry then 8388608 else roundedMantissa
    c9 sign (N3 (normalizedPosition-171)) (d0 normalizedMantissa)
open Model.NativeBinary32 in
lemma bo(n k:Ἀ)(h:n>>>k≠0):(Aq n)=k+(Aq (n>>>k)):=by rw[β]at h⊢;have e:=(mc (϶ k)).1 (ϰ h);have d:=(ϸ (϶ k)).1 (ϙ (n:=n/2^k));rw[mz (λz=>h (by rw[z,Nat.zero_div]))];simp only[Nat.pow_succ,m7]at d⊢;rw[Ϗ (2^k)];exact⟨e,by rwa[Nat.mul_right_comm]at d⟩
lemma d7(x:ἆ):(Aq (lq x))<64:=by
 by_cases h:(lq x)=0
 .simp[h]
 .exact(m9 h).mpr x.toNat_lt
lemma cx(j r x k:ἆ)(p:(lq k)<=32)(o:0<(lq k))
  (d:(Aq (lq j))=(lq r)+(Aq (lq x)))(m:(lq x)<2^(2*(lq k))):
  (Aq (lq j))=(lq (sr k x r))+(Aq (lq (sx k x)))/\(lq (sx k x))<2^(lq k):=by
 have e:(lq (x>>>k))=(lq x)>>>(lq k):=by rw[ν,γ (by (clear * - p o;omega))]
 rw[sx,sr];by_cases g:x>>>k=0;simp only[g,Az,Bool.cond_true];use d;simpa[e,β,Nat.div_eq_zero_iff]using congrArg lq g;simp only[(by simpa using g:(x>>>k==0)=ἄ),Bool.cond_false];have q:=bo (lq x) (lq k) λh=>g (ξ.mp (by simpa[e]using h));have l:=d7 j;use(by rw[η,γ (by clear*-d l p;omega),e];clear*-d q;omega);rw[e,β];exact Ϟ (m7 _ _ _▸m.trans_eq (by congr 1;exact Nat.two_mul k.toNat))
lemma dr(n:Ἀ)(h:n<2):(Aq n)=0:=by rw[Nat.log2_def,AC (by exact Nat.not_le_of_lt h)]
lemma aA(x:ἆ):ai x=x.log2:=by apply ξ.1;rw[show (lq x.log2)=(Aq (lq x))from ἅ];have d:(lq x)<2^(2*(lq (32:ἆ))):=x.toNat_lt;have⟨o,e⟩:=cx x 0 x 32 (by decide) (by decide) (by simp only [ϋ,Ai]) d;obtain⟨p,g⟩:=cx x _ _ 16 (by decide) (by decide) o (by simpa using e);obtain⟨q,h⟩:=cx x _ _ 8 (by decide) (by decide) p (by simpa using g);obtain⟨r,j⟩:=cx x _ _ 4 (by decide) (by decide) q (by simpa using h);obtain⟨t,l⟩:=cx x _ _ 2 (by decide) (by decide) r (by simpa using j);obtain⟨u,m⟩:=cx x _ _ 1 (by decide) (by decide) t (by simpa using l);rw[dr (lq (sx 1 (sx 2 (sx 4 (sx 8 (sx 16 (sx 32 x))))))) (by simpa using m)]at u;dsimp only[ai];clear*-u;omega
lemma gH_h1(a x k:Ἀ)(p:a<k)(m:k<2^(64:Ἀ)):a+x%2<2^(64:Ἀ):=(J1 (J2 (ϊ x ((by positivity):0<2))) _).trans_lt (C0 p m)
lemma gH_h2(q:ἆ)(R H b:Ἀ)(t:b<2):(q+if H<R+b then 1 else 0)=if R<H then q else if H<R then q+1 else if b=0 then q else q+1:=by
 rcases Nat.lt_trichotomy R H with c|rfl|c
 rw[if_neg (W3.2 ((J1 (J2 t) R).trans c)),if_pos c,UInt64.add_zero]
 rw[if_neg (AX _),if_neg (AX _)];rcases Nat.eq_zero_or_pos b with rfl|h
 rw[ϑ,if_neg (AX _),if_pos rfl,UInt64.add_zero]
 rw[if_pos (P5 h),if_neg (BD.1 h)]
 rw[if_pos (U1 c (V1 _ b)),if_neg (W3.2 c.le),if_pos c]
lemma gH(v s:ἆ)(o:s≠0)(h:s<64):de v s=f7 v s:=by
 have g:(lq s)<64:=h;have m:(2:Ἀ)^(lq s)<2^(64:Ἀ):=lu (by decide) g;have e:(lq (v&&&((1<<<s)-1)))=(lq v)%2^(lq s):=by have k:(lq ((1:ἆ)<<<s))=2^(lq s):=(by rw[ψ,pS,γ g,na,γ m]);rw[Ε,χ (1<<<s) 1 (θ.2 (BI.trans_eq k.symm)),k,pS,ζ]
 have d:(lq ((v>>>s)&&&1))=(lq (v>>>s))%2:=by rw[Ε,pS,mP]
 have p:=ϊ (lq v) (϶ (lq s));have j:(lq ((v&&&((1<<<s)-1))+((v>>>s)&&&1)))=(lq v)%2^(lq s)+(lq (v>>>s))%2:=by rw[η,e,d,γ (gH_h1 _ _ _ p m)]
 have l:((v>>>s)&&&1==0)=decide ((lq (v>>>s))%2=0):=by rw[<-d,ϓ,AQ,As,<-ξ,ϋ]
 simp only[de,f7,BC,μ,j,e,l,AJ];clear e d j l m p g;have:=ϊ (lq (v>>>s)) ((by decide):0<2);generalize (lq (v>>>s))%2=b at *;generalize (lq (1<<<(s-1)))=H at *;generalize (lq v)%2^(lq s)=R at *;exact gH_h2 _ _ _ _ this
lemma eP_h1(e d:ἆ):aW e d=(if d==0 then e else if d<64 then f7 e d else if d==64 then(if e>0x8000000000000000 then 1 else 0) else 0):=by
 rw[aW];by_cases a:(d==0)=true;rw[if_pos a,if_pos a];rw[if_neg a,if_neg a];by_cases b:d<64;rw[if_pos b,if_pos b,gH _ _ (fun h=>a (by subst h;rfl)) b];rw[if_neg b,if_neg b]
lemma eP(e d:ἆ):
  aW e d=lk e (lq d):=by
 rw[eP_h1,f7,lk];have p:=d.toNat_lt;by_cases g:d=0;subst g;rfl;have j:(lq d)≠0:=λh=>g (ξ.1 (h.trans ϋ.symm));simp only[g,AQ,j,reduceIte,UInt64.ofNat_toNat,(ξ.1 ((Α ((K4 _ _).trans_lt p)).trans (χ d 1 (θ.2 (Nat.one_le_iff_ne_zero.2 j))).symm):lh ((lq d)-1)=d-1),<-show(d<64)↔(lq d)<64 from μ,(by rw[<-ξ];rfl:d=64↔(lq d)=64)]
open Model.NativeBinary32 in
lemma eJ(d:Ἆ)(e f:Ἀ):
  c9 d (lA e) (lA f)=mkBits d e f:=by rw[c9,mkBits];cases d<;>rfl
lemma bY(x:ἆ):(N3 x)=lA (lq x):=(by rfl)
lemma c5(n:ἆ):d0 n=lA ((lq n)-8388608):=by
 rw[d0];split<;>rename_i h<;>rw[μ]at h<;>rw[bY]
 .rw[show (lq n)-8388608=0 by simp at h;omega];rfl
 .rw[χ _ _ (by rw[θ];simp at h⊢;exact l7 h)];rfl
open Model.NativeBinary32 in
lemma bE(g:Ἆ)(d e:ἆ):
  az g d e=roundProduct g d e:=by unfold az roundProduct;simp only[aA,eP,c5,bY];rw[show(1:ἇ)=lA 1 from rfl,show(0:ἇ)=lA 0 from rfl];simp only[eJ,FloatLib.Numerics.FixedWord.log2Word_eq_log2]
open Model.NativeBinary32 in
lemma bD(w:ἇ):kW (ofUInt32 w)=w:=by
 have h:=toUInt32_ofUInt32 w;unfold ps at h;unfold kW;conv_rhs=>rw[<-h]
 exact Β.mp (γ (BitVec.isLt _))
open Model.NativeBinary32 in
lemma aJ(g:Ἆ)(m d:ἆ)(h:(lq d)<=506):
  az g m d=kW (kE lU ⟨g,(lq m),((lq d):Ἁ)-298⟩):=by
 rw[bE,roundProduct_eq_roundDyadic _ _ _ h,show kE lU ⟨g,(lq m),((lq d):Ἁ)-298⟩=ofUInt32 (roundDyadic ⟨g,(lq m),Int.ofNat (lq d)-298⟩) from(ofUInt32_roundDyadic _).symm,bD]
noncomputable def aw(hi lo:ἆ):Ἁ:=ae hi*2^(64:Ἀ)+(lq lo)
@[inline]def aF(hi lo:ἆ):ἆ:=if hi>>>63!=0 then(0:ἆ)-lo else lo
@[noinline]def aQ(hi lo:ἆ):ἆ:=
 if hi>>>63!=0 then(if lo==0 then(0:ἆ)-hi else~~~hi)else hi
lemma gB(d:ἆ):(d>>>63!=0)=decide (2^(63:Ἀ)<=(lq d)):=by
 have e:(lq (d>>>63))=(lq d)/2^(63:Ἀ):=by rw[ν];simp[β]
 have:=d.toNat_lt;rw[ϓ,AM,Ak,<-ξ,e,As];simp
lemma bN(d e:ἆ):
  aw d e=((lq d)*2^(64:Ἀ)+(lq e):Ἁ)-(if 2^(63:Ἀ)<=(lq d)then 2^(128:Ἀ) else 0):=by rw[aw,ae];rw[lv,nR];omega
lemma fk_h1(e:Ἀ)(m:e<18446744073709551616):(I6 ((e:Ἁ)*18446744073709551616-340282366920938463463374607431768211456))=(18446744073709551616-e)*18446744073709551616/\(e:Ἁ)*18446744073709551616<340282366920938463463374607431768211456:=by
 have k:(e:Ἁ)*18446744073709551616-340282366920938463463374607431768211456=-(((18446744073709551616-e)*18446744073709551616:Ἀ):Ἁ):=by rw[Y4,pO m.le];push_cast;ring
 have p:(0:Ἁ)<(((18446744073709551616-e)*18446744073709551616:Ἀ):Ἁ):=by exact_mod_cast Ϯ (Nat.sub_pos_of_lt m) (϶ 64)
 exact⟨by rw[k,Int.natAbs_neg,Int.natAbs_cast],by rw[←sub_neg,k];exact neg_neg_of_pos p⟩
lemma fk_h2(e d:Ἀ)(m:e<18446744073709551616)(o:d<18446744073709551616):(I6 ((e:Ἁ)*18446744073709551616+d-340282366920938463463374607431768211456))=(18446744073709551615-e)*18446744073709551616+(18446744073709551616-d)/\(e:Ἁ)*18446744073709551616+d<340282366920938463463374607431768211456:=by
 have k:(e:Ἁ)*18446744073709551616+d-340282366920938463463374607431768211456=-(((18446744073709551615-e)*18446744073709551616+(18446744073709551616-d):Ἀ):Ἁ):=by rw[W5,Y4,pO (J2 m),pO o.le];push_cast;ring
 have p:(0:Ἁ)<(((18446744073709551615-e)*18446744073709551616+(18446744073709551616-d):Ἀ):Ἁ):=by exact_mod_cast (Nat.sub_pos_of_lt o).trans_le (K2 _ _)
 exact⟨by rw[k,Int.natAbs_neg,Int.natAbs_cast],by rw[←sub_neg,k];exact neg_neg_of_pos p⟩
lemma fk_h3(x:Ἀ)(h:0<x):18446744073709551616-x<18446744073709551616:=Nat.sub_lt (϶ 64) h
lemma fk(e d:ἆ):
  (I6 (aw e d))=(lq (aQ e d))*2^(64:Ἀ)+(lq (aF e d))/\
  decide (aw e d<0)=(e>>>63!=0):=by have m:=e.toNat_lt;have o:=d.toNat_lt;rw[aQ,aF,bN,gB];by_cases j:2^(63:Ἀ)<=(lq e);simp only[j,decide_true,ἁ];by_cases g:d=0;subst g;simp only[Az,ἁ,ϋ];rw[Ζ];simp only [Int.reducePow,CharP.cast_eq_zero,Av,nr,ϋ,sub_self,sub_neg,AJ];rw[γ (fk_h3 _ ((϶ 63).trans_le j))];exact fk_h1 _ m;have l:(lq d)≠0:=λh=>g (ξ.1 (by simpa using h));simp only[show(d==0)=false by simpa using g,υ,Am];rw[Ζ,UInt64.toNat_not];simp only [Int.reducePow,Nat.add_one_sub_one,nr,ϋ,Av,sub_neg,AJ];rw[γ (fk_h3 _ (ϵ l))];exact fk_h2 _ _ m o;simp only[j,decide_false,υ,ἂ];simp only [Int.reducePow,sub_zero,nr,decide_eq_false_iff_not,not_lt];exact⟨by norm_cast,(by positivity)⟩
@[noinline]def au(hi lo B:ἆ):ἆ:=
 if hi==0&&lo==0 then 4294967296 else
 if B>442 then 4294967296 else
 let neg:=hi>>>63!=0
 let mlo:=aF hi lo
 let mhi:=aQ hi lo
 if mhi==0 then(UInt32.toUInt64 (az neg mlo B))
 else if mhi>>>32==0 then
  let q:=(mhi<<<32)+(mlo>>>32)
  (UInt32.toUInt64 (az neg (if mlo&&&0xFFFFFFFF==0 then q else q|||1) (B+32)))
 else(UInt32.toUInt64 (az neg (if mlo==0 then mhi else mhi|||1) (B+64)))
lemma toUInt32_toUInt64(x:ἇ):(N3 (UInt32.toUInt64 x))=x:=(by simp only [UInt32.toUInt32_toUInt64])
lemma i6_h1(e d:Ἀ):(e*2^(64:Ἀ)+d)/2^(32:Ἀ)=e*2^(32:Ἀ)+d/2^(32:Ἀ)/\(e*2^(64:Ἀ)+d)%2^(32:Ἀ)=d%2^(32:Ἀ):=by
 have k:e*2^(64:Ἀ)+d=d+e*2^(32:Ἀ)*2^(32:Ἀ):=by rw[Ϛ];exact Τ _ _
 exact⟨by rw[k,Ρ _ _ (϶ 32),Τ],by rw[k,Μ]⟩
lemma i6_h2(e d:Ἀ)(h:e<2^(32:Ἀ))(j:d<2^(64:Ἀ)):e*4294967296+d/4294967296<18446744073709551616:=(O1 ((ϸ (϶ 32)).2 j) _).trans_le ((Ο _ _).symm.trans_le (ι _ h))
lemma i6_h3(X Y:ἆ)(m r:ℕ)(l:(lq X)=m)(k:(lq Y)=r):(lq (if Y==0 then X else X|||1))=if (r==0)=true then m else m|||1:=by
 by_cases c:r=0
 · rw[if_pos (AQ.mpr (ξ.mp (k.trans (c.trans ϋ.symm)))),if_pos (AQ.mpr c)];exact l
 · rw[if_neg (fun q=>c (k.symm.trans ((congrArg lq (AQ.mp q)).trans ϋ))),if_neg (fun q=>c (AQ.mp q)),my,l,pS]
lemma i6(e d:ἆ)(h:(lq e)<2^(32:Ἀ)):
  (lq (if d&&&0xFFFFFFFF==0 then(e<<<32)+(d>>>32)
   else((e<<<32)+(d>>>32))|||1))=
   kQ ((lq e)*2^(64:Ἀ)+(lq d)) 32:=by
 have l:(lq ((e<<<32)+(d>>>32)))=(lq e)*2^(32:Ἀ)+(lq d)/2^(32:Ἀ):=by rw[η,ψ,ν];simp only[kA,kH,ο,β,Nat.mod_add_mod];exact γ (i6_h2 _ _ h d.toNat_lt)
 obtain⟨q1,q2⟩:=i6_h1 (lq e) (lq d);simp only[kQ,q1,q2];exact i6_h3 _ _ _ _ l ((Ε _ _).trans (ζ _ 32))
lemma i7(e d:ἆ):
  (lq (if d==0 then e else e|||1))=
   kQ ((lq e)*2^(64:Ἀ)+(lq d)) 64:=by have l:=d.toNat_lt;have g:((lq e)*2^(64:Ἀ)+(lq d))/2^(64:Ἀ)=(lq e):=(by omega);have h:((lq e)*2^(64:Ἀ)+(lq d))%2^(64:Ἀ)=(lq d):=(by exact Nat.mul_add_mod_of_lt l);rw[kQ];rw[g,h];by_cases j:(lq d)=0<;>simp[j,my,<-ξ]
lemma eN(s:Ἆ)(q B J:ἆ)(M:Ἀ)
  (e:(lq q)=kQ M (lq J))(h:(lq B)<=442)(j:(lq J)<=64)
  (d:2^((lq J)+25)<=M):
  az s q (B+J)=kW (kE lU ⟨s,M,((lq B):Ἁ)-298⟩):=by
 have g:(lq (B+J))=(lq B)+(lq J):=by simp[η];(clear * - h j;omega)
 rw[aJ _ _ _ (by (clear * - h j g;omega)),e,g,<-cm s M _ (lq J) ((mu (BD.1 (AN (by positivity) d))).2 d)];congr 3;(clear * -;omega)
lemma dq_h3(d e:Ἀ)(hd:d<2^(64:Ἀ))(he:e<2^(64:Ἀ))(j:(d:Ἁ)*2^(64:Ἀ)+e-2^(128:Ἀ)=0):d=0/\e=0:=by
 have N:d*2^(64:Ἀ)+e<2^(128:Ἀ):=(O1 he _).trans_le ((Ο d _).symm.trans_le (ι _ hd))
 have h:((d*2^(64:Ἀ)+e:Ἀ):Ἁ)=((2^(128:Ἀ):Ἀ):Ἁ):=by push_cast;exact sub_eq_zero.1 j
 exact absurd N (by rw[Nat.cast_inj.1 h];exact AX _)
lemma dq_h4(d e:Ἀ)(j:(d:Ἁ)*2^(64:Ἀ)+e-0=0):d=0/\e=0:=by
 obtain⟨h1,h2⟩:=Nat.add_eq_zero_iff.1 (by exact_mod_cast (sub_zero ((d:Ἁ)*2^(64:Ἀ)+e)).symm.trans j:d*2^(64:Ἀ)+e=0)
 exact⟨(Nat.mul_eq_zero.1 h1).resolve_right (BD.1 (϶ 64)),h2⟩
lemma dq_h1(a b:Ἀ)(v:a≠0):2^(57:Ἀ)<=a*2^(64:Ἀ)+b:=((by decide):2^(57:Ἀ)<=2^(64:Ἀ)).trans ((lZ _ (ϵ v)).trans (V1 _ _))
lemma dq_h2(a b:Ἀ)(u:2^(32:Ἀ)<=a):2^(89:Ἀ)<=a*2^(64:Ἀ)+b:=((by decide):2^(89:Ἀ)<=2^(32:Ἀ)*2^(64:Ἀ)).trans ((ι _ u).trans (V1 _ _))
lemma dq(d e B:ἆ)(h:au d e B<4294967296):
  aw d e≠0/\(lq B)<=442/\
  (N3 (au d e B))=
   kW (kE lU (nb (aw d e) (((lq B):Ἁ)-298))):=by
 obtain⟨m,o⟩:=fk d e;rw[au]at h⊢;by_cases t:(d==0&&e==0)=ἃ
 · rw[if_pos t]at h;exact (UInt64.lt_irrefl _ h).elim
 rw[if_neg t]at h⊢;have w:aw d e≠0:=λj=>t (by rw[bN]at j;simp only[Η,AQ,<-ξ,ϋ];split at j;exact dq_h3 _ _ d.toNat_lt e.toNat_lt j;exact dq_h4 _ _ j);by_cases p:B>442
 · rw[if_pos p]at h;exact (UInt64.lt_irrefl _ h).elim
 rw[if_neg p]at h⊢;have g:(lq B)<=442:=θ.1 (UInt64.not_lt.1 p)
 refine⟨w,g,?_⟩;rw[nb,o,m];have r:=ν (aQ d e) 32;rw[β]at r
 by_cases j:(aQ d e==0)=true
 · rw[if_pos j,(congrArg lq (AQ.1 j)).trans ϋ,ϳ,Κ,toUInt32_toUInt64,aJ _ _ _ (g.trans ((by decide):442<=506))]
 rw[if_neg j];have v:(lq (aQ d e))≠0:=λl=>j (by simp[<-ξ,l])
 by_cases l:(aQ d e>>>32==0)=true
 · rw[if_pos l];have q:(lq (aQ d e))<2^(32:Ἀ):=O8 fun h=>BD.1 (Nat.div_pos h (϶ 32)) (r.symm.trans (congrArg lq (AQ.1 l)))
   rw[toUInt32_toUInt64];exact eN _ _ _ 32 _ (i6 _ _ q) g (by decide) (dq_h1 _ _ v)
 rw[if_neg l];have u:2^(32:Ἀ)<=(lq (aQ d e)):=W3.1 fun x=>l (AQ.2 (ξ.1 (r.trans (ώ x))))
 rw[toUInt32_toUInt64];exact eN _ _ _ 64 _ (i7 _ _) g (by decide) (dq_h2 _ _ u)
lemma aY(n:Ἀ)(e g:Ἀ->kF)(l m B:ἆ)
  (j:aG n e g)(d:af n e g=aw l m*2^(lq B))
  (h:au l m B<4294967296):ms n e g=(N3 (au l m B)):=by
 have⟨q,r,p⟩:=dq l m B h;have o:af n e g≠0:=d▸mul_ne_zero q (by positivity)
 rw[bT n e g j o,p];refine mf _ _ _ _ o ?_;rw[toRat_ofIntDyadic,d];push_cast;rw[Ac AL,AA];field_simp
@[inline]def ar(N:ἆ):ἆ:=(N+(0x3FFFFFFFFF:ἆ)+((N>>>39)&&&1))>>>39
noncomputable def a8(lb v B:ἆ):ἆ:=
 let m:=aF v v
 let L:=ai m
 let E:=L+B-172
 if m-lb<0x8000000000000000-lb/\E<253 then((v>>>63)<<<31)+((E<<<23)+ar (m<<<(62-L)))
 else 4294967296
noncomputable def bP(hi lo B:ἆ):ἆ:=
 let r:=if hi=(0:ἆ)-(lo>>>63)then a8 1 lo B else 4294967296
 if r=4294967296 then au hi lo B else r
lemma gA(n:Ἆ)(e f:ἇ):
  (mJ (c9 n e f))=(if n then 2^(31:Ἀ) else 0)+(mJ e)%2^(8:Ἀ)*2^(23:Ἀ)+(mJ f)%2^(23:Ἀ):=by
 have j:=ϊ (mJ e) ((by positivity):0<2^(8:Ἀ));have d:(mJ ((e&&&255)<<<23|||(f&&&8388607)))=2^(23:Ἀ)*((mJ e)%2^(8:Ἀ))+(mJ f)%2^(23:Ἀ):=by rw[nK,(by rw[ϴ,Ξ,show(mJ (255:ἇ))=2^(8:Ἀ)-1 from ἅ,ζ,ο,Ϗ];exact γ (by simp only [kN,kH,nr];omega):(mJ ((e&&&255)<<<23))=2^(23:Ἀ)*((mJ e)%2^(8:Ἀ))),Ξ,show(mJ (8388607:ἇ))=2^(23:Ἀ)-1 from ἅ,ζ,<-Ϥ (ϊ (mJ f) ((by positivity):0<2^(23:Ἀ)))]
 rw[c9,UInt32.or_assoc];cases n
 .simp only[υ,ἂ,UInt32.zero_or,d];clear*-;ring
 simp only[ἁ,nK,d];rw[show(mJ (0x80000000:ἇ))=2^(31:Ἀ)*1 from ἅ,<-Ϥ (by clear*-j;omega)];clear*-;ring
lemma bM_h1_h1(k r G:Ἀ)(x M:ἆ)(y:(lq x)=62+G+k)(w:110<=G)(v:G<363)(j:k<2)(m:8388608<=(lq M))(q:(lq M)<2^(24:Ἀ))(h:k*2^(23:Ἀ)+(lq M)=r):(mJ (N3 (x-171)))%2^(8:Ἀ)*2^(23:Ἀ)+(mJ (d0 M))%2^(23:Ἀ)=(G-110)*2^(23:Ἀ)+r:=by
 obtain⟨d,rfl⟩:=Nat.exists_eq_add_of_le w;obtain⟨e,f⟩:=Nat.exists_eq_add_of_le m
 have l:(lq (x-171))=d+k+1:=by rw[χ x 171 (θ.2 (by rw[y];show 171<=_;exact ((by decide):171<=62+110).trans ((J1 (V1 110 d) 62).trans (V1 _ k)))),y];show 62+(110+d)+k-171=_;rw[show 62+(110+d)+k=d+k+1+171 by ring,BK]
 have hdk:d+k+1<2^(8:Ἀ):=by clear*-v j;omega
 have hee:e<2^(23:Ἀ):=by rw[f]at q;exact Nat.lt_of_add_lt_add_left q
 rw[Ϫ,l,c5,lE,f,ϱ,ϱ,γ (hdk.trans (by decide):d+k+1<2^(32:Ἀ)),γ (hdk:d+k+1<2^(8:Ἀ)),γ (hee.trans (by decide):e<2^(32:Ἀ)),γ hee,<-h,f];simp only[Ϡ,n4,pA];rfl
theorem bM_h1(g R:ἆ)(w:110<=(lq g))(v:(lq g)<363)(t:2^(23:Ἀ)<=(lq R)/\(lq R)<=2^(24:Ἀ)):(mJ (N3 ((if (R==16777216)=ἃ then 62+g+1 else 62+g)-171)))%2^(8:Ἀ)*2^(23:Ἀ)+(mJ (d0 (if (R==16777216)=ἃ then 8388608 else R)))%2^(23:Ἀ)=((lq g)-110)*2^(23:Ἀ)+(lq R):=by
 have H:=bM_h1_h1
 have y:(lq (62+g))=62+(lq g):=(η 62 g).trans (γ ((O1 v 62).trans (by decide):62+(lq g)<2^(64:Ἀ)))
 split
 .rename_i c;have:R=16777216:=by simpa using c
  subst this;exact H 1 _ _ _ _ (by rw[η,y];exact γ ((X3 (O1 v 62) 1).trans (by decide):62+(lq g)+1<2^(64:Ἀ))) w v (by decide) (by decide) (by decide) ἅ
 rename_i c;have c:¬R=16777216:=by simpa using c
 exact H 0 _ _ _ _ y w v (by decide) t.1 (lt_of_le_of_ne t.2 (λe=>c (ξ.1 e))) (Κ _)
lemma bM_h2(x b:Ἀ)(h:b<2):(x+274877906943+b)/2^(39:Ἀ)=x/2^(39:Ἀ)+if 2^(38:Ἀ)<x%2^(39:Ἀ)+b then 1 else 0:=by
 have e:=Au x (2^(39:Ἀ));have s:=ϊ x (϶ 39);generalize x/2^(39:Ἀ)=q at e⊢;generalize x%2^(39:Ἀ)=r at e s⊢;subst e
 rw[show q*2^(39:Ἀ)+r+274877906943+b=(r+274877906943+b)+q*2^(39:Ἀ) by ring,Ρ _ _ (϶ 39),Τ _ q];congr 1;split<;>rename_i c
 · exact mG (by rw[lO];exact L1 c 274877906943) ((P6 (X3 s _) (J2 h)).trans_le (by decide))
 · exact ώ (by rw[lO];exact (L1 (not_lt.1 c) _).trans_lt (by decide))
lemma bM_h3(Q R b:Ἀ)(hR:R<2^(39:Ἀ))(hb:b<2)(hQ:Q<2^(24:Ἀ)):(Q+if 274877906944<(R+b)%2^(64:Ἀ) then 1 else 0)%2^(64:Ἀ)=Q+if 2^(38:Ἀ)<R+b then 1 else 0:=by
 show (Q+if 2^(38:Ἀ)<(R+b)%2^(64:Ἀ) then 1 else 0)%2^(64:Ἀ)=_
 rw[γ ((E6 hR hb).trans (by decide))];by_cases c:2^(38:Ἀ)<R+b
 · rw[if_pos c,γ ((l7 hQ).trans_lt (by decide))]
 · rw[if_neg c,ϑ,γ (hQ.trans (by decide))]
lemma bM_h4(g:Ἀ)(v:g<363):(62+g+1)%18446744073709551616<=425/\(62+g)%18446744073709551616<=425:=⟨(Ay _ _).trans (J1 v 62),(Ay _ _).trans (J1 v.le 62)⟩
lemma bM_h5(g:ἆ)(w:110<=(lq g))(v:(lq g)<363):¬62+g<172:=fun h=>by
 have e:(lq (62+g))=62+(lq g):=(η 62 g).trans (γ ((O1 v 62).trans (by decide):62+(lq g)<2^(64:Ἀ)));have:=μ.1 h;rw[e]at this;exact absurd this (not_lt.2 (J1 w 62))
lemma bM_h6(N:Ἀ)(p:N<2^(63:Ἀ)):N+274877906943<18446744073709551616/\N+274877906943+N/549755813888%2<18446744073709551616:=⟨(X3 p _).trans (by decide),(P6 (X3 p _) (J2 (ϊ _ ((by positivity):0<2)))).trans_le (by decide)⟩
lemma bM(n:Ἆ)(N g:ἆ)(o:2^(62:Ἀ)<=(lq N))(p:(lq N)<2^(63:Ἀ))(w:110<=(lq g))
  (v:(lq g)<363):
  (mJ (az n N g))=(if n then 2^(31:Ἀ) else 0)+(((lq g)-110)*2^(23:Ἀ)+(lq (ar N)))/\
   2^(23:Ἀ)<=(lq (ar N))/\(lq (ar N))<=2^(24:Ἀ):=by
 have l:(lq (N>>>39))=(lq N)/2^(39:Ἀ):=by simp[β]
 have j:(lq ((N>>>39)&&&1))=(lq N)/2^(39:Ἀ)%2:=by simp[l,mP]
 have m:(lq (ar N))=(lq N)/2^(39:Ἀ)+if 2^(38:Ἀ)<(lq N)%2^(39:Ἀ)+(lq N)/2^(39:Ἀ)%2 then 1 else 0:=by obtain⟨b1,b3⟩:=bM_h6 _ p;rw[ar];simp(disch:=assumption) only[ν,η,j,kA,β,nr,γ];exact bM_h2 _ _ (ϊ _ ((by positivity):0<2))
 have d:2^(23:Ἀ)<=(lq N)/2^(39:Ἀ)/\(lq N)/2^(39:Ἀ)<2^(24:Ἀ):=⟨(mc (϶ 39)).2 ((B3 ἅ).trans o),(ϸ (϶ 39)).2 (p.trans_eq ἅ)⟩
 have e:2^(23:Ἀ)<=(lq (ar N))/\(lq (ar N))<=2^(24:Ἀ):=by rw[m];split;exact⟨d.1.trans (E7 _),d.2⟩;exact⟨d.1,d.2.le⟩
 have h:(lq (aW N (62-23)))=(lq (ar N)):=by rw[m];show(lq (de N 39))=_;rw[de];simp only[show((1:ἆ)<<<39)-1=0x7FFFFFFFFF from ἅ,show(1:ἆ)<<<(39-1)=0x4000000000 from ἅ,BC,μ,η,(Ε _ _▸ζ _ 39:(lq (N&&&0x7FFFFFFFFF))=(lq N)%2^(39:Ἀ)),j,l,Ax lq,kA];clear*-d;exact bM_h3 _ _ _ (ϊ (lq N) (϶ 39)) (ϊ _ ((by positivity):0<2)) d.2
 have q:ai N=62:=by rw[aA];apply ξ.1;rw[show (lq N.log2)=(Aq (lq N))from ἅ,mz (BD.1 ((϶ 62).trans_le o))];exact⟨o,p⟩
 refine⟨?_,e⟩;rw[az];simp only[q,(by simp[<-ξ];exact Nat.ne_zero_of_lt o:(N==0)=ἄ),bM_h5 g w v,((by decide):¬(62:ἆ)<23),ἂ,υ];generalize aW N (62-23)=R at h⊢;have t:2^(23:Ἀ)<=(lq R)/\(lq R)<=2^(24:Ἀ):=h▸e;rw[<-h];clear l j m h q o p d e;rw[if_neg (by split<;>simp[μ]<;>first|exact (bM_h4 _ v).1|exact (bM_h4 _ v).2),gA,pA,Nat.add_left_cancel_iff];exact bM_h1 g R w v t
lemma cl(n:Ἆ)(M σ:ἆ)(e:(lq M)≠0)(d:(lq σ)<=506):
  az n M σ=md ((if n then-((lq M):Ἁ)else (lq M))*2^(lq σ)) (-298) ἄ:=by rw[aJ _ _ _ d];refine(mf _ _ _ _ (mul_ne_zero (by split<;>omega) (pow_ne_zero _ AL)) ?_).symm;cases n<;>simp[pR,lr,Ac,div_eq_mul_inv,Ap]
lemma f9(v:ἆ):(v>>>63=0/\(lq v)<2^(63:Ἀ))\/(v>>>63=1/\2^(63:Ἀ)<=(lq v)):=by have:=v.toNat_lt;simp only[<-ξ,ν];simp[β];omega
lemma dZ_h3(S:Ἀ)(hS:S<63+2^(64:Ἀ))(q:(18446744073709551444+S)%18446744073709551616<253):(18446744073709551444+S)%18446744073709551616=S-172/\172<=S/\S<425:=by
 rcases Z7 S 172 with h|h
 · rw[γ (O1 h _:18446744073709551444+S<18446744073709551616)]at q;exact absurd ((V1 _ _).trans_lt q) (by decide)
 · have e:18446744073709551444+S=(S-172)+18446744073709551616:=by rw[show (18446744073709551616:Ἀ)=18446744073709551444+172 from ἅ,Nat.add_left_comm,L3 h]
   have s1:S-172<18446744073709551616:=G4 h (hS.trans ((by decide):63+2^(64:Ἀ)<172+18446744073709551616))
   rw[e,Nat.add_mod_right,γ s1]at q⊢;exact⟨ἅ,h,(tsub_lt_tsub_iff_right h).1 (q:S-172<425-172)⟩
lemma dZ_h4(S a:Ἀ)(k:S<425)(N:a<=2^(24:Ἀ)):(S-172)*2^(23:Ἀ)+a<2^(64:Ἀ)/\2^(31:Ἀ)+((S-172)*2^(23:Ἀ)+a)%2^(64:Ἀ)<2^(64:Ἀ)/\0+((S-172)*2^(23:Ἀ)+a)%2^(64:Ἀ)<2^(64:Ἀ):=by
 have b:(S-172)*2^(23:Ἀ)+a<2^(31:Ἀ):=(Y3 (ι (2^(23:Ἀ)) (P4 k.le 172)) N).trans_lt (by decide)
 rw[γ (b.trans (by decide)),Κ];exact⟨b.trans (by decide),(O1 b _).trans (by decide),b.trans (by decide)⟩
lemma dZ_h1(l B:Ἀ)(y:172<=l+B)(k:l+B<425)(A:l<63):62-l<=B/\B-(62-l)=l+B-62/\110<=B-(62-l)/\B-(62-l)<363/\B<=506/\B-(62-l)<=506:=by
 have u2:B-(62-l)=l+B-62:=(Nat.sub_sub_right _ (J2 A)).trans (congrArg (·-62) (Τ _ _))
 rw[u2];exact⟨Nat.sub_le_iff_le_add.2 (((by decide):62<=172).trans (y.trans_eq (Τ _ _))),ἅ,G5 y,(P4 (J2 k) 62).trans_lt (by decide),(K2 _ _).trans (k.le.trans ((by decide):425<=506)),(K4 _ _).trans (k.le.trans ((by decide):425<=506))⟩
lemma dZ_h2(S:Ἀ)(k:S<425):(S-172)*2^(23:Ἀ)%2^(64:Ἀ)=(S-172)*2^(23:Ἀ):=γ ((ι (2^(23:Ἀ)) (P4 k.le 172)).trans_lt (by decide))
lemma dZ_h6(l B:Ἀ)(y:172<=l+B)(A:l<63):110<=B:=Nat.le_of_add_le_add_left (y.trans (L1 (J2 A) _):62+110<=62+B)
lemma dZ(p v B:ἆ)(r:1<=(lq p))(g:(lq p)<=2^(63:Ἀ))(h:a8 p v B≠4294967296):
  (lq p)<=(lq (aF v v))/\(lq (aF v v))<2^(63:Ἀ)/\110<=(lq B)/\(lq B)<=424/\
   a8 p v B=(UInt32.toUInt64 (az (v>>>63!=0) (aF v v) B)):=by
 dsimp only[a8]at h⊢;split at h;swap;exact(h ἅ).elim;rename_i E;rw[AD E];obtain⟨F,q⟩:=E;generalize aF v v=m at F q⊢;have j:(lq p)<=(lq m)/\(lq m)<2^(63:Ἀ):=by have H:=Nat.add_lt_of_lt_sub ((μ.1 F).trans_eq (χ 0x8000000000000000 p (θ.2 g)));have E:(lq m)=(lq (m-p))+(lq p):=(congrArg lq (UInt64.sub_add_cancel m p).symm).trans ((η _ _).trans (γ (H.trans (by decide))));exact⟨(K2 _ _).trans_eq E.symm,E.trans_lt H⟩
 have o:(lq m)≠0:=L6 (r.trans j.1);have r:(lq (ai m))=(Aq (lq m)):=by rw[aA];rfl
 have t:=ϰ o;have g:=@ϙ (lq m);have A:=(m9 o).2 j.2;generalize ai m=L at r q⊢;generalize (Aq (lq m))=l at r t g A;obtain⟨x,y,k⟩:(lq (L+B-172))=l+(lq B)-172/\172<=l+(lq B)/\l+(lq B)<425:=by rw[μ]at q;rw[Ζ,η,r,Nat.add_mod_mod]at q⊢;exact dZ_h3 _ (E6 A B.toNat_lt) q
 refine⟨j.1,j.2,dZ_h6 _ _ y A,J2 ((K2 _ _).trans_lt k),?_⟩;have D:(lq (62-L))=62-l:=by rw[χ _ _ (by rw[θ,r];exact J2 A),r]<;>rfl
 have z:(lq m)*2^(62-l)<2^(63:Ἀ):=AN (ϐ g (϶ _)) (by rw[<-AF];exact Θ (by decide) (B3 (by rw[lO,M4 (J2 A)])));have w:(lq (m<<<(62-L)))=(lq m)*2^(62-l):=by rw[ψ,D,ο,γ (AO (K4 _ _) (by decide):62-l<64),γ (z.trans ((by decide):2^(63:Ἀ)<2^(64:Ἀ)))]
 obtain⟨lb,u2,c1,c2,c3,c4⟩:=dZ_h1 l (lq B) y k A;have u:(lq (B-(62-L)))=(lq B)-(62-l):=by rw[χ _ _ (by rw[θ,D];exact lb),D]
 obtain⟨K,M,N⟩:=bM (v>>>63!=0) (m<<<(62-L)) (B-(62-L)) (w▸AZ (Eq.le (by rw[<-AF,M4 (J2 A)])) (ι _ t)) (w▸z) (by rw[u];exact c1) (by rw[u];exact c2);rw[<-ξ,ϩ,(by rw[cl _ m B o c3,cl _ (m<<<(62-L)) (B-(62-L)) (by rw[w];exact Nat.mul_ne_zero o (϶ _).ne') (by rw[u];exact c4),w,u];refine congrArg (md · (-298) ἄ) ?_;obtain⟨d,J⟩:∃d,(lq B)=62-l+d:=⟨(lq B)-(62-l),(M4 lb).symm⟩;rw[J,ϱ];push_cast;split<;>ring:az (v>>>63!=0) m B=az (v>>>63!=0) (m<<<(62-L)) (B-(62-L))),K,u,η,η,(by rcases f9 v with⟨e,-⟩|⟨e,-⟩<;>rw[e]<;>rfl:(lq ((v>>>63)<<<31))=(if(v>>>63!=0)then 2^(31:Ἀ) else 0)),(by rw[ψ,x];simp only[kA,kH,ο];exact dZ_h2 _ k:(lq ((L+B-172)<<<23))=(l+(lq B)-172)*2^(23:Ἀ))];rw[(by rw[u2,F7]:(lq B)-(62-l)-110=l+(lq B)-172)];obtain⟨d1,d2,d3⟩:=dZ_h4 _ _ k N;split<;>rw[γ,γ]<;>with_reducible first|exact d1|exact d2|exact d3
lemma fd(d v B:ἆ)(o:1<=(lq d))(j:(lq d)<=2^(63:Ἀ))(h:a8 d v B≠4294967296):
  a8 d v B=au ((0:ἆ)-(v>>>63)) v B:=by
 obtain⟨m,-,-,p,e⟩:=dZ d v B o j h;rw[e];have l:v≠0:=by rintro rfl;exact absurd m (by show¬(lq d)<=0;(clear * - o j;omega))
 have g:¬B>442:=by simp[μ];(clear * - p;omega)
 symm;unfold au aQ aF;rcases f9 v with⟨e,-⟩|⟨e,-⟩;simp[e,l,g];rw[e,show(0:ἆ)-1=0xFFFFFFFFFFFFFFFF from ἅ];simp[l,g,show(0xFFFFFFFFFFFFFFFF:ἆ)>>>63=1 from ἅ,show~~~(0xFFFFFFFFFFFFFFFF:ἆ)=0 from ἅ]
lemma dY(g d B:ἆ):bP g d B=au g d B:=by
 unfold bP;dsimp only;by_cases e:g=(0:ἆ)-(d>>>63)
 .rw[AD e];split
  .rfl
  .rename_i h;rw[e];exact fd 1 d B (by decide) (by decide) h
 .rw[AC e,AD rfl]
end Core open Core
lemma cn(w:kF):kW (lI w)=w:=by apply Β.mp;rw[kW,lI,Model.toNatBits,pu,lE];exact γ w.toBitVec.isLt
lemma aR(z:Model lU):lI (kW z)=z:=by have h:z.toNatBits<2^(32:Ἀ):=mQ z;rw[lI,kW,kT,lE,γ h];exact m3 z
namespace Dense
lemma ac{k n:Ἄ}(h:k<n):(J0 (k+1))=(J0 k)+1:=by have:=ε.mp h;have:=n.toNat_lt_size;have:=nT;rw[lJ,mA,γ (by omega)]
lemma aj{a b:Ἄ}(h:(J0 a)+(J0 b)<kX):(J0 (a+b))=(J0 a)+(J0 b):=by rw[lJ,γ h]
lemma aa{n:Ἀ}(h:n<kX):(J0 (G0 n))=n:=by rw[lX,J3 h]
lemma aC(x:ἇ):(J0 (UInt32.toUSize x))=(mJ x):=by simp[UInt32.toNat_toUSize]
lemma cC(A:Ἂ ἇ)(u:Ἄ)(h:(J0 u)<(O0 A)):A.uget u h=A[(J0 u)]!:=by simp_all only [Array.uget,AB]
lemma ay(p:ἆ):
  ae p=((lq p):Ἁ)-(if 2^(63:Ἀ)<=(lq p)then 2^(64:Ἀ) else 0):=by rw[ae];rw[lv,nR];split<;>split<;>push_cast<;>omega
noncomputable def bj(x:ἇ):ἇ:=x&&&0x7FFFFFFF
lemma aM(x:ἇ):(mJ (bj x))=(mJ x)%2^(31:Ἀ):=by rw[bj];rw[Ξ];exact ζ _ 31
lemma aN(x:ἇ):(mJ (kB x))=(mJ (bj x))/2^(23:Ἀ):=by
 have:(mJ (kB x))=(mJ x)/2^(23:Ἀ)%2^(8:Ἀ):=by unfold kB;rw[Ξ,Υ];simp only[kN,kH,β];exact ζ _ 8
 rw[this,aM,show 2^(31:Ἀ)=2^(23:Ἀ)*2^(8:Ἀ) by norm_num,Nat.mod_mul_right_div_self]
lemma eZ(x:ἇ):kB x=0↔(mJ (bj x))<2^(23:Ἀ):=by rw[<-Β,aN];simp only [nr,nj,Nat.div_eq_zero_iff,OfNat.ofNat_ne_zero,false_or];try omega
lemma jO(x:ἇ):kG x=(mJ (bj x))/2^(23:Ἀ)-1:=by
 have a:=aN x;rw[kG];split<;>rename_i h
 .rw[ώ ((eZ x).mp h)]
 .rw[a]
lemma aT(x:ἇ):kG x=(mJ (kB x))-1:=by rw[jO,aN]
lemma jW(w:ἇ):(mJ w)/2^(23:Ἀ)=(mJ (kB w))+256*((mJ w)/2^(31:Ἀ)):=by rw[aN,aM];have:=w.toNat_lt;(clear * -;omega)
lemma bm(x:ἇ):(mJ (kB x))<256:=by rw[aN,aM];omega
lemma fe(w:ἇ):(mJ (w&&&8388607))=(mJ w)%2^(23:Ἀ):=by rw[Ξ];exact ζ _ 23
lemma eR(w:ἇ):|sv w|<2^(24:Ἀ):=by have:=lB w;rw[sv];split<;>simp<;>exact I7 this
def lz(y n:ἆ):ἆ:=if y=0 then n else lz (y>>>1) (n+1)
termination_by (lq y)
decreasing_by
 have:(lq y)≠0:=λe=>by simp_all[<-ξ]
 rw[ν];simp[β];exact Nat.bitwise_rec_lemma this
abbrev bl(k:Ἀ):Ἀ:=if k=0 then 0 else (Aq k)+1
lemma iZ(y:ἆ):bl (lq y)<=64:=by
 rw[bl];split;. (clear * -;omega)
 rename_i h;have:=(m9 h).mpr y.toNat_lt;exact Order.add_one_le_iff.mpr this
lemma ed(y:ἆ):(lq (y>>>1))=(lq y)/2:=by rw[ν];simp[β]
lemma dA(k:Ἀ):bl (k/2)=(Aq k):=by
 rw[bl];rw[Nat.log2_def k];by_cases d:2<=k
 .rw[AC (by omega),AD d]
 .rw[AD (by omega),AC d]
lemma dD(y n:ἆ)(h:(lq n)+bl (lq y)<=64):(lq (lz y n))=(lq n)+bl (lq y):=by
 rw[lz];by_cases d:y=0
 .rw[AD d];subst d;simp
 .rw[AC d];have j:(lq (n+1))=(lq n)+1:=by rw[η];simp;(clear * - h;omega)
  have g:(lq y)≠0:=λe=>d (ξ.mp (by simpa using e));have l:bl (lq y)=bl ((lq y)/2)+1:=by rw[dA];simp[bl,g]
  rw[dD (y>>>1) (n+1) (by rwa[j,ed,lO,pA,←l]),j,ed,l,lO,pA]
termination_by (lq y)
decreasing_by rw[ed];exact Nat.bitwise_rec_lemma g
@[inline]def lg(x:ἆ):ἆ:=lz (x>>>1) 0
lemma i8(x:ἆ):lg x=x.log2:=by apply ξ.mp;rw[lg,dD _ _ (by have:=iZ (x>>>1);simpa using this),ed,dA,ϋ,Κ];rfl
@[inline]def c0(lb v B:ἆ):ἆ:=
 let m:=Core.aF v v
 let L:=lg m
 let E:=L+B-172
 if m-lb<0x8000000000000000-lb/\E<253 then((v>>>63)<<<31)+((E<<<23)+Core.ar (m<<<(62-L)))
 else 4294967296
lemma ez:c0=a8:=by funext d v B;rw[c0,a8,aA,i8]
@[inline]def b9(hi lo B:ἆ):ἆ:=
 let r:=if hi=(0:ἆ)-(lo>>>63)then c0 1 lo B else 4294967296
 if r=4294967296 then Core.au hi lo B else r
lemma dp:b9=bP:=by funext d e B;rw[b9,bP,ez]
lemma cD{k:Ἄ}{a:Ἂ ἇ}(h:k<a.usize):(J0 k)<(O0 a):=by have:=ε.mp h;rw[Array.usize,lX,USize.toNat_ofNat']at this;exact AN this (Ay _ _)
@[inline]def gd(h l m s:ἆ):ἆ×ἆ:=
 let l':=l+(m<<<s)
 (h+(m>>>1>>>(63-s))+(if l'<l then 1 else 0),l')
lemma fD_h1(H L M:Ἀ)(hL:L<2^(64:Ἀ))(g:H*2^(64:Ἀ)+L+M<2^(128:Ἀ))(c:(L+M%2^(64:Ἀ))%2^(64:Ἀ)<L):
  ((H+M/2^(64:Ἀ))%2^(64:Ἀ)+1)%2^(64:Ἀ)*2^(64:Ἀ)+(L+M%2^(64:Ἀ))%2^(64:Ἀ)=H*2^(64:Ἀ)+L+M:=by
 have hb:=ϊ M ((by decide):2^(64:Ἀ)>0);have hM:=ϛ M (2^(64:Ἀ))
 generalize M/2^(64:Ἀ)=a,M%2^(64:Ἀ)=b at *;subst hM
 have k:2^(64:Ἀ)<=L+b:=not_lt.1 λh=>absurd ((γ h)▸c) (not_lt.2 (V1 L b))
 have s:H+a+1<2^(64:Ἀ):=by clear*-g k;omega
 rw[P8 k,γ (G4 k (E6 hL hb)),γ ((I3 _).trans s),γ s];clear*-k;omega
lemma fD_h2(H L M:Ἀ)(hL:L<2^(64:Ἀ))(g:H*2^(64:Ἀ)+L+M<2^(128:Ἀ))(c:¬(L+M%2^(64:Ἀ))%2^(64:Ἀ)<L):
  ((H+M/2^(64:Ἀ))%2^(64:Ἀ)+0)%2^(64:Ἀ)*2^(64:Ἀ)+(L+M%2^(64:Ἀ))%2^(64:Ἀ)=H*2^(64:Ἀ)+L+M:=by
 have hb:=ϊ M ((by decide):2^(64:Ἀ)>0);have hM:=ϛ M (2^(64:Ἀ))
 generalize M/2^(64:Ἀ)=a,M%2^(64:Ἀ)=b at *;subst hM
 have k:L+b<2^(64:Ἀ):=O8 λh=>c (by rw[P8 h,γ (G4 h (E6 hL hb))];exact G4 h ((O1 hb L).trans_eq (Τ L _)))
 have s:H+a<2^(64:Ἀ):=by clear*-g;omega
 rw[γ k,ϑ,γ s,γ s];ring
lemma fD(h l m s:ἆ)(j:(lq s)<=60)
  (g:(lq h)*2^(64:Ἀ)+(lq l)+(lq m)*2^(lq s)<2^(128:Ἀ)):
  (lq (gd h l m s).1)*2^(64:Ἀ)+(lq (gd h l m s).2)=(lq h)*2^(64:Ἀ)+(lq l)+(lq m)*2^(lq s):=by
 have d:(lq (63-s))=63-(lq s):=by rw[χ _ _ (by simp[θ];exact AZ j (by decide))];rfl
 have e:(lq m)/2/2^(63-(lq s))=(lq m)*2^(lq s)/2^(64:Ἀ):=by rw[lY,show(2:Ἀ)^(64:Ἀ)=2^(1:Ἀ)*2^(63-(lq s))*2^(lq s) by rw[<-AF,<-AF];congr 1;rw[pA,L3 (AZ j (by decide))],pow_one,Nat.mul_div_mul_right _ _ ((by positivity):0<2^(lq s))]
 have:=l.toNat_lt;simp only[gd,μ,η,ψ,ν,d,ο,β,γ (AO j (by decide):(lq s)<64),γ (AO (K4 _ _) (by decide):63-(lq s)<64),kA,kH,pow_one,e];clear d e j;split<;>simp only[kA];exact fD_h1 _ _ _ ‹_› g ‹_›;exact fD_h2 _ _ _ ‹_› g ‹_›
@[specialize]def jG(fa fb:Ἀ->Ἃ)(n t:Ἀ)(B:ἆ):ἆ:=
 if t<n then
  jG fa fb n (t+1) (if Core.mW (fa t)*Core.mW (fb t)==0 then B else min B (Core.iW (fa t)+Core.iW (fb t)))
 else B
termination_by n-t
decreasing_by all_goals omega
@[specialize]def dl(fa fb:Ἀ->Ἃ)(n:Ἀ)(B:ἆ)(t:Ἀ)(pH pL nH nL:ἆ):ἆ:=
 if t<n then
  let a:=fa t
  let b:=fb t
  let m:=Core.mW a*Core.mW b
  let e:=Core.iW a+Core.iW b
  if expOf a==255||expOf b==255||m !=0&&(e<B||60<e-B)then 4294967296 else
  if m==0 then dl fa fb n B (t+1) pH pL nH nL else
  let g:=signOf a==signOf b
  let q:=gd (if g then pH else nH) (if g then pL else nL) m (e-B)
  dl fa fb n B (t+1) (if g then q.1 else pH) (if g then q.2 else pL) (if g then nH else q.1) (if g then nL else q.2)
 else Core.au (pH-nH-(if pL<nL then 1 else 0)) (pL-nL) B
termination_by n-t
decreasing_by all_goals omega
lemma ku_h1_h1(h g c:ἆ)(hh:(lq h)<2^(62:Ἀ))(hg:(lq g)<2^(62:Ἀ))(hc:(lq c)<=1):
  ((lq (h-(g+c))):Ἁ)*2^(64:Ἀ)-(if 2^(63:Ἀ)<=(lq (h-(g+c))) then 2^(128:Ἀ) else 0)=(((lq h):Ἁ)-(lq g)-(lq c))*2^(64:Ἀ):=by
 have e:(lq (g+c))=(lq g)+(lq c):=BitVec.toNat_add_of_lt (show (lq g)+(lq c)<2^(64:Ἀ) from(P6 hg hc).trans (by decide))
 by_cases q:g+c<=h
 · have r:(lq (h-(g+c)))=(lq h)-(lq (g+c)):=BitVec.toNat_sub_of_le q
   rw[θ,e] at q;rw[r,e,if_neg (not_le.2 ((K4 _ _).trans_lt (hh.trans (by decide)))),sub_zero,pO q,W5,sub_sub]
 · have r:(lq (h-(g+c)))=2^(64:Ἀ)-((lq (g+c))-(lq h)):=BitVec.toNat_sub_of_lt (UInt64.not_le.1 q)
   rw[θ,e,not_le] at q
   have s:((lq g)+(lq c))-(lq h)<=2^(63:Ἀ):=(K4 _ _).trans ((Y3 hg.le hc).trans (by decide))
   rw[r,e,if_pos (G5 ((J1 s _).trans (by decide))),pO (s.trans (by decide)),pO q.le]
   push_cast;ring
lemma ku_h1_h2(e d:ἆ):((lq (e-d)):Ἁ)=(lq e)-(lq d)+(lq (if e<d then (1:ἆ) else 0))*2^(64:Ἀ):=by
 by_cases p:e<d
 · have r:(lq (e-d))=2^(64:Ἀ)-((lq d)-(lq e)):=BitVec.toNat_sub_of_lt p
   rw[if_pos p,r,pO ((K4 _ _).trans d.toNat_lt.le),pO (μ.1 p).le,pS];push_cast;ring
 · have r:(lq (e-d))=(lq e)-(lq d):=BitVec.toNat_sub_of_le (UInt64.not_lt.1 p)
   rw[if_neg p,r,pO (θ.1 (UInt64.not_lt.1 p)),ϋ,Nat.cast_zero,zero_mul,Av]
lemma ku_h1(h e g d:ἆ)(l:(lq h)*2^(64:Ἀ)+(lq e)<2^(126:Ἀ))
  (j:(lq g)*2^(64:Ἀ)+(lq d)<2^(126:Ἀ)):
  aw (h-g-(if e<d then 1 else 0)) (e-d)=
   (((lq h)*2^(64:Ἀ)+(lq e):Ἀ):Ἁ)-(((lq g)*2^(64:Ἀ)+(lq d):Ἀ):Ἁ):=by
 rw[sub_sub,bN,add_sub_right_comm,ku_h1_h1 h g _ (Nat.lt_of_mul_lt_mul_right (a:=2^(64:Ἀ)) ((V1 _ _).trans_lt l)) (Nat.lt_of_mul_lt_mul_right (a:=2^(64:Ἀ)) ((V1 _ _).trans_lt j)) (by split<;>exact (by decide)),ku_h1_h2 e d];push_cast;ring
lemma ku(h e g d:ἆ)(l:(lq h)*2^(64:Ἀ)+(lq e)<2^(126:Ἀ))
  (j:(lq g)*2^(64:Ἀ)+(lq d)<2^(126:Ἀ)):
  aw (h-g-(if e<d then 1 else 0)) (e-d)=
   (((lq h)*2^(64:Ἀ)+(lq e):Ἀ):Ἁ)-(((lq g)*2^(64:Ἀ)+(lq d):Ἀ):Ἁ):=ku_h1 h e g d l j
lemma fE_h1(u v:Ἃ)(B:Ἀ)(h:B<=kG u+kG v):tm u v=(if k9 u==k9 v then((kM u*kM v*2^(kG u+kG v-B):Ἀ):Ἁ)else-((kM u*kM v*2^(kG u+kG v-B):Ἀ):Ἁ))*2^B:=by
 unfold tm sv;rw[show (2:Ἁ)^(kG u+kG v)=2^(kG u+kG v-B)*2^B by rw[<-AF,L3 h],Y4,Y4,I4,nZ];generalize kG u+kG v-B=e;generalize (kM u:Ἁ)=x;generalize (kM v:Ἁ)=y;cases k9 u<;>cases k9 v<;>simp only[Bool.true_beq,Bool.false_beq,Bool.not_true,Bool.not_false,υ,ἁ,ἂ,↓reduceIte,neg_mul,mul_neg,neg_neg,Ap]
lemma fE_h2(u v:Ἃ)(h:kM u*kM v=0):tm u v=0:=by
 unfold tm sv;rcases Nat.mul_eq_zero.1 h with h|h<;>rw[h]<;>simp only[Nat.cast_zero,neg_zero,ite_self,zero_mul,mul_zero]
lemma fE_h4{c:Prop}[Decidable c]{a b k:ἆ}(j:(if c then a else b)<k):c∧a<k∨¬c∧b<k:=by
 by_cases h:c;exact pZ ⟨h,by rwa[if_pos h]at j⟩;exact BA ⟨h,by rwa[if_neg h]at j⟩
lemma fE(d g:Ἀ->kF)(n:Ἀ)(B:ἆ)(hn:n<2^(18:Ἀ)):
  ∀k t (r u o p:ἆ),n-t=k->t<=n->aG t d g->
   (lq r)*2^(64:Ἀ)+(lq u)<=t*2^(108:Ἀ)->(lq o)*2^(64:Ἀ)+(lq p)<=t*2^(108:Ἀ)->
   af t d g=((((lq r)*2^(64:Ἀ)+(lq u):Ἀ):Ἁ)-(((lq o)*2^(64:Ἀ)+(lq p):Ἀ):Ἁ))*
    2^(lq B)->
   dl d g n B t r u o p<4294967296->ms n d g=(N3 (dl d g n B t r u o p)):=by
 intro k;induction k<;>intro t r u o p hk ht S I G J j
 .obtain rfl:t=n:=C4 ht (K3.1 hk);rw[dl,AC (AX _)]at j⊢;exact aY _ _ _ _ _ _ S (by rw[J,ku _ _ _ _ (I.trans_lt ((ϐ hn (by decide)).trans_le (by decide))) (G.trans_lt ((ϐ hn (by decide)).trans_le (by decide)))]) j
 rename_i k e;obtain⟨X,y⟩:t<n/\n-(t+1)=k:=⟨N1 hk,by rw[V4,hk,BK]⟩;rw[dl,AD X]at j⊢;dsimp only at j⊢;obtain⟨-,j⟩|⟨Y,j⟩:=fE_h4 j;exact absurd j (by decide);rw[AC Y];simp only[nM,AQ,Η,AM,Ak,AJ,not_or,not_and,μ,kA]at Y;obtain⟨⟨K,M⟩,hw⟩:=Y;have x:aG (t+1) d g:=λs E=>if h:s<t then S s h else(C4 (J2 E) (not_lt.1 h):s=t)▸⟨K,M⟩
 have v:kM (d t)*kM (g t)<2^(48:Ἀ):=(ϕ (lB _) (lB _)).trans_eq (by norm_num);have D:=aK t d g;have m:t*2^(108:Ἀ)<=(t+1)*2^(108:Ἀ):=ι _ t.le_succ;have U:(lq (mW (d t)*mW (g t)))=kM (d t)*kM (g t):=by rw[ύ,an,an,γ (v.trans (by norm_num))]
 rcases fE_h4 j with⟨F,j⟩|⟨F,j⟩
 .rw[AD F];refine e (t+1) r u o p y X x (I.trans m) (G.trans m) ?_ j;rw[D,J,fE_h2 _ _ (by rw[<-U,AQ.1 F];rfl),Av]
 rw[AC F];obtain⟨R,O⟩:=hw λh=>F (by simp[h]);have i1:=idxOf_le _ K;have i2:=idxOf_le _ M;have Q:(lq (iW (d t)+iW (g t)))=kG (d t)+kG (g t):=by rw[η,am,am,γ ((Y3 i1 i2).trans_lt (by decide))]
 rw[Q]at R;have E:(lq (iW (d t)+iW (g t)-B))=kG (d t)+kG (g t)-(lq B):=by rw[χ _ _ (θ.2 (Q▸not_lt.1 R)),Q]
 rw[E]at O;have l:kM (d t)*kM (g t)*2^(kG (d t)+kG (g t)-(lq B))<=2^(108:Ἀ):=(ϧ v.le (Θ (by norm_num) (not_lt.1 O))).trans_eq (by norm_num)
 generalize w:kM (d t)*kM (g t)*2^(kG (d t)+kG (g t)-(lq B))=T at l;have A:tm (d t) (g t)=(if k9 (d t)==k9 (g t)then(T:Ἁ)else-T)*2^(lq B):=w▸fE_h1 _ _ _ (not_lt.1 R)
 have V:=λH L (ah:(lq H)*2^(64:Ἀ)+(lq L)<=t*2^(108:Ἀ))=>fD H L _ _ (E▸not_lt.1 O) (by rw[U,E,w];exact (Y3 ah l).trans_lt (((Ο t _).symm.trans_le (ι _ X)).trans_lt ((ϐ hn (by decide)).trans (by decide))));rw[U,E,w]at V;cases c:(k9 (d t)==k9 (g t))<;>simp only[c,ἁ,Am,υ]at A j⊢
 .have q:=V o p G;refine e (t+1) r u _ _ y X x (I.trans m) (by rw[q];exact Ο t _▸Y3 G l) ?_ j;rw[D,J,q,A,W5 _ T];ring
 have q:=V r u I;refine e (t+1) _ _ o p y X x (by rw[q];exact Ο t _▸Y3 I l) (G.trans m) ?_ j;rw[D,J,q,A,W5 _ T];ring
@[inline]def gf(n:Ἀ)(fa fb:Ἀ->Ἃ):Ἃ:=
 let r:=if n<262144 then dl fa fb n (jG fa fb n 0 1023) 0 0 0 0 0 else 4294967296
 if r<4294967296 then (N3 r)else dotOf n fa fb
lemma bO(n:Ἀ)(d e:Ἀ->kF):gf n d e=ms n d e:=by
 unfold gf;dsimp only;by_cases g:n<262144
 .rw[AD g];split
  .exact(fE d e n _ (by exact I7 g) _ 0 0 0 0 0 rfl (by exact ϟ n) (λ_ h=>absurd h (by (clear * - h;omega)))
    ((by simp)) ((by simp)) (by simp[af]) ‹_›).symm
  .rfl
 .rw[AC g,AC (by decide)]
@[noinline]def aD(n:Ἀ)(relu:Ἆ)(X W b:@&Ἂ Ἃ)(xo s j:Ἀ):Ἃ:=
 act relu (gf (n+1) (λt=>if t<n then X[xo+t*s]!else oneW)
  (λt=>if t<n then W[j*n+t]!else b[j]!))
namespace F
noncomputable def rv(v:Ἅ):ℝ:=Model.toReal (BB v.toModel)
def fv(v:Ἅ):Prop:=Model.isFinite (BB v.toModel)=ἃ
lemma iL(v:Ἅ):rv v=lw v.toModel.unpack:=by rw[rv];rw[Model.toReal_eq_unpackedToReal_toModel (by decide),ϒ]
lemma fO(v:Ἅ):fv v↔v.toModel.unpack.isFinite=ἃ:=by rw[fv];rw[<-Model.toDyadic?_isSome_eq_isFinite,Model.toDyadic?_ieee_eq_model _ (by decide),ϒ];cases v.toModel.unpack<;>rfl
lemma gV(x y:Ἅ)(d:fv x)(e:fv y):
  kI (x+y).toModel=pb (kI x.toModel) (kI y.toModel):=by change kI (pW x.toModel y.toModel)=_;rw[pW,nu];have h:=Model.add_eq_ofModel_add_of_isFinite (fmt:=kC) (by decide) (kI x.toModel) (kI y.toModel) (by rw[ϒ];exact(fO x).1 d) (by rw[ϒ];exact(fO y).1 e);rw[ϒ,ϒ]at h;exact h.symm
lemma pm(E:Ἁ)(e:E<=971):(2:ℝ)^E<=ls (mR kC):=by have h:=ϫ kC ἄ 1 E (by decide) (e.trans_eq (by decide));simp only[υ,Am,pJ,one_mul]at h;change|(2:ℝ)^E|<=_ at h;simpa only[AW (zpow_pos ((by norm_num):(0:ℝ)<2) E)]using h
open FloatLib.Floats.Formats.Flocq in
lemma re(x:ℝ)(E:Ἁ)(g:-1021<=E)(e:|x|<(2:ℝ)^E):
  |Model.roundAt kC x-x|<=(2:ℝ)^(E-53)/2:=by
 by_cases d:x=0
 .subst x;rw[pD,sub_zero,A5];positivity
 have h:magnitude kO x<=E:=magnitude_le_of_abs_lt_bpow _ x E d e;apply(Model.abs_roundAt_sub_le kC x).trans;apply div_le_div_of_nonneg_right _ (by norm_num);change ulp kO (Model.fexpOf kC) x<=(2:ℝ)^(E-53);rw[ulp.of_ne_zero _ _ x d];change(2:ℝ)^(max (magnitude kO x-53) (-1074))<=(2:ℝ)^(E-53);apply AG (by norm_num);(clear * - g h;omega)
lemma ii(x y:Ἅ)(d:fv x)(e:fv y)(j:|rv x+rv y|<=ls (mR kC)):
  fv (x+y)/\rv (x+y)=ns kC (rv x+rv y):=by have h:=gV x y d e;have g:=Model.isFinite_add_of_abs_toReal_add_le_posMaxFinite (kI x.toModel) (kI y.toModel) (by decide) d e j;exact⟨by rwa[fv,h],by rw[rv,h];exact mH _ _ (by decide) g⟩
lemma e8(x y:Ἅ)(l:fv x)(m:fv y)(E:Ἁ)(j:-1021<=E)(d:E<=971)
  (e:|rv x+rv y|<(2:ℝ)^E):
  fv (x+y)/\|rv (x+y)-(rv x+rv y)|<=(2:ℝ)^(E-53)/2:=by have⟨g,h⟩:=ii x y l m (e.le.trans (pm E d));exact⟨g,by rw[h];exact re _ E j e⟩
lemma rx(d:kU)(l:d.significand<2^(53:Ἀ))(g:-1074<=d.exponent)
  (e:|d.toReal|<=(2:ℝ)^(971:Ἁ)):
  ns kC d.toReal=d.toReal:=by have⟨h,j⟩:=lL kC (by decide) d d.significand 0 ((by simp only [pow_zero,mul_one])) l g (e.trans (pm 971 AP));rw[<-ό _ (by decide) d h,j]
lemma jJ(x y:Ἅ)(o:fv x)(p:fv y)(d:kU)(g:d.toReal=rv x+rv y)
  (m:d.significand<2^(53:Ἀ))(l:-1074<=d.exponent)(e:|d.toReal|<=(2:ℝ)^(971:Ἁ)):
  fv (x+y)/\rv (x+y)=rv x+rv y:=by have⟨h,j⟩:=ii x y o p (by rw[<-g];exact e.trans (pm 971 AP));exact⟨h,by rw[j,<-g];exact rx d m l e⟩
lemma dB(i:Int64)(e:(I6 (K7 i))<2^(53:Ἀ)):fv i.toFloat/\rv i.toFloat=(K7 i):=by
 have g:kI i.toFloat.toModel=kE kC (kR (K7 i) 0):=by have h:=ma i;rw[m5]at h;rw[h,<-Model.ofModel_ofInt_eq_intToFloat kC (by decide)];exact Model.ofModel_ofInt_eq_roundDyadic kC (by decide) _
 have d:(kR (K7 i) 0).toReal=(K7 i):=by rw[<-ϖ,nD];simp
 have⟨j,l⟩:=lL kC (by decide) (kR (K7 i) 0) (I6 (K7 i)) 0 (by simp[kR]) e (by simp[kR];decide) (by rw[d];exact AZ (((by rw[<-mI,Ϲ];exact_mod_cast e.le:|((K7 i):ℝ)|<=(2:ℝ)^(53:Ἀ)).trans_eq (AA _ _).symm).trans (AG (by norm_num) (by norm_num))) (pm 971 AP));exact⟨by rw[fv,g];exact j,by rw[rv,g,l,d]⟩
lemma jP(x y:Ἅ):kI (x*y).toModel=
  pv kC
   (lc k0 x.toModel.unpack y.toModel.unpack):=by change kI (pX x.toModel y.toModel)=_;rw[pX,nu]
lemma iR(b:BitVec 64)(s:pK)(m:Ἀ)(e:Ἁ)(h:0< m)
  (d:kL k0 b=.finite s m e h):
  2^(52:Ἀ)<=m\/e=-1074:=by revert d;rw[kL];dsimp only;split;split<;>intro d<;>cases d;split;split;intro d;cases d;rename_i l _;intro d;injection d with _ _ j;right;rw[<-j,l];decide;intro d;injection d with _ g _;left;rw[<-g,BitVec.toNat_append];exact AZ (by decide) Nat.left_le_or
lemma jo_h1(X:Ἀ)(a:Ἁ)(h:52<=X):a<=(X:Ἁ)+1+a-53:=Int.le_sub_left_of_add_le (T8 (T8 (U4.2 h) 1) a)
lemma jo_h2:∀s t:pK,(if mS (s*t) then (-1:ℝ) else 1)=(if mS s then -1 else 1)*(if mS t then -1 else 1):=fun
 | .negative, .negative =>((neg_one_mul (-1:ℝ)).trans (neg_neg 1)).symm
 | .negative, .positive =>(mul_one (-1:ℝ)).symm
 | .positive, .negative =>(one_mul (-1:ℝ)).symm
 | .positive, .positive =>(one_mul (1:ℝ)).symm
lemma jo_h3(a b:ℝ)(O P:Ἀ)(g h:Ἁ):a*b*((O*P:Ἀ):ℝ)*(2:ℝ)^(g+h)=a*O*2^g*(b*P*2^h):=by rw[Y4,AK AL,mul_mul_mul_comm a b,mul_mul_mul_comm]
lemma jo(x y:Ἅ)(G:fv x)(H:fv y)(z:|rv x*rv y|<=(2:ℝ)^(971:Ἁ)):
  fv (x*y)/\rv (x*y)=ns kC (rv x*rv y):=by
 have t:=(fO x).1 G;have u:=(fO y).1 H;have E:x.toModel.unpack=kL k0 x.toModel.toBits.toBitVec:=ἅ;have F:y.toModel.unpack=kL k0 y.toModel.toBits.toBitVec:=ἅ;rw[iL x,iL y]at z⊢;unfold fv rv;rw[jP];generalize x.toModel.unpack=X at t E z⊢;generalize y.toModel.unpack=Y at u F z⊢;cases X<;>cases Y<;>simp only[pE,υ]at t u
 case finite.finite l o g v m p h w=>
  have A:=iR _ l o g v E.symm;have B:=iR _ m p h w F.symm;have r:g+h<=BV k0 (kY (o*p) (g+h)):=by change g+h<=max ((Aq (o*p):Ἁ)+1+(g+h)-53) (-1074);apply le_max_iff.2;rcases A with A|A;exact pZ (jo_h1 _ _ ((mu (Ϯ v w).ne').2 (AZ A (J4 o w))));rcases B with B|B;exact pZ (jo_h1 _ _ ((mu (Ϯ v w).ne').2 (AZ B (lZ p v))));exact BA ((congrArg₂ HAdd.hAdd A B).trans_le (by decide))
  let q:kU:=⟨mS (l*m),o*p,g+h⟩;have D:Model.modelSign (mS (l*m))=l*m:=by cases l<;>cases m<;>rfl
  have d:q.toReal=lw (.finite l o g v)*lw (.finite m p h w):=((congrArg (·*(2:ℝ)^(g+h)) ((nf q).trans (congrArg (·*((o*p:Ἀ):ℝ)) (jo_h2 l m)))).trans (jo_h3 _ _ o p g h)).trans (congrArg₂ (·*·) (N6 l o g v) (N6 m p h w)).symm
  have j:=Ϙ kC (by decide) q (by rw[d];exact z.trans (pm 971 AP));rw[(by rw[lc,Model.roundWithAccuracy_exact_eq_round_of_le_targetExponent _ _ _ _ r];simp only[kE,show kC.isIEEE=ἃ from ἅ,ἁ,Model.ieeeRoundDyadic,q,D];rfl:pv kC (lc k0 (.finite l o g v) (.finite m p h w))=kE kC q)];exact⟨j,by rw[ό _ (by decide) q j,d]⟩
 all_goals simp only[lc,Model.isFinite_ofModel_zero kC (by decide),Model.toReal_ofModel_zero kC (by decide),Model.unpackedToReal_zero,nv,mul_zero,and_self,pD]
lemma bJ(s:Ἆ)(m:Ἀ)(e:Ἁ):
  (⟨s,m,e⟩:kU).toReal=(if s then-(m:ℝ)else m)*(2:ℝ)^e:=by rw[mb,nf];split<;>simp
def pw:Ἀ->Ἅ
 | 0=>(1:Int64).toFloat
 | k+1=>let p:=pw k;p+p
def p149:Ἅ:=pw 149
lemma iJ(k:Ἀ)(g:k<=900):fv (pw k)/\rv (pw k)=(2:ℝ)^k:=by
 induction k with
 |zero=>have:=dB 1 (by decide);rw[F.pw];exact⟨this.1,by rw[this.2];simp only [Int64.reduceToInt,Int.cast_one,pow_zero]⟩
 |succ k e=>
  have⟨f,v⟩:=e (by exact X5 g);have d:(⟨ἄ,1,((k+1:Ἀ):Ἁ)⟩:kU).toReal=(2:ℝ)^(k+1):=by rw[bJ];simp only[υ,Am,pJ,one_mul];exact AA 2 (k+1)
  have:=jJ (F.pw k) (F.pw k) f f _ (by rw[d,v,A4];ring) ((by decide):(1:Ἀ)<2^(53:Ἀ)) ((by exact Int.neg_ofNat_le_natCast 1074 (k+1)):(-1074:Ἁ)<=((k+1:Ἀ):Ἁ)) (by rw[d,AW (by positivity),<-AA];exact AG (by norm_num) (by (clear * - g;omega)));rw[F.pw];exact⟨this.1,by rw[this.2,v,A4];ring⟩
def nm:Ἅ:=(-1:Int64).toFloat
def P:ἀ:=⟨Aj (n:=512) λi=>if (Fin.val i)<256 then pw ((Fin.val i)-1)else nm*pw ((Fin.val i)-256-1)⟩
lemma gL:(kS P)=512:=by simp[P,kS]
lemma iV(i:Ἄ)(h:(J0 i)<(kS P)):
  P.uget i h=if (J0 i)<256 then pw ((J0 i)-1)else nm*pw ((J0 i)-256-1):=by simp[P,nN]
lemma eS(w:ἇ):(mJ (kB w))=(mJ (w<<<1))/2^(24:Ἀ):=by rw[aN,aM,ϴ];simp only[kN,kH,ο];have:=w.toNat_lt;(clear * -;omega)
lemma dg(w:ἇ):(mJ (w>>>23))=(mJ w)/2^(23:Ἀ):=by rw[Υ];simp[β]
@[inline]def dw(w:ἇ):Ἅ:=
 (Core.mW w).toInt64.toFloat*P.uget (UInt32.toUSize (w>>>23)) (by
  rw[gL,aC,dg];have:=w.toNat_lt;exact Ϟ this)
lemma aV(z e:Ἁ):(nb z e).toReal=(z:ℝ)*(2:ℝ)^e:=by rw[<-ϖ,toRat_ofIntDyadic];push_cast;rfl
lemma aE(x y:Ἅ)(o:fv x)(p:fv y)(z e:Ἁ)(g:rv x*rv y=(z:ℝ)*(2:ℝ)^e)
  (m:(I6 z)<2^(53:Ἀ))(l:-1074<=e)(d:|(z:ℝ)*(2:ℝ)^e|<=(2:ℝ)^(971:Ἁ)):
  fv (x*y)/\rv (x*y)=(z:ℝ)*(2:ℝ)^e:=by have⟨h,j⟩:=jo x y o p (by rw[g];exact d);have:=rx (nb z e) m l (by rw[aV];exact d);rw[aV]at this;exact⟨h,by rw[j,g,this]⟩
lemma bg(w:ἇ):(I6 (sv w))<2^(24:Ἀ):=by have:=lB w;rw[sv];split<;>simp<;>exact I7 this
lemma bL(a b c:Ἀ)(e:a+b<=c)(h:c<=971)(z:Ἁ)(g:(I6 z)<2^a):
  |(z:ℝ)*(2:ℝ)^(b:Ἁ)|<=(2:ℝ)^(971:Ἁ):=by
 rw[Ae,AW ((by positivity):(0:ℝ)<2^(b:Ἁ)),AA];have d:|(z:ℝ)|<=2^a:=by rw[<-mI,Ϲ];exact_mod_cast g.le
 refine(AE d (by positivity)).trans ?_;rw[<-AF,<-AA];exact AG (by norm_num) (by (clear * - e h;omega))
lemma mV(u:ἆ)(n:Ἀ)(d:(lq u)=n)(e:n<2^(25:Ἀ)):
  fv u.toInt64.toFloat/\rv u.toInt64.toFloat=(n:Ἁ):=by
 have:(K7 u.toInt64)=n:=by rw[show (K7 u.toInt64)=ae u from ἅ,ay,d];split<;>omega
 rw[<-this];exact dB _ (by rw[this];simp only [Int.natAbs_natCast,nr];omega)
lemma qV_h3(x q:Ἀ)(H:x+256*q<256):q=0:=by omega
lemma qV_h4(x q:Ἀ)(H:¬x+256*q<256)(hb:x<256)(t:q<2):q=1:=by omega
lemma qV_h1(x c:Ἀ)(h:x<256):x-c<=900:=(K4 _ _).trans (h.le.trans (by decide))
lemma qV_h2(x c:Ἀ)(h:x<256):1+(x-c)<=971:=(J1 ((K4 _ _).trans h.le) 1).trans (by decide)
lemma qV_h5(x w:Ἀ)(H:x+256*(w/2^(31:Ἀ))<256):¬2^(31:Ἀ)<=w:=by omega
lemma qV_h6(x w:Ἀ)(H:¬x+256*(w/2^(31:Ἀ))<256)(hb:x<256):2^(31:Ἀ)<=w:=by omega
lemma qV(c:Ἀ)(A:ἀ)(m:∀i h,A.uget i h=if (J0 i)<256 then pw ((J0 i)-c)else nm*pw ((J0 i)-256-c))
  (w:ἇ)(h):fv (A.uget (UInt32.toUSize (w>>>23)) h)/\∃σ:Ἁ,
  (∀v,k9 v=k9 w->σ*kM v=sv v)/\(I6 σ)=1/\
   rv (A.uget (UInt32.toUSize (w>>>23)) h)=σ*2^((mJ (kB w))-c):=by
 have hb:=bm w;have hw:=w.toNat_lt;have d:=ap w;obtain⟨j,l⟩:=iJ ((mJ (kB w))-c) (qV_h1 _ _ hb);rw[m,aC,dg,jW];split
 .rename_i H;rw[(qV_h3 _ _ H),A7,ϑ];exact⟨j,1,λv g=>by rw[sv,g,AC (by rw[d];exact qV_h5 _ _ H)];ring,ἅ,by rw[l];push_cast;ring⟩
 .rename_i H;rw[(qV_h4 _ _ H hb (Ϟ hw)),ϻ,BK];have e:=dB (-1) (by decide);rw[((by decide):(K7 (-1:Int64))=-1)]at e;have:fv (nm*F.pw ((mJ (kB w))-c))/\rv (nm*F.pw ((mJ (kB w))-c))=((-1:Ἁ):ℝ)*2^(((mJ (kB w))-c:Ἀ):Ἁ):=aE _ _ e.1 j _ _ (by rw[e.2,l,AA]) (by decide) (by exact Int.neg_ofNat_le_natCast 1074 ((expOf w).toNat-c)) (bL 1 _ 971 (qV_h2 _ _ hb) AP _ (by decide));exact⟨this.1,-1,λv g=>by rw[sv,g,AD (d.mpr (qV_h6 _ _ H hb))];ring,ἅ,by rw[this.2,AA]⟩
lemma dG_h1(q K X c:Ἀ)(r:q<=2)(hK:K<2^(24:Ἀ))(hX:X<256):q*K<2^(25:Ἀ)∧-1074<=((X-c:Ἀ):Ἁ)∧25+(X-c)<=971:=⟨(ι K r).trans_lt ((M7 hK (by decide)).trans_le (by decide)),(by decide:(-1074:Ἁ)<=0).trans (M3 _),(J1 ((K4 _ _).trans hX.le) 25).trans (by decide)⟩
lemma dG(c q:Ἀ)(A:ἀ)(o:∀i h,A.uget i h=if (J0 i)<256 then pw ((J0 i)-c)else nm*pw ((J0 i)-256-c))
  (u:ἆ)(v w:ἇ)(h)(r:q<=2)(x:(lq u)=q*kM v)
  (t:k9 v=k9 w)(p:(q:ℝ)*2^((mJ (kB w))-c)=2^kG v):
  fv (u.toInt64.toFloat*A.uget (UInt32.toUSize (w>>>23)) h)/\
   rv (u.toInt64.toFloat*A.uget (UInt32.toUSize (w>>>23)) h)=(sv v:ℝ)*2^kG v:=by
 obtain⟨k,k2,k3⟩:=dG_h1 q (kM v) _ c r (lB v) (bm w);have⟨g,j⟩:=mV u _ x k;have⟨l,σ,d,m,y⟩:=qV c A o w h;have e:(I6 (((q*kM v:Ἀ):Ἁ)*σ))<2^(25:Ἀ):=by rw[Ϩ,m,ϻ];exact k
 have:=aE _ _ g l _ ((mJ (kB w))-c:Ἀ) (by rw[j,y,AA];push_cast;ring) (Ao e (by norm_num)) k2 (bL 25 _ 971 k3 AP _ e);exact⟨this.1,by rw[this.2,<-p,AA,<-d v t];push_cast;ring⟩
lemma cI(w:ἇ)(_:kB w≠255):fv (dw w)/\rv (dw w)=(sv w:ℝ)*2^kG w:=
 dG 1 1 P iV _ w w _ (by exact NeZero.one_le) (by simp[an]) rfl (by simp[aT])
lemma dJ_h1(a b:Ἀ)(h:a<256)(k:b<256):48+(a-1+(b-1))<=971:=AZ (J1 (Y3 ((K4 _ _).trans h.le) ((K4 _ _).trans k.le)) _) (by decide)
lemma dJ(x w:ἇ)(j:kB x≠255)(h:kB w≠255):
  fv (dw x*dw w)/\rv (dw x*dw w)=(tm x w:ℝ):=by
 have⟨e,l⟩:=cI x j;have⟨g,m⟩:=cI w h;have d:(I6 (sv x*sv w))<2^(48:Ἀ):=by rw[Ϩ];exact(ϕ (bg x) (bg w)).trans_eq (by norm_num)
 have:=aE _ _ e g (sv x*sv w) (kG x+kG w:Ἀ) (by rw[l,m,AA,AF];push_cast;ring) (Ao d (by norm_num)) (by linarith only []) (bL 48 _ 971 (by rw[aT,aT];exact dJ_h1 _ _ (bm x) (bm w)) AP _ d);refine⟨this.1,?_⟩;rw[this.2,tm,AA];push_cast;ring
@[inline]def eF(b:ἆ):ἆ:=(b>>>52)&&&2047
@[inline]def mF(b:ἆ):ἆ:=(b&&&0xfffffffffffff)|||0x10000000000000
lemma jT(u:ἆ):(u&&&0x8000000000000000!=0)=(u>>>63!=0):=by
 apply ϓ.2;simp only[AM,Ak,<-ξ];rw[Ε,ν];change¬(lq u)&&&2^(63:Ἀ)=0↔¬(lq u)>>>63=0;rw[Nat.and_two_pow,β,lW];have d:(lq u)/2^(63:Ἀ)<2:=Ϟ u.toNat_lt
 rw[γ d];generalize (lq u)/2^(63:Ἀ)=b at d⊢;interval_cases b<;>simp
lemma ij(v:Ἅ):kI v.toModel=Model.ofFields kC (v.toBits>>>63!=0)
  ((lq ((v.toBits>>>52)&&&2047))) ((lq (v.toBits&&&0xfffffffffffff))):=by
 have d:Model.expField (kI v.toModel)=(lq ((v.toBits>>>52)&&&2047)):=ἅ;have e:Model.fracField (kI v.toModel)=(lq (v.toBits&&&0xfffffffffffff)):=ἅ;have g:Model.signBit (kI v.toModel)=(v.toBits>>>63!=0):=by rw[<-jT];apply ϓ.mpr;simp only[Model.signBit,AM,Ak];rw[<-UInt64.toBitVec_inj];exact Iff.rfl
 rw[<-Model.ofFields_signBit_expField_fracField (kI v.toModel),d,e,g]
lemma fq(v:Ἅ):(lq ((v.toBits>>>52)&&&2047))<2^(11:Ἀ)/\(lq (v.toBits&&&0xfffffffffffff))<2^(52:Ἀ):=by rw[Ε];exact⟨AO ϝ (by decide),AO ϝ (by decide)⟩
lemma dV(v:Ἅ)(e:fv v)(g:¬eF v.toBits=0):
  rv v=(⟨v.toBits>>>63!=0,(lq (v.toBits&&&0xfffffffffffff))+2^(52:Ἀ),
   ((lq (eF v.toBits)):Ἁ)-1075⟩:kU).toReal:=by
 unfold F.eF at g⊢;have⟨j,l⟩:=fq v;have d:(lq ((v.toBits>>>52)&&&2047))≠0:=λh=>g (ξ.mp (by simpa using h));unfold fv at e;unfold rv;rw[ij]at e⊢;rw[BR,Model.toDyadic?_ofFields_of_isFinite _ _ _ _ j l e,AC d];simp only[show kC.fracWidth=52 from ἅ,show kC.exponentBias=1023 from ἅ,Model.pow2_eq_two_pow];congr 2<;>(try simp only[nY,nZ])<;>ring
lemma aR(z:Model lU):lI (kW z)=z:=by have h:z.toNatBits<2^(32:Ἀ):=mQ z;rw[lI,kW,kT,lE,γ h];exact m3 z
lemma ef(w:kF)(d:kB w≠255):ls (lI w)=(sv w:ℝ)*(2:ℝ)^((kG w:Ἁ)-149):=by rw[lI,BR,kV w d];change (nh w).toReal=_;rw[<-ϖ,lG];push_cast;rfl
lemma iD(w:kF)(j:kB w≠255)(m:w&&&0x7FFFFFFF≠0):ls (lI w)≠0:=by
 rw[ef w j];refine mul_ne_zero ?_ (zpow_ne_zero _ (by norm_num));have l:kM w≠0:=by rw[kM];split;rename_i d;have e:=(eZ w).1 d;rw[aM]at e;have g:(mJ (bj w))≠0:=λh=>m (Β.1 (h.trans nj.symm));rw[aM]at g;rw[lK,fe];(clear * - e g;omega);exact Ne.symm (NeZero.ne' (2^23+(fracOf w).toNat))
 rw[sv];split<;>simp[l]
lemma iN(d e g:kU)(p:d.toReal<=e.toReal)(q:e.toReal<=g.toReal)
  (E:kW (kE lU d)=kW (kE lU g))
  (w:kB (kW (kE lU d))≠255)(F:kW (kE lU d)&&&0x7FFFFFFF≠0)
  (D:|e.toReal|<=(2:ℝ)^(127:Ἁ)):
  kW (kE lU e)=kW (kE lU d):=by
 have j:=aR (kE lU d);have m:=aR (kE lU g);rw[<-E]at m;have o:mN (kE lU d)=ἃ:=by rw[<-j,lI,m6];simpa using w
 have v:=Ϙ lU (by decide) e (D.trans (by have h:=ϫ lU ἄ (2^(23:Ἀ)) (127-23) (by decide) (by decide);simp only[υ,ἂ,one_mul]at h;change|((2^(23:Ἀ):Ἀ):ℝ)*(2:ℝ)^((127:Ἁ)-23)|<=_ at h;rwa[((by norm_num):((2^(23:Ἀ):Ἀ):ℝ)*(2:ℝ)^((127:Ἁ)-23)=(2:ℝ)^(127:Ἁ)),AW (by positivity)]at h));have G:=ό lU (by decide) d o;have A:=ό lU (by decide) e v;have H:=ό lU (by decide) g (by rw[<-m,j];exact o);have y:=lM lU p;have z:=lM lU q;have l:=m.symm.trans j;rw[<-G,<-A]at y;rw[<-A,<-H,l]at z;have x:=le_antisymm z y;rw[Model.eq_of_toReal_eq_of_nonzero x (by rw[x,<-j];exact iD _ w F)]
lemma jx_h1(r q:Ἀ)(t:r+q*2^(29:Ἀ)<2^(53:Ἀ))(o:q%2<2):(r+q*2^(29:Ἀ))*2^(10:Ἀ)+274877906943<2^(64:Ἀ)∧(r+q*2^(29:Ἀ))*2^(10:Ἀ)+274877906943+q%2<2^(64:Ἀ):=⟨(X3 (ϐ t (by decide)) _).trans_le (by decide),(P6 (X3 (ϐ t (by decide)) _) (J2 o)).trans_le (by decide)⟩
lemma jx(x:Ἀ)(t:x<2^(53:Ἀ))(m:x%2^(29:Ἀ)≠2^(28:Ἀ)):
  (lq (ar (lh (x*2^(10:Ἀ)))))=(x+2^(28:Ἀ))/2^(29:Ἀ):=by have d:(lq (lh (x*2^(10:Ἀ))))=x*2^(10:Ἀ):=Α ((ϐ t (by decide)).trans_le (by decide));generalize lh (x*2^(10:Ἀ))=N at d;unfold ar;simp only[ν,η,Ε,d,kA,kH,β,mP];rw[(by rw[(by norm_num:(2:Ἀ)^(39:Ἀ)=2^(29:Ἀ)*2^(10:Ἀ)),Nat.mul_div_mul_right _ _ (by positivity)]:x*2^(10:Ἀ)/2^(39:Ἀ)=x/2^(29:Ἀ))];obtain⟨q,r,p,rfl⟩:∃q r,r<2^(29:Ἀ)/\x=r+q*2^(29:Ἀ):=⟨x/2^(29:Ἀ),x%2^(29:Ἀ),ϊ _ (by norm_num),(Nat.mod_add_div' x _).symm⟩;rw[Ρ _ _ (by norm_num),ώ p,Κ];rw[Μ,γ p]at m;have o:=ϊ q ((by decide):0<2);rw[γ (jx_h1 r q t o).1,γ (jx_h1 r q t o).2,show(r+q*2^(29:Ἀ))*2^(10:Ἀ)+274877906943+q%2=(r*2^(10:Ἀ)+274877906943+q%2)+2^(39:Ἀ)*q by ring,show r+q*2^(29:Ἀ)+2^(28:Ἀ)=(r+2^(28:Ἀ))+2^(29:Ἀ)*q by ring,mv _ _ (by norm_num),mv _ _ (by norm_num),(by grind:(r*2^(10:Ἀ)+274877906943+q%2)/2^(39:Ἀ)=(r+2^(28:Ἀ))/2^(29:Ἀ))]
@[inline,irreducible]def ak(S:Ἅ)(h:ἆ):ἆ:=
 let b:=S.toBits
 let f:=b&&&0xfffffffffffff
 let E:=eF b
 let d:=h+1075-E
 let g:=(1:ἆ)<<<d
 let u:=(f+268435456)&&&536870911
 let r:=((b&&&0x7FFFFFFFFFFFFFFF)+268435456)>>>29
 if d<=26&&E-1195<253&&u-(g+1)<536870911-(g+g)&&r&&&8388607!=0 then
  ((b>>>63)<<<31)+r-0x255000000
 else 4294967296
lemma rb_h1(x:Ἀ):x%2^(63:Ἀ)+268435456<2^(64:Ἀ)∧x%2^(63:Ἀ)+268435456=(x%2^(52:Ἀ)+2^(28:Ἀ))+x/2^(52:Ἀ)%2^(11:Ἀ)*2^(23:Ἀ)*2^(29:Ἀ):=⟨(X3 (ϊ _ (by decide)) _).trans_le (by decide),by rw[show (2:Ἀ)^(63:Ἀ)=2^(52:Ἀ)*2^(11:Ἀ) by norm_num,Nat.mod_mul];ring⟩
lemma rb(b:ἆ)(h:(((b&&&0x7FFFFFFFFFFFFFFF)+268435456)>>>29&&&8388607!=0)=true):
  2^(28:Ἀ)<=(lq (b&&&0xfffffffffffff))/\(lq (b&&&0xfffffffffffff))+2^(28:Ἀ)<2^(52:Ἀ):=by simp only[AM,Ak,<-ξ,Ε,ν,η,kA,kH,β]at h⊢;have:=b.toNat_lt;rw[ζ _ 63,ζ _ 23,γ (rb_h1 (lq b)).1,(rb_h1 (lq b)).2,Ρ _ _ (by norm_num),Μ]at h;rw[ζ _ 52];have:=ϊ (lq b) ((by norm_num):0<2^(52:Ἀ));clear*-h this;omega
lemma j3_h1(f G:Ἀ)(l:G<=2^(26:Ἀ))(t:2^(28:Ἀ)<=f∧f+2^(28:Ἀ)<2^(52:Ἀ))(x:G+1<=(f+2^(28:Ἀ))%2^(29:Ἀ)∧(f+2^(28:Ἀ))%2^(29:Ἀ)+G<2^(29:Ἀ)):G+1<=(f+2^(28:Ἀ))%2^(29:Ἀ)∧(f+2^(28:Ἀ))%2^(29:Ἀ)+G<2^(29:Ἀ)∧G<=f∧f+G<2^(52:Ἀ):=⟨x.1,x.2,l.trans ((by norm_num:(2:Ἀ)^(26:Ἀ)<=2^(28:Ἀ)).trans t.1),(J1 (l.trans (by norm_num:(2:Ἀ)^(26:Ἀ)<=2^(28:Ἀ))) _).trans_lt t.2⟩
lemma j3_h2(H E:Ἀ)(o:(2^(64:Ἀ)-E+(H+1075)%2^(64:Ἀ))%2^(64:Ἀ)<=26)(p:(2^(64:Ἀ)-1195+E)%2^(64:Ἀ)<253)(w:H<2^(63:Ἀ))(u:E<2048):(2^(64:Ἀ)-E+(H+1075)%2^(64:Ἀ))%2^(64:Ἀ)=H+1075-E∧H+1075-E<=26∧E<=H+1075∧1195<=E∧E<=1447:=by omega
lemma j3_h3(A b:ἆ)(G R:Ἀ)(ha:(lq A)=R)(hb:(lq b)=G)(l:G<=67108864)(q:A-(b+1)<536870911-(b+b)):G+1<=R∧R+G<536870912:=by
 have e:=congrArg lq (sub_add_cancel A (b+1))
 have q':=μ.1 q
 rw[χ (536870911:ἆ) (b+b) (θ.2 (by rw[η,hb,γ (C0 (Y3 l l) (by decide))];exact U2 (Y3 l l) (by decide))),η,hb,γ (C0 (Y3 l l) (by decide))] at q'
 rw[η,η,hb,pS,γ (C0 (L1 l 1) (by decide)),ha,γ (U1 (P6 (q'.trans_le (K4 _ _)) (L1 l 1)) (by decide))] at e
 rw[<-e];exact⟨K2 _ _,by rw[pA,lO G 1 G,<-pA];exact Nat.succ_lt_succ (Nat.add_lt_of_lt_sub q')⟩
lemma j3_h4(f:ἆ)(v:(lq f)<4503599627370496):(lq ((f+268435456)&&&536870911))=((lq f)+268435456)%536870912:=by rw[Ε,η,γ (U1 (X3 v _) (by decide))];exact ζ _ 29
lemma j3(h E f:ἆ)(w:(lq h)<2^(63:Ἀ))(u:(lq E)<2048)(v:(lq f)<2^(52:Ἀ))
  (o:h+1075-E<=26)(p:E-1195<253)
  (q:((f+268435456)&&&536870911)-(((1:ἆ)<<<(h+1075-E))+1)<
   536870911-(((1:ἆ)<<<(h+1075-E))+((1:ἆ)<<<(h+1075-E))))
  (t:2^(28:Ἀ)<=(lq f)/\(lq f)+2^(28:Ἀ)<2^(52:Ἀ)):
  (lq E)<=(lq h)+1075/\(lq h)+1075-(lq E)<=26/\1195<=(lq E)/\(lq E)<=1447/\
   2^((lq h)+1075-(lq E))+1<=((lq f)+2^(28:Ἀ))%2^(29:Ἀ)/\
   ((lq f)+2^(28:Ἀ))%2^(29:Ἀ)+2^((lq h)+1075-(lq E))<2^(29:Ἀ)/\
   2^((lq h)+1075-(lq E))<=(lq f)/\(lq f)+2^((lq h)+1075-(lq E))<2^(52:Ἀ):=by
 have m:=Ζ (h+1075) E;rw[η] at m;rw[θ,m]at o;rw[μ,Ζ]at p;obtain⟨n,d,a,b,c⟩:=j3_h2 _ _ o p w u;have g:=m.trans n
 refine⟨a,d,b,c,j3_h1 _ _ (Θ (by decide) d) t (j3_h3 _ _ _ _ (j3_h4 f v) (by rw[ψ,g,γ (AO d (by decide)),pS,na,γ (lu (by decide) (AO d (by decide)))]) (Θ (by decide) d) q)⟩
lemma j4(f G:Ἀ)(g:1<=G)(h:f<2^(52:Ἀ))(j:G+1<=(f+2^(28:Ἀ))%2^(29:Ἀ))
  (l:(f+2^(28:Ἀ))%2^(29:Ἀ)+G<2^(29:Ἀ))(d:G<=f)(e:f+G<2^(52:Ἀ)):
  (f+2^(52:Ἀ)-G)%2^(29:Ἀ)≠2^(28:Ἀ)/\(f+2^(52:Ἀ)+G)%2^(29:Ἀ)≠2^(28:Ἀ)/\
   (f+2^(52:Ἀ)-G+2^(28:Ἀ))/2^(29:Ἀ)=2^(23:Ἀ)+(f+2^(28:Ἀ))/2^(29:Ἀ)/\
   (f+2^(52:Ἀ)+G+2^(28:Ἀ))/2^(29:Ἀ)=2^(23:Ἀ)+(f+2^(28:Ἀ))/2^(29:Ἀ):=by (clear * - g j l d;omega)
lemma g0_h1(s E f:Ἀ)(m:1195<=E)(o:E<2^(11:Ἀ))(p:f<2^(52:Ἀ))(q:s<2):(2^(64:Ἀ)-0x255000000+(s*2^(31:Ἀ)+(E*2^(52:Ἀ)+f+268435456)/2^(29:Ἀ)))%2^(64:Ἀ)=s*2^(31:Ἀ)+((E-1085-110)*2^(23:Ἀ)+(2^(23:Ἀ)+(f+2^(28:Ἀ))/2^(29:Ἀ))):=by
 rw[show E*2^(52:Ἀ)+f+268435456=(f+2^(28:Ἀ))+2^(29:Ἀ)*(E*2^(23:Ἀ)) by ring,mv _ _ (by decide)]
 have r:(f+2^(28:Ἀ))/2^(29:Ἀ)<2^(23:Ἀ)+1:=Ϟ ((X3 p _).trans_le (by decide))
 generalize (f+2^(28:Ἀ))/2^(29:Ἀ)=k at *
 obtain⟨e,rfl⟩:=Nat.exists_eq_add_of_le m
 rw[show 1195+e-1085-110=e by clear*-;omega,show 2^(64:Ἀ)-0x255000000+(s*2^(31:Ἀ)+(k+(1195+e)*2^(23:Ἀ)))=2^(64:Ἀ)+(s*2^(31:Ἀ)+(e*2^(23:Ἀ)+(2^(23:Ἀ)+k))) by ring,Nat.add_mod_left,γ ((Y3 (ι (2^(31:Ἀ)) (J2 q)) (Y3 (ι (2^(23:Ἀ)) ((K2 e 1195).trans_lt o).le) (J1 (J2 r) _))).trans_lt (by decide))]
lemma g0_h2(s E f:Ἀ)(o:E<2^(11:Ἀ))(p:f<2^(52:Ἀ))(q:s<2):s*2^(31:Ἀ)<2^(64:Ἀ)∧E*2^(52:Ἀ)+f+268435456<2^(64:Ἀ):=⟨(ϐ q (by decide)).trans_le (by decide),(X3 (((O1 p (E*2^(52:Ἀ))).trans_eq (Ο E _).symm).trans_le (ι _ o)) 268435456).trans_le (by decide)⟩
lemma g0(b:ἆ)(m:1195<=(lq (eF b))):
  (UInt64.toNat (((b>>>63)<<<31)+(((b&&&0x7FFFFFFFFFFFFFFF)+268435456)>>>29)-0x255000000))=
   (lq b)/2^(63:Ἀ)*2^(31:Ἀ)+(((lq (eF b))-1085-110)*2^(23:Ἀ)+
    (2^(23:Ἀ)+((lq (b&&&0xfffffffffffff))+2^(28:Ἀ))/2^(29:Ἀ))):=by
 have h:=b.toNat_lt;have d:(lq b)%2^(63:Ἀ)=(lq b)/2^(52:Ἀ)%2^(11:Ἀ)*2^(52:Ἀ)+(lq b)%2^(52:Ἀ):=(by rw[(by norm_num:(2:Ἀ)^(63:Ἀ)=2^(52:Ἀ)*2^(11:Ἀ)),Nat.mod_mul];ring);have q:(lq b)/2^(63:Ἀ)<2:=Ϟ h;have e:(lq (F.eF b))=(lq b)/2^(52:Ἀ)%2^(11:Ἀ):=by unfold F.eF;simp only[Ε,ν,kA,kH,β];exact ζ _ 11
 have g:(lq (b&&&0xfffffffffffff))=(lq b)%2^(52:Ἀ):=by rw[Ε];exact ζ _ 52
 have j:(lq (b&&&0x7FFFFFFFFFFFFFFF))=(lq b)%2^(63:Ἀ):=by rw[Ε];exact ζ _ 63
 have l:(lq (b>>>63))=(lq b)/2^(63:Ἀ):=by simp only[ν,kA,kH,β]
 rw[e]at m⊢;rw[g,Ζ,η,ψ,l,ν,η,j,d];simp only[kA,kH,β,ο,Nat.add_mod_mod];have o:=ϊ ((lq b)/2^(52:Ἀ)) (϶ 11);have p:=ϊ (lq b) (϶ 52);generalize (lq b)/2^(52:Ἀ)%2^(11:Ἀ)=E at*;generalize (lq b)%2^(52:Ἀ)=f at*;generalize (lq b)/2^(63:Ἀ)=s at*;clear*-m o p q;rw[γ (g0_h2 s E f o p q).1,γ (g0_h2 s E f o p q).2];exact g0_h1 s E f m o p q
lemma ce_h1(g G:Ἀ)(v:g<2^(52:Ἀ))(I:G<=g)(J:g+G<2^(52:Ἀ)):2^(62:Ἀ)<=(g+2^(52:Ἀ)-G)*2^(10:Ἀ)/\(g+2^(52:Ἀ)-G)*2^(10:Ἀ)<2^(63:Ἀ)/\2^(62:Ἀ)<=(g+2^(52:Ἀ)+G)*2^(10:Ἀ)/\(g+2^(52:Ἀ)+G)*2^(10:Ἀ)<2^(63:Ἀ)/\G<=g+2^(52:Ἀ)/\G<g+2^(52:Ἀ)/\g+2^(52:Ἀ)-G<2^(53:Ἀ)/\g+2^(52:Ἀ)+G<2^(53:Ἀ):=by
 have b7:g+2^(52:Ἀ)-G<2^(53:Ἀ):=((K4 _ _).trans_lt (X3 v _)).trans_eq (by norm_num)
 have b9:g+2^(52:Ἀ)+G<2^(53:Ἀ):=by rw[lO];exact(X3 J _).trans_eq (by norm_num)
 have a1:2^(52:Ἀ)<=g+2^(52:Ἀ)-G:=G5 (by rw[Τ];exact L1 I _)
 have b1:2^(62:Ἀ)<=(g+2^(52:Ἀ)-G)*2^(10:Ἀ):=(show (2:Ἀ)^(62:Ἀ)=2^(52:Ἀ)*2^(10:Ἀ) by norm_num).trans_le (ι _ a1)
 have b3:2^(62:Ἀ)<=(g+2^(52:Ἀ)+G)*2^(10:Ἀ):=(show (2:Ἀ)^(62:Ἀ)=2^(52:Ἀ)*2^(10:Ἀ) by norm_num).trans_le (ι _ ((K2 _ g).trans (V1 _ G)))
 have b2:(g+2^(52:Ἀ)-G)*2^(10:Ἀ)<2^(63:Ἀ):=(ϐ b7 (by positivity)).trans_eq (by norm_num)
 have b4:(g+2^(52:Ἀ)+G)*2^(10:Ἀ)<2^(63:Ἀ):=(ϐ b9 (by positivity)).trans_eq (by norm_num)
 exact⟨b1,b2,b3,b4,I.trans (V1 _ _),AN (Z4 I) (J1 BI g),b7,b9⟩
lemma ce_h2(E:Ἆ)(k q:Ἀ)(al:k<=252)(hq:q<=2^(23:Ἀ)):((if E then 2^(31:Ἀ) else 0)+(k*2^(23:Ἀ)+(2^(23:Ἀ)+q)))/2^(23:Ἀ)%2^(8:Ἀ)≠255/\((if E then 2^(31:Ἀ) else 0)+(k*2^(23:Ἀ)+(2^(23:Ἀ)+q)))%2^(31:Ἀ)≠0:=by
 have a:k*2^(23:Ἀ)+(2^(23:Ἀ)+q)<2^(23:Ἀ)*255:=(Y3 (ι _ al) (J1 hq _)).trans_lt (by decide)
 have b:(k*2^(23:Ἀ)+(2^(23:Ἀ)+q))/2^(23:Ἀ)<255:=Ϟ a
 have c:0<k*2^(23:Ἀ)+(2^(23:Ἀ)+q):=Nat.add_pos_right _ (Nat.add_pos_left (by decide) _)
 cases E<;>simp only[υ,Am,ἁ]
 · rw[Κ,γ (b.trans (by decide)),γ (a.trans (by decide))];exact⟨b.ne,c.ne'⟩
 · refine⟨?_,by rw[Nat.add_mod_left,γ (a.trans (by decide))];exact c.ne'⟩
   rw[show (2^(31:Ἀ):Ἀ)=2^(23:Ἀ)*2^(8:Ἀ) from rfl,Τ,mv _ _ (by decide),Nat.add_mod_right,γ (b.trans (by decide))];exact b.ne
lemma ce(S:Ἅ)(hS:fv S)(h:ἆ)(hh:(lq h)<2^63)(T:Ἁ)
  (L:|rv S-(T:ℝ)|<=(2:ℝ)^((lq h):Ἁ))(aL:|(T:ℝ)|<=(2:ℝ)^(425:Ἁ))
  (hc:ak S h<4294967296):T≠0/\(N3 (ak S h))=md T (-298) false:=by
 unfold ak at hc⊢
 dsimp only at hc⊢
 split at hc
 swap
 exact absurd hc (by decide)
 rename_i o
 rw[AD o]
 simp only[Η,AJ]at o
 obtain⟨⟨⟨c1,c2⟩,c3⟩,c4⟩:=o
 have y:(lq (F.eF S.toBits))<2048:=(fq S).1
 have v:=(fq S).2
 obtain⟨d1,d2,e1,e2,X,Y,I,J⟩:=j3 h _ _ hh y v c1 c2 c3 (rb _ c4)
 have D:=g0 _ e1
 have x:=dV _ hS (λh0=>by simp[h0]at e1)
 have O:(if(S.toBits>>>63!=0)=true then 2^(31:Ἀ) else 0)=(lq S.toBits)/2^(63:Ἀ)*2^(31:Ἀ):=by
  clear*-
  simp only[AM,Ak,<-ξ,ν,kA,kH,β,ϋ,pM]
  have:(lq S.toBits)/2^(63:Ἀ)<2:=Ϟ S.toBits.toNat_lt
  generalize (lq S.toBits)/2^(63:Ἀ)=b at this⊢
  rcases b with _|_|b
  rfl
  rfl
  exact(U5 _ (Nat.lt_of_succ_lt_succ (Nat.lt_of_succ_lt_succ this))).elim
 rw[<-O]at D
 have d:(lq (F.eF S.toBits-1085))=(lq (F.eF S.toBits))-1085:=χ _ _ (θ.mpr (AZ (by decide) e1))
 have j:(lq (F.eF S.toBits-1085))<=506:=(by rw[d];exact(P4 e2 1085).trans (by decide))
 generalize (S.toBits>>>63!=0)=E,F.eF S.toBits-1085=e5 at D x d j
 generalize a2:(lq (F.eF S.toBits))=l at d j D x d1 d2 e1 e2 X Y I J y
 generalize hg:2^((lq h)+1075-l)=G at X Y I J
 generalize a3:(lq (S.toBits&&&0xfffffffffffff))=g at D x X Y I J v
 have as:1<=G:=hg▸BI
 obtain⟨b5,b6,al⟩:110<=l-1085/\l-1085<363/\l-1085-110<=252:=⟨G5 e1,(P4 e2 1085).trans_lt (by decide),(P4 (P4 e2 1085) 110).trans (by decide)⟩
 obtain⟨b1,b2,b3,b4,K,z,b7,b9⟩:=ce_h1 _ _ v I J
 have hq:(g+2^(28:Ἀ))/2^(29:Ἀ)<=2^(23:Ἀ):=(ϭ (L1 (Nat.le_sub_one_of_lt v) _)).trans (by decide)
 have U:=Α (b2.trans (by decide))
 have n3:=Α (b4.trans (by decide))
 obtain⟨F,-,-⟩:=bM E _ e5 (b1.trans_eq U.symm) (U.trans_lt b2) (b5.trans_eq d.symm) (d.trans_lt b6)
 obtain⟨w3,-,-⟩:=bM E _ e5 (b3.trans_eq n3.symm) (n3.trans_lt b4) (b5.trans_eq d.symm) (d.trans_lt b6)
 obtain⟨w,t3,k1,k3⟩:=j4 g G as v X Y I J
 rw[jx _ b7 w,k1,d]at F
 rw[jx _ b9 t3,k3,d]at w3
 generalize(g+2^(28:Ἀ))/2^(29:Ἀ)=q,l-1085-110=k at F w3 D hq al
 generalize hR:az E (lh ((g+2^(52:Ἀ)-G)*2^(10:Ἀ))) e5=R at F
 have r:az E (lh ((g+2^(52:Ἀ)+G)*2^(10:Ἀ))) e5=R:=by apply Β.mp;rw[F,w3]
 have qv:(N3 (((S.toBits>>>63)<<<31)+(((S.toBits&&&0x7FFFFFFFFFFFFFFF)+268435456)>>>29)-0x255000000))=R:=by apply Β.mp;rw[F,Ϫ,D,γ];exact N7 (by split<;>decide) ((Y3 (ι (2^(23:Ἀ)) al) (J1 hq (2^(23:Ἀ)))).trans_lt (by decide))
 have V:kB R≠255:=by unfold kB;intro Q;have p:=congrArg mJ Q;simp only[Ξ,Υ,F,kN,kH,β]at p;rw[ζ _ 8]at p;exact (ce_h2 E k q al hq).1 p
 have W:R&&&0x7FFFFFFF≠0:=by intro Q;have p:=congrArg mJ Q;simp only[Ξ,F,kN]at p;rw[ζ _ 31]at p;exact (ce_h2 E k q al hq).2 p
 rw[qv]
 subst hR
 rw[aJ _ _ _ j,U,d]at V W⊢
 rw[aJ _ _ _ j,aJ _ _ _ j,U,n3,d]at r
 generalize g+2^(52:Ἀ)=Mn at x K z V W r⊢
 generalize rv S=sr at x L
 rw[bJ]at x
 have hU:(2:ℝ)^((lq h):Ἁ)=(G:ℝ)*(2:ℝ)^((l:Ἁ)-1075):=by
  rw[<-hg,I4,nZ,<-AA]
  rw[<-AK AL,show (((lq h):Ἀ):Ἁ)=(((lq h)+1075-l:Ἀ):Ἁ)+((l:Ἁ)-1075) by rw[pO d1,W5,nZ];exact((sub_add_sub_cancel _ _ _).trans (add_sub_cancel_right _ _)).symm]
 have t(m:Ἀ):((m*2^(10:Ἀ):Ἀ):ℝ)*(2:ℝ)^(((l-1085:Ἀ):Ἁ)-298)=(m:ℝ)*(2:ℝ)^((l:Ἁ)-1075)*2^(-298:Ἁ):=by
  have e:(2:ℝ)^(((l-1085:Ἀ):Ἁ)-298)*(2:ℝ)^(10:Ἁ)=(2:ℝ)^((l:Ἁ)-1075)*(2:ℝ)^(-298:Ἁ):=by rw[<-AK AL,<-AK AL,(by rw[pO (AZ (by decide) e1),nZ];ring:(((l-1085:Ἀ):Ἁ)-298)+10=((l:Ἁ)-1075)+(-298))]
  have ah:((m*2^(10:Ἀ):Ἀ):ℝ)=(m:ℝ)*(2:ℝ)^(10:Ἁ):=by rw[Y4,I4,nZ,<-AA];rfl
  rw[ah,Ap (m:ℝ) ((2:ℝ)^(10:Ἁ)),mul_comm ((2:ℝ)^(10:Ἁ)),e,<-Ap]
 have q:(0:ℝ)<(2:ℝ)^(-298:Ἁ):=(by positivity)
 have N:|(nb T (-298)).toReal|<=(2:ℝ)^(127:Ἁ):=by
  rw[aV,Ae,AW q]
  exact(AE aL q.le).trans_eq (by rw[<-AK AL];rfl)
 have hu:(0:ℝ)<(2:ℝ)^((l:Ἁ)-1075):=(by positivity)
 have aH:(G:ℝ)<Mn:=mod_cast z
 rw[hU]at L
 generalize (2:ℝ)^((l:Ἁ)-1075)=u at x L t hu
 obtain⟨w,B⟩:=A3.mp L
 have aH':=AI aH hu
 cases E<;>simp only[υ,Am,ἁ]at x<;>rw[x]at w B
 · have M:T≠0:=λh0=>by rw[h0,lH,sub_zero]at B;exact not_lt.mpr B aH'
   refine⟨M,?_⟩
   unfold md
   rw[AC M]
   refine(iN _ _ _ ?_ ?_ r.symm V W N).symm<;>rw[aV,bJ]<;>simp only[υ,Am,ἁ,neg_mul]<;>rw[t]
   refine AE ?_ q.le
   rw[pO K,sub_mul]
   exact sub_le_comm.mp B
   refine AE ?_ q.le
   rw[W5,add_mul]
   exact neg_le_sub_iff_le_add.mp w
 · have M:T≠0:=by
    intro h0
    rw[h0,lH]at w
    exact not_le.mpr aH' (by linear_combination w)
   refine⟨M,?_⟩
   unfold md
   rw[AC M]
   rw[<-r]at V W
   refine((iN _ _ _ ?_ ?_ r V W N).trans r).symm<;>rw[aV,bJ]<;>simp only[υ,Am,ἁ,neg_mul]<;>rw[t]
   rw[W5]
   linear_combination AE B q.le
   rw[pO K]
   linear_combination AE w q.le
@[specialize]def ao(p:Ἀ->Ἅ)(n t:Ἀ)(a:Ἅ):Ἅ:=
 if t<n then ao p n (t+1) (a+p t)else a
termination_by n-t
decreasing_by all_goals omega
lemma i5(x y:Ἅ)(e:fv x)(g:fv y)(E:Ἀ)(d:E<=971)(j:|rv x+rv y|<(2:ℝ)^E):
  fv (x+y)/\|rv (x+y)-(rv x+rv y)|<=(2:ℝ)^E*2^(-54:Ἁ):=by have h:=e8 x y e g E (by exact Int.neg_ofNat_le_natCast 1021 E) (by exact_mod_cast d) (by rwa[AA]);refine⟨h.1,h.2.trans_eq ?_⟩;rw[show((E:Ἁ)-53)=(E:Ἁ)+(-54)+1 by ring,AK AL,AK AL,AA,zpow_one];ring
lemma cv(n:Ἀ)(z:Ἀ->Ἁ)(e:Ἁ)(B:ℝ)(g:∀t<n,|(z t:ℝ)|<=B)(d:|(e:ℝ)|<=B):
  ∀t≤n,|(e:ℝ)+∑s∈kP t,(z s:ℝ)|<=((t:ℝ)+1)*B:=by
 intro t;induction t with
 |zero=>intro _;simpa using d
 |succ t h=>
  intro j;rw[Ν,<-add_assoc];push_cast;linarith only [Ag ((e:ℝ)+∑s∈kP t,(z s:ℝ)) (z t),h (by exact X5 j),g t j,h,j,g]
structure A8 where
 a0:Ἅ
 a1:Ἅ
 a2:Ἅ
 a3:Ἅ
 a4:Ἅ
 a5:Ἅ
 a6:Ἅ
 a7:Ἅ
local instance:Inhabited A8:=⟨⟨0,0,0,0,0,0,0,0⟩⟩
noncomputable def A8.f(s:A8):Ἀ->Ἅ
 | 0=>s.a0 | 1=>s.a1 | 2=>s.a2 | 3=>s.a3 | 4=>s.a4 | 5=>s.a5 | 6=>s.a6 | _=>s.a7
structure A9 where
 a0:Ἅ
 a1:Ἅ
 a2:Ἅ
 a3:Ἅ
 a4:Ἅ
 a5:Ἅ
 a6:Ἅ
 a7:Ἅ
 o:Ἃ
noncomputable def A9.s(R:A9):A8:=⟨R.a0,R.a1,R.a2,R.a3,R.a4,R.a5,R.a6,R.a7⟩
local instance:Inhabited A9:=⟨⟨0,0,0,0,0,0,0,0,0⟩⟩
lemma uk{i k:Ἄ}{S:Ἀ}(d:(J0 k)<8)(h:(J0 i)+8<=S)(e:S<2^(32:Ἀ)):
  (J0 (i+k))<S:=by have:=ς;rw[aj (by omega)];(clear * - d h;omega)
lemma ug(X:ἀ)(i:Ἄ)(h:(J0 i)<(kS X)):X.uget i h=X.data[(J0 i)]!:=by rcases X with⟨d⟩;rw[nN];rw[AB d (J0 i) h];rfl
lemma bW(Y:ἀ)(v:Ἅ):(kS (Y.push v))=(kS Y)+1:=by rcases Y with⟨d⟩;simp[FloatArray.push,kS]
lemma pg(Y:ἀ)(v:Ἅ)(q:Ἀ)(g:q<(kS Y)+1):
  (Y.push v).data[q]! =if q<(kS Y)then Y.data[q]!else v:=by rcases Y with⟨d⟩;change q<(O0 d)+1 at g;by_cases h:q<(O0 d);rw[AD (show q<kS ⟨d⟩from h)];change (d.push v)[q]! =d[q]!;rw[AB _ _ (by simp only [ϣ,Order.lt_add_one_iff];exact J2 g),AB d q h,nU h];rw[AC (show¬q<kS ⟨d⟩from h)];change (d.push v)[q]! =v;have e:q=(O0 d):=(by exact Nat.eq_of_lt_succ_of_not_lt g h);subst e;rw[AB _ _ (by simp only [ϣ,lt_add_iff_pos_right,Order.lt_one_iff]),nV]
@[specialize]def bQ(f:Ἀ->Ἅ)(N:Ἀ)(Y:ἀ):ἀ:=
 if (kS Y)<N then bQ f N (Y.push (f (kS Y)))else Y
termination_by N-(kS Y)
decreasing_by rw[bW];omega
lemma bk_h1(a N:Ἀ)(l:N-a=0)(j:a<=N):¬a<N∧a=N:=⟨not_lt.2 (O4 l),C4 j (O4 l)⟩
lemma bk_h2(a N k b:Ἀ)(l:N-a=k+1)(r:b=a+1):a<N∧N-b=k∧b<=N∧(∀{q},q<a->q<b)∧∀{q},a<=q->¬q=a->b<=q:=⟨N1 l,by rw[r,V4,l,BK],r.trans_le (N1 l),λe=>(K6 e).trans_eq r.symm,λh e=>r.trans_le (N4 h (Ne.symm e))⟩
lemma bk(f:Ἀ->Ἅ)(N:Ἀ):∀k (Y:ἀ),N-(kS Y)=k->(kS Y)<=N->
  (kS (bQ f N Y))=N/\(∀q<(kS Y),(bQ f N Y).data[q]! =Y.data[q]!)/\
   ∀q,(kS Y)<=q->q<N->(bQ f N Y).data[q]! =f q:=by
 intro k;induction k<;>intro Y l j
 .obtain⟨c,e⟩:=bk_h1 _ _ l j;rw[bQ,AC c];exact⟨e,λq _=>ἅ,λq a b=>absurd (AO a b) c⟩
 rename_i p;have r:=bW Y (f (kS Y));obtain⟨u,v,w,x,y⟩:=bk_h2 _ _ _ _ l r;rw[bQ,AD u];have⟨m,g,o⟩:=p (Y.push (f (kS Y))) v w;refine⟨m,λq e=>?_,λq h d=>?_⟩;rw[g q (x e),pg _ _ _ e.step,AD e];by_cases e:q=(kS Y);subst e;rw[g _ (r▸(by simp only [lt_add_iff_pos_right,Order.lt_one_iff])),pg _ _ _ (by simp only [lt_add_iff_pos_right,Order.lt_one_iff]),AC (AX _)];exact o q (y h e) d
lemma jp(v:Ἅ)(h:(lq (eF v.toBits))<2047):fv v:=by rw[fv,ij,Model.isFinite_ofFields _ _ _ _ (fq v).1 (fq v).2];change(lq (v.toBits>>>52&&&2047))<2047 at h;rw[show kC.encoding=.ieee from ἅ,((by decide):kC.expAllOnesNat=2047),AM,Ak];exact R8 h
lemma fK(v:Ἅ)(o:fv v):|rv v|<(2:ℝ)^(((lq (eF v.toBits)):Ἁ)-1022):=by
 have H:∀(x:ℝ)(M k:ℕ)(e f:ℤ),|x|=M->M<2^k->e+k=f->|x*(2:ℝ)^e|<(2:ℝ)^f:=λx M k e f h d w=>by
   have p:(0:ℝ)<(2:ℝ)^e:=by positivity
   rw[Ae,h,AW p,<-w,AK AL,AA,mul_comm ((2:ℝ)^e)]
   exact AI ((Nat.cast_lt.mpr d).trans_eq (I4 2 k)) p
 have Q:∀(s:Ἆ)(M:ℕ),|(if s then-(M:ℝ)else M)|=M∧|(if s then(-1:ℝ)else 1)*M|=M:=λs M=>by
   cases s
   · exact⟨nE M,(congrArg abs (one_mul (M:ℝ))).trans (nE M)⟩
   · exact⟨(BE _).trans (nE M),(congrArg abs (neg_one_mul (M:ℝ))).trans ((BE _).trans (nE M))⟩
 by_cases g:F.eF v.toBits=0
 · have d:=(fq v).2
   have l:(lq ((v.toBits>>>52)&&&2047))=0:=by rw[F.eF]at g;rw[g];rfl
   rw[rv,ij,l]
   generalize (lq (v.toBits&&&0xfffffffffffff))=m at d⊢
   by_cases e:m=0
   · rw[e,Model.toReal_eq_zero_of_isZero _ ((Model.isZero_ofFields _ _ _ _ (϶ _) (϶ _)).trans rfl),A5]
     positivity
   rw[Model.toReal_ofFields_subnormal _ _ _ e d]
   exact H _ m 52 _ _ (Q _ m).2 d (by rw[g];decide)
 rw[dV v o g,bJ]
 exact H _ _ 53 _ _ (Q _ _).1 (X3 (fq v).2 _) (by (clear * -;omega))
lemma jz(x y:Ἅ)(r:fv x)(t:fv y)(q:fv (x+y))(E:Ἁ)(l:-1074<=E)(u:E<=971)
  (h:|rv (x+y)|<(2:ℝ)^E):|rv x+rv y|<(2:ℝ)^E:=by
 have p:=gV x y r t;have v:=mH (kI x.toModel) (kI y.toModel) (by decide) (by rw[fv]at q;rwa[<-p]);unfold rv at h;rw[p,v]at h
 have m:|(2:ℝ)^E|<=(2:ℝ)^(971:Ἁ):=by rw[AW (by positivity)];exact AG (by norm_num) u
 have a:(⟨ἄ,1,E⟩:kU).toReal=2^E:=by simp only[bJ,υ,ἂ,pJ,one_mul]
 have b:(⟨ἃ,1,E⟩:kU).toReal=-2^E:=by simp only[bJ,ἁ,pJ,one_mul,neg_mul]
 have g:=rx _ (by norm_num) l (a▸m);have e:=rx ⟨ἃ,1,E⟩ (by norm_num) l (by rw[b,BE];exact m);rw[a]at g;rw[b]at e;have w:=abs_lt.1 h
 exact abs_lt.2 ⟨not_le.1 λd=>(w.1.trans_le ((lM kC d).trans_eq e)).false,not_le.1 λd=>((g.symm.trans_le (lM kC d)).trans_lt w.2).false⟩
@[specialize]def sT(p:Ἀ->Ἅ)(n t:Ἀ)(a:Ἅ)(m:ἆ):Ἅ×ἆ:=
 if t<n then
  let a':=a+p t
  let e:=eF a'.toBits
  sT p n (t+1) a' (if m<e then e else m)
 else(a,m)
termination_by n-t
decreasing_by all_goals omega
lemma eK(m e:ἆ):(lq (if m<e then e else m))=max (lq m) (lq e):=by split<;>rename_i h<;>rw[μ]at h<;>omega
lemma jg(p:Ἀ->Ἅ)(n:Ἀ):∀k t a m,n-t=k->(lq m)<=(lq (sT p n t a m).2):=by intro k;induction k<;>intro t a m h<;>rw[sT];rw[AC (by omega)];rename_i d;rw[AD (N1 h)];dsimp only;have:=d (t+1) (a+p t) (if m<F.eF (a+p t).toBits then F.eF (a+p t).toBits else m) (by (clear * - h;omega));rw[F.eK]at this;exact le_of_max_le_left this
lemma iM_h1(K:Ἀ)(x:1<=K)(y:K<=1993):-1074<=(K:Ἁ)-1022∧(K:Ἁ)-1022<=971∧-1021<=(K:Ἁ)-1022:=⟨(by decide:(-1074:Ἁ)<=0-1022).trans (Int.sub_le_sub_right (M3 K) 1022),sub_le_iff_le_add.2 (U4.2 y),le_sub_iff_add_le.2 (U4.2 x)⟩
lemma iM(p:Ἀ->Ἅ)(z:Ἀ->Ἁ)(u:Ἁ)(n K:Ἀ)(x:1<=K)(y:K<=1993)
  (E:∀t<n,fv (p t)/\rv (p t)=z t):
  ∀m t a M,n-t=m->t<=n->fv a->
   |rv a-((u:ℝ)+∑s∈kP t,(z s:ℝ))|<=t*(2:ℝ)^((K:Ἁ)-1076)->
   (lq (sT p n t a M).2)<=K->
   fv (sT p n t a M).1/\
    |rv (sT p n t a M).1-((u:ℝ)+∑s∈kP n,(z s:ℝ))|<=n*(2:ℝ)^((K:Ἁ)-1076):=by
 intro m;induction m with
 |zero=>intro t a M D Q e o g;obtain rfl:t=n:=C4 Q (K3.1 D);rw[sT,AC (AX _)];exact⟨e,o⟩
 |succ m G=>
  intro t a M D Q e o g;have d:t<n:=N1 D;rw[sT,AD d]at g⊢;dsimp only at g⊢;obtain⟨q,N⟩:=E t d;have B:=jg p n _ (t+1) (a+p t) (if M<F.eF (a+p t).toBits then F.eF (a+p t).toBits else M) rfl;rw[F.eK]at B;have O:(lq (F.eF (a+p t).toBits))<=K:=(le_max_right _ _).trans (B.trans g);have r:=jp (a+p t) (O.trans_lt (y.trans_lt (by decide)));have j:=fK (a+p t) r;have l:|rv (a+p t)|<(2:ℝ)^((K:Ἁ)-1022):=AN j (AG (by norm_num) (Int.sub_le_sub_right (U4.2 O) 1022));have k:=iM_h1 K x y;have A:=jz a (p t) e q r _ k.1 k.2.1 l;obtain⟨w,v⟩:=e8 a (p t) e q _ k.2.2 k.2.1 A;have h:|rv (a+p t)-(rv a+rv (p t))|<=(2:ℝ)^((K:Ἁ)-1076):=by refine v.trans_eq ?_;rw[(by ring:(K:Ἁ)-1022-53=((K:Ἁ)-1076)+1),zpow_add_one₀ AL];ring
  refine G (t+1) (a+p t) _ (by rw[V4,D,BK]) d w ?_ g;obtain⟨H,I⟩:=A3.1 h;obtain⟨J,L⟩:=A3.1 o;rw[Ν,A3];push_cast;constructor<;>linarith only[H,I,J,L,N]
lemma cw(n:Ἀ)(d e:Ἀ->kF)(g:aG n d e)(S:Ἅ)(m:fv S)(h:ἆ)
  (q:(lq h)<2^(63:Ἀ))(o:|rv S-(af n d e:ℝ)|<=(2:ℝ)^((lq h):Ἁ))
  (l:|(af n d e:ℝ)|<=(2:ℝ)^(425:Ἁ))(p:ak S h<4294967296):
  (N3 (ak S h))=ms n d e:=by have⟨j,r⟩:=ce S m h q _ o l p;rw[r,bT n d e g j]
def aO(A:Ἂ Ἃ)(i:Ἄ)(m:ἇ):ἇ:=
 if h:i<A.usize then
  let e:=(A.uget i (cD h)>>>23)&&&255
  aO A (i+1) (if m<e then e else m)
 else m
termination_by (O0 A)-(J0 i)
decreasing_by have:=cD h;rw[ac h];exact Z5 A.size iб this
lemma eY(A:Ἂ kF)(p:(O0 A)<kX):∀k (i:Ἄ) (m:ἇ),(O0 A)-(J0 i)=k->
  (mJ m)<=(mJ (aO A i m))/\∀q,(J0 i)<=q->q<(O0 A)->(mJ (kB A[q]!))<=(mJ (aO A i m)):=by
 intro k;induction k with
 |zero=>intro i m l;have c:=not_lt.2 (O4 l);rw[aO,dif_neg (λh=>c (cD h))];exact⟨AP,λq g d=>absurd (AO g d) c⟩
 |succ k v=>
  intro i m l;have h:i<A.usize:=by rw[ε,Array.usize,Dense.aa p];exact N1 l
  rw[aO,AT h];dsimp only;have r:∀e:ἇ,(mJ (if m<e then e else m))=max (mJ m) (mJ e):=λe=>by split<;>rename_i j<;>rw[Ι]at j<;>clear*-j<;>omega
  obtain⟨t,u⟩:=v (i+1) (if m<(A.uget i (cD h)>>>23)&&&255 then(A.uget i (cD h)>>>23)&&&255 else m) (by rw[ac h];exact congrArg Nat.pred l);rw[r]at t;refine⟨AZ (le_max_left _ _) t,λq g d=>?_⟩;by_cases o:q=(J0 i);subst o;rw[<-cC A i (cD h),kB];exact AZ (le_max_right _ _) t;exact u q (by rw[ac h];exact g.lt_of_ne' o) d
lemma bd(A:Ἂ kF)(j:(O0 A)<kX)(h:aO A 0 0<255)(q:Ἀ):
  kB A[q]!≠255/\kG A[q]!<=(mJ (aO A 0 0))-1:=by
 have g:=Ι.mp h;have d:(mJ (kB A[q]!))<=(mJ (aO A 0 0)):=by
  by_cases d:q<(O0 A)
  .exact(eY A j _ 0 0 rfl).2 q ((by simp)) d
  .rw[An A q d];exact ϟ _
 refine⟨λe=>by simp[e]at d g;(clear * - g d;omega),by rw[aT];exact P4 d 1⟩
@[specialize]def bC(k:Ἀ->Ἆ)(p:Ἀ->Ἅ)(n t:Ἀ)(a:Ἅ):Ἅ:=
 if t<n then bC k p n (t+1) (if k t then a else a+p t)else a
termination_by n-t
decreasing_by all_goals omega
lemma dM_h1(p:Ἀ->Ἅ)(n:Ἀ):∀m t a,n-t=m->ao p n t a=bC (λ_=>ἄ) p n t a:=by
 intro m;induction m<;>intro t a h<;>rw[ao,bC]
 .have q:=W3.2 (K3.1 h);rw[if_neg q,if_neg q]
 .rename_i ih;have q:=N1 h;rw[if_pos q,if_pos q];exact ih _ _ ((Nat.sub_succ n t).trans (congrArg Nat.pred h))
lemma dM(p:Ἀ->Ἅ)(n:Ἀ):∀m t a,n-t=m->ao p n t a=bC (λ_=>ἄ) p n t a:=dM_h1 p n
lemma dL(k:Ἀ->Ἆ)(p:Ἀ->Ἅ)(z:Ἀ->Ἁ)(d:Ἁ)(P L n:Ἀ)
  (Q:∀t<n,fv (p t)/\rv (p t)=z t)(N:∀t<n,k t=ἃ->z t=0)(V:∀t<n,|(z t:ℝ)|<=2^P)(v:|(d:ℝ)|<=2^P)
  (Y:n+2<=2^L)(w:L<=27)(J:P+L<=971):
  ∀m t a,n-t=m->t<=n->fv a->
   |rv a-((d:ℝ)+∑s∈kP t,(z s:ℝ))|<=t*((2:ℝ)^(P+L)*2^(-54:Ἁ))->
   fv (bC k p n t a)/\
    |rv (bC k p n t a)-((d:ℝ)+∑s∈kP n,(z s:ℝ))|<=n*((2:ℝ)^(P+L)*2^(-54:Ἁ)):=by
 have K:=cv n z d _ V v
 have X:(2:ℝ)^L*(2^L*2^(-54:Ἁ))<=1:=by rw[<-Ap,<-AF,<-AA,<-AK AL];exact zpow_le_one_of_nonpos₀ one_le_two (by clear*-w;omega)
 have q:(0:ℝ)<2^(-54:Ἁ):=(by positivity)
 have u:(0:ℝ)<2^P:=(by positivity)
 have r:(0:ℝ)<2^L:=(by positivity)
 intro m
 induction m with
 |zero=>
  intro t a O S o j
  obtain rfl:t=n:=C4 S (K3.1 O)
  rw[bC,AC (AX _)]
  exact⟨o,j⟩
 |succ m y=>
  intro t a O S o j
  have h:t<n:=N1 O
  have i:=congrArg (·-1) O
  rw[bC,AD h]
  by_cases g:k t
  · rw[AD g]
    refine y (t+1) a i h o ?_
    rw[Ν,N t h g,lH,Av]
    exact j.trans (AE (R5.2 (E7 t)) (by positivity))
  rw[AC g]
  obtain⟨G,A⟩:=Q t h
  have e:=K (t+1) h
  rw[Ν,<-add_assoc]at e
  have I:((t+1:Ἀ):ℝ)+1+1<=2^L:=mod_cast((L1 h 2).trans Y:t+1+1+1<=2^L)
  have U:(t:ℝ)*(2^L*2^(-54:Ἁ))<1:=(AI (by exact_mod_cast (h.trans_le ((V1 n 2).trans Y)):(t:ℝ)<2^L) (mul_pos r q)).trans_le X
  have x:(t:ℝ)*((2:ℝ)^(P+L)*2^(-54:Ἁ))<2^P:=by
   rw[AF,Ap,mul_left_comm]
   exact (mul_lt_mul_of_pos_left U u).trans_eq (mul_one _)
  have M:|rv a+rv (p t)|<(2:ℝ)^(P+L):=by
   rw[A,← sub_neg_eq_add,AF,mul_comm]
   refine (abs_sub_le _ _ _).trans_lt ((add_lt_add_of_lt_of_le (j.trans_lt x) ((congrArg abs (sub_neg_eq_add _ _)).trans_le e)).trans_le ?_)
   rw[add_comm,← add_one_mul]
   exact AE I u.le
  obtain⟨F,D⟩:=i5 a (p t) o G (P+L) J M
  refine y (t+1) (a+p t) i h F ?_
  rw[Ν,← add_assoc]
  exact (abs_sub_le _ (rv a+rv (p t)) _).trans ((add_le_add D (by rw [A,add_sub_add_right_eq_sub]; exact j)).trans_eq ((add_comm _ _).trans ((add_one_mul _ _).symm.trans (congrArg (· * _) (X7 t).symm))))
lemma dh(a b:kF)(A B:Ἀ)(l:kG a<=A)(m:kG b<=B):
  |(tm a b:ℝ)|<=(2:ℝ)^(48+A+B):=by
 have d:=eR a;have g:=eR b;have e:(I6 (tm a b))<=2^(48+A+B):=by
  rw[tm,Ϩ,Ϩ,Int.natAbs_pow];rw[Ϲ]at d g;have h:(I6 (sv a))<=2^(24:Ἀ):=(by (clear * - d;omega));have j:(I6 (sv b))<=2^(24:Ἀ):=(by (clear * - g;omega));calc
   _<=2^(24:Ἀ)*2^(24:Ἀ)*2^(A+B):=ϧ (ϧ h j) (Θ (by norm_num) (by exact Y3 l m))
   _=2^(48+A+B):=by rw[<-AF,<-AF];congr 1;ring
 rw[<-mI,Ϲ];exact_mod_cast e
noncomputable def s8(Y:ἀ)(o:Ἄ)(e0 e1 e2 e3 e4 e5 e6 e7:Ἅ):ἀ:=
 (((((((Y.set! (J0 o) e0).set! (J0 (o+1)) e1).set! (J0 (o+2)) e2).set! (J0 (o+3)) e3).set! (J0 (o+4)) e4).set!
  (J0 (o+5)) e5).set! (J0 (o+6)) e6).set! (J0 (o+7)) e7
@[inline]def tS(relu:Ἆ)(L n:Ἀ)(p:Ἀ->Ἅ)(bpf:Unit->Ἅ)(aB:Unit->Ἃ):Ἃ:=
 let r:=sT p n 0 (bpf ()) 0
 let c:=if 1<=(lq r.2)/\(lq r.2)<=1993/\1076<=(lq r.2)+L then
  ak r.1 (I0 ((lq r.2)+L-1076))else 4294967296
 if c<4294967296 then act relu (N3 c)else aB ()
theorem kRV(w:ἇ)(A:Ἀ)(hw:w.toNat=2^(31:Ἀ)+A)(hA:A<255*2^(23:Ἀ)):reluW w=0:=by
 have a3:A<2^(31:Ἀ):=hA.trans (by decide)
 have h1:(w&&&0x80000000!=0)=true:=by
  rw[AM,Ak,<-Β,Ξ];change ¬w.toNat&&&2^(31:Ἀ)=0
  rw[Nat.and_two_pow,hw,Nat.testBit_two_pow_add_eq,Nat.testBit_lt_two_pow a3];decide
 have q:A/2^(23:Ἀ)<255:=(ϸ (by decide)).2 hA
 have h2:(expOf w=255)=False:=by
  apply eq_false;intro z;have t:=congrArg mJ z
  unfold expOf at t;rw[Ξ,Υ,β,hw] at t
  change (2^(31:Ἀ)+A)/2^(23:Ἀ)&&&(2^(8:Ἀ)-1)=255 at t
  rw[ζ,show 2^(31:Ἀ)+A=A+2^(23:Ἀ)*2^(8:Ἀ) from by rw[Τ];rfl,mv _ _ (by decide),Nat.add_mod_right,γ (q.trans (by decide))] at t
  exact absurd t (R8 q)
 unfold reluW;rw[if_pos (by simp only[h1,h2,decide_false,Bool.false_and,Bool.not_false,Bool.and_self])]
open FloatLib.Floats.Formats.BinaryInterchange.Model.NativeBinary32 in
theorem kMB(e f:Ἀ)(he:e<256):(mkBits true e f).toNat=2^(31:Ἀ)+e*2^(23:Ἀ)+f%2^(23:Ἀ):=by
 unfold mkBits;simp only[if_true]
 have a1:((1:ἇ)<<<31).toNat=2^(31:Ἀ):=by decide
 have a2:((lA e&&&0xff)<<<23).toNat=e<<<23:=by
  rw[ϴ,Ξ,lE]
  simp only[kN]
  rw[show (255:Ἀ)=2^(8:Ἀ)-1 from rfl,ζ,γ (by (clear * - he;omega):e%2^(32:Ἀ)<2^(8:Ἀ)),γ (by (clear * - he;omega):e<2^(32:Ἀ))]
  show (e<<<(23%32))%2^(32:Ἀ)=_;rw[ο];exact γ (by norm_num;(clear * - he;omega))
 have a3:(lA f&&&0x007fffff).toNat=f%2^(23:Ἀ):=by
  rw[Ξ,lE];simp only[kN]
  rw[show (8388607:Ἀ)=2^(23:Ἀ)-1 from rfl,ζ,Nat.mod_mod_of_dvd _ (by decide)]
 rw[nK,nK,a1,a2,a3]
 have r:f%2^(23:Ἀ)<2^(23:Ἀ):=ϊ _ (by decide)
 have b1:(2^(31:Ἀ)|||e<<<23)=(2^(8:Ἀ)*1+e)<<<23:=by
  rw[Ϥ (by (clear * - he;omega):e<2^(8:Ἀ)) 1,Nat.shiftLeft_or_distrib];congr 1
 rw[b1,←Nat.shiftLeft_add_eq_or_of_lt r,ο];ring
open FloatLib.Floats.Formats.BinaryInterchange.Model.NativeBinary32 in
theorem kRD_h1{c:Prop}{i:Decidable c}{a b:ἇ}(ha:c->reluW a=0)(hb:¬c->reluW b=0):reluW (@ite _ c i a b)=0:=by
 by_cases h:c
 · rw[if_pos h];exact ha h
 · rw[if_neg h];exact hb h
theorem kRD_h2(e f:Ἀ)(he:e<255):reluW (FloatLib.Floats.Formats.BinaryInterchange.Model.NativeBinary32.mkBits true e f)=0:=
  kRV _ _ ((kMB e f (K6 he)).trans (pA _ _ _))
   ((O1 (ϊ f (by decide)) _).trans_le (by rw[<-Ο];exact ι _ he))
theorem kRD_h3(x:Int)(h:¬127<x):F6 (x+127)<255:=by omega
open FloatLib.Floats.Formats.BinaryInterchange.Model.NativeBinary32 in
theorem kRD(d:kU)(hn:d.negative=true):reluW (roundDyadic d)=0:=by
 unfold roundDyadic;simp only[hn,if_true]
 exact kRD_h1 (fun _=>by decide) fun _=>kRD_h1 (fun _=>kRD_h1 (fun _=>by decide) fun _=>kRD_h1 (fun _=>kRD_h2 1 0 (by decide)) fun _=>kRD_h2 0 _ (by decide)) fun _=>kRD_h1 (fun _=>by decide) fun h=>kRD_h2 _ _ (kRD_h3 _ h)
open FloatLib.Floats.Formats.BinaryInterchange.Model.NativeBinary32 in
theorem kRN(T:Ἁ)(h:T<0):p8 true (md T (-298) false)=0:=by
 have hT:T≠0:=by omega
 unfold md;rw[if_neg hT]
 rw[show kE lU (ofIntDyadic T (-298))=ofUInt32 (roundDyadic (ofIntDyadic T (-298))) from (ofUInt32_roundDyadic _).symm,bD]
 unfold p8;rw[if_pos rfl]
 exact kRD _ (by simp[ofIntDyadic,h])
theorem kNg(S:Ἅ)(m:fv S)(h:ἆ)(q:(lq h)<2^(63:Ἀ))(T:Ἁ)(o:|rv S-(T:ℝ)|<=(2:ℝ)^((lq h):Ἁ))
  (c1:(S.toBits>>>63!=0)=true)(c2:h+1024<=eF S.toBits):T<0:=by
 have hE:(lq h)+1024<=(lq (eF S.toBits)):=by
  have t:=θ.1 c2;rwa[η,γ (by simp only[kA];(clear * - q;omega))] at t
 have g:¬eF S.toBits=0:=fun z=>by rw[z] at hE;simp at hE
 have r:=dV S m g;rw[bJ] at r;simp only[c1,if_true] at r
 generalize (lq (S.toBits&&&0xfffffffffffff))=F at r
 generalize (lq (eF S.toBits))=E at r hE
 generalize (lq h)=H at o hE
 have k1:(2:ℝ)^((H:Ἁ)+1)<=(2:ℝ)^(52:Ἁ)*(2:ℝ)^((E:Ἁ)-1075):=by
  rw[<-AK AL];exact AG one_le_two (by (clear * - hE;omega))
 have k3:(2:ℝ)^(52:Ἁ)<=((F+2^(52:Ἀ):Ἀ):ℝ):=by rw[W5,I4,nZ,zpow_ofNat];exact le_add_of_nonneg_left (M3 _)
 have k6:(0:ℝ)<(2:ℝ)^(H:Ἁ):=zpow_pos two_pos _
 have s1:rv S<=-((2:ℝ)^(H:Ἁ)*2):=by rw[r,neg_mul,<-zpow_add_one₀ AL];exact neg_le_neg (k1.trans (AE k3 (zpow_nonneg zero_le_two _)))
 exact Int.cast_lt_zero.1 ((neg_le_sub_iff_le_add.1 (A3.1 o).1).trans_lt (lt_neg_iff_add_neg.1 (s1.trans_lt (neg_lt_neg (lt_mul_of_one_lt_right k6 one_lt_two)))))
theorem kTF0(relu:Ἆ)(n:Ἀ)(d e:Ἀ->kF)(g:aG n d e)(S:Ἅ)(m:fv S)(h:ἆ)(q:(lq h)<2^(63:Ἀ))
  (o:|rv S-(af n d e:ℝ)|<=(2:ℝ)^((lq h):Ἁ))
  (c:(relu&&(S.toBits>>>63!=0)&&h+1024<=eF S.toBits)=true):(0:Ἃ)=p8 relu (ms n d e):=by
 simp only[Η,AJ] at c
 obtain⟨⟨r,c1⟩,c2⟩:=c
 have t:=kNg S m h q _ o c1 c2
 rw[r,bT n d e g (by (clear * - t;omega))];exact (kRN _ t).symm
theorem kEB(E:Ἀ)(m:E<2048)(l:(2^(64:Ἀ)-1195+E)%2^(64:Ἀ)<253):1195<=E∧E-1085-110<253:=by omega
theorem kAkN(S:Ἅ)(h:ἆ)(e:ak S h<4294967296)(c1:(S.toBits>>>63!=0)=true):p8 true (N3 (ak S h))=0:=by
 unfold p8;rw[if_pos rfl]
 unfold ak at e⊢;dsimp only at e⊢;split at e;swap;exact absurd e (by decide);rename_i d;rw[AD d];simp only[Η,AJ]at d;obtain⟨⟨⟨-,l⟩,-⟩,-⟩:=d
 have m:(lq (eF S.toBits))<2048:=by rw[eF,Ε];exact AO ϝ (by decide)
 rw[μ,Ζ]at l;simp only[kA]at l
 obtain⟨E1,E2⟩:=kEB _ m l
 have s0:(lq S.toBits)/2^(63:Ἀ)≠0:=by
  have t:=c1;simp only[AM,Ak,<-ξ,ν,kA]at t;simpa[β] using t
 have s2:(lq S.toBits)/2^(63:Ἀ)<2:=(ϸ (by decide)).2 (S.toBits.toNat_lt.trans_eq (by norm_num))
 have s1:(lq S.toBits)/2^(63:Ἀ)=1:=by (clear * - s0 s2;omega)
 have f0:(lq (S.toBits&&&0xfffffffffffff))<2^(52:Ἀ):=Ε S.toBits 0xfffffffffffff▸AO ϝ (by decide)
 have r:((lq (S.toBits&&&0xfffffffffffff))+2^(28:Ἀ))/2^(29:Ἀ)<=2^(23:Ἀ):=J2 ((ϸ (by decide)).2 (by (clear * - f0;omega)))
 rw[μ]at e;simp only[kA]at e
 refine kRV _ _ (by rw[Ϫ,γ e,g0 S.toBits E1,s1,n4]) ?_
 generalize ((lq S.toBits&&&0xfffffffffffff)+2^(28:Ἀ))/2^(29:Ἀ)=k at r
 have g:((lq (eF S.toBits))-1085-110)*2^(23:Ἀ)<=252*2^(23:Ἀ):=ι _ (J2 E2)
 (clear * - E2 r;omega)
@[inline]def tf(relu:Ἆ)(S:Ἅ)(hh:ἆ)(sl:Unit->Ἃ):Ἃ:=
 if relu&&(S.toBits>>>63!=0)&&hh+1024<=eF S.toBits then 0 else
 let c:=ak S hh
 if c<4294967296 then act relu (N3 c)else sl ()
@[inline]def tw(relu:Ἆ)(S:Ἅ)(hh:ἆ)(L n:Ἀ)(p:Ἀ->Ἅ)(bpf:Unit->Ἅ)
  (aB:Unit->Ἃ):Ἃ:=tf relu S hh λ_=>tS relu L n p bpf aB
@[inline]def hG(e:Ἃ)(ix ib L:Ἀ):Ἆ×ἆ:=
 let p:=48+ix+max ((mJ e)-1) 55
 (decide (e<255/\p+L<=425),(I0 (max p ib+2*L-54)))
@[inline]def tG(relu:Ἆ)(S:Ἅ)(e:Ἃ)(ix ib L:Ἀ)(sl fb:Unit->Ἃ):Ἃ:=
 let h:=hG e ix ib L
 if h.1 then tf relu S h.2 sl else fb ()
theorem mA11 (x:ℕ):x&&&2047=x%2048:=ζ x 11
theorem mA52 (x:ℕ):x&&&4503599627370495=x%4503599627370496:=ζ x 52
theorem mA29 (x:ℕ):x&&&536870911=x%536870912:=ζ x 29
theorem mA63 (x:ℕ):x&&&9223372036854775807=x%9223372036854775808:=ζ x 63
theorem mA23 (x:ℕ):x&&&8388607=x%8388608:=ζ x 23
theorem mA8 (x:ℕ):x&&&255=x%256:=ζ x 8
theorem kAkF(S:Ἅ)(h:ἆ)(e:ak S h<4294967296):kB (N3 (ak S h))≠255:=by
  unfold ak at e ⊢; unfold F.eF at e ⊢; dsimp only at e ⊢
  generalize S.toBits=b at e ⊢
  split at e
  · rename_i c; rw [if_pos c]
    simp only [Η,AJ] at c
    obtain ⟨⟨⟨-, l⟩, -⟩, -⟩:=c
    have hB:=b.toNat_lt
    simp only [μ,θ,BC,Ζ,η,Ε,ψ,ν,ϩ,kA,kH,nr,Nat.reduceMul,ο,β,mA11,mA52,mA29,mA63,mA23,true_or,or_true,true_and,and_true,le_refl] at l hB
    have hX:b.toNat%9223372036854775808/4503599627370496=b.toNat/4503599627370496%2048:=by (clear * -;omega)
    have hR:1195*8388608 ≤ (((b&&&0x7FFFFFFFFFFFFFFF)+268435456)>>>29).toNat∧(((b&&&0x7FFFFFFFFFFFFFFF)+268435456)>>>29).toNat ≤ 1448*8388608:=by
      simp only [μ,θ,BC,Ζ,η,Ε,ψ,ν,ϩ,kA,kH,nr,Nat.reduceMul,ο,β,mA11,mA52,mA29,mA63,mA23,true_or,or_true,true_and,and_true,le_refl]; (clear * - l hX;omega)
    have hs:((b>>>63)<<<31).toNat=0∨((b>>>63)<<<31).toNat=2147483648:=by simp only [μ,θ,BC,Ζ,η,Ε,ψ,ν,ϩ,kA,kH,nr,Nat.reduceMul,ο,β,mA11,mA52,mA29,mA63,mA23,true_or,or_true,true_and,and_true,le_refl]; (clear * - hB;omega)
    generalize (((b&&&0x7FFFFFFFFFFFFFFF)+268435456)>>>29)=R at hR ⊢
    generalize (b>>>63)<<<31=s at hs ⊢
    intro hq; have hq':=congrArg mJ hq
    unfold kB at hq'
    simp only [μ,θ,BC,Ζ,η,Ε,ψ,ν,ϩ,kA,kH,nr,Nat.reduceMul,ο,β,mA11,mA52,mA29,mA63,mA23,true_or,or_true,true_and,and_true,le_refl,Ξ,Υ,kN,Ϫ,mA8] at hq'
    (clear * - hR hs hq';omega)
  · exact absurd e (by decide)
lemma jE_h1(P L:Ἀ)(J:P+L<=425)(D:L<=27):P+2*L-54<2^(63:Ἀ)∧((L+P:Ἀ):Ἁ)<=425:=⟨by rw[Nat.two_mul,<-pA];exact (K4 _ _).trans_lt ((Y3 J D).trans_lt (by decide)),U4.2 ((Τ L P).trans_le J)⟩
lemma jE_h2(K L:Ἀ)(y:K<=1993)(D:L<=27):K+L-1076<2^(63:Ἀ):=(K4 _ _).trans_lt ((Y3 y D).trans_lt (by decide))
lemma jE(g:Ἆ)(n:Ἀ)(d e:Ἀ->kF)(p:Ἀ->Ἅ)(h:Unit->Ἅ)(u:ἆ)(P L:Ἀ)
  (B:Unit->kF) (K:∀s<n,fv (p s)/\rv (p s)=tm (d s) (e s))
  (q:fv (h ())/\rv (h ())=tm (d n) (e n)) (j:aG (n+1) d e)
  (E:∀s≤n,|(tm (d s) (e s):ℝ)|<=2^P) (N:n+2<=2^L) (D:L<=27) (J:P+L<=425)
  (x:(lq u)=P+2*L-54) (G:B ()=p8 g (ms (n+1) d e))
  (S:Ἅ) (I:fv S) (v:|rv S-(af (n+1) d e:ℝ)|<=(2:ℝ)^((lq u):Ἁ)):
  tw g S u L n p h B=p8 g (ms (n+1) d e):=by
 have F:(af (n+1) d e:ℝ)=(tm (d n) (e n):ℝ)+∑s∈kP n,(tm (d s) (e s):ℝ):=by rw[aK,af,Int.cast_add,Int.cast_sum,add_comm]
 have y:(n:ℝ)<=2^L:=mod_cast((V1 n 2).trans N:n<=2^L)
 have o:|(af (n+1) d e:ℝ)|<=(2:ℝ)^(425:Ἁ):=by rw[F];refine(cv n _ _ _ (λs M=>E s M.le) (E n AP) n AP).trans ?_;have y:(n:ℝ)+1<=2^L:=mod_cast((E7 _).trans N:n+1<=2^L);refine(AE y (pow_nonneg zero_le_two _)).trans ?_;rw[<-AF,<-AA];exact AG one_le_two (jE_h1 P L J D).2
 by_cases RZ:(g&&(S.toBits>>>63!=0)&&u+1024<=eF S.toBits)=true
 · unfold tw tf;rw[if_pos RZ];exact kTF0 g (n+1) d e j S I u (by rw[x];exact (jE_h1 P L J D).1) v RZ
 unfold tw tf tS;rw[if_neg RZ];dsimp only;split
 .rename_i w;rw[cw _ _ _ j _ I u (by rw[x];exact (jE_h1 P L J D).1) v o w]
 generalize A:sT p n 0 (h ()) 0=r;split
 .rename_i t;split
  .rename_i w;obtain⟨m,l⟩:=iM _ _ (tm (d n) (e n)) n (lq r.2) t.1 t.2.1 K n 0 _ 0 rfl (ϟ _) q.1 (by simp only[q.2,X8,Av,sub_self,A5,Nat.cast_zero,zero_mul,le_refl]) (by rw[A])
   rw[A,<-F]at l;rw[A]at m;have H:(lq ((I0 ((lq r.2)+L-1076))))=(lq r.2)+L-1076:=by rw[Ϣ,Α (by rw[n0];exact(M8 (J1 (D.trans (by decide)) _)).trans_lt r.2.toNat_lt)]
   refine congrArg _ (cw _ _ _ j _ m _ (by rw[H];exact jE_h2 _ L t.2.1 D) (l.trans ?_) o w);rw[H];refine(AE y (zpow_nonneg zero_le_two _)).trans_eq ?_;rw[<-AA,<-AK AL];exact congrArg (λz:Ἁ=>(2:ℝ)^z) (by (clear * - t;omega))
  .exact G
 .rw[AC (by decide)];exact G
lemma gg(X:ἀ)(i:Ἀ):X.get! i=X.data[i]!:=by rcases X with⟨d⟩;rfl
lemma f8(v:Ἅ)(r:Ἀ):A8.f ⟨v,v,v,v,v,v,v,v⟩ r=v:=by unfold A8.f;split<;>rfl
@[specialize]def da{α:Type}(f:Ἀ->α)(N:Ἀ)(a:Ἂ α):Ἂ α:=
 if (O0 a)<N then da f N (a.push (f (O0 a)))else a
termination_by N-(O0 a)
decreasing_by rw[ϣ];omega
theorem iC_h1{α:Type}(f:Ἀ->α)(N:Ἀ):∀k (a:Ἂ α),N-(O0 a)=k->(O0 a)<=N->(∀i (h:i<(O0 a)),a[i]=f i)->da f N a=kK (n:=N) λq=>f (Fin.val q):=by
 intro k;induction k with
 |zero=>intro a j g d;rw[da,AC (W3.2 (O4 j))];exact mi ((C4 g (O4 j)).trans Ϝ.symm) λi e e2=>(d i e).trans (Λ e2).symm
 |succ k l=>intro a j g d;rw[da,AD (N1 j)];exact l _ ((congrArg (N-·) (ϣ _)).trans ((V4 _ _ _).trans (congrArg (·-1) j))) ((ϣ _).trans_le (N1 j)) λi h=>if q:i<a.size then (nU q).trans (d i q) else by obtain rfl:i=a.size:=C4 (J2 (h.trans_eq (ϣ _))) (W3.1 q);exact nV
lemma iC{α:Type}(f:Ἀ->α)(N:Ἀ):
  da f N (lQ N)=kK (n:=N) λq=>f (Fin.val q):=by
 exact iC_h1 f N _ _ rfl (ϟ N) λi h=>(U5 i h).elim
lemma dN(n:Ἀ)(e d:Ἀ->kF):∀c i,i+c=n->
  nC (kK (n:=n+1) λt=>if (Fin.val t)<n then d (Fin.val t)else ld)
   (kK (n:=n+1) λt=>if (Fin.val t)<n then e (Fin.val t)else 0x80000000) i (c+1)=
  nC (kK (n:=n) λt=>e (Fin.val t)) (kK (n:=n) λt=>d (Fin.val t)) i c:=by
 intro c;induction c with
 |zero=>intro i h;obtain rfl:i=n:=(by exact N8 h);simp only[nC,ab (i+1) (λt=>if t< i then d t else ld) i (by exact lt_add_one i),ab (i+1) (λt=>if t< i then e t else 0x80000000) i (by exact lt_add_one i),AX,Am];rfl
 |succ c g=>intro i h;rw[nC,g (i+1) (by (clear * - h;omega)),nC,ab (n+1) (λt=>if t<n then d t else ld) i (by (clear * - h;omega)),ab (n+1) (λt=>if t<n then e t else 0x80000000) i (by (clear * - h;omega)),ab n e i (by (clear * - h;omega)),ab n d i (by (clear * - h;omega))];simp only[((by (clear * - h;omega)):i<n),↓reduceIte,Ϗ (kM (d i)),Bool.xor_comm (k9 (d i))]
lemma dz(n:Ἀ)(d e:Ἀ->kF)(h:0<n)(g:aG n d e):
  ms (n+1) (λt=>if t<n then e t else ld) (λt=>if t<n then d t else 0x80000000)=
   ms n d e:=by
 rw[bI _ _ _ λt _=>by split;exacts[(g t ‹_›).symm,(by decide)],bI _ _ _ g,aK,dN n d e n 0 (Κ n)];congr 1
 .simp only[AX,Am,((by decide):tm ld 0x80000000=0),Av];exact ϔ rfl λt j=>by simp only[AD (ϥ.mp j)];unfold tm;rw[Τ];ring
 simp[h.ne']
theorem kStep (a x:Ἅ) (A X ea:ℝ) (K:ℕ) (hK:K ≤ 971)
    (ha:fv a) (hx:fv x) (hX:rv x=X) (hea : |rv a-A| ≤ ea)
    (hB : |A+X| + ea < (2:ℝ)^K) :
    fv (a+x) ∧ |rv (a+x)-(A+X)| ≤ ea+(2:ℝ)^K*(2:ℝ)^(-54:ℤ):=by
  obtain ⟨h2,h3⟩:=i5 a x ha hx K hK (by rw [hX, ← sub_add_cancel (rv a) A,add_assoc]; exact (Ag _ _).trans_lt ((add_le_add hea AP).trans_lt ((add_comm _ _).trans_lt hB)))
  rw [hX] at h3
  refine ⟨h2,?_⟩
  rw [← sub_add_sub_cancel (rv (a+x)) (rv a+X) (A+X),add_sub_add_right_eq_sub,add_comm ea]
  exact (Ag _ _).trans (add_le_add h3 hea)
theorem kSeq (p:ℕ → Ἅ) (z:ℕ → ℤ) (u:ℝ) (n K:ℕ) (B:ℝ) (hK:K ≤ 971)
    (hp : ∀ t < n,fv (p t)∧rv (p t)=z t)
    (hS : ∀ t ≤ n, |u + ∑ s ∈ kP t,(z s:ℝ)| ≤ B)
    (hBK:B+n*((2:ℝ)^K*(2:ℝ)^(-54:ℤ)) < (2:ℝ)^K) :
    ∀ m t a,n-t=m → t ≤ n → fv a →
      |rv a-(u + ∑ s ∈ kP t,(z s:ℝ))| ≤ t*((2:ℝ)^K*(2:ℝ)^(-54:ℤ)) →
      fv (ao p n t a) ∧
        |rv (ao p n t a)-(u + ∑ s ∈ kP n,(z s:ℝ))| ≤ n*((2:ℝ)^K*(2:ℝ)^(-54:ℤ)):=by
  have hu:(0:ℝ) ≤ (2:ℝ)^K*(2:ℝ)^(-54:ℤ):=Af (pow_nonneg zero_le_two K) (zpow_nonneg zero_le_two _)
  intro m; induction m with
  | zero => intro t a D Q e o; obtain rfl:t=n:=C4 Q (K3.1 D)
            rw [ao,if_neg (AX _)]; exact ⟨e,o⟩
  | succ m G =>
    intro t a D Q e o
    have d:t < n:=N1 D
    rw [ao,if_pos d]
    obtain ⟨q,N⟩:=hp t d
    have hB1:=hS (t+1) d
    rw [Ν, ← add_assoc] at hB1
    have:=kStep a (p t) _ _ _ K hK e q N o ((add_le_add hB1 (AE (R5.2 Q) hu)).trans_lt hBK)
    refine G (t+1) (a+p t) (by rw[V4,D,BK]) d this.1 ?_
    rw [Ν, ← add_assoc,X7,add_one_mul]; exact this.2
theorem kE1 (n:ℕ) (d e:ℕ → Ἃ) (p:ℕ → Ἅ) (h0:Ἅ) (P L:ℕ)
    (K : ∀ s < n,fv (p s)∧rv (p s)=tm (d s) (e s))
    (q:fv h0∧rv h0=tm (d n) (e n))
    (E : ∀ s ≤ n, |(tm (d s) (e s):ℝ)| ≤ 2^P) (N:n+2 ≤ 2^L) (D:L ≤ 27) (J:P+L ≤ 425) :
    fv (ao p n 0 h0) ∧
      |rv (ao p n 0 h0)-(af (n+1) d e:ℝ)| ≤ (2:ℝ)^(((P+2*L-54:ℕ)):ℤ):=by
  have hL1:((n:ℝ)+1)+1 ≤ (2:ℝ)^L:=by exact_mod_cast N
  have pP:(0:ℝ) < 2^P:=by positivity
  have hU:(0:ℝ) < (2:ℝ)^(P+L)*(2:ℝ)^(-54:ℤ):=by positivity
  have hn:(n:ℝ) < 2^L:=by exact_mod_cast ((I3 n).trans_le ((E7 _).trans N))
  have hE:2^L*((2:ℝ)^(P+L)*(2:ℝ)^(-54:ℤ)) ≤ (2:ℝ)^(((P+2*L-54:ℕ)):ℤ) :=
    (((Ap _ _ _).symm.trans (congrArg (· * (2:ℝ)^(-54:ℤ)) ((AF (2:ℝ) L (P+L)).symm.trans (AA (2:ℝ) (L+(P+L))).symm))).trans (AK AL _ _).symm).trans_le (AG one_le_two (by clear * -; omega))
  have hE2:(2:ℝ)^(((P+2*L-54:ℕ)):ℤ) ≤ 2^P :=
    (AA (2:ℝ) _).trans_le (AU one_le_two (Nat.sub_le_iff_le_add.2 (J1 (U3 2 D) P)))
  have hS : ∀ t ≤ n, |(tm (d n) (e n):ℝ) + ∑ s ∈ kP t,(tm (d s) (e s):ℝ)| ≤ (n+1)*2^P:=fun t ht =>
    (Ag _ _).trans ((add_le_add (E n AP) ((K8 _ _).trans (((Finset.sum_le_sum (fun s hs=>E s ((ϥ.1 hs).le.trans ht))).trans_eq ((Finset.sum_const _).trans ((congrArg (· • (2:ℝ)^P) (Finset.card_range t)).trans (nsmul_eq_mul _ _)))).trans (AE (R5.2 ht) pP.le)))).trans_eq ((add_comm _ _).trans (add_one_mul _ _).symm))
  have hBK:(n+1)*(2:ℝ)^P+n*((2:ℝ)^(P+L)*(2:ℝ)^(-54:ℤ)) < (2:ℝ)^(P+L) :=
    (add_lt_add_of_le_of_lt AP ((AI hn hU).trans_le (hE.trans hE2))).trans_le ((add_one_mul _ _).symm.trans_le ((AE hL1 pP.le).trans_eq ((mul_comm _ _).trans (AF _ _ _).symm)))
  have R:=kSeq p (fun s=>tm (d s) (e s)) (tm (d n) (e n)) n (P+L) ((n+1)*2^P) (J.trans (by decide))
    K hS hBK n 0 h0 (pQ n) (ϟ n) q.1 (((congrArg abs (sub_eq_zero.2 (q.2.trans ((Av _).symm.trans (congrArg ((tm (d n) (e n):ℝ) + ·) (X8 _).symm))))).trans A5).trans_le (Af (M3 0) hU.le))
  refine ⟨R.1,?_⟩
  exact (congrArg (fun x => |rv (ao p n 0 h0)-x|) ((congrArg Int.cast (Ν (fun s=>tm (d s) (e s)) n)).trans ((Int.cast_add _ _).trans ((congrArg (· + ((tm (d n) (e n):ℤ):ℝ)) (Int.cast_sum _ _)).trans (add_comm _ _))))).trans_le (R.2.trans ((AE hn.le hU.le).trans hE))
theorem kHG (n:ℕ) (d e':ℕ → Ἃ) (e:Ἃ) (ix ib:ℕ)
    (hd : ∀ s < n,idxOf (d s) ≤ ix) (he : ∀ s < n,(expOf (e' s)).toNat ≤ e.toNat)
    (hb : |(tm (d n) (e' n):ℝ)| ≤ 2^ib) :
    ∀ s ≤ n, |(tm (d s) (e' s):ℝ)| ≤ 2^(max (48+ix+max (e.toNat-1) 55) ib):=by
  intro s hs
  rcases Z7 s n with h | h
  · have i1:=hd s h; have i2:=he s h
    have j2:idxOf (e' s) ≤ max (e.toNat-1) 55:=by unfold idxOf; split_ifs; exact ϟ _; exact (P4 i2 1).trans (le_max_left _ _)
    have A : ∀ w:Ἃ, |(sv w:ℝ)| ≤ 2^24:=fun w=>(show |(sv w:ℝ)| = mantOf w by unfold sv; split_ifs <;> simp only [Int.cast_neg,M5,BE,nE]).trans_le (by exact_mod_cast (mantOf_lt_24 w).le)
    rw [tm,Int.cast_mul,Int.cast_mul,Int.cast_pow,Int.cast_ofNat,Ae,Ae,AW (by positivity:(0:ℝ) < 2^(idxOf (d s)+idxOf (e' s)))]
    calc |(sv (d s):ℝ)| * |(sv (e' s):ℝ)| * 2^(idxOf (d s)+idxOf (e' s))
        ≤ 2^24*2^24*2^(ix+max (e.toNat-1) 55) :=
          mul_le_mul (mul_le_mul (A _) (A _) (Aa _) (by positivity)) (AU one_le_two (Y3 i1 j2)) (by positivity) (by positivity)
      _=2^(48+ix+max (e.toNat-1) 55):=by rw [← AF, ← AF, ← pA]
      _ ≤ _:=AU one_le_two (le_max_left _ _)
  · obtain rfl:s=n:=C4 hs h
    exact hb.trans (AU one_le_two (le_max_right _ _))
theorem kc0:(0:Ἄ).toNat=0:=by simp only [Ϧ]
theorem kc1:(1:Ἄ).toNat=1:=by simp only [kD]
theorem kc2:(2:Ἄ).toNat=2:=by simp only [kD]
theorem kc3:(3:Ἄ).toNat=3:=by simp only [kD]
theorem kc4:(4:Ἄ).toNat=4:=by simp only [kD]
theorem kc5:(5:Ἄ).toNat=5:=by simp only [kD]
theorem kc6:(6:Ἄ).toNat=6:=by simp only [kD]
theorem kc7:(7:Ἄ).toNat=7:=by simp only [kD]
theorem kc8:(8:Ἄ).toNat=8:=by simp only [kD]
theorem kSm (a b:Ἄ) (h:a.toNat+b.toNat < 2^(31:Ἀ)):(a+b).toNat=a.toNat+b.toNat:=by
  have:2^(32:Ἀ) ≤ 2^nx:=ς
  rw [lJ]; exact γ (by omega)
theorem aos (p:ℕ → Ἅ) (n t:ℕ) (a:Ἅ) (h:t < n):ao p n t a=ao p n (t+1) (a+p t):=by rw [ao,if_pos h]
theorem aoe (p:ℕ → Ἅ) (n t:ℕ) (a:Ἅ) (h:n ≤ t):ao p n t a=a:=by rw [ao,if_neg (by exact W3.mpr h)]
structure WcE {K V:Type} (f:K → V) where
  k:K
  v:V
  h:v=f k
@[inline] def wcAeq (a b:Ἂ ἇ):Ἆ :=
  withPtrEq a b (fun _=>a==b) (fun h=>by subst h; exact Az a)
theorem wcAeq_ok {a b:Ἂ ἇ} (h:wcAeq a b=true):a=b:=eq_of_beq h
@[inline] def wcK3eq (a b:Ἂ ἇ × Nat × Nat):Ἆ :=
  a.2.1==b.2.1&&a.2.2==b.2.2&&wcAeq a.1 b.1
theorem wcK3eq_ok (a b:Ἂ ἇ × Nat × Nat) (h:wcK3eq a b=true):a=b:=by
  obtain ⟨a1,a2,a3⟩:=a; obtain ⟨b1,b2,b3⟩:=b
  simp only [wcK3eq,Η,AQ] at h
  obtain ⟨⟨h2,h3⟩,h1⟩:=h; rw [wcAeq_ok h1,h2,h3]
def wcFind {K V:Type} {f:K → V} (eq:K → K → Ἆ) (ok : ∀ a b,eq a b=true → a=b)
    (a:K) (s:Ἂ (WcE f)):Option {v // v=f a} :=
  s.findSome? fun e=>if hk:eq e.k a=true then some ⟨e.v,by rw [e.h,ok _ _ hk]⟩ else none
theorem wcSub {β:Type} {b:β} (x y : {v // v=b}):x=y:=Subtype.ext (x.2.trans y.2.symm)
@[inline] def wcGet3 {K V:Type} {f:K → V} {α:Type} (eq:K → K → Ἆ) (ok : ∀ a b,eq a b=true → a=b)
    (r:ST.Ref Unit (Ἂ (WcE f))) (n:Nat) (a:K) (pk:K → α) : {v // v=f a} :=
  let o:=ST.Prim.Ref.get r (Void.mk ())
  match o.val.findIdx? (fun e=>eq e.k a) with
  | some i =>
    if h:i < o.val.size then
      if hk:eq o.val[i].k a=true then
        withPtrAddr (pk o.val[i].k) (fun p1=>withPtrAddr (pk a) (fun p2 =>
          if p1=p2 then (⟨o.val[i].v,by rw [o.val[i].h,ok _ _ hk]⟩ : {v // v=f a}) else
          let e2:WcE f:=⟨a,o.val[i].v,by rw [o.val[i].h,ok _ _ hk]⟩
          let o2:=ST.Prim.Ref.swap r (#[e2] ++ o.val.eraseIdx i h) o.state
          if o2.val.size=0 then ⟨f a,rfl⟩ else ⟨o.val[i].v,by rw [o.val[i].h,ok _ _ hk]⟩)
          (fun _ _=>wcSub _ _)) (fun _ _=>wcSub _ _)
      else ⟨f a,rfl⟩
    else ⟨f a,rfl⟩
  | none =>
    let e:WcE f:=⟨a,f a,rfl⟩
    let o2:=ST.Prim.Ref.swap r (#[e] ++ (if o.val.size < n then o.val else o.val.pop)) o.state
    match wcFind eq ok a o2.val with
    | some y=>y
    | none=>⟨e.v,e.h⟩
def tchPg (P:ἀ) (i:Nat) (s:Ἅ):Nat → Ἅ
  | 0=>s
  | f+1=>if i < P.size then tchPg P (i+8) (s+P[i]!) f else s
def tchP (P:ἀ):Ἅ:=tchPg P 0 0 (P.size/8+1)
@[inline] def wcGet3T {K V:Type} {f:K → V} {α:Type} (eq:K → K → Ἆ) (ok : ∀ a b,eq a b=true → a=b)
    (r:ST.Ref Unit (Ἂ (WcE f))) (n:Nat) (a:K) (pk:K → α) (tk:V → Ἅ) (ne:Nat) : {v // v=f a} :=
  let o:=ST.Prim.Ref.get r (Void.mk ())
  match o.val.findIdx? (fun e=>eq e.k a) with
  | some i =>
    if h:i < o.val.size then
      if hk:eq o.val[i].k a=true then
        withPtrAddr (pk o.val[i].k) (fun p1=>withPtrAddr (pk a) (fun p2 =>
          if p1=p2 then (⟨o.val[i].v,by rw [o.val[i].h,ok _ _ hk]⟩ : {v // v=f a}) else
          let e2:WcE f:=⟨a,o.val[i].v,by rw [o.val[i].h,ok _ _ hk]⟩
          let o2:=ST.Prim.Ref.swap r (#[e2] ++ o.val.eraseIdx i h) o.state
          let o3:=ST.Prim.Ref.get r o2.state
          if o2.val.size=0 then ⟨f a,rfl⟩ else
          if (o3.val.foldl (fun (t:Ἅ) e=>t+tk e.v) 0 0 (min ne o3.val.size)).toBits==0x7FF4000000000001 then ⟨f a,rfl⟩ else
          ⟨o.val[i].v,by rw [o.val[i].h,ok _ _ hk]⟩)
          (fun _ _=>wcSub _ _)) (fun _ _=>wcSub _ _)
      else ⟨f a,rfl⟩
    else ⟨f a,rfl⟩
  | none =>
    let e:WcE f:=⟨a,f a,rfl⟩
    let o2:=ST.Prim.Ref.swap r (#[e] ++ (if o.val.size < n then o.val else o.val.pop)) o.state
    let o3:=ST.Prim.Ref.get r o2.state
    if (o3.val.foldl (fun (t:Ἅ) e=>t+tk e.v) 0 0 (min ne o3.val.size)).toBits==0x7FF4000000000001 then ⟨f a,rfl⟩ else
    match wcFind eq ok a o2.val with
    | some y=>y
    | none=>⟨e.v,e.h⟩
theorem s5nb:2^(32:Ἀ) ≤ 2^nx:=by
  rcases System.Platform.numBits_eq with h|h <;> rw [h] <;> decide
theorem s5uA {a b:Ἄ} (h:a.toNat+b.toNat < 2^(32:Ἀ)):(a+b).toNat=a.toNat+b.toNat:=by
  rw [lJ,γ (by have:=s5nb; exact FloatLib.Numerics.StaticStorage.lt_capacity h this)]
theorem s5uL {W:Ἂ ἇ}{k:Ἄ}(h:k<W.usize):k.toNat<W.size:=by
  have:=ε.mp h; have:W.usize.toNat ≤ W.size:=by
    simp only [Array.usize,G0,USize.toNat_ofNat]; exact Ay _ _
  exact cD h
theorem s5uget (W:Ἂ ἇ)(k:Ἄ)(h:k.toNat<W.size): W.uget k h=W[k.toNat]! := by
  rw [AB W _ h]; rfl
theorem s5push_get (A:ἀ)(v:Ἅ)(i:Nat): (A.push v).data[i]! = if i<A.size then A.data[i]! else if i=A.size then v else A.data[i]! := by
  show (A.data.push v)[i]! = _
  by_cases h1:i < A.data.size
  · rw [AB _ i (by simp; exact X5 h1),nU h1,AB _ i h1]; simp [kS,h1]
  · by_cases h2:i=A.data.size
    · subst h2; rw [AB _ _ (by simp),nV]; simp [kS]
    · rw [An _ i (by simp; omega)]; simp only [kS,h1,h2,if_false]; rw [An _ i h1]
theorem s5exp (w:ἇ): ((w<<<1)>>>24).toNat=(kB w).toNat:=((Υ _ _).trans (β _ _)).trans (eS w).symm
theorem s5tu (k:Nat)(h:k<2^(32:Ἀ)):(k.toUSize).toNat=k:=by
  simp only [G0,USize.toNat_ofNat]; exact γ (by have:=s5nb; exact FloatLib.Numerics.StaticStorage.lt_capacity h this)
theorem s5fmax(l:List ἇ):∀ m0:ἇ,m0.toNat ≤ (l.foldl (fun (m v:ἇ)=>if m<v then v else m) m0).toNat ∧
    ∀ x ∈ l,x.toNat ≤ (l.foldl (fun (m v:ἇ)=>if m<v then v else m) m0).toNat:=by
 induction l with
 |nil=>exact fun _=>⟨le_refl _,fun _ h=>nomatch h⟩
 |cons y l ih=>
  intro m0;simp only[Ϸ];obtain⟨i1,i2⟩:=ih (if m0<y then y else m0)
  by_cases c:m0<y<;>simp only[c,if_true,if_false]at i1 i2⊢
  ·exact⟨AZ (Ι.mp c).le i1,fun x hx=>(List.mem_cons.mp hx).elim (fun h=>h▸i1) (i2 x)⟩
  ·exact⟨i1,fun x hx=>(List.mem_cons.mp hx).elim (fun h=>h▸AZ (W4 fun g=>c (Ι.mpr g)) i1) (i2 x)⟩
structure A16 where
 a0:Ἅ
 a1:Ἅ
 a2:Ἅ
 a3:Ἅ
 a4:Ἅ
 a5:Ἅ
 a6:Ἅ
 a7:Ἅ
 b0:Ἅ
 b1:Ἅ
 b2:Ἅ
 b3:Ἅ
 b4:Ἅ
 b5:Ἅ
 b6:Ἅ
 b7:Ἅ
local instance:Inhabited A16:=⟨⟨0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0⟩⟩
def rMu(W:Ἂ Ἃ)(k e:Ἄ)(m:Ἃ):Ἃ:=
 if h:k<e then
  have hk:=ε.mp h
  rMu W (k+1) e (let v:=(if hk:k<W.usize then W.uget k (s5uL hk) else 0)<<<1;if m<v then v else m)
 else m>>>24
termination_by (J0 e)-(J0 k)
decreasing_by all_goals (rw[ac h];exact Z5 eб kб h)
def tBi(W:Ἂ Ἃ)(n ow o8 t:Ἄ)(o k:Ἄ)(A:ἀ):ἀ:=
 if h:o<o8 then
  have ho:=ε.mp h
  tBi W n ow o8 t (o+1) (k+n) (A.push (if o<ow then (if hk:k<W.usize then dw (W.uget k (s5uL hk)) else 0) else (0:Ἅ)))
 else A
termination_by (J0 o8)-(J0 o)
decreasing_by all_goals (rw[ac h];exact Z5 o8б oб h)
def tPt(W:Ἂ Ἃ)(n ow g16 e16:Ἄ)(t:Ἄ)(A:ἀ):ἀ:=
 if h:t<n then tPt W n ow g16 e16 (t+1) (tBi W n ow e16 t g16 (g16*n+t) A) else A
termination_by (J0 n)-(J0 t)
decreasing_by all_goals (rw[ac h];have:=ε.mp h;exact Z5 nб tб h)
def wcDF(a:Ἂ Ἃ×Ἀ×Ἀ):ἀ:=bQ (λk=>dw a.1[k]!) (a.2.1*a.2.2) (AR (a.2.1*a.2.2))
@[noinline] def wcDS:ST.Ref Unit (Ἂ (WcE wcDF)):=(ST.Prim.mkRef #[] (Void.mk ())).val
theorem s5tBi_inv (W:Ἂ ἇ)(n ow o8 t:Ἄ)(B:Nat)
    (hB:(ow.toNat+16)*n.toNat<2^(31:Ἀ))(ho8:o8.toNat≤ow.toNat+16)(hW:ow.toNat*n.toNat≤W.size)(hWs:W.size<2^(32:Ἀ))(htn:t.toNat<n.toNat) :
    ∀ m (o k:Ἄ)(A:ἀ),o8.toNat-o.toNat=m → o.toNat≤o8.toNat → k.toNat=o.toNat*n.toNat+t.toNat → A.size=B+o.toNat →
    let R:=tBi W n ow o8 t o k A
    R.size=B+o8.toNat∧(∀ i<B+o.toNat,R.data[i]! = A.data[i]!) ∧
    (∀ p,o.toNat≤p → p<o8.toNat → R.data[B+p]! = if p<ow.toNat then dw W[p*n.toNat+t.toNat]! else 0):=by
  have hO:ow.toNat+16 ≤ (ow.toNat+16)*n.toNat:=J4 _ ((ϟ _).trans_lt htn)
  intro m; induction m with
  | zero => intro o k A hm ho hk hA; have z:=K3.1 hm; rw [tBi,dif_neg (by rw [ε]; exact not_lt.2 z)]; dsimp only
            exact ⟨by rw[hA,C4 ho z],fun i _=>rfl,fun p h1 h2=>absurd (h1.trans_lt h2) (not_lt.2 z)⟩
  | succ m ih =>
    intro o k A hm ho hk hA
    have h1:o.toNat < o8.toNat:=N1 hm
    have q1:(o.toNat+1)*n.toNat ≤ (ow.toNat+16)*n.toNat:=ι _ (l7 (h1.trans_le ho8))
    have e1:(o+1).toNat=o.toNat+1:=by rw [s5uA (a:=o) (b:=1) (by simp; clear*-h1 ho8 hO hB; omega)]; simp
    have q2:=nO o.toNat n.toNat
    have e2:(k+n).toNat=(o.toNat+1)*n.toNat+t.toNat:=by
      rw [s5uA (a:=k) (b:=n) (by rw [hk]; clear*-htn q1 q2 hB; omega),hk,q2]; exact lO _ _ _
    rw [tBi,dif_pos (by rw [ε]; exact h1)]
    have hA':(A.push (if o<ow then (if hk:k<W.usize then dw (W.uget k (s5uL hk)) else 0) else 0)).size=B+(o+1).toNat:=by
      have hp : ∀ v,(A.push v).size=A.size+1:=fun v=>by show (A.data.push v).size=A.data.size+1; simp
      rw [hp,e1,hA,pA]
    obtain ⟨r1,r2,r3⟩:=ih (o+1) (k+n) _ (by rw [e1,←F7,hm,BK]) (by rw [e1]; exact h1) (by rw [e2,e1]) hA'
    refine ⟨r1,fun i hi=>by rw [r2 i (by rw [e1]; exact K6 hi),s5push_get,if_pos (by rw[hA];exact hi)],fun p hp1 hp2 => ?_⟩
    rcases Z7 o.toNat p with hp | hp
    · exact r3 p (by rw [e1]; exact hp) hp2
    · have pe:p=o.toNat:=C4 hp hp1
      subst pe
      rw [r2 _ (by rw [e1]; exact I3 _),s5push_get,if_neg (by rw[hA];exact AX _),if_pos hA.symm]
      by_cases hw:o < ow
      · have hw':=ε.mp hw
        have kb:k.toNat < W.size:=by
          rw [hk]; have:=ι n.toNat (l7 hw'); rw [Ο] at this; exact (O1 htn _).trans_le (this.trans hW)
        have kl:k < W.usize:=by
          have uz:W.usize.toNat=W.size%kX:=by simp [Array.usize]
          rw [ε,uz,γ (hWs.trans_le ς)]; exact kb
        rw [if_pos hw,dif_pos kl,if_pos hw',s5uget,hk]
      · have hw':¬ o.toNat < ow.toNat:=fun h=>hw (ε.mpr h)
        rw [if_neg hw,if_neg hw']
theorem s5rMu_spec_h1 (x v:ἇ):x ≤ (if x<v then v else x)∧v ≤ (if x<v then v else x):=by
  by_cases h:x<v
  · rw [if_pos h]; exact ⟨V7.mpr (Ι.mp h).le,V7.mpr AP⟩
  · rw [if_neg h]; exact ⟨V7.mpr AP,V7.mpr (not_lt.1 (h ∘ Ι.mpr))⟩
theorem s5rMu_spec (W:Ἂ ἇ)(hWs:W.size<2^(32:Ἀ)) :
    ∀ m (k e:Ἄ)(x:ἇ),e.toNat-k.toNat=m → e.toNat≤W.size →
    (x>>>24).toNat ≤ (rMu W k e x).toNat ∧ ∀ j,k.toNat≤j → j<e.toNat → (kB W[j]!).toNat ≤ (rMu W k e x).toNat:=by
  intro m; induction m with
  | zero=>intro k e x hm he; have z:=O4 hm; rw [rMu,dif_neg (by rw [ε]; exact not_lt.2 z)]; exact ⟨le_refl _,fun j h1 h2=>absurd (h1.trans_lt h2) (not_lt.2 z)⟩
  | succ m ih =>
    intro k e x hm he
    have h1:k.toNat < e.toNat:=N1 hm
    have e1:(k+1).toNat=k.toNat+1:=by rw [s5uA (a:=k) (b:=1) (by rw[mA]; exact C0 h1 (he.trans_lt hWs))]; simp
    rw [rMu,dif_pos (by rw [ε]; exact h1)]
    have kl:k < W.usize:=by
      have uz:W.usize.toNat=W.size%kX:=by simp [Array.usize]
      rw [ε,uz,γ (hWs.trans_le ς)]; exact h1.trans_le he
    simp only [dif_pos kl]
    obtain ⟨i1,i2⟩:=ih (k+1) e _ (by rw[e1,←F7,hm,BK]) he
    have mono : ∀ a b:ἇ,a ≤ b → (a>>>24).toNat ≤ (b>>>24).toNat:=fun a b h=>by
      simp only [Υ,β]; exact ϭ (V7.mp h)
    set v:=W.uget k (s5uL kl)<<<1
    obtain ⟨mx1,mx2⟩:=s5rMu_spec_h1 x v
    refine ⟨AZ (mono _ _ mx1) i1,fun j hj1 hj2 => ?_⟩
    rcases Z7 k.toNat j with hj | hj
    · exact i2 j (by rw [e1]; exact hj) hj2
    · have je:j=k.toNat:=le_antisymm hj hj1
      subst je
      rw [← s5exp, ← s5uget W k (s5uL kl)]; exact AZ (mono _ _ mx2) i1
theorem s5da_spec{α:Type}[Inhabited α](f:Nat->α)(N:Nat):∀ m (a:Ἂ α),N-a.size=m→a.size≤N→(∀ i<a.size,a[i]! = f i)→
    (da f N a).size=N ∧ ∀ i<N,(da f N a)[i]! = f i:=by
  intro m; induction m with
  | zero=>intro a hm ha hi; have z:=O4 hm; rw [da,if_neg (not_lt.2 z)]; exact ⟨le_antisymm ha z,fun i h=>hi i (h.trans_le z)⟩
  | succ m ih =>
    intro a hm ha hi; rw [da,if_pos (N1 hm)]
    refine ih _ (by simp; (clear * - hm ha;omega)) (by simp; exact N1 hm) (fun i h => ?_)
    simp at h
    rcases Z7 i a.size with h1|h1
    · rw [AB _ i (by simp; exact X5 h1),nU h1, ← AB a i h1]; exact hi i h1
    · have:i=a.size:=le_antisymm h h1
      subst this; rw [AB _ _ (by simp),nV]
theorem s5wcGet3_val {K V:Type} {f:K → V} {a:K} (x : {v // v=f a}):x.1=f a:=x.2
theorem kMm (a b:Ἄ) (h:a.toNat*b.toNat < 2^(31:Ἀ)):(a*b).toNat=a.toNat*b.toNat:=by
  have:2^(32:Ἀ) ≤ 2^nx:=ς
  rw [BO]; exact γ (by omega)
theorem kFU (S:ἀ) (i:Ἄ) (h:i.toNat < S.size):S.uget i h=S.data[i.toNat]! := by
  rcases S with ⟨d⟩; simp only [nN,kS] at h ⊢; rw [AB d i.toNat h]; rfl
theorem kXU (X:Ἂ Ἃ)(i:Ἄ)(h:(J0 i)<(O0 X))(c:ℕ)(hc:(J0 i)=c):X.uget i h=X[c]! := by
  subst hc; rw [AB X (J0 i) h]; rfl
theorem kIx7 (a k:Ἄ) (c:ℕ) (hk:k.toNat=c) (hc:c ≤ 7) (S:ἀ) (h:a.toNat+7 < S.size) (h2:a.toNat+7 < 2^(31:Ἀ)) :
    (a+k).toNat < S.size:=by rw [kSm a k (by (clear * - hk hc h2;omega))]; (clear * - hk hc h;omega)
theorem kTB (w:Ἃ) (F E:ℕ) (K:idxOf w ≤ F) (al:174+F ≤ E) : |(tm ld w:ℝ)| ≤ 2^E :=
  have H1:(tm ld w).natAbs ≤ 2^23*2^24*2^(126+F):=(Ϩ _ _).trans_le (ϧ ((Ϩ _ _).trans_le (ϧ (by decide) (bg w).le)) ((Int.natAbs_pow 2 _).trans_le (Θ two_pos (Y3 (by decide) K))))
  have H2:2^23*2^24*2^(126+F) ≤ 2^E:=(m7 2 47 (126+F)).symm.trans_le (Θ two_pos (((pA 47 126 F).symm.trans_le (L1 (by decide:173 ≤ 174) F)).trans al))
  (mI.symm.trans (Nat.cast_natAbs _).symm:|(tm ld w:ℝ)|=_).trans_le (((R5 (α:=ℝ)).mpr (H1.trans H2)).trans_eq ((I4 2 E).trans (congrArg (·^E) nZ)))
theorem kLS (n c:ℕ) (I:ℕ → ℕ) (f:ℕ → ℤ) (hI : ∀ k < c,I k < n)
    (hm : ∀ a b,a < b → b < c → I a < I b) (hz : ∀ s < n,(∀ k < c,I k≠s) → f s=0) :
    ∑ k ∈ kP c,f (I k) = ∑ s ∈ kP n,f s:=by
  have inj:Set.InjOn I (kP c):=by
    intro a ha b hb e
    simp only [Finset.coe_range,Set.mem_Iio] at ha hb
    rcases Nat.lt_trichotomy a b with h | h | h
    · exact absurd e (R8 (hm a b h hb))
    · exact h
    · exact absurd e.symm (R8 (hm b a h ha))
  rw [← Finset.sum_image (fun a ha b hb e=>inj ha hb e)]
  apply Finset.sum_subset
  · intro s hs; simp only [Finset.mem_image,ϥ] at hs ⊢
    obtain ⟨k,hk,rfl⟩:=hs; exact hI k hk
  · intro s hs hn; simp only [Finset.mem_image,ϥ,not_exists,not_and] at hs hn
    exact hz s hs (fun k hk e=>hn k hk e)
theorem kGB1T (g:Ἆ) (n c j:ℕ) (l:ℕ → Ἃ) (W b:Ἂ Ἃ) (I:ℕ → ℕ) (p p':ℕ → Ἅ) (e:Ἃ)
    (t E L F:ℕ) (B:Unit → Ἃ)
    (A : ∀ s < n,expOf (l s)≠255∧idxOf (l s) ≤ t)
    (hw : ∀ s < n,(expOf W[j*n+s]!).toNat ≤ e.toNat)
    (K:expOf b[j]! ≠ 255∧idxOf b[j]! ≤ F) (al:174+F ≤ E) (H:n+2 ≤ 2^L) (r:L ≤ 27)
    (S:E+L ≤ 425) (hc:c ≤ n) (hI : ∀ k < c,I k < n) (hm : ∀ a b,a < b → b < c → I a < I b)
    (hz : ∀ s < n,(∀ k < c,I k≠s) → tm (l s) W[j*n+s]! = 0)
    (hp : ∀ k < c,expOf W[j*n+I k]! ≠ 255 → fv (p k)∧rv (p k)=tm (l (I k)) W[j*n+I k]!)
    (hp' : ∀ s < n,expOf W[j*n+s]! ≠ 255 → fv (p' s)∧rv (p' s)=tm (l s) W[j*n+s]!)
