import ContractExact
/-! -/
namespace FastAct
/-! -/
@[inline] def midW (a b:UInt64):UInt64:=
 (((a&&&(0xffffffff:UInt64))*(b&&&(0xffffffff:UInt64)))>>>(32:UInt64))+
   (((a&&&(0xffffffff:UInt64))*(b>>>(32:UInt64)))&&&(0xffffffff:UInt64))+
   (((a>>>(32:UInt64))*(b&&&(0xffffffff:UInt64)))&&&(0xffffffff:UInt64))
@[inline] def mulHiW (a b:UInt64):UInt64:=
 (a>>>(32:UInt64))*(b>>>(32:UInt64))+(((a&&&(0xffffffff:UInt64))*(b>>>(32:UInt64)))>>>(32:UInt64))+
   (((a>>>(32:UInt64))*(b&&&(0xffffffff:UInt64)))>>>(32:UInt64))+(midW a b>>>(32:UInt64))
@[inline] def mulLoW (a b:UInt64):UInt64:=
 (midW a b<<<(32:UInt64))+((a&&&(0xffffffff:UInt64))*(b&&&(0xffffffff:UInt64))&&&(0xffffffff:UInt64))
def mul128 (a b:UInt64):UInt64×UInt64:=(mulHiW a b,mulLoW a b)
lemma toNat_and_mask (x:UInt64):(x&&&0xffffffff).toNat=x.toNat%2^32:=by
 rw[UInt64.toNat_and]
 exact Nat.and_two_pow_sub_one_eq_mod x.toNat 32
lemma toNat_shr32 (x:UInt64):(x>>>32).toNat=x.toNat/2^32:=by
 rw[UInt64.toNat_shiftRight]
 simp only[UInt64.reduceToNat,Nat.reduceMod,Nat.shiftRight_eq_div_pow,Nat.reducePow]
lemma mul128_spec (a b:UInt64):
  (mul128 a b).1.toNat*2^64+(mul128 a b).2.toNat=a.toNat*b.toNat:=by
 have ha:=a.toNat_lt
 have hb:=b.toNat_lt
 have hk:a.toNat*b.toNat<2^128:=
  calc a.toNat*b.toNat<2^64*2^64:=Nat.mul_lt_mul'' ha hb
   _=2^128:=by norm_num
 simp only[mul128,mulHiW,mulLoW,midW,UInt64.toNat_add,UInt64.toNat_mul,
  UInt64.toNat_shiftLeft,toNat_and_mask,toNat_shr32]
 have h32:(32:UInt64).toNat%64=32:=by decide
 rw[h32,Nat.shiftLeft_eq]
 have hA:=(Nat.div_add_mod' a.toNat (2^32)).symm
 have hB:=(Nat.div_add_mod' b.toNat (2^32)).symm
 have hA0:a.toNat%2^32<2^32:=Nat.mod_lt _ (by norm_num)
 have hB0:b.toNat%2^32<2^32:=Nat.mod_lt _ (by norm_num)
 have hA1:a.toNat/2^32<2^32:=by omega
 have hB1:b.toNat/2^32<2^32:=by omega
 generalize a.toNat/2^32=A1 at*
 generalize a.toNat%2^32=A0 at*
 generalize b.toNat/2^32=B1 at*
 generalize b.toNat%2^32=B0 at*
 have prod_lt: ∀x y:Nat,x<2^32->y<2^32->x*y<2^64:= λx y hx hy=>
  calc x*y<2^32*2^32:=Nat.mul_lt_mul'' hx hy
   _=2^64:=by norm_num
 have m00:=prod_lt A0 B0 hA0 hB0
 have m01:=prod_lt A0 B1 hA0 hB1
 have m10:=prod_lt A1 B0 hA1 hB0
 have m11:=prod_lt A1 B1 hA1 hB1
 have hab:a.toNat*b.toNat=A1*B1*2^64+(A0*B1+A1*B0)*2^32+A0*B0:=by
  rw[hA,hB];ring
 simp only[Nat.mod_eq_of_lt m00,Nat.mod_eq_of_lt m01,Nat.mod_eq_of_lt m10,
  Nat.mod_eq_of_lt m11]
 rw[hab]
 generalize A0*B0=p00 at*
 generalize A0*B1=p01 at*
 generalize A1*B0=p10 at*
 generalize A1*B1=p11 at*
 omega
/-! -/
def one:Nat:=2^62
noncomputable def V (a:Nat):ℝ:=(a:ℝ)/2^62
lemma V_nonneg (a:Nat):0<=V a:=by unfold V;positivity
lemma V_one:V one=1:=by norm_num[V,one]
lemma V_mono {a b:Nat} (h:a<=b):V a<=V b:=by
 unfold V;gcongr
lemma V_add (a b:Nat):V (a+b)=V a+V b:=by unfold V;push_cast;ring
lemma V_sub_ge (a b:Nat):V a-V b<=V (a-b):=by
 unfold V
 rcases le_total b a with h | h
 · rw[Nat.cast_sub h];ring_nf;rfl
 · rw[Nat.sub_eq_zero_of_le h]
   have:(a:ℝ)<=b:=by exact_mod_cast h
   simp only[Nat.cast_zero,zero_div]
   rw[div_sub_div_same]
   exact div_nonpos_of_nonpos_of_nonneg (by linarith only[this]) (by positivity)
lemma V_sub_le_of_le {a b:Nat} (h:b<=a):V (a-b)=V a-V b:=by
 unfold V;rw[Nat.cast_sub h];ring
@[inline] def mulDnW (a b:UInt64):UInt64:=(mulHiW a b<<<2)+(mulLoW a b>>>62)
@[inline] def mulUpW (a b:UInt64):UInt64:=
 if mulLoW a b&&&0x3fffffffffffffff=0 then mulDnW a b else mulDnW a b+1
@[inline] def mulDn (a b:Nat):Nat:=(mulDnW a.toUInt64 b.toUInt64).toNat
@[inline] def mulUp (a b:Nat):Nat:=(mulUpW a.toUInt64 b.toUInt64).toNat
lemma toNat_toUInt64_le (a:Nat):a.toUInt64.toNat<=a:=by
 rw[UInt64.toNat_ofNat'];exact Nat.mod_le _ _
lemma toNat_toUInt64_of_lt {a:Nat} (h:a<2^64):a.toUInt64.toNat=a:=by
 rw[UInt64.toNat_ofNat'];exact Nat.mod_eq_of_lt h
lemma toNat_mulDnW (a b:UInt64):(mulDnW a b).toNat=
  ((mulHiW a b).toNat*4%2^64+(mulLoW a b).toNat/2^62)%2^64:=by
 unfold mulDnW
 rw[UInt64.toNat_add,UInt64.toNat_shiftLeft,UInt64.toNat_shiftRight]
 have h2:(2:UInt64).toNat%64=2:=by decide
 have h62:(62:UInt64).toNat%64=62:=by decide
 rw[h2,h62,Nat.shiftLeft_eq,Nat.shiftRight_eq_div_pow]
lemma low62_zero_iff (x:UInt64):(x&&&0x3fffffffffffffff=0)↔x.toNat%2^62=0:=by
 rw[<-UInt64.toNat_inj,UInt64.toNat_and]
 have:(0x3fffffffffffffff:UInt64).toNat=2^62-1:=by decide
 rw[this,Nat.and_two_pow_sub_one_eq_mod];simp only[Nat.reducePow,UInt64.toNat_zero]
lemma halves {a b:Nat} (ha:a<2^64) (hb:b<2^64):
  (mulHiW a.toUInt64 b.toUInt64).toNat*2^64+(mulLoW a.toUInt64 b.toUInt64).toNat=a*b:=by
 have hs:=mul128_spec a.toUInt64 b.toUInt64
 rwa[toNat_toUInt64_of_lt ha,toNat_toUInt64_of_lt hb] at hs
lemma mulDn_mul_le (a b:Nat):mulDn a b*2^62<=a*b:=by
 have hs:=mul128_spec a.toUInt64 b.toUInt64
 have hle:a.toUInt64.toNat*b.toUInt64.toNat<=a*b:=
  Nat.mul_le_mul (toNat_toUInt64_le a) (toNat_toUInt64_le b)
 simp only[mul128] at hs
 unfold mulDn
 rw[toNat_mulDnW]
 generalize (mulHiW a.toUInt64 b.toUInt64).toNat=h at*
 generalize (mulLoW a.toUInt64 b.toUInt64).toNat=l at*
 omega
lemma mulUp_bounds {a b:Nat} (ha:a<=one) (hb:b<=one):
  a*b<=mulUp a b*2^62/\mulUp a b*2^62<a*b+2^62:=by
 have hs:=halves (a:=a) (b:=b) (by unfold one at ha;omega) (by unfold one at hb;omega)
 have hab:a*b<=2^62*2^62:=Nat.mul_le_mul ha hb
 have hl:=(mulLoW a.toUInt64 b.toUInt64).toNat_lt
 unfold mulUp mulUpW
 split
 · rename_i h
   rw[low62_zero_iff] at h
   rw[toNat_mulDnW]
   generalize a*b=p at*
   generalize (mulHiW a.toUInt64 b.toUInt64).toNat=hi at*
   generalize (mulLoW a.toUInt64 b.toUInt64).toNat=l at*
   omega
 · rename_i h
   rw[low62_zero_iff] at h
   rw[UInt64.toNat_add,toNat_mulDnW,show (1:UInt64).toNat=1 from rfl]
   generalize a*b=p at*
   generalize (mulHiW a.toUInt64 b.toUInt64).toNat=hi at*
   generalize (mulLoW a.toUInt64 b.toUInt64).toNat=l at*
   omega
lemma mulUp_mul_ge {a b:Nat} (ha:a<=one) (hb:b<=one):
 a*b<=mulUp a b*2^62:=(mulUp_bounds ha hb).1
lemma mulUp_le {a b:Nat} (ha:a<=one) (hb:b<=one):mulUp a b<=one:=by
 have h:=(mulUp_bounds ha hb).2
 have hab:a*b<=2^62*2^62:=Nat.mul_le_mul ha hb
 unfold one at*
 generalize a*b=p at*
 omega
lemma V_mulDn (a b:Nat):V (mulDn a b)<=V a*V b:=by
 have h:=mulDn_mul_le a b
 unfold V
 rw[div_mul_div_comm,le_div_iff₀ (by positivity),div_mul_eq_mul_div,div_le_iff₀ (by positivity)]
 have:((mulDn a b*2^62:Nat):ℝ)<=((a*b:Nat):ℝ):=by exact_mod_cast h
 push_cast at this
 linarith only[this]
lemma V_mulUp {a b:Nat} (ha:a<=one) (hb:b<=one):V a*V b<=V (mulUp a b):=by
 have h:=mulUp_mul_ge ha hb
 unfold V
 rw[div_mul_div_comm,div_le_iff₀ (by positivity),div_mul_eq_mul_div,le_div_iff₀ (by positivity)]
 have:((a*b:Nat):ℝ)<=((mulUp a b*2^62:Nat):ℝ):=by exact_mod_cast h
 push_cast at this
 linarith only[this]
lemma V_mulDn_le {a b:Nat} {x y:ℝ} (ha:V a<=x) (hb:V b<=y):
  V (mulDn a b)<=x*y:=
 (V_mulDn a b).trans (mul_le_mul ha hb (V_nonneg b) ((V_nonneg a).trans ha))
lemma le_V_mulUp {a b:Nat} {x y:ℝ} (ha:x<=V a) (hb:y<=V b) (hy:0<=y)
  (ha1:a<=one) (hb1:b<=one):x*y<=V (mulUp a b):=
 (mul_le_mul ha hb hy (V_nonneg a)).trans (V_mulUp ha1 hb1)
lemma V_div_le (k:Nat):V (one/k)<=1/k:=by
 rcases Nat.eq_zero_or_pos k with rfl | hk
 · simp only[V,Nat.div_zero,CharP.cast_eq_zero,zero_div,div_zero,Std.le_refl]
 have h:one/k*k<=one:=Nat.div_mul_le_self one k
 have hc:((one/k:Nat):ℝ)*k<=(one:ℝ):=by exact_mod_cast h
 unfold V
 rw[div_le_div_iff₀ (by positivity) (by exact_mod_cast hk)]
 have:(one:ℝ)=2^62:=by norm_num[one]
 linarith only[hc,this]
lemma le_V_divUp (k:Nat) (hk:0<k):1/(k:ℝ)<=V ((one+k-1)/k):=by
 have h:one<=(one+k-1)/k*k:=by
  have h2:=Nat.div_add_mod (one+k-1) k
  have h3:=Nat.mod_lt (one+k-1) hk
  have:(one+k-1)/k*k=k*((one+k-1)/k):=Nat.mul_comm _ _
  omega
 have hc:(one:ℝ)<=(((one+k-1)/k:Nat):ℝ)*k:=by exact_mod_cast h
 unfold V
 rw[div_le_div_iff₀ (by exact_mod_cast hk) (by positivity)]
 have:(one:ℝ)=2^62:=by norm_num[one]
 linarith only[hc,this]
/-! -/
def f2l:Nat:=one/2
def f3l:Nat:=one/6
def f4l:Nat:=one/24
def f5l:Nat:=one/120
def f6l:Nat:=one/720
def f2h:Nat:=(one+2-1)/2
def f3h:Nat:=(one+6-1)/6
def f4h:Nat:=(one+24-1)/24
def f5h:Nat:=(one+120-1)/120
def f6h:Nat:=(one+720-1)/720
def remF:Nat:=(one+35280-1)/35280*8
@[inline] def expSmall (rl rh:Nat):Nat×Nat:=
 let p2l:=mulDn rl rl
 let p3l:=mulDn p2l rl
 let p4l:=mulDn p3l rl
 let p5l:=mulDn p4l rl
 let p6l:=mulDn p5l rl
 let p2h:=mulUp rh rh
 let p3h:=mulUp p2h rh
 let p4h:=mulUp p3h rh
 let p5h:=mulUp p4h rh
 let p6h:=mulUp p5h rh
 let p7h:=mulUp p6h rh
 let lo:=(one+mulDn p2l f2l+mulDn p4l f4l+mulDn p6l f6l)-
  (rh+mulUp p3h f3h+mulUp p5h f5h+mulUp p7h remF)
 let hi:=(one+mulUp p2h f2h+mulUp p4h f4h+mulUp p6h f6h+mulUp p7h remF)-
  (rl+mulDn p3l f3l+mulDn p5l f5l)
 (lo,min one hi)
lemma exp_neg_taylor {r:ℝ} (h0:0<=r) (h1:r<=1):
  |Real.exp (-r)-(1-r+r^2/2-r^3/6+r^4/24-r^5/120+r^6/720)| <=
    r^7*(8/35280):=by
 have h:=Real.exp_bound (x:=-r) (by rw[abs_neg,abs_of_nonneg h0];exact h1) (n:=7)
  (by norm_num)
 simp only[Finset.sum_range_succ,Finset.sum_range_zero,Nat.factorial] at h
 rw[abs_neg,abs_of_nonneg h0] at h
 convert h using 2
 · norm_num;ring
 · norm_num
lemma f_le_one {x:Nat} (k:Nat) (hk:0<k) (h:x=(one+k-1)/k):x<=one:=by
 subst h;unfold one;apply Nat.div_le_of_le_mul;omega
lemma expSmall_spec {rl rh:Nat} {r:ℝ} (hl:V rl<=r) (hh:r<=V rh) (hrh:rh<=one):
  V (expSmall rl rh).1<=Real.exp (-r)/\Real.exp (-r)<=V (expSmall rl rh).2/\
    (expSmall rl rh).2<=one:=by
 have h0:0<=r:=(V_nonneg rl).trans hl
 have h1:r<=1:=hh.trans (V_one▸V_mono hrh)
 have hT:=exp_neg_taylor h0 h1
 have hp: ∀n,0<=r^n:= λn=>pow_nonneg h0 n
 have l2:V (mulDn rl rl)<=r^2:=by rw[sq];exact V_mulDn_le hl hl
 have l3:V (mulDn (mulDn rl rl) rl)<=r^3:=by rw[pow_succ];exact V_mulDn_le l2 hl
 have l4:V (mulDn (mulDn (mulDn rl rl) rl) rl)<=r^4:=by
  rw[pow_succ];exact V_mulDn_le l3 hl
 have l5:V (mulDn (mulDn (mulDn (mulDn rl rl) rl) rl) rl)<=r^5:=by
  rw[pow_succ];exact V_mulDn_le l4 hl
 have l6:V (mulDn (mulDn (mulDn (mulDn (mulDn rl rl) rl) rl) rl) rl)<=r^6:=by
  rw[pow_succ];exact V_mulDn_le l5 hl
 have u2:r^2<=V (mulUp rh rh):=by rw[sq];exact le_V_mulUp hh hh h0 hrh hrh
 have b2:mulUp rh rh<=one:=mulUp_le hrh hrh
 have u3:r^3<=V (mulUp (mulUp rh rh) rh):=by
  rw[pow_succ];exact le_V_mulUp u2 hh h0 b2 hrh
 have b3:=mulUp_le b2 hrh
 have u4:r^4<=V (mulUp (mulUp (mulUp rh rh) rh) rh):=by
  rw[pow_succ];exact le_V_mulUp u3 hh h0 b3 hrh
 have b4:=mulUp_le b3 hrh
 have u5:r^5<=V (mulUp (mulUp (mulUp (mulUp rh rh) rh) rh) rh):=by
  rw[pow_succ];exact le_V_mulUp u4 hh h0 b4 hrh
 have b5:=mulUp_le b4 hrh
 have u6:r^6<=V (mulUp (mulUp (mulUp (mulUp (mulUp rh rh) rh) rh) rh) rh):=by
  rw[pow_succ];exact le_V_mulUp u5 hh h0 b5 hrh
 have b6:=mulUp_le b5 hrh
 have u7:r^7<=V (mulUp (mulUp (mulUp (mulUp (mulUp (mulUp rh rh) rh) rh) rh) rh) rh):=by
  rw[pow_succ];exact le_V_mulUp u6 hh h0 b6 hrh
 have b7:=mulUp_le b6 hrh
 have fl: ∀(k:Nat) (x:Nat),x=one/k->V x<=1/k:= λk x h=>h▸V_div_le k
 have fh: ∀(k:Nat) (x:Nat),0<k->x=(one+k-1)/k->1/(k:ℝ)<=V x:=
  λk x hk h=>h▸le_V_divUp k hk
 have hf2l:=fl 2 f2l rfl
 have hf3l:=fl 6 f3l rfl
 have hf4l:=fl 24 f4l rfl
 have hf5l:=fl 120 f5l rfl
 have hf6l:=fl 720 f6l rfl
 have hf2h:=fh 2 f2h (by norm_num) rfl
 have hf3h:=fh 6 f3h (by norm_num) rfl
 have hf4h:=fh 24 f4h (by norm_num) rfl
 have hf5h:=fh 120 f5h (by norm_num) rfl
 have hf6h:=fh 720 f6h (by norm_num) rfl
 have hremF:8/(35280:ℝ)<=V remF:=by
  have h:=fh 35280 ((one+35280-1)/35280) (by norm_num) rfl
  unfold remF V at*
  push_cast at*
  rw[mul_div_right_comm]
  linarith only[h]
 have bf2:=f_le_one 2 (by norm_num) (rfl:f2h=_)
 have bf3:=f_le_one 6 (by norm_num) (rfl:f3h=_)
 have bf4:=f_le_one 24 (by norm_num) (rfl:f4h=_)
 have bf5:=f_le_one 120 (by norm_num) (rfl:f5h=_)
 have bf6:=f_le_one 720 (by norm_num) (rfl:f6h=_)
 have brem:remF<=one:=by unfold remF one;norm_num
 have c2:=V_mulDn_le l2 hf2l
 have c3:=V_mulDn_le l3 hf3l
 have c4:=V_mulDn_le l4 hf4l
 have c5:=V_mulDn_le l5 hf5l
 have c6:=V_mulDn_le l6 hf6l
 have d2:=le_V_mulUp u2 hf2h (by norm_num) b2 bf2
 have d3:=le_V_mulUp u3 hf3h (by norm_num) b3 bf3
 have d4:=le_V_mulUp u4 hf4h (by norm_num) b4 bf4
 have d5:=le_V_mulUp u5 hf5h (by norm_num) b5 bf5
 have d6:=le_V_mulUp u6 hf6h (by norm_num) b6 bf6
 have d7:=le_V_mulUp u7 hremF (by norm_num) b7 brem
 have hexp1:Real.exp (-r)<=1:=Real.exp_le_one_iff.mpr (by linarith only[h0])
 have hab:=abs_le.mp hT
 refine ⟨?_,?_,min_le_left _ _⟩
 · simp only[expSmall]
   refine le_trans (b:=max 0 (V (one+mulDn (mulDn rl rl) f2l+
    mulDn (mulDn (mulDn (mulDn rl rl) rl) rl) f4l+
    mulDn (mulDn (mulDn (mulDn (mulDn (mulDn rl rl) rl) rl) rl) rl) f6l)-
    V (rh+mulUp (mulUp (mulUp rh rh) rh) f3h+
      mulUp (mulUp (mulUp (mulUp (mulUp rh rh) rh) rh) rh) f5h+
      mulUp (mulUp (mulUp (mulUp (mulUp (mulUp (mulUp rh rh) rh) rh) rh) rh) rh) remF))) ?_ ?_
   · unfold V
     rw[le_max_iff]
     rcases le_total (rh+mulUp (mulUp (mulUp rh rh) rh) f3h+
      mulUp (mulUp (mulUp (mulUp (mulUp rh rh) rh) rh) rh) f5h+
      mulUp (mulUp (mulUp (mulUp (mulUp (mulUp (mulUp rh rh) rh) rh) rh) rh) rh) remF)
      (one+mulDn (mulDn rl rl) f2l+
        mulDn (mulDn (mulDn (mulDn rl rl) rl) rl) f4l+
        mulDn (mulDn (mulDn (mulDn (mulDn (mulDn rl rl) rl) rl) rl) rl) f6l) with h | h
     · right;rw[Nat.cast_sub h];ring_nf;rfl
     · left;rw[Nat.sub_eq_zero_of_le h];simp only[CharP.cast_eq_zero,zero_div,Std.le_refl]
   · apply max_le (Real.exp_pos _).le
     simp only[V_add,V_one] at*
     linarith only[hh,c2,c4,c6,d3,d5,d7,hab]
 · simp only[expSmall]
   rcases le_total one (one+mulUp (mulUp rh rh) f2h+mulUp (mulUp (mulUp (mulUp rh rh) rh) rh) f4h+
    mulUp (mulUp (mulUp (mulUp (mulUp (mulUp rh rh) rh) rh) rh) rh) f6h+
    mulUp (mulUp (mulUp (mulUp (mulUp (mulUp (mulUp rh rh) rh) rh) rh) rh) rh) remF-
    (rl+mulDn (mulDn (mulDn rl rl) rl) f3l+
      mulDn (mulDn (mulDn (mulDn (mulDn rl rl) rl) rl) rl) f5l)) with h | h
   · rw[min_eq_left h,V_one];exact hexp1
   · rw[min_eq_right h]
     refine le_trans ?_ (V_sub_ge _ _)
     simp only[V_add,V_one]
     linarith only[hab.2,hl,c3,c5,d2,d4,d6,d7,hab]
/-! -/
def Encl (iv:Nat×Nat) (b:ℝ):Prop:=
 V iv.1<=Real.exp (-b)/\Real.exp (-b)<=V iv.2/\iv.2<=one
def powOf (base:Nat×Nat):Nat->Nat×Nat
 | 0=>(one,one)
 | j+1=>let p:=powOf base j;(mulDn p.1 base.1,min one (mulUp p.2 base.2))
lemma Encl.mul {x y:Nat×Nat} {a b:ℝ} (hx:Encl x a) (hy:Encl y b):
  Encl (mulDn x.1 y.1,min one (mulUp x.2 y.2)) (a+b):=by
 have he:Real.exp (-(a+b))=Real.exp (-a)*Real.exp (-b):=by
  rw[<-Real.exp_add];ring_nf
 refine ⟨?_,?_,min_le_left _ _⟩
 · rw[he];exact V_mulDn_le hx.1 hy.1
 · rw[he]
   rcases le_total one (mulUp x.2 y.2) with h | h
   · rw[min_eq_left h,V_one,<-he]
     exact Real.exp_le_one_iff.mpr (by
      have:=Real.exp_le_one_iff.mp (le_trans hx.2.1 (V_one▸V_mono hx.2.2))
      have:=Real.exp_le_one_iff.mp (le_trans hy.2.1 (V_one▸V_mono hy.2.2))
      linarith)
   · rw[min_eq_right h]
     exact le_V_mulUp hx.2.1 hy.2.1 (Real.exp_pos _).le hx.2.2 hy.2.2
lemma powOf_spec {base:Nat×Nat} {b:ℝ} (hb:Encl base b):
  ∀j:Nat,Encl (powOf base j) (j*b)
 | 0=>by
   refine ⟨?_,?_,le_rfl⟩<;>simp only[powOf,V_one,CharP.cast_eq_zero,zero_mul,neg_zero,Real.exp_zero,Std.le_refl]
 | j+1=>by
   have:=(powOf_spec hb j).mul hb
   simp only[powOf]
   convert this using 2;push_cast;ring
def sixtyFourth:Nat:=2^56
lemma V_sixtyFourth:V sixtyFourth=1/64:=by norm_num[V,sixtyFourth]
def base64:Nat×Nat:=expSmall sixtyFourth sixtyFourth
lemma base64_spec:Encl base64 (1/64):=by
 exact expSmall_spec (rl:=sixtyFourth) (rh:=sixtyFourth) (r:=1/64)
  V_sixtyFourth.le V_sixtyFourth.ge (by norm_num[sixtyFourth,one])
def fracTab:Array (Nat×Nat):=Array.ofFn (n:=64) λj=>powOf base64 j.val
def baseOne:Nat×Nat:=powOf base64 64
def intTab:Array (Nat×Nat):=Array.ofFn (n:=17) λn=>powOf baseOne n.val
lemma getElem!_ofFn' {α:Type} [Inhabited α] {n:Nat} (f:Fin n->α) {i:Nat}
  (h:i<n):(Array.ofFn f)[i]! =f ⟨i,h⟩:=by
 rw[getElem!_pos (Array.ofFn f) i (by simpa only[Array.size_ofFn] using h)]
 simp only[Array.getElem_ofFn]
lemma fracTab_spec {j:Nat} (h:j<64):Encl fracTab[j]! (j*(1/64)):=by
 rw[fracTab,getElem!_ofFn' _ h]
 exact powOf_spec base64_spec j
lemma intTab_spec {n:Nat} (h:n<17):Encl intTab[n]! n:=by
 rw[intTab,getElem!_ofFn' _ h]
 have h1:Encl baseOne 1:=by
  have:=powOf_spec base64_spec 64
  simp only[baseOne];convert this using 1;norm_num
 have:=powOf_spec h1 n
 simpa only[mul_one] using this
/-! -/
@[inline] def scaled (m:Nat) (e:Int):Nat×Nat:=
 if 0<=e+56 then let v:=m<<<(e+56).toNat;(v,v)
 else let v:=m>>>(-(e+56)).toNat;(v,v+1)
lemma scaled_spec (m:Nat) (e:Int):
  ((scaled m e).1:ℝ)<=m*(2:ℝ)^e*2^56/\
    (m:ℝ)*(2:ℝ)^e*2^56<=(scaled m e).2/\(scaled m e).1<=(scaled m e).2:=by
 unfold scaled
 have hz:(m:ℝ)*(2:ℝ)^e*2^56=m*(2:ℝ)^(e+56):=by
  rw[zpow_add₀ two_ne_zero];norm_num;ring
 split
 · rename_i h
   have hk:((2:ℝ)^(e+56))=2^(e+56).toNat:=by
    rw[<-zpow_natCast,Int.toNat_of_nonneg h]
   simp only[Nat.shiftLeft_eq,hz,hk];push_cast;simp only[Std.le_refl,and_self]
 · rename_i h
   set k:=(-(e+56)).toNat
   have hk:((2:ℝ)^(e+56))=1/2^k:=by
    rw[show e+56=-(k:Int) by omega,zpow_neg,zpow_natCast,one_div]
   simp only[Nat.shiftRight_eq_div_pow,hz,hk]
   have hp:(0:ℝ)<2^k:=by positivity
   have h1:=Nat.div_mul_le_self m (2^k)
   have h2:=Nat.lt_succ_iff.mpr (le_refl (m/2^k))
   have h3:m<(m/2^k+1)*2^k:=by
    have:=Nat.lt_mul_div_succ m (Nat.two_pow_pos k);linarith only[this]
   refine ⟨?_,?_,Nat.le_succ _⟩
   · rw[mul_one_div,le_div_iff₀ hp];exact_mod_cast h1
   · rw[mul_one_div,div_le_iff₀ hp];push_cast;exact_mod_cast h3.le
def lim17:Nat:=17*2^56
@[inline] def expNeg (m:Nat) (e:Int):Option (Nat×Nat):=
 let c:=scaled m e
 if lim17<=c.2 then none else
 let q:=c.1>>>50
 let rl:=(c.1-(q<<<50))<<<6
 let rh:=(c.2-(q<<<50))<<<6
 if one<rh then none else
 let P:=expSmall rl rh
 let F:=fracTab[q%64]!
 let I:=intTab[q>>>6]!
 let IF:=(mulDn I.1 F.1,min one (mulUp I.2 F.2))
 some (mulDn IF.1 P.1,min one (mulUp IF.2 P.2))
lemma expNeg_spec {m:Nat} {e:Int} {iv:Nat×Nat} (h:expNeg m e=some iv):
  Encl iv (m*(2:ℝ)^e):=by
 unfold expNeg at h
 simp only[Nat.shiftRight_eq_div_pow,Nat.shiftLeft_eq] at h
 obtain ⟨hlo,hhi,hle⟩:=scaled_spec m e
 generalize (scaled m e).1=cl at*
 generalize (scaled m e).2=ch at*
 split at h
 · exact absurd h (by simp only[reduceCtorEq,not_false_eq_true])
 rename_i hlim
 split at h
 · exact absurd h (by simp only[reduceCtorEq,not_false_eq_true])
 rename_i hrh
 simp only[Option.some.injEq] at h
 subst h
 set q:=cl/2^50
 have hq:q*2^50<=cl:=Nat.div_mul_le_self cl _
 have hqd:q=q/2^6*64+q%64:=by have:=Nat.div_add_mod q 64;omega
 have hn:q/2^6<17:=by unfold lim17 at hlim;omega
 have hj:q%64<64:=Nat.mod_lt _ (by norm_num)
 set C:=(m:ℝ)*(2:ℝ)^e
 set r:=C-(q/2^6:Nat)-(q%64:Nat)*(1/64:ℝ)
 have hr:C=(q/2^6:Nat)+(q%64:Nat)*(1/64:ℝ)+r:=by ring
 have hqr:((q:Nat):ℝ)*2^50=((q/2^6:Nat):ℝ)*2^56+((q%64:Nat):ℝ)*2^50:=by
  have:((q:Nat):ℝ)=((q/2^6*64+q%64:Nat):ℝ):=by rw[<-hqd]
  rw[this];push_cast;ring
 have hrl:V ((cl-q*2^50)*2^6)<=r:=by
  unfold V;rw[Nat.cast_mul,Nat.cast_sub hq];push_cast
  rw[div_le_iff₀ (by positivity)]
  simp only[r];linarith only[hqr,hlo]
 have hrh':r<=V ((ch-q*2^50)*2^6):=by
  unfold V;rw[Nat.cast_mul,Nat.cast_sub (hq.trans hle)];push_cast
  rw[le_div_iff₀ (by positivity)]
  simp only[r];linarith only[hqr,hhi]
 have hP:=expSmall_spec hrl hrh' (by omega)
 have hI:=intTab_spec hn
 have hF:=fracTab_spec hj
 rw[hr]
 exact (hI.mul hF).mul hP
/-! -/
def two62f:Float:=4611686018427387904.0
@[inline] def mulLe (q d num:Nat):Bool:=
 let h:=(mulHiW q.toUInt64 d.toUInt64).toNat
 let nh:=num>>>2
 h<nh || (h==nh&&mulLoW q.toUInt64 d.toUInt64<=(num%4).toUInt64<<<62)
@[inline] def mulGe (q d num:Nat):Bool:=
 let h:=(mulHiW q.toUInt64 d.toUInt64).toNat
 let nh:=num>>>2
 nh<h || (h==nh&&(num%4).toUInt64<<<62<=mulLoW q.toUInt64 d.toUInt64)
lemma num_split (num:Nat):
  ((num%4).toUInt64<<<62).toNat=num%4*2^62:=by
 have h4:num%4<4:=Nat.mod_lt _ (by norm_num)
 rw[UInt64.toNat_shiftLeft,toNat_toUInt64_of_lt (by omega)]
 have:(62:UInt64).toNat%64=62:=by decide
 rw[this,Nat.shiftLeft_eq,Nat.mod_eq_of_lt (by omega)]
lemma mulLe_spec {q d num:Nat} (hq:q<2^64) (hd:d<2^64) (h:mulLe q d num=true):
  q*d<=num*2^62:=by
 have hs:=halves hq hd
 unfold mulLe at h
 simp only[Nat.shiftRight_eq_div_pow,Bool.or_eq_true,decide_eq_true_eq,Bool.and_eq_true,
  beq_iff_eq,UInt64.le_iff_toNat_le,num_split] at h
 have hl:=(mulLoW q.toUInt64 d.toUInt64).toNat_lt
 generalize (mulHiW q.toUInt64 d.toUInt64).toNat=a at*
 generalize (mulLoW q.toUInt64 d.toUInt64).toNat=b at*
 omega
lemma mulGe_spec {q d num:Nat} (hq:q<2^64) (hd:d<2^64) (h:mulGe q d num=true):
  num*2^62<=q*d:=by
 have hs:=halves hq hd
 unfold mulGe at h
 simp only[Nat.shiftRight_eq_div_pow,Bool.or_eq_true,decide_eq_true_eq,Bool.and_eq_true,
  beq_iff_eq,UInt64.le_iff_toNat_le,num_split] at h
 have hl:=(mulLoW q.toUInt64 d.toUInt64).toNat_lt
 generalize (mulHiW q.toUInt64 d.toUInt64).toNat=a at*
 generalize (mulLoW q.toUInt64 d.toUInt64).toNat=b at*
 omega
@[inline] def guess (num den:Nat):Nat:=
 ((num.toUInt64.toFloat*two62f)/den.toUInt64.toFloat).toUInt64.toNat
def lim64:Nat:=2^64
@[inline] def divLo (num den:Nat):Option Nat:=
 let g:=guess num den
 let q:=g-((g>>>50)+2)
 if q<lim64&&den<lim64&&mulLe q den num then some q else none
@[inline] def divHi (num den:Nat):Option Nat:=
 let g:=guess num den
 let q:=g+((g>>>50)+2)
 if q<lim64&&den<lim64&&mulGe q den num then some q else none
lemma V_mul_iff (a b c:Nat):V a*V b<=V c↔a*b<=c*2^62:=by
 unfold V
 rw[div_mul_div_comm,div_le_div_iff₀ (by positivity) (by positivity)]
 constructor
 · intro h
   have:((a*b:Nat):ℝ)*2^62<=((c*2^62:Nat):ℝ)*2^62:=by
    push_cast;linarith only[h]
   exact_mod_cast le_of_mul_le_mul_right this (by positivity)
 · intro h
   have:((a*b:Nat):ℝ)<=((c*2^62:Nat):ℝ):=by exact_mod_cast h
   push_cast at this;linarith only[this]
lemma divLo_spec {num den q:Nat} (h:divLo num den=some q):V q*V den<=V num:=by
 unfold divLo at h
 dsimp only at h
 generalize guess num den=g at h
 split at h
 · rename_i hc
   simp only[Bool.and_eq_true,decide_eq_true_eq] at hc
   simp only[Option.some.injEq] at h;subst h
   exact (V_mul_iff _ _ _).mpr (mulLe_spec hc.1.1 hc.1.2 hc.2)
 · simp only[reduceCtorEq] at h
lemma divHi_spec {num den q:Nat} (h:divHi num den=some q):V num<=V q*V den:=by
 unfold divHi at h
 dsimp only at h
 generalize guess num den=g at h
 split at h
 · rename_i hc
   simp only[Bool.and_eq_true,decide_eq_true_eq] at hc
   simp only[Option.some.injEq] at h;subst h
   have h2:=mulGe_spec hc.1.1 hc.1.2 hc.2
   unfold V
   rw[div_mul_div_comm,div_le_div_iff₀ (by positivity) (by positivity)]
   have:((num*2^62:Nat):ℝ)<=(((g+(g>>>50+2))*den:Nat):ℝ):=by exact_mod_cast h2
   push_cast at this⊢;linarith only[this]
 · simp only[reduceCtorEq] at h
/-! -/
open ExactRuntime FloatLib.Floats.Formats.BinaryInterchange FloatLib.Floats FloatLib.Numerics
open ExactRuntime.FastRound
def settleInt (lo hi:Int):Option UInt32:=
 let a:=finishWord lo (-62) false
 if lo≠0&&hi≠0&&a==finishWord hi (-62) false&&expOf a != 255&&
     (decodeD a).significand != 0 then some a else none
@[inline] def settle (neg:Bool) (a b:Nat):Option UInt32:=
 let w:=roundW neg a (-62)
 if a≠0&&b≠0&&w==roundW neg b (-62)&&expOf w != 255&&
     (decodeD w).significand != 0 then some w else none
lemma settle_eq (neg:Bool) (a b:Nat):
  settle neg a b=
   settleInt (if neg then-(b:Int) else a) (if neg then-(a:Int) else b):=by
 unfold settle settleInt
 by_cases ha:a=0
 · subst ha;cases neg<;>simp only[ne_eq,not_true_eq_false,decide_false,decide_not,Bool.false_and,Int.reduceNeg,Bool.false_eq_true, ↓reduceIte,CharP.cast_eq_zero,Int.natCast_eq_zero,neg_eq_zero,neg_zero,Bool.and_false]
 by_cases hb:b=0
 · subst hb;cases neg<;>simp only[ne_eq,decide_not,not_true_eq_false,decide_false,Bool.and_false,Int.reduceNeg,Bool.false_and,Bool.false_eq_true, ↓reduceIte,Int.natCast_eq_zero,CharP.cast_eq_zero,neg_zero,neg_eq_zero]
 rw[roundW_signed neg a _ ha,roundW_signed neg b _ hb]
 cases neg
 · simp only[ne_eq,decide_not,Bool.false_eq_true, ↓reduceIte,Int.reduceNeg,Bool.and_eq_true,Bool.not_eq_eq_eq_not,Bool.not_true,decide_eq_false_iff_not,beq_iff_eq,bne_iff_ne,Int.natCast_eq_zero]
 · simp only[ite_true]
   by_cases he:finishWord (-(a:Int)) (-62) false=finishWord (-(b:Int)) (-62) false
   · simp only[ne_eq,ha,not_false_eq_true,decide_true,hb,Bool.and_self,Int.reduceNeg,he,BEq.rfl,Bool.true_and,Bool.and_eq_true,bne_iff_ne,neg_eq_zero,Int.natCast_eq_zero]
   · have he': ¬finishWord (-(b:Int)) (-62) false=finishWord (-(a:Int)) (-62) false:=
      λh=>he h.symm
     simp only[ne_eq,ha,not_false_eq_true,decide_true,hb,Bool.and_self,Int.reduceNeg,Bool.true_and,Bool.and_eq_true,beq_iff_eq,he,bne_iff_ne,false_and, ↓reduceIte,neg_eq_zero,Int.natCast_eq_zero,he']
lemma settleInt_spec {lo hi:Int} {a:UInt32} (h:settleInt lo hi=some a) {v:ℝ}
  (hl:(lo:ℝ)*2^(-62:Int)<=v) (hh:v<=(hi:ℝ)*2^(-62:Int)) {z:Model Fmt}
  (hz:Model.toReal z=Model.roundAt Fmt v):toWord z=a:=by
 unfold settleInt at h
 dsimp only at h
 split at h
 · rename_i hc
   simp only[Option.some.injEq] at h
   subst h
   simp only[Bool.and_eq_true,bne_iff_ne,ne_eq,decide_eq_true_eq,beq_iff_eq] at hc
   obtain ⟨⟨⟨⟨hlo,hhi⟩,heq⟩,hfin⟩,hsig⟩:=hc
   exact bracket_spec hlo hhi rfl heq.symm hfin hsig hl hh hz
 · simp only[reduceCtorEq] at h
lemma int_scale (n:Nat):((n:Int):ℝ)*2^(-62:Int)=V n:=by
 unfold V;push_cast;rw[zpow_neg,zpow_ofNat];ring
lemma toReal_ofW_decode {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
/-! -/
lemma realSigmoid_neg_eq (C:ℝ):
  realSigmoid (-C)=Real.exp (-C)/(1+Real.exp (-C)):=by
 unfold realSigmoid
 rw[neg_neg]
 have hp:=Real.exp_pos (-C)
 have h:Real.exp C=1/Real.exp (-C):=by rw[Real.exp_neg,one_div,inv_inv]
 rw[h];field_simp;ring
def sigmoidFast (w:UInt32):Option UInt32:=
 if expOf w=255 then none else
 let d:=decodeD w
 match expNeg d.significand d.exponent with
 | none=>none
 | some t=>
   if d.negative then
    match divLo t.1 (one+t.1),divHi t.2 (one+t.2) with
    | some ql,some qh=>settle false ql qh
    | _,_=>none
   else
    match divLo one (one+t.2),divHi one (one+t.1) with
    | some ql,some qh=>settle false ql qh
    | _,_=>none
lemma sigmoidFast_spec {w a:UInt32} (h:sigmoidFast w=some a):sigmoidCR w=a:=by
 unfold sigmoidFast at h
 split at h
 · simp only[reduceCtorEq] at h
 rename_i hfin
 dsimp only at h
 have hx:=toReal_ofW_decode hfin
 generalize decodeD w=d at h hx
 have hspec:=(toReal_sigmoidModel (fmt:=Fmt) (by decide)
  (input:=Model.ofUInt32 w) (by simp only[isFinite_ofUInt32,bne_iff_ne,ne_eq,hfin,not_false_eq_true])).2
 change Model.toReal (sigmoidModel (ofW w))=Model.roundAt Fmt (realSigmoid (Model.toReal (ofW w))) at hspec
 rw[hx] at hspec
 change toWord (sigmoidModel (ofW w))=a
 split at h
 · simp only[reduceCtorEq] at h
 rename_i t ht
 have hE:=expNeg_spec ht
 set C:=(d.significand:ℝ)*(2:ℝ)^d.exponent
 set T:=Real.exp (-C)
 have hT:0<T:=Real.exp_pos _
 obtain ⟨h1,h2,h3⟩:=hE
 have v1:=V_nonneg t.1
 have v2:=V_nonneg t.2
 split at h
 · rename_i hneg
   split at h
   · rename_i ql qh hql hqh
     have bl:=divLo_spec hql
     have bh:=divHi_spec hqh
     rw[V_add,V_one] at bl bh
     rw[settle_eq] at h
     simp only[Bool.false_eq_true,ite_false] at h
     refine settleInt_spec h (v:=realSigmoid (-C)) ?_ ?_ (by rw[hspec,hneg];simp only[↓reduceIte,neg_mul,one_mul])
     · rw[int_scale]
       have:realSigmoid (-C)=T/(1+T):=realSigmoid_neg_eq C
       rw[this,le_div_iff₀ (by linarith only[hT])]
       nlinarith only[bl,h1,v1,hT,V_nonneg ql]
     · rw[int_scale]
       have:realSigmoid (-C)=T/(1+T):=realSigmoid_neg_eq C
       rw[this,div_le_iff₀ (by linarith only[hT])]
       nlinarith only[bh,h2,v2,hT,V_nonneg qh]
   · simp only[reduceCtorEq] at h
 · rename_i hneg
   split at h
   · rename_i ql qh hql hqh
     have bl:=divLo_spec hql
     have bh:=divHi_spec hqh
     rw[V_add,V_one] at bl bh
     have hs:realSigmoid C=1/(1+T):=rfl
     rw[settle_eq] at h
     simp only[Bool.false_eq_true,ite_false] at h
     refine settleInt_spec h (v:=realSigmoid C) ?_ ?_ (by rw[hspec];simp only[hneg,Bool.false_eq_true, ↓reduceIte,one_mul])
     · rw[int_scale,hs,le_div_iff₀ (by linarith only[hT])]
       linarith only[bl,mul_le_mul_of_nonneg_left h2 (V_nonneg ql)]
     · rw[int_scale,hs,div_le_iff₀ (by linarith only[hT])]
       linarith only[bh,mul_le_mul_of_nonneg_left h1 (V_nonneg qh)]
   · simp only[reduceCtorEq] at h
def sigmoidF (w:UInt32):UInt32:=(sigmoidFast w).getD (sigmoidCR w)
lemma sigmoidF_eq:sigmoidF=sigmoidCR:=by
 funext w
 unfold sigmoidF
 cases h:sigmoidFast w with
 | none=>rfl
 | some a=>exact (sigmoidFast_spec h).symm
/-! -/
def tanhFast (w:UInt32):Option UInt32:=
 if expOf w=255 then none else
 let d:=decodeD w
 match expNeg d.significand (d.exponent+1) with
 | none=>none
 | some t=>
   if one<t.1 then none else
   match divLo (one-t.2) (one+t.2),divHi (one-t.1) (one+t.1) with
   | some ql,some qh=>
     settle d.negative ql qh
   | _,_=>none
lemma neg_int_scale (n:Nat):((-(n:Int):Int):ℝ)*2^(-62:Int)=-V n:=by
 rw[Int.cast_neg,neg_mul,int_scale]
lemma tanhFast_spec {w a:UInt32} (h:tanhFast w=some a):tanhCR w=a:=by
 unfold tanhFast at h
 split at h
 · simp only[reduceCtorEq] at h
 rename_i hfin
 dsimp only at h
 have hx:=toReal_ofW_decode hfin
 generalize decodeD w=d at h hx
 have hspec:=(toReal_tanhModel (fmt:=Fmt) (by decide)
  (input:=Model.ofUInt32 w) (by simp only[isFinite_ofUInt32,bne_iff_ne,ne_eq,hfin,not_false_eq_true])).2
 change Model.toReal (tanhModel (ofW w))=Model.roundAt Fmt (Real.tanh (Model.toReal (ofW w))) at hspec
 rw[hx] at hspec
 change toWord (tanhModel (ofW w))=a
 split at h
 · simp only[reduceCtorEq] at h
 rename_i t ht
 have hE:=expNeg_spec ht
 set C:=(d.significand:ℝ)*(2:ℝ)^d.exponent
 have h2C:(d.significand:ℝ)*(2:ℝ)^(d.exponent+1)=2*C:=by
  rw[zpow_add₀ two_ne_zero];simp only[C];ring
 rw[h2C] at hE
 set T:=Real.exp (-(2*C))
 have hT:0<T:=Real.exp_pos _
 have htanh:Real.tanh C=(1-T)/(1+T):=by
  rw[tanh_eq_exp_neg_two];simp only[T];ring_nf
 obtain ⟨h1,h2,h3⟩:=hE
 have v1:=V_nonneg t.1
 have v2:=V_nonneg t.2
 split at h
 · simp only[reduceCtorEq] at h
 rename_i ht1
 replace ht1:=not_lt.mp ht1
 split at h
 · rename_i ql qh hql hqh
   have bl:=divLo_spec hql
   have bh:=divHi_spec hqh
   rw[V_add,V_one,V_sub_le_of_le h3,V_one] at bl
   rw[V_add,V_one,V_sub_le_of_le ht1,V_one] at bh
   have lo:V ql<=Real.tanh C:=by
    rw[htanh,le_div_iff₀ (by linarith only[hT])];linarith only[bl,h2,mul_le_mul_of_nonneg_left h2 (V_nonneg ql)]
   have hi:Real.tanh C<=V qh:=by
    rw[htanh,div_le_iff₀ (by linarith only[hT])];linarith only[bh,h1,mul_le_mul_of_nonneg_left h1 (V_nonneg qh)]
   rw[settle_eq] at h
   cases hneg:d.negative<;>simp only[hneg,ite_true,Bool.false_eq_true,ite_false] at h
   · refine settleInt_spec h (v:=Real.tanh C) ?_ ?_ (by rw[hspec];simp only[hneg,Bool.false_eq_true, ↓reduceIte,one_mul])
     · rw[int_scale];exact lo
     · rw[int_scale];exact hi
   · refine settleInt_spec h (v:=Real.tanh (-C)) ?_ ?_ (by rw[hspec,hneg];simp only[↓reduceIte,neg_mul,one_mul,Real.tanh_neg,Model.roundAt_neg,Model.IEEERoundingMode.negate_nearest])
     · rw[neg_int_scale,Real.tanh_neg];linarith only[hi]
     · rw[neg_int_scale,Real.tanh_neg];linarith only[lo]
 · simp only[reduceCtorEq] at h
def tanhF (w:UInt32):UInt32:=(tanhFast w).getD (tanhCR w)
lemma tanhF_eq:tanhF=tanhCR:=by
 funext w
 unfold tanhF
 cases h:tanhFast w with
 | none=>rfl
 | some a=>exact (tanhFast_spec h).symm
/-! -/
end FastAct
namespace FexK
open ExactRuntime FloatLib.Floats.Formats.BinaryInterchange FloatLib.Floats
/-! -/
@[inline] def m64 (w:UInt32):UInt64:=
 if (w>>>23)&&&0xFF=0 then (w&&&0x7FFFFF).toUInt64 else (w&&&0x7FFFFF).toUInt64+0x800000
@[inline] def i32 (w:UInt32):UInt32:=if (w>>>23)&&&0xFF=0 then 0 else ((w>>>23)&&&0xFF)-1
structure FY where
 P:UInt64
 N:UInt64
 lo:UInt32
 ok:Bool
@[specialize] def fyLoop (fa fb:Nat->UInt32) (index:Nat):Nat->UInt64->UInt64->UInt32->FY
 | 0,P,N,lo=>⟨P,N,lo,true⟩
 | cnt+1,P,N,lo=>
   let a:=fa index
   let b:=fb index
   if expOf a=255\/expOf b=255 then ⟨P,N,lo,false⟩
   else
    let p:=m64 a*m64 b
    if p=0 then fyLoop fa fb (index+1) cnt P N lo
    else
     let k:=i32 a+i32 b
     if P=0/\N=0 then
      if Bool.xor (signOf a) (signOf b) then fyLoop fa fb (index+1) cnt 0 p k
      else fyLoop fa fb (index+1) cnt p 0 k
     else if lo<=k then
      let d:=k-lo
      if d<=15 then
       let q:=p<<<d.toUInt64
       if Bool.xor (signOf a) (signOf b) then
        if N<=N+q then fyLoop fa fb (index+1) cnt P (N+q) lo else ⟨P,N,lo,false⟩
       else
        if P<=P+q then fyLoop fa fb (index+1) cnt (P+q) N lo else ⟨P,N,lo,false⟩
      else ⟨P,N,lo,false⟩
     else
      let s:=lo-k
      if s<=15/\P<(1:UInt64)<<<(63-s).toUInt64/\N<(1:UInt64)<<<(63-s).toUInt64 then
       if Bool.xor (signOf a) (signOf b) then
        fyLoop fa fb (index+1) cnt (P<<<s.toUInt64) ((N<<<s.toUInt64)+p) k
       else fyLoop fa fb (index+1) cnt ((P<<<s.toUInt64)+p) (N<<<s.toUInt64) k
      else ⟨P,N,lo,false⟩
@[inline] def fexDot (n:Nat) (fa fb:Nat->UInt32):UInt32:=
 let r:=fyLoop fa fb 0 n 0 0 0
 if r.ok&&r.P != r.N then
  if r.P<r.N then FastRound.roundW true (r.N-r.P).toNat ((r.lo.toNat:Int)-298)
  else FastRound.roundW false (r.P-r.N).toNat ((r.lo.toNat:Int)-298)
 else Pinned.dotX (Array.ofFn (n:=n) λi=>fa i.val) (Array.ofFn (n:=n) λi=>fb i.val)
/-! -/
lemma m64_eq (w:UInt32):(m64 w).toNat=mantOf w:=by
 unfold m64 mantOf expOf fracOf
 have hf:(w&&&0x7FFFFF).toNat<2^23:=by
  rw[UInt32.toNat_and];exact Nat.lt_of_le_of_lt Nat.and_le_right (by decide)
 split
 · rw[UInt32.toNat_toUInt64]
 · rw[UInt64.toNat_add,UInt32.toNat_toUInt64]
   simp only[UInt64.reduceToNat];omega
lemma i32_eq (w:UInt32):(i32 w).toNat=idxOf w:=by
 unfold i32 idxOf expOf
 split
 · rfl
 · rename_i h
   have h1:(1:UInt32)<=(w>>>23)&&&0xFF:=by
    rw[UInt32.le_iff_toNat_le]
    have:((w>>>23)&&&0xFF).toNat≠0:= λh0=>h (UInt32.toNat_inj.mp (by simpa only[UInt32.toNat_and,UInt32.toNat_shiftRight,UInt32.reduceToNat,Nat.reduceMod,UInt32.toNat_zero] using h0))
    simp only[UInt32.reduceToNat];omega
   rw[UInt32.toNat_sub_of_le _ _ h1];rfl
lemma mant_lt (w:UInt32):mantOf w<2^24:=by
 unfold mantOf fracOf
 have hf:(w&&&0x7FFFFF).toNat<2^23:=by
  rw[UInt32.toNat_and];exact Nat.lt_of_le_of_lt Nat.and_le_right (by decide)
 split<;>omega
lemma p_eq (a b:UInt32):(m64 a*m64 b).toNat=mantOf a*mantOf b:=by
 rw[UInt64.toNat_mul,m64_eq,m64_eq]
 apply Nat.mod_eq_of_lt
 have:=mant_lt a;have:=mant_lt b
 calc mantOf a*mantOf b<2^24*2^24:=Nat.mul_lt_mul'' ‹_› ‹_›
  _<=2^64:=by decide
lemma k_eq (a b:UInt32) (ha:expOf a≠255) (hb:expOf b≠255):
  (i32 a+i32 b).toNat=idxOf a+idxOf b:=by
 rw[UInt32.toNat_add,i32_eq,i32_eq]
 have:=idxOf_le a ha;have:=idxOf_le b hb
 omega
lemma add_nov (x y:UInt64) (h:x<=x+y):(x+y).toNat=x.toNat+y.toNat:=by
 rw[UInt64.le_iff_toNat_le,UInt64.toNat_add] at h
 rw[UInt64.toNat_add]
 have:=y.toNat_lt;have:=x.toNat_lt
 omega
lemma shl_eq (x:UInt64) (s:UInt32) (hs:s.toNat<64) (hx:x.toNat*2^s.toNat<2^64):
  (x<<<s.toUInt64).toNat=x.toNat*2^s.toNat:=by
 rw[UInt64.toNat_shiftLeft,Nat.shiftLeft_eq,UInt32.toNat_toUInt64,Nat.mod_eq_of_lt hs,
  Nat.mod_eq_of_lt hx]
lemma bnd (x:UInt64) (s:UInt32) (hs:s<=15) (h:x<(1:UInt64)<<<(63-s).toUInt64):
  x.toNat*2^s.toNat<2^63:=by
 rw[UInt32.le_iff_toNat_le] at hs
 simp only[UInt32.reduceToNat] at hs
 have h63:(63-s).toNat=63-s.toNat:=by
  rw[UInt32.toNat_sub_of_le _ _ (by rw[UInt32.le_iff_toNat_le];simp only[UInt32.reduceToNat];omega)]
  rfl
 rw[UInt64.lt_iff_toNat_lt,UInt64.toNat_shiftLeft,UInt32.toNat_toUInt64,h63,
  Nat.mod_eq_of_lt (by omega:63-s.toNat<64),Nat.shiftLeft_eq] at h
 simp only[UInt64.reduceToNat,Nat.one_mul] at h
 rw[Nat.mod_eq_of_lt (Nat.pow_lt_pow_right (by norm_num) (by omega))] at h
 calc x.toNat*2^s.toNat<2^(63-s.toNat)*2^s.toNat:=
       Nat.mul_lt_mul_of_pos_right h (by positivity)
  _=2^63:=by rw[<-pow_add];congr 1;omega
lemma fyLoop_spec (fa fb:Nat->UInt32) (A B:Array UInt32)
  (cnt:Nat): ∀(index:Nat) (P N:UInt64) (lo:UInt32),
  (∀k,index<=k->k< index+cnt->A[k]! =fa k/\B[k]! =fb k)->
  (fyLoop fa fb index cnt P N lo).ok=true->
  (∀k,index<=k->k< index+cnt->expOf (fa k)≠255/\expOf (fb k)≠255)/\
  (((fyLoop fa fb index cnt P N lo).P.toNat:Int)-(fyLoop fa fb index cnt P N lo).N.toNat)*
      2^(fyLoop fa fb index cnt P N lo).lo.toNat=
    ((P.toNat:Int)-N.toNat)*2^lo.toNat+dsum A B index cnt:=by
 induction cnt with
 | zero=>
   intro index P N lo _ _
   refine ⟨λk h1 h2=>by omega,?_⟩
   simp only[fyLoop,dsum,add_zero]
 | succ cnt ih=>
   intro index P N lo hAB hok
   obtain ⟨hA,hB⟩:=hAB index le_rfl (by omega)
   have hAB': ∀k,index+1<=k->k< index+1+cnt->A[k]! =fa k/\B[k]! =fb k:=
    λk h1 h2=>hAB k (by omega) (by omega)
   have hterm:dsum A B index (cnt+1)=
     (if Bool.xor (signOf (fa index)) (signOf (fb index))
       then-((mantOf (fa index)*mantOf (fb index):Nat):Int)
       else ((mantOf (fa index)*mantOf (fb index):Nat):Int))*
       2^(idxOf (fa index)+idxOf (fb index))+dsum A B (index+1) cnt:=by
    rw[dsum,hA,hB,Pinned.sv_mul]
   by_cases hsp:expOf (fa index)=255\/expOf (fb index)=255
   · simp only[fyLoop,hsp, ↓reduceIte,Bool.false_eq_true] at hok
   · have hsp':=not_or.mp hsp
     have fin_of (h: ∀k,index+1<=k->k< index+1+cnt->
       expOf (fa k)≠255/\expOf (fb k)≠255):
       ∀k,index<=k->k< index+(cnt+1)->expOf (fa k)≠255/\expOf (fb k)≠255:=by
      intro k h1 h2
      rcases Nat.lt_or_ge k (index+1) with hk | hk
      · have:k=index:=by omega
        subst this;exact hsp'
      · exact h k hk (by omega)
     have hp:=p_eq (fa index) (fb index)
     have hk:=k_eq (fa index) (fb index) hsp'.1 hsp'.2
     have hpl:mantOf (fa index)*mantOf (fb index)<2^48:=by
      have:=mant_lt (fa index);have:=mant_lt (fb index)
      calc mantOf (fa index)*mantOf (fb index)<2^24*2^24:=Nat.mul_lt_mul'' ‹_› ‹_›
       _=2^48:=by norm_num
     generalize hpv:m64 (fa index)*m64 (fb index)=p at hp
     generalize hkv:i32 (fa index)+i32 (fb index)=k at hk
     generalize hng:Bool.xor (signOf (fa index)) (signOf (fb index))=ng at hterm
     generalize hM:mantOf (fa index)*mantOf (fb index)=M at hp hterm hpl
     generalize hI:idxOf (fa index)+idxOf (fb index)=I at hk hterm
     have unf:fyLoop fa fb index (cnt+1) P N lo=
       (if p=0 then fyLoop fa fb (index+1) cnt P N lo
       else if P=0/\N=0 then
        (if ng then fyLoop fa fb (index+1) cnt 0 p k
        else fyLoop fa fb (index+1) cnt p 0 k)
       else if lo<=k then
        (if k-lo<=15 then
          (if ng then
            (if N<=N+p<<<(k-lo).toUInt64 then
              fyLoop fa fb (index+1) cnt P (N+p<<<(k-lo).toUInt64) lo
            else ⟨P,N,lo,false⟩)
          else
           (if P<=P+p<<<(k-lo).toUInt64 then
             fyLoop fa fb (index+1) cnt (P+p<<<(k-lo).toUInt64) N lo
           else ⟨P,N,lo,false⟩))
        else ⟨P,N,lo,false⟩)
       else
        (if lo-k<=15/\P<(1:UInt64)<<<(63-(lo-k)).toUInt64/\
            N<(1:UInt64)<<<(63-(lo-k)).toUInt64 then
          (if ng then
            fyLoop fa fb (index+1) cnt (P<<<(lo-k).toUInt64) ((N<<<(lo-k).toUInt64)+p) k
          else fyLoop fa fb (index+1) cnt ((P<<<(lo-k).toUInt64)+p) (N<<<(lo-k).toUInt64) k)
        else ⟨P,N,lo,false⟩)):=by
      rw[fyLoop];simp only[hsp,ite_false,hpv,hkv,hng]
     rw[unf] at hok⊢
     by_cases c1:p=0
     · rw[if_pos c1] at hok⊢
       obtain ⟨h1,h2⟩:=ih (index+1) P N lo hAB' hok
       refine ⟨fin_of h1,?_⟩
       have hM0:M=0:=by rw[<-hp,c1];rfl
       rw[h2,hterm,hM0];simp only[CharP.cast_eq_zero,neg_zero,ite_self,zero_mul,zero_add]
     rw[if_neg c1] at hok⊢
     by_cases c2:P=0/\N=0
     · rw[if_pos c2] at hok⊢
       cases ng
       · rw[if_neg (by simp only[Bool.false_eq_true,not_false_eq_true])] at hok⊢
         obtain ⟨h1,h2⟩:=ih _ _ _ _ hAB' hok
         refine ⟨fin_of h1,?_⟩
         rw[h2,hterm,c2.1,c2.2,hp,hk];simp only[UInt64.toNat_zero,CharP.cast_eq_zero,sub_zero,sub_self,zero_mul,Bool.false_eq_true, ↓reduceIte,zero_add]
       · rw[if_pos rfl] at hok⊢
         obtain ⟨h1,h2⟩:=ih _ _ _ _ hAB' hok
         refine ⟨fin_of h1,?_⟩
         rw[h2,hterm,c2.1,c2.2,hp,hk];simp only[UInt64.toNat_zero,CharP.cast_eq_zero,zero_sub,neg_mul,sub_self,zero_mul, ↓reduceIte,zero_add]
     rw[if_neg c2] at hok⊢
     by_cases c4:lo<=k
     · rw[if_pos c4] at hok⊢
       have hle:lo.toNat<=k.toNat:=UInt32.le_iff_toNat_le.mp c4
       have hdk:(k-lo).toNat=k.toNat-lo.toNat:=UInt32.toNat_sub_of_le _ _ c4
       by_cases c5:k-lo<=15
       · rw[if_pos c5] at hok⊢
         have hd15:(k-lo).toNat<=15:=by
          have:=UInt32.le_iff_toNat_le.mp c5;simpa only[ge_iff_le,UInt32.reduceToNat] using this
         have hq:(p<<<(k-lo).toUInt64).toNat=p.toNat*2^(k-lo).toNat:=by
          apply shl_eq _ _ (by omega)
          calc p.toNat*2^(k-lo).toNat<2^48*2^15:=
                Nat.mul_lt_mul_of_lt_of_le (by omega) (Nat.pow_le_pow_right (by norm_num) hd15) (by positivity)
           _<=2^64:=by norm_num
         have e:(2:Int)^k.toNat=2^(k.toNat-lo.toNat)*2^lo.toNat:=by
          rw[<-pow_add];congr 1;omega
         cases ng
         · rw[if_neg (by simp only[Bool.false_eq_true,not_false_eq_true])] at hok⊢
           by_cases c8:P<=P+p<<<(k-lo).toUInt64
           · rw[if_pos c8] at hok⊢
             obtain ⟨h1,h2⟩:=ih _ _ _ _ hAB' hok
             refine ⟨fin_of h1,?_⟩
             rw[h2,hterm,add_nov _ _ c8,hq,hp,hdk,<-hk,e]
             simp only[Nat.cast_add,Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat,Bool.false_eq_true, ↓reduceIte];push_cast;ring
           · rw[if_neg c8] at hok;simp only[Bool.false_eq_true] at hok
         · rw[if_pos rfl] at hok⊢
           by_cases c7:N<=N+p<<<(k-lo).toUInt64
           · rw[if_pos c7] at hok⊢
             obtain ⟨h1,h2⟩:=ih _ _ _ _ hAB' hok
             refine ⟨fin_of h1,?_⟩
             rw[h2,hterm,add_nov _ _ c7,hq,hp,hdk,<-hk,e]
             simp only[Nat.cast_add,Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat, ↓reduceIte,neg_mul];push_cast;ring
           · rw[if_neg c7] at hok;simp only[Bool.false_eq_true] at hok
       · rw[if_neg c5] at hok;simp only[Bool.false_eq_true] at hok
     · rw[if_neg c4] at hok⊢
       by_cases c9:lo-k<=15/\P<(1:UInt64)<<<(63-(lo-k)).toUInt64/\
        N<(1:UInt64)<<<(63-(lo-k)).toUInt64
       · rw[if_pos c9] at hok⊢
         obtain ⟨s15,bP,bN⟩:=c9
         have hlt:k.toNat<lo.toNat:=by
          have:=UInt32.le_iff_toNat_le.not.mp c4;omega
         have hsk:(lo-k).toNat=lo.toNat-k.toNat:=
          UInt32.toNat_sub_of_le _ _ (UInt32.le_iff_toNat_le.mpr (by omega))
         have hs15:(lo-k).toNat<=15:=by
          have:=UInt32.le_iff_toNat_le.mp s15;simpa only[ge_iff_le,UInt32.reduceToNat] using this
         have eP:=shl_eq P (lo-k) (by omega) (by have:=bnd P _ s15 bP;omega)
         have eN:=shl_eq N (lo-k) (by omega) (by have:=bnd N _ s15 bN;omega)
         have e:(2:Int)^lo.toNat=2^(lo.toNat-k.toNat)*2^k.toNat:=by
          rw[<-pow_add];congr 1;omega
         cases ng
         · rw[if_neg (by simp only[Bool.false_eq_true,not_false_eq_true])] at hok⊢
           obtain ⟨h1,h2⟩:=ih _ _ _ _ hAB' hok
           refine ⟨fin_of h1,?_⟩
           have eP2:(P<<<(lo-k).toUInt64+p).toNat=P.toNat*2^(lo-k).toNat+p.toNat:=by
            rw[UInt64.toNat_add,eP];apply Nat.mod_eq_of_lt
            have:=bnd P _ s15 bP;omega
           rw[h2,hterm,eP2,eN,hp,hsk,<-hk,e]
           simp only[Nat.cast_add,Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat,Bool.false_eq_true, ↓reduceIte];push_cast;ring
         · rw[if_pos rfl] at hok⊢
           obtain ⟨h1,h2⟩:=ih _ _ _ _ hAB' hok
           refine ⟨fin_of h1,?_⟩
           have eN2:(N<<<(lo-k).toUInt64+p).toNat=N.toNat*2^(lo-k).toNat+p.toNat:=by
            rw[UInt64.toNat_add,eN];apply Nat.mod_eq_of_lt
            have:=bnd N _ s15 bN;omega
           rw[h2,hterm,eN2,eP,hp,hsk,<-hk,e]
           simp only[Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat,Nat.cast_add, ↓reduceIte,neg_mul];push_cast;ring
       · rw[if_neg c9] at hok;simp only[Bool.false_eq_true] at hok
lemma dotLoop_nsp (fa fb:Nat->UInt32) (cnt:Nat): ∀(index:Nat) (pos neg:Array Nat) (nz:Bool),
  (∀k,index<=k->k< index+cnt->expOf (fa k)≠255/\expOf (fb k)≠255)->
  pos.size=18->neg.size=18->
  (Pinned.dotLoop fa fb pos neg false nz index cnt).special=false:=by
 induction cnt with
 | zero=>intro index pos neg nz _ _ _;simp only[Pinned.dotLoop]
 | succ cnt ih=>
   intro index pos neg nz hf hp hn
   have he:=hf index le_rfl (by omega)
   obtain ⟨h1,h2,h3,-,-⟩:=
    Pinned.Acc.add_spec ⟨pos,neg,false,nz⟩ (mantOf (fa index)*mantOf (fb index))
     (idxOf (fa index)+idxOf (fb index)) (Bool.xor (signOf (fa index)) (signOf (fb index)))
     hp hn (ExactRuntime.mul_mant_lt _ _)
     (by have:=ExactRuntime.idxOf_le _ he.1;have:=ExactRuntime.idxOf_le _ he.2;omega)
   have hstep:Pinned.dotStep ⟨pos,neg,false,nz⟩ (fa index) (fb index)=
     Pinned.Acc.add ⟨pos,neg,false,nz⟩ (mantOf (fa index)*mantOf (fb index))
      (idxOf (fa index)+idxOf (fb index))
      (Bool.xor (signOf (fa index)) (signOf (fb index))):=by
    simp only[Pinned.dotStep,he.1,he.2,or_self, ↓reduceIte]
   rw[Pinned.dotLoop,hstep]
   generalize hr:Pinned.Acc.add ⟨pos,neg,false,nz⟩ (mantOf (fa index)*mantOf (fb index))
    (idxOf (fa index)+idxOf (fb index))
    (Bool.xor (signOf (fa index)) (signOf (fb index)))=r at h1 h2 h3
   obtain ⟨p',n',s',z'⟩:=r
   simp only at h1 h2 h3⊢
   subst h3
   exact ih (index+1) p' n' z' (λk k1 k2=>hf k (by omega) (by omega)) h1 h2
lemma dotOf_fin (n:Nat) (fa fb:Nat->UInt32)
  (hf: ∀k,k<n->expOf (fa k)≠255/\expOf (fb k)≠255):
  ∃nz:Bool,dotOf n fa fb=
   finishWord (ExactRuntime.dsum (Array.ofFn (n:=n) λi=>fa i.val)
    (Array.ofFn (n:=n) λi=>fb i.val) 0 n) (-298) nz:=by
 rw[<-Pinned.dotFn_eq_dotOf]
 unfold Pinned.dotFn
 have hsp:(Pinned.dotLoop fa fb (Array.replicate 18 0) (Array.replicate 18 0) false true 0 n).special
  =false:=dotLoop_nsp fa fb n 0 _ _ true (λk _ h2=>hf k (by omega)) (by simp only[Array.size_replicate]) (by simp only[Array.size_replicate])
 obtain ⟨-,-,hD,-⟩:=Pinned.dotLoopFacts fa fb (Array.ofFn (n:=n) λi=>fa i.val)
  (Array.ofFn (n:=n) λi=>fb i.val) n (Pinned.getElem!_ofFn n fa)
  (Pinned.getElem!_ofFn n fb) Pinned.Acc.empty 0 n (by omega) (by simp only[Pinned.Acc.empty,Array.size_replicate])
  (by simp only[Pinned.Acc.empty,Array.size_replicate]) hsp
 refine ⟨decide (n≠0)&&(Pinned.dotLoop fa fb (Array.replicate 18 0) (Array.replicate 18 0)
  false true 0 n).negZero,?_⟩
 dsimp only
 rw[if_neg (by simp only[hsp,Bool.false_eq_true,not_false_eq_true]),Pinned.finishFast_eq]
 congr 1
 have h0:=Pinned.Acc.total_empty
 simp only[Pinned.Acc.empty] at hD h0
 rw[h0,Int.zero_add] at hD
 exact hD
lemma fexDot_eq (n:Nat) (fa fb:Nat->UInt32):fexDot n fa fb=dotOf n fa fb:=by
 unfold fexDot
 dsimp only
 split
 · rename_i h
   simp only[Bool.and_eq_true,bne_iff_ne,ne_eq] at h
   obtain ⟨hok,hne⟩:=h
   obtain ⟨hf,hv⟩:=fyLoop_spec fa fb (Array.ofFn (n:=n) λi=>fa i.val)
    (Array.ofFn (n:=n) λi=>fb i.val) n 0 0 0 0
    (λk _ h2=>⟨Pinned.getElem!_ofFn n fa k (by omega),Pinned.getElem!_ofFn n fb k (by omega)⟩)
    hok
   obtain ⟨nz,hd⟩:=dotOf_fin n fa fb (λk hk=>hf k (by omega) (by omega))
   rw[hd]
   simp only[UInt64.toNat_zero,Nat.cast_zero,sub_zero,zero_mul,zero_add] at hv
   rw[<-hv]
   generalize (fyLoop fa fb 0 n 0 0 0)=r at hne⊢
   have hne':r.P.toNat≠r.N.toNat:= λh=>hne (UInt64.toNat_inj.mp h)
   rw[<-finishWord_scale]
   have e2:((r.lo.toNat:Int)-298)=-298+((r.lo.toNat:Nat):Int):=by ring
   split
   · rename_i hlt
     have hlt':=UInt64.lt_iff_toNat_lt.mp hlt
     have hs:(r.N-r.P).toNat=r.N.toNat-r.P.toNat:=UInt64.toNat_sub_of_le _ _ (UInt64.le_of_lt hlt)
     rw[FastRound.roundW_eq,<-Pinned.finishWord_signed _ _ _ nz (by omega),e2,hs]
     congr 1;simp only[ite_true];omega
   · rename_i hlt
     have hlt':r.N.toNat<=r.P.toNat:=by have:=UInt64.lt_iff_toNat_lt.not.mp hlt;omega
     have hs:(r.P-r.N).toNat=r.P.toNat-r.N.toNat:=
      UInt64.toNat_sub_of_le _ _ (UInt64.le_iff_toNat_le.mpr hlt')
     rw[FastRound.roundW_eq,<-Pinned.finishWord_signed _ _ _ nz (by omega),e2,hs]
     congr 1;simp only[Bool.false_eq_true,ite_false];omega
 · rw[Pinned.dotX_eq_spec];rfl
end FexK
namespace Submission
open ExactRuntime FloatLib.Floats.Formats.BinaryInterchange FloatLib.Floats FloatLib.Floats.ExecFloat.Binary
open _root_ renaming mul_le_mul_of_nonneg_right → ἀ, mul_le_mul_of_nonneg_left → ἁ, abs_of_nonneg → ἅ, abs_nonneg → Ἀ, dite_eq_left → Ἄ, mul_nonneg → Ἆ, ite_eq_right → ἐ, mul_le_mul → ἑ, zpow_le_zpow_right₀ → ἒ, two_ne_zero → ἓ, dite_eq_right → ἔ, abs_add_le → Ἐ, zpow_add₀ → Ἓ, getElem!_pos → Ἕ, abs_of_pos → ἡ, abs_mul → ἢ, abs_sub_comm → ἣ, zpow_natCast → Ἡ, lt_of_lt_of_le → Ἥ, mul_lt_mul_of_pos_right → Ἦ, ite_eq_left → ἱ, zpow_pos → ἵ, lt_of_le_of_lt → ἷ, pow_le_pow_left₀ → Ἳ, le_trans → ὂ, sq_nonneg → ὅ, getElem!_neg → ὑ, abs_sub → Ὕ, le_refl → Ὗ, FloatArray → ὤ, UInt32 → ὥ, USize → ὦ, Float → ὧ, UInt64 → Ὠ, Bool → Ὡ, Nat → Ὣ, abs_sub_abs_le_abs_sub → Ὥ, if_pos → Ὦ, if_neg → Ὧ, abs_le → ὸ, zpow_sub₀ → ᾂ
open ExactRuntime renaming rationalWord → ἂ, toDyadic_ofUInt32 → ἶ, Word → Ὢ
open UInt32 renaming le_iff_toNat_le → ἃ, toNat_inj → ἠ, toNat_and → Ἧ, lt_iff_toNat_lt → ἳ, toNat_toUSize → ὃ, toNat_sub_of_le → ὓ, le_refl → Ὓ, toNat_ofNat' → ὡ, ofNat → ὺ
open List renaming range → ἄ, mem_range → Ἅ, nodup_range → Ὂ, foldl_cons → ὔ, mem_cons_of_mem → ὖ, range_eq_range' → ὗ, map_congr_left → ὠ, mem_cons_self → ὣ
open FloatArray renaming emptyWithCapacity → ἆ
open Array renaming replicate → ἇ
open Nat renaming mod_eq_of_lt → Ἁ, and_two_pow_sub_one_eq_mod → Ἂ, zero_add → Ἇ, mul_le_mul_right → Ἒ, cast_nonneg → Ἔ, mul_comm → ἥ, div_lt_of_lt_mul → Ἰ, shiftRight_eq_div_pow → Ἱ, not_lt_zero → Ἲ, add_zero → Ἵ, mod_lt → ὀ, shiftLeft_eq → ὁ, div_eq_of_lt → Ὃ, le_mul_of_pos_right → Ὄ, zero_mul → ὒ, one_mul → Ὑ, add_assoc → ὢ, add_mul → ὰ, pos_of_ne_zero → ὼ, succ_mul → ᾀ
open Float renaming ofBits → Ἃ, sqrt → Ἤ
open FloatLib.Floats.Formats.BinaryInterchange.Model renaming roundDyadic → ἕ, isFinite → Ἑ, toReal_eq_unpackedToReal_toModel → ἦ, toReal → Ἢ, modelSignBit → Ἴ, toModel_ofFloatModel → Ὀ, modelSign → ὶ, roundAt → ᾁ
open UInt64 renaming toNat_add → ἤ, toNat_shiftRight → ἰ, lt_iff_toNat_lt → ἴ, toNat_and → Ὅ, le_iff_toNat_le → Ὤ, toNat_sub → ὲ
open Real renaming sqrt → ἧ
open FloatLib.Numerics.Dyadic renaming cast_toRat → Ἠ, toRat_eq_zero_iff → ὐ
open Finset renaming range → Ἣ, sum_congr → Ἷ, mem_range → Ὁ
open FloatLib.Numerics renaming binaryRadix → ἲ
open Float.Model.UnpackedFloat renaming unpack → Ἶ
open USize renaming toNat_one → ὄ, ofNat → ὴ
open Bool renaming and_eq_true_iff → ὕ
local notation"А"=>USize.lt_iff_toNat_lt local notation"В"=>0x7FFFFFFF local notation"Е"=>0xFFFFFFFF local notation"У"=>0x7F800000 local notation"ь"=>0x3CE0000000000000 local notation"س"=>0x80000000 local notation"ظ"=>0x3370000000000000
local notation"א"=>by omega local notation"ב"=>by norm_num local notation"ג"=>by simp
local notation"ѐ"=>Array UInt32 local notation"ё"=>Array Word local notation"ї"=>Array Nat local notation"љ"=>List Nat
open USize (toNat)
open UInt32 (toNat)
open UInt64 (toNat)
local notation:max"Ѡ"x:max=>toNat x
open Array (size)
open FloatArray (size)
local notation:max"ѡ"x:max=>size x
open Nat (toUSize)
open UInt32 (toUSize)
local notation:max"Ѣ"x:max=>toUSize x
local notation:max"Ѥ"x:max=>USize.toNat x
local notation:max"ѥ"x:max=>UInt32.toNat x
local notation:max"Ѧ"x:max=>UInt64.toNat x
local notation:max"ѧ"x:max=>Array.size x
local notation:max"Ѩ"x:max=>FloatArray.size x
local notation:max"ѩ"x:max=>Nat.toUSize x
local notation:max"Ѫ"x:max=>UInt32.toUSize x
local notation:max"ѣ"x:max=>List.length x
open ExactRuntime FloatLib.Floats.Formats.BinaryInterchange FloatLib.Floats FloatLib.Floats.ExecFloat.Binary
open FloatLib.Numerics.Dyadic (signedSignificand toRat_eq_zero_iff add_toRat mul_toRat mulFields)
open UInt64 (toNat_sub_of_le)
open Bool (false_eq_true and_eq_true or_eq_true)
open USize (toNat)
open UInt32 (toNat)
open UInt64 (toNat)
open Array (size)
open FloatArray (size)
open Nat (toUSize)
open UInt32 (toUSize)
local notation"Մ"=>Array.ext
local notation"ה"=>(by decide)
local notation"ו"=>(by positivity)
local notation"ז"=>(by simp;omega)
local notation"ט"=>(by rw[А];omega)
local notation:max x:max "ᶠ"=>Float.isFinite x=true
local notation "⟡"=>FloatFormat.binary64
local notation "⟠"=>Float.Model.Format.binary64
lemma omS3 {x0 x1:ℕ} (h0:x0/32+1<18) (h1:x1=18):x0/32+1<x1:=by subst_vars;omega
lemma omS4 {x0 x1 x2:ℕ} (h0:x0+(x2+1)<=x1):x0<x1:=א
lemma omS5 {x0 x1 x2:ℕ} (h0:x0+(x1+1)<=x2):x0+1+x1<=x2:=by subst_vars;simp only[ὢ,Nat.add_comm,Nat.add_left_comm,Ἇ,Ἵ] at*;assumption
lemma omS6 {x0 x1:ℕ} (h0:x1<=x0) (h1:x0<x1+1):x0=x1:=א
lemma omS7 {x0 x1 x2:ℕ} (h0:x0<x1+(x2+1)):x0<x1+1+x2:=by subst_vars;simp only[ὢ,Nat.add_comm,Nat.add_left_comm,Ἇ,Ἵ] at*;assumption
lemma omS8 {x0 x1:ℕ} (h0:x0<x1+1) (h1:x0>=x1):x0=x1:=א
lemma omS9 {x0:ℕ}:0<x0+1:=by subst_vars;simp only[lt_add_iff_pos_left,Order.lt_add_one_iff,zero_le]
lemma omS10 {x0:ℕ}:0+x0<=x0:=by subst_vars;simp only[zero_add,Std.le_refl]
lemma omS11 {x0 x1 x2:ℕ} (h0:x1=x2) (h1:x0=x2):x0=x1:=by subst_vars;rfl
lemma omS12 {x0 x1 x2:ℕ} (h0:x2=x1) (h1:x0<x2):x0<0+x1:=by subst_vars;simp only[ὢ,Nat.add_comm,Nat.add_left_comm,Ἇ,Ἵ] at*;assumption
lemma omS13 {x0:ℕ} (h0:x0<4294967296):x0/8388608<512:=א
lemma omS14 {x0 x1 x2 x3:ℕ} (h0:x0+x2<=x3) (h1:x3<2^32) (h2:x1<x2):x0+x1<2^32:=א
lemma omS15 {x0 x1 x2 x3:ℕ} (h0:x0+x3<=x2) (h1:x1<x3):x0+x1<x2:=א
lemma omS16 {x0 x1 x2 x3:ℕ} (h0:x1+8*x2<=x3) (h1:x3<2^32) (h2:x0<x2):8*x0<2^32:=א
lemma omS17 {x0 x1 x2 x3:ℕ} (h0:x0+8*x2<=x3) (h1:x3<2^32) (h2:x1<x2):x0+8*x1<2^32:=א
lemma omS18 {x0 x1 x2 x3 x4:ℕ} (h0:x2<8) (h1:x0+8*x3<=x4) (h2:x4<2^32) (h3:x1<x3):x0+8*x1+x2<2^32:=א
lemma omS19 {x0 x1 x2:ℕ} (h0:x0+5+8*x1<=x2) (h1:x2<=4294967295):x0+5<2^32:=א
lemma omS23 {x0 x1 x2 x3 x4:ℕ} (h0:x1<8) (h1:x2<x3) (h2:x0=8*x2+1) (h3:8*x3+1<=x4) (h4:x4<2^32):x0+x1<2^32:=by subst_vars;omega
lemma omS24 {x0 x1 x2 x3 x4:ℕ} (h0:x1<8) (h1:x3<x4) (h2:x0=8*x3+1) (h3:8*x4+1<=x2):x0+x1<x2:=by subst_vars;omega
lemma omS27 {x0:ℕ} {x1:ℤ} (h0: ↑(0:ℕ)-((1023:ℤ)+ ↑(52:ℕ))+(1:ℤ)=x1):(2:ℕ)^(52:ℕ)<=x0\/x1=(-1074:ℤ):=by simp_all only[CharP.cast_eq_zero,Nat.cast_ofNat,Int.reduceAdd,zero_sub,Int.reduceNeg,Nat.reducePow,or_true]
lemma omS28 {x0:ℕ}: ↑(x0%(256:ℕ))-(150:ℤ)<=(128:ℤ)/\(-150:ℤ)<= ↑(x0%(256:ℕ))-(150:ℤ):=א
lemma omS29 {x0:ℤ} (h0:x0<=(128:ℤ)/\(-150:ℤ)<=x0):x0<=(300:ℤ):=א
lemma omS30 {x0:ℤ} {x1:ℤ} (h0:(-900:ℤ)<=x0) (h1:x1<=(128:ℤ)/\(-150:ℤ)<=x1):(-1074:ℤ)<=x0+x1:=א
lemma omS31 {x0:ℕ}:x0/2^31%2=0\/x0/2^31%2=1:=א
lemma omS32 {x0:ℕ} (h0:x0<2^32): ¬x0/2^31%2*2^31=0↔2^31<=x0:=א
lemma omS33 {x0 x1:ℕ} (h0:x0=x1%2^31) (h1:(4286578688+x0)%4294967296<=2130706431):2^23<=x0/\x0<2139095040:=by subst_vars;omega
lemma omS34 {x0:ℕ} (h0:255=x0/2^23%2^8) (h1:x0%2^31=0\/
 2^23<=x0%2^31/\x0%2^31<2139095040):False:=א
lemma omS35 {x0:ℕ} (h0:x0%2^31=0):x0/2^23%2^8=0:=א
lemma omS36 {x0:ℕ} (h0:x0%2^31=0):x0%2^23=0:=א
lemma omS37 {x0:ℕ} (h0:x0%2^31=0\/
 2^23<=x0%2^31/\x0%2^31<2139095040) (h1: ¬x0%2^31=0) (h2:0=x0/2^23%2^8):False:=א
lemma omS38 {x0:ℕ}:x0%2^31%2^23=x0%2^23:=א
lemma omS39 {x0:ℕ}:x0%2^31/2^23=x0/2^23%2^8:=א
lemma omS40 {x0:ℕ}:2^23+x0%2^23<=2^24:=א
lemma omS41 {x0:ℕ} (h0:x0<(2139095040:ℕ)): ↑(x0/(2:ℕ)^(23:ℕ))-(150:ℤ)<=(104:ℤ):=א
lemma omS42 {x0:ℕ} (h0:2^23<=x0): ¬x0=0:=א
lemma omS43 {x0 x1:ℕ} (h0:x1<=x0) (h1:2^23<=x1): ¬x0=0:=א
lemma omS44 {x0 x1:ℕ} (h0:x0<=x1) (h1:x0/2^23=x1/2^23):2^23+x0%2^23<=2^23+x1%2^23:=א
lemma omS45 {x0:ℕ} {x1:ℕ} (h0:x0/(2:ℕ)^(23:ℕ)<x1/(2:ℕ)^(23:ℕ)): ↑(x0/(2:ℕ)^(23:ℕ))-(150:ℤ)+(1:ℤ)<= ↑(x1/(2:ℕ)^(23:ℕ))-(150:ℤ):=א
lemma omS46 {x0:ℕ}:2^23<=2^23+x0%2^23:=א
lemma omS47 {x0:ℕ}:2^23+x0%2^23<=2^23*2:=א
lemma omS48 {x0:ℕ}:2^23+x0%2^23<2^53:=א
lemma omS49 {x0:ℕ} (h0:x0<2^32):x0/2^23<512:=א
lemma omS51 {x0:ℕ}:2^23+x0%2^23<=2^128:=א
lemma omS52 {x0:ℕ} (h0:x0%2^31=0):x0/2^23%256=0:=א
lemma omS53 {x0:ℕ}:256<=x0/2^23↔2^31<=x0:=א
lemma omS54 {x0:ℕ}:x0%2^31/2^23=x0/2^23%256:=א
lemma omS55 {x0:ℕ} (h0:2^23<=x0%2^31): ¬x0/2^23%256=0:=א
lemma omS56 {x0:ℕ} (h0:2^23<=x0%2^31): ¬x0%2^31=0:=א
lemma omS57 {x0:ℕ} (h0:x0<2^20):x0+8<18446744073709551616:=א
lemma omS58 {x0:ℕ} (h0:x0<2^20):x0+8<2^53:=א
lemma omS60 {x0:ℕ} (h0:x0<2^20):x0+8<=2^128:=א
lemma omS61 {x0 x1:ℕ} (h0:x1<2^20) (h1:x0<=x1):x0<=2^20:=א
lemma omS62 {x0 x1:ℕ} (h0:x1<2^20) (h1:x0<=x1):x0<=2097152:=א
lemma omS63 {x0:ℕ}:(-900:ℤ)<= ↑(x0/(2:ℕ)^(23:ℕ)%(256:ℕ))-(150:ℤ):=א
lemma omS64 {x0:ℤ} {x1:ℤ} {x2:ℕ} (h0:x1=x0+ ↑(x2+(1:ℕ))):x0<=x1:=by subst_vars;omega
lemma omS65 {x0:ℤ} {x1:ℤ} {x2:ℕ} (h0:x0=x1+ ↑(x2+(1:ℕ))):(x0-x1).toNat=x2+(1:ℕ):=by simp_all only[Nat.cast_add,Nat.cast_one,add_sub_cancel_left,Int.toNat_natCast_add_one,Nat.add_left_cancel_iff]
lemma omS66 {x0 x1 x2:ℕ} (h0:x0<2*x1+x2) (h1:2*x1<=x0):x0-2*x1<x2:=א
lemma omS67 {x0 x1 x2:ℕ} (h0:2*x0<x1+x2) (h1:x1<=2*x0):2*x0-x1<x2:=א
lemma omS68 {x0:ℤ} (h0:(-149:ℤ)<=x0):(-126:ℤ)<=x0+(23:ℤ):=א
lemma omS69 {x0:ℤ} (h0:x0+(150:ℤ)<=(254:ℤ)): ¬(2:ℕ)^(8:ℕ)<=(x0+(150:ℤ)).toNat+(1:ℕ):=by subst_vars;simp only[Nat.reducePow,Nat.reduceLeDiff,not_le,Nat.ofNat_pos,Int.toNat_lt',Nat.cast_ofNat] at*;omega
lemma omS70 {x0:ℤ} (h0:x0+(150:ℤ)<=(254:ℤ)):(x0+(150:ℤ)).toNat<(2:ℕ)^(8:ℕ):=by subst_vars;simp only[Nat.reducePow,Nat.ofNat_pos,Int.toNat_lt',Nat.cast_ofNat] at*;omega
lemma omS71 {x0:ℕ} (h0:2^23<=x0):0<x0:=א
lemma omS72 {x0:ℤ} {x1:ℤ}:x0-x1+(2:ℤ)<= ↑(x0-x1+(2:ℤ)).toNat:=by subst_vars;simp only[Int.ofNat_toNat,le_max_iff,Std.le_refl,true_or]
lemma omS73 {x0:ℤ} {x1:ℕ} {x2:ℤ} (h0:x0-x2+(2:ℤ)<= ↑x1):x0- ↑x1+ ↑((x2-(x0- ↑x1)-(2:ℤ)).toNat+(2:ℕ))=x2:=א
lemma omS74 {x0:ℤ} {x1:ℕ}:x0= ↑x1+(x0- ↑x1):=by subst_vars;simp only[add_sub_cancel]
lemma omS75 {x0:ℤ} {x1:ℕ} {x2:ℤ} {x3:ℕ} (h0:x2- ↑x3+ ↑(x1+(2:ℕ))=x0):x0= ↑(x1+(2:ℕ))+(x2- ↑x3):=by subst_vars;ring
lemma omS76 {x0 x1 x2:ℕ} (h0:x0<4*x2+2*x1) (h1:x2<=(2^24-1)*x1):x0<2^25*x1*2:=א
lemma omS77 {x0:ℕ} {x1:ℤ} {x2:ℤ} (h0:(-149:ℤ)<=x2) (h1:x1+ ↑(x0+(2:ℕ))=x2):Max.max (↑((25:ℕ)+x0)+(1:ℤ)+x1-(24:ℤ)) (-149:ℤ)=x2:=by subst_vars;omega
lemma omS78 {x0 x1:ℕ} (h0:2^25*x0<=x1) (h1:x1<4*(2^23*x0)+2*x0):4*(2^23*x0)<x1+2*x0:=א
lemma omS79 {x0:ℕ} (h0:2^23<=x0) (h1: ¬x0=2^23):2^23+1<=x0:=א
lemma omS80 {x0 x1:ℕ} (h0:33554431*x0<x1):2^24*x0<=x1:=א
lemma omS81 {x0 x1:ℕ} (h0: ¬2^25*x1<=x0):x0<2^24*x1*2:=א
lemma omS82 {x0:ℕ} {x1:ℤ} {x2:ℤ} (h0:x1+ ↑(x0+(2:ℕ))=x2) (h1:(-148:ℤ)<=x2):Max.max (↑((24:ℕ)+x0)+(1:ℤ)+x1-(24:ℤ)) (-149:ℤ)=x2-(1:ℤ):=by subst_vars;omega
lemma omS83 {x0:ℤ} {x1:ℤ} {x2:ℕ} (h0:x1+ ↑(x2+(2:ℕ))=x0):x0-(1:ℤ)=x1+ ↑(x2+(1:ℕ)):=by subst_vars;push_cast;ring
lemma omS84 {x0 x1:ℕ} (h0:33554431*x0<x1):2*(2^24*x0)<x1+x0:=א
lemma omS85 {x0 x1:ℕ} (h0: ¬2^25*x1<=x0):x0<2*(2^24*x1)+x1:=א
lemma omS86 {x0:ℤ} (h0:(-148:ℤ)<=x0):(-126:ℤ)<=x0-(1:ℤ)+(23:ℤ):=א
lemma omS87 {x0:ℤ}:x0-(1:ℤ)+(23:ℤ)-(23:ℤ)=x0-(1:ℤ):=by subst_vars;simp only[add_sub_cancel_right]
lemma omS88 {x0:ℤ}:x0-(1:ℤ)+(23:ℤ)+(1:ℤ)-(23:ℤ)=x0:=by ring
lemma omS89 {x0:ℕ} {x1:ℤ} {x2:ℤ} (h0:(-149:ℤ)<=x2) (h1:x1+ ↑(x0+(2:ℕ))=x2) (h2: ¬(-148:ℤ)<=x2):Max.max (↑((24:ℕ)+x0)+(1:ℤ)+x1-(24:ℤ)) (-149:ℤ)=x2:=by subst_vars;omega
lemma omS90 {x0 x1:ℕ} (h0:33554431*x0<x1):2*(2^23*(2*x0))<x1+2*x0:=א
lemma omS91 {x0 x1:ℕ} (h0:x0<4*(2^23*x1)+2*x1):x0<2*(2^23*(2*x1))+2*x1:=א
lemma omS92 {x0:ℤ} (h0:(-149:ℤ)<=x0):(1:ℤ)<=x0+(150:ℤ):=א
lemma omS93 {x0:ℤ} (h0:x0+(150:ℤ)<=(254:ℤ)):x0+(150:ℤ)<=(254:ℤ):=א
lemma omS94 {x0:ℕ} (h0:x0<2047): ¬x0=2047:=א
lemma omS95 {x0:ℕ} (h0:1<=x0): ¬x0=0:=א
lemma omS96 {x0:ℕ}: ↑x0-(1023:ℤ)-(52:ℤ)= ↑x0-(1075:ℤ):=by ring
lemma omS97 {x0 x1:ℕ} (h0:x0=(x1+268435455)%2^64/2^29):x0<2^35:=by subst_vars;omega
lemma omS98 {x0 x1:ℕ} (h0:897<=x1/2^23%2^11) (h1:x1=(x0+268435455)%2^64/2^29) (h2:2^34<=x1):x0=2^63+(x0-2^63):=by subst_vars;omega
lemma omS99 {x0 x1:ℕ} (h0:x0=(x1+268435455)%2^64/2^29) (h1:2^34<=x0):(x0/2^23%2^11*2^23+x0%2^23)*2^29<
 x1-2^63+2^28:=by subst_vars;omega
lemma omS100 {x0 x1:ℕ} (h0:x0<2^64) (h1:x1=(x0+268435455)%2^64/2^29) (h2:2^34<=x1):x0-2^63<=
 (x1/2^23%2^11*2^23+x1%2^23)*2^29+
   2^28:=by subst_vars;omega
lemma omS101 {x0:ℕ}:x0=0+x0:=by subst_vars;simp only[zero_add]
lemma omS102 {x0 x1:ℕ} (h0:x0=(x1+268435455)%2^64/2^29):(x0/2^23%2^11*2^23+x0%2^23)*2^29<
 x1+2^28:=by subst_vars;omega
lemma omS103 {x0 x1:ℕ} (h0:897<=x1/2^23%2^11) (h1:x0<2^64) (h2:x1=(x0+268435455)%2^64/2^29) (h3: ¬2^34<=x1):x0<=
 (x1/2^23%2^11*2^23+x1%2^23)*2^29+
   2^28:=א
lemma omS107 {x0 x1:ℕ} (h0:x1=x0/2^23%2^11) (h1:(18446744073709550719+x1)%18446744073709551616<254):897<=x0/2^23%2^11:=by subst_vars;omega
lemma omS108 {x0 x1:ℕ} (h0:x1=x0/2^23%2^11) (h1:(18446744073709550719+x1)%18446744073709551616<254):x0/2^23%2^11<1151:=by subst_vars;omega
lemma omS109 {x0:ℕ}:x0%2^23<2^23:=א
lemma omS110 {x0:ℕ} (h0:x0<2^32) (h1:2^31<=x0):x0/2^31=1:=א
lemma omS111 {x0:ℕ} (h0:x0<2^32) (h1:2^31<=x0):x0/2147483648%2=1:=א
lemma omS112 {x0:ℕ} (h0: ¬2^31<=x0):x0/2^31=0:=א
lemma omS113 {x0:ℕ} (h0: ¬2^31<=x0):x0/2147483648%2=0:=א
lemma omS114 {x0:ℕ}:x0=x0/2^52*2^52+x0%2^52:=א
lemma omS115 {x0 x1 x2 x3:ℕ} (h0:x2<2^23) (h1:x3<2^52) (h2:(x1*2^23+x2)*2^29<x0*2^52+x3+2^28) (h3:x0*2^52+x3<=(x1*2^23+x2)*2^29+2^28):x0=x1\/x0+1=x1/\x2=0:=א
lemma omS116 {x0 x1 x2 x3:ℕ} (h0:897<=x1) (h1:x2<2^52) (h2:(x1*2^23+x3)*2^29<x0*2^52+x2+2^28):1<=x0:=א
lemma omS117 {x0 x1 x2 x3:ℕ} (h0:x1<1151) (h1:x2<2^23) (h2:x0*2^52+x3<=(x1*2^23+x2)*2^29+2^28):x0<2047:=א
lemma omS118 {x0 x1 x2:ℕ} (h0:x2*2^52+x0<=(x2*2^23+x1)*2^29+2^28):2*(2^52+x0)<=2^53+x1*2^30+2^29:=א
lemma omS119 {x0 x1 x2:ℕ} (h0:(x2*2^23+x0)*2^29<x2*2^52+x1+2^28):x0≠0->2^53+x0*2^30<2*(2^52+x1)+2^29:=א
lemma omS120 {x0:ℕ}:2^53<2*(2^52+x0)+2^28:=א
lemma omS121 {x0:ℕ} (h0:x0<2^52):2^52+x0<=2^53+0*2^30+2^29:=א
lemma omS122 {x0:ℕ}:0≠0->2^53+0*2^30<2^52+x0+2^29:=א
lemma omS123 {x0 x1:ℕ} (h0:((x1+1)*2^23+0)*2^29<x1*2^52+x0+2^28):2^53<2^52+x0+2^28:=א
lemma omS124 {x0:ℕ} (h0:(897:ℕ)<=x0):(-179:ℤ)<= ↑x0-(1076:ℤ):=א
lemma omS125 {x0:ℕ}:2^23<=2^23+x0:=א
lemma omS126 {x0:ℕ} (h0:x0<2^23):2^23+x0<2^24:=א
lemma omS127 {x0:ℕ} (h0:(897:ℕ)<=x0):(-149:ℤ)<= ↑x0-(1046:ℤ):=א
lemma omS128 {x0:ℕ} (h0:x0<(1151:ℕ)): ↑x0-(1046:ℤ)+(150:ℤ)<=(254:ℤ):=א
lemma omS129 {x0:ℕ} (h0:x0=0):2^23+x0=2^23:=by subst_vars;omega
lemma omS130 {x0:ℕ}:(↑x0-(1046:ℤ)+(150:ℤ)).toNat=x0-(896:ℕ):=א
lemma omS131 {x0:ℕ} (h0:x0<2^23):(2^23+x0)%2^23=x0:=א
lemma omS132 {x0 x1 x2:ℕ} (h0:x1<x2) (h1: ¬x0<x2): ¬x0=x1:=א
lemma omS133 {x0:ℕ} (h0: ѥ(8388607:ὥ)<=x0- ѥ(1:ὥ)) (h1: ѥ(8388607:ὥ)=8388607) (h2: ѥ(8388608:ὥ)=8388608) (h3: ѥ(1:ὥ)=1): ѥ(8388608:ὥ)<=x0:=א
lemma omS134 {x0:ℕ} (h0:x0< ѥ(2139095040:ὥ)) (h1: ѥ(2139095040:ὥ)=2139095040) (h2: ѥ(8388608:ὥ)=8388608) (h3: ѥ(2130706431:ὥ)=2130706431):x0- ѥ(8388608:ὥ)<= ѥ(2130706431:ὥ):=א
lemma omS135 {x0:Prop} {x1:Prop} {x2:ℕ} {x3:ℕ} {x4:ℕ} (h0:x2+(0:ℕ)=x3) (h1:x2<=x4) (h2:x4<x3):x0/\x1:=by subst_vars;simp only[add_zero] at*;omega
lemma omS136 {x0 x1 x2:ℕ} (h0:x0+(x2+1)=x1):x0<x1:=by subst_vars;simp only[lt_add_iff_pos_right,Order.lt_add_one_iff,zero_le]
lemma omS137 {x0 x1 x2:ℕ} (h0:x0+(x1+1)=x2):x0+1+x1=x2:=by subst_vars;simp+arith at*
lemma omS138 {x0 x1:ℕ} (h0:x0<=x1) (h1: ¬x1=x0):x0+1<=x1:=by subst_vars;simp only[Order.add_one_le_iff] at*;omega
lemma omS139 {x0:ℕ}:0+x0=x0:=by subst_vars;simp only[zero_add]
lemma omS140 {x0 x1 x2:ℕ} (h0:x1=x2) (h1:x0<x2):x0<x1:=by subst_vars;omega
lemma omS141 {x0:ℕ} (h0:x0≠0) (h1:8388607<=(2^32-1+x0)%2^32):8388608<=x0:=א
lemma omS142 {x0:ℕ} (h0:x0<=2^31-1):x0%2^31=x0:=א
lemma omS143 {x0 x1 x2:ℕ} (h0:x0+x1<=x2) (h1:x2<2^32):x0<2^32:=by exact Nat.lt_of_le_of_lt (Nat.le_trans (Nat.le_add_right _ _) ‹_›) ‹_›
lemma omS144 {x0 x1 x2 x3:ℕ} (h0:x2<=x1) (h1:x0+(x3+1)=x2):x0<x1:=by subst_vars;omega
lemma omS145 {x0 x1 x2 x3:ℕ} (h0:x2<2^32) (h1:x0+x3<=x2) (h2:x1<x3):x0+x1<2^32:=א
lemma omS146 {x0 x1:ℕ}:x0+1+x1=x0+(x1+1):=by simp+arith at*
lemma omS147 {x0 x1:ℕ} (h0:x1<x0):0<x0:=א
lemma omS148 {x0 x1:ℕ} (h0:x0<x1+1):x0=x1\/x0<x1:=א
lemma omS149 {x0 x1 x2:ℕ} (h0:x0=x2) (h1:x2+x1<2^32):x0+x1<2^32:=by subst_vars;omega
lemma omS150 {x0:ℕ} (h0:x0+3<2^32):x0+1<2^32:=א
lemma omS151 {x0 x1:ℕ} (h0:x1<2^32) (h1:x0+4<=x1):x0+3<2^32:=א
lemma omS152 {x0 x1:ℕ} (h0:x0+4<=x1):x0<x1:=א
lemma omS153 {x0 x1:ℕ} (h0:x0+4<=x1):x0+1<x1:=א
lemma omS154 {x0 x1 x2:ℕ} (h0:x1<2^32) (h1:x0+x2<=x1):x0<2^32:=by exact Nat.lt_of_le_of_lt (Nat.le_trans (Nat.le_add_right _ _) ‹_›) ‹_›
lemma omS155 {x0 x1:ℕ} (h0:x0+0=x1):x0=x1:=א
lemma omS156 {x0 x1 x2 x3 x4 x5:ℕ} (h0:x2<2^32) (h1:x3+x4<=x2) (h2:x0=x3) (h3:x1+(x5+1)=x4):x0+x1<2^32:=by subst_vars;omega
lemma omS157 {x0 x1 x2 x3:ℕ} (h0:x3+(x1+1)=x2) (h1:x0=x3+1):x0+x1=x2:=by subst_vars;simp+arith at*
lemma omS158 {x0 x1:ℕ} (h0:x0<x1):x0≠x1:=א
lemma omS159 {x0 x1:ℕ} (h0:x0<x1):x0≠x1+1:=א
lemma omS160 {x0 x1:ℕ} (h0:x0<x1):x0≠x1+1+1:=א
lemma omS161 {x0 x1:ℕ} (h0:x0<x1):x0≠x1+1+1+1:=by subst_vars;simp+arith at*<;>omega
lemma omS162 {x0 x1:ℕ} (h0:x0<x1):x0≠x1+1+1+1+1:=by subst_vars;simp+arith at*<;>omega
lemma omS163 {x0 x1:ℕ} (h0:x0<x1):x0≠x1+1+1+1+1+1:=by subst_vars;simp+arith at*<;>omega
lemma omS164 {x0 x1:ℕ} (h0:x0<x1):x0≠x1+1+1+1+1+1+1:=by subst_vars;simp+arith at*<;>omega
lemma omS165 {x0 x1:ℕ} (h0:x0<x1):x0≠x1+1+1+1+1+1+1+1:=by subst_vars;simp+arith at*<;>omega
lemma omS166 {x0 x1:ℕ} (h0:x0<x1):x0<x1+1+1+1+1+1+1+1+1:=by subst_vars;simp+arith at*<;>omega
lemma omS167 {x0 x1 x2 x3:ℕ} (h0:x2<8) (h1:x0=x1+8*(x3-x3)+x2):x0<x1+1+1+1+1+1+1+1+1:=by simp_all+arith
lemma omS168 {x0 x1 x2 x3:ℕ} (h0:x0=x1+8*(x3-x3)+x2):x0=x1+x2:=by simp_all+arith
lemma omS169 {x0 x1 x2 x3 x4:ℕ} (h0:x3<=x2) (h1:x0=x1+8*(x2-x3)+x4) (h2: ¬x2=x3):x0=
 x1+1+1+1+1+1+1+1+1+8*(x2-(x3+1))+
  x4:=by subst_vars;simp+arith at*<;>omega
lemma omS170 {x0 x1 x2:ℕ} (h0:x2<x0) (h1:x1<=x2):x0-(x1+1)<x0-x1:=א
lemma omS171 {x0 x1 x2:ℕ} (h0:x0<x1):x0<x1+8*(x2-0):=א
lemma omS172 {x0 x1 x2 x3 x4:ℕ} (h0:x3<x2) (h1:x4<8) (h2:x0=x1+0+8*x3+x4):x0<x1+8*(x2-0):=by subst_vars;simp only[add_zero,tsub_zero] at*;omega
lemma omS173 {x0 x1 x2 x3:ℕ} (h0:x0=x1+0+8*x2+x3):x0=x1+8*(x2-0)+x3:=א
lemma omS174 {x0 x1:ℕ} (h0:x1<=x0) (h1: ¬x0=x1):x0-x1=x0-(x1+1)+1:=א
lemma omS175 {x0 x1:ℕ} (h0:x1-x0=x1-(x0+1)+1):x0+1<=x1:=by subst_vars;simp only[Order.add_one_le_iff] at*;omega
lemma omS176 {x0 x1 x2 x3 x4 x5:ℕ} (h0:x0=x1+(x3+8*x2)+8*x4+x5):x0=x1+8*(x2-0)+x3+8*x4+x5:=by subst_vars;simp+arith at*
lemma omS177 {x0 x1:ℕ} (h0:x1<x0):x0-x1=x0-(x1+1)+1:=א
lemma omS178 {x0 x1:ℕ} (h0: ¬x1<x0):x0-x1=0:=by simp_all+arith
lemma omS179 {x0:ℕ} (h0:x0%8=0):8*(x0/8)=x0:=א
lemma omS180 {x0 x1:ℕ} (h0:x0+8<=x1):x0/8<x1/8:=א
lemma omS181 {x0 x1 x2:ℕ}:x0+5+8*x1+x2=5+x0+8*x1+x2:=by simp+arith at*
lemma omS182 {x0 x1:ℕ} (h0:x0+8<=x1):x0/8+1<=x1/8:=א
lemma omS183 {x0 x1 x2 x3:ℕ} (h0:x1<x3) (h1:x2<8):x0+8*x1+x2<x0+8*x3:=א
lemma omS184 {x0 x1 x2 x3:ℕ} (h0:x2<2^32) (h1:x1<x3) (h2:x0+x3<=x2):x0+x1<2^32:=א
lemma omS185 {x0:ℕ} (h0:x0<2^32):x0<=4294967295:=א
lemma omS186 {x0 x1:ℕ} (h0:x1<2^32) (h1:x0+4<=x1):x0<2^32:=by exact Nat.lt_of_le_of_lt (Nat.le_trans (Nat.le_add_right _ _) ‹_›) ‹_›
lemma omS187 {x0 x1:ℕ} (h0:x0+4<=x1):x0+1<=x1:=by subst_vars;simp only[Order.add_one_le_iff] at*;omega
lemma omS188 {x0 x1:ℕ} (h0:x0+4<=x1):x0+1+1<=x1:=by subst_vars;simp only[Order.add_one_le_iff] at*;omega
lemma omS189 {x0 x1:ℕ} (h0:x0+4<=x1):x0+2+1<=x1:=by subst_vars;exact le_of_lt ‹_›
lemma omS190 {x0 x1:ℕ} (h0:x0+4<=x1):x0+3+1<=x1:=א
lemma omS191 {x0 x1:ℕ} (h0:x1<=x0):x0=x1+(x0-x1):=by simp_all+arith
lemma omS192 {x0 x1 x2:ℕ} (h0:x0<=x1) (h1:x1+4<=x2):x0+(x1-x0)<x2:=א
lemma omS193 {x0 x1:ℕ} (h0:x1<=x0):x0+1=x1+(x0+1-x1):=by subst_vars;exact (Nat.add_sub_of_le (Nat.le_trans ‹_› (Nat.le_add_right _ _))).symm
lemma omS194 {x0 x1 x2:ℕ} (h0:x0<=x1) (h1:x1+4<=x2):x0+(x1+1-x0)<x2:=א
lemma omS195 {x0 x1:ℕ} (h0:x1<=x0):x0+2=x1+(x0+2-x1):=by subst_vars;exact (Nat.add_sub_of_le (Nat.le_trans ‹_› (Nat.le_add_right _ _))).symm
lemma omS196 {x0 x1 x2:ℕ} (h0:x0<=x1) (h1:x1+4<=x2):x0+(x1+2-x0)<x2:=א
lemma omS197 {x0 x1:ℕ} (h0:x1<=x0):x0+3=x1+(x0+3-x1):=by subst_vars;exact (Nat.add_sub_of_le (Nat.le_trans ‹_› (Nat.le_add_right _ _))).symm
lemma omS198 {x0 x1 x2:ℕ} (h0:x0<=x1) (h1:x1+4<=x2):x0+(x1+3-x0)<x2:=א
lemma omS199 {x0 x1 x2:ℕ} (h0:x0-x1=x2) (h1:x1+4<=x0):x0-(x1+4)<x2:=by subst_vars;omega
lemma omS200 {x0 x1:ℕ} (h0:x0<=x1):x0<=x1+4:=א
lemma omS201 {x0 x1:ℕ} (h0:x1<2^32) (h1:x0<x1):x0<2^32:=א
lemma omS202 {x0 x1:ℕ} (h0:x0<x1):x0+1<=x1:=by subst_vars;assumption
lemma omS203 {x0 x1 x2:ℕ} (h0:x0<=x1) (h1:x1<x2):x0+(x1-x0)<x2:=by simp_all+arith
lemma omS204 {x0 x1 x2:ℕ} (h0:x0-x1=x2) (h1:x1<x0):x0-(x1+1)<x2:=by subst_vars;exact Nat.sub_lt_sub_left ‹_› (Nat.lt_succ_self _)
lemma omS205 {x0 x1:ℕ} (h0:x0<=x1):x0<=x1+1:=א
lemma omS206 {x0 x1 x2:ℕ} (h0:x1=8*(x2/8)) (h1:x1+x0<x2):x0<8:=by subst_vars;omega
lemma omS207 {x0 x1:ℕ} (h0:x0=8*(x1/8)) (h1:x1<2^32):x0<2^32:=by subst_vars;omega
lemma omS208 {x0 x1:ℕ} (h0:x0=8*(x1/8)):x0<=x1:=by subst_vars;omega
lemma omS209 {x0 x1 x2 x3:ℕ} (h0:x0+x3<x2):Min.min (x0+x1) (x2-1)+1<=x2:=by subst_vars;simp only[Order.add_one_le_iff,min_lt_iff,tsub_lt_self_iff,Order.lt_one_iff,and_true] at*;omega
lemma omS210 {x0 x1 x2:ℕ} (h0:x0+x1<x2):x0+x1=Min.min (x0+x1) (x2-1):=by subst_vars;simp only[left_eq_inf] at*;omega
lemma omS211 {x0 x1:ℕ} (h0:x1<2^32) (h1:x0+8<=x1):x0<2^32:=by exact Nat.lt_of_le_of_lt (Nat.le_trans (Nat.le_add_right _ _) ‹_›) ‹_›
lemma omS212 {x0 x1:ℕ} (h0:x1<2^32) (h1:x0+8<=x1):x0+4<2^32:=א
lemma omS213 {x0 x1:ℕ} (h0:x1<2^32) (h1:x0+8<=x1):x0+3<2^32:=א
lemma omS214 {x0 x1:ℕ} (h0:x1<2^32) (h1:x0+8<=x1):x0+4+3<2^32:=א
lemma omS215 {x0 x1:ℕ} (h0:x0+8<=x1):x0+1<=x1:=by subst_vars;simp only[Order.add_one_le_iff] at*;omega
lemma omS216 {x0 x1:ℕ} (h0:x0+8<=x1):x0+1+1<=x1:=by subst_vars;simp only[Order.add_one_le_iff] at*;omega
lemma omS217 {x0 x1:ℕ} (h0:x0+8<=x1):x0+2+1<=x1:=by subst_vars;simp only[Order.add_one_le_iff] at*;omega
lemma omS218 {x0 x1:ℕ} (h0:x0+8<=x1):x0+3+1<=x1:=by subst_vars;simp only[Order.add_one_le_iff] at*;omega
lemma omS219 {x0 x1:ℕ} (h0:x0+8<=x1):x0+4+1<=x1:=by subst_vars;simp only[Order.add_one_le_iff] at*;omega
lemma omS220 {x0 x1:ℕ} (h0:x0+8<=x1):x0+4+1+1<=x1:=by subst_vars;simp only[Order.add_one_le_iff] at*;omega
lemma omS221 {x0:ℕ}:x0+5=x0+4+1:=by subst_vars;rfl
lemma omS222 {x0:ℕ}:x0+6=x0+4+2:=by subst_vars;rfl
lemma omS223 {x0:ℕ}:x0+7=x0+4+3:=by subst_vars;rfl
lemma omS224 {x0 x1:ℕ} (h0:x0+8<=x1):x0+4<=x1:=א
lemma omS226 {x0 x1 x2:ℕ} (h0:x0-x1=x2) (h1:x1+8<=x0):x0-(x1+8)<x2:=by subst_vars;omega
lemma omS227 {x0:ℕ} (h0:x0%8=0):(x0+8)%8=0:=א
lemma omS228 {x0 x1:ℕ} (h0:x0%8=0) (h1: ¬x0+8<=x1) (h2:x0<x1):x0=8*(x1/8):=א
lemma omS229 {x0 x1:ℕ} (h0:x1<x0) (h1:x1=8*(x0/8)):x0%8≠0:=by subst_vars;omega
lemma omS230 {x0 x1 x2 x3 x4:ℕ} (h0:x0<2^64) (h1:(x0+2097152)%2^64=x1+x4) (h2:(2^64-2097152+x0)%2^64=x1+x2) (h3:x3<x4+2^28) (h4:x4<=x3+2^28) (h5:x3<x2+2^28) (h6:x2<=x3+2^28):x0=x1+(x2+2097152)/\
 x3+2097152<x2+2097152+2^28/\
  x2+2097152+2097152<=x3+2^28:=א
lemma omS231 {x0 x1 x2:ℕ} (h0:x2+x1+2097152<x2+x0+2^28) (h1:x2+x0+2097152<=x2+x1+2^28):x0+2097152<=x1+2^28/\x1+2097152<x0+2^28:=א
lemma omS232 {x0 x1:ℕ} (h0:897<=x0):2^61<=(x0*2^23+x1)*2^29:=א
lemma omS233 {x0 x1:ℕ} (h0:x0<1151) (h1:x1<2^23):(x0*2^23+x1)*2^29<2^63-2^60:=א
lemma omS234 {x0 x1 x2 x3:ℕ} (h0:x2<2^23) (h1:x3<2^52) (h2:(x1*2^23+x2)*2^29+2097152<x0*2^52+x3+2^28) (h3:x0*2^52+x3+2097152<=(x1*2^23+x2)*2^29+2^28):x0=x1\/x0+1=x1/\x2=0:=א
lemma omS235 {x0:ℕ} {x1:ℕ} {x2:ℕ} {x3:ℕ} (h0:(897:ℕ)<=x1) (h1:x2<(2:ℕ)^(52:ℕ)) (h2:(x1*(2:ℕ)^(23:ℕ)+x3)*(2:ℕ)^(29:ℕ)+(2097152:ℕ)<x0*(2:ℕ)^(52:ℕ)+x2+(2:ℕ)^(28:ℕ)):(-1021:ℤ)<= ↑x0-(1075:ℤ):=א
lemma omS236 {x0:ℕ} (h0:x0<2^52):2^52+x0<=2^53:=א
lemma omS237 {x0 x1:ℕ} (h0:((x1+1)*2^23+0)*2^29+2097152<
 x1*2^52+x0+2^28):2^52+2097152<x0+2^28:=א
lemma omS238 {x0 x1 x2 x3:ℕ} (h0:897<=x1) (h1:x2<2^52) (h2:(x1*2^23+x3)*2^29+2097152<x0*2^52+x2+2^28):1<=x0:=א
lemma omS239 {x0 x1 x2 x3:ℕ} (h0:x1<1151) (h1:x2<2^23) (h2:x0*2^52+x3+2097152<=(x1*2^23+x2)*2^29+2^28):x0<2047:=א
lemma omS240 {x0 x1 x2 x3:ℕ} (h0:(x0*2^23+x1)*2^29+2097152<x2*2^52+x3+2^28):(x0*2^23+x1)*2^29+2097152<x2*2^52+x3+2^28:=א
lemma omS241 {x0 x1 x2 x3:ℕ} (h0:x0*2^52+x1+2097152<=(x2*2^23+x3)*2^29+2^28):x0*2^52+x1+2097152<=(x2*2^23+x3)*2^29+2^28:=א
lemma omS242 {x0 x1:ℕ} (h0:x0=x1/2^52%2^11) (h1:(18446744073709551615+x0)%18446744073709551616<1900):1<=x0/\x0<=1900:=by subst_vars;omega
lemma omS243 {x0 x1 x2:ℕ} (h0:1<=x0/\x0<=1900) (h1:x2<2^52) (h2:(x0+35)*9007199254740992%18446744073709551616<=2*(x1*2^52+x2)):x0+35<=x1:=א
lemma omS244 {x0:ℕ} (h0:1<=x0/\x0<=1900):x0<2047:=א
lemma omS245 {x0 x1 x2:ℕ} (h0:(18446744073709551615+x1)%18446744073709551616<1900) (h1:x2<2^52) (h2:(x1+35)*9007199254740992%18446744073709551616<=2*(x0*2^52+x2)):1<=x0:=א
lemma omS246 {x0 x1:ℕ} (h0:x0*2^52+x1<2047*2^52):x0<2047:=א
lemma omS247 {x0:ℕ} {x1:ℕ} (h0:x0+(35:ℕ)<=x1): ↑x0+(35:ℤ)-(1075:ℤ)<= ↑x1-(1075:ℤ):=by subst_vars;simp only[tsub_le_iff_right,sub_add_cancel] at*;omega
lemma omS248 {x0:ℕ}:2^52<=2^52+x0:=א
lemma omS249 {x0:ℕ} (h0:x0<2047*2^52) (h1:18446744073709551615<=2*x0):False:=א
lemma omS250 {x0 x1:ℕ} (h0:x0+2097152<x1+2^28) (h1:x1+2097152<=x0+2^28):x0<x1+2^28/\x1<=x0+2^28:=א
lemma omS251 {x0 x1 x2:ℕ} (h0:x1<1151) (h1:x2<2^23) (h2:x0+2097152<=(x1*2^23+x2)*2^29+2^28):x0<2047*2^52:=א
lemma omS252 {x0:ℕ} (h0:x0<(1151:ℕ)): ↑x0-(1076:ℤ)<=(74:ℤ):=א
lemma omS253 {x0 x1:ℕ} (h0:x0<10349271943697399808) (h1:(18446744073709551616-x0+x1)%18446744073709551616<
 (18446744073709551616-x0+10349271943697399808)%18446744073709551616):x0<=x1:=א
lemma omS254 {x0 x1:ℕ} (h0:x1<10349271943697399808) (h1:(18446744073709551616-x1+x0)%18446744073709551616<
 (18446744073709551616-x1+10349271943697399808)%18446744073709551616) (h2:x0<18446744073709551616):x0<10349271943697399808:=א
lemma omS255 {x0:ℕ} (h0:x0*2%18446744073709551616<10349271943697399808) (h1:x0<18446744073709551616):x0+266338303<18446744073709551616:=א
lemma omS256 {x0:ℕ} (h0:x0*2%18446744073709551616<10349271943697399808) (h1:x0<18446744073709551616) (h2:(x0+266338303)%536870912<532676608):(x0+2097152+268435455)%18446744073709551616/536870912=
 (x0+266338303)/536870912:=א
lemma omS257 {x0:ℕ} (h0:x0*2%18446744073709551616<10349271943697399808) (h1:x0<18446744073709551616):(18446744073707454464+x0+268435455)%18446744073709551616/536870912=
 (x0+266338303)/536870912:=א
lemma omS258 {x0:ℕ} (h0:8088464930757410816<=x0*2%18446744073709551616) (h1:x0*2%18446744073709551616<10349271943697399808):898<=(x0+266338303)/536870912/8388608%2048:=א
lemma omS259 {x0:ℕ} (h0:x0*2%18446744073709551616<10349271943697399808):(x0+266338303)/536870912/8388608%2048<1151:=א
lemma omS260 {x0:ℕ} (h0:x0*2%18446744073709551616<10349271943697399808) (h1:x0<18446744073709551616):(x0+266338303)/536870912<34359738368:=א
lemma omS261 {x0:ℕ} (h0:898<=x0/8388608%2048) (h1:x0/8388608%2048<1151):(18446744073709550719+x0/8388608%2048)%18446744073709551616<254:=א
lemma omS262 {x0 x1:ℕ} (h0:x0< Ѧ(8088464930757410816:Ὠ)) (h1: Ѧ(8088464930757410816:Ὠ)<=x1):x0<=x1:=א
lemma omS263 {x0 x1:ℕ} (h0: ¬x1< Ѧ(8088464930757410816:Ὠ)) (h1:x1<=x0): Ѧ(8088464930757410816:Ὠ)<=x0:=א
lemma omS264 {x0 x1 x2:ℕ} (h0:x1<1151) (h1:x2<2^23) (h2:x0+2097152<=(x1*2^23+x2)*2^29+2^28):x0<2^63:=א
lemma omS265 {x0 x1:ℕ} (h0:x1<2^52) (h1:8088464930757410816<=2*(x0*2^52+x1)):898<=x0:=א
lemma omS266 {x0:ℕ} (h0:898<=x0):1<=x0:=א
lemma omS267 {x0:ℕ} (h0:(898:ℕ)<=x0):(-125:ℤ)<=(52:ℤ)+(↑x0-(1075:ℤ)):=א
lemma omS268 {x0 x1:ℕ} (h0:x0<=x1) (h1:x1<2^20):x0<=2^20:=א
lemma omS269 {x0 x1 x2 x3:ℕ} (h0:x0+x1<=x2) (h1:x2=x3) (h2:x3<2^32):x0<2^32:=by subst_vars;omega
lemma omS270 {x0 x1 x2 x3:ℕ} (h0:x0+x1<=x3) (h1:x3=x2):x0+x1<=x2:=by subst_vars;omega
lemma omS271 {x0 x1 x2:ℕ} (h0:x1<2^32) (h1:x2+(x0+3)<x1):x0<2^32:=א
lemma omS272 {x0 x1 x2:ℕ} (h0:x1<2^32) (h1:x2+(x0+3)<x1):x0+3<2^32:=א
lemma omS273 {x0 x1 x2:ℕ} (h0:x2<2^32) (h1:x0+(x1+3)<x2):x0+x1<2^32:=א
lemma omS274 {x0 x1 x2:ℕ} (h0:x1+x0=x2) (h1:x1+4<=x2):x0-4<x0:=by subst_vars;simp only[add_le_add_iff_left,tsub_lt_self_iff,Nat.ofNat_pos,and_true] at*;omega
lemma omS275 {x0 x1 x2:ℕ} (h0:x0+x1=x2) (h1:x0+4<=x2):x0+4+(x1-4)=x2:=by subst_vars;simp_all+arith
lemma omS276 {x0 x1:ℕ}:x0+(x1+4)=x0+x1+4:=by subst_vars;rfl
lemma omS277 {x0 x1:ℕ} (h0:x0+4<=x1):x0+0<x1:=by subst_vars;simp only[add_zero] at*;omega
lemma omS278 {x0 x1 x2:ℕ} (h0:x1<2^32) (h1:x2+x0<x1):x0<2^32:=א
lemma omS279 {x0 x1 x2:ℕ} (h0:x2<2^32) (h1:x0+x1<x2):x0+x1<2^32:=א
lemma omS280 {x0 x1 x2:ℕ} (h0:x1+x0=x2) (h1:x1<x2):x0-1<x0:=by subst_vars;simp only[lt_add_iff_pos_right,tsub_lt_self_iff,Order.lt_one_iff,and_true] at*;omega
lemma omS281 {x0 x1 x2:ℕ} (h0:x0+x1=x2) (h1:x0<x2):x0+1+(x1-1)=x2:=by subst_vars;simp_all+arith
lemma omS282 {x0 x1 x2:ℕ} (h0:x0+x2=x1) (h1: ¬x0<x1):x0=x1:=by subst_vars;simp only[lt_add_iff_pos_right,not_lt,nonpos_iff_eq_zero,Nat.left_eq_add] at*;omega
lemma omS283 {x0 x1:ℕ} (h0:x0+x1<2^32):x0<2^32:=א
lemma omS284 {x0 x1:ℕ} (h0:x1<2^32) (h1:8*x0+9<=x1):8*x0<2^32:=by exact Nat.lt_of_le_of_lt (Nat.le_trans (Nat.le_add_right _ _) ‹_›) ‹_›
lemma omS285 {x0 x1:ℕ} (h0:x1<2^32) (h1:8*x0+9<=x1):8*x0+1<2^32:=א
lemma omS286 {x0 x1:ℕ} (h0:8*x0+9<=x1):8*x0+1+7<x1:=by subst_vars;assumption
lemma omS287 {x0 x1:ℕ} (h0:8*x0+9<=x1):8*x0+1+6<x1:=א
lemma omS288 {x0 x1:ℕ} (h0:8*x0+9<=x1):8*x0+1+5<x1:=א
lemma omS289 {x0 x1:ℕ} (h0:8*x0+9<=x1):8*x0+1+4<x1:=א
lemma omS290 {x0 x1:ℕ} (h0:8*x0+9<=x1):8*x0+1+3<x1:=א
lemma omS291 {x0 x1:ℕ} (h0:8*x0+9<=x1):8*x0+1+2<x1:=א
lemma omS292 {x0 x1:ℕ} (h0:8*x0+9<=x1):8*x0+1+1<x1:=א
lemma omS293 {x0 x1:ℕ} (h0:8*x0+9<=x1):8*x0+1+0<x1:=by subst_vars;simp only[add_zero] at*;omega
lemma omS294 {x0 x1 x2 x3 x4:ℕ} (h0:x0=8*x2+1) (h1:x2<x3) (h2:x4<2^32) (h3:8*x3+1<=x4) (h4:x1<8):x0+x1<2^32:=by subst_vars;omega
lemma omS295 {x0 x1:ℕ} (h0:x1<=128) (h1:x0+1=x1*2^23):x0<1073741824:=א
lemma omS296 {x0 x1:ℕ} (h0:2<=x1) (h1:x1<=128) (h2:x0+1=x1*2^23):8388608<=x0%2^31:=א
lemma omS297 {x0 x1:ℕ} (h0:x1<=128) (h1:x0+1=x1*2^23):x0%2^31<1073741824:=א
lemma omS298 {x0 x1:ℕ} (h0:x1<=128) (h1:x0+1=x1*2^23):x0<=2147483647:=א
lemma omS299 {x0 x1 x2:ℕ} (h0:x0<=x1/2^23%256/\x1/2^23%256<x2):x0*8388608<=x1%2^31:=א
lemma omS300 {x0 x1 x2 x3:ℕ} (h0:x1+1=x2*2^23) (h1:x3<=x0/2^23%256/\x0/2^23%256<x2):x0%2^31<=x1:=א
lemma omS309 {x0 x1:ℕ} (h0:x0<x1+1) (h1: ¬x0<x1):x0=x1:=א
lemma omS312 {x0 x1:ℕ} (h0:x1<1073741824) (h1:x0<=x1):x0<2139095040:=א
lemma omS313 {x0 x1 x2 x3 x4:ℕ} (h0:x2<2^32) (h1:x0+x3<=x2) (h2:x1+(x4+1)=x3):x0+x1<2^32:=by subst_vars;omega
lemma omS314 {x0 x1 x2 x3 x4:ℕ} (h0:8*x1+1<=x2) (h1:x2<2^32) (h2:x0=8*x3+1) (h3:x3+(x4+1)=x1):x0+8<2^32:=by subst_vars;omega
lemma omS315 {x0 x1:ℕ} (h0:x0=8*x1+1):x0+8=8*(x1+1)+1:=by subst_vars;rfl
lemma omS316 {x0 x1 x2 x3:ℕ} (h0:x2+x1<2^32) (h1:x2=x3+x0):x0+x1<2^32:=by subst_vars;omega
lemma omS317 {x0 x1 x2 x3:ℕ} (h0:x3+x2<2^32) (h1:x3=x0+x1):x0+(x1+x2)<2^32:=by subst_vars;omega
lemma omS318 {x0 x1:ℕ} (h0:x1<8):x0+x1+1<=x0+8:=by subst_vars;simp+arith at*<;>omega
lemma omS319 {x0 x1 x2:ℕ} (h0:x1<x2+8) (h1:x1=x2+x0):x0<8:=by subst_vars;simp only[add_lt_add_iff_left] at*;omega
lemma omS320 {x0 x1 x2 x3 x4:ℕ} (h0:x3<2^32) (h1:x4+x2<x3) (h2:x4=x0+x1):x0+(x1+x2)<2^32:=by subst_vars;omega
lemma omS321 {x0 x1 x2 x3:ℕ} (h0:x2<2^32) (h1:x3+x1<x2) (h2:x3=x0):x0+x1<2^32:=by subst_vars;omega
lemma omS322 {x0 x1:ℕ} (h0:1*x0+x1<2^32):x0+x1<2^32:=א
lemma omS323 {x0 x1 x2:ℕ} (h0:x1=x2) (h1:x2<2^32) (h2:x0<=x1):x0<2^32:=by subst_vars;omega
lemma omS324 {x0 x1 x2:ℕ} (h0:x1<2^32) (h1:x0+(x2+1)=x1):x0<2^32:=by subst_vars;omega
lemma omS325 {x0 x1:ℕ} (h0:x1+8<=x0):1<=x0:=א
lemma omS326 {x0 x1 x2:ℕ} (h0:x0+8<=x2) (h1:x1<8):x0+x1<x2:=א
lemma omS329 {x0 x1:ℕ}:x0+(x1+1)=x0+x1+1:=by subst_vars;rfl
lemma omS330 {x0 x1:ℕ}:x0+1+x1=x0+x1+1:=by simp+arith at*
lemma omS332 {x0 x1:ℕ} (h0:1*x0=x1):0*x0+x0<=x1:=by subst_vars;simp only[zero_mul,zero_add,one_mul,Std.le_refl]
lemma omS333 {x0:ℕ}:0*x0<2^32:=א
lemma omS334 {x0 x1:ℕ} (h0:x0<1*x1):x0<0*x1+x1:=by subst_vars;simp only[one_mul,zero_mul,zero_add] at*;omega
lemma omS335 {x0:ℕ}: ↑x0-(1046:ℤ)= ↑(30:ℕ)+(↑x0-(1076:ℤ)):=by ring
lemma omS336 {x0:ℕ} (h0:x0%2^31=0\/
 2^23<=x0%2^31/\x0%2^31<2139095040):x0%2^31=0\/2^23<=x0%2^31:=א
lemma omS337 {x0:ℕ}:x0%2^52<2^52:=א
lemma omS338 {x0 x1:ℕ} (h0:x0/2^52=x1):x0=x1*2^52+x0%2^52:=א
lemma omS339 {x0:ℕ}: ↑x0-(1023:ℤ)= ↑(52:ℕ)+(↑x0-(1075:ℤ)):=by ring
lemma omS340 {x0:ℕ}: ↑x0-(1022:ℤ)= ↑(53:ℕ)+(↑x0-(1075:ℤ)):=by ring
lemma omS341 {x0 x1:ℕ} (h0:x0+x1=1023):x0<2047:=א
lemma omS342 {x0:ℕ} {x1:ℕ} (h0:x0+x1=(1023:ℕ)): ↑x0-(1075:ℤ)=-((52:ℤ)+ ↑x1):=א
lemma omS343 {x0 x1:ℕ} (h0:x0/2^52=x1) (h1:x1<2047):x0<2^63:=by subst_vars;omega
lemma omS344 {x0:ℕ} (h0:x0<1151) (h1: ¬x0<=1149):x0=1150:=א
lemma omS345 {x0:ℕ} (h0:x0<=(1149:ℕ)): ↑x0-(1076:ℤ)<=(73:ℤ):=by subst_vars;simp only[tsub_le_iff_right,Int.reduceAdd,Nat.cast_le_ofNat] at*;omega
lemma omS346 {x0 x1 x2 x3 x4 x5:ℕ} (h0:x2<2^32) (h1:x0=x3) (h2:x3+x4<=x2) (h3:x1+(x5+1)=x4):x0+x1<2^32:=by subst_vars;omega
lemma omS347 {x0 x1 x2 x3 x4:ℕ} (h0:x1<2^32) (h1:x2+x3<=x1) (h2:x0+(x4+1)=x3):x0+1<2^32:=by subst_vars;omega
lemma omS348 {x0 x1 x2 x3 x4:ℕ} (h0:x0+x3<=x2) (h1:x1+(x4+1)=x3):x0+x1<x2:=by subst_vars;omega
lemma omS349 {x0:ℕ} (h0:x0<=1895825407): ¬2^31<=x0:=א
lemma omS350 {x0:ℕ} (h0:528482304<=x0) (h1:x0<=1895825407):528482304<=x0%2^31:=א
lemma omS351 {x0:ℕ} (h0:x0<2^20):x0+8<=2^21:=א
lemma omS352 {x0:ℕ} (h0:x0<(2:ℕ)^(20:ℕ)):(↑x0+(8:ℤ)).natAbs<(2:ℕ)^(53:ℕ):=by subst_vars;simp only[Nat.reducePow] at*;omega
lemma omS353 {x0:ℕ} (h0:x0<(1:ℕ)+(1222:ℕ)): ↑x0-(1022:ℤ)<=(200:ℤ):=by subst_vars;simp only[Nat.reduceAdd,tsub_le_iff_right,Int.reduceAdd,Nat.cast_le_ofNat] at*;omega
lemma omS354 {x0:ℕ} (h0:(972:ℕ)<=x0):(-51:ℤ)<= ↑x0-(1023:ℤ):=א
lemma omS355 {x0:ℕ} (h0:x0<(972:ℕ)+(2:ℕ)): ↑x0-(1022:ℤ)<=(-49:ℤ):=by subst_vars;simp only[Nat.reduceAdd,Int.reduceNeg,tsub_le_iff_right,Int.reduceAdd,Nat.cast_le_ofNat] at*;omega
lemma omS356 {x0:ℕ} (h0:x0<=(1222:ℕ)): ↑x0-(1022:ℤ)<=(200:ℤ):=by subst_vars;simp only[tsub_le_iff_right,Int.reduceAdd,Nat.cast_le_ofNat] at*;omega
lemma omS357 {x0 x1:ℕ} (h0:2*(x0*2^52+x1)<10349271943697399808):x0<=1148:=א
lemma omS358 {x0:ℕ} (h0:x0<=1148):x0<2047:=א
lemma omS359 {x0:ℕ} (h0:x0<=(1148:ℕ)):(53:ℤ)+(↑x0-(1075:ℤ))<=(126:ℤ):=א
lemma omS360 {x0 x1:ℕ} (h0:x1<2^32) (h1:x0=x1):x0<2^32:=by subst_vars;omega
lemma omS361 {x0 x1:ℕ} (h0:x1<2^20) (h1:x0=x1):x0<2^32:=by subst_vars;omega
lemma omS362 {x0 x1:ℕ} (h0:x0=x1) (h1:x1<2^20):x0<2^32:=by subst_vars;omega
lemma omS363 {x0:ℕ} (h0:x0<2^20):x0<2^32:=א
lemma omS364 {x0:ℕ} (h0:x0*2%18446744073709551616<10349271943697399808) (h1:x0<18446744073709551616):x0+268435453<18446744073709551616:=א
lemma omS365 {x0:ℕ} (h0:x0*2%18446744073709551616<10349271943697399808) (h1:x0<18446744073709551616) (h2:(x0+268435453)%536870912<536870908):(x0+2+268435455)%18446744073709551616/536870912=
 (x0+268435453)/536870912:=א
lemma omS366 {x0:ℕ} (h0:x0*2%18446744073709551616<10349271943697399808) (h1:x0<18446744073709551616):(18446744073709551614+x0+268435455)%18446744073709551616/536870912=
 (x0+268435453)/536870912:=א
lemma omS367 {x0:ℕ} (h0:8088464930757410816<=x0*2%18446744073709551616) (h1:x0*2%18446744073709551616<10349271943697399808):898<=(x0+268435453)/536870912/8388608%2048:=א
lemma omS368 {x0:ℕ} (h0:x0*2%18446744073709551616<10349271943697399808):(x0+268435453)/536870912/8388608%2048<1151:=א
lemma omS369 {x0:ℕ} (h0:x0*2%18446744073709551616<10349271943697399808) (h1:x0<18446744073709551616):(x0+268435453)/536870912<34359738368:=א
lemma omS370 {x0 x1:ℕ} (h0:x0<x1): ¬x0=x1:=א
lemma omS371 {x0 x1:ℕ} (h0: ¬x1<x0) (h1: ¬x1=x0):x0<x1:=א
lemma omS372 {x0 x1 x2 x3:ℕ} (h0:x2<2^32) (h1:x0+x1+x3<x2):x0+x1<2^32:=א
lemma omS373 {x0 x1 x2:ℕ} (h0:x0<2^32):x0-(x1+x2)<2^32:=א
lemma omS374 {x0 x1:ℕ} (h0:x0<x1+1) (h1: ¬x0=x1):x0<x1:=א
lemma omS375 {x0 x1:ℕ}:x0+(x1+1)=x0+1+x1:=by simp+arith at*
lemma omS376 {x0 x1:ℕ} (h0:x1<x0):x0>0:=א
lemma omS377 {x0 x1 x2:ℕ} (h0:x2+x0<x2+x1):x0<x1:=by subst_vars;simp only[add_lt_add_iff_left] at*;omega
lemma omS378 {x0 x1 x2:ℕ}:x0+(x1+2*x2)=x0+x2+x1+x2:=by simp+arith at*
lemma omS379 {x0 x1:ℕ} (h0:x1+2*x0<2^32):x0<2^32:=א
lemma omS380 {x0 x1:ℕ} (h0:x0+2*x1<2^32):x0<2^32:=א
lemma omS381 {x0 x1 x2:ℕ} (h0:x0<x2):x0<x1+2*x2:=א
lemma omS382 {x0 x1 x2 x3 x4:ℕ} (h0:x2<x1):x0<x1\/x2<x1\/x0-x1>=x3\/x2-x1>=x4:=by simp_all+arith
lemma omS383 {x0 x1 x2:ℕ} (h0:x1<x2):x0+x1<x2+2*x0:=by subst_vars;simp+arith at*<;>omega
lemma omS384 {x0 x1 x2 x3 x4:ℕ} (h0:x1<=x0) (h1:x0-x1<x3) (h2:x2<x4): ¬(x0<x1\/x1+x2<x1\/x0-x1>=x3\/x1+x2-x1>=x4):=by simp_all+arith
lemma omS385 {x0 x1 x2 x3:ℕ} (h0:x1<2^32) (h1:x1=x2) (h2:x0+x3<=x2):x0<2^32:=by subst_vars;omega
lemma omS386 {x0 x1 x2:ℕ} (h0:x2<x0):x0+(x1+x2)<x1+2*x0:=by subst_vars;simp+arith at*<;>omega
lemma omS387 {x0 x1 x2 x3 x4:ℕ}:x0<x1\/x1+(x2+x3)<x1\/x0-x1>=x4\/x1+(x2+x3)-x1>=x2:=by subst_vars;simp only[add_lt_iff_neg_left,not_lt_zero,ge_iff_le,add_tsub_cancel_left,le_add_iff_nonneg_right,zero_le,or_true]
lemma omS388 {x0 x1 x2 x3 x4:ℕ} (h0: ¬(x0<x1\/x3<x1\/x0-x1>=x2\/x3-x1>=x4)):x0-x1<x2:=by simp_all+arith
lemma omS389 {x0 x1 x2 x3 x4:ℕ} (h0: ¬(x1<x0\/x2<x0\/x1-x0>=x3\/x2-x0>=x4)):x0<=x1:=by simp_all+arith
lemma omS390 {x0 x1 x2 x3:ℕ} (h0:x0=x2) (h1:x1=x3) (h2:x2+x3<2^32):x0+x1<2^32:=by subst_vars;omega
lemma omS391 {x0 x1 x2:ℕ}:x0+(x1+x2)=x0+x1+x2:=by simp+arith at*
lemma omS392 {x0 x1 x2 x3 x4:ℕ} (h0:x1+x2<=x3) (h1:x3<2^32) (h2:x4+x0<x2):x0<2^32:=א
lemma omS393 {x0 x1 x2 x3 x4:ℕ} (h0:x1+x2<=x3) (h1:x3<2^32) (h2:x4+x0<=x2):x0<2^32:=א
lemma omS394 {x0 x1 x2 x3 x4:ℕ} (h0:x0+x3<=x4) (h1:x4<2^32) (h2:x1+x2<=x3):x0+x1+x2<2^32:=א
lemma omS395 {x0 x1:ℕ} (h0:x0<2^32):x0-x1<2^32:=א
lemma omS396 {x0 x1 x2:ℕ}:x0<x1-x2↔x2+x0<x1:=א
lemma omS397 {x0 x1 x2 x3 x4:ℕ} (h0:x0+x4<=x3) (h1:x1+x2<x4):x0+(x1+x2)<x3:=א
lemma omS398 {x0 x1 x2:ℕ} (h0: ¬x1+x2<x0):x0<=x1+x2+1:=א
lemma omS399 {x0 x1 x2:ℕ} (h0:x0-x1=x2) (h1:x1<x0):x0-(x1+8)<x2:=by subst_vars;omega
lemma omS400 {x0 x1:ℕ} (h0: ¬x1<x0):x0<=x1:=by subst_vars;simp only[not_lt] at*;omega
lemma omS401 {x0 x1 x2 x3:ℕ} (h0:x1+(x3+1)=x2):x0+x1<x0+x2:=by subst_vars;simp only[add_lt_add_iff_left,lt_add_iff_pos_right,Order.lt_add_one_iff,zero_le]
lemma omS402 {x0 x1 x2 x3:ℕ} (h0:x1<=x2) (h1:x2<2^32) (h2:x3+x0<=x1):x0<2^32:=א
lemma omS403 {x0 x1 x2 x3:ℕ} (h0:x2<=x3) (h1:x3<2^32) (h2:x0+x1<=x2):x0+x1<2^32:=א
lemma omS404 {x0 x1:ℕ} (h0:x0+0=x1): ¬x0<x1:=by subst_vars;simp only[add_zero,lt_self_iff_false,not_false_eq_true]
lemma omS405 {x0 x1 x2:ℕ} (h0:x0=x2) (h1:x2<=x1):x0<x1+1:=by subst_vars;simp only[Order.lt_add_one_iff] at*;omega
lemma omS406 {x0 x1 x2 x3:ℕ} (h0:x2=x1) (h1:x0+(x3+1)=x2):x0<x1:=by subst_vars;simp only[lt_add_iff_pos_right,Order.lt_add_one_iff,zero_le]
lemma omS407 {x0 x1:ℕ} (h0: ¬x0=x1) (h1: ¬x0<x1):x0-x1=x0-x1-1+1:=א
lemma omS408 {x0 x1 x2:ℕ} (h0:x1<2^32) (h1:x2+x0<=x1):x0<2^32:=א
lemma omS409 {x0 x1 x2:ℕ} (h0:x0=x1+x2):x0+1=x1+(x2+1):=by subst_vars;rfl
lemma omS410 {x0:ℕ} {x1:ℕ} {x2:ℕ} {x3:Prop} {x4:Prop} (h0:x0-x1=x2) (h1:x1<x0/\x3/\x4):x0-(x1+(8:ℕ))<x2:=by subst_vars;omega
lemma omS411 {x0 x1:ℕ} (h0:x1-x0=0): ¬x0<x1:=א
lemma omS412 {x0 x1 x2:ℕ} (h0:x1-x0=x2+1):x0<x1:=א
lemma omS413 {x0 x1 x2:ℕ} (h0:x0-x1=x2+1):x0-(x1+1)=x2:=א
lemma omS414 {x0 x1 x2 x3:ℕ} (h0:x1=x3) (h1:x2=x3) (h2:x0<=x3):x0<=x1/\x0<=x2:=by subst_vars;simp only[and_self] at*;omega
lemma omS415 {x0 x1 x2 x3:ℕ} (h0:x2+x3<=x1) (h1:x0=x2+x3):x0<=x1:=by subst_vars;omega
lemma omS416 {x0 x1 x2:ℕ} (h0: ¬x0<x1) (h1:x0<x1+x2):x0-x1<x2:=א
lemma omS417 {x0 x1:ℕ} (h0:x0=x1):x0<x1+1:=by subst_vars;simp only[lt_add_iff_pos_right,Order.lt_one_iff]
lemma omS418 {x0 x1:ℕ} (h0:x0<x1):x0<x1+1:=א
lemma omS419 {x0 x1:ℕ} (h0: ¬x0=x1) (h1: ¬x0<x1): ¬x0<x1+1:=א
lemma omS420 {x0 x1 x2 x3 x4:ℕ} (h0:x2<=x3) (h1:x3<2^32) (h2:x0+x1+x4+x1<=x2):x0+x1<2^32:=א
lemma omS421 {x0 x1 x2 x3 x4:ℕ} (h0:x4<=x3) (h1:x0+x1+x2+x1<=x4):x0+x1+x2<=x3:=א
lemma omS422 {x0 x1:ℕ} (h0:x0+7<=x1):x0<=x1:=א
lemma omS423 {x0 x1 x2:ℕ} (h0:x2+7<=x1) (h1:x0<x2):x0<x1:=א
lemma omS424 {x0:ℕ} (h0:0<x0):x0≠0:=א
lemma omS425 {x0:ℕ}:x0-x0/2^39*2^39=x0%2^39:=א
lemma omS426 {x0:ℤ} (h0:(-148:ℤ)<=x0+(39:ℤ)):Max.max (↑(62:ℕ)+(1:ℤ)+x0-(24:ℤ)) (-149:ℤ)=x0+(39:ℤ):=by subst_vars;simp only[Int.reduceNeg,Nat.cast_ofNat,Int.reduceAdd] at*;omega
lemma omS427 {x0:ℤ} (h0:x0<=(104:ℤ)):x0+(150:ℤ)<=(254:ℤ):=א
lemma omS429 {x0:ℤ}:x0<=x0+(39:ℤ):=by subst_vars;simp only[le_add_iff_nonneg_right,Nat.ofNat_nonneg]
lemma omS430 {x0:ℤ}:(x0+(39:ℤ)-x0).toNat=(39:ℕ):=by subst_vars;simp only[add_sub_cancel_left,Int.reduceToNat]
lemma omS431 {x0:ℤ} (h0:(-148:ℤ)<=x0+(39:ℤ)):(-126:ℤ)<=x0+(39:ℤ)+(23:ℤ):=א
lemma omS432 {x0:ℤ}:x0+(39:ℤ)+(23:ℤ)-(23:ℤ)=x0+(39:ℤ):=by subst_vars;simp only[add_sub_cancel_right]
lemma omS433 {x0:ℤ}:x0+(39:ℤ)+(23:ℤ)+(1:ℤ)-(23:ℤ)=x0+(40:ℤ):=by ring
lemma omS434 {x0:ℤ} (h0:(-148:ℤ)<=x0+(39:ℤ)):(-149:ℤ)<=x0+(40:ℤ):=א
lemma omS435 {x0:ℤ} (h0:x0+(39:ℤ)+(150:ℤ)<=(253:ℤ)):x0+(40:ℤ)<=(104:ℤ):=א
lemma omS436 {x0:ℤ} (h0:(-148:ℤ)<=x0+(39:ℤ)):(x0+(40:ℤ)+(150:ℤ)).toNat=(x0+(39:ℤ)+(149:ℤ)).toNat+(2:ℕ):=א
lemma omS437 {x0:ℤ}:((x0+(39:ℤ)+(149:ℤ)).toNat+(2:ℕ))*(2:ℕ)^(23:ℕ)+(2:ℕ)^(23:ℕ)%(2:ℕ)^(23:ℕ)=
 (x0+(39:ℤ)+(149:ℤ)).toNat*(2:ℕ)^(23:ℕ)+(2:ℕ)^(24:ℕ):=א
lemma omS438 {x0:ℕ} (h0:x0<=2^24) (h1: ¬x0=2^24):x0<2^24:=א
lemma omS439 {x0:ℤ} (h0:(-148:ℤ)<=x0+(39:ℤ)):(-149:ℤ)<=x0+(39:ℤ):=by subst_vars;exact le_of_lt ‹_›
lemma omS440 {x0:ℤ} (h0:x0+(39:ℤ)+(150:ℤ)<=(253:ℤ)):x0+(39:ℤ)<=(104:ℤ):=א
lemma omS441 {x0:ℤ} (h0:(-148:ℤ)<=x0+(39:ℤ)):(x0+(39:ℤ)+(150:ℤ)).toNat=(x0+(39:ℤ)+(149:ℤ)).toNat+(1:ℕ):=א
lemma omS442 {x0:ℤ} {x1:ℕ} (h0:(2:ℕ)^(23:ℕ)<=x1) (h1:x1<=(2:ℕ)^(24:ℕ)) (h2: ¬x1=(2:ℕ)^(24:ℕ)):((x0+(39:ℤ)+(149:ℤ)).toNat+(1:ℕ))*(2:ℕ)^(23:ℕ)+x1%(2:ℕ)^(23:ℕ)=
 (x0+(39:ℤ)+(149:ℤ)).toNat*(2:ℕ)^(23:ℕ)+x1:=א
lemma omS452 {x0:ℕ} (h0:x0<2^32) (h1:0=x0%2147483648):x0=0\/x0=2147483648:=א
lemma omS453 {x0 x1:ℕ} (h0:x1%2^31=0\/
 2^23<=x1%2^31/\x1%2^31<2139095040) (h1:x1%2^31≠0) (h2:x0=x1/2^23%2^8):1<=x0:=by subst_vars;omega
lemma omS454 {x0 x1:ℕ} (h0:x1%2^31=0\/
 2^23<=x1%2^31/\x1%2^31<2139095040) (h1:x0=x1/2^23%2^8):x0<=254:=by subst_vars;omega
lemma omS455 {x0 x1:ℕ} (h0:x0=x1%2^23):x0<2^23:=by subst_vars;omega
lemma omS456 {x0 x1:ℕ} (h0:x0=x1%2^23):x0+8388608<2^32:=by subst_vars;omega
lemma omS457 {x0:ℕ}:x0+8388608=2^23+x0:=א
lemma omS458 {x0 x1:ℕ} (h0:x0=x1%2^23):2^23+x0<=2^24:=by subst_vars;omega
lemma omS459 {x0 x1:ℕ} (h0:x0=x1%2^23):2^23+x0<=2^24-1:=by subst_vars;omega
lemma omS460 {x0:ℕ} {x1:ℕ} (h0:x1<=x0): ↑x0-(150:ℤ)= ↑(x0-x1)+(↑x1-(150:ℤ)):=by simp_all only[Nat.cast_sub,sub_add_sub_cancel]
lemma omS461 {x0 x1:ℕ} (h0:x1=2^61-1) (h1:x0<2^61):x0<=x1:=by subst_vars;omega
lemma omS462 {x0 x1 x2 x3:ℕ} (h0:x2+x3<2^63) (h1:x0<=x2) (h2:x1<2^61):x0+x1<2^64:=א
lemma omS463 {x0 x1 x2 x3:ℕ} (h0:x3=2^61-1) (h1:x0<=x2) (h2:x1<2^61):x0+x1<=x2+x3:=by subst_vars;omega
lemma omS464 {x0 x1 x2:ℕ} (h0:x0<=x1):x0<=x1+x2:=א
lemma omS465 {x0 x1 x2:ℕ} (h0:x1<x2) (h1:x0=x2-x1):x0≠0:=by subst_vars;omega
lemma omS466 {x0 x1 x2:ℕ} (h0:x1<2^63) (h1:x0=x1-x2):x0<2^63:=by subst_vars;omega
lemma omS467 {x0 x1 x2:ℕ} (h0:x1≠x2) (h1:x2<=x1) (h2:x0=x1-x2):x0≠0:=by subst_vars;omega
lemma omS468 {x0 x1:ℕ} (h0:x0<=255) (h1:x1<=62):x0+x1<2^64:=א
lemma omS469 {x0 x1:ℕ} (h0: ¬x0+x1<= Ѧ(24:Ὠ)) (h1: Ѧ(24:Ὠ)=24):24<x0+x1:=by subst_vars;simp only[UInt64.reduceToNat,not_le] at*;omega
lemma omS470 {x0 x1:ℕ} (h0: ¬Ѧ(277:Ὠ)<=x0+x1) (h1: Ѧ(277:Ὠ)=277):x0+x1<277:=by subst_vars;simp only[UInt64.reduceToNat,not_le] at*;omega
lemma omS471 {x0 x1 x2 x3 x4:ℕ} (h0:x0<2^64) (h1:(x0+2)%2^64=x1+x4) (h2:(2^64-2+x0)%2^64=x1+x2) (h3:x3<x4+2^28) (h4:x4<=x3+2^28) (h5:x3<x2+2^28) (h6:x2<=x3+2^28):x0=x1+(x2+2)/\
 x3+2<x2+2+2^28/\x2+2+2<=x3+2^28:=א
lemma omS472 {x0 x1 x2:ℕ} (h0:x2+x1+2<x2+x0+2^28) (h1:x2+x0+2<=x2+x1+2^28):x0+2<=x1+2^28/\x1+2<x0+2^28:=א
lemma omS473 {x0 x1 x2 x3:ℕ} (h0:x2<2^23) (h1:x3<2^52) (h2:(x1*2^23+x2)*2^29+2<x0*2^52+x3+2^28) (h3:x0*2^52+x3+2<=(x1*2^23+x2)*2^29+2^28):x0=x1\/x0+1=x1/\x2=0:=א
lemma omS474 {x0:ℕ} {x1:ℕ} {x2:ℕ} {x3:ℕ} (h0:(897:ℕ)<=x1) (h1:x2<(2:ℕ)^(52:ℕ)) (h2:(x1*(2:ℕ)^(23:ℕ)+x3)*(2:ℕ)^(29:ℕ)+(2:ℕ)<x0*(2:ℕ)^(52:ℕ)+x2+(2:ℕ)^(28:ℕ)):(-1021:ℤ)<= ↑x0-(1075:ℤ):=א
lemma omS475 {x0 x1:ℕ} (h0:((x1+1)*2^23+0)*2^29+2<
 x1*2^52+x0+2^28):2^52+2<x0+2^28:=א
lemma omS476 {x0 x1 x2 x3:ℕ} (h0:897<=x1) (h1:x2<2^52) (h2:(x1*2^23+x3)*2^29+2<x0*2^52+x2+2^28):1<=x0:=א
lemma omS477 {x0 x1 x2 x3:ℕ} (h0:x1<1151) (h1:x2<2^23) (h2:x0*2^52+x3+2<=(x1*2^23+x2)*2^29+2^28):x0<2047:=א
lemma omS478 {x0 x1 x2 x3:ℕ} (h0:(x0*2^23+x1)*2^29+2<x2*2^52+x3+2^28):(x0*2^23+x1)*2^29+2<x2*2^52+x3+2^28:=א
lemma omS479 {x0 x1 x2 x3:ℕ} (h0:x0*2^52+x1+2<=(x2*2^23+x3)*2^29+2^28):x0*2^52+x1+2<=(x2*2^23+x3)*2^29+2^28:=א
lemma omS480 {x0:ℕ}:(-160:ℤ)<= ↑(x0%(2:ℕ)^(31:ℕ)/(2:ℕ)^(23:ℕ))-(150:ℤ):=א
lemma omS481 {x0:ℕ}:2^23+x0%2^31%2^23<2^25:=א
lemma omS482 {x0:ℤ} {x1:ℤ} (h0:(-160:ℤ)<=x0) (h1:(-160:ℤ)<=x1):(-1074:ℤ)<=x0+x1:=א
lemma omS483 {x0 x1 x2:ℕ} (h0:x0+0=x2): ¬(x0<=x1/\x1<x2):=by subst_vars;simp only[add_zero,not_and,not_lt,imp_self]
lemma omS484 {x0 x1 x2:ℕ} (h0:x0<=x2) (h1:x1=x2):x0<=x1:=by subst_vars;omega
lemma omS485 {x0 x1:ℕ}: ¬(x0+1<=x0/\x0<x1):=by subst_vars;simp only[add_le_iff_nonpos_right,nonpos_iff_eq_zero,one_ne_zero,false_and,not_false_eq_true]
lemma omS486 {x0 x1 x2:ℕ} (h0:x0+(x2+1)=x1):x0<=x0/\x0<x1:=by subst_vars;simp only[Std.le_refl,lt_add_iff_pos_right,Order.lt_add_one_iff,zero_le,and_self]
lemma omS487 {x0 x1 x2:ℕ} (h0: ¬x1=x0):x0+1<=x1/\x1<x2↔x0<=x1/\x1<x2:=by subst_vars;simp only[Order.add_one_le_iff,and_congr_left_iff] at*;omega
lemma omS488 {x0:ℕ} (h0: ¬x0/2^23%256=0/\ ¬x0/2^23%256=255):8388608<=x0%2^31:=א
lemma omS489 {x0:ℕ} (h0: ¬x0/2^23%256=0/\ ¬x0/2^23%256=255):x0%2^31<2139095040:=א
lemma omS490 {x0 x1:ℕ} (h0:x0+2<x1+2^28) (h1:x1+2<=x0+2^28):x0<x1+2^28/\x1<=x0+2^28:=א
lemma omS491 {x0:ℕ}: ¬x0<x0+0:=by subst_vars;simp only[add_zero,lt_self_iff_false,not_false_eq_true]
lemma omS492 {x0 x1 x2:ℕ} (h0:x0+(x2+1)=x1):x0+1<=x1:=by subst_vars;simp only[add_le_add_iff_left,le_add_iff_nonneg_left,zero_le]
lemma omS493 {x0 x1 x2:ℕ} (h0:x1=x2) (h1:x1<2^32) (h2:x0<=x2):x0<2^32:=by subst_vars;omega
lemma omS494 {x0 x1 x2 x3 x4:ℕ} (h0:x1=x2) (h1:x1<2^32) (h2:x3<=x2) (h3:x3=x0+x4):x0<2^32:=by subst_vars;omega
structure OS where
 (cin h w cout kh kw stride pad:Nat)
def OS.ho (s:OS):Nat:=(s.h+2*s.pad-s.kh)/s.stride+1
def OS.wo (s:OS):Nat:=(s.w+2*s.pad-s.kw)/s.stride+1
@[inline] def windowAtO (s:OS) (inp:ё) (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 conv2dWithO (dot:DotFn) (s:OS) (relu:Ὡ) (inp wts b:ё):ё:=
 let k:=s.cin*s.kh*s.kw
 Array.ofFn (n:=s.cout*(s.ho*s.wo)) λq=>
  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) (λt=>if t<k then windowAtO s inp y x t else oneW)
   (λt=>if t<k then wts[o*k+t]! else b[o]!)
def conv2dSpecO:=conv2dWithO dotOf
@[specialize] def depthwiseWithO (dot:DotFn) (s:OS) (relu:Ὡ) (inp wts b:ё):ё:=
 let k:=s.kh*s.kw
 Array.ofFn (n:=s.cin*(s.ho*s.wo)) λq=>
  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) (λt=>if t<k then windowAtO s inp y x (c*k+t) else oneW)
   (λt=>if t<k then wts[c*k+t]! else b[c]!)
def depthwiseSpecO:=depthwiseWithO dotOf
@[inline] def poolAtO (h w k:Nat) (inp:ё) (ch y x t:Nat):Word:=
 inp[(ch*h+y*k+t/k)*w+x*k+t%k]!
@[specialize] def avgpoolWithO (dot:DotFn) (c h w k:Nat) (inp:ё):ё:=
 Array.ofFn (n:=c*(h/k*(w/k))) λq=>
  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) (poolAtO h w k inp ch y x) (λ_=>oneW)
  toWord (Model.div (ofW s) (ofW (natW (k*k))))
def avgpoolSpecO:=avgpoolWithO dotOf
def maxpoolSpecO (c h w k:Nat) (inp:ё):ё:=
 Array.ofFn (n:=c*(h/k*(w/k))) λq=>
  let ch:=q.val/(h/k*(w/k))
  let y:=q.val/(w/k)%(h/k)
  let x:=q.val%(w/k)
  toWord<| (ἄ (k*k)).foldl (λacc t=>
   Model.maximum acc (ofW (poolAtO h w k inp ch y x t))) (ofW (poolAtO h w k inp ch y x 0))
namespace C
lemma zw_mant {w:ὥ} (h:w&&&В=0):mantOf w=0:=by
 have h1:(w&&&В).toNat=0:=by rw[h];rfl
 rw[Ἧ] at h1
 have h3:(В:ὥ).toNat=2^31-1:=rfl
 rw[h3,Ἂ] at h1
 have hl:=w.toNat_lt
 unfold mantOf expOf fracOf
 have he:(w>>>23)&&&0xFF=0:=by
  apply UInt32.toNat_inj.mp
  rw[Ἧ,UInt32.toNat_shiftRight]
  have h3:(0xFF:ὥ).toNat=2^8-1:=rfl
  have h4:(23:ὥ).toNat%32=23:=rfl
  rw[h3,h4,Ἂ,Ἱ]
  show _=0
  exact omS52 h1
 rw[ἱ he]
 rw[Ἧ]
 have h5:(0x7FFFFF:ὥ).toNat=2^23-1:=rfl
 rw[h5,Ἂ]
 omega
def xLo (X:ѐ) (base i:Nat):Nat->Nat
 | 0=>i
 | n+1=>if X[base+i]!&&&В==0 then xLo X base (i+1) n else i
def xHi (X:ѐ) (base:Nat):Nat->Nat
 | 0=>0
 | n+1=>if X[base+n]!&&&В==0 then xHi X base n else n+1
lemma xLo_spec (X:ѐ) (base:Nat): ∀n i xx,i<=xx->xx<xLo X base i n->mantOf X[base+xx]! =0:=by
 intro n
 induction n with
 | zero=>intro i xx h1 h2;simp only[xLo] at h2;omega
 | succ n ih=>
  intro i xx h1 h2
  simp only[xLo] at h2
  split at h2
  · rename_i hz
    by_cases hx:xx=i
    · subst hx;exact zw_mant (by simpa only[beq_iff_eq] using hz)
    · exact ih (i+1) xx (by exact omS138 h1 hx) h2
  · omega
lemma xHi_spec (X:ѐ) (base:Nat): ∀n xx,xHi X base n<=xx->xx<n->mantOf X[base+xx]! =0:=by
 intro n
 induction n with
 | zero=>intro xx _ h2;omega
 | succ n ih=>
  intro xx h1 h2
  simp only[xHi] at h1
  split at h1
  · rename_i hz
    by_cases hx:xx=n
    · subst hx;exact zw_mant (by simpa only[beq_iff_eq] using hz)
    · exact ih xx h1 (by exact omS374 h2 hx)
  · omega
lemma xHi_le (X:ѐ) (base:Nat): ∀n,xHi X base n<=n:=by
 intro n
 induction n with
 | zero=>simp only[xHi,Std.le_refl]
 | succ n ih=>simp only[xHi];split<;>omega
lemma push2_getD (a:ї) (u v j:Nat):
  ((a.push u).push v).getD j 0=if j<a.size then a.getD j 0 else if j=a.size then u else if j=a.size+1 then v else 0:=by
 simp only[Array.getD,Array.size_push]
 by_cases h1:j<a.size
 · rw[Ἄ א,ἱ h1,Ἄ h1];simp only[Array.getInternal_eq_getElem,Array.getElem_push,Array.size_push,show j<a.size+1 by exact omS418 h1, ↓reduceDIte,h1]
 · rw[ἐ h1]
   by_cases h2:j=a.size
   · subst h2;rw[Ἄ א,ἱ rfl];simp only[Array.getInternal_eq_getElem,Array.getElem_push,Array.size_push,lt_add_iff_pos_right,Order.lt_one_iff, ↓reduceDIte,lt_self_iff_false]
   · rw[ἐ h2]
     by_cases h3:j=a.size+1
     · subst h3;rw[Ἄ א,ἱ rfl];simp only[Array.getInternal_eq_getElem,Array.getElem_push,Array.size_push,lt_self_iff_false, ↓reduceDIte]
     · rw[ἐ h3,ἔ א]
@[inline] def pixZ (s:OS) (X:ѐ) (c yy xx:Nat):Ὡ:=
 decide (yy<s.pad) || decide (xx<s.pad) || decide (s.h<=yy-s.pad) || decide (s.w<=xx-s.pad) ||
  mantOf X[(c*s.h+(yy-s.pad))*s.w+(xx-s.pad)]! ==0
@[inline] def rowE (X:ѐ) (w pad r:Nat):Nat×Nat:=
 let lo:=xLo X (r*w) 0 w
 if lo=w then (w+2*pad,w+2*pad) else (pad+lo,w+pad-xHi X (r*w) w)
def rowsE (X:ѐ) (w pad:Nat) (r:Nat):Nat->ї->ї
 | 0,a=>a
 | n+1,a=>let p:=rowE X w pad r;rowsE X w pad (r+1) n ((a.push p.1).push p.2)
def hullE (s:OS) (RE:ї) (c y0:Nat) (i:Nat) (L G:Nat):Nat->Nat×Nat
 | 0=>(L,G)
 | n+1=>
  if y0+i<s.pad\/s.h<=y0+i-s.pad then hullE s RE c y0 (i+1) L G n
  else hullE s RE c y0 (i+1) (min L (RE.getD (2*(c*s.h+(y0+i-s.pad))) 0))
   (min G (RE.getD (2*(c*s.h+(y0+i-s.pad))+1) 0)) n
def Sound3 (s:OS) (X:ѐ) (dw:Ὡ) (ho:Nat) (e:ї):Prop:=
 ∀c y xx i,c<s.cin->y<ho->i<s.kh->
  (xx<e.getD (2*((if dw then c else 0)*ho+y)) 0\/s.w+2*s.pad-e.getD (2*((if dw then c else 0)*ho+y)+1) 0<=xx)->
  pixZ s X c (y*s.stride+i) xx=true
def ex3G (s:OS) (RE:ї) (ho:Nat) (c y:Nat):Nat->ї->ї
 | 0,a=>a
 | n+1,a=>let p:=hullE s RE c (y*s.stride) 0 (s.w+2*s.pad) (s.w+2*s.pad) s.kh
   ex3G s RE ho c (y+1) n ((a.push p.1).push p.2)
def ex3D (s:OS) (RE:ї) (ho:Nat) (c:Nat):Nat->ї->ї
 | 0,a=>a
 | n+1,a=>ex3D s RE ho (c+1) n (ex3G s RE ho c 0 ho a)
lemma pixZ_pad (s:OS) (X:ѐ) (c yy xx:Nat) (h:s.w+2*s.pad<=xx):pixZ s X c yy xx=true:=by
 unfold pixZ;simp only[Bool.or_eq_true,decide_eq_true_eq];exact Or.inl (Or.inr א)
def RowOk (X:ѐ) (w pad R:Nat) (e:ї):Prop:=
 ∀r xx,r<R->(xx<e.getD (2*r) 0\/w+2*pad-e.getD (2*r+1) 0<=xx)->
  xx<pad\/w<=xx-pad\/mantOf X[r*w+(xx-pad)]! =0
lemma rowE_ok (X:ѐ) (w pad r xx:Nat) (h:xx<(rowE X w pad r).1\/w+2*pad-(rowE X w pad r).2<=xx):
  xx<pad\/w<=xx-pad\/mantOf X[r*w+(xx-pad)]! =0:=by
 by_cases q1:xx<pad
 · exact Or.inl q1
 by_cases q2:w<=xx-pad
 · exact Or.inr (Or.inl q2)
 refine Or.inr (Or.inr ?_)
 by_cases hl:xLo X (r*w) 0 w=w
 · simp only[rowE,hl, ↓reduceIte] at h
   exact xLo_spec X _ w 0 (xx-pad) (Nat.zero_le _) א
 · simp only[rowE,hl, ↓reduceIte] at h
   have hhl:=xHi_le X (r*w) w
   rcases h with h | h
   · exact xLo_spec X _ w 0 (xx-pad) (Nat.zero_le _) א
   · exact xHi_spec X _ w (xx-pad) א א
lemma rowsE_spec (X:ѐ) (w pad:Nat): ∀n r (a:ї),a.size=2*r->
  (rowsE X w pad r n a).size=2*(r+n)/\(∀j,j<a.size->(rowsE X w pad r n a).getD j 0=a.getD j 0)/\
  ∀r' xx,r<=r'->r'<r+n->(xx<(rowsE X w pad r n a).getD (2*r') 0\/w+2*pad-(rowsE X w pad r n a).getD (2*r'+1) 0<=xx)->
   xx<pad\/w<=xx-pad\/mantOf X[r'*w+(xx-pad)]! =0:=by
 intro n
 induction n with
 | zero=>intro r a hs;simp only[rowsE];exact ⟨א, λ_ _=>by simp only, λr' _ h1 h2=>א⟩
 | succ n ih=>
  intro r a hs
  simp only[rowsE]
  have hs2:((a.push (rowE X w pad r).1).push (rowE X w pad r).2).size=2*(r+1):=by simp only[Array.size_push];omega
  obtain ⟨a1,a2,a3⟩:=ih (r+1) _ hs2
  refine ⟨by rw[a1];omega, λj hj=>?_, λr' xx h1 h2 h3=>?_⟩
  · rw[a2 j (by simp only[Array.size_push,Order.lt_add_one_iff];omega),push2_getD,ἱ א]
  · by_cases hr:r'=r
    · subst hr
      rw[a2 _ (by simp only[Array.size_push,Order.lt_add_one_iff];omega),a2 _ (by simp only[Array.size_push,Order.lt_add_one_iff,add_le_add_iff_right];omega),push2_getD,push2_getD] at h3
      rw[ἐ א,ἱ א] at h3
      rw[ἐ א,ἐ א,ἱ א] at h3
      exact rowE_ok X w pad r' xx h3
    · exact a3 r' xx (by exact omS138 h1 hr) (by exact omS7 h2) h3
lemma hullE_spec (s:OS) (X:ѐ) (RE:ї) (hR:RowOk X s.w s.pad (s.cin*s.h) RE) (c y0:Nat) (hc:c<s.cin):
  ∀n i L G,(hullE s RE c y0 i L G n).1<=L/\(hullE s RE c y0 i L G n).2<=G/\
  ∀i' xx,i<=i'->i'< i+n->(xx<(hullE s RE c y0 i L G n).1\/s.w+2*s.pad-(hullE s RE c y0 i L G n).2<=xx)->
   pixZ s X c (y0+i') xx=true:=by
 intro n
 induction n with
 | zero=>intro i L G;simp only[hullE];exact ⟨Nat.le_refl _,Nat.le_refl _, λi' xx h1 h2 _=>א⟩
 | succ n ih=>
  intro i L G
  simp only[hullE]
  split
  · rename_i hp
    obtain ⟨a1,a2,a3⟩:=ih (i+1) L G
    refine ⟨a1,a2, λi' xx h1 h2 h3=>?_⟩
    by_cases hi:i'=i
    · subst hi;unfold pixZ;simp only[Bool.or_eq_true,decide_eq_true_eq]
      rcases hp with hp | hp
      · exact Or.inl (Or.inl (Or.inl (Or.inl hp)))
      · exact Or.inl (Or.inl (Or.inr hp))
    · exact a3 i' xx (by exact omS138 h1 hi) (by exact omS7 h2) h3
  · rename_i hp
    obtain ⟨a1,a2,a3⟩:=ih (i+1) (min L (RE.getD (2*(c*s.h+(y0+i-s.pad))) 0))
     (min G (RE.getD (2*(c*s.h+(y0+i-s.pad))+1) 0))
    refine ⟨Nat.le_trans a1 (Nat.min_le_left _ _),Nat.le_trans a2 (Nat.min_le_left _ _), λi' xx h1 h2 h3=>?_⟩
    by_cases hi:i'=i
    · subst hi
      have hm1:=Nat.min_le_right L (RE.getD (2*(c*s.h+(y0+i'-s.pad))) 0)
      have hm2:=Nat.min_le_right G (RE.getD (2*(c*s.h+(y0+i'-s.pad))+1) 0)
      have hrow:c*s.h+(y0+i'-s.pad)<s.cin*s.h:=by
       have:(c+1)*s.h<=s.cin*s.h:=Ἒ _ (by exact omS202 hc)
       rw[ὰ,Ὑ] at this;omega
      have r:=hR _ xx hrow (h3.imp (λh=>א) (λh=>א))
      unfold pixZ;simp only[Bool.or_eq_true,decide_eq_true_eq,beq_iff_eq]
      rcases r with r | r | r
      · exact Or.inl (Or.inl (Or.inl (Or.inr r)))
      · exact Or.inl (Or.inr r)
      · exact Or.inr r
    · exact a3 i' xx (by exact omS138 h1 hi) (by exact omS7 h2) h3
lemma ex3G_spec (s:OS) (X:ѐ) (RE:ї) (hR:RowOk X s.w s.pad (s.cin*s.h) RE) (ho c:Nat) (hc:c<s.cin):
  ∀n y (a:ї),a.size=2*(c*ho+y)->y+n<=ho->
  (ex3G s RE ho c y n a).size=2*(c*ho+y+n)/\
  (∀j,j<a.size->(ex3G s RE ho c y n a).getD j 0=a.getD j 0)/\
  ∀y' xx i,y<=y'->y'<y+n->i<s.kh->
   (xx<(ex3G s RE ho c y n a).getD (2*(c*ho+y')) 0\/s.w+2*s.pad-(ex3G s RE ho c y n a).getD (2*(c*ho+y')+1) 0<=xx)->
   pixZ s X c (y'*s.stride+i) xx=true:=by
 intro n
 induction n with
 | zero=>intro y a hs _;simp only[ex3G];exact ⟨א, λ_ _=>by simp only, λy' _ _ h1 h2=>א⟩
 | succ n ih=>
  intro y a hs hy
  simp only[ex3G]
  have hs2:((a.push (hullE s RE c (y*s.stride) 0 (s.w+2*s.pad) (s.w+2*s.pad) s.kh).1).push
   (hullE s RE c (y*s.stride) 0 (s.w+2*s.pad) (s.w+2*s.pad) s.kh).2).size=2*(c*ho+(y+1)):=by simp only[Array.size_push];omega
  obtain ⟨a1,a2,a3⟩:=ih (y+1) _ hs2 (by exact omS5 hy)
  obtain ⟨b1,b2,b3⟩:=hullE_spec s X RE hR c (y*s.stride) hc s.kh 0 (s.w+2*s.pad) (s.w+2*s.pad)
  refine ⟨by rw[a1];omega, λj hj=>?_, λy' xx i h1 h2 h4 h3=>?_⟩
  · rw[a2 j (by simp only[Array.size_push,Order.lt_add_one_iff];omega),push2_getD,ἱ א]
  · by_cases hy':y'=y
    · subst hy'
      rw[a2 _ (by simp only[Array.size_push,Order.lt_add_one_iff];omega),a2 _ (by simp only[Array.size_push,Order.lt_add_one_iff,add_le_add_iff_right];omega),push2_getD,push2_getD] at h3
      rw[ἐ א,ἱ א] at h3
      rw[ἐ א,ἐ א,ἱ א] at h3
      exact b3 i xx (Nat.zero_le _) א h3
    · exact a3 y' xx i (by exact omS138 h1 hy') (by exact omS7 h2) h4 h3
lemma ex3D_spec (s:OS) (X:ѐ) (RE:ї) (hR:RowOk X s.w s.pad (s.cin*s.h) RE) (ho:Nat):
  ∀n c (a:ї),a.size=2*(c*ho)->c+n<=s.cin->
  (ex3D s RE ho c n a).size=2*((c+n)*ho)/\(∀j,j<a.size->(ex3D s RE ho c n a).getD j 0=a.getD j 0)/\
  ∀c' y xx i,c<=c'->c'<c+n->y<ho->i<s.kh->
   (xx<(ex3D s RE ho c n a).getD (2*(c'*ho+y)) 0\/s.w+2*s.pad-(ex3D s RE ho c n a).getD (2*(c'*ho+y)+1) 0<=xx)->
   pixZ s X c' (y*s.stride+i) xx=true:=by
 intro n
 induction n with
 | zero=>intro c a hs _;simp only[ex3D];exact ⟨א, λ_ _=>by simp only, λc' _ _ _ h1 h2=>א⟩
 | succ n ih=>
  intro c a hs hn
  simp only[ex3D]
  obtain ⟨b1,b2,b3⟩:=ex3G_spec s X RE hR ho c (by exact omS4 hn) ho 0 a א (by exact omS10)
  obtain ⟨a1,a2,a3⟩:=ih (c+1) (ex3G s RE ho c 0 ho a) (by rw[b1,ὰ,Ὑ];omega) (by exact omS5 hn)
  refine ⟨by rw[a1];congr 1;rw[show c+(n+1)=c+1+n by exact omS375], λj hj=>?_, λc' y xx i h1 h2 hy hi h3=>?_⟩
  · rw[a2 j (by rw[b1];omega),b2 j hj]
  · by_cases hc:c'=c
    · subst hc
      have hlt:2*(c'*ho+y)+1<(ex3G s RE ho c' 0 ho a).size:=by
       rw[b1];omega
      rw[a2 _ א,a2 _ hlt] at h3
      exact b3 y xx i (Nat.zero_le _) א hi h3
    · exact a3 c' y xx i (by exact omS138 h1 hc) (by exact omS7 h2) hy hi h3
lemma sound3_empty (s:OS) (X:ѐ) (dw:Ὡ) (ho:Nat):Sound3 s X dw ho #[]:=by
 intro c y xx i _ _ _ h
 have e: ∀j,(#[]:ї).getD j 0=0:= λj=>rfl
 rw[e,e,Nat.sub_zero] at h
 rcases h with h | h
 · omega
 · exact pixZ_pad s X c _ xx h
lemma rowsE_ok (s:OS) (X:ѐ):RowOk X s.w s.pad (s.cin*s.h)
  (rowsE X s.w s.pad 0 (s.cin*s.h) (Array.emptyWithCapacity (2*(s.cin*s.h)))):=by
 intro r xx hr h
 exact (rowsE_spec X s.w s.pad (s.cin*s.h) 0 _ (by simp only[Array.emptyWithCapacity_eq,List.size_toArray,List.length_nil,mul_zero])).2.2 r xx (Nat.zero_le _) א h
@[noinline] def mkExS (s:OS) (X b:ѐ) (dw:Ὡ) (ho:Nat):{e:ї // Sound3 s X dw ho e}:=
 if dw then ⟨#[],sound3_empty s X dw ho⟩ else
 if hg:(b.all (· ==0)&&decide (1<s.kh*s.kw)&&(dw || s.cin==1))=true then
  ⟨ex3D s (rowsE X s.w s.pad 0 (s.cin*s.h) (Array.emptyWithCapacity (2*(s.cin*s.h)))) ho 0 s.cin
    (Array.emptyWithCapacity (2*(s.cin*ho))), λc y xx i hc hy hi h=>by
   have e:(if dw then c else 0)=c:=by
    cases hd:dw
    · simp only[hd,Bool.false_or,Bool.and_eq_true,beq_iff_eq] at hg
      simp only[Bool.false_eq_true,ite_false];omega
    · rfl
   rw[e] at h
   exact (ex3D_spec s X _ (rowsE_ok s X) ho s.cin 0 _ (by simp) (by omega)).2.2 c y xx i (Nat.zero_le _) (by omega) hy hi h⟩
 else ⟨#[],sound3_empty s X dw ho⟩
structure D_Sc where
 (lo hi:ὥ)
structure D_C where
 (dw relu f4:Ὡ) (s:OS) (X W b:ѐ) (k C ho wo wp:Ὣ)
 (step:ὦ) (hi:ὥ) (xd:ὤ) (nf:ὧ) (tab:ѐ)
 (ex:{e:ї // Sound3 s X dw ho e})
structure D_P where
 (g:D_C) (wd:ὤ) (off:ѐ) (e l1 l8:ὦ) (e0 bv ze:ὧ)
 (he:Ѥe<=ѧoff) (hw:Ѥe<=Ѩwd) (hs:g.xd.size<2^32)
 (h1:∀b:ὦ,b<l1->∀j (h:j< ѧoff),Ѥb+off[j].toNat+0<g.xd.size)
 (h8:∀b:ὦ,b<l8->∀j (h:j< ѧoff),Ѥb+off[j].toNat+7<g.xd.size)
 (hz:ze=0)
structure D_R where
 (t:UInt8) (A B C D E F G I:ὧ)
structure R2 where
 (off:ѐ) (buf:ὤ) (c:ὦ) (l:ὧ) (m:ὥ)
 (hm:∀j (h:j< ѧoff),off[j].toNat<=ѥm)
structure CV where
 (hW hb:ὥ) (tab:ѐ) (rs:Array R2)
end C
namespace D
open ExactRuntime FloatLib.Floats.Formats.BinaryInterchange FloatLib.Floats FloatLib.Floats.ExecFloat.Binary
open USize (toNat)
open UInt32 (toNat)
open UInt64 (toNat)
open Array (size)
open FloatArray (size)
open Nat (toUSize)
open UInt32 (toUSize)
section
open FloatLib.Numerics
end
structure LnS where
 fast:Ὡ
 m:ὧ
 ir:ὧ
 em:ὧ
 den:ὥ
deriving Inhabited
structure Acc where
 (pos neg:ї) (special negZero:Ὡ)
structure F1 where
 (s t:ὧ)
structure F4 where
 (a b c d:ὧ)
structure G8 where
 (a b c d e f g k:ὧ)
structure G24 where
 (a b c d e f g k a' b' c' d' e' f' g' k' a'' b'' c'' d'' e'' f'' g'' k'':ὧ)
structure G16 where
 (a b c d e f g k a' b' c' d' e' f' g' k':ὧ)
structure CN where
 (out:ѐ) (ok:Ὡ)
structure F8 where
 (u:UInt8) (s0 s1 s2 s3 s4 s5 s6 s7:ὧ) (p q:ὥ)
structure H8 where
 (xs:ὤ) (h0 h1 h2 h3 h4 h5 h6 h7:ὧ)
structure L8 where
 (l0 l1 l2 l3 l4 l5 l6 l7:Ὠ)
lemma qAb (n:Ὣ) (f:Ὣ->ὥ) (k:Ὣ) (hk:k<n):(Array.ofFn (n:=n) λi=>f i.val)[k]! =f k:=by rw[Ἕ _ k (by simpa only[Array.size_ofFn] using hk),Array.getElem_ofFn]
lemma qEb (z:Model Fmt):ofW (toWord z)=z:=by have h:z.toNatBits<2^32:=Model.toNatBits_lt_two_pow z;simp only[ofW,toWord,Model.ofUInt32,ὡ,Ἁ h];exact Model.ofNatBits_toNatBits z
lemma qGw {w:ὥ} (h:expOf w≠255):Ἑ (ofW w)=true:=by simp only[ofW,isFinite_ofUInt32,bne_iff_ne,ne_eq,h,not_false_eq_true]
lemma qEg (A B:ѐ) (hB:B.size=A.size) (hall:∀k,k<A.size->expOf A[k]!≠255/\expOf B[k]!≠255):finishWord (dsum A B 0 A.size) (-298) (decide (A.size≠0)&&dallNZ A B 0 A.size)=dotSpec A B:=by
 set L:=A.map (Model.ofUInt32 (fmt:=Fmt)) with hL
 set R:=B.map (Model.ofUInt32 (fmt:=Fmt)) with hR
 have hLR:L.size=R.size:=by simp only[hL,Array.size_map,hR,hB]
 have hLsize:L.size=A.size:=by simp only[hL,Array.size_map]
 have hLfin: ∀k (hk:k<L.size),Model.isFinite L[k]=true:=by
  intro k hk
  have hk':k<A.size:=by simpa only[hL,Array.size_map] using hk
  have:=(hall k hk').1
  rw[Ἕ A k hk'] at this
  simp only[hL,Array.getElem_map,isFinite_ofUInt32,bne_iff_ne,ne_eq,this,not_false_eq_true]
 have hRfin: ∀k (hk:k<R.size),Model.isFinite R[k]=true:=by
  intro k hk
  have hk':k<B.size:=by simpa only[hR,Array.size_map] using hk
  have:=(hall k א).2
  rw[Ἕ B k hk'] at this
  simp only[hR,Array.getElem_map,isFinite_ofUInt32,bne_iff_ne,ne_eq,this,not_false_eq_true]
 have hslice:Model.Reduction.Semantics.finiteDotSlice L R 0 L.size (by simp only[zero_add,Std.le_refl]) (by simp only[hLR,zero_add,Std.le_refl])=
   (dsum A B 0 A.size:ℚ)*2^(-298:ℤ):=by
  have:=finiteDotSlice_eq A B 0 L.size (by simp only[hL,Array.size_map,zero_add,Std.le_refl]) (by simp only[hL,Array.size_map,zero_add,hB,Std.le_refl])
   (by intro k k1 k2;exact hall k א)
  exact this.trans (by rw[hLsize])
 have hexact:=Model.Reduction.Internal.dotState_exact_toRat (destination:=Fmt) L R hLR
 unfold dotSpec
 rw[<-Model.Reduction.map_value_dot]
 by_cases hS0:dsum A B 0 A.size=0
 · have hz0:(Model.Reduction.Internal.dotState L R hLR:
    Model.Reduction.Internal.State Fmt).exact.significand=0:=by
    rw[<-ὐ,hexact,hslice,hS0];simp only[Int.cast_zero,Int.reduceNeg,zpow_neg,zpow_ofNat,zero_mul]
   rw[Model.Reduction.dot_status_eq_zero_of_finite Fmt L R .nearest hLR hLfin hRfin hz0,hS0,
    finishWord_zero]
   unfold Model.Reduction.Internal.dotState
   obtain ⟨h1,h2⟩:=dotStateLoop_flags A B {} 0 L.size (by simp only[hL,Array.size_map,zero_add,Std.le_refl]) (by simp only[hL,Array.size_map,zero_add,hB,Std.le_refl])
    (by intro k k1 k2;exact hall k א)
   have h1':(Model.Reduction.Internal.dotStateLoop L R
     ({}:Model.Reduction.Internal.State Fmt) 0 L.size (by simp only[zero_add,Std.le_refl])
     (by simp only[hLR,zero_add,Std.le_refl])).sawFiniteTerm=decide (L.size≠0):=by
    rw[show decide (L.size≠0)=(false || decide (L.size≠0)) by simp only[ne_eq,Array.size_eq_zero_iff,decide_not,Bool.false_or]];exact h1
   have h2':(Model.Reduction.Internal.dotStateLoop L R
     ({}:Model.Reduction.Internal.State Fmt) 0 L.size (by simp only[zero_add,Std.le_refl])
     (by simp only[hLR,zero_add,Std.le_refl])).allTermsNegativeZero=dallNZ A B 0 L.size:=by
    rw[show dallNZ A B 0 L.size=(true&&dallNZ A B 0 L.size) by simp only[Bool.true_and]];exact h2
   simp only[Except.map,h1',h2']
   simp only[ne_eq,Array.size_eq_zero_iff,decide_not,FloatFormat.isIEEE_binary32,Model.zero_eq_signedZero_of_isIEEE,and_eq_true,Bool.not_eq_eq_eq_not,Bool.not_true,decide_eq_false_iff_not,hLsize,or_eq_true,beq_iff_eq,reduceCtorEq,and_false,or_false]
 · have hnz:Model.Reduction.Semantics.finiteDot L.toList R.toList≠ .ok 0:=by
    rw[Model.Reduction.Semantics.finiteDot_toList_eq_slice L R hLR]
    intro h
    injection h with h
    rw[hslice] at h
    simp only[Int.reduceNeg,zpow_neg,zpow_ofNat,mul_eq_zero,Int.cast_eq_zero,inv_eq_zero,ne_eq,OfNat.ofNat_ne_zero,not_false_eq_true,pow_eq_zero_iff,or_false] at h;exact hS0 h
   rw[Model.Reduction.dot_status_eq_round_of_finite_nonzero Fmt L R .nearest hLR hLfin hRfin hnz]
   simp only[Except.map]
   exact finishWord_of_ne _ _ _ _ hS0 (by rw[hexact,hslice])
def Acc.total (acc:Acc):Int:=(limbVal acc.pos.toList:Int)-limbVal acc.neg.toList
@[inline] def Acc.add (acc:Acc) (m idx:Ὣ) (neg:Ὡ):Acc:=
 match acc with
 | ⟨pos,ng,sp,nz⟩=>
  if m=0 then ⟨pos,ng,sp,nz&&neg⟩
  else if neg then ⟨pos,addLimbs64 ng m.toUInt64 idx,sp,false⟩
  else ⟨addLimbs64 pos m.toUInt64 idx,ng,sp,false⟩
lemma Acc.add_spec (acc:Acc) (m idx:Ὣ) (neg:Ὡ) (hp:acc.pos.size=18) (hn:acc.neg.size=18) (hm:m<2^64) (hi:idx/32+1<18):(acc.add m idx neg).pos.size=18/\(acc.add m idx neg).neg.size=18/\(acc.add m idx neg).special=acc.special/\(acc.add m idx neg).negZero=(acc.negZero&&(decide (m=0)&&neg))/\(acc.add m idx neg).total=acc.total+(if neg then-(m:Int) else m)*2^idx:=by
 obtain ⟨pos,ng,sp,nz⟩:=acc;simp only at hp hn;by_cases hm0:m=0;. simp only[add,hm0, ↓reduceIte,hp,hn,decide_true,Bool.true_and,total,CharP.cast_eq_zero,neg_zero,ite_self,zero_mul,add_zero,and_self]
 . cases neg;. refine ⟨by simp only[add,hm0, ↓reduceIte,false_eq_true,Nat.toUInt64_eq,addLimbs64_size,hp],by simp only[add,hm0, ↓reduceIte,false_eq_true,Nat.toUInt64_eq,hn],by simp only[add,hm0, ↓reduceIte,false_eq_true,Nat.toUInt64_eq],by simp only[add,hm0, ↓reduceIte,false_eq_true,Nat.toUInt64_eq,decide_false,Bool.and_self,Bool.and_false],?_⟩;simp only[Acc.add,hm0,ite_false,Bool.false_eq_true,addLimbs64_eq _ _ _ hm,Acc.total];rw[addLimbs_val _ _ _ (by clear hm hn hm0;exact omS3 hi hp)];push_cast;ring
   . refine ⟨by simp only[add,hm0, ↓reduceIte,Nat.toUInt64_eq,hp],by simp only[add,hm0, ↓reduceIte,Nat.toUInt64_eq,addLimbs64_size,hn],by simp only[add,hm0, ↓reduceIte,Nat.toUInt64_eq],by simp only[add,hm0, ↓reduceIte,Nat.toUInt64_eq,decide_false,Bool.and_true,Bool.and_false],?_⟩;simp only[Acc.add,hm0,ite_false,ite_true,addLimbs64_eq _ _ _ hm,Acc.total];rw[addLimbs_val _ _ _ (by clear hm hp hm0;exact omS3 hi hn)];push_cast;ring
@[inline] def dotStep (acc:Acc) (wa wb:ὥ):Acc:=
 if expOf wa=255\/expOf wb=255 then ⟨acc.pos,acc.neg,true,acc.negZero⟩
 else acc.add (mantOf wa*mantOf wb) (idxOf wa+idxOf wb) (Bool.xor (signOf wa) (signOf wb))
@[specialize] def dotLoop (fa fb:Ὣ->ὥ) (pos neg:ї) (sp nz:Ὡ)
 (index:Ὣ):Ὣ->Acc
 | 0=>⟨pos,neg,sp,nz⟩
 | count+1=>
  match dotStep ⟨pos,neg,sp,nz⟩ (fa index) (fb index) with
  | ⟨p,n,s,z⟩=>dotLoop fa fb p n s z (index+1) count
lemma qCb (fa fb:Ὣ->ὥ) (pos neg:ї) (nz:Ὡ) (index count:Ὣ):(dotLoop fa fb pos neg true nz index count).special=true:=by
 induction count generalizing pos neg nz index with
 | zero=>rfl
 | succ count ih=>
  rw[dotLoop];have hs:(dotStep ⟨pos,neg,true,nz⟩ (fa index) (fb index)).special=true:=by unfold dotStep Acc.add;split_ifs<;>rfl
  generalize dotStep ⟨pos,neg,true,nz⟩ (fa index) (fb index)=r at hs⊢;obtain ⟨p,q,s,z⟩:=r;simp only at hs;subst hs;exact ih ..
lemma qC6 (fa fb:Ὣ->ὥ) (A B:ѐ) (n:Ὣ) (hA:∀k,k<n->A[k]! =fa k) (hB:∀k,k<n->B[k]! =fb k) (acc:Acc) (index remaining:Ὣ) (h:index+remaining<=n) (hp:acc.pos.size=18) (hn:acc.neg.size=18)
 (hfin:(dotLoop fa fb acc.pos acc.neg acc.special acc.negZero index remaining).special
  =false):
 (dotLoop fa fb acc.pos acc.neg acc.special acc.negZero index remaining).pos.size=18/\
 (dotLoop fa fb acc.pos acc.neg acc.special acc.negZero index remaining).neg.size=18/\
 acc.special=false/\
 (∀k,index<=k->k< index+remaining->expOf A[k]!≠255/\expOf B[k]!≠255)/\
 (dotLoop fa fb acc.pos acc.neg acc.special acc.negZero index remaining).total=acc.total+dsum A B index remaining/\
 (dotLoop fa fb acc.pos acc.neg acc.special acc.negZero index remaining).negZero=(acc.negZero&&dallNZ A B index remaining):=by
 induction remaining generalizing acc index with
 | zero=>obtain ⟨p,n,s,z⟩:=acc;simp_all only[add_zero,dotLoop,ne_eq,isEmpty_Prop,not_lt,IsEmpty.forall_iff,implies_true,dsum,dallNZ,Bool.and_true,and_self]
 | succ count ih=>
  obtain ⟨p,q,s,z⟩:=acc
  simp only at hp hn
  have ha:=hA index (by clear hp hn;exact omS4 h)
  have hb:=hB index (by clear hp hn ha;exact omS4 h)
  by_cases he:expOf (fa index)=255\/expOf (fb index)=255
  . exfalso;rw[dotLoop] at hfin;have hs:(dotStep ⟨p,q,s,z⟩ (fa index) (fb index)).special=true:=by simp only[dotStep,he, ↓reduceIte]
    generalize dotStep ⟨p,q,s,z⟩ (fa index) (fb index)=r at hs hfin;obtain ⟨p',q',s',z'⟩:=r;simp only at hs;subst hs;rw[qCb] at hfin;exact Bool.noConfusion hfin
  . have he':=not_or.mp he
    obtain ⟨h1,h2,h3,h4,h5⟩:=
     Acc.add_spec ⟨p,q,s,z⟩ (mantOf (fa index)*mantOf (fb index)) (idxOf (fa index)+idxOf (fb index)) (Bool.xor (signOf (fa index)) (signOf (fb index))) hp hn (mul_mant_lt _ _) (by have:=idxOf_le _ he'.1;have:=idxOf_le _ he'.2;(clear h hp hn ha hb he he';omega))
    have hstep:dotStep ⟨p,q,s,z⟩ (fa index) (fb index)=
     Acc.add ⟨p,q,s,z⟩ (mantOf (fa index)*mantOf (fb index)) (idxOf (fa index)+idxOf (fb index)) (Bool.xor (signOf (fa index)) (signOf (fb index))):=by
     simp only[dotStep,he, ↓reduceIte]
    rw[dotLoop,hstep] at hfin⊢;obtain ⟨hP,hN,hs,hall,hD,hz⟩:=ih _ (index+1) (by clear hp hn ha hb he he' h1 h2 h5;exact omS5 h) h1 h2 hfin;refine ⟨hP,hN,by rw[<-h3,hs],?_,?_,?_⟩
    . intro k k1 k2;rcases Nat.lt_or_ge k (index+1) with hk | hk;. have:k=index:=(by clear h hp hn ha hb he he' h1 h2 h5 hP hN hD k2;exact omS6 k1 hk);subst this;rw[ha,hb];exact he'
      . exact hall k hk (by clear h hp hn ha hb he he' h1 h2 h5 hP hN hD k1 hk;exact omS7 k2)
    . have sv2:(if Bool.xor (signOf (fa index)) (signOf (fb index)) then-((mantOf (fa index)*mantOf (fb index):Ὣ):Int) else ((mantOf (fa index)*mantOf (fb index):Ὣ):Int))=sv (fa index)*sv (fb index):=by unfold sv;cases signOf (fa index)<;>cases signOf (fb index)<;>simp only[bne_self_eq_false,false_eq_true, ↓reduceIte,Nat.cast_mul,Bool.bne_true,Bool.not_false,mul_neg,Bool.bne_false,neg_mul,neg_neg]
      rw[hD,h5,dsum,ha,hb,sv2];ring
    . rw[hz,h4,dallNZ,ha,hb];simp only[mul_eq_zero,Bool.decide_or,Bool.and_assoc]
def Y_low (p q:ї) (k:Ὣ):Ὣ->Ὣ
 | 0=>k
 | f+1=>if p[k]! =0&&q[k]! =0 then Y_low p q (k+1) f else k
lemma qD3 (p q:ї):∀f k,(∀j,j<k->p[j]! =0/\q[j]! =0)-> ∀j,j<Y_low p q k f->p[j]! =0/\q[j]! =0:=by
 intro f;induction f with
 | zero=>exact λk h j hj=>h j hj
 | succ f ih=>
  intro k h j hj;unfold Y_low at hj;split at hj
  . rename_i hk;simp only[Bool.and_eq_true,decide_eq_true_eq] at hk;refine ih (k+1) (λi hi=>?_) j hj;rcases Nat.lt_or_ge i k with h1 | h1;. exact h i h1
    . have:i=k:=(by clear hj hk;exact omS8 hi h1);subst this;exact hk
  . exact h j hj
lemma qHd (L:љ):∀l,(∀j,j<l->L[j]! =0)->limbVal L=2^(32*l)*limbVal (L.drop l):=by
 induction L with
 | nil=>simp only[List.length_nil,not_lt_zero,not_false_eq_true,ὑ,Nat.default_eq_zero,implies_true,limbVal,List.drop_nil,mul_zero,imp_self]
 | cons x xs ih=>
  intro l h;cases l with
  | zero=>simp only[mul_zero,pow_zero,List.drop_zero,one_mul]
  | succ l=>
   have h0:x=0:=by simpa only[List.length_cons,lt_add_iff_pos_left,Order.lt_add_one_iff,zero_le,Ἕ,List.getElem_cons_zero] using h 0 (by exact omS9)
   have:=ih l (λj hj=>by simpa only[List.getElem!_eq_getElem?_getD,Nat.default_eq_zero,List.getElem?_cons_succ] using h (j+1) (by exact Nat.lt_succ_of_le ‹_›));simp only[limbVal,List.drop_succ_cons,h0,this];rw[Nat.mul_add 32 l 1,pow_add];ring
lemma qFr (A:ї) (l:Ὣ) (h:∀j,j<l->A[j]! =0):limbVal A.toList=2^(32*l)*limbValA (A.extract l ѧA):=by rw[limbValA_eq,Array.toList_extract,List.extract_eq_drop_take,<-Array.length_toList];rw[List.take_of_length_le ג];refine qHd _ l λj hj=>?_;have:=h j hj;rw[getElem!_def] at this;simp only[getElem!_def] at*;cases e:A[j]?<;>simp_all only[Nat.default_eq_zero,getElem?_eq_none_iff,not_lt,Array.length_toList,getElem?_neg,Array.getElem?_toList]
lemma qIo (S:Int) (l:Ὣ) (e:Int) (z:Ὡ):finishWord S (e+32*l) z=finishWord (2^(32*l)*S) e z:=by
 by_cases hS:S=0;. simp only[finishWord,hS, ↓reduceIte,mul_zero]
 have h2:(2:Int)^(32*l)*S≠0:=mul_ne_zero ו hS
 rw[finishWord_of_ne (ofIntDyadic (2^(32*l)*S) e) S (e+32*l) z hS (by
  rw[toRat_ofIntDyadic,Ἓ ו,show (32:ℤ)*l=((32*l:ℕ):ℤ) by push_cast;ring,Ἡ];push_cast;ring)]
 simp only[finishWord,h2, ↓reduceIte]
@[inline] def dotFnO (n:Ὣ) (fa fb:Ὣ->ὥ):ὥ:=
 let acc:=dotLoop fa fb (ἇ 18 0) (ἇ 18 0) false true 0 n
 if acc.special then
  dotSpec (Array.ofFn (n:=n) λi=>fa i) (Array.ofFn (n:=n) λi=>fb i)
 else
  let l:=Y_low acc.pos acc.neg 0 18
  finishWord ((limbValA (acc.pos.extract l 18):Int)-limbValA (acc.neg.extract l 18))
   (-298+32*l) (n≠0&&acc.negZero)
lemma qAiO:dotFnO=dotOf:=by
 funext n fa fb;unfold dotFnO dotOf;set A:=Array.ofFn (n:=n) λi=>fa i with hA;set B:=Array.ofFn (n:=n) λi=>fb i with hB;set acc:=dotLoop fa fb (ἇ 18 0) (ἇ 18 0) false true 0 n with hacc;by_cases hsp:acc.special=true;. simp only[hsp, ↓reduceIte]
 have hsp':acc.special=false:=by simpa only[Bool.not_eq_true] using hsp
 have hF:=qC6 fa fb A B n (qAb n fa) (qAb n fb) ⟨ἇ 18 0,ἇ 18 0,false,true⟩ 0 n (by exact omS10) ג ג hsp';obtain ⟨hP,hN,-,hall,hD,hz⟩:=hF;have hAs:ѧA=n:=by simp only[hA,Array.size_ofFn]
 have hBs:ѧB=n:=by simp only[hB,Array.size_ofFn]
 have hL:=qD3 acc.pos acc.neg 18 0 (λj hj=>absurd hj (Ἲ _));simp only[hsp',Bool.false_eq_true,ite_false];have e0:(⟨ἇ 18 0,ἇ 18 0,false,true⟩:Acc).total=0:=ה
 rw[e0,Int.zero_add] at hD;rw[Bool.true_and] at hz;generalize Y_low acc.pos acc.neg 0 18=l at hL⊢;have h1:=qFr acc.pos l (λj hj=>(hL j hj).1);have h2:=qFr acc.neg l (λj hj=>(hL j hj).2);rw[hP] at h1;rw[hN] at h2;rw[qIo,<-qEg A B (by clear hP hN hD e0 h1 h2;exact omS11 hAs hBs) (λk hk=>hall k (by exact Nat.zero_le _) (by clear hP hN hD e0 h1 h2;exact omS12 hAs hk)),hAs,<-hD,<-hz];congr 1;have:acc.total=(limbVal acc.pos.toList:Int)-limbVal acc.neg.toList:=rfl;rw[this,h1,h2];push_cast;ring
@[inline] def dotFn (n:Ὣ) (fa fb:Ὣ->ὥ):ὥ:=FexK.fexDot n fa fb
lemma qAi:dotFn=dotOf:=by
 funext n fa fb;exact FexK.fexDot_eq n fa fb
lemma usz (t n:ὦ) (h:t<n):(t+1).toNat=Ѥt+1:=by have h1:Ѥt< Ѥn:=А.mp h;have h2:Ѥn<USize.size:=n.toNat_lt_size;have h3:USize.size=2^System.Platform.numBits:=rfl;rw[USize.toNat_add,ὄ];exact Ἁ (by exact Nat.lt_of_le_of_lt ‹_› ‹_›)
lemma qAf (a b:ὦ) (h:Ѥa+Ѥb<2^32):(a+b).toNat=Ѥa+Ѥb:=by have h1:2^32<=USize.size:=USize.le_size;have h3:USize.size=2^System.Platform.numBits:=rfl;rw[USize.toNat_add];exact Ἁ (by exact Nat.lt_of_lt_of_le ‹_› ‹_›)
lemma uk (k:Ὣ) (h:k<2^32):(ὴ k).toNat=k:=by have h1:2^32<=USize.size:=USize.le_size;have h3:USize.size=2^System.Platform.numBits:=rfl;rw[USize.toNat_ofNat'];exact Ἁ (by exact Nat.lt_of_lt_of_le ‹_› ‹_›)
lemma qDo (a:ὤ) (i:ὦ) (h:i<a.usize):Ѥi< Ѩa:=by
 have h1:=А.mp h;have h2:a.usize.toNat<=Ѩa:=by simp only[FloatArray.usize,Nat.toUSize_eq,USize.toNat_ofNat'];exact Nat.mod_le _ _
 (exact Nat.lt_of_lt_of_le ‹_› ‹_›)
lemma qBd (a:ѐ) (i:ὦ) (h:i<a.usize):Ѥi< ѧa:=by
 have h1:=А.mp h;have h2:a.usize.toNat<=ѧa:=by simp only[Array.usize,Nat.toUSize_eq,USize.toNat_ofNat'];exact Nat.mod_le _ _
 (exact Nat.lt_of_lt_of_le ‹_› ‹_›)
@[inline] def rdW (a:ѐ) (i:ὦ):ὥ:=
 if h:i<a.usize then a.uget i (qBd a i h) else 0
@[inline] def wrF (a:ὤ) (i:ὦ) (v:ὧ):ὤ:=
 if h:i<a.usize then a.uset i v (qDo a i h) else a
@[inline] def wrW (a:ѐ) (i:ὦ) (v:ὥ):ѐ:=
 if h:i<a.usize then a.uset i v (qBd a i h) else a
def pwEntry (i:Ὣ):ὧ:=
 let e:=(i%256).toUInt64
 let s:Ὠ:=if i>=256 then 0x8000000000000000 else 0
 if e==0 then Ἃ s else Ἃ (s ||| ((e+873)<<<(52:Ὠ)))
def pw:ὤ:=
 (ἄ 512).foldl (λa i=>a.push (pwEntry i)) (ἆ 512)
lemma qDx:pw.data=(ἄ 512).toArray.map pwEntry:=by
 have key:∀(l:љ) (a:ὤ),(l.foldl (λa i=>a.push (pwEntry i)) a).data=a.data++(l.toArray.map pwEntry):=by
  intro l;induction l with
  | nil=>intro a;simp only[List.foldl_nil,List.map_toArray,List.map_nil,Array.append_empty]
  | cons x l ih=>intro a;simp only[ὔ,ih];simp only[FloatArray.push,List.map_toArray,List.push_append_toArray,List.map_cons]
 simp only[pw,key];simp only[ἆ,List.toArray_range,Array.empty_append]
lemma qGe:Ѩpw=512:=by simp only[FloatArray.size,qDx,List.toArray_range,Array.size_map,Array.size_range]
lemma qCh (w:ὥ):(w>>>(23:ὥ)).toUSize.toNat< Ѩpw:=by rw[qGe];have h:=w.toNat_lt_size;simp only[ὃ,UInt32.toNat_shiftRight];norm_num[Ἱ] at h⊢;exact omS13 h
def pwEntryN (z k i:Ὣ):ὧ:=
 if i%256<z || k<=i%256 then Ἃ 0x7FF8000000000000 else pwEntry i
def pwTb (z k:Ὣ):ὤ:=
 (ἄ 512).foldl (λa i=>a.push (pwEntryN z k i)) (ἆ 512)
lemma qDj (z k:Ὣ):(pwTb z k).data=(ἄ 512).toArray.map (pwEntryN z k):=by
 have key:∀(l:љ) (a:ὤ),(l.foldl (λa i=>a.push (pwEntryN z k i)) a).data=a.data++(l.toArray.map (pwEntryN z k)):=by
  intro l;induction l with
  | nil=>intro a;simp only[List.foldl_nil,List.map_toArray,List.map_nil,Array.append_empty]
  | cons x l ih=>intro a;simp only[ὔ,ih];simp only[FloatArray.push,List.map_toArray,List.push_append_toArray,List.map_cons]
 simp only[pwTb,key];simp only[ἆ,List.toArray_range,Array.empty_append]
lemma qTd (z k:Ὣ):(pwTb z k).size=512:=by simp only[FloatArray.size,qDj,Array.size_map,List.size_toArray,List.length_range]
lemma qBx {T:ὤ} (hT:ѨT=512) (w:ὥ):(w>>>(23:ὥ)).toUSize.toNat< ѨT:=by
 rw[hT];have h:=w.toNat_lt_size;simp only[ὃ,UInt32.toNat_shiftRight];norm_num[Ἱ,UInt32.size] at h⊢;(clear hT;exact omS13 h)
def tZa:ὤ:=pwTb 0 125
def tZb:ὤ:=pwTb 0 126
def tZc:ὤ:=pwTb 0 127
def tZd:ὤ:=pwTb 0 128
@[inline] def tZ (n:Ὣ):ὤ:=if 256<=n then tZa else if 64<=n then tZb else if 16<=n then tZc else tZd
@[inline] def hvI (n:Ὣ):ὥ:=if 256<=n then 0x3E7FFFFF else if 64<=n then 0x3EFFFFFF else if 16<=n then 0x3F7FFFFF else 0x3FFFFFFF
lemma qFz (z k:Ὣ) (i:ὦ) (h:Ѥi<(pwTb z k).size):(pwTb z k).uget i h=pwEntryN z k Ѥi:=by simp only[FloatArray.uget,qDj,List.toArray_range,Array.ugetElem_eq_getElem,Array.getElem_map,Array.getElem_range]
def TOK (z:Ὣ) (T:ὤ) (h:ὥ):Prop:=∃k,ѨT=512/\(∀(i:ὦ) (p:Ѥi< ѨT),T.uget i p=pwEntryN z k Ѥi)/\2<=k/\k<=128/\ѥh+1=k*2^23
lemma qTs {z:Ὣ} {T:ὤ} {h:ὥ} (o:TOK z T h):ѨT=512:=by obtain ⟨k,e,-⟩:=o;exact e
lemma qTZ (n:Ὣ):TOK 0 (tZ n) (hvI n):=by
 unfold tZ hvI;split
 . exact ⟨125,qTd _ _,qFz _ _,by (norm_num1),by (norm_num1),rfl⟩
 . split
   . exact ⟨126,qTd _ _,qFz _ _,by (norm_num1),by (norm_num1),rfl⟩
   . split
     . exact ⟨127,qTd _ _,qFz _ _,by (norm_num1),by (norm_num1),rfl⟩
     . exact ⟨128,qTd _ _,qFz _ _,by (norm_num1),by (norm_num1),rfl⟩
@[inline] def dv (w:ὥ):ὧ:=
 ((w&&&0x7FFFFF) ||| 0x800000).toFloat*pw.uget (w>>>(23:ὥ)).toUSize (qCh w)
@[inline] def dvB (v:ὧ) (w:ὥ):ὧ:=
 (v*pw.uget (w>>>(23:ὥ)).toUSize (qCh w))*((w&&&0x7FFFFF) ||| 0x800000).toFloat
@[inline] def good (w:ὥ):Ὡ:=
 let v:=w&&&В
 v==0 || v-(0x00800000:ὥ)<=(0x7EFFFFFF:ὥ)
def scan (A:ѐ) (n i:ὦ) (hn:Ѥn<=ѧA) (lo hi:ὥ):ὥ:=
 if h:i<n then
  have ht:=А.mp h
  let v:=A.uget i א&&&В
  let u:=v-1
  scan A n (i+1) hn (if u<lo then u else lo) (if hi<v then v else hi)
 else if 0x007FFFFF<=lo/\hi<У then hi else Е
partial_fixpoint
def scanF (A:ѐ) (n i:ὦ) (hn:Ѥn<=ѧA) (o s:ὥ):ὥ:=
 if h:i<n then
  have ht:=А.mp h
  let v:=A.uget i א&&&В
  scanF A n (i+1) hn (o ||| v) (s&&&(v-1))
 else if 0x7FFFFF<=s/\o<У then o else Е
partial_fixpoint
structure C_Row (I:Ὣ) (X:ё) (i:Ὣ) (ix:ѐ) (x:ὤ)
 (c:ὦ) (L:љ) (l:ὧ):Prop where
 len:Ѥc=ѣL
 cle:ѣL<=I
 six:I<=ѧix
 sx:I<=Ѩx/\Ѩx<2^32
 nd:L.Nodup
 lt:∀t∈L,t<I
 hz:∀t,t<I->t∉L->X[i*I+t]!&&&В=0
 rd:∀k p:Ὣ,L[k]?=some p->
  (ix[k]!:ὥ).toNat=p/\x.data[k]! =dv X[i*I+p]!
 ok:l=L.foldl (λ(a:ὧ) t=>a+(dv X[i*I+t]!).abs) 0
lemma qAc (a:ѐ):a.usize.toNat<=ѧa:=by simp only[Array.usize,Nat.toUSize_eq,USize.toNat_ofNat'];exact Nat.mod_le _ _
@[inline] def scanH (r:Ὣ) (A:ѐ):ὥ:=
 let h:=if r==1 then scanF A A.usize 0 (qAc A) 0 Е else Е
 if h==Е then scan A A.usize 0 (qAc A) Е 0 else h
@[inline] def scanW (r:Ὣ) (A:ѐ):ὥ:=
 if r==8 || r==1 || r==4 then 0 else scanH r A
def pushZ (n k:ὦ) (a:ὤ):ὤ:=
 if h:k<n then pushZ n (k+1) (a.push 0) else a
partial_fixpoint
@[inline] def zivW (A B:Ὠ):ὥ:=
 let x:Ὠ:=(A+(0x0FFFFFFF:Ὠ))>>>(29:Ὠ)
 let y:Ὠ:=(B+(0x0FFFFFFF:Ὠ))>>>(29:Ὠ)
 let ex:Ὠ:=(x>>>(23:Ὠ))&&&(0x7FF:Ὠ)
 if x==y&&ex-(897:Ὠ)<(254:Ὠ) then
  (((x>>>(34:Ὠ))<<<(31:Ὠ)) |||
   ((x&&&(0x3FFFFFFFF:Ὠ))-((896:Ὠ)<<<(23:Ὠ)))).toUInt32
 else Е
@[inline] def ziv (s e:ὧ):ὥ:=
 zivW (s+e).toBits (s-e).toBits
@[noinline] def slowOut (inW:Ὣ) (relu:Ὡ) (X W b:@& ё) (i j:Ὣ):Ὢ:=
 act relu (dotFn (inW+1) (λt=>if t< inW then X[i*inW+t]! else oneW)
  (λt=>if t< inW then W[j*inW+t]! else b[j]!))
