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
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,Nat.le_sub_of_add_le (by rw[Τ];exact (J1 ((by decide):23<=25) j).trans h),(Nat.le_add_left 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:=⟨(Int.toNat (-(e+149)-1)),by omega,by omega⟩
 exact⟨a,v,by (clear * - v y;omega),(Nat.sub_add_comm ((V1 j 2).trans y)).symm,y.trans (Nat.le_succ 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) (Nat.le_succ 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 _ (Nat.le_sub_of_add_le 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 (Nat.lt_of_lt_of_eq 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:=Nat.le_of_eq (Κ _)
 have hb:0+(A.map (kT (fmt:=lU))).size<=(B.map (kT (fmt:=lU))).size:=Nat.le_of_eq ((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 (Nat.lt_of_lt_of_eq 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:=
 (UInt64.toUInt32 (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 (UInt64.toUInt32 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 (UInt64.toUInt32 (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 (Nat.le_of_lt_succ (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 (Nat.not_lt.2 ((J1 (Nat.le_of_lt_succ 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 (Nat.lt_add_of_pos_right h),if_neg (BD.1 h)]
 rw[if_pos (U1 c (V1 _ b)),if_neg (Nat.not_lt.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 ((Nat.sub_le _ _).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):(UInt64.toUInt32 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):(Int.natAbs ((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[Nat.cast_mul,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):(Int.natAbs ((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[Nat.cast_add,Nat.cast_mul,pO (Nat.le_of_lt_succ 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 (Nat.le_add_left _ _)
 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):
  (Int.natAbs (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):(UInt64.toUInt32 (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/\
  (UInt64.toUInt32 (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):=Nat.lt_of_not_le 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)):=Nat.not_lt.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=(UInt64.toUInt32 (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 (UInt64.toUInt32 (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 (UInt64.toUInt32 ((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 ((Nat.add_lt_add_right (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) ((Nat.add_lt_add_of_lt_of_le (Nat.add_lt_add_right s _) (Nat.le_of_lt_succ 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 ((Nat.add_lt_add 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:=⟨(Nat.add_lt_add_right p _).trans (by decide),(Nat.add_lt_add_of_lt_of_le (Nat.add_lt_add_right p _) (Nat.le_of_lt_succ (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 ((Nat.le_of_eq 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 (Nat.le_succ _),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 Nat.lt_or_ge 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,Nat.sub_add_cancel h]
   have s1:S-172<18446744073709551616:=Nat.sub_lt_left_of_lt_add 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):=(Nat.add_le_add (ι (2^(23:Ἀ)) (Nat.sub_le_sub_right 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 _ (Nat.le_of_lt_succ 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,Nat.le_sub_of_add_le y,(Nat.sub_le_sub_right (Nat.le_of_lt_succ k) 62).trans_lt (by decide),(Nat.le_add_left _ _).trans (k.le.trans ((by decide):425<=506)),(Nat.sub_le _ _).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)) (Nat.sub_le_sub_right 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 (Nat.le_of_lt_succ 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⟨(Nat.le_add_left _ _).trans_eq E.symm,E.trans_lt H⟩
 have o:(lq m)≠0:=Nat.ne_of_gt (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 _ (Nat.add_lt_add A B.toNat_lt) q
 refine⟨j.1,j.2,dZ_h6 _ _ y A,Nat.le_of_lt_succ ((Nat.le_add_left _ _).trans_lt k),?_⟩;have D:(lq (62-L))=62-l:=by rw[χ _ _ (by rw[θ,r];exact Nat.le_of_lt_succ A),r]<;>rfl
 have z:(lq m)*2^(62-l)<2^(63:Ἀ):=AN (ϐ g (϶ _)) (by rw[<-AF];exact Θ (by decide) (Nat.le_of_eq (by rw[lO,Nat.add_sub_cancel' (Nat.le_of_lt_succ A)])));have w:(lq (m<<<(62-L)))=(lq m)*2^(62-l):=by rw[ψ,D,ο,γ (AO (Nat.sub_le _ _) (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,Nat.add_sub_cancel' (Nat.le_of_lt_succ 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),(Nat.add_sub_cancel' 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,Nat.sub_sub]:(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,USize.toNat_ofNat_of_lt' 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 Nat.lt_of_succ_le 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[Nat.mod_eq_sub_mod k,γ (Nat.sub_lt_left_of_lt_add k (Nat.add_lt_add hL hb)),γ ((Nat.lt_succ_self _).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):=Nat.lt_of_not_le λh=>c (by rw[Nat.mod_eq_sub_mod h,γ (Nat.sub_lt_left_of_lt_add h (Nat.add_lt_add hL hb))];exact Nat.sub_lt_left_of_lt_add 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,Nat.sub_add_cancel (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 (Nat.sub_le _ _) (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(Nat.add_lt_add_of_lt_of_le 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 ((Nat.sub_le _ _).trans_lt (hh.trans (by decide)))),sub_zero,pO q,Nat.cast_add,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):=(Nat.sub_le _ _).trans ((Nat.add_le_add hg.le hc).trans (by decide))
   rw[r,e,if_pos (Nat.le_sub_of_add_le ((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 ((Nat.sub_le _ _).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,Nat.sub_add_cancel h],Nat.cast_mul,Nat.cast_mul,Nat.cast_pow,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=(UInt64.toUInt32 (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:=Nat.le_antisymm ht (Nat.sub_eq_zero_iff_le.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[Nat.sub_add_eq,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(Nat.le_antisymm (Nat.le_of_lt_succ 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 ((Nat.add_le_add 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 (Nat.add_le_add 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 _▸Nat.add_le_add G l) ?_ j;rw[D,J,q,A,Nat.cast_add _ T];ring
 have q:=V r u I;refine e (t+1) _ _ o p y X x (by rw[q];exact Ο t _▸Nat.add_le_add I l) (G.trans m) ?_ j;rw[D,J,q,A,Nat.cast_add _ 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 (UInt64.toUInt32 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 Nat.lt_of_succ_le 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:(Int.natAbs (Int64.toInt i))<2^(53:Ἀ)):fv i.toFloat/\rv i.toFloat=(Int64.toInt i):=by
 have g:kI i.toFloat.toModel=kE kC (kR (Int64.toInt 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 (Int64.toInt i) 0).toReal=(Int64.toInt i):=by rw[<-ϖ,nD];simp
 have⟨j,l⟩:=lL kC (by decide) (kR (Int64.toInt i) 0) (Int.natAbs (Int64.toInt i)) 0 (by simp[kR]) e (by simp[kR];decide) (by rw[d];exact AZ (((by rw[<-mI,Ϲ];exact_mod_cast e.le:|((Int64.toInt 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 (Int.add_le_add_right (Int.add_le_add_right (Int.ofNat_le.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[Nat.cast_mul,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 (Nat.le_mul_of_pos_right 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 (·*·) (Model.unpackedToReal_finite l o g v) (Model.unpackedToReal_finite 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 Nat.le_of_succ_le 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:(Int.natAbs 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):(Int.natAbs (sv w))<2^(24:Ἀ):=by have:=lB w;rw[sv];split<;>simp<;>exact Nat.lt_of_succ_le this
lemma bL(a b c:Ἀ)(e:a+b<=c)(h:c<=971)(z:Ἁ)(g:(Int.natAbs 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:(Int64.toInt u.toInt64)=n:=by rw[show (Int64.toInt 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:=(Nat.sub_le _ _).trans (h.le.trans (by decide))
lemma qV_h2(x c:Ἀ)(h:x<256):1+(x-c)<=971:=(J1 ((Nat.sub_le _ _).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)/\(Int.natAbs 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):(Int64.toInt (-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 ((Nat.mul_lt_mul_of_pos_left hK (by decide)).trans_le (by decide)),(by decide:(-1074:Ἁ)<=0).trans (Nat.cast_nonneg _),(J1 ((Nat.sub_le _ _).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:(Int.natAbs (((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 (Nat.add_le_add ((Nat.sub_le _ _).trans h.le) ((Nat.sub_le _ _).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:(Int.natAbs (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):=⟨(Nat.add_lt_add_right (ϐ t (by decide)) _).trans_le (by decide),(Nat.add_lt_add_of_lt_of_le (Nat.add_lt_add_right (ϐ t (by decide)) _) (Nat.le_of_lt_succ 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):=⟨(Nat.add_lt_add_right (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 (Nat.add_le_add l l) (by decide))];exact Nat.le_trans (Nat.add_le_add l l) (by decide))),η,hb,γ (C0 (Nat.add_le_add l l) (by decide))] at q'
 rw[η,η,hb,pS,γ (C0 (L1 l 1) (by decide)),ha,γ (U1 (Nat.add_lt_add_of_lt_of_le (q'.trans_le (Nat.sub_le _ _)) (L1 l 1)) (by decide))] at e
 rw[<-e];exact⟨Nat.le_add_left _ _,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 (Nat.add_lt_add_right 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 ((Nat.add_lt_add_right 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 ((Nat.add_le_add (ι (2^(31:Ἀ)) (Nat.le_of_lt_succ q)) (Nat.add_le_add (ι (2^(23:Ἀ)) ((Nat.le_add_left e 1195).trans_lt o).le) (J1 (Nat.le_of_lt_succ 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),(Nat.add_lt_add_right (((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):=((Nat.sub_le _ _).trans_lt (Nat.add_lt_add_right v _)).trans_eq (by norm_num)
 have b9:g+2^(52:Ἀ)+G<2^(53:Ἀ):=by rw[lO];exact(Nat.add_lt_add_right J _).trans_eq (by norm_num)
 have a1:2^(52:Ἀ)<=g+2^(52:Ἀ)-G:=Nat.le_sub_of_add_le (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 _ ((Nat.le_add_left _ 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 (Nat.lt_succ_of_le 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:=(Nat.add_le_add (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/\(UInt64.toUInt32 (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(Nat.not_lt_zero _ (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(Nat.sub_le_sub_right 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:=⟨Nat.le_sub_of_add_le e1,(Nat.sub_le_sub_right e2 1085).trans_lt (by decide),(Nat.sub_le_sub_right (Nat.sub_le_sub_right 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:(UInt64.toUInt32 (((S.toBits>>>63)<<<31)+(((S.toBits&&&0x7FFFFFFFFFFFFFFF)+268435456)>>>29)-0x255000000))=R:=by apply Β.mp;rw[F,Ϫ,D,γ];exact Nat.add_lt_add_of_le_of_lt (by split<;>decide) ((Nat.add_le_add (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,Nat.cast_pow,nZ,<-AA]
  rw[<-AK AL,show (((lq h):Ἀ):Ἁ)=(((lq h)+1075-l:Ἀ):Ἁ)+((l:Ἁ)-1075) by rw[pO d1,Nat.cast_add,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[Nat.cast_mul,Nat.cast_pow,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[Nat.cast_add,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[Nat.cast_add]
   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 Nat.le_of_succ_le 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 Nat.le_of_lt_succ 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 (Nat.le_of_sub_eq_zero l),Nat.le_antisymm j (Nat.le_of_sub_eq_zero 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,Nat.sub_add_eq,l,BK],r.trans_le (N1 l),λe=>(Nat.lt_succ_of_lt e).trans_eq r.symm,λh e=>r.trans_le (Nat.lt_of_le_of_ne 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 Nat.ne_of_lt 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 (Nat.cast_pow 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 (Nat.add_lt_add_right (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 (Nat.cast_nonneg K) 1022),sub_le_iff_le_add.2 (Int.ofNat_le.2 y),le_sub_iff_add_le.2 (Int.ofNat_le.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:=Nat.le_antisymm Q (Nat.sub_eq_zero_iff_le.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 (Int.ofNat_le.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[Nat.sub_add_eq,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):
  (UInt64.toUInt32 (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 Nat.sub_succ_lt_self 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 (Nat.le_of_sub_eq_zero 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 Nat.sub_le_sub_right 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:=Nat.not_lt.2 (Nat.sub_eq_zero_iff_le.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:=Nat.le_antisymm S (Nat.sub_eq_zero_iff_le.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 (Nat.cast_le.2 (Nat.le_succ 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 (· * _) (Nat.cast_succ 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:(Int.natAbs (tm a b))<=2^(48+A+B):=by
  rw[tm,Ϩ,Ϩ,Int.natAbs_pow];rw[Ϲ]at d g;have h:(Int.natAbs (sv a))<=2^(24:Ἀ):=(by (clear * - d;omega));have j:(Int.natAbs (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 Nat.add_le_add 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 (UInt64.toUInt32 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 (Nat.ne_of_lt 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 (Nat.lt_succ_of_lt he)).trans (pA _ _ _))
   ((O1 (ϊ f (by decide)) _).trans_le (by rw[<-Ο];exact ι _ he))
theorem kRD_h3(x:Int)(h:¬127<x):Int.toNat (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[Nat.cast_add,Nat.cast_pow,nZ,zpow_ofNat];exact le_add_of_nonneg_left (Nat.cast_nonneg _)
 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 (UInt64.toUInt32 (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):=Nat.le_of_lt_succ ((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 _ (Nat.le_of_lt_succ 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 (UInt64.toUInt32 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 (UInt64.toUInt32 (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 (Nat.sub_le _ _).trans_lt ((Nat.add_le_add J D).trans_lt (by decide)),Int.ofNat_le.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):=(Nat.sub_le _ _).trans_lt ((Nat.add_le_add 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((Nat.le_succ _).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,Finset.sum_range_zero,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(Nat.sub_le_of_le_add (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 (Nat.not_lt.2 (Nat.le_of_sub_eq_zero j))];exact mi ((Nat.le_antisymm g (Nat.le_of_sub_eq_zero 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 ((Nat.sub_add_eq _ _ _).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:=Nat.le_antisymm (Nat.le_of_lt_succ (h.trans_eq (ϣ _))) (Nat.not_lt.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=>(Nat.not_lt_zero 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 Nat.add_right_cancel 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:=Nat.le_antisymm Q (Nat.sub_eq_zero_iff_le.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 (Nat.cast_le.2 Q) hu)).trans_lt hBK)
    refine G (t+1) (a+p t) (by rw[Nat.sub_add_eq,D,BK]) d this.1 ?_
    rw [Ν, ← add_assoc,Nat.cast_succ,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 ((Nat.lt_succ_self n).trans_le ((Nat.le_succ _).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 (Nat.mul_le_mul_left 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) ((Finset.abs_sum_le_sum_abs _ _).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 (Nat.cast_le.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) + ·) (Finset.sum_range_zero _).symm))))).trans A5).trans_le (Af (Nat.cast_nonneg 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 Nat.lt_or_ge 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 (Nat.sub_le_sub_right 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,Int.cast_natCast,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 (Nat.add_le_add 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:=Nat.le_antisymm 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 Nat.not_lt.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 Nat.le_of_succ_le 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 (Nat.le_of_not_lt 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 Nat.sub_succ_lt_self 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 Nat.sub_succ_lt_self 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 Nat.sub_succ_lt_self 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:=Nat.le_mul_of_pos_right _ ((ϟ _).trans_lt htn)
  intro m; induction m with
  | zero => intro o k A hm ho hk hA; have z:=Nat.sub_eq_zero_iff_le.1 hm; rw [tBi,dif_neg (by rw [ε]; exact not_lt.2 z)]; dsimp only