    (hB:B ()=p8 g (ms (n+1) (fun s=>if s < n then l s else ld) (fun s=>if s < n then W[j*n+s]! else b[j]!))) :
    tG g (ao p c 0 (dw ld*dw b[j]!)) e t E L (fun _=>tS g L n p' (fun _=>dw ld*dw b[j]!) B) B =
      p8 g (ms (n+1) (fun s=>if s < n then l s else ld) (fun s=>if s < n then W[j*n+s]! else b[j]!)):=by
  unfold tG; dsimp only; split
  swap; · exact hB
  rename_i h1
  have h1':e < 255∧48+t+max (e.toNat-1) 55+L ≤ 425:=of_decide_eq_true h1
  obtain ⟨he,hpL⟩:=h1'
  let P:=max (48+t+max (e.toNat-1) 55) E
  have hPL:P+L ≤ 425:=(max_add_add_right _ _ L).symm.trans_le (max_le hpL S)
  have hq:(hG e t E L).2.toNat=P+2*L-54:=by
    unfold hG; dsimp only
    rw [Α (AO ((K4 _ _).trans ((congrArg (P + ·) (Nat.two_mul L)).trans_le ((pA P L L).symm.trans_le ((Y3 hPL r).trans (by decide)))):_ ≤ 1000) (by decide))]
  have Ew : ∀ s < n,expOf W[j*n+s]! ≠ 255:=fun s hs h=>Nat.not_le_of_lt (Ι.1 he) ((congrArg mJ h).symm.trans_le (hw s hs))
  have EB:=kHG n (fun s=>if s < n then l s else ld) (fun s=>if s < n then W[j*n+s]! else b[j]!) e t E
    (fun s hs=>(congrArg idxOf (if_pos hs)).trans_le (A s hs).2)
    (fun s hs=>(congrArg (fun w=>(expOf w).toNat) (if_pos hs)).trans_le (hw s hs))
    ((congrArg₂ (fun a b => |((tm a b:ℤ):ℝ)|) (if_neg (AX n)) (if_neg (AX n))).trans_le (kTB _ F E K.2 al))
  have EC:=kHG c (fun k=>if k < c then l (I k) else ld) (fun k=>if k < c then W[j*n+I k]! else b[j]!) e t E
    (fun k hk=>(congrArg idxOf (if_pos hk)).trans_le (A _ (hI k hk)).2)
    (fun k hk=>(congrArg (fun w=>(expOf w).toNat) (if_pos hk)).trans_le (hw _ (hI k hk)))
    ((congrArg₂ (fun a b => |((tm a b:ℤ):ℝ)|) (if_neg (AX c)) (if_neg (AX c))).trans_le (kTB _ F E K.2 al))
  have hg:aG (n+1) (fun s=>if s < n then l s else ld) (fun s=>if s < n then W[j*n+s]! else b[j]!):=fun k hk =>
    if hkn:k < n then ⟨fun H=>(A k hkn).1 ((congrArg expOf (if_pos hkn)).symm.trans H),fun H=>Ew k hkn ((congrArg expOf (if_pos hkn)).symm.trans H)⟩
    else ⟨fun H=>(by decide:expOf ld≠255) ((congrArg expOf (if_neg hkn)).symm.trans H),fun H=>K.1 ((congrArg expOf (if_neg hkn)).symm.trans H)⟩
  have R:=kE1 c (fun k=>if k < c then l (I k) else ld) (fun k=>if k < c then W[j*n+I k]! else b[j]!) p
    (dw ld*dw b[j]!) P L
    (fun k hk=>(hp k hk (Ew _ (hI k hk))).imp_right (fun H=>H.trans (congrArg Int.cast (congrArg₂ tm (if_pos hk) (if_pos hk))).symm))
    ((dJ ld b[j]! (by decide) K.1).imp_right (fun H=>H.trans (congrArg Int.cast (congrArg₂ tm (if_neg (AX c)) (if_neg (AX c)))).symm)) EC ((L1 hc 2).trans H) r hPL
  have hsum:(af (c+1) (fun k=>if k < c then l (I k) else ld) (fun k=>if k < c then W[j*n+I k]! else b[j]!):ℤ)
      = af (n+1) (fun s=>if s < n then l s else ld) (fun s=>if s < n then W[j*n+s]! else b[j]!) :=
    (Ν _ c).trans ((congrArg₂ (·+·) ((ϔ rfl fun k hk=>congrArg₂ tm (if_pos (ϥ.1 hk)) (if_pos (ϥ.1 hk))).trans ((kLS n c I (fun s=>tm (l s) W[j*n+s]!) hI hm hz).trans (ϔ rfl fun s hs=>congrArg₂ tm (if_pos (ϥ.1 hs)) (if_pos (ϥ.1 hs))).symm)) ((congrArg₂ tm (if_neg (AX c)) (if_neg (AX c))).trans (congrArg₂ tm (if_neg (AX n)) (if_neg (AX n))).symm)).trans (Ν _ n).symm)
  show tw g _ _ L n p' (fun _=>dw ld*dw b[j]!) B=_
  exact jE g n _ _ p' (fun _=>dw ld*dw b[j]!) (hG e t E L).2 P L B
    (fun s hs=>(hp' s hs (Ew s hs)).imp_right (fun H=>H.trans (congrArg Int.cast (congrArg₂ tm (if_pos hs) (if_pos hs))).symm))
    ((dJ ld b[j]! (by decide) K.1).imp_right (fun H=>H.trans (congrArg Int.cast (congrArg₂ tm (if_neg (AX n)) (if_neg (AX n)))).symm)) hg EB H r hPL hq hB _ R.1 (by rw [hq, ← hsum]; exact R.2)
theorem kAD (n:ℕ) (relu:Ἆ) (X W b:Ἂ Ἃ) (r o:ℕ) :
    aD n relu X W b (r*n) 1 o=act relu (dotOf (n+1) (fun t=>if t < n then X[r*n+t]! else oneW)
      (fun t=>if t < n then W[o*n+t]! else b[o]!)):=by
  unfold aD; rw [bO]; simp only [ϻ]
theorem kAddS (a b:ἆ) (h:a.toNat+b.toNat < 2^(63:Ἀ)):(a+b).toNat=a.toNat+b.toNat:=by
  rw [η]; exact γ (K5 h (by decide))
theorem kOfS (n:ℕ) (h:n < 2^(63:Ἀ)):n.toUInt64.toNat=n:=by
  rw [Ϣ,O6]; exact γ (K5 h (by decide))
theorem kE64 (e:Ἃ):e.toUInt64.toNat=e.toNat:=ϩ e
lemma s5gS(x y:Ἅ)(d:fv x)(e:fv y):
  kI (x-y).toModel=Model.sub (kI x.toModel) (kI y.toModel):=by
 change kI (Float.Model.sub x.toModel y.toModel)=_;rw[Float.Model.sub,nu]
 have h:=Model.sub_eq_ofModel_sub_of_isFinite (fmt:=kC) (by decide) (kI x.toModel) (kI y.toModel) (by rw[ϒ];exact(fO x).1 d) (by rw[ϒ];exact(fO y).1 e)
 rw[ϒ,ϒ]at h;exact h.symm
lemma s5iS(x y:Ἅ)(d:fv x)(e:fv y)(j:|rv x-rv y|<=ls (mR kC)):
  fv (x-y)/\rv (x-y)=ns kC (rv x-rv y):=by
 have h:=s5gS x y d e
 have g:=Model.isFinite_sub_of_abs_toReal_sub_le_posMaxFinite (kI x.toModel) (kI y.toModel) (by decide) d e j
 exact⟨by rwa[fv,h],by rw[rv,h];exact Model.toReal_sub_eq_roundAt _ _ (by decide) g⟩
lemma s5ofb(a:ἆ)(hv:¬(a>>>52)&&&2047=2047):kI (N5 a).toModel=C8 (fmt:=kC) a.toBitVec:=by
 have e1:kI (N5 a).toModel=Model.canonicalizeModel (C8 (fmt:=kC) a.toBitVec):=by
  show BB (Float.Model.ofBits a)=_
  unfold Float.Model.ofBits;exact Model.ofFloatModel_pack _
 rw[e1];apply Model.canonicalizeModel_eq_self
 refine ⟨fun hE _=>absurd hE ?_⟩
 intro hE;apply hv
 have:=congrArg BitVec.toNat hE
 have k1:FloatFormat.binary64.toModel.exponentBits=11:=rfl
 have k2:FloatFormat.binary64.toModel.mantissaBitsWithoutImplicit=52:=rfl
 simp [nX,Model.toModelBits,C8,k1,k2] at this
 change (BitVec.extractLsb' 52 11 (a.toBitVec:BitVec 64)).toNat=2047 at this
 rw[BitVec.extractLsb'_toNat] at this
 apply ξ.mp
 simp only[Ε,ν,kA]
 rw[show (2047:Nat)=2^(11:Ἀ)-1 from rfl,ζ]
 simp only[nR,β] at this ⊢
 have:=a.toNat_lt;omega
lemma s5fld(a:ἆ):C8 (fmt:=kC) a.toBitVec=Model.ofFields kC (a>>>63!=0) ((lq ((a>>>52)&&&2047))) ((lq (a&&&0xfffffffffffff))):=by
 have d:Model.expField (C8 (fmt:=kC) a.toBitVec)=(lq ((a>>>52)&&&2047)):=ἅ
 have e:Model.fracField (C8 (fmt:=kC) a.toBitVec)=(lq (a&&&0xfffffffffffff)):=ἅ
 have g:Model.signBit (C8 (fmt:=kC) a.toBitVec)=(a>>>63!=0):=by rw[<-jT];apply ϓ.mpr;simp only[Model.signBit,AM,Ak];rw[<-UInt64.toBitVec_inj];exact Iff.rfl
 rw[<-Model.ofFields_signBit_expField_fracField (C8 (fmt:=kC) a.toBitVec),d,e,g]
lemma s5pw(m:ℕ)(z:Ἁ):((2^m:ℕ):ℝ)*(2:ℝ)^(z-m)=(2:ℝ)^z:=by
 rw[I4,Nat.cast_two,<-AA,<-AK AL];congr 1;ring
lemma s5and(w:ἇ):(mJ (w&&&0x80000000))=(if (mJ w)/2^(31:Ἀ)%2=1 then 2^(31:Ἀ) else 0):=by
 rw[Ξ];simp only[kN];rw[show (2147483648:ℕ)=2^(31:Ἀ) from rfl,Nat.and_two_pow,lW]
 by_cases c:(mJ w)/2147483648%2=1<;>simp[c]
theorem kReluW(relu:Ἆ)(w:ἇ)(hw:expOf w≠255):
  w&&&((if relu then (0:ἇ) else (0xFFFFFFFF:ἇ))|||((w>>>31)-1))=act relu w:=by
 have l:=w.toNat_lt
 have b:(mJ (w>>>31))=(mJ w)/2^(31:Ἀ):=by rw[Υ];simp[β]
 have fa:w&&&(4294967295:ἇ)=w:=by
  apply Β.mp;rw[Ξ];simp only[kN]
  rw[show (4294967295:ℕ)=2^(32:Ἀ)-1 from rfl,ζ];exact γ l
 have s1:=s5and w
 unfold act reluW
 rcases Z7 (mJ w) (2^(31:Ἀ)) with c|c
 · have h0:w>>>31=0:=Β.mp (by rw[b,nj];exact ώ c)
   have hz:(w&&&0x80000000)=0:=Β.mp (by rw[s1,if_neg (by (clear * - l c;omega)),nj])
   cases relu<;>simp[h0,hz,fa]
 · have h1:w>>>31=1:=Β.mp (by rw[b,UInt32.toNat_one];exact mG c l)
   have hn:(w&&&0x80000000)≠0:=fun q=>by have:=congrArg mJ q;rw[s1,if_pos (by (clear * - l c;omega))]at this;simp at this
   cases relu<;>simp[h1,hn,hw,fa]
lemma zaa(n:Ἀ)(h:n<2^(32:Ἀ)):(J0 (G0 n))=n:=Dense.aa (AN h ς)
lemma jV(a:Ἂ ἇ):(J0 a.usize)<=(O0 a):=by exact Ay _ _
lemma uL{a:Ἂ ἇ}{i:Ἄ}(h:i<a.usize):(J0 i)<(O0 a):=by have:=jV a;rw[ε]at h;exact cD h
lemma kr(A:Ἂ kF)(u:Ἄ)(h:(J0 u)<(O0 A)):A.uget u h=A[(J0 u)]!:=by simp[Array.uget,AB A (J0 u) h]
def ZPq:Ἂ ἀ:=Aj (n:=10) λk=>⟨Ah (2^((Fin.val k)+7)) (Dense.F.dw 0)⟩
def zPq(n:Ἀ):ἀ:=
 let k:=Aq n+1-7
 if k<10 then ZPq[k]!else⟨Ah n (Dense.F.dw 0)⟩
lemma uU{Y:ἀ}{i:Ἄ}(g:i<Y.usize):(J0 i)<(kS Y):=by have h:=ε.mp g;have:(J0 Y.usize)<=(kS Y):=Ay _ _;exact U1 g this
lemma fuS(Y:ἀ)(i:Ἄ)(v:Ἅ)(h:(J0 i)<(kS Y))(q:Ἀ)(hq:q<(kS Y)):(Y.uset i v h).data[q]! =if (J0 i)=q then v else Y.data[q]!:=by cases Y;simp only[kS]at h hq;simp only[FloatArray.uset,Array.uset];simp[Array.getElem_set,hq]
lemma uZ(Y:ἀ)(i:Ἄ)(v:Ἅ)(h:(J0 i)<(kS Y)):(kS (Y.uset i v h))=(kS Y):=by cases Y;simp only[FloatArray.uset,kS,Array.uset];apply Array.size_set
lemma zPs_h1(N q:Ἀ)(h:q<N):(mB N (Dense.F.dw 0))[q]! =Dense.F.dw 0:=by
 have H:q<(mB N (Dense.F.dw 0)).size:=by rwa[O3]
 exact (AB _ q H).trans (Array.getElem_replicate H)
lemma zPs_h2(n:Ἀ):n<=(kS ⟨mB n (Dense.F.dw 0)⟩)/\((kS ⟨mB n (Dense.F.dw 0)⟩)<=2^(16:Ἀ)\/(kS ⟨mB n (Dense.F.dw 0)⟩)=n)/\∀q<(kS ⟨mB n (Dense.F.dw 0)⟩),(⟨mB n (Dense.F.dw 0)⟩:ἀ).data[q]! =Dense.F.dw 0:=
 ⟨O3.ge,.inr O3,λq h=>zPs_h1 n q (h.trans_eq O3)⟩
lemma zPs(n:Ἀ):n<=(kS (zPq n))/\((kS (zPq n))<=2^(16:Ἀ)\/(kS (zPq n))=n)/\∀q<(kS (zPq n)),(zPq n).data[q]! =Dense.F.dw 0:=by
 dsimp only[zPq];split;rw[AB ZPq _ (by rwa[ZPq,Ϝ])];simp only[ZPq,Λ,kS,O3];refine⟨(ϙ (n:=n)).le.trans (Θ (by decide) (by exact le_tsub_add)),.inl (Θ (by decide) (by omega)),zPs_h1 _⟩;exact zPs_h2 n
def wdU(K:Ἂ Ἃ)(i n:Ἄ)(Y:ἀ):ἀ:=
 if h:i<n then wdU K (i+1) n (if g:i<Y.usize then if k:i<K.usize then Y.uset i (Dense.F.dw (K.uget i (uL k))) (uU g) else Y else Y)else Y
termination_by (J0 n)-(J0 i)
decreasing_by have:=ε.mp h;rw[Dense.ac h];exact Z5 nб iб h
lemma wdUz(K:Ἂ kF)(n:Ἄ):∀m (i:Ἄ)(Y:ἀ),(J0 n)-(J0 i)=m->(kS (wdU K i n Y))=(kS Y):=by
 intro m;induction m with
 |zero=>intros;rw[wdU,dif_neg (λh=>by have:=ε.mp h;omega)]
 |succ m r=>intro i Y e;have h:i<n:=ε.mpr (by exact N1 e);rw[wdU,dif_pos h,r _ _ (by rw[Dense.ac h];(clear * - e;omega))];split_ifs<;>simp only[uZ]
lemma wdUs(K:Ἂ kF)(n:Ἄ)(hK:(O0 K)<2^(32:Ἀ)):∀m (i:Ἄ)(Y:ἀ),(J0 n)-(J0 i)=m->(J0 n)<=(kS Y)->(kS Y)<2^(32:Ἀ)->(∀q<(kS Y),Y.data[q]! =if q< (J0 i) then Dense.F.dw K[q]! else Dense.F.dw 0)->
  ∀q<(J0 n),(wdU K i n Y).data[q]! =Dense.F.dw K[q]!:=by
 intro m;induction m with
 |zero=>
  intro i Y e a b c q hq;rw[wdU,dif_neg (λh=>Z8 (ε.1 h) e),c q (AN hq a),if_pos (AN hq (O4 e))]
 |succ m ih=>
  intro i Y e a b c;have h:i<n:=ε.2 (N1 e);have hi:=ε.1 h;have:=ς;have kz:(J0 K.usize)=(O0 K):=zaa _ hK;have g:i<Y.usize:=ε.2 (by simp only[FloatArray.usize];rw[lX,J3 (AN b this)];exact AN hi a);have hE:(J0 n)-(J0 (i+1))=m:=by rw[Dense.ac h,<-F7,e,BK]
  rw[wdU,dif_pos h,dif_pos g];split
  .rename_i k;have sz:=uZ Y i (Dense.F.dw (K.uget i (uL k))) (uU g)
   refine ih (i+1) _ hE (sz▸a) (sz▸b) λq hq=>?_
   rw[sz]at hq;rw[fuS _ _ _ _ _ hq,Dense.ac h,c q hq];by_cases q1:(J0 i)=q
   .subst q1;rw[if_pos ἅ,if_pos (by exact lt_add_one iб),kr]
   .rw[if_neg q1];by_cases q2:q< (J0 i)
    .rw[if_pos q2,if_pos (K6 q2)]
    .rw[if_neg q2,if_neg (by (clear * - g k q1 q2;omega))]
  .rename_i k;refine ih (i+1) Y hE a b λq hq=>?_
   rw[c q hq,Dense.ac h];by_cases q2:q< (J0 i)
   .rw[if_pos q2,if_pos (K6 q2)]
   .by_cases q3:q=(J0 i)
    .subst q3;rw[if_neg q2,if_pos (by exact lt_add_one iб),An K _ (λhh=>k (ε.2 (kz▸hh)))];rfl
    .rw[if_neg q2,if_neg (by (clear * - q2 q3;omega))]
theorem Lk2(t n S:ℕ)(h:t < n)(hX:8*n≤S):8*t+7 < S:=by omega
theorem Lkd(R r:ℕ)(h:r < R):R-(r+1) < R-r:=C5 h (I3 r)
theorem k7d1:1≤7:=N2 rfl
theorem k7d2:2≤7:=N2 rfl
theorem k7d3:3≤7:=N2 rfl
theorem k7d4:4≤7:=N2 rfl
theorem k7d5:5≤7:=N2 rfl
theorem k7d6:6≤7:=N2 rfl
theorem k7d7:7≤7:=N2 rfl
def kX8(Wd XT:ἀ)(wo n:Ἄ)(hW:(J0 wo)+(J0 n)<=(kS Wd))(hX:8*(J0 n)<=(kS XT))(h31:(J0 wo)+(J0 n)<2^(31:Ἀ)∧8*(J0 n)<2^(31:Ἀ))(t:Ἄ)(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ):A8:=
 if h:t<n then
  have ht:=ε.mp h
  have h8:(J0 (8*t))=8*(J0 t):=by rw[kMm _ _ (by rw[kc8];exact AO (U3 8 (X4 ht)) h31.2),kc8]
  let w:=Wd.uget (wo+t) (by rw[kSm _ _ (AO (J1 (X4 ht) _) h31.1)];exact AN (O1 ht _) hW)
  let q:=8*t
  have q7:(J0 q)+7<(kS XT):=by show (J0 (8*t))+7<_;rw[h8];exact Lk2 _ _ _ ht hX
  have q2:(J0 q)+7<2^(31:Ἀ):=by show (J0 (8*t))+7<_;rw[h8];exact Lk2 _ _ _ ht (X4 h31.2)
  kX8 Wd XT wo n hW hX h31 (t+1) (a0+XT.uget q (AO (V1 _ 7) q7)*w) (a1+XT.uget (q+1) (kIx7 q 1 1 kc1 k7d1 XT q7 q2)*w) (a2+XT.uget (q+2) (kIx7 q 2 2 kc2 k7d2 XT q7 q2)*w)
   (a3+XT.uget (q+3) (kIx7 q 3 3 kc3 k7d3 XT q7 q2)*w) (a4+XT.uget (q+4) (kIx7 q 4 4 kc4 k7d4 XT q7 q2)*w) (a5+XT.uget (q+5) (kIx7 q 5 5 kc5 k7d5 XT q7 q2)*w)
   (a6+XT.uget (q+6) (kIx7 q 6 6 kc6 k7d6 XT q7 q2)*w) (a7+XT.uget (q+7) (kIx7 q 7 7 kc7 k7d7 XT q7 q2)*w)
 else ⟨a0,a1,a2,a3,a4,a5,a6,a7⟩
termination_by (J0 n)-(J0 t)
decreasing_by all_goals (rw[ac h];exact Lkd _ _ (ε.mp h))
theorem kX8_spec_h3(XT:ἀ)(q k:Ἄ)(c:Ἀ)(hk:(J0 k)=c)(q2:(J0 q)+7<2^(31:Ἀ))(hc:c<=7)(h:(J0 (q+k))<(kS XT)):XT.uget (q+k) h=XT.data[(J0 q)+c]!:=by
 rw [kFU,kSm _ _ (by rw [hk]; exact AO (J1 hc _) q2),hk]
theorem kX8_spec_h1(a0 a1 a2 a3 a4 a5 a6 a7 w:Ἅ)(XT:ἀ)(q:Ἄ)(q2:(J0 q)+7<2^(31:Ἀ))(h0:(J0 q)<(kS XT))(h1:(J0 (q+1))<(kS XT)) (h2:(J0 (q+2))<(kS XT)) (h3:(J0 (q+3))<(kS XT)) (h4:(J0 (q+4))<(kS XT)) (h5:(J0 (q+5))<(kS XT)) (h6:(J0 (q+6))<(kS XT)) (h7:(J0 (q+7))<(kS XT))(l:Ἀ)(hl:l<8):
  A8.f ⟨a0+XT.uget q h0*w,a1+XT.uget (q+1) h1*w,a2+XT.uget (q+2) h2*w,a3+XT.uget (q+3) h3*w,a4+XT.uget (q+4) h4*w,a5+XT.uget (q+5) h5*w,a6+XT.uget (q+6) h6*w,a7+XT.uget (q+7) h7*w⟩ l=A8.f ⟨a0,a1,a2,a3,a4,a5,a6,a7⟩ l+XT.data[(J0 q)+l]!*w:=by
 interval_cases l
 exact congrArg (HAdd.hAdd _) (congrArg (· * w) (kFU _ _ _))
 all_goals exact congrArg (HAdd.hAdd _) (congrArg (· * w) (kX8_spec_h3 _ _ _ _ (by first | exact kc1 | exact kc2 | exact kc3 | exact kc4 | exact kc5 | exact kc6 | exact kc7) q2 (by decide) _))
theorem kX8_spec_h2(z y:A8)(f:Ἀ->Ἅ)(c k k1 l:Ἀ)(u v:Ἅ)(Ez:z=y)(I:y.f l=ao f c k1 u)(hk1:k1=k+1)(hu:u=v+f k)(ht:k<c):z.f l=ao f c k v:=by
 subst Ez hk1; rw [I,hu,aos f c k v ht]
theorem kX8_spec(Wd XT:ἀ)(wo n:Ἄ)(hW:(J0 wo)+(J0 n)<=(kS Wd))(hX:8*(J0 n)<=(kS XT))(h31:(J0 wo)+(J0 n)<2^(31:Ἀ)∧8*(J0 n)<2^(31:Ἀ)) :
  ∀ m (t:Ἄ)(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ),(J0 n)-(J0 t)=m -> ∀ l,l<8 ->
   (kX8 Wd XT wo n hW hX h31 t a0 a1 a2 a3 a4 a5 a6 a7).f l =
    ao (fun s=>XT.data[8*s+l]!*Wd.data[(J0 wo)+s]!) (J0 n) (J0 t) (A8.f ⟨a0,a1,a2,a3,a4,a5,a6,a7⟩ l):=by
 intro m; induction m with
 | zero =>
   intro t a0 a1 a2 a3 a4 a5 a6 a7 H l hl
   have z:=O4 H;rw[(kX8.eq_1 Wd XT wo n hW hX h31 t a0 a1 a2 a3 a4 a5 a6 a7).trans (dif_neg (by rw[ε];exact W3.2 z)),aoe _ _ _ _ z]
 | succ m IH =>
   intro t a0 a1 a2 a3 a4 a5 a6 a7 H l hl
   have ht:(J0 t) < (J0 n):=N1 H
   have h:t < n:=ε.mpr ht
   have h8:(J0 (8*t))=8*(J0 t):=by rw[kMm _ _ (by rw[kc8];exact AO (U3 8 (X4 ht)) h31.2),kc8]
   have hw:(J0 (wo+t))=(J0 wo)+(J0 t):=kSm _ _ (AO (J1 (X4 ht) _) h31.1)
   exact kX8_spec_h2 _ _ _ _ _ _ _ _ _ ((kX8.eq_1 Wd XT wo n hW hX h31 t a0 a1 a2 a3 a4 a5 a6 a7).trans (dif_pos h)) (IH (t+1) _ _ _ _ _ _ _ _ (by rw[ac h,V4,H,BK]) l hl) (ac h)
     ((kX8_spec_h1 _ _ _ _ _ _ _ _ _ _ _ (by rw[h8];exact AO ((J1 ((by decide):7<=8) _).trans (U3 8 ht)) h31.2) _ _ _ _ _ _ _ _ l hl).trans (by rw [kFU,hw,h8])) ht
@[inline] def cEq(F a:Ἅ):ἆ:=let e:=eF (F*a).toBits;if (1078:ἆ)<e then e-(1078:ἆ) else (0:ἆ)
@[noinline] def zq2(v:Ἅ)(e:ἆ)(relu:Ἆ)(cq hb:ἆ):ἇ:=
 let p:ἆ:=(if e<(56:ἆ) then (56:ἆ) else e)+cq
 if e<(255:ἆ)&&p<=(372:ἆ) then
  let c:=ak v (if hb>p then hb else p)
  if c<(4294967296:ἆ) then (let w:=N3 c; w&&&((if relu then (0:ἇ) else (0xFFFFFFFF:ἇ))|||((w>>>31)-1))) else (0xFFFFFFFF:ἇ)
 else (0xFFFFFFFF:ἇ)
@[inline] def mx2(a c:ἆ):ἆ:=if a<c then c else a
theorem kDM (a c w j k:ℕ) (h:a*w+j=c*w+k) (hj:j<w) (hk:k<w):a=c∧j=k:=by
  have h1:(a*w+j)/w=a:=by rw [Τ,Ρ _ _ (by exact L7 hj),ώ hj,Κ]
  have h2:(c*w+k)/w=c:=by rw [Τ,Ρ _ _ (by exact L7 hj),ώ hk,Κ]
  have e:a=c:=by rw [← h1, ← h2,h]
  subst e; exact ⟨rfl,by exact J8 h⟩
theorem kBnd (ow n R:ℕ) (h1:(ow+16)*n < 2^(31:Ἀ)) (h2:R*n < 2^(31:Ἀ)) (h3:0 < n) :
    8*n < 2^(31:Ἀ)∧ow*n < 2^(31:Ἀ)∧ow < 2^(31:Ἀ)∧R < 2^(31:Ἀ):=by
  have a:=ι n (show 16 ≤ ow+16 by exact K2 16 ow)
  have b:=ι n (show ow ≤ ow+16 by exact V1 ow 16)
  have c:=J4 ow h3
  have d:=J4 R h3
  (clear * - h1 h2 a b c d;omega)
theorem kUb(a t n:Ἄ)(S:Ἀ)(h:(J0 t)<(J0 n))(hb:(J0 a)+(J0 n)<=S∧(J0 a)+(J0 n)<2^(31:Ἀ)):(J0 (a+t))<S∧(J0 (a+t))=(J0 a)+(J0 t):=by
 have l:=O1 h (J0 a)
 rw[kSm a t (l.trans hb.2)];exact⟨l.trans_le hb.1,rfl⟩
theorem kDc(t n:Ἄ)(h:t<n):(J0 n)-(J0 (t+1))<(J0 n)-(J0 t):=by rw[ac h];exact C5 (ε.mp h) (I3 _)
theorem kSt(t n:Ἄ)(k:Ἀ)(hk:(J0 n)-(J0 t)=k+1):t<n∧(J0 n)-(J0 (t+1))=k:=by
 have h1:(J0 t)<(J0 n):=N1 hk
 exact⟨ε.mpr h1,by rw[ac (ε.mpr h1),V4,hk,BK]⟩
theorem kS0(t n:Ἄ)(hk:(J0 n)-(J0 t)=0):¬t<n∧¬(J0 t)<(J0 n):=
 ⟨fun h=>Z8 (ε.mp h) hk,fun h=>Z8 h hk⟩
def sTb(X:Ἂ Ἃ)(Wd:ἀ)(xo wo n:Ἄ)(hX:(J0 xo)+(J0 n)<=(O0 X)∧(J0 xo)+(J0 n)<2^(31:Ἀ))
  (hW:(J0 wo)+(J0 n)<=(kS Wd)∧(J0 wo)+(J0 n)<2^(31:Ἀ))(t:Ἄ)(a:Ἅ)(m:ἆ):Ἅ×ἆ:=
 if h:t<n then
  let a':=a+dw (X.uget (xo+t) (kUb xo t n _ (ε.mp h) hX).1)*Wd.uget (wo+t) (kUb wo t n _ (ε.mp h) hW).1
  let e:=eF a'.toBits
  sTb X Wd xo wo n hX hW (t+1) a' (if m<e then e else m)
 else (a,m)
termination_by (J0 n)-(J0 t)
decreasing_by exact kDc t n h
theorem sTb_eq(X:Ἂ Ἃ)(Wd:ἀ)(xo wo n:Ἄ)(hX:(J0 xo)+(J0 n)<=(O0 X)∧(J0 xo)+(J0 n)<2^(31:Ἀ))
  (hW:(J0 wo)+(J0 n)<=(kS Wd)∧(J0 wo)+(J0 n)<2^(31:Ἀ)):
  ∀k (t:Ἄ)(a:Ἅ)(m:ἆ),(J0 n)-(J0 t)=k->
  sTb X Wd xo wo n hX hW t a m=sT (fun s=>dw X[(J0 xo)+s]!*Wd.get! ((J0 wo)+s)) (J0 n) (J0 t) a m:=by
 intro k;induction k with
 |zero=>intro t a m hk;rw[sTb,dif_neg (kS0 t n hk).1,sT,if_neg (kS0 t n hk).2]
 |succ k ih=>
  intro t a m hk;obtain⟨h1,h2⟩:=kSt t n k hk
  have q1:=(kUb xo t n _ (ε.mp h1) hX).2
  have q2:=(kUb wo t n _ (ε.mp h1) hW).2
  have u1:∀hh,X.uget (xo+t) hh=X[(J0 xo)+(J0 t)]! :=fun hh=>by rw[AB X _ (q1▸hh)];simp only[Array.uget,q1]
  have u2:∀hh,Wd.uget (wo+t) hh=Wd.get! ((J0 wo)+(J0 t)):=fun hh=>by rw[ug,gg,q2]
  rw[sTb,dif_pos h1,sT,if_pos (ε.mp h1)];dsimp only;rw[u1,u2,ih (t+1) _ _ h2,ac h1]
@[noinline]def tR(relu:Ἆ)(L:Ἀ)(r:Ἅ×ἆ)(aB:Unit->Ἃ):Ἃ:=
 let c:=if 1<=(lq r.2)/\(lq r.2)<=1993/\1076<=(lq r.2)+L then
  ak r.1 (I0 ((lq r.2)+L-1076))else 4294967296
 if c<4294967296 then act relu (N3 c)else aB ()
theorem sTB_h(a n:Ἀ)(h:a+n<2^(31:Ἀ)):(J0 (G0 a))=a∧(J0 (G0 n))=n:=
 ⟨J3 (AN (AN ((V1 a n).trans_lt h) (by decide)) ς),J3 (AN (AN ((K2 n a).trans_lt h) (by decide)) ς)⟩
@[noinline]def sTB(X:Ἂ Ἃ)(Wd:ἀ)(n:Ἀ)(row j:Ἄ)(a0:Ἅ):Ἅ×ἆ:=
 if h:(J0 row)*n+n<=(O0 X)∧(J0 row)*n+n<2^(31:Ἀ)∧(J0 j)*n+n<=(kS Wd)∧(J0 j)*n+n<2^(31:Ἀ) then
  sTb X Wd (G0 ((J0 row)*n)) (G0 ((J0 j)*n)) (G0 n)
   (by rw[(sTB_h _ _ h.2.1).1,(sTB_h _ _ h.2.1).2];exact⟨h.1,h.2.1⟩) (by rw[(sTB_h _ _ h.2.2.2).1,(sTB_h _ _ h.2.2.2).2];exact⟨h.2.2.1,h.2.2.2⟩) 0 a0 0
 else sT (fun s=>dw X[(J0 row)*n+s]!*Wd.get! ((J0 j)*n+s)) n 0 a0 0
theorem sTB_eq(X:Ἂ Ἃ)(Wd:ἀ)(n:Ἀ)(row j:Ἄ)(a0:Ἅ):sTB X Wd n row j a0=sT (fun s=>dw X[(J0 row)*n+s]!*Wd.get! ((J0 j)*n+s)) n 0 a0 0:=by
 unfold sTB;split
 · rename_i h;rw[sTb_eq _ _ _ _ _ _ _ _ 0 a0 0 rfl,(sTB_h _ _ h.2.1).1,(sTB_h _ _ h.2.1).2,(sTB_h _ _ h.2.2.2).1,Ϧ]
 · rfl
theorem kQb(xb t n Qu:Ἀ)(h:t<n)(hx:xb+8*n<=Qu+7):xb+8*t<Qu:=by omega
theorem kQ8(q Qu:Ἀ)(h:q<Qu)(h2:Qu+8<2^(31:Ἀ)):q+(J0 (8:Ἄ))<2^(31:Ἀ):=by simp only[AH];omega
def sTq(Y Q:ἀ)(vo n:Ἄ)(hY:(J0 vo)+(J0 n)<=(kS Y)∧(J0 vo)+(J0 n)<2^(31:Ἀ))(Qu:Ἄ)(hQ:(kS Q)=(J0 Qu)∧(J0 Qu)+8<2^(31:Ἀ))(t q:Ἄ)(a:Ἅ)(m:ἆ):Ἅ×ἆ:=
 if h:t<n∧q<Qu then
  let a':=a+Y.uget (vo+t) (kUb vo t n _ (ε.mp h.1) hY).1*Q.uget q ((ε.mp h.2).trans_eq hQ.1.symm)
  let e:=eF a'.toBits
  sTq Y Q vo n hY Qu hQ (t+1) (q+8) a' (if m<e then e else m)
 else (a,m)
termination_by (J0 n)-(J0 t)
decreasing_by exact kDc t n h.1
theorem sTq_eq(Y Q:ἀ)(vo n:Ἄ)(hY:(J0 vo)+(J0 n)<=(kS Y)∧(J0 vo)+(J0 n)<2^(31:Ἀ))(Qu:Ἄ)(hQ:(kS Q)=(J0 Qu)∧(J0 Qu)+8<2^(31:Ἀ))(xb:Ἀ)
  (hx:xb+8*(J0 n)<=(J0 Qu)+7):
  ∀k (t q:Ἄ)(a:Ἅ)(m:ἆ),(J0 n)-(J0 t)=k->(J0 q)=xb+8*(J0 t)->
  sTq Y Q vo n hY Qu hQ t q a m=sT (fun s=>Y.get! ((J0 vo)+s)*Q.get! (xb+8*s)) (J0 n) (J0 t) a m:=by
 intro k;induction k with
 |zero=>intro t q a m hk hq;rw[sTq,dif_neg (fun h=>(kS0 t n hk).1 h.1),sT,if_neg (kS0 t n hk).2]
 |succ k ih=>
  intro t q a m hk hq;obtain⟨h1,h2⟩:=kSt t n k hk
  have h3:q<Qu:=ε.mpr (hq▸kQb _ _ _ _ (ε.mp h1) hx)
  have q1:=(kUb vo t n _ (ε.mp h1) hY).2
  have q8:(J0 (q+8))=(J0 q)+8:=(kSm q 8 (kQ8 _ _ (ε.mp h3) hQ.2)).trans (by simp)
  have u1:∀hh,Y.uget (vo+t) hh=Y.get! ((J0 vo)+(J0 t)):=fun hh=>by rw[ug,gg,q1]
  have u2:∀hh,Q.uget q hh=Q.get! (xb+8*(J0 t)):=fun hh=>by rw[ug,gg,hq]
  rw[sTq,dif_pos ⟨h1,h3⟩,sT,if_pos (ε.mp h1)];dsimp only;rw[u1,u2,ih (t+1) (q+8) _ _ h2 (by rw[q8,ac h1,hq,Nat.mul_succ,pA]),ac h1]
@[noinline]def sTQ(Y Q:ἀ)(n vo xb:Ἀ)(a0:Ἅ):Ἅ×ἆ:=
 if h:vo+n<=(kS Y)∧vo+n<2^(31:Ἀ)∧xb+8*n<=(kS Q)+7∧(kS Q)+8<2^(31:Ἀ)∧xb<2^(31:Ἀ) then
  sTq Y Q (G0 vo) (G0 n) (by rw[(sTB_h _ _ h.2.1).1,(sTB_h _ _ h.2.1).2];exact⟨h.1,h.2.1⟩) (G0 (kS Q)) (by rw[(sTB_h _ _ h.2.2.2.1).1];exact⟨rfl,h.2.2.2.1⟩) 0 (G0 xb) a0 0
 else sT (fun s=>Y.get! (vo+s)*Q.get! (xb+8*s)) n 0 a0 0
theorem sTQ_eq(Y Q:ἀ)(n vo xb:Ἀ)(a0:Ἅ):sTQ Y Q n vo xb a0=sT (fun s=>Y.get! (vo+s)*Q.get! (xb+8*s)) n 0 a0 0:=by
 unfold sTQ;split
 · rename_i h;have e:=sTB_h _ _ h.2.1
   rw[sTq_eq _ _ _ _ _ _ _ xb (by rw[e.2,(sTB_h _ _ h.2.2.2.1).1];exact h.2.2.1) _ 0 _ a0 0 rfl (by rw[Ϧ,J3 (AN (AN h.2.2.2.2 (by decide)) ς)];rfl),e.1,e.2,Ϧ]
 · rfl
theorem tR_tS(relu:Ἆ)(L n:Ἀ)(p:Ἀ->Ἅ)(a0:Ἅ)(aB:Unit->Ἃ):tR relu L (sT p n 0 a0 0) aB=tS relu L n p (fun _=>a0) aB:=rfl
noncomputable def bXfb(relu:Ἆ)(X W b E:Ἂ Ἃ)(Wd:ἀ)(ix ib L N:Ἀ):Ἄ->Ἄ->Ἅ->Ἃ:=fun (row j:Ἄ)(v:Ἅ)=>tG relu v E[(J0 j)]! ix ib L
  (λ_=>tR relu L (sTB X Wd N row j (dw oneW*dw b[(J0 j)]!)) λ_=>aD N relu X W b ((J0 row)*N) 1 (J0 j))
  (λ_=>aD N relu X W b ((J0 row)*N) 1 (J0 j))
def kX16(Wd XT:ἀ)(wo wp n:Ἄ)(hW:(J0 wo)+(J0 n)<=(kS Wd))(hP:(J0 wp)+(J0 n)<=(kS Wd))(hX:8*(J0 n)<=(kS XT))(h31:(J0 wo)+(J0 n)<2^(31:Ἀ)∧(J0 wp)+(J0 n)<2^(31:Ἀ)∧8*(J0 n)<2^(31:Ἀ))(t:Ἄ)(a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7:Ἅ):A16:=
 if h:t<n then
  have ht:=ε.mp h
  have h8:(J0 (8*t))=8*(J0 t):=by rw[kMm _ _ (by rw[kc8];exact AO (U3 8 (X4 ht)) h31.2.2),kc8]
  let w:=Wd.uget (wo+t) (by rw[kSm _ _ (AO (J1 (X4 ht) _) h31.1)];exact AN (O1 ht _) hW)
  let z:=Wd.uget (wp+t) (by rw[kSm _ _ (AO (J1 (X4 ht) _) h31.2.1)];exact AN (O1 ht _) hP)
  let q:=8*t
  have q7:(J0 q)+7<(kS XT):=by show (J0 (8*t))+7<_;rw[h8];exact Lk2 _ _ _ ht hX
  have q2:(J0 q)+7<2^(31:Ἀ):=by show (J0 (8*t))+7<_;rw[h8];exact Lk2 _ _ _ ht (X4 h31.2.2)
  kX16 Wd XT wo wp n hW hP hX h31 (t+1) (a0+XT.uget q (AO (V1 _ 7) q7)*w) (a1+XT.uget (q+1) (kIx7 q 1 1 kc1 k7d1 XT q7 q2)*w) (a2+XT.uget (q+2) (kIx7 q 2 2 kc2 k7d2 XT q7 q2)*w) (a3+XT.uget (q+3) (kIx7 q 3 3 kc3 k7d3 XT q7 q2)*w) (a4+XT.uget (q+4) (kIx7 q 4 4 kc4 k7d4 XT q7 q2)*w) (a5+XT.uget (q+5) (kIx7 q 5 5 kc5 k7d5 XT q7 q2)*w) (a6+XT.uget (q+6) (kIx7 q 6 6 kc6 k7d6 XT q7 q2)*w) (a7+XT.uget (q+7) (kIx7 q 7 7 kc7 k7d7 XT q7 q2)*w)
   (c0+XT.uget q (AO (V1 _ 7) q7)*z) (c1+XT.uget (q+1) (kIx7 q 1 1 kc1 k7d1 XT q7 q2)*z) (c2+XT.uget (q+2) (kIx7 q 2 2 kc2 k7d2 XT q7 q2)*z) (c3+XT.uget (q+3) (kIx7 q 3 3 kc3 k7d3 XT q7 q2)*z) (c4+XT.uget (q+4) (kIx7 q 4 4 kc4 k7d4 XT q7 q2)*z) (c5+XT.uget (q+5) (kIx7 q 5 5 kc5 k7d5 XT q7 q2)*z) (c6+XT.uget (q+6) (kIx7 q 6 6 kc6 k7d6 XT q7 q2)*z) (c7+XT.uget (q+7) (kIx7 q 7 7 kc7 k7d7 XT q7 q2)*z)
 else ⟨a0,a1,a2,a3,a4,a5,a6,a7,c0,c1,c2,c3,c4,c5,c6,c7⟩
termination_by (J0 n)-(J0 t)
decreasing_by all_goals (rw[ac h];exact Lkd _ _ (ε.mp h))
theorem kX16_eq(Wd XT:ἀ)(wo wp n:Ἄ)(hW:(J0 wo)+(J0 n)<=(kS Wd))(hP:(J0 wp)+(J0 n)<=(kS Wd))(hX:8*(J0 n)<=(kS XT))(h31:(J0 wo)+(J0 n)<2^(31:Ἀ)∧(J0 wp)+(J0 n)<2^(31:Ἀ)∧8*(J0 n)<2^(31:Ἀ)) :
  ∀ m (t:Ἄ)(a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7:Ἅ),(J0 n)-(J0 t)=m ->
   (⟨(kX16 Wd XT wo wp n hW hP hX h31 t a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7).a0,(kX16 Wd XT wo wp n hW hP hX h31 t a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7).a1,(kX16 Wd XT wo wp n hW hP hX h31 t a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7).a2,(kX16 Wd XT wo wp n hW hP hX h31 t a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7).a3,(kX16 Wd XT wo wp n hW hP hX h31 t a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7).a4,(kX16 Wd XT wo wp n hW hP hX h31 t a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7).a5,(kX16 Wd XT wo wp n hW hP hX h31 t a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7).a6,(kX16 Wd XT wo wp n hW hP hX h31 t a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7).a7⟩:A8)=kX8 Wd XT wo n hW hX ⟨h31.1,h31.2.2⟩ t a0 a1 a2 a3 a4 a5 a6 a7 ∧
   (⟨(kX16 Wd XT wo wp n hW hP hX h31 t a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7).b0,(kX16 Wd XT wo wp n hW hP hX h31 t a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7).b1,(kX16 Wd XT wo wp n hW hP hX h31 t a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7).b2,(kX16 Wd XT wo wp n hW hP hX h31 t a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7).b3,(kX16 Wd XT wo wp n hW hP hX h31 t a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7).b4,(kX16 Wd XT wo wp n hW hP hX h31 t a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7).b5,(kX16 Wd XT wo wp n hW hP hX h31 t a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7).b6,(kX16 Wd XT wo wp n hW hP hX h31 t a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7).b7⟩:A8)=kX8 Wd XT wp n hP hX ⟨h31.2.1,h31.2.2⟩ t c0 c1 c2 c3 c4 c5 c6 c7:=by
 intro m; induction m with
 | zero =>
   intro t a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7 H
   have z:=O4 H
   rw[(kX16.eq_1 Wd XT wo wp n hW hP hX h31 t a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7).trans (dif_neg (by rw[ε];exact W3.2 z)),
    (kX8.eq_1 Wd XT wo n hW hX ⟨h31.1,h31.2.2⟩ t a0 a1 a2 a3 a4 a5 a6 a7).trans (dif_neg (by rw[ε];exact W3.2 z)),
    (kX8.eq_1 Wd XT wp n hP hX ⟨h31.2.1,h31.2.2⟩ t c0 c1 c2 c3 c4 c5 c6 c7).trans (dif_neg (by rw[ε];exact W3.2 z))]
   exact ⟨rfl,rfl⟩
 | succ m IH =>
   intro t a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7 H
   have ht:(J0 t) < (J0 n):=N1 H
   have h:t < n:=ε.mpr ht
   rw[(kX16.eq_1 Wd XT wo wp n hW hP hX h31 t a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7).trans (dif_pos h),
    (kX8.eq_1 Wd XT wo n hW hX ⟨h31.1,h31.2.2⟩ t a0 a1 a2 a3 a4 a5 a6 a7).trans (dif_pos h),
    (kX8.eq_1 Wd XT wp n hP hX ⟨h31.2.1,h31.2.2⟩ t c0 c1 c2 c3 c4 c5 c6 c7).trans (dif_pos h)]
   exact IH (t+1) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (by rw[ac h,V4,H,BK])
@[inline] def dwA(w:ἇ):Ἅ:=(Core.mW w).toInt64.toFloat*P.uget (UInt32.toUSize ((w>>>23)&&&255)) (by
  rw[gL,aC,Ξ];exact AO ϝ (by decide))
theorem k8i (k r:Ἄ)(n:ℕ)(hk:(J0 k)<=n)(h8:8*n+7<2^31)(hr:(J0 r)<8):(J0 (8*k+r))=8*(J0 k)+(J0 r):=by
  have e8k:(J0 (8*k))=8*(J0 k):=by rw [kMm _ _ (by rw [kc8]; exact (U3 8 hk).trans_lt (C0 (V1 _ _) h8)),kc8]
  rw [kSm _ _ (by rw [e8k]; exact C0 (Y3 (U3 8 hk) (J2 hr)) h8),e8k]
def dRw8(X:Ἂ ἇ)(i n r k:Ἄ)(Y:ἀ)(a:Ἅ)(hX:(J0 i)+(J0 n)<=(O0 X))(hY:8*(J0 n)+8<=(kS Y))(hB:(J0 i)+(J0 n)<2^31/\8*(J0 n)+7<2^31)(hr:(J0 r)<8):ἀ:=
 if h:k<n then
  let w:=X.uget (i+k) (by have:=ε.mp h;rw[kSm i k ((O1 this _).trans hB.1)];exact (O1 this _).trans_le hX)
  dRw8 X i n r (k+1) (Y.uset (8*k+r) (dw w) (by have:=ε.mp h;rw[k8i k r (J0 n) this.le hB.2 hr];exact (O1 hr _).trans_le ((L1 (U3 8 this.le) 8).trans hY))) (a+dwA w) hX (by rw[uZ];exact hY) hB hr
 else Y.uset (8*n+r) a (by rw[k8i n r (J0 n) (le_refl _) hB.2 hr];exact (O1 hr _).trans_le hY)
termination_by (J0 n)-(J0 k)
decreasing_by all_goals (rw[ac h];exact C5 (ε.mp h) (I3 _))
theorem dRw8_size (X:Ἂ ἇ)(i n r:Ἄ)(hX:(J0 i)+(J0 n)<=(O0 X))(hB:(J0 i)+(J0 n)<2^31/\8*(J0 n)+7<2^31)(hr:(J0 r)<8) :
    ∀ m (k:Ἄ)(Y:ἀ)(a:Ἅ)(hY:8*(J0 n)+8<=(kS Y)),(J0 n)-(J0 k)=m → (kS (dRw8 X i n r k Y a hX hY hB hr))=(kS Y):=by
  intro m; induction m with
  | zero => intro k Y a hY hm; rw [dRw8]; split
            · rename_i h; exact absurd hm (Z8 (ε.mp h))
            · exact uZ _ _ _ _
  | succ m ih => intro k Y a hY hm; have h1:(J0 k) < (J0 n):=N1 hm
                 rw [dRw8,dif_pos (ε.mpr h1)]; rw [ih (k+1) _ _ _ (by rw [ac (ε.mpr h1),V4,hm,BK]),uZ]
theorem kBRn (b r n:Ἄ)(l:ℕ)(hr:(J0 r)<l)(hB:(J0 b)+l*(J0 n)<2^31):(J0 (b+r*n))=(J0 b)+(J0 r)*(J0 n)∧(J0 b)+(J0 r)*(J0 n)+(J0 n) <= (J0 b)+l*(J0 n):=by
  have m1:(J0 r)*(J0 n)+(J0 n) <= l*(J0 n):=by have:=ι (J0 n) (show (J0 r)+1<=l from hr); rw [Ϡ,n4] at this; exact this
  have e:(J0 (r*n))=(J0 r)*(J0 n):=kMm _ _ ((((V1 _ _).trans m1).trans (K2 _ _)).trans_lt hB)
  exact ⟨by rw [kSm _ _ (by rw [e]; exact (J1 ((V1 _ _).trans m1) _).trans_lt hB),e],by rw [pA]; exact J1 m1 _⟩
def dX8(X:Ἂ ἇ)(b n lim r:Ἄ)(Y:ἀ)(hX:(J0 b)+(J0 lim)*(J0 n)<=(O0 X))(hY:8*(J0 n)+8<=(kS Y))(hB:(J0 b)+(J0 lim)*(J0 n)<2^31/\8*(J0 n)+7<2^31)(hl:(J0 lim)<=8):ἀ:=
 if h:r<lim then
  have hr:=ε.mp h
  have K:=kBRn b r n (J0 lim) hr hB.1
  dX8 X b n lim (r+1) (dRw8 X (b+r*n) n r 0 Y 0 (by rw[K.1];exact K.2.trans hX) hY ⟨by rw[K.1];exact K.2.trans_lt hB.1,hB.2⟩ (hr.trans_le hl)) hX
   (by rw[dRw8_size X _ _ _ _ _ _ _ 0 Y 0 hY rfl];exact hY) hB hl
 else Y
termination_by (J0 lim)-(J0 r)
decreasing_by all_goals (rw[ac h];exact C5 (ε.mp h) (I3 _))
theorem dX8_size (X:Ἂ ἇ)(b n lim:Ἄ)(hX:(J0 b)+(J0 lim)*(J0 n)<=(O0 X))(hB:(J0 b)+(J0 lim)*(J0 n)<2^31/\8*(J0 n)+7<2^31)(hl:(J0 lim)<=8) :
    ∀ m (r:Ἄ)(Y:ἀ)(hY:8*(J0 n)+8<=(kS Y)),(J0 lim)-(J0 r)=m → (kS (dX8 X b n lim r Y hX hY hB hl))=(kS Y):=by
  intro m; induction m with
  | zero=>intro r Y hY hm; rw [dX8,dif_neg (show ¬ r < lim from fun h=>Z8 (ε.mp h) hm)]
  | succ m ih =>
    intro r Y hY hm; have h1:(J0 r) < (J0 lim):=N1 hm
    rw [dX8,dif_pos (ε.mpr h1)]; rw [ih (r+1) _ _ (by rw [ac (ε.mpr h1),V4,hm,BK]),dRw8_size X _ _ _ _ _ _ _ 0 Y 0 hY rfl]
theorem wcDF_spec (W:Ἂ Ἃ)(ow n:Ἀ):(kS (wcDF (W,ow,n)))=ow*n ∧ ∀k,k<ow*n->(wcDF (W,ow,n)).data[k]! =dw W[k]! := by
  have B:=bk (λk=>dw W[k]!) (ow*n) _ (AR (ow*n)) rfl (by simp[AR,kS])
  unfold wcDF; exact ⟨B.1,fun k hk=>B.2.2 k (by simp[AR,kS]) hk⟩
def tPg8(W:Ἂ ἇ)(n ow G gi:Ἄ)(A:ἀ):ἀ:=
 if h:gi<G then tPg8 W n ow G (gi+1) (tPt W n ow (8*gi) (8*gi+8) 0 A) else A
termination_by (J0 G)-(J0 gi)
decreasing_by all_goals (rw[ac h];have:=ε.mp h;exact Z5 Gб giб h)
def wcT8F(a:Ἂ ἇ×Nat×Nat):ἀ×Ἂ ἇ×ἇ:=
 let E:=da (λo=>rMu a.1 (G0 (o*a.2.2)) (G0 (o*a.2.2+a.2.2)) 0) a.2.1 (AS a.2.1)
 (tPg8 a.1 (G0 a.2.2) (G0 a.2.1) (G0 ((a.2.1+7)/8)) 0 (AR (a.2.2*(8*((a.2.1+7)/8)))),E,E.foldl (fun (m v:ἇ)=>if m<v then v else m) (0:ἇ))
@[noinline] def wcT8S:ST.Ref Unit (Ἂ (WcE wcT8F)):=(ST.Prim.mkRef #[] (Void.mk ())).val
def wcBF(a:Ἂ ἇ×Nat×Nat):ἀ:=bQ (λo=>if o<a.2.1 then dw oneW*dw a.1[o]! else dw 0) a.2.2 (AR a.2.2)
@[noinline] def wcBS:ST.Ref Unit (Ἂ (WcE wcBF)):=(ST.Prim.mkRef #[] (Void.mk ())).val
@[noinline] def bBo(b:Ἂ ἇ)(ow N:Nat):{v:ἀ//v=wcBF (b,ow,N)}:=wcGet3 wcK3eq (fun a b=>wcK3eq_ok a b) wcBS 32 (b,ow,N) (fun k=>k.1)
def bBs(b:Ἂ ἇ)(ow N:Nat)(s:Ἂ (WcE wcBF))(i:Nat):{v:ἀ//v=wcBF (b,ow,N)}:=
 if h:i<s.size then
  if hk:wcK3eq s[i].k (b,ow,N)=true then
   withPtrAddr s[i].k.1 (fun p1=>withPtrAddr b (fun p2=>if p1=p2 then (⟨s[i].v,s[i].h.trans (congrArg wcBF (wcK3eq_ok _ _ hk))⟩:{v:ἀ//v=wcBF (b,ow,N)}) else bBo b ow N)
     (fun _ _=>wcSub _ _)) (fun _ _=>wcSub _ _)
  else bBs b ow N s (i+1)
 else bBo b ow N
termination_by s.size-i
@[inline] def bBg(b:Ἂ ἇ)(ow N:Nat):{v:ἀ//v=wcBF (b,ow,N)}:=bBs b ow N (ST.Prim.Ref.get wcBS (Void.mk ())).val 0
theorem kSh (w:ἇ):(w&&&0x7FFFFFFF)>>>23=(w>>>23)&&&255:=by
  apply Β.1; rw [Υ,Ξ,Ξ,Υ]
  simp only [β]
  rw [show (0x7FFFFFFF:ἇ).toNat=2^(31:Ἀ)-1 by decide,ζ,show (255:ἇ).toNat=2^(8:Ἀ)-1 by decide,ζ]
  simp only [nr,kN,kH]; omega
theorem kAbsDw (w:ἇ):dwA w=dw (w&&&0x7FFFFFFF):=by
  have h1:Core.mW (w&&&0x7FFFFFFF)=Core.mW w:=by
    unfold Core.mW Core.f5; rw [kSh,UInt32.and_assoc,UInt32.and_assoc,show (0x7FFFFFFF&&&8388607:ἇ)=8388607 by decide,show (255&&&255:ἇ)=255 by decide]
  unfold dwA dw; rw [h1]; congr 2; rw [kSh]
def dRw(X:Ἂ ἇ)(i n o k:Ἄ)(Y:ἀ)(a:Ἅ)(hX:(J0 i)+(J0 n)<=(O0 X))(hY:(J0 o)+(J0 n)<(kS Y))(hB:(J0 i)+(J0 n)<2^(31:Ἀ)/\(J0 o)+(J0 n)<2^(31:Ἀ)):ἀ:=
 if h:k<n then
  let w:=X.uget (i+k) (by have:=ε.mp h;rw[kSm i k (by omega)];omega)
  dRw X i n o (k+1) (Y.uset (o+k) (dw w) (by have:=ε.mp h;rw[kSm o k (by omega)];omega)) (a+dwA w) hX (by rw[uZ];exact hY) hB
 else Y.uset (o+n) a (by rw[kSm o n hB.2];exact hY)
termination_by (J0 n)-(J0 k)
decreasing_by all_goals (rw[ac h];have:=ε.mp h;exact Z5 nб kб h)
theorem dRw_size(X:Ἂ ἇ)(i n o:Ἄ)(hX:(J0 i)+(J0 n)<=(O0 X))(hB:(J0 i)+(J0 n)<2^(31:Ἀ)/\(J0 o)+(J0 n)<2^(31:Ἀ)):∀ m (k:Ἄ)(Y:ἀ)(a:Ἅ)(hY:(J0 o)+(J0 n)<(kS Y)),(J0 n)-(J0 k)=m->(kS (dRw X i n o k Y a hX hY hB))=(kS Y):=by
 intro m;induction m with
 | zero=>intro k Y a hY hm;rw[dRw,dif_neg (by rw[ε];(clear * - hm;omega))];exact uZ _ _ _ _
 | succ m ih=>intro k Y a hY hm;have h1:(J0 k)<(J0 n):=by exact N1 hm
              rw[dRw,dif_pos (ε.mpr h1)];dsimp only;rw[ih _ _ _ _ (by rw[ac (ε.mpr h1)];(clear * - hm h1;omega)),uZ]
def dOw(X:Ἂ ἇ)(n m R r:Ἄ)(Y:ἀ)(hX:(J0 R)*(J0 n)<=(O0 X))(hY:(J0 R)*(J0 m)<=(kS Y))(hm:(J0 m)=(J0 n)+1)(hB:(J0 R)*(J0 m)<2^(31:Ἀ)):ἀ:=
 if h:r<R then
  have hr:=ε.mp h
  have q1:((J0 r)+1)*(J0 m)<=(J0 R)*(J0 m):=ι _ hr
  have q2:((J0 r)+1)*(J0 n)<=(J0 R)*(J0 n):=ι _ hr
  have q3:(J0 R)*(J0 n)<=(J0 R)*(J0 m):=U3 _ (Nat.le.intro hm.symm)
  have e1:(J0 (r*n))=(J0 r)*(J0 n):=kMm _ _ (C0 (ϧ (X4 hr) (Nat.le.intro hm.symm)) hB)
  have e2:(J0 (r*m))=(J0 r)*(J0 m):=kMm _ _ (C0 (ι _ (X4 hr)) hB)
  dOw X n m R (r+1) (dRw X (r*n) n (r*m) 0 Y 0 (U2 (B3 ((congrArg (·+J0 n) e1).trans (nO _ _).symm)) (U2 q2 hX)) (U1 (O1 (B3 hm.symm) _) (U2 (B3 ((congrArg (·+J0 m) e2).trans (nO _ _).symm)) (U2 q1 hY))) ⟨C0 (U2 (B3 ((congrArg (·+J0 n) e1).trans (nO _ _).symm)) (U2 q2 q3)) hB,K5 (O1 (B3 hm.symm) _) (C0 (U2 (B3 ((congrArg (·+J0 m) e2).trans (nO _ _).symm)) q1) hB)⟩) hX
   (U2 hY (B3 (dRw_size X (r*n) n (r*m) _ _ _ 0 Y 0 _ rfl).symm)) hm hB
 else Y
termination_by (J0 R)-(J0 r)
decreasing_by exact ac h ▸ Z5 _ _ (ε.mp h)
def zrRw(X:Ἂ ἇ)(rn:Nat):Nat→Ἆ
|0=>true
|k+1=>let x:=X[rn+k]!;(x==0||x==0x80000000)&&zrRw X rn k
theorem zrRw_ok(X:Ἂ ἇ)(rn:Nat):∀k,zrRw X rn k=true→∀s<k,X[rn+s]! = 0∨X[rn+s]! = 0x80000000
|0,_,s,hs=>absurd hs (U5 _)
|k+1,h,s,hs=>by
 simp only[zrRw,Η,nM,AQ]at h
 by_cases e:s=k
 ·rw[e];exact h.1
 exact zrRw_ok X rn k h.2 s (by omega)
theorem zrNC(a q:Ἂ Ἃ):∀m i,(decide (kM a[i+m]!*kM q[i+m]! = 0)&&Bool.xor (k9 a[i+m]!) (k9 q[i+m]!))=false→nC a q i (m+1)=false
|0,i,h=>by simp only[ϑ]at h;simp only[nC,h,Bool.false_and]
|m+1,i,h=>by rw[nC,zrNC a q m (i+1) (by rw[show i+1+m=i+(m+1) by omega];exact h),Bool.and_false]
theorem zrDot(n:ℕ)(D Wf:ℕ→Ἃ)(hz:∀s<n,D s=0∨D s=0x80000000)(hW:∀s<n,kB (Wf s)≠255)(hDn:D n=ld)(hWn:Wf n=0):ms (n+1) D Wf=0:=by
 have g:aG (n+1) D Wf:=by
  intro k hk;by_cases e:k<n
  ·rcases hz k e with h|h<;>rw[h]<;>exact⟨by decide,hW k e⟩
  ·rw[show k=n by omega,hDn,hWn];exact⟨by decide,by decide⟩
 have hS:af (n+1) D Wf=0:=by
  unfold af;apply Finset.sum_eq_zero;intro k hk;have:=ϥ.mp hk;by_cases e:k<n
  ·have e0:sv (D k)=0:=by rcases hz k e with h|h<;>rw[h]<;>decide
   simp only[tm,e0,zero_mul]
  ·rw[show k=n by omega,hWn];simp only[tm,show sv (0:Ἃ)=0 by decide,mul_zero,zero_mul]
 rw[Core.bI (n+1) D Wf g,hS,zrNC _ _ n 0 (by rw[Κ,ab (n+1) D n (by omega),ab (n+1) Wf n (by omega),hDn,hWn];decide),Bool.and_false];decide
@[noinline] def sQ2(relu:Ἆ)(n:Nat)(X W b E:Ἂ ἇ)(ix ib L:Nat)(Y Q:ἀ)(vo rn j:Nat)(hh:ἆ)(v:Ἅ):ἇ:=
 if b[j]! == 0&&(E[j]!).toUInt64<(255:ἆ)&&zrRw X rn n then 0 else
 let tg:=fun (_:Unit)=>tG relu v E[j]! ix ib L (fun _=>tR relu L (sTQ Y Q n vo (j/8*n*8+j%8) (dw oneW*dw b[j]!)) fun _=>aD n relu X W b rn 1 j) (fun _=>aD n relu X W b rn 1 j)
 if (E[j]!).toUInt64<(255:ἆ)&&hh<=(372:ἆ) then tf relu v hh tg else tg ()
theorem dOw_size(X:Ἂ ἇ)(n m R:Ἄ)(hX:(J0 R)*(J0 n)<=(O0 X))(hm:(J0 m)=(J0 n)+1)(hB:(J0 R)*(J0 m)<2^(31:Ἀ)):∀ k (r:Ἄ)(Y:ἀ)(hY:(J0 R)*(J0 m)<=(kS Y)),(J0 R)-(J0 r)=k->(kS (dOw X n m R r Y hX hY hm hB))=(kS Y):=by
 intro k;induction k with
 | zero=>intro r Y hY hk;rw[dOw,dif_neg (by rw[ε];(clear * - hk;omega))]
 | succ k ih=>intro r Y hY hk;have h1:(J0 r)<(J0 R):=by exact N1 hk
              rw[dOw,dif_pos (ε.mpr h1)];dsimp only;rw[ih _ _ _ (by rw[ac (ε.mpr h1)];(clear * - hk h1;omega)),dRw_size _ _ _ _ _ _ _ 0 _ _ _ rfl]
theorem s5tPt8_inv_h1 (n g t:Ἄ)(ow:Nat)(hB:(ow+16)*n.toNat<2^(31:Ἀ))(h8:g.toNat+8≤ow+16)(h1:t.toNat<n.toNat):(g*n+t).toNat=g.toNat*n.toNat+t.toNat:=by
  have q1:g.toNat*n.toNat+t.toNat < 2^(31:Ἀ):=K5 (O1 h1 _) (by rw [← nO]; exact C0 (ι _ (U2 (J1 (K2 1 7) _) h8)) hB)
  have hgn:=C0 (V1 _ _) q1
  rw [kSm _ _ (by rw [kMm _ _ hgn]; exact q1),kMm _ _ hgn]
theorem s5tPt8_inv (W:Ἂ ἇ)(n ow g16:Ἄ)(B:Nat)
    (hB:(ow.toNat+16)*n.toNat<2^(31:Ἀ))(hO:ow.toNat+16<2^(31:Ἀ))(hg:g16.toNat≤ow.toNat)(hW:ow.toNat*n.toNat≤W.size)(hWs:W.size<2^(32:Ἀ))(hgB:g16.toNat≤B∨n.toNat=0) :
    ∀ m (t:Ἄ)(A:ἀ),n.toNat-t.toNat=m → t.toNat≤n.toNat → A.size=B+t.toNat*8 →
    let R:=tPt W n ow g16 (g16+8) t A
    R.size=B+n.toNat*8∧(∀ i<B+t.toNat*8,R.data[i]! = A.data[i]!) ∧
    (∀ s l,t.toNat ≤ s → s < n.toNat → l < 8 → R.data[B+s*8+l]! = if g16.toNat+l < ow.toNat then dw W[(g16.toNat+l)*n.toNat+s]! else 0):=by
  have h8:g16.toNat+8 ≤ ow.toNat+16:=Y3 hg (K2 8 8)
  have e16:(g16+8).toNat=g16.toNat+8:=by rw [kSm _ _ (by rw[kc8]; exact C0 h8 ‹_›),kc8]
  intro m; induction m with
  | zero => intro t A hm ht hA; have z:=K3.1 hm; rw [tPt,dif_neg (by rw [ε]; exact not_lt.2 z)]; dsimp only
            have:t.toNat=n.toNat:=le_antisymm ht z
            exact ⟨by rw [hA,this],fun i _=>rfl,fun s l h1 h2 _=>absurd (h1.trans_lt h2) (not_lt.2 z)⟩
  | succ m ih =>
    intro t A hm ht hA
    have h1:t.toNat < n.toNat:=N1 hm
    have e1:(t+1).toNat=t.toNat+1:=ac (ε.mpr h1)
    have hsc:B+t.toNat*8-g16.toNat+g16.toNat=B+t.toNat*8:=L3 (Nat.le_add_right_of_le (hgB.resolve_right (L7 h1).ne'))
    have e8:B+(t+1).toNat*8=B+t.toNat*8+8:=by rw [e1,Ϡ,n4,pA]
    rw [tPt,dif_pos (by rw [ε]; exact h1)]
    obtain ⟨b1,b2,b3⟩:=s5tBi_inv W n ow (g16+8) t (B+t.toNat*8-g16.toNat) hB (e16.trans_le h8) hW hWs h1 _ g16 (g16*n+t) A rfl ((V1 _ 8).trans_eq e16.symm) (s5tPt8_inv_h1 n g16 t _ hB h8 h1) (hA.trans hsc.symm)
    obtain ⟨r1,r2,r3⟩:=ih (t+1) _ (by rw[e1,←F7,hm,BK]) (by rw [e1]; exact h1) (by rw [b1,e16,e8, ← pA,hsc])
    refine ⟨r1,fun i hi=>(r2 i (e8 ▸ U1 hi (V1 _ _))).trans (b2 i (hsc.symm ▸ hi)),fun s l hs1 hs2 hl => ?_⟩
    rcases Z7 t.toNat s with hs | hs
    · exact r3 s l (by rw [e1]; exact hs) hs2 hl
    · have se:s=t.toNat:=le_antisymm hs hs1
      subst se
      have:=b3 (g16.toNat+l) (V1 _ _) (by rw[e16]; exact O1 hl _)
      rw [← pA,hsc] at this
      exact (r2 _ (by rw [e8]; exact O1 hl _)).trans this
theorem s5tPg8_inv (W:Ἂ ἇ)(n ow G:Ἄ)
    (hB:(ow.toNat+16)*n.toNat<2^(31:Ἀ))(hO:ow.toNat+16<2^(31:Ἀ))(hW:ow.toNat*n.toNat≤W.size)(hWs:W.size<2^(32:Ἀ))(hG:8*G.toNat≤ow.toNat+7) :
    ∀ m (gi:Ἄ)(A:ἀ),G.toNat-gi.toNat=m → gi.toNat≤G.toNat → A.size=8*gi.toNat*n.toNat →
    let R:=tPg8 W n ow G gi A
    R.size=8*G.toNat*n.toNat∧(∀ i<8*gi.toNat*n.toNat,R.data[i]! = A.data[i]!) ∧
    (∀ g s l,gi.toNat ≤ g → g < G.toNat → s < n.toNat → l < 8 → R.data[8*g*n.toNat+s*8+l]! = if 8*g+l < ow.toNat then dw W[(8*g+l)*n.toNat+s]! else 0):=by
  intro m; induction m with
  | zero => intro gi A hm hgi hA; have z:=K3.1 hm; rw [tPg8,dif_neg (by rw [ε]; exact not_lt.2 z)]; dsimp only
            have:gi.toNat=G.toNat:=le_antisymm hgi z
            exact ⟨by rw [hA,this],fun i _=>rfl,fun g s l h1 h2 _ _=>absurd (h1.trans_lt h2) (not_lt.2 z)⟩
  | succ m ih =>
    intro gi A hm hgi hA
    have h1:gi.toNat < G.toNat:=N1 hm
    have e1:(gi+1).toNat=gi.toNat+1:=ac (ε.mpr h1)
    have eg:(8*gi).toNat=8*gi.toNat:=by rw [kMm _ _ (by rw[kc8]; (clear * - hO hG h1 e1;omega)),kc8]
    have e16:(8*gi+8)=(8*gi)+8:=rfl
    rw [tPg8,dif_pos (by rw [ε]; exact h1)]
    obtain ⟨b1,b2,b3⟩:=s5tPt8_inv W n ow (8*gi) (8*gi.toNat*n.toNat) hB hO (by rw[eg]; (clear * - hG h1 e1;omega)) hW hWs
      (by rcases Nat.eq_zero_or_pos n.toNat with h|h
          · exact BA h
          · exact pZ (by rw[eg]; exact J4 _ h)) _ 0 A rfl (by simp only [Ϧ,zero_le]) (by simp only [Ϧ,zero_mul,Av]; exact hA)
    simp only [Ϧ,ϳ,ϑ] at b1 b2 b3
    have sz:8*gi.toNat*n.toNat+n.toNat*8=8*(gi.toNat+1)*n.toNat:=by ring
    obtain ⟨r1,r2,r3⟩:=ih (gi+1) _ (by rw[e1,←F7,hm,BK]) (by rw [e1]; exact h1) (by rw [b1,e1,sz])
    refine ⟨r1,fun i hi=>by rw [r2 i (by rw [e1, ← sz]; exact Nat.lt_add_right (nб*8) hi),b2 i hi],fun g s l hg1 hg2 hs hl => ?_⟩
    rcases Z7 gi.toNat g with hg | hg
    · exact r3 g s l (by rw [e1]; exact hg) hg2 hs hl
    · have ge:g=gi.toNat:=le_antisymm hg hg1
      subst ge
      rw [r2 _ (by rw [e1, ← sz]; have:=ι 8 (show s+1≤n.toNat by exact Order.add_one_le_iff.mpr hs); rw[Ϡ,n4] at this; (clear * - hs hl;omega)),b3 s l (ϟ _) hs hl,eg]
theorem tPg8_n0 (W:Ἂ ἇ)(n ow G:Ἄ)(hn:n.toNat=0) : ∀ m (gi:Ἄ)(A:ἀ),G.toNat-gi.toNat=m → (tPg8 W n ow G gi A).size=A.size:=by
  intro m; induction m with
  | zero=>intro gi A hm; rw [tPg8,dif_neg (by rw [ε]; (clear * - hm;omega))]
  | succ m ih =>
    intro gi A hm; have h1:gi.toNat < G.toNat:=by exact N1 hm
    rw [tPg8,dif_pos (by rw [ε]; exact h1),ih (gi+1) _ (by rw[ac (ε.mpr h1)]; (clear * - hm h1;omega)),tPt,dif_neg (by rw [ε]; exact Std.Rxo.size_eq_zero_iff_not_le.mp hn)]
theorem wcT8F_panel (W:Ἂ ἇ)(ow n:Nat)(hB:(ow+16)*n<2^(31:Ἀ))(hW:ow*n≤W.size)(hWs:W.size<2^(32:Ἀ)) :
    let WT:=(wcT8F (W,ow,n)).1
    WT.size=n*(8*((ow+7)/8)) ∧ ∀ t < n, ∀ g l,l < 8 → 8*g+l < 8*((ow+7)/8) → WT.data[(g*n+t)*8+l]! = if 8*g+l < ow then dw W[(8*g+l)*n+t]! else 0:=by
  rcases Nat.eq_zero_or_pos n with hO|hO
  · subst hO; refine ⟨?_,fun t ht=>by (clear * - ht;omega)⟩; simp only [wcT8F]; rw [tPg8_n0 _ _ _ _ (by simp only [lX,USize.toNat_ofNat',Nat.zero_mod]) _ _ _ rfl]; simp only [ϳ]; rfl
  · have e1:=s5tu n (by have:=lZ n (show 0<ow+16 by exact Nat.zero_lt_succ (ow+15)); (clear * - hB this;omega))
    have e2:=s5tu ow (by have:=J4 (ow+16) hO; (clear * - hB this;omega))
    have e3:=s5tu ((ow+7)/8) (by have:=J4 (ow+16) hO; (clear * - hB e2 this;omega))
    obtain ⟨r1,-,r3⟩:=s5tPg8_inv W n.toUSize ow.toUSize ((ow+7)/8).toUSize (by rw [e1,e2]; exact hB) (by rw [e2]; have:=J4 (ow+16) hO; (clear * - hB this;omega)) (by rw [e1,e2]; exact hW) hWs (by rw[e3,e2]; exact Nat.mul_div_le (ow+7) 8)
      _ 0 (kJ (n*(8*((ow+7)/8)))) rfl (by simp only [Ϧ,lX,USize.toNat_ofNat',zero_le]) (by simp only [Ϧ,mul_zero,lX,USize.toNat_ofNat',zero_mul]; rfl)
    rw [e1,e3] at r1; rw [e1,e3,e2] at r3
    refine ⟨by show (tPg8 W n.toUSize ow.toUSize ((ow+7)/8).toUSize 0 _).size=_; rw[r1]; ring,fun t ht g l hl hgl => ?_⟩
    show (tPg8 W n.toUSize ow.toUSize ((ow+7)/8).toUSize 0 _).data[_]! = _
    rw [show (g*n+t)*8+l=8*g*n+t*8+l by ring]
    exact r3 g t l (ϟ _) (by (clear * - hl hgl;omega)) ht hl
theorem wcT8F_exp (W:Ἂ ἇ)(ow n:Nat)(hB:(ow+16)*n<2^(31:Ἀ))(hW:ow*n≤W.size)(hWs:W.size<2^(32:Ἀ)) :
    let c:=wcT8F (W,ow,n)
    c.2.1.size=ow∧(∀ o<ow, ∀ s<n,(kB W[o*n+s]!).toNat ≤ (c.2.1[o]!).toNat)∧(∀ o<ow,(c.2.1[o]!).toNat ≤ c.2.2.toNat):=by
  obtain ⟨d1,d2⟩:=s5da_spec (fun o=>rMu W (o*n).toUSize (o*n+n).toUSize 0) ow _ (lQ ow) rfl (by simp) (fun i h=>by simp at h)
  simp only [wcT8F]
  refine ⟨d1,fun o ho s hs => ?_,fun o ho => ?_⟩
  · rw [d2 o ho]
    have q1:(o+1)*n ≤ ow*n:=ι _ ho
    have q2:=nO o n
    have e1:=s5tu (o*n) (by (clear * - hW hWs hs q1 q2;omega)); have e2:=s5tu (o*n+n) (by (clear * - hW hWs q1 q2;omega))
    exact (s5rMu_spec W hWs _ (o*n).toUSize (o*n+n).toUSize 0 rfl (by rw [e2]; (clear * - hW q1 q2 e2;omega))).2 (o*n+s) (by rw [e1]; exact V1 (o*n) s) (by rw [e2]; exact O1 hs (o*n))
  · have mem:(da (fun o=>rMu W (o*n).toUSize (o*n+n).toUSize 0) ow (lQ ow))[o]! ∈
        (da (fun o=>rMu W (o*n).toUSize (o*n+n).toUSize 0) ow (lQ ow)).toList:=by
      rw [AB _ o (by exact V6 ho (id (Eq.symm d1)))]; exact Array.getElem_mem_toList _
    rw [← Array.foldl_toList]; exact (s5fmax _ 0).2 _ mem
theorem kRR_h1 (n:ℤ):(0:ℝ) < (2:ℝ)^n:=zpow_pos two_pos n
theorem kRR (z:ℝ) : |ns kC z-z| ≤ (2:ℝ)^(-53:ℤ)*|z| + (2:ℝ)^(-1075:ℤ):=by
  by_cases hs : |z| < (2:ℝ)^(-1021:ℤ)
  · have h:=re z (-1021) AP hs
    rw [show ((-1021:ℤ)-53) = -1075+1 by decide,AK AL,zpow_one,mul_div_cancel_right₀ _ AL] at h
    exact h.trans (le_add_of_nonneg_left (Af (kRR_h1 _).le (Aa z)))
  · push_neg at hs
    have hz:0 < |z| := AN (kRR_h1 _) hs
    have h1 : |z| < (2:ℝ)^(ph 2 |z| + 1):=by
      have:=Int.lt_zpow_succ_log_self (R:=ℝ) (b:=2) (by norm_num) |z|; simpa using this
    have h2:(2:ℝ)^(ph 2 |z|) ≤ |z| := by
      have:=Int.zpow_log_le_self (R:=ℝ) (b:=2) (by norm_num) hz; simpa using this
    have h3:(-1021:ℤ) ≤ ph 2 |z| := by
      have:=(Int.zpow_le_iff_le_log (R:=ℝ) (b:=2) (x:=-1021) (by norm_num) hz).1 (by exact_mod_cast hs); exact this
    have h:=re z (ph 2 |z| + 1) (by exact Int.le_add_one h3) h1
    rw [sub_eq_add_neg (ph 2 |z| + 1),add_right_comm (ph 2 |z|),AK AL,AK AL,zpow_one,mul_div_cancel_right₀ _ AL] at h
    exact h.trans ((AE h2 (kRR_h1 _).le).trans ((mul_comm _ _).trans_le (le_add_of_nonneg_right (kRR_h1 _).le)))
theorem kRA (x y:Ἅ)(hx:fv x)(hy:fv y)(h:|rv x+rv y|<(2:ℝ)^(971:ℤ)) :
    fv (x+y) ∧ |rv (x+y)-(rv x+rv y)| ≤ (2:ℝ)^(-53:ℤ)*|rv x+rv y|+(2:ℝ)^(-1075:ℤ):=by
  have ⟨g,e⟩:=ii x y hx hy (h.le.trans (pm 971 (by norm_num)))
  exact ⟨g,by rw [e]; exact kRR _⟩
theorem kRM (x y:Ἅ)(hx:fv x)(hy:fv y)(h:|rv x*rv y|<=(2:ℝ)^(971:ℤ)) :
    fv (x*y) ∧ |rv (x*y)-(rv x*rv y)| ≤ (2:ℝ)^(-53:ℤ)*|rv x*rv y|+(2:ℝ)^(-1075:ℤ):=by
  have ⟨g,e⟩:=jo x y hx hy h
  exact ⟨g,by rw [e]; exact kRR _⟩
theorem kSR_h (A t c d e:ℝ) (hA:0 ≤ A) (ht:0 ≤ t) (htT:t ≤ 134217728)
    (hc:c=(1/9007199254740992+1/9444732965739290427392)*A+1/9007199254740992)
    (E:e ≤ 1/9007199254740992*(A+t*c)+1/1208925819614629174706176) (P:d ≤ t*c):d+e ≤ (t+1)*c:=by
  have k:t*c≤134217728*c:=AE htT (hc▸by positivity)
  linear_combination P+E+1/9007199254740992*k+66060287/633825300114114700748351602688*hA+(1/67108864-1)*hc
theorem kSR (p:ℕ→Ἅ)(z:ℕ→ℝ)(c0 A:ℝ)(n:ℕ)(hn:n≤134217728)
    (hp:∀ t<n,fv (p t)∧rv (p t)=z t)
    (hA:|c0| + ∑ s∈kP n, |z s| ≤ A) (hA2:A ≤ (2:ℝ)^(960:ℤ)) :
    ∀ m t a,n-t=m → t ≤ n → fv a →
      |rv a-(c0 + ∑ s∈kP t,z s)| ≤ t*((1/9007199254740992+1/9444732965739290427392)*A+1/9007199254740992) →
      fv (ao p n t a) ∧ |rv (ao p n t a)-(c0 + ∑ s∈kP n,z s)| ≤ n*((1/9007199254740992+1/9444732965739290427392)*A+1/9007199254740992):=by
  have hA0:0 ≤ A:=AZ (add_nonneg (Aa _) (D8 fun _ _=>Aa _)) hA
  generalize hc':(1/9007199254740992+1/9444732965739290427392)*A+1/9007199254740992=c
  have hc:=hc'.symm
  have hc0:0 ≤ c:=by rw [hc]; exact add_nonneg (Af (by norm_num) hA0) (by norm_num)
  intro m; induction m with
  | zero => intro t a D Q e o; obtain rfl:t=n:=C4 Q (K3.1 D)
            rw [ao,if_neg (AX _)]; exact ⟨e,o⟩
  | succ m G =>
    intro t a D Q e o
    have ht:t < n:=N1 D
    rw [ao,if_pos ht]
    obtain ⟨fp,rp⟩:=hp t ht
    have hS : |c0 + ∑ s∈kP (t+1),z s| ≤ A:=by
      refine AZ ?_ hA
      refine (Ag _ _).trans (add_le_add AP ((K8 _ _).trans ?_))
      exact Finset.sum_le_sum_of_subset_of_nonneg (fun x hx=>ϥ.2 (AN (ϥ.1 hx) ht)) (fun _ _ _=>Aa _)
    have hsucc:c0 + ∑ s∈kP (t+1),z s=(c0 + ∑ s∈kP t,z s)+z t:=by
      rw [Ν]; ring
    have htT:(t:ℝ) ≤ 134217728:=by exact_mod_cast (X4 ht).trans hn
    have htc:(t:ℝ)*c ≤ 134217728*c:=AE htT hc0
    rw [hsucc] at hS
    have hy : |rv a+rv (p t)| ≤ A+t*c:=by
      rw [rp,show rv a+z t=(rv a-(c0 + ∑ s∈kP t,z s))+((c0 + ∑ s∈kP t,z s)+z t) by ring,add_comm A]
      exact (Ag _ _).trans (add_le_add o hS)
    have h971:A+t*c < (2:ℝ)^(971:ℤ):=by
      have e1:(2:ℝ)^(971:ℤ)=2^(960:ℤ)*2^(11:ℤ):=AK AL 960 11
      have e3:(1:ℝ) ≤ 2^(960:ℤ):=one_le_zpow₀ one_le_two (by decide)
      have e4:(2:ℝ)^(11:ℤ)=2048:=by norm_num
      rw [e1,e4]; generalize (2:ℝ)^(960:ℤ)=X at hA2 e3 ⊢; linarith only [htc,hc,hA2,e3,hA0]
    obtain ⟨f2,r2⟩:=kRA a (p t) e fp (AO hy h971)
    refine G (t+1) _ (by rw [V4,D,BK]) ht f2 ?_
    have e53:(2:ℝ)^(-53:ℤ)=1/9007199254740992:=by norm_num [zpow_neg]
    have e80:(2:ℝ)^(-80:ℤ)=1/1208925819614629174706176:=by norm_num [zpow_neg]
    have e75:(2:ℝ)^(-1075:ℤ) ≤ 1/1208925819614629174706176 :=
      e80 ▸ AG one_le_two (by norm_num)
    rw [e53] at r2
    have E : |rv (a+p t)-(rv a+rv (p t))| ≤ 1/9007199254740992*(A+t*c)+1/1208925819614629174706176 :=
      r2.trans (add_le_add (mul_le_mul_of_nonneg_left hy (one_div_nonneg.2 (by norm_num))) e75)
    have:=(add_comm _ _).trans_le (kSR_h A t c _ _ hA0 (M3 _) htT hc E o)
    rw [X7,hsucc, ← rp]
    refine (abs_sub_le _ (rv a+rv (p t)) _).trans ?_
    rw [add_sub_add_right_eq_sub]; exact this
theorem kMsk (w:ἇ):kB (w&&&0x7FFFFFFF)=kB w∧kG (w&&&0x7FFFFFFF)=kG w∧sv (w&&&0x7FFFFFFF) = |sv w| := by
  have hs:k9 (w&&&0x7FFFFFFF)=false:=by
    unfold k9; rw [UInt32.and_assoc,show (0x7FFFFFFF&&&0x80000000:ἇ)=0 by decide,UInt32.and_zero]; decide
  have he:kB (w&&&0x7FFFFFFF)=kB w:=by
    unfold kB; apply Β.1; rw [Ξ,Ξ,Υ,Υ,Ξ]
    simp only [β]
    rw [show (0x7FFFFFFF:ἇ).toNat=2^(31:Ἀ)-1 by decide,ζ,show (0xFF:ἇ).toNat=2^(8:Ἀ)-1 by decide,ζ,ζ]
    simp only [nr,kN,kH]; (clear * -;omega)
  have hf:lK (w&&&0x7FFFFFFF)=lK w:=by
    unfold lK; rw [UInt32.and_assoc,show (0x7FFFFFFF&&&0x7FFFFF:ἇ)=0x7FFFFF by decide]
  have hm:kM (w&&&0x7FFFFFFF)=kM w:=by unfold kM; rw [he,hf]
  refine ⟨he,by unfold kG; rw [he],?_⟩
  unfold sv; rw [hs,hm]; simp only [υ,if_false]; by_cases h:k9 w=true <;> simp [h]
lemma mV53(u:ἆ)(n:Ἀ)(d:(lq u)=n)(e:n<2^(53:Ἀ)):
  fv u.toInt64.toFloat/\rv u.toInt64.toFloat=(n:Ἁ):=by
 have:(K7 u.toInt64)=n:=by rw[show (K7 u.toInt64)=ae u from ἅ,ay,d];split<;>omega
 rw[<-this];exact dB _ (by rw[this];simp only [Int.natAbs_natCast,nr];exact I7 e)
theorem kCE (F a:Ἅ)(hZ:fv (F*a)) : |rv (F*a)| < (2:ℝ)^(((cEq F a).toNat:ℤ)+56)∧(cEq F a).toNat ≤ 969:=by
  have h:=fK (F*a) hZ
  have hb:(eF (F*a).toBits).toNat < 2048:=by unfold eF; rw [Ε]; exact AO ϝ (by decide)
  unfold cEq; dsimp only
  split
  · rename_i hc
    have hc':1078 < (eF (F*a).toBits).toNat:=by rw [μ] at hc; simpa using hc
    have e:(eF (F*a).toBits-1078).toNat=(eF (F*a).toBits).toNat-1078:=by rw [χ _ _ (UInt64.le_of_lt hc)]; rfl
    rw [e]
    refine ⟨?_,by (clear * - hb hc';omega)⟩
    have:((((eF (F*a).toBits).toNat-1078:ℕ)):ℤ)+56=(((eF (F*a).toBits).toNat:ℕ):ℤ)-1022:=by (clear * - hb hc';omega)
    rw [this]; exact h
  · rename_i hc
    have hc':(eF (F*a).toBits).toNat ≤ 1078:=by rw [μ] at hc; simp at hc; (clear * - hc;omega)
    refine ⟨AN h (AG (by norm_num) (by simp; exact W4 hc)),by simp⟩
theorem kWinR (nn T Λ lf zr B S M X P Hb H Ib R v:ℝ)
    (hn1:1 ≤ nn) (hnT:nn+2 ≤ T) (hT:T ≤ 134217728) (hT4:4 ≤ T)
    (hΛ0:0 ≤ Λ) (hΛ : |lf-Λ| ≤ nn*((1/9007199254740992+1/9444732965739290427392)*Λ+1/9007199254740992))
    (hv0:0 ≤ v) (hv:v ≤ 1/1208925819614629174706176)
    (hZ : |zr-(9*(nn+1))*lf| ≤ 1/9007199254740992*|(9*(nn+1))*lf| + v)
    (hZb : |zr| < X) (hX:72057594037927936 ≤ X)
    (hB0:0 ≤ B) (hB:B*2 ≤ Ib) (hS0:0 ≤ S) (hS:S ≤ M*Λ) (hM:0 ≤ M)
    (hMX:M*X=P*72057594037927936) (hTI:T*Ib=Hb*1125899906842624)
    (hHP:P ≤ H) (hHb:Hb ≤ H) (hHbig:21267647932558653966460912964485513216*T ≤ Hb)
    (hR1:P*9007199254740992 ≤ R) (hR2:Ib*4 ≤ R) :
    nn*((1/9007199254740992+1/9444732965739290427392)*(B+S)+1/9007199254740992) ≤ H ∧
    B+S+H ≤ H*4503599627370496∧B+S ≤ R:=by
 have hnn0:=zero_le_one.trans hn1
 have hnT':nn ≤ T:=by linear_combination hnT
 have h27:=hnT'.trans hT
 have hT0:=hnn0.trans hnT'
 have hW0:(0:ℝ) ≤ 9*(nn+1):=Af (R6 _) (add_nonneg hnn0 zero_le_one)
 have G:=(le_abs_self _).trans ((Aw Λ lf).trans_le (hΛ.trans (AE h27 (add_nonneg (Af (by norm_num) hΛ0) (by norm_num)))))
 have A:=(sub_le_iff_le_add'.1 (AY _ zr)).trans (add_le_add hZb.le (((Aw _ zr).trans_le hZ).trans (add_le_add_right hv _)))
 have F1:9*(nn+1)*Λ ≤ X*(1+1/8388608):=by
  have w:=mul_le_mul_of_nonneg_left (add_le_add_left h27 1) (R6 9)
  generalize 9*(nn+1)=W at hW0 w A ⊢
  linear_combination (70368744177664/70368743129087:ℝ)*(mul_le_mul_of_nonneg_left G hW0+le_abs_self (W*lf))+(633825300114114700748351602688/633825290669372657441063305217:ℝ)*A+(1048576/70368743129087:ℝ)*w+(554597052006948792370591694849/5316911903911424829191363170649767936:ℝ)*hX
 have F3:9*(nn+1)*S ≤ P*72057594037927936*(1+1/8388608):=(mul_le_mul_of_nonneg_left hS hW0).trans ((mul_left_comm _ _ _).trans_le ((mul_le_mul_of_nonneg_left F1 hM).trans_eq (by rw [← Ap,hMX])))
 have TB:=(mul_le_mul_of_nonneg_left hB hT0).trans hTI.le
 have b2:=Af hB0 zero_le_two
 have hH1:=(Nat.one_le_ofNat.trans hT4).trans ((le_mul_of_one_le_left hT0 Nat.one_le_ofNat).trans (hHbig.trans hHb))
 have Q:=(AE (mul_le_mul_of_nonneg_left (add_le_add_left hn1 1) (R6 9)) hS0).trans F3
 have B8:=(AE hB zero_le_four).trans hR2
 refine ⟨?_,?_,?_⟩
 · linear_combination (1048577/18889465931478580854784:ℝ)*((AE hnT' b2).trans TB)+(1048577/85002596691653613846528:ℝ)*F3+(1048577/9444732965739290427392:ℝ)*hS0+(1/9007199254740992:ℝ)*h27+(2932034153131/3298534883328:ℝ)*hHP+(1048577/16777216:ℝ)*hHb+(160342103381/3298534883328:ℝ)*hH1
 · linear_combination (1/8:ℝ)*((AE hT4 b2).trans TB)+(1/18:ℝ)*Q+(12009600437977088/3:ℝ)*hHP+(140737488355328:ℝ)*hHb+(1078985979068413/3:ℝ)*hH1
 · linear_combination (1/8:ℝ)*B8+(1/18:ℝ)*Q+(2796203/6291456:ℝ)*hR1+(2708821/6291456:ℝ)*((Af b2 zero_le_four).trans B8)
theorem kZ0:fv (0:Ἅ)∧rv (0:Ἅ)=0:=by
  refine ⟨by rw [fO]; rfl,?_⟩
  rw [iL,show (0:Ἅ).toModel.unpack = .zero .positive from rfl]; simp [lw,Model.unpackedToDyadic?]
theorem kTB2 (w:Ἃ) (F E:ℕ) (K:idxOf w ≤ F) (al:174+F ≤ E) : |(tm ld w:ℝ)| * 2 ≤ 2^E:=by
  have h:(tm ld w).natAbs*2 ≤ 2^E:=by
    rw [tm,Ϩ,Ϩ,Int.natAbs_pow,(by decide:(sv ld).natAbs=2^23),(by decide:idxOf ld=126)]
    refine U2 (ι 2 (ϧ (U3 _ (bg w).le) (Θ (n:=2) (by decide) (J1 K 126)))) ?_
    rw [← m7, ← m7, ← Nat.pow_succ]
    exact Θ (by decide) (by (clear * - al;omega))
  rw [← mI,Ϲ,M5]
  exact_mod_cast h
theorem kTW (x w:Ἃ) (e:ℕ) (hw:(kB w).toNat ≤ e) :
    |(tm x w:ℝ)| ≤ (|(sv x:ℝ)| * 2^(kG x))*2^(max e 56+23):=by
  have a : |(sv w:ℝ)| ≤ 2^24:=by
    have:=bg w; rw [← mI,Ϲ]; exact_mod_cast this.le
  have b:24+kG w ≤ max e 56+23:=by rw [aT]; (clear * - hw;omega)
  have e1:(tm x w:ℝ)=(sv x:ℝ)*2^(kG x)*((sv w:ℝ)*2^(kG w)):=by unfold tm; push_cast; ring
  rw [e1,Ae,Ae,Ae,AW (by positivity:(0:ℝ) < 2^kG x),AW (by positivity:(0:ℝ) < 2^kG w)]
  refine mul_le_mul_of_nonneg_left ?_ (by positivity)
  calc |(sv w:ℝ)| * 2^kG w ≤ 2^(24:Ἀ)*2^kG w:=AE a (by positivity)
    _=2^(24+kG w):=by rw [AF]
    _ ≤ 2^(max e 56+23):=AU (by norm_num) b
theorem kL1 (n:ℕ)(l:ℕ→Ἃ)(Lf:Ἅ)(A:∀ s<n,kB (l s)≠255)(hn27:n ≤ 134217728)
    (hL:Lf=ao (fun k=>dw (l k&&&0x7FFFFFFF)) n 0 0) :
    fv Lf ∧ |rv Lf - ∑ s∈kP n, |(sv (l s):ℝ)| * 2^(kG (l s))| ≤
      n*((1/9007199254740992+1/9444732965739290427392)*(∑ s∈kP n, |(sv (l s):ℝ)| * 2^(kG (l s)))+1/9007199254740992) ∧
    ∑ s∈kP n, |(sv (l s):ℝ)| * 2^(kG (l s)) ≤ 2^(304:Ἀ) ∧ |rv Lf| ≤ 2^(306:Ἀ):=by
  have A' : ∀ s < n,(kB (l s)).toNat≠255:=fun s hs h=>A s hs (Β.1 (by rw [h]; rfl))
  have Hl : ∀ s < n,fv (dw (l s&&&0x7FFFFFFF))∧rv (dw (l s&&&0x7FFFFFFF)) = |(sv (l s):ℝ)| * 2^(kG (l s)):=by
    intro s hs; obtain ⟨m1,m2,m3⟩:=kMsk (l s)
    have:=cI (l s&&&0x7FFFFFFF) (by rw [m1]; exact A s hs)
    exact ⟨this.1,by rw [this.2,m2,m3,mI]⟩
  have hcast : ∀ s, |(sv (l s):ℝ)| * 2^(kG (l s))=(((sv (l s)).natAbs*2^(kG (l s)):ℕ):ℝ):=fun s=>by
    rw [← mI,Ϲ,M5,Y4,I4,nZ]
  have hS : ∑ s∈kP n,(sv (l s)).natAbs*2^(kG (l s)) ≤ 2^(304:Ἀ):=by
    refine (Finset.sum_le_card_nsmul _ _ (2^(277:Ἀ)) fun s hs => ?_).trans ?_
    · exact (ϧ (bg (l s)).le (Θ Nat.zero_lt_two ((aT (l s)).trans_le
        (M8 (J2 (N4 (J2 (bm (l s))) (A' s (ϥ.1 hs)))))))).trans_eq
        (m7 2 24 253).symm
    · rw [Finset.card_range,smul_eq_mul]; exact (ι _ hn27).trans_eq (m7 2 27 277).symm
  have hnR:(n:ℝ) ≤ 134217728:=R5.2 hn27
  have hΛS:((∑ s∈kP n,(sv (l s)).natAbs*2^(kG (l s)):ℕ):ℝ) = ∑ s∈kP n, |(sv (l s):ℝ)| * 2^(kG (l s)):=by
    rw [Nat.cast_sum]; exact ϔ rfl fun s _=>(hcast s).symm
  generalize hΛ : ∑ s∈kP n, |(sv (l s):ℝ)| * 2^(kG (l s))=Λ at hΛS ⊢
  have hΛ0:0 ≤ Λ:=hΛS ▸ M3 _
  have hΛb:Λ ≤ 2^(304:Ἀ):=hΛS ▸ (R5.2 hS).trans_eq (by rw [I4,nZ])
  have h304:(2:ℝ)^(304:Ἀ) ≤ (2:ℝ)^(960:ℤ):=by rw [← AA]; exact AG one_le_two (by decide)
  have RL:=kSR (fun k=>dw (l k&&&0x7FFFFFFF)) (fun s => |(sv (l s):ℝ)| * 2^(kG (l s))) 0 Λ n hn27 Hl
    (by rw [A5,Ai, ← hΛ]; exact (ϔ rfl (fun s _=>by rw [hcast s,nE])).le) (hΛb.trans h304)
    n 0 0 (pQ n) (ϟ _) kZ0.1 (by rw [kZ0.2,X8,Nat.cast_zero,zero_mul,Av,sub_self,A5])
  rw [← hL,hΛ,Ai] at RL
  refine ⟨RL.1,RL.2,hΛb,?_⟩
  have t:=AY (rv Lf) Λ
  rw [Ad hΛ0] at t
  have X1:(1:ℝ) ≤ 2^(304:Ἀ):=(pow_zero (2:ℝ)).symm.trans_le (AU one_le_two (ϟ _))
  rw [A4 (2:ℝ) 305,A4 (2:ℝ) 304]
  linear_combination t.trans (RL.2.trans (AE hnR (add_nonneg (Af (by positivity) hΛ0) (by positivity))))+(70368745226241/70368744177664:ℝ)*hΛb+(211106231484415/70368744177664:ℝ)*X1
theorem kZq (n:ℕ)(Lf:Ἅ)(hn27:n ≤ 134217728)(hf:fv Lf)(hb:|rv Lf| ≤ 2^(306:Ἀ)) :
    fv ((I0 (9*(n+1))).toInt64.toFloat*Lf) ∧
    |rv ((I0 (9*(n+1))).toInt64.toFloat*Lf)-(9*((n:ℝ)+1))*rv Lf| ≤ 1/9007199254740992*|(9*((n:ℝ)+1))*rv Lf| + (2:ℝ)^(-1075:ℤ) ∧
    |rv ((I0 (9*(n+1))).toInt64.toFloat*Lf)| < (2:ℝ)^((cEq (I0 (9*(n+1))).toInt64.toFloat Lf).toNat+56) ∧
    (cEq (I0 (9*(n+1))).toInt64.toFloat Lf).toNat ≤ 969 :=
 have a:=U3 9 (Nat.succ_le_succ hn27)
 have FN:=mV53 (I0 (9*(n+1))) (9*(n+1)) (kOfS _ (a.trans_lt (by decide))) (a.trans_lt (by decide))
 have e:((9*(n+1):ℕ):ℝ)=9*((n:ℝ)+1):=(Y4 _ _).trans (congrArg₂ (·*·) nZ (X7 n))
 have hFn:=FN.2.trans ((M5 _).trans e)
 have Zr:=kRM _ Lf FN.1 hf (((Ae _ _).trans (congrArg (·*|rv Lf|) ((congrArg abs hFn).trans (Ad ((M3 _).trans_eq e))))).trans_le ((mul_le_mul (e.symm.trans_le (((R5 (α:=ℝ)).mpr (a.trans (by decide))).trans_eq ((I4 2 31).trans (congrArg (·^31) nZ)))) hb (Aa _) (pow_nonneg zero_le_two 31)).trans ((AF (2:ℝ) 31 306).symm.trans_le ((AA (2:ℝ) (31+306)).symm.trans_le (AG one_le_two (by decide))))))
 have CE:=kCE _ Lf Zr.1
 ⟨Zr.1,hFn▸(by norm_num [zpow_neg]:(2:ℝ)^(-53:ℤ)=1/9007199254740992)▸Zr.2,CE.1.trans_eq ((congrArg (HPow.hPow (2:ℝ)) (W5 _ 56).symm).trans (AA _ _)),CE.2⟩
theorem kWinS (n L ib e c:ℕ) (Λ lf zr B Sm:ℝ) (hn:0<n) (Hn:n+2 ≤ 2^L) (r:L≤27) (S:ib+L≤425) (hib:174 ≤ ib) (hL2:2 ≤ L)
    (hc:max e 56+23+c ≤ 372) (hΛ0:0 ≤ Λ)
    (R2:|lf-Λ| ≤ n*((1/9007199254740992+1/9444732965739290427392)*Λ+1/9007199254740992))
    (Z2:|zr-(9*((n:ℝ)+1))*lf| ≤ 1/9007199254740992*|(9*((n:ℝ)+1))*lf| + (2:ℝ)^(-1075:ℤ)) (Z3:|zr| < (2:ℝ)^(c+56))
    (hB0:0 ≤ B) (hB2:B*2 ≤ 2^ib) (hS0:0 ≤ Sm) (hSM:Sm ≤ 2^(max e 56+23)*Λ) :
    n*((1/9007199254740992+1/9444732965739290427392)*(B+Sm)+1/9007199254740992) ≤ 2^(max (ib+L-50) (max e 56+23+c)) ∧
    B+Sm+2^(max (ib+L-50) (max e 56+23+c)) ≤ 2^(max (ib+L-50) (max e 56+23+c))*4503599627370496∧B+Sm ≤ 2^(425:Ἀ):=by
  have hT:(2:ℝ)^L ≤ 134217728:=by rw [show (134217728:ℝ)=2^(27:Ἀ) by norm_num]; exact AU (by norm_num) r
  have hT4:(4:ℝ) ≤ 2^L:=by rw [show (4:ℝ)=2^(2:Ἀ) by norm_num]; exact AU (by norm_num) hL2
  have hnT:(n:ℝ)+2 ≤ 2^L:=by exact_mod_cast Hn
  have hv:(2:ℝ)^(-1075:ℤ) ≤ 1/1208925819614629174706176:=by
    exact (AG (by norm_num) (by norm_num:(-1075:ℤ) ≤ -80)).trans_eq (by norm_num)
  exact kWinR (n:ℝ) (2^L) Λ lf zr B Sm (2^(max e 56+23)) (2^(c+56)) (2^(max e 56+23+c))
    (2^(ib+L-50)) (2^(max (ib+L-50) (max e 56+23+c))) (2^ib) (2^(425:Ἀ)) (2^(-1075:ℤ))
    (Nat.one_le_cast.2 hn) hnT hT hT4 hΛ0 R2 (by positivity) hv Z2 Z3
    (by rw [show (72057594037927936:ℝ)=2^(56:Ἀ) by norm_num]; exact AU (by norm_num) (by exact K2 56 c))
    hB0 hB2 hS0 hSM (by positivity)
    (by rw [show (72057594037927936:ℝ)=2^(56:Ἀ) by norm_num, ← AF, ← AF, ← pA])
    (by rw [show (1125899906842624:ℝ)=2^(50:Ἀ) by norm_num, ← AF, ← AF,L3 (Nat.le_add_right_of_le ((by norm_num:50 ≤ 174).trans hib)),Τ])
    (AU (by norm_num) (le_max_right _ _)) (AU (by norm_num) (le_max_left _ _))
    (by rw [show (21267647932558653966460912964485513216:ℝ)=2^(124:Ἀ) by norm_num, ← AF]; exact AU (by norm_num) (G5 ((lO 124 L 50).trans_le (L1 hib L))))
    (by rw [show (9007199254740992:ℝ)=2^(53:Ἀ) by norm_num, ← AF]; exact AU (by norm_num) (L1 hc 53))
    (by rw [show (4:ℝ)=2^(2:Ἀ) by norm_num, ← AF]; exact AU (by norm_num) ((J1 hL2 ib).trans S))
theorem kWin (n:ℕ)(l:ℕ→Ἃ)(W b:Ἂ Ἃ)(j e ib L H:ℕ)(p:ℕ→Ἅ)(Lf:Ἅ)
    (A:∀ s<n,kB (l s)≠255)
    (hw:∀ s<n,(kB W[j*n+s]!).toNat ≤ e)(he:e<255)
    (K:kB b[j]! ≠ 255∧kG b[j]! ≤ H)(al:174+H ≤ ib)(Hn:n+2 ≤ 2^L)(r:L≤27)(S:ib+L≤425)(hn:0<n)
    (hp:∀ s<n,fv (p s)∧rv (p s)=tm (l s) W[j*n+s]!)
    (hL:Lf=ao (fun k=>dw (l k&&&0x7FFFFFFF)) n 0 0)
    (hc: max e 56+23+(cEq (I0 (9*(n+1))).toInt64.toFloat Lf).toNat ≤ 372) :
    aG (n+1) (fun s=>if s < n then l s else ld) (fun s=>if s < n then W[j*n+s]! else b[j]!) ∧
    fv (ao p n 0 (dw ld*dw b[j]!)) ∧
    |rv (ao p n 0 (dw ld*dw b[j]!))-(af (n+1) (fun s=>if s < n then l s else ld) (fun s=>if s < n then W[j*n+s]! else b[j]!):ℝ)| ≤ (2:ℝ)^(((max (ib+L-50) (max e 56+23+(cEq (I0 (9*(n+1))).toInt64.toFloat Lf).toNat)):ℕ):ℤ) ∧
    |(af (n+1) (fun s=>if s < n then l s else ld) (fun s=>if s < n then W[j*n+s]! else b[j]!):ℝ)| ≤ (2:ℝ)^(425:ℤ) ∧
    |rv (ao p n 0 (dw ld*dw b[j]!))| ≤ (2:ℝ)^(((max (ib+L-50) (max e 56+23+(cEq (I0 (9*(n+1))).toInt64.toFloat Lf).toNat))+52:ℕ):ℤ):=by
  have hL2:2 ≤ L:=W4 fun hl=>absurd (Nat.le_of_add_le_add_right (Hn.trans (Θ Nat.zero_lt_two (J2 hl)):n+2 ≤ 0+2)) (Nat.not_le_of_lt hn)
  have hn27:n ≤ 134217728:=(V1 n 2).trans (Hn.trans (Θ Nat.zero_lt_two r))
  obtain ⟨R1,R2,R3,R4⟩:=kL1 n l Lf A hn27 hL
  obtain ⟨Z1,Z2,Z3, -⟩:=kZq n Lf hn27 R1 R4
  have Ew : ∀ s < n,kB W[j*n+s]! ≠ 255:=fun s hs h'=>absurd (hw s hs) (h' ▸ Nat.not_le_of_lt he)
  have hT:=K8 (fun s=>(tm (l s) W[j*n+s]!:ℝ)) (kP n)
  generalize hSm : ∑ s∈kP n, |(tm (l s) W[j*n+s]!:ℝ)| = Sm at hT
  have hS0:0 ≤ Sm:=(Aa _).trans hT
  have hSM:Sm ≤ 2^(max e 56+23) * ∑ s∈kP n, |(sv (l s):ℝ)| * 2^(kG (l s)):=by
    rw [← hSm,Finset.mul_sum]; exact Finset.sum_le_sum (fun s hs=>(kTW _ _ e (hw s (ϥ.1 hs))).trans_eq (mul_comm _ _))
  have hΛ0:=nonneg_of_mul_nonneg_right (hS0.trans hSM) (pow_pos (by norm_num) _)
  have Wn:=kWinS n L ib e _ _ _ _ _ Sm hn Hn r S (Nat.le_of_add_right_le al) hL2 hc hΛ0 R2 Z2 Z3 (Aa _) (kTB2 _ H ib K.2 al) hS0 hSM
  have hld:kB ld≠255:=by decide
  have hd:=dJ ld b[j]! hld K.1
  have RV:=kSR p (fun s=>(tm (l s) W[j*n+s]!:ℝ)) (tm ld b[j]!:ℝ) (|(tm ld b[j]!:ℝ)|+Sm) n hn27 hp (by rw [hSm]) (Wn.2.2.trans ((AU (by norm_num) (N2 rfl:425 ≤ 960)).trans_eq (AA (2:ℝ) 960).symm))
    n 0 _ (pQ n) (ϟ _) hd.1 (by rw [hd.2,X8,Av,sub_self,A5,Nat.cast_zero,zero_mul])
  have hAF:(af (n+1) (fun s=>if s < n then l s else ld) (fun s=>if s < n then W[j*n+s]! else b[j]!):ℝ)=(tm ld b[j]!:ℝ) + ∑ s∈kP n,(tm (l s) W[j*n+s]!:ℝ):=by
    rw [aK,Int.cast_add,af,Int.cast_sum,add_comm,if_neg (AX n),if_neg (AX n)]
    exact congrArg _ (ϔ rfl (fun s hs=>have h:=ϥ.1 hs; congrArg₂ (fun a b=>((tm a b:ℤ):ℝ)) (if_pos h) (if_pos h)))
  have hsum:=(Ag (tm ld b[j]!:ℝ) _).trans (add_le_add AP hT)
  rw [hAF]
  refine ⟨?_,RV.1,?_,?_,?_⟩
  · intro k hk; by_cases hkn:k < n
    · simp only [if_pos hkn]; exact ⟨A k hkn,Ew k hkn⟩
    · simp only [if_neg hkn]; exact ⟨hld,K.1⟩
  · exact (RV.2.trans Wn.1).trans_eq (AA _ _).symm
  · exact (hsum.trans Wn.2.2).trans_eq (AA (2:ℝ) 425).symm
  · rw [AA,AF,show (2:ℝ)^(52:Ἀ)=4503599627370496 by norm_num]
    exact ((sub_le_iff_le_add'.1 (AY _ _)).trans (add_le_add hsum (RV.2.trans Wn.1))).trans Wn.2.1
theorem kAD2 (n:ℕ) (relu:Ἆ) (X W b:Ἂ Ἃ) (xo o:ℕ) :
    aD n relu X W b xo 1 o=act relu (dotOf (n+1) (fun t=>if t < n then X[xo+t]! else oneW)
      (fun t=>if t < n then W[o*n+t]! else b[o]!)):=by
  unfold aD; rw [bO]; simp only [ϻ,ϻ]
theorem kZQ2 (v:Ἅ)(e cq hb:ἆ)(relu:Ἆ)(n:ℕ)(D Wf:ℕ→Ἃ)(Fb:Ἃ)(hfb:Fb=p8 relu (ms (n+1) D Wf))
    (hw:∀ p:ἆ,p=(if e<(56:ἆ) then (56:ἆ) else e)+cq → e<(255:ἆ) → p ≤ (372:ἆ) →
      aG (n+1) D Wf∧fv v ∧ |rv v-(af (n+1) D Wf:ℝ)| ≤ (2:ℝ)^(((if hb>p then hb else p).toNat:ℕ):ℤ) ∧
      |(af (n+1) D Wf:ℝ)| ≤ (2:ℝ)^(425:ℤ) ∧ |rv v| ≤ (2:ℝ)^((((if hb>p then hb else p).toNat)+52:ℕ):ℤ)∧(if hb>p then hb else p).toNat ≤ 600) :
    (if zq2 v e relu cq hb==(0xFFFFFFFF:Ἃ) then Fb else zq2 v e relu cq hb)=p8 relu (ms (n+1) D Wf):=by
  unfold zq2; dsimp only
  generalize hp:(if e<(56:ἆ) then (56:ἆ) else e)+cq=p
  by_cases W:e < 255∧p ≤ 372
  · obtain ⟨g,fv1,b1,b2,_,b6⟩:=hw p hp.symm W.1 W.2
    have cT:(decide (e < 255)&&decide (p ≤ 372))=true:=by simp [W.1,W.2]
    simp only [cT,ἁ]
    generalize hP:(if hb>p then hb else p)=P at b1 b6 ⊢
    by_cases hc:ak v P < (4294967296:ἆ)
    · have R:=cw (n+1) D Wf g v fv1 P (by (clear * - b6;omega)) b1 b2 hc
      rw [if_pos hc,kReluW relu _ (kAkF v P hc),R]
      by_cases hf:p8 relu (ms (n+1) D Wf)=0xFFFFFFFF
      · rw [hf]; simp only [Az,ἁ]; rw [hfb,hf]
      · have hn:(p8 relu (ms (n+1) D Wf)==0xFFFFFFFF)=false:=by simpa using hf
        simp only [hn,υ,ἂ]
    · simp only [if_neg hc,Az,ἁ]; exact hfb
  · have cF:(decide (e < 255)&&decide (p ≤ 372))=false:=Bool.and_eq_false_iff.2 ((not_and_or.1 W).imp decide_eq_false decide_eq_false)
    simp only [cF,υ,ἂ,Az,ἁ]; exact hfb
theorem kSQ2 (relu:Ἆ)(n:ℕ)(X W b E:Ἂ Ἃ)(ix ib L H:ℕ)(Y Q:ἀ)(vo rn j:ℕ)(hh:ἆ)(v:Ἅ)
    (F:∀ q:ℕ,kB X[q]! ≠ 255∧kG X[q]! ≤ ix)(hbF:∀ q:ℕ,kB b[q]! ≠ 255∧kG b[q]! ≤ H)(M:174+H ≤ ib)(Hn:n+2 ≤ 2^L)(x:L≤27)(z:ib+L≤425)
    (hEv:∀ s<n,(kB W[j*n+s]!).toNat ≤ (E[j]!).toNat)
    (hYv:∀ s<n,Y.data[vo+s]! = dw X[rn+s]!)
    (hQv:∀ s<n,Q.data[j/8*n*8+j%8+8*s]! = dw W[j*n+s]!)
    (hv:v=ao (fun s=>Y.data[vo+s]! * Q.data[j/8*n*8+j%8+8*s]!) n 0 (dw ld*dw b[j]!))
    (hw:(E[j]!).toUInt64 < 255 → hh ≤ 372 →
      aG (n+1) (fun s=>if s < n then X[rn+s]! else ld) (fun s=>if s < n then W[j*n+s]! else b[j]!)∧fv v ∧
      |rv v-(af (n+1) (fun s=>if s < n then X[rn+s]! else ld) (fun s=>if s < n then W[j*n+s]! else b[j]!):ℝ)| ≤ (2:ℝ)^((hh.toNat:ℕ):ℤ) ∧
      |(af (n+1) (fun s=>if s < n then X[rn+s]! else ld) (fun s=>if s < n then W[j*n+s]! else b[j]!):ℝ)| ≤ (2:ℝ)^(425:ℤ)) :
    sQ2 relu n X W b E ix ib L Y Q vo rn j hh v=p8 relu (ms (n+1) (fun s=>if s < n then X[rn+s]! else ld) (fun s=>if s < n then W[j*n+s]! else b[j]!)):=by
  by_cases hc:(b[j]! == 0&&(E[j]!).toUInt64<(255:ἆ)&&zrRw X rn n)=true
  ·unfold sQ2;rw[if_pos hc];simp only[Η,AQ,AJ]at hc;obtain⟨⟨hb0,hE⟩,hz⟩:=hc
   have hE':(E[j]!).toNat<255:=by have:=μ.1 hE;simpa using this
   rw[zrDot n _ _ (fun s hs=>by simp only[hs,if_true];exact zrRw_ok X rn n hz s hs) (fun s hs h=>by simp only[hs,if_true]at h;have:=hEv s hs;rw[h]at this;simp at this;omega) (by simp) (by simp[hb0])];cases relu<;>decide
  have hp : ∀ k < n,kB W[j*n+k]! ≠ 255 → fv ((fun s=>Y.data[vo+s]! * Q.data[j/8*n*8+j%8+8*s]!) k) ∧
      rv ((fun s=>Y.data[vo+s]! * Q.data[j/8*n*8+j%8+8*s]!) k)=tm ((fun s=>X[rn+s]!) k) W[j*n+k]! := by
    intro k hk h; simp only; rw [hYv k hk,hQv k hk]; exact dJ _ _ (F _).1 h
  have hp' : ∀ k < n,kB W[j*n+k]! ≠ 255 → fv ((fun s=>Y.get! (vo+s)*Q.get! (j/8*n*8+j%8+8*s)) k) ∧
      rv ((fun s=>Y.get! (vo+s)*Q.get! (j/8*n*8+j%8+8*s)) k)=tm ((fun s=>X[rn+s]!) k) W[j*n+k]! := by
    intro k hk h; simp only [gg]; rw [hYv k hk,hQv k hk]; exact dJ _ _ (F _).1 h
  have G:=kGB1T relu n n j (fun s=>X[rn+s]!) W b (fun k=>k) (fun s=>Y.data[vo+s]! * Q.data[j/8*n*8+j%8+8*s]!)
    (fun s=>Y.get! (vo+s)*Q.get! (j/8*n*8+j%8+8*s)) E[j]! ix ib L H (fun _=>aD n relu X W b rn 1 j)
    (fun s _=>F _) hEv (hbF j) M Hn x z AP (fun k hk=>hk) (fun _ _ h1 _=>h1) (fun s hs h=>absurd rfl (h s hs)) hp hp' (kAD2 _ _ _ _ _ _ _)
  rw [← hv] at G
  unfold sQ2; rw[if_neg hc]; dsimp only
  simp only [sTQ_eq,tR_tS]
  split
  · rename_i hg
    simp only [Η,AJ] at hg
    obtain ⟨g,fv1,b1,b2⟩:=hw hg.1 hg.2
    by_cases RZ:(relu&&(v.toBits>>>63!=0)&&hh+1024<=eF v.toBits)=true
    · unfold tf; rw [if_pos RZ]
      exact kTF0 relu (n+1) _ _ g v fv1 hh (by have:=θ.1 hg.2; simp at this; (clear * - this;omega)) b1 RZ
    unfold tf; rw [if_neg RZ]; dsimp only; split
    · rename_i hk
      rw [cw (n+1) _ _ g v fv1 hh (by have:=θ.1 hg.2; simp at this; (clear * - this;omega)) b1 b2 hk]
    · exact G
  · exact G
theorem kCEb (F a:Ἅ):(cEq F a).toNat ≤ 969:=by
  have hb:(eF (F*a).toBits).toNat < 2048:=by unfold eF; rw [Ε]; exact AO ϝ (by decide)
  unfold cEq; dsimp only; split
  · rename_i hc
    rw [χ _ _ (UInt64.le_of_lt hc)]; simp only [kA,tsub_le_iff_right,Nat.reduceAdd,ge_iff_le]; exact J2 hb
  · simp only [ϋ,zero_le]
theorem kOutW_h1 (e c:ἆ) (h:e.toNat < 255) (hc:c.toNat ≤ 969) :
    ((if e < (56:ἆ) then (56:ἆ) else e)+(23+c)).toNat=max e.toNat 56+23+c.toNat:=by
  have q:(23+c).toNat=23+c.toNat:=γ (C0 (J1 hc 23) (by decide))
  have m:(if e < (56:ἆ) then (56:ἆ) else e).toNat=max e.toNat 56:=by
    by_cases g:e < (56:ἆ)
    · rw [if_pos g]; exact (Nat.max_eq_right (X4 (show e.toNat < 56 from g))).symm
    · rw [if_neg g]; exact (Nat.max_eq_left (W4 (show ¬ e.toNat < 56 from g))).symm
  rw [η,m,q, ← pA]
  exact γ (C0 (Y3 (L1 (Nat.max_le.2 ⟨X4 h,by decide⟩) 23) hc) (by decide))
theorem kOutW_h2 (a b:ἆ):(if a > b then a else b).toNat=max a.toNat b.toNat:=by
  by_cases g:a > b
  · rw [if_pos g]; exact (Nat.max_eq_left (X4 (show b.toNat < a.toNat from g))).symm
  · rw [if_neg g]; exact (Nat.max_eq_right (W4 (show ¬ b.toNat < a.toNat from g))).symm
theorem kOutW (n:ℕ)(X W b E:Ἂ Ἃ)(ix ib L H:ℕ)(Y Q:ἀ)(vo rn j:ℕ)(cq hb p:ἆ)(v Lf:Ἅ)
    (F:∀ q:ℕ,kB X[q]! ≠ 255∧kG X[q]! ≤ ix)(hbF:∀ q:ℕ,kB b[q]! ≠ 255∧kG b[q]! ≤ H)(M:174+H ≤ ib)(Hn:n+2 ≤ 2^L)(x:L≤27)(z:ib+L≤425)(hn:0<n)
    (hEv:∀ s<n,(kB W[j*n+s]!).toNat ≤ (E[j]!).toNat)
    (hYv:∀ s<n,Y.data[vo+s]! = dw X[rn+s]!)
    (hQv:∀ s<n,Q.data[j/8*n*8+j%8+8*s]! = dw W[j*n+s]!)
    (hv:v=ao (fun s=>Y.data[vo+s]! * Q.data[j/8*n*8+j%8+8*s]!) n 0 (dw ld*dw b[j]!))
    (hLf:Lf=ao (fun k=>dw (X[rn+k]! &&& 0x7FFFFFFF)) n 0 0)
    (hcq:cq=23+cEq (I0 (9*(n+1))).toInt64.toFloat Lf) (hhb:hb=(ib+L-50).toUInt64)
    (hp:p=(if (E[j]!).toUInt64<(56:ἆ) then (56:ἆ) else (E[j]!).toUInt64)+cq) (he:(E[j]!).toUInt64 < 255) (hp372:p ≤ 372) :
    aG (n+1) (fun s=>if s < n then X[rn+s]! else ld) (fun s=>if s < n then W[j*n+s]! else b[j]!)∧fv v ∧
    |rv v-(af (n+1) (fun s=>if s < n then X[rn+s]! else ld) (fun s=>if s < n then W[j*n+s]! else b[j]!):ℝ)| ≤ (2:ℝ)^(((if hb>p then hb else p).toNat:ℕ):ℤ) ∧
    |(af (n+1) (fun s=>if s < n then X[rn+s]! else ld) (fun s=>if s < n then W[j*n+s]! else b[j]!):ℝ)| ≤ (2:ℝ)^(425:ℤ) ∧
    |rv v| ≤ (2:ℝ)^((((if hb>p then hb else p).toNat)+52:ℕ):ℤ)∧(if hb>p then hb else p).toNat ≤ 600:=by
  have eN:=kE64 E[j]!
  have c:=kCEb (I0 (9*(n+1))).toInt64.toFloat Lf
  generalize hcc:(cEq (I0 (9*(n+1))).toInt64.toFloat Lf)=cc at hcq c
  have e:(E[j]!).toNat < 255:=he
  have tp:p.toNat=max (E[j]!).toNat 56+23+cc.toNat:=by rw [hp,hcq]; exact kOutW_h1 _ _ e c
  have hk:max (E[j]!).toNat 56+23+cc.toNat ≤ 372:=tp ▸ hp372
  have Wn:=kWin n (fun s=>X[rn+s]!) W b j (E[j]!).toNat ib L H (fun s=>Y.data[vo+s]! * Q.data[j/8*n*8+j%8+8*s]!) Lf
    (fun s _=>(F _).1) hEv e (hbF j) M Hn x z hn (fun s hs=>by rw [hYv s hs,hQv s hs]; exact dJ _ _ (F _).1 (fun h'=>absurd (hEv s hs) (h' ▸ Nat.not_le_of_lt e))) hLf (hcc ▸ hk)
  rw [hcc] at Wn
  rw [kOutW_h2,hhb,kOfS _ (C0 (U2 (K4 _ _) z) (by decide)),tp,hv]
  exact ⟨Wn.1,Wn.2.1,Wn.2.2.1,Wn.2.2.2.1,Wn.2.2.2.2,Nat.max_le.2 ⟨U2 (K4 _ _) (U2 z (by decide)),U2 hk (by decide)⟩⟩
theorem kOut (relu:Ἆ)(n:ℕ)(X W b E:Ἂ Ἃ)(ix ib L H:ℕ)(Y Q:ἀ)(vo rn j:ℕ)(cq hb:ἆ)(v Lf:Ἅ)
    (F:∀ q:ℕ,kB X[q]! ≠ 255∧kG X[q]! ≤ ix)(hbF:∀ q:ℕ,kB b[q]! ≠ 255∧kG b[q]! ≤ H)(M:174+H ≤ ib)(Hn:n+2 ≤ 2^L)(x:L≤27)(z:ib+L≤425)(hn:0<n)
    (hEv:∀ s<n,(kB W[j*n+s]!).toNat ≤ (E[j]!).toNat)
    (hYv:∀ s<n,Y.data[vo+s]! = dw X[rn+s]!)
    (hQv:∀ s<n,Q.data[j/8*n*8+j%8+8*s]! = dw W[j*n+s]!)
    (hv:v=ao (fun s=>Y.data[vo+s]! * Q.data[j/8*n*8+j%8+8*s]!) n 0 (dw ld*dw b[j]!))
    (hLf:Lf=ao (fun k=>dw (X[rn+k]! &&& 0x7FFFFFFF)) n 0 0)
    (hcq:cq=23+cEq (I0 (9*(n+1))).toInt64.toFloat Lf) (hhb:hb=(ib+L-50).toUInt64) :
    (if zq2 v (E[j]!).toUInt64 relu cq hb==(0xFFFFFFFF:Ἃ) then
      sQ2 relu n X W b E ix ib L Y Q vo rn j (if hb > (if (E[j]!).toUInt64<(56:ἆ) then (56:ἆ) else (E[j]!).toUInt64)+cq then hb else (if (E[j]!).toUInt64<(56:ἆ) then (56:ἆ) else (E[j]!).toUInt64)+cq) v
     else zq2 v (E[j]!).toUInt64 relu cq hb) =
    p8 relu (ms (n+1) (fun s=>if s < n then X[rn+s]! else ld) (fun s=>if s < n then W[j*n+s]! else b[j]!)):=by
  refine kZQ2 v _ cq hb relu n _ _ _ ?_ (fun p hp he hp372=>kOutW n X W b E ix ib L H Y Q vo rn j cq hb p v Lf F hbF M Hn x z hn hEv hYv hQv hv hLf hcq hhb hp he hp372)
  refine kSQ2 relu n X W b E ix ib L H Y Q vo rn j _ v F hbF M Hn x z hEv hYv hQv hv (fun he hh => ?_)
  have hle:(if (E[j]!).toUInt64<(56:ἆ) then (56:ἆ) else (E[j]!).toUInt64)+cq ≤ 372:=by
    refine ϗ ?_ hh
    generalize (if (E[j]!).toUInt64<(56:ἆ) then (56:ἆ) else (E[j]!).toUInt64)+cq=a
    split
    · exact UInt64.le_of_lt (by assumption)
    · exact UInt64.le_refl _
  have:=kOutW n X W b E ix ib L H Y Q vo rn j cq hb _ v Lf F hbF M Hn x z hn hEv hYv hQv hv hLf hcq hhb rfl he hle
  exact ⟨this.1,this.2.1,this.2.2.1,this.2.2.2.1⟩
theorem kFS (Y:ἀ)(i:Ἄ)(v:Ἅ)(h:(J0 i)<(kS Y))(q:ℕ):(Y.uset i v h).data[q]! = if (J0 i)=q then v else Y.data[q]! := by
  by_cases hq:q < (kS Y)
  · exact fuS Y i v h q hq
  · have:(J0 i)≠q:=by omega
    rw [if_neg this]; cases Y; simp only [FloatArray.uset,kS] at hq h ⊢
    rw [An _ _ (by simp only [pP,Array.size_set,not_lt]; exact W4 hq),An _ _ (by exact Not.imp hq fun a=>a)]
theorem dRw_spec (X:Ἂ ἇ)(i n o:Ἄ)(hX:(J0 i)+(J0 n)<=(O0 X))(hB:(J0 i)+(J0 n)<2^(31:Ἀ)/\(J0 o)+(J0 n)<2^(31:Ἀ)) :
    ∀ m (k:Ἄ)(Y:ἀ)(a:Ἅ)(hY:(J0 o)+(J0 n)<(kS Y)),(J0 n)-(J0 k)=m → (J0 k) ≤ (J0 n) → ∀ q,
    (dRw X i n o k Y a hX hY hB).data[q]! = if q=(J0 o)+(J0 n) then ao (fun t=>dw (X[(J0 i)+t]! &&& 0x7FFFFFFF)) (J0 n) (J0 k) a
      else if (J0 o)+(J0 k) ≤ q∧q < (J0 o)+(J0 n) then dw X[(J0 i)+(q-(J0 o))]! else Y.data[q]! := by
  intro m; induction m with
  | zero =>
    intro k Y a hY hm hk q
    have e:=O4 hm
    rw[dRw,dif_neg (mt ε.mp (W3.mpr e)),kFS,kSm o n hB.2,ao,if_neg (W3.mpr e),if_neg (fun(g:_∧_)=>W3.mpr (U2 (J1 e _) g.1) g.2)]
    exact ite_cond_congr (propext eq_comm)
  | succ m ih =>
    intro k Y a hY hm hk q
    have h1:=N1 hm
    have h2:=ac (ε.mpr h1)
    rw[ao,if_pos h1,dRw,dif_pos (ε.mpr h1)]; dsimp only
    rw[ih (k+1) _ _ _ (by rw[h2,← F7,hm,BK]) (by rw[h2]; exact h1) q,h2,kFS,kSm o k (K5 (O1 h1 _) hB.2),kXU _ _ _ _ (kSm i k (K5 (O1 h1 _) hB.1)),kAbsDw]
    refine congrArg (ite _ _) ?_
    by_cases g2:oб+(kб+1)≤q∧q<oб+nб
    · rw[if_pos g2,if_pos (And.intro (U2 (J1 (E7 _) _) g2.1) g2.2)]
    · rw[if_neg g2]
      by_cases g3:oб+kб=q
      · rw[if_pos g3,← g3,if_pos (And.intro (Nat.le_refl _) (O1 h1 _)),ϱ]
      · rw[if_neg g3,if_neg (fun(g:oб+kб≤q∧_)=>g2 ⟨N4 g.1 g3,g.2⟩)]
theorem dRw8_spec (X:Ἂ ἇ)(i n r:Ἄ)(hX:(J0 i)+(J0 n)<=(O0 X))(hB:(J0 i)+(J0 n)<2^31/\8*(J0 n)+7<2^31)(hr:(J0 r)<8) :
    ∀ m (k:Ἄ)(Y:ἀ)(a:Ἅ)(hY:8*(J0 n)+8<=(kS Y)),(J0 n)-(J0 k)=m → (J0 k) ≤ (J0 n) → ∀ q,
    (dRw8 X i n r k Y a hX hY hB hr).data[q]! = if q=8*(J0 n)+(J0 r) then ao (fun t=>dw (X[(J0 i)+t]! &&& 0x7FFFFFFF)) (J0 n) (J0 k) a
      else if q%8=(J0 r)∧(J0 k) ≤ q/8∧q/8 < (J0 n) then dw X[(J0 i)+q/8]! else Y.data[q]! := by
  intro m; induction m with
  | zero =>
    intro k Y a hY hm hk q
    have e:(J0 k)=(J0 n):=C4 hk (O4 hm)
    have nl:¬ (J0 k) < (J0 n):=fun h=>B4 _ (h.trans_le e.ge)
    have eao:ao (fun t=>dw (X[(J0 i)+t]! &&& 0x7FFFFFFF)) (J0 n) (J0 k) a=a:=by rw [ao,if_neg nl]
    rw [dRw8,dif_neg (show ¬ k < n from fun h=>nl (ε.mp h)),kFS,k8i n r (J0 n) (le_refl _) hB.2 hr,eao,
      if_neg (show ¬(q%8=(J0 r)∧(J0 k) ≤ q/8∧q/8 < (J0 n)) from fun h=>nl (h.2.1.trans_lt h.2.2))]
    by_cases hq:q=8*(J0 n)+(J0 r)
    · rw [if_pos hq.symm,if_pos hq]
    · rw [if_neg (show ¬(8*(J0 n)+(J0 r)=q) from fun h=>hq h.symm),if_neg hq]
  | succ m ih =>
    intro k Y a hY hm hk q
    have h1:(J0 k) < (J0 n):=N1 hm
    have a1:=ac (ε.mpr h1)
    rw [dRw8,dif_pos (ε.mpr h1)]
    rw [ih (k+1) _ _ _ (by rw [a1,V4,hm,BK]) (by rw [a1]; exact h1) q,a1,kFS,k8i k r (J0 n) h1.le hB.2 hr]
    rw [kXU X (i+k) _ ((J0 i)+(J0 k)) (kSm i k ((O1 h1 _).trans hB.1))]
    have eao:ao (fun t=>dw (X[(J0 i)+t]! &&& 0x7FFFFFFF)) (J0 n) (J0 k) a=ao (fun t=>dw (X[(J0 i)+t]! &&& 0x7FFFFFFF)) (J0 n) ((J0 k)+1) (a+dw (X[(J0 i)+(J0 k)]! &&& 0x7FFFFFFF)):=by
      rw [ao,if_pos h1]
    rw [kAbsDw, ← eao]
    by_cases hq:q=8*(J0 n)+(J0 r)
    · rw [if_pos hq,if_pos hq]
    rw [if_neg hq,if_neg hq]
    by_cases hA:q%8=(J0 r)∧(J0 k)+1 ≤ q/8∧q/8 < (J0 n)
    · rw [if_pos hA,if_pos (show q%8=(J0 r)∧(J0 k) ≤ q/8∧q/8 < (J0 n) from ⟨hA.1,X5 hA.2.1,hA.2.2⟩)]
    rw [if_neg hA]
    by_cases hk8:8*(J0 k)+(J0 r)=q
    · subst hk8
      have d1:(8*(J0 k)+(J0 r))/8=(J0 k):=by rw [W8 (by decide),ώ hr,ϑ]
      have d2:(8*(J0 k)+(J0 r))%8=(J0 r):=by rw [O7,γ hr]
      rw [if_pos rfl,if_pos (show (8*(J0 k)+(J0 r))%8=(J0 r)∧(J0 k) ≤ (8*(J0 k)+(J0 r))/8∧(8*(J0 k)+(J0 r))/8 < (J0 n) from ⟨d2,d1.ge,by rw [d1]; exact h1⟩),d1]
    · have hn:¬(q%8=(J0 r)∧(J0 k) ≤ q/8∧q/8 < (J0 n)):=by
        intro h; apply hk8
        have e8:q/8=(J0 k):=C4 (J2 (O8 fun h'=>hA ⟨h.1,h',h.2.2⟩)) h.2.1
        rw [← e8, ← h.1]; exact ϛ q 8
      rw [if_neg hk8,if_neg hn]
theorem dX8_spec_h1{α:Type}(f:ℕ→ℕ→α)(g:ℕ→α)(Yq YD FR GR:α)(q N R L R1 k0 j s:ℕ)(hq:8*j+s=q)(hs8:s < 8)(hr8:R < 8)(h1:R < L)(hR1:R1=R+1)(hk0:k0=0)(hFR:FR=f R j)(hGR:GR=g R)(hYD:YD=if q=8*N+R then GR else if s=R∧k0 ≤ j∧j < N then FR else Yq):
  (if q < 8*N+8∧R1 ≤ s∧s < L then (if j < N then f s j else g s) else YD)=if q < 8*N+8∧R ≤ s∧s < L then (if j < N then f s j else g s) else Yq:=by
  subst hR1 hk0 hFR hGR hYD hq
  by_cases hs:s=R
  · subst hs
    rw [if_neg (fun h=>Nat.not_succ_le_self s h.2.1)]
    rcases Nat.lt_trichotomy j N with hj|hj|hj
    · rw [if_neg (fun e=>R8 hj (Nat.eq_of_mul_eq_mul_left (by decide) (N8 e))),if_pos ⟨rfl,ϟ _,hj⟩,if_pos ⟨(O1 hs8 _).trans_le (L1 (U3 8 hj.le) 8),le_refl s,h1⟩,if_pos hj]
    · subst hj
      rw [if_pos rfl,if_pos ⟨O1 hs8 _,le_refl s,h1⟩,if_neg (B4 j)]
    · rw [if_neg (fun e=>L6 hj (Nat.eq_of_mul_eq_mul_left (by decide) (N8 e))),if_neg (fun h=>Nat.lt_asymm h.2.2 hj),if_neg (fun h=>W3.2 (((Nat.mul_succ 8 N).symm.trans_le (U3 8 hj)).trans (V1 _ _)) h.1)]
  · by_cases hc:8*j+s < 8*N+8∧R ≤ s∧s < L
    · rw [if_pos ⟨hc.1,N4 hc.2.1 (Ne.symm hs),hc.2.2⟩,if_pos hc]
    · rw [if_neg (fun h=>hc ⟨h.1,X5 h.2.1,h.2.2⟩),if_neg (fun e=>hs (((O7 8 j s).trans (γ hs8)).symm.trans ((congrArg (·%8) e).trans ((O7 8 N R).trans (γ hr8))))),if_neg (fun h=>hs h.1),if_neg hc]
theorem dX8_spec (X:Ἂ ἇ)(b n lim:Ἄ)(hX:(J0 b)+(J0 lim)*(J0 n)<=(O0 X))(hB:(J0 b)+(J0 lim)*(J0 n)<2^31/\8*(J0 n)+7<2^31)(hl:(J0 lim)<=8) :
    ∀ m (r:Ἄ)(Y:ἀ)(hY:8*(J0 n)+8<=(kS Y)),(J0 lim)-(J0 r)=m → (J0 r) ≤ (J0 lim) → ∀ q,
    (dX8 X b n lim r Y hX hY hB hl).data[q]! = if q < 8*(J0 n)+8∧(J0 r) ≤ q%8∧q%8 < (J0 lim) then
      (if q/8 < (J0 n) then dw X[(J0 b)+(q%8)*(J0 n)+q/8]! else ao (fun t=>dw (X[(J0 b)+(q%8)*(J0 n)+t]! &&& 0x7FFFFFFF)) (J0 n) 0 0) else Y.data[q]! := by
  intro m; induction m with
  | zero =>
    intro r Y hY hm hr q
    have e:(J0 r)=(J0 lim):=C4 hr (O4 hm)
    rw [dX8,dif_neg (fun h=>B4 _ ((ε.mp h).trans_le e.ge))]; exact (if_neg (fun h=>B4 _ (h.2.1.trans_lt (h.2.2.trans_le e.ge)))).symm
  | succ m ih =>
    intro r Y hY hm hr q
    have h1:(J0 r) < (J0 lim):=N1 hm
    have a1:=ac (ε.mpr h1)
    have K:=kBRn b r n (J0 lim) h1 hB.1
    have hr8:(J0 r) < 8:=h1.trans_le hl
    rw [dX8,dif_pos (ε.mpr h1)]; exact (ih (r+1) _ _ ((congrArg ((J0 lim)-·) a1).trans ((Nat.sub_succ _ _).trans (congrArg Nat.pred hm))) (a1.trans_le h1) q).trans (dX8_spec_h1 (fun s t=>dw X[(J0 b)+s*(J0 n)+t]!) (fun s=>ao (fun t=>dw (X[(J0 b)+s*(J0 n)+t]! &&& 0x7FFFFFFF)) (J0 n) 0 0) _ _ _ _ q _ _ _ _ _ (q/8) (q%8) (ϛ q 8) (ϊ _ (by decide)) hr8 h1 a1 rfl (congrArg (fun i=>dw X[i+q/8]!) K.1) (congrArg (fun i=>ao (fun t=>dw (X[i+t]! &&& 0x7FFFFFFF)) (J0 n) (J0 0) 0) K.1) (dRw8_spec X _ n r _ _ hr8 _ 0 Y 0 hY rfl (ϟ _) q))
theorem kOutWm (n:ℕ)(l:ℕ→Ἃ)(W b E:Ἂ Ἃ)(ib L H:ℕ)(j:ℕ)(cq hb p:ἆ)(f:ℕ→Ἅ)(v Lf:Ἅ)
    (A:∀ s<n,kB (l s)≠255)(hbF:kB b[j]! ≠ 255∧kG b[j]! ≤ H)(M:174+H ≤ ib)(Hn:n+2 ≤ 2^L)(x:L≤27)(z:ib+L≤425)(hn:0<n)
    (hEv:∀ s<n,(kB W[j*n+s]!).toNat ≤ (E[j]!).toNat)
    (hpp0:(E[j]!).toNat < 255 → ∀ s<n,fv (f s)∧rv (f s)=tm (l s) W[j*n+s]!)
    (hv:v=ao f n 0 (dw ld*dw b[j]!))
    (hLf:Lf=ao (fun k=>dw (l k&&&0x7FFFFFFF)) n 0 0)
    (hcq:23+(cEq (I0 (9*(n+1))).toInt64.toFloat Lf).toNat ≤ cq.toNat)(hcq2:cq.toNat ≤ 992)(hhb:hb=(ib+L-50).toUInt64)
    (hp:p=(if (E[j]!).toUInt64<(56:ἆ) then (56:ἆ) else (E[j]!).toUInt64)+cq) (he:(E[j]!).toUInt64 < 255) (hp372:p ≤ 372) :
    aG (n+1) (fun s=>if s < n then l s else ld) (fun s=>if s < n then W[j*n+s]! else b[j]!)∧fv v ∧
    |rv v-(af (n+1) (fun s=>if s < n then l s else ld) (fun s=>if s < n then W[j*n+s]! else b[j]!):ℝ)| ≤ (2:ℝ)^(((if hb>p then hb else p).toNat:ℕ):ℤ) ∧
    |(af (n+1) (fun s=>if s < n then l s else ld) (fun s=>if s < n then W[j*n+s]! else b[j]!):ℝ)| ≤ (2:ℝ)^(425:ℤ) ∧
    |rv v| ≤ (2:ℝ)^((((if hb>p then hb else p).toNat)+52:ℕ):ℤ)∧(if hb>p then hb else p).toNat ≤ 600:=by
  have eN:=kE64 E[j]!
  have hc9:=kCEb (I0 (9*(n+1))).toInt64.toFloat Lf
  generalize hcc:(cEq (I0 (9*(n+1))).toInt64.toFloat Lf)=cc at hcq hc9
  have the:(E[j]!).toNat < 255:=by have:=μ.1 he; rw [eN] at this; exact this
  have hm:(if (E[j]!).toUInt64<(56:ἆ) then (56:ἆ) else (E[j]!).toUInt64).toNat=max (E[j]!).toNat 56:=by
    split
    · rename_i h; have:=μ.1 h; rw [eN] at this; exact (Nat.max_eq_right this.le).symm
    · rename_i h; have:=μ.not.1 h; rw [eN] at this; rw [eN]; exact (Nat.max_eq_left (W4 this)).symm
  have tp:p.toNat=max (E[j]!).toNat 56+cq.toNat:=by
    rw [hp,kAddS _ _ (by rw [hm]; exact C0 (Y3 (Nat.max_le.2 ⟨the.le,(by decide)⟩) hcq2) (by decide)),hm]
  have t372:p.toNat ≤ 372:=θ.1 hp372
  have thb:hb.toNat=ib+L-50:=by rw [hhb]; exact kOfS _ ((K4 _ _).trans_lt (C0 z (by decide)))
  have th:(if hb>p then hb else p).toNat=max (ib+L-50) p.toNat:=by
    rw [← thb]; split
    · rename_i h; exact (Nat.max_eq_left (μ.1 h).le).symm
    · rename_i h; exact (Nat.max_eq_right (W4 (μ.not.1 h))).symm
  have hk:max (E[j]!).toNat 56+23+cc.toNat ≤ max (E[j]!).toNat 56+cq.toNat:=by rw [pA]; exact J1 hcq _
  have Wn:=kWin n l W b j (E[j]!).toNat ib L H f Lf A hEv the hbF M Hn x z hn (hpp0 the) hLf (by rw [hcc]; exact hk.trans (tp.symm.trans_le t372))
  rw [hcc] at Wn
  have mono : ∀ a c:ℕ,a ≤ c → (2:ℝ)^((a:ℕ):ℤ) ≤ (2:ℝ)^((c:ℕ):ℤ):=fun a c h=>AG (by norm_num) (by exact_mod_cast h)
  have hXY:max (ib+L-50) (max (E[j]!).toNat 56+23+cc.toNat) ≤ (if hb>p then hb else p).toNat:=by
    rw [th,tp]; exact max_le_max_left _ hk
  rw [hv]
  exact ⟨Wn.1,Wn.2.1,Wn.2.2.1.trans (mono _ _ hXY),Wn.2.2.2.1,Wn.2.2.2.2.trans (mono _ _ (L1 hXY 52)),
    by rw [th]; exact Nat.max_le.2 ⟨(K4 _ _).trans (z.trans (by decide)),t372.trans (by decide)⟩⟩
theorem kMx2 (a c:ἆ):a.toNat ≤ (mx2 a c).toNat∧c.toNat ≤ (mx2 a c).toNat∧((mx2 a c).toNat=a.toNat∨(mx2 a c).toNat=c.toNat):=by
  unfold mx2; split
  · rename_i h; have:=μ.1 h; exact ⟨this.le,le_refl _,BA rfl⟩
  · rename_i h; have:=μ.not.1 h; exact ⟨le_refl _,W4 this,pZ rfl⟩
theorem kMx8 (f:ℕ→ἆ)(B:ℕ)(hf:∀ r<8,(f r).toNat ≤ B) :
    (∀ r<8,(f r).toNat ≤ (mx2 (mx2 (mx2 (f 0) (f 1)) (mx2 (f 2) (f 3))) (mx2 (mx2 (f 4) (f 5)) (mx2 (f 6) (f 7)))).toNat) ∧
    (mx2 (mx2 (mx2 (f 0) (f 1)) (mx2 (f 2) (f 3))) (mx2 (mx2 (f 4) (f 5)) (mx2 (f 6) (f 7)))).toNat ≤ B:=by
  have a01:=kMx2 (f 0) (f 1); have a23:=kMx2 (f 2) (f 3); have a45:=kMx2 (f 4) (f 5); have a67:=kMx2 (f 6) (f 7)
  have b0:=kMx2 (mx2 (f 0) (f 1)) (mx2 (f 2) (f 3)); have b1:=kMx2 (mx2 (f 4) (f 5)) (mx2 (f 6) (f 7))
  have c:=kMx2 (mx2 (mx2 (f 0) (f 1)) (mx2 (f 2) (f 3))) (mx2 (mx2 (f 4) (f 5)) (mx2 (f 6) (f 7)))
  have u2 : ∀ a c:ἆ,a.toNat ≤ B → c.toNat ≤ B → (mx2 a c).toNat ≤ B:=fun a c ha hc =>
    (kMx2 a c).2.2.elim (fun e=>e.trans_le ha) (fun e=>e.trans_le hc)
  have T0:=b0.1.trans c.1; have T1:=b0.2.1.trans c.1; have T2:=b1.1.trans c.2.1; have T3:=b1.2.1.trans c.2.1
  refine ⟨fun r hr => ?_,u2 _ _ (u2 _ _ (u2 _ _ (hf 0 (by decide)) (hf 1 (by decide))) (u2 _ _ (hf 2 (by decide)) (hf 3 (by decide)))) (u2 _ _ (u2 _ _ (hf 4 (by decide)) (hf 5 (by decide))) (u2 _ _ (hf 6 (by decide)) (hf 7 (by decide))))⟩
  interval_cases r
  exacts [a01.1.trans T0,a01.2.1.trans T0,a23.1.trans T1,a23.2.1.trans T1,a45.1.trans T2,a45.2.1.trans T2,a67.1.trans T3,a67.2.1.trans T3]
theorem kHB (a0 l0:ℕ)(h1:50 ≤ a0)(h2:a0+l0 ≤ 425):a0.toUInt64+l0.toUInt64-(50:ἆ)=(a0+l0-50).toUInt64:=by
  have t:a0+l0 < 2^63:=C0 h2 (by decide)
  have e1:a0.toUInt64.toNat=a0:=kOfS _ ((V1 _ _).trans_lt t)
  have e2:l0.toUInt64.toNat=l0:=kOfS _ ((K2 _ _).trans_lt t)
  have e3:(a0.toUInt64+l0.toUInt64).toNat=a0+l0:=by rw [kAddS _ _ (by rw [e1,e2]; exact t),e1,e2]
  apply ξ.1
  rw [χ _ _ (by rw [θ,e3]; exact h1.trans (V1 _ _)),e3,kOfS _ ((K4 _ _).trans_lt t)]; simp
theorem kCR (F a:Ἅ):((23:ἆ)+cEq F a).toNat=23+(cEq F a).toNat∧((23:ἆ)+cEq F a).toNat ≤ 992:=by
  have:=kCEb F a
  have e:((23:ἆ)+cEq F a).toNat=23+(cEq F a).toNat:=kAddS _ _ (C0 (J1 this 23) (by decide))
  exact ⟨e,by rw [e]; exact J1 this 23⟩
theorem gXb_spec_h2(Z Y:ἀ)(X:Ἂ Ἃ)(b nn l z r0 n:Ἀ)(hz:z=0)(hn:nn=n)(hb:b=r0*n)(hl:l<=8)(hY:∀q,q<8*n+8->Y.data[q]! =dw 0)
  (D:∀q,Z.data[q]! =if q<8*nn+8∧z<=q%8∧q%8<l then (if q/8<nn then dw X[b+(q%8)*nn+q/8]! else ao (fun t=>dw (X[b+(q%8)*nn+t]! &&& 0x7FFFFFFF)) nn 0 0) else Y.data[q]!):
  (∀s,s<nn->∀r,r<8->Z.data[8*s+r]! =if r<l then dw X[(r0+r)*n+s]! else dw 0)∧(∀r,r<l->Z.data[8*n+r]! =ao (fun k=>dw (X[(r0+r)*n+k]! &&& 0x7FFFFFFF)) n 0 0):=by
 subst hz hb hn
 refine ⟨fun s hs r hr=>?_,fun r hr=>?_⟩
 · have q1:(8*s+r)%8=r:=(O7 8 s r).trans (γ hr)
   have q2:(8*s+r)/8=s:=by rw[W8 (by decide),ώ hr,ϑ]
   have c1:8*s+r<8*nn+8:=(O1 hr _).trans_le (L1 (U3 8 hs.le) 8)
   refine (D _).trans ?_
   rw[q1,q2,if_pos hs,Ϡ]
   exact if_congr ⟨fun h=>h.2.2,fun h=>⟨c1,ϟ _,h⟩⟩ rfl (hY _ c1)
 · have r8:r<8:=hr.trans_le hl
   have q1:(8*nn+r)%8=r:=(O7 8 nn r).trans (γ r8)
   have q2:(8*nn+r)/8=nn:=by rw[W8 (by decide),ώ r8,ϑ]
   refine (D _).trans ?_
   rw[q1,q2,if_neg (B4 nn),Ϡ]
   exact if_pos ⟨O1 r8 _,ϟ _,hr⟩
theorem dOw_spec_h1 {r R n m x y B:Ἀ} (h1:Nat.lt r R) (hnm:Nat.lt n m) (hB:Nat.lt (Nat.mul R m) B) (hX:Nat.le (Nat.mul R n) x) (hY:Nat.le (Nat.mul R m) y) :
    Nat.lt (Nat.mul r n) B∧Nat.lt (Nat.mul r m) B∧(∀ a,a=Nat.mul r n → Nat.le (Nat.add a n) x)∧(∀ a,a=Nat.mul r m → Nat.lt (Nat.add a n) y)∧(∀ a,a=Nat.mul r n → Nat.lt (Nat.add a n) B)∧(∀ a,a=Nat.mul r m → Nat.lt (Nat.add a n) B):=by
  have q1:=U2 (B3 (Ο r m).symm) (ι m h1)
  have q2:=U2 (B3 (Ο r n).symm) (ι n h1)
  have q3:=U3 R (X4 hnm)
  exact ⟨C0 ((V1 _ _).trans (q2.trans q3)) hB,C0 ((V1 _ _).trans q1) hB,fun _ h=>h ▸ q2.trans hX,fun _ h=>h ▸ U1 (O1 hnm _) (q1.trans hY),fun _ h=>h ▸ C0 (q2.trans q3) hB,fun _ h=>h ▸ K5 (O1 hnm _) (C0 q1 hB)⟩
theorem dOw_spec (X:Ἂ ἇ)(n m R:Ἄ)(hX:(J0 R)*(J0 n)<=(O0 X))(hm:(J0 m)=(J0 n)+1)(hB:(J0 R)*(J0 m)<2^(31:Ἀ)) :
    ∀ k (r:Ἄ)(Y:ἀ)(hY:(J0 R)*(J0 m)<=(kS Y)),(J0 R)-(J0 r)=k → (J0 r) ≤ (J0 R) →
    (∀ q,q < (J0 r)*(J0 m) → (dOw X n m R r Y hX hY hm hB).data[q]! = Y.data[q]!) ∧
    ∀ r' t,r' < (J0 R) → (J0 r) ≤ r' → t ≤ (J0 n) → (dOw X n m R r Y hX hY hm hB).data[r'*(J0 m)+t]! =
      if t=(J0 n) then ao (fun s=>dw (X[r'*(J0 n)+s]! &&& 0x7FFFFFFF)) (J0 n) 0 0 else dw X[r'*(J0 n)+t]! := by
  intro k; induction k with
  | zero =>
    intro r Y hY hk hr
    exact ⟨fun q _=>by rw [dOw,dif_neg (mt ε.mp (W3.2 (K3.1 hk)))],fun r' t h1 h2 _=>(B4 r' (U1 h1 ((K3.1 hk).trans h2))).elim⟩
  | succ k ih =>
    intro r Y hY hk hr
    have h1:=N1 hk
    have hnm:=V6 (I3 _) hm.symm
    obtain ⟨a1,a2,a3,a4,a5,a6⟩:=dOw_spec_h1 h1 hnm hB hX hY
    have e1:=kMm r n a1
    have e2:=kMm r m a2
    have hr1:=ac (ε.mpr h1)
    have P1:=a3 _ e1
    have P2:=a4 _ e2
    have P3:=And.intro (a5 _ e1) (a6 _ e2)
    rw [dOw,dif_pos (ε.mpr h1)]; dsimp only
    obtain ⟨I1,I2⟩:=ih (r+1) (dRw X _ n _ 0 Y 0 P1 P2 P3) (U2 hY (B3 (dRw_size X _ n _ P1 P3 _ 0 Y 0 P2 rfl).symm)) ((congrArg (Nat.sub R.toNat) hr1).trans ((Nat.sub_succ _ _).trans (congrArg Nat.pred hk))) (U2 (B3 hr1) h1)
    have D:=dRw_spec X _ n _ P1 P3 _ 0 Y 0 P2 rfl (kc0 ▸ ϟ _)
    refine ⟨fun q hq => ?_,fun r' t h2 h3 h4 => ?_⟩
    · rw [I1 q (U1 hq (ι _ ((E7 _).trans (B3 hr1.symm)))),D q,e2,if_neg (by exact R8 (U1 hq (V1 _ _))),if_neg (by exact fun h=>W3.2 ((V1 _ _).trans h.1) hq)]
    rcases Z7 r.toNat r' with h5 | h5
    · exact I2 r' t h2 (U2 (B3 hr1) h5) h4
    · obtain rfl:=C4 h5 h3
      rw [I1 _ (by rw [hr1,Ϡ,n4]; exact O1 (C0 h4 hnm) _),D, ← e2, ← e1]
      by_cases ht:t=n.toNat
      · subst ht; rw [if_pos rfl,if_pos rfl,kc0]
      · rw [if_neg (by exact fun h=>ht (J8 h)),if_neg ht,if_pos (by exact ⟨J1 (ϟ _) _,O1 (N4 h4 ht) _⟩),ϱ]
noncomputable def oV(relu:Ἆ)(n:Nat)(X W b E:Ἂ ἇ)(ix ib L:Nat)(Y Q:ἀ)(vo rn:Nat)(cq hb:ἆ)(j:Nat)(v:Ἅ):ἇ:=
 if zq2 v (E[j]!).toUInt64 relu cq hb==(0xFFFFFFFF:Ἃ) then
  sQ2 relu n X W b E ix ib L Y Q vo rn j (if hb > (if (E[j]!).toUInt64<(56:ἆ) then (56:ἆ) else (E[j]!).toUInt64)+cq then hb else (if (E[j]!).toUInt64<(56:ἆ) then (56:ἆ) else (E[j]!).toUInt64)+cq) v
 else zq2 v (E[j]!).toUInt64 relu cq hb
noncomputable def lsel(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ)(k:Nat):Ἅ:=
 if k=0 then a0 else if k=1 then a1 else if k=2 then a2 else if k=3 then a3 else if k=4 then a4 else if k=5 then a5 else if k=6 then a6 else a7
theorem kAS (a:Ἂ ἇ)(i:Ἄ)(x:ἇ)(h:(J0 i)<(O0 a))(q:ℕ):(a.uset i x h)[q]! = if (J0 i)=q then x else a[q]! := by
  rw [pP]
  by_cases hq:q < (O0 a)
  · rw [AB (a.set (J0 i) x h) q (by simp only [Array.size_set]; exact hq),AB a q hq,Array.getElem_set]
  · have:(J0 i)≠q:=by omega
    rw [if_neg this,An (a.set (J0 i) x h) q (by simp only [Array.size_set,not_lt]; exact W4 hq),An a q hq]
theorem kOutV (relu:Ἆ)(n:ℕ)(X W b E:Ἂ Ἃ)(ix ib L H:ℕ)(Y Q:ἀ)(vo rn j:ℕ)(cq hb:ἆ)(v Lf:Ἅ)
    (F:∀ q:ℕ,kB X[q]! ≠ 255∧kG X[q]! ≤ ix)(hbF:∀ q:ℕ,kB b[q]! ≠ 255∧kG b[q]! ≤ H)(M:174+H ≤ ib)(Hn:n+2 ≤ 2^L)(x:L≤27)(z:ib+L≤425)(hn:0<n)
    (hEv:∀ s<n,(kB W[j*n+s]!).toNat ≤ (E[j]!).toNat)
    (hYv:∀ s<n,Y.data[vo+s]! = dw X[rn+s]!)
    (hQv:∀ s<n,Q.data[j/8*n*8+j%8+8*s]! = dw W[j*n+s]!)
    (hv:v=ao (fun s=>Y.data[vo+s]! * Q.data[j/8*n*8+j%8+8*s]!) n 0 (dw ld*dw b[j]!))
    (hLf:Lf=ao (fun k=>dw (X[rn+k]! &&& 0x7FFFFFFF)) n 0 0)
    (hcq:cq=23+cEq (I0 (9*(n+1))).toInt64.toFloat Lf) (hhb:hb=(ib+L-50).toUInt64) :
    oV relu n X W b E ix ib L Y Q vo rn cq hb j v=p8 relu (ms (n+1) (fun s=>if s < n then X[rn+s]! else ld) (fun s=>if s < n then W[j*n+s]! else b[j]!)):=by
  unfold oV; exact kOut relu n X W b E ix ib L H Y Q vo rn j cq hb v Lf F hbF M Hn x z hn hEv hYv hQv hv hLf hcq hhb
theorem wcBF_spec (b:Ἂ Ἃ)(ow N:Ἀ):(kS (wcBF (b,ow,N)))=N ∧ ∀k,k<N->(wcBF (b,ow,N)).data[k]! = if k<ow then dw oneW*dw b[k]! else dw 0:=by
  have B:=bk (λo=>if o<ow then dw oneW*dw b[o]! else dw 0) N _ (AR N) rfl (by simp[AR,kS])
  unfold wcBF; exact ⟨B.1,fun k hk=>B.2.2 k (by simp[AR,kS]) hk⟩
theorem sMb1(Q Y:ἀ)(vo nu i0:Ἄ)(hY:(J0 vo)+(J0 nu)<=(kS Y)/\(J0 vo)+(J0 nu)<2^(31:Ἀ))(hQ:(J0 i0)+8*(J0 nu)<=(kS Q)/\(J0 i0)+8*(J0 nu)<2^(31:Ἀ))(t:Ἄ)(h:(J0 t)<(J0 nu)) :
    (J0 (vo+t))=(J0 vo)+(J0 t)∧(J0 (i0+8*t))=(J0 i0)+8*(J0 t) ∧
    (J0 (vo+t))<(kS Y)∧(J0 (i0+8*t))+7<(kS Q)∧(J0 (i0+8*t))+7<2^(31:Ἀ)∧(J0 (t+1))=(J0 t)+1:=by
  have e8:(J0 (8*t))=8*(J0 t):=by rw[kMm 8 t (by rw[kc8];clear * - h hQ;omega),kc8]
  have a1:=kSm vo t (by clear * - h hY;omega); have a2:=kSm i0 (8*t) (by rw[e8];clear * - h hQ;omega)
  have e1:=kSm t 1 (by rw[kc1];clear * - h hY;omega); rw[kc1] at e1
  rw[e8] at a2
  exact ⟨a1,a2,by rw[a1];exact U1 (O1 h _) hY.1,by rw[a2];clear * - h hQ;omega,by rw[a2];clear * - h hQ;omega,e1⟩
theorem sMb(Q Y:ἀ)(vo nu i0:Ἄ)(hY:(J0 vo)+(J0 nu)<=(kS Y)/\(J0 vo)+(J0 nu)<2^(31:Ἀ))(hQ:(J0 i0)+8*(J0 nu)<=(kS Q)/\(J0 i0)+8*(J0 nu)<2^(31:Ἀ))(t:Ἄ)(h:(J0 t)<(J0 nu))(h2:(J0 t)+1<(J0 nu)) :
    (J0 (t+2))=(J0 t)+2 ∧
    (J0 (vo+(t+1)))=(J0 vo)+((J0 t)+1)∧(J0 (i0+8*(t+1)))=(J0 i0)+8*((J0 t)+1) ∧
    (J0 (vo+(t+1)))<(kS Y)∧(J0 (i0+8*(t+1)))+7<(kS Q)∧(J0 (i0+8*(t+1)))+7<2^(31:Ἀ):=by
  have e1:=kSm t 1 (by rw[kc1];clear * - h hY;omega); rw[kc1] at e1
  have e2:=kSm t 2 (by rw[kc2];clear * - h2 hY;omega); rw[kc2] at e2
  have e9:(J0 (8*(t+1)))=8*((J0 t)+1):=by rw[kMm 8 (t+1) (by rw[kc8,e1];clear * - h2 hQ;omega),kc8,e1]
  have a1:=kSm vo (t+1) (by rw[e1];clear * - h2 hY;omega); have a2:=kSm i0 (8*(t+1)) (by rw[e9];clear * - h2 hQ;omega)
  rw[e1] at a1; rw[e9] at a2
  exact ⟨e2,a1,a2,by rw[a1];exact U1 (O1 h2 _) hY.1,by rw[a2];clear * - h2 hQ;omega,by rw[a2];clear * - h2 hQ;omega⟩
theorem Lq0(R r:ℕ)(h:R-r=0):¬r < R:=W3.mpr (K3.mp h)
theorem Lqs(R r k:ℕ)(h:R-r=k+1):r < R∧R-(r+1)=k:=⟨N1 h,by rw[Nat.sub_succ,h];rfl⟩
theorem Lqd(R r:ℕ)(h:r < R):R-(r+1) < R-r:=C5 h (I3 r)
theorem Lq2(R r:ℕ)(h:r+1 < R):R-(r+1+1) < R-r:=C5 (B6 h) (K6 (I3 r))
theorem Lql(a j m c:ℕ)(h:j < m)(hc:a+m<=c):a+j < c:=U1 (O1 h a) hc
theorem Lqn(a j m q:ℕ)(h:m<=j):¬(a+j<=q∧q < a+m):=fun g=>B4 _ (C0 ((J1 h a).trans g.1) g.2)
theorem Lqa(a b m:ℕ):¬(a+(b+1)≤a+b∧a+b < a+m):=fun g=>Nat.not_succ_le_self _ g.1
theorem Lqb(a b m:ℕ)(h:b < m):a+b≤a+b∧a+b < a+m:=⟨Nat.le_refl _,O1 h a⟩
theorem Lqc(a b m q:ℕ)(g:a+(b+1)≤q∧q < a+m):a+b≤q∧q < a+m:=⟨X5 g.1,g.2⟩
theorem Lqe(a b m q:ℕ)(g:¬(a+(b+1)≤q∧q < a+m))(e:¬a+b=q):¬(a+b≤q∧q < a+m):=fun g'=>g ⟨N4 g'.1 e,g'.2⟩
theorem Lqf(a r r' w o o':ℕ)(g1:r+1≤r')(h3:o < w):(a+r')*w+o'≠(a+r)*w+o:=by
  have m:=ι w (J1 g1 a); rw [show a+(r+1)=(a+r)+1 from rfl,Ο] at m
  exact L6 (U1 (O1 h3 _) (m.trans (V1 _ _)))
theorem Lwq(a g l:ℕ)(hl:l < 8):(8*a ≤ 8*g+l∧8*g+l < 8*a+8) ↔ g=a:=by omega
theorem Lw1(o r:ℕ)(hr:r < 8):8*o+8≤8*(o+1)+r∧8*(o+1)+r < 8*o+8+8:=(Lwq (o+1) (o+1) r hr).mpr rfl
theorem Lw2(o r:ℕ):8*(o+1)+r-(8*o+8)=r:=ϱ (8*o+8) r
theorem Lw8(x o r:ℕ)(f:x=8*o+8):8*(o+1)+r-x=r:=by rw[f];exact Lw2 o r
theorem Lw3(o o' r:ℕ)(hr:r < 8)(e:¬o'=o+1):¬(8*o+8≤8*o'+r∧8*o'+r < 8*o+8+8):=fun g=>e ((Lwq (o+1) o' r hr).mp g)
theorem Lw4(o r:ℕ)(hr:r < 8):8*o≤8*o+r∧8*o+r < 8*o+8:=⟨V1 _ _,O1 hr _⟩
theorem Lw5(o o' r:ℕ)(hr:r < 8)(e:¬o'=o):¬(8*o≤8*o'+r∧8*o'+r < 8*o+8):=fun g=>e ((Lwq o o' r hr).mp g)
theorem Lw6(o o':ℕ)(h:o' < o+1+1)(e1:¬o'=o+1)(e0:¬o'=o):o' < o:=N4 (J2 (N4 (J2 h) e1)) e0
theorem Lw7(o o' w:ℕ)(ho:o' < w)(h2:¬o+1 < w)(e0:¬o'=o):o' < o:=N4 (J2 (U1 ho (W4 h2))) e0
theorem n5d0:0≤7:=N2 rfl
theorem n5d1:1≤7:=N2 rfl
theorem n5d2:2≤7:=N2 rfl
theorem n5d3:3≤7:=N2 rfl
theorem n5d4:4≤7:=N2 rfl
theorem n5d5:5≤7:=N2 rfl
theorem n5d6:6≤7:=N2 rfl
theorem n5d7:7≤7:=N2 rfl
theorem sMbT(Q Y:ἀ)(vo nu i0:Ἄ)(hY:(J0 vo)+(J0 nu)<=(kS Y)/\(J0 vo)+(J0 nu)<2^(31:Ἀ))(hQ:(J0 i0)+8*(J0 nu)<=(kS Q)/\(J0 i0)+8*(J0 nu)<2^(31:Ἀ))(t:Ἄ)(ht0:(J0 t)<=(J0 nu)):(J0 (t+1))=(J0 t)+1:=by
  rw[kSm t 1 (by rw[kc1];clear * - ht0 hQ;omega),kc1]
theorem fzPU(m:Float.Model):M6 m.unpack=m:=by
 apply FloatLib.Floats.Formats.BinaryInterchange.Model.ofFloatModel_injective
 rw[nu]
 change pv kC (FloatLib.Floats.Formats.BinaryInterchange.Model.toModel (kI m))=kI m
 exact FloatLib.Floats.Formats.BinaryInterchange.Model.canonicalizeModel_eq_self _ (FloatLib.Floats.Formats.BinaryInterchange.Model.isModelCanonical_ofFloatModel m)
open FloatLib.Floats.Formats.BinaryInterchange in
theorem fzRN(M E:Int):pp k0 (Model.modelSign (decide (M<0))) M.natAbs E=G6 k0 M E .positive:=by
 rcases lt_trichotomy M 0 with h|h|h
 ·have c:compare M 0=.lt:=Int.compare_eq_lt.mpr h;have ha:M.natAbs=(-M).toNat:=by omega
  simp only[G6,c,Model.modelSign,h,decide_true,if_true,ha]
 ·subst h;simp[G6,Model.modelSign]
 ·have c:compare M 0=.gt:=Int.compare_eq_gt.mpr h;have ha:M.natAbs=M.toNat:=by omega
  simp only[G6,c,Model.modelSign,not_lt.mpr h.le,decide_false,υ,↓reduceIte,ha]
theorem fzSS(M E:Int):(FloatLib.Numerics.Dyadic.mk (decide (M<0)) M.natAbs E).signedSignificand=M:=by
 unfold lr
 by_cases h:M<0
 ·simp[h,abs_of_neg h]
 ·simp[h,Ad (not_lt.mp h)]
open FloatLib.Floats.Formats.BinaryInterchange in
theorem fzVD(M1 E1 M2 E2:Int)(h:(M1:ℚ)*2^E1=(M2:ℚ)*2^E2):M6 (G6 k0 M1 E1 .positive)=M6 (G6 k0 M2 E2 .positive):=by
 by_cases h1:M1=0
 ·have h2:M2=0:=by
   subst h1;simp at h
   rcases h with h|h
   ·exact_mod_cast h
   ·exact absurd h (by positivity)
  subst h1;subst h2;rfl
 have h2:M2≠0:=by
  intro h2;subst h2;simp at h
  rcases h with h|h
  ·exact h1 (by exact_mod_cast h)
  ·exact absurd h (by positivity)
 apply Model.ofFloatModel_injective
 rw[Model.ofFloatModel_pack,Model.ofFloatModel_pack,← fzRN,← fzRN]
 have e:=ExactRuntime.roundDyadic_congr FloatFormat.binary64 (by decide) ⟨decide (M1<0),M1.natAbs,E1⟩ ⟨decide (M2<0),M2.natAbs,E2⟩ (by simpa using h1) (by simpa using h2) (by simp only[pR,fzSS];exact h)
 simp[Model.roundDyadic,Model.ieeeRoundDyadic] at e
 exact e
theorem fzZ(s:pK):(M6 (.zero s)).unpack=.zero s:=by cases s<;>rfl
open FloatLib.Floats.Formats.BinaryInterchange in
theorem fzSA(s:pK)(n:ℤ):((s.apply n:ℤ):ℝ)=(if J6 s=true then -1 else 1)*(n:ℝ):=by
 cases s<;>simp[Float.Model.UnpackedFloat.Sign.apply,J6]
open FloatLib.Floats.Formats.BinaryInterchange in
theorem fzSM(s1 s2:pK):(if J6 (s1*s2)=true then (-1:ℝ) else 1)=(if J6 s1=true then -1 else 1)*(if J6 s2=true then -1 else 1):=by
 cases s1<;>cases s2<;>simp[J6]
theorem fzBP(e:ℤ):kZ kO e=(2:ℝ)^e:=by
 simp[kZ,kO,mg]
theorem fzAL(m:ℕ)(e t:ℤ)(h:t≤e):((m<<<(e-t).toNat:ℕ):ℝ)*(2:ℝ)^t=(m:ℝ)*(2:ℝ)^e:=by
 rw[ο];push_cast
 rw[Ap,← AA,← AK (by norm_num:(2:ℝ)≠0)]
 congr 2;omega
open Float.Model.UnpackedFloat FloatLib.Floats.Formats.BinaryInterchange in
theorem fzFma(x y z:Ἅ)(hX:x.toModel.unpack.isFinite=true)(hY:y.toModel.unpack.isFinite=true)(hP:(x*y).toModel.unpack.isFinite=true)
  (hV:Model.unpackedToReal (x*y).toModel.unpack=Model.unpackedToReal x.toModel.unpack*Model.unpackedToReal y.toModel.unpack):Float.fma x y z=z+x*y:=by
 show (⟨x.toModel.fma y.toModel z.toModel⟩:Ἅ)=⟨z.toModel+(x*y).toModel⟩
 congr 1
 show M6 (fma .binary64 x.toModel.unpack y.toModel.unpack z.toModel.unpack)=M6 (add .binary64 z.toModel.unpack (x*y).toModel.unpack)
 have e3:(x*y).toModel=M6 (mul .binary64 x.toModel.unpack y.toModel.unpack):=rfl
 generalize hPd:(x*y).toModel.unpack=P at hP hV ⊢
 rw[e3] at hPd
 generalize x.toModel.unpack=X at hX hV hPd ⊢
 generalize y.toModel.unpack=Y at hY hV hPd ⊢
 generalize z.toModel.unpack=Z
 cases X with
 | notANumber=>simp[isFinite] at hX
 | infinity s1=>simp[isFinite] at hX
 | zero s1=>
  cases Y with
  | notANumber=>simp[isFinite] at hY
  | infinity s2=>simp[isFinite] at hY
  | zero s2=>
   have hp:P=.zero (s1*s2):=by rw[← hPd];exact fzZ _
   subst hp
   cases Z with
   | notANumber=>rfl
   | infinity s3=>rfl
   | zero s3=>cases s1<;>cases s2<;>cases s3<;>rfl
   | finite s3 m3 e3 h3=>rfl
  | finite s2 m2 e2 h2=>
   have hp:P=.zero (s1*s2):=by rw[← hPd];exact fzZ _
   subst hp
   cases Z with
   | notANumber=>rfl
   | infinity s3=>rfl
   | zero s3=>cases s1<;>cases s2<;>cases s3<;>rfl
   | finite s3 m3 e3 h3=>rfl
 | finite s1 m1 e1 h1=>
  cases Y with
  | notANumber=>simp[isFinite] at hY
  | infinity s2=>simp[isFinite] at hY
  | zero s2=>
   have hp:P=.zero (s1*s2):=by rw[← hPd];exact fzZ _
   subst hp
   cases Z with
   | notANumber=>rfl
   | infinity s3=>rfl
   | zero s3=>cases s1<;>cases s2<;>cases s3<;>rfl
   | finite s3 m3 e3 h3=>rfl
  | finite s2 m2 e2 h2=>
   have hm1:(0:ℝ) < m1:=by exact_mod_cast h1
   have hm2:(0:ℝ) < m2:=by exact_mod_cast h2
   have hVne:Model.unpackedToReal P≠0:=by
    rw[hV,N6,N6,fzBP,fzBP]
    cases s1<;>cases s2<;>simp[J6,hm1.ne',hm2.ne']<;>exact⟨zpow_ne_zero _ (by norm_num),zpow_ne_zero _ (by norm_num)⟩
   cases P with
   | notANumber=>simp[isFinite] at hP
   | infinity sP=>simp[isFinite] at hP
   | zero sP=>exact absurd (Model.unpackedToReal_zero sP) hVne
   | finite sP mP eP hmP=>
    cases Z with
    | notANumber=>rfl
    | infinity s3=>rfl
    | zero s3=>
     show M6 (mul .binary64 (.finite s1 m1 e1 h1) (.finite s2 m2 e2 h2))=M6 (.finite sP mP eP hmP)
     rw[← hPd,fzPU]
    | finite s3 m3 e3 h3=>
     apply fzVD
     rw[N6,N6,N6,fzBP,fzBP,fzBP] at hV
     have A:=fzAL (m1*m2) (e1+e2) (min (e1+e2) e3) (min_le_left _ _)
     have B:=fzAL m3 e3 (min (e1+e2) e3) (min_le_right _ _)
     have C:=fzAL m3 e3 (min e3 eP) (min_le_left _ _)
     have D:=fzAL mP eP (min e3 eP) (min_le_right _ _)
     have E:=fzSM s1 s2
     have F:(2:ℝ)^(e1+e2)=(2:ℝ)^e1*(2:ℝ)^e2:=AK (by norm_num) _ _
     push_cast at A
     rw[← Rat.cast_inj (α:=ℝ)]
     simp only[J7,Rat.cast_add,Rat.cast_intCast,S8,G7,Int.cast_add,fzSA,M5]
     linear_combination (if J6 (s1*s2)=true then (-1:ℝ) else 1)*A+(if J6 s3=true then (-1:ℝ) else 1)*B-(if J6 s3=true then (-1:ℝ) else 1)*C-(if J6 sP=true then (-1:ℝ) else 1)*D+((m1:ℝ)*m2*(2:ℝ)^(e1+e2))*E-hV+((if J6 s1=true then (-1:ℝ) else 1)*(if J6 s2=true then (-1:ℝ) else 1)*m1*m2)*F
def Yarm:Bool:=!(System.Platform.target.startsWith "x86")
@[inline] def fM(arm:Ἆ)(a x q:Ἅ):Ἅ:=if arm then Float.fma x q a else a+x*q
abbrev FMok(arm:Ἆ)(Q Y:ἀ)(vo i0 nu:Ἄ):Prop:=∀ s<(J0 nu),∀ l<8,∀ a:Ἅ,fM arm a Y.data[(J0 vo)+s]! Q.data[(J0 i0)+8*s+l]! = a+Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+l]!
theorem FMok_f(Q Y:ἀ)(vo i0 nu:Ἄ):FMok false Q Y vo i0 nu:=fun _ _ _ _ _=>rfl
theorem fzD(arm:Ἆ)(w:ἇ)(hw:kB w≠255)(q:Ἅ)(hq:(∃ v,kB v≠255∧q=dw v)∨q=0)(a:Ἅ):fM arm a (dw w) q=a+dw w*q:=by
 cases arm;·rfl
 show Float.fma (dw w) q a=a+dw w*q
 have X:=cI w hw
 rcases hq with ⟨v,hv,rfl⟩|rfl
 ·have V:=cI v hv;have P:=dJ w v hw hv
  apply fzFma _ _ _ ((fO _).1 X.1) ((fO _).1 V.1) ((fO _).1 P.1)
  rw[← iL,← iL,← iL,P.2,X.2,V.2];unfold tm;push_cast;ring
 ·have Z:=kZ0;have P:=jo (dw w) 0 X.1 Z.1 (by rw[Z.2,mul_zero,A5];positivity)
  apply fzFma _ _ _ ((fO _).1 X.1) ((fO _).1 Z.1) ((fO _).1 P.1)
  rw[← iL,← iL,← iL,P.2,Z.2,mul_zero];simp [pD]
@[inline] def sM1(arm:Ἆ)(Q Y:ἀ)(vo nu i0:Ἄ)(hY:(J0 vo)+(J0 nu)<=(kS Y)/\(J0 vo)+(J0 nu)<2^(31:Ἀ))(hQ:(J0 i0)+8*(J0 nu)<=(kS Q)/\(J0 i0)+8*(J0 nu)<2^(31:Ἀ))(t:Ἄ)(h:(J0 t)<(J0 nu))(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ):A8:=
 have C:=sMb1 Q Y vo nu i0 hY hQ t h
 let i:=i0+8*t
 let x:=Y.uget (vo+t) C.2.2.1
 ⟨fM arm a0 x (Q.uget (i+0) (kIx7 i 0 0 kc0 n5d0 Q C.2.2.2.1 C.2.2.2.2.1)),fM arm a1 x (Q.uget (i+1) (kIx7 i 1 1 kc1 n5d1 Q C.2.2.2.1 C.2.2.2.2.1)),fM arm a2 x (Q.uget (i+2) (kIx7 i 2 2 kc2 n5d2 Q C.2.2.2.1 C.2.2.2.2.1)),fM arm a3 x (Q.uget (i+3) (kIx7 i 3 3 kc3 n5d3 Q C.2.2.2.1 C.2.2.2.2.1)),fM arm a4 x (Q.uget (i+4) (kIx7 i 4 4 kc4 n5d4 Q C.2.2.2.1 C.2.2.2.2.1)),fM arm a5 x (Q.uget (i+5) (kIx7 i 5 5 kc5 n5d5 Q C.2.2.2.1 C.2.2.2.2.1)),fM arm a6 x (Q.uget (i+6) (kIx7 i 6 6 kc6 n5d6 Q C.2.2.2.1 C.2.2.2.2.1)),fM arm a7 x (Q.uget (i+7) (kIx7 i 7 7 kc7 n5d7 Q C.2.2.2.1 C.2.2.2.2.1))⟩
theorem sM1_spec(arm:Ἆ)(Q Y:ἀ)(vo nu i0:Ἄ)(hY:(J0 vo)+(J0 nu)<=(kS Y)/\(J0 vo)+(J0 nu)<2^(31:Ἀ))(hQ:(J0 i0)+8*(J0 nu)<=(kS Q)/\(J0 i0)+8*(J0 nu)<2^(31:Ἀ))(hF:FMok arm Q Y vo i0 nu)(t:Ἄ)(h:(J0 t)<(J0 nu))(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ) :
    sM1 arm Q Y vo nu i0 hY hQ t h a0 a1 a2 a3 a4 a5 a6 a7=⟨a0+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 i0)+8*(J0 t)+0]!,a1+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 i0)+8*(J0 t)+1]!,a2+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 i0)+8*(J0 t)+2]!,a3+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 i0)+8*(J0 t)+3]!,a4+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 i0)+8*(J0 t)+4]!,a5+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 i0)+8*(J0 t)+5]!,a6+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 i0)+8*(J0 t)+6]!,a7+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 i0)+8*(J0 t)+7]!⟩:=by
  have C:=sMb1 Q Y vo nu i0 hY hQ t h
  have ex : ∀ h,Y.uget (vo+t) h=Y.data[(J0 vo)+(J0 t)]! := fun h=>by rw [kFU,C.1]
  have eq : ∀ (k:Ἄ)(c:ℕ)(hk:(J0 k)=c)(hc:c≤7) h,Q.uget ((i0+8*t)+k) h=Q.data[(J0 i0)+8*(J0 t)+c]! := fun k c hk hc h=>by
    rw [kFU,kSm (i0+8*t) k (C0 (J1 (hk ▸ hc) _) C.2.2.2.2.1),hk,C.2.1]
  unfold sM1; simp (disch:=decide) only [ex,eq 0 0 kc0 n5d0,eq 1 1 kc1 n5d1,eq 2 2 kc2 n5d2,eq 3 3 kc3 n5d3,eq 4 4 kc4 n5d4,eq 5 5 kc5 n5d5,eq 6 6 kc6 n5d6,eq 7 7 kc7 n5d7,hF (J0 t) h]
def sM(Q Y:ἀ)(vo nu i0:Ἄ)(hY:(J0 vo)+(J0 nu)<=(kS Y)/\(J0 vo)+(J0 nu)<2^(31:Ἀ))(hQ:(J0 i0)+8*(J0 nu)<=(kS Q)/\(J0 i0)+8*(J0 nu)<2^(31:Ἀ))(t:Ἄ)(ht0:(J0 t)<=(J0 nu))(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ):A8:=
 if h:t+1<nu then
  have h1:(J0 t)+1<(J0 nu):=by rw[← sMbT Q Y vo nu i0 hY hQ t ht0];exact ε.mp h
  have C:=sMb1 Q Y vo nu i0 hY hQ t (B6 h1)
  have B:=sMb Q Y vo nu i0 hY hQ t (B6 h1) h1
  let r:=sM1 false Q Y vo nu i0 hY hQ t (B6 h1) a0 a1 a2 a3 a4 a5 a6 a7
  let r2:=sM1 false Q Y vo nu i0 hY hQ (t+1) (by rw[C.2.2.2.2.2];exact h1) r.a0 r.a1 r.a2 r.a3 r.a4 r.a5 r.a6 r.a7
  sM Q Y vo nu i0 hY hQ (t+2) (by rw[B.1];exact h1) r2.a0 r2.a1 r2.a2 r2.a3 r2.a4 r2.a5 r2.a6 r2.a7
else if h2:t<nu then sM1 false Q Y vo nu i0 hY hQ t (ε.mp h2) a0 a1 a2 a3 a4 a5 a6 a7
 else ⟨a0,a1,a2,a3,a4,a5,a6,a7⟩
termination_by (J0 nu)-(J0 t)
decreasing_by (have h1:(J0 t)+1<(J0 nu):=(by rw[← sMbT Q Y vo nu i0 hY hQ t ht0];exact ε.mp h);have B:=sMb Q Y vo nu i0 hY hQ t (B6 h1) h1;rw[B.1];exact C5 (B6 h1) (P5 (by decide)))
theorem sM_spec(Q Y:ἀ)(vo nu i0:Ἄ)(hY:(J0 vo)+(J0 nu)<=(kS Y)/\(J0 vo)+(J0 nu)<2^(31:Ἀ))(hQ:(J0 i0)+8*(J0 nu)<=(kS Q)/\(J0 i0)+8*(J0 nu)<2^(31:Ἀ)) :
    ∀ m (t:Ἄ)(ht0:(J0 t)<=(J0 nu))(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ),(J0 nu)-(J0 t)=m →
    sM Q Y vo nu i0 hY hQ t ht0 a0 a1 a2 a3 a4 a5 a6 a7 =
    ⟨ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+0]!) (J0 nu) (J0 t) a0,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+1]!) (J0 nu) (J0 t) a1,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+2]!) (J0 nu) (J0 t) a2,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+3]!) (J0 nu) (J0 t) a3,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+4]!) (J0 nu) (J0 t) a4,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+5]!) (J0 nu) (J0 t) a5,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+6]!) (J0 nu) (J0 t) a6,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+7]!) (J0 nu) (J0 t) a7⟩:=by
  intro m; induction m using C6 with
  | _ m ih =>
    intro t ht0 a0 a1 a2 a3 a4 a5 a6 a7 hm
    have T:=sMbT Q Y vo nu i0 hY hQ t ht0
    by_cases h3:(J0 t) < (J0 nu)
    · by_cases h2:(J0 t)+1 < (J0 nu)
      · have B:=sMb Q Y vo nu i0 hY hQ t h3 h2
        have st : ∀ (f:ℕ→Ἅ)(a:Ἅ),ao f (J0 nu) (J0 t) a=ao f (J0 nu) ((J0 t)+2) ((a+f (J0 t))+f ((J0 t)+1)):=fun f a=>by
          rw [aos f _ _ a h3,aos f _ _ _ h2]
        rw [sM,dif_pos (ε.mpr (by rw [T]; exact h2))]; simp only [sM1_spec false Q Y vo nu i0 hY hQ (FMok_f _ _ _ _ _)]
        rw [ih ((J0 nu)-(J0 (t+2))) (by rw [B.1, ← hm]; exact C5 h3 (P5 (by decide))) (t+2) _ _ _ _ _ _ _ _ _ rfl]
        simp only [B.1,T,st]
      · have st : ∀ (f:ℕ→Ἅ)(a:Ἅ),ao f (J0 nu) (J0 t) a=a+f (J0 t):=fun f a=>by
          rw [aos f _ _ a h3,aoe f _ _ _ (W4 h2)]
        rw [sM,dif_neg (fun h=>h2 (by rw [← T]; exact ε.mp h)),dif_pos (ε.mpr h3),sM1_spec false Q Y vo nu i0 hY hQ (FMok_f _ _ _ _ _)]
        simp only [st]
    · have st : ∀ (f:ℕ→Ἅ)(a:Ἅ),ao f (J0 nu) (J0 t) a=a:=fun f a=>aoe f _ _ a (W4 h3)
      rw [sM,dif_neg (fun h=>h3 (B6 (show (J0 t)+1 < (J0 nu) by rw [← T]; exact ε.mp h))),dif_neg (fun h=>h3 (ε.mp h))]
      simp only [st]
@[inline] def w8(S:ἀ)(o:Ἄ)(y:A8)(h:(J0 o)+7<(kS S))(h2:(J0 o)+7<2^(31:Ἀ)):ἀ:=
 let S1:=S.uset (o+0) y.a0 (kIx7 o 0 0 kc0 (by decide) S h h2)
 have g1:(J0 o)+7<(kS S1):=h.trans_eq (uZ _ _ _ _).symm
 let S2:=S1.uset (o+1) y.a1 (kIx7 o 1 1 kc1 (by decide) S1 g1 h2)
 have g2:(J0 o)+7<(kS S2):=g1.trans_eq (uZ _ _ _ _).symm
 let S3:=S2.uset (o+2) y.a2 (kIx7 o 2 2 kc2 (by decide) S2 g2 h2)
 have g3:(J0 o)+7<(kS S3):=g2.trans_eq (uZ _ _ _ _).symm
 let S4:=S3.uset (o+3) y.a3 (kIx7 o 3 3 kc3 (by decide) S3 g3 h2)
 have g4:(J0 o)+7<(kS S4):=g3.trans_eq (uZ _ _ _ _).symm
 let S5:=S4.uset (o+4) y.a4 (kIx7 o 4 4 kc4 (by decide) S4 g4 h2)
 have g5:(J0 o)+7<(kS S5):=g4.trans_eq (uZ _ _ _ _).symm
 let S6:=S5.uset (o+5) y.a5 (kIx7 o 5 5 kc5 (by decide) S5 g5 h2)
 have g6:(J0 o)+7<(kS S6):=g5.trans_eq (uZ _ _ _ _).symm
 let S7:=S6.uset (o+6) y.a6 (kIx7 o 6 6 kc6 (by decide) S6 g6 h2)
 have g7:(J0 o)+7<(kS S7):=g6.trans_eq (uZ _ _ _ _).symm
 S7.uset (o+7) y.a7 (kIx7 o 7 7 kc7 (by decide) S7 g7 h2)
theorem w8_size(S:ἀ)(o:Ἄ)(y:A8)(h:(J0 o)+7<(kS S))(h2:(J0 o)+7<2^(31:Ἀ)):(kS (w8 S o y h h2))=(kS S):=by
 unfold w8;dsimp only;simp only[uZ]
theorem w8_st (S S0:ἀ)(o k:Ἄ)(c:ℕ)(hk:(J0 k)=c)(hc:c≤7)(v:Ἅ)(lz:ℕ→Ἅ)(hv:v=lz c)(h:(J0 o)+7<(kS S))(h2:(J0 o)+7<2^(31:Ἀ))
    (H:∀ q,S.data[q]! = if (J0 o) ≤ q∧q < (J0 o)+c then lz (q-(J0 o)) else S0.data[q]!) :
    ∀ q,(S.uset (o+k) v (kIx7 o k c hk hc S h h2)).data[q]! = if (J0 o) ≤ q∧q < (J0 o)+(c+1) then lz (q-(J0 o)) else S0.data[q]! := by
  intro q; rw [kFS,kSm o k (by omega),hk,H]
  by_cases e:(J0 o)+c=q
  · subst e; rw [if_pos rfl,if_pos (by omega),hv,ϱ]
  · rw [if_neg e]; by_cases g:(J0 o) ≤ q∧q < (J0 o)+c
    · rw [if_pos g,if_pos (by omega)]
    · rw [if_neg g,if_neg (by omega)]
theorem w8_spec(S:ἀ)(o:Ἄ)(y:A8)(h:(J0 o)+7<(kS S))(h2:(J0 o)+7<2^(31:Ἀ))(q:ℕ):
    (w8 S o y h h2).data[q]! = if (J0 o) ≤ q∧q < (J0 o)+8 then lsel y.a0 y.a1 y.a2 y.a3 y.a4 y.a5 y.a6 y.a7 (q-(J0 o)) else S.data[q]! := by
  have H0 : ∀ q,S.data[q]! = if (J0 o) ≤ q∧q < (J0 o)+0 then lsel y.a0 y.a1 y.a2 y.a3 y.a4 y.a5 y.a6 y.a7 (q-(J0 o)) else S.data[q]! := fun q=>by rw [if_neg (by omega)]
  have H1:=w8_st S S o 0 0 kc0 (by decide) y.a0 _ (by simp [lsel]) h h2 H0
  have H2:=w8_st _ S o 1 1 kc1 (by decide) y.a1 _ (by simp [lsel]) (by simp only [uZ]; exact h) h2 H1
  have H3:=w8_st _ S o 2 2 kc2 (by decide) y.a2 _ (by simp [lsel]) (by simp only [uZ]; exact h) h2 H2
  have H4:=w8_st _ S o 3 3 kc3 (by decide) y.a3 _ (by simp [lsel]) (by simp only [uZ]; exact h) h2 H3
  have H5:=w8_st _ S o 4 4 kc4 (by decide) y.a4 _ (by simp [lsel]) (by simp only [uZ]; exact h) h2 H4
  have H6:=w8_st _ S o 5 5 kc5 (by decide) y.a5 _ (by simp [lsel]) (by simp only [uZ]; exact h) h2 H5
  have H7:=w8_st _ S o 6 6 kc6 (by decide) y.a6 _ (by simp [lsel]) (by simp only [uZ]; exact h) h2 H6
  have H8:=w8_st _ S o 7 7 kc7 (by decide) y.a7 _ (by simp [lsel]) (by simp only [uZ]; exact h) h2 H7
  exact H8 q
theorem sPb(n G g vo Ys Qs Bs Ss:ℕ)(hg:g<G)(hY:vo+n<=Ys)(hQ:Qs=n*(8*G))(hBd:8*G<=Bs)(hS:8*G<=Ss)(hB1:vo+n<2^(31:Ἀ))(hB2:n*(8*G)<2^(31:Ἀ))(hB3:8*G<2^(31:Ἀ)) :
    ((J0 (G0 vo))+(J0 (G0 n))<=Ys∧(J0 (G0 vo))+(J0 (G0 n))<2^(31:Ἀ)) ∧
    ((J0 (G0 (g*n*8)))+8*(J0 (G0 n))<=Qs∧(J0 (G0 (g*n*8)))+8*(J0 (G0 n))<2^(31:Ἀ)) ∧
    (J0 (G0 (8*g)))+7<Bs∧(J0 (G0 (8*g)))+7<Ss∧(J0 (G0 (8*g)))+7<2^(31:Ἀ) ∧
    (J0 (G0 vo))=vo∧(J0 (G0 n))=n∧(J0 (G0 (g*n*8)))=g*n*8∧(J0 (G0 (8*g)))=8*g:=by
  have A:g*n*8+8*n ≤ n*(8*G):=by
    have h1:(g+1)*n ≤ G*n:=ι n hg
    calc g*n*8+8*n=8*((g+1)*n):=by ring
      _ ≤ 8*(G*n):=U3 8 h1
      _=n*(8*G):=by ring
  rw [s5tu vo (by omega),s5tu n (by omega),s5tu (g*n*8) (by omega),s5tu (8*g) (by omega)]; omega
structure A24 where
 a:A8
 c:A8
 d:A8
theorem lselAo (F:ℕ→ℕ→Ἅ)(G:ℕ→Ἅ)(n t l:ℕ)(hl:l<8):lsel (ao (F 0) n t (G 0)) (ao (F 1) n t (G 1)) (ao (F 2) n t (G 2)) (ao (F 3) n t (G 3)) (ao (F 4) n t (G 4)) (ao (F 5) n t (G 5)) (ao (F 6) n t (G 6)) (ao (F 7) n t (G 7)) l=ao (F l) n t (G l):=by
  interval_cases l <;> rfl
theorem n5g(g G k:ℕ)(hk:k<3)(h:g+3<=G):g+k<G:=U1 (O1 hk g) h
theorem n5p(t nu:Ἄ)(f:ℕ)(ht:(J0 t)+2*(f+1)=(J0 nu)):(J0 t)+1<(J0 nu):=by omega
theorem n5q(t nu:Ἄ)(f:ℕ)(e:(J0 (t+2))=(J0 t)+2)(ht:(J0 t)+2*(f+1)=(J0 nu)):(J0 (t+2))+2*f=(J0 nu):=by omega
@[inline] def st24(arm:Ἆ)(Q Y:ἀ)(vo nu i0 j0 k0:Ἄ)(hY:(J0 vo)+(J0 nu)<=(kS Y)/\(J0 vo)+(J0 nu)<2^(31:Ἀ))(hI:((J0 i0)+8*(J0 nu)<=(kS Q)/\(J0 i0)+8*(J0 nu)<2^(31:Ἀ)))(hJ:((J0 j0)+8*(J0 nu)<=(kS Q)/\(J0 j0)+8*(J0 nu)<2^(31:Ἀ)))(hK:((J0 k0)+8*(J0 nu)<=(kS Q)/\(J0 k0)+8*(J0 nu)<2^(31:Ἀ)))(t:Ἄ)(h:(J0 t)<(J0 nu))(a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7 d0 d1 d2 d3 d4 d5 d6 d7:Ἅ):A24:=
  have C:=sMb1 Q Y vo nu i0 hY hI t h
  have D:=sMb1 Q Y vo nu j0 hY hJ t h
  have E:=sMb1 Q Y vo nu k0 hY hK t h
  let x:=Y.uget (vo+t) C.2.2.1
  ⟨⟨fM arm a0 x (Q.uget ((i0+8*t)+0) (kIx7 (i0+8*t) 0 0 kc0 n5d0 Q C.2.2.2.1 C.2.2.2.2.1)),fM arm a1 x (Q.uget ((i0+8*t)+1) (kIx7 (i0+8*t) 1 1 kc1 n5d1 Q C.2.2.2.1 C.2.2.2.2.1)),fM arm a2 x (Q.uget ((i0+8*t)+2) (kIx7 (i0+8*t) 2 2 kc2 n5d2 Q C.2.2.2.1 C.2.2.2.2.1)),fM arm a3 x (Q.uget ((i0+8*t)+3) (kIx7 (i0+8*t) 3 3 kc3 n5d3 Q C.2.2.2.1 C.2.2.2.2.1)),fM arm a4 x (Q.uget ((i0+8*t)+4) (kIx7 (i0+8*t) 4 4 kc4 n5d4 Q C.2.2.2.1 C.2.2.2.2.1)),fM arm a5 x (Q.uget ((i0+8*t)+5) (kIx7 (i0+8*t) 5 5 kc5 n5d5 Q C.2.2.2.1 C.2.2.2.2.1)),fM arm a6 x (Q.uget ((i0+8*t)+6) (kIx7 (i0+8*t) 6 6 kc6 n5d6 Q C.2.2.2.1 C.2.2.2.2.1)),fM arm a7 x (Q.uget ((i0+8*t)+7) (kIx7 (i0+8*t) 7 7 kc7 n5d7 Q C.2.2.2.1 C.2.2.2.2.1))⟩,⟨fM arm c0 x (Q.uget ((j0+8*t)+0) (kIx7 (j0+8*t) 0 0 kc0 n5d0 Q D.2.2.2.1 D.2.2.2.2.1)),fM arm c1 x (Q.uget ((j0+8*t)+1) (kIx7 (j0+8*t) 1 1 kc1 n5d1 Q D.2.2.2.1 D.2.2.2.2.1)),fM arm c2 x (Q.uget ((j0+8*t)+2) (kIx7 (j0+8*t) 2 2 kc2 n5d2 Q D.2.2.2.1 D.2.2.2.2.1)),fM arm c3 x (Q.uget ((j0+8*t)+3) (kIx7 (j0+8*t) 3 3 kc3 n5d3 Q D.2.2.2.1 D.2.2.2.2.1)),fM arm c4 x (Q.uget ((j0+8*t)+4) (kIx7 (j0+8*t) 4 4 kc4 n5d4 Q D.2.2.2.1 D.2.2.2.2.1)),fM arm c5 x (Q.uget ((j0+8*t)+5) (kIx7 (j0+8*t) 5 5 kc5 n5d5 Q D.2.2.2.1 D.2.2.2.2.1)),fM arm c6 x (Q.uget ((j0+8*t)+6) (kIx7 (j0+8*t) 6 6 kc6 n5d6 Q D.2.2.2.1 D.2.2.2.2.1)),fM arm c7 x (Q.uget ((j0+8*t)+7) (kIx7 (j0+8*t) 7 7 kc7 n5d7 Q D.2.2.2.1 D.2.2.2.2.1))⟩,⟨fM arm d0 x (Q.uget ((k0+8*t)+0) (kIx7 (k0+8*t) 0 0 kc0 n5d0 Q E.2.2.2.1 E.2.2.2.2.1)),fM arm d1 x (Q.uget ((k0+8*t)+1) (kIx7 (k0+8*t) 1 1 kc1 n5d1 Q E.2.2.2.1 E.2.2.2.2.1)),fM arm d2 x (Q.uget ((k0+8*t)+2) (kIx7 (k0+8*t) 2 2 kc2 n5d2 Q E.2.2.2.1 E.2.2.2.2.1)),fM arm d3 x (Q.uget ((k0+8*t)+3) (kIx7 (k0+8*t) 3 3 kc3 n5d3 Q E.2.2.2.1 E.2.2.2.2.1)),fM arm d4 x (Q.uget ((k0+8*t)+4) (kIx7 (k0+8*t) 4 4 kc4 n5d4 Q E.2.2.2.1 E.2.2.2.2.1)),fM arm d5 x (Q.uget ((k0+8*t)+5) (kIx7 (k0+8*t) 5 5 kc5 n5d5 Q E.2.2.2.1 E.2.2.2.2.1)),fM arm d6 x (Q.uget ((k0+8*t)+6) (kIx7 (k0+8*t) 6 6 kc6 n5d6 Q E.2.2.2.1 E.2.2.2.2.1)),fM arm d7 x (Q.uget ((k0+8*t)+7) (kIx7 (k0+8*t) 7 7 kc7 n5d7 Q E.2.2.2.1 E.2.2.2.2.1))⟩⟩
theorem st24_spec(arm:Ἆ)(Q Y:ἀ)(vo nu i0 j0 k0:Ἄ)(hY:(J0 vo)+(J0 nu)<=(kS Y)/\(J0 vo)+(J0 nu)<2^(31:Ἀ))(hI:((J0 i0)+8*(J0 nu)<=(kS Q)/\(J0 i0)+8*(J0 nu)<2^(31:Ἀ)))(hJ:((J0 j0)+8*(J0 nu)<=(kS Q)/\(J0 j0)+8*(J0 nu)<2^(31:Ἀ)))(hK:((J0 k0)+8*(J0 nu)<=(kS Q)/\(J0 k0)+8*(J0 nu)<2^(31:Ἀ)))(hFi:FMok arm Q Y vo i0 nu)(hFj:FMok arm Q Y vo j0 nu)(hFk:FMok arm Q Y vo k0 nu)(t:Ἄ)(h:(J0 t)<(J0 nu))(a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7 d0 d1 d2 d3 d4 d5 d6 d7:Ἅ) :
    st24 arm Q Y vo nu i0 j0 k0 hY hI hJ hK t h a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7 d0 d1 d2 d3 d4 d5 d6 d7 =
    ⟨⟨a0+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 i0)+8*(J0 t)+0]!,a1+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 i0)+8*(J0 t)+1]!,a2+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 i0)+8*(J0 t)+2]!,a3+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 i0)+8*(J0 t)+3]!,a4+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 i0)+8*(J0 t)+4]!,a5+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 i0)+8*(J0 t)+5]!,a6+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 i0)+8*(J0 t)+6]!,a7+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 i0)+8*(J0 t)+7]!⟩,⟨c0+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 j0)+8*(J0 t)+0]!,c1+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 j0)+8*(J0 t)+1]!,c2+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 j0)+8*(J0 t)+2]!,c3+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 j0)+8*(J0 t)+3]!,c4+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 j0)+8*(J0 t)+4]!,c5+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 j0)+8*(J0 t)+5]!,c6+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 j0)+8*(J0 t)+6]!,c7+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 j0)+8*(J0 t)+7]!⟩,⟨d0+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 k0)+8*(J0 t)+0]!,d1+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 k0)+8*(J0 t)+1]!,d2+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 k0)+8*(J0 t)+2]!,d3+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 k0)+8*(J0 t)+3]!,d4+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 k0)+8*(J0 t)+4]!,d5+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 k0)+8*(J0 t)+5]!,d6+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 k0)+8*(J0 t)+6]!,d7+Y.data[(J0 vo)+(J0 t)]! * Q.data[(J0 k0)+8*(J0 t)+7]!⟩⟩:=by
  have C:=sMb1 Q Y vo nu i0 hY hI t h
  have D:=sMb1 Q Y vo nu j0 hY hJ t h
  have E:=sMb1 Q Y vo nu k0 hY hK t h
  have ex : ∀ h,Y.uget (vo+t) h=Y.data[(J0 vo)+(J0 t)]! := fun h=>by rw [kFU,C.1]
  have eqi : ∀ (k:Ἄ)(c:ℕ)(hk:(J0 k)=c)(hc:c≤7) h,Q.uget ((i0+8*t)+k) h=Q.data[(J0 i0)+8*(J0 t)+c]! := fun k c hk hc h=>by
    rw [kFU,kSm (i0+8*t) k (C0 (J1 (hk ▸ hc) _) C.2.2.2.2.1),hk,C.2.1]
  have eqj : ∀ (k:Ἄ)(c:ℕ)(hk:(J0 k)=c)(hc:c≤7) h,Q.uget ((j0+8*t)+k) h=Q.data[(J0 j0)+8*(J0 t)+c]! := fun k c hk hc h=>by
    rw [kFU,kSm (j0+8*t) k (C0 (J1 (hk ▸ hc) _) D.2.2.2.2.1),hk,D.2.1]
  have eqk : ∀ (k:Ἄ)(c:ℕ)(hk:(J0 k)=c)(hc:c≤7) h,Q.uget ((k0+8*t)+k) h=Q.data[(J0 k0)+8*(J0 t)+c]! := fun k c hk hc h=>by
    rw [kFU,kSm (k0+8*t) k (C0 (J1 (hk ▸ hc) _) E.2.2.2.2.1),hk,E.2.1]
  unfold st24; simp (disch:=decide) only [hFi (J0 t) h,hFj (J0 t) h,hFk (J0 t) h,ex,eqi 0 0 kc0 n5d0,eqi 1 1 kc1 n5d1,eqi 2 2 kc2 n5d2,eqi 3 3 kc3 n5d3,eqi 4 4 kc4 n5d4,eqi 5 5 kc5 n5d5,eqi 6 6 kc6 n5d6,eqi 7 7 kc7 n5d7,eqj 0 0 kc0 n5d0,eqj 1 1 kc1 n5d1,eqj 2 2 kc2 n5d2,eqj 3 3 kc3 n5d3,eqj 4 4 kc4 n5d4,eqj 5 5 kc5 n5d5,eqj 6 6 kc6 n5d6,eqj 7 7 kc7 n5d7,eqk 0 0 kc0 n5d0,eqk 1 1 kc1 n5d1,eqk 2 2 kc2 n5d2,eqk 3 3 kc3 n5d3,eqk 4 4 kc4 n5d4,eqk 5 5 kc5 n5d5,eqk 6 6 kc6 n5d6,eqk 7 7 kc7 n5d7]
def sM24(arm:Ἆ)(Q Y:ἀ)(vo nu i0 j0 k0:Ἄ)(hY:(J0 vo)+(J0 nu)<=(kS Y)/\(J0 vo)+(J0 nu)<2^(31:Ἀ))(hI:((J0 i0)+8*(J0 nu)<=(kS Q)/\(J0 i0)+8*(J0 nu)<2^(31:Ἀ)))(hJ:((J0 j0)+8*(J0 nu)<=(kS Q)/\(J0 j0)+8*(J0 nu)<2^(31:Ἀ)))(hK:((J0 k0)+8*(J0 nu)<=(kS Q)/\(J0 k0)+8*(J0 nu)<2^(31:Ἀ))):(f:ℕ)→(t:Ἄ)→((J0 t)+2*f=(J0 nu))→(a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7 d0 d1 d2 d3 d4 d5 d6 d7:Ἅ)→A24
 | 0,_,_,a0,a1,a2,a3,a4,a5,a6,a7,c0,c1,c2,c3,c4,c5,c6,c7,d0,d1,d2,d3,d4,d5,d6,d7=>⟨⟨a0,a1,a2,a3,a4,a5,a6,a7⟩,⟨c0,c1,c2,c3,c4,c5,c6,c7⟩,⟨d0,d1,d2,d3,d4,d5,d6,d7⟩⟩
 | f+1,t,ht,a0,a1,a2,a3,a4,a5,a6,a7,c0,c1,c2,c3,c4,c5,c6,c7,d0,d1,d2,d3,d4,d5,d6,d7 =>
  have h2:=n5p t nu f ht
  have h:=B6 h2
  have C:=sMb1 Q Y vo nu i0 hY hI t h
  have C2:=sMb Q Y vo nu i0 hY hI t h h2
  let r:=st24 arm Q Y vo nu i0 j0 k0 hY hI hJ hK t h a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7 d0 d1 d2 d3 d4 d5 d6 d7
  let r2:=st24 arm Q Y vo nu i0 j0 k0 hY hI hJ hK (t+1) (by rw[C.2.2.2.2.2];exact h2) r.a.a0 r.a.a1 r.a.a2 r.a.a3 r.a.a4 r.a.a5 r.a.a6 r.a.a7 r.c.a0 r.c.a1 r.c.a2 r.c.a3 r.c.a4 r.c.a5 r.c.a6 r.c.a7 r.d.a0 r.d.a1 r.d.a2 r.d.a3 r.d.a4 r.d.a5 r.d.a6 r.d.a7
  sM24 arm Q Y vo nu i0 j0 k0 hY hI hJ hK f (t+2) (n5q t nu f C2.1 ht) r2.a.a0 r2.a.a1 r2.a.a2 r2.a.a3 r2.a.a4 r2.a.a5 r2.a.a6 r2.a.a7 r2.c.a0 r2.c.a1 r2.c.a2 r2.c.a3 r2.c.a4 r2.c.a5 r2.c.a6 r2.c.a7 r2.d.a0 r2.d.a1 r2.d.a2 r2.d.a3 r2.d.a4 r2.d.a5 r2.d.a6 r2.d.a7
theorem sM24_spec(arm:Ἆ)(Q Y:ἀ)(vo nu i0 j0 k0:Ἄ)(hY:(J0 vo)+(J0 nu)<=(kS Y)/\(J0 vo)+(J0 nu)<2^(31:Ἀ))(hI:((J0 i0)+8*(J0 nu)<=(kS Q)/\(J0 i0)+8*(J0 nu)<2^(31:Ἀ)))(hJ:((J0 j0)+8*(J0 nu)<=(kS Q)/\(J0 j0)+8*(J0 nu)<2^(31:Ἀ)))(hK:((J0 k0)+8*(J0 nu)<=(kS Q)/\(J0 k0)+8*(J0 nu)<2^(31:Ἀ)))(hFi:FMok arm Q Y vo i0 nu)(hFj:FMok arm Q Y vo j0 nu)(hFk:FMok arm Q Y vo k0 nu) :
    ∀ f (t:Ἄ)(ht:(J0 t)+2*f=(J0 nu))(a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7 d0 d1 d2 d3 d4 d5 d6 d7:Ἅ),
    sM24 arm Q Y vo nu i0 j0 k0 hY hI hJ hK f t ht a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7 d0 d1 d2 d3 d4 d5 d6 d7 =
    ⟨⟨ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+0]!) (J0 nu) (J0 t) a0,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+1]!) (J0 nu) (J0 t) a1,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+2]!) (J0 nu) (J0 t) a2,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+3]!) (J0 nu) (J0 t) a3,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+4]!) (J0 nu) (J0 t) a4,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+5]!) (J0 nu) (J0 t) a5,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+6]!) (J0 nu) (J0 t) a6,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+7]!) (J0 nu) (J0 t) a7⟩,⟨ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 j0)+8*s+0]!) (J0 nu) (J0 t) c0,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 j0)+8*s+1]!) (J0 nu) (J0 t) c1,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 j0)+8*s+2]!) (J0 nu) (J0 t) c2,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 j0)+8*s+3]!) (J0 nu) (J0 t) c3,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 j0)+8*s+4]!) (J0 nu) (J0 t) c4,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 j0)+8*s+5]!) (J0 nu) (J0 t) c5,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 j0)+8*s+6]!) (J0 nu) (J0 t) c6,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 j0)+8*s+7]!) (J0 nu) (J0 t) c7⟩,⟨ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 k0)+8*s+0]!) (J0 nu) (J0 t) d0,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 k0)+8*s+1]!) (J0 nu) (J0 t) d1,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 k0)+8*s+2]!) (J0 nu) (J0 t) d2,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 k0)+8*s+3]!) (J0 nu) (J0 t) d3,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 k0)+8*s+4]!) (J0 nu) (J0 t) d4,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 k0)+8*s+5]!) (J0 nu) (J0 t) d5,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 k0)+8*s+6]!) (J0 nu) (J0 t) d6,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 k0)+8*s+7]!) (J0 nu) (J0 t) d7⟩⟩:=by
  intro f; induction f with
  | zero =>
    intro t ht a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7 d0 d1 d2 d3 d4 d5 d6 d7
    have st : ∀ (f:ℕ→Ἅ)(a:Ἅ),ao f (J0 nu) (J0 t) a=a:=fun f a=>aoe f _ _ a (B3 (by omega))
    show (⟨⟨a0,a1,a2,a3,a4,a5,a6,a7⟩,⟨c0,c1,c2,c3,c4,c5,c6,c7⟩,⟨d0,d1,d2,d3,d4,d5,d6,d7⟩⟩:A24)=_; simp only [st]
  | succ f ih =>
    intro t ht a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7 d0 d1 d2 d3 d4 d5 d6 d7
    have h2:=n5p t nu f ht
    have h3:=B6 h2
    have C:=sMb1 Q Y vo nu i0 hY hI t h3
    have C2:=sMb Q Y vo nu i0 hY hI t h3 h2
    have st : ∀ (f:ℕ→Ἅ)(a:Ἅ),ao f (J0 nu) (J0 t) a=ao f (J0 nu) ((J0 t)+2) ((a+f (J0 t))+f ((J0 t)+1)):=fun f a=>by rw [aos f _ _ a h3,aos f _ _ _ h2]
    show (let r:=st24 arm Q Y vo nu i0 j0 k0 hY hI hJ hK t h3 a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7 d0 d1 d2 d3 d4 d5 d6 d7; let r2:=st24 arm Q Y vo nu i0 j0 k0 hY hI hJ hK (t+1) (by rw[C.2.2.2.2.2];exact h2) r.a.a0 r.a.a1 r.a.a2 r.a.a3 r.a.a4 r.a.a5 r.a.a6 r.a.a7 r.c.a0 r.c.a1 r.c.a2 r.c.a3 r.c.a4 r.c.a5 r.c.a6 r.c.a7 r.d.a0 r.d.a1 r.d.a2 r.d.a3 r.d.a4 r.d.a5 r.d.a6 r.d.a7; sM24 arm Q Y vo nu i0 j0 k0 hY hI hJ hK f (t+2) (n5q t nu f C2.1 ht) r2.a.a0 r2.a.a1 r2.a.a2 r2.a.a3 r2.a.a4 r2.a.a5 r2.a.a6 r2.a.a7 r2.c.a0 r2.c.a1 r2.c.a2 r2.c.a3 r2.c.a4 r2.c.a5 r2.c.a6 r2.c.a7 r2.d.a0 r2.d.a1 r2.d.a2 r2.d.a3 r2.d.a4 r2.d.a5 r2.d.a6 r2.d.a7)=_
    simp only [st24_spec arm Q Y vo nu i0 j0 k0 hY hI hJ hK hFi hFj hFk]
    rw [ih]
    simp only [C2.1,C.2.2.2.2.2,st]
theorem n5o(nu:Ἄ)(n:ℕ)(en:(J0 nu)=n)(h:n%2=1):(J0 0)<(J0 nu)∧(J0 1)+2*(n/2)=(J0 nu):=by rw[kc0,kc1,en];omega
theorem n5e(nu:Ἄ)(n:ℕ)(en:(J0 nu)=n)(h:¬n%2=1):(J0 0)+2*(n/2)=(J0 nu):=by rw[kc0,en];omega
@[inline] def sM24P(arm:Ἆ)(Q Y:ἀ)(vo nu i0 j0 k0:Ἄ)(hY:(J0 vo)+(J0 nu)<=(kS Y)/\(J0 vo)+(J0 nu)<2^(31:Ἀ))(hI:((J0 i0)+8*(J0 nu)<=(kS Q)/\(J0 i0)+8*(J0 nu)<2^(31:Ἀ)))(hJ:((J0 j0)+8*(J0 nu)<=(kS Q)/\(J0 j0)+8*(J0 nu)<2^(31:Ἀ)))(hK:((J0 k0)+8*(J0 nu)<=(kS Q)/\(J0 k0)+8*(J0 nu)<2^(31:Ἀ)))(n:ℕ)(en:(J0 nu)=n)(a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7 d0 d1 d2 d3 d4 d5 d6 d7:Ἅ):A24:=
 if h:n%2=1 then
  have H:=n5o nu n en h
  let r:=st24 arm Q Y vo nu i0 j0 k0 hY hI hJ hK 0 H.1 a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7 d0 d1 d2 d3 d4 d5 d6 d7
  sM24 arm Q Y vo nu i0 j0 k0 hY hI hJ hK (n/2) 1 H.2 r.a.a0 r.a.a1 r.a.a2 r.a.a3 r.a.a4 r.a.a5 r.a.a6 r.a.a7 r.c.a0 r.c.a1 r.c.a2 r.c.a3 r.c.a4 r.c.a5 r.c.a6 r.c.a7 r.d.a0 r.d.a1 r.d.a2 r.d.a3 r.d.a4 r.d.a5 r.d.a6 r.d.a7
 else sM24 arm Q Y vo nu i0 j0 k0 hY hI hJ hK (n/2) 0 (n5e nu n en h) a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7 d0 d1 d2 d3 d4 d5 d6 d7
theorem sM24P_spec(arm:Ἆ)(Q Y:ἀ)(vo nu i0 j0 k0:Ἄ)(hY:(J0 vo)+(J0 nu)<=(kS Y)/\(J0 vo)+(J0 nu)<2^(31:Ἀ))(hI:((J0 i0)+8*(J0 nu)<=(kS Q)/\(J0 i0)+8*(J0 nu)<2^(31:Ἀ)))(hJ:((J0 j0)+8*(J0 nu)<=(kS Q)/\(J0 j0)+8*(J0 nu)<2^(31:Ἀ)))(hK:((J0 k0)+8*(J0 nu)<=(kS Q)/\(J0 k0)+8*(J0 nu)<2^(31:Ἀ)))(hFi:FMok arm Q Y vo i0 nu)(hFj:FMok arm Q Y vo j0 nu)(hFk:FMok arm Q Y vo k0 nu)(n:ℕ)(en:(J0 nu)=n)(a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7 d0 d1 d2 d3 d4 d5 d6 d7:Ἅ) :
    sM24P arm Q Y vo nu i0 j0 k0 hY hI hJ hK n en a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7 d0 d1 d2 d3 d4 d5 d6 d7 =
    ⟨⟨ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+0]!) (J0 nu) 0 a0,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+1]!) (J0 nu) 0 a1,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+2]!) (J0 nu) 0 a2,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+3]!) (J0 nu) 0 a3,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+4]!) (J0 nu) 0 a4,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+5]!) (J0 nu) 0 a5,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+6]!) (J0 nu) 0 a6,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+7]!) (J0 nu) 0 a7⟩,⟨ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 j0)+8*s+0]!) (J0 nu) 0 c0,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 j0)+8*s+1]!) (J0 nu) 0 c1,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 j0)+8*s+2]!) (J0 nu) 0 c2,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 j0)+8*s+3]!) (J0 nu) 0 c3,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 j0)+8*s+4]!) (J0 nu) 0 c4,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 j0)+8*s+5]!) (J0 nu) 0 c5,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 j0)+8*s+6]!) (J0 nu) 0 c6,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 j0)+8*s+7]!) (J0 nu) 0 c7⟩,⟨ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 k0)+8*s+0]!) (J0 nu) 0 d0,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 k0)+8*s+1]!) (J0 nu) 0 d1,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 k0)+8*s+2]!) (J0 nu) 0 d2,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 k0)+8*s+3]!) (J0 nu) 0 d3,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 k0)+8*s+4]!) (J0 nu) 0 d4,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 k0)+8*s+5]!) (J0 nu) 0 d5,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 k0)+8*s+6]!) (J0 nu) 0 d6,ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 k0)+8*s+7]!) (J0 nu) 0 d7⟩⟩:=by
  unfold sM24P
  by_cases h:n%2=1
  · rw [dif_pos h]; dsimp only
    have H:=n5o nu n en h
    rw [sM24_spec arm Q Y vo nu i0 j0 k0 hY hI hJ hK hFi hFj hFk,st24_spec arm Q Y vo nu i0 j0 k0 hY hI hJ hK hFi hFj hFk]; dsimp only
    have st : ∀ (f:ℕ→Ἅ)(a:Ἅ),ao f (J0 nu) 0 a=ao f (J0 nu) 1 (a+f 0):=fun f a=>aos f _ _ a (by rw [kc0] at H; exact H.1)
    simp only [kc0,kc1,st]
  · rw [dif_neg h,sM24_spec arm Q Y vo nu i0 j0 k0 hY hI hJ hK hFi hFj hFk,kc0]
theorem wq(a g l:ℕ)(hl:l<8):(8*a ≤ 8*g+l∧8*g+l < 8*a+8) ↔ g=a:=by omega
@[inline] def zR(v:Ἅ)(relu:Ἆ)(H:ἆ):ἇ:=
 let c:=ak v H
 if c<(4294967296:ἆ) then (let w:=N3 c; w&&&((if relu then (0:ἇ) else (0xFFFFFFFF:ἇ))|||((w>>>31)-1))) else (0xFFFFFFFF:ἇ)
theorem kAkBig (S:Ἅ):ak S 4000=4294967296:=by
  unfold ak; unfold F.eF; dsimp only
  generalize S.toBits=b
  rw [if_neg]
  intro h
  simp only [Η,AJ] at h
  obtain ⟨⟨⟨h1, -⟩, -⟩, -⟩:=h
  have hE:((b>>>52)&&&2047).toNat ≤ 2047:=by rw [Ε]; exact ϝ
  have hl:(b>>>52)&&&2047 ≤ (4000:ἆ)+1075:=by rw [θ]; exact hE.trans (by decide)
  rw [θ,χ _ _ hl] at h1
  have:((4000:ἆ)+1075).toNat=5075:=by decide
  rw [this] at h1; have:(26:ἆ).toNat=26:=by decide
  omega
theorem kZRBig (v:Ἅ)(relu:Ἆ):zR v relu 4000=(0xFFFFFFFF:ἇ):=by
  unfold zR; rw [kAkBig]; rfl
theorem kZR (v:Ἅ)(H:ἆ)(relu:Ἆ)(n:ℕ)(D Wf:ℕ→Ἃ)(Fb:Ἃ)(hfb:Fb=p8 relu (ms (n+1) D Wf))
    (g:aG (n+1) D Wf)(fv1:fv v)(b1:|rv v-(af (n+1) D Wf:ℝ)| ≤ (2:ℝ)^((H.toNat:ℕ):ℤ))(b2:|(af (n+1) D Wf:ℝ)| ≤ (2:ℝ)^(425:ℤ))(b6:H.toNat ≤ 600) :
    (if zR v relu H==(0xFFFFFFFF:Ἃ) then Fb else zR v relu H)=p8 relu (ms (n+1) D Wf):=by
  unfold zR; dsimp only
  by_cases hc:ak v H < (4294967296:ἆ)
  · have R:=cw (n+1) D Wf g v fv1 H (by omega) b1 b2 hc
    rw [if_pos hc,kReluW relu _ (kAkF v H hc),R]
    by_cases hf:p8 relu (ms (n+1) D Wf)=0xFFFFFFFF
    · rw [hf]; simp only [Az,ἁ]; rw [hfb,hf]
    · have hn:(p8 relu (ms (n+1) D Wf)==0xFFFFFFFF)=false:=by simpa using hf
      simp only [hn,υ,ἂ]
  · simp only [if_neg hc,Az,ἁ]; exact hfb
noncomputable def oR(relu:Ἆ)(n:Nat)(X W b E:Ἂ ἇ)(ix ib L:Nat)(Y Q:ἀ)(vo rn:Nat)(em cq hb:ἆ)(j:Nat)(v:Ἅ):ἇ:=
 if zR v relu (if ((if em<(56:ἆ) then (56:ἆ) else em)+cq)<=(372:ἆ) then (if hb>((if em<(56:ἆ) then (56:ἆ) else em)+cq) then hb else ((if em<(56:ἆ) then (56:ἆ) else em)+cq)) else (4000:ἆ))==(0xFFFFFFFF:ἇ) then oV relu n X W b E ix ib L Y Q vo rn cq hb j v else zR v relu (if ((if em<(56:ἆ) then (56:ἆ) else em)+cq)<=(372:ἆ) then (if hb>((if em<(56:ἆ) then (56:ἆ) else em)+cq) then hb else ((if em<(56:ἆ) then (56:ἆ) else em)+cq)) else (4000:ἆ))
theorem kOutR (relu:Ἆ)(n:ℕ)(X W b E:Ἂ Ἃ)(ix ib L H:ℕ)(Y Q:ἀ)(vo rn j:ℕ)(em cq hb:ἆ)(v Lf:Ἅ)
    (F:∀ q:ℕ,kB X[q]! ≠ 255∧kG X[q]! ≤ ix)(hbF:∀ q:ℕ,kB b[q]! ≠ 255∧kG b[q]! ≤ H)(M:174+H ≤ ib)(Hn:n+2 ≤ 2^L)(x:L≤27)(z:ib+L≤425)(hn:0<n)
    (hEv:∀ s<n,(kB W[j*n+s]!).toNat ≤ (E[j]!).toNat)
    (hYv:∀ s<n,Y.data[vo+s]! = dw X[rn+s]!)