@[inline] def fin (inW:Ὣ) (relu:Ὡ) (X W b:ё) (i j:Ὣ)
 (s nf t:ὧ) (bw:ὥ):ὥ:=
 let bv:=dv bw
 let r:=ziv (s+bv) (nf*(t+bv.abs))
 if r==Е then slowOut inW relu X W b i j else act relu r
local instance ecNe0:Nonempty F1:=⟨by constructor <;> exact default⟩
def tightGo (X W:ѐ) (xo wo n t:ὦ) (s a:ὧ):F1:=
 if h:t<n then
  let q:=dv (rdW X (xo+t))*dv (rdW W (wo+t))
  tightGo X W xo wo n (t+1) (s+q) (a+q.abs)
 else ⟨s,a⟩
termination_by Ѥn-Ѥt
decreasing_by rw[usz t n h];exact Nat.sub_lt_sub_left (А.mp h) (Nat.lt_succ_self _)
@[noinline] def fin8s (inW:Ὣ) (relu:Ὡ) (X W b:@& ё) (i j:Ὣ) (nf:ὧ) (bw:ὥ) (n:ὦ):ὥ:=
 let q:=tightGo X W (i*inW).toUSize (j*inW).toUSize n 0 0 0
 fin inW relu X W b i j q.s nf q.t bw
@[inline] def fin8 (inW:Ὣ) (relu:Ὡ) (X W b:ё) (i j:Ὣ)
 (s nf a:ὧ) (hx bw:ὥ) (n:ὦ):ὥ:=
 let bv:=dv bw
 let r:=ziv (s+bv) (nf*(dv hx*a+bv.abs))
 if r==Е then fin8s inW relu X W b i j nf bw n
 else r&&&((if relu then (0:ὥ) else Е) ||| ((r>>>(31:ὥ))-1))
def thOf (e:ὧ):Ὠ:=
 let E:=(e.toBits>>>(52:Ὠ))&&&0x7FF
 if E-1<1900 then (E+35)<<<(53:Ὠ) else 0xFFFFFFFFFFFFFFFF
@[inline] def zivC (x:Ὠ):ὥ:=
 (((x>>>(34:Ὠ))<<<(31:Ὠ)) ||| ((x&&&(0x3FFFFFFFF:Ὠ))-((896:Ὠ)<<<(23:Ὠ)))).toUInt32
def loOf (e:ὧ):Ὠ:=
 let th:=thOf e
 if th<0x7040000000000000 then 0x7040000000000000 else th
def spOf (lo:Ὠ):Ὠ:=
 if lo<0x8FA0000000000000 then 0x8FA0000000000000-lo else 0
def spR (relu:Ὡ) (lo:Ὠ):Ὠ:=if relu then 0 else spOf lo
@[inline] def fin1 (lo sp:Ὠ) (s:ὧ):ὥ:=
 let w:=s.toBits
 let u:=w+0x0FDFFFFF
 if (w<<<(1:Ὠ))-lo<sp&&(u&&&0x1FFFFFFF)<0x1FC00000 then zivC (u>>>(29:Ὠ)) else Е
@[inline] def fz (relu:Ὡ) (r:ὥ):ὥ:=r&&&((if relu then (0:ὥ) else Е) ||| ((r>>>(31:ὥ))-1))
@[noinline] def fin8n (inW:Ὣ) (relu:Ὡ) (X W b:@& ё) (i j:Ὣ) (s nf a:ὧ) (hx bw:ὥ) (n:ὦ):ὥ:=fin8 inW relu X W b i j s nf a hx bw n
@[inline] def fin8L (inW:Ὣ) (relu:Ὡ) (X W b:ё) (i j:Ὣ) (s nf a:ὧ) (hx:ὥ) (lo:Ὠ) (bw:ὥ) (n:ὦ):ὥ:=
 let t:=fin1 lo (spOf lo) (s+dv bw)
 if t==Е then fin8n inW relu X W b i j s nf a hx bw n else fz relu t
structure R2 where
 (buf:ὤ) (l:ὧ) (ok:Ὡ)
local instance ecNe1:Nonempty R2:=⟨by constructor <;> exact default⟩
def decD (X:ѐ) (o n t:ὦ) (buf:ὤ) (l:ὧ):R2:=
 if h:t<n then
  let v:=dv (rdW X (o+t))
  decD X o n (t+1) (wrF buf t v) (l+v.abs)
 else ⟨buf,l,true⟩
partial_fixpoint
def idA:ѐ:=Array.ofFn (n:=8192) (λk=>k.val.toUInt32)
lemma wi (w i m:ὦ) (W:ѐ) (hw:Ѥw+Ѥm<=ѧW) (hs:ѧW<2^32) (hi:Ѥi< Ѥm):(w+i).toNat< ѧW:=by rw[qAf _ _ (by exact omS14 hw hs hi)];(clear hs;exact omS15 hw hi)
def armT:Ὡ:=System.Platform.target.startsWith "aarch64"
local instance ecNe2:Nonempty F4:=⟨by constructor <;> exact default⟩
def go4B (x:ὤ) (ix W:ѐ) (w0 w1 w2 w3 m n t:ὦ)
 (hx:Ѥn<=Ѩx) (hi:Ѥn<=ѧix) (hs:ѧW<2^32)
 (h0:Ѥw0+Ѥm<=ѧW) (h1:Ѥw1+Ѥm<=ѧW)
 (h2:Ѥw2+Ѥm<=ѧW) (h3:Ѥw3+Ѥm<=ѧW)
 (a b c d:ὧ):F4:=
 if h:t<n then
  have ht:=А.mp h
  let v:=x.uget t א
  let i:=(ix.uget t א).toUSize
  if hm:i< m then
   have him:=А.mp hm
   let q0:=v*dv (W.uget (w0+i) (wi w0 i m W h0 hs him))
   let q1:=dvB v (W.uget (w1+i) (wi w1 i m W h1 hs him))
   let q2:=v*dv (W.uget (w2+i) (wi w2 i m W h2 hs him))
   let q3:=dvB v (W.uget (w3+i) (wi w3 i m W h3 hs him))
   go4B x ix W w0 w1 w2 w3 m n (t+1) hx hi hs h0 h1 h2 h3 (a+q0) (b+q1) (c+q2) (d+q3)
  else go4B x ix W w0 w1 w2 w3 m n (t+1) hx hi hs h0 h1 h2 h3 a b c d
 else ⟨a,b,c,d⟩
termination_by Ѥn-Ѥt
decreasing_by all_goals (rw[usz t n h];omega)
def go1 (x:ὤ) (ix W:ѐ) (w0 m n t:ὦ)
 (hx:Ѥn<=Ѩx) (hi:Ѥn<=ѧix) (hs:ѧW<2^32)
 (h0:Ѥw0+Ѥm<=ѧW) (a:ὧ):ὧ:=
 if h:t<n then
  have ht:=А.mp h
  let i:=(ix.uget t א).toUSize
  if hm:i< m then
   have him:=А.mp hm
   let q:=x.uget t א*dv (W.uget (w0+i) (wi w0 i m W h0 hs him))
   go1 x ix W w0 m n (t+1) hx hi hs h0 (a+q)
  else go1 x ix W w0 m n (t+1) hx hi hs h0 a
 else a
partial_fixpoint
lemma qA9 {n:Ὣ} (h:n<=4294967295):n<2^32:=(by exact Nat.lt_succ_of_le ‹_›)
def cols1 (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (i:Ὣ)
 (ix:ѐ) (x wd:ὤ) (nf a:ὧ) (hx:ὥ) (lo:Ὠ) (m n:ὦ) (j:Ὣ)
 (base:ὦ) (out:ѐ):ѐ:=
 if h4:j+4<=outW then
  let ju:=ѩj
  let w0:=ju*m
  let w1:=(ju+1)*m
  let w2:=(ju+2)*m
  let w3:=(ju+3)*m
  if hg:Ѥn<=Ѩx/\Ѥn<=ѧix/\ѧW<=4294967295/\Ѥw0+Ѥm<=ѧW/\
   Ѥw1+Ѥm<=ѧW/\Ѥw2+Ѥm<=ѧW/\Ѥw3+Ѥm<=ѧW then
   let r:=go4B x ix W w0 w1 w2 w3 m n 0 hg.1 hg.2.1 (qA9 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 0 0 0 0
   let o0:=fin8L inW relu X W b i j r.a nf a hx lo (rdW b ju) m
   let o1:=fin8L inW relu X W b i (j+1) r.b nf a hx lo (rdW b (ju+1)) m
   let o2:=fin8L inW relu X W b i (j+2) r.c nf a hx lo (rdW b (ju+2)) m
   let o3:=fin8L inW relu X W b i (j+3) r.d nf a hx lo (rdW b (ju+3)) m
   let out:=wrW out (base+ju) o0
   let out:=wrW out (base+ju+1) o1
   let out:=wrW out (base+ju+2) o2
   let out:=wrW out (base+ju+3) o3
   cols1 inW outW relu X W b i ix x wd nf a hx lo m n (j+4) base out
  else out
 else if h1:j<outW then
  let ju:=ѩj
  let w0:=ju*m
  if hg:Ѥn<=Ѩx/\Ѥn<=ѧix/\ѧW<=4294967295/\Ѥw0+Ѥm<=ѧW then
   let r:=go1 x ix W w0 m n 0 hg.1 hg.2.1 (qA9 hg.2.2.1) hg.2.2.2 0
   let o:=fin8L inW relu X W b i j r nf a hx lo (rdW b ju) m
   cols1 inW outW relu X W b i ix x wd nf a hx lo m n (j+1) base (wrW out (base+ju) o)
  else out
 else out
partial_fixpoint
lemma qWm (a b:ὦ) (h:Ѥa*Ѥb<2^32):(a*b).toNat=Ѥa*Ѥb:=by have h1:2^32<=USize.size:=USize.le_size;have h3:USize.size=2^System.Platform.numBits:=rfl;rw[USize.toNat_mul];exact Ἁ (by exact Nat.lt_of_lt_of_le ‹_› ‹_›)
lemma wq (p i m:ὦ) (wb:ὤ) (k:Ὣ) (hk:k<8) (hb:Ѥp+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (him:Ѥi< Ѥm):(p+ὴ 8*i+ὴ k).toNat< Ѩwb:=by
 have e8:(ὴ 8*i).toNat=8*Ѥi:=by rw[qWm _ _ (by rw[uk 8 ה];(clear hk;exact omS16 hb hs him)),uk 8 ה]
 rw[qAf _ _ (by rw[qAf _ _ (by rw[e8];(clear hk e8;exact omS17 hb hs him)),e8,uk k (by exact Nat.lt_of_lt_of_le ‹_› ה)];(clear e8;exact omS18 hk hb hs him)),qAf _ _ (by rw[e8];(clear hk e8;exact omS17 hb hs him)),e8,uk k (by exact Nat.lt_of_lt_of_le ‹_› ה)];(clear hs e8;omega)
lemma wq0 (p i m:ὦ) (wb:ὤ) (hb:Ѥp+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (him:Ѥi< Ѥm):(p+ὴ 8*i+ὴ 0).toNat< Ѩwb:=wq p i m wb 0 ה hb hs him
lemma wq1 (p i m:ὦ) (wb:ὤ) (hb:Ѥp+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (him:Ѥi< Ѥm):(p+ὴ 8*i+ὴ 1).toNat< Ѩwb:=wq p i m wb 1 ה hb hs him
lemma wq2 (p i m:ὦ) (wb:ὤ) (hb:Ѥp+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (him:Ѥi< Ѥm):(p+ὴ 8*i+ὴ 2).toNat< Ѩwb:=wq p i m wb 2 ה hb hs him
lemma wq3 (p i m:ὦ) (wb:ὤ) (hb:Ѥp+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (him:Ѥi< Ѥm):(p+ὴ 8*i+ὴ 3).toNat< Ѩwb:=wq p i m wb 3 ה hb hs him
lemma wq4 (p i m:ὦ) (wb:ὤ) (hb:Ѥp+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (him:Ѥi< Ѥm):(p+ὴ 8*i+ὴ 4).toNat< Ѩwb:=wq p i m wb 4 ה hb hs him
lemma wq5 (p i m:ὦ) (wb:ὤ) (hb:Ѥp+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (him:Ѥi< Ѥm):(p+ὴ 8*i+ὴ 5).toNat< Ѩwb:=wq p i m wb 5 ה hb hs him
lemma wq6 (p i m:ὦ) (wb:ὤ) (hb:Ѥp+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (him:Ѥi< Ѥm):(p+ὴ 8*i+ὴ 6).toNat< Ѩwb:=wq p i m wb 6 ה hb hs him
lemma wq7 (p i m:ὦ) (wb:ὤ) (hb:Ѥp+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (him:Ѥi< Ѥm):(p+ὴ 8*i+ὴ 7).toNat< Ѩwb:=wq p i m wb 7 ה hb hs him
lemma pk (w m:ὦ) (wb:ὤ) (h:Ѥw+5+8*Ѥm<=Ѩwb) (hs:Ѩwb<=4294967295):Ѥ(w+ὴ 5)+8*Ѥm<=Ѩwb/\Ѥ(w+ὴ 5)=Ѥw+5:=by
 have e:Ѥ(w+ὴ 5)=Ѥw+5:=by rw[qAf _ _ (by rw[uk 5 ה];exact omS19 h hs),uk 5 ה]
 exact ⟨by rw[e];exact h,e⟩
def go8W (x wb:ὤ) (ix:ѐ) (p m n t:ὦ) (hx:Ѥn<=Ѩx) (hi:Ѥn<=ѧix) (hb:Ѥp+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (a b c d e f g k:ὧ):F4×F4:=
 if h:t<n then
  have ht:=А.mp h
  let v:=x.uget t (Nat.lt_of_lt_of_le ht hx)
  let i:=(ix.uget t (Nat.lt_of_lt_of_le ht hi)).toUSize
  if hm:i< m then
   have him:=А.mp hm
   let q:=p+ὴ 8*i
   go8W x wb ix p m n (t+1) hx hi hb hs
    (a+v*wb.uget (q+ὴ 0) (wq0 p i m wb hb hs him)) (b+v*wb.uget (q+ὴ 1) (wq1 p i m wb hb hs him))
    (c+v*wb.uget (q+ὴ 2) (wq2 p i m wb hb hs him)) (d+v*wb.uget (q+ὴ 3) (wq3 p i m wb hb hs him))
    (e+v*wb.uget (q+ὴ 4) (wq4 p i m wb hb hs him)) (f+v*wb.uget (q+ὴ 5) (wq5 p i m wb hb hs him))
    (g+v*wb.uget (q+ὴ 6) (wq6 p i m wb hb hs him)) (k+v*wb.uget (q+ὴ 7) (wq7 p i m wb hb hs him))
  else go8W x wb ix p m n (t+1) hx hi hb hs a b c d e f g k
 else (⟨a,b,c,d⟩,⟨e,f,g,k⟩)
termination_by Ѥn-Ѥt
decreasing_by all_goals (rw[usz t n h];omega)
local instance ecNe3:Nonempty G8:=⟨by constructor <;> exact default⟩
def go8V (x wb:ὤ) (p m n t:ὦ) (hx:Ѥn<=Ѩx) (hn:Ѥn<=Ѥm) (hb:Ѥp+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (a b c d e f g k:ὧ):G8:=
 if h:t<n then
  have ht:=А.mp h
  have him:Ѥt< Ѥm:=Nat.lt_of_lt_of_le ht hn
  let v:=x.uget t (Nat.lt_of_lt_of_le ht hx)
  let q:=p+ὴ 8*t
  go8V x wb p m n (t+1) hx hn hb hs
   (a+v*wb.uget (q+ὴ 0) (wq0 p t m wb hb hs him)) (b+v*wb.uget (q+ὴ 1) (wq1 p t m wb hb hs him))
   (c+v*wb.uget (q+ὴ 2) (wq2 p t m wb hb hs him)) (d+v*wb.uget (q+ὴ 3) (wq3 p t m wb hb hs him))
   (e+v*wb.uget (q+ὴ 4) (wq4 p t m wb hb hs him)) (f+v*wb.uget (q+ὴ 5) (wq5 p t m wb hb hs him))
   (g+v*wb.uget (q+ὴ 6) (wq6 p t m wb hb hs him)) (k+v*wb.uget (q+ὴ 7) (wq7 p t m wb hb hs him))
 else ⟨a,b,c,d,e,f,g,k⟩
termination_by Ѥn-Ѥt
decreasing_by all_goals (rw[usz t n h];omega)
def g8of (r:F4×F4):G8:=⟨r.1.a,r.1.b,r.1.c,r.1.d,r.2.a,r.2.b,r.2.c,r.2.d⟩
local instance ecNe4:Nonempty G24:=⟨by constructor <;> exact default⟩
def go24V (x wb:ὤ) (p p' p'' m n t:ὦ) (hx:Ѥn<=Ѩx) (hn:Ѥn<=Ѥm) (hb:Ѥp+8*Ѥm<=Ѩwb) (hb':Ѥp'+8*Ѥm<=Ѩwb) (hb'':Ѥp''+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (a b c d e f g k a' b' c' d' e' f' g' k' a'' b'' c'' d'' e'' f'' g'' k'':ὧ):G24:=
 if h:t<n then
  have ht:=А.mp h
  have him:Ѥt< Ѥm:=Nat.lt_of_lt_of_le ht hn
  let v:=x.uget t (Nat.lt_of_lt_of_le ht hx)
  let q:=p+ὴ 8*t
  let q':=p'+ὴ 8*t
  let q'':=p''+ὴ 8*t
  go24V x wb p p' p'' m n (t+1) hx hn hb hb' hb'' hs
   (a+v*wb.uget (q+ὴ 0) (wq0 p t m wb hb hs him)) (b+v*wb.uget (q+ὴ 1) (wq1 p t m wb hb hs him))
   (c+v*wb.uget (q+ὴ 2) (wq2 p t m wb hb hs him)) (d+v*wb.uget (q+ὴ 3) (wq3 p t m wb hb hs him))
   (e+v*wb.uget (q+ὴ 4) (wq4 p t m wb hb hs him)) (f+v*wb.uget (q+ὴ 5) (wq5 p t m wb hb hs him))
   (g+v*wb.uget (q+ὴ 6) (wq6 p t m wb hb hs him)) (k+v*wb.uget (q+ὴ 7) (wq7 p t m wb hb hs him))
   (a'+v*wb.uget (q'+ὴ 0) (wq0 p' t m wb hb' hs him)) (b'+v*wb.uget (q'+ὴ 1) (wq1 p' t m wb hb' hs him))
   (c'+v*wb.uget (q'+ὴ 2) (wq2 p' t m wb hb' hs him)) (d'+v*wb.uget (q'+ὴ 3) (wq3 p' t m wb hb' hs him))
   (e'+v*wb.uget (q'+ὴ 4) (wq4 p' t m wb hb' hs him)) (f'+v*wb.uget (q'+ὴ 5) (wq5 p' t m wb hb' hs him))
   (g'+v*wb.uget (q'+ὴ 6) (wq6 p' t m wb hb' hs him)) (k'+v*wb.uget (q'+ὴ 7) (wq7 p' t m wb hb' hs him))
   (a''+v*wb.uget (q''+ὴ 0) (wq0 p'' t m wb hb'' hs him)) (b''+v*wb.uget (q''+ὴ 1) (wq1 p'' t m wb hb'' hs him))
   (c''+v*wb.uget (q''+ὴ 2) (wq2 p'' t m wb hb'' hs him)) (d''+v*wb.uget (q''+ὴ 3) (wq3 p'' t m wb hb'' hs him))
   (e''+v*wb.uget (q''+ὴ 4) (wq4 p'' t m wb hb'' hs him)) (f''+v*wb.uget (q''+ὴ 5) (wq5 p'' t m wb hb'' hs him))
   (g''+v*wb.uget (q''+ὴ 6) (wq6 p'' t m wb hb'' hs him)) (k''+v*wb.uget (q''+ὴ 7) (wq7 p'' t m wb hb'' hs him))
 else ⟨a,b,c,d,e,f,g,k,a',b',c',d',e',f',g',k',a'',b'',c'',d'',e'',f'',g'',k''⟩
termination_by Ѥn-Ѥt
decreasing_by all_goals (rw[usz t n h];omega)
local instance ecNe5:Nonempty G16:=⟨by constructor <;> exact default⟩
def go16V (x wb:ὤ) (p p' m n t:ὦ) (hx:Ѥn<=Ѩx) (hn:Ѥn<=Ѥm) (hb:Ѥp+8*Ѥm<=Ѩwb) (hb':Ѥp'+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (a b c d e f g k a' b' c' d' e' f' g' k':ὧ):G16:=
 if h:t<n then
  have ht:=А.mp h
  have him:Ѥt< Ѥm:=Nat.lt_of_lt_of_le ht hn
  let v:=x.uget t (Nat.lt_of_lt_of_le ht hx)
  let q:=p+ὴ 8*t
  let q':=p'+ὴ 8*t
  go16V x wb p p' m n (t+1) hx hn hb hb' hs
   (a+v*wb.uget (q+ὴ 0) (wq0 p t m wb hb hs him)) (b+v*wb.uget (q+ὴ 1) (wq1 p t m wb hb hs him))
   (c+v*wb.uget (q+ὴ 2) (wq2 p t m wb hb hs him)) (d+v*wb.uget (q+ὴ 3) (wq3 p t m wb hb hs him))
   (e+v*wb.uget (q+ὴ 4) (wq4 p t m wb hb hs him)) (f+v*wb.uget (q+ὴ 5) (wq5 p t m wb hb hs him))
   (g+v*wb.uget (q+ὴ 6) (wq6 p t m wb hb hs him)) (k+v*wb.uget (q+ὴ 7) (wq7 p t m wb hb hs him))
   (a'+v*wb.uget (q'+ὴ 0) (wq0 p' t m wb hb' hs him)) (b'+v*wb.uget (q'+ὴ 1) (wq1 p' t m wb hb' hs him))
   (c'+v*wb.uget (q'+ὴ 2) (wq2 p' t m wb hb' hs him)) (d'+v*wb.uget (q'+ὴ 3) (wq3 p' t m wb hb' hs him))
   (e'+v*wb.uget (q'+ὴ 4) (wq4 p' t m wb hb' hs him)) (f'+v*wb.uget (q'+ὴ 5) (wq5 p' t m wb hb' hs him))
   (g'+v*wb.uget (q'+ὴ 6) (wq6 p' t m wb hb' hs him)) (k'+v*wb.uget (q'+ὴ 7) (wq7 p' t m wb hb' hs him))
 else ⟨a,b,c,d,e,f,g,k,a',b',c',d',e',f',g',k'⟩
termination_by Ѥn-Ѥt
decreasing_by all_goals (rw[usz t n h];omega)
def g16of (r s:G8):G16:=⟨r.a,r.b,r.c,r.d,r.e,r.f,r.g,r.k,s.a,s.b,s.c,s.d,s.e,s.f,s.g,s.k⟩
lemma g16V (x wb:ὤ) (p p' m n t:ὦ) (hx:Ѥn<=Ѩx) (hn:Ѥn<=Ѥm) (hb:Ѥp+8*Ѥm<=Ѩwb) (hb':Ѥp'+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (a b c d e f g k a' b' c' d' e' f' g' k':ὧ):
 go16V x wb p p' m n t hx hn hb hb' hs a b c d e f g k a' b' c' d' e' f' g' k'=g16of (go8V x wb p m n t hx hn hb hs a b c d e f g k) (go8V x wb p' m n t hx hn hb' hs a' b' c' d' e' f' g' k'):=by
 obtain ⟨N,hN⟩:∃N,Ѥn-Ѥt=N:=⟨_,rfl⟩
 induction N generalizing t a b c d e f g k a' b' c' d' e' f' g' k' with
 | zero=>
  have h:¬t<n:=λh=>by have:=А.mp h;omega
  rw[go16V,go8V,go8V,ἔ h,ἔ h,ἔ h];rfl
 | succ N ih=>
  have h:t<n:=А.mpr (by exact omS412 hN)
  rw[go16V,go8V,go8V,Ἄ h,Ἄ h,Ἄ h];dsimp only
  exact ih (t+1) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (by rw[usz t n h];exact omS413 hN)
def g24of (r s u:G8):G24:=⟨r.a,r.b,r.c,r.d,r.e,r.f,r.g,r.k,s.a,s.b,s.c,s.d,s.e,s.f,s.g,s.k,u.a,u.b,u.c,u.d,u.e,u.f,u.g,u.k⟩
lemma g24V (x wb:ὤ) (p p' p'' m n t:ὦ) (hx:Ѥn<=Ѩx) (hn:Ѥn<=Ѥm) (hb:Ѥp+8*Ѥm<=Ѩwb) (hb':Ѥp'+8*Ѥm<=Ѩwb) (hb'':Ѥp''+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (a b c d e f g k a' b' c' d' e' f' g' k' a'' b'' c'' d'' e'' f'' g'' k'':ὧ):
 go24V x wb p p' p'' m n t hx hn hb hb' hb'' hs a b c d e f g k a' b' c' d' e' f' g' k' a'' b'' c'' d'' e'' f'' g'' k''=g24of (go8V x wb p m n t hx hn hb hs a b c d e f g k) (go8V x wb p' m n t hx hn hb' hs a' b' c' d' e' f' g' k') (go8V x wb p'' m n t hx hn hb'' hs a'' b'' c'' d'' e'' f'' g'' k''):=by
 obtain ⟨N,hN⟩:∃N,Ѥn-Ѥt=N:=⟨_,rfl⟩
 induction N generalizing t a b c d e f g k a' b' c' d' e' f' g' k' a'' b'' c'' d'' e'' f'' g'' k'' with
 | zero=>
  have h:¬t<n:=λh=>by have:=А.mp h;omega
  rw[go24V,go8V,go8V,go8V,ἔ h,ἔ h,ἔ h,ἔ h];rfl
 | succ N ih=>
  have h:t<n:=А.mpr (by exact omS412 hN)
  rw[go24V,go8V,go8V,go8V,Ἄ h,Ἄ h,Ἄ h,Ἄ h];dsimp only
  exact ih (t+1) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (by rw[usz t n h];exact omS413 hN)
def go8WF (x wb:ὤ) (ix:ѐ) (p m n t:ὦ) (hx:Ѥn<=Ѩx) (hi:Ѥn<=ѧix) (hb:Ѥp+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (a b c d e f g k:ὧ):F4×F4:=
 if h:t<n then
  have ht:=А.mp h
  let v:=x.uget t (Nat.lt_of_lt_of_le ht hx)
  let i:=(ix.uget t (Nat.lt_of_lt_of_le ht hi)).toUSize
  if hm:i< m then
   have him:=А.mp hm
   let q:=p+ὴ 8*i
   go8WF x wb ix p m n (t+1) hx hi hb hs
    (Float.fma v (wb.uget (q+ὴ 0) (wq0 p i m wb hb hs him)) a) (Float.fma v (wb.uget (q+ὴ 1) (wq1 p i m wb hb hs him)) b)
    (Float.fma v (wb.uget (q+ὴ 2) (wq2 p i m wb hb hs him)) c) (Float.fma v (wb.uget (q+ὴ 3) (wq3 p i m wb hb hs him)) d)
    (Float.fma v (wb.uget (q+ὴ 4) (wq4 p i m wb hb hs him)) e) (Float.fma v (wb.uget (q+ὴ 5) (wq5 p i m wb hb hs him)) f)
    (Float.fma v (wb.uget (q+ὴ 6) (wq6 p i m wb hb hs him)) g) (Float.fma v (wb.uget (q+ὴ 7) (wq7 p i m wb hb hs him)) k)
  else go8WF x wb ix p m n (t+1) hx hi hb hs a b c d e f g k
 else (⟨a,b,c,d⟩,⟨e,f,g,k⟩)
termination_by Ѥn-Ѥt
decreasing_by all_goals (rw[usz t n h];omega)
def go8F (x wb:ὤ) (p m n t:ὦ) (hx:Ѥn<=Ѩx) (hn:Ѥn<=Ѥm) (hb:Ѥp+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (a b c d e f g k:ὧ):G8:=
 if h:t<n then
  have ht:=А.mp h
  have him:Ѥt< Ѥm:=Nat.lt_of_lt_of_le ht hn
  let v:=x.uget t (Nat.lt_of_lt_of_le ht hx)
  let q:=p+ὴ 8*t
  go8F x wb p m n (t+1) hx hn hb hs
   (Float.fma v (wb.uget (q+ὴ 0) (wq0 p t m wb hb hs him)) a) (Float.fma v (wb.uget (q+ὴ 1) (wq1 p t m wb hb hs him)) b)
   (Float.fma v (wb.uget (q+ὴ 2) (wq2 p t m wb hb hs him)) c) (Float.fma v (wb.uget (q+ὴ 3) (wq3 p t m wb hb hs him)) d)
   (Float.fma v (wb.uget (q+ὴ 4) (wq4 p t m wb hb hs him)) e) (Float.fma v (wb.uget (q+ὴ 5) (wq5 p t m wb hb hs him)) f)
   (Float.fma v (wb.uget (q+ὴ 6) (wq6 p t m wb hb hs him)) g) (Float.fma v (wb.uget (q+ὴ 7) (wq7 p t m wb hb hs him)) k)
 else ⟨a,b,c,d,e,f,g,k⟩
termination_by Ѥn-Ѥt
decreasing_by all_goals (rw[usz t n h];omega)
def go16F (x wb:ὤ) (p p' m n t:ὦ) (hx:Ѥn<=Ѩx) (hn:Ѥn<=Ѥm) (hb:Ѥp+8*Ѥm<=Ѩwb) (hb':Ѥp'+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (a b c d e f g k a' b' c' d' e' f' g' k':ὧ):G16:=
 if h:t<n then
  have ht:=А.mp h
  have him:Ѥt< Ѥm:=Nat.lt_of_lt_of_le ht hn
  let v:=x.uget t (Nat.lt_of_lt_of_le ht hx)
  let q:=p+ὴ 8*t
  let q':=p'+ὴ 8*t
  go16F x wb p p' m n (t+1) hx hn hb hb' hs
   (Float.fma v (wb.uget (q+ὴ 0) (wq0 p t m wb hb hs him)) a) (Float.fma v (wb.uget (q+ὴ 1) (wq1 p t m wb hb hs him)) b)
   (Float.fma v (wb.uget (q+ὴ 2) (wq2 p t m wb hb hs him)) c) (Float.fma v (wb.uget (q+ὴ 3) (wq3 p t m wb hb hs him)) d)
   (Float.fma v (wb.uget (q+ὴ 4) (wq4 p t m wb hb hs him)) e) (Float.fma v (wb.uget (q+ὴ 5) (wq5 p t m wb hb hs him)) f)
   (Float.fma v (wb.uget (q+ὴ 6) (wq6 p t m wb hb hs him)) g) (Float.fma v (wb.uget (q+ὴ 7) (wq7 p t m wb hb hs him)) k)
   (Float.fma v (wb.uget (q'+ὴ 0) (wq0 p' t m wb hb' hs him)) a') (Float.fma v (wb.uget (q'+ὴ 1) (wq1 p' t m wb hb' hs him)) b')
   (Float.fma v (wb.uget (q'+ὴ 2) (wq2 p' t m wb hb' hs him)) c') (Float.fma v (wb.uget (q'+ὴ 3) (wq3 p' t m wb hb' hs him)) d')
   (Float.fma v (wb.uget (q'+ὴ 4) (wq4 p' t m wb hb' hs him)) e') (Float.fma v (wb.uget (q'+ὴ 5) (wq5 p' t m wb hb' hs him)) f')
   (Float.fma v (wb.uget (q'+ὴ 6) (wq6 p' t m wb hb' hs him)) g') (Float.fma v (wb.uget (q'+ὴ 7) (wq7 p' t m wb hb' hs him)) k')
 else ⟨a,b,c,d,e,f,g,k,a',b',c',d',e',f',g',k'⟩
termination_by Ѥn-Ѥt
decreasing_by all_goals (rw[usz t n h];omega)
def go24F (x wb:ὤ) (p p' p'' m n t:ὦ) (hx:Ѥn<=Ѩx) (hn:Ѥn<=Ѥm) (hb:Ѥp+8*Ѥm<=Ѩwb) (hb':Ѥp'+8*Ѥm<=Ѩwb) (hb'':Ѥp''+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (a b c d e f g k a' b' c' d' e' f' g' k' a'' b'' c'' d'' e'' f'' g'' k'':ὧ):G24:=
 if h:t<n then
  have ht:=А.mp h
  have him:Ѥt< Ѥm:=Nat.lt_of_lt_of_le ht hn
  let v:=x.uget t (Nat.lt_of_lt_of_le ht hx)
  let q:=p+ὴ 8*t
  let q':=p'+ὴ 8*t
  let q'':=p''+ὴ 8*t
  go24F x wb p p' p'' m n (t+1) hx hn hb hb' hb'' hs
   (Float.fma v (wb.uget (q+ὴ 0) (wq0 p t m wb hb hs him)) a) (Float.fma v (wb.uget (q+ὴ 1) (wq1 p t m wb hb hs him)) b)
   (Float.fma v (wb.uget (q+ὴ 2) (wq2 p t m wb hb hs him)) c) (Float.fma v (wb.uget (q+ὴ 3) (wq3 p t m wb hb hs him)) d)
   (Float.fma v (wb.uget (q+ὴ 4) (wq4 p t m wb hb hs him)) e) (Float.fma v (wb.uget (q+ὴ 5) (wq5 p t m wb hb hs him)) f)
   (Float.fma v (wb.uget (q+ὴ 6) (wq6 p t m wb hb hs him)) g) (Float.fma v (wb.uget (q+ὴ 7) (wq7 p t m wb hb hs him)) k)
   (Float.fma v (wb.uget (q'+ὴ 0) (wq0 p' t m wb hb' hs him)) a') (Float.fma v (wb.uget (q'+ὴ 1) (wq1 p' t m wb hb' hs him)) b')
   (Float.fma v (wb.uget (q'+ὴ 2) (wq2 p' t m wb hb' hs him)) c') (Float.fma v (wb.uget (q'+ὴ 3) (wq3 p' t m wb hb' hs him)) d')
   (Float.fma v (wb.uget (q'+ὴ 4) (wq4 p' t m wb hb' hs him)) e') (Float.fma v (wb.uget (q'+ὴ 5) (wq5 p' t m wb hb' hs him)) f')
   (Float.fma v (wb.uget (q'+ὴ 6) (wq6 p' t m wb hb' hs him)) g') (Float.fma v (wb.uget (q'+ὴ 7) (wq7 p' t m wb hb' hs him)) k')
   (Float.fma v (wb.uget (q''+ὴ 0) (wq0 p'' t m wb hb'' hs him)) a'') (Float.fma v (wb.uget (q''+ὴ 1) (wq1 p'' t m wb hb'' hs him)) b'')
   (Float.fma v (wb.uget (q''+ὴ 2) (wq2 p'' t m wb hb'' hs him)) c'') (Float.fma v (wb.uget (q''+ὴ 3) (wq3 p'' t m wb hb'' hs him)) d'')
   (Float.fma v (wb.uget (q''+ὴ 4) (wq4 p'' t m wb hb'' hs him)) e'') (Float.fma v (wb.uget (q''+ὴ 5) (wq5 p'' t m wb hb'' hs him)) f'')
   (Float.fma v (wb.uget (q''+ὴ 6) (wq6 p'' t m wb hb'' hs him)) g'') (Float.fma v (wb.uget (q''+ὴ 7) (wq7 p'' t m wb hb'' hs him)) k'')
 else ⟨a,b,c,d,e,f,g,k,a',b',c',d',e',f',g',k',a'',b'',c'',d'',e'',f'',g'',k''⟩
termination_by Ѥn-Ѥt
decreasing_by all_goals (rw[usz t n h];omega)
lemma g16F (x wb:ὤ) (p p' m n t:ὦ) (hx:Ѥn<=Ѩx) (hn:Ѥn<=Ѥm) (hb:Ѥp+8*Ѥm<=Ѩwb) (hb':Ѥp'+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (a b c d e f g k a' b' c' d' e' f' g' k':ὧ):
 go16F x wb p p' m n t hx hn hb hb' hs a b c d e f g k a' b' c' d' e' f' g' k'=g16of (go8F x wb p m n t hx hn hb hs a b c d e f g k) (go8F x wb p' m n t hx hn hb' hs a' b' c' d' e' f' g' k'):=by
 obtain ⟨N,hN⟩:∃N,Ѥn-Ѥt=N:=⟨_,rfl⟩
 induction N generalizing t a b c d e f g k a' b' c' d' e' f' g' k' with
 | zero=>
  have h:¬t<n:=λh=>by have:=А.mp h;omega
  rw[go16F,go8F,go8F,ἔ h,ἔ h,ἔ h];rfl
 | succ N ih=>
  have h:t<n:=А.mpr (by exact omS412 hN)
  rw[go16F,go8F,go8F,Ἄ h,Ἄ h,Ἄ h];dsimp only
  exact ih (t+1) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (by rw[usz t n h];exact omS413 hN)
lemma g24F (x wb:ὤ) (p p' p'' m n t:ὦ) (hx:Ѥn<=Ѩx) (hn:Ѥn<=Ѥm) (hb:Ѥp+8*Ѥm<=Ѩwb) (hb':Ѥp'+8*Ѥm<=Ѩwb) (hb'':Ѥp''+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (a b c d e f g k a' b' c' d' e' f' g' k' a'' b'' c'' d'' e'' f'' g'' k'':ὧ):
 go24F x wb p p' p'' m n t hx hn hb hb' hb'' hs a b c d e f g k a' b' c' d' e' f' g' k' a'' b'' c'' d'' e'' f'' g'' k''=g24of (go8F x wb p m n t hx hn hb hs a b c d e f g k) (go8F x wb p' m n t hx hn hb' hs a' b' c' d' e' f' g' k') (go8F x wb p'' m n t hx hn hb'' hs a'' b'' c'' d'' e'' f'' g'' k''):=by
 obtain ⟨N,hN⟩:∃N,Ѥn-Ѥt=N:=⟨_,rfl⟩
 induction N generalizing t a b c d e f g k a' b' c' d' e' f' g' k' a'' b'' c'' d'' e'' f'' g'' k'' with
 | zero=>
  have h:¬t<n:=λh=>by have:=А.mp h;omega
  rw[go24F,go8F,go8F,go8F,ἔ h,ἔ h,ἔ h,ἔ h];rfl
 | succ N ih=>
  have h:t<n:=А.mpr (by exact omS412 hN)
  rw[go24F,go8F,go8F,go8F,Ἄ h,Ἄ h,Ἄ h,Ἄ h];dsimp only
  exact ih (t+1) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (by rw[usz t n h];exact omS413 hN)
@[inline] def go8WD (x wb:ὤ) (ix:ѐ) (p m n t:ὦ) (hx:Ѥn<=Ѩx) (hi:Ѥn<=ѧix) (hb:Ѥp+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (a b c d e f g k:ὧ):F4×F4:=
 if armT then go8WF x wb ix p m n t hx hi hb hs a b c d e f g k else go8W x wb ix p m n t hx hi hb hs a b c d e f g k
@[inline] def go8D (x wb:ὤ) (p m n t:ὦ) (hx:Ѥn<=Ѩx) (hn:Ѥn<=Ѥm) (hb:Ѥp+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (a b c d e f g k:ὧ):G8:=
 if armT then go8F x wb p m n t hx hn hb hs a b c d e f g k else go8V x wb p m n t hx hn hb hs a b c d e f g k
@[inline] def go16D (x wb:ὤ) (p p' m n t:ὦ) (hx:Ѥn<=Ѩx) (hn:Ѥn<=Ѥm) (hb:Ѥp+8*Ѥm<=Ѩwb) (hb':Ѥp'+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (a b c d e f g k a' b' c' d' e' f' g' k':ὧ):G16:=
 if armT then go16F x wb p p' m n t hx hn hb hb' hs a b c d e f g k a' b' c' d' e' f' g' k' else go16V x wb p p' m n t hx hn hb hb' hs a b c d e f g k a' b' c' d' e' f' g' k'
@[inline] def go24D (x wb:ὤ) (p p' p'' m n t:ὦ) (hx:Ѥn<=Ѩx) (hn:Ѥn<=Ѥm) (hb:Ѥp+8*Ѥm<=Ѩwb) (hb':Ѥp'+8*Ѥm<=Ѩwb) (hb'':Ѥp''+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (a b c d e f g k a' b' c' d' e' f' g' k' a'' b'' c'' d'' e'' f'' g'' k'':ὧ):G24:=
 if armT then go24F x wb p p' p'' m n t hx hn hb hb' hb'' hs a b c d e f g k a' b' c' d' e' f' g' k' a'' b'' c'' d'' e'' f'' g'' k'' else go24V x wb p p' p'' m n t hx hn hb hb' hb'' hs a b c d e f g k a' b' c' d' e' f' g' k' a'' b'' c'' d'' e'' f'' g'' k''
lemma g16D (x wb:ὤ) (p p' m n t:ὦ) (hx:Ѥn<=Ѩx) (hn:Ѥn<=Ѥm) (hb:Ѥp+8*Ѥm<=Ѩwb) (hb':Ѥp'+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (a b c d e f g k a' b' c' d' e' f' g' k':ὧ):
 go16D x wb p p' m n t hx hn hb hb' hs a b c d e f g k a' b' c' d' e' f' g' k'=g16of (go8D x wb p m n t hx hn hb hs a b c d e f g k) (go8D x wb p' m n t hx hn hb' hs a' b' c' d' e' f' g' k'):=by
 unfold go16D go8D;generalize armT=A;cases A
 . simp only[Bool.false_eq_true,ite_false];exact g16V x wb p p' m n t hx hn hb hb' hs a b c d e f g k a' b' c' d' e' f' g' k'
 . simp only[ite_true];exact g16F x wb p p' m n t hx hn hb hb' hs a b c d e f g k a' b' c' d' e' f' g' k'
lemma g24D (x wb:ὤ) (p p' p'' m n t:ὦ) (hx:Ѥn<=Ѩx) (hn:Ѥn<=Ѥm) (hb:Ѥp+8*Ѥm<=Ѩwb) (hb':Ѥp'+8*Ѥm<=Ѩwb) (hb'':Ѥp''+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (a b c d e f g k a' b' c' d' e' f' g' k' a'' b'' c'' d'' e'' f'' g'' k'':ὧ):
 go24D x wb p p' p'' m n t hx hn hb hb' hb'' hs a b c d e f g k a' b' c' d' e' f' g' k' a'' b'' c'' d'' e'' f'' g'' k''=g24of (go8D x wb p m n t hx hn hb hs a b c d e f g k) (go8D x wb p' m n t hx hn hb' hs a' b' c' d' e' f' g' k') (go8D x wb p'' m n t hx hn hb'' hs a'' b'' c'' d'' e'' f'' g'' k''):=by
 unfold go24D go8D;generalize armT=A;cases A
 . simp only[Bool.false_eq_true,ite_false];exact g24V x wb p p' p'' m n t hx hn hb hb' hb'' hs a b c d e f g k a' b' c' d' e' f' g' k' a'' b'' c'' d'' e'' f'' g'' k''
 . simp only[ite_true];exact g24F x wb p p' p'' m n t hx hn hb hb' hb'' hs a b c d e f g k a' b' c' d' e' f' g' k' a'' b'' c'' d'' e'' f'' g'' k''
def C_g8 (r:G8):Ὣ->ὧ
 | 0=>r.a
 | 1=>r.b
 | 2=>r.c
 | 3=>r.d
 | 4=>r.e
 | 5=>r.f
 | 6=>r.g
 | _=>r.k
def cols1T (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (i:Ὣ) (nf a:ὧ) (hx:ὥ) (lo:Ὠ) (m:ὦ) (rr:G8) (j0 j:Ὣ) (base:ὦ) (out:ѐ):ѐ:=
 if h4:j+4<=outW then
  let ju:=ѩj
  let o0:=fin8L inW relu X W b i j (C_g8 rr (j-j0)) nf a hx lo (rdW b ju) m
  let o1:=fin8L inW relu X W b i (j+1) (C_g8 rr (j+1-j0)) nf a hx lo (rdW b (ju+1)) m
  let o2:=fin8L inW relu X W b i (j+2) (C_g8 rr (j+2-j0)) nf a hx lo (rdW b (ju+2)) m
  let o3:=fin8L inW relu X W b i (j+3) (C_g8 rr (j+3-j0)) nf a hx lo (rdW b (ju+3)) m
  let out:=wrW out (base+ju) o0
  let out:=wrW out (base+ju+1) o1
  let out:=wrW out (base+ju+2) o2
  let out:=wrW out (base+ju+3) o3
  cols1T inW outW relu X W b i nf a hx lo m rr j0 (j+4) base out
 else if h1:j<outW then
  let ju:=ѩj
  let o:=fin8L inW relu X W b i j (C_g8 rr (j-j0)) nf a hx lo (rdW b ju) m
  cols1T inW outW relu X W b i nf a hx lo m rr j0 (j+1) base (wrW out (base+ju) o)
 else out
partial_fixpoint
def cols1W (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (i:Ὣ)
 (ix:ѐ) (x wd wb:ὤ) (nf a:ὧ) (hx:ὥ) (lo:Ὠ) (m n:ὦ) (j:Ὣ)
 (base:ὦ) (out:ѐ):ѐ:=
 if h8:j+8<=outW then
  let ju:=ѩj
  let jv:=(j+4).toUSize
  let w0:=ju*m
  let w1:=(ju+1)*m
  let w2:=(ju+2)*m
  let w3:=(ju+3)*m
  let v0:=jv*m
  let v1:=(jv+1)*m
  let v2:=(jv+2)*m
  let v3:=(jv+3)*m
  if hg:Ѥn<=Ѩx/\Ѥn<=ѧix/\ѧW<=4294967295/\Ѥw0+Ѥm<=ѧW/\Ѥw1+Ѥm<=ѧW/\Ѥw2+Ѥm<=ѧW/\Ѥw3+Ѥm<=ѧW/\
   Ѥv0+Ѥm<=ѧW/\Ѥv1+Ѥm<=ѧW/\Ѥv2+Ѥm<=ѧW/\Ѥv3+Ѥm<=ѧW/\Ѩwb<=4294967295/\Ѥw0+5+8*Ѥm<=Ѩwb then
   let rr:=go8WD x wb ix (w0+ὴ 5) m n 0 hg.1 hg.2.1 (pk w0 m wb hg.2.2.2.2.2.2.2.2.2.2.2.2 hg.2.2.2.2.2.2.2.2.2.2.2.1).1 (qA9 hg.2.2.2.2.2.2.2.2.2.2.2.1) 0 0 0 0 0 0 0 0
   let o0:=fin8L inW relu X W b i j rr.1.a nf a hx lo (rdW b ju) m
   let o1:=fin8L inW relu X W b i (j+1) rr.1.b nf a hx lo (rdW b (ju+1)) m
   let o2:=fin8L inW relu X W b i (j+2) rr.1.c nf a hx lo (rdW b (ju+2)) m
   let o3:=fin8L inW relu X W b i (j+3) rr.1.d nf a hx lo (rdW b (ju+3)) m
   let out:=wrW out (base+ju) o0
   let out:=wrW out (base+ju+1) o1
   let out:=wrW out (base+ju+2) o2
   let out:=wrW out (base+ju+3) o3
   let ju:=jv
   let o0:=fin8L inW relu X W b i (j+4) rr.2.a nf a hx lo (rdW b ju) m
   let o1:=fin8L inW relu X W b i ((j+4)+1) rr.2.b nf a hx lo (rdW b (ju+1)) m
   let o2:=fin8L inW relu X W b i ((j+4)+2) rr.2.c nf a hx lo (rdW b (ju+2)) m
   let o3:=fin8L inW relu X W b i ((j+4)+3) rr.2.d nf a hx lo (rdW b (ju+3)) m
   let out:=wrW out (base+ju) o0
   let out:=wrW out (base+ju+1) o1
   let out:=wrW out (base+ju+2) o2
   let out:=wrW out (base+ju+3) o3
   cols1W inW outW relu X W b i ix x wd wb nf a hx lo m n (j+8) base out
  else cols1 inW outW relu X W b i ix x wd nf a hx lo m n j base out
 else if ht:j<outW then
  if hg:Ѥn<=Ѩx/\Ѥn<=ѧix/\ѧW<=4294967295/\Ѩwb<=4294967295/\Ѥn<=Ѥm/\Ѥ(ѩj*m)+5+8*Ѥm<=Ѩwb then
   cols1T inW outW relu X W b i nf a hx lo m (g8of (go8W x wb ix (ѩj*m+ὴ 5) m n 0 hg.1 hg.2.1 (pk (ѩj*m) m wb hg.2.2.2.2.2 hg.2.2.2.1).1 (qA9 hg.2.2.2.1) 0 0 0 0 0 0 0 0)) j j base out
  else cols1 inW outW relu X W b i ix x wd nf a hx lo m n j base out
 else cols1 inW outW relu X W b i ix x wd nf a hx lo m n j base out
partial_fixpoint
def cols1V (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (i:Ὣ)
 (ix:ѐ) (x wd wb:ὤ) (nf a:ὧ) (hx:ὥ) (lo:Ὠ) (m n:ὦ) (j:Ὣ)
 (base:ὦ) (out:ѐ):ѐ:=
 if h8:j+8<=outW then
  let ju:=ѩj
  let jv:=(j+4).toUSize
  let w0:=ju*m
  let w1:=(ju+1)*m
  let w2:=(ju+2)*m
  let w3:=(ju+3)*m
  let v0:=jv*m
  let v1:=(jv+1)*m
  let v2:=(jv+2)*m
  let v3:=(jv+3)*m
  if hg:Ѥn<=Ѩx/\Ѥn<=ѧix/\ѧW<=4294967295/\Ѥw0+Ѥm<=ѧW/\Ѥw1+Ѥm<=ѧW/\Ѥw2+Ѥm<=ѧW/\Ѥw3+Ѥm<=ѧW/\
   Ѥv0+Ѥm<=ѧW/\Ѥv1+Ѥm<=ѧW/\Ѥv2+Ѥm<=ѧW/\Ѥv3+Ѥm<=ѧW/\Ѩwb<=4294967295/\Ѥn<=Ѥm/\Ѥw0+5+8*Ѥm<=Ѩwb then
   let rr:=go8D x wb (w0+ὴ 5) m n 0 hg.1 hg.2.2.2.2.2.2.2.2.2.2.2.2.1 (pk w0 m wb hg.2.2.2.2.2.2.2.2.2.2.2.2.2 hg.2.2.2.2.2.2.2.2.2.2.2.1).1 (qA9 hg.2.2.2.2.2.2.2.2.2.2.2.1) 0 0 0 0 0 0 0 0
   let o0:=fin8L inW relu X W b i j rr.a nf a hx lo (rdW b ju) m
   let o1:=fin8L inW relu X W b i (j+1) rr.b nf a hx lo (rdW b (ju+1)) m
   let o2:=fin8L inW relu X W b i (j+2) rr.c nf a hx lo (rdW b (ju+2)) m
   let o3:=fin8L inW relu X W b i (j+3) rr.d nf a hx lo (rdW b (ju+3)) m
   let out:=wrW out (base+ju) o0
   let out:=wrW out (base+ju+1) o1
   let out:=wrW out (base+ju+2) o2
   let out:=wrW out (base+ju+3) o3
   let ju:=jv
   let o0:=fin8L inW relu X W b i (j+4) rr.e nf a hx lo (rdW b ju) m
   let o1:=fin8L inW relu X W b i ((j+4)+1) rr.f nf a hx lo (rdW b (ju+1)) m
   let o2:=fin8L inW relu X W b i ((j+4)+2) rr.g nf a hx lo (rdW b (ju+2)) m
   let o3:=fin8L inW relu X W b i ((j+4)+3) rr.k nf a hx lo (rdW b (ju+3)) m
   let out:=wrW out (base+ju) o0
   let out:=wrW out (base+ju+1) o1
   let out:=wrW out (base+ju+2) o2
   let out:=wrW out (base+ju+3) o3
   cols1V inW outW relu X W b i ix x wd wb nf a hx lo m n (j+8) base out
  else cols1 inW outW relu X W b i ix x wd nf a hx lo m n j base out
 else if ht:j<outW then
  if hg:Ѥn<=Ѩx/\Ѥn<=ѧix/\ѧW<=4294967295/\Ѩwb<=4294967295/\Ѥn<=Ѥm/\Ѥ(ѩj*m)+5+8*Ѥm<=Ѩwb then
   cols1T inW outW relu X W b i nf a hx lo m (go8V x wb (ѩj*m+ὴ 5) m n 0 hg.1 hg.2.2.2.2.1 (pk (ѩj*m) m wb hg.2.2.2.2.2 hg.2.2.2.1).1 (qA9 hg.2.2.2.1) 0 0 0 0 0 0 0 0) j j base out
  else cols1 inW outW relu X W b i ix x wd nf a hx lo m n j base out
 else cols1 inW outW relu X W b i ix x wd nf a hx lo m n j base out
partial_fixpoint
@[noinline] def fw8 (inW:Ὣ) (relu:Ὡ) (X W b:@& ѐ) (i:Ὣ) (nf a:ὧ) (hx:ὥ) (lo:Ὠ) (m:ὦ) (j:Ὣ) (base:ὦ) (out:ѐ) (s0 s1 s2 s3 s4 s5 s6 s7:ὧ):ѐ:=
 let ju:=ѩj
 let jv:=(j+4).toUSize
 let o0:=fin8L inW relu X W b i j s0 nf a hx lo (rdW b ju) m
 let o1:=fin8L inW relu X W b i (j+1) s1 nf a hx lo (rdW b (ju+1)) m
 let o2:=fin8L inW relu X W b i (j+2) s2 nf a hx lo (rdW b (ju+2)) m
 let o3:=fin8L inW relu X W b i (j+3) s3 nf a hx lo (rdW b (ju+3)) m
 let out:=wrW out (base+ju) o0
 let out:=wrW out (base+ju+1) o1
 let out:=wrW out (base+ju+2) o2
 let out:=wrW out (base+ju+3) o3
 let ju:=jv
 let o0:=fin8L inW relu X W b i (j+4) s4 nf a hx lo (rdW b ju) m
 let o1:=fin8L inW relu X W b i ((j+4)+1) s5 nf a hx lo (rdW b (ju+1)) m
 let o2:=fin8L inW relu X W b i ((j+4)+2) s6 nf a hx lo (rdW b (ju+2)) m
 let o3:=fin8L inW relu X W b i ((j+4)+3) s7 nf a hx lo (rdW b (ju+3)) m
 let out:=wrW out (base+ju) o0
 let out:=wrW out (base+ju+1) o1
 let out:=wrW out (base+ju+2) o2
 let out:=wrW out (base+ju+3) o3
 out
def cols1VS (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (i:Ὣ) (ix:ѐ) (x wd wb:ὤ) (nf a:ὧ) (hx:ὥ) (lo:Ὠ) (m n:ὦ) (j:Ὣ) (base:ὦ) (out:ѐ):ѐ:=
 cols1V inW outW relu X W b i ix x wd wb nf a hx lo m n j base out
lemma c1VS' (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (i:Ὣ) (ix:ѐ) (x wd wb:ὤ) (nf a:ὧ) (hx:ὥ) (lo:Ὠ) (m n:ὦ) (j:Ὣ) (base:ὦ) (out:ѐ):
 cols1VS inW outW relu X W b i ix x wd wb nf a hx lo m n j base out=cols1V inW outW relu X W b i ix x wd wb nf a hx lo m n j base out:=rfl
@[inline] def dvN (T:ὤ) (hT:ѨT=512) (w:ὥ):ὧ:=
 ((w&&&0x7FFFFF) ||| 0x800000).toFloat*T.uget (w>>>(23:ὥ)).toUSize (qBx hT w)
lemma xi (xs:ὤ) (i t n:ὦ) (k:Ὣ) (hk:k<8) (ht:Ѥt< Ѥn) (hi:Ѥi=8*Ѥt+1) (hx:8*Ѥn+1<=Ѩxs) (hs:Ѩxs<2^32):(i+ὴ k).toNat< Ѩxs:=by rw[qAf _ _ (by rw[uk k (by exact Nat.lt_of_lt_of_le ‹_› ה)];exact omS23 hk ht hi hx hs),uk k (by exact Nat.lt_of_lt_of_le ‹_› ה)];(clear hs;exact omS24 hk ht hi hx)
local instance ecNe6:Nonempty F8:=⟨by constructor <;> exact default⟩
def go8Z (T:ὤ) (hT:ѨT=512) (xs:ὤ) (W:ѐ) (wo n t i:ὦ)
 (hx:8*Ѥn+1<=Ѩxs) (hsx:Ѩxs<2^32) (hs:ѧW<2^32)
 (hw:Ѥwo+Ѥn<=ѧW) (hi:Ѥi=8*Ѥt+1)
 (s0 s1 s2 s3 s4 s5 s6 s7:ὧ) (q:ὥ) (r:F8):F8:=
 if h:t<n then
  have ht:=А.mp h
  let w:=W.uget (wo+t) (wi wo t n W hw hs ht)
  let v:=dvN T hT w
  go8Z T hT xs W wo n (t+1) (i+ὴ 8) hx hsx hs hw
   (by
    rw[usz t n h,qAf _ _ (by rw[uk 8 (by decide)];omega),uk 8 (by decide)];omega)
   (s0+xs.uget (i+ὴ 0) (xi xs i t n 0 (by decide) ht hi hx hsx)*v)
   (s1+xs.uget (i+ὴ 1) (xi xs i t n 1 (by decide) ht hi hx hsx)*v)
   (s2+xs.uget (i+ὴ 2) (xi xs i t n 2 (by decide) ht hi hx hsx)*v)
   (s3+xs.uget (i+ὴ 3) (xi xs i t n 3 (by decide) ht hi hx hsx)*v)
   (s4+xs.uget (i+ὴ 4) (xi xs i t n 4 (by decide) ht hi hx hsx)*v)
   (s5+xs.uget (i+ὴ 5) (xi xs i t n 5 (by decide) ht hi hx hsx)*v)
   (s6+xs.uget (i+ὴ 6) (xi xs i t n 6 (by decide) ht hi hx hsx)*v)
   (s7+xs.uget (i+ὴ 7) (xi xs i t n 7 (by decide) ht hi hx hsx)*v) (q&&&((w&&&В)-1)) r
 else match r with
  | ⟨u,_,_,_,_,_,_,_,_,_,_⟩=>⟨u,s0,s1,s2,s3,s4,s5,s6,s7,0,q⟩
partial_fixpoint
def zA:ὤ:=pushZ 32 0 (ἆ 32)
def z0:ὤ:=pushZ 128 0 (ἆ 128)
def zB:ὤ:=pushZ 256 0 (ἆ 256)
def zC:ὤ:=pushZ 512 0 (ἆ 512)
def z1:ὤ:=pushZ 1024 0 (ἆ 1024)
def z4:ὤ:=pushZ 4096 0 (ἆ 4096)
def z8:ὤ:=pushZ 8192 0 (ἆ 8192)
@[inline] def zFor (m:Ὣ):ὤ:=
 if m<=32 then zA else if m<=128 then z0 else if m<=256 then zB else if m<=512 then zC else if m<=1024 then z1 else if m<=4096 then z4 else if m<=8192 then z8 else pushZ ѩm 0 (ἆ m)
@[inline] def wr8 (xs:ὤ) (i:ὦ) (a0 a1 a2 a3 a4 a5 a6 a7:ὧ):ὤ:=
 wrF (wrF (wrF (wrF (wrF (wrF (wrF (wrF xs (i+ὴ 0) a0) (i+ὴ 1) a1) (i+ὴ 2) a2)
  (i+ὴ 3) a3) (i+ὴ 4) a4) (i+ὴ 5) a5) (i+ὴ 6) a6) (i+ὴ 7) a7
def go8ZD (D:ὤ) (xs:ὤ) (W:ѐ) (hd:ѨD=ѧW) (wo n t i:ὦ)
 (hx:8*Ѥn+1<=Ѩxs) (hsx:Ѩxs<2^32) (hs:ѧW<2^32)
 (hw:Ѥwo+Ѥn<=ѧW) (hi:Ѥi=8*Ѥt+1)
 (s0 s1 s2 s3 s4 s5 s6 s7:ὧ) (q:ὥ) (r:F8):F8:=
 if h:t<n then
  have ht:=А.mp h
  let v:=D.uget (wo+t) (hd▸wi wo t n W hw hs ht)
  go8ZD D xs W hd wo n (t+1) (i+ὴ 8) hx hsx hs hw
   (by
    rw[usz t n h,qAf _ _ (by rw[uk 8 (by decide)];omega),uk 8 (by decide)];omega)
   (s0+xs.uget (i+ὴ 0) (xi xs i t n 0 (by decide) ht hi hx hsx)*v)
   (s1+xs.uget (i+ὴ 1) (xi xs i t n 1 (by decide) ht hi hx hsx)*v)
   (s2+xs.uget (i+ὴ 2) (xi xs i t n 2 (by decide) ht hi hx hsx)*v)
   (s3+xs.uget (i+ὴ 3) (xi xs i t n 3 (by decide) ht hi hx hsx)*v)
   (s4+xs.uget (i+ὴ 4) (xi xs i t n 4 (by decide) ht hi hx hsx)*v)
   (s5+xs.uget (i+ὴ 5) (xi xs i t n 5 (by decide) ht hi hx hsx)*v)
   (s6+xs.uget (i+ὴ 6) (xi xs i t n 6 (by decide) ht hi hx hsx)*v)
   (s7+xs.uget (i+ὴ 7) (xi xs i t n 7 (by decide) ht hi hx hsx)*v) q r
 else match r with
  | ⟨u,_,_,_,_,_,_,_,_,_,_⟩=>⟨u,s0,s1,s2,s3,s4,s5,s6,s7,0,q⟩
partial_fixpoint
def go8ZS (D:ὤ) (xs:ὤ) (W:ѐ) (hd:ѨD=ѧW) (wo n t i:ὦ)
 (hx:8*Ѥn+1<=Ѩxs) (hsx:Ѩxs<2^32) (hs:ѧW<2^32)
 (hw:Ѥwo+Ѥn<=ѧW) (hi:Ѥi=8*Ѥt+1)
 (s0 s1 s2 s3 s4 s5 s6 s7:ὧ) (q:ὥ) (r:F8):F8:=
 if h:armT/\t<n/\t+1<n then
  have ht:=А.mp h.2.1
  have hu:=А.mp h.2.2
  have hj:(i+ὴ 8).toNat=8*(t+1).toNat+1:=by rw[usz t n h.2.1,qAf _ _ (by rw[uk 8 (by decide)];omega),uk 8 (by decide)];omega
  let v:=D.uget (wo+t) (hd▸wi wo t n W hw hs ht)
  let u:=D.uget (wo+(t+1)) (hd▸wi wo (t+1) n W hw hs hu)
  go8ZS D xs W hd wo n (t+1+1) (i+ὴ 8+ὴ 8) hx hsx hs hw
   (by rw[usz (t+1) n h.2.2,qAf _ _ (by rw[uk 8 (by decide)];omega),uk 8 (by decide)];omega)
   (s0+xs.uget (i+ὴ 0) (xi xs i t n 0 (by decide) ht hi hx hsx)*v+xs.uget (i+ὴ 8+ὴ 0) (xi xs (i+ὴ 8) (t+1) n 0 (by decide) hu hj hx hsx)*u)
   (s1+xs.uget (i+ὴ 1) (xi xs i t n 1 (by decide) ht hi hx hsx)*v+xs.uget (i+ὴ 8+ὴ 1) (xi xs (i+ὴ 8) (t+1) n 1 (by decide) hu hj hx hsx)*u)
   (s2+xs.uget (i+ὴ 2) (xi xs i t n 2 (by decide) ht hi hx hsx)*v+xs.uget (i+ὴ 8+ὴ 2) (xi xs (i+ὴ 8) (t+1) n 2 (by decide) hu hj hx hsx)*u)
   (s3+xs.uget (i+ὴ 3) (xi xs i t n 3 (by decide) ht hi hx hsx)*v+xs.uget (i+ὴ 8+ὴ 3) (xi xs (i+ὴ 8) (t+1) n 3 (by decide) hu hj hx hsx)*u)
   (s4+xs.uget (i+ὴ 4) (xi xs i t n 4 (by decide) ht hi hx hsx)*v+xs.uget (i+ὴ 8+ὴ 4) (xi xs (i+ὴ 8) (t+1) n 4 (by decide) hu hj hx hsx)*u)
   (s5+xs.uget (i+ὴ 5) (xi xs i t n 5 (by decide) ht hi hx hsx)*v+xs.uget (i+ὴ 8+ὴ 5) (xi xs (i+ὴ 8) (t+1) n 5 (by decide) hu hj hx hsx)*u)
   (s6+xs.uget (i+ὴ 6) (xi xs i t n 6 (by decide) ht hi hx hsx)*v+xs.uget (i+ὴ 8+ὴ 6) (xi xs (i+ὴ 8) (t+1) n 6 (by decide) hu hj hx hsx)*u)
   (s7+xs.uget (i+ὴ 7) (xi xs i t n 7 (by decide) ht hi hx hsx)*v+xs.uget (i+ὴ 8+ὴ 7) (xi xs (i+ὴ 8) (t+1) n 7 (by decide) hu hj hx hsx)*u) q r
 else go8ZD D xs W hd wo n t i hx hsx hs hw hi s0 s1 s2 s3 s4 s5 s6 s7 q r
partial_fixpoint
lemma g8ZS (D:ὤ) (xs:ὤ) (W:ѐ) (hd:ѨD=ѧW) (wo n:ὦ) (hx:8*Ѥn+1<=Ѩxs) (hsx:Ѩxs<2^32) (hs:ѧW<2^32) (hw:Ѥwo+Ѥn<=ѧW) (r:F8) (t i:ὦ) (hi:Ѥi=8*Ѥt+1) (s0 s1 s2 s3 s4 s5 s6 s7:ὧ) (q:ὥ):
 go8ZS D xs W hd wo n t i hx hsx hs hw hi s0 s1 s2 s3 s4 s5 s6 s7 q r=go8ZD D xs W hd wo n t i hx hsx hs hw hi s0 s1 s2 s3 s4 s5 s6 s7 q r:=by
 generalize hm:Ѥn-Ѥt=m
 induction m using Nat.strong_induction_on generalizing t i hi s0 s1 s2 s3 s4 s5 s6 s7 q with
 | _ m ih=>
  rw[go8ZS];split
  . rename_i h;have:=usz t n h.2.1;have:=usz (t+1) n h.2.2;have:=А.mp h.2.2
    rw[ih _ (by clear hi h;omega) (t+1+1) _ _ _ _ _ _ _ _ _ _ _ rfl];simp only[go8ZD.eq_1 D xs W hd wo n t,go8ZD.eq_1 D xs W hd wo n (t+1),Ἄ h.2.1,Ἄ h.2.2]
  . rfl
def dnOf (T:ὤ) (hT:ѨT=512) (W:ѐ):ὤ:=⟨W.map (dvN T hT)⟩
lemma dnFoldL (f:ὥ->ὧ):∀(l:List ὥ) (a:ὤ),l.foldl (λa w=>a.push (f w)) a=⟨a.data++(l.map f).toArray⟩:=by
 intro l;induction l with
 | nil=>intro a;cases a;simp only[List.foldl_nil,List.map_nil,Array.append_empty]
 | cons x xs ih=>intro a;rw[ὔ,ih];cases a;simp only[FloatArray.push,List.push_append_toArray,List.map_cons]
lemma dnFold (f:ὥ->ὧ) (W:ѐ):W.foldl (λa w=>a.push (f w)) (ἆ W.size)=⟨W.map f⟩:=by
 rw[<-Array.foldl_toList,dnFoldL];simp only[ἆ,List.append_toArray,List.nil_append,FloatArray.mk.injEq];apply Array.ext';simp only[Array.toList_map]
@[inline] def nE (n:Ὣ) (W:ѐ):Ὣ:=let h:=scanH 2 W;if h<=hvI n then n else if h<=0x3EFFFFFF then 64 else if h<=0x3F7FFFFF then 16 else 0
def qScan (W:ѐ) (e i:ὦ) (q:ὥ):ὥ:=
 if h:i<e then qScan W e (i+1) (q&&&((rdW W i&&&В)-1)) else q
partial_fixpoint
def F8q (R:F8) (u:UInt8) (q:ὥ):F8:=⟨u,R.s0,R.s1,R.s2,R.s3,R.s4,R.s5,R.s6,R.s7,R.p,q⟩
abbrev WV8 (n:Ὣ) (W:ѐ):=({p:ὤ×ὥ×Ὣ//p=(dnOf (tZ (nE n W)) (qTs (qTZ (nE n W))) W,qScan W W.usize 0 Е,nE n W)})
local instance wv8Sub (n:Ὣ) (W:ѐ):Subsingleton (WV8 n W):=⟨λa b=>Subtype.ext (a.2.trans b.2.symm)⟩
structure WE8 where
 (w:ѐ) (n:Ὣ) (p:ὤ×ὥ×Ὣ) (g:ὦ) (ok:p=(dnOf (tZ (nE n w)) (qTs (qTZ (nE n w))) w,qScan w w.usize 0 Е,nE n w))
def wd8F (n:Ὣ) (W:ѐ) (g:ὦ) (s:Array WE8) (i:Ὣ):Option (WV8 n W):=
 if h:i<s.size then
  if s[i].g==g then
   if hn:s[i].n=n then
    if hk:withPtrEq s[i].w W (λ_=>s[i].w==W) (λe=>by rw[e];exact beq_self_eq_true W) then
     some ⟨s[i].p,by rw[<-hn,<-eq_of_beq hk];exact s[i].ok⟩
    else wd8F n W g s (i+1)
   else wd8F n W g s (i+1)
  else wd8F n W g s (i+1)
 else none
partial_fixpoint
@[noinline] def wd8S:ST.Ref Unit (Array WE8):=(ST.Prim.mkRef #[] (Void.mk ())).val
@[inline] def wd8Get (n:Ὣ) (W:ѐ):WV8 n W:=
 withPtrAddr W (λg=>
  let s:=(ST.Prim.Ref.get wd8S (Void.mk ())).val
  match wd8F n W g s 0 with
  | some v=>v
  | none=>
   let p:=(dnOf (tZ (nE n W)) (qTs (qTZ (nE n W))) W,qScan W W.usize 0 Е,nE n W)
   let old:=(ST.Prim.Ref.swap wd8S (#[⟨W,n,p,g,rfl⟩]++(if s.size<256 then s else s.pop)) (Void.mk ())).val
   match wd8F n W g old 0 with
   | some v=>v
   | none=>⟨p,rfl⟩) (λ_ _=>Subsingleton.elim _ _)
lemma dnU (T:ὤ) (hT:ѨT=512) (W:ѐ) (j:ὦ) (h1:Ѥj<(dnOf T hT W).size) (h2:Ѥj<ѧW):(dnOf T hT W).uget j h1=dvN T hT (W.uget j h2):=by
 simp only[FloatArray.uget,dnOf,Array.ugetElem_eq_getElem,Array.getElem_map,Array.uget]
lemma g8ZE (T:ὤ) (hT:ѨT=512) (xs:ὤ) (W:ѐ) (hd:(dnOf T hT W).size=ѧW) (wo n t i:ὦ) (hx:8*Ѥn+1<=Ѩxs) (hsx:Ѩxs<2^32) (hs:ѧW<2^32) (hw:Ѥwo+Ѥn<=ѧW) (hi:Ѥi=8*Ѥt+1) (s0 s1 s2 s3 s4 s5 s6 s7:ὧ) (q q':ὥ) (r r':F8):
 go8ZD (dnOf T hT W) xs W hd wo n t i hx hsx hs hw hi s0 s1 s2 s3 s4 s5 s6 s7 q' r'=F8q (go8Z T hT xs W wo n t i hx hsx hs hw hi s0 s1 s2 s3 s4 s5 s6 s7 q r) r'.u q':=by
 rw[go8ZD,go8Z];by_cases h:t<n
 . rw[Ἄ h,Ἄ h];dsimp only;rw[dnU T hT W (wo+t) _ (wi wo t n W hw hs (А.mp h))];exact g8ZE T hT xs W hd wo n (t+1) _ hx hsx hs hw _ _ _ _ _ _ _ _ _ _ _ r r'
 . rw[ἔ h,ἔ h];cases r;cases r';rfl
termination_by Ѥn-Ѥt
decreasing_by rw[usz t n h];have:=А.mp h;(clear h;omega)
/-! -/
def wbI (W:ѐ) (m r i:Ὣ) (a:ὤ):ὤ:=
 if i< m then wbI W m r (i+1) (((((((((a.push (dv W[r*m+i]!)).push (dv W[(r+1)*m+i]!)).push (dv W[(r+2)*m+i]!)).push (dv W[(r+3)*m+i]!)).push (dv W[(r+4)*m+i]!)).push (dv W[(r+5)*m+i]!)).push (dv W[(r+6)*m+i]!)).push (dv W[(r+7)*m+i]!)))
 else a
partial_fixpoint
def wbO (W:ѐ) (m nb c:Ὣ) (a:ὤ):ὤ:=
 if c<nb then wbO W m nb (c+1) (wbI W m (8*c) 0 a) else a
partial_fixpoint
def wbIT (W:ѐ) (m r o i:Ὣ) (a:ὤ):ὤ:=
 if i< m then wbIT W m r o (i+1) ((((((((a.push (dv W[(min (r+0) (o-1))*m+i]!)).push (dv W[(min (r+1) (o-1))*m+i]!)).push (dv W[(min (r+2) (o-1))*m+i]!)).push (dv W[(min (r+3) (o-1))*m+i]!)).push (dv W[(min (r+4) (o-1))*m+i]!)).push (dv W[(min (r+5) (o-1))*m+i]!)).push (dv W[(min (r+6) (o-1))*m+i]!)).push (dv W[(min (r+7) (o-1))*m+i]!))
 else a
partial_fixpoint
def wbZ (n:Ὣ):ὤ:=(((((ἆ (n+8)).push 0).push 0).push 0).push 0).push 0
def wbOf (m o:Ὣ) (W:ѐ):ὤ:=let a:=wbO W m (o/8) 0 (wbZ ѧW);if o%8=0 then a else wbIT W m (8*(o/8)) o 0 a
abbrev WVB (m o:Ὣ) (W:ѐ):=({p:ὤ×ὥ//p=(wbOf m o W,scanH 2 W)})
local instance wvbSub (m o:Ὣ) (W:ѐ):Subsingleton (WVB m o W):=⟨λa b=>Subtype.ext (a.2.trans b.2.symm)⟩
structure WEB where
 (w:ѐ) (m o:Ὣ) (p:ὤ×ὥ) (g:ὦ) (ok:p=(wbOf m o w,scanH 2 w))
def wbF (m o:Ὣ) (W:ѐ) (g:ὦ) (s:Array WEB) (i:Ὣ):Option (WVB m o W):=
 if h:i<s.size then
  if s[i].g==g then
   if ho:s[i].m=m/\s[i].o=o then
    if hk:withPtrEq s[i].w W (λ_=>s[i].w==W) (λe=>by rw[e];exact beq_self_eq_true W) then
     some ⟨s[i].p,by rw[<-ho.1,<-ho.2,<-eq_of_beq hk];exact s[i].ok⟩
    else wbF m o W g s (i+1)
   else wbF m o W g s (i+1)
  else wbF m o W g s (i+1)
 else none
partial_fixpoint
@[noinline] def wbS:ST.Ref Unit (Array WEB):=(ST.Prim.mkRef #[] (Void.mk ())).val
@[inline] def wbGet (m o:Ὣ) (W:ѐ):WVB m o W:=
 withPtrAddr W (λg=>
  let s:=(ST.Prim.Ref.get wbS (Void.mk ())).val
  match wbF m o W g s 0 with
  | some v=>v
  | none=>
   let p:=(wbOf m o W,scanH 2 W)
   let old:=(ST.Prim.Ref.swap wbS (#[⟨W,m,o,p,g,rfl⟩]++(if s.size<256 then s else s.pop)) (Void.mk ())).val
   match wbF m o W g old 0 with
   | some v=>v
   | none=>⟨p,rfl⟩) (λ_ _=>Subsingleton.elim _ _)
lemma qUs (a:ὤ) (i:ὦ) (v:ὧ) (h:Ѥi< Ѩa):(a.uset i v h).size=Ѩa:=by obtain ⟨ds⟩:=a;exact Array.size_uset _
def decE (X:ѐ) (o n t:ὦ) (buf:ὤ) (l:ὧ) (hX:Ѥo+Ѥn<=ѧX) (hs:ѧX<2^32) (hb:Ѥn<=Ѩbuf):R2:=
 if h:t<n then
  have ht:=А.mp h
  let v:=dv (X.uget (o+t) (wi o t n X hX hs ht))
  decE X o n (t+1) (buf.uset t v (by omega)) (l+v.abs) hX hs (by rw[qUs];exact hb)
 else ⟨buf,l,true⟩
partial_fixpoint
@[inline] def decF (X:ѐ) (o n:ὦ) (buf:ὤ) (l:ὧ):R2:=
 if hh:Ѥo+Ѥn<=ѧX/\ѧX<=4294967295/\Ѥn<=Ѩbuf/\Ѩbuf<=4294967295 then decE X o n 0 buf l hh.1 (by have:=hh.2.1;omega) hh.2.2.1 else decD X o n 0 buf l
structure R3 where
 (buf:ὤ) (l:ὧ) (lo hi:ὥ)
local instance ecNe7:Nonempty R3:=⟨by constructor <;> exact default⟩
def decV (X:ѐ) (o n t:ὦ) (buf:ὤ) (l:ὧ) (lo hi:ὥ) (hX:Ѥo+Ѥn<=ѧX) (hs:ѧX<2^32) (hb:Ѥn<=Ѩbuf):R3:=
 if h:t<n then
  have ht:=А.mp h
  let w:=X.uget (o+t) (wi o t n X hX hs ht)
  let v:=w&&&В
  let u:=v-1
  let d:=dv w
  decV X o n (t+1) (buf.uset t d (by omega)) (l+d.abs) (if u<lo then u else lo) (if hi<v then v else hi) hX hs (by rw[qUs];exact hb)
 else ⟨buf,l,lo,hi⟩
termination_by Ѥn-Ѥt
decreasing_by rw[usz t n h];exact Nat.sub_lt_sub_left (А.mp h) (Nat.lt_succ_self _)
def bOf (b:ѐ):ὤ:=⟨b.map dv⟩
def bOfF (b:ѐ):ὤ:=b.foldl (λa w=>a.push (dv w)) (ἆ b.size)
lemma bOfF_eq (b:ѐ):bOfF b=bOf b:=dnFold _ b
abbrev WVN (m o:Ὣ) (W b:ѐ):=({p:(ὤ×ὥ)×(ὤ×ὥ)//p=((wbOf m o W,scanH 2 W),(bOf b,scanH 1 b))})
structure WEN where
 (w b:ѐ) (m o:Ὣ) (p:(ὤ×ὥ)×(ὤ×ὥ)) (g:ὦ) (ok:p=((wbOf m o w,scanH 2 w),(bOf b,scanH 1 b)))
def wnF (m o:Ὣ) (W b:ѐ) (g:ὦ) (s:Array WEN) (i:Ὣ):Option (WVN m o W b):=
 if h:i<s.size then
  if s[i].g==g then
   if ho:s[i].m=m/\s[i].o=o then
    if hk:withPtrEq s[i].w W (λ_=>s[i].w==W) (λe=>by rw[e];exact beq_self_eq_true W) then
     if hk2:withPtrEq s[i].b b (λ_=>s[i].b==b) (λe=>by rw[e];exact beq_self_eq_true b) then
      some ⟨s[i].p,by rw[<-ho.1,<-ho.2,<-eq_of_beq hk,<-eq_of_beq hk2];exact s[i].ok⟩
     else wnF m o W b g s (i+1)
    else wnF m o W b g s (i+1)
   else wnF m o W b g s (i+1)
  else wnF m o W b g s (i+1)
 else none
partial_fixpoint
@[noinline] def wnS:ST.Ref Unit (Array WEN):=(ST.Prim.mkRef #[] (Void.mk ())).val
@[inline] def wnGet (m o:Ὣ) (W b:ѐ):WVN m o W b:=
 withPtrAddr W (λg=>
  let s:=(ST.Prim.Ref.get wnS (Void.mk ())).val
  match wnF m o W b g s 0 with
  | some v=>v
  | none=>
   let p:=((wbOf m o W,scanH 2 W),(bOfF b,scanH 1 b))
   let old:=(ST.Prim.Ref.swap wnS (#[⟨W,b,m,o,p,g,(congrArg (((wbOf m o W,scanH 2 W),·)) (congrArg (·,scanH 1 b) (bOfF_eq b)))⟩]++(if s.size<256 then s else s.pop)) (Void.mk ())).val
   match wnF m o W b g old 0 with
   | some v=>v
   | none=>⟨p,(congrArg (((wbOf m o W,scanH 2 W),·)) (congrArg (·,scanH 1 b) (bOfF_eq b)))⟩) (λ_ _=>Subsingleton.elim _ _)
@[inline] def rdB (a:ὤ) (i:ὦ):ὧ:=
 if h:i<a.usize then a.uget i (qDo a i h) else 0
@[noinline] def fin8u (inW:Ὣ) (relu:Ὡ) (X W b:@& ѐ) (i:Ὣ) (u:ὦ) (s nf a:ὧ) (hx:ὥ) (n:ὦ):ὥ:=
 fin8 inW relu X W b i Ѥu s nf a hx (rdW b u) n
@[inline] def fin8LU (inW:Ὣ) (relu:Ὡ) (X W b:ѐ) (i:Ὣ) (s nf a:ὧ) (hx:ὥ) (lo sp:Ὠ) (B:ὤ) (u:ὦ) (n:ὦ):ὥ:=
 let t:=fin1 lo sp (s+rdB B u)
 if t==Е then fin8u inW relu X W b i u s nf a hx n else fz relu t
@[noinline] def fN8 (inW:Ὣ) (relu:Ὡ) (X W b:@& ѐ) (i:Ὣ) (B:ὤ) (nf a:ὧ) (hx:ὥ) (lo sp:Ὠ) (m:ὦ) (j:Ὣ) (base:ὦ) (out:ѐ) (s0 s1 s2 s3 s4 s5 s6 s7:ὧ):ѐ:=
 let ju:=ѩj
 let jv:=(j+4).toUSize
 let o0:=fin8LU inW relu X W b i s0 nf a hx lo sp B ju m
 let o1:=fin8LU inW relu X W b i s1 nf a hx lo sp B (ju+1) m
 let o2:=fin8LU inW relu X W b i s2 nf a hx lo sp B (ju+2) m
 let o3:=fin8LU inW relu X W b i s3 nf a hx lo sp B (ju+3) m
 let out:=wrW out (base+ju) o0
 let out:=wrW out (base+ju+1) o1
 let out:=wrW out (base+ju+2) o2
 let out:=wrW out (base+ju+3) o3
 let ju:=jv
 let o0:=fin8LU inW relu X W b i s4 nf a hx lo sp B ju m
 let o1:=fin8LU inW relu X W b i s5 nf a hx lo sp B (ju+1) m
 let o2:=fin8LU inW relu X W b i s6 nf a hx lo sp B (ju+2) m
 let o3:=fin8LU inW relu X W b i s7 nf a hx lo sp B (ju+3) m
 let out:=wrW out (base+ju) o0
 let out:=wrW out (base+ju+1) o1
 let out:=wrW out (base+ju+2) o2
 let out:=wrW out (base+ju+3) o3
 out
def colsN2 (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (i:Ὣ) (ix:ѐ) (x wd wb B:ὤ) (nf a:ὧ) (hx:ὥ) (lo sp:Ὠ) (m n:ὦ) (j:Ὣ) (base:ὦ) (out:ѐ):ѐ:=
 if h16:j+16<=outW then
  let w0:=ѩj*m
  let yw0:=(j+8).toUSize*m
  if hg:Ѥn<=Ѩx/\Ѩwb<=4294967295/\Ѥn<=Ѥm/\Ѥw0+5+8*Ѥm<=Ѩwb/\Ѥyw0+5+8*Ѥm<=Ѩwb then
   let rr:=go16D x wb (w0+ὴ 5) (yw0+ὴ 5) m n 0 hg.1 hg.2.2.1 (pk w0 m wb hg.2.2.2.1 hg.2.1).1 (pk yw0 m wb hg.2.2.2.2 hg.2.1).1 (qA9 hg.2.1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
   let out:=fN8 inW relu X W b i B nf a hx lo sp m j base out rr.a rr.b rr.c rr.d rr.e rr.f rr.g rr.k
   let out:=fN8 inW relu X W b i B nf a hx lo sp m (j+8) base out rr.a' rr.b' rr.c' rr.d' rr.e' rr.f' rr.g' rr.k'
   colsN2 inW outW relu X W b i ix x wd wb B nf a hx lo sp m n (j+8+8) base out
  else cols1V inW outW relu X W b i ix x wd wb nf a hx lo m n j base out
 else cols1V inW outW relu X W b i ix x wd wb nf a hx lo m n j base out
partial_fixpoint
def colsN3 (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (i:Ὣ) (ix:ѐ) (x wd wb B:ὤ) (nf a:ὧ) (hx:ὥ) (lo sp:Ὠ) (m n:ὦ) (j:Ὣ) (base:ὦ) (out:ѐ):ѐ:=
 if h24:j+24<=outW then
  let w0:=ѩj*m
  let yw0:=(j+8).toUSize*m
  let zw0:=(j+8+8).toUSize*m
  if hg:Ѥn<=Ѩx/\Ѩwb<=4294967295/\Ѥn<=Ѥm/\Ѥw0+5+8*Ѥm<=Ѩwb/\Ѥyw0+5+8*Ѥm<=Ѩwb/\Ѥzw0+5+8*Ѥm<=Ѩwb then
   let rr:=go24D x wb (w0+ὴ 5) (yw0+ὴ 5) (zw0+ὴ 5) m n 0 hg.1 hg.2.2.1 (pk w0 m wb hg.2.2.2.1 hg.2.1).1 (pk yw0 m wb hg.2.2.2.2.1 hg.2.1).1 (pk zw0 m wb hg.2.2.2.2.2 hg.2.1).1 (qA9 hg.2.1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
   let out:=fN8 inW relu X W b i B nf a hx lo sp m j base out rr.a rr.b rr.c rr.d rr.e rr.f rr.g rr.k
   let out:=fN8 inW relu X W b i B nf a hx lo sp m (j+8) base out rr.a' rr.b' rr.c' rr.d' rr.e' rr.f' rr.g' rr.k'
   let out:=fN8 inW relu X W b i B nf a hx lo sp m (j+8+8) base out rr.a'' rr.b'' rr.c'' rr.d'' rr.e'' rr.f'' rr.g'' rr.k''
   colsN3 inW outW relu X W b i ix x wd wb B nf a hx lo sp m n (j+8+8+8) base out
  else colsN2 inW outW relu X W b i ix x wd wb B nf a hx lo sp m n j base out
 else colsN2 inW outW relu X W b i ix x wd wb B nf a hx lo sp m n j base out
partial_fixpoint
def linF0 (rows inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ):ѐ:=
 let hw:=scanW rows W
 if rows*inW=ѧX/\outW*inW=ѧW/\outW=ѧb/\0< inW/\0<outW/\inW<2^20/\
  ѧX<=4294967295/\ѧW<=4294967295/\ѧb<=4294967295/\rows*outW<=4294967295/\
  scanH 0 X≠Е/\hw≠Е/\scanH 1 b≠Е then
  let n:=ѩinW
  let nf:ὧ:=(inW.toUInt64+8).toFloat*Ἃ 0x3CA0000000000000
  let out:ѐ:=ἇ (rows*outW) 0
  if rows==1 then
   let p:=(wbGet inW outW W).1
   if p.2≠Е then
    let v:=decF X (0*inW).toUSize n (zFor inW) 0
    if v.ok&&inW<=8192 then cols1VS inW outW relu X W b 0 idA v.buf (ἆ 0) p.1 nf v.l p.2 (loOf (nf*(dv p.2*v.l+dv (scanH 1 b)))) n n 0 (0*outW).toUSize (ἇ (1*outW) 0)
    else linearWith dotFn rows inW outW relu X W b
   else linearWith dotFn rows inW outW relu X W b
  else
   linearWith dotFn rows inW outW relu X W b
 else linearWith dotFn rows inW outW relu X W b
def fz0:ὧ:=0
@[inline] def zU (lo v:ὥ):ὥ:=if v-1<lo then v-1 else lo
@[inline] def zV (hi v:ὥ):ὥ:=if hi<v then v else hi
def decZ (X:ѐ) (n:ὦ) (hX:Ѥn<=ѧX):Ὣ->ὦ->ὤ->ѐ->ὦ->ὧ->ὥ->ὥ->(ὤ×ѐ)×(ὦ×ὧ)×(ὥ×ὥ)
 | 0,_,buf,ix,c,l,lo,hi=>((buf,ix),(c,l),(lo,hi))
 | f+1,t,buf,ix,c,l,lo,hi=>
  if h:t<n then
   let w:=X.uget t (Nat.lt_of_lt_of_le (А.mp h) hX)
   let v:=w&&&В
   if v==0 then decZ X n hX f (t+1) buf ix c l (zU lo v) (zV hi v)
   else
    let d:=dv w
    decZ X n hX f (t+1) (wrF buf c d) (wrW ix c t.toUInt32) (c+1) (l+d.abs) (zU lo v) (zV hi v)
  else ((buf,ix),(c,l),(lo,hi))
def zgL (X:ѐ) (s i:ὦ) (nz:Ὣ):Ὣ->Ὡ
 | 0=>true
 | f+1=>if rdW X i&&&В==0 then zgL X s (i+s) nz f else if nz==0 then false else zgL X s (i+s) (nz-1) f
def linZc (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (c:(ὤ×ὥ)×(ὤ×ὥ)) (nf:ὧ) (n:ὦ) (z:ὤ) (hX:Ѥn<=ѧX):ѐ:=
 let r:=decZ X n hX Ѥn 0 z (ἇ Ѥn 0) 0 0 Е 0
 if 0x007FFFFF<=r.2.2.1/\r.2.2.2<У then
  cols1W inW outW relu X W b 0 r.1.2 r.1.1 (ἆ 0) c.1.1 nf r.2.1.2 c.1.2 (loOf (nf*(dv c.1.2*r.2.1.2+dv c.2.2))) n r.2.1.1 0 (0*outW).toUSize (ἇ (1*outW) 0)
 else linF0 1 inW outW relu X W b
@[noinline] def linNc (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (c:@& ((ὤ×ὥ)×(ὤ×ὥ))):ѐ:=
 if 1*inW=ѧX/\outW*inW=ѧW/\outW=ѧb/\0< inW/\0<outW/\inW<2^20/\ѧX<=4294967295/\ѧW<=4294967295/\ѧb<=4294967295/\1*outW<=4294967295/\scanW 1 W≠Е/\c.2.2≠Е/\c.1.2≠Е/\inW<=8192 then
  let n:=ѩinW
  let nf:ὧ:=(inW.toUInt64+8).toFloat*Ἃ 0x3CA0000000000000
  let o:=(0*inW).toUSize
  let z:=zFor inW
  if hz:Ѥo+Ѥn<=ѧX/\Ѥn<=Ѩz/\Ѩz<=4294967295/\ѧX<=4294967295 then
   if 128<=inW&&zgL X (ѩ(inW/32)) 0 10 32 then linZc inW outW relu X W b c nf n z (Nat.le_trans (Nat.le_add_left _ _) hz.1)
   else
   let v:=decV X o n 0 z fz0 Е 0 hz.1 (by have:=hz.2.2.2;omega) hz.2.1
   if 0x007FFFFF<=v.lo/\v.hi<У then
    let lo:=loOf (nf*(dv c.1.2*v.l+dv c.2.2))
    if 128<=inW then colsN3 inW outW relu X W b 0 idA v.buf (ἆ 0) c.1.1 c.2.1 nf v.l c.1.2 lo (spOf lo) n n 0 (0*outW).toUSize (ἇ (1*outW) 0)
    else colsN2 inW outW relu X W b 0 idA v.buf (ἆ 0) c.1.1 c.2.1 nf v.l c.1.2 lo (spOf lo) n n 0 (0*outW).toUSize (ἇ (1*outW) 0)
   else linF0 1 inW outW relu X W b
  else linF0 1 inW outW relu X W b
 else linF0 1 inW outW relu X W b
def linN (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ):ѐ:=linNc inW outW relu X W b (wnGet inW outW W b).1
def linF (rows inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ):ѐ:=
 if rows==1 then linN inW outW relu X W b else linF0 rows inW outW relu X W b
section
noncomputable def R (f:ὧ):ℝ:=Ἢ (toModel (ofFloat f:Binary64))
noncomputable def wv (w:ὥ):ℝ:=Ἢ (ofW w)
noncomputable def exactS (n:Ὣ) (xw ww:Ὣ->ὥ) (bw:ὥ):ℝ:=
 (Ἣ n).sum (λt=>wv (xw t)*wv (ww t))+wv bw
def nfOf (n:Ὣ):ὧ:=(n.toUInt64+8).toFloat*Ἃ 0x3CA0000000000000
structure C_G (R I O:Ὣ) (X W b:ё):Prop where
 hX:R*I=ѧX
 hW:O*I=ѧW
 hb:O=ѧb
 pi:0<I
 po:0<O
 ni:I<2^20
 sX:ѧX<2^32
 sW:ѧW<2^32
 sb:ѧb<2^32
 sO:R*O<2^32
 gX:∀q,q< ѧX->good X[q]! =true
 gW:∀q,q< ѧW->good W[q]! =true
 gb:∀q,q< ѧb->good b[q]! =true
lemma qAj:⟡.isIEEE=true:=ה
lemma qAy (a:Model ⟡):Ἑ a=(Model.toModel a).isFinite:=by rw[<-Model.toDyadic?_isSome_eq_isFinite,Model.toDyadic?_ieee_eq_model _ qAj];cases Model.toModel a<;>rfl
abbrev A_M (x:ὧ):Model ⟡:=Model.ofFloatModel x.toModel
lemma A_R (x:ὧ):R x=Ἢ (A_M x):=rfl
lemma qAh (x:ὧ):Ἑ (A_M x)=x.isFinite:=by rw[qAy,Ὀ];rfl
lemma qA3 {x:ὧ} (h:xᶠ):Ἑ (A_M x)=true:=(qAh x).trans h
lemma qA0 (e:ℤ):Formats.Flocq.bpow ἲ e=(2:ℝ)^e:=by simp only[Formats.Flocq.bpow,FloatLib.Numerics.Radix.toReal,ἲ,Nat.cast_ofNat]
lemma qCd:(2:ℝ)^(1000:ℤ)<=Ἢ (Model.posMaxFinite ⟡):=by
 rw[Model.toReal_posMaxFinite,show Model.pow2 ⟡.fracWidth+⟡.maxFiniteFracField=2^53-1 by decide+kernel,
  show ⟡.maxNormalExponent-Int.ofNat ⟡.fracWidth=971 by
   decide+kernel,Model.bpow,qA0,show (1000:ℤ)=29+971 from rfl,Ἓ ἓ]
 exact ἀ (by norm_num1) ו
open Float.Model.UnpackedFloat in
lemma qDv {b:BitVec 64} {s m e h} (a:unpack ⟠ b=.finite s m e h):2^52<=m\/e=-1074:=by
 unfold Ἶ at a;simp only at a;rw[show (⟠.exponentBias:ℤ)=1023 by decide] at a;split_ifs at a with h1 h2 h3;. injection a with _ _ he;rw[show (unpackExponent (spec:=⟠) b).toNat=0 by simp only[h3,BitVec.toNat_ofNat,Nat.reducePow,Nat.zero_mod]] at he;exact omS27 he
 . injection a with _ hm;rw[<-hm,BitVec.toNat_append];exact .inl (ὂ (by simp only[Nat.reducePow,BitVec.toNat_ofNat,pow_one,Nat.mod_succ,ὁ,one_mul,Std.le_refl]) Nat.left_le_or)
open Float.Model.UnpackedFloat in
lemma qH1 {s₁ s₂ m₁ m₂ e₁ e₂ h₁ h₂} {f g:BitVec 64} (e:unpack ⟠ f=.finite s₁ m₁ e₁ h₁) (b:unpack ⟠ g=.finite s₂ m₂ e₂ h₂):
 e₁+e₂<=⟠.targetExponent (Float.Model.totalExponent (m₁*m₂) (e₁+e₂)):=by
 have a:2^52<=m₁*m₂->52<=(m₁*m₂).log2:=(Nat.le_log2 (Nat.mul_ne_zero h₁.ne' h₂.ne')).2
 simp only[Float.Model.Format.targetExponent,Float.Model.totalExponent,Float.Model.Format.mantissaBits,Float.Model.Format.minExponent]
 (norm_num1)
 rcases qDv e with c | c
 . have:=a (c.trans (Ὄ _ h₂));(clear c;omega)
 rcases qDv b with d | d
 . have:=a (d.trans (Nat.le_mul_of_pos_left _ h₁));(clear d;omega)
 omega
lemma qB0 (x y:ὧ) (a:xᶠ) (b:yᶠ):A_M (x*y)=Model.mul (A_M x) (A_M y):=by
 change Model.ofFloatModel (Float.Model.mul x.toModel y.toModel)=_;rw[Float.Model.mul,Model.ofFloatModel_pack,Model.Proof.mul_eq_spec,Model.Spec.mul];simp only[Model.toDyadic?_ieee_eq_model _ qAj,Model.toModel_ofFloatModel];change x.toModel.unpack.isFinite=true at a;change y.toModel.unpack.isFinite=true at b;have c:∀s t:Float.Model.UnpackedFloat.Sign,Model.ofModel ⟡ (.zero (s*t))=Model.zero _ (Ἴ s ^^ Ἴ t):=λs t=>by cases s<;>cases t<;>simp only[Model.modelSignBit,bne_self_eq_false,FloatFormat.isIEEE_binary64,Model.zero_eq_signedZero_of_isIEEE,false_eq_true, ↓reduceIte,Bool.bne_false,Bool.bne_true,Bool.not_false]<;>first | exact (Model.negZero_eq_ofModel_zero _).symm | exact (Model.posZero_eq_ofModel_zero _).symm
 cases hxm:x.toModel.unpack<;>cases hym:y.toModel.unpack<;>simp only[hxm,hym,Float.Model.UnpackedFloat.isFinite,Bool.false_eq_true] at a b
 case finite.finite s₁ m₁ e₁ h₁ s₂ m₂ e₂ h₂=>
  have d:(m₁==0 || m₂==0)=false:=by simp only[Bool.or_eq_false_iff,beq_eq_false_iff_ne,ne_eq,h₁.ne',not_false_eq_true,h₂.ne',and_self]
  have e:Model.modelSign (Ἴ s₁ ^^ Ἴ s₂)=s₁*s₂:=by cases s₁<;>cases s₂<;>rfl
  simp only[Model.unpackedToDyadic?,Float.Model.UnpackedFloat.mul];rw[Model.roundWithAccuracy_exact_eq_round_of_le_targetExponent _ _ _ _ (qH1 hxm hym),d];simp only[Bool.false_eq_true,ite_false,Model.roundDyadic,qAj,ite_true,Model.ieeeRoundDyadic,e];rfl
 all_goals simp only[Float.Model.UnpackedFloat.mul,c,FloatFormat.isIEEE_binary64,Model.zero_eq_signedZero_of_isIEEE,bne_iff_ne,ne_eq,ite_not,Model.unpackedToDyadic?,BEq.rfl,Bool.or_self, ↓reduceIte,Bool.true_or,Bool.or_true]
lemma A_E (z:ὧ) {a:Model ⟡} {c:ℝ} (h:A_M z=a) (b:Ἑ a=true) (d:Ἢ a=Model.roundAt ⟡ c):zᶠ/\R z=Model.roundAt ⟡ c:=by subst h;exact ⟨(qAh z).symm.trans b,d⟩
lemma qEd (x y:ὧ) (hx:xᶠ) (hy:yᶠ) (hb: |R x| * |R y| <=2^(1000:ℤ)):
 (x*y)ᶠ /\R (x*y)=Model.roundAt ⟡ (R x*R y):=
 A_E _ (qB0 x y hx hy) (Model.isFinite_mul_of_abs_mul_le_posMaxFinite _ _ qAj (qA3 hx) (qA3 hy) (hb.trans qCd))
  (Model.toReal_mul_eq_roundAt_of_abs_mul_le_posMaxFinite _ _ qAj (qA3 hx) (qA3 hy) (hb.trans qCd))
noncomputable def A_z:ℝ:=2^(-1021:ℤ)
noncomputable def A_B:ℝ:=2^(300:ℤ)
lemma qAr:0<A_z:=by unfold A_z;positivity
lemma A_p (a b c d:ℤ) (h:a+b<=c+d):(2:ℝ)^a*2^b<=2^c*2^d:=by rw[<-Ἓ ἓ,<-Ἓ ἓ];exact ἒ one_le_two h
lemma qAw:A_z<=1/2^200*(1/2^200):=by simpa only[A_z,Int.reduceNeg,zpow_neg,zpow_ofNat,one_div,pow_zero,mul_one] using A_p (-1021) 0 (-200) (-200) (by norm_num1)
lemma qI1:0<A_B:=by unfold A_B;positivity
lemma qCc:(2:ℝ)^256*2^42<=A_B:=by simpa only[A_B,zpow_ofNat,pow_zero,mul_one] using A_p 256 42 300 0 (by norm_num1)
lemma qAn:(2:ℝ)^128<=A_B:=(by (norm_num1):(2:ℝ)^128<=2^256*2^42).trans qCc
lemma qE7 {k:ℕ} (h:k<=2^128):(k:ℝ)<=A_B:=(show (k:ℝ)<=2^128 by exact_mod_cast h).trans qAn
lemma qE3:A_B*2<=2^(1000:ℤ):=by simpa only[A_B,zpow_ofNat,pow_one,pow_zero,mul_one] using A_p 300 1 1000 0 (by norm_num1)
lemma qDe:A_B*A_B<=2^(1000:ℤ):=by simpa only[A_B,zpow_ofNat,pow_zero,mul_one] using A_p 300 300 1000 0 (by norm_num1)
lemma qFp (x:ℝ):
 |Model.roundAt ⟡ x-x| <=1/2^53*(|x| +A_z):=by
 rw[show (1/2^53*(|x| +A_z):ℝ)=2^(-53:ℤ)* |x| +2^(-1074:ℤ) by
  rw[show (1/2^53:ℝ)=2^(-53:ℤ) by (norm_num1),show (-1074:ℤ)=-53+-1021 from rfl,Ἓ ἓ (-53),mul_add]
  rfl]
 by_cases a:x=0
 . rw[a,Model.roundAt_zero,sub_self,abs_zero];positivity
 refine (Model.abs_roundAt_sub_le _ x).trans ?_
 change Formats.Flocq.ulp _ (Model.fexpOf ⟡) x/2<=_
 rw[Formats.Flocq.ulp.of_ne_zero _ _ x a]
 simp only[Formats.Flocq.cexp,Model.fexpOf,Formats.Flocq.fltExp]
 have b:=Formats.Flocq.bpow_magnitude_sub_one_le ἲ x a
 rw[qA0] at b
 rw[qA0,show ((⟡.fracWidth:ℤ)+1)=53 from rfl,show ⟡.minSubnormalExponent=-1074 from rfl]
 generalize Formats.Flocq.magnitude ἲ x=M at b⊢
 have d:(0:ℝ)<=2^(-53:ℤ)* |x|:=ו
 have c:(0:ℝ)<2^(-1074:ℤ):=ו
 rcases le_total (-1074) (M-53) with h | h
 . rw[max_eq_left h,show (2:ℝ)^(M-53)/2=2^(-53:ℤ)*2^(M-1) by
    rw[<-Ἓ ἓ,show (-53:ℤ)+(M-1)=M-53-1 by ring,ᾂ ἓ _ 1,zpow_one]]
   linarith only[ἁ b (show (0:ℝ)<=2^(-53:ℤ) by positivity),c,b]
 . rw[max_eq_right h];exact (half_le_self c.le).trans (le_add_of_nonneg_left d)
lemma qFc {z:ὧ} {c:ℝ} (h:zᶠ/\R z=Model.roundAt ⟡ c):zᶠ/\|R z-c| <=1/2^53*(|c| +A_z):=⟨h.1,h.2▸qFp c⟩
lemma qEe {a b:ℝ} (d: |a| <=A_B) (c: |b| <=A_B):
 |a| + |b| <=Ἢ (Model.posMaxFinite ⟡):=by
 have:=qE3
 have:=qCd
 linarith
section
variable (x y:ὧ) (hx:xᶠ) (hy:yᶠ) (h1 : |R x| ≤ A_B) (h2 : |R y| ≤ A_B)
include x y hx hy h1 h2
lemma qBf:(x+y)ᶠ/\|R (x+y)-(R x+R y)| <=1/2^53*(|R x+R y| +A_z):=
 qFc (A_E _ (toModel_ofFloat_add_of_isFinite x y hx hy) (Model.isFinite_add_of_abs_add_le_posMaxFinite _ _ qAj (qA3 hx) (qA3 hy) (qEe h1 h2))
  (Model.toReal_add_eq_roundAt_of_abs_add_le_posMaxFinite _ _ qAj (qA3 hx) (qA3 hy) (qEe h1 h2)))
lemma qDf:(x-y)ᶠ/\|R (x-y)-(R x-R y)| <=1/2^53*(|R x-R y| +A_z):=
 qFc (A_E _ (toModel_ofFloat_sub_of_isFinite x y hx hy) (Model.isFinite_sub_of_abs_add_le_posMaxFinite _ _ qAj (qA3 hx) (qA3 hy) (qEe h1 h2))
  (Model.toReal_sub_eq_roundAt_of_abs_add_le_posMaxFinite _ _ qAj (qA3 hx) (qA3 hy) (qEe h1 h2)))
lemma qCf:(x*y)ᶠ/\|R (x*y)-R x*R y| <=1/2^53*(|R x*R y| +A_z):=qFc (qEd x y hx hy ((ἑ h1 h2 (Ἀ _) qI1.le).trans qDe))
end
lemma qA2 (m k:ℤ) (hm:m.natAbs<2^53) (hk:-1074<=k):Model.roundAt ⟡ ((m:ℝ)*2^k)=(m:ℝ)*2^k:=by apply Formats.Flocq.round_preserves_generic;exact (Formats.Flocq.generic_format_FLT_iff (⟡.minSubnormalExponent) (⟡.fracWidth+1) ה _).2 ⟨ה,⟨m,k⟩,by simp only[Formats.Flocq.toReal,qA0],hm,hk⟩
lemma qBk (x:ὧ):R x=Model.unpackedToReal x.toModel.unpack:=by rw[A_R,ἦ qAj,Ὀ]
lemma qE0 {f:ὧ} {s m e h} (c:f.toModel.unpack=.finite s m e h):(Model.ofModel ⟡ (.finite .positive m e h)).toModel=.finite .positive m e h:=by
 have a:Model.toModel (A_M f)=.finite s m e h:=(Ὀ _).trans c;cases s;. have b:=Model.toModel_neg qAj (A_M f);rw[a] at b;change _=Float.Model.UnpackedFloat.finite .positive m e h at b;rw[<-b];exact Model.toModel_canonicalizeModel _
 . rw[<-a];exact Model.toModel_canonicalizeModel _
set_option maxRecDepth 8192 in
lemma qBl (f:ὧ) (hf:fᶠ):f.absᶠ/\R f.abs= |R f|:=by
 have a:A_M f.abs=Model.ofModel ⟡ f.toModel.unpack.abs:=Model.ofFloatModel_pack _;change f.toModel.unpack.isFinite=true at hf;cases hu:f.toModel.unpack with
 | notANumber | infinity _=>simp only[Float.Model.UnpackedFloat.isFinite,hu,false_eq_true] at hf
 | zero s=>rw[hu] at a;refine ⟨by rw[<-qAh,a];exact Model.isFinite_ofModel_zero _ qAj _,?_⟩;rw[A_R f.abs,a,qBk f,hu];simp only[Float.Model.UnpackedFloat.abs,Model.toReal_ofModel_zero _ (FloatFormat.supportsSignedZero_eq_true_of_isIEEE _ qAj),Model.unpackedToReal_zero,abs_zero]
 | finite s m e h=>have b:=qE0 hu;rw[hu] at a;have c:A_M f.abs=Model.ofModel ⟡ (.finite .positive m e h):=a;refine ⟨by rw[<-qAh,c,qAy,b];rfl,?_⟩;rw[A_R f.abs,c,qBk f,hu,ἦ qAj,b];simp only[Model.unpackedToReal_finite,qA0];cases s<;>simp only[Model.modelSignBit,false_eq_true, ↓reduceIte,one_mul,neg_mul,abs_neg,ἅ (by positivity:(0:ℝ)<=(m:ℝ)*2^e)]
lemma qCl (f:ὧ) (k:ℕ) (g:k<2^53) (e:f.toModel=Float.Model.pack (Float.Model.UnpackedFloat.ofNat ⟠ k)):fᶠ/\R f=k:=by
 obtain ⟨d,b,c⟩:∃d,A_M f=ἕ ⟡ d/\d.toReal=k:=by
  refine ⟨.ofScaledInt k 0,?_,?_⟩;. change Model.ofFloatModel f.toModel=_;rw[e,Model.ofFloatModel_pack];exact Model.ofModel_ofInt_eq_roundDyadic _ qAj (k:ℤ)
  . rw[<-Ἠ,FloatLib.Numerics.Dyadic.ofScaledInt_toRat];simp only[Rat.ofInt_eq_cast,Int.cast_natCast,zpow_ofNat,pow_zero,mul_one,Rat.cast_natCast]
 have a:Ἑ (A_M f)=true:=by rw[b];apply Model.isFinite_roundDyadic_of_isIEEE_of_abs_toReal_le_posMaxFinite _ qAj;rw[c,ἅ (Ἔ _)];exact ((qE7 ((by exact Nat.le_of_lt (Nat.lt_of_lt_of_le ‹_› ה)):k<=2^128)).trans ((le_mul_of_one_le_right qI1.le one_le_two).trans qE3)).trans qCd
 refine ⟨(qAh f).symm.trans a,?_⟩;rw[A_R,b,Model.toReal_roundDyadic_eq_roundAt _ qAj _ (b▸a),c];simpa only[Int.cast_natCast,zpow_ofNat,pow_zero,mul_one] using qA2 k 0 (by simpa only[Int.natAbs_natCast,Nat.reducePow] using g) (by norm_num1)
lemma qCk (m:Ὠ) (h:Ѧm<2^53):m.toFloatᶠ/\R m.toFloat=Ѧm:=qCl _ _ h rfl
lemma qHg:(0:ὧ)ᶠ/\R 0=0:=by simpa only[CharP.cast_eq_zero] using qCl 0 0 ו rfl
def A_bits (i:ℕ):Ὠ:=
 let e:=(i%256).toUInt64
 let s:Ὠ:=if i>=256 then 0x8000000000000000 else 0
 if e==0 then s else s ||| ((e+873)<<<(52:Ὠ))
lemma qCu (i:ℕ):pwEntry i=Ἃ (A_bits i):=by unfold pwEntry A_bits;dsimp only;split<;>rfl
def A_chk (i:ℕ):Ὡ:=
 match Ἶ ⟠ (A_bits i).toBitVec with
 | .finite s m e _=>(Ἴ s==decide (256<=i))&&m==2^52&&
  e==((i%256:ℕ):ℤ)-202&&(i%256 != 0)
 | .zero _=>i%256==0
 | _=>false
lemma qEm:∀i,i<512->A_chk i=true:=by decide+kernel
lemma qDa (i:ℕ) (b:i<512):
 (pwEntry i)ᶠ/\R (pwEntry i)=if i%256=0 then 0 else (if 256<=i then-1 else 1)*2^(((i%256:ℕ):ℤ)-150):=by
 have a:=qEm i b
 unfold A_chk at a
 rw[qCu,qBk]
 change (Ἃ (A_bits i)).toModel.unpack.isFinite=true/\_
 rw[show (Ἃ (A_bits i)).toModel.unpack=Ἶ ⟠ (A_bits i).toBitVec from Model.unpack_pack_unpack _ _]
 generalize Ἶ ⟠ (A_bits i).toBitVec=u at a
 cases u with
 | notANumber | infinity _=>simp only[false_eq_true] at a
 | zero s=>
  simp only[beq_iff_eq] at a
  exact ⟨rfl,by simp only[Model.unpackedToReal_zero,a, ↓reduceIte]⟩
 | finite s m e h=>
  simp only[Bool.and_eq_true,beq_iff_eq,bne_iff_ne] at a
  obtain ⟨⟨⟨g,d⟩,f⟩,c⟩:=a
  refine ⟨rfl,?_⟩
  rw[Model.unpackedToReal_finite,qA0,f,g,d,mul_assoc,
   show ((2^52:ℕ):ℝ)*2^(((i%256:ℕ):ℤ)-202)=2^(((i%256:ℕ):ℤ)-150) by
   rw[Nat.cast_pow,Nat.cast_ofNat,<-Ἡ,<-Ἓ ἓ]
   congr 1
   push_cast
   ring]
  simp only[decide_eq_true_eq,c,↓reduceIte]
lemma qCw (i:ℕ) (a:i<512):∃σ:ℤ,σ.natAbs<=1/\R (pwEntry i)=σ*2^(((i%256:ℕ):ℤ)-150):=by rw[(qDa i a).2];split_ifs;exacts [⟨0,by (positivity),by simp only[Int.cast_zero,Int.natCast_emod,Nat.cast_ofNat,zero_mul]⟩,⟨-1,by (norm_num1),by simp only[Int.natCast_emod,Nat.cast_ofNat,neg_mul,one_mul,Int.reduceNeg,Int.cast_neg,Int.cast_one]⟩,⟨1,by (norm_num1),by simp only[Int.natCast_emod,Nat.cast_ofNat,one_mul,Int.cast_one]⟩]
lemma qCm (f:ὧ) (i:ℕ) (g:i<512) (q:fᶠ) (c k:ℤ) (r:c.natAbs<2^53) (s:-900<=k) (m: |R f| <=A_B) (l:R f=c*2^k):(f*pwEntry i)ᶠ/\R (f*pwEntry i)=R f*R (pwEntry i):=by
 obtain ⟨b,a,d⟩:=qCw i g;have o:((i%256:ℕ):ℤ)-150<=128/\-150<=((i%256:ℕ):ℤ)-150:=(by clear g r s a;exact omS28);generalize ((i%256:ℕ):ℤ)-150=E at o d;have p: |(b:ℝ)| <=1:=by rw[<-Int.cast_abs,<-Nat.cast_natAbs];exact_mod_cast a
 have e:=ἵ (two_pos (α:=ℝ)) E;have h: |R (pwEntry i)| <=A_B:=by rw[d,ἢ,ἡ e,<-one_mul A_B];exact ἑ p (ἒ one_le_two (by clear g r s a;exact omS29 o)) e.le zero_le_one
 obtain ⟨j,n⟩:=qEd f _ q (qDa i g).1 ((ἑ m h (Ἀ _) qI1.le).trans qDe);refine ⟨j,?_⟩
 have:=qA2 (c*b) (k+E) (by
  rw[Int.natAbs_mul];have:=Nat.mul_le_mul_left c.natAbs a
  (clear g s a o;omega)) (by clear g r a;exact omS30 s o)
 rw[Ἓ ἓ] at this;push_cast at this;rw[n,d,l,show (c:ℝ)*2^k*(b*2^E)=c*b*(2^k*2^E) by ring];exact this
lemma qDc (i:ὦ) (h:Ѥi< Ѩpw):pw.uget i h=pwEntry Ѥi:=by simp only[FloatArray.uget,qDx,List.toArray_range,Array.ugetElem_eq_getElem,Array.getElem_map,Array.getElem_range]
section
variable (w:ὥ)
include w
lemma qAq:(w&&&В).toNat=ѥw%2^31:=by rw[Ἧ];exact Ἂ ѥw 31
lemma qCo:(w&&&0x7FFFFF).toNat=ѥw%2^23:=by rw[Ἧ];exact Ἂ ѥw 23
lemma qD8:(w>>>23).toNat=ѥw/2^23:=by rw[UInt32.toNat_shiftRight,Ἱ];rfl
lemma qCa:(expOf w).toNat=ѥw/2^23%2^8:=by unfold expOf;rw[Ἧ,qD8];exact Ἂ _ 8
lemma qGl:signOf w=decide (2^31<=ѥw):=by
 have b:=w.toNat_lt;unfold signOf;have h:((w&&&س).toNat)=(ѥw/2^31%2)*2^31:=by rw[Ἧ];show ѥw&&&2^31=_;rw[Nat.and_two_pow,Nat.testBit_eq_decide_div_mod_eq];have a:ѥw/2^31%2=0\/ѥw/2^31%2=1:=(by clear b;exact omS31);rcases a with a | a<;>rw[a]<;>rfl
 rw[Bool.eq_iff_iff];simp only[bne_iff_ne,ne_eq,decide_eq_true_eq];rw[<-ἠ,h];simp only[UInt32.toNat_zero];exact omS32 b
lemma qHa:((w&&&0x7FFFFF) ||| 0x800000).toNat=2^23+ѥw%2^23:=by rw[UInt32.toNat_or,qCo,Nat.or_comm];simpa only[UInt32.reduceToNat,Nat.reducePow,mul_one] using (Nat.two_pow_add_eq_or_of_lt (ὀ ѥw (by positivity:2^23>0)) 1).symm
lemma qAz (h:good w=true): ѥw%2^31=0\/(2^23<=ѥw%2^31/\ѥw%2^31<У):=by
 have a:=qAq w;have b:=(w&&&В).toNat_lt;unfold good at h;simp only[Bool.or_eq_true,beq_iff_eq,decide_eq_true_eq] at h;rcases h with h | h;. left;rw[<-a,h];rfl
 . right;rw[ἃ,UInt32.toNat_sub] at h;rw[<-a];generalize (w&&&В).toNat=v at*;simp only[Nat.reducePow,UInt32.reduceToNat,Nat.reduceSub] at h;(clear b;exact omS33 a h)
lemma qAe (h:good w=true):expOf w≠255:=by intro he;have a:=qCa w;rw[he] at a;change 255=_ at a;have:=qAz w h;have:=w.toNat_lt;(clear he this;exact omS34 a this)
end
noncomputable def A_mag (v:ℕ):ℝ:=
 if v=0 then 0 else ((2^23+v%2^23:ℕ):ℝ)*2^(((v/2^23:ℕ):ℤ)-150)
lemma qAs (v:ℕ):0<=A_mag v:=by unfold A_mag;split_ifs<;>positivity
lemma qA7 (w:ὥ) (hg:good w=true):wv w=(if 2^31<=ѥw then-1 else 1)*A_mag (ѥw%2^31):=by
 have g:=w.toNat_lt;have:=qAz w hg;have a:=qCa w;have d:(fracOf w).toNat=ѥw%2^23:=qCo w;have f:wv w=(decodeD w).toReal:=by unfold wv ofW;rw[Model.toReal_eq,ἶ w (qAe w hg)]
 rw[f];unfold decodeD A_mag;by_cases c:ѥw%2^31=0;. have b:expOf w=0:=ἠ.mp (by rw[a];show _=0;(clear g this;exact omS35 c));have e:fracOf w=0:=ἠ.mp (by rw[d];show _=0;(clear g this a b;exact omS36 c));simp only[b,e,c,↓reduceIte];simp only[FloatLib.Numerics.Dyadic.toReal,signedSignificand,Int.ofNat_eq_natCast,CharP.cast_eq_zero,neg_zero,ite_self,Int.cast_zero,zpow_ofNat,pow_zero,mul_one,Nat.reducePow,mul_zero]
 . have b:expOf w≠0:=λh=>by rw[h] at a;change 0=_ at a;(clear g h;exact omS37 this c a)
   simp only[b,c,↓reduceIte];rw[show ѥw%2^31%2^23=(fracOf w).toNat by rw[d];(clear g this a c b;exact omS38),show ѥw%2^31/2^23=(expOf w).toNat by rw[a];(clear g this c b;exact omS39)];simp only[FloatLib.Numerics.Dyadic.toReal,FloatLib.Numerics.Dyadic.signedSignificand,qGl];simp only[Nat.reducePow,decide_eq_true_eq,Int.ofNat_eq_natCast,Nat.cast_add,Nat.cast_ofNat,neg_add_rev,Int.reduceNeg,Int.cast_ite,Int.cast_add,Int.cast_neg,Int.cast_natCast,Int.cast_ofNat,ite_mul,neg_mul,one_mul];split_ifs<;>ring
lemma qFb (v:ℕ) (h:v<У):A_mag v<=2^128:=by
 unfold A_mag;split_ifs with h0;. positivity
 . exact (ἑ (show ((2^23+v%2^23:ℕ):ℝ)<=((2^24:ℕ):ℝ) from Nat.cast_le.2 (by clear h h0;exact omS40)) (ἒ one_le_two (by (clear h0;exact omS41 h):_<=(104:ℤ))) ו ו).trans_eq (by norm_num1)
lemma qA6 (a b:ℕ) (hab:a<=b) (ha:a=0\/2^23<=a):A_mag a<=A_mag b:=by
 rcases ha with rfl | ha;. simpa only[A_mag, ↓reduceIte,Nat.reducePow,Nat.cast_add,Nat.cast_ofNat,Int.natCast_ediv] using qAs b
 unfold A_mag;simp only[show ¬a=0 by (clear hab;exact omS42 ha),show ¬b=0 by exact omS43 hab ha,↓reduceIte];have c:a/2^23<=b/2^23:=Nat.div_le_div_right hab;rcases c.eq_or_lt with d | g;. rw[d];exact ἀ (Nat.cast_le.2 (by clear ha c;exact omS44 hab d)) ו
 . have e:(2:ℝ)^(((a/2^23:ℕ):ℤ)-150)*2<=2^(((b/2^23:ℕ):ℤ)-150):=by rw[<-zpow_add_one₀ ἓ];exact ἒ one_le_two (by clear hab ha c;exact omS45 g)
   have f:((2^23:ℕ):ℝ)<=((2^23+b%2^23:ℕ):ℝ):=Nat.cast_le.2 (by clear hab ha c g;exact omS46)
   calc _<=((2^23*2:ℕ):ℝ)*2^(((a/2^23:ℕ):ℤ)-150):=
         ἀ (Nat.cast_le.2 (by clear hab ha c g;exact omS47)) ו
    _=((2^23:ℕ):ℝ)*(2^(((a/2^23:ℕ):ℤ)-150)*2):=by push_cast;ring
    _<=_:=ἑ f e ו ו
abbrev A_mf (w:ὥ):ὧ:=((w&&&0x7FFFFF) ||| 0x800000).toFloat
lemma qD1 (w:ὥ):pw.uget (w>>>23).toUSize (qCh w)=pwEntry (ѥw/2^23)/\ѥw/2^23<512/\(A_mf w)ᶠ /\(A_mf w*pwEntry (ѥw/2^23))ᶠ /\R (A_mf w*pwEntry (ѥw/2^23))=R (A_mf w)*R (pwEntry (ѥw/2^23))/\R (A_mf w)=((2^23+ѥw%2^23:ℕ):ℝ):=by
 have g:=w.toNat_lt;have c:=qHa w;obtain ⟨d,a⟩:=qCl (A_mf w) _ (by rw[c];(clear g;(clear c;exact omS48))) rfl;rw[c] at a;have b:ѥw/2^23<512:=(by exact omS49 g)
 obtain ⟨f,e⟩:=qCm _ _ b d ((2^23+ѥw%2^23:ℕ):ℤ) 0 (by clear g b;(clear c;omega))
  (by norm_num1) (by
   rw[a,ἅ (Ἔ _)]
   exact qE7 (by clear g b;(clear c;exact omS51)))
  (by rw[a,zpow_zero,mul_one,Int.cast_natCast])
 refine ⟨?_,b,d,f,e,a⟩;rw[qDc,ὃ,qD8]
lemma qDvF (w:ὥ):(dv w)ᶠ:=by obtain ⟨g,-,-,e,-,-⟩:=qD1 w;unfold dv;rw[g];exact e
lemma qAm (w:ὥ) (hg:good w=true):(dv w)ᶠ/\R (dv w)=wv w:=by
 obtain ⟨g,a,-,e,d,c⟩:=qD1 w;have j:=w.toNat_lt;unfold dv;rw[g];refine ⟨e,?_⟩;rw[d,c,(qDa _ a).2,qA7 w hg];have b:=qAz w hg;generalize ѥw=v at*;rcases b with h | ⟨l,k⟩;. rw[h];simp only[show v/2^23%256=0 by (clear a j;exact omS52 h),↓reduceIte,mul_zero,show A_mag 0=0 by simp only[A_mag, ↓reduceIte]]
 . have f:(256<=v/2^23)↔2^31<=v:=(by exact omS53);unfold A_mag;rw[((by clear a j f;exact omS54):v%2^31/2^23=v/2^23%256),((by clear a j l k f;exact omS38):v%2^31%2^23=v%2^23),ἐ ((by clear a j f;exact omS55 l):¬v/2^23%256=0),ἐ ((by clear a j f;exact omS56 l):¬v%2^31=0)];by_cases i:2^31<=v<;>simp only[f,i,↓reduceIte]<;>ring
lemma zFn0 (m:ℤ) (e:ℤ) (k:ℕ) (z:Float.Model.UnpackedFloat.Sign):
 Float.Model.UnpackedFloat.normalize ⟠ (m*2^k) (e-k) z=Float.Model.UnpackedFloat.normalize ⟠ m e z:=by
 have hk:(0:ℤ)<2^k:=ו
 unfold Float.Model.UnpackedFloat.normalize
 rcases lt_trichotomy m 0 with h | h | h
 . have h1:m*2^k<0:=Int.mul_neg_of_neg_of_pos h hk
   have c1:compare m 0=.lt:=by simp only[compare,compareOfLessAndEq,h, ↓reduceIte]
   have c2:compare (m*2^k) 0=.lt:=by simp only[compare,compareOfLessAndEq,h1, ↓reduceIte]
   rw[c1,c2];dsimp only
   have e2:(-(m*2^k)).toNat=(-m).toNat*2^k:=by
    have e1:-(m*2^k)=(((-m).toNat*2^k:ℕ):ℤ):=by push_cast;rw[Int.toNat_of_nonneg א];ring
    rw[e1,Int.toNat_natCast]
   rw[e2];exact ExactRuntime.round_scale ⟠ .negative _ e k א
 . subst h;simp only[zero_mul,Std.compare_self]
 . have h1:0< m*2^k:=Int.mul_pos h hk
   have c1:compare m 0=.gt:=by simp only[compare,compareOfLessAndEq,Int.not_lt.2 h.le, ↓reduceIte,h.ne']
   have c2:compare (m*2^k) 0=.gt:=by simp only[compare,compareOfLessAndEq,Int.not_lt.2 h1.le, ↓reduceIte,h1.ne']
   rw[c1,c2];dsimp only
   have e2:(m*2^k).toNat=m.toNat*2^k:=by
    have e1:m*2^k=((m.toNat*2^k:ℕ):ℤ):=by push_cast;rw[Int.toNat_of_nonneg א]
    rw[e1,Int.toNat_natCast]
   rw[e2];exact ExactRuntime.round_scale ⟠ .positive _ e k א
lemma zFn1 (u u' f f':ℤ) (z:Float.Model.UnpackedFloat.Sign) (h:(u:ℝ)*2^f=(u':ℝ)*2^f'):
 Float.Model.UnpackedFloat.normalize ⟠ u f z=Float.Model.UnpackedFloat.normalize ⟠ u' f' z:=by
 obtain ⟨a,ha⟩:∃a:ℕ,f=min f f'+a:=⟨(f-min f f').toNat,א⟩
 obtain ⟨b,hb⟩:∃b:ℕ,f'=min f f'+b:=⟨(f'-min f f').toNat,א⟩
 have e1:=zFn0 u f a z
 have e2:=zFn0 u' f' b z
 rw[show f-a=min f f' by omega] at e1
 rw[show f'-b=min f f' by omega] at e2
 rw[<-e1,<-e2]
 congr 1
 have h2:((u*2^a:ℤ):ℝ)*2^(min f f')=((u'*2^b:ℤ):ℝ)*2^(min f f'):=by
  have h':(u:ℝ)*2^(min f f'+(a:ℤ))=(u':ℝ)*2^(min f f'+(b:ℤ)):=by rw[<-ha,<-hb];exact h
  rw[Ἓ ἓ,Ἓ ἓ,Ἡ,Ἡ] at h'
  push_cast
  linarith only[h']
 exact_mod_cast mul_right_cancel₀ ו h2
lemma zFd0 (m:ℕ) (e L:ℤ):
 (((Float.Model.UnpackedFloat.decreaseExponent m e L).1:ℕ):ℝ)*2^((Float.Model.UnpackedFloat.decreaseExponent m e L).2)=(m:ℝ)*2^e:=by
 unfold Float.Model.UnpackedFloat.decreaseExponent
 dsimp only
 rw[ὁ]
 push_cast
 rw[mul_assoc,<-Ἡ,<-Ἓ ἓ]
 congr 2
 ring
lemma zFd1 (m:ℕ) (e L:ℤ) (h:L<=e):(Float.Model.UnpackedFloat.decreaseExponent m e L).2=L:=by
 unfold Float.Model.UnpackedFloat.decreaseExponent;dsimp only;omega
lemma zFd2 (m:ℕ) (e L:ℤ) (h:L<=e):
 ((((Float.Model.UnpackedFloat.decreaseExponent m e L).1:ℕ):ℤ):ℝ)*2^L=(m:ℝ)*2^e:=by
 have h0:=zFd0 m e L;rw[zFd1 m e L h] at h0;rw[Int.cast_natCast];exact h0
lemma zFs (s:Float.Model.UnpackedFloat.Sign) (n:ℤ):((s.apply n:ℤ):ℝ)=(if s=.negative then-1 else 1)*(n:ℝ):=by
 cases s<;>simp only[Float.Model.UnpackedFloat.Sign.apply,Int.cast_neg, ↓reduceIte,neg_mul,one_mul,reduceCtorEq]
lemma zFz (s:Float.Model.UnpackedFloat.Sign):Ἶ ⟠ (Float.Model.UnpackedFloat.pack ⟠ (.zero s))=.zero s:=by
 cases s<;>rfl
lemma zFf (s1 s2 s3:Float.Model.UnpackedFloat.Sign) (m1 m2 m3:ℕ) (e1 e2 e3:ℤ) (h1 h2 h3):
 Float.Model.UnpackedFloat.fma ⟠ (.finite s1 m1 e1 h1) (.finite s2 m2 e2 h2) (.finite s3 m3 e3 h3)=
  Float.Model.UnpackedFloat.normalize ⟠ ((s1*s2).apply ((Float.Model.UnpackedFloat.decreaseExponent (m1*m2) (e1+e2) (min (e1+e2) e3)).1:ℤ)+
   s3.apply ((Float.Model.UnpackedFloat.decreaseExponent m3 e3 (min (e1+e2) e3)).1:ℤ)) (min (e1+e2) e3) .positive:=rfl
lemma zFa (s1 s2:Float.Model.UnpackedFloat.Sign) (m1 m2:ℕ) (e1 e2:ℤ) (h1 h2):
 Float.Model.UnpackedFloat.add ⟠ (.finite s1 m1 e1 h1) (.finite s2 m2 e2 h2)=
  Float.Model.UnpackedFloat.normalize ⟠ (s1.apply ((Float.Model.UnpackedFloat.decreaseExponent m1 e1 (min e1 e2)).1:ℤ)+
   s2.apply ((Float.Model.UnpackedFloat.decreaseExponent m2 e2 (min e1 e2)).1:ℤ)) (min e1 e2) .positive:=rfl
lemma zFp (v:Float.Model):Float.Model.pack v.unpack=v:=
 FloatLib.Floats.Formats.BinaryInterchange.Model.toFloatModel_ofFloatModel v
lemma zFM (x w a:Float.Model) (hx:x.unpack.isFinite=true) (hw:w.unpack.isFinite=true)
 (HP:∀s1 m1 e1 h1 s2 m2 e2 h2,x.unpack=.finite s1 m1 e1 h1->w.unpack=.finite s2 m2 e2 h2->
  ∃M E h,(Float.Model.mul x w).unpack=.finite (s1*s2) M E h/\(M:ℝ)*2^E=((m1*m2:ℕ):ℝ)*2^(e1+e2)):
 Float.Model.fma x w a=Float.Model.add a (Float.Model.mul x w):=by
 have hP:(Float.Model.mul x w).unpack=Ἶ ⟠ (Float.Model.UnpackedFloat.pack ⟠ (Float.Model.UnpackedFloat.mul ⟠ x.unpack w.unpack)):=by
  simp only[Float.Model.unpack,Float.Model.mul,Float.Model.pack,UInt64.toBitVec_ofBitVec]
 unfold Float.Model.fma Float.Model.add
 cases hxu:x.unpack<;>cases hwu:w.unpack<;>simp only[hxu,hwu,Float.Model.UnpackedFloat.isFinite,Bool.false_eq_true] at hx hw
 case zero.zero s1 s2=>
  rw[hP,hxu,hwu]
  show _=Float.Model.pack (Float.Model.UnpackedFloat.add ⟠ a.unpack (Ἶ ⟠ (Float.Model.UnpackedFloat.pack ⟠ (.zero (s1*s2)))))
  rw[zFz]
  cases a.unpack with
  | zero s3=>cases s1<;>cases s2<;>cases s3<;>rfl
  | _=>rfl
 case zero.finite s1 s2 m2 e2 h2=>
  rw[hP,hxu,hwu]
  show _=Float.Model.pack (Float.Model.UnpackedFloat.add ⟠ a.unpack (Ἶ ⟠ (Float.Model.UnpackedFloat.pack ⟠ (.zero (s1*s2)))))
  rw[zFz]
  cases a.unpack with
  | zero s3=>cases s1<;>cases s2<;>cases s3<;>rfl
  | _=>rfl
 case finite.zero s1 m1 e1 h1 s2=>
  rw[hP,hxu,hwu]
  show _=Float.Model.pack (Float.Model.UnpackedFloat.add ⟠ a.unpack (Ἶ ⟠ (Float.Model.UnpackedFloat.pack ⟠ (.zero (s1*s2)))))
  rw[zFz]
  cases a.unpack with
  | zero s3=>cases s1<;>cases s2<;>cases s3<;>rfl
  | _=>rfl
 case finite.finite s1 m1 e1 h1 s2 m2 e2 h2=>
  obtain ⟨M,E,hM,hPU,hv⟩:=HP s1 m1 e1 h1 s2 m2 e2 h2 hxu hwu
  rw[hPU]
  cases ha:a.unpack with
  | notANumber=>rfl
  | infinity s3=>rfl
  | zero s3=>
   show Float.Model.pack (Float.Model.UnpackedFloat.mul ⟠ (.finite s1 m1 e1 h1) (.finite s2 m2 e2 h2))=Float.Model.pack (.finite (s1*s2) M E hM)
   rw[<-hPU,zFp]
   unfold Float.Model.mul
   rw[hxu,hwu]
  | finite s3 m3 e3 h3=>
   rw[zFf,zFa]
   congr 1
   apply zFn1
   push_cast
   rw[zFs,zFs,zFs,zFs,add_mul,add_mul,mul_assoc,mul_assoc,mul_assoc,mul_assoc,
    zFd2 _ _ _ (min_le_left _ _),zFd2 _ _ _ (min_le_right _ _),zFd2 _ _ _ (min_le_left _ _),zFd2 _ _ _ (min_le_right _ _),hv]
   push_cast;ring
lemma zFR (f:ὧ) {s m e h} (hf:f.toModel.unpack=.finite s m e h):R f=(if Ἴ s then (-1:ℝ) else 1)*(m:ℝ)*2^e:=by
 rw[qBk,hf,FloatLib.Floats.Formats.BinaryInterchange.Model.unpackedToReal_finite,qA0]
lemma zFH (x w:ὧ) (hf:(x*w)ᶠ) (hr:R (x*w)=R x*R w): ∀s1 m1 e1 h1 s2 m2 e2 h2,x.toModel.unpack=.finite s1 m1 e1 h1->w.toModel.unpack=.finite s2 m2 e2 h2->
 ∃M E h,(Float.Model.mul x.toModel w.toModel).unpack=.finite (s1*s2) M E h/\(M:ℝ)*2^E=((m1*m2:ℕ):ℝ)*2^(e1+e2):=by
 intro s1 m1 e1 h1 s2 m2 e2 h2 hx hw
 have rx:=zFR x hx
 have rw':=zFR w hw
 have hm:(x*w).toModel=Float.Model.mul x.toModel w.toModel:=rfl
 change (x*w).toModel.unpack.isFinite=true at hf
 have q1:(0:ℝ)< m1:=by exact_mod_cast h1
 have q2:(0:ℝ)< m2:=by exact_mod_cast h2
 have p1:0<(m1:ℝ)*m2*2^(e1+e2):=ו
 cases hp:(Float.Model.mul x.toModel w.toModel).unpack with
 | notANumber=>rw[hm,hp] at hf;exact absurd hf (by simp only[Float.Model.UnpackedFloat.isFinite,false_eq_true,not_false_eq_true])
 | infinity s=>rw[hm,hp] at hf;exact absurd hf (by simp only[Float.Model.UnpackedFloat.isFinite,false_eq_true,not_false_eq_true])
 | zero s=>
  exfalso
  have r0:=qBk (x*w)
  rw[hm,hp,FloatLib.Floats.Formats.BinaryInterchange.Model.unpackedToReal_zero,hr,rx,rw'] at r0
  have k:(if Ἴ s1 then (-1:ℝ) else 1)*(if Ἴ s2 then (-1:ℝ) else 1)*((m1:ℝ)*m2*2^(e1+e2))=0:=by
   rw[Ἓ ἓ];linear_combination r0
  cases s1<;>cases s2<;>simp only[Ἴ,if_true,if_false,Bool.false_eq_true] at k<;>linarith only[p1,k]
 | finite s M E hM=>
  have rp:=zFR (x*w) (hm▸hp)
  rw[hr,rx,rw'] at rp
  have q3:(0:ℝ)<M:=by exact_mod_cast hM
  have p2:0<(M:ℝ)*2^E:=ו
  have k:(if Ἴ s then (-1:ℝ) else 1)*((M:ℝ)*2^E)=(if Ἴ s1 then (-1:ℝ) else 1)*(if Ἴ s2 then (-1:ℝ) else 1)*((m1:ℝ)*m2*2^(e1+e2)):=by
   rw[Ἓ ἓ];linear_combination-rp
  push_cast
  cases s<;>cases s1<;>cases s2<;>simp only[Ἴ,if_true,if_false,Bool.false_eq_true] at k<;>first|exact ⟨M,E,hM,rfl,by linarith only[k]⟩|(exfalso;linarith only[p1,p2,k])
lemma zFdr (u:ὥ):(dv u)ᶠ/\∃c k:ℤ,R (dv u)=(c:ℝ)*2^k/\c.natAbs<2^24/\-150<=k/\k<=105:=by
 obtain ⟨g,a,-,e,d,c⟩:=qD1 u
 obtain ⟨σ,hσ,hp⟩:=qCw _ a
 have E:dv u=A_mf u*pwEntry (ѥu/2^23):=by unfold dv;rw[g]
 rw[E];refine ⟨e,((2^23+ѥu%2^23:ℕ):ℤ)*σ,((ѥu/2^23%256:ℕ):ℤ)-150,?_,?_,א,?_⟩
 . rw[d,c,hp,Int.cast_mul,Int.cast_natCast];ring
 . rw[Int.natAbs_mul,Int.natAbs_natCast];have:=ὀ (ѥu) (show 2^23>0 by positivity);have:=Nat.mul_le_mul_left (2^23+ѥu%2^23) hσ;omega
 . have:=ὀ (ѥu/2^23) (show 256>0 by positivity);omega
lemma zFdv (u w:ὥ):(dv u*dv w)ᶠ/\R (dv u*dv w)=R (dv u)*R (dv w):=by
 obtain ⟨fu,c,k,ru,hc,hk0,hk⟩:=zFdr u
 obtain ⟨fw,c',k',rw',hc',hk0',hk'⟩:=zFdr w
 have B:∀(c k:ℤ),c.natAbs<2^24->k<=105->|(c:ℝ)*2^k|<=2^(129:ℤ):=λc k h1 h2=>by
  rw[ἢ,ἡ (by positivity:(0:ℝ)<2^k)]
  have h3:|(c:ℝ)|<=2^24:=by rw[<-Int.cast_abs,<-Int.natCast_natAbs];exact_mod_cast h1.le
  calc |(c:ℝ)| *2^k<=2^24*2^(105:ℤ):=ἑ h3 (ἒ one_le_two h2) ו ו
   _=2^(129:ℤ):=ב
 obtain ⟨f,r⟩:=qEd _ _ fu fw (by
  rw[ru,rw'];calc |(c:ℝ)*2^k| * |(c':ℝ)*2^k'|<=2^(129:ℤ)*2^(129:ℤ):=ἑ (B c k hc hk) (B c' k' hc' hk') (Ἀ _) ו
   _<=2^(1000:ℤ):=by rw[<-Ἓ ἓ];exact ἒ one_le_two ב)
 refine ⟨f,?_⟩
 have e2:(c:ℝ)*2^k*((c':ℝ)*2^k')=((c*c':ℤ):ℝ)*2^(k+k'):=by push_cast;rw[Ἓ ἓ];ring
 rw[r,ru,rw',e2]
 exact qA2 _ _ (by rw[Int.natAbs_mul];have:=Nat.mul_lt_mul'' hc hc';exact Ἥ this ב) א
lemma zFV (u w:ὥ) (a:ὧ):Float.fma (dv u) (dv w) a=a+dv u*dv w:=by
 obtain ⟨f,r⟩:=zFdv u w
 show Float.ofModel (Float.Model.fma (dv u).toModel (dv w).toModel a.toModel)=Float.ofModel (Float.Model.add a.toModel (Float.Model.mul (dv u).toModel (dv w).toModel))
 rw[zFM _ _ _ (qDvF u) (qDvF w) (zFH _ _ f r)]
lemma zFE {v w:ὧ} (hv:∃u,v=dv u) (hw:∃u,w=dv u) (a:ὧ):Float.fma v w a=a+v*w:=by
 obtain ⟨u,rfl⟩:=hv;obtain ⟨u',rfl⟩:=hw;exact zFV u u' a
def A_fsum (l:List ὧ):ὧ:=l.foldl (.+·) 0
noncomputable def A_sa (l:List ὧ):ℝ:=(l.map λq=>|R q|).sum
noncomputable def A_sr (l:List ὧ):ℝ:=(l.map R).sum
lemma qC4 {a:Type} (l:List a) (f:a->ℝ):0<=(l.map λq=>|f q|).sum:=List.sum_nonneg (by simp only[List.mem_map];rintro _ ⟨q,_,rfl⟩;exact Ἀ _)
lemma qCe (l:List ὧ):0<=A_sa l:=qC4 l R
lemma qGr (l:List ὧ) (q:ὧ):A_sa (l++[q])=A_sa l+ |R q| /\A_sr (l++[q])=A_sr l+R q/\A_fsum (l++[q])=A_fsum l+q:=by simp only[A_sa,List.map_append,List.map_cons,List.map_nil,List.sum_append,List.sum_cons,List.sum_nil,add_zero,A_sr,A_fsum,List.foldl_append,ὔ,List.foldl_nil,and_self]
lemma qC7 (l:List ὧ): |A_sr l| <=A_sa l:=by
 induction l with
 | nil=>simp only[A_sr,List.map_nil,List.sum_nil,abs_zero,A_sa,Std.le_refl]
 | cons q l ih=>simp only[A_sa,A_sr,List.map_cons,List.sum_cons] at ih⊢;exact (Ἐ _ _).trans (by linarith only[ih])
lemma qBo:A_z<=1:=qAw.trans (by norm_num1)
section
variable (l:List ὧ) (hf:∀ q∈l,qᶠ) (hlen:ѣl<=2^21) (hb:A_sa l<=A_B/4)
include l hf hlen hb
lemma qDd:
 (A_fsum l)ᶠ /\
  |R (A_fsum l)-A_sr l| <=(1+1/2^22)*ѣl*(1/2^53*(A_sa l+A_z)):=by
 have e2AB6:A_B/4<=A_B:=div_le_self qI1.le ב
 induction l using List.reverseRecOn with
 | nil=>exact ⟨qHg.1,by simp only[A_fsum,List.foldl_nil,qHg.2,A_sr,List.map_nil,List.sum_nil,sub_self,abs_zero,one_div,List.length_nil,CharP.cast_eq_zero,mul_zero,zero_mul,Std.le_refl]⟩
 | append_singleton l q ih=>
  obtain ⟨e,f,n⟩:=qGr l q
  rw[e] at hb
  rw[List.length_append,List.length_singleton] at hlen
  have t:=qCe l
  have j:=Ἀ (R q)
  obtain ⟨h,v⟩:=ih (λx i=>hf x (by simp only[List.mem_append,i,List.mem_cons,List.not_mem_nil,or_false,true_or])) (by exact Nat.le_of_lt ‹_›) ((le_add_of_nonneg_right j).trans hb)
  have u:(ѣl:ℝ)+1<=2^21:=by exact_mod_cast hlen
  have d:(0:ℝ)<=ѣl:=Ἔ _
  have y:=qC7 l
  have o:=qAr
  have m:=qBo
  have p:=qAn
  have s:=Ὥ (R (A_fsum l)) (A_sr l)
  have w:=ἀ ((le_add_of_nonneg_right zero_le_one).trans u:(ѣl:ℝ)<=2^21) (add_nonneg t o.le)
  obtain ⟨b,z⟩:=qBf _ q h (hf q ג) (by linarith only[hb,j,v,y,m,p,s,w]) ((le_add_of_nonneg_left t).trans (hb.trans e2AB6))
  rw[n,e,f,List.length_append,List.length_singleton]
  refine ⟨b,?_⟩
  push_cast
  generalize R (A_fsum l)=r at*
  generalize R q=a at*
  generalize A_sa l=P at*
  generalize A_sr l=S at*
  generalize (ѣl:ℝ)=k at*
  obtain ⟨W,g⟩:∃W:ℝ,W=1/2^53*(P+ |a| +A_z):=⟨_,rfl⟩
  have c:0<=W:=(Ἆ (one_div_nonneg.mpr (pow_nonneg zero_le_two 53)) (add_nonneg (add_nonneg t j) o.le)).trans_eq g.symm
  rw[<-g]
  refine (abs_sub_le _ (r+a) (S+a)).trans ?_
  rw[add_sub_add_right_eq_sub]
  linarith only[z,v,y,s,g,c,Ἐ r a,ἁ (show 1/2^53*(P+A_z)<=W from (ἁ (add_le_add (le_add_of_nonneg_right j) le_rfl) (one_div_nonneg.mpr (pow_nonneg zero_le_two 53))).trans_eq g.symm) d,ἀ ((le_add_of_nonneg_right zero_le_one).trans u:k<=2^21) c]
lemma qBz:
 (A_fsum l)ᶠ /\
  |R (A_fsum l)-A_sr l| <=2*ѣl*(1/2^53*(A_sa l+A_z)):=by
 obtain ⟨a,b⟩:=qDd l hf hlen hb
 have c:=Ἆ (Ἔ (α:=ℝ) ѣl) (add_nonneg (qCe l) qAr.le)
 exact ⟨a,b.trans (by linarith only[c])⟩
end
lemma qB7 (l:List ὧ) (c:∀q∈l,qᶠ) (d:ѣl<=2^21) (f:A_sa l<=A_B/4):
 (A_fsum (l.map Float.abs))ᶠ /\
  |R (A_fsum (l.map Float.abs))-A_sa l| <=2*ѣl*(1/2^53*(A_sa l+A_z)):=by
 have a:A_sa (l.map Float.abs)=A_sa l/\A_sr (l.map Float.abs)=A_sa l:=by
  constructor<;>simp only[A_sa,A_sr,List.map_map]<;>congr 1<;>exact ὠ λq b=>by simp only[Function.comp_apply,(qBl q (c q b)).2,abs_abs]
 have:=qBz (l.map Float.abs) (λq b=>by
  obtain ⟨x,e,rfl⟩:=List.mem_map.mp b
  exact (qBl x (c x e)).1) (by simpa only[List.length_map,Nat.reducePow]) (a.1.symm▸f)
 rwa[a.1,a.2,List.length_map] at this
lemma qIn (n:ℕ) (g:n<2^20):(nfOf n)ᶠ/\R (nfOf n)=((n:ℝ)+8)/2^53:=by
 have b:(n.toUInt64+8).toNat=n+8:=by have c:(8:Ὠ).toNat=8:=rfl;simp only[Nat.toUInt64_eq,ἤ,UInt64.toNat_ofNat',Nat.reducePow,c,Nat.mod_add_mod,Nat.mod_succ_eq_iff_lt,Nat.succ_eq_add_one,Nat.reduceAdd,gt_iff_lt];(clear c;exact omS57 g)
 obtain ⟨f,a⟩:=qCk (n.toUInt64+8) (by rw[b];exact omS58 g);rw[b] at a
 obtain ⟨e,d⟩:=qCm _ 97 ב f ((n+8:ℕ):ℤ) 0 א ב
  (by
   rw[a,ἅ (Ἔ _)]
   exact qE7 (by exact omS60 g))
  (by rw[a];simp only[Nat.cast_add,Nat.cast_ofNat,Int.cast_add,Int.cast_natCast,Int.cast_ofNat,zpow_ofNat,pow_zero,mul_one])
 push_cast at a;rw[show nfOf n=(n.toUInt64+8).toFloat*pwEntry 97 by rw[nfOf,qCu];rfl];refine ⟨e,?_⟩;rw[d,a,(qDa 97 ב).2];norm_num;ring
lemma qGb {n:ℕ} (l:n<2^20) {s t:ὧ} (k:sᶠ) (m:tᶠ) {X Y:ℝ} (e:0<=Y) (v:Y<=A_B/8) (y: |R s-X| <=((n:ℝ)+4)*(1/2^53*Y)+1/2^250) (u: |R t-Y| <=Y/2^30+1/2^250) (I: |R s| <=2*Y) (c:(2:ℝ)^(-127:ℤ)<= |R (s+nfOf n*t)|):R (s-nfOf n*t)<X/\X<R (s+nfOf n*t):=by
 have sck:∀{k a b:ℝ},0<k->a<=1/k*b->k*a<=b:=λhk h=>(le_div_iff₀' hk).mp (h.trans_eq (one_div_mul_eq_div _ _))
 have e2d:∀{x:ℝ} (n:ℕ),0<=x->1/2^n*x<=x:=λn h0=>mul_le_of_le_one_left h0 (div_le_one_of_le₀ (one_le_pow₀ one_le_two) (pow_nonneg zero_le_two n))
 have e2aa:∀(x y:ℝ),|x| <= |y| + |x-y|:=λx y=>sub_le_iff_le_add'.mp (Ὥ x y)
 have A:(n:ℝ)+1<=2^20:=by exact_mod_cast l
 have o:(0:ℝ)<=n:=Ἔ n;obtain ⟨h,b⟩:=qIn n l;have E:=qAr;have B:=qAw
 have B':(2:ℝ)^200*(2^200*A_z)<=1:=(le_div_iff₀' (by norm_num1)).mp ((le_div_iff₀' (by norm_num1)).mp (B.trans_eq (one_div_mul_eq_div _ _)))
 have b':(n:ℝ)+8=R (nfOf n)*2^53:=(div_eq_iff (by norm_num1)).mp b.symm
 have nf0:0<=R (nfOf n):=b▸div_nonneg (add_nonneg o (by norm_num1)) (by norm_num1)
 have c30:2^30*R (nfOf n)<=1:=by linarith only[b',A]
 have nf1:R (nfOf n)<=1:=by linarith only[c30,nf0]
 generalize hε:(1:ℝ)/2^250=ε at y u;have hε1:(2:ℝ)^250*ε=1:=by rw[<-hε];exact mul_one_div_cancel (by norm_num1)
 have hε0:0<=ε:=by linarith only[hε1]
 have hε2:ε<=1:=by linarith only[hε1]
 have y':2^53*|R s-X| <=((n:ℝ)+4)*Y+2^53*ε:=sck (by norm_num1) (y.trans_eq (by ring))
 have u':2^30*|R t-Y| <=Y+2^30*ε:=(le_div_iff₀' (by norm_num1)).mp (u.trans_eq (by rw[add_div,mul_div_cancel_left₀ _ (by norm_num1)]))
 have i: |R t| <=2*Y+ε:=by linarith only[e2aa (R t) Y,ἅ e,u',e,hε0]
 have x':2^30*|R (nfOf n)*R t| <=2*Y+ε:=by rw[ἢ,ἅ nf0,<-mul_assoc];exact (ἑ c30 i (Ἀ _) zero_le_one).trans_eq (one_mul _)
 have h8:(2:ℝ)^3<=A_B:=(pow_le_pow_right₀ one_le_two (Nat.le_of_ble_eq_true rfl:3<=128)).trans qAn
 have kAB:∀{x:ℝ},x<=(2*Y+1)+((2*Y+1)+1)->x<=A_B:=λh=>h.trans (by linarith only[v,h8])
 have nn':(0:ℝ)<=2*Y+1:=add_nonneg (Ἆ zero_le_two e) zero_le_one
 have nn:(0:ℝ)<=(2*Y+1)+1:=add_nonneg nn' zero_le_one
 obtain ⟨d,f⟩:=qCf _ _ h m ((ἅ nf0).le.trans (nf1.trans ((one_le_pow₀ one_le_two).trans qAn))) (kAB (i.trans (add_le_add (le_add_of_nonneg_right zero_le_one) (hε2.trans (le_add_of_nonneg_left nn')))))
 have Xa: |R (nfOf n)*R t| <=2*Y+1:=by linarith only[x',hε2,e,hε0]
 have G6:=kAB ((e2aa _ (R (nfOf n)*R t)).trans (add_le_add Xa ((f.trans (e2d 53 (add_nonneg (Ἀ _) E.le))).trans (add_le_add Xa qBo))))
 have D:=e2aa (R (nfOf n*t)) (R (nfOf n)*R t)
 obtain ⟨z,j⟩:=qBf s _ k d (kAB (I.trans (le_add_of_le_of_nonneg (le_add_of_nonneg_right zero_le_one) nn))) G6
 obtain ⟨r,g⟩:=qDf s _ k d (kAB (I.trans (le_add_of_le_of_nonneg (le_add_of_nonneg_right zero_le_one) nn))) G6
 have H:=e2aa (R (s+nfOf n*t)) (R s+R (nfOf n*t));have q:=Ἐ (R s) (R (nfOf n*t));have p:=Ὕ (R s) (R (nfOf n*t))
 rw[zpow_neg,zpow_ofNat,<-one_div] at c;have c':=(div_le_iff₀' (by norm_num1:(0:ℝ)<2^127)).mp c
 have f':=sck (by norm_num1) f;have j':=sck (by norm_num1) j;have g':=sck (by norm_num1) g
 have M:=ἀ ((le_add_of_nonneg_right zero_le_one).trans A:(n:ℝ)<=2^20) e
 have bT:((n:ℝ)+8)*R t=R (nfOf n)*2^53*R t:=by rw[b']
 have bY:((n:ℝ)+8)*Y=R (nfOf n)*2^53*Y:=by rw[b']
 have nT:=ἁ (neg_abs_le (R t-Y)) nf0;have nU:=ἁ u' nf0;have nE:=ἀ c30 hε0
 have pg:=le_abs_self (R (s-nfOf n*t)-(R s-R (nfOf n*t)));have nj:=neg_abs_le (R (s+nfOf n*t)-(R s+R (nfOf n*t)))
 have pf:=le_abs_self (R (nfOf n*t)-R (nfOf n)*R t);have nf':=neg_abs_le (R (nfOf n*t)-R (nfOf n)*R t)
 have py:=le_abs_self (R s-X);have ny:=neg_abs_le (R s-X);have nS:=Ἀ (R s);have nP:=Ἀ (R (nfOf n*t))
 constructor
 . linarith only[y',py,g',pg,p,I,f',nf',x',D,bT,bY,nT,nU,nE,M,A,o,e,B',E,hε1,hε0,nf0,c30,c',H,j',q,nS,nP]
 . linarith only[y',ny,j',nj,q,I,f',nf',x',D,bT,bY,nT,nU,nE,M,A,o,e,B',E,hε1,hε0,nf0,c30,c',H,p,nS,nP]
lemma qEv (g h:ℕ->ὧ) (l:List ℕ) (a b:ὧ):
 l.foldl (λ(d:F1) t=>F1.mk (d.s+g t) (d.t+h t)) ⟨a,b⟩=⟨l.foldl (λ(c:ὧ) t=>c+g t) a,l.foldl (λ(c:ὧ) t=>c+h t) b⟩:=by
 induction l generalizing a b with
 | nil=>rfl
 | cons x l ih=>simp only[ὔ,ih]
lemma qC9 (a:List ℕ) (f:ℕ->ℝ) (M:ℝ) (h:∀t∈a,f t<=M):(a.map f).sum<=ѣa*M:=by simpa only[List.map_const',List.sum_replicate,nsmul_eq_mul] using List.sum_le_sum h
section
variable (w:ὥ) (hg:good w=true)
include w hg
lemma qB5: |wv w| =A_mag (ѥw%2^31):=by rw[qA7 w hg,ἢ,ἅ (qAs _)];split_ifs<;>simp only[abs_neg,abs_one,Nat.reducePow,one_mul]
lemma qAo: |wv w| <=2^128:=by rw[qB5 w hg];apply qFb;rcases qAz w hg with h | h<;>omega
lemma qBr (h:w&&&В=0):wv w=0:=by rw[qA7 w hg,<-qAq w,h];simp only[Nat.reducePow,A_mag,UInt32.toNat_zero, ↓reduceIte,mul_zero]
end
lemma qHc (a b:ὥ) (ha:good a=true) (hb:good b=true):
 (dv a*dv b)ᶠ /\
  |R (dv a*dv b)-wv a*wv b| <=1/2^53*(|wv a*wv b| +A_z)/\
  |wv a*wv b| <=2^256:=by
 obtain ⟨j,g⟩:=qAm a ha
 obtain ⟨i,d⟩:=qAm b hb
 have c:=qAo a ha
 have e:=qAo b hb
 obtain ⟨h,f⟩:=qCf _ _ j i (g▸c.trans qAn) (d▸e.trans qAn)
 rw[g,d] at f
 exact ⟨h,f,by
  rw[ἢ]
  exact (ἑ c e (Ἀ _) ו).trans_eq (by norm_num1)⟩
lemma qBs {n:ℕ} {a b:ℕ->ὥ} {f:ὥ} (g:∀t,t<n->good (a t)=true) (j:∀t,t<n->good (b t)=true) {l:List ℕ} (k:l.Nodup) (d:∀t∈l,t<n) (i:∀t,t<n->t∉l->a t&&&В=0\/b t&&&В=0):exactS n a b f=(l.map λt=>wv (a t)*wv (b t)).sum+wv f:=by
 unfold exactS;congr 1;rw[<-List.sum_toFinset _ k];refine (Finset.sum_subset (λt c=>by simpa only[Ὁ] using d t (by simpa only[List.mem_toFinset] using c)) λt c h=>?_).symm;have e:=Ὁ.mp c;rcases i t e (by simpa only[List.mem_toFinset] using h) with h | h;. simp only[qBr (a t) (g t e) h,zero_mul]
 . simp only[qBr (b t) (j t e) h,mul_zero]
lemma qFu {n:ℕ} {l:List ℕ} (b:l.Nodup) (a:∀t∈l,t<n):ѣl<=n:=by simpa only[List.length_range] using (b.subperm λt h=>List.mem_range.2 (a t h)).length_le
abbrev A_G (ts:List ℕ) (g:ℕ->ὧ) (p:ℕ->ℝ):Prop:=
 ∀t∈ts,(g t)ᶠ/\|R (g t)-p t| <=1/2^53*(|p t| +A_z)/\|p t| <=2^256
lemma qFj {b c:ℕ->ὥ} {l:List ℕ} {n:ℕ} (a:∀t∈l,t<n) (e:∀t,t<n->good (b t)=true) (f:∀t,t<n->good (c t)=true):A_G l (λt=>dv (b t)*dv (c t)) λt=>wv (b t)*wv (c t):=λt d=>qHc _ _ (e t (a t d)) (f t (a t d))
lemma qCr {a:List ℕ} {g:ℕ->ὧ} {p:ℕ->ℝ} (b:ὧ) (c:A_G a g p):let A:=(a.map λt=>|p t|).sum
 let L:=a.map g++[b]
 |A_sr L-((a.map p).sum+R b)| <=1/2^53*(A+ѣa*A_z)/\A_sa L<=A+ |R b| +1/2^53*(A+ѣa*A_z)/\A<=2*A_sa L+ѣa*A_z:=by
 induction a with
 | nil=>simp only[A_sr,List.map_nil,List.nil_append,List.map_cons,List.sum_cons,List.sum_nil,add_zero,zero_add,sub_self,abs_zero,one_div,List.length_nil,CharP.cast_eq_zero,zero_mul,mul_zero,Std.le_refl,A_sa,Nat.ofNat_pos,mul_nonneg_iff_of_pos_left,Ἀ,and_self]
 | cons x a ih=>
  obtain ⟨d,i,j⟩:=ih (λt e=>c t (by simp only[List.mem_cons,e,or_true]))
  have f:=(c x ג).2.1
  simp only[A_sa,A_sr,List.map_cons,List.sum_cons,List.length_cons,List.cons_append] at d i j⊢
  have sck:∀{k a b:ℝ},0<k->a<=1/k*b->k*a<=b:=λhk h=>(le_div_iff₀' hk).mp (h.trans_eq (one_div_mul_eq_div _ _))
  have m:=qAr
  have f':=sck (by norm_num1) f
  have d':=sck (by norm_num1) d
  have i':=sck (by norm_num1) (sub_le_iff_le_add'.mpr i)
  have nf:=neg_abs_le (R (g x)-p x);have pf:=le_abs_self (R (g x)-p x)
  have nd:=neg_abs_le ((List.map R (List.map g a++[b])).sum-((List.map p a).sum+R b));have pd:=le_abs_self ((List.map R (List.map g a++[b])).sum-((List.map p a).sum+R b))
  have k:=Ἀ (p x)
  have h:=Ὥ (p x) (R (g x))
  have l:=Ὥ (R (g x)) (p x)
  rw[ἣ] at h
  refine ⟨abs_le.mpr ⟨?_,?_⟩,?_,?_⟩<;>push_cast
  . rw[one_div_mul_eq_div,<-neg_div,div_le_iff₀' (by norm_num1)];linarith only[nf,nd,f',d',m,k]
  . rw[one_div_mul_eq_div,le_div_iff₀' (by norm_num1)];linarith only[pf,pd,f',d',m,k]
  . rw[<-sub_le_iff_le_add',one_div_mul_eq_div,le_div_iff₀' (by norm_num1)];linarith only[i',l,f',m,k]
  . linarith only[j,h,f',m,k]
lemma qHh (l:List ℕ) (g:ℕ->ὧ) (b:ὧ):((l.map g++[b]).length:ℝ)=ѣl+1:=by simp only[List.length_append,List.length_map,List.length_cons,List.length_nil,zero_add,Nat.cast_add,Nat.cast_one]
lemma qHf {a:List ℕ} {g:ℕ->ὧ} {p:ℕ->ℝ} {b:ὧ} (d:A_G a g p) (c:bᶠ):∀q∈a.map g++[b],qᶠ:=by
 simp only[List.mem_append,List.mem_map,List.mem_singleton];rintro q (⟨t,e,rfl⟩ | rfl);. exact (d t e).1
 . exact c
lemma qCs {a:ὥ} (d:good a=true) (h:a<=В):
 0<=wv a/\wv a<=2^128/\ ∀w,good w=true->w&&&В<=a->|wv w| <=wv a:=by
 have g:ѥa<=2147483647:=ἃ.mp h
 have b:wv a=A_mag ѥa:=by
  rw[qA7 a d,Ἁ (by exact Nat.lt_succ_of_le ‹_›)]
  simp only[show ¬2^31<=ѥa by (exact Nat.not_lt.mpr ‹_›),↓reduceIte,one_mul]
 refine ⟨b▸qAs _,(le_abs_self _).trans (qAo a d),λw c f=>?_⟩
 have e:=ἃ.mp f
 rw[qAq] at e
 rw[qB5 _ c,b]
 exact qA6 _ _ e ((qAz _ c).imp_right (·.1))
section
variable {n:ℕ} {ts:List ℕ} {g:ℕ->ὧ} {p:ℕ->ℝ} {bw hh:ὥ} {X Y J:ℝ} {s t a e:ὧ}
 (A:n<2^20) (B:ѣts<=n) (C:A_G ts g p) (D:good bw=true)
 (E:X=(ts.map p).sum+wv bw) (F:s=ts.foldl (λ(c:ὧ) t=>c+g t) 0+dv bw)
 (G:(2:ℝ)^(-127:ℤ) ≤ |R (s+e)|)
include A B C D E F G
lemma qEz (k:tᶠ) (b:Y<=A_B/8) (o:A_sa (ts.map g++[dv bw])+A_z<=(1+1/2^22)*Y)
 (u:(1+1/2^22)*((ѣts:ℝ)+1)*(A_sa (ts.map g++[dv bw])+A_z)+((ts.map λt=>|p t|).sum+ѣts*A_z)<=((ѣts:ℝ)+4)*Y+1/2^240)
 (m: |R t-Y| <=Y/2^30+1/2^250) (h:e=nfOf n*t):
 R (s-e)<X/\X<R (s+e):=by
 have e2AB7:∀{Y:ℝ},0<=Y->Y<=A_B/8->(1+1/2^22)*Y<=A_B/4:=λhY0 hY=>by linarith only[hY0,hY]
 subst h
 obtain rfl:s=A_fsum (ts.map g++[dv bw]):=by simp only[F,A_fsum,List.foldl_append,List.foldl_map,ὔ,List.foldl_nil]
 obtain ⟨c,d⟩:=qAm bw D
 obtain ⟨j,-,-⟩:=qCr (dv bw) C
 rw[d,<-E] at j
 have i:=qCe (ts.map g++[dv bw])
 have r:=qAr
 have l:(ѣts:ℝ)<=n:=by exact_mod_cast B
 have q:(n:ℝ)+1<=2^20:=by exact_mod_cast A
 have a:0<=Y:=nonneg_of_mul_nonneg_right ((add_nonneg i r.le).trans o) ו
 obtain ⟨f,h⟩:=qDd _ (qHf C c) (by rw[List.length_append,List.length_map,List.length_singleton];exact (Nat.succ_le_of_lt (Nat.lt_of_le_of_lt B A)).trans (Nat.le_of_ble_eq_true rfl)) ((le_add_of_nonneg_right r.le).trans (o.trans (e2AB7 a b)))
 rw[qHh] at h
 generalize ts.map g++[dv bw]=L at*
 have x:=qC7 L
 have y:=ἀ l a
 have v:=ἑ ((add_le_add l le_rfl).trans q:(ѣts:ℝ)+1<=2^20) o (add_nonneg i r.le) ו
 have w:=abs_sub_le (R (A_fsum L)) (A_sr L) X
 have z:=Ὥ (R (A_fsum L)) (A_sr L)
 exact qGb A f k a b (by linarith only[u,j,h,y,w]) m (by linarith only[r,o,i,h,x,v,z]) G
lemma qDi (P:good hh=true) (hp:hh<=В) (U:(ts.map λt=>|p t|).sum<=wv hh*J) (m:aᶠ) (d:0<=J) (z:J<=2^149) (v: |R a-J| <=(J+A_z)/2^32) (he:e=nfOf n*(dv hh*a+(dv bw).abs)):R (s-e)<X/\X<R (s+e):=by
 have e2pa:∀{x y:ℝ} {a b j:ℕ},x<=2^a->y<=2^b->a<=j->b<=j->x+y<=2^(j+1):=λ{x y a b j} hx hy ha hb=>(add_le_add (hx.trans (pow_le_pow_right₀ one_le_two ha)) (hy.trans (pow_le_pow_right₀ one_le_two hb))).trans_eq ((two_mul _).symm.trans (pow_succ' 2 j).symm)
 have e2z0:A_z<=2^0:=qBo.trans_eq (pow_zero 2).symm
 have e2d:∀{x:ℝ} (n:ℕ),0<=x->1/2^n*x<=x:=λn h0=>mul_le_of_le_one_left h0 (div_le_one_of_le₀ (one_le_pow₀ one_le_two) (pow_nonneg zero_le_two n))
 have e2AB:∀{x:ℝ} {k:ℕ},x<=2^k->k+3<=256+42->x<=A_B/8:=λ{x k} h hk=>(le_div_iff₀ (by norm_num:(0:ℝ)<8)).mpr ((ἀ h (by norm_num:(0:ℝ)<=8)).trans ((((congrArg ((2:ℝ)^k*·) (by norm_num:(8:ℝ)=2^3)).trans (pow_add 2 k 3).symm).trans_le ((pow_le_pow_right₀ one_le_two hk).trans_eq (pow_add 2 256 42))).trans qCc))
 have e2AB1:∀{x:ℝ} {k:ℕ},x<=2^k->k<=256+42->x<=A_B:=λh hk=>h.trans (((pow_le_pow_right₀ one_le_two hk).trans_eq (pow_add 2 256 42)).trans qCc)
 have e2aa:∀(x y:ℝ),|x| <= |y| + |x-y|:=λx y=>sub_le_iff_le_add'.mp (Ὥ x y)
 obtain ⟨b,h,-⟩:=qCs P hp;obtain ⟨Q,N⟩:=qAm hh P;obtain ⟨q,j⟩:=qAm bw D;obtain ⟨-,K,-⟩:=qCr (dv bw) C;rw[j] at K;have S:=qAo bw D;generalize wv hh=H at*;have hk:(ѣts:ℝ)+1<=2^20:=by exact_mod_cast (by (exact Nat.lt_of_le_of_lt ‹_› ‹_›):ѣts+1<=2^20)
 have L:(0:ℝ)<=ѣts:=Ἔ _;have hkN:(ѣts:ℝ)<=2^20:=(le_add_of_nonneg_right zero_le_one).trans hk;have M:=qAr;have x:=qAw;have r:=qC4 ts p;have u:=Ὥ (R a) J;rw[ἅ d] at u;obtain ⟨e1,W⟩:=abs_le.mp v;have e3:=ἁ e1 b;have e4:=ἁ W b;have e5:=ἑ h x M.le ו
 have e6':=(e2aa (R a) J).trans (add_le_add (ἅ d).le v);have e6: |H*R a| <=H*(J+(J+A_z)/2^32):=by rw[ἢ,ἅ b];exact ἁ e6' b
 have bRa: |R a| <=2^151:=e6'.trans (e2pa (j:=150) z ((div_le_self (add_nonneg d M.le) (one_le_pow₀ one_le_two)).trans (e2pa (j:=149) z e2z0 le_rfl (Nat.le_of_ble_eq_true rfl))) (Nat.le_of_ble_eq_true rfl) le_rfl)
 have hRa: |H*R a| <=2^279:=by rw[ἢ,ἅ b];exact (ἑ h bRa (Ἀ _) ו).trans_eq (pow_add 2 128 151).symm
 have hW:=Ἆ b d;have hHJ:H*J<=2^128*2^149:=ἑ h z d ו;obtain ⟨o,i⟩:=qCf _ a Q m (by rw[N,ἅ b];exact h.trans qAn) (e2AB1 bRa (Nat.le_of_ble_eq_true rfl));rw[N] at i;have e7:=Ὥ (R (dv hh*a)) (H*R a);obtain ⟨w,f⟩:=qBl (dv bw) q;rw[j] at f
 have bRha: |R (dv hh*a)| <=2^281:=(e2aa _ (H*R a)).trans (e2pa (j:=280) hRa (i.trans ((e2d 53 (add_nonneg (Ἀ _) M.le)).trans (e2pa (j:=279) hRa e2z0 le_rfl (Nat.le_of_ble_eq_true rfl)))) (Nat.le_of_ble_eq_true rfl) le_rfl)
 obtain ⟨y,k⟩:=qBf _ _ o w (e2AB1 bRha (Nat.le_of_ble_eq_true rfl)) (by rw[f,abs_abs];exact S.trans qAn);rw[f] at k;have e8:=Ἐ (R (dv hh*a)) |wv bw|;rw[abs_abs] at e8;obtain ⟨e9,Z⟩:=abs_le.mp i;obtain ⟨e11,V⟩:=abs_le.mp k;have hkz:=ἀ hkN M.le;have O:=Ἀ (wv bw);obtain ⟨Y,hY⟩:∃Y,Y=H*J+ |wv bw| +A_z:=⟨_,rfl⟩;have c:A_sa (ts.map g++[dv bw])+A_z<=(1+1/2^22)*Y:=(by linarith only[hY,O,hkz,r,M,U,K]);have d2: |H*R a-H*J| <=H*((J+A_z)/2^32):=by rw[<-mul_sub,ἢ,ἅ b];exact ἁ v b
 have d3: |R (dv hh*a+(dv bw).abs)-Y| <= |R (dv hh*a+(dv bw).abs)-(R (dv hh*a)+|wv bw|)|+(|R (dv hh*a)-H*R a|+(|H*R a-H*J|+A_z)):=by
  rw[hY,show R (dv hh*a+(dv bw).abs)-(H*J+|wv bw|+A_z)=(R (dv hh*a+(dv bw).abs)-(R (dv hh*a)+|wv bw|))+((R (dv hh*a)-H*R a)+((H*R a-H*J)-A_z)) by ring]
  exact (Ἐ _ _).trans (add_le_add le_rfl ((Ἐ _ _).trans (add_le_add le_rfl ((Ὕ _ _).trans (add_le_add le_rfl (ἅ M.le).le)))))
 have l: |R (dv hh*a+(dv bw).abs)-Y| <=Y/2^30+1/2^250:=d3.trans (by linarith only[k,i,d2,e8,e7,e6,e5,hY,O,x,M,hW])
 have Y0:0<=Y:=(add_nonneg (add_nonneg hW O) M.le).trans_eq hY.symm
 refine qEz A B C D E F G y (e2AB (k:=279) (hY.trans_le (e2pa (j:=278) (e2pa (j:=277) (hHJ.trans_eq (pow_add 2 128 149).symm) S (Nat.le_of_ble_eq_true rfl) (Nat.le_of_ble_eq_true rfl)) e2z0 (Nat.le_of_ble_eq_true rfl) (Nat.le_of_ble_eq_true rfl))) (Nat.le_of_ble_eq_true rfl)) c ?_ l he
 have h3:=ἁ c (add_nonneg L zero_le_one);have h4:=ἀ hkN Y0
 linarith only[h4,h3,hY,O,hkz,r,x,M,U]
lemma qBm {a:ℕ->ὥ} (u:good hh=true) (w:hh<=В) (i:∀t∈ts,good (a t)=true) (o:∀t∈ts,|p t| <=wv hh* |wv (a t)|) (r:e=nfOf n*(dv hh*ts.foldl (λ(c:ὧ) t=>c+(dv (a t)).abs) 0+(dv bw).abs)):R (s-e)<X/\X<R (s+e):=by
 have f:(ѣts:ℝ)<=2^20:=by exact_mod_cast (by (clear w;exact omS61 A B):ѣts<=2^20)
 have k:A_sa (ts.map λt=>dv (a t))=(ts.map λt=>|wv (a t)|).sum:=by simp only[A_sa,List.map_map];congr 1;exact ὠ λt d=>by simp only[Function.comp_apply,(qAm _ (i t d)).2]
 have h:=(qC9 ts _ _ λt d=>qAo _ (i t d)).trans (ἀ f ו);rw[<-k] at h;have m:=qCc
 obtain ⟨j,b⟩:=qB7 _ (λq v=>by
  obtain ⟨t,d,rfl⟩:=List.mem_map.mp v
  exact (qAm _ (i t d)).1) (by simp only[List.length_map,Nat.reducePow];(clear w;exact omS62 A B)) (by linarith only[m,h,D])
 rw[List.length_map] at b;have g:=qCe (ts.map λt=>dv (a t));have l:=qAr;refine qDi A B C D E F G u w ?_ j g (h.trans (by norm_num1)) ?_ (r.trans (by simp only[A_fsum,List.map_map,List.foldl_map,Function.comp_apply]));. rw[k,<-List.sum_map_mul_left];exact List.sum_le_sum o
 . refine b.trans ?_;have:=ἀ f (add_nonneg g l.le);linarith only[this,D]
end
section
variable (n:Ὣ) (hn:n<2^20) (xw ww:Ὣ->ὥ)
include n hn xw ww
lemma qDm (bw:ὥ) (hx:∀t,t<n->good (xw t)=true) (hw:∀t,t<n->good (ww t)=true) (hb:good bw=true) (l:љ) (hl:l.Nodup) (hl2:∀t∈l,t<n) (hz:∀t,t<n->t∉l->xw t&&&В=0\/ww t&&&В=0) (s e:ὧ)
 (hs:let r:=l.foldl (λ(acc:F1) t=>let q:=dv (xw t)*dv (ww t)
  F1.mk (acc.s+q) (acc.t+q.abs)) ⟨0,0⟩;s=r.s+dv bw/\e=nfOf n*(r.t+(dv bw).abs))
 (hbig:(2:ℝ)^(-127:ℤ)<= |R (s+e)|):
 R (s-e)<exactS n xw ww bw/\exactS n xw ww bw<R (s+e):=by
 have e2pa:∀{x y:ℝ} {a b j:ℕ},x<=2^a->y<=2^b->a<=j->b<=j->x+y<=2^(j+1):=λ{x y a b j} hx hy ha hb=>(add_le_add (hx.trans (pow_le_pow_right₀ one_le_two ha)) (hy.trans (pow_le_pow_right₀ one_le_two hb))).trans_eq ((two_mul _).symm.trans (pow_succ' 2 j).symm)
 have e2z0:A_z<=2^0:=qBo.trans_eq (pow_zero 2).symm
 have e2d:∀{x:ℝ} (n:ℕ),0<=x->1/2^n*x<=x:=λn h0=>mul_le_of_le_one_left h0 (div_le_one_of_le₀ (one_le_pow₀ one_le_two) (pow_nonneg zero_le_two n))
 have e2AB:∀{x:ℝ} {k:ℕ},x<=2^k->k+3<=256+42->x<=A_B/8:=λ{x k} h hk=>(le_div_iff₀ (by norm_num:(0:ℝ)<8)).mpr ((ἀ h (by norm_num:(0:ℝ)<=8)).trans ((((congrArg ((2:ℝ)^k*·) (by norm_num:(8:ℝ)=2^3)).trans (pow_add 2 k 3).symm).trans_le ((pow_le_pow_right₀ one_le_two hk).trans_eq (pow_add 2 256 42))).trans qCc))
 have e2AB4:A_B/8<=A_B/4:=div_le_div_of_nonneg_left qI1.le ב ב
 have e2z250:A_z<=1/2^250:=qAw.trans ב
 simp only[qEv] at hs
 obtain ⟨hs,E⟩:=hs
 have a:=qFj hl2 hx hw
 have f:=qFu hl hl2
 obtain ⟨h,i⟩:=qAm bw hb
 obtain ⟨-,m,y⟩:=qCr (dv bw) a
 rw[i] at m
 have b:=qCe (l.map (λt=>dv (xw t)*dv (ww t))++[dv bw])
 have k:=qC9 l _ _ λt u=>(a t u).2.2
 have v:(ѣl:ℝ)+1<=2^20:=by exact_mod_cast (by (exact Nat.lt_of_le_of_lt ‹_› ‹_›):ѣl+1<=2^20)
 have o:(0:ℝ)<=ѣl:=Ἔ _
 have d:=qAr
 have D:=qAw
 have D':(2:ℝ)^200*(2^200*A_z)<=1:=(le_div_iff₀' (by norm_num1)).mp ((le_div_iff₀' (by norm_num1)).mp (qAw.trans_eq (one_div_mul_eq_div _ _)))
 have w:=qAo bw hb
 have z:=qCc
 have A:=qC4 l λt=>wv (xw t)*wv (ww t)
 have v':(ѣl:ℝ)<=2^20:=(le_add_of_nonneg_right zero_le_one).trans v
 have p:=(k.trans (ἀ v' ו)).trans_eq (pow_add 2 20 256).symm
 have C:=ἀ v' d.le
 have nz:=(C.trans (mul_le_of_le_one_right (pow_nonneg zero_le_two 20) qBo))
 have pa:=e2pa (j:=276) p nz (Nat.le_of_ble_eq_true rfl) (Nat.le_of_ble_eq_true rfl)
 have g:A_sa (l.map (λt=>dv (xw t)*dv (ww t))++[dv bw])+A_z<=A_B/8:=e2AB (k:=279) (e2pa (j:=278) (m.trans (e2pa (j:=277) (e2pa (j:=276) p w (Nat.le_of_ble_eq_true rfl) (Nat.le_of_ble_eq_true rfl)) ((e2d 53 (add_nonneg A (Ἆ o d.le))).trans pa) (Nat.le_of_ble_eq_true rfl) (Nat.le_of_ble_eq_true rfl))) e2z0 (Nat.le_of_ble_eq_true rfl) (Nat.le_of_ble_eq_true rfl)) (Nat.le_of_ble_eq_true rfl)
 obtain ⟨j,c⟩:=qB7 _ (qHf a h) (by rw[List.length_append,List.length_map,List.length_singleton];exact (Nat.succ_le_of_lt (Nat.lt_of_le_of_lt f hn)).trans (Nat.le_of_ble_eq_true rfl)) (((le_add_of_nonneg_right d.le).trans g).trans e2AB4)
 rw[qHh] at c
 have B:=Ἆ o b
 have x:=ἀ v (add_nonneg b d.le)
 have Y0:=add_nonneg b d.le
 exact qEz hn f a hb (qBs hx hw hl hl2 hz) hs hbig j g (le_mul_of_one_le_left Y0 (le_add_of_nonneg_right (one_div_nonneg.mpr (pow_nonneg zero_le_two 22)))) (by have G:=(abs_le.mp c).1;have F:=(abs_le.mp c).2;linarith only[y,D',C,x,G,F]) ((congrArg (λr:ℝ=>|r|) (sub_sub _ _ _).symm).trans_le ((Ὕ _ _).trans (add_le_add (c.trans (by linarith only[x,Y0]:_<=(A_sa (l.map (λt=>dv (xw t)*dv (ww t))++[dv bw])+A_z)/2^30)) ((ἡ d).trans_le e2z250)))) (E.trans (by simp only[A_fsum,List.map_append,List.map_map,List.map_cons,List.map_nil,List.foldl_append,List.foldl_map,Function.comp_apply,ὔ,List.foldl_nil]))
lemma qC8 (bw hv:ὥ) (hx:∀t,t<n->good (xw t)=true) (hw:∀t,t<n->good (ww t)=true) (hb:good bw=true) (hgv:good hv=true) (hpv:hv<=В) (l:љ) (hl:l.Nodup) (hl2:∀t∈l,t<n) (hmv:∀t∈l,ww t&&&В<=hv) (hz:∀t,t<n->t∉l->xw t&&&В=0) (s e:ὧ)
 (hs:s=l.foldl (λ(acc:ὧ) t=>acc+dv (xw t)*dv (ww t)) 0+dv bw/\e=nfOf n*(dv hv*(l.foldl (λ(acc:ὧ) t=>acc+(dv (xw t)).abs) 0)+(dv bw).abs))
 (hbig:(2:ℝ)^(-127:ℤ)<= |R (s+e)|):
 R (s-e)<exactS n xw ww bw/\exactS n xw ww bw<R (s+e):=by
 obtain ⟨hs,b⟩:=hs
 obtain ⟨-,-,c⟩:=qCs hgv hpv
 exact qBm hn (qFu hl hl2) (qFj hl2 hx hw) hb (qBs hx hw hl hl2 λt i z=>.inl (hz t i z)) hs hbig hgv hpv
  (λt a=>hx t (hl2 t a)) (λt a=>by
   rw[ἢ,mul_comm]
   exact ἀ (c _ (hw t (hl2 t a)) (hmv t a)) (Ἀ _)) b
end
end
section
noncomputable def A_r (b:Ὡ):ℝ:=if b then-1 else 1
lemma qF9 (s t:Ὡ):A_r (s ^^ t)=A_r s*A_r t:=by cases s<;>cases t<;>simp only[A_r,bne_self_eq_false,false_eq_true, ↓reduceIte,mul_one,Bool.bne_true,Bool.not_false,mul_neg,Bool.bne_false,neg_neg]
lemma qF8 (S T:Ὡ) (X Y:ℝ) (a:0<X) (b:0<Y) (h:A_r S*X=A_r T*Y):S=T:=by cases S<;>cases T<;>simp only[A_r,false_eq_true, ↓reduceIte,one_mul,neg_mul,Bool.true_eq_false,neg_inj] at h⊢<;>linarith only[h,b,a]
section
variable (d:FloatLib.Numerics.Dyadic)
include d
lemma qD4:d.toReal=A_r d.negative*((d.significand:ℝ)*2^d.exponent):=by obtain ⟨n,m,e⟩:=d;cases n<;>simp only[FloatLib.Numerics.Dyadic.toReal,signedSignificand,false_eq_true, ↓reduceIte,Int.ofNat_eq_natCast,Int.cast_natCast,A_r,one_mul,Int.cast_neg,neg_mul]
lemma A_s:d.toReal=0↔d.significand=0:=by have h:(2:ℝ)^d.exponent≠0:=zpow_ne_zero _ ἓ;rw[qD4];cases d.negative<;>simp only[A_r,false_eq_true, ↓reduceIte,one_mul,mul_eq_zero,Nat.cast_eq_zero,h,or_false,neg_mul,neg_eq_zero]
lemma qE9 (h:d.significand≠0):0<(d.significand:ℝ)*2^d.exponent:=mul_pos (Nat.cast_pos.mpr (ὼ h)) (ἵ two_pos _)
end
lemma qIe (a b:ὧ) (h:A_M a=A_M b):a=b:=by have:=Model.ofFloatModel_injective h;cases a;cases b;simp_all only
lemma qDk (f:ὧ) (h:fᶠ): ∃d,Model.toDyadic? (A_M f)=some d/\d.toReal=R f:=by obtain ⟨d,a⟩:=Model.exists_toDyadic?_of_isFinite ((qAh f).trans h);exact ⟨d,a,by rw[A_R,Model.toReal_eq,a]⟩
lemma qCq {a b:ὧ} {x y:FloatLib.Numerics.Dyadic} (f:Model.toDyadic? (A_M a)=some x) (d:Model.toDyadic? (A_M b)=some y) (e:aᶠ) (c:bᶠ):
 A_M (a*b)=if x.significand*y.significand=0 then
  Model.zero _ (x.negative ^^ y.negative) else ἕ _ ⟨x.negative ^^ y.negative,x.significand*y.significand,x.exponent+y.exponent⟩:=by
 rw[qB0 a b e c,Model.Proof.mul_eq_spec,Model.Spec.mul,f,d]
 simp only[or_eq_true,beq_iff_eq,FloatFormat.isIEEE_binary64,Model.zero_eq_signedZero_of_isIEEE,bne_iff_ne,ne_eq,ite_not,mul_eq_zero]
lemma qHe {a b:ὧ} {x y z:FloatLib.Numerics.Dyadic} (m:Model.toDyadic? (A_M a)=some x) (n:Model.toDyadic? (A_M b)=some y) (e:Model.toDyadic? (A_M (a*b))=some z) (k:aᶠ) (l:bᶠ) (c:z.toReal=x.toReal*y.toReal):z.negative=(x.negative ^^ y.negative):=by
 by_cases g:x.significand*y.significand=0;. have d:=qCq m n k l;simp only[g,↓reduceIte] at d;rw[d,Model.toDyadic?_zero] at e;cases e;show Model.signBit _=_;cases x.negative<;>cases y.negative<;>simp only[Model.zero,bne_self_eq_false,FloatFormat.isIEEE_binary64,FloatFormat.supportsSignedZero_eq_true_of_isIEEE,Bool.and_true,false_eq_true, ↓reduceIte,Model.signBit_posZero,Bool.bne_true,Bool.not_false,Bool.and_self,Model.signBit_negZero,Bool.bne_false]
 . obtain ⟨f,i⟩:=Nat.mul_ne_zero_iff.mp g;have j:z.significand≠0:=λh=>by rw[(A_s z).2 h] at c;exact mul_ne_zero (mt (A_s x).1 f) (mt (A_s y).1 i) c.symm
   rw[qD4 z,qD4 x,qD4 y,mul_mul_mul_comm,<-qF9] at c;exact qF8 _ _ _ _ (qE9 z j) (mul_pos (qE9 x f) (qE9 y i)) c
lemma qGg (v m p:ὧ) (f:vᶠ) (i:mᶠ) (h:pᶠ) (u:(v*p)ᶠ) (s:(m*p)ᶠ) (q:R (v*p)=R v*R p) (r:R (m*p)=R m*R p):v*p*m=v*(m*p):=by
 obtain ⟨a,o,n⟩:=qDk v f;obtain ⟨b,y,k⟩:=qDk p h;obtain ⟨c,w,A⟩:=qDk m i;obtain ⟨d,t,B⟩:=qDk _ u;obtain ⟨e,l,x⟩:=qDk _ s;have D:=qHe o y t f h (by rw[B,n,k,q]);have E:=qHe w y l i h (by rw[x,A,k,r]);have C:(d.negative ^^ c.negative)=(a.negative ^^ e.negative):=by rw[D,E];cases a.negative<;>cases b.negative<;>cases c.negative<;>rfl
 have j:d.toReal*c.toReal=a.toReal*e.toReal:=by rw[B,A,n,x,q,r];ring
 have g:d.significand*c.significand=0↔a.significand*e.significand=0:=by simp only[Nat.mul_eq_zero,<-A_s,<-mul_eq_zero,j]
 apply qIe;rw[qCq t w u i,qCq o l f s];by_cases z:d.significand*c.significand=0;. simp only[z,g.1 z,↓reduceIte,C]
 . simp only[z,mt g.2 z,↓reduceIte];refine ExactRuntime.roundDyadic_congr _ qAj _ _ z (mt g.2 z) (Rat.cast_injective (α:=ℝ) ?_);rw[Ἠ,Ἠ,Model.Dyadic.toReal_mul,Model.Dyadic.toReal_mul,j]
lemma qFw (x w:ὥ) (hx:good x=true):dvB (dv x) w=dv x*dv w:=by
 obtain ⟨e,o⟩:=qAm x hx;obtain ⟨p,f,j,q,g,-⟩:=qD1 w;obtain ⟨d,a,-,-,l,b⟩:=qD1 x;obtain ⟨m,c,k⟩:=qCw _ a;have r:=x.toNat_lt;have h:R (dv x)=(((2^23+ѥx%2^23:ℕ)*m:ℤ):ℝ)*2^(((ѥx/2^23%256:ℕ):ℤ)-150):=by unfold dv;rw[d,l,b,k,Int.cast_mul,Int.cast_natCast,mul_assoc]
 obtain ⟨n,i⟩:=qCm (dv x) _ f e _ _ (by
  rw[Int.natAbs_mul,Int.natAbs_natCast];have:=Nat.mul_le_mul_left (2^23+ѥx%2^23) c
  (clear f a c r;omega)) (by clear f a c r;exact omS63) (by rw[o];exact (qAo x hx).trans qAn) h
 show dv x*pw.uget _ _*_=dv x*(_*pw.uget _ _);rw[p];exact qGg _ _ _ e j (qDa _ f).1 n q i g
end
section
section
open Float.Model Float.Model.UnpackedFloat
lemma qFq (t:Int):Format.binary32.targetExponent t=max (t-24) (-149):=by simp only[Format.targetExponent,Format.mantissaBits,Nat.reduceAdd,Nat.cast_ofNat,Format.minExponent,Nat.add_one_sub_one,Int.reducePow,Int.reduceSub,Int.reduceNeg]
lemma B_z (a b:ℤ) (n:ℕ) (h:a=n+b):(2:ℝ)^a=2^n*2^b:=by rw[h,Ἓ ἓ,Ἡ]
lemma qDg (m L:Ὣ) (h1:2^L<=m) (h2:m<2^(L+1)):m.log2=L:=(Nat.log2_eq_iff (by have:=Nat.two_pow_pos L;(clear h2;omega))).2 ⟨h1,h2⟩
lemma qEa (M:Ὣ) (e T:Int) (K r:Ὣ) (d:T=e+(K+1:Ὣ)) (c:0<r) (f:2*(r*2^K)<M+2^K) (g:M<2*(r*2^K)+2^K):Model.roundMantissaAtExponentEven M e T=r:=by
 have b:r<<<(K+1)=2*(r*2^K):=by rw[ὁ,Nat.pow_succ];ring
 have a:Model.pow2 (K+1-1)=2^K:=Model.pow2_eq_two_pow K;rw[Model.roundMantissaAtExponentEven,ἱ (by clear c f g b a;exact omS64 d),show (T-e).toNat=K+1 by (clear c f g b a;exact omS65 d)];rcases le_total (2*(r*2^K)) M with h | h;. have:=Model.roundShiftRightEven_shiftLeft_add_of_lt_half r (M-2*(r*2^K)) (K+1) (by rw[a];(clear d c f b;exact omS66 g h));rwa[b,Nat.add_sub_cancel' h] at this
 . have:=Model.roundShiftRightEven_shiftLeft_sub_of_lt_half r (2*(r*2^K)-M) (K+1) (by rw[a];(clear d c g b;exact omS67 f h));rwa[b,Nat.sub_sub_self h] at this
lemma qEf (s:Sign) (P:Ὣ) (E:Int) (h1:2^23<=P) (h2:P<2^24) (hE:-149<=E) (hp:0<P):Model.finishRoundedMantissa Format.binary32 s (P,E)=.finite s P E hp:=by
 have:=Model.finishRoundedMantissa_normalized FloatFormat.binary32 s P (E+23) h1 h2 (by
  show-126<=E+23;(clear h1 h2 hp;exact omS68 hE))
 rwa[show E+23-(FloatFormat.binary32.fracWidth:ℤ)=E from Int.add_sub_cancel E 23] at this
lemma qBb (neg:Ὡ) (m:Ὣ) (e:Int):
 ἕ Fmt ⟨neg,m,e⟩=Model.ofModel Fmt (UnpackedFloat.round Format.binary32 (Model.modelSign neg) m e):=by
 simp only[Model.roundDyadic,FloatFormat.isIEEE_binary32, ↓reduceIte,Model.ieeeRoundDyadic]
 rfl
lemma qBn (s:Sign) (M:Ὣ) (E:Int) (h:0<M) (hM1:2^23<=M) (hM2:M<2^24) (hb1:1<=E+150) (hb2:E+150<=254):
 (UnpackedFloat.pack Format.binary32 (.finite s M E h)).toNat=(if s=.negative then 2^31 else 0)+(E+150).toNat*2^23+M%2^23:=by
 simp only[UnpackedFloat.pack,qDg M 23 hM1 hM2]
 rw[show (E+↑Format.binary32.exponentBias+ ↑Format.binary32.mantissaBitsWithoutImplicit).toNat=(E+150).toNat by
  simp only[Format.exponentBias];(norm_num1);ring_nf]
 have c:¬(2^Format.binary32.exponentBits<=(E+150).toNat+1):=by
  show ¬(2^8<=(E+150).toNat+1);(clear h hM1 hM2;exact omS69 hb2)
 have a:(23+1=Format.binary32.mantissaBits):=rfl
 have b:s.toBitVec.toNat=if s=.negative then 1 else 0:=by cases s<;>rfl
 simp only[c,ite_false,a,ite_true,packComponents,BitVec.toNat_append,b,BitVec.toNat_ofNat]
 rw[Ἁ (by (clear h hM1 hM2 c a b;exact omS70 hb2):(E+150).toNat<2^8),
  <-Nat.shiftLeft_add_eq_or_of_lt (by (clear h hM1 hM2 c a b;exact omS70 hb2):(E+150).toNat<2^8),
  <-Nat.shiftLeft_add_eq_or_of_lt (ὀ _ ו:M%2^23<2^23)]
 split_ifs<;>simp only[ὁ,Nat.reducePow,one_mul,Nat.add_right_cancel_iff,zero_mul,zero_add,Nat.add_left_cancel_iff]
 (clear h hM1 hM2 hb1 hb2 c a b;omega)
lemma qE4 (neg:Ὡ) (m:Ὣ) (e:Int) (P:Ὣ) (E:Int) (hm:m≠0) (hP1:2^23<=P) (hP2:P<2^24) (hE:-149<=E) (hE2:E+150<=254) (hhi:2*((m:ℝ)*2^e)<(2*P+1)*2^E) (hlo1:P≠2^23->(2*P-1)*2^E<2*((m:ℝ)*2^e)) (hlo2:P=2^23->33554431*2^E<4*((m:ℝ)*2^e)):
 toWord (ἕ Fmt ⟨neg,m,e⟩)=ὺ ((if neg then 2^31 else 0)+(E+150).toNat*2^23+P%2^23):=by
 have l:0<P:=(by clear hm hE hE2;exact omS71 hP1)
 obtain ⟨k,F⟩:∃k:Ὣ,e-E+2<=k:=⟨(e-E+2).toNat,by (clear hm hP1 hP2 hE hE2 l;exact omS72)⟩
 obtain ⟨g,z⟩:∃g:Ὣ,(e-k)+(g+2:Ὣ)=E:=⟨(E-(e-k)-2).toNat,by (clear hm hP1 hP2 hE hE2 l;exact omS73 F)⟩
 have u:(0:ℝ)<2^(e-k):=ἵ two_pos _
 have s:(m:ℝ)*2^e=((m*2^k:Ὣ):ℝ)*2^(e-k):=by
  push_cast
  rw[mul_assoc,<-B_z e _ k (by clear hm hP1 hP2 hE hE2 l F z;exact omS74)]
 have i:(2:ℝ)^E=4*((2^g:Ὣ):ℝ)*2^(e-k):=by
  rw[B_z E (e-k) (g+2) (by clear hm hP1 hP2 hE hE2 l F;exact omS75 z)]
  push_cast
  ring
 rw[qBb,<-ExactRuntime.round_scale _ _ m e k hm]
 have p:m*2^k≠0:=Nat.mul_ne_zero hm ו
 rw[s,i] at hhi
 rw[s,i] at hlo1 hlo2
 have H:0<2^g:=ו
 generalize m*2^k=M at*
 generalize e-k=q at*
 generalize (2:ℝ)^q=b at*
 have f:∀A B:ℕ,(A:ℝ)*b<B*b->A<B:=λA B h=>by exact_mod_cast lt_of_mul_lt_mul_right h u.le
 push_cast at hhi hlo1 hlo2
 have G:M<4*(P*2^g)+2*2^g:=f _ _ (by push_cast;linarith only[hhi])
 have t:P≠2^23->4*(P*2^g)<M+2*2^g:=λh=>f _ _ (by push_cast;linarith only[hlo1 h])
 have y:P=2^23->33554431*2^g<M:=λh=>f _ _ (by push_cast;linarith only[hlo2 h])
 have C:P*2^g<=(2^24-1)*2^g:=Ἒ _ (by exact Nat.le_of_lt_succ ‹_›)
 have r:2*(P*2^(g+1))=4*(P*2^g):=by rw[Nat.pow_succ];ring
 have n:2^(g+1)=2*2^g:=by rw[Nat.pow_succ];ring
 have w:UnpackedFloat.round Format.binary32 (Model.modelSign neg) M q=.finite (Model.modelSign neg) P E l:=by
  rw[Model.round_exact_eq_finishRoundedMantissa _ _ _ _]
  have o:∀(L:ℕ) (T:ℤ),2^L*2^g<=M->M<2^L*2^g*2->max (((L+g:ℕ):ℤ)+1+q-24) (-149)=T->Format.binary32.targetExponent (totalExponent M q)=T:=λL T a c d=>by
   rw[<-Nat.pow_add] at a c
   simp only[totalExponent,qDg M _ a c,qFq];exact d
  by_cases v:2^25*2^g<=M
  . rw[o 25 E v (by clear hm hP1 hP2 hE hE2 l F H p z r n v;exact omS76 G C) (by clear hm hP1 hP2 l F H p G C r n v;exact omS77 hE z),qEa M q E (g+1) P (by clear hm hP1 hP2 hE hE2 l F H p G C r n v;(exact Eq.symm ‹_›)) l ?_ (by rw[r,n];(exact Nat.succ_le_of_lt ‹_›))];. exact qEf _ P E hP1 hP2 hE l
    . rw[r,n];by_cases j:P=2^23;. subst j;(clear hm hE hE2 F H p z n hP1 hP2 l C r;exact omS78 v G)
      . have:=t j;(exact Nat.succ_le_of_lt ‹_›)
  have j:P=2^23:=by
   by_contra hne
   have:=t hne
   have:(2^23+1)*2^g<=P*2^g:=Ἒ _ (by clear hm hE hE2 l F H p z G C r n v this;exact omS79 hP1 hne)
   (clear hm hP1 hP2 hE hE2 l F H p z C r n hne;(clear G;omega))
  have D:=y j
  subst j
  by_cases x:-148<=E
  . rw[o 24 (E-1) (by clear hm hE hE2 F H p z n v hP1 hP2 l G C r x;exact omS80 D) (by clear hm hE hE2 F H p z n D hP1 hP2 l G C r x;exact omS81 v) (by clear hm F H p n v D hP1 hP2 l G C r;exact omS82 z x),qEa M q (E-1) g (2^24) (by clear hm hE hE2 F H p n v D hP1 hP2 l G C r x;exact omS83 z) ו (by clear hm hE hE2 F H p z n v hP1 hP2 l G C r x;exact omS84 D) (by clear hm hE hE2 F H p z n hP1 hP2 l G C r x;(clear D;exact omS85 v))];have:=Model.finishRoundedMantissa_normalized_carry FloatFormat.binary32 (Model.modelSign neg) (E-1+23) (by show-126<=E-1+23;(clear hm F H p n v D hP1 hP2 l G C r;exact omS86 x));rw[show (FloatFormat.binary32.fracWidth:ℤ)=23 from rfl] at this;rwa[((by clear hm hE hE2 F H p z n v D hP1 hP2 l G C r x;exact omS87):E-1+23-23=E-1),((by clear hm hE hE2 F H p z n v D hP1 hP2 l G C r x;exact omS88):E-1+23+1-23=E)] at this
  . rw[o 24 E (by clear hm hE hE2 F H p z n hP1 hP2 l G C r x;(clear v;exact omS80 D)) (by clear hm hE hE2 F H p z n D hP1 hP2 l G C r x;exact omS81 v) (by clear hm F H p n v D hP1 hP2 l G C r;exact omS89 hE z x),qEa M q E (g+1) (2^23) (by clear hm hE hE2 F H p n v D hP1 hP2 l G C r x;(exact Eq.symm ‹_›)) ו (by rw[n];(clear hm hE hE2 F H p z n hP1 hP2 l G C r x;exact omS90 D)) (by rw[n];(clear hm hE hE2 F H p z n v D hP1 hP2 l C r x;exact omS91 G))];exact qEf _ _ E (by norm_num1) (by norm_num1) hE l
 rw[w]
 simp only[toWord,Model.toNatBits,Model.ofModel,Model.ofModelBits]
 congr 1
 show (UnpackedFloat.pack Format.binary32 _).toNat=_
 rw[qBn _ _ _ _ hP1 hP2 (by clear hm hP1 hP2 F H p G C r n;exact omS92 hE) (by exact omS93 hE2)]
 cases neg<;>simp only[Model.modelSign,false_eq_true, ↓reduceIte,reduceCtorEq,Nat.reducePow,zero_add,Nat.add_left_cancel_iff]
end
abbrev B_D:=ExecFloat.Binary.descriptor (exponentBits:=11) (fractionBits:=52)
lemma qFs (f:ὧ):Model.signBit (toModel (ofFloat f:Binary64))=decide (2^63<=f.toBits.toNat):=by rw[Model.signBit_eq_msb];show f.toBits.toBitVec.msb=_;rw[BitVec.msb_eq_decide];rfl
lemma B_R (f:ὧ) (s:Ὡ) (E F:Ὣ) (hE1:1<=E) (hE:E<2047) (hF:F<2^52) (h:f.toBits.toNat=(if s then 2^63 else 0)+E*2^52+F):
 f.isFinite=true/\R f=(if s then-1 else 1)*((2^52+F:Ὣ):ℝ)*2^((E:Int)-1075):=by
 have g:Model.expField (toModel (ofFloat f:Binary64))=f.toBits.toNat/2^52%2^11:=by
  show ((f.toBits.toBitVec>>>52)&&&BitVec.ofNat 64 2047).toNat=_
  rw[BitVec.toNat_and,BitVec.toNat_ushiftRight,BitVec.toNat_ofNat,Ἱ]
  exact Ἂ _ 11
 have e:Model.fracField (toModel (ofFloat f:Binary64))=f.toBits.toNat%2^52:=by
  show (f.toBits.toBitVec&&&BitVec.ofNat 64 (2^52-1)).toNat=_
  rw[BitVec.toNat_and,BitVec.toNat_ofNat]
  exact Ἂ _ 52
 have i:Model.signBit (toModel (ofFloat f:Binary64))=s:=(qFs f).trans (by cases s<;>simp only[false_eq_true, ↓reduceIte,Nat.reducePow,zero_add,decide_eq_false_iff_not,not_le,decide_eq_true_eq] at h⊢<;>(clear g;omega))
 have b:E<2^11:=(by exact Nat.lt_of_lt_of_le ‹_› ה)
 have hE':f.toBits.toNat/2^52%2^11=E:=by
  rw[h];cases s
  . rw[ἐ Bool.false_ne_true,Ἇ,ἥ E,Nat.mul_add_div ה,Ὃ hF,Ἵ,Ἁ b]
  . rw[ἱ rfl,show (2:ℕ)^63=2^11*2^52 by decide,<-ὰ,ἥ (2^11+E),Nat.mul_add_div ה,Ὃ hF,Ἵ,Nat.add_mod_left,Ἁ b]
 have hF':f.toBits.toNat%2^52=F:=by
  rw[h];cases s
  . rw[ἐ Bool.false_ne_true,Ἇ,ἥ E,Nat.mul_add_mod,Ἁ hF]
  . rw[ἱ rfl,show (2:ℕ)^63=2^11*2^52 by decide,<-ὰ,ἥ (2^11+E),Nat.mul_add_mod,Ἁ hF]
 have c:=Model.ofFields_signBit_expField_fracField (toModel (ofFloat f:Binary64))
 rw[g,e,i,hE',hF'] at c
 have a:Ἑ (Model.ofFields B_D s E F)=true:=by
  rw[Model.isFinite_ofFields B_D _ _ _ b hF]
  show (E != 2047)=true
  simp only[bne_iff_ne,ne_eq];(clear hF h g e b;exact omS94 hE)
 have d:=Model.toDyadic?_ofFields_of_isFinite B_D _ _ _ b hF a
 rw[c,ἐ ((by clear hF h g e b;exact omS95 hE1):¬E=0)] at d
 rw[c] at a
 refine ⟨(qAh f).symm.trans a,?_⟩
 rw[R,Model.toReal_eq,d]
 simp only[show Model.pow2 B_D.fracWidth=2^52 from rfl,
  show Int.ofNat E-Int.ofNat B_D.exponentBias-Int.ofNat B_D.fracWidth=(E:ℤ)-1075 by
   show (E:ℤ)-1023-52=_;(clear hE1 hE hF h g e b;exact omS96)]
 cases s<;>simp only[FloatLib.Numerics.Dyadic.toReal,signedSignificand,false_eq_true, ↓reduceIte,Nat.reducePow,Int.ofNat_eq_natCast,Nat.cast_add,Nat.cast_ofNat,Int.cast_add,Int.cast_ofNat,Int.cast_natCast,one_mul,neg_add_rev,Int.reduceNeg,Int.cast_neg,neg_mul]
def B_near (A:Ὣ) (S:Ὡ) (ex fr:Ὣ):Prop:=
 ∃M,A=(if S then 2^63 else 0)+M/\(ex*2^23+fr)*2^29<M+2^28/\
  M<=(ex*2^23+fr)*2^29+2^28
lemma B_x (A:Ὠ) (x:Ὠ) (hx:(A+0x0FFFFFFF)>>>29=x) (h1:897<=Ѧx/2^23%2^11) (h2:Ѧx/2^23%2^11<1151):
 Ѧx<2^35/\B_near ѦA (decide (2^34<=Ѧx)) (Ѧx/2^23%2^11) (Ѧx%2^23):=by
 have a:=A.toNat_lt
 have e:Ѧx=(ѦA+268435455)%2^64/2^29:=by
  rw[<-hx,ἰ,ἤ]
  norm_num[Ἱ]
 refine ⟨by (clear hx h1 h2 a;exact omS97 e),?_⟩
 unfold B_near
 by_cases h:2^34<=Ѧx
 . simp only[h,decide_true,ite_true];exact ⟨ѦA-2^63,by (clear hx;exact omS98 h1 e h),by (clear hx;exact omS99 e h),by (clear hx h1 h2;exact omS100 a e h)⟩
 . simp only[h,decide_false,Bool.false_eq_true,ite_false];exact ⟨ѦA,by (clear hx h1 h2 a e h;exact omS101),by (clear hx h;exact omS102 e),by (clear hx;exact omS103 h1 a e h)⟩
lemma qHw (x:Ὠ) (hx:Ѧx<2^35) (h1:897<=Ѧx/2^23%2^11) (h2:Ѧx/2^23%2^11<1151):
 (((x>>>34)<<<31) ||| ((x&&&(0x3FFFFFFFF:Ὠ))-((896:Ὠ)<<<23))).toUInt32=
 ὺ ((if decide (2^34<=Ѧx) then 2^31 else 0)+(Ѧx/2^23%2^11-896)*2^23+Ѧx%2^23):=by
 have G1:Ѧx%2^34/2^23=Ѧx/2^23%2^11:=by rw[show (2:ℕ)^34=2^23*2^11 by decide];exact Nat.mod_mul_right_div_self _ _ _
 have F1:Ѧx%2^34%2^23=Ѧx%2^23:=Nat.mod_mod_of_dvd _ (Nat.pow_dvd_pow 2 ה)
 have h1':897<=Ѧx%2^34/2^23:=by rw[G1];exact h1
 have h2':Ѧx%2^34/2^23<1151:=by rw[G1];exact h2
 have L896:896*2^23<=Ѧx%2^34:=Nat.le_trans ה ((Nat.le_div_iff_mul_le ה).mp h1')
 have b:((x>>>34)<<<31).toNat=(Ѧx/2^34)*2^31:=by
  rw[UInt64.toNat_shiftLeft,ἰ]
  norm_num[Ἱ,ὁ]
  exact Nat.lt_of_lt_of_le (Nat.mul_lt_mul_of_pos_right (Ἰ (hx:Ѧx<17179869184*2)) ה) ה
 have a:(x&&&(0x3FFFFFFFF:Ὠ)).toNat=Ѧx%2^34:=by
  rw[Ὅ]
  exact Ἂ _ 34
 have d:((x&&&(0x3FFFFFFFF:Ὠ))-((896:Ὠ)<<<23)).toNat=Ѧx%2^34-896*2^23:=by
  rw[show ((896:Ὠ)<<<23)=7516192768 by decide,UInt64.toNat_sub_of_le,a]
  . rfl
  . rw[Ὤ,a];exact L896
 have c:(Ѧx/2^34)*2^31 ||| (Ѧx%2^34-896*2^23)=(Ѧx/2^34)*2^31+(Ѧx%2^34-896*2^23):=by
  rw[ἥ (Ѧx/2^34)]
  exact (Nat.two_pow_add_eq_or_of_lt (Nat.lt_of_lt_of_le (Nat.sub_lt_sub_right L896 ((Nat.div_lt_iff_lt_mul ה).mp h2')) ה) _).symm
 have GEF:Ѧx%2^34-896*2^23=(Ѧx/2^23%2^11-896)*2^23+Ѧx%2^23:=by
  rw[<-Nat.div_add_mod' (Ѧx%2^34) (2^23),G1,F1,Nat.sub_add_comm (Ἒ _ (Nat.le_of_lt h1)),<-Nat.sub_mul]
 apply ἠ.mp
 rw[UInt64.toNat_toUInt32,UInt64.toNat_or,b,d,c,ὡ]
 simp only[decide_eq_true_eq]
 split_ifs with hh
 . rw[Nat.div_eq_of_lt_le (k:=1) (by rw[Ὑ];exact hh) hx,GEF,Ὑ,ὢ]
 . rw[Ὃ (Nat.not_le.mp hh),GEF,ὒ,Ἇ,Ἇ]
lemma qEs (A B:Ὠ) (h:zivW A B≠Е):
 ∃(S:Ὡ) (ex fr:Ὣ),897<=ex/\ex<1151/\fr<2^23/\zivW A B=ὺ ((if S then 2^31 else 0)+(ex-896)*2^23+fr)/\B_near ѦA S ex fr/\B_near ѦB S ex fr:=by
 unfold zivW at h⊢
 dsimp only at h⊢
 generalize hx:(A+(0x0FFFFFFF:Ὠ))>>>(29:Ὠ)=x at h⊢
 generalize hy:(B+(0x0FFFFFFF:Ὠ))>>>(29:Ὠ)=y at h⊢
 have hex:((x>>>(23:Ὠ))&&&(0x7FF:Ὠ)).toNat=Ѧx/2^23%2^11:=by
  rw[Ὅ,ἰ]
  have:=Ἂ (Ѧx>>>23) 11
  norm_num[Ἱ] at this⊢
  exact this
 generalize (x>>>(23:Ὠ))&&&(0x7FF:Ὠ)=ex at h hex⊢
 split at h
 . rename_i hc;have hc0:=hc;obtain ⟨hxy,hc⟩:=ὕ.mp hc;have hxy:x=y:=by simpa only[beq_iff_eq] using hxy
   subst hxy;have hc:ex-(897:Ὠ)<(254:Ὠ):=by simpa only[BEq.rfl,Bool.true_and,decide_eq_true_eq] using hc
   rw[ἴ,ὲ] at hc;have hv:=ex.toNat_lt;norm_num at hc hv;have h1:897<=Ѧx/2^23%2^11:=(by clear hx h hy hv;exact omS107 hex hc);have h2:Ѧx/2^23%2^11<1151:=(by clear hx h hy hv;exact omS108 hex hc);simp only[hc0,ite_true];obtain ⟨hx35,ha⟩:=B_x A x hx h1 h2;exact ⟨_,_,_,h1,h2,by (clear hx hex h hy hc hv h1 h2 hx35;exact omS109),qHw x hx35 h1 h2,ha,(B_x B x hy h1 h2).2⟩
 . exact absurd rfl h
lemma qR0 (Y:Ὡ) (r:ὥ) (he:expOf r≠255):r&&&((if Y then (0:ὥ) else Е) ||| ((r>>>(31:ὥ))-1))=act Y r:=by
 have b:=r.toNat_lt
 have f:∀u:ὥ,u&&&Е=u:=λu=>UInt32.toNat_inj.mp (by rw[Ἧ];show ѥu&&&(2^32-1)=_;rw[Ἂ];exact Ἁ u.toNat_lt)
 by_cases h:2^31<=ѥr
 . have e:r>>>(31:ὥ)=1:=UInt32.toNat_inj.mp (by rw[UInt32.toNat_shiftRight];show ѥr>>>31=1;rw[Ἱ];(clear he;exact omS110 b h))
   have g:(r&&&س!=0)=true:=by
    have:(r&&&س).toNat≠0:=by rw[Ἧ];show ѥr&&&2^31≠0;rw[Nat.and_two_pow];simp only[Nat.testBit_eq_decide_div_mod_eq,Nat.reducePow,ne_eq,mul_eq_zero,Bool.toNat_eq_zero,decide_eq_false_iff_not,Nat.mod_two_not_eq_one,OfNat.ofNat_ne_zero,or_false,Nat.mod_two_not_eq_zero];(clear he e;exact omS111 b h)
    simp only[bne_iff_ne];intro c;rw[c] at this;exact this rfl
   cases Y<;>simp only[false_eq_true, ↓reduceIte,e,sub_self,UInt32.or_zero,f,act,UInt32.or_self,UInt32.and_zero,reluW,g,he,decide_false,Bool.false_and,Bool.not_false,Bool.and_self]
 . have e:r>>>(31:ὥ)=0:=UInt32.toNat_inj.mp (by rw[UInt32.toNat_shiftRight];show ѥr>>>31=0;rw[Ἱ];(clear he;exact omS112 h))
   have g:(r&&&س)=0:=UInt32.toNat_inj.mp (by rw[Ἧ];show ѥr&&&2^31=0;rw[Nat.and_two_pow];simp only[Nat.testBit_eq_decide_div_mod_eq,Nat.reducePow,mul_eq_zero,Bool.toNat_eq_zero,decide_eq_false_iff_not,Nat.mod_two_not_eq_one,OfNat.ofNat_ne_zero,or_false];(clear he e;exact omS113 h))
   cases Y<;>simp only[false_eq_true, ↓reduceIte,e,zero_sub,UInt32.or_neg_one,UInt32.and_neg_one,act,reluW,g,bne_self_eq_false,Bool.not_and,Bool.false_and]
lemma qRe (w:ὥ):(expOf w).toNat=ѥw/2^23%2^8:=by
 unfold expOf
 rw[Ἧ,UInt32.toNat_shiftRight]
 show (ѥw>>>23)&&&(2^8-1)=_
 rw[Ἂ,Ἱ]
lemma qR1 (r:ὥ) (S:Ὡ) (ex fr:Ὣ) (h1:897<=ex) (h2:ex<1151) (h3:fr<2^23) (hz:r=ὺ ((if S then 2^31 else 0)+(ex-896)*2^23+fr)):expOf r≠255:=by
 intro e;have a:=qRe r;rw[e] at a;change 255=_ at a;rw[hz,ὡ] at a;cases S<;>simp only[Bool.false_eq_true,ite_false,ite_true] at a<;>(clear e hz;omega)
lemma qB6 (s e:ὧ) (h:ziv s e≠Е):
 ∃(S:Ὡ) (ex fr:Ὣ),897<=ex/\ex<1151/\fr<2^23/\ziv s e=ὺ ((if S then 2^31 else 0)+(ex-896)*2^23+fr)/\B_near (s+e).toBits.toNat S ex fr/\B_near (s-e).toBits.toNat S ex fr:=
 qEs (s+e).toBits (s-e).toBits h
set_option maxRecDepth 8192 in
lemma qA4 (f:ὧ) (S:Ὡ) (ex fr:Ὣ) (h1:897<=ex) (h2:ex<1151) (hfr:fr<2^23) (hn:B_near f.toBits.toNat S ex fr):
 f.isFinite=true/\∃v:ℝ,R f=(if S then-v else v)/\v<=(2^53+fr*2^30+2^29)*2^((ex:Int)-1076)/\(fr≠0->(2^53+fr*2^30-2^29)*2^((ex:Int)-1076)<v)/\(2^53-2^28)*2^((ex:Int)-1076)<v:=by
 obtain ⟨M,k,o,n⟩:=hn
 have d:M%2^52<2^52:=ὀ _ ו
 have b:M=(M/2^52)*2^52+M%2^52:=(by clear h1 h2 hfr k o n d;exact omS114)
 generalize M/2^52=E at b
 generalize M%2^52=F at b d
 subst b
 have c:(0:ℝ)<2^((ex:Int)-1076):=ἵ ו _
 have m:E=ex\/(E+1=ex/\fr=0):=(by clear h1 h2;exact omS115 hfr d o n)
 obtain ⟨a,l⟩:=B_R f S E F (by clear m;exact omS116 h1 d o) (by clear m;exact omS117 h2 hfr n) d (by rw[k,ὢ])
 obtain ⟨w,j,i,e,g⟩:∃w:ℕ,((2^52+F:Ὣ):ℝ)*2^((E:Int)-1075)=w*2^((ex:Int)-1076)/\w<=2^53+fr*2^30+2^29/\(fr≠0->2^53+fr*2^30<w+2^29)/\2^53<w+2^28:=by
  rcases m with rfl | ⟨rfl,rfl⟩
  . refine ⟨2*(2^52+F),?_,by (clear hfr d h1 h2;(clear k;exact omS118 n)),by (clear hfr d h1 h2;(clear k;exact omS119 o)),by (clear hfr h1 h2 o n;exact omS120)⟩;rw[show (E:Int)-1075=1+((E:Int)-1076) by ring,Ἓ ἓ,zpow_one];push_cast;ring
  . exact ⟨2^52+F,by congr 2;push_cast;ring,by (clear h1 h2 hfr n;exact omS121 d),by (clear h1 h2 hfr n;exact omS122),by (clear h1 h2 hfr n;exact omS123 o)⟩
 refine ⟨a,(w:ℝ)*2^((ex:Int)-1076),?_,ἀ ?_ c.le,λh=>Ἦ ?_ c,Ἦ ?_ c⟩
 . rw[l,mul_assoc,j];cases S<;>simp only[false_eq_true, ↓reduceIte,one_mul,neg_mul]
 . exact_mod_cast i
 . rw[sub_lt_iff_lt_add];exact_mod_cast e h
 . rw[sub_lt_iff_lt_add];exact_mod_cast g
lemma qBw (s e:ὧ) (h:ziv s e≠Е):(s+e).isFinite=true/\(s-e).isFinite=true/\(2:ℝ)^(-127:ℤ)<= |R (s+e)|:=by
 obtain ⟨S,f,c,b,d,a,-,j,k⟩:=qB6 s e h;obtain ⟨g,v,i,-,-,l⟩:=qA4 _ S f c b d a j;refine ⟨g,(qA4 _ S f c b d a k).1,?_⟩
 exact (show (2:ℝ)^(-127:ℤ)<=(2^53-2^28)*2^(-179:ℤ) by (norm_num1)).trans
  ((ἁ (ἒ one_le_two (by clear h a;exact omS124 b)) ו).trans (l.le.trans (by rw[i];cases S<;>simp only[false_eq_true, ↓reduceIte,le_abs_self,abs_neg])))
lemma qAx (w:ὥ) (h:expOf w≠255):wv w=(ἂ w:ℝ):=by rw[wv,Model.toReal_eq,ofW,ἶ w h];simp only[rationalWord,Ἠ]
lemma qEl (a b:ѐ) (start n:Ὣ):
 (dsum a b start n:ℚ)*2^(-298:Int)=∑ j∈Ἣ n,ἂ a[start+j]!*ἂ b[start+j]!:=by
 have c:∀a b:ὥ,((sv a:ℚ)*sv b*(2:ℚ)^(idxOf a+idxOf b))*2^(-298:Int)=ἂ a*ἂ b:=λa b=>by
  simp only[rationalWord,decodeD_toRat]
  rw[show (-298:Int)=-149+-149 by (norm_num1),Ἓ ἓ]
  rw[ᾂ ἓ,ᾂ ἓ,Ἡ,Ἡ,pow_add]
  norm_num[zpow_neg,Ἡ]
  ring
 induction n generalizing start with
 | zero=>simp only[dsum,Int.cast_zero,Int.reduceNeg,zpow_neg,zpow_ofNat,zero_mul,Finset.range_zero,Finset.sum_empty]
 | succ n ih=>
  rw[dsum,Int.cast_add,Int.cast_mul,Int.cast_mul,Int.cast_pow,Int.cast_ofNat,add_mul,ih,Finset.sum_range_succ',Ἵ,c,add_comm]
  refine congrArg (· +_) ?_
  apply Ἷ rfl
  intro j _
  rw[ὢ,Nat.add_comm 1 j]
lemma qEc (n:Ὣ) (fa fb:Ὣ->ὥ) (hf:∀j,j<n->expOf (fa j)≠255/\expOf (fb j)≠255) (hn:(∑ j∈Ἣ n,ἂ (fa j)*ἂ (fb j))≠0):
 ∃d:FloatLib.Numerics.Dyadic,
  d.toRat=(∑ j∈Ἣ n,ἂ (fa j)*ἂ (fb j))/\dotOf n fa fb=toWord (ἕ Fmt d):=by
 let a:=Array.ofFn (n:=n) λi=>fa i
 let b:=Array.ofFn (n:=n) λi=>fb i
 have e:ѧa=n:=by simp only[Array.size_ofFn,a]
 have g:ѧb=n:=by simp only[Array.size_ofFn,b]
 have c:(dsum a b 0 n:ℚ)*2^(-298:Int)=∑ j∈Ἣ n,ἂ (fa j)*ἂ (fb j):=by
  rw[qEl]
  refine Ἷ rfl λj m=>?_
  have h:=Ὁ.mp m
  rw[Ἇ,qAb n fa j h,qAb n fb j h]
 have l:dsum a b 0 n≠0:=λh=>hn (by rw[<-c,h];simp only[Int.cast_zero,Int.reduceNeg,zpow_neg,zpow_ofNat,zero_mul])
 have Z:=qEg a b (by rw[e,g]) λk f=>by
  rw[e] at f
  rw[qAb n fa k f,qAb n fb k f];exact hf k f
 rw[e] at Z
 refine ⟨ofIntDyadic (dsum a b 0 n) (-298),by rw[toRat_ofIntDyadic,c],?_⟩
 show dotSpec a b=_
 rw[<-Z];simp only[finishWord,l, ↓reduceIte,Int.reduceNeg]
lemma qIg:ἂ oneW=1:=by norm_num[rationalWord,decodeD,show expOf oneW=127 by decide,show fracOf oneW=0 by decide,show signOf oneW=false by decide,FloatLib.Numerics.Dyadic.toRat,FloatLib.Numerics.Dyadic.signedSignificand]
lemma qM1 (y:ὧ) (hy:yᶠ):y*dv oneW=y:=by
 obtain ⟨h1,h2⟩:=qAm oneW ה
 obtain ⟨x,hx,-⟩:=qDk y hy
 obtain ⟨o,ho,eo⟩:=qDk _ h1
 have e1:o.toReal=1:=by rw[eo,h2,qAx oneW ה,qIg];norm_num
 have os:o.significand≠0:=λh=>by have:=(A_s o).2 h;rw[e1] at this;norm_num at this
 have on:o.negative=false:=by
  cases hn:o.negative
  · rfl
  · have q:=qD4 o;rw[hn,e1] at q;have:=qE9 o os;simp only[A_r,↓reduceIte] at q;linarith only[this,q]
 apply qIe
 rw[qCq hx ho hy h1]
 have hy':=Model.Internal.Directed.roundDyadic_toDyadic? qAj .nearest hx
 split
 · rename_i z
   have xs:x.significand=0:=by rcases Nat.mul_eq_zero.1 z with h|h;exact h;exact absurd h os
   rw[<-hy',on,Bool.xor_false]
   obtain ⟨n,m,e⟩:=x
   simp only at xs
   subst xs
   rw[Model.Internal.Directed.roundDyadic_zero]
 · rename_i z
   rw[<-hy']
   exact ExactRuntime.roundDyadic_congr _ qAj _ _ z (λh=>z (by rw[h,ὒ])) (Rat.cast_injective (α:=ℝ) (by rw[Ἠ,Ἠ,Model.Dyadic.toReal_mul,e1,mul_one]))
lemma qHv (n:Ὣ) (xw ww:Ὣ->ὥ) (bw:ὥ) (hx:∀t,t<n->good (xw t)=true) (hw:∀t,t<n->good (ww t)=true) (hb:good bw=true):
 ((∑ j∈Ἣ (n+1),ἂ (if j<n then xw j else oneW)*ἂ (if j<n then ww j else bw):ℚ):ℝ)=exactS n xw ww bw:=by
 rw[Finset.sum_range_succ]
 simp only[lt_self_iff_false,ite_false,qIg,one_mul]
 rw[Rat.cast_add,Rat.cast_sum,exactS,qAx bw (qAe bw hb)]
 congr 1
 refine Ἷ rfl λt b=>?_
 have a:=Ὁ.mp b
 simp only[a,ite_true]
 rw[qAx _ (qAe _ (hx t a)),qAx _ (qAe _ (hw t a)),Rat.cast_mul]
section
variable (s e:ὧ) (V:ℝ) (hlo:R (s-e)< V) (hhi:V< R (s+e)) (h:ziv s e≠Е)
include s e V hlo hhi h
lemma qB8:
 ∃(S:Ὡ) (ex fr:Ὣ) (W:ℝ),897<=ex/\ex<1151/\fr<2^23/\ziv s e=ὺ ((if S then 2^31 else 0)+(ex-896)*2^23+fr)/\V=(if S then-W else W)/\W<(2^53+fr*2^30+2^29)*2^((ex:Int)-1076)/\(fr≠0->(2^53+fr*2^30-2^29)*2^((ex:Int)-1076)<W)/\(2^53-2^28)*2^((ex:Int)-1076)<W:=by
 obtain ⟨S,ex,fr,h1,h2,h3,hz,ha,hb'⟩:=qB6 s e h
 obtain ⟨-,va,hva,hia,loa1,loa⟩:=qA4 _ S ex fr h1 h2 h3 ha
 obtain ⟨-,vb,hvb,hib,lob1,lob⟩:=qA4 _ S ex fr h1 h2 h3 hb'
 rw[hva] at hhi
 rw[hvb] at hlo
 refine ⟨S,ex,fr,if S then-V else V,h1,h2,h3,hz,?_,?_,?_,?_⟩
 all_goals cases S
 all_goals simp only[Bool.false_eq_true,ite_false,ite_true,neg_neg] at hhi hlo⊢
 . exact Ἥ hhi hia
 . linarith only[hib,hlo]
 . exact λh=>(lob1 h).trans hlo
 . exact λh=>by linarith only[loa1 h,hhi,loa1,h]
 . exact lob.trans hlo
 . linarith only[loa,hhi]
lemma qG5:V≠0:=by
 obtain ⟨S,ex,fr,W,-,-,-,-,hV,-,-,hW3⟩:=qB8 s e V hlo hhi h;have hb0:(0:ℝ)<2^((ex:Int)-1076):=ἵ ו _;have hA:(0:ℝ)<=(2^53-2^28)*2^((ex:Int)-1076):=Ἆ ו hb0.le;intro h0;rw[h0] at hV;cases S;. simp only[Bool.false_eq_true,ite_false] at hV;linarith only[hW3,hA,hV]
 . simp only[ite_true] at hV;linarith only[hW3,hA,hV]
end
lemma qEo (d:FloatLib.Numerics.Dyadic) (S:Ὡ) (ex fr:Ὣ) (h1:897<=ex) (h2:ex<1151) (h3:fr<2^23) (W:ℝ) (hd:(d.toRat:ℝ)=if S then-W else W) (hW1:W<(2^53+fr*2^30+2^29)*2^((ex:Int)-1076)) (hW2:fr≠0->(2^53+fr*2^30-2^29)*2^((ex:Int)-1076)<W) (hW3:(2^53-2^28)*2^((ex:Int)-1076)<W):
 toWord (ἕ Fmt d)=ὺ ((if S then 2^31 else 0)+(ex-896)*2^23+fr):=by
 have hb0:(0:ℝ)<2^((ex:Int)-1076):=ἵ ו _
 have hE:(2:ℝ)^((ex:Int)-1046)=2^30*2^((ex:Int)-1076):=by
  rw[<-Ἡ,<-Ἓ ἓ]
  congr 1;push_cast;ring
 have hW0:0<W:=ἷ (Ἆ ו hb0.le) hW3
 obtain ⟨neg,m,q⟩:=d
 have hdV:(FloatLib.Numerics.Dyadic.mk neg m q).toReal=if S then-W else W:=by
  rw[<-Ἠ,hd]
 have hq:(0:ℝ)<2^q:=ἵ ו _
 have hm0:(0:ℝ)<=m:=Ἔ _
 have hmq:=Ἆ hm0 hq.le
 have hMW:(m:ℝ)*2^q=W/\neg=S:=by
  simp only[FloatLib.Numerics.Dyadic.toReal,FloatLib.Numerics.Dyadic.signedSignificand] at hdV
  cases neg<;>cases S<;>
   simp only[Bool.false_eq_true,ite_false,ite_true,Int.ofNat_eq_natCast,Int.cast_neg,Int.cast_natCast] at hdV
  . exact ⟨by linarith only[hdV],rfl⟩
  . exfalso;linarith only[hW0,hmq,hdV]
  . exfalso;linarith only[hW0,hmq,hdV]
  . exact ⟨by linarith only[hdV],rfl⟩
 obtain ⟨hMW,rfl⟩:=hMW
 have hm:m≠0:=by
  rintro rfl
  simp only[CharP.cast_eq_zero,zero_mul] at hMW
  linarith only[hW0,hMW]
 have hword:=qE4 neg m q (2^23+fr) ((ex:Int)-1046) hm (by clear h1 h2 hm;exact omS125) (by clear h1 h2 hm;exact omS126 h3) (by clear h3 hm;exact omS127 h1)
  (by clear h3 hm;exact omS128 h2) (by
   rw[hMW,hE];push_cast;linarith only[hW0,hW1])
  (by
   intro hfr
   have hfr':fr≠0:=by intro h0;exact hfr (by clear h1 h2 h3 hm hfr;exact omS129 h0)
   have:=hW2 hfr'
   rw[hMW,hE];push_cast;linarith only[this])
  (by
   intro _
   rw[hMW,hE];linarith only[hW3])
 have e1:((ex:Int)-1046+150).toNat=ex-896:=(by clear h3 hm hword;exact omS130)
 have e2:(2^23+fr)%2^23=fr:=(by clear h1 h2 hm hword e1;exact omS131 h3)
 rw[hword,e1,e2]
lemma qIq (s e:ὧ) (d:FloatLib.Numerics.Dyadic) (hlo:R (s-e)<(d.toRat:ℝ)) (hhi:(d.toRat:ℝ)<R (s+e)) (h:ziv s e≠Е):ziv s e=toWord (ἕ Fmt d):=by obtain ⟨S,ex,fr,W,h1,h2,h3,hz,hV,hW1,hW2,hW3⟩:=qB8 s e _ hlo hhi h;rw[hz];exact (qEo d S ex fr h1 h2 h3 W hV hW1 hW2 hW3).symm
lemma qCn (n:Ὣ) (hn:n<2^20) (xw ww:Ὣ->ὥ) (bw:ὥ) (hx:∀t,t<n->good (xw t)=true) (hw:∀t,t<n->good (ww t)=true) (hb:good bw=true) (s e:ὧ) (hlo:R (s-e)<exactS n xw ww bw) (hhi:exactS n xw ww bw<R (s+e)) (h:ziv s e≠Е):
 ziv s e=dotOf (n+1) (λt=>if t<n then xw t else oneW) (λt=>if t<n then ww t else bw):=by
 have hsum:=qHv n xw ww bw hx hw hb
 obtain ⟨d,hd,hw'⟩:=qEc (n+1) (λt=>if t<n then xw t else oneW)
  (λt=>if t<n then ww t else bw) (λj _=>by
   by_cases hj:j<n
   . simp only[hj,ite_true];exact ⟨qAe _ (hx j hj),qAe _ (hw j hj)⟩
   . simp only[hj,ite_false]
     exact ⟨ה,qAe _ hb⟩) λh0=>qG5 s e _ hlo hhi h (by rw[<-hsum,h0,Rat.cast_zero])
 have q:(d.toRat:ℝ)=exactS n xw ww bw:=by rw[hd,hsum]
 rw[hw']
 exact qIq s e d (by rw[q];exact hlo) (by rw[q];exact hhi) h
end
section
lemma qDn (n:Ὣ) (hn:n<2^20) (xw ww:Ὣ->ὥ) (bw:ὥ) (hx:∀t,t<n->good (xw t)=true) (hw:∀t,t<n->good (ww t)=true) (hb:good bw=true) (l:љ) (hl:l.Nodup) (hl2:∀t∈l,t<n) (hz:∀t,t<n->t∉l->xw t&&&В=0\/ww t&&&В=0) (s e:ὧ)
 (hs:let r:=l.foldl (λ(acc:F1) t=>let q:=dv (xw t)*dv (ww t)
  F1.mk (acc.s+q) (acc.t+q.abs)) ⟨0,0⟩;s=r.s+dv bw/\e=nfOf n*(r.t+(dv bw).abs))
 (h:ziv s e≠Е):
 ziv s e=dotOf (n+1) (λt=>if t<n then xw t else oneW) (λt=>if t<n then ww t else bw):=by
 obtain ⟨-,-,hbig⟩:=qBw s e h
 obtain ⟨hlo,hhi⟩:=qDm n hn xw ww bw hx hw hb l hl hl2 hz s e hs hbig
 exact qCn n hn xw ww bw hx hw hb s e hlo hhi h
end
section
end
section
lemma C_z:(0:ὦ).toNat=0:=rfl
lemma qAd (k:Ὣ) (h:k<2^32):k.toUSize.toNat=k:=by rw[Nat.toUSize_eq];exact uk k h
lemma qBa (a b:ὦ) (h:Ѥa*Ѥb<2^32):(a*b).toNat=Ѥa*Ѥb:=by have h1:2^32<=USize.size:=USize.le_size;have h3:USize.size=2^System.Platform.numBits:=rfl;rw[USize.toNat_mul];exact Ἁ (by exact Nat.lt_of_lt_of_le ‹_› ‹_›)
lemma qBe {α:Type} (a:Array α) (h:ѧa<2^32):a.usize.toNat=ѧa:=qAd _ h
lemma qAg (a:ѐ) (i:ὦ) (hs:ѧa<2^32):rdW a i=a[Ѥi]!:=by
 unfold rdW;split;. rename_i h;have h2:=qBd a i h;rw[Ἕ a Ѥi h2];rfl
 . rename_i h;have h':¬ Ѥi< ѧa:=by intro h2;apply h;rw[А,qBe a hs];exact h2
   rw[ὑ a Ѥi h'];rfl
lemma qBh (a:ѐ) (i:ὦ) (v:ὥ):(wrW a i v).size=ѧa:=by unfold wrW;split<;>simp only[Array.uset_eq_set,Array.size_set]
lemma qFo {α:Type} [Inhabited α] (a:Array α) (i q:Ὣ) (v:α) (hi:i< ѧa):(a.set i v hi)[q]! =if q=i then v else a[q]!:=by
 by_cases hq:q< ѧa
 . rw[Ἕ _ q (by simpa only[Array.size_set] using hq),Ἕ a q hq,Array.getElem_set];by_cases e:q=i;. simp only[e, ↓reduceIte]
   . rw[ἐ e,ἐ (λh=>e h.symm)]
 . rw[ὑ _ q (by simpa only[Array.size_set,not_lt] using hq),ὑ a q hq,ἐ (by exact omS132 hi hq)]
lemma qBu (a:ѐ) (i:ὦ) (v:ὥ) (q:Ὣ) (hs:ѧa<2^32) (hi:Ѥi< ѧa):(wrW a i v)[q]! =if q=Ѥi then v else a[q]!:=by
 have h:i<a.usize:=by rw[А,qBe a hs];exact hi
 unfold wrW;rw[Ἄ h];exact qFo a _ q v hi
def C_Inv (N:Ὣ) (sp:Ὣ->Ὢ) (S:Ὣ->Prop) (out:ё):Prop:=
 ѧout=N/\∀q,q<N->S q->out[q]! =sp q
section
variable {N:Ὣ} {sp:Ὣ->Ὢ}
include N sp
lemma qAk {S:Ὣ->Prop} {out:ё} (hN:N<2^32) (h:C_Inv N sp S out) {u:ὦ} {p:Ὣ} {v:Ὢ} (hu:Ѥu=p) (hp:p<N) (hv:v=sp p):C_Inv N sp (λq=>q=p\/S q) (wrW out u v):=by
 obtain ⟨h1,h2⟩:=h;refine ⟨by rw[qBh,h1],λq hq hS=>?_⟩;rw[qBu _ _ _ _ (by clear hu hp hq;omega) (by clear hN hq;omega),hu];by_cases e:q=p;. rw[ἱ e,e,hv]
 . rw[ἐ e];exact h2 q hq (hS.resolve_left e)
lemma qA5 {S S':Ὣ->Prop} {out:ё} (h:C_Inv N sp S out) (hS:∀q,q<N->S' q->S q):C_Inv N sp S' out:=⟨h.1,λq hq h'=>h.2 q hq (hS q hq h')⟩
lemma qBv {S:Ὣ->Prop} {out:ё} (h:C_Inv N sp S out) (hS:∀q,q<N->S q):out=Array.ofFn (n:=N) λq=>sp q.val:=by
 obtain ⟨h1,h2⟩:=h;apply Մ;. simp only[h1,Array.size_ofFn]
 . intro q h3 h4;have h5:q<N:=(by clear h4;(exact Nat.lt_of_lt_of_eq ‹_› ‹_›));have:=h2 q h5 (hS q h5);rw[Ἕ out q h3] at this;simp only[this,Array.getElem_ofFn]
end
lemma qEn (v:ὥ) (h1:(0x7FFFFF:ὥ)<=v-1) (h2:v<У):(v==0 || v-(0x00800000:ὥ)<=(0x7EFFFFFF:ὥ))=true:=by
 have a1:(0x7FFFFF:ὥ).toNat=0x7FFFFF:=rfl;have a2:(У:ὥ).toNat=У:=rfl;have a3:(0x00800000:ὥ).toNat=0x00800000:=rfl;have a4:(0x7EFFFFFF:ὥ).toNat=0x7EFFFFFF:=rfl;have a5:(1:ὥ).toNat=1:=rfl;by_cases h0:v=0;. simp only[h0,BEq.rfl,zero_sub,Bool.true_or]
 . have h3:(1:ὥ)<=v:=by rw[ἃ];have:ѥv≠0:=λh=>h0 (ἠ.mp h);(clear h1 h2 a1 a2 a3 a4 h0;omega)
   rw[ἃ,ὓ _ _ h3] at h1;rw[ἳ] at h2;have h4:(0x00800000:ὥ)<=v:=by rw[ἃ];(clear h2 a2 a4 h0 h3;exact omS133 h1 a1 a3 a5)
   simp only[Bool.or_eq_true,decide_eq_true_eq];right;rw[ἃ,ὓ _ _ h4];(clear h1 a1 a5 h0 h3 h4;exact omS134 h2 a2 a3 a4)
lemma qHu (c u lo:ὥ) (h:c<=if u<lo then u else lo):c<=u/\c<=lo:=by simp only[ἃ,ἳ] at*;split at h<;>omega
lemma qH2 (c v hi:ὥ) (h:(if hi<v then v else hi)<c):v<c/\hi<c:=by simp only[ἳ] at*;split at h<;>omega
lemma qB2 (w:ὥ) (h:good w=true):good (w&&&В)=true:=by
 have e:(w&&&В)&&&В=w&&&В:=by rw[UInt32.and_assoc,UInt32.and_self]
 unfold good at h⊢;simpa only[e] using h
lemma qCx {h:ὥ} (h1:h<У):h<=В:=by have a1:(У:ὥ).toNat=У:=rfl;have a2:(В:ὥ).toNat=В:=rfl;rw[ἳ] at h1;rw[ἃ];(exact Nat.le_of_lt (Nat.lt_of_lt_of_le ‹_› ה))
def C_Hv (A:ё) (h:Ὢ):Prop:=
 good h=true/\h<=В/\(∀k,k< ѧA->good A[k]! =true)/\
  ∀k,k< ѧA->A[k]!&&&В<=h
lemma qFa (r v hi:ὥ) (h:(if hi<v then v else hi)<=r):v<=r/\hi<=r:=by simp only[ἃ,ἳ] at*;split at h<;>omega
lemma qHj (A:ѐ) (n:ὦ) (hn:Ѥn<=ѧA) (d:Ὣ):
 ∀(i:ὦ) (lo hi:ὥ),Ѥi+d=Ѥn->scan A n i hn lo hi≠Е->
  (0x7FFFFF:ὥ)<=lo/\hi<У/\hi<=scan A n i hn lo hi/\scan A n i hn lo hi<У/\(good hi=true->good (scan A n i hn lo hi)=true)/\
   ∀k,Ѥi<=k->k< Ѥn->good A[k]! =true/\A[k]!&&&В<=scan A n i hn lo hi:=by
 induction d with
 | zero=>
  intro i lo hi he h
  have e:scan A n i hn lo hi=if 0x007FFFFF<=lo/\hi<У then hi else Е:=by
   rw[scan,ἔ (by rw[А];(clear hn h;(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›)))))]
  rw[e] at h⊢
  by_cases c:0x007FFFFF<=lo/\hi<У
  . rw[ἱ c];exact ⟨c.1,c.2,Ὓ _,c.2,id,λk h1 h2=>by (clear hn h e c;exact omS135 he h1 h2)⟩
  . rw[ἐ c] at h;exact absurd rfl h
 | succ d ih=>
  intro i lo hi he h
  have hi':i<n:=by rw[А];(clear h;exact omS136 he)
  have hb:Ѥi< ѧA:=(by exact Nat.lt_of_lt_of_le ‹_› ‹_›)
  have e:scan A n i hn lo hi=scan A n (i+1) hn (if (A.uget i hb&&&В)-1<lo then (A.uget i hb&&&В)-1 else lo) (if hi<A.uget i hb&&&В then A.uget i hb&&&В else hi):=by
   rw[scan,Ἄ hi']
  rw[e] at h⊢
  obtain ⟨h1,h2,h3,h4,h5,h6⟩:=ih (i+1) _ _ (by rw[usz i n hi'];(clear hi' hb h e;exact omS137 he)) h
  obtain ⟨h7,h8⟩:=qHu _ _ _ h1
  obtain ⟨h9,h10⟩:=qH2 _ _ _ h2
  obtain ⟨h11,h12⟩:=qFa _ _ _ h3
  have hg:good (A.uget i hb)=true:=qEn _ h7 h9
  refine ⟨h8,h10,h12,h4,λg0=>h5 ?_,λk hk1 hk2=>?_⟩
  . split;. exact qB2 _ hg
    . exact g0
  . by_cases e2:k=Ѥi;. subst e2;rw[Ἕ A Ѥi hb];exact ⟨hg,h11⟩
    . exact h6 k (by rw[usz i n hi'];(clear he hi' h e h1 h2 h3 h4 h7 h8 h9 h10 h11 h12 hk2;exact omS138 hk1 e2)) hk2
lemma qBi (A:ѐ) (hs:ѧA<2^32) (h:scan A A.usize 0 (qAc A) Е 0≠Е):C_Hv A (scan A A.usize 0 (qAc A) Е 0):=by have e:=qBe A hs;have h0:(0:ὦ).toNat=0:=rfl;obtain ⟨-,-,-,h4,h5,h6⟩:=qHj A A.usize (qAc A) ѧA 0 _ _ (by rw[h0,e];(clear hs h e h0;exact omS139)) h;have h7:∀k,k< ѧA->good A[k]! =true/\A[k]!&&&В<=scan A A.usize 0 (qAc A) Е 0:=λk hk=>h6 k (by exact Nat.zero_le _) (by clear hs h h0 h4;exact omS140 e hk);exact ⟨h5 rfl,qCx h4,λk hk=>(h7 k hk).1,λk hk=>(h7 k hk).2⟩
lemma qF6 (a b:ὥ):a<=a ||| b/\b<=a ||| b:=by rw[ἃ,ἃ,UInt32.toNat_or];exact ⟨Nat.left_le_or,Nat.right_le_or⟩
lemma qBt (v:ὥ) (h1:v=0\/0x800000<=v) (h2:v<У):(v==0 || v-(0x00800000:ὥ)<=(0x7EFFFFFF:ὥ))=true:=by
 rcases h1 with rfl | h1;. rfl
 . have a1:(У:ὥ).toNat=У:=rfl;have a2:(0x00800000:ὥ).toNat=0x00800000:=rfl;have a3:(0x7EFFFFFF:ὥ).toNat=0x7EFFFFFF:=rfl;simp only[Bool.or_eq_true,decide_eq_true_eq];right;rw[ἃ,ὓ _ _ h1];rw[ἳ] at h2;rw[ἃ] at h1;exact omS134 h2 a1 a2 a3
lemma qFe (a b:ὥ):a&&&b<=a/\a&&&b<=b:=by rw[ἃ,ἃ,Ἧ];exact ⟨Nat.and_le_left,Nat.and_le_right⟩
lemma qGi (v:ὥ) (h:0x7FFFFF<=v-1):v=0\/0x800000<=v:=by
 by_cases z:v=0;. exact .inl z
 . right;have a1:(0x7FFFFF:ὥ).toNat=0x7FFFFF:=rfl;have a2:(0x800000:ὥ).toNat=0x800000:=rfl;have a3:(1:ὥ).toNat=1:=rfl;have a4:ѥv≠0:=λe=>z (ἠ.mp (by rw[e];rfl));have a5:=v.toNat_lt;have h1:=ἃ.mp h;rw[a1,UInt32.toNat_sub,a3] at h1;rw[ἃ,a2];(clear h z a1 a2 a3;exact omS141 a4 h1)
lemma qF3 (A:ѐ) (n:ὦ) (hn:Ѥn<=ѧA) (d:Ὣ):
 ∀(i:ὦ) (o s:ὥ),Ѥi+d=Ѥn->scanF A n i hn o s≠Е->
  o<=scanF A n i hn o s/\scanF A n i hn o s<У/\0x7FFFFF<=s/\((o=0\/0x800000<=o)->scanF A n i hn o s=0\/0x800000<=scanF A n i hn o s)/\
   ∀k,Ѥi<=k->k< Ѥn->good A[k]! =true/\A[k]!&&&В<=scanF A n i hn o s:=by
 induction d with
 | zero=>
  intro i o s he h
  rw[scanF,ἔ (by rw[А];(clear hn h;(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›)))))] at h⊢
  by_cases c:0x7FFFFF<=s/\o<У
  . rw[ἱ c];exact ⟨Ὓ _,c.2,c.1,id,λk h1 h2=>by (clear hn h c;exact omS135 he h1 h2)⟩
  . rw[ἐ c] at h;exact absurd rfl h
 | succ d ih=>
  intro i o s he h
  have hi':i<n:=by rw[А];(clear h;exact omS136 he)
  have hb:Ѥi< ѧA:=(by exact Nat.lt_of_lt_of_le ‹_› ‹_›)
  rw[scanF,Ἄ hi'] at h⊢
  obtain ⟨h1,h2,h3,h4,h5⟩:=ih (i+1) _ _ (by rw[usz i n hi'];(clear hi' h hb;exact omS137 he)) h
  obtain ⟨h6,h7⟩:=qF6 o _
  obtain ⟨h8,h9⟩:=qFe s ((A.uget i hb&&&В)-1)
  have hv:=qGi _ (UInt32.le_trans h3 h9)
  have hl:=UInt32.le_trans h7 h1
  refine ⟨UInt32.le_trans h6 h1,h2,UInt32.le_trans h3 h8,λg0=>h4 ?_,λk hk1 hk2=>?_⟩
  . rcases hv with hv | hv
    . rcases g0 with g0 | g0;. left;rw[g0,hv];rfl
      . right;exact UInt32.le_trans g0 h6
    . right;exact UInt32.le_trans hv h7
  . by_cases e2:k=Ѥi;. subst e2;rw[Ἕ A Ѥi hb];exact ⟨qBt _ hv (UInt32.lt_of_le_of_lt hl h2),hl⟩
    . exact h5 k (by rw[usz i n hi'];(clear he h hb h1 h2 h3 h6 h7 h8 h9 hv hl hk2;exact omS138 hk1 e2)) hk2
lemma qB1 (A:ѐ) (hs:ѧA<2^32) (h:scanF A A.usize 0 (qAc A) 0 Е≠Е):C_Hv A (scanF A A.usize 0 (qAc A) 0 Е):=by
 have e:=qBe A hs;have h0:(0:ὦ).toNat=0:=rfl;obtain ⟨-,h2,-,h4,h5⟩:=qF3 A A.usize (qAc A) ѧA 0 _ _ (by rw[h0,e];(clear hs h e h0;exact omS139)) h;generalize scanF A A.usize 0 (qAc A) 0 Е=S at*;have hp:=qCx h2;have hm:S&&&В=S:=by have a:(В:ὥ).toNat=2^31-1:=rfl;apply ἠ.mp;rw[ἃ,a] at hp;rw[Ἧ,a,Ἂ];(clear hs e h0 h h2 a;exact omS142 hp)
 have hg:good S=true:=by unfold good;dsimp only;rw[hm];exact qBt _ (h4 (Or.inl rfl)) h2
 exact ⟨hg,hp,λk hk=>(h5 k (by exact Nat.zero_le _) (by clear hs h0 h h2 hp hm;omega)).1,λk hk=>(h5 k (by exact Nat.zero_le _) (by clear hs h0 h h2 hp hm;omega)).2⟩
lemma qA8 (r:Ὣ) (A:ѐ) (hs:ѧA<2^32) (h:scanH r A≠Е):C_Hv A (scanH r A):=by
 revert h;unfold scanH;dsimp only;by_cases c:(r==1)=true
 . rw[ἱ c];split;. exact qBi A hs
   . exact qB1 A hs
 . rw[ἐ c];exact qBi A hs
lemma qGd (r:Ὣ) (A:ѐ) (h8:r≠8) (h1:r≠1) (h4:r≠4) (hs:ѧA<2^32) (h:scanW r A≠Е):C_Hv A (scanW r A):=by
 have e:scanW r A=scanH r A:=by unfold scanW;rw[ἐ (by simp only[or_eq_true,beq_iff_eq,h8,h1,or_self,h4,not_false_eq_true])]
 rw[e] at h⊢;exact qA8 r A hs h
def C_out (I:Ὣ) (Y:Ὡ) (X W b:ё) (i j:Ὣ):Ὢ:=
 act Y (dotOf (I+1) (λt=>if t<I then X[i*I+t]! else oneW)
  (λt=>if t<I then W[j*I+t]! else b[j]!))