            exact ⟨by rw[hA,Nat.le_antisymm 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,←Nat.sub_sub,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 Nat.lt_succ_of_lt hi),s5push_get,if_pos (by rw[hA];exact hi)],fun p hp1 hp2 => ?_⟩
    rcases Nat.lt_or_ge o.toNat p with hp | hp
    · exact r3 p (by rw [e1]; exact hp) hp2
    · have pe:p=o.toNat:=Nat.le_antisymm hp hp1
      subst pe
      rw [r2 _ (by rw [e1]; exact Nat.lt_succ_self _),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 ⟨UInt32.le_iff_toNat_le.mpr (Ι.mp h).le,UInt32.le_iff_toNat_le.mpr AP⟩
  · rw [if_neg h]; exact ⟨UInt32.le_iff_toNat_le.mpr AP,UInt32.le_iff_toNat_le.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:=Nat.le_of_sub_eq_zero 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,←Nat.sub_sub,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 (UInt32.le_iff_toNat_le.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 Nat.lt_or_ge 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:=Nat.le_of_sub_eq_zero 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 Nat.lt_or_ge i a.size with h1|h1
    · rw [AB _ i (by simp; exact Nat.le_of_succ_le 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 (Nat.add_le_add (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 (((Nat.cast_le (α:=ℝ)).mpr (H1.trans H2)).trans_eq ((Nat.cast_pow 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 (Nat.ne_of_lt (hm a b h hb))
    · exact h
    · exact absurd e.symm (Nat.ne_of_lt (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 ((Nat.sub_le _ _).trans ((congrArg (P + ·) (Nat.two_mul L)).trans_le ((pA P L L).symm.trans_le ((Nat.add_le_add 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 (Nat.lt_trans h (by decide))
theorem kOfS (n:ℕ) (h:n < 2^(63:Ἀ)):n.toUInt64.toNat=n:=by
  rw [Ϣ,UInt64.toNat_ofNat']; exact γ (Nat.lt_trans 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 (Float.ofBits a).toModel=Model.ofModelBits (fmt:=kC) a.toBitVec:=by
 have e1:kI (Float.ofBits a).toModel=Model.canonicalizeModel (Model.ofModelBits (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,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 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):Model.ofModelBits (fmt:=kC) a.toBitVec=Model.ofFields kC (a>>>63!=0) ((lq ((a>>>52)&&&2047))) ((lq (a&&&0xfffffffffffff))):=by
 have d:Model.expField (Model.ofModelBits (fmt:=kC) a.toBitVec)=(lq ((a>>>52)&&&2047)):=ἅ
 have e:Model.fracField (Model.ofModelBits (fmt:=kC) a.toBitVec)=(lq (a&&&0xfffffffffffff)):=ἅ
 have g:Model.signBit (Model.ofModelBits (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 (Model.ofModelBits (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[Nat.cast_pow,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 Nat.lt_or_ge (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[Array.size_replicate]
 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:=
 ⟨Array.size_replicate.ge,.inr Array.size_replicate,λq h=>zPs_h1 n q (h.trans_eq Array.size_replicate)⟩
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,Array.size_replicate];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 Nat.sub_succ_lt_self 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=>Nat.sub_ne_zero_of_lt (ε.1 h) e),c q (AN hq a),if_pos (AN hq (Nat.le_of_sub_eq_zero 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,USize.toNat_ofNat_of_lt' (AN b this)];exact AN hi a);have hE:(J0 n)-(J0 (i+1))=m:=by rw[Dense.ac h,<-Nat.sub_sub,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 (Nat.lt_succ_of_lt 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 (Nat.lt_succ_of_lt 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:=Nat.sub_lt_sub_left h (Nat.lt_succ_self r)
theorem k7d1:1≤7:=Nat.le_of_ble_eq_true rfl
theorem k7d2:2≤7:=Nat.le_of_ble_eq_true rfl
theorem k7d3:3≤7:=Nat.le_of_ble_eq_true rfl
theorem k7d4:4≤7:=Nat.le_of_ble_eq_true rfl
theorem k7d5:5≤7:=Nat.le_of_ble_eq_true rfl
theorem k7d6:6≤7:=Nat.le_of_ble_eq_true rfl
theorem k7d7:7≤7:=Nat.le_of_ble_eq_true 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 (Nat.mul_le_mul_left 8 (Nat.le_of_lt ht)) h31.2),kc8]
  let w:=Wd.uget (wo+t) (by rw[kSm _ _ (AO (J1 (Nat.le_of_lt 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 (Nat.le_of_lt 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:=Nat.le_of_sub_eq_zero 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 Nat.not_lt.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 (Nat.mul_le_mul_left 8 (Nat.le_of_lt ht)) h31.2),kc8]
   have hw:(J0 (wo+t))=(J0 wo)+(J0 t):=kSm _ _ (AO (J1 (Nat.le_of_lt 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,Nat.sub_add_eq,H,BK]) l hl) (ac h)
     ((kX8_spec_h1 _ _ _ _ _ _ _ _ _ _ _ (by rw[h8];exact AO ((J1 ((by decide):7<=8) _).trans (Nat.mul_le_mul_left 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:=UInt64.toUInt32 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 Nat.zero_lt_of_lt hj),ώ hj,Κ]
  have h2:(c*w+k)/w=c:=by rw [Τ,Ρ _ _ (by exact Nat.zero_lt_of_lt hj),ώ hk,Κ]
  have e:a=c:=by rw [← h1, ← h2,h]
  subst e; exact ⟨rfl,by exact Nat.add_left_cancel 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 Nat.le_add_left 16 ow)
  have b:=ι n (show ow ≤ ow+16 by exact V1 ow 16)
  have c:=Nat.le_mul_of_pos_right ow h3
  have d:=Nat.le_mul_of_pos_right 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 Nat.sub_lt_sub_left (ε.mp h) (Nat.lt_succ_self _)
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),Nat.sub_add_eq,hk,BK]⟩
theorem kS0(t n:Ἄ)(hk:(J0 n)-(J0 t)=0):¬t<n∧¬(J0 t)<(J0 n):=
 ⟨fun h=>Nat.sub_ne_zero_of_lt (ε.mp h) hk,fun h=>Nat.sub_ne_zero_of_lt 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 (UInt64.toUInt32 c)else aB ()
theorem sTB_h(a n:Ἀ)(h:a+n<2^(31:Ἀ)):(J0 (G0 a))=a∧(J0 (G0 n))=n:=
 ⟨USize.toNat_ofNat_of_lt' (AN (AN ((V1 a n).trans_lt h) (by decide)) ς),USize.toNat_ofNat_of_lt' (AN (AN ((Nat.le_add_left 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,USize.toNat_ofNat_of_lt' (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 (Nat.mul_le_mul_left 8 (Nat.le_of_lt ht)) h31.2.2),kc8]
  let w:=Wd.uget (wo+t) (by rw[kSm _ _ (AO (J1 (Nat.le_of_lt ht) _) h31.1)];exact AN (O1 ht _) hW)
  let z:=Wd.uget (wp+t) (by rw[kSm _ _ (AO (J1 (Nat.le_of_lt 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 (Nat.le_of_lt 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:=Nat.le_of_sub_eq_zero 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 Nat.not_lt.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 Nat.not_lt.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 Nat.not_lt.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,Nat.sub_add_eq,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 (Nat.mul_le_mul_left 8 hk).trans_lt (C0 (V1 _ _) h8)),kc8]
  rw [kSm _ _ (by rw [e8k]; exact C0 (Nat.add_le_add (Nat.mul_le_mul_left 8 hk) (Nat.le_of_lt_succ 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 (Nat.mul_le_mul_left 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 Nat.sub_lt_sub_left (ε.mp h) (Nat.lt_succ_self _))
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 (Nat.sub_ne_zero_of_lt (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),Nat.sub_add_eq,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 (Nat.le_add_left _ _)).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 Nat.sub_lt_sub_left (ε.mp h) (Nat.lt_succ_self _))
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=>Nat.sub_ne_zero_of_lt (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),Nat.sub_add_eq,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 Nat.sub_succ_lt_self 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 Nat.sub_succ_lt_self 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):=Nat.mul_le_mul_left _ (Nat.le.intro hm.symm)
  have e1:(J0 (r*n))=(J0 r)*(J0 n):=kMm _ _ (C0 (ϧ (Nat.le_of_lt hr) (Nat.le.intro hm.symm)) hB)
  have e2:(J0 (r*m))=(J0 r)*(J0 m):=kMm _ _ (C0 (ι _ (Nat.le_of_lt hr)) hB)
  dOw X n m R (r+1) (dRw X (r*n) n (r*m) 0 Y 0 (Nat.le_trans (Nat.le_of_eq ((congrArg (·+J0 n) e1).trans (nO _ _).symm)) (Nat.le_trans q2 hX)) (U1 (O1 (Nat.le_of_eq hm.symm) _) (Nat.le_trans (Nat.le_of_eq ((congrArg (·+J0 m) e2).trans (nO _ _).symm)) (Nat.le_trans q1 hY))) ⟨C0 (Nat.le_trans (Nat.le_of_eq ((congrArg (·+J0 n) e1).trans (nO _ _).symm)) (Nat.le_trans q2 q3)) hB,Nat.lt_trans (O1 (Nat.le_of_eq hm.symm) _) (C0 (Nat.le_trans (Nat.le_of_eq ((congrArg (·+J0 m) e2).trans (nO _ _).symm)) q1) hB)⟩) hX
   (Nat.le_trans hY (Nat.le_of_eq (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 ▸ Nat.sub_succ_lt_self _ _ (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 (Nat.not_lt_zero _)
|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):=Nat.lt_trans (O1 h1 _) (by rw [← nO]; exact C0 (ι _ (Nat.le_trans (J1 (Nat.le_add_left 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:=Nat.add_le_add hg (Nat.le_add_left 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:=Nat.sub_eq_zero_iff_le.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:=Nat.sub_add_cancel (Nat.le_add_right_of_le (hgB.resolve_right (Nat.zero_lt_of_lt 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,←Nat.sub_sub,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 Nat.lt_or_ge 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:=Nat.sub_eq_zero_iff_le.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 Nat.le_mul_of_pos_right _ 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,←Nat.sub_sub,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 Nat.lt_or_ge 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:=Nat.le_mul_of_pos_right (ow+16) hO; (clear * - hB this;omega))
    have e3:=s5tu ((ow+7)/8) (by have:=Nat.le_mul_of_pos_right (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:=Nat.le_mul_of_pos_right (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 Nat.lt_of_lt_of_eq 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 _) (Finset.sum_nonneg 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:=Nat.le_antisymm Q (Nat.sub_eq_zero_iff_le.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 ((Finset.abs_sum_le_sum_abs _ _).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 (Nat.le_of_lt 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 [Nat.sub_add_eq,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 (Nat.cast_nonneg _) htT hc E o)
    rw [Nat.cast_succ,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:(Int64.toInt u.toInt64)=n:=by rw[show (Int64.toInt u.toInt64)=ae u from ἅ,ay,d];split<;>omega
 rw[<-this];exact dB _ (by rw[this];simp only [Int.natAbs_natCast,nr];exact Nat.lt_of_succ_le 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 Nat.le_of_not_lt 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 (Nat.ofNat_nonneg _) (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) (Nat.ofNat_nonneg 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) (Nat.ofNat_nonneg 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 Nat.le_trans (ι 2 (ϧ (Nat.mul_le_mul_left _ (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,Int.cast_natCast]
  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,Int.cast_natCast,Nat.cast_mul,Nat.cast_pow,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
        (Nat.sub_le_of_le_add (Nat.le_of_lt_succ (Nat.lt_of_le_of_ne (Nat.le_of_lt_succ (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:=Nat.cast_le.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 ▸ Nat.cast_nonneg _
  have hΛb:Λ ≤ 2^(304:Ἀ):=hΛS ▸ (Nat.cast_le.2 hS).trans_eq (by rw [Nat.cast_pow,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,Finset.sum_range_zero,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:=Nat.mul_le_mul_left 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):=(Nat.cast_mul _ _).trans (congrArg₂ (·*·) nZ (Nat.cast_succ n))
 have hFn:=FN.2.trans ((Int.cast_natCast _).trans e)
 have Zr:=kRM _ Lf FN.1 hf (((Ae _ _).trans (congrArg (·*|rv Lf|) ((congrArg abs hFn).trans (Ad ((Nat.cast_nonneg _).trans_eq e))))).trans_le ((mul_le_mul (e.symm.trans_le (((Nat.cast_le (α:=ℝ)).mpr (a.trans (by decide))).trans_eq ((Nat.cast_pow 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)) (Nat.cast_add _ 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 Nat.le_add_left 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,Nat.sub_add_cancel (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) (Nat.le_sub_of_add_le ((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:=Nat.le_of_not_lt fun hl=>absurd (Nat.le_of_add_le_add_right (Hn.trans (Θ Nat.zero_lt_two (Nat.le_of_lt_succ 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:=Finset.abs_sum_le_sum_abs (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) (Nat.le_of_ble_eq_true rfl:425 ≤ 960)).trans_eq (AA (2:ℝ) 960).symm))
    n 0 _ (pQ n) (ϟ _) hd.1 (by rw [hd.2,Finset.sum_range_zero,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 Nat.le_of_lt_succ 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 (Nat.le_of_lt (show e.toNat < 56 from g))).symm
    · rw [if_neg g]; exact (Nat.max_eq_left (Nat.le_of_not_lt (show ¬ e.toNat < 56 from g))).symm
  rw [η,m,q, ← pA]
  exact γ (C0 (Nat.add_le_add (L1 (Nat.max_le.2 ⟨Nat.le_of_lt 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 (Nat.le_of_lt (show b.toNat < a.toNat from g))).symm
  · rw [if_neg g]; exact (Nat.max_eq_right (Nat.le_of_not_lt (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 (Nat.le_trans (Nat.sub_le _ _) 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 ⟨Nat.le_trans (Nat.sub_le _ _) (Nat.le_trans z (by decide)),Nat.le_trans 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 Nat.le_of_not_lt 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:=Nat.le_of_sub_eq_zero hm
    rw[dRw,dif_neg (mt ε.mp (Nat.not_lt.mpr e)),kFS,kSm o n hB.2,ao,if_neg (Nat.not_lt.mpr e),if_neg (fun(g:_∧_)=>Nat.not_lt.mpr (Nat.le_trans (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,← Nat.sub_sub,hm,BK]) (by rw[h2]; exact h1) q,h2,kFS,kSm o k (Nat.lt_trans (O1 h1 _) hB.2),kXU _ _ _ _ (kSm i k (Nat.lt_trans (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 (Nat.le_trans (J1 (Nat.le_succ _) _) 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 ⟨Nat.lt_of_le_of_ne 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):=Nat.le_antisymm hk (Nat.le_of_sub_eq_zero hm)
    have nl:¬ (J0 k) < (J0 n):=fun h=>Nat.lt_irrefl _ (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,Nat.sub_add_eq,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,Nat.le_of_succ_le 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 [Nat.mul_add_div (by decide),ώ hr,ϑ]
      have d2:(8*(J0 k)+(J0 r))%8=(J0 r):=by rw [Nat.mul_add_mod,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):=Nat.le_antisymm (Nat.le_of_lt_succ (Nat.lt_of_not_le 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=>Nat.ne_of_lt hj (Nat.eq_of_mul_eq_mul_left (by decide) (Nat.add_right_cancel e))),if_pos ⟨rfl,ϟ _,hj⟩,if_pos ⟨(O1 hs8 _).trans_le (L1 (Nat.mul_le_mul_left 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 (Nat.lt_irrefl j)]
    · rw [if_neg (fun e=>Nat.ne_of_gt hj (Nat.eq_of_mul_eq_mul_left (by decide) (Nat.add_right_cancel e))),if_neg (fun h=>Nat.lt_asymm h.2.2 hj),if_neg (fun h=>Nat.not_lt.2 (((Nat.mul_succ 8 N).symm.trans_le (Nat.mul_le_mul_left 8 hj)).trans (V1 _ _)) h.1)]
  · by_cases hc:8*j+s < 8*N+8∧R ≤ s∧s < L
    · rw [if_pos ⟨hc.1,Nat.lt_of_le_of_ne hc.2.1 (Ne.symm hs),hc.2.2⟩,if_pos hc]
    · rw [if_neg (fun h=>hc ⟨h.1,Nat.le_of_succ_le h.2.1,h.2.2⟩),if_neg (fun e=>hs (((Nat.mul_add_mod 8 j s).trans (γ hs8)).symm.trans ((congrArg (·%8) e).trans ((Nat.mul_add_mod 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):=Nat.le_antisymm hr (Nat.le_of_sub_eq_zero hm)
    rw [dX8,dif_neg (fun h=>Nat.lt_irrefl _ ((ε.mp h).trans_le e.ge))]; exact (if_neg (fun h=>Nat.lt_irrefl _ (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 (Nat.le_of_not_lt this)).symm
  have tp:p.toNat=max (E[j]!).toNat 56+cq.toNat:=by
    rw [hp,kAddS _ _ (by rw [hm]; exact C0 (Nat.add_le_add (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 _ ((Nat.sub_le _ _).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 (Nat.le_of_not_lt (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 ⟨(Nat.sub_le _ _).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 _,Nat.le_of_not_lt 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 _ ((Nat.le_add_left _ _).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 _ ((Nat.sub_le _ _).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:=(Nat.mul_add_mod 8 s r).trans (γ hr)
   have q2:(8*s+r)/8=s:=by rw[Nat.mul_add_div (by decide),ώ hr,ϑ]
   have c1:8*s+r<8*nn+8:=(O1 hr _).trans_le (L1 (Nat.mul_le_mul_left 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:=(Nat.mul_add_mod 8 nn r).trans (γ r8)
   have q2:(8*nn+r)/8=nn:=by rw[Nat.mul_add_div (by decide),ώ r8,ϑ]
   refine (D _).trans ?_
   rw[q1,q2,if_neg (Nat.lt_irrefl 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:=Nat.le_trans (Nat.le_of_eq (Ο r m).symm) (ι m h1)
  have q2:=Nat.le_trans (Nat.le_of_eq (Ο r n).symm) (ι n h1)
  have q3:=Nat.mul_le_mul_left R (Nat.le_of_lt 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 ▸ Nat.lt_trans (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 (Nat.not_lt.2 (Nat.sub_eq_zero_iff_le.1 hk)))],fun r' t h1 h2 _=>(Nat.lt_irrefl r' (U1 h1 ((Nat.sub_eq_zero_iff_le.1 hk).trans h2))).elim⟩