    (hQv:∀ s<n,Q.data[j/8*n*8+j%8+8*s]! = dw W[j*n+s]!)
    (hv:v=ao (fun s=>Y.data[vo+s]! * Q.data[j/8*n*8+j%8+8*s]!) n 0 (dw ld*dw b[j]!))
    (hLf:Lf=ao (fun k=>dw (X[rn+k]! &&& 0x7FFFFFFF)) n 0 0)
    (hcq:cq=23+cEq (I0 (9*(n+1))).toInt64.toFloat Lf) (hhb:hb=(ib+L-50).toUInt64)
    (hEm:(E[j]!).toNat ≤ em.toNat)(hem:em.toNat < 255) :
    oR relu n X W b E ix ib L Y Q vo rn em cq hb j v=p8 relu (ms (n+1) (fun s=>if s < n then X[rn+s]! else ld) (fun s=>if s < n then W[j*n+s]! else b[j]!)):=by
  have KV:=kOutV relu n X W b E ix ib L H Y Q vo rn j cq hb v Lf F hbF M Hn x z hn hEv hYv hQv hv hLf hcq hhb
  unfold oR
  by_cases hP:((if em<(56:ἆ) then (56:ἆ) else em)+cq)<=(372:ἆ)
  · rw [if_pos hP]
    have c:=kCEb (I0 (9*(n+1))).toInt64.toFloat Lf
    have ej:(E[j]!).toUInt64.toNat=(E[j]!).toNat:=ϩ _
    have hEl:(E[j]!).toNat < 255:=C0 hEm hem
    have tpj:((if (E[j]!).toUInt64<(56:ἆ) then (56:ἆ) else (E[j]!).toUInt64)+cq).toNat=max (E[j]!).toUInt64.toNat 56+23+(cEq (I0 (9*(n+1))).toInt64.toFloat Lf).toNat:=by
      rw [hcq]; exact kOutW_h1 _ _ (by rw [ej]; exact hEl) c
    have tP:((if em<(56:ἆ) then (56:ἆ) else em)+cq).toNat=max em.toNat 56+23+(cEq (I0 (9*(n+1))).toInt64.toFloat Lf).toNat:=by
      rw [hcq]; exact kOutW_h1 _ _ hem c
    have hP':((if em<(56:ἆ) then (56:ἆ) else em)+cq).toNat ≤ 372:=hP
    have mE:max (E[j]!).toUInt64.toNat 56+23+(cEq (I0 (9*(n+1))).toInt64.toFloat Lf).toNat ≤ max em.toNat 56+23+(cEq (I0 (9*(n+1))).toInt64.toFloat Lf).toNat :=
      L1 (L1 (max_le_max_right 56 (by rw [ej]; exact hEm)) 23) _
    have pj:((if (E[j]!).toUInt64<(56:ἆ) then (56:ἆ) else (E[j]!).toUInt64)+cq) ≤ (372:ἆ):=(show ((if (E[j]!).toUInt64<(56:ἆ) then (56:ἆ) else (E[j]!).toUInt64)+cq).toNat ≤ 372 by rw [tpj]; exact mE.trans (by rw [← tP]; exact hP'))
    obtain ⟨g,fv1,b1,b2,_,b6⟩:=kOutW n X W b E ix ib L H Y Q vo rn j cq hb _ v Lf F hbF M Hn x z hn hEv hYv hQv hv hLf hcq hhb rfl hEl pj
    have mono:(if hb>((if (E[j]!).toUInt64<(56:ἆ) then (56:ἆ) else (E[j]!).toUInt64)+cq) then hb else ((if (E[j]!).toUInt64<(56:ἆ) then (56:ἆ) else (E[j]!).toUInt64)+cq)).toNat ≤ (if hb>((if em<(56:ἆ) then (56:ἆ) else em)+cq) then hb else ((if em<(56:ἆ) then (56:ἆ) else em)+cq)).toNat:=by
      rw [kOutW_h2,kOutW_h2,tpj,tP]; exact max_le_max_left _ mE
    have h6:(if hb>((if em<(56:ἆ) then (56:ἆ) else em)+cq) then hb else ((if em<(56:ἆ) then (56:ἆ) else em)+cq)).toNat ≤ 600:=by
      rw [kOutW_h2] at b6 ⊢; exact max_le ((le_max_left _ _).trans b6) (hP'.trans (by decide))
    exact kZR v _ relu n _ _ _ KV g fv1 (AZ b1 (AG (by norm_num) (by exact_mod_cast mono))) b2 h6
  · rw [if_neg hP,kZRBig]; simp only [Az,ἁ]; exact KV
structure DC where
  relu:Ἆ
  n:Nat
  X:Ἂ ἇ
  W:Ἂ ἇ
  b:Ἂ ἇ
  E:Ἂ ἇ
  ix:Nat
  ib:Nat
  L:Nat
  Y:ἀ
  Q:ἀ
  Bd:ἀ
  hb:ἆ
  em:ἆ
  f9:Ἅ
  arm:Ἆ
abbrev FMc(c:DC)(vo n G:Nat):Prop:=∀ s<n,∀ g<G,∀ l<8,∀ a:Ἅ,fM c.arm a c.Y.data[vo+s]! c.Q.data[(g*n+s)*8+l]! = a+c.Y.data[vo+s]! * c.Q.data[(g*n+s)*8+l]!
theorem FMc_ok(c:DC)(vo n G g:Nat)(voU nU iU:Ἄ)(hv:voU.toNat=vo)(hn:nU.toNat=n)(hi:iU.toNat=g*n*8)(hg:g<G)(h:FMc c vo n G):FMok c.arm c.Q c.Y voU iU nU:=by
 intro s hs l hl a;rw[hv,hi];rw[hn]at hs;have:=h s hs g hg l hl a;rwa[show g*n*8+8*s+l=(g*n+s)*8+l by ring]
theorem FMd(c:DC)(n G ow vo xo:Nat)(hYv:∀ t<n,c.Y.data[vo+t]! = dw c.X[xo+t]!)(hQv:∀ t<n,∀ g l,l<8→8*g+l<8*G→c.Q.data[(g*n+t)*8+l]! = if 8*g+l<ow then dw c.W[(8*g+l)*n+t]! else 0)(hX:∀ q:Nat,kB c.X[q]!≠255)(hW:∀ o<ow,∀ s<n,kB c.W[o*n+s]!≠255):FMc c vo n G:=by
 intro s hs g hg l hl a
 rw[hYv s hs,hQv s hs g l hl (by omega)]
 split
 ·rename_i h;exact fzD c.arm _ (hX _) _ (pZ ⟨_,hW _ h s hs,rfl⟩) a
 ·exact fzD c.arm _ (hX _) _ (BA rfl) a
@[noinline] def dnS(c:@& DC)(vo rn:Nat)(cq:ἆ)(j:Nat)(v:Ἅ):ἇ:=
 if zq2 v (c.E[j]!).toUInt64 c.relu cq c.hb==(0xFFFFFFFF:ἇ) then
  sQ2 c.relu c.n c.X c.W c.b c.E c.ix c.ib c.L c.Y c.Q vo rn j (if c.hb > (if (c.E[j]!).toUInt64<(56:ἆ) then (56:ἆ) else (c.E[j]!).toUInt64)+cq then c.hb else (if (c.E[j]!).toUInt64<(56:ἆ) then (56:ἆ) else (c.E[j]!).toUInt64)+cq) v
 else zq2 v (c.E[j]!).toUInt64 c.relu cq c.hb
theorem dnS_eq(c:DC)(vo rn:Nat)(cq:ἆ)(j:Nat)(v:Ἅ):dnS c vo rn cq j v=oV c.relu c.n c.X c.W c.b c.E c.ix c.ib c.L c.Y c.Q vo rn cq c.hb j v:=rfl
@[inline] def dn1(c:@& DC)(vo rn:Nat)(cq H:ἆ)(j:Nat)(l cnt q s:Ἄ)(v:Ἅ)(out:Ἂ ἇ)(hin:∀ k:Ἄ,k<cnt → (q+k*s).toNat<out.size):{a:Ἂ ἇ // a.size=out.size}:=
 if hl:l<cnt then
  let z:=zR v c.relu H
  ⟨out.uset (q+l*s) (if z==(0xFFFFFFFF:ἇ) then dnS c vo rn cq (j+l.toNat) v else z) (hin l hl),BT ..⟩
 else ⟨out,rfl⟩
theorem dn1_sz(c:DC)(vo rn:Nat)(cq H:ἆ)(j:Nat)(l cnt q s:Ἄ)(v:Ἅ)(out:Ἂ ἇ)(hin:∀ k:Ἄ,k<cnt → (q+k*s).toNat<out.size):
    (dn1 c vo rn cq H j l cnt q s v out hin).1.size=out.size:=(dn1 c vo rn cq H j l cnt q s v out hin).2
theorem dn1_get(c:DC)(vo rn:Nat)(cq H:ἆ)(j:Nat)(l cnt q s:Ἄ)(v:Ἅ)(out:Ἂ ἇ)(hin:∀ k:Ἄ,k<cnt → (q+k*s).toNat<out.size)(i:Nat):
    (dn1 c vo rn cq H j l cnt q s v out hin).1[i]! = if l<cnt∧(q+l*s).toNat=i then
      (if zR v c.relu H==(0xFFFFFFFF:ἇ) then oV c.relu c.n c.X c.W c.b c.E c.ix c.ib c.L c.Y c.Q vo rn cq c.hb (j+l.toNat) v else zR v c.relu H) else out[i]!:=by
  unfold dn1; split
  · rename_i hl; dsimp only; rw [kAS,dnS_eq]
    by_cases e:(q+l*s).toNat=i
    · rw [if_pos e,if_pos (show l<cnt∧(q+l*s).toNat=i from ⟨hl,e⟩)]
    · rw [if_neg e,if_neg (show ¬(l<cnt∧(q+l*s).toNat=i) from fun h=>e h.2)]
  · rename_i hl; rw [if_neg (show ¬(l<cnt∧(q+l*s).toNat=i) from fun h=>hl h.1)]
@[noinline] def dn8(c:@& DC)(vo rn:Nat)(cq H:ἆ)(j:Nat)(cnt q s:Ἄ)(out:Ἂ ἇ)(hin:∀ k:Ἄ,k<cnt → (q+k*s).toNat<out.size)(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ):Ἂ ἇ:=
 let r0:=dn1 c vo rn cq H j 0 cnt q s a0 out hin
 let r1:=dn1 c vo rn cq H j 1 cnt q s a1 r0.1 (fun k hk=>by rw [r0.2]; exact hin k hk)
 let r2:=dn1 c vo rn cq H j 2 cnt q s a2 r1.1 (fun k hk=>by rw [r1.2,r0.2]; exact hin k hk)
 let r3:=dn1 c vo rn cq H j 3 cnt q s a3 r2.1 (fun k hk=>by rw [r2.2,r1.2,r0.2]; exact hin k hk)
 let r4:=dn1 c vo rn cq H j 4 cnt q s a4 r3.1 (fun k hk=>by rw [r3.2,r2.2,r1.2,r0.2]; exact hin k hk)
 let r5:=dn1 c vo rn cq H j 5 cnt q s a5 r4.1 (fun k hk=>by rw [r4.2,r3.2,r2.2,r1.2,r0.2]; exact hin k hk)
 let r6:=dn1 c vo rn cq H j 6 cnt q s a6 r5.1 (fun k hk=>by rw [r5.2,r4.2,r3.2,r2.2,r1.2,r0.2]; exact hin k hk)
 (dn1 c vo rn cq H j 7 cnt q s a7 r6.1 (fun k hk=>by rw [r6.2,r5.2,r4.2,r3.2,r2.2,r1.2,r0.2]; exact hin k hk)).1
theorem dn8_size(c:DC)(vo rn:Nat)(cq H:ἆ)(j:Nat)(cnt q s:Ἄ)(out:Ἂ ἇ)(hin)(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ):
    (dn8 c vo rn cq H j cnt q s out hin a0 a1 a2 a3 a4 a5 a6 a7).size=out.size:=by
  unfold dn8; dsimp only; simp only [dn1_sz]
noncomputable def dnV(c:DC)(vo rn:Nat)(cq H:ἆ)(j:Nat)(v:Ἅ):ἇ:=
 if zR v c.relu H==(0xFFFFFFFF:ἇ) then oV c.relu c.n c.X c.W c.b c.E c.ix c.ib c.L c.Y c.Q vo rn cq c.hb j v else zR v c.relu H
theorem dn1_ne(c:DC)(vo rn:Nat)(cq H:ἆ)(j:Nat)(l cnt q s:Ἄ)(v:Ἅ)(out:Ἂ ἇ)(hin)(i:Nat)
    (h:¬(l<cnt∧(q+l*s).toNat=i)):(dn1 c vo rn cq H j l cnt q s v out hin).1[i]! = out[i]!:=by
  rw [dn1_get,if_neg h]
theorem dn1_hit(c:DC)(vo rn:Nat)(cq H:ἆ)(j:Nat)(l cnt q s:Ἄ)(v:Ἅ)(out:Ἂ ἇ)(hin)(hl:l<cnt):
    (dn1 c vo rn cq H j l cnt q s v out hin).1[(q+l*s).toNat]! = dnV c vo rn cq H (j+l.toNat) v:=by
  rw [dn1_get,if_pos ⟨hl,rfl⟩]; rfl
theorem dnIx(q s k:Ἄ)(hk:k.toNat≤8)(hq:q.toNat+8*s.toNat<2^(31:Nat)):(q+k*s).toNat=q.toNat+k.toNat*s.toNat:=by
  have m1:k.toNat*s.toNat≤8*s.toNat:=ι _ hk
  have e:(k*s).toNat=k.toNat*s.toNat:=kMm _ _ (C0 (m1.trans (K2 _ _)) hq)
  rw [kSm _ _ (by rw [e]; exact C0 (J1 m1 _) hq),e]
theorem dn8_ne(c:DC)(vo rn:Nat)(cq H:ἆ)(j:Nat)(cnt q s:Ἄ)(out:Ἂ ἇ)(hin)(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ)(i:Nat)
    (hq:q.toNat+8*s.toNat<2^(31:Nat))(h:∀ l<cnt.toNat,q.toNat+l*s.toNat≠i):
    (dn8 c vo rn cq H j cnt q s out hin a0 a1 a2 a3 a4 a5 a6 a7)[i]! = out[i]!:=by
  have N:∀ k:Ἄ,k.toNat≤7 → ¬(k<cnt∧(q+k*s).toNat=i):=fun k hk g =>
    h k.toNat (ε.mp g.1) (by rw [← dnIx q s k (Nat.le_succ_of_le hk) hq]; exact g.2)
  unfold dn8; dsimp only
  rw [dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 7 (by rw [kc7]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 6 (by rw [kc6]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 5 (by rw [kc5]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 4 (by rw [kc4]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 3 (by rw [kc3]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 2 (by rw [kc2]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 1 (by rw [kc1]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 0 (by rw [kc0]; try decide))]
theorem dn8_hit(c:DC)(vo rn:Nat)(cq H:ἆ)(j:Nat)(cnt q s:Ἄ)(out:Ἂ ἇ)(hin)(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ)(l:Nat)
    (hl:l<cnt.toNat)(hl8:l<8)(hq:q.toNat+8*s.toNat<2^(31:Nat))(hs1:0<s.toNat):
    (dn8 c vo rn cq H j cnt q s out hin a0 a1 a2 a3 a4 a5 a6 a7)[q.toNat+l*s.toNat]! = dnV c vo rn cq H (j+l) (lsel a0 a1 a2 a3 a4 a5 a6 a7 l):=by
  have N:∀ k:Ἄ,k.toNat≤7 → k.toNat≠l → ¬(k<cnt∧(q+k*s).toNat=q.toNat+l*s.toNat):=fun k hk e g =>
    e (B7 hs1 (J8 ((dnIx q s k (Nat.le_succ_of_le hk) hq).symm.trans g.2)))
  have Hk:∀ k:Ἄ,k.toNat=l → k<cnt∧(q+k*s).toNat=q.toNat+l*s.toNat:=fun k e =>
    ⟨ε.mpr (e ▸ hl),by rw [dnIx q s k (by omega) hq,e]⟩
  unfold dn8; dsimp only
  have hl':l ≤ 7:=J2 hl8
  interval_cases l
  · rw [dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 7 (by rw [kc7]; try decide) (by rw [kc7]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 6 (by rw [kc6]; try decide) (by rw [kc6]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 5 (by rw [kc5]; try decide) (by rw [kc5]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 4 (by rw [kc4]; try decide) (by rw [kc4]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 3 (by rw [kc3]; try decide) (by rw [kc3]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 2 (by rw [kc2]; try decide) (by rw [kc2]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 1 (by rw [kc1]; try decide) (by rw [kc1]; try decide)), ← (Hk 0 kc0).2]
    rw [dn1_hit _ _ _ _ _ _ _ _ _ _ _ _ _ (Hk 0 kc0).1,kc0]; rfl
  · rw [dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 7 (by rw [kc7]; try decide) (by rw [kc7]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 6 (by rw [kc6]; try decide) (by rw [kc6]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 5 (by rw [kc5]; try decide) (by rw [kc5]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 4 (by rw [kc4]; try decide) (by rw [kc4]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 3 (by rw [kc3]; try decide) (by rw [kc3]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 2 (by rw [kc2]; try decide) (by rw [kc2]; try decide)), ← (Hk 1 kc1).2]
    rw [dn1_hit _ _ _ _ _ _ _ _ _ _ _ _ _ (Hk 1 kc1).1,kc1]; rfl
  · rw [dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 7 (by rw [kc7]; try decide) (by rw [kc7]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 6 (by rw [kc6]; try decide) (by rw [kc6]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 5 (by rw [kc5]; try decide) (by rw [kc5]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 4 (by rw [kc4]; try decide) (by rw [kc4]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 3 (by rw [kc3]; try decide) (by rw [kc3]; try decide)), ← (Hk 2 kc2).2]
    rw [dn1_hit _ _ _ _ _ _ _ _ _ _ _ _ _ (Hk 2 kc2).1,kc2]; rfl
  · rw [dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 7 (by rw [kc7]; try decide) (by rw [kc7]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 6 (by rw [kc6]; try decide) (by rw [kc6]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 5 (by rw [kc5]; try decide) (by rw [kc5]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 4 (by rw [kc4]; try decide) (by rw [kc4]; try decide)), ← (Hk 3 kc3).2]
    rw [dn1_hit _ _ _ _ _ _ _ _ _ _ _ _ _ (Hk 3 kc3).1,kc3]; rfl
  · rw [dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 7 (by rw [kc7]; try decide) (by rw [kc7]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 6 (by rw [kc6]; try decide) (by rw [kc6]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 5 (by rw [kc5]; try decide) (by rw [kc5]; try decide)), ← (Hk 4 kc4).2]
    rw [dn1_hit _ _ _ _ _ _ _ _ _ _ _ _ _ (Hk 4 kc4).1,kc4]; rfl
  · rw [dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 7 (by rw [kc7]; try decide) (by rw [kc7]; try decide)),dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 6 (by rw [kc6]; try decide) (by rw [kc6]; try decide)), ← (Hk 5 kc5).2]
    rw [dn1_hit _ _ _ _ _ _ _ _ _ _ _ _ _ (Hk 5 kc5).1,kc5]; rfl
  · rw [dn1_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ (N 7 (by rw [kc7]; try decide) (by rw [kc7]; try decide)), ← (Hk 6 kc6).2]
    rw [dn1_hit _ _ _ _ _ _ _ _ _ _ _ _ _ (Hk 6 kc6).1,kc6]; rfl
  · rw [← (Hk 7 kc7).2]
    rw [dn1_hit _ _ _ _ _ _ _ _ _ _ _ _ _ (Hk 7 kc7).1,kc7]; rfl
noncomputable def dnT(c:DC)(n vo rn:Nat)(cq H:ἆ)(o:Nat):ἇ:=
 dnV c vo rn cq H o (ao (fun s=>c.Y.data[vo+s]! * c.Q.data[o/8*n*8+8*s+o%8]!) n 0 (c.Bd.data[o]!))
theorem dnQ(q0 qs:Ἄ)(G g:Nat)(hgG:g+1<=G)(hq:q0.toNat+8*G*qs.toNat<2^(31:Nat))(eo:(G0 (8*g)).toNat=8*g):
    (q0+(G0 (8*g))*qs).toNat=q0.toNat+8*g*qs.toNat∧(q0+(G0 (8*g))*qs).toNat+8*qs.toNat<2^(31:Nat):=by
  have m:8*g*qs.toNat+8*qs.toNat<=8*G*qs.toNat:=by rw [← Ϡ]; exact ι _ (by omega)
  have e1:=kMm (G0 (8*g)) qs (by rw [eo]; exact C0 ((V1 (8*g*qs.toNat) (8*qs.toNat)).trans (m.trans (K2 _ _))) hq)
  rw [eo] at e1
  have e2:(q0+(G0 (8*g))*qs).toNat=q0.toNat+8*g*qs.toNat:=by
    rw [kSm _ _ (by rw [e1]; exact C0 (J1 ((V1 _ _).trans m) _) hq),e1]
  exact ⟨e2,by rw [e2,pA]; exact C0 (J1 m _) hq⟩
theorem dnHin(q0 qs:Ἄ)(ow G g:Nat)(out:Ἂ ἇ)(hgG:g+1<=G)(hq:q0.toNat+8*G*qs.toNat<2^(31:Nat))(eo:(G0 (8*g)).toNat=8*g)
    (hin:∀ o<ow,q0.toNat+o*qs.toNat<out.size)(cnt:Ἄ)(hc:cnt.toNat=min 8 (ow-8*g)) :
    ∀ k:Ἄ,k<cnt → (q0+(G0 (8*g))*qs+k*qs).toNat<out.size:=by
  intro k hk
  have QQ:=dnQ q0 qs G g hgG hq eo
  have hk':k.toNat< min 8 (ow-8*g):=by rw [← hc]; exact ε.mp hk
  rw [dnIx _ _ _ (by omega) QQ.2,QQ.1,pA, ← Ϡ]
  exact hin _ (by omega)
@[inline] def dn1r(c:@& DC)(rl:Ἆ)(hrl:rl=c.relu)(vo rn:Nat)(cq H:ἆ)(j:Nat)(l cnt q s:Ἄ)(v:Ἅ)(out:Ἂ ἇ)(hin:∀ k:Ἄ,k<cnt → (q+k*s).toNat<out.size):{a:Ἂ ἇ // a.size=out.size}:=
 if hl:l<cnt then
  let z:=zR v rl H
  ⟨out.uset (q+l*s) (if z==(0xFFFFFFFF:ἇ) then dnS c vo rn cq (j+l.toNat) v else z) (hin l hl),BT ..⟩
 else ⟨out,rfl⟩
@[inline] def dn8G(c:@& DC)(rl:Ἆ)(hrl:rl=c.relu)(vo rn:Nat)(cq H:ἆ)(j:Nat)(q:Ἄ)(out:Ἂ ἇ)(hin:∀ k:Ἄ,k<8 → (q+k*1).toNat<out.size)(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ):Ἂ ἇ:=
 let r0:=dn1r c rl hrl vo rn cq H j 0 8 q 1 a0 out hin
 let r1:=dn1r c rl hrl vo rn cq H j 1 8 q 1 a1 r0.1 (fun k hk=>by rw [r0.2]; exact hin k hk)
 let r2:=dn1r c rl hrl vo rn cq H j 2 8 q 1 a2 r1.1 (fun k hk=>by rw [r1.2,r0.2]; exact hin k hk)
 let r3:=dn1r c rl hrl vo rn cq H j 3 8 q 1 a3 r2.1 (fun k hk=>by rw [r2.2,r1.2,r0.2]; exact hin k hk)
 let r4:=dn1r c rl hrl vo rn cq H j 4 8 q 1 a4 r3.1 (fun k hk=>by rw [r3.2,r2.2,r1.2,r0.2]; exact hin k hk)
 let r5:=dn1r c rl hrl vo rn cq H j 5 8 q 1 a5 r4.1 (fun k hk=>by rw [r4.2,r3.2,r2.2,r1.2,r0.2]; exact hin k hk)
 let r6:=dn1r c rl hrl vo rn cq H j 6 8 q 1 a6 r5.1 (fun k hk=>by rw [r5.2,r4.2,r3.2,r2.2,r1.2,r0.2]; exact hin k hk)
 (dn1r c rl hrl vo rn cq H j 7 8 q 1 a7 r6.1 (fun k hk=>by rw [r6.2,r5.2,r4.2,r3.2,r2.2,r1.2,r0.2]; exact hin k hk)).1
theorem dn8G_eq(c:DC)(rl:Ἆ)(hrl:rl=c.relu)(vo rn:Nat)(cq H:ἆ)(j:Nat)(q:Ἄ)(out:Ἂ ἇ)(hin:∀ k:Ἄ,k<8 → (q+k*1).toNat<out.size)(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ):
    dn8G c rl hrl vo rn cq H j q out hin a0 a1 a2 a3 a4 a5 a6 a7=dn8 c vo rn cq H j 8 q 1 out hin a0 a1 a2 a3 a4 a5 a6 a7:=by subst hrl; rfl
@[noinline] def dn8GN(c:@& DC)(rl:Ἆ)(hrl:rl=c.relu)(vo rn:Nat)(cq H:ἆ)(j:Nat)(q:Ἄ)(out:Ἂ ἇ)(hin:∀ k:Ἄ,k<8 → (q+k*1).toNat<out.size)(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ):Ἂ ἇ:=dn8G c rl hrl vo rn cq H j q out hin a0 a1 a2 a3 a4 a5 a6 a7
@[inline] def dn8FR(c:@& DC)(h:true=c.relu)(vo rn:Nat)(cq H:ἆ)(j:Nat)(q:Ἄ)(out:Ἂ ἇ)(hin:∀ k:Ἄ,k<8 → (q+k*1).toNat<out.size)(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ):Ἂ ἇ:=dn8GN c true h vo rn cq H j q out hin a0 a1 a2 a3 a4 a5 a6 a7
@[inline] def dn8FN(c:@& DC)(h:false=c.relu)(vo rn:Nat)(cq H:ἆ)(j:Nat)(q:Ἄ)(out:Ἂ ἇ)(hin:∀ k:Ἄ,k<8 → (q+k*1).toNat<out.size)(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ):Ἂ ἇ:=dn8GN c false h vo rn cq H j q out hin a0 a1 a2 a3 a4 a5 a6 a7
@[inline] def dn8D(c:@& DC)(vo rn:Nat)(cq H:ἆ)(j:Nat)(cnt q s:Ἄ)(out:Ἂ ἇ)(hin:∀ k:Ἄ,k<cnt → (q+k*s).toNat<out.size)(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ):Ἂ ἇ:=
 if hc:cnt=8∧s=1 then
  (if hr:c.relu=true then dn8FR c hr.symm vo rn cq H j q out (fun k hk=>by have h2:=hin k (by rw [hc.1]; exact hk); rwa [hc.2] at h2) a0 a1 a2 a3 a4 a5 a6 a7
   else dn8FN c (F8.mpr hr).symm vo rn cq H j q out (fun k hk=>by have h2:=hin k (by rw [hc.1]; exact hk); rwa [hc.2] at h2) a0 a1 a2 a3 a4 a5 a6 a7)
 else dn8 c vo rn cq H j cnt q s out hin a0 a1 a2 a3 a4 a5 a6 a7
theorem dn8D_eq(c:DC)(vo rn:Nat)(cq H:ἆ)(j:Nat)(cnt q s:Ἄ)(out:Ἂ ἇ)(hin:∀ k:Ἄ,k<cnt → (q+k*s).toNat<out.size)(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ):
    dn8D c vo rn cq H j cnt q s out hin a0 a1 a2 a3 a4 a5 a6 a7=dn8 c vo rn cq H j cnt q s out hin a0 a1 a2 a3 a4 a5 a6 a7:=by
 unfold dn8D; split
 · rename_i hc; obtain ⟨h1,h2⟩:=hc; subst h1 h2
   split
   · unfold dn8FR dn8GN; exact dn8G_eq ..
   · unfold dn8FN dn8GN; exact dn8G_eq ..
 · rfl
theorem dn8D_size(c:DC)(vo rn:Nat)(cq H:ἆ)(j:Nat)(cnt q s:Ἄ)(out:Ἂ ἇ)(hin:∀ k:Ἄ,k<cnt → (q+k*s).toNat<out.size)(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ):
    (dn8D c vo rn cq H j cnt q s out hin a0 a1 a2 a3 a4 a5 a6 a7).size=out.size:=by rw [dn8D_eq]; exact dn8_size ..
def mc8(c:@& DC)(vN rN:Nat)(cq H:ἆ)(j:Nat)(cnt q s:Ἄ)(out:Ἂ ἇ)(hin:∀ k:Ἄ,k<cnt → (q+k*s).toNat<out.size)(Q Y:ἀ)(vo nu i0:Ἄ)(hY:(J0 vo)+(J0 nu)<=(kS Y)/\(J0 vo)+(J0 nu)<2^(31:Ἀ))(hQ:(J0 i0)+8*(J0 nu)<=(kS Q)/\(J0 i0)+8*(J0 nu)<2^(31:Ἀ))(t:Ἄ)(ht0:(J0 t)<=(J0 nu))(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ):Ἂ ἇ:=
 if h:t<nu then
   have h1:(J0 t)<(J0 nu):=ε.mp h
   let r:=sM1 c.arm Q Y vo nu i0 hY hQ t h1 a0 a1 a2 a3 a4 a5 a6 a7
  mc8 c vN rN cq H j cnt q s out hin Q Y vo nu i0 hY hQ (t+1) (by rw[sMbT Q Y vo nu i0 hY hQ t ht0];exact h1) r.a0 r.a1 r.a2 r.a3 r.a4 r.a5 r.a6 r.a7
 else dn8D c vN rN cq H j cnt q s out hin a0 a1 a2 a3 a4 a5 a6 a7
termination_by (J0 nu)-(J0 t)
decreasing_by (have h1:(J0 t)<(J0 nu):=ε.mp h;rw[sMbT Q Y vo nu i0 hY hQ t ht0];omega)
theorem mc8_ao(c:DC)(vN rN:Nat)(cq H:ἆ)(j:Nat)(cnt q s:Ἄ)(out:Ἂ ἇ)(hin:∀ k:Ἄ,k<cnt → (q+k*s).toNat<out.size)(Q Y:ἀ)(vo nu i0:Ἄ)(hY:(J0 vo)+(J0 nu)<=(kS Y)/\(J0 vo)+(J0 nu)<2^(31:Ἀ))(hQ:(J0 i0)+8*(J0 nu)<=(kS Q)/\(J0 i0)+8*(J0 nu)<2^(31:Ἀ))(hF:FMok c.arm Q Y vo i0 nu) :
    ∀ m (t:Ἄ)(ht0:(J0 t)<=(J0 nu))(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ),(J0 nu)-(J0 t)=m →
    mc8 c vN rN cq H j cnt q s out hin Q Y vo nu i0 hY hQ t ht0 a0 a1 a2 a3 a4 a5 a6 a7 =
    dn8D c vN rN cq H j cnt q s out hin (ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+0]!) (J0 nu) (J0 t) a0) (ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+1]!) (J0 nu) (J0 t) a1) (ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+2]!) (J0 nu) (J0 t) a2) (ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+3]!) (J0 nu) (J0 t) a3) (ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+4]!) (J0 nu) (J0 t) a4) (ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+5]!) (J0 nu) (J0 t) a5) (ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+6]!) (J0 nu) (J0 t) a6) (ao (fun s=>Y.data[(J0 vo)+s]! * Q.data[(J0 i0)+8*s+7]!) (J0 nu) (J0 t) a7):=by
  intro m; induction m using C6 with
  | _ m ih =>
  intro t ht0 a0 a1 a2 a3 a4 a5 a6 a7 hm
  rw [mc8]
  by_cases h:t<nu
  · simp only [dif_pos h]
    have h1:(J0 t)<(J0 nu):=ε.mp h
    have e1:(J0 (t+1))=(J0 t)+1:=sMbT Q Y vo nu i0 hY hQ t ht0
    rw [ih ((J0 nu)-(J0 (t+1))) (by rw [← hm,e1]; omega) (t+1) (by rw[e1];exact h1) _ _ _ _ _ _ _ _ rfl]
    have st : ∀ (f:ℕ→Ἅ)(a:Ἅ),ao f (J0 nu) (J0 t) a=ao f (J0 nu) ((J0 t)+1) (a+f (J0 t)):=fun f a=>aos f _ _ a h1
    simp only [sM1_spec c.arm Q Y vo nu i0 hY hQ hF,e1,st]
  · simp only [dif_neg h]
    have st : ∀ (f:ℕ→Ἅ)(a:Ἅ),ao f (J0 nu) (J0 t) a=a:=fun f a=>aoe f _ _ a (W4 (fun h3=>h (ε.mpr h3)))
    simp only [st]
theorem mc8_eq(c:DC)(vN rN:Nat)(cq H:ἆ)(j:Nat)(cnt q s:Ἄ)(out:Ἂ ἇ)(hin:∀ k:Ἄ,k<cnt → (q+k*s).toNat<out.size)(Q Y:ἀ)(vo nu i0:Ἄ)(hY:(J0 vo)+(J0 nu)<=(kS Y)/\(J0 vo)+(J0 nu)<2^(31:Ἀ))(hQ:(J0 i0)+8*(J0 nu)<=(kS Q)/\(J0 i0)+8*(J0 nu)<2^(31:Ἀ))(hF:FMok c.arm Q Y vo i0 nu) :
    ∀ m (t:Ἄ)(ht0:(J0 t)<=(J0 nu))(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ),(J0 nu)-(J0 t)=m → mc8 c vN rN cq H j cnt q s out hin Q Y vo nu i0 hY hQ t ht0 a0 a1 a2 a3 a4 a5 a6 a7 =
    (let y:=sM Q Y vo nu i0 hY hQ t ht0 a0 a1 a2 a3 a4 a5 a6 a7; dn8 c vN rN cq H j cnt q s out hin y.a0 y.a1 y.a2 y.a3 y.a4 y.a5 y.a6 y.a7):=by
  intro m t ht0 a0 a1 a2 a3 a4 a5 a6 a7 hm
  rw [mc8_ao c vN rN cq H j cnt q s out hin Q Y vo nu i0 hY hQ hF m t ht0 a0 a1 a2 a3 a4 a5 a6 a7 hm,sM_spec Q Y vo nu i0 hY hQ m t ht0 a0 a1 a2 a3 a4 a5 a6 a7 hm]
  simp only [dn8D_eq]
theorem mc8_sz(c:DC)(vN rN:Nat)(cq H:ἆ)(j:Nat)(cnt q s:Ἄ)(out:Ἂ ἇ)(hin:∀ k:Ἄ,k<cnt → (q+k*s).toNat<out.size)(Q Y:ἀ)(vo nu i0:Ἄ)(hY:(J0 vo)+(J0 nu)<=(kS Y)/\(J0 vo)+(J0 nu)<2^(31:Ἀ))(hQ:(J0 i0)+8*(J0 nu)<=(kS Q)/\(J0 i0)+8*(J0 nu)<2^(31:Ἀ)) :
    ∀ m (t:Ἄ)(ht0:(J0 t)<=(J0 nu))(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ),(J0 nu)-(J0 t)=m →
    (mc8 c vN rN cq H j cnt q s out hin Q Y vo nu i0 hY hQ t ht0 a0 a1 a2 a3 a4 a5 a6 a7).size=out.size:=by
  intro m;induction m using C6 with
  | _ m ih=>
  intro t ht0 a0 a1 a2 a3 a4 a5 a6 a7 hm
  rw[mc8];split
  ·rename_i h;have e1:=sMbT Q Y vo nu i0 hY hQ t ht0;exact ih _ (by rw[← hm,e1];have:=ε.mp h;omega) _ _ _ _ _ _ _ _ _ _ rfl
  ·exact dn8D_size ..
@[inline] def dnW8(c:@& DC)(n G ow vo rn:Nat)(cq H:ἆ)(q0 qs voU nU:Ἄ)(hY:vo+n<=c.Y.size)(hQ:c.Q.size=n*(8*G))(hBd:8*G<=c.Bd.size)(hB1:vo+n<2^(31:Nat))(hB2:n*(8*G)<2^(31:Nat))(hB3:8*G<2^(31:Nat))(hvo:voU=G0 vo)(hnU:nU=G0 n)(hq:q0.toNat+8*G*qs.toNat<2^(31:Nat))(g:Nat)(hg:g<G)(oU iU:Ἄ)(hoU:oU=G0 (8*g))(hiU:iU=G0 (g*n*8))(out:Ἂ ἇ)(hin:∀ o<ow,q0.toNat+o*qs.toNat<out.size):Ἂ ἇ:=
  have B:=sPb n G g vo _ _ _ _ hg hY hQ hBd (le_refl _) hB1 hB2 hB3
  mc8 c vo rn cq H (8*g) (G0 (min 8 (ow-8*g))) (q0+oU*qs) qs out (by rw [hoU]; exact dnHin q0 qs ow G g out hg hq B.2.2.2.2.2.2.2.2 hin _ (s5tu _ (C0 (P7 _ _) (by decide))))
    c.Q c.Y voU nU iU (by rw [hvo,hnU]; exact B.1) (by rw [hiU,hnU]; exact B.2.1) 0 (by rw[kc0];exact ϟ _)
    (c.Bd.uget (oU+0) (by rw [hoU]; exact kIx7 _ 0 0 kc0 n5d0 c.Bd B.2.2.1 B.2.2.2.2.1)) (c.Bd.uget (oU+1) (by rw [hoU]; exact kIx7 _ 1 1 kc1 n5d1 c.Bd B.2.2.1 B.2.2.2.2.1)) (c.Bd.uget (oU+2) (by rw [hoU]; exact kIx7 _ 2 2 kc2 n5d2 c.Bd B.2.2.1 B.2.2.2.2.1)) (c.Bd.uget (oU+3) (by rw [hoU]; exact kIx7 _ 3 3 kc3 n5d3 c.Bd B.2.2.1 B.2.2.2.2.1)) (c.Bd.uget (oU+4) (by rw [hoU]; exact kIx7 _ 4 4 kc4 n5d4 c.Bd B.2.2.1 B.2.2.2.2.1)) (c.Bd.uget (oU+5) (by rw [hoU]; exact kIx7 _ 5 5 kc5 n5d5 c.Bd B.2.2.1 B.2.2.2.2.1)) (c.Bd.uget (oU+6) (by rw [hoU]; exact kIx7 _ 6 6 kc6 n5d6 c.Bd B.2.2.1 B.2.2.2.2.1)) (c.Bd.uget (oU+7) (by rw [hoU]; exact kIx7 _ 7 7 kc7 n5d7 c.Bd B.2.2.1 B.2.2.2.2.1))
theorem dnW8_eq(c:DC)(n G ow vo rn:Nat)(cq H:ἆ)(q0 qs voU nU:Ἄ)(hY:vo+n<=c.Y.size)(hQ:c.Q.size=n*(8*G))(hBd:8*G<=c.Bd.size)(hB1:vo+n<2^(31:Nat))(hB2:n*(8*G)<2^(31:Nat))(hB3:8*G<2^(31:Nat))(hvo:voU=G0 vo)(hnU:nU=G0 n)(hq:q0.toNat+8*G*qs.toNat<2^(31:Nat))(g:Nat)(hg:g<G)(oU iU:Ἄ)(hoU:oU=G0 (8*g))(hiU:iU=G0 (g*n*8))(hF:FMok c.arm c.Q c.Y voU iU nU)(out:Ἂ ἇ)(hin:∀ o<ow,q0.toNat+o*qs.toNat<out.size):
    dnW8 c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq g hg oU iU hoU hiU out hin=(
  have B:=sPb n G g vo _ _ _ _ hg hY hQ hBd (le_refl _) hB1 hB2 hB3
  let y:=sM c.Q c.Y voU nU iU (by rw [hvo,hnU]; exact B.1) (by rw [hiU,hnU]; exact B.2.1) 0 (by rw[kc0];exact ϟ _)
    (c.Bd.uget (oU+0) (by rw [hoU]; exact kIx7 _ 0 0 kc0 n5d0 c.Bd B.2.2.1 B.2.2.2.2.1)) (c.Bd.uget (oU+1) (by rw [hoU]; exact kIx7 _ 1 1 kc1 n5d1 c.Bd B.2.2.1 B.2.2.2.2.1)) (c.Bd.uget (oU+2) (by rw [hoU]; exact kIx7 _ 2 2 kc2 n5d2 c.Bd B.2.2.1 B.2.2.2.2.1)) (c.Bd.uget (oU+3) (by rw [hoU]; exact kIx7 _ 3 3 kc3 n5d3 c.Bd B.2.2.1 B.2.2.2.2.1)) (c.Bd.uget (oU+4) (by rw [hoU]; exact kIx7 _ 4 4 kc4 n5d4 c.Bd B.2.2.1 B.2.2.2.2.1)) (c.Bd.uget (oU+5) (by rw [hoU]; exact kIx7 _ 5 5 kc5 n5d5 c.Bd B.2.2.1 B.2.2.2.2.1)) (c.Bd.uget (oU+6) (by rw [hoU]; exact kIx7 _ 6 6 kc6 n5d6 c.Bd B.2.2.1 B.2.2.2.2.1)) (c.Bd.uget (oU+7) (by rw [hoU]; exact kIx7 _ 7 7 kc7 n5d7 c.Bd B.2.2.1 B.2.2.2.2.1))
  dn8 c vo rn cq H (8*g) (G0 (min 8 (ow-8*g))) (q0+oU*qs) qs out (by rw [hoU]; exact dnHin q0 qs ow G g out hg hq B.2.2.2.2.2.2.2.2 hin _ (s5tu _ (C0 (P7 _ _) (by decide))))
    y.a0 y.a1 y.a2 y.a3 y.a4 y.a5 y.a6 y.a7):=by
  unfold dnW8; exact mc8_eq _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ hF _ _ _ _ _ _ _ _ _ _ _ rfl
theorem dnW8_spec(c:DC)(n G ow vo rn:Nat)(cq H:ἆ)(q0 qs voU nU:Ἄ)(hY:vo+n<=c.Y.size)(hQ:c.Q.size=n*(8*G))(hBd:8*G<=c.Bd.size)(hB1:vo+n<2^(31:Nat))(hB2:n*(8*G)<2^(31:Nat))(hB3:8*G<2^(31:Nat))(hvo:voU=G0 vo)(hnU:nU=G0 n)(hq:q0.toNat+8*G*qs.toNat<2^(31:Nat))(g:Nat)(hg:g<G)(oU iU:Ἄ)(hoU:oU=G0 (8*g))(hiU:iU=G0 (g*n*8))(out:Ἂ ἇ)(hin:∀ o<ow,q0.toNat+o*qs.toNat<out.size)
    (hF:FMc c vo n G)(hs1:0<qs.toNat) :
    (dnW8 c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq g hg oU iU hoU hiU out hin).size=out.size ∧
    (∀ i,(∀ o<ow,8*g<=o → o<8*g+8 → q0.toNat+o*qs.toNat≠i) → (dnW8 c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq g hg oU iU hoU hiU out hin)[i]! = out[i]!) ∧
    (∀ o<ow,8*g<=o → o<8*g+8 → (dnW8 c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq g hg oU iU hoU hiU out hin)[q0.toNat+o*qs.toNat]! = dnT c n vo rn cq H o):=by
  subst hvo hnU hoU hiU
  have B:=sPb n G g vo _ _ _ _ hg hY hQ hBd (le_refl _) hB1 hB2 hB3
  have QQ:=dnQ q0 qs G g hg hq B.2.2.2.2.2.2.2.2
  have ec:(G0 (min 8 (ow-8*g))).toNat=min 8 (ow-8*g):=s5tu _ (C0 (P7 _ _) (by decide))
  have F0:=FMc_ok c vo n G g _ _ _ B.2.2.2.2.2.1 B.2.2.2.2.2.2.1 B.2.2.2.2.2.2.2.1 hg hF
  rw [dnW8_eq (hF:=F0)]; dsimp only
  have hl8:∀ l,l<(G0 (min 8 (ow-8*g))).toNat → l<8∧8*g+l<ow:=fun l hl=>by rw [ec] at hl; omega
  have ix:∀ l,(q0+(G0 (8*g))*qs).toNat+l*qs.toNat=q0.toNat+(8*g+l)*qs.toNat:=fun l=>by rw [QQ.1,Ϡ,pA]
  refine ⟨dn8_size .., fun i hi=>dn8_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ i QQ.2 (fun l hl e=>hi (8*g+l) (hl8 l hl).2 (V1 _ _) (by have:=(hl8 l hl).1; omega) (by rw [← ix]; exact e)),fun o ho h1 h2 => ?_⟩
  have hl:o-8*g<(G0 (min 8 (ow-8*g))).toNat:=by rw [ec]; omega
  have eo:q0.toNat+o*qs.toNat=(q0+(G0 (8*g))*qs).toNat+(o-8*g)*qs.toNat:=by rw [ix,M4 h1]
  rw [eo,dn8_hit _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (o-8*g) hl (by omega) QQ.2 hs1]
  rw [sM_spec _ _ _ _ _ _ _ _ 0 _ _ _ _ _ _ _ _ _ rfl]
  have eb:∀ (k:Ἄ)(cc:Nat)(hk:k.toNat=cc)(hc:cc≤7) h,c.Bd.uget (G0 (8*g)+k) h=c.Bd.data[8*g+cc]!:=fun k cc hk hc h=>by
    rw [kFU,kSm _ k (C0 (J1 (hk ▸ hc) _) B.2.2.2.2.1),hk,B.2.2.2.2.2.2.2.2]
  simp only [B.2.2.2.2.2.1,B.2.2.2.2.2.2.1,B.2.2.2.2.2.2.2.1,kc0,eb 0 0 kc0 n5d0,eb 1 1 kc1 n5d1,eb 2 2 kc2 n5d2,eb 3 3 kc3 n5d3,eb 4 4 kc4 n5d4,eb 5 5 kc5 n5d5,eb 6 6 kc6 n5d6,eb 7 7 kc7 n5d7]
  rw [lselAo (fun l s=>c.Y.data[vo+s]! * c.Q.data[g*n*8+8*s+l]!) (fun l=>c.Bd.data[8*g+l]!) n 0 (o-8*g) (by omega)]
  have hd:o/8=g:=by omega
  have hm:o%8=o-8*g:=by omega
  unfold dnT; rw [hd,hm,M4 h1]
theorem dnTA(a b:Nat):G0 (a+b)=G0 a+G0 b:=by
  simp only [lX,mC]
@[inline] def fin24(c:@& DC)(vN rN:Nat)(cq H:ἆ)(p0 p1 p2:Nat)(k2 qa qb qc s:Ἄ)(out:Ἂ ἇ)(e0:∀ k:Ἄ,k<8 → (qa+k*s).toNat<out.size)(e1:∀ k:Ἄ,k<8 → (qb+k*s).toNat<out.size)(e2:∀ k:Ἄ,k<k2 → (qc+k*s).toNat<out.size)(a0 a1 a2 a3 a4 a5 a6 a7 c0 c1 c2 c3 c4 c5 c6 c7 d0 d1 d2 d3 d4 d5 d6 d7:Ἅ):Ἂ ἇ:=
  let r0:=dn8D c vN rN cq H p0 8 qa s out e0 a0 a1 a2 a3 a4 a5 a6 a7
  have s0:r0.size=out.size:=dn8D_size ..
  let r1:=dn8D c vN rN cq H p1 8 qb s r0 (fun k hk=>by rw [s0]; exact e1 k hk) c0 c1 c2 c3 c4 c5 c6 c7
  have s1:r1.size=out.size:=(dn8D_size ..).trans s0
  dn8D c vN rN cq H p2 k2 qc s r1 (fun k hk=>by rw [s1]; exact e2 k hk) d0 d1 d2 d3 d4 d5 d6 d7
noncomputable def dnW24o(c:@& DC)(n G ow vo rn:Nat)(cq H:ἆ)(q0 qs voU nU:Ἄ)(hY:vo+n<=c.Y.size)(hQ:c.Q.size=n*(8*G))(hBd:8*G<=c.Bd.size)(hB1:vo+n<2^(31:Nat))(hB2:n*(8*G)<2^(31:Nat))(hB3:8*G<2^(31:Nat))(hvo:voU=G0 vo)(hnU:nU=G0 n)(hq:q0.toNat+8*G*qs.toNat<2^(31:Nat))(hG2:8*G<=ow+7)(n8U:Ἄ)(hn8:n8U=G0 (n*8))(g:Nat)(h3:g+3<=G)(oU iU:Ἄ)(hoU:oU=G0 (8*g))(hiU:iU=G0 (g*n*8))(out:Ἂ ἇ)(hin:∀ o<ow,q0.toNat+o*qs.toNat<out.size):Ἂ ἇ:=
  have B0:=sPb n G g vo _ _ _ _ (n5g g G 0 (by decide) h3) hY hQ hBd (le_refl _) hB1 hB2 hB3
  have B1:=sPb n G (g+1) vo _ _ _ _ (n5g g G 1 (by decide) h3) hY hQ hBd (le_refl _) hB1 hB2 hB3
  have B2:=sPb n G (g+2) vo _ _ _ _ (n5g g G 2 (by decide) h3) hY hQ hBd (le_refl _) hB1 hB2 hB3
  have ho1:oU+8=G0 (8*(g+1)):=by rw [hoU,show 8*(g+1)=8*g+8 by ring,dnTA]; rfl
  have ho2:oU+16=G0 (8*(g+2)):=by rw [hoU,show 8*(g+2)=8*g+16 by ring,dnTA]; rfl
  have hi1:iU+n8U=G0 ((g+1)*n*8):=by rw [hiU,hn8, ← dnTA]; congr 1; ring
  have hi2:iU+n8U+n8U=G0 ((g+2)*n*8):=by rw [hi1,hn8, ← dnTA]; congr 1; ring
  let y:=sM24P c.arm c.Q c.Y voU nU iU (iU+n8U) (iU+n8U+n8U) (by rw [hvo,hnU]; exact B0.1) (by rw [hiU,hnU]; exact B0.2.1) (by rw [hi1,hnU]; exact B1.2.1) (by rw [hi2,hnU]; exact B2.2.1) n (by rw [hnU]; exact B0.2.2.2.2.2.2.1)
    (c.Bd.uget (oU+0) (by rw [hoU]; exact kIx7 _ 0 0 kc0 n5d0 c.Bd B0.2.2.1 B0.2.2.2.2.1)) (c.Bd.uget (oU+1) (by rw [hoU]; exact kIx7 _ 1 1 kc1 n5d1 c.Bd B0.2.2.1 B0.2.2.2.2.1)) (c.Bd.uget (oU+2) (by rw [hoU]; exact kIx7 _ 2 2 kc2 n5d2 c.Bd B0.2.2.1 B0.2.2.2.2.1)) (c.Bd.uget (oU+3) (by rw [hoU]; exact kIx7 _ 3 3 kc3 n5d3 c.Bd B0.2.2.1 B0.2.2.2.2.1)) (c.Bd.uget (oU+4) (by rw [hoU]; exact kIx7 _ 4 4 kc4 n5d4 c.Bd B0.2.2.1 B0.2.2.2.2.1)) (c.Bd.uget (oU+5) (by rw [hoU]; exact kIx7 _ 5 5 kc5 n5d5 c.Bd B0.2.2.1 B0.2.2.2.2.1)) (c.Bd.uget (oU+6) (by rw [hoU]; exact kIx7 _ 6 6 kc6 n5d6 c.Bd B0.2.2.1 B0.2.2.2.2.1)) (c.Bd.uget (oU+7) (by rw [hoU]; exact kIx7 _ 7 7 kc7 n5d7 c.Bd B0.2.2.1 B0.2.2.2.2.1)) (c.Bd.uget ((oU+8)+0) (by rw [ho1]; exact kIx7 _ 0 0 kc0 n5d0 c.Bd B1.2.2.1 B1.2.2.2.2.1)) (c.Bd.uget ((oU+8)+1) (by rw [ho1]; exact kIx7 _ 1 1 kc1 n5d1 c.Bd B1.2.2.1 B1.2.2.2.2.1)) (c.Bd.uget ((oU+8)+2) (by rw [ho1]; exact kIx7 _ 2 2 kc2 n5d2 c.Bd B1.2.2.1 B1.2.2.2.2.1)) (c.Bd.uget ((oU+8)+3) (by rw [ho1]; exact kIx7 _ 3 3 kc3 n5d3 c.Bd B1.2.2.1 B1.2.2.2.2.1)) (c.Bd.uget ((oU+8)+4) (by rw [ho1]; exact kIx7 _ 4 4 kc4 n5d4 c.Bd B1.2.2.1 B1.2.2.2.2.1)) (c.Bd.uget ((oU+8)+5) (by rw [ho1]; exact kIx7 _ 5 5 kc5 n5d5 c.Bd B1.2.2.1 B1.2.2.2.2.1)) (c.Bd.uget ((oU+8)+6) (by rw [ho1]; exact kIx7 _ 6 6 kc6 n5d6 c.Bd B1.2.2.1 B1.2.2.2.2.1)) (c.Bd.uget ((oU+8)+7) (by rw [ho1]; exact kIx7 _ 7 7 kc7 n5d7 c.Bd B1.2.2.1 B1.2.2.2.2.1)) (c.Bd.uget ((oU+16)+0) (by rw [ho2]; exact kIx7 _ 0 0 kc0 n5d0 c.Bd B2.2.2.1 B2.2.2.2.2.1)) (c.Bd.uget ((oU+16)+1) (by rw [ho2]; exact kIx7 _ 1 1 kc1 n5d1 c.Bd B2.2.2.1 B2.2.2.2.2.1)) (c.Bd.uget ((oU+16)+2) (by rw [ho2]; exact kIx7 _ 2 2 kc2 n5d2 c.Bd B2.2.2.1 B2.2.2.2.2.1)) (c.Bd.uget ((oU+16)+3) (by rw [ho2]; exact kIx7 _ 3 3 kc3 n5d3 c.Bd B2.2.2.1 B2.2.2.2.2.1)) (c.Bd.uget ((oU+16)+4) (by rw [ho2]; exact kIx7 _ 4 4 kc4 n5d4 c.Bd B2.2.2.1 B2.2.2.2.2.1)) (c.Bd.uget ((oU+16)+5) (by rw [ho2]; exact kIx7 _ 5 5 kc5 n5d5 c.Bd B2.2.2.1 B2.2.2.2.2.1)) (c.Bd.uget ((oU+16)+6) (by rw [ho2]; exact kIx7 _ 6 6 kc6 n5d6 c.Bd B2.2.2.1 B2.2.2.2.2.1)) (c.Bd.uget ((oU+16)+7) (by rw [ho2]; exact kIx7 _ 7 7 kc7 n5d7 c.Bd B2.2.2.1 B2.2.2.2.2.1))
  let r0:=dn8 c vo rn cq H (8*g) 8 (q0+oU*qs) qs out (by rw [hoU]; exact dnHin q0 qs ow G g out (by omega) hq B0.2.2.2.2.2.2.2.2 hin _ (by rw [kc8]; omega))
    y.a.a0 y.a.a1 y.a.a2 y.a.a3 y.a.a4 y.a.a5 y.a.a6 y.a.a7
  have s0:r0.size=out.size:=dn8_size ..
  let r1:=dn8 c vo rn cq H (8*(g+1)) 8 (q0+(oU+8)*qs) qs r0 (by rw [ho1,s0]; exact dnHin q0 qs ow G (g+1) out (by omega) hq B1.2.2.2.2.2.2.2.2 hin _ (by rw [kc8]; omega))
    y.c.a0 y.c.a1 y.c.a2 y.c.a3 y.c.a4 y.c.a5 y.c.a6 y.c.a7
  have s1:r1.size=out.size:=(dn8_size ..).trans s0
  dn8 c vo rn cq H (8*(g+2)) (G0 (min 8 (ow-8*(g+2)))) (q0+(oU+16)*qs) qs r1 (by rw [ho2,s1]; exact dnHin q0 qs ow G (g+2) out (by omega) hq B2.2.2.2.2.2.2.2.2 hin _ (s5tu _ (C0 (P7 _ _) (by decide))))
    y.d.a0 y.d.a1 y.d.a2 y.d.a3 y.d.a4 y.d.a5 y.d.a6 y.d.a7
@[inline] def dnW24(c:@& DC)(n G ow vo rn:Nat)(cq H:ἆ)(q0 qs voU nU:Ἄ)(hY:vo+n<=c.Y.size)(hQ:c.Q.size=n*(8*G))(hBd:8*G<=c.Bd.size)(hB1:vo+n<2^(31:Nat))(hB2:n*(8*G)<2^(31:Nat))(hB3:8*G<2^(31:Nat))(hvo:voU=G0 vo)(hnU:nU=G0 n)(hq:q0.toNat+8*G*qs.toNat<2^(31:Nat))(hG2:8*G<=ow+7)(n8U:Ἄ)(hn8:n8U=G0 (n*8))(g:Nat)(h3:g+3<=G)(oU iU:Ἄ)(hoU:oU=G0 (8*g))(hiU:iU=G0 (g*n*8))(out:Ἂ ἇ)(hin:∀ o<ow,q0.toNat+o*qs.toNat<out.size):Ἂ ἇ:=
  have B0:=sPb n G g vo _ _ _ _ (n5g g G 0 (by decide) h3) hY hQ hBd (le_refl _) hB1 hB2 hB3
  have B1:=sPb n G (g+1) vo _ _ _ _ (n5g g G 1 (by decide) h3) hY hQ hBd (le_refl _) hB1 hB2 hB3
  have B2:=sPb n G (g+2) vo _ _ _ _ (n5g g G 2 (by decide) h3) hY hQ hBd (le_refl _) hB1 hB2 hB3
  have ho1:oU+8=G0 (8*(g+1)):=by rw [hoU,show 8*(g+1)=8*g+8 by ring,dnTA]; rfl
  have ho2:oU+16=G0 (8*(g+2)):=by rw [hoU,show 8*(g+2)=8*g+16 by ring,dnTA]; rfl
  have hi1:iU+n8U=G0 ((g+1)*n*8):=by rw [hiU,hn8, ← dnTA]; congr 1; ring
  have hi2:iU+n8U+n8U=G0 ((g+2)*n*8):=by rw [hi1,hn8, ← dnTA]; congr 1; ring
  let y:=sM24P c.arm c.Q c.Y voU nU iU (iU+n8U) (iU+n8U+n8U) (by rw [hvo,hnU]; exact B0.1) (by rw [hiU,hnU]; exact B0.2.1) (by rw [hi1,hnU]; exact B1.2.1) (by rw [hi2,hnU]; exact B2.2.1) n (by rw [hnU]; exact B0.2.2.2.2.2.2.1)
    (c.Bd.uget (oU+0) (by rw [hoU]; exact kIx7 _ 0 0 kc0 n5d0 c.Bd B0.2.2.1 B0.2.2.2.2.1)) (c.Bd.uget (oU+1) (by rw [hoU]; exact kIx7 _ 1 1 kc1 n5d1 c.Bd B0.2.2.1 B0.2.2.2.2.1)) (c.Bd.uget (oU+2) (by rw [hoU]; exact kIx7 _ 2 2 kc2 n5d2 c.Bd B0.2.2.1 B0.2.2.2.2.1)) (c.Bd.uget (oU+3) (by rw [hoU]; exact kIx7 _ 3 3 kc3 n5d3 c.Bd B0.2.2.1 B0.2.2.2.2.1)) (c.Bd.uget (oU+4) (by rw [hoU]; exact kIx7 _ 4 4 kc4 n5d4 c.Bd B0.2.2.1 B0.2.2.2.2.1)) (c.Bd.uget (oU+5) (by rw [hoU]; exact kIx7 _ 5 5 kc5 n5d5 c.Bd B0.2.2.1 B0.2.2.2.2.1)) (c.Bd.uget (oU+6) (by rw [hoU]; exact kIx7 _ 6 6 kc6 n5d6 c.Bd B0.2.2.1 B0.2.2.2.2.1)) (c.Bd.uget (oU+7) (by rw [hoU]; exact kIx7 _ 7 7 kc7 n5d7 c.Bd B0.2.2.1 B0.2.2.2.2.1)) (c.Bd.uget ((oU+8)+0) (by rw [ho1]; exact kIx7 _ 0 0 kc0 n5d0 c.Bd B1.2.2.1 B1.2.2.2.2.1)) (c.Bd.uget ((oU+8)+1) (by rw [ho1]; exact kIx7 _ 1 1 kc1 n5d1 c.Bd B1.2.2.1 B1.2.2.2.2.1)) (c.Bd.uget ((oU+8)+2) (by rw [ho1]; exact kIx7 _ 2 2 kc2 n5d2 c.Bd B1.2.2.1 B1.2.2.2.2.1)) (c.Bd.uget ((oU+8)+3) (by rw [ho1]; exact kIx7 _ 3 3 kc3 n5d3 c.Bd B1.2.2.1 B1.2.2.2.2.1)) (c.Bd.uget ((oU+8)+4) (by rw [ho1]; exact kIx7 _ 4 4 kc4 n5d4 c.Bd B1.2.2.1 B1.2.2.2.2.1)) (c.Bd.uget ((oU+8)+5) (by rw [ho1]; exact kIx7 _ 5 5 kc5 n5d5 c.Bd B1.2.2.1 B1.2.2.2.2.1)) (c.Bd.uget ((oU+8)+6) (by rw [ho1]; exact kIx7 _ 6 6 kc6 n5d6 c.Bd B1.2.2.1 B1.2.2.2.2.1)) (c.Bd.uget ((oU+8)+7) (by rw [ho1]; exact kIx7 _ 7 7 kc7 n5d7 c.Bd B1.2.2.1 B1.2.2.2.2.1)) (c.Bd.uget ((oU+16)+0) (by rw [ho2]; exact kIx7 _ 0 0 kc0 n5d0 c.Bd B2.2.2.1 B2.2.2.2.2.1)) (c.Bd.uget ((oU+16)+1) (by rw [ho2]; exact kIx7 _ 1 1 kc1 n5d1 c.Bd B2.2.2.1 B2.2.2.2.2.1)) (c.Bd.uget ((oU+16)+2) (by rw [ho2]; exact kIx7 _ 2 2 kc2 n5d2 c.Bd B2.2.2.1 B2.2.2.2.2.1)) (c.Bd.uget ((oU+16)+3) (by rw [ho2]; exact kIx7 _ 3 3 kc3 n5d3 c.Bd B2.2.2.1 B2.2.2.2.2.1)) (c.Bd.uget ((oU+16)+4) (by rw [ho2]; exact kIx7 _ 4 4 kc4 n5d4 c.Bd B2.2.2.1 B2.2.2.2.2.1)) (c.Bd.uget ((oU+16)+5) (by rw [ho2]; exact kIx7 _ 5 5 kc5 n5d5 c.Bd B2.2.2.1 B2.2.2.2.2.1)) (c.Bd.uget ((oU+16)+6) (by rw [ho2]; exact kIx7 _ 6 6 kc6 n5d6 c.Bd B2.2.2.1 B2.2.2.2.2.1)) (c.Bd.uget ((oU+16)+7) (by rw [ho2]; exact kIx7 _ 7 7 kc7 n5d7 c.Bd B2.2.2.1 B2.2.2.2.2.1))
  fin24 c vo rn cq H (8*g) (8*(g+1)) (8*(g+2)) (G0 (min 8 (ow-8*(g+2)))) (q0+oU*qs) (q0+(oU+8)*qs) (q0+(oU+16)*qs) qs out
    (by rw [hoU]; exact dnHin q0 qs ow G g out (by omega) hq B0.2.2.2.2.2.2.2.2 hin _ (by rw [kc8]; omega))
    (by rw [ho1]; exact dnHin q0 qs ow G (g+1) out (by omega) hq B1.2.2.2.2.2.2.2.2 hin _ (by rw [kc8]; omega))
    (by rw [ho2]; exact dnHin q0 qs ow G (g+2) out (by omega) hq B2.2.2.2.2.2.2.2.2 hin _ (s5tu _ (C0 (P7 _ _) (by decide))))
    y.a.a0 y.a.a1 y.a.a2 y.a.a3 y.a.a4 y.a.a5 y.a.a6 y.a.a7 y.c.a0 y.c.a1 y.c.a2 y.c.a3 y.c.a4 y.c.a5 y.c.a6 y.c.a7 y.d.a0 y.d.a1 y.d.a2 y.d.a3 y.d.a4 y.d.a5 y.d.a6 y.d.a7
theorem dnW24_eq(c:DC)(n G ow vo rn:Nat)(cq H:ἆ)(q0 qs voU nU:Ἄ)(hY:vo+n<=c.Y.size)(hQ:c.Q.size=n*(8*G))(hBd:8*G<=c.Bd.size)(hB1:vo+n<2^(31:Nat))(hB2:n*(8*G)<2^(31:Nat))(hB3:8*G<2^(31:Nat))(hvo:voU=G0 vo)(hnU:nU=G0 n)(hq:q0.toNat+8*G*qs.toNat<2^(31:Nat))(hG2:8*G<=ow+7)(n8U:Ἄ)(hn8:n8U=G0 (n*8))(g:Nat)(h3:g+3<=G)(oU iU:Ἄ)(hoU:oU=G0 (8*g))(hiU:iU=G0 (g*n*8))(out:Ἂ ἇ)(hin:∀ o<ow,q0.toNat+o*qs.toNat<out.size):
    dnW24 c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq hG2 n8U hn8 g h3 oU iU hoU hiU out hin=dnW24o c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq hG2 n8U hn8 g h3 oU iU hoU hiU out hin:=by
  unfold dnW24 dnW24o fin24; simp only [dn8D_eq]
theorem dnW24o_spec(c:DC)(n G ow vo rn:Nat)(cq H:ἆ)(q0 qs voU nU:Ἄ)(hY:vo+n<=c.Y.size)(hQ:c.Q.size=n*(8*G))(hBd:8*G<=c.Bd.size)(hB1:vo+n<2^(31:Nat))(hB2:n*(8*G)<2^(31:Nat))(hB3:8*G<2^(31:Nat))(hvo:voU=G0 vo)(hnU:nU=G0 n)(hq:q0.toNat+8*G*qs.toNat<2^(31:Nat))(hG2:8*G<=ow+7)(n8U:Ἄ)(hn8:n8U=G0 (n*8))(g:Nat)(h3:g+3<=G)(oU iU:Ἄ)(hoU:oU=G0 (8*g))(hiU:iU=G0 (g*n*8))(out:Ἂ ἇ)(hin:∀ o<ow,q0.toNat+o*qs.toNat<out.size)
    (hF:FMc c vo n G)(hs1:0<qs.toNat) :
    (dnW24o c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq hG2 n8U hn8 g h3 oU iU hoU hiU out hin).size=out.size ∧
    (∀ i,(∀ o<ow,8*g<=o → o<8*g+24 → q0.toNat+o*qs.toNat≠i) → (dnW24o c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq hG2 n8U hn8 g h3 oU iU hoU hiU out hin)[i]! = out[i]!) ∧
    (∀ o<ow,8*g<=o → o<8*g+24 → (dnW24o c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq hG2 n8U hn8 g h3 oU iU hoU hiU out hin)[q0.toNat+o*qs.toNat]! = dnT c n vo rn cq H o):=by
  have B0:=sPb n G g vo _ _ _ _ (n5g g G 0 (by decide) h3) hY hQ hBd (le_refl _) hB1 hB2 hB3
  have B1:=sPb n G (g+1) vo _ _ _ _ (n5g g G 1 (by decide) h3) hY hQ hBd (le_refl _) hB1 hB2 hB3
  have B2:=sPb n G (g+2) vo _ _ _ _ (n5g g G 2 (by decide) h3) hY hQ hBd (le_refl _) hB1 hB2 hB3
  have ho1:oU+8=G0 (8*(g+1)):=by rw [hoU,show 8*(g+1)=8*g+8 by ring,dnTA]; rfl
  have ho2:oU+16=G0 (8*(g+2)):=by rw [hoU,show 8*(g+2)=8*g+16 by ring,dnTA]; rfl
  have hi1:iU+n8U=G0 ((g+1)*n*8):=by rw [hiU,hn8, ← dnTA]; congr 1; ring
  have hi2:iU+n8U+n8U=G0 ((g+2)*n*8):=by rw [hi1,hn8, ← dnTA]; congr 1; ring
  have Q0:=dnQ q0 qs G g (by omega) hq B0.2.2.2.2.2.2.2.2
  rw [← hoU] at Q0
  have ix0:∀ l,(q0+oU*qs).toNat+l*qs.toNat=q0.toNat+(8*g+l)*qs.toNat:=fun l=>by rw [Q0.1,Ϡ,pA]
  have Q1:=dnQ q0 qs G (g+1) (by omega) hq B1.2.2.2.2.2.2.2.2
  rw [← ho1] at Q1
  have ix1:∀ l,(q0+(oU+8)*qs).toNat+l*qs.toNat=q0.toNat+(8*(g+1)+l)*qs.toNat:=fun l=>by rw [Q1.1,Ϡ,pA]
  have Q2:=dnQ q0 qs G (g+2) (by omega) hq B2.2.2.2.2.2.2.2.2
  rw [← ho2] at Q2
  have ix2:∀ l,(q0+(oU+16)*qs).toNat+l*qs.toNat=q0.toNat+(8*(g+2)+l)*qs.toNat:=fun l=>by rw [Q2.1,Ϡ,pA]
  have e8:(8:Ἄ).toNat=8:=kc8
  have ec:(G0 (min 8 (ow-8*(g+2)))).toNat=min 8 (ow-8*(g+2)):=s5tu _ (C0 (P7 _ _) (by decide))
  unfold dnW24o; dsimp only
  refine ⟨by simp only [dn8_size],fun i hi => ?_,fun o ho h1 h2 => ?_⟩
  · rw [dn8_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ i Q2.2 (fun l hl e=>hi (8*(g+2)+l) (by rw [ec] at hl; omega) (by omega) (by rw [ec] at hl; omega) (by rw [← ix2]; exact e))]
    rw [dn8_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ i Q1.2 (fun l hl e=>hi (8*(g+1)+l) (by rw [e8] at hl; omega) (by omega) (by rw [e8] at hl; omega) (by rw [← ix1]; exact e))]
    rw [dn8_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ i Q0.2 (fun l hl e=>hi (8*g+l) (by rw [e8] at hl; omega) (by omega) (by rw [e8] at hl; omega) (by rw [← ix0]; exact e))]
  have hd:∀ k,8*k<=o → o<8*k+8 → o/8=k∧o%8=o-8*k:=fun k a b=>⟨by omega,by omega⟩
  have eb0:∀ (kk:Ἄ)(cc:Nat)(hk:kk.toNat=cc)(hc:cc≤7) h,c.Bd.uget (oU+kk) h=c.Bd.data[8*g+cc]!:=fun kk cc hk hc h=>by
    rw [kFU]; rw [hoU]; rw [kSm _ kk (C0 (J1 (hk ▸ hc) _) B0.2.2.2.2.1),hk,B0.2.2.2.2.2.2.2.2]
  have eb1:∀ (kk:Ἄ)(cc:Nat)(hk:kk.toNat=cc)(hc:cc≤7) h,c.Bd.uget ((oU+8)+kk) h=c.Bd.data[8*(g+1)+cc]!:=fun kk cc hk hc h=>by
    rw [kFU]; rw [ho1]; rw [kSm _ kk (C0 (J1 (hk ▸ hc) _) B1.2.2.2.2.1),hk,B1.2.2.2.2.2.2.2.2]
  have eb2:∀ (kk:Ἄ)(cc:Nat)(hk:kk.toNat=cc)(hc:cc≤7) h,c.Bd.uget ((oU+16)+kk) h=c.Bd.data[8*(g+2)+cc]!:=fun kk cc hk hc h=>by
    rw [kFU]; rw [ho2]; rw [kSm _ kk (C0 (J1 (hk ▸ hc) _) B2.2.2.2.2.1),hk,B2.2.2.2.2.2.2.2.2]
  have tv:voU.toNat=vo:=by rw [hvo]; exact B0.2.2.2.2.2.1
  have tn:nU.toNat=n:=by rw [hnU]; exact B0.2.2.2.2.2.2.1
  have ti0:iU.toNat=g*n*8:=by rw [hiU]; exact B0.2.2.2.2.2.2.2.1
  have ti1:(iU+n8U).toNat=(g+1)*n*8:=by rw [hi1]; exact B1.2.2.2.2.2.2.2.1
  have ti2:(iU+n8U+n8U).toNat=(g+2)*n*8:=by rw [hi2]; exact B2.2.2.2.2.2.2.2.1
  have F0:=FMc_ok c vo n G g voU nU iU tv tn ti0 (by omega) hF
  have F1:=FMc_ok c vo n G (g+1) voU nU (iU+n8U) tv tn ti1 (by omega) hF
  have F2:=FMc_ok c vo n G (g+2) voU nU (iU+n8U+n8U) tv tn ti2 (by omega) hF
  by_cases p2:8*(g+2)<=o
  · have hl:o-8*(g+2)<(G0 (min 8 (ow-8*(g+2)))).toNat:=by rw [ec]; omega
    have eo:q0.toNat+o*qs.toNat=(q0+(oU+16)*qs).toNat+(o-8*(g+2))*qs.toNat:=by rw [ix2,M4 (by omega)]
    rw [eo]
    rw [dn8_hit _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (o-8*(g+2)) hl (by omega) Q2.2 hs1]
    rw [sM24P_spec c.arm _ _ _ _ _ _ _ _ _ _ _ F0 F1 F2]; dsimp only
    simp only [tv,tn,ti0,ti1,ti2,kc0,eb0 0 0 kc0 n5d0,eb0 1 1 kc1 n5d1,eb0 2 2 kc2 n5d2,eb0 3 3 kc3 n5d3,eb0 4 4 kc4 n5d4,eb0 5 5 kc5 n5d5,eb0 6 6 kc6 n5d6,eb0 7 7 kc7 n5d7,eb1 0 0 kc0 n5d0,eb1 1 1 kc1 n5d1,eb1 2 2 kc2 n5d2,eb1 3 3 kc3 n5d3,eb1 4 4 kc4 n5d4,eb1 5 5 kc5 n5d5,eb1 6 6 kc6 n5d6,eb1 7 7 kc7 n5d7,eb2 0 0 kc0 n5d0,eb2 1 1 kc1 n5d1,eb2 2 2 kc2 n5d2,eb2 3 3 kc3 n5d3,eb2 4 4 kc4 n5d4,eb2 5 5 kc5 n5d5,eb2 6 6 kc6 n5d6,eb2 7 7 kc7 n5d7]
    rw [lselAo (fun l s=>c.Y.data[vo+s]! * c.Q.data[(g+2)*n*8+8*s+l]!) (fun l=>c.Bd.data[8*(g+2)+l]!) n 0 (o-8*(g+2)) (by omega)]
    unfold dnT; rw [(hd (g+2) (by omega) (by omega)).1,(hd (g+2) (by omega) (by omega)).2,M4 (by omega)]
  by_cases p1:8*(g+1)<=o
  · have hl:o-8*(g+1)<(8:Ἄ).toNat:=by rw [e8]; omega
    have eo:q0.toNat+o*qs.toNat=(q0+(oU+8)*qs).toNat+(o-8*(g+1))*qs.toNat:=by rw [ix1,M4 (by omega)]
    rw [eo]
    rw [dn8_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Q2.2 (fun l hl' e=>by rw [ix2,ix1] at e; rw [ec] at hl'; have:=B7 hs1 (J8 e); omega)]
    rw [dn8_hit _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (o-8*(g+1)) hl (by omega) Q1.2 hs1]
    rw [sM24P_spec c.arm _ _ _ _ _ _ _ _ _ _ _ F0 F1 F2]; dsimp only
    simp only [tv,tn,ti0,ti1,ti2,kc0,eb0 0 0 kc0 n5d0,eb0 1 1 kc1 n5d1,eb0 2 2 kc2 n5d2,eb0 3 3 kc3 n5d3,eb0 4 4 kc4 n5d4,eb0 5 5 kc5 n5d5,eb0 6 6 kc6 n5d6,eb0 7 7 kc7 n5d7,eb1 0 0 kc0 n5d0,eb1 1 1 kc1 n5d1,eb1 2 2 kc2 n5d2,eb1 3 3 kc3 n5d3,eb1 4 4 kc4 n5d4,eb1 5 5 kc5 n5d5,eb1 6 6 kc6 n5d6,eb1 7 7 kc7 n5d7,eb2 0 0 kc0 n5d0,eb2 1 1 kc1 n5d1,eb2 2 2 kc2 n5d2,eb2 3 3 kc3 n5d3,eb2 4 4 kc4 n5d4,eb2 5 5 kc5 n5d5,eb2 6 6 kc6 n5d6,eb2 7 7 kc7 n5d7]
    rw [lselAo (fun l s=>c.Y.data[vo+s]! * c.Q.data[(g+1)*n*8+8*s+l]!) (fun l=>c.Bd.data[8*(g+1)+l]!) n 0 (o-8*(g+1)) (by omega)]
    unfold dnT; rw [(hd (g+1) (by omega) (by omega)).1,(hd (g+1) (by omega) (by omega)).2,M4 (by omega)]
  · have hl:o-8*g<(8:Ἄ).toNat:=by rw [e8]; omega
    have eo:q0.toNat+o*qs.toNat=(q0+oU*qs).toNat+(o-8*g)*qs.toNat:=by rw [ix0,M4 (by omega)]
    rw [eo]
    rw [dn8_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Q2.2 (fun l hl' e=>by rw [ix2,ix0] at e; rw [ec] at hl'; have:=B7 hs1 (J8 e); omega)]
    rw [dn8_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Q1.2 (fun l hl' e=>by rw [ix1,ix0] at e; rw [e8] at hl'; have:=B7 hs1 (J8 e); omega)]
    rw [dn8_hit _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (o-8*g) hl (by omega) Q0.2 hs1]
    rw [sM24P_spec c.arm _ _ _ _ _ _ _ _ _ _ _ F0 F1 F2]; dsimp only
    simp only [tv,tn,ti0,ti1,ti2,kc0,eb0 0 0 kc0 n5d0,eb0 1 1 kc1 n5d1,eb0 2 2 kc2 n5d2,eb0 3 3 kc3 n5d3,eb0 4 4 kc4 n5d4,eb0 5 5 kc5 n5d5,eb0 6 6 kc6 n5d6,eb0 7 7 kc7 n5d7,eb1 0 0 kc0 n5d0,eb1 1 1 kc1 n5d1,eb1 2 2 kc2 n5d2,eb1 3 3 kc3 n5d3,eb1 4 4 kc4 n5d4,eb1 5 5 kc5 n5d5,eb1 6 6 kc6 n5d6,eb1 7 7 kc7 n5d7,eb2 0 0 kc0 n5d0,eb2 1 1 kc1 n5d1,eb2 2 2 kc2 n5d2,eb2 3 3 kc3 n5d3,eb2 4 4 kc4 n5d4,eb2 5 5 kc5 n5d5,eb2 6 6 kc6 n5d6,eb2 7 7 kc7 n5d7]
    rw [lselAo (fun l s=>c.Y.data[vo+s]! * c.Q.data[g*n*8+8*s+l]!) (fun l=>c.Bd.data[8*g+l]!) n 0 (o-8*g) (by omega)]
    unfold dnT; rw [(hd g (by omega) (by omega)).1,(hd g (by omega) (by omega)).2,M4 (by omega)]
theorem dnW24_spec(c:DC)(n G ow vo rn:Nat)(cq H:ἆ)(q0 qs voU nU:Ἄ)(hY:vo+n<=c.Y.size)(hQ:c.Q.size=n*(8*G))(hBd:8*G<=c.Bd.size)(hB1:vo+n<2^(31:Nat))(hB2:n*(8*G)<2^(31:Nat))(hB3:8*G<2^(31:Nat))(hvo:voU=G0 vo)(hnU:nU=G0 n)(hq:q0.toNat+8*G*qs.toNat<2^(31:Nat))(hG2:8*G<=ow+7)(n8U:Ἄ)(hn8:n8U=G0 (n*8))(g:Nat)(h3:g+3<=G)(oU iU:Ἄ)(hoU:oU=G0 (8*g))(hiU:iU=G0 (g*n*8))(out:Ἂ ἇ)(hin:∀ o<ow,q0.toNat+o*qs.toNat<out.size)
    (hF:FMc c vo n G)(hs1:0<qs.toNat) :
    (dnW24 c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq hG2 n8U hn8 g h3 oU iU hoU hiU out hin).size=out.size ∧
    (∀ i,(∀ o<ow,8*g<=o → o<8*g+24 → q0.toNat+o*qs.toNat≠i) → (dnW24 c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq hG2 n8U hn8 g h3 oU iU hoU hiU out hin)[i]! = out[i]!) ∧
    (∀ o<ow,8*g<=o → o<8*g+24 → (dnW24 c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq hG2 n8U hn8 g h3 oU iU hoU hiU out hin)[q0.toNat+o*qs.toNat]! = dnT c n vo rn cq H o):=by
  rw [dnW24_eq]; exact dnW24o_spec c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq hG2 n8U hn8 g h3 oU iU hoU hiU out hin hF hs1
theorem dnW8_size(c:DC)(n G ow vo rn:Nat)(cq H:ἆ)(q0 qs voU nU:Ἄ)(hY:vo+n<=c.Y.size)(hQ:c.Q.size=n*(8*G))(hBd:8*G<=c.Bd.size)(hB1:vo+n<2^(31:Nat))(hB2:n*(8*G)<2^(31:Nat))(hB3:8*G<2^(31:Nat))(hvo:voU=G0 vo)(hnU:nU=G0 n)(hq:q0.toNat+8*G*qs.toNat<2^(31:Nat))(g:Nat)(hg:g<G)(oU iU:Ἄ)(hoU:oU=G0 (8*g))(hiU:iU=G0 (g*n*8))(out:Ἂ ἇ)(hin:∀ o<ow,q0.toNat+o*qs.toNat<out.size):
    (dnW8 c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq g hg oU iU hoU hiU out hin).size=out.size:=by unfold dnW8; exact mc8_sz _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ rfl
theorem dnW24_size(c:DC)(n G ow vo rn:Nat)(cq H:ἆ)(q0 qs voU nU:Ἄ)(hY:vo+n<=c.Y.size)(hQ:c.Q.size=n*(8*G))(hBd:8*G<=c.Bd.size)(hB1:vo+n<2^(31:Nat))(hB2:n*(8*G)<2^(31:Nat))(hB3:8*G<2^(31:Nat))(hvo:voU=G0 vo)(hnU:nU=G0 n)(hq:q0.toNat+8*G*qs.toNat<2^(31:Nat))(hG2:8*G<=ow+7)(n8U:Ἄ)(hn8:n8U=G0 (n*8))(g:Nat)(h3:g+3<=G)(oU iU:Ἄ)(hoU:oU=G0 (8*g))(hiU:iU=G0 (g*n*8))(out:Ἂ ἇ)(hin:∀ o<ow,q0.toNat+o*qs.toNat<out.size):
    (dnW24 c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq hG2 n8U hn8 g h3 oU iU hoU hiU out hin).size=out.size:=by rw [dnW24_eq]; unfold dnW24o; dsimp only; simp only [dn8_size]
theorem dnLt1(G g:Nat)(h:g<G):G-(g+1)<G-g:=C5 h (I3 g)
theorem dnLt3(G g:Nat)(h:g+3<=G):G-(g+3)<G-g:=C5 (U1 (P5 (by decide)) h) (P5 (by decide))
theorem dnO(g:Nat)(oU:Ἄ)(k:Nat)(hoU:oU=G0 (8*g)):oU+G0 (8*k)=G0 (8*(g+k)):=by rw [hoU, ← dnTA]; congr 1; ring
theorem dnI(g n k:Nat)(iU n8U:Ἄ)(hiU:iU=G0 (g*n*8))(hn8:n8U=G0 (n*8)):iU+n8U*G0 k=G0 ((g+k)*n*8):=by
  rw [hiU,hn8,lX,lX,lX, ← BP, ← mC, ← lX]; congr 1; ring
def dnP(c:@& DC)(n G ow vo rn:Nat)(cq H:ἆ)(q0 qs voU nU:Ἄ)(hY:vo+n<=c.Y.size)(hQ:c.Q.size=n*(8*G))(hBd:8*G<=c.Bd.size)(hB1:vo+n<2^(31:Nat))(hB2:n*(8*G)<2^(31:Nat))(hB3:8*G<2^(31:Nat))(hvo:voU=G0 vo)(hnU:nU=G0 n)(hq:q0.toNat+8*G*qs.toNat<2^(31:Nat))(hG2:8*G<=ow+7)(n8U:Ἄ)(hn8:n8U=G0 (n*8))(g:Nat)(oU iU:Ἄ)(hoU:oU=G0 (8*g))(hiU:iU=G0 (g*n*8))(out:Ἂ ἇ)(hin:∀ o<ow,q0.toNat+o*qs.toNat<out.size):Ἂ ἇ:=
 if hg:g<G then
  if h3:128<=n∧g+3<=G then
   dnP c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq hG2 n8U hn8 (g+3) (oU+G0 (8*3)) (iU+n8U*G0 3) (dnO g oU 3 hoU) (dnI g n 3 iU n8U hiU hn8) (dnW24 c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq hG2 n8U hn8 g h3.2 oU iU hoU hiU out hin)
    (fun o ho=>by rw [dnW24_size]; exact hin o ho)
  else
   dnP c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq hG2 n8U hn8 (g+1) (oU+G0 (8*1)) (iU+n8U*G0 1) (dnO g oU 1 hoU) (dnI g n 1 iU n8U hiU hn8) (dnW8 c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq g hg oU iU hoU hiU out hin)
    (fun o ho=>by rw [dnW8_size]; exact hin o ho)
 else out
termination_by G-g
decreasing_by
 · exact dnLt3 G g h3.2
 · exact dnLt1 G g hg
theorem dnP_spec(c:DC)(n G ow vo rn:Nat)(cq H:ἆ)(q0 qs voU nU:Ἄ)(hY:vo+n<=c.Y.size)(hQ:c.Q.size=n*(8*G))(hBd:8*G<=c.Bd.size)(hB1:vo+n<2^(31:Nat))(hB2:n*(8*G)<2^(31:Nat))(hB3:8*G<2^(31:Nat))(hvo:voU=G0 vo)(hnU:nU=G0 n)(hq:q0.toNat+8*G*qs.toNat<2^(31:Nat))(hG2:8*G<=ow+7)(n8U:Ἄ)(hn8:n8U=G0 (n*8))(hG:ow<=8*G)(hF:FMc c vo n G)(hs1:0<qs.toNat) :
    ∀ k g (oU iU:Ἄ)(hoU:oU=G0 (8*g))(hiU:iU=G0 (g*n*8))(out:Ἂ ἇ)(hin:∀ o<ow,q0.toNat+o*qs.toNat<out.size),G-g=k →
    (dnP c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq hG2 n8U hn8 g oU iU hoU hiU out hin).size=out.size ∧
    (∀ i,(∀ o<ow,8*g<=o → q0.toNat+o*qs.toNat≠i) → (dnP c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq hG2 n8U hn8 g oU iU hoU hiU out hin)[i]! = out[i]!) ∧
    (∀ o<ow,8*g<=o → (dnP c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq hG2 n8U hn8 g oU iU hoU hiU out hin)[q0.toNat+o*qs.toNat]! = dnT c n vo rn cq H o):=by
  intro k; induction k using C6 with
  | _ k ih =>
    intro g oU iU hoU hiU out hin hk
    rw [dnP]
    split
    · rename_i hg
      split
      · rename_i h3
        have W:=dnW24_spec c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq hG2 n8U hn8 g h3.2 oU iU hoU hiU out hin hF hs1
        have I:=ih (G-(g+3)) (by rw [← hk]; exact dnLt3 G g h3.2) (g+3) _ _ (dnO g oU 3 hoU) (dnI g n 3 iU n8U hiU hn8) _ (fun o ho=>by rw [W.1]; exact hin o ho) rfl
        refine ⟨by rw [I.1,W.1],fun i hi => ?_,fun o ho h1 => ?_⟩
        · rw [I.2.1 i (fun o ho h e=>hi o ho (by omega) e),W.2.1 i (fun o ho h1 _ e=>hi o ho h1 e)]
        · by_cases p:o<8*g+24
          · rw [I.2.1 _ (fun o' ho' h' e=>by have:=B7 hs1 (J8 e); omega),W.2.2 o ho h1 p]
          · exact I.2.2 o ho (by omega)
      · rename_i h3
        have W:=dnW8_spec c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq g hg oU iU hoU hiU out hin hF hs1
        have I:=ih (G-(g+1)) (by rw [← hk]; exact dnLt1 G g hg) (g+1) _ _ (dnO g oU 1 hoU) (dnI g n 1 iU n8U hiU hn8) _ (fun o ho=>by rw [W.1]; exact hin o ho) rfl
        refine ⟨by rw [I.1,W.1],fun i hi => ?_,fun o ho h1 => ?_⟩
        · rw [I.2.1 i (fun o ho h e=>hi o ho (by omega) e),W.2.1 i (fun o ho h1 _ e=>hi o ho h1 e)]
        · by_cases p:o<8*g+8
          · rw [I.2.1 _ (fun o' ho' h' e=>by have:=B7 hs1 (J8 e); omega),W.2.2 o ho h1 p]
          · exact I.2.2 o ho (by omega)
    · rename_i hg
      exact ⟨rfl,fun i _=>rfl,fun o ho h1=>absurd ho (by omega)⟩
theorem dnP_sizeK(c:DC)(n G ow vo rn:Nat)(cq H:ἆ)(q0 qs voU nU:Ἄ)(hY:vo+n<=c.Y.size)(hQ:c.Q.size=n*(8*G))(hBd:8*G<=c.Bd.size)(hB1:vo+n<2^(31:Nat))(hB2:n*(8*G)<2^(31:Nat))(hB3:8*G<2^(31:Nat))(hvo:voU=G0 vo)(hnU:nU=G0 n)(hq:q0.toNat+8*G*qs.toNat<2^(31:Nat))(hG2:8*G<=ow+7)(n8U:Ἄ)(hn8:n8U=G0 (n*8)) :
    ∀ k g (oU iU:Ἄ)(hoU:oU=G0 (8*g))(hiU:iU=G0 (g*n*8))(out:Ἂ ἇ)(hin:∀ o<ow,q0.toNat+o*qs.toNat<out.size),G-g=k →
    (dnP c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq hG2 n8U hn8 g oU iU hoU hiU out hin).size=out.size:=by
  intro k; induction k using C6 with
  | _ k ih =>
    intro g oU iU hoU hiU out hin hk
    rw [dnP]
    split
    · rename_i hg
      split
      · rename_i h3; rw [ih _ (by rw [← hk]; exact dnLt3 G g h3.2) _ _ _ _ _ _ _ rfl,dnW24_size]
      · rw [ih _ (by rw [← hk]; exact dnLt1 G g hg) _ _ _ _ _ _ _ rfl,dnW8_size]
    · rfl
theorem dnP_size(c:DC)(n G ow vo rn:Nat)(cq H:ἆ)(q0 qs voU nU:Ἄ)(hY:vo+n<=c.Y.size)(hQ:c.Q.size=n*(8*G))(hBd:8*G<=c.Bd.size)(hB1:vo+n<2^(31:Nat))(hB2:n*(8*G)<2^(31:Nat))(hB3:8*G<2^(31:Nat))(hvo:voU=G0 vo)(hnU:nU=G0 n)(hq:q0.toNat+8*G*qs.toNat<2^(31:Nat))(hG2:8*G<=ow+7)(n8U:Ἄ)(hn8:n8U=G0 (n*8))(g:Nat)(oU iU:Ἄ)(hoU:oU=G0 (8*g))(hiU:iU=G0 (g*n*8))(out:Ἂ ἇ)(hin:∀ o<ow,q0.toNat+o*qs.toNat<out.size):
    (dnP c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq hG2 n8U hn8 g oU iU hoU hiU out hin).size=out.size:=dnP_sizeK c n G ow vo rn cq H q0 qs voU nU hY hQ hBd hB1 hB2 hB3 hvo hnU hq hG2 n8U hn8 _ g oU iU hoU hiU out hin rfl
theorem dnRb(R n r:Nat)(hr:r<R)(Ys:Nat)(hY:R*(n+1)<=Ys)(hRm:R*(n+1)<2^(31:Nat)):
    r*(n+1)+n<=Ys∧r*(n+1)+n<2^(31:Nat)∧(G0 (r*(n+1)+n)).toNat<Ys∧(G0 (r*(n+1)+n)).toNat=r*(n+1)+n:=by
  have m:(r+1)*(n+1)<=R*(n+1):=ι _ hr
  rw [Ο] at m
  have e:(G0 (r*(n+1)+n)).toNat=r*(n+1)+n:=s5tu _ (by omega)
  exact ⟨by omega,by omega,by rw [e]; omega,e⟩
@[inline] def dnWin(c:@& DC)(cq:ἆ):ἆ:=
 let P:=(if c.em<(56:ἆ) then (56:ἆ) else c.em)+cq
 if P<=(372:ἆ) then (if c.hb>P then c.hb else P) else (4000:ἆ)
@[inline] def dnCqE(c:@& DC)(i:Ἄ)(h:i.toNat<c.Y.size):ἆ:=(23:ἆ)+cEq c.f9 (c.Y.uget i h)
theorem dnQR(r:Nat)(qr qs:Ἄ)(G:Nat)(hq:r*qr.toNat+8*G*qs.toNat<2^(31:Nat))(hr31:r<2^(31:Nat)):(G0 r*qr).toNat=r*qr.toNat:=by
  have er:(G0 r).toNat=r:=s5tu _ (by omega)
  have:=kMm (G0 r) qr (by rw [er]; omega)
  rw [er] at this; exact this
def dnR(c:@& DC)(n ow G R:Nat)(nU n1U n8U qr qs:Ἄ)(hnU:nU=G0 n)(hn1:n1U=G0 (n+1))(hn8:n8U=G0 (n*8))
    (hY:R*(n+1)<=c.Y.size)(hQ:c.Q.size=n*(8*G))(hBd:8*G<=c.Bd.size)(hRm:R*(n+1)<2^(31:Nat))(hB2:n*(8*G)<2^(31:Nat))(hB3:8*G<2^(31:Nat))(hG2:8*G<=ow+7)
    (hq:∀ r<R,r*qr.toNat+8*G*qs.toNat<2^(31:Nat))
    (r:Nat)(voU q0U:Ἄ)(hvo:voU=G0 (r*(n+1)))(hq0:q0U=G0 r*qr)(out:Ἂ ἇ)(hin:∀ r'<R, ∀ o<ow,r'*qr.toNat+o*qs.toNat<out.size):Ἂ ἇ:=
 if hr:r<R then
  have V:=dnRb R n r hr _ hY hRm
  have hR1:R<=R*(n+1):=J4 R (Nat.succ_pos n)
  have eq0:q0U.toNat=r*qr.toNat:=by rw [hq0]; exact dnQR r qr qs G (hq r hr) (by omega)
  have hvn:voU+nU=G0 (r*(n+1)+n):=by rw [hvo,hnU,dnTA]
  let cq:=dnCqE c (voU+nU) (by rw [hvn]; exact V.2.2.1)
  dnR c n ow G R nU n1U n8U qr qs hnU hn1 hn8 hY hQ hBd hRm hB2 hB3 hG2 hq (r+1) (voU+n1U) (q0U+qr)
   (by rw [hvo,hn1, ← dnTA,Ο]) (by rw [hq0,dnTA,USize.add_mul,show G0 1=1 from rfl,USize.one_mul])
   (dnP c n G ow (r*(n+1)) (r*n) cq (dnWin c cq) q0U qs voU nU (J2 (Z4 V.1)) hQ hBd V.2.1 hB2 hB3 hvo hnU (by rw [eq0]; exact hq r hr) hG2 n8U hn8 0 0 0 (by rfl) (by rw [ϳ,ϳ]; rfl) out
     (fun o ho=>by rw [eq0]; exact hin r hr o ho))
   (fun r' h1 o ho=>by rw [dnP_size]; exact hin r' h1 o ho)
 else out
termination_by R-r
decreasing_by exact dnLt1 R r hr
noncomputable def dnCq(c:DC)(n r:Nat):ἆ:=(23:ἆ)+cEq c.f9 (c.Y.data[r*(n+1)+n]!)
noncomputable def dnH(c:DC)(n r:Nat):ἆ:=dnWin c (dnCq c n r)
theorem dnR_spec(c:DC)(n ow G R:Nat)(nU n1U n8U qr qs:Ἄ)(hnU:nU=G0 n)(hn1:n1U=G0 (n+1))(hn8:n8U=G0 (n*8))
    (hY:R*(n+1)<=c.Y.size)(hQ:c.Q.size=n*(8*G))(hBd:8*G<=c.Bd.size)(hRm:R*(n+1)<2^(31:Nat))(hB2:n*(8*G)<2^(31:Nat))(hB3:8*G<2^(31:Nat))(hG2:8*G<=ow+7)
    (hq:∀ r<R,r*qr.toNat+8*G*qs.toNat<2^(31:Nat))(hG:ow<=8*G)(hs1:0<qs.toNat)(hF:∀ r<R,FMc c (r*(n+1)) n G)
    (hD:∀ r r' o o',r<R → r'<R → o<ow → o'<ow → r*qr.toNat+o*qs.toNat=r'*qr.toNat+o'*qs.toNat → r=r') :
    ∀ k r (voU q0U:Ἄ)(hvo:voU=G0 (r*(n+1)))(hq0:q0U=G0 r*qr)(out:Ἂ ἇ)(hin:∀ r'<R, ∀ o<ow,r'*qr.toNat+o*qs.toNat<out.size),R-r=k →
    (dnR c n ow G R nU n1U n8U qr qs hnU hn1 hn8 hY hQ hBd hRm hB2 hB3 hG2 hq r voU q0U hvo hq0 out hin).size=out.size ∧
    (∀ i,(∀ r'<R,r<=r' → ∀ o<ow,r'*qr.toNat+o*qs.toNat≠i) → (dnR c n ow G R nU n1U n8U qr qs hnU hn1 hn8 hY hQ hBd hRm hB2 hB3 hG2 hq r voU q0U hvo hq0 out hin)[i]! = out[i]!) ∧
    (∀ r'<R,r<=r' → ∀ o<ow,(dnR c n ow G R nU n1U n8U qr qs hnU hn1 hn8 hY hQ hBd hRm hB2 hB3 hG2 hq r voU q0U hvo hq0 out hin)[r'*qr.toNat+o*qs.toNat]! = dnT c n (r'*(n+1)) (r'*n) (dnCq c n r') (dnH c n r') o):=by
  intro k; induction k using C6 with
  | _ k ih =>
    intro r voU q0U hvo hq0 out hin hk
    rw [dnR]
    split
    · rename_i hr
      have V:=dnRb R n r hr _ hY hRm
      have hR1:R<=R*(n+1):=J4 R (Nat.succ_pos n)
      have eq0:q0U.toNat=r*qr.toNat:=by rw [hq0]; exact dnQR r qr qs G (hq r hr) (by omega)
      have hvn:voU+nU=G0 (r*(n+1)+n):=by rw [hvo,hnU,dnTA]
      have ecq:∀ h,dnCqE c (voU+nU) h=dnCq c n r:=fun h=>by unfold dnCqE dnCq; rw [kFU,hvn,V.2.2.2]
      have lt:R-(r+1)<k:=by rw [← hk]; exact dnLt1 R r hr
      dsimp only
      refine ⟨?_,fun i hi => ?_,fun r' h1 h2 o ho => ?_⟩
      · rw [(ih _ lt (r+1) _ _ _ _ _ _ rfl).1,dnP_size]
      · rw [(ih _ lt (r+1) _ _ _ _ _ _ rfl).2.1 i (fun r' h1 h2 o ho e=>hi r' h1 (by omega) o ho e)]
        rw [(dnP_spec c n G ow (r*(n+1)) (r*n) _ _ q0U qs voU nU _ hQ hBd _ hB2 hB3 hvo hnU _ hG2 n8U hn8 hG (hF _ (by omega)) hs1 _ 0 0 0 _ _ out _ rfl).2.1 i (fun o ho _ e=>hi r hr (le_refl _) o ho (by rw [← eq0]; exact e))]
      · by_cases e:r'=r
        · subst e
          rw [(ih _ lt (r'+1) _ _ _ _ _ _ rfl).2.1 _ (fun r'' h1' h2' o' ho' e'=>by have:=hD _ _ _ _ h1' h1 ho' ho e'; omega)]
          rw [← eq0,(dnP_spec c n G ow (r'*(n+1)) (r'*n) _ _ q0U qs voU nU _ hQ hBd _ hB2 hB3 hvo hnU _ hG2 n8U hn8 hG (hF _ (by omega)) hs1 _ 0 0 0 _ _ out _ rfl).2.2 o ho (ϟ _)]
          unfold dnH; rw [ecq]
        · exact (ih _ lt (r+1) _ _ _ _ _ _ rfl).2.2 r' h1 (by omega) o ho
    · rename_i hr
      exact ⟨rfl,fun i _=>rfl,fun r' h1 h2=>absurd h1 (by omega)⟩
theorem dnT_ok(c:DC)(R ow G H:Nat)(r o:Nat)(hr:r<R)(ho:o<ow)(hn:0<c.n)(hG:ow<=8*G)
    (hYv:∀ r'<R, ∀ t≤c.n,c.Y.data[r'*(c.n+1)+t]! = if t=c.n then ao (fun s=>dw (c.X[r'*c.n+s]! &&& 0x7FFFFFFF)) c.n 0 0 else dw c.X[r'*c.n+t]!)
    (hQv:∀ t<c.n, ∀ g l,l<8 → 8*g+l<8*G → c.Q.data[(g*c.n+t)*8+l]! = if 8*g+l<ow then dw c.W[(8*g+l)*c.n+t]! else 0)
    (hBdv:∀ o<ow,c.Bd.data[o]! = dw oneW*dw c.b[o]!)
    (hEv:∀ o<ow, ∀ s<c.n,(kB c.W[o*c.n+s]!).toNat ≤ (c.E[o]!).toNat)(hEm:∀ o<ow,(c.E[o]!).toNat ≤ c.em.toNat)(hem:c.em.toNat<255)
    (F:∀ q:Nat,kB c.X[q]! ≠ 255∧kG c.X[q]! ≤ c.ix)(hbF:∀ q:Nat,kB c.b[q]! ≠ 255∧kG c.b[q]! ≤ H)(M:174+H ≤ c.ib)(Hn:c.n+2 ≤ 2^c.L)(x:c.L≤27)(z:c.ib+c.L≤425)
    (hhb:c.hb=(c.ib+c.L-50).toUInt64)(hf9:c.f9=(I0 (9*(c.n+1))).toInt64.toFloat) :
    dnT c c.n (r*(c.n+1)) (r*c.n) (dnCq c c.n r) (dnH c c.n r) o=p8 c.relu (ms (c.n+1) (fun s=>if s<c.n then c.X[r*c.n+s]! else ld) (fun s=>if s<c.n then c.W[o*c.n+s]! else c.b[o]!)):=by
  have e:dnT c c.n (r*(c.n+1)) (r*c.n) (dnCq c c.n r) (dnH c c.n r) o =
      oR c.relu c.n c.X c.W c.b c.E c.ix c.ib c.L c.Y c.Q (r*(c.n+1)) (r*c.n) c.em (dnCq c c.n r) c.hb o
        (ao (fun s=>c.Y.data[r*(c.n+1)+s]! * c.Q.data[o/8*c.n*8+8*s+o%8]!) c.n 0 (c.Bd.data[o]!)):=rfl
  rw [e]
  have hQ':∀ s<c.n,c.Q.data[o/8*c.n*8+o%8+8*s]! = dw c.W[o*c.n+s]!:=fun s hs=>by
    have h1:=hQv s hs (o/8) (o%8) (ϊ _ (by decide)) (by have:=ϛ o 8; omega)
    rw [show 8*(o/8)+o%8=o from ϛ o 8,if_pos ho] at h1
    rw [show o/8*c.n*8+o%8+8*s=(o/8*c.n+s)*8+o%8 by ring]; exact h1
  refine kOutR c.relu c.n c.X c.W c.b c.E c.ix c.ib c.L H c.Y c.Q (r*(c.n+1)) (r*c.n) o c.em (dnCq c c.n r) c.hb _ (c.Y.data[r*(c.n+1)+c.n]!) F hbF M Hn x z hn (hEv o ho)
    (fun s hs=>(hYv r hr s (X4 hs)).trans (if_neg (R8 hs))) hQ' ?_ ((hYv r hr c.n (le_refl _)).trans (if_pos rfl)) (by unfold dnCq; rw [hf9]) hhb (hEm o ho) hem
  rw [hBdv o ho]; congr 1; funext s; rw [show o/8*c.n*8+8*s+o%8=o/8*c.n*8+o%8+8*s by ring]
theorem dnB2(n ow:Nat)(hn:0<n)(hon:(ow+16)*n<2^(31:Nat)):n*(8*((ow+7)/8))<2^(31:Nat):=by
  have:n*(8*((ow+7)/8))<=n*(ow+16):=U3 _ (by omega)
  rw [Ϗ n (ow+16)] at this; omega
theorem dnB3(n ow:Nat)(hn:0<n)(hon:(ow+16)*n<2^(31:Nat)):8*((ow+7)/8)<2^(31:Nat):=by
  have:ow+16<=(ow+16)*n:=J4 _ hn
  omega
def bQS(R n ow:Nat)(relu:Ἆ)(X W b:Ἂ ἇ)(ix ib L:Nat)(Q:ἀ)(E:Ἂ ἇ)(em:ἆ)
  (hn:0<n)(hQ:(kS Q)=n*(8*((ow+7)/8)))(hon:(ow+16)*n<2^(31:Ἀ))(hEo:ow<=(O0 E))(hX:R*n<=(O0 X))(hRm:R*(n+1)<2^(31:Ἀ))(hRo:R*ow<2^(31:Ἀ)):Ἂ ἇ:=
 have q1:n<=(ow+16)*n:=lZ n (by omega)
 have q2:R<=R*(n+1):=J4 R (by omega)
 have e1:=s5tu n (by omega)
 have e2:=s5tu (n+1) (by omega)
 have e3:=s5tu R (by omega)
 have hY:R*(n+1)<=(kS (zPq (R*(n+1)))):=(zPs _).1
 let G:=(ow+7)/8
 let Y:=dOw X (G0 n) (G0 (n+1)) (G0 R) 0 (zPq (R*(n+1))) (by rw[e1,e3];exact hX) (by rw[e2,e3];exact hY) (by rw[e1,e2]) (by rw[e2,e3];exact hRm)
 have hBs:8*G<=kS (bBg b ow (8*G)).1:=by rw[s5wcGet3_val];exact (wcBF_spec b ow (8*G)).1.ge
 have hYs:R*(n+1)<=kS Y:=by rw[dOw_size _ _ _ _ _ _ _ _ _ _ _ rfl];exact hY
 if h7:R*ow+8<2^(31:Nat) then
  have hw1:ow+16<=(ow+16)*n:=J4 _ hn
  have eow:(G0 ow).toNat=ow:=s5tu _ (by omega)
  have hRo2:∀ r<R,r*ow+ow<=R*ow:=fun r hr=>by have:=ι ow (l7 hr); rw [Ο] at this; exact this
  dnR ⟨relu,n,X,W,b,E,ix,ib,L,Y,Q,(bBg b ow (8*G)).1,(ib+L-50).toUInt64,em,(I0 (9*(n+1))).toInt64.toFloat,Yarm⟩
   n ow G R (G0 n) (G0 (n+1)) (G0 (n*8)) (G0 ow) 1 rfl rfl rfl hYs hQ hBs hRm (dnB2 n ow hn hon) (dnB3 n ow hn hon) (by omega)
   (fun r hr=>by rw [eow,kc1]; have:=hRo2 r hr; omega) 0 0 0 (by rw [ϳ]; rfl) (by rw [show G0 0=0 from rfl,USize.zero_mul])
   (mB (R*ow) (0:ἇ)) (fun r hr o ho=>by rw [O3,eow,kc1,ϻ]; have:=hRo2 r hr; omega)
 else mj R n ow relu X W b
theorem bQS_spec (R n ow:Ἀ)(relu:Ἆ)(X W b:Ἂ Ἃ)(ix ib L H:Ἀ)(Q:ἀ)(E:Ἂ Ἃ)(em:ἆ)
    (hn:0<n)(hQ:(kS Q)=n*(8*((ow+7)/8)))(hon:(ow+16)*n<2^(31:Ἀ))(hEo:ow<=(O0 E))(hX:R*n<=(O0 X))(hRm:R*(n+1)<2^(31:Ἀ))(hRo:R*ow<2^(31:Ἀ))
    (hQv:∀ t<n, ∀ g l,l<8 → 8*g+l < 8*((ow+7)/8) → Q.data[(g*n+t)*8+l]! = if 8*g+l<ow then dw W[(8*g+l)*n+t]! else 0)
    (hEv:∀ o<ow, ∀ s<n,(kB W[o*n+s]!).toNat ≤ (E[o]!).toNat)(hEm:∀ o<ow,(E[o]!).toNat ≤ em.toNat)(hem:em.toNat < 255)
    (F:∀ q:ℕ,kB X[q]! ≠ 255∧kG X[q]! ≤ ix)(hbF:∀ q:ℕ,kB b[q]! ≠ 255∧kG b[q]! ≤ H)(M:174+H ≤ ib)(Hn:n+2 ≤ 2^L)(x:L≤27)(z:ib+L≤425) :
    bQS R n ow relu X W b ix ib L Q E em hn hQ hon hEo hX hRm hRo=mj R n ow relu X W b:=by
  by_cases h7:R*ow+8<2^(31:Nat)
  rotate_left
  · unfold bQS; dsimp only; rw [dif_neg h7]
  have q1:n<=(ow+16)*n:=lZ n (Nat.succ_pos _)
  have q2:R<=R*(n+1):=J4 R (Nat.succ_pos n)
  have h31:2^(31:Ἀ)<2^(32:Ἀ):=Nat.pow_lt_pow_succ Nat.one_lt_two
  have hn31:=C0 q1 hon
  have e1:=s5tu n (K5 hn31 h31)
  have e2:=s5tu (n+1) (C0 (l7 hn31) h31)
  have e3:=s5tu R (K5 (C0 q2 hRm) h31)
  have p1:(J0 (G0 R))*(J0 (G0 n))<=(O0 X):=by rw [e1,e3]; exact hX
  have p2:(J0 (G0 R))*(J0 (G0 (n+1)))<=(kS (zPq (R*(n+1)))):=by rw [e2,e3]; exact (zPs _).1
  have p3:(J0 (G0 (n+1)))=(J0 (G0 n))+1:=by rw [e1,e2]
  have p4:(J0 (G0 R))*(J0 (G0 (n+1)))<2^(31:Ἀ):=by rw [e2,e3]; exact hRm
  have DY:=dOw_spec X (G0 n) (G0 (n+1)) (G0 R) p1 p3 p4 _ 0 (zPq (R*(n+1))) p2 rfl (kc0.trans_le (ϟ _))
  have DS:=dOw_size X (G0 n) (G0 (n+1)) (G0 R) p1 p3 p4 _ 0 (zPq (R*(n+1))) p2 rfl
  generalize hYd:dOw X (G0 n) (G0 (n+1)) (G0 R) 0 (zPq (R*(n+1))) p1 p2 p3 p4=Y at DY DS
  rw [e1,e2,e3,kc0] at DY
  have hY:R*(n+1)<=(kS Y):=by rw [DS]; exact (zPs _).1
  have hYv : ∀ r'<R, ∀ t≤n,Y.data[r'*(n+1)+t]! = if t=n then ao (fun s=>dw (X[r'*n+s]! &&& 0x7FFFFFFF)) n 0 0 else dw X[r'*n+t]! :=
    fun r' h1 t ht=>DY.2 r' t h1 (ϟ _) ht
  have hBv:=wcBF_spec b ow (8*((ow+7)/8))
  have hBs:8*((ow+7)/8) <= kS ((bBg b ow (8*((ow+7)/8))).1):=by rw [s5wcGet3_val]; exact hBv.1.ge
  have hBdv : ∀ o<ow,((bBg b ow (8*((ow+7)/8))).1).data[o]! = dw oneW*dw b[o]! := by
    intro o ho; rw [s5wcGet3_val,hBv.2 o (by omega),if_pos ho]
  have hRo':R*ow<2^(32:Nat):=K5 hRo h31
  have hw1:ow+16<=(ow+16)*n:=J4 _ hn
  have hR1:R<2^(31:Nat):=C0 q2 hRm
  have eow:(G0 ow).toNat=ow:=s5tu _ (by omega)
  have e1u:(1:Ἄ).toNat=1:=kc1
  have T:=dnR_spec ⟨relu,n,X,W,b,E,ix,ib,L,Y,Q,(bBg b ow (8*((ow+7)/8))).1,(ib+L-50).toUInt64,em,(I0 (9*(n+1))).toInt64.toFloat,Yarm⟩
    n ow ((ow+7)/8) R (G0 n) (G0 (n+1)) (G0 (n*8)) (G0 ow) 1 rfl rfl rfl hY hQ hBs hRm (dnB2 n ow hn hon) (dnB3 n ow hn hon) (by omega)
    (fun r hr=>by rw [eow,e1u]; have:=ι ow (l7 hr); rw [Ο] at this; omega) (by omega) (by rw [e1u]; decide)
    (fun r hr=>FMd _ n _ ow (r*(n+1)) (r*n) (fun t ht=>(hYv r hr t ht.le).trans (if_neg (R8 ht))) hQv (fun q=>(F q).1) (fun o ho s hs h=>by have:=hEv o ho s hs;have:=hEm o ho;rw[h]at *;simp at *;omega))
    (fun r r' o o' hr hr' ho ho' e=>by rw [eow,e1u,ϻ,ϻ] at e; exact (kDM _ _ _ _ _ e ho ho').1)
    _ 0 0 0 (by rw [ϳ]; rfl) (by rw [show G0 0=0 from rfl,USize.zero_mul]) (mB (R*ow) (0:ἇ)) (fun r hr o ho=>by
      rw [O3,eow,e1u,ϻ]; have:=ι ow (l7 hr); rw [Ο] at this; omega) rfl
  subst hYd
  unfold bQS; dsimp only
  rw [dif_pos h7]
  unfold mj linearWith
  apply mi (by rw [T.1]; simp only [Ϝ,O3])
  intro q hq hq'
  rw [Ϝ] at hq'
  simp only [Λ]
  rw [← AB _ q hq]
  have hpos:0<ow:=ϵ (fun h=>by rw [h,A7] at hq'; exact U5 _ hq')
  have hrq:q/ow<R:=(ϸ hpos).2 hq'
  have V:=T.2.2 (q/ow) hrq (ϟ _) (q%ow) (ϊ _ hpos)
  have ei:(q/ow)*(G0 ow).toNat+(q%ow)*(1:Ἄ).toNat=q:=by rw [eow,e1u,ϻ,Ϗ (q/ow) ow,ϛ]
  rw [ei] at V
  exact V.trans (dnT_ok _ R ow ((ow+7)/8) H (q/ow) (q%ow) hrq (ϊ _ hpos) hn (by omega) hYv hQv hBdv hEv hEm hem F hbF M Hn x z rfl rfl)
theorem sFGb (S:ἀ)(st s0 mu j:Ἄ)(hS:(J0 s0)<(J0 st)∧(J0 st)*(J0 mu)<=(kS S)∧(J0 st)*(J0 mu)<2^(31:Ἀ))(h:(J0 j)<(J0 mu)) :
    (J0 (st*j+s0))=(J0 st)*(J0 j)+(J0 s0)∧(J0 (st*j+s0))<(kS S):=by
  have m1:(J0 st)*((J0 j)+1) ≤ (J0 st)*(J0 mu):=U3 _ h
  have m2:(J0 st)*(J0 j)+(J0 s0) < (J0 st)*((J0 j)+1):=by rw [Nat.mul_succ]; exact O1 hS.1 _
  have e1:(J0 (st*j))=(J0 st)*(J0 j):=kMm _ _ (C0 (X4 (C0 (V1 _ _) m2)) (C0 m1 hS.2.2))
  have e2:(J0 (st*j+s0))=(J0 st)*(J0 j)+(J0 s0):=by rw [kSm _ _ (by rw [e1]; exact K5 m2 (C0 m1 hS.2.2)),e1]
  exact ⟨e2,by rw [e2]; exact U1 m2 (m1.trans hS.2.1)⟩
theorem kTU (j:Ἄ):G0 (J0 j)=j:=USize.ofNat_toNat
def sFG(relu:Ἆ)(H:ἆ)(fb:Ἄ->Ἅ->Ἃ)(S:ἀ)(st s0 q0 mu:Ἄ)(hS:(J0 s0)<(J0 st)∧(J0 st)*(J0 mu)<=(kS S)∧(J0 st)*(J0 mu)<2^(31:Ἀ))(hB:(J0 q0)+(J0 mu)<2^(31:Ἀ))
  (j:Ἄ)(a:Ἂ Ἃ)(ha:(J0 q0)+(J0 mu)<=(O0 a)):Ἂ Ἃ:=
 if h:j < mu then
  have B:=sFGb S st s0 mu j hS (ε.mp h)
  let v:=S.uget (st*j+s0) B.2
  let z:=zR v relu H
  sFG relu H fb S st s0 q0 mu hS hB (j+1) (a.uset (q0+j) (if z==(0xFFFFFFFF:ἇ) then fb j v else z) (by rw[kSm q0 j (Lql _ _ _ _ (ε.mp h) (X4 hB))];exact Lql _ _ _ _ (ε.mp h) ha)) (by rw[BT];exact ha)
 else a
termination_by (J0 mu)-(J0 j)
decreasing_by all_goals (rw[ac h];exact Lqd _ _ (ε.mp h))
theorem sFG_size(relu:Ἆ)(H:ἆ)(fb:Ἄ->Ἅ->Ἃ)(S:ἀ)(st s0 q0 mu:Ἄ)(hS:(J0 s0)<(J0 st)∧(J0 st)*(J0 mu)<=(kS S)∧(J0 st)*(J0 mu)<2^(31:Ἀ))(hB:(J0 q0)+(J0 mu)<2^(31:Ἀ)) :
    ∀ m (j:Ἄ)(a:Ἂ Ἃ)(ha:(J0 q0)+(J0 mu)<=(O0 a)),(J0 mu)-(J0 j)=m →
    (O0 (sFG relu H fb S st s0 q0 mu hS hB j a ha))=(O0 a):=by
  intro m; induction m with
  | zero=>intro j a ha hm; rw [sFG,dif_neg (fun h=>Lq0 _ _ hm (ε.mp h))]
  | succ m ih => intro j a ha hm; have HQ:=Lqs _ _ _ hm; have h1:=HQ.1
                 rw [sFG,dif_pos (ε.mpr h1)]; dsimp only; rw [ih _ _ _ (by rw [ac (ε.mpr h1)]; exact HQ.2),BT]
theorem sFG_sz(relu:Ἆ)(H:ἆ)(fb:Ἄ->Ἅ->Ἃ)(S:ἀ)(st s0 q0 mu:Ἄ)(hS:(J0 s0)<(J0 st)∧(J0 st)*(J0 mu)<=(kS S)∧(J0 st)*(J0 mu)<2^(31:Ἀ))(hB:(J0 q0)+(J0 mu)<2^(31:Ἀ))
    (j:Ἄ)(a:Ἂ Ἃ)(ha:(J0 q0)+(J0 mu)<=(O0 a)) :
    (O0 (sFG relu H fb S st s0 q0 mu hS hB j a ha))=(O0 a):=sFG_size relu H fb S st s0 q0 mu hS hB _ j a ha rfl
theorem sFG_spec(relu:Ἆ)(H:ἆ)(fb:Ἄ->Ἅ->Ἃ)(S:ἀ)(st s0 q0 mu:Ἄ)(hS:(J0 s0)<(J0 st)∧(J0 st)*(J0 mu)<=(kS S)∧(J0 st)*(J0 mu)<2^(31:Ἀ))(hB:(J0 q0)+(J0 mu)<2^(31:Ἀ)) :
    ∀ m (j:Ἄ)(a:Ἂ Ἃ)(ha:(J0 q0)+(J0 mu)<=(O0 a)),(J0 mu)-(J0 j)=m → (J0 j) ≤ (J0 mu) → ∀ q,
    (sFG relu H fb S st s0 q0 mu hS hB j a ha)[q]! =
      if (J0 q0)+(J0 j) ≤ q∧q < (J0 q0)+(J0 mu) then
        (if zR (S.data[(J0 st)*(q-(J0 q0))+(J0 s0)]!) relu H==(0xFFFFFFFF:ἇ) then fb (G0 (q-(J0 q0))) (S.data[(J0 st)*(q-(J0 q0))+(J0 s0)]!)
         else zR (S.data[(J0 st)*(q-(J0 q0))+(J0 s0)]!) relu H) else a[q]! := by
  intro m; induction m with
  | zero=>intro j a ha hm hj q; rw [sFG,dif_neg (fun h=>Lq0 _ _ hm (ε.mp h)),if_neg (Lqn _ _ _ _ (K3.mp hm))]
  | succ m ih =>
    intro j a ha hm hj q
    have HQ:=Lqs _ _ _ hm; have h1:=HQ.1
    have e1:=ac (ε.mpr h1)
    have B:=sFGb S st s0 mu j hS h1
    rw [sFG,dif_pos (ε.mpr h1)]; dsimp only
    rw [ih _ _ _ (by rw [e1]; exact HQ.2) (by rw [e1]; exact h1),kAS,kSm q0 j (Lql _ _ _ _ h1 (X4 hB)),kFU,B.1,e1]
    by_cases e:(J0 q0)+(J0 j)=q
    · subst e
      rw [if_neg (Lqa _ _ _),if_pos rfl,if_pos (Lqb _ _ _ h1),ϱ,kTU]
    · by_cases g:(J0 q0)+((J0 j)+1) ≤ q∧q < (J0 q0)+(J0 mu)
      · rw [if_pos g,if_pos (Lqc _ _ _ _ g)]
      · rw [if_neg g,if_neg e,if_neg (Lqe _ _ _ _ g e)]
theorem sPXb(Wd S:ἀ)(n ow o:Ἄ)(hW:(J0 ow)*(J0 n)<=(kS Wd))(h31:(J0 ow)*(J0 n)<2^(31:Ἀ)∧8*(J0 n)<2^(31:Ἀ)∧8*(J0 ow)<2^(31:Ἀ))(hS:8*(J0 ow)<=(kS S))(h:(J0 o)<(J0 ow)) :
    (J0 (o*n))=(J0 o)*(J0 n)∧(J0 (o*n))+(J0 n)<=(kS Wd)∧(J0 (o*n))+(J0 n)<2^(31:Ἀ) ∧
    (J0 (8*o))=8*(J0 o)∧(J0 (8*o))+7<(kS S)∧(J0 (8*o))+7<2^(31:Ἀ)∧(J0 (o+1))=(J0 o)+1:=by
  have m1:((J0 o)+1)*(J0 n) ≤ (J0 ow)*(J0 n):=ι _ h
  rw [Ο] at m1
  have m2:8*((J0 o)+1) ≤ 8*(J0 ow):=U3 _ h
  rw [Nat.mul_succ] at m2
  have e1:(J0 (o*n))=(J0 o)*(J0 n):=kMm _ _ (C0 ((V1 _ _).trans m1) h31.1)
  have e8:(J0 (8*o))=8*(J0 o):=by rw [kMm _ _ (by rw [kc8]; exact C0 ((V1 _ _).trans m2) h31.2.2),kc8]
  have o1:(J0 (o+1))=(J0 o)+1:=by rw [kSm _ _ (by rw [kc1]; exact C0 (U2 h (lZ _ (by decide))) h31.2.2),kc1]
  rw [e1,e8]
  exact ⟨rfl,m1.trans hW,C0 m1 h31.1,rfl,U1 (O1 (by decide) _) (m2.trans hS),
    U1 (O1 (by decide) _) (m2.trans (X4 h31.2.2)),o1⟩
def sPX(Wd XT:ἀ)(b:Ἂ Ἃ)(n ow:Ἄ)(hW:(J0 ow)*(J0 n)<=(kS Wd))(hX:8*(J0 n)<=(kS XT))(h31:(J0 ow)*(J0 n)<2^(31:Ἀ)∧8*(J0 n)<2^(31:Ἀ)∧8*(J0 ow)<2^(31:Ἀ))
  (o:Ἄ)(S:ἀ)(hS:8*(J0 ow)<=(kS S)):ἀ:=
 if h:o < ow then
  have A:=sPXb Wd S n ow o hW h31 hS (ε.mp h)
  let bo:=dw oneW*dw b[(J0 o)]!
  if h2:o+1 < ow then
   have A2:=sPXb Wd S n ow (o+1) hW h31 hS (ε.mp h2)
   let bp:=dw oneW*dw b[(J0 (o+1))]!
   let y:=kX16 Wd XT (o*n) ((o+1)*n) n A.2.1 A2.2.1 hX ⟨A.2.2.1,A2.2.2.1,h31.2.1⟩ 0 bo bo bo bo bo bo bo bo bp bp bp bp bp bp bp bp
   let S1:=w8 S (8*o) ⟨y.a0,y.a1,y.a2,y.a3,y.a4,y.a5,y.a6,y.a7⟩ A.2.2.2.2.1 A.2.2.2.2.2.1
   sPX Wd XT b n ow hW hX h31 (o+1+1) (w8 S1 (8*(o+1)) ⟨y.b0,y.b1,y.b2,y.b3,y.b4,y.b5,y.b6,y.b7⟩ (by rw[w8_size];exact A2.2.2.2.2.1) A2.2.2.2.2.2.1) (by rw[w8_size,w8_size];exact hS)
  else w8 S (8*o) (kX8 Wd XT (o*n) n A.2.1 hX ⟨A.2.2.1,h31.2.1⟩ 0 bo bo bo bo bo bo bo bo) A.2.2.2.2.1 A.2.2.2.2.2.1
 else S
termination_by (J0 ow)-(J0 o)
decreasing_by (have A2:=sPXb Wd S n ow (o+1) hW h31 hS (ε.mp h2);rw[A2.2.2.2.2.2.2,(sPXb Wd S n ow o hW h31 hS (ε.mp h)).2.2.2.2.2.2];have:=ε.mp h2;rw[(sPXb Wd S n ow o hW h31 hS (ε.mp h)).2.2.2.2.2.2] at this;exact Lq2 _ _ this)
theorem sPX_size(Wd XT:ἀ)(b:Ἂ Ἃ)(n ow:Ἄ)(hW:(J0 ow)*(J0 n)<=(kS Wd))(hX:8*(J0 n)<=(kS XT))(h31:(J0 ow)*(J0 n)<2^(31:Ἀ)∧8*(J0 n)<2^(31:Ἀ)∧8*(J0 ow)<2^(31:Ἀ)) :
    ∀ m (o:Ἄ)(S:ἀ)(hS:8*(J0 ow)<=(kS S)),(J0 ow)-(J0 o)=m → (kS (sPX Wd XT b n ow hW hX h31 o S hS))=(kS S):=by
  intro m; induction m using C6 with
  | _ m ih =>
    intro o S hS hm
    by_cases h:(J0 o) < (J0 ow)
    · have A:=sPXb Wd S n ow o hW h31 hS h
      by_cases h2:(J0 o)+1 < (J0 ow)
      · have A2:=sPXb Wd S n ow (o+1) hW h31 hS (by rw [A.2.2.2.2.2.2]; exact h2)
        rw [sPX,dif_pos (ε.mpr h),dif_pos (ε.mpr (by rw [A.2.2.2.2.2.2]; exact h2))]; dsimp only
        rw [ih _ (by rw [A2.2.2.2.2.2.2,A.2.2.2.2.2.2, ← hm]; exact Lq2 _ _ h2) _ _ _ rfl,w8_size,w8_size]
      · rw [sPX,dif_pos (ε.mpr h),dif_neg (fun h'=>h2 (by rw [← A.2.2.2.2.2.2]; exact ε.mp h'))]; rw [w8_size]
    · rw [sPX,dif_neg (fun h'=>h (ε.mp h'))]
theorem kLs (a0 a1 a2 a3 a4 a5 a6 a7:Ἅ)(r:ℕ)(hr:r<8):lsel a0 a1 a2 a3 a4 a5 a6 a7 r=A8.f ⟨a0,a1,a2,a3,a4,a5,a6,a7⟩ r:=by
  interval_cases r <;> rfl
theorem sPX_v(Wd XT:ἀ)(b:Ἂ Ἃ)(wo n:Ἄ)(hW:(J0 wo)+(J0 n)<=(kS Wd))(hX:8*(J0 n)<=(kS XT))(h31:(J0 wo)+(J0 n)<2^(31:Ἀ)∧8*(J0 n)<2^(31:Ἀ))
    (o':ℕ)(hwo:(J0 wo)=o'*(J0 n))(c:Ἅ)(hc:c=dw oneW*dw b[o']!)(y:A8)(hy:y=kX8 Wd XT wo n hW hX h31 0 c c c c c c c c) :
    ∀ r<8,lsel y.a0 y.a1 y.a2 y.a3 y.a4 y.a5 y.a6 y.a7 r=ao (fun s=>XT.data[8*s+r]!*Wd.data[o'*(J0 n)+s]!) (J0 n) 0 (dw oneW*dw b[o']!):=by
  intro r hr
  rw [kLs _ _ _ _ _ _ _ _ r hr,show (⟨y.a0,y.a1,y.a2,y.a3,y.a4,y.a5,y.a6,y.a7⟩:A8)=y from rfl,hy,kX8_spec _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ rfl r hr,f8,hwo,hc]
  rfl
theorem sPX_spec(Wd XT:ἀ)(b:Ἂ Ἃ)(n ow:Ἄ)(hW:(J0 ow)*(J0 n)<=(kS Wd))(hX:8*(J0 n)<=(kS XT))(h31:(J0 ow)*(J0 n)<2^(31:Ἀ)∧8*(J0 n)<2^(31:Ἀ)∧8*(J0 ow)<2^(31:Ἀ)) :
    ∀ m (o:Ἄ)(S:ἀ)(hS:8*(J0 ow)<=(kS S)),(J0 ow)-(J0 o)=m →
    (∀ o'<(J0 o),o'<(J0 ow) → ∀ r<8,S.data[8*o'+r]! = ao (fun s=>XT.data[8*s+r]!*Wd.data[o'*(J0 n)+s]!) (J0 n) 0 (dw oneW*dw b[o']!)) →
    ∀ o'<(J0 ow), ∀ r<8,(sPX Wd XT b n ow hW hX h31 o S hS).data[8*o'+r]! = ao (fun s=>XT.data[8*s+r]!*Wd.data[o'*(J0 n)+s]!) (J0 n) 0 (dw oneW*dw b[o']!):=by
  intro m; induction m using C6 with
  | _ m ih =>
    intro o S hS hm I
    by_cases h:(J0 o) < (J0 ow)
    · have A:=sPXb Wd S n ow o hW h31 hS h
      by_cases h2:(J0 o)+1 < (J0 ow)
      · have A2:=sPXb Wd S n ow (o+1) hW h31 hS (by rw [A.2.2.2.2.2.2]; exact h2)
        rw [sPX,dif_pos (ε.mpr h),dif_pos (ε.mpr (by rw [A.2.2.2.2.2.2]; exact h2))]; dsimp only
        refine ih _ (by rw [A2.2.2.2.2.2.2,A.2.2.2.2.2.2, ← hm]; exact Lq2 _ _ h2) _ _ _ rfl ?_
        intro o' ho' ho'w r hr
        rw [A2.2.2.2.2.2.2,A.2.2.2.2.2.2] at ho'
        have f0:(J0 (8*o))=8*(J0 o):=A.2.2.2.1
        have f1:(J0 (8*(o+1)))=8*(J0 o)+8:=by rw [A2.2.2.2.1,A.2.2.2.2.2.2,Nat.mul_succ]
        have K:=kX16_eq Wd XT (o*n) ((o+1)*n) n A.2.1 A2.2.1 hX ⟨A.2.2.1,A2.2.2.1,h31.2.1⟩ _ 0 (dw oneW*dw b[(J0 o)]!) (dw oneW*dw b[(J0 o)]!) (dw oneW*dw b[(J0 o)]!) (dw oneW*dw b[(J0 o)]!) (dw oneW*dw b[(J0 o)]!) (dw oneW*dw b[(J0 o)]!) (dw oneW*dw b[(J0 o)]!) (dw oneW*dw b[(J0 o)]!)
          (dw oneW*dw b[(J0 (o+1))]!) (dw oneW*dw b[(J0 (o+1))]!) (dw oneW*dw b[(J0 (o+1))]!) (dw oneW*dw b[(J0 (o+1))]!) (dw oneW*dw b[(J0 (o+1))]!) (dw oneW*dw b[(J0 (o+1))]!) (dw oneW*dw b[(J0 (o+1))]!) (dw oneW*dw b[(J0 (o+1))]!) rfl
        rw [w8_spec,w8_spec]
        by_cases e1:o'=(J0 o)+1
        · subst e1
          rw [if_pos (show (J0 (8*(o+1))) ≤ 8*((J0 o)+1)+r∧8*((J0 o)+1)+r < (J0 (8*(o+1)))+8 by rw [f1]; exact Lw1 _ _ hr),Lw8 _ _ _ f1]
          exact sPX_v Wd XT b ((o+1)*n) n A2.2.1 hX ⟨A2.2.2.1,h31.2.1⟩ ((J0 o)+1) (by rw [A2.1,A.2.2.2.2.2.2]) _ (by rw [A.2.2.2.2.2.2]) _ K.2 r hr
        · rw [if_neg (by rw [f1]; exact Lw3 _ _ _ hr e1)]
          by_cases e0:o'=(J0 o)
          · subst e0
            rw [if_pos (by rw [f0]; exact Lw4 _ _ hr)]; simp only [f0,ϱ]
            exact sPX_v Wd XT b (o*n) n A.2.1 hX ⟨A.2.2.1,h31.2.1⟩ (J0 o) A.1 _ rfl _ K.1 r hr
          · rw [if_neg (by rw [f0]; exact Lw5 _ _ _ hr e0)]
            exact I o' (Lw6 _ _ ho' e1 e0) ho'w r hr
      · rw [sPX,dif_pos (ε.mpr h),dif_neg (fun h'=>h2 (by rw [← A.2.2.2.2.2.2]; exact ε.mp h'))]
        intro o' ho' r hr
        have f0:(J0 (8*o))=8*(J0 o):=A.2.2.2.1
        rw [w8_spec]
        by_cases e0:o'=(J0 o)
        · subst e0
          rw [if_pos (by rw [f0]; exact Lw4 _ _ hr)]; simp only [f0,ϱ]
          exact sPX_v Wd XT b (o*n) n A.2.1 hX ⟨A.2.2.1,h31.2.1⟩ (J0 o) A.1 _ rfl _ rfl r hr
        · rw [if_neg (by rw [f0]; exact Lw5 _ _ _ hr e0)]
          exact I o' (Lw7 _ _ _ ho' h2 e0) ho' r hr
    · rw [sPX,dif_neg (fun h'=>h (ε.mp h'))]
      intro o' ho' r hr; exact I o' (U1 ho' (W4 h)) ho' r hr
theorem sFBb(r0 ow lim r:Ἄ)(hl:(J0 lim)<=8)(hR:(J0 r0)+(J0 lim)<2^(31:Ἀ))(hB:((J0 r0)+(J0 lim))*(J0 ow)<2^(31:Ἀ))(h:(J0 r)<(J0 lim)) :
    (J0 ((r0+r)*ow))=((J0 r0)+(J0 r))*(J0 ow)∧(J0 (r0+r))=(J0 r0)+(J0 r) ∧
    ((J0 r0)+(J0 r))*(J0 ow)+(J0 ow)<=((J0 r0)+(J0 lim))*(J0 ow)∧(J0 r)<(J0 (8:Ἄ))∧(J0 (r+1))=(J0 r)+1:=by
  have m1:((J0 r0)+(J0 r)+1)*(J0 ow) ≤ ((J0 r0)+(J0 lim))*(J0 ow):=ι _ (O1 h _)
  rw [Ο] at m1
  have e1:(J0 (r0+r))=(J0 r0)+(J0 r):=kSm _ _ (Lql _ _ _ _ h (X4 hR))
  have e2:(J0 ((r0+r)*ow))=((J0 r0)+(J0 r))*(J0 ow):=by
    rw [kMm _ _ (by rw [e1]; exact C0 ((V1 _ _).trans m1) hB),e1]
  exact ⟨e2,e1,m1,by rw [kc8]; exact U1 h hl,ac (ε.mpr h)⟩
def sFB(relu:Ἆ)(H:ἆ)(fb:Ἄ->Ἄ->Ἅ->Ἃ)(S:ἀ)(r0 ow lim:Ἄ)(hS:8*(J0 ow)<=(kS S)∧8*(J0 ow)<2^(31:Ἀ))(hl:(J0 lim)<=8)(hR:(J0 r0)+(J0 lim)<2^(31:Ἀ))(hB:((J0 r0)+(J0 lim))*(J0 ow)<2^(31:Ἀ))
  (r:Ἄ)(a:Ἂ Ἃ)(ha:((J0 r0)+(J0 lim))*(J0 ow)<=(O0 a)):Ἂ Ἃ:=
 if h:r < lim then
  have F:=sFBb r0 ow lim r hl hR hB (ε.mp h)
  sFB relu H fb S r0 ow lim hS hl hR hB (r+1)
   (sFG relu H (fb (r0+r)) S 8 r ((r0+r)*ow) ow ⟨F.2.2.2.1,by rw[kc8];exact hS.1,by rw[kc8];exact hS.2⟩ (by rw[F.1];exact C0 F.2.2.1 hB) 0 a (by rw[F.1];exact F.2.2.1.trans ha))
   (by rw[sFG_sz];exact ha)
 else a
termination_by (J0 lim)-(J0 r)
decreasing_by (rw[ac h];exact Lqd _ _ (ε.mp h))
theorem sFB_spec(relu:Ἆ)(H:ἆ)(fb:Ἄ->Ἄ->Ἅ->Ἃ)(S:ἀ)(r0 ow lim:Ἄ)(hS:8*(J0 ow)<=(kS S)∧8*(J0 ow)<2^(31:Ἀ))(hl:(J0 lim)<=8)(hR:(J0 r0)+(J0 lim)<2^(31:Ἀ))(hB:((J0 r0)+(J0 lim))*(J0 ow)<2^(31:Ἀ)) :
    ∀ m (r:Ἄ)(a:Ἂ Ἃ)(ha:((J0 r0)+(J0 lim))*(J0 ow)<=(O0 a)),(J0 lim)-(J0 r)=m →
    (O0 (sFB relu H fb S r0 ow lim hS hl hR hB r a ha))=(O0 a) ∧
    (∀ q,(∀ r' o',(J0 r) ≤ r' → r'<(J0 lim) → o'<(J0 ow) → ((J0 r0)+r')*(J0 ow)+o'≠q) → (sFB relu H fb S r0 ow lim hS hl hR hB r a ha)[q]! = a[q]!) ∧
    (∀ r' o',(J0 r) ≤ r' → r'<(J0 lim) → o'<(J0 ow) → (sFB relu H fb S r0 ow lim hS hl hR hB r a ha)[((J0 r0)+r')*(J0 ow)+o']! =
      (if zR (S.data[8*o'+r']!) relu H==(0xFFFFFFFF:ἇ) then fb (r0+C3 r') (C3 o') (S.data[8*o'+r']!) else zR (S.data[8*o'+r']!) relu H)):=by
  intro m; induction m with
  | zero =>
    intro r a ha hm
    rw [sFB,dif_neg (fun h=>Lq0 _ _ hm (ε.mp h))]
    exact ⟨rfl,fun q _=>rfl,fun r' o' h1 h2 _=>absurd (C0 h1 h2) (Lq0 _ _ hm)⟩
  | succ m ih =>
    intro r a ha hm
    have HQ:=Lqs _ _ _ hm; have h:=HQ.1
    have F:=sFBb r0 ow lim r hl hR hB h
    rw [sFB,dif_pos (ε.mpr h)]; dsimp only
    obtain ⟨T1,T2,T3⟩:=ih (r+1) _ _ (by rw [F.2.2.2.2]; exact HQ.2)
    have G:=sFG_spec relu H (fb (r0+r)) S 8 r ((r0+r)*ow) ow ⟨F.2.2.2.1,by rw[kc8];exact hS.1,by rw[kc8];exact hS.2⟩ (by rw[F.1];exact C0 F.2.2.1 hB) _ 0 a (by rw[F.1];exact F.2.2.1.trans ha) rfl (ϟ _)
    rw [F.2.2.2.2] at T2 T3
    refine ⟨by rw [T1,sFG_sz],fun q hq => ?_,fun r' o' h1 h2 h3 => ?_⟩
    · rw [T2 q (fun r' o' h1 h2 h3=>hq r' o' (X5 h1) h2 h3),G q,if_neg]
      rw [F.1,kc0,ϑ]; intro ⟨g1,g2⟩
      exact hq (J0 r) (q-((J0 r0)+(J0 r))*(J0 ow)) (le_refl _) h (G4 g1 g2) (BQ g1)
    · by_cases e:r'=J0 r
      · subst e
        rw [T2 _ (fun r'' o'' g1 g2 g3=>Lqf _ _ _ _ _ _ g1 h3),
          G,F.1,kc0,ϑ,if_pos ⟨V1 _ _,O1 h3 _⟩,ϱ,kc8]
        have er:r0+C3 (J0 r)=r0+r:=by rw [show C3 (J0 r)=r from USize.ofNat_toNat]
        rw [er]
      · exact T3 r' o' (N4 h1 (fun x=>e x.symm)) h2 h3
theorem kPm (e c:ἆ) (h:e.toNat < 255) (hc:c.toNat ≤ 992) :
    ((if e < (56:ἆ) then (56:ἆ) else e)+c).toNat=max e.toNat 56+c.toNat:=by
  have m:(if e < (56:ἆ) then (56:ἆ) else e).toNat=max e.toNat 56:=by
    by_cases g:e < (56:ἆ)
    · rw [if_pos g]; exact (Nat.max_eq_right (X4 (show e.toNat < 56 from g))).symm
    · rw [if_neg g]; exact (Nat.max_eq_left (W4 (show ¬ e.toNat < 56 from g))).symm
  rw [η,m]
  exact γ (C0 (Y3 (Nat.max_le.2 ⟨X4 h,by decide⟩) hc) (by decide))
theorem kOutRb (v:Ἅ)(relu:Ἆ)(n:ℕ)(D Wf:ℕ→Ἃ)(Fb:Ἃ)(hfb:Fb=p8 relu (ms (n+1) D Wf))(e cq hb em:ἆ)
    (hw:∀ p:ἆ,p=(if e<(56:ἆ) then (56:ἆ) else e)+cq → e<(255:ἆ) → p ≤ (372:ἆ) →
      aG (n+1) D Wf∧fv v ∧ |rv v-(af (n+1) D Wf:ℝ)| ≤ (2:ℝ)^(((if hb>p then hb else p).toNat:ℕ):ℤ) ∧
      |(af (n+1) D Wf:ℝ)| ≤ (2:ℝ)^(425:ℤ) ∧ |rv v| ≤ (2:ℝ)^((((if hb>p then hb else p).toNat)+52:ℕ):ℤ)∧(if hb>p then hb else p).toNat ≤ 600)
    (hEm:e.toNat ≤ em.toNat)(hem:em.toNat < 255)(hc2:cq.toNat ≤ 992) :
    (if zR v relu (if (if em<(56:ἆ) then (56:ἆ) else em)+cq<=(372:ἆ) then (if hb>(if em<(56:ἆ) then (56:ἆ) else em)+cq then hb else (if em<(56:ἆ) then (56:ἆ) else em)+cq) else (4000:ἆ))==(0xFFFFFFFF:Ἃ) then Fb
     else zR v relu (if (if em<(56:ἆ) then (56:ἆ) else em)+cq<=(372:ἆ) then (if hb>(if em<(56:ἆ) then (56:ἆ) else em)+cq then hb else (if em<(56:ἆ) then (56:ἆ) else em)+cq) else (4000:ἆ)))=p8 relu (ms (n+1) D Wf):=by
  by_cases hP:(if em<(56:ἆ) then (56:ἆ) else em)+cq<=(372:ἆ)
  · rw [if_pos hP]
    have he:e.toNat < 255:=C0 hEm hem
    have tpe:=kPm e cq he hc2
    have tP:=kPm em cq hem hc2
    have hP':((if em<(56:ἆ) then (56:ἆ) else em)+cq).toNat ≤ 372:=hP
    have mE:max e.toNat 56+cq.toNat ≤ max em.toNat 56+cq.toNat:=L1 (max_le_max_right 56 hEm) _
    have pe:(if e<(56:ἆ) then (56:ἆ) else e)+cq ≤ (372:ἆ):=(show ((if e<(56:ἆ) then (56:ἆ) else e)+cq).toNat ≤ 372 by rw [tpe]; exact mE.trans (by rw [← tP]; exact hP'))
    obtain ⟨g,fv1,b1,b2,_,b6⟩:=hw _ rfl he pe
    have mono:(if hb>(if e<(56:ἆ) then (56:ἆ) else e)+cq then hb else (if e<(56:ἆ) then (56:ἆ) else e)+cq).toNat ≤ (if hb>(if em<(56:ἆ) then (56:ἆ) else em)+cq then hb else (if em<(56:ἆ) then (56:ἆ) else em)+cq).toNat:=by
      rw [kOutW_h2,kOutW_h2,tpe,tP]; exact max_le_max_left _ mE
    have h6:(if hb>(if em<(56:ἆ) then (56:ἆ) else em)+cq then hb else (if em<(56:ἆ) then (56:ἆ) else em)+cq).toNat ≤ 600:=by
      rw [kOutW_h2] at b6 ⊢; exact max_le ((le_max_left _ _).trans b6) (hP'.trans (by decide))
    exact kZR v _ relu n _ _ _ hfb g fv1 (AZ b1 (AG (by norm_num) (by exact_mod_cast mono))) b2 h6
  · rw [if_neg hP,kZRBig]; simp only [Az,ἁ]; exact hfb
def gY(Wd XT:ἀ)(b:Ἂ Ἃ)(relu:Ἆ)(cq hbq em:ἆ)(fb:Ἄ->Ἄ->Ἅ->Ἃ)(n ow r0 lim:Ἄ)
  (hW:(J0 ow)*(J0 n)<=(kS Wd))(hX:8*(J0 n)<=(kS XT))(h31:(J0 ow)*(J0 n)<2^(31:Ἀ)∧8*(J0 n)<2^(31:Ἀ)∧8*(J0 ow)<2^(31:Ἀ))
  (hl:(J0 lim)<=8)(hR:(J0 r0)+(J0 lim)<2^(31:Ἀ))(hB:((J0 r0)+(J0 lim))*(J0 ow)<2^(31:Ἀ))(acc:Ἂ Ἃ):Ἂ Ἃ:=
 if ha:((J0 r0)+(J0 lim))*(J0 ow)<=(O0 acc) then
 let S:=sPX Wd XT b n ow hW hX h31 0 (zPq (8*(J0 ow))) (zPs _).1
 let P:=(if em<(56:ἆ) then (56:ἆ) else em)+cq
 sFB relu (if P<=(372:ἆ) then (if hbq>P then hbq else P) else (4000:ἆ)) fb S r0 ow lim ⟨by rw[sPX_size _ _ _ _ _ _ _ _ _ _ _ _ rfl];exact (zPs _).1,h31.2.2⟩ hl hR hB 0 acc ha
 else acc
theorem gY_spec(Wd XT:ἀ)(E:Ἂ Ἃ)(X W b:Ἂ Ἃ)(relu:Ἆ)(ix ib L H N:Ἀ)(n ow r0 lim:Ἄ)
  (hW:(J0 ow)*(J0 n)<=(kS Wd))(hX:8*(J0 n)<=(kS XT))(h31:(J0 ow)*(J0 n)<2^(31:Ἀ)∧8*(J0 n)<2^(31:Ἀ)∧8*(J0 ow)<2^(31:Ἀ))(hn:(J0 n)=N)
  (hl:(J0 lim)<=8)(hR:(J0 r0)+(J0 lim)<2^(31:Ἀ))(hB:((J0 r0)+(J0 lim))*(J0 ow)<2^(31:Ἀ))
  (hWv:∀k,k<(J0 ow)*(J0 n)->Wd.data[k]! =dw W[k]!)
  (hXv:∀s,s<(J0 n)->∀r,r<8->XT.data[8*s+r]! =if r<(J0 lim) then dw X[((J0 r0)+r)*N+s]! else dw 0)
  (hEv:∀o,o<(J0 ow)->∀s,s<N->(expOf W[o*N+s]!).toNat<=(E[o]!).toNat)
  (F:∀q:Ἀ,kB X[q]!≠255/\kG X[q]!<=ix)(hb:∀q:Ἀ,kB b[q]!≠255/\kG b[q]!<=H)(M:174+H<=ib)(Y:N+2<=2^L)(x:L<=27)(z:ib+L<=425)(hix:ix<=425)(cq hbq em:ἆ)(hcq:∀ r,r<(J0 lim) → 23+(cEq (I0 (9*(N+1))).toInt64.toFloat (ao (fun k=>dw (X[((J0 r0)+r)*N+k]! &&& 0x7FFFFFFF)) N 0 0)).toNat ≤ cq.toNat)(hcq2:cq.toNat ≤ 992)(hhb:hbq=(ib+L-50).toUInt64)(hN:0<N)
  (hEm:∀o,o<(J0 ow)->(E[o]!).toNat<=em.toNat)(hem:em.toNat<255)
  (acc:Ἂ Ἃ)(ha:((J0 r0)+(J0 lim))*(J0 ow)<=(O0 acc)) :
  (O0 (gY Wd XT b relu cq hbq em (bXfb relu X W b E Wd ix ib L N) n ow r0 lim hW hX h31 hl hR hB acc))=(O0 acc) ∧
  (∀q,(∀r o',r<(J0 lim)->o'<(J0 ow)->((J0 r0)+r)*(J0 ow)+o'≠q)->(gY Wd XT b relu cq hbq em (bXfb relu X W b E Wd ix ib L N) n ow r0 lim hW hX h31 hl hR hB acc)[q]! =acc[q]!) ∧
  (∀r o',r<(J0 lim)->o'<(J0 ow)->(gY Wd XT b relu cq hbq em (bXfb relu X W b E Wd ix ib L N) n ow r0 lim hW hX h31 hl hR hB acc)[((J0 r0)+r)*(J0 ow)+o']! =act relu (dotOf (N+1) (fun t=>if t<N then X[((J0 r0)+r)*N+t]! else oneW) fun t=>if t<N then W[(o')*N+t]! else b[o']!)):=by
  unfold gY; rw [dif_pos ha]; dsimp only
  obtain ⟨T1,T2,T3⟩:=sFB_spec relu (if (if em<(56:ἆ) then (56:ἆ) else em)+cq<=(372:ἆ) then (if hbq>(if em<(56:ἆ) then (56:ἆ) else em)+cq then hbq else (if em<(56:ἆ) then (56:ἆ) else em)+cq) else (4000:ἆ)) (bXfb relu X W b E Wd ix ib L N) (sPX Wd XT b n ow hW hX h31 0 (zPq (8*(J0 ow))) (zPs _).1) r0 ow lim ⟨by rw[sPX_size _ _ _ _ _ _ _ _ _ _ _ _ rfl];exact (zPs _).1,h31.2.2⟩ hl hR hB _ 0 acc ha rfl
  have SP:=sPX_spec Wd XT b n ow hW hX h31 _ 0 (zPq (8*(J0 ow))) (zPs _).1 rfl (fun o' h _=>absurd h (by rw [kc0]; exact U5 _))
  refine ⟨T1,fun q hq=>T2 q (fun r' o' _ h2 h3=>hq r' o' h2 h3),fun r o' hr ho => ?_⟩
  rw [T3 r o' (by rw [kc0]; exact ϟ _) hr ho,SP o' ho r (U1 hr hl)]
  have hS : ∀k,k<2^(31:Ἀ)->k<kX:=fun k hk=>AN (AN hk (by decide)) ς
  have ho':(J0 (C3 o'))=o':=J3 (hS _ (U1 ho (X4 (C0 (lZ _ (by decide)) h31.2.2))))
  have hr8:r < kX:=hS _ (U1 (U1 hr hl) (by decide))
  have hrow:(J0 (r0+C3 r))=(J0 r0)+r:=by
    rw [kSm _ _ (by rw [J3 hr8]; exact Lql _ _ _ _ hr (X4 hR)),J3 hr8]
  have m1:o'*(J0 n)+(J0 n) ≤ (J0 ow)*(J0 n):=by
    have:=ι (J0 n) (show o'+1 ≤ (J0 ow) from ho); rwa [Ο] at this
  simp only [bXfb,hrow,ho',hn]
  simp only [sTB_eq,tR_tS,hrow,ho']
  have hpx : ∀ k < N,expOf W[o'*N+k]! ≠ 255 → fv ((fun k=>XT.data[8*k+r]! * Wd.data[o'*N+k]!) k)∧rv ((fun k=>XT.data[8*k+r]! * Wd.data[o'*N+k]!) k)=tm X[((J0 r0)+r)*N+k]! W[o'*N+k]! := fun k hk hw=>by
    beta_reduce at hw ⊢
    rw [hXv k (by rw [hn]; exact hk) r (U1 hr hl),if_pos hr,hWv _ (by rw [← hn]; rw [← hn] at hk; exact AN (O1 hk _) m1)]
    exact dJ _ _ (F _).1 hw
  have G:=kGB1T relu N N o' (fun s=>X[((J0 r0)+r)*N+s]!) W b (fun k=>k) (fun k=>XT.data[8*k+r]! * Wd.data[o'*N+k]!)
    (fun s=>dw X[((J0 r0)+r)*N+s]! * Wd.get! (o'*N+s)) E[o']! ix ib L H (fun _=>aD N relu X W b (((J0 r0)+r)*N) 1 o')
    (fun s _=>F _) (fun s hs=>hEv o' ho s hs) (hb _) M Y x z (le_refl _) (fun k hk=>hk) (fun a c h _=>h)
    (fun s hs hh=>absurd rfl (hh s hs)) hpx
    (fun s hs hw=>by beta_reduce at hw ⊢; rw [gg,hWv _ (by rw [← hn]; rw [← hn] at hs; exact AN (O1 hs _) m1)]; exact dJ _ _ (F _).1 hw)
    (kAD N relu X W b ((J0 r0)+r) o')
  exact kOutRb _ relu N _ _ _ G (E[o']!).toUInt64 cq hbq em (fun p hp he h372=>kOutWm N (fun s=>X[((J0 r0)+r)*N+s]!) W b E ib L H o' cq hbq p (fun k=>XT.data[8*k+r]! * Wd.data[o'*N+k]!) _ _
      (fun s _=>(F _).1) (hb _) M Y x z hN (fun s hs=>hEv o' ho s hs)
      (fun the s hs=>hpx s hs (fun h'=>by have:=hEv o' ho s hs; rw [h'] at this; simp at this; omega)) rfl rfl (hcq r hr) hcq2 hhb hp he h372)
    (by rw [ϩ]; exact hEm o' ho) hem hcq2
def gXb(X:Ἂ Ἃ)(Wd:ἀ)(E:Ἂ Ἃ)(b:Ἂ Ἃ)(relu:Ἆ)(bs Lq ib em:ἆ)(fb:Ἄ->Ἄ->Ἅ->Ἃ)(R n ow:Ἀ)
  (hXs:R*n<=(O0 X))(hW:ow*n<=(kS Wd))(hRn:R*n<2^31)(h8n:8*n<2^31)(hon:ow*n<2^31)(how:ow<2^31)(hR31:R<2^31)(hRo:R*ow<2^31)(h8o:8*ow<2^31)(rb:Ἀ)(acc:Ἂ Ἃ):Ἂ Ἃ:=
 if h:rb*8<R then
  let lim:=min 8 (R-rb*8)
  have hl8:lim<=8:=P7 _ _
  have hlR:rb*8+lim<=R:=(J1 (Nat.min_le_right 8 (R-rb*8)) _).trans (B3 (BQ (X4 h)))
  have hS:∀k,k<2^31->k<kX:=fun k hk=>AN (AN hk (by decide)) ς
  have hR1:rb*8*n+lim*n<=R*n:=by rw[← Ϡ];exact ι n hlR
  have e1:(J0 ((G0 (rb*8*n))))=rb*8*n:=J3 (hS _ (AO ((V1 _ _).trans hR1) hRn))
  have e2:(J0 ((G0 n)))=n:=J3 (hS _ (AO (lZ n (by decide)) h8n))
  have e3:(J0 ((G0 lim)))=lim:=J3 (hS _ (AO hl8 (by decide)))
  have hB:∀r t:ℕ,r<(J0 ((G0 lim)))->t<(J0 ((G0 n)))->(J0 ((G0 (rb*8*n))))+r*(J0 ((G0 n)))+t<(O0 X)∧(J0 ((G0 (rb*8*n))))+r*(J0 ((G0 n)))+t<2^31:=by
   intro r t hr ht;rw[e3] at hr;rw[e2] at ht;rw[e1,e2]
   have:=ι n (show r+1<=lim from hr);rw[Ϡ,n4] at this
   have k:rb*8*n+r*n+t<R*n:=AN (O1 ht _) (by rw[pA];exact (J1 this _).trans hR1)
   exact ⟨AN k hXs,K5 k hRn⟩
  have hXd:(J0 ((G0 (rb*8*n))))+(J0 ((G0 lim)))*(J0 ((G0 n)))<=(O0 X):=by rw[e1,e3,e2];exact hR1.trans hXs
  have hYd:8*(J0 ((G0 n)))+8<=(kS (zPq (8*n+8))):=by rw[e2];exact (zPs _).1
  have hBd:(J0 ((G0 (rb*8*n))))+(J0 ((G0 lim)))*(J0 ((G0 n)))<2^31/\8*(J0 ((G0 n)))+7<2^31:=⟨by rw[e1,e3,e2];exact AO hR1 hRn,by rw[e2];clear * - h8n;omega⟩
  have hld:(J0 ((G0 lim)))<=8:=by rw[e3];exact hl8
  let XT:=dX8 X (G0 (rb*8*n)) (G0 n) (G0 lim) 0 (zPq (8*n+8)) hXd hYd hBd hld
  have hXT:8*n<=(kS XT):=by show 8*n<=(kS (dX8 X (G0 (rb*8*n)) (G0 n) (G0 lim) 0 (zPq (8*n+8)) hXd hYd hBd hld));rw[dX8_size X _ _ _ hXd hBd hld _ 0 _ hYd rfl];exact AZ (V1 _ 8) (zPs _).1
  let F9:=(I0 (9*(n+1))).toInt64.toFloat
  let cr:=fun (r:Nat)=>if r<lim then (23:ἆ)+cEq F9 (XT.get! (8*n+r)) else (0:ἆ)
  let cqb:=mx2 (mx2 (mx2 (cr 0) (cr 1)) (mx2 (cr 2) (cr 3))) (mx2 (mx2 (cr 4) (cr 5)) (mx2 (cr 6) (cr 7)))
  let hbq:=ib+Lq-(50:ἆ)
  have e4:(J0 ((G0 ow)))=ow:=J3 (hS _ how)
  have e5:(J0 ((G0 (rb*8))))=rb*8:=J3 (hS _ (K5 h hR31))
  gXb X Wd E b relu bs Lq ib em fb R n ow hXs hW hRn h8n hon how hR31 hRo h8o (rb+1) (gY Wd XT b relu cqb hbq em fb (G0 n) (G0 ow) (G0 (rb*8)) (G0 lim) (by rw[e4,e2];exact hW) (by rw[e2];exact hXT) ⟨by rw[e4,e2];exact hon,by rw[e2];exact h8n,by rw[e4];exact h8o⟩ (by rw[e3];exact hl8) (by rw[e5,e3];exact C0 hlR hR31) (by rw[e5,e3,e4];exact C0 (ι ow hlR) hRo) acc)
 else acc
termination_by R-rb*8
decreasing_by all_goals (simp_wf;rw[Ϡ,n4];exact C5 h (P5 (by decide)))
theorem gXb_spec_h1(z y A acc:Ἂ Ἃ)(R ow rb a b c d0:Ἀ)(Fv:Ἀ->Ἀ->Ἃ)(Ez:z=y)
  (IH:(O0 A)=R*ow->(∀row j,row<(rb+1)*8->row<R->j<ow->A[row*ow+j]! =Fv row j)->(O0 y)=R*ow ∧ ∀row j,row<R->j<ow->y[row*ow+j]! =Fv row j)
  (G:(O0 A)=R*ow∧(∀q,(∀r o',r<c->d0<=o'->o'<a->(b+r)*a+o'≠q)->A[q]! =acc[q]!)∧(∀r o',r<c->d0<=o'->o'<a->A[(b+r)*a+o']! =Fv (b+r) o'))
  (ha:a=ow)(hb:b=rb*8)(hc:c=min 8 (R-rb*8))(hd:d0=0)
  (hI:∀row j,row<rb*8->row<R->j<ow->acc[row*ow+j]! =Fv row j):
  (O0 z)=R*ow ∧ ∀row j,row<R->j<ow->z[row*ow+j]! =Fv row j:=by
 subst Ez a b c d0; obtain ⟨G1,G2,G3⟩:=G
 refine IH G1 (fun row j h1 h2 h3=>?_)
 by_cases hr:row < rb*8
 · rw[G2 _ (fun r o' hr' _ ho' he=>W3.2 (Nat.le.intro (kDM _ _ _ _ _ he ho' h3).1) hr)];exact hI row j hr h2 h3
 · have:=G3 (row-rb*8) j (lt_min (G4 (W3.1 hr) (by rwa[Ϡ,n4] at h1)) (Nat.sub_lt_sub_right (W3.1 hr) h2)) (ϟ _) h3
   rw[BQ (W3.1 hr)] at this;exact this
def bX8(R n ow:Ἀ)(relu:Ἆ)(X W b:Ἂ Ἃ)(ix ib L:Ἀ)(Wd:ἀ)(E:Ἂ Ἃ)(em:ἆ)
  (hXs:R*n<=(O0 X))(hW:ow*n<=(kS Wd))(hRn:R*n<2^(31:Ἀ))(h8n:8*n<2^(31:Ἀ))(hon:ow*n<2^(31:Ἀ))(how:ow<2^(31:Ἀ))(hR31:R<2^(31:Ἀ))(hRo:R*ow<2^(31:Ἀ))(h8o:8*ow<2^(31:Ἀ)):Ἂ Ἃ:=
 let fb:=fun (row j:Ἄ)(v:Ἅ)=>tG relu v E[(J0 j)]! ix ib L
    (λ_=>tR relu L (sTB X Wd n row j (dw oneW*dw b[(J0 j)]!)) λ_=>aD n relu X W b ((J0 row)*n) 1 (J0 j))
    (λ_=>aD n relu X W b ((J0 row)*n) 1 (J0 j))
 gXb X Wd E b relu (48+ix).toUInt64 L.toUInt64 ib.toUInt64 em fb R n ow hXs hW hRn h8n hon how hR31 hRo h8o 0 (Ah (R*ow) 0)
theorem gXb_spec(X:Ἂ Ἃ)(Wd:ἀ)(E:Ἂ Ἃ)(W b:Ἂ Ἃ)(relu:Ἆ)(ix ib L H R n ow:Ἀ)(em:ἆ)
  (hXs:R*n<=(O0 X))(hW:ow*n<=(kS Wd))(hRn:R*n<2^31)(h8n:8*n<2^31)(hon:ow*n<2^31)(how:ow<2^31)(hR31:R<2^31)(hRo:R*ow<2^31)(h8o:8*ow<2^31)
  (hWv:∀k,k<ow*n->Wd.data[k]! =dw W[k]!)
  (hEv:∀o,o<ow->∀s,s<n->(expOf W[o*n+s]!).toNat<=(E[o]!).toNat)
  (F:∀q:Ἀ,kB X[q]!≠255/\kG X[q]!<=ix)(hb:∀q:Ἀ,kB b[q]!≠255/\kG b[q]!<=H)(M:174+H<=ib)(Y:n+2<=2^L)(x:L<=27)(z:ib+L<=425)(hix:ix<=425)(hN:0<n)(hEm:∀o,o<ow->(E[o]!).toNat<=em.toNat)(hem:em.toNat<255) :
  ∀ m rb (acc:Ἂ Ἃ),R-rb*8<=m -> (O0 acc)=R*ow ->
   (∀row j,row<rb*8->row<R->j<ow->acc[row*ow+j]! =act relu (dotOf (n+1) (fun t=>if t<n then X[(row)*n+t]! else oneW) fun t=>if t<n then W[(j)*n+t]! else b[j]!)) ->
   (O0 (gXb X Wd E b relu (48+ix).toUInt64 L.toUInt64 ib.toUInt64 em (bXfb relu X W b E Wd ix ib L n) R n ow hXs hW hRn h8n hon how hR31 hRo h8o rb acc))=R*ow ∧ ∀row j,row<R->j<ow->(gXb X Wd E b relu (48+ix).toUInt64 L.toUInt64 ib.toUInt64 em (bXfb relu X W b E Wd ix ib L n) R n ow hXs hW hRn h8n hon how hR31 hRo h8o rb acc)[row*ow+j]! =act relu (dotOf (n+1) (fun t=>if t<n then X[(row)*n+t]! else oneW) fun t=>if t<n then W[(j)*n+t]! else b[j]!):=by
 intro m;induction m with
 | zero =>
   intro rb acc HH hA hI;have hz:=K3.1 (Nat.le_zero.1 HH);rw[(gXb.eq_1 X Wd E b relu (48+ix).toUInt64 L.toUInt64 ib.toUInt64 em (bXfb relu X W b E Wd ix ib L n) R n ow hXs hW hRn h8n hon how hR31 hRo h8o rb acc).trans (dif_neg (W3.2 hz))]
   exact ⟨hA,fun row j h1 h2=>hI row j (AN h1 hz) h1 h2⟩
 | succ m IH =>
   intro rb acc HH hA hI
   by_cases h:rb*8 < R
   swap
   · exact IH rb acc (by rw[Nat.sub_eq_zero_of_le (W3.1 h)];exact ϟ _) hA hI
   have hl8:min 8 (R-rb*8)<=8:=P7 _ _
   have hlR:rb*8+min 8 (R-rb*8)<=R:=(J1 (Nat.min_le_right 8 (R-rb*8)) _).trans (B3 (BQ (X4 h)))
   have hS:∀k,k<2^31->k<kX:=fun k hk=>AN (AN hk (by decide)) ς
   have hR1:rb*8*n+min 8 (R-rb*8)*n<=R*n:=by rw[← Ϡ];exact ι n hlR
   have e1:(J0 ((G0 (rb*8*n))))=rb*8*n:=J3 (hS _ (AO ((V1 _ _).trans hR1) hRn))
   have e2:(J0 ((G0 n)))=n:=J3 (hS _ (AO (lZ n (by decide)) h8n))
   have e3:(J0 ((G0 (min 8 (R-rb*8)))))=min 8 (R-rb*8):=J3 (hS _ (AO hl8 (by decide)))
   have e4:(J0 ((G0 ow)))=ow:=J3 (hS _ how)
   have e5:(J0 ((G0 (rb*8))))=rb*8:=J3 (hS _ (K5 h hR31))
   have hXd:(J0 ((G0 (rb*8*n))))+(J0 ((G0 (min 8 (R-rb*8)))))*(J0 ((G0 n)))<=(O0 X):=by rw[e1,e3,e2];exact hR1.trans hXs
   have hYd:8*(J0 ((G0 n)))+8<=(kS (zPq (8*n+8))):=by rw[e2];exact (zPs _).1
   have hBd:(J0 ((G0 (rb*8*n))))+(J0 ((G0 (min 8 (R-rb*8)))))*(J0 ((G0 n)))<2^31/\8*(J0 ((G0 n)))+7<2^31:=⟨by rw[e1,e3,e2];exact AO hR1 hRn,by rw[e2];exact U1 (I3 _) ((Nat.mul_add_one 8 n).symm.trans_le (U3 8 (Nat.lt_of_mul_lt_mul_left (a:=8) (c:=268435456) h8n)))⟩
   have hld:(J0 ((G0 (min 8 (R-rb*8)))))<=8:=by rw[e3];exact hl8
   have e0:(J0 (0:Ἄ))=0:=rfl
   have D:=dX8_spec X (G0 (rb*8*n)) (G0 n) (G0 (min 8 (R-rb*8))) hXd hBd hld ((J0 ((G0 (min 8 (R-rb*8)))))-(J0 (0:Ἄ))) 0 (zPq (8*n+8)) hYd rfl (ϟ _)
   have DS:=dX8_size X (G0 (rb*8*n)) (G0 n) (G0 (min 8 (R-rb*8))) hXd hBd hld ((J0 ((G0 (min 8 (R-rb*8)))))-(J0 (0:Ἄ))) 0 (zPq (8*n+8)) hYd rfl
   have ZP:=zPs (8*n+8)
   have P:=gXb_spec_h2 _ (zPq (8*n+8)) X _ _ _ _ (J0 ((G0 (rb*8)))) n e0 e2 (by rw[e1,e5]) hld (fun q hq=>ZP.2.2 _ (hq.trans_le ZP.1)) D
   let F9:=(I0 (9*(n+1))).toInt64.toFloat
   let cr:=fun (r:Nat)=>if r< min 8 (R-rb*8) then (23:ἆ)+cEq F9 ((dX8 X (G0 (rb*8*n)) (G0 n) (G0 (min 8 (R-rb*8))) 0 (zPq (8*n+8)) hXd hYd hBd hld).get! (8*n+r)) else (0:ἆ)
   have K:=kMx8 cr 992 (fun r _=>if hr:r< min 8 (R-rb*8) then (congrArg lq (if_pos hr)).trans_le (kCR _ _).2 else (congrArg lq (if_neg hr)).trans_le (ϟ _))
   have hcq:∀r,r<(J0 ((G0 (min 8 (R-rb*8)))))->23+(cEq F9 (ao (fun k=>dw (X[((J0 ((G0 (rb*8))))+r)*n+k]! &&& 0x7FFFFFFF)) n 0 0)).toNat<=(mx2 (mx2 (mx2 (cr 0) (cr 1)) (mx2 (cr 2) (cr 3))) (mx2 (mx2 (cr 4) (cr 5)) (mx2 (cr 6) (cr 7)))).toNat:=fun r hr=>((congrArg (fun x=>23+(cEq F9 x).toNat) ((gg _ _).trans (P.2 r hr)).symm).trans ((kCR _ _).1.symm.trans (congrArg lq (if_pos (hr.trans_eq e3)).symm))).trans_le (K.1 r ((hr.trans_eq e3).trans_le hl8))
   have hOW:(J0 ((G0 ow)))*(J0 ((G0 n)))=ow*n:=by rw[e4,e2]
   exact gXb_spec_h1 _ _ _ _ _ _ _ _ _ _ _ _ ((gXb.eq_1 X Wd E b relu (48+ix).toUInt64 L.toUInt64 ib.toUInt64 em (bXfb relu X W b E Wd ix ib L n) R n ow hXs hW hRn h8n hon how hR31 hRo h8o rb acc).trans (dif_pos h)) (IH (rb+1) _ (by rw[Ϡ,n4,V4];exact (Nat.sub_le_sub_left (by decide) _).trans (M8 HH)))
    (by obtain ⟨T1,T2,T3⟩:=gY_spec Wd (dX8 X (G0 (rb*8*n)) (G0 n) (G0 (min 8 (R-rb*8))) 0 (zPq (8*n+8)) hXd hYd hBd hld) E X W b relu ix ib L H n (G0 n) (G0 ow) (G0 (rb*8)) (G0 (min 8 (R-rb*8)))
          (by rw[e4,e2];exact hW) (by rw[DS,e2];exact AZ (V1 _ 8) ZP.1) ⟨by rw[e4,e2];exact hon,by rw[e2];exact h8n,by rw[e4];exact h8o⟩ e2 (by rw[e3];exact hl8)
          (by rw[e5,e3];exact C0 hlR hR31) (by rw[e5,e3,e4];exact C0 (ι ow hlR) hRo) (fun k hk=>hWv k (by rw[e4,e2] at hk;exact hk))
          P.1
          (fun o ho s hs=>hEv o (by rw[e4] at ho;exact ho) s hs) F hb M Y x z hix
          _ _ em hcq K.2 (kHB ib L (U2 (U2 (by decide:50 ≤ 174) (V1 174 H)) M) z) hN (fun o ho=>hEm o (by rw[e4] at ho;exact ho)) hem acc (by rw[e5,e3,e4,hA];exact ι ow hlR)
        exact ⟨T1.trans hA,fun q hq=>T2 q (fun r o' h1 h3=>hq r o' h1 (ϟ _) h3),fun r o' h1 _ h3=>T3 r o' h1 h3⟩)
    e4 e5 e3 rfl hI
theorem bX8_spec(R n ow:Ἀ)(relu:Ἆ)(X W b:Ἂ Ἃ)(ix ib L H:Ἀ)(Wd:ἀ)(E:Ἂ Ἃ)(em:ἆ)
  (hXs:R*n<=(O0 X))(hW:ow*n<=(kS Wd))(hRn:R*n<2^(31:Ἀ))(h8n:8*n<2^(31:Ἀ))(hon:ow*n<2^(31:Ἀ))(how:ow<2^(31:Ἀ))(hR31:R<2^(31:Ἀ))(hRo:R*ow<2^(31:Ἀ))(h8o:8*ow<2^(31:Ἀ))
  (hWv:∀k,k<ow*n->Wd.data[k]! =dw W[k]!)