lemma qCj {R I i t:Ὣ} (hi:i<R) (ht:t<I):i*I+t<R*I:=by have h:(i+1)*I<=R*I:=Ἒ _ hi;rw[ὰ,Ὑ] at h;(clear hi;omega)
lemma qBc {R I i t:Ὣ} (hi:i<R) (ht:t<=I):i*I+t<=R*I:=by have h:(i+1)*I<=R*I:=Ἒ _ hi;rw[ὰ,Ὑ] at h;(clear hi;omega)
section
variable {R I O:Ὣ} {X W b:ё}
section
variable (G:C_G R I O X W b)
include G
lemma C_G.gx {i:Ὣ} (hi:i<R): ∀t,t<I->good X[i*I+t]! =true:=λ_ ht=>G.gX _ (by rw[<-G.hX];exact qCj hi ht)
lemma C_G.gw {j:Ὣ} (hj:j<O): ∀t,t<I->good W[j*I+t]! =true:=λ_ ht=>G.gW _ (by rw[<-G.hW];exact qCj hj ht)
lemma qHx (Y:Ὡ) {i j:Ὣ} (hi:i<R) (hj:j<O) {s e:ὧ}
 (hs:let r:=(ἄ I).foldl (λ(acc:F1) t=>
  let q:=dv X[i*I+t]!*dv W[j*I+t]!
  F1.mk (acc.s+q) (acc.t+q.abs)) ⟨0,0⟩;s=r.s/\e=r.t):
 fin I Y X W b i j s (nfOf I) e b[j]! =C_out I Y X W b i j:=by
 obtain ⟨h1,h2⟩:=hs
 unfold fin
 dsimp only
 split
 . unfold slowOut C_out;rw[qAi]
 . rename_i h;unfold C_out;congr 1;exact qDn I G.ni (λt=>X[i*I+t]!) (λt=>W[j*I+t]!) b[j]! (G.gx hi) (G.gw hj) (G.gb j (by rw[<-G.hb];exact hj)) _ Ὂ (λt ht=>Ἅ.mp ht) (λt ht h3=>absurd (Ἅ.mpr ht) h3) _ _ ⟨by rw[h1],by rw[h2]⟩ (by simpa only[ne_eq,beq_iff_eq] using h)
end
lemma qE2 {X W:ѐ} {xo wo n:ὦ} (hX:ѧX<2^32) (hW:ѧW<2^32) (h1:Ѥxo+Ѥn<=ѧX) (h2:Ѥwo+Ѥn<=ѧW)
 (d:Ὣ):∀(t:ὦ) (s a:ὧ),Ѥt+d=Ѥn->
  tightGo X W xo wo n t s a=(List.range' Ѥt d).foldl (λ(acc:F1) k=>let q:=dv X[Ѥxo+k]!*dv W[Ѥwo+k]!
   F1.mk (acc.s+q) (acc.t+q.abs)) ⟨s,a⟩:=by
 induction d with
 | zero=>
  intro t s a he
  rw[tightGo,ἔ (by rw[А];(clear hX hW h1 h2;(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›)))))]
  rfl
 | succ d ih=>
  intro t s a he
  have ht:t<n:=by rw[А];(clear hX hW h1 h2;exact omS136 he)
  rw[tightGo,Ἄ ht,List.range'_succ,ὔ]
  dsimp only
  rw[qAg _ _ hX,qAg _ _ hW,qAf _ _ (by clear hW h2 ht;exact omS313 hX h1 he),qAf _ _ (by clear hX h1 ht;exact omS313 hW h2 he)]
  rw[ih (t+1) _ _ (by rw[usz t n ht];(clear hX hW h1 h2 ht;exact omS137 he)),usz t n ht]
section
variable (G:C_G R I O X W b) (Y:Ὡ) {i j:Ὣ} (hi:i< R) (hj:j< O)
include G Y i j hi hj
lemma qDw {s a:ὧ} {hh:Ὢ}
 (hc:ziv (s+dv b[j]!) (nfOf I*(dv hh*a+(dv b[j]!).abs))≠Е->
  ziv (s+dv b[j]!) (nfOf I*(dv hh*a+(dv b[j]!).abs))=
   dotOf (I+1) (λt=>if t<I then X[i*I+t]! else oneW) (λt=>if t<I then W[j*I+t]! else b[j]!)):
 fin8 I Y X W b i j s (nfOf I) a hh b[j]! ѩI=C_out I Y X W b i j:=by
 have hin:I<2^32:=lt_trans G.ni (by norm_num1)
 have hX1:i*I+I<=ѧX:=by rw[<-G.hX];exact qBc hi (Ὗ _)
 have hW1:j*I+I<=ѧW:=by rw[<-G.hW];exact qBc hj (Ὗ _)
 have sX:=G.sX
 have sW:=G.sW
 have e1:(i*I).toUSize.toNat=i*I:=qAd _ (by clear hi hj hW1 sW;exact omS143 hX1 sX)
 have e2:(j*I).toUSize.toNat=j*I:=qAd _ (by clear hi hj hX1 sX e1;exact omS143 hW1 sW)
 have e3:I.toUSize.toNat=I:=qAd _ hin
 unfold fin8
 dsimp only
 split
 . unfold fin8s;dsimp only;have e:=qE2 (xo:=(i*I).toUSize) (wo:=(j*I).toUSize) (n:=ѩI) G.sX G.sW (by rw[e1,e3];exact hX1) (by rw[e2,e3];exact hW1) I 0 0 0 (by rw[e3,C_z,Ἇ]);rw[C_z,e1,e2,<-ὗ] at e;rw[qHx G Y hi hj (s:=(tightGo X W (i*I).toUSize (j*I).toUSize ѩI 0 0 0).s) (e:=(tightGo X W (i*I).toUSize (j*I).toUSize ѩI 0 0 0).t) ⟨by rw[e],by rw[e]⟩]
 . rename_i h
   have hn:=(by simpa only[ne_eq,beq_iff_eq] using h:ziv (s+dv b[j]!) (nfOf I*(dv hh*a+(dv b[j]!).abs))≠Е)
   obtain ⟨S,ex,fr,a1,a2,a3,hz,-⟩:=qB6 _ _ hn
   rw[qR0 _ _ (qR1 _ S ex fr a1 a2 a3 hz)];unfold C_out;congr 1;exact hc hn
lemma qDL {s:ὧ} {hv:Ὢ} (g1:good hv=true) (g2:hv<=В) (L:љ) (hl:L.Nodup) (hl2:∀t∈L,t<I) (g4:∀t∈L,W[j*I+t]!&&&В<=hv) (hz:∀t,t<I->t∉L->X[i*I+t]!&&&В=0) (hs:s=L.foldl (λ(acc:ὧ) t=>acc+dv X[i*I+t]!*dv W[j*I+t]!) 0):
 fin8 I Y X W b i j s (nfOf I) (L.foldl (λ(a:ὧ) t=>a+(dv X[i*I+t]!).abs) 0) hv b[j]! ѩI=C_out I Y X W b i j:=by
 have gb:=G.gb j (by rw[<-G.hb];exact hj)
 exact qDw G Y hi hj λh=>by
  obtain ⟨-,-,k⟩:=qBw _ _ h
  obtain ⟨c1,c2⟩:=qC8 I G.ni (λt=>X[i*I+t]!) (λt=>W[j*I+t]!) b[j]! hv (G.gx hi) (G.gw hj) gb g1 g2 L hl hl2 g4 hz _ _ ⟨by rw[hs],rfl⟩ k
  exact qCn I G.ni _ _ _ (G.gx hi) (G.gw hj) gb _ _ c1 c2 h
lemma qDu {s:ὧ} {hv:Ὢ} (g1:good hv=true) (g2:hv<=В) (g4:∀t,t<I->W[j*I+t]!&&&В<=hv) (L:љ) (hl:L.Nodup) (hl2:∀t∈L,t<I) (hz:∀t,t<I->t∉L->X[i*I+t]!&&&В=0) (hs:s=L.foldl (λ(acc:ὧ) t=>acc+dv X[i*I+t]!*dv W[j*I+t]!) 0):
 fin8 I Y X W b i j s (nfOf I) (L.foldl (λ(a:ὧ) t=>a+(dv X[i*I+t]!).abs) 0) hv b[j]! ѩI=C_out I Y X W b i j:=
 qDL G Y hi hj g1 g2 L hl hl2 (λt ht=>g4 t (hl2 t ht)) hz hs
end
end
lemma qC5 (x:ὤ) (i:ὦ) (h:Ѥi< Ѩx):x.uget i h=x.data[Ѥi]!:=by obtain ⟨ds⟩:=x;have h':Ѥi< ѧds:=h;rw[Ἕ ds Ѥi h'];rfl
lemma qC3 (a:ѐ) (i:ὦ) (h:Ѥi< ѧa):a.uget i h=a[Ѥi]!:=by rw[Ἕ a Ѥi h];rfl
lemma qD6 (a:ὤ) (i:ὦ) (v:ὧ):(wrF a i v).size=Ѩa:=by
 unfold wrF;split;. obtain ⟨ds⟩:=a;exact Array.size_uset _
 . rfl
lemma qEt (a:ὤ) (i:ὦ) (v:ὧ) (q:Ὣ) (hs:Ѩa<2^32) (hi:Ѥi< Ѩa):(wrF a i v).data[q]! =if q=Ѥi then v else a.data[q]!:=by
 have h:i<a.usize:=by rw[А,FloatArray.usize];rwa[qAd _ hs]
 unfold wrF;rw[Ἄ h];obtain ⟨ds⟩:=a;exact qFo ds _ q v hi
def C_keep (xw:Ὣ->Ὢ) (t:Ὣ):љ:=
 (ἄ t).filter (λp=>xw p&&&В != 0)
section
variable (xw:Ὣ->Ὢ) (t:Ὣ)
include xw t
lemma qBj:(C_keep xw t).length<=t:=by have h:=List.length_filter_le (λp=>xw p&&&В != 0) (ἄ t);simpa only[C_keep,ge_iff_le,List.length_range] using h
lemma qDp (h:xw t&&&В=0):C_keep xw (t+1)=C_keep xw t:=by simp only[C_keep,List.range_succ,List.filter_append,h,bne_self_eq_false,false_eq_true,not_false_eq_true,List.filter_cons_of_neg,List.filter_nil,List.append_nil]
lemma qD5 (h:¬xw t&&&В=0):C_keep xw (t+1)=C_keep xw t++[t]:=by simp only[C_keep,List.range_succ,List.filter_append,bne_iff_ne,ne_eq,h,not_false_eq_true,List.filter_cons_of_pos,List.filter_nil]
end
lemma qAu (a:ὤ) (v:ὧ):(a.push v).size=Ѩa+1:=by obtain ⟨ds⟩:=a;exact Array.size_push _
lemma qDy (n:ὦ) (d:Ὣ):∀(k:ὦ) (a:ὤ),Ѥk+d=Ѥn->(pushZ n k a).size=Ѩa+d:=by
 induction d with
 | zero=>intro k a he;rw[pushZ,ἔ (by rw[А];(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›))))];rfl
 | succ d ih=>
  intro k a he;have ht:k<n:=by rw[А];exact omS136 he
  rw[pushZ,Ἄ ht,ih (k+1) _ (by rw[usz k n ht];(clear ht;exact omS137 he)),qAu];(clear he ht;omega)
lemma qH6 {x:ὤ} {ix W:ѐ} {w0 m n:ὦ} (hx:Ѥn<=Ѩx) (hi:Ѥn<=ѧix) (hs:ѧW<2^32) (h0:Ѥw0+Ѥm<=ѧW) (xf:Ὣ->ὧ) (ww:Ὣ->Ὢ) (hww:∀p,p< Ѥm->W[Ѥw0+p]! =ww p)
 (L:љ):∀(t:ὦ) (a:ὧ),Ѥt+ѣL=Ѥn->
  (∀k p:Ὣ,L[k]?=some p->p< Ѥm/\(ix[Ѥt+k]!:ὥ).toNat=p/\x.data[Ѥt+k]! =xf p)->
  go1 x ix W w0 m n t hx hi hs h0 a=L.foldl (λ(acc:ὧ) p=>acc+xf p*dv (ww p)) a:=by
 induction L with
 | nil=>
  intro t a he _
  rw[go1,ἔ (by rw[А];simp only[List.length_nil,add_zero] at he;(clear hx hi hs h0;(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›)))))]
  rfl
 | cons p L ih=>
  intro t a he H
  simp only[List.length_cons] at he
  have ht:t<n:=by rw[А];exact omS136 he
  have htn:Ѥt< Ѥn:=(by exact Nat.succ_le_of_lt ‹_›)
  obtain ⟨h1,h2,h3⟩:=H 0 p rfl
  rw[Ἵ] at h2 h3
  have hit:Ѥt< ѧix:=(by exact Nat.lt_of_lt_of_le ‹_› ‹_›)
  have e0:(ix.uget t hit).toUSize.toNat=p:=by
   rw[ὃ,qC3];exact h2
  have hm:(ix.uget t hit).toUSize< m:=by
   rw[А,e0];exact h1
  have e1:x.uget t (by clear ht htn h1 h2 e0 hm;(clear hit;exact omS144 hx he))=xf p:=by rw[qC5,h3]
  have e2:∀pf,W.uget (w0+(ix.uget t hit).toUSize) pf=ww p:=by
   intro pf
   rw[qC3,qAf _ _ (by rw[e0];(clear ht htn hm pf;exact omS145 hs h0 h1)),e0,hww p h1]
  rw[go1,Ἄ ht]
  dsimp only
  rw[Ἄ hm]
  rw[e1,e2,ih (t+1) _ (by rw[usz t n ht];(clear ht htn h1 h2 e0 hm;exact omS137 he)),ὔ]
  intro k q hq
  have:=H (k+1) q (by simpa only[List.getElem?_cons_succ] using hq)
  rw[usz t n ht,show Ѥt+1+k=Ѥt+(k+1) by (clear ht htn h1 h2 e0 hm this;exact omS146)]
  exact this
def C_Dw (wd:ὤ) (W:ё):Prop:=
 Ѩwd=ѧW->∀q,q< ѧW->wd.data[q]! =dv W[q]!
section
variable {x:ὤ} {ix W:ѐ} {w0 w1 w2 w3 m n:ὦ}
 (hx:Ѥn<=Ѩx) (hi:Ѥn<=ѧix) (hs:ѧW<2^32)
 (h0:Ѥw0+Ѥm<=ѧW) (h1:Ѥw1+Ѥm<=ѧW)
 (h2:Ѥw2+Ѥm<=ѧW) (h3:Ѥw3+Ѥm<=ѧW)
lemma qH0 (hv:∀j,j< Ѥn->∃w,good w=true/\x.data[j]! =dv w) (k:Ὣ):
 ∀(t:ὦ) (a b c d:ὧ),Ѥt+k=Ѥn->
  go4B x ix W w0 w1 w2 w3 m n t hx hi hs h0 h1 h2 h3 a b c d=
   ⟨go1 x ix W w0 m n t hx hi hs h0 a,go1 x ix W w1 m n t hx hi hs h1 b,go1 x ix W w2 m n t hx hi hs h2 c,go1 x ix W w3 m n t hx hi hs h3 d⟩:=by
 induction k with
 | zero=>
  intro t a b c d he
  have hn:¬t<n:=by rw[А];(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›)))
  unfold go4B go1
  simp only[ἔ hn]
 | succ k ih=>
  intro t a b c d he
  have ht:t<n:=by rw[А];exact omS136 he
  have htn:Ѥt< Ѥn:=(by exact Nat.succ_le_of_lt ‹_›)
  unfold go4B go1
  simp only[Ἄ ht]
  obtain ⟨w,g,e⟩:=hv _ htn
  have B:∀(u:ὥ) p0,dvB (x.uget t p0) u=x.uget t p0*dv u:=by
   intro u p0
   rw[qC5,e]
   exact qFw w u g
  have J:=λa b c d=>ih (t+1) a b c d (by rw[usz t n ht];(clear ht htn;exact omS137 he))
  split<;>simp only[B,J]
end
def C_sp (I O:Ὣ) (Y:Ὡ) (X W b:ё) (q:Ὣ):Ὢ:=
 C_out I Y X W b (q/O) (q%O)
lemma qCi (I O:Ὣ) (Y:Ὡ) (X W b:ё) {i j:Ὣ} (hj:j<O):C_sp I O Y X W b (i*O+j)=C_out I Y X W b i j:=by unfold C_sp;have h0:0<O:=(by exact omS147 hj);rw[ἥ i O,Nat.mul_add_div h0,Nat.mul_add_mod,Ὃ hj,Ἁ hj,Ἵ]
lemma qAp {N:Ὣ} {sp:Ὣ->Ὢ} {p:Ὣ} {out:ё} (hN:N<2^32) (h:C_Inv N sp (λq=>q<p) out) {u:ὦ} {v:Ὢ} (hu:Ѥu=p) (hp:p<N) (hv:v=sp p):C_Inv N sp (λq=>q<p+1) (wrW out u v):=
 qA5 (qAk hN h hu hp hv) (λq _ (hq:q<p+1)=>(by exact omS148 hq:q=p\/q<p))
lemma qFk {u:ὦ} {p:Ὣ} (hu:Ѥu=p) (h:p+3<2^32):
 (u+1).toNat=p+1/\(u+2).toNat=p+2/\(u+3).toNat=p+3/\(p+4<2^32->(u+4).toNat=p+4):=by
 have l:∀k,k<5->p+k<2^32->(u+ὴ k).toNat=p+k:=λk hk h=>by
  rw[qAf _ _ (by rw[uk k (by exact Nat.lt_of_lt_of_le ‹_› ה)];exact omS149 hu h),uk k (by exact Nat.lt_of_lt_of_le ‹_› ה),hu]
 exact ⟨l 1 ה (by exact omS150 h),l 2 ה (by exact Nat.le_of_lt ‹_›),l 3 ה (by exact Nat.succ_le_of_lt ‹_›),l 4 (by exact Nat.le_refl _)⟩
lemma qEq {N:Ὣ} {sp:Ὣ->Ὢ} {p:Ὣ} {out:ё} (hN:N<2^32) (h:C_Inv N sp (λq=>q<p) out) {u:ὦ} {v0 v1 v2 v3:Ὢ} (hu:Ѥu=p) (hp:p+4<=N) (h0:v0=sp p) (h1:v1=sp (p+1)) (h2:v2=sp (p+2))
 (h3:v3=sp (p+3)):C_Inv N sp (λq=>q<p+4) (wrW (wrW (wrW (wrW out u v0) (u+1) v1) (u+2) v2) (u+3) v3):=by
 obtain ⟨a1,a2,a3,-⟩:=qFk hu (by clear hu;exact omS151 hN hp)
 exact qAp hN (qAp hN (qAp hN (qAp hN h hu (by clear hN hu;exact omS152 hp) h0) a1 (by clear hN hu;exact omS153 hp) h1) a2 (by exact Nat.le_of_lt ‹_›) h2) a3 (by exact Nat.succ_le_of_lt ‹_›) h3
lemma qIA (k:Ὣ) (h:k<8192):(idA[k]!).toNat=k:=by
 rw[Ἕ idA k (by simp only[idA,Nat.toUInt32_eq,Array.size_ofFn];exact h)];simp only[idA,Array.getElem_ofFn];rw[Nat.toUInt32_eq,ὡ];exact Ἁ (by (exact Nat.lt_of_lt_of_le ‹_› ה))
lemma qIs:ѧidA=8192:=by simp only[idA,Nat.toUInt32_eq,Array.size_ofFn]
lemma qRd (X:ѐ) (n:Ὣ) (buf:ὤ) (h:n<=8192) (hv:∀k,k<n->buf.data[k]! =dv X[0*n+k]!):∀k p:Ὣ,(ἄ n)[k]?=some p->(idA[k]!).toNat=p/\buf.data[k]! =dv X[0*n+p]!:=λk p hk=>by
 by_cases hk2:k<n
 . rw[List.getElem?_range hk2] at hk;injection hk with e;subst e;exact ⟨qIA k (by (exact Nat.lt_of_lt_of_le ‹_› ‹_›)),hv k hk2⟩
 . rw[List.getElem?_eq_none (by simpa only[List.length_range,not_lt] using hk2)] at hk;exact absurd hk (by simp only[reduceCtorEq,not_false_eq_true])
lemma qVz {X:ѐ} {a:Ὣ} {n:ὦ} (hX:ѧX<2^32) (hn:a+Ѥn<=ѧX) (d:Ὣ): ∀(t:ὦ) (buf:ὤ) (l:ὧ),Ѥt+d=Ѥn-> Ѥn<=Ѩbuf-> Ѩbuf<2^32->(∀k,k< Ѥt->buf.data[k]! =dv X[a+k]!)->l=(ἄ Ѥt).foldl (λ(acc:ὧ) p=>acc+(dv X[a+p]!).abs) 0->(decD X ѩa n t buf l).ok=true->
  Ѩ(decD X ѩa n t buf l).buf=Ѩbuf/\(∀k,k< Ѥn->(decD X ѩa n t buf l).buf.data[k]! =dv X[a+k]!)/\(decD X ѩa n t buf l).l=(ἄ Ѥn).foldl (λ(acc:ὧ) p=>acc+(dv X[a+p]!).abs) 0:=by
 have ea:Ѥ(ѩa)=a:=qAd a (by exact omS154 hX hn)
 induction d with
 | zero=>
  intro t buf l he hb hs hd hl hok
  have hn':¬t<n:=by rw[А];(clear hX hn ea hb hs;(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›))))
  rw[decD,ἔ hn'];have e:Ѥt=Ѥn:=by exact omS155 he
  rw[e] at hd hl;exact ⟨rfl,hd,hl⟩
 | succ d ih=>
  intro t buf l he hb hs hd hl hok
  have ht:t<n:=by rw[А];(clear hX hn ea hb hs;exact omS136 he)
  have h1:=А.mp ht
  have u:=usz t n ht
  have er:rdW X (ѩa+t)=X[a+Ѥt]!:=by rw[qAg _ _ hX,qAf _ _ (by (clear hb hs ht h1;exact omS156 hX hn ea he)),ea]
  rw[decD,Ἄ ht] at hok⊢;dsimp only at hok⊢;rw[er] at hok⊢
  have R:=ih (t+1) (wrF buf t (dv X[a+Ѥt]!)) (l+(dv X[a+Ѥt]!).abs) (by (clear hX hn ea hb hs ht h1 er;exact omS157 he u)) (by rw[qD6];exact hb) (by rw[qD6];exact hs) (λk hk=>by
   rw[u] at hk;rw[qEt buf t _ k hs (by (exact Nat.lt_of_lt_of_le ‹_› ‹_›))];split
   . rename_i e;rw[e]
   . exact hd k (by (clear hX hn ea he hb hs ht h1 er;(clear u;omega)))) (by rw[u,List.range_succ,List.foldl_append,hl];rfl) hok
  rw[qD6] at R;exact R
lemma qDs (xw:Ὣ->Ὢ) (n:Ὣ):(C_keep xw n).Nodup/\(∀t∈C_keep xw n,t<n)/\ ∀t,t<n->t∉C_keep xw n->xw t&&&В=0:=
 ⟨Ὂ.filter _,λt ht=>Ἅ.mp (List.mem_filter.mp ht).1,
  λt ht hn=>by
   by_contra hc
   exact hn (List.mem_filter.mpr ⟨Ἅ.mpr ht,by simpa only[bne_iff_ne,ne_eq] using hc⟩)⟩
lemma qW8p (a:ὤ) (v0 v1 v2 v3 v4 v5 v6 v7:ὧ) (q:Ὣ) (h:q< Ѩa):((((((((a.push v0).push v1).push v2).push v3).push v4).push v5).push v6).push v7).data[q]! =a.data[q]!:=by
 obtain ⟨ds⟩:=a;have h':q<ds.size:=h;simp only[FloatArray.push,getElem!_def,Array.getElem?_push,Array.size_push,show q≠ds.size+1+1+1+1+1+1+1 by (clear h;exact omS165 h'), ↓reduceIte,show q≠ds.size+1+1+1+1+1+1 by (clear h;exact omS164 h'),show q≠ds.size+1+1+1+1+1 by (clear h;exact omS163 h'),show q≠ds.size+1+1+1+1 by (clear h;exact omS162 h'),show q≠ds.size+1+1+1 by (clear h;exact omS161 h'),show q≠ds.size+1+1 by (clear h;exact omS160 h'),show q≠ds.size+1 by (clear h;exact omS159 h'),show q≠ds.size by (clear h;exact omS158 h')]
lemma wbIs (W:ѐ) (m r i:Ὣ) (a:ὤ):(wbI W m r i a).size=Ѩa+8*(m-i):=by
 rw[wbI];split
 . rw[wbIs];simp only[qAu];omega
 . omega
termination_by m-i
decreasing_by all_goals omega
lemma wbIp (W:ѐ) (m r i:Ὣ) (a:ὤ) (q:Ὣ) (hq:q< Ѩa):(wbI W m r i a).data[q]! =a.data[q]!:=by
 rw[wbI];split
 . rw[wbIp _ _ _ _ _ _ (by simp only[qAu];exact omS166 hq),qW8p _ _ _ _ _ _ _ _ _ _ hq]
 . rfl
termination_by m-i
decreasing_by all_goals omega
lemma qW8k (a:ὤ) (v0 v1 v2 v3 v4 v5 v6 v7:ὧ) (k:Nat) (hk:k<8):
 ((((((((a.push v0).push v1).push v2).push v3).push v4).push v5).push v6).push v7).data[a.size+k]! =(#[v0,v1,v2,v3,v4,v5,v6,v7]:Array ὧ)[k]!:=by
 have e:((((((((a.push v0).push v1).push v2).push v3).push v4).push v5).push v6).push v7).data=a.data++ #[v0,v1,v2,v3,v4,v5,v6,v7]:=by
  simp only[FloatArray.push,Array.push_eq_append,Array.append_assoc,List.append_toArray,List.cons_append,List.nil_append]
 rw[e,Ἕ _ _ (by simp only[FloatArray.size,Array.size_append,List.size_toArray,List.length_cons,List.length_nil,zero_add,Nat.reduceAdd,add_lt_add_iff_left];(exact Nat.succ_le_of_lt ‹_›)),Ἕ _ _ (by simp only[List.size_toArray,List.length_cons,List.length_nil,zero_add,Nat.reduceAdd];(exact Nat.succ_le_of_lt ‹_›)),Array.getElem_append_right (by simp only[FloatArray.size,le_add_iff_nonneg_right,zero_le])]
 simp only[FloatArray.size,add_tsub_cancel_left,List.getElem_toArray]
lemma wbIn (W:ѐ) (m r i:Ὣ) (a:ὤ) (i':Ὣ) (h1:i<=i') (h2:i'< m) (k:Ὣ) (hk:k<8) (q:Ὣ) (hq:q=Ѩa+8*(i'-i)+k):(wbI W m r i a).data[q]! =dv W[(r+k)*m+i']!:=by
 rw[wbI,ἱ (by (exact Nat.lt_of_le_of_lt ‹_› ‹_›):i< m)]
 by_cases e:i'=i
 . subst e;rw[wbIp _ _ _ _ _ _ (by simp only[qAu];(clear h2 h1;exact omS167 hk hq))];rw[show q=Ѩa+k by (clear h2 hk h1;exact omS168 hq)]
   rw[qW8k _ _ _ _ _ _ _ _ _ k hk];interval_cases k<;>rfl
 . exact wbIn W m r (i+1) _ i' (by (clear h2 hk hq;exact omS138 h1 e)) h2 k hk q (by simp only[qAu];(clear h2 hk;exact omS169 h1 hq e))
termination_by m-i
decreasing_by all_goals (clear hk e;exact omS170 h2 h1)
lemma wbOp (W:ѐ) (m nb c:Ὣ) (a:ὤ) (q:Ὣ) (hq:q< Ѩa):(wbO W m nb c a).data[q]! =a.data[q]!:=by
 rw[wbO];split
 . rw[wbOp _ _ _ _ _ _ (by rw[wbIs];exact omS171 hq),wbIp _ _ _ _ _ _ hq]
 . rfl
termination_by nb-c
decreasing_by all_goals omega
lemma wbOn (W:ѐ) (m nb c:Ὣ) (a:ὤ) (c':Ὣ) (h1:c<=c') (h2:c'<nb) (i:Ὣ) (hi:i< m) (k:Ὣ) (hk:k<8) (q:Ὣ) (hq:q=Ѩa+(c'-c)*(8*m)+8*i+k):(wbO W m nb c a).data[q]! =dv W[(8*c'+k)*m+i]!:=by
 rw[wbO,ἱ (by (exact Nat.lt_of_le_of_lt ‹_› ‹_›):c<nb)]
 by_cases e:c'=c
 . subst e;rw[Nat.sub_self,ὒ] at hq;rw[wbOp _ _ _ _ _ _ (by rw[wbIs];(clear h2 h1;exact omS172 hi hk hq))];exact wbIn W m (8*c') 0 a i (Nat.zero_le _) hi k hk q (by exact omS173 hq)
 . have e2:c'-c=(c'-(c+1))+1:=by (clear h2 hi hk hq;exact omS174 h1 e)
   rw[e2,ὰ,Ὑ] at hq
   exact wbOn W m nb (c+1) _ c' (by (clear h2 hi hk hq e;exact omS175 e2)) h2 i hi k hk q (by rw[wbIs];(clear h1 h2 hi hk e e2;exact omS176 hq))
termination_by nb-c
decreasing_by all_goals (clear hi hk e e2;exact omS170 h2 h1)
lemma wbITp (W:ѐ) (m r o i:Ὣ) (a:ὤ) (q:Ὣ) (hq:q< Ѩa):(wbIT W m r o i a).data[q]! =a.data[q]!:=by
 rw[wbIT];split
 . rw[wbITp _ _ _ _ _ _ _ (by simp only[qAu];exact omS166 hq),qW8p _ _ _ _ _ _ _ _ _ _ hq]
 . rfl
termination_by m-i
decreasing_by all_goals omega
lemma wbITn (W:ѐ) (m r o i:Ὣ) (a:ὤ) (i':Ὣ) (h1:i<=i') (h2:i'< m) (k:Ὣ) (hk:k<8) (q:Ὣ) (hq:q=Ѩa+8*(i'-i)+k):(wbIT W m r o i a).data[q]! =dv W[(min (r+k) (o-1))*m+i']!:=by
 rw[wbIT,ἱ (by (exact Nat.lt_of_le_of_lt ‹_› ‹_›):i< m)]
 by_cases e:i'=i
 . subst e;rw[wbITp _ _ _ _ _ _ _ (by simp only[qAu];(clear h2 h1;exact omS167 hk hq))];rw[show q=Ѩa+k by (clear h2 hk h1;exact omS168 hq)]
   rw[qW8k _ _ _ _ _ _ _ _ _ k hk];interval_cases k<;>rfl
 . exact wbITn W m r o (i+1) _ i' (by (clear h2 hk hq;exact omS138 h1 e)) h2 k hk q (by simp only[qAu];(clear h2 hk;exact omS169 h1 hq e))
termination_by m-i
decreasing_by all_goals (clear hk e;exact omS170 h2 h1)
lemma wbOs (W:ѐ) (m nb c:Ὣ) (a:ὤ):(wbO W m nb c a).size=Ѩa+(nb-c)*(8*m):=by
 rw[wbO];split
 . rename_i h;rw[wbOs,wbIs];have:(nb-c)=(nb-(c+1))+1:=by exact omS177 h
   rw[this,ὰ,Ὑ,Nat.sub_zero];omega
 . rename_i h;rw[show nb-c=0 by exact omS178 h,ὒ,Ἵ]
termination_by nb-c
decreasing_by all_goals omega
lemma wbZs (n:Ὣ):(wbZ n).size=5:=by simp only[wbZ,qAu];simp only[FloatArray.size,ἆ,List.size_toArray,List.length_nil,zero_add,Nat.reduceAdd]
lemma wbP0 (W:ѐ) (m o j i k:Ὣ) (hj:j%8=0) (ho:j+8<=o) (hi:i< m) (hk:k<8):(wbO W m (o/8) 0 (wbZ ѧW)).data[j*m+5+8*i+k]! =dv W[(j+k)*m+i]!:=by
 have e1:8*(j/8)=j:=by (clear ho hi hk;exact omS179 hj)
 have e0:(wbZ ѧW).size=5:=wbZs _
 have e3:j/8*(8*m)=j*m:=by rw[<-Nat.mul_assoc,ἥ (j/8) 8,e1]
 have:=wbOn W m (o/8) 0 (wbZ ѧW) (j/8) (Nat.zero_le _) (by (clear hi hk e1 e0 e3;exact omS180 ho)) i hi k hk (j*m+5+8*i+k) (by rw[e0,Nat.sub_zero,e3];(clear hj ho hi hk e1 e0 e3;exact omS181))
 rw[e1] at this;exact this
lemma wbP (W:ѐ) (m o j i k:Ὣ) (hj:j%8=0) (ho:j+8<=o) (hi:i< m) (hk:k<8):(wbOf m o W).data[j*m+5+8*i+k]! =dv W[(j+k)*m+i]!:=by
 unfold wbOf;dsimp only;split
 . exact wbP0 W m o j i k hj ho hi hk
 . rw[wbITp _ _ _ _ _ _ _ (by
    rw[wbOs];have e0:(wbZ ѧW).size=5:=wbZs _
    rw[e0,Nat.sub_zero]
    have e1:8*(j/8)=j:=by (clear ho hi hk;exact omS179 hj)
    have:j/8+1<=o/8:=by exact omS182 ho
    have:(j/8+1)*(8*m)<=(o/8)*(8*m):=Ἒ _ this
    have e3:j*m+8*m=(j/8+1)*(8*m):=by rw[ὰ,Ὑ,<-Nat.mul_assoc,ἥ (j/8) 8,e1]
    have:j*m+8*i+k<j*m+8*m:=by exact omS183 hi hk
    omega)]
   exact wbP0 W m o j i k hj ho hi hk
lemma wbPt (W:ѐ) (m o i k:Ὣ) (ho:o%8≠0) (hi:i< m) (hk:k<8):(wbOf m o W).data[(8*(o/8))*m+5+8*i+k]! =dv W[(min (8*(o/8)+k) (o-1))*m+i]!:=by
 unfold wbOf;dsimp only;rw[ἐ ho]
 have e0:(wbZ ѧW).size=5:=wbZs _
 exact wbITn W m (8*(o/8)) o 0 _ i (Nat.zero_le _) hi k hk _ (by rw[wbOs,e0,Nat.sub_zero,Nat.sub_zero];ring)
lemma g8WE (x wb:ὤ) (ix W:ѐ) (p w0 w1 w2 w3 v0 v1 v2 v3 m n:ὦ) (hx:Ѥn<=Ѩx) (hi:Ѥn<=ѧix) (hb:Ѥp+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (hs':ѧW<2^32)
 (h0:Ѥw0+Ѥm<=ѧW) (h1:Ѥw1+Ѥm<=ѧW) (h2:Ѥw2+Ѥm<=ѧW) (h3:Ѥw3+Ѥm<=ѧW)
 (g0:Ѥv0+Ѥm<=ѧW) (g1:Ѥv1+Ѥm<=ѧW) (g2:Ѥv2+Ѥm<=ѧW) (g3:Ѥv3+Ѥm<=ѧW)
 (HB:∀i,i< Ѥm->wb.data[Ѥp+8*i+0]! =dv W[Ѥw0+i]!/\wb.data[Ѥp+8*i+1]! =dv W[Ѥw1+i]!/\wb.data[Ѥp+8*i+2]! =dv W[Ѥw2+i]!/\wb.data[Ѥp+8*i+3]! =dv W[Ѥw3+i]!/\
   wb.data[Ѥp+8*i+4]! =dv W[Ѥv0+i]!/\wb.data[Ѥp+8*i+5]! =dv W[Ѥv1+i]!/\wb.data[Ѥp+8*i+6]! =dv W[Ѥv2+i]!/\wb.data[Ѥp+8*i+7]! =dv W[Ѥv3+i]!)
 (k:Ὣ):∀(t:ὦ) (a b c d e f g z:ὧ),Ѥt+k=Ѥn->
 go8W x wb ix p m n t hx hi hb hs a b c d e f g z=(⟨go1 x ix W w0 m n t hx hi hs' h0 a,go1 x ix W w1 m n t hx hi hs' h1 b,go1 x ix W w2 m n t hx hi hs' h2 c,go1 x ix W w3 m n t hx hi hs' h3 d⟩,⟨go1 x ix W v0 m n t hx hi hs' g0 e,go1 x ix W v1 m n t hx hi hs' g1 f,go1 x ix W v2 m n t hx hi hs' g2 g,go1 x ix W v3 m n t hx hi hs' g3 z⟩):=by
 induction k with
 | zero=>
  intro t a b c d e f g z he
  have hn:¬t<n:=by rw[А];(clear hx hi hb hs hs' h0 h1 h2 h3 g0 g1 g2 g3;(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›))))
  unfold go8W go1
  simp only[ἔ hn]
 | succ k ih=>
  intro t a b c d e f g z he
  have ht:t<n:=by rw[А];exact omS136 he
  unfold go8W go1
  simp only[Ἄ ht]
  have J:=λa b c d e f g z=>ih (t+1) a b c d e f g z (by rw[usz t n ht];(clear ht;exact omS137 he))
  split
  . rename_i hm
    simp only[qC5,qC3] at hm⊢
    generalize (ix[Ѥt]!).toUSize=i at hm⊢
    have him:=А.mp hm
    obtain ⟨E0,E1,E2,E3,E4,E5,E6,E7⟩:=HB _ him
    have e8:(ὴ 8*i).toNat=8*Ѥi:=by rw[qWm _ _ (by rw[uk 8 ה];(clear he ht hm;exact omS16 hb hs him)),uk 8 ה]
    have T:∀kk,kk<8->(p+ὴ 8*i+ὴ kk).toNat=Ѥp+8*Ѥi+kk:=λkk hk=>by
     rw[qAf _ _ (by rw[qAf _ _ (by rw[e8];(clear he ht hm e8;exact omS17 hb hs him)),e8,uk kk (by exact Nat.lt_of_lt_of_le ‹_› ה)];(clear he ht hm e8;omega)),qAf _ _ (by rw[e8];(clear he ht hm e8;exact omS17 hb hs him)),e8,uk kk (by exact Nat.lt_of_lt_of_le ‹_› ה)]
    have V:∀w:ὦ,Ѥw+Ѥm<=ѧW->(w+i).toNat=Ѥw+Ѥi:=λw hw=>qAf _ _ (by (clear he ht hm e8;exact omS184 hs' him hw))
    rw[T 0 ה,T 1 ה,T 2 ה,T 3 ה,T 4 ה,T 5 ה,T 6 ה,T 7 (by exact Nat.le_refl _),V w0 h0,V w1 h1,V w2 h2,V w3 h3,V v0 g0,V v1 g1,V v2 g2,V v3 g3,E0,E1,E2,E3,E4,E5,E6,E7]
    exact J _ _ _ _ _ _ _ _
  . exact J _ _ _ _ _ _ _ _
lemma qR4 {x:ὤ} {ix W:ѐ} {wd:ὤ} {w0 w1 w2 w3 m n:ὦ} (hx:Ѥn<=Ѩx) (hi:Ѥn<=ѧix) (hs:ѧW<2^32) (h0:Ѥw0+Ѥm<=ѧW) (h1:Ѥw1+Ѥm<=ѧW) (h2:Ѥw2+Ѥm<=ѧW) (h3:Ѥw3+Ѥm<=ѧW)
 (HX:∀j,j< Ѥn->∃w,good w=true/\x.data[j]! =dv w) (Hd:C_Dw wd W):
 go4B x ix W w0 w1 w2 w3 m n 0 hx hi hs h0 h1 h2 h3 0 0 0 0=⟨go1 x ix W w0 m n 0 hx hi hs h0 0,go1 x ix W w1 m n 0 hx hi hs h1 0,go1 x ix W w2 m n 0 hx hi hs h2 0,go1 x ix W w3 m n 0 hx hi hs h3 0⟩:=by
 exact qH0 hx hi hs h0 h1 h2 h3 HX Ѥn 0 0 0 0 0 (Ἇ _)
set_option maxRecDepth 8192 in
lemma cT_eq (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (i:Ὣ) (ix:ѐ) (x wd:ὤ) (nf a:ὧ) (hx:ὥ) (lo:Ὠ) (m n:ὦ) (rr:G8) (j0:Ὣ)
 (HX:∀j,j< Ѥn->∃w,good w=true/\x.data[j]! =dv w) (Hd:C_Dw wd W)
 (hxn:Ѥn<=Ѩx) (hin:Ѥn<=ѧix) (hsW:ѧW<2^32) (HO:outW*Ѥm<=ѧW) (HO2:outW<2^32)
 (HL:∀(k:Ὣ) (w:ὦ) (hw:Ѥw=(j0+k)*Ѥm) (h0:Ѥw+Ѥm<=ѧW),j0+k<outW->C_g8 rr k=go1 x ix W w m n 0 hxn hin hsW h0 0)
 (j:Ὣ) (hj:j0<=j) (base:ὦ) (out:ѐ):
 cols1T inW outW relu X W b i nf a hx lo m rr j0 j base out=cols1 inW outW relu X W b i ix x wd nf a hx lo m n j base out:=by
 induction hd:outW-j using Nat.strong_induction_on generalizing j out with
 | _ d ih=>
  have M:∀(u:ὦ) (r:Ὣ),Ѥu=r->r+1<=outW->Ѥ(u*m)=r*Ѥm:=λu r hu hr=>by
   rw[qWm _ _ (by rw[hu];exact ἷ (ὂ (Ἒ _ (by (exact Nat.le_of_lt ‹_›))) HO) hsW),hu]
  have B:∀(u:ὦ) (r:Ὣ),Ѥu=r->r+1<=outW->Ѥ(u*m)+Ѥm<=ѧW:=λu r hu hr=>by
   rw[M u r hu hr];have:=Ἒ Ѥm hr;rw[ὰ,Ὑ] at this;exact this.trans HO
  have hW32:ѧW<=4294967295:=by exact omS185 hsW
  rw[cols1T,cols1]
  by_cases h4:j+4<=outW
  . rw[Ἄ h4,Ἄ h4]
    dsimp only
    have ej:Ѥ(ѩj)=j:=uk j (by exact omS186 HO2 h4)
    obtain ⟨e1,e2,e3,-⟩:=qFk ej (by exact omS151 HO2 h4)
    rw[Ἄ ⟨hxn,hin,hW32,B _ _ ej (by exact omS187 h4),B _ _ e1 (by exact omS188 h4),B _ _ e2 (by exact omS189 h4),B _ _ e3 (by exact omS190 h4)⟩]
    rw[qR4 _ _ _ _ _ _ _ HX Hd]
    dsimp only
    rw[HL (j-j0) (ѩj*m) (by rw[M _ _ ej (by exact omS187 h4)];congr 1;exact omS191 hj) (B _ _ ej (by exact omS187 h4)) (by exact omS192 hj h4),
     HL (j+1-j0) ((ѩj+1)*m) (by rw[M _ _ e1 (by exact omS188 h4)];congr 1;exact omS193 hj) (B _ _ e1 (by exact omS188 h4)) (by exact omS194 hj h4),
     HL (j+2-j0) ((ѩj+2)*m) (by rw[M _ _ e2 (by exact omS189 h4)];congr 1;exact omS195 hj) (B _ _ e2 (by exact omS189 h4)) (by exact omS196 hj h4),
     HL (j+3-j0) ((ѩj+3)*m) (by rw[M _ _ e3 (by exact omS190 h4)];congr 1;exact omS197 hj) (B _ _ e3 (by exact omS190 h4)) (by exact omS198 hj h4)]
    exact ih _ (by exact omS199 hd h4) (j+4) (by exact omS200 hj) _ rfl
  . rw[ἔ h4,ἔ h4]
    by_cases h1:j<outW
    . rw[Ἄ h1,Ἄ h1]
      dsimp only
      have ej:Ѥ(ѩj)=j:=uk j (by exact omS201 HO2 h1)
      rw[Ἄ ⟨hxn,hin,hW32,B _ _ ej (by exact omS202 h1)⟩]
      rw[HL (j-j0) (ѩj*m) (by rw[M _ _ ej (by exact omS202 h1)];congr 1;exact omS191 hj) (B _ _ ej (by exact omS202 h1)) (by exact omS203 hj h1)]
      exact ih _ (by exact omS204 hd h1) (j+1) (by exact omS205 hj) _ rfl
    . rw[ἔ h1,ἔ h1]
set_option maxRecDepth 8192 in
set_option maxHeartbeats 1000000 in
lemma cTL (x wb:ὤ) (ix W:ѐ) (outW:Ὣ) (m n:ὦ) (j:Ὣ) (hj:j=8*(outW/8)) (hxn:Ѥn<=Ѩx) (hin:Ѥn<=ѧix) (hb:Ѥ(ѩj*m+ὴ 5)+8*Ѥm<=Ѩwb) (he:Ѥ(ѩj*m+ὴ 5)=Ѥ(ѩj*m)+5) (hs:Ѩwb<2^32) (hsW:ѧW<2^32) (HO:outW*Ѥm<=ѧW) (HO2:outW<2^32)
 (HBt:∀i k,i< Ѥm->k<8->wb.data[(8*(outW/8))*Ѥm+5+8*i+k]! =dv W[(min (8*(outW/8)+k) (outW-1))*Ѥm+i]!):
 ∀(k:Ὣ) (w:ὦ) (hw:Ѥw=(j+k)*Ѥm) (h0:Ѥw+Ѥm<=ѧW),j+k<outW->C_g8 (g8of (go8W x wb ix (ѩj*m+ὴ 5) m n 0 hxn hin hb hs 0 0 0 0 0 0 0 0)) k=go1 x ix W w m n 0 hxn hin hsW h0 0:=by
 intro k w hw h0 hk
 have hk8:k<8:=by exact omS206 hj hk
 have hjm:Ѥ(ѩj*m)=j*Ѥm:=by rw[qWm _ _ (by rw[qAd j (by exact omS207 hj HO2)];exact ἷ (ὂ (Ἒ _ (by exact omS208 hj)) HO) hsW),qAd j (by exact omS207 hj HO2)]
 have twk:∀kk,kk<8-> Ѥ((ѩ(min (j+kk) (outW-1)))*m)=(min (j+kk) (outW-1))*Ѥm:=λkk hkk=>by
  rw[qWm _ _ (by rw[qAd _ א];exact ἷ (ὂ (Ἒ _ א) HO) hsW),qAd _ א]
 have bwk:∀kk,kk<8-> Ѥ((ѩ(min (j+kk) (outW-1)))*m)+Ѥm<=ѧW:=λkk hkk=>by
  rw[twk kk hkk];have:min (j+kk) (outW-1)+1<=outW:=by exact omS209 hk
  have:=Ἒ Ѥm this;rw[ὰ,Ὑ] at this;exact this.trans HO
 have HB8:∀i,i< Ѥm->wb.data[Ѥ(ѩj*m+ὴ 5)+8*i+0]! =dv W[Ѥ((ѩ(min (j+0) (outW-1)))*m)+i]!/\wb.data[Ѥ(ѩj*m+ὴ 5)+8*i+1]! =dv W[Ѥ((ѩ(min (j+1) (outW-1)))*m)+i]!/\wb.data[Ѥ(ѩj*m+ὴ 5)+8*i+2]! =dv W[Ѥ((ѩ(min (j+2) (outW-1)))*m)+i]!/\wb.data[Ѥ(ѩj*m+ὴ 5)+8*i+3]! =dv W[Ѥ((ѩ(min (j+3) (outW-1)))*m)+i]!/\
   wb.data[Ѥ(ѩj*m+ὴ 5)+8*i+4]! =dv W[Ѥ((ѩ(min (j+4) (outW-1)))*m)+i]!/\wb.data[Ѥ(ѩj*m+ὴ 5)+8*i+5]! =dv W[Ѥ((ѩ(min (j+5) (outW-1)))*m)+i]!/\wb.data[Ѥ(ѩj*m+ὴ 5)+8*i+6]! =dv W[Ѥ((ѩ(min (j+6) (outW-1)))*m)+i]!/\wb.data[Ѥ(ѩj*m+ὴ 5)+8*i+7]! =dv W[Ѥ((ѩ(min (j+7) (outW-1)))*m)+i]!:=λi hi=>by
  rw[he,hjm,twk 0 ה,twk 1 ה,twk 2 ה,twk 3 ה,twk 4 ה,twk 5 ה,twk 6 ה,twk 7 ה]
  have Q:=λkk (hkk:kk<8)=>HBt i kk hi hkk
  subst hj
  exact ⟨Q 0 ה,Q 1 ה,Q 2 ה,Q 3 ה,Q 4 ה,Q 5 ה,Q 6 ה,Q 7 ה⟩
 rw[g8WE x wb ix W (ѩj*m+ὴ 5) _ _ _ _ _ _ _ _ m n hxn hin hb hs hsW (bwk 0 ה) (bwk 1 ה) (bwk 2 ה) (bwk 3 ה) (bwk 4 ה) (bwk 5 ה) (bwk 6 ה) (bwk 7 ה) HB8 Ѥn 0 0 0 0 0 0 0 0 0 (by simp only[USize.toNat_zero,zero_add])]
 have ew:w=(ѩ(min (j+k) (outW-1)))*m:=USize.toNat_inj.mp (by rw[hw,twk k hk8];congr 1;exact omS210 hk)
 subst ew
 interval_cases k<;>rfl
lemma zFq (p i m:ὦ) (wb:ὤ) (k:Ὣ) (hk:k<8) (hb:Ѥp+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32) (him:Ѥi< Ѥm):(p+ὴ 8*i+ὴ k).toNat=Ѥp+8*Ѥi+k:=by
 have e8:(ὴ 8*i).toNat=8*Ѥi:=by rw[qWm _ _ (by rw[uk 8 ה];(clear hk;exact omS16 hb hs him)),uk 8 ה]
 rw[qAf _ _ (by rw[qAf _ _ (by rw[e8];(clear hk e8;exact omS17 hb hs him)),e8,uk k (by exact Nat.lt_of_lt_of_le ‹_› ה)];(clear e8;exact omS18 hk hb hs him)),qAf _ _ (by rw[e8];(clear hk e8;exact omS17 hb hs him)),e8,uk k (by exact Nat.lt_of_lt_of_le ‹_› ה)]
lemma g8WF (x wb:ὤ) (ix:ѐ) (p m n:ὦ) (hx:Ѥn<=Ѩx) (hi:Ѥn<=ѧix) (hb:Ѥp+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32)
 (HX:∀j,j< Ѥn->∃w,x.data[j]! =dv w) (HB:∀i,i< Ѥm->∀k,k<8->∃w,wb.data[Ѥp+8*i+k]! =dv w) (k:Ὣ):
 ∀(t:ὦ) (a b c d e f g z:ὧ),Ѥn-Ѥt=k->go8WF x wb ix p m n t hx hi hb hs a b c d e f g z=go8W x wb ix p m n t hx hi hb hs a b c d e f g z:=by
 induction k with
 | zero=>
  intro t a b c d e f g z he
  have hn:¬t<n:=by rw[А];exact omS411 he
  unfold go8WF go8W
  simp only[ἔ hn]
 | succ k ih=>
  intro t a b c d e f g z he
  have ht:t<n:=by rw[А];exact omS412 he
  have htn:=А.mp ht
  have J:=λa b c d e f g z=>ih (t+1) a b c d e f g z (by rw[usz t n ht];omega)
  unfold go8WF go8W
  simp only[Ἄ ht]
  by_cases hm:(ix.uget t (Nat.lt_of_lt_of_le htn hi)).toUSize< m
  . simp only[Ἄ hm]
    simp only[qC5,qC3] at hm⊢
    generalize (ix[Ѥt]!).toUSize=i at hm⊢
    have him:=А.mp hm
    obtain ⟨u,hu⟩:=HX _ htn
    obtain ⟨w0,h0⟩:=HB _ him 0 ה
    obtain ⟨w1,h1⟩:=HB _ him 1 ה
    obtain ⟨w2,h2⟩:=HB _ him 2 ה
    obtain ⟨w3,h3⟩:=HB _ him 3 ה
    obtain ⟨w4,h4⟩:=HB _ him 4 ה
    obtain ⟨w5,h5⟩:=HB _ him 5 ה
    obtain ⟨w6,h6⟩:=HB _ him 6 ה
    obtain ⟨w7,h7⟩:=HB _ him 7 ה
    rw[zFq p i m wb 0 ה hb hs him,zFq p i m wb 1 ה hb hs him,zFq p i m wb 2 ה hb hs him,zFq p i m wb 3 ה hb hs him,
     zFq p i m wb 4 ה hb hs him,zFq p i m wb 5 ה hb hs him,zFq p i m wb 6 ה hb hs him,zFq p i m wb 7 ה hb hs him,
     hu,h0,h1,h2,h3,h4,h5,h6,h7,zFV,zFV,zFV,zFV,zFV,zFV,zFV,zFV]
    exact J _ _ _ _ _ _ _ _
  . simp only[ἔ hm]
    exact J _ _ _ _ _ _ _ _
lemma g8WDE (x wb:ὤ) (ix:ѐ) (p m n:ὦ) (hx:Ѥn<=Ѩx) (hi:Ѥn<=ѧix) (hb:Ѥp+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32)
 (HX:∀j,j< Ѥn->∃w,x.data[j]! =dv w) (HB:∀i,i< Ѥm->∀k,k<8->∃w,wb.data[Ѥp+8*i+k]! =dv w) (t:ὦ) (a b c d e f g z:ὧ):
 go8WD x wb ix p m n t hx hi hb hs a b c d e f g z=go8W x wb ix p m n t hx hi hb hs a b c d e f g z:=by
 unfold go8WD;generalize armT=A;cases A
 . simp only[Bool.false_eq_true,ite_false]
 . simp only[ite_true];exact g8WF x wb ix p m n hx hi hb hs HX HB _ t a b c d e f g z rfl
lemma zF8 {wb:ὤ} {P:Ὣ} {A0 A1 A2 A3 A4 A5 A6 A7:ὥ} (h:wb.data[P+0]! =dv A0/\wb.data[P+1]! =dv A1/\wb.data[P+2]! =dv A2/\wb.data[P+3]! =dv A3/\
 wb.data[P+4]! =dv A4/\wb.data[P+5]! =dv A5/\wb.data[P+6]! =dv A6/\wb.data[P+7]! =dv A7): ∀k,k<8->∃w,wb.data[P+k]! =dv w:=by
 intro k hk;interval_cases k
 exacts[⟨_,h.1⟩,⟨_,h.2.1⟩,⟨_,h.2.2.1⟩,⟨_,h.2.2.2.1⟩,⟨_,h.2.2.2.2.1⟩,⟨_,h.2.2.2.2.2.1⟩,⟨_,h.2.2.2.2.2.2.1⟩,⟨_,h.2.2.2.2.2.2.2⟩]
lemma c1WE (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (i:Ὣ) (ix:ѐ) (x wd wb:ὤ) (nf a:ὧ) (hx:ὥ) (lo:Ὠ) (m n:ὦ)
 (HX:∀j,j< Ѥn->∃w,good w=true/\x.data[j]! =dv w) (Hd:C_Dw wd W) (HO:outW*Ѥm<=ѧW) (HO2:outW<2^32)
 (HB:∀j i k,j%8=0->j+8<=outW->i< Ѥm->k<8->wb.data[j*Ѥm+5+8*i+k]! =dv W[(j+k)*Ѥm+i]!)
 (HBt:outW%8≠0-> ∀i k,i< Ѥm->k<8->wb.data[(8*(outW/8))*Ѥm+5+8*i+k]! =dv W[(min (8*(outW/8)+k) (outW-1))*Ѥm+i]!)
 (j:Ὣ) (hj:j%8=0) (base:ὦ) (out:ѐ):
 cols1W inW outW relu X W b i ix x wd wb nf a hx lo m n j base out=cols1 inW outW relu X W b i ix x wd nf a hx lo m n j base out:=by
 induction hd:outW-j using Nat.strong_induction_on generalizing j hj out with
 | _ d ih=>
   rw[cols1W];split
   . rename_i h8;dsimp only;split
     . rename_i hg
       have sW:ѧW<2^32:=qA9 hg.2.2.1
       have ej:Ѥ(ѩj)=j:=uk j (by (clear HO hd hg sW;exact omS211 HO2 h8))
       have ej4:Ѥ((j+4).toUSize)=j+4:=uk (j+4) (by (clear HO hd hg sW ej;exact omS212 HO2 h8))
       obtain ⟨e1,e2,e3,-⟩:=qFk ej (by (clear HO hd hg sW ej ej4;exact omS213 HO2 h8))
       obtain ⟨f1,f2,f3,-⟩:=qFk ej4 (by (clear HO hd hg sW ej ej4 e1 e2 e3;exact omS214 HO2 h8))
       have M:∀(u:ὦ) (r:Ὣ),Ѥu=r->r+1<=outW->Ѥ(u*m)=r*Ѥm:=λu r hu hr=>by
        rw[qWm _ _ (by rw[hu];exact ἷ (ὂ (Ἒ _ (by (exact Nat.le_of_lt ‹_›))) HO) sW),hu]
       have HB8:∀i',i'< Ѥm->wb.data[Ѥ(ѩj*m+ὴ 5)+8*i'+0]! =dv W[Ѥ(ѩj*m)+i']!/\wb.data[Ѥ(ѩj*m+ὴ 5)+8*i'+1]! =dv W[((ѩj+1)*m).toNat+i']!/\wb.data[Ѥ(ѩj*m+ὴ 5)+8*i'+2]! =dv W[((ѩj+2)*m).toNat+i']!/\wb.data[Ѥ(ѩj*m+ὴ 5)+8*i'+3]! =dv W[((ѩj+3)*m).toNat+i']!/\
         wb.data[Ѥ(ѩj*m+ὴ 5)+8*i'+4]! =dv W[((j+4).toUSize*m).toNat+i']!/\wb.data[Ѥ(ѩj*m+ὴ 5)+8*i'+5]! =dv W[(((j+4).toUSize+1)*m).toNat+i']!/\wb.data[Ѥ(ѩj*m+ὴ 5)+8*i'+6]! =dv W[(((j+4).toUSize+2)*m).toNat+i']!/\wb.data[Ѥ(ѩj*m+ὴ 5)+8*i'+7]! =dv W[(((j+4).toUSize+3)*m).toNat+i']!:=λi' hi'=>by
        rw[(pk (ѩj*m) m wb hg.2.2.2.2.2.2.2.2.2.2.2.2 hg.2.2.2.2.2.2.2.2.2.2.2.1).2,M _ _ ej (by (clear HO HO2 hd hg sW ej ej4 e1 e2 e3 f1 f2 f3 hi';exact omS215 h8)),M _ _ e1 (by (clear HO HO2 hd hg sW ej ej4 e1 e2 e3 f1 f2 f3 hi';exact omS216 h8)),M _ _ e2 (by (clear HO HO2 hd hg sW ej ej4 e1 e2 e3 f1 f2 f3 hi';exact omS217 h8)),M _ _ e3 (by (clear HO HO2 hd hg sW ej ej4 e1 e2 e3 f1 f2 f3 hi';exact omS218 h8)),M _ _ ej4 (by (clear HO HO2 hd hg sW ej ej4 e1 e2 e3 f1 f2 f3 hi';exact omS219 h8)),M _ _ f1 (by (clear HO HO2 hd hg sW ej ej4 e1 e2 e3 f1 f2 f3 hi';exact omS220 h8)),M _ _ f2 (by (exact Nat.le_of_lt ‹_›)),M _ _ f3 (by (exact Nat.succ_le_of_lt ‹_›))]
        have Q:=λk hk=>HB j i' k hj h8 hi' hk
        refine ⟨by simpa only[add_zero] using Q 0 ה,Q 1 ה,Q 2 ה,Q 3 ה,Q 4 ה,?_,?_,?_⟩
        . have:=Q 5 ה;rwa[show j+5=j+4+1 by (clear HO HO2 hj hd h8 hg sW ej ej4 e1 e2 e3 f1 f2 f3 hi';exact omS221)] at this
        . have:=Q 6 ה;rwa[show j+6=j+4+2 by (clear HO HO2 hj hd h8 hg sW ej ej4 e1 e2 e3 f1 f2 f3 hi';exact omS222)] at this
        . have:=Q 7 (by exact Nat.le_refl _);rwa[show j+7=j+4+3 by (clear HO HO2 hj hd h8 hg sW ej ej4 e1 e2 e3 f1 f2 f3 hi';exact omS223)] at this
       rw[g8WDE x wb ix (ѩj*m+ὴ 5) m n hg.1 hg.2.1 _ _ (λj hj=>let ⟨w,_,e⟩:=HX j hj;⟨w,e⟩) (λi' hi'=>zF8 (HB8 i' hi'))]
       rw[g8WE x wb ix W (ѩj*m+ὴ 5) (ѩj*m) ((ѩj+1)*m) ((ѩj+2)*m) ((ѩj+3)*m) ((j+4).toUSize*m) (((j+4).toUSize+1)*m) (((j+4).toUSize+2)*m) (((j+4).toUSize+3)*m) m n hg.1 hg.2.1 (pk (ѩj*m) m wb hg.2.2.2.2.2.2.2.2.2.2.2.2 hg.2.2.2.2.2.2.2.2.2.2.2.1).1 (qA9 hg.2.2.2.2.2.2.2.2.2.2.2.1) sW 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 hg.2.2.2.2.2.2.2.2.2.1 hg.2.2.2.2.2.2.2.2.2.2.1 HB8 Ѥn 0 0 0 0 0 0 0 0 0 (Ἇ _)]
       rw[cols1,Ἄ (by (clear HO HO2 hd hg sW ej ej4 e1 e2 e3 f1 f2 f3;exact omS224 h8):j+4<=outW)];dsimp only;rw[Ἄ ⟨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⟩]
       rw[qR4 _ _ _ _ _ _ _ HX Hd];dsimp only
       rw[cols1,Ἄ (by (exact Nat.succ_le_of_lt ‹_›):j+4+4<=outW)];dsimp only;rw[Ἄ ⟨hg.1,hg.2.1,hg.2.2.1,hg.2.2.2.2.2.2.2.1,hg.2.2.2.2.2.2.2.2.1,hg.2.2.2.2.2.2.2.2.2.1,hg.2.2.2.2.2.2.2.2.2.2.1⟩]
       rw[qR4 _ _ _ _ _ _ _ HX Hd];dsimp only
       exact ih (outW-(j+8)) (by (clear HO HO2 hg sW ej ej4 e1 e2 e3 f1 f2 f3;exact omS226 hd h8)) (j+8) (by (clear HO HO2 hd h8 hg sW ej ej4 e1 e2 e3 f1 f2 f3;exact omS227 hj)) _ rfl
     . rfl
   . rename_i h8
     by_cases ht:j<outW
     . rw[Ἄ ht]
       by_cases hg:Ѥn<=Ѩx/\Ѥn<=ѧix/\ѧW<=4294967295/\Ѩwb<=4294967295/\Ѥn<=Ѥm/\Ѥ(ѩj*m)+5+8*Ѥm<=Ѩwb
       . rw[Ἄ hg]
         have hj8:j=8*(outW/8):=by exact omS228 hj h8 ht
         have sW:ѧW<2^32:=qA9 hg.2.2.1
         exact cT_eq inW outW relu X W b i ix x wd nf a hx lo m n _ j HX Hd hg.1 hg.2.1 sW HO HO2 (cTL x wb ix W outW m n j hj8 hg.1 hg.2.1 (pk (ѩj*m) m wb hg.2.2.2.2.2 hg.2.2.2.1).1 (pk (ѩj*m) m wb hg.2.2.2.2.2 hg.2.2.2.1).2 (qA9 hg.2.2.2.1) sW HO HO2 (HBt (by exact omS229 ht hj8))) j (Ὗ _) base out
       . rw[ἔ hg]
     . rw[ἔ ht]
lemma g8VW (x wb:ὤ) (ix:ѐ) (p m n:ὦ) (hx:Ѥn<=Ѩx) (hi:Ѥn<=ѧix) (hn:Ѥn<=Ѥm) (hb:Ѥp+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32)
 (HI:∀k,k< Ѥn->(ix[k]!).toNat=k) (k:Ὣ):∀(t:ὦ) (a b c d e f g z:ὧ),Ѥt+k=Ѥn->
 go8V x wb p m n t hx hn hb hs a b c d e f g z=g8of (go8W x wb ix p m n t hx hi hb hs a b c d e f g z):=by
 induction k with
 | zero=>
  intro t a b c d e f g z he
  have hn':¬t<n:=by rw[А];(clear hx hi hn hb hs;(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›))))
  rw[go8V,go8W,ἔ hn',ἔ hn'];rfl
 | succ k ih=>
  intro t a b c d e f g z he
  have ht:t<n:=by rw[А];exact omS136 he
  have J:=λa b c d e f g z=>ih (t+1) a b c d e f g z (by rw[usz t n ht];(clear ht;exact omS137 he))
  have E:(ix[Ѥt]!).toUSize=t:=USize.toNat_inj.mp (by rw[ὃ];exact HI _ (А.mp ht))
  rw[go8V,go8W,Ἄ ht,Ἄ ht];dsimp only;split
  . rename_i hm
    simp only[qC3] at hm⊢
    rw[J];simp only[E]
  . rename_i hm
    simp only[qC3,E] at hm
    exact absurd (by rw[А];have:=А.mp ht;(exact Nat.lt_of_lt_of_le ‹_› ‹_›)) hm
lemma g8VWF (x wb:ὤ) (ix:ѐ) (p m n:ὦ) (hx:Ѥn<=Ѩx) (hi:Ѥn<=ѧix) (hn:Ѥn<=Ѥm) (hb:Ѥp+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32)
 (HI:∀k,k< Ѥn->(ix[k]!).toNat=k) (k:Ὣ):∀(t:ὦ) (a b c d e f g z:ὧ),Ѥt+k=Ѥn->
 go8F x wb p m n t hx hn hb hs a b c d e f g z=g8of (go8WF x wb ix p m n t hx hi hb hs a b c d e f g z):=by
 induction k with
 | zero=>
  intro t a b c d e f g z he
  have hn':¬t<n:=by rw[А];(clear hx hi hn hb hs;(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›))))
  rw[go8F,go8WF,ἔ hn',ἔ hn'];rfl
 | succ k ih=>
  intro t a b c d e f g z he
  have ht:t<n:=by rw[А];exact omS136 he
  have J:=λa b c d e f g z=>ih (t+1) a b c d e f g z (by rw[usz t n ht];(clear ht;exact omS137 he))
  have E:(ix[Ѥt]!).toUSize=t:=USize.toNat_inj.mp (by rw[ὃ];exact HI _ (А.mp ht))
  rw[go8F,go8WF,Ἄ ht,Ἄ ht];dsimp only;split
  . rename_i hm
    simp only[qC3] at hm⊢
    rw[J];simp only[E]
  . rename_i hm
    simp only[qC3,E] at hm
    exact absurd (by rw[А];have:=А.mp ht;(exact Nat.lt_of_lt_of_le ‹_› ‹_›)) hm
lemma g8VWD (x wb:ὤ) (ix:ѐ) (p m n:ὦ) (hx:Ѥn<=Ѩx) (hi:Ѥn<=ѧix) (hn:Ѥn<=Ѥm) (hb:Ѥp+8*Ѥm<=Ѩwb) (hs:Ѩwb<2^32)
 (HI:∀k,k< Ѥn->(ix[k]!).toNat=k) (k:Ὣ):∀(t:ὦ) (a b c d e f g z:ὧ),Ѥt+k=Ѥn->
 go8D x wb p m n t hx hn hb hs a b c d e f g z=g8of (go8WD x wb ix p m n t hx hi hb hs a b c d e f g z):=by
 intro t a b c d e f g z he;unfold go8D go8WD;generalize armT=A;cases A
 . simp only[Bool.false_eq_true,ite_false];exact g8VW x wb ix p m n hx hi hn hb hs HI k t a b c d e f g z he
 . simp only[ite_true];exact g8VWF x wb ix p m n hx hi hn hb hs HI k t a b c d e f g z he
lemma c1VW (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (i:Ὣ) (ix:ѐ) (x wd wb:ὤ) (nf a:ὧ) (hx:ὥ) (lo:Ὠ) (m n:ὦ)
 (HI:∀k,k< Ѥn->(ix[k]!).toNat=k) (hn:Ѥn<=Ѥm) (j:Ὣ) (base:ὦ) (out:ѐ):
 cols1V inW outW relu X W b i ix x wd wb nf a hx lo m n j base out=cols1W inW outW relu X W b i ix x wd wb nf a hx lo m n j base out:=by
 rw[cols1V];split
 . rename_i h8;rw[cols1W,Ἄ h8];dsimp only;split
   . rename_i hg
     obtain ⟨a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14⟩:=hg
     rw[Ἄ ⟨a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a14⟩]
     rw[g8VWD x wb ix _ m n a1 a2 a13 _ _ HI Ѥn 0 0 0 0 0 0 0 0 0 (Ἇ _)];dsimp only[g8of]
     exact c1VW inW outW relu X W b i ix x wd wb nf a hx lo m n HI hn (j+8) base _
   . rename_i hg
     rw[ἔ (λh=>hg (by obtain ⟨a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a14⟩:=h;exact ⟨a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,hn,a14⟩))]
 . rename_i h8;rw[cols1W,ἔ h8]
   by_cases ht:j<outW
   . rw[Ἄ ht,Ἄ ht]
     by_cases hg:Ѥn<=Ѩx/\Ѥn<=ѧix/\ѧW<=4294967295/\Ѩwb<=4294967295/\Ѥn<=Ѥm/\Ѥ(ѩj*m)+5+8*Ѥm<=Ѩwb
     . rw[Ἄ hg,Ἄ hg,g8VW x wb ix _ m n hg.1 hg.2.1 hg.2.2.2.2.1 _ _ HI Ѥn 0 0 0 0 0 0 0 0 0 (Ἇ _)]
     . rw[ἔ hg,ἔ hg]
   . rw[ἔ ht,ἔ ht]
termination_by outW-j
decreasing_by all_goals omega
@[inline] def zivT (th:Ὠ) (s:ὧ):ὥ:=
 let a:=s.toBits
 if th<=a<<<(1:Ὠ) then zivW (a+0x200000) (a-0x200000) else Е
lemma qI8 (A P Q T g:Ὣ) (hg:g=0\/g=2^63) (ha:A<2^64) (e1:(A+2097152)%2^64=g+P) (e2:(2^64-2097152+A)%2^64=g+Q) (t1:2^61<=T) (t2:T<2^63-2^60) (l1:T<P+2^28) (u1:P<=T+2^28) (l2:T<Q+2^28) (u2:Q<=T+2^28):
 A=g+(Q+2097152)/\T+2097152<Q+2097152+2^28/\Q+2097152+2097152<=T+2^28:=by
 (clear hg t1 t2;exact omS230 ha e1 e2 l1 u1 l2 u2)
lemma qJh (G f F:Ὣ) (hl:G+f+2097152<G+F+2^28) (hu:G+F+2097152<=G+f+2^28):F+2097152<=f+2^28/\f+2097152<F+2^28:=by exact omS231 hl hu
lemma kFh (th:Ὠ) (s:ὧ) (h:zivT th s≠Е):
 th<=s.toBits<<<(1:Ὠ)/\∃(S:Ὡ) (ex fr M:Ὣ),897<=ex/\ex<1151/\fr<2^23/\zivT th s=ὺ ((if S then 2^31 else 0)+(ex-896)*2^23+fr)/\s.toBits.toNat=(if S then 2^63 else 0)+M/\(ex*2^23+fr)*2^29+2097152<M+2^28/\M+2097152<=(ex*2^23+fr)*2^29+2^28:=by
 unfold zivT at h⊢
 dsimp only at h⊢
 split at h
 . rename_i ht;rw[ἱ ht];refine ⟨ht,?_⟩;obtain ⟨S,ex,fr,h1,h2,h3,hz,⟨P,e1,l1,u1⟩,⟨Q,e2,l2,u2⟩⟩:=qEs _ _ h;rw[ἤ] at e1;rw[ὲ] at e2;obtain ⟨a,b,c⟩:=qI8 _ P Q ((ex*2^23+fr)*2^29) _ (by cases S<;>simp only[false_eq_true, ↓reduceIte,Nat.reducePow,OfNat.zero_ne_ofNat,or_false,OfNat.ofNat_ne_zero,or_true]) s.toBits.toNat_lt e1 e2 (by clear ht h hz e1 l1 u1 e2 l2 u2;exact omS232 h1) (by clear ht h hz e1 l1 u1 e2 l2 u2;exact omS233 h2 h3) l1 u1 l2 u2;exact ⟨S,ex,fr,Q+2097152,h1,h2,h3,hz,a,b,c⟩
 . exact absurd rfl h
lemma qI5 (v W z u:ℝ) (hu:0<u) (hz:z<=u) (h0:0<=v) (hv:v<=2^53*u) (h: |v-W| <=1/2^33*(|W| +z)):v-2^21*u<W/\W<v+2^21*u:=by
 have h1: |W| <=v+ |v-W|:=by have a:=Ὥ W v;rw[ἣ W v,ἅ h0] at a;linarith only[a]
 have h2:=abs_lt.mp ((by linarith only[h1,h,hv,hz,hu]):|v-W| <2^21*u);constructor<;>linarith only[h2.1,h2.2,h2]
set_option maxRecDepth 8192 in
lemma qJi (ex fr E F:Ὣ) (h1:897<=ex) (h2:ex<1151) (hfr:fr<2^23) (hF:F<2^52) (hl:(ex*2^23+fr)*2^29+2097152<E*2^52+F+2^28) (hu:E*2^52+F+2097152<=(ex*2^23+fr)*2^29+2^28) (W:ℝ) (h: |((2^52+F:Ὣ):ℝ)*2^((E:Int)-1075)-W| <=1/2^33*(|W| +A_z)):
 W<(2^53+fr*2^30+2^29)*2^((ex:Int)-1076)/\(fr≠0->(2^53+fr*2^30-2^29)*2^((ex:Int)-1076)<W)/\(2^53-2^28)*2^((ex:Int)-1076)<W:=by
 have hE:E=ex\/(E+1=ex/\fr=0):=(by clear h1 h2;exact omS234 hfr hF hl hu)
 have hu0:(0:ℝ)<2^((E:Int)-1075):=ἵ ו _
 have hz:A_z<=2^((E:Int)-1075):=by
  have e:A_z=2^(-1021:ℤ):=rfl
  rw[e]
  exact ἒ (by norm_num1) (by clear hE;exact omS235 h1 hF hl)
 have hN:((2^52+F:Ὣ):ℝ)<=2^53:=by
  have:2^52+F<=2^53:=(by clear h1 h2 hfr hl hu hE;exact omS236 hF)
  exact_mod_cast this
 have hN0:(0:ℝ)<=((2^52+F:Ὣ):ℝ):=Ἔ _
 obtain ⟨a,b⟩:=qI5 _ W A_z _ hu0 hz (Ἆ hN0 hu0.le) (ἀ hN hu0.le) h
 push_cast at a b
 rcases hE with rfl | ⟨rfl,rfl⟩
 . have e:(2:ℝ)^((E:Int)-1075)=2*2^((E:Int)-1076):=by rw[show (E:Int)-1075=1+((E:Int)-1076) by ring,Ἓ ἓ,zpow_one]
   rw[e] at a b;have hb:(0:ℝ)<2^((E:Int)-1076):=ἵ ו _;generalize (2:ℝ)^((E:Int)-1076)=c at a b hb⊢;have e1:(E*2^23+fr)*2^29=E*2^52+fr*2^29:=by ring
   rw[e1] at hl hu;obtain ⟨n3,n4⟩:=qJh _ _ _ hl hu;have h3:(F:ℝ)+2097152<=fr*2^29+2^28:=by exact_mod_cast n3
   have h4:(fr:ℝ)*2^29+2097152<F+2^28:=by exact_mod_cast n4
   have h5:(0:ℝ)<=F:=Ἔ _;have h6:(0:ℝ)<=fr:=Ἔ _;have h7:=ἀ h3 hb.le;have h8:=Ἦ h4 hb;have h9:=Ἆ h5 hb.le;have h10:=Ἆ h6 hb.le;refine ⟨?_,?_,?_⟩;. linarith only[b,h7]
   . intro _;linarith only[a,h8]
   . linarith only[a,h8,h9,h10]
 . have e:(2:ℝ)^(((E+1:Ὣ):Int)-1076)=2^((E:Int)-1075):=by congr 1;push_cast;ring
   rw[e];generalize (2:ℝ)^((E:Int)-1075)=c at a b hu0⊢;have h3:(F:ℝ)<=2^52:=by exact_mod_cast hF.le
   have h4:(2:ℝ)^52+2097152<F+2^28:=by exact_mod_cast (by (clear h1 h2 hfr;exact omS237 hl):2^52+2097152<F+2^28)
   have h7:=ἀ h3 hu0.le;have h8:=Ἦ h4 hu0;refine ⟨?_,?_,?_⟩;. simp only[Nat.cast_zero,zero_mul,add_zero];linarith only[b,h7,h8]
   . intro h0;exact absurd rfl h0
   . linarith only[a,h8]
lemma kHv (th:Ὠ) (s:ὧ) (d:FloatLib.Numerics.Dyadic) (hV: |R s-(d.toRat:ℝ)| <=1/2^33*(|(d.toRat:ℝ)| +A_z)) (h:zivT th s≠Е):zivT th s=toWord (ἕ Fmt d):=by
 obtain ⟨-,S,ex,fr,M,h1,h2,h3,hz,hb,hl,hu⟩:=kFh th s h;have hM:M=(M/2^52)*2^52+M%2^52:=(by clear h h1 h2 h3 hz hb hl hu;exact omS114);have hF:M%2^52<2^52:=ὀ _ ו;generalize M/2^52=E at hM;generalize M%2^52=F at hM hF;subst hM;obtain ⟨-,hR⟩:=B_R s S E F (by clear h hz;exact omS238 h1 hF hl) (by clear h hz;exact omS239 h2 h3 hu) hF (by rw[hb,ὢ]);rw[hR] at hV
 obtain ⟨W,hVW,hW⟩:∃W:ℝ,(d.toRat:ℝ)=(if S then-W else W)/\
  |((2^52+F:Ὣ):ℝ)*2^((E:Int)-1075)-W| <=1/2^33*(|W| +A_z):=by
  cases S
  . simp only[false_eq_true,ite_false,one_mul] at hV⊢;exact ⟨_,rfl,hV⟩
  . simp only[ite_true] at hV⊢;refine ⟨-(d.toRat:ℝ),by simp only[Ἠ,neg_neg],?_⟩;rw[abs_neg,show ((2^52+F:Ὣ):ℝ)*2^((E:Int)-1075)- -(d.toRat:ℝ)=-(-1*((2^52+F:Ὣ):ℝ)*2^((E:Int)-1075)-(d.toRat:ℝ)) by ring,abs_neg];exact hV
 obtain ⟨a,b,c⟩:=qJi ex fr E F h1 h2 h3 hF (by exact omS240 hl) (by exact omS241 hu) W hW;rw[hz];exact (qEo d S ex fr h1 h2 h3 W hVW a b c).symm
lemma qJc (x:Ὠ):((x>>>(52:Ὠ))&&&(0x7FF:Ὠ)).toNat=Ѧx/2^52%2^11:=by rw[Ὅ,ἰ];have:=Ἂ (Ѧx>>>52) 11;norm_num[Ἱ] at this⊢;exact this
lemma qGj (b:Ὡ) (x y:ℝ) (hx:0<=x) (hy:0<=y):
 |(if b=true then (-1:ℝ) else 1)*x*y| =x*y:=by
 have h:=ἅ (Ἆ hx hy)
 cases b
 . rw[show (if false=true then (-1:ℝ) else 1)*x*y=x*y by simp only[false_eq_true, ↓reduceIte,one_mul]];exact h
 . rw[show (if true=true then (-1:ℝ) else 1)*x*y=-(x*y) by simp only[↓reduceIte,neg_mul,one_mul],abs_neg];exact h
lemma qJb (a:Ὠ) (S:Ὡ) (M:Ὣ) (hM:M<2^63) (h:Ѧa=(if S then 2^63 else 0)+M):(a<<<(1:Ὠ)).toNat=2*M:=by rw[UInt64.toNat_shiftLeft,h];norm_num[ὁ];cases S<;>simp only[false_eq_true, ↓reduceIte,zero_add]<;>omega
lemma kGw (e s:ὧ) (S:Ὡ) (M:Ὣ) (hM2:M<2047*2^52) (hs:s.toBits.toNat=(if S then 2^63 else 0)+M) (h:thOf e<=s.toBits<<<(1:Ὠ)):e.isFinite=true/\|R e| <=1/2^34* |R s|:=by
 have hl:=qJb s.toBits S M (by exact Nat.lt_of_lt_of_le ‹_› ה) hs;unfold thOf at h;dsimp only at h;have hx:=qJc e.toBits;generalize (e.toBits>>>(52:Ὠ))&&&(0x7FF:Ὠ)=E at h hx;rw[Ὤ,hl] at h;split at h
 . rename_i hc;rw[ἴ,ὲ] at hc;rw[UInt64.toNat_shiftLeft,ἤ] at h;have hA:=e.toBits.toNat_lt;norm_num[ὁ] at hc h hA;obtain ⟨sg,hsg⟩:∃sg:Ὡ,e.toBits.toNat=(if sg then 2^63 else 0)+ѦE*2^52+e.toBits.toNat%2^52:=by refine ⟨decide (2^63<=e.toBits.toNat),?_⟩;by_cases h6:2^63<=e.toBits.toNat<;>simp only[h6,decide_true,decide_false,ite_true,ite_false,false_eq_true]<;>(clear hM2 hs hl hc h;omega)
   have hFe:e.toBits.toNat%2^52<2^52:=ὀ _ ב;generalize e.toBits.toNat%2^52=Fe at hsg hFe;have hE1:1<=ѦE/\ѦE<=1900:=(by clear hM2 hs hl h hA hsg hFe;exact omS242 hx hc);have hEs:M=(M/2^52)*2^52+M%2^52:=(by clear hM2 hs hl hx hc h hA hsg hFe hE1;exact omS114);have hFs:M%2^52<2^52:=ὀ _ ב;generalize M/2^52=Es at hEs;generalize M%2^52=Fs at hEs hFs;subst hEs;have hle:ѦE+35<=Es:=(by clear hc hA hsg hFe hl;exact omS243 hE1 hFs h);obtain ⟨fe,re⟩:=B_R e sg ѦE Fe hE1.1 (by clear hc hA hsg hFe hFs hM2 hs hl h hle;exact omS244 hE1) hFe hsg;obtain ⟨-,rs⟩:=B_R s S Es Fs (by clear hA hsg hFe hE1 hM2 hl hle;exact omS245 hc hFs h) (by clear hx hc hA hsg hFe hE1 hl h hle;exact omS246 hM2) hFs (by rw[hs,ὢ]);refine ⟨fe,?_⟩;have hc0:(0:ℝ)<2^((ѦE:Int)-1075):=ἵ ב _;have h1:(2:ℝ)^((ѦE:Int)+35-1075)<=2^((Es:Int)-1075):=ἒ ב (by clear hc hA hsg hFe hE1 hFs hM2 hs hl h;exact omS247 hle);have h2:(2:ℝ)^((ѦE:Int)+35-1075)=2^35*2^((ѦE:Int)-1075):=by rw[show (ѦE:Int)+35-1075=35+((ѦE:Int)-1075) by ring,Ἓ ἓ];norm_num
   have hae: |R e| =((2^52+Fe:Ὣ):ℝ)*2^((ѦE:Int)-1075):=by rw[re];exact qGj _ _ _ (Ἔ _) hc0.le
   have has: |R s| =((2^52+Fs:Ὣ):ℝ)*2^((Es:Int)-1075):=by rw[rs];exact qGj _ _ _ (Ἔ _) (ἵ ב _).le
   rw[hae,has];have n1:((2^52+Fe:Ὣ):ℝ)<=2^53:=by have:2^52+Fe<=2^53:=(by clear hx hc hA hsg hE1 hFs hM2 hs hl h hle;exact omS236 hFe);exact_mod_cast this
   have n2:(2:ℝ)^52<=((2^52+Fs:Ὣ):ℝ):=by have:2^52<=2^52+Fs:=(by clear hx hc hA hsg hFe hE1 hM2 hl h hle;exact omS248);exact_mod_cast this
   generalize (2:ℝ)^((ѦE:Int)-1075)=c at*;generalize (2:ℝ)^((Es:Int)-1075)=d at*;rw[h2] at h1;have n3:=ἀ n1 hc0.le;have n0:0<=d:=(by linarith only[hc0,h1]);have n4:=ἁ (ἀ n2 n0) (by norm_num:(0:ℝ)<=1/2^34);have n5:(2:ℝ)^53*c<=1/2^34*(2^52*d):=(by linarith only[h1]);exact n3.trans (n5.trans n4)
 . exfalso;norm_num at h;exact omS249 hM2 h
lemma qHs (s e:ὧ) (V:ℝ) (fs:s.isFinite=true) (fe:e.isFinite=true) (hrel: |R e| <=1/2^34* |R s|) (hB: |R s| <=2^(130:ℤ)) (hlo:R (s-e)<V) (hhi:V<R (s+e)): |R s-V| <=1/2^33*(|V| +A_z):=by
 have sck:∀{k a b:ℝ},0<k->a<=1/k*b->k*a<=b:=λhk h=>(le_div_iff₀' hk).mp (h.trans_eq (one_div_mul_eq_div _ _))
 have hA:(2:ℝ)^(130:ℤ)<=A_B:=by unfold A_B;exact ἒ (by norm_num1) (by norm_num1)
 have h1: |R e| <=A_B:=hrel.trans ((mul_le_of_le_one_left (Ἀ _) (by norm_num1)).trans (hB.trans hA))
 obtain ⟨-,p⟩:=qBf s e fs fe (hB.trans hA) h1;obtain ⟨-,m⟩:=qDf s e fs fe (hB.trans hA) h1
 have p':=sck (by norm_num1) p;have m':=sck (by norm_num1) m;have p2:=le_abs_self (R (s+e)-(R s+R e));have m2:=neg_abs_le (R (s-e)-(R s-R e))
 have q1:=Ἐ (R s) (R e);have q2:=Ὕ (R s) (R e);have a1: |R s| <= |V| + |R s-V|:=sub_le_iff_le_add'.mp (Ὥ (R s) V)
 have a2:=le_abs_self (R e);have a3:=neg_abs_le (R e);have hz:(0:ℝ)<=A_z:=by show (0:ℝ)<=(2:ℝ)^(-1021:ℤ);exact (ἵ two_pos _).le
 have hQ2:|2^53*(R s-V)| <=2^53* |R e| +(|R s| + |R e| +A_z):=abs_le.mpr ⟨by linarith only[a2,a3,q1,p',p2,hhi],by linarith only[a2,a3,q2,m',m2,hlo]⟩
 rw[ἢ,ἡ (by norm_num1:(0:ℝ)<2^53)] at hQ2
 have hrel2:=sck (by norm_num1) hrel
 rw[one_div_mul_eq_div,le_div_iff₀' (by norm_num1)]
 linarith only[hQ2,hz,a1,hrel2,Ἀ V]
lemma U_w (T M:Ὣ) (h1:T+2097152<M+2^28) (h2:M+2097152<=T+2^28):T<M+2^28/\M<=T+2^28:=by exact omS250 h1 h2
set_option maxRecDepth 8192 in
lemma qI4 (e s:ὧ) (h:zivT (thOf e) s≠Е):
 s.isFinite=true/\e.isFinite=true/\|R e| <=1/2^34* |R s|/\|R s| <=2^(130:ℤ)/\(2:ℝ)^(-127:ℤ)<= |R (s+e)|/\(2:ℝ)^(-127:ℤ)<= |R s|:=by
 obtain ⟨ht,S,ex,fr,M,h1,h2,h3,-,hbt,hl',hu'⟩:=kFh _ s h
 have hnr:B_near s.toBits.toNat S ex fr:=⟨M,hbt,(U_w _ _ hl' hu').1,(U_w _ _ hl' hu').2⟩
 obtain ⟨fs,v,hv,hv1,-,hv3⟩:=qA4 _ S ex fr h1 h2 h3 hnr
 obtain ⟨fe,hrel⟩:=kGw e s S M (by clear h ht hbt;exact omS251 h2 h3 hu') hbt ht
 have hb0:(2:ℝ)^(-179:ℤ)<=2^((ex:Int)-1076):=ἒ (by norm_num1) (by clear h ht h3 hbt hl' hu';exact omS124 h1)
 have hb1:(2:ℝ)^((ex:Int)-1076)<=2^(74:ℤ):=ἒ (by norm_num1) (by clear h ht h3 hbt hl' hu';exact omS252 h2)
 have hpos:(0:ℝ)<2^(-179:ℤ):=ἵ ו _
 have e179:(2:ℝ)^(-127:ℤ)=2^52*2^(-179:ℤ):=by
  rw[show (-127:ℤ)=52+(-179) by (norm_num1),Ἓ ἓ];(norm_num1)
 have e130:(2:ℝ)^(130:ℤ)=2^56*2^(74:ℤ):=by
  rw[show (130:ℤ)=56+74 by (norm_num1),Ἓ ἓ];(norm_num1)
 have hc0:(0:ℝ)<2^((ex:Int)-1076):=ἵ ו _
 have hv0:(2^53-2^28:ℝ)*2^(-179:ℤ)<=v:=(ἁ hb0 ו).trans hv3.le
 have hfr:(fr:ℝ)<=2^23:=by exact_mod_cast h3.le
 have hfr0:(0:ℝ)<=fr:=Ἔ _
 have hzs:A_z<=2^(-179:ℤ):=by
  have e':A_z=2^(-1021:ℤ):=rfl
  rw[e'];exact ἒ (by norm_num1) (by norm_num1)
 generalize (2:ℝ)^(-179:ℤ)=u at*
 have hvpos:0<v:=(by linarith only[hpos,hv0])
 have hRs: |R s| =v:=by
  rw[hv];cases S
  . simp only[false_eq_true, ↓reduceIte,ἡ hvpos]
  . simp only[↓reduceIte,abs_neg,ἡ hvpos]
 have hF:(2^53+(fr:ℝ)*2^30+2^29)<=2^56:=(by linarith only[hfr])
 have hB: |R s| <=2^(130:ℤ):=by
  rw[hRs,e130]
  exact hv1.trans ((ἀ hF hc0.le).trans (ἁ hb1 ו))
 have hA:(2:ℝ)^(130:ℤ)<=A_B:=by
  unfold A_B;exact ἒ (by norm_num1) (by norm_num1)
 have hR: |R e| <=1/2^34*v:=by rw[<-hRs];exact hrel
 have hvB:v<=A_B:=by rw[<-hRs];exact hB.trans hA
 have hfin: |R s| <=A_B:=by rw[hRs];exact hvB
 obtain ⟨-,p⟩:=qBf s e fs fe hfin (hR.trans (by linarith only[hfin,hvB,hA,h3]))
 have q2: |R s| - |R e| <= |R s+R e|:=by
  have:=Ὥ (R s) (-R e)
  rwa[abs_neg,sub_neg_eq_add] at this
 have q1:=Ἐ (R s) (R e)
 have q3:=Ὥ (R s+R e) (R (s+e))
 rw[ἣ] at q3
 rw[hRs] at q1 q2
 refine ⟨fs,fe,hrel,hB,?_,?_⟩
 . rw[e179];linarith only[Ἀ (R e),hv0,hpos,hR,q1,q2,q3,p,hzs,hvpos]
 . rw[hRs,e179];linarith only[hv0,hR,q2,q1]
lemma qJr (s:ὧ) (V:ℝ) (hV: |R s-V| <=1/2^33*(|V| +A_z)) (h2:(2:ℝ)^(-127:ℤ)<= |R s|):V≠0:=by rintro rfl;rw[abs_zero,zero_add,sub_zero] at hV;have e':A_z=2^(-1021:ℤ):=rfl;have hle:(2:ℝ)^(-1021:ℤ)<=2^(-127:ℤ):=ἒ (by norm_num1) (by norm_num1);have h0:(0:ℝ)<2^(-1021:ℤ):=ἵ ו _;rw[e'] at hV;linarith only[h0,hle,hV,h2]
lemma kD1 (lo t:Ὠ) (h:t-lo<spOf lo):lo<=t/\t<0x8FA0000000000000:=by
 unfold spOf at h;split at h<;>rename_i hl
 . have a:=lo.toNat_lt;have b:=t.toNat_lt;rw[ἴ,ὲ,ὲ] at h;rw[ἴ] at hl;norm_num at h hl a b;constructor;. rw[Ὤ];(clear a;exact omS253 hl h)
   . rw[ἴ];norm_num;(clear a;exact omS254 hl h b)
 . exact absurd h ג
lemma qIj (a:Ὠ) (h0:(0x7040000000000000:Ὠ)<=a<<<(1:Ὠ)) (h1:a<<<(1:Ὠ)<0x8FA0000000000000) (h2:(a+0x0FDFFFFF)&&&0x1FFFFFFF<0x1FC00000):
 zivW (a+0x200000) (a-0x200000)=zivC ((a+0x0FDFFFFF)>>>(29:Ὠ))/\zivC ((a+0x0FDFFFFF)>>>(29:Ὠ))≠Е:=by
 have ha:=a.toNat_lt
 have e0:(a<<<(1:Ὠ)).toNat=Ѧa*2%2^64:=by
  rw[UInt64.toNat_shiftLeft];norm_num[ὁ]
 rw[Ὤ,e0] at h0
 rw[ἴ,e0] at h1
 norm_num at h0 h1 ha
 have eu:(a+0x0FDFFFFF).toNat=Ѧa+266338303:=by
  rw[ἤ];norm_num;(clear h2;exact omS255 h1 ha)
 have em:((a+0x0FDFFFFF)&&&0x1FFFFFFF).toNat=(Ѧa+266338303)%2^29:=by
  rw[Ὅ,eu]
  have:=Ἂ (Ѧa+266338303) 29
  norm_num at this⊢
  exact this
 rw[ἴ,em] at h2
 norm_num at h2
 generalize hx:(a+0x0FDFFFFF)>>>(29:Ὠ)=x
 have ex:Ѧx=(Ѧa+266338303)/2^29:=by
  rw[<-hx,ἰ,eu];norm_num[Ἱ]
 have e1:(a+0x200000+0x0FFFFFFF)>>>(29:Ὠ)=x:=by
  apply UInt64.toNat_inj.mp
  rw[ex,ἰ,ἤ,ἤ]
  norm_num[Ἱ]
  (clear hx;exact omS256 h1 ha h2)
 have e2:(a-0x200000+0x0FFFFFFF)>>>(29:Ὠ)=x:=by
  apply UInt64.toNat_inj.mp
  rw[ex,ἰ,ἤ,ὲ]
  norm_num[Ἱ]
  (clear hx e1;exact omS257 h1 ha)
 have hE:((x>>>(23:Ὠ))&&&(0x7FF:Ὠ)).toNat=Ѧx/2^23%2^11:=by
  rw[Ὅ,ἰ]
  have:=Ἂ (Ѧx>>>23) 11
  norm_num[Ἱ] at this⊢
  exact this
 have r1:898<=Ѧx/2^23%2^11:=by rw[ex];norm_num;(clear ha hx e1 e2 hE;exact omS258 h0 h1)
 have r2:Ѧx/2^23%2^11<1151:=by rw[ex];norm_num;(clear ha hx e1 e2 hE r1;exact omS259 h1)
 have r3:Ѧx<2^35:=by rw[ex];norm_num;(clear hx e1 e2 hE r1 r2;exact omS260 h1 ha)
 constructor
 . unfold zivW zivC;dsimp only;rw[e1,e2];have hc:((x>>>(23:Ὠ))&&&(0x7FF:Ὠ))-(897:Ὠ)<(254:Ὠ):=by rw[ἴ,ὲ,hE];norm_num;norm_num at r1 r2;(clear e0 h0 h1 ha eu em h2 hx ex e1 e2 hE r3;exact omS261 r1 r2)
   simp only[BEq.rfl,hc,decide_true,Bool.and_self, ↓reduceIte,UInt64.toUInt32_or,UInt64.reduceLT,UInt64.toUInt32_shiftLeft,UInt64.toUInt32_ofNat,UInt64.toUInt32_sub,UInt64.toUInt32_and]
 . unfold zivC;rw[show ((x>>>(34:Ὠ))<<<(31:Ὠ)) ||| ((x&&&(0x3FFFFFFFF:Ὠ))-((896:Ὠ)<<<(23:Ὠ)))=((x>>>34)<<<31) ||| ((x&&&(0x3FFFFFFFF:Ὠ))-((896:Ὠ)<<<23)) from rfl,qHw x r3 (by clear e0 h0 h1 ha eu em h2 hx ex e1 e2 r3;(exact Nat.le_of_lt ‹_›)) r2];intro h;have h':=congrArg UInt32.toNat h;rw[ὡ] at h';have f:=ὀ Ѧx (ב:0<2^23);split at h'<;>norm_num at h' f r1 r2<;>(clear e0 h0 h1 ha eu em h2 hx ex e1 e2 hE r3 h f;omega)
lemma qHy (e s:ὧ) (h1:(s.toBits<<<(1:Ὠ))-loOf e<spOf (loOf e)) (h2:(s.toBits+0x0FDFFFFF)&&&0x1FFFFFFF<0x1FC00000):
 zivT (thOf e) s=zivC ((s.toBits+0x0FDFFFFF)>>>(29:Ὠ))/\zivT (thOf e) s≠Е:=by
 obtain ⟨l1,l2⟩:=kD1 _ _ h1
 have l3:thOf e<=s.toBits<<<(1:Ὠ)/\(0x7040000000000000:Ὠ)<=s.toBits<<<(1:Ὠ):=by
  unfold loOf at l1
  dsimp only at l1
  rw[Ὤ] at l1⊢
  split at l1<;>rename_i hc
  . rw[ἴ] at hc;constructor;. (clear h1 h2 l2;exact omS262 hc l1)
    . exact l1
  . rw[ἴ] at hc;constructor;. exact l1
    . rw[Ὤ];(clear h1 h2 l2;exact omS263 hc l1)
 have b:=qIj s.toBits l3.2 l2 h2
 have e1:zivT (thOf e) s=zivC ((s.toBits+0x0FDFFFFF)>>>(29:Ὠ)):=by
  unfold zivT
  dsimp only
  rw[ἱ l3.1]
  exact b.1
 exact ⟨e1,e1▸b.2⟩
lemma qCnF (n:Ὣ) (hn:n<2^20) (xw ww:Ὣ->ὥ) (bw:ὥ) (hx:∀t,t<n->good (xw t)=true) (hw:∀t,t<n->good (ww t)=true) (hb:good bw=true) (s e E:ὧ) (hlo:R (s-e)<exactS n xw ww bw) (hhi:exactS n xw ww bw<R (s+e)) (fe:e.isFinite=true) (hE: |R e| <= |R E|) (h1:(s.toBits<<<(1:Ὠ))-loOf E<spOf (loOf E)) (h2:(s.toBits+0x0FDFFFFF)&&&0x1FFFFFFF<0x1FC00000):
 zivC ((s.toBits+0x0FDFFFFF)>>>(29:Ὠ))=dotOf (n+1) (λt=>if t<n then xw t else oneW) (λt=>if t<n then ww t else bw):=by
 have hsum:=qHv n xw ww bw hx hw hb
 obtain ⟨a,b⟩:=qHy E s h1 h2
 obtain ⟨fs,-,hr,hB,-,hs1⟩:=qI4 E s b
 have hV:=qHs s e _ fs fe (hE.trans hr) hB hlo hhi
 obtain ⟨d,hd,hw'⟩:=qEc (n+1) (λt=>if t<n then xw t else oneW)
  (λt=>if t<n then ww t else bw) (λj _=>by
   by_cases hj:j<n
   . simp only[hj,ite_true];exact ⟨qAe _ (hx j hj),qAe _ (hw j hj)⟩
   . simp only[hj,ite_false]
     exact ⟨ה,qAe _ hb⟩) λh0=>qJr s _ hV hs1 (by rw[<-hsum,h0,Rat.cast_zero])
 have q:(d.toRat:ℝ)=exactS n xw ww bw:=by rw[hd,hsum]
 rw[hw',<-a]
 exact kHv _ s d (by rw[q];exact hV) b
set_option maxRecDepth 8192 in
lemma qBgF (E s e:ὧ) (h1:(s.toBits<<<(1:Ὠ))-loOf E<spOf (loOf E)) (h2:(s.toBits+0x0FDFFFFF)&&&0x1FFFFFFF<0x1FC00000) (fe:e.isFinite=true) (hE: |R e| <= |R E|):(2:ℝ)^(-127:ℤ)<= |R (s+e)|:=by
 obtain ⟨-,h⟩:=qHy E s h1 h2
 obtain ⟨fs,-,hr,hB,-,-⟩:=qI4 E s h
 obtain ⟨-,S,ex,fr,M,a1,a2,a3,-,hb,hl,hu⟩:=kFh _ s h
 have hlo:=(kD1 _ _ h1).1
 have hl0:(0x7040000000000000:Ὠ)<=loOf E:=by
  unfold loOf;dsimp only;split
  . exact UInt64.le_refl _
  . rename_i hh;exact UInt64.not_lt.mp hh
 have hM63:M<2^63:=by clear h h1 h2 hlo hl0;exact omS264 a2 a3 hu
 have hsh:=qJb s.toBits S M hM63 hb
 have h898:0x7040000000000000<=2*M:=by have:=Ὤ.mp (UInt64.le_trans hl0 hlo);rw[hsh] at this;exact this
 have hM:M=(M/2^52)*2^52+M%2^52:=(by clear h h1 h2 hb hl hu hlo hl0 hsh h898;exact omS114)
 have hF:M%2^52<2^52:=ὀ _ ו
 generalize M/2^52=E' at hM
 generalize M%2^52=F at hM hF
 subst hM
 have hE1:898<=E':=by clear h h1 h2 hb hl hu hlo hl0 hsh;exact omS265 hF h898
 have hE2:E'<2047:=by clear h h1 h2 hb hl hlo hl0 hsh;exact omS239 a2 a3 hu
 obtain ⟨-,hR⟩:=B_R s S E' F (by exact omS266 hE1) hE2 hF (by rw[hb,ὢ])
 have hk:(2:ℝ)^(-125:ℤ)=4*2^(-127:ℤ):=by rw[show (-125:ℤ)=2+(-127) by norm_num,Ἓ ἓ];norm_num
 have hs125:(2:ℝ)^(-125:ℤ)<= |R s|:=by
  rw[hR]
  have p:(0:ℝ)<2^((E':ℤ)-1075):=ו
  have q2:(2:ℝ)^(-125:ℤ)<=2^(52:ℤ)*2^((E':ℤ)-1075):=by
   rw[<-Ἓ ב];exact ἒ ב (by exact omS267 hE1)
  have q3:(2^(52:ℤ):ℝ)<=((2^52+F:Ὣ):ℝ):=by push_cast;norm_num
  have q4:(2:ℝ)^(52:ℤ)*2^((E':ℤ)-1075)<=((2^52+F:Ὣ):ℝ)*2^((E':ℤ)-1075):=ἀ q3 p.le
  cases S<;>simp only[Bool.false_eq_true,ite_false,ite_true,one_mul,neg_mul,abs_neg,ἢ,ἡ p]<;>rw[ἅ ו]<;>linarith only[q2,q4]
 have heb: |R e| <=1/2^34* |R s|:=hE.trans hr
 have hA:(2:ℝ)^(130:ℤ)<=A_B:=by unfold A_B;exact ἒ (by norm_num1) (by norm_num1)
 have hsA: |R s| <=A_B:=hB.trans hA
 have heA: |R e| <=A_B:=heb.trans (ὂ (by linarith only[Ἀ (R s)]) hsA)
 obtain ⟨-,hadd⟩:=qBf s e fs fe hsA heA
 have hzk:A_z<=(2:ℝ)^(-127:ℤ):=by show (2:ℝ)^(-1021:ℤ)<=_;exact ἒ ב ב
 have t0:=Ὥ (R s) (-(R e))
 rw[sub_neg_eq_add,abs_neg] at t0
 have t1: |R s| *(1-1/2^34)<= |R s+R e|:=by linarith only[le_abs_self (|R s|-|R e|),t0,heb]
 have t3:=Ὥ (R s+R e) (R (s+e))
 rw[ἣ (R s+R e) (R (s+e))] at t3
 linarith only[le_abs_self (|R s+R e|-|R (s+e)|),t3,hadd,hzk,t1,hs125,hk,Ἀ (R s+R e)]
lemma qAddE (x y:ὧ) (hx:xᶠ) (hy:yᶠ) (h1: |R x| <=A_B) (h2: |R y| <=A_B):(x+y)ᶠ /\R (x+y)=Model.roundAt ⟡ (R x+R y):=
 A_E _ (toModel_ofFloat_add_of_isFinite x y hx hy) (Model.isFinite_add_of_abs_add_le_posMaxFinite _ _ qAj (qA3 hx) (qA3 hy) (qEe h1 h2))
  (Model.toReal_add_eq_roundAt_of_abs_add_le_posMaxFinite _ _ qAj (qA3 hx) (qA3 hy) (qEe h1 h2))
set_option maxRecDepth 8192 in
lemma qEmo (nf X v bm:ὧ) (fnf:nfᶠ) (fX:Xᶠ) (fv:vᶠ) (fbm:bmᶠ) (pnf:0<=R nf) (pX:0<=R X) (hvb: |R v| <=R bm) (bX:R X<=2^(290:ℤ)) (bb:R bm<=2^(290:ℤ)) (bnf:R nf<=1):
 (nf*(X+v.abs))ᶠ /\ |R (nf*(X+v.abs))| <= |R (nf*(X+bm))| :=by
 obtain ⟨fa,Ra⟩:=qBl v fv
 have hA:(2:ℝ)^(290:ℤ)<=A_B:=by unfold A_B;exact ἒ (by norm_num1) (by norm_num1)
 have pb:0<=R bm:=(Ἀ _).trans hvb
 obtain ⟨f1,R1⟩:=qAddE X v.abs fX fa (by rw[ἅ pX];exact bX.trans hA) (by rw[Ra,abs_abs];exact (hvb.trans bb).trans hA)
 obtain ⟨f2,R2⟩:=qAddE X bm fX fbm (by rw[ἅ pX];exact bX.trans hA) (by rw[ἅ pb];exact bb.trans hA)
 have m1:R (X+v.abs)<=R (X+bm):=by rw[R1,R2,Ra];exact Model.roundAt_mono _ (by linarith only[hvb])
 have p1:0<=R (X+v.abs):=by rw[R1,<-Model.roundAt_zero ⟡];exact Model.roundAt_mono _ (by rw[Ra];linarith only[pX,Ἀ (R v)])
 have b2:R (X+bm)<=2^(292:ℤ):=by
  obtain ⟨-,e2⟩:=qFc ⟨f2,R2⟩
  rw[ἅ (add_nonneg pX pb)] at e2
  have r1:=sub_le_iff_le_add'.mp ((le_abs_self _).trans e2)
  have hz:=qBo
  have q6:(1:ℝ)<=2^(290:ℤ):=one_le_zpow₀ (by norm_num1) (by norm_num1)
  rw[show (2:ℝ)^(292:ℤ)=4*2^(290:ℤ) by rw[show (292:ℤ)=2+290 by norm_num1,Ἓ ἓ];norm_num]
  generalize (2:ℝ)^(290:ℤ)=P at*
  linarith only[r1,bX,bb,q6,hz,pX,pb]
 have bp:|R nf| * |R (X+bm)|<=2^(1000:ℤ):=by
  rw[ἅ pnf,ἅ (p1.trans m1)]
  have:(2:ℝ)^(292:ℤ)<=2^(1000:ℤ):=ἒ ב ב
  have hm:=ἀ bnf (p1.trans m1)
  exact hm.trans ((one_mul _).le.trans (b2.trans this))
 have bp1:|R nf| * |R (X+v.abs)|<=2^(1000:ℤ):=by
  refine ὂ ?_ bp;rw[ἅ p1,ἅ (p1.trans m1)];exact ἁ m1 (Ἀ _)
 obtain ⟨f3,R3⟩:=qEd nf (X+v.abs) fnf f1 bp1
 obtain ⟨-,R4⟩:=qEd nf (X+bm) fnf f2 bp
 have m2:R (nf*(X+v.abs))<=R (nf*(X+bm)):=by rw[R3,R4];exact Model.roundAt_mono _ (ἁ m1 pnf)
 have p2:0<=R (nf*(X+v.abs)):=by rw[R3,<-Model.roundAt_zero ⟡];exact Model.roundAt_mono _ (Ἆ pnf p1)
 exact ⟨f3,by rw[ἅ p2,ἅ (p2.trans m2)];exact m2⟩
set_option maxRecDepth 8192 in
lemma qFoldA (g:ℕ->ὧ): ∀(l:List ℕ) (c:ὧ),(∀t∈l,(g t)ᶠ /\ |R (g t)| <=2^(128:ℤ))->cᶠ ->0<=R c->R c<=2^(150:ℤ)-(ѣl:ℝ)*2^(129:ℤ)->
  (l.foldl (λ(a:ὧ) t=>a+(g t).abs) c)ᶠ /\0<=R (l.foldl (λ(a:ὧ) t=>a+(g t).abs) c)/\R (l.foldl (λ(a:ὧ) t=>a+(g t).abs) c)<=2^(150:ℤ):=by
 intro l
 induction l with
 | nil=>intro c _ fc p b;simp only[List.foldl_nil,List.length_nil,Nat.cast_zero,zero_mul,sub_zero] at b⊢;exact ⟨fc,p,b⟩
 | cons t l ih=>
  intro c hg fc p b
  simp only[ὔ]
  obtain ⟨ft,bt⟩:=hg t (ὣ ..)