  | succ k ih =>
    intro r Y hY hk hr
    have h1:=N1 hk
    have hnm:=Nat.lt_of_lt_of_eq (Nat.lt_succ_self _) 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) (Nat.le_trans hY (Nat.le_of_eq (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))) (Nat.le_trans (Nat.le_of_eq 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 _ ((Nat.le_succ _).trans (Nat.le_of_eq hr1.symm)))),D q,e2,if_neg (by exact Nat.ne_of_lt (U1 hq (V1 _ _))),if_neg (by exact fun h=>Nat.not_lt.2 ((V1 _ _).trans h.1) hq)]
    rcases Nat.lt_or_ge r.toNat r' with h5 | h5
    · exact I2 r' t h2 (Nat.le_trans (Nat.le_of_eq hr1) h5) h4
    · obtain rfl:=Nat.le_antisymm 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 (Nat.add_left_cancel h)),if_neg ht,if_pos (by exact ⟨J1 (ϟ _) _,O1 (Nat.lt_of_le_of_ne 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 Nat.le_of_not_lt 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:=Nat.not_lt.mpr (Nat.sub_eq_zero_iff_le.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:=Nat.sub_lt_sub_left h (Nat.lt_succ_self r)
theorem Lq2(R r:ℕ)(h:r+1 < R):R-(r+1+1) < R-r:=Nat.sub_lt_sub_left (Nat.lt_of_succ_lt h) (Nat.lt_succ_of_lt (Nat.lt_succ_self 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=>Nat.lt_irrefl _ (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:=⟨Nat.le_of_succ_le 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 ⟨Nat.lt_of_le_of_ne 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 Nat.ne_of_gt (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:=Nat.lt_of_le_of_ne (Nat.le_of_lt_succ (Nat.lt_of_le_of_ne (Nat.le_of_lt_succ h) e1)) e0
theorem Lw7(o o' w:ℕ)(ho:o' < w)(h2:¬o+1 < w)(e0:¬o'=o):o' < o:=Nat.lt_of_le_of_ne (Nat.le_of_lt_succ (U1 ho (Nat.le_of_not_lt h2))) e0
theorem n5d0:0≤7:=Nat.le_of_ble_eq_true rfl
theorem n5d1:1≤7:=Nat.le_of_ble_eq_true rfl
theorem n5d2:2≤7:=Nat.le_of_ble_eq_true rfl
theorem n5d3:3≤7:=Nat.le_of_ble_eq_true rfl
theorem n5d4:4≤7:=Nat.le_of_ble_eq_true rfl
theorem n5d5:5≤7:=Nat.le_of_ble_eq_true rfl
theorem n5d6:6≤7:=Nat.le_of_ble_eq_true rfl
theorem n5d7:7≤7:=Nat.le_of_ble_eq_true 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):Float.Model.pack 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=Float.Model.UnpackedFloat.normalize 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[Float.Model.UnpackedFloat.normalize,c,Model.modelSign,h,decide_true,if_true,ha]
 ·subst h;simp[Float.Model.UnpackedFloat.normalize,Model.modelSign]
 ·have c:compare M 0=.gt:=Int.compare_eq_gt.mpr h;have ha:M.natAbs=M.toNat:=by omega
  simp only[Float.Model.UnpackedFloat.normalize,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):Float.Model.pack (Float.Model.UnpackedFloat.normalize k0 M1 E1 .positive)=Float.Model.pack (Float.Model.UnpackedFloat.normalize 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):(Float.Model.pack (.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 Model.modelSignBit s=true then -1 else 1)*(n:ℝ):=by
 cases s<;>simp[Float.Model.UnpackedFloat.Sign.apply,Model.modelSignBit]
open FloatLib.Floats.Formats.BinaryInterchange in
theorem fzSM(s1 s2:pK):(if Model.modelSignBit (s1*s2)=true then (-1:ℝ) else 1)=(if Model.modelSignBit s1=true then -1 else 1)*(if Model.modelSignBit s2=true then -1 else 1):=by
 cases s1<;>cases s2<;>simp[Model.modelSignBit]
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 Float.Model.pack (fma .binary64 x.toModel.unpack y.toModel.unpack z.toModel.unpack)=Float.Model.pack (add .binary64 z.toModel.unpack (x*y).toModel.unpack)
 have e3:(x*y).toModel=Float.Model.pack (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,Model.unpackedToReal_finite,Model.unpackedToReal_finite,fzBP,fzBP]
    cases s1<;>cases s2<;>simp[Model.modelSignBit,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 Float.Model.pack (mul .binary64 (.finite s1 m1 e1 h1) (.finite s2 m2 e2 h2))=Float.Model.pack (.finite sP mP eP hmP)
     rw[← hPd,fzPU]
    | finite s3 m3 e3 h3=>
     apply fzVD
     rw[Model.unpackedToReal_finite,Model.unpackedToReal_finite,Model.unpackedToReal_finite,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[Rat.cast_mul,Rat.cast_add,Rat.cast_intCast,Rat.cast_zpow,Rat.cast_ofNat,Int.cast_add,fzSA,Int.cast_natCast]
     linear_combination (if Model.modelSignBit (s1*s2)=true then (-1:ℝ) else 1)*A+(if Model.modelSignBit s3=true then (-1:ℝ) else 1)*B-(if Model.modelSignBit s3=true then (-1:ℝ) else 1)*C-(if Model.modelSignBit sP=true then (-1:ℝ) else 1)*D+((m1:ℝ)*m2*(2:ℝ)^(e1+e2))*E-hV+((if Model.modelSignBit s1=true then (-1:ℝ) else 1)*(if Model.modelSignBit 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 (Nat.lt_of_succ_lt h1)
  have B:=sMb Q Y vo nu i0 hY hQ t (Nat.lt_of_succ_lt h1) h1
  let r:=sM1 false Q Y vo nu i0 hY hQ t (Nat.lt_of_succ_lt 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 (Nat.lt_of_succ_lt h1) h1;rw[B.1];exact Nat.sub_lt_sub_left (Nat.lt_of_succ_lt h1) (Nat.lt_add_of_pos_right (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 Nat.strong_induction_on 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 Nat.sub_lt_sub_left h3 (Nat.lt_add_of_pos_right (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 _ _ _ (Nat.le_of_not_lt 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 (Nat.le_of_not_lt h3)
      rw [sM,dif_neg (fun h=>h3 (Nat.lt_of_succ_lt (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):=Nat.mul_le_mul_left 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:=Nat.lt_of_succ_lt 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 (Nat.le_of_eq (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:=Nat.lt_of_succ_lt 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:=UInt64.toUInt32 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
@[noinline] 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 (Nat.le_add_left _ _)) 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 (Nat.eq_of_mul_eq_mul_right hs1 (Nat.add_left_cancel ((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:=Nat.le_of_lt_succ 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 (Nat.le_add_left _ _))) 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 (Bool.eq_false_iff.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 Nat.strong_induction_on 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 (Nat.le_of_not_lt (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 Nat.strong_induction_on 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 (Nat.min_le_left _ _) (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 (Nat.min_le_left _ _) (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 (Nat.min_le_left _ _) (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,Nat.add_sub_cancel' 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,Nat.add_sub_cancel' 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 (Nat.min_le_left _ _) (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 (Nat.min_le_left _ _) (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 (Nat.min_le_left _ _) (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,Nat.add_sub_cancel' (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,Nat.add_sub_cancel' (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,Nat.add_sub_cancel' (by omega)]
    rw [eo]
    rw [dn8_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Q2.2 (fun l hl' e=>by rw [ix2,ix1] at e; rw [ec] at hl'; have:=Nat.eq_of_mul_eq_mul_right hs1 (Nat.add_left_cancel 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,Nat.add_sub_cancel' (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,Nat.add_sub_cancel' (by omega)]
    rw [eo]
    rw [dn8_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Q2.2 (fun l hl' e=>by rw [ix2,ix0] at e; rw [ec] at hl'; have:=Nat.eq_of_mul_eq_mul_right hs1 (Nat.add_left_cancel e); omega)]
    rw [dn8_ne _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Q1.2 (fun l hl' e=>by rw [ix1,ix0] at e; rw [e8] at hl'; have:=Nat.eq_of_mul_eq_mul_right hs1 (Nat.add_left_cancel 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,Nat.add_sub_cancel' (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:=Nat.sub_lt_sub_left h (Nat.lt_succ_self g)
theorem dnLt3(G g:Nat)(h:g+3<=G):G-(g+3)<G-g:=Nat.sub_lt_sub_left (U1 (Nat.lt_add_of_pos_right (by decide)) h) (Nat.lt_add_of_pos_right (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 Nat.strong_induction_on 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:=Nat.eq_of_mul_eq_mul_right hs1 (Nat.add_left_cancel 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:=Nat.eq_of_mul_eq_mul_right hs1 (Nat.add_left_cancel 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 Nat.strong_induction_on 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):=Nat.le_mul_of_pos_right 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 (Nat.le_of_lt_succ (Nat.lt_succ_of_le 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 Nat.strong_induction_on 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):=Nat.le_mul_of_pos_right 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 (Nat.le_of_lt hs)).trans (if_neg (Nat.ne_of_lt 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):=Nat.mul_le_mul_left _ (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:=Nat.le_mul_of_pos_right _ 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):=Nat.le_mul_of_pos_right 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:=Nat.le_mul_of_pos_right _ 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 [Array.size_replicate,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):=Nat.le_mul_of_pos_right 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 (Nat.lt_trans hn31 h31)
  have e2:=s5tu (n+1) (C0 (l7 hn31) h31)
  have e3:=s5tu R (Nat.lt_trans (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):=Nat.lt_trans hRo h31
  have hw1:ow+16<=(ow+16)*n:=Nat.le_mul_of_pos_right _ 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 (Nat.ne_of_lt 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 [Array.size_replicate,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,Array.size_replicate])
  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 Nat.not_lt_zero _ 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):=Nat.mul_le_mul_left _ 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 (Nat.le_of_lt (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 Nat.lt_trans 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) (Nat.le_of_lt 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 _ _ _ _ (Nat.sub_eq_zero_iff_le.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 (Nat.le_of_lt 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):=Nat.mul_le_mul_left _ 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 (Nat.le_trans 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 (Nat.le_of_lt 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 Nat.strong_induction_on 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 Nat.strong_induction_on 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' (Nat.le_of_not_lt 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 (Nat.le_of_lt 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+USize.ofNat r') (USize.ofNat 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' (Nat.le_of_succ_le 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 (Nat.sub_lt_left_of_lt_add 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+USize.ofNat (J0 r)=r0+r:=by rw [show USize.ofNat (J0 r)=r from USize.ofNat_toNat]
        rw [er]
      · exact T3 r' o' (Nat.lt_of_le_of_ne 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 (Nat.le_of_lt (show e.toNat < 56 from g))).symm
    · rw [if_neg g]; exact (Nat.max_eq_left (Nat.le_of_not_lt (show ¬ e.toNat < 56 from g))).symm
  rw [η,m]
  exact γ (C0 (Nat.add_le_add (Nat.max_le.2 ⟨Nat.le_of_lt 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 Nat.not_lt_zero _))
  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 (USize.ofNat o'))=o':=USize.toNat_ofNat_of_lt' (hS _ (U1 ho (Nat.le_of_lt (C0 (lZ _ (by decide)) h31.2.2))))
  have hr8:r < kX:=hS _ (U1 (U1 hr hl) (by decide))
  have hrow:(J0 (r0+USize.ofNat r))=(J0 r0)+r:=by
    rw [kSm _ _ (by rw [USize.toNat_ofNat_of_lt' hr8]; exact Lql _ _ _ _ hr (Nat.le_of_lt hR)),USize.toNat_ofNat_of_lt' 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:=Nat.min_le_left _ _
  have hlR:rb*8+lim<=R:=(J1 (Nat.min_le_right 8 (R-rb*8)) _).trans (Nat.le_of_eq (BQ (Nat.le_of_lt 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:=USize.toNat_ofNat_of_lt' (hS _ (AO ((V1 _ _).trans hR1) hRn))
  have e2:(J0 ((G0 n)))=n:=USize.toNat_ofNat_of_lt' (hS _ (AO (lZ n (by decide)) h8n))
  have e3:(J0 ((G0 lim)))=lim:=USize.toNat_ofNat_of_lt' (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,Nat.lt_trans 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:=USize.toNat_ofNat_of_lt' (hS _ how)
  have e5:(J0 ((G0 (rb*8))))=rb*8:=USize.toNat_ofNat_of_lt' (hS _ (Nat.lt_trans 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 Nat.sub_lt_sub_left h (Nat.lt_add_of_pos_right (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=>Nat.not_lt.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 (Nat.sub_lt_left_of_lt_add (Nat.not_lt.1 hr) (by rwa[Ϡ,n4] at h1)) (Nat.sub_lt_sub_right (Nat.not_lt.1 hr) h2)) (ϟ _) h3
   rw[BQ (Nat.not_lt.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:=Nat.sub_eq_zero_iff_le.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 (Nat.not_lt.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 (Nat.not_lt.1 h)];exact ϟ _) hA hI
   have hl8:min 8 (R-rb*8)<=8:=Nat.min_le_left _ _
   have hlR:rb*8+min 8 (R-rb*8)<=R:=(J1 (Nat.min_le_right 8 (R-rb*8)) _).trans (Nat.le_of_eq (BQ (Nat.le_of_lt 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:=USize.toNat_ofNat_of_lt' (hS _ (AO ((V1 _ _).trans hR1) hRn))
   have e2:(J0 ((G0 n)))=n:=USize.toNat_ofNat_of_lt' (hS _ (AO (lZ n (by decide)) h8n))
   have e3:(J0 ((G0 (min 8 (R-rb*8)))))=min 8 (R-rb*8):=USize.toNat_ofNat_of_lt' (hS _ (AO hl8 (by decide)))
   have e4:(J0 ((G0 ow)))=ow:=USize.toNat_ofNat_of_lt' (hS _ how)
   have e5:(J0 ((G0 (rb*8))))=rb*8:=USize.toNat_ofNat_of_lt' (hS _ (Nat.lt_trans 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 (Nat.lt_succ_self _) ((Nat.mul_add_one 8 n).symm.trans_le (Nat.mul_le_mul_left 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,Nat.sub_add_eq];exact (Nat.sub_le_sub_left (by decide) _).trans (Nat.sub_le_of_le_add 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 (Nat.le_trans (Nat.le_trans (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 [Array.size_replicate]) (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 Nat.not_lt_zero _ 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 Nat.lt_succ_of_le 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[UInt32.le_iff_toNat_le,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 (Nat.le_of_eq ((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 Nat.sub_succ_lt_self 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:=Nat.sub_lt_left_of_lt_add (not_lt.1 h) (Nat.add_lt_add_right Q _);apply K;rw[M,Nat.mod_eq_sub_mod (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 Nat.lt_or_ge r (2^(28:Ἀ)) with h|h
 · have h2:r+2<=2^(28:Ἀ):=Nat.lt_of_le_of_ne h v1
   have a:r+2^(28:Ἀ)<2^(29:Ἀ):=(Nat.add_lt_add_right h _).trans_eq ἅ
   have b:r+268435457<2^(29:Ἀ):=(L1 (Nat.le_sub_of_add_le h2) 268435457).trans_lt (by decide)
   rw[ώ b,ώ a,γ a];exact⟨ἅ,((by decide):2<=2^(28:Ἀ)).trans (Nat.le_add_left _ _),(lO _ _ _).trans_le ((L1 h2 _).trans_eq ἅ)⟩
 · obtain⟨t,rfl⟩:∃t,r=2^(28:Ἀ)+2+t:=Nat.exists_eq_add_of_le (Nat.lt_of_le_of_ne (Nat.lt_of_le_of_ne 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 (Nat.lt_of_succ_lt tl),Μ,γ (Nat.lt_of_succ_lt tl)]
   exact⟨ἅ,Nat.le_add_left _ _,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 (Nat.add_right_cancel (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:=Nat.le_of_lt_succ (Ϟ ((Nat.add_lt_add_right 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 ((Nat.le_add_left _ _).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 _ (Nat.le_of_lt_succ 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 (Nat.add_le_add a (Nat.add_le_add 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[Nat.sub_sub,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):=(Nat.lt_succ_of_lt (Nat.lt_succ_self f)).trans_le L;have fl:f+268435456<2^(64:Ἀ):=(Nat.add_lt_add_right f5 _).trans (by decide)