  (hEv:∀o,o<ow->∀s,s<n->(expOf W[o*n+s]!).toNat<=(E[o]!).toNat)
  (F:∀q:Ἀ,kB X[q]!≠255/\kG X[q]!<=ix)(hb:∀q:Ἀ,kB b[q]!≠255/\kG b[q]!<=H)(M:174+H<=ib)(Y:n+2<=2^L)(x:L<=27)(z:ib+L<=425)(hix:ix<=425)(hN:0<n)(hEm:∀o,o<ow->(E[o]!).toNat<=em.toNat)(hem:em.toNat<255) :
  bX8 R n ow relu X W b ix ib L Wd E em hXs hW hRn h8n hon how hR31 hRo h8o=mj R n ow relu X W b:=by
 change gXb X Wd E b relu (48+ix).toUInt64 L.toUInt64 ib.toUInt64 em (bXfb relu X W b E Wd ix ib L n) R n ow hXs hW hRn h8n hon how hR31 hRo h8o 0 (Ah (R*ow) 0)=mj R n ow relu X W b
 obtain ⟨T1,T2⟩:=gXb_spec X Wd E W b relu ix ib L H R n ow em hXs hW hRn h8n hon how hR31 hRo h8o hWv hEv F hb M Y x z hix hN hEm hem (R-0*8) 0 (Ah (R*ow) 0) (le_refl _) (by simp only [O3]) (fun row j h=>absurd h (by clear*-;omega))
 unfold mj linearWith
 apply mi (by rw[T1];simp only [Ϝ])
 intro q hq hq'
 rw[Ϝ]at hq'
 simp only[Λ]
 rw[← AB _ q hq]
 have hpos:0<ow:=ϵ (fun h=>by rw[h,A7]at hq';exact U5 _ hq')
 have hrq:q/ow<R:=(ϸ hpos).2 hq'
 have:=T2 (q/ow) (q%ow) hrq (ϊ _ hpos)
 rw [Au] at this
 exact this
def bQU(R n ow:Ἀ)(relu:Ἆ)(X W b:Ἂ Ἃ)(ix ib L:Ἀ)(hW:ow*n<=(O0 W))(hWs:(O0 W)<2^(32:Ἀ))(v:ἀ×Ἂ ἇ×ἇ)(hv:v=wcT8F (W,ow,n)):Ἂ Ἃ:=
 if hc:v.2.2<255/\(ow+16)*n<2^(31:Ἀ)/\0<n then
  if hq:R*n<=(O0 X)/\R*(n+1)<2^(31:Ἀ)/\R*ow<2^(31:Ἀ) then
   bQS R n ow relu X W b ix ib L v.1 v.2.1 v.2.2.toUInt64 hc.2.2
    (by rw[hv];exact (wcT8F_panel W ow n hc.2.1 hW hWs).1)
    hc.2.1 (by rw[hv];exact (wcT8F_exp W ow n hc.2.1 hW hWs).1.ge)
    hq.1 hq.2.1 hq.2.2
  else mj R n ow relu X W b
 else mj R n ow relu X W b
theorem bQU_v(R n ow:Ἀ)(relu:Ἆ)(X W b:Ἂ Ἃ)(ix ib L:Ἀ)(hW:ow*n<=(O0 W))(hWs:(O0 W)<2^(32:Ἀ))(v:ἀ×Ἂ ἇ×ἇ)(hv:v=wcT8F (W,ow,n)):
  bQU R n ow relu X W b ix ib L hW hWs v hv=bQU R n ow relu X W b ix ib L hW hWs (wcT8F (W,ow,n)) rfl:=by subst hv;rfl
@[noinline]def bTo(R n ow:Ἀ)(relu:Ἆ)(X W b:Ἂ Ἃ)(ix ib L:Ἀ)(hW:ow*n<=(O0 W))(hWs:(O0 W)<2^(32:Ἀ)):Ἂ Ἃ:=
 bQU R n ow relu X W b ix ib L hW hWs (wcGet3T wcK3eq (fun a b=>wcK3eq_ok a b) wcT8S 32 (W,ow,n) (fun k=>k.1) (fun v=>tchP v.1) 3).1 (s5wcGet3_val _)
theorem bTo_eq(R n ow:Ἀ)(relu:Ἆ)(X W b:Ἂ Ἃ)(ix ib L:Ἀ)(hW:ow*n<=(O0 W))(hWs:(O0 W)<2^(32:Ἀ)):
  bTo R n ow relu X W b ix ib L hW hWs=bQU R n ow relu X W b ix ib L hW hWs (wcT8F (W,ow,n)) rfl:=by
 unfold bTo;exact bQU_v _ _ _ _ _ _ _ _ _ _ _ _ _ _
def bTs(R n ow:Ἀ)(relu:Ἆ)(X W b:Ἂ Ἃ)(ix ib L:Ἀ)(hW:ow*n<=(O0 W))(hWs:(O0 W)<2^(32:Ἀ))(s:Ἂ (WcE wcT8F))(i:Nat):Ἂ Ἃ:=
 if h:i<s.size then
  if hk:wcK3eq s[i].k (W,ow,n)=true then
   withPtrAddr s[i].k.1 (fun p1=>withPtrAddr W (fun p2=>if p1=p2 then bQU R n ow relu X W b ix ib L hW hWs s[i].v (s[i].h.trans (congrArg wcT8F (wcK3eq_ok _ _ hk))) else bTo R n ow relu X W b ix ib L hW hWs)
     (fun _ _=>by simp only[bQU_v,bTo_eq,ite_self])) (fun _ _=>by simp only[withPtrAddr,bQU_v,bTo_eq,ite_self])
  else bTs R n ow relu X W b ix ib L hW hWs s (i+1)
 else bTo R n ow relu X W b ix ib L hW hWs
termination_by s.size-i
theorem bTs_eq(R n ow:Ἀ)(relu:Ἆ)(X W b:Ἂ Ἃ)(ix ib L:Ἀ)(hW:ow*n<=(O0 W))(hWs:(O0 W)<2^(32:Ἀ))(s:Ἂ (WcE wcT8F))(i:Nat):
  bTs R n ow relu X W b ix ib L hW hWs s i=bQU R n ow relu X W b ix ib L hW hWs (wcT8F (W,ow,n)) rfl:=by
 rw[bTs];split
 · split
   · simp only[withPtrAddr,bQU_v,bTo_eq,ite_self]
   · exact bTs_eq R n ow relu X W b ix ib L hW hWs s (i+1)
 · exact bTo_eq _ _ _ _ _ _ _ _ _ _ _ _
termination_by s.size-i
theorem bQU_spec(R n ow:Ἀ)(relu:Ἆ)(X W b:Ἂ Ἃ)(ix ib L H:Ἀ)(hW:ow*n<=(O0 W))(hWs:(O0 W)<2^32)
  (F:∀q:Ἀ,kB X[q]!≠255/\kG X[q]!<=ix)(hb:∀q:Ἀ,kB b[q]!≠255/\kG b[q]!<=H)(M:174+H<=ib)(Y:n+2<=2^L)(x:L<=27)(z:ib+L<=425)
  :bQU R n ow relu X W b ix ib L hW hWs (wcT8F (W,ow,n)) rfl=mj R n ow relu X W b:=by
 unfold bQU; split
 · rename_i hc
   have P:=wcT8F_panel W ow n hc.2.1 hW hWs
   have Q:=wcT8F_exp W ow n hc.2.1 hW hWs
   split
   · exact bQS_spec R n ow relu X W b ix ib L H _ _ _ _ _ _ _ _ _ _ P.2 Q.2.1 (fun o ho=>(Q.2.2 o ho).trans_eq (ϩ _).symm) (by rw[ϩ];exact hc.1) F hb M Y x z
   · rfl
 · rfl
def bTc16(R n ow:Ἀ)(relu:Ἆ)(X W b:Ἂ Ἃ)(ix ib L:Ἀ)(hW:ow*n<=(O0 W))(hWs:(O0 W)<2^(32:Ἀ))(dn:Ἆ):Ἂ Ἃ:=
 if dn=false then bTs R n ow relu X W b ix ib L hW hWs (ST.Prim.Ref.get wcT8S (Void.mk ())).val 0 else
 let c:=(wcGet3T wcK3eq (fun a b=>wcK3eq_ok a b) wcT8S 32 (W,ow,n) (fun k=>k.1) (fun v=>tchP v.1) 3).1
 if hc:c.2.2<255/\48+ix+max ((mJ c.2.2)-1) 55+L<=425/\(ow+16)*n<2^(31:Ἀ)/\R*n<2^(31:Ἀ)/\0<n then
  (if hd:dn=true/\R*n<=(O0 X)/\R*ow<2^(31:Ἀ)/\8*ow<2^(31:Ἀ) then bX8 R n ow relu X W b ix ib L (wcGet3T wcK3eq (fun a b=>wcK3eq_ok a b) wcDS 32 (W,ow,n) (fun k=>k.1) (fun v=>tchP v) 3).1 c.2.1 c.2.2.toUInt64
     hd.2.1 (by rw[s5wcGet3_val];exact (wcDF_spec W ow n).1.ge) hc.2.2.2.1 (kBnd ow n R hc.2.2.1 hc.2.2.2.1 hc.2.2.2.2).1 (kBnd ow n R hc.2.2.1 hc.2.2.2.1 hc.2.2.2.2).2.1
     (kBnd ow n R hc.2.2.1 hc.2.2.2.1 hc.2.2.2.2).2.2.1 (kBnd ow n R hc.2.2.1 hc.2.2.2.1 hc.2.2.2.2).2.2.2 hd.2.2.1 hd.2.2.2
   else if hq:R*n<=(O0 X)/\R*(n+1)<2^(31:Ἀ)/\R*ow<2^(31:Ἀ) then
    bQS R n ow relu X W b ix ib L c.1 c.2.1 c.2.2.toUInt64 hc.2.2.2.2
     (by show ((wcGet3T wcK3eq (fun a b=>wcK3eq_ok a b) wcT8S 32 (W,ow,n) (fun k=>k.1) (fun v=>tchP v.1) 3).1).1.size=_;rw[s5wcGet3_val];exact (wcT8F_panel W ow n hc.2.2.1 hW hWs).1)
     hc.2.2.1 (by show ow<=((wcGet3T wcK3eq (fun a b=>wcK3eq_ok a b) wcT8S 32 (W,ow,n) (fun k=>k.1) (fun v=>tchP v.1) 3).1).2.1.size;rw[s5wcGet3_val];exact (wcT8F_exp W ow n hc.2.2.1 hW hWs).1.ge)
     hq.1 hq.2.1 hq.2.2
   else mj R n ow relu X W b)
 else mj R n ow relu X W b
theorem bTc16_spec(R n ow:Ἀ)(relu:Ἆ)(X W b:Ἂ Ἃ)(ix ib L H:Ἀ)(hW:ow*n<=(O0 W))(hWs:(O0 W)<2^32)(dn:Ἆ)
  (F:∀q:Ἀ,kB X[q]!≠255/\kG X[q]!<=ix)(hb:∀q:Ἀ,kB b[q]!≠255/\kG b[q]!<=H)(M:174+H<=ib)(Y:n+2<=2^L)(x:L<=27)(z:ib+L<=425)
  (hXs:(O0 X)<2^32):bTc16 R n ow relu X W b ix ib L hW hWs dn=mj R n ow relu X W b:=by
 unfold bTc16; split
 · rename_i hdn; rw[bTs_eq]; exact bQU_spec R n ow relu X W b ix ib L H hW hWs F hb M Y x z
 · dsimp only; split
   · rename_i hc
     have hv:(wcGet3T wcK3eq (fun a b=>wcK3eq_ok a b) wcT8S 32 (W,ow,n) (fun k=>k.1) (fun v=>tchP v.1) 3).1=wcT8F (W,ow,n):=s5wcGet3_val _
     have P:=wcT8F_panel W ow n hc.2.2.1 hW hWs
     have Q:=wcT8F_exp W ow n hc.2.2.1 hW hWs
     split
     · rename_i hd
       have hd2:(wcGet3T wcK3eq (fun a b=>wcK3eq_ok a b) wcDS 32 (W,ow,n) (fun k=>k.1) (fun v=>tchP v) 3).1=wcDF (W,ow,n):=s5wcGet3_val _
       exact bX8_spec R n ow relu X W b ix ib L H _ _ _ _ _ _ _ _ _ _ _ _ (by rw[hd2];exact (wcDF_spec W ow n).2) (by rw[hv];exact Q.2.1) F hb M Y x z (by have:=hc.2.1;clear*-this;omega) hc.2.2.2.2 (by rw[hv];exact fun o ho=>(Q.2.2 o ho).trans_eq (ϩ _).symm) (by rw[ϩ];exact hc.1)
     · split
       · exact bQS_spec R n ow relu X W b ix ib L H _ _ _ _ _ _ _ _ _ _ (by rw[hv];exact P.2) (by rw[hv];exact Q.2.1) (by rw[hv];exact fun o ho=>(Q.2.2 o ho).trans_eq (ϩ _).symm) (by rw[ϩ];exact hc.1) F hb M Y x z
       · rfl
   · rfl
end F
lemma bc{a:Ἀ}(h:a<=4294967295):a<2^(32:Ἀ):=(by exact Z4 h)
def c7(rows inW outW:Ἀ)(relu:Ἆ)(X W b:Ἂ Ἃ):Ἂ Ἃ:=
 let ex:=F.aO X 0 0
 let eb:=F.aO b 0 0
 let ix:=(mJ ex)-1
 let ib:=174+((mJ eb)-1)
 let L:=Aq (inW+1)+1
 if h:ex<255/\eb<255/\ib+L<=425/\L<=27/\outW*inW<=(O0 W)/\(O0 W)<=4294967295/\
   8*inW<=4294967295/\(O0 X)<=4294967295/\(O0 b)<=4294967295 then
  if 0<rows then F.bTc16 rows inW outW relu X W b ix ib L h.2.2.2.2.1 (bc h.2.2.2.2.2.1) (decide (1<rows/\8192<=inW*outW))
  else linearSpec rows inW outW relu X W b
 else linearSpec rows inW outW relu X W b
theorem kITE {α:Type} {p:Prop} [Decidable p] {a b c:α} (ha:a=c) (hb:b=c):(if p then a else b)=c:=by
  split <;> assumption
lemma eD:c7=mj:=by
 funext e d g j X W b;unfold c7;dsimp only;split
 · rename_i h;have:=ς;have⟨l,m,o,p,y,z,A,B,u⟩:=h;have q:d+2<=2^(Aq (d+1)+1):=by have:=@ϙ (d+1);exact l7 this
   refine kITE ?_ rfl
   exact F.bTc16_spec _ _ _ _ _ _ _ _ _ _ _ _ _ _ (F.bd X (by exact C0 B this) l) (F.bd b (by exact C0 u this) m) AP q p o (by exact bc B)
 · rfl
end Dense
namespace Bn
namespace NativeAdd
lemma en(x:Float32):n5 x.toModel=lI x.toBits:=by exact congrArg Model.mk (pu x.toBits).symm
lemma ja(x:Float32):kW (n5 x.toModel)=x.toBits:=by change lA (mJ x.toBits)=x.toBits;exact UInt32.ofNat_toNat
lemma bK(a:kF)(h:kB a≠255):
  (lx a).toBits=a/\(lx a).isFinite=ἃ:=by
 have j:nX (spec:=li) a.toBitVec≠-1#8:=by
  intro g;apply h;apply Β.mp;have e:=congrArg BitVec.toNat g;rwa[(by rw[kB,Ξ,Υ,BitVec.extractLsb'_toNat];exact ζ _ 8:(mJ (kB a))=(a.toBitVec.extractLsb' 23 8).toNat)]
 have e:(lx a).toBits=a:=by change BG (pT li (kL li a.toBitVec))=a;rw[Model.pack_unpack_eq,AC (λd=>j d.1)]
 use e;change (kL li (lx a).toBits.toBitVec).isFinite=ἃ;rw[e];simp only[kL,j,ἂ];split<;>(try split)<;>rfl
@[noinline]def e9(relu:Ἆ)(a b:Ἃ):Ἃ:=
 act relu (toWord (Model.add (ofW a) (ofW b)))
@[inline]def mR(relu:Ἆ)(w:Ἃ):Ἃ:=w&&&~~~((0:Ἃ)-((if relu then w>>>31 else 0)&&&(if w&&&0x7FFFFFFF<=0x7F800000 then 1 else 0)))
theorem mR_eq(d:Ἆ)(w:Ἃ):mR d w=act d w:=by
 have a0:∀x:Ἃ,x&&&~~~((0:Ἃ)-0)=x:=λ_=>UInt32.and_neg_one
 have a1:∀x:Ἃ,x&&&~~~((0:Ἃ)-1)=0:=λ_=>UInt32.and_zero
 unfold mR act;cases d
 · simp only[υ,↓reduceIte,UInt32.zero_and,a0]
 · simp only[↓reduceIte];unfold reluW
   have q:(w&&&0x7FFFFFFF<=0x7F800000)↔w.toNat%2^(31:Ἀ)<=0x7F800000:=by
    rw[V7,Ξ];exact iff_of_eq (congrArg (·<=_) (ζ _ 31))
   have h31:(w>>>31).toNat=w.toNat/2^(31:Ἀ):=(Υ w 31).trans (β _ _)
   have e1:(w&&&0x80000000).toNat=(w.toNat.testBit 31).toNat*2^(31:Ἀ):=by rw[Ξ];exact Nat.and_two_pow _ 31
   have hE:(kB w).toNat=w.toNat%2^(31:Ἀ)/2^(23:Ἀ):=by
    unfold kB;rw[Ξ,Υ,β];exact(ζ _ 8).trans (Nat.mod_mul_right_div_self _ _ _).symm
   have hF:(lK w).toNat=w.toNat%2^(31:Ἀ)%2^(23:Ἀ):=by
    unfold lK;rw[Ξ];exact(ζ _ 23).trans (Nat.mod_mod_of_dvd _ (Nat.pow_dvd_pow 2 (by decide))).symm
   by_cases g:2^(31:Ἀ)<=w.toNat
   · have s:w.toNat/2^(31:Ἀ)=1:=mG g w.toNat_lt
     rw[Β.1 (show (w>>>31).toNat=(1:Ἃ).toNat from h31.trans s),Β.1 (show (w&&&0x80000000).toNat=(0x80000000:Ἃ).toNat by rw[e1,lW,s]<;>rfl)]
     by_cases j:w.toNat%2^(31:Ἀ)<=0x7F800000
     · rw[if_pos (q.2 j),UInt32.and_self,a1]
       by_cases h:kB w=255
       · rw[A6 h,(Β.1 (hF.trans (by rw[Nat.mod_def _ (2^(23:Ἀ)),<-hE,h];exact Nat.sub_eq_zero_of_le j)):lK w=0)]<;>rfl
       · rw[decide_eq_false h]<;>rfl
     · have Q:w.toNat%2^(31:Ἀ)/2^(23:Ἀ)=255:=mG (Nat.le_of_not_le j) (ϊ _ (϶ 31))
       rw[if_neg (mt q.1 j),UInt32.and_zero,a0,A6 (Β.1 (hE.trans Q):kB w=255),AM.2 (λ(h:lK w=0)=>j (B3 ((ϛ (w.toNat%2^(31:Ἀ)) (2^(23:Ἀ))).symm.trans (by rw[Q,<-hF,h]<;>rfl))))]<;>rfl
   · have s:w.toNat/2^(31:Ἀ)=0:=ώ (Nat.gt_of_not_le g)
     rw[Β.1 (show (w>>>31).toNat=(0:Ἃ).toNat from h31.trans s),Β.1 (show (w&&&0x80000000).toNat=(0:Ἃ).toNat by rw[e1,lW,s]<;>rfl),UInt32.zero_and,a0]<;>rfl
@[inline]def add(relu:Ἆ)(a b:Ἃ):Ἃ:=
 if expOf a !=255&&expOf b !=255 then
  mR relu (BL a+BL b).toBits
 else e9 relu a b
lemma gW(d:Ἆ)(a b:kF):
  add d a b=p8 d (kW (pb (lI a) (lI b))):=by
 rw[add];split;rename_i h;rw[mR_eq];simp only[Η,AM]at h;have g:=bK a h.1;have j:=bK b h.2;have e:=ExecFloat.Binary.toModel_ofFloat32_add_of_isFinite (lx a) (lx b) g.2 j.2;simp only[ExecFloat.Binary.toModel_ofFloat32,en,g.1,j.1]at e;have E:=congrArg kW e;rw[<-en,ja]at E;exact congrArg (act d) E;rfl
end NativeAdd
lemma eC(x s t:kF)(p:kB x≠255)(m:kB s≠255)(o:kB t≠255)
  (V:Ἁ) (l:V=sv x*sv s*2^(kG x+kG s)+sv t*2^(kG t+149)) (j:V≠0):
  kW (pd (lI x) (lI s) (lI t))=md V (-298) ἄ:=by
 have g:=kV x p;have d:=kV s m;have e:=kV t o;rw[Model.Proof.fma_eq_spec,Model.Spec.fma]
 simp only[lI,(Model.chooseNaN3_eq_none_iff _ _ _).2 ⟨Χ g,Χ d,Χ e⟩,ϡ g,ϡ d,ϡ e,Bool.or_false,υ,Am,g,d,e];refine(mf _ V _ _ j ?_).symm;rw[Model.addDyadic,pF];erw[pG];rw[lG,lG,lG,l,show (-298:ℤ)=-149-149 from rfl]
 simp only[Int.cast_add,Int.cast_mul,Int.cast_pow,Ac (AL (α:=ℚ)),AA,zpow_neg,zpow_ofNat];ring
@[noinline]def jq(relu:Ἆ)(w s t:Ἃ):Ἃ:=act relu (toWord (Model.fma (ofW w) (ofW s) (ofW t)))
@[inline]def ad(rb:ἇ)(a b:Ἃ)(_slow:Unit->Ἃ):Ἃ:=NativeAdd.add (rb !=0) a b
@[specialize]def cE(stop:Ἄ)(f:(i:Ἄ)->(J0 i)<(J0 stop)->Ἃ->Ἃ)(i:Ἄ)
  (x:Ἂ Ἃ) (h:(J0 stop)<=(O0 x)):Ἂ Ἃ:=
 if hi:i<stop then
  have hs:=ε.mp hi
  cE stop f (i+1) (x.uset i (f i hs (x.uget i (by exact U1 hi h))) (by exact U1 hi h)) (by rw[BT];exact h)
 else x
termination_by (J0 stop)-(J0 i)
decreasing_by rw[Dense.ac hi];exact Z5 stopб iб hi
lemma cL(d:Ἄ)(f:(i:Ἄ)->(J0 i)<(J0 d)->kF->kF)(F:Ἀ->kF->kF)
  (p:∀i e w,f i e w=F (J0 i) w)(i:Ἄ)(x:Ἂ kF)(h:(J0 d)<=(O0 x)):
  (O0 (cE d f i x h))=(O0 x)/\
  ∀j,(cE d f i x h)[j]?=if (J0 i)<=j/\j<(J0 d)then x[j]?.map (F j)else x[j]?:=by
 fun_induction cE
 next i x h e r q=>have o:(J0 i)<(O0 x):=(ε.1 e).trans_le h;obtain⟨g,l⟩:=q;refine⟨by rw[g,BT],λj=>?_⟩;rw[l,Dense.ac e,pP,Array.getElem?_set o];by_cases m:(J0 i)=j;subst m;rw[AC (by (clear * - h e r;omega)),AD rfl,AD (by (clear * - h e r;omega)),lC o,Option.map_some,p];rfl;rw[AC m];clear o g;split_ifs<;>first|rfl|omega
 next i x h e=>exact⟨ἅ,λj=>by rw[AC (by rw[ε]at e;(clear * - e;omega))]⟩
@[inline]def wX(x:ἆ):ἆ:=((x>>>34)<<<31)+((x&&&0x3FFFFFFFF)-0x255000000)
lemma xw_h1(A:Ἀ)(Q:A<2^(64:Ἀ)):A=A/2^(63:Ἀ)*2^(63:Ἀ)+A/2^(52:Ἀ)%2^(11:Ἀ)*2^(52:Ἀ)+A%2^(52:Ἀ)/\A/2^(63:Ἀ)<2:=by
 refine⟨?_,Ϟ (Q.trans_eq ἅ)⟩;conv_lhs=>rw[←Au A (2^(52:Ἀ)),←Au (A/2^(52:Ἀ)) (2^(11:Ἀ)),lY]
 rw[Ϡ,Ϛ];rfl
lemma xw_h2(s E q r k:Ἀ):(s*2^(63:Ἀ)+E*2^(52:Ἀ)+(q*2^(29:Ἀ)+r)+k)/2^(29:Ἀ)=(r+k)/2^(29:Ἀ)+(s*2^(34:Ἀ)+E*2^(23:Ἀ)+q):=by
 rw[show s*2^(63:Ἀ)+E*2^(52:Ἀ)+(q*2^(29:Ἀ)+r)+k=(r+k)+(s*2^(34:Ἀ)+E*2^(23:Ἀ)+q)*2^(29:Ἀ) by ring,Ρ _ _ (϶ 29)]
lemma xw_h3(r:Ἀ)(c:(r+268435457)/2^(29:Ἀ)=(r+268435453)/2^(29:Ἀ)):r+1≠2^(28:Ἀ)/\r≠2^(28:Ἀ)/\r≠2^(28:Ἀ)+1:=
 ⟨fun h=>by rw[Nat.eq_sub_of_add_eq h]at c;exact absurd c (by decide),fun h=>by rw[h]at c;exact absurd c (by decide),fun h=>by rw[h]at c;exact absurd c (by decide)⟩
lemma xw_h4(A X:Ἀ)(Q:A<2^(64:Ἀ))(M:X=(A+268435457)%2^(64:Ἀ)/2^(29:Ἀ))(K:X%2^(23:Ἀ)≠0):A+268435457<2^(64:Ἀ):=by
 by_contra h;have w:A+268435457-2^(64:Ἀ)<268435457:=G4 (not_lt.1 h) (X3 Q _);apply K;rw[M,P8 (not_lt.1 h),γ (w.trans ((by decide):268435457<2^(64:Ἀ))),ώ (w.trans ((by decide):268435457<2^(29:Ἀ)))];rfl
lemma xw_h5(q r d T X:Ἀ)(p:X=(q+d)+T*2^(23:Ἀ))(K:X%2^(23:Ἀ)≠0)(g:(r+2^(28:Ἀ))/2^(29:Ἀ)=d):1<=q*2^(29:Ἀ)+r:=ϵ fun h=>K (by obtain⟨h1,h2⟩:=Nat.add_eq_zero_iff.1 h;rw[p,←g,h2,(Nat.mul_eq_zero.1 h1).resolve_right (BD.1 (϶ 29)),Μ];rfl)
lemma xw_h6(q r d T X:Ἀ)(p:X=(q+d)+T*2^(23:Ἀ))(K:X%2^(23:Ἀ)≠0)(S:q*2^(29:Ἀ)+r<2^(52:Ἀ))(F:d<=1):q+d<2^(23:Ἀ):=
 lt_of_le_of_ne ((J1 F q).trans (l7 (Nat.lt_of_mul_lt_mul_right ((V1 (q*2^(29:Ἀ)) r).trans_lt S|>.trans_eq (ἅ:2^(52:Ἀ)=2^(23:Ἀ)*2^(29:Ἀ)))))) (fun h=>K (by rw[p,h,Μ,Nat.mod_self]))
lemma xw_h7(E:Ἀ)(J:(2^(64:Ἀ)-1195+E)%2^(64:Ἀ)<253)(D:E<2^(11:Ἀ)):1195<=E/\E<=1447:=by omega
lemma xw_h8(r:Ἀ)(w:r<2^(29:Ἀ))(v1:r+1≠2^(28:Ἀ))(v2:r≠2^(28:Ἀ))(v3:r≠2^(28:Ἀ)+1):(r+268435457)/2^(29:Ἀ)=(r+2^(28:Ἀ))/2^(29:Ἀ)/\2<=(r+2^(28:Ἀ))%2^(29:Ἀ)/\(r+2^(28:Ἀ))%2^(29:Ἀ)+2<=2^(29:Ἀ):=by
 rcases Z7 r (2^(28:Ἀ)) with h|h
 · have h2:r+2<=2^(28:Ἀ):=N4 h v1
   have a:r+2^(28:Ἀ)<2^(29:Ἀ):=(X3 h _).trans_eq ἅ
   have b:r+268435457<2^(29:Ἀ):=(L1 (G5 h2) 268435457).trans_lt (by decide)
   rw[ώ b,ώ a,γ a];exact⟨ἅ,((by decide):2<=2^(28:Ἀ)).trans (K2 _ _),(lO _ _ _).trans_le ((L1 h2 _).trans_eq ἅ)⟩
 · obtain⟨t,rfl⟩:∃t,r=2^(28:Ἀ)+2+t:=Nat.exists_eq_add_of_le (N4 (N4 h (Ne.symm v2)) (Ne.symm v3))
   have tl:t+3<2^(29:Ἀ):=((Τ t 3).trans_le (L1 ((by decide):3<=2^(28:Ἀ)+2) t)).trans_lt w
   rw[show 2^(28:Ἀ)+2+t+268435457=(t+3)+1*2^(29:Ἀ) by ring,show 2^(28:Ἀ)+2+t+2^(28:Ἀ)=(t+2)+1*2^(29:Ἀ) by ring,Ρ _ _ (϶ 29),Ρ _ _ (϶ 29),ώ tl,ώ (B6 tl),Μ,γ (B6 tl)]
   exact⟨ἅ,K2 _ _,tl⟩
lemma xw_h9(q r d:Ἀ)(j:q+d<2^(23:Ἀ))(g:(r+2^(28:Ἀ))/2^(29:Ἀ)=d):q*2^(29:Ἀ)+r+2<=2^(52:Ἀ):=by
 have a:r+2^(28:Ἀ)<d*2^(29:Ἀ)+2^(29:Ἀ):=g▸Nat.lt_div_mul_add (϶ 29)
 have b:q*2^(29:Ἀ)+r+2<=(q+d+1)*2^(29:Ἀ):=by rw[show (q+d+1)*2^(29:Ἀ)=q*2^(29:Ἀ)+(d*2^(29:Ἀ)+2^(29:Ἀ)) by ring,pA];exact J1 (le_of_lt ((J1 ((by decide):2<=2^(28:Ἀ)) r).trans_lt a)) _
 exact b.trans (ι _ j)
lemma xw(A X:Ἀ)(Q:A<2^(64:Ἀ))(M:X=(A+268435457)%2^(64:Ἀ)/2^(29:Ἀ))
  (H:X=(A+268435453)%2^(64:Ἀ)/2^(29:Ἀ))(J:(2^(64:Ἀ)-1195+X/2^(23:Ἀ)%2^(11:Ἀ))%2^(64:Ἀ)<253)
  (K:X%2^(23:Ἀ)≠0):
  1195<=A/2^(52:Ἀ)%2^(11:Ἀ)/\A/2^(52:Ἀ)%2^(11:Ἀ)<=1447/\2<=(A%2^(52:Ἀ)+2^(28:Ἀ))%2^(29:Ἀ)/\
   (A%2^(52:Ἀ)+2^(28:Ἀ))%2^(29:Ἀ)+2<=2^(29:Ἀ)/\1<=A%2^(52:Ἀ)/\A%2^(52:Ἀ)+2<=2^(52:Ἀ)/\
   X/2^(34:Ἀ)=A/2^(63:Ἀ)/\X%2^(34:Ἀ)=A/2^(52:Ἀ)%2^(11:Ἀ)*2^(23:Ἀ)+(A%2^(52:Ἀ)+2^(28:Ἀ))/2^(29:Ἀ):=by
 have u:=xw_h4 A X Q M K;rw[γ u]at M;rw[γ (AO (by simp only [add_le_add_iff_left,Nat.reduceLeDiff]) u:A+268435453<2^(64:Ἀ))]at H
 obtain⟨e,U⟩:=xw_h1 A Q;have D:A/2^(52:Ἀ)%2^(11:Ἀ)<2^(11:Ἀ):=ϊ _ (϶ 11);have S:A%2^(52:Ἀ)<2^(52:Ἀ):=ϊ _ (϶ 52)
 generalize A/2^(63:Ἀ)=s at e U ⊢;generalize A/2^(52:Ἀ)%2^(11:Ἀ)=E at e D ⊢;generalize A%2^(52:Ἀ)=f at e S ⊢;subst e;clear Q
 obtain⟨q,r,rfl,w⟩:∃q r,f=q*2^(29:Ἀ)+r/\r<2^(29:Ἀ):=⟨f/2^(29:Ἀ),f%2^(29:Ἀ),(Au f _).symm,ϊ _ (϶ 29)⟩
 rw[xw_h2]at M H;obtain⟨v1,v2,v3⟩:=xw_h3 r (N8 (M.symm.trans H))
 clear H;obtain⟨h8a,h8b,h8c⟩:=xw_h8 r w v1 v2 v3;rw[h8a]at M;have F:(r+2^(28:Ἀ))/2^(29:Ἀ)<=1:=J2 (Ϟ ((X3 w _).trans (by decide)));generalize g:(r+2^(28:Ἀ))/2^(29:Ἀ)=d at M F;have p:X=(q+d)+(E+s*2^(11:Ἀ))*2^(23:Ἀ):=by rw[M];ring
 have j:=xw_h6 q r d _ X p K S F
 rw[(by rw[p,Ρ _ _ (by norm_num),ώ j,Κ,Μ,γ D]:X/2^(23:Ἀ)%2^(11:Ἀ)=E)]at J;have m:=xw_h7 E J D;rw[(by rw[←g,pA,Τ,Ρ _ _ (϶ 29),Τ]:(q*2^(29:Ἀ)+r+2^(28:Ἀ))/2^(29:Ἀ)=q+d)];have:(q*2^(29:Ἀ)+r+2^(28:Ἀ))%2^(29:Ἀ)=(r+2^(28:Ἀ))%2^(29:Ἀ):=by rw[pA,Τ,Μ]
 refine⟨m.1,m.2,by rw[this];exact h8b,by rw[this];exact h8c,xw_h5 q r d _ X p K g,xw_h9 q r d j g,(Nat.div_mod_unique (϶ 34)).2 ⟨by rw[M];ring,(O1 j _).trans_le ((Ο E _).symm.trans_le (ι _ (l7 D)))⟩⟩
lemma wq_h1(d e Q:Ἀ)(b:d*2^(31:Ἀ)+(e*2^(23:Ἀ)+(2^(23:Ἀ)+Q))<2^(64:Ἀ)):d*2^(31:Ἀ)+(e*2^(23:Ἀ)+(2^(23:Ἀ)+Q))=
   (d*2^(31:Ἀ)%2^(64:Ἀ)+(2^(64:Ἀ)-10015997952+((1195+e)*2^(23:Ἀ)+Q))%2^(64:Ἀ))%2^(64:Ἀ):=by
 rw[show 2^(64:Ἀ)-10015997952+((1195+e)*2^(23:Ἀ)+Q)=e*2^(23:Ἀ)+(2^(23:Ἀ)+Q)+2^(64:Ἀ) by clear*-;omega,Nat.add_mod_right,γ ((K2 _ _).trans_lt b),γ ((V1 _ _).trans_lt b),γ b]
lemma wq_h2(d e Q:Ἀ)(m:d<2)(h:1195+e<=1447)(j:Q<=2^(23:Ἀ)):d*2^(31:Ἀ)+(e*2^(23:Ἀ)+(2^(23:Ἀ)+Q))<2^(64:Ἀ):=by
 have a:d*2^(31:Ἀ)<=1*2^(31:Ἀ):=ι _ (J2 m)
 have h2:e<=252:=Nat.le_of_add_le_add_left (show 1195+e<=1195+252 from h)
 have b:e*2^(23:Ἀ)<=252*2^(23:Ἀ):=ι _ h2
 exact (Y3 a (Y3 b (J1 j (2^(23:Ἀ))))).trans_lt (by decide)
lemma wq(d E Q:Ἀ)(m:d<2)(g:1195<=E)(h:E<=1447)(j:Q<=2^(23:Ἀ)):
  d*2^(31:Ἀ)+((E-1085-110)*2^(23:Ἀ)+(2^(23:Ἀ)+Q))=
   (d*2^(31:Ἀ)%2^(64:Ἀ)+(2^(64:Ἀ)-10015997952+(E*2^(23:Ἀ)+Q))%2^(64:Ἀ))%2^(64:Ἀ):=by obtain⟨e,rfl⟩:∃e,E=1195+e:=⟨E-1195,(BQ g).symm⟩;rw[F7,show 1085+110=1195 from ἅ,ϱ];exact wq_h1 d e Q (wq_h2 d e Q m h j)
lemma cX_h1(x:ἆ)(E f:Ἀ)(g:(lq x)%2^(23:Ἀ)≠0)(y:1195<=E)(z:E<=1447)(I:2<=(f+268435456)%2^(29:Ἀ))(J:(f+268435456)%2^(29:Ἀ)+2<=2^(29:Ἀ))(L:f+2<=2^(52:Ἀ))(A:(lq x)%2^(34:Ἀ)=E*2^(23:Ἀ)+(f+2^(28:Ἀ))/2^(29:Ἀ)):((2^(64:Ἀ)-1195+E)%2^(64:Ἀ)<253/\(2^(64:Ἀ)-2+(f+268435456)%2^(64:Ἀ)%2^(29:Ἀ))%2^(64:Ἀ)<536870909)/\¬(E*2^(52:Ἀ)+f+268435456)%2^(64:Ἀ)/2^(29:Ἀ)%2^(23:Ἀ)=0:=by
 have f5:f<2^(52:Ἀ):=(K6 (I3 f)).trans_le L;have fl:f+268435456<2^(64:Ἀ):=(X3 f5 _).trans (by decide)
 have el:E*2^(52:Ἀ)+f+268435456<2^(64:Ἀ):=(X3 (N7 (ι (2^(52:Ἀ)) z) f5) 268435456).trans (by decide)
 have hQ:(f+2^(28:Ἀ))/2^(29:Ἀ)%2^(23:Ἀ)≠0:=by rw[←Μ ((f+2^(28:Ἀ))/2^(29:Ἀ)) E (2^(23:Ἀ)),←(A.trans (Τ _ _)),Nat.mod_mod_of_dvd _ (pow_dvd_pow 2 (by decide:23<=34))];exact g
 refine⟨⟨by rw[←R7 ((by decide):1195<=2^(64:Ἀ)),Τ (2^(64:Ἀ)),R7 y,Nat.add_mod_right,γ ((P4 z 1195).trans_lt (by decide):E-1195<2^(64:Ἀ))];exact (P4 z 1195).trans_lt (by decide),by rw[γ fl];generalize (f+268435456)%2^(29:Ἀ)=R at I J⊢;rw[←R7 ((by decide):2<=2^(64:Ἀ)),Τ (2^(64:Ἀ)),R7 I,Nat.add_mod_right,γ ((K4 _ _).trans_lt ((V1 _ _).trans_lt (J.trans_lt ((by decide):2^(29:Ἀ)<2^(64:Ἀ)))))];clear*-J;omega⟩,?_⟩
 rw[γ el,show E*2^(52:Ἀ)+f+268435456=(f+268435456)+E*2^(23:Ἀ)*2^(29:Ἀ) by ring,Ρ _ _ (϶ 29),Μ];exact hQ
lemma cX_h2(S X:Ἀ)(r:S<2)(hX:X<255*2^(23:Ἀ)):(S*2^(31:Ἀ)+X)%2^(32:Ἀ)%2^(31:Ἀ)<255*2^(23:Ἀ)/\S*2^(31:Ἀ)+X<4294967296:=by
 have b:S*2^(31:Ἀ)+X<2^(32:Ἀ):=(N7 (ι (2^(31:Ἀ)) (J2 r)) (hX.trans ((by decide):255*2^(23:Ἀ)<2^(31:Ἀ)))).trans_le ((by decide):1*2^(31:Ἀ)+2^(31:Ἀ)<=2^(32:Ἀ))
 exact⟨by rw[γ b,Τ,Μ,γ (hX.trans ((by decide):255*2^(23:Ἀ)<2^(31:Ἀ)))];exact hX,b⟩
lemma cX(S:Ἅ)
  (h:((S.toBits+268435457)>>>29==(S.toBits+268435453)>>>29&&
   (((S.toBits+268435457)>>>29>>>23)&&&2047)-1195<253&&
   (S.toBits+268435457)>>>29&&&8388607!=0)=true):
  Dense.F.ak S (Dense.F.eF S.toBits-1075)=wX ((S.toBits+268435457)>>>29)/\
   1195<=(lq (Dense.F.eF S.toBits))/\(lq (Dense.F.eF S.toBits))<=1447/\
   (mJ (N3 (Dense.F.ak S (Dense.F.eF S.toBits-1075))))%2^(31:Ἀ)<255*2^(23:Ἀ)/\
   Dense.F.ak S (Dense.F.eF S.toBits-1075)<4294967296:=by
 unfold Dense.F.ak;simp only[UInt64.sub_add_cancel,UInt64.sub_self,((by decide):((1:ἆ)<<<(0:ἆ))=1),((by decide):decide ((0:ἆ)<=26)=true),Bool.true_and]
 generalize S.toBits=a at h⊢;generalize H:(a+268435457)>>>29=x at h⊢;have j(x:Ἀ):x&&&2047=x%2^(11:Ἀ):=ζ x 11;have l(x:Ἀ):x&&&8388607=x%2^(23:Ἀ):=ζ x 23;have t(x:Ἀ):x&&&536870911=x%2^(29:Ἀ):=ζ x 29;have u(x:Ἀ):x&&&17179869183=x%2^(34:Ἀ):=ζ x 34;have e:(lq x)=((lq a)+268435457)%2^(64:Ἀ)/2^(29:Ἀ):=by rw[<-H];simp only[ν,η,kA,kH,β]
 simp only[Η,AQ,AJ,AM,Ak]at h;obtain⟨⟨D,o⟩,E⟩:=h;have d:=congrArg lq D;rw[e]at d;simp only[ν,η,kA,kH,β]at d;rw[μ,Ζ]at o;simp only[Ε,ν,kA,kH,β,j]at o;have g:(lq x)%2^(23:Ἀ)≠0:=λB=>E (ξ.1 (by rw[Ε];simpa only[kA,l]))
 obtain⟨y,z,I,J,K,L,M,A⟩:=xw _ _ a.toNat_lt e (e.trans d) o g;have p:(lq (Dense.F.eF a))=(lq a)/2^(52:Ἀ)%2^(11:Ἀ):=by simp only[Dense.F.eF,Ε,ν,kA,kH,β,j]
 have w:(lq (a&&&0xfffffffffffff))=(lq a)%2^(52:Ἀ):=Ε _ _▸ζ _ 52
 rw[AD];have m:=Dense.F.g0 a (p▸y);rw[p,w]at m;have r:(lq a)/2^(63:Ἀ)<2:=Ϟ (a.toNat_lt.trans_eq ἅ);have q:((lq a)%2^(52:Ἀ)+2^(28:Ἀ))/2^(29:Ἀ)<=2^(23:Ἀ):=J2 ((ϸ (϶ 29)).2 ((X3 (ϊ _ (϶ 52)) _).trans (by decide)));have v:((lq a)/2^(52:Ἀ)%2^(11:Ἀ)-1085-110)*2^(23:Ἀ)<=252*2^(23:Ἀ):=ι _ (P4 (P4 z 1085) 110);have X2:=cX_h2 _ _ r (C0 (Y3 v (J1 q _)) ((by decide):252*2^(23:Ἀ)+(2^(23:Ἀ)+2^(23:Ἀ))<255*2^(23:Ἀ)));refine⟨?_,p▸y,p▸z,?_,?_⟩
 .apply ξ.1;rw[m];simp only[wX,η,ψ,ν,Ζ,Ε,kA,kH,β,ο,u,M,A];exact wq _ _ _ r y z q
 .rw[Ϫ,m];exact X2.1
 .rw[μ,m];simp only[kA];exact X2.2
 simp only[Η,AJ,UInt64.reduceAdd,UInt64.reduceSub,AM,Ak,<-ξ];simp only[μ,Ζ,Ε,η,ν,p,w,kA,kH,β,l,t];rw[ζ _ 63,(by rw[←Au ((lq a)%2^(63:Ἀ)) (2^(52:Ἀ)),show (2:Ἀ)^(63:Ἀ)=2^(52:Ἀ)*2^(11:Ἀ) from ἅ,Nat.mod_mul_right_div_self,Nat.mod_mul_right_mod]:(lq a)%2^(63:Ἀ)=(lq a)/2^(52:Ἀ)%2^(11:Ἀ)*2^(52:Ἀ)+(lq a)%2^(52:Ἀ))];clear*-y z I J K L A g;generalize (lq a)/2^(52:Ἀ)%2^(11:Ἀ)=E at *;generalize (lq a)%2^(52:Ἀ)=f at *;exact cX_h1 _ _ _ g y z I J L A
lemma jf(d:Ἆ)(o:ἇ)(j:(mJ o)%2^(31:Ἀ)<255*2^(23:Ἀ)):
  (if o&&&(if d then 0x80000000 else 0)!=0 then 0 else o)=p8 d o:=by
 have g:kB o≠255:=λh=>by have e:=congrArg mJ h;simp only[kB,Ξ,Υ,kN,kH,β]at e;rw[show(255:Ἀ)=2^(8:Ἀ)-1 from ἅ,ζ]at e;(clear * - j e;omega)
 cases d<;>simp[p8,reluW,g]
@[inline]def dwQ(Q:ἀ)(hQ:Q=Dense.F.P)(w:ἇ):Ἅ:=
 (Core.mW w).toInt64.toFloat*Q.uget (UInt32.toUSize (w>>>23)) (by
  rw[hQ,Dense.F.gL,Dense.aC,Dense.F.dg];have:=w.toNat_lt;exact Ϟ this)
lemma dwQ_e(w:ἇ):dwQ Dense.F.P rfl w=Dense.F.dw w:=rfl
lemma rB(a:ἆ)(h:3<(a+268435457)&&&536870911):(a+268435457)>>>29=(a+268435453)>>>29:=by
 have t(x:Ἀ):x&&&536870911=x%2^(29:Ἀ):=ζ x 29
 rw[μ,Ε,η]at h;simp only[kA,t]at h;apply ξ.1;simp only[ν,η,kA,kH,β];have:=a.toNat_lt;omega
lemma rBb(a:ἆ)(B C:Ἆ)(h:((3<(a+268435457)&&&536870911&&B)&&C)=true):
  (((a+268435457)>>>29==(a+268435453)>>>29&&B)&&C)=true:=by
 simp only[Η,AJ]at h ⊢;exact⟨⟨by rw[rB a h.1.1];exact Az _,h.1.2⟩,h.2⟩
@[inline]def eBf(Q:ἀ)(hQ:Q=Dense.F.P)(m:ἇ)(Y Z:Ἅ)(w:Ἃ)(slow:Unit->Ἃ):Ἃ:=
  let a:=(dwQ Q hQ w*Y+Z).toBits
  let u:=a+268435457
  let x:=u>>>29
  if 3<u&&&536870911&&((x>>>23)&&&2047)-1195<253&&x&&&8388607!=0 then
   let o:=(N3 (wX x))
   if o&&&m !=0 then 0 else o
  else slow ()
@[inline]def eB(Q:ἀ)(hQ:Q=Dense.F.P)(g:Ἆ)(m:ἇ)(Y Z:Ἅ)(w:Ἃ)(slow:Unit->Ἃ):Ἃ:=
 if g&&((w>>>23)&&&255)-1<254 then eBf Q hQ m Y Z w slow
 else if g&&(w==0||w==0x80000000) then eBf Q hQ m Y Z w slow
 else slow ()
lemma i4_h1(E:Ἀ)(p:1195<=E)(q:E<=1447):-1074<=(E:Ἁ)-1022/\(E:Ἁ)-1022<=971/\-1021<=(E:Ἁ)-1022:=⟨le_sub_iff_add_le.2 (((by decide):(-1074:Ἁ)+1022<=0).trans (M3 _)),sub_le_iff_le_add.2 (R5.2 (q.trans ((by decide):1447<=1993))),le_sub_iff_add_le.2 (((by decide):(-1021:Ἁ)+1022<=1195).trans (R5.2 p))⟩
lemma cE2_a{i stop:Ἄ}(hi:(J0 i)<=(J0 stop))(hj:1<stop-i):(J0 i)+1<(J0 stop):=by
 rw[ε,USize.toNat_sub_of_le _ _ (U8.2 hi)]at hj;simp only[kD]at hj;omega
lemma ug(x:Ἂ Ἃ)(i j:Ἄ)(v:Ἃ)(a:(J0 i)<(O0 x))(b:(J0 j)<(O0 (x.uset i v a)))(n:(J0 i)≠(J0 j)):(x.uset i v a).uget j b=x.uget j (by rw[BT]at b;exact b):=by
 simp only[Array.uget,pP,Array.getElem_set,n,↓reduceIte]
def bnL(Q:ἀ)(hQ:Q=Dense.F.P)(g:Ἆ)(m:ἇ)(Y Z:Ἅ)(relu:Ἆ)(sc tc:Ἃ)(stop i:Ἄ)
  (x:Ἂ Ἃ)(h:(J0 stop)<=(O0 x))(hi:(J0 i)<=(J0 stop)):Ἂ Ἃ:=
 if hj:1<stop-i then
  have a:=cE2_a hi hj
  have b:i<stop:=ε.2 (by omega)
  have c:=Dense.ac b
  have d:i+1<stop:=ε.2 (by omega)
  have e:=Dense.ac d
  let w0:=x.uget i (by omega)
  let w1:=x.uget (i+1) (by omega)
  let y0:=eB Q hQ g m Y Z w0 λ_=>jq relu w0 sc tc
  let y1:=eB Q hQ g m Y Z w1 λ_=>jq relu w1 sc tc
  bnL Q hQ g m Y Z relu sc tc stop (i+1+1) ((x.uset i y0 (by omega)).uset (i+1) y1 (by rw[BT];omega)) (by rw[BT,BT];exact h) (by omega)
 else if hk:i<stop then
  let w0:=x.uget i (by have:=ε.1 hk;omega)
  x.uset i (eB Q hQ g m Y Z w0 λ_=>jq relu w0 sc tc) (by have:=ε.1 hk;omega)
 else x
termination_by (J0 stop)-(J0 i)
decreasing_by omega
theorem bnL_eq(Q:ἀ)(hQ:Q=Dense.F.P)(g:Ἆ)(m:ἇ)(Y Z:Ἅ)(relu:Ἆ)(sc tc:Ἃ)(stop i:Ἄ)
  (x:Ἂ Ἃ)(h:(J0 stop)<=(O0 x))(hi:(J0 i)<=(J0 stop)):
  bnL Q hQ g m Y Z relu sc tc stop i x h hi=cE stop (λ_ _ w=>eB Q hQ g m Y Z w λ_=>jq relu w sc tc) i x h:=by
 fun_induction bnL Q hQ g m Y Z relu sc tc stop i x h hi
 next i x h hi hj a b c d e w0 w1 y0 y1 ih=>
  have n:(J0 i)≠(J0 (i+1)):=by omega
  rw[ih];conv_rhs=>rw[cE,dif_pos b];dsimp only;rw[cE,dif_pos d];dsimp only;simp only[ug x i (i+1) _ _ _ n]
 next i x h hi hj hk w0=>
  have g:¬i+1<stop:=by rw[ε,Dense.ac hk];have:=ε.1 hk;rw[ε,USize.toNat_sub_of_le _ _ (U8.2 hi)]at hj;simp only[kD]at hj;omega
  conv_rhs=>rw[cE,dif_pos hk];dsimp only;rw[cE,dif_neg g]
 next i x h hi hj hk=>rw[cE,dif_neg hk]
lemma i4f(d:Ἆ)(s t w:kF)(g:Unit->kF)
  (l:g ()=p8 d (kW (pd (lI w) (lI s) (lI t))))(G:((s>>>23)&&&255!=255&&(t>>>23)&&&255!=255&&(w>>>23)&&&255!=255)=true):
  eBf Dense.F.P rfl (if d then 0x80000000 else 0) (Dense.F.dw s)
   (Dense.F.dw t*Dense.F.p149) w g=p8 d (kW (pd (lI w) (lI s) (lI t))):=by open Dense Dense.F in
 rw[Bn.eBf,dwQ_e,l,F.p149];dsimp only;obtain⟨G,m⟩:=I5.1 G;obtain⟨x,y⟩:=I5.1 G;replace x:¬kB s=255:=AM.1 x;replace y:¬kB t=255:=AM.1 y;replace m:¬kB w=255:=λh=>by change (w>>>23)&&&255=255 at h;rw[h]at m;exact absurd m (by decide)
 refine ite_eq_right_iff.2 fun L=>?_;obtain⟨v,p,q,D,E⟩:=cX _ (rBb _ _ _ L);rw[<-v];have it:=aT t;have W:=bm t;obtain⟨A,M⟩:=dJ w s m x;obtain⟨H,N⟩:=cI t y;obtain⟨I,O⟩:=iJ 149 (by decide);obtain⟨B,Q⟩:=aE _ _ H I (sv t) ((kG t+149:Ἀ):Ἁ) (by rw[N,O,AA,AF,Ap]) (Ao (bg t) (by decide)) (((by decide):(-1074:Ἁ)<=0).trans (Int.ofNat_nonneg _)) (bL 24 _ 971 (by rw[it];exact (J1 (L1 ((K4 _ _).trans W.le) 149) 24).trans (by decide)) AP _ (bg t));have o:=jp (dw w*dw s+dw t*F.pw 149) (q.trans_lt (by decide));obtain⟨i1,i2,i3⟩:=i4_h1 _ p q;have j:=jz _ _ A B o _ i1 i2 (fK _ o);obtain⟨-,e⟩:=e8 _ _ A B _ i3 i2 j;have J:rv (dw w*dw s)+rv (dw t*F.pw 149)=((tm w s+sv t*2^(kG t+149):Ἁ):ℝ):=by rw[M,Q,AA,Int.cast_add,Int.cast_mul (sv t),Int.cast_pow,Int.cast_ofNat]
 rw[J]at j e;generalize u:(tm w s+sv t*2^(kG t+149):Ἁ)=V at j e;generalize dw w*dw s+dw t*F.pw 149=S at E p q D o j e⊢;have K0:=χ (F.eF S.toBits) 1075 (θ.2 (((by decide):1075<=1195).trans p));have K:((lq (F.eF S.toBits-1075)):Ἁ)=((lq (F.eF S.toBits)):Ἁ)-1075:=by rw[K0];exact pO (((by decide):1075<=1195).trans p)
 obtain⟨r,z⟩:=ce S o (F.eF S.toBits-1075) (K0.trans_lt ((K4 _ _).trans_lt (q.trans_lt (by decide)))) V (by have h1:=half_le_self (zpow_nonneg (zero_le_two (α:=ℝ)) (((lq (F.eF S.toBits)):Ἁ)-1022-53));have h2:=AG (one_le_two (α:=ℝ)) (by rw[K,sub_sub];rfl:((lq (F.eF S.toBits)):Ἁ)-1022-53=((lq (F.eF S.toBits-1075)):Ἁ)).le;exact e.trans (h1.trans h2)) (j.le.trans (AG one_le_two (Int.sub_le_sub_right (U4.2 q) 1022))) E
 rw[eC w s t m x y V u.symm r,<-z];exact jf d _ D
lemma e255(e:ἇ)(h:e-1<254):e≠255:=by intro he;rw[he]at h;exact absurd h (by decide)
lemma i4(d:Ἆ)(s t w:kF)(g:Unit->kF)
  (l:g ()=p8 d (kW (pd (lI w) (lI s) (lI t)))):
  eB Dense.F.P rfl ((s>>>23)&&&255!=255&&(t>>>23)&&&255!=255) (if d then 0x80000000 else 0) (Dense.F.dw s)
   (Dense.F.dw t*Dense.F.p149) w g=p8 d (kW (pd (lI w) (lI s) (lI t))):=by
 unfold eB;split
 ·rename_i G;rw[Η]at G
  exact i4f d s t w g l (I5.2 ⟨G.1,AM.2 (e255 _ (by simpa using G.2))⟩)
 ·split
  ·rename_i _ G;rw[Η]at G
   have h0:w=0∨w=0x80000000:=by simpa using G.2
   rcases h0 with rfl|rfl<;>exact i4f d s t _ g l (I5.2 ⟨G.1,by decide⟩)
  ·exact l
def bH(relu:Ἆ)(hw c:Ἀ)(s t:Ἂ Ἃ)(Q:ἀ)(hQ:Q=Dense.F.P)(ch:Ἀ)(x:Ἂ Ἃ):Ἂ Ἃ:=
 if ch<c then
  let sc:=s[ch]!
  let tc:=t[ch]!
  let i0:=(G0 (ch*hw))
  let stop:=(G0 (ch*hw+hw))
  let m:ἇ:=if relu then 0x80000000 else 0
  let g:=(sc>>>23)&&&255!=255&&(tc>>>23)&&&255!=255
  let Y:=Dense.F.dw sc
  let Z:=Dense.F.dw tc*Dense.F.p149
  bH relu hw c s t Q hQ (ch+1) (if h:(J0 stop)<=(O0 x)/\i0<=stop then
   bnL Q hQ g m Y Z relu sc tc stop i0 x h.1 (U8.1 h.2)
   else x)
 else x
termination_by c-ch
decreasing_by all_goals omega
lemma dn_h1{α:Type}(a b j:Ἀ)(o:Option α)(f h:α->α)(q:a<=j->j<b->f=h)(r:a<=b):(if b<=j then (if a<=j/\j<b then o.map f else o).map h else (if a<=j/\j<b then o.map f else o))=if a<=j then o.map h else o:=
 if u:b<=j then(if_pos u).trans ((congrArg _ (if_neg λv=>W3.2 u v.2)).trans (if_pos (r.trans u)).symm)
 else if v:a<=j then(if_neg u).trans ((if_pos ⟨v,O8 u⟩).trans ((congrArg (Option.map · o) (q v (O8 u))).trans (if_pos v).symm))
 else(if_neg u).trans ((if_neg λw=>v w.1).trans (if_neg v).symm)
lemma dn(g:Ἆ)(e c:Ἀ)(s t:Ἂ kF)(d:Ἀ)(x:Ἂ kF)
  (p:(O0 x)=c*e)(r:c*e<kX)(j:Ἀ):
  (bH g e c s t Dense.F.P rfl d x)[j]?=if d*e<=j then
   x[j]?.map (λw=>p8 g (kW (pd (lI w) (lI s[j/e]!) (lI t[j/e]!))))
   else x[j]?:=by
 induction hk:c-d generalizing d x
 unfold bH;rw[AC (not_lt.2 (O4 hk))];split
 .rw[l5 (p.trans_le ((ι _ (O4 hk)).trans ‹_›))];rfl
 rfl;rename_i v;have q:d<c:=N1 hk
 have u:d*e+e<=c*e:=(nO d e).symm.trans_le (ι _ q)
 have h:(J0 (G0 (d*e+e)))=d*e+e:=Dense.aa (u.trans_lt r);have l:(J0 (G0 (d*e)))=d*e:=Dense.aa (((V1 _ _).trans u).trans_lt r);unfold bH;rw[AD q];dsimp only;have z:=h.trans_le (u.trans_eq p.symm);rw[AT ⟨z,U8.2 (by rw[l,h];exact V1 _ _)⟩,bnL_eq];obtain⟨m,o⟩:=cL _ _ (λ_ w=>p8 g (kW (pd (lI w) (lI s[d]!) (lI t[d]!)))) (λ_ _ w=>i4 g s[d]! t[d]! w (λ_=>jq g w s[d]! t[d]!) rfl) (G0 (d*e)) x z;rw[v (d+1) _ (m.trans p) ((Nat.sub_succ c d).trans (congrArg Nat.pred hk)),o,l,h,nO d e];exact dn_h1 _ _ _ _ _ _ (λb c=>by rw[mG b (by rw[Ο];exact c)]) (V1 _ _)
def br(relu:Ἆ)(c hw:Ἀ)(x s t:Ἂ Ἃ):Ἂ Ἃ:=
 if (O0 x)=c*hw/\c*hw<Al then bH relu hw c s t Dense.F.P rfl 0 x
 else scaleShiftSpec relu c hw x s t
lemma cO:br=scaleShiftSpec:=by
 funext d c e x s t;rw[br];split
 .rename_i h
  apply mU;intro j;rw[dn d e c s t 0 x h.1 h.2 j,AD ((by simp only [zero_mul,zero_le])),scaleShiftSpec,mE];split
  .rw[lC (by omega),AB x j (by omega)];rfl
  .rw[l5 (by omega)];rfl
 .rfl
lemma iY(d:Ἆ)(a b:kF)(e:Unit->kF)
  (g:e ()=p8 d (kW (pb (lI a) (lI b)))):
  ad (if d then 1 else 0) a b e=p8 d (kW (pb (lI a) (lI b))):=by cases d<;>simp[ad,NativeAdd.gW]
@[noinline]def jN(relu:Ἆ)(x y:Ἃ):Ἃ:=act relu (toWord (Model.add (ofW x) (ofW y)))
def cq(relu:Ἆ)(a b:Ἂ Ἃ):Ἂ Ἃ:=
 if h:(O0 a)<=(O0 b)/\(O0 a)<Al then
  let rb:ἇ:=if relu then 1 else 0
  cE (G0 (O0 a)) (λi hi x=>
   let y:=b.uget i (by rw[Dense.aa h.2]at hi;omega)
   ad rb x y λ_=>jN relu x y) 0 a (by rw[Dense.aa h.2])
 else addReluSpec relu a b
lemma dT:cq=addReluSpec:=by
 funext d a b;rw[cq];dsimp only;split;rename_i h;have g:(J0 (G0 (O0 a)))=(O0 a):=Dense.aa h.2;apply mU;intro j;rw[(cL _ _ (λj x=>p8 d (kW (pb (lI x) (lI b[j]!)))) (λi l x=>?_) 0 a _).2 j,addReluSpec,mE];by_cases e:j<(O0 a);rw[AD ⟨(by simp only [Ϧ,zero_le]),by rwa[g]⟩,AT e,lC e];rfl;rw[AC (by rw[g];exact not_and.mpr fun a=>e),dif_neg e,l5 (by exact W4 e)];rw[iY d _ _ _ rfl,AB b (J0 i) (by omega)];rfl;rfl
end Bn
namespace Conv
open Core Finset
open Dense.F (jV uL kr ZPq zPq uU fuS uZ zPs wdU wdUz wdUs)
variable(C E V S:ἆ) (y z:kF) (t:Ἆ) (P W:Ἂ ἇ) (c h w k n:Ἀ) (Q:kF->Ἆ)
 (fa:Ἀ->kF) (bx:ἆ) (s:lR) (J:Ἆ) (I K b:Ἂ kF) (D:Ἆ) (nc Y:Ἀ)
 (r d:kF) (x:ἆ) (v:Ἁ) (B a:ἆ) (m:Ἀ) (e j:Ἄ) (bw:ἆ) (u:ἇ) {α:Type}
lemma mo{a b:Ἀ}(h:a<b):a%b=a:=γ h
lemma mr{a b:Ἀ}(h:a∈kP b):a<b:=ϥ.mp h
lemma fT:(ἄ=ἃ)=False:=υ
local instance:Inhabited Dense.F.A8:=⟨⟨0,0,0,0,0,0,0,0⟩⟩
lemma dj:2^(32:Ἀ)<=2^nx:=by rcases System.Platform.numBits_eq with h|h<;>rw[h]<;>norm_num
lemma j9{n:Ἀ}(h:n<2^(32:Ἀ)):n%2^nx=n:=mo (AN h dj)
@[inline]def ok(w:ἇ):Ἆ:=
 expOf w !=255&&(mW w==0||iW w-E<=C)
@[noinline]def eG(a:Ἂ ἇ):ἆ:=
 a.foldl (λm w=>if mW w !=0&&iW w< m then iW w else m) 255
@[inline]def ct(h l:ἆ):ἆ:=(h<<<25)+l
@[inline]def d4(h l:ἆ):ἆ:=
 ((h+0x8000000000000000)>>>(39:ἆ))-(16777216:ἆ)-(l>>>(63:ἆ))+
  (if ct h l<l then(1:ἆ)else 0)
@[inline]def sg(m:ἆ):ἆ:=if u>>>31!=0 then 0-m else m
@[specialize]def lp(H:Ἀ)(f:Ἀ->Ἀ->Ἂ ἇ->Ἂ ἇ):
  Ἀ->Ἀ->Ἀ->Ἂ ἇ->Ἂ ἇ
 | 0,_,_,a=>a
 | n+1,o,y,a=>
  let a:=f o y a
  if y+1<H then lp H f n o (y+1) a else lp H f n (o+1) 0 a
@[specialize]def dc(f:Ἀ->Ἄ->ἇ)(s:Ἄ)(x:Ἀ):Ἀ->Ἄ->Ἂ ἇ->Ἂ ἇ
 | 0,_,a=>a
 | m+1,e,a=>dc f s (x+1) m (e+s) (a.push (f x e))
@[noinline]def eT(O H W:Ἀ)(f:Ἀ->Ἀ->Ἀ->ἇ):Ἂ ἇ:=
 if 0<H then lp H (λo y=>dc (λx _=>f o y x) 0 0 W 0) (O*H) 0 0 (AS (O*(H*W)))
 else#[]
def zpA(kk k cb o y x:Ἀ):Ἀ→Ἆ
|0=>true
|i+1=>((windowAt s I y x (cb*kk+i)==0||windowAt s I y x (cb*kk+i)==0x80000000)&&kB K[o*k+i]! != 255)&&zpA kk k cb o y x i
theorem zpA_ok(kk k cb o y x:Ἀ):∀n,zpA s I K kk k cb o y x n=true→∀t<n,(windowAt s I y x (cb*kk+t)=0∨windowAt s I y x (cb*kk+t)=0x80000000)∧kB K[o*k+t]!≠255
|0,_,t,ht=>absurd ht (U5 _)
|n+1,h,t,ht=>by
 simp only[zpA,Η,nM,AQ,AM,Ak]at h
 by_cases e:t=n
 ·subst e;exact⟨h.1.1,h.1.2⟩
 ·exact zpA_ok kk k cb o y x n h.2 t (by omega)
@[noinline]def aB(kk k cb o y x:Ἀ):Ἃ:=
 if b[o]! == 0&&zpA s I K kk k cb o y x k then 0 else
 Dense.gf (k+1) (λi=>if i<k then windowAt s I y x (cb*kk+i)else oneW)
  (λi=>if i<k then K[o*k+i]!else b[o]!)
theorem aB_eq(kk k cb o y x:Ἀ):aB s I K b kk k cb o y x=Dense.gf (k+1) (λi=>if i<k then windowAt s I y x (cb*kk+i)else oneW)
  (λi=>if i<k then K[o*k+i]!else b[o]!):=by
 unfold aB;split
 ·rename_i hc
  simp only[Η,AQ]at hc
  obtain⟨hb,hz⟩:=hc
  have W:=zpA_ok s I K kk k cb o y x k hz
  rw[Dense.bO]
  exact (Dense.F.zrDot k _ _ (λt ht=>by simp only[if_pos ht];exact (W t ht).1) (λt ht=>by simp only[if_pos ht];exact (W t ht).2) (by simp) (by simp[hb])).symm
 ·rfl
def zpW(kk k cb o y x:Ἀ):Ἀ→Ἆ
|0=>true
|i+1=>(K[o*k+i]! == 0||K[o*k+i]! == 0x80000000||windowAt s I y x (cb*kk+i)==0||windowAt s I y x (cb*kk+i)==0x80000000)&&zpW kk k cb o y x i
theorem zpW_ok(kk k cb o y x:Ἀ):∀n,zpW s I K kk k cb o y x n=true→∀t<n,(K[o*k+t]! =0∨K[o*k+t]! =0x80000000)∨(windowAt s I y x (cb*kk+t)=0∨windowAt s I y x (cb*kk+t)=0x80000000)
|0,_,t,ht=>absurd ht (U5 _)
|n+1,h,t,ht=>by
 simp only[zpW,Η,nM,AQ]at h
 by_cases e:t=n
 ·subst e;have h1:=h.1;tauto
 ·exact zpW_ok kk k cb o y x n h.2 t (by omega)
theorem zrDot2(n:ℕ)(D Wf:ℕ→Ἃ)(hz:∀s<n,(Wf s=0∨Wf s=0x80000000)∨(D s=0∨D s=0x80000000))(hD:∀s<n,kB (D s)≠255)(hW:∀s<n,kB (Wf s)≠255)(hDn:D n=ld)(hWn:Wf n=0):ms (n+1) D Wf=0:=by
 have g:aG (n+1) D Wf:=by
  intro k hk;by_cases e:k<n
  ·exact⟨hD k e,hW k e⟩
  ·rw[show k=n by omega,hDn,hWn];exact⟨by decide,by decide⟩
 have hS:af (n+1) D Wf=0:=by
  unfold af;apply Finset.sum_eq_zero;intro k hk;have:=ϥ.mp hk;by_cases e:k<n
  ·rcases hz k e with (h|h)|(h|h)<;>simp only[tm,h,show sv (0:Ἃ)=0 by decide,show sv (0x80000000:Ἃ)=0 by decide,mul_zero,zero_mul]
  ·rw[show k=n by omega,hWn];simp only[tm,show sv (0:Ἃ)=0 by decide,mul_zero,zero_mul]
 rw[Core.bI (n+1) D Wf g,hS,Dense.F.zrNC _ _ n 0 (by rw[Κ,ab (n+1) D n (by omega),ab (n+1) Wf n (by omega),hDn,hWn];decide),Bool.and_false];decide
theorem aBz(kk k cb o y x:Ἀ)(hb:b[o]! =0)(hz:zpW s I K kk k cb o y x k=true)(hi:∀q:ℕ,kB I[q]!≠255)(hw:∀q:ℕ,kB K[q]!≠255):aB s I K b kk k cb o y x=0:=by
 have W:=zpW_ok s I K kk k cb o y x k hz
 rw[aB_eq,Dense.bO]
 refine zrDot2 k _ _ (λt ht=>by simp only[if_pos ht];exact W t ht) (λt ht=>by simp only[if_pos ht];unfold windowAt;dsimp only;split;exact (by decide);exact hi _) (λt ht=>by simp only[if_pos ht];exact hw _) (by simp) (by simp[hb])
@[inline]def wA1(I:Ἂ Ἃ)(h w y x t:Ἀ):Ἃ:=if y ≥ h∨x ≥ w then 0 else I[(t*h+y)*w+x]!
theorem wA1_eq(hk:s.kh=1)(hw:s.kw=1)(hs:s.stride=1)(hp:s.pad=0)(y x t:Ἀ):windowAt s I y x t=wA1 I s.h s.w y x t:=by
 unfold windowAt wA1;simp only[hk,hw,hs,hp,ϻ,F5,Nat.mod_one,ϑ,pQ,U5,false_or]
def zpW1(h w k cb o y x:Ἀ):Ἀ→Ἆ
|0=>true
|i+1=>(K[o*k+i]! == 0||K[o*k+i]! == 0x80000000||wA1 I h w y x (cb+i)==0||wA1 I h w y x (cb+i)==0x80000000)&&zpW1 h w k cb o y x i
theorem zpW1_ok(h w k cb o y x:Ἀ):∀n,zpW1 I K h w k cb o y x n=true→∀t<n,(K[o*k+t]! =0∨K[o*k+t]! =0x80000000)∨(wA1 I h w y x (cb+t)=0∨wA1 I h w y x (cb+t)=0x80000000)
|0,_,t,ht=>absurd ht (U5 _)
|n+1,hz,t,ht=>by
 simp only[zpW1,Η,nM,AQ]at hz
 by_cases e:t=n
 ·subst e;have h1:=hz.1;tauto
 ·exact zpW1_ok h w k cb o y x n hz.2 t (by omega)
theorem aBz2(kk k cb o y x:Ἀ)(hb:b[o]! =0)(hz:∀t<k,(K[o*k+t]! =0∨K[o*k+t]! =0x80000000)∨(windowAt s I y x (cb*kk+t)=0∨windowAt s I y x (cb*kk+t)=0x80000000))(hi:∀q:ℕ,kB I[q]!≠255)(hw:∀q:ℕ,kB K[q]!≠255):aB s I K b kk k cb o y x=0:=by
 rw[aB_eq,Dense.bO]
 refine zrDot2 k _ _ (λt ht=>by simp only[if_pos ht];exact hz t ht) (λt ht=>by simp only[if_pos ht];unfold windowAt;dsimp only;split;exact (by decide);exact hi _) (λt ht=>by simp only[if_pos ht];exact hw _) (by simp) (by simp[hb])
def dS:Ἂ Ἃ:=
 if D then depthwiseSpec s J I K b else conv2dSpec s J I K b
@[noinline]def sk:ἇ:=
 if u<0x80000000 then u+0x80000000 else if u==0x80000000 then 0x80000000 else 0xFFFFFFFF-u
@[noinline]def mx:Ἃ:=
 if expOf y≠255/\expOf z≠255/\(sk y<sk z\/sk z<sk y\/y=z)then
  (if sk y<sk z then z else y)
 else toWord (Model.maximum (ofW y) (ofW z))
@[specialize]def wl{α:Type}(g:α->Ἃ->α)(I:Ἂ Ἃ)(w k:Ἀ):Ἀ->Ἀ->Ἀ->α->α
 | 0,_,_,acc=>acc
 | n+1,i,dx,acc=>
  let acc:=g acc I[i]!
  if dx+1<k then wl g I w k n (i+1) (dx+1) acc else wl g I w k n (i+w+1-k) 0 acc
@[inline]def eX:Ἃ:=
 let q:=expOf d-127
 if fracOf d=0/\127<=expOf d/\q<expOf r/\expOf r<255 then r-(q<<<23)
 else if r&&&0x7FFFFFFF=0 then r
 else toWord (Ar (ofW r) (ofW d))
@[inline]def t1:ἆ:=sg u (mW u<<<(iW u-B))
@[reducible]def cr(f:Ἃ->ἆ):=wl λ(a:ἆ) x=>a+f x
@[reducible]def wa:=wl λ(b:Ἆ) x=>b&&Q x
@[noinline]def cU(ch y x:Ἀ):Ἃ:=
 dotOf (k*k) (poolAt h w k I ch y x) λ_=>oneW
@[inline]def wo(A H:ἆ)(g:Unit->Ἆ)(fb:Unit->Ἃ)(d:Ἃ):Ἃ:=
 if A=0/\H=0 then(if g ()then 0x80000000 else 0)
 else
  let r:=Dense.b9 (d4 H A) (ct H A) (B+149)
  eX (if r<4294967296 then (N3 r)else fb ()) d
@[inline]def aq(w:Ἃ):ἆ:=if mW w !=0 then iW w else 255
@[inline]def cV(x0 x1 x2 x3:Ἃ):Ἆ:=Q x0&&Q x1&&Q x2&&Q x3
@[inline]def eF(u:Ἃ):ἆ:=(UInt32.toUInt64 ((u>>>23)&&&255))
noncomputable def gT(ch y x:Ἀ)(e:Ἄ):Ἃ:=
 let v:=(G0 w)
 if hb:e<I.usize/\e+1<I.usize/\e+v<I.usize/\e+v+1<I.usize then
  let x0:=I.uget e (uL hb.1)
  let x1:=I.uget (e+1) (uL hb.2.1)
  let x2:=I.uget (e+v) (uL hb.2.2.1)
  let x3:=I.uget (e+v+1) (uL hb.2.2.2)
  let B:=min (min (aq x0) (aq x1)) (min (aq x2) (aq x3))
  if max (max (eF x0) (eF x1)) (max (eF x2) (eF x3))<=min (B+37) 254 then
   wo B (t1 B x0+t1 B x1+t1 B x2+t1 B x3) 0 (λ_=>cV (.==0x80000000) x0 x1 x2 x3)
    (λ_=>cU h w 2 I ch y x) d
  else toWord (Ar (ofW (cU h w 2 I ch y x)) (ofW d))
 else toWord (Ar (ofW (cU h w 2 I ch y x)) (ofW d))
def df(I:Ἂ Ἃ)(B:ἆ)(stop:Ἄ)(hs:(J0 stop)<=(O0 I))(e:Ἄ)(acc:ἆ):ἆ:=
 if h:e<stop then df I B stop hs (e+1) (acc+t1 B (I.uget e (by have:=ε.mp h;exact U1 h hs)))
 else acc
termination_by (J0 stop)-(J0 e)
decreasing_by have:=ε.mp h;rw[Dense.ac h];exact Z5 stopб eб h
@[inline]def fj(I:Ἂ Ἃ)(B:ἆ)(w k U:Ἀ):ἆ:=
 if h:0<k/\w=k/\U+k*k<=(O0 I)/\U+k*k<=4294967295 then
  df I B (G0 (U+k*k)) (by have:=ς;rw[Dense.aa (by omega)];exact h.2.2.1) (G0 U) 0
 else cr (t1 B) I w k (k*k) U 0 0
abbrev DQ:Prop:=signOf d=ἄ/\expOf d≠0/\expOf d<255
@[noinline]def cK(m:Ἀ):ἆ:=if m<=64 then 33 else if m<=256 then 31 else if m<=1024 then 29 else 25
lemma eQ:ae x=((lq x):Ἁ)-2^(64:Ἀ)*((lq x)/2^(63:Ἀ):Ἀ):=by rw[ae];rw[lv];have e:x.toBitVec.toNat=(lq x):=ἅ;have:=x.toNat_lt;split<;>omega
lemma jC(h:((lq x):Ἁ)≡v [ZMOD2^(64:Ἀ)])(d:-2^(63:Ἀ)<=v)
  (e:v<2^(63:Ἀ)):ae x=v:=by rw[eQ];have:=x.toNat_lt;rw[pe]at h;omega
noncomputable def Z:ZMod (2^(64:Ἀ)):=(lq x)
lemma iB:((lq x):Ἁ)≡v [ZMOD2^(64:Ἀ)]↔Z x=v:=by
 rw[Z]
 rw[show(((lq x):ℕ):ZMod (2^(64:Ἀ)))=((((lq x):ℤ)):ZMod (2^(64:Ἀ))) by push_cast;rfl,
  ZMod.intCast_eq_intCast_iff]
 rfl
lemma sZ(h:Z x=v)(d:-2^(63:Ἀ)<=v)(e:v<2^(63:Ἀ)):
  ae x=v:=jC x v ((iB x v).mpr h) d e
lemma fI(b:ἆ):Z (a+b)=Z a+Z b:=by unfold Z;rw[η,BN];push_cast;ring
lemma iX(b:ἆ):Z (a-b)=Z a-Z b:=by have d:=b.toNat_lt;unfold Z;rw[Ζ,BN,W5,pO (by exact X5 d)];rw[show((2^(64:Ἀ):Ἀ):ZMod (2^(64:Ἀ)))=0 from ZMod.natCast_self _];ring
lemma bF:Z 0=0:=by simp[Z]
lemma cB{B:ἆ}(h:B<=255):(lq B)<=255:=by simpa using θ.mp h
noncomputable def bU(e:Ἀ)(w:ἇ):Ἁ:=sv w*2^(idxOf w-e)
lemma cz(h:kM u=0):sv u=0:=by rw[sv];simp[h]
lemma kd(h:kM u=0):mW u=0:=
 ξ.mp (by rw[an,h];rfl)
lemma dI_h1(w:ἇ)(m:(lq E)<=255)(l:(lq C)<=62)(e:iW w-E<=C):(lq E)<=kG w/\kG w<=(lq E)+(lq C)/\(lq (iW w-E))=kG w-(lq E):=by
 have g:=θ.1 e;have q:=η (iW w-E) E;rw[UInt64.sub_add_cancel,am,γ ((Y3 (g.trans l) m).trans_lt (by decide))]at q
 rw[q];exact⟨K2 _ _,(Τ _ _).trans_le (J1 g _),(BK _ _).symm⟩
lemma dI(w:ἇ)(m:(lq E)<=255)(l:(lq C)<=62)(h:ok C E w=ἃ):
  kB w≠255/\(kM w=0\/((lq E)<=kG w/\kG w<=(lq E)+(lq C)/\
   (lq (iW w-E))=kG w-(lq E))):=
 ((Η _ _).mp h).imp AM.1 λb=>((nM _ _).mp b).imp (λd=>(an w).symm.trans (congrArg _ (AQ.1 d))) (λe=>dI_h1 C E w m l (As.1 e))
lemma pw(v s c:Ἀ)(e:v<2^(24:Ἀ))(d:s<=c):v*2^s<2^(24+c):=by rw[AF];exact ly e (Θ (by norm_num) d) (by positivity)
lemma fw(e:ἆ)(w:ἇ)(c:Ἀ)(p:c<=39)
  (h:kM w=0\/((lq e)<=kG w/\kG w<=(lq e)+c/\
   (lq (iW w-e))=kG w-(lq e))):
  (lq (mW w<<<(iW w-e)))=kM w*2^(kG w-(lq e))/\
   kM w*2^(kG w-(lq e))<2^(24+c):=by have j:=lB w;rcases h with d|⟨m,o,l⟩;rw[kd w d,d];simp only [UInt64.zero_shiftLeft,ϋ,zero_mul,Order.lt_two_iff,zero_le,pow_pos,and_self];have g:=pw _ _ c j ((by exact Nat.sub_le_iff_le_add'.mpr o):kG w-(lq e)<=c);refine⟨?_,g⟩;rw[ψ,an,l,ο,γ (a:=kG w-(lq e)) (by (clear * - p m o;omega))];exact γ (AN g (Θ (by norm_num) (by (clear * - p;omega))))
lemma jK(b:ἆ):((lq (a+b)):Ἁ)≡((lq a):Ἁ)+(lq b) [ZMOD2^(64:Ἀ)]:=by rw[pe];rw[η];omega
lemma g5(b:ἆ):((lq (a-b)):Ἁ)≡((lq a):Ἁ)-(lq b) [ZMOD2^(64:Ἀ)]:=by have:=b.toNat_lt;rw[pe];rw[Ζ];omega
lemma ji:
  ((lq ((a+(0x8000000000000000:ἆ))>>>(39:ἆ))):Ἁ)=
   ((lq a)/2^(39:Ἀ):Ἀ)+(if (lq a)<2^(63:Ἀ) then 2^(24:Ἀ) else-2^(24:Ἀ)):=by have d:=a.toNat_lt;rw[ν,η];simp only[β,mY,nr,kH];split<;>omega
theorem eU_h2(D E:ℕ)(l:E<2^(64:ℕ)):(if (D%2^(39:ℕ)*2^(25:ℕ)+E)%2^(64:ℕ)<E then (1:ℤ) else 0)=(((D%2^(39:ℕ)*2^(25:ℕ)+E)/2^(64:ℕ):ℕ):ℤ):=by
 have hb:D%2^(39:ℕ)*2^(25:ℕ)<2^(64:ℕ):=ϐ (ϊ _ (϶ 39)) (϶ 25)
 by_cases h:D%2^(39:ℕ)*2^(25:ℕ)+E<2^(64:ℕ)
 rw[γ h,ώ h];exact if_neg (W3.2 (K2 _ _))
 have h1:2^(64:ℕ)<=D%2^(39:ℕ)*2^(25:ℕ)+E:=W4 h
 have hi:D%2^(39:ℕ)*2^(25:ℕ)+E<2^(64:ℕ)+2^(64:ℕ):=E6 hb l
 rw[mG (show 1*2^(64:ℕ)<=D%2^(39:ℕ)*2^(25:ℕ)+E from (n4 _).trans_le h1) (show D%2^(39:ℕ)*2^(25:ℕ)+E<(1+1)*2^(64:ℕ) from hi),P8 h1,γ ((Nat.sub_lt_iff_lt_add h1).2 hi)];exact if_pos ((Nat.sub_lt_iff_lt_add h1).2 ((X3 hb E).trans_eq (Τ _ _)))
theorem eU_h4(a i h q m d e s r u v w:ℤ)(W:w=v*u)(F1:v*a+r=d)(F2:m+w*q=r*u+e)(F3:i=-u*s):(a+i-h+q)*w+m=(d-w*s)*u+(e-w*h):=by
 have F4:=eq_sub_of_add_eq F2;subst W F1 F3 F4;ring
theorem eU_h1(D E:ℕ)(j:D<2^(64:ℕ))(x w y z q c:ℤ)(hx:x=q)(hq:q=((D/2^(39:ℕ):ℕ):ℤ)+(if D<2^(63:ℕ) then 0 else-2^(25:ℕ))-((E/2^(63:ℕ):ℕ):ℤ)+c)(hc:c=(((D%2^(39:ℕ)*2^(25:ℕ)+E)/2^(64:ℕ):ℕ):ℤ))(hy:y=(D:ℤ)-2^(64:ℕ)*((D/2^(63:ℕ):ℕ):ℤ))(hz:z=(E:ℤ)-2^(64:ℕ)*((E/2^(63:ℕ):ℕ):ℤ))(hw:w=(((D%2^(39:ℕ)*2^(25:ℕ)+E)%2^(64:ℕ):ℕ):ℤ)):x*2^(64:ℕ)+w=y*2^(25:ℕ)+z:=by
 subst hx hq hc hy hz hw
 have F1:=congrArg (Nat.cast:ℕ→ℤ) (ϛ D (2^(39:ℕ)))
 have F2:=congrArg (Nat.cast:ℕ→ℤ) (Nat.mod_add_div (D%2^(39:ℕ)*2^(25:ℕ)+E) (2^(64:ℕ)))
 simp only[Int.natCast_add,Int.natCast_mul,Int.natCast_pow] at F1 F2
 have F3:(if D<2^(63:ℕ) then (0:ℤ) else-2^(25:ℕ))=-2^(25:ℕ)*((D/2^(63:ℕ):ℕ):ℤ):=by
  by_cases h:D<2^(63:ℕ)
  ·rw[if_pos h,ώ h];exact(Int.mul_zero _).symm
  ·rw[if_neg h,mG ((n4 _).trans_le (W4 h)) j];exact(Int.mul_one _).symm
 exact eU_h4 _ _ _ _ _ _ _ _ _ _ _ _ (AF 2 39 25) F1 F2 F3
theorem eU_h3(D E:ℕ)(b:Prop)[Decidable b](j:D<2^(64:ℕ))(l:E<2^(64:ℕ)):-2^(63:ℕ)<=((D/2^(39:ℕ):ℕ):ℤ)+(if D<2^(63:ℕ) then 0 else-2^(25:ℕ))-((E/2^(63:ℕ):ℕ):ℤ)+(if b then 1 else 0)∧((D/2^(39:ℕ):ℕ):ℤ)+(if D<2^(63:ℕ) then 0 else-2^(25:ℕ))-((E/2^(63:ℕ):ℕ):ℤ)+(if b then 1 else 0)<2^(63:ℕ):=by
 have p:0<=(if b then (1:ℤ) else 0)∧(if b then (1:ℤ) else 0)<=1:=by split<;>decide
 have hA:((D/2^(39:ℕ):ℕ):ℤ)<2^(25:ℕ):=(Int.ofNat_lt.2 (Ϟ (j.trans_eq (m7 2 39 25)))).trans_eq (Int.natCast_pow 2 25)
 have hH:((E/2^(63:ℕ):ℕ):ℤ)<=1:=U4.2 (Nat.lt_succ_iff.1 (Ϟ (l.trans_eq (Nat.pow_succ 2 63))))
 have hI:-2^(25:ℕ)<=(if D<2^(63:ℕ) then (0:ℤ) else-2^(25:ℕ))∧(if D<2^(63:ℕ) then (0:ℤ) else-2^(25:ℕ))<=0:=by split<;>decide
 exact⟨Int.le_trans (by decide) (Int.add_le_add (Int.sub_le_sub (Int.add_le_add (W6 _) hI.1) hH) p.1),Int.lt_of_lt_of_le (Int.add_lt_add_of_lt_of_le (Int.sub_lt_sub_of_lt_of_le (Int.add_lt_add_of_lt_of_le hA hI.2) (W6 _)) p.2) (by decide)⟩
lemma eU(d e:ἆ):
  aw (d4 d e) (ct d e)=ae d*2^(25:Ἀ)+ae e:=by
 have j:=d.toNat_lt;have l:=e.toNat_lt;have o:(lq (ct d e))=(((lq d)%2^(39:Ἀ))*2^(25:Ἀ)+(lq e))%2^(64:Ἀ):=by rw[ct,η,ψ];simp only[ο,mY,nr,kH];exact congrArg (·%18446744073709551616) (congrArg (·+(lq e)) (Nat.mul_mod_mul_right _ _ 549755813888))
 have k:=eU_h3 _ _ (ct d e<e) j l
 exact eU_h1 _ _ j _ _ _ _ _ _ (jC _ _ (by rw[d4];refine(jK _ _).trans (Int.ModEq.add ((g5 _ _).trans (Int.ModEq.sub ((g5 _ _).trans ?_) (by rw[ν];simp only[β,mY,nr,kH];rfl))) ?_);rw[ji,show((lq (16777216:ἆ)):Ἁ)=2^(24:Ἀ) from ἅ];split;exact congrArg (·%2^(64:Ἀ)) ((Int.add_sub_cancel _ _).trans (Int.add_zero _).symm);exact congrArg (·%2^(64:Ἀ)) ((Int.add_sub_assoc _ _ _).trans (congrArg _ ἅ));split<;>rfl) k.1 k.2) rfl (by simp only[μ,o];exact eU_h2 _ _ l) (eQ d) (eQ e) (congrArg Nat.cast o)
lemma fB(a b:ἇ)(d e:Ἀ)(g:kM a=0\/d<=kG a)(j:kM b=0\/e<=kG b):tm a b=bU d a*bU e b*2^(d+e):=by
 unfold tm bU;rcases g with g | g
 .simp[cz a g]
 rcases j with j | j
 .simp[cz b j]
 rw[((by omega):kG a+kG b=(kG a-d)+(kG b-e)+(d+e)),AF,AF];ring
lemma cJ(f:Ἀ->Ἁ)(M:Ἁ)(h:∀t<k,|f t|<=M):
  |∑t∈kP k,f t|<=k*M:=
 (K8 _ _).trans ((Finset.sum_le_sum λt d=>h t (mr d)).trans
  ((by simp only [sum_const,card_range,Int.nsmul_eq_mul,Std.le_refl])))
abbrev ag(g:Ἀ->ἇ):Ἂ ἇ:=Aj (n:=m) λq=>g (Fin.val q)
lemma ec(g:Ἀ->ἇ):(ag m g).push (g m)=ag (m+1) g:=by simp[ag];rw[Array.ofFn_succ (f:=λq:Fin (m+1)=>g (Fin.val q))];rfl
lemma cf(o y x H W:Ἀ)(g:y<H)(e:x<W):
  ((o*H+y)*W+x)/(H*W)=o/\((o*H+y)*W+x)/W%H=y/\
   ((o*H+y)*W+x)%W=x:=by
 have d:((o*H+y)*W+x)/W=o*H+y:=by rw[Τ,Ρ _ _ (by exact L7 e),ώ e,Κ]
 refine⟨?_,?_,by rw[Τ,Μ,mo e]⟩
 .rw[Ϗ H W,<-lY,d,Τ,Ρ _ _ (by exact L7 g),ώ g,Κ]
 .rw[d,Τ,Μ,mo g]
lemma iu(H:Ἀ)(f:Ἀ->Ἀ->Ἂ ἇ->Ἂ ἇ)(S:Ἀ->Ἂ ἇ->Prop)
  (j:∀o y a,y<H->S (o*H+y) a->S (o*H+y+1) (f o y a)):
  ∀n o y a,y<H->S (o*H+y) a->S (o*H+y+n) (lp H f n o y a):=by
 intro n;induction n<;>intro o y a l h;exact h;rename_i n e;have d:=j o y a l h;simp[lp];split_ifs with g;exact((by ring):o*H+y+(n+1)=o*H+(y+1)+n)▸e _ _ _ g d;obtain rfl:H=y+1:=(by (clear * - l g;omega));exact((by ring):o*(y+1)+y+(n+1)=(o+1)*(y+1)+0+n)▸e _ _ _ (by exact Nat.zero_lt_succ y) (((by ring):(o+1)*(y+1)+0=o*(y+1)+y+1)▸d)
lemma kb(H W n c:Ἀ)(d:Ἀ->Ἀ->Ἂ ἇ->Ἂ ἇ)(F:Ἀ->Ἀ->Ἀ->ἇ)(r:0<H)
  (u:∀o y M g,y<H->o*H+y<n->(∀j<W,g (M+j)=F o y j)->d o y (ag M g)=ag (M+W) g):
  lp H d n 0 0 (lQ c)=ag (n*W) λq=>F (q/(H*W)) (q/W%H) (q%W):=by
 simpa[A7,ϳ,Κ,Ϗ W n]using iu H d (λk a=>k<=n->a=ag (W*k) λq=>F (q/(H*W)) (q/W%H) (q%W)) (λo y a e v A=>v (by exact X5 A)▸u _ _ _ _ e (by exact I7 A) λj x=>by obtain⟨l,m,p⟩:=cf o y j H W e x;simp[((by ring):W*(o*H+y)+j=(o*H+y)*W+j),l,m,p,Κ]) n 0 0 _ r λ_=>ἅ
lemma iK(f:Ἀ->Ἄ->ἇ)(g:Ἀ->ἇ)(s:Ἄ):∀m x e M,
  (∀j< m,g (M+j)=f (x+j) (e+s*(G0 j)))->dc f s x m e (ag M g)=ag (M+m) g:=by intro m;induction m<;>intro x e M l;rfl;rename_i o;have d:=l 0 (by (clear * -;omega));rw[ϑ,ϑ,show(G0 (0:Ἀ))=0 from ἅ,USize.mul_zero,Av]at d;rw[dc,<-d,ec,o _ _ _ λj p=>by rw[((by exact Nat.succ_add_eq_add_succ M j):M+1+j=M+(j+1)),l _ (by exact X3 p 1),((by exact Eq.symm (Nat.succ_add_eq_add_succ x j)):x+(j+1)=x+1+j),show(G0 (j+1))=(G0 j)+1 from mC j 1,USize.mul_add,USize.mul_one,USize.add_assoc,USize.add_comm (s*_)]];congr 1;(clear * -;omega)
lemma d8(H W n c:Ἀ)(s:Ἄ)(E:Ἀ->Ἀ->Ἄ)(F:Ἀ->Ἀ->Ἀ->Ἄ->ἇ)(d:0<H):lp H (λo y=>dc (F o y) s 0 W (E o y)) n 0 0 (lQ c)=
   ag (n*W) λq=>F (q/(H*W)) (q/W%H) (q%W) (E (q/(H*W)) (q/W%H)+s*(G0 (q%W))):=kb H W n c _ (λo y x=>F o y x (E o y+s*(G0 x))) d λ_ _ _ _ _ _ e=>iK _ _ _ _ _ _ _ λj g=>by rw[e j g,Κ]
lemma iO(O H W:Ἀ)(f:Ἀ->Ἀ->Ἀ->ἇ):
  eT O H W f=ag (O*(H*W)) λq=>f (q/(H*W)) (q/W%H) (q%W):=by
 rw[eT];split_ifs with h
 .rw[d8 _ _ _ _ 0 (λ_ _=>0) (λo y x _=>f o y x) h,Ϛ]
 .simp[ag,((by exact Nat.eq_zero_of_not_pos h):H=0)]
lemma jl(g:α->kF->α)(I:Ἂ kF)(w k U:Ἀ):
  ∀n j e d,e<k->wl g I w k n (U+j*w+e) e d=
   (pV (j*k+e) n).foldl (λd t=>g d I[U+t/k*w+t%k]!) d:=by
 intro n;induction n<;>intro j e d l;rfl;rename_i m;rw[BS,Ϸ,wl,(by rw[Τ,Ρ _ _ (by exact L7 l),ώ l,Κ]:(j*k+e)/k=j),(by rw[Τ,Μ,Conv.mo l]:(j*k+e)%k=e)];split_ifs with h;exact m j (e+1) _ h;obtain rfl:k=e+1:=(by (clear * - l h;omega));rw[(by rw[nO];(clear * -;omega):U+j*w+e+w+1-(e+1)=U+(j+1)*w+0),m (j+1) 0 _ (by exact Nat.zero_lt_succ e)];congr 2;ring
lemma gc(g:α->kF->α)(I:Ἂ kF)(w k U n:Ἀ)(d:α)(e:0<k):
  wl g I w k n U 0 d=(lj n).foldl (λd t=>g d I[U+t/k*w+t%k]!) d:=by have:=jl g I w k U n 0 0 d e;simp[ϳ,ϑ,<-List.range_eq_range']at this;exact this
lemma aa(h:n<2^(32:Ἀ)):(J0 (G0 n))=n:=j9 h
lemma fJ:Z x=(((lq x):Ἁ):ZMod (2^(64:Ἀ))):=by rw[Z];norm_cast
lemma gS:Z x=((ae x:Ἁ):ZMod (2^(64:Ἀ))):=
 (iB x _).mp (by rw[eQ];rw[pe];omega)
lemma jj(m:ἆ)(h:(lq m)<2^(62:Ἀ)):
  ae (sg u m)=if k9 u then-((lq m):Ἁ)else (lq m):=by
 unfold sg;rw[fy];cases k9 u<;>simp only[fT,ἁ,ἂ]
 .exact sZ _ _ (fJ m) (by exact Int.neg_ofNat_le_natCast (2^63) m.toNat) (by omega)
 .exact sZ _ _ (by rw[iX,bF,fJ];push_cast;ring) (by omega) (by omega)
lemma gC:ok C E 0=ἃ:=ἅ
lemma gi(a:Ἂ ἇ)(p:ἇ->Ἆ)(h:a.all p=ἃ)(e:p 0=ἃ)(q:Ἀ):
  p a[q]! =ἃ:=by
 by_cases d:q<(O0 a)
 .rw[AB a q d];exact np.mp h q d
 .rw[An a q d];exact e
noncomputable def aI(s:ConvShape)(I:Ἂ Ἃ)(c y x:Ἀ):Ἃ:=
 if c<s.cin/\1<=y/\y<=s.h/\1<=x/\x<=s.w then I[(c*s.h+(y-1))*s.w+(x-1)]!else 0
lemma uj{p j n:Ἄ}{S:Ἀ}(d:S<2^(32:Ἀ))(e:(J0 p)+(J0 n)<=S)(h:j<n):(J0 (p+j))<S:=by have:=ς;have:=ε.mp h;rw[Dense.aj (by omega)];(clear * - e this;omega)
def fcU(I:Ἂ Ἃ)(p d n:Ἄ)(hs:(O0 I)<2^(32:Ἀ))(hp:(J0 p)+(J0 n)<=(O0 I))(j:Ἄ)(Y:ἀ)(hd:(J0 d)+(J0 n)<=(kS Y)/\(kS Y)<2^(32:Ἀ)):ἀ:=
 if h:j<n then fcU I p d n hs hp (j+1) (Y.uset (d+j) (Dense.F.dw (I.uget (p+j) (uj hs hp h))) (by have:=ε.1 h;have:=ς;rw[Dense.aj (by omega)];omega)) (by rw[uZ];exact hd)else Y
termination_by (J0 n)-(J0 j)
decreasing_by have:=ε.mp h;rw[Dense.ac h];exact Z5 nб jб h
lemma fcUs_h2{α:Type}(A B C q:ℕ)(F:ℕ->α)(V Y:α)(hB:B<C)(hV:V=F (A+B)):(if A+(B+1)<=q∧q<A+C then F q else if A+B=q then V else Y)=if A+B<=q∧q<A+C then F q else Y:=
 if c1:A+(B+1)<=q∧q<A+C then (if_pos c1).trans (if_pos ⟨(E7 _).trans c1.1,c1.2⟩).symm else (if_neg c1).trans (if c2:A+B=q then (if_pos c2).trans ((hV.trans (congrArg F c2)).trans (if_pos ⟨c2.le,c2▸O1 hB _⟩).symm) else (if_neg c2).trans (if_neg λh=>c1 ⟨N4 h.1 c2,h.2⟩).symm)
lemma fcUs(I:Ἂ kF)(p d n:Ἄ)(hs:(O0 I)<2^(32:Ἀ))(hp:(J0 p)+(J0 n)<=(O0 I)):∀m (j:Ἄ)(Y:ἀ)(hd:(J0 d)+(J0 n)<=(kS Y)/\(kS Y)<2^(32:Ἀ)),(J0 n)-(J0 j)=m->
  (kS (fcU I p d n hs hp j Y hd))=(kS Y)/\∀q<(kS Y),(fcU I p d n hs hp j Y hd).data[q]! =
   if (J0 d)+(J0 j)<=q/\q<(J0 d)+(J0 n) then Dense.F.dw I[(J0 p)+(q-(J0 d))]! else Y.data[q]!:=by
 intro m;induction m with
 | zero=>
  intro j Y hd e;have k:=O4 e;rw[fcU,dif_neg λh=>W3.2 k (ε.1 h)];exact⟨ἅ,λq _=>by rw[if_neg λh=>AX _ (h.2.trans_le ((J1 k _).trans h.1))]⟩
 | succ m ih=>
  intro j Y hd e;have h:j<n:=ε.2 (N1 e);have hj:=ε.1 h;have hdj:=(O1 hj (J0 d)).trans_le hd.1;have dj:(J0 (d+j))=(J0 d)+(J0 j):=Dense.aj (hdj.trans (hd.2.trans_le ς));rw[fcU,dif_pos h];have sz:=uZ Y (d+j) (Dense.F.dw (I.uget (p+j) (uj hs hp h))) (dj▸hdj);obtain⟨u,v⟩:=ih (j+1) _ (sz▸hd) (by rw[Dense.ac h,V4,e,BK]);refine⟨u.trans sz,λq hq=>?_⟩;rw[v q (hq.trans_eq sz.symm),Dense.ac h,Dense.F.fuS _ _ _ _ _ hq,dj];exact fcUs_h2 _ _ _ _ (λq=>Dense.F.dw I[(J0 p)+(q-(J0 d))]!) _ _ hj (by rw[kr,Dense.aj (((O1 hj _).trans_le hp).trans (hs.trans_le ς)),ϱ])
def fRU(I:Ἂ Ἃ)(hs:(O0 I)<2^(32:Ἀ))(w V U h n:Ἄ)(r p d y:Ἄ)(Y:ἀ):ἀ:=
 if hr:r<n then
  fRU I hs w V U h n (r+1) (p+w) (if y+1=h then d+U else d+V) (if y+1=h then 0 else y+1)
   (if hp:(J0 p)+(J0 w)<=(O0 I)/\(J0 d)+(J0 w)<=(kS Y)/\(kS Y)<=4294967295 then fcU I p d w hs hp.1 0 Y ⟨hp.2.1,by omega⟩ else Y)
 else Y
termination_by (J0 n)-(J0 r)
decreasing_by have:=ε.mp hr;rw[Dense.ac hr];exact Z5 nб rб hr
lemma qD(q H V:Ἀ):q=(q/(H*V)*H+q/V%H)*V+q%V:=by have a:=ϛ q V;have b:=ϛ (q/V) H;rw[lY,Ϗ V H]at b;rw[Ϗ (q/(H*V)) H,b,Ϗ (q/V) V];exact a.symm
theorem fRUs_h1(A W R:Ἀ)(h:(A+4)*(W+2)<=R):A*(W+2)+1+3*(W+2)<=R:=(pA _ _ _).trans_le ((J1 ((Τ 1 _).trans_le ((J1 (K2 1 (W+1)) _).trans_eq (Ο 3 (W+2)).symm)) _).trans ((Ϡ A 4 (W+2)).symm.trans_le h))
lemma fRUs(I:Ἂ kF)(s:lR)(hs:(O0 I)<2^(32:Ἀ))(w V U h n:Ἄ)(hw:(J0 w)=s.w)(hV:(J0 V)=s.w+2)(hU:(J0 U)=3*(s.w+2))(hh:(J0 h)=s.h)(hn:(J0 n)=s.cin*s.h)
  (g:s.cin*s.h*s.w<=(O0 I))(hN:(s.cin*(s.h+2)+3)*(s.w+2)<2^(32:Ἀ))(h0:0<s.h):
  ∀m (r p d y:Ἄ)(Y:ἀ),(J0 n)-(J0 r)=m->(J0 r)<=(J0 n)->(J0 y)=(J0 r)%s.h->(J0 p)=(J0 r)*s.w->(J0 d)=((J0 r)/s.h*(s.h+2)+(J0 r)%s.h+1)*(s.w+2)+1->
  (s.cin*(s.h+2)+3)*(s.w+2)<=(kS Y)->(kS Y)<2^(32:Ἀ)->
  (∀q<(kS Y),Y.data[q]! =Dense.F.dw (if q/((s.h+2)*(s.w+2))*s.h+(q/(s.w+2)%(s.h+2)-1)<(J0 r) then aI s I (q/((s.h+2)*(s.w+2))) (q/(s.w+2)%(s.h+2)) (q%(s.w+2)) else 0))->
  (kS (fRU I hs w V U h n r p d y Y))=(kS Y)/\∀q<(kS Y),(fRU I hs w V U h n r p d y Y).data[q]! =Dense.F.dw (aI s I (q/((s.h+2)*(s.w+2))) (q/(s.w+2)%(s.h+2)) (q%(s.w+2))):=by
 intro m;induction m with
 | zero=>
  intro r p d y Y e a b c f N O J;have E:=K3.1 e;have hr:¬r<n:=λh=>AX _ (U1 (ε.1 h) E);rw[fRU,dif_neg hr];refine⟨ἅ,λq hq=>?_⟩;rw[J q hq];refine congrArg Dense.F.dw ?_;split;rfl;rename_i k;rw[aI,if_neg];rintro⟨u,v,x,-⟩;have:(q/((s.h+2)*(s.w+2))+1)*s.h<=s.cin*s.h:=ι _ u;rw[Ϡ,n4]at this;exact k (U1 (O1 (U1 (Nat.sub_lt v (by decide)) x) _) (U2 this (hn▸E)))
 | succ m ih=>
  intro r p d y Y e a b c f N O J;have hr:r<n:=ε.2 (N1 e);have R:=ε.1 hr;have S:=ς;have rq:=ϛ (J0 r) s.h;have ym:=ϊ (J0 r) h0;have cl:(J0 r)/s.h<s.cin:=Ϟ (by rw[Ϗ,<-hn];exact R);have c1:((J0 r)/s.h+1)*(s.h+2)<=s.cin*(s.h+2):=ι _ cl;rw[Ϡ,n4]at c1;have hp:(J0 p)+(J0 w)<=(O0 I):=by
   rw[c,hw];have:((J0 r)+1)*s.w<=s.cin*s.h*s.w:=ι _ (by rw[<-hn];exact R);rw[Ϡ,n4]at this;exact U2 this g
  have hd:(J0 d)+(J0 w)<=(kS Y):=by
   have:((J0 r)/s.h*(s.h+2)+(J0 r)%s.h+1+1)*(s.w+2)<=(s.cin*(s.h+2)+3)*(s.w+2):=ι _ (Nat.le_add_right_of_le (U2 (J1 (L1 ym.le 2) ((J0 r)/s.h*(s.h+2))) c1));rw[nO]at this;rw[f,hw,lO _ 1];exact X5 (U2 this N)
  have dB:(J0 d)+3*(s.w+2)<=(s.cin*(s.h+2)+3)*(s.w+2):=by
   have:((J0 r)/s.h*(s.h+2)+(J0 r)%s.h+1+4)*(s.w+2)<=(s.cin*(s.h+2)+3)*(s.w+2):=ι _ (Nat.le_succ_of_le (U2 (L1 (J1 ym ((J0 r)/s.h*(s.h+2))) 4) (L1 c1 2)));rw[f];exact fRUs_h1 _ _ _ this
  rw[fRU,dif_pos hr,dif_pos ⟨hp,hd,J2 O⟩];obtain⟨u,v⟩:=fcUs I p d w hs hp _ 0 Y ⟨hd,O⟩ ἅ;have yh:y<h:=ε.2 (b▸hh▸ym);have y1:(J0 (y+1))=(J0 y)+1:=Dense.ac yh;have hR:(J0 (r+1))=(J0 r)+1:=Dense.ac hr;have hYd:∃y' d':Ἄ,(if y+1=h then 0 else y+1)=y'/\(if y+1=h then d+U else d+V)=d'/\
    (J0 y')=((J0 r)+1)%s.h/\(J0 d')=(((J0 r)+1)/s.h*(s.h+2)+((J0 r)+1)%s.h+1)*(s.w+2)+1:=by
   by_cases k:y+1=h
   .rw[if_pos k,if_pos k];have:=congrArg (·б) k;rw[y1]at this;have E:s.h=(J0 r)%s.h+1:=b▸hh▸this.symm
    obtain⟨q1,q2⟩:=(Nat.div_mod_unique h0).2 (⟨by rw[Nat.mul_succ,Κ];exact (congrArg (s.h*((J0 r)/s.h)+·) E).trans (congrArg (·+1) rq),h0⟩:0+s.h*((J0 r)/s.h+1)=(J0 r)+1/\0<s.h);refine⟨_,_,ἅ,ἅ,by rw[Ϧ,q2],?_⟩;rw[q1,q2,Dense.aj (by rw[hU];exact U1 (C0 dB hN) S),hU,f,show s.h+2=(J0 r)%s.h+3 from congrArg (·+2) E];ring
   .rw[if_neg k,if_neg k];have:(J0 y)+1≠s.h:=λj=>k (O5.1 (y1.trans (j.trans hh.symm)))
    obtain⟨q1,q2⟩:=(Nat.div_mod_unique h0).2 (⟨by rw[lO,Τ ((J0 r)%s.h),rq],N4 ym (b▸this)⟩:(J0 r)%s.h+1+s.h*((J0 r)/s.h)=(J0 r)+1/\(J0 r)%s.h+1<s.h);refine⟨_,_,ἅ,ἅ,by rw[y1,b,q2],?_⟩;rw[q1,q2,Dense.aj (by rw[hV];exact U1 (C0 (J1 (lZ _ (by decide)) _) (C0 dB hN)) S),hV,f];ring
  obtain⟨y',d',yE,ed,by',bd'⟩:=hYd;rw[yE,ed];obtain⟨u2,v2⟩:=ih (r+1) (p+w) d' y' (fcU I p d w hs hp 0 Y ⟨hd,O⟩) (by rw[hR,<-F7,e,BK]) (hR▸R) (by rw[by',hR]) (by rw[hR,Dense.aj (U1 (C0 hp hs) S),c,hw,Ϡ,n4])
   (by rw[bd',hR]) (N.trans_eq u.symm) (u.trans_lt O) (by
   rw[u,hR];intro q hq;rw[v q hq,Ϧ,ϑ];by_cases k1:(J0 d)<=q/\q<(J0 d)+(J0 w)
   .rw[if_pos k1]
    obtain⟨j,rfl⟩:∃j,q=(J0 d)+j:=⟨q-(J0 d),(BQ k1.1).symm⟩;have w1:j<s.w:=hw▸Nat.lt_of_add_lt_add_left k1.2;rw[ϱ];obtain⟨q1,q2,q3⟩:=cf ((J0 r)/s.h) ((J0 r)%s.h+1) (j+1) (s.h+2) (s.w+2) (Nat.succ_lt_succ (K6 ym)) (Nat.succ_lt_succ (K6 w1));have:=Ϗ ((J0 r)/s.h) s.h;rw[(by rw[f];ring:(J0 d)+j=((J0 r)/s.h*(s.h+2)+((J0 r)%s.h+1))*(s.w+2)+(j+1)),q1,q2,q3,if_pos (by rw[BK,this,rq];exact I3 _)];unfold aI;rw[if_pos ⟨cl,K2 1 _,ym,K2 1 _,w1⟩,BK,BK,c,
     show (J0 r)/s.h*s.h+(J0 r)%s.h=(J0 r) from this▸rq]
   .rw[if_neg k1,J q hq];refine congrArg Dense.F.dw ?_;clear*-k1 f hw h0
    rcases Nat.lt_trichotomy (q/((s.h+2)*(s.w+2))*s.h+(q/(s.w+2)%(s.h+2)-1)) (J0 r) with k|k|k
    .rw[if_pos k,if_pos (K6 k)]
    .rw[if_neg (by rw[k];exact B4 _),if_pos (by rw[k];exact I3 _)];rw[aI,if_neg];rintro⟨u1,u2,u3,u4,u5⟩
     have:=Ϗ (q/((s.h+2)*(s.w+2))) s.h;obtain⟨q1,q2⟩:=(Nat.div_mod_unique h0).2 (⟨by rw[<-k,this];exact Τ _ _,U1 (Nat.sub_lt u2 (by decide)) u3⟩:q/(s.w+2)%(s.h+2)-1+s.h*(q/((s.h+2)*(s.w+2)))=(J0 r)/\q/(s.w+2)%(s.h+2)-1<s.h);have Q:=qD q (s.h+2) (s.w+2);rw[<-q1]at Q;have:q/(s.w+2)%(s.h+2)=(J0 r)%s.h+1:=q2▸(L3 u2).symm
     rw[this,<-pA]at Q;apply k1;rw[f,hw,lO _ 1,Q];exact⟨J1 u4 _,O1 (Z4 u5) _⟩
    .rw[if_neg (Nat.lt_asymm k),if_neg (W3.2 k)])
  rw[u2,u];exact⟨ἅ,λq hq=>v2 q (hq.trans_eq u.symm)⟩
lemma iQ(A:Ἂ Ἀ)(u:Ἄ)(h:(J0 u)<(O0 A)):A.uget u h=A[(J0 u)]!:=by simp[Array.uget,AB A (J0 u) h]
lemma cj{i:Ἄ}{e M S:Ἀ}(d:S<2^(32:Ἀ))(j:(J0 i)+M+8<=S)(g:e<=M):
  (J0 (i+(G0 e)))=(J0 i)+e:=by have:=ς;rw[Dense.aj (by rw[Dense.aa (by omega)];omega),Dense.aa (by omega)]
lemma j2(A B:Dense.F.A8)(h:∀r<8,A.f r=B.f r):A=B:=by cases A;cases B;rw[Dense.F.A8.mk.injEq];exact⟨h 0 (by exact Nat.zero_lt_succ 7),h 1 (by exact Nat.one_lt_succ_succ 6),h 2 (by exact N1 rfl),h 3 (by exact N1 rfl),h 4 (by exact N1 rfl),h 5 (by exact N1 rfl),h 6 (by exact N1 rfl),h 7 (by exact Nat.lt_add_one 7)⟩
noncomputable def eH(f:Ἀ->Ἅ):Dense.F.A8:=⟨f 0,f 1,f 2,f 3,f 4,f 5,f 6,f 7⟩
lemma fW(f:Ἀ->Ἅ)(r:Ἀ)(d:r<8):(eH f).f r=f r:=by rw[eH];interval_cases r<;>rfl
lemma kq(p:Ἀ->Ἅ)(n t:Ἀ)(a:Ἅ)(h:t<n):
  Dense.F.ao p n t a=Dense.F.ao p n (t+1) (a+p t):=by rw[Dense.F.ao,AD h]
lemma kp(p:Ἀ->Ἅ)(n t:Ἀ)(a:Ἅ)(h:¬t<n):Dense.F.ao p n t a=a:=by rw[Dense.F.ao,AC h]
@[noinline]def pc(kk t:Ἀ):Ἀ:=if kk=9 then t/9 else t
@[noinline]def py(kk t:Ἀ):Ἀ:=if kk=9 then t/3%3 else 1
@[noinline]def px(kk t:Ἀ):Ἀ:=if kk=9 then t%3 else 1
lemma jd(d t:Ἀ):py d t<=2:=by rw[py];split<;>omega
lemma jc(d t:Ἀ):px d t<=2:=by rw[px];split<;>omega
lemma jb(d t e:Ἀ)(g:d=9\/d=1)(j:t<e*d):pc d t<e:=by rw[pc];rcases g with rfl|rfl<;>simp<;>omega
abbrev fP(s:ConvShape)(kk:Ἀ):Prop:=
 s.kh=3/\s.kw=3/\s.stride=1/\s.pad=1/\kk=9\/s.kh=1/\s.kw=1/\s.stride=1/\s.pad=0/\kk=1
structure CD where
 s:ConvShape
 I:Ἂ Ἃ
 K:Ἂ Ἃ
 b:Ἂ Ἃ
 X:ἀ
 T:Ἂ Ἀ
 D:Ἆ
 J:Ἆ
 k:Ἀ
 M:Ἀ
 L:Ἀ
 hh:ἆ
 Y:Ἀ
 kk:Ἀ
 hX:(kS X)<2^32
 lo:ἆ
 sp:ἆ
 hz:∀E:ἆ,E-lo<sp->hh+1075-E<=21/\E-1195<253
 z9:Ἆ
 hz9:z9=(s.kh==3&&s.kw==3&&s.stride==1&&s.pad==1&&kk==9)
 P9:Ἀ
 hP9:P9=(s.h+2)*(s.w+2)
 V9:Ἄ
 hV9:V9=C3 (s.w+2)
 k9:Ἀ
 hk9:k9=k/9
lemma cZ(n:Ἀ):(kS (kJ n))=0:=by simp[kJ,kS]
def fQ:Bool:=System.Platform.target.startsWith "aarch64"
structure CC(M:Ἀ)where
 Tz:Ἂ Ἀ
 Wz:ἀ
 wo:Ἀ
 n:Ἀ
 ni:Ἆ
 h1:n<=(O0 Tz)
 h2:wo+n<=(kS Wz)
 h3:wo+n<2^32
 h4:∀t<n,Tz[t]!<=M
 h5:(kS Wz)<2^32
@[noinline]def ow(d:@&CD)(o y x:@&Ἀ)(S:Ἅ):Ἃ:=
 Dense.F.tw d.J S d.hh d.L d.k
  (let q:=(if d.D then o else 0)*((d.s.h+2)*(d.s.w+2))+y*(d.s.w+2)+x;let ko:=o*d.k
   λt=>d.X.get! (q+d.T[t]!)*Dense.F.dw d.K[ko+t]!)
  (λ_=>Dense.F.dw oneW*Dense.F.dw d.b[o]!)
  λ_=>act d.J (aB d.s d.I d.K d.b d.kk d.k (if d.D then o else 0) o y x)
@[inline]def czZ(a:Ἅ):Ἆ:=a == -a
theorem czK(n:Nat)(h:n<2^32):(C3 n).toNat=n:=J3 (U1 h ς)
theorem czK1:(1:Ἄ).toNat=1:=czK 1 (by decide)
theorem czK2:(2:Ἄ).toNat=2:=czK 2 (by decide)
theorem czU(a b:Ἄ)(h:a.toNat+b.toNat<2^32):(a+b).toNat=a.toNat+b.toNat:=by rw[lJ,γ (U1 h ς)]
@[inline]def cz3(X:ἀ)(p:Ἄ)(h:p.toNat+2<X.size)(s:X.size<2^32):Ἆ:=czZ (X.uget p (by omega))&&czZ (X.uget (p+1) (by rw[czU p 1 (by rw[czK1];omega),czK1];omega))&&czZ (X.uget (p+2) (by rw[czU p 2 (by rw[czK2];omega),czK2];omega))
theorem cz3_ok(X:ἀ)(p:Ἄ)(h:p.toNat+2<X.size)(s:X.size<2^32)(e:cz3 X p h s=true):∀j<3,czZ X.data[p.toNat+j]! =true:=by
 simp only[cz3,Η,Dense.F.kFU]at e;rw[czU p 1 (by rw[czK1];omega),czK1,czU p 2 (by rw[czK2];omega),czK2]at e
 intro j hj;match j,hj with
 | 0,_=>simpa using e.1.1
 | 1,_=>exact e.1.2
 | 2,_=>exact e.2
@[inline]def cz9(X:ἀ)(b V:Ἄ)(h:b.toNat+V.toNat+V.toNat+2<X.size)(s:X.size<2^32):Ἆ:=cz3 X b (by omega) s&&cz3 X (b+V) (by rw[czU b V (by omega)];omega) s&&cz3 X (b+V+V) (by rw[czU (b+V) V (by rw[czU b V (by omega)];omega),czU b V (by omega)];omega) s
theorem cz9_ok(X:ἀ)(b V:Ἄ)(h:b.toNat+V.toNat+V.toNat+2<X.size)(s:X.size<2^32)(e:cz9 X b V h s=true):∀j<9,czZ X.data[b.toNat+j/3*V.toNat+j%3]! =true:=by
 have u:=czU b V (by omega);have w:(b+V+V).toNat=b.toNat+V.toNat+V.toNat:=by rw[czU (b+V) V (by omega),u]
 simp only[cz9,Η]at e;intro j hj
 rcases (by omega:j/3=0∨j/3=1∨j/3=2) with q|q|q<;>rw[q]
 ·simpa using cz3_ok X b _ s e.1.1 (j%3) (by omega)
 ·have k:=cz3_ok X (b+V) _ s e.1.2 (j%3) (by omega);rw[u]at k;simpa using k
 ·have k:=cz3_ok X (b+V+V) _ s e.2 (j%3) (by omega);rw[w]at k;rw[show b.toNat+2*V.toNat+j%3=b.toNat+V.toNat+V.toNat+j%3 by omega];exact k
def czX9(X:ἀ)(P:Nat)(V:Ἄ)(s:X.size<2^32):Nat→Nat→Ἆ
 | _,0=>true
 | b,c+1=>(if h:b+V.toNat+V.toNat+2<X.size then cz9 X (C3 b) V (by rw[czK b (by omega)];exact h) s else false)&&czX9 X P V s (b+P) c
theorem czX9_ok(X:ἀ)(P:Nat)(V:Ἄ)(s:X.size<2^32):∀c b,czX9 X P V s b c=true→∀i<c,∀j<9,czZ X.data[b+i*P+j/3*V.toNat+j%3]! =true
 | 0,_,_,i,hi=>absurd hi (U5 _)
 | c+1,b,e,i,hi=>by
  simp only[czX9,Η]at e
  rcases i with _|i
  ·split at e
   ·rename_i h;have k:=cz9_ok X _ V _ s e.1;rw[czK b (by omega)]at k;intro j hj;simpa using k j hj
   ·exact absurd e.1 (by decide)
  ·intro j hj;rw[show b+(i+1)*P+j/3*V.toNat+j%3=b+P+i*P+j/3*V.toNat+j%3 by rw[Ο];omega];exact czX9_ok X P V s c (b+P) e.2 i (by omega) j hj
@[inline]def zb(S:Ἅ):Ἆ:=(S.toBits&&&0x7FFFFFFFFFFFFFFF)==0
theorem cz3_of(X:ἀ)(p:Ἄ)(h:p.toNat+2<X.size)(s:X.size<2^32)(e:∀j<3,czZ X.data[p.toNat+j]! =true):cz3 X p h s=true:=by
 simp only[cz3,Η,Dense.F.kFU];rw[czU p 1 (by rw[czK1];omega),czK1,czU p 2 (by rw[czK2];omega),czK2]
 exact⟨⟨by simpa using e 0 (by decide),e 1 (by decide)⟩,e 2 (by decide)⟩
theorem cz9_of(X:ἀ)(b V:Ἄ)(h:b.toNat+V.toNat+V.toNat+2<X.size)(s:X.size<2^32)(e:∀j<9,czZ X.data[b.toNat+j/3*V.toNat+j%3]! =true):cz9 X b V h s=true:=by
 have u:=czU b V (by omega);have w:(b+V+V).toNat=b.toNat+V.toNat+V.toNat:=by rw[czU (b+V) V (by omega),u]
 simp only[cz9,Η]
 refine⟨⟨cz3_of X b _ s (fun j hj=>by have k:=e j (by omega);rw[show j/3=0 by omega,show j%3=j by omega]at k;simpa using k),cz3_of X (b+V) _ s (fun j hj=>by have k:=e (3+j) (by omega);rw[show (3+j)/3=1 by omega,show (3+j)%3=j by omega]at k;rw[u];simpa using k)⟩,cz3_of X (b+V+V) _ s (fun j hj=>by have k:=e (6+j) (by omega);rw[show (6+j)/3=2 by omega,show (6+j)%3=j by omega]at k;rw[w];rw[show b.toNat+V.toNat+V.toNat+j=b.toNat+2*V.toNat+j by omega];exact k)⟩
theorem czK3:(3:Ἄ).toNat=3:=czK 3 (by decide)
theorem czK6:(6:Ἄ).toNat=6:=czK 6 (by decide)
theorem czK7:(7:Ἄ).toNat=7:=czK 7 (by decide)
@[inline]def c10(X:ἀ)(p:Ἄ)(h:p.toNat+9<X.size)(s:X.size<2^32):Ἆ:=
 cz3 X p (by omega) s&&cz3 X (p+3) (by rw[czU p 3 (by rw[czK3];omega),czK3];omega) s&&cz3 X (p+6) (by rw[czU p 6 (by rw[czK6];omega),czK6];omega) s&&cz3 X (p+7) (by rw[czU p 7 (by rw[czK7];omega),czK7];omega) s
theorem c10_ok(X:ἀ)(p:Ἄ)(h:p.toNat+9<X.size)(s:X.size<2^32)(e:c10 X p h s=true):∀j<10,czZ X.data[p.toNat+j]! =true:=by
 simp only[c10,Η]at e
 have u3:=czU p 3 (by rw[czK3];omega);have u6:=czU p 6 (by rw[czK6];omega);have u7:=czU p 7 (by rw[czK7];omega)
 rw[czK3]at u3;rw[czK6]at u6;rw[czK7]at u7
 intro j hj
 rcases (by omega:j<3∨(3≤j∧j<6)∨(6≤j∧j<9)∨j=9) with q|q|q|q
 ·exact cz3_ok X p _ s e.1.1.1 j q
 ·have k:=cz3_ok X (p+3) _ s e.1.1.2 (j-3) (by omega);rw[u3,show p.toNat+3+(j-3)=p.toNat+j by omega]at k;exact k
 ·have k:=cz3_ok X (p+6) _ s e.1.2 (j-6) (by omega);rw[u6,show p.toNat+6+(j-6)=p.toNat+j by omega]at k;exact k
 ·have k:=cz3_ok X (p+7) _ s e.2 2 (by omega);rw[u7,show p.toNat+7+2=p.toNat+j by omega]at k;exact k
@[inline]def cr30(X:ἀ)(b V:Ἄ)(h:b.toNat+V.toNat+V.toNat+9<X.size)(s:X.size<2^32):Ἆ:=c10 X b (by omega) s&&c10 X (b+V) (by rw[czU b V (by omega)];omega) s&&c10 X (b+V+V) (by rw[czU (b+V) V (by rw[czU b V (by omega)];omega),czU b V (by omega)];omega) s
theorem cr30_ok(X:ἀ)(b V:Ἄ)(h:b.toNat+V.toNat+V.toNat+9<X.size)(s:X.size<2^32)(e:cr30 X b V h s=true):∀dy<3,∀j<10,czZ X.data[b.toNat+dy*V.toNat+j]! =true:=by
 have u:=czU b V (by omega);have w:(b+V+V).toNat=b.toNat+V.toNat+V.toNat:=by rw[czU (b+V) V (by omega),u]
 simp only[cr30,Η]at e;intro dy hdy j hj
 rcases (by omega:dy=0∨dy=1∨dy=2) with q|q|q<;>subst q
 ·simpa using c10_ok X b _ s e.1.1 j hj
 ·have k:=c10_ok X (b+V) _ s e.1.2 j hj;rw[u]at k;simpa using k
 ·have k:=c10_ok X (b+V+V) _ s e.2 j hj;rw[w]at k;rw[show b.toNat+2*V.toNat+j=b.toNat+V.toNat+V.toNat+j by omega];exact k
def czRX(X:ἀ)(P:Nat)(V:Ἄ)(s:X.size<2^32):Nat→Nat→Ἆ
 | _,0=>true
 | b,c+1=>(if h:b+V.toNat+V.toNat+9<X.size then cr30 X (C3 b) V (by rw[czK b (by omega)];exact h) s else false)&&czRX X P V s (b+P) c
theorem czRX_X9(X:ἀ)(P:Nat)(V:Ἄ)(s:X.size<2^32):∀c b,czRX X P V s b c=true→∀r<8,czX9 X P V s (b+r) c=true
 | 0,_,_,_,_=>by simp[czX9]
 | c+1,b,e,r,hr=>by
  simp only[czRX,Η]at e
  obtain⟨e1,e2⟩:=e
  split at e1
  ·rename_i h
   have k:=cr30_ok X _ V _ s e1;rw[czK b (by omega)]at k
   simp only[czX9,Η]
   refine⟨?_,by rw[show b+r+P=b+P+r by omega];exact czRX_X9 X P V s c (b+P) e2 r hr⟩
   rw[dif_pos (by omega)]
   apply cz9_of;rw[czK (b+r) (by omega)]
   intro j hj;have:=k (j/3) (by omega) (r+j%3) (by omega);rw[show b+j/3*V.toNat+(r+j%3)=b+r+j/3*V.toNat+j%3 by omega]at this;exact this
  ·exact absurd e1 (by decide)
open Float.Model.UnpackedFloat in
theorem czSg(u:Float.Model.UnpackedFloat)(hn:u≠.notANumber)(hz:∀s,u≠.zero s):unpackSign (pack .binary64 u)≠unpackSign (pack .binary64 u.neg):=by
 cases u with
 | notANumber=>exact absurd rfl hn
 | zero s=>exact absurd rfl (hz s)
 | infinity s=>simp only[neg,pack,packedInfinity,unpackSign_packComponents];cases s<;>decide
 | finite s m e hm=>simp only[neg,pack];split_ifs<;>simp only[packedInfinity,unpackSign_packComponents]<;>cases s<;>decide
theorem czNZ(a:Ἅ)(h:czZ a=true):∃s,a.toModel.unpack=.zero s:=by
 have hb:a.toModel.unpack.beq (-a).toModel.unpack=true:=h
 rw[Float.Model.UnpackedFloat.beq_iff_ne_notANumber_and_eq]at hb
 by_contra hz;simp only[not_exists]at hz
 obtain⟨h1,-,e|⟨e,_⟩|⟨e,_⟩⟩:=hb
 ·have E:M6 a.toModel.unpack=M6 a.toModel.unpack.neg:=by (conv_lhs=>rw[e,Dense.F.fzPU]);rfl
  have E4:=congrArg (fun m:Float.Model=>Float.Model.UnpackedFloat.unpackSign (spec:=.binary64) m.toBits.toBitVec) E
  simp only[M6,UInt64.toBitVec_ofBitVec]at E4;exact czSg _ h1 hz E4
 ·exact hz _ e
 ·exact hz _ e
theorem czW(w:ἇ)(h:sv w=0):w=0∨w=0x80000000:=by
 have m0:kM w=0:=by unfold sv at h;split at h<;>omega
 unfold kM at m0;split at m0
 ·rename_i he;have q:=congrArg mJ he
  simp only[kB,lK,Ξ,Υ,kN,kH,β]at q m0
  rw[show (255:Nat)=2^8-1 by norm_num,ζ]at q;rw[show (8388607:Nat)=2^23-1 by norm_num,ζ]at m0
  have l:=w.toNat_lt;have:w.toNat=0∨w.toNat=2^31:=by omega
  rcases this with h0|h0;exact .inl (Β.mp (by simpa using h0));exact .inr (Β.mp (by simpa using h0))
 ·omega
theorem czDZ(w:ἇ)(hw:kB w≠255)(h:czZ (Dense.F.dw w)=true):w=0∨w=0x80000000:=by
 obtain⟨s,hs⟩:=czNZ _ h;have r:=(Dense.F.cI w hw).2;rw[Dense.F.iL,hs,Model.unpackedToReal_zero s]at r
 exact czW w (by rcases mul_eq_zero.mp r.symm with h1|h1;exact_mod_cast h1;exact absurd h1 (by positivity))
theorem czWF(s:ConvShape)(I:Ἂ ἇ)(hx:∀q:Nat,kB I[q]!≠255)(y x t:Nat):kB (windowAt s I y x t)≠255:=by unfold windowAt;dsimp only;split;exact (by decide);exact hx _
@[inline]def zpC(d:@&CD)(o y x:Ἀ):Ἆ:=
 if d.z9 then czX9 d.X d.P9 d.V9 d.hX ((if d.D then o else 0)*d.P9+y*(d.s.w+2)+x) d.k9
 else if d.s.kh==1&&d.s.kw==1&&d.s.stride==1&&d.s.pad==0&&d.kk==1 then zpW1 d.I d.K d.s.h d.s.w d.k (if d.D then o else 0) o y x d.k
 else zpW d.s d.I d.K d.kk d.k (if d.D then o else 0) o y x d.k
theorem zpC_ok(d:CD)(v:Nat)(a2:d.k=v*d.kk)(aH:(d.s.cin*(d.s.h+2)+3)*(d.s.w+2)<=(kS d.X))(hx:∀q:Nat,kB d.I[q]!≠255)(o y x:Nat)
 (xw:∀t<d.k,d.X.data[(if d.D then o else 0)*((d.s.h+2)*(d.s.w+2))+y*(d.s.w+2)+x+(pc d.kk t*((d.s.h+2)*(d.s.w+2))+py d.kk t*(d.s.w+2)+px d.kk t)]! =Dense.F.dw (windowAt d.s d.I y x ((if d.D then o else 0)*d.kk+t)))
 (hz:zpC d o y x=true):∀t<d.k,(d.K[o*d.k+t]! =0∨d.K[o*d.k+t]! =0x80000000)∨(windowAt d.s d.I y x ((if d.D then o else 0)*d.kk+t)=0∨windowAt d.s d.I y x ((if d.D then o else 0)*d.kk+t)=0x80000000):=by
 unfold zpC at hz
 rw[d.hz9,d.hP9,d.hV9,d.hk9] at hz
 by_cases c3:(d.s.kh==3&&d.s.kw==3&&d.s.stride==1&&d.s.pad==1&&d.kk==9)=true
 ·rw[if_pos c3]at hz;simp only[Η,AQ]at c3;obtain⟨-,hkk⟩:=c3
  intro t ht;right;have hX:=d.hX;have:d.s.w+2<=kS d.X:=U2 (lZ _ (by omega)) aH
  have K:=czX9_ok d.X ((d.s.h+2)*(d.s.w+2)) (C3 (d.s.w+2)) d.hX (d.k/9) _ hz (t/9) (by have:=a2;rw[hkk]at this;omega) (t%9) (by omega)
  have W:=xw t ht;simp only[hkk,pc,py,px,if_pos]at W
  rw[czK _ (by omega),show t%9/3=t/3%3 by omega,show t%9%3=t%3 by omega]at K;simp only[pA]at K W;rw[W]at K;rw[hkk];exact czDZ _ (czWF d.s d.I hx y x _) K
 ·rw[if_neg c3]at hz
  by_cases c1:(d.s.kh==1&&d.s.kw==1&&d.s.stride==1&&d.s.pad==0&&d.kk==1)=true
  ·rw[if_pos c1]at hz
   simp only[Η,AQ]at c1
   obtain⟨⟨⟨⟨hk,hw⟩,hs⟩,hp⟩,hkk⟩:=c1
   intro t ht
   rw[wA1_eq d.s d.I hk hw hs hp,hkk,ϻ]
   exact Conv.zpW1_ok d.I d.K d.s.h d.s.w d.k _ o y x d.k hz t ht
  ·rw[if_neg c1]at hz
   exact Conv.zpW_ok d.s d.I d.K d.kk d.k _ o y x d.k hz
@[noinline]def ow2(d:@&CD)(o y x:@&Ἀ)(S:Ἅ):Ἃ:=
 if d.b[o]! == 0&&zpC d o y x then act d.J 0 else ow d o y x S
@[inline]def dR(relu:Ἆ)(w:Ἃ):Ἃ:=if relu&&w&&&0x80000000!=0 then 0 else w
lemma jI(d:Ἆ)(w:kF)(e:kB w≠255):dR d w=p8 d w:=by rw[dR,p8,reluW];cases d<;>simp[e]
lemma ep(S:Ἅ)(h:ἆ)(e:Dense.F.ak S h<4294967296):
  kB (N3 (Dense.F.ak S h))≠255:=by
 unfold Dense.F.ak at e⊢;dsimp only at e⊢;split at e;swap;exact absurd e (by decide);rename_i d;rw[AD d];simp only[Η,AJ]at d;obtain⟨⟨⟨-,l⟩,-⟩,-⟩:=d;have m:(lq (Dense.F.eF S.toBits))<2048:=by rw[Dense.F.eF,Ε];exact AO ϝ (by decide)
 have o:(lq (S.toBits&&&0xfffffffffffff))<2^(52:Ἀ):=Ε _ _▸AO ϝ (by decide)
 rw[μ,Ζ]at l;simp only[kA]at l;have q:(lq S.toBits)/2^(63:Ἀ)<2:=by have:=S.toBits.toNat_lt;clear*-this;exact Ϟ this
 rw[μ]at e;simp only[kA]at e;intro g;have j:=congrArg mJ g;rw[kB,Ξ,Υ,Ϫ]at j;simp only[kN,kH,β]at j;rw[ζ _ 8,γ e,Dense.F.g0 S.toBits ((by (clear * - l m;omega)):1195<=(lq (Dense.F.eF S.toBits)))]at j;(clear * - l m o j;omega)
@[inline]def fz(S:Ἅ)(lo sp h:ἆ):ἆ:=
 let b:=S.toBits
 let u:=((b&&&0xfffffffffffff)+268435456)&&&536870911
 let r:=((b&&&0x7FFFFFFFFFFFFFFF)+268435456)>>>29
 if Dense.F.eF b-lo<sp&&u-2097153<532676607&&r&&&8388607!=0 then ((b>>>63)<<<31)+r-0x255000000
 else Dense.F.ak S h
lemma fz_g(u D:ἆ)(c:u-2097153<532676607)(d:D<=21):u-(((1:ἆ)<<<D)+1)<536870911-(((1:ἆ)<<<D)+((1:ἆ)<<<D)):=by
 have dn:(lq D)<=21:=d
 have hg:(lq ((1:ἆ)<<<D))=2^(lq D):=by
  rw[ψ,γ (by omega:(lq D)<64),pS,na]
  exact γ (lu (by decide) (by omega))
 have g1:1<=2^(lq D):=BI
 have g2:2^(lq D)<=2^(21:Ἀ):=Θ (by decide) dn
 have hu:=u.toNat_lt
 rw[μ] at c ⊢
 simp only[Ζ,η,hg,pS,kA] at c ⊢
 generalize 2^(lq D)=G at g1 g2 ⊢
 omega
lemma fz_ak(S:Ἅ)(lo sp h:ἆ)(hz:∀E:ἆ,E-lo<sp->h+1075-E<=21/\E-1195<253):fz S lo sp h=Dense.F.ak S h:=by
 unfold fz;dsimp only;split
 · rename_i c;simp only[Η,AJ]at c;obtain⟨⟨c1,c2⟩,c3⟩:=c;obtain⟨d1,d2⟩:=hz _ c1
   unfold Dense.F.ak;dsimp only;rw[if_pos];simp only[Η,AJ]
   exact⟨⟨⟨ϗ d1 (by decide),d2⟩,fz_g _ _ c2 d1⟩,c3⟩
 · rfl
def wlo(h:Ἀ):ἆ:=I0 (max (h+1054) 1195)
def wsp(h:Ἀ):ἆ:=I0 ((min 1448 (h+1076))-(max (h+1054) 1195))
theorem wok_h1(a b:Ἀ)(hab:a<=b)(hb:b<18446744073709551616):(18446744073709551616-a+b)%18446744073709551616=b-a:=by rw[←R7 (hab.trans hb.le),Nat.add_sub_assoc hab,Nat.add_mod_left,γ ((K4 b a).trans_lt hb)]
theorem wok_h2(h e:Ἀ)(hh:h<1024)(hE:e<18446744073709551616)(c:(18446744073709551616-max (h+1054) 1195%18446744073709551616+e)%18446744073709551616 < (min 1448 (h+1076)-max (h+1054) 1195)%18446744073709551616):(18446744073709551616-e+(h%18446744073709551616+1075)%18446744073709551616)%18446744073709551616 ≤ 21∧(18446744073709551616-1195+e)%18446744073709551616 < 253:=by
 have l1:=le_max_left (h+1054) 1195
 have l2:=le_max_right (h+1054) 1195
 have l3:max (h+1054) 1195<=2078:=max_le (X4 (X3 hh 1054)) (by decide)
 have m1:=min_le_left 1448 (h+1076)
 have m2:=min_le_right 1448 (h+1076)
 rw[γ (l3.trans_lt (by decide)),γ (((K4 _ _).trans m1).trans_lt (by decide))]at c
 generalize max (h+1054) 1195=lo at l1 l2 l3 c
 generalize min 1448 (h+1076)=m at m1 m2 c
 have le:lo<=e:=W3.1 λlt=>absurd (((Nat.sub_le_sub_left l3 _).trans (V1 _ e)).trans_lt ((γ ((O1 lt _).trans_le (B3 (L3 (l3.trans (by decide)))))).symm.trans_lt c)) (W3.2 (((K4 _ _).trans m1).trans (by decide)))
 rw[wok_h1 lo e le hE]at c
 have he:=(Nat.sub_lt_sub_iff_right le).1 c
 have h2:h+1075<18446744073709551616:=(X3 hh 1075).trans (by decide)
 rw[γ (hh.trans (by decide)),γ h2,wok_h1 e (h+1075) (J2 (he.trans_le m2)) h2,wok_h1 1195 e (l2.trans le) hE]
 exact⟨M8 ((Nat.add_left_comm h 21 1054).trans_le (J1 (l1.trans le) 21):h+1075<=21+e),G4 (l2.trans le) (he.trans_le m1)⟩
lemma wok(h:Ἀ)(hh:h<1024):∀E:ἆ,E-wlo h<wsp h->(I0 h)+1075-E<=21/\E-1195<253:=by
 intro E c
 unfold wlo wsp at c
 simp only[μ,θ,Ζ,η,Ϣ,O6,mY,kH,nr] at c ⊢
 exact wok_h2 h _ hh E.toNat_lt c
lemma wlt(P L:Ἀ)(a:P+L<=425)(b:L<=27):P+2*L-54<1024:=by omega
noncomputable def bVo(d:CD)(o y x:Ἀ)(S:Ἅ):Ἃ:=
 let c:=Dense.F.ak S d.hh
 if h:c<4294967296 then dR d.J (N3 c)else ow d o y x S
noncomputable def bV(d:CD)(o y x:Ἀ)(S:Ἅ):Ἃ:=
 let c:=Dense.F.ak S d.hh
 if h:c<4294967296 then dR d.J (N3 c)else ow2 d o y x S
lemma iE(d:CD)(o y x:Ἀ)(S:Ἅ):bVo d o y x S=ow d o y x S:=by
 unfold bVo;dsimp only;split
 .rename_i e;rw[jI _ _ (ep _ _ e)];rw[ow,Dense.F.tw,Dense.F.tf];dsimp only;rw[AD e];split;swap;rfl;rename_i c;simp only[Η,AJ]at c;obtain⟨⟨r,c1⟩,-⟩:=c;rw[r];exact Dense.F.kAkN S d.hh e c1
 .rfl
@[inline]def ju(d:CD)(o y:Ἀ)(xb:Ἄ)(r:Ἀ)(S:Ἅ):Ἃ:=
 let c:=fz S d.lo d.sp d.hh
 if c<4294967296 then dR d.J (N3 c)else ow2 d o y ((J0 xb)+r) S
lemma iF(d:CD)(o y:Ἀ)(e:Ἄ)(r:Ἀ)(S:Ἅ):ju d o y e r S=bV d o y ((J0 e)+r) S:=by
 unfold ju;rw[fz_ak _ _ _ _ d.hz];rfl
@[inline]def su(a:Ἂ Ἃ)(u:Ἄ)(v:Ἃ):Ἂ Ἃ:=
 if h:u<a.usize then a.uset u v (AN (ε.mp h) (Ay _ _))else a.set! (J0 u) v
lemma jk(a:Ἂ kF)(u:Ἄ)(v:kF):su a u v=a.set! (J0 u) v:=by
 rw[su];split
 .rename_i h
  rw[pP,Array.set!_eq_setIfInBounds,Array.setIfInBounds,AT (AN (ε.mp h) (Ay _ _))]
 .rfl
@[specialize]def pl(g:Ἀ->Ἃ)(n r q:Ἀ)(a:Ἂ Ἃ):Ἂ Ἃ:=
 if r<n then pl g n (r+1) q (a.set! (q+r) (g r))else a
termination_by n-r
decreasing_by all_goals omega
lemma jS(g:Ἀ->kF)(n q:Ἀ):∀m r (a:Ἂ kF),n-r=m->r<=n->(O0 (pl g n r q a))=(O0 a)/\
  ∀i<(O0 a),(pl g n r q a)[i]! =if q+r<=i/\i<q+n then g (i-q)else a[i]!:=by
 intro m;induction m with
 |zero=>intro r a l o;rw[pl,AC (by (clear * - l;omega))];exact⟨ἅ,λi e=>by rw[AC (by (clear * - l o;omega))]⟩
 |succ m p=>
  intro r a l o;have h:=N1 l;obtain⟨u,j⟩:=p (r+1) (a.set! (q+r) (g r)) (by rw[<-F7,l,BK]) h;rw[pl,AD h,u,BU];refine⟨ἅ,λi e=>?_⟩;rw[j i (by rwa[BU])];by_cases d:i=q+r;subst d;rw[Array.getElem!_set!_self _ _ _ e];clear*-h;split_ifs<;>first|rfl|omega|(congr 1;exact Nat.eq_sub_of_add_eq' rfl)
  rw[Array.getElem!_set!_ne _ _ _ _ (Ne.symm d)];clear*-d;split_ifs<;>first|rfl|omega
@[inline]def ln(d:CD)(o y:Ἀ)(xb nq q r:Ἄ)(S:Ἅ)(a:Ἂ Ἃ):Ἂ Ἃ:=
 if r<nq then su a (q+r) (ju d o y xb (J0 r) S)else a
theorem zpC_R(d:CD)(o y xb:Nat)(hz9:d.z9=true)(hR:czRX d.X d.P9 d.V9 d.hX ((if d.D then o else 0)*d.P9+y*(d.s.w+2)+xb) d.k9=true)(r:Nat)(hr:r<8):zpC d o y (xb+r)=true:=by
 unfold zpC;rw[if_pos hz9]
 have k:=czRX_X9 d.X d.P9 d.V9 d.hX d.k9 _ hR r hr
 rw[show (if d.D then o else 0)*d.P9+y*(d.s.w+2)+(xb+r)=(if d.D then o else 0)*d.P9+y*(d.s.w+2)+xb+r by omega];exact k
theorem owZ(d:CD)(o y x:Nat)(S:Ἅ)(hb:d.b[o]! =0)(hz:zpC d o y x=true):ow2 d o y x S=act d.J 0:=by
 unfold ow2;rw[if_pos (by simp[hb,hz])]
@[inline]def zk(d:CD)(o y:Nat)(xb:Ἄ):Ἆ:=d.b[o]! == 0&&d.z9&&czRX d.X d.P9 d.V9 d.hX ((if d.D then o else 0)*d.P9+y*(d.s.w+2)+xb.toNat) d.k9
@[inline]def jz(d:CD)(o y:Nat)(xb:Ἄ)(z:Ἆ)(r:Nat)(S:Ἅ):Word:=
 if z&&r<8 then (let c:=fz S d.lo d.sp d.hh;if c<4294967296 then dR d.J (N3 c) else act d.J 0) else ju d o y xb r S
theorem jzS(d:CD)(o y:Nat)(xb:Ἄ):jz d o y xb (zk d o y xb)=ju d o y xb:=by
 funext r S;unfold jz;split
 ·rename_i z
  simp only[zk,Η,AQ,AJ]at z
  obtain⟨⟨⟨hb,h9⟩,hR⟩,hr⟩:=z
  unfold ju;dsimp only;split
  ·rfl
  ·exact (owZ d o y (xb.toNat+r) S hb (zpC_R d o y xb.toNat h9 hR r hr)).symm
 ·rfl
@[inline]def fC8g(d:CD)(o y:Ἀ)(xb nq q:Ἄ)(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ)(a:Ἂ Ἃ)(h:8<=nq/\(J0 q)+8<=(O0 a)/\(O0 a)<=4294967295)(f:Ἀ→Ἅ→Ἃ):Ἂ Ἃ:=
  ((((((((a.uset q (f 0 a0) (by omega)).uset (q+1) (f 1 a1) (by simp only[BT];exact Dense.F.uk (by simp) h.2.1 (by omega))).uset (q+2) (f 2 a2) (by simp only[BT];exact Dense.F.uk (by simp) h.2.1 (by omega))).uset (q+3) (f 3 a3) (by simp only[BT];exact Dense.F.uk (by simp) h.2.1 (by omega))).uset (q+4) (f 4 a4) (by simp only[BT];exact Dense.F.uk (by simp) h.2.1 (by omega))).uset (q+5) (f 5 a5) (by simp only[BT];exact Dense.F.uk (by simp) h.2.1 (by omega))).uset (q+6) (f 6 a6) (by simp only[BT];exact Dense.F.uk (by simp) h.2.1 (by omega))).uset (q+7) (f 7 a7) (by simp only[BT];exact Dense.F.uk (by simp) h.2.1 (by omega)))
@[noinline]def fC8(d:CD)(o y:Ἀ)(xb nq q:Ἄ)(a0 a1 a2 a3 a4 a5 a6 a7:Ἅ)(a:Ἂ Ἃ):Ἂ Ἃ:=
 if h:8<=nq/\(J0 q)+8<=(O0 a)/\(O0 a)<=4294967295 then
  if czZ a0 then fC8g d o y xb nq q a0 a1 a2 a3 a4 a5 a6 a7 a h (jz d o y xb (zk d o y xb))
  else fC8g d o y xb nq q a0 a1 a2 a3 a4 a5 a6 a7 a h (ju d o y xb)
 else
 ln d o y xb nq q 7 a7 (ln d o y xb nq q 6 a6 (ln d o y xb nq q 5 a5 (ln d o y xb nq q 4 a4
  (ln d o y xb nq q 3 a3 (ln d o y xb nq q 2 a2 (ln d o y xb nq q 1 a1 (ln d o y xb nq q 0 a0 a)))))))
@[inline]def a9(d:CD)(o y:Ἀ)(xb nq q:Ἄ)(A:Dense.F.A8)(a:Ἂ Ἃ):Ἂ Ἃ:=
 fC8 d o y xb nq q A.a0 A.a1 A.a2 A.a3 A.a4 A.a5 A.a6 A.a7 a
lemma gE(g:Ἀ->kF)(n q:Ἀ):∀m r (a:Ἂ kF),n-r=m->r<=n->
  pl g (n+1) r q a=(pl g n r q a).set! (q+n) (g n):=by
 intro m;induction m<;>intro r a d e;have:r=n:=(by omega);subst this;conv_lhs=>rw[pl,AD (by exact lt_add_one r),pl,AC (by exact B4 (r+1))]
 conv_rhs=>rw[pl,AC (by exact B4 r)]
 rename_i j;conv_lhs=>rw[pl,AD (by exact Order.lt_add_one_iff.mpr e)]
 rw[j (r+1) _ (by (clear * - d e;omega)) (by (clear * - d e;omega))];conv_rhs=>rw[pl,AD (by exact N1 d)]
lemma ka_h1{a b c k:Ἀ}(p:a+c<kX)(m:b=k)(j:b<c):a+b<kX/\min (k+1) c=min k c+1/\min k c=k:=
 have J:k<c:=m▸j
 ⟨AO (J1 j.le a) p,(Nat.min_eq_left J).trans (congrArg (·+1) (Nat.min_eq_left J.le).symm),Nat.min_eq_left J.le⟩
lemma ka_h2{b c k:Ἀ}(m:b=k)(j:¬b<c):min (k+1) c=min k c:=
 have K:c<=k:=m▸W4 j
 (Nat.min_eq_right (K.trans k.le_succ)).trans (Nat.min_eq_right K).symm
lemma ka(d:CD)(o y:Ἀ)(g e q:Ἄ)(A:Dense.F.A8)(a:Ἂ kF)(p:(J0 q)+(J0 e)<kX)
  (r:Ἄ)(k:Ἀ)(m:(J0 r)=k)(S:Ἅ)(l:S=A.f k):
  ln d o y g e q r S (pl (λr=>bV d o y ((J0 g)+r) (A.f r)) (po k (J0 e)) 0 (J0 q) a)=
  pl (λr=>bV d o y ((J0 g)+r) (A.f r)) (po (k+1) (J0 e)) 0 (J0 q) a:=by subst l;rw[ln];by_cases h:r<e;have j:=ε.1 h;have H:=ka_h1 p m j;rw[AD h,jk,Dense.aj H.1,iF,m,H.2.1,gE _ _ _ _ _ _ rfl (by exact ϟ (min k eб)),H.2.2];have j:=mt ε.2 h;rw[AC h,ka_h2 m j]
lemma iT(d:CD)(o y:Ἀ)(g e q:Ἄ)(A:Dense.F.A8)(a:Ἂ kF)(l:(J0 q)+(J0 e)<kX):
  a9 d o y g e q A a=pl (λr=>bV d o y ((J0 g)+r) (A.f r)) (po 8 (J0 e)) 0 (J0 q) a:=by
 unfold a9 fC8;split
 · rename_i h;rw[jzS d o y g,ite_self];unfold fC8g;obtain⟨m,v,x⟩:=h
   have p:=U8.mp m;simp only[kD] at p
   have E:∀k:Ἄ,(J0 k)<8->(J0 (q+k))=(J0 q)+(J0 k):=λk B=>Dense.aj (by (clear * - l p B;omega))
   rw[(by exact Nat.min_eq_left p:po 8 (J0 e)=8)]
   have s5ux:∀(a:Ἂ kF)(u:Ἄ)(v:kF)(h:(J0 u)<(O0 a)),a.uset u v h=a.set! (J0 u) v:=λa u v h=>by rw[pP,Array.set!_eq_setIfInBounds,Array.setIfInBounds,AT h]
   iterate 8 rw[s5ux]
   rw[E 1 (by simp only [kD,Nat.one_lt_ofNat]),E 2 (by simp only [kD,Nat.reduceLT]),E 3 (by simp only [kD,Nat.reduceLT]),E 4 (by simp only [kD,Nat.reduceLT]),E 5 (by simp only [kD,Nat.reduceLT]),E 6 (by simp only [kD,Nat.reduceLT]),E 7 (by simp only [kD,Nat.lt_add_one])]
   iterate 8 rw[pl,AD (by decide)]
   rw[pl,AC (by decide)];simp only[iF,kD,ϑ];rfl
 have L:=ka d o y g e q A a l;have j:=L 0 0 (by simp only [Ϧ]) A.a0 rfl;rw[Nat.zero_min,pl,AC (by exact U5 0)]at j
 simp only[j,L 1 1 (by simp only [kD]) A.a1 rfl,L 2 2 (by simp only [kD]) A.a2 rfl,L 3 3 (by simp only [kD]) A.a3 rfl,L 4 4 (by simp only [kD]) A.a4 rfl,L 5 5 (by simp only [kD]) A.a5 rfl,L 6 6 (by simp only [kD]) A.a6 rfl,L 7 7 (by simp only [kD]) A.a7 rfl,Nat.reduceAdd]
lemma l0(d:Ἄ)(h:0<d):0<(J0 d):=by simpa only[kD]using ε.mp h
lemma l8(d:Ἄ)(h:8<d):8<(J0 d):=by simpa only[kD]using ε.mp h
lemma u8(x:Ἄ)(h:(J0 x)+8<kX):(J0 (x+8))=(J0 x)+8:=by rw[Dense.aj (by simp only[kD];exact I7 h)];simp only[kD]
lemma s8(d:Ἄ)(h:8<d):(J0 (d-8))=(J0 d)-8:=by have:=l8 d h;simpa only[kD]using USize.toNat_sub_of_le d 8 (U8.mpr (by simp only[kD];exact Nat.le_of_add_left_le this))
theorem ey_h1{t n:Ἄ}{Tz:Ἂ Ἀ}(h:t<n)(hn:(J0 n)<=(O0 Tz)):(J0 t)<(O0 Tz):=U1 h hn
theorem ey_h2{X:ἀ}{Tz:Ἂ Ἀ}{n i t:Ἄ}{M:Ἀ}(hn:(J0 n)<=(O0 Tz))(hT:∀t<(J0 n),Tz[t]!<=M)(hi:(J0 i)+M+8<=(kS X))(h:t<n):(J0 (i+(G0 (Tz.uget t (ey_h1 h hn)))))+8<=(kS X):=by
 have g:=(iQ Tz t (ey_h1 h hn)).trans_le (hT _ (ε.mp h))
 have a:(J0 (i+(G0 (Tz.uget t (ey_h1 h hn)))))<=(J0 i)+(Tz.uget t (ey_h1 h hn)):=(lJ _ _).trans_le (U2 (Ay _ _) (J1 ((USize.toNat_ofNat).trans_le (Ay _ _)) _))
 exact U2 (L1 (U2 a (J1 g _)) 8) hi
theorem ey_h3{Wz:ἀ}{wo n t:Ἄ}(hW:(J0 wo)+(J0 n)<=(kS Wz))(h:t<n):(J0 (wo+t))<(kS Wz):=C0 ((lJ _ _).trans_le (Ay _ _)) (U1 (O1 (ε.mp h) _) hW)
theorem ey_h4{k:Ἀ}(j:Ἄ){X:ἀ}(hJ:(J0 j)+8<=(kS X)):(J0 (j+@OfNat.ofNat Ἄ k _))<(kS X)+(k-7):=by
 have a:(J0 (j+@OfNat.ofNat Ἄ k _))<=(J0 j)+k:=(lJ _ _).trans_le (U2 (Ay _ _) (J1 ((USize.toNat_ofNat).trans_le (Ay _ _)) _))
 exact C0 a (C0 (U2 (J1 (U2 (Nat.le_add_of_sub_le (Nat.le_refl (k-7))) (B3 (Τ _ _))) _) (B3 (pA _ 7 _).symm)) (U1 (X3 (I3 _) _) (L1 hJ _)))
@[specialize]def ey(W:CD->Ἀ->Ἀ->Ἄ->Ἄ->Ἄ->Dense.F.A8->Ἂ Ἃ->Ἂ Ἃ)(Wz X:@&ἀ)(Tz:@&Ἂ Ἀ)(wo n:Ἄ)(M:Ἀ)
  (hs:(kS X)<2^(32:Ἀ)/\(kS Wz)<2^(32:Ἀ))
  (hn:(J0 n)<=(O0 Tz)) (hT:∀t<(J0 n),Tz[t]!<=M) (hW:(J0 wo)+(J0 n)<=(kS Wz))
  (i:Ἄ) (hi:(J0 i)+M+8<=(kS X)) (d:@&CD) (o y:Ἀ) (xb nq q:Ἄ) (a:Ἂ Ἃ)
  (t:Ἄ) (a0 a1 a2 a3 a4 a5 a6 a7:Ἅ):Ἂ Ἃ:=
 if h:t<n then
  have ht:(J0 t)<(J0 n):=h
  have hJ:(J0 (i+(G0 (Tz.uget t (ey_h1 h hn)))))+8<=(kS X):=ey_h2 hn hT hi h
  let j:=i+(G0 (Tz.uget t (ey_h1 h hn)))
  let w:=Wz.uget (wo+t) (ey_h3 hW h)
  ey W Wz X Tz wo n M hs hn hT hW i hi d o y xb nq q a (t+1)
   (a0+X.uget (j+0) (ey_h4 j hJ)*w)
   (a1+X.uget (j+1) (ey_h4 j hJ)*w)
   (a2+X.uget (j+2) (ey_h4 j hJ)*w)
   (a3+X.uget (j+3) (ey_h4 j hJ)*w)
   (a4+X.uget (j+4) (ey_h4 j hJ)*w)
   (a5+X.uget (j+5) (ey_h4 j hJ)*w)
   (a6+X.uget (j+6) (ey_h4 j hJ)*w)
   (a7+X.uget (j+7) (ey_h4 j hJ)*w)
 else W d o y xb nq q ⟨a0,a1,a2,a3,a4,a5,a6,a7⟩ a
termination_by (J0 n)-(J0 t)
decreasing_by exact C5 (ε.mp h) (V6 (I3 _) (Dense.ac h).symm)
lemma g9(W:CD->Ἀ->Ἀ->Ἄ->Ἄ->Ἄ->Dense.F.A8->Ἂ kF->Ἂ kF)(g X:ἀ)(e:Ἂ Ἀ)(l n:Ἄ)(M:Ἀ)
  (U:(kS X)<2^(32:Ἀ)/\(kS g)<2^(32:Ἀ))
  (hn:(J0 n)<=(O0 e)) (N:∀t<(J0 n),e[t]!<=M) (hW:(J0 l)+(J0 n)<=(kS g))
  (i:Ἄ) (O:(J0 i)+M+8<=(kS X)) (d:CD) (o y:Ἀ) (xb nq q:Ἄ) (a:Ἂ kF) (m:Ἀ) (t:Ἄ)
  (p u v x B D G H:Ἅ) (A:Dense.F.A8) (hm:(J0 n)-(J0 t)=m)
  (K:∀r<8,A.f r=Dense.F.ao (λs=>X.data[(J0 i)+e[s]!+r]!*g.data[(J0 l)+s]!) (J0 n) (J0 t)
   (Dense.F.A8.f ⟨p,u,v,x,B,D,G,H⟩ r)):
  ey W g X e l n M U hn N hW i O d o y xb nq q a t p u v x B D G H=W d o y xb nq q A a:=by open Dense Dense.F in
 induction m generalizing t p u v x B D G H A
 have h:¬t<n:=λh=>Z8 (ε.mp h) hm
 rw[ey,dif_neg h];congr 1;exact j2 _ _ λr I=>by rw[K r I,kp _ _ _ _ (mt ε.mpr h)]
 rename_i ih;have h:t<n:=ε.mpr (N1 hm);have Y:=ε.mp h;rw[ey,AT h];have hM:=N _ Y;have L:=cj (i:=i) U.1 O hM;have rx:∀(k:Ἀ)(hk:k<8)(j:(J0 (i+(G0 (e.uget t (by exact U1 h hn)))+G0 k))<(kS X)),X.uget (i+(G0 (e.uget t (by exact U1 h hn)))+G0 k) j=X.data[(J0 i)+e[(J0 t)]!+k]!:=λk hk j=>by have:=Conv.aa k (hk.trans (by decide));rw[ug,iQ,aj (by rw[L,this];exact(((N7 (J1 hM _) hk).trans_le O).trans U.1).trans_le ς),L,this]
 have J:∀j,g.uget (l+t) j=g.data[(J0 l)+(J0 t)]!:=λj=>by rw[ug,aj (AN (K5 (AN (O1 Y _) hW) U.2) ς)]
 refine ih (t+1) _ _ _ _ _ _ _ _ A (by rw[ac h,<-F7,hm,BK]) λr I=>?_;rw[K r I,kq _ _ _ _ Y,ac h];congr 1;interval_cases r<;>exact congrArg (_+·) (congrArg₂ (·*·) (rx _ I _).symm (J _).symm)
noncomputable def dK(d:CD)(c:CC d.M)(o y xb r:Ἀ):Ἅ:=
 Dense.F.ao (λt=>d.X.data[(if d.D then o else 0)*((d.s.h+2)*(d.s.w+2))+y*(d.s.w+2)+xb+
  c.Tz[t]!+r]!*c.Wz.data[c.wo+t]!) c.n 0 (Dense.F.dw oneW*Dense.F.dw d.b[o]!)
@[inline]def gY(W:CD->Ἀ->Ἀ->Ἄ->Ἄ->Ἄ->Dense.F.A8->Ἂ Ἃ->Ἂ Ἃ)(d:@&CD)(c:@&CC d.M)(o y:Ἀ)(xb nq q i:Ἄ)
  (hi:(J0 i)+d.M+8<=(kS d.X)) (bp:Ἅ) (a:Ἂ Ἃ):Ἂ Ἃ:=
 ey W c.Wz d.X c.Tz (G0 c.wo) (G0 c.n) d.M ⟨d.hX,c.h5⟩
  (by have:=ς;have:=c.h3;rw[Dense.aa (by omega)];exact c.h1)
  (by have:=ς;have:=c.h3;rw[Dense.aa (by omega)];exact c.h4)
  (by have:=ς;have:=c.h3;rw[Dense.aa (by omega),Dense.aa (by omega)];exact c.h2)
  i hi d o y xb nq q a 0 bp bp bp bp bp bp bp bp
lemma gZ(d:CD)(c:CC d.M)(o y:Ἀ)(e m q i:Ἄ)(p:(J0 i)+d.M+8<=(kS d.X))(l:Ἅ)
  (a:Ἂ kF) (j:(J0 i)=(if d.D then o else 0)*((d.s.h+2)*(d.s.w+2))+y*(d.s.w+2)+(J0 e))
  (g:l=Dense.F.dw ld*Dense.F.dw d.b[o]!):
  gY a9 d c o y e m q i p l a=a9 d o y e m q (eH (dK d c o y (J0 e))) a:=by
 subst g;unfold gY;have:=ς;have:=c.h3;refine g9 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ rfl λr u=>?_;rw[fW _ _ u];unfold dK;simp only[j,Dense.aa ((by omega):c.wo<kX),Dense.aa ((by omega):c.n<kX),Ϧ,Dense.F.f8]
@[specialize]def rB(W:CD->Ἀ->Ἀ->Ἄ->Ἄ->Ἄ->Dense.F.A8->Ἂ Ἃ->Ἂ Ἃ)(d:@&CD)(c:@&CC d.M)(o y:Ἀ)(xb nq q i:Ἄ)(bp:Ἅ)(a:Ἂ Ἃ)
  (hI:(J0 i)+(J0 nq)+d.M+7<=(kS d.X)):Ἂ Ἃ:=
 if h:0<nq then
  let a':=gY W d c o y xb nq q i (by have:=l0 nq h;omega) bp a
  if h8:8<nq then
   rB W d c o y (xb+8) (nq-8) (q+8) (i+8) bp a'
    (by have:=l8 nq h8;have:=ς;have:=d.hX;rw[u8 i (by omega),s8 nq h8];omega)
  else a'
 else a
termination_by (J0 nq)
decreasing_by have:=l8 nq h8;rw[s8 nq h8];omega
lemma je_h1(d:CD)(c:CC d.M)(o y:Ἀ)(F:Ἀ->kF)(G:∀g x,g<=x->x<g+8->x<d.s.w->bV d o y x (dK d c o y g (x-g))=F x)(g e q:Ἀ)(hx:g+e=d.s.w)(j:Ἀ)(h1:q<=j)(h2:j<q+po 8 e):bV d o y (g+(j-q)) ((eH (dK d c o y g)).f (j-q))=F (g+(j-q)):=by
 have t:=G4 h1 h2;have t8:=t.trans_le (min_le_left _ _);rw[fW _ _ t8,<-G g _ (V1 _ _) (O1 t8 _) ((O1 (t.trans_le (min_le_right _ _)) _).trans_eq hx),ϱ]
lemma je_h2(q e g j:Ἀ)(B:8<e)(X:kF)(f:Ἀ->kF)(b:kF)(I:q<=j->j<q+po 8 e->b=f (g+(j-q))):(if q+8<=j∧j<q+8+(e-8) then f (g+8+(j-(q+8))) else if q+0<=j∧j<q+po 8 e then b else X)=if q<=j∧j<q+e then f (g+(j-q)) else X:=by
 split_ifs with A1 A2 A3 A4 A5
 ·congr 1;rw[pA,V4,M4 (Nat.le_sub_of_add_le' A1.1)]
 ·exact absurd ⟨(V1 _ _).trans A1.1,A1.2.trans_eq (by rw[pA,M4 B.le])⟩ A2
 ·exact I A3.1 A3.2
 ·exact absurd ⟨A3.1,A3.2.trans_le (J1 (min_le_right _ _) _)⟩ A4
 ·exact (Z7 j (q+8)).elim (λw=>absurd ⟨A5.1,w.trans_eq (by rw[min_eq_left B.le])⟩ A3) (λw=>absurd ⟨w,A5.2.trans_eq (by rw[pA,M4 B.le])⟩ A1)
 ·rfl
lemma je_h4(m e g q i a s E8 G8 Q8 I8 X Y:Ἀ)(hm:e<=m+1)(hx:g+e=s)(L:i=X+Y+g)(hq:q+e<=a)(B:8<e)(U:E8=e-8)(N:G8=g+8)(O:Q8=q+8)(e4:I8=i+8):E8<=m∧G8+E8=s∧I8=X+Y+G8∧Q8+E8<=a:=by
 subst U N O e4 L;exact⟨M8 (hm.trans (J1 ((by decide):1<=8) m)),by rw[pA,M4 B.le];exact hx,pA _ _ _,by rw[pA,M4 B.le];exact hq⟩
lemma je_h3(q e g j:Ἀ)(B:¬8<e)(X:kF)(f:Ἀ->kF)(b:kF)(I:q<=j->j<q+po 8 e->b=f (g+(j-q))):(if q+0<=j∧j<q+po 8 e then b else X)=if q<=j∧j<q+e then f (g+(j-q)) else X:=by
 split_ifs with A1 A2 A3
 ·exact I A1.1 A1.2
 ·exact absurd ⟨A1.1,A1.2.trans_le (J1 (min_le_right _ _) _)⟩ A2
 ·exact absurd ⟨A3.1,A3.2.trans_le (J1 (le_min (W4 B) AP) _)⟩ A1
 ·rfl
lemma je(d:CD)(c:CC d.M)(o y:Ἀ)(u:Ἅ)(H:u=Dense.F.dw ld*Dense.F.dw d.b[o]!)(F:Ἀ->kF)
  (G:∀g x,g<=x->x<g+8->x<d.s.w->bV d o y x (dK d c o y g (x-g))=F x):
  ∀m (g e q i:Ἄ) (a:Ἂ kF) (v:(J0 i)+(J0 e)+d.M+7<=(kS d.X)),(J0 e)<=m->
   (J0 g)+(J0 e)=d.s.w->
   (J0 i)=(if d.D then o else 0)*((d.s.h+2)*(d.s.w+2))+y*(d.s.w+2)+(J0 g)->
   (J0 q)+(J0 e)<=(O0 a)->(O0 a)<2^(32:Ἀ)->
   (O0 (rB a9 d c o y g e q i u a v))=(O0 a)/\∀j<(O0 a),(rB a9 d c o y g e q i u a v)[j]! =
    if (J0 q)<=j/\j<(J0 q)+(J0 e)then F ((J0 g)+(j-(J0 q)))else a[j]!:=by
 intro m;induction m with
 |zero=>intro g e q i a v hm hx L hq J;rw[rB,dif_neg (λh=>(l0 e h).ne' (Nat.eq_zero_of_le_zero hm))];exact⟨ἅ,λj _=>by rw[AC (by clear*-hm;omega)]⟩
 |succ m ih=>
  intro g e q i a v hm hx L hq J;have S:=ς;have:=d.hX;by_cases h:0<e
  .rw[rB,AT h];dsimp only;rw[gZ d c o y g e q i _ u a L H,iT _ _ _ _ _ _ _ _ ((hq.trans_lt J).trans_le S)];obtain⟨p,K⟩:=jS (λr=>bV d o y ((J0 g)+r) ((eH (dK d c o y (J0 g))).f r)) (po 8 (J0 e)) (J0 q) _ 0 a rfl (ϟ _);have I:=je_h1 d c o y F G (J0 g) (J0 e) (J0 q) hx
   by_cases l:8<e
   .have B:=l8 e l;rw[AT l];have W:=(V1 _ _).trans ((V1 _ _).trans v);have N:=u8 g (((N7 ((K2 _ _).trans_eq L.symm) B).trans_le W).trans (this.trans_le S));have O:=u8 q (((O1 B _).trans_le hq).trans (J.trans_le S));have U:=s8 e l;have e4:=u8 i (((O1 B _).trans_le W).trans (this.trans_le S));obtain⟨k1,k2,k3,k4⟩:=je_h4 _ _ _ _ _ _ _ _ _ _ _ _ _ hm hx L hq B U N O e4;obtain⟨s1,Y⟩:=ih (g+8) (e-8) (q+8) (i+8) _ _ k1 k2 k3 (by rw[p];exact k4) (by rwa[p]);refine⟨by rw[s1,p],λj A=>?_⟩;rw[Y j (by rwa[p]),K j A,N,O,U];exact je_h2 _ _ _ j B _ F _ (I j)
   rw[dif_neg l];refine⟨p,λj A=>?_⟩;rw[K j A];exact je_h3 _ _ _ j (λhh=>l (ε.2 (by simp only[kD];exact hh))) _ F _ (I j)
  have h0:(J0 e)=0:=by have:=mt ε.2 h;simp only[kD]at this;exact Nat.eq_zero_of_not_pos this
  rw[rB,dif_neg h];exact⟨ἅ,λj _=>by rw[AC (λhh=>B4 _ ((hh.1.trans_lt hh.2).trans_eq (congrArg _ h0)))]⟩
@[specialize]def c8(h len:Ἀ)(V:Ἄ)(i0:Ἀ)(hN:i0+h*(J0 V)<Al)
  (f:(y:Ἀ)->y<h->(q:Ἀ)->(i:Ἄ)->(J0 i)=i0+y*(J0 V)->Ἂ Ἃ->Ἂ Ἃ)
  (y q:Ἀ) (i:Ἄ) (hi:(J0 i)=i0+y*(J0 V)) (a:Ἂ Ἃ):Ἂ Ἃ:=
 if hy:y<h then
  c8 h len V i0 hN f (y+1) (q+len) (i+V)
   (by have:=ι (J0 V) hy;rw[Ο]at this⊢;rw[Dense.aj (by omega),hi];exact pA i0 (y*Vб) Vб)
   (f y hy q i hi a)
 else a
termination_by h-y
decreasing_by all_goals omega
lemma eE(h d:Ἀ)(V:Ἄ)(o:Ἀ)(x:o+h*(J0 V)<kX)
  (f:(y:Ἀ)->y<h->(q:Ἀ)->(i:Ἄ)->(J0 i)=o+y*(J0 V)->Ἂ kF->Ἂ kF) (r N:Ἀ)
  (G:Ἀ->kF)
  (F:∀y (g:y<h) q (i:Ἄ) (e:(J0 i)=o+y*(J0 V)) (a:Ἂ kF),q=r+y*d->(O0 a)=N->
   (O0 (f y g q i e a))=(O0 a)/\∀j<(O0 a),(f y g q i e a)[j]! =if q<=j/\j<q+d then G j else a[j]!):
  ∀m y q (i:Ἄ) (e:(J0 i)=o+y*(J0 V)) (a:Ἂ kF),h-y=m->y<=h->q=r+y*d->(O0 a)=N->
   (O0 (c8 h d V o x f y q i e a))=(O0 a)/\
   ∀j<(O0 a),(c8 h d V o x f y q i e a)[j]! =if q<=j/\j<r+h*d then G j else a[j]!:=by
 intro m;induction m<;>intro y q i e a H g p l
 .cases g.antisymm (O4 H);rw[c8,dif_neg (AX _)];exact⟨ἅ,λj _=>by rw[AC (by (clear * - p;omega))]⟩
 rename_i K;have g:y<h:=N1 H;rw[c8,AT g];obtain⟨v,B⟩:=F y g q i e a p l;have J:q+d=r+(y+1)*d:=by rw[p];ring
 obtain⟨L,D⟩:=K _ _ (i+V) _ (f y g q i e a) (congrArg Nat.pred H) g J (v.trans l);refine⟨L.trans v,λj A=>?_⟩;rw[D j (v▸A),B j A];have:=J▸J1 (ι d g) r;clear*-this;split_ifs<;>first|rfl|omega
@[specialize]def db(W:CD->Ἀ->Ἀ->Ἄ->Ἄ->Ἄ->Dense.F.A8->Ἂ Ἃ->Ἂ Ἃ)(d:@&CD)(c:@&CC d.M)(o q:Ἀ)(bp:Ἅ)(a:Ἂ Ἃ):Ἂ Ἃ:=
 let V:=d.s.w+2
 let i0:=(if d.D then o else 0)*((d.s.h+2)*V)
 if hb:i0+(d.s.h-1)*V+d.s.w+d.M+7<=(kS d.X)then
  c8 d.s.h d.s.w (G0 V) i0
   (by
    have K:=C0 (Y3 (V1 (i0+(d.s.h-1)*V+d.s.w) d.M) (N2 ἅ:2<=7)) (C0 hb (U1 d.hX ς))
    exact C0 (J1 (U3 d.s.h (B3 (Dense.aa (n:=V) (C0 (L1 (K2 d.s.w (i0+(d.s.h-1)*V)) 2) K)))) i0) (C0 (U2 (J1 (Nat.le_add_of_sub_le (B3 (Nat.sub_one_mul d.s.h V).symm)) i0) (B3 ((pA i0 _ V).symm.trans (pA _ d.s.w 2).symm))) K))
   (λy hy q i hi a=>rB W d c o y 0 (G0 d.s.w) (G0 q) i bp a (by
    have K:=C0 hb (U1 d.hX ς)
    exact U2 (B3 (congrArg (·+d.M+7) (congrArg₂ (·+·) (hi.trans (congrArg (i0+y*·) (Dense.aa (n:=V) (C0 (U2 (L1 (K2 d.s.w (i0+(d.s.h-1)*V)) 2) (Y3 (V1 (i0+(d.s.h-1)*V+d.s.w) d.M) (N2 ἅ:2<=7))) K)))) (Dense.aa (n:=d.s.w) (C0 (U2 (K2 d.s.w (i0+(d.s.h-1)*V)) (U2 (V1 (i0+(d.s.h-1)*V+d.s.w) d.M) (V1 _ 7))) K))))) (U2 (L1 (L1 (L1 (J1 (ι V (Nat.le_sub_one_of_lt hy)) i0) d.s.w) d.M) 7) hb)))
   0 q (G0 i0) ((Dense.aa (n:=i0) (C0 (U2 (U2 (V1 i0 ((d.s.h-1)*V)) (V1 _ d.s.w)) (U2 (V1 _ d.M) (V1 _ 7))) (C0 hb (U1 d.hX ς)))).trans (congrArg (i0+·) (ϳ _)).symm) a
 else a
lemma iG(d:CD)(c:CC d.M)(o q:Ἀ)(l:Ἅ)(a:Ἂ kF)(N:Ἀ)(G:Ἀ->kF)
  (m:∀y (v:y<d.s.h) q' (i:Ἄ) (e:Ἂ kF) (u:(J0 i)+(J0 (G0 d.s.w))+d.M+7<=(kS d.X)),
   (J0 i)=(if d.D then o else 0)*((d.s.h+2)*(d.s.w+2))+y*(d.s.w+2)->q'=q+y*d.s.w->(O0 e)=N->
   (O0 (rB a9 d c o y 0 (G0 d.s.w) (G0 q') i l e u))=(O0 e)/\
   ∀j<(O0 e),(rB a9 d c o y 0 (G0 d.s.w) (G0 q') i l e u)[j]! =if q'<=j/\j<q'+d.s.w then G j else e[j]!)
  (A:(O0 a)=N) (H:d.s.w+2<kX)
  (B:0<d.s.h->(if d.D then o else 0)*((d.s.h+2)*(d.s.w+2))+(d.s.h-1)*(d.s.w+2)+d.s.w+d.M+7<=(kS d.X)):
  (O0 (db a9 d c o q l a))=(O0 a)/\
  ∀j<(O0 a),(db a9 d c o q l a)[j]! =if q<=j/\j<q+d.s.h*d.s.w then G j else a[j]!:=by
 dsimp only[db];by_cases g:(if d.D then o else 0)*((d.s.h+2)*(d.s.w+2))+(d.s.h-1)*(d.s.w+2)+d.s.w+d.M+7<=(kS d.X);rw[AT g];exact eE _ _ _ _ _ _ q N G (λy v q' i F e r p=>m y v q' i e _ (by rw[F,Dense.aa H]) r p) _ 0 q _ _ a rfl (ϟ _) (by simp only [zero_mul,Av]) A;rw[dif_neg g];use ἅ,λj _=>by rw[(pM.mp λh=>g (B (ϵ h)):d.s.h=0),ϳ,ϑ,AC (by (clear * -;omega))]
def bA(K:Ἂ Ἃ)(T:Ἂ Ἀ)(M ok k:Ἀ)(hT:∀t<k,T[t]!<=M)(hk:k<2^(32:Ἀ))(t:Ἀ)(Tz:Ἂ Ἀ)
  (Wz:ἀ) (hs:(O0 Tz)=(kS Wz)) (hn:(O0 Tz)<=t) (ht:t<=k) (hz:∀j<(O0 Tz),Tz[j]!<=M):CC M:=
 if h:t<k then
  let w:=K[ok+t]!
  if Core.mW w=0 then bA K T M ok k hT hk (t+1) Tz Wz hs (Nat.le_add_right_of_le hn) h hz
  else bA K T M ok k hT hk (t+1) (Tz.push T[t]!) (Wz.push (Dense.F.dw w)) ((ϣ ..).trans ((congrArg (·+1) hs).trans (Dense.F.bW ..).symm))
   (U2 (B3 (ϣ ..)) (L1 hn 1)) h (λj hj=>(Nat.lt_or_eq_of_le (J2 (V6 hj (ϣ ..)))).elim
    (λhjs=>U2 (B3 (((AB (Tz.push T[t]!) j hj).trans (nU hjs)).trans (AB Tz j hjs).symm)) (hz j hjs))
    (λhe=>U2 (B3 ((congrArg (fun (i:Ἀ)=>(Tz.push T[t]!)[i]!) he).trans ((AB (Tz.push T[t]!) (O0 Tz) (V6 (I3 _) (ϣ ..).symm)).trans nV))) (hT t h)))
 else⟨Tz,Wz,0,(O0 Tz),fQ,AP,B3 ((Κ _).trans hs),C0 (U2 (B3 (Κ _)) (U2 hn ht)) hk,hz,C0 (U2 (B3 hs.symm) (U2 hn ht)) hk⟩
termination_by k-t
decreasing_by all_goals exact C5 h (I3 t)
def NZ(K:Ἂ Ἃ)(T:Ἂ Ἀ)(ok t:Ἀ)(Tz:Ἂ Ἀ)(Wz:ἀ):Prop:=
 (O0 Tz)=(kS Wz)/\(O0 Tz)<=t/\∃ix:Ἀ->Ἀ,
  (∀j<(O0 Tz),ix j<t/\Tz[j]! =T[ix j]!/\Wz.data[j]! =Dense.F.dw K[ok+ix j]!)/\
  ∀z:Ἀ->ℝ,(∀s<t,Core.mW K[ok+s]! =0->z s=0)->∑j∈range (O0 Tz),z (ix j)=∑s∈range t,z s
lemma ft(K:Ἂ kF)(T:Ἂ Ἀ)(M A k:Ἀ)(H:∀t<k,T[t]!<=M)(I:k<2^(32:Ἀ)):
  ∀m t Tz Wz p u v g,k-t=m->NZ K T A t Tz Wz->
  NZ K T A k (bA K T M A k H I t Tz Wz p u v g).Tz (bA K T M A k H I t Tz Wz p u v g).Wz:=by
 intro m;induction m with
 |zero=>intro t _ _ _ _ v _ L h;cases v.antisymm (O4 L);rwa[bA,dif_neg (AX _)]
 |succ m J=>intro t Tz Wz _ _ _ _ L h;obtain⟨p,u,x,o,l⟩:=h;rw[bA,AT (N1 L)];dsimp only;split<;>rename_i G;refine J (t+1) _ _ _ _ _ _ (congrArg Nat.pred L) ⟨p,Nat.le.step u,x,λj r=>(o j r).imp_left Nat.lt.step,λz g=>?_⟩;rw[l z (λs D q=>g s (Nat.lt.step D) q),Ν,g t (Nat.lt.base t) G,Av];refine J (t+1) _ _ _ _ _ _ (congrArg Nat.pred L) ⟨by rw[ϣ,Dense.F.bW,p],by rw[ϣ];exact Nat.succ_le_succ u,λj=>if j<(O0 Tz)then x j else t,λj r=>?_,λz g=>?_⟩;by_cases d:j<(O0 Tz);obtain⟨U,N,O⟩:=o j d;refine⟨?_,?_,?_⟩<;>simp only[AD d];exact Nat.lt.step U;rwa[AB _ _ (by exact r),nU d,<-AB Tz j d];rwa[Dense.F.pg _ _ _ (Nat.lt.step (d.trans_eq p)),AD (d.trans_eq p)];rw[ϣ]at r;cases (J2 r).antisymm (not_lt.1 d);refine⟨?_,?_,?_⟩<;>simp only[AC (AX _)];exact Nat.lt.base t;rw[AB _ _ ((by simp only [ϣ,lt_add_iff_pos_right,Order.lt_one_iff])),nV];rw[Dense.F.pg _ _ _ (Z4 p.le),AC (p▸AX _)];rw[ϣ,Ν,(ϔ ἅ λj r=>by rw[AD (ϥ.1 r)]:∑j∈kP (O0 Tz),z (if j<(O0 Tz)then x j else t)=∑j∈kP (O0 Tz),z (x j))];dsimp only;rw[AC (AX _),l z (λs D q=>g s (Nat.lt.step D) q),Ν]
def dF(d:CD)(Z:Ἆ)(Wd:@&ἀ)(hT:∀t<d.k,d.T[t]!<=d.M)(hk:d.k<2^(32:Ἀ))(hkT:d.k<=(O0 d.T))
  (hW:Z=ἃ\/d.Y*d.k<=(kS Wd)/\(kS Wd)<2^(32:Ἀ)) (hY:d.Y*d.k<2^(32:Ἀ)) (o:Ἀ) (ho:o<d.Y):CC d.M:=
 if hZ:Z=ἃ then
  bA d.K d.T d.M (o*d.k) d.k hT hk 0 (AS d.k) (AR d.k) (by rw[cZ];simp)
   (ϟ _) (ϟ _) (λj hj=>by simp at hj)
 else
  have hok:o*d.k+d.k<=d.Y*d.k:=by rw[<-Ο];exact ι _ ho
  ⟨d.T,Wd,o*d.k,d.k,fQ,hkT,hok.trans (hW.resolve_left hZ).1,(by exact C0 hok hY),hT,(hW.resolve_left hZ).2⟩
lemma eM_h1{k t:Ἀ}(a:k-t=0):¬t<k:=W3.2 (K3.1 a)
lemma eM_h2{k t m:Ἀ}(a:k-t=m+1):t<k/\k-(t+1)=m:=⟨N1 a,(Nat.sub_succ k t).trans (congrArg Nat.pred a)⟩
lemma eM(K:Ἂ kF)(T:Ἂ Ἀ)(M q k:Ἀ)(o:∀t<k,T[t]!<=M)(p:k<2^(32:Ἀ)):
  ∀m t Tz d g e j l,k-t=m->
  (bA K T M q k o p t Tz d g e j l).n=(O0 (bA K T M q k o p t Tz d g e j l).Tz)/\
  (bA K T M q k o p t Tz d g e j l).wo=0:=by
 intro m;induction m with
 |zero=>intros;rename_i a;rw[bA,dif_neg (eM_h1 a)];exact⟨ἅ,ἅ⟩
 |succ m r=>intros;rename_i a;have b:=eM_h2 a;rw[bA,AT b.1];dsimp only;split<;>exact r _ _ _ _ _ _ _ b.2
lemma fm(d:CD)(Z:Ἆ)(p:ἀ)(u:∀t<d.k,d.T[t]!<=d.M)(v:d.k<2^(32:Ἀ))(G:d.k<=(O0 d.T))
  (X:Z=ἃ\/d.Y*d.k<=(kS p)/\(kS p)<2^(32:Ἀ)) (N:d.Y*d.k<2^(32:Ἀ))
  (x:Z=false->∀q<d.Y*d.k,p.data[q]! =Dense.F.dw d.K[q]!) (o:Ἀ) (U:o<d.Y)
  (c:CC d.M) (O:dF d Z p u v G X N o U=c):
  (FloatArray.size c.Wz)<2^(32:Ἀ)/\∃m:Ἀ->Ἀ,c.n<=d.k/\
   (∀j<c.n,m j<d.k/\c.Tz[j]! =d.T[m j]!/\c.Wz.data[c.wo+j]! =Dense.F.dw d.K[o*d.k+m j]!)/\
   ∀z:Ἀ->ℝ,(∀s<d.k,mW d.K[o*d.k+s]! =0->z s=0)->
    ∑j∈kP c.n,z (m j)=∑s∈kP d.k,z s:=by
 by_cases D:Z=ἃ
 .have⟨⟨l,L⟩,J,I,m,g,r⟩:(c.n=(O0 c.Tz)/\c.wo=0)/\NZ d.K d.T (o*d.k) d.k c.Tz c.Wz:=by subst O;rw[dF,AT D];exact⟨eM _ _ _ _ _ u v _ 0 _ _ _ _ _ _ rfl,ft _ _ _ _ _ u v _ 0 _ _ _ _ _ _ rfl ⟨by rw[cZ];simp,(by simp),id,λj e=>by simp at e,λz _=>(by simp)⟩⟩
  refine⟨by rw[<-J];exact C0 I v,m,l▸I,λj e=>by rw[L,Κ];exact g j (l▸e),l▸r⟩
 have ah:o*d.k+d.k<=d.Y*d.k:=Ο _ _▸ι _ U;subst O;rw[dF,dif_neg D];use(by dsimp only;exact(X.resolve_left D).2),id,AP,(λj e=>by dsimp only at e⊢;exact⟨e,ἅ,x (by simpa using D) (o*d.k+j) (by exact Formats.DecimalInterchange.Bits.mul_add_lt_mul U e)⟩),λz _=>ἅ
lemma gG(s:lR)(I:Ἂ kF)(d:s.kh=3/\s.kw=3/\s.stride=1/\s.pad=1)
  (y x t:Ἀ)(e:t/9<s.cin):
  ll s I y x t=aI s I (t/9) (y+t/3%3) (x+t%3):=by unfold ll aI;simp only[d.1,d.2.1,d.2.2.1,d.2.2.2,ϻ,Nat.reduceMul];clear*-e;split_ifs<;>first|rfl|omega
lemma gF(s:lR)(I:Ἂ kF)(d:s.kh=1/\s.kw=1/\s.stride=1/\s.pad=0)
  (y x t:Ἀ)(j:y<s.h)(g:x<s.w)(e:t<s.cin):
  ll s I y x t=aI s I t (y+1) (x+1):=by rw[ll,aI];simp only[d,ϻ,F5,Nat.mod_one,ϑ,pQ,BK];split_ifs<;>first|rfl|omega
theorem iI_h1(d t:Ἀ):(d*9+t)/9=d+t/9∧(d*9+t)/3%3=t/3%3∧(d*9+t)%3=t%3:=
 ⟨by rw[Τ (d*9) t,Ρ t d ((by decide):0<9),Τ (t/9) d],by rw[Τ (d*9) t,((Ϛ d 3 3).symm:d*9=d*3*3),Ρ t _ ((by decide):0<3),Μ],by rw[Τ (d*9) t,((Ϛ d 3 3).symm:d*9=d*3*3),Μ]⟩
lemma iI(s:lR)(I:Ἂ kF)(e:Ἀ)(q:fP s e)(y x t d:Ἀ)(u:y<s.h)(r:x<s.w)
  (p:d+pc e t<s.cin):
  ll s I y x (d*e+t)=aI s I (d+pc e t) (y+py e t) (x+px e t):=by rcases q with⟨g,l,m,o,rfl⟩|⟨g,l,m,o,rfl⟩;simp only[pc,py,px,↓reduceIte]at p⊢;obtain⟨h,i,j⟩:=iI_h1 d t;rw[gG s I ⟨g,l,m,o⟩ y x _ (h▸p),h,i,j];simp only[pc,py,px,((by decide):(1:Ἀ)≠9),↓reduceIte,ϻ]at p⊢;exact gF s I ⟨g,l,m,o⟩ y x _ u r p
lemma iS(s:lR)(I:Ἂ kF)(d:Ἀ)(e:∀q:Ἀ,kB I[q]!≠255/\kG I[q]!<=d)
  (y x t:Ἀ):kB (ll s I y x t)≠255/\kG (ll s I y x t)<=d:=by
 unfold ll;dsimp only;split
 .exact⟨(by decide),by rw[((by decide):kG (0:kF)=0)];exact ϟ _⟩
 .exact e _
lemma jR(s:lR)(I:Ἂ kF)(d:Ἀ)(e:∀q:Ἀ,kB I[q]!≠255/\kG I[q]!<=d)
  (c y x:Ἀ):kB (aI s I c y x)≠255/\kG (aI s I c y x)<=d:=by
 rw[aI];split
 .exact e _
 .exact⟨(by decide),by rw[((by decide):kG (0:kF)=0)];exact ϟ _⟩
def xF(s:ConvShape)(I:Ἂ Ἃ)(hs:(O0 I)<2^(32:Ἀ))(hw:s.w<2^(32:Ἀ)):ἀ:=
 fRU I hs (G0 s.w) (G0 (s.w+2)) (G0 (3*(s.w+2))) (G0 s.h) (G0 (s.cin*s.h)) 0 0 (G0 (s.w+3)) 0 (zPq ((s.cin*(s.h+2)+3)*(s.w+2)))
lemma dP(s:lR)(I:Ἂ kF)(d:(O0 I)<2^(32:Ἀ))(e:s.w<2^(32:Ἀ))(g:s.cin*s.h*s.w<=(O0 I))(hN:(s.cin*(s.h+2)+3)*(s.w+2)<2^(32:Ἀ))(h0:0<s.h):
  (s.cin*(s.h+2)+3)*(s.w+2)<=(kS (xF s I d e))/\(kS (xF s I d e))<2^(32:Ἀ)/\
  ∀q<(kS (xF s I d e)),(xF s I d e).data[q]! =Dense.F.dw (aI s I (q/((s.h+2)*(s.w+2))) (q/(s.w+2)%(s.h+2)) (q%(s.w+2))):=by
 obtain⟨z1,z2,z3⟩:=zPs ((s.cin*(s.h+2)+3)*(s.w+2));have:=ς;have Z:(kS (zPq ((s.cin*(s.h+2)+3)*(s.w+2))))<2^(32:Ἀ):=by rcases z2 with z|z;exact z.trans_lt (by decide);rw[z];exact hN
 rcases Nat.eq_zero_or_pos s.cin with c|c
 .have n0:(G0 (s.cin*s.h))=0:=by rw[c,ϳ];rfl
  rw[xF,n0,fRU,dif_neg (by simp)];refine⟨z1,Z,λq hq=>?_⟩;rw[z3 q hq,aI,if_neg (by simp[c])]
 have W:3*(s.w+2)<2^(32:Ἀ):=AO (ι _ (K2 3 _)) hN;have V:s.cin*(s.h+2)+3<2^(32:Ἀ):=AO (J4 _ (Nat.succ_pos _)) hN;obtain⟨u,v⟩:=fRUs I s d (G0 s.w) (G0 (s.w+2)) (G0 (3*(s.w+2))) (G0 s.h) (G0 (s.cin*s.h)) (aa _ e) (aa _ (AO (lZ _ (by decide)) W)) (aa _ W) (aa _ (AO (AZ (AZ (V1 _ 2) (lZ _ c)) (V1 _ 3)) V)) (aa _ (AO (AZ (U3 _ (V1 _ 2)) (V1 _ 3)) V)) g hN h0 _ 0 0 (G0 (s.w+3)) 0 _ ἅ (by simp) (by simp) (by simp) (by rw[aa _ (by clear*-W;omega),Ϧ,Nat.zero_div,Nat.zero_mod];ring) z1 Z (λq hq=>by rw[z3 q hq,if_neg (by simp)]);rw[xF,u];exact⟨z1,Z,v⟩
lemma gX(h w d j l y m e V:Ἀ)(x:V=w+2)(g:e=(h+2)*V)(p:y<h)(B:m<w)
  (A:l+j<=d)(o:0<j):
  l*e+y*V+m+((j-1)*e+2*V+2)+8<=(d*(h+2)+3)*V:=by
 have q:l*e+(j-1)*e+e<=d*e:=Ϡ _ _ _▸Ο _ _▸ι e (by (clear * - A o;omega));have r:y*V+V<=h*V:=Ο _ _▸ι V p;have u:e=h*V+2*V:=g▸(by ring);have v:(d*(h+2)+3)*V=d*e+3*V:=g▸(by ring);(clear * - x g B q r u v;omega)
lemma jQ(d:CD)(c:CC d.M)(v A O N P:Ἀ)(l:Ἀ->Ἀ)
  (hs:fP d.s d.kk) (a3:0<v) (a2:d.k=v*d.kk)
  (hc:∀o<d.Y,(if d.D then o else 0)+v<=d.s.cin)
  (hM:d.M=(v-1)*((d.s.h+2)*(d.s.w+2))+2*(d.s.w+2)+2)
  (as:∀t<d.k,d.T[t]! =pc d.kk t*((d.s.h+2)*(d.s.w+2))+py d.kk t*(d.s.w+2)+px d.kk t)
  (hT:∀t<d.k,d.T[t]!<=d.M)
  (aH:(d.s.cin*(d.s.h+2)+3)*(d.s.w+2)<=(kS d.X))
  (aL:∀q<(kS d.X),d.X.data[q]! =Dense.F.dw (aI d.s d.I (q/((d.s.h+2)*(d.s.w+2)))
   (q/(d.s.w+2)%(d.s.h+2)) (q%(d.s.w+2))))
  (hx:∀q:Ἀ,kB d.I[q]!≠255/\kG d.I[q]!<=A) (hw:∀q:Ἀ,kB d.K[q]!≠255/\kG d.K[q]!<=O)
  (hb:∀q:Ἀ,kB d.b[q]!≠255/\kG d.b[q]!<=N) (U:48+A+O<=P) (bb:174+N<=P)
  (hL:d.k+2<=2^d.L) (m:d.L<=27) (ah:P+d.L<=425) (G:(lq d.hh)=P+2*d.L-54)
  (o y x:Ἀ) (ho:o<d.Y) (hy:y<d.s.h) (g:x<d.s.w)(xw:∀t<d.k,d.X.data[(if d.D then o else 0)*((d.s.h+2)*(d.s.w+2))+y*(d.s.w+2)+x+d.T[t]!]! =Dense.F.dw (windowAt d.s d.I y x ((if d.D then o else 0)*d.kk+t))) (bh:c.n<=d.k)
  (e:∀j<c.n,l j<d.k/\c.Tz[j]! =d.T[l j]!/\c.Wz.data[c.wo+j]! =Dense.F.dw d.K[o*d.k+l j]!)