  obtain ⟨fa,Ra⟩:=qBl (g t) ft
  have hA:(2:ℝ)^(150:ℤ)<=A_B:=by unfold A_B;exact ἒ (by norm_num1) (by norm_num1)
  have hA2:(2:ℝ)^(128:ℤ)<=A_B:=by unfold A_B;exact ἒ (by norm_num1) (by norm_num1)
  have h129:(0:ℝ)<=2^(129:ℤ):=(ἵ two_pos _).le
  have hl:(0:ℝ)<=(ѣl:ℝ):=Ἔ _
  simp only[List.length_cons,Nat.cast_add,Nat.cast_one] at b
  have bc:R c<=2^(150:ℤ):=b.trans (sub_le_self _ (Ἆ (add_nonneg hl zero_le_one) h129))
  obtain ⟨f1,R1⟩:=qAddE c (g t).abs fc fa (by rw[ἅ p];exact bc.trans hA) (by rw[Ra,abs_abs];exact bt.trans hA2)
  obtain ⟨-,e1⟩:=qFc ⟨f1,R1⟩
  have p1:0<=R (c+(g t).abs):=by rw[R1,<-Model.roundAt_zero ⟡];exact Model.roundAt_mono _ (by rw[Ra];linarith only[p,Ἀ (R (g t))])
  refine ih _ (λu hu=>hg u (ὖ _ hu)) f1 p1 ?_
  have hz:A_z<=1:=by show (2:ℝ)^(-1021:ℤ)<=1;exact zpow_le_one_of_nonpos₀ ב ב
  have q1:(2:ℝ)^(129:ℤ)=2*2^(128:ℤ):=by rw[show (129:ℤ)=1+128 by norm_num,Ἓ ἓ];norm_num
  have q2:(2:ℝ)^(150:ℤ)=2^22*2^(128:ℤ):=by rw[show (150:ℤ)=22+128 by norm_num,Ἓ ἓ];norm_num
  have q3:(1:ℝ)<=2^(128:ℤ):=one_le_zpow₀ ב ב
  rw[Ra,ἅ (show (0:ℝ)<=R c+|R (g t)| by linarith only[p,Ἀ (R (g t))])] at e1
  have:=le_abs_self (R (c+(g t).abs)-(R c+|R (g t)|))
  have e1':=(le_div_iff₀' (by norm_num1:(0:ℝ)<2^53)).mp (e1.trans_eq (one_div_mul_eq_div _ _))
  linarith only[e1',this,b,bt,hz,q1,q2,q3,p,Ἀ (R (g t)),hl]
section
variable {Rr I O:Ὣ} {X W b:ё}
set_option maxRecDepth 8192 in
lemma qDuF (G:C_G Rr I O X W b) (Y:Ὡ) {i j:Ὣ} (hi:i<Rr) (hj:j<O) {s:ὧ} {hv:Ὢ} (g1:good hv=true) (g2:hv<=В) (g4:∀t,t<I->W[j*I+t]!&&&В<=hv) (L:љ) (hl:L.Nodup) (hl2:∀t∈L,t<I) (hz:∀t,t<I->t∉L->X[i*I+t]!&&&В=0) (hs:s=L.foldl (λ(acc:ὧ) t=>acc+dv X[i*I+t]!*dv W[j*I+t]!) 0)
 (bm:ὧ) (fbm:bmᶠ) (hbm: |R (dv b[j]!)| <=R bm) (bb:R bm<=2^(290:ℤ)) (lo:Ὠ) (hlo:lo=loOf (nfOf I*(dv hv*(L.foldl (λ(a:ὧ) t=>a+(dv X[i*I+t]!).abs) 0)+bm))):
 (if fin1 lo (spOf lo) (s+dv b[j]!)==Е then fin8 I Y X W b i j s (nfOf I) (L.foldl (λ(a:ὧ) t=>a+(dv X[i*I+t]!).abs) 0) hv b[j]! ѩI else fz Y (fin1 lo (spOf lo) (s+dv b[j]!)))=C_out I Y X W b i j:=by
 split
 . exact qDu G Y hi hj g1 g2 g4 L hl hl2 hz hs
 . rename_i hne
   have hgx:=G.gx hi
   have hgw:=G.gw hj
   have gb:=G.gb j (by rw[<-G.hb];exact hj)
   have hlen:ѣL<=I:=qFu hl hl2
   have ni:=G.ni
   obtain ⟨fA,pA,bA⟩:=qFoldA (λt=>dv X[i*I+t]!) L 0 (λt ht=>⟨(qAm _ (hgx t (hl2 t ht))).1,by rw[(qAm _ (hgx t (hl2 t ht))).2];exact qAo _ (hgx t (hl2 t ht))⟩) qHg.1 (by rw[qHg.2]) (by
    have:(ѣL:ℝ)<=2^20:=by exact_mod_cast (by exact omS268 hlen ni:ѣL<=2^20)
    have q:(2:ℝ)^(150:ℤ)=2^21*2^(129:ℤ):=by rw[show (150:ℤ)=21+129 by norm_num,Ἓ ἓ];norm_num
    have q2:(0:ℝ)<2^(129:ℤ):=ἵ two_pos _
    rw[qHg.2,q,sub_nonneg];exact ἀ (this.trans (by norm_num1)) q2.le)
   generalize hA:L.foldl (λ(a:ὧ) t=>a+(dv X[i*I+t]!).abs) 0=A at hlo fA pA bA⊢
   obtain ⟨fh,Rh⟩:=qAm hv g1
   obtain ⟨ph,bh,-⟩:=qCs g1 g2
   have hp:|R (dv hv)| * |R A| <=2^(1000:ℤ):=by
    rw[Rh,ἅ ph,ἅ pA]
    have:(2:ℝ)^128*2^(150:ℤ)<=2^(1000:ℤ):=by rw[show (2:ℝ)^128=2^(128:ℤ) by norm_num,<-Ἓ ἓ];exact ἒ ב ב
    exact (ἑ bh bA pA ו).trans this
   obtain ⟨fX,RX⟩:=qEd (dv hv) A fh fA hp
   have pX:0<=R (dv hv*A):=by rw[RX,<-Model.roundAt_zero ⟡];exact Model.roundAt_mono _ (by rw[Rh];exact Ἆ ph pA)
   have bX:R (dv hv*A)<=2^(290:ℤ):=by
    obtain ⟨-,e⟩:=qFc ⟨fX,RX⟩
    have c0:R (dv hv)*R A<=2^(278:ℤ):=by
     have:(2:ℝ)^128*2^(150:ℤ)=2^(278:ℤ):=by rw[show (2:ℝ)^128=2^(128:ℤ) by norm_num,<-Ἓ ἓ];norm_num
     rw[Rh,<-this];exact ἑ bh bA pA ו
    have c1:(0:ℝ)<=R (dv hv)*R A:=by rw[Rh];exact Ἆ ph pA
    have hz:A_z<=1:=by show (2:ℝ)^(-1021:ℤ)<=1;exact zpow_le_one_of_nonpos₀ ב ב
    have q1:(2:ℝ)^(290:ℤ)=2^12*2^(278:ℤ):=by rw[show (290:ℤ)=12+278 by norm_num,Ἓ ἓ];norm_num
    have q2:(1:ℝ)<=2^(278:ℤ):=one_le_zpow₀ ב ב
    rw[ἅ c1] at e
    have:=le_abs_self (R (dv hv*A)-R (dv hv)*R A)
    rw[q1];clear hp;generalize (2:ℝ)^(278:ℤ)=P at*;generalize R (dv hv)*R A=Q at*;linarith only[e,this,c0,c1,hz,q2]
   obtain ⟨fnf,Rnf⟩:=qIn I ni
   have pnf:0<=R (nfOf I):=by rw[Rnf];exact div_nonneg (add_nonneg (Ἔ _) (by norm_num1)) (by norm_num1)
   have bnf:R (nfOf I)<=1:=by
    have hI:(I:ℝ)<2^20:=by exact_mod_cast ni
    rw[Rnf,div_le_one ו];linarith only[hI]
   obtain ⟨fv,-⟩:=qAm _ gb
   obtain ⟨fe,hE⟩:=qEmo (nfOf I) (dv hv*A) (dv b[j]!) bm fnf fX fv fbm pnf pX hbm bX bb bnf
   have key:fin1 lo (spOf lo) (s+dv b[j]!)≠Е:=by simpa only[ne_eq,beq_iff_eq] using hne
   unfold fin1 at key⊢
   dsimp only at key⊢
   split at key
   . rename_i ht
     simp only[Bool.and_eq_true,decide_eq_true_eq] at ht
     obtain ⟨h1,h2⟩:=ht
     rw[ἱ (by simp only[Bool.and_eq_true,decide_eq_true_eq];exact ⟨h1,h2⟩)]
     subst hlo
     have hbig:=qBgF _ _ _ h1 h2 fe hE
     obtain ⟨c1,c2⟩:=qC8 I ni (λt=>X[i*I+t]!) (λt=>W[j*I+t]!) b[j]! hv hgx hgw gb g1 g2 L hl hl2 (λt ht=>g4 t (hl2 t ht)) hz (s+dv b[j]!) (nfOf I*(dv hv*A+(dv b[j]!).abs)) ⟨by rw[hs],by rw[hA]⟩ hbig
     have e:=qCnF I ni _ _ _ hgx hgw gb _ _ _ c1 c2 fe hE h1 h2
     obtain ⟨a,b'⟩:=qHy _ _ h1 h2
     obtain ⟨-,S,ex,fr,M,a1,a2,a3,hz',-⟩:=kFh _ _ b'
     rw[a] at hz'
     have he:=qR1 _ S ex fr a1 a2 a3 hz'
     unfold fz C_out
     rw[qR0 Y _ he,e]
   . exact absurd rfl key
end
section
variable {R I O:Ὣ} {X W b:ё}
section
variable (G:C_G R I O X W b)
include G
lemma C_G.nm:I.toUSize.toNat=I:=qAd _ (lt_trans G.ni (by norm_num1))
lemma C_G.wj {j:Ὣ} {u:ὦ} (hu:Ѥu=j)
 (hj:j<O):(u*ѩI).toNat=j*I/\(u*ѩI).toNat+I.toUSize.toNat<=ѧW:=by
 have h1:j*I+I<=O*I:=qBc hj (Ὗ _)
 have h2:=G.hW
 have h3:=G.sW
 have e:(u*ѩI).toNat=j*I:=by
  rw[qBa _ _ (by rw[hu,G.nm];(clear hu hj;exact omS269 h1 h2 h3)),hu,G.nm]
 exact ⟨e,by rw[e,G.nm];(clear hu hj h3;exact omS270 h1 h2)⟩
lemma qB9 (Y:Ὡ) {i j:Ὣ} (hi:i<R) (hj:j<O) {hv:Ὢ} (Hv:C_Hv W hv) {ix:ѐ} {x:ὤ} {c:ὦ} {L:љ} {l:ὧ} (Rw:C_Row I X i ix x c L l) {u:ὦ} (hu:Ѥu=j) (hx:Ѥc<=Ѩx) (hi':Ѥc<=ѧix) (hs:ѧW<2^32) (h0:(u*ѩI).toNat+I.toUSize.toNat<=ѧW) (bm:ὧ) (fbm:bmᶠ) (hbm:∀j,j<O-> |D.R (dv b[j]!)| <=D.R bm) (bb:D.R bm<=2^(290:ℤ)) {lo:Ὠ} (hlo:lo=loOf (nfOf I*(dv hv*l+bm))):
 fin8L I Y X W b i j (go1 x ix W (u*ѩI) ѩI c 0 hx hi' hs h0 0) (nfOf I) l hv lo (rdW b u) ѩI=C_out I Y X W b i j:=by
 have e:=qH6 hx hi' hs h0 (λp=>dv X[i*I+p]!) (λp=>W[j*I+p]!) (λp _=>by rw[(G.wj hu hj).1]) L 0 0 (by rw[C_z,Ἇ];exact Rw.len.symm)
  (λk p hk=>by
   have h1:=Rw.rd k p hk
   rw[C_z,Ἇ,G.nm]
   exact ⟨Rw.lt p (List.mem_of_getElem? hk),h1⟩)
 rw[qAg _ _ G.sb,hu,Rw.ok]
 rw[Rw.ok] at hlo
 unfold fin8L fin8n;dsimp only
 exact qDuF G Y hi hj Hv.1 Hv.2.1 (λt ht=>Hv.2.2.2 _ (by rw[<-G.hW];exact qCj hj ht)) L Rw.nd Rw.lt Rw.hz e bm fbm (hbm j hj) bb lo hlo
lemma qD0 (Y:Ὡ) {i:Ὣ} (hi:i<R) {hv:Ὢ} (Hv:C_Hv W hv) {ix:ѐ} {x wd:ὤ} {c:ὦ} {L:љ} {l:ὧ} (Rw:C_Row I X i ix x c L l) {base:ὦ} (hb:Ѥbase=i*O)
 (Hd:C_Dw wd W) (bm:ὧ) (fbm:bmᶠ) (hbm:∀j,j<O-> |D.R (dv b[j]!)| <=D.R bm) (bb:D.R bm<=2^(290:ℤ)) {lo:Ὠ} (hlo:lo=loOf (nfOf I*(dv hv*l+bm))) (d:Ὣ):∀(j:Ὣ) (out:ё),j+d=O->C_Inv (R*O) (C_sp I O Y X W b) (λq=>q< i*O+j) out->
  C_Inv (R*O) (C_sp I O Y X W b) (λq=>q< i*O+O) (cols1 I O Y X W b i ix x wd (nfOf I) l hv lo ѩI c j base out):=by
 induction d using Nat.strong_induction_on with
 | _ d ih=>
  intro j out he H
  have hN:=G.sO
  have sW:ѧW<=4294967295:=by have:=G.sW;(exact Nat.le_of_lt_succ ‹_›)
  have hbs:=G.sb
  have hob:=G.hb
  have hle:=Rw.cle
  have h1c:Ѥc<=Ѩx:=by rw[Rw.len];exact hle.trans Rw.sx.1
  have h2c:Ѥc<=ѧix:=by rw[Rw.len];exact hle.trans Rw.six
  have hsp:∀k,j+k<O->C_out I Y X W b i (j+k)=C_sp I O Y X W b (i*O+j+k):=λk hk=>by
   rw[ὢ];exact (qCi I O Y X W b hk).symm
  by_cases h4:j+4<=O
  . have hP:=qCj hi ((by exact Nat.succ_le_of_lt ‹_›):j+3<O);have e0:j.toUSize.toNat=j:=qAd _ (by clear hi sW hbs hle h1c h2c h4;exact omS271 hN hP);obtain ⟨e1,e2,e3,-⟩:=qFk e0 (by clear hi sW hbs hle h1c h2c h4;exact omS272 hN hP);have g0:=(G.wj e0 (by clear hi hb hN sW hbs hle h1c h2c hP;exact omS152 h4)).2;have g1:=(G.wj e1 (by clear hi hb hN sW hbs hle h1c h2c hP g0;exact omS153 h4)).2;have g2:=(G.wj e2 (by exact Nat.le_of_lt ‹_›)).2;have g3:=(G.wj e3 (by exact Nat.succ_le_of_lt ‹_›)).2;have a0:(base+ѩj).toNat=i*O+j:=by rw[qAf _ _ (by rw[hb,e0];(clear hi hb sW hbs hle h1c h2c h4 g0 g1 g2 g3;exact omS273 hN hP)),hb,e0]
    have o0:=qB9 G Y hi ((by clear hi hb hN sW hbs hle h1c h2c hP g0 g1 g2 g3 a0;exact omS152 h4):j<O) Hv Rw e0 h1c h2c G.sW g0 bm fbm hbm bb hlo;have o1:=qB9 G Y hi ((by clear hi hb hN sW hbs hle h1c h2c hP g0 g1 g2 g3 a0 o0;exact omS153 h4):j+1<O) Hv Rw e1 h1c h2c G.sW g1 bm fbm hbm bb hlo;have o2:=qB9 G Y hi ((by exact Nat.le_of_lt ‹_›):j+2<O) Hv Rw e2 h1c h2c G.sW g2 bm fbm hbm bb hlo;have o3:=qB9 G Y hi ((by exact Nat.succ_le_of_lt ‹_›):j+3<O) Hv Rw e3 h1c h2c G.sW g3 bm fbm hbm bb hlo;rw[cols1,Ἄ h4];dsimp only;rw[Ἄ ⟨h1c,h2c,sW,g0,g1,g2,g3⟩]
    rw[qH0 _ _ _ _ _ _ _ (λj hj=>by
     rw[Rw.len] at hj;have hp:=List.getElem?_eq_getElem hj
     exact ⟨_,G.gx hi _ (Rw.lt _ (List.mem_of_getElem? hp)),(Rw.rd j _ hp).2⟩) Ѥc 0
     _ _ _ _ (Ἇ _)]
    dsimp only;rw[o0,o1,o2,o3];refine ih (d-4) (by clear hi hb hN sW hbs hle h1c h2c hP g0 g1 g2 g3 a0 o0 o1 o2 o3;exact omS274 he h4) (j+4) _ (by clear hi hb hN sW hbs hle h1c h2c hP g0 g1 g2 g3 a0 o0 o1 o2 o3;exact omS275 he h4) ?_;rw[show i*O+(j+4)=i*O+j+4 by (clear hi hb he hN sW hbs hob hle h1c h2c h4 hP e0 e1 e2 e3 g0 g1 g2 g3 a0 o0 o1 o2 o3;exact omS276)];exact qEq hN H a0 (by exact Nat.succ_le_of_lt ‹_›) (hsp 0 (by clear hi hb hN sW hbs hle h1c h2c hP g0 g1 g2 g3 a0 o0 o1 o2 o3;exact omS277 h4)) (hsp 1 (by clear hi hb hN sW hbs hle h1c h2c hP g0 g1 g2 g3 a0 o0 o1 o2 o3;exact omS153 h4)) (hsp 2 (by exact Nat.le_of_lt ‹_›)) (hsp 3 (by exact Nat.succ_le_of_lt ‹_›))
  . by_cases h1:j<O
    . have hP:=qCj hi h1;have e0:j.toUSize.toNat=j:=qAd _ (by clear hi sW hbs hle h1c h2c h4 h1;exact omS278 hN hP);have g0:=(G.wj e0 h1).2;have a0:(base+ѩj).toNat=i*O+j:=by rw[qAf _ _ (by rw[hb,e0];(clear hi hb sW hbs hle h1c h2c h4 h1 g0;exact omS279 hN hP)),hb,e0]
      have o0:=qB9 G Y hi h1 Hv Rw e0 h1c h2c G.sW g0 bm fbm hbm bb hlo;rw[cols1,ἔ h4,Ἄ h1];dsimp only;rw[Ἄ ⟨h1c,h2c,sW,g0⟩];rw[o0];exact ih (d-1) (by clear hi hb hN sW hbs hle h1c h2c hP g0 a0 o0;exact omS280 he h1) (j+1) _ (by clear hi hb hN sW hbs hle h1c h2c hP g0 a0 o0;exact omS281 he h1) (qAp hN H a0 hP (hsp 0 (by exact Nat.succ_le_of_lt ‹_›)))
    . rw[cols1,ἔ h4,ἔ h1];have e:j=O:=(by clear hi hb hN sW hbs hob hle h1c h2c h4;exact omS282 he h1);rw[e] at H;exact H
end
end
section
variable {rows inW outW:Ὣ} {X W b:ё}
def C_hf (d:H8):Ὣ->ὧ
 | 0=>d.h0
 | 1=>d.h1
 | 2=>d.h2
 | 3=>d.h3
 | 4=>d.h4
 | 5=>d.h5
 | 6=>d.h6
 | _=>d.h7
def C_fs (d:F8):Ὣ->ὧ
 | 0=>d.s0
 | 1=>d.s1
 | 2=>d.s2
 | 3=>d.s3
 | 4=>d.s4
 | 5=>d.s5
 | 6=>d.s6
 | _=>d.s7
def C_lo (d:L8):Ὣ->Ὠ
 | 0=>d.l0
 | 1=>d.l1
 | 2=>d.l2
 | 3=>d.l3
 | 4=>d.l4
 | 5=>d.l5
 | 6=>d.l6
 | _=>d.l7
@[inline] def mkL8 (nf hvv bm:ὧ) (d:H8):L8:=⟨loOf (nf*(hvv*d.h0+bm)),loOf (nf*(hvv*d.h1+bm)),loOf (nf*(hvv*d.h2+bm)),loOf (nf*(hvv*d.h3+bm)),loOf (nf*(hvv*d.h4+bm)),loOf (nf*(hvv*d.h5+bm)),loOf (nf*(hvv*d.h6+bm)),loOf (nf*(hvv*d.h7+bm))⟩
lemma qB3 {n t c:ὦ} {k:Ὣ} (hc:Ѥc=k) (h:k*Ѥn+Ѥt<2^32):(c*n+t).toNat=k*Ѥn+Ѥt:=by have h0:k*Ѥn<2^32:=(by clear hc;exact omS283 h);rw[qAf _ _ (by rw[qBa _ _ (by rw[hc];exact h0),hc];exact h),qBa _ _ (by rw[hc];exact h0),hc]
lemma qWc (a:ὤ) (i:ὦ) (j:Ὣ) (hj:Ѥi=j) (v:ὧ) (hs:Ѩa<2^32) (hi:j< Ѩa) (q:Ὣ):(wrF a i v).data[q]! =if q=j then v else a.data[q]!:=by
 rw[qEt a i v q hs (by rw[hj];exact hi),hj]
lemma qW8 (xs:ὤ) (t:ὦ) (a0 a1 a2 a3 a4 a5 a6 a7:ὧ) (hs:Ѩxs<2^32) (hb:8*Ѥt+9<=Ѩxs):
 (wr8 xs (8*t+ὴ 1) a0 a1 a2 a3 a4 a5 a6 a7).size=Ѩxs/\∀q,(wr8 xs (8*t+ὴ 1) a0 a1 a2 a3 a4 a5 a6 a7).data[q]! =
  if q=8*Ѥt+1+7 then a7 else if q=8*Ѥt+1+6 then a6 else if q=8*Ѥt+1+5 then a5 else if q=8*Ѥt+1+4 then a4 else
  if q=8*Ѥt+1+3 then a3 else if q=8*Ѥt+1+2 then a2 else if q=8*Ѥt+1+1 then a1 else if q=8*Ѥt+1+0 then a0 else xs.data[q]!:=by
 have l8:(8:ὦ).toNat=8:=uk 8 (by norm_num1)
 have e80:(8*t).toNat=8*Ѥt:=by rw[qBa _ _ (by rw[l8];(clear l8;exact omS284 hs hb)),l8]
 have e8:(8*t+ὴ 1).toNat=8*Ѥt+1:=by rw[qAf _ _ (by rw[e80,uk 1 ה];(clear l8 e80;exact omS285 hs hb)),e80,uk 1 ה]
 have E:∀k,k<8->(8*t+ὴ 1+ὴ k).toNat=8*Ѥt+1+k:=λk hk=>by rw[qAf _ _ (by rw[e8,uk k (by exact Nat.lt_of_lt_of_le ‹_› ה)];(clear l8 e8;omega)),e8,uk k (by exact Nat.lt_of_lt_of_le ‹_› ה)]
 unfold wr8
 refine ⟨by simp only[qD6],λq=>?_⟩
 rw[qWc _ _ _ (E 7 (by exact Nat.le_refl _)) _ (by simp only[qD6];exact hs) (by simp only[qD6];(clear l8 e8 e80;exact omS286 hb)) q,
  qWc _ _ _ (E 6 ה) _ (by simp only[qD6];exact hs) (by simp only[qD6];(clear l8 e8 e80;exact omS287 hb)) q,
  qWc _ _ _ (E 5 ה) _ (by simp only[qD6];exact hs) (by simp only[qD6];(clear hs l8 e8 e80;exact omS288 hb)) q,
  qWc _ _ _ (E 4 ה) _ (by simp only[qD6];exact hs) (by simp only[qD6];(clear hs l8 e8 e80;exact omS289 hb)) q,
  qWc _ _ _ (E 3 ה) _ (by simp only[qD6];exact hs) (by simp only[qD6];(clear hs l8 e8 e80;exact omS290 hb)) q,
  qWc _ _ _ (E 2 ה) _ (by simp only[qD6];exact hs) (by simp only[qD6];(clear hs l8 e8 e80;exact omS291 hb)) q,
  qWc _ _ _ (E 1 ה) _ (by simp only[qD6];exact hs) (by simp only[qD6];(clear hs l8 e8 e80;exact omS292 hb)) q,
  qWc _ _ _ (E 0 ה) _ hs (by (clear hs l8 e8 e80;exact omS293 hb)) q]
lemma qZs (m:Ὣ) (h:m<2^32):(pushZ (ὴ m) 0 (ἆ m)).size=m:=by
 rw[qDy (ὴ m) m 0 _ (by rw[uk m h,C_z,Ἇ])];exact Ἇ _
lemma qZf (m:Ὣ) (h:m<2^32):m<=(zFor m).size/\(zFor m).size<2^32:=by
 unfold zFor
 by_cases hc32:m<=32
 · rw[ἱ hc32];have e:ѨzA=32:=qZs 32 (by norm_num1);rw[e];omega
 rw[ἐ hc32]
 by_cases hc128:m<=128
 · rw[ἱ hc128];have e:Ѩz0=128:=qZs 128 (by norm_num1);rw[e];omega
 rw[ἐ hc128]
 by_cases hc256:m<=256
 · rw[ἱ hc256];have e:ѨzB=256:=qZs 256 (by norm_num1);rw[e];omega
 rw[ἐ hc256]
 by_cases hc512:m<=512
 · rw[ἱ hc512];have e:ѨzC=512:=qZs 512 (by norm_num1);rw[e];omega
 rw[ἐ hc512]
 by_cases hc1024:m<=1024
 · rw[ἱ hc1024];have e:Ѩz1=1024:=qZs 1024 (by norm_num1);rw[e];omega
 rw[ἐ hc1024]
 by_cases hc4096:m<=4096
 · rw[ἱ hc4096];have e:Ѩz4=4096:=qZs 4096 (by norm_num1);rw[e];omega
 rw[ἐ hc4096]
 by_cases hc8192:m<=8192
 · rw[ἱ hc8192];have e:Ѩz8=8192:=qZs 8192 (by norm_num1);rw[e];omega
 rw[ἐ hc8192]
 have e:(pushZ ѩm 0 (ἆ m)).size=m:=qZs m h;rw[e];omega
lemma qBq {xs:ὤ} {i t n:ὦ} {xf:Ὣ->Ὣ->ὧ} (hxf:∀r k:Ὣ,r<8->k< Ѥn->xs.data[8*k+1+r]! =xf r k) (hi:Ѥi=8*Ѥt+1) (ht:Ѥt< Ѥn) (hsx:Ѩxs<2^32) (hx:8*Ѥn+1<=Ѩxs) (r:Ὣ) (hr:r<8) (pf:(i+ὴ r).toNat< Ѩxs):xs.uget (i+ὴ r) pf=xf r Ѥt:=by rw[qC5,qAf _ _ (by rw[uk r (by exact Nat.lt_of_lt_of_le ‹_› ה)];(clear pf;exact omS294 hi ht hsx hx hr)),uk r (by exact Nat.lt_of_lt_of_le ‹_› ה),hi,hxf r _ hr ht]
end
section
open Float.Model Float.Model.UnpackedFloat
lemma qD2 (s:Sign):(Float.Model.pack (UnpackedFloat.infinity s)).unpack.isFinite=false:=by cases s<;>decide+kernel
lemma qFh:(Float.Model.pack UnpackedFloat.notANumber).unpack.isFinite=false:=by decide+kernel
lemma qC1 (x y:ὧ) (h:(x*y)ᶠ):xᶠ/\yᶠ:=by change (Float.Model.mul x.toModel y.toModel).unpack.isFinite=true at h;change x.toModel.unpack.isFinite=true/\y.toModel.unpack.isFinite=true;rw[Float.Model.mul] at h;cases a:x.toModel.unpack<;>cases b:y.toModel.unpack<;>simp only[a,b,UnpackedFloat.mul,qD2,qFh,Bool.false_eq_true] at h<;>exact ⟨rfl,rfl⟩
lemma qC2 (x y:ὧ) (h:(x+y)ᶠ):xᶠ/\yᶠ:=by
 change (Float.Model.add x.toModel y.toModel).unpack.isFinite=true at h;change x.toModel.unpack.isFinite=true/\y.toModel.unpack.isFinite=true;rw[Float.Model.add] at h
 cases a:x.toModel.unpack<;>cases b:y.toModel.unpack<;>simp only[a,b,UnpackedFloat.add,qD2,qFh,Bool.false_eq_true] at h<;>
  first
   | exact ⟨rfl,rfl⟩
   | (split at h<;>simp only[qD2,qFh,Bool.false_eq_true] at h)
@[inline] def okF (t:ὧ):Ὡ:=t-t==t-t
lemma fS1 (s:Sign):(Float.Model.pack (UnpackedFloat.sub ⟠ (UnpackedFloat.infinity s) (UnpackedFloat.infinity s))).unpack.beq (Float.Model.pack (UnpackedFloat.sub ⟠ (UnpackedFloat.infinity s) (UnpackedFloat.infinity s))).unpack=false:=by
 cases s<;>decide+kernel
lemma fS2:(Float.Model.pack (UnpackedFloat.sub ⟠ UnpackedFloat.notANumber UnpackedFloat.notANumber)).unpack.beq (Float.Model.pack (UnpackedFloat.sub ⟠ UnpackedFloat.notANumber UnpackedFloat.notANumber)).unpack=false:=by
 decide+kernel
lemma finSub (t:ὧ) (h:okF t=true):tᶠ :=by
 unfold okF at h
 change (Float.Model.beq (Float.Model.sub t.toModel t.toModel) (Float.Model.sub t.toModel t.toModel))=true at h
 change t.toModel.unpack.isFinite=true
 rw[Float.Model.sub,Float.Model.beq] at h
 cases a:t.toModel.unpack
 . rename_i s;rw[a,fS1] at h;exact absurd h ה
 . rw[a,fS2] at h;exact absurd h ה
 . rfl
 . rfl
lemma qHz:(Ἃ 0x7FF8000000000000).isFinite=false:=by decide+kernel
def N_ok (w:ὥ):Prop:=0x800000<=w&&&В/\w&&&В<0x40000000
lemma qEx {w:ὥ} (h:N_ok w):good w=true:=qBt _ (Or.inr h.1) (by have t:w&&&В<0x40000000:=h.2;have a:(0x40000000:ὥ).toNat=0x40000000:=rfl;have b:(У:ὥ).toNat=У:=rfl;rw[ἳ,a] at t;rw[ἳ,b];(exact Nat.lt_of_lt_of_le ‹_› ה))
def Z_ok (hv w:ὥ):Prop:=w&&&В<=hv
lemma qTb {z:Ὣ} {T:ὤ} {h:ὥ} (o:TOK z T h):ѥh<0x40000000:=by obtain ⟨k,-,-,k2,k3,e⟩:=o;exact omS295 k3 e
lemma qTg {z:Ὣ} {T:ὤ} {h:ὥ} (o:TOK z T h):good h=true/\h<=В:=by
 obtain ⟨k,-,-,k2,k3,e⟩:=o
 have a:=qAq h
 refine ⟨qEx ⟨?_,?_⟩,?_⟩
 . rw[ἃ,a,show (0x800000:ὥ).toNat=0x800000 from rfl];exact omS296 k2 k3 e
 . rw[ἳ,a,show (0x40000000:ὥ).toNat=0x40000000 from rfl];exact omS297 k3 e
 . rw[ἃ,show (В:ὥ).toNat=В from rfl];(clear a;exact omS298 k3 e)
lemma N_w {z:Ὣ} {T:ὤ} {h:ὥ} (o:TOK z T h) (hT:ѨT=512) (w:ὥ) (f:(T.uget (w>>>(23:ὥ)).toUSize (qBx hT w))ᶠ):
 T.uget (w>>>(23:ὥ)).toUSize (qBx hT w)=pw.uget (w>>>(23:ὥ)).toUSize (qCh w)/\z*0x800000<=(w&&&В).toNat/\w&&&В<=h:=by
 obtain ⟨k,-,eT,k2,k3,hk⟩:=o
 have s:=qD8 w
 rw[eT,ὃ,s] at f
 rw[eT,qDc,ὃ,s]
 unfold pwEntryN at f⊢
 have e:=qAq w
 have g:=w.toNat_lt
 by_cases c:(decide (ѥw/2^23%256<z) || decide (k<=ѥw/2^23%256))=true
 . rw[ἱ c,qHz] at f;exact absurd f ה
 . rw[ἐ c];simp only[Bool.or_eq_true,decide_eq_true_eq,not_or,not_lt,not_le] at c;refine ⟨rfl,?_,?_⟩
   . rw[e];(clear hT hk s g;exact omS299 c)
   . rw[ἃ,e];(clear hT s g;exact omS300 hk c)
lemma qZk {T:ὤ} {hT:ѨT=512} {hv:ὥ} (oT:TOK 0 T hv) {w:ὥ} (h:(dvN T hT w)ᶠ):dvN T hT w=dv w/\Z_ok hv w:=by obtain ⟨e,-,b⟩:=N_w oT hT w (qC1 _ _ h).2;exact ⟨by unfold dvN dv;rw[e],b⟩
def N_ff (f:Ὣ->ὧ) (L:љ) (a:ὧ):ὧ:=
 L.foldl (λacc s=>acc+f s) a
lemma qB4 (f:Ὣ->ὧ) (L:љ): ∀a,(N_ff f L a)ᶠ->aᶠ/\∀s∈L,(f s)ᶠ:=by
 induction L with
 | nil=>exact λa h=>⟨h,λs hs=>by simp only[List.not_mem_nil] at hs⟩
 | cons p L ih=>
  intro a h;obtain ⟨h1,h2⟩:=ih _ h;obtain ⟨h3,h4⟩:=qC2 _ _ h1;refine ⟨h3,λs hs=>?_⟩;rcases List.mem_cons.mp hs with e | e;. rw[e];exact h4
  . exact h2 s e
section
variable {T:ὤ} {hT:ѨT=512} {x:ὤ} {ix W:ѐ} {w0 w1 w2 w3 m n:ὦ} (f:ὧ->ὥ->ὧ)
end
section
variable {T:ὤ} {hT:ѨT=512} {hv:ὥ} {x:ὤ} {W:ѐ} {u:Ὣ} {L:љ}
include x W u L
end
section
variable {T:ὤ} {hT:ѨT=512} {hv:ὥ} {I O:Ὣ} {X W b:ё}
set_option maxRecDepth 8000 in
lemma qS8 (R:F8) (H:(R.s0+R.s1+R.s2+R.s3+R.s4+R.s5+R.s6+R.s7)ᶠ):∀r,r<8->(C_fs R r)ᶠ:=by
 obtain ⟨h1,h7⟩:=qC2 _ _ H
 obtain ⟨h2,h6⟩:=qC2 _ _ h1
 obtain ⟨h3,h5⟩:=qC2 _ _ h2
 obtain ⟨h4,h4'⟩:=qC2 _ _ h3
 obtain ⟨h9,h3'⟩:=qC2 _ _ h4
 obtain ⟨h10,h2'⟩:=qC2 _ _ h9
 obtain ⟨h0,h1'⟩:=qC2 _ _ h10
 intro r hr
 interval_cases r
 . exact h0
 . exact h1'
 . exact h2'
 . exact h3'
 . exact h4'
 . exact h5
 . exact h6
 . exact h7
lemma qZn (oT:TOK 0 T hv) (f:Ὣ->ὧ) (ww:Ὣ->ὥ) (L:љ) (h:(N_ff (λs=>f s*dvN T hT (ww s)) L 0)ᶠ):
 (∀s∈L,Z_ok hv (ww s))/\N_ff (λs=>f s*dvN T hT (ww s)) L 0=L.foldl (λacc s=>acc+f s*dv (ww s)) 0:=by
 have F:=(qB4 _ _ _ h).2
 have G:=λs hs=>qZk oT (qC1 _ _ (F s hs)).2
 exact ⟨λs hs=>(G s hs).2,List.foldl_ext _ _ _ λacc s hs=>by
  show acc+f s*dvN T hT (ww s)=_
  rw[(G s hs).1]⟩
lemma qZg {hv w:ὥ} (t:ѥhv<0x40000000) (h:Z_ok hv w) (g:w&&&В=0\/0x800000<=w&&&В):good w=true:=qBt _ g (by have c:=ἃ.mp h;have b:(У:ὥ).toNat=У:=rfl;rw[ἳ,b];(clear g b;exact omS312 t c))
lemma qZm (ww:Ὣ->ὥ) (L:љ):∀q,0x7FFFFF<=L.foldl (λq s=>q&&&((ww s&&&В)-1)) q->0x7FFFFF<=q/\∀s∈L,ww s&&&В=0\/0x800000<=ww s&&&В:=by
 induction L with
 | nil=>exact λq h=>⟨h,λs hs=>by simp only[List.not_mem_nil] at hs⟩
 | cons p L ih=>
  intro q h
  obtain ⟨h1,h2⟩:=ih _ h
  obtain ⟨h8,h9⟩:=qFe q ((ww p&&&В)-1)
  refine ⟨UInt32.le_trans h1 h8,λs hs=>?_⟩
  rcases List.mem_cons.mp hs with e | e
  . rw[e];exact qGi _ (UInt32.le_trans h1 h9)
  . exact h2 s e
lemma qZ1 {xs:ὤ} {W:ѐ} {wo n:ὦ} (hx:8*Ѥn+1<=Ѩxs) (hsx:Ѩxs<2^32) (hs:ѧW<2^32) (hw:Ѥwo+Ѥn<=ѧW) (ww:Ὣ->Ὢ) (hww:∀k,k< Ѥn->W[Ѥwo+k]! =ww k) (d:Ὣ):
 ∀(t i:ὦ) (hi:Ѥi=8*Ѥt+1) (s0 s1 s2 s3 s4 s5 s6 s7:ὧ) (q:ὥ) (r0:F8),Ѥt+d=Ѥn->
  (∀r,r<8->C_fs (go8Z T hT xs W wo n t i hx hsx hs hw hi s0 s1 s2 s3 s4 s5 s6 s7 q r0) r=
   N_ff (λs=>xs.data[8*s+1+r]!*dvN T hT (ww s)) (List.range' Ѥt d) (C_fs ⟨0,s0,s1,s2,s3,s4,s5,s6,s7,0,0⟩ r))/\
  (go8Z T hT xs W wo n t i hx hsx hs hw hi s0 s1 s2 s3 s4 s5 s6 s7 q r0).q=(List.range' Ѥt d).foldl (λq s=>q&&&((ww s&&&В)-1)) q:=by
 induction d with
 | zero=>
  intro t i hi s0 s1 s2 s3 s4 s5 s6 s7 q r0 he
  rw[go8Z,ἔ (by rw[А];(clear hx hsx hs hw hi;(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›)))))]
  cases r0
  exact ⟨λr hr=>by interval_cases r<;>rfl,rfl⟩
 | succ d ih=>
  intro t i hi s0 s1 s2 s3 s4 s5 s6 s7 q r0 he
  have ht:t<n:=by rw[А];(clear hi;exact omS136 he)
  have htn:Ѥt< Ѥn:=(by exact Nat.succ_le_of_lt ‹_›)
  have hu:=usz t n ht
  have e:W.uget (wo+t) (wi wo t n W hw hs htn)=ww Ѥt:=by
   rw[qC3,qAf _ _ (by clear hi ht htn;exact omS313 hs hw he),hww _ htn]
  have B:=λ(k:Ὣ) (hk:k<8)=>qBq (xf:=λr k=>xs.data[8*k+1+r]!) (λ_ _ _ _=>rfl) hi htn hsx hx k hk
  rw[go8Z,Ἄ ht]
  dsimp only
  rw[e,B 0 ה,B 1 ה,B 2 ה,B 3 ה,B 4 ה,B 5 ה,B 6 ה,B 7 (by exact Nat.le_refl _)]
  obtain ⟨J1,J2⟩:=ih (t+1) (i+ὴ 8) (by rw[hu,qAf _ _ (by rw[uk 8 ה];(clear ht htn e;exact omS314 hx hsx hi he)),uk 8 ה];(clear he ht htn hu e;exact omS315 hi)) _ _ _ _ _ _ _ _ _ r0 (by rw[hu];(clear hi ht htn hu e;exact omS137 he))
  have e5:List.range' (t+1).toNat d=List.range' (Ѥt+1) d:=by rw[hu]
  rw[e5] at J1 J2
  refine ⟨λr hr=>(J1 r hr).trans ?_,J2.trans ?_⟩
  . rw[List.range'_succ]
    unfold N_ff
    rw[ὔ]
    interval_cases r<;>rfl
  . rw[List.range'_succ];rfl
@[noinline] def fin8o (inW:Ὣ) (relu:Ὡ) (X W b:ё) (ri K j:Ὣ)
 (s nf a:ὧ) (hx bw:ὥ) (n:ὦ):ὥ:=
 let bv:=dv bw
 let r:=ziv (s+bv) (nf*(dv hx*a+bv.abs))
 if r==Е then
  let i:=ri+K
  let q:=tightGo X W (i*inW).toUSize (j*inW).toUSize n 0 0 0
  fin inW relu X W b i j q.s nf q.t bw
 else r&&&((if relu then (0:ὥ) else Е) ||| ((r>>>(31:ὥ))-1))
local instance ecNe8:Nonempty H8:=⟨by constructor <;> exact default⟩
def dec8o (X:ѐ) (o n t:ὦ) (xs:ὤ)
 (h0 h1 h2 h3 h4 h5 h6 h7:ὧ):H8:=
 if h:t<n then
  let v0:=dv (rdW X (o+(0*n+t)))
  let v1:=dv (rdW X (o+(1*n+t)))
  let v2:=dv (rdW X (o+(2*n+t)))
  let v3:=dv (rdW X (o+(3*n+t)))
  let v4:=dv (rdW X (o+(4*n+t)))
  let v5:=dv (rdW X (o+(5*n+t)))
  let v6:=dv (rdW X (o+(6*n+t)))
  let v7:=dv (rdW X (o+(7*n+t)))
  dec8o X o n (t+1) (wr8 xs (8*t+ὴ 1) v0 v1 v2 v3 v4 v5 v6 v7)
   (h0+v0.abs) (h1+v1.abs) (h2+v2.abs) (h3+v3.abs) (h4+v4.abs) (h5+v5.abs) (h6+v6.abs) (h7+v7.abs)
 else ⟨xs,h0,h1,h2,h3,h4,h5,h6,h7⟩
termination_by Ѥn-Ѥt
decreasing_by rw[usz t n h];exact Nat.sub_lt_sub_left (А.mp h) (Nat.lt_succ_self _)
local instance ecNe9:Nonempty CN:=⟨by constructor <;> exact default⟩
def cols8Zo (T:ὤ) (hT:ѨT=512) (hv:ὥ) (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (d:H8) (nf:ὧ)
 (n:ὦ) (ri:Ὣ) (ob:ὦ) (j:Ὣ) (out:ѐ) (q:ὥ) (th:L8) (r0:F8):CN:=
 if hj:j<outW then
  let ju:=ѩj
  let wo:=ju*n
  if hg:8*Ѥn+1<=d.xs.size/\d.xs.size<=4294967295/\ѧW<=4294967295/\Ѥwo+Ѥn<=ѧW then
   let r:=go8Z T hT d.xs W wo n 0 (ὴ 1) hg.1 (qA9 hg.2.1) (qA9 hg.2.2.1) hg.2.2.2
    (by rw[show (0:ὦ).toNat=0 from rfl,uk 1 (by decide)]) 0 0 0 0 0 0 0 0 q r0
   if okF (r.s0+r.s1+r.s2+r.s3+r.s4+r.s5+r.s6+r.s7) then
    let bw:=rdW b ju
    let ow:=ѩoutW
    let bv:=dv bw
    let t0:=fin1 th.l0 (spOf th.l0) (r.s0+bv)
    let out:=wrW out (ob+ju) (if t0==Е then fin8 inW relu X W b (ri+0) j r.s0 nf d.h0 hv bw n else fz relu t0)
    let t1:=fin1 th.l1 (spOf th.l1) (r.s1+bv)
    let out:=wrW out (ob+(ow+ju)) (if t1==Е then fin8 inW relu X W b (ri+1) j r.s1 nf d.h1 hv bw n else fz relu t1)
    let t2:=fin1 th.l2 (spOf th.l2) (r.s2+bv)
    let out:=wrW out (ob+(2*ow+ju)) (if t2==Е then fin8 inW relu X W b (ri+2) j r.s2 nf d.h2 hv bw n else fz relu t2)
    let t3:=fin1 th.l3 (spOf th.l3) (r.s3+bv)
    let out:=wrW out (ob+(3*ow+ju)) (if t3==Е then fin8 inW relu X W b (ri+3) j r.s3 nf d.h3 hv bw n else fz relu t3)
    let t4:=fin1 th.l4 (spOf th.l4) (r.s4+bv)
    let out:=wrW out (ob+(4*ow+ju)) (if t4==Е then fin8 inW relu X W b (ri+4) j r.s4 nf d.h4 hv bw n else fz relu t4)
    let t5:=fin1 th.l5 (spOf th.l5) (r.s5+bv)
    let out:=wrW out (ob+(5*ow+ju)) (if t5==Е then fin8 inW relu X W b (ri+5) j r.s5 nf d.h5 hv bw n else fz relu t5)
    let t6:=fin1 th.l6 (spOf th.l6) (r.s6+bv)
    let out:=wrW out (ob+(6*ow+ju)) (if t6==Е then fin8 inW relu X W b (ri+6) j r.s6 nf d.h6 hv bw n else fz relu t6)
    let t7:=fin1 th.l7 (spOf th.l7) (r.s7+bv)
    let out:=wrW out (ob+(7*ow+ju)) (if t7==Е then fin8 inW relu X W b (ri+7) j r.s7 nf d.h7 hv bw n else fz relu t7)
    cols8Zo T hT hv inW outW relu X W b d nf n ri ob (j+1) out r.q th r
   else ⟨out,false⟩
  else ⟨out,false⟩
 else ⟨out,decide (0x7FFFFF<=q)⟩
termination_by outW-j
decreasing_by all_goals omega
def cols8ZDo (D:ὤ) (hv:ὥ) (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (hd:ѨD=ѧW) (d:H8) (nf:ὧ)
 (n:ὦ) (ri:Ὣ) (ob:ὦ) (j:Ὣ) (out:ѐ) (q:ὥ) (th:L8) (r0:F8):CN:=
 if hj:j<outW then
  let ju:=ѩj
  let wo:=ju*n
  if hg:8*Ѥn+1<=d.xs.size/\d.xs.size<=4294967295/\ѧW<=4294967295/\Ѥwo+Ѥn<=ѧW then
   let r:=go8ZS D d.xs W hd wo n 0 (ὴ 1) hg.1 (qA9 hg.2.1) (qA9 hg.2.2.1) hg.2.2.2
    (by rw[show (0:ὦ).toNat=0 from rfl,uk 1 (by decide)]) 0 0 0 0 0 0 0 0 q r0
   if okF (r.s0+r.s1+r.s2+r.s3+r.s4+r.s5+r.s6+r.s7) then
    let bw:=rdW b ju
    let ow:=ѩoutW
    let bv:=dv bw
    let t0:=fin1 th.l0 (spOf th.l0) (r.s0+bv)
    let out:=wrW out (ob+ju) (if t0==Е then fin8o inW relu X W b ri 0 j r.s0 nf d.h0 hv bw n else fz relu t0)
    let t1:=fin1 th.l1 (spOf th.l1) (r.s1+bv)
    let out:=wrW out (ob+(ow+ju)) (if t1==Е then fin8o inW relu X W b ri 1 j r.s1 nf d.h1 hv bw n else fz relu t1)
    let t2:=fin1 th.l2 (spOf th.l2) (r.s2+bv)
    let out:=wrW out (ob+(2*ow+ju)) (if t2==Е then fin8o inW relu X W b ri 2 j r.s2 nf d.h2 hv bw n else fz relu t2)
    let t3:=fin1 th.l3 (spOf th.l3) (r.s3+bv)
    let out:=wrW out (ob+(3*ow+ju)) (if t3==Е then fin8o inW relu X W b ri 3 j r.s3 nf d.h3 hv bw n else fz relu t3)
    let t4:=fin1 th.l4 (spOf th.l4) (r.s4+bv)
    let out:=wrW out (ob+(4*ow+ju)) (if t4==Е then fin8o inW relu X W b ri 4 j r.s4 nf d.h4 hv bw n else fz relu t4)
    let t5:=fin1 th.l5 (spOf th.l5) (r.s5+bv)
    let out:=wrW out (ob+(5*ow+ju)) (if t5==Е then fin8o inW relu X W b ri 5 j r.s5 nf d.h5 hv bw n else fz relu t5)
    let t6:=fin1 th.l6 (spOf th.l6) (r.s6+bv)
    let out:=wrW out (ob+(6*ow+ju)) (if t6==Е then fin8o inW relu X W b ri 6 j r.s6 nf d.h6 hv bw n else fz relu t6)
    let t7:=fin1 th.l7 (spOf th.l7) (r.s7+bv)
    let out:=wrW out (ob+(7*ow+ju)) (if t7==Е then fin8o inW relu X W b ri 7 j r.s7 nf d.h7 hv bw n else fz relu t7)
    cols8ZDo D hv inW outW relu X W b hd d nf n ri ob (j+1) out q th r
   else ⟨out,false⟩
  else ⟨out,false⟩
 else ⟨out,decide (0x7FFFFF<=q)⟩
termination_by outW-j
decreasing_by all_goals omega
def C_D8o (X:ё) (ri m t:Ὣ) (d:H8):Prop:=
 (8*m+1<=d.xs.size/\d.xs.size<2^32)/\(∀k r:Ὣ,k<t->r<8->d.xs.data[8*k+1+r]! =dv X[(ri+r)*m+k]!)/\
  ∀r:Ὣ,r<8->C_hf d r=(ἄ t).foldl (λ(a:ὧ) k=>a+(dv X[(ri+r)*m+k]!).abs) 0
lemma qRo {X:ё} {o n t c:ὦ} {ri k:Ὣ} (hs:ѧX<2^32) (ho:Ѥo=ri*Ѥn) (hc:Ѥc=k) (h:(ri+k)*Ѥn+Ѥt<2^32):rdW X (o+(c*n+t))=X[(ri+k)*Ѥn+Ѥt]!:=by
 have h0:(ri+k)*Ѥn=ri*Ѥn+k*Ѥn:=ὰ _ _ _
 have h1:k*Ѥn+Ѥt<2^32:=by (clear hs hc;exact omS316 h h0)
 rw[qAg _ _ hs,qAf _ _ (by rw[qB3 hc h1,ho];(clear hs ho hc h1;exact omS317 h h0)),qB3 hc h1,ho,h0,ὢ]
lemma qGqo {X:ё} {o n:ὦ} {ri:Ὣ} (hs:ѧX<2^32) (ho:Ѥo=ri*Ѥn) (h8:(ri+8)*Ѥn<=ѧX) (d:Ὣ):
 ∀(t:ὦ) (xs:ὤ) (h0 h1 h2 h3 h4 h5 h6 h7:ὧ),Ѥt+d=Ѥn->C_D8o X ri Ѥn Ѥt ⟨xs,h0,h1,h2,h3,h4,h5,h6,h7⟩->C_D8o X ri Ѥn Ѥn (dec8o X o n t xs h0 h1 h2 h3 h4 h5 h6 h7):=by
 induction d with
 | zero=>
  intro t xs h0 h1 h2 h3 h4 h5 h6 h7 he H
  rw[dec8o,ἔ (by rw[А];(clear hs ho h8;(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›)))))]
  have e:Ѥt=Ѥn:=(by exact omS155 he)
  rw[e] at H;exact H
 | succ d ih=>
  intro t xs h0 h1 h2 h3 h4 h5 h6 h7 he H
  have ht:t<n:=by rw[А];(clear hs ho h8;exact omS136 he)
  have hu:=usz t n ht
  have bnd:∀k,k<8->(ri+k)*Ѥn+Ѥt<2^32:=λk hk=>by
   have a1:=Ἒ Ѥn (show ri+k+1<=ri+8 by (clear hs ho h8 he ht hu;exact omS318 hk))
   have a2:(ri+k+1)*Ѥn=(ri+k)*Ѥn+Ѥn:=by rw[ὰ,Ὑ]
   (clear ho ht hk;omega)
  rw[dec8o,Ἄ ht]
  dsimp only
  rw[qRo hs ho (show (0:ὦ).toNat=0 from uk 0 ו) (bnd 0 ו),qRo hs ho (show (1:ὦ).toNat=1 from uk 1 (by norm_num1)) (bnd 1 (by norm_num1)),qRo hs ho (show (2:ὦ).toNat=2 from uk 2 (by norm_num1)) (bnd 2 (by norm_num1)),qRo hs ho (show (3:ὦ).toNat=3 from uk 3 (by norm_num1)) (bnd 3 (by norm_num1)),qRo hs ho (show (4:ὦ).toNat=4 from uk 4 (by norm_num1)) (bnd 4 (by norm_num1)),qRo hs ho (show (5:ὦ).toNat=5 from uk 5 (by norm_num1)) (bnd 5 (by norm_num1)),qRo hs ho (show (6:ὦ).toNat=6 from uk 6 (by norm_num1)) (bnd 6 (by norm_num1)),qRo hs ho (show (7:ὦ).toNat=7 from uk 7 (by norm_num1)) (bnd 7 (by norm_num1))]
  obtain ⟨H1,H2,H3⟩:=H
  dsimp only at H1 H2
  obtain ⟨Z1,Z2⟩:=qW8 xs t (dv X[(ri+0)*Ѥn+Ѥt]!) (dv X[(ri+1)*Ѥn+Ѥt]!) (dv X[(ri+2)*Ѥn+Ѥt]!) (dv X[(ri+3)*Ѥn+Ѥt]!) (dv X[(ri+4)*Ѥn+Ѥt]!) (dv X[(ri+5)*Ѥn+Ѥt]!) (dv X[(ri+6)*Ѥn+Ѥt]!) (dv X[(ri+7)*Ѥn+Ѥt]!) H1.2 (by have:=H1.1;(clear hs ho h8 ht this;omega))
  refine ih (t+1) _ _ _ _ _ _ _ _ _ (by clear hs ho h8 ht H1 Z1;exact omS157 he hu) ⟨?_,?_,?_⟩
  . dsimp only;rw[Z1];exact H1
  . intro k r hk hr;rw[hu] at hk;dsimp only;rw[Z2];by_cases e:k=Ѥt
    . subst e;interval_cases r<;>simp only[add_zero,Nat.left_eq_add,OfNat.ofNat_ne_zero, ↓reduceIte,one_ne_zero,Nat.add_left_cancel_iff,OfNat.one_ne_ofNat,Nat.reduceEqDiff]
    . have hk':k< Ѥt:=Nat.lt_of_le_of_ne (Nat.le_of_lt_succ hk) e
      have hlt:8*k+1+r<8*Ѥt+1:=by (clear hs ho h8 ht H1 Z1 hk e;omega)
      rw[ἐ (Nat.ne_of_lt (hlt.trans_le (Nat.le_add_right _ _))),ἐ (Nat.ne_of_lt (hlt.trans_le (Nat.le_add_right _ _))),ἐ (Nat.ne_of_lt (hlt.trans_le (Nat.le_add_right _ _))),ἐ (Nat.ne_of_lt (hlt.trans_le (Nat.le_add_right _ _))),ἐ (Nat.ne_of_lt (hlt.trans_le (Nat.le_add_right _ _))),ἐ (Nat.ne_of_lt (hlt.trans_le (Nat.le_add_right _ _))),ἐ (Nat.ne_of_lt (hlt.trans_le (Nat.le_add_right _ _))),ἐ (Nat.ne_of_lt (hlt.trans_le (Nat.le_add_right _ _)))];exact H2 k r hk' hr
  . intro r hr;have h9:=H3 r hr;rw[hu,List.range_succ,List.foldl_append,<-h9];interval_cases r<;>rfl
def C_B (ri O j q:Ὣ):Prop:=q<ri*O\/(ri*O<=q/\q<(ri+8)*O/\q%O<j)
lemma qF4o {O q j ri:Ὣ} (hj:j<O) (h:C_B ri O (j+1) q):
 q=(ri+7)*O+j\/q=(ri+6)*O+j\/q=(ri+5)*O+j\/q=(ri+4)*O+j\/q=(ri+3)*O+j\/q=(ri+2)*O+j\/q=(ri+1)*O+j\/q=(ri+0)*O+j\/C_B ri O j q:=by
 unfold C_B at h⊢
 rcases h with h|⟨h1,h2,h3⟩
 . simp only[h,true_or,or_true]
 . by_cases e:q%O<j
   . simp only[h1,h2,e,and_self,or_true]
   . have e2:q%O=j:=(by clear hj h1 h2;exact omS309 h3 e)
     have hO:0<O:=by (clear h1 h2 e2;exact omS147 hj)
     have e3:=Nat.div_add_mod q O
     have e5:ri<=q/O:=(Nat.le_div_iff_mul_le hO).2 h1
     have e4:q/O<ri+8:=(Nat.div_lt_iff_lt_mul hO).2 h2
     rw[e2,ἥ] at e3
     obtain ⟨a,ha⟩:∃a,q/O=ri+a:=⟨q/O-ri,by (clear hj h1 h2 h3 e e2 hO e3;exact omS191 e5)⟩
     have a8:a<8:=by (clear hj h1 h2 h3 e e2 hO e3;exact omS319 e4 ha)
     rw[ha] at e3
     interval_cases a<;>simp only[e3.symm,true_or,or_true]
lemma qW9o {N O ri j:Ὣ} {sp:Ὣ->Ὢ} {out:ё} {ob:ὦ} (hN:N<2^32) (hR:(ri+8)*O<=N) (hob:Ѥob=ri*O) (hj:j<O) (H:C_Inv N sp (C_B ri O j) out):
 C_Inv N sp (C_B ri O (j+1)) (wrW (wrW (wrW (wrW (wrW (wrW (wrW (wrW out (ob+ѩj) (sp ((ri+0)*O+j))) (ob+(ѩO+ѩj)) (sp ((ri+1)*O+j)))
  (ob+(2*ѩO+ѩj)) (sp ((ri+2)*O+j))) (ob+(3*ѩO+ѩj)) (sp ((ri+3)*O+j))) (ob+(4*ѩO+ѩj)) (sp ((ri+4)*O+j)))
  (ob+(5*ѩO+ѩj)) (sp ((ri+5)*O+j))) (ob+(6*ѩO+ѩj)) (sp ((ri+6)*O+j))) (ob+(7*ѩO+ѩj)) (sp ((ri+7)*O+j))):=by
 have bk:∀k,k<8->(ri+k)*O+j<N/\k*O+j<2^32:=λk hk=>by
  have a1:=Ἒ O (show ri+k+1<=ri+8 by (clear hN hR hob hj;exact omS318 hk))
  have a2:(ri+k+1)*O=(ri+k)*O+O:=by rw[ὰ,Ὑ]
  have a3:(ri+k)*O=ri*O+k*O:=ὰ _ _ _
  constructor<;>(clear hk;omega)
 have e0:j.toUSize.toNat=j:=qAd _ (by have:=bk 0 ו;(clear hN hR hob hj;omega))
 have eo:O.toUSize.toNat=O:=qAd _ (by have:=bk 1 (by norm_num1);(clear hN hR hob;(clear hj;omega)))
 have pk:∀(c:ὦ) (k:Ὣ),k<8->Ѥc=k->(ob+(c*ѩO+ѩj)).toNat=(ri+k)*O+j:=λc k hk hc=>by
  obtain ⟨b1,b2⟩:=bk k hk
  have a3:(ri+k)*O=ri*O+k*O:=ὰ _ _ _
  have i1:(c*ѩO+ѩj).toNat=k*O+j:=by rw[qB3 hc (by rw[eo,e0];(exact Nat.succ_le_of_lt ‹_›)),eo,e0]
  rw[qAf _ _ (by rw[i1,hob];(clear hR hob hj eo hk hc b2;(clear e0;exact omS320 hN b1 a3))),i1,hob];(clear hN hR hob hj e0 eo hk hc b1 b2 i1;omega)
 have u0:(ob+ѩj).toNat=(ri+0)*O+j:=by
  obtain ⟨b1,b2⟩:=bk 0 ו
  have a0:(ri+0)*O=ri*O:=by rw[Ἵ]
  rw[qAf _ _ (by rw[hob,e0];(clear hR hob hj eo b2;(clear e0;exact omS321 hN b1 a0))),hob,e0];(clear hN hR hob hj e0 eo b1 b2;omega)
 have u1:(ob+(ѩO+ѩj)).toNat=(ri+1)*O+j:=by
  obtain ⟨b1,b2⟩:=bk 1 (by norm_num1)
  have a1:(ri+1)*O=ri*O+O:=by rw[ὰ,Ὑ]
  have i1:(ѩO+ѩj).toNat=O+j:=by rw[qAf _ _ (by rw[eo,e0];(clear hN hR hob hj b1 a1;(clear e0 eo;exact omS322 b2))),eo,e0]
  rw[qAf _ _ (by rw[i1,hob];(clear hR hob hj b2 i1;(clear e0 eo;exact omS320 hN b1 a1))),i1,hob];(clear hN hR hob hj e0 u0 b1 b2 i1;(clear eo;omega))
 have I0:=qAk hN H u0 (bk 0 ו).1 rfl
 have I1:=qAk hN I0 u1 (bk 1 (by norm_num1)).1 rfl
 have I2:=qAk hN I1 (pk 2 2 (by norm_num1) (uk 2 (by norm_num1))) (bk 2 (by norm_num1)).1 rfl
 have I3:=qAk hN I2 (pk 3 3 (by norm_num1) (uk 3 (by norm_num1))) (bk 3 (by norm_num1)).1 rfl
 have I4:=qAk hN I3 (pk 4 4 (by norm_num1) (uk 4 (by norm_num1))) (bk 4 (by norm_num1)).1 rfl
 have I5:=qAk hN I4 (pk 5 5 (by norm_num1) (uk 5 (by norm_num1))) (bk 5 (by norm_num1)).1 rfl
 have I6:=qAk hN I5 (pk 6 6 (by norm_num1) (uk 6 (by norm_num1))) (bk 6 (by norm_num1)).1 rfl
 have I7:=qAk hN I6 (pk 7 7 (by norm_num1) (uk 7 (by norm_num1))) (bk 7 (by norm_num1)).1 rfl
 exact qA5 I7 (λq _ hq'=>qF4o hj hq')
set_option maxHeartbeats 1000000 in
set_option maxRecDepth 8000 in
lemma qZ8o (oT:TOK 0 T hv) {Y:Ὡ} {d:H8} {R ri:Ὣ} {ob:ὦ} (hob:Ѥob=ri*O) (hR:ri+8<=R) (hW:O*I=ѧW) (pi:0<I) (ni:I<2^20) (sW:ѧW<2^32) (hd:8*I+1<=d.xs.size/\d.xs.size<2^32) (th:L8) (bm:ὧ) (fbm:bmᶠ) (hbm:∀j,j<O-> |Submission.D.R (dv b[j]!)| <=Submission.D.R bm) (bb:Submission.D.R bm<=2^(290:ℤ)) (hth:∀r,r<8->C_lo th r=loOf (nfOf I*(dv hv*C_hf d r+bm))) (k:Ὣ):
 ∀(j:Ὣ) (out:ё) (q:ὥ) (r0:F8),j+k=O->(cols8Zo T hT hv I O Y X W b d (nfOf I) ѩI ri ob j out q th r0).ok=true->
  0x7FFFFF<=q/\(∀j',j<=j'->j'<O->∀s,s<I->good W[j'*I+s]! =true/\W[j'*I+s]!&&&В<=hv)/\
  ∀(G:C_G R I O X W b),C_D8o X ri I I d->C_Inv (R*O) (C_sp I O Y X W b) (C_B ri O j) out->
   C_Inv (R*O) (C_sp I O Y X W b) (C_B ri O O) (cols8Zo T hT hv I O Y X W b d (nfOf I) ѩI ri ob j out q th r0).out:=by
 have hnm:I.toUSize.toNat=I:=qAd _ (by exact Nat.lt_of_lt_of_le ‹_› ה)
 have hO:O<2^32:=by
  have:=Ὄ O pi
  (clear hob hR pi ni hd hnm;omega)
 have W1:∀j,j<=O->(ѩj*ѩI).toNat=j*I:=λj hj=>by
  have h1:j*I<=O*I:=Ἒ _ hj
  rw[qBa _ _ (by rw[qAd _ (by exact Nat.lt_of_le_of_lt ‹_› ‹_›),hnm];(clear hob hR pi ni hd hnm hO hj;exact omS323 hW sW h1)),qAd _ (by exact Nat.lt_of_le_of_lt ‹_› ‹_›),hnm]
 induction k with
 | zero=>
  intro j out q r0 he ok
  rw[cols8Zo,ἔ (by clear hob hR hW pi ni sW hd hnm hO;(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›))))] at ok⊢
  have e:j=O:=(by exact omS155 he)
  rw[e]
  exact ⟨of_decide_eq_true ok,λj' a b'=>by (clear hob hR hW pi ni sW hd hnm hO e;(clear he;omega)),λ_ _ H=>H⟩
 | succ k ih=>
  intro j out q r0 he ok
  have hj:j<O:=(by clear hob hR hW pi ni sW hd hnm hO;exact omS136 he)
  have hlt:j*I+I<=ѧW:=by rw[<-hW];exact qBc hj (Ὗ _)
  have w1:=W1 j hj.le
  have hg:(ѩj*ѩI).toNat+I.toUSize.toNat<=ѧW:=by rw[w1,hnm];exact hlt
  have hx:8*I.toUSize.toNat+1<=d.xs.size:=by rw[hnm];exact hd.1
  have hsx:d.xs.size<2^32:=hd.2
  have hi0:(ὴ 1).toNat=8*(0:ὦ).toNat+1:=by rw[C_z,uk 1 ה]
  obtain ⟨key,kq⟩:=qZ1 (T:=T) (hT:=hT) hx hsx sW hg (λk=>W[j*I+k]!) (λk _=>by rw[w1]) I 0 (ὴ 1) hi0 0 0 0 0 0 0 0 0 q r0 (by rw[hnm,C_z,Ἇ])
  rw[C_z,<-ὗ] at key kq
  rw[cols8Zo,Ἄ hj] at ok⊢
  dsimp only at ok⊢
  rw[Ἄ ⟨hx,by (exact Nat.le_of_lt_succ ‹_›),by (exact Nat.le_of_lt_succ ‹_›),hg⟩] at ok⊢
  generalize go8Z T hT d.xs W (ѩj*ѩI) ѩI 0 (ὴ 1) hx hsx sW hg hi0 0 0 0 0 0 0 0 0 q r0=RR at key kq ok⊢
  by_cases c':okF (RR.s0+RR.s1+RR.s2+RR.s3+RR.s4+RR.s5+RR.s6+RR.s7)=true
  . rw[ἱ c'] at ok⊢
    have c:=finSub _ c'
    have F:=qS8 RR c
    have Ks:∀r,r<8->C_fs RR r=N_ff (λs=>d.xs.data[8*s+1+r]!*dvN T hT W[j*I+s]!) (ἄ I) 0:=λr hr=>by
     have k1:=key r hr
     have e1:C_fs ⟨0,0,0,0,0,0,0,0,0,0,0⟩ r=0:=by interval_cases r<;>rfl
     rw[e1] at k1
     exact k1
    have Zs:=λ(r:Ὣ) (hr:r<8)=>qZn oT (λs=>d.xs.data[8*s+1+r]!) (λs=>W[j*I+s]!) (ἄ I) (by have f:=F r hr;rw[Ks r hr] at f;exact f)
    obtain ⟨q1,IH,-⟩:=ih (j+1) _ _ _ (by clear hob hR hW pi ni sW hd hnm hO hj hlt w1 hg hx hsx hi0 kq;exact omS137 he) ok
    rw[kq] at q1
    obtain ⟨q2,q3⟩:=qZm (λs=>W[j*I+s]!) (ἄ I) q q1
    refine ⟨q2,λj' a b' s hs=>?_,λG D H=>?_⟩
    . by_cases e:j'=j
      . rw[e];exact ⟨qZg (qTb oT) ((Zs 0 ה).1 s (Ἅ.2 hs)) (q3 s (Ἅ.2 hs)),(Zs 0 ה).1 s (Ἅ.2 hs)⟩
      . exact IH j' (by clear hob hR hW pi ni sW hd hnm hO he hj hlt w1 hg hx hsx hi0 kq q1 q2 b' hs;exact omS138 a e) b' s hs
    . have hbs:=G.sb
      have e0:j.toUSize.toNat=j:=qAd _ (by clear hob hR hW pi ni sW hd hnm hj hlt w1 hg hx hsx hi0 kq q1 q2 hbs;exact omS324 hO he)
      have hN:R*O<2^32:=G.sO
      have hOR:O<=R*O:=Nat.le_mul_of_pos_left O (by (clear hob hW pi ni sW hd hnm hO he hj hlt w1 hg hx hsx hi0 kq q1 q2 hbs e0 hN;exact omS325 hR))
      have gv:=(qTg oT).1
      have hv1:=(qTg oT).2
      have O':∀r,r<8->(if fin1 (C_lo th r) (spOf (C_lo th r)) (C_fs RR r+dv b[j]!)==Е then fin8 I Y X W b (ri+r) j (C_fs RR r) (nfOf I) (C_hf d r) hv b[j]! ѩI else fz Y (fin1 (C_lo th r) (spOf (C_lo th r)) (C_fs RR r+dv b[j]!)))=C_sp I O Y X W b ((ri+r)*O+j):=by
       intro r hr
       have hl:=hth r hr
       rw[qCi _ _ _ _ _ _ hj,D.2.2 r hr]
       rw[D.2.2 r hr] at hl
       obtain ⟨z1,z2⟩:=Zs r hr
       exact qDuF G Y (by (clear hob hW pi ni sW hd hnm hO he hj hlt w1 hg hx hsx hi0 kq q1 q2 hbs e0 hN hOR hv1;exact omS326 hR hr):ri+r<R) hj gv hv1 (λt ht=>z1 t (Ἅ.2 ht)) (ἄ I) Ὂ (λt h=>Ἅ.1 h) (λt ht hn=>absurd (Ἅ.2 ht) hn) ((Ks r hr).trans (z2.trans (List.foldl_ext _ _ _ λacc t ht=>by
        show acc+d.xs.data[8*t+1+r]!*_=_
        rw[D.2.1 t r (Ἅ.1 ht) hr]))) bm fbm (hbm j hj) bb _ hl
      have o0:(if fin1 th.l0 (spOf th.l0) (RR.s0+dv b[j]!)==Е then fin8 I Y X W b (ri+0) j RR.s0 (nfOf I) d.h0 hv b[j]! ѩI else fz Y (fin1 th.l0 (spOf th.l0) (RR.s0+dv b[j]!)))=_:=O' 0 ה
      have o1:(if fin1 th.l1 (spOf th.l1) (RR.s1+dv b[j]!)==Е then fin8 I Y X W b (ri+1) j RR.s1 (nfOf I) d.h1 hv b[j]! ѩI else fz Y (fin1 th.l1 (spOf th.l1) (RR.s1+dv b[j]!)))=_:=O' 1 ה
      have o2:(if fin1 th.l2 (spOf th.l2) (RR.s2+dv b[j]!)==Е then fin8 I Y X W b (ri+2) j RR.s2 (nfOf I) d.h2 hv b[j]! ѩI else fz Y (fin1 th.l2 (spOf th.l2) (RR.s2+dv b[j]!)))=_:=O' 2 ה
      have o3:(if fin1 th.l3 (spOf th.l3) (RR.s3+dv b[j]!)==Е then fin8 I Y X W b (ri+3) j RR.s3 (nfOf I) d.h3 hv b[j]! ѩI else fz Y (fin1 th.l3 (spOf th.l3) (RR.s3+dv b[j]!)))=_:=O' 3 ה
      have o4:(if fin1 th.l4 (spOf th.l4) (RR.s4+dv b[j]!)==Е then fin8 I Y X W b (ri+4) j RR.s4 (nfOf I) d.h4 hv b[j]! ѩI else fz Y (fin1 th.l4 (spOf th.l4) (RR.s4+dv b[j]!)))=_:=O' 4 ה
      have o5:(if fin1 th.l5 (spOf th.l5) (RR.s5+dv b[j]!)==Е then fin8 I Y X W b (ri+5) j RR.s5 (nfOf I) d.h5 hv b[j]! ѩI else fz Y (fin1 th.l5 (spOf th.l5) (RR.s5+dv b[j]!)))=_:=O' 5 ה
      have o6:(if fin1 th.l6 (spOf th.l6) (RR.s6+dv b[j]!)==Е then fin8 I Y X W b (ri+6) j RR.s6 (nfOf I) d.h6 hv b[j]! ѩI else fz Y (fin1 th.l6 (spOf th.l6) (RR.s6+dv b[j]!)))=_:=O' 6 ה
      have o7:(if fin1 th.l7 (spOf th.l7) (RR.s7+dv b[j]!)==Е then fin8 I Y X W b (ri+7) j RR.s7 (nfOf I) d.h7 hv b[j]! ѩI else fz Y (fin1 th.l7 (spOf th.l7) (RR.s7+dv b[j]!)))=_:=O' 7 (by exact Nat.le_refl _)
      simp only[qAg _ _ hbs,e0,o0,o1,o2,o3,o4,o5,o6,o7] at ok⊢
      exact (ih (j+1) _ _ _ (by clear hob hR hW pi ni sW hd hnm hO hj hlt w1 hg hx hsx hi0 kq q1 q2 hbs e0 hN hOR hv1 o0 o1 o2 o3 o4 o5 o6 o7;exact omS137 he) ok).2.2 G D (qW9o hN (Ἒ O hR) hob hj H)
  . rw[ἐ c'] at ok;exact absurd ok ג
end
section
open Float.Model Float.Model.UnpackedFloat
lemma d8ix {o n t c:ὦ} {k:Ὣ} (hc:Ѥc=k) (hb:Ѥo+(k*Ѥn+Ѥt)<2^32): Ѥ(o+(c*n+t))=Ѥo+(k*Ѥn+Ѥt):=by
 rw[qAf _ _ (by rw[qB3 hc א];exact hb),qB3 hc א]
lemma d8rd {X:ѐ} {o n t c:ὦ} {k:Ὣ} (hc:Ѥc=k) (hb1:Ѥo+(k*Ѥn+Ѥt)< ѧX) (hb2:Ѥo+(k*Ѥn+Ѥt)<2^32): Ѥ(o+(c*n+t))< ѧX:=by
 rw[d8ix hc hb2];exact hb1
local instance ecNe10:Nonempty H8:=⟨by constructor <;> exact default⟩
lemma d8b {o n t:ὦ} {N:Ὣ} (k:Ὣ) (hk:k<8) (h:Ѥt<Ѥn) (hx:Ѥo+8*Ѥn<=N): Ѥo+(k*Ѥn+Ѥt)<N:=by
 have:=Ἒ (Ѥn) (show k+1<=8 from hk);rw[ᾀ] at this;omega
def dec8oI (X:ѐ) (o n t:ὦ) (xs:ὤ) (h0 h1 h2 h3 h4 h5 h6 h7:ὧ) (hx:Ѥo+8*Ѥn<=ѧX) (hs:Ѥo+8*Ѥn<2^32):H8:=
 if h:t<n then
  let v0:=dv (X.uget (o+(0*n+t)) (d8rd (k:=0) (uk 0 (by positivity)) (d8b 0 (by decide) (А.mp h) hx) (d8b 0 (by decide) (А.mp h) (Nat.le_of_lt hs))))
  let v1:=dv (X.uget (o+(1*n+t)) (d8rd (k:=1) (uk 1 (by norm_num1)) (d8b 1 (by decide) (А.mp h) hx) (d8b 1 (by decide) (А.mp h) (Nat.le_of_lt hs))))
  let v2:=dv (X.uget (o+(2*n+t)) (d8rd (k:=2) (uk 2 (by norm_num1)) (d8b 2 (by decide) (А.mp h) hx) (d8b 2 (by decide) (А.mp h) (Nat.le_of_lt hs))))
  let v3:=dv (X.uget (o+(3*n+t)) (d8rd (k:=3) (uk 3 (by norm_num1)) (d8b 3 (by decide) (А.mp h) hx) (d8b 3 (by decide) (А.mp h) (Nat.le_of_lt hs))))
  let v4:=dv (X.uget (o+(4*n+t)) (d8rd (k:=4) (uk 4 (by norm_num1)) (d8b 4 (by decide) (А.mp h) hx) (d8b 4 (by decide) (А.mp h) (Nat.le_of_lt hs))))
  let v5:=dv (X.uget (o+(5*n+t)) (d8rd (k:=5) (uk 5 (by norm_num1)) (d8b 5 (by decide) (А.mp h) hx) (d8b 5 (by decide) (А.mp h) (Nat.le_of_lt hs))))
  let v6:=dv (X.uget (o+(6*n+t)) (d8rd (k:=6) (uk 6 (by norm_num1)) (d8b 6 (by decide) (А.mp h) hx) (d8b 6 (by decide) (А.mp h) (Nat.le_of_lt hs))))
  let v7:=dv (X.uget (o+(7*n+t)) (d8rd (k:=7) (uk 7 (by norm_num1)) (d8b 7 (by decide) (А.mp h) hx) (d8b 7 (by decide) (А.mp h) (Nat.le_of_lt hs))))
  dec8oI X o n (t+1) (wr8 xs (8*t+ὴ 1) v0 v1 v2 v3 v4 v5 v6 v7)
   (h0+v0.abs) (h1+v1.abs) (h2+v2.abs) (h3+v3.abs) (h4+v4.abs) (h5+v5.abs) (h6+v6.abs) (h7+v7.abs) hx hs
 else ⟨xs,h0,h1,h2,h3,h4,h5,h6,h7⟩
termination_by Ѥn-Ѥt
decreasing_by rw[usz t n h];exact Nat.sub_lt_sub_left (А.mp h) (Nat.lt_succ_self _)
lemma aU (a:ѐ) (hs:ѧa<2^32) (i:ὦ) (h:Ѥi< ѧa):a.uget i h=rdW a i:=by
 unfold rdW;rw[Ἄ (by rw[А,Array.usize,qAd _ hs];exact h)]
lemma dec8oI_eq (X:ѐ) (o n:ὦ) (hx:Ѥo+8*Ѥn<=ѧX) (hs:Ѥo+8*Ѥn<2^32) (hxs:ѧX<2^32) (d:Ὣ):
 ∀(t:ὦ) (xs:ὤ) (h0 h1 h2 h3 h4 h5 h6 h7:ὧ),Ѥt+d=Ѥn->dec8oI X o n t xs h0 h1 h2 h3 h4 h5 h6 h7 hx hs=dec8o X o n t xs h0 h1 h2 h3 h4 h5 h6 h7:=by
 induction d with
 | zero=>
  intro t xs h0 h1 h2 h3 h4 h5 h6 h7 e
  rw[dec8oI,dec8o,ἔ (by rw[А];exact omS404 e),ἔ (by rw[А];exact omS404 e)]
 | succ d ih=>
  intro t xs h0 h1 h2 h3 h4 h5 h6 h7 e
  have ht:t<n:=А.2 (by exact omS136 e)
  rw[dec8oI,dec8o,Ἄ ht,Ἄ ht]
  dsimp only
  rw[ih (t+1) _ _ _ _ _ _ _ _ _ (by rw[usz t n ht];exact omS137 e)]
  simp only[aU X hxs]
@[inline] def dec8oC (X:ѐ) (o n:ὦ) (xs:ὤ):H8:=
 if h:Ѥo+8*Ѥn<=ѧX/\Ѥo+8*Ѥn<=4294967295 then dec8oI X o n 0 xs 0 0 0 0 0 0 0 0 h.1 (by omega) else dec8o X o n 0 xs 0 0 0 0 0 0 0 0
lemma dec8oC_eq (X:ѐ) (o n:ὦ) (xs:ὤ) (hxs:ѧX<2^32):dec8oC X o n xs=dec8o X o n 0 xs 0 0 0 0 0 0 0 0:=by
 unfold dec8oC;split
 . rename_i h;exact dec8oI_eq X o n h.1 א hxs (Ѥn) 0 xs 0 0 0 0 0 0 0 0 (by simp only[USize.toNat_zero,zero_add])
 . rfl
end
end
/-! -/
structure F16 where
 (u:UInt8) (s0 s1 s2 s3 s4 s5 s6 s7 t0 t1 t2 t3 t4 t5 t6 t7:ὧ)
def F16of (u:UInt8) (a b:F8):F16:=⟨u,a.s0,a.s1,a.s2,a.s3,a.s4,a.s5,a.s6,a.s7,b.s0,b.s1,b.s2,b.s3,b.s4,b.s5,b.s6,b.s7⟩
abbrev FSa (v:ὧ):Prop:=∀(u:ὥ) (a:ὧ),Float.fma (dv u) v a=a+dv u*v
lemma mI8 {xs:ὤ} {t n i:ὦ} (hx:8*Ѥn+1<=Ѩxs) (hsx:Ѩxs<2^32) (hi:Ѥi=8*Ѥt+1) (h:t<n):(i+ὴ 8).toNat=8*(t+1).toNat+1:=by
 have:=А.mp h;rw[usz t n h,qAf _ _ (by rw[uk 8 ה];omega),uk 8 ה];omega
lemma mK1 {k:Ὣ} {t n:ὦ} (h:2*(k+1)+Ѥt<=Ѥn/\Ѥn<=2*(k+1)+Ѥt+1):t<n/\t+1<n/\2*k+Ѥ(t+1+1)<=Ѥn/\Ѥn<=2*k+Ѥ(t+1+1)+1:=by
 have ht:t<n:=А.mpr א
 have e1:=usz t n ht
 have hu:t+1<n:=А.mpr א
 have e2:=usz (t+1) n hu
 exact ⟨ht,hu,א,א⟩
lemma mK0 (n:ὦ):2*(Ѥn/2)+Ѥ(0:ὦ)<=Ѥn/\Ѥn<=2*(Ѥn/2)+Ѥ(0:ὦ)+1:=by rw[C_z];omega
def go2B (D:ὤ) (xs:ὤ) (W:ѐ) (hd:ѨD=ѧW) (wo wo' n:ὦ) (hx:8*Ѥn+1<=Ѩxs) (hsx:Ѩxs<2^32) (hs:ѧW<2^32)
 (hw:Ѥwo+Ѥn<=ѧW) (hw':Ѥwo'+Ѥn<=ѧW) (k:Ὣ) (t i:ὦ) (hi:Ѥi=8*Ѥt+1) (hk:2*k+Ѥt<=Ѥn/\Ѥn<=2*k+Ѥt+1)
 (s0 s1 s2 s3 s4 s5 s6 s7 t0 t1 t2 t3 t4 t5 t6 t7:ὧ) (r:F16):F16:=
 match k,hk with
 | k+1,hk=>
  have K:=mK1 hk
  have ht:=А.mp K.1
  have hu:=А.mp K.2.1
  have hj:(i+ὴ 8).toNat=8*(t+1).toNat+1:=mI8 hx hsx hi K.1
  let v:=D.uget (wo+t) (hd▸wi wo t n W hw hs ht)
  let u:=D.uget (wo'+t) (hd▸wi wo' t n W hw' hs ht)
  let v':=D.uget (wo+(t+1)) (hd▸wi wo (t+1) n W hw hs hu)
  let u':=D.uget (wo'+(t+1)) (hd▸wi wo' (t+1) n W hw' hs hu)
  let x0:=xs.uget (i+ὴ 0) (xi xs i t n 0 (by decide) ht hi hx hsx)
  let x1:=xs.uget (i+ὴ 1) (xi xs i t n 1 (by decide) ht hi hx hsx)
  let x2:=xs.uget (i+ὴ 2) (xi xs i t n 2 (by decide) ht hi hx hsx)
  let x3:=xs.uget (i+ὴ 3) (xi xs i t n 3 (by decide) ht hi hx hsx)
  let x4:=xs.uget (i+ὴ 4) (xi xs i t n 4 (by decide) ht hi hx hsx)
  let x5:=xs.uget (i+ὴ 5) (xi xs i t n 5 (by decide) ht hi hx hsx)
  let x6:=xs.uget (i+ὴ 6) (xi xs i t n 6 (by decide) ht hi hx hsx)
  let x7:=xs.uget (i+ὴ 7) (xi xs i t n 7 (by decide) ht hi hx hsx)
  let y0:=xs.uget (i+ὴ 8+ὴ 0) (xi xs (i+ὴ 8) (t+1) n 0 (by decide) hu hj hx hsx)
  let y1:=xs.uget (i+ὴ 8+ὴ 1) (xi xs (i+ὴ 8) (t+1) n 1 (by decide) hu hj hx hsx)
  let y2:=xs.uget (i+ὴ 8+ὴ 2) (xi xs (i+ὴ 8) (t+1) n 2 (by decide) hu hj hx hsx)
  let y3:=xs.uget (i+ὴ 8+ὴ 3) (xi xs (i+ὴ 8) (t+1) n 3 (by decide) hu hj hx hsx)
  let y4:=xs.uget (i+ὴ 8+ὴ 4) (xi xs (i+ὴ 8) (t+1) n 4 (by decide) hu hj hx hsx)
  let y5:=xs.uget (i+ὴ 8+ὴ 5) (xi xs (i+ὴ 8) (t+1) n 5 (by decide) hu hj hx hsx)
  let y6:=xs.uget (i+ὴ 8+ὴ 6) (xi xs (i+ὴ 8) (t+1) n 6 (by decide) hu hj hx hsx)
  let y7:=xs.uget (i+ὴ 8+ὴ 7) (xi xs (i+ὴ 8) (t+1) n 7 (by decide) hu hj hx hsx)
  go2B D xs W hd wo wo' n hx hsx hs hw hw' k (t+1+1) (i+ὴ 8+ὴ 8) (mI8 hx hsx hj K.2.1) K.2.2
  (Float.fma y0 v' (Float.fma x0 v s0)) (Float.fma y1 v' (Float.fma x1 v s1)) (Float.fma y2 v' (Float.fma x2 v s2)) (Float.fma y3 v' (Float.fma x3 v s3)) (Float.fma y4 v' (Float.fma x4 v s4)) (Float.fma y5 v' (Float.fma x5 v s5)) (Float.fma y6 v' (Float.fma x6 v s6)) (Float.fma y7 v' (Float.fma x7 v s7))
  (Float.fma y0 u' (Float.fma x0 u t0)) (Float.fma y1 u' (Float.fma x1 u t1)) (Float.fma y2 u' (Float.fma x2 u t2)) (Float.fma y3 u' (Float.fma x3 u t3)) (Float.fma y4 u' (Float.fma x4 u t4)) (Float.fma y5 u' (Float.fma x5 u t5)) (Float.fma y6 u' (Float.fma x6 u t6)) (Float.fma y7 u' (Float.fma x7 u t7)) r
 | 0,_=>
  if h2:t<n then
   have ht:=А.mp h2
   let v:=D.uget (wo+t) (hd▸wi wo t n W hw hs ht)
   let u:=D.uget (wo'+t) (hd▸wi wo' t n W hw' hs ht)
   let x0:=xs.uget (i+ὴ 0) (xi xs i t n 0 (by decide) ht hi hx hsx)
   let x1:=xs.uget (i+ὴ 1) (xi xs i t n 1 (by decide) ht hi hx hsx)
   let x2:=xs.uget (i+ὴ 2) (xi xs i t n 2 (by decide) ht hi hx hsx)
   let x3:=xs.uget (i+ὴ 3) (xi xs i t n 3 (by decide) ht hi hx hsx)
   let x4:=xs.uget (i+ὴ 4) (xi xs i t n 4 (by decide) ht hi hx hsx)
   let x5:=xs.uget (i+ὴ 5) (xi xs i t n 5 (by decide) ht hi hx hsx)
   let x6:=xs.uget (i+ὴ 6) (xi xs i t n 6 (by decide) ht hi hx hsx)
   let x7:=xs.uget (i+ὴ 7) (xi xs i t n 7 (by decide) ht hi hx hsx)
   ⟨r.u,Float.fma x0 v s0,Float.fma x1 v s1,Float.fma x2 v s2,Float.fma x3 v s3,Float.fma x4 v s4,Float.fma x5 v s5,Float.fma x6 v s6,Float.fma x7 v s7,Float.fma x0 u t0,Float.fma x1 u t1,Float.fma x2 u t2,Float.fma x3 u t3,Float.fma x4 u t4,Float.fma x5 u t5,Float.fma x6 u t6,Float.fma x7 u t7⟩
  else ⟨r.u,s0,s1,s2,s3,s4,s5,s6,s7,t0,t1,t2,t3,t4,t5,t6,t7⟩
lemma mgB1 (D:ὤ) (xs:ὤ) (W:ѐ) (hd:ѨD=ѧW) (wo wo' n:ὦ) (hx:8*Ѥn+1<=Ѩxs) (hsx:Ѩxs<2^32) (hs:ѧW<2^32)
 (hw:Ѥwo+Ѥn<=ѧW) (hw':Ѥwo'+Ѥn<=ѧW) (k:Ὣ) (t i:ὦ) (hi:Ѥi=8*Ѥt+1) (hk:2*(k+1)+Ѥt<=Ѥn/\Ѥn<=2*(k+1)+Ѥt+1)
 (s0 s1 s2 s3 s4 s5 s6 s7 t0 t1 t2 t3 t4 t5 t6 t7:ὧ) (r:F16):
 go2B D xs W hd wo wo' n hx hsx hs hw hw' (k+1) t i hi hk s0 s1 s2 s3 s4 s5 s6 s7 t0 t1 t2 t3 t4 t5 t6 t7 r=(
  have K:=mK1 hk
  have ht:=А.mp K.1
  have hu:=А.mp K.2.1
  have hj:(i+ὴ 8).toNat=8*(t+1).toNat+1:=mI8 hx hsx hi K.1
  let v:=D.uget (wo+t) (hd▸wi wo t n W hw hs ht)
  let u:=D.uget (wo'+t) (hd▸wi wo' t n W hw' hs ht)
  let v':=D.uget (wo+(t+1)) (hd▸wi wo (t+1) n W hw hs hu)
  let u':=D.uget (wo'+(t+1)) (hd▸wi wo' (t+1) n W hw' hs hu)
  let x0:=xs.uget (i+ὴ 0) (xi xs i t n 0 ה ht hi hx hsx)
  let x1:=xs.uget (i+ὴ 1) (xi xs i t n 1 ה ht hi hx hsx)
  let x2:=xs.uget (i+ὴ 2) (xi xs i t n 2 ה ht hi hx hsx)
  let x3:=xs.uget (i+ὴ 3) (xi xs i t n 3 ה ht hi hx hsx)
  let x4:=xs.uget (i+ὴ 4) (xi xs i t n 4 ה ht hi hx hsx)
  let x5:=xs.uget (i+ὴ 5) (xi xs i t n 5 ה ht hi hx hsx)
  let x6:=xs.uget (i+ὴ 6) (xi xs i t n 6 ה ht hi hx hsx)
  let x7:=xs.uget (i+ὴ 7) (xi xs i t n 7 ה ht hi hx hsx)
  let y0:=xs.uget (i+ὴ 8+ὴ 0) (xi xs (i+ὴ 8) (t+1) n 0 ה hu hj hx hsx)
  let y1:=xs.uget (i+ὴ 8+ὴ 1) (xi xs (i+ὴ 8) (t+1) n 1 ה hu hj hx hsx)
  let y2:=xs.uget (i+ὴ 8+ὴ 2) (xi xs (i+ὴ 8) (t+1) n 2 ה hu hj hx hsx)
  let y3:=xs.uget (i+ὴ 8+ὴ 3) (xi xs (i+ὴ 8) (t+1) n 3 ה hu hj hx hsx)
  let y4:=xs.uget (i+ὴ 8+ὴ 4) (xi xs (i+ὴ 8) (t+1) n 4 ה hu hj hx hsx)
  let y5:=xs.uget (i+ὴ 8+ὴ 5) (xi xs (i+ὴ 8) (t+1) n 5 ה hu hj hx hsx)
  let y6:=xs.uget (i+ὴ 8+ὴ 6) (xi xs (i+ὴ 8) (t+1) n 6 ה hu hj hx hsx)
  let y7:=xs.uget (i+ὴ 8+ὴ 7) (xi xs (i+ὴ 8) (t+1) n 7 ה hu hj hx hsx)
  go2B D xs W hd wo wo' n hx hsx hs hw hw' k (t+1+1) (i+ὴ 8+ὴ 8) (mI8 hx hsx hj K.2.1) K.2.2
  (Float.fma y0 v' (Float.fma x0 v s0)) (Float.fma y1 v' (Float.fma x1 v s1)) (Float.fma y2 v' (Float.fma x2 v s2)) (Float.fma y3 v' (Float.fma x3 v s3)) (Float.fma y4 v' (Float.fma x4 v s4)) (Float.fma y5 v' (Float.fma x5 v s5)) (Float.fma y6 v' (Float.fma x6 v s6)) (Float.fma y7 v' (Float.fma x7 v s7))
  (Float.fma y0 u' (Float.fma x0 u t0)) (Float.fma y1 u' (Float.fma x1 u t1)) (Float.fma y2 u' (Float.fma x2 u t2)) (Float.fma y3 u' (Float.fma x3 u t3)) (Float.fma y4 u' (Float.fma x4 u t4)) (Float.fma y5 u' (Float.fma x5 u t5)) (Float.fma y6 u' (Float.fma x6 u t6)) (Float.fma y7 u' (Float.fma x7 u t7)) r):=rfl
lemma mgB0 (D:ὤ) (xs:ὤ) (W:ѐ) (hd:ѨD=ѧW) (wo wo' n:ὦ) (hx:8*Ѥn+1<=Ѩxs) (hsx:Ѩxs<2^32) (hs:ѧW<2^32)
 (hw:Ѥwo+Ѥn<=ѧW) (hw':Ѥwo'+Ѥn<=ѧW) (k:Ὣ) (t i:ὦ) (hi:Ѥi=8*Ѥt+1) (hk:2*0+Ѥt<=Ѥn/\Ѥn<=2*0+Ѥt+1)
 (s0 s1 s2 s3 s4 s5 s6 s7 t0 t1 t2 t3 t4 t5 t6 t7:ὧ) (r:F16):
 go2B D xs W hd wo wo' n hx hsx hs hw hw' 0 t i hi hk s0 s1 s2 s3 s4 s5 s6 s7 t0 t1 t2 t3 t4 t5 t6 t7 r=(
  if h2:t<n then
   have ht:=А.mp h2
   let v:=D.uget (wo+t) (hd▸wi wo t n W hw hs ht)
   let u:=D.uget (wo'+t) (hd▸wi wo' t n W hw' hs ht)
   let x0:=xs.uget (i+ὴ 0) (xi xs i t n 0 ה ht hi hx hsx)
   let x1:=xs.uget (i+ὴ 1) (xi xs i t n 1 ה ht hi hx hsx)
   let x2:=xs.uget (i+ὴ 2) (xi xs i t n 2 ה ht hi hx hsx)
   let x3:=xs.uget (i+ὴ 3) (xi xs i t n 3 ה ht hi hx hsx)
   let x4:=xs.uget (i+ὴ 4) (xi xs i t n 4 ה ht hi hx hsx)
   let x5:=xs.uget (i+ὴ 5) (xi xs i t n 5 ה ht hi hx hsx)
   let x6:=xs.uget (i+ὴ 6) (xi xs i t n 6 ה ht hi hx hsx)
   let x7:=xs.uget (i+ὴ 7) (xi xs i t n 7 ה ht hi hx hsx)
   ⟨r.u,Float.fma x0 v s0,Float.fma x1 v s1,Float.fma x2 v s2,Float.fma x3 v s3,Float.fma x4 v s4,Float.fma x5 v s5,Float.fma x6 v s6,Float.fma x7 v s7,Float.fma x0 u t0,Float.fma x1 u t1,Float.fma x2 u t2,Float.fma x3 u t3,Float.fma x4 u t4,Float.fma x5 u t5,Float.fma x6 u t6,Float.fma x7 u t7⟩
  else ⟨r.u,s0,s1,s2,s3,s4,s5,s6,s7,t0,t1,t2,t3,t4,t5,t6,t7⟩):=rfl
lemma mgZu (D:ὤ) (xs:ὤ) (W:ѐ) (hd:ѨD=ѧW) (wo n:ὦ) (hx:8*Ѥn+1<=Ѩxs) (hsx:Ѩxs<2^32) (hs:ѧW<2^32) (hw:Ѥwo+Ѥn<=ѧW) (q:ὥ) (r:F8) (d:Ὣ):
 ∀(t i:ὦ) (hi:Ѥi=8*Ѥt+1) (s0 s1 s2 s3 s4 s5 s6 s7:ὧ),Ѥn-Ѥt=d->
 (go8ZD D xs W hd wo n t i hx hsx hs hw hi s0 s1 s2 s3 s4 s5 s6 s7 q r).u=r.u/\(go8ZD D xs W hd wo n t i hx hsx hs hw hi s0 s1 s2 s3 s4 s5 s6 s7 q r).p=0/\(go8ZD D xs W hd wo n t i hx hsx hs hw hi s0 s1 s2 s3 s4 s5 s6 s7 q r).q=q:=by
 induction d with
 | zero=>
  intro t i hi s0 s1 s2 s3 s4 s5 s6 s7 he
  rw[go8ZD,ἔ (by rw[А];exact omS411 he)];cases r;exact ⟨rfl,rfl,rfl⟩
 | succ d ih=>
  intro t i hi s0 s1 s2 s3 s4 s5 s6 s7 he
  have ht:t<n:=by rw[А];exact omS412 he
  rw[go8ZD,Ἄ ht];exact ih _ _ _ _ _ _ _ _ _ _ _ (by rw[usz t n ht];exact omS413 he)
lemma mXF {x:ὧ} (hx:∃w,x=dv w) (V:ὧ) (hV:FSa V) (a:ὧ):Float.fma x V a=a+x*V:=by obtain ⟨w,rfl⟩:=hx;exact hV w a
lemma mg2 (D:ὤ) (xs:ὤ) (W:ѐ) (hd:ѨD=ѧW) (wo wo' n:ὦ) (hx:8*Ѥn+1<=Ѩxs) (hsx:Ѩxs<2^32) (hs:ѧW<2^32) (hw:Ѥwo+Ѥn<=ѧW) (hw':Ѥwo'+Ѥn<=ѧW)
 (HX:∀r k:Ὣ,r<8->k< Ѥn->∃w,xs.data[8*k+1+r]! =dv w) (HD:∀k,k< ѨD->FSa D.data[k]!) (q:ὥ) (r r':F8) (R:F16) (m:Ὣ):
 ∀(t i:ὦ) (hi:Ѥi=8*Ѥt+1) (hk:2*m+Ѥt<=Ѥn/\Ѥn<=2*m+Ѥt+1) (s0 s1 s2 s3 s4 s5 s6 s7 t0 t1 t2 t3 t4 t5 t6 t7:ὧ),
 go2B D xs W hd wo wo' n hx hsx hs hw hw' m t i hi hk s0 s1 s2 s3 s4 s5 s6 s7 t0 t1 t2 t3 t4 t5 t6 t7 R=
  F16of R.u (go8ZD D xs W hd wo n t i hx hsx hs hw hi s0 s1 s2 s3 s4 s5 s6 s7 q r) (go8ZD D xs W hd wo' n t i hx hsx hs hw' hi t0 t1 t2 t3 t4 t5 t6 t7 q r'):=by
 induction m with
 | zero=>
  intro t i hi hk s0 s1 s2 s3 s4 s5 s6 s7 t0 t1 t2 t3 t4 t5 t6 t7
  by_cases h2:t<n
  . have htn:=А.mp h2
    have hn:¬t+1<n:=by rw[А,usz t n h2];omega
    have B:=λ(k:Ὣ) (hk:k<8)=>qBq (xf:=λr k=>xs.data[8*k+1+r]!) (λ_ _ _ _=>rfl) hi htn hsx hx k hk
    have X:=λ(r:Ὣ) (hr:r<8)=>mXF (HX r _ hr htn)
    have hv:FSa (D.uget (wo+t) (hd▸wi wo t n W hw hs htn)):=by rw[qC5];exact HD _ (hd▸wi wo t n W hw hs htn)
    have hu0:FSa (D.uget (wo'+t) (hd▸wi wo' t n W hw' hs htn)):=by rw[qC5];exact HD _ (hd▸wi wo' t n W hw' hs htn)
    simp only[mgB0,go8ZD.eq_1 D xs W hd wo n t,go8ZD.eq_1 D xs W hd wo' n t,go8ZD.eq_1 D xs W hd wo n (t+1),go8ZD.eq_1 D xs W hd wo' n (t+1),Ἄ h2,ἔ hn,B 0 ה,B 1 ה,B 2 ה,B 3 ה,B 4 ה,B 5 ה,B 6 ה,B 7 ה,X 0 ה _ hv,X 0 ה _ hu0,X 1 ה _ hv,X 1 ה _ hu0,X 2 ה _ hv,X 2 ה _ hu0,X 3 ה _ hv,X 3 ה _ hu0,X 4 ה _ hv,X 4 ה _ hu0,X 5 ה _ hv,X 5 ה _ hu0,X 6 ה _ hv,X 6 ה _ hu0,X 7 ה _ hv,X 7 ה _ hu0]
    cases R;cases r;cases r';rfl
  . simp only[mgB0,go8ZD.eq_1 D xs W hd wo n t,go8ZD.eq_1 D xs W hd wo' n t,ἔ h2]
    cases R;cases r;cases r';rfl
 | succ m ih=>
  intro t i hi hk s0 s1 s2 s3 s4 s5 s6 s7 t0 t1 t2 t3 t4 t5 t6 t7
  have K:=mK1 hk
  have htn:=А.mp K.1
  have hu:=А.mp K.2.1
  have hj:=mI8 hx hsx hi K.1
  have B:=λ(k:Ὣ) (hk:k<8)=>qBq (xf:=λr k=>xs.data[8*k+1+r]!) (λ_ _ _ _=>rfl) hi htn hsx hx k hk
  have B':=λ(k:Ὣ) (hk:k<8)=>qBq (xf:=λr k=>xs.data[8*k+1+r]!) (λ_ _ _ _=>rfl) hj hu hsx hx k hk
  have X:=λ(r:Ὣ) (hr:r<8)=>mXF (HX r _ hr htn)
  have Y:=λ(r:Ὣ) (hr:r<8)=>mXF (HX r _ hr hu)
  have hv:FSa (D.uget (wo+t) (hd▸wi wo t n W hw hs htn)):=by rw[qC5];exact HD _ (hd▸wi wo t n W hw hs htn)
  have hu0:FSa (D.uget (wo'+t) (hd▸wi wo' t n W hw' hs htn)):=by rw[qC5];exact HD _ (hd▸wi wo' t n W hw' hs htn)
  have hv1:FSa (D.uget (wo+(t+1)) (hd▸wi wo (t+1) n W hw hs hu)):=by rw[qC5];exact HD _ (hd▸wi wo (t+1) n W hw hs hu)
  have hu1:FSa (D.uget (wo'+(t+1)) (hd▸wi wo' (t+1) n W hw' hs hu)):=by rw[qC5];exact HD _ (hd▸wi wo' (t+1) n W hw' hs hu)
  simp only[mgB1,go8ZD.eq_1 D xs W hd wo n t,go8ZD.eq_1 D xs W hd wo' n t,go8ZD.eq_1 D xs W hd wo n (t+1),go8ZD.eq_1 D xs W hd wo' n (t+1),Ἄ K.1,Ἄ K.2.1,B 0 ה,B 1 ה,B 2 ה,B 3 ה,B 4 ה,B 5 ה,B 6 ה,B 7 ה,B' 0 ה,B' 1 ה,B' 2 ה,B' 3 ה,B' 4 ה,B' 5 ה,B' 6 ה,B' 7 ה,X 0 ה _ hv,X 0 ה _ hu0,X 1 ה _ hv,X 1 ה _ hu0,X 2 ה _ hv,X 2 ה _ hu0,X 3 ה _ hv,X 3 ה _ hu0,X 4 ה _ hv,X 4 ה _ hu0,X 5 ה _ hv,X 5 ה _ hu0,X 6 ה _ hv,X 6 ה _ hu0,X 7 ה _ hv,X 7 ה _ hu0,Y 0 ה _ hv1,Y 0 ה _ hu1,Y 1 ה _ hv1,Y 1 ה _ hu1,Y 2 ה _ hv1,Y 2 ה _ hu1,Y 3 ה _ hv1,Y 3 ה _ hu1,Y 4 ה _ hv1,Y 4 ה _ hu1,Y 5 ה _ hv1,Y 5 ה _ hu1,Y 6 ה _ hv1,Y 6 ה _ hu1,Y 7 ה _ hv1,Y 7 ה _ hu1]
  exact ih (t+1+1) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
lemma mN1 (x z:Float.Model.UnpackedFloat):Float.Model.UnpackedFloat.fma ⟠ x .notANumber z= .notANumber:=by
 cases x<;>cases z<;>rfl
lemma mN2 (x:Float.Model.UnpackedFloat):Float.Model.UnpackedFloat.add ⟠ x .notANumber= .notANumber:=by
 cases x<;>rfl
lemma mNp:Ἶ ⟠ (Float.Model.UnpackedFloat.pack ⟠ .notANumber)=.notANumber:=rfl
lemma mN3 (x:Float.Model.UnpackedFloat):Float.Model.UnpackedFloat.mul ⟠ x .notANumber= .notANumber:=by
 cases x<;>rfl
lemma mFN (v:ὧ) (h:v.toModel.unpack=.notANumber):FSa v:=by
 intro u a
 show Float.ofModel (Float.Model.fma (dv u).toModel v.toModel a.toModel)=Float.ofModel (Float.Model.add a.toModel (Float.Model.mul (dv u).toModel v.toModel))
 have hP:(Float.Model.mul (dv u).toModel v.toModel).unpack=Ἶ ⟠ (Float.Model.UnpackedFloat.pack ⟠ (Float.Model.UnpackedFloat.mul ⟠ (dv u).toModel.unpack v.toModel.unpack)):=rfl
 unfold Float.Model.fma Float.Model.add
 rw[hP,h,mN1,mN3,mNp,mN2]
lemma mNb:(Ἃ 0x7FF8000000000000).toModel.unpack=.notANumber:=rfl
lemma mNm (x y:ὧ) (h:y.toModel.unpack=.notANumber):(x*y).toModel.unpack=.notANumber:=by
 have hP:(x*y).toModel.unpack=Ἶ ⟠ (Float.Model.UnpackedFloat.pack ⟠ (Float.Model.UnpackedFloat.mul ⟠ x.toModel.unpack y.toModel.unpack)):=rfl
 rw[hP,h,mN3];rfl
lemma mFS {T:ὤ} {h:ὥ} (hT:ѨT=512) (o:TOK 0 T h) (w:ὥ):FSa (dvN T hT w):=by
 obtain ⟨k,-,hu,-⟩:=o
 have ei:((w>>>(23:ὥ)).toUSize).toNat=ѥw/2^23:=by simp only[ὃ,UInt32.toNat_shiftRight,Ἱ];rfl
 unfold dvN;rw[hu,ei]
 unfold pwEntryN;split
 . exact mFN _ (mNm _ _ mNb)
 . have e:dv w=A_mf w*pwEntry (ѥw/2^23):=by unfold dv;rw[(qD1 w).1]
   rw[<-e];exact λu a=>zFV u w a
lemma mFD {T:ὤ} {h:ὥ} (hT:ѨT=512) (o:TOK 0 T h) (W:ѐ):∀k,k< Ѩ(dnOf T hT W)->FSa (dnOf T hT W).data[k]!:=by
 intro k hk
 have hk':k< ѧW:=by simpa only[FloatArray.size,dnOf,Array.size_map] using hk
 have:(dnOf T hT W).data[k]! =dvN T hT W[k]!:=by simp only[dnOf,Array.size_map,hk',Ἕ,Array.getElem_map]
 rw[this];exact mFS hT o _
lemma mGdv (X:ѐ) (o n:ὦ) (d:Ὣ): ∀(t:ὦ) (xs:ὤ) (h0 h1 h2 h3 h4 h5 h6 h7:ὧ),Ѥt+d=Ѥn-> Ѩxs<2^32->8*Ѥn+1<=Ѩxs->
 (∀r k:Ὣ,r<8->k< Ѥt->∃w,xs.data[8*k+1+r]! =dv w)->
 ∀r k:Ὣ,r<8->k< Ѥn->∃w,(dec8o X o n t xs h0 h1 h2 h3 h4 h5 h6 h7).xs.data[8*k+1+r]! =dv w:=by
 induction d with
 | zero=>
  intro t xs h0 h1 h2 h3 h4 h5 h6 h7 he _ _ H
  rw[dec8o,ἔ (by rw[А];exact omS404 he)]
  have e:Ѥt=Ѥn:=by exact omS155 he
  rw[e] at H;exact H
 | succ d ih=>
  intro t xs h0 h1 h2 h3 h4 h5 h6 h7 he hs hb H
  have ht:t<n:=by rw[А];exact omS136 he
  have hu:=usz t n ht
  rw[dec8o,Ἄ ht]
  dsimp only
  obtain ⟨Z1,Z2⟩:=qW8 xs t (dv (rdW X (o+(0*n+t)))) (dv (rdW X (o+(1*n+t)))) (dv (rdW X (o+(2*n+t)))) (dv (rdW X (o+(3*n+t)))) (dv (rdW X (o+(4*n+t)))) (dv (rdW X (o+(5*n+t)))) (dv (rdW X (o+(6*n+t)))) (dv (rdW X (o+(7*n+t)))) hs א
  refine ih (t+1) _ _ _ _ _ _ _ _ _ (by exact omS157 he hu) (by rw[Z1];exact hs) (by rw[Z1];exact hb) ?_
  intro r k hr hk;rw[hu] at hk;rw[Z2];by_cases e:k=Ѥt
  . subst e;interval_cases r<;>simp only[add_zero,Nat.left_eq_add,OfNat.ofNat_ne_zero, ↓reduceIte,one_ne_zero,Nat.add_left_cancel_iff,zero_mul,zero_add,exists_apply_eq_apply',OfNat.one_ne_ofNat,one_mul,Nat.reduceEqDiff]
  . have hk':k< Ѥt:=א
    have hlt:8*k+1+r<8*Ѥt+1:=א
    rw[ἐ (Nat.ne_of_lt (hlt.trans_le (Nat.le_add_right _ _))),ἐ (Nat.ne_of_lt (hlt.trans_le (Nat.le_add_right _ _))),ἐ (Nat.ne_of_lt (hlt.trans_le (Nat.le_add_right _ _))),ἐ (Nat.ne_of_lt (hlt.trans_le (Nat.le_add_right _ _))),ἐ (Nat.ne_of_lt (hlt.trans_le (Nat.le_add_right _ _))),ἐ (Nat.ne_of_lt (hlt.trans_le (Nat.le_add_right _ _))),ἐ (Nat.ne_of_lt (hlt.trans_le (Nat.le_add_right _ _))),ἐ (Nat.ne_of_lt (hlt.trans_le (Nat.le_add_right _ _)))];exact H r k hr hk'
@[inline] def mwc8 (hv:ὥ) (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (d:H8) (nf:ὧ) (n:ὦ) (ri:Ὣ) (ob:ὦ) (j:Ὣ) (out:ѐ) (th:L8) (s0 s1 s2 s3 s4 s5 s6 s7:ὧ):ѐ:=
 let ju:=ѩj
 let bw:=rdW b ju
 let ow:=ѩoutW
 let bv:=dv bw
 let t0:=fin1 th.l0 (spOf th.l0) (s0+bv)
 let out:=wrW out (ob+ju) (if t0==Е then fin8o inW relu X W b ri 0 j s0 nf d.h0 hv bw n else fz relu t0)
 let t1:=fin1 th.l1 (spOf th.l1) (s1+bv)
 let out:=wrW out (ob+(ow+ju)) (if t1==Е then fin8o inW relu X W b ri 1 j s1 nf d.h1 hv bw n else fz relu t1)
 let t2:=fin1 th.l2 (spOf th.l2) (s2+bv)
 let out:=wrW out (ob+(2*ow+ju)) (if t2==Е then fin8o inW relu X W b ri 2 j s2 nf d.h2 hv bw n else fz relu t2)
 let t3:=fin1 th.l3 (spOf th.l3) (s3+bv)
 let out:=wrW out (ob+(3*ow+ju)) (if t3==Е then fin8o inW relu X W b ri 3 j s3 nf d.h3 hv bw n else fz relu t3)
 let t4:=fin1 th.l4 (spOf th.l4) (s4+bv)
 let out:=wrW out (ob+(4*ow+ju)) (if t4==Е then fin8o inW relu X W b ri 4 j s4 nf d.h4 hv bw n else fz relu t4)
 let t5:=fin1 th.l5 (spOf th.l5) (s5+bv)
 let out:=wrW out (ob+(5*ow+ju)) (if t5==Е then fin8o inW relu X W b ri 5 j s5 nf d.h5 hv bw n else fz relu t5)
 let t6:=fin1 th.l6 (spOf th.l6) (s6+bv)
 let out:=wrW out (ob+(6*ow+ju)) (if t6==Е then fin8o inW relu X W b ri 6 j s6 nf d.h6 hv bw n else fz relu t6)
 let t7:=fin1 th.l7 (spOf th.l7) (s7+bv)
 let out:=wrW out (ob+(7*ow+ju)) (if t7==Е then fin8o inW relu X W b ri 7 j s7 nf d.h7 hv bw n else fz relu t7)
 out
lemma mPh {ph:Ὡ} {P:Prop} (hc:ph=true\/P) (hp:¬ph=true):P:=hc.resolve_left hp
@[inline] def F8x (r:F8) (s0 s1 s2 s3 s4 s5 s6 s7:ὧ) (q:ὥ):F8:=⟨r.u,s0,s1,s2,s3,s4,s5,s6,s7,0,q⟩
def cols8ZD2 (D:ὤ) (hv:ὥ) (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (hd:ѨD=ѧW) (d:H8) (nf:ὧ)
 (n:ὦ) (ri:Ὣ) (ob:ὦ) (q:ὥ) (th:L8):Ὣ->Ὣ->ѐ->F8->F16->Ὡ->CN
 | 0,j,out,r0,_,_=>cols8ZDo D hv inW outW relu X W b hd d nf n ri ob j out q th r0
 | k+1,j,out,r0,R,ph=>
  if hc:ph=true\/(j+1<outW/\8*Ѥn+1<=d.xs.size/\d.xs.size<=4294967295/\ѧW<=4294967295/\Ѥ(ѩj*n)+Ѥn<=ѧW/\Ѥ(ѩ(j+1)*n)+Ѥn<=ѧW) then
   let R':=if hp:ph=true then R else
    have G:=mPh hc hp
    go2B D d.xs W hd (ѩj*n) (ѩ(j+1)*n) n G.2.1 (qA9 G.2.2.1) (qA9 G.2.2.2.1) G.2.2.2.2.1 G.2.2.2.2.2 (Ѥn/2) 0 (ὴ 1)
     (by rw[show (0:ὦ).toNat=0 from rfl,uk 1 (by decide)]) (mK0 n) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 R
   let s0:=if ph then R'.t0 else R'.s0
   let s1:=if ph then R'.t1 else R'.s1
   let s2:=if ph then R'.t2 else R'.s2
   let s3:=if ph then R'.t3 else R'.s3
   let s4:=if ph then R'.t4 else R'.s4
   let s5:=if ph then R'.t5 else R'.s5
   let s6:=if ph then R'.t6 else R'.s6
   let s7:=if ph then R'.t7 else R'.s7
   if okF (s0+s1+s2+s3+s4+s5+s6+s7) then
    cols8ZD2 D hv inW outW relu X W b hd d nf n ri ob q th k (j+1) (mwc8 hv inW outW relu X W b d nf n ri ob j out th s0 s1 s2 s3 s4 s5 s6 s7) (F8x r0 s0 s1 s2 s3 s4 s5 s6 s7 q) R' (!ph)
   else ⟨out,false⟩
  else cols8ZDo D hv inW outW relu X W b hd d nf n ri ob j out q th r0
@[inline] def cols8ZDA (D:ὤ) (hv:ὥ) (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (hd:ѨD=ѧW) (d:H8) (nf:ὧ)
 (n:ὦ) (ri:Ὣ) (ob:ὦ) (j:Ὣ) (out:ѐ) (q:ὥ) (th:L8) (r0:F8):CN:=
 if armT then cols8ZD2 D hv inW outW relu X W b hd d nf n ri ob q th (outW+1) j out r0 ⟨0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0⟩ false
 else cols8ZDo D hv inW outW relu X W b hd d nf n ri ob j out q th r0
lemma mZo1 (D:ὤ) (hv:ὥ) (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (hd:ѨD=ѧW) (d:H8) (nf:ὧ) (n:ὦ) (ri:Ὣ) (ob:ὦ) (q:ὥ) (th:L8) (j:Ὣ) (out:ѐ) (r0 R:F8) (hj:j<outW)
 (hg:8*Ѥn+1<=d.xs.size/\d.xs.size<=4294967295/\ѧW<=4294967295/\(ѩj*n).toNat+Ѥn<=ѧW) (hi0:(ὴ 1).toNat=8*(0:ὦ).toNat+1)
 (hR:go8ZD D d.xs W hd (ѩj*n) n 0 (ὴ 1) hg.1 (qA9 hg.2.1) (qA9 hg.2.2.1) hg.2.2.2 hi0 0 0 0 0 0 0 0 0 q r0=R):
 cols8ZDo D hv inW outW relu X W b hd d nf n ri ob j out q th r0=
  (if okF (R.s0+R.s1+R.s2+R.s3+R.s4+R.s5+R.s6+R.s7) then
   cols8ZDo D hv inW outW relu X W b hd d nf n ri ob (j+1) (mwc8 hv inW outW relu X W b d nf n ri ob j out th R.s0 R.s1 R.s2 R.s3 R.s4 R.s5 R.s6 R.s7) q th R
  else ⟨out,false⟩):=by
 rw[cols8ZDo,Ἄ hj];dsimp only;rw[Ἄ hg,g8ZS,hR];rfl
def F16t (R:F16) (G:F8):Prop:=R.t0=G.s0/\R.t1=G.s1/\R.t2=G.s2/\R.t3=G.s3/\R.t4=G.s4/\R.t5=G.s5/\R.t6=G.s6/\R.t7=G.s7
lemma mF8x (r R:F8) (q:ὥ) (h0:R.u=r.u) (h1:R.p=0) (h2:R.q=q):F8x r R.s0 R.s1 R.s2 R.s3 R.s4 R.s5 R.s6 R.s7 q=R:=by
 cases r;cases R;simp only at h0 h1 h2;subst h0 h1 h2;rfl
lemma mc2E (D:ὤ) (hv:ὥ) (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (hd:ѨD=ѧW) (d:H8) (nf:ὧ) (n:ὦ) (ri:Ὣ) (ob:ὦ) (q:ὥ) (th:L8)
 (HX:8*Ѥn+1<=d.xs.size->d.xs.size<=4294967295->∀r k:Ὣ,r<8->k< Ѥn->∃w,d.xs.data[8*k+1+r]! =dv w) (HD:∀k,k< ѨD->FSa D.data[k]!) (m:Ὣ):
 ∀(j:Ὣ) (out:ѐ) (r0:F8) (R:F16) (ph:Ὡ),(ph=true->∃(hj:j<outW) (hg:8*Ѥn+1<=d.xs.size/\d.xs.size<=4294967295/\ѧW<=4294967295/\(ѩj*n).toNat+Ѥn<=ѧW) (hi0:(ὴ 1).toNat=8*(0:ὦ).toNat+1),F16t R (go8ZD D d.xs W hd (ѩj*n) n 0 (ὴ 1) hg.1 (qA9 hg.2.1) (qA9 hg.2.2.1) hg.2.2.2 hi0 0 0 0 0 0 0 0 0 q r0))->
  cols8ZD2 D hv inW outW relu X W b hd d nf n ri ob q th m j out r0 R ph=cols8ZDo D hv inW outW relu X W b hd d nf n ri ob j out q th r0:=by
 induction m with
 | zero=>intro j out r0 R ph _;rfl
 | succ m ih=>
  intro j out r0 R ph I
  rw[cols8ZD2]
  cases ph
  . by_cases hc:j+1<outW/\8*Ѥn+1<=d.xs.size/\d.xs.size<=4294967295/\ѧW<=4294967295/\Ѥ(ѩj*n)+Ѥn<=ѧW/\Ѥ(ѩ(j+1)*n)+Ѥn<=ѧW
    . rw[Ἄ (Or.inr hc)]
      simp only[Bool.false_eq_true,↓reduceDIte,↓reduceIte,Bool.not_false]
      have hi0:(ὴ 1).toNat=8*(0:ὦ).toNat+1:=by rw[C_z,uk 1 ה]
      have hj:j<outW:=א
      have hg:8*Ѥn+1<=d.xs.size/\d.xs.size<=4294967295/\ѧW<=4294967295/\(ѩj*n).toNat+Ѥn<=ѧW:=⟨hc.2.1,hc.2.2.1,hc.2.2.2.1,hc.2.2.2.2.1⟩
      have hg':8*Ѥn+1<=d.xs.size/\d.xs.size<=4294967295/\ѧW<=4294967295/\(ѩ(j+1)*n).toNat+Ѥn<=ѧW:=⟨hc.2.1,hc.2.2.1,hc.2.2.2.1,hc.2.2.2.2.2⟩
      have u1:=mgZu D d.xs W hd (ѩj*n) n hg.1 (qA9 hg.2.1) (qA9 hg.2.2.1) hg.2.2.2 q r0 _ 0 (ὴ 1) hi0 0 0 0 0 0 0 0 0 rfl
      generalize hR1:go8ZD D d.xs W hd (ѩj*n) n 0 (ὴ 1) hg.1 (qA9 hg.2.1) (qA9 hg.2.2.1) hg.2.2.2 hi0 0 0 0 0 0 0 0 0 q r0=R1 at u1
      rw[mg2 D d.xs W hd _ _ n hc.2.1 (qA9 hc.2.2.1) (qA9 hc.2.2.2.1) hc.2.2.2.2.1 hc.2.2.2.2.2 (HX hc.2.1 hc.2.2.1) HD q r0 R1 R _ 0 (ὴ 1) hi0 (mK0 n) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0,hR1]
      generalize hR2:go8ZD D d.xs W hd (ѩ(j+1)*n) n 0 (ὴ 1) hc.2.1 (qA9 hc.2.2.1) (qA9 hc.2.2.2.1) hc.2.2.2.2.2 hi0 0 0 0 0 0 0 0 0 q R1=R2
      rw[mZo1 D hv inW outW relu X W b hd d nf n ri ob q th j out r0 R1 hj hg hi0 hR1]
      simp only[F16of]
      rw[mF8x r0 R1 q u1.1 u1.2.1 u1.2.2,ih (j+1) _ R1 _ true (λ_=>⟨hc.1,hg',hi0,by rw[hR2];exact ⟨rfl,rfl,rfl,rfl,rfl,rfl,rfl,rfl⟩⟩)]
      rfl
    . rw[ἔ (by simp only[Bool.false_eq_true,false_or];exact hc)]
  . obtain ⟨hj,hg,hi0,F⟩:=I rfl
    rw[Ἄ (Or.inl rfl)]
    simp only[↓reduceDIte,↓reduceIte,Bool.not_true]
    have u1:=mgZu D d.xs W hd (ѩj*n) n hg.1 (qA9 hg.2.1) (qA9 hg.2.2.1) hg.2.2.2 q r0 _ 0 (ὴ 1) hi0 0 0 0 0 0 0 0 0 rfl
    generalize hR1:go8ZD D d.xs W hd (ѩj*n) n 0 (ὴ 1) hg.1 (qA9 hg.2.1) (qA9 hg.2.2.1) hg.2.2.2 hi0 0 0 0 0 0 0 0 0 q r0=R1 at u1 F
    obtain ⟨f0,f1,f2,f3,f4,f5,f6,f7⟩:=F
    rw[mZo1 D hv inW outW relu X W b hd d nf n ri ob q th j out r0 R1 hj hg hi0 hR1,f0,f1,f2,f3,f4,f5,f6,f7,mF8x r0 R1 q u1.1 u1.2.1 u1.2.2,ih (j+1) _ R1 _ false (λh=>absurd h ה)]
lemma mGsz (X:ѐ) (o n:ὦ) (d:Ὣ): ∀(t:ὦ) (xs:ὤ) (h0 h1 h2 h3 h4 h5 h6 h7:ὧ),Ѥn-Ѥt=d->(dec8o X o n t xs h0 h1 h2 h3 h4 h5 h6 h7).xs.size=Ѩxs:=by
 induction d with
 | zero=>
  intro t xs h0 h1 h2 h3 h4 h5 h6 h7 he
  rw[dec8o,ἔ (by rw[А];exact omS411 he)]
 | succ d ih=>
  intro t xs h0 h1 h2 h3 h4 h5 h6 h7 he
  have ht:t<n:=by rw[А];exact omS412 he
  rw[dec8o,Ἄ ht];dsimp only
  rw[ih _ _ _ _ _ _ _ _ _ _ (by rw[usz t n ht];exact omS413 he)];unfold wr8;simp only[qD6]
lemma mcA (D:ὤ) (hv:ὥ) (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (hd:ѨD=ѧW) (d:H8) (nf:ὧ) (n:ὦ) (ri:Ὣ) (ob:ὦ) (j:Ὣ) (out:ѐ) (q:ὥ) (th:L8) (r0:F8)
 (HX:8*Ѥn+1<=d.xs.size->d.xs.size<=4294967295->∀r k:Ὣ,r<8->k< Ѥn->∃w,d.xs.data[8*k+1+r]! =dv w) (HD:∀k,k< ѨD->FSa D.data[k]!):
 cols8ZDA D hv inW outW relu X W b hd d nf n ri ob j out q th r0=cols8ZDo D hv inW outW relu X W b hd d nf n ri ob j out q th r0:=by
 unfold cols8ZDA;split
 . exact mc2E D hv inW outW relu X W b hd d nf n ri ob q th HX HD _ j out r0 _ false (λh=>absurd h ה)
 . rfl
lemma mGX (X:ѐ) (o n:ὦ) (xs:ὤ):8*Ѥn+1<=(dec8o X o n 0 xs 0 0 0 0 0 0 0 0).xs.size->(dec8o X o n 0 xs 0 0 0 0 0 0 0 0).xs.size<=4294967295->
 ∀r k:Ὣ,r<8->k< Ѥn->∃w,(dec8o X o n 0 xs 0 0 0 0 0 0 0 0).xs.data[8*k+1+r]! =dv w:=by
 have e:=mGsz X o n Ѥn 0 xs 0 0 0 0 0 0 0 0 (by rw[C_z,Nat.sub_zero])
 rw[e];intro h8 h9
 exact mGdv X o n Ѥn 0 xs 0 0 0 0 0 0 0 0 (by rw[C_z,Ἇ]) א h8 (λr k _ hk=>absurd hk (by rw[C_z];exact Ἲ _))
lemma qZsh (W:ѐ) (c N:Ὣ):∀(a:Ὣ) (q:ὥ),(List.range' a N).foldl (λ(q:ὥ) (s:Ὣ)=>q&&&((W[c+s]!&&&В)-1)) q=(List.range' (c+a) N).foldl (λ(q:ὥ) (s:Ὣ)=>q&&&((W[s]!&&&В)-1)) q:=by
 induction N with
 | zero=>intro a q;rfl
 | succ N ih=>intro a q;rw[List.range'_succ,List.range'_succ,ὔ,ὔ,ih (a+1),show c+(a+1)=c+a+1 by exact omS329]
lemma qZsp (W:ѐ) (m N:Ὣ):∀(s:Ὣ) (q:ὥ),(List.range' s (m+N)).foldl (λ(q:ὥ) (s:Ὣ)=>q&&&((W[s]!&&&В)-1)) q=(List.range' (s+m) N).foldl (λ(q:ὥ) (s:Ὣ)=>q&&&((W[s]!&&&В)-1)) ((List.range' s m).foldl (λ(q:ὥ) (s:Ὣ)=>q&&&((W[s]!&&&В)-1)) q):=by
 induction m with
 | zero=>intro s q;rw[Ἇ,Ἵ];rfl
 | succ m ih=>intro s q;rw[show m+1+N=(m+N)+1 by exact omS330,List.range'_succ,List.range'_succ,ὔ,ὔ,ih (s+1),show s+1+m=s+(m+1) by exact omS146]
lemma qScanE (W:ѐ) (hs:ѧW<2^32) (d:Ὣ):∀(i:ὦ) (q:ὥ),Ѥi+d=ѧW->qScan W W.usize i q=(List.range' Ѥi d).foldl (λ(q:ὥ) (s:Ὣ)=>q&&&((W[s]!&&&В)-1)) q:=by
 have hu:W.usize.toNat=ѧW:=qBe _ hs
 induction d with
 | zero=>intro i q e;rw[qScan,ἔ (by rw[А,hu];(clear hs;(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›)))))];rfl
 | succ d ih=>
  intro i q e
  have hi:i<W.usize:=by rw[А,hu];(clear hs;(clear hu;exact omS136 e))
  rw[qScan,Ἄ hi,ih (i+1) _ (by rw[usz i _ hi];(clear hs hi;(clear hu;exact omS137 e))),usz i _ hi,List.range'_succ,ὔ,qAg W i hs]
set_option maxHeartbeats 1000000 in
lemma c8ZEo (T:ὤ) (hT:ѨT=512) (hv:ὥ) (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (hd:(dnOf T hT W).size=ѧW) (d:H8) (nf:ὧ) (n:ὦ) (ri:Ὣ) (ob:ὦ) (hn:Ѥn=inW) (hW:outW*inW=ѧW) (hs:ѧW<2^32) (pi:0< inW) (th:L8) (k:Ὣ):
 ∀(j:Ὣ) (out:ѐ) (q:ὥ) (r0 r0':F8),j+k=outW->
  cols8ZDo (dnOf T hT W) hv inW outW relu X W b hd d nf n ri ob j out ((List.range' (j*inW) (k*inW)).foldl (λ(q:ὥ) (s:Ὣ)=>q&&&((W[s]!&&&В)-1)) q) th r0'=cols8Zo T hT hv inW outW relu X W b d nf n ri ob j out q th r0:=by
 have hO:outW<2^32:=by have:=Ὄ outW pi;(clear hn pi;omega)
 have W1:∀j,j<=outW->(ѩj*n).toNat=j*inW:=λj hj=>by
  have h1:j*inW<=outW*inW:=Ἒ _ hj
  rw[qBa _ _ (by rw[qAd _ (by (exact Nat.lt_of_le_of_lt ‹_› ‹_›)),hn];(clear hn pi hO hj;exact omS323 hW hs h1)),qAd _ (by (exact Nat.lt_of_le_of_lt ‹_› ‹_›)),hn]
 induction k with
 | zero=>
  intro j out q r0 r0' e
  rw[cols8ZDo,cols8Zo,ἔ (by (clear hd hn hW hs pi hO;(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›))))),ἔ (by (clear hd hn hW hs pi hO;(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›))))),ὒ]
  rfl
 | succ k ih=>
  intro j out q r0 r0' e
  have hj:j<outW:=by (clear hn hW hs pi hO;exact omS136 e)
  rw[cols8ZDo,cols8Zo,Ἄ hj,Ἄ hj]
  dsimp only
  by_cases hg:8*Ѥn+1<=d.xs.size/\d.xs.size<=4294967295/\ѧW<=4294967295/\(ѩj*n).toNat+Ѥn<=ѧW
  . rw[Ἄ hg,Ἄ hg,g8ZS,g8ZE (q:=q) (r:=r0)]
    have hi0:(ὴ 1).toNat=8*(0:ὦ).toNat+1:=by rw[C_z,uk 1 ה]
    have hq:=(qZ1 (T:=T) (hT:=hT) hg.1 (qA9 hg.2.1) (qA9 hg.2.2.1) hg.2.2.2 (λk=>W[j*inW+k]!) (λk _=>by rw[W1 j hj.le]) Ѥn 0 (ὴ 1) hi0 0 0 0 0 0 0 0 0 q r0 (by rw[C_z,Ἇ])).2
    generalize go8Z T hT d.xs W (ѩj*n) n 0 (ὴ 1) hg.1 (qA9 hg.2.1) (qA9 hg.2.2.1) hg.2.2.2 hi0 0 0 0 0 0 0 0 0 q r0=R at hq⊢
    rw[C_z,hn,qZsh W (j*inW) inW 0 q,Ἵ] at hq
    rw[show (k+1)*inW=inW+k*inW by ring,qZsp W inW (k*inW) (j*inW) q,<-hq,show j*inW+inW=(j+1)*inW by ring]
    dsimp only[F8q]
    exact if_congr Iff.rfl (ih (j+1) _ R.q R _ (by (clear hn hW hs pi hO hj hg hi0 hq;exact omS137 e))) rfl
  . rw[ἔ hg,ἔ hg]
lemma c8ZSo (T:ὤ) (hT:ѨT=512) (hv:ὥ) (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (d:H8) (nf:ὧ) (n:ὦ) (ri:Ὣ) (ob:ὦ) (hn:Ѥn=inW) (hW:outW*inW=ѧW) (hs:ѧW<2^32) (pi:0< inW) (out:ѐ) (th:L8) (r0:F8):
 (if hd:(dnOf T hT W).size=ѧW then cols8ZDo (dnOf T hT W) hv inW outW relu X W b hd d nf n ri ob 0 out (qScan W W.usize 0 Е) th r0 else cols8Zo T hT hv inW outW relu X W b d nf n ri ob 0 out Е th r0)=cols8Zo T hT hv inW outW relu X W b d nf n ri ob 0 out Е th r0:=by
 split
 . rename_i hd
   have e:=c8ZEo T hT hv inW outW relu X W b hd d nf n ri ob hn hW hs pi th outW 0 out Е r0 r0 (by (clear hn hW hs pi hd;(clear hT;exact omS139)))
   rw[ὒ] at e
   rw[qScanE W hs ѧW 0 Е (by rw[C_z,Ἇ]),C_z,<-hW]
   exact e
 . rfl
lemma qDE (X:ѐ) (o n:ὦ) (hX:Ѥo+Ѥn<=ѧX) (hs:ѧX<2^32) (t:ὦ) (buf:ὤ) (l:ὧ) (hb:Ѥn<=Ѩbuf) (hbs:Ѩbuf<2^32):
 decE X o n t buf l hX hs hb=decD X o n t buf l:=by
 rw[decE,decD];by_cases h:t<n
 . rw[Ἄ h,Ἄ h];dsimp only
   have ht:=А.mp h
   have c1:o+t<X.usize:=by rw[А,qBe X hs];exact wi o t n X hX hs ht
   have c2:t<buf.usize:=by rw[А,FloatArray.usize,qAd _ hbs];omega
   have e1:rdW X (o+t)=X.uget (o+t) (wi o t n X hX hs ht):=by unfold rdW;rw[Ἄ c1]
   rw[e1]
   have e2:wrF buf t (dv (X.uget (o+t) (wi o t n X hX hs ht)))=buf.uset t (dv (X.uget (o+t) (wi o t n X hX hs ht))) א:=by unfold wrF;rw[Ἄ c2]
   rw[e2];exact qDE X o n hX hs (t+1) _ _ _ (by rw[qUs];exact hbs)
 . rw[ἔ h,ἔ h]
termination_by Ѥn-Ѥt
decreasing_by rw[usz t n h];have:=А.mp h;(clear h;omega)
lemma qDF (X:ѐ) (o n:ὦ) (buf:ὤ) (l:ὧ):decF X o n buf l=decD X o n 0 buf l:=by
 unfold decF;split
 . rename_i hh;exact qDE X o n hh.1 (by have:=hh.2.1;omega) 0 buf l hh.2.2.1 (by have:=hh.2.2.2;omega)
 . rfl
lemma qEr0:linF0=linearSpec:=by
 funext rows inW outW relu X W b;unfold linF0;simp only[qDF];(try dsimp only);have Hs:=qGd rows W;generalize scanW rows W=hw at Hs⊢;split
 . rename_i h;obtain ⟨h1,h2,h3,h4,h5,h6,h7,h8,h9,h10,h11,h12,h13⟩:=h;replace h7:=qA9 h7;replace h8:=qA9 h8;replace h9:=qA9 h9;replace h10:=qA9 h10
   have G0:=λ(A:∀q,q< ѧW->good W[q]! =true)=>
    (⟨h1,h2,h3,h4,h5,h6,h7,h8,h9,h10,(qA8 0 X h7 h11).2.2.1,A,(qA8 1 b h9 h13).2.2.1⟩:C_G rows inW outW X W b)
   have e:linearSpec rows inW outW relu X W b=Array.ofFn (n:=rows*outW) λq=>C_sp inW outW relu X W b q.val:=rfl;have hnf:(inW.toUInt64+8).toFloat*Ἃ 0x3CA0000000000000=nfOf inW:=rfl;rw[e,hnf];split
   . rename_i hr1;have e1:rows=1:=by simpa only[beq_iff_eq] using hr1
     subst e1;have GX:=(qA8 0 X h7 h11).2.2.1;have sX1:0*inW+inW<=ѧX:=(by clear h2 h3 h4 h5 h6 h11 h12 h13 h7 h8 h9 h10;exact omS332 h1);have nm:inW.toUSize.toNat=inW:=qAd _ (by exact Nat.lt_of_lt_of_le ‹_› ה);have P0:=qZf inW (by (exact Nat.lt_of_lt_of_le ‹_› ה))
     have H0:C_Inv (1*outW) (C_sp inW outW relu X W b) (λq=>q<0) (ἇ (1*outW) 0):=⟨by simp only[one_mul,Array.size_replicate],λq _ h=>absurd h (Ἲ _)⟩
     rw[(wbGet inW outW W).2];dsimp only;split
     . rename_i hv;have Hv:=qA8 2 W h8 hv;have G:=G0 Hv.2.2.1;have hb0:(0*outW).toUSize.toNat=0*outW:=qAd _ (by clear h2 h3 h4 h5 h6 h11 h12 h13 h7 h8 h9 h1 h10 sX1 nm P0 hv;exact omS333);have HdE:C_Dw (ἆ 0) W:=λh q hq=>by have e:0=ѧW:=h;(clear h2 h3 h4 h5 h6 h11 h12 h13 h7 h8 h9 h1 h10 sX1 nm P0 hv hb0;(clear h;omega))
       split
       . rename_i hok;obtain ⟨o1,o2⟩:=Bool.and_eq_true_iff.mp hok;have h8k:inW<=8192:=by simpa only[decide_eq_true_eq] using o2
         obtain ⟨v1,v2,v3⟩:=qVz (a:=0*inW) (n:=ѩinW) h7 (by rw[nm];exact sX1) inW 0 (zFor inW) 0 (by rw[C_z,Ἇ,nm]) (by rw[nm];exact P0.1) P0.2 (λk hk=>absurd hk (Ἲ _)) rfl o1
         rw[nm] at v2 v3
         have Rd:=qRd X inW _ h8k v2
         have Rw:C_Row inW X 0 idA (decD X (0*inW).toUSize ѩinW 0 (zFor inW) 0).buf ѩinW (ἄ inW) (decD X (0*inW).toUSize ѩinW 0 (zFor inW) 0).l:=⟨by rw[nm,List.length_range],by simp only[List.length_range,Std.le_refl],by rw[qIs];exact h8k,⟨by rw[v1];exact P0.1,by rw[v1];exact P0.2⟩,Ὂ,λt ht=>Ἅ.1 ht,λt ht hn=>absurd (Ἅ.2 ht) hn,Rd,v3⟩
         have HI:∀k,k< Ѥ(ѩinW)->(idA[k]!).toNat=k:=λk hk=>qIA k (Nat.lt_of_lt_of_le (by rwa[nm] at hk) h8k)
         rw[c1VS',c1VW _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ HI (Ὗ _)]
         rw[c1WE _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (λj hj=>by rw[Rw.len] at hj;have hp:=List.getElem?_eq_getElem hj;exact ⟨_,G.gx ((by exact Nat.le_refl _):0<1) _ (Rw.lt _ (List.mem_of_getElem? hp)),(Rw.rd j _ hp).2⟩) HdE (by rw[nm,h2]) (by rw[h3];exact h9) (λj i k a b c d=>by rw[nm] at c⊢;exact wbP W inW outW j i k a b c d) (λho i k hi hk=>by rw[nm] at hi⊢;exact wbPt W inW outW i k ho hi hk) 0 rfl]
         obtain ⟨gb1,gb2,-,gb4⟩:=qA8 1 b h9 h13
         obtain ⟨pb0,pb1,pb2⟩:=qCs gb1 gb2
         obtain ⟨fbm,Rbm⟩:=qAm _ gb1
         have hbm:∀j,j<outW-> |R (dv b[j]!)| <=R (dv (scanH 1 b)):=λj hj=>by
          have gj:=(qA8 1 b h9 h13).2.2.1 j (by rw[<-h3];exact hj)
          rw[(qAm _ gj).2,Rbm];exact pb2 _ gj (gb4 j (by rw[<-h3];exact hj))
         have bb:R (dv (scanH 1 b))<=2^(290:ℤ):=by rw[Rbm];exact pb1.trans (by rw[show (2:ℝ)^128=2^(128:ℤ) by norm_num];exact ἒ ב ב)
         refine qBv (qD0 G relu ((by exact Nat.le_refl _):0<1) Hv Rw hb0 HdE (dv (scanH 1 b)) fbm hbm bb rfl outW 0 _ (Ἇ _) (by rw[ὒ,Ἵ];exact H0)) (λq hq=>by (clear h2 h4 h5 h6 h11 h12 h13 h7 h8 h9 h1 h10 sX1 nm P0 hv hb0 h8k v1;exact omS334 hq))
       . rw[qAi];rfl
     . rw[qAi];rfl
   . rw[qAi];rfl
 . rw[qAi];rfl
end
lemma bOfS (b:ѐ):Ѩ(bOf b)=ѧb:=by simp only[FloatArray.size,bOf,Array.size_map]
lemma rdBOf (b:ѐ) (u:ὦ) (h:Ѥu< ѧb) (hs:ѧb<2^32):rdB (bOf b) u=dv (rdW b u):=by
 have h2:u<b.usize:=А.mpr (by rw[Array.usize,qAd _ hs];exact h)
 have h1:u<(bOf b).usize:=А.mpr (by rw[FloatArray.usize,bOfS,qAd _ hs];exact h)
 unfold rdB rdW;rw[Ἄ h1,Ἄ h2]
 simp only[FloatArray.uget,bOf,Array.ugetElem_eq_getElem,Array.getElem_map,Array.uget]
lemma fLU (inW:Ὣ) (relu:Ὡ) (X W b:ѐ) (i k:Ὣ) (s nf a:ὧ) (hx:ὥ) (lo:Ὠ) (u n:ὦ) (hu:Ѥu=k) (h:Ѥu< ѧb) (hs:ѧb<2^32):
 fin8LU inW relu X W b i s nf a hx lo (spOf lo) (bOf b) u n=fin8L inW relu X W b i k s nf a hx lo (rdW b u) n:=by
 unfold fin8LU fin8L fin8u fin8n;rw[rdBOf b u h hs,hu]
lemma lt8 (j k:Ὣ) (hk:k<=8) (h:j+8<2^32):j+k<2^32:=א
lemma lk8 (j c:Ὣ) (hc:c+1<=8):j+c+1<=j+8:=א
lemma fb (u:ὦ) (k j n:Ὣ) (hu:Ѥu=k) (hk:k+1<=j+8) (hj:j+8<=n):Ѥu<n:=א
lemma fNw (inW:Ὣ) (relu:Ὡ) (X W b:ѐ) (i:Ὣ) (nf a:ὧ) (hx:ὥ) (lo:Ὠ) (m:ὦ) (j:Ὣ) (base:ὦ) (out:ѐ) (s0 s1 s2 s3 s4 s5 s6 s7:ὧ) (hj:j+8<=ѧb) (hs:ѧb<2^32):
 fN8 inW relu X W b i (bOf b) nf a hx lo (spOf lo) m j base out s0 s1 s2 s3 s4 s5 s6 s7=fw8 inW relu X W b i nf a hx lo m j base out s0 s1 s2 s3 s4 s5 s6 s7:=by
 have hj32:j+8<2^32:=Nat.lt_of_le_of_lt hj hs
 have e0:(ѩj).toNat=j:=qAd _ (lt8 j 0 ה hj32)
 obtain ⟨e1,e2,e3,-⟩:=qFk e0 (lt8 j 3 ה hj32)
 have f0:((j+4).toUSize).toNat=j+4:=qAd _ (lt8 j 4 ה hj32)
 obtain ⟨f1,f2,f3,-⟩:=qFk f0 (lt8 j 7 ה hj32)
 unfold fN8 fw8
 simp only[fLU inW relu X W b i j _ nf a hx lo (ѩj) m e0 (fb _ _ j _ e0 (lk8 j 0 ה) hj) hs,fLU inW relu X W b i (j+1) _ nf a hx lo (ѩj+1) m e1 (fb _ _ j _ e1 (lk8 j 1 ה) hj) hs,
  fLU inW relu X W b i (j+2) _ nf a hx lo (ѩj+2) m e2 (fb _ _ j _ e2 (lk8 j 2 ה) hj) hs,fLU inW relu X W b i (j+3) _ nf a hx lo (ѩj+3) m e3 (fb _ _ j _ e3 (lk8 j 3 ה) hj) hs,
  fLU inW relu X W b i (j+4) _ nf a hx lo ((j+4).toUSize) m f0 (fb _ _ j _ f0 (lk8 j 4 ה) hj) hs,fLU inW relu X W b i (j+4+1) _ nf a hx lo ((j+4).toUSize+1) m f1 (fb _ _ j _ f1 (lk8 j 5 ה) hj) hs,
  fLU inW relu X W b i (j+4+2) _ nf a hx lo ((j+4).toUSize+2) m f2 (fb _ _ j _ f2 (lk8 j 6 ה) hj) hs,fLU inW relu X W b i (j+4+3) _ nf a hx lo ((j+4).toUSize+3) m f3 (fb _ _ j _ f3 (lk8 j 7 ה) hj) hs]
lemma rowB (u m:ὦ) (k outW:Ὣ) (W:ѐ) (hu:Ѥu=k) (hk:k+1<=outW) (hW:outW*Ѥm<=ѧW) (hs:ѧW<2^32):Ѥ(u*m)+Ѥm<=ѧW:=by
 have h1:(k+1)*Ѥm<=outW*Ѥm:=Ἒ _ hk
 rw[ᾀ] at h1
 have h2:Ѥ(u*m)=k*Ѥm:=by rw[qWm _ _ (by rw[hu];omega),hu]
 omega
lemma lk (j c outW:Ὣ) (hc:c+1<=8) (hj:j+8<=outW):j+c+1<=outW:=א
lemma tG0 (j outW m W:Ὣ) (hj:j+8<=outW) (hOW:outW*m<=W) (hW:W<=4294967295) (hm:0< m):j+8<2^32:=by
 have:=Ὄ outW hm;omega
lemma tileG (j outW:Ὣ) (m n:ὦ) (x wb:ὤ) (ix W:ѐ) (hj:j+8<=outW) (hx:Ѥn<=Ѩx) (hix:Ѥn<=ѧix) (hW:ѧW<=4294967295) (hOW:outW*Ѥm<=ѧW) (hwb:Ѩwb<=4294967295) (hnm:Ѥn<=Ѥm) (hp:Ѥ(ѩj*m)+5+8*Ѥm<=Ѩwb) (hm:0<Ѥm):
 Ѥn<=Ѩx/\Ѥn<=ѧix/\ѧW<=4294967295/\Ѥ(ѩj*m)+Ѥm<=ѧW/\Ѥ((ѩj+1)*m)+Ѥm<=ѧW/\Ѥ((ѩj+2)*m)+Ѥm<=ѧW/\Ѥ((ѩj+3)*m)+Ѥm<=ѧW/\
  Ѥ((j+4).toUSize*m)+Ѥm<=ѧW/\Ѥ(((j+4).toUSize+1)*m)+Ѥm<=ѧW/\Ѥ(((j+4).toUSize+2)*m)+Ѥm<=ѧW/\Ѥ(((j+4).toUSize+3)*m)+Ѥm<=ѧW/\Ѩwb<=4294967295/\Ѥn<=Ѥm/\Ѥ(ѩj*m)+5+8*Ѥm<=Ѩwb:=by
 have hs:ѧW<2^32:=Nat.lt_of_le_of_lt hW ה
 have hj32:=tG0 j outW (Ѥm) (ѧW) hj hOW hW hm
 have e0:(ѩj).toNat=j:=qAd _ (lt8 j 0 ה hj32)
 obtain ⟨e1,e2,e3,-⟩:=qFk e0 (lt8 j 3 ה hj32)
 have f0:((j+4).toUSize).toNat=j+4:=qAd _ (lt8 j 4 ה hj32)
 obtain ⟨f1,f2,f3,-⟩:=qFk f0 (lt8 j 7 ה hj32)
 exact ⟨hx,hix,hW,rowB _ m j outW W e0 (lk j 0 outW ה hj) hOW hs,rowB _ m (j+1) outW W e1 (lk j 1 outW ה hj) hOW hs,rowB _ m (j+2) outW W e2 (lk j 2 outW ה hj) hOW hs,
  rowB _ m (j+3) outW W e3 (lk j 3 outW ה hj) hOW hs,rowB _ m (j+4) outW W f0 (lk j 4 outW ה hj) hOW hs,rowB _ m (j+4+1) outW W f1 (lk j 5 outW ה hj) hOW hs,
  rowB _ m (j+4+2) outW W f2 (lk j 6 outW ה hj) hOW hs,rowB _ m (j+4+3) outW W f3 (lk j 7 outW ה hj) hOW hs,hwb,hnm,hp⟩
lemma cN2W (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (i:Ὣ) (ix:ѐ) (x wd wb:ὤ) (nf a:ὧ) (hx:ὥ) (lo:Ὠ) (m n:ὦ) (base:ὦ) (hob:outW<=ѧb) (hs:ѧb<2^32) (hix:Ѥn<=ѧix) (hW:ѧW<=4294967295) (hOW:outW*Ѥm<=ѧW) (hm:0<Ѥm):
 ∀(d j:Ὣ) (out:ѐ),outW-j=d->colsN2 inW outW relu X W b i ix x wd wb (bOf b) nf a hx lo (spOf lo) m n j base out=cols1V inW outW relu X W b i ix x wd wb nf a hx lo m n j base out:=by
 intro d;induction d using Nat.strong_induction_on with
 | _ d ih=>
  intro j out hd
  rw[colsN2];by_cases h16:j+16<=outW
  . rw[Ἄ h16];dsimp only
    by_cases hg:Ѥn<=Ѩx/\Ѩwb<=4294967295/\Ѥn<=Ѥm/\Ѥ(ѩj*m)+5+8*Ѥm<=Ѩwb/\Ѥ((j+8).toUSize*m)+5+8*Ѥm<=Ѩwb
    . rw[Ἄ hg,fNw _ _ _ _ _ _ _ _ _ _ _ j _ _ _ _ _ _ _ _ _ _ א hs,fNw _ _ _ _ _ _ _ _ _ _ _ (j+8) _ _ _ _ _ _ _ _ _ _ א hs]
      rw[g16D];dsimp only[g16of,fw8]
      rw[cols1V,Ἄ (by omega:j+8<=outW)];dsimp only;rw[Ἄ (tileG j outW m n x wb ix W א hg.1 hix hW hOW hg.2.1 hg.2.2.1 hg.2.2.2.1 hm)]
      rw[cols1V,Ἄ (by omega:j+8+8<=outW)];dsimp only;rw[Ἄ (tileG (j+8) outW m n x wb ix W א hg.1 hix hW hOW hg.2.1 hg.2.2.1 hg.2.2.2.2 hm)]
      exact ih _ א _ _ rfl
    . rw[ἔ hg]
  . rw[ἔ h16]
lemma cN3W (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ) (i:Ὣ) (ix:ѐ) (x wd wb:ὤ) (nf a:ὧ) (hx:ὥ) (lo:Ὠ) (m n:ὦ) (base:ὦ) (hob:outW<=ѧb) (hs:ѧb<2^32) (hix:Ѥn<=ѧix) (hW:ѧW<=4294967295) (hOW:outW*Ѥm<=ѧW) (hm:0<Ѥm):
 ∀(d j:Ὣ) (out:ѐ),outW-j=d->colsN3 inW outW relu X W b i ix x wd wb (bOf b) nf a hx lo (spOf lo) m n j base out=cols1V inW outW relu X W b i ix x wd wb nf a hx lo m n j base out:=by
 intro d;induction d using Nat.strong_induction_on with
 | _ d ih=>
  intro j out hd
  rw[colsN3];by_cases h24:j+24<=outW
  . rw[Ἄ h24];dsimp only
    by_cases hg:Ѥn<=Ѩx/\Ѩwb<=4294967295/\Ѥn<=Ѥm/\Ѥ(ѩj*m)+5+8*Ѥm<=Ѩwb/\Ѥ((j+8).toUSize*m)+5+8*Ѥm<=Ѩwb/\Ѥ((j+8+8).toUSize*m)+5+8*Ѥm<=Ѩwb
    . rw[Ἄ hg,fNw _ _ _ _ _ _ _ _ _ _ _ j _ _ _ _ _ _ _ _ _ _ א hs,fNw _ _ _ _ _ _ _ _ _ _ _ (j+8) _ _ _ _ _ _ _ _ _ _ א hs,fNw _ _ _ _ _ _ _ _ _ _ _ (j+8+8) _ _ _ _ _ _ _ _ _ _ א hs]
      rw[g24D];dsimp only[g24of,fw8]
      rw[cols1V,Ἄ (by omega:j+8<=outW)];dsimp only;rw[Ἄ (tileG j outW m n x wb ix W א hg.1 hix hW hOW hg.2.1 hg.2.2.1 hg.2.2.2.1 hm)]
      rw[cols1V,Ἄ (by omega:j+8+8<=outW)];dsimp only;rw[Ἄ (tileG (j+8) outW m n x wb ix W א hg.1 hix hW hOW hg.2.1 hg.2.2.1 hg.2.2.2.2.1 hm)]
      rw[cols1V,Ἄ (by omega:j+8+8+8<=outW)];dsimp only;rw[Ἄ (tileG (j+8+8) outW m n x wb ix W א hg.1 hix hW hOW hg.2.1 hg.2.2.1 hg.2.2.2.2.2 hm)]
      exact ih _ א _ _ rfl
    . rw[ἔ hg];exact cN2W inW outW relu X W b i ix x wd wb nf a hx lo m n base hob hs hix hW hOW hm _ j out rfl
  . rw[ἔ h24];exact cN2W inW outW relu X W b i ix x wd wb nf a hx lo m n base hob hs hix hW hOW hm _ j out rfl
lemma dVE (X:ѐ) (o n t:ὦ) (buf:ὤ) (l:ὧ) (lo hi:ὥ) (hX:Ѥo+Ѥn<=ѧX) (hs:ѧX<2^32) (hb:Ѥn<=Ѩbuf):
 (decV X o n t buf l lo hi hX hs hb).buf=(decE X o n t buf l hX hs hb).buf/\(decV X o n t buf l lo hi hX hs hb).l=(decE X o n t buf l hX hs hb).l:=by
 rw[decV,decE];by_cases h:t<n
 . rw[Ἄ h,Ἄ h];dsimp only;exact dVE X o n (t+1) _ _ _ _ hX hs _
 . rw[ἔ h,ἔ h];exact ⟨rfl,rfl⟩
termination_by Ѥn-Ѥt
decreasing_by rw[usz t n h];have:=А.mp h;(clear h;omega)
lemma dEok (X:ѐ) (o n t:ὦ) (buf:ὤ) (l:ὧ) (hX:Ѥo+Ѥn<=ѧX) (hs:ѧX<2^32) (hb:Ѥn<=Ѩbuf):(decE X o n t buf l hX hs hb).ok=true:=by
 rw[decE];by_cases h:t<n
 . rw[Ἄ h];dsimp only;exact dEok X o n (t+1) _ _ hX hs _
 . rw[ἔ h]
termination_by Ѥn-Ѥt
decreasing_by rw[usz t n h];have:=А.mp h;(clear h;omega)
lemma dVS (X:ѐ) (o n N:ὦ) (hNn:N=n) (ho:o=0) (hN:Ѥn<=ѧX) (hN':ѤN<=ѧX) (hX:Ѥo+Ѥn<=ѧX) (hs:ѧX<2^32) (t:ὦ) (buf:ὤ) (l:ὧ) (lo hi:ὥ) (hb:Ѥn<=Ѩbuf):
 scan X N t hN' lo hi=(if 0x007FFFFF<=(decV X o n t buf l lo hi hX hs hb).lo/\(decV X o n t buf l lo hi hX hs hb).hi<У then (decV X o n t buf l lo hi hX hs hb).hi else Е):=by
 rw[scan,decV];by_cases h:t<n
 . have h':t<N:=hNn▸h
   rw[Ἄ h',Ἄ h];dsimp only
   have e:o+t=t:=by rw[ho,USize.zero_add]
   simp only[e]
   exact dVS X o n N hNn ho hN hN' hX hs (t+1) _ _ _ _ _
 . have h':¬t<N:=hNn▸h
   rw[ἔ h',ἔ h]
termination_by Ѥn-Ѥt
decreasing_by rw[usz t n h];have:=А.mp h;(clear h;omega)
lemma ltUne (x:ὥ) (h:x<У):x≠Е:=by intro e;rw[e] at h;exact absurd h ה
section
variable {n:ℕ} {ts:List ℕ} {g:ℕ->ὧ} {p:ℕ->ℝ} {bw hh:ὥ} {X Y J:ℝ} {s t a e:ὧ}
 (A:n<2^20) (B:ѣts<=n) (C:A_G ts g p) (D:good bw=true)
 (E:X=(ts.map p).sum+wv bw) (F:s=ts.foldl (λ(c:ὧ) t=>c+g t) 0+dv bw)
 (G:(2:ℝ)^(-127:ℤ) ≤ |R (s+e)|)
include A B C D E F G
end
section
variable (n:Ὣ) (hn:n<2^20) (xw ww:Ὣ->ὥ)
include n hn xw ww
end
section
variable {R I O:Ὣ} {X W b:ё}
section
variable (G:C_G R I O X W b) (Y:Ὡ) {i j:Ὣ} (hi:i< R) (hj:j< O)
include G Y i j hi hj
end
end
section
variable {Rr I O:Ὣ} {X W b:ё}
end
section
variable {R I O:Ὣ} {X W b:ё}
section
variable (G:C_G R I O X W b)
include G
end
end
def zI (X:ѐ) (k:Ὣ) (buf:ὤ) (ix:ѐ) (c:ὦ) (l:ὧ):Prop:=Ѥc=ѣ(C_keep (λp=>X[p]!) k)/\(∀q p:Ὣ,(C_keep (λp=>X[p]!) k)[q]?=some p->(ix[q]!:ὥ).toNat=p/\buf.data[q]! =dv X[p]!)/\l=(C_keep (λp=>X[p]!) k).foldl (λ(a:ὧ) p=>a+(dv X[p]!).abs) 0
def zO (X:ѐ) (k b0 i0:Ὣ) (r:(ὤ×ѐ)×(ὦ×ὧ)×(ὥ×ὥ)):Prop:=Ѩr.1.1=b0/\ѧr.1.2=i0/\zI X k r.1.1 r.1.2 r.2.1.1 r.2.1.2
lemma zN1 {t d n:Ὣ} (h:t+(d+1)=n):t<n:=by exact omS136 h
lemma zN2 {t d n:Ὣ} (h:t+(d+1)=n):t+1+d=n:=by exact omS137 h
lemma zN3 {c t n b:Ὣ} (h1:c<=t) (h2:t<n) (h3:n<=b):c<b:=א
lemma zN4 {q L:Ὣ} (h:¬q<L) (h2:q-L≠0):1<=q-L:=א
lemma zN5 {q L c:Ὣ} (h:¬q<L) (h2:q-L=0) (h3:c=L):q=c:=א
lemma zN6 {t n x:Ὣ} (h1:t<n) (h2:n<=x) (h3:x<2^32):t<2^32:=א
lemma dZ (X:ѐ) (n:ὦ) (hX:Ѥn<=ѧX) (hs:ѧX<2^32) (d:Ὣ): ∀(t:ὦ) (buf:ὤ) (ix:ѐ) (c:ὦ) (l:ὧ) (lo hi:ὥ),Ѥt+d=Ѥn-> Ѥn<=Ѩbuf-> Ѩbuf<2^32-> Ѥn<=ѧix-> ѧix<2^32->
  zI X Ѥt buf ix c l->zO X Ѥn Ѩbuf ѧix (decZ X n hX d t buf ix c l lo hi):=by
 induction d with
 | zero=>
  intro t buf ix c l lo hi he hb hbs hix his hI
  rw[decZ];rw[Ἵ] at he;rw[he] at hI;exact ⟨rfl,rfl,hI⟩
 | succ d ih=>
  intro t buf ix c l lo hi he hb hbs hix his hI
  obtain ⟨hc,hr,hl⟩:=hI
  have h1:=zN1 he
  have ht:t<n:=А.mpr h1
  have u:=usz t n ht
  have he':Ѥ(t+1)+d=Ѥn:=by rw[u];exact zN2 he
  have hct:Ѥc<=Ѥt:=by rw[hc];exact qBj _ _
  rw[decZ,Ἄ ht];dsimp only;rw[qC3]
  split
  . rename_i hv
    have e:=qDp (λp=>X[p]!) Ѥt (by simpa only[beq_iff_eq] using hv)
    exact ih (t+1) buf ix c l _ _ he' hb hbs hix his (by rw[u,zI,e];exact ⟨hc,hr,hl⟩)
  . rename_i hv
    have e:=qD5 (λp=>X[p]!) Ѥt (by simpa only[beq_iff_eq] using hv)
    have hcn:c<n:=А.mpr (zN3 hct h1 (Ὗ _))
    have R:=ih (t+1) (wrF buf c (dv X[Ѥt]!)) (wrW ix c (t.toUInt32)) (c+1) (l+(dv X[Ѥt]!).abs) (zU lo (X[Ѥt]!&&&В)) (zV hi (X[Ѥt]!&&&В)) he' (by rw[qD6];exact hb) (by rw[qD6];exact hbs) (by rw[qBh];exact hix) (by rw[qBh];exact his) ?_
    . rw[qD6,qBh] at R;exact R
    rw[u,zI,e,List.length_append,List.foldl_append,<-hl,usz c n hcn,hc]
    refine ⟨rfl,λq p hq=>?_,rfl⟩
    rw[qEt _ _ _ _ hbs (zN3 hct h1 hb),qBu _ _ _ _ his (zN3 hct h1 hix)]
    by_cases hk2:q< ѣ(C_keep (λp=>X[p]!) Ѥt)
    . rw[List.getElem?_append_left hk2] at hq
      simp only[Nat.ne_of_lt (hc▸hk2),ite_false];exact hr q p hq
    . rw[List.getElem?_append_right (Nat.le_of_not_lt hk2)] at hq
      have e0:q-ѣ(C_keep (λp=>X[p]!) Ѥt)=0:=by
       by_contra hne;rw[List.getElem?_eq_none (by simpa only[List.length_cons,List.length_nil,zero_add] using zN4 hk2 hne)] at hq;exact absurd hq (by simp only[reduceCtorEq,not_false_eq_true])
      rw[e0] at hq;simp only[List.getElem?_cons_zero,Option.some.injEq] at hq;subst hq
      simp only[zN5 hk2 e0 hc,ite_true,USize.toNat_toUInt32];exact ⟨Ἁ (zN6 h1 hX hs),trivial⟩
lemma dZS (X:ѐ) (n N:ὦ) (hNn:N=n) (hN':ѤN<=ѧX) (hX:Ѥn<=ѧX) (d:Ὣ): ∀(t:ὦ) (buf:ὤ) (ix:ѐ) (c:ὦ) (l:ὧ) (lo hi:ὥ),Ѥt+d=Ѥn->
 scan X N t hN' lo hi=(if 0x007FFFFF<=(decZ X n hX d t buf ix c l lo hi).2.2.1/\(decZ X n hX d t buf ix c l lo hi).2.2.2<У then (decZ X n hX d t buf ix c l lo hi).2.2.2 else Е):=by
 induction d with
 | zero=>
  intro t buf ix c l lo hi he
  have h':¬t<N:=by rw[hNn,А];simpa only[not_lt,add_zero] using Nat.le_of_eq he.symm
  rw[scan,decZ,ἔ h']
 | succ d ih=>
  intro t buf ix c l lo hi he
  have h:t<n:=А.mpr (zN1 he)
  have h':t<N:=hNn▸h
  rw[scan,decZ,Ἄ h',Ἄ h];dsimp only
  have he':Ѥ(t+1)+d=Ѥn:=by rw[usz t n h];exact zN2 he
  rcases Bool.eq_false_or_eq_true (X.uget t (Nat.lt_of_lt_of_le (А.mp h) hX)&&&В==0) with hv|hv<;>simp only[hv,ite_true,Bool.false_eq_true,ite_false]<;>exact ih (t+1) _ _ _ _ _ _ he'
lemma zM1 {a:Ὣ} (h:a<2^20):a<2^32:=by exact omS363 h
lemma linZe (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ)
 (hh:1*inW=ѧX/\outW*inW=ѧW/\outW=ѧb/\0< inW/\0<outW/\inW<2^20/\ѧX<=4294967295/\ѧW<=4294967295/\ѧb<=4294967295/\1*outW<=4294967295/\scanW 1 W≠Е/\scanH 1 b≠Е/\scanH 2 W≠Е/\inW<=8192)
 (hX:Ѥ(ѩinW)<=ѧX) (hs:ѧX<2^32):
 linZc inW outW relu X W b ((wbOf inW outW W,scanH 2 W),(bOf b,scanH 1 b)) ((inW.toUInt64+8).toFloat*Ἃ 0x3CA0000000000000) (ѩinW) (zFor inW) hX=linF0 1 inW outW relu X W b:=by
 unfold linZc;dsimp only
 have hnf:(inW.toUInt64+8).toFloat*Ἃ 0x3CA0000000000000=nfOf inW:=rfl
 rw[hnf]
 obtain ⟨h1,h2,h3,h4,h5,h6,h7,h8,h9,h10,h11,h12,h13,h14⟩:=hh
 replace h7:=qA9 h7;replace h8:=qA9 h8;replace h9:=qA9 h9;replace h10:=qA9 h10
 have nm:inW.toUSize.toNat=inW:=qAd _ (zM1 h6)
 have P0:=qZf inW (zM1 h6)
 obtain ⟨z1,z2,z3,z4,z5⟩:=dZ X (ѩinW) hX hs Ѥ(ѩinW) 0 (zFor inW) (ἇ Ѥ(ѩinW) 0) 0 0 Е 0 (by simp only[USize.toNat_zero,Nat.toUSize_eq,USize.toNat_ofNat',zero_add]) (by rw[nm];exact P0.1) P0.2 (by simp only[Nat.toUSize_eq,USize.toNat_ofNat',Array.size_replicate,Std.le_refl]) (by rw[Array.size_replicate,nm];exact zM1 h6) ⟨by simp only[USize.toNat_zero,C_keep,List.range_zero,List.filter_nil,List.length_nil],λq p h=>by simp only[C_keep,USize.toNat_zero,List.range_zero,List.filter_nil,List.length_nil,not_lt_zero,not_false_eq_true,getElem?_neg,reduceCtorEq] at h,by simp only[C_keep,USize.toNat_zero,List.range_zero,List.filter_nil,List.foldl_nil]⟩
 have en:X.usize=ѩinW:=by rw[Array.usize,<-h1,Ὑ]
 have hS:=dZS X (ѩinW) X.usize en (qAc X) hX Ѥ(ѩinW) 0 (zFor inW) (ἇ Ѥ(ѩinW) 0) 0 0 Е 0 (by simp only[USize.toNat_zero,Nat.toUSize_eq,USize.toNat_ofNat',zero_add])
 generalize decZ X (ѩinW) hX Ѥ(ѩinW) 0 (zFor inW) (ἇ Ѥ(ѩinW) 0) 0 0 Е 0=r at z1 z2 z3 z4 z5 hS⊢
 obtain ⟨⟨rb,ri⟩,⟨rc,rl⟩,⟨rlo,rhi⟩⟩:=r
 dsimp only at z1 z2 z3 z4 z5 hS⊢
 rw[nm] at z2 z3 z4 z5
 split
 . rename_i hc
   have hsc:scanH 0 X≠Е:=by
    unfold scanH;simp only[show ((0:Ὣ)==1)=false by decide,Bool.false_eq_true,ite_false,beq_self_eq_true,ite_true]
    rw[hS,ἱ hc];exact ltUne _ hc.2
   have Hv:=qA8 2 W h8 h13
   have G:C_G 1 inW outW X W b:=⟨h1,h2,h3,h4,h5,h6,h7,h8,h9,h10,(qA8 0 X h7 hsc).2.2.1,Hv.2.2.1,(qA8 1 b h9 h12).2.2.1⟩
   have Rw:C_Row inW X 0 ri rb rc (C_keep (λp=>X[p]!) inW) rl:=⟨z3,qBj _ _,by rw[z2];simp only[Array.size_replicate,Std.le_refl],⟨by rw[z1];exact P0.1,by rw[z1];exact P0.2⟩,(qDs _ _).1,(qDs _ _).2.1,λt ht hn=>by simpa only[zero_mul,zero_add] using (qDs _ _).2.2 t ht hn,λk p hk=>by simpa only[zero_mul,zero_add] using z4 k p hk,by simpa only[zero_mul,zero_add] using z5⟩
   rw[qEr0]
   have e:linearSpec 1 inW outW relu X W b=Array.ofFn (n:=1*outW) λq=>C_sp inW outW relu X W b q.val:=rfl
   rw[e]
   have HdE:C_Dw (ἆ 0) W:=λh q hq=>absurd hq (by rw[<-(h:0=ѧW)];exact Ἲ _)
   have hb0:(0*outW).toUSize.toNat=0*outW:=qAd _ omS333
   rw[c1WE _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (λj hj=>by rw[Rw.len] at hj;have hp:=List.getElem?_eq_getElem hj;exact ⟨_,G.gx ((by exact Nat.le_refl _):0<1) _ (Rw.lt _ (List.mem_of_getElem? hp)),(Rw.rd j _ hp).2⟩) HdE (by rw[nm,h2]) (by rw[h3];exact h9) (λj i k a b c d=>by rw[nm] at c⊢;exact wbP W inW outW j i k a b c d) (λho i k hi hk=>by rw[nm] at hi⊢;exact wbPt W inW outW i k ho hi hk) 0 rfl]
   obtain ⟨gb1,gb2,-,gb4⟩:=qA8 1 b h9 h12
   obtain ⟨pb0,pb1,pb2⟩:=qCs gb1 gb2
   obtain ⟨fbm,Rbm⟩:=qAm _ gb1
   have hbm:∀j,j<outW-> |R (dv b[j]!)| <=R (dv (scanH 1 b)):=λj hj=>by
    have gj:=(qA8 1 b h9 h12).2.2.1 j (by rw[<-h3];exact hj)
    rw[(qAm _ gj).2,Rbm];exact pb2 _ gj (gb4 j (by rw[<-h3];exact hj))
   have bb:R (dv (scanH 1 b))<=2^(290:ℤ):=by rw[Rbm];exact pb1.trans (by rw[show (2:ℝ)^128=2^(128:ℤ) by norm_num];exact ἒ ב ב)
   have H0:C_Inv (1*outW) (C_sp inW outW relu X W b) (λq=>q<0) (ἇ (1*outW) 0):=⟨by simp only[one_mul,Array.size_replicate],λq _ h=>absurd h (Ἲ _)⟩
   exact qBv (qD0 G relu ((by exact Nat.le_refl _):0<1) Hv Rw hb0 HdE (dv (scanH 1 b)) fbm hbm bb rfl outW 0 _ (Ἇ _) (by rw[ὒ,Ἵ];exact H0)) (λq hq=>omS334 hq)
 . rfl
lemma linNe (inW outW:Ὣ) (relu:Ὡ) (X W b:ѐ):linN inW outW relu X W b=linF0 1 inW outW relu X W b:=by
 unfold linN linNc;rw[(wnGet inW outW W b).2];dsimp only;rw[show fz0=(0:ὧ) from rfl]
 split
 . rename_i hh
   split
   . rename_i hz
     split
     . exact linZe inW outW relu X W b hh _ (qA9 hh.2.2.2.2.2.2.1)
     split
     . rename_i hc
       obtain ⟨h1,h2,h3,h4,h5,h6,h7,h8,h9,h10,h11,h12,h13,h14⟩:=hh
       have hs:ѧX<2^32:=א
       have en:X.usize=ѩinW:=by rw[Array.usize];congr 1;omega
       have hsc:scan X X.usize 0 (qAc X) Е 0≠Е:=by
        rw[dVS X _ _ X.usize en (by rw[ὒ];rfl) (by have:=hz.1;omega) (qAc X) hz.1 hs 0 (zFor inW) 0 Е 0 hz.2.1]
        rw[ἱ hc];exact ltUne _ hc.2
       have hS:scanH 0 X≠Е:=by
        unfold scanH;simp only[show ((0:Ὣ)==1)=false by decide,Bool.false_eq_true,ite_false,beq_self_eq_true,ite_true];exact hsc
       unfold linF0;rw[(wbGet inW outW W).2];dsimp only
       have hG:1*inW=ѧX/\outW*inW=ѧW/\outW=ѧb/\0< inW/\0<outW/\inW<2^20/\ѧX<=4294967295/\ѧW<=4294967295/\ѧb<=4294967295/\1*outW<=4294967295/\scanH 0 X≠Е/\scanW 1 W≠Е/\scanH 1 b≠Е:=⟨h1,h2,h3,h4,h5,h6,h7,h8,h9,h10,hS,h11,h12⟩
       rw[ἱ hG]
       simp only[show ((1:Ὣ)==1)=true by decide,Bool.false_eq_true,ite_true]
       rw[ἱ h13]
       have hdF:decF X (0*inW).toUSize (ѩinW) (zFor inW) 0=decE X (0*inW).toUSize (ѩinW) 0 (zFor inW) 0 hz.1 hs hz.2.1:=by
        unfold decF;rw[Ἄ ⟨hz.1,h7,hz.2.1,hz.2.2.1⟩]
       rw[hdF,dEok,<-(dVE X _ _ 0 (zFor inW) 0 Е 0 hz.1 hs hz.2.1).1,<-(dVE X _ _ 0 (zFor inW) 0 Е 0 hz.1 hs hz.2.1).2]
       rw[ἱ (show (true&&decide (inW<=8192))=true by simp only[h14,decide_true,Bool.and_self])]
       have hix:Ѥ(ѩinW)<=ѧidA:=by rw[qIs,qAd _ א];omega
       have hOW:outW*Ѥ(ѩinW)<=ѧW:=by rw[qAd _ א];omega
       have hm0:0<Ѥ(ѩinW):=by rw[qAd _ א];omega
       (try split);(all_goals first|rw[cN3W inW outW relu X W b 0 idA _ (ἆ 0) _ _ _ _ _ _ _ _ א א hix h8 hOW hm0 _ 0 _ rfl,c1VS']|rw[cN2W inW outW relu X W b 0 idA _ (ἆ 0) _ _ _ _ _ _ _ _ א א hix h8 hOW hm0 _ 0 _ rfl,c1VS']|rw[cNV inW outW relu X W b 0 idA _ (ἆ 0) _ _ _ _ _ _ _ _ א א _ 0 _ rfl,c1VS']|rw[cN3V inW outW relu X W b 0 idA _ (ἆ 0) _ _ _ _ _ _ _ _ א א _ 0 _ rfl,c1VS']|rw[cN2V inW outW relu X W b 0 idA _ (ἆ 0) _ _ _ _ _ _ _ _ א א _ 0 _ rfl,c1VS'])
     . rfl
   . rfl
 . rfl
lemma qEr:linF=linearSpec:=by
 funext rows inW outW relu X W b;unfold linF;split
 . rename_i h;have e:rows=1:=by simpa only[beq_iff_eq] using h
   subst e;rw[linNe,qEr0]
 . rw[qEr0]
lemma qDz:linF=linearSpec:=qEr
@[inline] def sm (w:ὥ):Ὡ:=good w&&(w&&&В)<=0x70FFFFFF
section
open Float.Model Float.Model.UnpackedFloat
set_option maxRecDepth 8192 in
end
section
open Float.Model Float.Model.UnpackedFloat
end
section
lemma qGm (S:Ὡ) (ex fr m:Ὣ) (q:Int) (h1:897<=ex) (h2:ex<1151) (h3:fr<2^23) (hm:m≠0) (hW1:(m:ℝ)*2^q<(2^53+fr*2^30+2^29)*2^((ex:Int)-1076)) (hW2:fr≠0->(2^53+fr*2^30-2^29)*2^((ex:Int)-1076)<(m:ℝ)*2^q) (hW3:(2^53-2^28)*2^((ex:Int)-1076)<(m:ℝ)*2^q):
 toWord (ἕ Fmt ⟨S,m,q⟩)=ὺ ((if S then 2^31 else 0)+(ex-896)*2^23+fr):=by
 have hE:(2:ℝ)^((ex:Int)-1046)=2^30*2^((ex:Int)-1076):=by
  rw[<-Ἡ,<-Ἓ ἓ]
  congr 1;(clear h1 h2 h3 hm;exact omS335)
 rw[qE4 S m q (2^23+fr) ((ex:Int)-1046) hm (by clear h1 h2 hm;exact omS125) (by clear h1 h2 hm;exact omS126 h3) (by clear h3 hm;exact omS127 h1) (by clear h3 hm;exact omS128 h2) (by rw[hE];push_cast;linarith only[hW1]) (λ_=>by rw[hE];push_cast;linarith[hW2 (by clear h1 h2 h3 hm;omega)]) (λ_=>by rw[hE];linarith only[hW3]),show ((ex:Int)-1046+150).toNat=ex-896 by (clear h3 hm;exact omS130),show (2^23+fr)%2^23=fr by (clear h1 h2 hm;exact omS131 h3)]
lemma qFt (a b:ℝ) (ha:0<=a) (h:a<b): ∃(m:Ὣ) (q:Int),m≠0/\a<(m:ℝ)*2^q/\(m:ℝ)*2^q<=b:=by
 obtain ⟨k,hk⟩:=exists_pow_lt_of_lt_one (sub_pos.2 h) one_half_lt_one;have h2:(0:ℝ)<2^k:=ו
 rw[one_div_pow,div_lt_iff₀ h2] at hk;refine ⟨⌊b*2^k⌋₊,-k,?_⟩;rw[zpow_neg,Ἡ,<-div_eq_mul_inv,lt_div_iff₀ h2,div_le_iff₀ h2];have h1:=Nat.lt_floor_add_one (b*2^k);have h5:a*2^k<⌊b*2^k⌋₊ :=(by linarith only[hk,h1]);exact ⟨(Nat.cast_pos.1 ((Ἆ ha h2.le).trans_lt h5)).ne',h5,Nat.floor_le (Ἆ (ha.trans h.le) h2.le)⟩
lemma qGz (z:Model Fmt):ofW (toWord z)=z:=qEb z
lemma qIm (w:ὥ) (h0:expOf w≠0) (h1:expOf w≠255):wv w≠0:=by
 have:sv w≠0:=by unfold sv mantOf;split<;>simp only[Nat.reducePow,Nat.cast_add,Nat.cast_ofNat,neg_add_rev,Int.reduceNeg,ne_eq]<;>(clear h0 h1;omega)
 simp only[qAx w h1,rationalWord,decodeD_toRat,Rat.cast_mul,Rat.cast_intCast,Rat.cast_zpow,Rat.cast_ofNat,ne_eq,mul_eq_zero,Int.cast_eq_zero,this,OfNat.ofNat_ne_zero,not_false_eq_true,zpow_ne_zero,or_self]
lemma qDqW (S:Ὡ) (ex fr:Ὣ) (h1:897<=ex) (h2:ex<1151) (h3:fr<2^23) (V W:ℝ) (hV:V=if S then-W else W) (hW1:W<(2^53+fr*2^30+2^29)*2^((ex:Int)-1076)) (hW2:fr≠0->(2^53+fr*2^30-2^29)*2^((ex:Int)-1076)<W) (hW3:(2^53-2^28)*2^((ex:Int)-1076)<W) (M:Model Fmt) (hf:Ἑ M=true) (hM:Ἢ M=Model.roundAt Fmt V):toWord M=ὺ ((if S then 2^31 else 0)+(ex-896)*2^23+fr):=by
 have e2zw:(if S then 2^31 else 0)+(ex-896)*2^23+fr<2^32/\((if S then 2^31 else 0)+(ex-896)*2^23+fr)/2^23%2^8=ex-896:=by constructor<;>split<;>omega
 have hb0:(0:ℝ)<2^((ex:Int)-1076):=ἵ ו _;have hA2:(0:ℝ)<=(2^53-2^28)*2^((ex:Int)-1076):=Ἆ ו hb0.le
 obtain ⟨Lo,hLo0,hLoW,hLo1,hLo2⟩:∃Lo:ℝ,0<=Lo/\Lo<W/\(fr≠0->(2^53+fr*2^30-2^29)*2^((ex:Int)-1076)<=Lo)/\(2^53-2^28)*2^((ex:Int)-1076)<=Lo:=by
  by_cases hfr:fr=0;. exact ⟨_,hA2,hW3,λh=>absurd hfr h,le_rfl⟩
  . have hfr1:(1:ℝ)<=fr:=by exact_mod_cast Nat.one_le_iff_ne_zero.mpr hfr
    refine ⟨_,?_,hW2 hfr,λ_=>le_rfl,?_⟩;. exact Ἆ (sub_nonneg.mpr ((pow_le_pow_right₀ one_le_two (Nat.le_of_ble_eq_true rfl:29<=53)).trans (le_add_of_nonneg_right ו))) hb0.le
    . exact ἀ (by linarith only[hfr1]) hb0.le
 have hW0:0<W:=ἷ hLo0 hLoW;obtain ⟨m1,q1,hm1,hd1,hd1'⟩:=qFt Lo W hLo0 hLoW;set B:ℝ:=(2^53+fr*2^30+2^29)*2^((ex:Int)-1076) with hB;obtain ⟨m2,q2,hm2,hd2,hd2'⟩:=qFt W ((W+B)/2) hW0.le (left_lt_add_div_two.mpr hW1);have w1:=qGm S ex fr m1 q1 h1 h2 h3 hm1 (hd1'.trans_lt (left_lt_add_div_two.mp (hd2.trans_le hd2'))) (λh=>ἷ (hLo1 h) hd1) (ἷ hLo2 hd1)
 have w2:=qGm S ex fr m2 q2 h1 h2 h3 hm2 (hd2'.trans_lt (add_div_two_lt_right.mpr (left_lt_add_div_two.mp (hd2.trans_le hd2')))) (λh=>by
  exact ἷ (hLo1 h) ((hd1.trans_le hd1').trans hd2)) (ἷ hLo2 ((hd1.trans_le hd1').trans hd2))
 set z:=ὺ ((if S then 2^31 else 0)+(ex-896)*2^23+fr) with hzd;have hzn:ѥz=(if S then 2^31 else 0)+(ex-896)*2^23+fr:=by rw[hzd,ὡ,Ἁ];exact e2zw.1
 have hexp:(expOf z).toNat=ex-896:=by rw[qCa,hzn];exact e2zw.2
 have he0:expOf z≠0:=λh=>by rw[h] at hexp;exact absurd (Nat.sub_eq_zero_iff_le.mp hexp.symm) (Nat.not_le.mpr h1)
 have he1:expOf z≠255:=λh=>by rw[h] at hexp;exact absurd ((Nat.sub_eq_iff_eq_add (Nat.le_of_succ_le h1)).mp hexp.symm) (Nat.ne_of_lt h2)
 have r1:ἕ Fmt ⟨S,m1,q1⟩=ofW z:=by rw[<-w1,qGz]
 have r2:ἕ Fmt ⟨S,m2,q2⟩=ofW z:=by rw[<-w2,qGz]
 have fz:Ἑ (ofW z)=true:=by simp only[ofW,isFinite_ofUInt32,bne_iff_ne,ne_eq,he1,not_false_eq_true]
 have t1:=Model.toReal_roundDyadic_eq_roundAt Fmt ה ⟨S,m1,q1⟩ (by rw[r1];exact fz);have t2:=Model.toReal_roundDyadic_eq_roundAt Fmt ה ⟨S,m2,q2⟩ (by rw[r2];exact fz);rw[r1] at t1;rw[r2] at t2
 have hv:Ἢ (ofW z)=Model.roundAt Fmt V:=by
  apply le_antisymm
  . cases S;. rw[t1];apply Model.roundAt_mono;simp only[FloatLib.Numerics.Dyadic.toReal,signedSignificand,false_eq_true, ↓reduceIte,Int.ofNat_eq_natCast,Int.cast_natCast,hV];linarith only[hd1']
    . rw[t2];apply Model.roundAt_mono;simp only[FloatLib.Numerics.Dyadic.toReal,signedSignificand, ↓reduceIte,Int.ofNat_eq_natCast,Int.cast_neg,Int.cast_natCast,neg_mul,hV,neg_le_neg_iff];linarith only[hd2]
  . cases S;. rw[t2];apply Model.roundAt_mono;simp only[hV,false_eq_true, ↓reduceIte,FloatLib.Numerics.Dyadic.toReal,signedSignificand,Int.ofNat_eq_natCast,Int.cast_natCast];linarith only[hd2]
    . rw[t1];apply Model.roundAt_mono;simp only[hV, ↓reduceIte,FloatLib.Numerics.Dyadic.toReal,signedSignificand,Int.ofNat_eq_natCast,Int.cast_neg,Int.cast_natCast,neg_mul,neg_le_neg_iff];linarith only[hd1']
 have hnz:Ἢ M≠0:=by rw[hM,<-hv];exact qIm z he0 he1
 have:=Model.eq_of_toReal_eq_of_nonzero (hM.trans hv.symm) hnz;rw[this];have hr:toWord (ofW z)=z:=by rw[<-r1,w1]
 exact hr
set_option maxRecDepth 8192 in
lemma qDq (s e:ὧ) (V:ℝ) (hlo:R (s-e)<V) (hhi:V<R (s+e)) (h:ziv s e≠Е) (M:Model Fmt) (hf:Ἑ M=true) (hM:Ἢ M=Model.roundAt Fmt V):toWord M=ziv s e:=by
 obtain ⟨S,ex,fr,W,h1,h2,h3,hz,hV,hW1,hW2,hW3⟩:=qB8 s e V hlo hhi h;rw[hz];exact qDqW S ex fr h1 h2 h3 V W hV hW1 hW2 hW3 M hf hM
lemma qIx:(2:ℝ)^(127:ℤ)<=Ἢ (Model.posMaxFinite Fmt):=by rw[Model.toReal_posMaxFinite,show Model.pow2 Fmt.fracWidth+Fmt.maxFiniteFracField=2^24-1 by decide+kernel,show Fmt.maxNormalExponent-Int.ofNat Fmt.fracWidth=104 by decide+kernel,Model.bpow,qA0];(norm_num1)
lemma qFn (q:ℚ):Model.signedScaledRatToReal (decide (q<0)) q.num.natAbs q.den 0=(q:ℝ):=by
 rw[Model.signedScaledRatToReal,Model.scaledRatToReal,Nat.cast_natAbs,Rat.cast_def];by_cases hq:q<0;. simp only[hq,decide_true, ↓reduceIte,abs_of_neg (Rat.num_neg.2 hq),Int.cast_neg,Model.bpow_zero,mul_one];ring
 . simp only[hq,decide_false,false_eq_true, ↓reduceIte,ἅ (Rat.num_nonneg.2 (not_lt.1 hq)),Model.bpow_zero,mul_one]
lemma qH7 (s e:ὧ) (q:ℚ) (hlo:R (s-e)<(q:ℝ)) (hhi:(q:ℝ)<R (s+e)) (h:ziv s e≠Е) (hb: |(q:ℝ)| <=2^(127:ℤ)):roundRational q=ziv s e:=by have hf:=Model.isFinite_roundRatScaled_of_abs_le_posMaxFinite Fmt (decide (q<0)) q.num.natAbs q.den 0 ה q.den_nz (qFn q▸hb.trans qIx);have hM:=Model.toReal_roundRatScaled_eq_roundAt Fmt _ _ _ 0 ה hf;exact qDq s e q hlo hhi h _ hf (qFn q▸hM)
lemma qH4 (N d:Ὣ) (x:Int) (hr:sqrtLower N d≠0) (hd:0<d) (hs:x<=(FloatFormat.toModel Fmt).targetExponent (Float.Model.totalExponent (sqrtLower N d) x))
 (hb:ἧ ((N:ℝ)/d)*FloatLib.Floats.Formats.Flocq.bpow ἲ x<2^(127:ℤ)):
 Ἑ (Model.ofModel Fmt (Float.Model.UnpackedFloat.roundWithAccuracy (FloatFormat.toModel Fmt) .positive (sqrtLower N d) x (sqrtAccuracy N d)))=true:=
 Model.isFinite_ofModel_roundWithAccuracy_of_lt_bpow Fmt rfl .positive _ _ _ (ἧ ((N:ℝ)/d)) (sqrtAccuracy_correct N d hd) hs (hb.trans_eq (qA0 127).symm)
lemma qIp (q:ℚ) (hq:0<q) (hb:ἧ (q:ℝ)<2^(127:ℤ)):Ἑ (sqrtModel q)=true:=
 qH4 _ _ _ (ne_of_gt (sqrtRoot_pos q hq)) q.pos (sqrtStep_le_target q) (by rwa[sqrtScale_correct q hq])
lemma qGs (s e:ὧ) (q:ℚ) (hq:0<q) (hlo:R (s-e)<ἧ (q:ℝ)) (hhi:ἧ (q:ℝ)<R (s+e)) (h:ziv s e≠Е) (hb:ἧ (q:ℝ)<2^(127:ℤ)):sqrtRational q=ziv s e:=by have hf:=qIp q hq hb;exact (ἐ (not_le.2 hq)).trans (qDq s e _ hlo hhi h _ hf (sqrtModel_correct q hq hf))
lemma qA1 (w:ὥ) (h:sm w=true):good w=true/\|wv w| <=2^99:=by obtain ⟨hg,hl⟩:=ὕ.1 h;have h1:=ἃ.1 (of_decide_eq_true hl);rw[qAq] at h1;have h2:=qAz w hg;rw[qB5 w hg];refine ⟨hg,(qA6 _ _ h1 (by clear h1;exact omS336 h2)).trans ?_⟩;unfold A_mag;norm_num
set_option maxRecDepth 8192 in
end
section
section
open Float.Model Float.Model.UnpackedFloat
end
end
@[specialize] def fC {σ:Type} (f:σ->ὦ->ὦ->Ὢ) (s:σ) (b j:ὦ)
 (o:ё):Ὣ->ё
 | 0=>o
 | k+1=>fC f s b (j+1) (wrW o (b+j) (f s (b+j) j)) k
@[specialize] def fR {σ:Type} (h:Ὣ->σ) (f:σ->ὦ->ὦ->Ὢ) (n r:Ὣ)
 (o:ё):Ὣ->ё
 | 0=>o
 | k+1=>fR h f n (r+1) (fC f (h r) (r*n).toUSize 0 o n) k
def lnSum (x:ё) (base n i:ὦ) (s t:ὧ):F1:=
 if h:i<n then
  let v:=dv (rdW x (base+i))
  lnSum x base n (i+1) (s+v) (t+v.abs)
 else ⟨s,t⟩
partial_fixpoint
def lnVar (x:ё) (base n i:ὦ) (m v:ὧ):ὧ:=
 if h:i<n then
  let d:=dv (rdW x (base+i))-m
  lnVar x base n (i+1) m (v+d*d)
 else v
partial_fixpoint
@[inline] def invOK (a b:ὧ):Ὡ:=
 decide ((a.toBits>>>52)-1<1222)&&
  decide ((((a*b-Ἃ 0x3FF0000000000000)+Ἃ 0x3CD0000000000000).toBits>>>52)-972<2)
@[inline] def lnQ (b:Ὡ) (n:Ὣ) (x:ё) (r:Ὣ):ℚ:=if b then 0 else rowMean n x r
@[inline] def lnStat (b:Ὡ) (n:Ὣ) (eps:Ὢ) (x:ё) (r:Ὣ):LnS:=
 let base:=(r*n).toUSize
 let nu:=ѩn
 let nd:=n.toUInt64.toFloat
 let nf:=(n.toUInt64+8).toFloat*Ἃ ь
 let ind:=1.0/nd
 let a:=if b then F1.mk 0 0 else lnSum x base nu 0 0 0
 let m:=if b then 0 else a.s*ind
 let v:=lnVar x base nu 0 m 0
 let em:=(nf*a.t)*ind+Ἃ ظ
 let tt:=v*ind+dv eps
 let d:=Ἤ tt
 let z:=ziv d ((nf+nf)*d+em)
 let ir:=1.0/dv z
 if sm z&&decide ((0x1F800000:ὥ)<=z)&&decide (z<=(0x70FFFFFF:ὥ))&&decide ((tt.toBits>>>52)-1<1222)&&invOK ind nd&&invOK ir (dv z) then
  ⟨true,m,ir,em,z⟩
 else
  ⟨false,0,0,0,sqrtRational (rowVariance n x r (lnQ b n x r)+ἂ eps)⟩
/-! -/
def mKf (f:Nat->ὥ) (L:љ) (k:Nat):Nat:=
 L.foldl (λk j=>if mantOf (f j)=0 then k else min k (idxOf (f j))) k
def pw2T:ї:=Array.ofFn (n:=63) (λi=>2^i.val)
@[inline] def pw2 (k:Nat):Nat:=if k<63 then pw2T[k]! else 2^k
lemma pw2_eq (k:Nat):pw2 k=2^k:=by
 unfold pw2;split
 · rename_i h
   rw[Ἕ pw2T k (by simp only[pw2T,Array.size_ofFn];exact h)]
   simp only[pw2T,Array.getElem_ofFn]
 · rfl
def mSf (f:Nat->ὥ) (K:Nat) (L:љ) (t:Int):Int:=
 L.foldl (λt j=>t+sv (f j)*((pw2 (idxOf (f j)-K):Nat):Int)) t
lemma mKf_le (f:Nat->ὥ) (L:љ): ∀k,mKf f L k<=k:=by
 induction L with
 | nil=>intro k;exact Nat.le_refl k
 | cons p L ih=>
  intro k;unfold mKf;rw[ὔ]
  refine (ih _).trans ?_
  split
  · exact Nat.le_refl k
  · exact Nat.min_le_left _ _
lemma mKf_lo (f:Nat->ὥ) (L:љ): ∀k, ∀j∈L,mantOf (f j)=0\/mKf f L k<=idxOf (f j):=by
 induction L with
 | nil=>intro k j hj;simp only[List.not_mem_nil] at hj
 | cons p L ih=>
  intro k j hj
  rcases List.mem_cons.1 hj with rfl | hj
  · by_cases h:mantOf (f j)=0
    · exact Or.inl h
    · refine Or.inr ?_
      unfold mKf;rw[ὔ];simp only[h,ite_false]
      exact (mKf_le f L _).trans (Nat.min_le_right _ _)
  · unfold mKf;rw[ὔ];exact ih _ j hj
lemma sv_zero (w:ὥ) (h:mantOf w=0):sv w=0:=by
 unfold sv;split<;>simp only[h,CharP.cast_eq_zero,neg_zero]
lemma mSf_eq (f:Nat->ὥ) (K:Nat) (L:љ) (hL:∀j∈L,mantOf (f j)=0\/K<=idxOf (f j)):
 ∀t:Int,L.foldl (λt j=>t+sv (f j)*((2^idxOf (f j):Nat):Int)) (t*((2^K:Nat):Int))=mSf f K L t*((2^K:Nat):Int):=by
 induction L with
 | nil=>intro t;rfl
 | cons p L ih=>
  intro t
  have hp:=hL p (ὣ ..)