 have el:E*2^(52:Ἀ)+f+268435456<2^(64:Ἀ):=(Nat.add_lt_add_right (Nat.add_lt_add_of_le_of_lt (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[←Nat.sub_add_comm ((by decide):1195<=2^(64:Ἀ)),Τ (2^(64:Ἀ)),Nat.sub_add_comm y,Nat.add_mod_right,γ ((Nat.sub_le_sub_right z 1195).trans_lt (by decide):E-1195<2^(64:Ἀ))];exact (Nat.sub_le_sub_right z 1195).trans_lt (by decide),by rw[γ fl];generalize (f+268435456)%2^(29:Ἀ)=R at I J⊢;rw[←Nat.sub_add_comm ((by decide):2<=2^(64:Ἀ)),Τ (2^(64:Ἀ)),Nat.sub_add_comm I,Nat.add_mod_right,γ ((Nat.sub_le _ _).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):=(Nat.add_lt_add_of_le_of_lt (ι (2^(31:Ἀ)) (Nat.le_of_lt_succ 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 (UInt64.toUInt32 (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):=Nat.le_of_lt_succ ((ϸ (϶ 29)).2 ((Nat.add_lt_add_right (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 _ (Nat.sub_le_sub_right (Nat.sub_le_sub_right z 1085) 110);have X2:=cX_h2 _ _ r (C0 (Nat.add_le_add 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:=(UInt64.toUInt32 (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 (Nat.cast_nonneg _)),sub_le_iff_le_add.2 (Nat.cast_le.2 (q.trans ((by decide):1447<=1993))),le_sub_iff_add_le.2 (((by decide):(-1021:Ἁ)+1022<=1195).trans (Nat.cast_le.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 _ _ (USize.le_iff_toNat_le.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 _ _ (USize.le_iff_toNat_le.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⟩:=Bool.and_eq_true_iff.1 G;obtain⟨x,y⟩:=Bool.and_eq_true_iff.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 ((Nat.sub_le _ _).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 ((Nat.sub_le _ _).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 (Int.ofNat_le.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 (Bool.and_eq_true_iff.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 (Bool.and_eq_true_iff.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 (USize.le_iff_toNat_le.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=>Nat.not_lt.2 u v.2)).trans (if_pos (r.trans u)).symm)
 else if v:a<=j then(if_neg u).trans ((if_pos ⟨v,Nat.lt_of_not_le u⟩).trans ((congrArg (Option.map · o) (q v (Nat.lt_of_not_le 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 (Nat.le_of_sub_eq_zero hk))];split
 .rw[l5 (p.trans_le ((ι _ (Nat.le_of_sub_eq_zero 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,USize.le_iff_toNat_le.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 Nat.le_of_not_lt 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 (Nat.not_lt_zero _)
|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 (Nat.not_lt_zero _)
|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,Nat.div_one,Nat.mod_one,ϑ,pQ,Nat.not_lt_zero,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 (Nat.not_lt_zero _)
|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 (UInt64.toUInt32 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 Nat.sub_succ_lt_self 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,Nat.cast_add,pO (by exact Nat.le_of_succ_le 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 ((Nat.add_le_add (g.trans l) m).trans_lt (by decide))]at q
 rw[q];exact⟨Nat.le_add_left _ _,(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 (Nat.not_lt.2 (Nat.le_add_left _ _))
 have h1:2^(64:ℕ)<=D%2^(39:ℕ)*2^(25:ℕ)+E:=Nat.le_of_not_lt h
 have hi:D%2^(39:ℕ)*2^(25:ℕ)+E<2^(64:ℕ)+2^(64:ℕ):=Nat.add_lt_add 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),Nat.mod_eq_sub_mod h1,γ ((Nat.sub_lt_iff_lt_add h1).2 hi)];exact if_pos ((Nat.sub_lt_iff_lt_add h1).2 ((Nat.add_lt_add_right 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 (Nat.le_of_not_lt 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:=Int.ofNat_le.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 (Int.natCast_nonneg _) 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) (Int.natCast_nonneg _)) 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:=
 (Finset.abs_sum_le_sum_abs _ _).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 Nat.zero_lt_of_lt e),ώ e,Κ]
 refine⟨?_,?_,by rw[Τ,Μ,mo e]⟩
 .rw[Ϗ H W,<-lY,d,Τ,Ρ _ _ (by exact Nat.zero_lt_of_lt 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 Nat.le_of_succ_le A)▸u _ _ _ _ e (by exact Nat.lt_of_succ_le 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 Nat.add_lt_add_right 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 Nat.zero_lt_of_lt 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 Nat.sub_succ_lt_self 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 ⟨(Nat.le_succ _).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 ⟨Nat.lt_of_le_of_ne 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:=Nat.le_of_sub_eq_zero e;rw[fcU,dif_neg λh=>Nat.not_lt.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,Nat.sub_add_eq,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 Nat.sub_succ_lt_self 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 (Nat.le_add_left 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:=Nat.sub_eq_zero_iff_le.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) _) (Nat.le_trans 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 Nat.le_trans 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 (Nat.le_trans (J1 (L1 ym.le 2) ((J0 r)/s.h*(s.h+2))) c1));rw[nO]at this;rw[f,hw,lO _ 1];exact Nat.le_of_succ_le (Nat.le_trans 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 (Nat.le_trans (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,Nat.le_of_lt_succ 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 (USize.toNat_inj.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],Nat.lt_of_le_of_ne 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,<-Nat.sub_sub,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 (Nat.lt_succ_of_lt ym)) (Nat.succ_lt_succ (Nat.lt_succ_of_lt 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 Nat.lt_succ_self _)];unfold aI;rw[if_pos ⟨cl,Nat.le_add_left 1 _,ym,Nat.le_add_left 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 (Nat.lt_succ_of_lt k)]
    .rw[if_neg (by rw[k];exact Nat.lt_irrefl _),if_pos (by rw[k];exact Nat.lt_succ_self _)];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▸(Nat.sub_add_cancel u2).symm
     rw[this,<-pA]at Q;apply k1;rw[f,hw,lO _ 1,Q];exact⟨J1 u4 _,O1 (Nat.lt_succ_of_le u5) _⟩
    .rw[if_neg (Nat.lt_asymm k),if_neg (Nat.not_lt.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=USize.ofNat (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):(USize.ofNat n).toNat=n:=USize.toNat_ofNat_of_lt' (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 (USize.ofNat 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 (Nat.not_lt_zero _)
 | 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 (USize.ofNat 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:Float.Model.pack a.toModel.unpack=Float.Model.pack 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[Float.Model.pack,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:=Nat.le_trans (lZ _ (by omega)) aH
  have K:=czX9_ok d.X ((d.s.h+2)*(d.s.w+2)) (USize.ofNat (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 (UInt64.toUInt32 (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[←Nat.sub_add_comm (hab.trans hb.le),Nat.add_sub_assoc hab,Nat.add_mod_left,γ ((Nat.sub_le 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 (Nat.le_of_lt (Nat.add_lt_add_right 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 (((Nat.sub_le _ _).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:=Nat.not_lt.1 λlt=>absurd (((Nat.sub_le_sub_left l3 _).trans (V1 _ e)).trans_lt ((γ ((O1 lt _).trans_le (Nat.le_of_eq (Nat.sub_add_cancel (l3.trans (by decide)))))).symm.trans_lt c)) (Nat.not_lt.2 (((Nat.sub_le _ _).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:=(Nat.add_lt_add_right hh 1075).trans (by decide)
 rw[γ (hh.trans (by decide)),γ h2,wok_h1 e (h+1075) (Nat.le_of_lt_succ (he.trans_le m2)) h2,wok_h1 1195 e (l2.trans le) hE]
 exact⟨Nat.sub_le_of_le_add ((Nat.add_left_comm h 21 1054).trans_le (J1 (l1.trans le) 21):h+1075<=21+e),Nat.sub_lt_left_of_lt_add (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,UInt64.toNat_ofNat',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 (UInt64.toUInt32 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 (UInt64.toUInt32 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 (UInt64.toUInt32 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[<-Nat.sub_sub,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 (UInt64.toUInt32 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 Nat.lt_irrefl (r+1))]
 conv_rhs=>rw[pl,AC (by exact Nat.lt_irrefl 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▸Nat.le_of_not_lt 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:=USize.le_iff_toNat_le.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 Nat.not_lt_zero 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 Nat.lt_of_succ_le 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 (USize.le_iff_toNat_le.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 (Nat.le_trans (Ay _ _) (J1 ((USize.toNat_ofNat).trans_le (Ay _ _)) _))
 exact Nat.le_trans (L1 (Nat.le_trans 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 (Nat.le_trans (Ay _ _) (J1 ((USize.toNat_ofNat).trans_le (Ay _ _)) _))
 exact C0 a (C0 (Nat.le_trans (J1 (Nat.le_trans (Nat.le_add_of_sub_le (Nat.le_refl (k-7))) (Nat.le_of_eq (Τ _ _))) _) (Nat.le_of_eq (pA _ 7 _).symm)) (U1 (Nat.add_lt_add_right (Nat.lt_succ_self _) _) (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 Nat.sub_lt_sub_left (ε.mp h) (Nat.lt_of_lt_of_eq (Nat.lt_succ_self _) (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=>Nat.sub_ne_zero_of_lt (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(((Nat.add_lt_add_of_le_of_lt (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 (Nat.lt_trans (AN (O1 Y _) hW) U.2) ς)]
 refine ih (t+1) _ _ _ _ _ _ _ _ A (by rw[ac h,<-Nat.sub_sub,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:=Nat.sub_lt_left_of_lt_add 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,Nat.sub_add_eq,Nat.add_sub_cancel' (Nat.le_sub_of_add_le' A1.1)]
 ·exact absurd ⟨(V1 _ _).trans A1.1,A1.2.trans_eq (by rw[pA,Nat.add_sub_cancel' B.le])⟩ A2
 ·exact I A3.1 A3.2
 ·exact absurd ⟨A3.1,A3.2.trans_le (J1 (min_le_right _ _) _)⟩ A4
 ·exact (Nat.lt_or_ge 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,Nat.add_sub_cancel' 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⟨Nat.sub_le_of_le_add (hm.trans (J1 ((by decide):1<=8) m)),by rw[pA,Nat.add_sub_cancel' B.le];exact hx,pA _ _ _,by rw[pA,Nat.add_sub_cancel' 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 (Nat.le_of_not_lt 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 (((Nat.add_lt_add_of_le_of_lt ((Nat.le_add_left _ _).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=>Nat.lt_irrefl _ ((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 (Nat.le_of_sub_eq_zero 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 (Nat.add_le_add (V1 (i0+(d.s.h-1)*V+d.s.w) d.M) (Nat.le_of_ble_eq_true ἅ:2<=7)) (C0 hb (U1 d.hX ς))
    exact C0 (J1 (Nat.mul_le_mul_left d.s.h (Nat.le_of_eq (Dense.aa (n:=V) (C0 (L1 (Nat.le_add_left d.s.w (i0+(d.s.h-1)*V)) 2) K)))) i0) (C0 (Nat.le_trans (J1 (Nat.le_add_of_sub_le (Nat.le_of_eq (Nat.sub_one_mul d.s.h V).symm)) i0) (Nat.le_of_eq ((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 Nat.le_trans (Nat.le_of_eq (congrArg (·+d.M+7) (congrArg₂ (·+·) (hi.trans (congrArg (i0+y*·) (Dense.aa (n:=V) (C0 (Nat.le_trans (L1 (Nat.le_add_left d.s.w (i0+(d.s.h-1)*V)) 2) (Nat.add_le_add (V1 (i0+(d.s.h-1)*V+d.s.w) d.M) (Nat.le_of_ble_eq_true ἅ:2<=7))) K)))) (Dense.aa (n:=d.s.w) (C0 (Nat.le_trans (Nat.le_add_left d.s.w (i0+(d.s.h-1)*V)) (Nat.le_trans (V1 (i0+(d.s.h-1)*V+d.s.w) d.M) (V1 _ 7))) K))))) (Nat.le_trans (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 (Nat.le_trans (Nat.le_trans (V1 i0 ((d.s.h-1)*V)) (V1 _ d.s.w)) (Nat.le_trans (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))
   (Nat.le_trans (Nat.le_of_eq (ϣ ..)) (L1 hn 1)) h (λj hj=>(Nat.lt_or_eq_of_le (Nat.le_of_lt_succ (Nat.lt_of_lt_of_eq hj (ϣ ..)))).elim
    (λhjs=>Nat.le_trans (Nat.le_of_eq (((AB (Tz.push T[t]!) j hj).trans (nU hjs)).trans (AB Tz j hjs).symm)) (hz j hjs))
    (λhe=>Nat.le_trans (Nat.le_of_eq ((congrArg (fun (i:Ἀ)=>(Tz.push T[t]!)[i]!) he).trans ((AB (Tz.push T[t]!) (O0 Tz) (Nat.lt_of_lt_of_eq (Nat.lt_succ_self _) (ϣ ..).symm)).trans nV))) (hT t h)))
 else⟨Tz,Wz,0,(O0 Tz),fQ,AP,Nat.le_of_eq ((Κ _).trans hs),C0 (Nat.le_trans (Nat.le_of_eq (Κ _)) (Nat.le_trans hn ht)) hk,hz,C0 (Nat.le_trans (Nat.le_of_eq hs.symm) (Nat.le_trans hn ht)) hk⟩
termination_by k-t
decreasing_by all_goals exact Nat.sub_lt_sub_left h (Nat.lt_succ_self 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 (Nat.le_of_sub_eq_zero 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 (Nat.le_of_lt_succ 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 _ _ _ (Nat.lt_succ_of_le 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:=Nat.not_lt.2 (Nat.sub_eq_zero_iff_le.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,Nat.div_one,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 _ (Nat.le_add_left 3 _)) hN;have V:s.cin*(s.h+2)+3<2^(32:Ἀ):=AO (Nat.le_mul_of_pos_right _ (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 (Nat.mul_le_mul_left _ (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(Nat.le_trans bh (Nat.le_trans (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 (Nat.le_trans (L1 bh 2) hL) m (Nat.le_trans ah (by decide)) c.n 0 _ rfl
  (ϟ _) u.1 (by simp only[u.2,Finset.sum_range_zero,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 (Nat.le_trans (Nat.le_trans (Nat.le_trans (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)
   (Nat.add_lt_add_of_lt_of_le hy (jd d.kk t)) (Nat.add_lt_add_of_lt_of_le 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 (Nat.lt_add_of_pos_right (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)
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:=
 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
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 (Nat.le_of_eq (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 (Nat.le_of_eq ((congrArg (_+·+_+7) (Nat.sub_add_cancel 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 (Nat.lt_trans (AN (Nat.lt_add_of_pos_right 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 (Nat.sub_lt_left_of_lt_add 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 Array.size_replicate
 exact mi (aL.trans (Array.size_replicate.trans Ϝ.symm)) λi v bS=>(AB _ i v).symm.trans ((ba i (Nat.lt_of_lt_of_eq v aL)).trans ((AD ⟨ϟ _,Nat.lt_of_lt_of_eq (Nat.lt_of_lt_of_eq v (aL.trans Array.size_replicate)) (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];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,Nat.pair (Nat.pair A B) (Nat.pair 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)),Nat.pair (Nat.pair ((s.h+2)*(s.w+2)) (s.w+2)) (Nat.pair (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)),Nat.pair (Nat.pair ((s.h+2)*(s.w+2)) (s.w+2)) (Nat.pair (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)),Nat.pair (Nat.pair ((s.h+2)*(s.w+2)) (s.w+2)) (Nat.pair (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 Nat.lt_succ_of_le hIs0
  have hX32:(s.cin*H+3)*V<2^(32:Ἀ):=by exact Nat.lt_succ_of_le hX0
  have hw32:s.w<2^(32:Ἀ):=by exact Nat.lt_succ_of_le hw0
  have hY32:Y*k<2^(32:Ἀ):=by exact Nat.lt_succ_of_le hY0
  have hk32:k<2^(32:Ἀ):=by exact Nat.lt_succ_of_le 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,(USize.ofNat (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
def fL(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)),Nat.pair (Nat.pair ((s.h+2)*(s.w+2)) (s.w+2)) (Nat.pair (s.kh*s.kw) nc)) (fun k=>k.1))
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,Nat.div_one]<;>exact⟨Nat.sub_add_cancel a2,Nat.sub_add_cancel aH⟩
lemma ev_h2(P L:Ἀ)(H:P+L<=425)(p:L<=27):P+2*L-54<n0:=
 C0 (Nat.le_trans (Nat.sub_le _ _) (by rw[two_mul,<-pA];exact Nat.add_le_add 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)),Nat.pair (Nat.pair ((s.h+2)*(s.w+2)) (s.w+2)) (Nat.pair (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 Nat.le_of_lt_succ G),(by exact Nat.le_of_lt_succ aL),(by exact Nat.le_of_lt_succ a1),(by exact Nat.le_of_lt_succ u),(by exact Nat.le_of_lt_succ A),(by exact Nat.le_of_lt_succ v),(by exact Nat.le_of_lt_succ x),al,hI,(by exact Nat.le_of_lt_succ 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 Nat.le_of_eq (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 Nat.not_lt_zero _)]) 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
 unfold fL
 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)),Nat.pair (Nat.pair ((s.h+2)*(s.w+2)) (s.w+2)) (Nat.pair (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 Nat.lt_succ_of_le hg.2.2.2.2.2.2.2.2.2.1) (by exact Nat.lt_succ_of_le hg.2.2.2.2.2.2.2.2.2.2.1) (by exact Nat.lt_succ_of_le hg.2.2.2.2.2.2.2.2.2.2.2.1) (by exact Nat.lt_succ_of_le hg.2.2.2.2.2.2.2.2.2.2.2.2.1) (by exact Nat.lt_succ_of_le hg.2.2.2.2.2.2.2.2.2.2.2.2.2.1) (by exact Nat.lt_succ_of_le hg.2.2.2.2.2.2.2.2.2.2.2.2.2.2.1) (by exact Nat.lt_succ_of_le 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 Nat.lt_succ_of_le 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,Nat.div_one,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=>(Nat.mod_eq_sub_mod i).trans (γ (Nat.sub_lt_left_of_lt_add i j))
 have P:∀z,0<z->0<Kf z:=λz i=>C0 (ϟ _) (dv 0 z i)
 (Nat.lt_or_ge x N).elim (λhx=>(Nat.lt_or_ge 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 (Nat.not_lt.2 hy)));absurd (if e:y=N then lt_of_lt_of_eq h (if_pos e) else Nat.lt_trans (U1 (lt_of_lt_of_eq h (if_neg e)) (Nat.sub_le_sub_left hy _)) q) (Nat.not_lt.2 (Nat.le_add_left _ _)))
 λhx=>(Nat.lt_or_ge y N).elim