  (H:∀z:Ἀ->ℝ,(∀s<d.k,mW d.K[o*d.k+s]! =0->z s=0)->
   ∑j∈kP c.n,z (l j)=∑s∈kP d.k,z s)
  (S:Ἅ) (hS:S=Dense.F.ao (λj=>d.X.data[(if d.D then o else 0)*((d.s.h+2)*(d.s.w+2))+
   y*(d.s.w+2)+x+c.Tz[j]!]!*c.Wz.data[c.wo+j]!) c.n 0 (Dense.F.dw ld*Dense.F.dw d.b[o]!)):
  bVo d o y x S=p8 d.J (aB d.s d.I d.K d.b d.kk d.k (if d.D then o else 0) o y x):=by
 have p:∀t<d.k,Dense.F.fv (d.X.data[(if d.D then o else 0)*((d.s.h+2)*(d.s.w+2))+y*(d.s.w+2)+x+
   d.T[t]!]!*Dense.F.dw d.K[o*d.k+t]!)/\Dense.F.rv (d.X.data[(if d.D then o else 0)*((d.s.h+2)*(d.s.w+2))+
   y*(d.s.w+2)+x+d.T[t]!]!*Dense.F.dw d.K[o*d.k+t]!)=
   tm (ll d.s d.I y x ((if d.D then o else 0)*d.kk+t)) d.K[o*d.k+t]!:=by
  intro t B;rw[xw t B];exact Dense.F.dJ _ _ (iS d.s d.I A hx _ _ _).1 (hw _).1
 have l0:kB ld≠255:=by decide
 have rA:∀i j,|(tm (ll d.s d.I y x i) d.K[j]!:ℝ)|<=2^P:=λ_ _=>(Dense.F.dh _ _ A O (iS d.s d.I A hx _ _ _).2 (hw _).2).trans (AU one_le_two U)
 have rB:|(tm ld d.b[o]!:ℝ)|<=2^P:=(Dense.F.dh _ _ 126 N (by decide) (hb _).2).trans (AU one_le_two bb)
 have u:Dense.F.fv (Dense.F.dw ld*Dense.F.dw d.b[o]!)/\
   Dense.F.rv (Dense.F.dw ld*Dense.F.dw d.b[o]!)=tm ld d.b[o]!:=Dense.F.dJ ld d.b[o]! l0 (hb _).1
 have bp:(c.n:ℝ)<=2^d.L:=by exact_mod_cast(U2 bh (U2 (V1 _ 2) hL):c.n<=2^d.L)
 obtain⟨fS,eS⟩:=Dense.F.dM _ _ _ _ _ rfl▸Dense.F.dL (λ_=>false) (λj=>d.X.data[(if d.D then o else 0)*((d.s.h+2)*(d.s.w+2))+
   y*(d.s.w+2)+x+c.Tz[j]!]!*c.Wz.data[c.wo+j]!)
  (λj=>tm (ll d.s d.I y x ((if d.D then o else 0)*d.kk+l j)) d.K[o*d.k+l j]!) (tm ld d.b[o]!) P d.L c.n
  (λj g=>by
   obtain⟨i1,i2,i3⟩:=e j g;rw[i2,i3];exact p _ i1)
  (λ_ _ h=>absurd h lN) (λ_ _=>rA _ _)
  rB (U2 (L1 bh 2) hL) m (U2 ah (by decide)) c.n 0 _ rfl
  (ϟ _) u.1 (by simp only[u.2,X8,Av,sub_self,A5,Nat.cast_zero,zero_mul,le_refl])
 have ba:((tm ld d.b[o]!:Ἁ):ℝ)+∑j∈kP c.n,
   ((tm (ll d.s d.I y x ((if d.D then o else 0)*d.kk+l j)) d.K[o*d.k+l j]!:Ἁ):ℝ)=
   (af (d.k+1) (λt=>if t<d.k then ll d.s d.I y x ((if d.D then o else 0)*d.kk+t)else ld)
    (λt=>if t<d.k then d.K[o*d.k+t]!else d.b[o]!):ℝ):=by
  rw[aK,af,Int.cast_add,Int.cast_sum]
  exact (add_comm _ _).trans (congrArg₂ (·+·) ((ϔ rfl λj g=>by simp only[AD (e j (ϥ.mp g)).1]).trans (H _ λs a5 h0=>by simp only[AD a5,tm,cz d.K[o*d.k+s]! (by rw[<-an,h0];rfl),mul_zero,zero_mul,lH])) (by simp only[AX,Am]))
 rw[iE,hS];unfold ow;simp only[Dense.F.gg]
 refine(Dense.F.jE d.J d.k (λt=>if t<d.k then ll d.s d.I y x ((if d.D then o else 0)*d.kk+t)else ld)
  (λt=>if t<d.k then d.K[o*d.k+t]!else d.b[o]!) _ (λ_=>Dense.F.dw ld*Dense.F.dw d.b[o]!) d.hh P d.L
  (λ_=>p8 d.J (aB d.s d.I d.K d.b d.kk d.k (if d.D then o else 0) o y x))
  (λt B=>by simp only[AD B];exact p t B) (by simpa only[AX,Am]using u) (λt _=>?_) (λs _=>if h:s<d.k then by simp only[AD h];exact rA _ _ else by simp only[AC h];exact rB)
  hL m ah G (congrArg (p8 d.J) ((aB_eq _ _ _ _ _ _ _ _ _ _).trans (Dense.bO _ _ _))) _ fS ?_).trans (congrArg (p8 d.J) ((aB_eq _ _ _ _ _ _ _ _ _ _).trans (Dense.bO _ _ _))).symm
 ·dsimp only;by_cases h:t<d.k
  ·rw[AD h,AD h];exact⟨(iS d.s d.I A hx _ _ _).1,(hw _).1⟩
  ·rw[AC h,AC h];exact⟨l0,(hb _).1⟩
 ·rw[<-ba]
  refine eS.trans ?_
  have hE:((lq d.hh:ℕ):Ἁ)=((d.L+(P+d.L):Ἀ):Ἁ)+(-54):=by rw[G,Int.ofNat_sub (U2 (U2 (U2 (by decide) (V1 174 N)) bb) (V1 _ _)),(Nat.add_left_comm d.L P d.L).trans (congrArg (P+·) (two_mul d.L).symm)];rfl
  rw[hE,AK AL,AA,AF (2:ℝ) d.L (P+d.L),Ap];exact AE bp (by positivity)
lemma jQ2(d:CD)(c:CC d.M)(v A O N P:Ἀ)(l:Ἀ->Ἀ)
  (hs:fP d.s d.kk) (a3:0<v) (a2:d.k=v*d.kk)
  (hc:∀o<d.Y,(if d.D then o else 0)+v<=d.s.cin)
  (hM:d.M=(v-1)*((d.s.h+2)*(d.s.w+2))+2*(d.s.w+2)+2)
  (as:∀t<d.k,d.T[t]! =pc d.kk t*((d.s.h+2)*(d.s.w+2))+py d.kk t*(d.s.w+2)+px d.kk t)
  (hT:∀t<d.k,d.T[t]!<=d.M)
  (aH:(d.s.cin*(d.s.h+2)+3)*(d.s.w+2)<=(kS d.X))
  (aL:∀q<(kS d.X),d.X.data[q]! =Dense.F.dw (aI d.s d.I (q/((d.s.h+2)*(d.s.w+2)))
   (q/(d.s.w+2)%(d.s.h+2)) (q%(d.s.w+2))))
  (hx:∀q:Ἀ,kB d.I[q]!≠255/\kG d.I[q]!<=A) (hw:∀q:Ἀ,kB d.K[q]!≠255/\kG d.K[q]!<=O)
  (hb:∀q:Ἀ,kB d.b[q]!≠255/\kG d.b[q]!<=N) (U:48+A+O<=P) (bb:174+N<=P)
  (hL:d.k+2<=2^d.L) (m:d.L<=27) (ah:P+d.L<=425) (G:(lq d.hh)=P+2*d.L-54)
  (o y x:Ἀ) (ho:o<d.Y) (hy:y<d.s.h) (g:x<d.s.w) (bh:c.n<=d.k)
  (e:∀j<c.n,l j<d.k/\c.Tz[j]! =d.T[l j]!/\c.Wz.data[c.wo+j]! =Dense.F.dw d.K[o*d.k+l j]!)
  (H:∀z:Ἀ->ℝ,(∀s<d.k,mW d.K[o*d.k+s]! =0->z s=0)->
   ∑j∈kP c.n,z (l j)=∑s∈kP d.k,z s)
  (S:Ἅ) (hS:S=Dense.F.ao (λj=>d.X.data[(if d.D then o else 0)*((d.s.h+2)*(d.s.w+2))+
   y*(d.s.w+2)+x+c.Tz[j]!]!*c.Wz.data[c.wo+j]!) c.n 0 (Dense.F.dw ld*Dense.F.dw d.b[o]!)):
  bV d o y x S=p8 d.J (aB d.s d.I d.K d.b d.kk d.k (if d.D then o else 0) o y x):=by
 have xw:∀t<d.k,d.X.data[(if d.D then o else 0)*((d.s.h+2)*(d.s.w+2))+y*(d.s.w+2)+x+d.T[t]!]! =Dense.F.dw (windowAt d.s d.I y x ((if d.D then o else 0)*d.kk+t)):=by
  have a0:=hc o ho;have hB:=gX d.s.h d.s.w d.s.cin v (if d.D then o else 0) y x ((d.s.h+2)*(d.s.w+2)) (d.s.w+2) rfl rfl hy g a0 a3
  intro t B;have al:=as t B;have a6:t<v*d.kk:=a2▸B;have a1:d.kk=9\/d.kk=1:=by rcases hs with h | h<;>[exact pZ h.2.2.2.2;exact BA h.2.2.2.2]
  have a7:(if d.D then o else 0)+pc d.kk t<d.s.cin:=U1 (O1 (jb d.kk t v a1 a6) _) a0
  have h1:∀X a b u w e f k:Ἀ,X*(a*b)+u*b+w+(e*(a*b)+f*b+k)=((X+e)*a+(u+f))*b+(w+k):=λX a b u w e f k=>(Nat.add_add_add_comm _ _ _ _).trans (congrArg (·+(w+k)) ((Nat.add_add_add_comm _ _ _ _).trans ((congrArg₂ (·+·) ((Ϡ X e (a*b)).symm.trans (Ϛ (X+e) a b).symm) (Ϡ u f b).symm).trans (Ϡ ((X+e)*a) (u+f) b).symm)))
  have hq:(if d.D then o else 0)*((d.s.h+2)*(d.s.w+2))+y*(d.s.w+2)+x+d.T[t]! =
    (((if d.D then o else 0)+pc d.kk t)*(d.s.h+2)+(y+py d.kk t))*(d.s.w+2)+(x+px d.kk t):=by
     rw[al];exact h1 _ _ _ _ _ _ _ _
  obtain⟨q1,q2,q3⟩:=cf ((if d.D then o else 0)+pc d.kk t) (y+py d.kk t) (x+px d.kk t) (d.s.h+2) (d.s.w+2)
   (P6 hy (jd d.kk t)) (P6 g (jc d.kk t))
  have a4:(if d.D then o else 0)*((d.s.h+2)*(d.s.w+2))+y*(d.s.w+2)+x+d.T[t]!<(kS d.X):=
     U1 (C0 (J1 ((hT t B).trans_eq hM) _) (U1 (P5 (by decide)) hB)) aH
  rw[aL _ a4,hq,q1,q2,q3,iI d.s d.I d.kk hs y x t _ hy g a7]
 have J:=jQ d c v A O N P l hs a3 a2 hc hM as hT aH aL hx hw hb U bb hL m ah G o y x ho hy g xw bh e H S hS
 unfold bV;unfold bVo at J;dsimp only at J ⊢;split
 ·rename_i h;rw[dif_pos h]at J;exact J
 ·rename_i h;rw[dif_neg h]at J;unfold ow2
  by_cases hz:(d.b[o]! == 0&&zpC d o y x)=true
  ·rw[if_pos hz]
   simp only[Η,AQ]at hz
   rw[Conv.aBz2 d.s d.I d.K d.b d.kk d.k _ o y x hz.1 (zpC_ok d v a2 aH (fun q=>(hx q).1) o y x (fun t ht=>by have:=xw t ht;rwa[as t ht]at this) hz.2) (fun q=>(hx q).1) (fun q=>(hw q).1)]
  ·rw[if_neg hz];exact J
def czF(a:Ἂ Ἃ×Ἂ Ἀ×Ἀ×Ἀ×Ἀ):Ἂ (Ἂ Ἀ×ἀ):=
 if h:(∀t<a.2.2.2.1,a.2.1[t]!<=a.2.2.1)/\a.2.2.2.1<2^(32:Ἀ) then
  Dense.F.da (λo=>let c:=bA a.1 a.2.1 a.2.2.1 (o*a.2.2.2.1) a.2.2.2.1 h.1 h.2 0 (AS a.2.2.2.1) (AR a.2.2.2.1) (by rw[cZ];simp) (ϟ _) (ϟ _) (λj hj=>by simp at hj);(c.Tz,c.Wz)) a.2.2.2.2 (AS a.2.2.2.2)
 else #[]
@[inline]def czEq(a b:Ἂ Ἃ×Ἂ Ἀ×Ἀ×Ἀ×Ἀ):Ἆ:=a.2.2.1==b.2.2.1&&a.2.2.2.1==b.2.2.2.1&&a.2.2.2.2==b.2.2.2.2&&a.2.1==b.2.1&&Dense.F.wcAeq a.1 b.1
theorem czEq_ok(a b:Ἂ Ἃ×Ἂ Ἀ×Ἀ×Ἀ×Ἀ)(h:czEq a b=true):a=b:=by
 obtain⟨a1,a2,a3,a4,a5⟩:=a;obtain⟨b1,b2,b3,b4,b5⟩:=b
 simp only[czEq,Η,AQ]at h
 obtain⟨⟨⟨⟨rfl,rfl⟩,rfl⟩,rfl⟩,h1⟩:=h;rw[Dense.F.wcAeq_ok h1]
@[noinline]def czS:ST.Ref Unit (Ἂ (Dense.F.WcE czF)):=(ST.Prim.mkRef #[] (Void.mk ())).val
noncomputable def zcB(d:CD)(hT:∀t<d.k,d.T[t]!<=d.M)(hk:d.k<2^(32:Ἀ))(o:Ἀ):CC d.M:=
 bA d.K d.T d.M (o*d.k) d.k hT hk 0 (AS d.k) (AR d.k) (by rw[cZ];simp) (ϟ _) (ϟ _) (λj hj=>by simp at hj)
lemma zcN(K:Ἂ kF)(T:Ἂ Ἀ)(M q k:Ἀ)(o:∀t<k,T[t]!<=M)(p:k<2^(32:Ἀ)):
  ∀m t Tz d g e j l,k-t=m->(bA K T M q k o p t Tz d g e j l).ni=fQ:=by
 intro m;induction m with
 |zero=>intros;rw[bA,dif_neg (by omega)]
 |succ m r=>intros;rw[bA,AT (by omega)];dsimp only;split<;>exact r _ _ _ _ _ _ _ (by omega)
lemma zcE(d:CD)(hT:∀t<d.k,d.T[t]!<=d.M)(hk:d.k<2^(32:Ἀ))(o:Ἀ):
  (zcB d hT hk o).n=(O0 (zcB d hT hk o).Tz)/\(zcB d hT hk o).wo=0/\(zcB d hT hk o).ni=fQ:=
 ⟨(eM d.K d.T d.M (o*d.k) d.k hT hk _ 0 (AS d.k) (AR d.k) (by rw[cZ];simp) (ϟ _) (ϟ _) (λj hj=>by simp at hj) rfl).1,
  (eM d.K d.T d.M (o*d.k) d.k hT hk _ 0 (AS d.k) (AR d.k) (by rw[cZ];simp) (ϟ _) (ϟ _) (λj hj=>by simp at hj) rfl).2,
  zcN d.K d.T d.M (o*d.k) d.k hT hk _ 0 (AS d.k) (AR d.k) (by rw[cZ];simp) (ϟ _) (ϟ _) (λj hj=>by simp at hj) rfl⟩
lemma zcH{M:Ἀ}(c:CC M)(e1:c.n=(O0 c.Tz))(e2:c.wo=0):0+(O0 c.Tz)<=(kS c.Wz)/\0+(O0 c.Tz)<2^(32:Ἀ)/\(∀t<(O0 c.Tz),c.Tz[t]!<=M)/\(kS c.Wz)<2^(32:Ἀ):=by
 have a:=c.h2;have b:=c.h3;have f:=c.h4;rw[e1,e2]at a b;rw[e1]at f;exact⟨a,b,f,c.h5⟩
lemma zcV(d:CD)(hT:∀t<d.k,d.T[t]!<=d.M)(hk:d.k<2^(32:Ἀ))(v:Ἂ (Ἂ Ἀ×ἀ))(hv:v=czF (d.K,d.T,d.M,d.k,d.Y))(o:Ἀ)(ho:o<d.Y):
  v[o]! =((zcB d hT hk o).Tz,(zcB d hT hk o).Wz):=by
 subst hv;unfold czF;split
 .rw[Dense.F.iC,AB _ _ (by simpa using ho),Λ];try rfl
 .rename_i h;exact absurd ⟨hT,hk⟩ h
def zcD(d:CD)(hT:∀t<d.k,d.T[t]!<=d.M)(hk:d.k<2^(32:Ἀ))(v:Ἂ (Ἂ Ἀ×ἀ))(hv:v=czF (d.K,d.T,d.M,d.k,d.Y))(o:Ἀ)(ho:o<d.Y):CC d.M:=
 ⟨v[o]!.1,v[o]!.2,0,v[o]!.1.size,fQ,AP,
  by rw[zcV d hT hk v hv o ho];exact (zcH _ (zcE d hT hk o).1 (zcE d hT hk o).2.1).1,
  by rw[zcV d hT hk v hv o ho];exact (zcH _ (zcE d hT hk o).1 (zcE d hT hk o).2.1).2.1,
  by rw[zcV d hT hk v hv o ho];exact (zcH _ (zcE d hT hk o).1 (zcE d hT hk o).2.1).2.2.1,
  by rw[zcV d hT hk v hv o ho];exact (zcH _ (zcE d hT hk o).1 (zcE d hT hk o).2.1).2.2.2⟩
lemma zcX{M:Ἀ}(a b:CC M)(h1:a.Tz=b.Tz)(h2:a.Wz=b.Wz)(h3:a.wo=b.wo)(h4:a.n=b.n)(h5:a.ni=b.ni):a=b:=by
 cases a;cases b;simp only at h1 h2 h3 h4 h5;subst h1 h2 h3 h4 h5;rfl
lemma zcQ(d:CD)(hT:∀t<d.k,d.T[t]!<=d.M)(hk:d.k<2^(32:Ἀ))(v:Ἂ (Ἂ Ἀ×ἀ))(hv:v=czF (d.K,d.T,d.M,d.k,d.Y))(o:Ἀ)(ho:o<d.Y)
  (Wd:ἀ)(hkT:d.k<=(O0 d.T))(hW:ἃ=ἃ\/d.Y*d.k<=(kS Wd)/\(kS Wd)<2^(32:Ἀ))(hY:d.Y*d.k<2^(32:Ἀ)):
  zcD d hT hk v hv o ho=dF d ἃ Wd hT hk hkT hW hY o ho:=by
 have e:=zcV d hT hk v hv o ho;obtain⟨e1,e2,e3⟩:=zcE d hT hk o
 have q:dF d ἃ Wd hT hk hkT hW hY o ho=zcB d hT hk o:=by rw[dF,dif_pos rfl];rfl
 rw[q];exact zcX _ _ (by simp only[zcD,e]) (by simp only[zcD,e]) e2.symm (by simp only[zcD,e,e1]) e3.symm
def zcC(d:CD)(Z:Ἆ)(Wd:@&ἀ)(cs:Option {v:Ἂ (Ἂ Ἀ×ἀ)//v=czF (d.K,d.T,d.M,d.k,d.Y)})(hT:∀t<d.k,d.T[t]!<=d.M)(hk:d.k<2^(32:Ἀ))(hkT:d.k<=(O0 d.T))
  (hW:Z=ἃ\/d.Y*d.k<=(kS Wd)/\(kS Wd)<2^(32:Ἀ)) (hY:d.Y*d.k<2^(32:Ἀ)) (o:Ἀ) (ho:o<d.Y):CC d.M:=
 match cs with
 |some v=>if hZ:Z=ἃ then zcD d hT hk v.1 v.2 o ho else dF d Z Wd hT hk hkT hW hY o ho
 |none=>dF d Z Wd hT hk hkT hW hY o ho
lemma zcCq(d:CD)(Z:Ἆ)(Wd:ἀ)(cs:Option {v:Ἂ (Ἂ Ἀ×ἀ)//v=czF (d.K,d.T,d.M,d.k,d.Y)})(hT:∀t<d.k,d.T[t]!<=d.M)(hk:d.k<2^(32:Ἀ))(hkT:d.k<=(O0 d.T))
  (hW:Z=ἃ\/d.Y*d.k<=(kS Wd)/\(kS Wd)<2^(32:Ἀ)) (hY:d.Y*d.k<2^(32:Ἀ)) (o:Ἀ) (ho:o<d.Y):
  zcC d Z Wd cs hT hk hkT hW hY o ho=dF d Z Wd hT hk hkT hW hY o ho:=by
 unfold zcC;split
 .split
  .rename_i hZ;subst hZ;exact zcQ _ _ _ _ _ _ _ _ _ _ _
  .rfl
 .rfl
def g7c(d:CD)(Z:Ἆ)(Wd:ἀ)(cs:Option {v:Ἂ (Ἂ Ἀ×ἀ)//v=czF (d.K,d.T,d.M,d.k,d.Y)})(hT:∀t<d.k,d.T[t]!<=d.M)(hk:d.k<2^(32:Ἀ))(hkT:d.k<=(O0 d.T))
  (hW:Z=ἃ\/d.Y*d.k<=(kS Wd)/\(kS Wd)<2^(32:Ἀ)) (hY:d.Y*d.k<2^(32:Ἀ)):Ἂ Ἃ:=
 c8 d.Y (d.s.h*d.s.w) 0 0 (by have:=ς;simp only[AH,A7];exact Nat.lt_of_add_left_lt this) (λo ho q _ _ a=>
  let c:=zcC d Z Wd cs hT hk hkT hW hY o ho
  db a9 d c o q (Dense.F.dw oneW*Dense.F.dw d.b[o]!) a) 0 0 0 (by simp) (Ah (d.Y*(d.s.h*d.s.w)) 0)
@[inline]def ccw{M:Ἀ}(c:CC M)(w:Ἀ)(h2:w+c.n<=(kS c.Wz))(h3:w+c.n<2^32):CC M:=⟨c.Tz,c.Wz,w,c.n,c.ni,c.h1,h2,h3,c.h4,c.h5⟩
def g7w(d:@&CD)(Wd:@&ἀ)(hW:d.Y*d.k<=(kS Wd))(hY:d.Y*d.k<2^(32:Ἀ))(y q:Ἀ)(a:Ἂ Ἃ)(c:CC d.M)(hc:c.Tz=d.T∧c.Wz=Wd∧c.n=d.k∧c.ni=fQ):Ἂ Ἃ:=
 if hy:y<d.Y then
  g7w d Wd hW hY (y+1) (q+d.s.h*d.s.w) (db a9 d (ccw c (y*d.k) (by rw[hc.2.1,hc.2.2.1];exact (show y*d.k+d.k<=d.Y*d.k by rw[<-Ο];exact ι _ hy).trans hW) (by rw[hc.2.2.1];exact C0 (show y*d.k+d.k<=d.Y*d.k by rw[<-Ο];exact ι _ hy) hY)) y q (Dense.F.dw oneW*Dense.F.dw d.b[y]!) a)
   (ccw c (y*d.k) (by rw[hc.2.1,hc.2.2.1];exact (show y*d.k+d.k<=d.Y*d.k by rw[<-Ο];exact ι _ hy).trans hW) (by rw[hc.2.2.1];exact C0 (show y*d.k+d.k<=d.Y*d.k by rw[<-Ο];exact ι _ hy) hY)) hc
 else a
termination_by d.Y-y
decreasing_by all_goals omega
theorem g7w_eq(d:CD)(Z:Ἆ)(Wd:ἀ)(cs:Option {v:Ἂ (Ἂ Ἀ×ἀ)//v=czF (d.K,d.T,d.M,d.k,d.Y)})(hT:∀t<d.k,d.T[t]!<=d.M)(hk:d.k<2^(32:Ἀ))(hkT:d.k<=(O0 d.T))
  (hW':Z=ἃ\/d.Y*d.k<=(kS Wd)/\(kS Wd)<2^(32:Ἀ))(hZ:¬Z=ἃ)(hW:d.Y*d.k<=(kS Wd))(hY:d.Y*d.k<2^(32:Ἀ))(hN:0+d.Y*(J0 (0:Ἄ))<Al):
  ∀m y q (i:Ἄ) (hi:(J0 i)=0+y*(J0 (0:Ἄ))) (a:Ἂ Ἃ) (c:CC d.M) (hc:c.Tz=d.T∧c.Wz=Wd∧c.n=d.k∧c.ni=fQ),d.Y-y=m->
  g7w d Wd hW hY y q a c hc=c8 d.Y (d.s.h*d.s.w) 0 0 hN (λo ho q _ _ a=>
   let c:=zcC d Z Wd cs hT hk hkT hW' hY o ho
   db a9 d c o q (Dense.F.dw oneW*Dense.F.dw d.b[o]!) a) y q i hi a:=by
 intro m;induction m with
 |zero=>intro y q i hi a c hc H;rw[g7w,c8,dif_neg (by omega),dif_neg (by omega)]
 |succ m ih=>
  intro y q i hi a c hc H;have hy:y<d.Y:=by omega
  rw[g7w,c8,dif_pos hy,dif_pos hy]
  refine (ih (y+1) _ (i+0) ?_ _ _ _ (by omega)).trans ?_
  · simpa using hi
  dsimp only
  congr 1;congr 1;rw[zcCq];apply zcX<;>simp only[ccw,dF,dif_neg hZ,hc.1,hc.2.1,hc.2.2.1,hc.2.2.2]
def g7(d:CD)(Z:Ἆ)(Wd:ἀ)(hT:∀t<d.k,d.T[t]!<=d.M)(hk:d.k<2^(32:Ἀ))(hkT:d.k<=(O0 d.T))
  (hW:Z=ἃ\/d.Y*d.k<=(kS Wd)/\(kS Wd)<2^(32:Ἀ)) (hY:d.Y*d.k<2^(32:Ἀ)):Ἂ Ἃ:=
 if hz:Z=ἃ\/d.Y=0 then
  g7c d Z Wd (if Z=ἃ then some (Dense.F.wcGet3 czEq czEq_ok czS 32 (d.K,d.T,d.M,d.k,d.Y) (fun k=>k.1)) else none) hT hk hkT hW hY
 else
  g7w d Wd (hW.resolve_left (fun e=>hz (.inl e))).1 hY 0 0 (Ah (d.Y*(d.s.h*d.s.w)) 0)
   ⟨d.T,Wd,0,d.k,fQ,hkT,by rw[Κ];exact (lZ _ (ϵ (fun e=>hz (.inr e)))).trans (hW.resolve_left (fun e=>hz (.inl e))).1,
    by rw[Κ];exact C0 (lZ _ (ϵ (fun e=>hz (.inr e)))) hY,hT,(hW.resolve_left (fun e=>hz (.inl e))).2⟩ ⟨rfl,rfl,rfl,rfl⟩
theorem g7_eq(d:CD)(Z:Ἆ)(Wd:ἀ)(hT:∀t<d.k,d.T[t]!<=d.M)(hk:d.k<2^(32:Ἀ))(hkT:d.k<=(O0 d.T))
  (hW:Z=ἃ\/d.Y*d.k<=(kS Wd)/\(kS Wd)<2^(32:Ἀ)) (hY:d.Y*d.k<2^(32:Ἀ)):
  g7 d Z Wd hT hk hkT hW hY=g7c d Z Wd (if Z=ἃ then some (Dense.F.wcGet3 czEq czEq_ok czS 32 (d.K,d.T,d.M,d.k,d.Y) (fun k=>k.1)) else none) hT hk hkT hW hY:=by
 rw[g7];split
 · rfl
 · rename_i hz;rw[g7c];exact g7w_eq d Z Wd _ hT hk hkT hW (fun e=>hz (.inl e)) _ hY _ _ 0 0 0 (by simp) _ _ _ rfl
lemma g8c(d:CD)(Z:Ἆ)(p:ἀ)(cs:Option {v:Ἂ (Ἂ Ἀ×ἀ)//v=czF (d.K,d.T,d.M,d.k,d.Y)})(hW:Z=ἃ\/d.Y*d.k<=(kS p)/\(kS p)<2^(32:Ἀ))(G IX IW IB P:Ἀ)
  (hs:fP d.s d.kk) (al:0<G) (bq:d.k=G*d.kk)
  (hc:∀o<d.Y,(if d.D then o else 0)+G<=d.s.cin)
  (hM:d.M=(G-1)*((d.s.h+2)*(d.s.w+2))+2*(d.s.w+2)+2)
  (a3:∀t<d.k,d.T[t]! =pc d.kk t*((d.s.h+2)*(d.s.w+2))+py d.kk t*(d.s.w+2)+px d.kk t)
  (hT:∀t<d.k,d.T[t]!<=d.M) (m:d.k<=(O0 d.T)) (l:d.k<2^(32:Ἀ)) (j:d.Y*d.k<2^(32:Ἀ))
  (r:(d.s.cin*(d.s.h+2)+3)*(d.s.w+2)<=(kS d.X)) (bR:(kS d.X)<2^(32:Ἀ))
  (a5:∀q<(kS d.X),d.X.data[q]! =Dense.F.dw (aI d.s d.I (q/((d.s.h+2)*(d.s.w+2)))
   (q/(d.s.w+2)%(d.s.h+2)) (q%(d.s.w+2))))
  (a4:Z=false->∀q<d.Y*d.k,p.data[q]! =Dense.F.dw d.K[q]!)
  (hx:∀q:Ἀ,kB d.I[q]!≠255/\kG d.I[q]!<=IX) (hw:∀q:Ἀ,kB d.K[q]!≠255/\kG d.K[q]!<=IW)
  (hb:∀q:Ἀ,kB d.b[q]!≠255/\kG d.b[q]!<=IB) (a0:48+IX+IW<=P) (a1:174+IB<=P)
  (hL:d.k+2<=2^d.L) (N:d.L<=27) (a2:P+d.L<=425) (bh:(lq d.hh)=P+2*d.L-54)
  (bB:0<d.s.w) (U:d.Y*(d.s.h*d.s.w)<2^(32:Ἀ)):
  g7c d Z p cs hT l m hW j=ag (d.Y*(d.s.h*d.s.w)) λq=>p8 d.J (aB d.s d.I d.K d.b d.kk d.k
   (if d.D then q/(d.s.h*d.s.w)else 0) (q/(d.s.h*d.s.w)) (q/d.s.w%d.s.h) (q%d.s.w)):=by
 rw[g7c]
 obtain⟨aL,ba⟩:=eE d.Y (d.s.h*d.s.w) 0 0 (AO (B3 (congrArg (0+d.Y*·) Ϧ)) (AN (϶ 32) ς))
  (λo A q _ _ a=>
  let c:=zcC d Z p cs hT l m hW j o A
  db a9 d c o q (Dense.F.dw ld*Dense.F.dw d.b[o]!) a) 0 (d.Y*(d.s.h*d.s.w)) (λq=>p8 d.J (aB d.s d.I d.K d.b d.kk d.k
   (if d.D then q/(d.s.h*d.s.w)else 0) (q/(d.s.h*d.s.w)) (q/d.s.w%d.s.h) (q%d.s.w))) (λo A q _ _ a hq hN=>by
  try dsimp only
  generalize bv:zcC d Z p cs hT l m hW j o A=c;rw[zcCq]at bv;obtain⟨bG,ix,a7,bp,aH⟩:=fm d Z p hT l m hW j a4 o A c bv;have:=ς;have hV:d.s.w+2<=(kS d.X):=AZ (lZ _ (Nat.succ_pos _)) r;have hK:d.s.w+2<kX:=AN (AO hV bR) this
  have bs:(o+1)*(d.s.h*d.s.w)<=d.Y*(d.s.h*d.s.w):=ι _ A;rw[Ο]at bs
  refine iG d c o q _ a _ _ (λy hy H i a' hI hi bw O=>?_) hN hK (λbb=>by
   have a6:=gX _ d.s.w _ G _ (d.s.h-1) (d.s.w-1) _ _ rfl rfl (Nat.sub_lt bb Nat.one_pos) (Nat.sub_lt bB Nat.one_pos)
    (hc o A) al
   rw[<-hM]at a6;exact AZ (B3 ((congrArg (_+·+_+7) (L3 bB).symm).trans (congrArg (·+7) (lO _ 1 _)))) (AZ a6 r))
  try dsimp only
  have bt:(y+1)*d.s.w<=d.s.h*d.s.w:=ι _ hy;rw[Ο]at bt;have e:(o*d.s.h+y)*d.s.w=H:=by rw[Ϡ,Ϛ,bw,hq,Κ]
  have h2:H+d.s.w<=d.Y*(d.s.h*d.s.w):=AZ (by rw[bw,hq,Κ,pA];exact J1 bt _) bs
  have B:d.s.w<kX:=AO (V1 _ 2) hK;have as:H<kX:=AN (K5 (AN (P5 bB) h2) U) this
  have u:=je d c o y _ rfl (λx=>p8 d.J (aB d.s d.I d.K d.b d.kk d.k (if d.D then o else 0) o y x))
   (λxb x h1 h2 h3=>jQ2 d c G IX IW IB P ix hs al bq hc hM a3 hT r a5 hx hw hb a0 a1 hL N
    a2 bh o y x A hy h3 a7 bp aH _ (by
     rw[dK]
     exact congrArg (λf=>Dense.F.ao f c.n 0 _) (funext λt=>by
      rw[lO _ c.Tz[t]! (x-xb),pA _ xb (x-xb),BQ h1])))
   _ 0 (G0 d.s.w) (G0 H) i a' hI AP
   (by rw[Dense.aa B];simp only[kD,Κ])
   (by rw[hi];simp only[kD,ϑ])
   (by rw[Dense.aa as,Dense.aa B,O];exact h2)
   (by rw[O];exact U)
  refine⟨u.1,λi hi=>(u.2 i hi).trans ?_⟩
  rw[Dense.aa as,Dense.aa B,Ϧ,Κ];by_cases g:H<=i/\i<H+d.s.w
  .rw[AD g,AD g]
   obtain⟨q1,q2,q3⟩:=cf o y (i-H) d.s.h d.s.w hy (G4 g.1 g.2);rw[e,BQ g.1]at q1 q2 q3;rw[q1,q2,q3]
  .rw[AC g,AC g])
  d.Y 0 0 0 (Ϧ.trans ((Κ _).trans (ϳ _)).symm) (mB (d.Y*(d.s.h*d.s.w)) 0) rfl (ϟ _) ((Κ _).trans (ϳ _)).symm O3
 exact mi (aL.trans (O3.trans Ϝ.symm)) λi v bS=>(AB _ i v).symm.trans ((ba i (V6 v aL)).trans ((AD ⟨ϟ _,V6 (V6 v (aL.trans O3)) (Κ _).symm⟩).trans (Λ bS).symm))
lemma g8(d:CD)(Z:Ἆ)(p:ἀ)(hW:Z=ἃ\/d.Y*d.k<=(kS p)/\(kS p)<2^(32:Ἀ))(G IX IW IB P:Ἀ)
  (hs:fP d.s d.kk) (al:0<G) (bq:d.k=G*d.kk)
  (hc:∀o<d.Y,(if d.D then o else 0)+G<=d.s.cin)
  (hM:d.M=(G-1)*((d.s.h+2)*(d.s.w+2))+2*(d.s.w+2)+2)
  (a3:∀t<d.k,d.T[t]! =pc d.kk t*((d.s.h+2)*(d.s.w+2))+py d.kk t*(d.s.w+2)+px d.kk t)
  (hT:∀t<d.k,d.T[t]!<=d.M) (m:d.k<=(O0 d.T)) (l:d.k<2^(32:Ἀ)) (j:d.Y*d.k<2^(32:Ἀ))
  (r:(d.s.cin*(d.s.h+2)+3)*(d.s.w+2)<=(kS d.X)) (bR:(kS d.X)<2^(32:Ἀ))
  (a5:∀q<(kS d.X),d.X.data[q]! =Dense.F.dw (aI d.s d.I (q/((d.s.h+2)*(d.s.w+2)))
   (q/(d.s.w+2)%(d.s.h+2)) (q%(d.s.w+2))))
  (a4:Z=false->∀q<d.Y*d.k,p.data[q]! =Dense.F.dw d.K[q]!)
  (hx:∀q:Ἀ,kB d.I[q]!≠255/\kG d.I[q]!<=IX) (hw:∀q:Ἀ,kB d.K[q]!≠255/\kG d.K[q]!<=IW)
  (hb:∀q:Ἀ,kB d.b[q]!≠255/\kG d.b[q]!<=IB) (a0:48+IX+IW<=P) (a1:174+IB<=P)
  (hL:d.k+2<=2^d.L) (N:d.L<=27) (a2:P+d.L<=425) (bh:(lq d.hh)=P+2*d.L-54)
  (bB:0<d.s.w) (U:d.Y*(d.s.h*d.s.w)<2^(32:Ἀ)):
  g7 d Z p hT l m hW j=ag (d.Y*(d.s.h*d.s.w)) λq=>p8 d.J (aB d.s d.I d.K d.b d.kk d.k
   (if d.D then q/(d.s.h*d.s.w)else 0) (q/(d.s.h*d.s.w)) (q/d.s.w%d.s.h) (q%d.s.w)):=by
 rw[g7_eq];exact g8c d Z p _ hW G IX IW IB P hs al bq hc hM a3 hT m l j r bR a5 a4 hx hw hb a0 a1 hL N a2 bh bB U
def wd(Z:Ἆ)(K:Ἂ Ἃ)(n:Ἀ)(hn:n<2^(32:Ἀ)):{W:ἀ//Z=ἃ\/n<=(kS W)/\(kS W)<2^(32:Ἀ)}:=
 if hZ:Z=ἃ then⟨AR 0,pZ hZ⟩
 else⟨wdU K 0 (G0 n) (zPq n),BA ⟨by rw[wdUz _ _ _ _ _ rfl];exact(zPs n).1,by rw[wdUz _ _ _ _ _ rfl];rcases(zPs n).2.1 with z|z;omega;rw[z];exact hn⟩⟩
lemma tB(kk nc A B:Ἀ)(g:kk=9\/kk=1):∀t<nc*kk,(Dense.F.da (λt=>pc kk t*A+py kk t*B+px kk t) (nc*kk) (AS (nc*kk)))[t]!<=(nc-1)*A+2*B+2:=by
 intro t h;rw[Dense.F.iC,AB _ _ (by simpa using h),Λ];dsimp only;have:=jb kk t nc g h;have:=jd kk t;have:=jc kk t;have:pc kk t*A<=(nc-1)*A:=ι _ (by omega);have:py kk t*B<=2*B:=ι _ (by exact jd kk t);omega
def lwf(q:Ἂ Ἃ×Ἀ×Ἀ):ἇ×Ἆ×ἀ×Ἂ Ἀ:=
 let Z:=q.1.any λu=>Core.mW u==0
 let a:=Nat.unpair q.2.2
 let c:=Nat.unpair a.2
 (Dense.F.aO q.1 0 0,Z,(if h:q.2.1<2^(32:Ἀ) then (wd Z q.1 q.2.1 h).1 else AR 0),
  Dense.F.da (λt=>pc c.1 t*(Nat.unpair a.1).1+py c.1 t*(Nat.unpair a.1).2+px c.1 t) (c.2*c.1) (AS (c.2*c.1)))
@[noinline]def lwS:ST.Ref Unit (Ἂ (Dense.F.WcE lwf)):=(ST.Prim.mkRef #[] (Void.mk ())).val
lemma lwf_T(K:Ἂ Ἃ)(n A B kk nc:Ἀ):(lwf (K,n,X6 (X6 A B) (X6 kk nc))).2.2.2=Dense.F.da (λt=>pc kk t*A+py kk t*B+px kk t) (nc*kk) (AS (nc*kk)):=by
 simp only[lwf,Nat.unpair_pair]
lemma lwf_W(K:Ἂ Ἃ)(n q:Ἀ)(h:n<2^(32:Ἀ)):(lwf (K,n,q)).2.1=ἃ\/n<=(kS (lwf (K,n,q)).2.2.1)/\(kS (lwf (K,n,q)).2.2.1)<2^(32:Ἀ):=by
 simp only[lwf,dif_pos h];exact (wd _ K n h).2
lemma lwf_D(K:Ἂ Ἃ)(n q:Ἀ)(h:n<2^(32:Ἀ)):(lwf (K,n,q)).2.2.1=(wd (lwf (K,n,q)).2.1 K n h).1:=by
 simp only[lwf,dif_pos h]
lemma lwB(s:ConvShape)(nc Y:Ἀ)(K:Ἂ Ἃ)(lw:{v:ἇ×Ἆ×ἀ×Ἂ Ἀ//v=lwf (K,Y*(nc*(s.kh*s.kw)),X6 (X6 ((s.h+2)*(s.w+2)) (s.w+2)) (X6 (s.kh*s.kw) nc))})(g:s.kh*s.kw=9\/s.kh*s.kw=1):
  ∀t<nc*(s.kh*s.kw),lw.1.2.2.2[t]!<=(nc-1)*((s.h+2)*(s.w+2))+2*(s.w+2)+2:=by
 obtain⟨v,rfl⟩:=lw;rw[lwf_T];exact tB _ _ _ _ g
lemma lwW(s:ConvShape)(nc Y:Ἀ)(K:Ἂ Ἃ)(lw:{v:ἇ×Ἆ×ἀ×Ἂ Ἀ//v=lwf (K,Y*(nc*(s.kh*s.kw)),X6 (X6 ((s.h+2)*(s.w+2)) (s.w+2)) (X6 (s.kh*s.kw) nc))})(h:Y*(nc*(s.kh*s.kw))<2^(32:Ἀ)):
  lw.1.2.1=ἃ\/Y*(nc*(s.kh*s.kw))<=(kS lw.1.2.2.1)/\(kS lw.1.2.2.1)<2^(32:Ἀ):=by
 obtain⟨v,rfl⟩:=lw;exact lwf_W _ _ _ h
def fLc(s:ConvShape)(J:Ἆ)(I K b:Ἂ Ἃ)(D:Ἆ)(nc Y:Ἀ)(lw:@&{v:ἇ×Ἆ×ἀ×Ἂ Ἀ//v=lwf (K,Y*(nc*(s.kh*s.kw)),X6 (X6 ((s.h+2)*(s.w+2)) (s.w+2)) (X6 (s.kh*s.kw) nc))}):Ἂ Ἃ:=
 let H:=s.h+2
 let V:=s.w+2
 let HV:=H*V
 let kk:=s.kh*s.kw
 let k:=nc*kk
 let ex:=Dense.F.aO I 0 0
 let ew:=lw.1.1
 let eb:=Dense.F.aO b 0 0
 let P:=max (48+((mJ ex)-1)+((mJ ew)-1)) (174+((mJ eb)-1))
 let L:=Aq (k+1)+1
 let M:=(nc-1)*HV+2*V+2
 let T:=lw.1.2.2.2
 if hg:fP s kk/\0<s.h/\0<s.w/\0<nc/\ex<255/\ew<255/\
   eb<255/\P+L<=425/\L<=27/\(O0 I)<=4294967295/\(O0 K)<=4294967295/\(O0 b)<=4294967295/\
   (s.cin*H+3)*V<=4294967295/\s.w<=4294967295/\Y*k<=4294967295/\k<=4294967295/\k<=(O0 T)/\s.cin*s.h*s.w<=(O0 I)/\
   Y*(s.h*s.w)<=4294967295 then
  have⟨hs,h0,_,_,_,_,_,hPL,hL,hIs0,_,_,hX0,hw0,hY0,hk0,hkT,hI,_⟩:=hg
  have hIs:(O0 I)<2^(32:Ἀ):=by exact Z4 hIs0
  have hX32:(s.cin*H+3)*V<2^(32:Ἀ):=by exact Z4 hX0
  have hw32:s.w<2^(32:Ἀ):=by exact Z4 hw0
  have hY32:Y*k<2^(32:Ἀ):=by exact Z4 hY0
  have hk32:k<2^(32:Ἀ):=by exact Z4 hk0
  have hT:∀t<k,T[t]!<=M:=lwB s nc Y K lw (by rcases hs with h|h<;>[exact pZ h.2.2.2.2;exact BA h.2.2.2.2])
  g7 ⟨s,I,K,b,xF s I hIs hw32,T,D,J,k,M,L,(I0 (P+2*L-54)),Y,kk,
    (dP s I hIs hw32 hI hX32 h0).2.1,wlo (P+2*L-54),wsp (P+2*L-54),wok (P+2*L-54) (wlt P L hPL hL),(s.kh==3&&s.kw==3&&s.stride==1&&s.pad==1&&kk==9),rfl,((s.h+2)*(s.w+2)),rfl,(C3 (s.w+2)),rfl,(k/9),rfl⟩ lw.1.2.1 lw.1.2.2.1 hT hk32 hkT (lwW s nc Y K lw hY32) hY32
 else dS s J I K b D
@[noinline]def fLo(s:ConvShape)(J:Ἆ)(I K b:Ἂ Ἃ)(D:Ἆ)(nc Y:Ἀ):Ἂ Ἃ:=
 fLc s J I K b D nc Y (Dense.F.wcGet3 Dense.F.wcK3eq Dense.F.wcK3eq_ok lwS 32 (K,Y*(nc*(s.kh*s.kw)),X6 (X6 ((s.h+2)*(s.w+2)) (s.w+2)) (X6 (s.kh*s.kw) nc)) (fun k=>k.1))
def fLs(s:ConvShape)(J:Ἆ)(I K b:Ἂ Ἃ)(D:Ἆ)(nc Y n q:Ἀ)(hn:n=Y*(nc*(s.kh*s.kw)))(hq:q=X6 (X6 ((s.h+2)*(s.w+2)) (s.w+2)) (X6 (s.kh*s.kw) nc))
  (st:Ἂ (Dense.F.WcE lwf))(i:Nat):{r:Ἂ Ἃ//r=fLo s J I K b D nc Y}:=
 if h:i<st.size then
  withPtrAddr st[i].k.1 (fun p1=>withPtrAddr K (fun p2=>
   if p1=p2 then
    (if hk:Dense.F.wcK3eq st[i].k (K,n,q)=true then ⟨fLc s J I K b D nc Y ⟨st[i].v,by rw[st[i].h,Dense.F.wcK3eq_ok _ _ hk,hn,hq]⟩,by unfold fLo;exact congrArg (fLc s J I K b D nc Y) (Dense.F.wcSub _ _)⟩
     else ⟨fLo s J I K b D nc Y,rfl⟩)
   else fLs s J I K b D nc Y n q hn hq st (i+1)) (fun _ _=>Dense.F.wcSub _ _)) (fun _ _=>Dense.F.wcSub _ _)
 else ⟨fLo s J I K b D nc Y,rfl⟩
termination_by st.size-i
def fL(s:ConvShape)(J:Ἆ)(I K b:Ἂ Ἃ)(D:Ἆ)(nc Y:Ἀ):Ἂ Ἃ:=
 (fLs s J I K b D nc Y (Y*(nc*(s.kh*s.kw))) (X6 (X6 ((s.h+2)*(s.w+2)) (s.w+2)) (X6 (s.kh*s.kw) nc)) rfl rfl (ST.Prim.Ref.get lwS (Void.mk ())).val 0).1
theorem fL_eq(s:ConvShape)(J:Ἆ)(I K b:Ἂ Ἃ)(D:Ἆ)(nc Y:Ἀ):fL s J I K b D nc Y=fLo s J I K b D nc Y:=(fLs s J I K b D nc Y _ _ rfl rfl _ 0).2
lemma ev_h1(s:lR)(kk:Ἀ)(a2:0<s.h)(aH:0<s.w)(hs:fP s kk):s.ho=s.h∧s.wo=s.w:=by
 rcases hs with⟨e,j,l,m,-⟩ | ⟨e,j,l,m,-⟩<;>simp only[ConvShape.ho,ConvShape.wo,e,j,l,m,F5]<;>exact⟨L3 a2,L3 aH⟩
lemma ev_h2(P L:Ἀ)(H:P+L<=425)(p:L<=27):P+2*L-54<n0:=
 C0 (U2 (K4 _ _) (by rw[two_mul,<-pA];exact Y3 H p:P+2*L<=452)) (by decide)
lemma ev_h3(s:lR)(J:Ἆ)(I K b:Ἂ kF)(D:Ἆ)(d Y:Ἀ)
  (hd:D=false/\d=s.cin/\Y=s.cout\/D=true/\d=1/\Y=s.cin)(lw:{v:ἇ×Ἆ×ἀ×Ἂ Ἀ//v=lwf (K,Y*(d*(s.kh*s.kw)),X6 (X6 ((s.h+2)*(s.w+2)) (s.w+2)) (X6 (s.kh*s.kw) d))})
  (hs:fP s (s.kh*s.kw))(a2:0<s.h)(aH:0<s.w)(as:0<d)(ah:Dense.F.aO I 0 0<255)(X:lw.1.1<255)(U:Dense.F.aO b 0 0<255)(H:max (48+((mJ (Dense.F.aO I 0 0))-1)+((mJ lw.1.1)-1)) (174+((mJ (Dense.F.aO b 0 0))-1))+(Aq (d*(s.kh*s.kw)+1)+1)<=425)(p:Aq (d*(s.kh*s.kw)+1)+1<=27)(G:(O0 I)<2^(32:Ἀ))(aL:(O0 K)<2^(32:Ἀ))(a1:(O0 b)<2^(32:Ἀ))(u:(s.cin*(s.h+2)+3)*(s.w+2)<2^(32:Ἀ))(A:s.w<2^(32:Ἀ))(v:Y*(d*(s.kh*s.kw))<2^(32:Ἀ))(x:d*(s.kh*s.kw)<2^(32:Ἀ))(al:d*(s.kh*s.kw)<=(O0 lw.1.2.2.2))(hI:s.cin*s.h*s.w<=(O0 I))(r:Y*(s.h*s.w)<2^(32:Ἀ)):
  fLc s J I K b D d Y lw=dS s J I K b D:=by
 have E1:lw.1.1=Dense.F.aO K 0 0:=by rw[lw.2];rfl
 have E3:lw.1.2.2.2=Dense.F.da (λt=>pc (s.kh*s.kw) t*((s.h+2)*(s.w+2))+py (s.kh*s.kw) t*(s.w+2)+px (s.kh*s.kw) t) (d*(s.kh*s.kw)) (AS (d*(s.kh*s.kw))):=by rw[lw.2];exact lwf_T _ _ _ _ _ _
 have E4:lw.1.2.2.1=(wd lw.1.2.1 K (Y*(d*(s.kh*s.kw))) v).1:=by rw[lw.2];exact lwf_D _ _ _ v
 refine (dif_pos ⟨hs,a2,aH,as,ah,X,U,H,p,(by exact J2 G),(by exact J2 aL),(by exact J2 a1),(by exact J2 u),(by exact J2 A),(by exact J2 v),(by exact J2 x),al,hI,(by exact J2 r)⟩).trans ?_
 dsimp only
 obtain⟨hA,hB⟩:=ev_h1 s _ a2 aH hs
 have hc:∀o<Y,(if D then o else 0)+d<=s.cin:=λo ho=>by
  rcases hd with⟨rfl,rfl,rfl⟩ | ⟨rfl,rfl,rfl⟩;exact B3 (Κ _);exact ho
 refine (g8 _ _ _ _ d _ _ _ _ (by exact hs) as (by rfl) (by exact hc) (by rfl) (λt ht=>by dsimp only;rw[E3,Dense.F.iC,AB _ _ (by rw[Ϝ];exact ht),Λ]) (lwB s d Y K lw (by rcases hs with h|h<;>[exact pZ h.2.2.2.2;exact BA h.2.2.2.2])) (by exact al)
  (by exact x) (by exact v) (by exact (dP s I G A hI u a2).1) (by exact (dP s I G A hI u a2).2.1) (by exact (dP s I G A hI u a2).2.2)
  (λhZ=>by
   dsimp only;rw[E4,hZ];unfold wd;rw[dif_neg lN];dsimp only
   obtain⟨z1,z2,z3⟩:=zPs (Y*(d*(s.kh*s.kw)));have:=aa _ v
   intro q hq;exact wdUs K _ aL _ 0 _ rfl (by rw[this];exact z1) (by exact z2.elim (λz=>C0 z (by decide)) (λz=>z.trans_lt v))
    (λq hq=>by rw[z3 q hq,if_neg (by rw[Ϧ];exact U5 _)]) q (by rw[this];exact hq))
  (by exact Dense.F.bd I (U1 G ς) ah) (by have g:=Dense.F.bd K (U1 aL ς) (E1▸X);rw[<-E1] at g;exact g)
  (by exact Dense.F.bd b (U1 a1 ς) U) (le_max_left _ _) (le_max_right _ _)
  ϙ (by exact p) (by exact H)
  (by dsimp only;rw[Ϣ,Α (by exact ev_h2 _ _ H p)]) (by exact aH)
  (by exact r)).trans ?_
 rcases hd with⟨rfl,rfl,rfl⟩ | ⟨rfl,rfl,rfl⟩<;>simp only[dS,fT,ἁ,ἂ]
 all_goals
  simp only[conv2dSpec,conv2dWith,depthwiseSpec,depthwiseWith,hA,hB]
  refine mi (by rw[Ϝ,Ϝ,hA,hB]) λq e _=>?_
  simp only[ag,Λ]
  refine congrArg (p8 _) ?_
  rw[aB_eq];simp only[Dense.bO,Ϛ,n4,ϳ,Κ]
lemma ev(s:lR)(J:Ἆ)(I K b:Ἂ kF)(D:Ἆ)(d Y:Ἀ)
  (hd:D=false/\d=s.cin/\Y=s.cout\/D=true/\d=1/\Y=s.cin):
  fL s J I K b D d Y=dS s J I K b D:=by
 rw[fL_eq];unfold fLo
 refine (Classical.em _).elim (λhg=>?_) (λhn=>by exact dif_neg hn)
 exact (ev_h3 s J I K b D d Y hd (Dense.F.wcGet3 Dense.F.wcK3eq Dense.F.wcK3eq_ok lwS 32 (K,Y*(d*(s.kh*s.kw)),X6 (X6 ((s.h+2)*(s.w+2)) (s.w+2)) (X6 (s.kh*s.kw) d)) (fun k=>k.1)) hg.1 hg.2.1 hg.2.2.1 hg.2.2.2.1 hg.2.2.2.2.1 hg.2.2.2.2.2.1 hg.2.2.2.2.2.2.1 hg.2.2.2.2.2.2.2.1 hg.2.2.2.2.2.2.2.2.1 (by exact Z4 hg.2.2.2.2.2.2.2.2.2.1) (by exact Z4 hg.2.2.2.2.2.2.2.2.2.2.1) (by exact Z4 hg.2.2.2.2.2.2.2.2.2.2.2.1) (by exact Z4 hg.2.2.2.2.2.2.2.2.2.2.2.2.1) (by exact Z4 hg.2.2.2.2.2.2.2.2.2.2.2.2.2.1) (by exact Z4 hg.2.2.2.2.2.2.2.2.2.2.2.2.2.2.1) (by exact Z4 hg.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.1) hg.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.1 hg.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.1 (by exact Z4 hg.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2))
theorem s5c1(s:ConvShape)(J:Ἆ)(I K b:Ἂ Word)(e:s.kh=1)(g:s.kw=1)(j:s.stride=1)(l:s.pad=0)(p:0<s.h)(r:0<s.w):
  conv2dSpec s J I K b=kK (n:=s.cout*(s.h*s.w)) fun q=>act J (dotOf (s.cin+1)
   (fun t=>if t<s.cin then I[t*(s.h*s.w)+q.val%(s.h*s.w)]! else oneW)
   fun t=>if t<s.cin then K[q.val/(s.h*s.w)*s.cin+t]! else b[q.val/(s.h*s.w)]!):=by
 have m:s.ho=s.h:=by simp only[ConvShape.ho,e,j,l];(clear * - p;omega)
 have o:s.wo=s.w:=by simp only[ConvShape.wo,g,j,l];(clear * - r;omega)
 simp only[conv2dSpec,conv2dWith,e,g,ϻ]
 apply mi
 · simp only[Ϝ,m,o]
 intro q h1 h2;simp only[Λ];simp only[m,o];congr 2;funext t;split_ifs with h;swap;rfl
 have y:=ϊ (q/s.w) r;have y':=ϊ (q/s.w) p;have x:=ϊ q r
 simp only[windowAt,e,g,j,l,F5,Nat.mod_one,ϻ,ϑ,pQ]
 rw[if_neg (by (clear * - m o y' x;omega))];congr 1
 rw[Ϗ s.h s.w,Nat.mod_mul];ring
def s5rs(s:ConvShape):ConvShape:=
 if s.kh=1/\s.kw=1/\s.stride=1/\s.pad=0/\0<s.h/\s.w%8≠0 then
  if s.h*s.w%16=0 then {s with h:=s.h*s.w/16,w:=16}
  else if s.h*s.w%20=0/\s.w<20 then {s with h:=s.h*s.w/20,w:=20} else {s with h:=1,w:=s.h*s.w}
 else s
theorem s5rsE(s:ConvShape)(J:Ἆ)(I K b:Ἂ Word):conv2dSpec (s5rs s) J I K b=conv2dSpec s J I K b:=by
 unfold s5rs;split
 · rename_i h;obtain⟨e,g,j,l,p,w8⟩:=h
   have r:0<s.w:=ϵ (fun z=>by rw[z] at w8;exact w8 rfl)
   have hw:0<s.h*s.w:=Ϯ p r
   rw[s5c1 s J I K b e g j l p r]
   split
   · rename_i c;have d:=Nat.dvd_of_mod_eq_zero c
     rw[s5c1 {s with h:=s.h*s.w/16,w:=16} J I K b e g j l (Nat.div_pos (Nat.le_of_dvd hw d) (by decide)) (Nat.succ_pos _)]
     dsimp only;rw[Nat.div_mul_cancel d]
   split
   · rename_i c;have d:=Nat.dvd_of_mod_eq_zero c.1
     rw[s5c1 {s with h:=s.h*s.w/20,w:=20} J I K b e g j l (Nat.div_pos (Nat.le_of_dvd hw d) (by decide)) (Nat.succ_pos _)]
     dsimp only;rw[Nat.div_mul_cancel d]
   rw[s5c1 {s with h:=1,w:=s.h*s.w} J I K b e g j l Nat.one_pos hw];dsimp only;rw[n4]
 · rfl
def er:Ἂ Ἃ:=
 fL (s5rs s) J I K b ἄ (s5rs s).cin (s5rs s).cout
def cG:Ἂ Ἃ:=
 fL s J I K b ἃ 1 s.cin
lemma eu:er=conv2dSpec:=by funext s J I K b;rw[<-s5rsE s J I K b];exact ev (s5rs s) J I K b ἄ (s5rs s).cin (s5rs s).cout (.inl ⟨ἅ,ἅ,ἅ⟩)
lemma cY:cG=depthwiseSpec:=by funext s J I K b;exact ev s J I K b ἃ 1 s.cin (.inr ⟨ἅ,ἅ,ἅ⟩)
lemma aZ:(mJ (kB u))=(mJ u)/2^(23:Ἀ)%256:=by rw[kB];rw[Ξ,Υ,show(mJ (255:ἇ))=2^(8:Ἀ)-1 from ἅ,ζ];simp[β]
lemma aP:(mJ (lK u))=(mJ u)%2^(23:Ἀ):=by rw[lK];rw[Ξ,show(mJ (8388607:ἇ))=2^(23:Ἀ)-1 from ἅ,ζ]
noncomputable def Kf:Ἀ:=
 (if m/2^(23:Ἀ)=0 then m%2^(23:Ἀ) else 2^(23:Ἀ)+m%2^(23:Ἀ))*2^(if m/2^(23:Ἀ)=0 then 0 else m/2^(23:Ἀ)-1)
lemma iA:kM u*2^kG u=Kf ((mJ u)%2^(31:Ἀ)):=by
 have d:(mJ u)%2^(31:Ἀ)/2^(23:Ἀ)=(mJ (kB u)):=by rw[aZ];omega
 have e:(mJ u)%2^(31:Ἀ)%2^(23:Ἀ)=(mJ (lK u)):=by rw[aP];exact Nat.mod_mod_of_dvd _ (by decide)
 unfold kM kG Kf;rw[d,e];simp only[<-Β,nj]
lemma iU:Kf m<2^(m/2^(23:Ἀ)+23):=by
 unfold Kf;split_ifs with h
 .rw[h];simpa using ϊ m (show 0<2^(23:Ἀ) by norm_num)
 .have:=ϐ ((by (clear * -;omega)):2^(23:Ἀ)+m%2^(23:Ἀ)<2^(24:Ἀ)) (϶ (m/2^(23:Ἀ)-1));rwa[<-AF,((by (clear * - h;omega)):24+(m/2^(23:Ἀ)-1)=m/2^(23:Ἀ)+23)]at this
lemma dv_h1(e d:Ἀ)(m:e/2^(23:Ἀ)<d/2^(23:Ἀ)):Kf e<Kf d:=by
 have hd:d/2^(23:Ἀ)≠0:=Nat.ne_zero_of_lt m
 obtain⟨q,hq⟩:=Nat.exists_eq_succ_of_ne_zero hd
 have l:Kf d=(2^(23:Ἀ)+d%2^(23:Ἀ))*2^q:=by unfold Kf;rw[if_neg hd,if_neg hd,hq,Nat.succ_sub_one]
 rw[l];exact (iU e).trans_le ((Θ (by decide) ((Τ _ 23).trans_le (J1 (Nat.lt_succ_iff.1 (m.trans_eq hq)) 23))).trans ((m7 2 23 q).trans_le (ι _ (V1 _ _))))
lemma dv(e d:Ἀ)(h:e<d):Kf e<Kf d:=by
 rcases BJ (ϭ (c:=2^(23:Ἀ)) h.le)with g|m;have k:e%2^(23:Ἀ)<d%2^(23:Ἀ):=Nat.lt_of_add_lt_add_left (show 2^(23:Ἀ)*(d/2^(23:Ἀ))+e%2^(23:Ἀ)<_ by rw[ϛ d,←g,ϛ e];exact h)
 unfold Kf;rw[g];split_ifs;exact ϐ k (϶ _);exact ϐ (O1 k _) (϶ _);exact dv_h1 e d m
noncomputable def kv(w:Ἃ):Ἁ:=sv w*2^idxOf w
lemma im:
  kv u=if 2^(31:Ἀ)<=(mJ u)then-(Kf ((mJ u)%2^(31:Ἀ)):Ἁ)else(Kf ((mJ u)%2^(31:Ἀ)):Ἁ):=by unfold kv sv;rw[<-iA];simp only[<-ap];split_ifs<;>push_cast<;>ring
lemma bf:
  (mJ (sk u))=if (mJ u)<2^(31:Ἀ) then (mJ u)+2^(31:Ἀ) else if (mJ u)=2^(31:Ἀ) then 2^(31:Ἀ)
   else 2^(32:Ἀ)-1-(mJ u):=by have:=u.toNat_lt;rw[sk];simp only[Ι,AQ,<-Β,kN,nr];split_ifs<;>simp only[mh,pU,kN,nr];exact γ (by omega);omega
lemma ga_h1(x y N:Ἀ)(d:x<N+N)(g:y<N+N)(q:N+N-1-N<N)(h:(if x<N then x+N else if x=N then N else N+N-1-x)<(if y<N then y+N else if y=N then N else N+N-1-y)):(if N<=x then-(Kf (x%N):Ἁ)else(Kf (x%N):Ἁ))<(if N<=y then-(Kf (y%N):Ἁ)else(Kf (y%N):Ἁ)):=
 have M:∀z,N<=z->z<N+N->z%N=z-N:=λz i j=>(P8 i).trans (γ (G4 i j))
 have P:∀z,0<z->0<Kf z:=λz i=>C0 (ϟ _) (dv 0 z i)
 (Z7 x N).elim (λhx=>(Z7 y N).elim
  (λhy=>lt_of_eq_of_lt (if_neg (Nat.not_le.2 hx)) (lt_of_lt_of_eq (Int.ofNat_lt.2 (dv _ _ (lt_of_eq_of_lt (γ hx) (lt_of_lt_of_eq (Nat.lt_of_add_lt_add_right (lt_of_eq_of_lt (if_pos hx).symm (lt_of_lt_of_eq h (if_pos hy)))) (γ hy).symm)))) (if_neg (Nat.not_le.2 hy)).symm))
  λhy=>have h:=lt_of_eq_of_lt (if_pos hx).symm (lt_of_lt_of_eq h (if_neg (W3.2 hy)));absurd (if e:y=N then lt_of_lt_of_eq h (if_pos e) else K5 (U1 (lt_of_lt_of_eq h (if_neg e)) (Nat.sub_le_sub_left hy _)) q) (W3.2 (K2 _ _)))
 λhx=>(Z7 y N).elim
  (λhy=>have h:=lt_of_eq_of_lt (if_neg (W3.2 hx)).symm (lt_of_lt_of_eq h (if_pos hy));lt_of_eq_of_lt (if_pos hx) (lt_of_lt_of_eq (if e:x=N then Int.lt_of_le_of_lt (Int.neg_nonpos_of_nonneg (W6 _)) (Int.natCast_pos.2 (P _ (lt_of_lt_of_eq (Nat.lt_add_left_iff_pos.1 (lt_of_eq_of_lt (if_pos e).symm h)) (γ hy).symm))) else Int.lt_of_lt_of_le (Int.neg_neg_of_pos (Int.natCast_pos.2 (P _ (lt_of_lt_of_eq (Nat.sub_pos_of_lt (N4 hx (Ne.symm e))) (M x hx d).symm)))) (W6 _)) (if_neg (Nat.not_le.2 hy)).symm))
  λhy=>have h:=lt_of_eq_of_lt (if_neg (W3.2 hx)).symm (lt_of_lt_of_eq h (if_neg (W3.2 hy)));lt_of_eq_of_lt (if_pos hx) (lt_of_lt_of_eq (Int.neg_lt_neg (Int.ofNat_lt.2 (dv _ _ (lt_of_eq_of_lt (M y hy g) (lt_of_lt_of_eq (if e:y=N then lt_of_eq_of_lt ((congrArg (·-N) e).trans (Nat.sub_self N)) (Nat.sub_pos_of_lt (N4 hx λf=>B4 _ (lt_of_eq_of_lt (if_pos f.symm).symm (lt_of_lt_of_eq h (if_pos e))))) else have h:=lt_of_lt_of_eq h (if_neg e);if f:x=N then absurd (K5 (U1 (lt_of_eq_of_lt (if_pos f).symm h) (Nat.sub_le_sub_left hy _)) q) (B4 _) else Nat.sub_lt_sub_right hy (O8 λl=>W3.2 (Nat.sub_le_sub_left l _) (lt_of_eq_of_lt (if_neg f).symm h))) (M x hx d).symm))))) (if_pos hy).symm)
lemma ga(a b:ἇ)(h:sk a<sk b):kv a<kv b:=(congrArg₂ (·<·) (im a) (im b)).mpr (ga_h1 _ _ _ a.toNat_lt b.toNat_lt (N2 ἅ) ((congrArg₂ (·<·) (bf a) (bf b)).mp (Ι.mp h)))
lemma g6:Model.cmpDyadic (nh y) (nh z)=compare (kv y) (kv z):=by
 have d:∀w,(nh w).toReal=(kv w:ℝ)*2^(-149:Ἁ):=λw=>by rw[<-ϖ,lG,kv];push_cast;rw[Ac AL,AA];ring
 have e:(0:ℝ)<2^(-149:Ἁ):=(by positivity);cases h:compare (kv y) (kv z);rw[Model.cmpDyadic_lt_iff,d,d];exact AI (mod_cast compare_lt_iff_lt.1 h) e;rw[Model.cmpDyadic_eq_iff,d,d,compare_eq_iff_eq.1 h];rw[Model.cmpDyadic_gt_iff,d,d];exact AI (mod_cast compare_gt_iff_gt.1 h) e
lemma co(e:kM u=0):u=0\/u=0x80000000:=by
 have j:=u.toNat_lt;rw[kM]at e;split_ifs at e with g
 .have d:=aZ u;have l:=aP u;rw[g]at d;rw[<-Β,<-Β];simp at d⊢;omega
 .omega
lemma be:lI (mx y z)=nt (lI y) (lI z):=by
 rw[Conv.mx];split
 .rename_i h;have e:=kV y h.1;have g:=kV z h.2.1;have j:Model.compare (lI y) (lI z)=pj _:=Model.compare_eq_some_cmpDyadic_of_toDyadic? _ _ e g;rw[g6]at j;rw[nt,Model.Internal.Value.maximum,Model.withNaNSelection_of_none _ _ ((Model.chooseNaN2_eq_none_iff _ _).2 ⟨Χ e,Χ g⟩:Model.chooseNaN2 (lI y) (lI z)=none),Model.compareNonNaN_eq_of_compare_eq_some _ _ _ _ j];rcases h.2.2 with d|d|rfl;rw[AD d,compare_lt_iff_lt.2 (ga y z d)];rw[AC (nP d),compare_gt_iff_gt.2 (ga z y d)];rw[AC (UInt32.lt_irrefl _),compare_eq_iff_eq.2 rfl];by_cases l:kM y=0;rcases co y l with rfl|rfl<;>decide;simp[(by rw[lI,Model.isZero_eq_beq_zero_of_toDyadic?_some e,decodeD_sig_zero];simpa using l:Model.isZero (lI y)=ἄ)]
 exact aR _
lemma fn(w k U n:Ἀ)(e:0<k)(d:kF):
  lI (wl mx I w k n U 0 d)=
   (lj n).foldl (λm t=>nt m (lI I[U+t/k*w+t%k]!)) (lI d):=by rw[gc _ _ _ _ _ _ _ e,List.foldl_hom lI];exact λ_ _=>(be _ _).symm
lemma iP:
  ps (kT u:Model lU)=u:=by show BG (kT u:Model lU).bits=u;rw[pu]
lemma aS(d:kB u≠0):
  nh u=⟨k9 u,2^(23:Ἀ)+(mJ (lK u)),((mJ (kB u)):Ἁ)-150⟩:=by rw[nh];rw[AC d]
lemma cM(f:Ἀ)(e:f<2^(23:Ἀ)):
  lf (2^(23:Ἀ)+f) (2^(23:Ἀ))=0:=by
 have d:(Aq (2^(23:Ἀ)+f))=23:=by rw[mz (by exact Ne.symm (NeZero.ne' (2^23+f)))];constructor<;>omega
 rw[lf];rw[d,Nat.log2_two_pow];simp[mM,mZ,ο]<;>exact Nat.add_lt_of_lt_sub' e
lemma eO:la (m*2^(23:Ἀ)) (2^(23:Ἀ))=m:=by rw[la];simp only [nr,Nat.mul_mod_left,mul_zero,Nat.ofNat_pos, ↓reduceIte,Ak,OfNat.ofNat_ne_zero,not_false_eq_true,mul_div_cancel_right₀]
lemma hO_h1(er ed q:Ἀ)(M:ed=q+127)(x:q<er)(b:er<255):¬((er:Ἁ)-150-((ed:Ἁ)-150)>127)∧¬((er:Ἁ)-150-((ed:Ἁ)-150)< -150)∧¬((er:Ἁ)-150-((ed:Ἁ)-150)< -126)∧(F6 ((er:Ἁ)-150-((ed:Ἁ)-150)+127))=er-q:=by subst M;refine⟨?_,?_,?_,?_⟩<;>omega
lemma hO_h2(r q e:ἇ)(N:(mJ (kB r))=(mJ r)/2^(23:Ἀ)%256)(O:(mJ (lK r))=(mJ r)%2^(23:Ἀ))(x:(mJ q)<(mJ (kB r)))(hle:(mJ q)*2^(23:Ἀ)<=(mJ r))(g:(mJ e)=(mJ r)-(mJ q)*2^(23:Ἀ)):k9 e=k9 r∧(mJ (kB e))=(mJ (kB r))-(mJ q)∧(mJ (lK e))=(mJ (lK r)):=by
 refine⟨?_,?_,by rw[aP e,g,Ϗ (mJ q),Nat.sub_mul_mod ((Ϗ (2^(23:Ἀ)) (mJ q)).trans_le hle),O]⟩
 ·rw[ϓ,ap,ap,g,((by decide):2^(31:Ἀ)=256*2^(23:Ἀ)),<-mc (϶ 23),<-mc (϶ 23),Ϗ (mJ q),Nat.sub_mul_div];rw[N]at x;generalize (mJ r)/2^(23:Ἀ)=u at x⊢
  exact⟨λh=>h.trans (K4 _ _),λh=>by rw[<-Nat.mod_add_div u 256,R7 x.le];exact (J4 256 (Nat.div_pos h ((by decide):0<256))).trans (K2 _ _)⟩
 ·rw[aZ e,g,Ϗ (mJ q),Nat.sub_mul_div,N];rw[N]at x;generalize (mJ r)/2^(23:Ἀ)=a at x⊢
  have h:=Nat.mod_add_div a 256;have t:=ϊ a ((by decide):256>0)
  generalize a%256=s at x h t ⊢;generalize a/256=k at h;subst h
  rw[R7 x.le,Nat.add_mul_mod_self_left,γ ((K4 _ _).trans_lt t)]
lemma hO(q:ἇ)(p:lK d=0)(u:k9 d=ἄ)
  (M:(mJ (kB d))=(mJ q)+127)(v:kB d≠255)
  (x:(mJ q)<(mJ (kB r)))(j:kB r≠255):
  kW (pf (lI r) (lI d))=r-(q<<<23):=by
 have N:=aZ r
 have O:=aP r
 have U:(mJ (kB r))≠255:=λh=>j (Β.1 (by simpa using h))
 have E8:(mJ (kB r))<256:=N.trans_lt (ϊ _ ((by decide):256>0))
 have hh:=hO_h1 _ _ _ M x (N4 (J2 E8) U)
 rw[ny]
 rw[p0,lI,lI,kV r j,kV d v,aS r (λh=>by simp[h]at x),aS d (λh=>by simp[h]at M),p,u]
 have l:((2:Ἀ)^(23:Ἀ)==0)=ἄ:=(by decide)
 have m:(2^(23:Ἀ)+(mJ (lK r))==0)=ἄ:=(by simp)
 simp only[nj,ϑ,Bool.xor_false,l,m,fT,ἂ]
 change kW (n6 FloatFormat.binary32 (k9 r) (2^(23:Ἀ)+(mJ (lK r))) (2^(23:Ἀ)) _)=_
 rw[<-nA,bD]
 rw[pC,cM _ (fracOf_toNat_lt r)]
 simp only[l,m,fT,ἂ,Ai,sub_zero]
 rw[AC hh.1,AC hh.2.1,AC hh.2.2.1]
 have D:(2^(23:Ἀ)+(mJ (lK r))).shiftLeft 23=(2^(23:Ἀ)+(mJ (lK r)))*2^(23:Ἀ):=ο _ _
 rw[show(23:Ἁ)=Int.ofNat 23 from ἅ,mw,D,eO]
 simp only[(beq_eq_false_iff_ne.2 (R8 ((O1 (fracOf_toNat_lt r) _).trans_eq ((by decide):2^(23:Ἀ)+2^(23:Ἀ)=p7 24))):((2^(23:Ἀ)+(mJ (lK r)):Ἀ)==p7 24)=ἄ),fT,ἂ]
 rw[AC hh.1]
 set e:=r-(q<<<23)with F
 have o:(mJ (q<<<23))=(mJ q)*2^(23:Ἀ):=by rw[ϴ];simp only[kN,kH,ο];exact γ ((ϐ (x.trans E8) (϶ 23)).trans_le (by decide))
 have hle:(mJ q)*2^(23:Ἀ)<=(mJ r):=(ι _ (x.le.trans (N.trans_le (Ay _ _)))).trans (pr _ _)
 have g:(mJ e)=(mJ r)-(mJ q)*2^(23:Ἀ):=by rw[F,nG _ _ (by rw[V7,o];exact hle),o]
 rw[<-n7]
 conv_rhs=>rw[<-iP e,ofUInt32_eq_ofFields]
 obtain⟨z1,z2,z3⟩:=hO_h2 r q e N O x hle g
 rw[z1,z2,z3,hh.2.2.2,((by decide):p7 23=2^(23:Ἀ)),ϱ]
lemma i9(s:kF)(h:s&&&0x7FFFFFFF=0):s=0\/s=0x80000000:=by
 have e:=s.toNat_lt;have d:(mJ (s&&&0x7FFFFFFF))=(mJ s)%2^(31:Ἀ):=by rw[Ξ,show(mJ (0x7FFFFFFF:ἇ))=2^(31:Ἀ)-1 from ἅ,ζ]
 rw[h]at d;rw[<-Β,<-Β];simp at d⊢;(clear * - e d;omega)
lemma cg(e:r=0\/r=0x80000000)(g:kB d≠0)(j:kB d≠255)(l:k9 d=ἄ):
  kW (pf (lI r) (lI d))=r:=by
 rw[ny,p0,lI,lI,kV r (by rcases e with rfl|rfl<;>decide),kV d j,aS d g,l,(by rcases e with rfl|rfl<;>rfl:nh r=⟨k9 r,0,0⟩)];simp[(by simp:(2^(23:Ἀ)+(mJ (lK d))==0)=ἄ),fT,ἂ,Az,ἁ,Bool.xor_false];rcases e with rfl|rfl<;>rfl
lemma aX{a b:ἇ}(h:a<b):a≠b:=λe=>by have:=Ι.mp h;rw[e]at this;exact (lt_self_iff_false b.toNat).mp this
abbrev av:Ἃ:=toWord (Ar (ofW (dotOf n fa λ_=>oneW)) (ofW d))
lemma gl(v:DQ d):eX r d=kW (pf (lI r) (lI d)):=by
 obtain⟨e,l,m⟩:=v;have g:=aX m;dsimp only[eX];split;obtain⟨q,j,o,p⟩:=‹_/\_›;exact(hO r d _ q e (by rw[nG _ _ j];have:=V7.1 j;simp at this⊢;(clear * - this;omega)) g (Ι.1 o) (aX p)).symm;split;exact(cg r d (i9 r ‹_›) l g e).symm;rfl
lemma gp(g:Ἀ->ἆ)(m:Ἀ):∀d:ἆ,
  Z ((lj m).foldl (λd t=>d+g t) d)=Z d+∑t∈kP m,Z (g t):=by
 induction m with
 |zero=>intro d;simp only [nr,List.range_zero,mT,range_zero,sum_empty,Av]
 |succ m e=>
  intro d;rw[mD,l6,Ϸ,mT,fI,e,Ν];ring
lemma jB:ae 0=0:=by rw[eQ];rfl
lemma gK:
  (decide (kM u*kM ld=0)&&Bool.xor (k9 u) (k9 ld))=(u==0x80000000):=by
 by_cases d:kM u=0
 .rcases co u d with rfl|rfl<;>decide
 have:(u==0x80000000)=ἄ:=by simp;rintro rfl;exact d (by decide)
 rw[this,((by decide):kM ld=2^(23:Ἀ)),((by decide):k9 ld=ἄ)];simp[d]
lemma dW(c:Ἀ):∀i,i+c<=n->
  (nC (kK (n:=n) λj=>fa (Fin.val j)) (kK (n:=n) λ_=>ld) i c=ἃ↔
   ∀t,i<=t->t< i+c->fa t=0x80000000):=by
 induction c with
 |zero=>intro i _;simp[nC];intro t d e;(clear * - d e;omega)
 |succ c l=>
  intro i m
  rw[nC,ab n fa i (by exact Nat.lt_of_add_right_lt m),ab n (λ_=>ld) i (by exact Nat.lt_of_add_right_lt m),
   gK,Η,l (i+1) (by (clear * - m;omega)),AQ]
  constructor
  .rintro⟨d,e⟩ t g o
   rcases BJ g with h|h
   .subst h;exact d
   .exact e t (by exact Order.add_one_le_iff.mpr h) (by (clear * - o;omega))
  .intro h;exact⟨h i (le_refl _) (by (clear * -;omega)),λt d e=>h t (by exact X5 d) (by (clear * - e;omega))⟩
lemma eW(O:0<n)(v:∀t<n,kB (fa t)≠255)
  (I:af n fa (λ_=>ld)=0)(g:Ἆ)(L:g=ἃ↔∀t<n,fa t=0x80000000):
  ms n fa (λ_=>ld)=if g then 0x80000000 else 0:=by
 have G:=cH n fa (λ_=>ld);have H:=dW n fa n 0 (by (clear * -;omega));have D:=ab n fa;have F:=ab n λ_=>ld;have d:=Ϝ (f:=λi:Fin n=>fa (Fin.val i));have p:=Ϝ (f:=λ_:Fin n=>ld);rw[ms];unfold dotSpec;rw[←Model.Reduction.map_value_dot];generalize (kK (n:=n) λi=>fa (Fin.val i))=A,(kK (n:=n) λ_:Fin n=>ld)=B at*
 have l:∀k,0<=k->k<0+(O0 (A.map (kT (fmt:=lU))))->kB A[k]!≠255/\kB B[k]!≠255:=λk _ e=>by rw[m0,d]at e;rw[D k (by (clear * - d p e;omega)),F k (by (clear * - d p e;omega))];exact⟨v k (by (clear * - d p e;omega)),(by decide)⟩
 have q(X:Ἂ ἇ):(O0 X)=n->(∀k<n,kB X[k]!≠255)->∀k (e:k<(O0 (X.map (kT (fmt:=lU))))),mN (X.map (kT (fmt:=lU)))[k]=ἃ:=by intro N K k e;rw[m0]at e;have j:=K k (by exact V6 e N);rw[AB X k e]at j;simp[m6,j]
 have m:(O0 (A.map (kT (fmt:=lU))))=(O0 (B.map (kT (fmt:=lU)))):=by simp[d,p]
 have r:=q A d λk e=>(l k (by exact ϟ k) (by simp[d];exact I7 e)).1;have u:=q B p λk e=>(l k (by exact ϟ k) (by simp[d];exact I7 e)).2;have o:=ϯ (destination:=lU) _ _ m;rw[finiteDotSlice_eq A B 0 _ (by simp) (by rw[<-m];simp) l,m0,d,G,I,lH,nv]at o;rw[Ϭ lU _ _ .nearest m r u ((ϲ _).1 o)];simp only[Except.map]
 obtain⟨j,x⟩:=dotStateLoop_flags A B ({}:me lU) 0 _ (by simp) (by rw[<-m];simp) l;rw[lT,j,x];simp only[m0,d,(by rw[ϓ,H,L];exact⟨λh t M=>h t (by exact ϟ t) (by exact Nat.lt_add_left 0 M),λh t _ M=>h t (by (clear * - d p M;omega))⟩:nC A B 0 n=g)];cases g<;>simp[show n≠0 by omega]<;>rfl
lemma dk(w k U m:Ἀ)(l:0<k):
  wa Q I w k m U 0 ἃ=ἃ↔∀t< m,Q I[U+t/k*w+t%k]! =ἃ:=by
 rw[wa,gc _ _ _ _ _ _ _ l];induction m;simp only [List.range_zero,mT,not_lt_zero,IsEmpty.forall_iff,implies_true];rename_i m o;rw[mD,l6,Ϸ,mT,Η,o];use λ⟨g,j⟩ t d=>(Z6.1 d).elim (g t) λe=>e▸j,λh=>⟨λt d=>h t (by exact Nat.lt_add_right 1 d),h m (by exact lt_add_one m)⟩
lemma cu(d y x t:Ἀ):
  I[(d*h+y*k)*w+x*k+t/k*w+t%k]! =nI h w k I d y x t:=by congr 1;ring
lemma eg(B c A H:ἆ)(D:(lq B)<=255)(G:(lq c)<=62)(J:0<n)
  (u:∀t<n,ok c B (fa t)=true)(l j:kF->ἆ)(L M:Ἁ)
  (T:∀t<n,ae (j (fa t))*2^(25:Ἀ)+ae (l (fa t))=bU (lq B) (fa t)/\
   |ae (l (fa t))|<=L/\|ae (j (fa t))|<=M)(O:n*L<2^(63:Ἀ))(U:n*M<2^(63:Ἀ))
  (x:Z A=∑t∈kP n,Z (l (fa t)))(F:Z H=∑t∈kP n,Z (j (fa t)))
  (g:Unit->Ἆ)(I:g ()=true↔∀t<n,fa t=0x80000000)(q:Unit->kF)
  (v:q ()=ms n fa λ_=>ld)(d:kF)(m:DQ d):
  wo B A H g q d=av n fa d:=by
 have p:=λt e=>dI c B _ D G (u t e);have K:=A3.1 (cJ n _ _ λt e=>(T t e).2.1);have N:=A3.1 (cJ n _ _ λt e=>(T t e).2.2);have o:af n fa (λ_=>ld)=(∑t∈kP n,bU (lq B) (fa t))*2^((lq B)+149):=by rw[n8];refine ϔ ἅ λt e=>?_;rw[fB _ ld _ 126 ((p t (mr e)).2.imp_right (·.1)) (BA (by decide)),((by decide):bU 126 ld=2^(23:Ἀ))];ring
 rw[<-ϔ ἅ λt e=>(T t (mr e)).1,nJ,<-n8,<-sZ A (∑t∈kP n,ae (l (fa t))) (by simp only[x,gS,Int.cast_sum]) ((neg_le_neg O.le).trans K.1) (K.2.trans_lt O),<-sZ H (∑t∈kP n,ae (j (fa t))) (by simp only[F,gS,Int.cast_sum]) ((neg_le_neg U.le).trans N.1) (N.2.trans_lt U)]at o;rw[wo,av];split;next r=>rw[r.1,r.2,jB]at o;rw[eW n fa J (λt e=>(p t e).1) (by rw[o];ring) _ I,cg _ d (by split<;>simp) m.2.1 (aX m.2.2) m.1]
 .rw[gl _ d m];refine congrArg (λs=>kW (pf (lI s) (lI d))) ?_;rw[Dense.dp,dY];split
  .refine(aY n fa _ _ _ _ (λt e=>⟨(p t e).1,(by decide)⟩) ?_ ‹_›).symm;rw[o,eU,(by simp[η];(clear * - D;omega):(lq (B+149))=(lq B)+149)]
  .exact v
lemma ee(c:ἆ)(x:kF)(g:(lq B)<=255)(m:(lq c)<=37)(h:ok c B x=true):
  ae 0*2^(25:Ἀ)+ae (t1 B x)=bU (lq B) x/\|ae (t1 B x)|<=2^(24+(lq c))-1/\
   |ae 0|<=0:=by
 obtain⟨-,j⟩:=dI c B x g (by (clear * - m;omega)) h;obtain⟨l,d⟩:=fw B x c.toNat (by (clear * - m;omega)) j;have k:=Int.le_sub_one_of_lt (Int.ofNat_lt.mpr d);rw[<-nE (kM x*_),Y4,I4,I4,nZ]at k;rw[jB,(by rw[t1,bU,sv,jj _ _ (by rw[l];exact d.trans_le (Θ (by decide) (by (clear * - m;omega)))),l];cases k9 x<;>simp only[fT,ἁ,Am,neg_mul,Y4,I4,nZ]:ae (t1 B x)=bU B.toNat x)];refine⟨by rw[nv,Ai],?_,A5.le⟩;rw[bU,sv];cases k9 x<;>simp only[fT,ἁ,Am,neg_mul,BE,k]
lemma em(f:kF->ἆ)(h w k:Ἀ)(I:Ἂ kF)(d y x:Ἀ)(e:0<k):
  Z (cr f I w k (k*k) ((d*h+y*k)*w+x*k) 0 0)=
   ∑t∈kP (k*k),Z (f (nI h w k I d y x t)):=by rw[cr,gc _ _ _ _ _ _ _ e,gp,bF,Ai];simp only[cu]
lemma cW:
  cV Q (fa 0) (fa 1) (fa 2) (fa 3)=true↔∀t<2*2,Q (fa t)=true:=by
 simp only[cV,Η]
 exact⟨λ⟨⟨⟨a,b⟩,c⟩,e⟩ t d=>by
  interval_cases t<;>assumption,
  λH=>⟨⟨⟨H 0 (by decide),H 1 (by decide)⟩,H 2 (by decide)⟩,H 3 (by decide)⟩⟩
lemma ia:(lq (eF u))=(mJ (kB u)):=by rw[eF,kB];simp only [UInt32.toUInt64_and,UInt32.toUInt64_shiftRight,Nat.reduceLT,UInt32.toUInt64_ofNat,UInt64.reduceMod,Ε,ν,ϩ,kA,kH,Ξ,Υ,kN]
lemma ik(x:kF):aq x<=255:=by
 rw[aq];split
 .rw[θ,am];rw[kG];have:=mO x;split<;>simp<;>exact X5 this
 .exact UInt64.le_refl _
lemma mn(a b:ἆ):(lq (po a b))=po (lq a) (lq b):=by
 show(lq (if a<=b then a else b))=_;split<;>rename_i h
 .have:=θ.mp h;exact Eq.symm (Nat.min_eq_left h)
 .have:=mt θ.mpr h;exact Eq.symm (Std.LawfulOrderLeftLeaningMin.min_eq_right a.toNat b.toNat h)
lemma dH(a b:ἆ):po a b<=a:=θ.mpr (by rw[mn];exact P7 a.toNat b.toNat)
lemma fX(a b:ἆ):po a b<=b:=θ.mpr (by rw[mn];exact Nat.min_le_right a.toNat b.toNat)
lemma fZ(a b:ἆ):a<=max a b:=by show a<=if a<=b then b else a;split<;>simp_all
lemma f0(a b:ἆ):b<=max a b:=by show b<=if a<=b then b else a;split<;>rename_i h<;>simp_all[θ];exact X5 h
lemma cQ(x:kF)(B M:ἆ)(d:B<=aq x)(r:eF x<=M)(p:M<=po (B+37) 254)
  (o:B<=255):ok 37 B x=ἃ:=by
 have q:=(ia x).ge.trans (θ.1 r);rw[θ,mn,η,Nat.le_min]at p;simp only[ok,Η,AM,nM,AQ]
 refine⟨λe=>absurd ((congrArg mJ e).ge.trans (q.trans p.2)) (by decide),?_⟩
 by_cases m:mW x=0;exact pZ m;right;rw[aq,if_pos (AM.2 m)]at d;apply A6
 rw[θ,χ _ _ d,am];exact tsub_le_iff_left.2 ((Dense.aT x).trans_le ((K4 _ _).trans (q.trans (p.1.trans (Ay _ _)))))
lemma ki(d e g j:kF)
  (h:max (max (eF d) (eF e)) (max (eF g) (eF j))<=po (po (po (aq d) (aq e)) (po (aq g) (aq j))+37) 254):
  po (po (aq d) (aq e)) (po (aq g) (aq j))<=255/\
   cV (ok 37 (po (po (aq d) (aq e)) (po (aq g) (aq j)))) d e g j=ἃ:=by
 have l:po (po (aq d) (aq e)) (po (aq g) (aq j))<=255:= ϗ (ϗ (dH _ _) (dH _ _)) (ik d);simp only[cV,Η];use l,⟨⟨cQ _ _ _ (ϗ (dH _ _) (dH _ _)) (ϗ (fZ _ _) (fZ _ _)) h l,cQ _ _ _ (ϗ (dH _ _) (fX _ _)) (ϗ (f0 _ _) (fZ _ _)) h l⟩,cQ _ _ _ (ϗ (fX _ _) (dH _ _)) (ϗ (fZ _ _) (f0 _ _)) h l⟩,cQ _ _ _ (ϗ (fX _ _) (fX _ _)) (ϗ (f0 _ _) (f0 _ _)) h l
lemma gU_h1{P w:Ἀ}(G:P+w+1<2^(31:Ἀ)):(P+1)%2^nx=P+1/\(P+w)%2^nx=P+w/\(P+w+1)%2^nx=P+w+1/\w<2^(32:Ἀ):=
 have c:2^(31:Ἀ)<=2^(32:Ἀ):=by decide
 have K:=G.trans_le (c.trans dj)
 ⟨γ ((L1 (V1 P w) 1).trans_lt K),γ ((Nat.lt_add_one _).trans K),γ K,(K2 w P).trans_lt ((Nat.lt_add_one _).trans (G.trans_le c))⟩
lemma gU(B:DQ d)(j y x:Ἀ)(e:Ἄ)
  (D:(J0 e)=(j*h+y*2)*w+x*2)(G:(j*h+y*2)*w+x*2+w+1<2^(31:Ἀ)):
  gT h w I d j y x e=av (2*2) (nI h w 2 I j y x) d:=by have q:=cu h w 2 I j y x;have p:=q 0;have r:=q 1;have u:=q 2;have v:=q 3;simp only[Nat.reduceDiv,kH,nv,n4,Av]at p r u v;have m:kW (pf (lI (cU h w 2 I j y x)) (lI d))=av (2*2) (nI h w 2 I j y x) d:=ἅ;have H:=gU_h1 G;have g:∀i o,I.uget i o=I[(J0 i)]!:=λi o=>(AB I _ o).symm;dsimp only[gT];split;swap;exact m;simp only[g,lJ,mA,aa _ H.2.2.2,H,D,p,r,u,v];split;swap;exact m;rename_i l;have l:=ki _ _ _ _ l;have A:=(cW _ (nI h w 2 I j y x)).1 l.2;exact eg _ _ _ 37 _ _ (cB l.1) (by decide) (by decide) A _ (λ_=>0) _ 0 (λt F=>ee _ _ _ (cB l.1) (by decide) (A t F)) (by decide) (by decide) (by simp only[fI,show 2*2=4 from ἅ,Ν,X8,Ai]) (by simp[bF]) _ (by rw[cW];simp only[AQ]) _ ἅ d B
lemma dO(f:Ἀ->Ἀ->Ἀ->kF)
  (e:∀d y x,f d y x=av (k*k) (nI h w k I d y x) (natW (k*k))):
  eT c (h/k) (w/k) f=n9 c h w k I:=by
 rw[iO];rw[n9];refine mi (by simp[pa]) λq _ _=>?_
 simp only[ag,Λ,pa];exact e _ _ _
theorem gI_h1(a b n:Ἀ)(q:a-b=n+1):b<a∧a-(b+1)=n:=⟨N1 q,by rw[←F7,q,BK]⟩
theorem gI_h2(a b:Ἀ)(q:a-b=0):¬b<a:=W3.2 (K3.1 q)
lemma gI(B:ἆ)(d:Ἄ)(r:(J0 d)<=(O0 I)):∀m (e:Ἄ) (l:ἆ),
  (J0 d)-(J0 e)=m->df I B d r e l=(lj m).foldl (λa j=>a+t1 B I[(J0 e)+j]!) l:=by have g:∀i o,I.uget i o=I[(J0 i)]!:=λi o=>(AB I _ o).symm;intro m;induction m<;>intro e l q;rw[df,dif_neg (mt ε.1 (gI_h2 _ _ q))];rfl;rename_i u;obtain⟨h0,h1⟩:=gI_h1 _ _ _ q;have h:e<d:=ε.2 h0;have p:=Dense.ac h;rw[df,AT h,u (e+1) _ (p▸h1),List.range_succ_eq_map,Ϸ,List.foldl_map];simp only[g,p,ϑ,Nat.succ_eq_add_one,pA,Τ 1]
lemma gx(B:ἆ)(U:Ἀ):fj I B w k U=cr (t1 B) I w k (k*k) U 0 0:=by
 rw[fj];split
 .rename_i h;have⟨d,e,g,j⟩:=h
  rw[gI I B _ _ (k*k) _ 0 (by rw[aa _ (Dense.bc j),aa _ (by (clear * - h;omega))];exact Nat.add_sub_self_left U (k*k)),cr,
   gc _ _ _ _ _ _ _ d,aa _ (by (clear * - h;omega))]
  exact congrArg (List.foldl · 0 _) (funext λa=>funext λt=>by rw[e,pA,Au])
 .rfl
lemma jL(m:Ἀ)(h:m<16384):(lq (cK m))<=37/\(m:Ἁ)*(2^(24+(lq (cK m)))-1)<2^(63:Ἀ):=by rw[cK];split_ifs<;>simp only [kA,Nat.reduceLeDiff,Nat.reduceAdd,Int.reducePow,Int.reduceSub,true_and]<;>omega
@[noinline]def cUe(w:Ἀ)(I:Ἂ Ἃ)(e:Ἀ):Ἃ:=dotOf (2*2) (λt=>I[e+t/2*w+t%2]!) λ_=>oneW
lemma cUq(h w:Ἀ)(I:Ἂ Ἃ)(ch y x:Ἀ):cU h w 2 I ch y x=cUe w I ((ch*h+y*2)*w+x*2):=by
 unfold cU cUe;congr 1;funext t;exact (cu h w 2 I ch y x t).symm
@[inline]def gTm(w:Ἀ)(ww:Ἄ)(I:@&Ἂ Ἃ)(d:Ἃ)(e:Ἄ):Ἃ:=
 if hb:e<I.usize/\e+1<I.usize/\e+ww<I.usize/\e+ww+1<I.usize then
  let x0:=I.uget e (uL hb.1)
  let x1:=I.uget (e+1) (uL hb.2.1)
  let x2:=I.uget (e+ww) (uL hb.2.2.1)
  let x3:=I.uget (e+ww+1) (uL hb.2.2.2)
  let B:=min (min (aq x0) (aq x1)) (min (aq x2) (aq x3))
  if max (max (eF x0) (eF x1)) (max (eF x2) (eF x3))<=min (B+37) 254 then
   wo B (t1 B x0+t1 B x1+t1 B x2+t1 B x3) 0 (λ_=>cV (.==0x80000000) x0 x1 x2 x3)
    (λ_=>cUe w I (J0 e)) d
  else toWord (Ar (ofW (cUe w I (J0 e))) (ofW d))
 else toWord (Ar (ofW (cUe w I (J0 e))) (ofW d))
lemma gTe(h w:Ἀ)(I:Ἂ Ἃ)(d:Ἃ)(ch y x:Ἀ)(e:Ἄ)(u:(ch*h+y*2)*w+x*2=(J0 e)):gT h w I d ch y x e=gTm w (G0 w) I d e:=by
 rw[gT,gTm];simp only[cUq,u]
lemma qVm(r x W:Ἀ)(hx:x<W):(r*W+x)/W=r/\(r*W+x)%W=x:=
 ⟨by rw[Τ,Ρ _ _ (by exact L7 hx),ώ hx,Κ],by rw[Τ,Μ,γ hx]⟩
lemma uA{a b:Ἄ}(h:(J0 a)+(J0 b)<2^(32:Ἀ)):(J0 (a+b))=(J0 a)+(J0 b):=Dense.aj (by have:=ς;exact FloatLib.Numerics.StaticStorage.lt_capacity h this)
@[specialize]def pQ(f:Ἄ->Ἃ)(W2 H2 st sk:Ἄ)(q e ox oy:Ἄ)(a:Ἂ Ἃ):Ἂ Ἃ:=
 if hq:q<a.usize then
  let a:=a.uset q (f e) (uL hq)
  if ox+1<W2 then pQ f W2 H2 st sk (q+1) (e+2) (ox+1) oy a
  else if oy+1<H2 then pQ f W2 H2 st sk (q+1) (e+st) 0 (oy+1) a
  else pQ f W2 H2 st sk (q+1) (e+sk) 0 0 a
 else a
termination_by (O0 a)-(J0 q)
decreasing_by all_goals simp only[BT];have:=uL hq;rw[Dense.ac hq];omega
lemma pqG(a b x H W:Ἀ):a*H*W<=(a*H+b)*W+x:=(ι W (V1 _ b)).trans (V1 _ x)
lemma pQl(f:Ἄ->Ἃ)(F:Ἀ->Ἃ)(E:Ἀ->Ἀ->Ἀ->Ἀ)(C H W R S:Ἀ)(Wu Hu st sk:Ἄ)
  (hW:(J0 Wu)=W)(hH:(J0 Hu)=H)(hst:(J0 st)=R)(hsk:(J0 sk)=S)(W0:0<W)(hN:C*H*W<2^(32:Ἀ))
  (hX:∀a b x,E a b x+2=E a b (x+1))(hR:∀a b,E a b (W-1)+R=E a (b+1) 0)(hS:∀a,E a (H-1) (W-1)+S=E (a+1) 0 0)
  (hE:∀a b x,a<=C->b<H->x<W->E a b x<2^(32:Ἀ))
  (hf:∀a b x (e:Ἄ),a<C->b<H->x<W->(J0 e)=E a b x->f e=F ((a*H+b)*W+x)):
  ∀m(a b x:Ἀ)(q e ox oy:Ἄ)(A:Ἂ Ἃ),C*H*W-(J0 q)=m->(O0 A)=C*H*W->(J0 q)=(a*H+b)*W+x->a<=C->b<H->x<W->
  (J0 ox)=x->(J0 oy)=b->(J0 e)=E a b x->(∀j<(J0 q),A[j]! =F j)->pQ f Wu Hu st sk q e ox oy A=ag (C*H*W) F:=by
 have hU:(A:Ἂ Ἃ)->(O0 A)<2^(32:Ἀ)->(J0 A.usize)=(O0 A):=λA g=>by change (J0 (G0 ((O0 A))))=(O0 A);exact aa _ g
 intro m;induction m with
 |zero=>
  intro a b x q e ox oy A hm hA hq ha hb hx ho hy he ht
  have hz:=K3.1 hm
  rw[pQ,dif_neg (by rw[ε,hU A (hA.trans_lt hN)];exact W3.2 (hA.trans_le hz))]
  apply mi;·simp[hA,ag]
  intro j h1 h2;simp only[ag,Λ];rw[<-AB A j h1];exact ht j (U1 (h1.trans_eq hA) hz)
 |succ m ih=>
  intro a b x q e ox oy A hm hA hq ha hb hx ho hy he ht
  have hqc:(J0 q)<C*H*W:=N1 hm
  have hC:a<C:=O8 λh=>Nat.not_le_of_lt hqc ((ι _ (ι _ h)).trans ((pqG a b x H W).trans_eq hq.symm))
  have qa:q<A.usize:=by rw[ε,hU A (hA.trans_lt hN)];exact hqc.trans_eq hA.symm
  have q1:(J0 (q+1))=(J0 q)+1:=Dense.ac qa
  have t2:(J0 (2:Ἄ))=2:=by simp
  have hv:=hf a b x e hC hb hx he
  have hs:(O0 (A.uset q (f e) (uL qa)))=C*H*W:=by rw[BT,hA]
  have hj:∀j<(J0 q)+1,(A.uset q (f e) (uL qa))[j]! =F j:=by
   intro j g;have hjA:j<(O0 A):=U1 g (uL qa);simp only[pP];rw[AB _ j (by simp only[Array.size_set,hjA]),Array.getElem_set]
   split;·rename_i k;rw[<-k,hq,<-hv]
   rename_i k;rw[<-AB A j hjA];exact ht j (N4 (J2 g) (Ne.symm k))
  have o1:(J0 (ox+1))=x+1:=by rw[uA (by rw[ho,mA];exact C0 ((K2 x _).trans_lt (hq.symm.trans_lt hqc)) hN),ho,mA]
  have b2:b<=(J0 q):=hq▸(K2 b (a*H)).trans ((J4 _ W0).trans (V1 _ x))
  have y1:(J0 (oy+1))=b+1:=by rw[uA (by rw[hy,mA];exact C0 (C0 b2 hqc) hN),hy,mA]
  have hm1:C*H*W-(J0 (q+1))=m:=by rw[q1,V4,hm,BK]
  rw[pQ,dif_pos qa];dsimp only
  split_ifs with g1 g2
  ·rw[ε,o1,hW]at g1
   exact ih a b (x+1) _ _ _ _ _ hm1 hs (by rw[q1,hq,pA]) ha hb g1 o1 hy (by rw[uA (by rw[he,t2];exact (hX a b x).trans_lt (hE a b (x+1) ha hb g1)),he,t2,hX]) (by rw[q1];exact hj)
  ·rw[ε,o1,hW]at g1;rw[ε,y1,hH]at g2
   have xw:x+1=W:=C4 hx (W3.1 g1)
   rw[Nat.eq_sub_of_add_eq xw]at he
   exact ih a (b+1) 0 _ _ _ _ _ hm1 hs (by rw[q1,hq,pA,xw,<-pA (a*H),nO,ϑ]) ha g2 W0 Ϧ y1 (by rw[uA (by rw[he,hst];exact (hR a b).trans_lt (hE a (b+1) 0 ha g2 W0)),he,hst,hR]) (by rw[q1];exact hj)
  ·rw[ε,o1,hW]at g1;rw[ε,y1,hH]at g2
   have xw:x+1=W:=C4 hx (W3.1 g1)
   have yb:b+1=H:=C4 hb (W3.1 g2)
   rw[Nat.eq_sub_of_add_eq xw,Nat.eq_sub_of_add_eq yb]at he
   exact ih (a+1) 0 0 _ _ _ _ _ hm1 hs (by rw[q1,hq,pA,xw,<-nO (a*H+b) W,pA (a*H) b 1,yb,ϑ,ϑ,nO a H]) hC (L7 hb) W0 Ϧ Ϧ (by rw[uA (by rw[he,hsk];exact (hS a).trans_lt (hE (a+1) 0 0 hC (L7 hb) W0)),he,hsk,hS]) (by rw[q1];exact hj)
lemma pqE(a b x c h w:Ἀ)(ha:a<=c)(hb:b<h/2)(hx:x<w/2)(c0:0<c):(a*h+b*2)*w+x*2<2*(c*h*w):=by
 have s2:(a*h+b*2+2)*w<=(c*h+h)*w:=ι _ (by have:=ι h ha;(clear * - hb this;omega))
 have s3:=ι w (lZ h c0)
 rw[show (a*h+b*2+2)*w=(a*h+b*2)*w+2*w by ring,Ϡ (c*h) h w]at s2;(clear * - hx s2 s3;omega)
noncomputable def aT(h w:Ἀ)(I:Ἂ Ἃ)(d:Ἃ)(q:Ἀ):Ἃ:=gT h w I d (q/(h/2*(w/2))) (q/(w/2)%(h/2)) (q%(w/2)) ((G0 ((q/(h/2*(w/2))*h+q/(w/2)%(h/2)*2)*w))+2*(G0 (q%(w/2))))
lemma aQe_h0(n m:Ἀ)(B:m<2^(30:Ἀ))(h:n<=2*m):n<2^(32:Ἀ):=by omega
lemma aQe_h1(n m:Ἀ)(B:m<2^(30:Ἀ))(h:n<=m):(J0 (G0 n))=n:=aa _ (aQe_h0 _ _ B (h.trans (lZ _ (by decide))))
lemma aQe_h2(w:Ἀ)(W:0<w/2):(w/2-1)*2+(w+(w-2*(w/2))+2)=2*w/\w+(w-2*(w/2))+2<=2*w:=by omega
lemma aQe_h3(h:Ἀ)(H:0<h/2):(h/2-1)*2+(2+(h-2*(h/2)))=h/\2+(h-2*(h/2))<=h:=by omega
lemma aQe_h4(h w:Ἀ)(W:0<w/2):∀a b:Ἀ,(a*h+b*2)*w+(w/2-1)*2+(w+(w-2*(w/2))+2)=(a*h+(b+1)*2)*w+0*2:=
 λa b=>by rw[pA,(aQe_h2 w W).1,ϳ,ϑ,nO b 2,<-pA,Ϡ (a*h+b*2) 2 w]
lemma aQe_h5(h w:Ἀ)(H:0<h/2)(W:0<w/2):∀a:Ἀ,(a*h+(h/2-1)*2)*w+(w/2-1)*2+(w+(w-2*(w/2))+2+(h-2*(h/2))*w)=((a+1)*h+0*2)*w+0*2:=
 λa=>by rw[pA,<-pA ((w/2-1)*2),(aQe_h2 w W).1,<-Ϡ,<-Ϡ,pA (a*h),(aQe_h3 h H).1,ϳ,ϑ,ϑ,nO a h]
lemma aQe_h6(c h w:Ἀ)(H:0<h/2)(W:0<w/2)(c0:0<c)(B:c*h*w<2^(30:Ἀ)):(J0 (G0 (w/2)))=w/2/\(J0 (G0 (h/2)))=h/2/\(J0 (G0 (w+(w-2*(w/2))+2)))=w+(w-2*(w/2))+2/\(J0 (G0 (w+(w-2*(w/2))+2+(h-2*(h/2))*w)))=w+(w-2*(w/2))+2+(h-2*(h/2))*w/\c*(h/2)*(w/2)<2^(32:Ἀ):=by
 obtain⟨-,s3⟩:=aQe_h2 w W
 obtain⟨-,s4⟩:=aQe_h3 h H
 have hl:h*w<=c*h*w:=ι w (lZ h c0)
 have wl:w<=c*h*w:=(lZ w (H.trans_le (L5 h 2))).trans hl
 exact⟨aQe_h1 _ _ B ((L5 w 2).trans wl),aQe_h1 _ _ B ((L5 h 2).trans ((J4 _ (W.trans_le (L5 w 2))).trans hl)),aQe_h1 _ _ B (s3.trans ((ι w ((V1 2 _).trans s4)).trans hl)),aQe_h1 _ _ B ((L1 s3 _).trans (((Ϡ 2 _ w).symm.trans_le (ι w s4)).trans hl)),aQe_h0 _ _ B ((ϧ (U3 _ (L5 _ _)) (L5 _ _)).trans (lZ _ (by decide)))⟩
lemma aQe(c h w:Ἀ)(I:Ἂ Ἃ)(d:Ἃ)(H:0<h/2)(W:0<w/2)(B:c*h*w<2^(30:Ἀ)):
  pQ (gTm w (G0 w) I d) (G0 (w/2)) (G0 (h/2)) (G0 (w+(w-2*(w/2))+2)) (G0 (w+(w-2*(w/2))+2+(h-2*(h/2))*w)) 0 0 0 0 (Ah (c*(h/2)*(w/2)) 0)=ag (c*(h/2)*(w/2)) (aT h w I d):=by
 rcases Nat.eq_zero_or_pos c with rfl|c0
 ·rw[pQ,dif_neg (by rw[Array.usize,O3,ϳ,ϳ];exact USize.lt_irrefl 0)];simp[ag]
 obtain⟨y1,y2,y3,y4,y5⟩:=aQe_h6 c h w H W c0 B
 refine pQl (gTm w (G0 w) I d) (aT h w I d) (λa b x=>(a*h+b*2)*w+x*2) c (h/2) (w/2) (w+(w-2*(w/2))+2) (w+(w-2*(w/2))+2+(h-2*(h/2))*w) _ _ _ _ y1 y2 y3 y4 W y5
  (λa b x=>by rw[nO x 2,pA]) (aQe_h4 h w W) (aQe_h5 h w H W) (λa b x ha hb hx=>aQe_h0 _ _ B (pqE a b x c h w ha hb hx c0).le) ?_ _ 0 0 0 0 0 0 0 _ rfl (by rw[O3]) (by simp only[Ϧ,ϳ,ϑ]) (ϟ _) H W Ϧ Ϧ (by simp only[Ϧ,ϳ,ϑ]) (λj g=>absurd g (by rw[Ϧ];exact U5 j))
 intro a b x e ha hb hx he
 obtain⟨d1,m1⟩:=qVm (a*(h/2)+b) x (w/2) hx
 obtain⟨d2,m2⟩:=qVm a b (h/2) hb
 have hE:((G0 ((a*h+b*2)*w))+2*(G0 x))=(G0 ((a*h+b*2)*w+x*2)):=by rw[show(2:Ἄ)=C3 2 from ἅ,<-BP,<-mC,Ϗ 2,<-lX]
 have u:(J0 (G0 ((a*h+b*2)*w+x*2)))=(a*h+b*2)*w+x*2:=aa _ (aQe_h0 _ _ B (pqE a b x c h w ha.le hb hx c0).le)
 unfold aT;rw[Ϗ (h/2),<-lY,d1,m1,d2,m2,hE,gTe h w I d _ _ _ _ (by rw[u])]
 congr 1;apply O5.mp;rw[u,he]
def dU:Ἂ Ἃ:=
 let d:=natW (k*k)
 if 0<k/\0<h/k/\0<w/k/\k*k<16384/\DQ d then
  if k=2/\c*h*w<2^(30:Ἀ) then
   pQ (gTm w (G0 w) I d) (G0 (w/2)) (G0 (h/2)) (G0 (w+(w-2*(w/2))+2)) (G0 (w+(w-2*(w/2))+2+(h-2*(h/2))*w)) 0 0 0 0 (Ah (c*(h/2)*(w/2)) 0)
  else
   let q:=cK (k*k)
   let bx:=eG I
   if bx<=255/\I.all (ok q bx)=ἃ then eT c (h/k) (w/k) λch y x=>
    let U:=(ch*h+y*k)*w+x*k
    wo bx (fj I bx w k U) 0 (λ_=>wa (.==0x80000000) I w k (k*k) U 0 ἃ)
     (λ_=>cU h w k I ch y x) d
   else eT c (h/k) (w/k) λch y x=>
    let U:=(ch*h+y*k)*w+x*k
    let B:=wl (λ(m:ἆ) v=>min m (aq v)) I w k (k*k) U 0 255
    if B<=255/\wa (ok q B) I w k (k*k) U 0 ἃ=ἃ then
     wo B (cr (t1 B) I w k (k*k) U 0 0) 0 (λ_=>wa (.==0x80000000) I w k (k*k) U 0 ἃ)
      (λ_=>cU h w k I ch y x) d
    else toWord (Ar (ofW (cU h w k I ch y x)) (ofW d))
 else avgpoolSpec c h w k I
lemma eV_h1{α:Type}{C:Prop}[Decidable C]{X Y R:α}(h1:C->X=R)(h2:¬C->Y=R):ite C X Y=R:=if c:C then (if_pos c).trans (h1 c) else (if_neg c).trans (h2 c)
lemma eV_h2(X w m H:Ἀ)(A:X*w+w<=H)(G:H<1073741824)(t:m<w/2):X*w+m*2+w+1<2147483648:=by omega
lemma eV:dU=n9:=by
 funext c h w k I;unfold dU;dsimp only;refine eV_h1 (λd=>?_) (λ_=>(by rfl))
 .obtain⟨e,o,u,B,m⟩:=d;refine eV_h1 (λD=>?_) (λD=>?_)
  .obtain⟨rfl,G⟩:=D;rw[aQe c h w I _ o u G,Ϛ];unfold n9;refine mi (by simp only[ag,pa,Ϝ]) λq r _=>?_;simp only[ag,Λ,pa,Ϝ,aT]at r⊢;have F:=(ϸ (Ϯ o u)).mpr r;have a:=ϊ (q/(w/2)) o;have:=ϊ q u;have v:q/(h/2*(w/2))*h+q/(w/2)%(h/2)*2+1<=c*h:=by have:=ι h (l7 F);rw[Ο]at this;have b:=(mc (by decide)).1 a;rw[nO]at b;exact AZ (J1 (X5 b) _) this
   have A:=ι w v;rw[nO]at A;have Q:=eV_h2 _ _ _ _ A G this
   refine gU h w I _ m _ _ _ _ ?_ Q;rw[show(2:Ἄ)=C3 2 from rfl,<-BP,<-mC,Ϗ 2,<-lX,aa _ (by clear*-Q;omega)]
  .have l:=jL _ B;refine eV_h1 (λd=>?_) (λd=>?_)
   .refine dO _ _ _ _ _ _ λj y x=>?_;have g:∀t<k*k,ok (cK (k*k)) (eG I) (nI h w k I j y x t)=true:=λt _=>gi I _ d.2 (gC _ _) _;exact eg _ _ _ (cK (k*k)) _ _ (cB d.1) (by (clear * - l;omega)) (Ϯ e e) g _ (λ_=>0) _ 0 (λt p=>ee _ _ _ (cB d.1) l.1 (g t p)) l.2 (by simp) (by rw[gx];exact em _ h w k I j y x e) (by simp[bF]) _ (by simp only[dk _ _ _ _ _ _ e,cu,AQ]) _ rfl _ m
   .refine dO _ _ _ _ _ _ λj y x=>?_;refine eV_h1 (λd=>?_) (λ_=>(by rfl))
    have g:=(dk _ _ _ _ _ _ e).1 d.2;simp only[cu]at g;exact eg _ _ _ (cK (k*k)) _ _ (cB d.1) (by (clear * - l;omega)) (Ϯ e e) g _ (λ_=>0) _ 0 (λt p=>ee _ _ _ (cB d.1) l.1 (g t p)) l.2 (by simp) (em _ h w k I j y x e) (by simp[bF]) _ (by simp only[dk _ _ _ _ _ _ e,cu,AQ]) _ rfl _ m
@[inline]def ky(u:Ἃ):ἇ:=if u<0x80000000 then u+0x80000000 else 0xFFFFFFFF-u
@[inline]def bu(a b:Ἃ):Ἃ:=if ky a<ky b then b else a
@[inline]def nn(u:Ἃ):Ἆ:=expOf u !=255
lemma ci:(mJ (ky u))=if (mJ u)<2^(31:Ἀ) then (mJ u)+2^(31:Ἀ) else 2^(32:Ἀ)-1-(mJ u):=by have:=u.toNat_lt;rw[ky];simp only[Ι,kN,nr];split_ifs<;>simp only[mh,pU,kN,nr];exact γ (by omega);omega
lemma il(a b:kF):(mJ (ky (bu a b)))=max (mJ (ky a)) (mJ (ky b)):=by rw[bu];split<;>rename_i h<;>rw[Ι]at h<;>omega
lemma io(a b:kF)(h:(mJ (ky a))=(mJ (ky b))):a=b:=by rw[ci,ci]at h;have:=a.toNat_lt;have:=b.toNat_lt;rw[<-Β];split_ifs at h<;>omega
lemma fY_h0(b:ℕ)(h:2147483648<b):4294967295-b<2147483647:=C0 (Nat.sub_le_sub_left h 4294967295) (by decide)
lemma fY_h1(u:Ἃ):(mJ u)<2147483648∧(mJ (sk u))=(mJ u)+2147483648∧(mJ (ky u))=(mJ u)+2147483648∨(mJ u)=2147483648∧(mJ (sk u))=2147483648∧(mJ (ky u))=2147483647∨2147483648<(mJ u)∧(mJ (sk u))=4294967295-(mJ u)∧(mJ (ky u))=4294967295-(mJ u)∧(mJ u)<4294967296:=by
 rw[bf,ci];rcases Nat.lt_trichotomy (mJ u) 2147483648 with h|h|h
 exact pZ ⟨h,if_pos h,if_pos h⟩
 exact BA (pZ ⟨h,(if_neg (W3.2 (B3 h.symm))).trans (if_pos h),(if_neg (W3.2 (B3 h.symm))).trans (congrArg (2^(32:Ἀ)-1-·) h)⟩)
 exact BA (BA ⟨h,(if_neg (W3.2 h.le)).trans (if_neg h.ne'),if_neg (W3.2 h.le),u.toNat_lt⟩)
lemma fY_h2(a b:Ἃ)(h:(mJ (sk a))<(mJ (sk b))):(mJ (ky a))<(mJ (ky b)):=by
 rcases fY_h1 a with⟨a1,a2,a3⟩|⟨a1,a2,a3⟩|⟨a1,a2,a3,a4⟩<;>rcases fY_h1 b with⟨b1,b2,b3⟩|⟨b1,b2,b3⟩|⟨b1,b2,b3,b4⟩<;>rw[a2,b2]at h
 rwa[a3,b3]
 exact absurd h (W3.2 (K2 _ _))
 exact absurd (h.trans (fY_h0 _ b1)) (W3.2 ((E7 _).trans (K2 _ _)))
 rw[a3,b3];exact U1 (I3 _) (K2 _ _)
 exact absurd h (B4 _)
 exact absurd (h.trans (fY_h0 _ b1)) (W3.2 (E7 _))
 rw[a3,b3];exact (fY_h0 _ a1).trans (U1 (I3 _) (K2 _ _))
 rw[a3,b3];exact fY_h0 _ a1
 rwa[a3,b3]
lemma fY_h3(a b:Ἃ)(d:¬(mJ (sk a))<(mJ (sk b)))(g:¬(mJ (sk b))<(mJ (sk a)))(m:(mJ a)≠(mJ b)):((mJ a)=0∧(mJ b)=2147483648)∨((mJ a)=2147483648∧(mJ b)=0):=by
 have e:=C4 (W3.1 g) (W3.1 d);rcases fY_h1 a with⟨a1,a2,a3⟩|⟨a1,a2,a3⟩|⟨a1,a2,a3,a4⟩<;>rcases fY_h1 b with⟨b1,b2,b3⟩|⟨b1,b2,b3⟩|⟨b1,b2,b3,b4⟩
 exact absurd (N8 (a2.symm.trans (e.trans b2))) m
 exact pZ ⟨N8 ((a2.symm.trans (e.trans b2)).trans (Κ _).symm),b1⟩
 exact absurd (a2.symm.trans (e.trans b2)) (L6 ((fY_h0 _ b1).trans (U1 (I3 _) (K2 _ _))))
 exact BA ⟨a1,N8 ((b2.symm.trans (e.symm.trans a2)).trans (Κ _).symm)⟩
 exact absurd (a1.trans b1.symm) m
 exact absurd (a2.symm.trans (e.trans b2)) (L6 ((fY_h0 _ b1).trans (I3 _)))
 exact absurd (b2.symm.trans (e.symm.trans a2)) (L6 ((fY_h0 _ a1).trans (U1 (I3 _) (K2 _ _))))
 exact absurd (b2.symm.trans (e.symm.trans a2)) (L6 ((fY_h0 _ a1).trans (I3 _)))
 exact absurd ((Nat.sub_sub_self (Nat.le_sub_one_of_lt a4)).symm.trans ((congrArg (4294967295-·) (a2.symm.trans (e.trans b2))).trans (Nat.sub_sub_self (Nat.le_sub_one_of_lt b4)))) m
lemma fY(o:kB y≠255)(p:kB z≠255):mx y z=bu y z:=by
 rw[Conv.mx,bu];by_cases h:sk y<sk z\/sk z<sk y\/y=z;rw[AD ⟨o,p,h⟩];rcases h with h|h|rfl;rw[if_pos h,if_pos (Ι.2 (fY_h2 _ _ (Ι.1 h)))];rw[if_neg (nP h),if_neg (nP (Ι.2 (fY_h2 _ _ (Ι.1 h))))];exact(ite_self _).trans (ite_self _).symm
 rw[AC λj=>h j.2.2];simp only[not_or,Ι]at h;obtain⟨d,g,l⟩:=h;obtain⟨rfl,rfl⟩|⟨rfl,rfl⟩:(y=0/\z=0x80000000)\/(y=0x80000000/\z=0):=(fY_h3 _ _ d g (λe=>l (Β.1 e))).imp (λ⟨a,b⟩=>⟨Β.1 a,Β.1 b⟩) (λ⟨a,b⟩=>⟨Β.1 a,Β.1 b⟩)<;>decide
lemma m4(g j l m:kF):(if nn g&&nn j&&nn l&&nn m then bu (bu g j) (bu l m)
  else mx (mx (mx (mx g g) j) l) m)=mx (mx (mx (mx g g) j) l) m:=by
 split
 .rename_i h;simp only[nn,Η,AM,Ak]at h;obtain⟨⟨⟨d,o⟩,p⟩,q⟩:=h;have e:∀a b,kB a≠255->kB b≠255->kB (bu a b)≠255:=λa b r u=>by rw[bu];split<;>assumption
  rw[fY _ _ d d,fY _ _ (e _ _ d d) o,fY _ _ (e _ _ (e _ _ d d) o) p,fY _ _ (e _ _ (e _ _ (e _ _ d d) o) p) q];apply io;simp only[il,max_self,max_assoc]
 .rfl
noncomputable def iy(ch y x:Ἀ)(e:Ἄ):Ἃ:=
 let v:=(G0 w)
 if hb:e<I.usize/\e+1<I.usize/\e+v<I.usize/\e+v+1<I.usize then
  let x0:=I.uget e (uL hb.1)
  let x1:=I.uget (e+1) (uL hb.2.1)
  let x2:=I.uget (e+v) (uL hb.2.2.1)
  let x3:=I.uget (e+v+1) (uL hb.2.2.2)
  if nn x0&&nn x1&&nn x2&&nn x3 then bu (bu x0 x1) (bu x2 x3)else mx (mx (mx (mx x0 x0) x1) x2) x3
 else
  let U:=(ch*h+y*2)*w+x*2
  wl mx I w 2 4 U 0 I[U]!
lemma iz_h1(I:Ἂ kF)(A w x:Ἀ):kW ((lj (2*2)).foldl (λj t=>nt j (lI I[A*w+x*2+t/2*w+t%2]!)) (lI I[A*w+x*2]!))=kW ((lj (2*2)).foldl (λj t=>nt j (lI I[(A+t/2)*w+x*2+t%2]!)) (lI I[(A+0/2)*w+x*2+0%2]!)):=by
 rw[Nat.zero_div,Nat.zero_mod,ϑ,ϑ];exact congrArg kW (congrArg (List.foldl · _ _) (funext λj=>funext λt=>by rw[Ϡ A (t/2) w,lO (A*w) (t/2*w) (x*2)]))
lemma iz_h2(A w N P Q:Ἀ)(c:P<Q)(t:Q<=N)(v:A+w+1<P):w<Q∧A+1<N∧A+w<N∧A+w+1<N:=
 have a:=K5 v c;have k3:=U1 a t
 ⟨C0 (U2 (K2 w A) (E7 _)) a,C0 (L1 (V1 A w) 1) k3,K5 (I3 _) k3,k3⟩
lemma iz(d y x:Ἀ)(e:Ἄ)(u:(J0 e)=(d*h+y*2)*w+x*2)
  (v:(d*h+y*2)*w+x*2+w+1<2^(31:Ἀ)):
  iy h w I d y x e=kW ((lj (2*2)).foldl (λj t=>nt j (lI (nI h w 2 I d y x t)))
   (lI (nI h w 2 I d y x 0))):=by
 have q:=cu h w 2 I d y x;have m:=q 0;have o:=q 1;have p:=q 2;have r:=q 3;simp only[Nat.reduceDiv,kH,nv,n4,Av]at m o p r;have:=dj;have g:∀i l,I.uget i l=I[(J0 i)]!:=λi l=>(AB I _ l).symm;dsimp only[iy];split;rw[m4];obtain⟨k0,k1,k2,k3⟩:=iz_h2 _ _ _ _ _ (by decide) this v;simp only[g,lJ,mA,aa w k0,u,γ k1,γ k2,γ k3];rw[<-cn (Conv.mx _ _),be,be,be,be,m,o,p,r];simp only[mD,List.range_zero,l6,Ϸ,mT];rw[<-cn (wl _ _ _ _ _ _ _ _),fn I w 2 _ (2*2) (by decide)];simp only[nI];exact iz_h1 I _ w x
@[inline]def iYm(w:Ἀ)(ww:Ἄ)(I:@&Ἂ Ἃ)(e:Ἄ):Ἃ:=
 if hb:e<I.usize/\e+1<I.usize/\e+ww<I.usize/\e+ww+1<I.usize then
  let x0:=I.uget e (uL hb.1)
  let x1:=I.uget (e+1) (uL hb.2.1)
  let x2:=I.uget (e+ww) (uL hb.2.2.1)
  let x3:=I.uget (e+ww+1) (uL hb.2.2.2)
  if nn x0&&nn x1&&nn x2&&nn x3 then bu (bu x0 x1) (bu x2 x3)else mx (mx (mx (mx x0 x0) x1) x2) x3
 else wl mx I w 2 4 (J0 e) 0 I[(J0 e)]!
lemma iYe(h w:Ἀ)(I:Ἂ Ἃ)(ch y x:Ἀ)(e:Ἄ)(u:(ch*h+y*2)*w+x*2=(J0 e)):iy h w I ch y x e=iYm w (G0 w) I e:=by
 rw[iy,iYm];simp only[u]
noncomputable def mT(h w:Ἀ)(I:Ἂ Ἃ)(q:Ἀ):Ἃ:=iy h w I (q/(h/2*(w/2))) (q/(w/2)%(h/2)) (q%(w/2)) ((G0 ((q/(h/2*(w/2))*h+q/(w/2)%(h/2)*2)*w))+2*(G0 (q%(w/2))))
lemma mQe_h1(A B w q r:Ἀ)(hw:w=2*(q+1)+r):(A+B*2)*w+q*2+(w+r+2)=(A+(B+1)*2)*w+0*2:=by subst hw;ring
lemma mQe_h2(a h w p q s r:Ἀ)(hw:w=2*(q+1)+r)(hh:h=2*(p+1)+s):(a*h+p*2)*w+q*2+(w+r+2+s*w)=((a+1)*h+0*2)*w+0*2:=by subst hw hh;ring
lemma mQe_h3(w r s:Ἀ)(hw:w<2^(30:Ἀ))(hr:r<=1)(hs:s<=w):w+r+2<2^(32:Ἀ)∧w+r+2+s<2^(32:Ἀ):=by omega
lemma mQe_h4(w:Ἀ)(W:0<w/2):w=2*(w/2-1+1)+(w-2*(w/2))∧w-2*(w/2)<=1:=⟨by rw[L3 W];exact (M4 (Nat.mul_div_le w 2)).symm,(Nat.mod_def w 2).symm.trans_le (J2 (ϊ w two_pos))⟩
lemma mQe(c h w:Ἀ)(I:Ἂ Ἃ)(H:0<h/2)(W:0<w/2)(B:c*h*w<2^(30:Ἀ)):
  pQ (iYm w (G0 w) I) (G0 (w/2)) (G0 (h/2)) (G0 (w+(w-2*(w/2))+2)) (G0 (w+(w-2*(w/2))+2+(h-2*(h/2))*w)) 0 0 0 0 (Ah (c*(h/2)*(w/2)) 0)=ag (c*(h/2)*(w/2)) (mT h w I):=by
 rcases Nat.eq_zero_or_pos c with rfl|c0
 ·rw[pQ,dif_neg (by simp;rfl)];simp[ag]
 have wl:w<=c*h*w:=lZ w (Ϯ c0 (H.trans_le (L5 _ _)))
 have hl:h<=c*h*w:=(lZ h c0).trans (J4 _ (W.trans_le (L5 _ _)))
 obtain⟨hw,rw⟩:=mQe_h4 w W
 obtain⟨hh,rh⟩:=mQe_h4 h H
 have nl:c*(h/2)*(w/2)<=c*h*w:=ϧ (U3 _ (L5 _ _)) (L5 _ _)
 have qL:=wl.trans_lt B
 obtain⟨x3,x4⟩:=mQe_h3 w _ _ qL rw ((ι w rh).trans_eq (n4 w))
 have k2:2*(c*h*w)<2^(32:Ἀ):=(M7 B two_pos).trans_le (by decide)
 refine pQl (iYm w (G0 w) I) (mT h w I) (λa b x=>(a*h+b*2)*w+x*2) c (h/2) (w/2) (w+(w-2*(w/2))+2) (w+(w-2*(w/2))+2+(h-2*(h/2))*w) _ _ _ _ (aa _ ((L5 _ _).trans_lt (qL.trans (by decide)))) (aa _ ((L5 _ _).trans_lt ((hl.trans_lt B).trans (by decide)))) (aa _ x3) (aa _ x4) W (nl.trans_lt (B.trans (by decide)))
  (λa b x=>(pA _ _ _).trans (congrArg _ (nO x 2).symm)) (λa b=>mQe_h1 (a*h) b w _ _ hw)
  (λa=>mQe_h2 a h w _ _ _ _ hw hh)
  (λa b x ha hb hx=>(pqE a b x c h w ha hb hx c0).trans k2) ?_ _ 0 0 0 0 0 0 0 _ rfl (by simp) (by simp) (ϟ _) H W Ϧ Ϧ (by simp) (λj q=>(U5 j (q.trans_eq Ϧ)).elim)
 intro a b x e ha hb hx he
 obtain⟨d1,m1⟩:=qVm (a*(h/2)+b) x (w/2) hx
 obtain⟨d2,m2⟩:=qVm a b (h/2) hb
 have hE:((G0 ((a*h+b*2)*w))+2*(G0 x))=(G0 ((a*h+b*2)*w+x*2)):=by rw[show(2:Ἄ)=C3 2 from ἅ,<-BP,<-mC,Ϗ 2,<-lX]
 have u:(J0 (G0 ((a*h+b*2)*w+x*2)))=(a*h+b*2)*w+x*2:=aa _ ((pqE a b x c h w ha.le hb hx c0).trans k2)
 unfold mT;rw[Ϗ (h/2),<-lY,d1,m1,d2,m2,hE,iYe h w I _ _ _ _ (by rw[u])]
 congr 1;apply O5.mp;rw[u,he]
def d9:Ἂ Ἃ:=
 if k=2/\0<h/2/\0<w/2/\c*h*w<2^(30:Ἀ) then
  pQ (iYm w (G0 w) I) (G0 (w/2)) (G0 (h/2)) (G0 (w+(w-2*(w/2))+2)) (G0 (w+(w-2*(w/2))+2+(h-2*(h/2))*w)) 0 0 0 0 (Ah (c*(h/2)*(w/2)) 0)
 else maxpoolSpec c h w k I
lemma fo:d9=maxpoolSpec:=by
 funext c h w k I;unfold d9;split;obtain⟨rfl,d,g,p⟩:=‹_›;rw[mQe c h w I d g p,Ϛ];unfold maxpoolSpec;refine mi (by simp only [Ϝ,Nat.reduceMul]) λq e _=>?_;simp only[ag,Λ,Ϝ,mT]at e⊢;have o:=(ϸ (Ϯ d g)).mpr e;have:=ϊ (q/(w/2)) d;have:=ϊ q g;have j:q/(h/2*(w/2))*h+q/(w/2)%(h/2)*2+1<=c*h:=by have:=ι h (l7 o);rw[Ο]at this;omega
 have l:=ι w j;rw[nO]at l;refine iz h w I _ _ _ _ ?_ (by omega);rw[show(2:Ἄ)=C3 2 from ἅ,<-BP,<-mC,Ϗ 2,<-lX,aa _ (by omega)];rfl
end Conv
section
open Float.Model.UnpackedFloat
open ExactRuntime renaming rationalWord->ZA,sqrtLower->ZK,mantOf_lt_24->ZO,roundRational->ZT,mantOf->ZW,signOf->ZX,idxOf->ZZ,rowVariance->XH
open Submission.Core renaming mW->ZC,iW->ZD,an->ZI,ai->ZR,am->ZS
open _root_ renaming zpow_natCast->ZF,decide_eq_true_eq->ZH,zpow_add₀->ZL,two_ne_zero->ZQ,beq_iff_eq->ZY,ite_false->XA,zpow_sub₀->XB,two_ne_zero'->XC,mul_assoc->XD
open UInt64 renaming toNat_ofNat'->ZG,ofNat->XF
open Nat renaming div_eq_zero_iff_lt->ZJ,div_add_mod'->ZU,div_le_self->ZV
open Finset renaming range->ZM
open Float.Model.UnpackedFloat.ExtendedMantissa renaming ofMantissaAndAccuracy->XE
open Bool renaming xor->XG
namespace Elem
lemma VV(w:kF):(lq (ZC w))<2^(24:Ἀ):=by rw[ZI];exact ZO w
lemma YB(a b:kF):(lq (ZC a*ZC b))=ZW a*ZW b:=by rw[ύ,ZI,ZI];exact γ (mul_mant_lt a b)
end Elem open Elem
namespace Norm
def Ya:Accuracy->Ἀ
 | .exact=>0
 | .inexact _=>1
lemma Yc{α:Type}(f:α->α)(n:Ἀ)(a:α):
  Nat.repeat f (n+1) a=Nat.repeat f n (f a):=by
 induction n generalizing a with
 |zero=>rfl
 |succ n d=>exact congrArg f (d a)
theorem QA_h1(R y:Ἀ)(h:y<2):(2*R+y)/2=R∧(2*R+y)%2=y:=⟨by rw[W8 ((by decide):2>0),ώ h,ϑ],by rw[O7,γ h]⟩
theorem QA_h2(R:Ἀ)(d:Accuracy):lm (lm (XE (2*R+Ya d) .exact))=lm (XE R d):=by
 obtain⟨a,b⟩:=QA_h1 R (Ya d) (by rcases d with _|⟨_|_|_⟩<;>decide)
 show (⟨(2*R+Ya d)/2/2,(2*R+Ya d)/2%2 != 0,(2*R+Ya d)%2 != 0||(false||false)⟩:ExtendedMantissa)=lm (XE R d)
 rw[a,b];rcases d with _|⟨_|_|_⟩<;>rfl
lemma QA(R k:Ἀ)(d:Accuracy):
  XE (2*R+Ya d) .exact>>>(k+2)=
   XE R d>>>(k+1):=by
 show Nat.repeat lm (k+2) _= Nat.repeat lm (k+1) _;rw[Yc,Yc,Yc,QA_h2]
open FloatLib.Numerics in
lemma VY(q r D t:Ἀ)(e:0<D)(d:r<D)(g:1<=t):
  la (q*D+r) (D*2^t)=
   la (2*q+(if r=0 then 0 else 1)) (2^(t+1)):=by
 obtain⟨k,rfl⟩:∃k,t=k+1:=⟨t-1,(L3 g).symm⟩;have h:=lF (2*q+Ya (lP r D)) 1 (k+1+1) one_ne_zero;rw[n4,F5,Nat.mod_one,show lP 0 1=.exact from ἅ,QA]at h;rw[<-lF _ _ _ e.ne',(by rw[lP];split<;>rfl:(if r=0 then 0 else 1)=Ya (lP r D)),<-h,(by rw[Τ,Ρ _ _ e,ώ d,Ai]:(q*D+r)/D=q),(by rw[Τ,Μ,γ d]:(q*D+r)%D=r)]
lemma Y5(e:Ἁ):Model.bpow e=(2:ℝ)^e:=(by rfl)
lemma Vi(q b:Ἀ)(d:q≠0)(h:b<=1):(Aq (2*q+b))=(Aq q)+1:=by rw[Nat.log2_def,if_pos (by omega:2<=2*q+b),W8 (by decide),ώ (AO h one_lt_two)];rfl
open FloatLib.Numerics in
lemma V0(s:Ἆ)(P d:Ἀ)(o:P≠0)(l:d≠0)(q r D:Ἀ)
  (x:0<D)(t:r<D)(e:Ἁ)(h:(P:ℝ)/d=(q+r/D)*(2:ℝ)^e)
  (z:2^(25:Ἀ)<=q)(A:-126<=((Aq q):Ἁ)+e)(B:((Aq q):Ἁ)+e<=126):
  Model.roundRat lU s P d=
   kE lU ⟨s,2*q+(if r=0 then 0 else 1),e-1⟩:=by
 have j:q≠0:=(AN (by decide) z).ne';have m:=(mu j).2 z;have u:(2:ℝ)^(Aq q)<=q:=mod_cast ϰ j;have v:(q:ℝ)+1<=2^((Aq q)+1):=mod_cast ϙ (n:=q);have g:=Vi q _ j (by split<;>decide:(if r=0 then 0 else 1)<=1);have G:(r:ℝ)/D<1:=(div_lt_one ((by positivity))).2 (mod_cast t);have E:(0:ℝ)<=r/D:=((by positivity));have w:lf P d=((Aq q):Ἁ)+e:=by
  apply Model.floorLog2_eq_of_bounds P d _
  .rw[h,Y5,ZL ZQ,ZF];exact AE (le_add_of_le_of_nonneg u E) (by positivity)
  .rw[h,Y5,(by rw[X7];exact add_right_comm _ _ _:((Aq q):Ἁ)+e+1=(((Aq q)+1:Ἀ):Ἁ)+e),ZL ZQ,ZF];exact AI ((add_lt_add_of_le_of_lt AP G).trans_le v) (by positivity)
 have p:la (l9 P d (23-(((Aq q):Ἁ)+e))).1 (l9 P d (23-(((Aq q):Ἁ)+e))).2=roundShiftRightEven (2*q+if r=0 then 0 else 1) ((Aq (2*q+if r=0 then 0 else 1))-23):=by
  rw[roundShiftRightEven_eq_roundQuotientEven,g,(R7 (U2 (by decide) m):(Aq q)+1-23=((Aq q)-23)+1),<-VY q r D _ x t (by exact G5 (AZ (by decide) m))]
  apply Model.roundQuotientEven_eq_of_rat_eq _ _ _ _ (RationalBinary.scaleByPowerOfTwo_snd_ne_zero _ _ _ l) ((by positivity));rw[Model.scaleByPowerOfTwo_real,Model.scaledRatToReal,h,Y5,XD,<-ZL ZQ,(by rw[pO (U2 (by decide) m),nZ];ring:e+(23-(((Aq q):Ἁ)+e))=-(((Aq q)-23:Ἀ):Ἁ)),zpow_neg,ZF];push_cast;rw[←div_eq_mul_inv,←div_div];congr 1;rw[add_div,mul_div_cancel_right₀ _ (Nat.cast_pos.2 x).ne']
 generalize F:2*q+(if r=0 then 0 else 1)=J at g p⊢;rw[kE,AD (by decide),Model.ieeeRoundDyadic_eq_internal];simp only[Model.roundRat,n6,((by decide):Fmt.isIEEE=ἃ),↓reduceDIte,Model.ieeeRoundRatScaled,Model.Internal.ieeeRoundDyadic,ZY,o,l,XA,w,Av,nY,g,((by decide):(FloatFormat.ieeeMaxNormalExponent lU:Ἁ)=127),((by decide):FloatFormat.ieeeMinNormalExponent lU=-126),((by decide):(FloatFormat.normalMantissaExpOffset lU:Ἁ)=150),((by decide):Fmt.fracWidth=23),show J≠0 by rw[←F];exact L6 (U1 (Ϯ (by decide) (ϵ j)) (V1 _ _)),(by rw[X7,add_add_sub_cancel]:(((Aq q)+1:Ἀ):Ἁ)+(e-1)=((Aq q):Ἁ)+e),show¬(((Aq q):Ἁ)+e>127) from not_lt.2 (AZ B (by decide)),show¬(((Aq q):Ἁ)+e< -150) from not_lt.2 (AZ (by decide) A),show¬(((Aq q):Ἁ)+e< -126) from not_lt.2 A,(Nat.le_succ_of_le (U2 (by decide) m):23<=(Aq q)+1),ἁ,nZ,p]
lemma Y8(v:ℚ)(w:v≠0)(s:Ἆ)(u:decide (v<0)=s)
  (q:Ἀ)(ρ:ℚ)(o:0<=ρ)(g:ρ<1)(e:Ἁ)
  (j:|v|=((q:ℚ)+ρ)*(2:ℚ)^e)
  (p:2^(25:Ἀ)<=q)(l:-126<=((Aq q):Ἁ)+e)(m:((Aq q):Ἁ)+e<=126):
  ZT v=
   kW (kE lU ⟨s,2*q+(if ρ=0 then 0 else 1),e-1⟩):=by
 have h:ρ=((F6 ρ.num):ℚ)/ρ.den:=by(conv_lhs=>rw[<-Rat.num_div_den ρ]);congr 1;exact_mod_cast(Int.toNat_of_nonneg (Rat.num_nonneg.2 o)).symm
 have d:=ρ.den_pos;generalize ρ.den=D,(F6 ρ.num)=r at h d;subst h;rw[roundRational,u,V0 s _ _ ((by positivity)) v.den_nz q r D d (by rwa[div_lt_one (mod_cast d),Nat.cast_lt]at g) e ?_ p l m];simp[d.ne'];have:=congrArg (λz:ℚ=>(z:ℝ)) j;rw[Rat.abs_def,Rat.divInt_eq_div]at this;push_cast at this;rwa[Nat.cast_natAbs,mI]
lemma Q7_h1(R:Ἀ)(s c:Ἁ)(h:2^(25:Ἀ)<=R):-s+1<=max ((Aq R:Ἁ)+1+-s-((FloatFormat.toModel Fmt).mantissaBits:Ἁ)) c:=le_max_of_le_left (by linarith only[(U4.2 ((by decide:(FloatFormat.toModel Fmt).mantissaBits<=25).trans ((mu (U1 (϶ 25) h).ne').2 h)):((FloatFormat.toModel Fmt).mantissaBits:Ἁ)<=Aq R)])
lemma Q7(R:Ἀ)(s:Ἁ)(h:2^(25:Ἀ)<=R):
  -s+1<=((FloatFormat.toModel Fmt)).targetExponent (kY R (-s)):=by rw[BV,kY];exact Q7_h1 R s _ h
lemma Vd(v:ℚ)(q:0<v)(e d:Ἀ)(t:0<d)(s:Ἁ)
  (p:(e:ℝ)/d=(v:ℝ)*(2:ℝ)^(2*s))
  (o:2^(25:Ἀ)<=ZK e d):
  kW (pv lU (nk ((FloatFormat.toModel Fmt)) .positive
   (ZK e d) (-s) (sqrtAccuracy e d)))=mK v:=by
 have g:=sqrtAccuracy_correct e d t;have h:ZK e d≠0:=(U1 (϶ 25) o).ne';have j:=AZ (le_add_of_nonneg_right zero_le_one:-s<=-s+1) (Q7 _ s o);have l:=Model.cexp_accuracy_mul_bpow_eq_targetExponent lU _ (-s) _ _ (by decide) h g;have m:=Model.scaledMantissa_accuracy_mul_bpow lU _ (-s) _ _ (by decide) h g;rw[(by rw[show kZ kO (-s)=2^(-s) from Y5 _,p,(by rw[<-ZF,<-zpow_mul,mul_comm];rfl:(2:ℝ)^(2*s)=(2^s)^(2:Ἀ)),Real.sqrt_mul (Rat.cast_nonneg.2 q.le),Real.sqrt_sq (zpow_pos two_pos s).le,XD,<-ZL ZQ];simp only [add_neg_cancel,zpow_ofNat,pow_zero,mul_one]:O2 (e/d)*kZ kO (-s)=O2 v)]at l m;rw[sqrtRational_eq_math,Model.roundWithAccuracy_eq_finishRoundedMantissa];simp only[sqrtMathWord,not_le.mpr q,XA,l,m,<-Model.roundedMantissa_shiftToTargetExponent_eq_nearestEven lU _ (-s) _ _ g j];rw[Model.shiftToTargetExponent_eq_of_le_targetExponent lU _ _ _ j];rfl
lemma Y1(w:kF):
  ZA w=(sv w:ℚ)*2^((ZZ w:Ἁ)-149):=by rw[rationalWord];rw[lG]
lemma VT(w:kF):|(sv w:ℚ)|=ZW w:=by rw[sv];split_ifs<;>simp
lemma YN(w:kF):(sv w:ℚ)=0↔ZW w=0:=by rw[sv];split_ifs<;>simp
lemma Vr(w:kF):(sv w:ℚ)=(if ZX w then-1 else 1)*ZW w:=by rw[sv];simp only [Int.cast_ite,Int.cast_neg,M5,ite_mul,neg_mul,one_mul]
lemma Y7(x:ἆ):(lq (ZR x))=(Aq (lq x)):=by rw[aA];rfl
@[specialize]def Yh(x:(Ἂ Word))(i stop u:Ἀ):Ἀ:=
 if i<stop then
  let w:=x[i]!
  Yh x (i+1) stop (if ZC w==0 then u else min u (lq (ZD w)))
 else u
termination_by stop-i
decreasing_by all_goals omega
lemma YK(w:kF):(ZC w==0)=decide (ZW w=0):=by
 by_cases h:ZW w=0
 .have:ZC w=0:=ξ.mp (by rw[ZI,h];rfl);grind
 .have:ZC w≠0:=λd=>h (by rw[<-ZI,d];rfl)
  grind
theorem Yi_h1(d i:ℕ)(h:d-i=0):¬i<d:=W3.2 (K3.1 h)
lemma Yi_h2(x:(Ἂ Word))(d:Ἀ):
  ∀k i u,d-i=k->Yh x i d u<=u/\
   ∀j,i<=j->j<d->ZW x[j]! =0\/Yh x i d u<=ZZ x[j]!:=by
 intro k;induction k<;>intro i u h<;>rw[Yh]
 .rw[AC (Yi_h1 _ _ h)];exact⟨Nat.le_refl _,λj a b=>absurd (C0 a b) (Yi_h1 _ _ h)⟩
 .rename_i e;rw[AD (N1 h)];dsimp only;rw[YK,ZS];split<;>rename_i hz
  .have E:=e (i+1) u (by rw[V4,h,BK]);exact⟨E.1,λj a b=>(BJ a).elim (λq=>by subst q;exact .inl (of_decide_eq_true hz)) (λa=>E.2 j a b)⟩
  .have E:=e (i+1) (min u (ZZ x[i]!)) (by rw[V4,h,BK]);exact⟨E.1.trans (min_le_left _ _),λj a b=>(BJ a).elim (λq=>by subst q;exact .inr (E.1.trans (min_le_right _ _))) (λa=>E.2 j a b)⟩
lemma Yi(x:(Ἂ Word))(d:Ἀ):
  ∀k i u,d-i=k->Yh x i d u<=u/\
   ∀j,i<=j->j<d->ZW x[j]! =0\/Yh x i d u<=ZZ x[j]!:=Yi_h2 x d
def VJ(x:(Ἂ Word))(u j:Ἀ):Ἀ:=
 (ZW x[j]!*ZW x[j]!)<<<(2*(ZZ x[j]!-u))
@[specialize]def Yl(x:(Ἂ Word))(u i stop:Ἀ)(l h:ἆ)(c big:Ἀ):Ἀ:=
 if i<stop then
  let w:=x[i]!
  let mm:=ZC w*ZC w
  let sh:=(lq (ZD w))-u
  if sh<=39 then
   let k:=(I0 (2*sh))
   let vl:=if k<64 then mm<<<k else 0
   let vh:=if k<64 then(mm>>>1)>>>(63-k)else mm<<<(k-64)
   let l':=l+vl
   let h':=h+vh+(if l'<l then 1 else 0)
   Yl x u (i+1) stop l' h' (c+(if h'<h then 1 else 0)) big
  else Yl x u (i+1) stop l h c (big+(lq mm)<<<(2*sh))
 else big+(lq l)+((lq h)+c<<<64)<<<64
termination_by stop-i
decreasing_by all_goals omega
lemma VL(d:ἆ)(k:Ἀ)(o:(lq d)<2^(48:Ἀ))(p:k<=78):
  (UInt64.toNat (if (I0 k)<64 then d<<<(I0 k)else 0))+
   (lq (if (I0 k)<64 then(d>>>1)>>>(63-(I0 k))
    else d<<<((I0 k)-64)))*2^(64:Ἀ)=(lq d)*2^k/\
  (UInt64.toNat (if (I0 k)<64 then(d>>>1)>>>(63-(I0 k))
    else d<<<((I0 k)-64)))<2^(62:Ἀ):=by
 have g:(lq (I0 k))=k:=by rw[Ϣ,ZG,γ (C0 p (by decide))]
 by_cases h:k<64
 simp only[(by rw[μ,g];simpa:(I0 k)<64),ἁ,ψ,ν,χ _ _ (show (I0 k)<=63 by rw[θ,g];simp only [kA];exact Nat.lt_succ.1 h),g,kA,ο,β,γ h,γ (AO (K4 _ _) (by decide):63-k<64),kH,lY]
 rw[(by rw[<-AF];congr 1;rw[Τ];exact (R7 (Nat.lt_succ.1 h)).symm:2^(1:Ἀ)*2^(63-k)=2^(64-k)),(by rw[<-AF,L3 h.le]:2^(64:Ἀ)=2^(64-k)*2^k),Nat.mul_mod_mul_right,<-Ϛ,<-Ϡ,Nat.mod_add_div'];exact⟨ἅ,AO (ZV _ _) ((by omega))⟩
 have m:(lq d)*2^(k-64)<2^(62:Ἀ):=calc
  _<2^(48:Ἀ)*2^(14:Ἀ):=ly o (Θ ((by positivity)) (P4 p 64)) ((by positivity))
  _=_:=ἅ
 simp only[(by rw[μ,g];simpa using h:¬(I0 k)<64),XA,ϋ,Κ,ψ,χ _ _ (show(64:ἆ)<=(I0 k) by rw[θ,g];simp only [kA];exact not_lt.1 h),g,kA,γ (C0 (P4 p 64) (by decide):k-64<64),ο,γ (m.trans (by decide):(lq d)*2^(k-64)<2^(64:Ἀ))]
 refine⟨?_,m⟩;rw[Ϛ,<-AF];rw[L3 (W4 h)]
lemma Vb_h2(x y:ἆ):(lq (x+y))+(if x+y<x then 2^(64:Ἀ) else 0)=(lq x)+(lq y):=by
 have hx:=x.toNat_lt;have hy:=y.toNat_lt;rw[η];by_cases hc:(lq x)+(lq y)<2^(64:Ἀ)
 ·rw[γ hc,if_neg (by rw[μ,η,γ hc];exact W3.2 (V1 _ _)),ϑ]
 ·have hc:=W4 hc;have e:((lq x)+(lq y))%2^(64:Ἀ)=(lq x)+(lq y)-2^(64:Ἀ):=by rw[P8 hc,γ (G4 hc (E6 hx hy))]
  rw[e,if_pos (by rw[μ,η,e];exact G4 hc ((O1 hy _).trans_eq (Τ _ _))),L3 hc]
lemma Vb_h3(h w c:ἆ)(hw:(lq w)<2^(62:Ἀ))(hc:(lq c)<=1):(lq (h+w+c))+(if h+w+c<h then 2^(64:Ἀ) else 0)=(lq h)+(lq w)+(lq c):=by
 have:=h.toNat_lt;have e:(lq (h+w+c))=((lq h)+(lq w)+(lq c))%2^(64:Ἀ):=by rw[η,η,Nat.mod_add_mod]
 by_cases T:(lq h)+(lq w)+(lq c)<2^(64:Ἀ)
 ·rw[e,γ T,if_neg (by rw[μ,e,γ T];exact W3.2 ((V1 _ _).trans (V1 _ _))),ϑ]
 ·have T:=W4 T;have wc:(lq w)+(lq c)<2^(64:Ἀ):=(J1 hc _).trans_lt ((X3 hw 1).trans (by decide));have e2:((lq h)+(lq w)+(lq c))%2^(64:Ἀ)=(lq h)+(lq w)+(lq c)-2^(64:Ἀ):=by rw[P8 T,γ (G4 T ((pA _ _ _).trans_lt (E6 this wc)))]
  rw[e,e2,if_pos (by rw[μ,e,e2];exact G4 T ((pA _ _ _).trans_lt ((O1 wc _).trans_eq (Τ _ _)))),L3 T]
lemma Vb_h4(P:Prop)[Decidable P]:(if P then 1 else 0)*2^(64:Ἀ)=if P then 2^(64:Ἀ) else 0:=by split<;>decide
lemma Vb_h1(l v h w:ἆ)(hw:(lq w)<2^(62:Ἀ)):(lq (l+v))+(lq (h+w+(if l+v<l then 1 else 0)))*2^(64:Ἀ)+(if h+w+(if l+v<l then 1 else 0)<h then 1 else 0)*2^(128:Ἀ)=(lq l)+(lq v)+((lq h)+(lq w))*2^(64:Ἀ):=by
 have E1:=Vb_h2 l v;have E2:=Vb_h3 h w (if l+v<l then 1 else 0) hw (by split<;>decide)
 have X1:(lq (if l+v<l then (1:ἆ) else 0))*2^(64:Ἀ)=if l+v<l then 2^(64:Ἀ) else 0:=by split<;>decide
 linear_combination E1+2^(64:Ἀ)*E2+X1+2^(64:Ἀ)*Vb_h4 (h+w+(if l+v<l then 1 else 0)<h)
theorem Vb_h5(b L H C c S l v h w M X Y:Ἀ)(E1:L+H*X+C*Y=l+v+(h+w)*X)(E2:v+w*X=M):b+L+H*X+(c+C)*Y+S=b+l+h*X+c*Y+(S+M):=by
 rw[Ϡ c C,<-pA (b+L+H*X) (c*Y),lO (b+L+H*X) (c*Y),pA b L,pA b (L+H*X),E1,Ϡ h w,Nat.add_add_add_comm l v,<-pA b (l+h*X),<-pA b l,lO (b+l+h*X) (v+w*X),lO (b+l+h*X+c*Y) (v+w*X),pA (b+l+h*X+c*Y) S,E2]
theorem Vb_h6(b M l H C S:Ἀ):b+M+l+H+C+S=b+l+H+C+(S+M):=by
 rw[lO b M l,lO (b+l) M H,lO (b+l+H) M C,pA (b+l+H+C) M S,Τ M S]
lemma Vb(x:(Ἂ Word))(u d:Ἀ):
  ∀k i (l h:ἆ) c big,d-i=k->
   Yl x u i d l h c big=big+(lq l)+(lq h)*2^(64:Ἀ)+c*2^(128:Ἀ)+
    ∑j∈ZM k,VJ x u (i+j):=by
 intro k;induction k<;>intro i l h c big q<;>rw[Yl]
 .rw[AC (W3.2 (K3.1 q))];simp only[ο,Finset.range_zero,Finset.sum_empty];rw[Av,Ϡ,Ϛ,<-m7];exact (pA _ _ _).symm
 rename_i p;rw[AD (N1 q),lV];simp only[(λj=>by rw[lO];rfl:∀j,VJ x u (i+(j+1))=VJ x u (i+1+j)),ϑ];have o:VJ x u i=(lq (ZC x[i]!*ZC x[i]!))*2^(2*((lq (ZD x[i]!))-u)):=by rw[VJ,ο,YB,ZS]
 have r:(lq (ZC x[i]!*ZC x[i]!))<2^(48:Ἀ):=by rw[YB];have:=ZO x[i]!;exact(ϕ this this).trans_eq ἅ
 rw[o];generalize (lq (ZD x[i]!))-u=g at*;generalize ZC x[i]!*ZC x[i]! =m at*;by_cases t:g<=39
 .have:=VL m (2*g) r (U3 2 t);generalize (if(I0 (2*g))<64 then m<<<(I0 (2*g))else 0)=v at*;generalize (if(I0 (2*g))<64 then(m>>>1)>>>(63-(I0 (2*g))) else m<<<((I0 (2*g))-64))=w at*;rw[p (i+1) _ _ _ _ (by rw[V4,q,BK])];rw[AD t];exact Vb_h5 _ _ _ _ _ _ _ _ _ _ _ _ _ (Vb_h1 l v h w this.2) this.1
 rw[AC t,p (i+1) _ _ _ _ (by rw[V4,q,BK]),ο];exact Vb_h6 _ _ _ _ _ _
lemma Yk(n:Ἀ)(f:Ἀ->ℚ)(a:ℚ):
  (lj n).foldl (λt j=>t+f j) a=a+∑j∈ZM n,f j:=by
 induction n generalizing a with
 |zero=>simp only [List.range_zero,mT,Finset.range_zero,Finset.sum_empty,Av]
 |succ n d=>
  rw[mD,l6,d,Ν];simp only [Ϸ,mT];ring
lemma VG_h1(d u:Ἀ)(h:u<=d):((d:Ἁ)-149)*((2:Ἀ):Ἁ)=((2*(d-u):Ἀ):Ἁ)+(2*(u:Ἁ)-298):=by rw[Y4,pO h,nZ,mul_sub,sub_add_sub_cancel,sub_mul,mul_comm];rfl
lemma VG(d u:Ἀ)(h:u<=d):
  ((2:ℚ)^((d:Ἁ)-149))^(2:Ἀ)=(2:ℚ)^(2*(d-u))*(2:ℚ)^(2*(u:Ἁ)-298):=by rw[<-ZF ((2:ℚ)^((d:Ἁ)-149)) 2,<-zpow_mul,<-ZF (2:ℚ) (2*(d-u)),<-ZL ZQ];exact congrArg _ (VG_h1 d u h)
lemma Ye_h1(a k:Ἀ)(y:ℚ):(a:ℚ)*a*((2:ℚ)^k*y)=((a*a*2^k:Ἀ):ℚ)*y:=by rw[Y4,Y4,I4,nZ,<-Ap]
lemma Ye(w:kF)(u:Ἀ)(h:ZW w=0\/u<=ZZ w):
  ZA w^(2:Ἀ)=
   (((ZW w*ZW w)<<<(2*(ZZ w-u)):Ἀ):ℚ)*2^(2*(u:Ἁ)-298):=by rw[Y1,ο];rcases h with h|h;rw[(YN w).2 h,h];simp only [zero_mul,Ak,OfNat.ofNat_ne_zero,not_false_eq_true,zero_pow,mul_zero,CharP.cast_eq_zero];rw[mul_pow,(by rw[<-sq_abs,VT,sq]:(sv w:ℚ)^(2:Ἀ)=(ZW w:ℚ)*ZW w),VG _ _ h];exact Ye_h1 _ _ _
def Yf(num den:Ἀ)(s:Ἁ):Ἃ:=
 toWord (Model.ofModel Fmt (roundWithAccuracy ((FloatFormat.toModel Fmt)) .positive
  (ZK num den) (-s) (sqrtAccuracy num den)))
open Float.Model in
lemma Q4_h1(e T:Ἁ)(p:e+1<=T):(F6 (e-1-T))=0:=Int.toNat_of_nonpos (Int.sub_nonpos_of_le ((Int.sub_le_self _ (by decide)).trans ((Int.le_add_one (le_refl _)).trans p)))
lemma Q4_h2(e T:Ἁ)(p:e+1<=T):∃k,(F6 (T-e))=k+1:=⟨(F6 (T-e))-1,(L3 ((Int.lt_toNat (m:=0)).2 (Int.sub_pos.2 (Int.lt_of_add_one_le p)))).symm⟩
lemma Q4_h3(e T:Ἁ)(k:Ἀ)(p:e+1<=T)(j:(F6 (T-e))=k+1):(F6 (T-(e-1)))=k+2:=by
 rw[sub_sub_eq_add_sub,add_sub_right_comm,Int.toNat_add (Int.sub_nonneg.2 (le_of_lt (Int.lt_of_add_one_le p))) (by decide),j]<;>rfl
open Float.Model in
lemma Q4(spec:Format)(g:pK)(R:Ἀ)(e:Ἁ)(d:Accuracy)(l:R≠0)
  (p:e+1<=spec.targetExponent (kY R e)):
  nk spec g R e d=pp spec g (2*R+Ya d) (e-1):=by
 have h:kY (2*R+Ya d) (e-1)=kY R e:=by rw[kY,kY,Vi R _ l (by cases d<;>simp[Ya])];grind
 simp only[pp,decreaseExponent,h];rw[Q4_h1 _ _ p];simp only [BH,CharP.cast_eq_zero,sub_zero];unfold nk shiftToTargetExponent shiftToExponent;rw[h];generalize spec.targetExponent (kY R e)=T at*;obtain⟨k,j⟩:=Q4_h2 _ _ p;dsimp only;rw[j,Q4_h3 _ _ _ p j,QA,(by push_cast;ring:e-1+((k+2:Ἀ):Ἁ)=e+((k+1:Ἀ):Ἁ))]
lemma Vh(e d:Ἀ):
  Ya (sqrtAccuracy e d)=if e=ZK e d^(2:Ἀ)*d then 0 else 1:=by unfold sqrtAccuracy;dsimp only;split_ifs<;>rfl
lemma V9_h1(s:Ἁ)(m:-209<=s):(F6 (297-s))<2^(64:Ἀ):=by
 have h:(F6 (297-s))<=506:=Int.toNat_le.2 ((Int.sub_le_sub_left m 297).trans_eq (by decide))
 exact C0 h (by decide)
lemma V9_h2(s:Ἁ)(o:s<=297):(((F6 (297-s)):Ἀ):Ἁ)-298=-s-1:=by rw[Int.toNat_of_nonneg (Int.sub_nonneg.2 o)];ring
lemma V9_h3(R c:Ἀ)(l:R>>>31=0)(q:c<=1):2*R+c<2^(64:Ἀ):=by
 have hR:R<2^(31:Ἀ):=(Nat.div_eq_zero_iff.1 ((β R 31).symm.trans l)).resolve_left (BD.1 (϶ 31))
 exact(J1 q _).trans_lt ((X3 (M7 hR (by decide)) 1).trans (by decide))
lemma V9(e d:Ἀ)(s:Ἁ)(p:33554432<=ZK e d)
  (l:ZK e d>>>31=0)(m:-209<=s)(o:s<=297):
  Yf e d s=az false (XF (2*ZK e d+
   (if e=ZK e d*ZK e d*d then 0 else 1)))
   (XF (F6 (297-s))):=by
 set R:=ZK e d with g;have h:(lq (XF (F6 (297-s))))=(F6 (297-s)):=by rw[ZG,γ (V9_h1 s m)]
 have q:(if e=R*R*d then 0 else 1)<=1:=by split_ifs<;>decide
 have j:(lq (XF (2*R+(if e=R*R*d then 0 else 1))))=2*R+(if e=R*R*d then 0 else 1):=by rw[ZG,γ (V9_h3 _ _ l q)]
 rw[aJ _ _ _ (by rw[h];exact Int.toNat_le.2 (Int.sub_le_sub_left m 297)),h,j,Yf,<-g,Q4 _ _ R (-s) _ (L6 (AN (by decide) p)) (Q7 R s p),Vh,<-g,sq,V9_h2 s o];rfl
@[inline]def YG(num den R:Ἀ)(s:Ἁ):Ἃ:=
 if R>>>31==0&&-209<=s&&s<=297 then
  Core.az ἄ (XF (2*R+(if num==R*R*den then 0 else 1)))
   (XF (F6 (297-s)))
 else Yf num den s
lemma Vc(e d:Ἀ)(s:Ἁ)(g:33554432<=ZK e d):
  YG e d (ZK e d) s=Yf e d s:=by
 rw[YG];split
 next h=>
  simp at h;rw[V9 e d s g h.1.1 h.1.2 h.2];simp
 simp
def V8(B:Ἀ)(c:Ἁ):Ἀ:=if 0<=c then B<<<(F6 c)else B
def YF(Dn:Ἀ)(c:Ἁ):Ἀ:=if 0<=c then Dn else Dn<<<(F6 (-c))
lemma VE(B d:Ἀ)(e:0<d)(c:Ἁ):
  0<YF d c/\((V8 B c:ℚ)/YF d c=(B:ℚ)/d*2^c):=by
 rw[V8,YF];split_ifs with h<;>rw[ο,Y4,I4,nZ,<-AA];exact⟨e,by rw[Int.toNat_of_nonneg h];exact mul_div_right_comm _ _ _⟩
 rw[Int.toNat_of_nonneg (Int.neg_nonneg_of_nonpos (Int.le_of_not_le h)),zpow_neg,div_mul_eq_div_div,div_inv_eq_mul];exact⟨Ϯ e (϶ _),rfl⟩
def V5(A Dn u:Ἀ)(eps:Ἃ):Ἃ:=
 let E:=ZW eps
 let ie:=ZZ eps
 let w0:=min (2*u) (ie+149)
 let B:=(A<<<(2*u-w0))+((Dn*E)<<<(ie+149-w0))
 let s:Ἁ:=(350-(w0:Ἁ)-(((Aq B):Ἁ)-((Aq Dn):Ἁ)))/2
 let c:Ἁ:=(w0:Ἁ)-298+2*s
 let num:=V8 B c
 let den:=YF Dn c
 let R:=ZK num den
 if 33554432<=R then YG num den R s
 else sqrtRational ((A:ℚ)*2^(2*(u:Ἁ)-298)/Dn+ZA eps)
lemma YW(A e u:Ἀ)(d:kF)(t:0<e)(o:0<ZA d):
  V5 A e u d=
   mK ((A:ℚ)*2^(2*(u:Ἁ)-298)/e+ZA d):=by
 dsimp only[V5];split_ifs with x;swap;rfl;rw[Vc _ _ _ x];have z:=zpow_pos (two_pos:(0:ℚ)<2);have p:(0:ℚ)<A*2^(2*(u:Ἁ)-298)/e+ZA d:=add_pos_of_nonneg_of_pos (div_nonneg (Af A.cast_nonneg (z _).le) e.cast_nonneg) o;have hE:(sv d:ℚ)=ZW d:=by rw[<-VT,AW (pos_of_mul_pos_left (o.trans_eq (Y1 d)) (z _).le)]
 rw[Y1,hE]at p⊢;generalize ZW d=E at*;generalize ZZ d=l at*;generalize y:po (2*u) (l+149)=g at*;generalize v:A<<<(2*u-g)+(e*E)<<<(l+149-g)=B at*;generalize (350-(g:Ἁ)-((B.log2:Ἁ)-(e.log2:Ἁ)))/2=s at*;obtain⟨q,w⟩:=VE B e t ((g:Ἁ)-298+2*s);refine Vd _ p _ _ q s ?_ ((by omega))
 have:(_:ℝ)=_:=congrArg Rat.cast (w.trans (by rw[<-v,ο,ο,(by rw[pO (y▸P7 (2*u) (l+149)),Y4,nZ,sub_add_sub_cancel]:2*(u:Ἁ)-298=((2*u-g:Ἀ):Ἁ)+((g:Ἁ)-298)),(by rw[pO (y▸Nat.min_le_right (2*u) (l+149)),sub_add_sub_cancel,W5];ring:(l:Ἁ)-149=((l+149-g:Ἀ):Ἁ)+((g:Ἁ)-298)),ZL ZQ,ZL ZQ,ZL ZQ,ZF,ZF];push_cast;rw[add_div,Ap (e:ℚ),mul_div_cancel_left₀ _ (by positivity),<-Ap (A:ℚ),<-Ap (E:ℚ),mul_div_right_comm ((A:ℚ)*2^(2*u-g)),<-add_mul,Ap _ ((2:ℚ)^((g:Ἁ)-298))]:_=((A:ℚ)*2^(2*(u:Ἁ)-298)/e+E*2^((l:Ἁ)-149))*2^(2*s)))
 rwa[Rat.cast_div,T7,T7,J7,S8,G7]at this
@[inline]def VO(x k:Ἀ):Ἀ:=
 if k<63&&x>>>(63-k)==0 then(lq ((I0 x)<<<(I0 k)))else x<<<k
lemma Yz(x k:Ἀ):VO x k=x<<<k:=by
 rw[VO];by_cases h:(k<63&&x>>>(63-k)==0)=true
 .obtain⟨l,d⟩:=(Η ..).mp h;have l:=of_decide_eq_true l
  have e:x<2^(63-k):=(ZJ (϶ _)).1 ((β ..).symm.trans (AQ.mp d))
  have g:x*2^k<2^(63:Ἀ):=by rw[(L3 l.le).symm,m7];exact ϐ e (϶ _)
  rw[if_pos h,ψ,Ϣ,Ϣ,ZG,ZG,ο,ο,γ (AN e (Θ (by decide) ((K4 63 k).trans (by decide))):x<2^(64:Ἀ)),γ (AN l (by decide):k<2^(64:Ἀ)),γ (K6 l:k<64)]
  exact γ (K5 g (by decide))
 .exact if_neg h
@[inline]def Q2(x:Ἀ):Ἀ:=if x>>>63==0 then(lq (ZR (I0 x)))else (Aq x)
@[inline]def V2(r:Ἀ)(st:Ἆ):Ἀ:=if r !=0||st then 1 else 0
lemma VI(N M:Ἀ)(e:0<M)(δ:ℚ)(g:0<=δ)(j:δ<1):
  (N:ℚ)+δ=((N/M:Ἀ)+((N%M:Ἀ)+δ)/M)*M/\0<=((N%M:Ἀ)+δ)/M/\
   ((N%M:Ἀ)+δ)/M<1/\(((N%M:Ἀ)+δ)/M=0↔N%M=0/\δ=0):=by
 have d:(0:ℚ)<M:=Nat.cast_pos.2 e
 refine⟨?_,div_nonneg (add_nonneg (M3 _) g) d.le,(div_lt_one d).2 ((add_lt_add_right j _).trans_le ((X7 _).symm.trans_le (R5.2 (ϊ N e)))),(div_eq_zero_iff.trans (or_iff_left d.ne')).trans ((add_eq_zero_iff_of_nonneg (M3 _) g).trans (and_congr_left' Nat.cast_eq_zero))⟩
 rw[add_mul,div_mul_cancel₀ _ d.ne',<-add_assoc,<-Y4,<-W5,ZU]
theorem Qx_h1(r:Ἀ)(l:Ἆ)(δ y:ℚ)(g:y=0↔r=0/\δ=0)(w:l=ἃ↔δ≠0):V2 r l=if y=0 then 0 else 1:=by
 by_cases h:y=0
 · obtain⟨rfl,rfl⟩:=g.1 h;rw[if_pos h];cases l;rfl;exact absurd rfl (w.1 rfl)
 · rw[if_neg h];by_cases a:r=0
   · cases l;exact absurd (w.2 (λb=>h (g.2 ⟨a,b⟩))) lN;subst a;rfl
   · rw[V2,if_pos (by rw[nM,AM];exact pZ a)]
theorem Qx_h2(X:ℚ)(d:Ἀ)(e:Ἁ)(x:0<d):X*d*2^e/d=X*2^e:=by rw[mul_right_comm,mul_div_cancel_right₀ _ (Nat.cast_ne_zero.2 x.ne')]
theorem Qx_h3(W d:Ἀ)(Y:ℚ)(e:Ἁ)(z:2^(25:Ἀ)<=W/d)(m:0<=Y):0<(((W/d:Ἀ):ℚ)+Y)*2^e:=mul_pos (add_pos_of_pos_of_nonneg (Nat.cast_pos.2 (AN (by decide) z)) m) (zpow_pos two_pos e)
theorem Qx_h4(a:Ἀ)(e:Ἁ)(B:(a:Ἁ)=e+297):(a:Ἁ)-298=e-1:=by omega
theorem Qx_h5(a:Ἀ)(e L:Ἁ)(B:(a:Ἁ)=e+297)(L0:0<=L)(u:L+e<=126):a<=506:=U4.1 (B.trans_le ((add_le_add ((le_add_of_nonneg_left L0).trans u) AP).trans (by decide)))
lemma Qx(s:Ἆ)(W:Ἀ)(l:Ἆ)(d:Ἀ)(e:Ἁ)(v δ:ℚ)(M σ:ἆ)
  (A:v≠0->decide (v<0)=s) (p:0<=δ) (q:δ<1) (w:l=ἃ↔δ≠0) (x:0<d)
  (r:|v|=((W:ℚ)+δ)*2^e/d) (z:2^(25:Ἀ)<=W/d) (t:-126<=((Aq (W/d)):Ἁ)+e)
  (u:((Aq (W/d)):Ἁ)+e<=126) (y:(lq M)=2*(W/d)+V2 (W%d) l)
  (B:((lq σ):Ἁ)=e+297):
  az s M σ=ZT v:=by
 obtain⟨E,m,o,g⟩:=VI W d x δ p q;have h:|v|=((W/d:Ἀ)+((W%d:Ἀ)+δ)/d)*2^e:=by rw[r,E,Qx_h2 _ _ _ x]
 have j:v≠0:=abs_pos.1 ((Qx_h3 _ _ _ _ z m).trans_eq h.symm);rw[aJ _ _ _ (Qx_h5 _ _ _ B (W6 _) u),Y8 v j s (A j) _ _ m o _ h z t u,y,Qx_h1 _ _ _ _ g w,Qx_h4 _ _ B]
@[specialize]def Ym(x:(Ἂ Word))(u i stop P N:Ἀ):Ἀ×Ἀ:=
 if i<stop then
  let w:=x[i]!