  have hL': ∀j∈L,mantOf (f j)=0\/K<=idxOf (f j):= λj hj=>hL j (ὖ _ hj)
  unfold mSf;rw[ὔ,ὔ,pw2_eq]
  have e:t*((2^K:Nat):Int)+sv (f p)*((2^idxOf (f p):Nat):Int)=(t+sv (f p)*((2^(idxOf (f p)-K):Nat):Int))*((2^K:Nat):Int):=by
   rcases hp with h | h
   · rw[sv_zero _ h];ring
   · have:idxOf (f p)=(idxOf (f p)-K)+K:=(Nat.sub_add_cancel h).symm
     conv_lhs=>rw[this]
     push_cast;rw[pow_add];ring
  rw[e];exact ih hL' _
def meanI (n:Ὣ) (x:ё) (r:Ὣ):ℤ:=
 (ἄ n).foldl (λt j=>t+sv x[r*n+j]!*((2^idxOf x[r*n+j]!:ℕ):ℤ)) 0
def meanIK (n:Nat) (x:ё) (r K:Nat):Int:=
 mSf (λj=>x[r*n+j]!) K (ἄ n) 0*((2^K:Nat):Int)
def meanIF (n:Nat) (x:ё) (r:Nat):Int:=
 meanIK n x r (mKf (λj=>x[r*n+j]!) (ἄ n) 1000)
lemma meanIK_eq (n:Nat) (x:ё) (r K:Nat) (hK:∀j∈ἄ n,mantOf x[r*n+j]! =0\/K<=idxOf x[r*n+j]!):
 meanIK n x r K=meanI n x r:=by
 have h:=mSf_eq (λj=>x[r*n+j]!) K (ἄ n) hK 0
 rw[Int.zero_mul] at h
 exact h.symm
lemma meanIF_eq (n:Nat) (x:ё) (r:Nat):meanIF n x r=meanI n x r:=
 meanIK_eq n x r _ (λj hj=>mKf_lo (λj=>x[r*n+j]!) (ἄ n) 1000 j hj)
def meanQ (n:Ὣ) (x:ё) (r:Ὣ):ℚ:=(meanIF n x r:ℚ)/((2^149*n:ℕ):ℚ)
@[noinline] def lnSlow (x g b den:Ὢ) (mean:Unit->ℚ):Ὢ:=
 roundRational ((ἂ x-mean ())*ἂ g/ἂ den+ἂ b)
@[inline] def lnFa (m ir em:ὧ) (den w gw bw:Ὢ) (gv bv:ὧ) (lo:Ὠ) (mean:Unit->ℚ):Ὢ:=
 let gd:=gv*ir
 let q:=(dv w-m)*gd
 let t:=fin1 lo (spOf lo) (q+bv)
 if t==Е then
  let r:=ziv (q+bv) (Ἃ 0x3D10000000000000*(q.abs+bv.abs)+em*gd.abs)
  if 0x7EFFFFFF<(r&&&В) then lnSlow w gw bw den mean else r
 else t
@[inline] def lnFb (m ir em:ὧ) (den:Ὢ) (x g b:ё) (p j:ὦ) (gv bv:ὧ) (lo sp:Ὠ) (mean:Unit->ℚ):Ὢ:=
 let w:=rdW x p
 let gd:=gv*ir
 let q:=(dv w-m)*gd
 let t:=fin1 lo sp (q+bv)
 if t==Е then
  let r:=ziv (q+bv) (Ἃ 0x3D10000000000000*(q.abs+bv.abs)+em*gd.abs)
  if 0x7EFFFFFF<(r&&&В) then lnSlow w (rdW g j) (rdW b j) den mean else r
 else t
@[inline] def lnFc (m ir em:ὧ) (den w:Ὢ) (g:ё) (j:ὦ) (gv:ὧ) (lo sp:Ὠ):Ὢ:=
 let gd:=gv*ir
 let q:=(dv w-m)*gd
 let t:=fin1 lo sp (q+dv 0)
 if t==Е then
  let r:=ziv (q+dv 0) (Ἃ 0x3D10000000000000*(q.abs+(dv 0).abs)+em*gd.abs)
  if 0x7EFFFFFF<(r&&&В) then lnSlow w (rdW g j) 0 den (λ_=>0) else r
 else t
@[inline] def lnOutF (s:LnS) (lo:Ὠ) (w gw bw:Ὢ) (gv bv:ὧ) (mean:Unit->ℚ):Ὢ:=
 if s.fast then lnFa s.m s.ir s.em s.den w gw bw gv bv lo mean else lnSlow w gw bw s.den mean
@[inline] def lnOut (s:LnS) (lo:Ὠ) (w gw bw:Ὢ) (mean:Unit->ℚ):Ὢ:=lnOutF s lo w gw bw (dv gw) (dv bw) mean
@[inline] def lnEB (s:LnS) (xm gm bm:ὧ):ὧ:=Ἃ 0x3D10000000000000*((xm+s.m.abs)*(gm*s.ir)+bm)+s.em*(gm*s.ir)
@[inline] def rdF (a:ὤ) (i:ὦ):ὧ:=
 if h:i<a.usize then a.uget i (qDo a i h) else 0
def decG (A:ё) (n t:ὦ) (o:ὤ):ὤ:=
 if h:t<n then decG A n (t+1) (wrF o t (dv (rdW A t))) else o
partial_fixpoint
@[inline] def gdF (r:Ὣ) (g:ё) (n:Ὣ):ὤ:=
 if r==1 then ἆ 0 else decG g ѩn 0 (zFor n)
@[inline] def gdR (r:Ὣ) (g:ё) (G:ὤ) (j:ὦ):ὧ:=
 if r==1 then dv (rdW g j) else rdF G j
@[inline] def smA (A:ё):Ὡ:=
 let h:=scan A A.usize 0 (qAc A) Е 0
 h != Е&&decide (h<=0x70FFFFFF)
@[inline] def normOK (rows n:Ὣ) (eps:Ὢ) (x g:ё):Ὡ:=
 decide (0<rows)&&decide (0<n)&& ѧx==rows*n&& ѧg==n&&decide (n<2^20)&&decide (rows*n<=4294967295)&&
  smA x&&smA g&&sm eps&&!signOf eps&&decide (0< mantOf eps)
def lnRF (x g b:ё) (G B:ὤ) (m ir em:ὧ) (den:Ὢ) (mean:Unit->ℚ) (lo sp:Ὠ) (base n j:ὦ) (o:ё):ё:=
 if h:j<n then
  lnRF x g b G B m ir em den mean lo sp base n (j+1)
   (wrW o (base+j) (lnFb m ir em den x g b (base+j) j (rdF G j) (rdF B j) lo sp mean))
 else o
partial_fixpoint
def lnRS (x g b:ё) (den:Ὢ) (mean:Unit->ℚ) (base n j:ὦ) (o:ё):ё:=
 if h:j<n then
  lnRS x g b den mean base n (j+1) (wrW o (base+j) (lnSlow (rdW x (base+j)) (rdW g j) (rdW b j) den mean))
 else o
partial_fixpoint
@[noinline] def lnSlowM (w gw bw den:Ὢ) (nN:Ὣ) (x:ё) (rR:Ὣ):Ὢ:=lnSlow w gw bw den (λ_=>meanQ nN x rR)
@[noinline] def lnRSM (x g b:ё) (den:Ὢ) (nN rR:Ὣ) (base n j:ὦ) (o:ё):ё:=lnRS x g b den (λ_=>meanQ nN x rR) base n j o
@[noinline] def lnRFM (x g b:ё) (G B:ὤ) (m ir em:ὧ) (den:Ὢ) (nN rR:Ὣ) (lo sp:Ὠ) (base n j:ὦ) (o:ё):ё:=lnRF x g b G B m ir em den (λ_=>meanQ nN x rR) lo sp base n j o
def lnRows (x g b:ё) (G B:ὤ) (eps:Ὢ) (xm gm bm:ὧ) (n r:Ὣ) (o:ё):Ὣ->ё
 | 0=>o
 | d+1=>
   let s:=lnStat false n eps x r
   lnRows x g b G B eps xm gm bm n (r+1)
   (if s.fast then lnRF x g b G B s.m s.ir s.em s.den (λ_=>meanQ n x r) (loOf (lnEB s xm gm bm)) (spOf (loOf (lnEB s xm gm bm))) (r*n).toUSize ѩn 0 o
    else lnRS x g b s.den (λ_=>meanQ n x r) (r*n).toUSize ѩn 0 o) d
def lnDecB (x:ё) (base n j:ὦ) (buf:ὤ) (t s:ὧ):ὤ:=
 if h:j<n then
  let v:=dv (rdW x (base+j))
  lnDecB x base n (j+1) (wrF buf j v) (t+v.abs) (s+v)
 else wrF (wrF buf n t) (n+1) s
partial_fixpoint
lemma lnDecB_size (x:ё) (base n:ὦ) (k:Ὣ): ∀(j:ὦ) (buf:ὤ) (t s:ὧ),Ѥj+k=Ѥn->(lnDecB x base n j buf t s).size=Ѩbuf:=by
 induction k with
 | zero=>
  intro j buf t s e
  rw[lnDecB,ἔ (by rw[А];exact omS404 e),qD6,qD6]
 | succ k ih=>
  intro j buf t s e
  have hj:j<n:=А.2 (by exact omS136 e)
  rw[lnDecB,Ἄ hj];dsimp only;rw[ih (j+1) _ _ _ (by rw[usz j n hj];exact omS137 e),qD6]
lemma lnDecB_sz0 (x:ё) (base:ὦ) (n:Ὣ) (buf:ὤ):(lnDecB x base ѩn 0 buf 0 0).size=Ѩbuf:=lnDecB_size x base ѩn (Ѥ(ѩn)) 0 buf 0 0 (by simp only[USize.toNat_zero,Nat.toUSize_eq,USize.toNat_ofNat',zero_add])
lemma uS1 (n:ὦ) (h:Ѥn+1<2^32):Ѥ(n+1)=Ѥn+1:=by rw[qAf n 1 (by rw[ὄ];exact h),ὄ]
def lnDecI (x:ё) (base n j:ὦ) (buf:ὤ) (t s:ὧ) (hx:Ѥbase+Ѥn<=ѧx) (hs:Ѥbase+Ѥn<2^32):ὤ:=
 if h:j<n then
  let v:=dv (x.uget (base+j) (by rw[qAf base j (by have:=А.mp h;omega)];have:=А.mp h;omega))
  lnDecI x base n (j+1) (wrF buf j v) (t+v.abs) (s+v) hx hs
 else wrF (wrF buf n t) (n+1) s
partial_fixpoint
lemma lnDecI_eq (x:ё) (base n:ὦ) (hx:Ѥbase+Ѥn<=ѧx) (hs:Ѥbase+Ѥn<2^32) (hxs:ѧx<2^32) (k:Ὣ):
 ∀(j:ὦ) (buf:ὤ) (t s:ὧ),Ѥj+k=Ѥn->lnDecI x base n j buf t s hx hs=lnDecB x base n j buf t s:=by
 induction k with
 | zero=>
  intro j buf t s e
  rw[lnDecI,lnDecB,ἔ (by rw[А];exact omS404 e),ἔ (by rw[А];exact omS404 e)]
 | succ k ih=>
  intro j buf t s e
  have hj:j<n:=А.2 (by exact omS136 e)
  rw[lnDecI,lnDecB,Ἄ hj,Ἄ hj]
  dsimp only
  rw[ih (j+1) _ _ _ (by rw[usz j n hj];exact omS137 e)]
  simp only[aU x hxs]
@[inline] def lnDecC (x:ё) (base n:ὦ) (buf:ὤ):ὤ:=
 if h:Ѥbase+Ѥn<=ѧx/\ѧx<=4294967295 then lnDecI x base n 0 buf 0 0 h.1 (by omega) else lnDecB x base n 0 buf 0 0
lemma lnDecC_eq (x:ё) (base n:ὦ) (buf:ὤ):lnDecC x base n buf=lnDecB x base n 0 buf 0 0:=by
 unfold lnDecC;split
 . rename_i h;exact lnDecI_eq x base n h.1 א א (Ѥn) 0 buf 0 0 (by simp only[USize.toNat_zero,zero_add])
 . rfl
lemma lnDecC_sz0 (x:ё) (base:ὦ) (n:Ὣ) (buf:ὤ):(lnDecC x base ѩn buf).size=Ѩbuf:=by rw[lnDecC_eq];exact lnDecB_sz0 x base n buf
def lnVarB (buf:ὤ) (n j:ὦ) (m v:ὧ) (hb:Ѥn<=Ѩbuf):ὧ:=
 if h:j<n then
  let d:=buf.uget j (by have:=А.mp h;omega)-m
  lnVarB buf n (j+1) m (v+d*d) hb
 else v
partial_fixpoint
def oneF:ὧ:=1.0
@[inline] def lnStatB (n:Ὣ) (eps:Ὢ) (x:ё) (r:Ὣ) (ind:ὧ) (okI:Ὡ) (buf:ὤ) (hb:n+2<=Ѩbuf) (hbs:Ѩbuf<2^32):LnS:=
 let nu:=ѩn
 let nf:=(n.toUInt64+8).toFloat*Ἃ ь
 let s:=buf.uget (ѩ(n+1)) (by rw[qAd (n+1) (by omega)];omega)
 let t:=buf.uget nu (by rw[qAd n (by omega)];omega)
 let m:=s*ind
 let v:=lnVarB buf nu 0 m 0 (by rw[qAd n (by omega)];omega)
 let em:=(nf*t)*ind+Ἃ ظ
 let tt:=v*ind+dv eps
 let d:=Ἤ tt
 let z:=ziv d ((nf+nf)*d+em)
 let ir:=oneF/dv z
 if sm z&&decide ((0x1F800000:ὥ)<=z)&&decide (z<=(0x70FFFFFF:ὥ))&&decide ((tt.toBits>>>52)-1<1222)&&okI&&invOK ir (dv z) then
  ⟨true,m,ir,em,z⟩
 else
  ⟨false,0,0,0,sqrtRational (rowVariance n x r (lnQ false n x r)+ἂ eps)⟩
@[inline] def lnFvM (m ir em:ὧ) (den w:Ὢ) (v:ὧ) (g b:ё) (j:ὦ) (gv bv:ὧ) (lo sp:Ὠ) (nN:Ὣ) (x:ё) (rR:Ὣ):Ὢ:=
 let gd:=gv*ir
 let q:=(v-m)*gd
 let t:=fin1 lo sp (q+bv)
 if t==Е then
  let r:=ziv (q+bv) (Ἃ 0x3D10000000000000*(q.abs+bv.abs)+em*gd.abs)
  if 0x7EFFFFFF<(r&&&В) then lnSlowM w (rdW g j) (rdW b j) den nN x rR else r
 else t
def lnRB (x g b:ё) (buf G B:ὤ) (m ir em:ὧ) (den:Ὢ) (nN rR:Ὣ) (lo sp:Ὠ) (base n j:ὦ) (o:ё)
 (hb:Ѥn<=Ѩbuf) (hG:Ѥn<=ѨG) (hB:Ѥn<=ѨB):ё:=
 if h:j<n then
  lnRB x g b buf G B m ir em den nN rR lo sp base n (j+1)
   (wrW o (base+j) (lnFvM m ir em den (rdW x (base+j)) (buf.uget j (by have:=А.mp h;omega)) g b j
     (G.uget j (by have:=А.mp h;omega)) (B.uget j (by have:=А.mp h;omega)) lo sp nN x rR)) hb hG hB
 else o
partial_fixpoint
def lnRowsB (x g b:ё) (G B:ὤ) (eps:Ὢ) (xm gm bm:ὧ) (n:Ὣ) (ind:ὧ) (okI:Ὡ) (r:Ὣ) (buf:ὤ) (o:ё):Ὣ->ё
 | 0=>o
 | d+1=>
   if h:n+2<=Ѩbuf/\Ѩbuf<=4294967295/\n<=ѨG/\ѨG<=4294967295/\n<=ѨB/\ѨB<=4294967295/\r*n+n<=ѧx/\ѧx<=4294967295 then
    let base:=(r*n).toUSize
    let bf:=lnDecC x base ѩn buf
    let s:=lnStatB n eps x r ind okI bf (by rw[lnDecC_sz0];omega) (by rw[lnDecC_sz0];omega)
    lnRowsB x g b G B eps xm gm bm n ind okI (r+1) bf
     (if s.fast then lnRB x g b bf G B s.m s.ir s.em s.den n r (loOf (lnEB s xm gm bm)) (spOf (loOf (lnEB s xm gm bm))) base ѩn 0 o
       (by rw[lnDecC_sz0,qAd n (by omega)];omega) (by rw[qAd n (by omega)];omega) (by rw[qAd n (by omega)];omega)
      else lnRSM x g b s.den n r base ѩn 0 o) d
   else
    let s:=lnStat false n eps x r
    lnRowsB x g b G B eps xm gm bm n ind okI (r+1) buf
     (if s.fast then lnRFM x g b G B s.m s.ir s.em s.den n r (loOf (lnEB s xm gm bm)) (spOf (loOf (lnEB s xm gm bm))) (r*n).toUSize ѩn 0 o
      else lnRSM x g b s.den n r (r*n).toUSize ѩn 0 o) d
def rmsRF (rows:Ὣ) (x g:ё) (G:ὤ) (m ir em:ὧ) (den:Ὢ) (lo sp:Ὠ) (base n j:ὦ) (o:ё):ё:=
 if h:j<n then
  let w:=rdW x (base+j)
  rmsRF rows x g G m ir em den lo sp base n (j+1) (wrW o (base+j) (if w&&&В==0 then 0 else lnFc m ir em den w g j (gdR rows g G j) lo sp))
 else o
partial_fixpoint
def rmsRows (rows:Ὣ) (x g:ё) (G:ὤ) (eps:Ὢ) (xm gm:ὧ) (n r:Ὣ) (o:ё):Ὣ->ё
 | 0=>o
 | d+1=>
   let s:=lnStat true n eps x r
   rmsRows rows x g G eps xm gm n (r+1)
   (if s.fast then rmsRF rows x g G s.m s.ir s.em s.den (loOf (lnEB s xm gm (dv 0))) (spOf (loOf (lnEB s xm gm (dv 0)))) (r*n).toUSize ѩn 0 o
    else fC (λs p j=>let w:=rdW x p
     if w&&&В==0 then 0 else lnOutF s.1 s.2 w (rdW g j) 0 (gdR rows g G j) (dv 0) λ_=>0) (s,loOf (lnEB s xm gm (dv 0))) (r*n).toUSize 0 o n) d
@[inline] def okX (h:ὥ):Ὡ:=h != Е&&decide (h<=0x70FFFFFF)
@[inline] def normOK2 (rows n:Ὣ) (eps:Ὢ) (x g:ё) (hx hg:ὥ):Ὡ:=
 decide (0<rows)&&decide (0<n)&& ѧx==rows*n&& ѧg==n&&decide (n<2^20)&&decide (rows*n<=4294967295)&&
  okX hx&&okX hg&&sm eps&&!signOf eps&&decide (0< mantOf eps)
abbrev LNV (n:Ὣ) (g b:ё):=({p:(ὤ×ὤ)×(ὥ×ὥ)//p=((decG g ѩn 0 (zFor n),decG b ѩn 0 (zFor n)),(scan g g.usize 0 (qAc g) Е 0,scan b b.usize 0 (qAc b) Е 0))})
structure LNE where
 (g b:ё) (n:Ὣ) (p:(ὤ×ὤ)×(ὥ×ὥ)) (k:ὦ) (ok:p=((decG g ѩn 0 (zFor n),decG b ѩn 0 (zFor n)),(scan g g.usize 0 (qAc g) Е 0,scan b b.usize 0 (qAc b) Е 0)))
def lnFd (n:Ὣ) (g b:ё) (k:ὦ) (s:Array LNE) (i:Ὣ):Option (LNV n g b):=
 if h:i<s.size then
  if s[i].k==k then
   if ho:s[i].n=n then
    if hk:withPtrEq s[i].g g (λ_=>s[i].g==g) (λe=>by rw[e];exact beq_self_eq_true g) then
     if hk2:withPtrEq s[i].b b (λ_=>s[i].b==b) (λe=>by rw[e];exact beq_self_eq_true b) then
      some ⟨s[i].p,by rw[<-ho,<-eq_of_beq hk,<-eq_of_beq hk2];exact s[i].ok⟩
     else lnFd n g b k s (i+1)
    else lnFd n g b k s (i+1)
   else lnFd n g b k s (i+1)
  else lnFd n g b k s (i+1)
 else none
partial_fixpoint
@[noinline] def lnS:ST.Ref Unit (Array LNE):=(ST.Prim.mkRef #[] (Void.mk ())).val
@[inline] def lnGet (n:Ὣ) (g b:ё):LNV n g b:=
 withPtrAddr g (λk=>
  let s:=(ST.Prim.Ref.get lnS (Void.mk ())).val
  match lnFd n g b k s 0 with
  | some v=>v
  | none=>
   let p:=((decG g ѩn 0 (zFor n),decG b ѩn 0 (zFor n)),(scan g g.usize 0 (qAc g) Е 0,scan b b.usize 0 (qAc b) Е 0))
   let old:=(ST.Prim.Ref.swap lnS (#[⟨g,b,n,p,k,rfl⟩]++(if s.size<256 then s else s.pop)) (Void.mk ())).val
   match lnFd n g b k old 0 with
   | some v=>v
   | none=>⟨p,rfl⟩) (λ_ _=>Subsingleton.elim _ _)
abbrev RNV (g:ё):=({h:ὥ//h=scan g g.usize 0 (qAc g) Е 0})
structure RNE where
 (g:ё) (h:ὥ) (k:ὦ) (ok:h=scan g g.usize 0 (qAc g) Е 0)
def rnFd (g:ё) (k:ὦ) (s:Array RNE) (i:Ὣ):Option (RNV g):=
 if h:i<s.size then
  if s[i].k==k then
   if hk:withPtrEq s[i].g g (λ_=>s[i].g==g) (λe=>by rw[e];exact beq_self_eq_true g) then
    some ⟨s[i].h,by rw[<-eq_of_beq hk];exact s[i].ok⟩
   else rnFd g k s (i+1)
  else rnFd g k s (i+1)
 else none
partial_fixpoint
@[noinline] def rnS:ST.Ref Unit (Array RNE):=(ST.Prim.mkRef #[] (Void.mk ())).val
@[inline] def rnGet (g:ё):RNV g:=
 withPtrAddr g (λk=>
  let s:=(ST.Prim.Ref.get rnS (Void.mk ())).val
  match rnFd g k s 0 with
  | some v=>v
  | none=>
   let h:=scan g g.usize 0 (qAc g) Е 0
   let old:=(ST.Prim.Ref.swap rnS (#[⟨g,h,k,rfl⟩]++(if s.size<256 then s else s.pop)) (Void.mk ())).val
   match rnFd g k old 0 with
   | some v=>v
   | none=>⟨h,rfl⟩) (λ_ _=>Subsingleton.elim _ _)
def lnFq (rows n:Ὣ) (eps:Ὢ) (x g b:ё) (hx:ὥ) (c:(ὤ×ὤ)×(ὥ×ὥ)):ё:=
 if normOK2 rows n eps x g hx c.2.1&& ѧb==n&&okX c.2.2 then
  let nd:=n.toUInt64.toFloat
  let ind:=oneF/nd
  lnRowsB x g b c.1.1 c.1.2 eps (dv hx) (dv c.2.1) (dv c.2.2) n ind (invOK ind nd) 0 (zFor (n+2)) (ἇ (rows*n) 0) rows
 else layernormSpec rows n eps x g b
def lnF (rows n:Ὣ) (eps:Ὢ) (x g b:ё):ё:=lnFq rows n eps x g b (scan x x.usize 0 (qAc x) Е 0) (lnGet n g b).1
def rmsFz (rows n:Ὣ) (eps:Ὢ) (x g:ё) (hx hg:ὥ):ё:=
 if normOK2 rows n eps x g hx hg then
  rmsRows rows x g (gdF rows g n) eps (dv hx) (dv hg) n 0 (ἇ (rows*n) 0) rows
 else rmsnormSpec rows n eps x g
def rmsF (rows n:Ὣ) (eps:Ὢ) (x g:ё):ё:=rmsFz rows n eps x g (scan x x.usize 0 (qAc x) Е 0) (rnGet g).1
section
lemma qFv (x:Ὠ):(x>>>52).toNat=Ѧx/2^52:=by rw[ἰ,Ἱ];rfl
lemma qEj (x k N:Ὠ) (h:x-k<N) (hN:Ѧk+ѦN<2^64): Ѧk<=Ѧx/\Ѧx< Ѧk+ѦN:=by have h1:=ἴ.mp h;rw[ὲ] at h1;have h2:=x.toNat_lt;have h3:=k.toNat_lt;(clear h;omega)
lemma qEi (a b:ℤ) (n:ℕ) (h:a=n+b):(2:ℝ)^a=2^n*2^b:=by rw[h,Ἓ ἓ,Ἡ]
lemma qCz {f:ὧ} {E:Ὣ} (h:(f.toBits>>>52).toNat=E) (hE1:1<=E) (hE:E<2047):f.isFinite=true/\(2:ℝ)^((E:ℤ)-1023)<=R f/\R f<2^((E:ℤ)-1022):=by
 rw[qFv] at h;obtain ⟨F,hF,hM⟩:∃F<2^52,f.toBits.toNat=E*2^52+F:=⟨f.toBits.toNat%2^52,by (clear hE1 hE;exact omS337),by (clear hE1 hE;exact omS338 h)⟩;obtain ⟨h1,h2⟩:=B_R f false E F hE1 hE hF (by simpa only[false_eq_true, ↓reduceIte,Nat.reducePow,zero_add] using hM);have hb:(0:ℝ)<2^((E:ℤ)-1075):=ἵ ו _;have hF0:(0:ℝ)<=F:=Ἔ _;have hF2:(F:ℝ)<2^52:=by exact_mod_cast hF
 simp only[Bool.false_eq_true,ite_false,one_mul] at h2;push_cast at h2;rw[h2,qEi ((E:ℤ)-1023) (E-1075) 52 (by clear h hE1 hE hF hM;exact omS339),qEi ((E:ℤ)-1022) (E-1075) 53 (by clear h hE1 hE hF hM;exact omS340)];exact ⟨h1,ἀ (by linarith only[hF]) hb.le,Ἦ (by linarith only[hF2]) hb⟩
lemma L_k (c:Ὠ) (E n:Ὣ) (hE1:1<=E) (hn:E+n=1023) (h:(Ἃ c).toBits.toNat=E*2^52):(Ἃ c).isFinite=true/\R (Ἃ c)=1/2^n:=by obtain ⟨h1,h2⟩:=B_R (Ἃ c) false E 0 hE1 (by clear h;exact omS341 hn) ב (by simpa only[false_eq_true, ↓reduceIte,Nat.reducePow,zero_add,add_zero] using h);refine ⟨h1,?_⟩;rw[h2,show (E:ℤ)-1075=-(52+(n:ℤ)) by (clear hE1 h;exact omS342 hn),zpow_neg,Ἓ ἓ];norm_num;field_simp
lemma qGk {f:ὧ} {E:Ὣ} (h:(f.toBits>>>52).toNat=E) (hE1:1<=E) (hE:E<2047):
 (Ἤ f).isFinite=true/\0<R f/\
  |R (Ἤ f)-ἧ (R f)| <=1/2^53*(ἧ (R f)+A_z):=by
 obtain ⟨hf,hl,-⟩:=qCz h hE1 hE
 have hp:0<R f:=Ἥ (ἵ ו _) hl
 rw[qFv] at h
 have hs:Model.signBit (toModel (ofFloat f:Binary64))=false:=by
  rw[qFs];simp only[decide_eq_false_iff_not,not_le];exact omS343 h hE
 have hm:toModel (ofFloat f:Binary64)=A_M f:=toModel_ofFloat f
 have hfm:Ἑ (A_M f)=true:=by rw[qAh,hf]
 rw[hm] at hs
 have hr:=Model.toReal_sqrt_eq_roundAt (A_M f) ה hfm (Or.inr hs)
 have hq:=Model.isFinite_sqrt_of_isFinite (A_M f) ה hfm (Or.inr hs)
 have hd:A_M (Ἤ f)=Model.canonicalizeModel (Model.sqrt (A_M f)):=by
  have h3:=toModel_ofFloat_sqrt f
  rw[hm,toModel_ofFloat] at h3
  exact h3
 have ht:Model.toModel (A_M (Ἤ f))=Model.toModel (Model.sqrt (A_M f)):=by
  rw[hd,Model.toModel_canonicalizeModel]
 have hR:R (Ἤ f)=Model.roundAt ⟡ (ἧ (R f)):=by
  rw[A_R,ἦ qAj,ht,<-ἦ qAj]
  exact hr
 refine ⟨?_,hp,?_⟩
 . rw[<-qAh,qAy,ht,<-qAy];exact hq
 . rw[hR];have:=qFp (ἧ (R f));rwa[ἅ (Real.sqrt_nonneg _)] at this
lemma qAa (x y:ℝ): |x| <= |y| + |x-y|:=by have:=Ὥ x y;linarith only[this]
lemma qAt {k:ℝ} (h:k<=2^256*2^44):k<=A_B:=h.trans_eq ((qEi 300 44 256 rfl).trans (by norm_num1)).symm
lemma qBg (a:ѐ) (p:ὥ->Ὡ) (h:a.all p=true) (i:ℕ) (hi:i< ѧa):p a[i]! =true:=by rw[Ἕ a i hi];simpa only using Array.all_eq_true.1 h i hi
lemma qE6 (w:ὥ) (g:good w=true) (h:w&&&В=0):ἂ w=0:=by exact_mod_cast (qAx w (qAe _ g)).symm.trans (qBr w g h)
lemma qIu (s e:ὧ) (V:ℝ) (hlo:R (s-e)<V) (hhi:V<R (s+e)) (h0:ziv s e≠Е) (h:¬(0x7EFFFFFF:ὥ)<(ziv s e&&&В)):
 |V| <=2^(127:ℤ):=by
 obtain ⟨S,ex,fr,W,h1,h2,h3,hz,hV,hW1,-,hW3⟩:=qB8 s e V hlo hhi h0
 have hb:(0:ℝ)<2^((ex:Int)-1076):=ἵ ו _
 have hW0:0<W:=lt_trans (mul_pos ו hb) hW3
 have hN:((if S then 2^31 else 0)+(ex-896)*2^23+fr)<2^32:=by
  split<;>(clear h0 h hz;omega)
 have hex:ex<=1149:=by
  by_contra hc
  apply h
  apply ἳ.mpr
  rw[qAq,hz,UInt32.toNat_ofNat_of_lt' hN,((by clear h0 h h3 hz hN;exact omS344 h2 hc):ex=1150)]
  show 2130706431<_
  split<;>(clear h0 h h1 h2 hz hN hc;omega)
 have hF:(fr:ℝ)+1<=2^23:=by exact_mod_cast ((by clear h0 h h1 h2 hz hN hex;(exact Nat.succ_le_of_lt ‹_›)):fr+1<=2^23)
 rw[hV]
 have hW:W<=2^54*2^(73:ℤ):=hW1.le.trans (ἑ (by linarith only[hF]) (ἒ (by norm_num1) (by clear h0 h h3 hz hN;exact omS345 hex)) hb.le ו)
 rw[<-qEi 127 73 54 rfl] at hW
 split<;>simp only[abs_neg,ἡ hW0,hW]
lemma L_I {n I:ℝ} (hn:1<=n) (h3:0<I) (h4: |I*n-1| <=1/2^48):I<=2:=by have:=ἁ hn h3.le;linarith only[(abs_le.mp h4).2,this,h4]
lemma qF0 (n I v t Sb E T:ℝ) (h3:0<I) (h4: |I*n-1| <=1/2^48) (ht: |t| <=Sb) (hv: |v-n*t| <=n*E+T) (hE:0<=E):
 |v*I-t| <=E*(1+1/2^48)+T*I+Sb*(1/2^48):=by
 have a1: |(v-n*t)*I| <=(n*E+T)*I:=by
  rw[ἢ,ἡ h3];exact ἀ hv h3.le
 have a2:=ἁ (abs_le.mp h4).2 hE
 have a3: |t*(I*n-1)| <=Sb*(1/2^48):=by
  rw[ἢ];exact ἑ ht h4 (Ἀ _) ((Ἀ _).trans ht)
 have a4:=Ἐ ((v-n*t)*I) (t*(I*n-1))
 rw[show (v-n*t)*I+t*(I*n-1)=v*I-t by ring] at a4
 linarith only[a4,a3,a2,a1]
section
variable {z:ℝ} (hz0:0<=z) (hz:z<=1/2^200*(1/2^200))
include hz0 hz
lemma qD9 {P rp rs rc:ℝ} (h1: |rp-P| <=1/2^53*(|P| +z)) (h2: |rs-(rp-1)| <=1/2^53*(|rp-1| +z)) (h3: |rc-(rs+1/2^50)| <=1/2^53*(|rs+1/2^50| +z)) (lo:1/2^51<=rc) (hi:rc<=1/2^49): |P-1| <=1/2^48:=by
 have sck:∀{k a b:ℝ},0<k->a<=1/k*b->k*a<=b:=λhk h=>(le_div_iff₀' hk).mp (h.trans_eq (one_div_mul_eq_div _ _))
 have hz1:z<=1:=hz.trans (by norm_num1)
 have lo':1<=2^51*rc:=(div_le_iff₀' (by norm_num1)).mp lo;have hi':2^49*rc<=1:=(le_div_iff₀' (by norm_num1)).mp hi
 have a1:=qAa (rs+1/2^50) rc;rw[ἣ (rs+1/2^50) rc] at a1;have a2: |rc| =rc:=ἅ (ὂ (by norm_num1) lo)
 generalize hq:(1:ℝ)/2^50=q at h3 a1;have hq':(2:ℝ)^50*q=1:=by rw[<-hq];exact mul_one_div_cancel (by norm_num1)
 have h1':=sck (by norm_num1) h1;have h2':=sck (by norm_num1) h2;have h3':=sck (by norm_num1) h3
 have c1:=neg_abs_le (rc-(rs+q));have c2:=le_abs_self (rc-(rs+q))
 have a3:2^49*|rs| <=1:=by rw[<-ἡ (by norm_num1:(0:ℝ)<2^49),<-ἢ];exact abs_le.mpr ⟨by linarith only[c1,c2,a1,a2,h3',lo',hi',hq',hz1,hz0],by linarith only[c1,c2,a1,a2,h3',lo',hi',hq',hz1,hz0]⟩
 have b1:=qAa (rp-1) rs;rw[ἣ (rp-1) rs] at b1;have b2:2^50*|rp-1| <=3:=by linarith only[b1,a3,h2',hz1,hz0]
 have c3:=qAa (P-1) (rp-1);rw[sub_sub_sub_cancel_right,ἣ P rp] at c3;have c4:=qAa P 1;rw[abs_one] at c4
 exact (le_div_iff₀' (by norm_num1)).mpr (by linarith only[c3,c4,b2,h1',hz1,hz0])
lemma qG0 {a Mr Rm Em g D b I RA Rg Rq RY RE Rsa Rcs Reg Hi Lo:ℝ} (hE0:0<=Em) (hE1:Em<=2^120) (hM:2* |Mr-Rm| <=Em) (hD:1/2^64<=D) (ha: |a| <=2^99) (hg: |g| <=2^99) (hm: |Rm| <=2^110) (h1: |RA-(a-Rm)| <=1/2^53*(|a-Rm| +z)) (h2: |I*D-1| <=1/2^48) (h3: |Rg-g*I| <=1/2^53*(|g*I| +z)) (h4: |Rq-RA*Rg| <=1/2^53*(|RA*Rg| +z)) (h5: |RY-(Rq+b)| <=1/2^53*(|Rq+b| +z)) (h6: |Rsa-(|Rq| + |b|)| <=1/2^53*(|(|Rq| + |b|)| +z)) (h7: |Rcs-1/2^46*Rsa| <=1/2^53*(|1/2^46*Rsa| +z)) (h8: |Reg-Em*(|Rg|)| <=1/2^53*(|Em*(|Rg|)| +z)) (h9: |RE-(Rcs+Reg)| <=1/2^53*(|Rcs+Reg| +z)) (h10: |Hi-(RY+RE)| <=1/2^53*(|RY+RE| +z)) (h11: |Lo-(RY-RE)| <=1/2^53*(|RY-RE| +z)) (hbig:1/2^127<= |Hi|):Lo<(a-Mr)*g/D+b/\(a-Mr)*g/D+b<Hi:=by
 have sck:∀{k a b:ℝ},0<k->a<=1/k*b->k*a<=b:=λhk h=>(le_div_iff₀' hk).mp (h.trans_eq (one_div_mul_eq_div _ _))
 have scu:∀{a b:ℝ},a<=1/2^53*b->2^53*a<=b:=sck (by norm_num1)
 have hDp:0<D:=Ἥ (div_pos one_pos (pow_pos two_pos 64)) hD
 rw[mul_div_assoc,<-sub_add_sub_cancel a Rm Mr,add_mul]
 have hGb: |g/D| <=2^163:=by rw[abs_div,ἡ hDp,div_le_iff₀ hDp];exact hg.trans ((by norm_num1:(2:ℝ)^99=2^163*(1/2^64)).trans_le (ἁ hD (pow_nonneg zero_le_two 163)))
 have hXb: |a-Rm| <=2^111:=(Ὕ _ _).trans ((add_le_add ha hm).trans (by norm_num1))
 have hgI:2^48*|g*I-g/D| <= |g/D|:=by rw[show g*I-g/D=g/D*(I*D-1) by rw[mul_sub,mul_one,mul_comm I D,<-mul_assoc,div_mul_cancel₀ g hDp.ne'],ἢ,mul_left_comm];exact (ἁ (sck (by norm_num1) (h2.trans_eq (mul_one _).symm)) (Ἀ _)).trans_eq (mul_one _)
 generalize g/D=G at hGb hgI⊢;generalize a-Rm=X at h1 hXb⊢;generalize g*I=Y at h3 hgI;clear hDp hD ha hg hm h2
 have rU:∀{R x:ℝ},|R-x| <=1/2^53*(|x| +z)->|R| <=2*|x|+z:=λ{R x} h=>((qAa R x).trans (add_le_add le_rfl (h.trans (mul_le_of_le_one_left (add_nonneg (Ἀ x) hz0) (by norm_num1))))).trans_eq (by rw[<-add_assoc,two_mul])
 have hz':(2:ℝ)^200*(2^200*z)<=1:=(le_div_iff₀' (by norm_num1)).mp ((le_div_iff₀' (by norm_num1)).mp (hz.trans_eq (one_div_mul_eq_div _ _)))
 have hz1:z<=1:=hz.trans (by norm_num1)
 have S1:=scu h1;have S3:=scu h3;have S4:=scu h4;have S5:=scu h5;have S6:=scu h6;have S9:=scu h9;have S10:=scu h10;have S11:=scu h11
 have nX:=Ἀ X;have nG:=Ἀ G;have nA:=Ἀ RA;have nR:=Ἀ Rg;have a1:=qAa X RA;rw[ἣ X RA] at a1
 have a4:2^53*|RA-X| <=2*|RA|+2*z:=by linarith only[a1,nX,S1,hz0]
 have a5: |RA| <=2^165:=((rU h1).trans (add_le_add (ἁ hXb zero_le_two) hz1)).trans (by norm_num1)
 have b1:=qAa Y G;have b2:=qAa G Rg;have b3:=abs_sub_le G Y Rg;rw[ἣ G Y,ἣ Y Rg] at b3
 have b4:2^53*|G| <=(2^53+2^6)*|Rg|+2*z:=by linarith only[b3,b2,b1,nG,hgI,S3,hz0]
 have b5:2^53*|G-Rg| <=2^6*|Rg|+2*z:=by linarith only[b3,b2,b1,nR,hgI,S3,hz0]
 have c1: |Rg| <=2^165:=by have:=qAa Rg G;rw[ἣ] at this;linarith only[this,b3,b1,hgI,S3,hGb,hz']
 have f1:=Ἐ ((X-RA)*G) (RA*(G-Rg));rw[ἢ,ἢ,show (X-RA)*G+RA*(G-Rg)=X*G-RA*Rg by rw[sub_mul,mul_sub,sub_add_sub_cancel],ἣ X RA] at f1
 have f2:=ἑ a4 b4 (Ἆ (by norm_num1) nG) (add_nonneg (Ἆ (by norm_num1) nA) (Ἆ (by norm_num1) hz0));have f3:=ἁ b5 nA
 have zr:z* |RA| <=z*2^165:=ἁ a5 hz0;have zg:z* |Rg| <=z*2^165:=ἁ c1 hz0;have zz:z*z<=z*2^165:=ἁ (hz1.trans (by norm_num1)) hz0;have ze:z*Em<=z*2^165:=ἁ (hE1.trans (by norm_num1)) hz0;have nP:=Ἆ nA nR
 have f4:2^230*|X*G-RA*Rg| <=9*2^180*(|RA| * |Rg|)+1:=by linarith only[nP,zz,zg,zr,f3,f2,f1,hz']
 have p1:2^53*|RA*Rg| <=(2^53+2)*|Rq|+2*z:=by have:=qAa (RA*Rg) Rq;rw[ἣ (RA*Rg) Rq] at this;have nq:=Ἀ (RA*Rg);linarith only[nq,this,S4,hz0]
 rw[<-ἢ] at f4;have q1:2^229*|X*G-Rq| <=5*2^180*|Rq|+1:=by have:=Ὕ (X*G-RA*Rg) (Rq-RA*Rg);rw[sub_sub_sub_cancel_right] at this;have nq:=Ἀ Rq;linarith only[nq,this,p1,f4,S4,hz']
 generalize hT:Em* |Rg| =T at h8;have nT:0<=T:=hT▸Ἆ hE0 nR
 have e1:2^230*|(Rm-Mr)*G| <=5*2^227*T+1:=by rw[ἢ,ἣ];have h:=ἑ hM b4 (Ἆ (by norm_num1) nG) hE0;subst hT;linarith only[nT,h,ze,hz']
 have s3:=rU h8;rw[ἅ nT] at h8 s3;have S8:=scu h8;have nq:=Ἀ Rq;have nb:=Ἀ b;have nS:0<= |Rq| + |b| :=add_nonneg nq nb;have s1:=rU h6;rw[ἅ nS] at h6 s1 S6;have s0: |Rq+b| <= |Rq| + |b|:=Ἐ _ _
 generalize hV:(1:ℝ)/2^46*Rsa=V at h7;have hV':2^46*|V| = |Rsa|:=by rw[<-hV,ἢ,ἡ (div_pos one_pos (pow_pos two_pos 46):(0:ℝ)<1/2^46),<-mul_assoc,mul_one_div_cancel (by norm_num1),one_mul]
 have S7:=scu h7;have hV2:2^46*V=Rsa:=by rw[<-hV,<-mul_assoc,mul_one_div_cancel (by norm_num1),one_mul]
 have s2:2^44*|Rcs| <=(|Rq| + |b|)+2^44*z:=by have:=qAa Rcs V;linarith only[this,s1,nb,nq,S7,hV',hz0]
 have s4: |RY| <=2*(|Rq| + |b|)+z:=(rU h5).trans (add_le_add (ἁ s0 zero_le_two) le_rfl)
 have s5:=Ἐ Rcs Reg;have s6:2^43*|RE| <=(|Rq| + |b|)+2^45*T+2^45*z:=by have:=qAa RE (Rcs+Reg);linarith only[this,s5,s3,s2,nb,nq,nT,S9,hz0]
 have s7:=Ἐ RY RE;have s8:=Ὕ RY RE;have hb0:=qAa Hi (RY+RE)
 have hbig':1<=2^127*|Hi|:=(div_le_iff₀' (by norm_num1)).mp hbig
 have n9:=neg_abs_le (RE-(Rcs+Reg));have n8:=neg_abs_le (Reg-T);have n7:=neg_abs_le (Rcs-V);have n6:=neg_abs_le (Rsa-(|Rq|+|b|));have n10:=neg_abs_le (Hi-(RY+RE));have p10:=le_abs_self (Hi-(RY+RE));have n11:=neg_abs_le (Lo-(RY-RE));have p11:=le_abs_self (Lo-(RY-RE));have nq1:=neg_abs_le (X*G-Rq);have pq1:=le_abs_self (X*G-Rq);have ne1:=neg_abs_le ((Rm-Mr)*G);have pe1:=le_abs_self ((Rm-Mr)*G);have n5:=neg_abs_le (RY-(Rq+b));have p5:=le_abs_self (RY-(Rq+b))
 have M:5*2^181*|Rq|+3+5*2^227*T+2^177*(|Rq+b|+z)+2^177*(|RY|+|RE|+z)<2^230*RE:=by linarith only[S9,n9,S8,n8,S7,n7,hV',hV2,S6,n6,hb0,s7,S10,n10,hbig',s6,s5,s4,s3,s2,s1,s0,nb,nq,nT,hz',hz0]
 constructor
 . linarith only[S11,p11,s8,S5,n5,p5,q1,nq1,e1,ne1,M]
 . linarith only[S10,n10,s7,S5,n5,p5,q1,pq1,e1,pe1,M]
lemma qG3 {n Rs Rt I Rm N Y Em c:ℝ} (r Sb:ℝ) (hn1:1<=n) (hn2:n<=2^20) (hc:c=(n+8)/2^49) (hr: |r| <=Sb) (hS:Sb<=2^99) (h1: |Rs-n*r| <=2*n*(1/2^53*(n*Sb+z))) (h2: |Rt-n*Sb| <=2*n*(1/2^53*(n*Sb+z))) (h3:0<I) (h4: |I*n-1| <=1/2^48) (h5: |Rm-Rs*I| <=1/2^53*(|Rs*I| +z)) (h6: |N-c*Rt| <=1/2^53*(|c*Rt| +z)) (h7: |Y-N*I| <=1/2^53*(|N*I| +z)) (h8: |Em-(Y+1/2^200)| <=1/2^53*(|Y+1/2^200| +z)):2* |r-Rm| <=Em/\Em<=2^120/\1/2^201<=Em/\|Rm| <=2^110:=by
 have scu:∀{a b:ℝ},a<=1/2^53*b->2^53*a<=b:=λh=>(le_div_iff₀' (by norm_num1)).mp (h.trans_eq (one_div_mul_eq_div _ _))
 have hS0:0<=Sb:=ὂ (Ἀ _) hr;have n0:0<=n:=ὂ zero_le_one hn1;have nS:0<=n*Sb:=Ἆ n0 hS0;have nSb:n*Sb<=2^20*2^99:=ἑ hn2 hS hS0 (pow_nonneg zero_le_two 20);generalize hW:n*Sb+z=W at h1 h2;have nW:0<=W:=hW▸add_nonneg nS hz0;have hI2:=L_I hn1 h3 h4;generalize hE:201/100/2^53*W+Sb/2^48=E
 have hE':100*2^53*E=201*W+3200*Sb:=by rw[<-hE];ring
 have key:∀v t:ℝ,|t| <=Sb->|v-n*t| <=2*n*(1/2^53*W)->|v*I-t| <=E:=
  λv t ht hv=>by
   have:=qF0 n I v t Sb (2*(1/2^53*W)) 0 h3 h4 ht (hv.trans_eq (((mul_assoc 2 n _).trans (mul_left_comm 2 n _)).trans (add_zero _).symm)) (Ἆ zero_le_two (Ἆ (by norm_num1) nW));linarith only[this,hE',nW]
 have hr1:=key Rs r hr h1;have hA1:=key Rt Sb (by rw[ἅ hS0]) h2;generalize hA:Rt*I=A at hA1
 have hc0:0<=c:=hc▸div_nonneg (add_nonneg n0 (by norm_num1)) (by norm_num1)
 have hc':n+8=c*2^49:=(div_eq_iff (by norm_num1)).mp hc.symm
 have hc1:2^28*c<=1:=by linarith only[hc',hn2]
 have b0:=qAa A Sb;rw[ἅ hS0] at b0
 have b1:2^53*|N*I-c*A| <= |c*A|+2*z:=by rw[<-hA,show N*I-c*(Rt*I)=(N-c*Rt)*I by rw[sub_mul,mul_assoc],ἢ,ἡ h3,show |c*(Rt*I)| = |c*Rt| *I by rw[<-mul_assoc,ἢ (c*Rt),ἡ h3]];have u1:=ἀ (scu h6) h3.le;have u2:=ἁ hI2 hz0;linarith only[u1,u2]
 have b2:2^28*|c*A| <=Sb+E:=by rw[ἢ,ἅ hc0,<-mul_assoc];exact (ἑ hc1 (b0.trans (add_le_add le_rfl hA1)) (Ἀ _) zero_le_one).trans_eq (one_mul _)
 have h9:2^28*|c*(A-Sb)| <=E:=by rw[ἢ,ἅ hc0,<-mul_assoc];exact (ἑ hc1 hA1 (Ἀ _) zero_le_one).trans_eq (one_mul _)
 have b4:(n+8)*Sb=c*2^49*Sb:=by rw[hc']
 have c1:=qAa (N*I) (c*A);have c2:=qAa Y (N*I);have c3:=Ἐ Y (1/2^200);rw[ἡ (by norm_num1:(0:ℝ)<1/2^200)] at c3;generalize hq:(1:ℝ)/2^200=q at h8 c3;have hq':2^200*q=1:=by rw[<-hq];exact mul_one_div_cancel (by norm_num1)
 have d1:=qAa Rm (Rs*I);have d2:=qAa (Rs*I) r;have d3:=abs_sub_le r (Rs*I) Rm;rw[ἣ r (Rs*I),ἣ (Rs*I) Rm] at d3
 have s5:=scu h5;have s7:=scu h7;have s8:=scu h8;have hz':(2:ℝ)^200*(2^200*z)<=1:=(le_div_iff₀' (by norm_num1)).mp ((le_div_iff₀' (by norm_num1)).mp (hz.trans_eq (one_div_mul_eq_div _ _)))
 have g1:=le_abs_self (c*A);have n1:=neg_abs_le (N*I-c*A);have p1:=le_abs_self (N*I-c*A);have n7:=neg_abs_le (Y-N*I);have p7:=le_abs_self (Y-N*I);have n8:=neg_abs_le (Em-(Y+q));have p8:=le_abs_self (Em-(Y+q));have n9:=neg_abs_le (c*(A-Sb))
 refine ⟨?_,?_,(div_le_iff₀' (by norm_num1)).mpr ?_,?_⟩
 . linarith only[h9,n9,b2,hr1,s8,n8,s7,n7,b1,n1,d3,d2,c3,c2,c1,b4,nS,hS0,hr,hz',hq',hE',hW,s5]
 . linarith only[g1,b2,s8,p8,s7,p7,b1,p1,c3,c2,c1,nSb,hS,hz',hq',hE',hW]
 . linarith only[h9,n9,b2,s8,n8,s7,n7,b1,n1,c3,c2,c1,b4,nS,hS0,hz',hq',hE',hW]
 . linarith only[hr1,d2,d1,nSb,s5,hS,hr,hz',hW,hE']
lemma qHr {s c Em dd Rn2 Rmu ee hi lo:ℝ} (hs0:0<=s) (hs1:s<=2^101) (hc0:9/2^49<=c) (hc1:c<=1/2^20) (hE:1/2^201<=Em) (hE1:Em<=2^120) (h1: |dd-s| <=1/2^53*(s+z)) (h2: |Rn2-(c+c)| <=1/2^53*(|c+c| +z)) (h3: |Rmu-Rn2*dd| <=1/2^53*(|Rn2*dd| +z)) (h4: |ee-(Rmu+Em)| <=1/2^53*(|Rmu+Em| +z)) (h5: |hi-(dd+ee)| <=1/2^53*(|dd+ee| +z)) (h6: |lo-(dd-ee)| <=1/2^53*(|dd-ee| +z)):s*(1+3/2*c)+9/10*Em<=hi/\lo<=s*(1-3/2*c)-9/10*Em:=by
 have sc:∀{a b:ℝ},a<=1/2^53*b->2^53*a<=b:=λh=>(le_div_iff₀' (by norm_num1)).mp (h.trans_eq (one_div_mul_eq_div _ _))
 have hcp:0<c:=Ἥ (div_pos (Nat.ofNat_pos) (pow_pos two_pos 49)) hc0;have hcs:0<=c*s:=Ἆ hcp.le hs0;have hcs2:=ἀ ((div_le_iff₀' (by norm_num1:(0:ℝ)<2^49)).mp hc0) hs0;have hcc: |c+c| =c+c:=ἡ (add_pos hcp hcp);rw[hcc] at h2
 have hz':(2^200*2^200:ℝ)*z<=1:=(le_div_iff₀' (mul_pos (pow_pos two_pos 200) (pow_pos two_pos 200))).mp (hz.trans_eq (one_div_mul_one_div _ _));have hE':=(div_le_iff₀' (by norm_num1:(0:ℝ)<2^201)).mp hE
 have h1':=sc h1;have d1:=(qAa dd s).trans (add_le_add (ἅ hs0).le le_rfl);have d2:=d1.trans (add_le_add le_rfl ((le_mul_of_one_le_left (Ἀ _) (by norm_num1)).trans h1'))
 have m1:=Ἐ ((Rn2-(c+c))*dd) ((c+c)*(dd-s));rw[ἢ,ἢ,hcc,show (Rn2-(c+c))*dd+(c+c)*(dd-s)=Rn2*dd-(c+c)*s by rw[sub_mul,mul_sub,sub_add_sub_cancel]] at m1;have m2:=ἑ (sc h2) d2 (Ἀ _) (add_nonneg (add_nonneg hcp.le hcp.le) hz0);have m3:=ἁ h1' (add_nonneg hcp.le hcp.le);have zs:=ἑ hz' hs1 hs0 zero_le_one;have zc:z*c<=z*1:=ἁ (hc1.trans (by norm_num1)) hz0;have zz:z*z<=z*1:=ἁ (hz.trans (by norm_num1)) hz0
 have m4:2^230*|Rn2*dd-(c+c)*s| <=2^181*(c*s)+1:=by linarith only[zs,zz,zc,m3,m2,m1,hcs,hz']
 generalize Rn2*dd=M at h3 m4;rw[show (c+c)*s=2*(c*s) by rw[<-two_mul,mul_assoc]] at m4;have n1:=(qAa M (2*(c*s))).trans (add_le_add (ἅ (Ἆ zero_le_two hcs)).le le_rfl);have n2:=qAa Rmu M;have n3:=(Ἐ Rmu Em).trans (add_le_add le_rfl (ἡ (Ἥ (by norm_num1) hE)).le);have n4:=qAa ee (Rmu+Em)
 have Mg:10*2^53*(|dd-s|+|ee-(Rmu+Em)|+|Rmu-M|+|M-2*(c*s)|)+10*(|dd|+|ee|+z)<=2^53*(5*(c*s)+Em):=by linarith only[h1',sc h4,sc h3,m4,d1,n4,n3,n2,n1,hcs2,hcs,hE',hz',hz0,hs0]
 constructor
 . linarith only[neg_abs_le (hi-(dd+ee)),neg_abs_le (dd-s),neg_abs_le (ee-(Rmu+Em)),neg_abs_le (Rmu-M),neg_abs_le (M-2*(c*s)),(sc h5).trans (add_le_add (Ἐ dd ee) le_rfl),Mg]
 . linarith only[le_abs_self (lo-(dd-ee)),le_abs_self (dd-s),neg_abs_le (ee-(Rmu+Em)),neg_abs_le (Rmu-M),neg_abs_le (M-2*(c*s)),(sc h6).trans (add_le_add (Ὕ dd ee) le_rfl),Mg]
lemma qIw {x Rd RQ:ℝ} (hx: |x| <=2^111) (h1: |Rd-x| <=1/2^53*(|x| +z)) (h2: |RQ-Rd*Rd| <=1/2^53*(|Rd*Rd| +z)):
 |RQ-x^2| <=4/2^53*x^2+1/2^250:=by
 have sck:∀{k a b:ℝ},0<k->a<=1/k*b->k*a<=b:=λhk h=>(le_div_iff₀' hk).mp (h.trans_eq (one_div_mul_eq_div _ _))
 have nA:=Ἀ x
 have hz':(2:ℝ)^200*(2^200*z)<=1:=(le_div_iff₀' (by norm_num1)).mp ((le_div_iff₀' (by norm_num1)).mp (hz.trans_eq (one_div_mul_eq_div _ _)))
 have zA:=ἑ hz' hx (Ἀ x) zero_le_one
 have zz:z*z<=z*1:=ἁ (hz.trans (by norm_num1)) hz0
 have hsq: |x| ^2=x^2:=sq_abs x
 generalize he:Rd-x=e at h1
 have h1':=sck (by norm_num1) h1;have h2':=sck (by norm_num1) h2
 have q1:=Ἳ (Ἆ (by norm_num1) (Ἀ e)) h1' 2
 have q3:2^53* |2*x*e| <=2* |x| *(|x| +z):=by
  rw[ἢ,ἢ,abs_two,mul_left_comm]
  exact ἁ h1' (Ἆ zero_le_two nA)
 have q6: |Rd*Rd-x^2| <= |2*x*e| + |e| ^2:=by
  rw[show Rd*Rd-x^2=2*x*e+e^2 by rw[<-he];ring,<-abs_pow]
  exact Ἐ _ _
 have q8:=qAa (Rd*Rd) (x^2)
 rw[ἅ (ὅ x)] at q8
 have q9:=qAa (RQ-x^2) (RQ-Rd*Rd)
 rw[show RQ-x^2-(RQ-Rd*Rd)=Rd*Rd-x^2 by ring] at q9
 have nX:=ὅ x
 have G:2^250*|RQ-x^2| <=2^199*x^2+1:=by linarith only[nX,q9,q8,q6,q3,q1,hsq,zz,zA,h2',hz',hz0,nA]
 exact ((le_div_iff₀' (by norm_num1)).mpr G).trans_eq (by ring)
lemma qH9 {n Rv I Rp Rt ep c:ℝ} (Ws tau:ℝ) (hn1:1<=n) (hn2:n<=2^20) (hc:c=(n+8)/2^49) (hWs:0<=Ws) (hep:1/2^126<=ep) (ht0:0<=tau) (ht:tau<=1/2^229) (h1: |Rv-n*Ws| <=(2*n+6)/2^53*(n*Ws)+tau) (h3:0<I) (h4: |I*n-1| <=1/2^48) (h5: |Rp-Rv*I| <=1/2^53*(|Rv*I| +z)) (h6: |Rt-(Rp+ep)| <=1/2^53*(|Rp+ep| +z)):
 |Rt-(Ws+ep)| <=c/2*(Ws+ep):=by
 have hep0:0<ep:=Ἥ ו hep
 have hI2:=L_I hn1 h3 h4
 have hX:0<=(2*n+6)/2^53*Ws:=Ἆ (by linarith only[hn1]) hWs
 have b1:=qF0 n I Rv Ws Ws _ tau h3 h4 (by rw[ἅ hWs]) (by linarith only[h1]) hX
 have a3:=ἁ hI2 ht0
 have nnW:=Ἆ (by linarith only[hn1]:0<=n) hWs
 have nnE:=Ἆ (by linarith only[hn1]:0<=n) hep0.le
 have nnW2:=ἀ hn2 hWs
 have b2:=qAa (Rv*I) Ws
 rw[ἅ hWs] at b2
 have b3:=Ἐ Rp ep
 rw[ἡ hep0] at b3
 have b4:=qAa Rp (Rv*I)
 have b5:=Ἐ (Rt-(Rp+ep)) ((Rp-Rv*I)+(Rv*I-Ws))
 have b6:=Ἐ (Rp-Rv*I) (Rv*I-Ws)
 rw[show Rt-(Rp+ep)+(Rp-Rv*I+(Rv*I-Ws))=Rt-(Ws+ep) by ring] at b5
 rw[hc]
 linarith only[b6,b5,b4,b3,b2,nnE,a3,b1,hX,hep0,h6,h5,ht,hep,hWs,hz]
lemma qG8 {n W Rv Sq Si:ℝ} (hn1:1<=n) (hn2:n<=2^20) (hW:0<=W) (h1: |Rv-Sq| <=2*n*(1/2^53*(Si+z))) (h2: |Sq-W| <=4/2^53*W+n*(1/2^250)) (h3:Si<=(1+4/2^53)*W+n*(1/2^250)):
 |Rv-W| <=(2*n+6)/2^53*W+1/2^229:=by
 have hn0:0<=n:=(by linarith only[hn1])
 have a1:=ἁ h3 hn0
 have a3:=ἀ hn2 hW
 have a4:=ἑ hn2 hn2 hn0 ו
 have a5:=ἑ hn2 hz hz0 ו
 have a6:=abs_sub_le Rv Sq W
 linarith only[a6,a5,a4,a3,a1,h2,h1,hW,hn2]
end
lemma qHl {T s c Em lo hi:ℝ} (hT:0<T) (hs:0<s) (hc0:0<c) (hc1:c<=1/2^20) (hE:0<=Em) (h1:T<=s^2*(1+c)) (h2:s^2*(1-c)-Em^2/4<=T) (h3:s*(1+3/2*c)+9/10*Em<=hi) (h4:lo<=s*(1-3/2*c)-9/10*Em):lo<ἧ T/\ἧ T<hi:=by
 have hsT:=Real.sqrt_pos.mpr hT;have hq:0<s^2*c:=mul_pos (pow_pos hs 2) hc0
 have hc15:(0:ℝ)<=1+3/2*c:=(add_pos_of_pos_of_nonneg one_pos (Ἆ (by norm_num1) hc0.le)).le
 have h5:0<s*(1+3/2*c):=mul_pos hs (add_pos_of_pos_of_nonneg one_pos (Ἆ (by norm_num1) hc0.le))
 constructor
 . by_cases hl:lo<0;. linarith only[hl,hsT]
   . rw[not_lt] at hl;rw[Real.lt_sqrt hl];generalize hk:s*(1-3/2*c)=k at h4;have hk0:0<=k-9/10*Em:=(by linarith only[h4,hl]);have q1:=Ἳ hl h4 2;have q3:=Ἆ hE hk0
     have q4:k^2<=s^2*(1-3/2*c):=by
      have h6:(1-3/2*c)^2<=1-3/2*c:=by have:=ἁ hc1 hc0.le;linarith only[this,hc0]
      calc k^2=s^2*(1-3/2*c)^2:=by rw[<-hk,mul_pow]
       _<=_:=ἁ h6 (ὅ _)
     have q5:=ὅ Em;linarith only[h2,hq,q1,q3,q4,q5]
 . have hh:0<hi:=by linarith only[h5,h3,hE]
   rw[Real.sqrt_lt' hh];have q1:=Ἳ (add_nonneg h5.le (Ἆ (by norm_num1) hE)) h3 2;have r1:0<=s^2*c^2:=Ἆ (ὅ s) (ὅ c)
   have r2:0<=s*Em*(1+3/2*c):=Ἆ (Ἆ hs.le hE) hc15
   have r3:=ὅ Em;linarith only[h1,hq,q1,r1,r2,r3]
lemma qHp (l:List ℕ) (a:ℕ->ℝ) (c:ℝ):
 (l.map λj=>(a j-c)^2).sum=(l.map λj=>a j^2).sum-2*c*(l.map a).sum+ѣl*c^2:=by
 induction l with
 | nil=>simp only[List.map_nil,List.sum_nil,mul_zero,sub_self,List.length_nil,CharP.cast_eq_zero,zero_mul,add_zero]
 | cons j l ih=>
  simp only[List.map_cons,List.sum_cons,List.length_cons,ih]
  push_cast
  ring
lemma qGh (l:List ℕ) (a:ℕ->ℝ) (m:ℝ) (hl:(ѣl:ℝ)≠0):
 (l.map λj=>(a j-m)^2).sum=
  (l.map λj=>(a j-(l.map a).sum/ѣl)^2).sum+ ѣl*((l.map a).sum/ѣl-m)^2:=by
 rw[qHp,qHp]
 field_simp
 ring
lemma qGx (l:List ℕ) (f w:ℕ->ℝ) (u t:ℝ) (hw:∀j∈l,0<=w j) (h:∀j∈l,|f j-w j| <=u*w j+t) (hu:0<=u) (ht:0<=t):
 |(l.map f).sum-(l.map w).sum| <=u*(l.map w).sum+ѣl*t/\(l.map λj=>|f j|).sum<=(1+u)*(l.map w).sum+ѣl*t/\0<=(l.map w).sum:=by
 induction l with
 | nil=>simp only[List.map_nil,List.sum_nil,sub_self,abs_zero,mul_zero,List.length_nil,CharP.cast_eq_zero,zero_mul,add_zero,Std.le_refl,and_self]
 | cons j l ih=>
  obtain ⟨i1,i2,i3⟩:=ih (λk hk=>hw k (ὖ _ hk)) (λk hk=>h k (ὖ _ hk))
  have h0:=hw j ὣ
  have h1:=h j ὣ
  simp only[List.map_cons,List.sum_cons,List.length_cons]
  push_cast
  have a1:=Ἐ (f j-w j) ((l.map f).sum-(l.map w).sum)
  rw[show f j-w j+((l.map f).sum-(l.map w).sum)=f j+(l.map f).sum-(w j+(l.map w).sum) by ring] at a1
  have a2:=qAa (f j) (w j)
  rw[ἅ h0] at a2
  exact ⟨by linarith only[i1,h1,a1],by linarith only[i2,h1,a2],by linarith only[i3,h0]⟩
lemma qIr {σ:Type} {N n c:Ὣ} {sp:Ὣ->Ὢ} {f:σ->ὦ->ὦ->Ὢ} {s:σ} {b:ὦ} (hN:N<2^32) (hb:Ѥb=c) (hn:c+n<=N) (hf:∀p u:ὦ,Ѥp=c+Ѥu->Ѥu<n->f s p u=sp Ѥp) (k:Ὣ):
 ∀(j:ὦ) (o:ё),Ѥj+k=n->C_Inv N sp (λq=>q<c+Ѥj) o->C_Inv N sp (λq=>q<c+n) (fC f s b j o k):=by
 induction k with
 | zero=>
  intro j o e h
  rw[fC,<-e];exact h
 | succ k ih=>
  intro j o e h
  have a0:(b+j).toNat=c+Ѥj:=by rw[<-hb];exact qAf _ _ (by exact omS346 hN hb hn e)
  have a1:(j+1).toNat=Ѥj+1:=by
   rw[qAf _ _ (by rw[ὄ];exact omS347 hN hn e),ὄ]
  rw[fC]
  refine ih _ _ (by clear hN hb hn a0;exact omS157 e a1) ?_
  rw[a1]
  exact qAp hN h a0 (by clear hN hb a1;exact omS348 hn e) (by rw[hf _ _ a0 (by clear hN hb hn a0 a1;exact omS136 e),a0])
lemma qDb {σ:Type} {R n:Ὣ} (F:Ὣ->Ὣ->Ὣ->Ὢ) {h:Ὣ->σ} {f:σ->ὦ->ὦ->Ὢ} (hN:R*n<2^32) (hn:0<n)
 (hf:∀(r:Ὣ) (p u:ὦ),r<R->Ѥp=r*n+Ѥu->Ѥu<n->f (h r) p u=F r Ѥu Ѥp):
 fR h f n 0 (ἇ (R*n) 0) R=Array.ofFn (n:=R*n) λq=>F (q.val/n) (q.val%n) q.val:=by
 let sp q:=F (q/n) (q%n) q
 have K:∀(d r:Ὣ) (o:ё),r+d=R->C_Inv (R*n) sp (λq=>q<r*n) o->C_Inv (R*n) sp (λq=>q<R*n) (fR h f n r o d):=λd=>by
  induction d with
  | zero=>
   intro r o e H
   subst e
   exact H
  | succ d ih=>
   intro r o e H
   have h1:r*n+n<=R*n:=qBc ((by clear hN hn;exact omS136 e):r<R) (Ὗ _)
   rw[fR]
   refine ih _ _ (by clear hN hn h1;exact omS137 e) ?_
   rw[ᾀ]
   exact qIr hN (qAd _ (by clear e;exact omS154 hN h1)) h1 (λp u h2 h3=>by
    rw[hf r p u (by clear hN hn h1 h2 h3;exact omS136 e) h2 h3]
    show F _ _ _=F _ _ _
    rw[h2,ἥ r n,Nat.mul_add_div hn,Nat.mul_add_mod,Ὃ h3,Ἁ h3,Ἵ]) n 0 o (Ἇ _) H
 exact qBv (K R 0 _ (by clear hN hn;exact omS139) ⟨by simp only[Array.size_replicate],λq _ (h:q<0*n)=>א⟩) λq hq=>hq
section
variable (x:ѐ) (hx:ѧx<2^32) (base nu:ὦ)
include x hx base nu
lemma qG9 (hb:Ѥbase+Ѥnu<2^32) (d:ℕ) (i:ὦ) (s t:ὧ) (h:Ѥi+d=Ѥnu):
 lnSum x base nu i s t=
  ⟨((List.range' Ѥi d).map λj=>dv x[Ѥbase+j]!).foldl (.+·) s,((List.range' Ѥi d).map λj=>(dv x[Ѥbase+j]!).abs).foldl (.+·) t⟩:=by
 induction d generalizing i s t with
 | zero=>
  rw[lnSum,ἔ (by rw[А];(clear hx hb;(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›)))))];simp only[List.range'_zero,List.map_nil,List.foldl_nil]
 | succ d ih=>
  have hn:i<nu:=by rw[А];(clear hx hb;exact omS136 h)
  have e1:=usz i nu hn
  rw[lnSum,Ἄ hn,ih (i+1) _ _ (by rw[e1];(clear hx hb hn e1;exact omS137 h)),e1,qAg x _ hx,qAf base i (by clear hx hn;omega)]
  simp only[List.range'_succ,List.map_cons,ὔ]
lemma qG2 (m:ὧ) (hb:Ѥbase+Ѥnu<2^32) (d:ℕ) (i:ὦ) (v:ὧ) (h:Ѥi+d=Ѥnu):
 lnVar x base nu i m v=
  ((List.range' Ѥi d).map λj=>(dv x[Ѥbase+j]!-m)*(dv x[Ѥbase+j]!-m)).foldl (.+·) v:=by
 induction d generalizing i v with
 | zero=>
  rw[lnVar,ἔ (by rw[А];(clear hx hb;(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›)))))];simp only[List.range'_zero,List.map_nil,List.foldl_nil]
 | succ d ih=>
  have hn:i<nu:=by rw[А];(clear hx hb;exact omS136 h)
  have e1:=usz i nu hn
  rw[lnVar,Ἄ hn,ih (i+1) _ (by rw[e1];(clear hx hb hn e1;exact omS137 h)),e1,qAg x _ hx,qAf base i (by clear hx hn;omega)]
  simp only[List.range'_succ,List.map_cons,ὔ]
end
lemma qEp (l:List ℕ) (f:ℕ->ℚ) (u:ℕ->ℝ) (c:ℚ) (h:∀j∈l,(f j:ℝ)=u j):((l.foldl (λt j=>t+f j) c:ℚ):ℝ)=c+(l.map u).sum:=by
 induction l generalizing c with
 | nil=>simp only[List.foldl_nil,List.map_nil,List.sum_nil,add_zero]
 | cons j l ih=>simp only[ὔ,List.map_cons,List.sum_cons];rw[ih _ λk hk=>h k (ὖ _ hk),<-h j ὣ];push_cast;ring
lemma qIb (w:ὥ) (hg:good w=true) (h1:(0x1F800000:ὥ)<=w) (h2:w<=(0x70FFFFFF:ὥ)):1/2^64<=wv w:=by
 have a1:528482304<=ѥw:=ἃ.mp h1;have a2:ѥw<=1895825407:=ἃ.mp h2;rw[qA7 w hg,ἐ (by clear h1 h2;exact omS349 a2),one_mul];have h3:=qA6 528482304 (ѥw%2^31) (by clear h1 h2;exact omS350 a1 a2) ה;have:A_mag 528482304=1/2^64:=by unfold A_mag;norm_num
 rwa[this] at h3
lemma qH3 (w:ὥ) (hg:good w=true) (h0:0<wv w):1/2^126<=wv w:=by
 have hn:=qAz w hg;rw[qA7 w hg] at h0⊢;rcases hn with hn | ⟨hn,-⟩;. rw[hn] at h0;simp only[Nat.reducePow,A_mag, ↓reduceIte,mul_zero,lt_self_iff_false] at h0
 . have hs:¬2^31<=ѥw:=λhc=>by rw[ἱ hc] at h0;linarith only[qAs (ѥw%2^31),h0]
   rw[ἐ hs,one_mul];have h3:=qA6 (2^23) (ѥw%2^31) hn ה;have:A_mag (2^23)=1/2^126:=by unfold A_mag;norm_num
   rwa[this] at h3
lemma qIi (n:ℕ) (hn:n<2^20):
 ((n.toUInt64+8).toFloat*Ἃ ь).isFinite=true/\R ((n.toUInt64+8).toFloat*Ἃ ь)=((n:ℝ)+8)/2^49:=by
 have ht:(n.toUInt64+8).toNat=n+8:=by
  have h8:(8:Ὠ).toNat=8:=rfl
  simp only[Nat.toUInt64_eq,ἤ,UInt64.toNat_ofNat',Nat.reducePow,h8,Nat.mod_add_mod,Nat.mod_succ_eq_iff_lt,Nat.succ_eq_add_one,Nat.reduceAdd,gt_iff_lt]
  (clear h8;exact omS57 hn)
 have h1:=qCk (n.toUInt64+8) (by rw[ht];exact omS58 hn)
 obtain ⟨h2,h3⟩:=L_k ь 974 49 (by norm_num1) rfl (by rfl)
 have h4:(n:ℝ)+8<=2^21:=by exact_mod_cast ((by exact omS351 hn):n+8<=2^21)
 obtain ⟨hf,he⟩:=qEd _ _ h1.1 h2 (by
  rw[h1.2,h3,ht]
  refine ὂ (ἑ (qAt ?_) (qAt (by rw[ἅ ו];(norm_num1))) (Ἀ _) qI1.le) qDe
  rw[Nat.cast_add,ἅ ו]
  push_cast
  linarith only[h4])
 refine ⟨hf,?_⟩
 rw[he,h1.2,h3,ht]
 have:=qA2 ((n:ℤ)+8) (-49) (by exact omS352 hn) (by norm_num1)
 push_cast at this⊢
 rw[show (2:ℝ)^(-49:ℤ)=1/2^49 by (norm_num1)] at this
 rw[this]
 ring
lemma qD7 {r x B:ℝ} (e: |r-x| <=1/2^53*(|x| +A_z)) (h: |x| <=B):
 |r| <=9/8*B+A_z:=by
 linarith only[qAa r x,Ἀ x,qAr,e,h]
section
variable (x y:ὧ) (B:ℝ) (hx:x.isFinite=true) (hy:y.isFinite=true) (h1 : |R x| ≤ B) (h2 : |R y| ≤ B) (hB:B<=2^256*2^44:=by norm_num)
include x y B hx hy h1 h2 hB
lemma qAv:
 (x+y).isFinite=true/\|R (x+y)-(R x+R y)| <=1/2^53*(|R x+R y| +A_z)/\
  |R (x+y)| <=9/8*(B+B)+A_z:=by
 obtain ⟨f,e⟩:=qBf x y hx hy (h1.trans (qAt hB)) (h2.trans (qAt hB))
 exact ⟨f,e,qD7 e ((Ἐ _ _).trans (add_le_add h1 h2))⟩
lemma qCg:
 (x-y).isFinite=true/\|R (x-y)-(R x-R y)| <=1/2^53*(|R x-R y| +A_z)/\
  |R (x-y)| <=9/8*(B+B)+A_z:=by
 obtain ⟨f,e⟩:=qDf x y hx hy (h1.trans (qAt hB)) (h2.trans (qAt hB))
 exact ⟨f,e,qD7 e ((Ὕ _ _).trans (add_le_add h1 h2))⟩
end
lemma qAl (x y:ὧ) (B C:ℝ) (hx:x.isFinite=true) (hy:y.isFinite=true) (h1: |R x| <=B) (h2: |R y| <=C) (hB:B<=2^256*2^44:=by norm_num) (hC:C<=2^256*2^44:=by norm_num):
 (x*y).isFinite=true/\|R (x*y)-R x*R y| <=1/2^53*(|R x*R y| +A_z)/\
  |R (x*y)| <=9/8*(B*C)+A_z:=by
 obtain ⟨f,e⟩:=qCf x y hx hy (h1.trans (qAt hB)) (h2.trans (qAt hC))
 exact ⟨f,e,qD7 e (ἢ (R x) (R y)▸ἑ h1 h2 (Ἀ _) ((Ἀ _).trans h1))⟩
lemma qFy (a b:ὧ) (hb:b.isFinite=true) (hb2: |R b| <=2^99) (h:invOK a b=true):a.isFinite=true/\0<R a/\R a<=2^200/\|R a*R b-1| <=1/2^48:=by
 simp only[invOK,Bool.and_eq_true,decide_eq_true_eq] at h;obtain ⟨r1,r2⟩:1<=_/\_<1+1222:=qEj _ 1 1222 h.1 ה;obtain ⟨r3,r4⟩:972<=_/\_<972+2:=qEj _ 972 2 h.2 ה;obtain ⟨f1,l1,u1⟩:=qCz rfl r1 (by exact Nat.lt_of_lt_of_le ‹_› ה);have hp:0<R a:=Ἥ (ἵ ו _) l1;have ha2:R a<=2^200:=u1.le.trans ((ἒ (by norm_num1) (by clear h r3 r4;exact omS353 r2)).trans_eq (by (norm_num1):(2:ℝ)^(200:ℤ)=2^200));obtain ⟨k1,k2⟩:=L_k 0x3FF0000000000000 1023 0 (by norm_num1) rfl (by rfl);obtain ⟨k3,k4⟩:=L_k 0x3CD0000000000000 973 50 (by norm_num1) rfl (by rfl);rw[pow_zero,div_one] at k2;have hz1:=qBo;obtain ⟨f2,g1,q1⟩:=qAl a b (2^200) (2^99) f1 hb (by rwa[ἡ hp]) hb2 (by norm_num1) (by norm_num1);obtain ⟨f3,g2⟩:=qDf _ _ f2 k1 (qAt (q1.trans ((add_le_add le_rfl hz1).trans (by norm_num1)))) (qAt (by rw[k2];(norm_num1)));rw[k2] at g2;have q3:=Ὕ (R (a*b)) 1;rw[abs_one] at q3;have q2:=qD7 g2 q3;obtain ⟨f4,g3⟩:=qBf _ _ f3 k3 (qAt (q2.trans ((add_le_add (ἁ (add_le_add (q1.trans (add_le_add le_rfl hz1)) le_rfl) (by norm_num1)) hz1).trans (by norm_num1)))) (qAt (by rw[k4];(norm_num1)));rw[k4] at g3;obtain ⟨-,l2,u2⟩:=qCz rfl (ὂ (by norm_num1) r3) (by exact Nat.lt_of_lt_of_le ‹_› ה);refine ⟨f1,hp,ha2,qD9 qAr.le qAw g1 g2 g3 ?_ ?_⟩;. exact (by (norm_num1):(1/2^51:ℝ)=2^(-51:ℤ)).trans_le ((ἒ (by norm_num1) (by clear h r1 r2;exact omS354 r3)).trans l2)
 . exact u2.le.trans ((ἒ (by norm_num1) (by clear h r1 r2;exact omS355 r4)).trans_eq (by (norm_num1):(2:ℝ)^(-49:ℤ)=1/2^49))
lemma qDr (n:ℕ) (f:ℕ->ℝ) (B:ℝ) (h:∀j,j<n->f j<=B):((ἄ n).map f).sum<=n*B:=by
 simpa only[List.length_map,List.length_range,nsmul_eq_mul] using List.sum_le_length_nsmul ((ἄ n).map f) B (by simpa only[List.mem_map,Ἅ,forall_exists_index,and_imp,forall_apply_eq_imp_iff₂] using λj hj=>h j hj)
lemma qFm (n:ℕ) (hn:n<2^20) (f:ℕ->ὧ) (a b:ℕ->ℝ) (hf:∀j,j<n->(f j).isFinite=true/\R (f j)=a j/\|a j| =b j) (hb:((ἄ n).map b).sum<=A_B/4):(A_fsum ((ἄ n).map f)).isFinite=true/\
 |R (A_fsum ((ἄ n).map f))-((ἄ n).map a).sum| <=2*n*(1/2^53*(((ἄ n).map b).sum+A_z)):=by
 have s:∀(u:ὧ->ℝ) (v:ℕ->ℝ),(∀j,j<n->u (f j)=v j)->(((ἄ n).map f).map u).sum=((ἄ n).map v).sum:=λu v h=>by
  simp only[List.map_map]
  exact congrArg _ (ὠ λj hj=>h j (Ἅ.1 hj))
 have e1:=s (λq=>|R q|) b λj h=>by simp only[(hf j h).2.1,(hf j h).2.2]
 have e2:=s R a λj h=>(hf j h).2.1
 have:=qBz ((ἄ n).map f)
  (λq hq=>by
   obtain ⟨j,hj,rfl⟩:=List.mem_map.1 hq
   exact (hf j (Ἅ.1 hj)).1)
  (by simp only[List.length_map,List.length_range,Nat.reducePow];(exact Nat.le_of_lt (Nat.lt_of_lt_of_le ‹_› ה))) (by rw[A_sa,e1];exact hb)
 rwa[A_sa,A_sr,e1,e2,List.length_map,List.length_range] at this
lemma qGp (n:ℕ) (hn:n<2^20) (hn0:0<n) (g:ℕ->ὧ) (m v:ὧ) (W:ℝ) (hv:v=A_fsum ((ἄ n).map λj=>(g j-m)*(g j-m))) (hW:W=((ἄ n).map λj=>(R (g j)-R m)^2).sum) (hg:∀j,j<n->(g j).isFinite=true/\|R (g j)| <=2^99) (hm:m.isFinite=true) (hm2: |R m| <=2^110):v.isFinite=true/\|R v-W| <=(2*n+6)/2^53*W+1/2^229/\W<=n*2^222/\0<=W:=by
 have hz0:=qAr.le;have hzl:=qAw;have hzb:=qBo
 have hq:∀j,j<n->((g j-m)*(g j-m)).isFinite=true/\
  |R ((g j-m)*(g j-m))-(R (g j)-R m)^2| <=4/2^53*(R (g j)-R m)^2+1/2^250/\(R (g j)-R m)^2<=2^222:=by
  intro j hj
  obtain ⟨f,b⟩:=hg j hj
  have B1: |R (g j)| <=2^110:=b.trans (by norm_num1)
  obtain ⟨f1,e1,b1⟩:=qCg (g j) m (2^110) f hm B1 hm2 (by norm_num1)
  have b2: |R (g j-m)| <=2^112:=b1.trans ((add_le_add le_rfl hzb).trans (by norm_num1))
  obtain ⟨f2,e2,-⟩:=qAl _ _ (2^112) (2^112) f1 f1 b2 b2 (by norm_num1) (by norm_num1)
  have hx: |R (g j)-R m| <=2^111:=(Ὕ _ _).trans ((add_le_add B1 hm2).trans (by norm_num1))
  refine ⟨f2,qIw hz0 hzl hx e1 e2,?_⟩
  have:=Ἳ (Ἀ _) hx 2
  rw[sq_abs] at this
  exact this.trans (by norm_num1)
 obtain ⟨a1,a2,a3⟩:=qGx (ἄ n) (λj=>R ((g j-m)*(g j-m))) (λj=>(R (g j)-R m)^2) (4/2^53) (1/2^250) (λj _=>ὅ _) (λj hj=>(hq j (Ἅ.1 hj)).2.1) ו ו;simp only[List.length_range] at a1 a2;rw[<-hW] at a1 a2 a3;have hW':W<=n*2^222:=hW▸qDr n _ _ λj h=>(hq j h).2.2;have hnR:(n:ℝ)<=2^20:=by exact_mod_cast hn.le
 have hnR1:(1:ℝ)<=n:=by exact_mod_cast hn0
 have h1:(n:ℝ)*2^222<=2^20*2^222:=ἀ hnR (pow_nonneg zero_le_two 222)
 have h2:(n:ℝ)*(1/2^250)<=2^20*(1/2^250):=ἀ hnR (div_nonneg zero_le_one (pow_nonneg zero_le_two 250))
 obtain ⟨f2,e2⟩:=qFm n hn _ _ (λj=>|R ((g j-m)*(g j-m))|)
  (λj h=>⟨(hq j h).1,rfl,rfl⟩) (by
   exact ὂ a2 ((add_le_add (ἁ (hW'.trans h1) (by norm_num1)) h2).trans (ὂ (by norm_num1) (div_le_div_of_nonneg_right (qAt (k:=2^256*2^44) le_rfl) (by norm_num1)))))
 rw[<-hv] at e2 f2;exact ⟨f2,qG8 hz0 hzl hnR1 hnR a3 e2 a1 a2,hW',a3⟩
lemma qF2 (l:List ℕ) (f:ℕ->ℝ): |(l.map f).sum| <=(l.map λj=>|f j|).sum:=by
 induction l with
 | nil=>simp only[List.map_nil,List.sum_nil,abs_zero,Std.le_refl]
 | cons j l ih=>simp only[List.map_cons,List.sum_cons];exact (Ἐ _ _).trans (add_le_add le_rfl ih)
def L_oF (s:LnS) (M:ℝ):Prop:=
 s.fast=true->s.m.isFinite=true/\s.ir.isFinite=true/\s.em.isFinite=true/\
  R s.em<=2^120/\1/2^201<=R s.em/\2* |M-R s.m| <=R s.em/\
  |R s.m| <=2^110/\good s.den=true/\1/2^64<=wv s.den/\
  |R s.ir*wv s.den-1| <=1/2^48
def L_ok (b:Ὡ) (n:ℕ) (eps:Ὢ) (x:ё) (r:ℕ) (s:LnS):Prop:=
 s.den=sqrtRational (rowVariance n x r (lnQ b n x r)+ἂ eps)/\
  L_oF s (lnQ b n x r:ℚ)
lemma qHi (n r:ℕ) (x:ё) (g:ℕ->ὧ) (h:∀j,j<n->R (g j)=(ἂ x[r*n+j]!:ℝ)) (q:ℚ):((rowMean n x r:ℚ):ℝ)=((ἄ n).map λj=>R (g j)).sum/n/\
 ((rowVariance n x r q:ℚ):ℝ)=((ἄ n).map λj=>(R (g j)-(q:ℝ))^2).sum/n:=by
 constructor
 . rw[rowMean,Rat.cast_div,qEp _ _ (λj=>R (g j)) 0 λj hj=>(h j (Ἅ.1 hj)).symm];simp only[Rat.cast_zero,zero_add,Rat.cast_natCast]
 . rw[rowVariance,Rat.cast_div,qEp _ _ (λj=>(R (g j)-(q:ℝ))^2) 0 λj hj=>by
    rw[h j (Ἅ.1 hj)];push_cast;rfl]
   simp only[Rat.cast_zero,zero_add,Rat.cast_natCast]
section
variable (n:ℕ) (hnR1:(1:ℝ)<=n) (hnR2:(n:ℝ)<=2^20)
include n hnR1 hnR2
lemma qG1 (Fs Ft ind nf m em:ὧ) (S Sa c:ℝ) (hc:c=((n:ℝ)+8)/2^49) (o1:Fs.isFinite=true) (o2:Ft.isFinite=true) (o3: |S| <=Sa) (o4:Sa<=n*2^99) (o5: |R Fs-S| <=2*n*(1/2^53*(Sa+A_z))) (o6: |R Ft-Sa| <=2*n*(1/2^53*(Sa+A_z))) (fnf:nf.isFinite=true) (Rnf:R nf=c) (fi:ind.isFinite=true) (pi:0<R ind) (ei: |R ind*n-1| <=1/2^48) (hm:Fs*ind=m) (hem:nf*Ft*ind+Ἃ ظ=em):
 m.isFinite=true/\em.isFinite=true/\R em<=2^120/\1/2^201<=R em/\2* |S/n-R m| <=R em/\|R m| <=2^110:=by
 have hz0:=qAr.le
 have hzb:=qBo
 have hn0:(0:ℝ)<=n:=ὂ zero_le_one hnR1
 have hc0:0<c:=hc▸div_pos (add_pos_of_nonneg_of_pos hn0 (by norm_num1)) (by norm_num1)
 have hc1:c<=1/2^20:=by rw[hc];exact (div_le_div_of_nonneg_right (add_le_add hnR2 le_rfl) (pow_nonneg zero_le_two 49)).trans (by norm_num1)
 have bI: |R ind| <=2:=by rw[ἡ pi];exact L_I hnR1 pi ei
 have hSa0:0<=Sa:=(Ἀ _).trans o3
 have hSb:=ἀ hnR2 (pow_nonneg zero_le_two 99:(0:ℝ)<=2^99)
 have hSA:Sa+A_z<=2^119+1:=add_le_add (o4.trans (hSb.trans_eq (by norm_num1))) hzb
 have E5:2*(n:ℝ)*(1/2^53*(Sa+A_z))<=2*2^20*(1/2^53*(2^119+1)):=ἑ (ἁ hnR2 zero_le_two) (ἁ hSA (by norm_num1)) (Ἆ (by norm_num1) (add_nonneg hSa0 hz0)) (by norm_num1)
 have bs: |R Fs| <=2^120:=(qAa (R Fs) S).trans ((add_le_add (o3.trans (o4.trans hSb)) (o5.trans E5)).trans (by norm_num1))
 have bt: |R Ft| <=2^120:=(qAa (R Ft) Sa).trans ((add_le_add ((ἅ hSa0).le.trans (o4.trans hSb)) (o6.trans E5)).trans (by norm_num1))
 have bnf: |R nf| <=1:=by rw[Rnf,ἡ hc0];exact hc1.trans (by norm_num1)
 obtain ⟨fm,em1,-⟩:=qAl Fs ind (2^120) 2 o1 fi bs bI (by norm_num1) (by norm_num1)
 rw[hm] at fm em1
 obtain ⟨fN,emN,bN⟩:=qAl nf Ft 1 (2^120) fnf o2 bnf bt (by norm_num1) (by norm_num1)
 rw[Rnf] at emN
 obtain ⟨fY,emY,bY⟩:=qAl (nf*Ft) ind (2^122) 2 fN fi (bN.trans ((add_le_add le_rfl hzb).trans (by norm_num1))) bI (by norm_num1) (by norm_num1)
 obtain ⟨k1,k2⟩:=L_k ظ 823 200 (by norm_num1) rfl (by rfl)
 obtain ⟨fE,emE,-⟩:=qAv (nf*Ft*ind) _ (2^125) fY k1 (bY.trans ((add_le_add le_rfl hzb).trans (by norm_num1))) (by rw[k2];(norm_num1)) (by norm_num1)
 rw[k2,hem] at emE
 rw[hem] at fE
 have hnpos:(0:ℝ)<n:=zero_lt_one.trans_le hnR1
 have hnp:=hnpos.ne'
 have eS:(n:ℝ)*(S/n)=S:=mul_div_cancel₀ S hnp
 have eA:(n:ℝ)*(Sa/n)=Sa:=mul_div_cancel₀ Sa hnp
 obtain ⟨m1,m2,m3,m4⟩:=qG3 hz0 qAw (S/n) (Sa/n) hnR1 hnR2 hc (by rw[abs_div,ἡ hnpos];exact div_le_div_of_nonneg_right o3 hnpos.le) (by rw[div_le_iff₀ hnpos,mul_comm];exact o4) (by rw[eS,eA];exact o5) (by rw[eA];exact o6) pi ei em1 emN emY emE
 exact ⟨fm,fE,m2,m3,m1,m4⟩
lemma qIv (v ind tt:ὧ) (eps:Ὢ) (W c:ℝ) (hc:c=((n:ℝ)+8)/2^49) (p1:v.isFinite=true) (p2: |R v-W| <=(2*n+6)/2^53*W+1/2^229) (p3:W<=n*2^222) (hW0:0<=W) (fi:ind.isFinite=true) (pi:0<R ind) (ei: |R ind*n-1| <=1/2^48) (gE:good eps=true) (bE: |wv eps| <=2^99) (hep0:0<wv eps) (htt:v*ind+dv eps=tt):tt.isFinite=true/\|R tt-(W/n+wv eps)| <=c/2*(W/n+wv eps):=by
 have hnpos:(0:ℝ)<n:=zero_lt_one.trans_le hnR1;have hnp:=hnpos.ne';have hzb:=qBo;have bI: |R ind| <=2:=by rw[ἡ pi];exact L_I hnR1 pi ei
 have bW:=ἀ hnR2 (by (positivity):(0:ℝ)<=2^222);have hW2:=p3.trans bW;have bv: |R v| <=2^250:=(qAa (R v) W).trans ((add_le_add ((ἅ hW0).le.trans hW2) (p2.trans (add_le_add ((ἀ (div_le_div_of_nonneg_right (add_le_add (ἁ hnR2 zero_le_two) le_rfl) (by norm_num1)) hW0).trans (ἁ hW2 (by norm_num1))) le_rfl))).trans (by norm_num1))
 obtain ⟨fp,ep1,bp⟩:=qAl v ind (2^250) 2 p1 fi bv bI (by norm_num1) (by norm_num1);obtain ⟨de1,de2⟩:=qAm eps gE;obtain ⟨ft,et,-⟩:=qAv (v*ind) (dv eps) (2^253) fp de1 (bp.trans ((add_le_add le_rfl hzb).trans (by norm_num1))) (by rw[de2];exact bE.trans (by norm_num1)) (by norm_num1);rw[de2,htt] at et;rw[htt] at ft;exact ⟨ft,qH9 qAr.le qAw (W/n) (1/2^229) hnR1 hnR2 hc (div_nonneg hW0 hnpos.le) (qH3 eps gE hep0) ו le_rfl (by rw[mul_div_cancel₀ W hnp];exact p2) pi ei ep1 et⟩
end
lemma qIk (tt nf em:ὧ) (E:ℕ) (hE1:1<=E) (hE:E<=1222) (hEe:(tt.toBits>>>52).toNat=E) (c P δ T:ℝ) (hc1:c<=1/2^20) (hc2:9/2^49<=c) (fnf:nf.isFinite=true) (Rnf:R nf=c) (fE:em.isFinite=true) (m2:R em<=2^120) (m3:1/2^201<=R em) (hP0:0<P) (hq1: |R tt-P| <=c/2*P) (m1:2* |δ| <=R em) (hT:T=P-δ^2) (hT0:0<T):
 R (Ἤ tt-((nf+nf)*Ἤ tt+em))<ἧ T/\ἧ T<R (Ἤ tt+((nf+nf)*Ἤ tt+em))/\ἧ T<2^(127:ℤ):=by
 have e2le:∀{k:ℕ},k<=300->(2:ℝ)^k<=2^256*2^44:=λh=>(pow_le_pow_right₀ one_le_two h).trans_eq (pow_add 2 256 44)
 have e2pf:∀{x P:ℝ},x<=9/8*P+A_z->1<=P->x<=2^2*P:=λ{x P} h h1=>h.trans ((add_le_add (ἀ (by norm_num1:(9:ℝ)/8<=3) (zero_le_one.trans h1)) (qBo.trans h1)).trans ((add_one_mul (3:ℝ) P).symm.trans (congrArg (·*P) (by norm_num1:(3:ℝ)+1=2^2))).le)
 have e2pk:∀{x:ℝ} {a k:ℕ},x<=9/8*((2:ℝ)^a+2^a)+A_z->2+(a+1)<=k->x<=2^k:=λ{x a k} h hk=>(e2pf (h.trans_eq (congrArg (λP:ℝ=>9/8*P+A_z) ((two_mul _).symm.trans (pow_succ' 2 a).symm))) (one_le_pow₀ one_le_two)).trans ((pow_add 2 2 (a+1)).symm.trans_le (pow_le_pow_right₀ one_le_two hk))
 have e2pm:∀{x:ℝ} {a b k:ℕ},x<=9/8*((2:ℝ)^a*2^b)+A_z->2+(a+b)<=k->x<=2^k:=λ{x a b k} h hk=>(e2pf (h.trans_eq (congrArg (λP:ℝ=>9/8*P+A_z) (pow_add 2 a b).symm)) (one_le_pow₀ one_le_two)).trans ((pow_add 2 2 (a+b)).symm.trans_le (pow_le_pow_right₀ one_le_two hk))
 have e2pa:∀{x y:ℝ} {a b j:ℕ},x<=2^a->y<=2^b->a<=j->b<=j->x+y<=2^(j+1):=λ{x y a b j} hx hy ha hb=>(add_le_add (hx.trans (pow_le_pow_right₀ one_le_two ha)) (hy.trans (pow_le_pow_right₀ one_le_two hb))).trans_eq ((two_mul _).symm.trans (pow_succ' 2 j).symm)
 have e2z0:A_z<=2^0:=qBo.trans_eq (pow_zero 2).symm
 have e2d:∀{x:ℝ} (n:ℕ),0<=x->1/2^n*x<=x:=λn h0=>mul_le_of_le_one_left h0 (div_le_one_of_le₀ (one_le_pow₀ one_le_two) (pow_nonneg zero_le_two n))
 have e2c1:∀{c:ℝ} {n:ℕ},c<=1/2^n->c<=1:=λ{c n} h=>h.trans (div_le_one_of_le₀ (one_le_pow₀ one_le_two) (pow_nonneg zero_le_two n))
 have hzb:=qBo
 have hc0:0<c:=Ἥ ו hc2
 obtain ⟨-,-,u1⟩:=qCz hEe hE1 (by exact Nat.lt_of_le_of_lt ‹_› ה)
 obtain ⟨fd,ptt,ed⟩:=qGk hEe hE1 (by exact Nat.lt_of_le_of_lt ‹_› ה)
 generalize Ἤ tt=d at fd ed⊢
 have ht2:R tt<=2^200:=u1.le.trans ((ἒ (by norm_num1) (by exact omS356 hE)).trans_eq (by (norm_num1):(2:ℝ)^(200:ℤ)=2^200))
 generalize hs':ἧ (R tt)=s' at ed
 have hs0:0<s':=by rw[<-hs'];exact Real.sqrt_pos.mpr ptt
 have hsq:s'^2=R tt:=by rw[<-hs'];exact Real.sq_sqrt ptt.le
 have hs1:s'<=2^100:=by
  rw[<-hs',Real.sqrt_le_iff]
  exact ⟨by (positivity),ht2.trans_eq (pow_mul 2 100 2)⟩
 have bd: |R d| <=2^102:=by
  have a1:=qAa (R d) s'
  rw[ἡ hs0] at a1
  exact a1.trans (e2pa (j:=101) hs1 (ed.trans ((e2d 53 (add_nonneg hs0.le qAr.le)).trans (e2pa (j:=100) hs1 e2z0 le_rfl (Nat.le_of_ble_eq_true rfl)))) (Nat.le_of_ble_eq_true rfl) le_rfl)
 have bnf: |R nf| <=2^0:=by rw[Rnf,ἡ hc0,pow_zero];exact e2c1 hc1
 obtain ⟨f2n,e2n,b2n⟩:=qAv nf nf (2^0) fnf fnf bnf bnf (e2le (Nat.le_of_ble_eq_true rfl))
 rw[Rnf] at e2n
 obtain ⟨fmu,emu,bmu⟩:=qAl (nf+nf) d (2^3) (2^102) f2n fd (e2pk b2n le_rfl) bd (e2le (Nat.le_of_ble_eq_true rfl)) (e2le (Nat.le_of_ble_eq_true rfl))
 have hEm0:0<=R em:=ὂ ו m3
 have bEm: |R em| <=2^120:=by rw[ἅ hEm0];exact m2
 obtain ⟨fe,ee,be⟩:=qAv ((nf+nf)*d) em (2^120) fmu fE (e2pm bmu (Nat.le_of_ble_eq_true rfl)) bEm (e2le (Nat.le_of_ble_eq_true rfl))
 have bd3:=bd.trans (pow_le_pow_right₀ one_le_two (Nat.le_of_ble_eq_true rfl:102<=123))
 have be3:=e2pk (k:=123) be le_rfl
 obtain ⟨fh,eh,-⟩:=qAv d _ (2^123) fd fe bd3 be3 (e2le (Nat.le_of_ble_eq_true rfl))
 obtain ⟨fl,el,-⟩:=qCg d _ (2^123) fd fe bd3 be3 (e2le (Nat.le_of_ble_eq_true rfl))
 obtain ⟨hhi,hlo⟩:=qHr qAr.le qAw hs0.le (hs1.trans (pow_le_pow_right₀ one_le_two (Nat.le_of_ble_eq_true rfl))) hc2 hc1 m3 m2 ed e2n emu ee eh el
 obtain ⟨b1,b2⟩:=abs_le.mp hq1
 have hδ2:δ^2*4<=R em^2:=by
  have a1:(2* |δ|)^2<=R em^2:=Ἳ (Ἆ zero_le_two (Ἀ δ)) m1 2
  rw[mul_pow,sq_abs] at a1
  linarith only[a1]
 have nP:0<=c*P:=(mul_pos hc0 hP0).le
 have a3:=ἀ (e2c1 hc1) nP
 have a4:=Ἆ hc0.le nP
 have nd2:=ὅ δ
 have h1':T<=s'^2*(1+c):=by
  rw[hsq,hT]
  have a1:=ἀ ((div_le_iff₀' two_pos).mp (by rwa[div_mul_eq_mul_div,<-neg_div] at b1):-(c*P)<=2*(R tt-P)) (add_pos one_pos hc0).le
  linarith only[a3,nd2,a1]
 have h2':s'^2*(1-c)-R em^2/4<=T:=by
  rw[hsq,hT]
  have a1:=ἀ ((le_div_iff₀' two_pos).mp (by rwa[div_mul_eq_mul_div] at b2):2*(R tt-P)<=c*P) (sub_nonneg.mpr (e2c1 hc1))
  linarith only[hδ2,a3,a4,a1]
 obtain ⟨l1,l2⟩:=qHl hT0 hs0 hc0 hc1 hEm0 h1' h2' hhi hlo
 refine ⟨l1,l2,?_⟩
 rw[Real.sqrt_lt' ו,zpow_ofNat,<-pow_mul]
 have a1:=ἑ (hsq.trans_le ht2) ((add_le_add le_rfl (e2c1 hc1)).trans_eq one_add_one_eq_two:1+c<=2) (add_pos one_pos hc0).le ו
 exact h1'.trans_lt (a1.trans_lt ((pow_succ 2 200).symm.trans_lt (pow_lt_pow_right₀ one_lt_two (Nat.le_of_ble_eq_true rfl))))
lemma qE1 (b:Ὡ) (rows n:ℕ) (eps:Ὢ) (x:ё) (r:ℕ) (hr:r<rows) (hn0:0<n) (hn:n<2^20) (hs:ѧx=rows*n) (hrn:rows*n<2^32) (hx:x.all sm=true) (he:sm eps=true) (he0:0<ἂ eps):L_ok b n eps x r (lnStat b n eps x r):=by
 have e2AB:∀{x:ℝ} {k:ℕ},x<=2^k->k+3<=256+42->x<=A_B/8:=λ{x k} h hk=>(le_div_iff₀ (by norm_num:(0:ℝ)<8)).mpr ((ἀ h (by norm_num:(0:ℝ)<=8)).trans ((((congrArg ((2:ℝ)^k*·) (by norm_num:(8:ℝ)=2^3)).trans (pow_add 2 k 3).symm).trans_le ((pow_le_pow_right₀ one_le_two hk).trans_eq (pow_add 2 256 42))).trans qCc))
 have e2AB4:A_B/8<=A_B/4:=div_le_div_of_nonneg_left qI1.le ב ב
 have hmul:r*n+n<=rows*n:=by have:=Ἒ n hr;rwa[ᾀ] at this
 generalize hgd:(λj:ℕ=>dv x[r*n+j]!)=g;have hg0:∀j,j<n->(g j).isFinite=true/\|R (g j)| <=2^99/\R (g j)=(ἂ x[r*n+j]!:ℝ):=λj h=>by obtain ⟨a1,a2⟩:=qA1 _ (qBg x sm hx (r*n+j) (by rw[hs];exact (Nat.add_lt_add_left h _).trans_le hmul));obtain ⟨a3,a4⟩:=qAm _ a1;rw[<-hgd];beta_reduce;rw[a4];exact ⟨a3,a2,qAx _ (qAe _ a1)⟩
 have hnR1:(1:ℝ)<=n:=by exact_mod_cast hn0
 have hnR2:(n:ℝ)<=2^20:=by exact_mod_cast hn.le
 have o4:((ἄ n).map λj=>|R (g j)|).sum<=n*2^99:=qDr n _ _ λj h=>(hg0 j h).2.1;have hA2:((ἄ n).map λj=>|R (g j)|).sum<=A_B/4:=(e2AB (k:=20+99) (o4.trans ((ἀ hnR2 (by (positivity):(0:ℝ)<=2^99)).trans_eq (pow_add 2 20 99).symm)) (Nat.le_of_ble_eq_true rfl)).trans e2AB4
 obtain ⟨o1,o5⟩:=qFm n hn g (λj=>R (g j)) (λj=>|R (g j)|) (λj h=>⟨(hg0 j h).1,rfl,rfl⟩) hA2
 obtain ⟨o2,o6⟩:=qFm n hn (λj=>(g j).abs) (λj=>|R (g j)|) (λj=>|R (g j)|)
  (λj h=>by
   obtain ⟨p,q⟩:=qBl _ (hg0 j h).1
   exact ⟨p,q,abs_abs _⟩) hA2
 have o3:=qF2 (ἄ n) λj=>R (g j);obtain ⟨hMc,hVc⟩:=qHi n r x g (λj h=>(hg0 j h).2.2) (lnQ b n x r);generalize hQ:((lnQ b n x r:ℚ):ℝ)=Q at hVc;generalize hS:((ἄ n).map λj=>R (g j)).sum=S at o3 o5 hMc;generalize ((ἄ n).map λj=>|R (g j)|).sum=Sa at o3 o4 o5 o6;unfold lnStat;dsimp only;generalize hnu:ѩn=nu;generalize hbase:(r*n).toUSize=base;generalize hnd:n.toUInt64.toFloat=nd;generalize hnf:(n.toUInt64+8).toFloat*Ἃ ь=nf;generalize hind:(1.0:ὧ)/nd=ind;generalize ha:(if b=true then F1.mk 0 0 else lnSum x base nu 0 0 0)=a;generalize hm:(if b=true then (0:ὧ) else a.s*ind)=m;generalize hv:lnVar x base nu 0 m 0=v;generalize hem:nf*a.t*ind+Ἃ ظ=em;generalize htt:v*ind+dv eps=tt;generalize hd:Ἤ tt=d;generalize hz:ziv d ((nf+nf)*d+em)=z;generalize hir:(1.0:ὧ)/dv z=ir;split
 . rename_i hc;simp only[Bool.and_eq_true,decide_eq_true_eq] at hc;obtain ⟨⟨⟨⟨⟨c1,c2⟩,c3⟩,c4⟩,c5⟩,c6⟩:=hc;have hnp:(n:ℝ)≠0:=(zero_lt_one.trans_le hnR1).ne';have hzb:=qBo;have hz0:=qAr.le;have hu:Ѥnu=n:=by rw[<-hnu];exact qAd n (by exact Nat.lt_of_lt_of_le ‹_› ה)
   have hb:Ѥbase=r*n:=by rw[<-hbase];exact qAd _ (((Nat.le_add_right _ _).trans hmul).trans_lt hrn)
   have hxs:ѧx<2^32:=hs.trans_lt hrn;obtain ⟨fnf,Rnf⟩:=hnf▸qIi n hn;obtain ⟨c,hc⟩:∃c:ℝ,c=((n:ℝ)+8)/2^49:=⟨_,rfl⟩;rw[<-hc] at Rnf;have hc1:c<=1/2^20:=by rw[hc];exact (div_le_div_of_nonneg_right (add_le_add hnR2 le_rfl) (by norm_num1)).trans (by norm_num1)
   have hc2:9/2^49<=c:=by rw[hc];exact div_le_div_of_nonneg_right ((show (9:ℝ)=1+8 by norm_num1).trans_le (add_le_add hnR1 le_rfl)) (by norm_num1)
   obtain ⟨fnd,Rnd⟩:nd.isFinite=true/\R nd=n:=by
    rw[<-hnd];have e:n.toUInt64.toNat=n:=by simp only[Nat.toUInt64_eq,UInt64.toNat_ofNat',Nat.reducePow,Nat.mod_succ_eq_iff_lt,Nat.succ_eq_add_one,Nat.reduceAdd];(exact Nat.lt_of_lt_of_le ‹_› ה)
    have:=qCk n.toUInt64 (by rw[e];exact hn.trans ה);rwa[e] at this
   obtain ⟨fi,pi,-,ei⟩:=qFy ind nd fnd (by
    rw[Rnd,ἅ (zero_le_one.trans hnR1)];exact hnR2.trans (by norm_num1)) c5
   rw[Rnd] at ei;obtain ⟨z1,z2⟩:=qHg
   obtain ⟨fm,fE,m2,m3,m1,m4⟩:m.isFinite=true/\em.isFinite=true/\R em<=2^120/\1/2^201<=R em/\2* |Q-R m| <=R em/\|R m| <=2^110:=by
    cases b with
    | true=>
     have e1:a=F1.mk 0 0:=ha.symm;have e2:m=0:=hm.symm;have e3:Q=0:=by rw[<-hQ];simp only[lnQ, ↓reduceIte,Rat.cast_zero]
     subst e1;obtain ⟨-,fE,m2,m3,-,-⟩:=qG1 n hnR1 hnR2 0 0 ind nf _ em 0 0 c hc z1 z1 ג ו (by rw[z2,sub_self,abs_zero];exact Ἆ (Ἆ zero_le_two (Ἔ n)) (Ἆ (div_nonneg zero_le_one (pow_nonneg zero_le_two 53)) (add_nonneg le_rfl hz0))) (by rw[z2,sub_self,abs_zero];exact Ἆ (Ἆ zero_le_two (Ἔ n)) (Ἆ (div_nonneg zero_le_one (pow_nonneg zero_le_two 53)) (add_nonneg le_rfl hz0))) fnf Rnf fi pi ei rfl hem;rw[e2,e3,z2];refine ⟨z1,fE,m2,m3,?_,by (norm_num1)⟩;simp only[sub_zero,abs_zero,mul_zero];exact ὂ ו m3
    | false=>
     replace ha:lnSum x base nu 0 0 0=a:=ha;replace hm:a.s*ind=m:=hm;have hA:a=⟨A_fsum ((ἄ n).map g),A_fsum ((ἄ n).map λj=>(g j).abs)⟩:=by rw[<-ha,qG9 x hxs base nu (by rw[hb,hu];exact hmul.trans_lt hrn) n 0 0 0 (by simp only[USize.toNat_zero,zero_add,hu])];simp only[USize.toNat_zero,hb,<-hgd,A_fsum,ὗ]
     subst hA;rw[hQ.symm.trans hMc];exact qG1 n hnR1 hnR2 _ _ ind nf m em S Sa c hc o1 o2 o3 o4 o5 o6 fnf Rnf fi pi ei hm hem
   obtain ⟨p1,p2,p3,hW0⟩:=qGp n hn hn0 g m v _ (by
    rw[<-hv,qG2 x hxs base nu m (by rw[hb,hu];exact hmul.trans_lt hrn) n 0 0 (by simp only[USize.toNat_zero,zero_add,hu])]
    simp only[USize.toNat_zero,hb,<-hgd,A_fsum,ὗ]) rfl
    (λj h=>⟨(hg0 j h).1,(hg0 j h).2.1⟩) fm m4
   generalize hW:((ἄ n).map λj=>(R (g j)-R m)^2).sum=W at p2 p3 hW0;generalize hV0:((ἄ n).map λj=>(R (g j)-Q)^2).sum=V0 at hVc;generalize hδ:Q-R m=δ at m1;have hnpos:(0:ℝ)<n:=zero_lt_one.trans_le hnR1
   have hWs:W/n=V0/n+δ^2:=by
    cases b with
    | true=>
     have e2:m=0:=hm.symm;have e3:Q=0:=by rw[<-hQ];simp only[lnQ, ↓reduceIte,Rat.cast_zero]
     rw[<-hδ,<-hW,<-hV0,e2,e3,z2];simp only[sub_zero,sub_self,ne_eq,OfNat.ofNat_ne_zero,not_false_eq_true,zero_pow,add_zero]
    | false=>have hs2:=qGh (ἄ n) (λj=>R (g j)) (R m) (by rw[List.length_range];exact hnp);simp only[List.length_range] at hs2;rw[hS,hW,show S/n=Q from (hQ.symm.trans hMc).symm,hV0,hδ] at hs2;rw[hs2,add_div,mul_div_cancel_left₀ _ hnp]
   obtain ⟨gE,bE⟩:=qA1 eps he;have epsv:wv eps=(ἂ eps:ℝ):=qAx _ (qAe _ gE);have hep0:0<wv eps:=by rw[epsv];exact_mod_cast he0
   obtain ⟨ft,hq1⟩:=qIv n hnR1 hnR2 v ind tt eps W c hc p1 p2 p3 hW0 fi pi ei gE bE hep0 htt;generalize hq:rowVariance n x r (lnQ b n x r)+ἂ eps=q;have hV1:0<=V0/n:=by rw[<-hV0];refine div_nonneg (List.sum_nonneg λy hy=>?_) hnpos.le;obtain ⟨j,-,rfl⟩:=List.mem_map.1 hy;exact ὅ _
   have hT:(q:ℝ)=(W/n+wv eps)-δ^2:=by rw[<-hq,Rat.cast_add,hVc,epsv,hWs];ring
   have hT0:(0:ℝ)<q:=by rw[hT,hWs,add_sub_right_comm,add_sub_cancel_right];exact add_pos_of_nonneg_of_pos hV1 hep0
   have hP0:0<W/n+wv eps:=add_pos_of_nonneg_of_pos (div_nonneg hW0 hnpos.le) hep0
   obtain ⟨r1,r2⟩:1<=_/\_<1+1222:=qEj _ 1 1222 c4 ה;obtain ⟨l1,l2,hb127⟩:=qIk tt nf em _ r1 (by exact Nat.le_of_lt_succ ‹_›) rfl c _ δ (q:ℝ) hc1 hc2 fnf Rnf fE m2 m3 hP0 hq1 m1 hT hT0;rw[hd] at l1 l2;have hq0:0<q:=by exact_mod_cast hT0
   have hzne:ziv d ((nf+nf)*d+em)≠Е:=by rw[hz];intro e0;rw[e0] at c3;exact absurd c3 ה
   have hden:=((qGs d _ q hq0 l1 l2 hzne hb127).trans hz).symm;rw[<-hq] at hden;obtain ⟨gZ,bZ⟩:=qA1 z c1;obtain ⟨dz1,dz2⟩:=qAm z gZ;obtain ⟨fi2,-,-,ei2⟩:=qFy ir (dv z) dz1 (by rw[dz2];exact bZ) c6;rw[dz2] at ei2;refine ⟨hden,λ_=>⟨fm,fi2,fE,m2,m3,?_,m4,gZ,qIb z gZ c2 c3,ei2⟩⟩;dsimp only;rw[hQ,hδ];exact m1
 . exact ⟨rfl,λh=>Bool.noConfusion h⟩
lemma qI0b (m ir em:ὧ) (w gw bw:Ὢ) (M D:ℝ) (fm:m.isFinite=true) (fi:ir.isFinite=true) (fe:em.isFinite=true) (e1:R em<=2^120) (e2:1/2^201<=R em) (e3:2* |M-R m| <=R em) (e4: |R m| <=2^110) (hD:1/2^64<=D) (hI: |R ir*D-1| <=1/2^48) (hw:sm w=true) (hg:sm gw=true) (hb:sm bw=true) (s e:ὧ) (hs:s=(dv w-m)*(dv gw*ir)+dv bw)
 (he:e=Ἃ 0x3D10000000000000*(((dv w-m)*(dv gw*ir)).abs+(dv bw).abs)+em*(dv gw*ir).abs) (big0:(2:ℝ)^(-127:ℤ)<= |R (s+e)|):
 R (s-e)<(wv w-M)*wv gw/D+wv bw/\(wv w-M)*wv gw/D+wv bw<R (s+e):=by
 have e2le:∀{k:ℕ},k<=300->(2:ℝ)^k<=2^256*2^44:=λh=>(pow_le_pow_right₀ one_le_two h).trans_eq (pow_add 2 256 44)
 have e2pf:∀{x P:ℝ},x<=9/8*P+A_z->1<=P->x<=2^2*P:=λ{x P} h h1=>h.trans ((add_le_add (ἀ (by norm_num1:(9:ℝ)/8<=3) (zero_le_one.trans h1)) (qBo.trans h1)).trans ((add_one_mul (3:ℝ) P).symm.trans (congrArg (·*P) (by norm_num1:(3:ℝ)+1=2^2))).le)
 have e2pk:∀{x:ℝ} {a k:ℕ},x<=9/8*((2:ℝ)^a+2^a)+A_z->2+(a+1)<=k->x<=2^k:=λ{x a k} h hk=>(e2pf (h.trans_eq (congrArg (λP:ℝ=>9/8*P+A_z) ((two_mul _).symm.trans (pow_succ' 2 a).symm))) (one_le_pow₀ one_le_two)).trans ((pow_add 2 2 (a+1)).symm.trans_le (pow_le_pow_right₀ one_le_two hk))
 have e2pm:∀{x:ℝ} {a b k:ℕ},x<=9/8*((2:ℝ)^a*2^b)+A_z->2+(a+b)<=k->x<=2^k:=λ{x a b k} h hk=>(e2pf (h.trans_eq (congrArg (λP:ℝ=>9/8*P+A_z) (pow_add 2 a b).symm)) (one_le_pow₀ one_le_two)).trans ((pow_add 2 2 (a+b)).symm.trans_le (pow_le_pow_right₀ one_le_two hk))
 have e2p1:∀{x:ℝ} {b k:ℕ},x<=9/8*(1*(2:ℝ)^b)+A_z->2+b<=k->x<=2^k:=λ{x b k} h hk=>(e2pf (h.trans_eq (congrArg (λP:ℝ=>9/8*P+A_z) (one_mul _))) (one_le_pow₀ one_le_two)).trans ((pow_add 2 2 b).symm.trans_le (pow_le_pow_right₀ one_le_two hk))
 have hzb:=qBo
 obtain ⟨gw1,bw1⟩:=qA1 w hw
 obtain ⟨gg1,bg1⟩:=qA1 gw hg
 obtain ⟨gb1,bb1⟩:=qA1 bw hb
 obtain ⟨fw,Rw⟩:=qAm w gw1
 obtain ⟨fg,Rg⟩:=qAm gw gg1
 obtain ⟨fb,Rb⟩:=qAm bw gb1
 have hD0:0<D:=Ἥ ו hD
 have bI: |R ir| <=2^65:=by
  have a1:=qAa (R ir*D) 1
  rw[abs_one,ἢ,ἡ hD0] at a1
  have a2:=ἁ hD (Ἀ (R ir))
  have a3:=(a2.trans (a1.trans (add_le_add le_rfl (hI.trans (div_le_one_of_le₀ (one_le_pow₀ one_le_two) (pow_nonneg zero_le_two 48)))))).trans_eq one_add_one_eq_two
  rw[mul_one_div] at a3
  exact ((div_le_iff₀ (pow_pos two_pos 64)).mp a3).trans_eq (pow_succ' 2 64).symm
 obtain ⟨f1,h1,b1⟩:=qCg (dv w) m (2^111) fw fm (by rw[Rw];exact bw1.trans (pow_le_pow_right₀ one_le_two (Nat.le_of_ble_eq_true rfl))) (e4.trans (pow_le_pow_right₀ one_le_two (Nat.le_of_ble_eq_true rfl))) (e2le (Nat.le_of_ble_eq_true rfl))
 rw[Rw] at h1
 obtain ⟨f2,h3,b2⟩:=qAl (dv gw) ir (2^99) (2^65) fg fi (by rw[Rg];exact bg1) bI (e2le (Nat.le_of_ble_eq_true rfl)) (e2le (Nat.le_of_ble_eq_true rfl))
 rw[Rg] at h3
 obtain ⟨f3,h4,b3⟩:=qAl _ _ (2^114) (2^166) f1 f2 (e2pk b1 le_rfl) (e2pm b2 (Nat.le_of_ble_eq_true rfl)) (e2le (Nat.le_of_ble_eq_true rfl)) (e2le (Nat.le_of_ble_eq_true rfl))
 generalize hQ:(dv w-m)*(dv gw*ir)=Q at hs he f3 h4 b3
 generalize dv gw*ir=G at he f2 h3 b2 h4 hQ
 have bQ:=e2pm (k:=282) b3 (Nat.le_of_ble_eq_true rfl)
 have bB: |R (dv bw)| <=2^282:=by rw[Rb];exact bb1.trans (pow_le_pow_right₀ one_le_two (Nat.le_of_ble_eq_true rfl))
 obtain ⟨f4,h5,b4⟩:=qAv Q (dv bw) (2^282) f3 fb bQ bB (e2le (Nat.le_of_ble_eq_true rfl))
 rw[Rb,<-hs] at h5
 rw[<-hs] at f4 b4
 obtain ⟨fa1,Ra1⟩:=qBl Q f3
 obtain ⟨fa2,Ra2⟩:=qBl (dv bw) fb
 rw[Rb] at Ra2
 obtain ⟨fa3,Ra3⟩:=qBl G f2
 obtain ⟨f5,h6,b5⟩:=qAv Q.abs (dv bw).abs (2^282) fa1 fa2 (by rw[Ra1,abs_abs];exact bQ) (by rw[Ra2,abs_abs,<-Rb];exact bB) (e2le (Nat.le_of_ble_eq_true rfl))
 rw[Ra1,Ra2] at h6
 obtain ⟨k1,k2⟩:=L_k 0x3D10000000000000 977 46 (Nat.le_of_ble_eq_true rfl) rfl (by rfl)
 obtain ⟨f6,h7,b6⟩:=qAl _ _ 1 (2^285) k1 f5 (by rw[k2,ἅ ו];exact div_le_one_of_le₀ (one_le_pow₀ one_le_two) ו) (e2pk b5 le_rfl) ((pow_zero (2:ℝ)).symm.trans_le (e2le (Nat.le_of_ble_eq_true rfl))) (e2le (Nat.le_of_ble_eq_true rfl))
 rw[k2] at h7
 have bE: |R em| <=2^120:=by
  rw[ἅ (ὂ ו e2)];exact e1
 obtain ⟨f7,h8,b7⟩:=qAl em G.abs (2^120) (2^166) fe fa3 bE (by rw[Ra3,abs_abs];exact e2pm b2 (Nat.le_of_ble_eq_true rfl)) (e2le (Nat.le_of_ble_eq_true rfl)) (e2le (Nat.le_of_ble_eq_true rfl))
 rw[Ra3] at h8
 obtain ⟨f8,h9,b8⟩:=qAv _ _ (2^288) f6 f7 (e2p1 b6 (Nat.le_of_ble_eq_true rfl)) (e2pm b7 (Nat.le_of_ble_eq_true rfl)) (e2le (Nat.le_of_ble_eq_true rfl))
 rw[<-he] at f8 h9 b8
 have bs:=e2pk (k:=291) b4 (Nat.le_of_ble_eq_true rfl)
 have be:=e2pk (k:=291) b8 (Nat.le_of_ble_eq_true rfl)
 obtain ⟨f9,h10,-⟩:=qAv s e (2^291) f4 f8 bs be (e2le (Nat.le_of_ble_eq_true rfl))
 obtain ⟨f10,h11,-⟩:=qCg s e (2^291) f4 f8 bs be (e2le (Nat.le_of_ble_eq_true rfl))
 have big:=big0
 rw[zpow_neg,zpow_ofNat,<-one_div] at big
 exact qG0 qAr.le qAw (ὂ ו e2) e1 e3 hD bw1 bg1 e4 h1 hI h3 h4 h5 h6 h7 h8 h9 h10 h11 big
lemma kHq (th:Ὠ) (s:ὧ) (q:ℚ) (hV: |R s-(q:ℝ)| <=1/2^33*(|(q:ℝ)| +A_z)) (hb: |(q:ℝ)| <=2^(127:ℤ)) (h:zivT th s≠Е):roundRational q=zivT th s:=by
 obtain ⟨-,S,ex,fr,M,h1,h2,h3,hz,hbt,hl,hu⟩:=kFh th s h
 have hM:M=(M/2^52)*2^52+M%2^52:=(by clear h h1 h2 h3 hz hbt hl hu;exact omS114)
 have hF:M%2^52<2^52:=ὀ _ ו
 generalize M/2^52=E at hM
 generalize M%2^52=F at hM hF
 subst hM
 obtain ⟨-,hR⟩:=B_R s S E F (by clear h hz;exact omS238 h1 hF hl) (by clear h hz;exact omS239 h2 h3 hu) hF (by rw[hbt,ὢ])
 rw[hR] at hV
 obtain ⟨W,hVW,hW⟩:∃W:ℝ,(q:ℝ)=(if S then-W else W)/\ |((2^52+F:Ὣ):ℝ)*2^((E:Int)-1075)-W| <=1/2^33*(|W| +A_z):=by
  cases S
  . simp only[false_eq_true,ite_false,one_mul] at hV⊢;exact ⟨_,rfl,hV⟩
  . simp only[ite_true] at hV⊢;refine ⟨-(q:ℝ),by simp only[neg_neg],?_⟩;rw[abs_neg,show ((2^52+F:Ὣ):ℝ)*2^((E:Int)-1075)- -(q:ℝ)=-(-1*((2^52+F:Ὣ):ℝ)*2^((E:Int)-1075)-(q:ℝ)) by ring,abs_neg];exact hV
 obtain ⟨a,b,c⟩:=qJi ex fr E F h1 h2 h3 hF (by exact omS240 hl) (by exact omS241 hu) W hW
 have hb0:(0:ℝ)<2^((ex:Int)-1076):=ἵ ו _
 have hW0:0<W:=ἷ (Ἆ ב hb0.le) c
 have hq0:(q:ℝ)≠0:=by intro h0;rw[h0] at hVW;cases S<;>simp only[false_eq_true, ↓reduceIte,zero_eq_neg] at hVW<;>linarith only[hW0,hVW]
 have hf:=Model.isFinite_roundRatScaled_of_abs_le_posMaxFinite Fmt (decide (q<0)) q.num.natAbs q.den 0 ה q.den_nz (qFn q▸hb.trans qIx)
 have hMr:=Model.toReal_roundRatScaled_eq_roundAt Fmt _ _ _ 0 ה hf
 rw[hz]
 exact qDqW S ex fr h1 h2 h3 (q:ℝ) W hVW a b c _ hf (qFn q▸hMr)
lemma qSubE (x y:ὧ) (hx:xᶠ) (hy:yᶠ) (h1: |R x| <=A_B) (h2: |R y| <=A_B):(x-y)ᶠ /\R (x-y)=Model.roundAt ⟡ (R x-R y):=
 A_E _ (toModel_ofFloat_sub_of_isFinite x y hx hy) (Model.isFinite_sub_of_abs_add_le_posMaxFinite _ _ qAj (qA3 hx) (qA3 hy) (qEe h1 h2))
  (Model.toReal_sub_eq_roundAt_of_abs_add_le_posMaxFinite _ _ qAj (qA3 hx) (qA3 hy) (qEe h1 h2))
lemma rAbs (x:ℝ): |Model.roundAt ⟡ x| =Model.roundAt ⟡ |x| :=by
 rcases le_or_gt 0 x with h|h
 . rw[ἅ h,ἅ];rw[<-Model.roundAt_zero ⟡];exact Model.roundAt_mono _ h
 . rw[abs_of_neg h,abs_of_nonpos];. rw[Model.roundAt_neg ⟡ x .nearest];rfl
   . rw[<-Model.roundAt_zero ⟡];exact Model.roundAt_mono _ h.le
lemma rUp (y:ℝ) (h:0<=y):Model.roundAt ⟡ y<=2*y+1:=by
 have e:=qFp y
 have hz:A_z<=1:=by show (2:ℝ)^(-1021:ℤ)<=1;exact zpow_le_one_of_nonpos₀ ב ב
 rw[ἅ h] at e
 have:=le_abs_self (Model.roundAt ⟡ y-y)
 linarith only[e,this,hz,h]
set_option maxRecDepth 8192 in
lemma mMul (a b c d:ὧ) (fa:aᶠ) (fb:bᶠ) (fc:cᶠ) (fd:dᶠ) (ha: |R a| <=R c) (hb: |R b| <=R d) (Bc Bd:ℝ) (hc:R c<=Bc) (hd:R d<=Bd) (hB:Bc*Bd<=2^256*2^256*2^256):
 (a*b)ᶠ /\(c*d)ᶠ /\ |R (a*b)| <=R (c*d)/\R (c*d)<=2*(Bc*Bd)+1:=by
 have pc:0<=R c:=(Ἀ _).trans ha
 have pd:0<=R d:=(Ἀ _).trans hb
 have k9:(2:ℝ)^256*2^256*2^256<=2^(1000:ℤ):=by rw[<-pow_add,<-pow_add,<-Ἡ];exact ἒ ב ב
 have hcd:R c*R d<=Bc*Bd:=ἑ hc hd pd (pc.trans hc)
 obtain ⟨f1,R1⟩:=qEd a b fa fb ((ἑ ha hb (Ἀ _) pc).trans (hcd.trans (hB.trans k9)))
 obtain ⟨f2,R2⟩:=qEd c d fc fd (by rw[ἅ pc,ἅ pd];exact hcd.trans (hB.trans k9))
 refine ⟨f1,f2,?_,?_⟩
 . rw[R1,R2,rAbs,ἢ];exact Model.roundAt_mono _ (ἑ ha hb (Ἀ _) pc)
 . rw[R2];exact (rUp _ (Ἆ pc pd)).trans (by linarith only[hcd])
set_option maxRecDepth 8192 in
lemma mAdd (a b c d:ὧ) (fa:aᶠ) (fb:bᶠ) (fc:cᶠ) (fd:dᶠ) (ha: |R a| <=R c) (hb: |R b| <=R d) (Bc Bd:ℝ) (hc:R c<=Bc) (hd:R d<=Bd) (hB:Bc+Bd<=2^256*2^40):
 (a+b)ᶠ /\(c+d)ᶠ /\ |R (a+b)| <=R (c+d)/\R (c+d)<=2*(Bc+Bd)+1:=by
 have pc:0<=R c:=(Ἀ _).trans ha
 have pd:0<=R d:=(Ἀ _).trans hb
 have k1:Bc<=2^256*2^44:=by linarith only[hB,pc,pd,hc,hd]
 have k2:Bd<=2^256*2^44:=by linarith only[hB,pc,pd,hc,hd]
 obtain ⟨f1,R1⟩:=qAddE a b fa fb (qAt (ha.trans (hc.trans k1))) (qAt (hb.trans (hd.trans k2)))
 obtain ⟨f2,R2⟩:=qAddE c d fc fd (qAt (by rw[ἅ pc];exact hc.trans k1)) (qAt (by rw[ἅ pd];exact hd.trans k2))
 refine ⟨f1,f2,?_,?_⟩
 . rw[R1,R2,rAbs];exact Model.roundAt_mono _ ((Ἐ _ _).trans (add_le_add ha hb))
 . rw[R2];exact (rUp _ (add_nonneg pc pd)).trans (by linarith only[hc,hd])
set_option maxRecDepth 8192 in
lemma mSub (a b c d:ὧ) (fa:aᶠ) (fb:bᶠ) (fc:cᶠ) (fd:dᶠ) (ha: |R a| <=R c) (hb: |R b| <=R d) (Bc Bd:ℝ) (hc:R c<=Bc) (hd:R d<=Bd) (hB:Bc+Bd<=2^256*2^40):
 (a-b)ᶠ /\(c+d)ᶠ /\ |R (a-b)| <=R (c+d)/\R (c+d)<=2*(Bc+Bd)+1:=by
 have pc:0<=R c:=(Ἀ _).trans ha
 have pd:0<=R d:=(Ἀ _).trans hb
 have k1:Bc<=2^256*2^44:=by linarith only[hB,pc,pd,hc,hd]
 have k2:Bd<=2^256*2^44:=by linarith only[hB,pc,pd,hc,hd]
 obtain ⟨f1,R1⟩:=qSubE a b fa fb (qAt (ha.trans (hc.trans k1))) (qAt (hb.trans (hd.trans k2)))
 obtain ⟨f2,R2⟩:=qAddE c d fc fd (qAt (by rw[ἅ pc];exact hc.trans k1)) (qAt (by rw[ἅ pd];exact hd.trans k2))
 refine ⟨f1,f2,?_,?_⟩
 . rw[R1,R2,rAbs];exact Model.roundAt_mono _ ((Ὕ _ _).trans (add_le_add ha hb))
 . rw[R2];exact (rUp _ (add_nonneg pc pd)).trans (by linarith only[hc,hd])
set_option maxRecDepth 8192 in
lemma qLmo (m ir em xm gm bm:ὧ) (w gw bw:Ὢ) (fm:mᶠ) (fi:irᶠ) (fe:emᶠ) (fxm:xmᶠ) (fgm:gmᶠ) (fbm:bmᶠ) (hw:sm w=true) (hg:sm gw=true) (hb:sm bw=true) (pir:0<R ir) (bir:R ir<=2^65) (pem:0<=R em) (bem:R em<=2^120) (bmm: |R m| <=2^110) (hxm: |R (dv w)| <=R xm) (hgm: |R (dv gw)| <=R gm) (hbm: |R (dv bw)| <=R bm) (bxm:R xm<=2^99) (bgm:R gm<=2^99) (bbm:R bm<=2^99):
 (Ἃ 0x3D10000000000000*(((dv w-m)*(dv gw*ir)).abs+(dv bw).abs)+em*(dv gw*ir).abs)ᶠ /\
 |R (Ἃ 0x3D10000000000000*(((dv w-m)*(dv gw*ir)).abs+(dv bw).abs)+em*(dv gw*ir).abs)| <= |R (Ἃ 0x3D10000000000000*((xm+m.abs)*(gm*ir)+bm)+em*(gm*ir))| :=by
 obtain ⟨fw,-⟩:=qAm w (qA1 w hw).1
 obtain ⟨fg,-⟩:=qAm gw (qA1 gw hg).1
 obtain ⟨fb,-⟩:=qAm bw (qA1 bw hb).1
 obtain ⟨k1,k2⟩:=L_k 0x3D10000000000000 977 46 (by norm_num1) rfl (by rfl)
 obtain ⟨fma,Rma⟩:=qBl m fm
 obtain ⟨fG1,fG2,hG,bG⟩:=mMul (dv gw) ir gm ir fg fi fgm fi hgm (le_of_eq (ἡ pir)) (2^99) (2^65) bgm bir ב
 obtain ⟨fD,fP,hD,bP⟩:=mSub (dv w) m xm m.abs fw fm fxm fma hxm (by rw[Rma]) (2^99) (2^110) bxm (by rw[Rma];exact bmm) ב
 obtain ⟨fQ1,fQ2,hQ,bQ⟩:=mMul _ _ _ _ fD fG1 fP fG2 hD hG _ _ bP bG ב
 obtain ⟨fa1,Ra1⟩:=qBl _ fQ1
 obtain ⟨fa2,Ra2⟩:=qBl (dv bw) fb
 obtain ⟨fS1,fS2,hS,bS⟩:=mAdd _ _ _ _ fa1 fa2 fQ2 fbm (by rw[Ra1,abs_abs];exact hQ) (by rw[Ra2,abs_abs];exact hbm) _ _ bQ bbm ב
 obtain ⟨fT1,fT2,hT,bT⟩:=mMul _ _ _ _ k1 fS1 k1 fS2 (by rw[k2];norm_num) hS _ _ (le_of_eq k2) bS ב
 obtain ⟨fa3,Ra3⟩:=qBl _ fG1
 obtain ⟨fU1,fU2,hU,bU⟩:=mMul _ _ _ _ fe fa3 fe fG2 (by rw[ἅ pem]) (by rw[Ra3,abs_abs];exact hG) _ _ bem bG ב
 obtain ⟨fE1,fE2,hE,-⟩:=mAdd _ _ _ _ fT1 fU1 fT2 fU2 hT hU _ _ bT bU ב
 exact ⟨fE1,hE.trans (le_abs_self _)⟩
lemma qSmx (A:ё) (hs:ѧA<2^32) (h:smA A=true):(dv (scan A A.usize 0 (qAc A) Е 0))ᶠ /\R (dv (scan A A.usize 0 (qAc A) Е 0))<=2^99/\ ∀k,k< ѧA-> |R (dv A[k]!)| <=R (dv (scan A A.usize 0 (qAc A) Е 0)):=by
 unfold smA at h;simp only[Bool.and_eq_true,bne_iff_ne,ne_eq,decide_eq_true_eq] at h
 obtain ⟨h1,h2⟩:=h
 obtain ⟨g1,g2,g3,g4⟩:=qBi A hs h1
 have hsm:sm (scan A A.usize 0 (qAc A) Е 0)=true:=by unfold sm;simp only[Bool.and_eq_true,decide_eq_true_eq];exact ⟨g1,UInt32.le_trans UInt32.and_le_left h2⟩
 obtain ⟨-,b99⟩:=qA1 _ hsm
 obtain ⟨f,Rf⟩:=qAm _ g1
 obtain ⟨p0,-,mono⟩:=qCs g1 g2
 refine ⟨f,?_,λk hk=>?_⟩
 . rw[Rf];exact (le_abs_self _).trans b99
 . obtain ⟨-,Rk⟩:=qAm _ (g3 k hk);rw[Rk,Rf];exact mono _ (g3 k hk) (g4 k hk)
set_option maxRecDepth 8192 in
lemma qUb (E s:ὧ) (h1:(s.toBits<<<(1:Ὠ))-loOf E<spOf (loOf E)) (h2:(s.toBits+0x0FDFFFFF)&&&0x1FFFFFFF<0x1FC00000): |R s| <=2^(126:ℤ):=by
 obtain ⟨-,h⟩:=qHy E s h1 h2
 obtain ⟨-,S,ex,fr,M,a1,a2,a3,-,hb,hl,hu⟩:=kFh _ s h
 have hhi:=(kD1 _ _ h1).2
 have hM63:M<2^63:=by clear h h1 h2 hhi;exact omS264 a2 a3 hu
 have hsh:=qJb s.toBits S M hM63 hb
 have hlt:2*M<0x8FA0000000000000:=by have:=ἴ.mp hhi;rw[hsh] at this;exact this
 have hM:M=(M/2^52)*2^52+M%2^52:=(by clear h h1 h2 hb hl hu hhi hsh hlt;exact omS114)
 have hF:M%2^52<2^52:=ὀ _ ו
 generalize M/2^52=E' at hM
 generalize M%2^52=F at hM hF
 subst hM
 have hE1:E'<=1148:=by clear h h1 h2 hb hl hu hhi hsh;exact omS357 hlt
 have hE0:1<=E':=by clear h h1 h2 hb hhi hsh hlt;exact omS238 a1 hF hl
 obtain ⟨-,hR⟩:=B_R s S E' F hE0 (by exact omS358 hE1) hF (by rw[hb,ὢ])
 rw[hR]
 have p:(0:ℝ)<2^((E':ℤ)-1075):=ו
 have q3:((2^52+F:Ὣ):ℝ)<=2^(53:ℤ):=by push_cast;have:(F:ℝ)<2^52:=by exact_mod_cast hF
                                      rw[show (2:ℝ)^(53:ℤ)=2^52+2^52 by norm_num];linarith only[this]
 have q4:(2:ℝ)^(53:ℤ)*2^((E':ℤ)-1075)<=2^(126:ℤ):=by rw[<-Ἓ ב];exact ἒ ב (by exact omS359 hE1)
 have q5:((2^52+F:Ὣ):ℝ)*2^((E':ℤ)-1075)<=2^(126:ℤ):=(ἀ q3 p.le).trans q4
 cases S<;>simp only[Bool.false_eq_true,ite_false,ite_true,one_mul,neg_mul,abs_neg,ἢ,ἡ p]<;>rw[ἅ ו]<;>exact q5
set_option maxRecDepth 8192 in
lemma qEkF (s:LnS) (M:ℚ) (mean:Unit->ℚ) (hmean:mean ()=M) (w gw bw:Ὢ) (hw:sm w=true) (hg:sm gw=true) (hb:sm bw=true) (h:L_oF s (M:ℝ))
 (xm gm bm:ὧ) (fxm:xmᶠ) (fgm:gmᶠ) (fbm:bmᶠ) (hxm: |R (dv w)| <=R xm) (hgm: |R (dv gw)| <=R gm) (hbm: |R (dv bw)| <=R bm) (bxm:R xm<=2^99) (bgm:R gm<=2^99) (bbm:R bm<=2^99):
 lnOut s (loOf (lnEB s xm gm bm)) w gw bw mean=roundRational ((ἂ w-M)*ἂ gw/ἂ s.den+ἂ bw):=by
 unfold lnOut lnOutF
 split
 . rename_i hf;obtain ⟨fm,fi,fe,e1,e2,e3,e4,gd,hD,hI⟩:=h hf
   have ew:=qAx w (qAe _ (qA1 w hw).1);have eg:=qAx gw (qAe _ (qA1 gw hg).1);have eb:=qAx bw (qAe _ (qA1 bw hb).1);have ed:=qAx s.den (qAe _ gd);have hV:(((ἂ w-M)*ἂ gw/ἂ s.den+ἂ bw:ℚ):ℝ)=(wv w-M)*wv gw/wv s.den+wv bw:=by rw[ew,eg,eb,ed];push_cast;ring
   unfold lnFa;dsimp only;split
   . split;. unfold lnSlow;rw[hmean]
     . rename_i _ hc
       have hne:ziv ((dv w-s.m)*(dv gw*s.ir)+dv bw)
        (Ἃ 0x3D10000000000000*(((dv w-s.m)*(dv gw*s.ir)).abs+(dv bw).abs)+s.em*(dv gw*s.ir).abs)≠Е:=λe0=>by
        rw[e0] at hc
        exact hc ה
       obtain ⟨l1,l2⟩:=qI0b s.m s.ir s.em w gw bw M (wv s.den) fm fi fe e1 e2 e3 e4 hD hI hw hg hb
        _ _ rfl rfl (qBw _ _ hne).2.2
       rw[<-hV] at l1 l2;exact (qH7 _ _ _ l1 l2 hne (qIu _ _ _ l1 l2 hne hc)).symm
   . rename_i hne
     have key:fin1 (loOf (lnEB s xm gm bm)) (spOf (loOf (lnEB s xm gm bm))) ((dv w-s.m)*(dv gw*s.ir)+dv bw)≠Е:=by simpa only[ne_eq,beq_iff_eq] using hne
     unfold fin1 at key⊢
     dsimp only at key⊢
     split at key
     . rename_i ht
       simp only[Bool.and_eq_true,decide_eq_true_eq] at ht
       obtain ⟨h1,h2⟩:=ht
       rw[Ὦ (by simp only[Bool.and_eq_true,decide_eq_true_eq];exact ⟨h1,h2⟩)]
       have hD0:0<wv s.den:=Ἥ ו hD
       have pir:0<R s.ir:=by
        have q:0<R s.ir*wv s.den:=Ἥ (by norm_num1:(0:ℝ)<1-1/2^48) (sub_le_iff_le_add.mpr (neg_le_sub_iff_le_add.mp (abs_le.mp hI).1))
        exact pos_of_mul_pos_left q hD0.le
       have bir:R s.ir<=2^65:=by
        have:=abs_le.mp hI;have q:R s.ir*wv s.den<=2:=by linarith only[this.2]
        have q2:R s.ir*(1/2^64)<=R s.ir*wv s.den:=ἁ hD pir.le
        rw[mul_one_div] at q2
        exact ((div_le_iff₀ (by norm_num1)).mp (q2.trans q)).trans_eq (by norm_num1)
       obtain ⟨fe',hE⟩:=qLmo s.m s.ir s.em xm gm bm w gw bw fm fi fe fxm fgm fbm hw hg hb pir bir ((by positivity:(0:ℝ)<=1/2^201).trans e2) e1 e4 hxm hgm hbm bxm bgm bbm
       have hbig:=qBgF _ _ _ h1 h2 fe' hE
       obtain ⟨l1,l2⟩:=qI0b s.m s.ir s.em w gw bw M (wv s.den) fm fi fe e1 e2 e3 e4 hD hI hw hg hb _ _ rfl rfl hbig
       rw[<-hV] at l1 l2
       obtain ⟨a,b⟩:=qHy _ _ h1 h2
       obtain ⟨fs,-,hr,hB,-,-⟩:=qI4 _ _ b
       have hVq:=qHs _ _ _ fs fe' (hE.trans hr) hB l1 l2
       have hub:=qUb _ _ h1 h2
       have hz:A_z<=1:=by show (2:ℝ)^(-1021:ℤ)<=1;exact zpow_le_one_of_nonpos₀ ב ב
       have hq127: |(((ἂ w-M)*ἂ gw/ἂ s.den+ἂ bw:ℚ):ℝ)| <=2^(127:ℤ):=by
        have t1:=Ὥ (((ἂ w-M)*ἂ gw/ἂ s.den+ἂ bw:ℚ):ℝ) (R ((dv w-s.m)*(dv gw*s.ir)+dv bw))
        rw[ἣ] at t1
        have q:(2:ℝ)^(127:ℤ)=2*2^(126:ℤ):=by rw[show (127:ℤ)=1+126 by norm_num,Ἓ ἓ];norm_num
        have q1:(1:ℝ)<=2^(126:ℤ):=one_le_zpow₀ ב ב
        rw[q];generalize (2:ℝ)^(126:ℤ)=P at*
        generalize |(((ἂ w-M)*ἂ gw/ἂ s.den+ἂ bw:ℚ):ℝ)|=Y at*
        linarith only[t1,hVq,hub,hz,q1,Ἀ (R ((dv w-s.m)*(dv gw*s.ir)+dv bw))]
       rw[<-a];exact (kHq _ _ _ hVq hq127 b).symm
     . exact absurd rfl key
 . unfold lnSlow;rw[hmean]
lemma qBp {α:Type} [Inhabited α] (rows k:ℕ) (f:ℕ->α) (hk:k<rows):(Array.ofFn (n:=rows) λr=>f r.val)[k]! =f k:=by rw[Ἕ _ k (by simpa only[Array.size_ofFn] using hk),Array.getElem_ofFn]
lemma qMf (f:Ὣ->Ὢ) (L:љ) (a:ℤ):
 L.foldl (λt j=>t+ἂ (f j)) ((a:ℚ)/2^149)=((L.foldl (λt j=>t+sv (f j)*((2^idxOf (f j):ℕ):ℤ)) a:ℤ):ℚ)/2^149:=by
 induction L generalizing a with
 | nil=>rfl
 | cons p L ih=>
  simp only[ὔ]
  rw[<-ih]
  congr 1
  have e:ἂ (f p)=(sv (f p):ℚ)*2^((idxOf (f p):ℤ)-149):=by rw[rationalWord,decodeD_toRat]
  rw[e,ᾂ (by (positivity):(2:ℚ)≠0),Ἡ]
  push_cast
  ring
lemma qMq (n:Ὣ) (x:ё) (r:Ὣ):meanQ n x r=rowMean n x r:=by
 have h:=qMf (λj=>x[r*n+j]!) (ἄ n) 0
 simp only[Int.cast_zero,zero_div] at h
 unfold meanQ;rw[meanIF_eq];unfold meanI rowMean
 rw[h]
 by_cases c:(n:ℚ)=0
 . simp only[Nat.cast_pow,Nat.cast_ofNat,Nat.reducePow,Nat.cast_mul,c,mul_zero,div_zero]
 . push_cast;rw[div_div];norm_num
lemma qGD {A:ё} {n:ὦ} (hs:ѧA<2^32) (d:Ὣ):
 ∀(t:ὦ) (o:ὤ),Ѥt+d=Ѥn->(Ѥn<=Ѩo/\Ѩo<2^32)->(∀k,k< Ѥt->o.data[k]! =dv A[k]!)->
 (Ѥn<=(decG A n t o).size/\(decG A n t o).size<2^32)/\∀k,k< Ѥn->(decG A n t o).data[k]! =dv A[k]!:=by
 induction d with
 | zero=>
  intro t o he H1 H2
  rw[decG,ἔ (by rw[А];(clear hs H1;(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›)))))]
  have e:Ѥt=Ѥn:=(by exact omS155 he)
  rw[e] at H2
  exact ⟨H1,H2⟩
 | succ d ih=>
  intro t o he H1 H2
  have ht:t<n:=by rw[А];(clear hs H1;exact omS136 he)
  have hu:=usz t n ht
  have r0:rdW A t=A[Ѥt]!:=qAg _ _ hs
  rw[decG,Ἄ ht,r0]
  refine ih (t+1) _ (by rw[hu];(clear hs H1 ht hu r0;exact omS137 he)) ⟨by rw[qD6];exact H1.1,by rw[qD6];exact H1.2⟩ ?_
  intro k hk
  rw[hu] at hk
  rw[qEt _ _ _ _ H1.2 (by have:=H1.1;have:=А.mp ht;(exact Nat.lt_of_lt_of_le ‹_› ‹_›))]
  by_cases e:k=Ѥt
  . rw[ἱ e,e]
  . rw[ἐ e];exact H2 k (by (clear hs H1 ht r0;omega))
lemma qGR {A:ё} {n:Ὣ} (hs:ѧA<2^32) (hn:n<2^32) (u:ὦ) (hu:Ѥu<n):rdF (decG A ѩn 0 (zFor n)) u=dv A[Ѥu]!:=by
 have ek:n.toUSize.toNat=n:=qAd _ (by exact Nat.succ_le_of_lt ‹_›)
 have P0:=qZf n hn
 obtain ⟨⟨D1,D2⟩,D3⟩:=qGD (A:=A) (n:=ѩn) hs n 0 (zFor n) (by rw[ek,C_z,Ἇ]) (by rw[ek];exact P0) (λk hk=>by rw[C_z] at hk;(clear hs hn hu ek P0;omega))
 rw[ek] at D1 D3
 generalize decG A ѩn 0 (zFor n)=a at D1 D2 D3⊢
 have h:u<a.usize:=by
  rw[А];have e:a.usize.toNat=Ѩa:=by
   show (a.size.toUSize).toNat=Ѩa
   exact qAd _ D2
  rw[e];(exact Nat.lt_of_lt_of_le ‹_› ‹_›)
 unfold rdF
 rw[Ἄ h,qC5,D3 _ hu]
lemma qGS {r:Ὣ} {A:ё} {n:Ὣ} (hs:ѧA<2^32) (hn:n<2^32) (u:ὦ) (hu:Ѥu<n):gdR r A (gdF r A n) u=dv A[Ѥu]!:=by
 unfold gdR gdF
 by_cases c:(r==1)=true
 . rw[ἱ c,qAg _ _ hs]
 . rw[ἐ c,ἐ c];exact qGR hs hn u hu
lemma qSm (A:ё) (hs:ѧA<2^32) (h:smA A=true):A.all sm=true/\A.all finiteWord=true:=by
 unfold smA at h
 dsimp only at h
 rw[Bool.and_eq_true] at h
 obtain ⟨h1,h2⟩:=h
 have hh:=of_decide_eq_true h2
 obtain ⟨-,-,c3,c4⟩:=qBi A hs (by simpa only[Array.usize,Nat.toUSize_eq,ne_eq,bne_iff_ne] using h1)
 constructor
 . rw[Array.all_eq_true]
   intro i hi
   have g:=c3 i hi
   have l:=c4 i hi
   rw[Ἕ A i hi] at g l
   unfold sm
   rw[Bool.and_eq_true]
   exact ⟨g,decide_eq_true (UInt32.le_trans l hh)⟩
 . rw[Array.all_eq_true]
   intro i hi
   have g:=c3 i hi
   rw[Ἕ A i hi] at g
   simp only[finiteWord,bne_iff_ne,ne_eq,qAe _ g,not_false_eq_true]
lemma qEq0 (w:ὥ) (h1:signOf w=false) (h2:0< mantOf w):0<rationalWord w:=by
 rw[rationalWord,decodeD_toRat]
 have s:sv w=(mantOf w:ℤ):=by simp only[sv,h1,false_eq_true, ↓reduceIte]
 rw[s]
 have m:(0:ℚ)<(mantOf w:ℚ):=by exact_mod_cast h2
 push_cast
 exact mul_pos m (ἵ (by (positivity)) _)
lemma qNO {rows n:Ὣ} {eps:Ὢ} {x g:ё} (h:normOK rows n eps x g=true):stableNormDomain rows n eps x g=true/\x.all sm=true/\g.all sm=true/\sm eps=true/\n<2^20/\rows*n<2^32:=by
 unfold normOK at h
 simp only[Bool.and_eq_true,decide_eq_true_eq,beq_iff_eq,Bool.not_eq_true'] at h
 obtain ⟨⟨⟨⟨⟨⟨⟨⟨⟨⟨h1,h2⟩,h3⟩,h4⟩,h5⟩,h6⟩,h7⟩,h8⟩,h9⟩,h10⟩,h11⟩:=h
 replace h6:=Nat.lt_of_le_of_lt h6 (show 4294967295<2^32 by norm_num)
 obtain ⟨a1,a2⟩:=qSm x (by (clear h11 h5 h4 h1 h2;exact omS360 h6 h3)) h7
 obtain ⟨b1,b2⟩:=qSm g (by (clear h11 h6 h3 h1;exact omS361 h5 h4)) h8
 have e0:=qEq0 eps h10 h11
 refine ⟨?_,a1,b1,h9,h5,h6⟩
 unfold stableNormDomain
 simp only[Bool.and_eq_true,beq_iff_eq,gt_iff_lt,decide_eq_true_eq]
 exact ⟨⟨⟨⟨⟨⟨⟨h1,h2⟩,h3⟩,h4⟩,a2⟩,b2⟩,by simp only[finiteWord,bne_iff_ne,ne_eq,qAe _ (qA1 eps h9).1,not_false_eq_true]⟩,e0⟩
lemma lnFa_eq (s:LnS) (h:s.fast=true) (lo:Ὠ) (w gw bw:Ὢ) (gv bv:ὧ) (mean:Unit->ℚ):lnOutF s lo w gw bw gv bv mean=lnFa s.m s.ir s.em s.den w gw bw gv bv lo mean:=by
 unfold lnOutF;rw[ἱ h]
lemma lnRF_eq {rows:Ὣ} {x g b:ё} {G B G0 B0:ὤ} {s:LnS} {mean:Unit->ℚ} {lo:Ὠ} {base n:ὦ} (hf:s.fast=true)
 (hG:∀u:ὦ,Ѥu<Ѥn->rdF G u=gdR rows g G0 u) (hB:∀u:ὦ,Ѥu<Ѥn->rdF B u=gdR rows b B0 u) (k:Ὣ):
 ∀(j:ὦ) (o:ё),Ѥj+k=Ѥn->lnRF x g b G B s.m s.ir s.em s.den mean lo (spOf lo) base n j o=
  fC (λs p j=>lnOutF s.1 s.2.2 (rdW x p) (rdW g j) (rdW b j) (gdR rows g G0 j) (gdR rows b B0 j) s.2.1) (s,mean,lo) base j o k:=by
 induction k with
 | zero=>
  intro j o e
  rw[lnRF,ἔ (by rw[А];(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›)))),fC]
 | succ k ih=>
  intro j o e
  have hj:j<n:=А.2 (by exact omS136 e)
  rw[lnRF,Ἄ hj,fC,ih (j+1) _ (by rw[usz j n hj];(clear hj;exact omS137 e))]
  dsimp only
  rw[lnFa_eq s hf lo,hG j (by (exact Nat.succ_le_of_lt ‹_›)),hB j (by (exact Nat.succ_le_of_lt ‹_›))]
  try rfl
lemma lnRS_eq {rows:Ὣ} {x g b:ё} {G0 B0:ὤ} {s:LnS} {mean:Unit->ℚ} {lo:Ὠ} {base n:ὦ} (hf:s.fast=false) (k:Ὣ):
 ∀(j:ὦ) (o:ё),Ѥj+k=Ѥn->lnRS x g b s.den mean base n j o=
  fC (λs p j=>lnOutF s.1 s.2.2 (rdW x p) (rdW g j) (rdW b j) (gdR rows g G0 j) (gdR rows b B0 j) s.2.1) (s,mean,lo) base j o k:=by
 have e:∀(w gw bw:Ὢ) (gv bv:ὧ),lnOutF s lo w gw bw gv bv mean=lnSlow w gw bw s.den mean:=by
  intro w gw bw gv bv;unfold lnOutF;rw[ἐ (by rw[hf];decide)]
 induction k with
 | zero=>
  intro j o e
  rw[lnRS,ἔ (by rw[А];(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›)))),fC]
 | succ k ih=>
  intro j o e'
  have hj:j<n:=А.2 (by exact omS136 e')
  rw[lnRS,Ἄ hj,fC,ih (j+1) _ (by rw[usz j n hj];(clear hj;exact omS137 e'))]
  dsimp only
  rw[e]
lemma lnRows_eq {rows n:Ὣ} {x g b:ё} {G B G0 B0:ὤ} {eps:Ὢ} {xm gm bm:ὧ} (hn:n<2^32)
 (hG:∀u:ὦ,Ѥu<n->rdF G u=gdR rows g G0 u) (hB:∀u:ὦ,Ѥu<n->rdF B u=gdR rows b B0 u) (d:Ὣ):
 ∀(r:Ὣ) (o:ё),lnRows x g b G B eps xm gm bm n r o d=fR (λr=>(lnStat false n eps x r,λ_:Unit=>meanQ n x r,loOf (lnEB (lnStat false n eps x r) xm gm bm)))
  (λs p j=>lnOutF s.1 s.2.2 (rdW x p) (rdW g j) (rdW b j) (gdR rows g G0 j) (gdR rows b B0 j) s.2.1) n r o d:=by
 have hu:(ѩn:ὦ).toNat=n:=qAd n hn
 induction d with
 | zero=>intro r o;rfl
 | succ d ih=>
  intro r o
  rw[lnRows,fR,ih]
  try dsimp only
  congr 1
  split
  . rename_i hf
    exact lnRF_eq hf (λu h=>hG u (by rw[hu] at h;exact h)) (λu h=>hB u (by rw[hu] at h;exact h)) n 0 o (by rw[hu];simp only[USize.toNat_zero,zero_add])
  . rename_i hf
    exact lnRS_eq (by simpa only[Bool.not_eq_true] using hf) n 0 o (by rw[hu];simp only[USize.toNat_zero,zero_add])
lemma lnDecB_dat (x:ё) (base n:ὦ) (hs:Ѥbase+Ѥn<2^32) (hxs:ѧx<2^32) (k:Ὣ):
 ∀(j:ὦ) (buf:ὤ) (t s:ὧ),Ѥn+1< Ѩbuf-> Ѩbuf<2^32-> Ѥj+k=Ѥn-> ∀q,q< Ѥn->(lnDecB x base n j buf t s).data[q]! =if Ѥj<=q then dv x[Ѥbase+q]! else buf.data[q]!:=by
 induction k with
 | zero=>
  intro j buf t s hb hbs e q hq
  have e1:=uS1 n א
  rw[lnDecB,ἔ (by rw[А];exact omS404 e),qEt _ _ _ _ (by rw[qD6];exact hbs) (by rw[qD6,e1];exact hb),ἐ (by rw[e1];omega),qEt _ _ _ _ hbs א,ἐ א,ἐ א]
 | succ k ih=>
  intro j buf t s hb hbs e q hq
  have hj:j<n:=А.2 (by exact omS136 e)
  have hj1:=А.mp hj
  rw[lnDecB,Ἄ hj];dsimp only
  rw[ih (j+1) _ _ _ (by rw[qD6];exact hb) (by rw[qD6];exact hbs) (by rw[usz j n hj];exact omS137 e) q hq,qEt _ _ _ _ hbs א,usz j n hj,qAg _ _ hxs,qAf base j א]
  by_cases h1:q=Ѥj
  . subst h1;rw[ἐ א,ἱ rfl,ἱ א]
  . by_cases h2:Ѥj+1<=q
    . rw[ἱ h2,ἱ א]
    . rw[ἐ h2,ἐ h1,ἐ א]
lemma lnDecB_t (x:ё) (base n:ὦ) (k:Ὣ):
 ∀(j:ὦ) (buf:ὤ) (s t:ὧ),Ѥn+1< Ѩbuf-> Ѩbuf<2^32-> Ѥj+k=Ѥn->(lnDecB x base n j buf t s).data[Ѥn]! =(lnSum x base n j s t).t:=by
 induction k with
 | zero=>
  intro j buf s t hb hbs e
  have e1:=uS1 n א
  rw[lnDecB,lnSum,ἔ (by rw[А];exact omS404 e),ἔ (by rw[А];exact omS404 e),qEt _ _ _ _ (by rw[qD6];exact hbs) (by rw[qD6,e1];exact hb),ἐ (by rw[e1];omega),qEt _ _ _ _ hbs א,ἱ rfl]
 | succ k ih=>
  intro j buf s t hb hbs e
  have hj:j<n:=А.2 (by exact omS136 e)
  rw[lnDecB,lnSum,Ἄ hj,Ἄ hj];dsimp only
  exact ih (j+1) _ _ _ (by rw[qD6];exact hb) (by rw[qD6];exact hbs) (by rw[usz j n hj];exact omS137 e)
lemma lnDecB_s (x:ё) (base n:ὦ) (k:Ὣ):
 ∀(j:ὦ) (buf:ὤ) (s t:ὧ),Ѥn+1< Ѩbuf-> Ѩbuf<2^32-> Ѥj+k=Ѥn->(lnDecB x base n j buf t s).data[Ѥn+1]! =(lnSum x base n j s t).s:=by
 induction k with
 | zero=>
  intro j buf s t hb hbs e
  have e1:=uS1 n א
  rw[lnDecB,lnSum,ἔ (by rw[А];exact omS404 e),ἔ (by rw[А];exact omS404 e),qEt _ _ _ _ (by rw[qD6];exact hbs) (by rw[qD6,e1];exact hb),ἱ (by rw[e1])]
 | succ k ih=>
  intro j buf s t hb hbs e
  have hj:j<n:=А.2 (by exact omS136 e)
  rw[lnDecB,lnSum,Ἄ hj,Ἄ hj];dsimp only
  exact ih (j+1) _ _ _ (by rw[qD6];exact hb) (by rw[qD6];exact hbs) (by rw[usz j n hj];exact omS137 e)
lemma lnVarB_eq (x:ё) (buf:ὤ) (base n:ὦ) (hb:Ѥn<=Ѩbuf) (hs:Ѥbase+Ѥn<2^32) (hxs:ѧx<2^32) (hc:∀q,q< Ѥn->buf.data[q]! =dv x[Ѥbase+q]!) (m:ὧ) (k:Ὣ):
 ∀(j:ὦ) (v:ὧ),Ѥj+k=Ѥn->lnVarB buf n j m v hb=lnVar x base n j m v:=by
 induction k with
 | zero=>
  intro j v e
  rw[lnVarB,lnVar,ἔ (by rw[А];exact omS404 e),ἔ (by rw[А];exact omS404 e)]
 | succ k ih=>
  intro j v e
  have hj:j<n:=А.2 (by exact omS136 e)
  have hj1:=А.mp hj
  rw[lnVarB,lnVar,Ἄ hj,Ἄ hj]
  dsimp only
  rw[qC5,hc _ hj1,qAg _ _ hxs,qAf base j א,ih (j+1) _ (by rw[usz j n hj];exact omS137 e)]
lemma lnStatB_eq (n:Ὣ) (eps:Ὢ) (x:ё) (r:Ὣ) (buf:ὤ) (hb:n+2<=Ѩbuf) (hbs:Ѩbuf<2^32) (hs:r*n+n<2^32) (hxs:ѧx<2^32)
 (hc:∀q,q<n->buf.data[q]! =dv x[r*n+q]!) (ht:buf.data[n]! =(lnSum x ((r*n).toUSize) ѩn 0 0 0).t)
 (hs2:buf.data[n+1]! =(lnSum x ((r*n).toUSize) ѩn 0 0 0).s):
 lnStatB n eps x r (oneF/n.toUInt64.toFloat) (invOK (oneF/n.toUInt64.toFloat) n.toUInt64.toFloat) buf hb hbs=lnStat false n eps x r:=by
 have e1:Ѥ(ѩn)=n:=qAd n א
 have e2:Ѥ((r*n).toUSize)=r*n:=qAd _ א
 have hs':Ѥ((r*n).toUSize)+Ѥ(ѩn)<2^32:=by rw[e1,e2];exact hs
 have hc':∀q,q< Ѥ(ѩn)->buf.data[q]! =dv x[Ѥ((r*n).toUSize)+q]!:=λq h=>by rw[e2];exact hc q (by rw[e1] at h;exact h)
 have e3:Ѥ(ѩ(n+1))=n+1:=qAd (n+1) א
 unfold lnStatB lnStat oneF
 simp only[Bool.false_eq_true,ite_false,qC5,e1,e3,ht,hs2]
 rw[lnVarB_eq x buf _ _ (by rw[e1];omega) hs' hxs hc' _ (Ѥ(ѩn)) 0 0 (by simp only[USize.toNat_zero,Nat.toUSize_eq,USize.toNat_ofNat',zero_add])]
lemma lnRB_eq (x g b:ё) (buf G B:ὤ) (m ir em:ὧ) (den:Ὢ) (nN rR:Ὣ) (lo sp:Ὠ) (base n:ὦ)
 (hb:Ѥn<=Ѩbuf) (hG:Ѥn<=ѨG) (hB:Ѥn<=ѨB) (hs:Ѥbase+Ѥn<2^32) (hxs:ѧx<2^32) (hGs:ѨG<2^32) (hBs:ѨB<2^32)
 (hc:∀q,q< Ѥn->buf.data[q]! =dv x[Ѥbase+q]!) (k:Ὣ):
 ∀(j:ὦ) (o:ё),Ѥj+k=Ѥn->lnRB x g b buf G B m ir em den nN rR lo sp base n j o hb hG hB=lnRF x g b G B m ir em den (λ_=>meanQ nN x rR) lo sp base n j o:=by
 induction k with
 | zero=>
  intro j o e
  rw[lnRB,lnRF,ἔ (by rw[А];exact omS404 e),ἔ (by rw[А];exact omS404 e)]
 | succ k ih=>
  intro j o e
  have hj:j<n:=А.2 (by exact omS136 e)
  have hj1:=А.mp hj
  have p3:j<G.usize:=by rw[А,FloatArray.usize,qAd _ hGs];exact omS144 hG e
  have p4:j<B.usize:=by rw[А,FloatArray.usize,qAd _ hBs];exact omS144 hB e
  have ev:lnFvM m ir em den (rdW x (base+j)) (buf.uget j (by exact omS144 hb e)) g b j (G.uget j (by exact omS144 hG e)) (B.uget j (by exact omS144 hB e)) lo sp nN x rR=lnFb m ir em den x g b (base+j) j (rdF G j) (rdF B j) lo sp (λ_=>meanQ nN x rR):=by
   unfold lnFb lnFvM lnSlowM rdF
   dsimp only
   rw[Ἄ p3,Ἄ p4,qC5,hc _ hj1,qAg x _ hxs,qAf base j א]
  rw[lnRB,lnRF,Ἄ hj,Ἄ hj,ev,ih (j+1) _ (by rw[usz j n hj];exact omS137 e)]
lemma lnRowsB_eq (x g b:ё) (G B:ὤ) (eps:Ὢ) (xm gm bm:ὧ) (n:Ὣ) (d:Ὣ):
 ∀(r:Ὣ) (buf:ὤ) (o:ё),lnRowsB x g b G B eps xm gm bm n (oneF/n.toUInt64.toFloat) (invOK (oneF/n.toUInt64.toFloat) n.toUInt64.toFloat) r buf o d=lnRows x g b G B eps xm gm bm n r o d:=by
 induction d with
 | zero=>intro r buf o;rfl
 | succ d ih=>
  intro r buf o
  rw[lnRowsB,lnRows]
  simp only[lnRSM,lnRFM]
  split
  . rename_i h
    obtain ⟨h1,h2,h3,h3',h4,h4',h5,h6⟩:=h
    have e1:Ѥ(ѩn)=n:=qAd n א
    have e2:Ѥ((r*n).toUSize)=r*n:=qAd _ א
    have hbs:Ѩbuf<2^32:=א
    have hxs:ѧx<2^32:=א
    have hs':Ѥ((r*n).toUSize)+Ѥ(ѩn)<2^32:=by rw[e1,e2];omega
    have hc:∀q,q<n->(lnDecB x ((r*n).toUSize) ѩn 0 buf 0 0).data[q]! =dv x[r*n+q]!:=λq hq=>by
     rw[lnDecB_dat x _ _ hs' hxs (Ѥ(ѩn)) 0 buf 0 0 (by rw[e1];omega) hbs (by simp only[USize.toNat_zero,Nat.toUSize_eq,USize.toNat_ofNat',zero_add]) q (by rw[e1];exact hq),ἱ (by simp only[USize.toNat_zero,zero_le]),e2]
    have hc':∀q,q< Ѥ(ѩn)->(lnDecB x ((r*n).toUSize) ѩn 0 buf 0 0).data[q]! =dv x[Ѥ((r*n).toUSize)+q]!:=λq hq=>by rw[e2];exact hc q (by rw[e1] at hq;exact hq)
    have ht:(lnDecB x ((r*n).toUSize) ѩn 0 buf 0 0).data[n]! =(lnSum x ((r*n).toUSize) ѩn 0 0 0).t:=by
     have:=lnDecB_t x ((r*n).toUSize) ѩn (Ѥ(ѩn)) 0 buf 0 0 (by rw[e1];omega) hbs (by simp only[USize.toNat_zero,Nat.toUSize_eq,USize.toNat_ofNat',zero_add]);rw[e1] at this;exact this
    have hs2:(lnDecB x ((r*n).toUSize) ѩn 0 buf 0 0).data[n+1]! =(lnSum x ((r*n).toUSize) ѩn 0 0 0).s:=by
     have:=lnDecB_s x ((r*n).toUSize) ѩn (Ѥ(ѩn)) 0 buf 0 0 (by rw[e1];omega) hbs (by simp only[USize.toNat_zero,Nat.toUSize_eq,USize.toNat_ofNat',zero_add]);rw[e1] at this;exact this
    simp only[lnDecC_eq]
    try dsimp only
    rw[lnStatB_eq n eps x r _ (by rw[lnDecB_sz0];omega) (by rw[lnDecB_sz0];omega) א hxs hc ht hs2,ih]
    congr 1
    split
    . rw[lnRB_eq x g b _ G B _ _ _ _ _ _ _ _ _ ѩn _ _ _ hs' hxs א א hc' (Ѥ(ѩn)) 0 o (by simp only[USize.toNat_zero,Nat.toUSize_eq,USize.toNat_ofNat',zero_add])]
    . rfl
  . rw[ih]
lemma rmsRF_eq {rows:Ὣ} {x g:ё} {G:ὤ} {s:LnS} {lo:Ὠ} {base n:ὦ} (hf:s.fast=true) (k:Ὣ):
 ∀(j:ὦ) (o:ё),Ѥj+k=Ѥn->rmsRF rows x g G s.m s.ir s.em s.den lo (spOf lo) base n j o=
  fC (λs p j=>let w:=rdW x p
       if w&&&В==0 then 0 else lnOutF s.1 s.2 w (rdW g j) 0 (gdR rows g G j) (dv 0) λ_=>0) (s,lo) base j o k:=by
 induction k with
 | zero=>
  intro j o e
  rw[rmsRF,ἔ (by rw[А];(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›)))),fC]
 | succ k ih=>
  intro j o e
  have hj:j<n:=А.2 (by exact omS136 e)
  rw[rmsRF,Ἄ hj,fC,ih (j+1) _ (by rw[usz j n hj];(clear hj;exact omS137 e))]
  dsimp only
  rw[lnFa_eq s hf lo]
  try rfl
lemma rmsRows_eq {rows n:Ὣ} {x g:ё} {G:ὤ} {eps:Ὢ} {xm gm:ὧ} (hn:n<2^32) (d:Ὣ):
 ∀(r:Ὣ) (o:ё),rmsRows rows x g G eps xm gm n r o d=fR (λr=>(lnStat true n eps x r,loOf (lnEB (lnStat true n eps x r) xm gm (dv 0))))
  (λs p j=>let w:=rdW x p
       if w&&&В==0 then 0 else lnOutF s.1 s.2 w (rdW g j) 0 (gdR rows g G j) (dv 0) λ_=>0) n r o d:=by
 have hu:(ѩn:ὦ).toNat=n:=qAd n hn
 induction d with
 | zero=>intro r o;rfl
 | succ d ih=>
  intro r o
  rw[rmsRows,fR,ih]
  try dsimp only
  congr 1
  split
  . rename_i hf
    exact rmsRF_eq hf n 0 o (by rw[hu];simp only[USize.toNat_zero,zero_add])
  . rfl
lemma qGc:lnF=layernormSpec:=by
 funext rows n eps x g b;unfold lnF;rw[(lnGet n g b).2];unfold lnFq;try simp only[]
 simp only[lnRowsB_eq]
 split
 . rename_i hc;simp only[Bool.and_eq_true,beq_iff_eq] at hc;obtain ⟨⟨h0',h2⟩,hb'⟩:=hc;have h0:normOK rows n eps x g=true:=h0';have hb:smA b=true:=hb';obtain ⟨hD,h4,h5,h7,h8,h9⟩:=qNO h0;obtain ⟨h6,h3⟩:=qSm b (by (clear h9;exact omS362 h2 h8)) hb;have hs:(stableNormDomain rows n eps x g&& ѧb==n&&b.all finiteWord)=true:=by simp only[Bool.and_eq_true,beq_iff_eq];exact ⟨⟨hD,h2⟩,h3⟩
   unfold layernormSpec layernormWith;rw[ἱ hs];unfold layernormStable;dsimp only;unfold stableNormDomain at hD;simp only[Bool.and_eq_true,beq_iff_eq,gt_iff_lt,decide_eq_true_eq] at hD;obtain ⟨⟨⟨⟨⟨⟨⟨h10,h11⟩,h12⟩,h13⟩,-⟩,-⟩,-⟩,h14⟩:=hD
   have hsx:smA x=true:=by unfold normOK at h0;simp only[Bool.and_eq_true] at h0;exact h0.1.1.1.1.2
   have hsg:smA g=true:=by unfold normOK at h0;simp only[Bool.and_eq_true] at h0;exact h0.1.1.1.2
   obtain ⟨fxm,bxm,hxm⟩:=qSmx x (by rw[h12];exact h9) hsx
   obtain ⟨fgm,bgm,hgm⟩:=qSmx g (by rw[h13];(clear h9 h12 h10;exact omS363 h8)) hsg
   obtain ⟨fbm,bbm,hbm⟩:=qSmx b (by rw[h2];(clear h9 h12 h10;exact omS363 h8)) hb
   rw[lnRows_eq (G0:=gdF rows g n) (B0:=gdF rows b n) (by (exact Nat.lt_of_lt_of_le ‹_› ה)) (λu hu=>(qGR (by (clear h9 h12 h10 hu;omega)) (by (exact Nat.lt_of_lt_of_le ‹_› ה)) u hu).trans (qGS (by (clear h9 h12 h10 hu;omega)) (by (exact Nat.lt_of_lt_of_le ‹_› ה)) u hu).symm) (λu hu=>(qGR (by (clear h9 h12 h10 hu;exact omS362 h2 h8)) (by (exact Nat.lt_of_lt_of_le ‹_› ה)) u hu).trans (qGS (by (clear h9 h12 h10 hu;exact omS362 h2 h8)) (by (exact Nat.lt_of_lt_of_le ‹_› ה)) u hu).symm)]
   simp only[qMq]
   rw[qDb (λr c q=>lnOut (lnStat false n eps x r) (loOf (lnEB (lnStat false n eps x r) (dv (scan x x.usize 0 (qAc x) Е 0)) (dv (scan g g.usize 0 (qAc g) Е 0)) (dv (scan b b.usize 0 (qAc b) Е 0)))) x[q]! g[c]! b[c]! λ_=>rowMean n x r) h9 h11 λr p u hr hp hu=>by
    rw[qGS (by clear h9 h12 h10 hr hp hu;omega) (by exact Nat.lt_of_lt_of_le ‹_› ה) u hu,qGS (by clear h9 h12 h10 hr hp hu;exact omS362 h2 h8) (by exact Nat.lt_of_lt_of_le ‹_› ה) u hu]
    repeat rw[qAg _ _ (by clear h10 hr hp hu;omega)]
    rfl]
   apply Մ ג;intro q h1 h1';simp only[Array.getElem_ofFn,Array.size_ofFn] at h1 h1'⊢;have hr:q/n<rows:=Ἰ (by rwa[ἥ]);have hj:q%n<n:=ὀ _ h11;have e2:=qBp rows (q/n) (λr=>rowMean n x r) hr
   have e3:=qBp rows (q/n)
    (λr=>sqrtRational (rowVariance n x r ((Array.ofFn (n:=rows) λr=>rowMean n x r.val)[r]!)+ἂ eps)) hr
   beta_reduce at e2 e3;rw[e3,e2];obtain ⟨o1,o2⟩:=qE1 false rows n eps x (q/n) hr h11 h8 h12 h9 h4 h7 h14;rw[qEkF _ (rowMean n x (q/n)) _ rfl _ _ _ (qBg x sm h4 q (by clear h2 h8 h9 h13 h10 h11 h1' hr hj e3 o1;exact omS140 h12 h1)) (qBg g sm h5 _ (by clear h2 h8 h9 h12 h10 h11 h1 h1' hr e3 o1;exact omS140 h13 hj)) (qBg b sm h6 _ (by clear h8 h9 h12 h10 h11 h1 h1' hr e3 o1;omega)) o2 _ _ _ fxm fgm fbm (hxm q (by clear h2 h8 h9 h13 h10 h11 h1' hr hj e3 o1;omega)) (hgm _ (by clear h2 h8 h9 h12 h10 h11 h1 h1' hr e3 o1;omega)) (hbm _ (by clear h8 h9 h12 h10 h11 h1 h1' hr e3 o1;exact omS140 h2 hj)) bxm bgm bbm,o1];rfl
 . rfl
lemma qFf:rmsF=rmsnormSpec:=by
 funext rows n eps x g;unfold rmsF;rw[(rnGet g).2];unfold rmsFz;split
 . rename_i hc';have hc:normOK rows n eps x g=true:=hc';obtain ⟨hD,h4,h5,h7,h8,h9⟩:=qNO hc;unfold rmsnormSpec rmsnormWith;rw[ἱ hD];unfold rmsnormStable;dsimp only;unfold stableNormDomain at hD;simp only[Bool.and_eq_true,beq_iff_eq,gt_iff_lt,decide_eq_true_eq] at hD;obtain ⟨⟨⟨⟨⟨⟨⟨h10,h11⟩,h12⟩,h13⟩,-⟩,-⟩,-⟩,h14⟩:=hD
   have hsx:smA x=true:=by unfold normOK at hc;simp only[Bool.and_eq_true] at hc;exact hc.1.1.1.1.2
   have hsg:smA g=true:=by unfold normOK at hc;simp only[Bool.and_eq_true] at hc;exact hc.1.1.1.2
   obtain ⟨fxm,bxm,hxm⟩:=qSmx x (by rw[h12];exact h9) hsx
   obtain ⟨fgm,bgm,hgm⟩:=qSmx g (by rw[h13];(clear h9 h12 h10;exact omS363 h8)) hsg
   obtain ⟨f0,R0⟩:=qAm 0 ה
   have z0:R (dv 0)=0:=by rw[R0,qAx 0 (qAe _ ה)];exact_mod_cast qE6 0 ה rfl
   rw[rmsRows_eq (by (exact Nat.lt_of_lt_of_le ‹_› ה))]
   rw[qDb (λr c q=>if x[q]!&&&В==0 then 0 else lnOut (lnStat true n eps x r) (loOf (lnEB (lnStat true n eps x r) (dv (scan x x.usize 0 (qAc x) Е 0)) (dv (scan g g.usize 0 (qAc g) Е 0)) (dv 0))) x[q]! g[c]! 0 λ_=>0) h9 h11 λr p u hr hp hu=>by
    rw[qGS (by clear h9 h12 h10 hr hp hu;omega) (by exact Nat.lt_of_lt_of_le ‹_› ה) u hu]
    repeat rw[qAg _ _ (by clear h10 hr hp hu;omega)]
    rfl]
   apply Մ ג;intro q h1 h1';simp only[Array.getElem_ofFn,Array.size_ofFn] at h1 h1'⊢;have hr:q/n<rows:=Ἰ (by rwa[ἥ]);have hj:q%n<n:=ὀ _ h11;have e3:=qBp rows (q/n) (λr=>sqrtRational (rowVariance n x r 0+ἂ eps)) hr;beta_reduce at e3;rw[e3];have hw:=qBg x sm h4 q (by clear h8 h9 h13 h10 h11 h1' hr hj e3;exact omS140 h12 h1);split;. rename_i h0;rw[qE6 _ (qA1 _ hw).1 (by simpa only[beq_iff_eq] using h0),zero_mul,zero_div];rfl
   . obtain ⟨o1,o2⟩:=qE1 true rows n eps x (q/n) hr h11 h8 h12 h9 h4 h7 h14;rw[qEkF _ 0 _ rfl _ _ _ hw (qBg g sm h5 _ (by clear h8 h9 h12 h10 h11 h1 h1' hr e3 o1;exact omS140 h13 hj)) ה o2 _ _ _ fxm fgm f0 (hxm q (by clear h8 h9 h13 h10 h11 h1' hr hj e3 o1;omega)) (hgm _ (by clear h8 h9 h12 h10 h11 h1 h1' hr e3 o1;omega)) (by rw[z0,abs_zero]) bxm bgm (by rw[z0];norm_num),o1,qE6 0 ה rfl,sub_zero,add_zero];rfl
 . rfl
end
end D
namespace C
open ExactRuntime FloatLib.Floats.Formats.BinaryInterchange FloatLib.Floats FloatLib.Floats.ExecFloat.Binary
open FloatLib.Numerics.Dyadic (signedSignificand toRat_eq_zero_iff add_toRat mul_toRat mulFields)
open UInt64 (toNat_sub_of_le)
open Bool (false_eq_true and_eq_true or_eq_true)
open USize (toNat)
open UInt32 (toNat)
open UInt64 (toNat)
open Array (size)
open FloatArray (size)
open Nat (toUSize)
open UInt32 (toUSize)
open Submission.D renaming Acc → Acc, Acc.total → Acc.total, Acc.add → Acc.add, dotStep → dotStep, dotLoop → dotLoop, Y_low → Y_low, dotFn → dotFn, rdW → rdW, wrF → wrF, wrW → wrW, pwEntry → pwEntry, pw → pw, dv → dv, dvB → dvB, good → good, scan → scan, scanF → scanF, pushZ → pushZ, zivW → zivW, ziv → ziv, F1 → F1, F4 → F4, go4B → go4B, go1 → go1, R → R, wv → wv, exactS → exactS, nfOf → nfOf, A_M → A_M, A_z → A_z, A_B → A_B, A_bits → A_bits, A_chk → A_chk, A_mag → A_mag, A_mf → A_mf, A_fsum → A_fsum, A_sa → A_sa, A_sr → A_sr, A_G → A_G, B_D → B_D, B_near → B_near, C_Inv → C_Inv, C_Hv → C_Hv, C_keep → C_keep, qAb → qAb, qEb → qAy, qGw → qBi, qEg → qEk, Acc.add_spec → Acc.add_spec, qCb → qCd, qC6 → qC0, qD3 → qEm, qHd → qH1, qFr → qGq, qIo → qJz, qAi → qAg, usz → usz, qAf → qAv, uk → uk, qDo → qDm, qBd → qAp, qDx → qDy, qGe → qG8, qCh → qB4, qAc → qAd, wi → wi, qWm → qJu, qAj → qAj, qAy → qBy, A_R → A_R, qAh → qAl, qA3 → qAu, qA0 → qBf, qCd → qB6, qDv → qD3, qH1 → qIz, qB0 → qDk, A_E → A_E, qEd → qD7, qAr → qAn, A_p → A_p, qAw → qB3, qI1 → qJ0, qCc → qBl, qAn → qAe, qE7 → qEo, qE3 → qFj, qDe → qDi, qFp → qIw, qFc → qF9, qEe → qEp, qBf → qA4, qDf → qC9, qCf → qB8, qA2 → qAt, qBk → qBj, qE0 → qE3, qBl → qCx, qCl → qCo, qCk → qDz, qHg → qJ1, qCu → qBb, qEm → qFk, qDa → qCm, qCw → qEu, qCm → qBu, qDc → qC2, qAq → qA0, qCo → qBt, qD8 → qEc, qCa → qAx, qGl → qEx, qHa → qH3, qAz → qBd, qAe → qAc, qAs → qA3, qA7 → qBe, qFb → qFs, qA6 → qD8, qD1 → qI2, qAm → qAs, qC4 → qC6, qCe → qB2, qGr → qHt, qC7 → qDc, qBo → qJm, qDd → qCy, qBz → qDj, qB7 → qB5, qIn → qJp, qGb → qGx, qEv → qHm, qC9 → qDb, qB5 → qCz, qAo → qAh, qBr → qA8, qHc → qEv, qBs → qCq, qFu → qGf, qFj → qH9, qCr → qCj, qHh → qH8, qHf → qId, qCs → qDw, qEz → qFg, qDi → qDe, qBm → qCw, qFq → qGo, B_z → B_z, qDg → qC8, qEa → qEs, qEf → qEf, qBb → qA5, qBn → qBk, qE4 → qHj, qFs → qIr, B_R → B_R, B_x → B_x, qHw → qEl, qEs → qCg, qB6 → qCa, qA4 → qAk, qBw → qCp, qAx → qA6, qEl → qE8, qEc → qBa, qIg → qGk, qHv → qGh, qB8 → qDd, qG5 → qH6, qEo → qDh, qIq → qEa, qCn → qD4, qDn → qDq, C_z → C_z, qAd → qAa, qBe → qBs, qAg → qAo, qBh → qAm, qFo → qGi, qBu → qAq, qAk → qF2, qA5 → qBg, qBv → qCe, qEn → qFl, qHu → qIs, qH2 → qI7, qB2 → qDt, qCx → qCt, qFa → qF1, qHj → qIa, qBi → qBh, qF6 → qGy, qBt → qBo, qFe → qFr, qGi → qHf, qF3 → qHe, qB1 → qDl, qCj → qDf, qBc → qBv, qC5 → qIg, qC3 → qFx, qD6 → qB9, qEt → qCr, qBj → qCb, qDp → qDs, qD5 → qEb, qAu → qA2, qDy → qG6, qAp → qBn, qDs → qG5, qIm → qFm, qD2 → qEt, qFh → qFn, qC1 → qGl, qC2 → qIu, qHz → qI6, qBg → qI1, qE6 → qJk, Acc.pos → Acc.pos, Acc.neg → Acc.neg, Acc.special → Acc.special, Acc.negZero → Acc.negZero, Acc.mk → Acc.mk, F1.s → F1.s, F1.t → F1.t, F1.mk → F1.mk, F4.a → F4.a, F4.b → F4.b, F4.c → F4.c, F4.d → F4.d, F4.mk → F4.mk, zivT → zivT, thOf → thOf, zivC → zivC, loOf → loOf, spOf → spOf, spR → spR, qI8 → qI8, qJh → qJh, kFh → qFh, qI5 → qI5, qJi → qJi, kHv → qHv, qJc → qJc, qGj → qGj, qJb → qJb, kGw → qGw, qHs → qHs, U_w → U_w, qI4 → qI4, qJr → qJr, kD1 → qD1, qIj → qIj, qHy → qHy
section
open FloatLib.Numerics
end
lemma qAw (w:Ὢ):toWord (ofW w)=w:=by apply ἠ.mp;simp only[toWord,ofW,Model.toNatBits,ofUInt32_bits,ὡ];exact Ἁ w.toBitVec.isLt
@[inline] def keyOf (w:Ὢ):Int:=
 if signOf w then-((mantOf w<<<idxOf w:Ὣ):Int) else ((mantOf w<<<idxOf w:Ὣ):Int)
lemma qGb (w:Ὢ):keyOf w=sv w*2^idxOf w:=by unfold keyOf sv;split<;>simp only[ὁ,Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat,neg_mul]
lemma qAf (w:Ὢ):(decodeD w).toReal=(keyOf w:ℝ)*2^(-149:Int):=by rw[<-Ἠ,decodeD_toRat,qGb];push_cast;rw[ᾂ ἓ,Ἡ];ring
@[inline] def maxW (a b:Ὢ):Ὢ:=
 if expOf a=255\/expOf b=255 then toWord (Model.maximum (ofW a) (ofW b))
 else match compare (keyOf a) (keyOf b) with
  | .lt=>b
  | .gt=>a
  | .eq=>toWord (if Model.isZero (ofW a)&&Model.isZero (ofW b) then
   Model.zero Fmt (Model.signBit (ofW a)&&Model.signBit (ofW b)) else ofW a)
lemma qDo (a b:Ὢ):Model.cmpDyadic (decodeD a) (decodeD b)=compare (keyOf a) (keyOf b):=by
 have hs:(0:ℝ)<2^(-149:Int):=ἵ two_pos _;cases h:compare (keyOf a) (keyOf b);. rw[Model.cmpDyadic_lt_iff,qAf,qAf];exact Ἦ (by exact_mod_cast compare_lt_iff_lt.mp h) hs
 . rw[Model.cmpDyadic_eq_iff,qAf,qAf,compare_eq_iff_eq.mp h]
 . rw[Model.cmpDyadic_gt_iff,qAf,qAf];exact Ἦ (by exact_mod_cast compare_gt_iff_gt.mp h) hs
lemma qFd (a b:Ὢ):ofW (maxW a b)=Model.maximum (ofW a) (ofW b):=by
 unfold maxW;split;. exact qAy _
 . rename_i hs;have ha:=ἶ a (λh=>hs (Or.inl h));have hb:=ἶ b (λh=>hs (Or.inr h));have hna:=Model.isNaN_eq_false_of_toDyadic?_some ha;have hnb:=Model.isNaN_eq_false_of_toDyadic?_some hb;have hc:Model.compare (ofW a) (ofW b)=some _:=Model.compare_eq_some_cmpDyadic_of_toDyadic? _ _ ha hb;rw[qDo] at hc;have hnone:Model.chooseNaN2 (ofW a) (ofW b)=none:=(Model.chooseNaN2_eq_none_iff _ _).2 ⟨hna,hnb⟩;unfold Model.maximum Model.Internal.Value.maximum;rw[Model.withNaNSelection_of_none _ _ hnone];dsimp only;rw[Model.compareNonNaN_eq_of_compare_eq_some _ _ _ _ hc];cases compare (keyOf a) (keyOf b)<;>simp only[qAy]
def maxpoolW (c h w k:Ὣ) (inp:ё):ё:=
 Array.ofFn (n:=c*(h/k*(w/k))) λq=>
  let ch:=q.val/(h/k*(w/k))
  let y:=q.val/(w/k)%(h/k)
  let x:=q.val%(w/k)
  (ἄ (k*k)).foldl (λacc t=>maxW acc (poolAtO h w k inp ch y x t))
   (poolAtO h w k inp ch y x 0)
lemma qEj (p:Ὣ->Ὢ) (l:љ) (acc:Ὢ):
 l.foldl (λm t=>Model.maximum m (ofW (p t))) (ofW acc)=ofW (l.foldl (λa t=>maxW a (p t)) acc):=by
 induction l generalizing acc with
 | nil=>rfl
 | cons t l ih=>simp only[ὔ,<-qFd,ih]
lemma qDn:maxpoolW=maxpoolSpecO:=by funext c h w k inp;simp only[maxpoolW,maxpoolSpecO,qEj,qAw]
@[noinline] def E_div (n:Ὣ) (s:ὥ):ὥ:=
 toWord (Model.div (ofW s) (ofW (natW n)))
lemma qIt:∀k:Fin 13,0<k.val->expOf (natW (k.val*k.val))≠255/\signOf (natW (k.val*k.val))=false/\mantOf (natW (k.val*k.val))≠0/\mantOf (natW (k.val*k.val))*2^idxOf (natW (k.val*k.val))=k.val*k.val*2^149:=by decide+kernel
lemma qJo:∀k,k<13->0<k->E_div (k*k) (dotFn (k*k) (λ_=>0) (λ_=>oneW))=0:=by
 rw[show dotFn=Submission.D.dotFnO from Submission.D.qAi.trans Submission.D.qAiO.symm];decide+kernel
lemma qJo14:E_div (14*14) (dotFn (14*14) (λ_=>0) (λ_=>oneW))=0:=by
 rw[show dotFn=Submission.D.dotFnO from Submission.D.qAi.trans Submission.D.qAiO.symm];decide+kernel
def pwEntryN (i:Ὣ):ὧ:=
 if i%256==0 || i%256==255 then Ἃ 0x7FF8000000000000 else pwEntry i
def pwN:ὤ:=
 (ἄ 512).foldl (λa i=>a.push (pwEntryN i)) (ἆ 512)
lemma qDg:pwN.data=(ἄ 512).toArray.map pwEntryN:=by
 have key:∀(l:љ) (a:ὤ),(l.foldl (λa i=>a.push (pwEntryN i)) a).data=a.data++(l.toArray.map pwEntryN):=by
  intro l;induction l with
  | nil=>intro a;simp only[List.foldl_nil,List.map_toArray,List.map_nil,Array.append_empty]
  | cons x l ih=>intro a;simp only[ὔ,ih];simp only[FloatArray.push,List.map_toArray,List.push_append_toArray,List.map_cons]
 simp only[pwN,key];simp only[ἆ,List.toArray_range,Array.empty_append]
lemma qCl (w:ὥ):(w>>>(23:ὥ)).toUSize.toNat< ѨpwN:=by
 have e:ѨpwN=512:=by simp only[FloatArray.size,qDg,List.toArray_range,Array.size_map,Array.size_range]
 rw[e];have h:=w.toNat_lt_size;simp only[ὃ,UInt32.toNat_shiftRight];norm_num[Ἱ] at h⊢;(clear e;exact omS13 h)
@[inline] def dvN (w:ὥ):ὧ:=
 ((w&&&0x7FFFFF) ||| 0x800000).toFloat*pwN.uget (w>>>(23:ὥ)).toUSize (qCl w)
@[inline] def scanH (A:ѐ):ὥ:=
 let h:=scanF A A.usize 0 (qAd A) 0 Е
 if h==Е then scan A A.usize 0 (qAd A) Е 0 else h
@[inline] def zivS (s:ὧ):ὥ:=
 let a:=s.toBits
 zivW (a+2) (a-2)
section
open Float.Model.UnpackedFloat in
open Float.Model.UnpackedFloat in
section
variable (x y:ὧ) (hx:xᶠ) (hy:yᶠ) (h1 : |R x| ≤ A_B) (h2 : |R y| ≤ A_B)
include x y hx hy h1 h2
end
section
variable (w:ὥ)
include w
end
section
variable (l:List ὧ) (hf:∀ q∈l,qᶠ) (hlen:ѣl<=2^21) (hb:A_sa l<=A_B/4)
include l hf hlen hb
end
section
variable (w:ὥ) (hg:good w=true)
include w hg
end
section
variable {n:ℕ} {ts:List ℕ} {g:ℕ->ὧ} {p:ℕ->ℝ} {bw hh:ὥ} {X Y J:ℝ} {s t a e:ὧ}
 (A:n<2^20) (B:ѣts<=n) (C:A_G ts g p) (D:good bw=true)
 (E:X=(ts.map p).sum+wv bw) (F:s=ts.foldl (λ(c:ὧ) t=>c+g t) 0+dv bw)
 (G:(2:ℝ)^(-127:ℤ) ≤ |R (s+e)|)
include A B C D E F G
end
section
variable (n:Ὣ) (hn:n<2^20) (xw ww:Ὣ->ὥ) (bw:ὥ) (hx:∀ t,t< n->good (xw t)=true) (hw:∀ t,t< n->good (ww t)=true) (hb:good bw=true) (l:љ) (hl:l.Nodup) (hl2:∀ t∈l,t< n)
include n hn xw ww bw hx hw hb l hl hl2
lemma qBx (hz:∀t,t<n->t∉l->xw t&&&В=0\/ww t&&&В=0) (s e:ὧ)
 (hs:let r:=l.foldl (λ(acc:F1) t=>let q:=dv (xw t)*dv (ww t)
  F1.mk (acc.s+q) (acc.t+q.abs)) ⟨0,0⟩;s=r.s+dv bw/\e=nfOf n*(r.t+(dv bw).abs))
 (hbig:(2:ℝ)^(-127:ℤ)<= |R (s+e)|):
 R (s-e)<exactS n xw ww bw/\exactS n xw ww bw<R (s+e):=
 Submission.D.qDm n hn xw ww bw hx hw hb l hl hl2 hz s e hs hbig
lemma qEd (hz:∀t,t<n->t∉l->xw t&&&В=0) (s e:ὧ)
 (hs:let r:=l.foldl (λ(acc:F1) t=>let q:=dv (xw t)*dv (ww t)
  F1.mk (acc.s+q) (acc.t+q.abs)) ⟨0,0⟩;s=r.s+dv bw/\e=nfOf n*(r.t+(dv bw).abs))
 (hbig:(2:ℝ)^(-127:ℤ)<= |R (s+e)|):
 R (s-e)<exactS n xw ww bw/\exactS n xw ww bw<R (s+e):=
 qBx n hn xw ww bw hx hw hb l hl hl2 (λt b a=>.inl (hz t b a)) s e hs hbig
end
lemma qBr (n:Ὣ) (hn:n<2^20) (xw ww:Ὣ->ὥ) (bw hh:ὥ) (hx:∀t,t<n->good (xw t)=true) (hw:∀t,t<n->good (ww t)=true) (hb:good bw=true) (hg:good hh=true) (hm:∀t,t<n->xw t&&&В<=hh) (hp:hh<=В) (l:љ) (hl:l.Nodup) (hl2:∀t∈l,t<n) (hz:∀t,t<n->t∉l->xw t&&&В=0\/ww t&&&В=0) (s e:ὧ)
 (hs:s=l.foldl (λ(acc:ὧ) t=>acc+dv (xw t)*dv (ww t)) 0+dv bw/\e=nfOf n*(dv hh*(l.foldl (λ(acc:ὧ) t=>acc+(dv (ww t)).abs) 0)+(dv bw).abs))
 (hbig:(2:ℝ)^(-127:ℤ)<= |R (s+e)|):
 R (s-e)<exactS n xw ww bw/\exactS n xw ww bw<R (s+e):=by
 obtain ⟨hs,b⟩:=hs
 obtain ⟨-,-,c⟩:=qDw hg hp
 exact qCw hn (qGf hl hl2) (qH9 hl2 hx hw) hb
  (qCq hx hw hl hl2 hz) hs hbig hg hp (λt a=>hw t (hl2 t a)) (λt a=>by
   rw[ἢ]
   exact ἀ (c _ (hx t (hl2 t a)) (hm t (hl2 t a))) (Ἀ _)) b
end
section
end
section
section
open Float.Model Float.Model.UnpackedFloat
lemma qC1 (H a b:Ὣ) (hb:b<H):(H*a+b)/H=a/\(H*a+b)%H=b:=by
 have c:0<H:=(by exact omS147 hb);constructor;. rw[Nat.mul_add_div c,Ὃ hb,Ἵ]
 . rw[Nat.mul_add_mod,Ἁ hb]
end
set_option maxRecDepth 8192 in
section
variable (s e:ὧ) (V:ℝ) (hlo:R (s-e)< V) (hhi:V< R (s+e)) (h:ziv s e≠Е)
include s e V hlo hhi h
end
end
section
lemma qC3 (n:Ὣ) (hn:n<2^20) (xw ww:Ὣ->ὥ) (bw hh:ὥ) (hx:∀t,t<n->good (xw t)=true) (hw:∀t,t<n->good (ww t)=true) (hb:good bw=true) (hg:good hh=true) (hm:∀t,t<n->xw t&&&В<=hh) (hp:hh<=В) (l:љ) (hl:l.Nodup) (hl2:∀t∈l,t<n) (hz:∀t,t<n->t∉l->xw t&&&В=0\/ww t&&&В=0) (s e:ὧ)
 (hs:s=l.foldl (λ(acc:ὧ) t=>acc+dv (xw t)*dv (ww t)) 0+dv bw/\e=nfOf n*(dv hh*(l.foldl (λ(acc:ὧ) t=>acc+(dv (ww t)).abs) 0)+(dv bw).abs))
 (h:ziv s e≠Е):
 ziv s e=dotOf (n+1) (λt=>if t<n then xw t else oneW) (λt=>if t<n then ww t else bw):=by
 obtain ⟨-,-,hbig⟩:=qCp s e h
 obtain ⟨hlo,hhi⟩:=qBr n hn xw ww bw hh hx hw hb hg hm hp l hl hl2 hz s e hs hbig
 exact qD4 n hn xw ww bw hx hw hb s e hlo hhi h
end
section
lemma qDa (n:Ὣ) (hn:n<2^20) (xw ww:Ὣ->ὥ) (bw hh:ὥ) (hx:∀t,t<n->good (xw t)=true) (hw:∀t,t<n->good (ww t)=true) (hb:good bw=true) (hg:good hh=true) (hm:∀t,t<n->xw t&&&В<=hh) (hp:hh<=В) (l:љ) (hl:l.Nodup) (hl2:∀t∈l,t<n) (hz:∀t,t<n->t∉l->xw t&&&В=0\/ww t&&&В=0) (s e:ὧ)
 (hs:s=l.foldl (λ(acc:ὧ) t=>acc+dv (xw t)*dv (ww t)) 0+dv bw/\e=nfOf n*(dv hh*(l.foldl (λ(acc:ὧ) t=>acc+(dv (ww t)).abs) 0)+(dv bw).abs))
 (h:zivT (thOf e) s≠Е):
 zivT (thOf e) s=dotOf (n+1) (λt=>if t<n then xw t else oneW) (λt=>if t<n then ww t else bw):=by
 obtain ⟨fs,fe,hrel,hB,hbig,hl2'⟩:=qI4 e s h
 obtain ⟨hlo,hhi⟩:=qBr n hn xw ww bw hh hx hw hb hg hm hp l hl hl2 hz s e hs hbig
 have hV:=qHs s e _ fs fe hrel hB hlo hhi
 have hsum:=qGh n xw ww bw hx hw hb
 have hne:(∑ j∈Ἣ (n+1),ἂ (if j<n then xw j else oneW)*ἂ (if j<n then ww j else bw):ℚ)≠0:=by
  intro h0
  rw[h0,Rat.cast_zero] at hsum
  exact qJr s _ hV hl2' hsum.symm
 obtain ⟨d,hd,hw'⟩:=qBa (n+1) (λt=>if t<n then xw t else oneW) (λt=>if t<n then ww t else bw)
  (by
   intro j _
   have hone:expOf oneW≠255:=ה
   by_cases hj:j<n
   . simp only[hj,ite_true];exact ⟨qAc _ (hx j hj),qAc _ (hw j hj)⟩
   . simp only[hj,ite_false]
     exact ⟨hone,qAc _ hb⟩) hne
 rw[hw']
 refine qHv _ s d ?_ h
 rw[hd,hsum]
 exact hV
lemma qH2 (a:Ὠ) (h0:(0x7040000000000000:Ὠ)<=a<<<(1:Ὠ)) (h1:a<<<(1:Ὠ)<0x8FA0000000000000) (h2:(a+0x0FFFFFFD)&&&0x1FFFFFFF<0x1FFFFFFC):
 zivW (a+2) (a-2)=zivC ((a+0x0FFFFFFD)>>>(29:Ὠ))/\zivC ((a+0x0FFFFFFD)>>>(29:Ὠ))≠Е:=by
 have ha:=a.toNat_lt
 have e0:(a<<<(1:Ὠ)).toNat=Ѧa*2%2^64:=by
  rw[UInt64.toNat_shiftLeft];norm_num[ὁ]
 rw[Ὤ,e0] at h0
 rw[ἴ,e0] at h1
 norm_num at h0 h1 ha
 have eu:(a+0x0FFFFFFD).toNat=Ѧa+268435453:=by
  rw[ἤ];norm_num;(clear h2;exact omS364 h1 ha)
 have em:((a+0x0FFFFFFD)&&&0x1FFFFFFF).toNat=(Ѧa+268435453)%2^29:=by
  rw[Ὅ,eu]
  have:=Ἂ (Ѧa+268435453) 29
  norm_num at this⊢
  exact this
 rw[ἴ,em] at h2
 norm_num at h2
 generalize hx:(a+0x0FFFFFFD)>>>(29:Ὠ)=x
 have ex:Ѧx=(Ѧa+268435453)/2^29:=by
  rw[<-hx,ἰ,eu];norm_num[Ἱ]
 have e1:(a+2+0x0FFFFFFF)>>>(29:Ὠ)=x:=by
  apply UInt64.toNat_inj.mp
  rw[ex,ἰ,ἤ,ἤ]
  norm_num[Ἱ]
  (clear hx;exact omS365 h1 ha h2)
 have e2:(a-2+0x0FFFFFFF)>>>(29:Ὠ)=x:=by
  apply UInt64.toNat_inj.mp
  rw[ex,ἰ,ἤ,ὲ]
  norm_num[Ἱ]
  (clear hx e1;exact omS366 h1 ha)
 have hE:((x>>>(23:Ὠ))&&&(0x7FF:Ὠ)).toNat=Ѧx/2^23%2^11:=by
  rw[Ὅ,ἰ]
  have:=Ἂ (Ѧx>>>23) 11
  norm_num[Ἱ] at this⊢
  exact this
 have r1:898<=Ѧx/2^23%2^11:=by rw[ex];norm_num;(clear ha hx e1 e2 hE;exact omS367 h0 h1)
 have r2:Ѧx/2^23%2^11<1151:=by rw[ex];norm_num;(clear ha hx e1 e2 hE r1;exact omS368 h1)
 have r3:Ѧx<2^35:=by rw[ex];norm_num;(clear hx e1 e2 hE r1 r2;exact omS369 h1 ha)
 constructor
 . unfold zivW zivC;dsimp only;rw[e1,e2];have hc:((x>>>(23:Ὠ))&&&(0x7FF:Ὠ))-(897:Ὠ)<(254:Ὠ):=by rw[ἴ,ὲ,hE];norm_num;norm_num at r1 r2;(clear e0 h0 h1 ha eu em h2 hx ex e1 e2 hE r3;exact omS261 r1 r2)
   simp only[BEq.rfl,hc,decide_true,Bool.and_self, ↓reduceIte,UInt64.toUInt32_or,UInt64.reduceLT,UInt64.toUInt32_shiftLeft,UInt64.toUInt32_ofNat,UInt64.toUInt32_sub,UInt64.toUInt32_and]
 . unfold zivC;rw[show ((x>>>(34:Ὠ))<<<(31:Ὠ)) ||| ((x&&&(0x3FFFFFFFF:Ὠ))-((896:Ὠ)<<<(23:Ὠ)))=((x>>>34)<<<31) ||| ((x&&&(0x3FFFFFFFF:Ὠ))-((896:Ὠ)<<<23)) from rfl,qEl x r3 (by clear e0 h0 h1 ha eu em h2 hx ex e1 e2 r3;(exact Nat.le_of_lt ‹_›)) r2];intro h;have h':=congrArg UInt32.toNat h;rw[ὡ] at h';have f:=ὀ Ѧx (ב:0<2^23);split at h'<;>norm_num at h' f r1 r2<;>(clear e0 h0 h1 ha eu em h2 hx ex e1 e2 hE r3 h f;omega)
lemma qIf (s:ὧ) (h1:(s.toBits<<<(1:Ὠ))-0x7040000000000000<0x1F60000000000000) (h2:(s.toBits+0x0FFFFFFD)&&&0x1FFFFFFF<0x1FFFFFFC):zivS s=zivC ((s.toBits+0x0FFFFFFD)>>>(29:Ὠ))/\zivS s≠Е:=by
 have sp0:spOf 0x7040000000000000=0x1F60000000000000:=ה
 obtain ⟨l1,l2⟩:=qD1 _ _ (sp0.symm▸h1);have b:=qH2 s.toBits l1 l2 h2;have e1:zivS s=zivC ((s.toBits+0x0FFFFFFD)>>>(29:Ὠ)):=b.1;exact ⟨e1,e1▸b.2⟩
end
section
lemma qIb (a b:ὦ) (h:Ѥa*Ѥb<2^32):(a*b).toNat=Ѥa*Ѥb:=qJu a b h
section
variable {N:Ὣ} {sp:Ὣ->Ὢ}
include N sp
end
lemma qD0 (A:ѐ) (hs:ѧA<2^32) (h:scanH A≠Е):C_Hv A (scanH A):=by
 revert h;unfold scanH;dsimp only;split;. exact qBh A hs
 . exact qDl A hs
section
variable {R I O:Ὣ} {X W b:ё}
end
section
variable (xw:Ὣ->Ὢ) (t:Ὣ)
include xw t
end
section
variable {x:ὤ} {ix W:ѐ} {w0 w1 w2 w3 m n:ὦ}
 (hx:Ѥn<=Ѩx) (hi:Ѥn<=ѧix) (hs:ѧW<2^32)
 (h0:Ѥw0+Ѥm<=ѧW) (h1:Ѥw1+Ѥm<=ѧW)
 (h2:Ѥw2+Ѥm<=ѧW) (h3:Ѥw3+Ѥm<=ѧW)
end
section
variable {R I O:Ὣ} {X W b:ё}
end
lemma qGr (xs:ὤ) (v:ὧ) (q:Ὣ):(xs.push v).data[q]! =if q=Ѩxs then v else xs.data[q]!:=by
 obtain ⟨ds⟩:=xs;show (ds.push v)[q]! =if q=ѧds then v else ds[q]!;by_cases hq:q< ѧds
 . have h3:q<(ds.push v).size:=by simp only[Array.size_push,Order.lt_add_one_iff];(exact Nat.le_of_lt ‹_›)
   rw[Ἕ _ q h3,Ἕ ds q hq,ἐ (by clear h3;exact omS370 hq),Array.getElem_push_lt]
 . by_cases e:q=ѧds
   . subst e;have h3:ѧds<(ds.push v).size:=by simp only[Array.size_push,lt_add_iff_pos_right,Order.lt_one_iff]
     rw[Ἕ (ds.push v) ѧds h3,ἱ rfl,Array.getElem_push_eq]
   . have h3:¬q<(ds.push v).size:=by simp only[Array.size_push,Order.lt_add_one_iff,not_le];exact omS371 hq e
     rw[ὑ _ q h3,ὑ ds q hq,ἐ e]
section
variable {I O:Ὣ} {X W b:ё}
end
end
section
local instance ecNe11:Nonempty D_Sc:=⟨by constructor <;> exact default⟩
def D_scan (A:ѐ) (n i:ὦ) (hn:Ѥn<=ѧA) (lo hi:ὥ):D_Sc:=
 if h:i<n then
  have ht:=А.mp h
  let v:=A.uget i א&&&В
  let u:=v-1
  D_scan A n (i+1) hn (if u<lo then u else lo) (if hi<v then v else hi)
 else ⟨lo,hi⟩
partial_fixpoint
def D_fill (v:ὧ) (n k:ὦ) (a:ὤ):ὤ:=
 if h:k<n then D_fill v n (k+1) (a.push v) else a
partial_fixpoint
def D_copy (X:ѐ) (o n t:ὦ) (hs:ѧX<2^32)
 (ho:Ѥo+Ѥn<=ѧX) (a:ὤ):ὤ:=
 if h:t<n then
  D_copy X o n (t+1) hs ho
   (a.push (dv (X.uget (o+t) (wi o t n X ho hs (А.mp h)))))
 else a
partial_fixpoint
def D_z0:ὤ:=D_fill (dv 0) 512 0 (ἆ 512)
def D_z1:ὤ:=D_fill (dv 0) 1024 0 (ἆ 1024)
def D_z2:ὤ:=D_fill (dv 0) 2048 0 (ἆ 2048)
def D_z3:ὤ:=D_fill (dv 0) 4096 0 (ἆ 4096)
def D_z4:ὤ:=D_fill (dv 0) 8192 0 (ἆ 8192)
def D_z5:ὤ:=D_fill (dv 0) 16384 0 (ἆ 16384)
@[inline] def D_zs (n:Ὣ):ὤ:=
 if n<=512 then D_z0 else if n<=1024 then D_z1 else if n<=2048 then D_z2 else
 if n<=4096 then D_z3 else if n<=8192 then D_z4 else D_z5
def D_cpz (X:ѐ) (o n t q:ὦ) (hs:ѧX<2^32)
 (ho:Ѥo+Ѥn<=ѧX) (a:ὤ):ὤ:=
 if h:t<n then
  D_cpz X o n (t+1) (q+1) hs ho
   (wrF a q (dv (X.uget (o+t) (wi o t n X ho hs (А.mp h)))))
 else a
partial_fixpoint
def D_imz (s:OS) (X:ѐ) (hs:ѧX<2^32) (wp R r:Ὣ)
 (a:ὤ):ὤ:=
 if hr:r<R then
  let hp:=s.h+2*s.pad
  let yy:=r%hp
  let o:=((r/hp*s.h+(yy-s.pad))*s.w).toUSize
  if hg:s.pad<=yy/\yy-s.pad<s.h/\Ѥo+s.w.toUSize.toNat<=ѧX then
   D_imz s X hs wp R (r+1) (D_cpz X o s.w.toUSize 0 (r*wp+s.pad).toUSize hs hg.2.2 a)
  else D_imz s X hs wp R (r+1) a
 else a
partial_fixpoint
def D_tq (k kw kh hp wp t c r q:Ὣ) (a:ѐ):ѐ:=
 if t<k then
  let a:=a.push ((c*hp+r)*wp+q).toUInt32
  if q+1<kw then D_tq k kw kh hp wp (t+1) c r (q+1) a
  else if r+1<kh then D_tq k kw kh hp wp (t+1) c (r+1) 0 a
  else D_tq k kw kh hp wp (t+1) (c+1) 0 0 a
 else a
partial_fixpoint
def D_img (s:OS) (X:ѐ) (hs:ѧX<2^32) (z:ὧ) (R r:Ὣ)
 (a:ὤ):ὤ:=
 if hr:r<R then
  let hp:=s.h+2*s.pad
  let yy:=r%hp
  let o:=((r/hp*s.h+(yy-s.pad))*s.w).toUSize
  let pu:=s.pad.toUSize
  if hg:s.pad<=yy/\yy-s.pad<s.h/\Ѥo+s.w.toUSize.toNat<=ѧX then
   D_img s X hs z R (r+1) (D_fill z pu 0 (D_copy X o s.w.toUSize 0 hs hg.2.2 (D_fill z pu 0 a)))
  else D_img s X hs z R (r+1) (D_fill z (s.w+2*s.pad).toUSize 0 a)
 else a
partial_fixpoint
def D_rw2 (X:ѐ) (hs:X.size<2^32) (w wp:ὦ) (o q:ὦ) (k:Nat) (a:ὤ):ὤ:=
 match k with
 | 0=>a
 | k+1=>
  if hg:USize.toNat o+w.toNat<=X.size then D_rw2 X hs w wp (o+w) (q+wp) k (D_cpz X o w 0 q hs hg a)
  else D_rw2 X hs w wp (o+w) (q+wp) k a
def D_ch2 (s:OS) (X:ѐ) (hs:X.size<2^32) (wp:Nat) (c k:Nat) (a:ὤ):ὤ:=
 match k with
 | 0=>a
 | k+1=>D_ch2 s X hs wp (c+1) k (D_rw2 X hs s.w.toUSize wp.toUSize ((c*s.h)*s.w).toUSize
    ((c*(s.h+2*s.pad)+s.pad)*wp+s.pad).toUSize s.h a)
lemma imz_skip (s:OS) (X:ѐ) (hs:X.size<2^32) (wp R r:Nat) (a:ὤ) (hr:r<R)
  (hn:¬(s.pad<=r%(s.h+2*s.pad)/\r%(s.h+2*s.pad)-s.pad<s.h)):
  D_imz s X hs wp R r a=D_imz s X hs wp R (r+1) a:=by
 rw[D_imz,dif_pos hr];dsimp only
 rw[dif_neg (λh=>hn ⟨h.1,h.2.1⟩)]
lemma imz_skipk (s:OS) (X:ѐ) (hs:X.size<2^32) (wp R:Nat) (k:Nat): ∀(r:Nat) (a:ὤ),r+k<=R->
  (∀i,i<k-> ¬(s.pad<=(r+i)%(s.h+2*s.pad)/\(r+i)%(s.h+2*s.pad)-s.pad<s.h))->
  D_imz s X hs wp R r a=D_imz s X hs wp R (r+k) a:=by
 induction k with
 | zero=>intro r a _ _;rfl
 | succ k ih=>
  intro r a hk hi
  rw[imz_skip s X hs wp R r a (by exact omS4 hk) (by simpa only[not_and,not_lt,add_zero] using hi 0 (by exact omS9))]
  rw[ih (r+1) a (by exact omS5 hk) (λi hi'=>by have:=hi (i+1) א;rwa[show r+(i+1)=r+1+i by exact omS375] at this)]
  rw[show r+1+k=r+(k+1) by exact omS146]
lemma cpz_congr (X:ѐ) (o1 o2 n t q:ὦ) (hs:X.size<2^32) (h1:USize.toNat o1+n.toNat<=X.size)
 (h2:USize.toNat o2+n.toNat<=X.size) (a:ὤ) (e:o1=o2):
 D_cpz X o1 n t q hs h1 a=D_cpz X o2 n t q hs h2 a:=by subst e;rfl
lemma imz_rows (s:OS) (X:ѐ) (hs:X.size<2^32) (wp R c:Nat) (k:Nat):
  ∀(yy:Nat) (a:ὤ),s.pad<=yy->yy+k=s.pad+s.h->(c*(s.h+2*s.pad)+yy)+k<=R->
  D_imz s X hs wp R (c*(s.h+2*s.pad)+yy) a=
   D_imz s X hs wp R (c*(s.h+2*s.pad)+yy+k) (D_rw2 X hs s.w.toUSize wp.toUSize ((c*s.h+(yy-s.pad))*s.w).toUSize
    ((c*(s.h+2*s.pad)+yy)*wp+s.pad).toUSize k a):=by
 induction k with
 | zero=>intro yy a _ _ _;rfl
 | succ k ih=>
  intro yy a h1 h2 h3
  have hp:0<s.h+2*s.pad:=א
  have ey:(c*(s.h+2*s.pad)+yy)%(s.h+2*s.pad)=yy:=by
   rw[ἥ,Nat.mul_add_mod];exact Ἁ א
  have ec:(c*(s.h+2*s.pad)+yy)/(s.h+2*s.pad)=c:=by
   rw[ἥ,Nat.mul_add_div hp,Ὃ א,Ἵ]
  have eo1:(((c*(s.h+2*s.pad)+yy)/(s.h+2*s.pad)*s.h+((c*(s.h+2*s.pad)+yy)%(s.h+2*s.pad)-s.pad))*s.w).toUSize
   =((c*s.h+(yy-s.pad))*s.w).toUSize:=by rw[ey,ec]
  have eo:((c*s.h+(yy-s.pad))*s.w).toUSize+s.w.toUSize=((c*s.h+(yy+1-s.pad))*s.w).toUSize:=by
   rw[<-USize.ofNat_add];congr 1;rw[show yy+1-s.pad=(yy-s.pad)+1 by omega];simp only[ὰ,Ὑ,ὢ]
  have eq:((c*(s.h+2*s.pad)+yy)*wp+s.pad).toUSize+wp.toUSize=((c*(s.h+2*s.pad)+(yy+1))*wp+s.pad).toUSize:=by
   rw[<-USize.ofNat_add];congr 1;simp only[ὰ,Ὑ];omega
  have e3:c*(s.h+2*s.pad)+yy+(k+1)=c*(s.h+2*s.pad)+(yy+1)+k:=א
  rw[D_imz,dif_pos א]
  dsimp only
  simp only[D_rw2]
  by_cases hg:USize.toNat ((c*s.h+(yy-s.pad))*s.w).toUSize+s.w.toUSize.toNat<=X.size
  · have hc:s.pad<=(c*(s.h+2*s.pad)+yy)%(s.h+2*s.pad)/\(c*(s.h+2*s.pad)+yy)%(s.h+2*s.pad)-s.pad<s.h/\
     USize.toNat (((c*(s.h+2*s.pad)+yy)/(s.h+2*s.pad)*s.h+((c*(s.h+2*s.pad)+yy)%(s.h+2*s.pad)-s.pad))*s.w).toUSize
      +s.w.toUSize.toNat<=X.size:=⟨א,א,by rw[eo1];exact hg⟩
    rw[dif_pos hc,dif_pos hg,eo,eq,e3,cpz_congr X _ _ _ _ _ hs _ hg a eo1]
    exact ih (yy+1) _ א א א
  · have hc: ¬(s.pad<=(c*(s.h+2*s.pad)+yy)%(s.h+2*s.pad)/\(c*(s.h+2*s.pad)+yy)%(s.h+2*s.pad)-s.pad<s.h/\
     USize.toNat (((c*(s.h+2*s.pad)+yy)/(s.h+2*s.pad)*s.h+((c*(s.h+2*s.pad)+yy)%(s.h+2*s.pad)-s.pad))*s.w).toUSize
      +s.w.toUSize.toNat<=X.size):= λh=>hg (by rw[<-eo1];exact h.2.2)
    rw[dif_neg hc,dif_neg hg,eo,eq,e3]
    exact ih (yy+1) _ א א א
lemma imz_chan (s:OS) (X:ѐ) (hs:X.size<2^32) (wp R c:Nat) (a:ὤ) (hR:(c+1)*(s.h+2*s.pad)<=R):
  D_imz s X hs wp R (c*(s.h+2*s.pad)) a=D_imz s X hs wp R ((c+1)*(s.h+2*s.pad))
   (D_rw2 X hs s.w.toUSize wp.toUSize ((c*s.h)*s.w).toUSize ((c*(s.h+2*s.pad)+s.pad)*wp+s.pad).toUSize s.h a):=by
 have hR':c*(s.h+2*s.pad)+s.h+2*s.pad<=R:=by rw[ὰ,Ὑ] at hR;omega
 rw[imz_skipk s X hs wp R s.pad (c*(s.h+2*s.pad)) a א (λi hi=>by
  rw[ἥ,Nat.mul_add_mod,Ἁ (by exact omS381 hi)];omega)]
 rw[imz_rows s X hs wp R c s.h s.pad a (Nat.le_refl _) א א]
 rw[Nat.sub_self,Ἵ]
 rw[imz_skipk s X hs wp R s.pad _ _ א (λi hi=>by
  rw[show c*(s.h+2*s.pad)+s.pad+s.h+i=(s.pad+s.h+i)+(s.h+2*s.pad)*c by ring,Nat.add_mul_mod_self_left,
   Ἁ א];omega)]
 congr 1
 rw[ὰ,Ὑ];omega
lemma ch2_eq (s:OS) (X:ѐ) (hs:X.size<2^32) (wp R:Nat) (hR:R=s.cin*(s.h+2*s.pad)) (k:Nat):
  ∀(c:Nat) (a:ὤ),c+k=s.cin->D_ch2 s X hs wp c k a=D_imz s X hs wp R (c*(s.h+2*s.pad)) a:=by
 induction k with
 | zero=>
  intro c a h
  simp only[D_ch2]
  rw[D_imz,dif_neg (by subst hR;rw[<-h,Ἵ];exact omS491)]
 | succ k ih=>
  intro c a h
  simp only[D_ch2]
  rw[ih (c+1) _ (by exact omS137 h),imz_chan s X hs wp R c a (by subst hR;exact Ἒ _ (by exact omS492 h))]
@[inline] def D_xs (s:OS) (X:ѐ) (hs:ѧX<2^32) (wp R:Ὣ):ὤ:=
 if R*wp+7<=16384 then D_ch2 s X hs wp 0 s.cin (D_zs (R*wp+7))
 else D_fill (dv 0) 7 0 (D_img s X hs (dv 0) R 0 (ἆ (R*wp+7)))
def D_off (s:OS) (t:Ὣ):Ὣ:=
 (t/(s.kh*s.kw)*(s.h+2*s.pad)+t/s.kw%s.kh)*(s.w+2*s.pad)+t%s.kw
lemma qIo {n d:Ὣ} (b:ὦ) (u:ὥ) (hs:n<2^32) (h:Ѥb+ѥu+d<n):(b+Ѫu).toNat+d<n:=by rw[qAv _ _ (by rw[ὃ];exact omS372 hs h),ὃ];exact h
lemma qB1 {n:Ὣ} (p:ὦ) (k:Ὣ) (hs:n<2^32) (h:Ѥp+k<n):(p+ὴ k).toNat<n:=by rw[qAv _ _ (by rw[uk k (by exact omS278 hs h)];exact omS279 hs h),uk k (by exact omS278 hs h)];exact h
local instance ecNe12:Nonempty D_R:=⟨by constructor <;> exact default⟩
def D_go8R1 (xd wd:ὤ) (off:ѐ) (b e i:ὦ)
 (he:Ѥe<=ѧoff) (hw:Ѥe<=Ѩwd) (hs:Ѩxd<2^32)
 (hb:∀j (h:j< ѧoff),Ѥb+off[j].toNat+7< Ѩxd)
 (A B C D E F G I:ὧ) (r:D_R):D_R:=
 if h:i<e then
  have ht:=А.mp h
  let p:=b+(off.uget i (Nat.lt_of_lt_of_le ht he)).toUSize
  have hp:Ѥp+7< Ѩxd:=qIo _ _ hs (hb Ѥi (Nat.lt_of_lt_of_le ht he))
  let v:=wd.uget i (Nat.lt_of_lt_of_le ht hw)
  D_go8R1 xd wd off b e (i+1) he hw hs hb (A+xd.uget p (Nat.lt_of_le_of_lt (Nat.le_add_right _ 7) hp)*v)
   (B+xd.uget (p+ὴ 1) (qB1 p 1 hs (Nat.lt_of_le_of_lt (Nat.add_le_add_left ה _) hp))*v)
   (C+xd.uget (p+ὴ 2) (qB1 p 2 hs (Nat.lt_of_le_of_lt (Nat.add_le_add_left ה _) hp))*v)
   (D+xd.uget (p+ὴ 3) (qB1 p 3 hs (Nat.lt_of_le_of_lt (Nat.add_le_add_left ה _) hp))*v)
   (E+xd.uget (p+ὴ 4) (qB1 p 4 hs (Nat.lt_of_le_of_lt (Nat.add_le_add_left ה _) hp))*v)
   (F+xd.uget (p+ὴ 5) (qB1 p 5 hs (Nat.lt_of_le_of_lt (Nat.add_le_add_left ה _) hp))*v)
   (G+xd.uget (p+ὴ 6) (qB1 p 6 hs (Nat.lt_of_le_of_lt (Nat.add_le_add_left ה _) hp))*v)
   (I+xd.uget (p+ὴ 7) (qB1 p 7 hs hp)*v) r
 else match r with
  | ⟨t,_,_,_,_,_,_,_,_⟩=>⟨t,A,B,C,D,E,F,G,I⟩
partial_fixpoint
def D_go8RF (xd wd:ὤ) (off:ѐ) (b e i:ὦ)
 (he:Ѥe<=ѧoff) (hw:Ѥe<=Ѩwd) (hs:Ѩxd<2^32)
 (hb:∀j (h:j< ѧoff),Ѥb+off[j].toNat+7< Ѩxd)
 (A B C D E F G I:ὧ) (r:D_R):D_R:=
 if h:i<e/\i+1<e then
  have ht:=А.mp h.1
  have hu:=А.mp h.2
  let p:=b+(off.uget i (Nat.lt_of_lt_of_le ht he)).toUSize
  have hp:Ѥp+7< Ѩxd:=qIo _ _ hs (hb Ѥi (Nat.lt_of_lt_of_le ht he))
  let v:=wd.uget i (Nat.lt_of_lt_of_le ht hw)
  let p':=b+(off.uget (i+1) (Nat.lt_of_lt_of_le hu he)).toUSize
  have hp':Ѥp'+7< Ѩxd:=qIo _ _ hs (hb Ѥ(i+1) (Nat.lt_of_lt_of_le hu he))
  let u:=wd.uget (i+1) (Nat.lt_of_lt_of_le hu hw)
  D_go8RF xd wd off b e (i+1+1) he hw hs hb
   (Float.fma (xd.uget p (Nat.lt_of_le_of_lt (Nat.le_add_right _ 7) hp)) v A+xd.uget p' (Nat.lt_of_le_of_lt (Nat.le_add_right _ 7) hp')*u)
   (Float.fma (xd.uget (p+ὴ 1) (qB1 p 1 hs (Nat.lt_of_le_of_lt (Nat.add_le_add_left ה _) hp))) v B+xd.uget (p'+ὴ 1) (qB1 p' 1 hs (Nat.lt_of_le_of_lt (Nat.add_le_add_left ה _) hp'))*u)
   (Float.fma (xd.uget (p+ὴ 2) (qB1 p 2 hs (Nat.lt_of_le_of_lt (Nat.add_le_add_left ה _) hp))) v C+xd.uget (p'+ὴ 2) (qB1 p' 2 hs (Nat.lt_of_le_of_lt (Nat.add_le_add_left ה _) hp'))*u)
   (Float.fma (xd.uget (p+ὴ 3) (qB1 p 3 hs (Nat.lt_of_le_of_lt (Nat.add_le_add_left ה _) hp))) v D+xd.uget (p'+ὴ 3) (qB1 p' 3 hs (Nat.lt_of_le_of_lt (Nat.add_le_add_left ה _) hp'))*u)
   (Float.fma (xd.uget (p+ὴ 4) (qB1 p 4 hs (Nat.lt_of_le_of_lt (Nat.add_le_add_left ה _) hp))) v E+xd.uget (p'+ὴ 4) (qB1 p' 4 hs (Nat.lt_of_le_of_lt (Nat.add_le_add_left ה _) hp'))*u)
   (Float.fma (xd.uget (p+ὴ 5) (qB1 p 5 hs (Nat.lt_of_le_of_lt (Nat.add_le_add_left ה _) hp))) v F+xd.uget (p'+ὴ 5) (qB1 p' 5 hs (Nat.lt_of_le_of_lt (Nat.add_le_add_left ה _) hp'))*u)
   (Float.fma (xd.uget (p+ὴ 6) (qB1 p 6 hs (Nat.lt_of_le_of_lt (Nat.add_le_add_left ה _) hp))) v G+xd.uget (p'+ὴ 6) (qB1 p' 6 hs (Nat.lt_of_le_of_lt (Nat.add_le_add_left ה _) hp'))*u)
   (Float.fma (xd.uget (p+ὴ 7) (qB1 p 7 hs hp)) v I+xd.uget (p'+ὴ 7) (qB1 p' 7 hs hp')*u) r
 else D_go8R1 xd wd off b e i he hw hs hb A B C D E F G I r
partial_fixpoint
lemma vFgf (xd wd:ὤ) (off:ѐ) (b e:ὦ) (he:Ѥe<=ѧoff) (hw:Ѥe<=Ѩwd) (hs:Ѩxd<2^32)
 (hb:∀j (h:j< ѧoff),Ѥb+off[j].toNat+7< Ѩxd) (HX:∀(p:ὦ) (h:Ѥp< Ѩxd),∃u,xd.uget p h=dv u)
 (HW:∀(q:ὦ) (h:Ѥq< Ѩwd),Ѥq< Ѥe->∃u,wd.uget q h=dv u) (r:D_R) (i:ὦ) (A B C D E F G I:ὧ):
 D_go8RF xd wd off b e i he hw hs hb A B C D E F G I r=D_go8R1 xd wd off b e i he hw hs hb A B C D E F G I r:=by
 generalize hm:Ѥe-Ѥi=m
 induction m using Nat.strong_induction_on generalizing i A B C D E F G I with
 | _ m ih=>
  rw[D_go8RF];split
  . rename_i h;have:=usz i e h.1;have:=usz (i+1) e h.2;have:=А.mp h.2
    have Ev:∀(p:ὦ) (hp:Ѥp< Ѩxd) (hq) (a:ὧ),Float.fma (xd.uget p hp) (wd.uget i hq) a=a+xd.uget p hp*wd.uget i hq:=
     λp hp hq a=>Submission.D.zFE (HX p hp) (HW i hq (А.mp h.1)) a
    simp only[Ev]
    rw[ih _ (by clear h;omega) (i+1+1) _ _ _ _ _ _ _ _ rfl];simp only[D_go8R1.eq_1 xd wd off b e i,D_go8R1.eq_1 xd wd off b e (i+1),Ἄ h.1,Ἄ h.2]
  . rfl
@[inline] def D_go8R (xd wd:ὤ) (off:ѐ) (b e i:ὦ)
 (he:Ѥe<=ѧoff) (hw:Ѥe<=Ѩwd) (hs:Ѩxd<2^32)
 (hb:∀j (h:j< ѧoff),Ѥb+off[j].toNat+7< Ѩxd)
 (A B C D E F G I:ὧ) (r:D_R):D_R:=
 if Submission.D.armT then D_go8RF xd wd off b e i he hw hs hb A B C D E F G I r else D_go8R1 xd wd off b e i he hw hs hb A B C D E F G I r
lemma qGR (xd wd:ὤ) (off:ѐ) (b e:ὦ) (he:Ѥe<=ѧoff) (hw:Ѥe<=Ѩwd) (hs:Ѩxd<2^32)
 (hb:∀j (h:j< ѧoff),Ѥb+off[j].toNat+7< Ѩxd) (HX:∀(p:ὦ) (h:Ѥp< Ѩxd),∃u,xd.uget p h=dv u)
 (HW:∀(q:ὦ) (h:Ѥq< Ѩwd),Ѥq< Ѥe->∃u,wd.uget q h=dv u) (r:D_R) (i:ὦ) (A B C D E F G I:ὧ):
 D_go8R xd wd off b e i he hw hs hb A B C D E F G I r=D_go8R1 xd wd off b e i he hw hs hb A B C D E F G I r:=by
 unfold D_go8R;split
 . exact vFgf xd wd off b e he hw hs hb HX HW r i A B C D E F G I
 . rfl
local instance ecNe13:Nonempty F1:=⟨by constructor <;> exact default⟩
def D_goT (xd wd:ὤ) (off:ѐ) (b e i:ὦ)
 (he:Ѥe<=ѧoff) (hw:Ѥe<=Ѩwd) (hs:Ѩxd<2^32)
 (hb:∀j (h:j< ѧoff),Ѥb+off[j].toNat+0< Ѩxd) (a t:ὧ):F1:=
 if h:i<e then
  have ht:=А.mp h
  let q:=xd.uget (b+(off.uget i (Nat.lt_of_lt_of_le ht he)).toUSize)
   (qIo (d:=0) _ _ hs (hb Ѥi (Nat.lt_of_lt_of_le ht he)))*wd.uget i (Nat.lt_of_lt_of_le ht hw)
  D_goT xd wd off b e (i+1) he hw hs hb (a+q) (t+q.abs)
 else ⟨a,t⟩
termination_by Ѥe-Ѥi
decreasing_by rw[usz i e h];omega
@[specialize] def zAll (fa fb:Nat->ὥ) (i:Nat):Nat->Ὡ
 | 0=>true
 | n+1=>(let a:=fa i;let b:=fb i;expOf a != 255&&expOf b != 255&&mantOf a*mantOf b==0)&&zAll fa fb (i+1) n
@[specialize] def zNeg (fa fb:Nat->ὥ) (i:Nat):Nat->Ὡ
 | 0=>true
 | n+1=>(let a:=fa i;let b:=fb i;Bool.xor (signOf a) (signOf b))&&zNeg fa fb (i+1) n
@[inline] def zdotFn (n:Nat) (fa fb:Nat->ὥ):ὥ:=
 if zAll fa fb 0 n then (if n≠0&&zNeg fa fb 0 n then س else 0) else dotFn n fa fb
def zCols (s:OS) (X:ѐ) (c yy x0:Nat) (i:Nat):Nat->Ὡ
 | 0=>true
 | n+1=>pixZ s X c yy (x0+i)&&zCols s X c yy x0 (i+1) n
def zRows (s:OS) (X:ѐ) (c y0 x0:Nat) (i:Nat):Nat->Ὡ
 | 0=>true
 | n+1=>zCols s X c (y0+i) x0 0 s.kw&&zRows s X c y0 x0 (i+1) n
def wFin (W:ѐ) (base:Nat) (i:Nat):Nat->Ὡ
 | 0=>true
 | n+1=>expOf W[base+i]! != 255&&wFin W base (i+1) n
def zWin (g:D_C) (o y x:Nat):Ὡ:=
 g.dw&&g.k==g.s.kh*g.s.kw&&decide (0<g.s.kw)&&g.b[o]! ==0&&wFin g.W (o*g.k) 0 g.k&&
  zRows g.s g.X o (y*g.s.stride) (x*g.s.stride) 0 g.s.kh
def zChans (s:OS) (X:ѐ) (y0 x0:Nat) (i:Nat):Nat->Ὡ
 | 0=>true
 | n+1=>zRows s X i y0 x0 0 s.kh&&zChans s X y0 x0 (i+1) n
def zWinC (g:D_C) (o y x:Nat):Ὡ:=
 !g.dw&&g.k==g.s.cin*(g.s.kh*g.s.kw)&&decide (0<g.s.kw)&&g.b[o]! ==0&&wFin g.W (o*g.k) 0 g.k&&
  zChans g.s g.X (y*g.s.stride) (x*g.s.stride) 0 g.s.cin
@[noinline] def D_slow (g:D_C) (o y x:Ὣ):ὥ:=
 act g.relu (dotFn (g.k+1)
  (λt=>if t<g.k then windowAtO g.s g.X y x ((if g.dw then o*g.k else 0)+t) else oneW)
  (λt=>if t<g.k then g.W[o*g.k+t]! else g.b[o]!))