  (λhy=>have h:=lt_of_eq_of_lt (if_neg (Nat.not_lt.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 (Int.natCast_nonneg _)) (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 (Nat.lt_of_le_of_ne hx (Ne.symm e))) (M x hx d).symm)))) (Int.natCast_nonneg _)) (if_neg (Nat.not_le.2 hy)).symm))
  λhy=>have h:=lt_of_eq_of_lt (if_neg (Nat.not_lt.2 hx)).symm (lt_of_lt_of_eq h (if_neg (Nat.not_lt.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 (Nat.lt_of_le_of_ne hx λf=>Nat.lt_irrefl _ (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 (Nat.lt_trans (U1 (lt_of_eq_of_lt (if_pos f).symm h) (Nat.sub_le_sub_left hy _)) q) (Nat.lt_irrefl _) else Nat.sub_lt_sub_right hy (Nat.lt_of_not_le λl=>Nat.not_lt.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 (Nat.le_of_ble_eq_true 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)∧(Int.toNat ((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 (Nat.sub_le _ _),λh=>by rw[<-Nat.mod_add_div u 256,Nat.sub_add_comm x.le];exact (Nat.le_mul_of_pos_right 256 (Nat.div_pos h ((by decide):0<256))).trans (Nat.le_add_left _ _)⟩
 ·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[Nat.sub_add_comm x.le,Nat.add_mul_mod_self_left,γ ((Nat.sub_le _ _).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 (Nat.lt_of_le_of_ne (Nat.le_of_lt_succ 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 (Nat.ne_of_lt ((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[UInt32.le_iff_toNat_le,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:=UInt32.le_iff_toNat_le.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 Nat.le_of_succ_le 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 Nat.lt_of_lt_of_eq 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 Nat.lt_of_succ_le e)).1;have u:=q B p λk e=>(l k (by exact ϟ k) (by simp[d];exact Nat.lt_of_succ_le 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=>(Nat.lt_succ_iff_lt_or_eq.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*_),Nat.cast_mul,Nat.cast_pow,Nat.cast_pow,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,Nat.cast_mul,Nat.cast_pow,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 Nat.le_of_succ_le 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 Nat.min_le_left 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 Nat.le_of_succ_le 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 ((Nat.sub_le _ _).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,(Nat.le_add_left 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,Finset.sum_range_zero,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[←Nat.sub_sub,q,BK]⟩
theorem gI_h2(a b:Ἀ)(q:a-b=0):¬b<a:=Nat.not_lt.2 (Nat.sub_eq_zero_iff_le.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 Nat.zero_lt_of_lt 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:=Nat.sub_eq_zero_iff_le.1 hm
  rw[pQ,dif_neg (by rw[ε,hU A (hA.trans_lt hN)];exact Nat.not_lt.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:=Nat.lt_of_not_le 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 (Nat.lt_of_le_of_ne (Nat.le_of_lt_succ g) (Ne.symm k))
  have o1:(J0 (ox+1))=x+1:=by rw[uA (by rw[ho,mA];exact C0 ((Nat.le_add_left x _).trans_lt (hq.symm.trans_lt hqc)) hN),ho,mA]
  have b2:b<=(J0 q):=hq▸(Nat.le_add_left b (a*H)).trans ((Nat.le_mul_of_pos_right _ 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,Nat.sub_add_eq,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:=Nat.le_antisymm hx (Nat.not_lt.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:=Nat.le_antisymm hx (Nat.not_lt.1 g1)
   have yb:b+1=H:=Nat.le_antisymm hb (Nat.not_lt.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 (Nat.zero_lt_of_lt hb) W0 Ϧ Ϧ (by rw[uA (by rw[he,hsk];exact (hS a).trans_lt (hE (a+1) 0 0 hC (Nat.zero_lt_of_lt 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 (Nat.div_le_self h 2))).trans hl
 exact⟨aQe_h1 _ _ B ((Nat.div_le_self w 2).trans wl),aQe_h1 _ _ B ((Nat.div_le_self h 2).trans ((Nat.le_mul_of_pos_right _ (W.trans_le (Nat.div_le_self 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 (Nat.mul_le_mul_left _ (Nat.div_le_self _ _)) (Nat.div_le_self _ _)).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,Array.size_replicate,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[Array.size_replicate]) (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 Nat.not_lt_zero 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)=USize.ofNat 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 USize.toNat_inj.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 (Nat.le_of_succ_le 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)=USize.ofNat 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 (Nat.not_lt.2 (Nat.le_of_eq h.symm))).trans (if_pos h),(if_neg (Nat.not_lt.2 (Nat.le_of_eq h.symm))).trans (congrArg (2^(32:Ἀ)-1-·) h)⟩)
 exact BA (BA ⟨h,(if_neg (Nat.not_lt.2 h.le)).trans (if_neg h.ne'),if_neg (Nat.not_lt.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 (Nat.not_lt.2 (Nat.le_add_left _ _))
 exact absurd (h.trans (fY_h0 _ b1)) (Nat.not_lt.2 ((Nat.le_succ _).trans (Nat.le_add_left _ _)))
 rw[a3,b3];exact U1 (Nat.lt_succ_self _) (Nat.le_add_left _ _)
 exact absurd h (Nat.lt_irrefl _)
 exact absurd (h.trans (fY_h0 _ b1)) (Nat.not_lt.2 (Nat.le_succ _))
 rw[a3,b3];exact (fY_h0 _ a1).trans (U1 (Nat.lt_succ_self _) (Nat.le_add_left _ _))
 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:=Nat.le_antisymm (Nat.not_lt.1 g) (Nat.not_lt.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 (Nat.add_right_cancel (a2.symm.trans (e.trans b2))) m
 exact pZ ⟨Nat.add_right_cancel ((a2.symm.trans (e.trans b2)).trans (Κ _).symm),b1⟩
 exact absurd (a2.symm.trans (e.trans b2)) (Nat.ne_of_gt ((fY_h0 _ b1).trans (U1 (Nat.lt_succ_self _) (Nat.le_add_left _ _))))
 exact BA ⟨a1,Nat.add_right_cancel ((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)) (Nat.ne_of_gt ((fY_h0 _ b1).trans (Nat.lt_succ_self _)))
 exact absurd (b2.symm.trans (e.symm.trans a2)) (Nat.ne_of_gt ((fY_h0 _ a1).trans (U1 (Nat.lt_succ_self _) (Nat.le_add_left _ _))))
 exact absurd (b2.symm.trans (e.symm.trans a2)) (Nat.ne_of_gt ((fY_h0 _ a1).trans (Nat.lt_succ_self _)))
 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:=Nat.lt_trans v c;have k3:=U1 a t
 ⟨C0 (Nat.le_trans (Nat.le_add_left w A) (Nat.le_succ _)) a,C0 (L1 (V1 A w) 1) k3,Nat.lt_trans (Nat.lt_succ_self _) 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[Nat.sub_add_cancel W];exact (Nat.add_sub_cancel' (Nat.mul_div_le w 2)).symm,(Nat.mod_def w 2).symm.trans_le (Nat.le_of_lt_succ (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 (Nat.div_le_self _ _)))
 have hl:h<=c*h*w:=(lZ h c0).trans (Nat.le_mul_of_pos_right _ (W.trans_le (Nat.div_le_self _ _)))
 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 (Nat.mul_le_mul_left _ (Nat.div_le_self _ _)) (Nat.div_le_self _ _)
 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):=(Nat.mul_lt_mul_of_pos_left 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 _ ((Nat.div_le_self _ _).trans_lt (qL.trans (by decide)))) (aa _ ((Nat.div_le_self _ _).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=>(Nat.not_lt_zero 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)=USize.ofNat 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 USize.toNat_inj.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)=USize.ofNat 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[Nat.mul_add_div ((by decide):2>0),ώ h,ϑ],by rw[Nat.mul_add_mod,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,(Nat.sub_add_cancel g).symm⟩;have h:=lF (2*q+Ya (lP r D)) 1 (k+1+1) one_ne_zero;rw[n4,Nat.div_one,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),Nat.mul_add_div (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[Nat.cast_succ];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,(Nat.sub_add_comm (Nat.le_trans (by decide) m):(Aq q)+1-23=((Aq q)-23)+1),<-VY q r D _ x t (by exact Nat.le_sub_of_add_le (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 (Nat.le_trans (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 Nat.ne_of_gt (U1 (Ϯ (by decide) (ϵ j)) (V1 _ _)),(by rw[Nat.cast_succ,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 (Nat.le_trans (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=((Int.toNat 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,(Int.toNat 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[(Int.ofNat_le.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]:Real.sqrt (e/d)*kZ kO (-s)=Real.sqrt 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,Int.cast_natCast,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:=Nat.not_lt.2 (Nat.sub_eq_zero_iff_le.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[Nat.sub_add_eq,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[Nat.sub_add_eq,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 (Nat.sub_le _ _) (by decide):63-k<64),kH,lY]
 rw[(by rw[<-AF];congr 1;rw[Τ];exact (Nat.sub_add_comm (Nat.lt_succ.1 h)).symm:2^(1:Ἀ)*2^(63-k)=2^(64-k)),(by rw[<-AF,Nat.sub_add_cancel 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)) (Nat.sub_le_sub_right 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 (Nat.sub_le_sub_right 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[Nat.sub_add_cancel (Nat.le_of_not_lt 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 Nat.not_lt.2 (V1 _ _)),ϑ]
 ·have hc:=Nat.le_of_not_lt hc;have e:((lq x)+(lq y))%2^(64:Ἀ)=(lq x)+(lq y)-2^(64:Ἀ):=by rw[Nat.mod_eq_sub_mod hc,γ (Nat.sub_lt_left_of_lt_add hc (Nat.add_lt_add hx hy))]
  rw[e,if_pos (by rw[μ,η,e];exact Nat.sub_lt_left_of_lt_add hc ((O1 hy _).trans_eq (Τ _ _))),Nat.sub_add_cancel 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 Nat.not_lt.2 ((V1 _ _).trans (V1 _ _))),ϑ]
 ·have T:=Nat.le_of_not_lt T;have wc:(lq w)+(lq c)<2^(64:Ἀ):=(J1 hc _).trans_lt ((Nat.add_lt_add_right 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[Nat.mod_eq_sub_mod T,γ (Nat.sub_lt_left_of_lt_add T ((pA _ _ _).trans_lt (Nat.add_lt_add this wc)))]
  rw[e,e2,if_pos (by rw[μ,e,e2];exact Nat.sub_lt_left_of_lt_add T ((pA _ _ _).trans_lt ((O1 wc _).trans_eq (Τ _ _)))),Nat.sub_add_cancel 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 (Nat.not_lt.2 (Nat.sub_eq_zero_iff_le.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 (Nat.mul_le_mul_left 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[Nat.sub_add_eq,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[Nat.sub_add_eq,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[Nat.cast_mul,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[Nat.cast_mul,Nat.cast_mul,Nat.cast_pow,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):(Int.toNat (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,(Int.toNat (T-e))=k+1:=⟨(Int.toNat (T-e))-1,(Nat.sub_add_cancel ((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:(Int.toNat (T-e))=k+1):(Int.toNat (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):(Int.toNat (297-s))<2^(64:Ἀ):=by
 have h:(Int.toNat (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):(((Int.toNat (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 ((Nat.add_lt_add_right (Nat.mul_lt_mul_of_pos_left 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 (Int.toNat (297-s))):=by
 set R:=ZK e d with g;have h:(lq (XF (Int.toNat (297-s))))=(Int.toNat (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) _ (Nat.ne_of_gt (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 (Int.toNat (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<<<(Int.toNat c)else B
def YF(Dn:Ἀ)(c:Ἁ):Ἀ:=if 0<=c then Dn else Dn<<<(Int.toNat (-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,Nat.cast_mul,Nat.cast_pow,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▸Nat.min_le_left (2*u) (l+149)),Nat.cast_mul,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,Nat.cast_add];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,Rat.cast_natCast,Rat.cast_natCast,Rat.cast_mul,Rat.cast_zpow,Rat.cast_ofNat]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[(Nat.sub_add_cancel l.le).symm,m7];exact ϐ e (϶ _)
  rw[if_pos h,ψ,Ϣ,Ϣ,ZG,ZG,ο,ο,γ (AN e (Θ (by decide) ((Nat.sub_le 63 k).trans (by decide))):x<2^(64:Ἀ)),γ (AN l (by decide):k<2^(64:Ἀ)),γ (Nat.lt_succ_of_lt l:k<64)]
  exact γ (Nat.lt_trans 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 (Nat.cast_nonneg _) g) d.le,(div_lt_one d).2 ((add_lt_add_right j _).trans_le ((Nat.cast_succ _).symm.trans_le (Nat.cast_le.2 (ϊ N e)))),(div_eq_zero_iff.trans (or_iff_left d.ne')).trans ((add_eq_zero_iff_of_nonneg (Nat.cast_nonneg _) g).trans (and_congr_left' Nat.cast_eq_zero))⟩
 rw[add_mul,div_mul_cancel₀ _ d.ne',<-add_assoc,<-Nat.cast_mul,<-Nat.cast_add,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:=Int.ofNat_le.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 (Int.natCast_nonneg _) 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:=Nat.not_lt.2 (Nat.sub_eq_zero_iff_le.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[Nat.sub_add_eq,h,BK]),Nat.cast_add];ring
 ·rw[ih (i+1) _ _ (by rw[Nat.sub_add_eq,h,BK]),Nat.cast_add];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[Nat.cast_mul,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[Nat.cast_mul,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 (Finset.sum_nonneg λj _=>sq_nonneg _) (Nat.cast_nonneg _)
 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,Int.cast_natCast]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 ((Nat.cast_le (α:=ℚ)).1 (by rw[YO,Nat.cast_mul,o];exact sub_nonneg.1 v)),YO,o,l,Nat.cast_mul,Nat.cast_mul]
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 (Nat.le_of_not_lt h)];exact pO (Nat.le_of_not_lt h)
lemma cP(w:kF):kG w<=254:=by rw[kG];split;exacts[ϟ _,Nat.sub_le_sub_right (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,Nat.ofNat_nonneg,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:=
 (UInt64.toUInt32 ((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):=Nat.add_le_add a hy
 have d:c+(x*2^(23:Ἀ)+y)<=2^(31:Ἀ)+(252*2^(23:Ἀ)+2^(24:Ἀ)):=Nat.add_le_add 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):=Nat.mul_lt_mul_of_pos_left l (by decide)
 rw[γ (Nat.lt_trans 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 (Nat.mul_lt_mul_of_pos_left 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) (Nat.sub_le_sub_right 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(Nat.sub_le _ _).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)),Nat.cast_mul,Nat.cast_pow,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,Nat.sub_sub]
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 (Int.ofNat_le.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 ((Nat.le_add_left 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),Nat.add_sub_cancel' (Nat.min_le_left _ _)⟩
theorem jv_h2(e:Ἀ)(l:e<=508):e-po e 254<=254:=Nat.sub_le_iff_le_add.2 ((Nat.le_min.2 ⟨Nat.le_add_left 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 (Nat.sub_le_sub_right 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,Nat.sub_add_cancel (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) ((Nat.add_le_add (L1 u 1) (Nat.sub_le_sub_right 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 (Nat.add_le_add x w) (by decide)),(by rw[<-Ϛ,<-Ϡ,Nat.mod_add_div,<-AF];exact congrArg (2^·) ((congrArg (52+·) (Nat.add_sub_cancel' 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 (Nat.le_sub_of_add_le ((J1 q.le _).trans (by decide))),pO (q.trans (by decide)).le,Nat.cast_pow,nZ,sub_sub_cancel_left]
lemma ke_h2(j:ℕ):((2^j:ℕ):ℝ)=2^j:=by rw[Nat.cast_pow,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,Nat.sub_add_cancel (Nat.sub_le 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 Nat.mul_le_mul_left _ al.le
 have bs:=cP d
 have bA:kG d+(lq x)+52<=342:=L1 (Nat.add_le_add 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) _) (Nat.cast_nonneg (α:=ℝ) _),?_,?_,(bA.trans (by decide)),?_,?_,?_,?_,?_⟩
 have hz:(lq (I0 e))=e:=Α (q.trans (by decide))
 have W:(Int64.toInt (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:(Int.natAbs (if X then(e:ℤ)else-(e:ℤ)))<2^(53:Ἀ):=by rw[Ax Int.natAbs,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,Int.cast_natCast]
 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,<-Nat.cast_mul,Nat.cast_le];exact pr _ _)
 have a2:(S:ℝ)<e*2^j+2^j:=(by rw[<-m,β,<-ke_h2,<-Nat.cast_mul,<-Nat.cast_add,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(Int.natAbs (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 (Nat.cast_le.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),<-Nat.cast_add,<-Nat.cast_mul,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,<-Nat.cast_mul,Nat.cast_le];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 (Nat.cast_nonneg _) (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 (Nat.ne_of_gt (AN (by decide) ba))
 rw[Y1,V,Rat.cast_mul,Rat.cast_zpow,Rat.cast_intCast,Rat.cast_ofNat,Int.cast_natCast,Nat.cast_mul,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:(Int64.toInt (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,Int.cast_natCast]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):=Nat.le_of_not_lt 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 ((Nat.sub_le _ _).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:(Int.natAbs (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 _ (Int.cast_natCast 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 (Nat.add_le_add 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 ((Nat.add_le_add (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 (Nat.add_lt_add_right eb 10) (by decide)),(γ (AO (Nat.add_le_add ib (L1 Jb 1056)) (by decide))).trans (pA _ _ _).symm,γ (AO (Nat.add_le_add 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⟨(Nat.add_lt_add_right e1 _).trans_eq (by decide),Nat.le_add_left _ _⟩