  let a:=VO (lq (ZC w)) ((lq (ZD w))-u)
  if ZX w then Ym x u (i+1) stop P (N+a)else Ym x u (i+1) stop (P+a) N
 else(P,N)
termination_by stop-i
decreasing_by all_goals omega
def YS(x:(Ἂ Word))(u j:Ἀ):Ἁ:=
 if ZX x[j]!then-((ZW x[j]!<<<(ZZ x[j]!-u):Ἀ):Ἁ)
 else((ZW x[j]!<<<(ZZ x[j]!-u):Ἀ):Ἁ)
theorem VN_h1(d i:ℕ)(h:d-i=0):¬i<d:=W3.2 (K3.1 h)
lemma VN_h2(x:(Ἂ Word))(u d:Ἀ):
  ∀k i P N,d-i=k->
   ((Ym x u i d P N).1:Ἁ)-(Ym x u i d P N).2=
    (P:Ἁ)-N+∑j∈ZM k,YS x u (i+j):=by
 intro k;induction k<;>intro i P N h<;>rw[Ym]
 .rw[AC (VN_h1 _ _ h)];simp only [Finset.range_zero,Finset.sum_empty,Av]
 rename_i n ih;rw[AD (N1 h)];simp only[Yz,ZI,ZS,lV];have e:∀j,YS x u (i+(j+1))=YS x u (i+1+j):=λj=>by rw[lO];rfl
 simp only[e,ϑ];rw[YS];split<;>rename_i hz
 ·rw[ih (i+1) _ _ (by rw[V4,h,BK]),W5];ring
 ·rw[ih (i+1) _ _ (by rw[V4,h,BK]),W5];ring
lemma VN(x:(Ἂ Word))(u d:Ἀ):
  ∀k i P N,d-i=k->
   ((Ym x u i d P N).1:Ἁ)-(Ym x u i d P N).2=
    (P:Ἁ)-N+∑j∈ZM k,YS x u (i+j):=VN_h2 x u d
lemma Yx(x:(Ἂ Word))(u j:Ἀ)
  (h:ZW x[j]! =0\/u<=ZZ x[j]!):
  ZA x[j]! =(YS x u j:ℚ)*2^((u:Ἁ)-149):=by
 rw[Y1,YS];rcases h with h|h;simp[(YN _).2 h,h];rw[(by rw[<-ZF,<-ZL ZQ,pO h,sub_add_sub_cancel]:(2:ℚ)^((ZZ x[j]!:Ἁ)-149)=2^(ZZ x[j]!-u)*2^((u:Ἁ)-149)),Vr,ο];split_ifs<;>push_cast<;>ring
lemma VW(n:Ἀ)(x:(Ἂ Word))(r u:Ἀ)
  (d:∀j,r*n<=j->j<r*n+n->ZW x[j]! =0\/u<=ZZ x[j]!):
  (∑j∈ZM n,ZA x[r*n+j]! =
   ((((Ym x u (r*n) (r*n+n) 0 0).1:Ἁ)-(Ym x u (r*n) (r*n+n) 0 0).2:Ἁ):ℚ)*
    2^((u:Ἁ)-149))/\
  (∑j∈ZM n,ZA x[r*n+j]!^(2:Ἀ)=
   ((Yl x u (r*n) (r*n+n) 0 0 0 0:Ἀ):ℚ)*2^(2*(u:Ἁ)-298)):=by
 rw[VN x u _ n _ 0 0 ((by exact Nat.add_sub_self_left (r*n) n)),Vb x u _ n _ 0 0 0 0 ((by exact Nat.add_sub_self_left (r*n) n))];push_cast[sub_zero,Ai,ϋ,ϳ,ϑ,Κ,n8];constructor<;>refine ϔ rfl λj e=>?_<;>rw[ϥ]at e;exacts[Yx x u _ (d _ ((by exact V1 (r*n) j)) ((by exact O1 e (r*n)))),Ye _ u (d _ ((by exact V1 (r*n) j)) ((by exact O1 e (r*n))))]
@[inline]def Y0(a b:Ἀ):Ἀ:=if b<=a then a-b else b-a
lemma YO_h1(a b:Ἀ)(h:b<=a):(((a-b)*(a-b):Ἀ):ℚ)=((a:ℚ)-b)^(2:Ἀ):=by rw[Y4,pO h,sq]
lemma YO_h2(a b:Ἀ)(h:a<=b):(((b-a)*(b-a):Ἀ):ℚ)=((a:ℚ)-b)^(2:Ἀ):=by rw[Y4,pO h,<-neg_sub (a:ℚ),neg_mul_neg,sq]
lemma YO(a b:Ἀ):((Y0 a b*Y0 a b:Ἀ):ℚ)=((a:ℚ)-b)^(2:Ἀ):=by
 rw[Y0];split_ifs with h
 .exact YO_h1 a b h
 .exact YO_h2 a b (Nat.le_of_not_le h)
lemma VF_h1(n:Ἀ)(x:(Ἂ Word))(r:Ἀ)(m Y c:ℚ)(w:0<n)(hc:0<c)(h:XH n x r m=Y*c^(2:Ἀ)/(n*n)):0<=Y:=by
 have e:0<=XH n x r m:=by rw[rowVariance,Yk,Ai];exact div_nonneg (D8 λj _=>sq_nonneg _) (M3 _)
 rw[h,le_div_iff₀ (mul_pos ((Nat.cast_pos (α:=ℚ)).2 w) ((Nat.cast_pos (α:=ℚ)).2 w)),nv]at e;exact(mul_nonneg_iff_of_pos_right (pow_pos hc 2)).1 e
lemma VF_h2(u:ℤ):(2:ℚ)^(2*u-298)=(2^(u-149))^(2:Ἀ):=by rw[<-ZF,<-zpow_mul,Int.sub_mul,Int.mul_comm];rfl
theorem VF_h3(Q S c n:ℚ)(h:n≠0):(Q*c^(2:Ἀ)-2*(S*c)*(S*c/n)+n*(S*c/n)^(2:Ἀ))/n=(n*Q-S^(2:Ἀ))*c^(2:Ἀ)/(n*n):=by have ht:n*(S*c/n)=S*c:=mul_div_cancel₀ _ h;rw[sub_mul,<-mul_pow];generalize S*c/n=t at ht⊢;rw[<-ht,<-mul_div_mul_left _ n h];generalize c^(2:Ἀ)=C;exact congrArg (·/(n*n)) (by ring)
lemma VF(n:Ἀ)(x:(Ἂ Word))(r u:Ἀ)(w:0<n)
  (t:∀j,r*n<=j->j<r*n+n->ZW x[j]! =0\/u<=ZZ x[j]!):
  let pn:=Ym x u (r*n) (r*n+n) 0 0
  let Q:=Yl x u (r*n) (r*n+n) 0 0 0 0
  no n x r=((pn.1:ℚ)-pn.2)*2^((u:Ἁ)-149)/n/\
  XH n x r (no n x r)=
   ((n*Q-Y0 pn.1 pn.2*Y0 pn.1 pn.2:Ἀ):ℚ)*
    2^(2*(u:Ἁ)-298)/((n*n:Ἀ):ℚ):=by
 intro pn Q;obtain⟨d,m⟩:=VW n x r u t;simp only[Int.cast_sub,M5]at d;have g:no n x r=(pn.1-pn.2)*2^((u:Ἁ)-149)/n:=by rw[no,Yk,Ai,d]
 have l:=VF_h2 u;generalize q:(2:ℚ)^((u:Ἁ)-149)=c at d g l⊢
 generalize o:(pn.1:ℚ)-pn.2=S at d g⊢;have h:XH n x r (no n x r)=(n*Q-S^(2:Ἀ))*c^(2:Ἀ)/(n*n):=by rw[rowVariance,Yk,Ai,g,ϔ ἅ λj _=>sub_sq _ _,nJ,nF,<-n8,<-Finset.mul_sum,d,m,l,Finset.sum_const,Finset.card_range,nsmul_eq_mul];exact VF_h3 _ _ _ _ ((Nat.cast_ne_zero (R:=ℚ)).2 w.ne')
 have v:0<=n*Q-S^(2:Ἀ):=VF_h1 n x r _ _ c w (q▸zpow_pos two_pos _) h
 refine⟨g,?_⟩;rw[h,pO ((R5 (α:=ℚ)).1 (by rw[YO,Y4,o];exact sub_nonneg.1 v)),YO,o,l,Y4,Y4]
def sg(s:Ἆ)(M:ℚ):ℚ:=if s then-M else M
lemma Q6(w:kF):(ZC w !=0)=!decide (ZW w=0):=by rw[show(ZC w !=0)=!(ZC w==0)from ἅ,YK]
lemma Y2(w:kF):ZA w=sg (ZX w) (ZW w)*2^((ZZ w:Ἁ)-149):=by rw[Y1,Vr];rw[sg];simp only [ite_mul,neg_mul,one_mul]
lemma Yn_h1(a b:Ἀ)(h:a<b):-(((b-a:Ἀ)):ℚ)=(a:ℚ)-b:=by rw[pO h.le,neg_sub]
lemma Yn(a b:Ἀ):sg (decide (a<b)) (Y0 a b)=(a:ℚ)-b:=by
 unfold sg Y0;by_cases h:a<b
 .rw[A6 h,if_pos (rfl:true=true),if_neg (Nat.not_le.2 h)];exact Yn_h1 a b h
 .rw[decide_eq_false h,if_neg lN,if_pos (W4 h)];exact pO (W4 h)
lemma cP(w:kF):kG w<=254:=by rw[kG];split;exacts[ϟ _,P4 (mO w) 2]
namespace NF
open Dense.F
open Dense (ay)
open FloatLib.Floats.Formats.Flocq in
lemma jy(z:ℝ):|Model.roundAt kC z-z|<=|z|/2^(53:Ἀ)+1/2^(1075:Ἀ):=by
 by_cases g:z=0
 .subst g;simp only [Model.roundAt_zero,generic_format_zero,round_preserves_generic,sub_self,A5,zero_div,one_div,Ai,inv_nonneg,R6,pow_succ_nonneg]
 have l:0< |z|:=abs_pos.2 g
 have d:|z|<(2:ℝ)^(ph 2 |z|+1):=Int.lt_zpow_succ_log_self (R:=ℝ) (b:=2) (by norm_num) |z|
 have m:(2:ℝ)^(ph 2 |z|)<=|z|:=Int.zpow_log_le_self (R:=ℝ) (b:=2) (by norm_num) l
 by_cases j:-1021<=ph 2 |z|+1
 .have:=re z _ j d;have e:(2:ℝ)^(ph 2 |z|+1-53)/2=(2:ℝ)^(ph 2 |z|)/2^(53:Ἀ):=by rw[((by ring):ph 2 |z|+1-53=ph 2 |z|+(-52)),AK (by norm_num)];ring
  exact this.trans (e.trans_le ((div_le_div_of_nonneg_right m (by positivity)).trans (le_add_of_nonneg_right (by positivity))))
 .have h:|z|<(2:ℝ)^(-1021:ℤ):=AN d (AG (by norm_num) (not_le.1 j).le);have:=re z (-1021) AP h;have e:(2:ℝ)^((-1021:ℤ)-53)/2=1/2^(1075:Ἀ):=by rw[((by norm_num):((-1021:ℤ)-53)=-((1074:ℕ):ℤ)),zpow_neg,AA,inv_eq_one_div,div_div,<-A4]<;>rfl
  exact this.trans (e.trans_le (by linarith only[Aa z]))
lemma fA(x y:Ἅ)(h:fv x)(j:fv y)(g:|rv x+rv y|<=(2:ℝ)^(971:Ἁ)):
  fv (x+y)/\|rv (x+y)-(rv x+rv y)|<=|rv x+rv y|/2^(53:Ἀ)+1/2^(1075:Ἀ):=by have⟨d,e⟩:=ii x y h j (g.trans (pm 971 AP));exact⟨d,by rw[e];exact jy _⟩
lemma fM(x y:Ἅ)(h:fv x)(j:fv y)(g:|rv x*rv y|<=(2:ℝ)^(971:Ἁ)):
  fv (x*y)/\|rv (x*y)-rv x*rv y|<=|rv x*rv y|/2^(53:Ἀ)+1/2^(1075:Ἀ):=by have⟨d,e⟩:=jo x y h j g;exact⟨d,by rw[e];exact jy _⟩
lemma kc(b:ἆ):(lq (mF b))=(lq (b&&&0xfffffffffffff))+2^(52:Ἀ):=by
 have d:(lq (b&&&0xfffffffffffff))<2^(52:Ἀ):=by rw[Ε];exact AO ϝ (by decide)
 rw[mF];rw[my,show(lq (0x10000000000000:ἆ))=2^(52:Ἀ)*1 from ἅ,Nat.or_comm,<-Ϥ d,ϻ,Τ]
lemma jh(v Y U:ℝ)(d:Ἆ)(M L c:ℕ)(o:0<U)(j:v=(if d then-(M:ℝ)else M)*U)
  (u:L=M%2^(27:Ἀ)) (g:2^c<L) (h:L+2^c<2^(27:Ἀ)) (e:|v-Y|<=2^c*U):
  ((M/2^(27:Ἀ):ℕ):ℝ)*2^(27:Ἀ)*U<(if d then-Y else Y)/\
   (if d then-Y else Y)<(((M/2^(27:Ἀ):ℕ):ℝ)+1)*2^(27:Ἀ)*U:=by
 have l:(2:ℝ)^c<L:=mod_cast g;have m:(L:ℝ)+2^c<2^(27:Ἀ):=mod_cast h
 have⟨r,t⟩:=A3.1 e;rw[(by rw[u];exact_mod_cast(Au M (2^(27:Ἀ))).symm:(M:ℝ)=((M/2^(27:Ἀ):ℕ):ℝ)*2^(27:Ἀ)+L)]at j;cases d<;>simp only[ἁ,υ,Am]at j⊢<;>constructor;linear_combination t+AI l o-j;linear_combination r+AI m o+j;linear_combination r+AI l o+j;linear_combination t+AI m o-j
end NF
namespace NE
open NF Dense.F
open Dense (ay)
@[inline]def mx(a b:ἆ):ἆ:=if a<b then b else a
lemma eK(m e:ἆ):(lq (mx m e))=max (lq m) (lq e):=by rw[mx];split<;>rename_i h<;>rw[μ]at h<;>omega
@[inline]def pk(s:Ἆ)(q σ:ἆ):Ἃ:=
 (N3 ((if s then(0x80000000:ἆ)else 0)+((σ-146)<<<23+Core.ar (((2:ἆ)*q+1)<<<36))))
@[inline]def el(x g b:Ἃ)(nd nS R pz:Ἅ)(A1 A2 C1:ἆ)(slow:Unit->Ἃ):Ἃ:=
 let v:=(dw x*nd+nS)*dw g*R+dw b*pz
 let B:=v.toBits
 let E:=eF B
 let σ:=E-C1
 let M:=mF B
 let L:=M&&&0x7FFFFFF
 let c:=mx (mx (E+10) (Core.iW b+A1)) (Core.iW g+A2)-E
 if 146<=σ&&σ<=398&&c<=25&&(1<<<c)<L&&L+(1<<<c)<0x8000000 then
  pk (B>>>63!=0) (M>>>27) σ
 else slow ()
theorem kj_h1(c x y:ℕ)(hc:c<=2^(31:Ἀ))(hx:x<=252)(hy:y<=2^(24:Ἀ)):(c+(x*2^(23:Ἀ)%2^(64:Ἀ)+y)%2^(64:Ἀ))%2^(64:Ἀ)%2^(32:Ἀ)=c+(x*2^(23:Ἀ)+y):=by
 have a:x*2^(23:Ἀ)<=252*2^(23:Ἀ):=ι _ hx
 have b:x*2^(23:Ἀ)+y<=252*2^(23:Ἀ)+2^(24:Ἀ):=Y3 a hy
 have d:c+(x*2^(23:Ἀ)+y)<=2^(31:Ἀ)+(252*2^(23:Ἀ)+2^(24:Ἀ)):=Y3 hc b
 rw[γ (C0 a (by decide):x*2^(23:Ἀ)<2^(64:Ἀ)),γ (C0 b (by decide):x*2^(23:Ἀ)+y<2^(64:Ἀ)),γ (C0 d (by decide):c+(x*2^(23:Ἀ)+y)<2^(64:Ἀ)),γ (C0 d (by decide))]
theorem kj_h2(x:ℕ)(l:x<2^(26:Ἀ)):(2*x%2^(64:Ἀ)+1)%2^(64:Ἀ)=2*x+1:=by
 have a:2*x<2*2^(26:Ἀ):=M7 l (by decide)
 rw[γ (K5 a (by decide):2*x<2^(64:Ἀ)),γ (C0 (l7 a) (by decide):2*x+1<2^(64:Ἀ))]
theorem kj_h3(N S x y:ℕ)(j:2^(25:Ἀ)<=x)(l:x<2^(26:Ἀ))(m:146<=y)(o:y<=398)(g:N=(2*x+1)*2^(36:Ἀ))(h:S=y-36):2^(62:Ἀ)<=N/\N<2^(63:Ἀ)/\110<=S/\S<363:=by subst g h;omega
theorem kj_h4(c:Ἆ)(A B P:ℤ):(if c then -A else A)*(B*P)=(if c then -(A*B) else A*B)*P:=by
 cases c;exact (Ap A B P).symm;exact (neg_mul A (B*P)).trans ((congrArg Neg.neg (Ap A B P).symm).trans (neg_mul (A*B) P).symm)
lemma kj(s:Ἆ)(q σ:ἆ)(j:2^(25:Ἀ)<=(lq q))(l:(lq q)<2^(26:Ἀ))(m:146<=(lq σ))
  (o:(lq σ)<=398):pk s q σ=az s ((2:ἆ)*q+1) σ:=by
 have d:(lq ((2:ἆ)*q+1))=2*(lq q)+1:=by rw[η,ύ];simp only[kA];exact kj_h2 _ l
 have g:(lq (((2:ἆ)*q+1)<<<36))=(2*(lq q)+1)*2^(36:Ἀ):=by rw[ψ,d];simp only[kA,kH,ο];exact γ ((ι (2^(36:Ἀ)) (l7 (M7 l (by decide)))).trans_lt (by decide))
 have h:(lq (σ-36))=(lq σ)-36:=by rw[χ σ 36 (AZ (by decide) m:36<=(lq σ))];rfl
 have H:=kj_h3 _ _ _ _ j l m o g h;obtain⟨r,-,t⟩:=bM s (((2:ἆ)*q+1)<<<36) (σ-36) H.1 H.2.1 H.2.2.1 H.2.2.2;have p:(mJ (pk s q σ))=(if s then 2^(31:Ἀ) else 0)+(((lq σ)-146)*2^(23:Ἀ)+(lq (ar (((2:ἆ)*q+1)<<<36)))):=by rw[pk,Ϫ,η,η,ψ,χ σ 146 m];cases s<;>simp only[ἁ,ἂ,υ,kA,kH,ο]<;>exact kj_h1 _ _ _ (by decide) (P4 o 146) t
 rw[(by clear p r t;rw[cl s _ _ (by rw[d];exact Nat.succ_ne_zero _) (o.trans (by decide)),cl s _ _ (by rw[g];exact Nat.mul_ne_zero (Nat.succ_ne_zero _) (by decide)) (by rw[h];exact(K4 _ _).trans (o.trans (by decide))),d,g,h,(by rw[<-AF,BQ (AZ (by decide) m:36<=(lq σ))]:(2:Ἁ)^(lq σ)=2^(36:Ἀ)*2^((lq σ)-36)),Y4,I4,nZ,kj_h4]:az s ((2:ἆ)*q+1) σ=az s (((2:ἆ)*q+1)<<<36) (σ-36))];apply Β.1;rw[p,r,h,F7]
lemma pB(a b:ℝ)(p q:ℕ)(d:|a|<=2^p)(e:|b|<=2^q):|a*b|<=2^(p+q):=by rw[Ae,AF];exact mul_le_mul d e (Aa _) (by positivity)
theorem sB_h1(a b c:ℝ)(d:|a|<=c)(e:|b|<=c):|a+b|<=c*2:=(Ag a b).trans ((add_le_add d e).trans_eq (mul_two c).symm)
lemma sB(a b:ℝ)(p:ℕ)(d:|a|<=2^p)(e:|b|<=2^p):|a+b|<=2^(p+1):=(sB_h1 a b _ d e).trans_eq (A4 _ _).symm
lemma eB(x y:ℝ)(p:ℕ)(h:|x-y|<=|y|/2^(53:Ἀ)+1/2^(1075:Ἀ))(g:|y|<=2^p):|x|<=2^(p+1):=by have e:(1:ℝ)<=2^p:=one_le_pow₀ (by norm_num);have d:(1:ℝ)/2^(1075:Ἀ)<=1/2^(1:Ἀ):=AV (by norm_num) (by norm_num);rw[pow_one]at d;have:=AY x y;rw[A4];grind
lemma mo(a:ℝ)(p q:ℕ)(h:|a|<=2^p)(d:p<=q):|a|<=2^q:=
 h.trans (AU (by norm_num) d)
lemma z9(a:ℝ)(p:ℕ)(h:|a|<=2^p)(d:p<=971):|a|<=(2:ℝ)^(971:ℤ):=by rw[show(971:ℤ)=((971:ℕ):ℤ)from ἅ,AA];exact mo a p 971 h d
lemma wB(w:kF):|(sv w:ℝ)*2^kG w|<=2^(24+kG w):=by
 rw[Ae,AW (by positivity:(0:ℝ)<2^idxOf w),AF];have:|(sv w:ℝ)|<=2^(24:Ἀ):=by rw[<-mI,Ϲ];exact_mod_cast(bg w).le
 exact AE this (by positivity)
theorem jM_h1(x y e τ:ℝ)(h:x-y<=e)(k:e<=x/2^(53:Ἀ)+τ)(hv:0<=y)(m:0<=τ):x<=2*y+2*τ:=by linarith
theorem jM_h2(n c o:ℕ)(h:n+1056<=c+o):(2:ℝ)^n<=2^c*2^((o:ℤ)-1075)*2^(19:Ἀ):=by rw[<-AA (2:ℝ) n,<-AA (2:ℝ) c,<-AA (2:ℝ) 19,<-AK AL,<-AK AL,<-add_sub_assoc,sub_add];exact AG one_le_two (le_sub_iff_add_le.2 (U4.2 h))
lemma jM(A H w p G R κ l a t j v q:ℝ)(o c E u y x:ℕ)
  (hA:A=H+w) (he:|w+p|<=q)
  (z:|a-A|<=|A|/2^(53:Ἀ)+1/2^(1075:Ἀ))
  (B:|t-a*G|<=|a*G|/2^(53:Ἀ)+1/2^(1075:Ἀ))
  (D:|j-t*R|<=|t*R|/2^(53:Ἀ)+1/2^(1075:Ἀ))
  (V:|v-(j+l)|<=|j+l|/2^(53:Ἀ)+1/2^(1075:Ἀ))
  (g:0<=R) (hR:|R-κ|<=R/2^(50:Ἀ)) (hκ:0<=κ)
  (Y:|v|<=2^(53:Ἀ)*(2:ℝ)^((o:ℤ)-1075)) (ah:|l|<=2^(24+E+x)) (hG:|G|<=2^(24+u))
  (a0:u<=254) (a6:κ<=2^(53:Ἀ)) (al:R<=2^(52:Ἀ))
  (as:10<=c) (aH:E+x+1056<=c+o) (N:q=0\/(q=2^y/\y+u+1154<=c+o)):
  |v-((H-p)*G*κ+l)|<=2^c*(2:ℝ)^((o:ℤ)-1075):=by
 refine(?_:_<=(|v|+|l|)/2^(45:Ἀ)+q*|G| *κ+(4*(1/2^(1075:Ἀ))+4*(1/2^(1075:Ἀ)*R)+4*(1/2^(1075:Ἀ)*(|G| *R)))).trans ?_;obtain⟨τ,hτ⟩:∃τ:ℝ,τ=1/2^(1075:Ἀ):=⟨_,rfl⟩;rw[<-hτ]at z B D V⊢;have m:0<=τ:=hτ▸(by positivity);have r:=Aa G;rw[Ae a G]at B;rw[Ae t R,Ad g]at D;have X:|j+l|<=2*|v|+2*τ:=jM_h1 _ _ _ _ ((AY (j+l) v).trans_eq (Aw _ _)) V (Aa v) m
 have K:|a| *|G|<=2*|t|+2*τ:=by have:=AY (a*G) t;rw[Aw,Ae]at this;exact jM_h1 _ _ _ _ this B (Aa t) m
 have J:|A|<=2*|a|+2*τ:=jM_h1 _ _ _ _ ((AY A a).trans_eq (Aw _ _)) z (Aa a) m
 rw[show v-((H-p)*G*κ+l)=(v-(j+l))+(j-t*R)+(t-a*G)*R+(a-A)*(G*R)+A*G*(R-κ)+(w+p)*G*κ by rw[hA];ring];have T:∀a b c d e f:ℝ,|a+b+c+d+e+f|<=|a|+|b|+|c|+|d|+|e|+|f|:=λa b c d e f=>(Ag _ f).trans (add_le_add ((Ag _ e).trans (add_le_add ((Ag _ d).trans (add_le_add ((Ag _ c).trans (add_le_add (Ag a b) AP)) AP)) AP)) AP)
 refine(T _ _ _ _ _ _).trans ?_;simp only[Ae,Ad g,Ad hκ];linear_combination V+X/2^(53:Ἀ)+D+AE B g+19*AE K g/2^(53:Ἀ)+AE z (Af r g)+9*AE J (Af r g)/2^(53:Ἀ)+mul_le_mul_of_nonneg_left hR (Af (Aa A) r)+AE (AE he r) hκ+78*(by have:=Ag (j+l) (-l);rw[add_neg_cancel_right,BE]at this;exact this.trans (add_le_add X AP):|j|<=2*|v|+2*τ+|l|)/2^(53:Ἀ)+39*(by have:=AY (t*R) j;rw[Aw,Ae,Ad g]at this;exact jM_h1 _ _ _ _ this D (Aa j) m:|t| *R<=2*|j|+2*τ)/2^(53:Ἀ)+98*Aa v/2^(53:Ἀ)+178*Aa l/2^(53:Ἀ)+18014398509481748*m/2^(53:Ἀ)+27021597764222938*Af m g/2^(53:Ἀ)+27021597764222958*Af m (Af r g)/2^(53:Ἀ)
 have Pp:(0:ℝ)<=2^((o:ℤ)-1075):=by positivity
 have F1:=jM_h2 _ c o aH;have F3:=jM_h2 0 c o ((K2 1056 (E+x)).trans aH);rw[AF (2:ℝ) E x]at F1;rw[AF (2:ℝ) (24+E) x,AF (2:ℝ) 24 E]at ah
 have t3:q*|G| *κ<=2^c*2^((o:ℤ)-1075)/4:=by rcases N with I|⟨I,aL⟩;rw[I,nv,nv];positivity;rw[I];have h:=jM_h2 (y+u+98) c o aL;rw[AF (2:ℝ) (y+u) 98,AF (2:ℝ) y u]at h;generalize(2:ℝ)^((o:ℤ)-1075)=P at h⊢;linear_combination mul_le_mul (mul_le_mul_of_nonneg_left (hG.trans_eq (AF (2:ℝ) 24 u)) (pow_nonneg zero_le_two y)) a6 hκ (by positivity)+h/2^(21:Ἀ)
 have g1:|G|<=2^(278:Ἀ):=hG.trans (AU one_le_two (J1 a0 24));have a3:(1:ℝ)/2^(1075:Ἀ)*(|G| *R)<=1/2^(25:Ἀ):=calc
  _<=(1:ℝ)/2^(1075:Ἀ)*(2^(278:Ἀ)*2^(52:Ἀ)):=mul_le_mul_of_nonneg_left (mul_le_mul g1 al g (by positivity)) (by positivity)
  _=1/2^(745:Ἀ):=by rw[div_mul_eq_mul_div,one_mul,<-AF,div_eq_div_iff (by positivity) (by positivity),one_mul,<-AF]<;>rfl
  _<=_:=AV (by norm_num) (by norm_num)
 have a2:(1:ℝ)/2^(1075:Ἀ)*R<=1/2^(25:Ἀ):=calc
  _<=(1:ℝ)/2^(1075:Ἀ)*2^(52:Ἀ):=mul_le_mul_of_nonneg_left al (by positivity)
  _=1/2^(1023:Ἀ):=by rw[div_mul_eq_mul_div,one_mul,div_eq_div_iff (by positivity) (by positivity),one_mul,<-AF]<;>rfl
  _<=_:=AV (by norm_num) (by norm_num)
 generalize(2:ℝ)^((o:ℤ)-1075)=P at Y Pp F1 F3 t3⊢;linear_combination Y/2^(45:Ἀ)+AE (AU (one_le_two:(1:ℝ)<=2) as) Pp/4+ah/2^(45:Ἀ)+F1/2^(21:Ἀ)+t3+4*a2+4*a3+4*(AV (by norm_num) (by norm_num):(1:ℝ)/2^(1075:Ἀ)<=1/2^(25:Ἀ))+F3/2^(21:Ἀ)
end NE
namespace NR
open NF Dense.F
open Dense (ay)
@[inline]def pt(j:Ἀ):Ἅ:=P.get! (j+1)
@[inline]def p2(e:Ἀ):Ἅ:=pt (min e 254)*pt (e-min e 254)
lemma jw(j:Ἀ)(g:j<=254):fv (pt j)/\rv (pt j)=(2:ℝ)^j:=by
 have d:P.get! (j+1)=P.data[j+1]!:=by cases P;rfl
 have k:j+1<256:=Nat.succ_le_succ (Nat.succ_le_succ g)
 rw[pt,d,AB P.data (j+1) (by simp only[P,Ϝ];exact k.trans (by decide))];simp only[P,Λ,if_pos k];exact iJ j (g.trans (by decide))
theorem jv_h1(e:Ἀ)(l:e<=508):po e 254<=254/\(-1074:ℤ)<=e/\1+e<=971/\po e 254+(e-po e 254)=e:=⟨Nat.min_le_right _ _,AZ (by decide) (Int.ofNat_zero_le _),(J1 l 1).trans (by decide),M4 (P7 _ _)⟩
theorem jv_h2(e:Ἀ)(l:e<=508):e-po e 254<=254:=Nat.sub_le_iff_le_add.2 ((Nat.le_min.2 ⟨K2 e 254,l⟩).trans_eq (Nat.add_min_add_left 254 e 254))
lemma jv(e:Ἀ)(l:e<=508):fv (p2 e)/\rv (p2 e)=(2:ℝ)^e:=by
 have H:=jv_h1 e l;obtain⟨d,h⟩:=jw (po e 254) H.1;obtain⟨g,j⟩:=jw _ (jv_h2 e l);have:=aE _ _ d g 1 (e:ℤ) (by rw[h,j,<-AF,AA];push_cast;rw[one_mul];congr 1;exact H.2.2.2) (by decide) H.2.1 (bL 1 e 971 H.2.2.1 (by norm_num) 1 (by decide));rw[p2];refine⟨this.1,?_⟩;rw[this.2,AA];simp only [Int.cast_one,one_mul]
structure fH where
 td:Ἃ
 ok:Ἆ
 nS:Ἅ
 R:Ἅ
 pz:Ἅ
 A1:ἆ
 A2:ἆ
 C1:ἆ
@[noinline]def eI(n:Ἀ)(t:Ἀ×Ἀ×Ἆ×Ἀ×Ἀ×Ἃ):fH:=
 let k2:=t.2.1-21
 let Sm:=VO t.2.2.2.1 (t.2.1-k2)+(t.2.2.2.2.1>>>k2)
 let es:=k2+t.1
 let d:=t.2.2.2.2.2
 let D:=(I0 n)*Core.mW d
 let lD:=Core.ai D
 let s:=lD-11
 let R0:=((0x8000000000000000/D)<<<s)+(((0x8000000000000000%D)<<<s)/D)
 let Zr:=Core.iW d+lD+52
 let z:=(I0 Sm)
 ⟨d,
  decide (n<4096)&&!signOf d&&expOf d !=0&&11<=lD&&lD<=36&&Sm>>>53==0&&
   decide (es<=500),
  (if t.2.2.1 then z.toInt64 else(0-z).toInt64).toFloat*p2 es,
  R0.toInt64.toFloat,p2 (lq Zr),Zr+1056,if k2==0 then 0 else (I0 es)+1154,Zr+900⟩
lemma kn(D s:ἆ)(d L:Ἀ)(m:(lq D)=d)(h:2^L<=d)(j:d<2^(L+1))
  (y:(lq s)=L-11)(t:11<=L)(u:L<=36):
  (lq (((0x8000000000000000/D)<<<s)+(((0x8000000000000000%D)<<<s)/D)))=2^(52+L)/d:=by
 have g:0<d:=AN (by positivity) h;have e:(lq s)%64=L-11:=(by rw[y];exact γ (AO (P4 u 11) (by decide)));clear y;have x:2^(63:Ἀ)/d*2^(L-11)<=2^(52:Ἀ):=by have:2^(63:Ἀ)/d<=2^(63:Ἀ)/2^L:=pq h (by positivity);rw[Nat.pow_div (AZ u (by decide)) (by norm_num)]at this;exact(ι _ this).trans_eq (by rw[<-AF,<-Nat.add_sub_assoc t,L3 (AZ u (by decide))])
 have v:2^(63:Ἀ)%d*2^(L-11)<2^(63:Ἀ):=(ϐ (Ao (ϊ _ g) j) (by positivity)).trans_le (by rw[<-AF];exact Θ (by norm_num) ((Y3 (L1 u 1) (P4 u 11)).trans (by decide)))
 have o:(lq (0x8000000000000000:ἆ))=2^(63:Ἀ):=ἅ;have w:2^(63:Ἀ)%d*2^(L-11)/d<=2^(63:Ἀ):=AZ (ZV _ _) v.le;rw[η,(by rw[ψ,ml,o,m,e,ο,γ (AO x (by decide))]:(lq ((0x8000000000000000/D)<<<s))=2^(63:Ἀ)/d*2^(L-11)),(by rw[ml,ψ,nB,o,m,e,ο,γ (v.trans (by decide))]:(lq (((0x8000000000000000%D)<<<s)/D))=2^(63:Ἀ)%d*2^(L-11)/d),γ (AO (Y3 x w) (by decide)),(by rw[<-Ϛ,<-Ϡ,Nat.mod_add_div,<-AF];exact congrArg (2^·) ((congrArg (52+·) (M4 t)).symm.trans (pA 52 11 _).symm):2^(52+L)=2^(63:Ἀ)%d*2^(L-11)+d*(2^(63:Ἀ)/d*2^(L-11))),mv _ _ g];ring
lemma ke_h1(e:Ἀ)(q:e<2^(53:Ἀ)):ae (0-(I0 e))=-(e:ℤ):=by
 rw[ay,Ζ,Α (q.trans (by decide)),ϋ,ϑ];by_cases e0:e=0
 subst e0;rfl
 rw[γ (Nat.sub_lt (϶ 64) (ϵ e0)),if_pos (G5 ((J1 q.le _).trans (by decide))),pO (q.trans (by decide)).le,I4,nZ,sub_sub_cancel_left]
lemma ke_h2(j:ℕ):((2^j:ℕ):ℝ)=2^j:=by rw[I4,nZ]
lemma ke(n u k o v:ℕ)(X:Ἆ)(d:kF)(ρ:fH)(hρ:ρ=eI n (u,k,X,o,v,d))(a0:0<n)(h:ρ.ok=ἃ):
  ∃(e F E g:ℕ) (κ:ℝ),n<2^(29:Ἀ)/\
   (fv ρ.nS/\
    rv ρ.nS=(if X then(e:ℝ)else-(e:ℝ))*2^F)/\
   e<2^(53:Ἀ)/\F<=500/\(∃eS:ℝ,|(e:ℝ)*2^F-((o*2^k+v:ℕ):ℝ)*2^u|<=eS/\
    (eS=0\/(eS=2^F/\(lq ρ.A2)=F+1154)))/\
   (fv ρ.R/\rv ρ.R=g)/\g<=2^(52:Ἀ)/\0<=κ/\
   |(g:ℝ)-κ|<=g/2^(50:Ἀ)/\
   (fv ρ.pz/\rv ρ.pz=2^E)/\E<=500/\
   (ZA d:ℝ)≠0/\(2:ℝ)^E=κ*n*(ZA d:ℝ)*2^(149:Ἀ)/\
   (lq ρ.A1)=E+1056/\(lq ρ.A2)<=2000/\
   (lq ρ.C1)=900+E:=by
 subst hρ
 dsimp only[eI]at h⊢
 have e3:(o<<<(k-(k-21)))+(v>>>(k-21))=((o<<<k)+v)>>>(k-21):=by simp only[ο,β];rw[show o*2^k=o*2^(k-(k-21))*2^(k-21) by rw[Ap,<-AF,L3 (K4 k 21)],Τ _ v,Ρ _ _ (϶ _),Τ]
 rw[Yz,e3]at h⊢
 clear e3
 simp only[Η,AJ,BM,AM,Ak,θ,kA,AQ]at h
 obtain⟨⟨⟨⟨⟨⟨hn,a4⟩,aH⟩,t⟩,A⟩,a7⟩,Q⟩:=h
 have q:((o<<<k)+v)>>>(k-21)<2^(53:Ἀ):=(ZJ (϶ 53)).1 ((β _ 53).symm.trans a7)
 generalize hS:(o<<<k)+v=S at q⊢
 have as:S=o*2^k+v:=(by rw[<-hS,ο])
 generalize k-21=j at q Q⊢
 generalize m:S>>>j=e at q⊢
 have ba:2^(23:Ἀ)<=kM d:=by rw[kM,AC aH];exact V1 _ _
 have p:(lq ((I0 n)*mW d))=n*kM d:=by rw[ύ,Α (hn.trans (by decide)),an,γ];exact(ϕ hn (lB d)).trans (by decide)
 generalize (I0 n)*mW d=D at t A p⊢
 have hL:=Y7 D
 generalize ai D=x at t A hL⊢
 have w:n*kM d≠0:=Nat.mul_ne_zero a0.ne' (AN (϶ 23) ba).ne'
 obtain⟨I,al⟩:2^(lq x)<=n*kM d/\n*kM d<2^((lq x)+1):=by rw[hL,p];exact⟨ϰ w,ϙ⟩
 have J:=kn _ (x-11) _ _ p I al (by rw[χ _ _ (t:11<=x)];rfl) t A
 generalize((0x8000000000000000/D)<<<(x-11))+(((0x8000000000000000%D)<<<(x-11))/D)=g at J⊢
 have l:0<(n*kM d:Ἀ):=ϵ w
 have K:(lq g)<=2^(52:Ἀ):=by rw[J,AF];exact(pq I (϶ (lq x))).trans (Nat.mul_div_cancel _ (϶ (lq x))).le
 have M:2^(51:Ἀ)<=(lq g):=by rw[J,mc l,show 52+(lq x)=51+((lq x)+1) from Nat.succ_add 51 (lq x),AF];exact U3 _ al.le
 have bs:=cP d
 have bA:kG d+(lq x)+52<=342:=L1 (Y3 bs A) 52
 have r:(lq (iW d+x+52))=kG d+(lq x)+52:=by rw[η,η,am d];simp only[kA];rw[γ (AO ((V1 _ 52).trans bA) (by decide)),γ (AO bA (by decide))]
 generalize iW d+x+52=Zu at r⊢
 have V:sv d=kM d:=by rw[sv,AC (by simp[a4])]
 refine⟨e,j+u,kG d+(lq x)+52,(lq g),(2:ℝ)^(52+(lq x))/(n*kM d:Ἀ),(AN hn (by decide)),?_,q,Q,?_,?_,K,div_nonneg (pow_nonneg (zero_le_two:(0:ℝ)<=2) _) (M3 (α:=ℝ) _),?_,?_,(bA.trans (by decide)),?_,?_,?_,?_,?_⟩
 have hz:(lq (I0 e))=e:=Α (q.trans (by decide))
 have W:(K7 (if X then (I0 e).toInt64 else(0-(I0 e)).toInt64))=if X then(e:ℤ)else-(e:ℤ):=by split;show ae (I0 e)=_;rw[ay,hz,AC (Nat.not_le.2 (q.trans (by decide))),Int.sub_zero];exact ke_h1 e q
 have z5:(I6 (if X then(e:ℤ)else-(e:ℤ)))<2^(53:Ἀ):=by rw[Ax I6,Int.natAbs_neg,ite_self];exact q
 have B:=dB _ (by rw[W];exact z5)
 rw[W]at B
 obtain⟨f2,v2⟩:=jv (j+u) (Q.trans (by decide))
 have:=aE _ _ B.1 f2 _ _ (by rw[B.2,v2,AA]) z5 (AZ (by decide) (Int.ofNat_zero_le _)) (bL 53 (j+u) 971 ((J1 Q 53).trans (by decide)) AP _ z5)
 refine⟨this.1,?_⟩
 rw[this.2,AA,Ax (Int.cast:ℤ->ℝ),Int.cast_neg,M5]
 rw[<-as]
 by_cases z:j=0
 exact⟨0,by rw[<-m,z,Nat.shiftRight_zero,Κ,sub_self,A5],pZ ἅ⟩
 refine⟨2^(j+u),?_,BA ⟨ἅ,by simp only[z,AQ,Am];rw[η,Α (AO Q (by decide):j+u<2^(64:Ἀ))];simp only[kA];exact γ (AO (L1 Q 1154) (by decide))⟩⟩
 have a1:(e:ℝ)*2^j<=S:=(by rw[<-m,β,<-ke_h2,<-Y4,R5];exact pr _ _)
 have a2:(S:ℝ)<e*2^j+2^j:=(by rw[<-m,β,<-ke_h2,<-Y4,<-W5,Nat.cast_lt];exact Nat.lt_div_mul_add (϶ j))
 have hu:(0:ℝ)<2^u:=(by positivity)
 rw[AF]
 exact A3.2 ⟨neg_le_sub_iff_le_add.2 ((AI a2 hu).le.trans_eq (by rw[add_mul,Ap])),(sub_nonpos.2 ((Ap _ _ _).symm.trans_le (AE a1 hu.le))).trans (by positivity)⟩
 have B:=dB g.toInt64 (by show(I6 (ae g))<2^(53:Ἀ);rw[ay,AC (Nat.not_le.2 (AO K (by decide)))];exact K.trans_lt (by decide))
 refine⟨B.1,?_⟩
 rw[B.2]
 show((ae g:ℤ):ℝ)=(lq g)
 rw[ay,AC (Nat.not_le.2 (AO K (by decide)))]
 simp
 have H:(0:ℝ)<(n*kM d:Ἀ):=Nat.cast_pos.2 l
 have:(2:ℝ)^(50:Ἀ)<=(lq g):=(ke_h2 50).symm.trans_le (R5.2 ((Θ Nat.zero_lt_two ((by decide):50<=51)).trans M))
 have o:=(one_le_div₀ ((by positivity):(0:ℝ)<2^(50:Ἀ))).2 this
 have u:(2:ℝ)^(52+(lq x))<((lq g)+1)*(n*kM d:Ἀ):=by rw[J,mul_comm,<-ke_h2,<-pJ (R:=ℝ),<-W5,<-Y4,Nat.cast_lt];exact Nat.lt_mul_div_succ _ l
 have w:((lq g):ℝ)*(n*kM d:Ἀ)<=2^(52+(lq x)):=by rw[J,<-ke_h2,<-Y4,R5];exact pr _ _
 have u2:=(div_lt_iff₀ H).2 u
 have w2:=(le_div_iff₀ H).2 w
 have p:(0:ℝ)<=(lq g)/2^(50:Ἀ):=div_nonneg (M3 _) (by positivity)
 have a:=(neg_le_neg o).trans (neg_le_sub_iff_le_add.2 u2.le)
 have b:=(sub_nonpos.2 w2).trans p
 exact A3.2 ⟨a,b⟩
 rw[r]
 exact jv _ (bA.trans (by decide))
 rw[Rat.cast_ne_zero,Y1,V]
 exact mul_ne_zero (Int.cast_ne_zero.2 (Nat.cast_ne_zero.2 (AN (by decide) ba).ne')) (zpow_ne_zero _ AL)
 have:(n:ℝ)≠0:=Nat.cast_ne_zero.2 a0.ne'
 have:(kM d:ℝ)≠0:=Nat.cast_ne_zero.2 (L6 (AN (by decide) ba))
 rw[Y1,V,J7,S8,Rat.cast_intCast,G7,M5,Y4,Ac AL,AA]
 field_simp
 ring
 swap
 split
 simp
 rw[η,Α (AO Q (by decide):j+u<2^(64:Ἀ))]
 simp only[kA]
 rw[γ (AO (L1 Q 1154) (by decide))]
 exact(L1 Q 1154).trans (by decide)
 all_goals rw[η,r];simp only[kA];rw[γ (AO (L1 bA _) (by decide))];try exact Τ _ _
lemma kg(n:Ἀ)(g:n<2^(29:Ἀ)):fv (I0 n).toInt64.toFloat/\rv (I0 n).toInt64.toFloat=n:=by
 have d:(lq (I0 n))=n:=Α (g.trans (by decide));have e:(K7 (I0 n).toInt64)=n:=by show ae (I0 n)=n;rw[ay,d,if_neg (not_le.2 (g.trans (by decide))),sub_zero]
 have:=dB _ (by rw[e,Int.natAbs_natCast];exact g.trans (by decide));rwa[e,M5]at this
lemma ic(a:(Ἂ Word))(h:∀i (d:i<(O0 a)),finiteWord a[i]=ἃ)(i:Ἀ)(d:i<(O0 a)):kB a[i]!≠255:=by have:=h i d;rw[AB a i d];rw[finiteWord]at this;grind
end NR
namespace NE
open NF Dense.F
theorem jn_h1(a b:ἆ)(h:(lq (a-b))<=398)(hb:(lq b)<=1400):(lq b)<=(lq a):=W4 λg=>by
 have e:(lq (a-b))=2^(64:Ἀ)-((lq b)-(lq a)):=BitVec.toNat_sub_of_lt (x:=a.toBitVec) (y:=b.toBitVec) (μ.2 g);rw[e]at h;exact absurd h (Nat.not_le.2 (AN (by decide) (Nat.sub_le_sub_left ((K4 _ _).trans hb) _)))
theorem jn_h2(X G P Bs Bk κ n D c:ℝ)(hn:n≠0)(hD:D≠0)(hc:c≠0):(X*n-P)*G*κ+Bs*(Bk*(κ*n*D*c))=((X/c-P/n/c)*(G/c)/D+Bs*Bk/c)*(c*(κ*n*D*c)):=by
 have h1:(X/c-P/n/c)*(G/c)/D*(c*(κ*n*D*c))=(X*n-P)*G*κ:=by
  rw[←sub_div,div_mul_div_comm,div_div,mul_comm (c*c) D,div_mul_eq_mul_div,mul_comm c (κ*n*D*c),Ap (κ*n*D) c c,Ap (κ*n) D (c*c),mul_div_assoc,mul_div_cancel_right₀ _ (mul_ne_zero hD (mul_ne_zero hc hc)),mul_comm κ n,←Ap ((X-P/n)*G) n κ,mul_right_comm (X-P/n) G n,sub_mul,div_mul_cancel₀ _ hn]
 have h2:Bs*Bk/c*(c*(κ*n*D*c))=Bs*(Bk*(κ*n*D*c)):=(div_mul_eq_mul_div _ _ _).trans ((congrArg (·/c) (mul_left_comm _ _ _)).trans ((mul_div_cancel_left₀ _ hc).trans (Ap _ _ _)))
 exact ((add_mul _ _ _).trans (congrArg₂ (·+·) h1 h2)).symm
theorem jn_h4(A J:ℤ):A-(900+J)=A-1075+27-149-J+297:=by
 rw[sub_add (A-1075+27-149) J 297,sub_sub (A-1075+27) 149 (J-297),sub_add A 1075 27,sub_sub A (1075-27) _,←add_assoc,add_sub (1075-27+149) J 297,add_sub_right_comm];rfl
lemma jn(x g b d:kF)(y:Unit->kF)(n u k J K:ℕ)(sS:Ἆ)(ρ:NR.fH)(l:ℚ)
  (hρ:ρ=NR.eI n (u,k,sS,J,K,d))(hx:kB x≠255)(hg:kB g≠255)(hb:kB b≠255)(b6:0<n)
  (h:ρ.ok=ἃ)(o:(l:ℝ)=(if sS then-((J*2^k+K:ℕ):ℝ)else((J*2^k+K:ℕ):ℝ))*2^u/n/2^(149:Ἀ))
  (a2:y ()=ZT ((ZA x-l)*ZA g/ZA d+ZA b)):
  el x g b (I0 n).toInt64.toFloat ρ.nS ρ.R ρ.pz ρ.A1 ρ.A2 ρ.C1 y=ZT ((ZA x-l)*ZA g/ZA d+ZA b):=by
 obtain⟨_,es,j,g9,κ,hn,E,b1,hF,⟨eS,Q,b2⟩,hR,b0,dd,hK,a1,Jb,hD,hZ,bh,Ab,bp⟩:=NR.ke n u k J K sS d ρ hρ b6 h;have F:=NR.kg n hn;generalize J*2^k+K=S at Q o;generalize (I0 n).toInt64.toFloat=p at F⊢;dsimp only[el];split_ifs with hc;swap;exact a2;obtain⟨fx,vx⟩:=cI x hx;obtain⟨fg,vg⟩:=cI g hg;obtain⟨fb,vb⟩:=cI b hb;have wg:=wB g;have ix:=cP x;have ig:=cP g;have ib:=cP b;have a5:(I6 (sv x*n:ℤ))<2^(53:Ἀ):=by rw[Ϩ,Int.natAbs_natCast];exact AN (ϕ (bg x) hn) (by decide)
 have c0:((sv x*n:ℤ):ℝ)=(sv x:ℝ)*n:=(Int.cast_mul _ _).trans (congrArg _ (M5 n))
 have hX:=aE (dw x) p fx F.1 (sv x*n) (kG x) (by rw[vx,F.2,AA,c0,mul_right_comm]) a5 (AZ (by decide) (Int.ofNat_zero_le _)) (bL 53 (kG x) 971 ((J1 ix 53).trans (by decide)) AP _ a5);have r:=aE (dw b) ρ.pz fb a1.1 (sv b) ((kG b+j:ℕ):ℤ) (by rw[vb,a1.2,AA,AF,Ap]) (AN (bg b) (by norm_num)) (AZ (by decide) (Int.ofNat_zero_le _)) (bL 24 (kG b+j) 971 ((J1 (Y3 ib Jb) 24).trans (by decide)) AP _ (bg b));rw[AA]at hX r;have hXn:rv (dw x*p)=(sv x:ℝ)*2^kG x*n:=hX.2.trans ((congrArg (·*_) c0).trans (mul_right_comm _ _ _))
 have bX:|rv (dw x*p)|<=2^(553:Ἀ):=by rw[hXn];exact mo _ (24+kG x+29) 553 (pB _ _ _ _ (wB x) (by rw[nE];exact_mod_cast hn.le)) ((L1 (J1 ix 24) 29).trans (by decide))
 have bS:|rv ρ.nS|<=2^(553:Ἀ):=by rw[E.2];apply mo _ (53+es) 553 _ ((J1 hF 53).trans (by decide));apply pB _ _ _ _ _ (by rw[AW (by positivity)]);rw[Ax _root_.abs,BE,ite_self,nE];exact_mod_cast b1.le
 have ha:=fA (dw x*p) ρ.nS hX.1 E.1 (z9 _ _ (sB _ _ _ bX bS) (by decide));have ba:|rv (dw x*p+ρ.nS)|<=2^(555:Ἀ):=eB _ _ _ ha.2 (sB _ _ _ bX bS);have bG:|rv (dw g)|<=2^(278:Ἀ):=vg▸mo _ _ _ wg (J1 ig 24);have ht:=fM (dw x*p+ρ.nS) (dw g) ha.1 fg (z9 _ _ (pB _ _ _ _ ba bG) (by decide));have bt:|rv ((dw x*p+ρ.nS)*dw g)|<=2^(834:Ἀ):=eB _ _ _ ht.2 (pB _ _ _ _ ba bG);have bR:|rv ρ.R|<=2^(52:Ἀ):=by rw[hR.2,nE];exact_mod_cast b0
 have G:=fM ((dw x*p+ρ.nS)*dw g) ρ.R ht.1 hR.1 (z9 _ _ (pB _ _ _ _ bt bR) (by decide));have bB:|rv (dw b*ρ.pz)|<=2^(24+kG b+j):=by rw[r.2,AF,<-Ap,Ae,AW ((by positivity):(0:ℝ)<2^j),AF (2:ℝ) (24+kG b)];exact AE (wB b) (by positivity)
 have hv:=fA ((dw x*p+ρ.nS)*dw g*ρ.R) (dw b*ρ.pz) G.1 r.1 (z9 _ _ (sB _ _ _ (eB _ _ _ G.2 (pB _ _ _ _ bt bR)) (mo _ _ _ bB ((Y3 (J1 ib 24) Jb).trans (by decide)))) (by decide));generalize(dw x*p+ρ.nS)*dw g*ρ.R+dw b*ρ.pz=V at hc hv;simp only[Η,AJ,θ,μ,kA]at hc;obtain⟨⟨⟨⟨ch,c1⟩,X⟩,B⟩,as⟩:=hc;have eb:(lq (eF V.toBits))<2048:=by rw[eF,Ε];exact AO ϝ (by decide)
 have ds:(lq (eF V.toBits-ρ.C1))=(lq (eF V.toBits))-(900+j):=by rw[χ _ _ (θ.2 (jn_h1 _ _ c1 (by rw[bp];exact J1 Jb 900))),bp]
 have M:146+(900+j)<=(lq (eF V.toBits)):=(have c:146<=(lq (eF V.toBits))-(900+j):=ds▸ch;Nat.add_le_of_le_sub (Nat.lt_of_sub_pos (AN (by decide) c)).le c);have bs:¬eF V.toBits=0:=λh0=>absurd M (by rw[h0];exact Nat.not_le.2 (U1 (by decide) (V1 _ _)))
 have aL:=eK (eF V.toBits+10) (iW b+ρ.A1);have D:=eK (mx (eF V.toBits+10) (iW b+ρ.A1)) (iW g+ρ.A2);obtain⟨b4,A,b5⟩:(lq (eF V.toBits+10))=(lq (eF V.toBits))+10/\(lq (iW b+ρ.A1))=kG b+j+1056/\(lq (iW g+ρ.A2))=kG g+(lq ρ.A2):=by rw[η,η,η,am,am,bh];simp only[kA];exact⟨γ (AN (X3 eb 10) (by decide)),(γ (AO (Y3 ib (L1 Jb 1056)) (by decide))).trans (pA _ _ _).symm,γ (AO (Y3 ig Ab) (by decide))⟩
 rw[b4,A]at aL;rw[aL,b5]at D;have ah:eF V.toBits<=mx (mx (eF V.toBits+10) (iW b+ρ.A1)) (iW g+ρ.A2):=by rw[θ,D];exact le_max_of_le_left (le_max_of_le_left (V1 _ _))
 have z:=χ _ _ ah;rw[D]at z;generalize mx (mx (eF V.toBits+10) (iW b+ρ.A1)) (iW g+ρ.A2)-eF V.toBits=cc at z X B as;have L:(lq ((1:ἆ)<<<cc))=2^(lq cc):=by rw[ψ,pS,γ (AO X (by decide):(lq cc)<64),na,γ (lu (by decide) (AO X (by decide):(lq cc)<64))]
 have hL:(lq (mF V.toBits&&&0x7FFFFFF))=(lq (mF V.toBits))%2^(27:Ἀ):=Ε _ _▸ζ _ 27;have bw:=kc V.toBits;have e1:(lq (V.toBits&&&0xfffffffffffff))<2^(52:Ἀ):=Ε _ _▸AO ϝ (by decide);rw[L,hL]at B;rw[η,L,hL]at as;have a3:=dV V hv.1 bs;rw[bJ,<-bw]at a3;have W:0<((2:ℝ)^(((lq (eF V.toBits)):ℤ)-1075)):=((by positivity));obtain⟨N,d1⟩:(lq (mF V.toBits))<2^(53:Ἀ)/\2^(52:Ἀ)<=(lq (mF V.toBits)):=by rw[bw];exact⟨(X3 e1 _).trans_eq (by decide),K2 _ _⟩
 have bv:|rv V|<=2^(53:Ἀ)*((2:ℝ)^(((lq (eF V.toBits)):ℤ)-1075)):=by rw[a3,Ae,AW W,Ax _root_.abs,BE,nE,ite_self];exact AE (by exact_mod_cast N.le) W.le
 have al:eS=0\/(eS=2^es/\es+kG g+1154<=(lq cc)+(lq (eF V.toBits))):=b2.imp id λ⟨h0,hA⟩=>⟨h0,by rw[z,L3 (le_max_of_le_left (le_max_of_le_left (V1 _ _))),hA];exact le_max_of_le_right (B3 ((congrArg (·+1154) (Τ es (kG g))).trans (pA _ _ _)))⟩
 have m:=jM _ (rv (dw x*p)) _ ((if sS then-(S:ℝ)else S)*2^u) _ _ κ (rv (dw b*ρ.pz)) _ _ _ _ eS (lq (eF V.toBits)) (lq cc) _ _ es j ἅ (by rw[E.2];split;rw[neg_mul,<-sub_eq_add_neg];exact Q;rw[neg_mul,neg_add_eq_sub,Aw];exact Q) ha.2 ht.2 G.2 hv.2 (hR.2▸M3 _) (hR.2▸hK) dd bv bB (vg▸wg) ig ((sub_le_iff_le_add'.1 ((le_abs_self _).trans ((Aw _ _).trans_le hK))).trans ((add_le_add (hR.2▸(le_abs_self _).trans bR) (div_le_div_of_nonneg_right (hR.2▸(le_abs_self _).trans bR) (by positivity))).trans ((add_le_add AP (div_le_self (pow_nonneg (zero_le_two:(0:ℝ)<=2) 52) (one_le_pow₀ (one_le_two:(1:ℝ)<=2)))).trans_eq ((two_mul _).symm.trans (pow_succ' (2:ℝ) 52).symm)))) ((le_abs_self _).trans bR) (z▸G5 (le_max_of_le_left (le_max_of_le_left (Τ _ _).le))) (by rw[z,L3 (le_max_of_le_left (le_max_of_le_left (V1 _ _)))];exact le_max_of_le_left (le_max_right _ _)) al
 have t2:(2:ℝ)≠0:=AL
 have hY:((sv x:ℝ)*2^kG x*n-(if sS then-(S:ℝ)else S)*2^u)*((sv g:ℝ)*2^kG g)*κ+(sv b:ℝ)*2^(kG b+j)=(((ZA x-l)*ZA g/ZA d+ZA b:ℚ):ℝ)*(2^(149:Ἀ)*2^j):=by
  have e:∀w:kF,((ZA w:ℚ):ℝ)=(sv w:ℝ)*2^kG w/2^(149:Ἀ):=λw=>by rw[Y1,J7,S8,Rat.cast_intCast,G7,Ac t2,AA,zpow_ofNat,mul_div_assoc]
  rw[Rat.cast_add,Rat.cast_div,J7,Rat.cast_sub,e x,e g,e b,o,AF,hZ]
  exact jn_h2 _ _ _ _ _ _ _ _ _ (Nat.cast_ne_zero.2 b6.ne') hD (pow_ne_zero _ t2)
 rw[hXn,r.2,vg,hY]at m
 generalize(ZA x-l)*ZA g/ZA d+ZA b=Y at m a2⊢
 obtain⟨lo,hi⟩:=jh (rv V) ((Y:ℝ)*(2^(149:Ἀ)*2^j)) ((2:ℝ)^(((lq (eF V.toBits)):ℤ)-1075)) _ (lq (mF V.toBits)) ((lq (mF V.toBits))%2^(27:Ἀ)) (lq cc) W a3 ἅ B ((γ ((E6 (ϊ _ (϶ 27)) (B.trans (ϊ _ (϶ 27)))).trans (by decide):(lq (mF V.toBits))%2^(27:Ἀ)+2^(lq cc)<2^(64:Ἀ))).symm.trans_lt as) m
 obtain⟨q,hq⟩:∃q,q=(lq (mF V.toBits))/2^(27:Ἀ):=⟨_,rfl⟩
 rw[<-hq]at lo hi
 have b7:2^(25:Ἀ)<=q:=hq▸(mc (϶ 27)).2 (AZ (by decide) d1)
 have cb:q<2^(26:Ἀ):=hq▸Ϟ (N.trans_eq (by decide))
 have q0:q≠0:=(AN (by decide) b7).ne'
 obtain⟨e,he⟩:∃e:ℤ,e=((lq (eF V.toBits)):ℤ)-1075+27-149-j:=⟨_,rfl⟩
 have e4:(2:ℝ)^(27:Ἀ)*((2:ℝ)^(((lq (eF V.toBits)):ℤ)-1075))=(2:ℝ)^e*(2^(149:Ἀ)*2^j):=by rw[he,<-AA (2:ℝ) 149,<-AA (2:ℝ) j,<-AA (2:ℝ) 27,<-AK t2,<-AK t2,<-AK t2];exact congrArg (fun t:ℤ=>(2:ℝ)^t) (by rw[nZ,nZ,sub_sub,sub_add_cancel,add_comm])
 have hP:(0:ℝ)<2^(149:Ἀ)*2^j:=((by positivity))
 rw[Ap,e4,<-Ap]at lo
 rw[Ap,e4,<-Ap ((q:ℝ)+1)]at hi
 obtain⟨s,hs⟩:∃s,s=(V.toBits>>>63!=0):=⟨_,rfl⟩
 rw[<-hs]at lo hi a3⊢
 generalize(2:ℝ)^(149:Ἀ)*2^j=K at lo hi hP
 have kq:(if s then-((Y:ℝ)*K)else(Y:ℝ)*K)=(((if s then-Y else Y:ℚ)):ℝ)*K:=by by_cases h:s=true;rw[if_pos h,if_pos h,Rat.cast_neg,neg_mul];rw[if_neg h,if_neg h]
 rw[kq]at lo hi
 have a6:(q:ℚ)*2^e<(if s then-Y else Y):=Rat.cast_lt.1 (show (((q:ℚ)*2^e:ℚ):ℝ)<((if s then-Y else Y:ℚ):ℝ) by rw[J7,S8,T7,G7];exact BF lo hP.le)
 have A:(if s then-Y else Y)<((q:ℚ)+1)*2^e:=Rat.cast_lt.1 (show ((if s then-Y else Y:ℚ):ℝ)<((((q:ℚ)+1)*2^e:ℚ):ℝ) by rw[J7,S8,Rat.cast_add,T7,Rat.cast_one,G7];exact BF hi hP.le)
 have a0:(0:ℚ)<2^e:=((by positivity))
 have v:0<(if s then-Y else Y):=AO (Af (M3 _) a0.le) a6
 have tA:|Y|=(if s then-Y else Y)∧(Y≠0->decide (Y<0)=s):=by by_cases h:s=true;rw[if_pos h]at v⊢;exact⟨abs_of_neg (neg_pos.1 v),λ_=>h▸A6 (neg_pos.1 v)⟩;rw[if_neg h]at v⊢;exact⟨AW v,λ_=>(F8.2 h)▸decide_eq_false (not_lt.2 v.le)⟩
 obtain⟨δ,hδ⟩:∃δ,δ=(if s then-Y else Y)/2^e-q:=⟨_,rfl⟩;have c3:0<δ:=hδ▸sub_pos.2 ((lt_div_iff₀ a0).2 a6)
 have d2:(Aq q)=25:=(mz q0).2 ⟨b7,cb⟩;have e3:(lq (mF V.toBits>>>27))=q:=(ν _ _).trans ((β _ _).trans hq.symm)
 have E5:((lq (eF V.toBits-ρ.C1)):ℤ)=e+297:=by rw[ds,he,pO (AZ (K2 _ _) M),W5,nZ];exact jn_h4 _ _
 have k1:-126<=((25:ℕ):ℤ)+e:=AZ (show (-126:ℤ)<=((25:ℕ):ℤ)+(((146:ℕ):ℤ)-297) by decide) (add_le_add AP (sub_le_iff_le_add.2 ((U4.2 ch).trans_eq E5)))
 have k2:((25:ℕ):ℤ)+e<=126:=AZ (add_le_add AP (le_sub_iff_add_le.2 (E5.symm.trans_le (U4.2 c1)))) (show ((25:ℕ):ℤ)+(((398:ℕ):ℤ)-297)<=126 by decide)
 rw[kj _ _ _ (e3▸b7) (e3▸cb) ch c1,<-Qx s q ἃ 1 e Y δ ((2:ἆ)*(mF V.toBits>>>27)+1) (eF V.toBits-ρ.C1) tA.2 c3.le (hδ▸sub_lt_iff_lt_add'.2 ((div_lt_iff₀ a0).2 A)) ⟨λ_=>c3.ne',λ_=>ἅ⟩ one_pos (by rw[tA.1,hδ,add_sub_cancel,pJ,div_one,div_mul_cancel₀ _ a0.ne']) (by rw[F5];exact b7) (by rw[F5,d2];exact k1) (by rw[F5,d2];exact k2) (by rw[η,ύ,e3];simp only[kA,V2,Bool.or_true,ἁ,F5];exact kj_h2 _ cb) E5]
end NE
local instance:Inhabited NR.fH:=⟨⟨0,ἄ,0,0,0,0,0,0⟩⟩
lemma uS{i:Ἄ}(h:(J0 i)+1<kX):(J0 (i+1))=(J0 i)+1:=by rw[lJ,mA];exact γ h
@[inline]def ck(a:(Ἂ Word))(i:Ἄ):Ἃ:=if h:i<a.usize then a.uget i (Dense.F.uL h)else a[(J0 i)]!
lemma f6(a:(Ἂ Word))(i:Ἄ):ck a i=a[(J0 i)]!:=by
 rw[ck];split
 .exact(AB a (J0 i) (Dense.F.uL ‹_›)).symm
 .rfl
@[specialize]def Yq{σ:Type}(f:σ->Ἄ->Ἄ->Ἃ)(s:σ)(lo i e:Ἄ)(out:(Ἂ Word)):(Ἂ Word):=
 if h:i<e then Yq f s lo (i+1) e (if g:i<out.usize then out.uset i (f s i (i-lo)) (Dense.F.uL g) else out)else out
termination_by (J0 e)-(J0 i)
decreasing_by have:=ε.mp h;have:=e.toNat_lt_size;rw[uS ((by exact C0 h this))];exact Z5 eб iб h
lemma ju0{σ:Type}(f:σ->Ἄ->Ἄ->kF)(s:σ)(g e:Ἄ)(i:Ἄ)(d:(Ἂ Word))(he:(J0 e)<=(O0 d))(hd:(O0 d)<2^(32:Ἀ)):
  (O0 (Yq f s g i e d))=(O0 d)/\∀q<(O0 d),(Yq f s g i e d)[q]! =if (J0 i)<=q/\q<(J0 e) then f s (G0 q) ((G0 q)-g) else d[q]!:=by
 have:=e.toNat_lt_size
 induction h:(J0 e)-(J0 i) generalizing i d with
 |zero=>
  have z:=W3.mpr (K3.mp h)
  rw[Yq,dif_neg (fun x=>z (ε.mp x))];exact ⟨rfl,fun q _=>(if_neg (fun x=>z (C0 x.1 x.2))).symm⟩
 |succ k ih=>
  have hi:(J0 i)<(J0 e):=N1 h
  have hX:(J0 i)+1<kX:=C0 hi this
  have hu:i<d.usize:=by rw[ε];show (J0 i)<(J0 (G0 ((O0 d))));rw[Conv.aa _ hd];exact U1 hi he
  rw[Yq,dif_pos (ε.mpr hi),dif_pos hu]
  have hs:(O0 (d.uset i (f s i (i-g)) (Dense.F.uL hu)))=(O0 d):=BT ..
  obtain⟨a,b⟩:=ih (i+1) _ (by rw[hs];exact he) (by rw[hs];exact hd) (by rw[uS hX,V4,h,BK])
  refine ⟨a.trans hs,fun q hq=>?_⟩
  rw[b q (by rw[hs];exact hq),uS hX,pP,AB _ _ (by rwa[Array.size_set])]
  by_cases hq1:(J0 i)=q
  · subst hq1;rw[if_neg (fun c=>Nat.not_succ_le_self _ c.1),if_pos ⟨Nat.le_refl _,hi⟩,Array.getElem_set_self,lX,USize.ofNat_toNat]
  · rw[Array.getElem_set,if_neg hq1,<-AB d q hq]
    by_cases c:(J0 i)+1<=q/\q<(J0 e)
    · rw[if_pos c,if_pos ⟨X5 c.1,c.2⟩]
    · rw[if_neg c,if_neg (fun x=>c ⟨N4 x.1 hq1,x.2⟩)]
@[specialize]def Yr{σ:Type}(g:Ἀ->σ)(f:σ->Ἄ->Ἄ->Ἃ)(n rows r:Ἀ)(out:(Ἂ Word)):(Ἂ Word):=
 if r<rows then Yr g f n rows (r+1) (Yq f (g r) (G0 (r*n)) (G0 (r*n)) (G0 (r*n+n)) out)else out
termination_by rows-r
decreasing_by all_goals omega
lemma ju1{σ:Type}(g:Ἀ->σ)(f:σ->Ἄ->Ἄ->kF)(n d r:Ἀ)(e:(Ἂ Word))(m:d*n<2^(32:Ἀ))(he:(O0 e)=d*n):
  (O0 (Yr g f n d r e))=(O0 e)/\∀q<(O0 e),(Yr g f n d r e)[q]! =if r*n<=q then f (g (q/n)) (G0 q) ((G0 q)-(G0 (q/n*n))) else e[q]!:=by
 induction h:d-r generalizing r e with
 |zero=>have k0:=O4 h;rw[Yr,if_neg (W3.2 k0)];exact ⟨rfl,fun q hq=>by rw[if_neg (Nat.not_le.2 ((hq.trans_eq he).trans_le (ι n k0)))]⟩
 |succ k ih=>
  have hr:=N1 h
  have e1:(r+1)*n=r*n+n:=add_one_mul r n
  have h1:r*n+n<=d*n:=e1▸ι n hr
  rw[Yr,if_pos hr]
  have ha:=Conv.aa _ (((V1 _ n).trans h1).trans_lt m);have hb:=Conv.aa _ (h1.trans_lt m)
  obtain⟨a0,b0⟩:=ju0 f (g r) (G0 (r*n)) (G0 (r*n+n)) (G0 (r*n)) e (by rw[hb,he];exact h1) (he.trans_lt m)
  obtain⟨a1,b1⟩:=ih (r+1) _ (by rw[a0,he]) (by rw[V4,h,BK])
  refine ⟨a1.trans a0,fun q hq=>?_⟩
  rw[b1 q (by rw[a0];exact hq),b0 q hq,ha,hb,e1]
  by_cases c1:r*n+n<=q
  · rw[if_pos c1,if_pos ((V1 _ _).trans c1)]
  · rw[if_neg c1];by_cases c2:r*n<=q
    · rw[if_pos ⟨c2,Nat.not_le.1 c1⟩,if_pos c2,mG c2 ((Nat.not_le.1 c1).trans_eq e1.symm)]
    · rw[if_neg (c2∘And.left),if_neg c2]
lemma Ys{σ:Type}(g:Ἀ->σ)(f:σ->Ἄ->Ἄ->kF)(n d:Ἀ)(j:d*n<2^(32:Ἀ))(F:Fin (d*n)->kF)
  (h:∀q (h:q<d*n),f (g (q/n)) (G0 q) (G0 (q%n))=F ⟨q,h⟩):
  Yr g f n d 0 (Ah (d*n) 0)=kK F:=by
 obtain⟨a,b⟩:=ju1 g f n d 0 (Ah (d*n) 0) j (by simp only [O3])
 apply mi (by rw[a];simp only [O3,Ϝ])
 intro q h1 h2
 have hq:q<d*n:=by simpa using h2
 have u:=b q (by simpa using hq)
 rw[if_pos ((ϳ n).trans_le (ϟ q))]at u
 rw[<-AB (Yr g f n d 0 (Ah (d*n) 0)) q h1,u,Λ,(by rw[(by rw[Nat.mod_add_div']:(G0 q)=(G0 (q%n+q/n*n))),lX,mC,USize.add_sub_cancel]:(G0 q)-(G0 (q/n*n))=(G0 (q%n)))]
 exact h q hq
def cS(x:(Ἂ Word))(i e:Ἄ)(u:ἆ):ἆ:=
 if h:i<e then
  let w:=ck x i
  cS x (i+1) e (if ZC w==0 then u else if ZD w<u then ZD w else u)
 else u
termination_by (J0 e)-(J0 i)
decreasing_by have:=ε.mp h;have:=e.toNat_lt_size;rw[uS ((by exact C0 h this))];exact Z5 eб iб h
lemma gO_h1(w:kF)(u:ἆ):(lq (if ZC w==0 then u else if ZD w<u then ZD w else u))=if ZC w==0 then (lq u) else min (lq u) (lq (ZD w)):=by
 split_ifs with h2 d;rfl;exact(min_eq_right (μ.1 d).le).symm;exact(min_eq_left (W3.1 (d∘μ.2))).symm
lemma gO_h2(i e:Ἄ)(n:ℕ)(h:(J0 e)-(J0 i)=n+1):(J0 i)<(J0 e)∧(J0 (i+1))=(J0 i)+1:=⟨N1 h,uS (C0 (N1 h) e.toNat_lt_size)⟩
lemma gO(x:(Ἂ Word))(e i:Ἄ)(u:ἆ):(lq (cS x i e u))=Yh x (J0 i) (J0 e) (lq u):=by
 induction h:(J0 e)-(J0 i)generalizing i u<;>rw[cS,Yh];rw[dif_neg (by rw[ε];exact VN_h1 _ _ h),AC (VN_h1 _ _ h)];rename_i g;obtain⟨k,m⟩:=gO_h2 i e _ h;rw[AT (ε.2 k),AD k,g _ _ (by rw[m,V4,h,BK]),m,f6];congr 1;exact gO_h1 _ _
structure jY where
 l:ἆ
 h:ἆ
 c:Ἀ
 big:Ἀ
 P:Ἀ
 N:Ἀ
@[inline]def cA(P:Ἀ)(q p:ἆ):Ἀ:=if q<p then P+2^(64:Ἀ) else P
lemma cy_h1(P x y N:Ἀ)(hx:x<N)(hy:y<N):(if (x+y)%N<x then P+N else P)+(x+y)%N=P+x+y:=by
 by_cases h:N<=x+y
 · rw[P8 h,γ (G4 h (E6 hx hy)),if_pos (G4 h (Τ x N▸O1 hy x)),pA,M4 h,pA]
 · rw[γ (O8 h),if_neg (Nat.not_lt_of_le (V1 x y)),pA]
lemma cy(P:Ἀ)(p a:ἆ):cA P (p+a) p+(lq (p+a))=P+(lq p)+(lq a):=by unfold cA;simp only[μ,η];exact cy_h1 _ _ _ _ p.toNat_lt a.toNat_lt
lemma j5(P:Ἀ)(p:ἆ):cA P p p=P:=by simp[cA]
lemma jZ(w:kF)(d:Ἀ)(h:d<=39):(lq (ZC w<<<(I0 d)))=VO (lq (ZC w)) d:=by have:=VV w;have e:(lq (ZC w))*2^d<2^(63:Ἀ):=AN (ϐ this ((by positivity))) (by rw[<-AF];exact Θ ((by positivity)) ((by omega)));rw[Yz,ψ,Ϣ,ZG,γ ((by omega):d<2^(64:Ἀ)),γ ((by omega):d<64),ο];omega
@[specialize]def cF(x:(Ἂ Word))(u:ἆ)(i e:Ἄ)(l h:ἆ)(c big P N:Ἀ)(pl nl:ἆ):jY:=
 if hi:i<e then
  let w:=ck x i
  let mm:=ZC w*ZC w
  let s:=if u<=ZD w then ZD w-u else 0
  if s<=39 then
   let a:=ZC w<<<s
   let p:=if ZX w then pl else pl+a
   let q:=if ZX w then nl+a else nl
   let k:=2*s
   let vl:=if k<64 then mm<<<k else 0
   let vh:=if k<64 then(mm>>>1)>>>(63-k)else mm<<<(k-64)
   let l':=l+vl
   let h':=h+vh+(if l'<l then 1 else 0)
   cF x u (i+1) e l' h' (c+(if h'<h then 1 else 0)) big (cA P p pl) (cA N q nl) p q
  else
   let a:=VO (lq (ZC w)) (lq s)
   cF x u (i+1) e l h c (big+(lq mm)<<<(2*(lq s))) (if ZX w then P else P+a) (if ZX w then N+a else N) pl nl
 else⟨l,h,c,big,P+(lq pl),N+(lq nl)⟩
termination_by (J0 e)-(J0 i)
decreasing_by all_goals (have:=ε.mp hi;have:=e.toNat_lt_size;rw[uS ((by exact C0 hi this))];exact Z5 eб iб hi)
lemma ek(x:(Ἂ Word))(u:ἆ)(e i:Ἄ) l h c big P N pl nl:
  let s:=cF x u i e l h c big P N pl nl
  s.big+(lq s.l)+((lq s.h)+s.c<<<64)<<<64=Yl x (lq u) (J0 i) (J0 e) l h c big/\(s.P,s.N)=Ym x (lq u) (J0 i) (J0 e) (P+(lq pl)) (N+(lq nl)):=by
 induction hk:(J0 e)-(J0 i)generalizing i l h c big P N pl nl<;>dsimp only<;>unfold cF Yl Ym<;>have:=e.toNat_lt_size
 .rw[dif_neg (by rw[ε];exact W3.2 (K3.1 hk))];simp only[show¬(J0 i)<(J0 e) from W3.2 (K3.1 hk),XA,and_self]
 rename_i j;have m:=uS (i:=i) (C0 (N1 hk) this);rw[AT (ε.2 (N1 hk))];simp only[(N1 hk),ἁ,f6,<-m]
 generalize x[(J0 i)]! =w;have:=(ZD w).toNat_lt;have o:(if u<=ZD w then ZD w-u else 0)=(I0 ((lq (ZD w))-(lq u))):=ξ.1 (by split<;>rename_i g;rw[χ _ _ g,Ϣ,ZG,γ ((K4 _ _).trans_lt this)];rw[θ]at g;rw[ϋ,Nat.sub_eq_zero_of_le (Nat.le_of_not_le g)];rfl);rw[o];generalize q:(lq (ZD w))-(lq u)=d;have t:(lq (I0 d))=d:=by rw[Ϣ,ZG];exact γ (q▸(K4 _ _).trans_lt this)
 have p:(2:ἆ)*(I0 d)=(I0 (2*d)):=(by rw[Ϣ,Ϣ,UInt64.ofNat_mul];rfl);simp only[t,p,show (I0 d)<=39↔d<=39 by rw[θ,t];rfl];clear q o t p;by_cases hd:d<=39<;>simp only[hd,ἁ,ἂ]<;>refine⟨(j _ _ _ _ _ _ _ _ _ (by rw[m,V4,hk,BK])).1,(j _ _ _ _ _ _ _ _ _ (by rw[m,V4,hk,BK])).2.trans ?_⟩<;>split<;>simp(disch:=assumption) only[j5,cy,jZ,lO _ (VO _ _)]
def nS201:Ἅ:=Dense.F.pw 201
def nQs(x:(Ἂ Word))(i e:Ἄ)(v:Ἅ):Ἅ:=
 if h:i<e then let d:=Dense.F.dw (ck x i);nQs x (i+1) e (v+d*d) else v
termination_by (J0 e)-(J0 i)
decreasing_by have:=ε.mp h;have:=e.toNat_lt_size;rw[uS ((by omega))];omega
section
open NF Dense.F NE
@[inline]def nW(B:ἆ):Ἃ:=pk (B>>>63!=0) (mF B>>>27) (eF B-900)
@[inline]def nWs(sh B:ἆ):Ἃ:=pk (B>>>63!=0) (mF B>>>27) (eF B-(900+sh))
@[inline]def nOK(B c:ἆ):Ἆ:=let E:=(B>>>52)&&&2047;let L:=B&&&0x7FFFFFF;let t:=(1:ἆ)<<<c;1047<=E&&E<=1298&&c<=25&&t<L&&L+t<0x8000000
theorem nOK_dec_h1(B c:ἆ)(a3:(lq c)<=25)(a4:(1:ἆ)<<<c<B&&&0x7FFFFFF)(a5:(B&&&0x7FFFFFF)+((1:ἆ)<<<c)<0x8000000):2^(lq c)<(lq B)%2^(27:Ἀ)∧(lq B)%2^(27:Ἀ)+2^(lq c)<2^(27:Ἀ):=by
 have hc:=C0 a3 (N2 rfl:25<64)
 have hL:(lq ((1:ἆ)<<<c))=2^(lq c):=by rw[ψ,pS,γ hc,na];exact γ (lu Nat.one_lt_two hc)
 have hl:(lq (B&&&0x7FFFFFF))=(lq B)%2^(27:Ἀ):=(Ε _ _).trans (ζ _ 27)
 have a4:2^(lq c)<(lq B)%2^(27:Ἀ):=hL▸hl▸a4
 have a5:(lq ((B&&&0x7FFFFFF)+((1:ἆ)<<<c)))<2^(27:Ἀ):=a5
 rw[η,hL,hl,γ (U1 (E6 (ϊ _ (϶ 27)) (K5 a4 (ϊ _ (϶ 27)))) (N2 rfl))]at a5
 exact⟨a4,a5⟩
theorem nOK_dec(B c:ἆ)(h:nOK B c=ἃ):1047<=(lq (eF B))∧(lq (eF B))<=1298∧(lq c)<=25∧2^(lq c)<(lq B)%2^(27:Ἀ)∧(lq B)%2^(27:Ἀ)+2^(lq c)<2^(27:Ἀ):=by
 obtain⟨h,a5⟩:=I5.1 h
 obtain⟨h,a4⟩:=I5.1 h
 obtain⟨h,a3⟩:=I5.1 h
 obtain⟨a1,a2⟩:=I5.1 h
 exact⟨of_decide_eq_true a1,of_decide_eq_true a2,of_decide_eq_true a3,nOK_dec_h1 B c (of_decide_eq_true a3) (of_decide_eq_true a4) (of_decide_eq_true a5)⟩
theorem nOK_win(y:ℝ)(v:Ἅ)(c:ℕ)(hv:fv v)(hE0:(lq (eF v.toBits))≠0)(a4:2^c<(lq (v.toBits))%2^(27:Ἀ))(a5:(lq (v.toBits))%2^(27:Ἀ)+2^c<2^(27:Ἀ))
  (he:|y-rv v|<2^((c:ℤ)-1)*2^(((lq (eF v.toBits)):ℤ)-1075)):
  2^(25:Ἀ)<=(lq (mF v.toBits>>>27))∧(lq (mF v.toBits>>>27))<2^(26:Ἀ)∧
  ((lq (mF v.toBits>>>27)):ℝ)*2^(27:Ἀ)*2^(((lq (eF v.toBits)):ℤ)-1075)<(if v.toBits>>>63!=0 then-y else y)∧
  (if v.toBits>>>63!=0 then-y else y)<(((lq (mF v.toBits>>>27)):ℝ)+1)*2^(27:Ἀ)*2^(((lq (eF v.toBits)):ℤ)-1075):=by
 have bw:=kc v.toBits
 have hm:(lq (v.toBits&&&4503599627370495))=(lq (v.toBits))%2^(52:Ἀ):=by simp only[Ε,kA];exact ζ _ 52
 have hq:(lq (mF v.toBits>>>27))=(lq (mF v.toBits))/2^(27:Ἀ):=by simp only[ν,kA,β]
 rw[hm]at bw
 have bs:¬eF v.toBits=0:=λh0=>hE0 (by rw[h0];rfl)
 have a6:=dV v hv bs
 rw[bJ,show(lq (v.toBits&&&0xfffffffffffff))+2^(52:Ἀ)=(lq (mF v.toBits)) by rw[bw,hm]]at a6
 have W:0<((2:ℝ)^(((lq (eF v.toBits)):ℤ)-1075)):=zpow_pos (by norm_num) _
 have a7:rv v=(if v.toBits>>>63!=0 then-((lq (mF v.toBits)):ℝ)else (lq (mF v.toBits)))*((2:ℝ)^(((lq (eF v.toBits)):ℤ)-1075)):=by rw[a6]
 have a8:|rv v-y|<=2^c*2^(((lq (eF v.toBits)):ℤ)-1075):=by
  rw[Aw];refine he.le.trans (AE ?_ W.le)
  rw[<-AA];exact AG one_le_two (sub_le_self _ zero_le_one)
 obtain⟨lo,hi⟩:=jh (rv v) y _ _ _ _ c W a7 (by rw[bw];exact (Nat.mod_mod_of_dvd _ (pow_dvd_pow 2 (by decide))).symm.trans (Μ _ (2^(25:Ἀ)) _).symm) a4 a5 a8
 rw[<-hq]at lo hi
 exact⟨by rw[hq,mc (by decide),bw];exact K2 _ _,by rw[hq,bw];exact Ϟ (X3 (ϊ _ (by decide)) _),lo,hi⟩
theorem nOK_spec_h1(s:Ἆ)(y:ℚ)(P:ℝ):(if s then-((y:ℝ)*P)else(y:ℝ)*P)=(((if s then-y else y:ℚ)):ℝ)*P:=by cases s<;>[rfl;rw[if_pos rfl,if_pos rfl,Rat.cast_neg,neg_mul]]
theorem nOK_spec(y:ℚ)(v:Ἅ)(c:ἆ)(hv:fv v)(hok:nOK v.toBits c=ἃ)
  (he:|(y:ℝ)*2^(149:Ἀ)-rv v|<2^(((lq c):ℤ)-1)*2^(((lq (eF v.toBits)):ℤ)-1075)):nW v.toBits=ZT y:=by
 obtain⟨a1,a2,a3,a4,a5⟩:=nOK_dec _ c hok
 obtain⟨b7,cb,lo,hi⟩:=nOK_win _ v (lq c) hv (λh0=>by rw[h0]at a1;exact absurd a1 (by decide)) a4 a5 he
 unfold nW
 have h9:(lq (eF v.toBits-900))=(lq (eF v.toBits))-900:=χ _ _ (by rw[θ];simp only[kA];exact (by decide:900<=1047).trans a1)
 have ch:146<=(lq (eF v.toBits-900)):=by rw[h9];exact G5 ((by decide:146+900<=1047).trans a1)
 have c1:(lq (eF v.toBits-900))<=398:=by rw[h9];exact M8 (a2.trans (by decide))
 generalize hq:(lq (mF v.toBits>>>27))=q at b7 cb lo hi
 obtain⟨e,hE⟩:∃e:ℤ,e=((lq (eF v.toBits)):ℤ)-1197:=⟨_,rfl⟩
 have e4:(2:ℝ)^(27:Ἀ)*((2:ℝ)^(((lq (eF v.toBits)):ℤ)-1075))=(2:ℝ)^e*2^(149:Ἀ):=by
  rw[<-AA,<-AA,<-AK AL,<-AK AL,hE];exact congrArg _ (by ring)
 rw[Ap,e4,<-Ap]at lo
 rw[Ap ((q:ℝ)+1),e4,<-Ap]at hi
 have hP:(0:ℝ)<2^(149:Ἀ):=by positivity
 obtain⟨s,hs⟩:∃s,s=(v.toBits>>>63!=0):=⟨_,rfl⟩
 rw[<-hs]at lo hi⊢
 rw[nOK_spec_h1 s y]at lo hi
 have z2:(((2:ℚ)^e:ℚ):ℝ)=(2:ℝ)^e:=by rw[S8,G7]
 have a6:(q:ℚ)*2^e<(if s then-y else y):=Rat.cast_lt.1 (((J7 _ _).trans (congrArg₂ (·*·) (T7 q) z2)).trans_lt (BF lo hP.le))
 have A:(if s then-y else y)<((q:ℚ)+1)*2^e:=Rat.cast_lt.1 ((BF hi hP.le).trans_eq ((J7 _ _).trans (congrArg₂ (·*·) ((Rat.cast_add _ _).trans (congrArg₂ (·+·) (T7 q) Rat.cast_one)) z2)).symm)
 have a0:(0:ℚ)<2^e:=by positivity
 have q0:q≠0:=BD.mp (U1 (by decide) b7)
 have w:0<(if s then-y else y):=AO (Af (M3 _) a0.le) a6
 have t:|y|=(if s then-y else y):=by cases s<;>[exact AW w;exact abs_of_neg (neg_pos.1 w)]
 obtain⟨δ,hδ⟩:∃δ,δ=(if s then-y else y)/2^e-q:=⟨_,rfl⟩
 have c3:0<δ:=hδ▸sub_pos.mpr ((lt_div_iff₀ a0).mpr a6)
 have d2:(Aq q)=25:=(mz q0).2 ⟨b7,cb⟩
 have E5:((lq (eF v.toBits-900)):ℤ)=e+297:=by rw[h9,hE];rw[pO ((by decide:900<=1047).trans a1)];ring
 have n1:((1047:ℕ):ℤ)<=(lq (eF v.toBits)):=U4.2 a1
 have n2:((lq (eF v.toBits)):ℤ)<=((1298:ℕ):ℤ):=U4.2 a2
 have k1:-126<=((25:ℕ):ℤ)+e:=hE▸(by decide:(-126:ℤ)<=((25:ℕ):ℤ)+(((1047:ℕ):ℤ)-1197)).trans (Int.add_le_add_left (Int.sub_le_sub_right n1 1197) _)
 have k2:((25:ℕ):ℤ)+e<=126:=hE▸(Int.add_le_add_left (Int.sub_le_sub_right n2 1197) _).trans (by decide:((25:ℕ):ℤ)+(((1298:ℕ):ℤ)-1197)<=126)
 rw[kj _ _ _ (hq▸b7) (hq▸cb) ch c1,<-Qx s q ἃ 1 e y δ ((2:ἆ)*(mF v.toBits>>>27)+1) (eF v.toBits-900) (λ_=>by cases s<;>[exact decide_eq_false (not_lt.2 w.le);exact A6 (neg_pos.1 w)]) c3.le (by rw[hδ,sub_lt_iff_lt_add',div_lt_iff₀ a0];exact A) ⟨λ_=>c3.ne',λ_=>rfl⟩ one_pos (by rw[t,hδ,add_sub_cancel,pJ,div_one,div_mul_cancel₀ _ a0.ne']) (by rw[F5];exact b7) (by rw[F5,d2];exact k1) (by rw[F5,d2];exact k2) (by rw[η,ύ,hq];simp only[kA,V2,Bool.or_true,ἁ,F5,Nat.mod_one];exact kj_h2 _ cb) E5]
theorem nEl_e1_h1(a b c d e f:ℝ):|a+b+c+d+e+f|<=|a|+|b|+|c|+|d|+|e|+|f|:=
 (Ag _ _).trans (add_le_add ((Ag _ _).trans (add_le_add ((Ag _ _).trans (add_le_add ((Ag _ _).trans (add_le_add (Ag _ _) AP)) AP)) AP)) AP)
theorem nEl_e1_h2(t p Bb a b X m M g ρ γ:ℝ)(h:g*ρ=γ):t-((X-M)*g+Bb)=(t-(p+Bb))+(p-a*b)+(X-m)*g*(ρ-1)+(M-m)*g+(X-m)*(b-γ)+(a-(X-m))*b:=by linear_combination (m-X)*h
theorem nEl_e1(X m M G r Bb D a b p t Q Γ w:ℝ)(hD:0<D)(hr:|r*D-1|<=1/2^(50:Ἀ))(hm:|m-M|<=Q)
  (hS:(|X|+|m|)/2^(53:Ἀ)<=Q/2)(hG:|G|/D<=Γ)(hw0:0<=w)(hw1:w<=1)
  (ha:|a-(X-m)|<=|X-m|/2^(53:Ἀ)+w)
  (hb:|b-G*r|<=|G*r|/2^(53:Ἀ)+w)
  (hp:|p-a*b|<=|a*b|/2^(53:Ἀ)+w)
  (ht:|t-(p+Bb)|<=|p+Bb|/2^(53:Ἀ)+w):
  |t-((X-M)*(G/D)+Bb)|<=12*(Q*Γ)+|t|/2^(52:Ἀ)+6*((|X|+|m|)+Γ+1)*w:=by
 have hQ0:0<=Q:=AZ (Aa _) hm
 have hGa:|G/D|<=Γ:=by rw[abs_div,AW hD];exact hG
 have hΓ0:0<=Γ:=(Aa _).trans hGa
 obtain⟨S,hSdef⟩:∃S,S=|X|+|m|:=⟨_,rfl⟩
 rw[<-hSdef]at hS ⊢
 have hS0:0<=S:=hSdef▸add_nonneg (Aa _) (Aa _)
 have hK0:(0:ℝ)<=2^(53:Ἀ):=pow_nonneg zero_le_two _
 have hK1:(1:ℝ)<=2^(53:Ἀ):=one_le_pow₀ one_le_two
 have hXm:|X-m|<=S:=hSdef▸abs_sub _ _
 have hrD:|r*D|<=2:=((sub_le_iff_le_add.1 (AY (r*D) 1)).trans (add_le_add (hr.trans (div_le_one_of_le₀ (one_le_pow₀ one_le_two) (pow_nonneg zero_le_two _))) abs_one.le)).trans_eq one_add_one_eq_two
 have hGD:G/D*(r*D)=G*r:=by rw[mul_comm r D,← Ap,div_mul_cancel₀ _ hD.ne']
 have hGr:|G*r|<=2*Γ:=(congrArg (abs:ℝ→ℝ) hGD).symm.trans_le ((Ae _ _).trans_le ((mul_le_mul hGa hrD (Aa _) hΓ0).trans_eq (mul_comm _ _)))
 have hid:t-((X-M)*(G/D)+Bb)=(t-(p+Bb))+(p-a*b)+(X-m)*(G/D)*(r*D-1)+(M-m)*(G/D)+(X-m)*(b-G*r)+(a-(X-m))*b:=nEl_e1_h2 _ _ _ _ _ _ _ _ _ _ _ hGD
 have ea:|a-(X-m)|<=S/2^(53:Ἀ)+w:=ha.trans (add_le_add (div_le_div_of_nonneg_right hXm hK0) AP)
 have eb:|b-G*r|<=2*Γ/2^(53:Ἀ)+w:=hb.trans (add_le_add (div_le_div_of_nonneg_right hGr hK0) AP)
 have Am:|a|<=S+w+S:=(sub_le_iff_le_add.1 (AY a (X-m))).trans (add_le_add (ea.trans (add_le_add (div_le_self hS0 hK1) AP)) hXm)
 have Bm:|b|<=2*Γ+w+2*Γ:=(sub_le_iff_le_add.1 (AY b (G*r))).trans (add_le_add (eb.trans (add_le_add (div_le_self (Af zero_le_two hΓ0) hK1) AP)) hGr)
 have hab:|a*b|/2^(53:Ἀ)<=(S+w+S)*(2*Γ+w+2*Γ)/2^(53:Ἀ):=div_le_div_of_nonneg_right ((Ae a b).trans_le (mul_le_mul Am Bm (Aa _) (add_nonneg (add_nonneg hS0 hw0) hS0))) hK0
 have e:|p+Bb|<=|t|+|t-(p+Bb)|:=by have:=abs_sub t (t-(p+Bb));rwa[sub_sub_cancel]at this
 have T1:|(X-m)*(G/D)*(r*D-1)|<=S*Γ*(1/2^(50:Ἀ)):=(Ae _ _).trans_le (mul_le_mul ((Ae _ _).trans_le (mul_le_mul hXm hGa (Aa _) hS0)) hr (Aa _) (Af hS0 hΓ0))
 have T2:|(M-m)*(G/D)|<=Q*Γ:=(Ae _ _).trans_le (mul_le_mul ((Aw M m).trans_le hm) hGa (Aa _) hQ0)
 have T3:|(X-m)*(b-G*r)|<=S*(2*Γ/2^(53:Ἀ)+w):=(Ae _ _).trans_le (mul_le_mul hXm eb (Aa _) hS0)
 have T4:|(a-(X-m))*b|<=(S/2^(53:Ἀ)+w)*(2*Γ+w+2*Γ):=(Ae _ _).trans_le (mul_le_mul ea Bm (Aa _) (add_nonneg (div_nonneg hS0 hK0) hw0))
 have X0:|t-(p+Bb)|<=|t|/2^(52:Ἀ)+2*w:=by linear_combination 2*ht+(1/4503599627370496:ℝ)*e+(4503599627370495/4503599627370496:ℝ)*Aa (t-(p+Bb))
 rw[hid]
 linear_combination nEl_e1_h1 (t-(p+Bb)) (p-a*b) ((X-m)*(G/D)*(r*D-1)) ((M-m)*(G/D)) ((X-m)*(b-G*r)) ((a-(X-m))*b)+X0+hp+hab+T1+T2+T3+T4+22*AE hS hΓ0+(9007199254740993/9007199254740992:ℝ)*mul_le_of_le_one_left hw0 hw1+(45035996273704957/9007199254740992:ℝ)*Af hS0 hw0+(4503599627370495/2251799813685248:ℝ)*Af hΓ0 hw0+(18014398509481983/9007199254740992:ℝ)*hw0
theorem nXb(w:kF)(h:kB w≠255):|rv (dw w)|<2^((mJ (kB w))+23):=by
 obtain⟨-,e⟩:=cI w h
 rw[e,Ae,AW (by positivity:(0:ℝ)<2^kG w)]
 have hs:|(sv w:ℝ)|=(kM w:ℝ):=by rw[sv];split_ifs<;>simp
 rw[hs]
 by_cases z:kB w=0
 · have hk:kG w=0:=by simp[idxOf,z]
   have hm:kM w<2^(23:Ἀ):=by simp only[mantOf,z,ἁ];exact fracOf_toNat_lt w
   rw[hk,z];simp;exact_mod_cast hm
 · have hk:kG w=(mJ (kB w))-1:=by simp[idxOf,z]
   have e1:1<=(mJ (kB w)):=ϵ fun q=>z (Β.1 q)
   have hm:=lB w
   calc (kM w:ℝ)*2^kG w<2^(24:Ἀ)*2^kG w:=AI (by exact_mod_cast hm) (by positivity)
    _=2^((mJ (kB w))+23):=by rw[hk,<-AF];exact congrArg _ ((Τ 24 _).trans ((pA _ 1 23).symm.trans (congrArg (·+23) (L3 e1))))
theorem nDb(d:kF)(hd:0<(mJ (kB d))∧kB d≠255∧ZX d=false):(2:ℝ)^((mJ (kB d))+22)<=rv (dw d):=by
 obtain⟨-,e⟩:=cI d hd.2.1
 rw[e]
 have z:kB d≠0:=λq=>L6 hd.1 (congrArg mJ q)
 have hs:(sv d:ℝ)=(kM d:ℝ):=by rw[sv,hd.2.2,if_neg lN,M5]
 have hm:2^(23:Ἀ)<=kM d:=by rw[mantOf,if_neg z];exact V1 _ _
 have hk:kG d=(mJ (kB d))-1:=by rw[idxOf,if_neg z]
 rw[hs,hk]
 calc (2:ℝ)^((mJ (kB d))+22)=2^(23:Ἀ)*2^((mJ (kB d))-1):=by rw[<-AF,<-Nat.add_sub_assoc hd.1,Τ 23];rfl
  _<=(kM d:ℝ)*2^((mJ (kB d))-1):=AE (by exact_mod_cast hm) (pow_nonneg zero_le_two _)
theorem nEl_fin(E s c:ℤ)(R V S' Γ:ℝ)(hE:1047<=E)(hc:c=if s<=E+2 then 3 else s-E)
  (hV:V<2^(E-1022))(hS':S'<=2^(548:Ἀ))(hΓ:Γ<=2^(256:Ἀ))(hΓ0:0<=Γ)(hS0:0<=S')
  (hR:R<=12*2^(s-1081)+V/2^(52:Ἀ)+6*(S'+Γ+1)*(1/2^(1075:Ἀ))):
  R<2^(c-1)*2^(E-1075):=by
 have two:(0:ℝ)<2:=two_pos
 have hU:(0:ℝ)<2^(E-1075):=zpow_pos two _
 have e0:(2:ℝ)^(E-1022)=2^(53:Ἀ)*2^(E-1075):=by rw[← AA,← AK two.ne'];congr 1;ring
 have hsm:6*(S'+Γ+1)*(1/2^(1075:Ἀ))<=2^(E-1075)/4:=by
  have a:(2:ℝ)^(256:Ἀ)<=2^(548:Ἀ):=AU one_le_two (by norm_num)
  have b:(1:ℝ)<=2^(548:Ἀ):=one_le_pow₀ one_le_two
  have e:(2:ℝ)^(548:Ἀ)*(1/2^(1075:Ἀ))=2^(-527:ℤ):=by rw[← AA,← AA,one_div,← zpow_neg,← AK two.ne'];congr 1
  have h7:(2:ℝ)^(-527:ℤ)<=2^(E-1082):=AG one_le_two (by linarith only[hE])
  have e5:(2:ℝ)^(E-1075)=2^(7:Ἀ)*2^(E-1082):=by rw[← AA,← AK two.ne'];congr 1;ring
  linear_combination (6/2^(1075:Ἀ):ℝ)*hS'+(6/2^(1075:Ἀ):ℝ)*hΓ+(6/2^(1075:Ἀ):ℝ)*a+(6/2^(1075:Ἀ):ℝ)*b+18*e+18*h7-(1/4:ℝ)*e5+14*(zpow_pos two (E-1082)).le
 have hV2:V/2^(52:Ἀ)<2*2^(E-1075):=by linear_combination (1/2^(52:Ἀ):ℝ)*hV+(1/2^(52:Ἀ):ℝ)*e0
 split at hc
 · rename_i h
   subst hc
   have h4:(2:ℝ)^(s-1081)<=2^(E-1079):=AG one_le_two (by linarith only[h])
   have e3:(2:ℝ)^(E-1075)=2^(4:Ἀ)*2^(E-1079):=by rw[← AA,← AK two.ne'];congr 1;ring
   have e4:(2:ℝ)^((3:ℤ)-1)=4:=by norm_num
   linear_combination hR+hV2+hsm+12*h4-(3/4:ℝ)*e3-(2:ℝ)^(E-1075)*e4+hU
 · rename_i h
   subst hc
   have e1:(2:ℝ)^(s-1081)=2^(s-E-6)*2^(E-1075):=by rw[<-AK two.ne'];congr 1;ring
   have e2:(2:ℝ)^(s-E-1)=32*2^(s-E-6):=by rw[show (32:ℝ)=2^(5:ℤ) by norm_num,<-AK two.ne'];congr 1;ring
   have hZ:(1:ℝ)/8<=2^(s-E-6):=by rw[show (1:ℝ)/8=2^(-3:ℤ) by norm_num];exact AG one_le_two (by linarith only[not_le.1 h])
   linear_combination hR+hV2+hsm+20*AE hZ hU.le+(1/4:ℝ)*hU+12*e1-(2:ℝ)^(E-1075)*e2
lemma nfS(x y:Ἅ)(h:fv x)(j:fv y)(g:|rv x-rv y|<=(2:ℝ)^(971:Ἁ)):
  fv (x-y)/\|rv (x-y)-(rv x-rv y)|<=|rv x-rv y|/2^(53:Ἀ)+1/2^(1075:Ἀ):=by have⟨d,e⟩:=s5iS x y h j (g.trans (pm 971 AP));exact⟨d,by rw[e];exact jy _⟩
@[inline]def nfe(w:Ἃ):ἆ:=((w>>>23)&&&255).toUInt64
@[inline]def nC(s E:ἆ):ἆ:=if s<=E+2 then 3 else s-E
@[inline]def nElB(xw gw bw:Ἃ)(m ri:Ἅ):ἆ:=((Dense.F.dw xw-m)*(Dense.F.dw gw*ri)+Dense.F.dw bw).toBits
@[inline]def nElK(m:Ἅ)(L Dl ed:Ἀ):ἆ:=((max ((L:ℤ)+Dl-30) ((((m.toBits>>>52)&&&2047).toNat:ℤ)-1073))+1082-ed).toNat.toUInt64
@[inline]def nElOK(xw gw:Ἃ)(Kp Ke B:ἆ):Ἆ:=nOK B (nC (nfe gw+(if Kp<nfe xw+Ke then nfe xw+Ke else Kp)) ((B>>>52)&&&2047))
lemma nC_toNat_h1(E:ἆ)(hE:(lq E)<2048):(lq E)+(lq 2)<2^(64:Ἀ):=(X3 hE 2).trans (by decide)
lemma nC_toNat_h2(a b:Ἀ)(h:¬a<=b+2):b<=a:=(V1 _ 2).trans (Nat.le_of_not_le h)
theorem nC_toNat(s E:ἆ)(hs:(lq s)<2^(60:Ἀ))(hE:(lq E)<2048):(lq (nC s E))=if (lq s)<=(lq E)+2 then 3 else (lq s)-(lq E):=by
 unfold nC;split<;>rename_i h<;>rw[θ,η,γ (nC_toNat_h1 E hE)]at h<;>simp only[kA]at h
 · simp only[kA,if_pos h]
 · rw[if_neg h,χ _ _ (θ.2 (nC_toNat_h2 _ _ h))]
theorem nS_toNat_h1{p q:Ἀ}(hp:p<256)(hq:q<256+2^(40:Ἀ)):p+q<2^64:=U1 (E6 hp hq) (N2 rfl)
theorem nS_toNat_h2{x y:ἆ}(h:(lq x)+(lq y)<2^64):(lq (x+y))=(lq x)+(lq y):=(η x y).trans (γ h)
theorem nS_toNat(a b K Ke:ἆ)(ha:(lq a)<256)(hb:(lq b)<256)(hK:(lq K)<2^(40:Ἀ))(hKe:(lq Ke)<2^(40:Ἀ)):
  (lq (a+(if K<b+Ke then b+Ke else K)))=(lq a)+max (lq K) ((lq b)+(lq Ke)):=by
 have e:=nS_toNat_h2 (nS_toNat_h1 hb (Nat.lt_add_left _ hKe))
 by_cases h:K<b+Ke
 · rw[if_pos h,nS_toNat_h2 (e▸nS_toNat_h1 ha (E6 hb hKe)),e,Nat.max_eq_right (X4 (e▸h))]
 · rw[if_neg h,nS_toNat_h2 (nS_toNat_h1 ha (Nat.lt_add_left _ hK)),Nat.max_eq_left (W4 (e▸h))]
theorem nEl_spec_h1(X G Dd E L Dl:Ἀ)(A:Ἁ)(hX:X<256)(hG:G<256)(hD:Dd<256)(hE:E<2048)(hL:3<=L∧L<=14)(hDl:Dl<=254)(hA:A=max ((L:ℤ)+Dl-30) ((E:ℤ)-1073)):(A+1082-Dd).toNat<2^(64:Ἀ)∧(A+1082-Dd).toNat<2^(40:Ἀ)∧1054-Dd<2^(40:Ἀ)∧1054-Dd<2^(64:Ἀ)∧G+max (A+1082-Dd).toNat (X+(1054-Dd))<2^(60:Ἀ):=by
 have a:A<=974:=hA▸max_le (sub_le_iff_le_add.mpr ((Int.add_le_add (U4.mpr hL.2) (U4.mpr hDl)).trans (by decide))) (sub_le_iff_le_add.mpr ((U4.mpr (J2 hE)).trans (by decide)))
 have b:(A+1082-Dd).toNat<=2056:=Int.toNat_le.mpr ((Int.sub_le_self _ (W6 Dd)).trans ((T8 a 1082).trans (by decide)))
 have c:1054-Dd<=1054:=K4 _ _
 exact⟨b.trans_lt (by decide),b.trans_lt (by decide),c.trans_lt (by decide),c.trans_lt (by decide),(P6 hG (max_le b ((Y3 (X4 hX) c).trans (by decide)))).trans (by decide)⟩
theorem nEl_spec_h2(X G Dd Bw E T Sn Cn L Dl:Ἀ)(A μ:Ἁ)(hX:X<256)(hG:G<256)(hD:Dd<256)(hB:Bw<256)(hA:A=max ((L:ℤ)+Dl-30) ((E:ℤ)-1073))(hμ:μ=(Sn:ℤ)-G+Dd-1082)(hs:Sn=G+max (A+1082-Dd).toNat (X+(1054-Dd)))(hc:Cn=if Sn<=T+2 then 3 else Sn-T)(a1:1047<=T)(a2:T<=1298)(a3:Cn<=25):
  X+23<=547∧1<=Dd+22∧G+23<=278∧Bw+23<=548+1+(278+0+1)+1∧((Cn:ℤ)=if (Sn:ℤ)<=(T:ℤ)+2 then 3 else (Sn:ℤ)-T)∧(1047:ℤ)<=T∧((L:ℤ)+Dl-30<=μ∧(E:ℤ)-1022<=((547:Ἀ):ℤ)∧(((X+23:Ἀ)):ℤ)<=μ+51∧(E:ℤ)-1022<=μ+51∧(((G+23:Ἀ)):ℤ)=(G:ℤ)-Dd+1+(((Dd+22:Ἀ)):ℤ)∧μ+((G:ℤ)-Dd+1)=(Sn:ℤ)-1081∧(G:ℤ)-Dd+1<=((256:Ἀ):ℤ)):=by
 have k1:G+(A+1082-Dd).toNat<=Sn:=hs▸J1 (le_max_left _ _) G
 have k2:G+(X+(1054-Dd))<=Sn:=hs▸J1 (le_max_right _ _) G
 have u:(L:ℤ)+Dl-30<=A:=hA▸le_max_left _ _
 have v:(E:ℤ)-1073<=A:=hA▸le_max_right _ _
 have k3:A+1082-Dd<=((A+1082-Dd).toNat:ℤ):=Int.self_le_toNat _
 have k4:Dd+(1054-Dd)=1054:=BQ ((X4 hD).trans (by decide))
 clear hs hA
 generalize (A+1082-Dd).toNat=K at k1 k3
 generalize 1054-Dd=Q at k2 k4
 have wq:Sn<=T+25∧(Cn:ℤ)=if (Sn:ℤ)<=(T:ℤ)+2 then 3 else (Sn:ℤ)-T:=by
  by_cases h:Sn<=T+2
  · rw[if_pos h]at hc;exact⟨h.trans (J1 (by decide) T),by rw[if_pos (show (Sn:ℤ)<=(T:ℤ)+2 from U4.mpr h),hc];rfl⟩
  · rw[if_neg h]at hc;exact⟨Nat.sub_le_iff_le_add'.mp (hc▸a3),by rw[if_neg (show ¬(Sn:ℤ)<=(T:ℤ)+2 from λh'=>h (U4.mp h')),hc,pO ((V1 T 2).trans (X4 (O8 h)))]⟩
 obtain⟨w,q⟩:=wq
 have k1':(G:ℤ)+K<=Sn:=U4.mpr k1
 have k2':(G:ℤ)+(X+Q)<=Sn:=U4.mpr k2
 have k4':(Dd:ℤ)+Q=1054:=congrArg Int.ofNat k4
 have w':(Sn:ℤ)<=T+25:=U4.mpr w
 have a2':(T:ℤ)<=1298:=U4.mpr a2
 have hG':(G:ℤ)<=255:=U4.mpr (J2 hG)
 have hD':(Dd:ℤ)<=255:=U4.mpr (J2 hD)
 have Aμ:A<=μ:=(Int.le_sub_right_of_add_le ((Int.le_add_of_sub_right_le k3).trans (T8 (Int.le_sub_left_of_add_le k1') Dd))).trans_eq hμ.symm
 have μ4:μ<=496:=hμ▸Int.sub_le_sub_right (Int.add_le_add ((Int.sub_le_self _ (W6 G)).trans (w'.trans (T8 a2' 25))) hD') 1082
 have g3:(X:ℤ)+23<=μ+51:=by linear_combination k2'-k4'-hμ
 have g4:(E:ℤ)-1022<=μ+51:=Int.sub_right_le_of_le_add ((Int.le_add_of_sub_right_le (v.trans Aμ)).trans_eq (Int.add_assoc μ 51 1022).symm)
 have g2:(E:ℤ)-1022<=496+51:=g4.trans (T8 μ4 51)
 have g5:(G:ℤ)+23=G-Dd+1+((Dd:ℤ)+22):=by ring
 have g6:μ+((G:ℤ)-Dd+1)=Sn-1081:=by linear_combination hμ
 have g7:(G:ℤ)-Dd+1<=255+1:=T8 ((Int.sub_le_self _ (W6 _)).trans hG') 1
 exact⟨(L1 (X4 hX) 23).trans (by decide),(by decide:1<=22).trans (K2 22 Dd),(L1 (J2 hG) 23).trans (by decide),(L1 (X4 hB) 23).trans (by decide),q,U4.mpr a1,u.trans Aμ,g2,g3,g4,g5,g6,g7⟩
theorem nEl_spec_h3(a b:Ἁ)(h:a<=b):(2:ℝ)^a<=2^b:=AG one_le_two h
theorem nEl_spec_h4(a b:Ἀ)(h:a<=b):(2:ℝ)^a<=2^b:=AU one_le_two h
theorem nEl_spec(xw gw bw d:Ἃ)(m ri:Ἅ)(M:ℚ)(L Dl:Ἀ)
  (hx:kB xw≠255)(hg:kB gw≠255)(hb:kB bw≠255)
  (hd:0<(mJ (kB d))∧kB d≠255∧ZX d=false)
  (hri:fv ri∧|rv ri*rv (dw d)-1|<=1/2^(50:Ἀ))
  (hm:fv m∧|rv m-(M:ℝ)*2^(149:Ἀ)|<=2^((L:ℤ)+Dl-30))
  (hxD:(mJ (kB xw))<=Dl)(hL:3<=L∧L<=14)(hDl:Dl<=254)
  (hok:nElOK xw gw (nElK m L Dl (mJ (kB d))) (1054-(mJ (kB d))).toUInt64 (nElB xw gw bw m ri)=ἃ):
  nW (nElB xw gw bw m ri)=ZT ((ZA xw-M)*ZA gw/ZA d+ZA bw):=by
 obtain⟨fX,vX⟩:=cI xw hx
 obtain⟨fG,vG⟩:=cI gw hg
 obtain⟨fB,vB⟩:=cI bw hb
 obtain⟨fD,vD⟩:=cI d hd.2.1
 have bX:=nXb xw hx;have bG:=nXb gw hg;have bB:=nXb bw hb;have bD:=nDb d hd
 have bm:=fK m hm.1
 have ex:=mO xw;have eg:=mO gw;have ed:=mO d;have eb2:=mO bw
 unfold nElOK nElB at hok
 generalize ht:(dw xw-m)*(dw gw*ri)+dw bw=t at hok
 have hEt:(t.toBits>>>52)&&&2047=eF t.toBits:=rfl
 rw[hEt]at hok
 have e1:(lq (nfe xw))=(mJ (kB xw)):=ϩ _
 have e2:(lq (nfe gw))=(mJ (kB gw)):=ϩ _
 obtain⟨A,hA⟩:∃A:ℤ,A=max ((L:ℤ)+Dl-30) (((lq (eF m.toBits)):ℤ)-1073):=⟨_,rfl⟩
 have hEm:(lq (eF m.toBits))<2048:=by simp only[eF,Ε,kA];exact C0 ϝ (by decide)
 obtain⟨p1,p2,p3,p4,p5⟩:=nEl_spec_h1 _ _ _ _ _ _ _ ex eg ed hEm hL hDl hA
 have e3:(lq ((1054-(mJ (kB d))).toUInt64))=1054-(mJ (kB d)):=by show (lh _).toNat=_;rw[O6];exact γ p4
 have e4:(lq (nElK m L Dl (mJ (kB d))))=(A+1082-(mJ (kB d))).toNat:=by
  show (lh (max ((L:ℤ)+Dl-30) (((lq (eF m.toBits)):ℤ)-1073)+1082-(mJ (kB d))).toNat).toNat=_
  rw[O6,<-hA];exact γ p1
 have hs:=nS_toNat (nfe gw) (nfe xw) (nElK m L Dl (mJ (kB d))) ((1054-(mJ (kB d))).toUInt64) (by rw[e2];exact eg) (by rw[e1];exact ex) (by rw[e4];exact p2) (by rw[e3];exact p3)
 rw[e1,e2,e3,e4]at hs
 generalize hS:nfe gw+(if nElK m L Dl (mJ (kB d))<nfe xw+(1054-(mJ (kB d))).toUInt64 then nfe xw+(1054-(mJ (kB d))).toUInt64 else nElK m L Dl (mJ (kB d)))=S at hok hs
 have hEt2:(lq (eF t.toBits))<2048:=by simp only[eF,Ε,kA];exact C0 ϝ (by decide)
 have hc:=nC_toNat S (eF t.toBits) (by rw[hs];exact p5) hEt2
 generalize nC S (eF t.toBits)=c at hok hc
 have hok2:=hok
 simp only[nOK,Η,AJ,θ,μ,kA]at hok2
 obtain⟨⟨⟨⟨a1,a2⟩,a3⟩,-⟩,-⟩:=hok2
 change 1047<=(lq (eF t.toBits)) at a1;change (lq (eF t.toBits))<=1298 at a2
 obtain⟨μ,hμ⟩:∃μ:ℤ,μ=((lq S):ℤ)-(mJ (kB gw))+(mJ (kB d))-1082:=⟨_,rfl⟩
 obtain⟨q2,q4,q5,q6,q11,q13,q1,q3,q7,q8,q9,q10,q12⟩:=nEl_spec_h2 _ _ _ _ _ _ _ _ _ _ _ _ ex eg ed eb2 hA hμ hs hc a1 a2 a3
 have two:(0:ℝ)<2:=two_pos
 have hD0:(0:ℝ)<rv (dw d):=AN (pow_pos two _) bD
 have aX:|rv (dw xw)|<=2^(547:Ἀ):=bX.le.trans (nEl_spec_h4 _ _ q2)
 have am:|rv m|<=2^(547:Ἀ):=bm.le.trans ((nEl_spec_h3 _ _ q3).trans_eq (AA _ _))
 have aS:|rv (dw xw)|+|rv m|<=2^(548:Ἀ):=(add_le_add aX am).trans_eq ((mul_two _).symm.trans (A4 _ _).symm)
 have aXm:|rv (dw xw)-rv m|<=2^(548:Ἀ):=(abs_sub _ _).trans aS
 obtain⟨fa,ea⟩:=nfS (dw xw) m fX hm.1 (z9 _ _ aXm (by decide))
 have aa:=eB _ _ _ ea aXm
 have ar:|rv ri|<=1:=by
  have h1:|rv ri*rv (dw d)|<=2:=((sub_le_iff_le_add.mp (AY (rv ri*rv (dw d)) 1)).trans (add_le_add (hri.2.trans (div_le_one_of_le₀ ((pow_zero (2:ℝ)).symm.trans_le (nEl_spec_h4 0 _ (ϟ _))) (pow_pos two _).le)) abs_one.le)).trans_eq one_add_one_eq_two
  have h2:(2:ℝ)<=rv (dw d):=(pow_one (2:ℝ)).ge.trans ((nEl_spec_h4 _ _ q4).trans bD)
  exact le_of_mul_le_mul_right (((Ae _ _).symm.trans_le h1).trans ((h2.trans (le_abs_self _)).trans_eq (one_mul _).symm)) (hD0.trans_le (le_abs_self _))
 have aG:|rv (dw gw)|<=2^(278:Ἀ):=bG.le.trans (nEl_spec_h4 _ _ q5)
 have aGr:|rv (dw gw)*rv ri|<=2^(278+0):=pB _ _ _ _ aG (ar.trans_eq (pow_zero 2).symm)
 obtain⟨fb,eb⟩:=fM (dw gw) ri fG hri.1 (z9 _ _ aGr (by decide))
 have ab:=eB _ _ _ eb aGr
 have aab:=pB _ _ _ _ aa ab
 obtain⟨fp,ep⟩:=fM _ _ fa fb (z9 _ _ aab (by decide))
 have ap:=eB _ _ _ ep aab
 have aB:=bB.le.trans (nEl_spec_h4 _ _ q6)
 have apB:=sB _ _ _ ap aB
 obtain⟨ft,et⟩:=fA _ _ fp fB (z9 _ _ apB (by decide))
 rw[ht]at ft et
 have hQ:|rv m-(M:ℝ)*2^(149:Ἀ)|<=(2:ℝ)^μ:=hm.2.trans (nEl_spec_h3 _ _ q1)
 have cX:|rv (dw xw)|<=(2:ℝ)^(μ+51):=bX.le.trans ((AA _ _).symm.trans_le (nEl_spec_h3 _ _ q7))
 have cm:|rv m|<=(2:ℝ)^(μ+51):=bm.le.trans (nEl_spec_h3 _ _ q8)
 have hS:(|rv (dw xw)|+|rv m|)/2^(53:Ἀ)<=(2:ℝ)^μ/2:=by
  have e:(2:ℝ)^(μ+51)=2^μ*2^(51:Ἀ):=(AK two.ne' μ 51).trans (congrArg (2^μ*·) (AA 2 51))
  rw[e]at cX cm;calc (|rv (dw xw)|+|rv m|)/2^(53:Ἀ)<=(2^μ*2^(51:Ἀ)+2^μ*2^(51:Ἀ))/2^(53:Ἀ):=div_le_div_of_nonneg_right (add_le_add cX cm) (pow_pos two _).le
   _=(2:ℝ)^μ/2:=by ring
 have hG:|rv (dw gw)|/rv (dw d)<=(2:ℝ)^(((mJ (kB gw)):ℤ)-(mJ (kB d))+1):=by
  rw[div_le_iff₀ hD0];refine bG.le.trans ?_;refine AZ ?_ (mul_le_mul_of_nonneg_left bD (zpow_pos two _).le)
  rw[<-AA,<-AA,<-AK two.ne',q9]
 have E1:=nEl_e1 _ _ _ _ _ _ _ _ _ _ _ _ _ (1/2^(1075:Ἀ)) hD0 hri.2 hQ hS hG (div_nonneg zero_le_one (pow_pos two _).le) (div_le_one_of_le₀ ((pow_zero (2:ℝ)).symm.trans_le (nEl_spec_h4 0 _ (ϟ _))) (pow_pos two _).le) ea eb ep et
 have eZ:∀w:kF,kB w≠255->((ZA w:ℚ):ℝ)=rv (dw w)/2^(149:Ἀ):=λw h=>by rw[(cI w h).2,Y1,J7,S8,Rat.cast_intCast,G7,<-AA,mul_div_assoc,(AA (2:ℝ) 149).symm,<-Ac two.ne'];rfl
 have hT:((((ZA xw-M)*ZA gw/ZA d+ZA bw:ℚ)):ℝ)*2^(149:Ἀ)=(rv (dw xw)-(M:ℝ)*2^(149:Ἀ))*(rv (dw gw)/rv (dw d))+rv (dw bw):=by
  rw[Rat.cast_add,Rat.cast_div,J7,Rat.cast_sub,eZ xw hx,eZ gw hg,eZ d hd.2.1,eZ bw hb,mul_div_assoc,div_div_div_cancel_right₀ (pow_pos two 149).ne',add_mul,div_mul_cancel₀ (rv (dw bw)) (pow_pos two 149).ne',mul_right_comm,sub_mul,div_mul_cancel₀ (rv (dw xw)) (pow_pos two 149).ne']
 unfold nElB;rw[ht]
 refine nOK_spec _ t c ft hok ?_
 rw[hT,Aw]
 have hQΓ:(2:ℝ)^μ*2^(((mJ (kB gw)):ℤ)-(mJ (kB d))+1)=2^(((lq S):ℤ)-1081):=by rw[<-AK two.ne',q10]
 rw[hQΓ]at E1
 refine nEl_fin _ _ _ _ |rv t| _ _ q13 q11 (fK t ft) aS ((nEl_spec_h3 _ _ q12).trans_eq (AA _ _)) (zpow_pos two _).le (add_nonneg (Aa _) (Aa _)) E1
end
lemma nSg(x:Model kC)(h:Model.isFinite x=ἃ)(p:0<=Model.toReal x):Model.isZero x=ἃ\/Model.signBit x=ἄ:=by
 by_cases z:Model.isZero x=ἃ
 · exact pZ z
 right;obtain⟨d,hd⟩:=Model.exists_toDyadic?_of_isFinite h
 have hm:=Model.significand_ne_zero_of_toDyadic?_some_of_isZero_eq_false hd (F8.2 z)
 rw[<-Model.sign_eq_signBit_of_toDyadic?_some hd]
 refine F8.2 fun hs=>?_
 have hx:Model.toReal x=d.toReal:=by rw[BR,hd]
 have q:=Int.cast_nonneg_iff.1 (nonneg_of_mul_nonneg_left (hx▸p:0<=d.toReal) (zpow_pos two_pos _))
 rw[lr,hs,if_pos rfl]at q;exact hm (Int.ofNat.inj (le_antisymm (Int.neg_nonneg.1 q) (Int.ofNat_nonneg _)))
lemma fS(v:Ἅ)(h:Dense.F.fv v)(p:0<=Dense.F.rv v):Dense.F.fv (W7 v)/\Dense.F.rv (W7 v)=ns kC (O2 (Dense.F.rv v)):=by
 have e:=FloatLib.Floats.ExecFloat.Binary.toModel_ofFloat_sqrt v
 simp only[m5]at e
 have hd:=nSg _ h p
 have q:=Model.toReal_sqrt_eq_roundAt (kI v.toModel) (by decide) h hd
 have f:=Model.isFinite_sqrt_of_isFinite (kI v.toModel) (by decide) h hd
 have c:Model.canonicalizeModel (Model.sqrt (kI v.toModel))=Model.sqrt (kI v.toModel):=by
  by_cases z:Model.isZero (kI v.toModel)=ἃ
  · have hs:Model.sqrt (kI v.toModel)=kI v.toModel:=by
     rw[Model.Proof.sqrt_eq_spec];simp[Model.Spec.sqrt,Model.chooseNaN1_none_of_isFinite _ h,Model.isInf_eq_false_of_isFinite_eq_true _ h,z]
    rw[hs];exact Model.canonicalizeModel_eq_self _ (Model.isModelCanonical_ofFloatModel _)
  · have s:=hd.resolve_left z
    rw[Model.Proof.sqrt_eq_spec,Model.Spec.sqrt_eq_model (by decide) _ h s,Model.canonicalizeModel_ofModel]
 refine⟨?_,?_⟩
 · unfold Dense.F.fv;rw[e];exact (congrArg Model.isFinite c).trans f
 · unfold Dense.F.rv;rw[e];exact (congrArg Model.toReal c).trans q
lemma nQs_t(w:kF)(hw:kB w≠255):Dense.F.fv (Dense.F.dw w*Dense.F.dw w)/\Dense.F.rv (Dense.F.dw w*Dense.F.dw w)=((sv w:ℝ)*2^kG w)^(2:Ἀ):=by
 obtain⟨f,h⟩:=Dense.F.cI w hw
 have m:(sv w*sv w).natAbs<2^(48:Ἀ):=by rw[Ϩ];exact (ϕ (Dense.F.bg w) (Dense.F.bg w)).trans_eq (m7 2 24 24).symm
 have r:=Dense.F.aE _ _ f f (sv w*sv w) ((2*kG w:ℕ):ℤ) (by rw[h,AA,Nat.two_mul,AF,Int.cast_mul];exact mul_mul_mul_comm _ _ _ _) (m.trans (by decide)) (Int.le_trans (by decide) (W6 _)) (Dense.F.bL 48 (2*kG w) 556 (J1 (U3 2 (cP w)) 48) (by decide) _ m)
 exact⟨r.1,by rw[r.2,AA,Nat.two_mul,AF,Int.cast_mul,sq];exact (mul_mul_mul_comm _ _ _ _).symm⟩
lemma nQs_e0(a r A t u w k:ℝ)(hA:0<=A)(ht:0<=t)(hu:0<=u)(hw:0<=w)(k0:0<=k)(hk:2*k*u<=1)
  (H1:|a-A|<=2*k*u*A+2*k*w)(H2:|r-(a+t)|<=u*|a+t|+w):
  |r-(A+t)|<=2*(k+1)*u*(A+t)+2*(k+1)*w:=by
 obtain⟨h1,h2⟩:=A3.1 H1
 have g:|a+t|<=(A+t)+2*k*u*A+2*k*w:=A3.2 ⟨by linear_combination h1+2*hA+2*ht,by linear_combination h2⟩
 have T:=abs_sub_le r (a+t) (A+t)
 rw[add_sub_add_right_eq_sub]at T
 refine T.trans ?_
 linear_combination H2+H1+mul_le_mul_of_nonneg_left g hu+AE hk (Af hu hA)+AE hk hw+Af hu ht+2*Af (Af k0 hu) ht
lemma nQs_e(a r A t:ℝ)(k:ℕ)(hA:0<=A)(ht:0<=t)(hk:(k:ℝ)<=2^(20:Ἀ))
  (H1:|a-A|<=2*k*(1/2^(53:Ἀ))*A+2*k*(1/2^(1075:Ἀ)))(H2:|r-(a+t)|<=|a+t|/2^(53:Ἀ)+1/2^(1075:Ἀ)):
  |r-(A+t)|<=2*((k:ℝ)+1)*(1/2^(53:Ἀ))*(A+t)+2*((k:ℝ)+1)*(1/2^(1075:Ἀ)):=
 nQs_e0 a r A t (1/2^(53:Ἀ)) (1/2^(1075:Ἀ)) k hA ht (by positivity) (by positivity) (M3 k)
  ((AE (mul_le_mul_of_nonneg_left hk zero_le_two) (by positivity)).trans (by norm_num)) H1 (H2.trans_eq (congrArg (·+_) (one_div_mul_eq_div _ _).symm))
lemma nP(a b:ℕ)(h:a<=b):(2:ℝ)^a*(1/2^b)<=1:=by rw[mul_one_div,div_le_one (pow_pos two_pos b)];exact AU one_le_two h
lemma nN(b:ℕ):(0:ℝ)<=1/2^b:=one_div_nonneg.2 (pow_pos two_pos b).le
lemma nP2(a b c:ℕ)(h:a+c<=b):(2:ℝ)^a*(1/2^b)<=1/2^c:=by rw[mul_one_div,div_le_div_iff₀ (pow_pos two_pos b) (pow_pos two_pos c),one_mul,<-AF];exact AU one_le_two h
theorem nSm_h1(a A X:ℝ)(s:X<=1*A+1)(h1:-X<=a-A)(h2:a-A<=X):a<=2*A+1/\-1<=a:=⟨((sub_le_iff_le_add'.1 h2).trans (add_le_add AP s)).trans_eq (by ring),(by ring:(-1:ℝ)=-(1*A+1)+A).trans_le ((add_le_add (neg_le_neg s) AP).trans (le_sub_iff_add_le.1 h1))⟩
lemma nSm(a A k u w K:ℝ)(hA:0<=A)(k0:0<=k)(kk:k<=K)(u0:0<=u)(w0:0<=w)(cu:2*K*u<=1)(cw:2*K*w<=1)(h1:-(2*k*u*A+2*k*w)<=a-A)(h2:a-A<=2*k*u*A+2*k*w):a<=2*A+1/\-1<=a:=by
 have q:=mul_le_mul_of_nonneg_left kk zero_le_two;exact nSm_h1 _ _ _ (add_le_add (AE ((AE q u0).trans cu) hA) ((AE q w0).trans cw)) h1 h2
lemma nFw0(v Ts n q P Ep W u w D:ℝ)(Ts0:0<=Ts)(n0:0<=n)(nb:n<=2^(12:Ἀ))(u0:0<=u)(w0:0<=w)(q0:0<=q)(qb:q<=2^(52:Ἀ))(cu:2*2^(12:Ἀ)*u<=1)(cw:2^(68:Ἀ)*w<=1)(D2:2<=D)
  (h1:-(2*n*u*Ts+2*n*w)<=v-Ts)(pe:|P-v*q|<=|v*q|/D+w)(hEp:1<=Ep)(we:|W-(P+Ep)|<=|P+Ep|/D+w):0<=W:=by
 have D1:1<=D:=one_le_two.trans D2
 have m1:=AE (AE nb u0) Ts0
 have c1:=AE cu Ts0
 have m2:=AE nb w0
 have vl:-(2*2^(12:Ἀ)*w)<=v:=by linear_combination h1+2*m1+c1+2*m2
 have a1:=AE vl q0
 have a2:=mul_le_mul_of_nonneg_left qb w0
 have s8:-(v*q+1/4)<=v*q:=by linear_combination 2*a1+16384*a2+cw/4
 have w8:w<=1/8:=by linear_combination cw/8+36893488147419103231*w0
 obtain⟨p1,_⟩:=A3.1 pe
 have ab:|v*q|<=v*q+1/4:=A3.2 ⟨s8,by linear_combination⟩
 have Pl:-(1/2)<=P:=by linear_combination p1+div_le_self (Aa (v*q)) D1+ab+w8
 obtain⟨r1,_⟩:=A3.1 we
 have Y0:0<P+Ep:=by linear_combination Pl+hEp
 linear_combination r1+div_le_div_of_nonneg_left (Aa (P+Ep)) two_pos D2+(1/2)*AW Y0+Pl/2+hEp/2+w8
lemma nBd(a A T Rs:ℝ)(k:ℕ)(hA:0<=A)(tn:0<=T)(rest:0<=Rs)(kk:(k:ℝ)<=2^(20:Ἀ))(h1:-(2*k*(1/2^(53:Ἀ))*A+2*k*(1/2^(1075:Ἀ)))<=a-A)(h2:a-A<=2*k*(1/2^(53:Ἀ))*A+2*k*(1/2^(1075:Ἀ)))(hb:A+(T+Rs)<=2^(960:Ἀ)):|a+T|<=(2:ℝ)^(971:ℤ):=by
 have cw:=nP 21 1075 (by decide);have cu:=nP 21 53 (by decide)
 rw[pow_succ']at cw cu
 obtain⟨v1,v2⟩:=nSm _ _ _ _ _ _ hA k.cast_nonneg kk (nN _) (nN _) cu cw h1 h2
 have c8:(1:ℝ)<=2^960:=one_le_pow₀ one_le_two
 have c1:(2:ℝ)^960*2^11=2^971:=(AF _ _ _).symm
 rw[zpow_ofNat,A3]
 generalize (2:ℝ)^971=P at c1 ⊢;generalize (2:ℝ)^960=Q at c1 c8 hb
 have s:=(add_le_add AP (le_add_of_nonneg_right rest)).trans hb
 exact⟨(show -P<=-1 by linear_combination c1+2048*c8).trans (le_add_of_le_of_nonneg v2 tn),(add_le_add (v1.trans (add_le_add (mul_le_mul_of_nonneg_left ((le_add_of_nonneg_right tn).trans s) zero_le_two) c8)) ((le_add_of_nonneg_left hA).trans s)).trans (by linear_combination c1+2044*c8)⟩
lemma nQs_spec(x:(Ἂ Word))(e:Ἄ)(hX:∀j,j<(J0 e)->kB x[j]!≠255):∀m (i:Ἄ)(v:Ἅ)(A:ℝ)(k:ℕ),(J0 e)-(J0 i)=m->(J0 i)<=(J0 e)->Dense.F.fv v->0<=A->
  |Dense.F.rv v-A|<=2*k*(1/2^(53:Ἀ))*A+2*k*(1/2^(1075:Ἀ))->k+m<=2^(20:Ἀ)->
  A+∑j∈B5 (J0 i) (J0 e),((sv x[j]!:ℝ)*2^kG x[j]!)^(2:Ἀ)<=2^(960:Ἀ)->
  Dense.F.fv (nQs x i e v)/\|Dense.F.rv (nQs x i e v)-(A+∑j∈B5 (J0 i) (J0 e),((sv x[j]!:ℝ)*2^kG x[j]!)^(2:Ἀ))|<=
   2*((k+m:ℕ):ℝ)*(1/2^(53:Ἀ))*(A+∑j∈B5 (J0 i) (J0 e),((sv x[j]!:ℝ)*2^kG x[j]!)^(2:Ἀ))+2*((k+m:ℕ):ℝ)*(1/2^(1075:Ἀ)):=by
 intro m;induction m with
 |zero=>
  intro i v A k hm hie hf hA he hk hb
  have q:(J0 i)=(J0 e):=C4 hie (K3.1 hm)
  rw[nQs,dif_neg (by rw[ε,q];exact B4 _),q,Finset.Ico_self,Finset.sum_empty,Av];exact⟨hf,he⟩
 |succ m ih=>
  intro i v A k hm hie hf hA he hk hb
  have hl:(J0 i)<(J0 e):=N1 hm
  rw[nQs,dif_pos (ε.2 hl)];dsimp only
  rw[f6 x i]
  obtain⟨tf,tr⟩:=nQs_t x[(J0 i)]! (hX _ hl)
  have hs:∑j∈B5 (J0 i) (J0 e),((sv x[j]!:ℝ)*2^kG x[j]!)^(2:Ἀ)=((sv x[(J0 i)]!:ℝ)*2^kG x[(J0 i)]!)^(2:Ἀ)+∑j∈B5 ((J0 i)+1) (J0 e),((sv x[j]!:ℝ)*2^kG x[j]!)^(2:Ἀ):=Finset.sum_eq_sum_Ico_succ_bot hl _
  have i1:(J0 (i+1))=(J0 i)+1:=Dense.ac (ε.2 hl)
  have tn:0<=((sv x[(J0 i)]!:ℝ)*2^kG x[(J0 i)]!)^(2:Ἀ):=sq_nonneg _
  have rest:0<=∑j∈B5 ((J0 i)+1) (J0 e),((sv x[j]!:ℝ)*2^kG x[j]!)^(2:Ἀ):=D8 (fun _ _=>sq_nonneg _)
  obtain⟨h1,h2⟩:=A3.1 he
  have kk:(k:ℝ)<=2^(20:Ἀ):=by exact_mod_cast (show k<=2^(20:Ἀ) from (V1 _ _).trans hk)
  have bd:|Dense.F.rv v+Dense.F.rv (Dense.F.dw x[(J0 i)]!*Dense.F.dw x[(J0 i)]!)|<=(2:ℝ)^(971:ℤ):=by rw[tr];rw[hs]at hb;exact nBd _ A _ _ k hA tn rest kk h1 h2 hb
  obtain⟨af,ae⟩:=NF.fA _ _ hf tf bd
  rw[tr]at ae
  have step:=nQs_e _ _ A _ k hA tn kk he ae
  have:=ih (i+1) _ (A+((sv x[(J0 i)]!:ℝ)*2^kG x[(J0 i)]!)^(2:Ἀ)) (k+1) (by rw[i1,<-F7,hm,BK]) (by rw[i1];exact hl) af (add_nonneg hA tn) (by rw[X7];exact step) ((lO k 1 m).trans_le hk) (by rw[i1,add_assoc,<-hs];exact hb)
  rw[i1,show k+1+m=k+(m+1) from lO k 1 m,add_assoc,<-hs]at this;exact this
lemma nD_w(Wr Z e:ℝ)(hZ:0<Z)(hW:0<=Wr)(h:|Wr-Z|<=e):|O2 Wr-O2 Z|<=e/O2 Z:=by
 have sZ:0<O2 Z:=Real.sqrt_pos.2 hZ
 have sW:0<=O2 Wr:=Real.sqrt_nonneg _
 have k:(O2 Wr-O2 Z)*(O2 Wr+O2 Z)=Wr-Z:=by linear_combination Real.sq_sqrt hW-Real.sq_sqrt hZ.le
 have q:|O2 Wr-O2 Z| *(O2 Wr+O2 Z)=|Wr-Z|:=by rw[<-k,Ae,AW (add_pos_of_nonneg_of_pos sW sZ)]
 rw[le_div_iff₀ sZ];exact ((mul_le_mul_of_nonneg_left (le_add_of_nonneg_left sW) (Aa _)).trans_eq q).trans h
lemma nD_chainK(Ts Ep N q v Pr Wr Dr u w K c:ℝ)(hu0:0<u)(hw0:0<=w)(hNu:(c+8)*u<=1/2^(30:Ἀ))(hwu:2^(60:Ἀ)*w<=u)
  (hN:1<=N)(hTs:0<=Ts)(hEp:1<=Ep)(hq1:N*q<=2^(52:Ἀ))(hq2:2^(52:Ἀ)-N<N*q)(hq0:0<=q)(hu2:2^(52:Ἀ)*u*2=1)(hK0:0<=K)(hK:K<=16*N)(hc:c=K+2*N)
  (h1:|v-Ts|<=K*u*Ts+K*w)(h2:|Pr-v*q|<=u*|v*q|+w)(h3:|Wr-(Pr+Ep)|<=u*|Pr+Ep|+w)(hW:0<=Wr)
  (h4:|Dr-O2 Wr|<=u*O2 Wr+w):
  |Dr-O2 (Ts*2^(52:Ἀ)/N+Ep)|<=(c+7)*u*O2 (Ts*2^(52:Ἀ)/N+Ep):=by
 have N0:0<N:=zero_lt_one.trans_le hN
 obtain⟨F,hF⟩:∃F,F=Ts*2^(52:Ἀ)/N:=⟨_,rfl⟩
 obtain⟨Z,hZ⟩:∃Z,Z=F+Ep:=⟨_,rfl⟩
 rw[<-hF,<-hZ]
 have qF:q<=2^(52:Ἀ)/N:=by rw[le_div_iff₀ N0,mul_comm];exact hq1
 have qF2:2^(52:Ἀ)/N-1<q:=by rw[div_sub_one N0.ne',div_lt_iff₀ N0,mul_comm];exact hq2
 have Tq1:Ts*q<=F:=by rw[hF,mul_div_assoc];exact mul_le_mul_of_nonneg_left qF hTs
 have Tq2:F-Ts<=Ts*q:=by have:=mul_le_mul_of_nonneg_left qF2.le hTs;rw[hF,mul_div_assoc];rw[mul_sub,mul_one]at this;exact this
 have F0:0<=F:=(Af hTs hq0).trans Tq1
 have Z1:1<=Z:=(le_add_of_nonneg_of_le F0 hEp).trans_eq hZ.symm
 have Y1:1<=O2 Z:=by rw[show (1:ℝ)=O2 1 from Real.sqrt_one.symm];exact Real.sqrt_le_sqrt Z1
 have FN:F*N=Ts*2^(52:Ἀ):=by rw[hF];exact div_mul_cancel₀ _ N0.ne'
 have Ku0:0<=K*u:=Af hK0 hu0.le
 have cu1:c*u+8*u<=1:=by linear_combination hNu
 have b1:|v*q-Ts*q|<=(K*u*Ts+K*w)*q:=by rw[<-sub_mul,Ae,Ad hq0];exact AE h1 hq0
 have b2:=mul_le_mul_of_nonneg_left Tq1 Ku0
 have a1:=mul_le_mul_of_nonneg_left hq1 (Af (pow_nonneg zero_le_two 4) hw0)
 have a2:=AE hK (Af hw0 hq0)
 have tF:|Ts*q-F|<=Ts:=abs_sub_le_iff.2 ⟨(sub_nonpos.2 Tq1).trans hTs,sub_le_comm.1 Tq2⟩
 have bvq:|v*q-F|<=c*u*F+2^(56:Ἀ)*w:=by linear_combination (abs_sub_le (v*q) (Ts*q) F)+b1+tF+b2+a1+a2+(-2*u)*FN-Ts*hu2-(u*F)*hc
 have cu0:0<=c*u:=Af ((add_nonneg hK0 (Af zero_le_two N0.le)).trans_eq hc.symm) hu0.le
 have uF0:0<=u*F:=Af hu0.le F0
 have g1:(c*u+8*u)*F<=F:=(AE cu1 F0).trans_eq (one_mul F)
 have vq:|v*q|<=2*F+2^(56:Ἀ)*w:=by linear_combination AY (v*q) F+bvq+g1+Ad F0+8*uF0
 have c3:u*|v*q|<=u*(2*F+2^(56:Ἀ)*w):=mul_le_mul_of_nonneg_left vq hu0.le
 have cP:|Pr+Ep-Z|<=(c*u+2*u)*F+2^(57:Ἀ)*w:=by rw[hZ,add_sub_add_right_eq_sub];linear_combination (abs_sub_le Pr (v*q) F)+h2+c3+bvq+(8*w)*hu2+72057594037927927*hw0
 have Z0:0<=Z:=zero_le_one.trans Z1
 have Ep0:0<=Ep:=zero_le_one.trans hEp
 have d3:|Pr+Ep|<=2*Z+2^(57:Ἀ)*w:=by linear_combination AY (Pr+Ep) Z+cP+Ad Z0+g1+6*uF0-hZ+Ep0
 have d4:u*|Pr+Ep|<=u*(2*Z+2^(57:Ἀ)*w):=mul_le_mul_of_nonneg_left d3 hu0.le
 have g4:0<=(c*u+2*u)*Ep:=Af (add_nonneg cu0 (Af zero_le_two hu0.le)) Ep0
 have dW:|Wr-Z|<=(c*u+4*u)*Z+2^(58:Ἀ)*w:=by linear_combination (abs_sub_le Wr (Pr+Ep) Z)+h3+d4+cP+g4+(16*w)*hu2-(c*u+2*u)*hZ+144115188075855855*hw0
 have e1:=nD_w Wr Z _ (zero_lt_one.trans_le Z1) hW dW
 have sZ:0<O2 Z:=zero_lt_one.trans_le Y1
 have sq:O2 Z*O2 Z=Z:=Real.mul_self_sqrt Z0
 have g3:w<=w*O2 Z:=le_mul_of_one_le_right hw0 Y1
 have e2:((c*u+4*u)*Z+2^(58:Ἀ)*w)/O2 Z<=(c*u+4*u)*O2 Z+2^(58:Ἀ)*w:=by
  rw[div_le_iff₀ sZ];linear_combination 288230376151711744*g3-(c*u+4*u)*sq
 have e3:|O2 Wr-O2 Z|<=(c*u+4*u)*O2 Z+2^(58:Ἀ)*w:=e1.trans e2
 have g5:(c*u+8*u)*O2 Z<=O2 Z:=(AE cu1 sZ.le).trans_eq (one_mul _)
 have sW:O2 Wr<=2*O2 Z+2^(58:Ἀ)*w:=by linear_combination le_abs_self (O2 Wr-O2 Z)+e3+g5+4*Af hu0.le sZ.le
 have f5:u*O2 Wr<=u*(2*O2 Z+2^(58:Ἀ)*w):=mul_le_mul_of_nonneg_left sW hu0.le
 have g6:u<=u*O2 Z:=le_mul_of_one_le_right hu0.le Y1
 have g7:2^(60:Ἀ)*w<=u*O2 Z:=hwu.trans g6
 linear_combination (abs_sub_le Dr (O2 Wr) (O2 Z))+h4+e3+f5+g7+(32*w)*hu2+864691128455135199*hw0
lemma nD_chain(Ts Ep N q v Pr Wr Dr u w:ℝ)(hu0:0<u)(hw0:0<=w)(hNu:(4*N+8)*u<=1/2^(30:Ἀ))(hwu:2^(60:Ἀ)*w<=u)
  (hN:1<=N)(hTs:0<=Ts)(hEp:1<=Ep)(hq1:N*q<=2^(52:Ἀ))(hq2:2^(52:Ἀ)-N<N*q)(hq0:0<=q)(hu2:2^(52:Ἀ)*u*2=1)
  (h1:|v-Ts|<=2*N*u*Ts+2*N*w)(h2:|Pr-v*q|<=u*|v*q|+w)(h3:|Wr-(Pr+Ep)|<=u*|Pr+Ep|+w)(hW:0<=Wr)
  (h4:|Dr-O2 Wr|<=u*O2 Wr+w):
  |Dr-O2 (Ts*2^(52:Ἀ)/N+Ep)|<=(4*N+7)*u*O2 (Ts*2^(52:Ἀ)/N+Ep):=
 nD_chainK Ts Ep N q v Pr Wr Dr u w (2*N) (4*N) hu0 hw0 hNu hwu hN hTs hEp hq1 hq2 hq0 hu2 (by linarith only[hN]) (by linarith only[hN]) (by ring) h1 h2 h3 hW h4
lemma nD_sq(X:ℚ)(hX:0<X)(q:Ἀ)(s:ℤ)(hq1:2^(25:Ἀ)<=q)(hq2:q<2^(26:Ἀ))(hs1:-209<=s)(hs2:s<=297)
  (w1:(q:ℝ)<O2 (X:ℝ)*2^s)(w2:O2 (X:ℝ)*2^s<q+1):
  mK X=az false (XF (2*q+1)) (XF (297-s).toNat):=by
 set a:=X.num.natAbs with ha
 set b:=X.den with hb
 have a0:0<a:=by rw[ha];exact Int.natAbs_pos.2 (Rat.num_pos.2 hX).ne'
 have b0:0<b:=X.den_pos
 have hn:(a:ℝ)=(X.num:ℝ):=by rw[ha,Nat.cast_natAbs,mI,Ad (Int.cast_nonneg (Rat.num_nonneg.2 hX.le))]
 have hXab:(X:ℝ)=a/b:=by rw[hn,hb];exact Rat.cast_def X
 set e:=a*2^(2*s.toNat) with he
 set d:=b*2^(2*(-s).toNat) with hd
 have d0:0<d:=by positivity
 have hed:(e:ℝ)/d=(X:ℝ)*(2:ℝ)^(2*s):=by
  rw[he,hd,hXab];push_cast
  rcases le_or_gt 0 s with h|h
  · rw[Int.toNat_of_nonpos (Int.neg_nonpos_of_nonneg h),show (2:ℝ)^(2*s)=2^(2*s.toNat) by rw[<-AA,Y4,nZ,Int.toNat_of_nonneg h],mul_zero,pow_zero,mul_one,mul_div_right_comm]
  · rw[Int.toNat_of_nonpos h.le,show (2:ℝ)^(2*s)=1/2^(2*(-s).toNat) by rw[<-AA,Y4,nZ,Int.toNat_of_nonneg (Int.neg_nonneg_of_nonpos h.le),one_div,<-zpow_neg,mul_neg,neg_neg],mul_zero,pow_zero,mul_one,mul_one_div,div_div]
 have sX:0<O2 (X:ℝ):=Real.sqrt_pos.2 (Rat.cast_pos.2 hX)
 have p2:(0:ℝ)<2^s:=zpow_pos two_pos s
 have sq:(O2 (X:ℝ)*2^s)^(2:Ἀ)=(X:ℝ)*2^(2*s):=by rw[mul_pow,Real.sq_sqrt (Rat.cast_nonneg.2 hX.le),<-AA,<-zpow_mul,nZ,mul_comm s]
 have q0:(0:ℝ)<=q:=M3 q
 have l1:(q:ℝ)^(2:Ἀ)<(e:ℝ)/d:=by rw[hed,<-sq];exact pow_lt_pow_left₀ w1 q0 AL
 have l2:(e:ℝ)/d<((q:ℝ)+1)^(2:Ἀ):=by rw[hed,<-sq];exact pow_lt_pow_left₀ w2 (Af sX.le p2.le) AL
 have dR:(0:ℝ)<d:=Nat.cast_pos.2 d0
 have n1:q*q*d<e:=by rw[lt_div_iff₀ dR,pow_two]at l1;exact_mod_cast l1
 have n2:e<(q+1)*(q+1)*d:=by rw[div_lt_iff₀ dR,pow_two]at l2;exact_mod_cast l2
 have hZK:sqrtLower e d=q:=by
  unfold sqrtLower;symm;rw[Nat.eq_sqrt];exact⟨(mc d0).2 n1.le,(ϸ d0).2 n2⟩
 have Vd':=Vd X hX e d d0 s (by exact_mod_cast hed) (by rw[hZK];exact hq1)
 have V9':=V9 e d s (by rw[hZK];exact hq1) (by rw[hZK,β];exact ώ (U1 hq2 (Θ two_pos (N2 rfl)))) hs1 hs2
 have ne:e≠sqrtLower e d*sqrtLower e d*d:=by rw[hZK];exact n1.ne'
 rw[if_neg ne,hZK]at V9'
 rw[<-Vd'];exact V9'
lemma kg2(n:Ἀ)(g:n<2^(53:Ἀ)):Dense.F.fv (I0 n).toInt64.toFloat/\Dense.F.rv (I0 n).toInt64.toFloat=n:=by
 have d:(lq (I0 n))=n:=Α (g.trans (by decide));have e:(K7 (I0 n).toInt64)=n:=by show Core.ae (I0 n)=n;rw[Dense.ay,d,if_neg (Nat.not_le.2 (g.trans (by decide))),Int.sub_zero]
 have:=Dense.F.dB _ (by rw[e,Int.natAbs_natCast];exact g);rw[e,M5]at this;exact this
lemma nD_eps(eps:kF)(he:kB eps≠255):Dense.F.fv (Dense.F.dw eps*nS201)/\Dense.F.rv (Dense.F.dw eps*nS201)=(sv eps:ℝ)*2^kG eps*2^(201:Ἀ):=by
 obtain⟨ef,er⟩:=Dense.F.cI eps he
 obtain⟨sf,sr⟩:=Dense.F.iJ 201 (N2 rfl)
 unfold nS201
 have r:=Dense.F.aE _ _ ef sf (sv eps) ((kG eps+201:ℕ):ℤ) (by rw[er,sr,AA,AF,Ap]) ((Dense.F.bg eps).trans (N2 rfl)) ((by decide:(-1074:ℤ)<=0).trans (W6 _)) (Dense.F.bL 24 (kG eps+201) 479 (J1 (L1 (cP eps) 201) 24) (N2 rfl) _ (Dense.F.bg eps))
 refine⟨r.1,?_⟩;rw[r.2,AA,AF,Ap]
lemma nI(a b:ℕ)(h:a<=b):(1:ℝ)/2^b<=1/2^a:=one_div_le_one_div_of_le (by positivity) (AU (by norm_num) h)
lemma nD_tb(w:kF):((sv w:ℝ)*2^kG w)^(2:Ἀ)<=2^(556:Ἀ):=by
 have a:=NE.wB w;have k:=cP w
 have b:|(sv w:ℝ)*2^kG w|<=2^(278:Ἀ):=a.trans (AU (by norm_num) (by omega))
 rw[<-sq_abs];calc |(sv w:ℝ)*2^kG w|^(2:Ἀ)<=(2^(278:Ἀ))^(2:Ἀ):=pow_le_pow_left₀ (Aa _) b 2
  _=2^(556:Ἀ):=by rw[<-pow_mul]
theorem nVb_h1(a T G:ℝ)(Tsb:T<=G)(g1:1<=G)(v1:a<=2*T+1)(v2:-1<=a):-(G*2^2)<=a/\a<=G*2^2:=⟨(neg_le_neg (g1.trans (le_mul_of_one_le_right (zero_le_one.trans g1) (one_le_pow₀ one_le_two)))).trans v2,v1.trans ((add_le_add (mul_le_mul_of_nonneg_left Tsb zero_le_two) ((one_le_two.trans_eq (mul_one 2).symm).trans (mul_le_mul_of_nonneg_left g1 zero_le_two))).trans_eq (by ring))⟩
lemma nVb(a Ts:ℝ)(n:ℕ)(Ts0:0<=Ts)(Tsb:Ts<=2^(568:Ἀ))(nb:(n:ℝ)<=2^(12:Ἀ))(h1:-(2*(n:ℝ)*(1/2^(53:Ἀ))*Ts+2*n*(1/2^(1075:Ἀ)))<=a-Ts)(h2:a-Ts<=2*(n:ℝ)*(1/2^(53:Ἀ))*Ts+2*n*(1/2^(1075:Ἀ))):|a|<=2^(570:Ἀ):=by
 have e:(2:ℝ)^(13:Ἀ)=2*2^(12:Ἀ):=pow_succ' 2 12
 obtain⟨v1,v2⟩:=nSm a Ts n (1/2^(53:Ἀ)) (1/2^(1075:Ἀ)) (2^(12:Ἀ)) Ts0 (M3 n) nb (nN _) (nN _) (e▸nP 13 53 (by decide)) (e▸nP 13 1075 (by decide)) h1 h2
 rw[AF 2 568 2,A3];exact nVb_h1 a Ts _ Tsb (one_le_pow₀ one_le_two) v1 v2
lemma nFw(v Ts:ℝ)(n:ℕ)(q P Ep W:ℝ)(Ts0:0<=Ts)(nb:(n:ℝ)<=2^(12:Ἀ))(q0:0<=q)(qb:|q|<=2^(52:Ἀ))(h1:-(2*(n:ℝ)*(1/2^(53:Ἀ))*Ts+2*n*(1/2^(1075:Ἀ)))<=v-Ts)(pe:|P-v*q|<=|v*q|/2^(53:Ἀ)+1/2^(1075:Ἀ))(hEp1:1<=Ep)(we:|W-(P+Ep)|<=|P+Ep|/2^(53:Ἀ)+1/2^(1075:Ἀ)):0<=W:=
 nFw0 v Ts n q P Ep W (1/2^(53:Ἀ)) (1/2^(1075:Ἀ)) (2^(53:Ἀ)) Ts0 (M3 n) nb (nN _) (nN _) q0 ((le_abs_self q).trans qb) (by norm_num) (nP 68 1075 (by norm_num)) (by norm_num) h1 pe hEp1 we
lemma nD_float(n:Ἀ)(eps:kF)(x:(Ἂ Word))(r:Ἀ)(hn:0<n)(hn2:n<=2^(12:Ἀ))(hb:r*n+n<2^(32:Ἀ))(hX:∀j,j<r*n+n->kB x[j]!≠255)(he:kB eps≠255)(hs:0<sv eps):
  Dense.F.fv (W7 (nQs x (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat*(I0 (4503599627370496/n)).toInt64.toFloat+Dense.F.dw eps*nS201))/\
  |Dense.F.rv (W7 (nQs x (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat*(I0 (4503599627370496/n)).toInt64.toFloat+Dense.F.dw eps*nS201))-
   O2 ((∑j∈B5 (r*n) (r*n+n),((sv x[j]!:ℝ)*2^kG x[j]!)^(2:Ἀ))*2^(52:Ἀ)/n+(sv eps:ℝ)*2^kG eps*2^(201:Ἀ))|<=
   (4*n+7)*(1/2^(53:Ἀ))*O2 ((∑j∈B5 (r*n) (r*n+n),((sv x[j]!:ℝ)*2^kG x[j]!)^(2:Ἀ))*2^(52:Ἀ)/n+(sv eps:ℝ)*2^kG eps*2^(201:Ἀ)):=by
 set Ts:=∑j∈B5 (r*n) (r*n+n),((sv x[j]!:ℝ)*2^kG x[j]!)^(2:Ἀ) with hTs
 set Ep:=(sv eps:ℝ)*2^kG eps*2^(201:Ἀ) with hEp
 have lo:(J0 ((G0 (r*n))))=r*n:=Conv.aa _ ((V1 _ n).trans_lt hb)
 have hi:(J0 ((G0 (r*n+n))))=r*n+n:=Conv.aa _ hb
 obtain⟨z0f,z0r⟩:=kg2 0 (϶ _)
 have Ts0:0<=Ts:=D8 (fun _ _=>sq_nonneg _)
 have nb:(n:ℝ)<=2^(12:Ἀ):=(R5.2 hn2).trans_eq (by rw[I4,nZ])
 have Tsb:Ts<=2^(568:Ἀ):=by
  have h0:=Finset.sum_le_card_nsmul (B5 (r*n) (r*n+n)) (fun j=>((sv x[j]!:ℝ)*2^kG x[j]!)^(2:Ἀ)) (2^(556:Ἀ)) (fun j _=>nD_tb _)
  rw[Nat.card_Ico,ϱ,nsmul_eq_mul]at h0
  have h2:(n:ℝ)*2^(556:Ἀ)<=2^(12:Ἀ)*2^(556:Ἀ):=AE nb (pow_pos two_pos _).le
  have h3:(2:ℝ)^(12:Ἀ)*2^(556:Ἀ)=2^(568:Ἀ):=by rw[<-AF]
  exact h0.trans (h2.trans_eq h3)
 obtain⟨vf,ve⟩:=nQs_spec x ((G0 (r*n+n))) (fun j hj=>hX j (by rwa[hi]at hj)) n ((G0 (r*n))) _ 0 0 (by rw[hi,lo];exact Nat.add_sub_self_left _ _) (by rw[hi,lo];exact V1 _ _) z0f AP (by rw[z0r];simp only [CharP.cast_eq_zero,sub_self,A5,mul_zero,one_div,zero_mul,Av,Std.le_refl]) ((Κ n).trans_le (hn2.trans (by decide))) (by rw[lo,hi,Ai];exact Tsb.trans (AU one_le_two (by decide)))
 rw[lo,hi]at ve;simp only[Ai,Κ,Nat.cast_zero]at ve
 obtain⟨qf,qr⟩:=kg2 (4503599627370496/n) ((L5 _ _).trans_lt (by decide))
 set q:ℝ:=((4503599627370496/n:ℕ):ℝ) with hq
 have q0:0<=q:=M3 _
 have h52:((4503599627370496:ℕ):ℝ)=2^(52:Ἀ):=by rw[show (4503599627370496:ℕ)=2^(52:Ἀ) from rfl,I4,nZ]
 have hq1:(n:ℝ)*q<=2^(52:Ἀ):=by
  have h:=R5 (α:=ℝ)|>.2 (pr 4503599627370496 n)
  rw[Y4]at h;rw[mul_comm];exact h.trans_eq h52
 have hq2:2^(52:Ἀ)-(n:ℝ)<n*q:=by
  have h0:((n*(4503599627370496/n)+4503599627370496%n:ℕ):ℝ)=((4503599627370496:ℕ):ℝ):=congrArg Nat.cast (ϛ 4503599627370496 n)
  rw[W5,Y4,h52]at h0
  exact (sub_lt_sub_left (Nat.cast_lt.2 (ϊ _ hn)) _).trans_eq (by rw[<-h0,add_sub_cancel_right])
 have qn:q<=n*q:=le_mul_of_one_le_left q0 (Nat.one_le_cast.2 hn)
 have qb:|q|<=2^(52:Ἀ):=by rw[Ad q0];exact qn.trans hq1
 obtain⟨ef,er⟩:=nD_eps eps he
 obtain⟨h1,h2⟩:=A3.1 ve
 have vb:|Dense.F.rv (nQs x ((G0 (r*n))) ((G0 (r*n+n))) (I0 (0:Ἀ)).toInt64.toFloat)|<=2^(570:Ἀ):=nVb _ Ts n Ts0 Tsb nb h1 h2
 have pb:=NE.pB _ _ 570 52 vb qb
 obtain⟨pf,pe⟩:=NF.fM _ _ vf qf (by rw[qr];exact NE.z9 _ _ pb (by decide))
 rw[qr]at pe
 set v2:=nQs x ((G0 (r*n))) ((G0 (r*n+n))) (I0 (0:Ἀ)).toInt64.toFloat with hv2
 set P:=v2*(I0 (4503599627370496/n)).toInt64.toFloat with hP
 have Pb:=NE.eB _ _ 622 pe pb
 have kE:=cP eps
 have Ewb:=NE.wB eps
 have E1:|(sv eps:ℝ)*2^kG eps|<=2^(278:Ἀ):=Ewb.trans (AU one_le_two (J1 kE 24))
 have E2:|(2:ℝ)^(201:Ἀ)|<=2^(201:Ἀ):=(Ad (pow_pos two_pos _).le).le
 have Epb:|Ep|<=2^(623:Ἀ):=NE.mo _ _ _ (NE.pB _ _ 278 201 E1 E2) (by decide)
 have Sb:=NE.sB _ _ 623 Pb Epb
 obtain⟨wf,we⟩:=NF.fA _ _ pf ef (by rw[er];exact NE.z9 _ _ Sb (by decide))
 rw[er]at we
 set W:=P+Dense.F.dw eps*nS201 with hW'
 have s1:(1:ℝ)<=sv eps:=Int.cast_one_le_of_pos hs
 have a1:(1:ℝ)<=2^kG eps:=one_le_pow₀ one_le_two
 have a2:(1:ℝ)<=2^(201:Ἀ):=one_le_pow₀ one_le_two
 have hEp1:1<=Ep:=one_le_mul_of_one_le_of_one_le (one_le_mul_of_one_le_of_one_le s1 a1) a2
 have Wl:0<=Dense.F.rv W:=nFw _ Ts n q _ Ep _ Ts0 nb q0 qb h1 pe hEp1 we
 obtain⟨df,dr⟩:=fS W wf Wl
 refine⟨df,?_⟩
 have h4:|Dense.F.rv (W7 W)-O2 (Dense.F.rv W)|<=(1/2^(53:Ἀ))*O2 (Dense.F.rv W)+1/2^(1075:Ἀ):=by
  rw[dr];have:=NF.jy (O2 (Dense.F.rv W));rw[Ad (Real.sqrt_nonneg _)]at this;exact this.trans_eq (by rw[one_div_mul_eq_div])
 have hu2:(2:ℝ)^(52:Ἀ)*(1/2^(53:Ἀ))*2=1:=by rw[mul_one_div,div_mul_eq_mul_div,<-A4];exact div_self (pow_ne_zero _ AL)
 have n14:4*(n:ℝ)<=2^(14:Ἀ):=(mul_le_mul (by norm_num:(4:ℝ)<=2^(2:Ἀ)) nb (M3 _) (pow_pos two_pos _).le).trans_eq (by rw[<-AF])
 have hNu:(4*(n:ℝ)+8)*(1/2^(53:Ἀ))<=1/2^(30:Ἀ):=(AE ((add_le_add n14 (by norm_num:(8:ℝ)<=2^(14:Ἀ))).trans_eq (by rw[<-mul_two,<-A4]):4*(n:ℝ)+8<=2^(15:Ἀ)) (nN 53)).trans (nP2 15 53 30 (by decide))
 exact nD_chain Ts Ep n q (Dense.F.rv v2) (Dense.F.rv P) (Dense.F.rv W) _ (1/2^(53:Ἀ)) (1/2^(1075:Ἀ)) (one_div_pos.2 (pow_pos two_pos _)) (nN _) hNu (nP2 60 1075 53 (by decide)) (Nat.one_le_cast.2 hn) Ts0 hEp1 hq1 hq2 q0 hu2 ve (pe.trans_eq (by rw[one_div_mul_eq_div])) (we.trans_eq (by rw[one_div_mul_eq_div])) Wl h4
lemma nXH(n:Ἀ)(x:(Ἂ Word))(r:Ἀ)(eps:kF)(hn:0<n):
  ((∑j∈B5 (r*n) (r*n+n),((sv x[j]!:ℝ)*2^kG x[j]!)^(2:Ἀ))*2^(52:Ἀ)/n+(sv eps:ℝ)*2^kG eps*2^(201:Ἀ))=
   2^(350:Ἀ)*(((XH n x r 0+ZA eps:ℚ)):ℝ):=by
 have h2:∀w:kF,(sv w:ℝ)*2^kG w=((ZA w:ℚ):ℝ)*2^(149:Ἀ):=by
  intro w;rw[Y1,J7,S8,Rat.cast_intCast,G7,Ac AL,AA,zpow_ofNat,mul_div_assoc',div_mul_cancel₀ _ (pow_ne_zero _ AL)]
 rw[rowVariance,Yk,Ai,Rat.cast_add,Rat.cast_div,Rat.cast_sum,T7,Finset.sum_Ico_eq_sum_range,ϱ]
 simp only[h2,sub_zero,mul_pow,<-n8,Rat.cast_pow]
 rw[show (2:ℝ)^(350:Ἀ)=(2^(149:Ἀ))^(2:Ἀ)*2^(52:Ἀ) by rw[<-pow_mul,<-AF],show (2:ℝ)^(201:Ἀ)=2^(149:Ἀ)*2^(52:Ἀ) by rw[<-AF]]
 generalize (2:ℝ)^(149:Ἀ)=a;generalize (2:ℝ)^(52:Ἀ)=b
 ring
section
open NF Dense.F NE
@[inline]def nOKs(B c:ἆ):Ἆ:=let E:=(B>>>52)&&&2047;let L:=B&&&0x7FFFFFF;let t:=(1:ἆ)<<<c;1073<=E&&E<=1324&&c<=25&&t<L&&L+t<0x8000000
theorem nOKs_dec_h1(B c:ἆ)(a3:(lq c)<=25)(a4:(1:ἆ)<<<c<B&&&0x7FFFFFF)(a5:(B&&&0x7FFFFFF)+((1:ἆ)<<<c)<0x8000000):2^(lq c)<(lq B)%2^(27:Ἀ)∧(lq B)%2^(27:Ἀ)+2^(lq c)<2^(27:Ἀ):=by
 have hc:=C0 a3 (N2 rfl:25<64)
 have hL:(lq ((1:ἆ)<<<c))=2^(lq c):=by rw[ψ,pS,γ hc,na];exact γ (lu Nat.one_lt_two hc)
 have hl:(lq (B&&&0x7FFFFFF))=(lq B)%2^(27:Ἀ):=(Ε _ _).trans (ζ _ 27)
 have a4:2^(lq c)<(lq B)%2^(27:Ἀ):=hL▸hl▸a4
 have a5:(lq ((B&&&0x7FFFFFF)+((1:ἆ)<<<c)))<2^(27:Ἀ):=a5
 rw[η,hL,hl,γ (U1 (E6 (ϊ _ (϶ 27)) (K5 a4 (ϊ _ (϶ 27)))) (N2 rfl))]at a5
 exact⟨a4,a5⟩
theorem nOKs_dec(B c:ἆ)(h:nOKs B c=ἃ):1073<=(lq (eF B))∧(lq (eF B))<=1324∧(lq c)<=25∧2^(lq c)<(lq B)%2^(27:Ἀ)∧(lq B)%2^(27:Ἀ)+2^(lq c)<2^(27:Ἀ):=by
 obtain⟨h,a5⟩:=I5.1 h
 obtain⟨h,a4⟩:=I5.1 h
 obtain⟨h,a3⟩:=I5.1 h
 obtain⟨a1,a2⟩:=I5.1 h
 exact⟨of_decide_eq_true a1,of_decide_eq_true a2,of_decide_eq_true a3,nOKs_dec_h1 B c (of_decide_eq_true a3) (of_decide_eq_true a4) (of_decide_eq_true a5)⟩
end
lemma nK(n:ℕ)(h:n<=2^(12:Ἀ)):(4*(n:ℝ)+7)/2^(53:Ἀ)<=1/2^(30:Ἀ)/\2*((4*(n:ℝ)+7)/2^(53:Ἀ))*(4*n+7)<=1:=by
 have nR:(n:ℝ)<=4096:=by exact_mod_cast (show n<=4096 from h)
 have n0:(0:ℝ)<=n:=M3 n
 exact⟨by linear_combination (4/2^(53:Ἀ))*nR,by linear_combination (2/2^(53:Ἀ))*(16*mul_le_mul nR nR n0 (n0.trans nR)+56*nR)⟩
lemma nC8_h1(n:ℕ):4*n+8<=2^(Aq (n+4)+3):=by
 have a:=M7 (ϙ (n:=n+4)) (show 0<4 by decide)
 rw[Nat.mul_add]at a
 have b:2^(Aq (n+4)+3)=4*2^(Aq (n+4)+1):=by rw[Ϗ];exact m7 2 _ 2
 rw[b];exact (J1 (show 8<=4*4 by decide) _).trans a.le
lemma nC8(n:ℕ):(4*(n:ℝ)+8)<=2^(((Aq (n+4)+4:ℕ):ℤ)-1):=by
 rw[show ((Aq (n+4)+4:ℕ):ℤ)-1=((Aq (n+4)+3:ℕ):ℤ) from sub_eq_of_eq_add (X7 _),AA]
 exact_mod_cast nC8_h1 n
section
open NF Dense.F NE
lemma nQ_err2(Y Dr Uu e k:ℝ)(hU:0<Uu)(hk0:0<=k)(k1:k<=1/2)(hY:0<=Y)(hD:|Dr|<2^(53:Ἀ)*Uu)(hku:k*2^(53:Ἀ)<=e)(hke:2*k*e<=1)
  (hd:|Dr-Y|<=k*Y):|Y-Dr|<(e+1)*Uu:=by
 obtain⟨a1,-⟩:=A3.1 hd
 have Yb2:(1-k)*Y<=|Dr|:=by linarith only[a1,le_abs_self Dr]
 have kDu:k*(2^(53:Ἀ)*Uu)<=e*Uu:=by rw[<-Ap];exact AE hku hU.le
 have m1:k*((1-k)*Y)<=e*Uu:=(mul_le_mul_of_nonneg_left (Yb2.trans hD.le) hk0).trans kDu
 have kk:k*(k*Y)<=(1/2)*(k*Y):=AE k1 (Af hk0 hY)
 have kY2:k*Y<=2*(e*Uu):=by linear_combination 2*m1+2*kk
 have m3:k*(2*(e*Uu))<=Uu:=by rw[mul_left_comm];have:=AE hke hU.le;rwa[Ap,Ap,one_mul] at this
 have eq:k*Y=k*((1-k)*Y)+k*(k*Y):=by ring
 rw[Aw,add_one_mul];refine AO hd ?_
 rcases eq_or_lt_of_le hk0 with h|h
 · rw[<-h] at kY2 ⊢;linarith only[kY2,hU]
 · rw[eq];exact add_lt_add_of_lt_of_le ((mul_lt_mul_of_pos_left (Yb2.trans_lt hD) h).trans_le kDu) ((mul_le_mul_of_nonneg_left kY2 hk0).trans m3)
lemma nQ_tail_h1(σ:ἆ)(h1:1073<=(lq σ))(h2:(lq σ)<=1324):146<=(lq (σ-(900+26)))∧(lq (σ-(900+26)))<=398∧(-209:ℤ)<=1223-((lq σ):ℤ)∧1223-((lq σ):ℤ)<=297∧σ-(900+26)=XF (F6 (297-(1223-((lq σ):ℤ)))):=by
 have k:lq (900+26:ἆ)=926:=rfl
 have l0:926<=lq σ:=(N2 rfl:926<=1073).trans h1
 have e:(lq (σ-(900+26)))=(lq σ)-926:=(χ _ _ (θ.2 (k.trans_le l0))).trans (congrArg (lq σ-·) k)
 have i1:((lq σ):ℤ)<=1223-(-209):=(R5.2 (h2.trans (N2 rfl:1324<=1432))).trans_eq nZ
 have i2:(1223-297:ℤ)<=((lq σ):ℤ):=nZ.symm.trans_le (R5.2 l0)
 have q:(297:ℤ)-(1223-((lq σ):ℤ))=((lq σ):ℤ)-((926:ℕ):ℤ):=by rw[sub_sub_eq_add_sub,add_comm,add_sub_assoc];rfl
 refine⟨(G5 (X5 h1:146+926<=_)).trans_eq e.symm,e.trans_le (M8 (h2:_<=398+926)),le_sub_comm.2 i1,sub_le_comm.2 i2,ξ.1 ?_⟩
 rw[e,O6,q,Int.toNat_sub]
 exact (γ (C0 (K4 _ _) (C0 h2 (N2 rfl)))).symm
lemma nQ_tail(D:Ἅ)(df:Dense.F.fv D)(X:ℚ)(X0:0<X)(c:ℕ)(hc:c<64)(hok:nOKs D.toBits c.toUInt64=ἃ)(k e:ℝ)(hk0:0<=k)(k1:k<=1/2)(hku:k*2^(53:Ἀ)<=e)(hke:2*k*e<=1)(hec:e+1<=2^((c:ℤ)-1))
  (dd:|Dense.F.rv D-2^(175:Ἀ)*O2 (X:ℝ)|<=k*(2^(175:Ἀ)*O2 (X:ℝ))):nWs 26 D.toBits=mK X:=by
 set Y:=2^(175:Ἀ)*O2 (X:ℝ) with hY
 have Y0:0<=Y:=by positivity
 obtain⟨d1,d2,d3,d4,d5⟩:=nOKs_dec _ _ hok
 have hc':(lq (c.toUInt64))=c:=by simp only[Ϣ,mY];exact γ (U1 hc (N2 rfl))
 rw[hc']at d4 d5
 set E:=(lq (eF D.toBits)) with hE
 set Uu:=(2:ℝ)^((E:ℤ)-1075) with hU
 have U0:0<Uu:=by positivity
 have fk:=Dense.F.fK D df
 have hD53:|Dense.F.rv D|<2^(53:Ἀ)*Uu:=by rw[hU,<-AA,<-AK AL];exact fk.trans_eq (congrArg _ (by simp only [nZ];ring))
 have er:=nQ_err2 Y (Dense.F.rv D) Uu e k U0 hk0 k1 Y0 hD53 hku hke dd
 have he':|Y-Dense.F.rv D|<2^((c:ℤ)-1)*Uu:=er.trans_le (AE hec U0.le)
 obtain⟨q1,q2,w1,w2⟩:=nOK_win Y D c df (L6 (U1 (Nat.zero_lt_succ 1072) d1)) d4 d5 he'
 set qU:=mF D.toBits>>>27 with hq
 have qpos:(0:ℝ)<(qU.toNat:ℝ)*2^(27:Ἀ)*Uu:=mul_pos (mul_pos (Nat.cast_pos.2 (AN (by norm_num) q1)) (by norm_num)) U0
 have sg:(D.toBits>>>63!=0)=false:=by
  by_contra h;rw[Bool.not_eq_false]at h;rw[if_pos h]at w1;exact absurd (qpos.trans w1) (not_lt.2 (neg_nonpos.2 Y0))
 rw[if_neg (by rw[sg];decide)]at w1 w2
 set s:ℤ:=1223-(E:ℤ) with hs
 have key:(2:ℝ)^(27:Ἀ)*Uu*2^s=2^(175:Ἀ):=by
  rw[hU,hs,<-AA (2:ℝ) 27,<-AK (by norm_num),<-AK (by norm_num),<-AA (2:ℝ) 175];exact congrArg _ (by simp only [nZ];ring)
 have m:(O2 (X:ℝ)*2^s)*(2^(27:Ἀ)*Uu)=Y:=by rw[Ap,mul_comm ((2:ℝ)^s),key,hY];exact mul_comm _ _
 have pU:(0:ℝ)<2^(27:Ἀ)*Uu:=mul_pos (by norm_num) U0
 have w1':(qU.toNat:ℝ)<O2 (X:ℝ)*2^s:=by
  apply BF _ pU.le;rw[m,<-Ap];exact w1
 have w2':O2 (X:ℝ)*2^s<qU.toNat+1:=by
  apply BF _ pU.le;rw[m,<-Ap];exact w2
 have hh:=nQ_tail_h1 (eF D.toBits) d1 d2
 have sq:=nD_sq X X0 qU.toNat s q1 q2 hh.2.2.1 hh.2.2.2.1 w1' w2'
 have kj':=kj false qU (eF D.toBits-(900+26)) q1 q2 hh.1 hh.2.1
 have l1:nWs 26 D.toBits=pk false qU (eF D.toBits-(900+26)):=by unfold nWs;rw[sg]
 rw[l1,kj',sq];refine congrArg₂ (az false) ?_ hh.2.2.2.2
 apply ξ.1;simp only [η,ύ,kA,nr,Nat.mod_add_mod,UInt64.ofNat_add,UInt64.ofNat_mul,UInt64.reduceOfNat,UInt64.ofNat_toNat,UInt64.ofNat_one]
lemma nQ_cert(n:Ἀ)(eps:kF)(x:(Ἂ Word))(r:Ἀ)(hn:0<n)(hn2:n<=2^(12:Ἀ))(hb:r*n+n<2^(32:Ἀ))(hX:∀j,j<r*n+n->kB x[j]!≠255)(he:kB eps≠255)(hs:0<sv eps)
  (hok:nOKs (W7 (nQs x (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat*(I0 (4503599627370496/n)).toInt64.toFloat+Dense.F.dw eps*nS201)).toBits ((Aq (n+4)+4).toUInt64)=ἃ):
  nWs 26 (W7 (nQs x (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat*(I0 (4503599627370496/n)).toInt64.toFloat+Dense.F.dw eps*nS201)).toBits=mK (XH n x r 0+ZA eps):=by
 obtain⟨df,dd⟩:=nD_float n eps x r hn hn2 hb hX he hs
 have hxh:=nXH n x r eps hn
 rw[hxh]at dd
 set D:=W7 (nQs x (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat*(I0 (4503599627370496/n)).toInt64.toFloat+Dense.F.dw eps*nS201) with hD
 set X:ℚ:=XH n x r 0+ZA eps with hXd
 have sv0:(0:ℝ)<sv eps:=by exact_mod_cast hs
 have Ep0:0<(sv eps:ℝ)*2^kG eps*2^(201:Ἀ):=mul_pos (mul_pos sv0 (pow_pos (by norm_num) _)) (by norm_num)
 have T0:(0:ℝ)<=(∑j∈B5 (r*n) (r*n+n),((sv x[j]!:ℝ)*2^kG x[j]!)^(2:Ἀ))*2^(52:Ἀ)/n:=div_nonneg (Af (D8 (fun _ _=>sq_nonneg _)) (by norm_num)) (M3 n)
 have L0:=add_pos_of_nonneg_of_pos T0 Ep0
 rw[hxh]at L0
 have XR:(0:ℝ)<(X:ℝ):=pos_of_mul_pos_right L0 (by norm_num)
 have X0:0<X:=by exact_mod_cast XR
 have Yq:O2 (2^(350:Ἀ)*(X:ℝ))=2^(175:Ἀ)*O2 (X:ℝ):=by
  rw[Real.sqrt_mul (by positivity),show (2:ℝ)^(350:Ἀ)=(2^(175:Ἀ))^(2:Ἀ) by rw[<-pow_mul],Real.sqrt_sq (by positivity)]
 rw[Yq]at dd
 obtain⟨k1,k2⟩:=nK n hn2
 have hc:Aq (n+4)+4<64:=by
  have t:=(m9 (by omega)).2 (show n+4<2^(13:Ἀ) by have:(2:Ἀ)^(12:Ἀ)=4096:=rfl;have:(2:Ἀ)^(13:Ἀ)=8192:=rfl;omega)
  omega
 exact nQ_tail D df X X0 _ hc hok _ _ (by positivity) (k1.trans (by norm_num)) (div_mul_cancel₀ _ (by positivity)).le k2 (by linear_combination nC8 n) (dd.trans (le_of_eq (by ring)))
end
lemma Qg{α:Type}[Inhabited α]{m k:Ἀ}(f:Fin m->α)(h:k< m):
  (kK f)[k]! =f ⟨k,h⟩:=by rw[AB _ k (by simpa using h),Λ]
lemma YP_h1{c1 c2:Prop}[Decidable c1][Decidable c2](a b r:kF)(ha:c1→a=r)(hb:c2→b=r):(if c1 then a else if c2 then b else r)=r:=if h:c1 then (if_pos h).trans (ha h) else (if_neg h).trans (ite_eq_right_iff.2 hb)
@[inline]def SN(rows n:Ἀ)(eps:Ἃ)(x g:(Ἂ Word)):Ἆ:=
 rows>0&&n>0&&(O0 x)==rows*n&&(O0 g)==n&&x.all finiteWord&&g.all finiteWord&&finiteWord eps&&(!ZX eps&&ZC eps !=0)
lemma fG(e n:Ἀ)(d:kF)(x g:(Ἂ Word)):SN e n d x g=mq e n d x g:=by rw[SN,mq];congr 1;rw[Q6,Y2];have:(0:ℚ)<2^((ZZ d:Ἁ)-149):=((by positivity));rw[sg];cases ZX d<;>simp[this,BD]
lemma jm{d n:Ἀ}{e:kF}{x g:(Ἂ Word)}(h:mq d n e x g=ἃ):
  0<n/\0<ZA e/\(∀i,i<d*n->kB x[i]!≠255)/\∀j,j<n->kB g[j]!≠255:=by
 simp only[mq,Η,ZH,BC]at h;exact⟨h.1.1.1.1.1.1.2,h.2,λi m=>NR.ic x (np.mp h.1.1.1.2) i ((ZY.mp h.1.1.1.1.1.2)▸m),λj o=>NR.ic g (np.mp h.1.1.2) j ((ZY.mp h.1.1.1.1.2)▸o)⟩
def YU(n:Ἀ)(eps:Ἃ)(x:(Ἂ Word))(r:Ἀ):Ἀ×Ἀ×Ἆ×Ἀ×Ἀ×Ἃ:=
 let lo:=r*n
 let u:=Yh x lo (lo+n) 255
 let Q:=Yl x u lo (lo+n) 0 0 0 0
 let pn:=Ym x u lo (lo+n) 0 0
 let S:=Y0 pn.1 pn.2
 let k:=Q2 (max pn.1 pn.2)-30
 (u,k,decide (pn.1<pn.2),S>>>k,S&&&(VO 1 k-1),V5 (n*Q-S*S) (n*n) u eps)
lemma Vn(n:Ἀ)(d:kF)(x:(Ἂ Word))(r:Ἀ)(e:0<n)
  (h:0<ZA d):
  let t:=YU n d x r
  no n x r=sg t.2.2.1 (t.2.2.2.1*2^t.2.1+t.2.2.2.2.1:Ἀ)*2^((t.1:Ἁ)-149)/n/\
  t.2.2.2.2.2=mK (XH n x r (no n x r)+ZA d)/\
  (∀j,r*n<=j->j<r*n+n->ZW x[j]! =0\/t.1<=ZZ x[j]!):=by
 have g:=(Yi x (r*n+n) n (r*n) 255 (ϱ ..)).2;have⟨l,m⟩:=VF n x r _ e g;use(by rw[l,<-Yn];simp only[YU,Yz,ο,n4,ζ,β,ZU]),(by simp only[YU];rw[YW _ _ _ _ (Ϯ e e) h,m]),g
@[noinline]def gM(n:Ἀ)(eps:Ἃ)(x:(Ἂ Word))(r:Ἀ):Ἀ×Ἀ×Ἆ×Ἀ×Ἀ×Ἃ:=
 let lo:=r*n
 let u:=cS x (G0 lo) (G0 (lo+n)) 255
 let st:=cF x u (G0 lo) (G0 (lo+n)) 0 0 0 0 0 0 0 0
 let S:=Y0 st.P st.N
 let k:=Q2 (max st.P st.N)-30
 ((lq u),k,decide (st.P<st.N),S>>>k,S&&&(VO 1 k-1),
  V5 (n*(st.big+(lq st.l)+((lq st.h)+st.c<<<64)<<<64)-S*S) (n*n) (lq u) eps)
lemma gN(n:Ἀ)(eps:kF)(x:(Ἂ Word))(r:Ἀ)(hn:0<n)(he:0<ZA eps)(hb:r*n+n<2^(32:Ἀ)):gM n eps x r=YU n eps x r:=by
 dsimp only[gM,YU];have A:=Conv.aa (r*n) ((V1 _ _).trans_lt hb);have B:=Conv.aa _ hb;have U:=gO x (G0 (r*n+n)) (G0 (r*n)) 255;simp only[A,B,kA]at U;have E:=ek x (cS x (G0 (r*n)) (G0 (r*n+n)) 255) (G0 (r*n+n)) (G0 (r*n)) 0 0 0 0 0 0 0 0;simp only[kA,ϑ,A,B,U]at E;rw[U,<-E.1,<-E.2]
@[noinline]def gP(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(i j:Ἀ):Ἃ:=
 let t:=YU n eps x (i/n);ZT ((ZA x[i]!-sg t.2.2.1 (t.2.2.2.1*2^t.2.1+t.2.2.2.2.1:Ἀ)*2^((t.1:Ἁ)-149)/n)*ZA g[j]!/ZA t.2.2.2.2.2+ZA b[j]!)