@[noinline] def D_slowZ (g:D_C) (o y x:Nat):ὥ:=if zWin g o y x || zWinC g o y x then 0 else D_slow g o y x
def zWinX (g:D_C) (o y x:Nat):Ὡ:=
 (x*g.s.stride+g.s.kw<=g.ex.1.getD (2*((if g.dw then o else 0)*g.ho+y)) 0 ||
  g.wp<=x*g.s.stride+g.ex.1.getD (2*((if g.dw then o else 0)*g.ho+y)+1) 0)&&g.b[o]! ==0
@[noinline] def zBlkG (g:D_C) (o y x:Nat):Ὡ:=
 (x*g.s.stride+g.s.kw+7<=g.ex.1.getD (2*((if g.dw then o else 0)*g.ho+y)) 0 ||
  g.wp<=x*g.s.stride+g.ex.1.getD (2*((if g.dw then o else 0)*g.ho+y)+1) 0)&&g.b[o]! ==0
@[inline] def zBlk (g:D_C) (o y x:Nat):Ὡ:=!g.ex.1.isEmpty&&zBlkG g o y x
def D_zt (xd:ὤ) (off:ѐ) (b:Nat) (i:Nat):Nat->Ὡ
 | 0=>true
 | n+1=>(decide (b+(off[i]!).toNat<xd.size)&&(xd.get! (b+(off[i]!).toNat)).toBits<<<(1:Ὠ)==0)&&
   D_zt xd off b (i+1) n
def D_ztF (xd:ὤ) (off:ѐ) (b:ὦ) (i e:ὦ):Ὡ:=
 if h:i<e then
  if h2:i.toNat<off.size then
   let p:=b+(off.uget i h2).toUSize
   if hp:b<=p/\p.toNat<xd.size then
    ((xd.uget p hp.2).toBits<<<(1:Ὠ)==0)&&D_ztF xd off b (i+1) e
   else false
  else false
 else true
termination_by e.toNat-i.toNat
decreasing_by
 have h1:=USize.lt_iff_toNat_lt.mp h
 have h3:=e.toNat_lt_size
 have:(i+1).toNat=i.toNat+1:=by
  rw[USize.toNat_add,ὄ];exact Ἁ (by rw[<-USize.size_eq_two_pow];omega)
 omega
lemma uszNW (b o:ὦ) (h:b<=b+o):(b+o).toNat=b.toNat+o.toNat:=by
 have hb:=b.toNat_lt_size;have ho:=o.toNat_lt_size
 rw[USize.size_eq_two_pow] at hb ho
 have h1:=USize.le_iff_toNat_le.mp h
 rw[USize.toNat_add] at h1⊢
 by_cases hc:b.toNat+o.toNat<2^System.Platform.numBits
 · exact Ἁ hc
 · exfalso
   have hc':2^System.Platform.numBits<=b.toNat+o.toNat:=א
   rw[Nat.mod_eq_sub_mod hc',Ἁ א] at h1
   omega
lemma D_ztF_imp (xd:ὤ) (off:ѐ) (b e:ὦ):
  ∀n (i:ὦ),i.toNat+n=e.toNat->D_ztF xd off b i e=true->D_zt xd off b.toNat i.toNat n=true:=by
 intro n
 induction n with
 | zero=>intro i _ _;rfl
 | succ n ih=>
   intro i hn h
   have hie:i<e:=USize.lt_iff_toNat_lt.mpr (by exact omS136 hn)
   have hi1:(i+1).toNat=i.toNat+1:=by
    have h3:=e.toNat_lt_size
    rw[USize.toNat_add,ὄ];exact Ἁ (by rw[<-USize.size_eq_two_pow];omega)
   rw[D_ztF,dif_pos hie] at h
   by_cases h2:i.toNat<off.size
   · rw[dif_pos h2] at h
     dsimp only at h
     have hu:(off.uget i h2)=off[i.toNat]!:=by
      simp only[Array.uget,Ἕ off i.toNat h2]
     by_cases h3:b<=b+(off.uget i h2).toUSize/\(b+(off.uget i h2).toUSize).toNat<xd.size
     · have hp:(b+(off.uget i h2).toUSize).toNat=b.toNat+(off[i.toNat]!).toNat:=by
        rw[uszNW _ _ h3.1,ὃ,hu]
       rw[dif_pos h3] at h
       simp only[Bool.and_eq_true,beq_iff_eq] at h
       obtain ⟨h4,h5⟩:=h
       simp only[D_zt,Bool.and_eq_true,decide_eq_true_eq,beq_iff_eq]
       refine ⟨⟨by rw[<-hp];exact h3.2,?_⟩,?_⟩
       · have g:xd.get! (b.toNat+(off[i.toNat]!).toNat)=xd.uget (b+(off.uget i h2).toUSize) h3.2:=by
          rw[<-hp];cases xd with | mk d=>simp only[FloatArray.get!,FloatArray.uget];exact Ἕ d _ h3.2
         rw[g];exact h4
       · rw[<-hi1];exact ih (i+1) א h5
     · rw[dif_neg h3] at h;exact absurd h (by simp only[false_eq_true,not_false_eq_true])
   · rw[dif_neg h2] at h;exact absurd h (by simp only[false_eq_true,not_false_eq_true])
lemma D_ztF_imp0 (xd:ὤ) (off:ѐ) (b e:ὦ) (h:D_ztF xd off b 0 e=true):
  D_zt xd off b.toNat 0 e.toNat=true:=by
 have:=D_ztF_imp xd off b e e.toNat 0 (by simp only[USize.toNat_zero,zero_add]) h
 simpa only[USize.toNat_zero] using this
@[inline] def zbits (xd:ὤ) (p:ὦ):Ὠ:=
 if h:p.toNat<xd.size then (xd.uget p h).toBits else 2
def D_ztB (xd:ὤ) (off:ѐ) (b:ὦ) (i e:ὦ):Ὡ:=
 if h:i<e then
  if h2:i.toNat<off.size then
   let p:=b+(off.uget i h2).toUSize
   if b<=p/\p<=p+7 then
    ((zbits xd p ||| zbits xd (p+1) ||| zbits xd (p+2) ||| zbits xd (p+3) ||| zbits xd (p+4) |||
      zbits xd (p+5) ||| zbits xd (p+6) ||| zbits xd (p+7))<<<(1:Ὠ)==0)&&D_ztB xd off b (i+1) e
   else false
  else false
 else true
termination_by e.toNat-i.toNat
decreasing_by
 have h1:=USize.lt_iff_toNat_lt.mp h
 have h3:=e.toNat_lt_size
 have:(i+1).toNat=i.toNat+1:=by
  rw[USize.toNat_add,ὄ];exact Ἁ (by rw[<-USize.size_eq_two_pow];omega)
 omega
lemma zbOr0 (a b:Ὠ) (h:a ||| b=0):a=0/\b=0:=by
 have h1:(a ||| b).toBitVec=0:=by rw[h];rfl
 rw[UInt64.toBitVec_or] at h1
 have:=BitVec.or_eq_zero_iff.mp h1
 exact ⟨UInt64.toBitVec_inj.mp (by rw[this.1];rfl),UInt64.toBitVec_inj.mp (by rw[this.2];rfl)⟩
lemma zbOrS (a b:Ὠ) (h:(a ||| b)<<<(1:Ὠ)=0):a<<<(1:Ὠ)=0/\b<<<(1:Ὠ)=0:=by
 rw[UInt64.shiftLeft_or] at h;exact zbOr0 _ _ h
lemma zbP7 (p:ὦ) (h:p<=p+7) (c:ὦ) (hc:c.toNat<=7):(p+c).toNat=p.toNat+c.toNat:=by
 have h7:=uszNW p 7 h
 have e7:(7:ὦ).toNat=7:=by simp only[USize.reduceToNat]
 rw[e7] at h7
 have hl:=(p+7).toNat_lt_size
 rw[h7,USize.size_eq_two_pow] at hl
 rw[USize.toNat_add]
 exact Ἁ א
lemma zbOr8 (z0 z1 z2 z3 z4 z5 z6 z7:Ὠ)
  (h:(z0 ||| z1 ||| z2 ||| z3 ||| z4 ||| z5 ||| z6 ||| z7)<<<(1:Ὠ)=0):
  z0<<<(1:Ὠ)=0/\z1<<<(1:Ὠ)=0/\z2<<<(1:Ὠ)=0/\z3<<<(1:Ὠ)=0/\
  z4<<<(1:Ὠ)=0/\z5<<<(1:Ὠ)=0/\z6<<<(1:Ὠ)=0/\z7<<<(1:Ὠ)=0:=by
 obtain ⟨h,a7⟩:=zbOrS _ _ h
 obtain ⟨h,a6⟩:=zbOrS _ _ h
 obtain ⟨h,a5⟩:=zbOrS _ _ h
 obtain ⟨h,a4⟩:=zbOrS _ _ h
 obtain ⟨h,a3⟩:=zbOrS _ _ h
 obtain ⟨h,a2⟩:=zbOrS _ _ h
 obtain ⟨a0,a1⟩:=zbOrS _ _ h
 exact ⟨a0,a1,a2,a3,a4,a5,a6,a7⟩
lemma zbits_ok (xd:ὤ) (p:ὦ) (h:zbits xd p<<<(1:Ὠ)=0):
  p.toNat<xd.size/\(xd.get! p.toNat).toBits<<<(1:Ὠ)=0:=by
 unfold zbits at h
 split at h
 · rename_i hp
   refine ⟨hp,?_⟩
   have g:xd.get! p.toNat=xd.uget p hp:=by
    cases xd with | mk d=>simp only[FloatArray.get!,FloatArray.uget];exact Ἕ d _ hp
   rw[g];exact h
 · exact absurd h ה
lemma D_ztB_imp (xd:ὤ) (off:ѐ) (b e:ὦ):
  ∀n (i:ὦ),i.toNat+n=e.toNat->D_ztB xd off b i e=true->
   ∀k,k<8->D_zt xd off (b.toNat+k) i.toNat n=true:=by
 intro n
 induction n with
 | zero=>intro i _ _ k _;rfl
 | succ n ih=>
   intro i hn h k hk
   have hie:i<e:=USize.lt_iff_toNat_lt.mpr (by exact omS136 hn)
   have hi1:(i+1).toNat=i.toNat+1:=by
    have h3:=e.toNat_lt_size
    rw[USize.toNat_add,ὄ];exact Ἁ (by rw[<-USize.size_eq_two_pow];omega)
   rw[D_ztB,dif_pos hie] at h
   by_cases h2:i.toNat<off.size
   · rw[dif_pos h2] at h
     dsimp only at h
     have hu:(off.uget i h2)=off[i.toNat]!:=by
      simp only[Array.uget,Ἕ off i.toNat h2]
     by_cases h3:b<=b+(off.uget i h2).toUSize/\b+(off.uget i h2).toUSize<=b+(off.uget i h2).toUSize+7
     · rw[Ὦ h3] at h
       simp only[Bool.and_eq_true,beq_iff_eq] at h
       obtain ⟨h4,h5⟩:=h
       generalize hP:b+(off.uget i h2).toUSize=P at h3 h4
       have hp:P.toNat=b.toNat+(off[i.toNat]!).toNat:=by
        rw[<-hP,uszNW _ _ (hP▸h3.1),ὃ,hu]
       obtain ⟨a0,a1,a2,a3,a4,a5,a6,a7⟩:=zbOr8 _ _ _ _ _ _ _ _ h4
       have lane: ∀c:ὦ,c.toNat<8->zbits xd (P+c)<<<(1:Ὠ)=0:=by
        intro c hc
        have hc':c=(c.toNat:Nat).toUSize:=by simp only[Nat.toUSize_eq,USize.ofNat_toNat]
        generalize hcn:c.toNat=m at hc hc'
        subst hc'
        interval_cases m
        · have e0:Nat.toUSize 0=0:=rfl
          rw[e0,USize.add_zero];exact a0
        · exact a1
        · exact a2
        · exact a3
        · exact a4
        · exact a5
        · exact a6
        · exact a7
       have kU:(k.toUSize).toNat=k:=by
        rw[Nat.toUSize_eq];exact USize.toNat_ofNat_of_lt' (by have:=USize.le_size;omega)
       obtain ⟨z1,z2⟩:=zbits_ok xd _ (lane k.toUSize א)
       rw[zbP7 P h3.2 _ א,kU] at z1 z2
       simp only[D_zt,Bool.and_eq_true,decide_eq_true_eq,beq_iff_eq]
       refine ⟨⟨by rw[Nat.add_right_comm,<-hp];exact z1,by rw[Nat.add_right_comm,<-hp];exact z2⟩,?_⟩
       rw[<-hi1];exact ih (i+1) (by exact omS157 hn hi1) h5 k hk
     · rw[Ὧ h3] at h;exact absurd h (by simp only[false_eq_true,not_false_eq_true])
   · rw[dif_neg h2] at h;exact absurd h (by simp only[false_eq_true,not_false_eq_true])
lemma D_ztB_imp0 (xd:ὤ) (off:ѐ) (b e:ὦ) (h:D_ztB xd off b 0 e=true) (k:Nat) (hk:k<8):
  D_zt xd off (b.toNat+k) 0 e.toNat=true:=by
 have:=D_ztB_imp xd off b e e.toNat 0 (by simp only[USize.toNat_zero,zero_add]) h k hk
 simpa only[USize.toNat_zero] using this
@[noinline] def D_slowY (c:D_P) (o y x:Nat) (b:ὦ):ὥ:=
 if zWinX c.g o y x then 0
 else if c.g.b[o]! ==0&&D_zt c.g.xd c.off b.toNat 0 c.e.toNat then 0
 else D_slowZ c.g o y x
@[inline] def D_zl (q n:ὦ) (k:Ὣ) (out:ѐ):ѐ:=if ὴ k<n then wrW out (q+ὴ k) 0 else out
@[inline] def D_z8 (q n:ὦ) (out:ѐ):ѐ:=
 D_zl q n 7 (D_zl q n 6 (D_zl q n 5 (D_zl q n 4 (D_zl q n 3 (D_zl q n 2 (D_zl q n 1 (D_zl q n 0 out)))))))
@[noinline] def D_finT (c:D_P) (o y x:Ὣ) (b:ὦ):ὥ:=
 if hb:b<c.l1 then
  let q:=D_goT c.g.xd c.wd c.off b c.e 0 c.he c.hw c.hs (c.h1 b hb) c.ze c.ze
  let r:=ziv (q.s+c.bv) (c.g.nf*(q.t+c.bv.abs))
  if r==Е then (if q.t.toBits==0 then D_slowY c o y x b else D_slow c.g o y x) else act c.g.relu r
 else D_slow c.g o y x
@[inline] def D_fin (c:D_P) (o y x:Ὣ) (b:ὦ) (a:ὧ):ὥ:=
 let r:=ziv (a+c.bv) c.e0
 if r==Е then (if (a+c.bv).toBits==0 then D_slowY c o y x b else D_finT c o y x b) else act c.g.relu r
@[noinline] def D_finK (c:D_P) (o y x k:Ὣ) (b:ὦ) (a:ὧ):ὥ:=
 D_fin c o y (x+k) (b+ὴ k) a
@[inline] def D_w (c:D_P) (lo sp:Ὠ) (o y x:Ὣ) (b:ὦ) (k:Ὣ) (a:ὧ):ὥ:=
 let w:=(a+c.bv).toBits
 let u:=w+0x0FDFFFFF
 if (w<<<(1:Ὠ))-lo<sp&&u&&&0x1FFFFFFF<0x1FC00000 then
  zivC (u>>>(29:Ὠ))
 else if w<<<(1:Ὠ)==0 then (if c.g.b[o]! ==0&&D_ztF c.g.xd c.off (b+ὴ k) 0 c.e then 0 else D_slowY c o y (x+k) (b+ὴ k)) else D_finK c o y x k b a
@[inline] def D_l (c:D_P) (lo sp:Ὠ) (o y x:Ὣ) (b q n:ὦ) (k:Ὣ) (a:ὧ) (out:ѐ):
 ѐ:=
 if ὴ k<n then wrW out (q+ὴ k) (D_w c lo sp o y x b k a) else out
lemma lcGq {a:ѐ} {q:ὦ} (h:q<a.usize/\8<=a.usize-q) (k:Ὣ) (hk:k<8):(q+ὴ k).toNat< ѧa:=by
 have h1:=А.mp h.1
 have h2:=USize.le_iff_toNat_le.mp h.2
 have h8:(8:ὦ).toNat=8:=uk 8 ה
 rw[USize.toNat_sub_of_le _ _ (USize.le_iff_toNat_le.mpr (Nat.le_of_lt h1)),h8] at h2
 have h3:a.usize.toNat<=ѧa:=by simp only[Array.usize,Nat.toUSize_eq,USize.toNat_ofNat'];exact Nat.mod_le _ _
 have h4:=a.usize.toNat_lt_size;have h5:USize.size=2^System.Platform.numBits:=rfl
 rw[USize.toNat_add,uk k א,Ἁ א]
 omega
lemma lcN8 {n:ὦ} (h8:8<=n) (k:Ὣ) (hk:k<8):ὴ k<n:=by
 have h:=USize.le_iff_toNat_le.mp h8;rw[show (8:ὦ).toNat=8 from uk 8 ה] at h;exact А.mpr (by rw[uk k א];omega)
lemma lcWU (a:ѐ) (i:ὦ) (v:ὥ) (h:Ѥi< ѧa) (hs:ѧa<2^32):a.uset i v h=wrW a i v:=by
 have h2:=USize.le_size;have h3:USize.size=2^System.Platform.numBits:=rfl
 unfold wrW;rw[Ἄ (А.mpr (by simp only[Array.usize,Nat.toUSize_eq,USize.toNat_ofNat'];rw[Ἁ א];exact h))]
@[inline] def D_g8a (c:D_P) (lo sp:Ὠ) (o y x:Ὣ) (b q n:ὦ) (r:D_R) (out:ѐ):ѐ:=
 if hq:8<=n/\q<out.usize/\8<=out.usize-q then
  ((((((((out.uset (q+ὴ 0) (D_w c lo sp o y x b 0 r.A) (lcGq hq.2 0 ה)).uset (q+ὴ 1) (D_w c lo sp o y x b 1 r.B) (by simp only[Array.size_uset];exact lcGq hq.2 1 ה)).uset (q+ὴ 2) (D_w c lo sp o y x b 2 r.C) (by simp only[Array.size_uset];exact lcGq hq.2 2 ה)).uset (q+ὴ 3) (D_w c lo sp o y x b 3 r.D) (by simp only[Array.size_uset];exact lcGq hq.2 3 ה)).uset (q+ὴ 4) (D_w c lo sp o y x b 4 r.E) (by simp only[Array.size_uset];exact lcGq hq.2 4 ה)).uset (q+ὴ 5) (D_w c lo sp o y x b 5 r.F) (by simp only[Array.size_uset];exact lcGq hq.2 5 ה)).uset (q+ὴ 6) (D_w c lo sp o y x b 6 r.G) (by simp only[Array.size_uset];exact lcGq hq.2 6 ה)).uset (q+ὴ 7) (D_w c lo sp o y x b 7 r.I) (by simp only[Array.size_uset];exact lcGq hq.2 7 ה))
 else
  let out:=D_l c lo sp o y x b q n 0 r.A out
  let out:=D_l c lo sp o y x b q n 1 r.B out
  let out:=D_l c lo sp o y x b q n 2 r.C out
  let out:=D_l c lo sp o y x b q n 3 r.D out
  let out:=D_l c lo sp o y x b q n 4 r.E out
  let out:=D_l c lo sp o y x b q n 5 r.F out
  let out:=D_l c lo sp o y x b q n 6 r.G out
  let out:=D_l c lo sp o y x b q n 7 r.I out
  out
@[noinline] def D_zbk (c:D_P) (o:Nat) (b:ὦ):Ὡ:=c.g.b[o]! ==0&&D_ztB c.g.xd c.off b 0 c.e
@[inline] def D_g8 (c:D_P) (lo sp:Ὠ) (o y x:Ὣ) (b q n:ὦ) (r:D_R) (out:ѐ):ѐ:=
 if r.A.toBits<<<(1:Ὠ)==0&&D_zbk c o b then D_z8 q n out
 else D_g8a c lo sp o y x b q n r out
def D_cols (c:D_P) (lo sp:Ὠ) (o y x:Ὣ) (b q:ὦ) (rr:D_R) (out:ѐ):
 ѐ:=
 if h8:x<c.g.wo/\c.g.f4=true/\b<c.l8 then
  let r:=D_go8R c.g.xd c.wd c.off b c.e 0 c.he c.hw c.hs (c.h8 b h8.2.2) c.ze c.ze c.ze c.ze
   c.ze c.ze c.ze c.ze rr
  let n:=(c.g.wo-x).toUSize
  D_cols c lo sp o y (x+8) (b+8) (q+8) r (D_g8 c lo sp o y x b q n r out)
 else if x<c.g.wo then
  D_cols c lo sp o y (x+1) (b+c.g.step) (q+1) rr (wrW out q (D_finT c o y x b))
 else out
partial_fixpoint
def D_rc (c:D_P) (lo sp:Ὠ) (o y x:Ὣ) (b q q0:ὦ) (rr:D_R) (out:ѐ):
 ѐ:=
 if hy:y<c.g.ho then
  if h8:x<c.g.wo/\c.g.f4=true/\b<c.l8 then
   let r:=D_go8R c.g.xd c.wd c.off b c.e 0 c.he c.hw c.hs (c.h8 b h8.2.2) c.ze c.ze c.ze c.ze
    c.ze c.ze c.ze c.ze rr
   let n:=(c.g.wo-x).toUSize
   D_rc c lo sp o y (x+8) (b+8) (q+8) q0 r (D_g8 c lo sp o y x b q n r out)
  else if x<c.g.wo then
   D_rc c lo sp o y (x+1) (b+c.g.step) (q+1) q0 rr (wrW out q (D_finT c o y x b))
  else
   D_rc c lo sp o (y+1) 0 ((y+1).toUSize*c.g.step*c.g.wp.toUSize) (q0+c.g.wo.toUSize)
    (q0+c.g.wo.toUSize) rr out
 else out
partial_fixpoint
def D_rows (c:D_P) (lo sp:Ὠ) (o y:Ὣ) (q:ὦ) (rs:List D_R) (out:ѐ):ѐ:=
 if y<c.g.ho then
  D_rows c lo sp o (y+1) (q+c.g.wo.toUSize) rs.tail
   (D_cols c lo sp o y 0 (ѩy*c.g.step*c.g.wp.toUSize) q
    (rs.headD ⟨0,0,0,0,0,0,0,0,0⟩) out)
 else out
partial_fixpoint
def D_colsZ (c:D_P) (lo sp:Ὠ) (o y x:Ὣ) (b q:ὦ) (rr:D_R) (out:ѐ):
 ѐ:=
 if h8:x<c.g.wo/\c.g.f4=true/\b<c.l8 then
  if zBlk c.g o y x then D_colsZ c lo sp o y (x+8) (b+8) (q+8) rr (D_z8 q (c.g.wo-x).toUSize out) else
  let r:=D_go8R c.g.xd c.wd c.off b c.e 0 c.he c.hw c.hs (c.h8 b h8.2.2) c.ze c.ze c.ze c.ze
   c.ze c.ze c.ze c.ze rr
  let n:=(c.g.wo-x).toUSize
  D_colsZ c lo sp o y (x+8) (b+8) (q+8) r (D_g8a c lo sp o y x b q n r out)
 else if x<c.g.wo then
  D_colsZ c lo sp o y (x+1) (b+c.g.step) (q+1) rr (wrW out q (D_finT c o y x b))
 else out
partial_fixpoint
def D_rcZ (c:D_P) (lo sp:Ὠ) (o y x:Ὣ) (b q q0:ὦ) (rr:D_R) (out:ѐ):
 ѐ:=
 if hy:y<c.g.ho then
  if h8:x<c.g.wo/\c.g.f4=true/\b<c.l8 then
   if zBlk c.g o y x then D_rcZ c lo sp o y (x+8) (b+8) (q+8) q0 rr (D_z8 q (c.g.wo-x).toUSize out) else
   let r:=D_go8R c.g.xd c.wd c.off b c.e 0 c.he c.hw c.hs (c.h8 b h8.2.2) c.ze c.ze c.ze c.ze
    c.ze c.ze c.ze c.ze rr
   let n:=(c.g.wo-x).toUSize
   D_rcZ c lo sp o y (x+8) (b+8) (q+8) q0 r (D_g8a c lo sp o y x b q n r out)
  else if x<c.g.wo then
   D_rcZ c lo sp o y (x+1) (b+c.g.step) (q+1) q0 rr (wrW out q (D_finT c o y x b))
  else
   D_rcZ c lo sp o (y+1) 0 ((y+1).toUSize*c.g.step*c.g.wp.toUSize) (q0+c.g.wo.toUSize)
    (q0+c.g.wo.toUSize) rr out
 else out
partial_fixpoint
def D_rowsZ (c:D_P) (lo sp:Ὠ) (o y:Ὣ) (q:ὦ) (rs:List D_R) (out:ѐ):ѐ:=
 if y<c.g.ho then
  D_rowsZ c lo sp o (y+1) (q+c.g.wo.toUSize) rs.tail
   (D_colsZ c lo sp o y 0 (ѩy*c.g.step*c.g.wp.toUSize) q
    (rs.headD ⟨0,0,0,0,0,0,0,0,0⟩) out)
 else out
partial_fixpoint
lemma qGn {n m:Ὣ} (d:Ὣ) (off:ѐ) (hs:n<2^32) (hm:∀j (h:j< ѧoff),off[j].toNat<=m) (b:ὦ) (hb:b<(n-(m+d)).toUSize) (j:Ὣ) (h:j< ѧoff):Ѥb+off[j].toNat+d<n:=by have h1:=А.1 hb;rw[qAa _ (by clear hb h h1;exact omS373 hs)] at h1;have:=hm j h;(clear hs hb;omega)
lemma qJv {off:ѐ} {m f:ὥ} {c:ὦ} (hm:∀j (h:j< ѧoff),off[j].toNat<=ѥm): ∀j (h:j<(wrW off c f).size),(wrW off c f)[j].toNat<=(if m<f then f else m).toNat:=by
 intro j h;have e:(if m<f then f else m).toNat=max ѥm ѥf:=by split<;>rename_i hc<;>rw[ἳ] at hc<;>(clear h;omega)
 rw[e];by_cases hc:c<off.usize
 . have e2:wrW off c f=off.uset c f (qAp off c hc):=by unfold wrW;rw[Ἄ hc]
   have h0:j< ѧoff:=by rw[e2] at h;simpa only[Array.uset_eq_set,Array.size_set] using h
   have g:=hm j h0;simp only[e2,Array.uset,Array.getElem_set];split<;>omega
 . have e2:wrW off c f=off:=by unfold wrW;rw[ἔ hc]
   have h0:j< ѧoff:=by rw[e2] at h;exact h
   have g:=hm j h0;simp only[e2];(clear e hc;omega)
lemma qF0 (k:Ὣ):∀j (h:j<(ἇ k (0:ὥ)).size),(ἇ k (0:ὥ))[j].toNat<=(0:ὥ).toNat:=by intro j h;simp only[Array.getElem_replicate,UInt32.toNat_zero,Std.le_refl]
def D_dec (tab W:ѐ) (base:ὥ) (o n t:ὦ) (off:ѐ) (buf:ὤ)
 (c:ὦ) (l:ὧ) (m:ὥ) (hm:∀j (h:j< ѧoff),off[j].toNat<=ѥm):R2:=
 if h:t<n then
  let w:=rdW W (o+t)
  if w&&&В==0 then D_dec tab W base o n (t+1) off buf c l m hm
  else
   let v:=dv w
   let f:=base+rdW tab t
   D_dec tab W base o n (t+1) (wrW off c f) (wrF buf c v) (c+1) (l+v.abs)
    (if m<f then f else m) (qJv hm)
 else ⟨off,buf,c,l,m,hm⟩
termination_by Ѥn-Ѥt
decreasing_by all_goals (rw[usz t n h];have:=А.mp h;omega)
local instance:Inhabited R2:=⟨⟨#[],FloatArray.empty,0,0,0,λ_ h=>absurd h (Ἲ _)⟩⟩
def D_decAll (tab W:ѐ) (dw:Ὡ) (s:OS) (wp k C:Ὣ):Array R2:=
 Array.ofFn (n:=C) λo=>D_dec tab W (if dw then (o.val*((s.h+2*s.pad)*wp)).toUInt32 else 0)
  (o.val*k).toUSize ѩk 0 (ἇ k 0) (pushZ ѩk 0 (ἆ k))
  0 0 0 (qF0 k)
def mkCV (W b:ѐ) (s:OS) (dw:Ὡ) (k C:Ὣ):CV:=
 let tab:=D_tq k s.kw s.kh (s.h+2*s.pad) (s.w+2*s.pad) 0 0 0 0 (Array.emptyWithCapacity k)
 ⟨scanH W,scanH b,tab,D_decAll tab W dw s (s.w+2*s.pad) k C⟩
abbrev CVV (W b:ѐ) (s:OS) (dw:Ὡ) (k C:Ὣ):=
 {v:CV // v=mkCV W b s dw k C}
local instance cvSub (W b:ѐ) (s:OS) (dw:Ὡ) (k C:Ὣ) :
 Subsingleton (CVV W b s dw k C):=⟨λa b=>Subtype.ext (a.2.trans b.2.symm)⟩
structure CE where
 (W b:ѐ) (s:OS) (dw:Ὡ) (k C:Ὣ) (v:CV) (g:ὦ)
 (ok:v=mkCV W b s dw k C)
@[inline] def sEq (a b:OS):Ὡ:=
 a.cin==b.cin&&a.h==b.h&&a.w==b.w&&a.cout==b.cout&&a.kh==b.kh&&a.kw==b.kw&&
  a.stride==b.stride&&a.pad==b.pad
lemma qH4 {a b:OS} (h:sEq a b=true):a=b:=by cases a;cases b;simp only[sEq,and_eq_true,beq_iff_eq] at h;obtain ⟨⟨⟨⟨⟨⟨⟨h1,h2⟩,h3⟩,h4⟩,h5⟩,h6⟩,h7⟩,h8⟩:=h;subst h1 h2 h3 h4 h5 h6 h7 h8;rfl
def cvF (W b:ѐ) (s:OS) (dw:Ὡ) (k C:Ὣ) (g:ὦ) (es:Array CE)
 (i:Ὣ):Option (CVV W b s dw k C):=
 if h:i< ѧes then
  if es[i].g==g then
   if hk:withPtrEq es[i].W W (λ_=>es[i].W==W) (λe=>by rw[e];exact beq_self_eq_true W) then
    if hb:withPtrEq es[i].b b (λ_=>es[i].b==b) (λe=>by rw[e];exact beq_self_eq_true b) then
     if hs:sEq es[i].s s&&es[i].dw==dw&&es[i].k==k&&es[i].C==C then
      some ⟨es[i].v,by
       simp only[and_eq_true,beq_iff_eq] at hs
       obtain ⟨⟨⟨h1,h2⟩,h3⟩,h4⟩:=hs
       rw[<-eq_of_beq hk,<-eq_of_beq hb,<-qH4 h1,<-h2,<-h3,<-h4];exact es[i].ok⟩
     else cvF W b s dw k C g es (i+1)
    else cvF W b s dw k C g es (i+1)
   else cvF W b s dw k C g es (i+1)
  else cvF W b s dw k C g es (i+1)
 else none
partial_fixpoint
@[noinline] def cvS:ST.Ref Unit (Array CE):=(ST.Prim.mkRef #[] (Void.mk ())).val
@[inline] def cvGet (W b:ѐ) (s:OS) (dw:Ὡ) (k C:Ὣ):CVV W b s dw k C:=
 withPtrAddr W (λg=>
  let es:=(ST.Prim.Ref.get cvS (Void.mk ())).val
  match cvF W b s dw k C g es 0 with
  | some v=>v
  | none=>
   let v:=mkCV W b s dw k C
   let old:=(ST.Prim.Ref.swap cvS (#[⟨W,b,s,dw,k,C,v,g,rfl⟩]++
    (if ѧes<256 then es else es.pop)) (Void.mk ())).val
   match cvF W b s dw k C g old 0 with
   | some v=>v
   | none=>⟨v,rfl⟩) (λ_ _=>Subsingleton.elim _ _)
def D_outsC (g:D_C) (hs:g.xd.size<2^32) (o:Ὣ) (q:ὦ) (rs:Array R2)
 (out:ѐ):ѐ:=
 if o<g.C then
  let r:=rs[o]!
  if h:r.c.toNat<=r.off.size/\r.c.toNat<=r.buf.size then
   let c:D_P:=⟨g,r.buf,r.off,r.c,(g.xd.size-(r.m.toNat+0)).toUSize,
    (g.xd.size-(r.m.toNat+7)).toUSize,g.nf*(dv g.hi*r.l+(dv (rdW g.b ѩo)).abs),
    dv (rdW g.b ѩo),0,h.1,h.2,hs,
    qGn 0 r.off hs r.hm,qGn 7 r.off hs r.hm,rfl⟩
   D_outsC g hs (o+1) (q+(g.ho*g.wo).toUSize) rs
    (if g.ex.1.isEmpty then D_rc c (loOf c.e0) (spR c.g.relu (loOf c.e0)) o 0 0 ((0:Ὣ).toUSize*c.g.step*c.g.wp.toUSize) q q
     ⟨0,0,0,0,0,0,0,0,0⟩ out else D_rcZ c (loOf c.e0) (spR c.g.relu (loOf c.e0)) o 0 0 ((0:Ὣ).toUSize*c.g.step*c.g.wp.toUSize) q q
     ⟨0,0,0,0,0,0,0,0,0⟩ out)
  else out
 else out
partial_fixpoint
def D_Fc (dw:Ὡ) (s:OS) (relu:Ὡ) (X W b:ѐ) (k C:Ὣ) (cv:CV):
 ѐ:=
 let hp:=s.h+2*s.pad
 let wp:=s.w+2*s.pad
 let rx:=D_scan X X.usize 0 (qAd X) Е 0
 if hg:ѧX<=4294967295/\0x007FFFFF<=rx.lo/\rx.hi<У/\ѧX=s.cin*s.h*s.w/\
  ѧW=C*k/\ѧb=C/\0<s.stride/\s.stride<=4294967295/\0<k/\k<2^20/\
  s.kh<=hp/\s.kw<=wp/\hp<=4294967295/\wp<=4294967295/\ѧW<=4294967295/\ѧb<=4294967295/\
  C*(s.ho*s.wo)<=4294967295/\s.cin*hp*wp<=4294967295/\
  cv.hW≠Е/\
  cv.hb≠Е then
  let tight:=(rx.hi>>>(23:ὥ))-((rx.lo+1)>>>(23:ὥ))>99
  let R:=s.cin*hp
  let xd:=D_xs s X (Nat.lt_of_le_of_lt hg.1 (by norm_num)) wp R
  if hs:Ѩxd<=4294967295 then
   let g:D_C:=⟨dw,relu,s.stride==1&&!tight,s,X,W,b,k,C,s.ho,s.wo,wp,
    s.stride.toUSize,rx.hi,xd,(k.toUInt64+8).toFloat*Ἃ 0x3CA0000000000000,
    cv.tab,mkExS s X b dw s.ho⟩
   D_outsC g (Nat.lt_of_le_of_lt hs (by norm_num)) 0 0 cv.rs (ἇ (C*(s.ho*s.wo)) 0)
  else if dw then depthwiseWithO dotFn s relu X W b else conv2dWithO dotFn s relu X W b
 else if dw then depthwiseWithO dotFn s relu X W b else conv2dWithO dotFn s relu X W b
def D_F (dw:Ὡ) (s:OS) (relu:Ὡ) (X W b:ѐ) (k C:Ὣ):
 ѐ:=
 D_Fc dw s relu X W b k C (cvGet W b s dw k C).1
def Cn1 (s:OS):OS:=
 if s.kh=1/\s.kw=1/\s.stride=1/\s.pad=0/\0<s.h/\0<s.w then {s with h:=1,w:=s.h*s.w} else s
def convF (s:OS) (relu:Ὡ) (X W b:ѐ):ѐ:=
 D_F false (Cn1 s) relu X W b (s.cin*s.kh*s.kw) s.cout
def depthF (s:OS) (relu:Ὡ) (X W b:ѐ):ѐ:=
 D_F true s relu X W b (s.kh*s.kw) s.cin
def D_px (s:OS) (X:ё) (c yy xx:Ὣ):Ὢ:=
 if yy<s.pad\/xx<s.pad\/yy-s.pad>=s.h\/xx-s.pad>=s.w then 0
 else X[(c*s.h+(yy-s.pad))*s.w+(xx-s.pad)]!
def D_Fx (s:OS) (X:ё) (q:Ὣ):ὧ:=
 dv (D_px s X (q/(s.w+2*s.pad)/(s.h+2*s.pad)) (q/(s.w+2*s.pad)%(s.h+2*s.pad))
  (q%(s.w+2*s.pad)))
def D_Pre (F:Ὣ->ὧ) (m:Ὣ) (a:ὤ):Prop:=
 Ѩa=m/\∀q,q< m->a.data[q]! =F q
lemma qCh {F:Ὣ->ὧ} {m:Ὣ} {a:ὤ} {v:ὧ} (h:D_Pre F m a) (hv:v=F m):D_Pre F (m+1) (a.push v):=by
 refine ⟨by rw[qA2,h.1],λq hq=>?_⟩;rw[qGr,h.1];split<;>rename_i e;. rw[e,hv]
 exact h.2 q (by exact omS374 hq e)
lemma qFp (m d:Ὣ):m+(d+1)=m+1+d:=(by exact omS375)
def D_Sp (F:Ὣ->ὧ) (m:Ὣ) (a:ὤ):Prop:=
 m<=Ѩa/\∀q,q< m->a.data[q]! =F q
structure D_Ok (g:D_C):Prop where
 hk:g.k=if g.dw then g.s.kh*g.s.kw else g.s.cin*g.s.kh*g.s.kw
 hC:g.C=if g.dw then g.s.cin else g.s.cout
 ho:g.ho=g.s.ho
 wo:g.wo=g.s.wo
 wp:g.wp=g.s.w+2*g.s.pad
 st:g.step.toNat=g.s.stride
 nf:g.nf=nfOf g.k
 f4:g.f4=true->g.s.stride=1
 p0:0<g.s.stride
 pk:0<g.k
 nk:g.k<2^20
 kh:g.s.kh<=g.s.h+2*g.s.pad
 kw:g.s.kw<=g.s.w+2*g.s.pad
 xd:D_Sp (D_Fx g.s g.X) (g.s.cin*(g.s.h+2*g.s.pad)*(g.s.w+2*g.s.pad)) g.xd
 xs:g.xd.size<2^32
 Hx:C_Hv g.X g.hi
 HW:∀q:Ὣ,good g.W[q]! =true
 Hb:∀q:Ὣ,good g.b[q]! =true
 sW:g.W.size=g.C*g.k
 sW2:g.W.size<2^32
 sb:g.b.size<2^32
 sN:g.C*(g.ho*g.wo)<2^32
 wp2:g.wp<2^32
 tb:g.tab=Array.ofFn (n:=g.k) λt=>(D_off g.s t.val).toUInt32
 xv:∀q,q< Ѩg.xd->∃u,g.xd.data[q]! =dv u
lemma qCk {i e:ὦ} (h:Ѥe<=Ѥi):¬i<e:=mt А.1 (Nat.not_lt.2 h)
lemma qBz {i e:ὦ} {d:Ὣ} (h:Ѥi+(d+1)=Ѥe):i<e/\(i+1).toNat+d=Ѥe:=by have t:=А.2 ((by exact omS136 h):Ѡi< Ѡe);exact ⟨t,by rw[usz i e t];(clear t;exact omS137 h)⟩
lemma qCc {F:Ὣ->ὧ} {v:ὧ} {n:ὦ} (d:Ὣ):
 ∀(k:ὦ) (m:Ὣ) (a:ὤ),Ѥk+d=Ѥn->D_Pre F m a->(∀i,m<=i->i< m+d->F i=v)->D_Pre F (m+d) (D_fill v n k a):=by
 induction' d with d ih<;>intro k m a he H hv
 . rw[D_fill,ἔ (qCk he.ge)];exact H
 obtain ⟨ht,hu⟩:=qBz he
 rw[D_fill,Ἄ ht,qFp]
 exact ih (k+1) (m+1) _ hu (qCh H (hv m (by exact Nat.le_refl _) (by clear he ht;(clear hu;omega))).symm) λi h1 h2=>hv i (by exact Nat.le_of_lt ‹_›) (by clear he ht h1;(clear hu;omega))
lemma qEr {F:Ὣ->ὧ} {X:ѐ} {o n:ὦ} (hs:ѧX<2^32) (ho:Ѥo+Ѥn<=ѧX) (d:Ὣ):
 ∀(t:ὦ) (m:Ὣ) (a:ὤ),Ѥt+d=Ѥn->D_Pre F m a->(∀j,j<d->F (m+j)=dv X[Ѥo+Ѥt+j]!)->D_Pre F (m+d) (D_copy X o n t hs ho a):=by
 induction' d with d ih<;>intro t m a he H hv
 . rw[D_copy,ἔ (qCk he.ge)];exact H
 obtain ⟨ht,hu⟩:=qBz he
 rw[D_copy,Ἄ ht,qFp]
 refine ih (t+1) (m+1) _ hu (qCh H ?_) λj hj=>?_
 . rw[qFx,qAv _ _ (by clear ht;omega)];exact (hv 0 (by clear he ht hu;exact omS9)).symm
 simpa only[usz t n ht,ὢ,Nat.add_comm 1] using hv (j+1) (by exact Nat.lt_succ_of_le ‹_›)
lemma qCn {c:Ὣ} (a:Ὣ) {b:Ὣ} (h:b<c):(a*c+b)/c=a/\(a*c+b)%c=b:=by rw[ἥ a c,Nat.mul_add_div (by exact omS376 h),Nat.mul_add_mod,Ὃ h,Ἁ h];exact ⟨rfl,rfl⟩
lemma qAi (s:OS) (X:ё) (r:Ὣ) {i:Ὣ} (hi:i<s.w+2*s.pad):
 D_Fx s X (r*(s.w+2*s.pad)+i)=dv (D_px s X (r/(s.h+2*s.pad)) (r%(s.h+2*s.pad)) i):=by
 simp only[D_Fx,qCn r hi]
lemma qFo {F:Ὣ->ὧ} {v:ὧ} {n:ὦ} {m d:Ὣ} {a:ὤ} (hn:Ѥn=d) (H:D_Pre F m a) (hv:∀u,u<d->F (m+u)=v):D_Pre F (m+d) (D_fill v n 0 a):=
 qCc d 0 m a (by rw[hn];exact Ἇ d) H λi h1 h2=>by
  obtain ⟨u,rfl⟩:=Nat.exists_eq_add_of_le h1
  exact hv u (by clear hn h1;exact omS377 h2)
lemma qF5 (m p w:Ὣ):m+(w+2*p)=m+p+w+p:=(by exact omS378)
lemma qE0 (s:OS) (X:ѐ) (hs:ѧX<2^32) (hX:ѧX=s.cin*s.h*s.w) (hh:s.h+2*s.pad<2^32) (hw:s.w+2*s.pad<2^32) {R:Ὣ} (hR:R=s.cin*(s.h+2*s.pad)) (d:Ὣ):
 ∀(r:Ὣ) (a:ὤ),r+d=R->D_Pre (D_Fx s X) (r*(s.w+2*s.pad)) a->D_Pre (D_Fx s X) (R*(s.w+2*s.pad)) (D_img s X hs (dv 0) R r a):=by
 induction' d with d ih<;>intro r a he H
 . rw[D_img,ἔ (by clear hs hX hh hw;(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›)))),<-he];exact H
 have hr:r<R:=(by clear hX hh hw;exact omS136 he)
 have hc:=Ἰ (hr.trans_eq (hR.trans (ἥ _ _)))
 have ep:=qAa s.pad (by clear hX hw hR he hr hc;exact omS379 hh)
 have ew:=qAa s.w (by clear hX hR he hr hc;exact omS380 hw)
 have hO:=λg:r%(s.h+2*s.pad)-s.pad<s.h=>qBv (qDf hc g) (Nat.le_refl s.w)
 rw[D_img,Ἄ hr]
 dsimp only
 split<;>rename_i g
 . obtain ⟨g1,g2,g3⟩:=g;have:=hO g2;apply ih (r+1) _ (by clear hX hh hw hr hc ep ew g1 g2 g3 this;exact omS137 he);rw[ᾀ,qF5];refine qFo ep (qEr hs g3 s.w 0 _ _ (by rw[ew];exact Ἇ _) (qFo ep H λu hu=>?_) λj hj=>?_) λu hu=>?_;. rw[qAi _ _ _ (by clear hX hh hw hR he hr hc g1 g2 g3 this;(clear ep;exact omS381 hu)),D_px,ἱ (by clear hX hh hw hR he hr hc ew g1 g2 g3 this;(clear ep;exact omS382 hu))]
   . rw[ὢ,qAi _ _ _ (by clear hX hw hR he hr hc g1 g2 g3 this;(clear ew;exact omS383 hj)),D_px,ἐ (by clear hX hh hw hR he hr hc g3 this;(clear ep ew;exact omS384 g1 g2 hj)),Nat.add_sub_cancel_left,qAa _ (by clear hR he hr hc g1 g2 g3 hj;exact omS385 hs hX this)];rfl
   . rw[ὢ,ὢ,qAi _ _ _ (by clear hX hh hw hR he hr hc ew g1 g2 g3 this;(clear ep;exact omS386 hu)),D_px,ἱ (by clear hh hw hR he hr hc ep g1 g2 this hu;exact omS387)]
 apply ih (r+1) _ (by clear hX hh hw hr hc ep ew g;exact omS137 he)
 rw[ᾀ]
 exact qFo (qAa _ hw) H λu hu=>by
  rw[qAi _ _ _ hu,D_px,ἱ]
  by_contra n
  have:=hO (by clear hX hh hw hR he hr hc ep ew g hu;exact omS388 n)
  exact g ⟨by (clear hX hh hw hR he hr hc ew g hu this;(clear ep;exact omS389 n)),by (clear hX hh hw hR he hr hc ep ew g hu this;exact omS388 n),by rw[qAa _ (by clear hh hw hR he hr hc ep g hu n;exact omS385 hs hX this),ew,hX];exact this⟩
def D_t0 (g:D_C) (o:Ὣ):Ὣ:=if g.dw then o*g.k else 0
def D_xw (g:D_C) (o y x t:Ὣ):Ὢ:=windowAtO g.s g.X y x (D_t0 g o+t)
def D_out (g:D_C) (o y x:Ὣ):Ὢ:=
 act g.relu (dotOf (g.k+1) (λt=>if t<g.k then D_xw g o y x t else oneW)
  (λt=>if t<g.k then g.W[o*g.k+t]! else g.b[o]!))
structure D_PO (c:D_P) (o:Ὣ) (L:љ):Prop where
 ok:D_Ok c.g
 hC:o<c.g.C
 len:c.e.toNat=ѣL
 nd:L.Nodup
 lt:∀t∈L,t<c.g.k
 hz:∀t,t<c.g.k->t∉L->c.g.W[o*c.g.k+t]!&&&В=0
 rd:∀j p:Ὣ,L[j]?=some p->
  (c.off[j]!:ὥ).toNat=D_off c.g.s (D_t0 c.g o+p)/\
   c.wd.data[j]! =dv c.g.W[o*c.g.k+p]!
 e0:c.e0=nfOf c.g.k*(dv c.g.hi*L.foldl (λacc t=>acc+(dv c.g.W[o*c.g.k+t]!).abs) 0+
  (dv c.g.b[o]!).abs)
 bv:c.bv=dv c.g.b[o]!
lemma qCv (g:D_C) (o y x:Ὣ):D_slow g o y x=D_out g o y x:=by unfold D_slow D_out D_xw D_t0;rw[qAg]
lemma zAll_spec (fa fb:Nat->ὥ) (N:Nat): ∀n i,i+n<=N->zAll fa fb i n=true->
  dsum (Array.ofFn (n:=N) λj=>fa j) (Array.ofFn (n:=N) λj=>fb j) i n=0/\
  dallNZ (Array.ofFn (n:=N) λj=>fa j) (Array.ofFn (n:=N) λj=>fb j) i n=zNeg fa fb i n/\
  (∀j,i<=j->j< i+n->expOf (fa j)≠255/\expOf (fb j)≠255):=by
 intro n
 induction n with
 | zero=>intro i _ _;exact ⟨rfl,rfl, λj h1 h2=>absurd h2 א⟩
 | succ n ih=>
  intro i hi h
  simp only[zAll,Bool.and_eq_true,bne_iff_ne,ne_eq,beq_iff_eq] at h
  obtain ⟨⟨⟨h1,h2⟩,h3⟩,h4⟩:=h
  obtain ⟨r1,r2,r3⟩:=ih (i+1) (by exact omS5 hi) h4
  have ea:(Array.ofFn (n:=N) λj=>fa j)[i]! =fa i:=Pinned.getElem!_ofFn N fa i (by exact omS4 hi)
  have eb:(Array.ofFn (n:=N) λj=>fb j)[i]! =fb i:=Pinned.getElem!_ofFn N fb i (by exact omS4 hi)
  have hsv:sv (fa i)*sv (fb i)=0:=by
   rcases Nat.mul_eq_zero.1 h3 with h | h<;>simp only[sv,h,CharP.cast_eq_zero,neg_zero,ite_self,mul_ite,mul_neg,zero_mul,mul_zero]
  refine ⟨?_,?_,?_⟩
  · simp only[dsum,ea,eb,hsv,zero_mul,r1,add_zero]
  · simp only[dallNZ,ea,eb,h3,r2,zNeg,decide_true,Bool.true_and]
  · intro j hj1 hj2
    by_cases hji:j=i
    · subst hji;exact ⟨h1,h2⟩
    · exact r3 j (by exact omS138 hj1 hji) (by exact omS7 hj2)
lemma zdotFn_eq:zdotFn=dotFn:=by
 funext n fa fb
 unfold zdotFn
 split
 · rename_i hz
   obtain ⟨r1,r2,r3⟩:=zAll_spec fa fb n n 0 (by exact omS10) hz
   rw[D.qAi]
   unfold dotOf
   have hs:(Array.ofFn (n:=n) λi=>fb i).size=(Array.ofFn (n:=n) λi=>fa i).size:=by simp only[Array.size_ofFn]
   have hall: ∀k,k<(Array.ofFn (n:=n) λi=>fa i).size->
     expOf (Array.ofFn (n:=n) λi=>fa i)[k]!≠255/\expOf (Array.ofFn (n:=n) λi=>fb i)[k]!≠255:=by
    intro k hk
    simp only[Array.size_ofFn] at hk
    rw[Pinned.getElem!_ofFn n fa k hk,Pinned.getElem!_ofFn n fb k hk]
    exact r3 k א א
   rw[<-D.qEg _ _ hs hall]
   simp only[Array.size_ofFn,r1,r2]
   unfold finishWord
   simp only[ne_eq,decide_not,and_eq_true,Bool.not_eq_eq_eq_not,Bool.not_true,decide_eq_false_iff_not, ↓reduceIte]
 · rfl
lemma zCols_spec (s:OS) (X:ѐ) (c yy x0:Nat): ∀n i,zCols s X c yy x0 i n=true->
  ∀d,i<=d->d< i+n->pixZ s X c yy (x0+d)=true:=by
 intro n
 induction n with
 | zero=>intro i _ d h1 h2;omega
 | succ n ih=>
  intro i h d h1 h2
  simp only[zCols,Bool.and_eq_true] at h
  by_cases hd:d=i
  · subst hd;exact h.1
  · exact ih (i+1) h.2 d (by exact omS138 h1 hd) (by exact omS7 h2)
lemma zRows_spec (s:OS) (X:ѐ) (c y0 x0:Nat): ∀n i,zRows s X c y0 x0 i n=true->
  ∀e,i<=e->e< i+n->zCols s X c (y0+e) x0 0 s.kw=true:=by
 intro n
 induction n with
 | zero=>intro i _ e h1 h2;omega
 | succ n ih=>
  intro i h e h1 h2
  simp only[zRows,Bool.and_eq_true] at h
  by_cases he:e=i
  · subst he;exact h.1
  · exact ih (i+1) h.2 e (by exact omS138 h1 he) (by exact omS7 h2)
lemma wFin_spec (W:ѐ) (base:Nat): ∀n i,wFin W base i n=true->
  ∀d,i<=d->d< i+n->expOf W[base+d]!≠255:=by
 intro n
 induction n with
 | zero=>intro i _ d h1 h2;omega
 | succ n ih=>
  intro i h d h1 h2
  simp only[wFin,Bool.and_eq_true,bne_iff_ne,ne_eq] at h
  by_cases hd:d=i
  · subst hd;exact h.1
  · exact ih (i+1) h.2 d (by exact omS138 h1 hd) (by exact omS7 h2)
lemma mantOf_fin {w:ὥ} (h:mantOf w=0):expOf w≠255:=by
 unfold mantOf at h
 split at h
 · rename_i he;rw[he];decide
 · omega
lemma win_mant (s:OS) (X:ѐ) (o y x:Nat) (hk:0<s.kw)
  (h:zRows s X o (y*s.stride) (x*s.stride) 0 s.kh=true) (t:Nat) (ht:t<s.kh*s.kw):
  mantOf (windowAtO s X y x (o*(s.kh*s.kw)+t))=0:=by
 have hdy:t/s.kw<s.kh:=(Nat.div_lt_iff_lt_mul hk).2 ht
 have hr:=zRows_spec s X o (y*s.stride) (x*s.stride) s.kh 0 h (t/s.kw) (Nat.zero_le _) א
 have hp:=zCols_spec s X o (y*s.stride+t/s.kw) (x*s.stride) s.kw 0 hr (t%s.kw) (Nat.zero_le _)
  (by have:=ὀ t hk;omega)
 have hkk:0<s.kh*s.kw:=by exact omS147 ht
 have e1:(o*(s.kh*s.kw)+t)/(s.kh*s.kw)=o:=by
  rw[ἥ,Nat.mul_add_div hkk,Ὃ ht,Ἵ]
 have e2:(o*(s.kh*s.kw)+t)/s.kw%s.kh=t/s.kw:=by
  rw[show o*(s.kh*s.kw)+t=t+s.kw*(o*s.kh) by ring,Nat.add_mul_div_left _ _ hk,
   Nat.add_mul_mod_self_right,Ἁ hdy]
 have e3:(o*(s.kh*s.kw)+t)%s.kw=t%s.kw:=by
  rw[show o*(s.kh*s.kw)+t=t+s.kw*(o*s.kh) by ring,Nat.add_mul_mod_self_left]
 simp only[windowAtO,e1,e2,e3]
 simp only[pixZ,Bool.or_eq_true,decide_eq_true_eq,beq_iff_eq] at hp
 split
 · decide
 · rename_i hc
   rcases hp with (((h1 | h1) | h1) | h1) | h1
   · exact absurd (Or.inl h1) hc
   · exact absurd (Or.inr (Or.inl h1)) hc
   · exact absurd (Or.inr (Or.inr (Or.inl h1))) hc
   · exact absurd (Or.inr (Or.inr (Or.inr h1))) hc
   · exact h1
lemma zAll_of (fa fb:Nat->ὥ): ∀n i,(∀t,i<=t->t< i+n->
  expOf (fa t)≠255/\expOf (fb t)≠255/\mantOf (fa t)*mantOf (fb t)=0)->zAll fa fb i n=true:=by
 intro n
 induction n with
 | zero=>intro i _;rfl
 | succ n ih=>
  intro i h
  obtain ⟨h1,h2,h3⟩:=h i (Nat.le_refl _) א
  simp only[zAll,Bool.and_eq_true,bne_iff_ne,ne_eq,beq_iff_eq]
  exact ⟨⟨⟨h1,h2⟩,h3⟩,ih (i+1) (λt a b=>h t א א)⟩
lemma zNeg_spec (fa fb:Nat->ὥ): ∀n i,zNeg fa fb i n=true->
  ∀t,i<=t->t< i+n->Bool.xor (signOf (fa t)) (signOf (fb t))=true:=by
 intro n
 induction n with
 | zero=>intro i _ t h1 h2;omega
 | succ n ih=>
  intro i h t h1 h2
  simp only[zNeg,Bool.and_eq_true] at h
  by_cases ht:t=i
  · subst ht;exact h.1
  · exact ih (i+1) h.2 t (by exact omS138 h1 ht) (by exact omS7 h2)
lemma zWin_out (g:D_C) (o y x:Nat) (h:zWin g o y x=true):D_out g o y x=0:=by
 simp only[zWin,Bool.and_eq_true,beq_iff_eq,decide_eq_true_eq] at h
 obtain ⟨⟨⟨⟨⟨hdw,hk⟩,hkw⟩,hb⟩,hw⟩,hr⟩:=h
 have hza:zAll (λt=>if t<g.k then D_xw g o y x t else oneW)
   (λt=>if t<g.k then g.W[o*g.k+t]! else g.b[o]!) 0 (g.k+1)=true:=by
  apply zAll_of
  intro t _ ht
  by_cases htk:t<g.k
  · have hm:mantOf (D_xw g o y x t)=0:=by
     simp only[D_xw,D_t0,hdw,ite_true,hk]
     exact win_mant g.s g.X o y x hkw hr t (hk▸htk)
    simp only[htk,ite_true,hm,ὒ,and_true]
    exact ⟨mantOf_fin hm,wFin_spec g.W (o*g.k) g.k 0 hw t (Nat.zero_le _) א⟩
  · have htk':t=g.k:=א
    subst htk'
    simp only[Nat.lt_irrefl,ite_false,hb]
    decide
 have hzn:zNeg (λt=>if t<g.k then D_xw g o y x t else oneW)
   (λt=>if t<g.k then g.W[o*g.k+t]! else g.b[o]!) 0 (g.k+1)=false:=by
  cases hz:zNeg (λt=>if t<g.k then D_xw g o y x t else oneW)
   (λt=>if t<g.k then g.W[o*g.k+t]! else g.b[o]!) 0 (g.k+1)
  · rfl
  · have:=zNeg_spec _ _ (g.k+1) 0 hz g.k (Nat.zero_le _) א
    simp only[Nat.lt_irrefl,ite_false,hb] at this
    exact absurd this ה
 unfold D_out
 rw[<-D.qAi,<-zdotFn_eq]
 unfold zdotFn
 rw[Ὦ hza]
 simp only[hzn,Bool.and_false]
 cases g.relu<;>rfl
lemma zChans_spec (s:OS) (X:ѐ) (y0 x0:Nat): ∀n i,zChans s X y0 x0 i n=true->
  ∀c,i<=c->c< i+n->zRows s X c y0 x0 0 s.kh=true:=by
 intro n
 induction n with
 | zero=>intro i _ c h1 h2;omega
 | succ n ih=>
  intro i h c h1 h2
  simp only[zChans,Bool.and_eq_true] at h
  by_cases hc:c=i
  · subst hc;exact h.1
  · exact ih (i+1) h.2 c (by exact omS138 h1 hc) (by exact omS7 h2)
lemma zWinC_out (g:D_C) (o y x:Nat) (h:zWinC g o y x=true):D_out g o y x=0:=by
 simp only[zWinC,Bool.and_eq_true,beq_iff_eq,decide_eq_true_eq,Bool.not_eq_true'] at h
 obtain ⟨⟨⟨⟨⟨hdw,hk⟩,hkw⟩,hb⟩,hw⟩,hr⟩:=h
 have hza:zAll (λt=>if t<g.k then D_xw g o y x t else oneW)
   (λt=>if t<g.k then g.W[o*g.k+t]! else g.b[o]!) 0 (g.k+1)=true:=by
  apply zAll_of
  intro t _ ht
  by_cases htk:t<g.k
  · have hkk:0<g.s.kh*g.s.kw:=by
     rcases Nat.eq_zero_or_pos (g.s.kh*g.s.kw) with h0 | h0
     · rw[hk,h0,Nat.mul_zero] at htk;omega
     · exact h0
    have hc:t/(g.s.kh*g.s.kw)<g.s.cin:=by
     rw[hk] at htk
     exact (Nat.div_lt_iff_lt_mul hkk).2 htk
    have ht':t%(g.s.kh*g.s.kw)<g.s.kh*g.s.kw:=ὀ _ hkk
    have hdec:t=t/(g.s.kh*g.s.kw)*(g.s.kh*g.s.kw)+t%(g.s.kh*g.s.kw):=by
     rw[ἥ];exact (Nat.div_add_mod t _).symm
    have hm:mantOf (D_xw g o y x t)=0:=by
     simp only[D_xw,D_t0,hdw,Bool.false_eq_true,ite_false,Ἇ]
     rw[hdec]
     exact win_mant g.s g.X (t/(g.s.kh*g.s.kw)) y x hkw
      (zChans_spec g.s g.X _ _ g.s.cin 0 hr _ (Nat.zero_le _) א) _ ht'
    simp only[htk,ite_true,hm,ὒ,and_true]
    exact ⟨mantOf_fin hm,wFin_spec g.W (o*g.k) g.k 0 hw t (Nat.zero_le _) א⟩
  · have htk':t=g.k:=א
    subst htk'
    simp only[Nat.lt_irrefl,ite_false,hb]
    decide
 have hzn:zNeg (λt=>if t<g.k then D_xw g o y x t else oneW)
   (λt=>if t<g.k then g.W[o*g.k+t]! else g.b[o]!) 0 (g.k+1)=false:=by
  cases hz:zNeg (λt=>if t<g.k then D_xw g o y x t else oneW)
   (λt=>if t<g.k then g.W[o*g.k+t]! else g.b[o]!) 0 (g.k+1)
  · rfl
  · have:=zNeg_spec _ _ (g.k+1) 0 hz g.k (Nat.zero_le _) א
    simp only[Nat.lt_irrefl,ite_false,hb] at this
    exact absurd this ה
 unfold D_out
 rw[<-D.qAi,<-zdotFn_eq]
 unfold zdotFn
 rw[Ὦ hza]
 simp only[hzn,Bool.and_false]
 cases g.relu<;>rfl
lemma qCvZ (g:D_C) (o y x:Ὣ):D_slowZ g o y x=D_out g o y x:=by
 unfold D_slowZ;split
 · rename_i h;rcases Bool.or_eq_true_iff.1 h with h|h
   · exact (zWin_out g o y x h).symm
   · exact (zWinC_out g o y x h).symm
 · exact qCv ..
lemma zCols_of (s:OS) (X:ѐ) (c yy x0:Nat): ∀n i,
  (∀d,i<=d->d< i+n->pixZ s X c yy (x0+d)=true)->zCols s X c yy x0 i n=true:=by
 intro n
 induction n with
 | zero=>intro i _;rfl
 | succ n ih=>
  intro i h
  simp only[zCols,Bool.and_eq_true]
  exact ⟨h i (Nat.le_refl _) א,ih (i+1) (λd h1 h2=>h d א א)⟩
lemma zRows_of (s:OS) (X:ѐ) (c y0 x0:Nat): ∀n i,
  (∀e,i<=e->e< i+n->zCols s X c (y0+e) x0 0 s.kw=true)->zRows s X c y0 x0 i n=true:=by
 intro n
 induction n with
 | zero=>intro i _;rfl
 | succ n ih=>
  intro i h
  simp only[zRows,Bool.and_eq_true]
  exact ⟨h i (Nat.le_refl _) א,ih (i+1) (λd h1 h2=>h d א א)⟩
lemma zChans_of (s:OS) (X:ѐ) (y0 x0:Nat): ∀n i,
  (∀c,i<=c->c< i+n->zRows s X c y0 x0 0 s.kh=true)->zChans s X y0 x0 i n=true:=by
 intro n
 induction n with
 | zero=>intro i _;rfl
 | succ n ih=>
  intro i h
  simp only[zChans,Bool.and_eq_true]
  exact ⟨h i (Nat.le_refl _) א,ih (i+1) (λd h1 h2=>h d א א)⟩
lemma wFin_of (W:ѐ) (base:Nat) (hW:∀q:Nat,good W[q]! =true): ∀n i,wFin W base i n=true:=by
 intro n
 induction n with
 | zero=>intro i;rfl
 | succ n ih=>
  intro i
  simp only[wFin,Bool.and_eq_true,bne_iff_ne,ne_eq]
  exact ⟨D.qAe _ (hW _),ih (i+1)⟩
lemma win_hull (g:D_C) (hwp:g.wp=g.s.w+2*g.s.pad) (c y x:Nat) (hc:c<g.s.cin) (hy:y<g.ho)
  (h:(x*g.s.stride+g.s.kw<=g.ex.1.getD (2*((if g.dw then c else 0)*g.ho+y)) 0 ||
    g.wp<=x*g.s.stride+g.ex.1.getD (2*((if g.dw then c else 0)*g.ho+y)+1) 0)=true):
  zRows g.s g.X c (y*g.s.stride) (x*g.s.stride) 0 g.s.kh=true:=by
 simp only[Bool.or_eq_true,decide_eq_true_eq] at h
 apply zRows_of
 intro i _ hi
 apply zCols_of
 intro d _ hd
 apply g.ex.2 c y _ i hc hy א
 rcases h with h | h
 · exact Or.inl א
 · exact Or.inr א
lemma zWinX_out (g:D_C) (hk:g.k=if g.dw then g.s.kh*g.s.kw else g.s.cin*g.s.kh*g.s.kw) (pk:0<g.k)
  (HW:∀q:Nat,good g.W[q]! =true) (hwp:g.wp=g.s.w+2*g.s.pad) (o y x:Nat) (hC:g.dw=true->o<g.s.cin) (hy:y<g.ho)
  (h:zWinX g o y x=true):D_out g o y x=0:=by
 unfold zWinX at h
 simp only[Bool.and_eq_true,beq_iff_eq] at h
 obtain ⟨hr,hb⟩:=h
 cases hd:g.dw
 · rw[hd] at hk
   simp only[Bool.false_eq_true,ite_false] at hk
   apply zWinC_out g o y x
   have kw:0<g.s.kw:=by rw[hk] at pk;exact Nat.pos_of_mul_pos_left pk
   simp only[zWinC,hd,Bool.not_false,Bool.true_and,Bool.and_eq_true,beq_iff_eq,decide_eq_true_eq]
   refine ⟨⟨⟨⟨by rw[hk,Nat.mul_assoc],kw⟩,hb⟩,wFin_of _ _ HW _ _⟩,?_⟩
   apply zChans_of
   intro c _ hc
   apply win_hull g hwp c y x א hy
   simp only[hd,Bool.false_eq_true,ite_false] at hr⊢
   exact hr
 · rw[hd] at hk
   simp only[ite_true] at hk
   apply zWin_out g o y x
   have kw:0<g.s.kw:=by rw[hk] at pk;exact Nat.pos_of_mul_pos_left pk
   simp only[zWin,hd,Bool.true_and,Bool.and_eq_true,beq_iff_eq,decide_eq_true_eq]
   refine ⟨⟨⟨⟨hk,kw⟩,hb⟩,wFin_of _ _ HW _ _⟩,?_⟩
   apply win_hull g hwp o y x (hC hd) hy
   simp only[hd,ite_true] at hr⊢
   exact hr
lemma zBlk_lane (g:D_C) (hs:g.s.stride=1) (o y x k:Nat) (hk:k<8) (h:zBlk g o y x=true):zWinX g o y (x+k)=true:=by
 unfold zBlk zBlkG at h
 unfold zWinX
 simp only[Bool.and_eq_true,beq_iff_eq,Bool.or_eq_true,decide_eq_true_eq] at h⊢
 obtain ⟨_,hr,hb⟩:=h
 refine ⟨?_,hb⟩
 simp only[hs,Nat.mul_one] at hr⊢
 rcases hr with hr | hr
 · exact Or.inl א
 · exact Or.inr א
def D_B (g:D_C) (y x:Ὣ):Ὣ:=y*g.s.stride*g.wp+x*g.s.stride
lemma qE1 {A:ё} {h:Ὢ}:good (0:Ὢ)=true/\(0:Ὢ)&&&В<=h:=⟨rfl,ἃ.2 (Nat.zero_le _)⟩
lemma qGp {A:ё} (H:∀k:Ὣ,k< ѧA->good A[k]! =true) (q:Ὣ):good A[q]! =true:=by
 by_cases h:q< ѧA;. exact H q h
 rw[ὑ A q h];rfl
lemma qBc {s:OS} {X:ё} {hi:Ὢ} (H:C_Hv X hi) (c yy xx:Ὣ):good (D_px s X c yy xx)=true/\D_px s X c yy xx&&&В<=hi:=by
 unfold D_px;split;. exact qE1 (A:=X)
 generalize (c*s.h+(yy-s.pad))*s.w+(xx-s.pad)=n;by_cases h:n< ѧX;. exact ⟨H.2.2.1 n h,H.2.2.2 n h⟩
 rw[ὑ X n h];exact qE1 (A:=X)
lemma qHk {g:D_C} (ok:D_Ok g) {o t:Ὣ} (hC:o<g.C) (ht:t<g.k):D_t0 g o+t<g.s.cin*(g.s.kh*g.s.kw):=by
 have h1:=ok.hk;have h2:=ok.hC;unfold D_t0;cases hd:g.dw<;>simp only[hd,false_eq_true, ↓reduceIte,zero_add,gt_iff_lt] at h1 h2⊢;. rwa[<-Nat.mul_assoc,<-h1]
 exact h1▸qDf (h2▸hC) (h1▸ht)
lemma qHl {n a b c:Ὣ} (h:n<a*(b*c)):0<b/\0<c:=by constructor<;>refine ὼ λe=>?_<;>simp only[e,zero_mul,mul_zero,not_lt_zero] at h
lemma qGd {a b st n m:Ὣ} (h0:0<st) (hk:b<=a) (hy:n<(a-b)/st+1) (hm:m<b):n*st+m<a:=by have:=Ἒ st (show n<=(a-b)/st by (exact Nat.le_of_lt_succ ‹_›));have:=Nat.div_mul_le_self (a-b) st;(clear h0 hy;omega)
lemma qJs (a p w v u y x:Ὣ):v*w+u+((a*p+y)*w+x)=(a*p+(v+y))*w+(u+x):=by ring
lemma qCf {g:D_C} (ok:D_Ok g) {o y x t:Ὣ} (hC:o<g.C) (hy:y<g.ho) (hx:x<g.wo) (ht:t<g.k):
 D_B g y x+D_off g.s (D_t0 g o+t)<g.xd.size/\g.xd.data[D_B g y x+D_off g.s (D_t0 g o+t)]! =dv (D_xw g o y x t):=by
 have hb:=qHk ok hC ht
 have hk:=qHl hb
 unfold D_xw
 generalize D_t0 g o+t=u at hb⊢
 have hyy:=qGd ok.p0 ok.kh (hy.trans_eq ok.ho) (ὀ (u/g.s.kw) hk.1)
 have hxx:=qGd ok.p0 ok.kw (hx.trans_eq ok.wo) (ὀ u hk.2)
 have hq:=qDf (qDf (Ἰ (hb.trans_eq (ἥ _ _))) hyy) hxx
 rw[D_B,D_off,ok.wp,qJs]
 refine ⟨Nat.lt_of_lt_of_le hq ok.xd.1,?_⟩
 rw[ok.xd.2 _ hq]
 simp only[D_Fx,qCn _ hxx,qCn _ hyy]
 rfl
lemma qBm {u v:ὦ} {a b:Ὣ} (hu:Ѥu=a) (hv:Ѥv=b) (h:a+b<2^32):(u+v).toNat=a+b:=by rw[qAv _ _ (by exact omS390 hu hv h),hu,hv]
lemma qH0 {n d:Ὣ} {b i:ὦ} {off:ѐ} (hs:n<2^32) (hb:∀j (h:j< ѧoff),Ѥb+off[j].toNat+d<n) (hi:Ѥi< ѧoff):
 (b+(off.uget i hi).toUSize).toNat=Ѥb+(off[Ѥi]!:ὥ).toNat/\ Ѥb+(off[Ѥi]!:ὥ).toNat+d<n:=by
 have h:Ѥb+(off.uget i hi).toNat+d<n:=hb _ hi
 rw[<-qFx _ _ hi]
 exact ⟨qBm rfl (ὃ _) (by exact omS372 hs h),h⟩
lemma qGc {c:D_P} {b:ὦ} (hb:∀j (h:j<c.off.size),Ѥb+c.off[j].toNat+0<c.g.xd.size) (xf wf:Ὣ->ὧ)
 (L:љ):∀(i:ὦ) (a t:ὧ),Ѥi+ѣL=c.e.toNat->
  (∀j p:Ὣ,L[j]?=some p->c.g.xd.data[Ѥb+(c.off[Ѥi+j]!:ὥ).toNat]! =xf p/\c.wd.data[Ѥi+j]! =wf p)->
  D_goT c.g.xd c.wd c.off b c.e i c.he c.hw c.hs hb a t=L.foldl (λ(acc:F1) p=>let q:=xf p*wf p
   F1.mk (acc.s+q) (acc.t+q.abs)) ⟨a,t⟩:=by
 have:=c.he
 induction' L with p L ih<;>intro i a t h0 H
 . rw[D_goT,ἔ (qCk h0.ge)];rfl
 rw[List.length_cons] at h0
 obtain ⟨ht,hu⟩:=qBz h0
 have h1:=H 0 p rfl
 rw[Ἵ] at h1
 rw[D_goT,Ἄ ht]
 simp only[qIg]
 rw[(qH0 c.hs hb (by exact Nat.lt_of_lt_of_le ‹_› ‹_›)).1,h1.1,h1.2,ih (i+1) _ _ hu,@ὔ]
 intro j q hq
 rw[usz i _ ht,Nat.add_right_comm,ὢ]
 exact H (j+1) q hq
def D_ff (xf:Ὣ->Ὣ->ὧ) (wf:Ὣ->ὧ) (L:љ) (k:Ὣ) (a:ὧ):ὧ:=
 L.foldl (λacc p=>acc+xf k p*wf p) a
lemma qF7o {c:D_P} {b:ὦ} (hb:∀j (h:j<c.off.size),Ѥb+c.off[j].toNat+7<c.g.xd.size) (r:D_R) (xf:Ὣ->Ὣ->ὧ)
 (wf:Ὣ->ὧ) (L:љ):∀(i:ὦ) (A B C D E F G I:ὧ),Ѥi+ѣL=c.e.toNat->
  (∀j p:Ὣ,L[j]?=some p->c.wd.data[Ѥi+j]! =wf p/\∀d,d<8->c.g.xd.data[Ѥb+(c.off[Ѥi+j]!:ὥ).toNat+d]! =xf d p)->
  D_go8R1 c.g.xd c.wd c.off b c.e i c.he c.hw c.hs hb A B C D E F G I r=⟨r.t,D_ff xf wf L 0 A,D_ff xf wf L 1 B,D_ff xf wf L 2 C,D_ff xf wf L 3 D,D_ff xf wf L 4 E,D_ff xf wf L 5 F,D_ff xf wf L 6 G,D_ff xf wf L 7 I⟩:=by
 have:=c.he
 have:=c.hs
 induction' L with p L ih<;>intro i A B C D E F G I h0 J
 . rw[D_go8R1,ἔ (qCk h0.ge)];cases r;rfl
 rw[List.length_cons] at h0
 obtain ⟨ht,hu⟩:=qBz h0
 obtain ⟨h1,h2⟩:=J 0 p rfl
 rw[Ἵ] at h1 h2
 obtain ⟨ep,hbd⟩:=qH0 c.hs hb ((by exact Nat.lt_of_lt_of_le ‹_› ‹_›):Ѡi<c.off.size)
 have U:=λ(k:Ὣ) (hk:k<8)=>(qBm ep (uk k (by exact Nat.lt_of_lt_of_le ‹_› ה)) (by clear h0 hu;omega)).symm▸h2 k hk
 rw[D_go8R1,Ἄ ht]
 simp only[qIg]
 rw[ep,h1,U 1 ה,U 2 ה,U 3 ה,U 4 ה,U 5 ה,U 6 ה,U 7 (by exact Nat.le_refl _),<-Ἵ (_+_),h2 0 ה]
 refine ih (i+1) _ _ _ _ _ _ _ _ hu λj q hq=>?_
 rw[usz i _ ht,Nat.add_right_comm,ὢ]
 exact J (j+1) q hq
lemma vFug (a:ὤ) (p:ὦ) (h:Ѥp< Ѩa):a.uget p h=a.data[Ѥp]!:=by
 cases a with | mk d=>simp only[FloatArray.uget];exact (Ἕ d (Ѥp) h).symm
lemma vFxv {g:D_C} (ok:D_Ok g) (p:ὦ) (hp:Ѥp< Ѩg.xd):∃u,g.xd.uget p hp=dv u:=by rw[vFug];exact ok.xv _ hp
lemma vFwv {c:D_P} {o:Ὣ} {L:љ} (h:D_PO c o L) (q:ὦ) (hq:Ѥq< Ѩc.wd) (he:Ѥq< Ѥc.e):∃u,c.wd.uget q hq=dv u:=by
 rw[vFug];have hl:Ѥq< ѣL:=h.len▸he;exact ⟨_,(h.rd (Ѥq) L[Ѥq] (List.getElem?_eq_getElem hl)).2⟩
lemma qF7 {c:D_P} {b:ὦ} (hb:∀j (h:j<c.off.size),Ѥb+c.off[j].toNat+7<c.g.xd.size) (r:D_R) (xf:Ὣ->Ὣ->ὧ)
 (wf:Ὣ->ὧ) (L:љ) (HX:∀(p:ὦ) (h:Ѥp< Ѩc.g.xd),∃u,c.g.xd.uget p h=dv u)
 (HW:∀(q:ὦ) (h:Ѥq< Ѩc.wd),Ѥq< Ѥc.e->∃u,c.wd.uget q h=dv u):∀(i:ὦ) (A B C D E F G I:ὧ),Ѥi+ѣL=c.e.toNat->
  (∀j p:Ὣ,L[j]?=some p->c.wd.data[Ѥi+j]! =wf p/\∀d,d<8->c.g.xd.data[Ѥb+(c.off[Ѥi+j]!:ὥ).toNat+d]! =xf d p)->
  D_go8R c.g.xd c.wd c.off b c.e i c.he c.hw c.hs hb A B C D E F G I r=⟨r.t,D_ff xf wf L 0 A,D_ff xf wf L 1 B,D_ff xf wf L 2 C,D_ff xf wf L 3 D,D_ff xf wf L 4 E,D_ff xf wf L 5 F,D_ff xf wf L 6 G,D_ff xf wf L 7 I⟩:=by
 intro i A B C D E F G I h0 J
 rw[qGR _ _ _ _ _ _ _ _ _ HX HW];exact qF7o hb r xf wf L i A B C D E F G I h0 J
lemma D_zt_spec (xd:ὤ) (off:ѐ) (b:Nat): ∀n i,D_zt xd off b i n=true->
  ∀j,i<=j->j< i+n->(xd.data[b+(off[j]!).toNat]!).toBits<<<(1:Ὠ)=0:=by
 intro n
 induction n with
 | zero=>intro i _ j h1 h2;omega
 | succ n ih=>
  intro i h j h1 h2
  simp only[D_zt,Bool.and_eq_true,decide_eq_true_eq,beq_iff_eq] at h
  by_cases hj:j=i
  · subst hj
    have e:=h.1.2
    have g:xd.get! (b+(off[j]!).toNat)=xd.data[b+(off[j]!).toNat]!:=by cases xd;rfl
    rw[g] at e
    exact e
  · exact ih (i+1) h.2 j (by exact omS138 h1 hj) (by exact omS7 h2)
lemma Rz0 (f:ὧ) (h:f.toBits<<<(1:Ὠ)=0):D.R f=0:=by
 have h0:(f.toBits<<<(1:Ὠ)).toNat=0:=by rw[h];rfl
 rw[UInt64.toNat_shiftLeft] at h0
 have hl:=f.toBits.toNat_lt
 have h1:(1:Ὠ).toNat%64=1:=rfl
 rw[h1,ὁ] at h0
 have g:Model.expField (toModel (ofFloat f:Binary64))=f.toBits.toNat/2^52%2^11:=by
  show ((f.toBits.toBitVec>>>52)&&&BitVec.ofNat 64 2047).toNat=_
  rw[BitVec.toNat_and,BitVec.toNat_ushiftRight,BitVec.toNat_ofNat,Ἱ]
  exact Ἂ _ 11
 have e:Model.fracField (toModel (ofFloat f:Binary64))=f.toBits.toNat%2^52:=by
  show (f.toBits.toBitVec&&&BitVec.ofNat 64 (2^52-1)).toNat=_
  rw[BitVec.toNat_and,BitVec.toNat_ofNat]
  exact Ἂ _ 52
 have g0:f.toBits.toNat/2^52%2^11=0:=א
 have e0:f.toBits.toNat%2^52=0:=א
 have c:=Model.ofFields_signBit_expField_fracField (toModel (ofFloat f:Binary64))
 rw[g,e,g0,e0] at c
 have a:Model.isFinite (Model.ofFields D.B_D (Model.signBit (toModel (ofFloat f:Binary64))) 0 0)=true:=by
  rw[Model.isFinite_ofFields D.B_D _ _ _ ה ה]
  rfl
 have d:=Model.toDyadic?_ofFields_of_isFinite D.B_D _ _ _ ה ה a
 rw[c] at d
 rw[D.R,Model.toReal_eq,d]
 simp only[↓reduceIte,Nat.reduceAdd,toModel_ofFloat,FloatLib.Numerics.Dyadic.toReal,signedSignificand,Int.ofNat_eq_natCast,CharP.cast_eq_zero,neg_zero,ite_self,Int.cast_zero,zpow_ofNat,pow_zero,mul_one]
lemma mant_of_dv0 (w:ὥ) (hg:D.good w=true) (h:(D.dv w).toBits<<<(1:Ὠ)=0):mantOf w=0:=by
 have h1:=(D.qAm w hg).2
 rw[Rz0 _ h,D.qA7 w hg] at h1
 have h2:D.A_mag (w.toNat%2^31)=0:=by
  split at h1<;>linarith only[h1]
 unfold D.A_mag at h2
 split at h2
 · rename_i hv
   apply zw_mant
   apply UInt32.toNat_inj.mp
   rw[Ἧ]
   have h3:(В:ὥ).toNat=2^31-1:=rfl
   rw[h3,Ἂ,hv];rfl
 · exfalso
   have:(0:ℝ)<((2^23+w.toNat%2^31%2^23:ℕ):ℝ)*2^((((w.toNat%2^31)/2^23:ℕ):ℤ)-150):=ו
   linarith only[h2,this]
lemma zt_out {c:D_P} {o:Nat} {L:љ} (h:D_PO c o L) {y x:Nat} {b:ὦ} (hy:y<c.g.ho) (hx:x<c.g.wo)
  (hb:b.toNat=D_B c.g y x) (hz:D_zt c.g.xd c.off b.toNat 0 c.e.toNat=true) (hbz:c.g.b[o]! =0):
  D_out c.g o y x=0:=by
 have hzt:=D_zt_spec _ _ _ _ 0 hz
 have hza:zAll (λt=>if t<c.g.k then D_xw c.g o y x t else oneW)
   (λt=>if t<c.g.k then c.g.W[o*c.g.k+t]! else c.g.b[o]!) 0 (c.g.k+1)=true:=by
  apply zAll_of
  intro t _ ht
  by_cases htk:t<c.g.k
  · simp only[htk,ite_true]
    have gx:D.good (D_xw c.g o y x t)=true:=(qBc h.ok.Hx _ _ _).1
    have gw:D.good c.g.W[o*c.g.k+t]! =true:=h.ok.HW _
    refine ⟨D.qAe _ gx,D.qAe _ gw,?_⟩
    by_cases hm:t∈L
    · obtain ⟨j,hj⟩:=List.mem_iff_getElem?.mp hm
      have hjl:j<L.length:=(List.getElem?_eq_some_iff.mp hj).1
      have r:=h.rd j t hj
      have e1:=hzt j (Nat.zero_le _) (by rw[h.len];omega)
      rw[hb,r.1,(qCf h.ok h.hC hy hx (h.lt t hm)).2] at e1
      rw[mant_of_dv0 _ gx e1,ὒ]
    · rw[zw_mant (h.hz t htk hm),Nat.mul_zero]
  · have htk':t=c.g.k:=א
    subst htk'
    simp only[Nat.lt_irrefl,ite_false,hbz]
    decide
 have hzn:zNeg (λt=>if t<c.g.k then D_xw c.g o y x t else oneW)
   (λt=>if t<c.g.k then c.g.W[o*c.g.k+t]! else c.g.b[o]!) 0 (c.g.k+1)=false:=by
  cases hq:zNeg (λt=>if t<c.g.k then D_xw c.g o y x t else oneW)
   (λt=>if t<c.g.k then c.g.W[o*c.g.k+t]! else c.g.b[o]!) 0 (c.g.k+1)
  · rfl
  · have:=zNeg_spec _ _ (c.g.k+1) 0 hq c.g.k (Nat.zero_le _) א
    simp only[Nat.lt_irrefl,ite_false,hbz] at this
    exact absurd this ה
 unfold D_out
 rw[<-D.qAi,<-zdotFn_eq]
 unfold zdotFn
 rw[Ὦ hza]
 simp only[hzn,Bool.and_false]
 cases c.g.relu<;>rfl
section
variable {c:D_P} {o:Ὣ} {L:љ} (h:D_PO c o L) {y x:Ὣ} {b:ὦ} (hy:y< c.g.ho) (hx:x< c.g.wo) (hb:Ѥb=D_B c.g y x)
include c o L h y x b hy hx hb
lemma qCvY:D_slowY c o y x b=D_out c.g o y x:=by
 unfold D_slowY
 split
 · rename_i hz;exact (zWinX_out c.g h.ok.hk h.ok.pk h.ok.HW h.ok.wp o y x (λhd=>by have e:=h.ok.hC;simp only[hd,ite_true] at e;exact e▸h.hC) hy hz).symm
 split
 · rename_i hz;simp only[Bool.and_eq_true,beq_iff_eq] at hz;exact (zt_out h hy hx hb hz.2 hz.1).symm
 · exact qCvZ ..
lemma qCu:D_finT c o y x b=D_out c.g o y x:=by
 unfold D_finT;split<;>rename_i hl;swap;. exact qCv ..
 have e:=qGc (c.h1 b hl) (dv∘D_xw c.g o y x)
  (λp=>dv c.g.W[o*c.g.k+p]!) L 0 0 0 ((Ἇ _).trans h.len.symm) λj p hj=>by
   have h1:=h.rd j p hj;rw[C_z,Ἇ,h1.1,hb];exact ⟨(qCf h.ok h.hC hy hx (h.lt p (List.mem_of_getElem? hj))).2,h1.2⟩
 simp only[c.hz,e,h.bv,h.ok.nf];split<;>rename_i hz;. split;exact qCvY h hy hx hb;exact qCv ..
 exact congrArg (act _) (qDq c.g.k h.ok.nk _ _ _ (λt _=>(qBc h.ok.Hx _ _ _).1) (λt _=>h.ok.HW _) (h.ok.Hb _) L h.nd h.lt (λt a b=>.inr (h.hz t a b)) _ _ ⟨rfl,rfl⟩ (by simpa only[Function.comp_apply,ne_eq,beq_iff_eq] using hz))
lemma qF6 {a:ὧ} (ha:a=L.foldl (λacc t=>acc+dv (D_xw c.g o y x t)*dv c.g.W[o*c.g.k+t]!) 0):D_fin c o y x b a=D_out c.g o y x:=by
 subst ha;simp only[D_fin,h.bv,h.e0];split<;>rename_i hz;. split;exact qCvY h hy hx hb;exact qCu h hy hx hb
 exact congrArg (act _) (qC3 c.g.k h.ok.nk _ _ _ _ (λt _=>(qBc h.ok.Hx _ _ _).1) (λt _=>h.ok.HW _) (h.ok.Hb _) h.ok.Hx.1 (λt _=>(qBc h.ok.Hx _ _ _).2) h.ok.Hx.2.1 L h.nd h.lt (λt a b=>.inr (h.hz t a b)) _ _ ⟨rfl,rfl⟩ (by simpa only[ne_eq,beq_iff_eq] using hz))
end
lemma qIk {c:D_P} {o:Ὣ} {L:љ} (h:D_PO c o L) {y x k:Ὣ} {b:ὦ} (hy:y<c.g.ho) (hx:x+k<c.g.wo) (hb:(b+ὴ k).toNat=D_B c.g y (x+k)) {a:ὧ} (ha:a=L.foldl (λacc t=>acc+dv (D_xw c.g o y (x+k) t)*dv c.g.W[o*c.g.k+t]!) 0):D_w c (loOf c.e0) (spR c.g.relu (loOf c.e0)) o y x b k a=D_out c.g o y (x+k):=by
 unfold D_w;dsimp only;split<;>rename_i hc
 . obtain ⟨h1,h2⟩:=ὕ.mp hc;have h1:((a+c.bv).toBits<<<(1:Ὠ))-loOf c.e0<spR c.g.relu (loOf c.e0):=by simpa only[decide_eq_true_eq] using h1
   have hr:c.g.relu=false:=by
    cases hq:c.g.relu;. rfl
    . rw[hq] at h1;exact absurd h1 (by simp only[spR, ↓reduceIte,UInt64.not_lt_zero,not_false_eq_true])
   rw[hr] at h1;have h1:((a+c.bv).toBits<<<(1:Ὠ))-loOf c.e0<spOf (loOf c.e0):=by simpa only[spR,false_eq_true, ↓reduceIte] using h1
   have h2:((a+c.bv).toBits+0x0FDFFFFF)&&&0x1FFFFFFF<0x1FC00000:=by simpa only[decide_eq_true_eq] using h2
   obtain ⟨e1,e2⟩:=qHy _ _ h1 h2;rw[<-e1];refine (show act c.g.relu (zivT (thOf c.e0) (a+c.bv))=zivT (thOf c.e0) (a+c.bv) by rw[hr];rfl).symm.trans ?_;subst ha;simp only[h.bv,h.e0] at e2⊢;exact congrArg (act _) (qDa c.g.k h.ok.nk _ _ _ _ (λt _=>(qBc h.ok.Hx _ _ _).1) (λt _=>h.ok.HW _) (h.ok.Hb _) h.ok.Hx.1 (λt _=>(qBc h.ok.Hx _ _ _).2) h.ok.Hx.2.1 L h.nd h.lt (λt a b=>.inr (h.hz t a b)) _ _ ⟨rfl,rfl⟩ e2)
 . split
   . split
     . rename_i hz;simp only[Bool.and_eq_true,beq_iff_eq] at hz
       exact (zt_out h hy hx hb (D_ztF_imp0 _ _ _ _ hz.2) hz.1).symm
     . exact qCvY h hy hx hb
   . unfold D_finK;exact qF6 h hy hx hb ha
def D_sp (g:D_C) (q:Ὣ):Ὢ:=
 D_out g (q/(g.ho*g.wo)) (q/g.wo%g.ho) (q%g.wo)
lemma qG9 (g:D_C) {o y x:Ὣ} (hy:y<g.ho) (hx:x<g.wo):D_out g o y x=D_sp g ((o*g.ho+y)*g.wo+x):=by simp only[D_sp,ἥ g.ho g.wo,<-Nat.div_div_eq_div_mul,qCn _ hx,qCn _ hy]
lemma qGa {g:D_C} (ok:D_Ok g) {o y x:Ὣ} (hC:o<g.C) (hy:y<g.ho) (hx:x<g.wo):D_B g y x<2^32:=by have h:=(qCf ok hC hy hx ok.pk).1;have:=ok.xs;(clear hC hy hx;omega)
lemma qIn (g:D_C) (y x k:Ὣ):D_B g y (x+k)=D_B g y x+k*g.s.stride:=by unfold D_B;rw[ὰ];exact omS391
def D_xf (c:D_P) (o:Ὣ) (b:ὦ) (k p:Ὣ):ὧ:=
 c.g.xd.data[Ѥb+D_off c.g.s (D_t0 c.g o+p)+k]!
def D_I (c:D_P) (Q x:Ὣ) (out:ё):Prop:=
 C_Inv (c.g.C*(c.g.ho*c.g.wo)) (D_sp c.g) (λp=>p<Q+min x c.g.wo) out
lemma qFe {c:D_P} {o:Ὣ} {L:љ} (h:D_PO c o L) {y Q:Ὣ} (hy:y<c.g.ho) (hQ:Q=(o*c.g.ho+y)*c.g.wo) (hN:Q+c.g.wo<=c.g.C*(c.g.ho*c.g.wo)) (d:Ὣ):
 ∀(x:Ὣ) (b q:ὦ) (rr:D_R) (out:ё),c.g.wo-x=d->(x<c.g.wo->Ѥb=D_B c.g y x)->(x<c.g.wo->Ѥq=Q+x)->D_I c Q x out->
  C_Inv (c.g.C*(c.g.ho*c.g.wo)) (D_sp c.g) (λp=>p<Q+c.g.wo) (D_cols c (loOf c.e0) (spR c.g.relu (loOf c.e0)) o y x b q rr out):=by
 induction' d using Nat.strong_induction_on with d ih
 intro x b q rr out he hb hq H
 have sN:=h.ok.sN
 have S:∀x',x'<c.g.wo->D_out c.g o y x'=D_sp c.g (Q+x'):=λ_ hx=>hQ▸qG9 c.g hy hx
 rw[D_cols]
 split<;>rename_i h8
 . obtain ⟨hx,hf,hl⟩:=h8;have hb0:=hb hx;have hq0:=hq hx;have hB:=λk=>(qIn c.g y x k).trans (by rw[h.ok.f4 hf,Nat.mul_one]);have G:=λk (hk:x+k<c.g.wo)=>qBm hb0 (uk k (by clear hy he hx hl hb0;exact omS392 hN sN hk)) (hB k▸qGa h.ok h.hC hy hk);have V:=λk (hk:x+k<=c.g.wo)=>(qBm hq0 (uk k (by clear hy he hx hl hb0;exact omS393 hN sN hk)) (by clear hy he hx hl hb0;exact omS394 hN sN hk)).trans (ὢ Q x k)
   have R:=qF7 (c.h8 b hl) rr (D_xf c o b) (λp=>dv c.g.W[o*c.g.k+p]!) L (vFxv h.ok) (vFwv h) 0 0 0 0 0 0 0 0 0 ((Ἇ _).trans h.len.symm)
    λj p hj=>by
     have h1:=h.rd j p hj;rw[C_z,Ἇ,h1.1];exact ⟨h1.2,λ_ _=>rfl⟩
   have wL:c.g.wo<2^32:=by have:=Nat.le_mul_of_pos_left c.g.wo ((by clear hQ hN he sN hx hb0 hq0;exact omS147 hy):0<c.g.ho);have:=Nat.le_mul_of_pos_left (c.g.ho*c.g.wo) (show 0<c.g.C by have:=h.hC;omega);omega
   have A:∀k,k<8->∀u,D_I c Q (x+k) u->D_I c Q (x+k+1) (D_l c (loOf c.e0) (spR c.g.relu (loOf c.e0)) o y x b q (c.g.wo-x).toUSize k (D_ff (D_xf c o b) (λp=>dv c.g.W[o*c.g.k+p]!) L k 0) u):=λk k8 u H=>by
    have hN:(ὴ k<(c.g.wo-x).toUSize)↔x+k<c.g.wo:=by rw[А,uk k (by exact Nat.lt_of_lt_of_le ‹_› ה),qAa _ (by clear hy hQ hN sN hb0 hq0 k8;exact omS395 wL)];(clear hy hQ hN sN hb0 hq0 wL k8;exact omS396)
    unfold D_l D_I at*;simp only[hN];split<;>rename_i hk;. rw[Nat.min_eq_left hk.le] at H;rw[Nat.min_eq_left hk];refine qBn sN H (V k (by exact Nat.le_of_lt ‹_›)) (by clear hy he sN hx hb0 wL k8 hN;exact omS397 hN hk) ((qIk h hy hk ((G k hk).trans (hB k).symm) (List.foldl_ext _ _ _ λ_ p hp=>?_)).trans (S _ hk));simp only[D_xf];rw[Nat.add_right_comm,hb0,<-hB,(qCf h.ok h.hC hy hk (h.lt p hp)).2]
    . rw[Nat.min_eq_right (Nat.not_lt.1 hk)] at H;rwa[Nat.min_eq_right (by exact omS398 hk)]
   simp only[c.hz,R];refine ih (c.g.wo-(x+8)) (by clear hy hQ hN sN hb0 hq0 wL;exact omS399 he hx) (x+8) _ _ _ _ rfl (λhk=>(G 8 hk).trans (hB 8).symm) (λhk=>V 8 hk.le) ?_
   have A':∀k,k<8->8<=(c.g.wo-x).toUSize->∀u (hu:(q+ὴ k).toNat< ѧu),D_I c Q (x+k) u->D_I c Q (x+k+1) (u.uset (q+ὴ k) (D_w c (loOf c.e0) (spR c.g.relu (loOf c.e0)) o y x b k (D_ff (D_xf c o b) (λp=>dv c.g.W[o*c.g.k+p]!) L k 0)) hu):=λk k8 h8 u hu H0=>by
    have h0:=A k k8 u H0;unfold D_l at h0;rw[ἱ (lcN8 h8 k k8)] at h0;unfold D_I C_Inv at H0;rwa[lcWU _ _ _ hu (by rw[H0.1];exact sN)]
   unfold D_g8;split
   . rename_i hzb;simp only[Bool.and_eq_true,beq_iff_eq,D_zbk] at hzb
     have B0:∀k,k<8->∀u,D_I c Q (x+k) u->D_I c Q (x+k+1) (D_zl q (c.g.wo-x).toUSize k u):=λk k8 u H=>by
      have hN:(ὴ k<(c.g.wo-x).toUSize)↔x+k<c.g.wo:=by rw[А,uk k (by exact Nat.lt_of_lt_of_le ‹_› ה),qAa _ (by clear hy hQ hN sN hb0 hq0 k8;exact omS395 wL)];(clear hy hQ hN sN hb0 hq0 wL k8;exact omS396)
      unfold D_zl D_I at*;simp only[hN];split<;>rename_i hk
      . rw[Nat.min_eq_left hk.le] at H;rw[Nat.min_eq_left hk];exact qBn sN H (V k (by exact Nat.le_of_lt ‹_›)) (by clear hy he sN hx hb0 wL k8 hN;exact omS397 hN hk) ((zt_out h hy hk ((G k hk).trans (hB k).symm) (by rw[G k hk,<-hb0];exact D_ztB_imp0 _ _ _ _ hzb.2.2 k k8) hzb.2.1).symm.trans (S _ hk))
      . rw[Nat.min_eq_right (Nat.not_lt.1 hk)] at H;rwa[Nat.min_eq_right (by exact omS398 hk)]
     exact B0 7 ה _ (B0 6 ה _ (B0 5 ה _ (B0 4 ה _ (B0 3 ה _ (B0 2 ה _ (B0 1 ה _ (B0 0 ה _ H)))))))
   . unfold D_g8a;split
     . rename_i hq;exact A' 7 ה hq.1 _ _ (A' 6 ה hq.1 _ _ (A' 5 ה hq.1 _ _ (A' 4 ה hq.1 _ _ (A' 3 ה hq.1 _ _ (A' 2 ה hq.1 _ _ (A' 1 ה hq.1 _ _ (A' 0 ה hq.1 _ _ H)))))))
     . exact (A 7 (by exact Nat.le_refl _) _ (A 6 ה _ (A 5 ה _ (A 4 ה _ (A 3 ה _ (A 2 ה _ (A 1 ה _ (A 0 ה _ H))))))))
 split<;>rename_i h1
 . have hb0:=hb h1;unfold D_I at*;rw[Nat.min_eq_left h1.le] at H;refine ih (c.g.wo-(x+1)) (by clear hy hQ hN sN hb0;exact omS204 he h1) (x+1) _ _ _ _ rfl (λhx=>?_) (λ_=>(qBm (hq h1) ὄ (by clear hy he h8 h1 hb0;omega)).trans rfl) ?_;. have:=qGa h.ok h.hC hy hx;rw[qIn,Ὑ] at this⊢;exact qBm hb0 h.ok.st this
   . rw[qCu h hy h1 hb0,Nat.min_eq_left h1];exact qBn sN H (hq h1) (by clear hy he sN h8 hb0;exact omS15 hN h1) (S _ h1)
 unfold D_I at H
 rwa[Nat.min_eq_right (by clear hy hQ hN he sN h8;exact omS400 h1)] at H
lemma qFeZ {c:D_P} {o:Ὣ} {L:љ} (h:D_PO c o L) {y Q:Ὣ} (hy:y<c.g.ho) (hQ:Q=(o*c.g.ho+y)*c.g.wo) (hN:Q+c.g.wo<=c.g.C*(c.g.ho*c.g.wo)) (d:Ὣ):
 ∀(x:Ὣ) (b q:ὦ) (rr:D_R) (out:ё),c.g.wo-x=d->(x<c.g.wo->Ѥb=D_B c.g y x)->(x<c.g.wo->Ѥq=Q+x)->D_I c Q x out->
  C_Inv (c.g.C*(c.g.ho*c.g.wo)) (D_sp c.g) (λp=>p<Q+c.g.wo) (D_colsZ c (loOf c.e0) (spR c.g.relu (loOf c.e0)) o y x b q rr out):=by
 induction' d using Nat.strong_induction_on with d ih
 intro x b q rr out he hb hq H
 have sN:=h.ok.sN
 have S:∀x',x'<c.g.wo->D_out c.g o y x'=D_sp c.g (Q+x'):=λ_ hx=>hQ▸qG9 c.g hy hx
 rw[D_colsZ]
 split<;>rename_i h8
 . obtain ⟨hx,hf,hl⟩:=h8;have hb0:=hb hx;have hq0:=hq hx;have hB:=λk=>(qIn c.g y x k).trans (by rw[h.ok.f4 hf,Nat.mul_one]);have G:=λk (hk:x+k<c.g.wo)=>qBm hb0 (uk k (by clear hy he hx hl hb0;exact omS392 hN sN hk)) (hB k▸qGa h.ok h.hC hy hk);have V:=λk (hk:x+k<=c.g.wo)=>(qBm hq0 (uk k (by clear hy he hx hl hb0;exact omS393 hN sN hk)) (by clear hy he hx hl hb0;exact omS394 hN sN hk)).trans (ὢ Q x k)
   have R:=qF7 (c.h8 b hl) rr (D_xf c o b) (λp=>dv c.g.W[o*c.g.k+p]!) L (vFxv h.ok) (vFwv h) 0 0 0 0 0 0 0 0 0 ((Ἇ _).trans h.len.symm)
    λj p hj=>by
     have h1:=h.rd j p hj;rw[C_z,Ἇ,h1.1];exact ⟨h1.2,λ_ _=>rfl⟩
   have wL:c.g.wo<2^32:=by have:=Nat.le_mul_of_pos_left c.g.wo ((by clear hQ hN he sN hx hb0 hq0;exact omS147 hy):0<c.g.ho);have:=Nat.le_mul_of_pos_left (c.g.ho*c.g.wo) (show 0<c.g.C by have:=h.hC;omega);omega
   by_cases hz:zBlk c.g o y x=true
   . simp only[hz,↓reduceIte]
     have A0:∀k,k<8->∀u,D_I c Q (x+k) u->D_I c Q (x+k+1) (D_zl q (c.g.wo-x).toUSize k u):=λk k8 u H=>by
      have hN:(ὴ k<(c.g.wo-x).toUSize)↔x+k<c.g.wo:=by rw[А,uk k (by exact Nat.lt_of_lt_of_le ‹_› ה),qAa _ (by clear hy hQ hN sN hb0 hq0 k8;exact omS395 wL)];(clear hy hQ hN sN hb0 hq0 wL k8;exact omS396)
      unfold D_zl D_I at*;simp only[hN];split<;>rename_i hk
      . rw[Nat.min_eq_left hk.le] at H;rw[Nat.min_eq_left hk];exact qBn sN H (V k (by exact Nat.le_of_lt ‹_›)) (by clear hy he sN hx hb0 wL k8 hN;exact omS397 hN hk) ((zWinX_out c.g h.ok.hk h.ok.pk h.ok.HW h.ok.wp o y (x+k) (λhd=>by have e:=h.ok.hC;simp only[hd,ite_true] at e;exact e▸h.hC) hy (zBlk_lane c.g (h.ok.f4 hf) o y x k k8 hz)).symm.trans (S _ hk))
      . rw[Nat.min_eq_right (Nat.not_lt.1 hk)] at H;rwa[Nat.min_eq_right (by exact omS398 hk)]
     exact ih (c.g.wo-(x+8)) (by clear hy hQ hN sN hb0 hq0 wL;exact omS399 he hx) (x+8) _ _ _ _ rfl (λhk=>(G 8 hk).trans (hB 8).symm) (λhk=>V 8 hk.le) (A0 7 ה _ (A0 6 ה _ (A0 5 ה _ (A0 4 ה _ (A0 3 ה _ (A0 2 ה _ (A0 1 ה _ (A0 0 ה _ H))))))))
   simp only[hz,Bool.false_eq_true,↓reduceIte]
   have A:∀k,k<8->∀u,D_I c Q (x+k) u->D_I c Q (x+k+1) (D_l c (loOf c.e0) (spR c.g.relu (loOf c.e0)) o y x b q (c.g.wo-x).toUSize k (D_ff (D_xf c o b) (λp=>dv c.g.W[o*c.g.k+p]!) L k 0) u):=λk k8 u H=>by
    have hN:(ὴ k<(c.g.wo-x).toUSize)↔x+k<c.g.wo:=by rw[А,uk k (by exact Nat.lt_of_lt_of_le ‹_› ה),qAa _ (by clear hy hQ hN sN hb0 hq0 k8;exact omS395 wL)];(clear hy hQ hN sN hb0 hq0 wL k8;exact omS396)
    unfold D_l D_I at*;simp only[hN];split<;>rename_i hk;. rw[Nat.min_eq_left hk.le] at H;rw[Nat.min_eq_left hk];refine qBn sN H (V k (by exact Nat.le_of_lt ‹_›)) (by clear hy he sN hx hb0 wL k8 hN;exact omS397 hN hk) ((qIk h hy hk ((G k hk).trans (hB k).symm) (List.foldl_ext _ _ _ λ_ p hp=>?_)).trans (S _ hk));simp only[D_xf];rw[Nat.add_right_comm,hb0,<-hB,(qCf h.ok h.hC hy hk (h.lt p hp)).2]
    . rw[Nat.min_eq_right (Nat.not_lt.1 hk)] at H;rwa[Nat.min_eq_right (by exact omS398 hk)]
   simp only[c.hz,R];refine ih (c.g.wo-(x+8)) (by clear hy hQ hN sN hb0 hq0 wL;exact omS399 he hx) (x+8) _ _ _ _ rfl (λhk=>(G 8 hk).trans (hB 8).symm) (λhk=>V 8 hk.le) ?_
   have A':∀k,k<8->8<=(c.g.wo-x).toUSize->∀u (hu:(q+ὴ k).toNat< ѧu),D_I c Q (x+k) u->D_I c Q (x+k+1) (u.uset (q+ὴ k) (D_w c (loOf c.e0) (spR c.g.relu (loOf c.e0)) o y x b k (D_ff (D_xf c o b) (λp=>dv c.g.W[o*c.g.k+p]!) L k 0)) hu):=λk k8 h8 u hu H0=>by
    have h0:=A k k8 u H0;unfold D_l at h0;rw[ἱ (lcN8 h8 k k8)] at h0;unfold D_I C_Inv at H0;rwa[lcWU _ _ _ hu (by rw[H0.1];exact sN)]
   unfold D_g8a;split
   . rename_i hq;exact A' 7 ה hq.1 _ _ (A' 6 ה hq.1 _ _ (A' 5 ה hq.1 _ _ (A' 4 ה hq.1 _ _ (A' 3 ה hq.1 _ _ (A' 2 ה hq.1 _ _ (A' 1 ה hq.1 _ _ (A' 0 ה hq.1 _ _ H)))))))
   . exact (A 7 (by exact Nat.le_refl _) _ (A 6 ה _ (A 5 ה _ (A 4 ה _ (A 3 ה _ (A 2 ה _ (A 1 ה _ (A 0 ה _ H))))))))
 split<;>rename_i h1
 . have hb0:=hb h1;unfold D_I at*;rw[Nat.min_eq_left h1.le] at H;refine ih (c.g.wo-(x+1)) (by clear hy hQ hN sN hb0;exact omS204 he h1) (x+1) _ _ _ _ rfl (λhx=>?_) (λ_=>(qBm (hq h1) ὄ (by clear hy he h8 h1 hb0;omega)).trans rfl) ?_;. have:=qGa h.ok h.hC hy hx;rw[qIn,Ὑ] at this⊢;exact qBm hb0 h.ok.st this
   . rw[qCu h hy h1 hb0,Nat.min_eq_left h1];exact qBn sN H (hq h1) (by clear hy he sN h8 hb0;exact omS15 hN h1) (S _ h1)
 unfold D_I at H
 rwa[Nat.min_eq_right (by clear hy hQ hN he sN h8;exact omS400 h1)] at H
lemma qHu {u v:ὦ} {a b:Ὣ} (hu:Ѥu=a) (hv:Ѥv=b) (h:a*b<2^32):(u*v).toNat=a*b:=by rw[qIb _ _ (by rw[hu,hv];exact h),hu,hv]
lemma qEy {c:D_P} {o:Ὣ} {L:љ} (h:D_PO c o L) (hN:(o*c.g.ho+c.g.ho)*c.g.wo<=c.g.C*(c.g.ho*c.g.wo)) (d:Ὣ):
 ∀(y:Ὣ) (q:ὦ) (rs:List D_R) (out:ё),y+d=c.g.ho-> Ѥq=(o*c.g.ho+y)*c.g.wo->C_Inv (c.g.C*(c.g.ho*c.g.wo)) (D_sp c.g) (λp=>p<(o*c.g.ho+y)*c.g.wo) out->
  C_Inv (c.g.C*(c.g.ho*c.g.wo)) (D_sp c.g) (λp=>p<(o*c.g.ho+c.g.ho)*c.g.wo) (D_rows c (loOf c.e0) (spR c.g.relu (loOf c.e0)) o y q rs out):=by
 induction' d with d ih<;>intro y q rs out he hq H
 . rw[D_rows,ἐ (by clear hN hq;(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›))))];rwa[((by exact omS155 he):y=c.g.ho)] at H
 have hy:y<c.g.ho:=(by clear hN hq;exact omS136 he)
 have sN:=h.ok.sN
 have hA:=qBv ((by clear hN hq sN;(clear hy;exact omS401 he)):o*c.g.ho+y<o*c.g.ho+c.g.ho) (Nat.le_refl c.g.wo)
 have eW:=qAa c.g.wo (by clear he hy;exact omS402 hN sN hA)
 have bB:=qGa h.ok h.hC hy (x:=0) (h.ok.wo▸Nat.succ_pos _)
 have hwp:0<c.g.wp:=h.ok.wp▸(qHl (qHk h.ok h.hC h.ok.pk)).2.trans_le h.ok.kw
 simp only[D_B,zero_mul,add_zero,Nat.reducePow] at bB
 have:=Ὄ (y*c.g.s.stride) hwp
 have:=Ὄ y h.ok.p0
 rw[D_rows,ἱ hy]
 refine ih (y+1) _ _ _ (by exact omS137 he) ?_ ?_<;>rw[<-ὢ,Nat.add_one_mul]
 . exact qBm hq eW (by exact omS403 hN sN hA)
 exact qFe h hy rfl (by exact Nat.le_trans ‹_› ‹_›) c.g.wo 0 _ _ _ _ (Nat.sub_zero _) (λ_=>(qHu (qHu (qAa y א) h.ok.st (by exact Nat.lt_of_le_of_lt ‹_› ‹_›)) (qAa _ h.ok.wp2) bB).trans (by simp only[D_B,zero_mul,add_zero])) (λ_=>hq) (by unfold D_I;rw[Nat.zero_min];exact H)
lemma qEyZ {c:D_P} {o:Ὣ} {L:љ} (h:D_PO c o L) (hN:(o*c.g.ho+c.g.ho)*c.g.wo<=c.g.C*(c.g.ho*c.g.wo)) (d:Ὣ):
 ∀(y:Ὣ) (q:ὦ) (rs:List D_R) (out:ё),y+d=c.g.ho-> Ѥq=(o*c.g.ho+y)*c.g.wo->C_Inv (c.g.C*(c.g.ho*c.g.wo)) (D_sp c.g) (λp=>p<(o*c.g.ho+y)*c.g.wo) out->
  C_Inv (c.g.C*(c.g.ho*c.g.wo)) (D_sp c.g) (λp=>p<(o*c.g.ho+c.g.ho)*c.g.wo) (D_rowsZ c (loOf c.e0) (spR c.g.relu (loOf c.e0)) o y q rs out):=by
 induction' d with d ih<;>intro y q rs out he hq H
 . rw[D_rowsZ,ἐ (by clear hN hq;(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›))))];rwa[((by exact omS155 he):y=c.g.ho)] at H
 have hy:y<c.g.ho:=(by clear hN hq;exact omS136 he)
 have sN:=h.ok.sN
 have hA:=qBv ((by clear hN hq sN;(clear hy;exact omS401 he)):o*c.g.ho+y<o*c.g.ho+c.g.ho) (Nat.le_refl c.g.wo)
 have eW:=qAa c.g.wo (by clear he hy;exact omS402 hN sN hA)
 have bB:=qGa h.ok h.hC hy (x:=0) (h.ok.wo▸Nat.succ_pos _)
 have hwp:0<c.g.wp:=h.ok.wp▸(qHl (qHk h.ok h.hC h.ok.pk)).2.trans_le h.ok.kw
 simp only[D_B,zero_mul,add_zero,Nat.reducePow] at bB
 have:=Ὄ (y*c.g.s.stride) hwp
 have:=Ὄ y h.ok.p0
 rw[D_rowsZ,ἱ hy]
 refine ih (y+1) _ _ _ (by exact omS137 he) ?_ ?_<;>rw[<-ὢ,Nat.add_one_mul]
 . exact qBm hq eW (by exact omS403 hN sN hA)
 exact qFeZ h hy rfl (by exact Nat.le_trans ‹_› ‹_›) c.g.wo 0 _ _ _ _ (Nat.sub_zero _) (λ_=>(qHu (qHu (qAa y א) h.ok.st (by exact Nat.lt_of_le_of_lt ‹_› ‹_›)) (qAa _ h.ok.wp2) bB).trans (by simp only[D_B,zero_mul,add_zero])) (λ_=>hq) (by unfold D_I;rw[Nat.zero_min];exact H)
lemma qGt (s:OS) (o:Ὣ) {p:Ὣ} (hp:p<s.kh*s.kw):
 D_off s (o*(s.kh*s.kw)+p)=o*((s.h+2*s.pad)*(s.w+2*s.pad))+D_off s p:=by
 unfold D_off
 rw[(qCn o hp).1,Ὃ hp,<-Nat.mul_assoc,Nat.add_comm (o*s.kh*s.kw),Nat.add_mul_div_right _ _ (Nat.pos_of_mul_pos_left (Nat.zero_lt_of_lt hp)),Nat.add_mul_mod_self_right,Nat.add_mul_mod_self_right]
 ring
lemma qIq (a b:Ὣ):a.toUInt32+b.toUInt32=(a+b).toUInt32:=(UInt32.ofNat_add a b).symm
lemma qHp {g:D_C} (ok:D_Ok g) (o:Ὣ) {t:ὦ} (hp:Ѥt<g.k):
 (if g.dw then (o*((g.s.h+2*g.s.pad)*g.wp)).toUInt32 else 0)+rdW g.tab t=(D_off g.s (D_t0 g o+Ѥt)).toUInt32:=by
 have h2:=ok.nk
 rw[qAo _ _ (by simp only[ok.tb,Nat.toUInt32_eq,Array.size_ofFn,Nat.reducePow];(exact Nat.lt_of_lt_of_le ‹_› ה)),ok.tb]
 generalize Ѥt=p at hp⊢
 rw[Ἕ _ p (by simpa only[Nat.toUInt32_eq,Array.size_ofFn] using hp),Array.getElem_ofFn]
 have h1:=ok.hk
 unfold D_t0
 cases hd:g.dw
 . simp only[false_eq_true, ↓reduceIte,Nat.toUInt32_eq,zero_add]
 simp only[hd,ite_true] at h1⊢
 rw[h1,qGt g.s o (h1▸hp),ok.wp,qIq]
def D_Dc (g:D_C) (o:Ὣ) (xw:Ὣ->Ὢ) (sz t:Ὣ) (off:ѐ) (buf:ὤ)
 (c:ὦ) (l:ὧ):Prop:=
 Ѥc=(C_keep xw t).length/\ѧoff=sz/\Ѩbuf=sz/\
  (∀k p:Ὣ,(C_keep xw t)[k]?=some p->
   (off[k]!:ὥ)=(D_off g.s (D_t0 g o+p)).toUInt32/\buf.data[k]! =dv (xw p))/\
  l=(C_keep xw t).foldl (λ(a:ὧ) p=>a+(dv (xw p)).abs) 0
lemma qHn {g:D_C} (ok:D_Ok g) {o' n:ὦ} (o:Ὣ) (xw:Ὣ->Ὢ) (hk:Ѥn=g.k) (hn:Ѥn<2^32) (hxw:∀t:ὦ,t<n->rdW g.W (o'+t)=xw Ѥt) (d:Ὣ):
 ∀(t:ὦ) (off:ѐ) (buf:ὤ) (c:ὦ) (l:ὧ) (m:ὥ) (hm:∀j (h:j< ѧoff),off[j].toNat<=ѥm),Ѥt+d=Ѥn->D_Dc g o xw Ѥn Ѥt off buf c l->
  D_Dc g o xw Ѥn Ѥn (D_dec g.tab g.W (if g.dw then (o*((g.s.h+2*g.s.pad)*g.wp)).toUInt32 else 0) o' n t off buf c l m hm).off (D_dec g.tab g.W (if g.dw then (o*((g.s.h+2*g.s.pad)*g.wp)).toUInt32 else 0) o' n t off buf c l m hm).buf (D_dec g.tab g.W (if g.dw then (o*((g.s.h+2*g.s.pad)*g.wp)).toUInt32 else 0) o' n t off buf c l m hm).c (D_dec g.tab g.W (if g.dw then (o*((g.s.h+2*g.s.pad)*g.wp)).toUInt32 else 0) o' n t off buf c l m hm).l:=by
 induction d with
 | zero=>
  intro t off buf c l m hm he H
  rw[D_dec,ἔ (by rw[А];exact omS404 he)]
  have e:Ѥt=Ѥn:=(by exact omS155 he)
  rw[e] at H;exact H
 | succ d ih=>
  intro t off buf c l m hm he H
  have ht:t<n:=by rw[А];(clear hn;exact omS136 he)
  have hu:=usz t n ht
  rw[D_dec,Ἄ ht]
  dsimp only
  rw[hxw t ht]
  obtain ⟨h1,h2,h3,h4,h5⟩:=H
  by_cases hz:xw Ѥt&&&В=0
  . rw[ἱ (by simpa only[beq_iff_eq] using hz)];apply ih (t+1) _ _ _ _ _ _ (by clear hn ht h1 h2 h3;exact omS157 he hu);rw[hu,D_Dc,qDs xw _ hz];exact ⟨h1,h2,h3,h4,h5⟩
  . rw[ἐ (by simpa only[beq_iff_eq] using hz)];apply ih (t+1) _ _ _ _ _ _ (by clear hn ht h1 h2 h3;exact omS157 he hu);have hl:=qCb xw Ѥt;have hc:Ѥc< Ѥt+1:=(by clear hk hn he ht h2 h3;(clear hu;exact omS405 h1 hl));rw[hu,D_Dc,qEb xw _ hz];have hcn:Ѥc< Ѥn:=(by exact Nat.lt_of_lt_of_le ‹_› ‹_›);refine ⟨?_,by rw[qAm];exact h2,by rw[qB9];exact h3,λk p hk2=>?_,by rw[List.foldl_append,<-h5];rfl⟩;. rw[qAv _ _ (by rw[ὄ];(exact Nat.lt_of_le_of_lt ‹_› ‹_›)),ὄ,h1];simp only[List.length_append,List.length_cons,List.length_nil,zero_add,Nat.add_left_cancel_iff]
    . rw[qAq _ _ _ _ (by clear ht h1 hl hc hcn;exact omS360 hn h2) (by clear hn ht hl hc;exact omS140 h2 hcn),qCr _ _ _ _ (by clear ht h1 hl hc hcn;exact omS360 hn h3) (by clear hn ht hl hc;exact omS140 h3 hcn)];rw[List.getElem?_append,<-h1] at hk2;by_cases e:k=Ѥc;. subst e;simp only[lt_self_iff_false, ↓reduceIte,tsub_self,List.length_cons,List.length_nil,zero_add,Order.lt_one_iff,getElem?_pos,List.getElem_cons_zero,Option.some.injEq] at hk2;subst hk2;rw[ἱ rfl,ἱ rfl];exact ⟨qHp ok o (by clear hn ht h1 hl hc hcn;exact omS406 hk he),rfl⟩
      . rw[ἐ e,ἐ e];by_cases e2:k< Ѥc;. rw[ἱ e2] at hk2;exact h4 k p hk2
        . rw[ἐ e2] at hk2;obtain ⟨z,hz'⟩:∃z,k-Ѥc=z+1:=⟨k-Ѥc-1,by (clear hk hn he ht hu h2 h3 hl hc hcn;exact omS407 e e2)⟩;rw[hz'] at hk2;simp only[List.length_cons,List.length_nil,zero_add,Order.lt_one_iff,Nat.add_eq_zero_iff,one_ne_zero,and_false,not_false_eq_true,getElem?_neg,reduceCtorEq] at hk2
lemma qIi {g:D_C} (ok:D_Ok g) {o:Ὣ} (hC:o<g.C) {off:ѐ} {buf:ὤ} (h2:ѧoff=g.k) (h3:Ѩbuf=g.k) (m:ὥ) (hm:∀j (h:j< ѧoff),off[j].toNat<=ѥm):
 D_Dc g o (λp=>g.W[o*g.k+p]!) g.k g.k (D_dec g.tab g.W (if g.dw then (o*((g.s.h+2*g.s.pad)*g.wp)).toUInt32 else 0) (o*g.k).toUSize g.k.toUSize 0 off buf 0 0 m hm).off (D_dec g.tab g.W (if g.dw then (o*((g.s.h+2*g.s.pad)*g.wp)).toUInt32 else 0) (o*g.k).toUSize g.k.toUSize 0 off buf 0 0 m hm).buf (D_dec g.tab g.W (if g.dw then (o*((g.s.h+2*g.s.pad)*g.wp)).toUInt32 else 0) (o*g.k).toUSize g.k.toUSize 0 off buf 0 0 m hm).c (D_dec g.tab g.W (if g.dw then (o*((g.s.h+2*g.s.pad)*g.wp)).toUInt32 else 0) (o*g.k).toUSize g.k.toUSize 0 off buf 0 0 m hm).l:=by
 have hX:=ok.sW2
 have hP:=(qBv hC (Nat.le_refl g.k)).trans_eq ok.sW.symm
 have eo:(o*g.k).toUSize.toNat=o*g.k:=qAa _ (by clear hC;exact omS154 hX hP)
 have ek:g.k.toUSize.toNat=g.k:=qAa _ (by clear hC h2;(clear h3;exact omS408 hX hP))
 have D:=qHn (o':=(o*g.k).toUSize) (n:=g.k.toUSize) ok o (λp=>g.W[o*g.k+p]!) ek (by rw[ek];(clear hC h2;(clear h3;exact omS408 hX hP)))
  (λt ht=>by
   have h4:=А.mp ht
   rw[ek] at h4
   rw[qAo _ _ hX,qAv _ _ (by rw[eo];(clear hC h2 ht;(clear h3;exact omS145 hX hP h4))),eo])
  g.k 0 off buf 0 0 m hm
  (by rw[C_z,ek,Ἇ])
  ⟨by simp only[USize.toNat_zero,C_keep,List.range_zero,List.filter_nil,List.length_nil],by rw[ek];exact h2,by rw[ek];exact h3,λk p hk=>by simp only[C_keep,USize.toNat_zero,List.range_zero,List.filter_nil,List.length_nil,not_lt_zero,not_false_eq_true,getElem?_neg,reduceCtorEq] at hk,by simp only[C_keep,USize.toNat_zero,List.range_zero,List.filter_nil,List.foldl_nil]⟩
 rw[ek] at D
 exact D
lemma qFi (A:ѐ) (n:ὦ) (hn:Ѥn<=ѧA) (d:Ὣ):
 ∀(i:ὦ) (lo hi:ὥ),Ѥi+d=Ѥn->scan A n i hn lo hi=
  if 0x007FFFFF<=(D_scan A n i hn lo hi).lo/\(D_scan A n i hn lo hi).hi<У then (D_scan A n i hn lo hi).hi else Е:=by
 induction' d with d ih<;>intro i lo hi he<;>rw[scan,D_scan]
 . simp only[ἔ (qCk he.ge)]
 simp only[Ἄ (qBz he).1]
 exact ih (i+1) _ _ (qBz he).2
lemma qFw (s:OS) {c r q:Ὣ} (hr:r<s.kh) (hq:q<s.kw):D_off s ((c*s.kh+r)*s.kw+q)=(c*(s.h+2*s.pad)+r)*(s.w+2*s.pad)+q:=by
 have k0:0<s.kw:=(by clear hr;exact omS147 hq);have h0:0<s.kh:=(by clear hq k0;exact omS147 hr);have e1:((c*s.kh+r)*s.kw+q)/s.kw=c*s.kh+r:=by rw[Nat.add_comm,Nat.add_mul_div_right _ _ k0,Ὃ hq,Ἇ]
 have e2:((c*s.kh+r)*s.kw+q)%s.kw=q:=by rw[Nat.add_comm,Nat.add_mul_mod_self_right,Ἁ hq]
 have e3:(c*s.kh+r)%s.kh=r:=by rw[Nat.add_comm,Nat.add_mul_mod_self_right,Ἁ hr]
 have e4:(c*s.kh+r)/s.kh=c:=by rw[Nat.add_comm,Nat.add_mul_div_right _ _ h0,Ὃ hr,Ἇ]
 unfold D_off;rw[ἥ s.kh s.kw,<-Nat.div_div_eq_div_mul,e1,e4,e3,e2]
lemma qGz (s:OS) (k d:Ὣ):∀(t c r q:Ὣ) (a:ѐ),t+d=k->t=(c*s.kh+r)*s.kw+q->r<s.kh->q<s.kw->a=(Array.ofFn (n:=t) λi=>(D_off s i.val).toUInt32)->
 D_tq k s.kw s.kh (s.h+2*s.pad) (s.w+2*s.pad) t c r q a=Array.ofFn (n:=k) λi=>(D_off s i.val).toUInt32:=by
 induction d with
 | zero=>
  intro t c r q a h1 _ _ _ h5
  have e:t=k:=h1
  subst e
  rw[D_tq,ἐ (Nat.lt_irrefl _)]
  exact h5
 | succ d ih=>
  intro t c r q a h1 h2 h3 h4 h5
  have hp:a.push ((c*(s.h+2*s.pad)+r)*(s.w+2*s.pad)+q).toUInt32=Array.ofFn (n:=t+1) λi=>(D_off s i.val).toUInt32:=by
   rw[<-qFw s h3 h4,<-h2,h5,Array.ofFn_succ]
   rfl
  rw[D_tq,ἱ (by clear h3 h4;exact omS136 h1)]
  dsimp only
  rw[hp]
  split
  . exact ih (t+1) c r (q+1) _ (by exact omS137 h1) (by exact omS409 h2) h3 (by assumption) rfl
  split
  . have e:(c*s.kh+(r+1))*s.kw=(c*s.kh+r)*s.kw+s.kw:=by ring
    exact ih (t+1) c (r+1) 0 _ (by exact omS137 h1) א (by assumption) א rfl
  . have e2:s.kh=r+1:=א;have e:((c+1)*s.kh+0)*s.kw=(c*s.kh+r)*s.kw+s.kw:=by rw[e2];ring
    exact ih (t+1) (c+1) 0 0 _ (by exact omS137 h1) א א א rfl
lemma qFz (c:D_P) (lo sp:Ὠ) (o y:Ὣ) (q0:ὦ) (hy:y<c.g.ho) (d:Ὣ):
 ∀(x:Ὣ) (b q:ὦ) (rr:D_R) (out:ѐ),c.g.wo-x=d->
  ∃rr':D_R,D_rc c lo sp o y x b q q0 rr out=
   D_rc c lo sp o (y+1) 0 ((y+1).toUSize*c.g.step*c.g.wp.toUSize) (q0+c.g.wo.toUSize) (q0+c.g.wo.toUSize) rr' (D_cols c lo sp o y x b q rr out):=by
 induction' d using Nat.strong_induction_on with d ih
 intro x b q rr out he
 rw[D_rc,D_cols,Ἄ hy]
 by_cases h8:x<c.g.wo/\c.g.f4=true/\b<c.l8
 . simp only[Ἄ h8];exact ih _ (by clear hy;exact omS410 he h8) _ _ _ _ _ rfl
 . simp only[ἔ h8];by_cases h1:x<c.g.wo;. simp only[ἱ h1];exact ih _ (by clear hy;exact omS204 he h1) _ _ _ _ _ rfl
   . simp only[ἐ h1];exact ⟨rr,rfl⟩
lemma qFa (c:D_P) (lo sp:Ὠ) (o:Ὣ) (d:Ὣ):
 ∀(y:Ὣ) (q0:ὦ) (rr:D_R) (out:ѐ),c.g.ho-y=d->
  ∃rs:List D_R,D_rc c lo sp o y 0 (ѩy*c.g.step*c.g.wp.toUSize) q0 q0 rr out=D_rows c lo sp o y q0 rs out:=by
 induction d with
 | zero=>
  intro y q0 rr out he
  have hy:¬y<c.g.ho:=(by exact omS411 he)
  exact ⟨[],by rw[D_rc,ἔ hy,D_rows,ἐ hy]⟩
 | succ d ih=>
  intro y q0 rr out he
  have hy:y<c.g.ho:=(by exact omS412 he)
  obtain ⟨rr',h⟩:=qFz c lo sp o y q0 hy _ 0 (ѩy*c.g.step*c.g.wp.toUSize) q0 rr out rfl
  obtain ⟨rs,h2⟩:=ih (y+1) (q0+c.g.wo.toUSize) rr' _ (by exact omS413 he)
  refine ⟨rr::rs,?_⟩
  rw[h,h2]
  conv_rhs=>rw[D_rows,ἱ hy]
  rfl
lemma qF8 {c:D_P} {o:Ὣ} {L:љ} (h:D_PO c o L) (hN:(o*c.g.ho+c.g.ho)*c.g.wo<=c.g.C*(c.g.ho*c.g.wo)) {q:ὦ} {out:ё} (hq:Ѥq=(o*c.g.ho+0)*c.g.wo) (H:C_Inv (c.g.C*(c.g.ho*c.g.wo)) (D_sp c.g) (λp=>p<(o*c.g.ho+0)*c.g.wo) out):
 C_Inv (c.g.C*(c.g.ho*c.g.wo)) (D_sp c.g) (λp=>p<(o*c.g.ho+c.g.ho)*c.g.wo)
  (D_rc c (loOf c.e0) (spR c.g.relu (loOf c.e0)) o 0 0 ((0:Ὣ).toUSize*c.g.step*c.g.wp.toUSize) q q ⟨0,0,0,0,0,0,0,0,0⟩ out):=by
 obtain ⟨rs,hrs⟩:=qFa c (loOf c.e0) (spR c.g.relu (loOf c.e0)) o _ 0 q ⟨0,0,0,0,0,0,0,0,0⟩ out rfl
 rw[hrs]
 exact qEy h hN c.g.ho 0 q rs out (Ἇ _) hq H
lemma qFzZ (c:D_P) (lo sp:Ὠ) (o y:Ὣ) (q0:ὦ) (hy:y<c.g.ho) (d:Ὣ):
 ∀(x:Ὣ) (b q:ὦ) (rr:D_R) (out:ѐ),c.g.wo-x=d->
  ∃rr':D_R,D_rcZ c lo sp o y x b q q0 rr out=
   D_rcZ c lo sp o (y+1) 0 ((y+1).toUSize*c.g.step*c.g.wp.toUSize) (q0+c.g.wo.toUSize) (q0+c.g.wo.toUSize) rr' (D_colsZ c lo sp o y x b q rr out):=by
 induction' d using Nat.strong_induction_on with d ih
 intro x b q rr out he
 rw[D_rcZ,D_colsZ,Ἄ hy]
 by_cases h8:x<c.g.wo/\c.g.f4=true/\b<c.l8
 . simp only[Ἄ h8];by_cases hz:zBlk c.g o y x=true
   . simp only[hz,↓reduceIte];exact ih _ (by clear hy;exact omS410 he h8) _ _ _ _ _ rfl
   . simp only[hz,Bool.false_eq_true,↓reduceIte];exact ih _ (by clear hy;exact omS410 he h8) _ _ _ _ _ rfl
 . simp only[ἔ h8];by_cases h1:x<c.g.wo;. simp only[ἱ h1];exact ih _ (by clear hy;exact omS204 he h1) _ _ _ _ _ rfl
   . simp only[ἐ h1];exact ⟨rr,rfl⟩
lemma qFaZ (c:D_P) (lo sp:Ὠ) (o:Ὣ) (d:Ὣ):
 ∀(y:Ὣ) (q0:ὦ) (rr:D_R) (out:ѐ),c.g.ho-y=d->
  ∃rs:List D_R,D_rcZ c lo sp o y 0 (ѩy*c.g.step*c.g.wp.toUSize) q0 q0 rr out=D_rowsZ c lo sp o y q0 rs out:=by
 induction d with
 | zero=>
  intro y q0 rr out he
  have hy:¬y<c.g.ho:=(by exact omS411 he)
  exact ⟨[],by rw[D_rcZ,ἔ hy,D_rowsZ,ἐ hy]⟩
 | succ d ih=>
  intro y q0 rr out he
  have hy:y<c.g.ho:=(by exact omS412 he)
  obtain ⟨rr',h⟩:=qFzZ c lo sp o y q0 hy _ 0 (ѩy*c.g.step*c.g.wp.toUSize) q0 rr out rfl
  obtain ⟨rs,h2⟩:=ih (y+1) (q0+c.g.wo.toUSize) rr' _ (by exact omS413 he)
  refine ⟨rr::rs,?_⟩
  rw[h,h2]
  conv_rhs=>rw[D_rowsZ,ἱ hy]
  rfl
lemma qF8Z {c:D_P} {o:Ὣ} {L:љ} (h:D_PO c o L) (hN:(o*c.g.ho+c.g.ho)*c.g.wo<=c.g.C*(c.g.ho*c.g.wo)) {q:ὦ} {out:ё} (hq:Ѥq=(o*c.g.ho+0)*c.g.wo) (H:C_Inv (c.g.C*(c.g.ho*c.g.wo)) (D_sp c.g) (λp=>p<(o*c.g.ho+0)*c.g.wo) out):
 C_Inv (c.g.C*(c.g.ho*c.g.wo)) (D_sp c.g) (λp=>p<(o*c.g.ho+c.g.ho)*c.g.wo)
  (D_rcZ c (loOf c.e0) (spR c.g.relu (loOf c.e0)) o 0 0 ((0:Ὣ).toUSize*c.g.step*c.g.wp.toUSize) q q ⟨0,0,0,0,0,0,0,0,0⟩ out):=by
 obtain ⟨rs,hrs⟩:=qFaZ c (loOf c.e0) (spR c.g.relu (loOf c.e0)) o _ 0 q ⟨0,0,0,0,0,0,0,0,0⟩ out rfl
 rw[hrs]
 exact qEyZ h hN c.g.ho 0 q rs out (Ἇ _) hq H
section
variable {g:D_C} (ok:D_Ok g)
include g ok
lemma qC4 {o:Ὣ} (hC:o<g.C):
 (D_decAll g.tab g.W g.dw g.s g.wp g.k g.C)[o]! =
  D_dec g.tab g.W (if g.dw then (o*((g.s.h+2*g.s.pad)*g.wp)).toUInt32 else 0) (o*g.k).toUSize g.k.toUSize 0 (ἇ g.k 0) (pushZ g.k.toUSize 0 (ἆ g.k)) 0 0 0 (qF0 g.k):=by
 unfold D_decAll
 rw[Ἕ _ _ (by simp only[Nat.toUInt32_eq,UInt32.ofNat_mul,UInt32.ofNat_add,UInt32.reduceOfNat,Nat.toUSize_eq,USize.ofNat_mul,Array.size_ofFn];exact hC),Array.getElem_ofFn]
lemma qEg (hs:g.xd.size<2^32) (d:Ὣ):
 ∀(o:Ὣ) (q:ὦ) (out:ё),o+d=g.C->Ѥq=o*(g.ho*g.wo)->C_Inv (g.C*(g.ho*g.wo)) (D_sp g) (λp=>p<o*(g.ho*g.wo)) out->
  C_Inv (g.C*(g.ho*g.wo)) (D_sp g) (λp=>p<g.C*(g.ho*g.wo)) (D_outsC g hs o q (D_decAll g.tab g.W g.dw g.s g.wp g.k g.C) out):=by
 induction' d with d ih<;>intro o q out he hq H
 . rw[D_outsC,ἐ (by clear hs hq;(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›))))];rwa[((by exact omS155 he):o=g.C)] at H
 have hC:o<g.C:=(by clear hq;exact omS136 he)
 have hkn:=ok.nk
 have sW2:=ok.sW2
 have sN:=ok.sN
 have hP:=(qBv hC (Nat.le_refl g.k)).trans_eq ok.sW.symm
 have hO:=qBv hC (Nat.le_refl (g.ho*g.wo))
 have eB:rdW g.b ѩo=g.b[o]!:=by
  rw[qAo _ _ ok.sb,qAa _ (by have:=Ὄ o ok.pk;(clear hq hC sN hO;omega))]
 have h2:(ἇ g.k (0:ὥ)).size=g.k:=by simp only[Array.size_replicate]
 have h3:(pushZ g.k.toUSize 0 (ἆ g.k)).size=g.k:=
  (qG6 g.k.toUSize g.k 0 _ ((Ἇ _).trans (qAa g.k (ok.nk.trans (by norm_num1))).symm)).trans (Ἇ _)
 obtain ⟨d1,d2,d3,d4,d5⟩:=qIi ok hC h2 h3 0 (qF0 g.k)
 obtain ⟨k1,k2,k3⟩:=qG5 (λp=>g.W[o*g.k+p]!) g.k
 have hlen:=d1.trans_le (qCb _ _)
 rw[D_outsC,ἱ hC]
 dsimp only
 simp only[qC4 ok hC]
 rw[Ἄ (by clear he hq hC hkn sW2 sN hP hO eB;(clear h2;exact omS414 d2 d3 hlen)),eB]
 refine ih (o+1) _ _ (by clear hq hC hkn sW2 sN hP hO eB h2 h3 d1 d2 d3 hlen;exact omS137 he) (Nat.add_one_mul o _▸qBm hq (qAa _ (by clear he hC hkn sW2 hP eB h2 h3 d1 d2 d3 hlen;exact omS408 sN hO)) (by exact Nat.lt_of_le_of_lt ‹_› ‹_›)) ?_
 have E:(o*g.ho+g.ho)*g.wo=o*(g.ho*g.wo)+g.ho*g.wo:=by ring
 rw[Nat.add_one_mul,<-E]
 rw[<-Nat.mul_assoc o] at hq H
 by_cases hE:g.ex.1.isEmpty=true
 . rw[ἱ hE]
   refine qF8 (L:=C_keep _ g.k) ⟨ok,hC,d1,k1,k2,k3,λj p hj=>?_,by dsimp only;rw[ok.nf,d5],rfl⟩ (by dsimp only;(clear he hq hC hkn sW2 sN hP eB h2 h3 d1 d2 d3 hlen;exact omS415 hO E)) hq H
   have hjl:=(List.getElem?_eq_some_iff.mp hj).1
   refine ⟨?_,(d4 j p hj).2⟩
   dsimp only
   rw[(d4 j p hj).1]
   simpa only[Nat.toUInt32_eq,ὡ,Nat.reducePow,Nat.mod_succ_eq_iff_lt,Nat.succ_eq_add_one,Nat.reduceAdd] using Ἁ ((Nat.lt_of_add_left_lt (qCf ok hC (y:=0) (x:=0) (ok.ho▸Nat.succ_pos _) (ok.wo▸Nat.succ_pos _) (k2 _ (List.mem_of_getElem? hj))).1).trans ok.xs)
 . rw[ἐ hE]
   refine qF8Z (L:=C_keep _ g.k) ⟨ok,hC,d1,k1,k2,k3,λj p hj=>?_,by dsimp only;rw[ok.nf,d5],rfl⟩ (by dsimp only;(clear he hq hC hkn sW2 sN hP eB h2 h3 d1 d2 d3 hlen;exact omS415 hO E)) hq H
   have hjl:=(List.getElem?_eq_some_iff.mp hj).1
   refine ⟨?_,(d4 j p hj).2⟩
   dsimp only
   rw[(d4 j p hj).1]
   simpa only[Nat.toUInt32_eq,ὡ,Nat.reducePow,Nat.mod_succ_eq_iff_lt,Nat.succ_eq_add_one,Nat.reduceAdd] using Ἁ ((Nat.lt_of_add_left_lt (qCf ok hC (y:=0) (x:=0) (ok.ho▸Nat.succ_pos _) (ok.wo▸Nat.succ_pos _) (k2 _ (List.mem_of_getElem? hj))).1).trans ok.xs)
lemma qHh (hs:g.xd.size<2^32):
 D_outsC g hs 0 0 (D_decAll g.tab g.W g.dw g.s g.wp g.k g.C) (ἇ (g.C*(g.ho*g.wo)) 0)=if g.dw then depthwiseSpecO g.s g.relu g.X g.W g.b else conv2dSpecO g.s g.relu g.X g.W g.b:=by
 refine (qCe (qEg ok hs g.C 0 0 _ (Ἇ _) ג ⟨by simp only[Array.size_replicate],λq _ h=>א⟩) λq h=>h).trans ?_
 obtain ⟨dw,_,_,s,_,_,_,k,C,ho,wo⟩:=g
 obtain ⟨h1,h2,h3,h4⟩:=ok
 cases dw<;>simp only[false_eq_true, ↓reduceIte] at h1 h2 h3 h4<;>subst h1 h2 h3 h4<;>refine congrArg (Array.ofFn (n:=_)) (funext λq=>?_)<;>simp only[D_sp,D_out,D_xw,D_t0,false_eq_true, ↓reduceIte,zero_add]
end
def D_Zs (F:Ὣ->ὧ) (S m:Ὣ) (a:ὤ):Prop:=
 Ѩa=S/\S<2^32/\∀q,q<S->a.data[q]! =if q< m then F q else dv 0
section
variable {F:Ὣ->ὧ} {S m:Ὣ} {a:ὤ}
include F S m a
lemma qBq (d:Ὣ) (H:D_Zs F S m a) (hv:∀u,u<d->F (m+u)=dv 0):D_Zs F S (m+d) a:=by
 refine ⟨H.1,H.2.1,λq hq=>?_⟩;rw[H.2.2 q hq];by_cases h1:q< m;. rw[ἱ h1,ἱ (Nat.lt_add_right d h1)]
 rw[ἐ h1];by_cases h2:q< m+d;. have e:=hv (q-m) (by clear hq;exact omS416 h1 h2);rw[Nat.add_sub_cancel' (Nat.not_lt.1 h1)] at e;rw[ἱ h2,e]
 rw[ἐ h2]
lemma qGs (H:D_Zs F S m a) (i:ὦ) (hi:Ѥi=m) (hm:m<S):D_Zs F S (m+1) (wrF a i (F m)):=by
 have hz:Ѩa<2^32:=H.1▸H.2.1;refine ⟨(qB9 ..).trans H.1,H.2.1,λq hq=>?_⟩;rw[qCr _ _ _ _ hz (by rw[hi,H.1];exact hm),hi];by_cases e:q=m;. rw[ἱ e,ἱ (by clear hi hm hz hq;exact omS417 e),e]
 rw[ἐ e,H.2.2 q hq];by_cases h:q< m;. rw[ἱ h,ἱ (by clear hi hm hz hq e;exact omS418 h)]
 rw[ἐ h,ἐ (by clear hi hm hz hq;exact omS419 e h)]
end
lemma qE5 {F:Ὣ->ὧ} {S:Ὣ} {X:ѐ} {o n:ὦ} (hs:ѧX<2^32) (ho:Ѥo+Ѥn<=ѧX) (d:Ὣ):
 ∀(t q:ὦ) (m:Ὣ) (a:ὤ),Ѥt+d=Ѥn->Ѥq=m->m+d<=S->D_Zs F S m a->(∀j,j<d->F (m+j)=dv X[Ѥo+Ѥt+j]!)->D_Zs F S (m+d) (D_cpz X o n t q hs ho a):=by
 induction' d with d ih<;>intro t q m a he hq hm H hv
 . rw[D_cpz,ἔ (qCk he.ge)];exact H
 obtain ⟨ht,hu⟩:=qBz he
 rw[D_cpz,Ἄ ht,qFp]
 have hS:=H.2.1
 have e:∀h',dv (X.uget (o+t) h')=F m:=λh'=>by
  rw[qFx,qAv _ _ (by clear hq hm ht hS h';omega)];exact (hv 0 (by clear he hq hm ht hu hS h';exact omS9)).symm
 rw[e]
 refine ih (t+1) (q+1) (m+1) _ hu ?_ (by clear he hq ht hu hS;exact omS5 hm) (qGs H q hq (by clear he hq ht hu hS;exact omS4 hm)) λj hj=>?_
 . exact (qAv q 1 (by rw[ὄ];(clear he ht hu;omega))).trans (by rw[ὄ,hq])
 simpa only[usz t n ht,ὢ,Nat.add_comm 1] using hv (j+1) (by exact Nat.lt_succ_of_le ‹_›)
lemma qE6 (s:OS) (X:ѐ) (hs:ѧX<2^32) (hX:ѧX=s.cin*s.h*s.w) (hh:s.h+2*s.pad<2^32) (hw:s.w+2*s.pad<2^32) {R S:Ὣ} (hR:R=s.cin*(s.h+2*s.pad)) (hS:R*(s.w+2*s.pad)<=S) (d:Ὣ):
 ∀(r:Ὣ) (a:ὤ),r+d=R->D_Zs (D_Fx s X) S (r*(s.w+2*s.pad)) a->D_Zs (D_Fx s X) S (R*(s.w+2*s.pad)) (D_imz s X hs (s.w+2*s.pad) R r a):=by
 induction' d with d ih<;>intro r a he H
 . rw[D_imz,ἔ (by clear hs hX hh hw hS;(exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›)))),<-he];exact H
 have hr:r<R:=(by clear hX hh hw hS;exact omS136 he)
 have hc:=Ἰ (hr.trans_eq (hR.trans (ἥ _ _)))
 have ep:=qAa s.pad (by clear hX hw hR hS he hr hc;exact omS379 hh)
 have ew:=qAa s.w (by clear hX hR hS he hr hc;exact omS380 hw)
 have hO:=λg:r%(s.h+2*s.pad)-s.pad<s.h=>qBv (qDf hc g) (Nat.le_refl s.w)
 have hT:=H.2.1
 rw[D_imz,Ἄ hr]
 dsimp only
 split<;>rename_i g
 . obtain ⟨g1,g2,g3⟩:=g;have:=hO g2;have hl:(r+1)*(s.w+2*s.pad)<=R*(s.w+2*s.pad):=Ἒ _ hr;rw[ᾀ,qF5] at hl;apply ih (r+1) _ (by clear hX hh hw hS hr hc ep ew hT g1 g2 g3 this hl;exact omS137 he);rw[ᾀ,qF5];refine qBq s.pad (qE5 hs g3 s.w 0 _ _ _ (by rw[ew];exact Ἇ _) (qAa _ (by clear hX hR he hr hc g1 g2 g3 this;(clear hw;exact omS420 hS hT hl))) (by clear hX hw hR he hr hc hT g1 g2 g3 this;(clear ew;exact omS421 hS hl)) (qBq s.pad H λu hu=>?_) λj hj=>?_) λu hu=>?_;. rw[qAi _ _ _ (by clear hX hh hw hR hS he hr hc hT g1 g2 g3 this hl;(clear ep;exact omS381 hu)),D_px,ἱ (by clear hX hh hw hR hS he hr hc ew hT g1 g2 g3 this hl;(clear ep;exact omS382 hu))]
   . rw[ὢ,qAi _ _ _ (by clear hX hw hR hS he hr hc hT g1 g2 g3 this hl;(clear ew;exact omS383 hj)),D_px,ἐ (by clear hX hh hw hR hS he hr hc hT g3 this hl;(clear ep ew;exact omS384 g1 g2 hj)),Nat.add_sub_cancel_left,qAa _ (by clear hh hw hR hS he hr hc ep hT g1 g2 g3 hl;(clear hj;exact omS385 hs hX this))];rfl
   . rw[ὢ,ὢ,qAi _ _ _ (by clear hX hh hw hR hS he hr hc ew hT g1 g2 g3 this hl;(clear ep;exact omS386 hu)),D_px,ἱ (by clear hh hw hR hS he hr hc ep hT g1 g2 this hl hu;exact omS387)]
 apply ih (r+1) _ (by clear hX hh hw hS hr hc ep ew hT g;exact omS137 he)
 rw[ᾀ]
 exact qBq _ H λu hu=>by
  rw[qAi _ _ _ hu,D_px,ἱ]
  by_contra n
  have:=hO (by clear hX hh hw hR hS he hr hc ep ew hT g hu;exact omS388 n)
  exact g ⟨by (clear hX hh hw hR hS he hr hc ew hT g hu this;(clear ep;exact omS389 n)),by (clear hX hh hw hR hS he hr hc ep ew hT g hu this;exact omS388 n),by rw[qAa _ (by clear hh hw hR hS he hr hc ep hT g hu;(clear n;exact omS385 hs hX this)),ew,hX];exact this⟩
lemma qA1 (F:Ὣ->ὧ) {S:Ὣ} (N:ὦ) (c:Ὣ) (hN:ѤN=S) (hS:S<2^32):D_Zs F S 0 (D_fill (dv 0) N 0 (ἆ c)):=by have h:=qCc (F:=λ_=>dv 0) (v:=dv 0) (n:=N) S 0 0 (ἆ c) (by rw[C_z,hN,Ἇ]) ⟨rfl,λq hq=>absurd hq (Ἲ q)⟩ λ_ _ _=>rfl;rw[Ἇ] at h;exact ⟨h.1,hS,λq hq=>by rw[ἐ (Ἲ q)];exact h.2 q hq⟩
lemma qHd (F:Ὣ->ὧ) (n:Ὣ) (h:n<=16384):∃S,n<=S/\D_Zs F S 0 (D_zs n):=by
 unfold D_zs;split;. exact ⟨512,‹_›,qA1 F 512 512 (uk 512 (by norm_num1)) (by norm_num1)⟩
 split;. exact ⟨1024,‹_›,qA1 F 1024 1024 (uk 1024 (by norm_num1)) (by norm_num1)⟩
 split;. exact ⟨2048,‹_›,qA1 F 2048 2048 (uk 2048 (by norm_num1)) (by norm_num1)⟩
 split;. exact ⟨4096,‹_›,qA1 F 4096 4096 (uk 4096 (by norm_num1)) (by norm_num1)⟩
 split;. exact ⟨8192,‹_›,qA1 F 8192 8192 (uk 8192 (by norm_num1)) (by norm_num1)⟩
 exact ⟨16384,h,qA1 F 16384 16384 (uk 16384 (by norm_num1)) (by norm_num1)⟩
lemma qG7 (s:OS) (X:ѐ) (a:ѧX<2^32) (e:ѧX=s.cin*s.h*s.w) (p:s.h+2*s.pad<2^32) (q:s.w+2*s.pad<2^32):
 D_Sp (D_Fx s X) (s.cin*(s.h+2*s.pad)*(s.w+2*s.pad)) (D_xs s X a (s.w+2*s.pad) (s.cin*(s.h+2*s.pad)))/\
 ∀t,t< Ѩ(D_xs s X a (s.w+2*s.pad) (s.cin*(s.h+2*s.pad)))->∃u,(D_xs s X a (s.w+2*s.pad) (s.cin*(s.h+2*s.pad))).data[t]! =dv u:=by
 unfold D_xs
 split<;>rename_i hz
 . rw[ch2_eq s X a (s.w+2*s.pad) (s.cin*(s.h+2*s.pad)) rfl s.cin 0 _ (Ἇ _),ὒ]
   obtain ⟨S,h1,h2⟩:=qHd (D_Fx s X) _ hz;have P:=qE6 s X a e p q (S:=S) rfl (by clear a e p q hz;exact omS422 h1) _ 0 _ (Ἇ _) (by rw[ὒ];exact h2)
   refine ⟨⟨by rw[P.1];(clear e p q hz;exact omS422 h1),λt ht=>by rw[P.2.2 t (by clear e p q hz;exact omS423 h1 ht),ἱ ht]⟩,λt ht=>?_⟩
   rw[P.2.2 t (P.1▸ht)];split;. exact ⟨_,rfl⟩
   . exact ⟨0,rfl⟩
 have Pz:=qE0 s X a e p q rfl _ 0 (ἆ (s.cin*(s.h+2*s.pad)*(s.w+2*s.pad)+7)) (Ἇ _) ⟨by rw[ὒ];rfl,λq hq=>א⟩
 have Pf:D_Pre (λt=>if t<s.cin*(s.h+2*s.pad)*(s.w+2*s.pad) then D_Fx s X t else dv 0) (s.cin*(s.h+2*s.pad)*(s.w+2*s.pad)) (D_img s X a (dv 0) (s.cin*(s.h+2*s.pad)) 0 (ἆ (s.cin*(s.h+2*s.pad)*(s.w+2*s.pad)+7))):=
  ⟨Pz.1,λt ht=>by dsimp only;rw[Pz.2 t ht,ἱ ht]⟩
 have P:=qCc (v:=dv 0) (n:=7) 7 0 _ _ (by rw[C_z,Ἇ];exact (uk 7 (by norm_num1)).symm) Pf λi h1 h2=>by rw[ἐ א]
 refine ⟨⟨by rw[P.1];omega,λt ht=>by rw[P.2 t א];dsimp only;rw[ἱ ht]⟩,λt ht=>?_⟩
 rw[P.2 t (P.1▸ht)];dsimp only;split;. exact ⟨_,rfl⟩
 . exact ⟨0,rfl⟩
lemma qFb (dw:Ὡ) (s:OS) (relu:Ὡ) (X W b:ѐ) {k C:Ὣ} (hk:k=if dw then s.kh*s.kw else s.cin*s.kh*s.kw) (hC:C=if dw then s.cin else s.cout):
 D_F dw s relu X W b k C=if dw then depthwiseSpecO s relu X W b else conv2dSpecO s relu X W b:=by
 unfold D_F
 rw[(cvGet W b s dw k C).2]
 unfold D_Fc
 dsimp only
 rw[qAg]
 split;swap;rfl
 split;swap;rfl
 rename_i hg hs
 obtain ⟨a,c,d,e,f,-,i,j,l,m,n,o,p,q,r,t,u,-,v,x⟩:=hg
 replace hs:=Nat.lt_of_le_of_lt hs (show 4294967295<2^32 by norm_num)
 replace a:=Nat.lt_of_le_of_lt a (show 4294967295<2^32 by norm_num)
 replace j:=Nat.lt_of_le_of_lt j (show 4294967295<2^32 by norm_num)
 replace p:=Nat.lt_of_le_of_lt p (show 4294967295<2^32 by norm_num)
 replace q:=Nat.lt_of_le_of_lt q (show 4294967295<2^32 by norm_num)
 replace r:=Nat.lt_of_le_of_lt r (show 4294967295<2^32 by norm_num)
 replace t:=Nat.lt_of_le_of_lt t (show 4294967295<2^32 by norm_num)
 replace u:=Nat.lt_of_le_of_lt u (show 4294967295<2^32 by norm_num)
 have y:=qFi X X.usize (qAd X) X.usize.toNat 0 Е 0 (Ἇ _)
 rw[ἱ ⟨c,d⟩] at y
 have Hx:=y▸qBh X a (by rw[y];intro h;rw[h] at d;exact absurd d ה)
 refine (qHh ?_ _).trans ?_
 have hk0:k≠0:=(by clear hC hs y;exact omS424 l)
 rw[hk] at hk0
 have k1:0<s.kh:=ὼ λz=>hk0 (by rw[z];simp only[zero_mul,mul_zero,ite_self])
 have k2:0<s.kw:=ὼ λz=>hk0 (by rw[z];simp only[mul_zero,ite_self])
 exact ⟨hk,hC,rfl,rfl,rfl,qAa _ j,rfl,λh=>beq_iff_eq.1 (ὕ.1 h).1,i,l,m,n,o,(qG7 s X a e p q).1,hs,Hx,qGp (qD0 W r v).2.2.1,qGp (qD0 b t x).2.2.1,f,r,t,u,q,qGz s k k 0 0 0 0 _ (Ἇ _) ג k1 k2 (by simp only[Array.emptyWithCapacity_eq,Nat.toUInt32_eq,Array.ofFn_zero]),(qG7 s X a e p q).2⟩
 rfl
lemma qC1x (d:DotFn) (s:OS) (r:Ὡ) (X W b:ё):conv2dWithO d (Cn1 s) r X W b=conv2dWithO d s r X W b:=by
 unfold Cn1
 split
 · rename_i H
   obtain ⟨cin,h,w,co,kh,kw,st,pd⟩:=s
   obtain ⟨rfl,rfl,rfl,rfl,u,v⟩:=H
   simp only at u v
   have P:=Nat.mul_pos u v
   have e:∀n,0<n->(n+2*0-1)/1+1=n:=λn h=>א
   simp only[conv2dWithO,OS.ho,OS.wo,e _ u,e _ v,e _ P,e 1 Nat.one_pos,Ὑ,Nat.mul_one]
   refine Մ (by simp only[Nat.mul_zero,Ἵ,Nat.add_one_sub_one,Nat.reduceDiv,Nat.reduceAdd,Array.size_ofFn,Nat.div_one,Nat.sub_add_cancel P,one_mul,Nat.sub_add_cancel u,Nat.sub_add_cancel v]) λq _ _=>?_
   simp only[Array.getElem_ofFn,windowAtO,Nat.mul_one,Nat.div_one,Nat.mod_one,Ἵ,Nat.sub_zero,Ἲ,false_or]
   have:=ὀ q P
   have m:q%(h*w)=q%w+w*(q/w%h):=by rw[ἥ h w,Nat.mod_mul]
   have:=ὀ (q/w) u
   have:=ὀ q v
   congr 2
   funext t
   split
   · rw[ἐ א,ἐ א,m]
     congr 1
     ring
   · rfl
 · rfl
lemma qB7:convF=conv2dSpecO:=by funext s r X W b;exact (qFb false (Cn1 s) r X W b (k:=s.cin*s.kh*s.kw) (C:=s.cout) (by unfold Cn1;split<;>split<;>simp_all only[false_eq_true,not_false_eq_true,mul_one,not_and,not_lt,nonpos_iff_eq_zero]) (by unfold Cn1;split<;>split<;>simp_all only[false_eq_true,not_false_eq_true,not_and,not_lt,nonpos_iff_eq_zero])).trans (qC1x _ s r X W b)
lemma qDr:depthF=depthwiseSpecO:=by funext s r X W b;exact qFb true s r X W b rfl rfl
end
section
open Float.Model Float.Model.UnpackedFloat
@[noinline] def P39:Ὣ:=2^39
@[noinline] def P38:Ὣ:=2^38
def T_r (N:Ὣ):Ὣ:=
 N/P39+if P38<N%P39\/(N%P39=P38/\(N/P39)%2=1) then 1 else 0
lemma qIv (N:Ὣ):FloatLib.Numerics.roundShiftRightEven N 39=T_r N:=by
 have a:Nat.shiftRight N 39=N/2^39:=Ἱ N 39;have b:N-Nat.shiftLeft (N/2^39) 39=N%2^39:=by rw[show Nat.shiftLeft (N/2^39) 39=N/2^39*2^39 from ὁ _ _];(clear a;exact omS425)
 rw[FloatLib.Numerics.roundShiftRightEven_def];simp only[show (39==0)=false from rfl,false_eq_true,ite_false,a,b,beq_iff_eq];unfold T_r P39 P38;split_ifs<;>omega
set_option maxRecDepth 8192 in
lemma qIc (neg:Ὡ) (N:Ὣ) (e':Int) (hN1:2^62<=N) (hN2:N<2^63) (h1:-148<=e'+39) (h2:e'+39+150<=253):
 toWord (Model.ofModel Fmt (UnpackedFloat.round Format.binary32 (ὶ neg) N e'))=ὺ ((if neg then 2^31 else 0)+(e'+39+149).toNat*2^23+T_r N):=by
 have hr1:2^23<=T_r N:=by unfold T_r P39 P38;split<;>omega
 have hr2:T_r N<=2^24:=by unfold T_r P39 P38;split<;>omega
 have hT:Format.binary32.targetExponent (totalExponent N e')=e'+39:=by
  simp only[totalExponent,qC8 N 62 hN1 hN2,qGo];(clear hN1 hN2 hr1 hr2;exact omS426 h1)
 have K:∀(P:Ὣ) (E:Int) (hp:0<P),2^23<=P->P<2^24->-149<=E->E<=104->(E+150).toNat*2^23+P%2^23=(e'+39+149).toNat*2^23+T_r N->
  toWord (Model.ofModel Fmt (.finite (ὶ neg) P E hp))=ὺ ((if neg then 2^31 else 0)+(e'+39+149).toNat*2^23+T_r N):=by
  intro P E hp a b c d e
  simp only[toWord,Model.toNatBits,Model.ofModel,Model.ofModelBits]
  congr 1
  show (UnpackedFloat.pack Format.binary32 _).toNat=_
  rw[qBk _ _ _ _ a b (by clear hN1 hN2 h1 h2 hr1 hr2 hT hp a b e;exact omS92 c) (by clear hN1 hN2 h1 h2 hr1 hr2 hT hp a b e;exact omS427 d)]
  cases neg<;>simp only[Model.modelSign,false_eq_true, ↓reduceIte,reduceCtorEq,Nat.reducePow,zero_add]<;>(clear hN1 hN2 h1 h2 hr1 hr2 hT hp a b c d;omega)
 rw[Model.round_exact_eq_finishRoundedMantissa _ _ _ _,hT,Model.roundMantissaAtExponentEven,ἱ (by clear hN1 hN2 h1 h2 hr1 hr2 hT;exact omS429),show (e'+39-e').toNat=39 by (clear hN1 hN2 h1 h2 hr1 hr2 hT;exact omS430),qIv]
 by_cases hc:T_r N=2^24
 . have h:=Model.finishRoundedMantissa_normalized_carry FloatFormat.binary32 (ὶ neg) (e'+39+23) (by show-126<=e'+39+23;(clear hN1 hN2 hr1 hr2 hc;exact omS431 h1));rw[show e'+39+23-(FloatFormat.binary32.fracWidth:ℤ)=e'+39 by show e'+39+23-23=_;(clear hN1 hN2 h1 h2 hr1 hr2 hT hc;exact omS432),show e'+39+23+1-(FloatFormat.binary32.fracWidth:ℤ)=e'+40 by show e'+39+23+1-23=_;(clear hN1 hN2 h1 h2 hr1 hr2 hT hc;exact omS433)] at h;rw[show Model.finishRoundedMantissa Format.binary32 (ὶ neg) (T_r N,e'+39)=.finite (ὶ neg) (2^23) (e'+40) ו by rw[hc];exact h]
   exact K (2^23) (e'+40) ו le_rfl (by norm_num1) (by clear hN1 hN2 hr1 hr2 hc;exact omS434 h1) (by clear hN1 hN2 hr1 hr2 hc;exact omS435 h2) (by
    rw[hc,show (e'+40+150).toNat=(e'+39+149).toNat+2 by (clear hN1 hN2 hr1 hr2 hc;exact omS436 h1)];(clear hN1 hN2 h1 h2 hr1 hr2 hT hc;exact omS437))
 . rw[qEf _ _ _ hr1 (by clear hN1 hN2 h1 h2 hT;exact omS438 hr2 hc) (by clear hN1 hN2 hr1 hr2 hc;exact omS439 h1) (by clear hN1 hN2 h1 h2 hT hc;exact omS71 hr1)]
   exact K _ _ _ hr1 (by clear hN1 hN2 h1 h2 hT;exact omS438 hr2 hc) (by clear hN1 hN2 hr1 hr2 hc;exact omS439 h1) (by clear hN1 hN2 hr1 hr2 hc;exact omS440 h2) (by
    rw[show (e'+39+150).toNat=(e'+39+149).toNat+1 by (clear hN1 hN2 hr1 hr2 hc;exact omS441 h1)];(clear hN1 hN2;exact omS442 hr1 hr2 hc))
@[inline] def T_rd (neg:Ὡ) (mag e0 L:Ὠ):ὥ:=
 let n:=mag<<<(62-L)
 let q:=n>>>39
 let rm:=n&&&0x7FFFFFFFFF
 let w:=((e0+L-24)<<<23)+q+
  (if rm>0x4000000000 || (rm==0x4000000000&&q&&&1==1) then 1 else 0)
 (if neg then w ||| س else w).toUInt32
lemma qG4 (neg:Ὡ) (mag e0:Ὠ) (h0:Ѧmag≠0) (h1:Ѧmag<2^63) (h2:24< Ѧe0+mag.toNat.log2) (h3:Ѧe0+mag.toNat.log2<277):T_rd neg mag e0 mag.log2=toWord (ἕ Fmt ⟨neg,Ѧmag,(Ѧe0:Int)-150⟩):=by
 have l0:=Nat.log2_self_le h0;have l1:Ѧmag<2^(mag.toNat.log2+1):=Nat.lt_log2_self;have hL:mag.toNat.log2<=62:=Nat.lt_succ_iff.1 ((Nat.log2_lt h0).2 h1);obtain ⟨L,hLd⟩:∃L,L=mag.toNat.log2:=⟨_,rfl⟩;rw[<-hLd] at l0 l1 hL h2 h3;have eL:mag.log2.toNat=L:=hLd.symm;obtain ⟨N,hN⟩:∃N,N=Ѧmag*2^(62-L):=⟨_,rfl⟩;have p:2^L*2^(62-L)=2^62:=by rw[<-Nat.pow_add,Nat.add_sub_of_le hL]
 have hN1:2^62<=N:=by rw[hN,<-p];exact Ἒ _ l0
 have hN2:N<2^63:=by have:=Nat.mul_lt_mul_of_pos_right l1 (Nat.two_pow_pos (62-L));rwa[<-hN,<-Nat.pow_add,show L+1+(62-L)=63 from (Nat.add_right_comm L 1 (62-L)).trans (congrArg (·+1) (Nat.add_sub_of_le hL))] at this
 have s4:Ѧe0+L-24<=252:=Nat.sub_le_iff_le_add.mpr (Nat.le_of_lt_succ h3)
 obtain ⟨i1,i2,i3⟩:(-148:Int)<=(Ѧe0:Int)-150-((62-L:Ὣ):Int)+39/\(Ѧe0:Int)-150-((62-L:Ὣ):Int)+39+150<=253/\((Ѧe0:Int)-150-((62-L:Ὣ):Int)+39+149).toNat=Ѧe0+L-24:=by (clear h0 h1 l1 l0 hN p hN1 hN2 hLd eL;omega)
 rw[qA5,<-ExactRuntime.round_scale _ _ _ _ (62-L) h0,<-hN,qIc neg N _ hN1 hN2 i1 i2];have eT:((Ѧe0:Int)-150-((62-L:Ὣ):Int)+39+149).toNat=Ѧe0+L-24:=i3
 rw[eT];unfold T_rd;dsimp only;have e1:(62-mag.log2).toNat=62-L:=by rw[toNat_sub_of_le _ _ (Ὤ.2 (by rw[eL];exact hL)),eL];rfl
 have a1:(mag<<<(62-mag.log2)).toNat=N:=by rw[UInt64.toNat_shiftLeft,e1,Ἁ (Nat.lt_of_le_of_lt (Nat.sub_le 62 L) ה:62-L<64),ὁ,<-hN,Ἁ (by exact Nat.lt_of_lt_of_le ‹_› ה)]
 generalize mag<<<(62-mag.log2)=n at a1;have a2:(n>>>39).toNat=N/2^39:=by rw[ἰ,a1];simp only[UInt64.reduceToNat,Nat.reduceMod,Ἱ,Nat.reducePow]
 have a3:(n&&&0x7FFFFFFFFF).toNat=N%2^39:=by rw[Ὅ,a1];exact Ἂ N 39
 generalize n>>>39=q at a2;generalize n&&&0x7FFFFFFFFF=rm at a3;have a4:(q&&&1).toNat=Ѧq%2:=by rw[Ὅ];exact Ἂ _ 1
 have a5:(e0+mag.log2-24).toNat=Ѧe0+L-24:=by
  rw[toNat_sub_of_le _ _ (Ὤ.2 (by
   rw[ἤ,eL,Ἁ (by exact Nat.lt_of_lt_of_le ‹_› ה)];show 24<=_;(exact Nat.le_of_lt ‹_›))),
   ἤ,eL,Ἁ (by exact Nat.lt_of_lt_of_le ‹_› ה)]
  rfl
 have a6:((e0+mag.log2-24)<<<23).toNat=(Ѧe0+L-24)*2^23:=by rw[UInt64.toNat_shiftLeft,a5,ὁ];show _*2^23%_=_;rw[Ἁ (Nat.lt_of_le_of_lt (Ἒ (2^23) s4) ה)]
 generalize (e0+mag.log2-24)<<<23=hi at a6;have hup:(decide (rm>0x4000000000) || (rm==0x4000000000&&q&&&1==1))=decide (2^38<N%2^39\/(N%2^39=2^38/\(N/2^39)%2=1)):=by apply Bool.eq_iff_iff.2;simp only[or_eq_true,and_eq_true,decide_eq_true_eq,beq_iff_eq,gt_iff_lt,ἴ,<-UInt64.toNat_inj,a3,a4,a2];rfl
 rw[hup];obtain ⟨c,hc⟩:∃c,c=decide (2^38<N%2^39\/(N%2^39=2^38/\(N/2^39)%2=1)):=⟨_,rfl⟩;rw[<-hc];have hr:T_r N=N/2^39+if c=true then 1 else 0:=by rw[hc,T_r,P39,P38];simp only[decide_eq_true_eq]
 have hq:N/2^39<2^24:=Ἰ (Nat.lt_of_lt_of_le hN2 ה);have U':(Ѧe0+L-24)*2^23+(N/2^39+1)<2^31:=Nat.lt_of_le_of_lt (Nat.add_le_add (Ἒ (2^23) s4) (hq:N/2^39+1<=2^24)) ה;have U:(Ѧe0+L-24)*2^23+N/2^39+1<2^31:=lt_of_eq_of_lt (ὢ _ _ _) U';have W1:(Ѧe0+L-24)*2^23+N/2^39+1<2^64:=Nat.lt_trans U ה;have W0:(Ѧe0+L-24)*2^23+N/2^39<2^64:=Nat.lt_of_succ_lt W1
 have hw:(hi+q+(if c=true then (1:Ὠ) else 0)).toNat=(Ѧe0+L-24)*2^23+T_r N:=by
  rw[ἤ,ἤ,a6,a2,hr];cases c<;>simp only[false_eq_true,ite_false,ite_true]<;>(try rw[show (1:Ὠ).toNat=1 from rfl])<;>(try rw[show (0:Ὠ).toNat=0 from rfl])
  . rw[Ἵ,Ἵ,Ἁ W0,Ἁ W0]
  . rw[Ἁ W0,Ἁ W1,ὢ]
 generalize hi+q+(if c=true then (1:Ὠ) else 0)=w at hw;have hw2:Ѧw<2^31:=by rw[hw,hr];split;exact U';exact Nat.lt_of_succ_lt U
 apply ἠ.mp;rw[ὡ];cases neg;. simp only[false_eq_true,ite_false,UInt64.toNat_toUInt32,Ἇ,hw]
 . simp only[ite_true,UInt64.toNat_toUInt32,UInt64.toNat_or];have h5:Ѧw ||| (2147483648:Ὠ).toNat=2^31+Ѧw:=by have:=Nat.two_pow_add_eq_or_of_lt hw2 1;simp only[Nat.mul_one] at this;rw[this,Nat.or_comm];rfl
   rw[h5];congr 1;rw[hw];exact (ὢ _ _ _).symm
lemma qJf (w:ὥ) (h:(w&&&В)=0):ἂ w=0:=by
 have h1:=qA0 w;rw[h] at h1;have h2:=w.toNat_lt
 have h3:w=0\/w=س:=by
  have:ѥw=0\/ѥw=2147483648:=by simp only[UInt32.toNat_zero,Nat.reducePow] at h1;(clear h;exact omS452 h2 h1)
  rcases this with e | e;. exact Or.inl (ἠ.mp (by simpa only[UInt32.toNat_zero] using e))
  . exact Or.inr (ἠ.mp (by simpa only[UInt32.reduceToNat] using e))
 have h4:expOf w=0/\fracOf w=0:=by rcases h3 with rfl | rfl<;>decide
 simp only[rationalWord,decodeD,h4, ↓reduceIte,FloatLib.Numerics.Dyadic.toRat_mk_zero]
@[inline] def T_e (w:ὥ):Ὠ:=
 if w&&&В==0 then 255 else (expOf w).toUInt64
@[inline] def T_m (w:ὥ):Ὠ:=
 if w&&&В==0 then 0 else (fracOf w+0x800000).toUInt64
@[inline] def T_s (e0:Ὠ) (w:ὥ):Ὠ:=T_m w<<<(T_e w-e0)
@[inline] def T_ok (e0:Ὠ) (w:ὥ):Ὡ:=
 good w&&(w&&&В==0 || (e0<=T_e w&&T_e w-e0<=37))
@[inline] def T_p (e0:Ὠ) (w:ὥ) (p:Ὠ):Ὠ:=
 if signOf w then p else p+T_s e0 w
@[inline] def T_n (e0:Ὠ) (w:ὥ) (n:Ὠ):Ὠ:=
 if signOf w then n+T_s e0 w else n
lemma qJa (e0:Ὠ) (w:ὥ) (h:T_ok e0 w=true):
 (T_s e0 w).toNat<2^61/\ἂ w=
  (if signOf w then-((T_s e0 w).toNat:ℚ) else ((T_s e0 w).toNat:ℚ))*2^((Ѧe0:Int)-150):=by
 unfold T_ok at h
 simp only[and_eq_true,or_eq_true,beq_iff_eq,decide_eq_true_eq] at h
 obtain ⟨hg,hz⟩:=h
 by_cases h0:w&&&В=0
 . have hs:T_s e0 w=0:=by unfold T_s T_m;simp only[h0,BEq.rfl, ↓reduceIte,UInt64.zero_shiftLeft]
   rw[hs,qJf w h0];simp only[UInt64.toNat_zero,Nat.reducePow,Nat.ofNat_pos,CharP.cast_eq_zero,neg_zero,ite_self,zero_mul,and_self]
 . obtain ⟨h1,h2⟩:=hz.resolve_left h0;have g:=qBd w hg;have a0:(w&&&В).toNat≠0:=λe=>h0 (ἠ.mp (by simpa only[Ἧ,UInt32.reduceToNat,UInt32.toNat_zero] using e));rw[qA0] at a0;have hx:=qAx w;have hf:(fracOf w).toNat=ѥw%2^23:=qBt w;have hl:=w.toNat_lt;have x1:1<=(expOf w).toNat:=(by clear hz h0 h1 h2 hl;exact omS453 g a0 hx);have x2:(expOf w).toNat<=254:=(by clear hz h0 h1 h2 hl;exact omS454 g hx);have f1:(fracOf w).toNat<2^23:=(by clear hz h0 h1 h2 g a0 hx hl x1 x2;exact omS455 hf);have ee:(T_e w).toNat=(expOf w).toNat:=by unfold T_e;simp only[beq_iff_eq,h0, ↓reduceIte,UInt32.toNat_toUInt64]
   have em:(T_m w).toNat=2^23+(fracOf w).toNat:=by unfold T_m;simp only[h0,beq_iff_eq,ite_false,UInt32.toNat_toUInt64];rw[UInt32.toNat_add,Ἁ (by show _+8388608<_;(clear hz h0 h1 h2 g a0 hx hl x1 x2 f1 ee;exact omS456 hf))];show _+8388608=_;(clear hz h0 h1 h2 g a0 hx hf hl x1 x2 f1 ee;exact omS457)
   have l1:=Ὤ.1 h1;have l2:=Ὤ.1 h2;rw[toNat_sub_of_le _ _ h1,ee] at l2;rw[ee] at l1;have l3:(expOf w).toNat-Ѧe0<=37:=l2
   have es:(T_s e0 w).toNat=(2^23+(fracOf w).toNat)*2^((expOf w).toNat-Ѧe0):=by
    unfold T_s;have sh:(T_e w-e0).toNat%64=(expOf w).toNat-Ѧe0:=by rw[toNat_sub_of_le _ _ h1,ee];exact Ἁ (by exact Nat.lt_of_le_of_lt ‹_› ה)
    rw[UInt64.toNat_shiftLeft,em,sh,ὁ];apply Ἁ;have:2^((expOf w).toNat-Ѧe0)<=2^37:=Nat.pow_le_pow_right ו l3;have:(2^23+(fracOf w).toNat)*2^((expOf w).toNat-Ѧe0)<=2^24*2^37:=Nat.mul_le_mul (by clear hz h0 h1 h2 g a0 hx hl x1 x2 f1 ee l1 l2 l3 sh this;exact omS458 hf) this;(exact Nat.lt_of_le_of_lt ‹_› ה)
   refine ⟨?_,?_⟩;. have a:2^((expOf w).toNat-Ѧe0)<=2^37:=Nat.pow_le_pow_right ו l3;have b:(2^23+(fracOf w).toNat)*2^((expOf w).toNat-Ѧe0)<=(2^24-1)*2^37:=Nat.mul_le_mul (by clear hz h0 h1 h2 g a0 hx hl x1 x2 f1 ee l1 l2 l3 es a;exact omS459 hf) a;rw[es];(exact Nat.lt_of_le_of_lt ‹_› ה)
   . have nz:expOf w≠0:=λe=>by rw[e] at x1;simp only[UInt32.toNat_zero,nonpos_iff_eq_zero,one_ne_zero] at x1
     have ew:(2:ℚ)^(((expOf w).toNat:Int)-150)=2^((expOf w).toNat-Ѧe0)*2^((Ѧe0:Int)-150):=by rw[<-Ἡ,<-Ἓ ἓ];congr 1;(clear hz h0 h1 h2 g a0 hf hl x1 x2 f1 em l2 l3 es nz;exact omS460 l1)
     rw[es];simp only[rationalWord,decodeD,nz,ite_false,FloatLib.Numerics.Dyadic.toRat,signedSignificand,ew];cases signOf w<;>simp only[false_eq_true, ↓reduceIte,Nat.reducePow,Int.ofNat_eq_natCast,Nat.cast_add,Nat.cast_ofNat,Rat.ofInt_eq_cast,Int.cast_add,Int.cast_ofNat,Int.cast_natCast,Nat.cast_mul,Nat.cast_pow,neg_add_rev,Int.reduceNeg,Int.cast_neg,neg_mul]<;>ring
lemma qEn (e0:Ὠ) (w:ὥ) (p n:Ὠ) (V:ℚ) (B C:Ὣ) (hC:C=2^61-1) (hB:B+C<2^63) (hw:T_ok e0 w=true) (hp:Ѧp<=B) (hn:Ѧn<=B) (hV:V=((Ѧp:ℚ)-Ѧn)*2^((Ѧe0:Int)-150)):
 (T_p e0 w p).toNat<=B+C/\(T_n e0 w n).toNat<=B+C/\
  V+ἂ w=(((T_p e0 w p).toNat:ℚ)-(T_n e0 w n).toNat)*2^((Ѧe0:Int)-150):=by
 obtain ⟨h1,h2⟩:=qJa e0 w hw
 have h3:(T_s e0 w).toNat<=C:=(by clear hB hp hn;exact omS461 hC h1)
 unfold T_p T_n
 rw[h2,hV]
 cases signOf w
 . simp only[false_eq_true,ite_false];have e:(p+T_s e0 w).toNat=Ѧp+(T_s e0 w).toNat:=by rw[ἤ];exact Ἁ (by clear hn h3;exact omS462 hB hp h1)
   rw[e];refine ⟨by (clear hB hn h3;exact omS463 hC hp h1),by (clear hB hp h1 h3 e;exact omS464 hn),?_⟩;push_cast;ring
 . simp only[ite_true];have e:(n+T_s e0 w).toNat=Ѧn+(T_s e0 w).toNat:=by rw[ἤ];exact Ἁ (by clear hp h3;exact omS462 hB hn h1)
   rw[e];refine ⟨by (clear hB hn h1 h3 e;exact omS464 hp),by (clear hB hp h3;exact omS463 hC hn h1),?_⟩;push_cast;ring
@[inline] def T_lg (x:Ὠ):Ὠ:=
 let a:Ὠ:=if 0x100000000<=x then 32 else 0
 let y:=x>>>a
 let b:Ὠ:=if 0x10000<=y then 16 else 0
 let y:=y>>>b
 let c:Ὠ:=if 0x100<=y then 8 else 0
 let y:=y>>>c
 let d:Ὠ:=if 0x10<=y then 4 else 0
 let y:=y>>>d
 let e:Ὠ:=if 4<=y then 2 else 0
 let y:=y>>>e
 a+b+c+d+e+(if 2<=y then 1 else 0)
lemma lgS (n s:Nat) (h:2^s<=n):n.log2=(n/2^s).log2+s:=by
 have hn:n≠0:=by have:=Nat.one_le_two_pow (n:=s);omega
 have h1:=Nat.log2_self_le hn
 have h2:n<2^(n.log2+1):=Nat.lt_log2_self
 have hsL:s<=n.log2:=(Nat.le_log2 hn).2 h
 have q1:2^(n.log2-s)<=n/2^s:=by
  rw[Nat.le_div_iff_mul_le (Nat.two_pow_pos s),<-Nat.pow_add,Nat.sub_add_cancel hsL];exact h1
 have q2:n/2^s<2^(n.log2-s+1):=by
  rw[Nat.div_lt_iff_lt_mul (Nat.two_pow_pos s),<-Nat.pow_add,show n.log2-s+1+s=n.log2+1 by omega];exact h2
 have q0:n/2^s≠0:=by have:=Nat.one_le_two_pow (n:=n.log2-s);omega
 have:=(Nat.log2_eq_iff q0).2 ⟨q1,q2⟩
 omega
lemma lgU (y T c:Ὠ) (hT:T.toNat=2^c.toNat) (hc:c.toNat<64) (hy:y.toNat<2^(2*c.toNat)):
 (y>>>(if T<=y then c else 0)).toNat<2^c.toNat/\
 y.toNat.log2=(y>>>(if T<=y then c else 0)).toNat.log2+(if T<=y then c else 0).toNat:=by
 by_cases hTy:T<=y
 . rw[ἱ hTy]
   have e:(y>>>c).toNat=y.toNat/2^c.toNat:=by
    rw[ἰ,Ἁ hc];exact Ἱ _ _
   rw[e];constructor
   . rw[Nat.div_lt_iff_lt_mul (Nat.two_pow_pos _),<-Nat.pow_add];rw[show c.toNat+c.toNat=2*c.toNat by omega];exact hy
   . exact lgS _ _ (by rw[<-hT];exact UInt64.le_iff_toNat_le.mp hTy)
 . rw[ἐ hTy]
   have e:(y>>>(0:Ὠ)).toNat=y.toNat:=by simp only[UInt64.shiftRight_zero]
   rw[e];refine ⟨?_,by simp only[UInt64.toNat_zero,add_zero]⟩
   have:=UInt64.lt_iff_toNat_lt.mp (UInt64.not_le.mp hTy);omega
lemma lgF (y:Ὠ) (h:y.toNat<2^2):y.toNat.log2=(if 2<=y then (1:Ὠ) else 0).toNat:=by
 have h4:y.toNat<4:=by simpa only[Nat.reducePow] using h
 by_cases h2:(2:Ὠ)<=y
 . rw[ἱ h2];have:=UInt64.le_iff_toNat_le.mp h2
   have e:(2:Ὠ).toNat=2:=rfl;rw[e] at this
   exact (Nat.log2_eq_iff א).2 ⟨by simpa only[UInt64.reduceToNat,pow_one] using this,by simpa only[UInt64.reduceToNat,Nat.reduceAdd,Nat.reducePow] using h4⟩
 . rw[ἐ h2];have:=UInt64.lt_iff_toNat_lt.mp (UInt64.not_le.mp h2)
   have e:(2:Ὠ).toNat=2:=rfl;rw[e] at this
   have:y.toNat=0\/y.toNat=1:=א
   rcases this with e|e<;>rw[e]<;>decide
lemma qLg (x:Ὠ):T_lg x=x.log2:=by
 apply UInt64.toNat_inj.mp
 have eL:x.log2.toNat=x.toNat.log2:=rfl
 rw[eL];unfold T_lg;dsimp only
 obtain ⟨y1,e1⟩:=lgU x 0x100000000 32 ה ה (by have:=x.toNat_lt;simpa only[UInt64.reduceToNat,Nat.reduceMul,Nat.reducePow,gt_iff_lt] using this)
 generalize (if (0x100000000:Ὠ)<=x then (32:Ὠ) else 0)=a at y1 e1⊢
 generalize x>>>a=x1 at y1 e1⊢
 obtain ⟨y2,e2⟩:=lgU x1 0x10000 16 ה ה (by simpa only[UInt64.reduceToNat,Nat.reduceMul,Nat.reducePow] using y1)
 generalize (if (0x10000:Ὠ)<=x1 then (16:Ὠ) else 0)=b at y2 e2⊢
 generalize x1>>>b=x2 at y2 e2⊢
 obtain ⟨y3,e3⟩:=lgU x2 0x100 8 ה ה (by simpa only[UInt64.reduceToNat,Nat.reduceMul,Nat.reducePow] using y2)
 generalize (if (0x100:Ὠ)<=x2 then (8:Ὠ) else 0)=c at y3 e3⊢
 generalize x2>>>c=x3 at y3 e3⊢
 obtain ⟨y4,e4⟩:=lgU x3 0x10 4 ה ה (by simpa only[UInt64.reduceToNat,Nat.reduceMul,Nat.reducePow] using y3)
 generalize (if (0x10:Ὠ)<=x3 then (4:Ὠ) else 0)=d at y4 e4⊢
 generalize x3>>>d=x4 at y4 e4⊢
 obtain ⟨y5,e5⟩:=lgU x4 4 2 ה ה (by simpa only[UInt64.reduceToNat,Nat.reduceMul,Nat.reducePow] using y4)
 generalize (if (4:Ὠ)<=x4 then (2:Ὠ) else 0)=e at y5 e5⊢
 generalize x4>>>e=x5 at y5 e5⊢
 have e6:=lgF x5 (by simpa only[Nat.reducePow,UInt64.reduceToNat] using y5)
 generalize (if (2:Ὠ)<=x5 then (1:Ὠ) else 0)=f at e6⊢
 have hb:x.toNat.log2<2^64:=Nat.lt_of_le_of_lt (Nat.log2_le_self _) x.toNat_lt
 simp only[ἤ]
 omega
@[inline] def T_fin (p n e0:Ὠ) (f:Unit->ὥ):ὥ:=
 if p==n then f () else
  let mag:=if p<n then n-p else p-n
  let L:=T_lg mag
  if e0+L<=24 || e0+L>=277 then f () else T_rd (decide (p<n)) mag e0 L
lemma qFy (p n e0:Ὠ) (f:Unit->ὥ) (hp:Ѧp<2^63) (hn:Ѧn<2^63) (he:Ѧe0<=255) (R:ὥ) (hf:f ()=R)
 (hR:p≠n->∃d:FloatLib.Numerics.Dyadic,
  d.toRat=((Ѧp:ℚ)-Ѧn)*2^((Ѧe0:Int)-150)/\R=toWord (ἕ Fmt d)):T_fin p n e0 f=R:=by
 unfold T_fin
 simp only[qLg]
 split
 . exact hf
 . rename_i hne;have hne':p≠n:=λe=>hne (by simp only[e,BEq.rfl]);have hnn:Ѧp≠Ѧn:=λe=>hne' (UInt64.toNat_inj.mp e);obtain ⟨d,hd,hR⟩:=hR hne';(try dsimp only);obtain ⟨mag,hm⟩:∃m,m=if p<n then n-p else p-n:=⟨_,rfl⟩;rw[<-hm]
   have hm1:Ѧmag≠0/\Ѧmag<2^63/\
    (((if decide (p<n)=true then-(Ѧmag:ℚ) else (Ѧmag:ℚ)))=(Ѧp:ℚ)-Ѧn):=by
    by_cases c:p<n
    . have c':=ἴ.1 c;have:Ѧmag=Ѧn-Ѧp:=by rw[hm,ἱ c,toNat_sub_of_le _ _ (Ὤ.2 c'.le)]
      refine ⟨by (clear hp hn he hf hne' hnn hR hm c;exact omS465 c' this),by (clear he hf hne' hnn hR hm c;exact omS466 hn this),?_⟩;rw[this,Nat.cast_sub c'.le];simp only[c,decide_true, ↓reduceIte,neg_sub]
    . have c':Ѧn<=Ѧp:=by have:=ἴ.not.1 c;(clear hp hn he hf hne' hnn hR hm c;omega)
      have:Ѧmag=Ѧp-Ѧn:=by rw[hm,ἐ c,toNat_sub_of_le _ _ (Ὤ.2 c')]
      refine ⟨by (clear hp hn he hf hne' hR hm c;exact omS467 hnn c' this),by (clear he hf hne' hnn hR hm c;exact omS466 hp this),?_⟩;rw[this,Nat.cast_sub c'];simp only[c,decide_false,false_eq_true, ↓reduceIte]
   obtain ⟨m0,m1,m2⟩:=hm1;split;. exact hf
   . rename_i hb;simp only[or_eq_true,decide_eq_true_eq,not_or,ge_iff_le] at hb;obtain ⟨b1,b2⟩:=hb;have eL:mag.log2.toNat=mag.toNat.log2:=rfl;have hL:mag.toNat.log2<=62:=by have:=(Nat.log2_lt m0).2 m1;(exact Nat.le_of_lt_succ ‹_›)
     have eA:(e0+mag.log2).toNat=Ѧe0+mag.toNat.log2:=by rw[ἤ,eL];exact Ἁ (by clear hp hn hf hne' hnn hR hm m0 m1 b1 b2;exact omS468 he hL)
     have b3:¬(e0+mag.log2).toNat<=(24:Ὠ).toNat:=λh=>b1 (Ὤ.2 h);have b4:¬(277:Ὠ).toNat<=(e0+mag.log2).toNat:=λh=>b2 (Ὤ.2 h);rw[eA] at b3 b4;have t1:(24:Ὠ).toNat=24:=rfl;have t2:(277:Ὠ).toNat=277:=rfl;rw[qG4 _ _ _ m0 m1 (by clear hp hn he hf hne' hnn hR hm m0 m1 b1 b2 eL hL b4 t2;exact omS469 b3 t1) (by clear hp hn hf hne' hnn hR hm m0 m1 b1 b2 b3 t1;exact omS470 b4 t2),hR];refine congrArg toWord ?_
     have s0:d.significand≠0:=by
      intro h0;have:d.toRat=0:=(ὐ d).2 h0;rw[hd] at this;have q:(Ѧp:ℚ)-Ѧn≠0:=by rw[sub_ne_zero];exact_mod_cast hnn
      exact (mul_ne_zero q (zpow_ne_zero _ ἓ)) this
     refine roundDyadic_congr Fmt ה _ _ m0 s0 ?_;rw[hd,<-m2];simp only[FloatLib.Numerics.Dyadic.toRat,signedSignificand];cases decide (p<n)<;>simp only[false_eq_true, ↓reduceIte,Int.ofNat_eq_natCast,Rat.ofInt_eq_cast,Int.cast_natCast,Int.cast_neg,neg_mul]
lemma qHq (w:ὥ):(T_e w).toNat<=255:=by
 unfold T_e;split;. decide
 . have:=qAx w;simp only[UInt32.toNat_toUInt64];omega
@[inline] def T_min (x y:Ὠ):Ὠ:=if x<y then x else y
lemma qC5 (x y:Ὠ):(T_min x y).toNat<=Ѧx:=by
 unfold T_min;split;. exact le_rfl
 . have:=ἴ.not.1 ‹_›;omega
lemma qGv (e0:Ὠ) (w0 w1:ὥ) (h0:T_ok e0 w0=true) (h1:T_ok e0 w1=true):
 (T_p e0 w1 (T_p e0 w0 0)).toNat<=2*(2^61-1)/\(T_n e0 w1 (T_n e0 w0 0)).toNat<=2*(2^61-1)/\
  ἂ w0+ἂ w1=(((T_p e0 w1 (T_p e0 w0 0)).toNat:ℚ)-(T_n e0 w1 (T_n e0 w0 0)).toNat)*2^((Ѧe0:Int)-150):=by
 obtain ⟨a1,a2,a3⟩:=qEn e0 w0 0 0 0 0 _ rfl (by norm_num1) h0 ג ג ג
 obtain ⟨b1,b2,b3⟩:=qEn e0 w1 _ _ _ _ _ rfl (by norm_num1) h1 a1 a2 a3
 rw[zero_add] at b3
 exact ⟨by (exact Nat.le_trans ‹_› ה),by (exact Nat.le_trans ‹_› ה),b3⟩
lemma qF4 (e0:Ὠ) (w0 w1:ὥ) (p n:Ὠ) (V:ℚ) (hp:Ѧp<=2*(2^61-1)) (hn:Ѧn<=2*(2^61-1)) (hV:V=((Ѧp:ℚ)-Ѧn)*2^((Ѧe0:Int)-150)) (h0:T_ok e0 w0=true) (h1:T_ok e0 w1=true):
 (T_p e0 w1 (T_p e0 w0 p)).toNat<2^63/\(T_n e0 w1 (T_n e0 w0 n)).toNat<2^63/\
  V+ἂ w0+ἂ w1=(((T_p e0 w1 (T_p e0 w0 p)).toNat:ℚ)-(T_n e0 w1 (T_n e0 w0 n)).toNat)*2^((Ѧe0:Int)-150):=by
 obtain ⟨a1,a2,a3⟩:=qEn e0 w0 p n V _ _ rfl (by norm_num1) h0 hp hn hV
 obtain ⟨b1,b2,b3⟩:=qEn e0 w1 _ _ _ _ _ rfl (by norm_num1) h1 a1 a2 a3
 exact ⟨by (exact Nat.lt_of_le_of_lt ‹_› ה),by (exact Nat.lt_of_le_of_lt ‹_› ה),b3⟩
lemma qCi (e0:Ὠ) (w:ὥ) (h:T_ok e0 w=true):expOf w≠255:=by unfold T_ok at h;simp only[and_eq_true] at h;exact qAc w h.1
end
@[noinline] def addSlow (relu:Ὡ) (x y:Ὢ):Ὢ:=
 act relu (toWord (FloatLib.Floats.Formats.BinaryInterchange.Model.add (ofW x) (ofW y)))
section
open Float.Model Float.Model.UnpackedFloat
end
@[inline] def E_act (relu:Ὡ) (r:ὥ):ὥ:=
 if relu&&signOf r then 0 else r
@[inline] def addOne (relu:Ὡ) (x y:Ὢ):Ὢ:=
 if expOf x != 255&&expOf y != 255 then
  let r:=(Float32.ofBits x+Float32.ofBits y).toBits
  if expOf r != 255 then E_act relu r else act relu r
 else addSlow relu x y
section
lemma qJe (A P Q T g:Ὣ) (hg:g=0\/g=2^63) (ha:A<2^64) (e1:(A+2)%2^64=g+P) (e2:(2^64-2+A)%2^64=g+Q) (t1:2^61<=T) (t2:T<2^63-2^60) (l1:T<P+2^28) (u1:P<=T+2^28) (l2:T<Q+2^28) (u2:Q<=T+2^28):A=g+(Q+2)/\T+2<Q+2+2^28/\Q+2+2<=T+2^28:=by (clear hg t1 t2;exact omS471 ha e1 e2 l1 u1 l2 u2)
lemma qJg (G f F:Ὣ) (hl:G+f+2<G+F+2^28) (hu:G+F+2<=G+f+2^28):F+2<=f+2^28/\f+2<F+2^28:=by exact omS472 hl hu
lemma qEw (s:ὧ) (h:zivS s≠Е):
 ∃(S:Ὡ) (ex fr M:Ὣ),897<=ex/\ex<1151/\fr<2^23/\zivS s=ὺ ((if S then 2^31 else 0)+(ex-896)*2^23+fr)/\s.toBits.toNat=(if S then 2^63 else 0)+M/\(ex*2^23+fr)*2^29+2<M+2^28/\M+2<=(ex*2^23+fr)*2^29+2^28:=by
 obtain ⟨S,ex,fr,h1,h2,h3,hz,⟨P,e1,l1,u1⟩,⟨Q,e2,l2,u2⟩⟩:=qCg _ _ h
 rw[ἤ] at e1
 rw[ὲ] at e2
 obtain ⟨a,b,c⟩:=qJe _ P Q ((ex*2^23+fr)*2^29) _ (by cases S<;>simp only[false_eq_true, ↓reduceIte,Nat.reducePow,OfNat.zero_ne_ofNat,or_false,OfNat.ofNat_ne_zero,or_true]) s.toBits.toNat_lt e1 e2 (by clear h hz e1 l1 u1 e2 l2 u2;exact omS232 h1) (by clear h hz e1 l1 u1 e2 l2 u2;exact omS233 h2 h3) l1 u1 l2 u2
 exact ⟨S,ex,fr,Q+2,h1,h2,h3,hz,a,b,c⟩
lemma qJd (v W z u:ℝ) (hu:0<u) (hz:z<=u) (h0:0<=v) (hv:v<=2^53*u) (h: |v-W| <=1/2^53*(|W| +z)):v-2*u<W/\W<v+2*u:=by
 have h1: |W| <=v+ |v-W|:=by
  have:=Ἐ (v-(v-W)) (-(v-W)+(v-W));have e:W=v-(v-W):=by ring
  have a:=Ὥ W v;rw[ἣ W v,ἅ h0] at a;linarith only[a]
 have h2:=abs_lt.mp (show |v-W| <2*u by linarith only[hu,hz,hv,h,h1]);constructor<;>linarith only[h2.1,h2.2,h2]
set_option maxRecDepth 8192 in
lemma qJq (ex fr E F:Ὣ) (h1:897<=ex) (h2:ex<1151) (hfr:fr<2^23) (hF:F<2^52) (hl:(ex*2^23+fr)*2^29+2<E*2^52+F+2^28) (hu:E*2^52+F+2<=(ex*2^23+fr)*2^29+2^28) (W:ℝ) (h: |((2^52+F:Ὣ):ℝ)*2^((E:Int)-1075)-W| <=1/2^53*(|W| +A_z)):
 W<(2^53+fr*2^30+2^29)*2^((ex:Int)-1076)/\(fr≠0->(2^53+fr*2^30-2^29)*2^((ex:Int)-1076)<W)/\(2^53-2^28)*2^((ex:Int)-1076)<W:=by
 have hE:E=ex\/(E+1=ex/\fr=0):=(by clear h1 h2;exact omS473 hfr hF hl hu)
 have hu0:(0:ℝ)<2^((E:Int)-1075):=ἵ ו _
 have hz:A_z<=2^((E:Int)-1075):=by
  have e:A_z=2^(-1021:ℤ):=rfl
  rw[e]
  exact ἒ (by norm_num1) (by clear hE;exact omS474 h1 hF hl)
 have hN:((2^52+F:Ὣ):ℝ)<=2^53:=by
  have:2^52+F<=2^53:=(by clear h1 h2 hfr hl hu hE;exact omS236 hF)
  exact_mod_cast this
 have hN0:(0:ℝ)<=((2^52+F:Ὣ):ℝ):=Ἔ _
 obtain ⟨a,b⟩:=qJd _ W A_z _ hu0 hz (Ἆ hN0 hu0.le) (ἀ hN hu0.le) h
 push_cast at a b
 rcases hE with rfl | ⟨rfl,rfl⟩
 . have e:(2:ℝ)^((E:Int)-1075)=2*2^((E:Int)-1076):=by rw[show (E:Int)-1075=1+((E:Int)-1076) by ring,Ἓ ἓ,zpow_one]
   rw[e] at a b;have hb:(0:ℝ)<2^((E:Int)-1076):=ἵ ו _;generalize (2:ℝ)^((E:Int)-1076)=c at a b hb⊢;have e1:(E*2^23+fr)*2^29=E*2^52+fr*2^29:=by ring
   rw[e1] at hl hu;obtain ⟨n3,n4⟩:=qJg _ _ _ hl hu;have h3:(F:ℝ)+2<=fr*2^29+2^28:=by exact_mod_cast n3
   have h4:(fr:ℝ)*2^29+2<F+2^28:=by exact_mod_cast n4
   have h5:(0:ℝ)<=F:=Ἔ _;have h7:=ἀ h3 hb.le;have h8:=Ἦ h4 hb;have h9:=Ἆ h5 hb.le;refine ⟨?_,?_,?_⟩;. linarith only[b,h7]
   . intro _;linarith only[a,h8]
   . linarith only[a,h9,hb]
 . have e:(2:ℝ)^(((E+1:Ὣ):Int)-1076)=2^((E:Int)-1075):=by congr 1;push_cast;ring
   rw[e];generalize (2:ℝ)^((E:Int)-1075)=c at a b hu0⊢;have h3:(F:ℝ)<=2^52:=by exact_mod_cast hF.le
   have h4:(2:ℝ)^52+2<F+2^28:=by exact_mod_cast (by (clear h1 h2 hfr;exact omS475 hl):2^52+2<F+2^28)
   have h7:=ἀ h3 hu0.le;have h8:=Ἦ h4 hu0;refine ⟨?_,?_,?_⟩;. rw[Nat.cast_zero,zero_mul,add_zero];linarith only[b,h7,hu0]
   . intro h0;exact absurd rfl h0
   . linarith only[a,h8]
lemma qEz (s:ὧ) (d:FloatLib.Numerics.Dyadic) (hV: |R s-(d.toRat:ℝ)| <=1/2^53*(|(d.toRat:ℝ)| +A_z)) (h:zivS s≠Е):zivS s=toWord (ἕ Fmt d):=by
 obtain ⟨S,ex,fr,M,h1,h2,h3,hz,hb,hl,hu⟩:=qEw s h;have hM:M=(M/2^52)*2^52+M%2^52:=(by clear h h1 h2 h3 hz hb hl hu;exact omS114);have hF:M%2^52<2^52:=ὀ _ ו;generalize M/2^52=E at hM;generalize M%2^52=F at hM hF;subst hM;obtain ⟨-,hR⟩:=B_R s S E F (by clear h hz;exact omS476 h1 hF hl) (by clear h hz;exact omS477 h2 h3 hu) hF (by rw[hb,ὢ]);rw[hR] at hV
 obtain ⟨W,hVW,hW⟩:∃W:ℝ,(d.toRat:ℝ)=(if S then-W else W)/\
  |((2^52+F:Ὣ):ℝ)*2^((E:Int)-1075)-W| <=1/2^53*(|W| +A_z):=by
  cases S
  . simp only[false_eq_true,ite_false,one_mul] at hV⊢;exact ⟨_,rfl,hV⟩
  . simp only[ite_true] at hV⊢;refine ⟨-(d.toRat:ℝ),by simp only[Ἠ,neg_neg],?_⟩;rw[abs_neg,show ((2^52+F:Ὣ):ℝ)*2^((E:Int)-1075)- -(d.toRat:ℝ)=-(-1*((2^52+F:Ὣ):ℝ)*2^((E:Int)-1075)-(d.toRat:ℝ)) by ring,abs_neg];exact hV
 obtain ⟨a,b,c⟩:=qJq ex fr E F h1 h2 h3 hF (by exact omS478 hl) (by exact omS479 hu) W hW;rw[hz];exact (qDh d S ex fr h1 h2 h3 W hVW a b c).symm
lemma qE7 (w:ὥ) (hg:good w=true): ∃(m k:ℤ),wv w=(m:ℝ)*2^k/\m.natAbs<2^25/\-160<=k:=by
 have hN:=w.toNat_lt;rw[qBe w hg];unfold A_mag;by_cases h0:ѥw%2^31=0;. exact ⟨0,0,by rw[ἱ h0];simp only[Nat.reducePow,mul_zero,Int.cast_zero,zpow_ofNat,pow_zero,mul_one],by (positivity),by (norm_num1)⟩
 . simp only[h0,ite_false];have hk:-160<=(((ѥw%2^31/2^23:ℕ):ℤ)-150):=(by clear hN h0;exact omS480);have hM:2^23+ѥw%2^31%2^23<2^25:=(by clear hN h0 hk;exact omS481);generalize ((ѥw%2^31/2^23:ℕ):ℤ)-150=k at hk⊢;generalize 2^23+ѥw%2^31%2^23=M at hM⊢;by_cases hs:2^31<=ѥw;. exact ⟨-(M:ℤ),k,by rw[ἱ hs];push_cast;ring,by simpa only[Int.natAbs_neg,Int.natAbs_natCast,Nat.reducePow] using hM,hk⟩
   . exact ⟨(M:ℤ),k,by rw[ἐ hs];push_cast;ring,by simpa only[Int.natAbs_natCast,Nat.reducePow] using hM,hk⟩
lemma qHw (a b:ὥ) (ha:good a=true) (hb:good b=true):(dv a*dv b).isFinite=true/\R (dv a*dv b)=wv a*wv b:=by
 obtain ⟨fa,ra⟩:=qAs a ha;obtain ⟨fb,rb⟩:=qAs b hb;have la:=(qAh a ha).trans qAe;have lb:=(qAh b hb).trans qAe
 obtain ⟨h1,h2⟩:=qD7 _ _ fa fb (by
  rw[ra,rb]
  exact (ἑ la lb (Ἀ _) qJ0.le).trans qDi)
 refine ⟨h1,?_⟩;obtain ⟨m,k,e1,h3,h4⟩:=qE7 a ha;obtain ⟨n,l,e2,h5,h6⟩:=qE7 b hb;have e:wv a*wv b=((m*n:ℤ):ℝ)*2^(k+l):=by rw[e1,e2,Ἓ ἓ];push_cast;ring
 rw[h2,ra,rb,e];refine qAt _ _ ?_ (by clear h3 h5;exact omS482 h4 h6);rw[Int.natAbs_mul];have:=Nat.mul_lt_mul'' h3 h5;(exact Nat.lt_of_lt_of_le ‹_› ה)
lemma S_E (x y z:ὥ) (hx:good x=true) (hy:good y=true) (hz:good z=true) (d:FloatLib.Numerics.Dyadic) (hd:(d.toRat:ℝ)=wv x*wv y+wv z):
 |R (dv x*dv y+dv z)-(d.toRat:ℝ)| <=1/2^53*(|(d.toRat:ℝ)| +A_z):=by
 obtain ⟨f1,r1⟩:=qHw x y hx hy
 obtain ⟨-,-,l⟩:=qEv x y hx hy
 obtain ⟨f2,r2⟩:=qAs z hz
 have h256:(2:ℝ)^256<=A_B:=by
  have:(2:ℝ)^256<=2^256*2^42:=(by norm_num1)
  linarith only[qBl]
 obtain ⟨-,he⟩:=qA4 _ _ f1 f2 (by rw[r1];exact l.trans h256) (by rw[r2];exact (qAh z hz).trans qAe)
 rw[r1,r2] at he
 rw[hd]
 exact he
lemma qDx (relu:Ὡ) (r:ὥ) (hf:expOf r≠255):E_act relu r=act relu r:=by cases relu<;>simp only[E_act,signOf,Bool.false_and,false_eq_true, ↓reduceIte,act,Bool.true_and,bne_iff_ne,ne_eq,ite_not,reluW,hf,decide_false,Bool.not_false,Bool.and_true]
lemma qGg (relu:Ὡ) (s:ὧ) (h:zivS s≠Е):E_act relu (zivS s)=act relu (zivS s):=by
 apply qDx;obtain ⟨S,ex,fr,h1,h2,h3,hz,-,-⟩:=qCg _ _ h;have hz':zivS s=_:=hz;intro h0;have h4:(expOf (zivS s)).toNat=255:=by rw[h0];rfl
 rw[qAx,hz',ὡ] at h4;split at h4<;>(clear h hz hz' h0;omega)
lemma qAr (z:Model FloatFormat.binary32):Ἑ z=(Model.toModel z).isFinite:=by
 have he:=Model.unpackExponent_toNat z;have ha:=Model.toNat_neg_one_exponentBits FloatFormat.binary32;have hm:Ἑ z=(Model.expField z != FloatFormat.expAllOnesNat FloatFormat.binary32):=rfl;rw[hm];unfold Model.toModel Ἶ;simp only[];by_cases h:Float.Model.UnpackedFloat.unpackExponent (spec:=FloatFormat.toModel FloatFormat.binary32) (Model.toModelBits z)=-BitVec.ofNat _ 1
 . rw[ἱ h];have h2:Model.expField z=FloatFormat.expAllOnesNat FloatFormat.binary32:=by rw[<-he,h,ha]
   rw[h2];split<;>rfl
 . rw[ἐ h];have h2:Model.expField z≠FloatFormat.expAllOnesNat FloatFormat.binary32:=by intro h3;apply h;apply BitVec.eq_of_toNat_eq;rw[he,h3,<-ha]
   have h4:(Model.expField z != FloatFormat.expAllOnesNat FloatFormat.binary32)=true:=by simpa only[bne_iff_ne,ne_eq] using h2
   rw[h4];split;. split<;>rfl
   . rfl
def m32 (f:Float32):Model FloatFormat.binary32:=Model.ofFloat32Model f.toModel
lemma qDu (f:Float32):Ἑ (m32 f)=f.isFinite:=by rw[qAr];rfl
lemma qG3 (y:Model FloatFormat.binary32) (h:Ἑ y=true):Model.IsModelCanonical y:=by refine ⟨λhe _=>?_⟩;have hf:=qAr y;rw[h] at hf;unfold Model.toModel Ἶ at hf;simp only[] at hf;rw[ἱ he] at hf;split at hf<;>simp only[Float.Model.UnpackedFloat.isFinite,Bool.true_eq_false] at hf
lemma qAz (w:Ὢ) (h:expOf w≠255):m32 (Float32.ofBits w)=ofW w:=by
 have hf:Ἑ (ofW w)=true:=by simp only[ofW,isFinite_ofUInt32,bne_iff_ne,ne_eq,h,not_false_eq_true]
 rw[<-Model.canonicalizeModel_eq_self _ (qG3 _ hf)];unfold m32 Model.canonicalizeModel;change Model.ofFloat32Model (Float32.Model.pack _)=_;rw[Model.ofFloat32Model_pack];congr 1;unfold Model.toModel;congr 1;apply BitVec.eq_of_toNat_eq;simp only[UInt32.toNat_toBitVec,Model.toModelBits,ofW,Model.ofUInt32,Model.ofNatBits,Model.ofBits,FloatFormat.ofWordNat];show ѥw=ѥw%2^32;have:=w.toNat_lt;(clear h;omega)
lemma qEi (f:Float32):f.toBits=toWord (m32 f):=by unfold toWord m32 Model.ofFloat32Model Model.ofModelBits Model.toNatBits;simp only[Float32.toBits];exact UInt32.ofNat_toNat.symm
lemma qD2 (x y:Ὢ) (hx:expOf x≠255) (hy:expOf y≠255):(Float32.ofBits x+Float32.ofBits y).toBits=toWord (Model.add (ofW x) (ofW y)):=by
 have hfx:Ἑ (ofW x)=true:=by simp only[ofW,isFinite_ofUInt32,bne_iff_ne,ne_eq,hx,not_false_eq_true]
 have hfy:Ἑ (ofW y)=true:=by simp only[ofW,isFinite_ofUInt32,bne_iff_ne,ne_eq,hy,not_false_eq_true]
 have fx:(Float32.ofBits x).isFinite=true:=by rw[<-qDu,qAz x hx];exact hfx
 have fy:(Float32.ofBits y).isFinite=true:=by rw[<-qDu,qAz y hy];exact hfy
 rw[qEi,show m32 (Float32.ofBits x+Float32.ofBits y)=
  Model.add (m32 (Float32.ofBits x)) (m32 (Float32.ofBits y)) from toModel_ofFloat32_add_of_isFinite _ _ fx fy,qAz x hx,qAz y hy]
lemma qCs (relu:Ὡ) (x y:Ὢ):addOne relu x y=act relu (toWord (Model.add (ofW x) (ofW y))):=by
 unfold addOne;split
 . rename_i hc;simp only[and_eq_true,bne_iff_ne,ne_eq] at hc;dsimp only;split;. rename_i h2;rw[qDx _ _ (by simpa only[ne_eq,bne_iff_ne] using h2),qD2 x y hc.1 hc.2]
   . rw[qD2 x y hc.1 hc.2]
 . rfl
lemma qE4 {L:ὦ->ѐ->ѐ} (f:ὦ->ὥ->ὥ) {e:ὦ} (hL:∀i x,L i x=if i<e then L (i+1) (wrW x i (f i (rdW x i))) else x) (d:Ὣ):
 ∀(i:ὦ) (x:ѐ),Ѥi+d=Ѥe->Ѥe<=ѧx->ѧx<2^32->
  (L i x).size=ѧx/\∀q,(L i x)[q]! =if Ѥi<=q/\q< Ѥe then f (ὴ q) x[q]! else x[q]!:=by
 induction d with
 | zero=>
  intro i x h1 _ _
  rw[hL,ἐ (by rw[А];exact omS404 h1)]
  exact ⟨rfl,λq=>(ἐ (by exact omS483 h1)).symm⟩
 | succ d ih=>
  intro i x h1 h2 h3
  have hi:i<e:=А.2 (by clear h2 h3;exact omS136 h1)
  have hs:=qAm x i (f i (rdW x i))
  rw[hL,ἱ hi]
  obtain ⟨h4,h5⟩:=ih (i+1) (wrW x i (f i (rdW x i))) (by rw[usz i e hi];(clear h2 h3 hi hs;exact omS137 h1)) (by clear h1 h3 hi;exact omS484 h2 hs) (by clear h1 h2 hi;exact omS360 h3 hs)
  refine ⟨h4.trans hs,λq=>?_⟩
  rw[h5,qAq _ _ _ _ h3 (by exact Nat.lt_of_lt_of_le ‹_› ‹_›),usz i e hi,qAo _ _ h3]
  by_cases h6:q=Ѥi
  . subst h6;rw[ἐ (by clear h1 h2 h3 hi hs h4;exact omS485),ἱ rfl,ἱ (by clear h2 h3 hi hs h4;exact omS486 h1),USize.ofNat_toNat]
  . rw[ἐ h6];exact if_congr (by clear h2 h3 hi hs h4;exact omS487 h6) rfl rfl
def addGo (relu:Ὡ) (y:ё) (e i:ὦ) (x:ё):
 ё:=
 if h:i<e then
  addGo relu y e (i+1) (wrW x i (addOne relu (rdW x i) (rdW y i)))
 else x
partial_fixpoint
def addGu (relu:Ὡ) (y:ё) (e i:ὦ) (x:ё) (hx:ѧx=Ѥe) (hy:ѧy=Ѥe):
 ё:=
 if h:i<e then
  addGu relu y e (i+1) (x.uset i (addOne relu (x.uget i (hx▸А.mp h)) (y.uget i (hy▸А.mp h))) (hx▸А.mp h)) ((Array.size_uset ..).trans hx) hy
 else x
partial_fixpoint
lemma addGu_eq (relu:Ὡ) (y:ё) (e:ὦ) (k:Ὣ): ∀(i:ὦ) (x:ё) (hx:ѧx=Ѥe) (hy:ѧy=Ѥe),Ѥi+k=Ѥe->addGu relu y e i x hx hy=addGo relu y e i x:=by
 induction k with
 | zero=>
  intro i x hx hy h0
  rw[addGu,addGo,ἔ ט,ἔ ט]
 | succ k ih=>
  intro i x hx hy h0
  have hi:i<e:=А.2 א
  have hxu:Ѥx.usize=ѧx:=by simp only[Array.usize,Nat.toUSize_eq,USize.toNat_ofNat'];exact Ἁ (by rw[hx];exact e.toNat_lt_size)
  have hyu:Ѥy.usize=ѧy:=by simp only[Array.usize,Nat.toUSize_eq,USize.toNat_ofNat'];exact Ἁ (by rw[hy];exact e.toNat_lt_size)
  have hi1:i<x.usize:=by rw[А,hxu,hx];exact А.mp hi
  have hi2:i<y.usize:=by rw[А,hyu,hy];exact А.mp hi
  rw[addGu,addGo,Ἄ hi,Ἄ hi,ih (i+1) _ (by rw[Array.size_uset];exact hx) hy (by rw[usz i e hi];omega)]
  unfold wrW rdW
  rw[Ἄ hi1,Ἄ hi2,Ἄ hi1]
@[noinline] def addFb (relu:Ὡ) (a b:ё):ё:=Array.ofFn (n:=ѧa) λi=>addOne relu a[i] b[i.val]!