 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,Nat.sub_add_cancel (le_max_of_le_left (le_max_of_le_left (V1 _ _))),hA];exact le_max_of_le_right (Nat.le_of_eq ((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▸Nat.cast_nonneg _) (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▸Nat.le_sub_of_add_le (le_max_of_le_left (le_max_of_le_left (Τ _ _).le))) (by rw[z,Nat.sub_add_cancel (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,Rat.cast_mul,Rat.cast_zpow,Rat.cast_intCast,Rat.cast_ofNat,Ac t2,AA,zpow_ofNat,mul_div_assoc]
  rw[Rat.cast_add,Rat.cast_div,Rat.cast_mul,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 ((Nat.add_lt_add (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[Rat.cast_mul,Rat.cast_zpow,Rat.cast_natCast,Rat.cast_ofNat];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[Rat.cast_mul,Rat.cast_zpow,Rat.cast_add,Rat.cast_natCast,Rat.cast_one,Rat.cast_ofNat];exact BF hi hP.le)
 have a0:(0:ℚ)<2^e:=((by positivity))
 have v:0<(if s then-Y else Y):=AO (Af (Nat.cast_nonneg _) 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_=>(Bool.eq_false_iff.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 (Nat.le_add_left _ _) M),Nat.cast_add,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 ((Int.ofNat_le.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 (Int.ofNat_le.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[Nat.div_one];exact b7) (by rw[Nat.div_one,d2];exact k1) (by rw[Nat.div_one,d2];exact k2) (by rw[η,ύ,e3];simp only[kA,V2,Bool.or_true,ἁ,Nat.div_one];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 Nat.sub_succ_lt_self 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:=Nat.not_lt.mpr (Nat.sub_eq_zero_iff_le.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,Nat.sub_add_eq,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 ⟨Nat.le_of_succ_le c.1,c.2⟩]
    · rw[if_neg c,if_neg (fun x=>c ⟨Nat.lt_of_le_of_ne 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:=Nat.le_of_sub_eq_zero h;rw[Yr,if_neg (Nat.not_lt.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[Nat.sub_add_eq,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 [Array.size_replicate])
 apply mi (by rw[a];simp only [Array.size_replicate,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 Nat.sub_succ_lt_self 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 (Nat.not_lt.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,Nat.sub_add_eq,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[Nat.mod_eq_sub_mod h,γ (Nat.sub_lt_left_of_lt_add h (Nat.add_lt_add hx hy)),if_pos (Nat.sub_lt_left_of_lt_add h (Τ x N▸O1 hy x)),pA,Nat.add_sub_cancel' h,pA]
 · rw[γ (Nat.lt_of_not_le 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 Nat.sub_succ_lt_self 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 Nat.not_lt.2 (Nat.sub_eq_zero_iff_le.1 hk))];simp only[show¬(J0 i)<(J0 e) from Nat.not_lt.2 (Nat.sub_eq_zero_iff_le.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 ((Nat.sub_le _ _).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▸(Nat.sub_le _ _).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,Nat.sub_add_eq,hk,BK])).1,(j _ _ _ _ _ _ _ _ _ (by rw[m,Nat.sub_add_eq,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 (Nat.le_of_ble_eq_true 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 (Nat.add_lt_add (ϊ _ (϶ 27)) (Nat.lt_trans a4 (ϊ _ (϶ 27)))) (Nat.le_of_ble_eq_true 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⟩:=Bool.and_eq_true_iff.1 h
 obtain⟨h,a4⟩:=Bool.and_eq_true_iff.1 h
 obtain⟨h,a3⟩:=Bool.and_eq_true_iff.1 h
 obtain⟨a1,a2⟩:=Bool.and_eq_true_iff.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 Nat.le_add_left _ _,by rw[hq,bw];exact Ϟ (Nat.add_lt_add_right (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 Nat.le_sub_of_add_le ((by decide:146+900<=1047).trans a1)
 have c1:(lq (eF v.toBits-900))<=398:=by rw[h9];exact Nat.sub_le_of_le_add (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[Rat.cast_zpow,Rat.cast_ofNat]
 have a6:(q:ℚ)*2^e<(if s then-y else y):=Rat.cast_lt.1 (((Rat.cast_mul _ _).trans (congrArg₂ (·*·) (Rat.cast_natCast 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 ((Rat.cast_mul _ _).trans (congrArg₂ (·*·) ((Rat.cast_add _ _).trans (congrArg₂ (·+·) (Rat.cast_natCast 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 (Nat.cast_nonneg _) 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)):=Int.ofNat_le.2 a1
 have n2:((lq (eF v.toBits)):ℤ)<=((1298:ℕ):ℤ):=Int.ofNat_le.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[Nat.div_one];exact b7) (by rw[Nat.div_one,d2];exact k1) (by rw[Nat.div_one,d2];exact k2) (by rw[η,ύ,hq];simp only[kA,V2,Bool.or_true,ἁ,Nat.div_one,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) (Nat.sub_add_cancel 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=>Nat.ne_of_gt hd.1 (congrArg mJ q)
 have hs:(sv d:ℝ)=(kM d:ℝ):=by rw[sv,hd.2.2,if_neg lN,Int.cast_natCast]
 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):=(Nat.add_lt_add_right 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 (Nat.add_lt_add hp hq) (Nat.le_of_ble_eq_true 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 (Nat.add_lt_add hb hKe)),e,Nat.max_eq_right (Nat.le_of_lt (e▸h))]
 · rw[if_neg h,nS_toNat_h2 (nS_toNat_h1 ha (Nat.lt_add_left _ hK)),Nat.max_eq_left (Nat.le_of_not_lt (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 (Int.ofNat_le.mpr hL.2) (Int.ofNat_le.mpr hDl)).trans (by decide))) (sub_le_iff_le_add.mpr ((Int.ofNat_le.mpr (Nat.le_of_lt_succ hE)).trans (by decide)))
 have b:(A+1082-Dd).toNat<=2056:=Int.toNat_le.mpr ((Int.sub_le_self _ (Int.natCast_nonneg Dd)).trans ((Int.add_le_add_right a 1082).trans (by decide)))
 have c:1054-Dd<=1054:=Nat.sub_le _ _
 exact⟨b.trans_lt (by decide),b.trans_lt (by decide),c.trans_lt (by decide),c.trans_lt (by decide),(Nat.add_lt_add_of_lt_of_le hG (max_le b ((Nat.add_le_add (Nat.le_of_lt 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 ((Nat.le_of_lt 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 Int.ofNat_le.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 (Int.ofNat_le.mp h')),hc,pO ((V1 T 2).trans (Nat.le_of_lt (Nat.lt_of_not_le h)))]⟩
 obtain⟨w,q⟩:=wq
 have k1':(G:ℤ)+K<=Sn:=Int.ofNat_le.mpr k1
 have k2':(G:ℤ)+(X+Q)<=Sn:=Int.ofNat_le.mpr k2
 have k4':(Dd:ℤ)+Q=1054:=congrArg Int.ofNat k4
 have w':(Sn:ℤ)<=T+25:=Int.ofNat_le.mpr w
 have a2':(T:ℤ)<=1298:=Int.ofNat_le.mpr a2
 have hG':(G:ℤ)<=255:=Int.ofNat_le.mpr (Nat.le_of_lt_succ hG)
 have hD':(Dd:ℤ)<=255:=Int.ofNat_le.mpr (Nat.le_of_lt_succ hD)
 have Aμ:A<=μ:=(Int.le_sub_right_of_add_le ((Int.le_add_of_sub_right_le k3).trans (Int.add_le_add_right (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 _ (Int.natCast_nonneg G)).trans (w'.trans (Int.add_le_add_right 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 (Int.add_le_add_right 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:=Int.add_le_add_right ((Int.sub_le_self _ (Int.natCast_nonneg _)).trans hG') 1
 exact⟨(L1 (Nat.le_of_lt hX) 23).trans (by decide),(by decide:1<=22).trans (Nat.le_add_left 22 Dd),(L1 (Nat.le_of_lt_succ hG) 23).trans (by decide),(L1 (Nat.le_of_lt hB) 23).trans (by decide),q,Int.ofNat_le.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[UInt64.toNat_ofNat'];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[UInt64.toNat_ofNat',<-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,Rat.cast_mul,Rat.cast_zpow,Rat.cast_intCast,Rat.cast_ofNat,<-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,Rat.cast_mul,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 (Bool.eq_false_iff.2 z)
 rw[<-Model.sign_eq_signBit_of_toDyadic?_some hd]
 refine Bool.eq_false_iff.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 (Float.sqrt v)/\Dense.F.rv (Float.sqrt v)=ns kC (Real.sqrt (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) (Int.natCast_nonneg _)) (Dense.F.bL 48 (2*kG w) 556 (J1 (Nat.mul_le_mul_left 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) (Nat.cast_nonneg 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∈Finset.Ico (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∈Finset.Ico (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∈Finset.Ico (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):=Nat.le_antisymm hie (Nat.sub_eq_zero_iff_le.1 hm)
  rw[nQs,dif_neg (by rw[ε,q];exact Nat.lt_irrefl _),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∈Finset.Ico (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∈Finset.Ico ((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∈Finset.Ico ((J0 i)+1) (J0 e),((sv x[j]!:ℝ)*2^kG x[j]!)^(2:Ἀ):=Finset.sum_nonneg (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,<-Nat.sub_sub,hm,BK]) (by rw[i1];exact hl) af (add_nonneg hA tn) (by rw[Nat.cast_succ];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):|Real.sqrt Wr-Real.sqrt Z|<=e/Real.sqrt Z:=by
 have sZ:0<Real.sqrt Z:=Real.sqrt_pos.2 hZ
 have sW:0<=Real.sqrt Wr:=Real.sqrt_nonneg _
 have k:(Real.sqrt Wr-Real.sqrt Z)*(Real.sqrt Wr+Real.sqrt Z)=Wr-Z:=by linear_combination Real.sq_sqrt hW-Real.sq_sqrt hZ.le
 have q:|Real.sqrt Wr-Real.sqrt Z| *(Real.sqrt Wr+Real.sqrt 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-Real.sqrt Wr|<=u*Real.sqrt Wr+w):
  |Dr-Real.sqrt (Ts*2^(52:Ἀ)/N+Ep)|<=(c+7)*u*Real.sqrt (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<=Real.sqrt Z:=by rw[show (1:ℝ)=Real.sqrt 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<Real.sqrt Z:=zero_lt_one.trans_le Y1
 have sq:Real.sqrt Z*Real.sqrt Z=Z:=Real.mul_self_sqrt Z0
 have g3:w<=w*Real.sqrt Z:=le_mul_of_one_le_right hw0 Y1
 have e2:((c*u+4*u)*Z+2^(58:Ἀ)*w)/Real.sqrt Z<=(c*u+4*u)*Real.sqrt Z+2^(58:Ἀ)*w:=by
  rw[div_le_iff₀ sZ];linear_combination 288230376151711744*g3-(c*u+4*u)*sq
 have e3:|Real.sqrt Wr-Real.sqrt Z|<=(c*u+4*u)*Real.sqrt Z+2^(58:Ἀ)*w:=e1.trans e2
 have g5:(c*u+8*u)*Real.sqrt Z<=Real.sqrt Z:=(AE cu1 sZ.le).trans_eq (one_mul _)
 have sW:Real.sqrt Wr<=2*Real.sqrt Z+2^(58:Ἀ)*w:=by linear_combination le_abs_self (Real.sqrt Wr-Real.sqrt Z)+e3+g5+4*Af hu0.le sZ.le
 have f5:u*Real.sqrt Wr<=u*(2*Real.sqrt Z+2^(58:Ἀ)*w):=mul_le_mul_of_nonneg_left sW hu0.le
 have g6:u<=u*Real.sqrt Z:=le_mul_of_one_le_right hu0.le Y1
 have g7:2^(60:Ἀ)*w<=u*Real.sqrt Z:=hwu.trans g6
 linear_combination (abs_sub_le Dr (Real.sqrt Wr) (Real.sqrt 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-Real.sqrt Wr|<=u*Real.sqrt Wr+w):
  |Dr-Real.sqrt (Ts*2^(52:Ἀ)/N+Ep)|<=(4*N+7)*u*Real.sqrt (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)<Real.sqrt (X:ℝ)*2^s)(w2:Real.sqrt (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,Nat.cast_mul,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,Nat.cast_mul,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<Real.sqrt (X:ℝ):=Real.sqrt_pos.2 (Rat.cast_pos.2 hX)
 have p2:(0:ℝ)<2^s:=zpow_pos two_pos s
 have sq:(Real.sqrt (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:=Nat.cast_nonneg 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 (Nat.le_of_ble_eq_true 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:(Int64.toInt (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,Int.cast_natCast]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 (Nat.le_of_ble_eq_true 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 (Nat.le_of_ble_eq_true rfl)) ((by decide:(-1074:ℤ)<=0).trans (Int.natCast_nonneg _)) (Dense.F.bL 24 (kG eps+201) 479 (J1 (L1 (cP eps) 201) 24) (Nat.le_of_ble_eq_true 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 (Nat.cast_nonneg 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 (Nat.cast_nonneg 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 (Float.sqrt (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 (Float.sqrt (nQs x (G0 (r*n)) (G0 (r*n+n)) (I0 (0:Ἀ)).toInt64.toFloat*(I0 (4503599627370496/n)).toInt64.toFloat+Dense.F.dw eps*nS201))-
   Real.sqrt ((∑j∈Finset.Ico (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))*Real.sqrt ((∑j∈Finset.Ico (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∈Finset.Ico (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:=Finset.sum_nonneg (fun _ _=>sq_nonneg _)
 have nb:(n:ℝ)<=2^(12:Ἀ):=(Nat.cast_le.2 hn2).trans_eq (by rw[Nat.cast_pow,nZ])
 have Tsb:Ts<=2^(568:Ἀ):=by
  have h0:=Finset.sum_le_card_nsmul (Finset.Ico (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) ((Nat.div_le_self _ _).trans_lt (by decide))
 set q:ℝ:=((4503599627370496/n:ℕ):ℝ) with hq
 have q0:0<=q:=Nat.cast_nonneg _
 have h52:((4503599627370496:ℕ):ℝ)=2^(52:Ἀ):=by rw[show (4503599627370496:ℕ)=2^(52:Ἀ) from rfl,Nat.cast_pow,nZ]
 have hq1:(n:ℝ)*q<=2^(52:Ἀ):=by
  have h:=Nat.cast_le (α:=ℝ)|>.2 (pr 4503599627370496 n)
  rw[Nat.cast_mul]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[Nat.cast_add,Nat.cast_mul,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 (Float.sqrt W)-Real.sqrt (Dense.F.rv W)|<=(1/2^(53:Ἀ))*Real.sqrt (Dense.F.rv W)+1/2^(1075:Ἀ):=by
  rw[dr];have:=NF.jy (Real.sqrt (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 (Nat.cast_nonneg _) (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∈Finset.Ico (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,Rat.cast_mul,Rat.cast_zpow,Rat.cast_intCast,Rat.cast_ofNat,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,Rat.cast_natCast,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 (Nat.le_of_ble_eq_true 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 (Nat.add_lt_add (ϊ _ (϶ 27)) (Nat.lt_trans a4 (ϊ _ (϶ 27)))) (Nat.le_of_ble_eq_true 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⟩:=Bool.and_eq_true_iff.1 h
 obtain⟨h,a4⟩:=Bool.and_eq_true_iff.1 h
 obtain⟨h,a3⟩:=Bool.and_eq_true_iff.1 h
 obtain⟨a1,a2⟩:=Bool.and_eq_true_iff.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:=Nat.cast_nonneg 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:=Nat.mul_lt_mul_of_pos_left (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 (Nat.cast_succ _),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 (Int.toNat (297-(1223-((lq σ):ℤ)))):=by
 have k:lq (900+26:ἆ)=926:=rfl
 have l0:926<=lq σ:=(Nat.le_of_ble_eq_true 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):=(Nat.cast_le.2 (h2.trans (Nat.le_of_ble_eq_true rfl:1324<=1432))).trans_eq nZ
 have i2:(1223-297:ℤ)<=((lq σ):ℤ):=nZ.symm.trans_le (Nat.cast_le.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⟨(Nat.le_sub_of_add_le (Nat.le_of_succ_le h1:146+926<=_)).trans_eq e.symm,e.trans_le (Nat.sub_le_of_le_add (h2:_<=398+926)),le_sub_comm.2 i1,sub_le_comm.2 i2,ξ.1 ?_⟩
 rw[e,UInt64.toNat_ofNat',q,Int.toNat_sub]
 exact (γ (C0 (Nat.sub_le _ _) (C0 h2 (Nat.le_of_ble_eq_true 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)*Real.sqrt (X:ℝ)|<=k*(2^(175:Ἀ)*Real.sqrt (X:ℝ))):nWs 26 D.toBits=mK X:=by
 set Y:=2^(175:Ἀ)*Real.sqrt (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 (Nat.le_of_ble_eq_true 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 (Nat.ne_of_gt (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:(Real.sqrt (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)<Real.sqrt (X:ℝ)*2^s:=by
  apply BF _ pU.le;rw[m,<-Ap];exact w1
 have w2':Real.sqrt (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 (Float.sqrt (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 (Float.sqrt (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:=Float.sqrt (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∈Finset.Ico (r*n) (r*n+n),((sv x[j]!:ℝ)*2^kG x[j]!)^(2:Ἀ))*2^(52:Ἀ)/n:=div_nonneg (Af (Finset.sum_nonneg (fun _ _=>sq_nonneg _)) (by norm_num)) (Nat.cast_nonneg 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:Real.sqrt (2^(350:Ἀ)*(X:ℝ))=2^(175:Ἀ)*Real.sqrt (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*Float.ofBits ((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*Float.ofBits ((971-lb)<<<52))
  let v:=nS2 x m lo hi z0
  let D:=Float.sqrt (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 (Nat.sub_eq_zero_iff_le.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,Nat.sub_add_eq,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 (Nat.sub_eq_zero_iff_le.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,Nat.sub_add_eq,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∈Finset.Ico (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):=Nat.le_antisymm hie (Nat.sub_eq_zero_iff_le.1 hm)
  rw[nS1,dif_neg (by rw[ε,q];exact Nat.lt_irrefl _),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:=Nat.cast_nonneg 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[Nat.cast_succ,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,Nat.sub_add_eq,hm,BK]) (i1.le.trans (l7 hl)) af hA2 (by rw[Nat.cast_succ];exact step) (by rw[pA,Τ 1 m];exact hk) (fun j h1 h2=>hb j (Nat.le_of_succ_le (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:=Nat.ne_of_gt h1
 have l3:Aq n<=11:=Nat.le_of_lt_succ ((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,UInt64.toNat_ofNat']