@[noinline]def nFBz(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(nd:Ἅ)(z:NR.fH)(i j:Ἄ):Ἃ:=
 if z.ok then NE.el (ck x i) (ck g j) (ck b j) nd z.nS z.R z.pz z.A1 z.A2 z.C1 (λ_=>gP n eps x g b (J0 i) (J0 j)) else gP n eps x g b (J0 i) (J0 j)
def nS1(x:(Ἂ Word))(i e:Ἄ)(s:Ἅ)(Dl:ἆ):Ἅ×ἆ:=
 if h:i<e then let w:=ck x i;nS1 x (i+1) e (s+Dense.F.dw w) (if Dl<nfe w then nfe w else Dl) else (s,Dl)
termination_by (J0 e)-(J0 i)
decreasing_by have:=ε.mp h;have:=e.toNat_lt_size;rw[uS ((by omega))];omega
def nS2(x:(Ἂ Word))(m:Ἅ)(i e:Ἄ)(v:Ἅ):Ἅ:=
 if h:i<e then let d:=Dense.F.dw (ck x i)-m;nS2 x m (i+1) e (v+d*d) else v
termination_by (J0 e)-(J0 i)
decreasing_by have:=ε.mp h;have:=e.toNat_lt_size;rw[uS ((by omega))];omega
structure nLS where
 m:Ἅ
 ri:Ἅ
 Kp:ἆ
 Ke:ἆ
 fast:Ἆ
 th:Thunk NR.fH
 dW:Ἃ
@[inline]def nRi(dW:Ἃ)(ed:ἆ):Ἅ:=
 let m:ἆ:=(dW&&&0x7FFFFF).toUInt64+0x800000
 let t63:=(1:ἆ)<<<63
 let q1:=t63/m
 let q2:=((t63-q1*m)<<<12)/m
 ((q1<<<12)+q2).toInt64.toFloat*N5 ((949-ed)<<<52)
@[inline]def nLc(n Dl ef:Ἀ):Ἀ:=max (Aq (n+4)+5) (2*(Aq (n+4)+1)+2*Dl-(ef+178))+2
@[noinline]def nEI(n:Ἀ)(eps:Ἃ)(x:(Ἂ Word))(r:Ἀ):NR.fH:=NR.eI n (gM n eps x r)
noncomputable def nLR(n:Ἀ)(eps:Ἃ)(x:(Ἂ Word))(r:Ἀ):nLS:=
 let th:=Thunk.mk (λ_=>nEI n eps x r)
 let lo:=G0 (r*n);let hi:=G0 (r*n+n)
 if 4<=n&&n<=2048&&kB eps != 0&&kB eps != 255 then
  let z0:=(I0 (0:Ἀ)).toInt64.toFloat
  let p:=nS1 x lo hi z0 0
  let n6:=I0 n
  let lb:=(Aq n).toUInt64
  let m:=p.1*((((1:ἆ)<<<(52+lb))/n6).toInt64.toFloat*N5 ((971-lb)<<<52))
  let v:=nS2 x m lo hi z0
  let D:=W7 (v*(((1:ἆ)<<<52)/n6).toInt64.toFloat+Dense.F.dw eps*nS201)
  let Dl:=p.2.toNat
  let B:=D.toBits
  let dW:=nWs 26 B
  let ed:=kB dW
  ⟨m,nRi dW ed.toUInt64,nElK m (Aq (n+4)+1) Dl ed.toNat,(1054-ed.toNat).toUInt64,
   nOKs B (nLc n Dl (kB eps).toNat).toUInt64&&0<ed&&ed<255&&!ZX dW,th,dW⟩
 else ⟨0,0,0,0,ἄ,th,0⟩
noncomputable def nLo(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(nd:Ἅ)(e:nLS)(i j:Ἄ):Ἃ:=
 let z:=e.th.get
 if z.ok then NE.el (ck x i) (ck g j) (ck b j) nd z.nS z.R z.pz z.A1 z.A2 z.C1 λ_=>gP n eps x g b (J0 i) (J0 j)
 else gP n eps x g b (J0 i) (J0 j)
lemma nM_e(a r A t B u w k:ℝ)(u0:0<=u)(w0:0<=w)(k0:0<=k)(B0:0<=B)(ht:|t|<=B)(hA:|A|<=k*B)
  (sm:(k^(2:Ἀ)/2+k)*u<=1/4)(sw:k*w<=B/4)
  (H1:|a-A|<=(k^(2:Ἀ)/2+k)*B*u+k*w)(H2:|r-(a+t)|<=u*|a+t|+w):
  |r-(A+t)|<=((k+1)^(2:Ἀ)/2+(k+1))*B*u+(k+1)*w:=by
 have h:=AY (a+t) (A+t);rw[add_sub_add_right_eq_sub]at h
 have E2:=mul_le_mul_of_nonneg_left sm B0
 have ua:=mul_le_mul_of_nonneg_left (show |a+t|<=(k+1)*B+B/2 by linear_combination h+Ag A t+hA+ht+H1+E2+sw) u0
 have e3:=abs_sub_le r (a+t) (A+t);rw[add_sub_add_right_eq_sub]at e3
 linear_combination e3+H2+H1+ua
lemma nM_b_h1(P Q B:ℝ)(hP:P<=1/4*B)(hQ:Q<=B/4)(B0:0<=B):P+Q+B<=2*B:=by linear_combination hP+hQ+(1/2:ℝ)*B0
lemma nM_b(a A t B u w k:ℝ)(u0:0<=u)(B0:0<=B)(ht:|t|<=B)(hA:|A|<=k*B)
  (sm:(k^(2:Ἀ)/2+k)*u<=1/4)(sw:k*w<=B/4)(H1:|a-A|<=(k^(2:Ἀ)/2+k)*B*u+k*w):|a+t|<=(k+2)*B:=
 (Ag a t).trans ((add_le_add ((sub_le_iff_le_add'.1 (AY a A)).trans (add_le_add hA H1)) ht).trans (((add_assoc _ _ _).trans_le (add_le_add AP (nM_b_h1 _ _ _ ((mul_right_comm _ B u).trans_le (AE sm B0)) sw B0))).trans_eq (add_mul _ _ _).symm))
lemma nKb(w:kF)(h:kB w≠255):(mJ (kB w))<=254:=by
 have a:(mJ (kB w))<=255:=by
  show (mJ ((w>>>23)&&&0xFF))<=255;rw[Ξ];exact ϝ.trans (by decide)
 have b:(mJ (kB w))≠255:=fun q=>h (Β.1 (by rw[q];rfl))
 omega
lemma nS1_dl(x:(Ἂ Word))(e:Ἄ)(hX:∀j,j<(J0 e)->kB x[j]!≠255):∀m (i:Ἄ)(s:Ἅ)(Dl:ἆ),(J0 e)-(J0 i)=m->(J0 i)<=(J0 e)->(lq Dl)<=254->
  (lq Dl)<=(lq (nS1 x i e s Dl).2)/\(lq (nS1 x i e s Dl).2)<=254/\∀j,(J0 i)<=j->j<(J0 e)->(mJ (kB x[j]!))<=(lq (nS1 x i e s Dl).2):=by
 intro m;induction m with
 |zero=>
  intro i s Dl hm hie hD
  have q:=not_lt.2 (K3.1 hm)
  rw[nS1,dif_neg (fun h=>q (ε.1 h))];exact⟨AP,hD,fun j a b=>(q (AO a b)).elim⟩
 |succ m ih=>
  intro i s Dl hm hie hD
  have hl:(J0 i)<(J0 e):=Nat.lt_of_sub_pos (hm▸m.succ_pos)
  rw[nS1,dif_pos (ε.2 hl)];dsimp only;rw[f6 x i]
  have i1:=Dense.ac (ε.2 hl)
  have e1:(lq (nfe x[(J0 i)]!))=(mJ (kB x[(J0 i)]!)):=ϩ _
  have mx:(lq (if Dl<nfe x[(J0 i)]! then nfe x[(J0 i)]! else Dl))=max (lq Dl) (mJ (kB x[(J0 i)]!)):=by
   split_ifs with c<;>rw[μ,e1]at c
   · rw[e1];exact (max_eq_right c.le).symm
   · exact (max_eq_left (not_lt.1 c)).symm
  obtain⟨a1,a2,a3⟩:=ih (i+1) (s+Dense.F.dw x[(J0 i)]!) _ (by rw[i1,V4,hm,BK]) (i1▸hl) (mx▸max_le hD (nKb _ (hX _ hl)))
  rw[mx]at a1
  refine⟨AZ (le_max_left ..) a1,a2,fun j b c=>?_⟩
  rcases BJ b with q|q
  · rw[<-q];exact AZ (le_max_right ..) a1
  · exact a3 j (i1▸q) c
lemma nS1_ge(x:(Ἂ Word))(e:Ἄ):∀m (i:Ἄ)(s:Ἅ)(Dl:ἆ),(J0 e)-(J0 i)=m->(J0 i)<=(J0 e)->
  (lq Dl)<=(lq (nS1 x i e s Dl).2)/\∀j,(J0 i)<=j->j<(J0 e)->(mJ (kB x[j]!))<=(lq (nS1 x i e s Dl).2):=by
 intro m;induction m with
 |zero=>
  intro i s Dl hm hie
  have q:=not_lt.2 (K3.1 hm)
  rw[nS1,dif_neg (fun h=>q (ε.1 h))];exact⟨AP,fun j a b=>(q (AO a b)).elim⟩
 |succ m ih=>
  intro i s Dl hm hie
  have hl:(J0 i)<(J0 e):=Nat.lt_of_sub_pos (hm▸m.succ_pos)
  rw[nS1,dif_pos (ε.2 hl)];dsimp only;rw[f6 x i]
  have i1:=Dense.ac (ε.2 hl)
  have e1:(lq (nfe x[(J0 i)]!))=(mJ (kB x[(J0 i)]!)):=ϩ _
  have mx:(lq (if Dl<nfe x[(J0 i)]! then nfe x[(J0 i)]! else Dl))=max (lq Dl) (mJ (kB x[(J0 i)]!)):=by
   split_ifs with c<;>rw[μ,e1]at c
   · rw[e1];exact (max_eq_right c.le).symm
   · exact (max_eq_left (not_lt.1 c)).symm
  obtain⟨a1,a3⟩:=ih (i+1) (s+Dense.F.dw x[(J0 i)]!) _ (by rw[i1,V4,hm,BK]) (i1▸hl)
  rw[mx]at a1
  refine⟨AZ (le_max_left ..) a1,fun j b c=>?_⟩
  rcases BJ b with q|q
  · rw[<-q];exact AZ (le_max_right ..) a1
  · exact a3 j (i1▸q) c
lemma nS1_s(x:(Ἂ Word))(e:Ἄ)(hX:∀j,j<(J0 e)->kB x[j]!≠255)(B:ℝ)(B0:2^(23:Ἀ)<=B)(B1:B<=2^(300:Ἀ)):∀m (i:Ἄ)(s:Ἅ)(Dl:ἆ)(A:ℝ)(k:ℕ),(J0 e)-(J0 i)=m->(J0 i)<=(J0 e)->Dense.F.fv s->|A|<=k*B->
  |Dense.F.rv s-A|<=((k:ℝ)^(2:Ἀ)/2+k)*B*(1/2^(53:Ἀ))+k*(1/2^(1075:Ἀ))->k+m<=2^(12:Ἀ)->(∀j,(J0 i)<=j->j<(J0 e)->|(sv x[j]!:ℝ)*2^kG x[j]!|<=B)->
  Dense.F.fv (nS1 x i e s Dl).1/\|Dense.F.rv (nS1 x i e s Dl).1-(A+∑j∈B5 (J0 i) (J0 e),(sv x[j]!:ℝ)*2^kG x[j]!)|<=(((k+m:ℕ):ℝ)^(2:Ἀ)/2+((k+m:ℕ):ℝ))*B*(1/2^(53:Ἀ))+((k+m:ℕ):ℝ)*(1/2^(1075:Ἀ)):=by
 have Bp:(0:ℝ)<=B:=AZ (by norm_num) B0
 intro m;induction m with
 |zero=>
  intro i s Dl A k hm hie hf hA he hk hb
  have q:(J0 i)=(J0 e):=C4 hie (K3.1 hm)
  rw[nS1,dif_neg (by rw[ε,q];exact B4 _),q,Finset.Ico_self,Finset.sum_empty,Av,ϑ];exact⟨hf,he⟩
 |succ m ih=>
  intro i s Dl A k hm hie hf hA he hk hb
  have hl:(J0 i)<(J0 e):=N1 hm
  rw[nS1,dif_pos (ε.2 hl)];dsimp only;rw[f6 x i]
  obtain⟨tf,tr⟩:=Dense.F.cI x[(J0 i)]! (hX _ hl)
  have hs:=Finset.sum_eq_sum_Ico_succ_bot hl (fun j=>(sv x[j]!:ℝ)*2^kG x[j]!)
  have i1:(J0 (i+1))=(J0 i)+1:=Dense.ac (ε.2 hl)
  have ht:=hb _ AP hl
  have kk:(k:ℝ)<=2^(12:Ἀ):=by exact_mod_cast (show k<=2^(12:Ἀ) from (V1 _ _).trans hk)
  have k0:(0:ℝ)<=k:=M3 k
  have sm:((k:ℝ)^(2:Ἀ)/2+k)*(1/2^(53:Ἀ))<=1/4:=by
   have h1:(k:ℝ)^(2:Ἀ)<=2^(12:Ἀ)*2^(12:Ἀ):=by rw[sq];exact mul_le_mul kk kk k0 (by norm_num)
   have h2:((k:ℝ)^(2:Ἀ)/2+k)<=2^(23:Ἀ)+2^(12:Ἀ):=by linear_combination h1/2+kk
   exact (AE h2 (nN 53)).trans (by norm_num)
  have sw:(k:ℝ)*(1/2^(1075:Ἀ))<=B/4:=by
   have h1:(k:ℝ)*(1/2^(1075:Ἀ))<=2^(12:Ἀ)*(1/2^(1075:Ἀ)):=AE kk (nN _)
   have h2:(2:ℝ)^(12:Ἀ)*(1/2^(1075:Ἀ))<=1:=nP 12 1075 (by decide)
   exact h1.trans (h2.trans (by linear_combination B0/4))
  have ab:=nM_b (Dense.F.rv s) A _ B (1/2^(53:Ἀ)) (1/2^(1075:Ἀ)) k (nN _) Bp ht hA sm sw he
  have bd:|Dense.F.rv s+Dense.F.rv (Dense.F.dw x[(J0 i)]!)|<=(2:ℝ)^(971:ℤ):=by
   rw[tr,zpow_ofNat];refine ab.trans ((mul_le_mul_of_nonneg_left B1 (by linear_combination k0)).trans ?_)
   have h1:((k:ℝ)+2)*2^(300:Ἀ)<=2^(13:Ἀ)*2^(300:Ἀ):=AE (by linear_combination kk) (by positivity)
   exact h1.trans (by rw[<-AF];exact AU one_le_two (by decide))
  obtain⟨af,ae⟩:=NF.fA _ _ hf tf bd
  rw[tr]at ae
  have step:=nM_e _ _ A _ B (1/2^(53:Ἀ)) (1/2^(1075:Ἀ)) k (nN _) (nN _) k0 Bp ht hA sm sw he (ae.trans_eq (by rw[one_div_mul_eq_div]))
  have hA2:|A+(sv x[(J0 i)]!:ℝ)*2^kG x[(J0 i)]!|<=((k+1:ℕ):ℝ)*B:=by
   rw[X7,add_one_mul];exact (Ag _ _).trans (add_le_add hA ht)
  have:=ih (i+1) _ (if Dl<nfe x[(J0 i)]! then nfe x[(J0 i)]! else Dl) (A+(sv x[(J0 i)]!:ℝ)*2^kG x[(J0 i)]!) (k+1) (by rw[i1,V4,hm,BK]) (i1.le.trans (l7 hl)) af hA2 (by rw[X7];exact step) (by rw[pA,Τ 1 m];exact hk) (fun j h1 h2=>hb j (X5 (i1.symm.le.trans h1)) h2)
  rw[i1,pA k 1 m,Τ 1 m,add_assoc,<-hs]at this;exact this
lemma nMn(s Q Ri m n B u w L2:ℝ)(n4:4<=n)(B0:0<=B)(u0:0<=u)(w0:0<=w)(hQ:|Q|<=B)
  (hs:|s-n*Q|<=(n^(2:Ἀ)/2+n)*B*u+n*w)(R0:0<=Ri)(hR1:n*Ri<=1)(hR2:1-4*u<=n*Ri)
  (hm:|m-s*Ri|<=u*|s*Ri|+w)(sm:(n/2+5)*u<=1/8)(sw:w<=B*u/8)(hL:n+5<=L2):
  |m-Q|<=L2*B*u:=by
 have id:Q-s*Ri=Q*(1-n*Ri)+(n*Q-s)*Ri:=by ring
 have n0:(0:ℝ)<=n:=zero_le_four.trans n4
 have r0:(0:ℝ)<=1-n*Ri:=sub_nonneg.2 hR1
 have t1:|Q*(1-n*Ri)|<=B*(4*u):=by rw[Ae,Ad r0];exact mul_le_mul hQ (sub_le_comm.1 hR2) r0 B0
 have t2:|(n*Q-s)*Ri|<=(n/2+1)*B*u+w:=by
  rw[Ae,Aw,Ad R0]
  have e:(n^(2:Ἀ)/2+n)*B*u+n*w=n*((n/2+1)*B*u+w):=by ring
  have p0:0<=(n/2+1)*B*u+w:=add_nonneg (Af (Af (add_nonneg (div_nonneg n0 zero_le_two) zero_le_one) B0) u0) w0
  exact (AE (hs.trans_eq e) R0).trans ((mul_right_comm _ _ _).trans_le (mul_le_of_le_one_left p0 hR1))
 have d1:|s*Ri-Q|<=B*(4*u)+((n/2+1)*B*u+w):=by rw[Aw,id];exact (Ag _ _).trans (add_le_add t1 t2)
 have e1:=AE sm B0
 have u1:u<=1:=(le_mul_of_one_le_left u0 (le_add_of_nonneg_of_le (div_nonneg n0 zero_le_two) (by norm_num:(1:ℝ)<=5))).trans (sm.trans (by norm_num))
 have e0:B*u<=B:=mul_le_of_le_one_right B0 u1
 have d2:B*(4*u)+((n/2+1)*B*u+w)<=B/4:=by linear_combination e1+sw+(1/8)*e0
 have sr:|s*Ri|<=B+B/4:=(sub_le_iff_le_add'.1 (AY (s*Ri) Q)).trans (add_le_add hQ (d1.trans d2))
 have F:=(abs_sub_le m (s*Ri) Q).trans (add_le_add (hm.trans (add_le_add (mul_le_mul_of_nonneg_left sr u0) AP)) d1)
 have e2:(n+5)*B*u<=L2*B*u:=AE (AE hL B0) u0
 have e7:=Af B0 u0
 have e4:=AE n4 e7
 exact F.trans (by linear_combination e2+2*sw+(1/2)*e4+(1/2)*e7)
lemma nR1(n:Ἀ)(h1:1<=n)(h2:n<=2048):
  (lq (((1:ἆ)<<<(52+(Aq n).toUInt64))/I0 n))=2^(52+Aq n)/n/\2^(Aq n)<=n/\n<2^(Aq n+1)/\Aq n<=11:=by
 have hn:=L6 h1
 have l3:Aq n<=11:=J2 ((m9 hn).2 (AO h2 (by decide)))
 refine⟨?_,ϰ hn,ϙ,l3⟩
 have a:(lq (I0 n))=n:=Α (AO h2 (by decide))
 have c:=J1 l3 52
 have b:(lq (52+(Aq n).toUInt64))=52+Aq n:=by
  rw[η];simp only[Ϣ,mY,O6]
  rw[γ (AO l3 (by decide)),γ (show 52<2^(64:Ἀ) by decide),γ (AO c (by decide))]
 rw[ml,a];simp only[ψ,b,pS]
 rw[γ (Z4 c),na,γ (lu Nat.one_lt_two (Z4 c))]
lemma nMeq(n:Ἀ)(x:(Ἂ Word))(r:Ἀ)(hn:0<n):((no n x r:ℚ):ℝ)*2^(149:Ἀ)=(∑j∈B5 (r*n) (r*n+n),(sv x[j]!:ℝ)*2^kG x[j]!)/n:=by
 have h2:∀w:kF,((ZA w:ℚ):ℝ)*2^(149:Ἀ)=(sv w:ℝ)*2^kG w:=by
  intro w;rw[Y1,J7,S8,Rat.cast_intCast,G7,Ap,<-zpow_ofNat,<-AK AL,sub_add_cancel,AA]
 unfold rowMean
 rw[Yk,Ai,Rat.cast_div,Rat.cast_sum,T7,div_mul_eq_mul_div,n8,Finset.sum_Ico_eq_sum_range,ϱ]
 exact congrArg (fun z:ℝ=>z/n) (ϔ rfl fun j _=>h2 _)
section
open NF Dense.F NE
lemma nPwF(f:ἆ)(h1:1<=(lq f))(h2:(lq f)<=2046):
  (lq (((f<<<52)>>>52)&&&2047))=(lq f)/\((f<<<52)>>>63!=0)=false/\(lq ((f<<<52)&&&0xfffffffffffff))=0:=by
 have m:(lq f)*2^(52:Ἀ)<=2046*2^(52:Ἀ):=ι _ h2
 have q:(lq (f<<<52))=(lq f)*2^(52:Ἀ):=by
  rw[ψ];simp only[kA,ο,kH];exact γ (m.trans_lt (by decide))
 have d:(lq ((f<<<52)>>>52))=(lq f):=by
  rw[ν,q];simp only[β,kA,kH];exact Nat.mul_div_cancel _ (϶ _)
 refine⟨?_,?_,?_⟩
 · rw[Ε,d];simp only[kA]
   rw[show (2047:Nat)=2^(11:Ἀ)-1 from rfl,ζ];exact γ (h2.trans_lt (by decide))
 · have:(lq ((f<<<52)>>>63))=0:=by
     rw[ν,q];simp only[β,kA,kH]
     exact ώ (m.trans_lt (by decide))
   have g:(f<<<52)>>>63=0:=ξ.mp (by rw[this];rfl)
   rw[g];rfl
 · rw[Ε,q];simp only[kA]
   rw[show (4503599627370495:Nat)=2^(52:Ἀ)-1 from rfl,ζ];simp only [nr,Nat.mul_mod_left]
lemma nPw(f:ἆ)(h1:1<=(lq f))(h2:(lq f)<=2046):fv (N5 (f<<<52))/\rv (N5 (f<<<52))=(2:ℝ)^(((lq f):ℤ)-1023):=by
 obtain⟨f1,f2,f3⟩:=nPwF f h1 h2
 have ne:=R8 (Z4 h2)
 have K:kI (N5 (f<<<52)).toModel=Model.ofFields kC false (lq f) 0:=by rw[s5ofb _ fun g=>ne (f1.symm.trans (congrArg lq g)),s5fld,f1,f2,f3]
 have b1:(lq f)<2^kC.expWidth:=K6 (Z4 h2)
 have b2:0<2^kC.fracWidth:=϶ _
 have fin:Model.isFinite (Model.ofFields kC false (lq f) 0)=true:=by
  rw[Model.isFinite_ofFields kC _ _ _ b1 b2];exact AM.2 ne
 refine⟨by unfold fv;rw[K];exact fin,?_⟩
 unfold rv;rw[K,BR,Model.toDyadic?_ofFields_of_isFinite _ _ _ _ b1 b2 fin,if_neg (L6 h1)]
 simp only[show kC.fracWidth=52 from rfl,show kC.exponentBias=1023 from rfl,Model.pow2_eq_two_pow]
 rw[bJ];simp only[υ,if_false,ϑ]
 exact s5pw 52 _
end
lemma nZb(z k:ℕ)(h:(z:ℝ)<=2^(52:Ἀ)):|((z:ℤ):ℝ)*(2:ℝ)^(-(k:ℤ))|<=(2:ℝ)^(971:ℤ):=by
 rw[M5,zpow_neg,AA,zpow_ofNat]
 have p:(0:ℝ)<2^k:=pow_pos two_pos k
 rw[Ad (Af (M3 z) (inv_nonneg.2 p.le))]
 exact (mul_le_of_le_one_right (M3 z) (inv_le_one_of_one_le₀ (one_le_pow₀ one_le_two))).trans (h.trans (AU one_le_two (by norm_num)))
lemma nRiF(n:Ἀ)(h1:1<=n)(h2:n<=2048):
  Dense.F.fv ((((1:ἆ)<<<(52+(Aq n).toUInt64))/I0 n).toInt64.toFloat*N5 ((971-(Aq n).toUInt64)<<<52))/\
  Dense.F.rv ((((1:ἆ)<<<(52+(Aq n).toUInt64))/I0 n).toInt64.toFloat*N5 ((971-(Aq n).toUInt64)<<<52))=((2^(52+Aq n)/n:ℕ):ℝ)/2^(52+Aq n):=by
 obtain⟨a,l1,l2,l3⟩:=nR1 n h1 h2
 have R2:2^(52+Aq n)/n<=2^(52:Ἀ):=by
  have:2^(52+Aq n)/n<=2^(52+Aq n)/2^(Aq n):=pq l1 (϶ _)
  rw[AF,Nat.mul_div_cancel _ (϶ _)]at this;rwa[AF]
 have R3:2^(52+Aq n)/n<2^(53:Ἀ):=R2.trans_lt (by decide)
 have ue:((1:ἆ)<<<(52+(Aq n).toUInt64))/I0 n=I0 (2^(52+Aq n)/n):=by
  apply ξ.1;rw[a];simp only[Ϣ,O6];exact (γ (R3.trans (by decide))).symm
 obtain⟨rf,rr⟩:=kg2 (2^(52+Aq n)/n) R3
 have pt:(lq (971-(Aq n).toUInt64))=971-Aq n:=by
  have b:(lq (Aq n).toUInt64)=Aq n:=by simp only[Ϣ,O6];exact γ (l3.trans_lt (by decide))
  rw[χ _ _ (by rw[θ,b];simp only[kA];exact l3.trans (by decide)),b];simp only[kA]
 obtain⟨pf,pr⟩:=nPw (971-(Aq n).toUInt64) (by rw[pt];exact G5 ((J1 l3 1).trans (by decide))) (by rw[pt];exact (K4 _ _).trans (by decide))
 rw[pt]at pr
 rw[ue]
 have ez:((((971-Aq n):ℕ):ℤ)-1023)=-(((52+Aq n):ℕ):ℤ):=by rw[pO (l3.trans (by decide)),W5];ring
 have r1:((2^(52+Aq n)/n:ℕ):ℝ)<=2^(52:Ἀ):=by have:=(R5 (α:=ℝ)).2 R2;rwa[I4,nZ]at this
 have q:=Dense.F.aE _ _ rf pf ((2^(52+Aq n)/n:ℕ):ℤ) (-(((52+Aq n):ℕ):ℤ)) (by rw[rr,pr,ez];push_cast;rfl) R3 (Int.neg_le_neg (U4.2 ((J1 l3 52).trans (by decide)))) (nZb _ _ r1)
 refine⟨q.1,?_⟩;rw[q.2,zpow_neg,AA,M5,div_eq_mul_inv]
lemma nRb_h1(nR R1 P T:ℝ)(h:P<nR*(R1+1))(hn:nR<T)(hT:T*2^(51:Ἀ)=P):(1-4*(1/2^(53:Ἀ)))*P<=nR*R1:=by
 have q:(4:ℝ)*(1/2^(53:Ἀ))=1/2^(51:Ἀ):=by norm_num
 rw[q,sub_mul,one_mul,one_div_mul_eq_div,div_eq_of_eq_mul (pow_ne_zero _ AL) hT.symm];linear_combination h+hn
lemma nRb(nR R1 P T:ℝ)(P0:0<P)(h:P<nR*(R1+1))(hn:nR<T)(hT:T*2^(51:Ἀ)=P):1-4*(1/2^(53:Ἀ))<=nR*(R1/P):=
 ((le_div_iff₀ P0).2 (nRb_h1 _ _ _ _ h hn hT)).trans_eq (mul_div_assoc _ _ _)
lemma nMean(n:Ἀ)(x:(Ἂ Word))(r:Ἀ)(hn4:4<=n)(hn:n<=2048)(hb:r*n+n<2^(32:Ἀ))(hX:∀j,j<r*n+n->kB x[j]!≠255):
  Dense.F.fv ((nS1 x (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat 0).1*((((1:ἆ)<<<(52+(Aq n).toUInt64))/I0 n).toInt64.toFloat*N5 ((971-(Aq n).toUInt64)<<<52)))/\
  |Dense.F.rv ((nS1 x (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat 0).1*((((1:ἆ)<<<(52+(Aq n).toUInt64))/I0 n).toInt64.toFloat*N5 ((971-(Aq n).toUInt64)<<<52)))-((no n x r:ℚ):ℝ)*2^(149:Ἀ)|
   <=2^(((Aq (n+4)+1:ℕ):ℤ)+(lq (nS1 x (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat 0).2)-30)/\
  (∀j,r*n<=j->j<r*n+n->(mJ (kB x[j]!))<=(lq (nS1 x (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat 0).2))/\
  (lq (nS1 x (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat 0).2)<=254/\
  |((no n x r:ℚ):ℝ)*2^(149:Ἀ)|<=2^((lq (nS1 x (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat 0).2)+23)/\
  (∀j,r*n<=j->j<r*n+n->|(sv x[j]!:ℝ)*2^kG x[j]!|<=2^((lq (nS1 x (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat 0).2)+23)):=by
 have h1n:1<=n:=U2 (K2 1 3) hn4
 have lo:(J0 (G0 (r*n)))=r*n:=Conv.aa _ (C0 (V1 _ _) hb)
 have hi:(J0 (G0 (r*n+n)))=r*n+n:=Conv.aa _ hb
 have hX':∀j,j<(J0 (G0 (r*n+n)))->kB x[j]!≠255:=fun j h=>hX j (by rwa[hi]at h)
 obtain⟨z0f,z0r⟩:=kg2 0 (by norm_num)
 obtain⟨-,d2,d3⟩:=nS1_dl x _ hX' n (G0 (r*n)) (I0 (0:Ἀ)).toInt64.toFloat 0 ((congrArg₂ (·-·) hi lo).trans (ϱ _ _)) (lo.trans_le ((V1 _ _).trans_eq hi.symm)) (by simp only [ϋ,zero_le])
 rw[lo,hi]at d3
 obtain⟨rif,rir⟩:=nRiF n h1n hn
 obtain⟨-,-,l2,-⟩:=nR1 n h1n hn
 have SS:=fun (B:ℝ) (B0:2^(23:Ἀ)<=B) (B1:B<=2^(300:Ἀ)) (hB:∀j,(J0 (G0 (r*n)))<=j->j<(J0 (G0 (r*n+n)))->|(sv x[j]!:ℝ)*2^kG x[j]!|<=B)=>nS1_s x _ hX' B B0 B1 n (G0 (r*n)) (I0 (0:Ἀ)).toInt64.toFloat 0 0 0 ((congrArg₂ (·-·) hi lo).trans (ϱ _ _)) (lo.trans_le ((V1 _ _).trans_eq hi.symm)) z0f (by rw[A5,Nat.cast_zero,zero_mul]) (by rw[z0r,Nat.cast_zero,sub_self,A5,zero_pow AL,zero_div,Av,zero_mul,zero_mul,zero_mul,Av]) (U2 (B3 (Κ n)) (U2 hn (N2 rfl))) hB
 generalize nS1 x (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat 0=P at d2 d3 SS ⊢
 generalize lq P.2=Dl at d2 d3 ⊢
 have hB:∀j,r*n<=j->j<r*n+n->|(sv x[j]!:ℝ)*2^kG x[j]!|<=2^(Dl+23):=by
  intro j h1 h2;obtain⟨-,cr⟩:=Dense.F.cI x[j]! (hX j h2);rw[<-cr]
  exact (nXb _ (hX j h2)).le.trans (AU one_le_two (L1 (d3 j h1 h2) 23))
 have B0:(2:ℝ)^(23:Ἀ)<=2^(Dl+23):=AU one_le_two (K2 23 Dl)
 have B1:(2:ℝ)^(Dl+23)<=2^(300:Ἀ):=AU one_le_two (U2 (L1 d2 23) (N2 rfl))
 have Bp:(0:ℝ)<=2^(Dl+23):=pow_nonneg zero_le_two _
 obtain⟨sf,se⟩:=SS _ B0 B1 (fun j h1 h2=>hB j (by rwa[lo]at h1) (by rwa[hi]at h2))
 rw[lo,hi,Ai,Κ]at se
 obtain⟨S,hS⟩:∃S,∑j∈B5 (r*n) (r*n+n),(sv x[j]!:ℝ)*2^kG x[j]! = S:=⟨_,rfl⟩
 rw[hS]at se
 have nR:(0:ℝ)<n:=Nat.cast_pos.2 h1n
 have nR4:(4:ℝ)<=n:=by exact_mod_cast hn4
 have nR2:(n:ℝ)<=2048:=by exact_mod_cast hn
 have Sb:|S|<=n*2^(Dl+23):=by
  have h:=Finset.sum_le_card_nsmul (B5 (r*n) (r*n+n)) (fun j=>|(sv x[j]!:ℝ)*2^kG x[j]!|) (2^(Dl+23)) (fun j hj=>by rw[Finset.mem_Ico]at hj;exact hB j hj.1 hj.2)
  rw[Nat.card_Ico,ϱ,nsmul_eq_mul]at h
  exact hS▸(K8 _ _).trans h
 have Qe:=nMeq n x r h1n
 rw[hS]at Qe
 have hQ:|((no n x r:ℚ):ℝ)*2^(149:Ἀ)|<=2^(Dl+23):=by rw[Qe,abs_div,AW nR,div_le_iff₀ nR];exact Sb.trans_eq (mul_comm _ _)
 have nQ:(n:ℝ)*(((no n x r:ℚ):ℝ)*2^(149:Ἀ))=S:=by rw[Qe,mul_div_cancel₀ _ nR.ne']
 have sm:((n:ℝ)^(2:Ἀ)/2+n)*(1/2^(53:Ἀ))<=1/4:=by
  have h1:(n:ℝ)^(2:Ἀ)<=2048*2048:=by rw[sq];exact mul_le_mul nR2 nR2 nR.le (by norm_num)
  have h2:((n:ℝ)^(2:Ἀ)/2+n)<=2048*2048/2+2048:=add_le_add (div_le_div_of_nonneg_right h1 zero_le_two) nR2
  exact (AE h2 (nN 53)).trans (by norm_num)
 have sw:(n:ℝ)*(1/2^(1075:Ἀ))<=2^(Dl+23)/4:=by
  have h1:(n:ℝ)*(1/2^(1075:Ἀ))<=2^(12:Ἀ)*(1/2^(1075:Ἀ)):=AE (by linarith only[nR2]) (nN _)
  have h2:(2:ℝ)^(12:Ἀ)*(1/2^(1075:Ἀ))<=1:=nP 12 1075 (by norm_num)
  have h3:(1:ℝ)<=2^(23:Ἀ)/4:=by norm_num
  exact h1.trans (h2.trans (h3.trans (div_le_div_of_nonneg_right B0 (by norm_num))))
 have sb:=nM_b (Dense.F.rv P.1) S 0 (2^(Dl+23)) (1/2^(53:Ἀ)) (1/2^(1075:Ἀ)) n (nN _) Bp (by rw[A5];exact Bp) Sb sm sw se
 rw[Av]at sb
 obtain⟨R1,hR1d⟩:∃R1,2^(52+Aq n)/n=R1:=⟨_,rfl⟩
 rw[hR1d]at rir
 have P0:(0:ℝ)<2^(52+Aq n):=pow_pos two_pos _
 have R0:(0:ℝ)<=Dense.F.rv ((((1:ἆ)<<<(52+(Aq n).toUInt64))/I0 n).toInt64.toFloat*N5 ((971-(Aq n).toUInt64)<<<52)):=by rw[rir];exact div_nonneg (M3 _) P0.le
 have nR1:=(R5 (α:=ℝ)).2 (hR1d▸pr (2^(52+Aq n)) n)
 rw[Y4,I4,nZ]at nR1
 have hR1:(n:ℝ)*Dense.F.rv ((((1:ἆ)<<<(52+(Aq n).toUInt64))/I0 n).toInt64.toFloat*N5 ((971-(Aq n).toUInt64)<<<52))<=1:=by
  rw[rir,mul_div_assoc',mul_comm (n:ℝ),div_le_one P0];exact nR1
 have lt1:=(Nat.cast_lt (α:=ℝ)).2 (hR1d▸Nat.lt_mul_div_succ (2^(52+Aq n)) h1n)
 rw[Y4,I4,nZ,W5,pJ]at lt1
 have lt2:(n:ℝ)<2^(Aq n+1):=by exact_mod_cast l2
 have hR2:=nRb n R1 _ _ P0 lt1 lt2 (by rw[<-AF,pA,Τ])
 rw[<-rir]at hR2
 have Ri1:Dense.F.rv ((((1:ἆ)<<<(52+(Aq n).toUInt64))/I0 n).toInt64.toFloat*N5 ((971-(Aq n).toUInt64)<<<52))<=1:=by
  exact (le_mul_of_one_le_left R0 ((show (1:ℝ)<=4 by norm_num).trans nR4)).trans hR1
 have fb:|Dense.F.rv P.1*Dense.F.rv ((((1:ἆ)<<<(52+(Aq n).toUInt64))/I0 n).toInt64.toFloat*N5 ((971-(Aq n).toUInt64)<<<52))|<=(2:ℝ)^(971:ℤ):=by
  rw[Ae,Ad R0,zpow_ofNat]
  refine (mul_le_of_le_one_right (Aa _) Ri1).trans (sb.trans ((mul_le_mul_of_nonneg_left B1 (by linarith only[nR])).trans ?_))
  have h1:((n:ℝ)+2)*2^(300:Ἀ)<=2^(12:Ἀ)*2^(300:Ἀ):=AE ((add_le_add nR2 (le_refl 2)).trans (show (2048:ℝ)+2<=2^(12:Ἀ) by norm_num)) (pow_nonneg zero_le_two _)
  exact h1.trans (by rw[<-AF];exact AU one_le_two (by norm_num))
 obtain⟨mf,me⟩:=NF.fM _ _ sf rif fb
 have sm2:((n:ℝ)/2+5)*(1/2^(53:Ἀ))<=1/8:=(AE ((add_le_add (div_le_div_of_nonneg_right nR2 zero_le_two) (le_refl 5)).trans (show (2048:ℝ)/2+5<=1029 by norm_num)) (nN 53)).trans (show (1029:ℝ)*(1/2^(53:Ἀ))<=1/8 by norm_num)
 have sw2:(1:ℝ)/2^(1075:Ἀ)<=2^(Dl+23)*(1/2^(53:Ἀ))/8:=by
  have h1:(2:ℝ)^(23:Ἀ)*(1/2^(53:Ἀ))/8<=2^(Dl+23)*(1/2^(53:Ἀ))/8:=div_le_div_of_nonneg_right (AE B0 (nN 53)) (by norm_num)
  exact ((nI 33 1075 (by norm_num)).trans (by norm_num)).trans h1
 have hL:(n:ℝ)+5<=2^(Aq (n+4)+1):=by
  have h:n+5<=2^(Aq (n+4)+1):=ϙ
  simpa only [W5,I4,nZ] using (R5 (α:=ℝ)).2 h
 have fin:=nMn _ _ _ _ n (2^(Dl+23)) (1/2^(53:Ἀ)) (1/2^(1075:Ἀ)) _ nR4 Bp (nN _) (nN _) hQ (by rw[nQ];exact se) R0 hR1 hR2 (me.trans_eq (by rw[one_div_mul_eq_div])) sm2 sw2 hL
 refine⟨mf,fin.trans_eq ?_,d3,d2,hQ,hB⟩
 rw[Ac AL,←W5,AA,zpow_ofNat,mul_one_div,←AF,←pA,AF _ _ 23,(AF (2:ℝ) 30 23:(2:ℝ)^(53:Ἀ)=_)]
 exact mul_div_mul_right _ _ (pow_ne_zero _ AL)
lemma nV_t(y d dd u w Y:ℝ)(u0:0<=u)(u1:u<=1)(w0:0<=w)(w1:w<=1)(hY:|y|<=Y)
  (hd:|d-y|<=u*|y|+w)(hdd:|dd-d*d|<=u*|d*d|+w):|dd-y*y|<=7*u*(y*y)+(8*Y+3)*w:=by
 have h0:=Ag (d-y) (y+y);rw[sub_add_add_cancel]at h0
 have s1:|d+y|<=2*|y|+(u*|y|+w):=by linear_combination h0+Ag y y+hd
 have q:=mul_le_mul s1 hd (Aa (d-y)) ((Aa (d+y)).trans s1);rw[<-Ae,<-mul_self_sub_mul_self]at q
 have uy:=mul_le_of_le_one_left (Aa y) u1
 linear_combination abs_sub_le dd (d*d) (y*y)+hdd+mul_le_mul_of_nonneg_left (AY (d*d) (y*y)) u0+mul_le_of_le_one_left (Aa (d*d-y*y)) u1+2*(q+AE uy (Af u0 (Aa y))+2*AE uy w0+4*mul_le_mul_of_nonneg_left hY w0+mul_le_of_le_one_left w0 w1)+6*u*abs_mul_abs_self y+u*abs_mul_self y
lemma nV_e(a r A t t' u W k:ℝ)(u0:0<=u)(W0:0<=W)(A0:0<=A)(t0:0<=t)(k0:0<=k)(hk:10*(k+1)*u<=1)
  (H1:|a-A|<=10*k*u*A+10*k*W)(Ht:|t'-t|<=7*u*t+W)(H2:|r-(a+t')|<=u*|a+t'|+W):
  |r-(A+t)|<=10*(k+1)*u*(A+t)+10*(k+1)*W:=by
 have d:|a+t'-(A+t)|<=|a-A|+|t'-t|:=by rw[add_sub_add_comm];exact Ag _ _
 have hat:|a+t'|<=(A+t)+(|a-A|+|t'-t|):=by
  have:=AY (a+t') (A+t);rw[Ad (add_nonneg A0 t0)]at this;linear_combination this+d
 have ku:10*k*u<=1:=by linear_combination hk+10*u0
 have u7:7*u<=1:=by linear_combination hk+3*u0+10*Af k0 u0
 have uA:=Af u0 A0;have ut:=Af u0 t0
 have m1:(10*k*u)*(u*A)<=1*(u*A):=AE ku uA
 have m2:(10*k*u)*W<=1*W:=AE ku W0
 have m3:(7*u)*(u*t)<=1*(u*t):=AE u7 ut
 have m4:u*W<=1*W:=AE (by linear_combination u7+6*u0) W0
 have E:|a-A|+|t'-t|<=10*k*u*A+10*k*W+(7*u*t+W):=add_le_add H1 Ht
 have uE:u*(|a-A|+|t'-t|)<=u*(10*k*u*A+10*k*W+(7*u*t+W)):=mul_le_mul_of_nonneg_left E u0
 have ua:u*|a+t'|<=u*((A+t)+(|a-A|+|t'-t|)):=mul_le_mul_of_nonneg_left hat u0
 have tr:=(abs_sub_le r (a+t') (A+t)).trans (add_le_add AP d)
 linear_combination tr+H2+ua+uE+m1+m2+m3+m4+E+8*uA+ut+10*Af k0 ut+6*W0
section
open Dense.F NF
lemma fSb(x y:Ἅ)(h:fv x)(j:fv y)(g:|rv x-rv y|<=(2:ℝ)^(971:Ἁ)):
  fv (x-y)/\|rv (x-y)-(rv x-rv y)|<=|rv x-rv y|/2^(53:Ἀ)+1/2^(1075:Ἀ):=by have⟨d,e⟩:=s5iS x y h j (g.trans (pm 971 AP));exact⟨d,by rw[e];exact NF.jy _⟩
end
lemma nS2_s_h1(B:ℝ)(h:0<=B):B+(B+B)<=(B+B)+(B+B):=add_le_add (le_add_of_nonneg_left h) AP
lemma nS2_s(x:(Ἂ Word))(m:Ἅ)(e:Ἄ)(hX:∀j,j<(J0 e)->kB x[j]!≠255)(hm:Dense.F.fv m)(Y:ℝ)(Y0:0<=Y)(Y1:Y<=2^(302:Ἀ)):
  ∀M (i:Ἄ)(v:Ἅ)(A:ℝ)(k:ℕ),(J0 e)-(J0 i)=M->(J0 i)<=(J0 e)->Dense.F.fv v->0<=A->A<=k*(Y*Y)->
  |Dense.F.rv v-A|<=10*k*(1/2^(53:Ἀ))*A+10*k*((8*Y+3)*(1/2^(1075:Ἀ)))->k+M<=2^(12:Ἀ)->
  (∀j,(J0 i)<=j->j<(J0 e)->|(sv x[j]!:ℝ)*2^kG x[j]!-Dense.F.rv m|<=Y)->
  Dense.F.fv (nS2 x m i e v)/\|Dense.F.rv (nS2 x m i e v)-(A+∑j∈B5 (J0 i) (J0 e),((sv x[j]!:ℝ)*2^kG x[j]!-Dense.F.rv m)*((sv x[j]!:ℝ)*2^kG x[j]!-Dense.F.rv m))|
   <=10*((k+M:ℕ):ℝ)*(1/2^(53:Ἀ))*(A+∑j∈B5 (J0 i) (J0 e),((sv x[j]!:ℝ)*2^kG x[j]!-Dense.F.rv m)*((sv x[j]!:ℝ)*2^kG x[j]!-Dense.F.rv m))+10*((k+M:ℕ):ℝ)*((8*Y+3)*(1/2^(1075:Ἀ))):=by
 have W0:(0:ℝ)<=(8*Y+3)*(1/2^(1075:Ἀ)):=Af (add_nonneg (Af (R6 8) Y0) zero_le_three) (nN _)
 have q1:(1:ℝ)<=2^(302:Ἀ):=one_le_pow₀ one_le_two
 have e8:(8:ℝ)<=2^(3:Ἀ):=by norm_num
 have Y5:8*Y<=(2:ℝ)^(305:Ἀ):=(mul_le_mul e8 Y1 Y0 (pow_pos two_pos _).le).trans_eq (by rw[<-AF])
 have T5:(3:ℝ)<=2^(305:Ἀ):=(by norm_num:(3:ℝ)<=8).trans (e8.trans (AU one_le_two (by decide)))
 have W1:(8*Y+3)*(1/2^(1075:Ἀ))<=(2:ℝ)^(306:Ἀ)*(1/2^(1075:Ἀ)):=AE ((add_le_add Y5 T5).trans_eq (by rw[<-mul_two,<-A4])) (nN _)
 have w1:(1:ℝ)/2^(1075:Ἀ)<=(8*Y+3)*(1/2^(1075:Ἀ)):=le_mul_of_one_le_left (nN _) ((by norm_num:(1:ℝ)<=3).trans (le_add_of_nonneg_left (Af (R6 8) Y0)))
 have YY:Y*Y<=2^(604:Ἀ):=(mul_le_mul Y1 Y1 Y0 (pow_pos two_pos _).le).trans_eq (by rw[<-AF])
 have YY0:0<=Y*Y:=Af Y0 Y0
 have u0:=nN 53
 have u1:(1:ℝ)/2^(53:Ἀ)<=1:=div_le_one_of_le₀ (one_le_pow₀ one_le_two) (pow_pos two_pos _).le
 have w0:=nN 1075
 have w2:(1:ℝ)/2^(1075:Ἀ)<=1:=div_le_one_of_le₀ (one_le_pow₀ one_le_two) (pow_pos two_pos _).le
 intro M;induction M with
 |zero=>
  intro i v A k hM hie hf A0 hA he hk hb
  have q:(J0 i)=(J0 e):=C4 hie (K3.1 hM)
  rw[nS2,dif_neg (by rw[ε,q];exact B4 _),q,Finset.Ico_self,Finset.sum_empty,Av,ϑ];exact⟨hf,he⟩
 |succ M ih=>
  intro i v A k hM hie hf A0 hA he hk hb
  have hl:(J0 i)<(J0 e):=N1 hM
  rw[nS2,dif_pos (ε.2 hl)];dsimp only;rw[f6 x i]
  obtain⟨tf,tr⟩:=Dense.F.cI x[(J0 i)]! (hX _ hl)
  have hs:=Finset.sum_eq_sum_Ico_succ_bot hl (fun j=>((sv x[j]!:ℝ)*2^kG x[j]!-Dense.F.rv m)*((sv x[j]!:ℝ)*2^kG x[j]!-Dense.F.rv m))
  have i1:(J0 (i+1))=(J0 i)+1:=Dense.ac (ε.2 hl)
  have hy:=hb _ AP hl
  have kk:(k:ℝ)<=2^(12:Ἀ):=by exact_mod_cast (show k<=2^(12:Ἀ) from (V1 _ _).trans hk)
  have k0:(0:ℝ)<=k:=M3 k
  obtain⟨df,de⟩:=fSb _ m tf hm (by rw[tr,zpow_ofNat];exact hy.trans (Y1.trans (AU one_le_two (by decide))))
  rw[tr]at de
  generalize hyd:(sv x[(J0 i)]!:ℝ)*2^kG x[(J0 i)]!-Dense.F.rv m=y at de hy
  generalize Dense.F.dw x[(J0 i)]!-m=dF at df de ⊢
  have hd:|Dense.F.rv dF-y|<=(1/2^(53:Ἀ))*|y|+1/2^(1075:Ἀ):=by rw[one_div_mul_eq_div];exact de
  have uy:(1/2^(53:Ἀ))*|y|<=|y|:=mul_le_of_le_one_left (Aa _) u1
  have aD:|Dense.F.rv dF|<=2^(304:Ἀ):=by
   have yP:|y|<=2^(302:Ἀ):=hy.trans Y1
   rw[A4,A4,mul_two,mul_two]
   exact (sub_le_iff_le_add.1 (AY _ y)).trans (add_le_add (hd.trans (add_le_add (uy.trans yP) (w2.trans q1))) (yP.trans (le_add_of_nonneg_left (pow_pos two_pos _).le)))
  have aDD:|Dense.F.rv dF*Dense.F.rv dF|<=(2:ℝ)^(971:Ἁ):=by
   rw[Ae,zpow_ofNat];exact (mul_le_mul aD aD (Aa _) (pow_pos two_pos _).le).trans (by rw[<-AF];exact AU one_le_two (by decide))
  obtain⟨ddf,dde⟩:=NF.fM dF dF df df aDD
  have yy:y*y<=Y*Y:=by rw[<-abs_mul_abs_self y];exact mul_le_mul hy hy (Aa _) Y0
  have y0:=mul_self_nonneg y
  have Ht:=nV_t y (Dense.F.rv dF) (Dense.F.rv (dF*dF)) (1/2^(53:Ἀ)) (1/2^(1075:Ἀ)) Y u0 u1 w0 w2 hy hd (by rw[one_div_mul_eq_div];exact dde)
  have k16:10*(k:ℝ)<=2^(16:Ἀ):=(AE (by norm_num:(10:ℝ)<=2^(4:Ἀ)) k0).trans ((mul_le_mul_of_nonneg_left kk (pow_pos two_pos _).le).trans_eq (by rw[<-AF]))
  have kW:10*(k:ℝ)*((8*Y+3)*(1/2^(1075:Ἀ)))<=1:=by
   have:=mul_le_mul k16 W1 W0 (pow_pos two_pos _).le
   exact this.trans (by rw[<-Ap,<-AF];exact nP _ _ (by decide))
  have ku:10*(k:ℝ)*(1/2^(53:Ἀ))<=1:=(AE k16 u0).trans (nP 16 53 (by decide))
  have kuA:10*(k:ℝ)*(1/2^(53:Ἀ))*A<=A:=mul_le_of_le_one_left A0 ku
  have Ab:A<=2^(616:Ἀ):=hA.trans ((mul_le_mul kk YY YY0 (pow_pos two_pos _).le).trans_eq (by rw[<-AF]))
  have Y6:(2:ℝ)^(604:Ἀ)<=2^(616:Ἀ):=AU one_le_two (by decide)
  have q6:(1:ℝ)<=2^(616:Ἀ):=one_le_pow₀ one_le_two
  have av:|Dense.F.rv v|<=A+(10*k*(1/2^(53:Ἀ))*A+10*k*((8*Y+3)*(1/2^(1075:Ἀ)))):=by
   have:=AY (Dense.F.rv v) A;rw[Ad A0]at this;exact sub_le_iff_le_add'.mp (this.trans he)
  have u7:7*(1/2^(53:Ἀ):ℝ)*(y*y)<=y*y:=mul_le_of_le_one_left y0 ((AE (by norm_num:(7:ℝ)<=2^(3:Ἀ)) u0).trans (nP 3 53 (by decide)))
  have W2:(8*Y+3)*(1/2^(1075:Ἀ):ℝ)<=1:=W1.trans (nP _ _ (by decide))
  have ad:|Dense.F.rv (dF*dF)|<=y*y+(7*(1/2^(53:Ἀ))*(y*y)+(8*Y+3)*(1/2^(1075:Ἀ))):=by
   have:=AY (Dense.F.rv (dF*dF)) (y*y);rw[Ad y0]at this;exact sub_le_iff_le_add'.mp (this.trans Ht)
  have aS:|Dense.F.rv v+Dense.F.rv (dF*dF)|<=(2:ℝ)^(971:Ἁ):=by
   rw[zpow_ofNat]
   have BB:=nS2_s_h1 _ (pow_pos two_pos 616).le
   have yB:y*y<=2^(616:Ἀ):=yy.trans (YY.trans Y6)
   refine (Ag _ _).trans ((add_le_add ((av.trans (add_le_add Ab (add_le_add (kuA.trans Ab) (kW.trans q6)))).trans BB) ((ad.trans (add_le_add yB (add_le_add (u7.trans yB) (W2.trans q6)))).trans BB)).trans ?_)
   rw[<-mul_two,<-mul_two,<-mul_two,<-A4,<-A4,<-A4];exact AU one_le_two (by decide)
  obtain⟨rf,re⟩:=NF.fA v (dF*dF) hf ddf aS
  have H2:|Dense.F.rv (v+dF*dF)-(Dense.F.rv v+Dense.F.rv (dF*dF))|<=(1/2^(53:Ἀ))*|Dense.F.rv v+Dense.F.rv (dF*dF)|+(8*Y+3)*(1/2^(1075:Ἀ)):=by
   rw[one_div_mul_eq_div];exact re.trans (add_le_add AP w1)
  have step:=nV_e _ _ A (y*y) _ (1/2^(53:Ἀ)) ((8*Y+3)*(1/2^(1075:Ἀ))) k u0 W0 A0 y0 k0 ((AE ((mul_add 10 (k:ℝ) 1).trans_le ((add_le_add k16 (by norm_num:(10:ℝ)*1<=2^(16:Ἀ))).trans_eq (by rw[<-mul_two,<-A4]))) u0).trans (nP 17 53 (by decide))) he Ht H2
  have hA2:A+y*y<=((k+1:ℕ):ℝ)*(Y*Y):=by rw[X7,add_one_mul];exact add_le_add hA yy
  have:=ih (i+1) (v+dF*dF) (A+y*y) (k+1) (by rw[i1,V4,hM,BK]) (i1.le.trans (l7 hl)) rf (add_nonneg A0 y0) hA2 (by rw[X7];exact step) (by rw[pA,Τ 1 M];exact hk) (fun j h1 h2=>hb j (X5 (i1.symm.le.trans h1)) h2)
  rw[<-hyd,i1,pA k 1 M,Τ 1 M,add_assoc,<-hs]at this;exact this
section
open NF Dense.F NE
end
lemma nXHm(n:Ἀ)(x:(Ἂ Word))(r:Ἀ)(eps:kF)(hn:0<n):
  ((∑j∈B5 (r*n) (r*n+n),((sv x[j]!:ℝ)*2^kG x[j]!-((no n x r:ℚ):ℝ)*2^(149:Ἀ))^(2:Ἀ))*2^(52:Ἀ)/n+(sv eps:ℝ)*2^kG eps*2^(201:Ἀ))=
   2^(350:Ἀ)*(((XH n x r (no n x r)+ZA eps:ℚ)):ℝ):=by
 have h2:∀w:kF,(sv w:ℝ)*2^kG w=((ZA w:ℚ):ℝ)*2^(149:Ἀ):=by
  intro w;rw[Y1,J7,S8,Rat.cast_intCast,G7,Ac AL,AA,zpow_ofNat,mul_div_assoc',div_mul_cancel₀ _ (pow_ne_zero _ AL)]
 rw[rowVariance,Yk,Ai,Rat.cast_add,Rat.cast_div,Rat.cast_sum,T7,Finset.sum_Ico_eq_sum_range,ϱ]
 simp only[h2,Rat.cast_pow,Rat.cast_sub,<-sub_mul,mul_pow,<-n8]
 rw[show (2:ℝ)^(350:Ἀ)=(2^(149:Ἀ))^(2:Ἀ)*2^(52:Ἀ) by rw[<-pow_mul,<-AF],show (2:ℝ)^(201:Ἀ)=2^(149:Ἀ)*2^(52:Ἀ) by rw[<-AF]]
 generalize (2:ℝ)^(149:Ἀ)=a;generalize (2:ℝ)^(52:Ἀ)=b
 ring
theorem nTm_h1(a b M:ℝ):(a-b)*(a-b)=(a-M)^(2:Ἀ)+(b-M)^(2:Ἀ)-2*(a-M)*(b-M):=by rw[←sub_sub_sub_cancel_right a b M,←pow_two,sub_sq']
lemma nTm(s:Finset ℕ)(f:ℕ->ℝ)(M mr:ℝ)(h:∑j∈s,f j=s.card*M):
  ∑j∈s,(f j-mr)*(f j-mr)=∑j∈s,(f j-M)^(2:Ἀ)+s.card*((mr-M)*(mr-M)):=by
 simp only[nTm_h1 _ mr M]
 rw[nF,nJ,Finset.sum_const,nsmul_eq_mul,←n8,←Finset.mul_sum,nF,h,Finset.sum_const,nsmul_eq_mul,sub_self,mul_zero,zero_mul,sub_zero,pow_two (mr-M)]
lemma nL_rel(Dr Zm Zt cu η:ℝ)(Zt0:0<Zt)(η0:0<=η)(cu0:0<=cu)(h1:Zt<=Zm)(h2:Zm<=Zt+η*Zt)
  (hd:|Dr-O2 Zm|<=cu*O2 Zm):|Dr-O2 Zt|<=(cu*(1+η)+η)*O2 Zt:=by
 have a2:O2 Zm<=(1+η)*O2 Zt:=by
  rw[<-(Real.sqrt_mul (sq_nonneg (1+η)) Zt).trans (congrArg (·*O2 Zt) (Real.sqrt_sq (add_nonneg zero_le_one η0)))]
  exact Real.sqrt_le_sqrt (h2.trans ((one_add_mul η Zt).symm.trans_le (AE (le_self_pow₀ (le_add_of_nonneg_right η0) AL) Zt0.le)))
 exact ((abs_sub_le Dr (O2 Zm) (O2 Zt)).trans (add_le_add (hd.trans (mul_le_mul_of_nonneg_left a2 cu0)) ((Ad (sub_nonneg.2 (Real.sqrt_le_sqrt h1))).trans_le (sub_le_iff_le_add'.2 (a2.trans_eq (one_add_mul _ _)))))).trans_eq ((add_mul _ _ _).trans (congrArg (·+η*O2 Zt) (Ap _ _ _))).symm
lemma nEp(w:kF)(h0:kB w≠0)(hs:0<sv w):(2:ℝ)^((mJ (kB w))+223)<=(sv w:ℝ)*2^kG w*2^(201:Ἀ):=by
 have m1:sv w=((kM w:ℕ):ℤ):=by
  rw[sv,if_neg];intro g;rw[sv,if_pos g]at hs;exact absurd hs (Int.not_lt.2 (Int.neg_nonpos_of_nonneg (W6 _)))
 have m2:2^(23:Ἀ)<=kM w:=by unfold mantOf;rw[if_neg h0];exact V1 _ _
 have g1:kG w=(mJ (kB w))-1:=by unfold idxOf;rw[if_neg h0]
 rw[m1,g1,M5]
 have m3:(2:ℝ)^(23:Ἀ)<=(kM w:ℝ):=by exact_mod_cast m2
 exact ((congrArg ((2:ℝ)^·) (((lO 23 _ 201).trans ((Τ 224 _).trans ((pA _ 1 223).symm.trans (congrArg (·+223) (L3 (ϵ fun q=>h0 (Β.1 q))))))).symm)).trans ((AF _ _ _).trans (congrArg (·*(2:ℝ)^(201:Ἀ)) (AF _ _ _)))).trans_le (AE (AE m3 (pow_nonneg zero_le_two _)) (pow_nonneg zero_le_two _))
lemma nQ52(n:Ἀ)(h1:1<=n)(h2:n<=2048):((1:ἆ)<<<52)/I0 n=I0 (2^(52:Ἀ)/n):=by
 apply ξ.1
 have a:(lq (I0 n))=n:=by simp only[Ϣ,mY];exact γ (by omega)
 have b:2^(52:Ἀ)/n<2^(64:Ἀ):=AO (L5 _ _) (by norm_num)
 rw[ml,a];simp only[Ϣ,O6,γ b]
 rfl
section
open NF Dense.F NE
lemma nL_float_h5(k:ℕ):(0:ℝ)<2^k/\(1:ℝ)<=2^k:=⟨pow_pos two_pos k,one_le_pow₀ one_le_two⟩
lemma nL_float_h6(a b:ℕ)(h:a<=b):(2:ℝ)^a<=2^b:=AU one_le_two h
lemma nL_float_h7(Y:ℝ)(k:ℕ)(h:Y<=2^k):8*Y+3<=2^(k+4):=by
 have P0:=(nL_float_h5 k).1.le
 have a:8*Y<=2^(k+3):=(mul_le_mul_of_nonneg_left h (R6 8)).trans ((mul_comm _ _).trans_le ((mul_le_mul_of_nonneg_left (by norm_num:(8:ℝ)<=2^3) P0).trans_eq (AF (2:ℝ) k 3).symm))
 have b:(3:ℝ)<=2^(k+3):=(by norm_num:(3:ℝ)<=2^3).trans (AU one_le_two (K2 3 k))
 rw[A4,mul_two];exact add_le_add a b
lemma nL_float_h8(s M m:ℝ)(k:ℕ)(a:|s|<=2^k)(hQ:|M|<=2^k)(t4:|m-M|<=2^k):|s-m|<=2^(k+2):=by
 rw[A4,A4,mul_two,mul_two]
 exact (abs_sub s m).trans (add_le_add (a.trans (le_add_of_nonneg_right (nL_float_h5 k).1.le)) ((sub_le_iff_le_add.1 (AY m M)).trans (add_le_add t4 hQ)))
lemma nL_float_h1(Tm Tt N d E Z K:ℝ)(hN:N≠0)(h1:Tm=Tt+N*(d*d))(h2:Tt*K/N+E=Z):Tm*K/N+E=Z+d*d*K:=by
 rw[<-h2,h1,add_mul,add_div,Ap N,mul_div_cancel_left₀ _ hN];exact add_right_comm _ _ _
lemma nL_float_h2(x T R e:ℝ)(k:ℕ)(h:|x-T|<=R)(hR:R<=T+e)(he:e<=1/16)(hT:T<=2^k):-e<=x/\|x|<=2^(k+2):=by
 obtain⟨h1,h2⟩:=abs_sub_le_iff.1 h
 have P0:=(nL_float_h5 k).1.le
 have eP:e<=2^k:=he.trans ((by norm_num:(1:ℝ)/16<=1).trans (nL_float_h5 k).2)
 have hP2:(2:ℝ)^k<=2^k+2^k:=le_add_of_nonneg_right P0
 have lo:-e<=x:=(sub_le_sub_iff_left T).1 ((h2.trans hR).trans_eq (sub_neg_eq_add T e).symm)
 rw[A4,A4,mul_two,mul_two]
 exact⟨lo,A3.2 ⟨(neg_le_neg (eP.trans (hP2.trans (le_add_of_nonneg_right (add_nonneg P0 P0))))).trans lo,(sub_le_iff_le_add.1 h1).trans (add_le_add (hR.trans (add_le_add hT eP)) (hT.trans hP2))⟩⟩
lemma nL_float_h3(P x q e c Q:ℝ)(h:|P-x*q|<=|x*q|/2^(53:Ἀ)+c)(hx:-e<=x)(hq0:0<=q)(hq:q<=Q)(he0:0<=e)(heQ:e*Q<=1/16)(hc:c<=1/16):-(1/4)<=P:=by
 have y1:-e*q<=x*q:=AE hx hq0
 have zlo:-(1/16)<=x*q:=(neg_le_neg ((mul_le_mul_of_nonneg_left hq he0).trans heQ)).trans ((neg_mul e q).symm.le.trans y1)
 have u:|x*q|/2^(53:Ἀ)<=|x*q|:=div_le_self (Aa _) (nL_float_h5 _).2
 have a:|x*q|<=x*q+1/8:=A3.2 ⟨by linear_combination 2*zlo,le_add_of_nonneg_right (by norm_num)⟩
 have k:-(1/4:ℝ)<=x*q-(x*q+1/8+1/16):=by rw[add_assoc,sub_add_cancel_left];norm_num
 exact k.trans ((sub_le_sub_left (add_le_add (u.trans a) hc) _).trans (sub_le_comm.1 (abs_sub_le_iff.1 h).2))
lemma nL_float_h4(W S c:ℝ)(h:|W-S|<=|S|/2^(53:Ἀ)+c)(hS:1/2<=S)(hc:c<=1/16):0<=W:=by
 have S0:|S|=S:=Ad (one_half_pos.le.trans hS)
 have u:|S|/2^(53:Ἀ)<=|S|/2:=div_le_div_of_nonneg_left (Aa S) two_pos (le_self_pow₀ one_le_two (by norm_num))
 rw[S0]at h u
 have hc':c<=S/2:=hc.trans ((by norm_num:(1:ℝ)/16<=1/2/2).trans (div_le_div_of_nonneg_right hS zero_le_two))
 exact (sub_nonneg.2 ((add_le_add u hc').trans_eq (add_halves S))).trans (sub_le_comm.1 (abs_sub_le_iff.1 h).2)
lemma nL_float(n:Ἀ)(eps:kF)(x:(Ἂ Word))(r:Ἀ)(hn4:4<=n)(hn:n<=2048)(hb:r*n+n<2^(32:Ἀ))(hX:∀j,j<r*n+n->kB x[j]!≠255)(he:kB eps≠255)(hs:0<sv eps)
  (P:Ἅ×ἆ)(hP:P=nS1 x (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat 0)
  (m:Ἅ)(hm:m=P.1*((((1:ἆ)<<<(52+(Aq n).toUInt64))/I0 n).toInt64.toFloat*N5 ((971-(Aq n).toUInt64)<<<52)))
  (D:Ἅ)(hD:D=W7 (nS2 x m (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat*(((1:ἆ)<<<52)/I0 n).toInt64.toFloat+Dense.F.dw eps*nS201)):
  Dense.F.fv D/\(lq P.2)<=254/\∃δ:ℝ,|δ|<=2^(((Aq (n+4)+1:ℕ):ℤ)+(lq P.2)-30)/\
   |Dense.F.rv D-O2 (2^(350:Ἀ)*((XH n x r (no n x r)+ZA eps:ℚ):ℝ))|<=((12*n+7)*(1/2^(53:Ἀ))*(1+δ*δ*2^(52:Ἀ)/(2^(350:Ἀ)*((XH n x r (no n x r)+ZA eps:ℚ):ℝ)))+δ*δ*2^(52:Ἀ)/(2^(350:Ἀ)*((XH n x r (no n x r)+ZA eps:ℚ):ℝ)))*O2 (2^(350:Ἀ)*((XH n x r (no n x r)+ZA eps:ℚ):ℝ)):=by
 have H:=nMean n x r hn4 hn hb hX
 rw[<-hP,<-hm]at H;obtain⟨mf,mE,-,d2,hQ,hB⟩:=H;clear hP hm
 set Dl:=lq P.2 with hDl
 set M:=((no n x r:ℚ):ℝ)*2^(149:Ἀ) with hM
 have n0:0<n:=U1 (Nat.zero_lt_succ 3) hn4
 have lo:(J0 (G0 (r*n)))=r*n:=Conv.aa _ ((V1 _ n).trans_lt hb)
 have hi:(J0 (G0 (r*n+n)))=r*n+n:=Conv.aa _ hb
 have hX':∀j,j<(J0 (G0 (r*n+n)))->kB x[j]!≠255:=fun j h=>hX j (by rwa[hi]at h)
 clear hX
 obtain⟨z0f,z0r⟩:=kg2 0 (϶ _)
 have lL:Aq (n+4)<=11:=J2 ((m9 (Nat.succ_ne_zero _)).2 (C0 (L1 hn 4) (by decide)))
 have dB:(2:ℝ)^(((Aq (n+4)+1:ℕ):ℤ)+(Dl:ℤ)-30)<=2^(Dl+23):=by
  rw[<-AA];exact AG one_le_two (sub_le_iff_le_add.2 (U4.2 ((Τ _ Dl).trans_le (J1 ((Nat.succ_le_succ lL).trans (by decide:12<=53)) Dl)):((Aq (n+4)+1+Dl:ℕ):ℤ)<=((Dl+23+30:ℕ):ℤ)))
 set Y:=(2:ℝ)^(Dl+25) with hY
 have Y0:0<=Y:=(nL_float_h5 _).1.le
 have Y1:Y<=2^(279:Ἀ):=nL_float_h6 _ _ (L1 d2 25)
 have t4:=mE.trans dB
 have hYj:∀j,(J0 (G0 (r*n)))<=j->j<(J0 (G0 (r*n+n)))->|(sv x[j]!:ℝ)*2^kG x[j]!-Dense.F.rv m|<=Y:=
  fun j h1 h2=>nL_float_h8 _ _ _ _ (hB j (by rwa[lo]at h1) (by rwa[hi]at h2)) hQ t4
 obtain⟨vf,ve⟩:=nS2_s x m _ hX' mf Y Y0 (Y1.trans (nL_float_h6 _ _ (by decide))) n (G0 (r*n)) _ 0 0 (by rw[hi,lo];exact ϱ _ _) (by rw[hi,lo];exact V1 _ _) z0f AP (Af (M3 _) (mul_self_nonneg _)) (by rw[z0r];simp only [CharP.cast_eq_zero,sub_self,A5,mul_zero,one_div,zero_mul,Av,Std.le_refl]) ((Κ n).trans_le (hn.trans (by decide))) hYj
 rw[lo,hi]at ve;simp only[Ai]at ve
 generalize hv:nS2 x m (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat=v at vf ve hD
 set Tm:=∑j∈B5 (r*n) (r*n+n),((sv x[j]!:ℝ)*2^kG x[j]!-Dense.F.rv m)*((sv x[j]!:ℝ)*2^kG x[j]!-Dense.F.rv m) with hTm
 set Tt:=∑j∈B5 (r*n) (r*n+n),((sv x[j]!:ℝ)*2^kG x[j]!-M)^(2:Ἀ) with hTt
 set δ:=Dense.F.rv m-M with hδ
 have nR:(0:ℝ)<n:=Nat.cast_pos.2 n0
 have n1:(1:ℝ)<=n:=Nat.one_le_cast.2 n0
 have n11:(n:ℝ)<=2^(11:Ἀ):=(R5.2 (show n<=2^(11:Ἀ) from hn)).trans_eq (by rw[I4,nZ])
 have n15:10*(n:ℝ)<=2^(15:Ἀ):=(AE (by norm_num:(10:ℝ)<=2^(4:Ἀ)) nR.le).trans ((mul_le_mul_of_nonneg_left n11 (nL_float_h5 _).1.le).trans_eq (by rw[<-AF]))
 have cI:(B5 (r*n) (r*n+n)).card=n:=by rw[Nat.card_Ico];exact ϱ _ _
 have SM:∑j∈B5 (r*n) (r*n+n),(sv x[j]!:ℝ)*2^kG x[j]! =((B5 (r*n) (r*n+n)).card:ℝ)*M:=by
  rw[cI,hM,nMeq n x r n0,mul_div_cancel₀ _ nR.ne']
 have TmE:Tm=Tt+n*(δ*δ):=by rw[hTm,hTt,nTm _ _ M _ SM,cI]
 have Tt0:0<=Tt:=D8 (fun _ _=>sq_nonneg _)
 have Tm0:0<=Tm:=D8 (fun _ _=>mul_self_nonneg _)
 have Tmb:Tm<=n*(Y*Y):=by
  have h:=Finset.sum_le_card_nsmul (B5 (r*n) (r*n+n)) (fun j=>((sv x[j]!:ℝ)*2^kG x[j]!-Dense.F.rv m)*((sv x[j]!:ℝ)*2^kG x[j]!-Dense.F.rv m)) (Y*Y) (fun j hj=>by
   rw[Finset.mem_Ico]at hj;have:=hYj j (by rw[lo];exact hj.1) (by rw[hi];exact hj.2)
   rw[<-abs_mul_abs_self];exact mul_le_mul this this (Aa _) Y0)
  rwa[cI,nsmul_eq_mul]at h
 have YY:Y*Y<=2^(558:Ἀ):=(mul_le_mul Y1 Y1 Y0 (nL_float_h5 _).1.le).trans_eq (by rw[<-AF])
 have Tmb2:Tm<=2^(570:Ἀ):=Tmb.trans ((mul_le_mul (n11.trans (nL_float_h6 11 12 (by decide))) YY (mul_self_nonneg Y) (nL_float_h5 _).1.le).trans_eq (by rw[<-AF]))
 have Wb:(8*Y+3)*(1/2^(1075:Ἀ):ℝ)<=1/2^(700:Ἀ):=by
  have q3:(8*Y+3:ℝ)<=2^(375:Ἀ):=(nL_float_h7 _ _ Y1).trans (nL_float_h6 _ _ (by decide))
  exact (AE q3 (nN _)).trans (nP2 375 1075 700 (by decide))
 have nW:10*((n:ℕ):ℝ)*((8*Y+3)*(1/2^(1075:Ἀ)))<=10*n*(1/2^(700:Ἀ)):=mul_le_mul_of_nonneg_left Wb (Af (R6 _) nR.le)
 have nu1:10*((n:ℕ):ℝ)*(1/2^(53:Ἀ))<=1:=(AE n15 (nN 53)).trans (nP 15 53 (by decide))
 have e15:10*(n:ℝ)*(1/2^(700:Ἀ))<=1/2^(600:Ἀ):=(AE n15 (nN 700)).trans (nP2 15 700 600 (by decide))
 have i16:(1:ℝ)/2^(4:Ἀ)<=1/16:=by norm_num
 have w6:(1:ℝ)/2^(600:Ἀ)<=1/16:=(nI 4 600 (by decide)).trans i16
 obtain⟨vlo,vb⟩:=nL_float_h2 _ _ _ _ _ ve (add_le_add (mul_le_of_le_one_left Tm0 nu1) (nW.trans e15)) w6 Tmb2
 rw[nQ52 n n0 hn]at hD
 obtain⟨qf,qr⟩:=kg2 (2^(52:Ἀ)/n) ((L5 _ _).trans_lt (Nat.pow_lt_pow_succ (by decide)))
 have q0:0<=((2^(52:Ἀ)/n:ℕ):ℝ):=M3 _
 have h0:((n*(2^(52:Ἀ)/n)+2^(52:Ἀ)%n:ℕ):ℝ)=((2^(52:Ἀ):ℕ):ℝ):=congrArg Nat.cast (ϛ (2^(52:Ἀ)) n)
 have h3:((2^(52:Ἀ)%n:ℕ):ℝ)<n:=Nat.cast_lt.2 (ϊ _ n0)
 rw[W5,Y4,I4,nZ]at h0
 generalize hq:((2^(52:Ἀ)/n:ℕ):ℝ)=q at qr q0 h0
 have hq1:(n:ℝ)*q<=2^(52:Ἀ):=(le_add_of_nonneg_right (M3 _)).trans_eq h0
 have hq2:2^(52:Ἀ)-(n:ℝ)<n*q:=(sub_lt_sub_left h3 _).trans_eq (by rw[<-h0,add_sub_cancel_right])
 have qb:|q|<=2^(52:Ἀ):=(Ad q0).trans_le ((le_mul_of_one_le_left q0 n1).trans hq1)
 obtain⟨ef,er⟩:=nD_eps eps he
 have pb:=NE.pB _ _ 572 52 vb qb
 obtain⟨pf,pe⟩:=NF.fM _ _ vf qf (by rw[qr];exact NE.z9 _ _ pb (by decide))
 rw[qr]at pe
 generalize hP':v*(I0 (2^(52:Ἀ)/n)).toInt64.toFloat=P' at pf pe hD
 have Pb:=NE.eB _ _ 624 pe pb
 have Ewb:=NE.wB eps
 have E1:|(sv eps:ℝ)*2^kG eps|<=2^(278:Ἀ):=Ewb.trans (nL_float_h6 _ _ (J1 (cP eps) 24))
 have E2:|(2:ℝ)^(201:Ἀ)|<=2^(201:Ἀ):=le_of_eq (AW (nL_float_h5 _).1)
 have Epb:|(sv eps:ℝ)*2^kG eps*2^(201:Ἀ)|<=2^(625:Ἀ):=NE.mo _ _ _ (NE.pB _ _ 278 201 E1 E2) (by decide)
 have s1:(1:ℝ)<=sv eps:=(Int.cast_one (R:=ℝ)).symm.le.trans (Int.cast_le.2 hs)
 have a1:(1:ℝ)<=2^kG eps:=(nL_float_h5 _).2
 have a2:(1:ℝ)<=2^(201:Ἀ):=(nL_float_h5 _).2
 have hEp1:1<=(sv eps:ℝ)*2^kG eps*2^(201:Ἀ):=one_le_mul_of_one_le_of_one_le (one_le_mul_of_one_le_of_one_le s1 a1) a2
 generalize hEp:(sv eps:ℝ)*2^kG eps*2^(201:Ἀ)=Ep at er Epb hEp1
 have Sb:=NE.sB _ _ 625 Pb Epb
 obtain⟨wf,we⟩:=NF.fA _ _ pf ef (by rw[er];exact NE.z9 _ _ Sb (by decide))
 rw[er]at we
 generalize hW':P'+Dense.F.dw eps*nS201=W at wf we hD
 have w75:(1:ℝ)/2^(1075:Ἀ)<=1/2^(700:Ἀ):=nI 700 1075 (by decide)
 have w70:(1:ℝ)/2^(700:Ἀ)<=1/2^(600:Ἀ):=nI 600 700 (by decide)
 have w60:(1:ℝ)/2^(600:Ἀ)*2^(52:Ἀ)<=1/16:=by rw[mul_comm];exact (nP2 52 600 548 (by decide)).trans ((nI 4 548 (by decide)).trans i16)
 have w0:(1:ℝ)/2^(1075:Ἀ)<=1/16:=w75.trans (w70.trans w6)
 have Plo:-(1/4)<=Dense.F.rv P':=nL_float_h3 _ _ _ _ _ _ pe vlo q0 ((le_abs_self q).trans qb) (nN 600) w60 w0
 have Wl:0<=Dense.F.rv W:=nL_float_h4 _ _ _ we ((by norm_num:(1/2:ℝ)<=-(1/4)+1).trans (add_le_add Plo hEp1)) w0
 obtain⟨df,dr⟩:=fS W wf Wl
 rw[<-hD]at df dr
 refine⟨df,d2,δ,mE,?_⟩
 have h4:|Dense.F.rv D-O2 (Dense.F.rv W)|<=(1/2^(53:Ἀ))*O2 (Dense.F.rv W)+1/2^(700:Ἀ):=by
  rw[dr];have:=NF.jy (O2 (Dense.F.rv W));rw[Ad (Real.sqrt_nonneg _)]at this;exact this.trans (add_le_add (one_div_mul_eq_div _ _).ge w75)
 have hu2:(2:ℝ)^(52:Ἀ)*(1/2^(53:Ἀ))*2=1:=by rw[mul_one_div,div_mul_eq_mul_div,<-A4];exact div_self (pow_ne_zero _ AL)
 have n15b:12*(n:ℝ)<=2^(15:Ἀ):=(AE (by norm_num:(12:ℝ)<=2^(4:Ἀ)) nR.le).trans ((mul_le_mul_of_nonneg_left n11 (nL_float_h5 _).1.le).trans_eq (by rw[<-AF]))
 have hNu:(12*(n:ℝ)+8)*(1/2^(53:Ἀ))<=1/2^(30:Ἀ):=(AE ((add_le_add n15b (by norm_num:(8:ℝ)<=2^(15:Ἀ))).trans_eq (by rw[<-mul_two,<-A4]):12*(n:ℝ)+8<=2^(16:Ἀ)) (nN 53)).trans (nP2 16 53 30 (by decide))
 have ch:=nD_chainK Tm Ep n q (Dense.F.rv v) (Dense.F.rv P') (Dense.F.rv W) (Dense.F.rv D) (1/2^(53:Ἀ)) (1/2^(700:Ἀ)) (10*n) (12*n) (one_div_pos.2 (nL_float_h5 _).1) (nN _) hNu (nP2 60 700 53 (by decide)) n1 Tm0 hEp1 hq1 hq2 q0 hu2 (Af (R6 _) nR.le) (AE (by norm_num) nR.le) (by ring)
  (ve.trans (add_le_add AP nW)) (pe.trans (add_le_add (one_div_mul_eq_div _ _).ge w75)) (we.trans (add_le_add (one_div_mul_eq_div _ _).ge w75)) Wl h4
 have hZ:=nXHm n x r eps n0
 rw[<-hTt,hEp]at hZ
 generalize hZt:2^(350:Ἀ)*((XH n x r (no n x r)+ZA eps:ℚ):ℝ)=Zt at hZ ⊢
 have Zt0:0<Zt:=by rw[<-hZ];exact add_pos_of_nonneg_of_pos (div_nonneg (Af Tt0 (nL_float_h5 _).1.le) nR.le) (one_pos.trans_le hEp1)
 have ZmE:Tm*2^(52:Ἀ)/n+Ep=Zt+δ*δ*2^(52:Ἀ):=nL_float_h1 _ _ _ _ _ _ _ nR.ne' TmE hZ
 rw[ZmE]at ch
 have dd2:0<=δ*δ*2^(52:Ἀ):=Af (mul_self_nonneg δ) (nL_float_h5 _).1.le
 exact nL_rel _ _ _ _ _ Zt0 (div_nonneg dd2 Zt0.le) (Af (add_nonneg (Af (R6 _) nR.le) (R6 _)) (nN 53)) (le_add_of_nonneg_right dd2) (by rw[div_mul_cancel₀ _ Zt0.ne']) ch
end
section
open NF Dense.F NE
lemma nL_cert_h1(n Dl ef:Ἀ)(δ X R k:ℝ)(hn4:4<=n)(hn:n<=2048)(d2:Dl<=254)(d3:lq (nLc n Dl ef).toUInt64<=25)
  (dδ:|δ|<=2^(((Aq (n+4)+1:ℕ):ℤ)+(Dl:ℤ)-30))(ZtE:(2:ℝ)^(ef+223)<=2^(350:Ἀ)*X)
  (hk:k=(12*n+7)*(1/2^(53:Ἀ))*(1+δ*δ*2^(52:Ἀ)/(2^(350:Ἀ)*X))+δ*δ*2^(52:Ἀ)/(2^(350:Ἀ)*X))
  (dd:|R-O2 (2^(350:Ἀ)*X)|<=k*O2 (2^(350:Ἀ)*X)):
  nLc n Dl ef<64∧0<X∧0<=k∧k<=1/2∧2*k*(k*2^(53:Ἀ))<=1∧k*2^(53:Ἀ)+1<=2^(((nLc n Dl ef:ℕ):ℤ)-1)∧|R-2^(175:Ἀ)*O2 X|<=k*(2^(175:Ἀ)*O2 X):=by
 have p2:∀m:ℕ,(0:ℝ)<2^m:=fun m=>pow_pos two_pos m
 have Zt0:0<2^(350:Ἀ)*X:=(p2 _).trans_le ZtE
 have X0:0<X:=pos_of_mul_pos_right Zt0 (p2 _).le
 have Yq:O2 (2^(350:Ἀ)*X)=2^(175:Ἀ)*O2 X:=by
  rw[Real.sqrt_mul (p2 _).le,show (2:ℝ)^(350:Ἀ)=(2^(175:Ἀ))^(2:Ἀ) by rw[<-pow_mul],Real.sqrt_sq (p2 _).le]
 rw[Yq]at dd
 have nL:n+4<2^(Aq (n+4)+1):=ϙ
 have lL:Aq (n+4)<12:=(m9 (Nat.succ_ne_zero _)).2 (C0 (L1 hn 4) (by decide))
 have c3:max (Aq (n+4)+5) (2*(Aq (n+4)+1)+2*Dl-(ef+178))<=532:=max_le (U2 (L1 lL.le 5) (by decide)) ((K4 _ _).trans (Y3 (U3 2 (lL:Aq (n+4)+1<=12)) (U3 2 d2)))
 have c2:2*(Aq (n+4)+1)+2*Dl<=max (Aq (n+4)+5) (2*(Aq (n+4)+1)+2*Dl-(ef+178))+(ef+178):=le_tsub_add.trans (L1 (le_max_right _ _) _)
 have c1:=le_max_left (Aq (n+4)+5) (2*(Aq (n+4)+1)+2*Dl-(ef+178))
 have hc:nLc n Dl ef=max (Aq (n+4)+5) (2*(Aq (n+4)+1)+2*Dl-(ef+178))+2:=rfl
 rw[hc]at d3⊢
 generalize max (Aq (n+4)+5) (2*(Aq (n+4)+1)+2*Dl-(ef+178))=c at c1 c2 c3 d3⊢
 have c25:c+2<=25:=(Α (C0 (L1 c3 2) (by decide))).symm.trans_le d3
 have p5:2^(Aq (n+4)+5)=2^(Aq (n+4)+1)*16:=AF 2 (Aq (n+4)+1) 4
 have hA:(n+4)*16<=2^c:=(ι 16 nL.le).trans (p5.symm.trans_le (Θ two_pos c1))
 rw[show ((c+2:ℕ):ℤ)-1=((c+1:ℕ):ℤ) from add_sub_cancel_right ((c+1:ℕ):ℤ) 1,AA,A4 (2:ℝ) c]
 generalize he1:((Aq (n+4)+1:ℕ):ℤ)+(Dl:ℤ)-30=e1 at dδ
 have two0:(2:ℝ)≠0:=AL
 have h1:δ*δ<=2^e1*2^e1:=by rw[<-abs_mul_abs_self δ];exact mul_self_le_mul_self (Aa δ) dδ
 have h3:(2:ℝ)^(e1+e1+52+53)<=2^(((c+(ef+223)):ℕ):ℤ):=AG one_le_two (show e1+e1+52+53<=(c:ℤ)+((ef:ℤ)+223) by have H:2*((Aq (n+4)+1:ℕ):ℤ)+2*(Dl:ℤ)<=(c:ℤ)+((ef:ℤ)+178):=R5.2 c2;linear_combination H-2*he1)
 rw[AK two0,AK two0,AK two0,zpow_ofNat,zpow_ofNat,AA,AF]at h3
 have ηA:δ*δ*2^(52:Ἀ)/(2^(350:Ἀ)*X)*2^(53:Ἀ)<=2^c:=by
  rw[div_mul_eq_mul_div,div_le_iff₀ Zt0]
  exact (AE (AE h1 (p2 52).le) (p2 53).le).trans (h3.trans (mul_le_mul_of_nonneg_left ZtE (p2 c).le))
 have η0:0<=δ*δ*2^(52:Ἀ)/(2^(350:Ἀ)*X):=div_nonneg (Af (mul_self_nonneg δ) (p2 _).le) Zt0.le
 generalize δ*δ*2^(52:Ἀ)/(2^(350:Ἀ)*X)=η at hk ηA η0
 have A23:(2:ℝ)^c<=2^(23:Ἀ):=AU one_le_two (Nat.le_of_add_le_add_right (c25:c+2<=23+2))
 have hA':((n:ℝ)+4)*16<=2^c:=by exact_mod_cast hA
 have nR2:(n:ℝ)<=2048:=by exact_mod_cast hn
 generalize (2:ℝ)^c=A at ηA A23 hA'⊢
 have t:(12*(n:ℝ)+7)*η<=24583*η:=AE (by linear_combination 12*nR2) η0
 have K1:k*2^(53:Ἀ)<=(12*n+7)+1+A:=by linear_combination 2^(53:Ἀ)*hk+t+(1+24583/2^(53:Ἀ))*ηA+(24583/2^(53:Ἀ))*A23
 have k0:0<=k:=by rw[hk];exact add_nonneg (Af (Af (add_nonneg (Af (R6 _) (M3 n)) (R6 _)) (nN 53)) (add_nonneg zero_le_one η0)) η0
 have kb:k<=1/2^(29:Ἀ):=by linear_combination (1/2^(53:Ἀ))*(K1+12*nR2+A23)
 have kk:k*k<=1/2^(29:Ἀ)*(1/2^(29:Ἀ)):=mul_le_mul kb kb k0 (nN 29)
 exact⟨C0 c25 (by decide),X0,k0,by linear_combination kb,by linear_combination 2^(54:Ἀ)*kk,by linear_combination K1+hA'+4*(M3 n:(0:ℝ)<=n),dd⟩
lemma nL_cert(n:Ἀ)(eps:kF)(x:(Ἂ Word))(r:Ἀ)(hn4:4<=n)(hn:n<=2048)(hb:r*n+n<2^(32:Ἀ))(hX:∀j,j<r*n+n->kB x[j]!≠255)(he:kB eps≠255)(he0:kB eps≠0)(hs:0<sv eps)
  (P:Ἅ×ἆ)(hP:P=nS1 x (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat 0)
  (m:Ἅ)(hm:m=P.1*((((1:ἆ)<<<(52+(Aq n).toUInt64))/I0 n).toInt64.toFloat*N5 ((971-(Aq n).toUInt64)<<<52)))
  (D:Ἅ)(hD:D=W7 (nS2 x m (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat*(((1:ἆ)<<<52)/I0 n).toInt64.toFloat+Dense.F.dw eps*nS201))
  (hok:nOKs D.toBits (nLc n (lq P.2) (mJ (kB eps))).toUInt64=ἃ):
  nWs 26 D.toBits=mK (XH n x r (no n x r)+ZA eps):=by
 obtain⟨df,d2,δ,dδ,dd⟩:=nL_float n eps x r hn4 hn hb hX he hs P hP m hm D hD
 have hZ:=nXHm n x r eps ((by decide:0<4).trans_le hn4)
 obtain⟨c64,X0,k0,k1,k2,k3,dd⟩:=nL_cert_h1 n _ _ δ _ _ _ hn4 hn d2 (nOKs_dec _ _ hok).2.2.1 dδ (by rw[<-hZ];exact le_add_of_nonneg_of_le (div_nonneg (Af (D8 fun _ _=>sq_nonneg _) (pow_nonneg zero_le_two _)) (M3 _)) (nEp eps he0 hs)) rfl dd
 exact nQ_tail D df _ (Rat.cast_pos.1 X0) _ c64 hok _ _ k0 k1 AP k2 k3 dd
end
lemma nRiN_h1(M:ℕ)(M1:2^(23:Ἀ)<=M):2^(63:Ἀ)/M*2^(12:Ἀ)+2^(63:Ἀ)%M*2^(12:Ἀ)/M=2^(75:Ἀ)/M:=by
 have dm:=Au (2^(63:Ἀ)) M
 generalize 2^(63:Ἀ)/M=Q at dm ⊢
 generalize 2^(63:Ἀ)%M=R at dm ⊢
 have k:2^(75:Ἀ)=R*2^(12:Ἀ)+Q*2^(12:Ἀ)*M:=by rw[show 2^(75:Ἀ)=2^(63:Ἀ)*2^(12:Ἀ) from m7 2 63 12,<-dm,Ϡ,Nat.mul_right_comm Q M,Τ]
 rw[k,Ρ _ _ (U1 (϶ 23) M1),Τ]
lemma nRiN(f:Ἃ)(f1:(mJ f)<2^(23:Ἀ)):
  (lq (((((1:ἆ)<<<63)/(f.toUInt64+0x800000))<<<12)+((((1:ἆ)<<<63)-((1:ἆ)<<<63)/(f.toUInt64+0x800000)*(f.toUInt64+0x800000))<<<12)/(f.toUInt64+0x800000)))=2^(75:Ἀ)/((mJ f)+2^(23:Ἀ)):=by
 generalize hmm:f.toUInt64+0x800000=m
 have mE:(lq m)=(mJ f)+2^(23:Ἀ):=by rw[<-hmm,η,ϩ];exact γ ((X3 f1 _).trans (by decide))
 generalize hM:(mJ f)+2^(23:Ἀ)=M at mE
 have M1:2^(23:Ἀ)<=M:=hM▸K2 _ _
 have M2:M<2^(24:Ἀ):=hM▸(X3 f1 (2^(23:Ἀ))).trans_le (by decide)
 have t63:(lq ((1:ἆ)<<<63))=2^(63:Ἀ):=by rfl
 generalize (1:ἆ)<<<63=A at t63 ⊢
 have q1:(A/m).toNat=2^(63:Ἀ)/M:=by rw[ml,t63,mE]
 generalize A/m=a at q1 ⊢
 have q1b:2^(63:Ἀ)/M<=2^(40:Ἀ):=Nat.div_le_of_le_mul (U2 (by decide) (ι (2^(40:Ἀ)) M1))
 have dm:=Au (2^(63:Ἀ)) M
 have pm:(a*m).toNat=2^(63:Ἀ)/M*M:=by rw[ύ,q1,mE];exact γ ((pr _ _).trans_lt (by decide))
 have r1:(A-a*m).toNat=2^(63:Ἀ)%M:=by
  rw[χ _ _ (by rw[θ,pm,t63];exact pr _ _),pm,t63];exact Nat.sub_eq_of_eq_add ((Τ _ _).trans dm).symm
 generalize A-a*m=b at r1 ⊢
 have r1b:2^(63:Ἀ)%M<M:=ϊ _ (U1 (϶ 23) M1)
 have q2:((b<<<12)/m).toNat=2^(63:Ἀ)%M*2^(12:Ἀ)/M:=by
  rw[ml,ψ,r1,mE];simp only[kA,kH,ο];rw[γ ((ϐ (r1b.trans M2) (϶ 12)).trans_le (by decide))]
 have q2b:2^(63:Ἀ)%M*2^(12:Ἀ)/M<2^(12:Ἀ):=Ϟ (ϐ r1b (϶ 12))
 have q1s:(a<<<12).toNat=2^(63:Ἀ)/M*2^(12:Ἀ):=by
  rw[ψ,q1];simp only[kA,kH,ο];exact γ ((ι _ q1b).trans_lt (by decide))
 rw[η,q1s,q2,γ ((N7 (ι _ q1b) q2b).trans_le (by decide))]
 exact nRiN_h1 M M1
section
open Dense.F
lemma nRiV(dW:Ἃ)(ed:ἆ)(he:(lq ed)<=254):
  fv (nRi dW ed)/\rv (nRi dW ed)=((2^(75:Ἀ)/((mJ (dW&&&0x7FFFFF))+2^(23:Ἀ)):ℕ):ℝ)*2^(-((74+(lq ed):ℕ):ℤ)):=by
 have f1:(mJ (dW&&&0x7FFFFF))<2^(23:Ἀ):=by rw[Ξ];exact AO ϝ (by decide)
 have RN:=nRiN _ f1
 have R2:2^(75:Ἀ)/((mJ (dW&&&0x7FFFFF))+2^(23:Ἀ))<=2^(52:Ἀ):=Nat.div_le_of_le_mul (U2 (show 2^(75:Ἀ)<=2^(23:Ἀ)*2^(52:Ἀ) by decide) (ι (2^(52:Ἀ)) (K2 (2^(23:Ἀ)) _)))
 have ue:((((1:ἆ)<<<63)/((dW&&&0x7FFFFF).toUInt64+0x800000))<<<12)+((((1:ἆ)<<<63)-((1:ἆ)<<<63)/((dW&&&0x7FFFFF).toUInt64+0x800000)*((dW&&&0x7FFFFF).toUInt64+0x800000))<<<12)/((dW&&&0x7FFFFF).toUInt64+0x800000)=I0 (2^(75:Ἀ)/((mJ (dW&&&0x7FFFFF))+2^(23:Ἀ))):=by
  apply ξ.1;rw[RN];simp only[Ϣ,O6];exact (γ (R2.trans_lt (by decide))).symm
 have hn:nRi dW ed=(I0 (2^(75:Ἀ)/((mJ (dW&&&0x7FFFFF))+2^(23:Ἀ)))).toInt64.toFloat*N5 ((949-ed)<<<52):=by
  simp only[nRi,ue]
 rw[hn]
 obtain⟨R,hR⟩:∃R,R=2^(75:Ἀ)/((mJ (dW&&&0x7FFFFF))+2^(23:Ἀ)):=⟨_,rfl⟩
 rw[<-hR]at R2⊢
 obtain⟨rf,rr⟩:=kg2 R (R2.trans_lt (by decide))
 have pt:(lq (949-ed))=949-(lq ed):=by
  rw[χ _ _ (θ.2 (he.trans (by decide)))];rfl
 obtain⟨pf,pr⟩:=nPw (949-ed) (by rw[pt];exact G5 ((J1 he 1).trans (by decide))) (by rw[pt];exact (K4 949 _).trans (by decide))
 rw[pt]at pr
 have ez:((((949-(lq ed)):ℕ):ℤ)-1023)=-(((74+(lq ed)):ℕ):ℤ):=by rw[pO (he.trans (by decide)),W5];ring
 have r1:((R:ℕ):ℝ)<=2^(52:Ἀ):=by have:=R5 (α:=ℝ)|>.2 R2;rwa[I4,nZ]at this
 have q:=Dense.F.aE _ _ rf pf (R:ℤ) (-(((74+(lq ed)):ℕ):ℤ)) (by rw[rr,pr,ez,M5]) (by rw[Int.natAbs_cast];exact R2.trans_lt (by decide)) (Int.neg_le_neg (U4.2 ((J1 he 74).trans (by decide)):((74+(lq ed):ℕ):ℤ)<=((1074:ℕ):ℤ))) (nZb _ _ r1)
 refine⟨q.1,?_⟩;rw[q.2,M5]
lemma nRiOK_h1(M:ℕ)(M0:0 < M)(M2:M<=2^(24:Ἀ)):|((2^(75:Ἀ)/M:ℕ):ℝ)*(M:ℝ)*((2:ℝ)^(75:Ἀ))⁻¹-1|<=1/2^(50:Ἀ):=by
 have a1n:=pr (2^(75:Ἀ)) M
 have a2n:=Nat.lt_div_mul_add (a:=2^(75:Ἀ)) M0
 generalize 2^(75:Ἀ)/M=Q at a1n a2n ⊢
 have a1:(Q:ℝ)*M<=2^(75:Ἀ):=by have:=R5 (α:=ℝ)|>.2 a1n;rwa[Y4,I4,nZ]at this
 have a2:(2:ℝ)^(75:Ἀ)<=Q*M+M:=by have:=R5 (α:=ℝ)|>.2 a2n.le;rwa[W5,Y4,I4,nZ]at this
 have a3:(M:ℝ)<=2^(24:Ἀ):=by have:=R5 (α:=ℝ)|>.2 M2;rwa[I4,nZ]at this
 exact A3.2 ⟨by linear_combination (1/2^(75:Ἀ):ℝ)*a2+(1/2^(75:Ἀ):ℝ)*a3,by linear_combination (1/2^(75:Ἀ):ℝ)*a1⟩
lemma nRiOK(dW:Ἃ)(hd:0<(mJ (kB dW))∧kB dW≠255∧ZX dW=false):
  fv (nRi dW (kB dW).toUInt64)/\|rv (nRi dW (kB dW).toUInt64)*rv (dw dW)-1|<=1/2^(50:Ἀ):=by
 have ee:(lq (kB dW).toUInt64)=(mJ (kB dW)):=ϩ _
 have e254:(mJ (kB dW))<=254:=nKb _ hd.2.1
 obtain⟨rf,rr⟩:=nRiV dW (kB dW).toUInt64 (by rw[ee];exact e254)
 refine⟨rf,?_⟩
 obtain⟨-,e⟩:=cI dW hd.2.1
 rw[rr,e,ee]
 have z:kB dW≠0:=λq=>by rw[q]at hd;exact absurd hd.1 (by decide)
 have hs:(sv dW:ℝ)=(kM dW:ℝ):=by rw[sv,hd.2.2];simp
 have hm:kM dW=(mJ (dW&&&0x7FFFFF))+2^(23:Ἀ):=by simp only[mantOf,z,ἂ,fracOf];omega
 have hk:kG dW=(mJ (kB dW))-1:=by simp[idxOf,z]
 rw[hs,hk,hm]
 have f1:(mJ (dW&&&0x7FFFFF))<2^(23:Ἀ):=by rw[Ξ];exact AO ϝ (by decide)
 have h1:=hd.1
 generalize mJ (kB dW)=E at h1 ⊢
 rw[mul_mul_mul_comm,<-AA (2:ℝ) (E-1),<-AK AL,show -((74+E:ℕ):ℤ)+((E-1:ℕ):ℤ)=-((75:ℕ):ℤ) by omega,zpow_neg,AA]
 exact nRiOK_h1 _ (U1 (϶ 23) (K2 _ _)) ((L1 f1.le _).trans (by decide))
end
lemma nLR_th(n:Ἀ)(eps:Ἃ)(x:(Ἂ Word))(r:Ἀ):(nLR n eps x r).th.get=NR.eI n (gM n eps x r):=by
 unfold nLR;split<;>rfl
lemma nSv(w:kF)(h:0<ZA w):0<sv w:=by
 rw[Y1]at h;by_contra c;push_neg at c
 have a:(sv w:ℚ)<=0:=by exact_mod_cast c
 have:(sv w:ℚ)*2^((ZZ w:Ἁ)-149)<=0:=mul_nonpos_of_nonpos_of_nonneg a (by positivity)
 linarith
section
open Dense.F
lemma nLR_el(n:Ἀ)(eps:kF)(x:(Ἂ Word))(r:Ἀ)(hb:r*n+n<2^(32:Ἀ))(hX:∀j,j<r*n+n->kB x[j]!≠255)(hs:0<sv eps)(j:Ἀ)(hj1:r*n<=j)(hj2:j<r*n+n)
  (gw bw:kF)(hg:kB gw≠255)(hbw:kB bw≠255)(hf:(nLR n eps x r).fast=true)
  (hok:nElOK x[j]! gw (nLR n eps x r).Kp (nLR n eps x r).Ke (nElB x[j]! gw bw (nLR n eps x r).m (nLR n eps x r).ri)=true):
  nW (nElB x[j]! gw bw (nLR n eps x r).m (nLR n eps x r).ri)=ZT ((ZA x[j]!-no n x r)*ZA gw/ZA (mK (XH n x r (no n x r)+ZA eps))+ZA bw):=by
 unfold nLR at hf hok ⊢
 split_ifs at hf hok ⊢ with C
 · dsimp only at hf hok ⊢
   simp only[Η,AJ,AM,Ak]at C
   obtain⟨⟨⟨hn4,hn⟩,he0⟩,he⟩:=C
   simp only[Η,AJ,BM]at hf
   obtain⟨⟨⟨hO,h0⟩,h255⟩,hz⟩:=hf
   have hc:=nL_cert n eps x r hn4 hn hb hX he he0 hs _ rfl _ rfl _ rfl hO
   rw[<-hc]
   obtain⟨mf,mE,hxD,d2,-,-⟩:=nMean n x r hn4 hn hb hX
   have hd1:=Ι.1 h0
   rw[nj]at hd1
   have hd2:=Ι.1 h255
   have hd:=And.intro hd1 (And.intro (ne_of_apply_ne mJ (R8 hd2)) hz)
   obtain⟨rf,rr⟩:=nRiOK _ hd
   exact nEl_spec _ _ _ _ _ _ (no n x r) (Aq (n+4)+1) _ (hX j hj2) hg hbw hd ⟨rf,rr⟩ ⟨mf,mE⟩ (hxD j hj1 hj2) ⟨Nat.succ_le_succ ((mu (Nat.succ_ne_zero _)).2 (K2 4 n)),(m9 (Nat.succ_ne_zero _)).2 (C0 (L1 hn 4) (N2 rfl))⟩ d2 hok
end
lemma nCa{a b:Ἄ}(h:(J0 a)+(J0 b)<2^(32:Ἀ)):(J0 (a+b))=(J0 a)+(J0 b):=by
 have:2^(32:Ἀ)≤2^nx:=ς
 rw[lJ];exact γ (U1 h this)
lemma nCl{a n s:Ἀ}(h:a<n)(k:n+2<=s):a<s:=U1 h ((V1 n 2).trans k)
lemma nO1{t n:Ἄ}(hk:(J0 n)-(J0 t)=0):¬t<n:=fun h=>Z8 (ε.mp h) hk
lemma nOq{a t n:ℕ}(hb:a+n<2^(32:Ἀ))(h:t<n):a+t<2^(32:Ἀ):=C0 (J1 h.le a) hb
lemma nOz{i0 t n:Ἄ}(hb:(J0 i0)+(J0 n)<2^(32:Ἀ))(hk:(J0 n)-(J0 t)=0)(ht:(J0 t)<=(J0 n)):¬(i0+t < i0+n):=by
 intro h;have e:=C4 ht (K3.1 hk);have:=ε.mp h;rw[nCa (e▸hb),nCa hb,e] at this;exact B4 _ this
lemma nOs{t n:Ἄ}{k:ℕ}(hk:(J0 n)-(J0 t)=k+1):(J0 t)<(J0 n)∧(J0 n)-(J0 (t+1))=k∧(J0 (t+1))<=(J0 n):=by
 have h1:=N1 hk;rw[Dense.ac (ε.mpr h1)];exact⟨h1,by rw[V4,hk,BK],h1⟩
lemma nYqC{σ τ:Type}(f:σ->Ἄ->Ἄ->Ἃ)(s:σ)(f':τ->Ἄ->Ἄ->Ἃ)(s':τ)(lo e:Ἄ)(h:∀i j,f s i j=f' s' i j):∀k (i:Ἄ)(out:(Ἂ Word)),(J0 e)-(J0 i)=k->Yq f s lo i e out=Yq f' s' lo i e out:=by
 intro k;induction k with
 |zero=>intro i out hk;rw[Yq.eq_def f s,Yq.eq_def f' s',dif_neg (nO1 hk),dif_neg (nO1 hk)]
 |succ k ih=>intro i out hk;obtain⟨h1,hk',-⟩:=nOs hk;rw[Yq.eq_def f s,Yq.eq_def f' s',dif_pos (ε.mpr h1),dif_pos (ε.mpr h1),h];exact ih _ _ hk'
lemma nH1{i0 t n:Ἄ}{s:ℕ}(ht:(J0 t)<(J0 n))(hb:(J0 i0)+(J0 n)<2^(32:Ἀ))(hx:(J0 i0)+(J0 n)<=s):(J0 (i0+t))<s:=by
 rw[nCa (nOq hb ht)];exact U1 (O1 ht _) hx
lemma nH2{k:ℕ}(Xr:ἀ)(t:Ἄ)(v:Ἅ)(h:(J0 t)<(kS Xr))(hr:k<=(kS Xr)):k<=(kS (Xr.uset t v h)):=by rw[Dense.F.uZ];exact hr
lemma nH3{k:ℕ}(o:(Ἂ Word))(t:Ἄ)(v:Ἃ)(h:(J0 t)<(O0 o))(ho:k<=(O0 o)):k<=(O0 (o.uset t v h)):=by rw[BT];exact ho
lemma nHd{t n:Ἄ}(h:t<n):(J0 n)-(J0 (t+1))<(J0 n)-(J0 t):=by rw[Dense.ac h];exact Z5 _ _ (ε.mp h)
lemma nHk2{r k rows n:ℕ}(h:r+(k+1)=rows):r*n+n<=rows*n:=h▸(by rw[Ϡ,Ο];exact J1 (K2 n (k*n)) (r*n))
lemma nHk3{r k rows:ℕ}(h:r+(k+1)=rows):r+1+k=rows:=((lO r 1 k).trans (pA r k 1)).trans h
structure nDP where
 a:ἀ
 d:ἆ
noncomputable def nLoN(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(nd:Ἅ)(e:nLS)(i j:Ἄ):Ἃ:=nLo n eps x g b nd e i j
noncomputable def nEO(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(nd:Ἅ)(e:nLS)(i j:Ἄ):Ἃ:=
 if e.fast then
  let xw:=ck x i;let gw:=ck g j
  let B:=nElB xw gw (ck b j) e.m e.ri
  if nElOK xw gw e.Kp e.Ke B then nW B else nLoN n eps x g b nd e i j
 else nLoN n eps x g b nd e i j
noncomputable def nLQ(n:Ἀ)(eps:Ἃ)(s:Ἅ)(Dl:ἆ)(vf:Ἅ->Ἅ)(th:Thunk NR.fH):nLS:=
 let n6:=I0 n
 let lb:=(Aq n).toUInt64
 let m:=s*((((1:ἆ)<<<(52+lb))/n6).toInt64.toFloat*N5 ((971-lb)<<<52))
 let v:=vf m
 let D:=W7 (v*(((1:ἆ)<<<52)/n6).toInt64.toFloat+Dense.F.dw eps*nS201)
 let B:=D.toBits
 let dW:=nWs 26 B
 let ed:=kB dW
 ⟨m,nRi dW ed.toUInt64,nElK m (Aq (n+4)+1) Dl.toNat ed.toNat,(1054-ed.toNat).toUInt64,
  nOKs B (nLc n Dl.toNat (kB eps).toNat).toUInt64&&0<ed&&ed<255&&!ZX dW,th,dW⟩
lemma nLQ_e(n:Ἀ)(eps:Ἃ)(x:(Ἂ Word))(r:Ἀ)(h:(4<=n&&n<=2048&&kB eps != 0&&kB eps != 255)=true):nLR n eps x r=nLQ n eps (nS1 x (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat 0).1 (nS1 x (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat 0).2 (λm=>nS2 x m (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat) (Thunk.mk (λ_=>NR.eI n (gM n eps x r))):=by
 unfold nLR;rw[if_pos h];rfl
@[noinline]def gM0(n:Ἀ)(x:(Ἂ Word))(r:Ἀ)(w:Ἃ):Ἀ×Ἀ×Ἆ×Ἀ×Ἀ×Ἃ:=
 let lo:=r*n
 let u:=cS x (G0 lo) (G0 (lo+n)) 255
 let st:=cF x u (G0 lo) (G0 (lo+n)) 0 0 0 0 0 0 0 0
 let S:=Y0 st.P st.N
 let k:=Q2 (max st.P st.N)-30
 ((lq u),k,decide (st.P<st.N),S>>>k,S&&&(VO 1 k-1),w)
theorem gM0_e(n:Ἀ)(eps:Ἃ)(x:(Ἂ Word))(r:Ἀ):gM0 n x r (gM n eps x r).2.2.2.2.2=gM n eps x r:=rfl
theorem nLR_dW(n:Ἀ)(eps:kF)(x:(Ἂ Word))(r:Ἀ)(hb:r*n+n<2^(32:Ἀ))(hX:∀j,j<r*n+n->kB x[j]!≠255)(q:0<ZA eps)(hf:(nLR n eps x r).fast=true):(nLR n eps x r).dW=(gM n eps x r).2.2.2.2.2:=by
 unfold nLR at hf ⊢
 split_ifs at hf ⊢ with C
 · dsimp only at hf ⊢
   simp only[Η,AJ,AM,Ak]at C
   obtain⟨⟨⟨hn4,hn⟩,he0⟩,he⟩:=C
   simp only[Η,AJ,BM]at hf
   obtain⟨⟨⟨hO,-⟩,-⟩,-⟩:=hf
   rw[nL_cert n eps x r hn4 hn hb hX he he0 (nSv eps q) _ rfl _ rfl _ rfl hO,gN n eps x r (U1 (by decide) hn4) q hb]
   exact (Vn n eps x r (U1 (by decide) hn4) q).2.1.symm
@[noinline]def nFB(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(nd:Ἅ)(r:Ἀ)(dW:Ἃ)(ex:Option NR.fH)(i j:Ἄ):Ἃ×NR.fH:=
 let z:=(match ex with | some z=>z | none=>NR.eI n (gM0 n x r dW))
 (nFBz n eps x g b nd z i j,z)
def nDR(x:(Ἂ Word))(Xr:ἀ)(i0 t n:Ἄ)(hx:(J0 i0)+(J0 n)<=(O0 x))(hb:(J0 i0)+(J0 n)<2^(32:Ἀ))(hr:(J0 n)+2<=(kS Xr))(s:Ἅ)(Dl:ἆ):nDP:=
 if h:t<n then
  have ht:=ε.mp h
  let w:=x.uget (i0+t) (nH1 ht hb hx)
  let v:=Dense.F.dw w
  nDR x (Xr.uset t v (nCl ht hr)) i0 (t+1) n hx hb (nH2 _ _ _ _ hr) (s+v) (if Dl<nfe w then nfe w else Dl)
 else ⟨Xr.uset n s (U1 (I3 _) ((E7 _).trans hr)),Dl⟩
termination_by (J0 n)-(J0 t)
decreasing_by exact nHd h
lemma nDR_sz(x:(Ἂ Word))(i0 n:Ἄ)(hx:(J0 i0)+(J0 n)<=(O0 x))(hb:(J0 i0)+(J0 n)<2^(32:Ἀ)):∀k (t:Ἄ)(Xr:ἀ)(hr:(J0 n)+2<=(kS Xr))(s:Ἅ)(Dl:ἆ),(J0 n)-(J0 t)=k->(kS (nDR x Xr i0 t n hx hb hr s Dl).a)=(kS Xr):=by
 intro k;induction k with
 |zero=>
  intro t Xr hr s Dl hk;rw[nDR.eq_def,dif_neg (nO1 hk)];exact Dense.F.uZ ..
 |succ k ih=>
  intro t Xr hr s Dl hk;obtain⟨ht,hk',-⟩:=nOs hk
  rw[nDR.eq_def,dif_pos (ε.mpr ht)];dsimp only;rw[ih _ _ _ _ _ hk',Dense.F.uZ]
def nVR(Xr:ἀ)(m:Ἅ)(t n:Ἄ)(hr:(J0 n)<=(kS Xr))(v:Ἅ):Ἅ:=
 if h:t<n then let d:=Xr.uget t (U1 (ε.mp h) hr)-m;nVR Xr m (t+1) n hr (v+d*d) else v
termination_by (J0 n)-(J0 t)
decreasing_by exact nHd h
def nFE(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(Xr gd bd:ἀ)(ge:ByteArray)(nd m ri:Ἅ)(Kp:ἆ)(r:Ἀ)(dW:Ἃ)(ex:Option NR.fH)(lo t nn:Ἄ)
 (hr:(J0 nn)<=(kS Xr))(hg:(J0 nn)<=(kS gd))(hd:(J0 nn)<=(kS bd))(he:(J0 nn)<=ge.size)
 (hb:(J0 lo)+(J0 nn)<2^(32:Ἀ))(out:(Ἂ Word))(ho:(J0 lo)+(J0 nn)<=(O0 out)):(Ἂ Word):=
 if h:t<nn then
  have ht:=ε.mp h
  let B:=((Xr.uget t (U1 ht hr)-m)*(gd.uget t (U1 ht hg)*ri)+bd.uget t (U1 ht hd)).toBits
  have hu:(J0 (lo+t))<(O0 out):=nH1 ht hb ho
  if nOK B (nC ((ge.uget t (U1 ht he)).toUInt64+Kp) ((B>>>52)&&&2047)) then
   nFE n eps x g b Xr gd bd ge nd m ri Kp r dW ex lo (t+1) nn hr hg hd he hb (out.uset (lo+t) (nW B) hu) (nH3 _ _ _ _ ho)
  else
   let p:=nFB n eps x g b nd r dW ex (lo+t) t
   nFE n eps x g b Xr gd bd ge nd m ri Kp r dW (some p.2) lo (t+1) nn hr hg hd he hb (out.uset (lo+t) p.1 hu) (nH3 _ _ _ _ ho)
 else out
termination_by (J0 nn)-(J0 t)
decreasing_by all_goals exact nHd h
lemma nFE_sz(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(Xr gd bd:ἀ)(ge:ByteArray)(nd m ri:Ἅ)(Kp:ἆ)(r:Ἀ)(dW:Ἃ)(lo nn:Ἄ)
 (hr:(J0 nn)<=(kS Xr))(hg:(J0 nn)<=(kS gd))(hd:(J0 nn)<=(kS bd))(he:(J0 nn)<=ge.size)
 (hb:(J0 lo)+(J0 nn)<2^(32:Ἀ)):∀k (ex:Option NR.fH)(t:Ἄ)(out:(Ἂ Word))(ho:(J0 lo)+(J0 nn)<=(O0 out)),(J0 nn)-(J0 t)=k->
 (O0 (nFE n eps x g b Xr gd bd ge nd m ri Kp r dW ex lo t nn hr hg hd he hb out ho))=(O0 out):=by
 intro k;induction k with
 |zero=>
  intro ex t out ho hk;rw[nFE.eq_def,dif_neg (nO1 hk)]
 |succ k ih=>
  intro ex t out ho hk;obtain⟨ht,hk',-⟩:=nOs hk
  rw[nFE.eq_def,dif_pos (ε.mpr ht)];dsimp only;split<;>rw[ih _ _ _ _ hk',BT]
lemma nYq_sz{σ:Type}(f:σ->Ἄ->Ἄ->Ἃ)(s:σ)(lo e:Ἄ):∀k (i:Ἄ)(out:(Ἂ Word)),(J0 e)-(J0 i)=k->(O0 (Yq f s lo i e out))=(O0 out):=by
 intro k;induction k with
 |zero=>
  intro i out hk;rw[Yq,dif_neg (nO1 hk)]
 |succ k ih=>
  intro i out hk;obtain⟨h1,hk',-⟩:=nOs hk
  rw[Yq,dif_pos (ε.mpr h1),ih _ _ hk'];split;exact BT ..;rfl
@[inline]def nLc2(L Dl ef:Ἀ):Ἀ:=max (L+4) (2*L+2*Dl-(ef+178))+2
lemma nLc_e(n Dl ef:Ἀ):nLc2 (Aq (n+4)+1) Dl ef=nLc n Dl ef:=by unfold nLc2 nLc;omega
lemma kQ52(n:Ἀ)(h:n<2^(64:Ἀ)):((1:ἆ)<<<52)/I0 n=I0 (4503599627370496/n):=by
 apply ξ.1
 rw[ml,show ((1:ἆ)<<<52)=4503599627370496 from rfl]
 simp only[I0,O6]
 rw[γ h,γ (C0 (L5 _ _) (by decide))]
 rfl
lemma kO0(e:ἇ):e.toUInt64>>>32=I0 0:=by
 apply ξ.1;rw[ν,ϩ,show (lq 32)%64=32 from rfl,β,ώ (e.toNat_lt)];rfl
lemma kO1(e:ἇ):e.toUInt64>>>32+1=I0 1:=by rw[kO0];rfl
lemma nElK_n(m:Ἅ)(L Dl ed:Ἀ):nElK m L Dl ed=(max (L+Dl+1052) (((m.toBits>>>52)&&&2047).toNat+9)-ed).toUInt64:=by
 unfold nElK;congr 1;generalize ((m.toBits>>>52)&&&2047).toNat=E;omega
@[inline]def nElK3(m:Ἅ)(L Dl ed:Ἀ):ἆ:=(max (L+Dl+1052) (((m.toBits>>>52)&&&2047).toNat+9)-ed).toUInt64
lemma nElK3_e(m:Ἅ)(L Dl ed:Ἀ):nElK3 m L Dl ed=nElK m L Dl ed:=(nElK_n m L Dl ed).symm
lemma kZ0(n:Ἀ)(h:n<2^(40:Ἀ)):ZR (I0 n)=(Aq n).toUInt64:=by
 apply ξ.1;have l:=Nat.log2_le_self n
 rw[Y7,Dense.F.kOfS n (by omega),Dense.F.kOfS _ (by omega)]
lemma kZ1(n:Ἀ)(h:n<2^(40:Ἀ)):ZR (I0 n)+1=(Aq n+1).toUInt64:=by
 apply ξ.1;have l:=Nat.log2_le_self n
 rw[η,Y7,Dense.F.kOfS n (by omega),Dense.F.kOfS _ (by omega),show (1:ἆ).toNat=1 from rfl];omega
lemma kZ4(n:Ἀ)(h:n<2^(40:Ἀ)):ZR (I0 n)+4=(Aq n+4).toUInt64:=by
 apply ξ.1;have l:=Nat.log2_le_self n
 rw[η,Y7,Dense.F.kOfS n (by omega),Dense.F.kOfS _ (by omega),show (4:ἆ).toNat=4 from rfl];omega
def nCK(n:Ἀ)(eps:Ἃ)(rm nf ep z0:Ἅ)(L:ἆ):Prop:=rm=(((1:ἆ)<<<(52+ZR (I0 n)))/I0 n).toInt64.toFloat*N5 ((971-ZR (I0 n))<<<52)∧nf=(((1:ἆ)<<<52)/I0 n).toInt64.toFloat∧ep=Dense.F.dw eps*nS201∧z0=(I0 (0:Ἀ)).toInt64.toFloat∧L=ZR (I0 (n+4))+1
@[inline]def nLQk{β:Type}(eps:Ἃ)(s:Ἅ)(Dl:ἆ)(vf:Ἅ->Ἅ)(rm nf ep:Ἅ)(L:ἆ)(k:Ἅ->Ἅ->ἆ->Ἃ->Ἆ->β):β:=
 let m:=s*rm
 let v:=vf m
 let D:=W7 (v*nf+ep)
 let B:=D.toBits
 let dW:=nWs 26 B
 let ed:=kB dW
 k m (nRi dW ed.toUInt64) (nElK3 m L.toNat Dl.toNat ed.toNat) dW (nOKs B (nLc2 L.toNat Dl.toNat (kB eps).toNat).toUInt64&&0<ed&&ed<255&&!ZX dW)
lemma nLQk_e{β:Type}(n:Ἀ)(eps:Ἃ)(s:Ἅ)(Dl:ἆ)(vf:Ἅ->Ἅ)(rm nf ep z0:Ἅ)(L:ἆ)(hC:nCK n eps rm nf ep z0 L)(hn:n<2^(32:Ἀ))(th:Thunk NR.fH)(k:Ἅ->Ἅ->ἆ->Ἃ->Ἆ->β):nLQk eps s Dl vf rm nf ep L k=k (nLQ n eps s Dl vf th).m (nLQ n eps s Dl vf th).ri (nLQ n eps s Dl vf th).Kp (nLQ n eps s Dl vf th).dW (nLQ n eps s Dl vf th).fast:=by
 obtain⟨h1,h2,h3,-,h4⟩:=hC;subst h1 h2 h3 h4
 rw[kZ0 n (by omega),kZ1 (n+4) (by omega)]
 have h63:Aq (n+4)+1<2^(63:Ἀ):=C0 (Nat.succ_le_succ (Nat.log2_le_self _)) (by omega)
 simp only[nLQk,nLQ,nLc_e,nElK3_e,Dense.F.kOfS _ h63]
lemma nLo_z(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(nd:Ἅ)(e:nLS)(i j:Ἄ):nLo n eps x g b nd e i j=nFBz n eps x g b nd e.th.get i j:=rfl
@[noinline]def nRowS(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(nd:Ἅ)(r:Ἀ)(lo hi:Ἄ)(out:(Ἂ Word)):(Ἂ Word):=Yq (nFBz n eps x g b nd) (nEI n eps x r) lo lo hi out
lemma nRowS_e(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(nd:Ἅ)(r:Ἀ)(lo hi:Ἄ)(out:(Ἂ Word))(hf:(nLR n eps x r).fast=false):nRowS n eps x g b nd r lo hi out=Yq (nEO n eps x g b nd) (nLR n eps x r) lo lo hi out:=
 nYqC _ _ _ _ lo hi (fun i j=>by simp only[nEO,hf,υ,↓reduceIte,nLoN,nLo_z,nLR_th,nEI]) _ lo out rfl
lemma nLR_f(n:Ἀ)(eps:Ἃ)(x:(Ἂ Word))(r:Ἀ)(h:¬(4<=n&&n<=2048&&kB eps != 0&&kB eps != 255)=true):(nLR n eps x r).fast=false:=by unfold nLR;rw[if_neg h]
def nFE2(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(nd m ri:Ἅ)(Kp:ἆ)(r:Ἀ)(dW:Ἃ)(ex:Option NR.fH)(lo t nn:Ἄ)
 (hg:(J0 nn)<=(O0 g))(hd:(J0 nn)<=(O0 b))
 (hx:(J0 lo)+(J0 nn)<=(O0 x))(hb:(J0 lo)+(J0 nn)<2^(32:Ἀ))(out:(Ἂ Word))(ho:(J0 lo)+(J0 nn)<=(O0 out)):(Ἂ Word):=
 if h:t<nn then
  have ht:=ε.mp h
  let gw:=g.uget t (U1 ht hg)
  let B:=((Dense.F.dw (x.uget (lo+t) (nH1 ht hb hx))-m)*(Dense.F.dw gw*ri)+Dense.F.dw (b.uget t (U1 ht hd))).toBits
  have hu:(J0 (lo+t))<(O0 out):=nH1 ht hb ho
  if nOK B (nC (nfe gw+Kp) ((B>>>52)&&&2047)) then
   nFE2 n eps x g b nd m ri Kp r dW ex lo (t+1) nn hg hd hx hb (out.uset (lo+t) (nW B) hu) (nH3 _ _ _ _ ho)
  else
   let p:=nFB n eps x g b nd r dW ex (lo+t) t
   nFE2 n eps x g b nd m ri Kp r dW (some p.2) lo (t+1) nn hg hd hx hb (out.uset (lo+t) p.1 hu) (nH3 _ _ _ _ ho)
 else out
termination_by (J0 nn)-(J0 t)
decreasing_by all_goals exact nHd h
@[inline]def nRow(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(Xr gd bd:ἀ)(ge:ByteArray)(nd:Ἅ)(rm nf ep z0:Ἅ)(L:ἆ)(hC:nCK n eps rm nf ep z0 L)(r:Ἀ)(out:(Ἂ Word))
 (hr:n+2<=(kS Xr))(hg:n<=(kS gd))(hd:n<=(kS bd))(he:n<=ge.size)
 (hx:r*n+n<=(O0 x))(ho:r*n+n<=(O0 out))(hb:r*n+n<2^(32:Ἀ)):ἀ×(Ἂ Word):=
 let lo:=G0 (r*n);let hi:=G0 (r*n+n);let nn:=G0 n
 have q1:(J0 lo)=r*n:=Conv.aa _ (C0 (V1 _ _) hb)
 have q2:(J0 nn)=n:=Conv.aa _ (C0 (K2 _ _) hb)
 have hx':(J0 lo)+(J0 nn)<=(O0 x):=by rw[q1,q2];exact hx
 have hb':(J0 lo)+(J0 nn)<2^(32:Ἀ):=by rw[q1,q2];exact hb
 have hr':(J0 nn)+2<=(kS Xr):=by rw[q2];exact hr
 let pD:=nDR x Xr lo 0 nn hx' hb' hr' z0 0
 have hs:(kS pD.a)=(kS Xr):=nDR_sz x lo nn hx' hb' _ 0 Xr hr' z0 0 rfl
 have hr2:(J0 nn)+2<=(kS pD.a):=by rw[hs];exact hr'
 if pD.d<255 then
  let s:=pD.a.uget nn (U1 (I3 _) ((E7 _).trans hr2))
  nLQk eps s pD.d (λm=>nVR pD.a m 0 nn ((V1 _ 2).trans hr2) z0) rm nf ep L (λm ri Kp dW fast=>
   if fast then
    (pD.a,nFE n eps x g b pD.a gd bd ge nd m ri Kp r dW none lo 0 nn
     ((V1 _ 2).trans hr2) (by rw[q2];exact hg) (by rw[q2];exact hd) (by rw[q2];exact he) hb' out (by rw[q1,q2];exact ho))
   else (pD.a,nRowS n eps x g b nd r lo hi out))
 else (pD.a,#[])
lemma nRow_sz(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(Xr gd bd:ἀ)(ge:ByteArray)(nd:Ἅ)(rm nf ep z0:Ἅ)(L:ἆ)(hC:nCK n eps rm nf ep z0 L)(r:Ἀ)(out:(Ἂ Word))
 (hr:n+2<=(kS Xr))(hg:n<=(kS gd))(hd:n<=(kS bd))(he:n<=ge.size)
 (hx:r*n+n<=(O0 x))(ho:r*n+n<=(O0 out))(hb:r*n+n<2^(32:Ἀ)):
 (kS (nRow n eps x g b Xr gd bd ge nd rm nf ep z0 L hC r out hr hg hd he hx ho hb).1)=(kS Xr)∧((nRow n eps x g b Xr gd bd ge nd rm nf ep z0 L hC r out hr hg hd he hx ho hb).2=#[]∨(O0 (nRow n eps x g b Xr gd bd ge nd rm nf ep z0 L hC r out hr hg hd he hx ho hb).2)=(O0 out)):=by
 unfold nRow nLQk nRowS;dsimp only;split
 · split
   · exact⟨nDR_sz _ _ _ _ _ _ _ _ _ _ _ rfl,BA (nFE_sz _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ rfl)⟩
   · exact⟨nDR_sz _ _ _ _ _ _ _ _ _ _ _ rfl,BA (nYq_sz _ _ _ _ _ _ _ rfl)⟩
 · exact⟨nDR_sz _ _ _ _ _ _ _ _ _ _ _ rfl,pZ rfl⟩
@[noinline]def nRow1(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(nd:Ἅ)(rm nf ep z0:Ἅ)(L:ἆ)(hC:nCK n eps rm nf ep z0 L)(r:Ἀ)(out:(Ἂ Word))
 (hg:n<=(O0 g))(hd:n<=(O0 b))
 (hx:r*n+n<=(O0 x))(ho:r*n+n<=(O0 out))(hb:r*n+n<2^(32:Ἀ)):(Ἂ Word):=
 let lo:=G0 (r*n);let hi:=G0 (r*n+n);let nn:=G0 n
 have q1:(J0 lo)=r*n:=Conv.aa _ (C0 (V1 _ _) hb)
 have q2:(J0 nn)=n:=Conv.aa _ (C0 (K2 _ _) hb)
 have hx':(J0 lo)+(J0 nn)<=(O0 x):=by rw[q1,q2];exact hx
 have hb':(J0 lo)+(J0 nn)<2^(32:Ἀ):=by rw[q1,q2];exact hb
 let S:=nS1 x lo hi z0 0
 if S.2<255 then
  nLQk eps S.1 S.2 (λm=>nS2 x m lo hi z0) rm nf ep L (λm ri Kp dW fast=>
   if fast then
    nFE2 n eps x g b nd m ri Kp r dW none lo 0 nn (by rw[q2];exact hg) (by rw[q2];exact hd) hx' hb' out (by rw[q1,q2];exact ho)
   else nRowS n eps x g b nd r lo hi out)
 else #[]
def nRows(rows n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(gd bd:ἀ)(ge:ByteArray)(nd:Ἅ)(rm nf ep z0:Ἅ)(L:ἆ)(hC:nCK n eps rm nf ep z0 L)(hg:n<=(kS gd))(hd:n<=(kS bd))(he:n<=ge.size)
 (hx:rows*n<=(O0 x))(hb:rows*n<2^(32:Ἀ)):(k r:Ἀ)->(Xr:ἀ)->(out:(Ἂ Word))->n+2<=(kS Xr)->rows*n<=(O0 out)->r+k=rows->(Ἂ Word)
 | 0,_,_,out,_,_,_=>out
 | k+1,r,Xr,out,hr,ho,hk=>
  have kk:r*n+n<=rows*n:=nHk2 hk
  let p:=nRow n eps x g b Xr gd bd ge nd rm nf ep z0 L hC r out hr hg hd he (kk.trans hx) (kk.trans ho) (C0 kk hb)
  have q:(kS p.1)=(kS Xr):=(nRow_sz n eps x g b Xr gd bd ge nd rm nf ep z0 L hC r out hr hg hd he (kk.trans hx) (kk.trans ho) (C0 kk hb)).1
  if hs:rows*n<=(O0 p.2) then nRows rows n eps x g b gd bd ge nd rm nf ep z0 L hC hg hd he hx hb k (r+1) p.1 p.2 (q▸hr) hs (nHk3 hk)
  else #[]
lemma nGEu(w:ἇ):(((w>>>23)&&&255).toUInt8).toUInt64=nfe w:=by
 rw[nfe];apply ξ.1;rw[UInt8.toNat_toUInt64,UInt32.toNat_toUInt8,ϩ];apply γ;rw[Ξ];exact Z4 (ϝ.trans (by decide))
@[inline]def nWF(a:(Ἂ Word)):ἀ×ByteArray:=(⟨a.map Dense.F.dw⟩,⟨a.map (fun (w:Word)=>((w>>>23)&&&255).toUInt8)⟩)
lemma nWF_sp(a:(Ἂ Word)):(kS (nWF a).1)=a.size∧(∀t,t<a.size->(nWF a).1.data[t]! =Dense.F.dw a[t]!)∧(nWF a).2.size=a.size∧∀t,t<a.size->((nWF a).2.data[t]!).toUInt64=nfe a[t]!:=by
 refine⟨m0 ..,fun t ht=>?_,m0 ..,fun t ht=>?_⟩
 · show (a.map Dense.F.dw)[t]! =_;rw[AB _ _ (by rw[m0];exact ht),AB a t ht,Array.getElem_map]
 · show ((a.map (fun (w:Word)=>((w>>>23)&&&255).toUInt8))[t]!).toUInt64=_;rw[AB _ _ (by rw[m0];exact ht),AB a t ht,Array.getElem_map];exact nGEu _
@[inline]def nWF2(k:(Ἂ Word)×(Ἂ Word)):(ἀ×ByteArray)×ἀ:=(nWF k.1,(nWF k.2).1)
@[inline]def nK2eq(a b:(Ἂ Word)×(Ἂ Word)):Ἆ:=Dense.F.wcAeq a.1 b.1&&Dense.F.wcAeq a.2 b.2
theorem nK2eq_ok(a b:(Ἂ Word)×(Ἂ Word))(h:nK2eq a b=true):a=b:=by
 obtain⟨a1,a2⟩:=a;obtain⟨b1,b2⟩:=b;simp only[nK2eq,Η]at h;rw[Dense.F.wcAeq_ok h.1,Dense.F.wcAeq_ok h.2]
@[noinline]def nWS2:ST.Ref Unit (Ἂ (Dense.F.WcE nWF2)):=(ST.Prim.mkRef #[] (Void.mk ())).val
@[noinline]def nWGo(g b:(Ἂ Word)):{v//v=nWF2 (g,b)}:=Dense.F.wcGet3 nK2eq (fun a b=>nK2eq_ok a b) nWS2 32 (g,b) (fun k=>k.1)
def nWGs(g b:(Ἂ Word))(s:Ἂ (Dense.F.WcE nWF2))(i:Nat):{v//v=nWF2 (g,b)}:=
 if h:i<s.size then
  withPtrAddr s[i].k.1 (fun p1=>withPtrAddr g (fun p2=>
   if p1=p2 then (if hk:nK2eq s[i].k (g,b)=true then ⟨s[i].v,by rw[s[i].h,nK2eq_ok _ _ hk]⟩ else nWGo g b)
   else nWGs g b s (i+1)) (fun _ _=>Dense.F.wcSub _ _)) (fun _ _=>Dense.F.wcSub _ _)
 else nWGo g b
termination_by s.size-i
@[noinline]def nWG2(g b:(Ἂ Word)):(ἀ×ByteArray)×ἀ:=(nWGs g b (ST.Prim.Ref.get nWS2 (Void.mk ())).val 0).1
theorem nWG2_v(g b:(Ἂ Word)):nWG2 g b=nWF2 (g,b):=(nWGs g b _ 0).2
def nRowsS(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(nd:Ἅ)(rows r:Ἀ)(out:(Ἂ Word)):(Ἂ Word):=
 if r<rows then nRowsS n eps x g b nd rows (r+1) (nRowS n eps x g b nd r (G0 (r*n)) (G0 (r*n+n)) out) else out
termination_by rows-r
decreasing_by all_goals omega
lemma nRowsS_e(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(nd:Ἅ)(rows:Ἀ)(hf:∀r,(nLR n eps x r).fast=false):∀k r (out:(Ἂ Word)),rows-r=k->nRowsS n eps x g b nd rows r out=Yr (λr=>nLR n eps x r) (nEO n eps x g b nd) n rows r out:=by
 intro k;induction k with
 |zero=>
  intro r out hk;have c:¬r<rows:=by omega
  rw[nRowsS,Yr,if_neg c,if_neg c]
 |succ k ih=>
  intro r out hk;have c:r<rows:=by omega
  rw[nRowsS,Yr,if_pos c,if_pos c,nRowS_e _ _ _ _ _ _ _ _ _ _ (hf r)];exact ih (r+1) _ (by omega)
def Yw2(rows n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word)):(Ἂ Word):=
 let nd:=(I0 n).toInt64.toFloat
 let lg:=ZR (I0 n)
 let rm:=(((1:ἆ)<<<(52+lg))/I0 n).toInt64.toFloat*N5 ((971-lg)<<<52)
 let nf:=(((1:ἆ)<<<52)/I0 n).toInt64.toFloat
 let ep:=Dense.F.dw eps*nS201;let z0:=(eps.toUInt64>>>32).toInt64.toFloat;let L:=ZR (I0 (n+4))+1
 have hC:nCK n eps rm nf ep z0 L:=⟨rfl,rfl,rfl,congrArg (fun u:ἆ=>u.toInt64.toFloat) (kO0 eps),rfl⟩
 if h:(O0 x)=rows*n∧(O0 g)=n∧(O0 b)=n∧rows*n<=4294967295 then
  if 4<=n&&n<=2048&&kB eps != 0&&kB eps != 255 then
   if h1:rows=1 then
    nRow1 n eps x g b nd rm nf ep z0 L hC 0 (Ah (rows*n) 0) h.2.1.ge h.2.2.1.ge (by rw[h.1,h1];omega) (by simp only[O3];rw[h1];omega) (by have:=h.2.2.2;rw[h1]at this;omega)
   else
    let W:=nWG2 g b
    have hW:W=nWF2 (g,b):=nWG2_v g b
    nRows rows n eps x g b W.1.1 W.2 W.1.2 nd rm nf ep z0 L hC (by simp only[hW,nWF2,(nWF_sp g).1,h.2.1,le_refl]) (by simp only[hW,nWF2,(nWF_sp b).1,h.2.2.1,le_refl]) (by simp only[hW,nWF2,(nWF_sp g).2.2.1,h.2.1,le_refl])
     h.1.ge (Z4 h.2.2.2) rows 0 (Dense.F.zPq (n+2)) (Ah (rows*n) 0) (Dense.F.zPs (n+2)).1 (by simp only[O3];exact Nat.le_refl _) (Κ rows)
  else if x.all finiteWord then nRowsS n eps x g b nd rows 0 (Ah (rows*n) 0) else #[]
 else Ah (rows*n) 0
lemma nBu(a:ByteArray)(i:Ἄ)(h:(J0 i)<a.size):a.uget i h=a.data[(J0 i)]!:=by
 rcases a with⟨d⟩;simp only[ByteArray.uget,ByteArray.size]at h ⊢;rw[AB d _ h];rfl
lemma nFB_v(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(nd:Ἅ)(r:Ἀ)(dW:Ἃ)(ex:Option NR.fH)(hdW:dW=(gM n eps x r).2.2.2.2.2)(hx:ex=none∨ex=some (NR.eI n (gM n eps x r)))(i j:Ἄ):
 (nFB n eps x g b nd r dW ex i j).1=nLo n eps x g b nd (nLR n eps x r) i j∧(nFB n eps x g b nd r dW ex i j).2=NR.eI n (gM n eps x r):=by
 have E:NR.eI n (gM0 n x r dW)=NR.eI n (gM n eps x r):=by rw[hdW,gM0_e]
 rcases hx with h|h<;>subst h
 · exact⟨by rw[nFB,nFBz,nLo,nLR_th,E],E⟩
 · exact⟨by rw[nFB,nFBz,nLo,nLR_th],rfl⟩
lemma nUc(lo t:Ἄ):lo+t-lo=t:=by rw[USize.add_comm,USize.add_sub_cancel]
lemma nFE_eq(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(Xr gd bd:ἀ)(ge:ByteArray)(nd:Ἅ)(r:Ἀ)(lo nn:Ἄ)
 (hr:(J0 nn)<=(kS Xr))(hg:(J0 nn)<=(kS gd))(hd:(J0 nn)<=(kS bd))(he:(J0 nn)<=ge.size)
 (hb:(J0 lo)+(J0 nn)<2^(32:Ἀ))(hf:(nLR n eps x r).fast=true)
 (hK:∀t,t<(J0 nn)->¬((nLR n eps x r).Kp<nfe x[(J0 lo)+t]!+(nLR n eps x r).Ke))
 (hX:∀t,t<(J0 nn)->Xr.data[t]! =Dense.F.dw x[(J0 lo)+t]!)
 (hG:∀t,t<(J0 nn)->gd.data[t]! =Dense.F.dw g[t]!)
 (hB:∀t,t<(J0 nn)->bd.data[t]! =Dense.F.dw b[t]!)
 (hE:∀t,t<(J0 nn)->(ge.data[t]!).toUInt64=nfe g[t]!)(hq:r*n+n<2^(32:Ἀ))(hXf:∀j,j<r*n+n->kB x[j]!≠255)(heps:0<ZA eps):
 ∀k (ex:Option NR.fH)(t:Ἄ)(out:(Ἂ Word))(ho:(J0 lo)+(J0 nn)<=(O0 out)),(O0 out)<2^(32:Ἀ)->(J0 nn)-(J0 t)=k->(J0 t)<=(J0 nn)->(ex=none∨ex=some (NR.eI n (gM n eps x r)))->
 nFE n eps x g b Xr gd bd ge nd (nLR n eps x r).m (nLR n eps x r).ri (nLR n eps x r).Kp r (nLR n eps x r).dW ex lo t nn hr hg hd he hb out ho=Yq (nEO n eps x g b nd) (nLR n eps x r) lo (lo+t) (lo+nn) out:=by
 have hdW:=nLR_dW n eps x r hq hXf heps hf
 have hn:=nCa hb
 intro k;induction k with
 |zero=>
  intro ex t out ho hos hk ht hx
  rw[nFE.eq_def,dif_neg (nO1 hk),Yq,dif_neg (nOz hb hk ht)]
 |succ k ih=>
  intro ex t out ho hos hk ht hx
  obtain⟨h1,hk',ht'⟩:=nOs hk
  have h2:(J0 (lo+t))=(J0 lo)+(J0 t):=nCa (nOq hb h1)
  have h3:lo+t<out.usize:=by rw[ε,h2,Array.usize,Conv.aa _ hos];exact U1 (O1 h1 _) ho
  have e1:lo+t+1=lo+(t+1):=USize.add_assoc ..
  rw[nFE.eq_def,dif_pos (ε.mpr h1),Yq,dif_pos (by rw[ε,h2,hn];exact O1 h1 _),dif_pos h3,e1,nUc]
  simp only[Dense.F.kFU,nBu,hX _ h1,hG _ h1,hB _ h1,hE _ h1]
  have hV:nEO n eps x g b nd (nLR n eps x r) (lo+t) t=if nOK (nElB x[(J0 lo)+(J0 t)]! g[(J0 t)]! b[(J0 t)]! (nLR n eps x r).m (nLR n eps x r).ri) (nC (nfe g[(J0 t)]!+(nLR n eps x r).Kp) (((nElB x[(J0 lo)+(J0 t)]! g[(J0 t)]! b[(J0 t)]! (nLR n eps x r).m (nLR n eps x r).ri)>>>52)&&&2047)) then nW (nElB x[(J0 lo)+(J0 t)]! g[(J0 t)]! b[(J0 t)]! (nLR n eps x r).m (nLR n eps x r).ri) else nLo n eps x g b nd (nLR n eps x r) (lo+t) t:=by
   rw[nEO,if_pos hf];simp only[f6,h2,nElOK,if_neg (hK _ h1),nLoN]
  rw[hV];simp only[nElB]
  split
  · rename_i hc;simp only[hc,↓reduceIte];refine ih ex (t+1) _ _ ?_ hk' ht' hx;rw[BT];exact hos
  · rename_i hc;simp only[hc,υ,↓reduceIte];have q:=nFB_v n eps x g b nd r (nLR n eps x r).dW ex hdW hx (lo+t) t;rw[<-q.1];refine ih (some (nFB n eps x g b nd r (nLR n eps x r).dW ex (lo+t) t).2) (t+1) _ _ ?_ hk' ht' (BA (by rw[q.2]));rw[BT];exact hos
lemma nFE2_eq(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(nd:Ἅ)(r:Ἀ)(lo nn:Ἄ)
 (hg:(J0 nn)<=(O0 g))(hd:(J0 nn)<=(O0 b))(hx:(J0 lo)+(J0 nn)<=(O0 x))
 (hb:(J0 lo)+(J0 nn)<2^(32:Ἀ))(hf:(nLR n eps x r).fast=true)
 (hK:∀t,t<(J0 nn)->¬((nLR n eps x r).Kp<nfe x[(J0 lo)+t]!+(nLR n eps x r).Ke))
(hq:r*n+n<2^(32:Ἀ))(hXf:∀j,j<r*n+n->kB x[j]!≠255)(heps:0<ZA eps):
 ∀k (ex:Option NR.fH)(t:Ἄ)(out:(Ἂ Word))(ho:(J0 lo)+(J0 nn)<=(O0 out)),(O0 out)<2^(32:Ἀ)->(J0 nn)-(J0 t)=k->(J0 t)<=(J0 nn)->(ex=none∨ex=some (NR.eI n (gM n eps x r)))->
 nFE2 n eps x g b nd (nLR n eps x r).m (nLR n eps x r).ri (nLR n eps x r).Kp r (nLR n eps x r).dW ex lo t nn hg hd hx hb out ho=Yq (nEO n eps x g b nd) (nLR n eps x r) lo (lo+t) (lo+nn) out:=by
 have hdW:=nLR_dW n eps x r hq hXf heps hf
 have hn:=nCa hb
 intro k;induction k with
 |zero=>
  intro ex t out ho hos hk ht hx
  rw[nFE2.eq_def,dif_neg (nO1 hk),Yq,dif_neg (nOz hb hk ht)]
 |succ k ih=>
  intro ex t out ho hos hk ht hx
  obtain⟨h1,hk',ht'⟩:=nOs hk
  have h2:(J0 (lo+t))=(J0 lo)+(J0 t):=nCa (nOq hb h1)
  have h3:lo+t<out.usize:=by rw[ε,h2,Array.usize,Conv.aa _ hos];exact U1 (O1 h1 _) ho
  have e1:lo+t+1=lo+(t+1):=USize.add_assoc ..
  rw[nFE2.eq_def,dif_pos (ε.mpr h1),Yq,dif_pos (by rw[ε,h2,hn];exact O1 h1 _),dif_pos h3,e1,nUc]
  simp only[Dense.cC,h2]
  have hV:nEO n eps x g b nd (nLR n eps x r) (lo+t) t=if nOK (nElB x[(J0 lo)+(J0 t)]! g[(J0 t)]! b[(J0 t)]! (nLR n eps x r).m (nLR n eps x r).ri) (nC (nfe g[(J0 t)]!+(nLR n eps x r).Kp) (((nElB x[(J0 lo)+(J0 t)]! g[(J0 t)]! b[(J0 t)]! (nLR n eps x r).m (nLR n eps x r).ri)>>>52)&&&2047)) then nW (nElB x[(J0 lo)+(J0 t)]! g[(J0 t)]! b[(J0 t)]! (nLR n eps x r).m (nLR n eps x r).ri) else nLo n eps x g b nd (nLR n eps x r) (lo+t) t:=by
   rw[nEO,if_pos hf];simp only[f6,h2,nElOK,if_neg (hK _ h1),nLoN]
  rw[hV];simp only[nElB]
  split
  · rename_i hc;simp only[hc,↓reduceIte];refine ih ex (t+1) _ _ ?_ hk' ht' hx;rw[BT];exact hos
  · rename_i hc;simp only[hc,υ,↓reduceIte];have q:=nFB_v n eps x g b nd r (nLR n eps x r).dW ex hdW hx (lo+t) t;rw[<-q.1];refine ih (some (nFB n eps x g b nd r (nLR n eps x r).dW ex (lo+t) t).2) (t+1) _ _ ?_ hk' ht' (BA (by rw[q.2]));rw[BT];exact hos
lemma nDR_sp(x:(Ἂ Word))(i0 n:Ἄ)(hx:(J0 i0)+(J0 n)<=(O0 x))(hb:(J0 i0)+(J0 n)<2^(32:Ἀ)):∀k (t:Ἄ)(Xr:ἀ)(hr:(J0 n)+2<=(kS Xr))(s:Ἅ)(Dl:ἆ),(J0 n)-(J0 t)=k->(J0 t)<=(J0 n)->(∀j,j<(J0 t)->Xr.data[j]! =Dense.F.dw x[(J0 i0)+j]!)->
 (∀j,j<(J0 n)->(nDR x Xr i0 t n hx hb hr s Dl).a.data[j]! =Dense.F.dw x[(J0 i0)+j]!)∧(nDR x Xr i0 t n hx hb hr s Dl).a.data[(J0 n)]! =(nS1 x (i0+t) (i0+n) s Dl).1∧(nDR x Xr i0 t n hx hb hr s Dl).d=(nS1 x (i0+t) (i0+n) s Dl).2:=by
 have hn:=nCa hb
 intro k;induction k with
 |zero=>
  intro t Xr hr s Dl hk ht hX
  have q:(J0 t)=(J0 n):=C4 ht (K3.mp hk)
  rw[nDR.eq_def,dif_neg (nO1 hk),nS1,dif_neg (nOz hb hk ht)]
  refine⟨fun j hj=>?_,?_,rfl⟩
  · rw[Dense.F.fuS _ _ _ _ _ (U1 hj ((V1 _ 2).trans hr)),if_neg (L6 hj)];exact hX j (by rw[q];exact hj)
  · rw[Dense.F.fuS _ _ _ _ _ (U1 (K6 (I3 _)) hr),if_pos rfl]
 |succ k ih=>
  intro t Xr hr s Dl hk ht hX
  obtain⟨h1,hk',ht'⟩:=nOs hk
  have h2:(J0 (i0+t))=(J0 i0)+(J0 t):=nCa (nOq hb h1)
  rw[nDR.eq_def,dif_pos (ε.mpr h1),nS1,dif_pos (by rw[ε,h2,hn];exact O1 h1 _)]
  simp only[Dense.cC,f6,USize.add_assoc]
  refine ih _ _ _ _ _ hk' ht' (fun j hj=>?_)
  rw[Dense.ac (ε.mpr h1)] at hj
  rw[Dense.F.fuS _ _ _ _ _ (U1 hj ((l7 h1).trans ((V1 _ 2).trans hr)))];split
  · rename_i e;rw[<-e,h2]
  · exact hX j (N4 (J2 hj) (fun h=>by rename_i e;exact e h.symm))
lemma nVR_eq(x:(Ἂ Word))(Xr:ἀ)(m:Ἅ)(i0 n:Ἄ)(hr:(J0 n)<=(kS Xr))(hb:(J0 i0)+(J0 n)<2^(32:Ἀ))(hX:∀j,j<(J0 n)->Xr.data[j]! =Dense.F.dw x[(J0 i0)+j]!):
 ∀k (t:Ἄ)(v:Ἅ),(J0 n)-(J0 t)=k->(J0 t)<=(J0 n)->nVR Xr m t n hr v=nS2 x m (i0+t) (i0+n) v:=by
 have hn:=nCa hb
 intro k;induction k with
 |zero=>
  intro t v hk ht
  rw[nVR.eq_def,dif_neg (nO1 hk),nS2,dif_neg (nOz hb hk ht)]
 |succ k ih=>
  intro t v hk ht
  obtain⟨h1,hk',ht'⟩:=nOs hk
  have h2:(J0 (i0+t))=(J0 i0)+(J0 t):=nCa (nOq hb h1)
  rw[nVR.eq_def,dif_pos (ε.mpr h1),nS2,dif_pos (by rw[ε,h2,hn];exact O1 h1 _)]
  dsimp only;rw[Dense.F.kFU,hX _ h1,f6,h2,USize.add_assoc]
  exact ih _ _ hk' ht'
lemma nKpA(y d l q e:ℤ)(h1:y≤d)(h2:2≤l)(h3:l+d-30≤q):y+(1054-e)≤q+1082-e:=by omega
lemma nKpB(l d u:ℤ)(h1:l≤64)(h2:d≤254)(h3:u≤2047):max (l+d-30) (u-1073)≤2047:=max_le (by omega) (by omega)
lemma nKpC(m e:ℤ)(h:m≤2047)(h2:0≤e):m+1082-e≤3129:=by omega
lemma nKp_h(Y:ἇ)(D ed L Em:Ἀ)(hY:(mJ Y)<=D)(hD:D<=254)(he:ed<255)(hL:2<=L∧L<=64)(hEm:Em<=2047):
 ¬((max ((L:ℤ)+D-30) ((Em:ℤ)-1073)+1082-ed).toNat.toUInt64<Y.toUInt64+(1054-ed).toUInt64):=by
 have a:=le_max_left ((L:ℤ)+D-30) ((Em:ℤ)-1073);have c:=nKpB _ _ _ (U4.mpr hL.2) (U4.mpr hD) (U4.mpr hEm)
 generalize max ((L:ℤ)+D-30) ((Em:ℤ)-1073)=M at a c
 have e4:ed<=1054:=X4 (K5 he (by decide))
 have i1:(((mJ Y)+(1054-ed):Ἀ):ℤ)<=M+1082-ed:=by
  rw[W5,pO e4];exact nKpA _ _ _ _ _ (U4.mpr hY) (U4.mpr hL.1) a
 have k1:(mJ Y)+(1054-ed)<=(M+1082-ed).toNat:=(Int.toNat_natCast _).symm.le.trans (Int.toNat_le_toNat i1)
 have i2:M+1082-ed<=3129:=nKpC _ _ c (W6 ed)
 have k2:(M+1082-ed).toNat<2^(63:Ἀ):=C0 (Int.toNat_le_toNat i2) (by decide)
 have y:=Y.toNat_lt
 have k3:(mJ Y)+(1054-ed)<2^64:=C0 k1 (K5 k2 (by decide))
 rw[μ,η,ϩ,Dense.F.kOfS _ k2,Dense.F.kOfS _ (C0 (K4 1054 ed) (by decide)),γ k3]
 exact W3.2 k1
lemma nKp(n:Ἀ)(eps:Ἃ)(x:(Ἂ Word))(r:Ἀ)(hb:r*n+n<2^(32:Ἀ))(hX:∀j,j<r*n+n->kB x[j]!≠255)(hf:(nLR n eps x r).fast=true):
 ∀t,t<n->¬((nLR n eps x r).Kp<nfe x[r*n+t]!+(nLR n eps x r).Ke):=by
 intro t ht
 by_cases hc:(4<=n&&n<=2048&&kB eps != 0&&kB eps != 255)=true
 · rw[nLQ_e n eps x r hc] at hf ⊢;unfold nLQ at hf ⊢;dsimp only at hf ⊢
   simp only[Η,AJ] at hc hf
   have q1:(J0 (G0 (r*n)))=r*n:=Conv.aa _ (C0 (V1 _ _) hb)
   have q2:(J0 (G0 (r*n+n)))=r*n+n:=Conv.aa _ hb
   obtain⟨-,d2,d3⟩:=nS1_dl x (G0 (r*n+n)) (by rw[q2];exact hX) _ (G0 (r*n)) (I0 (0:Ἀ)).toInt64.toFloat 0 rfl (by rw[q1,q2];exact V1 _ _) (by decide)
   have d4:=d3 (r*n+t) (by rw[q1];exact V1 _ _) (by rw[q2];exact O1 ht _)
   have hed:=hf.1.2;rw[Ι]at hed
   have l1:2<=Aq (n+4):=(mu (Nat.succ_ne_zero _)).2 (K2 4 n)
   have l2:Aq (n+4)<13:=(m9 (Nat.succ_ne_zero _)).2 (C0 (L1 hc.1.1.2 4) (by decide:2048+4<2^13))
   have hL:2<=Aq (n+4)+1∧Aq (n+4)+1<=64:=⟨Nat.le_succ_of_le l1,l7 (U1 l2 (by decide))⟩
   exact nKp_h _ _ _ _ _ d4 d2 hed hL (by rw[Ε];exact ϝ.trans (by decide))
 · unfold nLR at hf;rw[if_neg hc]at hf;simp at hf
lemma nRow_eq(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(Xr gd bd:ἀ)(ge:ByteArray)(nd:Ἅ)(rm nf ep z0:Ἅ)(L:ἆ)(hC:nCK n eps rm nf ep z0 L)(r:Ἀ)(out:(Ἂ Word))
 (hr:n+2<=(kS Xr))(hg:n<=(kS gd))(hd:n<=(kS bd))(he:n<=ge.size)
 (hx:r*n+n<=(O0 x))(ho:r*n+n<=(O0 out))(hb:r*n+n<2^(32:Ἀ))(hos:(O0 out)<2^(32:Ἀ))
 (hX:∀j,j<r*n+n->kB x[j]!≠255)(heps:0<ZA eps)
 (hG:∀t,t<n->gd.data[t]! =Dense.F.dw g[t]!)(hB:∀t,t<n->bd.data[t]! =Dense.F.dw b[t]!)(hE:∀t,t<n->(ge.data[t]!).toUInt64=nfe g[t]!)(hN:(4<=n&&n<=2048&&kB eps != 0&&kB eps != 255)=ἃ):
 (nRow n eps x g b Xr gd bd ge nd rm nf ep z0 L hC r out hr hg hd he hx ho hb).2=Yq (nEO n eps x g b nd) (nLR n eps x r) (G0 (r*n)) (G0 (r*n)) (G0 (r*n+n)) out:=by
 have hz:=hC.2.2.2.1;subst hz
 have q1:(J0 (G0 (r*n)))=r*n:=Conv.aa _ (C0 (V1 _ _) hb)
 have q2:(J0 (G0 n))=n:=Conv.aa _ (C0 (K2 _ _) hb)
 have q3:(J0 (G0 (r*n+n)))=r*n+n:=Conv.aa _ hb
 have hb':(J0 (G0 (r*n)))+(J0 (G0 n))<2^(32:Ἀ):=by rw[q1,q2];exact hb
 have hx':(J0 (G0 (r*n)))+(J0 (G0 n))<=(O0 x):=by rw[q1,q2];exact hx
 have hr':(J0 (G0 n))+2<=(kS Xr):=by rw[q2];exact hr
 have hlo:G0 (r*n)+G0 n=G0 (r*n+n):=O5.1 (by rw[nCa hb',q1,q2,q3])
 have h0:(J0 (0:Ἄ))=0:=Ϧ
 obtain⟨A1,A2,A3⟩:=nDR_sp x (G0 (r*n)) (G0 n) hx' hb' _ 0 Xr hr' (I0 (0:Ἀ)).toInt64.toFloat 0 rfl (by rw[h0];exact ϟ _) (fun j hj=>by rw[h0]at hj;exact absurd hj (U5 _))
 have hs:=nDR_sz x (G0 (r*n)) (G0 n) hx' hb' _ 0 Xr hr' (I0 (0:Ἀ)).toInt64.toFloat 0 rfl
 have A4:=fun m (hh:(J0 (G0 n))<=(kS (nDR x Xr (G0 (r*n)) 0 (G0 n) hx' hb' hr' (I0 (0:Ἀ)).toInt64.toFloat 0).a))=>nVR_eq x _ m (G0 (r*n)) (G0 n) hh hb' A1 _ 0 (I0 (0:Ἀ)).toInt64.toFloat rfl (by rw[h0];exact ϟ _)
 rw[USize.add_zero,hlo]at A2 A3 A4
 have D:(nDR x Xr (G0 (r*n)) 0 (G0 n) hx' hb' hr' (I0 (0:Ἀ)).toInt64.toFloat 0).d<255:=by
  rw[A3,μ];exact C0 (nS1_dl x (G0 (r*n+n)) (fun j hj=>hX j (q3▸hj)) _ (G0 (r*n)) _ 0 rfl (by rw[q1,q3];omega) (by decide)).2.1 (by decide)
 unfold nRow;dsimp only;split
 · rename_i hc0
   rw[nLQk_e n eps _ _ _ rm nf ep _ L hC (C0 (K2 _ _) hb) (Thunk.mk (λ_=>NR.eI n (gM n eps x r)))]
   simp only[Dense.F.kFU,A2,A3,A4]
   rw[<-nLQ_e n eps x r hN];split
   · rename_i hf
     have K:=nFE_eq n eps x g b (nDR x Xr (G0 (r*n)) 0 (G0 n) hx' hb' hr' (I0 (0:Ἀ)).toInt64.toFloat 0).a gd bd ge nd r (G0 (r*n)) (G0 n) (by rw[hs];exact (V1 _ 2).trans hr') (by rw[q2];exact hg) (by rw[q2];exact hd) (by rw[q2];exact he) hb' hf
       (fun t ht=>by rw[q1];rw[q2]at ht;exact nKp n eps x r hb hX hf t ht) (fun t ht=>by rw[A1 t ht]) (fun t ht=>by rw[q2]at ht;exact hG t ht) (fun t ht=>by rw[q2]at ht;exact hB t ht) (fun t ht=>by rw[q2]at ht;exact hE t ht) hb hX heps
       _ none 0 out (by rw[q1,q2];exact ho) hos rfl (by rw[h0];exact ϟ _) (pZ rfl)
     rw[USize.add_zero,hlo]at K;exact K
   · rename_i hf;exact nRowS_e n eps x g b nd r _ _ out (F8.mpr hf)
 · rename_i hc;exact (hc D).elim
lemma nRow1_eq(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(nd:Ἅ)(rm nf ep z0:Ἅ)(L:ἆ)(hC:nCK n eps rm nf ep z0 L)(r:Ἀ)(out:(Ἂ Word))
 (hg:n<=(O0 g))(hd:n<=(O0 b))
 (hx:r*n+n<=(O0 x))(ho:r*n+n<=(O0 out))(hb:r*n+n<2^(32:Ἀ))(hos:(O0 out)<2^(32:Ἀ))
 (hX:∀j,j<r*n+n->kB x[j]!≠255)(heps:0<ZA eps)(hN:(4<=n&&n<=2048&&kB eps != 0&&kB eps != 255)=ἃ):
 nRow1 n eps x g b nd rm nf ep z0 L hC r out hg hd hx ho hb=Yq (nEO n eps x g b nd) (nLR n eps x r) (G0 (r*n)) (G0 (r*n)) (G0 (r*n+n)) out:=by
 have hz:=hC.2.2.2.1;subst hz
 have q1:(J0 (G0 (r*n)))=r*n:=Conv.aa _ (C0 (V1 _ _) hb)
 have q2:(J0 (G0 n))=n:=Conv.aa _ (C0 (K2 _ _) hb)
 have q3:(J0 (G0 (r*n+n)))=r*n+n:=Conv.aa _ hb
 have hb':(J0 (G0 (r*n)))+(J0 (G0 n))<2^(32:Ἀ):=by rw[q1,q2];exact hb
 have hx':(J0 (G0 (r*n)))+(J0 (G0 n))<=(O0 x):=by rw[q1,q2];exact hx
 have hlo:G0 (r*n)+G0 n=G0 (r*n+n):=O5.1 (by rw[nCa hb',q1,q2,q3])
 have h0:(J0 (0:Ἄ))=0:=Ϧ
 have D:(nS1 x (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat 0).2<255:=by
  rw[μ];exact C0 (nS1_dl x (G0 (r*n+n)) (fun j hj=>hX j (q3▸hj)) _ (G0 (r*n)) _ 0 rfl (by rw[q1,q3];omega) (by decide)).2.1 (by decide)
 unfold nRow1;dsimp only;split
 · rename_i hc0
   rw[nLQk_e n eps _ _ _ rm nf ep _ L hC (C0 (K2 _ _) hb) (Thunk.mk (λ_=>NR.eI n (gM n eps x r)))]
   rw[<-nLQ_e n eps x r hN];split
   · rename_i hf
     have K:=nFE2_eq n eps x g b nd r (G0 (r*n)) (G0 n) (by rw[q2];exact hg) (by rw[q2];exact hd) hx' hb' hf
       (fun t ht=>by rw[q1];rw[q2]at ht;exact nKp n eps x r hb hX hf t ht) hb hX heps
       _ none 0 out (by rw[q1,q2];exact ho) hos rfl (by rw[h0];exact ϟ _) (pZ rfl)
     rw[USize.add_zero,hlo]at K;exact K
   · rename_i hf;exact nRowS_e n eps x g b nd r _ _ out (F8.mpr hf)
 · rename_i hc;exact (hc D).elim
lemma nRows_eq(rows n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(gd bd:ἀ)(ge:ByteArray)(nd:Ἅ)(rm nf ep z0:Ἅ)(L:ἆ)(hC:nCK n eps rm nf ep z0 L)
 (hg:n<=(kS gd))(hd:n<=(kS bd))(he:n<=ge.size)
 (hx:rows*n<=(O0 x))(hb:rows*n<2^(32:Ἀ))(hX:∀j,j<rows*n->kB x[j]!≠255)(heps:0<ZA eps)
 (hG:∀t,t<n->gd.data[t]! =Dense.F.dw g[t]!)(hB:∀t,t<n->bd.data[t]! =Dense.F.dw b[t]!)(hE:∀t,t<n->(ge.data[t]!).toUInt64=nfe g[t]!)(hN:(4<=n&&n<=2048&&kB eps != 0&&kB eps != 255)=ἃ):
 ∀k (r:Ἀ)(Xr:ἀ)(out:(Ἂ Word))(hr:n+2<=(kS Xr))(ho:rows*n<=(O0 out))(hk:r+k=rows),(O0 out)=rows*n->
 nRows rows n eps x g b gd bd ge nd rm nf ep z0 L hC hg hd he hx hb k r Xr out hr ho hk=Yr (λr=>nLR n eps x r) (nEO n eps x g b nd) n rows r out:=by
 intro k;induction k with
 |zero=>
  intro r Xr out hr ho hk hos;rw[Yr,if_neg (by rw[<-hk];exact B4 r)];rfl
 |succ k ih=>
  intro r Xr out hr ho hk hos
  have kk:=nHk2 (n:=n) hk
  rw[Yr,if_pos (hk▸P5 (Nat.succ_pos k))]
  have E:=nRow_eq n eps x g b Xr gd bd ge nd rm nf ep z0 L hC r out hr hg hd he (kk.trans hx) (kk.trans ho) (C0 kk hb) (by rw[hos];exact hb) (fun j hj=>hX j (U1 hj kk)) heps hG hB hE hN
  have S2:(O0 (nRow n eps x g b Xr gd bd ge nd rm nf ep z0 L hC r out hr hg hd he (kk.trans hx) (kk.trans ho) (C0 kk hb)).2)=rows*n:=by rw[E,nYq_sz _ _ _ _ _ _ _ rfl,hos]
  rw[nRows,dif_pos S2.ge]
  refine (ih (r+1) _ _ _ _ (nHk3 hk) S2).trans ?_
  rw[E]
lemma nRow_bad(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(Xr gd bd:ἀ)(ge:ByteArray)(nd:Ἅ)(rm nf ep z0:Ἅ)(L:ἆ)(hC:nCK n eps rm nf ep z0 L)(r:Ἀ)(out:(Ἂ Word))
 (hr:n+2<=(kS Xr))(hg:n<=(kS gd))(hd:n<=(kS bd))(he:n<=ge.size)
 (hx:r*n+n<=(O0 x))(ho:r*n+n<=(O0 out))(hb:r*n+n<2^(32:Ἀ))(j:Ἀ)(j1:r*n<=j)(j2:j<r*n+n)(hj:kB x[j]! =255):
 (nRow n eps x g b Xr gd bd ge nd rm nf ep z0 L hC r out hr hg hd he hx ho hb).2=#[]:=by
 have q1:(J0 (G0 (r*n)))=r*n:=Conv.aa _ (C0 (V1 _ _) hb)
 have q2:(J0 (G0 n))=n:=Conv.aa _ (C0 (K2 _ _) hb)
 have q3:(J0 (G0 (r*n+n)))=r*n+n:=Conv.aa _ hb
 have hb':(J0 (G0 (r*n)))+(J0 (G0 n))<2^(32:Ἀ):=by rw[q1,q2];exact hb
 have hx':(J0 (G0 (r*n)))+(J0 (G0 n))<=(O0 x):=by rw[q1,q2];exact hx
 have hr':(J0 (G0 n))+2<=(kS Xr):=by rw[q2];exact hr
 have hlo:G0 (r*n)+G0 n=G0 (r*n+n):=O5.1 (by rw[nCa hb',q1,q2,q3])
 have h0:(J0 (0:Ἄ))=0:=Ϧ
 obtain⟨-,-,A3⟩:=nDR_sp x (G0 (r*n)) (G0 n) hx' hb' _ 0 Xr hr' z0 0 rfl (by rw[h0];exact ϟ _) (fun j hj=>by rw[h0]at hj;exact absurd hj (U5 _))
 rw[USize.add_zero,hlo]at A3
 have G:=(nS1_ge x (G0 (r*n+n)) _ (G0 (r*n)) z0 0 rfl (by rw[q1,q3];omega)).2 j (by rw[q1];exact j1) (by rw[q3];exact j2)
 rw[hj]at G
 have D:¬(nDR x Xr (G0 (r*n)) 0 (G0 n) hx' hb' hr' z0 0).d<255:=by rw[A3,μ];exact W3.2 (AZ (by decide) G)
 unfold nRow;dsimp only;rw[if_neg D]
lemma nRow1_bad(n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(nd:Ἅ)(rm nf ep z0:Ἅ)(L:ἆ)(hC:nCK n eps rm nf ep z0 L)(r:Ἀ)(out:(Ἂ Word))
 (hg:n<=(O0 g))(hd:n<=(O0 b))
 (hx:r*n+n<=(O0 x))(ho:r*n+n<=(O0 out))(hb:r*n+n<2^(32:Ἀ))(j:Ἀ)(j1:r*n<=j)(j2:j<r*n+n)(hj:kB x[j]! =255):
 nRow1 n eps x g b nd rm nf ep z0 L hC r out hg hd hx ho hb=#[]:=by
 have q1:(J0 (G0 (r*n)))=r*n:=Conv.aa _ (C0 (V1 _ _) hb)
 have q3:(J0 (G0 (r*n+n)))=r*n+n:=Conv.aa _ hb
 have G:=(nS1_ge x (G0 (r*n+n)) _ (G0 (r*n)) z0 0 rfl (by rw[q1,q3];omega)).2 j (by rw[q1];exact j1) (by rw[q3];exact j2)
 rw[hj]at G
 have D:¬(nS1 x (G0 (r*n)) (G0 (r*n+n)) z0 0).2<255:=by rw[μ];exact W3.2 (AZ (by decide) G)
 unfold nRow1;dsimp only;rw[if_neg D]
lemma nRows_bad(rows n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word))(gd bd:ἀ)(ge:ByteArray)(nd:Ἅ)(rm nf ep z0:Ἅ)(L:ἆ)(hC:nCK n eps rm nf ep z0 L)
 (hg:n<=(kS gd))(hd:n<=(kS bd))(he:n<=ge.size)
 (hx:rows*n<=(O0 x))(hb:rows*n<2^(32:Ἀ))(hn:0<n):
 ∀k (r:Ἀ)(Xr:ἀ)(out:(Ἂ Word))(hr:n+2<=(kS Xr))(ho:rows*n<=(O0 out))(hk:r+k=rows)(j:Ἀ),r*n<=j->j<rows*n->kB x[j]! =255->
 nRows rows n eps x g b gd bd ge nd rm nf ep z0 L hC hg hd he hx hb k r Xr out hr ho hk=#[]:=by
 intro k;induction k with
 |zero=>intro r Xr out hr ho hk j j1 j2 hj;rw[ϑ]at hk;subst hk;omega
 |succ k ih=>
  intro r Xr out hr ho hk j j1 j2 hj
  have kk:=nHk2 (n:=n) hk
  rw[nRows];split
  · rename_i S
    by_cases c:j<r*n+n
    · rw[nRow_bad n eps x g b Xr gd bd ge nd rm nf ep z0 L hC r out hr hg hd he (kk.trans hx) (kk.trans ho) (C0 kk hb) j j1 c hj]at S
      simp only[Array.size_empty]at S;omega
    · exact ih (r+1) _ _ _ _ (nHk3 hk) j (by rw[Ϡ,n4];omega) j2 hj
  · rfl
theorem V3L_h1(s:Ἆ)(N u n:ℕ)(l:ℚ)(v:l=sg s (N:ℚ)*2^((u:Ἁ)-149)/n):(l:ℝ)=(if s then-(N:ℝ) else (N:ℝ))*2^u/n/2^(149:Ἀ):=by
 rw[v,Ac AL,AA];unfold sg;split<;>simp only[Rat.cast_div,J7,Rat.cast_neg,Rat.cast_pow,T7,G7]<;>ring
lemma nV2{c:Prop}[Decidable c]{α:Type}(p q r:α)(hp:c→p=r)(hq:¬c→q=r):(if c then p else q)=r:=if h:c then (if_pos h).trans (hp h) else (if_neg h).trans (hq h)
lemma nV3{c0 c1:Prop}[Decidable c0][Decidable c1]{α:Type}(s a r:α)(hs:s=r)(ha:c0→c1→a=r):(if c0 then (if c1 then a else s) else s)=r:=
 if h0:c0 then (if_pos h0).trans (if h1:c1 then (if_pos h1).trans (ha h0 h1) else (if_neg h1).trans hs) else (if_neg h0).trans hs
lemma V3Lo(e n:Ἀ)(h:kF)(x g b:(Ἂ Word))(m:0<n)(B:e*n<2^(32:Ἀ))
  (q:0<ZA h)(H:∀i,i<e*n->kB x[i]!≠255)(F:∀j,j<n->kB g[j]!≠255)
  (D:∀j,j<n->kB b[j]!≠255):
  Yr (λr=>nLR n h x r) (nEO n h x g b (I0 n).toInt64.toFloat) n e 0 (Ah (e*n) 0)=layernormStable e n h x g b:=by
 apply Ys _ _ _ _ B;intro i G;have w:=Ϟ (by rwa[Ϗ]);have r:=ϊ i m;have z:=ι n (l7 w);rw[Ο]at z;dsimp only[nEO,nLoN]
 have hs:=nSv h q
 have hb:i/n*n+n<2^(32:Ἀ):=AO z B
 have hX:∀j,j< i/n*n+n->kB x[j]!≠255:=fun j hj=>H j (U1 hj z)
 have hj1:i/n*n<=i:=pr i n
 have hj2:i< i/n*n+n:=Nat.lt_div_mul_add m
 simp only[nLo,nLR_th,f6,Conv.aa _ (G.trans B:i<2^(32:Ἀ)),Conv.aa _ ((r.trans_le ((K2 n _).trans z)).trans B:i%n<2^(32:Ἀ)),gP,Qg _ w,Fin.is_lt,Qg,gN _ _ _ _ m q hb]
 obtain⟨v,A,-⟩:=Vn n h x (i/n) m q
 refine nV3 _ _ _ (nV2 _ _ _ (fun E3=>?_) (fun E3=>?_)) (fun E1 E2=>?_)
 · rw[<-v,<-A];generalize YU n h x (i/n)=t at*;obtain⟨u,k,y,o,p,d⟩:=t;dsimp only at*
   exact NE.jn _ _ _ d _ n u k o p y _ _ rfl (H i G) (F _ r) (D _ r) m ‹_› (V3L_h1 _ _ _ _ _ v) rfl
 · rw[<-v,<-A]
 · rw[nLR_el n h x (i/n) hb hX hs i hj1 hj2 _ _ (F _ r) (D _ r) E1 E2]
lemma V3L(e n:Ἀ)(h:kF)(x g b:(Ἂ Word))(m:0<n)(B:e*n<2^(32:Ἀ))
  (q:0<ZA h)(H:∀i,i<e*n->kB x[i]!≠255)(F:∀j,j<n->kB g[j]!≠255)
  (D:∀j,j<n->kB b[j]!≠255)(sx:(O0 x)=e*n)(sg:(O0 g)=n)(sb:(O0 b)=n):
  Yw2 e n h x g b=layernormStable e n h x g b:=by
 rw[Yw2];dsimp only;split
 · rename_i hs;split
   · rename_i hN;split
     · rename_i h1;subst h1
       have Y:=V3Lo 1 n h x g b m B q H F D
       rw[Yr,if_pos Nat.zero_lt_one,Κ,Yr,if_neg (B4 1)]at Y
       refine Eq.trans (nRow1_eq _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (by simp only[O3];exact B) (by rw[ϳ,Κ];intro j hj;exact H j (by rw[n4];exact hj)) q hN) ?_
       simp only[ϳ,Κ] at Y ⊢;exact Y
     have G:=nWF_sp g;have Bq:=nWF_sp b;have e1:(nWG2 g b).1=nWF g:=by rw[nWG2_v];rfl
     have e2:(nWG2 g b).2=(nWF b).1:=by rw[nWG2_v];rfl
     rw[<-e1] at G;rw[<-e2] at Bq
     exact (nRows_eq _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ H q (fun t ht=>G.2.1 t (by rw[hs.2.1];exact ht)) (fun t ht=>Bq.2.1 t (by rw[hs.2.2.1];exact ht)) (fun t ht=>G.2.2.2 t (by rw[hs.2.1];exact ht)) hN _ 0 _ _ _ _ _ (by simp only[O3])).trans (V3Lo e n h x g b m B q H F D)
   · rename_i hN
     have X:x.all finiteWord=ἃ:=np.mpr (fun i hi=>by have t:=H i (sx▸hi);rw[AB x i hi]at t;simp only[finiteWord,AM,Ak];exact t)
     rw[if_pos X]
     exact (nRowsS_e n h x g b _ e (fun r=>nLR_f n h x r hN) _ 0 _ rfl).trans (V3Lo e n h x g b m B q H F D)
 · rename_i hn;exact absurd ⟨sx,sg,sb,J2 B⟩ hn
lemma V3L_bad(e n:Ἀ)(h:kF)(x g b:(Ἂ Word))(m:0<n)(B:e*n<2^(32:Ἀ))(sx:(O0 x)=e*n)(sg:(O0 g)=n)(sb:(O0 b)=n)(hX:x.all finiteWord=ἄ):Yw2 e n h x g b=#[]:=by
 obtain⟨j,j2,hj⟩:∃j,j<e*n∧kB x[j]! =255:=by
  by_contra c;simp only[not_exists,not_and]at c
  have:x.all finiteWord=ἃ:=np.mpr (fun i hi=>by have t:=c i (sx▸hi);rw[AB x i hi]at t;simp only[finiteWord,AM,Ak];exact t)
  rw[this]at hX;exact absurd hX (by decide)
 rw[Yw2];dsimp only;rw[dif_pos ⟨sx,sg,sb,J2 B⟩];split
 · split
   · rename_i h1;subst h1;exact nRow1_bad _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ j (by omega) (by omega) hj
   · exact nRows_bad _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ m _ 0 _ _ _ _ _ j (by omega) j2 hj
 · simp only[hX,υ,↓reduceIte]
@[noinline]def nOEr(x g:(Ἂ Word))(nd:Ἅ)(e:NR.fH)(i j:Ἄ):Ἃ:=
 let xw:=ck x i;let gw:=ck g j
 if ZC xw==0 then ZT 0 else if e.ok then NE.el xw gw 0 nd e.nS e.R e.pz e.A1 e.A2 e.C1 (λ_=>ZT (ZA xw*ZA gw/ZA e.td))
 else ZT (ZA xw*ZA gw/ZA e.td)
@[noinline]def nFBr(x g:(Ἂ Word))(nd:Ἅ)(td:Ἃ)(i j:Ἄ):Ἃ:=if ZC (ck x i)==0 then ZT 0 else nOEr x g nd (NR.eI 1 (0,0,ἄ,0,0,td)) i j
lemma nFBr_e(x g:(Ἂ Word))(nd:Ἅ)(td:Ἃ)(i j:Ἄ):nFBr x g nd td i j=nOEr x g nd (NR.eI 1 (0,0,ἄ,0,0,td)) i j:=by
 unfold nFBr nOEr;dsimp only;split<;>simp only[*,↓reduceIte]
def bn(rows n:Ἀ)(eps:Ἃ)(x g b:(Ἂ Word)):(Ἂ Word):=
 if rows>0∧n>0∧(O0 x)=rows*n∧(O0 g)=n∧(O0 b)=n∧rows*n<=4294967295 then
  if g.all finiteWord&&b.all finiteWord&&finiteWord eps&&(!ZX eps&&ZC eps !=0) then
   let r:=Yw2 rows n eps x g b
   if (O0 r)==rows*n then r else layernormSpec rows n eps x g b
  else layernormSpec rows n eps x g b
 else layernormSpec rows n eps x g b
lemma cN:bn=layernormSpec:=by
 funext l n m x g b;rw[bn];split
 · rename_i h;obtain⟨l0,n0,sx,sg,sb,B⟩:=h;split
   · rename_i h2
     by_cases hX:x.all finiteWord=ἃ
     · have E:Yw2 l n m x g b=layernormSpec l n m x g b:=by
        have h:(SN l n m x g&&(O0 b)==n&&b.all finiteWord)=ἃ:=by
         simp only[Η]at h2
         rw[SN];simp only[Η,AQ,AJ]
         exact⟨⟨⟨⟨⟨⟨⟨⟨⟨l0,n0⟩,sx⟩,sg⟩,hX⟩,h2.1.1.1⟩,h2.1.2⟩,h2.2⟩,sb⟩,h2.1.1.2⟩
        have sz:=h;simp only[SN,Η,AQ]at sz;rw[fG]at h;rw[layernormSpec,layernormWith,AD h];simp only[Η]at h;obtain⟨a,e,c,d⟩:=jm h.1.1;have o:(O0 b)=n:=by simpa using h.1.2
        exact V3L l n m x g b a (by omega) e c d (λj p=>NR.ic b (by simpa[np]using h.2) j (by exact V6 p (id (Eq.symm o)))) sz.1.1.1.1.1.1.1.2 sz.1.1.1.1.1.1.2 o
       dsimp only;rw[E];simp only[ite_self]
     · have Z:=V3L_bad l n m x g b n0 (by omega) sx sg sb (F8.mpr hX)
       have N:¬((O0 (#[]:(Ἂ Word)))==l*n)=ἃ:=by simp only[Array.size_empty,AQ];exact R8 (Ϯ l0 n0)
       dsimp only;rw[Z,if_neg N]
   · rfl
 · rfl
end Norm
end
def gC(batch m k n:Nat)(A B:Ἂ Word):Ἂ Word:=
 if batch==1&&A.size==m*k&&B.size==k*n&&decide (0 < n)&&decide (0 < k)&&decide (0 < m)&&A.all (fun w=>(w>>>23)&&&255!=255)&&B.all (fun w=>(w>>>23)&&&255!=255) then
  Conv.er ⟨k,1,n,m,1,1,1,0⟩ false B A (mB m (0x80000000:Word))
 else gemmSpec batch m k n A B
theorem gCv:gC=gemmSpec:=by
 funext batch m k n A B
 unfold gC
 split
 · rename_i hc
   simp only [Η,AQ,AJ] at hc
   obtain ⟨⟨⟨⟨⟨⟨⟨h1,h2⟩,h3⟩,h4⟩,h5⟩,h6⟩,hA⟩,hB⟩:=hc
   subst h1
   have fA:=np.1 hA
   have fB:=np.1 hB
   rw [Conv.eu,Conv.s5c1 _ _ _ _ _ rfl rfl rfl rfl Nat.one_pos h4]
   unfold gemmSpec gemmWith
   apply mi
   · simp only [Ϝ,n4]
   intro q q1 q2
   simp only [Ϝ,n4] at q1
   have hi:q/n < m:=Ϟ (by rw [Ϗ]; exact q1)
   have hj:q%n < n:=ϊ _ h4
   have e1:q/(m*n)=0:=ώ q1
   have e2:q/n%m=q/n:=γ hi
   have gz:(mB m (0x80000000:Word))[q/n]! = 0x80000000:=by
     rw [AB _ _ (by simpa using hi)]; simp
   simp only [Λ,n4,act,υ, ↓reduceIte,gz,e1,e2,ϳ,Κ]
   refine Dense.F.dz k (fun t=>A[q/n*k+t]!) (fun t=>B[t*n+q%n]!) h5 (fun t ht=>⟨Norm.NR.ic A fA _ ?_,Norm.NR.ic B fB _ ?_⟩)
   · rw [h2]; calc q/n*k+t < q/n*k+k:=O1 ht _
       _=(q/n+1)*k:=by ring
       _ ≤ m*k:=ι k hi
   · rw [h3]; calc t*n+q%n < t*n+n:=O1 hj _
       _=(t+1)*n:=by ring
       _ ≤ k*n:=ι n ht
 · rfl
@[inline]def cvOk(s:L4):Ἆ:=
 s.strideH==s.strideW&&s.padT==s.padL&&s.padT==s.padB&&s.padT==s.padR&&s.dilH==1&&s.dilW==1&&s.groups==1&&0<s.kh&&0<s.kw
@[inline]def cvO(s:L4):ConvShape:=⟨s.cin,s.h,s.w,s.cout,s.kh,s.kw,s.strideH,s.padT⟩
theorem cvB(s:L4)(h:cvOk s=true)(relu:Ἆ)(inp wts b:Ἂ Word):
  ExactRuntime.conv2dSpec s relu inp wts b=conv2dSpec (cvO s) relu inp wts b:=by
 simp only[cvOk,Η,AQ,AJ]at h
 obtain⟨⟨⟨⟨⟨⟨⟨⟨hst,hp1⟩,hp2⟩,hp3⟩,hd1⟩,hd2⟩,hg⟩,hkh⟩,hkw⟩:=h
 have e1:s.ho=(cvO s).ho:=by
  simp only[ExactRuntime.ConvShape.ho,ExactRuntime.ConvShape.ekh,ConvShape.ho,cvO,hd1,n4]
  rw[show s.h+s.padT+s.padB-(s.kh-1+1)=s.h+2*s.padT-s.kh by omega]
 have e2:s.wo=(cvO s).wo:=by
  simp only[ExactRuntime.ConvShape.wo,ExactRuntime.ConvShape.ekw,ConvShape.wo,cvO,hd2,n4,hst]
  rw[show s.w+s.padL+s.padR-(s.kw-1+1)=s.w+2*s.padT-s.kw by omega]
 have e3:s.taps=s.cin*s.kh*s.kw:=by simp only[ExactRuntime.ConvShape.taps,T6,hg,F5]
 have e4:s.opg=s.cout:=by simp only[ExactRuntime.ConvShape.opg,hg,F5]
 unfold ExactRuntime.conv2dSpec ExactRuntime.conv2dWith conv2dSpec conv2dWith
 apply mi
 · simp only[Ϝ,e1,e2,cvO]
 intro q h1 h2
 simp only[Ϝ]at h1
 simp only[Λ,e3,e4]
 have hq:q/(s.ho*s.wo)<s.cout:=Ϟ (by rw[Ϗ];exact h1)
 have g0:q/(s.ho*s.wo)/s.cout=0:=ώ hq
 rw[g0,e1,e2]
 congr 2
 funext t
 simp only[cvO]
 by_cases ht:t<s.cin*s.kh*s.kw
 · simp only[ht,ἁ,U6,windowAt,ϳ,Κ,hd1,hd2,ϻ,hst,hp1]
 · simp only[ht,ἂ]
@[inline]def dwOk(s:L4):Ἆ:=
 s.strideH==s.strideW&&s.padT==s.padL&&s.padT==s.padB&&s.padT==s.padR&&s.dilH==1&&s.dilW==1&&0<s.kh&&0<s.kw
theorem dwB(s:L4)(h:dwOk s=true)(relu:Ἆ)(inp wts b:Ἂ Word):
  J9 s relu inp wts b=depthwiseSpec (cvO s) relu inp wts b:=by
 simp only[dwOk,Η,AQ,AJ]at h
 obtain⟨⟨⟨⟨⟨⟨⟨hst,hp1⟩,hp2⟩,hp3⟩,hd1⟩,hd2⟩,hkh⟩,hkw⟩:=h
 have e1:s.depthwise.ho=(cvO s).ho:=by
  simp only[ExactRuntime.ConvShape.ho,ExactRuntime.ConvShape.ekh,J5,ConvShape.ho,cvO,hd1,n4]
  rw[show s.h+s.padT+s.padB-(s.kh-1+1)=s.h+2*s.padT-s.kh by omega]
 have e2:s.depthwise.wo=(cvO s).wo:=by
  simp only[ExactRuntime.ConvShape.wo,ExactRuntime.ConvShape.ekw,J5,ConvShape.wo,cvO,hd2,n4,hst]
  rw[show s.w+s.padL+s.padR-(s.kw-1+1)=s.w+2*s.padT-s.kw by omega]
 unfold J9 ExactRuntime.depthwiseWith ExactRuntime.conv2dWith depthwiseSpec depthwiseWith
 apply mi
 · simp only[Ϝ];rw[e1,e2];rfl
 intro q h1 h2
 simp only[Ϝ]at h1
 change q<s.cin*(s.depthwise.ho*s.depthwise.wo) at h1
 have hc:0<s.cin:=ϵ (fun z=>by simp[z] at h1)
 have e3:s.depthwise.taps=s.kh*s.kw:=by simp only[ExactRuntime.ConvShape.taps,T6,J5,Nat.div_self hc,n4]
 have e4:s.depthwise.opg=1:=by simp only[ExactRuntime.ConvShape.opg,J5,Nat.div_self hc]
 have e5:s.depthwise.cpg=1:=by simp only[T6,J5,Nat.div_self hc]
 simp only[Λ,e3,e4,F5]
 rw[e1,e2]
 generalize q/((cvO s).ho*(cvO s).wo)=o
 congr 2
 funext t
 simp only[cvO]
 by_cases ht:t<s.kh*s.kw
 · have K:0<s.kh*s.kw:=Ϯ hkh hkw
   have a1:(o*(s.kh*s.kw)+t)/(s.kh*s.kw)=o:=by rw[Ϗ,W8 K,ώ ht,ϑ]
   have a2:(o*(s.kh*s.kw)+t)/s.kw%s.kh=t/s.kw%s.kh:=by
    rw[show o*(s.kh*s.kw)+t=s.kw*(o*s.kh)+t by ring,W8 hkw,Ϗ o,O7]
   have a3:(o*(s.kh*s.kw)+t)%s.kw=t%s.kw:=by
    rw[show o*(s.kh*s.kw)+t=s.kw*(o*s.kh)+t by ring,O7]
   have a4:t/(s.kh*s.kw)=0:=ώ ht
   simp only[ht,ἁ,U6,windowAt,e5,a1,a2,a3,a4,ϑ,hd1,hd2,hst,hp1,J5,T6,Nat.div_self hc,ϻ]
 · simp only[ht,ἂ]
@[inline]def apOk(s:Q9):Ἆ:=
 s.padT==0&&s.padL==0&&s.padB==0&&s.padR==0&&!s.ceilMode&&!s.countIncludePad&&s.kh==s.kw&&s.strideH==s.kh&&s.strideW==s.kw&&0<s.kh&&s.kh<=s.h&&s.kw<=s.w
theorem apF(s:Q9)(h:apOk s=true):s.padT=0∧s.padL=0∧s.padB=0∧s.padR=0∧s.ceilMode=false∧s.countIncludePad=false∧s.kw=s.kh∧s.strideH=s.kh∧s.strideW=s.kh∧0<s.kh∧s.kh<=s.h∧s.kh<=s.w:=by
 simp only[apOk,Η,AQ,AJ,BM]at h
 obtain⟨⟨⟨⟨⟨⟨⟨⟨⟨⟨⟨hT,hL⟩,hB⟩,hR⟩,hc⟩,hi⟩,hk⟩,hsh⟩,hsw⟩,hk0⟩,hkh⟩,hkw⟩:=h
 exact⟨hT,hL,hB,hR,hc,hi,hk.symm,hsh,hsw.trans hk.symm,hk0,hkh,by rw[hk];exact hkw⟩
theorem apHW(s:Q9)(h:apOk s=true):s.ho=s.h/s.kh∧s.wo=s.w/s.kh:=by
 obtain⟨hT,hL,hB,hR,hc,hi,hk,hsh,hsw,hk0,hkh,hkw⟩:=apF s h
 constructor
 ·simp only[ExactRuntime.PoolShape.ho,ExactRuntime.poolOut,hT,hB,hc,hsh,υ,ἂ,ϑ]
  rw[←Nat.add_div_right _ hk0,L3 hkh]
 ·simp only[ExactRuntime.PoolShape.wo,ExactRuntime.poolOut,hL,hR,hc,hsw,hk,υ,ἂ,ϑ]
  rw[←Nat.add_div_right _ hk0,L3 hkw]
theorem apW(s:Q9)(h:apOk s=true)(y x:Nat)(hy:y<s.h/s.kh)(hx:x<s.w/s.kh):
  s.window y x=⟨y*s.kh,x*s.kh,s.kh,s.kh⟩∧s.divisor y x=s.kh*s.kh:=by
 obtain⟨hT,hL,hB,hR,hc,hi,hk,hsh,hsw,hk0,hkh,hkw⟩:=apF s h
 have a:y*s.kh+s.kh<=s.h:=by have:=(mc hk0).1 hy;rw[Ο]at this;exact this
 have b:x*s.kh+s.kh<=s.w:=by have:=(mc hk0).1 hx;rw[Ο]at this;exact this
 have W:s.window y x=⟨y*s.kh,x*s.kh,s.kh,s.kh⟩:=by
  simp only[ExactRuntime.PoolShape.window,ExactRuntime.poolSpan,hT,hL,hsh,hsw,hk,pQ,Κ,Nat.min_eq_left a,Nat.min_eq_left b,ϱ]
 refine⟨W,?_⟩
 simp only[ExactRuntime.PoolShape.divisor,hi,υ,ἂ,W]
theorem apB(s:Q9)(h:apOk s=true)(inp:Ἂ Word):ExactRuntime.avgpoolSpec s inp=avgpoolSpec s.c s.h s.w s.kh inp:=by
 obtain⟨eh,ew⟩:=apHW s h
 unfold ExactRuntime.avgpoolSpec ExactRuntime.avgpoolWith avgpoolSpec avgpoolWith
 apply mi
 · simp only[Ϝ,eh,ew]
 intro q h1 h2
 simp only[Ϝ]at h1
 simp only[Λ]
 rw[eh,ew]at h1⊢
 have p1:0<s.h/s.kh*(s.w/s.kh):=ϵ (fun z=>by simp[z]at h1)
 have hy:q/(s.w/s.kh)%(s.h/s.kh)<s.h/s.kh:=ϊ _ (Nat.pos_of_mul_pos_right p1)
 have hx:q%(s.w/s.kh)<s.w/s.kh:=ϊ _ (Nat.pos_of_mul_pos_left p1)
 obtain⟨W,D⟩:=apW s h _ _ hy hx
 simp only[W,D,ExactRuntime.poolCell,poolAt]
 congr 3
theorem mpB(s:Q9)(h:apOk s=true)(inp:Ἂ Word):ExactRuntime.maxpoolSpec s inp=maxpoolSpec s.c s.h s.w s.kh inp:=by
 obtain⟨eh,ew⟩:=apHW s h
 unfold ExactRuntime.maxpoolSpec maxpoolSpec
 apply mi
 · simp only[Ϝ,eh,ew]
 intro q h1 h2
 simp only[Ϝ]at h1
 simp only[Λ]
 rw[eh,ew]at h1⊢
 have p1:0<s.h/s.kh*(s.w/s.kh):=ϵ (fun z=>by simp[z]at h1)
 have hy:q/(s.w/s.kh)%(s.h/s.kh)<s.h/s.kh:=ϊ _ (Nat.pos_of_mul_pos_right p1)
 have hx:q%(s.w/s.kh)<s.w/s.kh:=ϊ _ (Nat.pos_of_mul_pos_left p1)
 obtain⟨W,-⟩:=apW s h _ _ hy hx
 simp only[W,ExactRuntime.poolCell,poolAt]
theorem daE{α:Type}(f:Nat->α)(N:Nat):Dense.F.da f N (lQ N)=kK (n:=N) fun q=>f q.val:=
 Dense.F.iC_h1 f N N _ (by simp) (by simp) (fun i h=>absurd h (by simp))
@[inline]def im2cF(h kh k ho:Nat)(inp:Ἂ Word):Ἂ Word:=Dense.F.da (fun i=>inp[(i%k/kh)*h+i/k+i%k%kh]!) (ho*k) (lQ (ho*k))
theorem im2cF_eq(h kh k ho:Nat)(inp:Ἂ Word):im2cF h kh k ho inp=Cv1.im2c h kh k ho inp:=daE _ _
@[inline]def trnF(rows cols:Nat)(Y:Ἂ Word):Ἂ Word:=Dense.F.da (fun i=>Y[(i%rows)*cols+i/rows]!) (cols*rows) (lQ (cols*rows))
theorem trnF_eq(rows cols:Nat)(Y:Ἂ Word):trnF rows cols Y=Cv1.trn rows cols Y:=daE _ _
@[inline]def im2dF(h kh ho c:Nat)(inp:Ἂ Word):Ἂ Word:=Dense.F.da (fun i=>inp[c*h+i/kh+i%kh]!) (ho*kh) (lQ (ho*kh))
theorem im2dF_eq(h kh ho c:Nat)(inp:Ἂ Word):im2dF h kh ho c inp=Cv1.im2d h kh ho c inp:=daE _ _
def dwRowsF(s:L4)(relu:Ἆ)(inp wts b:Ἂ Word)(lin:Nat->Nat->Nat->Ἆ->Ἂ Word->Ἂ Word->Ἂ Word->Ἂ Word):Ἂ Word:=
 let ho:=s.h+1-s.kh
 let Ys:=kK (n:=s.cin) fun c=>lin ho s.kh 1 relu (im2dF s.h s.kh ho c.val inp) (kK (n:=s.kh) fun t=>wts[c.val*s.kh+t.val]!) #[b[c.val]!]
 Dense.F.da (fun i=>(Ys[i/ho]!)[i%ho]!) (s.cin*ho) (lQ (s.cin*ho))
theorem dwRowsF_eq(s:L4)(relu:Ἆ)(inp wts b:Ἂ Word)(lin:Nat->Nat->Nat->Ἆ->Ἂ Word->Ἂ Word->Ἂ Word->Ἂ Word):
  dwRowsF s relu inp wts b lin=Cv1.dwRows s relu inp wts b lin:=by
 unfold dwRowsF Cv1.dwRows;simp only[im2dF_eq];exact daE _ _
@[inline]def fpOk(s:L4):Ἆ:=s.strideH==1&&(s.kh==3&&s.kw==3&&s.padT==1||s.kh==1&&s.kw==1&&s.padT==0)
@[inline]def s31(cin H co:Nat):ConvShape:=⟨cin,1,H,co,3,3,1,1⟩
def k33(n:Nat)(K:Ἂ Word):Ἂ Word:=Dense.F.da (fun j=>if j%9/3=1 then K[j/9*3+j%3]! else 0) (n*9) (lQ (n*9))
def sl31(co H:Nat)(O:Ἂ Word):Ἂ Word:=Dense.F.da (fun i=>O[i/(H-2)*H+i%(H-2)+1]!) (co*(H-2)) (lQ (co*(H-2)))
@[inline]def c31Ok(s:L4)(inp wts b:Ἂ Word):Ἆ:=Cv1.c1Ok s&&s.kh==3&&b.all (· == 0)&&inp.all (fun w=>(w>>>23)&&&255!=255)&&wts.all (fun w=>(w>>>23)&&&255!=255)
theorem dE(n:Nat)(D E D' E':Nat→Word)
 (hD:∀t<3*n,expOf (D t)≠255)(hE:∀t<3*n,expOf (E t)≠255)
 (hD':∀t<9*n,expOf (D' t)≠255)(hE':∀t<9*n,expOf (E' t)≠255)
 (h1:∀c<n,∀j<3,D' (9*c+3+j)=D (3*c+j)∧E' (9*c+3+j)=E (3*c+j))
 (h0:∀c<n,∀j<9,j/3≠1→E' (9*c+j)=0):
 dotOf (9*n+1) (fun t=>if t<9*n then D' t else oneW) (fun t=>if t<9*n then E' t else 0)=
 dotOf (3*n+1) (fun t=>if t<3*n then D t else oneW) (fun t=>if t<3*n then E t else 0):=by
 have g1:aG (9*n+1) (fun t=>if t<9*n then D' t else oneW) (fun t=>if t<9*n then E' t else 0):=by
  intro k hk;by_cases e:k<9*n
  ·simp only[if_pos e];exact⟨hD' k e,hE' k e⟩
  ·simp only[if_neg e];exact⟨by decide,by decide⟩
 have g2:aG (3*n+1) (fun t=>if t<3*n then D t else oneW) (fun t=>if t<3*n then E t else 0):=by
  intro k hk;by_cases e:k<3*n
  ·simp only[if_pos e];exact⟨hD k e,hE k e⟩
  ·simp only[if_neg e];exact⟨by decide,by decide⟩
 have z0:tm oneW 0=0:=by simp only[tm,show sv (0:Word)=0 by decide,mul_zero,zero_mul]
 have tz:∀x:Word,tm x 0=0:=fun x=>by simp only[tm,show sv (0:Word)=0 by decide,mul_zero,zero_mul]
 have S:∀m,m≤n→∑t∈kP (9*m),tm (if t<9*n then D' t else oneW) (if t<9*n then E' t else 0)=
   ∑t∈kP (3*m),tm (if t<3*n then D t else oneW) (if t<3*n then E t else 0):=by
  intro m;induction m with
  |zero=>intro _;simp
  |succ m ih=>
   intro hm
   rw[show 9*(m+1)=9*m+9 by ring,show 3*(m+1)=3*m+3 by ring,Finset.sum_range_add,Finset.sum_range_add,ih (by omega)]
   congr 1
   have a:∀j<9,9*m+j<9*n:=fun j hj=>by omega
   have c:∀j<3,3*m+j<3*n:=fun j hj=>by omega
   simp only[Ν,X8,Ai]
   simp only[if_pos (a _ (by decide:0<9)),if_pos (a _ (by decide:1<9)),if_pos (a _ (by decide:2<9)),if_pos (a _ (by decide:3<9)),if_pos (a _ (by decide:4<9)),
    if_pos (a _ (by decide:5<9)),if_pos (a _ (by decide:6<9)),if_pos (a _ (by decide:7<9)),if_pos (a _ (by decide:8<9)),
    if_pos (c _ (by decide:0<3)),if_pos (c _ (by decide:1<3)),if_pos (c _ (by decide:2<3))]
   have hm':m<n:=by omega
   rw[h0 m hm' 0 (by decide) (by decide),h0 m hm' 1 (by decide) (by decide),h0 m hm' 2 (by decide) (by decide),
    h0 m hm' 6 (by decide) (by decide),h0 m hm' 7 (by decide) (by decide),h0 m hm' 8 (by decide) (by decide)]
   have e3:=h1 m hm' 0 (by decide);have e4:=h1 m hm' 1 (by decide);have e5:=h1 m hm' 2 (by decide)
   simp only[ϑ]at e3
   rw[show 9*m+4=9*m+3+1 by ring,show 9*m+5=9*m+3+2 by ring,e3.1,e3.2,e4.1,e4.2,e5.1,e5.2,tz,tz,tz,tz,tz,tz]
   simp only[ϑ,Ai,Av]
 rw[Core.bI _ _ _ g1,Core.bI _ _ _ g2]
 unfold af
 rw[Ν,Ν,S n AP]
 simp only[AX,if_false]
 rw[Dense.F.zrNC _ _ (9*n) 0 (by rw[Κ,ab (9*n+1) (fun t=>if t<9*n then D' t else oneW) (9*n) (by omega),ab (9*n+1) (fun t=>if t<9*n then E' t else (0:Word)) (9*n) (by omega)];simp only[AX,if_false];decide),
  Dense.F.zrNC _ _ (3*n) 0 (by rw[Κ,ab (3*n+1) (fun t=>if t<3*n then D t else oneW) (3*n) (by omega),ab (3*n+1) (fun t=>if t<3*n then E t else (0:Word)) (3*n) (by omega)];simp only[AX,if_false];decide)]
 simp
theorem fwA(a:Ἂ Word)(h:a.all finiteWord=true)(q:Nat):expOf a[q]!≠255:=by
 by_cases d:q<a.size
 ·exact Norm.NR.ic a (fun i hi=>np.1 h i hi) q d
 ·rw[An a q d];decide
theorem k33g(n:Nat)(K:Ἂ Word)(j:Nat)(hj:j<n*9):(k33 n K)[j]! =if j%9/3=1 then K[j/9*3+j%3]! else 0:=by
 unfold k33;rw[daE,AB _ j (by simpa using hj),Λ]
theorem k33f(n:Nat)(K:Ἂ Word)(hK:K.all finiteWord=true)(j:Nat):expOf (k33 n K)[j]!≠255:=by
 by_cases hj:j<n*9
 ·rw[k33g n K j hj];split
  ·exact fwA K hK _
  ·decide
 ·unfold k33;rw[daE,An _ j (by simpa using hj)];decide
theorem wOf(s:ConvShape)(inp:Ἂ Word)(hi:inp.all finiteWord=true)(y x t:Nat):expOf (windowAt s inp y x t)≠255:=by
 unfold windowAt;dsimp only;split
 ·decide
 ·exact fwA inp hi _
theorem wNf(s:L4)(inp:Ἂ Word)(hi:inp.all finiteWord=true)(g y x t:Nat):expOf (U6 s inp g y x t)≠255:=by
 unfold U6;dsimp only;split
 ·decide
 ·exact fwA inp hi _
theorem oldW(cin H co:Nat)(inp:Ἂ Word)(c p j:Nat)(hj:j<3)(hp:p+2<H):
  windowAt (s31 cin H co) inp 0 (p+1) (c*9+(3+j))=inp[c*H+(p+j)]!:=by
 have a1:(c*9+(3+j))/(3*3)=c:=by omega
 have a2:(c*9+(3+j))/3%3=1:=by omega
 have a3:(c*9+(3+j))%3=j:=by omega
 unfold windowAt
 simp only[s31,a1,a2,a3,ϻ,ϳ,Κ,Nat.sub_self,ϑ]
 split
 ·rename_i h;exfalso;omega
 ·congr 1;omega
theorem newWd(s:L4)(inp:Ἂ Word)(hk:s.kh=3)(hkw:s.kw=1)(hw1:s.w=1)(hsh:s.strideH=1)(hsw:s.strideW=1)
  (hdh:s.dilH=1)(hdw:s.dilW=1)(hpt:s.padT=0)(hpl:s.padL=0)(hcin:0<s.cin)(c p j:Nat)(hj:j<3)(hp:p+2<s.h):
  U6 s.depthwise inp c p 0 j=inp[c*s.h+(p+j)]!:=by
 have a1:j/(3*1)=0:=by omega
 have a2:j/1%3=j:=by omega
 have a3:j%1=0:=by omega
 unfold U6
 simp only[J5,T6,Nat.div_self hcin,hk,hkw,hw1,hsh,hsw,hdh,hdw,hpt,hpl,a1,a2,a3,ϻ,ϳ,Κ,pQ,ϑ]
 split
 ·rename_i h;exfalso;omega
 ·rfl
theorem d31_eq(s:L4)(relu:Ἆ)(inp wts b:Ἂ Word)(h:c31Ok s inp wts b=true):
  sl31 s.cin s.h (depthwiseSpec (s31 s.cin s.h s.cin) relu inp (k33 s.cin wts) b)=J9 s relu inp wts b:=by
 simp only[c31Ok,Η,AQ]at h
 obtain⟨⟨⟨⟨hc,hk⟩,hb⟩,hi⟩,hw⟩:=h
 obtain⟨hw1,hkw,hsh,hsw,hpt,hpl,hpb,hpr,hdh,hdw,hk0,hkh⟩:=Cv1.c1F s hc
 have e1:s.depthwise.ho=s.h-2:=by simp only[ExactRuntime.ConvShape.ho,ExactRuntime.ConvShape.ekh,J5,hpt,hpb,hdh,hsh,hk];omega
 have e2:s.depthwise.wo=1:=by simp only[ExactRuntime.ConvShape.wo,ExactRuntime.ConvShape.ekw,J5,hpl,hpr,hdw,hsw,hw1,hkw];try omega
 have hb0:∀q,b[q]! =(0:Word):=fun q=>by
  by_cases d:q<b.size
  ·have:=np.1 hb q d;rw[AB b q d];simpa using this
  ·rw[An b q d];rfl
 unfold sl31 J9 ExactRuntime.depthwiseWith ExactRuntime.conv2dWith
 rw[daE]
 apply mi
 ·simp only[Ϝ,e1,e2,ϻ];rfl
 intro q h1 h2
 have hq:q<s.cin*(s.h-2):=by simpa using h1
 have hH:3<=s.h:=by omega
 have hc:q/(s.h-2)<s.cin:=Ϟ (by rw[Ϗ];exact hq)
 have hp:q%(s.h-2)<s.h-2:=ϊ _ (by omega)
 have o1:(s31 s.cin s.h s.cin).ho=1:=by simp only[s31,ConvShape.ho]
 have o2:(s31 s.cin s.h s.cin).wo=s.h:=by simp only[s31,ConvShape.wo];omega
 rw[Λ]
 unfold depthwiseSpec depthwiseWith
 have hq':q/(s.h-2)*s.h+q%(s.h-2)+1<(s31 s.cin s.h s.cin).cin*((s31 s.cin s.h s.cin).ho*(s31 s.cin s.h s.cin).wo):=by
  rw[o1,o2,n4];simp only[s31]
  calc q/(s.h-2)*s.h+q%(s.h-2)+1<q/(s.h-2)*s.h+s.h:=by rw[pA];exact O1 (by omega) _
   _=(q/(s.h-2)+1)*s.h:=by ring
   _<=s.cin*s.h:=ι _ hc
 rw[AB _ _ (by simpa using hq'),Λ]
 simp only[Λ,o1,o2,n4,e1,e2,ϻ,F5,Nat.mod_one]
 have hcin:0<s.cin:=C0 (ϟ _) hc
 generalize hc2:q/(s.h-2)=c at *
 generalize hp2:q%(s.h-2)=p at *
 have m1:(c*s.h+p+1)%s.h=p+1:=by rw[pA,Ϗ,O7,γ (by omega)]
 have m2:(c*s.h+p+1)/s.h=c:=by rw[pA,Ϗ,W8 (by omega),ώ (by omega),ϑ]
 have t3:s.depthwise.taps=3:=by simp only[ExactRuntime.ConvShape.taps,T6,J5,hk,hkw,Nat.div_self hcin]
 have g1:s.depthwise.opg=1:=by simp only[ExactRuntime.ConvShape.opg,J5,Nat.div_self hcin]
 have k9:(s31 s.cin s.h s.cin).kh*(s31 s.cin s.h s.cin).kw=9:=rfl
 rw[m1,m2,t3,g1,k9,F5,hb0]
 congr 1
 have key:=dE 1 (fun t=>U6 s.depthwise inp c p 0 t) (fun t=>wts[c*3+t]!) (fun t=>windowAt (s31 s.cin s.h s.cin) inp 0 (p+1) (c*9+t)) (fun t=>(k33 s.cin wts)[c*9+t]!)
  (fun t _=>wNf _ _ hi _ _ _ _) (fun t _=>fwA _ hw _) (fun t _=>wOf _ _ hi _ _ _) (fun t _=>k33f _ _ hw _) ?h1 ?h0
 ·simpa using key
 case h1=>
  intro c' hc' j hj
  obtain rfl:c'=0:=by omega
  simp only[A7,Κ]
  constructor
  ·rw[oldW _ _ _ _ c p j hj (by omega),newWd s inp hk hkw hw1 hsh hsw hdh hdw hpt hpl hcin c p j hj (by omega)]
  ·rw[k33g _ _ _ (by omega),if_pos (by omega)];congr 1;omega
 case h0=>
  intro c' hc' j hj hj3
  obtain rfl:c'=0:=by omega
  simp only[A7,Κ]
  rw[k33g _ _ _ (by omega),if_neg (by omega)]
theorem newWc(s:L4)(inp:Ἂ Word)(hg:s.groups=1)(hk:s.kh=3)(hkw:s.kw=1)(hw1:s.w=1)(hsh:s.strideH=1)(hsw:s.strideW=1)
  (hdh:s.dilH=1)(hdw:s.dilW=1)(hpt:s.padT=0)(hpl:s.padL=0)(c p j:Nat)(hj:j<3)(hp:p+2<s.h):
  U6 s inp 0 p 0 (3*c+j)=inp[c*s.h+(p+j)]!:=by
 have a1:(3*c+j)/(3*1)=c:=by omega
 have a2:(3*c+j)/1%3=j:=by omega
 have a3:(3*c+j)%1=0:=by omega
 have a4:(3*c+j)%3=j:=by omega
 unfold U6
 simp only[T6,hg,F5,hk,hkw,hw1,hsh,hsw,hdh,hdw,hpt,hpl,a1,a2,a3,a4,ϻ,ϳ,Κ,pQ,ϑ]
 split
 ·rename_i h;exfalso;omega
 ·rfl
theorem c31_eq(s:L4)(relu:Ἆ)(inp wts b:Ἂ Word)(h:(c31Ok s inp wts b&&s.groups==1)=true):
  sl31 s.cout s.h (conv2dSpec (s31 s.cin s.h s.cout) relu inp (k33 (s.cout*s.cin) wts) b)=ExactRuntime.conv2dSpec s relu inp wts b:=by
 simp only[c31Ok,Η,AQ]at h
 obtain⟨⟨⟨⟨⟨hc,hk⟩,hb⟩,hi⟩,hw⟩,hg⟩:=h
 obtain⟨hw1,hkw,hsh,hsw,hpt,hpl,hpb,hpr,hdh,hdw,hk0,hkh⟩:=Cv1.c1F s hc
 have e1:s.ho=s.h-2:=by simp only[ExactRuntime.ConvShape.ho,ExactRuntime.ConvShape.ekh,hpt,hpb,hdh,hsh,hk];omega
 have e2:s.wo=1:=by simp only[ExactRuntime.ConvShape.wo,ExactRuntime.ConvShape.ekw,hpl,hpr,hdw,hsw,hw1,hkw];try omega
 have hb0:∀q,b[q]! =(0:Word):=fun q=>by
  by_cases d:q<b.size
  ·have:=np.1 hb q d;rw[AB b q d];simpa using this
  ·rw[An b q d];rfl
 unfold sl31 ExactRuntime.conv2dSpec ExactRuntime.conv2dWith
 rw[daE]
 apply mi
 ·simp only[Ϝ,e1,e2,ϻ]
 intro q h1 h2
 have hq:q<s.cout*(s.h-2):=by simpa using h1
 have hH:3<=s.h:=by omega
 have hc:q/(s.h-2)<s.cout:=Ϟ (by rw[Ϗ];exact hq)
 have hp:q%(s.h-2)<s.h-2:=ϊ _ (by omega)
 have o1:(s31 s.cin s.h s.cout).ho=1:=by simp only[s31,ConvShape.ho]
 have o2:(s31 s.cin s.h s.cout).wo=s.h:=by simp only[s31,ConvShape.wo];omega
 rw[Λ]
 unfold conv2dSpec conv2dWith
 have hq':q/(s.h-2)*s.h+q%(s.h-2)+1<(s31 s.cin s.h s.cout).cout*((s31 s.cin s.h s.cout).ho*(s31 s.cin s.h s.cout).wo):=by
  rw[o1,o2,n4];simp only[s31]
  calc q/(s.h-2)*s.h+q%(s.h-2)+1<q/(s.h-2)*s.h+s.h:=by rw[pA];exact O1 (by omega) _
   _=(q/(s.h-2)+1)*s.h:=by ring
   _<=s.cout*s.h:=ι _ hc
 rw[AB _ _ (by simpa using hq'),Λ]
 simp only[Λ,o1,o2,n4,e1,e2,ϻ,F5,Nat.mod_one]
 have hcout:0<s.cout:=C0 (ϟ _) hc
 generalize hc2:q/(s.h-2)=o at *
 generalize hp2:q%(s.h-2)=p at *
 have m1:(o*s.h+p+1)%s.h=p+1:=by rw[pA,Ϗ,O7,γ (by omega)]
 have m2:(o*s.h+p+1)/s.h=o:=by rw[pA,Ϗ,W8 (by omega),ώ (by omega),ϑ]
 have t3:s.taps=s.cin*3:=by simp only[ExactRuntime.ConvShape.taps,T6,hg,F5,hk,hkw,ϻ]
 have g1:o/s.opg=0:=by simp only[ExactRuntime.ConvShape.opg,hg,F5];exact ώ hc
 have k9:(s31 s.cin s.h s.cout).cin*(s31 s.cin s.h s.cout).kh*(s31 s.cin s.h s.cout).kw=s.cin*9:=by simp only[s31];ring
 rw[m1,m2,t3,g1,k9,hb0]
 congr 1
 have key:=dE s.cin (fun t=>U6 s inp 0 p 0 t) (fun t=>wts[o*(s.cin*3)+t]!) (fun t=>windowAt (s31 s.cin s.h s.cout) inp 0 (p+1) t) (fun t=>(k33 (s.cout*s.cin) wts)[o*(s.cin*9)+t]!)