def addF (relu:Ὡ) (a b:ё):ё:=
 if h:ѧa=ѧb/\ѧa<=4294967295 then addGu relu b a.usize 0 a (by have:=h.2;simp only[Array.usize,Nat.toUSize_eq];exact (uk _ (by omega)).symm) (by have:=h.2;rw[<-h.1];simp only[Array.usize,Nat.toUSize_eq];exact (uk _ (by omega)).symm)
 else addFb relu a b
lemma qFu:addF=addReluSpec:=by
 funext relu a b;unfold addF;split
 . rename_i h;obtain ⟨h1,h2'⟩:=h;have h2:ѧa<2^32:=א;have hu:a.usize.toNat=ѧa:=qAa _ h2;rw[addGu_eq relu b a.usize ѧa 0 a _ _ (by rw[hu];simp only[USize.toNat_zero,zero_add])];have hg:=qE4 (L:=addGo relu b a.usize) (λi w=>addOne relu w (rdW b i)) (λ_ _=>by rw[addGo];rfl) ѧa 0 a (by rw[hu,show (0:ὦ).toNat=0 from rfl,Ἇ]) (by clear h2;(exact Nat.le_of_eq ‹_›)) h2;apply Մ (by rw[hg.1];simp only[addReluSpec,Fin.getElem_fin,Array.size_ofFn]);intro i h3 h4;have hi:i< ѧa:=by rwa[hg.1] at h3
   rw[<-Ἕ _ i h3,hg.2 i,ἱ ⟨Nat.zero_le _,by (clear h2 h4 hi;omega)⟩,qCs,Ἕ a i hi,qAo _ _ (by clear h3 h4 hi;omega),uk i (by clear h4 hi;omega)];simp only[addReluSpec,Fin.getElem_fin,Array.getElem_ofFn]
 . simp only[addFb,addReluSpec,qCs]
end
section
@[noinline] def E_fma (relu:Ὡ) (w sw tw:ὥ):ὥ:=
 act relu (toWord (Model.fma (ofW w) (ofW sw) (ofW tw)))
@[noinline] def E_fz (relu g:Ὡ) (sw tw:ὥ) (y z:ὧ) (w:ὥ):ὥ:=
 if g&&good w then
  let r:=zivS (dv w*y+z)
  if r==Е then E_fma relu w sw tw else E_act relu r
 else E_fma relu w sw tw
@[inline] def E_el (relu g:Ὡ) (sw tw:ὥ) (y z:ὧ) (w:ὥ):ὥ:=
 let a:=((if w==0 then dv 0 else dvN w)*y+z).toBits
 let u:=a+0x0FFFFFFD
 if g&&(a<<<(1:Ὠ))-0x7040000000000000<0x1F60000000000000&&u&&&0x1FFFFFFF<0x1FFFFFFC then
  E_act relu (zivC (u>>>(29:Ὠ)))
 else E_fz relu g sw tw y z w
@[specialize] def E_map (f:ὥ->ὥ) (e i:ὦ) (x:ѐ):ѐ:=
 if h:i<e then E_map f e (i+1) (wrW x i (f (rdW x i))) else x
partial_fixpoint
@[specialize] def E_mU (f:ὥ->ὥ) (e i:ὦ) (x:ѐ) (he:Ѥe<=ѧx):ѐ:=
 if h:i<e then E_mU f e (i+1) (x.uset i (f (x.uget i (Nat.lt_of_lt_of_le (А.mp h) he))) (Nat.lt_of_lt_of_le (А.mp h) he)) (by rw[Array.size_uset];exact he) else x
partial_fixpoint
lemma lcRU (a:ѐ) (i:ὦ) (h:Ѥi< ѧa) (hs:ѧa<2^32):a.uget i h=rdW a i:=by
 have h2:=USize.le_size;have h3:USize.size=2^System.Platform.numBits:=rfl
 unfold rdW;rw[Ἄ (А.mpr (by simp only[Array.usize,Nat.toUSize_eq,USize.toNat_ofNat'];rw[Ἁ א];exact h))]
lemma lcMU (f:ὥ->ὥ) (e:ὦ) (d:Ὣ): ∀(i:ὦ) (x:ѐ) (he:Ѥe<=ѧx),Ѥe-Ѥi=d->ѧx<2^32->E_mU f e i x he=E_map f e i x:=by
 induction d using Nat.strong_induction_on with
 | _ d ih=>
  intro i x he hd hs
  rw[E_mU,E_map]
  by_cases h:i<e
  . rw[Ἄ h,Ἄ h]
    have e1:=usz i e h;have t:=А.mp h
    have hs':ѧ(x.uset i (f (x.uget i (Nat.lt_of_lt_of_le t he))) (Nat.lt_of_lt_of_le t he))<2^32:=by rw[Array.size_uset];exact hs
    rw[ih _ א (i+1) _ _ rfl hs',lcRU _ _ _ hs,lcWU _ _ _ _ hs]
  . rw[ἔ h,ἔ h]
@[inline] def E_elTT (sw tw:ὥ) (y z:ὧ) (w:ὥ):ὥ:=E_el true true sw tw y z w
@[inline] def E_elTF (sw tw:ὥ) (y z:ὧ) (w:ὥ):ὥ:=E_el true false sw tw y z w
@[inline] def E_elFT (sw tw:ὥ) (y z:ὧ) (w:ὥ):ὥ:=E_el false true sw tw y z w
@[inline] def E_elFF (sw tw:ὥ) (y z:ὧ) (w:ὥ):ὥ:=E_el false false sw tw y z w
@[noinline] def E_mTT (sw tw:ὥ) (y z:ὧ) (e i:ὦ) (x:ѐ) (he:Ѥe<=ѧx):ѐ:=E_mU (E_elTT sw tw y z) e i x he
@[noinline] def E_mTF (sw tw:ὥ) (y z:ὧ) (e i:ὦ) (x:ѐ) (he:Ѥe<=ѧx):ѐ:=E_mU (E_elTF sw tw y z) e i x he
@[noinline] def E_mFT (sw tw:ὥ) (y z:ὧ) (e i:ὦ) (x:ѐ) (he:Ѥe<=ѧx):ѐ:=E_mU (E_elFT sw tw y z) e i x he
@[noinline] def E_mFF (sw tw:ὥ) (y z:ὧ) (e i:ὦ) (x:ѐ) (he:Ѥe<=ѧx):ѐ:=E_mU (E_elFF sw tw y z) e i x he
@[inline] def E_bnU (relu g:Ὡ) (sw tw:ὥ) (y z:ὧ) (e i:ὦ) (x:ѐ) (he:Ѥe<=ѧx):ѐ:=
 if relu then (if g then E_mTT sw tw y z e i x he else E_mTF sw tw y z e i x he)
 else (if g then E_mFT sw tw y z e i x he else E_mFF sw tw y z e i x he)
lemma lcBU (relu g:Ὡ) (sw tw:ὥ) (y z:ὧ) (e i:ὦ) (x:ѐ) (he:Ѥe<=ѧx):E_bnU relu g sw tw y z e i x he=E_mU (E_el relu g sw tw y z) e i x he:=by
 cases relu<;>cases g<;>rfl
def E_chs (relu:Ὡ) (c hw:Ὣ) (s t:ѐ) (ch:Ὣ)
 (x:ѐ):ѐ:=
 if ch<c then
  let sw:=s[ch]!
  let tw:=t[ch]!
  let y:=dv sw
  let z:=dv tw
  E_chs relu c hw s t (ch+1) (if hE:Ѥ((ch+1)*hw).toUSize<=ѧx then E_bnU relu (good sw&&good tw) sw tw y z
   ((ch+1)*hw).toUSize (ch*hw).toUSize x hE else E_map (E_el relu (good sw&&good tw) sw tw y z)
   ((ch+1)*hw).toUSize (ch*hw).toUSize x)
 else x
partial_fixpoint
def scaleF (relu:Ὡ) (c hw:Ὣ) (x s t:ѐ):ѐ:=
 if ѧx=c*hw/\ѧx<=4294967295 then E_chs relu c hw s t 0 x
 else scaleShiftSpec relu c hw x s t
@[noinline] def E_dot (X:ѐ) (b:ὦ) (k w:Ὣ):ὥ:=
 dotFn (k*k) (λt=>X[Ѥb+(t/k*w+t%k)]!) (λ_=>oneW)
section
open Float.Model Float.Model.UnpackedFloat
def E_dotT (X:ѐ) (b:ὦ) (k w:Ὣ):ὥ:=
 if k=2 then
  let i:=Ѥb
  let a0:=X[i]!
  let a1:=X[i+1]!
  let a2:=X[i+w]!
  let a3:=X[i+(w+1)]!
  let e0:=T_min (T_min (T_e a0) (T_e a1)) (T_min (T_e a2) (T_e a3))
  if T_ok e0 a0&&T_ok e0 a1&&T_ok e0 a2&&T_ok e0 a3 then
   T_fin (T_p e0 a3 (T_p e0 a2 (T_p e0 a1 (T_p e0 a0 0))))
    (T_n e0 a3 (T_n e0 a2 (T_n e0 a1 (T_n e0 a0 0)))) e0 λ_=>E_dot X b k w
  else E_dot X b k w
 else E_dot X b k w
lemma qE9 (X:ѐ) (b:ὦ) (k w:Ὣ):E_dotT X b k w=E_dot X b k w:=by unfold E_dotT;split;swap;rfl;subst_vars;dsimp only;split;swap;rfl;rename_i hc;simp only[and_eq_true] at hc;obtain ⟨⟨⟨h0,h1⟩,h2⟩,h3⟩:=hc;obtain ⟨p1,n1,v⟩:=qGv _ _ _ h0 h1;obtain ⟨p2,n2,v⟩:=qF4 _ _ _ _ _ _ p1 n1 v h2 h3;refine qFy _ _ _ _ p2 n2 ((qC5 _ _).trans ((qC5 _ _).trans (qHq _))) _ rfl λe=>?_;have B:=qBa (2*2) (λj=>X[Ѥb+(j/2*w+j%2)]!) (λ_=>oneW) λj g=>⟨by interval_cases j<;>simp only[Nat.zero_div,zero_mul,Nat.zero_mod,add_zero,ne_eq,qCi _ _ h0,not_false_eq_true,Nat.reduceDiv,Nat.mod_succ,zero_add,qCi _ _ h1,Order.lt_two_iff,zero_le,Nat.div_self,one_mul,Nat.mod_self,qCi _ _ h2,Nat.reduceMod,qCi _ _ h3],ה⟩;simp only[Nat.reduceMul,D.qIg,mul_one,Finset.sum_range_succ,Finset.range_one,Finset.sum_singleton,Nat.zero_div,zero_mul,Nat.zero_mod,add_zero,Nat.reduceDiv,Nat.mod_succ,zero_add,Order.lt_two_iff,zero_le,Nat.div_self,one_mul,Nat.mod_self,Nat.reduceMod,ne_eq] at B;rw[v] at B;rw[E_dot,qAg];exact B (mul_ne_zero (sub_ne_zero.2 λh=>e (UInt64.toNat_inj.1 (Nat.cast_injective h))) (zpow_ne_zero _ ἓ))
end
def E_dq (n:Ὣ) (s:ὥ):ὥ:=
 if 1<n/\n<=144/\n.sqrt*n.sqrt=n/\expOf s≠255/\expOf s≠0 then
  let d:=dv s
  let r:=ziv (d*(((0x20000000000000:Ὠ)/n.toUInt64).toFloat*Ἃ 0x3CA0000000000000))
   (d.abs*Ἃ 0x3CC0000000000000)
  if r==Е then E_div n s else r
 else E_div n s
@[inline] def E_quo (n:Ὣ) (j s:ὥ):ὥ:=
 if j<expOf s/\expOf s≠255 then s-(j<<<(23:ὥ)) else E_dq n s
@[inline] def E_tl (X:ѐ) (b:ὦ) (k w n:Ὣ) (j z:ὥ) (nf zb s a:ὧ)
 (o:ὥ):ὥ:=
 if o=0 then z
 else
  let r:=ziv (s+zb) (nf*(a+zb.abs))
  E_quo n j (if r==Е then E_dotT X b k w else r)
def E_win (X off:ѐ) (k w n:Ὣ) (j z:ὥ) (u nf zb:ὧ) (b m t:ὦ)
 (s a:ὧ) (o:ὥ):ὥ:=
 if h:t< m then
  let v:=rdW X (b+(rdW off t).toUSize)
  let d:=dv v*u
  E_win X off k w n j z u nf zb b m (t+1) (s+d) (a+d.abs) (o ||| v)
 else E_tl X b k w n j z nf zb s a o
partial_fixpoint
@[specialize] def G_x (f:ὦ->ὥ) (sx:ὦ) (W x:Ὣ) (b q:ὦ)
 (out:ѐ):ѐ:=
 if x<W then G_x f sx W (x+1) (b+sx) (q+1) (wrW out q (f b)) else out
partial_fixpoint
@[specialize] def G_y (f:ὦ->ὥ) (sy sx:ὦ) (W H y:Ὣ) (b q:ὦ)
 (out:ѐ):ѐ:=
 if y<H then G_y f sy sx W H (y+1) (b+sy) (q+ѩW) (G_x f sx W 0 b q out) else out
partial_fixpoint
@[specialize] def G_c (f:ὦ->ὥ) (sc sy sx:ὦ) (W H c ch:Ὣ) (b q:ὦ)
 (out:ѐ):ѐ:=
 if ch<c then
  G_c f sc sy sx W H c (ch+1) (b+sc) (q+(H*W).toUSize) (G_y f sy sx W H 0 b q out)
 else out
partial_fixpoint
def E_off (k w:Ὣ):ѐ:=
 Array.ofFn (n:=k*k) λt=>(t.val/k*w+t.val%k).toUInt32
@[noinline] def E_grid (c h w k n:Ὣ) (X off:ѐ) (j z:ὥ) (u nf zb zr:ὧ):
 ѐ:=
 G_c (λb=>E_win X off k w n j z u nf zb b ѩn 0 zr zr 0) (h*w).toUSize (k*w).toUSize
  ѩk (w/k) (h/k) c 0 0 0 (ἇ (c*(h/k*(w/k))) 0)
@[inline] def E_w4 (X:ѐ) (k w n:Ὣ) (j z:ὥ) (u nf zb zr:ὧ)
 (a0 a1 a2 a3 b:ὦ):ὥ:=
 let v0:=rdW X (b+a0)
 let d0:=dv v0*u
 let v1:=rdW X (b+a1)
 let d1:=dv v1*u
 let v2:=rdW X (b+a2)
 let d2:=dv v2*u
 let v3:=rdW X (b+a3)
 let d3:=dv v3*u
 E_tl X b k w n j z nf zb (zr+d0+d1+d2+d3) (zr+d0.abs+d1.abs+d2.abs+d3.abs)
  (0 ||| v0 ||| v1 ||| v2 ||| v3)
@[inline] def E_w4u (X:ѐ) (k w n:Ὣ) (j z:ὥ) (nf zb zr:ὧ)
 (a0 a1 a2 a3 b:ὦ):ὥ:=
 let v0:=rdW X (b+a0)
 let d0:=dv v0
 let v1:=rdW X (b+a1)
 let d1:=dv v1
 let v2:=rdW X (b+a2)
 let d2:=dv v2
 let v3:=rdW X (b+a3)
 let d3:=dv v3
 E_tl X b k w n j z nf zb (zr+d0+d1+d2+d3) (zr+d0.abs+d1.abs+d2.abs+d3.abs)
  (0 ||| v0 ||| v1 ||| v2 ||| v3)
lemma qW4u (X:ѐ) (k w n:Ὣ) (j z:ὥ) (nf zb zr:ὧ) (a0 a1 a2 a3 b:ὦ):
 E_w4u X k w n j z nf zb zr a0 a1 a2 a3 b=E_w4 X k w n j z (dv oneW) nf zb zr a0 a1 a2 a3 b:=by
 unfold E_w4u E_w4;simp only[D.qM1 _ (D.qDvF _)]
@[noinline] def E_gridK (c h w k n:Ὣ) (X off:ѐ) (j z:ὥ) (u nf zb zr:ὧ):
 ѐ:=
 if k==2 then
  let a0:=(rdW off 0).toUSize
  let a1:=(rdW off (0+1)).toUSize
  let a2:=(rdW off (0+1+1)).toUSize
  let a3:=(rdW off (0+1+1+1)).toUSize
  G_c (λb=>E_w4 X k w n j z u nf zb zr a0 a1 a2 a3 b) (h*w).toUSize (k*w).toUSize
   ѩk (w/k) (h/k) c 0 0 0 (ἇ (c*(h/k*(w/k))) 0)
 else E_grid c h w k n X off j z u nf zb zr
def E_winC (X:ѐ) (k w n:Ὣ) (j z:ὥ) (u nf zb:ὧ) (b m t:ὦ)
 (s a:ὧ) (o:ὥ):ὥ:=
 if h:t< m then
  let v:=rdW X (b+t)
  let d:=dv v
  E_winC X k w n j z u nf zb b m (t+1) (s+d) (a+d.abs) (o ||| v)
 else E_tl X b k w n j z nf zb s a o
partial_fixpoint
@[noinline] def E_gridC (c h w k n:Ὣ) (X:ѐ) (j z:ὥ) (u nf zb zr:ὧ):
 ѐ:=
 G_c (λb=>E_winC X k w n j z u nf zb b ѩn 0 zr zr 0) (h*w).toUSize (k*w).toUSize
  ѩk (w/k) (h/k) c 0 0 0 (ἇ (c*(h/k*(w/k))) 0)
def A_x (X:ѐ) (w:Ὣ) (wu:ὦ) (u nf zb zr:ὧ) (b q e:ὦ) (out:ѐ):ѐ:=
 if h:q<e then A_x X w wu u nf zb zr (b+2) (q+1) e (wrW out q (E_w4u X 2 w 4 2 0 nf zb zr 0 1 wu (wu+1) b)) else out
partial_fixpoint
def A_y (X:ѐ) (w:Ὣ) (wu:ὦ) (u nf zb zr:ὧ) (W H y:Ὣ) (b q:ὦ) (out:ѐ):ѐ:=
 if y<H then A_y X w wu u nf zb zr W H (y+1) (b+wu+wu) (q+ѩW) (A_x X w wu u nf zb zr b q (q+ѩW) out) else out
partial_fixpoint
def A_c (X:ѐ) (w:Ὣ) (wu:ὦ) (u nf zb zr:ὧ) (sc:ὦ) (W H c ch:Ὣ) (b q:ὦ) (out:ѐ):ѐ:=
 if ch<c then A_c X w wu u nf zb zr sc W H c (ch+1) (b+sc) (q+(H*W).toUSize) (A_y X w wu u nf zb zr W H 0 b q out) else out
partial_fixpoint
@[noinline] def E_grid2 (c h w:Ὣ) (X:ѐ) (u nf zb zr:ὧ):ѐ:=
 A_c X w ѩw u nf zb zr (h*w).toUSize (w/2) (h/2) c 0 0 0 (ἇ (c*(h/2*(w/2))) 0)
def E_gridX (c h w k n:Ὣ) (X:ѐ) (j z:ὥ) (u nf zb zr:ὧ):ѐ:=
 if k=w then E_gridC c h w k n X j z u nf zb zr
 else if k=2/\w<2^31/\c*(h/2*(w/2))<=4294967295 then E_grid2 c h w X u nf zb zr
 else E_gridK c h w k n X (E_off k w) j z u nf zb zr
def avgF (c h w k:Ὣ) (X:ѐ):ѐ:=
 if c*h*w<=ѧX/\ѧX<=4294967295/\k*k<2^20/\
  scan X X.usize 0 (qAd X) Е 0≠Е then
  let n:=k*k
  let j:ὥ:=if n=4 then 2 else if n=64 then 6 else 255
  let z:ὥ:=if 0<k/\k<13 then 0 else if k=14 then 0 else E_div n (dotFn n (λ_=>0) (λ_=>oneW))
  E_gridX c h w k n X j z (dv oneW) (nfOf n) (dv 0) D.fz0
 else avgpoolSpecO c h w k X
def E_d (x y z:ὥ):FloatLib.Numerics.Dyadic:=
 Model.addDyadic ⟨Bool.xor (decodeD x).negative (decodeD y).negative,
  (decodeD x).significand*(decodeD y).significand,
  (decodeD x).exponent+(decodeD y).exponent⟩ (decodeD z)
section
variable (x y z:ὥ) (hx:expOf x≠255) (hy:expOf y≠255) (hz:expOf z≠255)
include x y z hx hy hz
lemma qC7:Model.fma (ofW x) (ofW y) (ofW z)=ἕ Fmt (E_d x y z):=by rw[Model.Proof.fma_eq_spec];have e:=ἶ x hx;have f:=ἶ y hy;have g:=ἶ z hz;simp only[Model.Spec.fma,ofW,(Model.chooseNaN3_eq_none_iff _ _ _).2 ⟨Model.isNaN_eq_false_of_toDyadic?_some e,Model.isNaN_eq_false_of_toDyadic?_some f,Model.isNaN_eq_false_of_toDyadic?_some g⟩,Model.isInf_eq_false_of_toDyadic?_some e,Model.isInf_eq_false_of_toDyadic?_some f,Bool.or_self,false_eq_true, ↓reduceIte,Model.isInf_eq_false_of_toDyadic?_some g,e,f,g];rfl
lemma qDv:((E_d x y z).toRat:ℝ)=wv x*wv y+wv z:=by have j:=mul_toRat (decodeD x) (decodeD y);simp only[FloatLib.Numerics.Dyadic.mul,mulFields] at j;rw[E_d,Model.addDyadic,add_toRat,j,qA6 x hx,qA6 y hy,qA6 z hz];simp only[Rat.cast_add,Rat.cast_mul,Ἠ,rationalWord]
end
section
open Float.Model Float.Model.UnpackedFloat
lemma qGe (i:ὦ) (h:Ѥi< ѨpwN):pwN.uget i h=pwEntryN Ѥi:=by simp only[FloatArray.uget,qDg,List.toArray_range,Array.ugetElem_eq_getElem,Array.getElem_map,Array.getElem_range]
def N_ok (w:ὥ):Prop:=0x800000<=w&&&В/\w&&&В<У
lemma qHr {w:ὥ} (h:N_ok w):good w=true:=qBo _ (Or.inr h.1) h.2
lemma N_w (w:ὥ) (h:(pwN.uget (w>>>(23:ὥ)).toUSize (qCl w))ᶠ):
 pwN.uget (w>>>(23:ὥ)).toUSize (qCl w)=pw.uget (w>>>(23:ὥ)).toUSize (qB4 w)/\N_ok w:=by
 have s:=qEc w
 rw[qGe,ὃ,s] at h
 rw[qGe,qC2,ὃ,s]
 unfold pwEntryN at h⊢
 have e:=qA0 w
 have g:=w.toNat_lt
 by_cases c:(ѥw/2^23%256==0 || ѥw/2^23%256==255)=true
 . rw[ἱ c,qI6] at h;exact absurd h ה
 . rw[ἐ c];simp only[or_eq_true,beq_iff_eq,not_or] at c;refine ⟨rfl,?_,?_⟩;. rw[ἃ,e,show (0x800000:ὥ).toNat=0x800000 from rfl];(clear s g;exact omS488 c)
   . rw[ἳ,e,show (У:ὥ).toNat=У from rfl];(clear s g;exact omS489 c)
lemma qJy {w:ὥ} (h:(dvN w)ᶠ):dvN w=dv w/\N_ok w:=by obtain ⟨e,o⟩:=N_w w (qGl _ _ h).2;exact ⟨by unfold dvN dv;rw[e],o⟩
end
lemma S_w (T M:Ὣ) (h1:T+2<M+2^28) (h2:M+2<=T+2^28):T<M+2^28/\M<=T+2^28:=by exact omS490 h1 h2
lemma qIx (s:ὧ) (h:zivS s≠Е):s.isFinite=true:=by obtain ⟨S,ex,fr,M,h1,h2,h3,-,hb,hl,hu⟩:=qEw s h;exact (qAk _ S ex fr h1 h2 h3 ⟨M,hb,(S_w _ _ hl hu).1,(S_w _ _ hl hu).2⟩).1
section
variable (relu g:Ὡ) (sw tw w:ὥ) (hg:g=true->good sw=true∧good tw=true)
include relu g sw tw w hg
lemma qGu:E_fz relu g sw tw (dv sw) (dv tw) w=E_fma relu w sw tw:=by unfold E_fz;split;swap;rfl;rename_i hc;obtain ⟨h1,h2⟩:=ὕ.1 hc;obtain ⟨h3,h4⟩:=hg h1;dsimp only;split;rfl;rename_i hr;have r:=mt beq_iff_eq.2 hr;have hf:=qAc _ h2;have g1:=qAc _ h3;have g2:=qAc _ h4;rw[qGg _ _ r,qEz _ (E_d w sw tw) (S_E w sw tw h2 h3 h4 _ (qDv _ _ _ hf g1 g2)) r,E_fma,qC7 _ _ _ hf g1 g2]
lemma qHa:E_el relu g sw tw (dv sw) (dv tw) w=E_fma relu w sw tw:=by
 by_cases hw:w=0
 . subst hw;unfold E_el;dsimp only;simp only[beq_self_eq_true,ite_true];split
   . rename_i hc;obtain ⟨hA,h9⟩:=ὕ.1 hc;obtain ⟨h1,h8⟩:=ὕ.1 hA;obtain ⟨h3,h4⟩:=hg h1
     obtain ⟨e9,hr0⟩:=qIf (dv 0*dv sw+dv tw) (by simpa only[decide_eq_true_eq] using h8) (by simpa only[decide_eq_true_eq] using h9);rw[<-e9]
     have hr':=hr0;have h2:good 0=true:=ה
     have hf:=qAc _ h2;have g1:=qAc _ h3;have g2:=qAc _ h4
     rw[qGg _ _ hr',qEz _ (E_d 0 sw tw) (S_E 0 sw tw h2 h3 h4 _ (qDv _ _ _ hf g1 g2)) hr',E_fma,qC7 _ _ _ hf g1 g2]
   . exact qGu relu g sw tw 0 hg
 . have hw':(w==0)=false:=by simpa only[beq_eq_false_iff_ne,ne_eq] using hw
   unfold E_el;dsimp only;simp only[hw',Bool.false_eq_true,ite_false];split
   . rename_i hc;obtain ⟨hA,h9⟩:=ὕ.1 hc;obtain ⟨h1,h8⟩:=ὕ.1 hA;obtain ⟨h3,h4⟩:=hg h1;obtain ⟨e9,hr0⟩:=qIf (dvN w*dv sw+dv tw) (by simpa only[decide_eq_true_eq] using h8) (by simpa only[decide_eq_true_eq] using h9);rw[<-e9];obtain ⟨e,o⟩:=qJy (qGl _ _ (qIu _ _ (qIx _ hr0)).1).1;rw[e] at hr0⊢;have hr':=hr0;have h2:=qHr o;have hf:=qAc _ h2;have g1:=qAc _ h3;have g2:=qAc _ h4;rw[qGg _ _ hr',qEz _ (E_d w sw tw) (S_E w sw tw h2 h3 h4 _ (qDv _ _ _ hf g1 g2)) hr',E_fma,qC7 _ _ _ hf g1 g2]
   . exact qGu relu g sw tw w hg
end
lemma qEh (relu:Ὡ) (c hw:Ὣ) (s t:ѐ) (d:Ὣ): ∀(ch:Ὣ) (x:ѐ),ch+d=c->ѧx=c*hw->ѧx<2^32->
 (E_chs relu c hw s t ch x).size=ѧx/\∀q,q<c*hw->
  (E_chs relu c hw s t ch x)[q]! =if ch*hw<=q then E_fma relu x[q]! s[q/hw]! t[q/hw]! else x[q]!:=by
 induction d with
 | zero=>
  rintro ch x rfl _ _
  rw[E_chs,ἐ (by exact omS491)]
  exact ⟨rfl,λq hq=>(ἐ (by simpa only[not_le,add_zero] using hq)).symm⟩
 | succ d ih=>
  intro ch x h1 h2 h3
  rw[E_chs,ἱ (by clear h2 h3;exact omS136 h1)]
  dsimp only
  have h4:=Ἒ hw ((by clear h2 h3;exact omS492 h1):ch+1<=c)
  have h5:=Nat.add_one_mul ch hw
  have e1:=qAa ((ch+1)*hw) (by clear h1;exact omS493 h2 h3 h4)
  have e0:=qAa (ch*hw) (by clear h1;exact omS494 h2 h3 h4 h5)
  rw[Ἄ (show Ѥ((ch+1)*hw).toUSize<=ѧx by rw[e1,h2];exact h4),lcBU,lcMU _ _ _ _ _ _ rfl h3]
  obtain ⟨h6,h7⟩:=qE4 (λ_=>E_el relu (good s[ch]!&&good t[ch]!) s[ch]! t[ch]! (dv s[ch]!) (dv t[ch]!)) (L:=E_map _ ((ch+1)*hw).toUSize) (λ_ _=>by rw[E_map];rfl) hw (ch*hw).toUSize x (by clear h1 h2 h3 h4;omega) (by clear h1 h3 e0;omega) h3
  obtain ⟨h8,h9⟩:=ih (ch+1) _ (by clear h2 h3 h4 h5 e1 e0 h6;exact omS137 h1) (h6.trans h2) (by clear h1 h4 h5 e1 e0;omega)
  refine ⟨h8.trans h6,λq hq=>?_⟩
  rw[h9 q hq,h7,e0,e1,qHa _ _ _ _ _ ὕ.1]
  by_cases a:(ch+1)*hw<=q
  . rw[ἱ a,ἐ (by clear h1 h2 h3 h4 e0 h6 h8 hq;omega),ἱ (by clear h1 h2 h3 h4 h6 h8 hq;omega)]
  . rw[ἐ a];by_cases b:ch*hw<=q;. rw[ἱ (by clear h1 h2 h3 h4 h6 h8 hq;omega),ἱ b,Nat.div_eq_of_lt_le b (by clear h1 h2 h3 h4 e0 h6 h8 hq b;omega)]
    . rw[ἐ (by clear h1 h2 h3 h4 h6 h8 hq a;omega),ἐ b]
lemma qFf:scaleF=scaleShiftSpec:=by
 funext relu c hw x s t;unfold scaleF;split
 . rename_i h;obtain ⟨h1,h2⟩:=h;replace h2:=Nat.lt_of_le_of_lt h2 (show 4294967295<2^32 by norm_num);obtain ⟨h3,h4⟩:=qEh relu c hw s t c 0 x (by clear h1 h2;exact omS139) h1 h2;apply Մ;. simp only[h3,h1,scaleShiftSpec,Fin.is_lt,Ἕ,Array.size_ofFn]
   . intro i hi hj;have hi':i<c*hw:=by rwa[h3,h1] at hi
     rw[<-Ἕ _ i hi,h4 i hi',ἱ ג];simp only[E_fma,scaleShiftSpec,Array.getElem_ofFn]
 . rfl
lemma qBp (w:ὥ) (h:expOf w≠255):Ἢ (ofW w)=(sv w:ℝ)*2^((idxOf w:ℤ)-149):=by simp only[show Model.toReal (ofW w)=wv w from rfl,qA6 w h,rationalWord,decodeD_toRat,Rat.cast_mul,Rat.cast_intCast,Rat.cast_zpow,Rat.cast_ofNat]
lemma qI3 (s j:ὥ) (hj:ѥj<=6) (a:ѥj<(expOf s).toNat):
 (expOf (s-(j<<<(23:ὥ)))).toNat=(expOf s).toNat-ѥj/\fracOf (s-(j<<<(23:ὥ)))=fracOf s/\signOf (s-(j<<<(23:ὥ)))=signOf s:=by
 rw[qAx] at a
 have hs:=s.toNat_lt
 have e:(j<<<(23:ὥ)).toNat=ѥj*2^23:=by
  rw[UInt32.toNat_shiftLeft,ὁ]
  exact Ἁ (by simp only[UInt32.reduceToNat,Nat.reduceMod,Nat.reducePow];omega)
 generalize j<<<(23:ὥ)=J at e⊢
 have d:=ὓ s J (by rw[ἃ,e];(clear e;omega))
 rw[e] at d
 simp only[qAx,fracOf,<-ἠ,qBt,qEx,d,decide_eq_decide]
 (clear e;omega)
lemma qE2 (s y j:ὥ) (f:ѥj<=6) (a:expOf s≠255) (b:ѥj<(expOf s).toNat) (d:expOf y≠255) (g:signOf y=false) (e:mantOf y=2^23) (k:idxOf y=126+ѥj):toWord (Model.div (ofW s) (ofW y))=s-(j<<<(23:ὥ)):=by
 have B:Model.isZero (ofW y)=false:=by rw[ofW,Model.isZero_eq_beq_zero_of_toDyadic?_some (ἶ y d),decodeD_sig_zero,e];decide
 obtain ⟨C,D,E⟩:=qI3 s j f b;generalize s-(j<<<(23:ὥ))=r at C D E⊢;have N:=expOf_toNat_lt s;have O:=mt ἠ.1 a;simp only[UInt32.reduceToNat] at O;have l:expOf r≠255:=λh=>by simp only[h,UInt32.reduceToNat] at C;(clear a d e D h;omega)
 have q:expOf r≠0:=λh=>by simp only[h,UInt32.toNat_zero] at C;(clear f a d e D N O l h;omega)
 have t:expOf s≠0:=λh=>by simp only[h,UInt32.toNat_zero,not_lt_zero] at b
 have x:Ἢ (ofW y)=2^ѥj:=by rw[qBp _ d,sv,g,e,k];push_cast;rw[show (126+ѥj-149:ℤ)=ѥj-23 by ring,ᾂ ἓ,Ἡ];field_simp;(norm_num1)
 have I:Ἢ (ofW r)=Ἢ (ofW s)/Ἢ (ofW y):=by rw[x,qBp _ l,qBp _ a,show sv r=sv s by unfold sv mantOf;rw[E,D,ἐ q,ἐ t],show (idxOf r:ℤ)=idxOf s-ѥj by unfold idxOf;rw[ἐ q,ἐ t,C];(clear a d e C D O l q t;omega),sub_right_comm,ᾂ ἓ,Ἡ,mul_div_assoc]
 have o:=qBi a;have u:=qBi d;have v:=qBi l;have z:=Model.isFinite_div_of_abs_div_le_posMaxFinite _ _ ה o u B (I▸Model.abs_toReal_le_posMaxFinite_of_isFinite _ v);have p:=Model.toReal_div_eq_roundAt _ _ ה u z;rw[<-I,Model.roundAt_toReal_eq _ v] at p;rw[Model.eq_of_toReal_eq_of_nonzero p (p▸qFm r q l),qAw]
lemma qH5 {a b:ℝ} (h:a<b): ∃d:FloatLib.Numerics.Dyadic,a<(d.toRat:ℝ)/\(d.toRat:ℝ)<b:=by
 obtain ⟨n,hn⟩:=exists_pow_lt_of_lt_one (sub_pos.2 h) (one_half_lt_one:(1/2:ℝ)<1);have h2:(0:ℝ)<2^n:=ו
 have hm1:a*2^n<((⌊a*2^n⌋+1:ℤ):ℝ):=by push_cast;exact Int.lt_floor_add_one _
 have hm2:((⌊a*2^n⌋+1:ℤ):ℝ)<=a*2^n+1:=by have:=Int.floor_le (a*2^n);push_cast;linarith only[this]
 generalize (⌊a*2^n⌋+1:ℤ)=m at hm1 hm2;refine ⟨⟨decide (m<0),m.natAbs,-(n:ℤ)⟩,?_⟩
 have e:((⟨decide (m<0),m.natAbs,-(n:ℤ)⟩:FloatLib.Numerics.Dyadic).toRat:ℝ)=m/2^n:=by
  rw[Ἠ,FloatLib.Numerics.Dyadic.toReal,FloatLib.Numerics.Dyadic.cast_signedSignificand,zpow_neg,Ἡ];by_cases g:m<0
  . have c:(m.natAbs:ℝ)=-(m:ℝ):=by rw[Nat.cast_natAbs,abs_of_neg g,Int.cast_neg]
    simp only[g,decide_true,ite_true,c];ring
  . have c:(m.natAbs:ℝ)=(m:ℝ):=by rw[Nat.cast_natAbs,ἅ (not_lt.1 g)]
    simp only[g,decide_false,false_eq_true,ite_false,c];ring
 rw[e];constructor;. rwa[lt_div_iff₀ h2]
 . rw[div_lt_iff₀ h2];have u:(1/2:ℝ)^n*2^n=1:=by rw[<-mul_pow];norm_num
   have:=Ἦ hn h2;rw[sub_mul,u] at this;linarith only[this,hm2]
lemma qHx (q e:ὧ) (x y:Model Fmt) (hlo:R (q-e)<Ἢ x/Ἢ y) (hhi:Ἢ x/Ἢ y<R (q+e)) (h:ziv q e≠Е) (hx:Ἑ x=true) (hy:Ἑ y=true) (hy0:Model.isZero y=false) (hb: |Ἢ x/Ἢ y| <=Ἢ (Model.posMaxFinite Fmt)):toWord (Model.div x y)=ziv q e:=by
 generalize hV:Ἢ x/Ἢ y=V at hlo hhi hb;obtain ⟨d1,a1,a2⟩:=qH5 hlo;obtain ⟨d2,b1,b2⟩:=qH5 hhi;have w1:=qEa q e d1 a1 (a2.trans hhi) h;have w2:=qEa q e d2 (hlo.trans b1) b2 h;obtain ⟨S,ex,fr,W,h1,h2,h3,hz,-⟩:=qDd q e V hlo hhi h;generalize ziv q e=r at w1 w2 hz⊢;have l:(expOf r).toNat=ex-896:=by rw[qAx,hz,ὡ];cases S<;>simp only[false_eq_true,ite_true,ite_false]<;>(norm_num1)<;>(clear h w1 w2 hz;omega)
 have n255:expOf r≠255:=λg=>by rw[g] at l;simp only[UInt32.reduceToNat] at l;(clear h h3 w1 w2 hz g;omega)
 have n0:expOf r≠0:=λg=>by rw[g] at l;simp only[UInt32.toNat_zero] at l;(clear h h3 w1 w2 hz n255 g;omega)
 have f:=qBi n255;have m1:ἕ Fmt d1=ofW r:=by rw[w1,qAy]
 have m2:ἕ Fmt d2=ofW r:=by rw[w2,qAy]
 have t1:=Model.toReal_roundDyadic_eq_roundAt Fmt ה d1 (m1▸f);have t2:=Model.toReal_roundDyadic_eq_roundAt Fmt ה d2 (m2▸f);rw[m1] at t1;rw[m2] at t2;rw[Ἠ] at a2 b1;have c1:=Model.roundAt_mono Fmt a2.le;have c2:=Model.roundAt_mono Fmt b1.le;have E:ᾁ Fmt V=Ἢ (ofW r):=le_antisymm (t2▸c2) (t1▸c1);have z:=Model.isFinite_div_of_abs_div_le_posMaxFinite x y ה hx hy hy0 (hV▸hb);have p:=Model.toReal_div_eq_roundAt _ _ ה hy z;rw[hV,E] at p;rw[Model.eq_of_toReal_eq_of_nonzero p (p▸qFm r n0 n255),qAw]
lemma qJx {a V W Q E P P' z:ℝ} (ha:0<a) (hz0:0<=z) (hz:z<=a/2^53) (hW: |W| <=a/2) (h4: |V-W| <=a/2^53) (h2: |Q-W| <=1/2^53*(|W| +z)) (hE:E=a/2^51) (hP: |P-(Q+E)| <=1/2^53*(|Q+E| +z)) (hN: |P'-(Q-E)| <=1/2^53*(|Q-E| +z)):P'<V/\V<P:=by
 have sck:∀{k a b:ℝ},0<k->a<=1/k*b->k*a<=b:=λhk h=>(le_div_iff₀' hk).mp (h.trans_eq (one_div_mul_eq_div _ _))
 have hz':(2:ℝ)^53*z<=a:=(le_div_iff₀' (by norm_num1)).mp hz;have hW':(2:ℝ)*|W| <=a:=(le_div_iff₀' (by norm_num1)).mp hW;have h4':(2:ℝ)^53*|V-W| <=a:=(le_div_iff₀' (by norm_num1)).mp h4
 have h2':=sck (by norm_num1) h2;have hP':=sck (by norm_num1) hP;have hN':=sck (by norm_num1) hN;have hE':a=E*2^51:=(div_eq_iff (by norm_num1)).mp hE.symm
 have pQ:=le_abs_self (Q-W);have nQ:=neg_abs_le (Q-W);have pW:=le_abs_self W;have nW:=neg_abs_le W
 have b1: |Q+E| <=a:=ὸ.2 ⟨by linarith only[nQ,nW,h2',hW',hz',hE',ha,hz0],by linarith only[pQ,pW,h2',hW',hz',hE',ha,hz0]⟩
 have b2: |Q-E| <=a:=ὸ.2 ⟨by linarith only[nQ,nW,h2',hW',hz',hE',ha,hz0],by linarith only[pQ,pW,h2',hW',hz',hE',ha,hz0]⟩
 constructor
 . linarith only[le_abs_self (P'-(Q-E)),hN',b2,pQ,pW,h2',hW',neg_abs_le (V-W),h4',hz',hE',ha,hz0]
 . linarith only[neg_abs_le (P-(Q+E)),hP',b1,nQ,nW,h2',hW',le_abs_self (V-W),h4',hz',hE',ha,hz0]
lemma qJn (s:ὥ) (h1:expOf s≠255) (h0:expOf s≠0):good s=true:=by
 have e:=qAx s;have b:=s.toNat_lt;have n1:(expOf s).toNat≠255:=λg=>h1 (ἠ.1 g);have n0:(expOf s).toNat≠0:=λg=>h0 (ἠ.1 g);have v:(s&&&В).toNat=ѥs%2^31:=by rw[Ἧ];exact Ἂ ѥs 31
 unfold good;simp only[or_eq_true,decide_eq_true_eq,beq_iff_eq];right;have g:(0x00800000:ὥ)<=s&&&В:=by rw[ἃ,v];show 8388608<=_;(clear h1 h0 b n1;omega)
 rw[ἃ,ὓ _ _ g,v];show _-8388608<=2130706431;(clear h1 h0 b g;omega)
lemma qG0 (n:Ὣ) (s:ὥ):E_dq n s=E_div n s:=by
 have e2pa:∀{x y:ℝ} {a b j:ℕ},x<=2^a->y<=2^b->a<=j->b<=j->x+y<=2^(j+1):=λ{x y a b j} hx hy ha hb=>(add_le_add (hx.trans (pow_le_pow_right₀ one_le_two ha)) (hy.trans (pow_le_pow_right₀ one_le_two hb))).trans_eq ((two_mul _).symm.trans (pow_succ' 2 j).symm)
 have e2z0:A_z<=2^0:=Submission.D.qBo.trans_eq (pow_zero 2).symm
 have e2d:∀{x:ℝ} (n:ℕ),0<=x->1/2^n*x<=x:=λn h0=>mul_le_of_le_one_left h0 (div_le_one_of_le₀ (one_le_pow₀ one_le_two) (pow_nonneg zero_le_two n))
 have e2AB1:∀{x:ℝ} {k:ℕ},x<=2^k->k<=256+42->x<=A_B:=λh hk=>h.trans (((pow_le_pow_right₀ one_le_two hk).trans_eq (pow_add 2 256 42)).trans Submission.D.qCc)
 have e2aa:∀(x y:ℝ),|x| <= |y| + |x-y|:=λx y=>sub_le_iff_le_add'.mp (Ὥ x y)
 unfold E_dq;split;swap;. rfl
 rename_i h;obtain ⟨n1,n2,nsq,h255,h0⟩:=h;dsimp only;split;. rfl
 rename_i hr;have hr':ziv (dv s*(((0x20000000000000:Ὠ)/n.toUInt64).toFloat*Ἃ 0x3CA0000000000000)) ((dv s).abs*Ἃ 0x3CC0000000000000)≠Е:=by simpa only[Nat.toUInt64_eq,ne_eq,beq_iff_eq] using hr
 have hg:=qJn s h255 h0;obtain ⟨df,dR⟩:=qAs s hg;have wl:=qAh s hg;have wn:=qFm s h0 h255;have k13:n.sqrt<13:=by by_contra hc;have:=Nat.mul_le_mul (Nat.le_of_not_lt hc) (Nat.le_of_not_lt hc);(clear h255 h0 hr' hc;omega)
 have k0:0<n.sqrt:=by
  rcases Nat.eq_zero_or_pos n.sqrt with h | h;. rw[h] at nsq;(clear h255 h0 hr' k13 h;omega)
  . exact h
 obtain ⟨y1,y2,y3,y4⟩:=qIt ⟨n.sqrt,k13⟩ k0;simp only[nsq] at y1 y2 y3 y4;have ys:(sv (natW n):ℝ)=mantOf (natW n):=by rw[sv,y2];simp only[false_eq_true, ↓reduceIte,Int.cast_natCast]
 have hq:(mantOf (natW n):ℝ)*2^idxOf (natW n)=n*2^149:=by exact_mod_cast y4
 have e149:(2:ℝ)^(149:ℤ)=2^149:=ב;have yV:Ἢ (ofW (natW n))=n:=by rw[qBp _ y1,ys,ᾂ ἓ,Ἡ,e149,<-mul_div_assoc,hq];exact mul_div_cancel_right₀ _ (pow_ne_zero _ ἓ)
 have hy0:Model.isZero (ofW (natW n))=false:=by rw[ofW,Model.isZero_eq_beq_zero_of_toDyadic?_some (ἶ _ y1),decodeD_sig_zero];simp only[y3,decide_false]
 generalize hm:((0x20000000000000:Ὠ)/n.toUInt64)=m at hr'
 have mN:Ѧm=2^53/n:=by
  rw[<-hm,UInt64.toNat_div];have:(n.toUInt64).toNat=n:=by simp only[Nat.toUInt64_eq,UInt64.toNat_ofNat',Nat.reducePow,Nat.mod_succ_eq_iff_lt,Nat.succ_eq_add_one,Nat.reduceAdd];(exact Nat.lt_of_le_of_lt ‹_› ה)
  rw[this];rfl
 have m1:n*(2^53/n)<=2^53:=Nat.mul_div_le _ _;have m2:2^53<n*(2^53/n+1):=Nat.lt_mul_div_succ _ (by exact Nat.le_of_lt ‹_›);have m3:Ѧm<2^53:=by rw[mN];exact Nat.div_lt_self ב (by exact Nat.succ_le_of_lt ‹_›)
 obtain ⟨f,a⟩:=qDz m m3
 obtain ⟨ue,ud⟩:=qBu _ 97 ב f (Ѧm:ℤ) 0 (by exact Nat.succ_le_of_lt ‹_›) ב
  (by
   rw[a,ἅ (Ἔ (Ѧm))]
   exact qEo (by exact Nat.le_of_lt (Nat.lt_of_lt_of_le ‹_› ה)))
  (by rw[a];simp only[Int.cast_natCast,zpow_ofNat,pow_zero,mul_one])
 rw[show m.toFloat*Ἃ 0x3CA0000000000000=m.toFloat*pwEntry 97 by rw[qBb];rfl,show Ἃ 0x3CC0000000000000=pwEntry 99 by rw[qBb];rfl] at hr';have Ru:R (m.toFloat*pwEntry 97)=(Ѧm:ℝ)/2^53:=by rw[ud,a,(qCm 97 ב).2];norm_num;ring
 have Rp:R (pwEntry 99)=1/2^51:=by rw[(qCm 99 ב).2];norm_num
 have sA: |wv s| =(mantOf s:ℝ)*2^((idxOf s:ℤ)-149):=by rw[show wv s=_ from qBp s h255,ἢ,ἡ (ἵ (two_pos:(0:ℝ)<2) _)];congr 1;unfold sv;split<;>simp only[Int.cast_neg,Int.cast_natCast,abs_neg,Nat.abs_cast]
 have a0:0< |wv s|:=abs_pos.2 wn
 have mS:1<=mantOf s:=by
  rcases Nat.eq_zero_or_pos (mantOf s) with g | g;. rw[g] at sA;simp only[CharP.cast_eq_zero,zero_mul,abs_eq_zero] at sA;exact absurd sA wn
  . exact g
 have aL:(2:ℝ)^(-149:ℤ)<= |wv s|:=by
  rw[sA];have c:(1:ℝ)<=mantOf s:=by exact_mod_cast mS
  have d:(2:ℝ)^(-149:ℤ)<=2^((idxOf s:ℤ)-149):=ἒ one_le_two (neg_le_sub_iff_le_add.mpr (le_add_of_nonneg_left (Int.natCast_nonneg _)));exact d.trans (le_mul_of_one_le_left (ἵ (show (0:ℝ)<2 by norm_num) _).le c)
 obtain ⟨af,aR⟩:=qCx (dv s) df;have aB: |R (dv s).abs| <=A_B:=by rw[aR,dR,abs_abs];exact wl.trans qAe
 obtain ⟨ee,ed⟩:=qBu (dv s).abs 99 ב af (mantOf s:ℤ) ((idxOf s:ℤ)-149) (by have:=mantOf_lt_24 s;(exact Nat.lt_of_lt_of_le ‹_› ה)) (neg_le_sub_iff_le_add.mpr ((by norm_num1:(149:ℤ)<=900).trans (le_add_of_nonneg_left (Int.natCast_nonneg _)))) aB
  (by rw[aR,dR,sA];simp only[Int.cast_natCast])
 have Re:R ((dv s).abs*pwEntry 99)= |wv s| /2^51:=by rw[ed,aR,dR,Rp];ring
 have dB: |R (dv s)| <=A_B:=by rw[dR];exact wl.trans qAe
 have uB0:(0:ℝ)<=(Ѧm:ℝ)/2^53:=div_nonneg (Ἔ _) (by norm_num1)
 have n2R:(2:ℝ)<=n:=by exact_mod_cast n1
 have m1R:(n:ℝ)*Ѧm<=2^53:=by rw[mN];exact_mod_cast m1
 have m2R:(2:ℝ)^53<n*(Ѧm+1):=by rw[mN];exact_mod_cast m2
 have uH:(Ѧm:ℝ)/2^53<=1/2:=by rw[div_le_iff₀ ב];have g:=ἀ n2R (show (0:ℝ)<=Ѧm from Ἔ _);linarith only[g,m1R]
 have uB: |R (m.toFloat*pwEntry 97)| <=A_B:=by rw[Ru,ἅ uB0];exact uH.trans ((by norm_num1:(1/2:ℝ)<=2^128).trans qAe)
 obtain ⟨qf,qE⟩:=qB8 _ _ df ue dB uB;rw[dR,Ru] at qE;have W0: |wv s*((Ѧm:ℝ)/2^53)| <= |wv s| /2:=by rw[ἢ,ἅ uB0];exact (ἁ uH (Ἀ (wv s))).trans_eq (mul_one_div _ _)
 have V0: |wv s/n-wv s*((Ѧm:ℝ)/2^53)| <= |wv s| /2^53:=by
  have n0R:(0:ℝ)<n:=zero_lt_two.trans_le n2R;have c1:wv s/n-wv s*((Ѧm:ℝ)/2^53)=wv s*(1/n-(Ѧm:ℝ)/2^53):=by ring
  have c2:0<=1/(n:ℝ)-(Ѧm:ℝ)/2^53:=by rw[sub_nonneg,div_le_div_iff₀ ב n0R];linarith only[m1R]
  have c3:1/(n:ℝ)-(Ѧm:ℝ)/2^53<=1/2^53:=by rw[sub_le_iff_le_add,<-add_div,div_le_div_iff₀ n0R ב];linarith only[m2R]
  rw[c1,ἢ,ἅ c2];have g:=ἁ c3 (Ἀ (wv s));rwa[mul_one_div] at g
 have zA:A_z<= |wv s| /2^53:=by have g:=qB3;have k:(2:ℝ)^(-149:ℤ)/2^53<= |wv s| /2^53:=div_le_div_of_nonneg_right aL ב;refine g.trans (ὂ ?_ k);norm_num
 have lim: |wv s| <=A_B:=wl.trans qAe;have qB: |R (dv s*(m.toFloat*pwEntry 97))| <=A_B:=by have hc:=W0.trans ((half_le_self (Ἀ _)).trans wl);exact e2AB1 ((e2aa _ _).trans (e2pa (j:=129) hc (qE.trans ((e2d 53 (add_nonneg (Ἀ _) qAn.le)).trans (e2pa (j:=128) hc e2z0 le_rfl (Nat.le_of_ble_eq_true rfl)))) (Nat.le_of_ble_eq_true rfl) le_rfl)) (Nat.le_of_ble_eq_true rfl)
 have eB: |R ((dv s).abs*pwEntry 99)| <=A_B:=by rw[Re,ἅ (div_nonneg (Ἀ (wv s)) (pow_nonneg zero_le_two 51))];exact (div_le_self (Ἀ (wv s)) (one_le_pow₀ one_le_two:(1:ℝ)<=2^51)).trans lim
 obtain ⟨-,pE⟩:=qA4 _ _ qf ee qB eB;obtain ⟨-,nE⟩:=qC9 _ _ qf ee qB eB;obtain ⟨lo,hi⟩:=qJx a0 qAn.le zA W0 V0 qE Re pE nE;refine (qHx _ _ (ofW s) (ofW (natW n)) ?_ ?_ hr' (qBi h255) (qBi y1) hy0 ?_).symm;. rw[yV];exact lo
 . rw[yV];exact hi
 . rw[yV,abs_div,Nat.abs_cast];exact (div_le_self (Ἀ _) (one_le_two.trans n2R)).trans (Model.abs_toReal_le_posMaxFinite_of_isFinite _ (qBi h255))
lemma qFt (n:Ὣ) (j s:ὥ) (hj:j=255\/(j=2/\n=4)\/(j=6/\n=64)):E_quo n j s=E_div n s:=by
 unfold E_quo;split;swap;exact qG0 n s;rename_i h;have h1:=ἳ.1 h.1;have h3:=expOf_toNat_lt s;rcases hj with rfl | ⟨rfl,rfl⟩ | ⟨rfl,rfl⟩;. simp only[UInt32.reduceToNat] at h1;(clear h;omega)
 all_goals exact (qE2 s _ _ ה h.2 h1 ה ה ה ה).symm
def E_f (xw:Ὣ->ὥ) (l:љ) (c:F1):F1:=
 l.foldl (λ(acc:F1) t=>let q:=dv (xw t)*dv oneW
  F1.mk (acc.s+q) (acc.t+q.abs)) c
lemma qG2 (X off:ѐ) (k w n:Ὣ) (j z:ὥ) (u nf zb zr:ὧ) (b:ὦ) (hn:n.toUSize.toNat=4):
 E_win X off k w n j z u nf zb b ѩn 0 zr zr 0=E_w4 X k w n j z u nf zb zr (rdW off 0).toUSize (rdW off (0+1)).toUSize (rdW off (0+1+1)).toUSize (rdW off (0+1+1+1)).toUSize b:=by
 have h0:(0:ὦ)< ѩn:=by rw[А,hn];decide
 have e1:=usz _ _ h0
 have h1:(0+1:ὦ)< ѩn:=by
  rw[А,hn,e1];decide
 have e2:=usz _ _ h1
 have h2:(0+1+1:ὦ)< ѩn:=by
  rw[А,hn,e2,e1];decide
 have e3:=usz _ _ h2
 have h3:(0+1+1+1:ὦ)< ѩn:=by
  rw[А,hn,e3,e2,e1];decide
 have e4:=usz _ _ h3
 have h4:¬(0+1+1+1+1:ὦ)< ѩn:=by
  rw[А,hn,e4,e3,e2,e1];decide
 rw[E_win,Ἄ h0]
 dsimp only
 rw[E_win,Ἄ h1]
 dsimp only
 rw[E_win,Ἄ h2]
 dsimp only
 rw[E_win,Ἄ h3]
 dsimp only
 rw[E_win,ἔ h4]
 rfl
lemma qEq (c h w k:Ὣ) (X off:ѐ) (j z:ὥ) (u nf zb zr:ὧ):E_gridK c h w k (k*k) X off j z u nf zb zr=E_grid c h w k (k*k) X off j z u nf zb zr:=by
 unfold E_gridK;split;. rename_i hk;have hk:=eq_of_beq hk;subst hk;have e:(λb=>E_w4 X 2 w (2*2) j z u nf zb zr (rdW off 0).toUSize (rdW off (0+1)).toUSize (rdW off (0+1+1)).toUSize (rdW off (0+1+1+1)).toUSize b)=λb=>E_win X off 2 w (2*2) j z u nf zb b (2*2).toUSize 0 zr zr 0:=funext λb=>(qG2 X off 2 w (2*2) j z u nf zb zr b (qAa _ (by norm_num1))).symm;dsimp only;rw[e,E_grid]
 . rfl
lemma qF3 (X off:ѐ) (k w n:Ὣ) (j z:ὥ) (nf zb:ὧ) (b m:ὦ) (xw:Ὣ->ὥ) (hx:∀t:ὦ,t< m->rdW X (b+(rdW off t).toUSize)=xw Ѥt)
 (d:Ὣ):∀(t:ὦ) (s a:ὧ) (o:ὥ),Ѥt+d=Ѥm->
  E_win X off k w n j z (dv oneW) nf zb b m t s a o=E_tl X b k w n j z nf zb (E_f xw (List.range' Ѥt d) ⟨s,a⟩).s (E_f xw (List.range' Ѥt d) ⟨s,a⟩).t ((List.range' Ѥt d).foldl (λc i=>c ||| xw i) o):=by
 induction d with
 | zero=>
  intro t s a o h
  rw[E_win,ἔ (by rw[А];exact omS404 h)]
  rfl
 | succ d ih=>
  intro t s a o h
  have ht:t< m:=А.2 (by exact omS136 h)
  rw[E_win,Ἄ ht]
  dsimp only
  rw[hx t ht,ih (t+1) _ _ _ (by rw[usz t m ht];(clear ht;exact omS137 h)),usz t m ht,List.range'_succ]
  rfl
lemma qIy (xw:Ὣ->ὥ) (l:љ):∀o:ὥ,l.foldl (λc i=>c ||| xw i) o=0->o=0/\∀i∈l,xw i=0:=by
 induction l with
 | nil=>simp only[List.foldl_nil,List.not_mem_nil,IsEmpty.forall_iff,implies_true,and_true,imp_self]
 | cons i l ih=>intro o h;obtain ⟨h1,h2⟩:=ih _ h;obtain ⟨h3,h4⟩:=UInt32.or_eq_zero_iff.1 h1;exact ⟨h3,List.forall_mem_cons.2 ⟨h4,h2⟩⟩
lemma qIm (n:Ὣ) (hn:n<2^20) (xw:Ὣ->ὥ) (hg:∀t,t<n->good (xw t)=true)
 (r:ziv ((E_f xw (ἄ n) ⟨0,0⟩).s+dv 0) (nfOf n*((E_f xw (ἄ n) ⟨0,0⟩).t+(dv 0).abs))≠Е):
 ziv ((E_f xw (ἄ n) ⟨0,0⟩).s+dv 0) (nfOf n*((E_f xw (ἄ n) ⟨0,0⟩).t+(dv 0).abs))=dotOf n xw (λ_=>oneW):=by
 obtain ⟨-,-,hbig⟩:=qCp _ _ r
 obtain ⟨hlo,hhi⟩:=qEd n hn xw (λ_=>oneW) 0 hg (λ_ _=>ה) ה (ἄ n) Ὂ ג (λt h1 h2=>absurd (Ἅ.mpr h1) h2) _ _ ⟨rfl,rfl⟩ hbig
 have hsum:((∑ j∈Ἣ n,ἂ (xw j)*ἂ oneW:ℚ):ℝ)=exactS n xw (λ_=>oneW) 0:=by
  rw[exactS,qA8 0 ה ה,add_zero,Rat.cast_sum]
  refine Ἷ rfl λt ht=>?_
  rw[Rat.cast_mul,qA6 _ (qAc _ (hg t (Ὁ.mp ht))),qA6 oneW ה]
 obtain ⟨d,hd,hw⟩:=qBa n xw (λ_=>oneW) (λj hj=>⟨qAc _ (hg j hj),ה⟩) λh=>qH6 _ _ _ hlo hhi r (by rw[<-hsum,h,Rat.cast_zero])
 have q:(d.toRat:ℝ)=exactS n xw (λ_=>oneW) 0:=by rw[hd,hsum]
 rw[hw]
 exact qEa _ _ d (by rw[q];exact hlo) (by rw[q];exact hhi) r
lemma qHc (X:ѐ) (b:ὦ) (k w n:Ὣ) (hn:n<2^20) (j:ὥ) (hj:j=255\/(j=2/\n=4)\/(j=6/\n=64)) (xw:Ὣ->ὥ) (hg:∀t,t<n->good (xw t)=true) (hs:E_dot X b k w=dotOf n xw (λ_=>oneW)):
 E_tl X b k w n j (E_div n (dotFn n (λ_=>0) (λ_=>oneW))) (nfOf n) (dv 0) (E_f xw (ἄ n) ⟨0,0⟩).s (E_f xw (ἄ n) ⟨0,0⟩).t ((ἄ n).foldl (λc i=>c ||| xw i) 0)=E_div n (dotOf n xw (λ_=>oneW)):=by
 unfold E_tl
 split
 next h1=>
  rw[qAg]
  unfold dotOf
  congr 3
  funext i
  exact ((qIy xw _ _ h1).2 _ (Ἅ.2 i.2)).symm
 . dsimp only;rw[qFt _ _ _ hj];congr 1;split;. rwa[qE9]
   next hr=>exact qIm _ hn xw hg (mt beq_iff_eq.2 hr)
lemma qEe {c h w k ch y x t:Ὣ} (hc:ch<c) (hy:y<h/k) (hx:x<w/k) (ht:t<k*k):(ch*h+y*k+t/k)*w+x*k+t%k<c*h*w:=by have hk:0<k:=ὼ (by rintro rfl;(clear hc hx ht;omega));have h1:=Ἰ ht;have h2:=ὀ t hk;have h3:=(Nat.le_div_iff_mul_le hk).1 hy;have h4:=(Nat.le_div_iff_mul_le hk).1 hx;have h5:=Ἒ h hc;rw[ᾀ] at h3 h4 h5;have h6:=Ἒ w ((by clear hc hy hx ht hk h2 h4;omega):ch*h+y*k+t/k+1<=c*h);rw[ᾀ] at h6;(clear hc hy hx ht hk h1 h3 h5;omega)
def G_P (N:Ὣ) (sp:Ὣ->ὥ) (out:ѐ) (p:Ὣ):Prop:=
 ѧout=N/\N<2^32/\∀i,i<p->out[i]! =sp i
lemma qJl {N p:Ὣ} {sp:Ὣ->ὥ} {out:ѐ} (h:G_P N sp out p) (hp:p<N) (q:ὦ) (hq:Ѥq=p):G_P N sp (wrW out q (sp p)) (p+1):=by
 obtain ⟨h1,h2,h3⟩:=h;refine ⟨(qAm _ _ _).trans h1,h2,λi hi=>?_⟩;rw[qAq _ _ _ _ (by clear hp hq hi;omega) (by clear h2 hi;omega),hq];by_cases e:i=p;. rw[ἱ e,e]
 . rw[ἐ e];exact h3 i (by clear hp h1 h2;(clear hq;exact omS374 hi e))
lemma qH7 (a s i:Ὣ):(a+(i+1)*s).toUSize=(a+i*s).toUSize+ѩs:=by rw[show a+(i+1)*s=a+i*s+s by ring];simp only[Nat.toUSize_eq,USize.ofNat_add]
lemma qD5 {L:Ὣ->ὦ->ὦ->ѐ->ѐ} {g:ὦ->ὦ->ѐ->ѐ} {sb S n N q0 b0:Ὣ} {sp:Ὣ->ὥ}
 (hL:∀i b q o,L i b q o=if i<n then L (i+1) (b+ѩsb) (q+ѩS) (g b q o) else o)
 (hs:∀i o,i<n->G_P N sp o (q0+i*S)->G_P N sp (g (b0+i*sb).toUSize (q0+i*S).toUSize o) (q0+(i+1)*S)) (d:Ὣ):
 ∀i o,i+d=n->G_P N sp o (q0+i*S)->G_P N sp (L i (b0+i*sb).toUSize (q0+i*S).toUSize o) (q0+n*S):=by
 induction d with
 | zero=>
  intro i o h1 h2
  rw[hL,ἐ (by exact Nat.not_lt.mpr (Nat.le_of_eq (Eq.symm ‹_›))),<-h1]
  exact h2
 | succ d ih=>
  intro i o h1 h2
  rw[hL,ἱ (by exact omS136 h1),<-qH7,<-qH7]
  exact ih (i+1) _ (by exact omS137 h1) (hs i o (by exact omS136 h1) h2)
lemma qIe {f:ὦ->ὥ} {sc sy sx W H c N:Ὣ} {sp:Ὣ->ὥ} (hN:c*(H*W)=N)
 (hs:∀ch y t,ch<c->y<H->t<W->f (ch*sc+y*sy+t*sx).toUSize=sp (ch*(H*W)+y*W+t))
 (o:ѐ) (ho:G_P N sp o 0):
 G_P N sp (G_c f ѩsc ѩsy ѩsx W H c 0 0 0 o) N:=by
 have:=qD5 (L:=G_c f ѩsc ѩsy ѩsx W H c) (sb:=sc) (S:=H*W) (n:=c)
  (N:=N) (q0:=0) (b0:=0) (sp:=sp) (λ_ _ _ _=>by rw[G_c]) (λch o a1 h=>by
   have h4:(ch+1)*(H*W)<=N:=hN▸Ἒ _ a1
   have:=qD5 (L:=G_y f ѩsy ѩsx W H) (sb:=sy) (S:=W) (n:=H) (N:=N) (q0:=ch*(H*W)) (b0:=ch*sc) (sp:=sp) (λ_ _ _ _=>by rw[G_y])
    (λy o a2 h=>by
     have h5:(y+1)*W<=H*W:=Ἒ _ a2
     rw[ᾀ] at h4 h5
     have:=qD5 (L:=G_x f ѩsx W) (g:=λb q o=>wrW o q (f b)) (sb:=sx) (S:=1) (n:=W) (N:=N) (q0:=ch*(H*W)+y*W) (b0:=ch*sc+y*sy) (sp:=sp)
      (λ_ _ _ _=>by rw[G_x];rfl) (λi o a3 h=>by
       have e1:ch*(H*W)+y*W+(i+1)*1=ch*(H*W)+y*W+i*1+1:=(by clear hN a1 h4 a2 h5 a3;omega)
       have e2:ch*(H*W)+y*W+i=ch*(H*W)+y*W+i*1:=(by clear hN a1 h4 a2 h5 a3 e1;omega)
       rw[e1,hs ch y i a1 a2 a3,e2]
       exact qJl h (by clear hN a1 a2 e1 e2;omega) _ (qAa _ (by have:=h.2.1;(clear a1 a2 e1 e2;omega))))
      W 0 o (by clear hN a1 h4 a2 h5;exact omS139) (by simpa only[mul_one,add_zero] using h)
     simpa only[Nat.toUSize_eq,USize.ofNat_add,USize.ofNat_mul,ᾀ,zero_mul,add_zero,mul_one,ὢ] using this) H 0 o (by clear hN a1 h4;exact omS139) (by simpa only[zero_mul,add_zero,zero_add] using h)
   simpa only[Nat.toUSize_eq,zero_add,USize.ofNat_mul,ᾀ,zero_mul,add_zero] using this) c 0 o (by clear hN;exact omS139) (by simpa only[zero_mul,add_zero] using ho)
 simp only[ὒ,Ἇ,Ἵ,hN] at this
 exact this
lemma qHz {c H W q:Ὣ} (hq:q<c*(H*W)):q/(H*W)<c/\q/W%H<H/\q%W<W:=by have hp:0<H*W:=ὼ (by rintro e;simp only[e,mul_zero,not_lt_zero] at hq);exact ⟨Ἰ (by rwa[ἥ]),ὀ _ (Nat.pos_of_mul_pos_right hp),ὀ _ (Nat.pos_of_mul_pos_left hp)⟩
lemma qIl (f:ὦ->ὥ) (c h w k:Ὣ) (hs:c*h*w<2^32):
 G_c f (h*w).toUSize (k*w).toUSize ѩk (w/k) (h/k) c 0 0 0 (ἇ (c*(h/k*(w/k))) 0)=
 Array.ofFn (n:=c*(h/k*(w/k))) λq=>let ch:=q.val/(h/k*(w/k))
  let y:=q.val/(w/k)%(h/k)
  let x:=q.val%(w/k)
  f ((ch*h+y*k)*w+x*k).toUSize:=by
 have hN:c*(h/k*(w/k))<2^32:=by
  have h1:h/k*(w/k)<=h*w:=Nat.mul_le_mul (Nat.div_le_self _ _) (Nat.div_le_self _ _)
  exact ἷ ((Nat.mul_le_mul_left c h1).trans (Nat.mul_assoc c h w).symm.le) hs
 generalize hH:h/k=H at*
 generalize hW:w/k=W at*
 have hp:=qIe (f:=f) (sc:=h*w) (sy:=k*w) (sx:=k) (W:=W) (H:=H) (c:=c) (sp:=λq=>f ((q/(H*W)*h+q/W%H*k)*w+q%W*k).toUSize) rfl
  (λch y t h1 h2 h3=>by
   obtain ⟨d1,d2⟩:=qC1 W (H*ch+y) t h3
   obtain ⟨d3,d4⟩:=qC1 H ch y h2
   have e:ch*(H*W)+y*W+t=W*(H*ch+y)+t:=by ring
   rw[e,d2,ἥ H W,<-Nat.div_div_eq_div_mul,d1,d3,d4]
   congr 2
   ring) (ἇ (c*(H*W)) 0) ⟨by simp only[Array.size_replicate],hN,by simp only[not_lt_zero,Nat.toUSize_eq,USize.ofNat_add,USize.ofNat_mul,IsEmpty.forall_iff,implies_true]⟩
 apply Մ (by rw[hp.1,Array.size_ofFn])
 intro i h1 h2
 rw[<-Ἕ _ i h1,hp.2.2 i (by simpa only[Nat.toUSize_eq,USize.ofNat_add,USize.ofNat_mul,Array.size_ofFn] using h2),Array.getElem_ofFn]
lemma qGm {c h w k ch y x:Ὣ} {X:ѐ} (h1:c*h*w<2^32) (h2:ѧX<2^32) (h3:k*k<2^32) (hc:ch<c) (hy:y<h/k) (hx:x<w/k) (t:ὦ) (ht:Ѥt<k*k):
 rdW X (((ch*h+y*k)*w+x*k).toUSize+(rdW (E_off k w) t).toUSize)=poolAtO h w k X ch y x Ѥt:=by
 have e0:=ὰ (ch*h+y*k) (Ѥt/k) w
 have h4:=qEe hc hy hx ht
 rw[qAo _ _ h2,qAo _ _ (by rwa[E_off,Array.size_ofFn]),E_off,qAb _ (λt=>(t/k*w+t%k).toUInt32) _ ht,poolAtO]
 generalize hu:Ѥt/k*w+Ѥt%k=u
 have e1:u.toUInt32.toUSize.toNat=u:=by
  rw[ὃ,Nat.toUInt32_eq,ὡ,Ἁ (by clear h2 h3 hc hy hx ht;omega)]
 rw[qAv _ _ (by rw[e1,qAa _ (by clear h2 h3 hc hy hx ht;omega)];(clear h2 h3 hc hy hx ht;omega)),e1,qAa _ (by clear h2 h3 hc hy hx ht;omega)]
 congr 1
 (clear h1 h2 h3 hc hy hx ht h4 e1;omega)
lemma qI9 {c h w k ch y x:Ὣ} {X:ѐ} (h1:c*h*w<=ѧX) (h2:ѧX<2^32) (h3:k*k<2^20) (hg:∀i,i< ѧX->good X[i]! =true) (hc:ch<c) (hy:y<h/k) (hx:x<w/k) (j:ὥ) (hj:j=255\/(j=2/\k*k=4)\/(j=6/\k*k=64)):
 E_win X (E_off k w) k w (k*k) j (E_div (k*k) (dotFn (k*k) (λ_=>0) (λ_=>oneW))) (dv oneW) (nfOf (k*k)) (dv 0) ((ch*h+y*k)*w+x*k).toUSize (k*k).toUSize 0 0 0 0=
 E_div (k*k) (dotOf (k*k) (poolAtO h w k X ch y x) (λ_=>oneW)):=by
 have hn:=qAa (k*k) (by exact Nat.lt_of_lt_of_le ‹_› ה)
 have hk:0<k:=ὼ (by rintro rfl;simp only[Nat.div_zero,not_lt_zero] at hy)
 have hI t (ht:t<k*k):=qEe hc hy hx ht
 have e2:=qAa ((ch*h+y*k)*w+x*k) (by have:=hI 0 (Nat.mul_pos hk hk);simp only[Nat.zero_div,add_zero,Nat.zero_mod] at this;(clear h3 hc hy hx hj hn hk;omega))
 rw[qF3 X _ k w (k*k) j _ _ _ _ _ (poolAtO h w k X ch y x) (λt ht=>qGm (by exact Nat.lt_of_le_of_lt ‹_› ‹_›) h2 (by exact Nat.lt_of_lt_of_le ‹_› ה) hc hy hx t (hn▸А.1 ht)) (k*k) 0 0 0 0 ((Ἇ _).trans hn.symm),USize.toNat_zero,<-ὗ]
 refine qHc X _ k w _ h3 j hj _ (λt ht=>hg _ (by have:=hI t ht;(exact Nat.lt_of_lt_of_le ‹_› ‹_›))) ?_
 rw[E_dot,qAg]
 congr 1
 funext t
 rw[e2,poolAtO]
 congr 1
 ring
lemma qWc (X off:ѐ) (k w n:Ὣ) (j z:ὥ) (u nf zb:ὧ) (hu:u=dv oneW) (b m:ὦ) (hx:∀t:ὦ,t< m->(rdW off t).toUSize=t)
 (d:Ὣ):∀(t:ὦ) (s a:ὧ) (o:ὥ),Ѥt+d=Ѥm->E_winC X k w n j z u nf zb b m t s a o=E_win X off k w n j z u nf zb b m t s a o:=by
 subst hu
 induction d with
 | zero=>
  intro t s a o h
  rw[E_winC,E_win,ἔ (by rw[А];exact omS404 h),ἔ (by rw[А];exact omS404 h)]
 | succ d ih=>
  intro t s a o h
  have ht:t< m:=А.2 (by exact omS136 h)
  rw[E_winC,E_win,Ἄ ht,Ἄ ht]
  dsimp only
  rw[hx t ht,D.qM1 _ (D.qDvF _),ih (t+1) _ _ _ (by rw[usz t m ht];exact omS137 h)]
lemma qOC (k:Ὣ) (hk:k*k<2^32) (t:ὦ) (ht:Ѥt<k*k):(rdW (E_off k k) t).toUSize=t:=by
 rw[qAo _ _ (by rwa[E_off,Array.size_ofFn]),E_off,qAb _ (λt=>(t/k*k+t%k).toUInt32) _ ht,Nat.div_add_mod']
 apply USize.toNat_inj.mp
 rw[ὃ,Nat.toUInt32_eq,ὡ,Ἁ (by exact omS201 hk ht)]
lemma qAX (X:ѐ) (w:Ὣ) (wu:ὦ) (u nf zb zr:ὧ) (hu:u=dv oneW) (W:Ὣ) (e:ὦ) (d:Ὣ): ∀(x:Ὣ) (b q:ὦ) (out:ѐ),W-x=d->x<=W-> Ѥq+(W-x)=Ѥe->
 A_x X w wu u nf zb zr b q e out=G_x (λb=>E_w4 X 2 w 4 2 0 u nf zb zr 0 1 wu (wu+1) b) 2 W x b q out:=by
 subst hu
 induction d with
 | zero=>
  intro x b q out h1 h2 h3
  rw[A_x,G_x,ἔ ט,ἐ (by exact omS411 h1)]
 | succ d ih=>
  intro x b q out h1 h2 h3
  have hq:q<e:=А.2 א
  rw[A_x,G_x,Ἄ hq,ἱ א,qW4u]
  exact ih (x+1) _ _ _ א א (by rw[usz q e hq];omega)
lemma qAY (X:ѐ) (w:Ὣ) (u nf zb zr:ὧ) (hu:u=dv oneW) (W H:Ὣ) (d:Ὣ): ∀(y:Ὣ) (b q:ὦ) (out:ѐ),H-y=d-> Ѥq+(H-y)*W<2^32->
 A_y X w ѩw u nf zb zr W H y b q out=G_y (λb=>E_w4 X 2 w 4 2 0 u nf zb zr 0 1 ѩw (ѩw+1) b) (2*w).toUSize 2 W H y b q out:=by
 induction d with
 | zero=>
  intro y b q out h1 h2
  rw[A_y,G_y,ἐ (by exact omS411 h1),ἐ (by exact omS411 h1)]
 | succ d ih=>
  intro y b q out h1 h2
  rw[h1,ᾀ] at h2
  rw[A_y,G_y,ἱ (by exact omS412 h1),ἱ (by exact omS412 h1)]
  have e1:b+ѩw+ѩw=b+(2*w).toUSize:=by rw[add_assoc,two_mul];simp only[Nat.toUSize_eq,USize.ofNat_add]
  have hW:Ѥ(q+ѩW)=Ѥq+W:=by rw[qAv _ _ (by rw[qAa _ א];omega),qAa _ א]
  rw[e1,qAX X w ѩw u nf zb zr hu W (q+ѩW) W 0 b q out א א (by rw[hW];omega)]
  exact ih (y+1) _ _ _ (by exact omS413 h1) (by rw[hW,show H-(y+1)=d by exact omS413 h1];omega)
lemma qAC (X:ѐ) (w:Ὣ) (u nf zb zr:ὧ) (hu:u=dv oneW) (sc:ὦ) (W H c:Ὣ) (d:Ὣ): ∀(ch:Ὣ) (b q:ὦ) (out:ѐ),c-ch=d-> Ѥq+(c-ch)*(H*W)<2^32->
 A_c X w ѩw u nf zb zr sc W H c ch b q out=G_c (λb=>E_w4 X 2 w 4 2 0 u nf zb zr 0 1 ѩw (ѩw+1) b) sc (2*w).toUSize 2 W H c ch b q out:=by
 induction d with
 | zero=>
  intro ch b q out h1 h2
  rw[A_c,G_c,ἐ (by exact omS411 h1),ἐ (by exact omS411 h1)]
 | succ d ih=>
  intro ch b q out h1 h2
  rw[h1,ᾀ] at h2
  rw[A_c,G_c,ἱ (by exact omS412 h1),ἱ (by exact omS412 h1)]
  have hHW:Ѥ(q+(H*W).toUSize)=Ѥq+H*W:=by rw[qAv _ _ (by rw[qAa _ א];omega),qAa _ א]
  rw[qAY X w u nf zb zr hu W H H 0 b q out א (by rw[Nat.sub_zero];omega)]
  exact ih (ch+1) _ _ _ (by exact omS413 h1) (by rw[hHW,show c-(ch+1)=d by exact omS413 h1];omega)
lemma qO2 (w t:Ὣ) (ht:t<4) (hw:t/2*w+t%2<2^32) (i:ὦ) (hi:Ѥi=t):(rdW (E_off 2 w) i).toUSize=ѩ(t/2*w+t%2):=by
 rw[qAo _ _ (by rw[E_off,Array.size_ofFn];norm_num),hi,E_off,qAb _ (λt=>(t/2*w+t%2).toUInt32) _ ht]
 apply USize.toNat_inj.mp
 rw[ὃ,Nat.toUInt32_eq,ὡ,Ἁ hw,qAa _ hw]
lemma qZ2 (c h w:Ὣ) (X:ѐ) (u nf zb zr:ὧ) (hu:u=dv oneW) (hN:c*(h/2*(w/2))<2^32) (hw:w<2^31):
 E_grid2 c h w X u nf zb zr=E_gridK c h w 2 (2*2) X (E_off 2 w) 2 0 u nf zb zr:=by
 have i1:Ѥ(0+1:ὦ)=1:=by rw[qAv _ _ (by rw[USize.toNat_zero,ὄ];norm_num),USize.toNat_zero,ὄ]
 have i2:Ѥ(0+1+1:ὦ)=2:=by rw[qAv _ _ (by rw[i1,ὄ];norm_num),i1,ὄ]
 have i3:Ѥ(0+1+1+1:ὦ)=3:=by rw[qAv _ _ (by rw[i2,ὄ];norm_num),i2,ὄ]
 have o0:(rdW (E_off 2 w) 0).toUSize=0:=by rw[qO2 w 0 ב (by exact omS333) 0 ה,show 0/2*w+0%2=0 by omega];decide
 have o1:(rdW (E_off 2 w) (0+1)).toUSize=1:=by rw[qO2 w 1 ב א (0+1) i1,show 1/2*w+1%2=1 by omega];simp only[Nat.toUSize_eq];rfl
 have o2:(rdW (E_off 2 w) (0+1+1)).toUSize=ѩw:=by rw[qO2 w 2 ב א (0+1+1) i2,show 2/2*w+2%2=w by omega]
 have o3:(rdW (E_off 2 w) (0+1+1+1)).toUSize=ѩw+1:=by
  rw[qO2 w 3 ב א (0+1+1+1) i3,show 3/2*w+3%2=w+1 by omega];simp only[Nat.toUSize_eq,USize.ofNat_add];rfl
 unfold E_grid2 E_gridK
 rw[qAC X w u nf zb zr hu _ (w/2) (h/2) c c 0 0 0 _ א (by simp only[USize.toNat_zero,tsub_zero,zero_add,Nat.reducePow];omega)]
 simp only[beq_self_eq_true,ite_true]
 rw[o0,o1,o2,o3]
 rfl
lemma qGX (c h w k:Ὣ) (X:ѐ) (j z:ὥ) (u nf zb zr:ὧ) (hu:u=dv oneW) (hk:k*k<2^32) (hj:k=2->j=2) (hz:k=2->z=0):
 E_gridX c h w k (k*k) X j z u nf zb zr=E_gridK c h w k (k*k) X (E_off k w) j z u nf zb zr:=by
 unfold E_gridX;split
 . rename_i e;subst e;rw[qEq];unfold E_gridC E_grid;congr 1;funext b
   exact qWc X (E_off k k) k k (k*k) j z u nf zb hu b ѩ(k*k) (λt ht=>qOC k hk t (by have:=А.mp ht;rwa[qAa _ hk] at this)) (k*k) 0 zr zr 0 (by rw[qAa _ hk];simp only[USize.toNat_zero,zero_add])
 . split
   . rename_i _ e;obtain ⟨rfl,h1,h2⟩:=e;rw[hj rfl,hz rfl];exact qZ2 c h w X u nf zb zr hu (Nat.lt_of_le_of_lt h2 (show 4294967295<2^32 by norm_num)) h1
   . rfl
lemma qG1:avgF=avgpoolSpecO:=by
 funext c h w k X;unfold avgF;rw[show D.fz0=(0:ὧ) from rfl];split
 . rename_i hc;obtain ⟨h1,h2,h3,h4⟩:=hc;replace h2:=Nat.lt_of_le_of_lt h2 (show 4294967295<2^32 by norm_num);simp only[avgpoolSpecO,avgpoolWithO]
   rw[show (if 0<k/\k<13 then (0:ὥ) else if k=14 then (0:ὥ) else E_div (k*k) (dotFn (k*k) (λ_=>0) (λ_=>oneW)))=E_div (k*k) (dotFn (k*k) (λ_=>0) (λ_=>oneW)) by
    split;. rename_i e;exact (qJo k e.2 e.1).symm
    . split;. rename_i e;subst e;exact qJo14.symm
      . rfl]
   rw[qGX _ _ _ _ _ _ _ _ _ _ _ rfl (by exact omS363 h3) (λe=>by subst e;rfl) (λe=>by subst e;exact qJo 2 ה ה),qEq,E_grid,qIl _ c h w k (by exact Nat.lt_of_le_of_lt ‹_› ‹_›)];apply congrArg Array.ofFn;funext ⟨q,hq⟩;obtain ⟨a1,a2,a3⟩:=qHz hq;dsimp only;refine qI9 h1 h2 h3 (qBh X h2 h4).2.2.1 a1 a2 a3 _ ?_;split_ifs with e1 e2;. exact Or.inr (Or.inl ⟨rfl,e1⟩)
   . exact Or.inr (Or.inr ⟨rfl,e2⟩)
   . exact Or.inl rfl
 . rfl
end
section
@[inline] def maxF (a b:Ὢ):Ὢ:=
 if a<У/\b<У then (if a<b then b else a) else maxW a b
def Mk (m:Ὣ):Ὣ:=if m/2^23=0 then m%2^23 else (2^23+m%2^23)*2^(m/2^23-1)
lemma Mk1:StrictMono Mk:=by
 refine strictMono_nat_of_lt_succ λm=>?_;unfold Mk;by_cases hc:m%2^23+1<2^23
 . rw[show (m+1)/2^23=m/2^23 by omega,show (m+1)%2^23=m%2^23+1 by omega];split;. (exact Nat.le_refl _)
   . exact Ἦ (by exact Nat.le_refl _) (Nat.two_pow_pos _)
 . rw[show (m+1)/2^23=m/2^23+1 by omega,show (m+1)%2^23=0 by omega,(א:m%2^23=2^23-1)];cases m/2^23 with
   | zero=>norm_num
   | succ e=>have:=Nat.two_pow_pos e;rw[ἐ (by clear hc this;omega),ἐ (by clear hc this;omega),Nat.add_sub_cancel,Nat.add_sub_cancel,pow_succ];(clear hc;omega)
lemma Mk2 (w:Ὢ) (h:ѥw<2^31):(expOf w).toNat=ѥw/2^23/\(fracOf w).toNat=ѥw%2^23/\signOf w=false:=by have a2:(0xFF:ὥ).toNat=2^8-1:=rfl;have a3:(0x7FFFFF:ὥ).toNat=2^23-1:=rfl;have a4:(23:ὥ).toNat%32=23:=rfl;have a5:(س:ὥ).toNat=2^31:=rfl;unfold expOf fracOf signOf;rw[Ἧ,Ἧ,UInt32.toNat_shiftRight,a2,a3,a4,Ἂ,Ἂ,Ἱ];refine ⟨by (clear a2 a3 a4 a5;omega),rfl,?_⟩;rw[bne_eq_false_iff_eq,<-ἠ,Ἧ,a5,Nat.and_two_pow,Nat.testBit_lt_two_pow h];rfl
lemma Mk3 (w:Ὢ) (h:ѥw<2^31):keyOf w=(Mk ѥw:Int):=by
 obtain ⟨h1,h2,h3⟩:=Mk2 w h;unfold keyOf;rw[ἐ (by simp only[h3,false_eq_true,not_false_eq_true])];congr 1;unfold mantOf idxOf Mk;rw[ὁ];by_cases e:expOf w=0;. rw[ἱ e,ἱ e,ἱ (by rw[<-h1,e];rfl),h2,pow_zero,Nat.mul_one]
 . rw[ἐ e,ἐ e,ἐ (λz=>e (ἠ.mp (by rw[h1,z];rfl))),h1,h2]
lemma Mk4 (a b:Ὢ) (h:ѥa<ѥb) (hb:ѥb<2^31):keyOf a<keyOf b:=by rw[Mk3 a א,Mk3 b hb];exact_mod_cast Mk1 h
lemma Mk5 (w:Ὢ) (h:w<У):expOf w≠255/\ѥw<2^31:=by have e:ѥw<У:=ἳ.mp h;refine ⟨λz=>?_,by (exact Nat.lt_of_lt_of_le ‹_› ה)⟩;have:=(Mk2 w (by exact Nat.lt_of_lt_of_le ‹_› ה)).1;rw[z] at this;have u:(255:ὥ).toNat=255:=rfl;(clear z;omega)
lemma Mk7 (a:Ὢ) (h:ѥa<2^31) (n:¬a=0) (hz:Model.isZero (ofW a)=true):False:=by obtain ⟨h1,h2,-⟩:=Mk2 a h;have e:ofW a=_:=ofUInt32_eq_ofFields a;rw[e,Model.isZero_ofFields Fmt _ _ _ (expOf_toNat_lt a) (fracOf_toNat_lt a)] at hz;have y:((expOf a).toNat==0&&(fracOf a).toNat==0)=true:=hz;simp only[and_eq_true,beq_iff_eq] at y;rw[h1,h2] at y;exact n (ἠ.mp (by rw[UInt32.toNat_zero];(clear h h1;(clear h2;omega))))
lemma Mk8 (a:Ὢ) (e:expOf a≠255) (h:ѥa<2^31) (n:¬a=0):maxW a a=a:=by unfold maxW;rw[ἐ (by simpa only[or_self,ne_eq] using e),compare_eq_iff_eq.mpr rfl];dsimp only;rw[ἐ (by simpa only[Bool.and_self,Bool.not_eq_true,imp_false] using Mk7 a h n)];exact qAw a
lemma qHi (a b:Ὢ):maxF a b=maxW a b:=by
 unfold maxF;split
 . rename_i h;obtain ⟨ha,hb⟩:=h;obtain ⟨e1,a1⟩:=Mk5 a ha;obtain ⟨e2,b1⟩:=Mk5 b hb;have d:¬(expOf a=255\/expOf b=255):=by simp only[e1,e2,or_self,not_false_eq_true]
   by_cases c:a<b;. rw[ἱ c,maxW,ἐ d,compare_lt_iff_lt.mpr (Mk4 a b (ἳ.mp c) b1)]
   . rw[ἐ c];by_cases c2:b<a;. rw[maxW,ἐ d,compare_gt_iff_gt.mpr (Mk4 b a (ἳ.mp c2) a1)]
     . have c3:=ἳ.not.mp c;have c4:=ἳ.not.mp c2;have y:a=b:=ἠ.mp (by clear e1 a1 e2 b1 d;omega);subst y;by_cases z:a=0;. rw[z];decide
       . exact (Mk8 a e1 a1 z).symm
 . rfl
def mpWin (X off:ѐ) (b m t:ὦ) (acc:ὥ):ὥ:=
 if h:t< m then mpWin X off b m (t+1) (maxF acc (rdW X (b+(rdW off t).toUSize))) else acc
partial_fixpoint
def maxpoolF (c h w k:Ὣ) (inp:ё):ё:=
 if c*h*w<=4294967295/\ѧinp<=4294967295/\k*k<=4294967295 then
  let off:=E_off k w
  G_c (λb=>mpWin inp off b (k*k).toUSize 0 (rdW inp b)) (h*w).toUSize (k*w).toUSize
   ѩk (w/k) (h/k) c 0 0 0 (ἇ (c*(h/k*(w/k))) 0)
 else maxpoolW c h w k inp
lemma qFv (X off:ѐ) (b m:ὦ) (xw:Ὣ->ὥ) (hx:∀t:ὦ,t< m->rdW X (b+(rdW off t).toUSize)=xw Ѥt) (d:Ὣ):
 ∀(t:ὦ) (a:ὥ),Ѥt+d=Ѥm->mpWin X off b m t a=(List.range' Ѥt d).foldl (λa i=>maxF a (xw i)) a:=by
 induction d with
 | zero=>
  intro t a h
  rw[mpWin,ἔ (by rw[А];exact omS404 h)]
  rfl
 | succ d ih=>
  intro t a h
  have ht:t< m:=by rw[А];exact omS136 h
  rw[mpWin,Ἄ ht,hx t ht,ih (t+1) _ (by rw[usz t m ht];(clear ht;exact omS137 h)),usz t m ht,List.range'_succ,ὔ]
@[inline] def mp4 (X:ѐ) (w b:ὦ):ὥ:=
 let v0:=D.rdW X b
 maxF (maxF (maxF (maxF v0 v0) (D.rdW X (b+1))) (D.rdW X (b+w))) (D.rdW X (b+w+1))
lemma rdOff (w k:Nat) (hk:k<4):D.rdW (E_off 2 w) k.toUSize=(k/2*w+k%2).toUInt32:=by
 have hs:(E_off 2 w).size=4:=by simp only[E_off,Nat.reduceMul,Nat.toUInt32_eq,UInt32.ofNat_add,UInt32.ofNat_mul,Array.size_ofFn]
 have hku:(k.toUSize).toNat=k:=D.uk k א
 have hlt:k.toUSize<(E_off 2 w).usize:=by
  rw[А,hku];simp only[Array.usize,hs];rw[D.uk 4 ב];exact hk
 unfold D.rdW;rw[dif_pos hlt]
 simp only[Array.uget,E_off,Nat.reduceMul,Nat.toUInt32_eq,UInt32.ofNat_add,UInt32.ofNat_mul,Nat.toUSize_eq,hku,Array.getElem_ofFn]
lemma mp4_eq (X:ѐ) (w:Nat) (hw:w<2^31) (b:ὦ):
  mp4 X w.toUSize b=mpWin X (E_off 2 w) b (2*2:Nat).toUSize 0 (D.rdW X b):=by
 have hm:USize.toNat (2*2:Nat).toUSize=4:=D.uk 4 ב
 rw[qFv X (E_off 2 w) b (2*2:Nat).toUSize (λj=>D.rdW X (b+(D.rdW (E_off 2 w) j.toUSize).toUSize))
  (λt _=>by simp only[USize.ofNat_toNat]) 4 0 (D.rdW X b) (by rw[hm];rfl)]
 have o0:D.rdW (E_off 2 w) (0:Nat).toUSize=0:=by rw[rdOff w 0 ב];simp only[Nat.zero_div,zero_mul,Nat.zero_mod,add_zero,Nat.toUInt32_eq,UInt32.reduceOfNat]
 have o1:D.rdW (E_off 2 w) (1:Nat).toUSize=1:=by rw[rdOff w 1 ב];simp only[Nat.reduceDiv,zero_mul,Nat.mod_succ,zero_add,Nat.toUInt32_eq,UInt32.ofNat_one]
 have o2:D.rdW (E_off 2 w) (2:Nat).toUSize=w.toUInt32:=by rw[rdOff w 2 ב];simp only[Order.lt_two_iff,zero_le,Nat.div_self,one_mul,Nat.mod_self,add_zero,Nat.toUInt32_eq]
 have o3:D.rdW (E_off 2 w) (3:Nat).toUSize=(w+1).toUInt32:=by rw[rdOff w 3 ב];simp only[Nat.reduceDiv,one_mul,Nat.reduceMod,Nat.toUInt32_eq,UInt32.ofNat_add,UInt32.ofNat_one]
 have e2:w.toUInt32.toUSize=w.toUSize:=by
  apply USize.toNat_inj.mp;rw[ὃ,ὡ,D.uk w א];omega
 have e3:(w+1).toUInt32.toUSize=w.toUSize+1:=by
  apply USize.toNat_inj.mp;rw[ὃ,ὡ,USize.toNat_add,D.uk w א,ὄ]
  have:2^32<=USize.size:=USize.le_size
  have h1:(w+1)%2^32=w+1:=Ἁ א
  have h2:(w+1)%USize.size=w+1:=Ἁ א
  simp only[h1,h2]
 have z0:USize.toNat 0=0:=rfl
 have u0:(0:ὥ).toUSize=0:=rfl
 have u1:(1:ὥ).toUSize=1:=by apply USize.toNat_inj.mp;rw[ὃ,ὄ];rfl
 simp only[List.range',List.foldl,mp4,z0,Nat.reduceAdd,o0,o1,o2,o3,e2,e3,u0,u1,add_zero,USize.add_assoc]
def maxpoolF2 (c h w k:Nat) (inp:ѐ):ѐ:=
 if k=2/\w<2^31/\c*h*w<=4294967295/\inp.size<=4294967295 then
  G_c (λb=>mp4 inp w.toUSize b) (h*w).toUSize (2*w).toUSize (2:Nat).toUSize (w/2) (h/2) c 0 0 0
   (ἇ (c*(h/2*(w/2))) 0)
 else maxpoolF c h w k inp
lemma maxpoolF2_eq:maxpoolF2=maxpoolF:=by
 funext c h w k inp
 unfold maxpoolF2
 split
 · rename_i hc
   obtain ⟨rfl,hw,h1,h2⟩:=hc
   unfold maxpoolF
   rw[Ὦ ⟨h1,h2,ב⟩]
   dsimp only
   simp only[mp4_eq inp w hw]
 · rfl
lemma qDp:maxpoolF=maxpoolSpecO:=by
 rw[<-qDn];funext c h w k inp;unfold maxpoolF;split
 . rename_i hc;obtain ⟨h1,h2,h3⟩:=hc;replace h1:=Nat.lt_of_le_of_lt h1 (show 4294967295<2^32 by norm_num);replace h2:=Nat.lt_of_le_of_lt h2 (show 4294967295<2^32 by norm_num);replace h3:=Nat.lt_of_le_of_lt h3 (show 4294967295<2^32 by norm_num);rw[qIl _ c h w k h1];simp only[maxpoolW];apply congrArg Array.ofFn;funext ⟨q,hq⟩;obtain ⟨a1,a2,a3⟩:=qHz hq;dsimp only;have hn:(k*k).toUSize.toNat=k*k:=qAa _ h3
   have e:∀{ch y x:Ὣ},ch<c->y<h/k->x<w/k->rdW inp ((ch*h+y*k)*w+x*k).toUSize=poolAtO h w k inp ch y x 0:=by
    intro ch y x b1 b2 b3
    have h4:=qEe (t:=0) b1 b2 b3 (by
     rcases Nat.eq_zero_or_pos k with rfl | h0;. simp only[Nat.div_zero,not_lt_zero] at b2
     . exact Nat.mul_pos h0 h0)
    simp only[Nat.zero_div,Nat.zero_mod,Ἵ] at h4;rw[qAo _ _ h2,qAa _ (by clear h2 h3 hq a1 a2 a3 hn b1 b2 b3;exact omS279 h1 h4)];simp only[poolAtO,Nat.zero_div,add_zero,Nat.zero_mod]
   rw[e a1 a2 a3,qFv inp _ _ _ (poolAtO h w k inp _ _ _) (λt ht=>qGm h1 h2 h3 a1 a2 a3 t (by rw[А,hn] at ht;exact ht)) (k*k) 0 _ (by rw[hn,show (0:ὦ).toNat=0 from rfl,Ἇ]),show (0:ὦ).toNat=0 from rfl,<-ὗ];simp only[qHi]
 . rfl
end
section
section
variable {z:ℝ} (hz0:0<=z) (hz:z<=1/2^200*(1/2^200))
include hz0 hz
end
lemma qHo (a b:ѐ) (hs:ѧa=ѧb) (hf:∀j,j< ѧa->expOf a[j]!≠255/\expOf b[j]!≠255) (hz:(dsum a b 0 ѧa:ℚ)*2^(-298:Int)=0):dotSpec a b=toWord (Model.zero Fmt (decide (ѧa≠0)&&dallNZ a b 0 ѧa)):=by rw[<-qEk a b hs.symm hf,show dsum a b 0 ѧa=0 by simpa only[Int.reduceNeg,zpow_neg,zpow_ofNat,mul_eq_zero,Int.cast_eq_zero,inv_eq_zero,ne_eq,OfNat.ofNat_ne_zero,not_false_eq_true,pow_eq_zero_iff,or_false] using hz,finishWord_zero]
lemma qD6 (n:Ὣ) (a b:Ὣ->ὥ) (hf:∀t,t<n->expOf (a t)≠255/\expOf (b t)≠255) (hS:∑ j∈Ἣ n,ἂ (a j)*ἂ (b j)=0):
 dotOf n a b=toWord (Model.zero Fmt (decide (n≠0)&&dallNZ (Array.ofFn (n:=n) λi=>a i.val) (Array.ofFn (n:=n) λi=>b i.val) 0 n)):=by
 have gA:=qAb n a
 have gB:=qAb n b
 have s:(Array.ofFn (n:=n) λi=>a i.val).size=n:=by simp only[Array.size_ofFn]
 have e:=qHo (Array.ofFn (n:=n) λi=>a i.val) (Array.ofFn (n:=n) λi=>b i.val) ג (λj hj=>by rw[s] at hj;rw[gA j hj,gB j hj];exact hf j hj)
  (by
   rw[s,qE8,<-hS]
   refine Ἷ rfl λj hj=>?_
   have h:=Ὁ.mp hj
   rw[Ἇ,gA j h,gB j h])
 rw[s] at e
 exact e
lemma qJw (a b:ѐ) (r i:Ὣ):
 dallNZ a b i (r+1)=(dallNZ a b i r&&(decide (mantOf a[i+r]!*mantOf b[i+r]! =0)&&Bool.xor (signOf a[i+r]!) (signOf b[i+r]!))):=by
 induction r generalizing i with
 | zero=>simp only[dallNZ,mul_eq_zero,Bool.decide_or,Bool.and_true,add_zero,Bool.true_and]
 | succ r ih=>
  rw[dallNZ,ih (i+1),dallNZ]
  simp only[Bool.and_assoc,show i+1+r=i+(r+1) by exact omS146]
lemma qIp (a b u v:ѐ) (r i:Ὣ) (h:∀j,i<=j->j< i+r->a[j]! =u[j]!/\b[j]! =v[j]!):dallNZ a b i r=dallNZ u v i r:=by
 induction r generalizing i with
 | zero=>rfl
 | succ r ih=>rw[dallNZ,dallNZ,ih (i+1) (λj h1 h2=>h j (by exact Nat.le_of_lt ‹_›) (by clear h1;omega)),(h i (by exact Nat.le_refl _) א).1,(h i (by exact Nat.le_refl _) א).2]
lemma qI0 (k:Ὣ) (a b:Ὣ->ὥ) (hk:0<k) (hf:∀t,t<k->expOf (a t)≠255/\expOf (b t)≠255):
 dotOf (k+1) (λt=>if t<k then a t else oneW) (λt=>if t<k then b t else س)=dotOf k a b:=by
 have hf':∀j,j<k+1->expOf (if j<k then a j else oneW)≠255/\expOf (if j<k then b j else س)≠255:=λj _=>by
  by_cases h:j<k
  . simpa only[h, ↓reduceIte,ne_eq] using hf j h
  . simp only[h,ite_false];exact ⟨ה,ה⟩
 have hsum:(∑ j∈Ἣ (k+1),ἂ (if j<k then a j else oneW)*ἂ (if j<k then b j else س))=∑ j∈Ἣ k,ἂ (a j)*ἂ (b j):=by
  rw[Finset.sum_range_succ]
  simp only[lt_self_iff_false,ite_false,qJk س ה ה,mul_zero,add_zero]
  exact Ἷ rfl λj hj=>by simp only[Finset.mem_range.mp hj, ↓reduceIte]
 by_cases hS:(∑ j∈Ἣ k,ἂ (a j)*ἂ (b j))=0
 . have g1:=qAb (k+1) λt=>if t<k then a t else oneW;have g2:=qAb (k+1) λt=>if t<k then b t else س
   rw[qD6 _ _ _ hf' (hsum.trans hS),qD6 k a b hf hS,qJw,Ἇ,g1 k (by exact Nat.le_refl _),g2 k (by exact Nat.le_refl _),
    qIp _ _ (Array.ofFn (n:=k) λi=>a i.val) (Array.ofFn (n:=k) λi=>b i.val) k 0 (λj _ h2=>by
     rw[g1 j א,g2 j א,qAb k a j א,qAb k b j א]
     simp only[show j<k by omega, ↓reduceIte,and_self])]
   simp only[ne_eq,Nat.add_eq_zero_iff,hk.ne',one_ne_zero,and_self,not_false_eq_true,decide_true,lt_self_iff_false, ↓reduceIte,show mantOf (س:ὥ)=0 by decide,mul_zero,show signOf oneW=false by decide,show signOf (س:ὥ)=true by decide,Bool.bne_true,Bool.not_false,Bool.and_self,Bool.and_true,Bool.true_and,FloatFormat.isIEEE_binary32,Model.zero_eq_signedZero_of_isIEEE]
 . obtain ⟨d,hd,hw⟩:=qBa k a b hf hS;obtain ⟨e,he,hw'⟩:=qBa (k+1) _ _ hf' (by rw[hsum];exact hS);have t:=@toRat_eq_zero_iff;have h3:e.toRat=d.toRat:=by rw[he,hd,hsum]
   rw[hw,hw',roundDyadic_congr Fmt ה e d (λh=>hS (hd.symm.trans (h3.symm.trans ((t _).2 h)))) (λh=>hS (hd.symm.trans ((t _).2 h))) h3]
lemma qD9 (A:ѐ) (h:A.all finiteWord=true) (t:Ὣ):expOf A[t]!≠255:=by
 by_cases ht:t< ѧA;. simpa only[ne_eq,finiteWord,bne_iff_ne] using qI1 A _ h t ht
 . rw[ὑ A t ht];decide
end
def linRg (inW m outW:Ὣ) (relu:Ὡ) (X W b:ё) (Dw:ὤ) (hd:ѨDw=ѧW) (q:ὥ) (nf bm:ὧ) (n:ὦ):Ὣ->Ὣ->ё->ὤ->D.CN
 | 0,_,out,_=>⟨out,true⟩
 | k+1,r0,out,xs=>
   let d:=D.dec8oC X (r0*inW).toUSize n xs
   let c:=D.cols8ZDA Dw (D.hvI m) inW outW relu X W b hd d nf n r0 (r0*outW).toUSize 0 out q (D.mkL8 nf (D.dv (D.hvI m)) bm d) ⟨0,0,0,0,0,0,0,0,0,0,0⟩
   if c.ok then linRg inW m outW relu X W b Dw hd q nf bm n k (r0+8) c.out d.xs else ⟨c.out,false⟩
def linR (rows inW outW:Ὣ) (relu:Ὡ) (X W b:ё):ё:=
 if rows*inW=ѧX/\outW*inW=ѧW/\outW=ѧb/\0< inW/\0<outW/\inW<2^20/\
  ѧX<=4294967295/\ѧW<=4294967295/\ѧb<=4294967295/\rows*outW<=4294967295/\
  D.scanH 0 X≠Е/\D.scanH 1 b≠Е/\8<=rows then
  let n:=ѩinW
  let nf:ὧ:=(inW.toUInt64+8).toFloat*Ἃ 0x3CA0000000000000
  let p:=(D.wd8Get inW W).1
  if hd:p.1.size=ѧW then
   let bm:=D.dv (D.scanH 1 b)
   let c:=linRg inW p.2.2 outW relu X W b p.1 hd p.2.1 nf bm n (rows/8) 0 (ἇ (rows*outW) 0) (D.zFor (8*inW+1))
   if c.ok then
    if rows%8=0 then c.out
    else if 1<=rows%8 then
     let c2:=linRg inW p.2.2 outW relu X W b p.1 hd p.2.1 nf bm n 1 (rows-8) c.out (D.zFor (8*inW+1))
     if c2.ok then c2.out else D.linF rows inW outW relu X W b
    else D.linF rows inW outW relu X W b
   else D.linF rows inW outW relu X W b
  else D.linF rows inW outW relu X W b
 else D.linF rows inW outW relu X W b
def linG (rows inW outW:Ὣ) (relu:Ὡ) (X W b:ё):ё:=
 if rows<=1 then D.linF rows inW outW relu X W b
 else if ѧX=rows*inW/\rows*outW+8*outW<=4294967295/\rows*inW+8*inW<=4294967295 then
  linR rows inW outW relu X W b
 else D.linF rows inW outW relu X W b
def linSp (inW outW:Ὣ) (relu:Ὡ) (X W b:ё) (q:Ὣ):Ὢ:=
 act relu (dotOf (inW+1) (λt=>if t< inW then X[q/outW*inW+t]! else oneW)
  (λt=>if t< inW then W[q%outW*inW+t]! else b[q%outW]!))
lemma qRgE {m:Ὣ} (inW outW:Ὣ) (relu:Ὡ) (X W b:ё) (HD:(D.dnOf (D.tZ m) (D.qTs (D.qTZ m)) W).size=ѧW) (o:ὦ) (xs:ὤ) (n:ὦ) (hn:Ѥn=inW) (hW:outW*inW=ѧW) (hs:ѧW<2^32) (pi:0< inW) (ri:Ὣ) (ob:ὦ) (out:ё) (th:D.L8) (r0:D.F8):
 D.cols8ZDA (D.dnOf (D.tZ m) (D.qTs (D.qTZ m)) W) (D.hvI m) inW outW relu X W b HD (D.dec8o X o n 0 xs 0 0 0 0 0 0 0 0) (D.nfOf inW) n ri ob 0 out (D.qScan W W.usize 0 Е) th r0
  =D.cols8Zo (D.tZ m) (D.qTs (D.qTZ m)) (D.hvI m) inW outW relu X W b (D.dec8o X o n 0 xs 0 0 0 0 0 0 0 0) (D.nfOf inW) n ri ob 0 out Е th r0:=by
 rw[D.mcA (HX:=D.mGX X o n xs) (HD:=D.mFD (D.qTs (D.qTZ m)) (D.qTZ m) W)]
 have e:=D.c8ZSo (D.tZ m) (D.qTs (D.qTZ m)) (D.hvI m) inW outW relu X W b (D.dec8o X o n 0 xs 0 0 0 0 0 0 0 0) (D.nfOf inW) n ri ob hn hW hs pi out th r0
 rw[Ἄ HD] at e;exact e
lemma qRg {rows inW outW m:Ὣ} {relu:Ὡ} {X W b:ё} (G:D.C_G rows inW outW X W b) (HD:(D.dnOf (D.tZ m) (D.qTs (D.qTZ m)) W).size=ѧW) (bm:ὧ) (fbm:bmᶠ) (hbm:∀j,j<outW-> |R (D.dv b[j]!)| <=R bm) (bb:R bm<=2^(290:ℤ)):
 ∀k r0 (out:ё) (xs:ὤ),8*inW+1<=xs.size->xs.size<2^32->r0+8*k<=rows->D.C_Inv (rows*outW) (D.C_sp inW outW relu X W b) (λq=>q<r0*outW) out->
  (linRg inW m outW relu X W b (D.dnOf (D.tZ m) (D.qTs (D.qTZ m)) W) HD (D.qScan W W.usize 0 Е) (D.nfOf inW) bm (ѩinW) k r0 out xs).ok=true->
  D.C_Inv (rows*outW) (D.C_sp inW outW relu X W b) (λq=>q<(r0+8*k)*outW) (linRg inW m outW relu X W b (D.dnOf (D.tZ m) (D.qTs (D.qTZ m)) W) HD (D.qScan W W.usize 0 Е) (D.nfOf inW) bm (ѩinW) k r0 out xs).out:=by
 intro k
 induction k with
 | zero=>
  intro r0 out xs _ _ _ H _
  simp only[linRg,Nat.mul_zero,Ἵ];exact H
 | succ k ih=>
  intro r0 out xs hx1 hx2 hk H ok
  have hni:=G.ni
  have hpi:=G.pi
  have hpo:=G.po
  have hr8:r0+8<=rows:=by (clear hni hpi hpo;omega)
  have nm:inW.toUSize.toNat=inW:=D.qAd _ (by (exact Nat.lt_of_lt_of_le ‹_› ה))
  have hX32:ѧX<2^32:=G.sX
  have hrO:(r0+8)*outW<=rows*outW:=Ἒ _ hr8
  have hrI:(r0+8)*inW<=rows*inW:=Ἒ _ hr8
  have hRO:=G.sO
  have hRI:rows*inW<2^32:=by rw[G.hX];exact hX32
  have e8:(r0+8)*outW=r0*outW+8*outW:=ὰ _ _ _
  have e8i:(r0+8)*inW=r0*inW+8*inW:=ὰ _ _ _
  have hob:((r0*outW).toUSize).toNat=r0*outW:=D.qAd _ (by (clear hk hni hpi hr8 nm hX32 hrI hRI e8i;omega))
  have ho:((r0*inW).toUSize).toNat=r0*(ѩinW).toNat:=by rw[nm];exact D.qAd _ (by (clear hk hpo hr8 hX32 hrO hRO e8 hob;omega))
  have h8X:(r0+8)*(ѩinW).toNat<=ѧX:=by rw[nm,<-G.hX];exact hrI
  have P0:8*inW+1<=xs.size/\xs.size<2^32:=⟨hx1,hx2⟩
  have D0:D.C_D8o X r0 (ѩinW).toNat (0:ὦ).toNat ⟨xs,0,0,0,0,0,0,0,0⟩:=by
   rw[nm,D.C_z];exact ⟨P0,λk r hk=>absurd hk (Ἲ _),λr hr=>by interval_cases r<;>rfl⟩
  have Dd:=D.qGqo hX32 ho h8X (ѩinW).toNat 0 _ 0 0 0 0 0 0 0 0 (by rw[D.C_z,Ἇ]) D0
  rw[nm] at Dd
  simp only[linRg] at ok⊢
  simp only[D.dec8oC_eq _ _ _ _ hX32] at ok⊢
  rw[qRgE inW outW relu X W b HD _ _ _ nm G.hW G.sW hpi] at ok⊢
  by_cases hc:(D.cols8Zo (D.tZ m) (D.qTs (D.qTZ m)) (D.hvI m) inW outW relu X W b (D.dec8o X (r0*inW).toUSize (ѩinW) 0 xs 0 0 0 0 0 0 0 0) (D.nfOf inW) (ѩinW) r0 (r0*outW).toUSize 0 out Е (D.mkL8 (D.nfOf inW) (D.dv (D.hvI m)) bm (D.dec8o X (r0*inW).toUSize (ѩinW) 0 xs 0 0 0 0 0 0 0 0)) ⟨0,0,0,0,0,0,0,0,0,0,0⟩).ok=true
  . rw[ἱ hc] at ok⊢
    obtain ⟨-,-,A2⟩:=D.qZ8o (D.qTZ m) hob hr8 G.hW hpi hni G.sW Dd.1 _ bm fbm hbm bb (λr hr=>by interval_cases r<;>rfl) outW 0 _ _ _ (Ἇ _) hc
    have H1:=A2 G Dd (D.qA5 H (λq _ hq=>by unfold D.C_B at hq;rcases hq with h|⟨_,_,h⟩;exact h;exact absurd h (Ἲ _)))
    have H2:D.C_Inv (rows*outW) (D.C_sp inW outW relu X W b) (λq=>q<(r0+8)*outW) _:=D.qA5 H1 (λq _ hq=>by
     unfold D.C_B;by_cases e:q<r0*outW;. exact Or.inl e
     . exact Or.inr ⟨א,hq,ὀ _ hpo⟩)
    have R:=ih (r0+8) _ _ Dd.1.1 Dd.1.2 (by (clear hni hpi hpo hr8 nm hX32 hrO hrI hRO hRI e8 e8i hob ho h8X P0;omega)) H2 ok
    have e:r0+8*(k+1)=r0+8+8*k:=by (clear hk hni hpi hpo hr8 nm hX32 hrO hrI hRO hRI e8 e8i hob ho h8X P0;omega)
    rw[e];exact R
  . rw[ἐ hc] at ok;exact absurd ok (by simp only[false_eq_true,not_false_eq_true])
lemma qRg1 {inW outW m:Ὣ} {relu:Ὡ} {X W b:ё} (HD:(D.dnOf (D.tZ m) (D.qTs (D.qTZ m)) W).size=ѧW) (hW:outW*inW=ѧW) (hs:ѧW<2^32) (pi:0< inW) (nm:inW.toUSize.toNat=inW) (bm:ὧ) (k r0:Ὣ) (out:ё) (xs:ὤ) (hX32:ѧX<2^32)
 (ok:(linRg inW m outW relu X W b (D.dnOf (D.tZ m) (D.qTs (D.qTZ m)) W) HD (D.qScan W W.usize 0 Е) (D.nfOf inW) bm (ѩinW) (k+1) r0 out xs).ok=true):
 (D.cols8Zo (D.tZ m) (D.qTs (D.qTZ m)) (D.hvI m) inW outW relu X W b (D.dec8o X (r0*inW).toUSize (ѩinW) 0 xs 0 0 0 0 0 0 0 0) (D.nfOf inW) (ѩinW) r0 (r0*outW).toUSize 0 out Е (D.mkL8 (D.nfOf inW) (D.dv (D.hvI m)) bm (D.dec8o X (r0*inW).toUSize (ѩinW) 0 xs 0 0 0 0 0 0 0 0)) ⟨0,0,0,0,0,0,0,0,0,0,0⟩).ok=true:=by
 simp only[linRg] at ok
 simp only[D.dec8oC_eq _ _ _ _ hX32] at ok
 rw[qRgE inW outW relu X W b HD _ _ _ nm hW hs pi] at ok
 by_cases hc:(D.cols8Zo (D.tZ m) (D.qTs (D.qTZ m)) (D.hvI m) inW outW relu X W b (D.dec8o X (r0*inW).toUSize (ѩinW) 0 xs 0 0 0 0 0 0 0 0) (D.nfOf inW) (ѩinW) r0 (r0*outW).toUSize 0 out Е (D.mkL8 (D.nfOf inW) (D.dv (D.hvI m)) bm (D.dec8o X (r0*inW).toUSize (ѩinW) 0 xs 0 0 0 0 0 0 0 0)) ⟨0,0,0,0,0,0,0,0,0,0,0⟩).ok=true
 . exact hc
 . rw[ἐ hc] at ok;exact absurd ok (by simp only[false_eq_true,not_false_eq_true])
lemma qRR (rows inW outW:Ὣ) (relu:Ὡ) (X W b:ё) (hX:ѧX=rows*inW) (hN:rows*outW+8*outW<2^32) (hNi:rows*inW+8*inW<2^32):
 linR rows inW outW relu X W b=Array.ofFn (n:=rows*outW) λq=>linSp inW outW relu X W b q.val:=by
 have FB:D.linF rows inW outW relu X W b=Array.ofFn (n:=rows*outW) λq=>linSp inW outW relu X W b q.val:=by rw[D.qDz];rfl
 unfold linR;dsimp only
 by_cases h:rows*inW=ѧX/\outW*inW=ѧW/\outW=ѧb/\0< inW/\0<outW/\inW<2^20/\ѧX<=4294967295/\ѧW<=4294967295/\ѧb<=4294967295/\rows*outW<=4294967295/\D.scanH 0 X≠Е/\D.scanH 1 b≠Е/\8<=rows
 . rw[ἱ h];obtain ⟨h1,h2,h3,h4,h5,h6,h7,h8,h9,h10,h11,h13,hr8⟩:=h;replace h7:=D.qA9 h7;replace h8:=D.qA9 h8;replace h9:=D.qA9 h9;replace h10:=D.qA9 h10
   have hnf:(inW.toUInt64+8).toFloat*Ἃ 0x3CA0000000000000=D.nfOf inW:=rfl
   have nm:inW.toUSize.toNat=inW:=D.qAd _ (by (exact Nat.lt_of_lt_of_le ‹_› ה))
   have e:linSp inW outW relu X W b=D.C_sp inW outW relu X W b:=rfl
   rw[hnf,(D.wd8Get inW W).2];dsimp only;by_cases HD:(D.dnOf (D.tZ (D.nE inW W)) (D.qTs (D.qTZ (D.nE inW W))) W).size=ѧW
   . rw[Ἄ HD]
     by_cases hok:(linRg inW (D.nE inW W) outW relu X W b (D.dnOf (D.tZ (D.nE inW W)) (D.qTs (D.qTZ (D.nE inW W))) W) HD (D.qScan W W.usize 0 Е) (D.nfOf inW) (D.dv (D.scanH 1 b)) (ѩinW) (rows/8) 0 (ἇ (rows*outW) 0) (D.zFor (8*inW+1))).ok=true
     . rw[ἱ hok]
       have hk:rows/8=(rows/8-1)+1:=by (clear hX hN hNi h1 h2 h3 h4 h5 h6 h11 h13 h7 h8 h9 h10 nm;omega)
       have ok1:=hok;rw[hk] at ok1
       have c1:=qRg1 HD h2 h8 h4 nm _ (rows/8-1) 0 _ (D.zFor (8*inW+1)) h7 ok1
       have hob0:((0*outW).toUSize).toNat=0*outW:=D.qAd _ (by (clear hX hN hNi h1 h2 h3 h4 h5 h6 h11 h13 hr8 h7 h8 h9 h10 nm hk;exact omS333))
       have P0:=D.qZf (8*inW+1) (by (clear hX hN hNi h1 h2 h3 h5 h11 h13 hr8 h7 h8 h9 h10 hk hob0;omega))
       have ho0:((0*inW).toUSize).toNat=0*(ѩinW).toNat:=by rw[nm];exact D.qAd _ (by (clear hX hN hNi h1 h2 h3 h4 h5 h6 h11 h13 hr8 h7 h8 h9 h10 nm hk hob0 P0;exact omS333))
       have h8X0:(0+8)*(ѩinW).toNat<=ѧX:=by rw[nm,<-h1];exact Ἒ _ hr8
       have D00:D.C_D8o X 0 (ѩinW).toNat (0:ὦ).toNat ⟨D.zFor (8*inW+1),0,0,0,0,0,0,0,0⟩:=by
        rw[nm,D.C_z];exact ⟨P0,λk r hk=>absurd hk (Ἲ _),λr hr=>by interval_cases r<;>rfl⟩
       have Dd0:=D.qGqo h7 ho0 h8X0 (ѩinW).toNat 0 _ 0 0 0 0 0 0 0 0 (by rw[D.C_z,Ἇ]) D00
       rw[nm] at Dd0
       obtain ⟨gb1,gb2,-,gb4⟩:=D.qA8 1 b h9 h13
       obtain ⟨pb0,pb1,pb2⟩:=D.qCs gb1 gb2
       obtain ⟨fbm,Rbm⟩:=D.qAm _ gb1
       have hbm:∀j,j<outW-> |R (D.dv b[j]!)| <=R (D.dv (D.scanH 1 b)):=λj hj=>by
        have gj:=(D.qA8 1 b h9 h13).2.2.1 j (by rw[<-h3];exact hj)
        rw[(D.qAm _ gj).2,Rbm];exact pb2 _ gj (gb4 j (by rw[<-h3];exact hj))
       have bb:R (D.dv (D.scanH 1 b))<=2^(290:ℤ):=by rw[Rbm];exact pb1.trans (by rw[show (2:ℝ)^128=2^(128:ℤ) by norm_num];exact ἒ ב ב)
       obtain ⟨-,A1,-⟩:=D.qZ8o (R:=rows) (D.qTZ (D.nE inW W)) hob0 (by (exact Nat.succ_le_of_lt ‹_›)) h2 h4 h6 h8 Dd0.1 _ (D.dv (D.scanH 1 b)) fbm hbm bb (λr hr=>by interval_cases r<;>rfl) outW 0 _ _ _ (Ἇ _) c1
       have A:∀q,q< ѧW->D.good W[q]! =true:=λq hq=>by
        have e1:=(A1 (q/inW) (Nat.zero_le _) (Ἰ (by rw[ἥ,h2];exact hq)) (q%inW) (ὀ _ h4)).1
        rwa[Nat.div_add_mod' q inW] at e1
       have G:D.C_G rows inW outW X W b:=⟨h1,h2,h3,h4,h5,h6,h7,h8,h9,h10,(D.qA8 0 X h7 h11).2.2.1,A,(D.qA8 1 b h9 h13).2.2.1⟩
       have R1:=qRg G HD (D.dv (D.scanH 1 b)) fbm hbm bb (rows/8) 0 _ (D.zFor (8*inW+1)) P0.1 P0.2 (by (clear hX hN hNi h1 h2 h3 h4 h5 h6 h11 h13 hr8 h7 h8 h9 h10 nm hob0 P0 ho0 h8X0;omega)) ⟨by simp only[Array.size_replicate],λq _ hq=>absurd hq (by simp only[zero_mul,not_lt_zero,not_false_eq_true])⟩ hok
       rw[Ἇ] at R1
       rw[e]
       by_cases hm:rows%8=0
       . rw[ἱ hm];have e2:8*(rows/8)=rows:=by (clear hX hN hNi h1 h2 h3 h4 h5 h6 h11 h13 hr8 h7 h8 h9 h10 nm hob0 P0 ho0 h8X0;omega)
         rw[e2] at R1;exact D.qBv R1 (λq hq=>hq)
       . rw[ἐ hm];by_cases h5:1<=rows%8
         . rw[ἱ h5]
           split
           . rename_i hok2
             have R2:=qRg G HD (D.dv (D.scanH 1 b)) fbm hbm bb 1 (rows-8) _ (D.zFor (8*inW+1)) P0.1 P0.2 א (D.qA5 R1 (λq _ hq=>by have:=Ἒ outW (show rows-8<=8*(rows/8) by omega);(exact Nat.lt_of_lt_of_le ‹_› ‹_›))) hok2
             have e3:rows-8+8*1=rows:=א
             rw[e3] at R2;exact D.qBv R2 (λq hq=>hq)
           . rw[<-e];exact FB
         . rw[ἐ h5]
           rw[<-e];exact FB
     . rw[ἐ hok];exact FB
   . rw[ἔ HD];exact FB
 . rw[ἐ h];exact FB
lemma qHb:linG=linearSpec:=by
 funext rows inW outW relu X W b;unfold linG;split;. rw[D.qDz]
 . split;. rename_i _ h;obtain ⟨hs,hN,hNi⟩:=h;rw[qRR rows inW outW relu X W b hs (Nat.lt_of_le_of_lt hN (show 4294967295<2^32 by norm_num)) (Nat.lt_of_le_of_lt hNi (show 4294967295<2^32 by norm_num))];rfl
   . rw[D.qDz]
def linRb (inW outW:Ὣ) (relu:Ὡ) (X W b:ё) (c:(ὤ×ὥ)×(ὤ×ὥ)) (nf:ὧ) (n:ὦ):Ὣ->Ὣ->ὤ->ё->D.CN
 | 0,_,_,out=>⟨out,true⟩
 | k+1,i,buf,out=>
   let v:=D.decF X (i*inW).toUSize n buf 0
   if v.ok then
    let lo:=D.loOf (nf*(D.dv c.1.2*v.l+D.dv c.2.2))
    linRb inW outW relu X W b c nf n k (i+1) v.buf
     (if 128<=inW then D.colsN3 inW outW relu X W b i D.idA v.buf (ἆ 0) c.1.1 c.2.1 nf v.l c.1.2 lo (D.spOf lo) n n 0 (i*outW).toUSize out
      else D.colsN2 inW outW relu X W b i D.idA v.buf (ἆ 0) c.1.1 c.2.1 nf v.l c.1.2 lo (D.spOf lo) n n 0 (i*outW).toUSize out)
   else ⟨out,false⟩
def linH (rows inW outW:Ὣ) (relu:Ὡ) (X W b:ё):ё:=
 let cb:=(D.wnGet inW outW W b).1
 if rows*inW=ѧX/\outW*inW=ѧW/\outW=ѧb/\0< inW/\0<outW/\inW<2^20/\
  ѧX<=4294967295/\ѧW<=4294967295/\ѧb<=4294967295/\rows*outW<=4294967295/\
  D.scanH 0 X≠Е/\cb.2.2≠Е/\cb.1.2≠Е/\inW<=8192 then
  let n:=ѩinW
  let nf:ὧ:=(inW.toUInt64+8).toFloat*Ἃ 0x3CA0000000000000
  let p:=(D.wd8Get inW W).1
  if hd:p.1.size=ѧW then
   let c:=linRg inW p.2.2 outW relu X W b p.1 hd p.2.1 nf (D.dv cb.2.2) n (rows/8) 0 (ἇ (rows*outW) 0) (D.zFor (8*inW+1))
   if c.ok then
    let c2:=linRb inW outW relu X W b cb nf n (rows%8) (rows/8*8) (D.zFor inW) c.out
    if c2.ok then c2.out else D.linF rows inW outW relu X W b
   else D.linF rows inW outW relu X W b
  else D.linF rows inW outW relu X W b
 else D.linF rows inW outW relu X W b
def linG2 (rows inW outW:Ὣ) (relu:Ὡ) (X W b:ё):ё:=
 if rows<=1 then D.linF rows inW outW relu X W b
 else if rows%8≠0/\(rows%8<=5\/32<=inW) then linH rows inW outW relu X W b
 else linG rows inW outW relu X W b
lemma qRdI (X:ѐ) (n i:Ὣ) (buf:ὤ) (h:n<=8192) (hv:∀k,k<n->buf.data[k]! =D.dv X[i*n+k]!):∀k p:Ὣ,(ἄ n)[k]?=some p->(D.idA[k]!).toNat=p/\buf.data[k]! =D.dv X[i*n+p]!:=λk p hk=>by
 by_cases hk2:k<n
 . rw[List.getElem?_range hk2] at hk;injection hk with e;subst e;exact ⟨D.qIA k (by (exact Nat.lt_of_lt_of_le ‹_› ‹_›)),hv k hk2⟩
 . rw[List.getElem?_eq_none (by simpa only[List.length_range,not_lt] using hk2)] at hk;exact absurd hk (by simp only[reduceCtorEq,not_false_eq_true])
lemma qRb {rows inW outW:Ὣ} {relu:Ὡ} {X W b:ё} (G:D.C_G rows inW outW X W b) (h8k:inW<=8192) (hv:D.scanH 2 W≠Е)
 (fbm:(D.dv (D.scanH 1 b))ᶠ) (hbm:∀j,j<outW-> |R (D.dv b[j]!)| <=R (D.dv (D.scanH 1 b))) (bb:R (D.dv (D.scanH 1 b))<=2^(290:ℤ)):
 ∀k i (buf:ὤ) (out:ё),i+k<=rows->inW<=Ѩbuf-> Ѩbuf<2^32->D.C_Inv (rows*outW) (D.C_sp inW outW relu X W b) (λq=>q< i*outW) out->
  (linRb inW outW relu X W b ((D.wbOf inW outW W,D.scanH 2 W),(D.bOf b,D.scanH 1 b)) (D.nfOf inW) (ѩinW) k i buf out).ok=true->
  D.C_Inv (rows*outW) (D.C_sp inW outW relu X W b) (λq=>q<(i+k)*outW) (linRb inW outW relu X W b ((D.wbOf inW outW W,D.scanH 2 W),(D.bOf b,D.scanH 1 b)) (D.nfOf inW) (ѩinW) k i buf out).out:=by
 intro k
 induction k with
 | zero=>
  intro i buf out _ _ _ H _
  simp only[linRb,Ἵ];exact H
 | succ k ih=>
  intro i buf out hk hb1 hb2 H ok
  have hi:i<rows:=by exact omS4 hk
  have nm:inW.toUSize.toNat=inW:=D.qAd _ (by have:=G.ni;omega)
  have hX32:=G.sX
  have hRO:=G.sO
  have hiI:i*inW+inW<=ѧX:=by
   rw[<-G.hX];have:=Ἒ inW (show i+1<=rows by omega);rw[ὰ,Ὑ] at this;exact this
  have hiO:i*outW+outW<=rows*outW:=by
   have:=Ἒ outW (show i+1<=rows by omega);rw[ὰ,Ὑ] at this;exact this
  simp only[linRb] at ok⊢
  rw[D.qDF] at ok⊢
  by_cases hc:(D.decD X (i*inW).toUSize (ѩinW) 0 buf 0).ok=true
  . rw[ἱ hc] at ok⊢
    obtain ⟨v1,v2,v3⟩:=D.qVz (a:=i*inW) (n:=ѩinW) hX32 (by rw[nm];exact hiI) inW 0 buf 0 (by rw[D.C_z,Ἇ,nm]) (by rw[nm];exact hb1) hb2 (λk hk=>absurd hk (Ἲ _)) rfl hc
    rw[nm] at v2 v3
    have Rd:=qRdI X inW i _ h8k v2
    have Rw:D.C_Row inW X i D.idA (D.decD X (i*inW).toUSize (ѩinW) 0 buf 0).buf (ѩinW) (ἄ inW) (D.decD X (i*inW).toUSize (ѩinW) 0 buf 0).l:=⟨by rw[nm,List.length_range],by simp only[List.length_range,Std.le_refl],by rw[D.qIs];exact h8k,⟨by rw[v1];exact hb1,by rw[v1];exact hb2⟩,Ὂ,λt ht=>Ἅ.1 ht,λt ht hn=>absurd (Ἅ.2 ht) hn,Rd,v3⟩
    have Hv:=D.qA8 2 W G.sW hv
    have HdE:D.C_Dw (ἆ 0) W:=λh q hq=>by have e:0=ѧW:=h;omega
    have hob:((i*outW).toUSize).toNat=i*outW:=D.qAd _ א
    have hix:Ѥ(ѩinW)<=ѧD.idA:=by rw[D.qIs,nm];omega
    have hOW:outW*Ѥ(ѩinW)<=ѧW:=by rw[nm,G.hW]
    have hm0:0<Ѥ(ѩinW):=by rw[nm];exact G.pi
    have hBo:outW<=ѧb:=by rw[G.hb]
    have hW8:ѧW<=4294967295:=by have:=G.sW;omega
    have HI:∀k,k< Ѥ(ѩinW)->(D.idA[k]!).toNat=k:=λk hk=>D.qIA k (Nat.lt_of_lt_of_le (by rwa[nm] at hk) h8k)
    have HX:∀j,j< Ѥ(ѩinW)->∃w,D.good w=true/\(D.decD X (i*inW).toUSize (ѩinW) 0 buf 0).buf.data[j]! =D.dv w:=λj hj=>by
     rw[Rw.len] at hj;have hp:=List.getElem?_eq_getElem hj;exact ⟨_,G.gx hi _ (Rw.lt _ (List.mem_of_getElem? hp)),(Rw.rd j _ hp).2⟩
    have HB:∀j i k,j%8=0->j+8<=outW->i< Ѥ(ѩinW)->k<8->(D.wbOf inW outW W).data[j*Ѥ(ѩinW)+5+8*i+k]! =D.dv W[(j+k)*Ѥ(ѩinW)+i]!:=λj i k a b c d=>by rw[nm] at c⊢;exact D.wbP W inW outW j i k a b c d
    have HBt:outW%8≠0-> ∀i k,i< Ѥ(ѩinW)->k<8->(D.wbOf inW outW W).data[(8*(outW/8))*Ѥ(ѩinW)+5+8*i+k]! =D.dv W[(min (8*(outW/8)+k) (outW-1))*Ѥ(ѩinW)+i]!:=λho i k hi hk=>by rw[nm] at hi⊢;exact D.wbPt W inW outW i k ho hi hk
    have HO2:outW<2^32:=by rw[G.hb];exact G.sb
    have E:(if 128<=inW then D.colsN3 inW outW relu X W b i D.idA (D.decD X (i*inW).toUSize (ѩinW) 0 buf 0).buf (ἆ 0) (D.wbOf inW outW W) (D.bOf b) (D.nfOf inW) (D.decD X (i*inW).toUSize (ѩinW) 0 buf 0).l (D.scanH 2 W) (D.loOf (D.nfOf inW*(D.dv (D.scanH 2 W)*(D.decD X (i*inW).toUSize (ѩinW) 0 buf 0).l+D.dv (D.scanH 1 b)))) (D.spOf (D.loOf (D.nfOf inW*(D.dv (D.scanH 2 W)*(D.decD X (i*inW).toUSize (ѩinW) 0 buf 0).l+D.dv (D.scanH 1 b))))) (ѩinW) (ѩinW) 0 (i*outW).toUSize out
      else D.colsN2 inW outW relu X W b i D.idA (D.decD X (i*inW).toUSize (ѩinW) 0 buf 0).buf (ἆ 0) (D.wbOf inW outW W) (D.bOf b) (D.nfOf inW) (D.decD X (i*inW).toUSize (ѩinW) 0 buf 0).l (D.scanH 2 W) (D.loOf (D.nfOf inW*(D.dv (D.scanH 2 W)*(D.decD X (i*inW).toUSize (ѩinW) 0 buf 0).l+D.dv (D.scanH 1 b)))) (D.spOf (D.loOf (D.nfOf inW*(D.dv (D.scanH 2 W)*(D.decD X (i*inW).toUSize (ѩinW) 0 buf 0).l+D.dv (D.scanH 1 b))))) (ѩinW) (ѩinW) 0 (i*outW).toUSize out)
      =D.cols1 inW outW relu X W b i D.idA (D.decD X (i*inW).toUSize (ѩinW) 0 buf 0).buf (ἆ 0) (D.nfOf inW) (D.decD X (i*inW).toUSize (ѩinW) 0 buf 0).l (D.scanH 2 W) (D.loOf (D.nfOf inW*(D.dv (D.scanH 2 W)*(D.decD X (i*inW).toUSize (ѩinW) 0 buf 0).l+D.dv (D.scanH 1 b)))) (ѩinW) (ѩinW) 0 (i*outW).toUSize out:=by
     split
     . rw[D.cN3W inW outW relu X W b i D.idA _ (ἆ 0) _ _ _ _ _ _ _ _ hBo G.sb hix hW8 hOW hm0 _ 0 _ rfl]
       rw[D.c1VW _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ HI (Ὗ _),D.c1WE _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ HX HdE hOW HO2 HB HBt 0 rfl]
     . rw[D.cN2W inW outW relu X W b i D.idA _ (ἆ 0) _ _ _ _ _ _ _ _ hBo G.sb hix hW8 hOW hm0 _ 0 _ rfl]
       rw[D.c1VW _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ HI (Ὗ _),D.c1WE _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ HX HdE hOW HO2 HB HBt 0 rfl]
    rw[E] at ok⊢
    have R1:=D.qD0 G relu hi Hv Rw hob HdE (D.dv (D.scanH 1 b)) fbm hbm bb rfl outW 0 out (Ἇ _) (by rw[Ἵ];exact H)
    have R2:=ih (i+1) _ _ א (by rw[v1];exact hb1) (by rw[v1];exact hb2) (D.qA5 R1 (λq _ hq=>by rw[ὰ,Ὑ] at hq;exact hq)) ok
    rw[show i+(k+1)=i+1+k by omega];exact R2
  . rw[ἐ hc] at ok;exact absurd ok (by simp only[false_eq_true,not_false_eq_true])
lemma qRH (rows inW outW:Ὣ) (relu:Ὡ) (X W b:ё):
 linH rows inW outW relu X W b=Array.ofFn (n:=rows*outW) λq=>linSp inW outW relu X W b q.val:=by
 have FB:D.linF rows inW outW relu X W b=Array.ofFn (n:=rows*outW) λq=>linSp inW outW relu X W b q.val:=by rw[D.qDz];rfl
 unfold linH;rw[(D.wnGet inW outW W b).2];dsimp only
 by_cases h:rows*inW=ѧX/\outW*inW=ѧW/\outW=ѧb/\0< inW/\0<outW/\inW<2^20/\ѧX<=4294967295/\ѧW<=4294967295/\ѧb<=4294967295/\rows*outW<=4294967295/\D.scanH 0 X≠Е/\D.scanH 1 b≠Е/\D.scanH 2 W≠Е/\inW<=8192
 . rw[ἱ h];obtain ⟨h1,h2,h3,h4,h5,h6,h7,h8,h9,h10,h11,h13,hv,h8k⟩:=h;replace h7:=D.qA9 h7;replace h8:=D.qA9 h8;replace h9:=D.qA9 h9;replace h10:=D.qA9 h10
   have hnf:(inW.toUInt64+8).toFloat*Ἃ 0x3CA0000000000000=D.nfOf inW:=rfl
   have e:linSp inW outW relu X W b=D.C_sp inW outW relu X W b:=rfl
   have G:D.C_G rows inW outW X W b:=⟨h1,h2,h3,h4,h5,h6,h7,h8,h9,h10,(D.qA8 0 X h7 h11).2.2.1,(D.qA8 2 W h8 hv).2.2.1,(D.qA8 1 b h9 h13).2.2.1⟩
   obtain ⟨gb1,gb2,-,gb4⟩:=D.qA8 1 b h9 h13
   obtain ⟨pb0,pb1,pb2⟩:=D.qCs gb1 gb2
   obtain ⟨fbm,Rbm⟩:=D.qAm _ gb1
   have hbm:∀j,j<outW-> |R (D.dv b[j]!)| <=R (D.dv (D.scanH 1 b)):=λj hj=>by
    have gj:=(D.qA8 1 b h9 h13).2.2.1 j (by rw[<-h3];exact hj)
    rw[(D.qAm _ gj).2,Rbm];exact pb2 _ gj (gb4 j (by rw[<-h3];exact hj))
   have bb:R (D.dv (D.scanH 1 b))<=2^(290:ℤ):=by rw[Rbm];exact pb1.trans (by rw[show (2:ℝ)^128=2^(128:ℤ) by norm_num];exact ἒ ב ב)
   rw[hnf,(D.wd8Get inW W).2];dsimp only;by_cases HD:(D.dnOf (D.tZ (D.nE inW W)) (D.qTs (D.qTZ (D.nE inW W))) W).size=ѧW
   . rw[Ἄ HD]
     have P0:=D.qZf (8*inW+1) א
     by_cases hok:(linRg inW (D.nE inW W) outW relu X W b (D.dnOf (D.tZ (D.nE inW W)) (D.qTs (D.qTZ (D.nE inW W))) W) HD (D.qScan W W.usize 0 Е) (D.nfOf inW) (D.dv (D.scanH 1 b)) (ѩinW) (rows/8) 0 (ἇ (rows*outW) 0) (D.zFor (8*inW+1))).ok=true
     . rw[ἱ hok]
       have R1:=qRg G HD (D.dv (D.scanH 1 b)) fbm hbm bb (rows/8) 0 _ (D.zFor (8*inW+1)) P0.1 P0.2 א ⟨by simp only[Array.size_replicate],λq _ hq=>absurd hq (by simp only[zero_mul,not_lt_zero,not_false_eq_true])⟩ hok
       have P1:=D.qZf inW א
       by_cases hok2:(linRb inW outW relu X W b ((D.wbOf inW outW W,D.scanH 2 W),(D.bOf b,D.scanH 1 b)) (D.nfOf inW) (ѩinW) (rows%8) (rows/8*8) (D.zFor inW) (linRg inW (D.nE inW W) outW relu X W b (D.dnOf (D.tZ (D.nE inW W)) (D.qTs (D.qTZ (D.nE inW W))) W) HD (D.qScan W W.usize 0 Е) (D.nfOf inW) (D.dv (D.scanH 1 b)) (ѩinW) (rows/8) 0 (ἇ (rows*outW) 0) (D.zFor (8*inW+1))).out).ok=true
       . rw[ἱ hok2]
         have R2:=qRb G h8k hv fbm hbm bb (rows%8) (rows/8*8) _ _ א P1.1 P1.2 (D.qA5 R1 (λq _ hq=>by rw[Ἇ,ἥ 8 (rows/8)];exact hq)) hok2
         rw[show rows/8*8+rows%8=rows by omega] at R2
         rw[e];exact D.qBv R2 (λq hq=>hq)
       . rw[ἐ hok2];exact FB
     . rw[ἐ hok];exact FB
   . rw[ἔ HD];exact FB
 . rw[ἐ h];exact FB
lemma qHb2:linG2=linearSpec:=by
 funext rows inW outW relu X W b;unfold linG2;split
 . rw[D.qDz]
 . split
   . rw[qRH];rfl
   . rw[qHb]
lemma qJj (a b:ѐ) (i r:Ὣ):dallNZ a b i r=dallNZ b a i r:=by
 induction r generalizing i with
 | zero=>rfl
 | succ r ih=>rw[dallNZ,dallNZ,ih (i+1),ἥ (mantOf a[i]!),Bool.xor_comm]
lemma qIh (n:Ὣ) (fa fb:Ὣ->ὥ) (hf:∀j,j<n->expOf (fa j)≠255/\expOf (fb j)≠255):dotOf n fa fb=dotOf n fb fa:=by
 have hf':∀j,j<n->expOf (fb j)≠255/\expOf (fa j)≠255:=λj h=>(hf j h).symm;have hs:(∑ j∈Ἣ n,ἂ (fb j)*ἂ (fa j))=∑ j∈Ἣ n,ἂ (fa j)*ἂ (fb j):=Ἷ rfl (λj _=>mul_comm _ _);by_cases hS:(∑ j∈Ἣ n,ἂ (fa j)*ἂ (fb j))=0;. rw[qD6 n fa fb hf hS,qD6 n fb fa hf' (hs.trans hS),qJj]
 . obtain ⟨d,hd,hw⟩:=qBa n fa fb hf hS;obtain ⟨e,he,hw'⟩:=qBa n fb fa hf' (by rw[hs];exact hS);have t:=@toRat_eq_zero_iff;have h3:d.toRat=e.toRat:=by rw[hd,he,hs]
   rw[hw,hw',roundDyadic_congr Fmt ה d e (λh=>hS (hd.symm.trans ((t _).2 h))) (λh=>hS (hd.symm.trans (h3.trans ((t _).2 h)))) h3]
def gemmC (batch m k n:Ὣ) (A B:ё):ё:=
 if batch==1&& ѧA==m*k&& ѧB==k*n&&decide (0<n)&&decide (0<k)&&
  decide (0< m)&&A.all finiteWord&&B.all finiteWord then
  convF ⟨k,1,n,m,1,1,1,0⟩ false B A (ἇ m س)
 else gemmSpec batch m k n A B
lemma qFq:gemmC=gemmSpec:=by
 funext batch m k n A B;unfold gemmC;split
 . rename_i hc;simp only[and_eq_true,beq_iff_eq,decide_eq_true_eq] at hc;obtain ⟨⟨⟨⟨⟨⟨⟨h1,h2⟩,h3⟩,h4⟩,h5⟩,hm0⟩,h6⟩,h7⟩:=hc;subst h1;rw[qB7];unfold conv2dSpecO conv2dWithO gemmSpec gemmWith;have eho:OS.ho ⟨k,1,n,m,1,1,1,0⟩=1:=by simp only[OS.ho,mul_zero,add_zero,tsub_self,Nat.div_one,zero_add]
   have ewo:OS.wo ⟨k,1,n,m,1,1,1,0⟩=n:=by simp only[OS.wo,mul_zero,add_zero,Nat.div_one];(clear hm0 h5 h3 h2 eho;omega)
   apply Մ (by simp only[mul_one,ewo,eho,one_mul,Array.size_ofFn]);intro q hq1 hq2;simp only[Array.getElem_ofFn,Array.size_ofFn] at hq1 hq2⊢;have hq:q< m*n:=by simpa only[one_mul] using hq2
   simp only[eho,ewo,Nat.mul_one,Ὑ,Nat.mod_one];have hi:q/n< m:=Ἰ (by rwa[ἥ]);have hj:q%n<n:=ὀ _ h4;have e1:q/(m*n)=0:=Ὃ hq;have e2:q/n%m=q/n:=Ἁ hi;have gz:(ἇ m (س:ὥ))[q/n]! =س:=by rw[Ἕ _ _ (by simpa only[Array.size_replicate] using hi)];simp only[Array.getElem_replicate]
   have hx:(λt=>if t<k then windowAtO ⟨k,1,n,m,1,1,1,0⟩ B 0 (q%n) t else oneW)=(λt=>if t<k then B[t*n+q%n]! else oneW):=by
    funext t;split;. simp only[windowAtO,Nat.mod_one,Nat.div_one,Nat.mul_one,Ἵ,Nat.sub_zero];rw[ἐ א]
    . rfl
   have hw:(λt=>if t<k then A[q/n*k+t]! else (ἇ m (س:ὥ))[q/n]!)=(λt=>if t<k then A[q/n*k+t]! else س):=by funext t;rw[gz]
   simp only[act,false_eq_true,ite_false];rw[hx,hw,qI0 k (λt=>B[t*n+q%n]!) (λt=>A[q/n*k+t]!) h5 (λt _=>⟨qD9 B h7 _,qD9 A h6 _⟩),qIh k _ _ (λt _=>⟨qD9 B h7 _,qD9 A h6 _⟩)];simp only[e1,e2,ὒ,Ἇ]
 . rfl
lemma tDm {a b c d n:Nat} (hb:b<n) (hd:d<n) (h:a*n+b=c*n+d):a=c/\b=d:=by
 have e1:(a*n+b)/n=a:=by rw[Nat.add_comm,Nat.add_mul_div_right _ _ א,Ὃ hb,Ἇ]
 have e2:(c*n+d)/n=c:=by rw[Nat.add_comm,Nat.add_mul_div_right _ _ א,Ὃ hd,Ἇ]
 have:a=c:=by rw[<-e1,<-e2,h]
 subst this;omega
lemma tBd {i j r c:Nat} (hi:i<r) (hj:j<c):j*r+i<r*c/\i*c+j<r*c:=by
 constructor
 · have:=Ἒ r (show j+1<=c by omega);rw[ᾀ] at this;rw[ἥ r c];omega
 · have:=Ἒ c (show i+1<=r by omega);rw[ᾀ] at this;omega
lemma tUa {a b:ὦ} {x:Nat} (h:a.toNat+b.toNat=x) (hx:x<2^32):(a+b).toNat=x:=by rw[D.qAf a b א];exact h
lemma tUm {a b:ὦ} {x:Nat} (h:a.toNat*b.toNat=x) (hx:x<2^32):(a*b).toNat=x:=by
 have h3:USize.size=2^System.Platform.numBits:=rfl
 have:=USize.le_size
 rw[USize.toNat_mul,<-h3,Ἁ (by rw[h];omega)];exact h
lemma tUma (a b c:ὦ) (h:a.toNat*b.toNat+c.toNat<2^32):(a*b+c).toNat=a.toNat*b.toNat+c.toNat:=
 tUa (x:=a.toNat*b.toNat+c.toNat) (by rw[tUm rfl א]) h
def tJ (A:ѐ) (r c i:ὦ) (N:Nat) (hA:A.size=N) (hN:r.toNat*c.toNat=N) (h2:N<2^32) (hi:i.toNat<r.toNat)
 (j k:ὦ) (o:ѐ) (ho:o.size=N) (ek:k.toNat=i.toNat*c.toNat+j.toNat):ѐ:=
 if h:j<c then
  have hj:=USize.lt_iff_toNat_lt.mp h
  have b:=tBd hi hj
  have ep:(j*r+i).toNat=j.toNat*r.toNat+i.toNat:=tUma j r i (by omega)
  tJ A r c i N hA hN h2 hi (j+1) (k+1) (o.uset (j*r+i) (A.uget k (by omega)) (by omega)) (by simp[ho])
   (by rw[tUa (b:=1) (x:=k.toNat+1) (by simp) (by omega),tUa (b:=1) (x:=j.toNat+1) (by simp) (by omega)];omega)
 else o
termination_by c.toNat-j.toNat
decreasing_by rw[D.usz j c h];exact Nat.sub_lt_sub_left (USize.lt_iff_toNat_lt.mp h) (Nat.lt_succ_self _)
lemma tJ_spec (A:ѐ) (r c i:ὦ) (N:Nat) (hA:A.size=N) (hN:r.toNat*c.toNat=N) (h2:N<2^32) (hi:i.toNat<r.toNat) (d:Nat):
 ∀(j k:ὦ) (o:ѐ) (ho:o.size=N) (ek:k.toNat=i.toNat*c.toNat+j.toNat),c.toNat-j.toNat=d->
 (tJ A r c i N hA hN h2 hi j k o ho ek).size=N/\
 (∀ja,j.toNat<=ja->ja<c.toNat->(tJ A r c i N hA hN h2 hi j k o ho ek)[ja*r.toNat+i.toNat]! =A[i.toNat*c.toNat+ja]!)/\
 ∀q,(∀ja,j.toNat<=ja->ja<c.toNat->q≠ja*r.toNat+i.toNat)->(tJ A r c i N hA hN h2 hi j k o ho ek)[q]! = o[q]!:=by
 induction d with
 | zero=>
  intro j k o ho ek hd
  rw[tJ,dif_neg (by rw[А];omega)]
  exact ⟨ho,λja h1 h2=>א,λ_ _=>rfl⟩
 | succ d ih=>
  intro j k o ho ek hd
  have h:j<c:=by rw[А];omega
  have hj:=USize.lt_iff_toNat_lt.mp h
  have b:=tBd hi hj
  have ep:(j*r+i).toNat=j.toNat*r.toNat+i.toNat:=tUma j r i א
  have e1:=D.usz j c h
  rw[tJ,dif_pos h]
  dsimp only
  have hs':(o.uset (j*r+i) (A.uget k א) א).size=N:=by rw[Array.size_uset];exact ho
  obtain ⟨a1,a2,a3⟩:=ih (j+1) (k+1) _ hs' (by rw[tUa (b:=1) (x:=k.toNat+1) (by simp only[USize.reduceToNat,Nat.add_left_cancel_iff]) א,tUa (b:=1) (x:=j.toNat+1) (by simp only[USize.reduceToNat,Nat.add_left_cancel_iff]) א];omega) (by rw[e1];omega)
  have rd:∀q,(o.uset (j*r+i) (A.uget k א) א)[q]! =if q=j.toNat*r.toNat+i.toNat then A[i.toNat*c.toNat+j.toNat]! else o[q]!:=by
   intro q;rw[lcWU _ _ _ _ א,D.qBu o _ _ q א א,ep]
   split
   · simp only[Array.uget];rw[Ἕ A _ א];simp only[ek]
   · rfl
  refine ⟨a1,λja h1 h3=>?_,λq hq=>?_⟩
  · by_cases e:ja=j.toNat
    · subst e;rw[a3 _ (λjb hb1 hb2 he=>by rw[e1] at hb1;obtain ⟨x,-⟩:=tDm hi hi he;omega),rd,Ὦ rfl]
    · exact a2 ja (by rw[e1];omega) h3
  · rw[a3 q (λjb hb1 hb2=>hq jb (by rw[e1] at hb1;omega) hb2),rd,Ὧ (hq _ (Nat.le_refl _) hj)]
def tI (A:ѐ) (r c:ὦ) (N:Nat) (hA:A.size=N) (hN:r.toNat*c.toNat=N) (h2:N<2^32) (i:ὦ) (o:ѐ) (ho:o.size=N):ѐ:=
 if h:i<r then
  have hi:=USize.lt_iff_toNat_lt.mp h
  tI A r c N hA hN h2 (i+1) (tJ A r c i N hA hN h2 hi 0 (i*c) o ho (by rw[tUm rfl (by have:=Ἒ c.toNat (show i.toNat<=r.toNat by omega);omega)];simp))
   (tJ_spec A r c i N hA hN h2 hi _ 0 (i*c) o ho _ rfl).1
 else o
termination_by r.toNat-i.toNat
decreasing_by rw[D.usz i r h];exact Nat.sub_lt_sub_left (USize.lt_iff_toNat_lt.mp h) (Nat.lt_succ_self _)
lemma tI_spec (A:ѐ) (r c:ὦ) (N:Nat) (hA:A.size=N) (hN:r.toNat*c.toNat=N) (h2:N<2^32) (d:Nat):
 ∀(i:ὦ) (o:ѐ) (ho:o.size=N),r.toNat-i.toNat=d->
 (tI A r c N hA hN h2 i o ho).size=N/\
 (∀ia ja,i.toNat<=ia->ia<r.toNat->ja<c.toNat->(tI A r c N hA hN h2 i o ho)[ja*r.toNat+ia]! =A[ia*c.toNat+ja]!)/\
 ∀q,(∀ia ja,i.toNat<=ia->ia<r.toNat->ja<c.toNat->q≠ja*r.toNat+ia)->(tI A r c N hA hN h2 i o ho)[q]! = o[q]!:=by
 induction d with
 | zero=>
  intro i o ho hd
  rw[tI,dif_neg (by rw[А];omega)]
  exact ⟨ho,λia ja h1 h3 h4=>א,λ_ _=>rfl⟩
 | succ d ih=>
  intro i o ho hd
  have h:i<r:=by rw[А];omega
  have hi:=USize.lt_iff_toNat_lt.mp h
  have e1:=D.usz i r h
  rw[tI,dif_pos h]
  dsimp only
  have ek:(i*c).toNat=i.toNat*c.toNat+(0:ὦ).toNat:=by rw[tUm rfl (by have:=Ἒ c.toNat (show i.toNat<=r.toNat by omega);omega)];simp only[USize.toNat_zero,add_zero]
  obtain ⟨j1,j2,j3⟩:=tJ_spec A r c i N hA hN h2 hi _ 0 (i*c) o ho ek rfl
  obtain ⟨a1,a2,a3⟩:=ih (i+1) _ j1 (by rw[e1];omega)
  refine ⟨a1,λia ja h1 h3 h4=>?_,λq hq=>?_⟩
  · by_cases e:ia=i.toNat
    · subst e
      rw[a3 _ (λib jb h5 h6 h7 he=>by rw[e1] at h5;obtain ⟨-,x⟩:=tDm hi h6 he;omega),j2 ja (by simp only[USize.toNat_zero,zero_le]) h4]
    · exact a2 ia ja (by rw[e1];omega) h3 h4
  · rw[a3 q (λib jb h5 h6 h7=>hq ib jb (by rw[e1] at h5;omega) h6 h7),j3 q (λjb h5 h6=>hq i.toNat jb (Nat.le_refl _) hi h6)]
@[inline] def trT (r c:Nat) (A:ѐ):ѐ:=
 if h:A.size=r*c/\r<2^32/\c<2^32/\r*c<2^32 then
  tI A r.toUSize c.toUSize (r*c) h.1 (by rw[D.qAd r h.2.1,D.qAd c h.2.2.1]) h.2.2.2 0 (ἇ (r*c) 0) (by simp)
 else A
lemma trT_spec (A:ѐ) (r c:Nat) (hA:A.size=r*c) (hN:r*c<2^32) (hr0:0<r) (hc:0<c):
 (trT r c A).size=r*c/\ ∀ia ja,ia<r->ja<c->(trT r c A)[ja*r+ia]! =A[ia*c+ja]!:=by
 have h1:=Ὄ r hc
 have h3:=Nat.le_mul_of_pos_left c hr0
 have er:=D.qAd r א
 have ec:=D.qAd c א
 unfold trT;rw[dif_pos ⟨hA,א,א,hN⟩]
 obtain ⟨a1,a2,-⟩:=tI_spec A r.toUSize c.toUSize (r*c) hA (by rw[er,ec]) hN _ 0 (ἇ (r*c) 0) (by simp only[Array.size_replicate]) rfl
 refine ⟨a1,λia ja h4 h5=>?_⟩
 have:=a2 ia ja (by simp only[USize.toNat_zero,zero_le]) (by rw[er];exact h4) (by rw[ec];exact h5)
 rw[show ja*r+ia=ja*r.toUSize.toNat+ia by rw[er],show ia*c+ja=ia*c.toUSize.toNat+ja by rw[ec]];exact this
lemma qGT (m k n:Nat) (A B:ѐ) (hB:B.size=k*n) (hk:0<k) (hn:0<n) (hm:0< m) (hs:m*n<2^32) (hsB:k*n<2^32)
 (fA:∀t:Nat,expOf A[t]!≠255) (fB:∀t:Nat,expOf B[t]!≠255):
 trT n m (linearSpec n k m false (trT k n B) A (ἇ m س))=gemmSpec 1 m k n A B:=by
 obtain ⟨_,T1⟩:=trT_spec B k n hB hsB hk hn
 have hL:(linearSpec n k m false (trT k n B) A (ἇ m س)).size=n*m:=by simp only[linearSpec,linearWith,Array.size_ofFn]
 obtain ⟨S2,T2⟩:=trT_spec (linearSpec n k m false (trT k n B) A (ἇ m س)) n m hL (by rw[ἥ];exact hs) hn hm
 apply Մ (by rw[S2];simp only[ἥ,gemmSpec,gemmWith,Array.size_ofFn,one_mul])
 intro q h1 h2
 have hq:q< m*n:=by simpa only[gemmSpec,gemmWith,Array.size_ofFn,one_mul] using h2
 have hi:q/n< m:=Ἰ (by rwa[ἥ])
 have hc:q%n<n:=ὀ _ hn
 have eq:q=q/n*n+q%n:=(Nat.div_add_mod' q n).symm
 rw[<-Ἕ _ q h1]
 conv_lhs=>rw[eq,T2 _ _ hc hi]
 have b1:(q%n*m+q/n)/m=q%n:=by rw[Nat.add_comm,Nat.add_mul_div_right _ _ א,Ὃ hi,Ἇ]
 have b2:(q%n*m+q/n)%m=q/n:=by rw[Nat.add_comm,Nat.add_mul_mod_self_right,Ἁ hi]
 have hlt:q%n*m+q/n<n*m:=by have:=(tBd hi hc).1;rw[ἥ n m];exact this
 rw[Ἕ _ _ (by rw[hL];exact hlt)]
 simp only[linearSpec,linearWith,gemmSpec,gemmWith,Array.getElem_ofFn,b1,b2]
 have e1:q/(m*n)=0:=Ὃ hq
 have e2:q/n%m=q/n:=Ἁ hi
 have gz:(ἇ m (س:ὥ))[q/n]! =س:=by rw[Ἕ _ _ (by simpa only[Array.size_replicate] using hi)];simp only[Array.getElem_replicate]
 have hx:(λt=>if t<k then (trT k n B)[q%n*k+t]! else oneW)=(λt=>if t<k then B[t*n+q%n]! else oneW):=by
  funext t;split
  · rename_i ht;exact T1 t (q%n) ht hc
  · rfl
 have hw:(λt=>if t<k then A[q/n*k+t]! else (ἇ m (س:ὥ))[q/n]!)=(λt=>if t<k then A[q/n*k+t]! else س):=by funext t;rw[gz]
 simp only[act,Bool.false_eq_true,ite_false,e1,e2,ὒ,Ἇ]
 rw[hx,hw,qI0 k (λt=>B[t*n+q%n]!) (λt=>A[q/n*k+t]!) hk (λt _=>⟨fB _,fA _⟩),qIh k _ _ (λt _=>⟨fB _,fA _⟩)]
lemma oAm (a b m:ὥ) (h:(a ||| b)&&&m=0):b&&&m=0/\a&&&m=0:=by
 rw[show (a ||| b)&&&m=(a&&&m) ||| (b&&&m) from UInt32.toBitVec_inj.1 (by simp only[UInt32.toBitVec_and,UInt32.toBitVec_or,BitVec.and_or_distrib_right]),UInt32.or_eq_zero_iff] at h;exact ⟨h.2,h.1⟩
def fsc (A:ѐ) (c:ὥ) (i:ὦ) (acc:ὥ):ὥ:=
 if h:i<A.usize then fsc A c (i+1) (acc ||| (expOf (A.uget i (D.qBd A i h))+c)) else acc
termination_by A.usize.toNat-i.toNat
decreasing_by rw[D.usz i A.usize h];exact Nat.sub_lt_sub_left (USize.lt_iff_toNat_lt.mp h) (Nat.lt_succ_self _)
lemma fsc_z (A:ѐ) (c m:ὥ) (hm:(255+c)&&&m≠0) (hs:A.size<2^32) (d:Nat): ∀(i:ὦ) (acc:ὥ),A.usize.toNat-i.toNat=d->fsc A c i acc&&&m=0->
 acc&&&m=0/\ ∀k,i.toNat<=k->k<A.size->expOf A[k]!≠255:=by
 have hu:=D.qBe A hs
 induction d with
 | zero=>
  intro i acc hd h
  rw[fsc,dif_neg (by rw[А];omega)] at h
  exact ⟨h,λk h1 h2=>א⟩
 | succ d ih=>
  intro i acc hd h
  have hi:i<A.usize:=by rw[А];omega
  have e1:=D.usz i A.usize hi
  rw[fsc,dif_pos hi] at h
  obtain ⟨a1,a2⟩:=ih (i+1) _ (by rw[e1];omega) h
  obtain ⟨b1,b2⟩:=oAm _ _ _ a1
  refine ⟨b2,λk h1 h2=>?_⟩
  by_cases hk:k=i.toNat
  · subst hk;intro e;rw[Ἕ A _ h2] at e;simp only[Array.uget] at b1;rw[e] at b1;exact hm b1
  · exact a2 k (by rw[e1];omega) h2
lemma fsc_fin (A:ѐ) (c m:ὥ) (hm:(255+c)&&&m≠0) (hs:A.size<2^32) (h:fsc A c 0 0&&&m=0) (t:Nat):expOf A[t]!≠255:=by
 by_cases ht:t<A.size
 · exact (fsc_z A c m hm hs _ 0 0 rfl h).2 t (by simp only[USize.toNat_zero,zero_le]) ht
 · rw[ὑ A t ht];decide
def gemmC2 (batch m k n:Nat) (A B:ѐ):ѐ:=
 if batch==1&&A.size==m*k&&B.size==k*n&&decide (0<n)&&decide (0<k)&&decide (0< m)&&
   decide (k*n<2^32)&&decide (m*n<2^32)&&decide (A.size<2^32)&&fsc A 1 0 0&&&0x100==0&&fsc B 1 0 0&&&0x100==0 then
  if 8<=n/\3*n<=m/\fsc A 0 0 0&&&0x80=0 then trT n m (linG n k m false (trT k n B) A (ἇ m س))
  else convF ⟨k,1,n,m,1,1,1,0⟩ false B A (ἇ m س)
 else gemmC batch m k n A B
lemma qGm2:gemmC2=gemmSpec:=by
 funext batch m k n A B;unfold gemmC2;split
 · rename_i hc;simp only[Bool.and_eq_true,beq_iff_eq,decide_eq_true_eq] at hc
   obtain ⟨⟨⟨⟨⟨⟨⟨⟨⟨⟨h1,h2⟩,h3⟩,h4⟩,h5⟩,hm0⟩,h6⟩,h7⟩,h8⟩,h9⟩,h10⟩:=hc
   subst h1
   have fA:=fsc_fin A 1 0x100 ה h8 h9
   have fB:=fsc_fin B 1 0x100 ה (by rw[h3];exact h6) h10
   split
   · rw[qHb];exact qGT m k n A B h3 h5 h4 hm0 h7 h6 fA fB
   · have aA:A.all finiteWord=true:=by
      rw[Array.all_eq_true];intro i hi;have:=fA i;rw[Ἕ A i hi] at this;simpa only[finiteWord,bne_iff_ne,ne_eq] using this
     have aB:B.all finiteWord=true:=by
      rw[Array.all_eq_true];intro i hi;have:=fB i;rw[Ἕ B i hi] at this;simpa only[finiteWord,bne_iff_ne,ne_eq] using this
     have hg:gemmC 1 m k n A B=convF ⟨k,1,n,m,1,1,1,0⟩ false B A (ἇ m س):=by
      unfold gemmC;rw[Ὦ (by simp only[aA,aB,Bool.and_true];simp only[BEq.rfl,h2,Bool.and_self,h3,h4,decide_true,h5,hm0])]
     rw[<-hg,qFq]
 · rw[qFq]
end C
open ExactRuntime
open ExactRuntime
/-! -/
namespace ConvFast
open ExactRuntime FloatLib.Floats.Formats.BinaryInterchange FloatLib.Floats
def toOld (s:ConvShape):OS:=⟨s.cin,s.h,s.w,s.cout,s.kh,s.kw,s.strideH,s.padT⟩
def swapO (s:OS):OS:=⟨s.cin,1,s.h,s.cout,1,s.kh,s.stride,s.pad⟩
def fixO (s:OS):OS:=if s.w=1/\s.kw=1/\s.pad=0 then swapO s else s
def conv2dF (s:ConvShape) (relu:Ὡ) (inp wts b:ё):ё:=
 if s.strideW=s.strideH/\s.padL=s.padT/\s.padB=s.padT/\s.padR=s.padT/\
     s.dilH=1/\s.dilW=1/\s.groups=1/\0<s.kh/\0<s.kw then
  C.convF (fixO (toOld s)) relu inp wts b
 else conv2dWith Pinned.dotFn s relu inp wts b
def depthwiseF (s:ConvShape) (relu:Ὡ) (inp wts b:ё):ё:=
 if s.strideW=s.strideH/\s.padL=s.padT/\s.padB=s.padT/\s.padR=s.padT/\
     s.dilH=1/\s.dilW=1/\0<s.kh/\0<s.kw then
  C.depthF (fixO (toOld s)) relu inp wts b
 else depthwiseWith Pinned.dotFn s relu inp wts b
def avgpoolF (s:PoolShape) (inp:ё):ё:=
 if s.kw=s.kh/\s.strideH=s.kh/\s.strideW=s.kh/\s.padT=0/\s.padL=0/\s.padB=0/\
     s.padR=0/\s.ceilMode=false/\s.countIncludePad=false/\0<s.kh/\s.kh<=s.h/\
     s.kh<=s.w then
  C.avgF s.c s.h s.w s.kh inp
 else avgpoolWith Pinned.dotFn s inp
def maxpoolF (s:PoolShape) (inp:ё):ё:=
 if s.kw=s.kh/\s.strideH=s.kh/\s.strideW=s.kh/\s.padT=0/\s.padL=0/\s.padB=0/\
     s.padR=0/\s.ceilMode=false/\0<s.kh/\s.kh<=s.h/\s.kh<=s.w then
  C.maxpoolF2 s.c s.h s.w s.kh inp
 else Pinned.maxpoolW s inp
/-! -/
lemma conv_bridge (s:ConvShape) (relu:Ὡ) (inp wts b:ё)
  (h:s.strideW=s.strideH/\s.padL=s.padT/\s.padB=s.padT/\s.padR=s.padT/\
    s.dilH=1/\s.dilW=1/\s.groups=1/\0<s.kh/\0<s.kw):
  conv2dSpecO (toOld s) relu inp wts b=conv2dSpec s relu inp wts b:=by
 obtain ⟨h1,h2,h3,h4,h5,h6,h7,h8,h9⟩:=h
 have hho:(toOld s).ho=s.ho:=by
  simp only[OS.ho,ConvShape.ho,ConvShape.ekh,toOld,h3,h5]
  congr 2;omega
 have hwo:(toOld s).wo=s.wo:=by
  simp only[OS.wo,ConvShape.wo,ConvShape.ekw,toOld,h2,h4,h6,h1]
  congr 2;omega
 have htaps:s.taps=s.cin*s.kh*s.kw:=by simp only[ConvShape.taps,ConvShape.cpg,h7,Nat.div_one]
 have hopg:s.opg=s.cout:=by simp only[ConvShape.opg,h7,Nat.div_one]
 have hcpg:s.cpg=s.cin:=by simp only[ConvShape.cpg,h7,Nat.div_one]
 apply Մ
 · simp only[conv2dSpecO,conv2dWithO,conv2dSpec,conv2dWith,Array.size_ofFn,hho,hwo]
   simp only[toOld]
 · intro q hq1 hq2
   have hq:q<s.cout*(s.ho*s.wo):=by simpa only[conv2dSpec,conv2dWith,Array.size_ofFn] using hq2
   have hp:0<s.ho*s.wo:=Nat.mul_pos (by simp only[ConvShape.ho,lt_add_iff_pos_left,Order.lt_add_one_iff,zero_le]) (by simp only[ConvShape.wo,lt_add_iff_pos_left,Order.lt_add_one_iff,zero_le])
   have hg:q/(s.ho*s.wo)/s.cout=0:=
    Ὃ ((Nat.div_lt_iff_lt_mul hp).2 hq)
   simp only[conv2dSpecO,conv2dWithO,conv2dSpec,conv2dWith,Array.getElem_ofFn]
   simp only[hho,hwo,htaps,hopg,hg]
   simp only[toOld]
   congr 2
   funext t
   by_cases ht:t<s.cin*s.kh*s.kw
   · simp only[ht, ↓reduceIte]
     simp only[windowAtO,windowAt,hcpg,h1,h2,h5,h6,Nat.mul_one,ὒ,Ἇ]
   · simp only[ht, ↓reduceIte]
lemma depth_bridge (s:ConvShape) (relu:Ὡ) (inp wts b:ё)
  (h:s.strideW=s.strideH/\s.padL=s.padT/\s.padB=s.padT/\s.padR=s.padT/\
    s.dilH=1/\s.dilW=1/\0<s.kh/\0<s.kw):
  depthwiseSpecO (toOld s) relu inp wts b=depthwiseSpec s relu inp wts b:=by
 obtain ⟨h1,h2,h3,h4,h5,h6,h8,h9⟩:=h
 have hho:(toOld s).ho=s.depthwise.ho:=by
  simp only[OS.ho,ConvShape.ho,ConvShape.ekh,ConvShape.depthwise,toOld,h3,h5]
  congr 2;omega
 have hwo:(toOld s).wo=s.depthwise.wo:=by
  simp only[OS.wo,ConvShape.wo,ConvShape.ekw,ConvShape.depthwise,toOld,h2,h4,h6,
   h1]
  congr 2;omega
 apply Մ
 · simp only[depthwiseSpecO,depthwiseWithO,depthwiseSpec,depthwiseWith,conv2dWith,
    Array.size_ofFn,hho,hwo]
   simp only[toOld,ConvShape.depthwise]
 · intro q hq1 hq2
   have hq:q<s.cin*(s.depthwise.ho*s.depthwise.wo):=by
    simpa only[ConvShape.depthwise,depthwiseSpec,depthwiseWith,conv2dWith,Array.size_ofFn] using hq2
   have hcin:0<s.cin:=by
    rcases Nat.eq_zero_or_pos s.cin with h0 | h0
    · simp only[h0,zero_mul,not_lt_zero] at hq
    · exact h0
   have hcpg:s.depthwise.cpg=1:=by
    simp only[ConvShape.cpg,ConvShape.depthwise,Nat.div_self hcin]
   have hopg:s.depthwise.opg=1:=by
    simp only[ConvShape.opg,ConvShape.depthwise,Nat.div_self hcin]
   have htaps:s.depthwise.taps=s.kh*s.kw:=by
    simp only[ConvShape.taps,hcpg,Ὑ];rfl
   simp only[depthwiseSpecO,depthwiseWithO,depthwiseSpec,depthwiseWith,conv2dWith,
    Array.getElem_ofFn]
   simp only[hho,hwo,htaps,hopg,Nat.div_one]
   simp only[toOld]
   congr 2
   funext t
   by_cases ht:t<s.kh*s.kw
   · simp only[ht, ↓reduceIte]
     have hk:0<s.kh*s.kw:=Nat.mul_pos h8 h9
     have e1:(q/(s.depthwise.ho*s.depthwise.wo)*(s.kh*s.kw)+t)/(s.kh*s.kw)=
       q/(s.depthwise.ho*s.depthwise.wo):=by
      rw[ἥ,Nat.mul_add_div hk,Ὃ ht,Ἵ]
     have e2:(q/(s.depthwise.ho*s.depthwise.wo)*(s.kh*s.kw)+t)/s.kw%s.kh=
       t/s.kw%s.kh:=by
      rw[show q/(s.depthwise.ho*s.depthwise.wo)*(s.kh*s.kw)+t=
         t+s.kw*(q/(s.depthwise.ho*s.depthwise.wo)*s.kh) by ring,
       Nat.add_mul_div_left _ _ h9,Nat.add_mul_mod_self_right]
     have e3:(q/(s.depthwise.ho*s.depthwise.wo)*(s.kh*s.kw)+t)%s.kw=t%s.kw:=by
      rw[show q/(s.depthwise.ho*s.depthwise.wo)*(s.kh*s.kw)+t=
         t+s.kw*(q/(s.depthwise.ho*s.depthwise.wo)*s.kh) by ring,
       Nat.add_mul_mod_self_left]
     have f1:s.depthwise.kh=s.kh:=rfl
     have f2:s.depthwise.kw=s.kw:=rfl
     have f3:s.depthwise.h=s.h:=rfl
     have f4:s.depthwise.w=s.w:=rfl
     have f5:s.depthwise.strideH=s.strideH:=rfl
     have f6:s.depthwise.strideW=s.strideH:=h1
     have f7:s.depthwise.padT=s.padT:=rfl
     have f8:s.depthwise.padL=s.padT:=h2
     have f9:s.depthwise.dilH=1:=h5
     have f10:s.depthwise.dilW=1:=h6
     simp only[windowAtO,windowAt,e1,e2,e3,hcpg,Nat.mul_one,Ὃ ht,
      Ἵ,f1,f2,f3,f4,f5,f6,f7,f8,f9,f10]
   · simp only[ht, ↓reduceIte]
lemma pool_out (n k:Nat) (hk:0<k) (hn:k<=n):poolOut n 0 0 k k false=n/k:=by
 simp only[poolOut,Ἵ,Bool.false_eq_true,ite_false]
 exact (Nat.div_eq_sub_div hk hn).symm
lemma pool_span (i k n:Nat) (hi:i<n/k):poolSpan i k 0 k n=(i*k,i*k+k):=by
 have:(i+1)*k<=n/k*k:=Ἒ _ hi
 have:=Nat.div_mul_le_self n k
 simp only[poolSpan,Nat.sub_zero,Ἇ,Prod.mk.injEq,true_and]
 rw[Nat.min_eq_left (by rw[ᾀ] at*;omega)]
lemma pool_window (s:PoolShape) (y x:Nat)
  (h:s.kw=s.kh/\s.strideH=s.kh/\s.strideW=s.kh/\s.padT=0/\s.padL=0)
  (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⟩:=by
 obtain ⟨h1,h2,h3,h4,h5⟩:=h
 simp only[PoolShape.window,h1,h2,h3,h4,h5,pool_span y s.kh s.h hy,pool_span x s.kh s.w hx]
 simp only[add_tsub_cancel_left]
lemma avg_bridge (s:PoolShape) (inp:ё)
  (h:s.kw=s.kh/\s.strideH=s.kh/\s.strideW=s.kh/\s.padT=0/\s.padL=0/\s.padB=0/\
    s.padR=0/\s.ceilMode=false/\s.countIncludePad=false/\0<s.kh/\s.kh<=s.h/\
    s.kh<=s.w):
  avgpoolSpecO s.c s.h s.w s.kh inp=avgpoolSpec s inp:=by
 obtain ⟨h1,h2,h3,h4,h5,h6,h7,h8,h9,h10,h11,h12⟩:=h
 have hho:s.ho=s.h/s.kh:=by
  simp only[PoolShape.ho,h2,h4,h6,h8];exact pool_out _ _ h10 h11
 have hwo:s.wo=s.w/s.kh:=by
  simp only[PoolShape.wo,h3,h5,h7,h8,h1];exact pool_out _ _ h10 h12
 apply Մ
 · simp only[avgpoolSpecO,avgpoolWithO,Array.size_ofFn,avgpoolSpec,avgpoolWith,hwo,hho]
 · intro q hq1 hq2
   simp only[avgpoolSpecO,avgpoolWithO,avgpoolSpec,avgpoolWith,Array.getElem_ofFn]
   rw[hho,hwo]
   have hy:=ὀ (q/(s.w/s.kh)) (Nat.div_pos h11 h10)
   have hx:=ὀ q (Nat.div_pos h12 h10)
   have hwin:=pool_window s _ _ ⟨h1,h2,h3,h4,h5⟩ hy hx
   have hdiv:s.divisor (q/(s.w/s.kh)%(s.h/s.kh)) (q%(s.w/s.kh))=s.kh*s.kh:=by
    simp only[PoolShape.divisor,h9,Bool.false_eq_true, ↓reduceIte,hwin]
   rw[hwin,hdiv]
   rfl
lemma max_bridge (s:PoolShape) (inp:ё)
  (h:s.kw=s.kh/\s.strideH=s.kh/\s.strideW=s.kh/\s.padT=0/\s.padL=0/\s.padB=0/\
    s.padR=0/\s.ceilMode=false/\0<s.kh/\s.kh<=s.h/\s.kh<=s.w):
  maxpoolSpecO s.c s.h s.w s.kh inp=maxpoolSpec s inp:=by
 obtain ⟨h1,h2,h3,h4,h5,h6,h7,h8,h10,h11,h12⟩:=h
 have hho:s.ho=s.h/s.kh:=by
  simp only[PoolShape.ho,h2,h4,h6,h8];exact pool_out _ _ h10 h11
 have hwo:s.wo=s.w/s.kh:=by
  simp only[PoolShape.wo,h3,h5,h7,h8,h1];exact pool_out _ _ h10 h12
 apply Մ
 · simp only[maxpoolSpecO,Array.size_ofFn,maxpoolSpec,hho,hwo]
 · intro q hq1 hq2
   simp only[maxpoolSpecO,maxpoolSpec,Array.getElem_ofFn]
   rw[hho,hwo]
   have hy:=ὀ (q/(s.w/s.kh)) (Nat.div_pos h11 h10)
   have hx:=ὀ q (Nat.div_pos h12 h10)
   rw[pool_window s _ _ ⟨h1,h2,h3,h4,h5⟩ hy hx]
   rfl
lemma swap_win (s:OS) (inp:ё) (hw:s.w=1) (hkw:s.kw=1) (hp:s.pad=0)
  (y t:Nat):windowAtO (swapO s) inp 0 y t=windowAtO s inp y 0 t:=by
 simp only[windowAtO,swapO,hw,hkw,hp,Nat.mul_one,Ὑ,Nat.mod_one,Nat.div_one,ὒ,
  Ἵ,Nat.sub_zero,Ἲ,false_or,Nat.lt_irrefl]
 by_cases h1:s.h<=y*s.stride+t%s.kh
 · simp only[ge_iff_le,nonpos_iff_eq_zero,one_ne_zero,h1,or_true, ↓reduceIte,or_false]
 · simp only[ge_iff_le,nonpos_iff_eq_zero,one_ne_zero,h1,or_self, ↓reduceIte]
lemma conv_swap (s:OS) (relu:Ὡ) (inp wts b:ё) (hw:s.w=1) (hkw:s.kw=1)
  (hp:s.pad=0):conv2dSpecO (swapO s) relu inp wts b=conv2dSpecO s relu inp wts b:=by
 have hho:(swapO s).ho=1:=by simp only[OS.ho,swapO,hp,mul_zero,add_zero,tsub_self,Nat.zero_div,zero_add]
 have hwo:(swapO s).wo=s.ho:=by simp only[OS.wo,swapO,hp,mul_zero,add_zero,OS.ho,Nat.add_left_cancel_iff]
 have hwo':s.wo=1:=by simp only[OS.wo,hw,hp,mul_zero,add_zero,hkw,tsub_self,Nat.zero_div,zero_add]
 have hk:(swapO s).cin*(swapO s).kh*(swapO s).kw=s.cin*s.kh*s.kw:=by
  simp only[swapO,mul_one,hkw]
 apply Մ
 · simp only[conv2dSpecO,conv2dWithO,Array.size_ofFn,hho,hwo,hwo'];simp only[swapO,one_mul,mul_one]
 · intro q hq1 hq2
   simp only[conv2dSpecO,conv2dWithO,Array.getElem_ofFn]
   simp only[hho,hwo,hwo',hk,Ὑ,Nat.mul_one,Nat.div_one,Nat.mod_one]
   simp only[swapO]
   congr 2
   funext t
   by_cases ht:t<s.cin*s.kh*s.kw
   · simp only[ht, ↓reduceIte]
     have:=swap_win s inp hw hkw hp (q%s.ho) t
     simp only[swapO] at this
     exact this
   · simp only[ht, ↓reduceIte]
lemma depth_swap (s:OS) (relu:Ὡ) (inp wts b:ё) (hw:s.w=1) (hkw:s.kw=1)
  (hp:s.pad=0):depthwiseSpecO (swapO s) relu inp wts b=depthwiseSpecO s relu inp wts b:=by
 have hho:(swapO s).ho=1:=by simp only[OS.ho,swapO,hp,mul_zero,add_zero,tsub_self,Nat.zero_div,zero_add]
 have hwo:(swapO s).wo=s.ho:=by simp only[OS.wo,swapO,hp,mul_zero,add_zero,OS.ho,Nat.add_left_cancel_iff]
 have hwo':s.wo=1:=by simp only[OS.wo,hw,hp,mul_zero,add_zero,hkw,tsub_self,Nat.zero_div,zero_add]
 have hk:(swapO s).kh*(swapO s).kw=s.kh*s.kw:=by simp only[swapO,one_mul,hkw,mul_one]
 apply Մ
 · simp only[depthwiseSpecO,depthwiseWithO,Array.size_ofFn,hho,hwo,hwo'];simp only[swapO,one_mul,mul_one]
 · intro q hq1 hq2
   simp only[depthwiseSpecO,depthwiseWithO,Array.getElem_ofFn]
   simp only[hho,hwo,hwo',hk,Ὑ,Nat.mul_one,Nat.div_one,Nat.mod_one]
   simp only[swapO]
   congr 2
   funext t
   by_cases ht:t<s.kh*s.kw
   · simp only[ht, ↓reduceIte]
     have:=swap_win s inp hw hkw hp (q%s.ho) (q/s.ho*(s.kh*s.kw)+t)
     simp only[swapO] at this
     exact this
   · simp only[ht, ↓reduceIte]
lemma conv_fix (s:OS) (relu:Ὡ) (inp wts b:ё):
  conv2dSpecO (fixO s) relu inp wts b=conv2dSpecO s relu inp wts b:=by
 unfold fixO;split_ifs with h
 · exact conv_swap s relu inp wts b h.1 h.2.1 h.2.2
 · rfl
lemma depth_fix (s:OS) (relu:Ὡ) (inp wts b:ё):
  depthwiseSpecO (fixO s) relu inp wts b=depthwiseSpecO s relu inp wts b:=by
 unfold fixO;split_ifs with h
 · exact depth_swap s relu inp wts b h.1 h.2.1 h.2.2
 · rfl
lemma conv2dF_eq:conv2dF=conv2dSpec:=by
 funext s relu inp wts b
 unfold conv2dF
 split_ifs with h
 · rw[C.qB7,conv_fix];exact conv_bridge s relu inp wts b h
 · rw[Pinned.dotFn_eq_dotOf];rfl
lemma depthwiseF_eq:depthwiseF=depthwiseSpec:=by
 funext s relu inp wts b
 unfold depthwiseF
 split_ifs with h
 · rw[C.qDr,depth_fix];exact depth_bridge s relu inp wts b h
 · rw[Pinned.dotFn_eq_dotOf];rfl
lemma avgpoolF_eq:avgpoolF=avgpoolSpec:=by
 funext s inp
 unfold avgpoolF
 split_ifs with h
 · rw[C.qG1];exact avg_bridge s inp h
 · rw[Pinned.dotFn_eq_dotOf];rfl
lemma maxpoolF_eq:maxpoolF=maxpoolSpec:=by
 funext s inp
 unfold maxpoolF
 split_ifs with h
 · rw[C.maxpoolF2_eq,C.qDp];exact max_bridge s inp h
 · rw[Pinned.maxpoolW_eq]
end ConvFast
namespace RnnFast
open D (dv good ziv nfOf F1 qDo wv exactS A_mag R qBw qDm qIg qAx qAe qG5 qIq qB5 qAq qA6 qAz qA0 qDq qAf uk qEl qAb)
lemma dotLoop_special_false (fa fb:Nat->ὥ):
  ∀(count index:Nat) (pos neg:ї) (nz:Ὡ),
   (∀k,index<=k->k< index+count->expOf (fa k)≠255/\expOf (fb k)≠255)->
   (Pinned.dotLoop fa fb pos neg false nz index count).special=false
 | 0,_,_,_,_,_=>by simp only[Pinned.dotLoop]
 | count+1,index,pos,neg,nz,hfin=>by
   have h:=hfin index le_rfl א
   simp only[Pinned.dotLoop]
   have hstep:(Pinned.dotStep ⟨pos,neg,false,nz⟩ (fa index) (fb index)).special=false:=by
    unfold Pinned.dotStep
    rw[Ὧ (by push_neg;exact h)]
    simp only[Pinned.Acc.add]
    split_ifs<;>rfl
   generalize Pinned.dotStep ⟨pos,neg,false,nz⟩ (fa index) (fb index)=acc at hstep
   obtain ⟨p,n,sp,z⟩:=acc
   simp only at hstep
   subst hstep
   exact dotLoop_special_false fa fb count (index+1) p n z
    (λk h1 h2=>hfin k א א)
lemma dotFn_finite (n:Nat) (fa fb:Nat->ὥ)
  (hfin: ∀k,k<n->expOf (fa k)≠255/\expOf (fb k)≠255):
  Pinned.dotFn n fa fb=
   finishWord (dsum (Array.ofFn (n:=n) λi=>fa i) (Array.ofFn (n:=n) λi=>fb i) 0 n)
    (-298) (n≠0&&dallNZ (Array.ofFn (n:=n) λi=>fa i)
      (Array.ofFn (n:=n) λi=>fb i) 0 n):=by
 unfold Pinned.dotFn
 have hsp:=dotLoop_special_false fa fb n 0 (ἇ 18 0) (ἇ 18 0) true
  (λk _ hk=>hfin k א)
 obtain ⟨-,-,hD,hz⟩:=Pinned.dotLoopFacts fa fb (Array.ofFn (n:=n) λi=>fa i)
  (Array.ofFn (n:=n) λi=>fb i) n (Pinned.getElem!_ofFn n fa) (Pinned.getElem!_ofFn n fb) Pinned.Acc.empty 0 n
  א (by simp only[Pinned.Acc.empty,Array.size_replicate]) (by simp only[Pinned.Acc.empty,Array.size_replicate]) hsp
 simp only[Pinned.Acc.empty] at hD hz hsp
 simp only[hsp,Bool.false_eq_true,ite_false]
 rw[Pinned.finishFast_eq]
 change finishWord (Pinned.Acc.total _) _ _=_
 have h0:(⟨ἇ 18 0,ἇ 18 0,false,true⟩:Pinned.Acc).total=0:=
  Pinned.Acc.total_empty
 rw[hD,hz,h0]
 simp only[zero_add,Int.reduceNeg,ne_eq,decide_not,Bool.true_and]
lemma qEcP (n:Nat) (fa fb:Nat->ὥ)
  (hf: ∀j,j<n->expOf (fa j)≠255/\expOf (fb j)≠255)
  (hn:(∑ j∈Ἣ n,rationalWord (fa j)*rationalWord (fb j))≠0):
  ∃d:FloatLib.Numerics.Dyadic,
   d.toRat=(∑ j∈Ἣ n,rationalWord (fa j)*rationalWord (fb j))/\
    Pinned.dotFn n fa fb=toWord (Model.roundDyadic Fmt d):=by
 let a:=Array.ofFn (n:=n) λi=>fa i
 let b:=Array.ofFn (n:=n) λi=>fb i
 have c:(dsum a b 0 n:ℚ)*2^(-298:Int)=
   ∑ j∈Ἣ n,rationalWord (fa j)*rationalWord (fb j):=by
  rw[qEl]
  refine Ἷ rfl λj m=>?_
  have h:=Finset.mem_range.mp m
  rw[Ἇ,qAb n fa j h,qAb n fb j h]
 have l:dsum a b 0 n≠0:= λh=>hn (by rw[<-c,h];simp only[Int.cast_zero,Int.reduceNeg,zpow_neg,zpow_ofNat,zero_mul])
 refine ⟨ofIntDyadic (dsum a b 0 n) (-298),by rw[toRat_ofIntDyadic,c],?_⟩
 rw[dotFn_finite n fa fb hf]
 exact finishWord_of_ne _ _ _ _ l (toRat_ofIntDyadic _ _)
/-! -/
/-! -/
section RnnK
@[inline] def stp (acc:F1) (q:ὧ):F1:=⟨acc.s+q,acc.t+q.abs⟩
@[specialize] def fpush (f:Nat->ὧ):Nat->Nat->ὤ->ὤ
 | _,0,a=>a
 | i,k+1,a=>fpush f (i+1) k (a.push (f i))
@[specialize] def allLt (f:Nat->Ὡ):Nat->Ὡ
 | 0=>true
 | k+1=>f k&&allLt f k
@[specialize] def build2 (f:Nat->Word×Word):Nat->Nat->ё->ё->
   ё×ё
 | _,0,a,b=>(a,b)
 | k,n+1,a,b=>
   let p:=f k
   build2 f (k+1) n (a.push p.1) (b.push p.2)
@[specialize] def rowsB (row:Nat->ὤ->ὤ):Nat->Nat->ὤ->ὤ
 | _,0,acc=>acc
 | r,n+1,acc=>rowsB row (r+1) n (row r acc)
@[specialize] def recFlat (step:Nat->ё×ё->ё×ё):
   Nat->Nat->ё×ё->ё->ё
 | _,0,_,out=>out
 | t,n+1,st,out=>
   let st':=step t st
   recFlat step (t+1) n st' (out++st'.1)
def zF0:ὧ:=0
def zFd:ὧ:=dv 0
def nfG1:ὧ:=nfOf 1
def nfG2:ὧ:=nfOf 2
def nfG3:ὧ:=nfOf 3
def oneD:ὧ:=dv oneW
def zFa:ὧ:=zFd.abs
@[noinline] def cellSlow (f i c g:Word):Word:=
 Pinned.dotFn 2 (λj=>if j=0 then f else i) (λj=>if j=0 then c else g)
@[inline] def cellF (f i c g:Word):Word:=
 if good f&&good i&&good c&&good g then
  let r:=stp (stp ⟨zF0,zF0⟩ (dv f*dv c)) (dv i*dv g)
  let z:=zFd
  let w:=ziv (r.s+z) (nfG2*(r.t+zFa))
  if w==Е then cellSlow f i c g else w
 else cellSlow f i c g
@[inline] def small (w:Word):Ὡ:=(w&&&В)<0x40000000
@[noinline] def mulSlow (o t:Word):Word:=toWord (Model.mul (ofW o) (ofW t))
@[inline] def mulF (o t:Word):Word:=
 if good o&&good t&&small o&&small t then
  let r:=stp ⟨zF0,zF0⟩ (dv o*dv t)
  let z:=zFd
  let w:=ziv (r.s+z) (nfG1*(r.t+zFa))
  if w==Е then mulSlow o t else w
 else mulSlow o t
@[noinline] def candSlow (inW H:Nat) (x wih bih:ё) (t u:Nat) (hu ru:Word):Word:=
 Pinned.dotFn (inW+2)
  (λj=>if j< inW then x[t*inW+j]! else if j=inW then oneW else hu)
  (λj=>if j< inW then wih[(2*H+u)*inW+j]! else if j=inW then bih[2*H+u]! else ru)
@[noinline] def upd3Slow (nu zu hu:Word):Word:=
 Pinned.dotFn 3 (λj=>if j=2 then hu else nu)
  (λj=>if j=0 then oneW else if j=1 then negW zu else zu)
@[inline] def updF (nu zu hu:Word):Word:=
 if good nu&&good zu&&good hu&&good (negW zu) then
  let r:=stp (stp (stp ⟨zF0,zF0⟩ (dv nu*oneD)) (dv nu*dv (negW zu))) (dv hu*dv zu)
  let z:=zFd
  let w:=ziv (r.s+z) (nfG3*(r.t+zFa))
  if w==Е then upd3Slow nu zu hu else w
 else upd3Slow nu zu hu
end RnnK
section RnnP
lemma fa_push_get (a:ὤ) (v:ὧ) (j:Nat):
  (a.push v).get! j=if j<a.size then a.get! j else if j=a.size then v else default:=by
 obtain ⟨ds⟩:=a
 show (ds.push v)[j]! =if j<ds.size then ds[j]! else if j=ds.size then v else default
 split_ifs with h1 h2
 · rw[Ἕ (ds.push v) j (by simp only[Array.size_push,Order.lt_add_one_iff];omega),Ἕ ds j h1,Array.getElem_push_lt h1]
 · subst h2;rw[Ἕ (ds.push v) ds.size (by simp only[Array.size_push,lt_add_iff_pos_right,Order.lt_one_iff]),Array.getElem_push_eq]
 · rw[ὑ (ds.push v) j (by simp only[Array.size_push,Order.lt_add_one_iff,not_le];omega)]
lemma fa_push_size (a:ὤ) (v:ὧ):(a.push v).size=a.size+1:=by
 obtain ⟨ds⟩:=a
 simp only[FloatArray.size,FloatArray.push,Array.size_push,Nat.add_left_cancel_iff]
lemma fpush_size (f:Nat->ὧ): ∀(i k:Nat) (a:ὤ),(fpush f i k a).size=a.size+k
 | _,0,_=>rfl
 | i,k+1,a=>by
   rw[fpush,fpush_size f (i+1) k,fa_push_size];omega
lemma fpush_get (f:Nat->ὧ): ∀(i k:Nat) (a:ὤ) (j:Nat),
  (j<a.size->(fpush f i k a).get! j=a.get! j)/\
  (a.size<=j->j<a.size+k->(fpush f i k a).get! j=f (i+(j-a.size)))
 | _,0,a,j=>⟨λ_=>rfl, λh1 h2=>א⟩
 | i,k+1,a,j=>by
   obtain ⟨ih1,ih2⟩:=fpush_get f (i+1) k (a.push (f i)) j
   rw[fa_push_size] at ih1 ih2
   refine ⟨λh=>?_, λh1 h2=>?_⟩
   · rw[fpush,ih1 א,fa_push_get,Ὦ h]
   · rw[fpush]
     by_cases hj:j=a.size
     · subst hj
       rw[ih1 א,fa_push_get,Ὧ א,Ὦ rfl];simp only[tsub_self,add_zero]
     · rw[ih2 א א];congr 1;omega
lemma foldl_stp_congr (g g':Nat->ὧ): ∀(l:љ) (acc:F1),
  (∀t∈l,g t=g' t)->
    l.foldl (λacc t=>stp acc (g t)) acc=l.foldl (λacc t=>stp acc (g' t)) acc:=
 λ_ _ h=>List.foldl_ext _ _ _ λ_ t ht=>by rw[h t ht]
lemma allLt_spec (f:Nat->Ὡ): ∀n,allLt f n=true-> ∀j,j<n->f j=true
 | 0,_,j,hj=>א
 | n+1,h,j,hj=>by
   simp only[allLt,Bool.and_eq_true] at h
   rcases Nat.lt_succ_iff_lt_or_eq.mp hj with hj | hj
   · exact allLt_spec f n h.2 j hj
   · subst hj;exact h.1
lemma good_zero:good 0=true:=ה
lemma good_one:good oneW=true:=ה
lemma all_good (a:ё) (h:a.all good=true) (i:Nat):good a[i]! =true:=by
 by_cases hi:i<a.size
 · rw[Ἕ a i hi]
   exact (Array.all_eq_true.mp h) i hi
 · rw[ὑ a i hi];exact good_zero
lemma rat_zero:rationalWord (0:Word)=0:=by
 norm_num[rationalWord,decodeD,show expOf 0=0 by decide,show fracOf 0=0 by decide,
  FloatLib.Numerics.Dyadic.toRat,FloatLib.Numerics.Dyadic.signedSignificand]
lemma fin_ofW {w:Word} (h:expOf w≠255):Model.isFinite (ofW w)=true:=by
 simp only[ofW,isFinite_ofUInt32,bne_iff_ne,ne_eq,h,not_false_eq_true]
lemma fastNB (N:Nat) (hN:N<2^20) (fa fb:Nat->Word)
  (hx: ∀t,t<N->good (fa t)=true) (hw: ∀t,t<N->good (fb t)=true)
  (g:Nat->ὧ) (hg: ∀t,t<N->g t=dv (fa t)*dv (fb t))
  (h:ziv (((ἄ N).foldl (λacc t=>stp acc (g t)) ⟨0,0⟩).s+dv 0)
    (nfOf N*(((ἄ N).foldl (λacc t=>stp acc (g t)) ⟨0,0⟩).t+(dv 0).abs))≠Е):
  ziv (((ἄ N).foldl (λacc t=>stp acc (g t)) ⟨0,0⟩).s+dv 0)
   (nfOf N*(((ἄ N).foldl (λacc t=>stp acc (g t)) ⟨0,0⟩).t+(dv 0).abs))=
   Pinned.dotFn N fa fb:=by
 have hr:(ἄ N).foldl (λacc t=>stp acc (g t)) ⟨0,0⟩=
   (ἄ N).foldl (λacc t=>stp acc (dv (fa t)*dv (fb t))) ⟨0,0⟩:=
  foldl_stp_congr _ _ _ _ λt ht=>hg t (List.mem_range.mp ht)
 rw[hr] at h⊢
 obtain ⟨-,-,hbig⟩:=qBw _ _ h
 obtain ⟨hlo,hhi⟩:=qDm N hN fa fb 0 hx hw good_zero (ἄ N) Ὂ
  (λt ht=>List.mem_range.mp ht) (λt ht hn=>absurd (List.mem_range.mpr ht) hn) _ _
  ⟨rfl,rfl⟩ hbig
 have hsum:((∑ j∈Ἣ N,rationalWord (fa j)*rationalWord (fb j):ℚ):ℝ)=exactS N fa fb 0:=by
  rw[Rat.cast_sum,exactS,qAx 0 (qAe _ good_zero),rat_zero,Rat.cast_zero,add_zero]
  refine Ἷ rfl λt ht=>?_
  have a:=Finset.mem_range.mp ht
  rw[qAx _ (qAe _ (hx t a)),qAx _ (qAe _ (hw t a)),Rat.cast_mul]
 have hne:(∑ j∈Ἣ N,rationalWord (fa j)*rationalWord (fb j))≠0:= λh0=>
  qG5 _ _ _ hlo hhi h (by rw[<-hsum,h0,Rat.cast_zero])
 obtain ⟨d,hd,hD⟩:=qEcP N fa fb (λj hj=>⟨qAe _ (hx j hj),qAe _ (hw j hj)⟩) hne
 rw[hD]
 exact qIq _ _ d (by rw[hd,hsum];exact hlo) (by rw[hd,hsum];exact hhi) h
lemma small_le (w:Word) (hg:good w=true) (hs:small w=true): |wv w| <=2:=by
 have h1:=qB5 w hg
 have hq:=qAq w
 have hlt:(w&&&В).toNat<2^30:=by
  have:(w&&&В)<0x40000000:=by simpa only[small,decide_eq_true_eq] using hs
  exact (UInt32.lt_iff_toNat_lt.mp this)
 rw[hq] at hlt
 rw[h1]
 have h2:=qA6 (w.toNat%2^31) (2^30) hlt.le (by rcases qAz w hg with h | h<;>omega)
 refine h2.trans (le_of_eq ?_)
 norm_num[A_mag]
lemma posMax32:(4:ℝ)<=Model.toReal (Model.posMaxFinite Fmt):=by
 rw[Model.toReal_posMaxFinite,show Model.pow2 Fmt.fracWidth+Fmt.maxFiniteFracField=2^24-1 by
  decide+kernel,show Fmt.maxNormalExponent-Int.ofNat Fmt.fracWidth=104 by decide+kernel,
  Model.bpow,qA0]
 norm_num
lemma fastMul (o t:Word) (ho:good o=true) (ht:good t=true) (so:small o=true)
  (st:small t=true)
  (h:ziv ((stp ⟨0,0⟩ (dv o*dv t)).s+dv 0) (nfOf 1*((stp ⟨0,0⟩ (dv o*dv t)).t+(dv 0).abs))≠Е):
  ziv ((stp ⟨0,0⟩ (dv o*dv t)).s+dv 0) (nfOf 1*((stp ⟨0,0⟩ (dv o*dv t)).t+(dv 0).abs))=
   toWord (Model.mul (ofW o) (ofW t)):=by
 have hl:(ἄ 1).foldl (λacc j=>stp acc (dv ((λ_=>o) j)*dv ((λ_=>t) j))) ⟨0,0⟩=
   stp ⟨0,0⟩ (dv o*dv t):=by
  simp only[List.range_one,ὔ,List.foldl_nil]
 rw[<-hl] at h⊢
 obtain ⟨-,-,hbig⟩:=qBw _ _ h
 obtain ⟨hlo,hhi⟩:=qDm 1 ב (λ_=>o) (λ_=>t) 0 (λ_ _=>ho) (λ_ _=>ht)
  good_zero (ἄ 1) Ὂ (λt ht=>List.mem_range.mp ht)
  (λt ht hn=>absurd (List.mem_range.mpr ht) hn) _ _ ⟨rfl,rfl⟩ hbig
 have hv:exactS 1 (λ_=>o) (λ_=>t) 0=Model.toReal (ofW o)*Model.toReal (ofW t):=by
  have h0:Model.toReal (ofW 0)=0:=by
   have:=qAx 0 (qAe _ good_zero)
   rw[wv] at this;rw[this,rat_zero,Rat.cast_zero]
  simp only[exactS,Finset.range_one,wv,Finset.sum_const,Finset.card_singleton,one_smul,h0,add_zero]
 rw[hv] at hlo hhi
 have hfo:=fin_ofW (qAe _ ho)
 have hft:=fin_ofW (qAe _ ht)
 have hbound: |Model.toReal (ofW o)| * |Model.toReal (ofW t)| <=Model.toReal (Model.posMaxFinite Fmt):=by
  have a:=small_le o ho so
  have b:=small_le t ht st
  simp only[wv] at a b
  exact ὂ (ἑ a b (Ἀ _) zero_le_two) (ὂ (le_of_eq ב) posMax32)
 have hf:=Model.isFinite_mul_of_abs_mul_le_posMaxFinite (ofW o) (ofW t) ה hfo hft hbound
 have hM:=Model.toReal_mul_eq_roundAt_of_abs_mul_le_posMaxFinite (ofW o) (ofW t) ה hfo hft
  hbound
 exact (qDq _ _ _ hlo hhi h _ hf hM).symm
lemma ofFn_get! (n:Nat) (g:Nat->Word) (i:Nat) (h:i<n):
  (Array.ofFn (n:=n) λj=>g j.val)[i]! =g i:=by
 rw[Ἕ _ i (by simpa only[Array.size_ofFn] using h),Array.getElem_ofFn]
lemma push_ofFnA {α:Type} (g:Nat->α) (k:Nat):
  (Array.ofFn (n:=k) λi=>g i.val).push (g k)=Array.ofFn (n:=k+1) λi=>g i.val:=by
 apply Մ
 · simp only[Array.size_push,Array.size_ofFn,Nat.add_left_cancel_iff]
 · intro i h1 h2
   simp only[Array.size_push,Array.size_ofFn] at h1
   rw[Array.getElem_ofFn]
   by_cases hi:i<k
   · rw[Array.getElem_push_lt (by simpa only[Array.size_ofFn] using hi),Array.getElem_ofFn]
   · have:i=k:=א
     subst this
     simp only[Array.getElem_push,Array.size_ofFn,lt_self_iff_false, ↓reduceDIte]
lemma ofFn_get!A {α:Type} [Inhabited α] (n:Nat) (g:Nat->α) (i:Nat) (h:i<n):
  (Array.ofFn (n:=n) λj=>g j.val)[i]! =g i:=by
 rw[Ἕ _ i (by simpa only[Array.size_ofFn] using h),Array.getElem_ofFn]
lemma ofFn_get!F {α:Type} [Inhabited α] {n:Nat} (f:Fin n->α) (i:Nat) (h:i<n):
  (Array.ofFn f)[i]! =f ⟨i,h⟩:=by
 rw[Ἕ _ i (by simpa only[Array.size_ofFn] using h),Array.getElem_ofFn]
lemma build2_spec (f:Nat->Word×Word): ∀(n k:Nat) (a b:ё),
  a=Array.ofFn (n:=k) (λi=>(f i.val).1)->b=Array.ofFn (n:=k) (λi=>(f i.val).2)->
  build2 f k n a b=
   (Array.ofFn (n:=k+n) (λi=>(f i.val).1),Array.ofFn (n:=k+n) (λi=>(f i.val).2))
 | 0,k,a,b,ha,hb=>by rw[build2,ha,hb];rfl
 | n+1,k,a,b,ha,hb=>by
   rw[build2]
   try dsimp only
   rw[build2_spec f n (k+1) _ _ (by rw[ha];exact push_ofFnA (λi=>(f i).1) k)
    (by rw[hb];exact push_ofFnA (λi=>(f i).2) k),show k+1+n=k+(n+1) by omega]
lemma build2_zero (f:Nat->Word×Word) (n c:Nat):
  build2 f 0 n (Array.emptyWithCapacity c) (Array.emptyWithCapacity c)=
   (Array.ofFn (n:=n) (λi=>(f i.val).1),Array.ofFn (n:=n) (λi=>(f i.val).2)):=by
 rw[build2_spec f n 0 _ _ (by simp only[Array.emptyWithCapacity_eq,Array.ofFn_zero]) (by simp only[Array.emptyWithCapacity_eq,Array.ofFn_zero]),Ἇ]
lemma rowsB_spec (N:Nat) (row:Nat->ὤ->ὤ) (g:Nat->Nat->ὧ)
  (hsz: ∀r acc,(row r acc).size=acc.size+N)
  (hold: ∀r acc j,j<acc.size->(row r acc).get! j=acc.get! j)
  (hnew: ∀r acc j,j<N->(row r acc).get! (acc.size+j)=g r j):
  ∀(n r:Nat) (acc:ὤ),acc.size=r*N->
   (∀r' j,r'<r->j<N->acc.get! (r'*N+j)=g r' j)->
   (rowsB row r n acc).size=(r+n)*N/\
   ∀r' j,r'<r+n->j<N->(rowsB row r n acc).get! (r'*N+j)=g r' j
 | 0,r,acc,hs,hg=>by rw[rowsB,Ἵ];exact ⟨hs,hg⟩
 | n+1,r,acc,hs,hg=>by
   rw[rowsB]
   have:=rowsB_spec N row g hsz hold hnew n (r+1) (row r acc)
    (by rw[hsz,hs,ὰ,Ὑ])
    (λr' j hr hj=>by
      rcases Nat.lt_succ_iff_lt_or_eq.mp hr with hr | hr
      · have hlt:r'*N+j<acc.size:=by
         rw[hs];have:=Ἒ N (show r'+1<=r by omega)
         rw[ὰ,Ὑ] at this;omega
        rw[hold r acc _ hlt,hg r' j hr hj]
      · rw[hr,<-hs,hnew r acc j hj])
   rw[show r+1+n=r+(n+1) by omega] at this
   exact this
def traj (step:Nat->ё×ё->ё×ё) (iniS:ё×ё):
    Nat->ё×ё
 | 0=>iniS
 | t+1=>step t (traj step iniS t)
lemma foldl_traj (step:Nat->ё×ё->ё×ё)
  (iniS:ё×ё): ∀T,(ἄ T).foldl
    (λ(acc:(ё×ё)×Array (ё)) t=>
      let st:=step t acc.1
      (st,acc.2.push st.1)) (iniS, #[])=
    (traj step iniS T,Array.ofFn (n:=T) λt=>(traj step iniS (t.val+1)).1)
 | 0=>by simp only[List.range_zero,List.foldl_nil,traj,Array.ofFn_zero]
 | T+1=>by
   rw[List.range_succ,List.foldl_append,foldl_traj step iniS T]
   simp only[ὔ,List.foldl_nil]
   rw[show step T (traj step iniS T)=traj step iniS (T+1) from rfl,
    push_ofFnA (λt=>(traj step iniS (t+1)).1) T]
lemma divH (q t H:Nat) (h1:t*H<=q) (h2:q<(t+1)*H):q/H=t/\q%H=q-t*H:=by
 have hH:0<H:=by
  rcases Nat.eq_zero_or_pos H with h | h
  · subst h;simp only[mul_zero,not_lt_zero] at h2
  · exact h
 have e:q/H=t:=Nat.div_eq_of_lt_le h1 h2
 refine ⟨e,?_⟩
 have:=Nat.mod_add_div q H
 rw[e,ἥ] at this
 omega
lemma recFlat_traj (step:Nat->ё×ё->ё×ё)
  (iniS:ё×ё) (H:Nat) (hsz: ∀t st,(step t st).1.size=H):
  ∀(n t:Nat) (out:ё),
   out=Array.ofFn (n:=t*H) (λq=>(traj step iniS (q.val/H+1)).1[q.val%H]!)->
   recFlat step t n (traj step iniS t) out=
    Array.ofFn (n:=(t+n)*H) (λq=>(traj step iniS (q.val/H+1)).1[q.val%H]!)