  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 (Nat.lt_succ_of_le c),na,γ (lu Nat.one_lt_two (Nat.lt_succ_of_le c))]
lemma nMeq(n:Ἀ)(x:(Ἂ Word))(r:Ἀ)(hn:0<n):((no n x r:ℚ):ℝ)*2^(149:Ἀ)=(∑j∈Finset.Ico (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,Rat.cast_mul,Rat.cast_zpow,Rat.cast_intCast,Rat.cast_ofNat,Ap,<-zpow_ofNat,<-AK AL,sub_add_cancel,AA]
 unfold rowMean
 rw[Yk,Ai,Rat.cast_div,Rat.cast_sum,Rat.cast_natCast,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 (Float.ofBits (f<<<52))/\rv (Float.ofBits (f<<<52))=(2:ℝ)^(((lq f):ℤ)-1023):=by
 obtain⟨f1,f2,f3⟩:=nPwF f h1 h2
 have ne:=Nat.ne_of_lt (Nat.lt_succ_of_le h2)
 have K:kI (Float.ofBits (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:=Nat.lt_succ_of_lt (Nat.lt_succ_of_le 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 (Nat.ne_of_gt 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[Int.cast_natCast,zpow_neg,AA,zpow_ofNat]
 have p:(0:ℝ)<2^k:=pow_pos two_pos k
 rw[Ad (Af (Nat.cast_nonneg z) (inv_nonneg.2 p.le))]
 exact (mul_le_of_le_one_right (Nat.cast_nonneg 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*Float.ofBits ((971-(Aq n).toUInt64)<<<52))/\
  Dense.F.rv ((((1:ἆ)<<<(52+(Aq n).toUInt64))/I0 n).toInt64.toFloat*Float.ofBits ((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,UInt64.toNat_ofNat'];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,UInt64.toNat_ofNat'];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 Nat.le_sub_of_add_le ((J1 l3 1).trans (by decide))) (by rw[pt];exact (Nat.sub_le _ _).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)),Nat.cast_add];ring
 have r1:((2^(52+Aq n)/n:ℕ):ℝ)<=2^(52:Ἀ):=by have:=(Nat.cast_le (α:=ℝ)).2 R2;rwa[Nat.cast_pow,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 (Int.ofNat_le.2 ((J1 l3 52).trans (by decide)))) (nZb _ _ r1)
 refine⟨q.1,?_⟩;rw[q.2,zpow_neg,AA,Int.cast_natCast,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*Float.ofBits ((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*Float.ofBits ((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:=Nat.le_trans (Nat.le_add_left 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]) (Nat.le_trans (Nat.le_of_eq (Κ n)) (Nat.le_trans hn (Nat.le_of_ble_eq_true 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 (Nat.le_add_left 23 Dl)
 have B1:(2:ℝ)^(Dl+23)<=2^(300:Ἀ):=AU one_le_two (Nat.le_trans (L1 d2 23) (Nat.le_of_ble_eq_true 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∈Finset.Ico (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 (Finset.Ico (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▸(Finset.abs_sum_le_sum_abs _ _).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*Float.ofBits ((971-(Aq n).toUInt64)<<<52)):=by rw[rir];exact div_nonneg (Nat.cast_nonneg _) P0.le
 have nR1:=(Nat.cast_le (α:=ℝ)).2 (hR1d▸pr (2^(52+Aq n)) n)
 rw[Nat.cast_mul,Nat.cast_pow,nZ]at nR1
 have hR1:(n:ℝ)*Dense.F.rv ((((1:ἆ)<<<(52+(Aq n).toUInt64))/I0 n).toInt64.toFloat*Float.ofBits ((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[Nat.cast_mul,Nat.cast_pow,nZ,Nat.cast_add,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*Float.ofBits ((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*Float.ofBits ((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 [Nat.cast_add,Nat.cast_pow,nZ] using (Nat.cast_le (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,←Nat.cast_add,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∈Finset.Ico (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∈Finset.Ico (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 (Nat.ofNat_nonneg 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 (Nat.ofNat_nonneg 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):=Nat.le_antisymm hie (Nat.sub_eq_zero_iff_le.1 hM)
  rw[nS2,dif_neg (by rw[ε,q];exact Nat.lt_irrefl _),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:=Nat.cast_nonneg 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[Nat.cast_succ,add_one_mul];exact add_le_add hA yy
  have:=ih (i+1) (v+dF*dF) (A+y*y) (k+1) (by rw[i1,Nat.sub_add_eq,hM,BK]) (i1.le.trans (l7 hl)) rf (add_nonneg A0 y0) hA2 (by rw[Nat.cast_succ];exact step) (by rw[pA,Τ 1 M];exact hk) (fun j h1 h2=>hb j (Nat.le_of_succ_le (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∈Finset.Ico (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,Rat.cast_mul,Rat.cast_zpow,Rat.cast_intCast,Rat.cast_ofNat,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,Rat.cast_natCast,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-Real.sqrt Zm|<=cu*Real.sqrt Zm):|Dr-Real.sqrt Zt|<=(cu*(1+η)+η)*Real.sqrt Zt:=by
 have a2:Real.sqrt Zm<=(1+η)*Real.sqrt Zt:=by
  rw[<-(Real.sqrt_mul (sq_nonneg (1+η)) Zt).trans (congrArg (·*Real.sqrt 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 (Real.sqrt Zm) (Real.sqrt 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*Real.sqrt 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 (Int.natCast_nonneg _)))
 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,Int.cast_natCast]
 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) (Nat.sub_add_cancel (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 (Nat.div_le_self _ _) (by norm_num)
 rw[ml,a];simp only[Ϣ,UInt64.toNat_ofNat',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 (Nat.ofNat_nonneg 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 (Nat.le_add_left 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*Float.ofBits ((971-(Aq n).toUInt64)<<<52)))
  (D:Ἅ)(hD:D=Float.sqrt (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-Real.sqrt (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)))*Real.sqrt (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:=Nat.le_of_lt_succ ((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 (Int.ofNat_le.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 (Nat.cast_nonneg _) (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∈Finset.Ico (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∈Finset.Ico (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):=(Nat.cast_le.2 (show n<=2^(11:Ἀ) from hn)).trans_eq (by rw[Nat.cast_pow,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:(Finset.Ico (r*n) (r*n+n)).card=n:=by rw[Nat.card_Ico];exact ϱ _ _
 have SM:∑j∈Finset.Ico (r*n) (r*n+n),(sv x[j]!:ℝ)*2^kG x[j]! =((Finset.Ico (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:=Finset.sum_nonneg (fun _ _=>sq_nonneg _)
 have Tm0:0<=Tm:=Finset.sum_nonneg (fun _ _=>mul_self_nonneg _)
 have Tmb:Tm<=n*(Y*Y):=by
  have h:=Finset.sum_le_card_nsmul (Finset.Ico (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 (Nat.ofNat_nonneg _) 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) ((Nat.div_le_self _ _).trans_lt (Nat.pow_lt_pow_succ (by decide)))
 have q0:0<=((2^(52:Ἀ)/n:ℕ):ℝ):=Nat.cast_nonneg _
 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[Nat.cast_add,Nat.cast_mul,Nat.cast_pow,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 (Nat.cast_nonneg _)).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-Real.sqrt (Dense.F.rv W)|<=(1/2^(53:Ἀ))*Real.sqrt (Dense.F.rv W)+1/2^(700:Ἀ):=by
  rw[dr];have:=NF.jy (Real.sqrt (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 (Nat.ofNat_nonneg _) 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 (Nat.ofNat_nonneg _) nR.le) (Nat.ofNat_nonneg _)) (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-Real.sqrt (2^(350:Ἀ)*X)|<=k*Real.sqrt (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)*Real.sqrt X|<=k*(2^(175:Ἀ)*Real.sqrt 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:Real.sqrt (2^(350:Ἀ)*X)=2^(175:Ἀ)*Real.sqrt 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 (Nat.le_trans (L1 lL.le 5) (by decide)) ((Nat.sub_le _ _).trans (Nat.add_le_add (Nat.mul_le_mul_left 2 (lL:Aq (n+4)+1<=12)) (Nat.mul_le_mul_left 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):=Nat.cast_le.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 (Nat.ofNat_nonneg _) (Nat.cast_nonneg n)) (Nat.ofNat_nonneg _)) (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*(Nat.cast_nonneg 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*Float.ofBits ((971-(Aq n).toUInt64)<<<52)))
  (D:Ἅ)(hD:D=Float.sqrt (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 (Finset.sum_nonneg fun _ _=>sq_nonneg _) (pow_nonneg zero_le_two _)) (Nat.cast_nonneg _)) (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 ((Nat.add_lt_add_right f1 _).trans (by decide))
 generalize hM:(mJ f)+2^(23:Ἀ)=M at mE
 have M1:2^(23:Ἀ)<=M:=hM▸Nat.le_add_left _ _
 have M2:M<2^(24:Ἀ):=hM▸(Nat.add_lt_add_right 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 (Nat.le_trans (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 ((Nat.add_lt_add_of_le_of_lt (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 (Nat.le_trans (show 2^(75:Ἀ)<=2^(23:Ἀ)*2^(52:Ἀ) by decide) (ι (2^(52:Ἀ)) (Nat.le_add_left (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,UInt64.toNat_ofNat'];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*Float.ofBits ((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 Nat.le_sub_of_add_le ((J1 he 1).trans (by decide))) (by rw[pt];exact (Nat.sub_le 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)),Nat.cast_add];ring
 have r1:((R:ℕ):ℝ)<=2^(52:Ἀ):=by have:=Nat.cast_le (α:=ℝ)|>.2 R2;rwa[Nat.cast_pow,nZ]at this
 have q:=Dense.F.aE _ _ rf pf (R:ℤ) (-(((74+(lq ed)):ℕ):ℤ)) (by rw[rr,pr,ez,Int.cast_natCast]) (by rw[Int.natAbs_cast];exact R2.trans_lt (by decide)) (Int.neg_le_neg (Int.ofNat_le.2 ((J1 he 74).trans (by decide)):((74+(lq ed):ℕ):ℤ)<=((1074:ℕ):ℤ))) (nZb _ _ r1)
 refine⟨q.1,?_⟩;rw[q.2,Int.cast_natCast]
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:=Nat.cast_le (α:=ℝ)|>.2 a1n;rwa[Nat.cast_mul,Nat.cast_pow,nZ]at this
 have a2:(2:ℝ)^(75:Ἀ)<=Q*M+M:=by have:=Nat.cast_le (α:=ℝ)|>.2 a2n.le;rwa[Nat.cast_add,Nat.cast_mul,Nat.cast_pow,nZ]at this
 have a3:(M:ℝ)<=2^(24:Ἀ):=by have:=Nat.cast_le (α:=ℝ)|>.2 M2;rwa[Nat.cast_pow,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) (Nat.le_add_left _ _)) ((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 (Nat.ne_of_lt 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 (Nat.le_add_left 4 n)),(m9 (Nat.succ_ne_zero _)).2 (C0 (L1 hn 4) (Nat.le_of_ble_eq_true 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=>Nat.sub_ne_zero_of_lt (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:=Nat.le_antisymm ht (Nat.sub_eq_zero_iff_le.1 hk);have:=ε.mp h;rw[nCa (e▸hb),nCa hb,e] at this;exact Nat.lt_irrefl _ 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[Nat.sub_add_eq,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 Nat.sub_succ_lt_self _ _ (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 (Nat.le_add_left 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*Float.ofBits ((971-lb)<<<52))
 let v:=vf m
 let D:=Float.sqrt (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 (Nat.lt_succ_self _) ((Nat.le_succ _).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,UInt64.toNat_ofNat']
 rw[γ h,γ (C0 (Nat.div_le_self _ _) (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*Float.ofBits ((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:=Float.sqrt (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 (Nat.le_add_left _ _) 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 (Nat.lt_succ_self _) ((Nat.le_succ _).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 (Nat.le_add_left _ _) 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 Nat.lt_succ_of_le (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 nWG2(g b:(Ἂ Word)):(ἀ×ByteArray)×ἀ:=(Dense.F.wcGet3 nK2eq (fun a b=>nK2eq_ok a b) nWS2 32 (g,b) (fun k=>k.1)).1
theorem nWG2_v(g b:(Ἂ Word)):nWG2 g b=nWF2 (g,b):=Dense.F.s5wcGet3_val _
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*Float.ofBits ((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[Array.size_replicate];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 (Nat.lt_succ_of_le h.2.2.2) rows 0 (Dense.F.zPq (n+2)) (Ah (rows*n) 0) (Dense.F.zPs (n+2)).1 (by simp only[Array.size_replicate];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):=Nat.le_antisymm ht (Nat.sub_eq_zero_iff_le.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 (Nat.ne_of_gt hj)];exact hX j (by rw[q];exact hj)
  · rw[Dense.F.fuS _ _ _ _ _ (U1 (Nat.lt_succ_of_lt (Nat.lt_succ_self _)) 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 (Nat.lt_of_le_of_ne (Nat.le_of_lt_succ 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 _ _ _ (Int.ofNat_le.mpr hL.2) (Int.ofNat_le.mpr hD) (Int.ofNat_le.mpr hEm)
 generalize max ((L:ℤ)+D-30) ((Em:ℤ)-1073)=M at a c
 have e4:ed<=1054:=Nat.le_of_lt (Nat.lt_trans he (by decide))
 have i1:(((mJ Y)+(1054-ed):Ἀ):ℤ)<=M+1082-ed:=by
  rw[Nat.cast_add,pO e4];exact nKpA _ _ _ _ _ (Int.ofNat_le.mpr hY) (Int.ofNat_le.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 (Int.natCast_nonneg 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 (Nat.lt_trans k2 (by decide))
 rw[μ,η,ϩ,Dense.F.kOfS _ k2,Dense.F.kOfS _ (C0 (Nat.sub_le 1054 ed) (by decide)),γ k3]
 exact Nat.not_lt.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 (Nat.le_add_left 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 (Nat.le_add_left _ _) 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):=USize.toNat_inj.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 (Nat.not_lt_zero _))
 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 (Nat.le_add_left _ _) 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 (Bool.eq_false_iff.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 (Nat.le_add_left _ _) 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):=USize.toNat_inj.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 (Nat.le_add_left _ _) 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 (Bool.eq_false_iff.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 Nat.lt_irrefl r)];rfl
 |succ k ih=>
  intro r Xr out hr ho hk hos
  have kk:=nHk2 (n:=n) hk
  rw[Yr,if_pos (hk▸Nat.lt_add_of_pos_right (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 (Nat.le_add_left _ _) 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):=USize.toNat_inj.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 (Nat.not_lt_zero _))
 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 Nat.not_lt.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 Nat.not_lt.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,Rat.cast_mul,Rat.cast_neg,Rat.cast_pow,Rat.cast_natCast,Rat.cast_ofNat]<;>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 ((Nat.le_add_left 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 (Nat.lt_irrefl 1)]at Y
       refine Eq.trans (nRow1_eq _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (by simp only[Array.size_replicate];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[Array.size_replicate])).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,Nat.le_of_lt_succ 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,Nat.le_of_lt_succ 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 Nat.lt_of_lt_of_eq 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 (Bool.eq_false_iff.mpr hX)
       have N:¬((O0 (#[]:(Ἂ Word)))==l*n)=ἃ:=by simp only[Array.size_empty,AQ];exact Nat.ne_of_lt (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:ExactRuntime.ConvShape):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:ExactRuntime.ConvShape):ConvShape:=⟨s.cin,s.h,s.w,s.cout,s.kh,s.kw,s.strideH,s.padT⟩
theorem cvB(s:ExactRuntime.ConvShape)(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,ExactRuntime.ConvShape.cpg,hg,Nat.div_one]
 have e4:s.opg=s.cout:=by simp only[ExactRuntime.ConvShape.opg,hg,Nat.div_one]
 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,ExactRuntime.windowAt,windowAt,ϳ,Κ,hd1,hd2,ϻ,hst,hp1]
 · simp only[ht,ἂ]
@[inline]def dwOk(s:ExactRuntime.ConvShape):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:ExactRuntime.ConvShape)(h:dwOk s=true)(relu:Ἆ)(inp wts b:Ἂ Word):
  ExactRuntime.depthwiseSpec 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,ExactRuntime.ConvShape.depthwise,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,ExactRuntime.ConvShape.depthwise,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 ExactRuntime.depthwiseSpec 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,ExactRuntime.ConvShape.cpg,ExactRuntime.ConvShape.depthwise,Nat.div_self hc,n4]
 have e4:s.depthwise.opg=1:=by simp only[ExactRuntime.ConvShape.opg,ExactRuntime.ConvShape.depthwise,Nat.div_self hc]
 have e5:s.depthwise.cpg=1:=by simp only[ExactRuntime.ConvShape.cpg,ExactRuntime.ConvShape.depthwise,Nat.div_self hc]
 simp only[Λ,e3,e4,Nat.div_one]
 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,Nat.mul_add_div 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,Nat.mul_add_div hkw,Ϗ o,Nat.mul_add_mod]
   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,Nat.mul_add_mod]
   have a4:t/(s.kh*s.kw)=0:=ώ ht
   simp only[ht,ἁ,ExactRuntime.windowAt,windowAt,e5,a1,a2,a3,a4,ϑ,hd1,hd2,hst,hp1,ExactRuntime.ConvShape.depthwise,ExactRuntime.ConvShape.cpg,Nat.div_self hc,ϻ]
 · simp only[ht,ἂ]
@[inline]def apOk(s:ExactRuntime.PoolShape):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:ExactRuntime.PoolShape)(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:ExactRuntime.PoolShape)(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,Nat.sub_add_cancel hkh]