  (fun t _=>wNf _ _ hi _ _ _ _) (fun t _=>fwA _ hw _) (fun t _=>wOf _ _ hi _ _ _) (fun t _=>k33f _ _ hw _) ?h1 ?h0
 ·simpa[Ϗ 9 s.cin,Ϗ 3 s.cin] using key
 case h1=>
  intro c hc' j hj
  constructor
  ·rw[show 9*c+3+j=c*9+(3+j) by ring,oldW _ _ _ _ c p j hj (by omega),newWc s inp hg hk hkw hw1 hsh hsw hdh hdw hpt hpl c p j hj (by omega)]
  ·have l1:o*(s.cin*9)+(9*c+3+j)<s.cout*s.cin*9:=by
    have u1:9*c+3+j<s.cin*9:=by omega
    calc o*(s.cin*9)+(9*c+3+j)<o*(s.cin*9)+s.cin*9:=O1 u1 _
     _=(o+1)*(s.cin*9):=by ring
     _<=s.cout*(s.cin*9):=ι _ hc
     _=s.cout*s.cin*9:=by ring
   rw[k33g _ _ _ l1]
   generalize hA:o*s.cin=A
   have r1:o*(s.cin*9)=A*9:=by rw[←hA];ring
   have r2:o*(s.cin*3)=A*3:=by rw[←hA];ring
   rw[r1,r2,if_pos (by omega)];congr 1;omega
 case h0=>
  intro c hc' j hj hj3
  have l1:o*(s.cin*9)+(9*c+j)<s.cout*s.cin*9:=by
   have u1:9*c+j<s.cin*9:=by omega
   calc o*(s.cin*9)+(9*c+j)<o*(s.cin*9)+s.cin*9:=O1 u1 _
    _=(o+1)*(s.cin*9):=by ring
    _<=s.cout*(s.cin*9):=ι _ hc
    _=s.cout*s.cin*9:=by ring
  rw[k33g _ _ _ l1]
  generalize hA:o*s.cin=A
  have r1:o*(s.cin*9)=A*9:=by rw[←hA];ring
  rw[r1,if_neg (by omega)]
def cvK(s:L4)(relu:Ἆ)(inp wts b:Ἂ Word):Ἂ Word:=
 if cvOk s&&fpOk s then Conv.er (cvO s) relu inp wts b
 else if c31Ok s inp wts b&&s.groups==1 then sl31 s.cout s.h (Conv.er (s31 s.cin s.h s.cout) relu inp (k33 (s.cout*s.cin) wts) b)
 else if Cv1.c1Ok s&&s.groups==1 then trnF (s.h+1-s.kh) s.cout (Dense.c7 (s.h+1-s.kh) (s.cin*s.kh) s.cout relu (im2cF s.h s.kh (s.cin*s.kh) (s.h+1-s.kh) inp) wts b)
 else U7.conv.conv2d s relu inp wts b
theorem cvK_eq:cvK=ExactRuntime.conv2dSpec:=by
 funext s relu inp wts b;unfold cvK;split
 ·rename_i h;rw[Η]at h;rw[Conv.eu,cvB s h.1]
 ·split
  ·rename_i h;rw[Conv.eu];exact c31_eq s relu inp wts b h
  ·split
   ·rename_i h;rw[Η,AQ]at h;rw[trnF_eq,im2cF_eq,Dense.eD];exact Cv1.lin_conv1 s h.1 h.2 relu inp wts b
   ·simp only[U7,Pinned.dotFn_eq_dotOf];rfl
def dwK(s:L4)(relu:Ἆ)(inp wts b:Ἂ Word):Ἂ Word:=
 if dwOk s&&fpOk s then Conv.cG (cvO s) relu inp wts b
 else if c31Ok s inp wts b then sl31 s.cin s.h (Conv.cG (s31 s.cin s.h s.cin) relu inp (k33 s.cin wts) b)
 else if Cv1.d1Ok s then dwRowsF s relu inp wts b Dense.c7
 else U7.conv.depthwise s relu inp wts b
def apK(s:PoolShape)(inp:Ἂ Word):Ἂ Word:=
 if apOk s then Conv.dU s.c s.h s.w s.kh inp else U7.elem.avgpool s inp
def mpK(s:PoolShape)(inp:Ἂ Word):Ἂ Word:=
 if apOk s then Conv.d9 s.c s.h s.w s.kh inp else U7.elem.maxpool s inp
namespace Rnn
open ExactRuntime FloatLib.Floats.Formats.BinaryInterchange FloatLib.Floats
abbrev LinFn:=Nat → Nat → Nat → Ἆ → Ἂ Word → Ἂ Word → Ἂ Word → Ἂ Word
@[inline] def fA (A:Ἂ Word) (ao n:Nat) (a1 a2:Word) (t:Nat):Word :=
  if t < n then A[ao+t]! else if t=n then a1 else a2
/-! (S4 11-11) ad2: Ἄ loops aJu (jG) and aDu (dl) over the array part, two word terms in the tail. -/
@[inline] def aGt (A:Ἂ Word) (k:Ἄ):Word:=if h:k < A.usize then A.uget k (Dense.cD h) else 0
theorem aGt_eq (A:Ἂ Word) (k:Ἄ) (hA:A.size < kX):aGt A k=A[k.toNat]!:=by
  unfold aGt; split
  · exact Dense.cC A k _
  · rename_i h
    have h1:¬ k.toNat < A.size:=by
      intro h2; apply h; rw [ε]
      rw [Array.usize, lX, J3 hA]; exact h2
    rw [An A _ h1]; rfl
theorem aUs {k:Ἄ} (h:k.toNat+1 < kX):(k+1).toNat=k.toNat+1:=by
  rw [lJ, mA]; exact γ h
theorem aUd {ka ea:Ἄ} (h:ka < ea):ea.toNat-(ka+1).toNat < ea.toNat-ka.toNat:=by
  have h1:=ε.mp h
  rw [aUs (C0 h1 (USize.toNat_lt_size ea))]
  omega
@[inline] def aJs (a b:Word) (Bm:ἆ):ἆ:=
  if Core.mW a*Core.mW b==0 then Bm else min Bm (Core.iW a+Core.iW b)
def aJu (A B:Ἂ Word) (ka kb ea:Ἄ) (Bm:ἆ):ἆ:=
  if h:ka < ea then aJu A B (ka+1) (kb+1) ea (aJs (aGt A ka) (aGt B kb) Bm) else Bm
termination_by ea.toNat-ka.toNat
decreasing_by exact aUd h
@[inline] def aDs (a b:Word) (Bm pH pL nH nL:ἆ) (k:ἆ → ἆ → ἆ → ἆ → ἆ):ἆ:=
  let m:=Core.mW a*Core.mW b
  let e:=Core.iW a+Core.iW b
  if expOf a==255||expOf b==255||m != 0&&(e < Bm||60 < e-Bm) then 4294967296 else
  if m==0 then k pH pL nH nL else
  let g:=signOf a==signOf b
  let q:=Dense.gd (if g then pH else nH) (if g then pL else nL) m (e-Bm)
  k (if g then q.1 else pH) (if g then q.2 else pL) (if g then nH else q.1) (if g then nL else q.2)
theorem aGd0 (h l s:ἆ):Dense.gd h l 0 s=(h,l):=by
  simp [Dense.gd]
@[inline] def aDs2 (a b:Word) (Bm pH pL nH nL:ἆ) (k:ἆ → ἆ → ἆ → ἆ → ἆ):ἆ:=
  let m:=Core.mW a*Core.mW b
  let e:=Core.iW a+Core.iW b
  if expOf a==255||expOf b==255||m != 0&&(e < Bm||60 < e-Bm) then 4294967296 else
  let g:=signOf a==signOf b
  let p:=Dense.gd pH pL (if g then m else 0) (e-Bm)
  let q:=Dense.gd nH nL (if g then 0 else m) (e-Bm)
  k p.1 p.2 q.1 q.2
theorem aDs2_eq (a b:Word) (Bm pH pL nH nL:ἆ) (k:ἆ → ἆ → ἆ → ἆ → ἆ):
    aDs2 a b Bm pH pL nH nL k=aDs a b Bm pH pL nH nL k:=by
  unfold aDs2 aDs
  dsimp only
  split
  · rfl
  · by_cases hm:(Core.mW a*Core.mW b==0)=true
    · have h0:Core.mW a*Core.mW b=0:=by simpa using hm
      rw [if_pos hm, h0]
      simp only [ite_self, aGd0]
    · rw [if_neg hm]
      cases signOf a==signOf b <;> simp only [aGd0, if_true, if_false, υ]
def aDu (A B:Ἂ Word) (ka kb ea:Ἄ) (a1 b1 a2 b2:Word) (Bm pH pL nH nL:ἆ):ἆ:=
  if h:ka < ea then
    aDs2 (aGt A ka) (aGt B kb) Bm pH pL nH nL fun pH pL nH nL=>aDu A B (ka+1) (kb+1) ea a1 b1 a2 b2 Bm pH pL nH nL
  else aDs2 a1 b1 Bm pH pL nH nL fun pH pL nH nL=>aDs2 a2 b2 Bm pH pL nH nL fun pH pL nH nL=>
    Core.au (pH-nH-(if pL < nL then 1 else 0)) (pL-nL) Bm
termination_by ea.toNat-ka.toNat
decreasing_by exact aUd h
theorem aDls (fa fb:Nat → Word) (n:Nat) (Bm:ἆ) (t:Nat) (ht:t < n) (pH pL nH nL:ἆ):
    Dense.dl fa fb n Bm t pH pL nH nL=aDs (fa t) (fb t) Bm pH pL nH nL fun pH pL nH nL=>Dense.dl fa fb n Bm (t+1) pH pL nH nL:=by
  rw [Dense.dl, if_pos ht]; rfl
theorem aDle (fa fb:Nat → Word) (n:Nat) (Bm:ἆ) (t:Nat) (ht:¬ t < n) (pH pL nH nL:ἆ):
    Dense.dl fa fb n Bm t pH pL nH nL=Core.au (pH-nH-(if pL < nL then 1 else 0)) (pL-nL) Bm:=by
  rw [Dense.dl, if_neg ht]
theorem aJgs (fa fb:Nat → Word) (n t:Nat) (ht:t < n) (Bm:ἆ):
    Dense.jG fa fb n t Bm=Dense.jG fa fb n (t+1) (aJs (fa t) (fb t) Bm):=by
  rw [Dense.jG, if_pos ht]; rfl
theorem aJge (fa fb:Nat → Word) (n t:Nat) (ht:¬ t < n) (Bm:ἆ):Dense.jG fa fb n t Bm=Bm:=by
  rw [Dense.jG, if_neg ht]
theorem aFn (A:Ἂ Word) (ao n:Nat) (a1 a2:Word):fA A ao n a1 a2 n=a1:=by simp [fA]
theorem aFm (A:Ἂ Word) (ao n:Nat) (a1 a2:Word):fA A ao n a1 a2 (n+1)=a2:=by simp [fA]
theorem aFl (A:Ἂ Word) (ao n:Nat) (a1 a2:Word) (t:Nat) (h:t < n):fA A ao n a1 a2 t=A[ao+t]!:=by simp [fA, h]
theorem aJu_eq (A B:Ἂ Word) (ao bo n:Nat) (a1 b1 a2 b2:Word) (ea:Ἄ)
    (hA:A.size < kX) (hB:B.size < kX) (he:ea.toNat=ao+n) (hb:bo+n < kX):
    ∀ k t (ka kb:Ἄ) Bm,n-t=k → t ≤ n → ka.toNat=ao+t → kb.toNat=bo+t →
    aJs a2 b2 (aJs a1 b1 (aJu A B ka kb ea Bm))=Dense.jG (fA A ao n a1 a2) (fA B bo n b1 b2) (n+2) t Bm:=by
  have hs:=USize.toNat_lt_size ea
  intro k
  induction k with
  | zero =>
    intro t ka kb Bm hk ht h1 h2
    obtain rfl:t=n:=by omega
    have h0:¬ ka < ea:=by rw [ε, h1, he]; exact B4 _
    rw [aJu, dif_neg h0, aJgs _ _ _ _ (by omega), aJgs _ _ _ _ (by omega), aJge _ _ _ _ (by omega),
      aFn, aFn, aFm, aFm]
  | succ k ih =>
    intro t ka kb Bm hk ht h1 h2
    have htn:t < n:=by omega
    have h0:ka < ea:=by rw [ε, h1, he]; omega
    rw [aJu, dif_pos h0, aJgs _ _ _ _ (by omega), aGt_eq A ka hA, aGt_eq B kb hB, h1, h2,
      aFl _ _ _ _ _ _ htn, aFl _ _ _ _ _ _ htn]
    exact ih (t+1) (ka+1) (kb+1) _ (by omega) (by omega) (by rw [aUs (by omega), h1]; rfl) (by rw [aUs (by omega), h2]; rfl)
theorem aDu_eq (A B:Ἂ Word) (ao bo n:Nat) (a1 b1 a2 b2:Word) (ea:Ἄ) (Bm:ἆ)
    (hA:A.size < kX) (hB:B.size < kX) (he:ea.toNat=ao+n) (hb:bo+n < kX):
    ∀ k t (ka kb:Ἄ) pH pL nH nL,n-t=k → t ≤ n → ka.toNat=ao+t → kb.toNat=bo+t →
    aDu A B ka kb ea a1 b1 a2 b2 Bm pH pL nH nL=Dense.dl (fA A ao n a1 a2) (fA B bo n b1 b2) (n+2) Bm t pH pL nH nL:=by
  have hs:=USize.toNat_lt_size ea
  intro k
  induction k with
  | zero =>
    intro t ka kb pH pL nH nL hk ht h1 h2
    obtain rfl:t=n:=by omega
    have h0:¬ ka < ea:=by rw [ε, h1, he]; exact B4 _
    rw [aDu, dif_neg h0]; simp only [aDs2_eq]; rw [aDls _ _ _ _ _ (by omega), aFn, aFn]
    simp only [aDls _ _ (t+2) Bm (t+1) (by omega), aDle _ _ (t+2) Bm (t+2) (by omega), aFm]
  | succ k ih =>
    intro t ka kb pH pL nH nL hk ht h1 h2
    have htn:t < n:=by omega
    have h0:ka < ea:=by rw [ε, h1, he]; omega
    rw [aDu, dif_pos h0]; simp only [aDs2_eq]; rw [aDls _ _ _ _ _ (by omega), aGt_eq A ka hA, aGt_eq B kb hB, h1, h2,
      aFl _ _ _ _ _ _ htn, aFl _ _ _ _ _ _ htn]
    have e:=fun pH pL nH nL=>ih (t+1) (ka+1) (kb+1) pH pL nH nL (by omega) (by omega)
      (by rw [aUs (by omega), h1]; rfl) (by rw [aUs (by omega), h2]; rfl)
    simp only [e]
@[inline] def ad2x (A B:Ἂ Word) (ao bo n:Nat) (a1 b1 a2 b2:Word):Word:=
  let r:=if n+2 < 262144 ∧ A.size ≤ 4294967295 ∧ B.size ≤ 4294967295 ∧ ao+n ≤ 4294967295 ∧ bo+n ≤ 4294967295 then
      aDu A B (C3 ao) (C3 bo) (C3 (ao+n)) a1 b1 a2 b2
        (aJs a2 b2 (aJs a1 b1 (aJu A B (C3 ao) (C3 bo) (C3 (ao+n)) 1023))) 0 0 0 0
    else 4294967296
  if r < 4294967296 then N3 r else dotOf (n+2) (fA A ao n a1 a2) (fA B bo n b1 b2)
theorem ad2x_eq (A B:Ἂ Word) (ao bo n:Nat) (a1 b1 a2 b2:Word):
    ad2x A B ao bo n a1 b1 a2 b2=dotOf (n+2) (fA A ao n a1 a2) (fA B bo n b1 b2):=by
  unfold ad2x
  by_cases h:n+2 < 262144 ∧ A.size ≤ 4294967295 ∧ B.size ≤ 4294967295 ∧ ao+n ≤ 4294967295 ∧ bo+n ≤ 4294967295
  · rw [if_pos h]
    obtain ⟨h0,hA,hB,ha,hb⟩:=h
    have s:=ς
    have hA':A.size < kX:=by omega
    have hB':B.size < kX:=by omega
    have he:(C3 (ao+n)).toNat=ao+n:=J3 (by omega)
    have h1:(C3 ao).toNat=ao+0:=J3 (by omega)
    have h2:(C3 bo).toNat=bo+0:=J3 (by omega)
    rw [aJu_eq A B ao bo n a1 b1 a2 b2 _ hA' hB' he (by omega) n 0 _ _ 1023 (by omega) (by omega) h1 h2,
      aDu_eq A B ao bo n a1 b1 a2 b2 _ _ hA' hB' he (by omega) n 0 _ _ 0 0 0 0 (by omega) (by omega) h1 h2]
    have g:=Dense.bO (n+2) (fA A ao n a1 a2) (fA B bo n b1 b2)
    unfold Dense.gf at g
    rw [if_pos h0] at g
    exact g
  · rw [if_neg h]; rfl
theorem aDok (fa fb:Nat → Word) (N:Nat) (Bm:ἆ) (g:N < 262144)
    (h:Dense.dl fa fb N Bm 0 0 0 0 0 < 4294967296):dotOf N fa fb=N3 (Dense.dl fa fb N Bm 0 0 0 0 0):=
  Dense.fE fa fb N Bm (by exact I7 g) _ 0 0 0 0 0 rfl (by exact ϟ N)
    (λ_ h=>absurd h (by (clear * - h;omega))) ((by simp)) ((by simp)) (by simp[af]) h
@[inline] def ad4 (g:ἆ) (A B:Ἂ Word) (ao bo n:Nat) (a1 b1 a2 b2:Word):Word:=
  let r:=if n+2 < 262144 ∧ A.size ≤ 4294967295 ∧ B.size ≤ 4294967295 ∧ ao+n ≤ 4294967295 ∧ bo+n ≤ 4294967295 then
      aDu A B (C3 ao) (C3 bo) (C3 (ao+n)) a1 b1 a2 b2 (aJs a2 b2 (aJs a1 b1 g)) 0 0 0 0
    else 4294967296
  if r < 4294967296 then N3 r else ad2x A B ao bo n a1 b1 a2 b2
theorem ad4_eq (g:ἆ) (A B:Ἂ Word) (ao bo n:Nat) (a1 b1 a2 b2:Word):
    ad4 g A B ao bo n a1 b1 a2 b2=dotOf (n+2) (fA A ao n a1 a2) (fA B bo n b1 b2):=by
  unfold ad4
  by_cases h:n+2 < 262144 ∧ A.size ≤ 4294967295 ∧ B.size ≤ 4294967295 ∧ ao+n ≤ 4294967295 ∧ bo+n ≤ 4294967295
  · rw [if_pos h]
    obtain ⟨h0,hA,hB,ha,hb⟩:=h
    have s:=ς
    have hA':A.size < kX:=by omega
    have hB':B.size < kX:=by omega
    have he:(C3 (ao+n)).toNat=ao+n:=J3 (by omega)
    have h1:(C3 ao).toNat=ao+0:=J3 (by omega)
    have h2:(C3 bo).toNat=bo+0:=J3 (by omega)
    rw [aDu_eq A B ao bo n a1 b1 a2 b2 _ _ hA' hB' he (by omega) n 0 _ _ 0 0 0 0 (by omega) (by omega) h1 h2]
    dsimp only
    split
    · rename_i hr; exact (aDok _ _ _ _ h0 hr).symm
    · exact ad2x_eq A B ao bo n a1 b1 a2 b2
  · rw [if_neg h]; exact ad2x_eq A B ao bo n a1 b1 a2 b2
/-! (S5 11-11) GRU candidate/update and LSTM cell dots: the double sum of the exact products
(`Dense.F.kE1`, term bound `Dense.F.kHG`), rounded through the window `Dense.F.ak` (`Dense.F.cw`);
the exact path when the window or a guard fails. -/
theorem gqT (a b:Word):|(Core.tm a b:ℝ)| ≤ 2^(48+idxOf a+idxOf b):=by
  have A:∀ w:Word,|(sv w:ℝ)| ≤ 2^24:=fun w=>(show |(sv w:ℝ)|=mantOf w by unfold sv; split_ifs <;> simp).trans_le
    (by exact_mod_cast (mantOf_lt_24 w).le)
  unfold Core.tm; push_cast
  calc |(sv a:ℝ)*sv b*2^(idxOf a+idxOf b)|=|(sv a:ℝ)| * |(sv b:ℝ)| * 2^(idxOf a+idxOf b):=by
        rw [Ae,Ae,AW (show (0:ℝ) < 2^(idxOf a+idxOf b) by positivity)]
    _ ≤ 2^24*2^24*2^(idxOf a+idxOf b):=by gcongr <;> exact A _
    _=2^(48+idxOf a+idxOf b):=by rw [← AF,← AF]; congr 1; omega
theorem gqI {w:Word} (h:idxOf w ≤ 253):expOf w ≠ 255:=by intro e; simp [idxOf,e] at h
theorem gqE {w:Word} (h:(expOf w).toNat ≤ 254):expOf w ≠ 255:=by intro e; simp [e] at h
theorem gqW (n:ℕ) (d e:ℕ → Word) (p:ℕ → Ἅ) (h0:Ἅ) (P L:ℕ)
    (K:∀ s < n,Dense.F.fv (p s)∧Dense.F.rv (p s)=Core.tm (d s) (e s))
    (q:Dense.F.fv h0∧Dense.F.rv h0=Core.tm (d n) (e n))
    (E:∀ s ≤ n,|(Core.tm (d s) (e s):ℝ)| ≤ 2^P) (N:n+2 ≤ 2^L) (D:L ≤ 27) (J:P+L ≤ 425)
    (g:Core.aG (n+1) d e)
    (c:Dense.F.ak (Dense.F.ao p n 0 h0) (lh (P+2*L-54)) < 4294967296) :
    N3 (Dense.F.ak (Dense.F.ao p n 0 h0) (lh (P+2*L-54)))=dotOf (n+1) d e:=by
  obtain ⟨f,r⟩:=Dense.F.kE1 n d e p h0 P L K q E N D J
  have hq:(lh (P+2*L-54)).toNat=P+2*L-54:=Α (by unfold n0; omega)
  have hL:(n:ℝ)+1 ≤ 2^L:=by exact_mod_cast (show n+1 ≤ 2^L by omega)
  have hA:|(Core.af (n+1) d e:ℝ)| ≤ (2:ℝ)^(425:ℤ):=by
    unfold Core.af; push_cast
    calc |∑ k ∈ kP (n+1),(Core.tm (d k) (e k):ℝ)| ≤ ∑ k ∈ kP (n+1),|(Core.tm (d k) (e k):ℝ)|:=
          K8 _ _
      _ ≤ ∑ k ∈ kP (n+1),(2:ℝ)^P:=Finset.sum_le_sum fun k hk=>E k (Nat.lt_succ_iff.1 (ϥ.1 hk))
      _=((n:ℝ)+1)*2^P:=by simp
      _ ≤ 2^L*2^P:=by gcongr
      _ ≤ (2:ℝ)^(425:ℤ):=by rw [← AF,← AA]; exact AG one_le_two (by omega)
  exact Dense.F.cw (n+1) d e g _ f _ (by rw [hq]; norm_num; omega) (by rw [hq]; exact r) hA c
theorem gqB (n:ℕ) (d e:ℕ → Word) (p:ℕ → Ἅ) (i E L:ℕ) (u v:Word) (hu:d (n+1)=u) (hv:e (n+1)=v)
    (hp:∀ s < n+1,p s=Dense.F.dw (d s)*Dense.F.dw (e s))
    (hb:∀ s < n+1,idxOf (d s) ≤ i∧(expOf (e s)).toNat ≤ E) (hi:i ≤ 253) (hE:E ≤ 254)
    (ha:expOf u ≠ 255) (hc:expOf v ≠ 255) (N:n+3 ≤ 2^L) (D:L ≤ 27)
    (J:max (48+i+max (E-1) 55) (48+idxOf u+idxOf v)+L ≤ 425)
    (c:Dense.F.ak (Dense.F.ao p (n+1) 0 (Dense.F.dw u*Dense.F.dw v))
      (lh (max (48+i+max (E-1) 55) (48+idxOf u+idxOf v)+2*L-54)) < 4294967296) :
    UInt64.toUInt32 (Dense.F.ak (Dense.F.ao p (n+1) 0 (Dense.F.dw u*Dense.F.dw v))
      (lh (max (48+i+max (E-1) 55) (48+idxOf u+idxOf v)+2*L-54)))=dotOf (n+2) d e:=by
  subst hu hv
  have hE':(lA E).toNat=E:=UInt32.toNat_ofNat_of_lt' (by unfold UInt32.size; omega)
  have K:=Dense.F.kHG (n+1) d e (lA E) i (48+idxOf (d (n+1))+idxOf (e (n+1)))
    (fun s hs=>(hb s hs).1) (fun s hs=>by rw [hE']; exact (hb s hs).2) (gqT _ _)
  rw [hE'] at K
  exact gqW (n+1) d e p _ _ L
    (fun s hs=>by rw [hp s hs]; exact Dense.F.dJ _ _ (gqI ((hb s hs).1.trans hi)) (gqE ((hb s hs).2.trans hE)))
    (Dense.F.dJ _ _ ha hc) K (by omega) D J
    (fun k hk=>by
      rcases Z6.1 hk with hk|hk
      · exact ⟨gqI ((hb k hk).1.trans hi),gqE ((hb k hk).2.trans hE)⟩
      · subst hk; exact ⟨ha,hc⟩) c
theorem gqL (p:ℕ → Ἅ) (n:ℕ):∀ k t a,n-t=k → t ≤ n → Dense.F.ao p (n+1) t a=Dense.F.ao p n t a+p n:=by
  intro k; induction k with
  | zero =>
    intro t a hk ht; obtain rfl:t=n:=by omega
    rw [Dense.F.aos _ _ _ _ (I3 t),Dense.F.aoe _ _ _ _ AP,Dense.F.aoe _ _ _ _ AP]
  | succ k ih =>
    intro t a hk ht; have h:t < n:=by omega
    rw [Dense.F.aos _ _ _ _ (by omega),Dense.F.aos _ _ _ _ h]; exact ih (t+1) _ (by omega) h
/-- Exact fallbacks kept out of line: inlined, their `dotOf` closures were built before the branch. -/
@[noinline] def gqY (A B:Ἂ Word) (ao bo n:Nat) (a1 b1 a2 b2:Word):Word:=ad2x A B ao bo n a1 b1 a2 b2
@[noinline] def gqX (g:ἆ) (A B:Ἂ Word) (ao bo n:Nat) (a1 b1 a2 b2:Word):Word:=ad4 g A B ao bo n a1 b1 a2 b2
@[inline] def gq1 (n x y:Nat):Nat:=if n=1 then max x y else y
@[inline] def gq2 (n:Nat) (h a b:Ἅ):Ἅ:=if n=1 then h+a*b else h
/-- The fast path for `n ≤ 1` array terms (LSTM cell, GRU update): two or three products. -/
@[inline] def ad2 (A B:Ἂ Word) (ao bo n:Nat) (a1 b1 a2 b2:Word):Word:=
  let a0:=if h:ao < A.size then A[ao] else 0
  let b0:=if h:bo < B.size then B[bo] else 0
  let i:=gq1 n (idxOf a0) (idxOf a1)
  let e:=gq1 n (expOf b0).toNat (expOf b1).toNat
  let P:=max (48+i+max (e-1) 55) (48+idxOf a2+idxOf b2)
  if n ≤ 1∧i ≤ 253∧e ≤ 254∧expOf a2 ≠ 255∧expOf b2 ≠ 255∧P+2 ≤ 425 then
    let o:=Dense.F.ak (gq2 n (Dense.F.dw a2*Dense.F.dw b2) (Dense.F.dw a0) (Dense.F.dw b0)+Dense.F.dw a1*Dense.F.dw b1)
      (lh (P+2*2-54))
    if o < 4294967296 then o.toUInt32 else gqY A B ao bo n a1 b1 a2 b2
  else gqY A B ao bo n a1 b1 a2 b2
theorem ad2_eq (A B:Ἂ Word) (ao bo n:Nat) (a1 b1 a2 b2:Word):
    ad2 A B ao bo n a1 b1 a2 b2=dotOf (n+2) (fA A ao n a1 a2) (fA B bo n b1 b2):=by
  have e0:(if h:ao < A.size then A[ao] else 0)=A[ao+0]!:=by
    by_cases h:ao < A.size
    · rw [dif_pos h]; exact (AB A ao h).symm
    · rw [dif_neg h]; exact (An A ao h).symm
  have e1:(if h:bo < B.size then B[bo] else 0)=B[bo+0]!:=by
    by_cases h:bo < B.size
    · rw [dif_pos h]; exact (AB B bo h).symm
    · rw [dif_neg h]; exact (An B bo h).symm
  unfold ad2; dsimp only; rw [e0,e1]
  split
  · rename_i hc; obtain ⟨c0,c1,c2,c3,c4,c5⟩:=hc
    split
    · rename_i ho
      let p:ℕ → Ἅ:=fun s=>if s < n then Dense.F.dw A[ao+s]!*Dense.F.dw B[bo+s]! else Dense.F.dw a1*Dense.F.dw b1
      have hS:gq2 n (Dense.F.dw a2*Dense.F.dw b2) (Dense.F.dw A[ao+0]!) (Dense.F.dw B[bo+0]!)
          +Dense.F.dw a1*Dense.F.dw b1=Dense.F.ao p (n+1) 0 (Dense.F.dw a2*Dense.F.dw b2):=by
        rcases (by omega:n=0∨n=1) with rfl|rfl
        · rw [Dense.F.aos _ _ _ _ (by omega),Dense.F.aoe _ _ _ _ AP]; simp [gq2,p]
        · rw [Dense.F.aos _ _ _ _ (by omega),Dense.F.aos _ _ _ _ (by omega),Dense.F.aoe _ _ _ _ AP]; simp [gq2,p]
      rw [hS] at ho ⊢
      refine gqB n _ _ p _ _ 2 a2 b2 (by simp [fA]) (by simp [fA]) (fun s hs=>?_) (fun s hs=>?_) c1 c2 c3 c4
        (by simp; omega) (by omega) c5 ho
      · rcases Z6.1 hs with hs|rfl
        · simp [p,fA,hs]
        · simp [p,fA]
      · rcases Z6.1 hs with hs|rfl
        · obtain rfl:n=1:=by omega
          obtain rfl:s=0:=by omega
          simp [fA,gq1]
        · simp only [fA,AX,if_false,if_true,gq1]; constructor <;> split <;> simp
    · exact ad2x_eq A B ao bo n a1 b1 a2 b2
  · exact ad2x_eq A B ao bo n a1 b1 a2 b2
theorem gqH {A:ἀ} {i:Ἄ} (h:i < A.usize):i.toNat < A.size:=by
  have:=ε.mp h
  rw [FloatArray.usize,lX,USize.toNat_ofNat'] at this
  exact U1 this (Ay _ _)
@[inline] def gqF (A:ἀ) (i:Ἄ):Ἅ:=if h:i < A.usize then A.uget i (gqH h) else 0
theorem gqV (A:ἀ) (i:Ἄ) (hA:A.size < kX) (h:i.toNat < A.size):gqF A i=A.data[i.toNat]!:=by
  have g:i < A.usize:=by
    rw [ε,FloatArray.usize,lX,USize.toNat_ofNat',γ hA]; exact h
  unfold gqF; rw [dif_pos g]
  rcases A with ⟨d⟩
  simp only [nN]
  exact (AB d i.toNat h).symm
def gqS (X W:ἀ) (i j e:Ἄ) (v:Ἅ):Ἅ:=
  if h:i < e then gqS X W (i+1) (j+1) e (v+gqF X i*gqF W j) else v
termination_by e.toNat-i.toNat
decreasing_by exact aUd h
theorem gqS_eq (X W:ἀ) (p:ℕ → Ἅ) (xo wo n:ℕ) (e:Ἄ) (hX:X.size < kX)
    (hW:W.size < kX) (he:e.toNat=xo+n) (hx:xo+n ≤ X.size) (hw:wo+n ≤ W.size)
    (hp:∀ s < n,p s=X.data[xo+s]!*W.data[wo+s]!) :
    ∀ k t (i j:Ἄ) v,n-t=k → t ≤ n → i.toNat=xo+t → j.toNat=wo+t →
    gqS X W i j e v=Dense.F.ao p n t v:=by
  intro k; induction k with
  | zero =>
    intro t i j v hk ht h1 h2; obtain rfl:t=n:=by omega
    have h0:¬ i < e:=by rw [ε,h1,he]; omega
    rw [gqS,dif_neg h0,Dense.F.aoe _ _ _ _ AP]
  | succ k ih =>
    intro t i j v hk ht h1 h2
    have htn:t < n:=by omega
    have h0:i < e:=by rw [ε,h1,he]; omega
    rw [gqS,dif_pos h0,Dense.F.aos _ _ _ _ htn,gqV X i hX (by omega),gqV W j hW (by omega),h1,h2,← hp t htn]
    exact ih (t+1) (i+1) (j+1) _ (by omega) (by omega) (by rw [aUs (by omega),h1]; rfl) (by rw [aUs (by omega),h2]; rfl)
/-- Words decoded as doubles (`Dense.F.dw`), once per call. -/
def gqD (A:Ἂ Word):ἀ:=A.foldl (fun r w=>r.push (Dense.F.dw w)) (kJ A.size)
theorem gqD_s (A:Ἂ Word):(gqD A).size=A.size∧∀ q < A.size,(gqD A).data[q]! =Dense.F.dw A[q]!:=by
  unfold gqD
  apply Array.foldl_induction (motive:=fun i (r:ἀ)=>r.size=i∧∀ q < i,r.data[q]! =Dense.F.dw A[q]!)
  · exact ⟨rfl,fun q h=>absurd h (U5 _)⟩
  · intro i r ⟨h1,h2⟩
    refine ⟨by rw [Dense.F.bW,h1],fun q hq=>?_⟩
    rw [Dense.F.s5push_get]
    rcases Z6.1 hq with hq|rfl
    · rw [if_pos (h1 ▸ hq)]; exact h2 _ hq
    · rw [if_neg (by omega),if_pos h1.symm,AB A _ i.2]; rfl
def gqM (A:Ἂ Word):Nat:=A.foldl (fun m w=>max m (idxOf w)) 0
theorem gqM_le (A:Ἂ Word) (q:Nat):idxOf A[q]! ≤ gqM A:=by
  by_cases hq:q < A.size
  · exact Array.foldl_induction (as:=A) (motive:=fun i (m:Nat)=>∀ q < i,idxOf A[q]! ≤ m)
      (fun q h=>absurd h (U5 _))
      (fun i m h q hq=>by
        rcases Z6.1 hq with hq|rfl
        · exact (h q hq).trans (le_max_left _ _)
        · rw [AB A _ i.2]; exact le_max_right _ _) q hq
  · rw [An A q hq,show idxOf (default:Word)=0 by decide]; exact ϟ _
def gqN (A:Ἂ Word):Nat:=A.foldl (fun m w=>max m (expOf w).toNat) 0
theorem gqN_le (A:Ἂ Word) (q:Nat):(expOf A[q]!).toNat ≤ gqN A:=by
  by_cases hq:q < A.size
  · exact Array.foldl_induction (as:=A) (motive:=fun i (m:Nat)=>∀ q < i,(expOf A[q]!).toNat ≤ m)
      (fun q h=>absurd h (U5 _))
      (fun i m h q hq=>by
        rcases Z6.1 hq with hq|rfl
        · exact (h q hq).trans (le_max_left _ _)
        · rw [AB A _ i.2]; exact le_max_right _ _) q hq
  · rw [An A q hq,show (expOf (default:Word)).toNat=0 by decide]; exact ϟ _
/-- The GRU candidate dot: `A`,`B` are `x`,`wih`, decoded once per call as `Xd`,`Wd`. -/
@[inline] def gqC (Xd Wd:ἀ) (ix E:Nat) (g:ἆ) (A B:Ἂ Word) (ao bo n:Nat) (a1 b1 a2 b2:Word):Word:=
  let i:=max ix (idxOf a1)
  let e:=max E (expOf b1).toNat
  let L:=Aq (n+3)+1
  let P:=max (48+i+max (e-1) 55) (48+idxOf a2+idxOf b2)
  if i ≤ 253∧e ≤ 254∧expOf a2 ≠ 255∧expOf b2 ≠ 255∧P+L ≤ 425∧L ≤ 27∧ao+n ≤ Xd.size∧bo+n ≤ Wd.size∧
      Xd.size ≤ 4294967295∧Wd.size ≤ 4294967295 then
    let o:=Dense.F.ak (gqS Xd Wd (C3 ao) (C3 bo) (C3 (ao+n)) (Dense.F.dw a2*Dense.F.dw b2)
      +Dense.F.dw a1*Dense.F.dw b1) (lh (P+2*L-54))
    if o < 4294967296 then o.toUInt32 else gqX g A B ao bo n a1 b1 a2 b2
  else gqX g A B ao bo n a1 b1 a2 b2
theorem gqC_eq (x w:Ἂ Word) (ix E:Nat) (hi:∀ q:ℕ,idxOf x[q]! ≤ ix) (he:∀ q:ℕ,(expOf w[q]!).toNat ≤ E) (g:ἆ)
    (ao bo n:Nat) (a1 b1 a2 b2:Word) :
    gqC (gqD x) (gqD w) ix E g x w ao bo n a1 b1 a2 b2=dotOf (n+2) (fA x ao n a1 a2) (fA w bo n b1 b2):=by
  obtain ⟨sx,vx⟩:=gqD_s x; obtain ⟨sw,vw⟩:=gqD_s w
  unfold gqC; dsimp only
  split
  · rename_i hc
    obtain ⟨c1,c2,c3,c4,c5,c6,c7,c8,c9,c10⟩:=hc
    split
    · rename_i ho
      have s:=ς
      let p:ℕ → Ἅ:=fun s=>if s < n then (gqD x).data[ao+s]!*(gqD w).data[bo+s]! else Dense.F.dw a1*Dense.F.dw b1
      have hS:gqS (gqD x) (gqD w) (C3 ao) (C3 bo) (C3 (ao+n)) (Dense.F.dw a2*Dense.F.dw b2)
          +Dense.F.dw a1*Dense.F.dw b1=Dense.F.ao p (n+1) 0 (Dense.F.dw a2*Dense.F.dw b2):=by
        have h1:(C3 ao).toNat=ao+0:=J3 (by omega)
        have h2:(C3 bo).toNat=bo+0:=J3 (by omega)
        rw [gqL p n n 0 _ (by omega) (ϟ _),
          gqS_eq (gqD x) (gqD w) p ao bo n _ (by omega) (by omega) (J3 (by omega)) c7 c8
            (fun s hs=>if_pos hs) n 0 _ _ _ (by omega) (ϟ _) h1 h2]
        simp [p]
      rw [hS] at ho ⊢
      refine gqB n _ _ p _ _ _ a2 b2 (by simp [fA]) (by simp [fA]) (fun s hs=>?_) (fun s hs=>?_) c1 c2 c3 c4
        (ϙ (n:=n+3)).le c6 c5 ho
      · rcases Z6.1 hs with hs|rfl
        · simp only [p,fA,if_pos hs]; rw [vx _ (by omega),vw _ (by omega)]
        · simp [p,fA]
      · rcases Z6.1 hs with hs|rfl
        · simp only [fA,if_pos hs]
          exact ⟨(hi _).trans (le_max_left _ _),(he _).trans (le_max_left _ _)⟩
        · simp only [fA,AX,if_false,if_true]
          exact ⟨le_max_right _ _,le_max_right _ _⟩
    · exact ad4_eq g x w ao bo n a1 b1 a2 b2
  · exact ad4_eq g x w ao bo n a1 b1 a2 b2
def gMn (A:Ἂ Word) (k e:Ἄ) (m:ἆ):ἆ:=
  if h:k < e then gMn A (k+1) e (let a:=aGt A k; if a &&& 0x7FFFFFFF==0 then m else min m (Core.iW a)) else m
termination_by e.toNat-k.toNat
decreasing_by exact aUd h
@[noinline] def gG (inW H:Nat) (x wih:Ἂ Word):ἆ:=
  gMn x 0 x.usize 511+gMn wih (C3 (2*H*inW)) (C3 (3*H*inW)) 511
abbrev AdFn:=Ἂ Word → Ἂ Word → Nat → Nat → Nat → Word → Word → Word → Word → Word
@[inline] def aKd {n:Nat} (f:Fin n → Word):Ἂ Word:=Dense.F.da (fun i=>if h:i < n then f ⟨i,h⟩ else 0) n (lQ n)
theorem aKd_eq {n:Nat} (f:Fin n → Word):aKd f=kK f:=by
  unfold aKd; rw [Dense.F.iC]; congr 1; funext q; exact dif_pos q.isLt
@[inline] def xh1 (inW H:Nat) (x h:Ἂ Word) (t:Nat):Ἂ Word :=
  aKd (n:=inW+H+1) fun j =>
    if j.val < inW then x[t*inW+j.val]! else if j.val < inW+H then h[j.val-inW]! else oneW
def wcat (G inW H:Nat) (wih whh bih:Ἂ Word):Ἂ Word :=
  Array.ofFn (n:=G*H*(inW+H+1)) fun q =>
    if q.val%(inW+H+1) < inW then wih[q.val/(inW+H+1)*inW+q.val%(inW+H+1)]!
    else if q.val%(inW+H+1) < inW+H then
      whh[q.val/(inW+H+1)*H+(q.val%(inW+H+1)-inW)]!
    else bih[q.val/(inW+H+1)]!
def wsub (H:Nat) (whh:Ἂ Word):Ἂ Word :=
  aKd (n:=H*H) fun q=>whh[2*H*H+q.val]!
def bsub (H:Nat) (bhh:Ἂ Word):Ἂ Word :=
  aKd (n:=H) fun q=>bhh[2*H+q.val]!
def oneA:Ἂ Word := #[oneW]
@[inline] def mulN (a b:Word):Word :=
  if (expOf a != 255 && expOf b != 255) = true then (lx a * lx b).toBits
  else toWord (Model.mul (ofW a) (ofW b))
theorem mulN_eq (a b:Word):mulN a b=toWord (Model.mul (ofW a) (ofW b)):=by
  unfold mulN; split
  · rename_i h
    simp only [Η, AM, Ak] at h
    have g:=Bn.NativeAdd.bK a h.1
    have j:=Bn.NativeAdd.bK b h.2
    have e:=FloatLib.Floats.ExecFloat.Binary.toModel_ofFloat32_mul_of_isFinite (lx a) (lx b) g.2 j.2
    simp only [FloatLib.Floats.ExecFloat.Binary.toModel_ofFloat32, Bn.NativeAdd.en, g.1, j.1] at e
    have E:=congrArg toWord e
    rw [← Bn.NativeAdd.en, Bn.NativeAdd.ja] at E
    exact E
  · rfl
@[inline] def wcF (k:(Nat × Nat × Nat) × Ἂ Word × Ἂ Word × Ἂ Word):Ἂ Word :=
  wcat k.1.1 k.1.2.1 k.1.2.2 k.2.1 k.2.2.1 k.2.2.2
@[inline] def wcEq (a b:(Nat × Nat × Nat) × Ἂ Word × Ἂ Word × Ἂ Word):Ἆ :=
  a.1 == b.1 && Dense.F.wcAeq a.2.1 b.2.1 && Dense.F.wcAeq a.2.2.1 b.2.2.1 && Dense.F.wcAeq a.2.2.2 b.2.2.2
theorem wcEq_ok (a b:(Nat × Nat × Nat) × Ἂ Word × Ἂ Word × Ἂ Word) (h:wcEq a b=true):a=b:=by
  obtain ⟨a1,a2,a3,a4⟩:=a; obtain ⟨b1,b2,b3,b4⟩:=b
  simp only [wcEq, Η, AQ] at h
  obtain ⟨⟨⟨h1,h2⟩,h3⟩,h4⟩:=h
  rw [h1, Dense.F.wcAeq_ok h2, Dense.F.wcAeq_ok h3, Dense.F.wcAeq_ok h4]
@[noinline] def wcS:ST.Ref Unit (Ἂ (Dense.F.WcE wcF)):=(ST.Prim.mkRef #[] (Void.mk ())).val
@[noinline] def wcG (G inW H:Nat) (wih whh bih:Ἂ Word):Ἂ Word :=
  (Dense.F.wcGet3 wcEq wcEq_ok wcS 32 ((G,inW,H),wih,whh,bih) (fun k=>k.2.1)).1
theorem wcG_v (G inW H:Nat) (wih whh bih:Ἂ Word):wcG G inW H wih whh bih=wcat G inW H wih whh bih:=
  Dense.F.s5wcGet3_val _
@[inline] def lstmTail (ad:AdFn) (sig tanh:Word → Word) (H:Nat) (pre cs:Ἂ Word) :
    Ἂ Word × Ἂ Word :=
  let cell:=aKd (n:=H) fun u =>
    ad #[] #[] 0 0 0 (sig pre[H+u.val]!) (cs[u.val]!) (sig pre[u.val]!) (tanh pre[2*H+u.val]!)
  let hidden:=aKd (n:=H) fun u =>
    mulN (sig pre[3*H+u.val]!) (tanh (cell[u.val]!))
  (hidden,cell)
@[inline] def lstmStep (lin:LinFn) (ad:AdFn) (sig tanh:Word → Word) (inW H:Nat)
    (x Wc bhh:Ἂ Word) (t:Nat) (st:Ἂ Word × Ἂ Word):Ἂ Word × Ἂ Word :=
  lstmTail ad sig tanh H (lin 1 (inW+H+1) (4*H) false (xh1 inW H x st.1 t) Wc bhh) st.2
@[specialize] def lstmK (lin:LinFn) (ad:AdFn) (sig tanh:Word → Word) (T inW H:Nat)
    (x wih whh bih bhh:Ἂ Word):Ἂ Word :=
  recurWith T H (lstmStep lin ad sig tanh inW H x (wcG 4 inW H wih whh bih) bhh)
    (mB H 0,mB H 0)
@[inline] def gruTail (ad adc:AdFn) (tanh:Word → Word) (inW H:Nat)
    (x wih bih:Ἂ Word) (t:Nat) (h r z hn:Ἂ Word):Ἂ Word :=
  let n:=aKd (n:=H) fun u=>tanh <|
    adc x wih (t*inW) ((2*H+u.val)*inW) inW oneW bih[2*H+u.val]! hn[u.val]! r[u.val]!
  aKd (n:=H) fun u=>ad n oneA u.val 0 1 n[u.val]! (negW z[u.val]!) h[u.val]! z[u.val]!
@[inline] def gruStep (lin:LinFn) (ad adc:AdFn) (sig tanh:Word → Word) (inW H:Nat)
    (x wih bih bhh Wrz Whn bhn:Ἂ Word) (t:Nat) (st:Ἂ Word × Ἂ Word) :
    Ἂ Word × Ἂ Word :=
  let a:=lin 1 (inW+H+1) (2*H) false (xh1 inW H x st.1 t) Wrz bhh
  (gruTail ad adc tanh inW H x wih bih t st.1
    (aKd (n:=H) fun u=>sig a[u.val]!)
    (aKd (n:=H) fun u=>sig a[H+u.val]!)
    (lin 1 H H false st.1 Whn bhn),st.2)
@[specialize] def gruK (lin:LinFn) (ad:AdFn) (sig tanh:Word → Word) (T inW H:Nat)
    (x wih whh bih bhh:Ἂ Word):Ἂ Word :=
  let g:=gG inW H x wih
  recurWith T H (gruStep lin ad (gqC (gqD x) (gqD wih) (gqM x) (gqN wih) g) sig tanh inW H x wih bih bhh (wcG 2 inW H wih whh bih)
    (wsub H whh) (bsub H bhh)) (mB H 0, #[])
theorem gofn {n:Nat} (f:Fin n → Word) (i:Nat) (h:i < n) :
    (kK f)[i]! = f ⟨i,h⟩:=by
  rw [AB (kK f) i (by simpa using h)]
  simp
theorem dotOf_congr {n m:Nat} (hnm:n=m) {fa fb ga gb:Nat → Word}
    (ha : ∀ j,j < n → fa j=ga j) (hb : ∀ j,j < n → fb j=gb j) :
    dotOf n fa fb=dotOf m ga gb:=by
  subst hnm
  unfold dotOf
  have e1:(kK (n:=n) fun i=>fa i.val)=kK (n:=n) fun i=>ga i.val :=
    congrArg kK (funext fun i=>ha i.val i.isLt)
  have e2:(kK (n:=n) fun i=>fb i.val)=kK (n:=n) fun i=>gb i.val :=
    congrArg kK (funext fun i=>hb i.val i.isLt)
  rw [e1,e2]
theorem pre_eq (G inW H:Nat) (x wih whh bih bhh h:Ἂ Word) (t:Nat) :
    linearSpec 1 (inW+H+1) (G*H) false (xh1 inW H x h t) (wcat G inW H wih whh bih) bhh =
    kK (n:=G*H) fun k=>gateWith dotOf inW H x wih whh bih bhh h t k.val:=by
  apply mi
  · simp [linearSpec,linearWith]
  · intro i h1 h2
    have hi:i < G*H:=by simpa using h2
    simp only [linearSpec,linearWith,Λ,ώ hi,
      γ hi,ϳ,Κ,act]
    simp only [υ,if_false]
    unfold gateWith
    apply dotOf_congr (by omega)
    · intro j hj
      by_cases hj1:j < inW+H+1
      · rw [if_pos hj1,xh1,aKd_eq,gofn _ _ hj1]
      · rw [if_neg hj1]
        have:¬ j < inW:=by omega
        have:¬ j < inW+H:=by omega
        simp only [*, if_false]
    · intro j hj
      by_cases hj1:j < inW+H+1
      · rw [if_pos hj1,wcat,gofn _ _ (by
            have:i*(inW+H+1)+j < G*H*(inW+H+1):=by
              calc i*(inW+H+1)+j < i*(inW+H+1)+(inW+H+1):=by omega
                _=(i+1)*(inW+H+1):=by ring
                _ ≤ G*H*(inW+H+1):=ι _ hi
            exact this)]
        have hp:0 < inW+H+1:=by omega
        have e1:(i*(inW+H+1)+j)/(inW+H+1)=i:=by
          rw [Ϗ,W8 hp,ώ hj1,ϑ]
        have e2:(i*(inW+H+1)+j)%(inW+H+1)=j:=by
          rw [Ϗ,O7,γ hj1]
        simp only [e1,e2]
        by_cases a1:j < inW
        · simp only [a1,if_true]
        · by_cases a2:j < inW+H
          · simp only [a1,a2,if_true,if_false]
          · have:j=inW+H:=by omega
            simp only [a1,a2,this,if_true,if_false]
      · rw [if_neg hj1]
        have:¬ j < inW:=by omega
        have:¬ j < inW+H:=by omega
        have:j≠inW+H:=by omega
        simp only [*, if_false]
theorem hn_eq (H:Nat) (whh bhh h:Ἂ Word) :
    linearSpec 1 H H false h (wsub H whh) (bsub H bhh) =
    Array.ofFn (n:=H) fun u=>dotOf (H+1) (fun j=>if j < H then h[j]! else oneW)
      (fun j=>if j < H then whh[(2*H+u.val)*H+j]! else bhh[2*H+u.val]!):=by
  apply mi
  · simp [linearSpec,linearWith]
  · intro i h1 h2
    have hi:i < H:=by simpa using h2
    simp only [linearSpec,linearWith,Λ,ώ hi,
      γ hi,ϳ,Κ,act]
    simp only [υ,if_false]
    apply dotOf_congr rfl
    · intro j hj; rfl
    · intro j hj
      by_cases hj1:j < H
      · have hq:i*H+j < H*H:=by
          calc i*H+j < i*H+H:=by omega
            _=(i+1)*H:=by ring
            _ ≤ H*H:=ι _ hi
        rw [if_pos hj1,if_pos hj1,wsub,aKd_eq,gofn _ _ hq]
        congr 1; ring
      · rw [if_neg hj1,if_neg hj1,bsub,aKd_eq,gofn _ _ hi]
theorem gqcand (adc:AdFn) (x wih:Ἂ Word)
    (hadc:∀ ao bo n a1 b1 a2 b2,adc x wih ao bo n a1 b1 a2 b2=dotOf (n+2) (fA x ao n a1 a2) (fA wih bo n b1 b2))
    (tanh:Word → Word) (inW H:Nat) (bih:Ἂ Word) (t:Nat) (R HN:Ἂ Word) :
    (Array.ofFn (n:=H) fun u=>tanh <|
      adc x wih (t*inW) ((2*H+u.val)*inW) inW oneW bih[2*H+u.val]! HN[u.val]! R[u.val]!) =
    Array.ofFn (n:=H) fun u=>tanh <| dotOf (inW+2)
      (fun j=>if j < inW then x[t*inW+j]! else if j=inW then oneW else HN[u.val]!)
      (fun j=>if j < inW then wih[(2*H+u.val)*inW+j]!
        else if j=inW then bih[2*H+u.val]! else R[u.val]!):=by
  congr 1; funext u; rw [hadc]; rfl
section
variable (ad:AdFn) (had : ∀ A B ao bo n a1 b1 a2 b2,
  ad A B ao bo n a1 b1 a2 b2=dotOf (n+2) (fA A ao n a1 a2) (fA B bo n b1 b2))
include had
theorem cell_eq (sig tanh:Word → Word) (H:Nat) (P cs:Ἂ Word) :
    (Array.ofFn (n:=H) fun u =>
      ad #[] #[] 0 0 0 (sig P[H+u.val]!) (cs[u.val]!) (sig P[u.val]!) (tanh P[2*H+u.val]!)) =
    Array.ofFn (n:=H) fun u =>
      dotOf 2 (fun j=>if j=0 then sig P[H+u.val]! else sig P[u.val]!)
        (fun j=>if j=0 then cs[u.val]! else tanh P[2*H+u.val]!):=by
  congr 1; funext u; rw [had]
  apply dotOf_congr rfl <;> intro j hj <;> simp [fA]
theorem lstmStep_eq (sig tanh:Word → Word) (inW H:Nat) (x wih whh bih bhh:Ἂ Word)
    (t:Nat) (st:Ἂ Word × Ἂ Word) :
    lstmStep linearSpec ad sig tanh inW H x (wcat 4 inW H wih whh bih) bhh t st =
    lstmStepWith dotOf sig tanh inW H x wih whh bih bhh t st:=by
  unfold lstmStep lstmTail
  simp -zeta only [aKd_eq]
  rw [pre_eq,cell_eq ad had]
  simp only [mulN_eq]
  rfl
theorem upd_eq (H:Nat) (N Z h:Ἂ Word) :
    (kK (n:=H) fun u=>ad N oneA u.val 0 1 N[u.val]! (negW Z[u.val]!) h[u.val]! Z[u.val]!) =
    Array.ofFn (n:=H) fun u=>dotOf 3 (fun j=>if j=2 then h[u.val]! else N[u.val]!)
      (fun j=>if j=0 then oneW else if j=1 then negW Z[u.val]! else Z[u.val]!):=by
  congr 1; funext u; rw [had]
  apply dotOf_congr rfl
  · intro j hj
    match j,hj with
    | 0,_=>simp [fA]
    | 1,_=>simp [fA]
    | 2,_=>simp [fA]
  · intro j hj
    match j,hj with
    | 0,_=>simp [fA,oneA]
    | 1,_=>simp [fA]
    | 2,_=>simp [fA]
theorem gruStep_eq (adc:AdFn) (sig tanh:Word → Word) (inW H:Nat) (x wih whh bih bhh:Ἂ Word)
    (hadc:∀ ao bo n a1 b1 a2 b2,adc x wih ao bo n a1 b1 a2 b2=dotOf (n+2) (fA x ao n a1 a2) (fA wih bo n b1 b2))
    (t:Nat) (st:Ἂ Word × Ἂ Word) :
    gruStep linearSpec ad adc sig tanh inW H x wih bih bhh (wcat 2 inW H wih whh bih)
      (wsub H whh) (bsub H bhh) t st=gruStepWith dotOf sig tanh inW H x wih whh bih bhh t st:=by
  unfold gruStep gruTail
  simp -zeta only [aKd_eq]
  rw [pre_eq,hn_eq]
  have hr:(kK (n:=H) fun u =>
      sig (kK (n:=2*H) fun k=>gateWith dotOf inW H x wih whh bih bhh st.1 t k.val)[u.val]!)
      = kK (n:=H) fun u=>sig (gateWith dotOf inW H x wih whh bih bhh st.1 t u.val):=by
    congr 1; funext u; rw [gofn _ _ (by omega)]
  have hz:(kK (n:=H) fun u =>
      sig (kK (n:=2*H) fun k=>gateWith dotOf inW H x wih whh bih bhh st.1 t k.val)[H+u.val]!)
      = kK (n:=H) fun u=>sig (gateWith dotOf inW H x wih whh bih bhh st.1 t (H+u.val)):=by
    congr 1; funext u; rw [gofn _ _ (by omega)]
  simp only [hr,hz,gqcand adc x wih hadc,upd_eq ad had]
  rfl
theorem lstmK_eq (lin:LinFn) (hl:lin=linearSpec) (sig tanh:Word → Word) :
    lstmK lin ad sig tanh=lstmWith dotOf sig tanh:=by
  subst hl
  funext T inW H x wih whh bih bhh
  unfold lstmK lstmWith
  rw [wcG_v]
  have hs:lstmStep linearSpec ad sig tanh inW H x (wcat 4 inW H wih whh bih) bhh =
      lstmStepWith dotOf sig tanh inW H x wih whh bih bhh:=by
    funext t st; exact lstmStep_eq ad had sig tanh inW H x wih whh bih bhh t st
  rw [hs]
theorem gruK_eq (lin:LinFn) (hl:lin=linearSpec) (sig tanh:Word → Word) :
    gruK lin ad sig tanh=gruWith dotOf sig tanh:=by
  subst hl
  funext T inW H x wih whh bih bhh
  unfold gruK gruWith
  dsimp only
  rw [wcG_v]
  have hs:gruStep linearSpec ad (gqC (gqD x) (gqD wih) (gqM x) (gqN wih) (gG inW H x wih)) sig tanh inW H x wih bih bhh
      (wcat 2 inW H wih whh bih) (wsub H whh) (bsub H bhh)=gruStepWith dotOf sig tanh inW H x wih whh bih bhh:=by
    funext t st; exact gruStep_eq ad had _ sig tanh inW H x wih whh bih bhh
      (gqC_eq x wih _ _ (gqM_le x) (gqN_le wih) _) t st
  rw [hs]
end
theorem lstmK_spec (sig tanh:Word → Word):lstmK linearSpec ad2 sig tanh=lstmWith dotOf sig tanh :=
  lstmK_eq ad2 ad2_eq _ rfl sig tanh
theorem gruK_spec (sig tanh:Word → Word):gruK linearSpec ad2 sig tanh=gruWith dotOf sig tanh :=
  gruK_eq ad2 ad2_eq _ rfl sig tanh
end Rnn
namespace NX
open ExactRuntime
theorem pkF1 (q:Nat) (hq:q < 2^26) :
    ((2*q+1)*2^36+274877906943+(2*q+1)*2^36/2^39%2)/2^39=(q+2)/4:=by
  have h4:=ϛ q 4
  generalize q/4=a at h4
  have hr:=ϊ q (by decide:4 > 0)
  generalize q%4=r at h4 hr
  subst h4
  interval_cases r <;> simp only [nr] <;> omega
theorem pkF2 (X:Nat):(X/2^27+2)/4=(X+2^28)/2^29:=by
  rw [show (2:Nat)^29=2^27*4 by norm_num, ← lY,
    show X+2^28=X+2*2^27 by norm_num,Ρ _ _ (by positivity)]
theorem pkF3 (b:Nat):(b%2^63+2^28)/2^29=(b/2^52%2^11)*2^23+(b%2^52+2^28)/2^29:=by
  rw [show (2:Nat)^63=2^52*2^11 by norm_num,Nat.mod_mul,show b%2^52+2^52*(b/2^52%2^11)+2^28
    = (b%2^52+2^28)+(b/2^52%2^11)*2^23*2^29 by ring,Ρ _ _ (by positivity)]
  ring
@[inline] def pk2 (B:ἆ):Word :=
  (((((B<<<1)>>>1)+0x10000000)>>>29)-((1045:ἆ)<<<23)).toUInt32|||((B>>>63).toUInt32<<<31)
theorem pk2_nW (B:ἆ) (h1:1047 ≤ ((B>>>52)&&&2047).toNat) (h2:((B>>>52)&&&2047).toNat ≤ 1298) :
    pk2 B=Norm.nW B:=by
  unfold pk2 Norm.nW Norm.NE.pk Core.ar Dense.F.eF
  apply Β.1
  have hm:=Norm.NF.kc B
  have hb:=B.toNat_lt
  generalize Dense.F.mF B=M at *
  generalize hbB:B.toNat=b at *
  have e2047:(2047:Nat)=2^11-1:=rfl
  have e52:(4503599627370495:Nat)=2^52-1:=rfl
  have hE:((B>>>52)&&&2047).toNat=b/2^52%2^11:=by
    simp only [Ε,ν,kA,hbB,β,
      kH,e2047,ζ]
  rw [hE] at h1 h2
  have hM:M.toNat=b%2^52+2^52:=by
    rw [hm]; simp only [Ε,kA,hbB,e52,ζ]
  have hq:(M>>>27).toNat=(b%2^52+2^52)/2^27:=by
    simp only [ν,kA,kH,β,hM]
  have hq1:(b%2^52+2^52)/2^27 < 2^26:=by omega
  have hN:(((2:ἆ)*(M>>>27)+1)<<<36).toNat=(2*((b%2^52+2^52)/2^27)+1)*2^36:=by
    simp only [ψ,η,ύ,kA,kH,hq,
      ο]
    omega
  generalize ((2:ἆ)*(M>>>27)+1)<<<36=N at *
  generalize hQ:(b%2^52+2^52)/2^27=q at *
  have hA:((N+(274877906943:ἆ)+((N>>>39)&&&1))>>>39).toNat=(N.toNat+274877906943+N.toNat/2^39%2)/2^39:=by
    simp only [ν,η,Ε,kA,kH,
      β,mP]
    rw [hN]; omega
  have k1:(900:ἆ) ≤ (B>>>52&&&2047):=by
    rw [θ,hE]; simp only [kA]; omega
  have k2:(146:ἆ) ≤ (B>>>52&&&2047)-900:=by
    rw [θ,χ _ _ k1,hE]; simp only [kA]; omega
  have hS2:((B>>>52&&&2047)-900-146).toNat=b/2^52%2^11-1046:=by
    rw [χ _ _ k2,χ _ _ k1,hE]; simp only [kA]; omega
  generalize (B>>>52&&&2047)-900-146=sg at *
  have hL1:(((B<<<1)>>>1)+268435456).toNat=b%2^63+2^28:=by
    simp only [η,ν,ψ,kA,kH,
      ο,β,hbB]
    omega
  have k3:((1045:ἆ)<<<23) ≤ (((B<<<1)>>>1)+268435456)>>>29:=by
    rw [θ,ν,hL1]
    simp only [ψ,kA,kH,ο,β]
    omega
  have hL:((((B<<<1)>>>1)+268435456)>>>29-(1045:ἆ)<<<23).toNat=(b%2^63+2^28)/2^29-1045*2^23:=by
    rw [χ _ _ k3,ν,hL1]
    simp only [ψ,kA,kH,ο,β]
    omega
  have hS:((B>>>63).toUInt32<<<(31:ἇ)).toNat=b/2^63*2^31:=by
    simp only [ϴ,Ϫ,ν,kN,kA,
      kH,ο,β,hbB]
    omega
  have hif:(if (B>>>63 != 0)=true then (2147483648:ἆ) else 0).toNat=b/2^63*2^31:=by
    by_cases hs:b < 2^63
    · have e:B>>>63=0:=by
        apply ξ.1
        simp only [ν,kA,kH,β,hbB]
        omega
      rw [e]; simp only [bne_self_eq_false,υ, ↓reduceIte,kA]; omega
    · have e:B>>>63≠0:=by
        intro h; have h':=congrArg lq h
        simp only [ν,kA,kH,β,hbB] at h'
        omega
      simp only [AM,Ak,e,not_false_eq_true, ↓reduceIte,kA]; omega
  have hor : ∀ a c:Nat,c < 2^31 → c|||a*2^31=c+a*2^31:=by
    intro a c hc; rw [Nat.lor_comm,Ϗ, ← Ϥ hc]; omega
  rw [nK,Ϫ,hL,hS,Ϫ,η,η,hif,
    ψ,hS2,hA]
  simp only [kA,kH,ο]
  rw [hN,pkF1 q hq1, ← hQ,pkF2,pkF3 b]
  clear hm hM hq hN hA hL hS hif hE hS2 k1 k2 k3 hL1 hQ hq1 e2047 e52 hbB
  have hY:(b%2^52+2^52+2^28)/2^29=2^23+(b%2^52+2^28)/2^29:=by
    rw [show b%2^52+2^52+2^28=(b%2^52+2^28)+2^23*2^29 by ring,Ρ _ _ (by positivity)]; ring
  rw [hY]
  have hYb:(b%2^52+2^28)/2^29 ≤ 2^23:=J2 (Ϟ (by have:=ϊ b (by positivity:2^52 > 0); norm_num at this ⊢; omega))
  generalize (b%2^52+2^28)/2^29=Y at *
  have hsb:b/2^63 ≤ 1:=J2 (Ϟ (by norm_num at hb ⊢; omega))
  generalize b/2^63=sb at *
  generalize b/2^52%2^11=E at *
  clear hb
  simp only [nr] at *
  rw [hor]
  · omega
  · omega
@[inline] def okF (B T:ἆ):Ἆ :=
  ((B<<<1)>>>53)-T ≤ 1298-T&&(B&&&0x7FFFFFF)-33 < 0x8000000-65
theorem okF_nOK (B T:ἆ) (h1:1047 ≤ T.toNat) (h2:T.toNat ≤ 1298) (h:okF B T=true) :
    Norm.nOK B 5=true∧T.toNat ≤ (Dense.F.eF B).toNat:=by
  unfold okF at h
  unfold Norm.nOK Dense.F.eF
  simp only [Η,AJ] at h ⊢
  obtain ⟨hE,hL⟩:=h
  have hb:=B.toNat_lt
  have eE:((B<<<1)>>>53).toNat=B.toNat/2^52%2048:=by
    simp only [ν,ψ,kA,kH,ο,
      β]
    omega
  have eE2:((B>>>52)&&&2047).toNat=B.toNat/2^52%2048:=by
    simp only [Ε,ν,kA,kH,β,
      show (2047:Nat)=2^11-1 from rfl,ζ]
  have eL:(B&&&0x7FFFFFF).toNat=B.toNat%2^27:=by
    simp only [Ε,kA,show (134217727:Nat)=2^27-1 from rfl,ζ]
  rw [θ] at hE
  rw [μ] at hL
  simp only [Ζ,kA,eE,eL] at hE hL
  simp only [θ,μ,η,ψ,kA,
    kH,ο,eE2,eL]
  simp only [nr] at *
  omega
theorem fastEl (y:ℚ) (v:Ἅ) (T:ἆ) (h1:1047 ≤ T.toNat) (h2:T.toNat ≤ 1298) (hv:Dense.F.fv v)
    (hok:okF v.toBits T=true)
    (he : |(y:ℝ)*2^(149:ℕ)-Dense.F.rv v| < 2^(4:ℤ)*2^(((Dense.F.eF v.toBits).toNat:ℤ)-1075)) :
    pk2 v.toBits=C7 y:=by
  obtain ⟨hk,hT⟩:=okF_nOK _ _ h1 h2 hok
  have hE:1047 ≤ ((v.toBits>>>52)&&&2047).toNat∧((v.toBits>>>52)&&&2047).toNat ≤ 1298:=by
    have:=Norm.nOK_dec _ _ hk; exact ⟨this.1,this.2.1⟩
  rw [pk2_nW _ hE.1 hE.2]
  exact Norm.nOK_spec y v 5 hv hk (by simpa using he)
theorem relMul (a b:Ἅ) (ha:Dense.F.fv a) (hb:Dense.F.fv b)
    (hup : |Dense.F.rv a*Dense.F.rv b| ≤ (2:ℝ)^(971:ℤ)) (hlo:(1:ℝ)/2^(300:ℕ) ≤ |Dense.F.rv a*Dense.F.rv b|) :
    Dense.F.fv (a*b)∧SAct.Rel (Dense.F.rv (a*b)) (Dense.F.rv a*Dense.F.rv b) (1/2^(52:ℕ)):=by
  obtain ⟨h1,h2⟩:=Norm.NF.fM a b ha hb hup
  refine ⟨h1,?_⟩
  unfold SAct.Rel
  have k:(1:ℝ)/2^(1075:ℕ) ≤ |Dense.F.rv a*Dense.F.rv b| / 2^(53:ℕ):=by
    rw [le_div_iff₀ (by positivity)]
    calc (1:ℝ)/2^(1075:ℕ)*2^(53:ℕ)=1/2^(1022:ℕ):=by rw [div_mul_eq_mul_div,one_mul,show (2:ℝ)^(1075:ℕ)=2^(1022:ℕ)*2^(53:ℕ) by rw [← AF]]; field_simp
      _ ≤ 1/2^(300:ℕ):=SAct.inv2le (by norm_num)
      _ ≤ _:=hlo
  have : |Dense.F.rv a*Dense.F.rv b| / 2^(53:ℕ) + |Dense.F.rv a*Dense.F.rv b| / 2^(53:ℕ)=1/2^(52:ℕ) * |Dense.F.rv a*Dense.F.rv b| := by
    rw [show (2:ℝ)^(53:ℕ)=2*2^(52:ℕ) by rw [A4]; ring]; field_simp; ring
  linarith
theorem rvEq (x:Ἅ):SAct.rv x=Dense.F.rv x:=rfl
theorem fvEq (x:Ἅ):SAct.fv x=Dense.F.fv x:=rfl
theorem relInv (D:Ἅ) (hD:Dense.F.fv D) (h0:0 < Dense.F.rv D)
    (hup:1/Dense.F.rv D ≤ 4) (hlo:(1:ℝ)/2^(300:ℕ) ≤ 1/Dense.F.rv D) :
    Dense.F.fv ((I0 1).toInt64.toFloat/D) ∧
      SAct.Rel (Dense.F.rv ((I0 1).toInt64.toFloat/D)) (1/Dense.F.rv D) (1/2^(52:ℕ)):=by
  obtain ⟨o1,o2⟩:=Dense.F.mV (I0 1) 1 rfl (by norm_num)
  have hdiv:=SAct.divF ((I0 1).toInt64.toFloat) D o1 hD (ne_of_gt h0)
    (by rw [rvEq,rvEq,o2]; push_cast; rw [AW (by positivity)]; exact hup)
  rw [rvEq,rvEq,rvEq,o2,fvEq] at hdiv; push_cast at hdiv
  refine ⟨hdiv.1,?_⟩
  unfold SAct.Rel
  rw [hdiv.2]
  have j:=Norm.NF.jy (1/Dense.F.rv D)
  have k:(1:ℝ)/2^(1075:ℕ) ≤ |1/Dense.F.rv D| / 2^(53:ℕ):=by
    rw [AW (by positivity),le_div_iff₀ (by positivity)]
    calc (1:ℝ)/2^(1075:ℕ)*2^(53:ℕ)=1/2^(1022:ℕ):=by rw [div_mul_eq_mul_div,one_mul,show (2:ℝ)^(1075:ℕ)=2^(1022:ℕ)*2^(53:ℕ) by rw [← AF]]; field_simp
      _ ≤ 1/2^(300:ℕ):=SAct.inv2le (by norm_num)
      _ ≤ _:=hlo
  have : |1/Dense.F.rv D| / 2^(53:ℕ) + |1/Dense.F.rv D| / 2^(53:ℕ)=1/2^(52:ℕ) * |1/Dense.F.rv D| := by
    rw [show (2:ℝ)^(53:ℕ)=2*2^(52:ℕ) by rw [A4]; ring]; field_simp; ring
  linarith
theorem dwZA (w:Word) (h:kB w≠255) :
    Dense.F.fv (Dense.F.dw w)∧Dense.F.rv (Dense.F.dw w)=(V3 w:ℝ)*2^(149:ℕ):=by
  obtain ⟨f,v⟩:=Dense.F.cI w h
  refine ⟨f,?_⟩
  rw [v,Norm.Y1 w]
  push_cast
  rw [Ap, ← AA (2:ℝ) 149, ← AK (by norm_num), ← AA]
  congr 2; ring
theorem mant_ne (w:Word) (h0:w<<<1≠0):kM w≠0:=by
  intro hm
  apply h0
  unfold kM at hm
  split_ifs at hm with he
  · apply Β.1
    have h1:(kB w).toNat=0:=by rw [he]; rfl
    unfold kB at h1; unfold lK at hm
    simp only [Ξ,Υ,ϴ,kN,kH,
      β,ο,show (255:Nat)=2^8-1 from rfl,show (8388607:Nat)=2^23-1 from rfl,
      ζ] at h1 hm ⊢
    have:=w.toNat_lt
    simp only [nr] at *
    omega
  · omega
theorem dwOne (w:Word) (h:kB w≠255) (h0:w<<<1≠0):1 ≤ |Dense.F.rv (Dense.F.dw w)| := by
  obtain ⟨-, v⟩:=Dense.F.cI w h
  rw [v,Ae,AW (by positivity:(0:ℝ) < 2^(kG w))]
  have hm:=mant_ne w h0
  have:(1:ℝ) ≤ |(L8 w:ℝ)| := by
    unfold L8; split_ifs <;> simp <;> exact_mod_cast Nat.one_le_iff_ne_zero.mpr hm
  calc (1:ℝ)=1*1:=by ring
    _ ≤ |(L8 w:ℝ)| * 2^(kG w):=mul_le_mul this (one_le_pow₀ (by norm_num)) (by norm_num) (by positivity)
theorem relSelf (a:ℝ):SAct.Rel a a 0:=by unfold SAct.Rel; simp
theorem relWin (v:Ἅ) (Y:ℝ) (hv:Dense.F.fv v) (h:SAct.Rel (Dense.F.rv v) Y (1/2^(50:ℕ))) :
    |Y-Dense.F.rv v| < 2^(4:ℤ)*2^(((Dense.F.eF v.toBits).toNat:ℤ)-1075):=by
  have hk:=Dense.F.fK v hv
  unfold SAct.Rel at h
  have hY : |Y| ≤ |Dense.F.rv v| * (2^(50:ℕ)/(2^(50:ℕ)-1)):=by
    have t:=AY Y (Dense.F.rv v)
    rw [Aw] at t
    rw [mul_div_assoc',le_div_iff₀ (by norm_num)]
    have : |Y| ≤ |Dense.F.rv v| + 1/2^(50:ℕ) * |Y| := by linarith
    nlinarith [Aa Y]
  rw [Aw]
  calc |Dense.F.rv v-Y| ≤ 1/2^(50:ℕ) * |Y| := h
    _ ≤ 1/2^(50:ℕ)*(|Dense.F.rv v| * (2^(50:ℕ)/(2^(50:ℕ)-1))):=by gcongr
    _ = |Dense.F.rv v| * (1/(2^(50:ℕ)-1)):=by
      rw [mul_comm (1/2^(50:ℕ):ℝ),Ap]; congr 1; field_simp
    _ < 2^(((Dense.F.eF v.toBits).toNat:ℤ)-1022)*(1/(2^(50:ℕ)-1)):=AI hk (by norm_num)
    _ ≤ 2^(((Dense.F.eF v.toBits).toNat:ℤ)-1022)*(2^(-(49:ℤ))):=mul_le_mul_of_nonneg_left (by norm_num) (by positivity)
    _=2^(4:ℤ)*2^(((Dense.F.eF v.toBits).toNat:ℤ)-1075):=by
      rw [← AK (by norm_num), ← AK (by norm_num)]; congr 1; ring
theorem rmsNum (xw gw td:Word) (hx:kB xw≠255) (hg:kB gw≠255)
    (hx0:xw<<<1≠0) (hg0:gw<<<1≠0)
    (hd:0 < (kB td).toNat∧kB td≠255∧k9 td=false)
    (hd2:(kB td).toNat ≤ 170) :
    Dense.F.fv (Dense.F.dw xw*(Dense.F.dw gw*((I0 1).toInt64.toFloat/Dense.F.dw td))) ∧
    SAct.Rel (Dense.F.rv (Dense.F.dw xw*(Dense.F.dw gw*((I0 1).toInt64.toFloat/Dense.F.dw td))))
      (((V3 xw*V3 gw/V3 td:ℚ):ℝ)*2^(149:ℕ))
      (1/2^(50:ℕ)):=by
  obtain ⟨fX,vX⟩:=dwZA xw hx
  obtain ⟨fG,vG⟩:=dwZA gw hg
  obtain ⟨fD,vD⟩:=dwZA td hd.2.1
  have X1:=dwOne xw hx hx0
  have G1:=dwOne gw hg hg0
  have Xb:=Norm.nXb xw hx
  have Gb:=Norm.nXb gw hg
  have Dlo:=Norm.nDb td hd
  have Dhi:=Norm.nXb td hd.2.1
  have ex:=mO xw
  have eg:=mO gw
  set X:=Dense.F.rv (Dense.F.dw xw) with hXd
  set G:=Dense.F.rv (Dense.F.dw gw) with hGd
  set Dd:=Dense.F.rv (Dense.F.dw td) with hDd
  have X2 : |X| ≤ 2^(278:ℕ):=Xb.le.trans (AU (by norm_num) (by omega))
  have G2 : |G| ≤ 2^(278:ℕ):=Gb.le.trans (AU (by norm_num) (by omega))
  have D1:(2:ℝ)^(23:ℕ) ≤ Dd:=AZ (AU (by norm_num) (by omega)) Dlo
  have D2:Dd ≤ 2^(193:ℕ):=(abs_lt.1 Dhi).2.le.trans (AU (by norm_num) (by omega))
  have D0:0 < Dd:=AN (by positivity) D1
  have iD1:1/Dd ≤ 4:=by rw [div_le_iff₀ D0]; nlinarith
  have iD2:(1:ℝ)/2^(300:ℕ) ≤ 1/Dd:=one_div_le_one_div_of_le D0 (D2.trans (AU (by norm_num) (by norm_num)))
  have iD3:1/Dd ≤ 1/2^(23:ℕ):=one_div_le_one_div_of_le (by positivity) D1
  have iD4:(1:ℝ)/2^(193:ℕ) ≤ 1/Dd:=one_div_le_one_div_of_le (by positivity) D2
  obtain ⟨fr,rr⟩:=relInv (Dense.F.dw td) fD D0 iD1 iD2
  set R:=Dense.F.rv ((I0 1).toInt64.toFloat/Dense.F.dw td) with hRd
  have R1:=rr.le
  have R2:=rr.ge
  rw [AW (by positivity:(0:ℝ) < 1/Dd)] at R1 R2
  have e0:(0:ℝ) ≤ 1/2^(52:ℕ):=by positivity
  have e1:(1:ℝ)/2^(52:ℕ) ≤ 1/2^(10:ℕ):=SAct.inv2le (by norm_num)
  have hR1 : |R| ≤ 1/2^(22:ℕ):=by
    refine R1.trans ?_
    calc (1+1/2^(52:ℕ))*(1/Dd) ≤ 2*(1/2^(23:ℕ)):=by gcongr; norm_num
      _=1/2^(22:ℕ):=by norm_num
  have hR2:(1:ℝ)/2^(194:ℕ) ≤ |R| := by
    refine AZ ?_ R2
    calc (1:ℝ)/2^(194:ℕ)=(1/2)*(1/2^(193:ℕ)):=by norm_num
      _ ≤ (1-1/2^(52:ℕ))*(1/Dd):=by gcongr; norm_num
  have p971:(2:ℝ)^(535:ℕ) ≤ (2:ℝ)^(971:ℤ):=by rw [← AA]; exact AG (by norm_num) (by norm_num)
  have p300:(1:ℝ)/2^(300:ℕ) ≤ 1/2^(195:ℕ):=SAct.inv2le (by norm_num)
  have GRu : |G*R| ≤ (2:ℝ)^(971:ℤ):=by
    rw [Ae]
    calc |G| * |R| ≤ 2^(278:ℕ)*1:=by gcongr; exact hR1.trans (by norm_num)
      _ ≤ 2^(535:ℕ):=by rw [mul_one]; exact AU (by norm_num) (by norm_num)
      _ ≤ _:=p971
  have GRl:(1:ℝ)/2^(300:ℕ) ≤ |G*R| := by
    rw [Ae]
    calc (1:ℝ)/2^(300:ℕ) ≤ 1/2^(194:ℕ):=SAct.inv2le (by norm_num)
      _=1*(1/2^(194:ℕ)):=by ring
      _ ≤ |G| * |R| := mul_le_mul G1 hR2 (by positivity) (Aa _)
  obtain ⟨fp,rp⟩:=relMul (Dense.F.dw gw) _ fG fr GRu GRl
  set Pp:=Dense.F.rv (Dense.F.dw gw*((I0 1).toInt64.toFloat/Dense.F.dw td)) with hPd
  have P1:=rp.le
  have P2:=rp.ge
  have hGR1 : |G*R| ≤ 2^(256:ℕ):=by
    rw [Ae]
    calc |G| * |R| ≤ 2^(278:ℕ)*(1/2^(22:ℕ)):=by gcongr
      _=2^(256:ℕ):=by rw [show (278:ℕ)=256+22 from rfl,AF]; field_simp
  have hP1 : |Pp| ≤ 2^(257:ℕ):=by
    refine P1.trans ?_
    calc (1+1/2^(52:ℕ)) * |G*R| ≤ 2*2^(256:ℕ):=by gcongr; norm_num
      _=2^(257:ℕ):=(pow_succ' (2:ℝ) 256).symm
  have hP2:(1:ℝ)/2^(195:ℕ) ≤ |Pp| := by
    refine AZ ?_ P2
    calc (1:ℝ)/2^(195:ℕ)=(1/2)*(1*(1/2^(194:ℕ))):=by rw [A4]; field_simp
      _ ≤ (1-1/2^(52:ℕ)) * |G*R| := by
          gcongr
          · norm_num
          · rw [Ae]; exact mul_le_mul G1 hR2 (by positivity) (Aa _)
  have XPu : |X*Pp| ≤ (2:ℝ)^(971:ℤ):=by
    rw [Ae]
    calc |X| * |Pp| ≤ 2^(278:ℕ)*2^(257:ℕ):=by gcongr
      _=2^(535:ℕ):=by rw [← AF]
      _ ≤ _:=p971
  have XPl:(1:ℝ)/2^(300:ℕ) ≤ |X*Pp| := by
    rw [Ae]
    calc (1:ℝ)/2^(300:ℕ) ≤ 1/2^(195:ℕ):=p300
      _=1*(1/2^(195:ℕ)):=by ring
      _ ≤ |X| * |Pp| := mul_le_mul X1 hP2 (by positivity) (Aa _)
  obtain ⟨fv,rv⟩:=relMul (Dense.F.dw xw) _ fX fp XPu XPl
  refine ⟨fv,?_⟩
  have r1:=SAct.Rel.mul (relSelf G) rr AP
  have r2:=SAct.Rel.trans rp r1 e0
  have r3:=SAct.Rel.mul (relSelf X) r2 AP
  have r4:=SAct.Rel.trans rv r3 e0
  have hT:X*(G*(1/Dd))=(((V3 xw*V3 gw/V3 td:ℚ):ℝ)*2^(149:ℕ)):=by
    have hD0:(V3 td:ℝ)≠0:=by intro h0; rw [vD,h0,zero_mul] at D0; exact AX _ D0
    rw [vX,vG,vD]; push_cast; field_simp
  rw [hT] at r4
  refine r4.mono ?_
  have:(1:ℝ)/2^(52:ℕ)*4 ≤ 1/2^(50:ℕ):=by norm_num
  nlinarith [e0,e1]
theorem zt0:C7 0=0:=by decide
theorem za0 (w:Word) (h0:w<<<1=0):V3 w=0:=by
  rw [Norm.Y1 w]
  have hm:kM w=0:=by
    have:=congrArg mJ h0
    unfold kM
    have hw:=w.toNat_lt
    have e1:(kB w).toNat=w.toNat/2^23%256:=by
      unfold kB
      simp only [Ξ,Υ,kN,kH,β,
        show (255:Nat)=2^8-1 from rfl,ζ]
    have e2:(lK w).toNat=w.toNat%2^23:=by
      unfold lK
      simp only [Ξ,kN,show (8388607:Nat)=2^23-1 from rfl,ζ]
    simp only [ϴ,kN,kH,ο] at this
    have ee:kB w=0:=by
      apply Β.1; rw [e1]; simp only [nr] at *; simp only [kN]; omega
    rw [if_pos ee,e2]; simp only [nr] at *; omega
  unfold L8; rw [hm]; simp
theorem nFBr_val (x g:Ἂ Word) (nd:Ἅ) (td:Word) (i j:Ἄ)
    (hnd:nd=(I0 1).toInt64.toFloat)
    (hx:kB (Norm.ck x i)≠255) (hg:kB (Norm.ck g j)≠255) :
    Norm.nFBr x g nd td i j=C7
      (V3 (Norm.ck x i)*V3 (Norm.ck g j)/V3 td):=by
  rw [Norm.nFBr_e]; unfold Norm.nOEr; dsimp only
  have p:(V3 (Norm.ck x i)-0)*V3 (Norm.ck g j)/V3 td
      + V3 0=V3 (Norm.ck x i)*V3 (Norm.ck g j)/V3 td:=by
    rw [Norm.Y1 0,(by decide:L8 0=0)]; simp
  refine Norm.YP_h1 _ _ _ (fun t => ?_) (fun y => ?_)
  · rw [Norm.Y1 (Norm.ck x i),(Norm.YN _).2 (of_decide_eq_true ((Norm.YK _).symm.trans t))]; simp
  · subst hnd
    have etd:(Norm.NR.eI 1 (0,0,false,0,0,td)).td=td:=by first | rfl | simp [Norm.NR.eI]
    exact (Norm.NE.jn _ _ 0 td _ 1 0 0 0 0 false _ 0 rfl hx hg (by decide) one_pos y (by simp)
      (by rw [etd]; exact (congrArg C7 p).symm)).trans (congrArg C7 p)
@[inline] def wR (x g:Ἂ Word) (nd:Ἅ) (td:Word) (xw gw:Word) (xv ri:Ἅ) (t:Ἄ):Word :=
  if (xw<<<1==0||gw<<<1==0)=true then 0 else
  let B:=(xv*(Dense.F.dw gw*ri)).toBits
  if okF B 1047=true then pk2 B else Norm.nFBr x g nd td t t
theorem elR (x g:Ἂ Word) (nd:Ἅ) (td:Word) (t:Ἄ) (xw gw:Word)
    (hxw:Norm.ck x t=xw) (hgw:Norm.ck g t=gw)
    (hx:kB xw≠255) (hg:kB gw≠255)
    (hnd:nd=(I0 1).toInt64.toFloat)
    (hd:0 < (kB td).toNat∧kB td≠255∧k9 td=false)
    (hd2:(kB td).toNat ≤ 170) :
    wR x g nd td xw gw (Dense.F.dw xw) ((I0 1).toInt64.toFloat/Dense.F.dw td) t =
      C7 (V3 xw*V3 gw/V3 td):=by
  unfold wR
  split
  · rename_i hz
    simp only [nM,AQ] at hz
    rcases hz with hz | hz
    · rw [za0 xw hz,zero_mul,zero_div,zt0]
    · rw [za0 gw hz,mul_zero,zero_div,zt0]
  · rename_i hz
    simp only [nM,AQ,not_or] at hz
    obtain ⟨fv,rl⟩:=rmsNum xw gw td hx hg hz.1 hz.2 hd hd2
    dsimp only
    split
    · rename_i hok
      exact fastEl _ _ 1047 (by decide) (by decide) fv hok (relWin _ _ fv rl)
    · rw [nFBr_val x g nd td t t hnd (hxw ▸ hx) (hgw ▸ hg),hxw,hgw]
theorem aset (o:Ἂ Word) (t:Ἄ) (v:Word) (h:t.toNat < o.size) (j:Nat) (hj:j < o.size) :
    (o.uset t v h)[j]! = if t.toNat=j then v else o[j]! := by
  have hs:(o.uset t v h).size=o.size:=BT ..
  rw [AB (o.uset t v h) j (by rw [hs]; exact hj),AB o j hj]
  simp only [Array.uset,Array.getElem_set]
theorem ugetF (o:ἀ) (t:Ἄ) (h:t.toNat < o.size):o.uget t h=o.data[t.toNat]! := by
  rcases o with ⟨d⟩
  simp only [kS] at h
  simp only [nN]
  rw [AB d t.toNat h]
  rfl
structure RQ where
  a:ἀ
  q:Ἅ
  dl:ἆ
def qR (x : @& Ἂ Word) (Xr:ἀ) (t n:Ἄ) (hx:n.toNat ≤ x.size) (hr:n.toNat ≤ Xr.size)
    (q:Ἅ) (Dl:ἆ):RQ :=
  if h:t < n then
    have ht:t.toNat < n.toNat:=ε.mp h
    let w:=x.uget t (U1 ht hx)
    let v:=Dense.F.dw w
    let e:=Norm.nfe w
    qR x (Xr.uset t v (U1 ht hr)) (t+1) n hx (by rw [Dense.F.uZ]; exact hr) (q+v*v) (if Dl < e then e else Dl)
  else ⟨Xr,q,Dl⟩
termination_by n.toNat-t.toNat
decreasing_by exact Norm.nHd h
theorem qR_spec (x:Ἂ Word) (n:Ἄ) (hx:n.toNat ≤ x.size) :
    ∀ k (t:Ἄ) (Xr:ἀ) (hr:n.toNat ≤ Xr.size) (q:Ἅ) (Dl:ἆ),
      n.toNat-t.toNat=k → t.toNat ≤ n.toNat →
      (qR x Xr t n hx hr q Dl).a.size=Xr.size ∧
      (∀ j,j < Xr.size → (qR x Xr t n hx hr q Dl).a.data[j]! =
          if t.toNat ≤ j∧j < n.toNat then Dense.F.dw x[j]! else Xr.data[j]!) ∧
      (qR x Xr t n hx hr q Dl).q=Norm.nQs x t n q ∧
      Dl.toNat ≤ (qR x Xr t n hx hr q Dl).dl.toNat ∧
      (∀ j,t.toNat ≤ j → j < n.toNat → (Norm.nfe x[j]!).toNat ≤ (qR x Xr t n hx hr q Dl).dl.toNat):=by
  intro k; induction k with
  | zero =>
    intro t Xr hr q Dl hk ht
    have hnt:¬ t < n:=Norm.nO1 hk
    rw [qR.eq_def,dif_neg hnt,Norm.nQs.eq_def,dif_neg hnt]
    refine ⟨rfl,fun j hj => ?_,rfl,AP,fun j h1 h2=>by omega⟩
    rw [if_neg (by omega)]
  | succ k ih =>
    intro t Xr hr q Dl hk ht
    obtain ⟨h1,hk',ht'⟩:=Norm.nOs hk
    have htn:t < n:=ε.mpr h1
    have hxt:t.toNat < x.size:=U1 h1 hx
    have hrt:t.toNat < Xr.size:=U1 h1 hr
    have e1:(t+1).toNat=t.toNat+1:=Dense.ac htn
    rw [qR.eq_def,dif_pos htn,Norm.nQs.eq_def,dif_pos htn]
    simp only [Dense.cC,Norm.f6]
    obtain ⟨a1,a2,a3,a4,a5⟩:=ih (t+1) (Xr.uset t (Dense.F.dw x[t.toNat]!) hrt) (by rw [Dense.F.uZ]; exact hr) (q+Dense.F.dw x[t.toNat]! * Dense.F.dw x[t.toNat]!)
      (if Dl < Norm.nfe x[t.toNat]! then Norm.nfe x[t.toNat]! else Dl) hk' ht'
    refine ⟨by rw [a1,Dense.F.uZ],fun j hj => ?_,a3,?_,fun j hj1 hj2 => ?_⟩
    · rw [a2 j (by rw [Dense.F.uZ]; exact hj),Dense.F.fuS _ _ _ _ _ hj,e1]
      by_cases hjt:t.toNat=j
      · subst hjt; rw [if_neg (by omega),if_pos rfl,if_pos ⟨AP,h1⟩]
      · simp only [hjt,if_false]
        by_cases hc:t.toNat ≤ j∧j < n.toNat
        · rw [if_pos (by omega),if_pos hc]
        · rw [if_neg (by omega),if_neg hc]
    · refine AZ ?_ a4
      split <;> rename_i hc
      · exact (μ.mp hc).le
      · exact AP
    · by_cases hjt:j=t.toNat
      · subst hjt; refine AZ ?_ a4
        split <;> rename_i hc
        · exact AP
        · exact θ.mp (UInt64.not_lt.mp hc)
      · exact a5 j (by omega) hj2
def eR (x g : @& Ἂ Word) (nd:Ἅ) (td:Word) (xb : @& ἀ) (ri:Ἅ) (t nn:Ἄ)
    (hx:nn.toNat ≤ x.size) (hg:nn.toNat ≤ g.size) (hr:nn.toNat ≤ xb.size)
    (out:Ἂ Word) (ho:nn.toNat ≤ out.size):Ἂ Word :=
  if h:t < nn then
    have ht:t.toNat < nn.toNat:=ε.mp h
    let gw:=g.uget t (U1 ht hg)
    if Norm.nfe gw < 255 then
      let w:=wR x g nd td (x.uget t (U1 ht hx)) gw (xb.uget t (U1 ht hr)) ri t
      eR x g nd td xb ri (t+1) nn hx hg hr (out.uset t w (U1 ht ho)) (by rw [BT]; exact ho)
    else #[]
  else out
termination_by nn.toNat-t.toNat
decreasing_by exact Norm.nHd h
theorem eR_spec (x g:Ἂ Word) (nd:Ἅ) (td:Word) (xb:ἀ) (ri:Ἅ) (nn:Ἄ)
    (hx:nn.toNat ≤ x.size) (hg:nn.toNat ≤ g.size) (hr:nn.toNat ≤ xb.size) :
    ∀ k (t:Ἄ) (out:Ἂ Word) (ho:nn.toNat ≤ out.size),nn.toNat-t.toNat=k → t.toNat ≤ nn.toNat →
      eR x g nd td xb ri t nn hx hg hr out ho = #[] ∨
      ((eR x g nd td xb ri t nn hx hg hr out ho).size=out.size ∧
       (∀ j,t.toNat ≤ j → j < nn.toNat → Norm.nfe g[j]! < 255) ∧
       ∀ j,j < out.size → (eR x g nd td xb ri t nn hx hg hr out ho)[j]! =
         if t.toNat ≤ j∧j < nn.toNat then wR x g nd td x[j]! g[j]! xb.data[j]! ri (C3 j) else out[j]!):=by
  intro k; induction k with
  | zero =>
    intro t out ho hk ht
    have hnt:¬ t < nn:=Norm.nO1 hk
    rw [eR.eq_def,dif_neg hnt]
    exact BA ⟨rfl,fun j h1 h2=>by omega,fun j hj=>by rw [if_neg (by omega)]⟩
  | succ k ih =>
    intro t out ho hk ht
    obtain ⟨h1,hk',ht'⟩:=Norm.nOs hk
    have htn:t < nn:=ε.mpr h1
    have e1:(t+1).toNat=t.toNat+1:=Dense.ac htn
    rw [eR.eq_def,dif_pos htn]
    simp only [Dense.cC]
    split
    · rename_i hgf
      have hot:t.toNat < out.size:=U1 h1 ho
      rcases ih (t+1) (out.uset t (wR x g nd td x[t.toNat]! g[t.toNat]! (xb.uget t (U1 h1 hr)) ri t) hot)
          (by rw [BT]; exact ho) hk' ht' with hz | ⟨a1,a2,a3⟩
      · exact pZ hz
      · refine BA ⟨by rw [a1,BT],fun j hj1 hj2 => ?_,fun j hj => ?_⟩
        · by_cases hjt:j=t.toNat
          · subst hjt; exact hgf
          · exact a2 j (by omega) hj2
        · rw [a3 j (by rw [BT]; exact hj),aset _ _ _ _ _ hj,e1]
          by_cases hjt:t.toNat=j
          · subst hjt
            rw [if_neg (by omega),if_pos rfl,if_pos ⟨AP,h1⟩,ugetF]
            congr 1; exact USize.ofNat_toNat.symm
          · simp only [hjt,if_false]
            by_cases hc:t.toNat ≤ j∧j < nn.toNat
            · rw [if_pos (by omega),if_pos hc]
            · rw [if_neg (by omega),if_neg hc]
    · exact pZ rfl
theorem usz (n:Nat) (h:n < 2^32):n < kX:=by
  rw [nT]; rcases System.Platform.numBits_eq with h' | h' <;> rw [h'] <;> omega
theorem qR0 (x:Ἂ Word) (n:Nat) (hx:n ≤ x.size) (hn:n < 2^32) :
    (n.toUSize).toNat ≤ x.size∧(n.toUSize).toNat ≤ (Dense.F.zPq n).size:=by
  have q2:(n.toUSize).toNat=n:=Dense.aa (usz n (by omega))
  rw [q2]; exact ⟨hx,(Dense.F.zPs n).1⟩
def rmsF (n:Nat) (eps:Word) (x g:Ἂ Word) (hx:n ≤ x.size) (hg:n ≤ g.size) (hn:n < 2^32):Ἂ Word :=
  let nn:=n.toUSize
  let o:=eps.toUInt64>>>32
  let one:=(o+1).toInt64.toFloat
  let Qr:=qR x (Dense.F.zPq n) 0 nn (qR0 x n hx hn).1 (qR0 x n hx hn).2 o.toInt64.toFloat 0
  if Qr.dl < 255 then
    let BD:=(W7 (Qr.q*(((1:ἆ)<<<52)/n.toUInt64).toInt64.toFloat+Dense.F.dw eps*Norm.nS201)).toBits
    if Norm.nOKs BD (Core.ai (n+4).toUInt64+4) then
      let td:=Norm.nWs 26 BD
      if 0 < (kB td).toUInt64&&(kB td).toUInt64 ≤ 170 && !ExactRuntime.signOf td then
        eR x g one td Qr.a (one/Dense.F.dw td) 0 nn (qR0 x n hx hn).1
          (by rw [Dense.aa (usz n (by omega))]; exact hg)
          (by rw [(qR_spec x nn (qR0 x n hx hn).1 _ 0 _ (qR0 x n hx hn).2 _ 0 rfl (ϟ _)).1]; exact (qR0 x n hx hn).2)
          (mB n 0) (by rw [O3,Dense.aa (usz n (by omega))])
      else #[]
    else #[]
  else #[]
def rms (rows n:Nat) (eps:Word) (x g:Ἂ Word):Ἂ Word :=
  if h:rows=1∧1 ≤ n∧n ≤ 4096∧x.size=n∧g.size=n∧kB eps≠0 ∧
      kB eps≠255∧k9 eps=false then
    let r:=rmsF n eps x g (by omega) (by omega) (by omega)
    if r.size=n then r else rmsnormSpec rows n eps x g
  else rmsnormSpec rows n eps x g
theorem eR_full (n:Nat) (eps:Word) (x g:Ἂ Word) (td:Word) (xb:ἀ) (nn:Ἄ)
    (hnn:nn.toNat=n) (hx:nn.toNat ≤ x.size) (hg:nn.toNat ≤ g.size) (hr:nn.toNat ≤ xb.size)
    (ho:nn.toNat ≤ (mB n (0:Word)).size)
    (hn1:1 ≤ n) (hn:n < 2^32) (hxs:x.size=n) (hgs:g.size=n)
    (hX : ∀ j,j < n → kB x[j]! ≠ 255)
    (hxb : ∀ j,j < n → xb.data[j]! = Dense.F.dw x[j]!)
    (htd:td=mK (ExactRuntime.rowVariance n x 0 0+V3 eps))
    (hd:0 < (kB td).toNat∧kB td≠255∧k9 td=false)
    (hd2:(kB td).toNat ≤ 170)
    (he1:ExactRuntime.finiteWord eps=true) (hepos:0 < V3 eps)
    (hsz:(eR x g (I0 1).toInt64.toFloat td xb ((I0 1).toInt64.toFloat/Dense.F.dw td) 0 nn
      hx hg hr (mB n 0) ho).size=n) :
    eR x g (I0 1).toInt64.toFloat td xb ((I0 1).toInt64.toFloat/Dense.F.dw td) 0 nn
      hx hg hr (mB n 0) ho=ExactRuntime.rmsnormSpec 1 n eps x g:=by
  rcases eR_spec x g _ td xb _ nn hx hg hr nn.toNat 0 (mB n 0) ho (by simp) (ϟ _)
    with hz | ⟨E1,E2,E3⟩
  · rw [hz] at hsz; simp at hsz; omega
  have nfeE : ∀ w:Word,(Norm.nfe w).toNat=(kB w).toNat:=fun w=>by
    unfold Norm.nfe kB; simp
  have hG : ∀ j,j < n → kB g[j]! ≠ 255:=fun j hj h=>by
    have:=E2 j (ϟ _) (by omega)
    rw [μ,nfeE,h] at this; simp at this
  have hdom:mq 1 n eps x g=true:=by
    unfold mq
    simp only [Η,AJ,AQ,np]
    refine ⟨⟨⟨⟨⟨⟨⟨by omega,by omega⟩,by omega⟩,hgs⟩,fun i hi => ?_⟩,fun i hi => ?_⟩,he1⟩,hepos⟩
    · have:=hX i (by omega); rw [AB x i hi] at this
      unfold ExactRuntime.finiteWord; simpa using this
    · have:=hG i (by omega); rw [AB g i hi] at this
      unfold ExactRuntime.finiteWord; simpa using this
  rw [ExactRuntime.rmsnormSpec,ExactRuntime.rmsnormWith,if_pos hdom]
  apply mi
  · rw [E1]; simp [ExactRuntime.rmsnormStable]
  · intro j h1 h2
    have hj:j < n:=by rw [E1] at h1; simpa using h1
    have e3:=E3 j (by simpa using hj)
    rw [if_pos ⟨ϟ _,by rw [hnn]; exact hj⟩,AB _ j h1] at e3
    rw [e3]
    simp only [ExactRuntime.rmsnormStable,Λ]
    rw [hxb j hj]
    have hck : ∀ a:Ἂ Word,Norm.ck a (C3 j)=a[j]! := fun a=>by
      rw [Norm.f6,J3 (usz j (by omega))]
    rw [elR x g _ td (C3 j) x[j]! g[j]! (hck x) (hck g) (hX j hj) (hG j hj) rfl hd hd2]
    congr 2
    · rw [γ hj]
    · rw [ώ hj,htd,AB _ 0 (by simp)]; simp
theorem epsPos (eps:Word) (he0:kB eps≠0) (hes:k9 eps=false) :
    0 < L8 eps∧0 < V3 eps:=by
  have hm:0 < kM eps:=by unfold kM; rw [if_neg he0]; positivity
  have hs:0 < L8 eps:=by unfold L8; rw [hes]; simp; exact_mod_cast hm
  refine ⟨hs,?_⟩
  rw [Norm.Y1]; exact mul_pos (by exact_mod_cast hs) (by positivity)
theorem rmsF_eq (n:Nat) (eps:Word) (x g:Ἂ Word) (hx:n ≤ x.size) (hg:n ≤ g.size) (hn:n < 2^32)
    (hn1:1 ≤ n) (hn2:n ≤ 4096) (hxs:x.size=n) (hgs:g.size=n)
    (he0:kB eps≠0) (he1:kB eps≠255) (hes:k9 eps=false)
    (hsz:(rmsF n eps x g hx hg hn).size=n) :
    rmsF n eps x g hx hg hn=ExactRuntime.rmsnormSpec 1 n eps x g:=by
  have q2:(n.toUSize).toNat=n:=Dense.aa (usz n hn)
  obtain ⟨Q1,Q2,Q3, -, Q5⟩:=qR_spec x n.toUSize (qR0 x n hx hn).1 n 0 (Dense.F.zPq n) (qR0 x n hx hn).2
    ((eps.toUInt64>>>32).toInt64.toFloat) 0 (by rw [q2]; rfl) (ϟ _)
  obtain ⟨hsv,hep⟩:=epsPos eps he0 hes
  unfold rmsF at hsz ⊢
  dsimp only at hsz ⊢
  split <;> rename_i hdl
  · rw [if_pos hdl] at hsz
    split <;> rename_i hok
    · rw [if_pos hok] at hsz
      split <;> rename_i htd
      · rw [if_pos htd] at hsz
        have hX : ∀ j,j < n → kB x[j]! ≠ 255:=fun j hj h=>by
          have h5:=Q5 j (ϟ _) (by rw [q2]; exact hj)
          have e:(Norm.nfe x[j]!).toNat=(kB x[j]!).toNat:=by unfold Norm.nfe kB; simp
          rw [e,h] at h5
          have h6:=μ.mp hdl
          simp only [kN,kA] at h5 h6; omega
        have hxb : ∀ j,j < n → (qR x (Dense.F.zPq n) 0 n.toUSize (qR0 x n hx hn).1 (qR0 x n hx hn).2
            (eps.toUInt64>>>32).toInt64.toFloat 0).a.data[j]! = Dense.F.dw x[j]! := fun j hj=>by
          rw [Q2 j (AN hj (Dense.F.zPs n).1),if_pos ⟨ϟ _,by rw [q2]; exact hj⟩]
        have hok':=hok
        rw [Q3,Norm.kO0,Norm.kQ52 n (by omega),Norm.kZ4 (n+4) (by omega)] at hok'
        have hcert:=Norm.nQ_cert n eps x 0 (by omega) (by norm_num; omega) (by omega)
          (fun j hj=>hX j (by omega)) he1 hsv (by convert hok' using 4 <;> simp <;> rfl)
        have htd':Norm.nWs 26 (W7 ((qR x (Dense.F.zPq n) 0 n.toUSize (qR0 x n hx hn).1 (qR0 x n hx hn).2
            (eps.toUInt64>>>32).toInt64.toFloat 0).q*(((1:ἆ)<<<52)/n.toUInt64).toInt64.toFloat
            + Dense.F.dw eps*Norm.nS201)).toBits
            = mK (ExactRuntime.rowVariance n x 0 0+V3 eps):=by
          rw [Q3,Norm.kO0,Norm.kQ52 n (by omega)]; convert hcert using 4 <;> simp <;> rfl
        simp only [Η,AJ,BM] at htd
        obtain ⟨⟨hd1,hd2⟩,hd3⟩:=htd
        rw [μ] at hd1; rw [θ] at hd2
        simp only [ϩ,kA] at hd1 hd2
        have hne:kB (Norm.nWs 26 (W7 ((qR x (Dense.F.zPq n) 0 n.toUSize (qR0 x n hx hn).1 (qR0 x n hx hn).2
            (eps.toUInt64>>>32).toInt64.toFloat 0).q*(((1:ἆ)<<<52)/n.toUInt64).toInt64.toFloat
            + Dense.F.dw eps*Norm.nS201)).toBits)≠255:=fun h=>by rw [h] at hd2; simp at hd2
        simp only [Norm.kO1] at hsz ⊢
        exact eR_full n eps x g _ _ n.toUSize q2 _ _ _ _ hn1 hn hxs hgs hX hxb htd' ⟨hd1,hne,hd3⟩ hd2
          (by unfold ExactRuntime.finiteWord; simpa using he1) hep hsz
      · rw [if_neg htd] at hsz; simp at hsz; omega
    · rw [if_neg hok] at hsz; simp at hsz; omega
  · rw [if_neg hdl] at hsz; simp at hsz; omega
theorem rms_eq:rms=ExactRuntime.rmsnormSpec:=by
  funext rows n eps x g
  unfold rms
  split
  · rename_i h
    obtain ⟨hr1,hn1,hn2,hxs,hgs,he0,he1,hes⟩:=h
    dsimp only
    split
    · rename_i hsz; subst hr1; exact rmsF_eq n eps x g _ _ _ hn1 hn2 hxs hgs he0 he1 hes hsz
    · rfl
  · rfl
end NX
namespace NXc
open ExactRuntime
theorem ez0 (w:Word) (h0:w<<<1=0):expOf w=0:=by
  have:=congrArg mJ h0
  have hw:=w.toNat_lt
  have e1:(expOf w).toNat=w.toNat/2^23%256:=by
    unfold kB
    simp only [Ξ,Υ,kN,kH,β,
      show (255:Nat)=2^8-1 from rfl,ζ]
  simp only [ϴ,kN,kH,ο] at this
  apply Β.1; rw [e1]; simp only [nr] at *; simp only [kN]; omega
theorem za0 (w:Word) (h0:w<<<1=0):rationalWord w=0:=by
  rw [Norm.Y1 w]
  have hm:mantOf w=0:=by
    have:=congrArg mJ h0
    unfold kM
    have hw:=w.toNat_lt
    have e2:(fracOf w).toNat=w.toNat%2^23:=by
      unfold lK
      simp only [Ξ,kN,show (8388607:Nat)=2^23-1 from rfl,ζ]
    simp only [ϴ,kN,kH,ο] at this
    rw [if_pos (ez0 w h0),e2]; simp only [nr] at *; omega
  unfold L8; rw [hm]; simp
theorem nfeL (w:Word) (h:Norm.nfe w < 255):expOf w≠255:=by
  intro c
  have e:Norm.nfe w=(expOf w).toUInt64:=rfl
  rw [e,c] at h; exact absurd h (by decide)
def cQ (x : @& Ἂ Word) (w0:Word) (i e:Ἄ):Ἆ :=
  if h:i < e then
    let w:=Norm.ck x i
    if (w==w0||(w<<<1==0&&w0<<<1==0)) then cQ x w0 (i+1) e else false
  else true
termination_by e.toNat-i.toNat
decreasing_by have:=ε.mp h; have:=e.toNat_lt_size; rw [Norm.uS (by omega)]; omega
theorem cQ_spec (x:Ἂ Word) (w0:Word) (e:Ἄ) : ∀ k (i:Ἄ),e.toNat-i.toNat=k → cQ x w0 i e=true →
    ∀ j,i.toNat ≤ j → j < e.toNat → (x[j]! = w0∨(x[j]! <<< 1=0∧w0<<<1=0)):=by
  intro k; induction k with
  | zero=>intro i hk _ j h1 h2; omega
  | succ k ih =>
    intro i hk hc j h1 h2
    have hi:i < e:=ε.mpr (by omega)
    rw [cQ,dif_pos hi] at hc
    dsimp only at hc
    split at hc
    · rename_i hw
      have hs:(i+1).toNat=i.toNat+1:=Norm.uS (by have:=e.toNat_lt_size; omega)
      by_cases hj:j=i.toNat
      · subst hj; rw [Norm.f6] at hw; simpa only [nM,AQ,Η] using hw
      · exact ih (i+1) (by omega) hc j (by omega) h2
    · exact absurd hc lN
theorem foldC (f:ℕ → ℚ) (c:ℚ) : ∀ (l:List ℕ) (a:ℚ),(∀ j ∈ l,f j=c) →
    l.foldl (fun t j=>t+f j) a=a+l.length*c:=by
  intro l; induction l with
  | nil=>intro a _; simp
  | cons h t ih =>
    intro a hf
    rw [Ϸ,ih _ (fun j hj=>hf j (List.mem_cons_of_mem _ hj)),hf h List.mem_cons_self]
    simp only [List.length_cons,W5,pJ]; ring
theorem meanC (n:ℕ) (x:Ἂ Word) (r:ℕ) (c:ℚ) (hn:0 < n)
    (h : ∀ j,j < n → rationalWord x[r*n+j]! = c):rowMean n x r=c:=by
  unfold rowMean
  have e:=foldC (fun j=>rationalWord x[r*n+j]!) c (lj n) 0 (fun j hj=>h j (List.mem_range.mp hj))
  rw [e,List.length_range,Ai]
  have:(n:ℚ)≠0:=by exact_mod_cast (BD.mp hn)
  field_simp
theorem eoV (n:ℕ) (h:Word) (x g b:Ἂ Word) (m:0 < n) (q:0 < rationalWord h) (i:ℕ)
    (hb:i/n*n+n < 2^(32:ℕ)) (hX : ∀ j,j < i/n*n+n → expOf x[j]! ≠ 255)
    (F : ∀ j,j < n → expOf g[j]! ≠ 255) (D : ∀ j,j < n → expOf b[j]! ≠ 255) :
    Norm.nEO n h x g b (I0 n).toInt64.toFloat (Norm.nLR n h x (i/n)) (G0 i) (G0 (i%n))
      = roundRational ((rationalWord x[i]! - rowMean n x (i/n))*rationalWord g[i%n]! /
          rationalWord (sqrtRational (rowVariance n x (i/n) (rowMean n x (i/n))+rationalWord h))+rationalWord b[i%n]!):=by
  have r:=ϊ i m
  have hj1:i/n*n ≤ i:=pr i n
  have hj2:i < i/n*n+n:=Nat.lt_div_mul_add m
  dsimp only [Norm.nEO,Norm.nLoN]
  have hs:=Norm.nSv h q
  simp only [Norm.nLo,Norm.nLR_th,Norm.f6,Conv.aa _ (K5 hj2 hb:i < 2^(32:ℕ)),
    Conv.aa _ (U1 r (U2 (K2 n _) (X4 hb)):i%n < 2^(32:ℕ)),
    Norm.gP,Norm.gN _ _ _ _ m q hb]
  obtain ⟨v,A, -⟩:=Norm.Vn n h x (i/n) m q
  refine Norm.nV3 _ _ _ (Norm.nV2 _ _ _ (fun E3 => ?_) (fun E3 => ?_)) (fun E1 E2 => ?_)
  · rw [← v, ← A]; generalize Norm.YU n h x (i/n)=t at *; obtain ⟨u,k,y,o,p,d⟩:=t; dsimp only at *
    exact Norm.NE.jn _ _ _ d _ n u k o p y _ _ rfl (hX i hj2) (F _ r) (D _ r) m ‹_› (Norm.V3L_h1 _ _ _ _ _ v) rfl
  · rw [← v, ← A]
  · rw [Norm.nLR_el n h x (i/n) hb hX hs i hj1 hj2 _ _ (F _ r) (D _ r) E1 E2]
theorem specC (n:ℕ) (x g b:Ἂ Word) (eps:Word) (k:ℕ) (hn:0 < n) (w0:Word)
    (hK : ∀ j,k/n*n ≤ j → j < k/n*n+n → (x[j]! = w0∨(x[j]! <<< 1=0∧w0<<<1=0))) :
    roundRational ((rationalWord x[k]! - rowMean n x (k/n))*rationalWord g[k%n]! /
          rationalWord (sqrtRational (rowVariance n x (k/n) (rowMean n x (k/n))+rationalWord eps))+rationalWord b[k%n]!)
      = roundRational (rationalWord b[k%n]!):=by
  have hZ : ∀ j,k/n*n ≤ j → j < k/n*n+n → rationalWord x[j]! = rationalWord w0:=by
    intro j h1 h2; rcases hK j h1 h2 with e | ⟨e1,e2⟩
    · rw [e]
    · rw [za0 _ e1,za0 _ e2]
  have hM:rowMean n x (k/n)=rationalWord w0:=meanC n x (k/n) _ hn (fun j hj=>hZ _ (by omega) (by omega))
  rw [hM,hZ k (pr k n) (Nat.lt_div_mul_add hn),sub_self,zero_mul,zero_div,Ai]
@[inline] def cbF (b:Ἂ Word):Ἂ Word:=b.map (fun w=>roundRational (rationalWord w))
@[noinline] def cbS:ST.Ref Unit (Ἂ (Dense.F.WcE cbF)):=(ST.Prim.mkRef #[] (Void.mk ())).val
@[noinline] def cbGo (b:Ἂ Word):{v//v=cbF b}:=Dense.F.wcGet3 Dense.F.wcAeq (fun a c h=>Dense.F.wcAeq_ok h) cbS 32 b (fun k=>k)
def cbGs (b:Ἂ Word) (s:Ἂ (Dense.F.WcE cbF)) (i:Nat):{v//v=cbF b}:=
  if h:i<s.size then
    withPtrAddr s[i].k (fun p1=>withPtrAddr b (fun p2=>
      if p1=p2 then (if hk:Dense.F.wcAeq s[i].k b=true then ⟨s[i].v,by rw [s[i].h,Dense.F.wcAeq_ok hk]⟩ else cbGo b)
      else cbGs b s (i+1)) (fun _ _=>Dense.F.wcSub _ _)) (fun _ _=>Dense.F.wcSub _ _)
  else cbGo b
termination_by s.size-i
@[noinline] def cbG (b:Ἂ Word):Ἂ Word:=(cbGs b (ST.Prim.Ref.get cbS (Void.mk ())).val 0).1
theorem cbG_v (b:Ἂ Word):cbG b=cbF b:=(cbGs b _ 0).2
theorem cbG_sp (b:Ἂ Word) (n:ℕ) (sb:b.size=n) :
    ∀ j,j < n → (cbG b)[j]! = roundRational (rationalWord b[j]!):=by
  intro j hj
  rw [cbG_v,cbF,AB _ j (by rw [m0]; omega),Array.getElem_map,AB b j (by omega)]
theorem cB (n:ℕ) (eps:Word) (x g b:Ἂ Word) (cb:Ἂ Word) (r:ℕ) (out:Ἂ Word)
    (ho:r*n+n ≤ out.size) (hb:r*n+n < 2^(32:ℕ)) (hos:out.size < 2^(32:ℕ))
    (hX : ∀ j,j < r*n+n → expOf x[j]! ≠ 255) (heps:0 < rationalWord eps)
    (hN:(4 ≤ n&&n ≤ 2048&&expOf eps != 0&&expOf eps != 255)=true)
    (F : ∀ j,j < n → expOf g[j]! ≠ 255) (D : ∀ j,j < n → expOf b[j]! ≠ 255)
    (hcb : ∀ j,j < n → cb[j]! = roundRational (rationalWord b[j]!)) (hc:(Norm.nfe (Norm.ck x (G0 (r*n))) < 255&&cQ x (Norm.ck x (G0 (r*n))) (G0 (r*n)) (G0 (r*n+n)))=true) :
    Norm.Yq (fun (c:Ἂ Word) (_:Ἄ) (j:Ἄ)=>Norm.ck c j) cb (G0 (r*n)) (G0 (r*n)) (G0 (r*n+n)) out
      = Norm.Yq (Norm.nEO n eps x g b (I0 n).toInt64.toFloat) (Norm.nLR n eps x r)
          (G0 (r*n)) (G0 (r*n)) (G0 (r*n+n)) out:=by
  have hm:0 < n:=by
    simp only [Η,AJ] at hN; omega
  simp only [Η,AJ] at hc
  obtain ⟨-, hq⟩:=hc
  have q1:(G0 (r*n)).toNat=r*n:=Conv.aa _ (C0 (V1 _ _) hb)
  have q3:(G0 (r*n+n)).toNat=r*n+n:=Conv.aa _ hb
  have hK:=cQ_spec x _ _ _ _ rfl hq
  rw [q1,q3] at hK
  obtain ⟨a1,b1⟩:=Norm.ju0 (fun (c:Ἂ Word) (_:Ἄ) (j:Ἄ)=>Norm.ck c j) cb (G0 (r*n))
    (G0 (r*n+n)) (G0 (r*n)) out (by rw [q3]; exact ho) hos
  obtain ⟨a2,b2⟩:=Norm.ju0 (Norm.nEO n eps x g b (I0 n).toInt64.toFloat) (Norm.nLR n eps x r) (G0 (r*n))
    (G0 (r*n+n)) (G0 (r*n)) out (by rw [q3]; exact ho) hos
  apply mi (by rw [a1,a2])
  intro k k1 k2
  rw [← AB _ k k1, ← AB _ k k2,b1 k (a1 ▸ k1),b2 k (a2 ▸ k2),q1,q3]
  split
  · rename_i hk
    have kr:k/n=r:=mG hk.1 (by first | exact hk.2 | (rw [Ο]; exact hk.2) | (rw [Ϡ,n4]; exact hk.2))
    have km:k%n=k-r*n:=by rw [Nat.mod_eq_sub_mul_div,kr,Ϗ]
    have k32:k < 2^(32:ℕ):=K5 hk.2 hb
    have hsub:G0 k-G0 (r*n)=G0 (k%n):=by
      apply O5.1
      rw [USize.toNat_sub_of_le _ _ (by rw [U8,q1,Conv.aa _ k32]; exact hk.1),q1,Conv.aa _ k32,
        Conv.aa _ (C0 (Ay k n) k32),km]
    rw [hsub,Norm.f6,Conv.aa _ (C0 (Ay k n) k32),hcb _ (ϊ _ hm), ← kr,
      eoV n eps x g b hm heps k (by rw [kr]; exact hb) (by rw [kr]; exact hX) F D]
    exact (specC n x g b eps k hm _ (by rw [kr]; exact hK)).symm
  · rfl
theorem cBb (n:ℕ) (x:Ἂ Word) (r:ℕ) (hb:r*n+n < 2^(32:ℕ)) (j:ℕ) (j1:r*n ≤ j) (j2:j < r*n+n)
    (hj:expOf x[j]! = 255) (hc:(Norm.nfe (Norm.ck x (G0 (r*n))) < 255&&cQ x (Norm.ck x (G0 (r*n))) (G0 (r*n)) (G0 (r*n+n)))=true) : False:=by
  simp only [Η,AJ] at hc
  obtain ⟨hf0,hq⟩:=hc
  have q1:(G0 (r*n)).toNat=r*n:=Conv.aa _ (C0 (V1 _ _) hb)
  have q3:(G0 (r*n+n)).toNat=r*n+n:=Conv.aa _ hb
  rcases cQ_spec x _ _ _ _ rfl hq j (by rw [q1]; exact j1) (by rw [q3]; exact j2) with e | ⟨e1, -⟩
  · rw [e] at hj; exact nfeL _ hf0 hj
  · rw [ez0 _ e1] at hj; exact absurd hj (by decide)
@[inline] def cRow (n:ℕ) (eps:Word) (x g b:Ἂ Word) (Xr gd bd:ἀ) (ge:ByteArray) (nd:Ἅ)
    (rm nf ep z0:Ἅ) (L:ἆ) (hC:Norm.nCK n eps rm nf ep z0 L) (cb:Ἂ Word) (r:ℕ) (out:Ἂ Word)
    (hr:n+2 ≤ Xr.size) (hg:n ≤ gd.size) (hd:n ≤ bd.size) (he:n ≤ ge.size)
    (hx:r*n+n ≤ x.size) (ho:r*n+n ≤ out.size) (hb:r*n+n < 2^(32:ℕ)):ἀ × Ἂ Word :=
  let lo:=G0 (r*n); let hi:=G0 (r*n+n)
  let w0:=Norm.ck x lo
  if (Norm.nfe w0 < 255&&cQ x w0 lo hi)=true then
    (Xr,Norm.Yq (fun (c:Ἂ Word) (_:Ἄ) (j:Ἄ)=>Norm.ck c j) cb lo lo hi out)
  else Norm.nRow n eps x g b Xr gd bd ge nd rm nf ep z0 L hC r out hr hg hd he hx ho hb
theorem cRow_sz (n:ℕ) (eps:Word) (x g b:Ἂ Word) (Xr gd bd:ἀ) (ge:ByteArray) (nd:Ἅ)
    (rm nf ep z0:Ἅ) (L:ἆ) (hC:Norm.nCK n eps rm nf ep z0 L) (cb:Ἂ Word) (r:ℕ) (out:Ἂ Word)
    (hr:n+2 ≤ Xr.size) (hg:n ≤ gd.size) (hd:n ≤ bd.size) (he:n ≤ ge.size)
    (hx:r*n+n ≤ x.size) (ho:r*n+n ≤ out.size) (hb:r*n+n < 2^(32:ℕ)) :
    (cRow n eps x g b Xr gd bd ge nd rm nf ep z0 L hC cb r out hr hg hd he hx ho hb).1.size=Xr.size ∧
    ((cRow n eps x g b Xr gd bd ge nd rm nf ep z0 L hC cb r out hr hg hd he hx ho hb).2 = #[] ∨
     (cRow n eps x g b Xr gd bd ge nd rm nf ep z0 L hC cb r out hr hg hd he hx ho hb).2.size=out.size):=by
  unfold cRow; dsimp only; split
  · exact ⟨rfl,BA (Norm.nYq_sz _ _ _ _ _ _ _ rfl)⟩
  · exact Norm.nRow_sz n eps x g b Xr gd bd ge nd rm nf ep z0 L hC r out hr hg hd he hx ho hb
def cRows (rows n:ℕ) (eps:Word) (x g b:Ἂ Word) (gd bd:ἀ) (ge:ByteArray) (nd:Ἅ)
    (rm nf ep z0:Ἅ) (L:ἆ) (hC:Norm.nCK n eps rm nf ep z0 L) (cb:Ἂ Word)
    (hg:n ≤ gd.size) (hd:n ≤ bd.size) (he:n ≤ ge.size) (hx:rows*n ≤ x.size) (hb:rows*n < 2^(32:ℕ)) :
    (k r:ℕ) → (Xr:ἀ) → (out:Ἂ Word) → n+2 ≤ Xr.size → rows*n ≤ out.size → r+k=rows → Ἂ Word
  | 0,_,_,out,_,_,_=>out
  | k+1,r,Xr,out,hr,ho,hk =>
    have kk:r*n+n ≤ rows*n:=Norm.nHk2 hk
    let p:=cRow n eps x g b Xr gd bd ge nd rm nf ep z0 L hC cb r out hr hg hd he (kk.trans hx) (kk.trans ho) (C0 kk hb)
    have q:p.1.size=Xr.size:=(cRow_sz n eps x g b Xr gd bd ge nd rm nf ep z0 L hC cb r out hr hg hd he (kk.trans hx) (kk.trans ho) (C0 kk hb)).1
    if hs:rows*n ≤ p.2.size then cRows rows n eps x g b gd bd ge nd rm nf ep z0 L hC cb hg hd he hx hb k (r+1) p.1 p.2 (q ▸ hr) hs (Norm.nHk3 hk)
    else #[]
theorem cRow_eq (n:ℕ) (eps:Word) (x g b:Ἂ Word) (Xr gd bd:ἀ) (ge:ByteArray)
    (rm nf ep z0:Ἅ) (L:ἆ) (hC:Norm.nCK n eps rm nf ep z0 L) (cb:Ἂ Word) (r:ℕ) (out:Ἂ Word)
    (hr:n+2 ≤ Xr.size) (hg:n ≤ gd.size) (hd:n ≤ bd.size) (he:n ≤ ge.size)
    (hx:r*n+n ≤ x.size) (ho:r*n+n ≤ out.size) (hb:r*n+n < 2^(32:ℕ)) (hos:out.size < 2^(32:ℕ))
    (hX : ∀ j,j < r*n+n → expOf x[j]! ≠ 255) (heps:0 < rationalWord eps)
    (hG : ∀ t,t < n → gd.data[t]! = Dense.F.dw g[t]!) (hB : ∀ t,t < n → bd.data[t]! = Dense.F.dw b[t]!)
    (hE : ∀ t,t < n → (ge.data[t]!).toUInt64=Norm.nfe g[t]!)
    (hN:(4 ≤ n&&n ≤ 2048&&expOf eps != 0&&expOf eps != 255)=true)
    (F : ∀ j,j < n → expOf g[j]! ≠ 255) (D : ∀ j,j < n → expOf b[j]! ≠ 255)
    (hcb : ∀ j,j < n → cb[j]! = roundRational (rationalWord b[j]!)) :
    (cRow n eps x g b Xr gd bd ge (I0 n).toInt64.toFloat rm nf ep z0 L hC cb r out hr hg hd he hx ho hb).2
      = Norm.Yq (Norm.nEO n eps x g b (I0 n).toInt64.toFloat) (Norm.nLR n eps x r)
          (G0 (r*n)) (G0 (r*n)) (G0 (r*n+n)) out:=by
  have hm:0 < n:=by
    simp only [Η,AJ] at hN; omega
  unfold cRow; dsimp only; split
  · exact cB n eps x g b cb r out ho hb hos hX heps hN F D hcb (by assumption)
  · exact Norm.nRow_eq n eps x g b Xr gd bd ge _ rm nf ep z0 L hC r out hr hg hd he hx ho hb hos hX heps hG hB hE hN
theorem cRow_bad (n:ℕ) (eps:Word) (x g b:Ἂ Word) (Xr gd bd:ἀ) (ge:ByteArray) (nd:Ἅ)
    (rm nf ep z0:Ἅ) (L:ἆ) (hC:Norm.nCK n eps rm nf ep z0 L) (cb:Ἂ Word) (r:ℕ) (out:Ἂ Word)
    (hr:n+2 ≤ Xr.size) (hg:n ≤ gd.size) (hd:n ≤ bd.size) (he:n ≤ ge.size)
    (hx:r*n+n ≤ x.size) (ho:r*n+n ≤ out.size) (hb:r*n+n < 2^(32:ℕ))
    (j:ℕ) (j1:r*n ≤ j) (j2:j < r*n+n) (hj:expOf x[j]! = 255) :
    (cRow n eps x g b Xr gd bd ge nd rm nf ep z0 L hC cb r out hr hg hd he hx ho hb).2 = #[]:=by
  unfold cRow; dsimp only; split
  · exact (cBb n x r hb j j1 j2 hj (by assumption)).elim
  · exact Norm.nRow_bad n eps x g b Xr gd bd ge nd rm nf ep z0 L hC r out hr hg hd he hx ho hb j j1 j2 hj
theorem cRows_eq (rows n:ℕ) (eps:Word) (x g b:Ἂ Word) (gd bd:ἀ) (ge:ByteArray)
    (rm nf ep z0:Ἅ) (L:ἆ) (hC:Norm.nCK n eps rm nf ep z0 L) (cb:Ἂ Word)
    (hg:n ≤ gd.size) (hd:n ≤ bd.size) (he:n ≤ ge.size) (hx:rows*n ≤ x.size) (hb:rows*n < 2^(32:ℕ))
    (hX : ∀ j,j < rows*n → expOf x[j]! ≠ 255) (heps:0 < rationalWord eps)
    (hG : ∀ t,t < n → gd.data[t]! = Dense.F.dw g[t]!) (hB : ∀ t,t < n → bd.data[t]! = Dense.F.dw b[t]!)
    (hE : ∀ t,t < n → (ge.data[t]!).toUInt64=Norm.nfe g[t]!)
    (hN:(4 ≤ n&&n ≤ 2048&&expOf eps != 0&&expOf eps != 255)=true)
    (F : ∀ j,j < n → expOf g[j]! ≠ 255) (D : ∀ j,j < n → expOf b[j]! ≠ 255)
    (hcb : ∀ j,j < n → cb[j]! = roundRational (rationalWord b[j]!)) :
    ∀ k (r:ℕ) (Xr:ἀ) (out:Ἂ Word) (hr:n+2 ≤ Xr.size) (ho:rows*n ≤ out.size) (hk:r+k=rows),
    out.size=rows*n →
    cRows rows n eps x g b gd bd ge (I0 n).toInt64.toFloat rm nf ep z0 L hC cb hg hd he hx hb k r Xr out hr ho hk
      = Norm.Yr (fun r=>Norm.nLR n eps x r) (Norm.nEO n eps x g b (I0 n).toInt64.toFloat) n rows r out:=by
  intro k; induction k with
  | zero =>
    intro r Xr out hr ho hk hos; rw [Norm.Yr,if_neg (by rw [← hk]; exact B4 r)]; rfl
  | succ k ih =>
    intro r Xr out hr ho hk hos
    have kk:=Norm.nHk2 (n:=n) hk
    rw [Norm.Yr,if_pos (hk ▸ P5 (Nat.succ_pos k))]
    have E:=cRow_eq n eps x g b Xr gd bd ge rm nf ep z0 L hC cb r out hr hg hd he (kk.trans hx) (kk.trans ho)
      (C0 kk hb) (by rw [hos]; exact hb) (fun j hj=>hX j (U1 hj kk)) heps hG hB hE hN F D hcb
    have S2:(cRow n eps x g b Xr gd bd ge (I0 n).toInt64.toFloat rm nf ep z0 L hC cb r out hr hg hd he (kk.trans hx)
        (kk.trans ho) (C0 kk hb)).2.size=rows*n:=by rw [E,Norm.nYq_sz _ _ _ _ _ _ _ rfl,hos]
    rw [cRows,dif_pos S2.ge]
    refine (ih (r+1) _ _ _ _ (Norm.nHk3 hk) S2).trans ?_
    rw [E]
theorem cRows_bad (rows n:ℕ) (eps:Word) (x g b:Ἂ Word) (gd bd:ἀ) (ge:ByteArray) (nd:Ἅ)
    (rm nf ep z0:Ἅ) (L:ἆ) (hC:Norm.nCK n eps rm nf ep z0 L) (cb:Ἂ Word)
    (hg:n ≤ gd.size) (hd:n ≤ bd.size) (he:n ≤ ge.size) (hx:rows*n ≤ x.size) (hb:rows*n < 2^(32:ℕ)) (hn:0 < n) :
    ∀ k (r:ℕ) (Xr:ἀ) (out:Ἂ Word) (hr:n+2 ≤ Xr.size) (ho:rows*n ≤ out.size) (hk:r+k=rows) (j:ℕ),
    r*n ≤ j → j < rows*n → expOf x[j]! = 255 →
    cRows rows n eps x g b gd bd ge nd rm nf ep z0 L hC cb hg hd he hx hb k r Xr out hr ho hk = #[]:=by
  intro k; induction k with
  | zero=>intro r Xr out hr ho hk j j1 j2 hj; rw [ϑ] at hk; subst hk; omega
  | succ k ih =>
    intro r Xr out hr ho hk j j1 j2 hj
    have kk:=Norm.nHk2 (n:=n) hk
    rw [cRows]; split
    · rename_i S
      by_cases c:j < r*n+n
      · rw [cRow_bad n eps x g b Xr gd bd ge nd rm nf ep z0 L hC cb r out hr hg hd he (kk.trans hx) (kk.trans ho)
          (C0 kk hb) j j1 c hj] at S
        simp only [Array.size_empty] at S; omega
      · exact ih (r+1) _ _ _ _ (Norm.nHk3 hk) j (by rw [Ϡ,n4]; omega) j2 hj
    · rfl
theorem s1S {x:Ἂ Word} {i e:Ἄ} {s:Ἅ} {Dl:ἆ} (h:i < e) :
    Norm.nS1 x i e s Dl=Norm.nS1 x (i+1) e (s+Dense.F.dw (Norm.ck x i))
      (if Dl < Norm.nfe (Norm.ck x i) then Norm.nfe (Norm.ck x i) else Dl):=by
  rw [Norm.nS1.eq_def x i e s Dl]; simp only [dif_pos h]
theorem s1E {x:Ἂ Word} {i e:Ἄ} {s:Ἅ} {Dl:ἆ} (h:¬i < e) : Norm.nS1 x i e s Dl=(s,Dl):=by
  rw [Norm.nS1.eq_def x i e s Dl]; simp only [dif_neg h]
theorem s2S {x:Ἂ Word} {m:Ἅ} {i e:Ἄ} {v:Ἅ} (h:i < e) :
    Norm.nS2 x m i e v=Norm.nS2 x m (i+1) e (v+(Dense.F.dw (Norm.ck x i)-m)*(Dense.F.dw (Norm.ck x i)-m)):=by
  rw [Norm.nS2.eq_def x m i e v]; simp only [dif_pos h]
theorem s2E {x:Ἂ Word} {m:Ἅ} {i e:Ἄ} {v:Ἅ} (h:¬i < e) : Norm.nS2 x m i e v=v:=by
  rw [Norm.nS2.eq_def x m i e v]; simp only [dif_neg h]
@[inline] def oE {β:Type} (n:ℕ) (eps:Word) (x g b:Ἂ Word) (nd:Ἅ) (r:ℕ) (dW:Word) (m ri:Ἅ) (Kp:ἆ)
    (v gw bw:Ἅ) (gx:ἆ) (i j:Ἄ) (ex:Option Norm.NR.fH) (k:Word → Option Norm.NR.fH → β):β:=
  let B:=((v-m)*(gw*ri)+bw).toBits
  if Norm.nOK B (Norm.nC (gx+Kp) ((B>>>52)&&&2047)) then k (Norm.nW B) ex
  else let p:=Norm.nFB n eps x g b nd r dW ex i j; k p.1 (some p.2)
theorem oE_c {β:Type} {n:ℕ} {eps:Word} {x g b:Ἂ Word} {nd:Ἅ} {r:ℕ} {dW:Word} {m ri:Ἅ} {Kp:ἆ}
    {v gw bw v' gw' bw':Ἅ} {gx gx':ἆ} {i j:Ἄ} {ex:Option Norm.NR.fH} {k k':Word → Option Norm.NR.fH → β}
    (h1:v=v') (h2:gw=gw') (h3:bw=bw') (h4:gx=gx') (h5:∀ o e,k o e=k' o e) :
    oE n eps x g b nd r dW m ri Kp v gw bw gx i j ex k=oE n eps x g b nd r dW m ri Kp v' gw' bw' gx' i j ex k':=by
  subst h1 h2 h3 h4; rw [show k=k' from funext fun o=>funext fun e=>h5 o e]
theorem f2S {n:ℕ} {eps:Word} {x g b:Ἂ Word} {nd m ri:Ἅ} {Kp:ἆ} {r:ℕ} {dW:Word} {ex:Option Norm.NR.fH} {lo t nn:Ἄ}
    {hg:nn.toNat ≤ g.size} {hd:nn.toNat ≤ b.size} {hx:lo.toNat+nn.toNat ≤ x.size} {hb:lo.toNat+nn.toNat < 2^(32:ℕ)}
    {out:Ἂ Word} {ho:lo.toNat+nn.toNat ≤ out.size} (h:t < nn) :
    Norm.nFE2 n eps x g b nd m ri Kp r dW ex lo t nn hg hd hx hb out ho
      = oE n eps x g b nd r dW m ri Kp (Dense.F.dw (x.uget (lo+t) (Norm.nH1 (ε.mp h) hb hx)))
          (Dense.F.dw (g.uget t (U1 (ε.mp h) hg))) (Dense.F.dw (b.uget t (U1 (ε.mp h) hd)))
          (Norm.nfe (g.uget t (U1 (ε.mp h) hg))) (lo+t) t ex
          (fun o e=>Norm.nFE2 n eps x g b nd m ri Kp r dW e lo (t+1) nn hg hd hx hb
            (out.uset (lo+t) o (Norm.nH1 (ε.mp h) hb ho)) (Norm.nH3 _ _ _ _ ho)):=by
  rw [Norm.nFE2.eq_def n eps x g b nd m ri Kp r dW ex lo t nn hg hd hx hb out ho]; simp only [dif_pos h,oE]
theorem f2E {n:ℕ} {eps:Word} {x g b:Ἂ Word} {nd m ri:Ἅ} {Kp:ἆ} {r:ℕ} {dW:Word} {ex:Option Norm.NR.fH} {lo t nn:Ἄ}
    {hg:nn.toNat ≤ g.size} {hd:nn.toNat ≤ b.size} {hx:lo.toNat+nn.toNat ≤ x.size} {hb:lo.toNat+nn.toNat < 2^(32:ℕ)}
    {out:Ἂ Word} {ho:lo.toNat+nn.toNat ≤ out.size} (h:¬t < nn) :
    Norm.nFE2 n eps x g b nd m ri Kp r dW ex lo t nn hg hd hx hb out ho=out:=by
  rw [Norm.nFE2.eq_def n eps x g b nd m ri Kp r dW ex lo t nn hg hd hx hb out ho]; simp only [dif_neg h]
theorem cku {x:Ἂ Word} {i j:Ἄ} (hij:i=j) (h:j.toNat < x.size) : Norm.ck x i=x.uget j h:=by
  subst hij; rw [Norm.f6,Dense.cC]
theorem k4c {g b:Ἂ Word} {gd bd:ἀ} {ge:ByteArray} (t:Ἄ) {n:ℕ} (ht:t.toNat < n)
    (hG : ∀ t,t < n → gd.data[t]! = Dense.F.dw g[t]!) (hB : ∀ t,t < n → bd.data[t]! = Dense.F.dw b[t]!)
    (hE : ∀ t,t < n → (ge.data[t]!).toUInt64=Norm.nfe g[t]!)
    (p1:n ≤ gd.size) (p2:n ≤ bd.size) (p3:n ≤ ge.size) (p4:n ≤ g.size) (p5:n ≤ b.size) :
    gd.uget t (U1 ht p1)=Dense.F.dw (g.uget t (U1 ht p4)) ∧
    bd.uget t (U1 ht p2)=Dense.F.dw (b.uget t (U1 ht p5)) ∧
    (ge.uget t (U1 ht p3)).toUInt64=Norm.nfe (g.uget t (U1 ht p4)):=by
  refine ⟨?_,?_,?_⟩
  · rw [Dense.F.kFU,hG _ ht,Dense.cC]
  · rw [Dense.F.kFU,hB _ ht,Dense.cC]
  · rw [Norm.nBu,hE _ ht,Dense.cC]
@[inline] def sLQ {β:Type} (eps:Word) (s:Ἅ) (Dl:ἆ) (vf:Ἅ->Ἅ) (rm nf ep:Ἅ) (L:ἆ) (k:Ἅ->Ἅ->ἆ->Word->Ἆ->β):β:=
  let m:=s*rm
  let v:=vf m
  let D:=W7 (v*nf+ep)
  let B:=D.toBits
  let dW:=Norm.nWs 26 B
  let ed:=expOf dW
  let E:=((m.toBits>>>52)&&&2047)+9
  let a:=L+Dl+1052
  let c:=2*L+2*Dl
  let f:=(expOf eps).toUInt64+178
  let d:=if f ≤ c then c-f else 0
  k m (Norm.nRi dW ed.toUInt64) ((if a ≤ E then E else a)-ed.toUInt64) dW
    (Norm.nOKs B ((if L+4 ≤ d then d else L+4)+2)&&0<ed&&ed<255&&!signOf dW)
theorem e8 (w:Word):(expOf w).toNat ≤ 255:=by
  unfold expOf; rw [Ξ]; exact ϝ
theorem kpU (L Dl Em:ἆ) (ed:Word) (hL:L.toNat ≤ 64) (hD:Dl.toNat < 255) (hE:Em.toNat ≤ 2047) (he:ed.toNat ≤ 255) :
    (if L+Dl+1052 ≤ Em+9 then Em+9 else L+Dl+1052)-ed.toUInt64
      = (max (L.toNat+Dl.toNat+1052) (Em.toNat+9)-ed.toNat).toUInt64:=by
  have a1:(L+Dl+1052).toNat=L.toNat+Dl.toNat+1052:=by simp only [η,mY]; omega
  have a2:(Em+9).toNat=Em.toNat+9:=by simp only [η,mY]; omega
  have a3:ed.toUInt64.toNat=ed.toNat:=ϩ ed
  apply ξ.mp
  rw [Dense.F.kOfS _ (by omega)]
  split
  · rename_i h; rw [θ,a1,a2] at h
    rw [χ _ _ (θ.mpr (by rw [a2,a3]; omega)),a2,a3]; omega
  · rename_i h; rw [θ,a1,a2] at h
    rw [χ _ _ (θ.mpr (by rw [a1,a3]; omega)),a1,a3]; omega
theorem lcU (L Dl:ἆ) (ef:Word) (hL:L.toNat ≤ 64) (hD:Dl.toNat < 255) (he:ef.toNat ≤ 255) :
    (if L+4 ≤ (if ef.toUInt64+178 ≤ 2*L+2*Dl then 2*L+2*Dl-(ef.toUInt64+178) else 0)
      then (if ef.toUInt64+178 ≤ 2*L+2*Dl then 2*L+2*Dl-(ef.toUInt64+178) else 0) else L+4)+2
      = (Norm.nLc2 L.toNat Dl.toNat ef.toNat).toUInt64:=by
  unfold Norm.nLc2
  have a1:(2*L+2*Dl).toNat=2*L.toNat+2*Dl.toNat:=by simp only [η,ύ,mY]; omega
  have a2:(ef.toUInt64+178).toNat=ef.toNat+178:=by simp only [η,mY,ϩ]; omega
  have a3:(L+4).toNat=L.toNat+4:=by simp only [η,mY]; omega
  apply ξ.mp
  rw [Dense.F.kOfS _ (by omega)]
  by_cases c1:ef.toUInt64+178 ≤ 2*L+2*Dl
  · have c1':=θ.mp c1; rw [a1,a2] at c1'
    have a4:(2*L+2*Dl-(ef.toUInt64+178)).toNat=2*L.toNat+2*Dl.toNat-(ef.toNat+178):=by
      rw [χ _ _ c1,a1,a2]
    simp only [c1,↓reduceIte]
    split
    · rename_i h; rw [θ,a3,a4] at h
      simp only [η,mY,a4]; omega
    · rename_i h; rw [θ,a3,a4] at h
      simp only [η,mY,a3]; omega
  · have c1':=mt θ.mpr c1; rw [a1,a2] at c1'
    simp only [c1,↓reduceIte]
    split
    · rename_i h; rw [θ,a3] at h; simp at h
    · simp only [η,mY,a3]; omega
theorem sLQ_e {β:Type} (eps:Word) (s:Ἅ) (Dl:ἆ) (vf:Ἅ->Ἅ) (rm nf ep:Ἅ) (L:ἆ) (k:Ἅ->Ἅ->ἆ->Word->Ἆ->β)
    (hL:L.toNat ≤ 64) (hD:Dl.toNat < 255) : sLQ eps s Dl vf rm nf ep L k=Norm.nLQk eps s Dl vf rm nf ep L k:=by
  unfold sLQ Norm.nLQk Norm.nElK3
  dsimp only
  rw [kpU _ _ _ _ hL hD (by rw [Ε]; exact ϝ) (e8 _),lcU _ _ _ hL hD (e8 _)]
theorem nCKL {n:ℕ} {eps:Word} {rm nf ep z0:Ἅ} {L:ἆ} (hC:Norm.nCK n eps rm nf ep z0 L) (hn:n < 2^(32:ℕ)) : L.toNat ≤ 64:=by
  obtain ⟨-,-,-,-,h⟩:=hC; subst h
  have h1:Aq (n+4) < 33:=(m9 (by omega)).mpr (by omega)
  rw [Norm.kZ1 (n+4) (by omega),Dense.F.kOfS _ (by omega)]; omega
theorem kU {t:Ἄ} {k n N s:ℕ} (ht:t.toNat=k) (hN:n=N) (hk:k < N) (h:n ≤ s):t.toNat < s:=by omega
theorem kQ {r n k s:ℕ} {t:Ἄ} (ht:t.toNat=k) (hk:k < n) (hb:r*n+n < 2^(32:ℕ)) (ho:r*n+n ≤ s) :
    (G0 (r*n)+t).toNat < s:=by
  have q1:(G0 (r*n)).toNat=r*n:=Conv.aa _ (C0 (V1 _ _) hb)
  rw [Norm.nCa (by rw [q1,ht]; omega),q1,ht]; omega
theorem cQS {x:Ἂ Word} {w0:Word} {i e:Ἄ} (h:i < e) :
    cQ x w0 i e=if (Norm.ck x i==w0||(Norm.ck x i<<<1==0&&w0<<<1==0)) then cQ x w0 (i+1) e else false:=by
  rw [cQ.eq_def x w0 i e]; simp only [dif_pos h]
theorem cQE {x:Ἂ Word} {w0:Word} {i e:Ἄ} (h:¬i < e) : cQ x w0 i e=true:=by
  rw [cQ.eq_def x w0 i e]; simp only [dif_neg h]
theorem kQl {r n k s:ℕ} {lo t:Ἄ} (hlo:lo=G0 (r*n)) (ht:t.toNat=k) (hk:k < n) (hb:r*n+n < 2^(32:ℕ)) (ho:r*n+n ≤ s) :
    (lo+t).toNat < s:=by subst hlo; exact kQ ht hk hb ho
theorem loS {r n:ℕ} {lo nn:Ἄ} (hlo:lo.toNat=r*n) (hnn:nn.toNat=n) (hb:r*n+n < 2^(32:ℕ)) : (lo+nn).toNat=(r+1)*n:=by
  rw [Norm.nCa (by rw [hlo,hnn]; exact hb),hlo,hnn,Ο]
theorem lE {lo:Ἄ} {m:ℕ} (h:lo.toNat=m) (hm:m < 2^(32:ℕ)) : lo=G0 m:=O5.1 (h.trans (Conv.aa _ hm).symm)
def cRw (n:ℕ) (eps:Word) (x g b:Ἂ Word) (nd rm nf ep z0:Ἅ) (L:ἆ) (hC:Norm.nCK n eps rm nf ep z0 L) (cb:Ἂ Word) (r:ℕ) (out:Ἂ Word)
    (hg':n ≤ g.size) (hd':n ≤ b.size) (hx:r*n+n ≤ x.size) (ho:r*n+n ≤ out.size) (hb:r*n+n < 2^(32:ℕ)):Ἂ Word:=
  if (Norm.nfe (Norm.ck x (G0 (r*n))) < 255&&cQ x (Norm.ck x (G0 (r*n))) (G0 (r*n)) (G0 (r*n+n)))=true then
    Norm.Yq (fun (c:Ἂ Word) (_:Ἄ) (j:Ἄ)=>Norm.ck c j) cb (G0 (r*n)) (G0 (r*n)) (G0 (r*n+n)) out
  else Norm.nRow1 n eps x g b nd rm nf ep z0 L hC r out hg' hd' hx ho hb
theorem uT0:(0:Ἄ).toNat=0:=Ϧ
theorem uT1:(0+1:Ἄ).toNat=1:=by rw [Norm.nCa (by rw [uT0,mA]; decide),uT0,mA]
theorem uT2:(0+1+1:Ἄ).toNat=2:=by rw [Norm.nCa (by rw [uT1,mA]; decide),uT1,mA]
theorem uT3:(0+1+1+1:Ἄ).toNat=3:=by rw [Norm.nCa (by rw [uT2,mA]; decide),uT2,mA]
theorem uT4:(0+1+1+1+1:Ἄ).toNat=4:=by rw [Norm.nCa (by rw [uT3,mA]; decide),uT3,mA]
theorem uT5:(0+1+1+1+1+1:Ἄ).toNat=5:=by rw [Norm.nCa (by rw [uT4,mA]; decide),uT4,mA]
theorem uT6:(0+1+1+1+1+1+1:Ἄ).toNat=6:=by rw [Norm.nCa (by rw [uT5,mA]; decide),uT5,mA]
theorem uT7:(0+1+1+1+1+1+1+1:Ἄ).toNat=7:=by rw [Norm.nCa (by rw [uT6,mA]; decide),uT6,mA]
theorem uT8:(0+1+1+1+1+1+1+1+1:Ἄ).toNat=8:=by rw [Norm.nCa (by rw [uT7,mA]; decide),uT7,mA]
@[inline] def k4O (out:Ἂ Word) (lo:Ἄ) (o0 o1 o2 o3:Word) (h0:(lo+0).toNat < out.size) (h1:(lo+(0+1)).toNat < out.size) (h2:(lo+(0+1+1)).toNat < out.size) (h3:(lo+(0+1+1+1)).toNat < out.size):Ἂ Word:=
  ((((out.uset (lo+0) o0 h0).uset (lo+(0+1)) o1 (by rw [BT]; exact h1)).uset (lo+(0+1+1)) o2 (by rw [BT,BT]; exact h2)).uset (lo+(0+1+1+1)) o3 (by rw [BT,BT,BT]; exact h3))
@[inline] def k4 (n:ℕ) (hN:n=4) (eps:Word) (x g b:Ἂ Word) (gd bd:ἀ) (ge:ByteArray) (nd:Ἅ)
    (rm nf ep z0:Ἅ) (L:ἆ) (cb:Ἂ Word) (r:ℕ) (lo:Ἄ) (hlo:lo=G0 (r*n)) (out:Ἂ Word)
    (hg:n ≤ gd.size) (hd:n ≤ bd.size) (he:n ≤ ge.size) (ho:r*n+n ≤ out.size) (hb:r*n+n < 2^(32:ℕ)):Ἂ Word:=
  let w0:=Norm.ck x (lo); let w1:=Norm.ck x (lo+1); let w2:=Norm.ck x (lo+1+1); let w3:=Norm.ck x (lo+1+1+1)
  if (Norm.nfe w0 < 255&&(if (w0==w0||(w0<<<1==0&&w0<<<1==0)) then if (w1==w0||(w1<<<1==0&&w0<<<1==0)) then if (w2==w0||(w2<<<1==0&&w0<<<1==0)) then if (w3==w0||(w3<<<1==0&&w0<<<1==0)) then true else false else false else false else false))=true then
    Norm.Yq (fun (c:Ἂ Word) (_:Ἄ) (j:Ἄ)=>Norm.ck c j) cb lo lo (G0 (r*n+n)) out
  else
  let v0:=Dense.F.dw w0; let v1:=Dense.F.dw w1; let v2:=Dense.F.dw w2; let v3:=Dense.F.dw w3
  let D0:=if (0:ἆ) < Norm.nfe w0 then Norm.nfe w0 else 0
  let D1:=if D0 < Norm.nfe w1 then Norm.nfe w1 else D0
  let D2:=if D1 < Norm.nfe w2 then Norm.nfe w2 else D1
  let D3:=if D2 < Norm.nfe w3 then Norm.nfe w3 else D2
  if D3 < 255 then
    sLQ eps ((((z0+v0)+v1)+v2)+v3) D3
      (fun m=>((((z0+(v0-m)*(v0-m))+(v1-m)*(v1-m))+(v2-m)*(v2-m))+(v3-m)*(v3-m))) rm nf ep L
      (fun m ri Kp dW fast=>
        if fast then
          oE n eps x g b nd r dW m ri Kp v0 (gd.uget (0) (kU uT0 hN (by decide) hg)) (bd.uget (0) (kU uT0 hN (by decide) hd))
            (ge.uget (0) (kU uT0 hN (by decide) he)).toUInt64 (lo+0) (0) none (fun o0 e1=>
          oE n eps x g b nd r dW m ri Kp v1 (gd.uget (0+1) (kU uT1 hN (by decide) hg)) (bd.uget (0+1) (kU uT1 hN (by decide) hd))
            (ge.uget (0+1) (kU uT1 hN (by decide) he)).toUInt64 (lo+(0+1)) (0+1) e1 (fun o1 e2=>
          oE n eps x g b nd r dW m ri Kp v2 (gd.uget (0+1+1) (kU uT2 hN (by decide) hg)) (bd.uget (0+1+1) (kU uT2 hN (by decide) hd))
            (ge.uget (0+1+1) (kU uT2 hN (by decide) he)).toUInt64 (lo+(0+1+1)) (0+1+1) e2 (fun o2 e3=>
          oE n eps x g b nd r dW m ri Kp v3 (gd.uget (0+1+1+1) (kU uT3 hN (by decide) hg)) (bd.uget (0+1+1+1) (kU uT3 hN (by decide) hd))
            (ge.uget (0+1+1+1) (kU uT3 hN (by decide) he)).toUInt64 (lo+(0+1+1+1)) (0+1+1+1) e3 (fun o3 _=>
            k4O out lo o0 o1 o2 o3 (kQl hlo uT0 (by rw [hN]; decide) hb ho) (kQl hlo uT1 (by rw [hN]; decide) hb ho) (kQl hlo uT2 (by rw [hN]; decide) hb ho) (kQl hlo uT3 (by rw [hN]; decide) hb ho)))))
        else Norm.nRowS n eps x g b nd r lo (G0 (r*n+n)) out)
  else #[]
theorem k4_eq (n:ℕ) (hN:n=4) (eps:Word) (x g b:Ἂ Word) (gd bd:ἀ) (ge:ByteArray) (nd:Ἅ)
    (rm nf ep z0:Ἅ) (L:ἆ) (cb:Ἂ Word) (r:ℕ) (lo:Ἄ) (hlo:lo=G0 (r*n)) (out:Ἂ Word)
    (hg:n ≤ gd.size) (hd:n ≤ bd.size) (he:n ≤ ge.size) (ho:r*n+n ≤ out.size) (hb:r*n+n < 2^(32:ℕ))
    (hC:Norm.nCK n eps rm nf ep z0 L) (hg':n ≤ g.size) (hd':n ≤ b.size) (hx:r*n+n ≤ x.size)
    (hG : ∀ t,t < n → gd.data[t]! = Dense.F.dw g[t]!) (hB : ∀ t,t < n → bd.data[t]! = Dense.F.dw b[t]!)
    (hE : ∀ t,t < n → (ge.data[t]!).toUInt64=Norm.nfe g[t]!) :
    k4 n hN eps x g b gd bd ge nd rm nf ep z0 L cb r lo hlo out hg hd he ho hb=cRw n eps x g b nd rm nf ep z0 L hC cb r out hg' hd' hx ho hb:=by
  subst hN hlo
  have q1:(G0 (r*4)).toNat=r*4:=Conv.aa _ (C0 (V1 _ _) hb)
  have q3:(G0 (r*4+4)).toNat=r*4+4:=Conv.aa _ hb
  have q2:(G0 4).toNat=4:=Conv.aa _ (C0 (K2 _ _) hb)
  have o1:(1:Ἄ).toNat=1:=mA
  have b1:(G0 (r*4)+1).toNat=r*4+1:=by rw [Norm.nCa (by rw [q1,o1]; omega),q1,o1]
  have b2:(G0 (r*4)+1+1).toNat=r*4+2:=by rw [Norm.nCa (by rw [b1,o1]; omega),b1,o1]
  have b3:(G0 (r*4)+1+1+1).toNat=r*4+3:=by rw [Norm.nCa (by rw [b2,o1]; omega),b2,o1]
  have b4:(G0 (r*4)+1+1+1+1).toNat=r*4+4:=by rw [Norm.nCa (by rw [b3,o1]; omega),b3,o1]
  have c0:(G0 (r*4)+0).toNat=r*4:=by rw [Norm.nCa (by rw [q1,uT0]; omega),q1,uT0]; rfl
  have c1:(G0 (r*4)+(0+1)).toNat=r*4+1:=by rw [Norm.nCa (by rw [q1,uT1]; omega),q1,uT1]
  have c2:(G0 (r*4)+(0+1+1)).toNat=r*4+2:=by rw [Norm.nCa (by rw [q1,uT2]; omega),q1,uT2]
  have c3:(G0 (r*4)+(0+1+1+1)).toNat=r*4+3:=by rw [Norm.nCa (by rw [q1,uT3]; omega),q1,uT3]
  have l0:G0 (r*4) < G0 (r*4+4):=ε.mpr (by rw [q1,q3]; exact P5 (by decide))
  have l1:G0 (r*4)+1 < G0 (r*4+4):=ε.mpr (by rw [b1,q3]; exact O1 (by decide) _)
  have l2:G0 (r*4)+1+1 < G0 (r*4+4):=ε.mpr (by rw [b2,q3]; exact O1 (by decide) _)
  have l3:G0 (r*4)+1+1+1 < G0 (r*4+4):=ε.mpr (by rw [b3,q3]; exact O1 (by decide) _)
  have l4:¬G0 (r*4)+1+1+1+1 < G0 (r*4+4):=fun h=>by have:=ε.mp h; rw [b4,q3] at this; exact B4 _ this
  have m0:(0:Ἄ) < G0 4:=ε.mpr (by rw [uT0,q2]; decide)
  have m1:(0+1:Ἄ) < G0 4:=ε.mpr (by rw [uT1,q2]; decide)
  have m2:(0+1+1:Ἄ) < G0 4:=ε.mpr (by rw [uT2,q2]; decide)
  have m3:(0+1+1+1:Ἄ) < G0 4:=ε.mpr (by rw [uT3,q2]; decide)
  have m4:¬(0+1+1+1+1:Ἄ) < G0 4:=fun h=>by have:=ε.mp h; rw [uT4,q2] at this; exact B4 _ this
  unfold k4 cRw Norm.nRow1; dsimp only
  rw [cQS l0,cQS l1,cQS l2,cQS l3,cQE l4]
  refine ite_congr rfl (fun _=>rfl) (fun _=>?_)
  rw [s1S l0,s1S l1,s1S l2,s1S l3,s1E l4]
  simp only [s2S l0,s2S l1,s2S l2,s2S l3,s2E l4]
  refine ite_congr rfl (fun hD=>?_) (fun _=>rfl)
  rw [sLQ_e _ _ _ _ _ _ _ _ _ (nCKL hC (by decide)) (by have:=μ.mp hD; simpa using this)]
  congr 1; funext m ri Kp dW fast
  cases fast
  · simp only [υ,↓reduceIte]
  · simp only [↓reduceIte]
    obtain ⟨e01,e02,e03⟩:=k4c (0) (n:=4) (by rw [uT0]; decide) hG hB hE hg hd he hg' hd'
    obtain ⟨e11,e12,e13⟩:=k4c (0+1) (n:=4) (by rw [uT1]; decide) hG hB hE hg hd he hg' hd'
    obtain ⟨e21,e22,e23⟩:=k4c (0+1+1) (n:=4) (by rw [uT2]; decide) hG hB hE hg hd he hg' hd'
    obtain ⟨e31,e32,e33⟩:=k4c (0+1+1+1) (n:=4) (by rw [uT3]; decide) hG hB hE hg hd he hg' hd'
    rw [f2S m0]; refine oE_c (congrArg _ (cku (O5.1 (by rw [q1,c0])) _)) e01 e02 e03 (fun o0 e1=>?_)
    rw [f2S m1]; refine oE_c (congrArg _ (cku (O5.1 (by rw [b1,c1])) _)) e11 e12 e13 (fun o1 e2=>?_)
    rw [f2S m2]; refine oE_c (congrArg _ (cku (O5.1 (by rw [b2,c2])) _)) e21 e22 e23 (fun o2 e3=>?_)
    rw [f2S m3]; refine oE_c (congrArg _ (cku (O5.1 (by rw [b3,c3])) _)) e31 e32 e33 (fun o3 e4=>?_)
    rw [f2E m4]; rfl
@[inline] def k8O (out:Ἂ Word) (lo:Ἄ) (o0 o1 o2 o3 o4 o5 o6 o7:Word) (h0:(lo+0).toNat < out.size) (h1:(lo+(0+1)).toNat < out.size) (h2:(lo+(0+1+1)).toNat < out.size) (h3:(lo+(0+1+1+1)).toNat < out.size) (h4:(lo+(0+1+1+1+1)).toNat < out.size) (h5:(lo+(0+1+1+1+1+1)).toNat < out.size) (h6:(lo+(0+1+1+1+1+1+1)).toNat < out.size) (h7:(lo+(0+1+1+1+1+1+1+1)).toNat < out.size):Ἂ Word:=
  ((((((((out.uset (lo+0) o0 h0).uset (lo+(0+1)) o1 (by rw [BT]; exact h1)).uset (lo+(0+1+1)) o2 (by rw [BT,BT]; exact h2)).uset (lo+(0+1+1+1)) o3 (by rw [BT,BT,BT]; exact h3)).uset (lo+(0+1+1+1+1)) o4 (by rw [BT,BT,BT,BT]; exact h4)).uset (lo+(0+1+1+1+1+1)) o5 (by rw [BT,BT,BT,BT,BT]; exact h5)).uset (lo+(0+1+1+1+1+1+1)) o6 (by rw [BT,BT,BT,BT,BT,BT]; exact h6)).uset (lo+(0+1+1+1+1+1+1+1)) o7 (by rw [BT,BT,BT,BT,BT,BT,BT]; exact h7))
@[inline] def k8 (n:ℕ) (hN:n=8) (eps:Word) (x g b:Ἂ Word) (gd bd:ἀ) (ge:ByteArray) (nd:Ἅ)
    (rm nf ep z0:Ἅ) (L:ἆ) (cb:Ἂ Word) (r:ℕ) (lo:Ἄ) (hlo:lo=G0 (r*n)) (out:Ἂ Word)
    (hg:n ≤ gd.size) (hd:n ≤ bd.size) (he:n ≤ ge.size) (ho:r*n+n ≤ out.size) (hb:r*n+n < 2^(32:ℕ)):Ἂ Word:=
  let w0:=Norm.ck x (lo); let w1:=Norm.ck x (lo+1); let w2:=Norm.ck x (lo+1+1); let w3:=Norm.ck x (lo+1+1+1); let w4:=Norm.ck x (lo+1+1+1+1); let w5:=Norm.ck x (lo+1+1+1+1+1); let w6:=Norm.ck x (lo+1+1+1+1+1+1); let w7:=Norm.ck x (lo+1+1+1+1+1+1+1)
  if (Norm.nfe w0 < 255&&(if (w0==w0||(w0<<<1==0&&w0<<<1==0)) then if (w1==w0||(w1<<<1==0&&w0<<<1==0)) then if (w2==w0||(w2<<<1==0&&w0<<<1==0)) then if (w3==w0||(w3<<<1==0&&w0<<<1==0)) then if (w4==w0||(w4<<<1==0&&w0<<<1==0)) then if (w5==w0||(w5<<<1==0&&w0<<<1==0)) then if (w6==w0||(w6<<<1==0&&w0<<<1==0)) then if (w7==w0||(w7<<<1==0&&w0<<<1==0)) then true else false else false else false else false else false else false else false else false))=true then
    Norm.Yq (fun (c:Ἂ Word) (_:Ἄ) (j:Ἄ)=>Norm.ck c j) cb lo lo (G0 (r*n+n)) out
  else
  let v0:=Dense.F.dw w0; let v1:=Dense.F.dw w1; let v2:=Dense.F.dw w2; let v3:=Dense.F.dw w3; let v4:=Dense.F.dw w4; let v5:=Dense.F.dw w5; let v6:=Dense.F.dw w6; let v7:=Dense.F.dw w7
  let D0:=if (0:ἆ) < Norm.nfe w0 then Norm.nfe w0 else 0
  let D1:=if D0 < Norm.nfe w1 then Norm.nfe w1 else D0
  let D2:=if D1 < Norm.nfe w2 then Norm.nfe w2 else D1
  let D3:=if D2 < Norm.nfe w3 then Norm.nfe w3 else D2
  let D4:=if D3 < Norm.nfe w4 then Norm.nfe w4 else D3
  let D5:=if D4 < Norm.nfe w5 then Norm.nfe w5 else D4
  let D6:=if D5 < Norm.nfe w6 then Norm.nfe w6 else D5
  let D7:=if D6 < Norm.nfe w7 then Norm.nfe w7 else D6
  if D7 < 255 then
    sLQ eps ((((((((z0+v0)+v1)+v2)+v3)+v4)+v5)+v6)+v7) D7
      (fun m=>((((((((z0+(v0-m)*(v0-m))+(v1-m)*(v1-m))+(v2-m)*(v2-m))+(v3-m)*(v3-m))+(v4-m)*(v4-m))+(v5-m)*(v5-m))+(v6-m)*(v6-m))+(v7-m)*(v7-m))) rm nf ep L
      (fun m ri Kp dW fast=>
        if fast then
          oE n eps x g b nd r dW m ri Kp v0 (gd.uget (0) (kU uT0 hN (by decide) hg)) (bd.uget (0) (kU uT0 hN (by decide) hd))
            (ge.uget (0) (kU uT0 hN (by decide) he)).toUInt64 (lo+0) (0) none (fun o0 e1=>
          oE n eps x g b nd r dW m ri Kp v1 (gd.uget (0+1) (kU uT1 hN (by decide) hg)) (bd.uget (0+1) (kU uT1 hN (by decide) hd))
            (ge.uget (0+1) (kU uT1 hN (by decide) he)).toUInt64 (lo+(0+1)) (0+1) e1 (fun o1 e2=>
          oE n eps x g b nd r dW m ri Kp v2 (gd.uget (0+1+1) (kU uT2 hN (by decide) hg)) (bd.uget (0+1+1) (kU uT2 hN (by decide) hd))
            (ge.uget (0+1+1) (kU uT2 hN (by decide) he)).toUInt64 (lo+(0+1+1)) (0+1+1) e2 (fun o2 e3=>
          oE n eps x g b nd r dW m ri Kp v3 (gd.uget (0+1+1+1) (kU uT3 hN (by decide) hg)) (bd.uget (0+1+1+1) (kU uT3 hN (by decide) hd))
            (ge.uget (0+1+1+1) (kU uT3 hN (by decide) he)).toUInt64 (lo+(0+1+1+1)) (0+1+1+1) e3 (fun o3 e4=>
          oE n eps x g b nd r dW m ri Kp v4 (gd.uget (0+1+1+1+1) (kU uT4 hN (by decide) hg)) (bd.uget (0+1+1+1+1) (kU uT4 hN (by decide) hd))
            (ge.uget (0+1+1+1+1) (kU uT4 hN (by decide) he)).toUInt64 (lo+(0+1+1+1+1)) (0+1+1+1+1) e4 (fun o4 e5=>
          oE n eps x g b nd r dW m ri Kp v5 (gd.uget (0+1+1+1+1+1) (kU uT5 hN (by decide) hg)) (bd.uget (0+1+1+1+1+1) (kU uT5 hN (by decide) hd))
            (ge.uget (0+1+1+1+1+1) (kU uT5 hN (by decide) he)).toUInt64 (lo+(0+1+1+1+1+1)) (0+1+1+1+1+1) e5 (fun o5 e6=>
          oE n eps x g b nd r dW m ri Kp v6 (gd.uget (0+1+1+1+1+1+1) (kU uT6 hN (by decide) hg)) (bd.uget (0+1+1+1+1+1+1) (kU uT6 hN (by decide) hd))
            (ge.uget (0+1+1+1+1+1+1) (kU uT6 hN (by decide) he)).toUInt64 (lo+(0+1+1+1+1+1+1)) (0+1+1+1+1+1+1) e6 (fun o6 e7=>
          oE n eps x g b nd r dW m ri Kp v7 (gd.uget (0+1+1+1+1+1+1+1) (kU uT7 hN (by decide) hg)) (bd.uget (0+1+1+1+1+1+1+1) (kU uT7 hN (by decide) hd))
            (ge.uget (0+1+1+1+1+1+1+1) (kU uT7 hN (by decide) he)).toUInt64 (lo+(0+1+1+1+1+1+1+1)) (0+1+1+1+1+1+1+1) e7 (fun o7 _=>
            k8O out lo o0 o1 o2 o3 o4 o5 o6 o7 (kQl hlo uT0 (by rw [hN]; decide) hb ho) (kQl hlo uT1 (by rw [hN]; decide) hb ho) (kQl hlo uT2 (by rw [hN]; decide) hb ho) (kQl hlo uT3 (by rw [hN]; decide) hb ho) (kQl hlo uT4 (by rw [hN]; decide) hb ho) (kQl hlo uT5 (by rw [hN]; decide) hb ho) (kQl hlo uT6 (by rw [hN]; decide) hb ho) (kQl hlo uT7 (by rw [hN]; decide) hb ho)))))))))
        else Norm.nRowS n eps x g b nd r lo (G0 (r*n+n)) out)
  else #[]
theorem k8_eq (n:ℕ) (hN:n=8) (eps:Word) (x g b:Ἂ Word) (gd bd:ἀ) (ge:ByteArray) (nd:Ἅ)
    (rm nf ep z0:Ἅ) (L:ἆ) (cb:Ἂ Word) (r:ℕ) (lo:Ἄ) (hlo:lo=G0 (r*n)) (out:Ἂ Word)
    (hg:n ≤ gd.size) (hd:n ≤ bd.size) (he:n ≤ ge.size) (ho:r*n+n ≤ out.size) (hb:r*n+n < 2^(32:ℕ))
    (hC:Norm.nCK n eps rm nf ep z0 L) (hg':n ≤ g.size) (hd':n ≤ b.size) (hx:r*n+n ≤ x.size)
    (hG : ∀ t,t < n → gd.data[t]! = Dense.F.dw g[t]!) (hB : ∀ t,t < n → bd.data[t]! = Dense.F.dw b[t]!)
    (hE : ∀ t,t < n → (ge.data[t]!).toUInt64=Norm.nfe g[t]!) :
    k8 n hN eps x g b gd bd ge nd rm nf ep z0 L cb r lo hlo out hg hd he ho hb=cRw n eps x g b nd rm nf ep z0 L hC cb r out hg' hd' hx ho hb:=by
  subst hN hlo
  have q1:(G0 (r*8)).toNat=r*8:=Conv.aa _ (C0 (V1 _ _) hb)
  have q3:(G0 (r*8+8)).toNat=r*8+8:=Conv.aa _ hb
  have q2:(G0 8).toNat=8:=Conv.aa _ (C0 (K2 _ _) hb)
  have o1:(1:Ἄ).toNat=1:=mA
  have b1:(G0 (r*8)+1).toNat=r*8+1:=by rw [Norm.nCa (by rw [q1,o1]; omega),q1,o1]
  have b2:(G0 (r*8)+1+1).toNat=r*8+2:=by rw [Norm.nCa (by rw [b1,o1]; omega),b1,o1]
  have b3:(G0 (r*8)+1+1+1).toNat=r*8+3:=by rw [Norm.nCa (by rw [b2,o1]; omega),b2,o1]
  have b4:(G0 (r*8)+1+1+1+1).toNat=r*8+4:=by rw [Norm.nCa (by rw [b3,o1]; omega),b3,o1]
  have b5:(G0 (r*8)+1+1+1+1+1).toNat=r*8+5:=by rw [Norm.nCa (by rw [b4,o1]; omega),b4,o1]
  have b6:(G0 (r*8)+1+1+1+1+1+1).toNat=r*8+6:=by rw [Norm.nCa (by rw [b5,o1]; omega),b5,o1]
  have b7:(G0 (r*8)+1+1+1+1+1+1+1).toNat=r*8+7:=by rw [Norm.nCa (by rw [b6,o1]; omega),b6,o1]
  have b8:(G0 (r*8)+1+1+1+1+1+1+1+1).toNat=r*8+8:=by rw [Norm.nCa (by rw [b7,o1]; omega),b7,o1]
  have c0:(G0 (r*8)+0).toNat=r*8:=by rw [Norm.nCa (by rw [q1,uT0]; omega),q1,uT0]; rfl
  have c1:(G0 (r*8)+(0+1)).toNat=r*8+1:=by rw [Norm.nCa (by rw [q1,uT1]; omega),q1,uT1]
  have c2:(G0 (r*8)+(0+1+1)).toNat=r*8+2:=by rw [Norm.nCa (by rw [q1,uT2]; omega),q1,uT2]
  have c3:(G0 (r*8)+(0+1+1+1)).toNat=r*8+3:=by rw [Norm.nCa (by rw [q1,uT3]; omega),q1,uT3]
  have c4:(G0 (r*8)+(0+1+1+1+1)).toNat=r*8+4:=by rw [Norm.nCa (by rw [q1,uT4]; omega),q1,uT4]
  have c5:(G0 (r*8)+(0+1+1+1+1+1)).toNat=r*8+5:=by rw [Norm.nCa (by rw [q1,uT5]; omega),q1,uT5]
  have c6:(G0 (r*8)+(0+1+1+1+1+1+1)).toNat=r*8+6:=by rw [Norm.nCa (by rw [q1,uT6]; omega),q1,uT6]
  have c7:(G0 (r*8)+(0+1+1+1+1+1+1+1)).toNat=r*8+7:=by rw [Norm.nCa (by rw [q1,uT7]; omega),q1,uT7]
  have l0:G0 (r*8) < G0 (r*8+8):=ε.mpr (by rw [q1,q3]; exact P5 (by decide))
  have l1:G0 (r*8)+1 < G0 (r*8+8):=ε.mpr (by rw [b1,q3]; exact O1 (by decide) _)
  have l2:G0 (r*8)+1+1 < G0 (r*8+8):=ε.mpr (by rw [b2,q3]; exact O1 (by decide) _)
  have l3:G0 (r*8)+1+1+1 < G0 (r*8+8):=ε.mpr (by rw [b3,q3]; exact O1 (by decide) _)
  have l4:G0 (r*8)+1+1+1+1 < G0 (r*8+8):=ε.mpr (by rw [b4,q3]; exact O1 (by decide) _)
  have l5:G0 (r*8)+1+1+1+1+1 < G0 (r*8+8):=ε.mpr (by rw [b5,q3]; exact O1 (by decide) _)
  have l6:G0 (r*8)+1+1+1+1+1+1 < G0 (r*8+8):=ε.mpr (by rw [b6,q3]; exact O1 (by decide) _)
  have l7:G0 (r*8)+1+1+1+1+1+1+1 < G0 (r*8+8):=ε.mpr (by rw [b7,q3]; exact O1 (by decide) _)
  have l8:¬G0 (r*8)+1+1+1+1+1+1+1+1 < G0 (r*8+8):=fun h=>by have:=ε.mp h; rw [b8,q3] at this; exact B4 _ this
  have m0:(0:Ἄ) < G0 8:=ε.mpr (by rw [uT0,q2]; decide)
  have m1:(0+1:Ἄ) < G0 8:=ε.mpr (by rw [uT1,q2]; decide)
  have m2:(0+1+1:Ἄ) < G0 8:=ε.mpr (by rw [uT2,q2]; decide)
  have m3:(0+1+1+1:Ἄ) < G0 8:=ε.mpr (by rw [uT3,q2]; decide)
  have m4:(0+1+1+1+1:Ἄ) < G0 8:=ε.mpr (by rw [uT4,q2]; decide)
  have m5:(0+1+1+1+1+1:Ἄ) < G0 8:=ε.mpr (by rw [uT5,q2]; decide)
  have m6:(0+1+1+1+1+1+1:Ἄ) < G0 8:=ε.mpr (by rw [uT6,q2]; decide)
  have m7:(0+1+1+1+1+1+1+1:Ἄ) < G0 8:=ε.mpr (by rw [uT7,q2]; decide)
  have m8:¬(0+1+1+1+1+1+1+1+1:Ἄ) < G0 8:=fun h=>by have:=ε.mp h; rw [uT8,q2] at this; exact B4 _ this
  unfold k8 cRw Norm.nRow1; dsimp only
  rw [cQS l0,cQS l1,cQS l2,cQS l3,cQS l4,cQS l5,cQS l6,cQS l7,cQE l8]
  refine ite_congr rfl (fun _=>rfl) (fun _=>?_)
  rw [s1S l0,s1S l1,s1S l2,s1S l3,s1S l4,s1S l5,s1S l6,s1S l7,s1E l8]
  simp only [s2S l0,s2S l1,s2S l2,s2S l3,s2S l4,s2S l5,s2S l6,s2S l7,s2E l8]
  refine ite_congr rfl (fun hD=>?_) (fun _=>rfl)
  rw [sLQ_e _ _ _ _ _ _ _ _ _ (nCKL hC (by decide)) (by have:=μ.mp hD; simpa using this)]
  congr 1; funext m ri Kp dW fast
  cases fast
  · simp only [υ,↓reduceIte]
  · simp only [↓reduceIte]
    obtain ⟨e01,e02,e03⟩:=k4c (0) (n:=8) (by rw [uT0]; decide) hG hB hE hg hd he hg' hd'
    obtain ⟨e11,e12,e13⟩:=k4c (0+1) (n:=8) (by rw [uT1]; decide) hG hB hE hg hd he hg' hd'
    obtain ⟨e21,e22,e23⟩:=k4c (0+1+1) (n:=8) (by rw [uT2]; decide) hG hB hE hg hd he hg' hd'
    obtain ⟨e31,e32,e33⟩:=k4c (0+1+1+1) (n:=8) (by rw [uT3]; decide) hG hB hE hg hd he hg' hd'
    obtain ⟨e41,e42,e43⟩:=k4c (0+1+1+1+1) (n:=8) (by rw [uT4]; decide) hG hB hE hg hd he hg' hd'
    obtain ⟨e51,e52,e53⟩:=k4c (0+1+1+1+1+1) (n:=8) (by rw [uT5]; decide) hG hB hE hg hd he hg' hd'
    obtain ⟨e61,e62,e63⟩:=k4c (0+1+1+1+1+1+1) (n:=8) (by rw [uT6]; decide) hG hB hE hg hd he hg' hd'
    obtain ⟨e71,e72,e73⟩:=k4c (0+1+1+1+1+1+1+1) (n:=8) (by rw [uT7]; decide) hG hB hE hg hd he hg' hd'
    rw [f2S m0]; refine oE_c (congrArg _ (cku (O5.1 (by rw [q1,c0])) _)) e01 e02 e03 (fun o0 e1=>?_)
    rw [f2S m1]; refine oE_c (congrArg _ (cku (O5.1 (by rw [b1,c1])) _)) e11 e12 e13 (fun o1 e2=>?_)
    rw [f2S m2]; refine oE_c (congrArg _ (cku (O5.1 (by rw [b2,c2])) _)) e21 e22 e23 (fun o2 e3=>?_)
    rw [f2S m3]; refine oE_c (congrArg _ (cku (O5.1 (by rw [b3,c3])) _)) e31 e32 e33 (fun o3 e4=>?_)
    rw [f2S m4]; refine oE_c (congrArg _ (cku (O5.1 (by rw [b4,c4])) _)) e41 e42 e43 (fun o4 e5=>?_)
    rw [f2S m5]; refine oE_c (congrArg _ (cku (O5.1 (by rw [b5,c5])) _)) e51 e52 e53 (fun o5 e6=>?_)
    rw [f2S m6]; refine oE_c (congrArg _ (cku (O5.1 (by rw [b6,c6])) _)) e61 e62 e63 (fun o6 e7=>?_)
    rw [f2S m7]; refine oE_c (congrArg _ (cku (O5.1 (by rw [b7,c7])) _)) e71 e72 e73 (fun o7 e8=>?_)
    rw [f2E m8]; rfl
@[inline] def kS (n:ℕ) (hK:n=4∨n=8) (eps:Word) (x g b:Ἂ Word) (gd bd:ἀ) (ge:ByteArray) (nd:Ἅ)
    (rm nf ep z0:Ἅ) (L:ἆ) (cb:Ἂ Word) (r:ℕ) (lo:Ἄ) (hlo:lo=G0 (r*n)) (out:Ἂ Word)
    (hg:n ≤ gd.size) (hd:n ≤ bd.size) (he:n ≤ ge.size) (ho:r*n+n ≤ out.size) (hb:r*n+n < 2^(32:ℕ)):Ἂ Word:=
  if h:n=4 then k4 n h eps x g b gd bd ge nd rm nf ep z0 L cb r lo hlo out hg hd he ho hb else k8 n (by omega) eps x g b gd bd ge nd rm nf ep z0 L cb r lo hlo out hg hd he ho hb
theorem kS_eq (n:ℕ) (hK:n=4∨n=8) (eps:Word) (x g b:Ἂ Word) (gd bd:ἀ) (ge:ByteArray) (nd:Ἅ)
    (rm nf ep z0:Ἅ) (L:ἆ) (cb:Ἂ Word) (r:ℕ) (lo:Ἄ) (hlo:lo=G0 (r*n)) (out:Ἂ Word)
    (hg:n ≤ gd.size) (hd:n ≤ bd.size) (he:n ≤ ge.size) (ho:r*n+n ≤ out.size) (hb:r*n+n < 2^(32:ℕ))
    (hC:Norm.nCK n eps rm nf ep z0 L) (hg':n ≤ g.size) (hd':n ≤ b.size) (hx:r*n+n ≤ x.size)
    (hG : ∀ t,t < n → gd.data[t]! = Dense.F.dw g[t]!) (hB : ∀ t,t < n → bd.data[t]! = Dense.F.dw b[t]!)
    (hE : ∀ t,t < n → (ge.data[t]!).toUInt64=Norm.nfe g[t]!) : kS n hK eps x g b gd bd ge nd rm nf ep z0 L cb r lo hlo out hg hd he ho hb=cRw n eps x g b nd rm nf ep z0 L hC cb r out hg' hd' hx ho hb:=by
  unfold kS; split
  · exact k4_eq n _ eps x g b gd bd ge nd rm nf ep z0 L cb r lo hlo out hg hd he ho hb hC hg' hd' hx hG hB hE
  · exact k8_eq n _ eps x g b gd bd ge nd rm nf ep z0 L cb r lo hlo out hg hd he ho hb hC hg' hd' hx hG hB hE
def cRowsK (rows n:ℕ) (hK:n=4∨n=8) (eps:Word) (x g b:Ἂ Word) (gd bd:ἀ) (ge:ByteArray) (nd:Ἅ)
    (rm nf ep z0:Ἅ) (L:ἆ) (cb:Ἂ Word) (hg:n ≤ gd.size) (hd:n ≤ bd.size) (he:n ≤ ge.size) (hx:rows*n ≤ x.size) (hb:rows*n < 2^(32:ℕ))
    (nn:Ἄ) (hnn:nn.toNat=n) (T:ℕ) (hT:T ≤ rows*n∧rows*n ≤ T) :
    (k r:ℕ) → (lo:Ἄ) → lo.toNat=r*n → (out:Ἂ Word) → rows*n ≤ out.size → r+k=rows → Ἂ Word
  | 0,_,_,_,out,_,_=>out
  | k+1,r,lo,hlo,out,ho,hk =>
    have kk:r*n+n ≤ rows*n:=Norm.nHk2 hk
    let p:=kS n hK eps x g b gd bd ge nd rm nf ep z0 L cb r lo (lE hlo (C0 (V1 _ _) (C0 kk hb))) out hg hd he (kk.trans ho) (C0 kk hb)
    if hs:T ≤ p.size then cRowsK rows n hK eps x g b gd bd ge nd rm nf ep z0 L cb hg hd he hx hb nn hnn T hT k (r+1)
      (lo+nn) (loS hlo hnn (C0 kk hb)) p (hT.2.trans hs) (Norm.nHk3 hk)
    else #[]
theorem cRowsK_eq (rows n:ℕ) (hK:n=4∨n=8) (eps:Word) (x g b:Ἂ Word) (gd bd:ἀ) (ge:ByteArray)
    (rm nf ep z0:Ἅ) (L:ἆ) (hC:Norm.nCK n eps rm nf ep z0 L) (cb:Ἂ Word)
    (hg:n ≤ gd.size) (hd:n ≤ bd.size) (he:n ≤ ge.size) (hg':n ≤ g.size) (hd':n ≤ b.size)
    (hx:rows*n ≤ x.size) (hb:rows*n < 2^(32:ℕ)) (nn:Ἄ) (hnn:nn.toNat=n) (T:ℕ) (hT:T ≤ rows*n∧rows*n ≤ T)
    (hX : ∀ j,j < rows*n → expOf x[j]! ≠ 255) (heps:0 < rationalWord eps)
    (hG : ∀ t,t < n → gd.data[t]! = Dense.F.dw g[t]!) (hB : ∀ t,t < n → bd.data[t]! = Dense.F.dw b[t]!)
    (hE : ∀ t,t < n → (ge.data[t]!).toUInt64=Norm.nfe g[t]!)
    (hN:(4 ≤ n&&n ≤ 2048&&expOf eps != 0&&expOf eps != 255)=true)
    (F : ∀ j,j < n → expOf g[j]! ≠ 255) (D : ∀ j,j < n → expOf b[j]! ≠ 255)
    (hcb : ∀ j,j < n → cb[j]! = roundRational (rationalWord b[j]!)) :
    ∀ k (r:ℕ) (lo:Ἄ) (hlo:lo.toNat=r*n) (out:Ἂ Word) (ho:rows*n ≤ out.size) (hk:r+k=rows),
    out.size=rows*n →
    cRowsK rows n hK eps x g b gd bd ge (I0 n).toInt64.toFloat rm nf ep z0 L cb hg hd he hx hb nn hnn T hT k r lo hlo out ho hk
      = Norm.Yr (fun r=>Norm.nLR n eps x r) (Norm.nEO n eps x g b (I0 n).toInt64.toFloat) n rows r out:=by
  intro k; induction k with
  | zero =>
    intro r lo hlo out ho hk hos; rw [Norm.Yr,if_neg (by rw [← hk]; exact B4 r)]; rfl
  | succ k ih =>
    intro r lo hlo out ho hk hos
    have kk:=Norm.nHk2 (n:=n) hk
    rw [Norm.Yr,if_pos (hk ▸ P5 (Nat.succ_pos k))]
    have E:kS n hK eps x g b gd bd ge (I0 n).toInt64.toFloat rm nf ep z0 L cb r lo (lE hlo (C0 (V1 _ _) (C0 kk hb))) out hg hd he (kk.trans ho) (C0 kk hb)
        =Norm.Yq (Norm.nEO n eps x g b (I0 n).toInt64.toFloat) (Norm.nLR n eps x r)
          (G0 (r*n)) (G0 (r*n)) (G0 (r*n+n)) out:=by
      rw [kS_eq n hK eps x g b gd bd ge _ rm nf ep z0 L cb r lo _ out hg hd he (kk.trans ho) (C0 kk hb) hC hg' hd' (kk.trans hx) hG hB hE]
      unfold cRw; split
      · exact cB n eps x g b cb r out (kk.trans ho) (C0 kk hb) (by rw [hos]; exact hb) (fun j hj=>hX j (U1 hj kk)) heps hN F D hcb (by assumption)
      · exact Norm.nRow1_eq n eps x g b _ rm nf ep z0 L hC r out hg' hd' (kk.trans hx) (kk.trans ho) (C0 kk hb) (by rw [hos]; exact hb) (fun j hj=>hX j (U1 hj kk)) heps hN
    have S2:(kS n hK eps x g b gd bd ge (I0 n).toInt64.toFloat rm nf ep z0 L cb r lo (lE hlo (C0 (V1 _ _) (C0 kk hb))) out hg hd he
        (kk.trans ho) (C0 kk hb)).size=rows*n:=by rw [E,Norm.nYq_sz _ _ _ _ _ _ _ rfl,hos]
    rw [cRowsK,dif_pos (hT.1.trans S2.ge)]
    refine (ih (r+1) _ _ _ _ (Norm.nHk3 hk) S2).trans ?_
    rw [E]
theorem cRowsK_bad (rows n:ℕ) (hK:n=4∨n=8) (eps:Word) (x g b:Ἂ Word) (gd bd:ἀ) (ge:ByteArray) (nd:Ἅ)
    (rm nf ep z0:Ἅ) (L:ἆ) (hC:Norm.nCK n eps rm nf ep z0 L) (cb:Ἂ Word)
    (hg:n ≤ gd.size) (hd:n ≤ bd.size) (he:n ≤ ge.size) (hg':n ≤ g.size) (hd':n ≤ b.size)
    (hx:rows*n ≤ x.size) (hb:rows*n < 2^(32:ℕ)) (nn:Ἄ) (hnn:nn.toNat=n) (T:ℕ) (hT:T ≤ rows*n∧rows*n ≤ T) (hn:0 < n)
    (hG : ∀ t,t < n → gd.data[t]! = Dense.F.dw g[t]!) (hB : ∀ t,t < n → bd.data[t]! = Dense.F.dw b[t]!)
    (hE : ∀ t,t < n → (ge.data[t]!).toUInt64=Norm.nfe g[t]!) :
    ∀ k (r:ℕ) (lo:Ἄ) (hlo:lo.toNat=r*n) (out:Ἂ Word) (ho:rows*n ≤ out.size) (hk:r+k=rows) (j:ℕ),
    r*n ≤ j → j < rows*n → expOf x[j]! = 255 →
    cRowsK rows n hK eps x g b gd bd ge nd rm nf ep z0 L cb hg hd he hx hb nn hnn T hT k r lo hlo out ho hk = #[]:=by
  intro k; induction k with
  | zero=>intro r lo hlo out ho hk j j1 j2 hj; rw [ϑ] at hk; subst hk; omega
  | succ k ih =>
    intro r lo hlo out ho hk j j1 j2 hj
    have kk:=Norm.nHk2 (n:=n) hk
    rw [cRowsK]; split
    · rename_i S
      by_cases c:j < r*n+n
      · have E:kS n hK eps x g b gd bd ge nd rm nf ep z0 L cb r lo (lE hlo (C0 (V1 _ _) (C0 kk hb))) out hg hd he (kk.trans ho) (C0 kk hb)=#[]:=by
          rw [kS_eq n hK eps x g b gd bd ge nd rm nf ep z0 L cb r lo _ out hg hd he (kk.trans ho) (C0 kk hb) hC hg' hd' (kk.trans hx) hG hB hE]
          unfold cRw; split
          · exact (cBb n x r (C0 kk hb) j j1 c hj (by assumption)).elim
          · exact Norm.nRow1_bad n eps x g b nd rm nf ep z0 L hC r out hg' hd' (kk.trans hx) (kk.trans ho) (C0 kk hb) j j1 c hj
        rw [E] at S; simp only [Array.size_empty] at S; omega
      · exact ih (r+1) _ _ _ _ (Norm.nHk3 hk) j (by rw [Ϡ,n4]; omega) j2 hj
    · rfl
def Yw3 (rows n:ℕ) (eps:Word) (x g b:Ἂ Word):Ἂ Word :=
  let nd:=(I0 n).toInt64.toFloat
  let lg:=Core.ai (I0 n)
  let rm:=(((1:ἆ)<<<(52+lg))/I0 n).toInt64.toFloat*N5 ((971-lg)<<<52)
  let nf:=(((1:ἆ)<<<52)/I0 n).toInt64.toFloat
  let ep:=Dense.F.dw eps*Norm.nS201; let z0:=(eps.toUInt64>>>32).toInt64.toFloat
  let L:=Core.ai (I0 (n+4))+1
  have hC:Norm.nCK n eps rm nf ep z0 L:=⟨rfl,rfl,rfl,congrArg (fun u:ἆ=>u.toInt64.toFloat) (Norm.kO0 eps),rfl⟩
  if h:x.size=rows*n∧g.size=n∧b.size=n∧rows*n ≤ 4294967295 then
    if (4 ≤ n&&n ≤ 2048&&expOf eps != 0&&expOf eps != 255)=true then
      if h1:rows=1 then
        Norm.nRow1 n eps x g b nd rm nf ep z0 L hC 0 (mB (rows*n) 0) h.2.1.ge h.2.2.1.ge (by rw [h.1,h1]; omega)
          (by simp only [O3]; rw [h1]; omega) (by have:=h.2.2.2; rw [h1] at this; omega)
      else
        let W:=Norm.nWG2 g b
        have hW:W=Norm.nWF2 (g,b):=Norm.nWG2_v g b
        if hK:n=4∨n=8 then
          cRowsK rows n hK eps x g b W.1.1 W.2 W.1.2 nd rm nf ep z0 L (cbG b)
            (by simp only [hW,Norm.nWF2,(Norm.nWF_sp g).1,h.2.1,le_refl])
            (by simp only [hW,Norm.nWF2,(Norm.nWF_sp b).1,h.2.2.1,le_refl])
            (by simp only [hW,Norm.nWF2,(Norm.nWF_sp g).2.2.1,h.2.1,le_refl])
            h.1.ge (Z4 h.2.2.2) (G0 n) (Conv.aa _ (by omega)) (rows*n) ⟨le_refl _,le_refl _⟩ rows 0 0 (Ϧ.trans (ϳ n).symm) (mB (rows*n) 0)
            (by simp only [O3]; exact Nat.le_refl _) (Κ rows)
        else
          cRows rows n eps x g b W.1.1 W.2 W.1.2 nd rm nf ep z0 L hC (cbG b)
            (by simp only [hW,Norm.nWF2,(Norm.nWF_sp g).1,h.2.1,le_refl])
            (by simp only [hW,Norm.nWF2,(Norm.nWF_sp b).1,h.2.2.1,le_refl])
            (by simp only [hW,Norm.nWF2,(Norm.nWF_sp g).2.2.1,h.2.1,le_refl])
            h.1.ge (Z4 h.2.2.2) rows 0 (Dense.F.zPq (n+2)) (mB (rows*n) 0) (Dense.F.zPs (n+2)).1
            (by simp only [O3]; exact Nat.le_refl _) (Κ rows)
    else if x.all finiteWord then Norm.nRowsS n eps x g b nd rows 0 (mB (rows*n) 0) else #[]
  else mB (rows*n) 0
theorem V3L3 (e n:ℕ) (h:Word) (x g b:Ἂ Word) (m:0 < n) (B:e*n < 2^(32:ℕ))
    (q:0 < rationalWord h) (H : ∀ i,i < e*n → expOf x[i]! ≠ 255) (F : ∀ j,j < n → expOf g[j]! ≠ 255)
    (D : ∀ j,j < n → expOf b[j]! ≠ 255) (sx:x.size=e*n) (sg:g.size=n) (sb:b.size=n) :
    Yw3 e n h x g b=layernormStable e n h x g b:=by
  rw [Yw3]; dsimp only; split
  · rename_i hs; split
    · rename_i hN; split
      · rename_i h1; subst h1
        have Y:=Norm.V3Lo 1 n h x g b m B q H F D
        rw [Norm.Yr,if_pos Nat.zero_lt_one,Κ,Norm.Yr,if_neg (B4 1)] at Y
        refine Eq.trans (Norm.nRow1_eq _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (by simp only [O3]; exact B)
          (by rw [ϳ,Κ]; intro j hj; exact H j (by rw [n4]; exact hj)) q hN) ?_
        simp only [ϳ,Κ] at Y ⊢; exact Y
      · have G:=Norm.nWF_sp g; have Bq:=Norm.nWF_sp b
        have e1:(Norm.nWG2 g b).1=Norm.nWF g:=by rw [Norm.nWG2_v]; rfl
        have e2:(Norm.nWG2 g b).2=(Norm.nWF b).1:=by rw [Norm.nWG2_v]; rfl
        rw [← e1] at G; rw [← e2] at Bq
        split
        · rename_i hK
          exact (cRowsK_eq _ _ hK _ _ _ _ _ _ _ _ _ _ _ _ ⟨rfl,rfl,rfl,congrArg (fun u:ἆ=>u.toInt64.toFloat) (Norm.kO0 h),rfl⟩
            _ _ _ _ sg.ge sb.ge _ _ _ _ _ _ H q (fun t ht=>G.2.1 t (by rw [hs.2.1]; exact ht))
            (fun t ht=>Bq.2.1 t (by rw [hs.2.2.1]; exact ht)) (fun t ht=>G.2.2.2 t (by rw [hs.2.1]; exact ht)) hN F D
            (cbG_sp b n sb) _ 0 _ _ _ _ _ (by simp only [O3])).trans (Norm.V3Lo e n h x g b m B q H F D)
        · exact (cRows_eq _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ H q (fun t ht=>G.2.1 t (by rw [hs.2.1]; exact ht))
            (fun t ht=>Bq.2.1 t (by rw [hs.2.2.1]; exact ht)) (fun t ht=>G.2.2.2 t (by rw [hs.2.1]; exact ht)) hN F D
            (cbG_sp b n sb) _ 0 _ _ _ _ _ (by simp only [O3])).trans (Norm.V3Lo e n h x g b m B q H F D)
    · rename_i hN
      have X:x.all finiteWord=true:=np.mpr (fun i hi=>by
        have t:=H i (sx ▸ hi); rw [AB x i hi] at t; simp only [finiteWord,AM,Ak]; exact t)
      rw [if_pos X]
      exact (Norm.nRowsS_e n h x g b _ e (fun r=>Norm.nLR_f n h x r hN) _ 0 _ rfl).trans (Norm.V3Lo e n h x g b m B q H F D)
  · rename_i hn; exact absurd ⟨sx,sg,sb,J2 B⟩ hn
theorem V3L3_bad (e n:ℕ) (h:Word) (x g b:Ἂ Word) (m:0 < n) (B:e*n < 2^(32:ℕ)) (sx:x.size=e*n)
    (sg:g.size=n) (sb:b.size=n) (hX:x.all finiteWord=false):Yw3 e n h x g b = #[]:=by
  obtain ⟨j,j2,hj⟩ : ∃ j,j < e*n∧expOf x[j]! = 255:=by
    by_contra c; simp only [not_exists,not_and] at c
    have:x.all finiteWord=true:=np.mpr (fun i hi=>by
      have t:=c i (sx ▸ hi); rw [AB x i hi] at t; simp only [finiteWord,AM,Ak]; exact t)
    rw [this] at hX; exact absurd hX (by decide)
  rw [Yw3]; dsimp only; rw [dif_pos ⟨sx,sg,sb,J2 B⟩]; split
  · split
    · rename_i h1; subst h1; exact Norm.nRow1_bad _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ j (by omega) (by omega) hj
    · split
      · rename_i hK
        have G:=Norm.nWF_sp g; have Bq:=Norm.nWF_sp b
        have e1:(Norm.nWG2 g b).1=Norm.nWF g:=by rw [Norm.nWG2_v]; rfl
        have e2:(Norm.nWG2 g b).2=(Norm.nWF b).1:=by rw [Norm.nWG2_v]; rfl
        rw [← e1] at G; rw [← e2] at Bq
        exact cRowsK_bad _ _ hK _ _ _ _ _ _ _ _ _ _ _ _ _ ⟨rfl,rfl,rfl,congrArg (fun u:ἆ=>u.toInt64.toFloat) (Norm.kO0 h),rfl⟩ _ _ _ _ sg.ge sb.ge _ _ _ _ _ _ m
          (fun t ht=>G.2.1 t (by rw [sg]; exact ht)) (fun t ht=>Bq.2.1 t (by rw [sb]; exact ht)) (fun t ht=>G.2.2.2 t (by rw [sg]; exact ht))
          _ 0 _ _ _ _ _ j (by omega) j2 hj
      · exact cRows_bad _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ m _ 0 _ _ _ _ _ j (by omega) j2 hj
  · simp only [hX,υ, ↓reduceIte]
theorem Yw3_eq (l n:ℕ) (m:Word) (x g b:Ἂ Word) (l0:0 < l) (n0:0 < n) (sx:x.size=l*n) (sg:g.size=n)
    (sb:b.size=n) (B:l*n ≤ 4294967295)
    (h2:(g.all finiteWord&&b.all finiteWord&&finiteWord m&&(!signOf m&&Core.mW m != 0))=true) :
    Yw3 l n m x g b=Norm.Yw2 l n m x g b:=by
  by_cases hX:x.all finiteWord=true
  · have h:(Norm.SN l n m x g&&b.size==n&&b.all finiteWord)=true:=by
      simp only [Η] at h2
      rw [Norm.SN]; simp only [Η,AQ,AJ]
      exact ⟨⟨⟨⟨⟨⟨⟨⟨⟨l0,n0⟩,sx⟩,sg⟩,hX⟩,h2.1.1.1⟩,h2.1.2⟩,h2.2⟩,sb⟩,h2.1.1.2⟩
    rw [Norm.fG] at h; simp only [Η] at h
    obtain ⟨a,e,c,d⟩:=Norm.jm h.1.1
    have D : ∀ j,j < n → expOf b[j]! ≠ 255:=fun j p =>
      Norm.NR.ic b (by simpa [np] using h.2) j (by omega)
    rw [V3L3 l n m x g b a (by omega) e c d D sx sg sb,Norm.V3L l n m x g b a (by omega) e c d D sx sg sb]
  · have hX':x.all finiteWord=false:=F8.mpr hX
    rw [V3L3_bad l n m x g b n0 (by omega) sx sg sb hX',Norm.V3L_bad l n m x g b n0 (by omega) sx sg sb hX']
def bnC (rows n:ℕ) (eps:Word) (x g b:Ἂ Word):Ἂ Word :=
  if rows > 0∧n > 0∧x.size=rows*n∧g.size=n∧b.size=n∧rows*n ≤ 4294967295 then
    if (g.all finiteWord&&b.all finiteWord&&finiteWord eps&&(!signOf eps&&Core.mW eps != 0))=true then
      let r:=Yw3 rows n eps x g b
      if (r.size==rows*n)=true then r else layernormSpec rows n eps x g b
    else layernormSpec rows n eps x g b
  else layernormSpec rows n eps x g b
theorem bnC_eq:bnC=layernormSpec:=by
  funext l n m x g b
  rw [← Norm.cN]
  unfold bnC Norm.bn
  by_cases h:l > 0∧n > 0∧x.size=l*n∧g.size=n∧b.size=n∧l*n ≤ 4294967295
  · rw [if_pos h,if_pos h]
    by_cases h2:(g.all finiteWord&&b.all finiteWord&&finiteWord m&&(!signOf m&&Core.mW m != 0))=true
    · rw [if_pos h2,if_pos h2]; obtain ⟨l0,n0,sx,sg,sb,B⟩:=h
      dsimp only; rw [Yw3_eq l n m x g b l0 n0 sx sg sb B h2]
    · rw [if_neg h2,if_neg h2]
  · rw [if_neg h,if_neg h]
end NXc
def kernels:Kernels:=
 { U7 with
   dense:=⟨Dense.c7,gC⟩
   conv:=⟨cvK,dwK,U7.conv.convTranspose⟩
   elem:=⟨apK,mpK,Bn.cq,Bn.br⟩
   rnn:=⟨Rnn.lstmK Dense.c7 Rnn.ad2 SAct.sg SAct.th,Rnn.gruK Dense.c7 Rnn.ad2 SAct.sg SAct.th⟩
   norm:=⟨NX.rms,NXc.bnC⟩
 }
theorem dwK_eq:dwK=J9:=by
 funext s relu inp wts b;unfold dwK;split
 ·rename_i h;rw[Η]at h;rw[Conv.cY,dwB s h.1]
 ·split
  ·rename_i h;rw[Conv.cY];exact d31_eq s relu inp wts b h
  ·split
   ·rename_i h;rw[dwRowsF_eq,Dense.eD];exact Cv1.lin_dw1 s h relu inp wts b
   ·simp only[U7,Pinned.dotFn_eq_dotOf];rfl
theorem apK_eq:apK=ExactRuntime.avgpoolSpec:=by
 funext s inp;unfold apK;split
 ·rename_i h;rw[Conv.eV,apB s h]
 ·simp only[U7,Pinned.dotFn_eq_dotOf];rfl
theorem mpK_eq:mpK=ExactRuntime.maxpoolSpec:=by
 funext s inp;unfold mpK;split
 ·rename_i h;rw[Conv.fo,mpB s h]
 ·simp only[U7,Pinned.maxpoolW_eq]
theorem correct:Obligation kernels:=by
 have h:=Pinned.correct
 simp only[Obligation,U7]at h
 simp only[Obligation,kernels,U7,Dense.eD,gCv,cvK_eq,dwK_eq,apK_eq,mpK_eq,Bn.dT,Bn.cO,NX.rms_eq,NXc.bnC_eq,
  SAct.sg_eq,SAct.th_eq,Pinned.dotFn_eq_dotOf,Rnn.lstmK_spec,Rnn.gruK_spec]
 rw[←h]
 simp only[specAttn,specRnn,scoresSpec,mixSpec,softmaxSpec,siluSpec,lstmSpec,gruSpec,Pinned.dotFn_eq_dotOf,Pinned.maxpoolW_eq]
 all_goals rfl
end Submission