 ·simp only[ExactRuntime.PoolShape.wo,ExactRuntime.poolOut,hL,hR,hc,hsw,hk,υ,ἂ,ϑ]
  rw[←Nat.add_div_right _ hk0,Nat.sub_add_cancel hkw]
theorem apW(s:ExactRuntime.PoolShape)(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:ExactRuntime.PoolShape)(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:ExactRuntime.PoolShape)(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:ExactRuntime.ConvShape)(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:ExactRuntime.ConvShape)(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:ExactRuntime.ConvShape):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:ExactRuntime.ConvShape)(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,Finset.sum_range_zero,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:ExactRuntime.ConvShape)(inp:Ἂ Word)(hi:inp.all finiteWord=true)(g y x t:Nat):expOf (ExactRuntime.windowAt s inp g y x t)≠255:=by
 unfold ExactRuntime.windowAt;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:ExactRuntime.ConvShape)(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):
  ExactRuntime.windowAt 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 ExactRuntime.windowAt
 simp only[ExactRuntime.ConvShape.depthwise,ExactRuntime.ConvShape.cpg,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:ExactRuntime.ConvShape)(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)=ExactRuntime.depthwiseSpec 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,ExactRuntime.ConvShape.depthwise,hpt,hpb,hdh,hsh,hk];omega
 have e2:s.depthwise.wo=1:=by simp only[ExactRuntime.ConvShape.wo,ExactRuntime.ConvShape.ekw,ExactRuntime.ConvShape.depthwise,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.depthwiseSpec 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,Nat.div_one,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,Nat.mul_add_mod,γ (by omega)]
 have m2:(c*s.h+p+1)/s.h=c:=by rw[pA,Ϗ,Nat.mul_add_div (by omega),ώ (by omega),ϑ]
 have t3:s.depthwise.taps=3:=by simp only[ExactRuntime.ConvShape.taps,ExactRuntime.ConvShape.cpg,ExactRuntime.ConvShape.depthwise,hk,hkw,Nat.div_self hcin]
 have g1:s.depthwise.opg=1:=by simp only[ExactRuntime.ConvShape.opg,ExactRuntime.ConvShape.depthwise,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,Nat.div_one,hb0]
 congr 1
 have key:=dE 1 (fun t=>ExactRuntime.windowAt 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:ExactRuntime.ConvShape)(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):
  ExactRuntime.windowAt 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 ExactRuntime.windowAt
 simp only[ExactRuntime.ConvShape.cpg,hg,Nat.div_one,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:ExactRuntime.ConvShape)(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,Nat.div_one,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,Nat.mul_add_mod,γ (by omega)]
 have m2:(o*s.h+p+1)/s.h=o:=by rw[pA,Ϗ,Nat.mul_add_div (by omega),ώ (by omega),ϑ]
 have t3:s.taps=s.cin*3:=by simp only[ExactRuntime.ConvShape.taps,ExactRuntime.ConvShape.cpg,hg,Nat.div_one,hk,hkw,ϻ]
 have g1:o/s.opg=0:=by simp only[ExactRuntime.ConvShape.opg,hg,Nat.div_one];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=>ExactRuntime.windowAt 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:ExactRuntime.ConvShape)(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 Pinned.kernels.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[Pinned.kernels,Pinned.dotFn_eq_dotOf];rfl
def dwK(s:ExactRuntime.ConvShape)(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 Pinned.kernels.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 Pinned.kernels.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 Pinned.kernels.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: USize loops aJu (jG) and aDu (dl) over the array part, two word terms in the tail. -/
@[inline] def aGt (A:Array Word) (k:USize):Word:=if h:k < A.usize then A.uget k (Dense.cD h) else 0
theorem aGt_eq (A:Array Word) (k:USize) (hA:A.size < USize.size):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 [USize.lt_iff_toNat_lt]
      rw [Array.usize, Nat.toUSize_eq, USize.toNat_ofNat_of_lt' hA]; exact h2
    rw [getElem!_neg A _ h1]; rfl
theorem aUs {k:USize} (h:k.toNat+1 < USize.size):(k+1).toNat=k.toNat+1:=by
  rw [USize.toNat_add, USize.toNat_one]; exact Nat.mod_eq_of_lt h
theorem aUd {ka ea:USize} (h:ka < ea):ea.toNat-(ka+1).toNat < ea.toNat-ka.toNat:=by
  have h1:=USize.lt_iff_toNat_lt.mp h
  rw [aUs (Nat.lt_of_le_of_lt h1 (USize.toNat_lt_size ea))]
  omega
@[inline] def aJs (a b:Word) (Bm:UInt64):UInt64:=
  if Core.mW a*Core.mW b==0 then Bm else min Bm (Core.iW a+Core.iW b)
def aJu (A B:Array Word) (ka kb ea:USize) (Bm:UInt64):UInt64:=
  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:UInt64) (k:UInt64 → UInt64 → UInt64 → UInt64 → UInt64):UInt64:=
  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:UInt64):Dense.gd h l 0 s=(h,l):=by
  simp [Dense.gd]
@[inline] def aDs2 (a b:Word) (Bm pH pL nH nL:UInt64) (k:UInt64 → UInt64 → UInt64 → UInt64 → UInt64):UInt64:=
  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:UInt64) (k:UInt64 → UInt64 → UInt64 → UInt64 → UInt64):
    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, Bool.false_eq_true]
def aDu (A B:Array Word) (ka kb ea:USize) (a1 b1 a2 b2:Word) (Bm pH pL nH nL:UInt64):UInt64:=
  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:UInt64) (t:Nat) (ht:t < n) (pH pL nH nL:UInt64):
    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:UInt64) (t:Nat) (ht:¬ t < n) (pH pL nH nL:UInt64):
    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:UInt64):
    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:UInt64):Dense.jG fa fb n t Bm=Bm:=by
  rw [Dense.jG, if_neg ht]
theorem aFn (A:Array Word) (ao n:Nat) (a1 a2:Word):fA A ao n a1 a2 n=a1:=by simp [fA]
theorem aFm (A:Array Word) (ao n:Nat) (a1 a2:Word):fA A ao n a1 a2 (n+1)=a2:=by simp [fA]
theorem aFl (A:Array 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:Array Word) (ao bo n:Nat) (a1 b1 a2 b2:Word) (ea:USize)
    (hA:A.size < USize.size) (hB:B.size < USize.size) (he:ea.toNat=ao+n) (hb:bo+n < USize.size):
    ∀ k t (ka kb:USize) 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 [USize.lt_iff_toNat_lt, h1, he]; exact Nat.lt_irrefl _
    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 [USize.lt_iff_toNat_lt, 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:Array Word) (ao bo n:Nat) (a1 b1 a2 b2:Word) (ea:USize) (Bm:UInt64)
    (hA:A.size < USize.size) (hB:B.size < USize.size) (he:ea.toNat=ao+n) (hb:bo+n < USize.size):
    ∀ k t (ka kb:USize) 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 [USize.lt_iff_toNat_lt, h1, he]; exact Nat.lt_irrefl _
    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 [USize.lt_iff_toNat_lt, 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 ad2 (A B:Array 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 (USize.ofNat ao) (USize.ofNat bo) (USize.ofNat (ao+n)) a1 b1 a2 b2
        (aJs a2 b2 (aJs a1 b1 (aJu A B (USize.ofNat ao) (USize.ofNat bo) (USize.ofNat (ao+n)) 1023))) 0 0 0 0
    else 4294967296
  if r < 4294967296 then UInt64.toUInt32 r else dotOf (n+2) (fA A ao n a1 a2) (fA B bo n b1 b2)
theorem ad2_eq (A B:Array 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
  unfold ad2
  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:=USize.le_size
    have hA':A.size < USize.size:=by omega
    have hB':B.size < USize.size:=by omega
    have he:(USize.ofNat (ao+n)).toNat=ao+n:=USize.toNat_ofNat_of_lt' (by omega)
    have h1:(USize.ofNat ao).toNat=ao+0:=USize.toNat_ofNat_of_lt' (by omega)
    have h2:(USize.ofNat bo).toNat=bo+0:=USize.toNat_ofNat_of_lt' (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:UInt64) (g:N < 262144)
    (h:Dense.dl fa fb N Bm 0 0 0 0 0 < 4294967296):dotOf N fa fb=UInt64.toUInt32 (Dense.dl fa fb N Bm 0 0 0 0 0):=
  Dense.fE fa fb N Bm (by exact Nat.lt_of_succ_le 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:UInt64) (A B:Array 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 (USize.ofNat ao) (USize.ofNat bo) (USize.ofNat (ao+n)) a1 b1 a2 b2 (aJs a2 b2 (aJs a1 b1 g)) 0 0 0 0
    else 4294967296
  if r < 4294967296 then UInt64.toUInt32 r else ad2 A B ao bo n a1 b1 a2 b2
theorem ad4_eq (g:UInt64) (A B:Array 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:=USize.le_size
    have hA':A.size < USize.size:=by omega
    have hB':B.size < USize.size:=by omega
    have he:(USize.ofNat (ao+n)).toNat=ao+n:=USize.toNat_ofNat_of_lt' (by omega)
    have h1:(USize.ofNat ao).toNat=ao+0:=USize.toNat_ofNat_of_lt' (by omega)
    have h2:(USize.ofNat bo).toNat=bo+0:=USize.toNat_ofNat_of_lt' (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 ad2_eq A B ao bo n a1 b1 a2 b2
  · rw [if_neg h]; exact ad2_eq A B ao bo n a1 b1 a2 b2
def gMn (A:Array Word) (k e:USize) (m:UInt64):UInt64:=
  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:Array Word):UInt64:=
  gMn x 0 x.usize 511+gMn wih (USize.ofNat (2*H*inW)) (USize.ofNat (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 (Float32.ofBits a * Float32.ofBits 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 [Bool.and_eq_true, bne_iff_ne, ne_eq] 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 (Float32.ofBits a) (Float32.ofBits 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) × Array Word × Array Word × Array Word):Array 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) × Array Word × Array Word × Array Word):Bool :=
  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) × Array Word × Array Word × Array Word) (h:wcEq a b=true):a=b:=by
  obtain ⟨a1,a2,a3,a4⟩:=a; obtain ⟨b1,b2,b3,b4⟩:=b
  simp only [wcEq, Bool.and_eq_true, beq_iff_eq] 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 (Array (Dense.F.WcE wcF)):=(ST.Prim.mkRef #[] (Void.mk ())).val
@[noinline] def wcG (G inW H:Nat) (wih whh bih:Array Word):Array 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:Array 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 (ad4 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,Nat.mul_add_div hp,ώ hj1,ϑ]
        have e2:(i*(inW+H+1)+j)%(inW+H+1)=j:=by
          rw [Ϗ,Nat.mul_add_mod,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]
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 cand_eq (tanh:Word → Word) (inW H:Nat) (x wih bih:Ἂ Word) (t:Nat)
    (R HN:Ἂ Word) :
    (Array.ofFn (n:=H) fun u=>tanh <|
      ad 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 [had]; 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) (hadc:∀ A B ao bo n a1 b1 a2 b2,
    adc A B ao bo n a1 b1 a2 b2=dotOf (n+2) (fA A ao n a1 a2) (fA B bo n b1 b2))
    (sig tanh:Word → Word) (inW H:Nat) (x wih whh bih bhh:Ἂ Word)
    (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,cand_eq adc 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 (ad4 (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 (ad4 (gG inW H x wih)) (ad4_eq (gG inW H x wih)) sig tanh inW H x wih whh bih bhh 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:=Nat.le_of_lt_succ (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:=Nat.le_of_lt_succ (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=ExactRuntime.roundRational 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)=(ExactRuntime.rationalWord 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) ≤ |(ExactRuntime.sv w:ℝ)| := by
    unfold ExactRuntime.sv; split_ifs <;> simp <;> exact_mod_cast Nat.one_le_iff_ne_zero.mpr hm
  calc (1:ℝ)=1*1:=by ring
    _ ≤ |(ExactRuntime.sv 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))))
      (((ExactRuntime.rationalWord xw*ExactRuntime.rationalWord gw/ExactRuntime.rationalWord 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))=(((ExactRuntime.rationalWord xw*ExactRuntime.rationalWord gw/ExactRuntime.rationalWord td:ℚ):ℝ)*2^(149:ℕ)):=by
    have hD0:(ExactRuntime.rationalWord 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:ExactRuntime.roundRational 0=0:=by decide
theorem za0 (w:Word) (h0:w<<<1=0):ExactRuntime.rationalWord 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 ExactRuntime.sv; 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=ExactRuntime.roundRational
      (ExactRuntime.rationalWord (Norm.ck x i)*ExactRuntime.rationalWord (Norm.ck g j)/ExactRuntime.rationalWord td):=by
  rw [Norm.nFBr_e]; unfold Norm.nOEr; dsimp only
  have p:(ExactRuntime.rationalWord (Norm.ck x i)-0)*ExactRuntime.rationalWord (Norm.ck g j)/ExactRuntime.rationalWord td
      + ExactRuntime.rationalWord 0=ExactRuntime.rationalWord (Norm.ck x i)*ExactRuntime.rationalWord (Norm.ck g j)/ExactRuntime.rationalWord td:=by
    rw [Norm.Y1 0,(by decide:ExactRuntime.sv 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 ExactRuntime.roundRational p).symm)).trans (congrArg ExactRuntime.roundRational 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 =
      ExactRuntime.roundRational (ExactRuntime.rationalWord xw*ExactRuntime.rationalWord gw/ExactRuntime.rationalWord 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 (USize.ofNat 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:=(Float.sqrt (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 [Array.size_replicate,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+ExactRuntime.rationalWord eps))
    (hd:0 < (kB td).toNat∧kB td≠255∧k9 td=false)
    (hd2:(kB td).toNat ≤ 170)
    (he1:ExactRuntime.finiteWord eps=true) (hepos:0 < ExactRuntime.rationalWord 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 (USize.ofNat j)=a[j]! := fun a=>by
      rw [Norm.f6,USize.toNat_ofNat_of_lt' (usz j (by omega))]
    rw [elR x g _ td (USize.ofNat 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 < ExactRuntime.sv eps∧0 < ExactRuntime.rationalWord eps:=by
  have hm:0 < kM eps:=by unfold kM; rw [if_neg he0]; positivity
  have hs:0 < ExactRuntime.sv eps:=by unfold ExactRuntime.sv; 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 (Float.sqrt ((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+ExactRuntime.rationalWord 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 (Float.sqrt ((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 ExactRuntime.sv; 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,Nat.cast_add,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 _ (Nat.lt_trans hj2 hb:i < 2^(32:ℕ)),
    Conv.aa _ (U1 r (Nat.le_trans (Nat.le_add_left n _) (Nat.le_of_lt 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 cbG (b:Ἂ Word):Ἂ Word :=
  (Dense.F.wcGet3 Dense.F.wcAeq (fun a c h=>Dense.F.wcAeq_ok h) cbS 32 b (fun k=>k)).1
theorem cbG_v (b:Ἂ Word):cbG b=cbF b:=Dense.F.s5wcGet3_val _
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)]
@[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
  · rename_i hc
    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):=Nat.lt_trans hk.2 hb
      have hsub:G0 k-G0 (r*n)=G0 (k%n):=by
        apply USize.toNat_inj.1
        rw [USize.toNat_sub_of_le _ _ (by rw [USize.le_iff_toNat_le,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
  · 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
  · rename_i hc
    exfalso
    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)
  · 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 Nat.lt_irrefl 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 ▸ Nat.lt_add_of_pos_right (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
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*Float.ofBits ((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 [Array.size_replicate]; 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
        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 (Nat.lt_succ_of_le h.2.2.2) rows 0 (Dense.F.zPq (n+2)) (mB (rows*n) 0) (Dense.F.zPs (n+2)).1
          (by simp only [Array.size_replicate]; 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 (Nat.lt_irrefl 1)] at Y
        refine Eq.trans (Norm.nRow1_eq _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (by simp only [Array.size_replicate]; 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
        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 [Array.size_replicate])).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,Nat.le_of_lt_succ 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,Nat.le_of_lt_succ B⟩]; split
  · split
    · rename_i h1; subst h1; exact Norm.nRow1_bad _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ j (by omega) (by omega) 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:=Bool.eq_false_iff.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:=
 { Pinned.kernels with
   dense:=⟨Dense.c7,gC⟩
   conv:=⟨cvK,dwK,Pinned.kernels.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=ExactRuntime.depthwiseSpec:=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[Pinned.kernels,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[Pinned.kernels,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[Pinned.kernels,Pinned.maxpoolW_eq]
theorem correct:Obligation kernels:=by
 have h:=Pinned.correct
 simp only[Obligation,Pinned.kernels]at h
 simp only[Obligation,kernels,Pinned.kernels,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