 | 0,t,out,ho=>by rw[recFlat,ho,Ἵ]
 | n+1,t,out,ho=>by
   rw[recFlat]
   try dsimp only
   rw[show step t (traj step iniS t)=traj step iniS (t+1) from rfl]
   rw[recFlat_traj step iniS H hsz n (t+1) _ ?_,show t+1+n=t+(n+1) by omega]
   rw[ho]
   have hs:(traj step iniS (t+1)).1.size=H:=hsz _ _
   apply Մ
   · simp only[Array.size_append,Array.size_ofFn,hs,ὰ,one_mul,Nat.add_left_cancel_iff]
   · intro q h1 h2
     simp only[Array.size_append,Array.size_ofFn,hs] at h1
     rw[Array.getElem_ofFn]
     by_cases hq:q<t*H
     · rw[Array.getElem_append_left (by simpa only[Array.size_ofFn] using hq),Array.getElem_ofFn]
     · rw[Array.getElem_append_right (by simp only[Array.size_ofFn];omega)]
       obtain ⟨d1,d2⟩:=divH q t H א (by rw[ὰ,Ὑ];omega)
       simp only[Array.size_ofFn]
       rw[d1,d2,Ἕ _ _ (by rw[hs];omega)]
lemma recFlat_eq (T H:Nat) (step:Nat->ё×ё->ё×ё)
  (iniS:ё×ё) (hsz: ∀t st,(step t st).1.size=H):
  recFlat step 0 T iniS (Array.emptyWithCapacity (T*H))=recurWith T H step iniS:=by
 have:=recFlat_traj step iniS H hsz T 0 (Array.emptyWithCapacity (T*H)) (by simp only[Array.emptyWithCapacity_eq,Array.empty_eq,Array.ofFn_eq_empty_iff,zero_mul])
 rw[show traj step iniS 0=iniS from rfl,Ἇ] at this
 rw[this,recurWith]
 simp only[foldl_traj]
 congr 1
 funext q
 have hq:q.val/H<T:=Ἰ (Nat.lt_of_lt_of_eq q.isLt (ἥ T H))
 rw[ofFn_get!A T (λt=>(traj step iniS (t+1)).1) _ hq]
end RnnP
section RnnS
lemma cellF_eq (f i c g:Word):
  cellF f i c g=Pinned.dotFn 2 (λj=>if j=0 then f else i) (λj=>if j=0 then c else g):=by
 unfold cellF cellSlow zFa zF0 zFd nfG2
 simp only[]
 split_ifs with hg hw
 · rfl
 · rw[beq_iff_eq] at hw
   simp only[Bool.and_eq_true] at hg
   obtain ⟨⟨⟨h1,h2⟩,h3⟩,h4⟩:=hg
   have hl:(ἄ 2).foldl (λacc j=>stp acc (dv ((λj=>if j=0 then f else i) j)*
     dv ((λj=>if j=0 then c else g) j))) ⟨0,0⟩=stp (stp ⟨0,0⟩ (dv f*dv c)) (dv i*dv g):=by
    simp only[List.range_succ,List.range_zero,List.nil_append,List.cons_append,ὔ, ↓reduceIte,one_ne_zero,List.foldl_nil]
   rw[<-hl] at hw⊢
   exact fastNB 2 ה _ _
    (λj hj=>by rcases (by omega:j=0\/j=1) with rfl | rfl<;>simp only[↓reduceIte,h1,one_ne_zero,h2])
    (λj hj=>by rcases (by omega:j=0\/j=1) with rfl | rfl<;>simp only[↓reduceIte,h3,one_ne_zero,h4])
    _ (λ_ _=>rfl) hw
 · rfl
lemma mulF_eq (o t:Word):mulF o t=toWord (Model.mul (ofW o) (ofW t)):=by
 unfold mulF mulSlow zFa zF0 zFd nfG1
 simp only[]
 split_ifs with hg hw
 · rfl
 · rw[beq_iff_eq] at hw
   simp only[Bool.and_eq_true] at hg
   obtain ⟨⟨⟨h1,h2⟩,h3⟩,h4⟩:=hg
   exact fastMul o t h1 h2 h3 h4 hw
 · rfl
lemma updF_eq (nu zu hu:Word):
  updF nu zu hu=Pinned.dotFn 3 (λj=>if j=2 then hu else nu)
   (λj=>if j=0 then oneW else if j=1 then negW zu else zu):=by
 unfold updF upd3Slow zFa zF0 zFd nfG3 oneD
 simp only[]
 split_ifs with hg hw
 · rfl
 · rw[beq_iff_eq] at hw
   simp only[Bool.and_eq_true] at hg
   obtain ⟨⟨⟨h1,h2⟩,h3⟩,h4⟩:=hg
   have hl:(ἄ 3).foldl (λacc j=>stp acc (dv ((λj=>if j=2 then hu else nu) j)*
     dv ((λj=>if j=0 then oneW else if j=1 then negW zu else zu) j))) ⟨0,0⟩=
     stp (stp (stp ⟨0,0⟩ (dv nu*dv oneW)) (dv nu*dv (negW zu))) (dv hu*dv zu):=by
    simp only[List.range_succ,List.range_zero,List.nil_append,List.cons_append,ὔ,OfNat.zero_ne_ofNat, ↓reduceIte,OfNat.one_ne_ofNat,one_ne_zero,OfNat.ofNat_ne_zero,OfNat.ofNat_ne_one,List.foldl_nil]
   rw[<-hl] at hw⊢
   exact fastNB 3 ה _ _
    (λj hj=>by rcases (by omega:j=0\/j=1\/j=2) with rfl | rfl | rfl<;>simp only[OfNat.zero_ne_ofNat, ↓reduceIte,h1,OfNat.one_ne_ofNat,h3])
    (λj hj=>by
      rcases (by omega:j=0\/j=1\/j=2) with rfl | rfl | rfl<;>simp only[↓reduceIte,good_one,one_ne_zero,h4,OfNat.ofNat_ne_zero,OfNat.ofNat_ne_one,h2])
    _ (λ_ _=>rfl) hw
 · rfl
end RnnS
section RnnT
lemma stepIdx4 (H u:Nat) (hu:u<H):
  H+u<4*H/\u<4*H/\2*H+u<4*H/\3*H+u<4*H:=by
 omega
end RnnT
section ActF
open FastAct in
def sigmoidFastX (w:ὥ):Option ὥ:=
 if expOf w=255 then none else
 let d:=decodeD w
 match expNeg d.significand (d.exponent-1) with
 | none=>none
 | some e1=>
   let t:=(mulDn e1.1 e1.1,min one (mulUp e1.2 e1.2))
   if d.negative then
    match divLo t.1 (one+t.1),divHi t.2 (one+t.2) with
    | some ql,some qh=>settle false ql qh
    | _,_=>none
   else none
open FastAct in
lemma sigmoidFastX_spec {w a:ὥ} (h:sigmoidFastX w=some a):sigmoidCR w=a:=by
 unfold sigmoidFastX at h
 split at h
 · simp only[reduceCtorEq] at h
 rename_i hfin
 dsimp only at h
 have hx:=toReal_ofW_decode hfin
 generalize decodeD w=d at h hx
 have hspec:=(toReal_sigmoidModel (fmt:=Fmt) ה
  (input:=Model.ofUInt32 w) (by simp only[isFinite_ofUInt32,bne_iff_ne,ne_eq,hfin,not_false_eq_true])).2
 change Model.toReal (sigmoidModel (ofW w))=Model.roundAt Fmt (realSigmoid (Model.toReal (ofW w))) at hspec
 rw[hx] at hspec
 change toWord (sigmoidModel (ofW w))=a
 split at h
 · simp only[reduceCtorEq] at h
 rename_i e1 he1
 have hE1:=expNeg_spec he1
 set C:=(d.significand:ℝ)*(2:ℝ)^d.exponent
 have hE:=hE1.mul hE1
 have hC:(d.significand:ℝ)*(2:ℝ)^(d.exponent-1)+
   (d.significand:ℝ)*(2:ℝ)^(d.exponent-1)=C:=by
  simp only[C];rw[ᾂ ἓ,zpow_one];ring
 rw[hC] at hE
 set T:=Real.exp (-C)
 have hT:0<T:=Real.exp_pos _
 obtain ⟨h1,h2,h3⟩:=hE
 simp only at h1 h2 h3
 set t1:=mulDn e1.1 e1.1 with ht1d
 set t2:=min one (mulUp e1.2 e1.2) with ht2d
 have v1:=V_nonneg t1
 have v2:=V_nonneg t2
 split at h
 · rename_i hneg
   split at h
   · rename_i ql qh hql hqh
     have bl:=divLo_spec hql
     have bh:=divHi_spec hqh
     rw[V_add,V_one] at bl bh
     rw[settle_eq] at h
     simp only[Bool.false_eq_true,ite_false] at h
     refine settleInt_spec h (v:=realSigmoid (-C)) ?_ ?_ (by rw[hspec,hneg];simp only[↓reduceIte,neg_mul,one_mul])
     · rw[int_scale]
       have:realSigmoid (-C)=T/(1+T):=realSigmoid_neg_eq C
       rw[this,le_div_iff₀ (by linarith only[hT])]
       nlinarith only[bl,h1,v1,hT,V_nonneg ql]
     · rw[int_scale]
       have:realSigmoid (-C)=T/(1+T):=realSigmoid_neg_eq C
       rw[this,div_le_iff₀ (by linarith only[hT])]
       nlinarith only[bh,h2,v2,hT,V_nonneg qh]
   · simp only[reduceCtorEq] at h
 · simp only[reduceCtorEq] at h
@[noinline] def sigmoidF9 (w:ὥ):ὥ:=
 match sigmoidFastX w with
 | some a=>a
 | none=>FastAct.sigmoidF w
@[noinline] def tanhF9 (w:ὥ):ὥ:=FastAct.tanhF w
lemma sigmoidF9_eq (w:ὥ):sigmoidF9 w=sigmoidCR w:=by
 unfold sigmoidF9
 cases h:sigmoidFastX w with
 | some a=>exact (sigmoidFastX_spec h).symm
 | none=>exact congrFun FastAct.sigmoidF_eq w
lemma tanhF9_eq (w:ὥ):tanhF9 w=tanhCR w:=by
 unfold tanhF9;exact congrFun FastAct.tanhF_eq w
end ActF
/-! -/
section ActG
def enclK (k:Nat):Nat×Nat:=
 let n:=k/64
 let F:=FastAct.fracTab[k%64]!
 let I:=if n<=16 then FastAct.intTab[n]! else
  (FastAct.mulDn FastAct.intTab[16]!.1 FastAct.intTab[n-16]!.1,
    min FastAct.one (FastAct.mulUp FastAct.intTab[16]!.2 FastAct.intTab[n-16]!.2))
 (FastAct.mulDn I.1 F.1,min FastAct.one (FastAct.mulUp I.2 F.2))
def toF62 (lo:Nat):ὧ:=
 let sh:=lo.log2-52
 (lo>>>sh).toUInt64.toFloat*D.pwEntry (88+sh)
def expTabG:ὤ:=⟨Array.ofFn (n:=1280) λk=>toF62 (enclK k.val).1⟩
def tabOK:Ὡ:=(ἄ 1280).all λk=>
 let e:=enclK k
 decide (e.2-e.1<=2^14)&&(decide (1088<=k) || decide ((e.2-e.1)*2^36<=e.1))
def oneG:ὧ:=(1:Ὠ).toFloat
def halfG:ὧ:=oneG/(2:Ὠ).toFloat
def c6G:ὧ:=oneG/(6:Ὠ).toFloat
def c24G:ὧ:=oneG/(24:Ὠ).toFloat
def c120G:ὧ:=oneG/(120:Ὠ).toFloat
def m34G:ὧ:=D.pwEntry 116
def m41G:ὧ:=D.pwEntry 109
def m26G:ὧ:=D.pwEntry 124
def negOneG:ὧ:=0-oneG
def loA:Ὠ:=D.loOf m41G
def spA:Ὠ:=D.spOf loA
def e0G:ὧ:=m34G*m34G
def lo0:Ὠ:=D.loOf e0G
def sp0:Ὠ:=D.spOf lo0
@[inline] def zA (lo sp:Ὠ) (q e:ὧ):ὥ:=
 let r:=D.fin1 lo sp q
 if r==Е then D.ziv q e else r
@[inline] def polyG (r:ὧ):ὧ:=
 let r2:=r*r
 (oneG-r)+r2*((halfG-r*c6G)+r2*(c24G-r*c120G))
lemma tabG_size:expTabG.size=1280:=by
 simp only[FloatArray.size,expTabG,Array.size_ofFn]
lemma kidx {k b:ὥ} (h:k<b) (hb:b.toNat<=1280):k.toUSize.toNat<expTabG.size:=by
 rw[tabG_size];simp only[ὃ];have:=UInt32.lt_iff_toNat_lt.mp h;omega
lemma kLt {k b:ὥ} (h:k<b) (hb:b<=1280):k<1280:=by
 rw[ἳ] at*;rw[ἃ] at hb;simp only[UInt32.reduceToNat] at*;omega
lemma eidx (a:ὥ):(a>>>23).toUSize.toNat<D.pw.size:=D.qCh a
@[inline] def sOfG (F:ὥ):ὥ:=if 144-F<31 then 144-F else 31
@[inline] def kOfG (m F:ὥ):ὥ:=m>>>sOfG F
@[inline] def rOfG (m F:ὥ) (h:F.toUSize.toNat<D.pw.size):ὧ:=
 (m-(kOfG m F<<<sOfG F)).toUInt64.toFloat*D.pw.uget F.toUSize h
@[inline] def tOfG (m F:ὥ) (h:F.toUSize.toNat<D.pw.size) (hk:kOfG m F<1280):ὧ:=
 expTabG.uget (kOfG m F).toUSize (kidx hk (by decide))*polyG (rOfG m F h)
@[inline] def sigG (w:ὥ):ὥ:=
 let a:=w&&&В
 let F:=a>>>23
 if F-1<131 then
  let m:=(a&&&0x7FFFFF) ||| 0x800000
  if w<س then
   if h:kOfG m F<1216 then
    let t:=tOfG m F (eidx a) (kLt h (by decide))
    let q:=oneG/(oneG+t)
    zA loA spA q m41G
   else ziv oneG m26G
  else
   if h:kOfG m F<1088 then
    let t:=tOfG m F (eidx a) (kLt h (by decide))
    let q:=t/(oneG+t)
    zA lo0 sp0 q (q*m34G)
   else Е
 else if 131<F&&F<255&&w<س then ziv oneG m26G
 else Е
@[inline] def tanhG (w:ὥ):ὥ:=
 let a:=w&&&В
 let F:=(a+0x800000)>>>23
 if F-2<131 then
  let m:=(a&&&0x7FFFFF) ||| 0x800000
  if h:kOfG m F<1280 then
   let t:=tOfG m F (eidx _) h
   let q:=(if w<س then oneG-t else t-oneG)/(oneG+t)
   zA loA spA q m41G
  else ziv (if w<س then oneG else negOneG) m26G
 else if 132<F&&F<256 then ziv (if w<س then oneG else negOneG) m26G
 else Е
def sigmoidF2 (w:ὥ):ὥ:=
 let v:=if tabOK then sigG w else Е
 if v==Е then sigmoidF9 w else v
def tanhF2 (w:ὥ):ὥ:=
 let v:=if tabOK then tanhG w else Е
 if v==Е then tanhF9 w else v
end ActG
/-! -/
section GTool
def gAp (x:ὧ) (X ε:ℝ):Prop:=x.isFinite=true/\ |D.R x-X| <=ε
lemma gAz:D.A_z<=1/2^100:=by
 unfold D.A_z
 rw[show (1:ℝ)/2^100=(2:ℝ)^(-100:ℤ) by norm_num]
 exact ἒ ב ב
lemma gAB:(2:ℝ)^100<=D.A_B:=by
 unfold D.A_B
 rw[show (2:ℝ)^100=(2:ℝ)^(100:ℤ) by norm_num]
 exact ἒ ב ב
lemma gAp.mono {x:ὧ} {X ε ε':ℝ} (h:gAp x X ε) (he:ε<=ε'):gAp x X ε':=
 ⟨h.1,h.2.trans he⟩
lemma gAp.absR {x:ὧ} {X ε:ℝ} (h:gAp x X ε): |D.R x| <= |X| +ε:=by
 have:=Ὥ (D.R x) X
 linarith only[h.2,this,h]
lemma gAp.bnd {x:ὧ} {X ε M:ℝ} (h:gAp x X ε) (hX: |X| <=M) (hM:M+ε<=2^100):
  |D.R x| <=D.A_B:=by
 have:=h.absR;linarith only[gAB,hX,hM,this]
lemma gAp_add {x y:ὧ} {X Y ε δ M N:ℝ} (hx:gAp x X ε) (hy:gAp y Y δ)
  (hX: |X| <=M) (hY: |Y| <=N) (hM:M+ε<=2^100) (hN:N+δ<=2^100):
  gAp (x+y) (X+Y) (ε+δ+1/2^53*(M+N+ε+δ)+1/2^153):=by
 obtain ⟨f,e⟩:=D.qBf x y hx.1 hy.1 (hx.bnd hX hM) (hy.bnd hY hN)
 refine ⟨f,?_⟩
 have h1:=hx.absR;have h2:=hy.absR
 have hz:=gAz
 have ha: |D.R x+D.R y| <=M+N+ε+δ:=(Ἐ _ _).trans (by linarith only[hX,hY,h1,h2])
 have hb: |D.R x+D.R y-(X+Y)| <=ε+δ:=by
  have:=Ἐ (D.R x-X) (D.R y-Y)
  rw[show D.R x-X+(D.R y-Y)=D.R x+D.R y-(X+Y) by ring] at this
  linarith only[hx.2,hy.2,this,hx,hy]
 have hc:=abs_sub_le (D.R (x+y)) (D.R x+D.R y) (X+Y)
 have:1/2^53*(|D.R x+D.R y| +D.A_z)<=1/2^53*(M+N+ε+δ)+1/2^153:=by
  have:(1:ℝ)/2^153=1/2^53*(1/2^100):=ב
  rw[this,<-mul_add]
  exact ἁ (by linarith only[hz,ha]) ב
 linarith only[e,hz,ha,hb,hc]
lemma gAp_sub {x y:ὧ} {X Y ε δ M N:ℝ} (hx:gAp x X ε) (hy:gAp y Y δ)
  (hX: |X| <=M) (hY: |Y| <=N) (hM:M+ε<=2^100) (hN:N+δ<=2^100):
  gAp (x-y) (X-Y) (ε+δ+1/2^53*(M+N+ε+δ)+1/2^153):=by
 obtain ⟨f,e⟩:=D.qDf x y hx.1 hy.1 (hx.bnd hX hM) (hy.bnd hY hN)
 refine ⟨f,?_⟩
 have h1:=hx.absR;have h2:=hy.absR
 have hz:=gAz
 have ha: |D.R x-D.R y| <=M+N+ε+δ:=(Ὕ _ _).trans (by linarith only[hX,hY,h1,h2])
 have hb: |D.R x-D.R y-(X-Y)| <=ε+δ:=by
  have:=Ὕ (D.R x-X) (D.R y-Y)
  rw[show D.R x-X-(D.R y-Y)=D.R x-D.R y-(X-Y) by ring] at this
  linarith only[hx.2,hy.2,this,hx,hy]
 have hc:=abs_sub_le (D.R (x-y)) (D.R x-D.R y) (X-Y)
 have:1/2^53*(|D.R x-D.R y| +D.A_z)<=1/2^53*(M+N+ε+δ)+1/2^153:=by
  have:(1:ℝ)/2^153=1/2^53*(1/2^100):=ב
  rw[this,<-mul_add]
  exact ἁ (by linarith only[hz,ha]) ב
 linarith only[e,hz,ha,hb,hc]
lemma gAp_mul {x y:ὧ} {X Y ε δ M N:ℝ} (hx:gAp x X ε) (hy:gAp y Y δ)
  (hX: |X| <=M) (hY: |Y| <=N) (hM:M+ε<=2^100) (hN:N+δ<=2^100):
  gAp (x*y) (X*Y) ((M+ε)*δ+N*ε+1/2^53*((M+ε)*(N+δ))+1/2^153):=by
 obtain ⟨f,e⟩:=D.qCf x y hx.1 hy.1 (hx.bnd hX hM) (hy.bnd hY hN)
 refine ⟨f,?_⟩
 have h1:=hx.absR;have h2:=hy.absR
 have hz:=gAz
 have e0:0<=ε:=(Ἀ _).trans hx.2
 have d0:0<=δ:=(Ἀ _).trans hy.2
 have ha: |D.R x*D.R y| <=(M+ε)*(N+δ):=by
  rw[ἢ];exact ἑ (by linarith only[hX,h1]) (by linarith only[hY,h2]) (Ἀ _) (by linarith only[Ἀ X,hX,e0])
 have hb: |D.R x*D.R y-X*Y| <=(M+ε)*δ+N*ε:=by
  have:D.R x*D.R y-X*Y=D.R x*(D.R y-Y)+Y*(D.R x-X):=by ring
  rw[this]
  refine (Ἐ _ _).trans ?_
  rw[ἢ,ἢ]
  exact add_le_add (ἑ (by linarith only[hX,h1]) hy.2 (Ἀ _) (by linarith only[Ἀ X,hX,e0]))
   (ἑ hY hx.2 (Ἀ _) (by linarith only[Ἀ Y,hY]))
 have hc:=abs_sub_le (D.R (x*y)) (D.R x*D.R y) (X*Y)
 have:1/2^53*(|D.R x*D.R y| +D.A_z)<=1/2^53*((M+ε)*(N+δ))+1/2^153:=by
  have:(1:ℝ)/2^153=1/2^53*(1/2^100):=ב
  rw[this,<-mul_add]
  exact ἁ (by linarith only[hz,ha]) ב
 linarith only[e,hz,ha,hb,hc]
lemma gDiv (x y:ὧ) (hx:x.isFinite=true) (hy:y.isFinite=true) (hy0:D.R y≠0)
  (hb: |D.R x/D.R y| <=D.A_B):
  (x/y).isFinite=true/\ |D.R (x/y)-D.R x/D.R y| <=1/2^53*(|D.R x/D.R y| +D.A_z):=by
 have hz:Model.isZero (D.A_M y)=false:=by
  cases h:Model.isZero (D.A_M y) with
  | false=>rfl
  | true=>exact absurd (Model.toReal_eq_zero_of_isZero _ h) hy0
 have hB: |Model.toReal (D.A_M x)/Model.toReal (D.A_M y)| <=Model.toReal (Model.posMaxFinite ⟡):=
  have h1:=ὂ (by linarith only[gAB]:D.A_B<=D.A_B*2) D.qE3
  hb.trans (h1.trans D.qCd)
 exact D.qFc (D.A_E _ (toModel_ofFloat_div_of_isFinite x y hx hy)
  (Model.isFinite_div_of_abs_div_le_posMaxFinite _ _ D.qAj (D.qA3 hx) (D.qA3 hy) hz hB)
  (Model.toReal_div_eq_roundAt_of_abs_div_le_posMaxFinite _ _ D.qAj (D.qA3 hx) (D.qA3 hy) hz hB))
end GTool
section GTool2
lemma gAp_div {x y:ὧ} {X Y ε δ M m:ℝ} (hx:gAp x X ε) (hy:gAp y Y δ)
  (hX: |X| <=M) (hm:m<=Y) (hδ:δ< m)
  (hq:(M+ε)/(m-δ)<=2^100):
  gAp (x/y) (X/Y) (ε/(m-δ)+M*δ/((m-δ)*m)+1/2^53*((M+ε)/(m-δ))+1/2^153):=by
 have h1:=hx.absR
 have e0:0<=ε:=(Ἀ _).trans hx.2
 have d0:0<=δ:=(Ἀ _).trans hy.2
 have hmd:0< m-δ:=by linarith only[hδ]
 have hm0:0< m:=by linarith only[hδ,d0]
 have hY0:0<Y:=by linarith only[hm,hδ,d0]
 have hRy:m-δ<=D.R y:=by have:=abs_le.mp hy.2;linarith only[hm,this]
 have hRy0:0<D.R y:=by linarith only[hδ,hRy]
 have hqa: |D.R x/D.R y| <=(M+ε)/(m-δ):=by
  rw[abs_div,ἡ hRy0]
  exact div_le_div₀ (by linarith only[Ἀ X,hX,e0]) (by linarith only[hX,h1]) hmd hRy
 obtain ⟨f,e⟩:=gDiv x y hx.1 hy.1 hRy0.ne' (hqa.trans (hq.trans gAB))
 refine ⟨f,?_⟩
 have hz:=gAz
 have hb: |D.R x/D.R y-X/Y| <=ε/(m-δ)+M*δ/((m-δ)*m):=by
  rw[div_sub_div _ _ hRy0.ne' hY0.ne',abs_div,ἡ (mul_pos hRy0 hY0)]
  have hn: |D.R x*Y-D.R y*X| <=ε*Y+M*δ:=by
   have:D.R x*Y-D.R y*X=(D.R x-X)*Y+X*(Y-D.R y):=by ring
   rw[this]
   refine (Ἐ _ _).trans ?_
   rw[ἢ,ἢ,ἡ hY0]
   exact add_le_add (ἀ hx.2 hY0.le)
    (ἑ hX (by rw[ἣ];exact hy.2) (Ἀ _) (by linarith only[Ἀ X,hX]))
  rw[div_le_iff₀ (mul_pos hRy0 hY0)]
  have hA:ε*Y<=ε/(m-δ)*(D.R y*Y):=by
   rw[div_mul_eq_mul_div,le_div_iff₀ hmd]
   have:=ἁ hRy (Ἆ e0 hY0.le)
   linarith only[this]
  have hB:M*δ<=M*δ/((m-δ)*m)*(D.R y*Y):=by
   have hM0:0<=M:=(Ἀ _).trans hX
   rw[div_mul_eq_mul_div,le_div_iff₀ (mul_pos hmd hm0)]
   have:=ἑ hRy hm hm0.le hRy0.le
   have:=ἁ this (Ἆ hM0 d0)
   linarith only[this]
  rw[add_mul];linarith only[hn,hA,hB]
 have hc:=abs_sub_le (D.R (x/y)) (D.R x/D.R y) (X/Y)
 have:1/2^53*(|D.R x/D.R y| +D.A_z)<=1/2^53*((M+ε)/(m-δ))+1/2^153:=by
  have:(1:ℝ)/2^153=1/2^53*(1/2^100):=ב
  rw[this,<-mul_add]
  exact ἁ (by linarith only[hqa,hz]) ב
 linarith only[hqa,e,hz,hb,hc]
lemma gAp_exact {x:ὧ} {X:ℝ} (hf:x.isFinite=true) (h:D.R x=X):gAp x X 0:=
 ⟨hf,by rw[h,sub_self,abs_zero]⟩
lemma gAp_inv (n:Ὠ) (hn0:0<n.toNat) (hn:n.toNat<2^53):
  gAp ((1:Ὠ).toFloat/n.toFloat) (1/n.toNat) (1/2^53*(1/n.toNat)+1/2^153):=by
 obtain ⟨f1,r1⟩:=D.qCk 1 ה
 obtain ⟨fn,rn⟩:=D.qCk n hn
 have hn0':(0:ℝ)<n.toNat:=by exact_mod_cast hn0
 have h1n:(1:ℝ)<=n.toNat:=by exact_mod_cast hn0
 have hb: |D.R (1:Ὠ).toFloat/D.R n.toFloat| <=D.A_B:=by
  rw[r1,rn];simp only[UInt64.toNat_one,Nat.cast_one]
  rw[ἡ ו]
  exact ((div_le_one hn0').mpr h1n).trans (ὂ ב gAB)
 obtain ⟨f,e⟩:=gDiv _ _ f1 fn (by rw[rn];exact hn0'.ne') hb
 refine ⟨f,?_⟩
 rw[r1,rn] at e;simp only[UInt64.toNat_one,Nat.cast_one] at e
 rw[ἡ (show (0:ℝ)<1/(n.toNat:ℝ) by positivity)] at e
 have hz:=gAz
 have:1/2^53*(1/(n.toNat:ℝ)+D.A_z)<=1/2^53*(1/n.toNat)+1/2^153:=by
  have:(1:ℝ)/2^153=1/2^53*(1/2^100):=ב
  rw[this,<-mul_add]
  exact ἁ (by linarith only[hz]) ב
 linarith only[e,hz]
lemma gOne:gAp oneG 1 0:=by
 obtain ⟨f,r⟩:=D.qCk 1 ה
 exact gAp_exact f (by rw[oneG,r];simp only[UInt64.reduceToNat,Nat.cast_one])
lemma gHalf:gAp halfG (1/2) (1/2^53):=by
 have h:=gAp_inv 2 ה ה
 have e:((UInt64.toNat 2:Nat):ℝ)=2:=ב
 rw[e] at h
 exact h.mono ב
end GTool2
section GPoly
lemma gC6:gAp c6G (1/6) (1/2^55):=by
 have h:=gAp_inv 6 ה ה
 have e:((UInt64.toNat 6:Nat):ℝ)=6:=ב
 rw[e] at h;exact h.mono ב
lemma gC24:gAp c24G (1/24) (1/2^57):=by
 have h:=gAp_inv 24 ה ה
 have e:((UInt64.toNat 24:Nat):ℝ)=24:=ב
 rw[e] at h;exact h.mono ב
lemma gC120:gAp c120G (1/120) (1/2^59):=by
 have h:=gAp_inv 120 ה ה
 have e:((UInt64.toNat 120:Nat):ℝ)=120:=ב
 rw[e] at h;exact h.mono ב
lemma exp_neg_taylor5 {r:ℝ} (h0:0<=r) (h1:r<=1):
  |Real.exp (-r)-(1-r*(1-r*(1/2-r*(1/6-r*(1/24-r*(1/120))))))| <=
    r^6*(7/4320):=by
 have h:=Real.exp_bound (x:=-r) (by rw[abs_neg,ἅ h0];exact h1) (n:=6)
  ב
 simp only[Finset.sum_range_succ,Finset.sum_range_zero,Nat.factorial] at h
 rw[abs_neg,ἅ h0] at h
 convert h using 2
 · norm_num;ring
 · norm_num
lemma gPoly {r:ὧ} {ρ:ℝ} (hr:gAp r ρ 0) (h0:0<=ρ) (h1:ρ<=1/64):
  gAp (polyG r) (Real.exp (-ρ)) (1/2^45):=by
 have hρ: |ρ| <=1/64:=by rw[ἅ h0];exact h1
 have r2:=(gAp_mul hr hr hρ hρ ב ב).mono (show _<=(1:ℝ)/2^64 by norm_num)
 have b2: |ρ*ρ| <=1/4096:=by
  rw[ἢ,ἅ h0]
  calc ρ*ρ<=(1/64)*(1/64):=ἑ h1 h1 h0 ב
   _=1/4096:=ב
 have a:=(gAp_sub gOne hr (by norm_num: |(1:ℝ)| <=1) hρ ב ב).mono
  (show _<=(1:ℝ)/2^52 by norm_num)
 have ba: |1-ρ| <=1:=by rw[ὸ];constructor<;>linarith only[h1,h0]
 have m6:=(gAp_mul hr gC6 hρ (by norm_num: |(1:ℝ)/6| <=1/6) ב ב).mono
  (show _<=(1:ℝ)/2^60 by norm_num)
 have bm6: |ρ*(1/6)| <=1/384:=by
  rw[ἢ,ἅ h0];norm_num;linarith only[h1]
 have b:=(gAp_sub gHalf m6 (by norm_num: |(1:ℝ)/2| <=1/2) bm6 ב ב).mono
  (show _<=(1:ℝ)/2^52 by norm_num)
 have bb: |1/2-ρ*(1/6)| <=1/2:=by rw[ὸ];constructor<;>linarith only[h1,h0]
 have m120:=(gAp_mul hr gC120 hρ (by norm_num: |(1:ℝ)/120| <=1/120) ב ב).mono
  (show _<=(1:ℝ)/2^64 by norm_num)
 have bm120: |ρ*(1/120)| <=1/7680:=by
  rw[ἢ,ἅ h0];norm_num;linarith only[h1]
 have c:=(gAp_sub gC24 m120 (by norm_num: |(1:ℝ)/24| <=1/24) bm120 ב ב).mono
  (show _<=(1:ℝ)/2^56 by norm_num)
 have bc: |1/24-ρ*(1/120)| <=1/24:=by rw[ὸ];constructor<;>linarith only[h1,h0]
 have rc:=(gAp_mul r2 c b2 bc ב ב).mono (show _<=(1:ℝ)/2^60 by norm_num)
 have brc: |ρ*ρ*(1/24-ρ*(1/120))| <=1/98304:=by
  rw[ἢ]
  calc |ρ*ρ| * |1/24-ρ*(1/120)| <=(1/4096)*(1/24):=
        ἑ b2 bc (Ἀ _) ב
   _=1/98304:=ב
 have s:=(gAp_add b rc bb brc ב ב).mono (show _<=(1:ℝ)/2^51 by norm_num)
 have bs: |1/2-ρ*(1/6)+ρ*ρ*(1/24-ρ*(1/120))| <=1:=by
  have:=Ἐ (1/2-ρ*(1/6)) (ρ*ρ*(1/24-ρ*(1/120)))
  linarith only[bb,brc,this]
 have rs:=(gAp_mul r2 s b2 bs ב ב).mono (show _<=(1:ℝ)/2^59 by norm_num)
 have brs: |ρ*ρ*(1/2-ρ*(1/6)+ρ*ρ*(1/24-ρ*(1/120)))| <=1/4096:=by
  rw[ἢ]
  calc |ρ*ρ| * |1/2-ρ*(1/6)+ρ*ρ*(1/24-ρ*(1/120))| <=(1/4096)*1:=
        ἑ b2 bs (Ἀ _) ב
   _=1/4096:=ב
 have p:=(gAp_add a rs ba brs ב ב).mono (show _<=(1:ℝ)/2^51 by norm_num)
 refine ⟨p.1,?_⟩
 have hT:=exp_neg_taylor5 h0 (by linarith only[h1])
 have h6:ρ^6<=(1/64)^6:=Ἳ h0 h1 6
 have eq:1-ρ+ρ*ρ*(1/2-ρ*(1/6)+ρ*ρ*(1/24-ρ*(1/120)))=
  1-ρ*(1-ρ*(1/2-ρ*(1/6-ρ*(1/24-ρ*(1/120))))):=by ring
 have e2:D.R (polyG r)=D.R ((oneG-r)+r*r*((halfG-r*c6G)+r*r*(c24G-r*c120G))):=rfl
 have hp:=p.2
 rw[<-e2,eq] at hp
 have:=abs_sub_le (D.R (polyG r)) (1-ρ*(1-ρ*(1/2-ρ*(1/6-ρ*(1/24-ρ*(1/120))))))
  (Real.exp (-ρ))
 rw[ἣ (1-_) (Real.exp _)] at this
 have h7:=ἀ h6 (by norm_num:(0:ℝ)<=7/4320)
 have h8:(1/2^51:ℝ)+(1/64)^6*(7/4320)<=1/2^45:=ב
 linarith only[hT,h6,hp,this]
end GPoly
section GTab
lemma gPw (i:Nat) (h0:0< i) (h1:i<256):
  (D.pwEntry i).isFinite=true/\D.R (D.pwEntry i)=(2:ℝ)^((i:ℤ)-150):=by
 obtain ⟨f,e⟩:=D.qDa i א
 refine ⟨f,?_⟩
 rw[e,Ὧ א,Ὧ א,Ἁ h1,one_mul]
lemma toF62_spec (lo:Nat) (h:lo<=2^62):
  (toF62 lo).isFinite=true/\D.R (toF62 lo)<=FastAct.V lo/\
    FastAct.V lo-D.R (toF62 lo)<=FastAct.V lo/2^52:=by
 unfold toF62
 generalize hsh:lo.log2-52=sh
 have hL:lo.log2<=62:=by
  rcases Nat.eq_zero_or_pos lo with h0 | h0
  · subst h0;simp only[Nat.log2_zero,zero_le]
  · exact Nat.lt_succ_iff.mp ((Nat.log2_lt h0.ne').mpr (ἷ h ב))
 have hsh10:sh<=10:=א
 have hq:lo>>>sh=lo/2^sh:=Ἱ lo sh
 have hlt:=(Nat.lt_log2_self (n:=lo))
 have hq53:lo/2^sh<2^53:=by
  rw[Nat.div_lt_iff_lt_mul ו,<-pow_add]
  exact Ἥ hlt (Nat.pow_le_pow_right ב א)
 have hqu:(lo>>>sh).toUInt64.toNat=lo/2^sh:=by
  rw[hq,Nat.toUInt64,UInt64.toNat_ofNat'];exact Ἁ (lt_trans hq53 ב)
 obtain ⟨fq,rq⟩:=D.qCk (lo>>>sh).toUInt64 (by rw[hqu];exact hq53)
 rw[hqu] at rq
 obtain ⟨fp,rp⟩:=gPw (88+sh) א א
 obtain ⟨f,e⟩:=D.qCm _ (88+sh) א fq ((lo/2^sh:Nat):ℤ) 0 א ב
  (by rw[rq,ἅ ו]
      exact ὂ (show ((lo/2^sh:Nat):ℝ)<=2^53 by exact_mod_cast hq53.le)
       (ὂ ב gAB))
  (by rw[rq,zpow_zero,mul_one,Int.cast_natCast])
 refine ⟨f,?_⟩
 rw[e,rq,rp]
 have hval:((lo/2^sh:Nat):ℝ)*(2:ℝ)^(((88+sh:Nat):ℤ)-150)=
   ((lo/2^sh*2^sh:Nat):ℝ)/2^62:=by
  rw[show (((88+sh:Nat):ℤ)-150)=(sh:ℤ)-62 by push_cast;ring,ᾂ ב,
   Ἡ]
  push_cast;ring
 rw[hval]
 unfold FastAct.V
 have hm:lo/2^sh*2^sh<=lo:=Nat.div_mul_le_self lo _
 have hr:lo-lo/2^sh*2^sh<2^sh:=by
  have:=ὀ lo (show 0<2^sh by positivity)
  have:=Nat.div_add_mod lo (2^sh)
  rw[ἥ] at this;omega
 refine ⟨div_le_div_of_nonneg_right (by exact_mod_cast hm) ו,?_⟩
 rcases Nat.eq_zero_or_pos sh with h0 | h0
 · subst h0;simp only[pow_zero,Nat.div_one,mul_one,sub_self]
   positivity
 · have hlo:2^(sh+52)<=lo:=by
    have hne:lo≠0:=by rintro rfl;simp only[Nat.log2_zero,zero_tsub] at hsh;omega
    exact (Nat.le_log2 hne).mp א
   rw[div_div,<-sub_div,div_le_div_iff₀ ו ו]
   have h1:((lo:ℝ)-((lo/2^sh*2^sh:Nat):ℝ))<=2^sh:=by
    rw[<-Nat.cast_sub hm];exact_mod_cast hr.le
   have h2:(2:ℝ)^(sh+52)<=lo:=by exact_mod_cast hlo
   rw[pow_add] at h2
   linarith only[h1,h2]
lemma Encl_congr {iv:Nat×Nat} {a b:ℝ} (h:FastAct.Encl iv a) (e:a=b):FastAct.Encl iv b:=
 e▸h
lemma enclK_spec {k:Nat} (hk:k<1280):FastAct.Encl (enclK k) (k*(1/64)):=by
 unfold enclK
 have hj:k%64<64:=ὀ _ ב
 have hF:=FastAct.fracTab_spec hj
 have hkd:(k:ℝ)=((k/64:Nat):ℝ)*64+((k%64:Nat):ℝ):=by
  have:=Nat.div_add_mod k 64;exact_mod_cast (by omega:k=k/64*64+k%64)
 dsimp only
 split
 · rename_i hn
   exact Encl_congr ((FastAct.intTab_spec א).mul hF) (by rw[hkd];ring)
 · rename_i hn
   have hI:=(FastAct.intTab_spec (n:=16) ב).mul (FastAct.intTab_spec (n:=k/64-16) א)
   refine Encl_congr (hI.mul hF) ?_
   rw[hkd,Nat.cast_sub א];push_cast;ring
lemma tabOK_spec (h:tabOK=true) {k:Nat} (hk:k<1280):
  (enclK k).2-(enclK k).1<=2^14/\(k<1088->((enclK k).2-(enclK k).1)*2^36<=(enclK k).1):=by
 unfold tabOK at h
 have:=List.all_eq_true.mp h k (List.mem_range.mpr hk)
 simp only[Bool.and_eq_true,Bool.or_eq_true,decide_eq_true_eq] at this
 refine ⟨this.1, λh2=>?_⟩
 rcases this.2 with h3 | h3
 · omega
 · exact h3
lemma tabG_get {k:ὥ} (hk:k.toUSize.toNat<expTabG.size):
  expTabG.uget k.toUSize hk=toF62 (enclK k.toNat).1:=by
 have hk':k.toNat<1280:=by rw[tabG_size] at hk;simpa only[ὃ] using hk
 have e:expTabG.uget k.toUSize hk=(Array.ofFn (n:=1280) λk=>toF62 (enclK k.val).1)[k.toNat]'(by simpa only[Array.size_ofFn] using hk'):=rfl
 rw[e]
 simp only[Array.getElem_ofFn]
lemma tabG_spec (hok:tabOK=true) {k:ὥ} (hk:k.toUSize.toNat<expTabG.size):
  gAp (expTabG.uget k.toUSize hk) (Real.exp (-(k.toNat*(1/64)))) (1/2^48+1/2^52)/\
   D.R (expTabG.uget k.toUSize hk)<=Real.exp (-(k.toNat*(1/64)))/\
   (k.toNat<1088->Real.exp (-(k.toNat*(1/64)))-D.R (expTabG.uget k.toUSize hk)<=
     (1/2^36+1/2^52)*Real.exp (-(k.toNat*(1/64)))):=by
 have hk':k.toNat<1280:=by rw[tabG_size] at hk;simpa only[ὃ] using hk
 rw[tabG_get]
 obtain ⟨e1,e2,e3⟩:=enclK_spec hk'
 obtain ⟨w1,w2⟩:=tabOK_spec hok hk'
 generalize enclK k.toNat=iv at e1 e2 e3 w1 w2
 obtain ⟨lo,hi⟩:=iv
 simp only at e1 e2 e3 w1 w2⊢
 have hlohi:lo<=hi:=by
  by_contra hc
  have:FastAct.V hi<FastAct.V lo:=by unfold FastAct.V;exact div_lt_div_of_pos_right (by exact_mod_cast (not_le.mp hc)) ו
  linarith only[e1,e2,this]
 have hhi:hi<=2^62:=e3
 obtain ⟨f,r1,r2⟩:=toF62_spec lo (ὂ hlohi hhi)
 set E:=Real.exp (-(k.toNat*(1/64:ℝ)))
 have hVlo:FastAct.V lo<=1:=by
  unfold FastAct.V;rw[div_le_one ו];exact_mod_cast ὂ hlohi hhi
 have hw:FastAct.V hi-FastAct.V lo=((hi-lo:Nat):ℝ)/2^62:=by
  unfold FastAct.V;rw[Nat.cast_sub hlohi];ring
 have hw1:FastAct.V hi-FastAct.V lo<=1/2^48:=by
  rw[hw]
  have:((hi-lo:Nat):ℝ)<=2^14:=by exact_mod_cast w1
  calc ((hi-lo:Nat):ℝ)/2^62<=2^14/2^62:=div_le_div_of_nonneg_right this ו
   _=1/2^48:=ב
 have hR:D.R (toF62 lo)<=E:=r1.trans e1
 refine ⟨⟨f,?_⟩,hR, λh2=>?_⟩
 · rw[ἣ,ἅ (by linarith only[r1,e1])]
   have:FastAct.V lo/2^52<=1/2^52:=div_le_div_of_nonneg_right hVlo ו
   linarith only[r2,e2,hVlo,hw1]
 · have w3:=w2 h2
   have hw2:FastAct.V hi-FastAct.V lo<=FastAct.V lo/2^36:=by
    rw[hw];unfold FastAct.V
    have:((hi-lo:Nat):ℝ)<=lo/2^36:=by
     rw[le_div_iff₀ ו];exact_mod_cast w3
    calc ((hi-lo:Nat):ℝ)/2^62<=(lo/2^36)/2^62:=div_le_div_of_nonneg_right this ו
     _=lo/2^62/2^36:=by ring
   have:FastAct.V lo/2^36<=E/2^36:=div_le_div_of_nonneg_right e1 ו
   have:FastAct.V lo/2^52<=E/2^52:=div_le_div_of_nonneg_right e1 ו
   linarith only[r1,r2,e2,hw2]
end GTab
section GSplit
lemma sOfG_toNat (F:ὥ) (hF:F.toNat<=144):(sOfG F).toNat=min (144-F.toNat) 31:=by
 unfold sOfG
 have e:(144-F).toNat=144-F.toNat:=by
  rw[ὓ _ _ (UInt32.le_iff_toNat_le.mpr (by simpa only[UInt32.reduceToNat] using hF))];simp only[UInt32.reduceToNat]
 split
 · rename_i h
   rw[ἳ,e] at h
   simp only[UInt32.reduceToNat] at h
   rw[e];omega
 · rename_i h
   rw[ἳ,e] at h
   simp only[UInt32.reduceToNat,not_lt] at h
   simp only[UInt32.reduceToNat,right_eq_inf,ge_iff_le];omega
lemma kOfG_toNat (m F:ὥ) (hF:F.toNat<=144):
  (kOfG m F).toNat=m.toNat/2^(sOfG F).toNat:=by
 unfold kOfG
 have hs:=sOfG_toNat F hF
 rw[UInt32.toNat_shiftRight,Ἱ,Ἁ א]
lemma split_spec (m F:ὥ) (hm2:m.toNat<2^24) (hF1:1<=F.toNat) (hF2:F.toNat<=144)
  (h:F.toUSize.toNat<D.pw.size):
  gAp (rOfG m F h) ((m.toNat:ℝ)*2^((F.toNat:ℤ)-150)-(kOfG m F).toNat*(1/64)) 0/\
   0<=(m.toNat:ℝ)*2^((F.toNat:ℤ)-150)-(kOfG m F).toNat*(1/64)/\
   (m.toNat:ℝ)*2^((F.toNat:ℤ)-150)-(kOfG m F).toNat*(1/64)<1/64:=by
 have hs:=sOfG_toNat F hF2
 have hk:=kOfG_toNat m F hF2
 set s:=(sOfG F).toNat with hsdef
 set M:=m.toNat
 set f:=F.toNat
 have hs31:s<=31:=א
 have hks:(kOfG m F<<<sOfG F).toNat=M/2^s*2^s:=by
  rw[UInt32.toNat_shiftLeft,hk,ὁ]
  have e32:s%32=s:=א
  rw[e32]
  exact Ἁ (ἷ (Nat.div_mul_le_self M _) א)
 have hle:kOfG m F<<<sOfG F<=m:=by
  rw[ἃ,hks];exact Nat.div_mul_le_self M _
 have hrr:(m-(kOfG m F<<<sOfG F)).toUInt64.toNat=M%2^s:=by
  rw[UInt32.toNat_toUInt64,ὓ _ _ hle,hks]
  have:=Nat.div_add_mod M (2^s);rw[ἥ] at this;omega
 have hmod:M%2^s<2^24:=ἷ (Nat.mod_le _ _) hm2
 obtain ⟨fq,rq⟩:=D.qCk (m-(kOfG m F<<<sOfG F)).toUInt64 (by rw[hrr];omega)
 rw[hrr] at rq
 have hpw:D.pw.uget F.toUSize h=D.pwEntry f:=by
  rw[D.qDc];simp only[ὃ,f]
 obtain ⟨fp,rp⟩:=gPw f hF1 א
 obtain ⟨fr,er⟩:=D.qCm _ f א fq ((M%2^s:Nat):ℤ) 0 א ב
  (by rw[rq,ἅ ו]
      exact ὂ (show ((M%2^s:Nat):ℝ)<=2^24 by exact_mod_cast hmod.le)
       (ὂ ב gAB))
  (by rw[rq,zpow_zero,mul_one,Int.cast_natCast])
 have hr:D.R (rOfG m F h)=((M%2^s:Nat):ℝ)*2^((f:ℤ)-150):=by
  unfold rOfG;rw[hpw,er,rq,rp]
 replace fr:(rOfG m F h).isFinite=true:=by unfold rOfG;rw[hpw];exact fr
 have hdm:(M:ℝ)=((M/2^s:Nat):ℝ)*2^s+((M%2^s:Nat):ℝ):=by
  have:=Nat.div_add_mod M (2^s);rw[ἥ] at this;exact_mod_cast this.symm
 have hp0:(0:ℝ)<2^((f:ℤ)-150):=ו
 rcases Nat.lt_or_ge (144-f) 31 with hc | hc
 · have hse:s=144-f:=א
   have h6:(2:ℝ)^s*2^((f:ℤ)-150)=1/64:=by
    rw[<-Ἡ,<-Ἓ ב,hse,Nat.cast_sub א]
    norm_num
   have hid:(M:ℝ)*2^((f:ℤ)-150)-((kOfG m F).toNat:ℝ)*(1/64)=
     ((M%2^s:Nat):ℝ)*2^((f:ℤ)-150):=by
    rw[hk,hdm,<-h6];ring
   rw[hid]
   refine ⟨gAp_exact fr hr,Ἆ (Ἔ _) hp0.le,?_⟩
   rw[<-h6]
   exact Ἦ (by exact_mod_cast ὀ _ ו) hp0
 · have hse:s=31:=א
   have hk0:(kOfG m F).toNat=0:=by rw[hk,hse];exact Ὃ א
   have hm0:M%2^s=M:=Ἁ (by rw[hse];omega)
   rw[hk0]
   rw[hm0] at hr
   simp only[Nat.cast_zero,zero_mul,sub_zero]
   refine ⟨gAp_exact fr hr,Ἆ (Ἔ _) hp0.le,?_⟩
   have h1:(M:ℝ)<2^24:=by exact_mod_cast hm2
   have h2:(2:ℝ)^((f:ℤ)-150)<=2^(-37:ℤ):=
    ἒ ב א
   calc (M:ℝ)*2^((f:ℤ)-150)<=2^24*2^(-37:ℤ):=
         ἑ h1.le h2 hp0.le ב
    _<1/64:=ב
end GSplit
section GCore
lemma tOfG_def (m F:ὥ) (h:F.toUSize.toNat<D.pw.size) (hk:kOfG m F<1280):
 tOfG m F h hk=expTabG.uget (kOfG m F).toUSize (kidx hk ה)*polyG (rOfG m F h):=by
 unfold tOfG;rfl
lemma exp17:(1:ℝ)/2^25<=Real.exp (-17):=by
 rw[Real.exp_neg,one_div,inv_le_inv₀ ב (Real.exp_pos _)]
 have h1:=Real.exp_one_lt_d9
 have h2:Real.exp 17=Real.exp 1^17:=by rw[<-Real.exp_nat_mul];norm_num
 rw[h2]
 calc Real.exp 1^17<=2.7182818286^17:=Ἳ (Real.exp_pos 1).le h1.le 17
  _<=2^25:=ב
lemma tOfG_spec (hok:tabOK=true) (m F:ὥ) (hm2:m.toNat<2^24) (hF1:1<=F.toNat)
  (hF2:F.toNat<=144) (h:F.toUSize.toNat<D.pw.size) (hk:kOfG m F<1280):
  gAp (tOfG m F h hk) (Real.exp (-((m.toNat:ℝ)*2^((F.toNat:ℤ)-150)))) (1/2^44)/\
  ((kOfG m F).toNat<1088->
    |D.R (tOfG m F h hk)-Real.exp (-((m.toNat:ℝ)*2^((F.toNat:ℤ)-150)))| <=
      (1/2^36+1/2^44)*Real.exp (-((m.toNat:ℝ)*2^((F.toNat:ℤ)-150))))/\
  (kOfG m F).toNat*(1/64:ℝ)<=(m.toNat:ℝ)*2^((F.toNat:ℤ)-150)/\
  (m.toNat:ℝ)*2^((F.toNat:ℤ)-150)<((kOfG m F).toNat+1)*(1/64):=by
 obtain ⟨hr,r0,r1⟩:=split_spec m F hm2 hF1 hF2 h
 set C:=(m.toNat:ℝ)*2^((F.toNat:ℤ)-150)
 set kk:=(kOfG m F).toNat
 set ρ:=C-kk*(1/64) with hρ
 have r0':(kk:ℝ)*(1/64)<=C:=by rw[hρ] at r0;linarith only[r0]
 have r1':C<((kk:ℝ)+1)*(1/64):=by rw[hρ] at r1;linarith only[r1]
 have hP:=gPoly hr r0 r1.le
 obtain ⟨hT,hT1,hT2⟩:=tabG_spec hok (k:=kOfG m F) (kidx hk ה)
 set E:=Real.exp (-(kk*(1/64:ℝ)))
 have hE1:E<=1:=Real.exp_le_one_iff.mpr (by have:(0:ℝ)<=kk:=Ἔ _;linarith only[])
 have hE0:0<E:=Real.exp_pos _
 have hp1:Real.exp (-ρ)<=1:=Real.exp_le_one_iff.mpr (by linarith only[r0])
 have hp0:0<Real.exp (-ρ):=Real.exp_pos _
 have hp63:63/64<=Real.exp (-ρ):=by linarith only[Real.add_one_le_exp (-ρ),r1]
 have hX:E*Real.exp (-ρ)=Real.exp (-C):=by
  rw[hρ,<-Real.exp_add];ring_nf
 have hm:gAp (tOfG m F h hk) (Real.exp (-C)) (1/2^44):=by
  have:=gAp_mul hT hP (M:=1) (N:=1) (by rw[ἡ hE0];exact hE1)
   (by rw[ἡ hp0];exact hp1) ב ב
  rw[hX,<-tOfG_def] at this;exact this.mono ב
 refine ⟨hm, λh1088=>?_,r0',r1'⟩
 have hC17:C<17:=by
  have:(kk:ℝ)+1<=1088:=by exact_mod_cast h1088
  linarith only[r1,hρ,this]
 have hEC:1/2^25<=Real.exp (-C):=exp17.trans (Real.exp_le_exp.mpr (by linarith only[hC17]))
 have hrel:=hT2 h1088
 set T:=expTabG.uget (kOfG m F).toUSize (kidx hk ה)
 set P:=polyG (rOfG m F h)
 have hPab:=abs_le.mp hP.2
 have hRP0:0<=D.R P:=by linarith only[hp0,hp63,hPab]
 obtain ⟨-,eR⟩:=D.qCf T P hT.1 hP.1 (hT.bnd (M:=1) (by rw[ἡ hE0];exact hE1) ב)
  (hP.bnd (M:=1) (by rw[ἡ hp0];exact hp1) ב)
 have hTP: |D.R T*D.R P-E*Real.exp (-ρ)| <=(1/2^36+1/2^52)*E*(Real.exp (-ρ)+1/2^45)+
   E*(1/2^45):=by
  have:D.R T*D.R P-E*Real.exp (-ρ)=(D.R T-E)*D.R P+E*(D.R P-Real.exp (-ρ)):=by ring
  rw[this]
  refine (Ἐ _ _).trans (add_le_add ?_ ?_)
  · rw[ἢ,ἅ hRP0,ἣ,ἅ (by linarith only[hT1])]
    exact ἑ hrel (by linarith only[hPab]) hRP0 ו
  · rw[ἢ,ἡ hE0];exact ἁ hP.2 hE0.le
 have hEQ:E<=(64/63)*(E*Real.exp (-ρ)):=by
  have:=ἁ hp63 hE0.le;linarith only[this]
 have hTPa: |D.R T*D.R P| <=2*Real.exp (-C):=by
  have:=Ὥ (D.R T*D.R P) (E*Real.exp (-ρ))
  rw[ἡ (mul_pos hE0 hp0)] at this
  have:(1/2^36+1/2^52)*E*(Real.exp (-ρ)+1/2^45)+E*(1/2^45)<=E*Real.exp (-ρ):=by
   linarith only[hE0,hEQ]
  rw[<-hX];linarith
 have hz:=gAz
 have hc:=abs_sub_le (D.R (T*P)) (D.R T*D.R P) (Real.exp (-C))
 rw[<-hX] at hc hEC hTPa⊢
 have key:(1/2^36+1/2^52)*E*(Real.exp (-ρ)+1/2^45)+E*(1/2^45)+
   1/2^53*(2*(E*Real.exp (-ρ))+1/2^100)<=(1/2^36+1/2^44)*(E*Real.exp (-ρ)):=by
  have h100:(1:ℝ)/2^100<=1/2^75*(E*Real.exp (-ρ)):=by
   have:(1:ℝ)/2^100=1/2^75*(1/2^25):=ב
   rw[this];exact ἁ hEC ב
  linarith only[hE0,hEC,hEQ]
 have e1:1/2^53*(|D.R T*D.R P| +D.A_z)<=1/2^53*(2*(E*Real.exp (-ρ))+1/2^100):=
  ἁ (by linarith only[hTPa,hz]) ב
 rw[tOfG_def]
 linarith only[hk,h,hE0,hEC,eR,hTP,hEQ,hTPa,hz,hc]
end GCore
section GFin
lemma gZiv {q e:ὧ} {V δ:ℝ} (hq:gAp q V δ) (he:e.isFinite=true)
  (hqB: |D.R q| <=2^90) (heB: |D.R e| <=2^90)
  (hm:δ+1/2^53*(|D.R q| + |D.R e|)+1/2^150<D.R e) (hz:D.ziv q e≠Е)
  {M:Model Fmt} (hf:M.isFinite=true) (hM:M.toReal=Model.roundAt Fmt V):
  toWord M=D.ziv q e:=by
 have hz0:=gAz
 have hqA: |D.R q| <=D.A_B:=hqB.trans (ὂ ב gAB)
 have heA: |D.R e| <=D.A_B:=heB.trans (ὂ ב gAB)
 obtain ⟨-,e1⟩:=D.qBf q e hq.1 he hqA heA
 obtain ⟨-,e2⟩:=D.qDf q e hq.1 he hqA heA
 have hqv:=abs_le.mp hq.2
 have a1: |D.R q+D.R e| <= |D.R q| + |D.R e| :=Ἐ _ _
 have a2: |D.R q-D.R e| <= |D.R q| + |D.R e| :=Ὕ _ _
 have b1:=abs_le.mp e1
 have b2:=abs_le.mp e2
 have c0:1/2^53*(|D.R q| + |D.R e| +1/2^100)<=1/2^53*(|D.R q| + |D.R e|)+1/2^150:=by
  rw[mul_add];have:(1:ℝ)/2^53*(1/2^100)<=1/2^150:=ב
  linarith only[]
 have c1:1/2^53*(|D.R q+D.R e| +D.A_z)<=1/2^53*(|D.R q| + |D.R e|)+1/2^150:=
  ὂ (ἁ (by linarith only[hz0,a1]) ב) c0
 have c2:1/2^53*(|D.R q-D.R e| +D.A_z)<=1/2^53*(|D.R q| + |D.R e|)+1/2^150:=
  ὂ (ἁ (by linarith only[hz0,a2]) ב) c0
 exact D.qDq q e V (by linarith only[hm,hz0,hqv,a2,b2]) (by linarith only[hm,hz0,hqv,a1,b1]) hz M hf hM
/-! -/
lemma kHvR (th:Ὠ) (s:ὧ) (V:ℝ) (hV: |D.R s-V| <=1/2^33*(|V| +D.A_z))
  (h:D.zivT th s≠Е) (M:Model Fmt) (hf:M.isFinite=true)
  (hM:M.toReal=Model.roundAt Fmt V):toWord M=D.zivT th s:=by
 obtain ⟨-,S,ex,fr,M',h1,h2,h3,hz,hb,hl,hu⟩:=D.kFh th s h
 have hM':M'=(M'/2^52)*2^52+M'%2^52:=omS114
 have hF:M'%2^52<2^52:=ὀ _ ו
 generalize M'/2^52=E at hM'
 generalize M'%2^52=F at hM' hF
 subst hM'
 obtain ⟨-,hR⟩:=D.B_R s S E F (omS238 h1 hF hl) (omS239 h2 h3 hu) hF (by rw[hb,ὢ])
 rw[hR] at hV
 obtain ⟨W,hVW,hW⟩: ∃W:ℝ,V=(if S then-W else W)/\
   |((2^52+F:Nat):ℝ)*2^((E:Int)-1075)-W| <=1/2^33*(|W| +D.A_z):=by
  cases S
  · simp only[Bool.false_eq_true,ite_false,one_mul] at hV⊢;exact ⟨_,rfl,hV⟩
  · simp only[ite_true] at hV⊢
    refine ⟨-V,by simp only[neg_neg],?_⟩
    have e: |((2^52+F:Nat):ℝ)*2^((E:Int)-1075)- -V| =
      |-1*((2^52+F:Nat):ℝ)*2^((E:Int)-1075)-V| :=by
     rw[<-abs_neg];ring_nf
    rw[e,abs_neg];exact hV
 obtain ⟨a,b,c⟩:=D.qJi ex fr E F h1 h2 h3 hF (omS240 hl) (omS241 hu) W hW
 rw[hz]
 exact D.qDqW S ex fr h1 h2 h3 V W hVW a b c M hf hM
lemma fin1T (E s:ὧ) (h:D.fin1 (D.loOf E) (D.spOf (D.loOf E)) s≠Е):
  D.fin1 (D.loOf E) (D.spOf (D.loOf E)) s=D.zivT (D.thOf E) s/\D.zivT (D.thOf E) s≠Е:=by
 unfold D.fin1 at h⊢
 dsimp only at h⊢
 split at h
 · rename_i ht
   simp only[Bool.and_eq_true,decide_eq_true_eq] at ht
   obtain ⟨h1,h2⟩:=ht
   rw[Ὦ (by simp only[Bool.and_eq_true,decide_eq_true_eq];exact ⟨h1,h2⟩)]
   obtain ⟨a,b⟩:=D.qHy E s h1 h2
   exact ⟨a.symm,b⟩
 · exact absurd rfl h
lemma gAz0:(0:ℝ)<=D.A_z:=by show (0:ℝ)<=(2:ℝ)^(-1021:ℤ);positivity
lemma gZA {lo sp:Ὠ} {q e E:ὧ} {V δ:ℝ} (hlo:lo=D.loOf E) (hsp:sp=D.spOf lo)
  (hq:gAp q V δ) (he:e.isFinite=true)
  (hqB: |D.R q| <=2^90) (heB: |D.R e| <=2^90)
  (hm:δ+1/2^53*(|D.R q| + |D.R e|)+1/2^150<D.R e)
  (hrel: |D.R E| <=1/2^34* |D.R q| ->δ<=1/2^33*(|V| +D.A_z))
  (hz:zA lo sp q e≠Е)
  {M:Model Fmt} (hf:M.isFinite=true) (hM:M.toReal=Model.roundAt Fmt V):
  toWord M=zA lo sp q e:=by
 subst hlo hsp
 unfold zA at hz⊢
 dsimp only at hz⊢
 by_cases hr:D.fin1 (D.loOf E) (D.spOf (D.loOf E)) q=Е
 · rw[Ὦ (by rw[hr];rfl)] at hz⊢
   exact gZiv hq he hqB heB hm hz hf hM
 · rw[Ὧ (by simpa only[beq_iff_eq] using hr)]
   obtain ⟨a,b⟩:=fin1T E q hr
   obtain ⟨-,-,h34,-,-,-⟩:=D.qI4 E q b
   rw[a]
   exact kHvR _ q V (hq.2.trans (hrel h34)) b M hf hM
def gApR (x:ὧ) (X a:ℝ):Prop:=x.isFinite=true/\ |D.R x-X| <=a*X
lemma gApR_div {x y:ὧ} {X Y a b:ℝ} (hx:gApR x X a) (hy:gApR y Y b) (hX:0<X) (hY:0<Y)
  (ha:0<=a) (hb0:0<=b) (hb:b<=1/2) (hXY:1/2^30<=X/Y) (hB:X/Y<=2^80)
  (hab:a<=1/2):
  gApR (x/y) (X/Y) ((a+b)/(1-b)+1/2^53*(1+(a+b)/(1-b))+1/2^120):=by
 have hz:=gAz
 have hb1:0<1-b:=by linarith only[hb]
 have hRy:(1-b)*Y<=D.R y:=by have:=abs_le.mp hy.2;rw[sub_mul,one_mul];linarith only[this]
 have hRy0:0<D.R y:=Ἥ (mul_pos hb1 hY) hRy
 have hRx: |D.R x| <=(1+a)*X:=by
  have:=Ὥ (D.R x) X;rw[ἡ hX] at this;rw[add_mul,one_mul];linarith only[hx.2,this,hx]
 set Q:=X/Y
 have hQ0:0<Q:=div_pos hX hY
 have hd: |D.R x/D.R y-Q| <=(a+b)/(1-b)*Q:=by
  simp only[Q]
  rw[div_sub_div _ _ hRy0.ne' hY.ne',abs_div,ἡ (mul_pos hRy0 hY)]
  rw[div_le_iff₀ (mul_pos hRy0 hY)]
  have hn: |D.R x*Y-D.R y*X| <=(a+b)*X*Y:=by
   have:D.R x*Y-D.R y*X=(D.R x-X)*Y+X*(Y-D.R y):=by ring
   rw[this]
   refine (Ἐ _ _).trans ?_
   rw[ἢ,ἢ,ἡ hY,ἡ hX,ἣ Y]
   have p1:=ἀ hx.2 hY.le
   have p2:=ἁ hy.2 hX.le
   linarith only[p1,p2]
  have:(a+b)*X*Y<=(a+b)/(1-b)*(X/Y)*(D.R y*Y):=by
   rw[show (a+b)/(1-b)*(X/Y)*(D.R y*Y)=(a+b)*X*(D.R y/(1-b)) by
    field_simp]
   apply ἁ _ ו
   rw[le_div_iff₀ hb1];linarith only[hRy]
  linarith only[hn,this]
 have hqa: |D.R x/D.R y| <=(1+(a+b)/(1-b))*Q:=by
  have:=Ὥ (D.R x/D.R y) Q
  rw[ἡ hQ0] at this;rw[add_mul,one_mul];linarith only[hd,this]
 have hfac:(a+b)/(1-b)<=2:=by rw[div_le_iff₀ hb1];linarith only[hb,hab]
 obtain ⟨f,e⟩:=gDiv x y hx.1 hy.1 hRy0.ne' (hqa.trans (by
  have:(1+(a+b)/(1-b))*Q<=3*2^80:=by
   have:=ἀ hfac hQ0.le
   rw[add_mul,one_mul];linarith only[hB,this]
  linarith only[gAB,show (3:ℝ)*2^80<=2^100 by norm_num,this]))
 refine ⟨f,?_⟩
 have hc:=abs_sub_le (D.R (x/y)) (D.R x/D.R y) Q
 have:1/2^53*(|D.R x/D.R y| +D.A_z)<=
   1/2^53*(1+(a+b)/(1-b))*Q+1/2^120*Q:=by
  have h1:D.A_z<=1/2^67*Q:=by
   calc D.A_z<=1/2^100:=hz
    _=1/2^67*(1/2^30)*(1/2^3):=ב
    _<=1/2^67*Q*1:=by gcongr;norm_num
    _=_:=by ring
  have:1/2^53*(|D.R x/D.R y| +D.A_z)<=1/2^53*((1+(a+b)/(1-b))*Q+1/2^67*Q):=
   ἁ (by linarith only[hz,hXY,hQ0,hqa]) ב
  have e3:1/2^53*((1+(a+b)/(1-b))*Q+1/2^67*Q)=
   1/2^53*(1+(a+b)/(1-b))*Q+1/2^120*Q:=by ring
  linarith only[hz,hXY,hQ0,hqa]
 linarith only[hz,hXY,hQ0,hd,hqa,e,hc]
lemma decW (w:ὥ) (h1:1<=((w&&&В)>>>23).toNat)
  (h2:((w&&&В)>>>23).toNat<=254):
  expOf w≠255/\2^23<=(((w&&&В)&&&0x7FFFFF) ||| 0x800000).toNat/\
   (((w&&&В)&&&0x7FFFFF) ||| 0x800000).toNat<2^24/\
   D.wv w=(if w<س then (1:ℝ) else-1)*
    ((((w&&&В)&&&0x7FFFFF) ||| 0x800000).toNat*
      (2:ℝ)^((((w&&&В)>>>23).toNat:ℤ)-150)):=by
 have ha:=D.qAq w
 have hm:=D.qHa (w&&&В)
 have hF:((w&&&В)>>>23).toNat=w.toNat%2^31/2^23:=by
  rw[UInt32.toNat_shiftRight,Ἱ,ha];rfl
 rw[ha] at hm
 rw[hm]
 rw[hF] at h1 h2⊢
 have hw:=w.toNat_lt
 have hg:D.good w=true:=by
  unfold D.good
  simp only[Bool.or_eq_true,beq_iff_eq,decide_eq_true_eq]
  right
  rw[ἃ,ὓ _ _ (by rw[ἃ,ha];simp only[UInt32.reduceToNat,Nat.reducePow];omega),ha]
  simp only[Nat.reducePow,UInt32.reduceToNat,tsub_le_iff_right,Nat.reduceAdd];omega
 have he:=D.qAe w hg
 refine ⟨he,א,א,?_⟩
 rw[D.qA7 w hg]
 unfold D.A_mag
 rw[Ὧ (show ¬(w.toNat%2^31=0) by omega)]
 have hs:(w<س)↔ ¬(2^31<=w.toNat):=by
  rw[ἳ];simp only[UInt32.reduceToNat,Nat.reducePow,not_le]
 by_cases hneg:2^31<=w.toNat
 · rw[Ὦ hneg,Ὧ (hs.not.mpr (not_not.mpr hneg))]
 · rw[Ὧ hneg,Ὦ (hs.mpr hneg)]
end GFin
section GSig
lemma gOneR:oneG.isFinite=true/\D.R oneG=1:=by
 obtain ⟨f,r⟩:=D.qCk 1 ה
 exact ⟨f,by rw[oneG,r];simp only[UInt64.reduceToNat,Nat.cast_one]⟩
lemma gM34:(m34G).isFinite=true/\D.R m34G=1/2^34:=by
 obtain ⟨f,e⟩:=gPw 116 ב ב
 exact ⟨f,by rw[m34G,e];norm_num⟩
lemma gM41:(m41G).isFinite=true/\D.R m41G=1/2^41:=by
 obtain ⟨f,e⟩:=gPw 109 ב ב
 exact ⟨f,by rw[m41G,e];norm_num⟩
lemma gM26:(m26G).isFinite=true/\D.R m26G=1/2^26:=by
 obtain ⟨f,e⟩:=gPw 124 ב ב
 exact ⟨f,by rw[m26G,e];norm_num⟩
lemma sigPos (hok:tabOK=true) (m F:ὥ) (hm2:m.toNat<2^24) (hF1:1<=F.toNat)
  (hF2:F.toNat<=144) (h:F.toUSize.toNat<D.pw.size) (hk:kOfG m F<1280) {M:Model Fmt}
  (hf:M.isFinite=true)
  (hM:M.toReal=Model.roundAt Fmt (realSigmoid ((m.toNat:ℝ)*2^((F.toNat:ℤ)-150))))
  (hz:zA loA spA (oneG/(oneG+tOfG m F h hk)) m41G≠Е):
  toWord M=zA loA spA (oneG/(oneG+tOfG m F h hk)) m41G:=by
 obtain ⟨ht,-,-,-⟩:=tOfG_spec hok m F hm2 hF1 hF2 h hk
 set C:=(m.toNat:ℝ)*2^((F.toNat:ℤ)-150)
 set T:=Real.exp (-C)
 have hC0:0<=C:=ו
 have hT0:0<T:=Real.exp_pos _
 have hT1:T<=1:=Real.exp_le_one_iff.mpr (by linarith only[hC0])
 have hd:=(gAp_add gOne ht (M:=1) (N:=1) ב (by rw[ἡ hT0];exact hT1)
  ב ב).mono (show _<=(1:ℝ)/2^43 by norm_num)
 have hq:=(gAp_div gOne hd (M:=1) (m:=1) ב (by linarith only[hT0]) ב
  ב).mono (show _<=(1:ℝ)/2^42 by norm_num)
 obtain ⟨f41,r41⟩:=gM41
 have hqa:=hq.absR
 have hV: |(1:ℝ)/(1+T)| <=1:=by
  rw[ἡ ו,div_le_one (by linarith only[hT0])];linarith only[hT0]
 refine gZA (E:=m41G) rfl rfl hq f41 (by linarith only[h,hk,hqa,hV]) (by rw[r41];norm_num) ?_ ?_ hz hf hM
 · rw[r41,ἡ (show (0:ℝ)<1/2^41 by norm_num)]
   have: |D.R (oneG/(oneG+tOfG m F h hk))| <=1+1/2^42:=by linarith only[h,hk,hqa,hV]
   have:1/2^53*(|D.R (oneG/(oneG+tOfG m F h hk))| +1/2^41)<=
    1/2^53*(1+1/2^42+1/2^41):=by gcongr
   have:(1:ℝ)/2^42+1/2^53*(1+1/2^42+1/2^41)+1/2^150<1/2^41:=ב
   linarith only[h,hk,hqa,hV]
 · intro _
   have h2:(1:ℝ)/2<= |(1:ℝ)/(1+T)| :=by
    rw[ἡ ו,le_div_iff₀ (by linarith only[hT0])];linarith only[hT1]
   have h3:=gAz0
   linarith only[h2,h3]
lemma sigNeg (hok:tabOK=true) (m F:ὥ) (hm2:m.toNat<2^24) (hF1:1<=F.toNat)
  (hF2:F.toNat<=144) (h:F.toUSize.toNat<D.pw.size) (hk:kOfG m F<1280)
  (h1088:(kOfG m F).toNat<1088) {M:Model Fmt} (hf:M.isFinite=true)
  (hM:M.toReal=Model.roundAt Fmt (realSigmoid (-((m.toNat:ℝ)*2^((F.toNat:ℤ)-150)))))
  (hz:zA lo0 sp0 (tOfG m F h hk/(oneG+tOfG m F h hk)) (tOfG m F h hk/(oneG+tOfG m F h hk)*m34G)≠
    Е):
  toWord M=zA lo0 sp0 (tOfG m F h hk/(oneG+tOfG m F h hk)) (tOfG m F h hk/(oneG+tOfG m F h hk)*m34G):=by
 obtain ⟨ht,hrel,-,hCk⟩:=tOfG_spec hok m F hm2 hF1 hF2 h hk
 have hrel':=hrel h1088
 clear hrel
 obtain ⟨ρ,hρdef⟩: ∃ρ:ℝ,ρ=1/2^36+1/2^44:=⟨_,rfl⟩
 rw[<-hρdef] at hrel'
 set C:=(m.toNat:ℝ)*2^((F.toNat:ℤ)-150)
 set T:=Real.exp (-C)
 set t:=tOfG m F h hk
 have hz0:=gAz
 have hC0:0<=C:=ו
 have hT0:0<T:=Real.exp_pos _
 have hT1:T<=1:=Real.exp_le_one_iff.mpr (by linarith only[hC0])
 have hC17:C<17:=by
  have:((kOfG m F).toNat:ℝ)+1<=1088:=by exact_mod_cast h1088
  linarith only[hCk,this]
 have hT25:1/2^25<=T:=exp17.trans (Real.exp_le_exp.mpr (by linarith only[hC17]))
 have hρ0:0<=ρ:=by norm_num[hρdef]
 have hρ1:ρ<=1/2^35:=by norm_num[hρdef]
 have hρT:ρ*T<=ρ:=by have:=ἁ hT1 hρ0;linarith only[this]
 have hρT0:0<=ρ*T:=Ἆ hρ0 hT0.le
 have htR:gApR t T ρ:=⟨ht.1,hrel'⟩
 have hRt:=abs_le.mp hrel'
 obtain ⟨fd,ed⟩:=D.qBf oneG t gOneR.1 ht.1 (by rw[gOneR.2];norm_num;linarith only[gAB])
  (ht.bnd (M:=1) (by rw[ἡ hT0];exact hT1) ב)
 rw[gOneR.2] at ed
 obtain ⟨b,hbdef⟩: ∃b:ℝ,b=ρ/2+1/2^51:=⟨_,rfl⟩
 have hdR:gApR (oneG+t) (1+T) b:=by
  refine ⟨fd,?_⟩
  have h1: |1+D.R t| <=2+ρ:=by rw[ὸ];constructor<;>linarith only[hT0,hρT,hRt,hT1]
  have h2:=abs_sub_le (D.R (oneG+t)) (1+D.R t) (1+T)
  have h3: |1+D.R t-(1+T)| <=ρ*T:=by rw[show 1+D.R t-(1+T)=D.R t-T by ring];exact hrel'
  have h4:1/2^53*(|1+D.R t| +D.A_z)<=1/2^53*(2+ρ+1/2^100):=
   ἁ (by linarith only[hz0,h1]) ב
  have h6:1/2^53*(2+ρ+1/2^100)<=1/2^51:=by
   have:1/2^53*(2+1/2^35+1/2^100)<=(1:ℝ)/2^51:=ב
   have:1/2^53*(2+ρ+1/2^100)<=1/2^53*(2+1/2^35+1/2^100):=
    ἁ (by linarith only[hρdef]) ב
   linarith only[hρdef]
  have h7:(0:ℝ)<=1/2^51*T:=ו
  have:b*(1+T)=ρ/2+ρ*T/2+1/2^51+1/2^51*T:=by rw[hbdef];ring
  linarith only[hρdef,hz0,hT0,hρT,ed,h1,h2,h3,this]
 have hb0:0<=b:=by rw[hbdef];positivity
 have hb1:b<=1/2:=by rw[hbdef];linarith only[hρdef]
 have hq:=gApR_div htR hdR hT0 (by linarith only[hT0]) hρ0 hb0 hb1
  (by rw[le_div_iff₀ (by linarith only[hT0])];linarith only[hT0,hT25])
  (by rw[div_le_iff₀ (by linarith only[hT0])];linarith only[hT0]) (by linarith only[hρdef])
 obtain ⟨r,hrdef⟩: ∃r:ℝ,r=(ρ+b)/(1-b)+1/2^53*(1+(ρ+b)/(1-b))+1/2^120:=
  ⟨_,rfl⟩
 rw[<-hrdef] at hq
 have hr:r<=(4/5)/2^35:=by rw[hrdef,hbdef,hρdef];norm_num
 have hV:realSigmoid (-C)=T/(1+T):=FastAct.realSigmoid_neg_eq C
 rw[hV] at hM
 set V:=T/(1+T)
 have hV0:0<V:=ו
 have hV1:V<=1:=by rw[div_le_one (by linarith only[hT0])];linarith only[]
 have hV26:1/2^26<=V:=by rw[le_div_iff₀ (by linarith only[hT0])];linarith only[hT0,hT25]
 set q:=t/(oneG+t)
 have hqA:gAp q V (r*V):=hq
 have hqv:=abs_le.mp hq.2
 have k7:r*V<=(4/5)/2^35*V:=ἀ hr hV0.le
 have hr0:0<=r*V:=(Ἀ _).trans hq.2
 have hRq0:0<D.R q:=by linarith only[hV0,hqv,k7]
 obtain ⟨f34,r34⟩:=gM34
 obtain ⟨fe,ee⟩:=D.qCf q m34G hq.1 f34
  (hqA.bnd (M:=1) (by rw[ἡ hV0];exact hV1) (by linarith only[hV1,k7]))
  (by rw[r34];norm_num;linarith only[gAB])
 rw[r34] at ee
 have he:=abs_le.mp ee
 rw[ἡ (mul_pos hRq0 ב:0<D.R q*(1/2^34))] at he
 have hRe0:0<D.R (q*m34G):=by linarith only[hz0,hV0,hV26,hqv,k7,he]
 refine gZA (E:=e0G) rfl rfl hqA fe (by rw[ἡ hRq0];linarith only[hV1,hqv,k7]) (by rw[ἡ hRe0];linarith only[hz0,hV1,hqv,k7,he])
  ?_ ?_ hz hf hM
 · rw[ἡ hRq0,ἡ hRe0]
   have k5:(1:ℝ)/2^150<=1/2^124*V:=by
    have:(1:ℝ)/2^150=1/2^124*(1/2^26):=ב
    rw[this];exact ἁ hV26 ב
   have k6:(1:ℝ)/2^153<=1/2^127*V:=by
    have:(1:ℝ)/2^153=1/2^127*(1/2^26):=ב
    rw[this];exact ἁ hV26 ב
   linarith only[hz0,hV26,hqv,k7,he,hRe0]
 · intro _
   have h3:=gAz0
   rw[ἡ hV0]
   linarith only[hqv,k7,he,hRe0,h3]
lemma sigSat {C:ℝ} (hC:19<=C) {M:Model Fmt} (hf:M.isFinite=true)
  (hM:M.toReal=Model.roundAt Fmt (realSigmoid C)) (hz:D.ziv oneG m26G≠Е):
  toWord M=D.ziv oneG m26G:=by
 have hT0:=Real.exp_pos (-C)
 have hT:Real.exp (-C)<=1/2^27:=exp_neg_le_inv_two_pow 27 (by norm_num;linarith only[hC])
 have hq:gAp oneG (realSigmoid C) (1/2^27):=by
  refine ⟨gOneR.1,?_⟩
  rw[gOneR.2];unfold realSigmoid
  rw[ἅ (by rw[sub_nonneg,div_le_one (by linarith only[hT0,hT])];linarith only[hT0])]
  rw[show (1:ℝ)-1/(1+Real.exp (-C))=Real.exp (-C)/(1+Real.exp (-C)) by field_simp;ring]
  rw[div_le_iff₀ (by linarith only[hT0,hT])];linarith only[hT0,hT]
 obtain ⟨f26,r26⟩:=gM26
 refine gZiv hq f26 (by rw[gOneR.2];norm_num) (by rw[r26];norm_num) ?_ hz hf hM
 rw[r26,gOneR.2];norm_num
end GSig
section GTanh
lemma tanhPos (hok:tabOK=true) (m F:ὥ) (hm2:m.toNat<2^24) (hF1:1<=F.toNat)
  (hF2:F.toNat<=144) (h:F.toUSize.toNat<D.pw.size) (hk:kOfG m F<1280) {C:ℝ}
  (hC:(m.toNat:ℝ)*2^((F.toNat:ℤ)-150)=2*C) {M:Model Fmt}
  (hf:M.isFinite=true) (hM:M.toReal=Model.roundAt Fmt (Real.tanh C))
  (hz:zA loA spA ((oneG-tOfG m F h hk)/(oneG+tOfG m F h hk)) m41G≠Е):
  toWord M=zA loA spA ((oneG-tOfG m F h hk)/(oneG+tOfG m F h hk)) m41G:=by
 obtain ⟨ht,-,-,-⟩:=tOfG_spec hok m F hm2 hF1 hF2 h hk
 rw[hC] at ht
 have hC0:0<=2*C:=by rw[<-hC];positivity
 have hT0:=Real.exp_pos (-(2*C))
 have hT1:Real.exp (-(2*C))<=1:=Real.exp_le_one_iff.mpr (by linarith only[hC0])
 have hn:=(gAp_sub gOne ht (M:=1) (N:=1) ב (by rw[ἡ hT0];exact hT1)
  ב ב).mono (show _<=(1:ℝ)/2^44+1/2^51 by norm_num)
 have hd:=(gAp_add gOne ht (M:=1) (N:=1) ב (by rw[ἡ hT0];exact hT1)
  ב ב).mono (show _<=(1:ℝ)/2^44+1/2^51 by norm_num)
 have hq:=(gAp_div hn hd (M:=1) (m:=1)
  (by rw[ὸ];constructor<;>linarith only[hT1,hT0]) (by linarith only[hT0]) ב
  ב).mono (show _<=(1:ℝ)/2^42 by norm_num)
 rw[show-(2*C)=-2*C by ring,<-tanh_eq_exp_neg_two] at hq
 obtain ⟨f41,r41⟩:=gM41
 have hqa:=hq.absR
 have hV: |Real.tanh C| <=1:=abs_le.mpr ⟨(Real.neg_one_lt_tanh C).le,(Real.tanh_lt_one C).le⟩
 refine gZA (E:=m41G) rfl rfl hq f41 (by linarith only[h,hk,hqa,hV]) (by rw[r41];norm_num) ?_ ?_ hz hf hM
 rotate_left
 · intro h34
   rw[r41,ἡ (show (0:ℝ)<1/2^41 by norm_num)] at h34
   have h3:=gAz0
   have h5:=Ὥ (D.R ((oneG-tOfG m F h hk)/(oneG+tOfG m F h hk))) (Real.tanh C)
   have h6:=hq.2
   linarith only[h,hk,hqa,h34,h3]
 rw[r41,ἡ (show (0:ℝ)<1/2^41 by norm_num)]
 have:1/2^53*(|D.R ((oneG-tOfG m F h hk)/(oneG+tOfG m F h hk))| +1/2^41)<=
  1/2^53*(1+1/2^42+1/2^41):=by gcongr;linarith only[h,hk,hqa,hV]
 have:(1:ℝ)/2^42+1/2^53*(1+1/2^42+1/2^41)+1/2^150<1/2^41:=ב
 linarith only[h,hk,hqa,hV]
lemma tanhNeg (hok:tabOK=true) (m F:ὥ) (hm2:m.toNat<2^24) (hF1:1<=F.toNat)
  (hF2:F.toNat<=144) (h:F.toUSize.toNat<D.pw.size) (hk:kOfG m F<1280) {C:ℝ}
  (hC:(m.toNat:ℝ)*2^((F.toNat:ℤ)-150)=2*C) {M:Model Fmt}
  (hf:M.isFinite=true) (hM:M.toReal=Model.roundAt Fmt (Real.tanh (-C)))
  (hz:zA loA spA ((tOfG m F h hk-oneG)/(oneG+tOfG m F h hk)) m41G≠Е):
  toWord M=zA loA spA ((tOfG m F h hk-oneG)/(oneG+tOfG m F h hk)) m41G:=by
 obtain ⟨ht,-,-,-⟩:=tOfG_spec hok m F hm2 hF1 hF2 h hk
 rw[hC] at ht
 have hC0:0<=2*C:=by rw[<-hC];positivity
 have hT0:=Real.exp_pos (-(2*C))
 have hT1:Real.exp (-(2*C))<=1:=Real.exp_le_one_iff.mpr (by linarith only[hC0])
 have hn:=(gAp_sub ht gOne (M:=1) (N:=1) (by rw[ἡ hT0];exact hT1) ב
  ב ב).mono (show _<=(1:ℝ)/2^44+1/2^51 by norm_num)
 have hd:=(gAp_add gOne ht (M:=1) (N:=1) ב (by rw[ἡ hT0];exact hT1)
  ב ב).mono (show _<=(1:ℝ)/2^44+1/2^51 by norm_num)
 have hq:=(gAp_div hn hd (M:=1) (m:=1)
  (by rw[ὸ];constructor<;>linarith only[hT0,hT1]) (by linarith only[hT0]) ב
  ב).mono (show _<=(1:ℝ)/2^42 by norm_num)
 have hV:(Real.exp (-(2*C))-1)/(1+Real.exp (-(2*C)))=Real.tanh (-C):=by
  rw[Real.tanh_neg,tanh_eq_exp_neg_two,show-2*C=-(2*C) by ring,<-neg_div];ring_nf
 rw[hV] at hq
 obtain ⟨f41,r41⟩:=gM41
 have hqa:=hq.absR
 have hV1: |Real.tanh (-C)| <=1:=abs_le.mpr ⟨(Real.neg_one_lt_tanh _).le,(Real.tanh_lt_one _).le⟩
 refine gZA (E:=m41G) rfl rfl hq f41 (by linarith only[h,hk,hqa,hV1]) (by rw[r41];norm_num) ?_ ?_ hz hf hM
 rotate_left
 · intro h34
   rw[r41,ἡ (show (0:ℝ)<1/2^41 by norm_num)] at h34
   have h3:=gAz0
   have h5:=Ὥ (D.R ((tOfG m F h hk-oneG)/(oneG+tOfG m F h hk))) (Real.tanh (-C))
   have h6:=hq.2
   linarith only[h,hk,hqa,h34,h3]
 rw[r41,ἡ (show (0:ℝ)<1/2^41 by norm_num)]
 have:1/2^53*(|D.R ((tOfG m F h hk-oneG)/(oneG+tOfG m F h hk))| +1/2^41)<=
  1/2^53*(1+1/2^42+1/2^41):=by gcongr;linarith only[h,hk,hqa,hV1]
 have:(1:ℝ)/2^42+1/2^53*(1+1/2^42+1/2^41)+1/2^150<1/2^41:=ב
 linarith only[h,hk,hqa,hV1]
lemma tanh_sat {C:ℝ} (hC:10<=C):1-1/2^27<=Real.tanh C/\Real.tanh C<1:=by
 refine ⟨?_,Real.tanh_lt_one C⟩
 have hT0:=Real.exp_pos (-(2*C))
 have hT:Real.exp (-(2*C))<=1/2^28:=exp_neg_le_inv_two_pow 28 (by norm_num;linarith only[hC])
 rw[tanh_eq_exp_neg_two,show-2*C=-(2*C) by ring,le_div_iff₀ (by linarith only[hT0,hT])]
 linarith only[hT0,hT]
lemma gNegOne:gAp negOneG (-1) (1/2^52):=by
 have h0:gAp (0:ὧ) 0 0:=gAp_exact D.qHg.1 D.qHg.2
 have:=gAp_sub h0 gOne (M:=0) (N:=1) ב ב ב ב
 exact (by simpa only[one_div,zero_sub] using this.mono (show _<=(1:ℝ)/2^52 by norm_num):gAp (0-oneG) (-1) (1/2^52))
lemma tanhSatPos {C:ℝ} (hC:10<=C) {M:Model Fmt} (hf:M.isFinite=true)
  (hM:M.toReal=Model.roundAt Fmt (Real.tanh C)) (hz:D.ziv oneG m26G≠Е):
  toWord M=D.ziv oneG m26G:=by
 obtain ⟨t1,t2⟩:=tanh_sat hC
 have hq:gAp oneG (Real.tanh C) (1/2^27):=
  ⟨gOneR.1,by rw[gOneR.2,ὸ];constructor<;>linarith only[t1,t2]⟩
 obtain ⟨f26,r26⟩:=gM26
 refine gZiv hq f26 (by rw[gOneR.2];norm_num) (by rw[r26];norm_num) ?_ hz hf hM
 rw[r26,gOneR.2];norm_num
lemma tanhSatNeg {C:ℝ} (hC:10<=C) {M:Model Fmt} (hf:M.isFinite=true)
  (hM:M.toReal=Model.roundAt Fmt (Real.tanh (-C))) (hz:D.ziv negOneG m26G≠Е):
  toWord M=D.ziv negOneG m26G:=by
 obtain ⟨t1,t2⟩:=tanh_sat hC
 have hn:=gNegOne
 have hq:gAp negOneG (Real.tanh (-C)) (1/2^27+1/2^52):=by
  refine ⟨hn.1,?_⟩
  rw[Real.tanh_neg]
  have:=abs_le.mp hn.2
  rw[ὸ];constructor<;>linarith only[t1,this,t2]
 have hqa:=hq.absR
 have hV1: |Real.tanh (-C)| <=1:=abs_le.mpr ⟨(Real.neg_one_lt_tanh _).le,(Real.tanh_lt_one _).le⟩
 obtain ⟨f26,r26⟩:=gM26
 refine gZiv hq f26 (by linarith only[t1,t2,hqa,hV1]) (by rw[r26];norm_num) ?_ hz hf hM
 rw[r26,ἡ (show (0:ℝ)<1/2^26 by norm_num)]
 have:1/2^53*(|D.R negOneG| +1/2^26)<=1/2^53*(2+1/2^26):=by gcongr;linarith only[t1,t2,hqa,hV1]
 have:(1:ℝ)/2^27+1/2^52+1/2^53*(2+1/2^26)+1/2^150<1/2^26:=ב
 linarith only[t1,t2,hqa,hV1]
end GTanh
section GMain
lemma sigModel (w:ὥ) (hfin:expOf w≠255):
  (sigmoidModel (ofW w)).isFinite=true/\
    (sigmoidModel (ofW w)).toReal=Model.roundAt Fmt (realSigmoid (D.wv w)):=
 toReal_sigmoidModel (fmt:=Fmt) ה (input:=Model.ofUInt32 w)
  (by simp only[isFinite_ofUInt32,bne_iff_ne,ne_eq,hfin,not_false_eq_true])
lemma tanhModel' (w:ὥ) (hfin:expOf w≠255):
  (tanhModel (ofW w)).isFinite=true/\
    (tanhModel (ofW w)).toReal=Model.roundAt Fmt (Real.tanh (D.wv w)):=
 toReal_tanhModel (fmt:=Fmt) ה (input:=Model.ofUInt32 w)
  (by simp only[isFinite_ofUInt32,bne_iff_ne,ne_eq,hfin,not_false_eq_true])
lemma sigCR_eq (w:ὥ):sigmoidCR w=toWord (sigmoidModel (ofW w)):=rfl
lemma tanhCR_eq (w:ὥ):tanhCR w=toWord (tanhModel (ofW w)):=rfl
lemma fRange {F:ὥ} (h:F-1<131):1<=F.toNat/\F.toNat<=131:=by
 rw[ἳ,UInt32.toNat_sub] at h
 have:=F.toNat_lt;simp only[Nat.reducePow,UInt32.reduceToNat,Nat.add_one_sub_one] at h;omega
lemma fRange2 {F:ὥ} (h:F-2<131):2<=F.toNat/\F.toNat<=132:=by
 rw[ἳ,UInt32.toNat_sub] at h
 have:=F.toNat_lt;simp only[Nat.reducePow,UInt32.reduceToNat,Nat.reduceSub] at h;omega
lemma sigG_spec (hok:tabOK=true) (w:ὥ):sigG w≠Е->sigG w=sigmoidCR w:=by
 unfold sigG
 dsimp only
 rw[sigCR_eq]
 split
 · rename_i hE
   obtain ⟨hF1,hF2⟩:=fRange hE
   obtain ⟨he,hm1,hm2,hx⟩:=decW w hF1 א
   obtain ⟨hf,hM⟩:=sigModel w he
   rw[hx] at hM
   split
   · rename_i hs
     rw[Ὦ hs,one_mul] at hM
     split
     · rename_i hk
       intro hz
       exact (sigPos hok _ _ hm2 hF1 א _ _ hf hM hz).symm
     · rename_i hk
       intro hz
       refine (sigSat ?_ hf hM hz).symm
       obtain ⟨-,hlo,-⟩:=split_spec _ _ hm2 hF1 א (eidx _)
       rw[ἳ] at hk;simp only[UInt32.reduceToNat,not_lt] at hk
       have:(1216:ℝ)<=((kOfG ((w&&&2147483647)&&&8388607 ||| 8388608) ((w&&&2147483647)>>>23)).toNat:ℝ):=by
        exact_mod_cast hk
       linarith only[hlo,this]
   · rename_i hs
     rw[Ὧ hs,neg_one_mul] at hM
     split
     · rename_i hk
       intro hz
       exact (sigNeg hok _ _ hm2 hF1 א _ _ (by rw[ἳ] at hk;simpa only[UInt32.reduceToNat] using hk) hf hM hz).symm
     · intro hz;exact absurd rfl hz
 · rename_i hE
   split
   · rename_i hc
     simp only[Bool.and_eq_true,decide_eq_true_eq] at hc
     obtain ⟨⟨h1,h2⟩,hs⟩:=hc
     rw[ἳ] at h1 h2
     simp only[UInt32.reduceToNat] at h1 h2
     obtain ⟨he,hm1,hm2,hx⟩:=decW w א א
     obtain ⟨hf,hM⟩:=sigModel w he
     rw[hx,Ὦ hs,one_mul] at hM
     intro hz
     refine (sigSat ?_ hf hM hz).symm
     have hp:(2:ℝ)^(5:ℤ)<=2^((((w&&&2147483647)>>>23).toNat:ℤ)-150+23):=
      ἒ ב א
     have hm:(2:ℝ)^23<=(((w&&&2147483647)&&&8388607 ||| 8388608).toNat:ℝ):=by exact_mod_cast hm1
     rw[Ἓ ב] at hp
     have:(2:ℝ)^((((w&&&2147483647)>>>23).toNat:ℤ)-150)*2^23<=
       (((w&&&2147483647)&&&8388607 ||| 8388608).toNat:ℝ)*2^((((w&&&2147483647)>>>23).toNat:ℤ)-150):=by
      rw[mul_comm];exact ἀ hm ו
     norm_num at hp this⊢
     linarith only[hp,this]
   · intro hz;exact absurd rfl hz
end GMain
section GMain2
lemma fPlus (w:ὥ):
  (((w&&&В)+0x800000)>>>23).toNat=((w&&&В)>>>23).toNat+1:=by
 have ha:=D.qAq w
 have hw:=w.toNat_lt
 rw[UInt32.toNat_shiftRight,UInt32.toNat_shiftRight,Ἱ,Ἱ,
  UInt32.toNat_add,ha]
 simp only[UInt32.reduceToNat,Nat.reduceMod,Nat.reducePow]
 have:w.toNat%2^31<2^31:=ὀ _ ב
 rw[Ἁ א]
 omega
lemma twoC (mm:ℝ) (E F:Nat) (h:F=E+1):
  mm*(2:ℝ)^((F:ℤ)-150)=2*(mm*2^((E:ℤ)-150)):=by
 subst h
 rw[show ((E+1:Nat):ℤ)-150=((E:ℤ)-150)+1 by push_cast;ring,zpow_add_one₀ ב]
 ring
lemma bigC (mm E:Nat) (hm:2^23<=mm) (hE:132<=E):(10:ℝ)<=(mm:ℝ)*2^((E:ℤ)-150):=by
 have h1:(2:ℝ)^23<=(mm:ℝ):=by exact_mod_cast hm
 have h2:(2:ℝ)^(-18:ℤ)<=2^((E:ℤ)-150):=ἒ ב א
 have h3:(2:ℝ)^23*2^(-18:ℤ)<=(mm:ℝ)*2^((E:ℤ)-150):=
  ἑ h1 h2 ו ו
 have h4:(2:ℝ)^23*2^(-18:ℤ)=32:=ב
 linarith only[h3,h4]
lemma tanhG_spec (hok:tabOK=true) (w:ὥ):tanhG w≠Е->tanhG w=tanhCR w:=by
 unfold tanhG
 dsimp only
 rw[tanhCR_eq]
 have hFE:=fPlus w
 split
 · rename_i hE
   obtain ⟨hF1,hF2⟩:=fRange2 hE
   obtain ⟨he,hm1,hm2,hx⟩:=decW w א א
   obtain ⟨hf,hM⟩:=tanhModel' w he
   rw[hx] at hM
   have hC:=twoC ((((w&&&2147483647)&&&8388607 ||| 8388608).toNat:ℝ)) _ _ hFE
   split
   · rename_i hk
     split
     · rename_i hs
       rw[Ὦ hs,one_mul] at hM
       intro hz
       refine (tanhPos hok _ _ hm2 ?_ ?_ _ _ hC hf hM hz).symm<;>omega
     · rename_i hs
       rw[Ὧ hs,neg_one_mul] at hM
       intro hz
       refine (tanhNeg hok _ _ hm2 ?_ ?_ _ _ hC hf hM hz).symm<;>omega
   · rename_i hk
     obtain ⟨-,hlo,-⟩:=split_spec ((w&&&2147483647)&&&8388607 ||| 8388608)
      (((w&&&2147483647)+8388608)>>>23) hm2 א א (eidx _)
     rw[ἳ] at hk;simp only[UInt32.reduceToNat,not_lt] at hk
     have h1280:(1280:ℝ)<=((kOfG ((w&&&2147483647)&&&8388607 ||| 8388608) (((w&&&2147483647)+8388608)>>>23)).toNat:ℝ):=by
      exact_mod_cast hk
     rw[hC] at hlo
     have hC10:(10:ℝ)<=(((w&&&2147483647)&&&8388607 ||| 8388608).toNat:ℝ)*
      (2:ℝ)^((((w&&&2147483647)>>>23).toNat:ℤ)-150):=by linarith only[hlo,h1280]
     split
     · rename_i hs
       rw[Ὦ hs,one_mul] at hM
       intro hz
       exact (tanhSatPos hC10 hf hM hz).symm
     · rename_i hs
       rw[Ὧ hs,neg_one_mul] at hM
       intro hz
       exact (tanhSatNeg hC10 hf hM hz).symm
 · rename_i hE
   split
   · rename_i hc
     simp only[Bool.and_eq_true,decide_eq_true_eq] at hc
     obtain ⟨h1,h2⟩:=hc
     rw[ἳ] at h1 h2
     simp only[UInt32.reduceToNat] at h1 h2
     obtain ⟨he,hm1,hm2,hx⟩:=decW w א א
     obtain ⟨hf,hM⟩:=tanhModel' w he
     rw[hx] at hM
     have hC10:=bigC _ _ hm1 (show 132<=((w&&&2147483647)>>>23).toNat by omega)
     split
     · rename_i hs
       rw[Ὦ hs,one_mul] at hM
       intro hz
       exact (tanhSatPos hC10 hf hM hz).symm
     · rename_i hs
       rw[Ὧ hs,neg_one_mul] at hM
       intro hz
       exact (tanhSatNeg hC10 hf hM hz).symm
   · intro hz;exact absurd rfl hz
lemma sigmoidF2_eq' (w:ὥ):sigmoidF2 w=sigmoidCR w:=by
 unfold sigmoidF2
 dsimp only
 split
 · rename_i hok
   split
   · rw[sigmoidF9_eq]
   · rename_i hne
     exact sigG_spec hok w (by simpa only[ne_eq,beq_iff_eq] using hne)
 · simp only[beq_self_eq_true,ite_true];exact sigmoidF9_eq w
lemma tanhF2_eq' (w:ὥ):tanhF2 w=tanhCR w:=by
 unfold tanhF2
 dsimp only
 split
 · rename_i hok
   split
   · rw[tanhF9_eq]
   · rename_i hne
     exact tanhG_spec hok w (by simpa only[ne_eq,beq_iff_eq] using hne)
 · simp only[beq_self_eq_true,ite_true];exact tanhF9_eq w
lemma sigmoidF2_eq:sigmoidF2=FastAct.sigmoidF:=by
 rw[FastAct.sigmoidF_eq];funext w;exact sigmoidF2_eq' w
lemma tanhF2_eq:tanhF2=FastAct.tanhF:=by
 rw[FastAct.tanhF_eq];funext w;exact tanhF2_eq' w
end GMain2
section RnnX
/-! -/
def xNZ:ὧ:=dv س
def xNA:ὧ:=xNZ.abs
@[inline] def gT (lo sp:Ὠ) (e s0:ὧ) (sl:Unit->Word):Word:=
 let s:=s0+xNZ
 let w:=D.fin1 lo sp s
 if w==Е then
  let w2:=ziv s e
  if w2==Е then sl () else w2
 else w
lemma good_nz:good س=true:=ה
lemma fin1_split (lo sp:Ὠ) (s:ὧ) (h:D.fin1 lo sp s≠Е):
  (s.toBits<<<(1:Ὠ))-lo<sp/\(s.toBits+0x0FDFFFFF)&&&0x1FFFFFFF<0x1FC00000/\
    D.fin1 lo sp s=D.zivC ((s.toBits+0x0FDFFFFF)>>>(29:Ὠ)):=by
 unfold D.fin1 at h⊢
 dsimp only at h⊢
 split at h
 · rename_i hc
   simp only[Bool.and_eq_true,decide_eq_true_eq] at hc
   exact ⟨hc.1,hc.2,by rw[Ὦ (by simp only[Bool.and_eq_true,decide_eq_true_eq];exact hc)]⟩
 · exact absurd rfl h
lemma gT_eq (n:Nat) (hn:n<2^20) (hn0:0<n) (xw ww:Nat->Word)
  (hx: ∀t,t<n->good (xw t)=true) (hw: ∀t,t<n->good (ww t)=true)
  (hv:Word) (hgv:good hv=true) (hpv:hv<=В)
  (hmv: ∀t,t<n->ww t&&&В<=hv) (s0 a:ὧ)
  (hs0:s0=(ἄ n).foldl (λacc t=>acc+dv (xw t)*dv (ww t)) 0)
  (ha:a=(ἄ n).foldl (λacc t=>acc+(dv (xw t)).abs) 0)
  (sl:Unit->Word) (hsl:sl ()=dotOf n xw ww):
  gT (D.loOf (nfOf n*(dv hv*a+xNA))) (D.spOf (D.loOf (nfOf n*(dv hv*a+xNA))))
   (nfOf n*(dv hv*a+xNA)) s0 sl=dotOf n xw ww:=by
 have hf: ∀t,t<n->expOf (xw t)≠255/\expOf (ww t)≠255:=
  λt ht=>⟨qAe _ (hx t ht),qAe _ (hw t ht)⟩
 have hI:=C.qI0 n xw ww hn0 hf
 have hs:s0+xNZ=(ἄ n).foldl (λ(acc:ὧ) t=>acc+dv (xw t)*dv (ww t)) 0+dv س/\
   nfOf n*(dv hv*a+xNA)=
    nfOf n*(dv hv*((ἄ n).foldl (λ(acc:ὧ) t=>acc+(dv (xw t)).abs) 0)+
     (dv س).abs):=by
  rw[hs0,ha];exact ⟨rfl,rfl⟩
 have br:= λhbig=>D.qC8 n hn xw ww س hv hx hw good_nz hgv hpv (ἄ n)
  (Ὂ) (λt ht=>List.mem_range.mp ht) (λt ht=>hmv t (List.mem_range.mp ht))
  (λt ht hnot=>absurd (List.mem_range.mpr ht) hnot) (s0+xNZ) _ hs hbig
 unfold gT
 dsimp only
 split
 · rename_i h1
   split
   · exact hsl
   · rename_i h2
     have hz:ziv (s0+xNZ) (nfOf n*(dv hv*a+xNA))≠Е:= λh=>h2 (by rw[h];rfl)
     obtain ⟨-,-,hbig⟩:=D.qBw _ _ hz
     obtain ⟨hlo,hhi⟩:=br hbig
     rw[D.qCn n hn xw ww س hx hw good_nz _ _ hlo hhi hz,hI]
 · rename_i h1
   have hz:D.fin1 (D.loOf (nfOf n*(dv hv*a+xNA))) (D.spOf (D.loOf (nfOf n*(dv hv*a+xNA))))
    (s0+xNZ)≠Е:= λh=>h1 (by rw[h];rfl)
   obtain ⟨c1,c2,c3⟩:=fin1_split _ _ _ hz
   obtain ⟨-,hT⟩:=D.qHy _ _ c1 c2
   obtain ⟨-,fe,-⟩:=D.qI4 _ _ hT
   have hbig:=D.qBgF _ _ _ c1 c2 fe le_rfl
   obtain ⟨hlo,hhi⟩:=br hbig
   rw[c3,D.qCnF n hn xw ww س hx hw good_nz _ _ _ hlo hhi fe le_rfl c1 c2,hI]
end RnnX
section RnnX2
/-! -/
@[inline] def pW (H K:Nat) (wf:Nat->Nat->Nat->Word) (c r0:Nat):Word:=
 if 4*(c/K/2)+(8*(c/K%2)+r0)/4<H then
  wf (4*(c/K/2)+(8*(c/K%2)+r0)/4) ((8*(c/K%2)+r0)%4) (c%K)
 else 0
def pnOf (H K:Nat) (wf:Nat->Nat->Nat->Word):ὤ:=
 rowsB (λc acc=>fpush (λr0=>dv (pW H K wf c r0)) 0 8 acc) 0 ((H+3)/4*2*K)
  (ἆ ((H+3)/4*16*K))
lemma pnOf_spec (H K:Nat) (wf:Nat->Nat->Nat->Word):
  (pnOf H K wf).size=(H+3)/4*2*K*8/\
    ∀c r0,c<(H+3)/4*2*K->r0<8->(pnOf H K wf).get! (c*8+r0)=dv (pW H K wf c r0):=by
 have:=rowsB_spec 8 (λc acc=>fpush (λr0=>dv (pW H K wf c r0)) 0 8 acc)
  (λc r0=>dv (pW H K wf c r0)) (λr acc=>fpush_size _ 0 8 acc)
  (λr acc j hj=>(fpush_get _ 0 8 acc j).1 hj)
  (λr acc j hj=>by rw[(fpush_get _ 0 8 acc (acc.size+j)).2 א א];congr 2;omega)
  ((H+3)/4*2*K) 0 (ἆ ((H+3)/4*16*K)) (by rw[ὒ];rfl)
  (λr' j h _=>absurd h (Ἲ _))
 rw[Ἇ] at this
 exact this
lemma ugetE (a:ὤ) (i:ὦ) (h:i.toNat<a.size):a.uget i h=a.get! i.toNat:=by
 obtain ⟨ds⟩:=a
 show ds[i.toNat]=ds[i.toNat]!
 rw[Ἕ ds i.toNat h]
lemma qixE (p i m:ὦ) (P:ὤ) (k:Nat) (hk:k<8) (hb:p.toNat+8*m.toNat<=P.size)
  (hs:P.size<2^32) (him:i.toNat< m.toNat):
  (p+ὴ 8*i+ὴ k).toNat=p.toNat+8*i.toNat+k:=by
 have e8:(ὴ 8*i).toNat=8*i.toNat:=by
  rw[D.qWm _ _ (by rw[D.uk 8 ה];omega),D.uk 8 ה]
 rw[D.qAf _ _ (by rw[D.qAf _ _ (by rw[e8];omega),e8,D.uk k א];omega),
  D.qAf _ _ (by rw[e8];omega),e8,D.uk k א]
def mxS (V P:ὤ) (p r t n:Nat) (a:ὧ):ὧ:=
 (List.range' t (n-t)).foldl (λacc i=>acc+V.get! i*P.get! (p+8*i+r)) a
lemma mxS_step (V P:ὤ) (p r t n:Nat) (a:ὧ) (h:t<n):
  mxS V P p r t n a=mxS V P p r (t+1) n (a+V.get! t*P.get! (p+8*t+r)):=by
 unfold mxS
 rw[show n-t=(n-(t+1))+1 by omega,List.range'_succ,ὔ]
lemma mxS_end (V P:ὤ) (p r t n:Nat) (a:ὧ) (h: ¬t<n):mxS V P p r t n a=a:=by
 unfold mxS;rw[show n-t=0 by omega];rfl
def mxV (V P:ὤ) (p p' m n t:ὦ) (hx:n.toNat<=V.size) (hn:n.toNat<=m.toNat)
    (hb:p.toNat+8*m.toNat<=P.size) (hb':p'.toNat+8*m.toNat<=P.size) (hs:P.size<2^32)
    (a0 a1 a2 a3 a4 a5 a6 a7 b0 b1 b2 b3 b4 b5 b6 b7:ὧ) (S:ὤ):ὤ:=
 if h:t<n then
  have ht:=USize.lt_iff_toNat_lt.mp h
  have him:t.toNat< m.toNat:=Nat.lt_of_lt_of_le ht hn
  let v:=V.uget t (Nat.lt_of_lt_of_le ht hx)
  let q:=p+ὴ 8*t
  let q':=p'+ὴ 8*t
  mxV V P p p' m n (t+1) hx hn hb hb' hs
   (a0+v*P.uget (q+ὴ 0) (D.wq0 p t m P hb hs him)) (a1+v*P.uget (q+ὴ 1) (D.wq1 p t m P hb hs him))
   (a2+v*P.uget (q+ὴ 2) (D.wq2 p t m P hb hs him)) (a3+v*P.uget (q+ὴ 3) (D.wq3 p t m P hb hs him))
   (a4+v*P.uget (q+ὴ 4) (D.wq4 p t m P hb hs him)) (a5+v*P.uget (q+ὴ 5) (D.wq5 p t m P hb hs him))
   (a6+v*P.uget (q+ὴ 6) (D.wq6 p t m P hb hs him)) (a7+v*P.uget (q+ὴ 7) (D.wq7 p t m P hb hs him))
   (b0+v*P.uget (q'+ὴ 0) (D.wq0 p' t m P hb' hs him)) (b1+v*P.uget (q'+ὴ 1) (D.wq1 p' t m P hb' hs him))
   (b2+v*P.uget (q'+ὴ 2) (D.wq2 p' t m P hb' hs him)) (b3+v*P.uget (q'+ὴ 3) (D.wq3 p' t m P hb' hs him))
   (b4+v*P.uget (q'+ὴ 4) (D.wq4 p' t m P hb' hs him)) (b5+v*P.uget (q'+ὴ 5) (D.wq5 p' t m P hb' hs him))
   (b6+v*P.uget (q'+ὴ 6) (D.wq6 p' t m P hb' hs him)) (b7+v*P.uget (q'+ὴ 7) (D.wq7 p' t m P hb' hs him)) S
 else
  (((((((((((((((S.push a0).push a1).push a2).push a3).push a4).push a5).push a6).push a7).push b0).push b1).push b2).push b3).push b4).push b5).push b6).push b7
termination_by n.toNat-t.toNat
decreasing_by all_goals (rw[D.usz t n h];omega)
def mxF (V P:ὤ) (p p' m n t:ὦ) (hx:n.toNat<=V.size) (hn:n.toNat<=m.toNat)
    (hb:p.toNat+8*m.toNat<=P.size) (hb':p'.toNat+8*m.toNat<=P.size) (hs:P.size<2^32)
    (a0 a1 a2 a3 a4 a5 a6 a7 b0 b1 b2 b3 b4 b5 b6 b7:ὧ) (S:ὤ):ὤ:=
 if h:t<n then
  have ht:=USize.lt_iff_toNat_lt.mp h
  have him:t.toNat< m.toNat:=Nat.lt_of_lt_of_le ht hn
  let v:=V.uget t (Nat.lt_of_lt_of_le ht hx)
  let q:=p+ὴ 8*t
  let q':=p'+ὴ 8*t
  mxF V P p p' m n (t+1) hx hn hb hb' hs
   (Float.fma v (P.uget (q+ὴ 0) (D.wq0 p t m P hb hs him)) a0) (Float.fma v (P.uget (q+ὴ 1) (D.wq1 p t m P hb hs him)) a1)
   (Float.fma v (P.uget (q+ὴ 2) (D.wq2 p t m P hb hs him)) a2) (Float.fma v (P.uget (q+ὴ 3) (D.wq3 p t m P hb hs him)) a3)
   (Float.fma v (P.uget (q+ὴ 4) (D.wq4 p t m P hb hs him)) a4) (Float.fma v (P.uget (q+ὴ 5) (D.wq5 p t m P hb hs him)) a5)
   (Float.fma v (P.uget (q+ὴ 6) (D.wq6 p t m P hb hs him)) a6) (Float.fma v (P.uget (q+ὴ 7) (D.wq7 p t m P hb hs him)) a7)
   (Float.fma v (P.uget (q'+ὴ 0) (D.wq0 p' t m P hb' hs him)) b0) (Float.fma v (P.uget (q'+ὴ 1) (D.wq1 p' t m P hb' hs him)) b1)
   (Float.fma v (P.uget (q'+ὴ 2) (D.wq2 p' t m P hb' hs him)) b2) (Float.fma v (P.uget (q'+ὴ 3) (D.wq3 p' t m P hb' hs him)) b3)
   (Float.fma v (P.uget (q'+ὴ 4) (D.wq4 p' t m P hb' hs him)) b4) (Float.fma v (P.uget (q'+ὴ 5) (D.wq5 p' t m P hb' hs him)) b5)
   (Float.fma v (P.uget (q'+ὴ 6) (D.wq6 p' t m P hb' hs him)) b6) (Float.fma v (P.uget (q'+ὴ 7) (D.wq7 p' t m P hb' hs him)) b7) S
 else
  (((((((((((((((S.push a0).push a1).push a2).push a3).push a4).push a5).push a6).push a7).push b0).push b1).push b2).push b3).push b4).push b5).push b6).push b7
termination_by n.toNat-t.toNat
decreasing_by all_goals (rw[D.usz t n h];omega)
lemma ugQ (p i m:ὦ) (P:ὤ) (k:Nat) (hk:k<8) (hb:p.toNat+8*m.toNat<=P.size)
  (hs:P.size<2^32) (him:i.toNat< m.toNat) (h:(p+ὴ 8*i+ὴ k).toNat<P.size):
  P.uget (p+ὴ 8*i+ὴ k) h=P.get! (p.toNat+8*i.toNat+k):=by
 rw[ugetE,qixE p i m P k hk hb hs him]
lemma ugQ0 (p i m:ὦ) (P:ὤ) (hb:p.toNat+8*m.toNat<=P.size) (hs:P.size<2^32)
 (him:i.toNat< m.toNat) (h:(p+ὴ 8*i+ὴ 0).toNat<P.size):
 P.uget (p+ὴ 8*i+ὴ 0) h=P.get! (p.toNat+8*i.toNat+0):=ugQ p i m P 0 ה hb hs him h
lemma ugQ1 (p i m:ὦ) (P:ὤ) (hb:p.toNat+8*m.toNat<=P.size) (hs:P.size<2^32)
 (him:i.toNat< m.toNat) (h:(p+ὴ 8*i+ὴ 1).toNat<P.size):
 P.uget (p+ὴ 8*i+ὴ 1) h=P.get! (p.toNat+8*i.toNat+1):=ugQ p i m P 1 ה hb hs him h
lemma ugQ2 (p i m:ὦ) (P:ὤ) (hb:p.toNat+8*m.toNat<=P.size) (hs:P.size<2^32)
 (him:i.toNat< m.toNat) (h:(p+ὴ 8*i+ὴ 2).toNat<P.size):
 P.uget (p+ὴ 8*i+ὴ 2) h=P.get! (p.toNat+8*i.toNat+2):=ugQ p i m P 2 ה hb hs him h
lemma ugQ3 (p i m:ὦ) (P:ὤ) (hb:p.toNat+8*m.toNat<=P.size) (hs:P.size<2^32)
 (him:i.toNat< m.toNat) (h:(p+ὴ 8*i+ὴ 3).toNat<P.size):
 P.uget (p+ὴ 8*i+ὴ 3) h=P.get! (p.toNat+8*i.toNat+3):=ugQ p i m P 3 ה hb hs him h
lemma ugQ4 (p i m:ὦ) (P:ὤ) (hb:p.toNat+8*m.toNat<=P.size) (hs:P.size<2^32)
 (him:i.toNat< m.toNat) (h:(p+ὴ 8*i+ὴ 4).toNat<P.size):
 P.uget (p+ὴ 8*i+ὴ 4) h=P.get! (p.toNat+8*i.toNat+4):=ugQ p i m P 4 ה hb hs him h
lemma ugQ5 (p i m:ὦ) (P:ὤ) (hb:p.toNat+8*m.toNat<=P.size) (hs:P.size<2^32)
 (him:i.toNat< m.toNat) (h:(p+ὴ 8*i+ὴ 5).toNat<P.size):
 P.uget (p+ὴ 8*i+ὴ 5) h=P.get! (p.toNat+8*i.toNat+5):=ugQ p i m P 5 ה hb hs him h
lemma ugQ6 (p i m:ὦ) (P:ὤ) (hb:p.toNat+8*m.toNat<=P.size) (hs:P.size<2^32)
 (him:i.toNat< m.toNat) (h:(p+ὴ 8*i+ὴ 6).toNat<P.size):
 P.uget (p+ὴ 8*i+ὴ 6) h=P.get! (p.toNat+8*i.toNat+6):=ugQ p i m P 6 ה hb hs him h
lemma ugQ7 (p i m:ὦ) (P:ὤ) (hb:p.toNat+8*m.toNat<=P.size) (hs:P.size<2^32)
 (him:i.toNat< m.toNat) (h:(p+ὴ 8*i+ὴ 7).toNat<P.size):
 P.uget (p+ὴ 8*i+ὴ 7) h=P.get! (p.toNat+8*i.toNat+7):=ugQ p i m P 7 ה hb hs him h
lemma mxV_eq (V P:ὤ) (p p' m n:ὦ) (hx:n.toNat<=V.size) (hn:n.toNat<=m.toNat)
  (hb:p.toNat+8*m.toNat<=P.size) (hb':p'.toNat+8*m.toNat<=P.size) (hs:P.size<2^32):
  ∀(N:Nat) (t:ὦ) (a0 a1 a2 a3 a4 a5 a6 a7 b0 b1 b2 b3 b4 b5 b6 b7:ὧ) (S:ὤ),
  n.toNat-t.toNat=N->
  mxV V P p p' m n t hx hn hb hb' hs a0 a1 a2 a3 a4 a5 a6 a7 b0 b1 b2 b3 b4 b5 b6 b7 S=
   (((((((((((((((S.push (mxS V P p.toNat 0 t.toNat n.toNat a0)).push (mxS V P p.toNat 1 t.toNat n.toNat a1)).push
     (mxS V P p.toNat 2 t.toNat n.toNat a2)).push (mxS V P p.toNat 3 t.toNat n.toNat a3)).push
     (mxS V P p.toNat 4 t.toNat n.toNat a4)).push (mxS V P p.toNat 5 t.toNat n.toNat a5)).push
     (mxS V P p.toNat 6 t.toNat n.toNat a6)).push (mxS V P p.toNat 7 t.toNat n.toNat a7)).push
     (mxS V P p'.toNat 0 t.toNat n.toNat b0)).push (mxS V P p'.toNat 1 t.toNat n.toNat b1)).push
     (mxS V P p'.toNat 2 t.toNat n.toNat b2)).push (mxS V P p'.toNat 3 t.toNat n.toNat b3)).push
     (mxS V P p'.toNat 4 t.toNat n.toNat b4)).push (mxS V P p'.toNat 5 t.toNat n.toNat b5)).push
     (mxS V P p'.toNat 6 t.toNat n.toNat b6)).push (mxS V P p'.toNat 7 t.toNat n.toNat b7):=by
 intro N
 induction N with
 | zero=>
   intro t a0 a1 a2 a3 a4 a5 a6 a7 b0 b1 b2 b3 b4 b5 b6 b7 S hN
   have h: ¬t<n:= λh=>by have:=USize.lt_iff_toNat_lt.mp h;omega
   have h': ¬t.toNat<n.toNat:=א
   rw[mxV,dif_neg h]
   simp only[mxS_end _ _ _ _ _ _ _ h']
 | succ N ih=>
   intro t a0 a1 a2 a3 a4 a5 a6 a7 b0 b1 b2 b3 b4 b5 b6 b7 S hN
   have h:t<n:=USize.lt_iff_toNat_lt.mpr א
   have ht:t.toNat<n.toNat:=א
   have him:t.toNat< m.toNat:=א
   rw[mxV,dif_pos h]
   dsimp only
   rw[ih (t+1) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ S (by rw[D.usz t n h];omega)]
   have hv:V.uget t (Nat.lt_of_lt_of_le ht hx)=V.get! t.toNat:=ugetE _ _ _
   simp only[D.usz t n h,mxS_step _ _ _ _ _ _ _ ht,hv,ugQ0 p t m P hb hs him,ugQ1 p t m P hb hs him,
    ugQ2 p t m P hb hs him,ugQ3 p t m P hb hs him,ugQ4 p t m P hb hs him,ugQ5 p t m P hb hs him,
    ugQ6 p t m P hb hs him,ugQ7 p t m P hb hs him,ugQ0 p' t m P hb' hs him,ugQ1 p' t m P hb' hs him,
    ugQ2 p' t m P hb' hs him,ugQ3 p' t m P hb' hs him,ugQ4 p' t m P hb' hs him,ugQ5 p' t m P hb' hs him,
    ugQ6 p' t m P hb' hs him,ugQ7 p' t m P hb' hs him,Ἵ]
lemma fmaE (u w:Word) (a:ὧ):Float.fma (dv u) (dv w) a=a+dv u*dv w:=D.zFV u w a
lemma mxF_eq (V P:ὤ) (p p' m n:ὦ) (hx:n.toNat<=V.size) (hn:n.toNat<=m.toNat)
  (hb:p.toNat+8*m.toNat<=P.size) (hb':p'.toNat+8*m.toNat<=P.size) (hs:P.size<2^32)
  (hV: ∀i,i<V.size-> ∃w,V.get! i=dv w) (hP: ∀i,i<P.size-> ∃w,P.get! i=dv w):
  ∀(N:Nat) (t:ὦ) (a0 a1 a2 a3 a4 a5 a6 a7 b0 b1 b2 b3 b4 b5 b6 b7:ὧ) (S:ὤ),
  n.toNat-t.toNat=N->
  mxF V P p p' m n t hx hn hb hb' hs a0 a1 a2 a3 a4 a5 a6 a7 b0 b1 b2 b3 b4 b5 b6 b7 S=
   mxV V P p p' m n t hx hn hb hb' hs a0 a1 a2 a3 a4 a5 a6 a7 b0 b1 b2 b3 b4 b5 b6 b7 S:=by
 intro N
 induction N with
 | zero=>
   intro t a0 a1 a2 a3 a4 a5 a6 a7 b0 b1 b2 b3 b4 b5 b6 b7 S hN
   have h: ¬t<n:= λh=>by have:=USize.lt_iff_toNat_lt.mp h;omega
   rw[mxV,mxF,dif_neg h,dif_neg h]
 | succ N ih=>
   intro t a0 a1 a2 a3 a4 a5 a6 a7 b0 b1 b2 b3 b4 b5 b6 b7 S hN
   have h:t<n:=USize.lt_iff_toNat_lt.mpr א
   have ht:t.toNat<n.toNat:=א
   have him:t.toNat< m.toNat:=א
   rw[mxV,mxF,dif_pos h,dif_pos h]
   dsimp only
   rw[ih (t+1) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ S (by rw[D.usz t n h];omega)]
   obtain ⟨wv,hwv⟩:=hV t.toNat א
   have hv:V.uget t (Nat.lt_of_lt_of_le ht hx)=dv wv:=(ugetE _ _ _).trans hwv
   have hq: ∀(pp:ὦ) (k:Nat) (hk:k<8) (hbb:pp.toNat+8*m.toNat<=P.size)
     (hq:(pp+ὴ 8*t+ὴ k).toNat<P.size), ∃w,
     P.uget (pp+ὴ 8*t+ὴ k) hq=dv w:= λpp k hk hbb hq=>
    let ⟨w,hw⟩:=hP _ hq
    ⟨w,(ugetE _ _ _).trans hw⟩
   obtain ⟨w0,e0⟩:=hq p 0 ה hb (D.wq0 p t m P hb hs him)
   obtain ⟨w1,e1⟩:=hq p 1 ה hb (D.wq1 p t m P hb hs him)
   obtain ⟨w2,e2⟩:=hq p 2 ה hb (D.wq2 p t m P hb hs him)
   obtain ⟨w3,e3⟩:=hq p 3 ה hb (D.wq3 p t m P hb hs him)
   obtain ⟨w4,e4⟩:=hq p 4 ה hb (D.wq4 p t m P hb hs him)
   obtain ⟨w5,e5⟩:=hq p 5 ה hb (D.wq5 p t m P hb hs him)
   obtain ⟨w6,e6⟩:=hq p 6 ה hb (D.wq6 p t m P hb hs him)
   obtain ⟨w7,e7⟩:=hq p 7 ה hb (D.wq7 p t m P hb hs him)
   obtain ⟨w8,e8⟩:=hq p' 0 ה hb' (D.wq0 p' t m P hb' hs him)
   obtain ⟨w9,e9⟩:=hq p' 1 ה hb' (D.wq1 p' t m P hb' hs him)
   obtain ⟨w10,e10⟩:=hq p' 2 ה hb' (D.wq2 p' t m P hb' hs him)
   obtain ⟨w11,e11⟩:=hq p' 3 ה hb' (D.wq3 p' t m P hb' hs him)
   obtain ⟨w12,e12⟩:=hq p' 4 ה hb' (D.wq4 p' t m P hb' hs him)
   obtain ⟨w13,e13⟩:=hq p' 5 ה hb' (D.wq5 p' t m P hb' hs him)
   obtain ⟨w14,e14⟩:=hq p' 6 ה hb' (D.wq6 p' t m P hb' hs him)
   obtain ⟨w15,e15⟩:=hq p' 7 ה hb' (D.wq7 p' t m P hb' hs him)
   rw[hv,e0,e1,e2,e3,e4,e5,e6,e7,e8,e9,e10,e11,e12,e13,e14,e15]
   simp only[fmaE]
lemma usz2 (t n:ὦ) (h:t<n/\t+1<n):t.toNat+1<n.toNat/\(t+1).toNat=t.toNat+1:=by
 have e:=D.usz t n h.1
 exact ⟨by have:=USize.lt_iff_toNat_lt.mp h.2;omega,e⟩
def mxF2 (V P:ὤ) (p p' m n t:ὦ) (hx:n.toNat<=V.size) (hn:n.toNat<=m.toNat)
    (hb:p.toNat+8*m.toNat<=P.size) (hb':p'.toNat+8*m.toNat<=P.size) (hs:P.size<2^32)
    (a0 a1 a2 a3 a4 a5 a6 a7 b0 b1 b2 b3 b4 b5 b6 b7:ὧ) (S:ὤ):ὤ:=
 if h:t<n/\t+1<n then
  have ht:=(usz2 t n h)
  have him:t.toNat< m.toNat:=by omega
  have him1:(t+1).toNat< m.toNat:=by omega
  let v:=V.uget t (by omega)
  let v1:=V.uget (t+1) (by omega)
  let q:=p+ὴ 8*t
  let q':=p'+ὴ 8*t
  let q1:=p+ὴ 8*(t+1)
  let q1':=p'+ὴ 8*(t+1)
  mxF2 V P p p' m n (t+1+1) hx hn hb hb' hs
   (Float.fma v1 (P.uget (q1+ὴ 0) (D.wq0 p (t+1) m P hb hs him1)) (Float.fma v (P.uget (q+ὴ 0) (D.wq0 p t m P hb hs him)) a0)) (Float.fma v1 (P.uget (q1+ὴ 1) (D.wq1 p (t+1) m P hb hs him1)) (Float.fma v (P.uget (q+ὴ 1) (D.wq1 p t m P hb hs him)) a1))
   (Float.fma v1 (P.uget (q1+ὴ 2) (D.wq2 p (t+1) m P hb hs him1)) (Float.fma v (P.uget (q+ὴ 2) (D.wq2 p t m P hb hs him)) a2)) (Float.fma v1 (P.uget (q1+ὴ 3) (D.wq3 p (t+1) m P hb hs him1)) (Float.fma v (P.uget (q+ὴ 3) (D.wq3 p t m P hb hs him)) a3))
   (Float.fma v1 (P.uget (q1+ὴ 4) (D.wq4 p (t+1) m P hb hs him1)) (Float.fma v (P.uget (q+ὴ 4) (D.wq4 p t m P hb hs him)) a4)) (Float.fma v1 (P.uget (q1+ὴ 5) (D.wq5 p (t+1) m P hb hs him1)) (Float.fma v (P.uget (q+ὴ 5) (D.wq5 p t m P hb hs him)) a5))
   (Float.fma v1 (P.uget (q1+ὴ 6) (D.wq6 p (t+1) m P hb hs him1)) (Float.fma v (P.uget (q+ὴ 6) (D.wq6 p t m P hb hs him)) a6)) (Float.fma v1 (P.uget (q1+ὴ 7) (D.wq7 p (t+1) m P hb hs him1)) (Float.fma v (P.uget (q+ὴ 7) (D.wq7 p t m P hb hs him)) a7))
   (Float.fma v1 (P.uget (q1'+ὴ 0) (D.wq0 p' (t+1) m P hb' hs him1)) (Float.fma v (P.uget (q'+ὴ 0) (D.wq0 p' t m P hb' hs him)) b0)) (Float.fma v1 (P.uget (q1'+ὴ 1) (D.wq1 p' (t+1) m P hb' hs him1)) (Float.fma v (P.uget (q'+ὴ 1) (D.wq1 p' t m P hb' hs him)) b1))
   (Float.fma v1 (P.uget (q1'+ὴ 2) (D.wq2 p' (t+1) m P hb' hs him1)) (Float.fma v (P.uget (q'+ὴ 2) (D.wq2 p' t m P hb' hs him)) b2)) (Float.fma v1 (P.uget (q1'+ὴ 3) (D.wq3 p' (t+1) m P hb' hs him1)) (Float.fma v (P.uget (q'+ὴ 3) (D.wq3 p' t m P hb' hs him)) b3))
   (Float.fma v1 (P.uget (q1'+ὴ 4) (D.wq4 p' (t+1) m P hb' hs him1)) (Float.fma v (P.uget (q'+ὴ 4) (D.wq4 p' t m P hb' hs him)) b4)) (Float.fma v1 (P.uget (q1'+ὴ 5) (D.wq5 p' (t+1) m P hb' hs him1)) (Float.fma v (P.uget (q'+ὴ 5) (D.wq5 p' t m P hb' hs him)) b5))
   (Float.fma v1 (P.uget (q1'+ὴ 6) (D.wq6 p' (t+1) m P hb' hs him1)) (Float.fma v (P.uget (q'+ὴ 6) (D.wq6 p' t m P hb' hs him)) b6)) (Float.fma v1 (P.uget (q1'+ὴ 7) (D.wq7 p' (t+1) m P hb' hs him1)) (Float.fma v (P.uget (q'+ὴ 7) (D.wq7 p' t m P hb' hs him)) b7)) S
 else mxF V P p p' m n t hx hn hb hb' hs a0 a1 a2 a3 a4 a5 a6 a7 b0 b1 b2 b3 b4 b5 b6 b7 S
termination_by n.toNat-t.toNat
decreasing_by all_goals (have:=usz2 t n h;rw[D.usz (t+1) n (USize.lt_iff_toNat_lt.mpr (by omega))];omega)
lemma fmaU (V P:ὤ) (hV: ∀i,i<V.size-> ∃w,V.get! i=dv w)
  (hP: ∀i,i<P.size-> ∃w,P.get! i=dv w) (i j:ὦ) (h:i.toNat<V.size) (h':j.toNat<P.size)
  (a:ὧ):Float.fma (V.uget i h) (P.uget j h') a=a+V.uget i h*P.uget j h':=by
 obtain ⟨w1,e1⟩:=hV _ h
 obtain ⟨w2,e2⟩:=hP _ h'
 rw[ugetE V i h,ugetE P j h',e1,e2,fmaE]
lemma mxF2_eq (V P:ὤ) (p p' m n:ὦ) (hx:n.toNat<=V.size) (hn:n.toNat<=m.toNat)
  (hb:p.toNat+8*m.toNat<=P.size) (hb':p'.toNat+8*m.toNat<=P.size) (hs:P.size<2^32)
  (hV: ∀i,i<V.size-> ∃w,V.get! i=dv w) (hP: ∀i,i<P.size-> ∃w,P.get! i=dv w):
  ∀(N:Nat) (t:ὦ) (a0 a1 a2 a3 a4 a5 a6 a7 b0 b1 b2 b3 b4 b5 b6 b7:ὧ) (S:ὤ),
  n.toNat-t.toNat<=N->
  mxF2 V P p p' m n t hx hn hb hb' hs a0 a1 a2 a3 a4 a5 a6 a7 b0 b1 b2 b3 b4 b5 b6 b7 S=
   mxV V P p p' m n t hx hn hb hb' hs a0 a1 a2 a3 a4 a5 a6 a7 b0 b1 b2 b3 b4 b5 b6 b7 S:=by
 intro N
 induction N with
 | zero=>
   intro t a0 a1 a2 a3 a4 a5 a6 a7 b0 b1 b2 b3 b4 b5 b6 b7 S hN
   have h: ¬(t<n/\t+1<n):= λh=>by have:=USize.lt_iff_toNat_lt.mp h.1;omega
   rw[mxF2,dif_neg h,mxF_eq V P p p' m n hx hn hb hb' hs hV hP _ t _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ S rfl]
 | succ N ih=>
   intro t a0 a1 a2 a3 a4 a5 a6 a7 b0 b1 b2 b3 b4 b5 b6 b7 S hN
   by_cases h:t<n/\t+1<n
   · have ht:=usz2 t n h
     have h1:t+1<n:=h.2
     rw[mxF2,dif_pos h,ih _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ S
      (by rw[D.usz (t+1) n h1];omega)]
     conv_rhs=>rw[mxV,dif_pos h.1]
     dsimp only
     conv_rhs=>rw[mxV,dif_pos h1]
     dsimp only
     simp only[fmaU V P hV hP]
   · rw[mxF2,dif_neg h,mxF_eq V P p p' m n hx hn hb hb' hs hV hP _ t _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ S rfl]
@[inline] def mxD (V P:ὤ) (p p' m n t:ὦ) (hx:n.toNat<=V.size) (hn:n.toNat<=m.toNat)
   (hb:p.toNat+8*m.toNat<=P.size) (hb':p'.toNat+8*m.toNat<=P.size) (hs:P.size<2^32)
   (S:ὤ):ὤ:=
 if D.armT then mxF2 V P p p' m n t hx hn hb hb' hs 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S
 else mxV V P p p' m n t hx hn hb hb' hs 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S
end RnnX2
section RnnX3
lemma mxV_zero (V P:ὤ) (p p' m n:ὦ) (hx:n.toNat<=V.size) (hn:n.toNat<=m.toNat)
  (hb:p.toNat+8*m.toNat<=P.size) (hb':p'.toNat+8*m.toNat<=P.size) (hs:P.size<2^32)
  (S:ὤ):
  mxV V P p p' m n 0 hx hn hb hb' hs 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S=
   fpush (λr=>if r<8 then mxS V P p.toNat r 0 n.toNat 0 else mxS V P p'.toNat (r-8) 0 n.toNat 0) 0 16 S:=by
 rw[mxV_eq V P p p' m n hx hn hb hb' hs _ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S rfl]
 rfl
lemma mxD_eq (V P:ὤ) (p p' m n:ὦ) (hx:n.toNat<=V.size) (hn:n.toNat<=m.toNat)
  (hb:p.toNat+8*m.toNat<=P.size) (hb':p'.toNat+8*m.toNat<=P.size) (hs:P.size<2^32)
  (hV: ∀i,i<V.size-> ∃w,V.get! i=dv w) (hP: ∀i,i<P.size-> ∃w,P.get! i=dv w)
  (S:ὤ):
  mxD V P p p' m n 0 hx hn hb hb' hs S=
   fpush (λr=>if r<8 then mxS V P p.toNat r 0 n.toNat 0 else mxS V P p'.toNat (r-8) 0 n.toNat 0) 0 16 S:=by
 unfold mxD
 split
 · rw[mxF2_eq V P p p' m n hx hn hb hb' hs hV hP _ 0 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ S le_rfl,mxV_zero]
 · rw[mxV_zero]
lemma blk_le (b K P:Nat) (h:(b+1)*16*K<=P):
  b*16*K+8*K+8*K<=P/\b*16*K+8*K<=P:=by
 have e:(b+1)*16*K=b*16*K+16*K:=by ring
 omega
@[inline] def mxRow (V P:ὤ) (K:ὦ) (hx:K.toNat<=V.size) (hs:P.size<2^32) (b:Nat)
   (S:ὤ):ὤ:=
 if h:(b+1)*16*K.toNat<=P.size then
  mxD V P (ὴ (b*16*K.toNat)) (ὴ (b*16*K.toNat+8*K.toNat)) K K 0 hx (Nat.le_refl _)
   (by rw[D.uk _ (by have:=blk_le b K.toNat P.size h;omega)];have:=blk_le b K.toNat P.size h;omega)
   (by rw[D.uk _ (by have:=blk_le b K.toNat P.size h;omega)];have:=blk_le b K.toNat P.size h;omega) hs S
 else fpush (λ_=>0) 0 16 S
def mxB (V P:ὤ) (K b sl:Nat):ὧ:=
 if (b+1)*16*K<=P.size then
  (if sl<8 then mxS V P (b*16*K) sl 0 K 0 else mxS V P (b*16*K+8*K) (sl-8) 0 K 0)
 else 0
lemma mxRow_spec (V P:ὤ) (K:ὦ) (hx:K.toNat<=V.size) (hs:P.size<2^32)
  (hV: ∀i,i<V.size-> ∃w,V.get! i=dv w) (hP: ∀i,i<P.size-> ∃w,P.get! i=dv w) (n:Nat):
  (rowsB (mxRow V P K hx hs) 0 n (ἆ (n*16))).size=n*16/\
    ∀b sl,b<n->sl<16->(rowsB (mxRow V P K hx hs) 0 n (ἆ (n*16))).get! (b*16+sl)=
     mxB V P K.toNat b sl:=by
 have:=rowsB_spec 16 (mxRow V P K hx hs) (λb sl=>mxB V P K.toNat b sl)
  (λr acc=>by unfold mxRow;split<;>(try rw[mxD_eq _ _ _ _ _ _ _ _ _ _ _ hV hP])<;>exact fpush_size _ 0 16 acc)
  (λr acc j hj=>by unfold mxRow;split<;>(try rw[mxD_eq _ _ _ _ _ _ _ _ _ _ _ hV hP])<;>exact (fpush_get _ 0 16 acc j).1 hj)
  (λr acc j hj=>by
    unfold mxRow mxB
    split
    · rename_i h
      rw[mxD_eq _ _ _ _ _ _ _ _ _ _ _ hV hP,(fpush_get _ 0 16 acc (acc.size+j)).2 א א,
       D.uk _ (by have:=blk_le r K.toNat P.size h;omega),D.uk _ (by have:=blk_le r K.toNat P.size h;omega)]
      simp only[Ἇ,Nat.add_sub_cancel_left]
    · rename_i h
      rw[(fpush_get _ 0 16 acc (acc.size+j)).2 א א])
  n 0 (ἆ (n*16)) (by rw[ὒ];rfl)
  (λr' j h _=>absurd h (Ἲ _))
 rw[Ἇ] at this
 exact this
@[inline] def vOf (inW H:Nat) (x h:ё) (t:Nat):ὤ:=
 ((fpush (λq=>dv h[q]!) 0 H (fpush (λq=>dv x[q]!) (t*inW) inW
   (ἆ (inW+H+2)))).push oneD).push oneD
@[inline] def gX (inW H:Nat) (x h:ё) (t j:Nat):Word:=
 if j< inW then x[t*inW+j]! else if j< inW+H then h[j-inW]! else oneW
@[inline] def gWt (inW H:Nat) (wih whh bih bhh:ё) (k j:Nat):Word:=
 if j< inW then wih[k*inW+j]! else if j< inW+H then whh[k*H+(j-inW)]!
 else if j=inW+H then bih[k]! else bhh[k]!
lemma vOf_spec (inW H:Nat) (x h:ё) (t:Nat):
  (vOf inW H x h t).size=inW+H+2/\
    ∀j,j< inW+H+2->(vOf inW H x h t).get! j=dv (gX inW H x h t j):=by
 have e0:(ἆ (inW+H+2)).size=0:=rfl
 have s1:=fpush_size (λq=>dv x[q]!) (t*inW) inW (ἆ (inW+H+2))
 have s2:=fpush_size (λq=>dv h[q]!) 0 H (fpush (λq=>dv x[q]!) (t*inW) inW
  (ἆ (inW+H+2)))
 rw[s1,e0,Ἇ] at s2
 refine ⟨by unfold vOf;rw[fa_push_size,fa_push_size,s2], λj hj=>?_⟩
 unfold vOf gX
 rw[fa_push_get,fa_push_size,s2]
 by_cases j1:j< inW
 · rw[Ὦ א,fa_push_get,s2,Ὦ א,(fpush_get _ 0 H _ j).1 (by rw[s1,e0];omega),
    (fpush_get _ _ inW _ j).2 (by rw[e0];omega) (by rw[e0];omega),Ὦ j1,e0]
   simp only[Nat.sub_zero]
 by_cases j2:j< inW+H
 · rw[Ὦ א,fa_push_get,s2,Ὦ א,(fpush_get _ 0 H _ j).2 (by rw[s1,e0];omega)
    (by rw[s1,e0];omega),Ὧ j1,Ὦ j2,s1,e0]
   simp only[Ἇ]
 · rw[Ὧ j1,Ὧ j2]
   by_cases j3:j=inW+H+1
   · rw[Ὧ א,Ὦ j3];rfl
   · rw[Ὦ א,fa_push_get,s2,Ὧ א,Ὦ א];rfl
def aSN (V:ὤ):Nat->Nat->ὧ->ὧ
 | _,0,a=>a
 | i,k+1,a=>aSN V (i+1) k (a+(V.get! i).abs)
lemma aSN_eq (V:ὤ): ∀k i a,aSN V i k a=(List.range' i k).foldl (λacc j=>acc+(V.get! j).abs) a
 | 0,_,_=>rfl
 | k+1,i,a=>by rw[aSN,aSN_eq V k (i+1),List.range'_succ,ὔ]
def aU (V:ὤ) (n i:ὦ) (hn:n.toNat<=V.size) (a:ὧ):ὧ:=
 if h:i<n then
  aU V n (i+1) hn (a+(V.uget i (Nat.lt_of_lt_of_le (USize.lt_iff_toNat_lt.mp h) hn)).abs)
 else a
termination_by n.toNat-i.toNat
decreasing_by all_goals (have:=USize.lt_iff_toNat_lt.mp h;rw[D.usz i n h];omega)
lemma aU_eq (V:ὤ) (n:ὦ) (hn:n.toNat<=V.size): ∀(N:Nat) (i:ὦ) (a:ὧ),
  n.toNat-i.toNat=N->aU V n i hn a=aSN V i.toNat (n.toNat-i.toNat) a:=by
 intro N
 induction N with
 | zero=>
   intro i a hN
   have h: ¬i<n:= λh=>by have:=USize.lt_iff_toNat_lt.mp h;omega
   rw[aU,dif_neg h,hN];rfl
 | succ N ih=>
   intro i a hN
   have h:i<n:=USize.lt_iff_toNat_lt.mpr א
   rw[aU,dif_pos h,ih (i+1) _ (by rw[D.usz i n h];omega),D.usz i n h,ugetE,
    show n.toNat-i.toNat=(n.toNat-(i.toNat+1))+1 by omega,aSN]
@[inline] def aS (V:ὤ) (i n:Nat) (a:ὧ):ὧ:=
 if h:i+n<=V.size/\V.size<2^32 then
  aU V (ὴ (i+n)) (ὴ i) (by rw[D.uk _ (by omega)];exact h.1) a
 else aSN V i n a
lemma aS_eq (V:ὤ) (k i:Nat) (a:ὧ):aS V i k a=(List.range' i k).foldl (λacc j=>acc+(V.get! j).abs) a:=by
 unfold aS
 split
 · rename_i h
   rw[aU_eq V _ _ _ _ a rfl,D.uk _ א,D.uk _ א,Nat.add_sub_cancel_left,aSN_eq]
 · rw[aSN_eq]
lemma foldl_ext {α:Type} (f g:α->Nat->α): ∀(l:љ) (a:α),(∀j∈l, ∀b,f b j=g b j)->
  l.foldl f a=l.foldl g a
 | [],_,_=>rfl
 | j::l,a,h=>by
   rw[ὔ,ὔ,h j (ὣ ..) a]
   exact foldl_ext f g l _ λu hu=>h u (ὖ _ hu)
lemma divmodK (m i K:Nat) (hi:i<K):(m*K+i)/K=m/\(m*K+i)%K=i:=by
 have hK:0<K:=א
 constructor
 · rw[Nat.add_comm,Nat.add_mul_div_right _ _ hK,Ὃ hi,Ἇ]
 · rw[Nat.add_comm,Nat.add_mul_mod_self_right,Ἁ hi]
lemma slot_eq (H K:Nat) (wf:Nat->Nat->Nat->Word) (V:ὤ) (hK:K<2^32)
  (hVs:V.size=K) (xw:Nat->Word) (hVx: ∀j,j<K->V.get! j=dv (xw j))
  (hs:(pnOf H K wf).size<2^32) (hx:(ὴ K).toNat<=V.size) (u g:Nat) (hu:u<H) (hg:g<4):
  (rowsB (mxRow V (pnOf H K wf) (ὴ K) hx hs) 0 ((H+3)/4)
    (ἆ ((H+3)/4*16))).get! (4*u+g)=
    (ἄ K).foldl (λacc j=>acc+dv (xw j)*dv (wf u g j)) 0:=by
 obtain ⟨hPs,hPg⟩:=pnOf_spec H K wf
 have hV: ∀i,i<V.size-> ∃w,V.get! i=dv w:= λi hi=>⟨xw i,hVx i א⟩
 have hP: ∀i,i<(pnOf H K wf).size-> ∃w,(pnOf H K wf).get! i=dv w:= λi hi=>by
  rw[hPs] at hi
  refine ⟨pW H K wf (i/8) (i%8),?_⟩
  rw[<-hPg (i/8) (i%8) א א,Nat.div_add_mod']
 have hKn:(ὴ K).toNat=K:=D.uk K hK
 have hb4:u/4<(H+3)/4:=א
 obtain ⟨-,hS⟩:=mxRow_spec V (pnOf H K wf) (ὴ K) hx hs hV hP ((H+3)/4)
 rw[show 4*u+g=u/4*16+(4*(u%4)+g) by omega,hS _ _ hb4 א]
 unfold mxB
 have hbl:(u/4+1)*16*K<=(pnOf H K wf).size:=by
  rw[hPs];have:=Ἒ K (Ἒ 16 (show u/4+1<=(H+3)/4 by omega))
  calc (u/4+1)*16*K<=(H+3)/4*16*K:=this
   _=(H+3)/4*2*K*8:=by ring
 rw[hKn,Ὦ hbl]
 have hs8: ∀(hf:Nat),hf<2->4*(u%4)+g=8*hf+(4*(u%4)+g-8*hf)->
   (4*(u%4)+g-8*hf)<8->
   mxS V (pnOf H K wf) (u/4*16*K+8*K*hf) (4*(u%4)+g-8*hf) 0 K 0=
    (ἄ K).foldl (λacc j=>acc+dv (xw j)*dv (wf u g j)) 0:=by
  intro hf hf2 hsl hr
  unfold mxS
  rw[Nat.sub_zero,<-ὗ]
  apply foldl_ext
  intro j hj b
  have hjK:=List.mem_range.mp hj
  have hc:(2*(u/4)+hf)*K+j<(H+3)/4*2*K:=by
   have:=Ἒ K (show 2*(u/4)+hf+1<=(H+3)/4*2 by omega)
   rw[ὰ] at this;omega
  rw[hVx j hjK,show u/4*16*K+8*K*hf+8*j+(4*(u%4)+g-8*hf)=
   ((2*(u/4)+hf)*K+j)*8+(4*(u%4)+g-8*hf) by ring,hPg _ _ hc hr]
  unfold pW
  obtain ⟨d1,d2⟩:=divmodK (2*(u/4)+hf) j K hjK
  rw[d1,d2,show (2*(u/4)+hf)/2=u/4 by omega,show (2*(u/4)+hf)%2=hf by omega,
   show 8*hf+(4*(u%4)+g-8*hf)=4*(u%4)+g by omega,
   show 4*(u/4)+(4*(u%4)+g)/4=u by omega,show (4*(u%4)+g)%4=g by omega,Ὦ hu]
 by_cases h8:4*(u%4)+g<8
 · rw[Ὦ h8]
   have:=hs8 0 ה א א
   rw[Nat.mul_zero,Ἵ,Nat.mul_zero,Nat.sub_zero] at this
   exact this
 · rw[Ὧ h8]
   have:=hs8 1 ה א א
   rw[Nat.mul_one,Nat.mul_one] at this
   exact this
end RnnX3
section RnnX4
def mxW (m:Word) (a:ё):Word:=
 a.foldl (λm w=>if m<(w&&&В) then (w&&&В) else m) m
@[inline] def wHN (inW H:Nat) (whh bhh:ё) (u j:Nat):Word:=
 if j< inW then 0 else if j< inW+H then whh[(2*H+u)*H+(j-inW)]!
 else if j=inW+H then 0 else bhh[2*H+u]!
@[inline] def wCD (inW H:Nat) (wih bih:ё) (u j:Nat):Word:=
 if j< inW then wih[(2*H+u)*inW+j]! else if j< inW+H then 0
 else if j=inW+H then bih[2*H+u]! else 0
@[inline] def wfX (gru:Ὡ) (inW H:Nat) (wih whh bih bhh:ё) (u g j:Nat):Word:=
 if gru then
  (if g<2 then gWt inW H wih whh bih bhh (g*H+u) j
  else if g=2 then wHN inW H whh bhh u j else wCD inW H wih bih u j)
 else gWt inW H wih whh bih bhh (g*H+u) j
def xpOf (gru:Ὡ) (inW H:Nat) (wih whh bih bhh:ё):ὤ×Word×Ὡ:=
 let hv:=mxW (mxW (mxW (mxW (if gru then 0x3F800000 else 0) wih) whh) bih) bhh
 let P:=pnOf H (inW+H+2) (wfX gru inW H wih whh bih bhh)
 (P,hv,wih.all good&&whh.all good&&bih.all good&&bhh.all good&&good hv&&decide (hv<=В)&&
   wih.all (λw=>w&&&В<=hv)&&whh.all (λw=>w&&&В<=hv)&&
   bih.all (λw=>w&&&В<=hv)&&bhh.all (λw=>w&&&В<=hv)&&
   decide (inW+H+3<2^20)&&decide (P.size<2^32)&&decide (H<2^20)&&
   decide (gru=false\/0x3F800000<=hv))
abbrev XPc (gru:Ὡ) (inW H:Nat) (wih whh bih bhh:ё):=
 {p:ὤ×Word×Ὡ // p=xpOf gru inW H wih whh bih bhh}
local instance xpcSub (gru : Bool) (inW H : Nat) (wih whh bih bhh : Array Word) :
    Subsingleton (XPc gru inW H wih whh bih bhh) :=
  ⟨fun a b => Subtype.ext (a.2.trans b.2.symm)⟩
structure XPE where
 (k:Ὡ) (nI nH:Nat) (a b c d:ё) (g:ὦ) (p:ὤ×Word×Ὡ)
 (ok:p=xpOf k nI nH a b c d)
def xpF (gru:Ὡ) (inW H:Nat) (wih whh bih bhh:ё) (g:ὦ) (s:Array XPE) (i:Nat):
    Option (XPc gru inW H wih whh bih bhh):=
 if h:i<s.size then
  if s[i].g==g then
   if hn:s[i].k=gru/\s[i].nI=inW/\s[i].nH=H then
    if ha:withPtrEq s[i].a wih (λ_=>s[i].a==wih) (λe=>by rw[e];exact beq_self_eq_true wih) then
     if hb:withPtrEq s[i].b whh (λ_=>s[i].b==whh) (λe=>by rw[e];exact beq_self_eq_true whh) then
      if hc:withPtrEq s[i].c bih (λ_=>s[i].c==bih) (λe=>by rw[e];exact beq_self_eq_true bih) then
       if hd:withPtrEq s[i].d bhh (λ_=>s[i].d==bhh) (λe=>by rw[e];exact beq_self_eq_true bhh) then
        some ⟨s[i].p,by
         rw[<-hn.1,<-hn.2.1,<-hn.2.2,<-eq_of_beq ha,<-eq_of_beq hb,<-eq_of_beq hc,<-eq_of_beq hd]
         exact s[i].ok⟩
       else xpF gru inW H wih whh bih bhh g s (i+1)
      else xpF gru inW H wih whh bih bhh g s (i+1)
     else xpF gru inW H wih whh bih bhh g s (i+1)
    else xpF gru inW H wih whh bih bhh g s (i+1)
   else xpF gru inW H wih whh bih bhh g s (i+1)
  else xpF gru inW H wih whh bih bhh g s (i+1)
 else none
partial_fixpoint
@[noinline] def xpS:ST.Ref Unit (Array XPE):=(ST.Prim.mkRef #[] (Void.mk ())).val
@[inline] def xpGet (gru:Ὡ) (inW H:Nat) (wih whh bih bhh:ё):XPc gru inW H wih whh bih bhh:=
 withPtrAddr wih (λg=>
  let s:=(ST.Prim.Ref.get xpS (Void.mk ())).val
  match xpF gru inW H wih whh bih bhh g s 0 with
  | some v=>v
  | none=>
    let p:=xpOf gru inW H wih whh bih bhh
    let old:=(ST.Prim.Ref.swap xpS (#[⟨gru,inW,H,wih,whh,bih,bhh,g,p,rfl⟩]++
     (if s.size<64 then s else s.pop)) (Void.mk ())).val
    match xpF gru inW H wih whh bih bhh g old 0 with
    | some v=>v
    | none=>⟨p,rfl⟩) (λ_ _=>Subsingleton.elim _ _)
lemma all_le (a:ё) (hv:Word) (h:a.all (λw=>w&&&В<=hv)=true) (i:Nat):
  a[i]!&&&В<=hv:=by
 by_cases hi:i<a.size
 · rw[Ἕ a i hi]
   exact of_decide_eq_true ((Array.all_eq_true.mp h) i hi)
 · rw[ὑ a i hi,show (default:Word)&&&В=0 from rfl];exact UInt32.zero_le
structure XQ (inW H:Nat) (wih whh bih bhh:ё) (hv:Word):Prop where
 gw: ∀i:Nat,good wih[i]! =true
 gu: ∀i:Nat,good whh[i]! =true
 gb: ∀i:Nat,good bih[i]! =true
 gc: ∀i:Nat,good bhh[i]! =true
 gv:good hv=true
 pv:hv<=В
 lw: ∀i:Nat,wih[i]!&&&В<=hv
 lu: ∀i:Nat,whh[i]!&&&В<=hv
 lb: ∀i:Nat,bih[i]!&&&В<=hv
 lc: ∀i:Nat,bhh[i]!&&&В<=hv
 hN:inW+H+3<2^20
lemma xq_of (gru:Ὡ) (inW H:Nat) (wih whh bih bhh:ё) (h:(xpOf gru inW H wih whh bih bhh).2.2=true):
  XQ inW H wih whh bih bhh (xpOf gru inW H wih whh bih bhh).2.1/\
   (xpOf gru inW H wih whh bih bhh).1.size<2^32/\H<2^20/\
   (gru=false\/0x3F800000<=(xpOf gru inW H wih whh bih bhh).2.1):=by
 unfold xpOf at h⊢
 simp only[Bool.and_eq_true,decide_eq_true_eq] at h
 obtain ⟨⟨⟨⟨⟨⟨⟨⟨⟨⟨⟨⟨⟨g1,g2⟩,g3⟩,g4⟩,g5⟩,g6⟩,l1⟩,l2⟩,l3⟩,l4⟩,n1⟩,n2⟩,n3⟩,n4⟩:=h
 exact ⟨⟨all_good _ g1,all_good _ g2,all_good _ g3,all_good _ g4,g5,g6,all_le _ _ l1,all_le _ _ l2,
  all_le _ _ l3,all_le _ _ l4,n1⟩,n2,n3,n4⟩
lemma gWt_good {inW H:Nat} {wih whh bih bhh:ё} {hv:Word} (q:XQ inW H wih whh bih bhh hv)
  (k j:Nat):good (gWt inW H wih whh bih bhh k j)=true/\gWt inW H wih whh bih bhh k j&&&В<=hv:=by
 unfold gWt
 split
 · exact ⟨q.gw _,q.lw _⟩
 split
 · exact ⟨q.gu _,q.lu _⟩
 split
 · exact ⟨q.gb _,q.lb _⟩
 · exact ⟨q.gc _,q.lc _⟩
lemma gX_good (inW H:Nat) (x h:ё) (t:Nat) (hx: ∀i:Nat,good x[i]! =true)
  (hc:h.all good=true) (j:Nat):
  good (gX inW H x h t j)=true:=by
 unfold gX
 split
 · exact hx _
 split
 · exact all_good h hc (j-inW)
 · exact good_one
@[noinline] def xSlow (inW H:Nat) (x wih whh bih bhh h:ё) (t k:Nat):Word:=
 Pinned.dotFn (inW+H+2) (gX inW H x h t) (gWt inW H wih whh bih bhh k)
@[inline] def xSum (gru:Ὡ) (inW H:Nat) (wih whh bih bhh:ё) (V P:ὤ)
   (hx:(ὴ (inW+H+2)).toNat<=V.size) (hs:P.size<2^32):ὤ:=
 rowsB (mxRow V P (ὴ (inW+H+2)) hx hs) 0 ((H+3)/4) (ἆ ((H+3)/4*16))
@[inline] def lstmUX (sig tanh:Word->Word) (inW H:Nat) (x wih whh bih bhh h cs:ё) (t:Nat)
   (S:ὤ) (lo sp:Ὠ) (e:ὧ) (u:Nat):Word×Word:=
 let gi:=gT lo sp e (S.get! (4*u)) (λ_=>xSlow inW H x wih whh bih bhh h t u)
 let gf:=gT lo sp e (S.get! (4*u+1)) (λ_=>xSlow inW H x wih whh bih bhh h t (H+u))
 let gg:=gT lo sp e (S.get! (4*u+2)) (λ_=>xSlow inW H x wih whh bih bhh h t (2*H+u))
 let go:=gT lo sp e (S.get! (4*u+3)) (λ_=>xSlow inW H x wih whh bih bhh h t (3*H+u))
 let c':=cellF (sig gf) (sig gi) cs[u]! (tanh gg)
 (mulF (sig go) (tanh c'),c')
@[specialize] def lsU (sig tanh:Word->Word) (inW H:Nat) (x wih whh bih bhh h cs:ё) (t:Nat)
   (S:ὤ) (lo sp:Ὠ) (e:ὧ):Nat->Nat->ё->ё->ё×ё
 | _,0,a,b=>(a,b)
 | u,n+1,a,b=>
   let p:=lstmUX sig tanh inW H x wih whh bih bhh h cs t S lo sp e u
   lsU sig tanh inW H x wih whh bih bhh h cs t S lo sp e (u+1) n (a.push p.1) (b.push p.2)
lemma lsU_eq (sig tanh:Word->Word) (inW H:Nat) (x wih whh bih bhh h cs:ё) (t:Nat)
  (S:ὤ) (lo sp:Ὠ) (e:ὧ): ∀n u a b,
  lsU sig tanh inW H x wih whh bih bhh h cs t S lo sp e u n a b=
   build2 (lstmUX sig tanh inW H x wih whh bih bhh h cs t S lo sp e) u n a b
 | 0,_,_,_=>rfl
 | n+1,u,a,b=>by
   rw[lsU,build2,lsU_eq sig tanh inW H x wih whh bih bhh h cs t S lo sp e n]
lemma oK (S:ὤ) (o:ὦ) (u k:Nat) (hk:k<4) (ho:o.toNat=4*u)
  (hS:4*u+3<S.size) (hs:S.size<2^32):(o+ὴ k).toNat=4*u+k/\
    (o+ὴ k).toNat<S.size:=by
 have e:(o+ὴ k).toNat=4*u+k:=by
  rw[D.qAf _ _ (by rw[ho,D.uk k א];omega),ho,D.uk k א]
 exact ⟨e,by rw[e];omega⟩
@[inline] def lstmUX2 (sig tanh:Word->Word) (inW H:Nat) (x wih whh bih bhh h cs:ё) (t:Nat)
   (S:ὤ) (lo sp:Ὠ) (e:ὧ) (hs:S.size<2^32) (u:Nat) (o:ὦ) (ho:o.toNat=4*u)
   (hS:4*u+3<S.size):Word×Word:=
 let gi:=gT lo sp e (S.uget (o+ὴ 0) (oK S o u 0 (by decide) ho hS hs).2) (λ_=>xSlow inW H x wih whh bih bhh h t u)
 let gf:=gT lo sp e (S.uget (o+ὴ 1) (oK S o u 1 (by decide) ho hS hs).2) (λ_=>xSlow inW H x wih whh bih bhh h t (H+u))
 let gg:=gT lo sp e (S.uget (o+ὴ 2) (oK S o u 2 (by decide) ho hS hs).2) (λ_=>xSlow inW H x wih whh bih bhh h t (2*H+u))
 let go:=gT lo sp e (S.uget (o+ὴ 3) (oK S o u 3 (by decide) ho hS hs).2) (λ_=>xSlow inW H x wih whh bih bhh h t (3*H+u))
 let c':=cellF (sig gf) (sig gi) cs[u]! (tanh gg)
 (mulF (sig go) (tanh c'),c')
lemma o4 (S:ὤ) (o:ὦ) (u:Nat) (ho:o.toNat=4*u) (hS:4*u+3<S.size)
  (hs:S.size<2^32):(o+ὴ 4).toNat=4*(u+1):=by
 rw[D.qAf _ _ (by rw[ho,D.uk 4 ה];omega),ho,D.uk 4 ה];omega
@[specialize] def lsU2 (sig tanh:Word->Word) (inW H:Nat) (x wih whh bih bhh h cs:ё) (t:Nat)
   (S:ὤ) (lo sp:Ὠ) (e:ὧ) (hs:S.size<2^32):
   (n u:Nat)->(o:ὦ)->o.toNat=4*u->4*(u+n)<=S.size->ё->ё->ё×ё
 | 0,_,_,_,_,a,b=>(a,b)
 | n+1,u,o,ho,hn,a,b=>
   let p:=lstmUX2 sig tanh inW H x wih whh bih bhh h cs t S lo sp e hs u o ho (by omega)
   lsU2 sig tanh inW H x wih whh bih bhh h cs t S lo sp e hs n (u+1) (o+ὴ 4) (o4 S o u ho (by omega) hs)
    (by omega) (a.push p.1) (b.push p.2)
lemma lsU2_eq (sig tanh:Word->Word) (inW H:Nat) (x wih whh bih bhh h cs:ё) (t:Nat)
  (S:ὤ) (lo sp:Ὠ) (e:ὧ) (hs:S.size<2^32): ∀n u o (ho:o.toNat=4*u)
  (hn:4*(u+n)<=S.size) a b,
  lsU2 sig tanh inW H x wih whh bih bhh h cs t S lo sp e hs n u o ho hn a b=lsU sig tanh inW H x wih whh bih bhh h cs t S lo sp e u n a b
 | 0,_,_,_,_,_,_=>rfl
 | n+1,u,o,ho,hn,a,b=>by
   have hS:4*u+3<S.size:=א
   have e2:lstmUX2 sig tanh inW H x wih whh bih bhh h cs t S lo sp e hs u o ho hS=
     lstmUX sig tanh inW H x wih whh bih bhh h cs t S lo sp e u:=by
    unfold lstmUX2 lstmUX
    simp only[ugetE,(oK S o u 0 ה ho hS hs).1,(oK S o u 1 ה ho hS hs).1,
     (oK S o u 2 ה ho hS hs).1,(oK S o u 3 ה ho hS hs).1,Ἵ]
   rw[lsU2,lsU,e2,lsU2_eq sig tanh inW H x wih whh bih bhh h cs t S lo sp e hs n (u+1)]
lemma dite_lsU (sig tanh:Word->Word) (inW H:Nat) (x wih whh bih bhh h cs:ё) (t:Nat)
  (S:ὤ) (lo sp:Ὠ) (e:ὧ):
  (if hS:S.size<2^32/\4*(0+H)<=S.size then
      lsU2 sig tanh inW H x wih whh bih bhh h cs t S lo sp e hS.1 H 0 0 rfl hS.2
       (Array.emptyWithCapacity H) (Array.emptyWithCapacity H)
    else lsU sig tanh inW H x wih whh bih bhh h cs t S lo sp e 0 H
     (Array.emptyWithCapacity H) (Array.emptyWithCapacity H))=
    lsU sig tanh inW H x wih whh bih bhh h cs t S lo sp e 0 H (Array.emptyWithCapacity H) (Array.emptyWithCapacity H):=by
 split
 · exact lsU2_eq _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
 · rfl
lemma vx_le (inW H:Nat) (x h:ё) (t:Nat) (hK:inW+H+3<2^20):
  (ὴ (inW+H+2)).toNat<=(vOf inW H x h t).size:=by
 rw[D.uk _ א,(vOf_spec inW H x h t).1]
lemma xS_size (gru:Ὡ) (inW H:Nat) (x wih whh bih bhh h:ё) (t:Nat)
  (hk:inW+H+3<2^20/\(xpOf gru inW H wih whh bih bhh).1.size<2^32):
  (xSum gru inW H wih whh bih bhh (vOf inW H x h t) (xpOf gru inW H wih whh bih bhh).1 (vx_le inW H x h t hk.1) hk.2).size=
    (H+3)/4*16:=by
 obtain ⟨hVs,hVg⟩:=vOf_spec inW H x h t
 obtain ⟨hPs,hPg⟩:=pnOf_spec H (inW+H+2) (wfX gru inW H wih whh bih bhh)
 have hV: ∀i,i<(vOf inW H x h t).size-> ∃w,(vOf inW H x h t).get! i=dv w:=
  λi hi=>⟨gX inW H x h t i,hVg i א⟩
 have hP: ∀i,i<(xpOf gru inW H wih whh bih bhh).1.size-> ∃w,(xpOf gru inW H wih whh bih bhh).1.get! i=dv w:=by
  intro i hi
  show ∃w,(pnOf H (inW+H+2) (wfX gru inW H wih whh bih bhh)).get! i=dv w
  have hi':i<(pnOf H (inW+H+2) (wfX gru inW H wih whh bih bhh)).size:=hi
  rw[hPs] at hi'
  refine ⟨pW H (inW+H+2) (wfX gru inW H wih whh bih bhh) (i/8) (i%8),?_⟩
  rw[<-hPg (i/8) (i%8) א א,Nat.div_add_mod']
 exact (mxRow_spec _ _ _ _ _ hV hP _).1
@[specialize] def lstmStepX (sig tanh:Word->Word) (inW H:Nat) (x wih whh bih bhh:ё)
   (P:ὤ) (hv:Word) (t:Nat) (st:ё×ё):ё×ё:=
 if hc:st.1.all good=true then
  if hk:inW+H+3<2^20/\P.size<2^32 then
   let V:=vOf inW H x st.1 t
   let e:=nfOf (inW+H+2)*(dv hv*aS V 0 (inW+H+2) 0+xNA)
   let lo:=D.loOf e
   let S:=xSum false inW H wih whh bih bhh V P (vx_le inW H x st.1 t hk.1) hk.2
   if hS:S.size<2^32/\4*(0+H)<=S.size then
    lsU2 sig tanh inW H x wih whh bih bhh st.1 st.2 t S lo (D.spOf lo) e hS.1 H 0 0 rfl hS.2
     (Array.emptyWithCapacity H) (Array.emptyWithCapacity H)
   else lsU sig tanh inW H x wih whh bih bhh st.1 st.2 t S lo (D.spOf lo) e 0 H
    (Array.emptyWithCapacity H) (Array.emptyWithCapacity H)
  else lstmStepWith Pinned.dotFn sig tanh inW H x wih whh bih bhh t st
 else lstmStepWith Pinned.dotFn sig tanh inW H x wih whh bih bhh t st
lemma gateX_eq (gru:Ὡ) (inW H:Nat) (x wih whh bih bhh h:ё) (t:Nat)
  (hq:(xpOf gru inW H wih whh bih bhh).2.2=true)
  (hx: ∀i:Nat,good x[i]! =true) (hc:h.all good=true)
  (hk:inW+H+3<2^20/\(xpOf gru inW H wih whh bih bhh).1.size<2^32)
  (u g:Nat) (hu:u<H) (hg:g<4) (ww:Nat->Word) (hww: ∀j,ww j=wfX gru inW H wih whh bih bhh u g j)
  (sl:Unit->Word) (hsl:sl ()=dotOf (inW+H+2) (gX inW H x h t) ww):
  let V:=vOf inW H x h t
  let e:=nfOf (inW+H+2)*(dv (xpOf gru inW H wih whh bih bhh).2.1*aS V 0 (inW+H+2) 0+xNA)
  let S:=xSum gru inW H wih whh bih bhh V (xpOf gru inW H wih whh bih bhh).1 (vx_le inW H x h t hk.1) hk.2
  (∀j,good (ww j)=true/\ww j&&&В<=(xpOf gru inW H wih whh bih bhh).2.1)->
  gT (D.loOf e) (D.spOf (D.loOf e)) e (S.get! (4*u+g)) sl=dotOf (inW+H+2) (gX inW H x h t) ww:=by
 intro V e S hwv
 obtain ⟨q,-,-,-⟩:=xq_of gru inW H wih whh bih bhh hq
 obtain ⟨hVs,hVg⟩:=vOf_spec inW H x h t
 have hS:S.get! (4*u+g)=(ἄ (inW+H+2)).foldl
   (λacc j=>acc+dv (gX inW H x h t j)*dv (ww j)) 0:=by
  have:=slot_eq H (inW+H+2) (wfX gru inW H wih whh bih bhh) V א hVs (gX inW H x h t) hVg hk.2
   (vx_le inW H x h t hk.1) u g hu hg
  simp only[<-hww] at this
  exact this
 have ha:aS V 0 (inW+H+2) 0=(ἄ (inW+H+2)).foldl (λacc j=>acc+(dv (gX inW H x h t j)).abs) 0:=by
  rw[aS_eq,<-ὗ]
  exact foldl_ext _ _ _ _ λj hj b=>by rw[hVg j (List.mem_range.mp hj)]
 exact gT_eq (inW+H+2) א א (gX inW H x h t) ww (λj _=>gX_good inW H x h t hx hc j)
  (λj _=>(hwv j).1) _ q.gv q.pv (λj _=>(hwv j).2) _ _ hS ha sl hsl
lemma lstmStepX_eq (sig tanh:Word->Word) (inW H:Nat) (x wih whh bih bhh:ё)
  (hq:(xpOf false inW H wih whh bih bhh).2.2=true) (hx: ∀i:Nat,good x[i]! =true):
  lstmStepX sig tanh inW H x wih whh bih bhh (xpOf false inW H wih whh bih bhh).1 (xpOf false inW H wih whh bih bhh).2.1=
   lstmStepWith dotOf sig tanh inW H x wih whh bih bhh:=by
 funext t st
 obtain ⟨q,-,-,-⟩:=xq_of false inW H wih whh bih bhh hq
 unfold lstmStepX
 split
 · rename_i hc
   split
   · rename_i hk
     have hg: ∀g u,u<H->g<4->
       gT (D.loOf (nfOf (inW+H+2)*(dv (xpOf false inW H wih whh bih bhh).2.1*
           aS (vOf inW H x st.1 t) 0 (inW+H+2) 0+xNA)))
        (D.spOf (D.loOf (nfOf (inW+H+2)*(dv (xpOf false inW H wih whh bih bhh).2.1*
          aS (vOf inW H x st.1 t) 0 (inW+H+2) 0+xNA))))
        (nfOf (inW+H+2)*(dv (xpOf false inW H wih whh bih bhh).2.1*
          aS (vOf inW H x st.1 t) 0 (inW+H+2) 0+xNA))
        ((xSum false inW H wih whh bih bhh (vOf inW H x st.1 t) (xpOf false inW H wih whh bih bhh).1
          (vx_le inW H x st.1 t hk.1) hk.2).get! (4*u+g))
        (λ_=>xSlow inW H x wih whh bih bhh st.1 t (g*H+u))=
       gateWith dotOf inW H x wih whh bih bhh st.1 t (g*H+u):= λg u hu hg4=>
      gateX_eq false inW H x wih whh bih bhh st.1 t hq hx hc hk u g hu hg4 (gWt inW H wih whh bih bhh (g*H+u))
       (λj=>rfl) _ (by unfold xSlow;rw[Pinned.dotFn_eq_dotOf])
       (λj=>gWt_good q _ j)
     dsimp only
     rw[dite_lsU,lsU_eq,build2_zero]
     unfold lstmStepWith
     simp only[]
     congr 1
     · congr 1;funext u
       obtain ⟨i1,i2,i3,i4⟩:=stepIdx4 H u.val u.isLt
       unfold lstmUX
       simp only[cellF_eq,mulF_eq,Pinned.dotFn_eq_dotOf]
       rw[ofFn_get!F _ u.val u.isLt]
       try dsimp only
       have h0:=hg 0 u.val u.isLt ה
       have h1:=hg 1 u.val u.isLt ה
       have h2:=hg 2 u.val u.isLt ה
       have h3:=hg 3 u.val u.isLt ה
       simp only[ὒ,Ἇ,Ὑ,Ἵ] at h0 h1 h2 h3
       rw[h0,h1,h2,h3,ofFn_get! _ _ _ i4,ofFn_get! _ _ _ i1,ofFn_get! _ _ _ i2,ofFn_get! _ _ _ i3]
     · congr 1;funext u
       obtain ⟨i1,i2,i3,i4⟩:=stepIdx4 H u.val u.isLt
       unfold lstmUX
       simp only[cellF_eq,Pinned.dotFn_eq_dotOf]
       have h0:=hg 0 u.val u.isLt ה
       have h1:=hg 1 u.val u.isLt ה
       have h2:=hg 2 u.val u.isLt ה
       simp only[ὒ,Ἇ,Ὑ,Ἵ] at h0 h1 h2
       rw[h0,h1,h2,ofFn_get! _ _ _ i1,ofFn_get! _ _ _ i2,ofFn_get! _ _ _ i3]
   · rw[Pinned.dotFn_eq_dotOf]
 · rw[Pinned.dotFn_eq_dotOf]
@[specialize] def lstmLX (sig tanh:Word->Word) (T inW H:Nat) (x wih whh bih bhh:ё):ё:=
 let p:=(xpGet false inW H wih whh bih bhh).1
 if p.2.2&&x.all good then
  recFlat (lstmStepX sig tanh inW H x wih whh bih bhh p.1 p.2.1) 0 T (ἇ H 0,ἇ H 0)
   (Array.emptyWithCapacity (T*H))
 else lstmWith Pinned.dotFn sig tanh T inW H x wih whh bih bhh
lemma lstmLX_eq (sig tanh:Word->Word):lstmLX sig tanh=lstmWith Pinned.dotFn sig tanh:=by
 funext T inW H x wih whh bih bhh
 unfold lstmLX
 rw[(xpGet false inW H wih whh bih bhh).2]
 dsimp only
 split
 · rename_i hq
   simp only[Bool.and_eq_true] at hq
   rw[lstmStepX_eq sig tanh inW H x wih whh bih bhh hq.1 (all_good x hq.2),
    recFlat_eq T H _ _ (λt st=>by unfold lstmStepWith;exact Array.size_ofFn),Pinned.dotFn_eq_dotOf]
   rfl
 · rfl
end RnnX4
section RnnX5
/-! -/
lemma dotOf_sum (n1 n2:Nat) (a1 b1 a2 b2:Nat->Word)
  (hf1: ∀j,j<n1->expOf (a1 j)≠255/\expOf (b1 j)≠255)
  (hf2: ∀j,j<n2->expOf (a2 j)≠255/\expOf (b2 j)≠255)
  (hS:(∑ j∈Ἣ n1,rationalWord (a1 j)*rationalWord (b1 j))=
    ∑ j∈Ἣ n2,rationalWord (a2 j)*rationalWord (b2 j))
  (h0:(∑ j∈Ἣ n1,rationalWord (a1 j)*rationalWord (b1 j))≠0):
  dotOf n1 a1 b1=dotOf n2 a2 b2:=by
 obtain ⟨d1,e1,w1⟩:=D.qEc n1 a1 b1 hf1 h0
 obtain ⟨d2,e2,w2⟩:=D.qEc n2 a2 b2 hf2 (hS▸h0)
 have t:= @ὐ
 rw[w1,w2,ExactRuntime.roundDyadic_congr Fmt ה d1 d2
  (λh=>h0 (e1.symm.trans ((t _).2 h))) (λh=>h0 (hS.trans (e2.symm.trans ((t _).2 h))))
  (by rw[e1,e2,hS])]
lemma dotOf_zero (n:Nat) (a b:Nat->Word) (hf: ∀t,t<n->expOf (a t)≠255/\expOf (b t)≠255)
  (hS:(∑ j∈Ἣ n,rationalWord (a j)*rationalWord (b j))=0):
  dotOf n a b&&&В=0:=by
 rw[C.qD6 n a b hf hS]
 generalize (decide (n≠0)&&_)=s
 cases s<;>decide
lemma gT_or (n:Nat) (hn:n<2^20) (hn0:0<n) (xw ww:Nat->Word)
  (hx: ∀t,t<n->good (xw t)=true) (hw: ∀t,t<n->good (ww t)=true)
  (hv:Word) (hgv:good hv=true) (hpv:hv<=В)
  (hmv: ∀t,t<n->ww t&&&В<=hv) (s0 a:ὧ)
  (hs0:s0=(ἄ n).foldl (λacc t=>acc+dv (xw t)*dv (ww t)) 0)
  (ha:a=(ἄ n).foldl (λacc t=>acc+(dv (xw t)).abs) 0) (sl:Unit->Word):
  gT (D.loOf (nfOf n*(dv hv*a+xNA))) (D.spOf (D.loOf (nfOf n*(dv hv*a+xNA))))
   (nfOf n*(dv hv*a+xNA)) s0 sl=sl ()\/
  gT (D.loOf (nfOf n*(dv hv*a+xNA))) (D.spOf (D.loOf (nfOf n*(dv hv*a+xNA))))
   (nfOf n*(dv hv*a+xNA)) s0 sl=dotOf n xw ww:=by
 by_cases h:gT (D.loOf (nfOf n*(dv hv*a+xNA))) (D.spOf (D.loOf (nfOf n*(dv hv*a+xNA))))
  (nfOf n*(dv hv*a+xNA)) s0 sl=sl ()
 · exact Or.inl h
 · refine Or.inr ?_
   rw[<-gT_eq n hn hn0 xw ww hx hw hv hgv hpv hmv s0 a hs0 ha (λ_=>dotOf n xw ww) rfl]
   revert h
   unfold gT
   dsimp only
   split
   · split
     · intro h;exact absurd rfl h
     · intro _;rfl
   · intro _;rfl
@[inline] def gZ (s0 e:ὧ) (sl:Unit->Word):Word:=
 let w:=ziv (s0+xNZ) e
 if w==Е then sl () else w
lemma gZ_or (n:Nat) (hn:n<2^20) (xw ww:Nat->Word)
  (hx: ∀t,t<n->good (xw t)=true) (hw: ∀t,t<n->good (ww t)=true)
  (hv:Word) (hgv:good hv=true) (hpv:hv<=В)
  (hmv: ∀t,t<n->ww t&&&В<=hv) (s0 a:ὧ)
  (hs0:s0=(ἄ n).foldl (λacc t=>acc+dv (xw t)*dv (ww t)) 0)
  (ha:a=(ἄ n).foldl (λacc t=>acc+(dv (xw t)).abs) 0) (sl:Unit->Word):
  gZ s0 (nfOf n*(dv hv*a+xNA)) sl=sl ()\/
   gZ s0 (nfOf n*(dv hv*a+xNA)) sl=dotOf (n+1) (λt=>if t<n then xw t else oneW)
    (λt=>if t<n then ww t else س):=by
 have hs:s0+xNZ=(ἄ n).foldl (λ(acc:ὧ) t=>acc+dv (xw t)*dv (ww t)) 0+dv س/\
   nfOf n*(dv hv*a+xNA)=
    nfOf n*(dv hv*((ἄ n).foldl (λ(acc:ὧ) t=>acc+(dv (xw t)).abs) 0)+
     (dv س).abs):=by
  rw[hs0,ha];exact ⟨rfl,rfl⟩
 unfold gZ
 dsimp only
 split
 · exact Or.inl rfl
 · rename_i h2
   refine Or.inr ?_
   have hz:ziv (s0+xNZ) (nfOf n*(dv hv*a+xNA))≠Е:= λh=>h2 (by rw[h];rfl)
   obtain ⟨-,-,hbig⟩:=D.qBw _ _ hz
   obtain ⟨hlo,hhi⟩:=D.qC8 n hn xw ww س hv hx hw good_nz hgv hpv (ἄ n)
    (Ὂ) (λt ht=>List.mem_range.mp ht) (λt ht=>hmv t (List.mem_range.mp ht))
    (λt ht hnot=>absurd (List.mem_range.mpr ht) hnot) (s0+xNZ) _ hs hbig
   exact D.qCn n hn xw ww س hx hw good_nz _ _ hlo hhi hz
@[noinline] def hnSlow (H:Nat) (whh bhh h:ё) (u:Nat):Word:=
 Pinned.dotFn (H+1) (λj=>if j<H then h[j]! else oneW)
  (λj=>if j<H then whh[(2*H+u)*H+j]! else bhh[2*H+u]!)
@[inline] def hnX (H:Nat) (whh bhh h:ё) (u:Nat) (w:Word):Word:=
 if w&&&В==0 then hnSlow H whh bhh h u else w
@[inline] def cdX (inW H:Nat) (x wih bih:ё) (t u:Nat) (sc a:ὧ) (hv:Word) (hn r:Word):Word:=
 if good hn&&good r&&decide (r&&&В<=hv) then
  let w:=gZ (sc+dv hn*dv r) (nfOf (inW+H+3)*(dv hv*(a+(dv hn).abs)+xNA))
   (λ_=>candSlow inW H x wih bih t u hn r)
  if w&&&В==0 then candSlow inW H x wih bih t u hn r else w
 else candSlow inW H x wih bih t u hn r
@[inline] def gruUX (sig tanh:Word->Word) (inW H:Nat) (x wih whh bih bhh h:ё) (t:Nat)
   (S:ὤ) (lo sp:Ὠ) (e a:ὧ) (hv:Word) (u:Nat):Word:=
 let gr:=gT lo sp e (S.get! (4*u)) (λ_=>xSlow inW H x wih whh bih bhh h t u)
 let gz:=gT lo sp e (S.get! (4*u+1)) (λ_=>xSlow inW H x wih whh bih bhh h t (H+u))
 let ghn:=hnX H whh bhh h u (gT lo sp e (S.get! (4*u+2)) (λ_=>hnSlow H whh bhh h u))
 let r:=sig gr
 updF (tanh (cdX inW H x wih bih t u (S.get! (4*u+3)) a hv ghn r)) (sig gz) h[u]!
@[specialize] def grU (sig tanh:Word->Word) (inW H:Nat) (x wih whh bih bhh h:ё) (t:Nat)
   (S:ὤ) (lo sp:Ὠ) (e a:ὧ) (hv:Word):Nat->Nat->ё->ё
 | _,0,acc=>acc
 | u,n+1,acc=>grU sig tanh inW H x wih whh bih bhh h t S lo sp e a hv (u+1) n
     (acc.push (gruUX sig tanh inW H x wih whh bih bhh h t S lo sp e a hv u))
lemma grU_eq (sig tanh:Word->Word) (inW H:Nat) (x wih whh bih bhh h:ё) (t:Nat)
  (S:ὤ) (lo sp:Ὠ) (e a:ὧ) (hv:Word): ∀n k (acc:ё),
  acc=Array.ofFn (n:=k) (λi=>gruUX sig tanh inW H x wih whh bih bhh h t S lo sp e a hv i.val)->
  grU sig tanh inW H x wih whh bih bhh h t S lo sp e a hv k n acc=
   Array.ofFn (n:=k+n) (λi=>gruUX sig tanh inW H x wih whh bih bhh h t S lo sp e a hv i.val)
 | 0,k,acc,h0=>by rw[grU,h0];rfl
 | n+1,k,acc,h0=>by
   rw[grU,grU_eq sig tanh inW H x wih whh bih bhh h t S lo sp e a hv n (k+1) _
    (by rw[h0];exact push_ofFnA (λi=>gruUX sig tanh inW H x wih whh bih bhh h t S lo sp e a hv i) k),
    show k+1+n=k+(n+1) by omega]
@[inline] def gruUX2 (sig tanh:Word->Word) (inW H:Nat) (x wih whh bih bhh h:ё) (t:Nat)
   (S:ὤ) (lo sp:Ὠ) (e a:ὧ) (hv:Word) (hs:S.size<2^32) (u:Nat) (o:ὦ)
   (ho:o.toNat=4*u) (hS:4*u+3<S.size):Word:=
 let gr:=gT lo sp e (S.uget (o+ὴ 0) (oK S o u 0 (by decide) ho hS hs).2) (λ_=>xSlow inW H x wih whh bih bhh h t u)
 let gz:=gT lo sp e (S.uget (o+ὴ 1) (oK S o u 1 (by decide) ho hS hs).2) (λ_=>xSlow inW H x wih whh bih bhh h t (H+u))
 let ghn:=hnX H whh bhh h u (gT lo sp e (S.uget (o+ὴ 2) (oK S o u 2 (by decide) ho hS hs).2) (λ_=>hnSlow H whh bhh h u))
 let r:=sig gr
 updF (tanh (cdX inW H x wih bih t u (S.uget (o+ὴ 3) (oK S o u 3 (by decide) ho hS hs).2) a hv ghn r)) (sig gz) h[u]!
@[specialize] def grU2 (sig tanh:Word->Word) (inW H:Nat) (x wih whh bih bhh h:ё) (t:Nat)
   (S:ὤ) (lo sp:Ὠ) (e a:ὧ) (hv:Word) (hs:S.size<2^32):
   (n u:Nat)->(o:ὦ)->o.toNat=4*u->4*(u+n)<=S.size->ё->ё
 | 0,_,_,_,_,acc=>acc
 | n+1,u,o,ho,hn,acc=>grU2 sig tanh inW H x wih whh bih bhh h t S lo sp e a hv hs n (u+1) (o+ὴ 4)
    (o4 S o u ho (by omega) hs) (by omega)
    (acc.push (gruUX2 sig tanh inW H x wih whh bih bhh h t S lo sp e a hv hs u o ho (by omega)))
lemma grU2_eq (sig tanh:Word->Word) (inW H:Nat) (x wih whh bih bhh h:ё) (t:Nat)
  (S:ὤ) (lo sp:Ὠ) (e a:ὧ) (hv:Word) (hs:S.size<2^32): ∀n u o (ho:o.toNat=4*u)
  (hn:4*(u+n)<=S.size) acc,
  grU2 sig tanh inW H x wih whh bih bhh h t S lo sp e a hv hs n u o ho hn acc=
   grU sig tanh inW H x wih whh bih bhh h t S lo sp e a hv u n acc
 | 0,_,_,_,_,_=>rfl
 | n+1,u,o,ho,hn,acc=>by
   have hS:4*u+3<S.size:=א
   have e2:gruUX2 sig tanh inW H x wih whh bih bhh h t S lo sp e a hv hs u o ho hS=
     gruUX sig tanh inW H x wih whh bih bhh h t S lo sp e a hv u:=by
    unfold gruUX2 gruUX
    simp only[ugetE,(oK S o u 0 ה ho hS hs).1,(oK S o u 1 ה ho hS hs).1,
     (oK S o u 2 ה ho hS hs).1,(oK S o u 3 ה ho hS hs).1,Ἵ]
   rw[grU2,grU,e2,grU2_eq sig tanh inW H x wih whh bih bhh h t S lo sp e a hv hs n (u+1)]
lemma dite_grU (sig tanh:Word->Word) (inW H:Nat) (x wih whh bih bhh h:ё) (t:Nat)
  (S:ὤ) (lo sp:Ὠ) (e a:ὧ) (hv:Word):
  (if hS:S.size<2^32/\4*(0+H)<=S.size then
      grU2 sig tanh inW H x wih whh bih bhh h t S lo sp e a hv hS.1 H 0 0 rfl hS.2 (Array.emptyWithCapacity H)
    else grU sig tanh inW H x wih whh bih bhh h t S lo sp e a hv 0 H (Array.emptyWithCapacity H))=
    grU sig tanh inW H x wih whh bih bhh h t S lo sp e a hv 0 H (Array.emptyWithCapacity H):=by
 split
 · exact grU2_eq _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
 · rfl
@[specialize] def gruStepX (sig tanh:Word->Word) (inW H:Nat) (x wih whh bih bhh:ё)
   (P:ὤ) (hv:Word) (t:Nat) (st:ё×ё):ё×ё:=
 if hc:st.1.all good=true then
  if hk:inW+H+3<2^20/\P.size<2^32 then
   let V:=vOf inW H x st.1 t
   let a:=aS V 0 (inW+H+2) 0
   let e:=nfOf (inW+H+2)*(dv hv*a+xNA)
   let lo:=D.loOf e
   let S:=xSum true inW H wih whh bih bhh V P (vx_le inW H x st.1 t hk.1) hk.2
   ((if hS:S.size<2^32/\4*(0+H)<=S.size then
     grU2 sig tanh inW H x wih whh bih bhh st.1 t S lo (D.spOf lo) e a hv hS.1 H 0 0 rfl hS.2 (Array.emptyWithCapacity H)
   else grU sig tanh inW H x wih whh bih bhh st.1 t S lo (D.spOf lo) e a hv 0 H (Array.emptyWithCapacity H)),st.2)
  else gruStepWith Pinned.dotFn sig tanh inW H x wih whh bih bhh t st
 else gruStepWith Pinned.dotFn sig tanh inW H x wih whh bih bhh t st
lemma rw_zero (w:Word):rationalWord w*rationalWord 0=0:=by rw[rat_zero,mul_zero]
lemma gX_x (inW H:Nat) (x h:ё) (t j:Nat) (hj:j< inW):gX inW H x h t j=x[t*inW+j]!:=by
 unfold gX;rw[Ὦ hj]
lemma gX_h (inW H:Nat) (x h:ё) (t j:Nat) (hj:j<H):gX inW H x h t (inW+j)=h[j]!:=by
 unfold gX;rw[Ὧ א,Ὦ א,Nat.add_sub_cancel_left]
lemma gX_1 (inW H:Nat) (x h:ё) (t j:Nat) (hj:inW+H<=j):gX inW H x h t j=oneW:=by
 unfold gX;rw[Ὧ א,Ὧ א]
lemma wHN_x (inW H:Nat) (whh bhh:ё) (u j:Nat) (hj:j< inW):wHN inW H whh bhh u j=0:=by
 unfold wHN;rw[Ὦ hj]
lemma wHN_h (inW H:Nat) (whh bhh:ё) (u j:Nat) (hj:j<H):
  wHN inW H whh bhh u (inW+j)=whh[(2*H+u)*H+j]!:=by
 unfold wHN;rw[Ὧ א,Ὦ א,Nat.add_sub_cancel_left]
lemma wHN_1 (inW H:Nat) (whh bhh:ё) (u:Nat):wHN inW H whh bhh u (inW+H)=0:=by
 unfold wHN;rw[Ὧ א,Ὧ א,Ὦ rfl]
lemma wHN_2 (inW H:Nat) (whh bhh:ё) (u:Nat):wHN inW H whh bhh u (inW+H+1)=bhh[2*H+u]!:=by
 unfold wHN;rw[Ὧ א,Ὧ א,Ὧ א]
lemma wCD_x (inW H:Nat) (wih bih:ё) (u j:Nat) (hj:j< inW):wCD inW H wih bih u j=wih[(2*H+u)*inW+j]!:=by
 unfold wCD;rw[Ὦ hj]
lemma wCD_h (inW H:Nat) (wih bih:ё) (u j:Nat) (hj:j<H):wCD inW H wih bih u (inW+j)=0:=by
 unfold wCD;rw[Ὧ א,Ὦ א]
lemma wCD_1 (inW H:Nat) (wih bih:ё) (u:Nat):wCD inW H wih bih u (inW+H)=bih[2*H+u]!:=by
 unfold wCD;rw[Ὧ א,Ὧ א,Ὦ rfl]
lemma wCD_2 (inW H:Nat) (wih bih:ё) (u:Nat):wCD inW H wih bih u (inW+H+1)=0:=by
 unfold wCD;rw[Ὧ א,Ὧ א,Ὧ א]
lemma hn_sum (inW H:Nat) (x whh bhh h:ё) (t u:Nat):
  (∑ j∈Ἣ (inW+H+2),rationalWord (gX inW H x h t j)*rationalWord (wHN inW H whh bhh u j))=
  ∑ j∈Ἣ (H+1),rationalWord (if j<H then h[j]! else oneW)*
    rationalWord (if j<H then whh[(2*H+u)*H+j]! else bhh[2*H+u]!):=by
 rw[show inW+H+2=inW+H+1+1 by omega,Finset.sum_range_succ,Finset.sum_range_succ,
  Finset.sum_range_add,Finset.sum_range_succ (n:=H)]
 have e1: ∑ j∈Ἣ inW,rationalWord (gX inW H x h t j)*rationalWord (wHN inW H whh bhh u j)=0:=
  Finset.sum_eq_zero λj hj=>by rw[wHN_x _ _ _ _ _ _ (Finset.mem_range.mp hj)];exact rw_zero _
 rw[e1,zero_add,wHN_1,rw_zero,add_zero,wHN_2,gX_1 _ _ _ _ _ _ א,Ὧ (Nat.lt_irrefl H),
  Ὧ (Nat.lt_irrefl H)]
 congr 1
 refine Ἷ rfl λj hj=>?_
 have:=Finset.mem_range.mp hj
 rw[gX_h _ _ _ _ _ _ this,wHN_h _ _ _ _ _ _ this,Ὦ this,Ὦ this]
lemma cd_sum (inW H:Nat) (x wih bih h:ё) (t u:Nat) (hn r:Word):
  (∑ j∈Ἣ (inW+H+2+1),
    rationalWord (if j< inW+H+2 then gX inW H x h t j else hn)*
    rationalWord (if j< inW+H+2 then wCD inW H wih bih u j else r))=
  ∑ j∈Ἣ (inW+2),
    rationalWord (if j< inW then x[t*inW+j]! else if j=inW then oneW else hn)*
    rationalWord (if j< inW then wih[(2*H+u)*inW+j]! else if j=inW then bih[2*H+u]! else r):=by
 rw[Finset.sum_range_succ,Ὧ (Nat.lt_irrefl _),Ὧ (Nat.lt_irrefl _),
  show inW+H+2=inW+H+1+1 by omega,Finset.sum_range_succ,Finset.sum_range_succ,
  Finset.sum_range_add,show inW+2=inW+1+1 by omega,Finset.sum_range_succ,Finset.sum_range_succ]
 have e1: ∑ j∈Ἣ H,rationalWord (if inW+j< inW+H+1+1 then gX inW H x h t (inW+j) else hn)*
   rationalWord (if inW+j< inW+H+1+1 then wCD inW H wih bih u (inW+j) else r)=0:=
  Finset.sum_eq_zero λj hj=>by
   have:=Finset.mem_range.mp hj
   rw[Ὦ א,Ὦ א,wCD_h _ _ _ _ _ _ this];exact rw_zero _
 rw[e1,add_zero,Ὦ א,Ὦ א,Ὦ א,Ὦ א,wCD_2,rw_zero,
  add_zero,wCD_1,gX_1 _ _ _ _ _ _ א,Ὧ (Nat.lt_irrefl _),Ὦ rfl,Ὧ (Nat.lt_irrefl _),
  Ὦ rfl,Ὧ א,Ὧ א,Ὧ א,Ὧ א]
 congr 2
 refine Ἷ rfl λj hj=>?_
 have:=Finset.mem_range.mp hj
 rw[Ὦ א,Ὦ א,gX_x _ _ _ _ _ _ this,wCD_x _ _ _ _ _ _ this,Ὦ this,Ὦ this]
end RnnX5
section RnnX6
lemma slotF (gru:Ὡ) (inW H:Nat) (x wih whh bih bhh h:ё) (t:Nat)
  (hk:inW+H+3<2^20/\(xpOf gru inW H wih whh bih bhh).1.size<2^32)
  (u g:Nat) (hu:u<H) (hg:g<4) (ww:Nat->Word) (hww: ∀j,ww j=wfX gru inW H wih whh bih bhh u g j):
  (xSum gru inW H wih whh bih bhh (vOf inW H x h t) (xpOf gru inW H wih whh bih bhh).1 (vx_le inW H x h t hk.1) hk.2).get!
    (4*u+g)=(ἄ (inW+H+2)).foldl (λacc j=>acc+dv (gX inW H x h t j)*dv (ww j)) 0/\
  aS (vOf inW H x h t) 0 (inW+H+2) 0=
   (ἄ (inW+H+2)).foldl (λacc j=>acc+(dv (gX inW H x h t j)).abs) 0:=by
 obtain ⟨hVs,hVg⟩:=vOf_spec inW H x h t
 refine ⟨?_,?_⟩
 · have:=slot_eq H (inW+H+2) (wfX gru inW H wih whh bih bhh) (vOf inW H x h t) א hVs
    (gX inW H x h t) hVg hk.2 (vx_le inW H x h t hk.1) u g hu hg
   simp only[<-hww] at this
   exact this
 · rw[aS_eq,<-ὗ]
   exact foldl_ext _ _ _ _ λj hj b=>by rw[hVg j (List.mem_range.mp hj)]
lemma hnX_eq (inW H:Nat) (x wih whh bih bhh h:ё) (t:Nat)
  (hq:(xpOf true inW H wih whh bih bhh).2.2=true) (hx: ∀i:Nat,good x[i]! =true)
  (hc:h.all good=true)
  (hk:inW+H+3<2^20/\(xpOf true inW H wih whh bih bhh).1.size<2^32) (u:Nat) (hu:u<H):
  let V:=vOf inW H x h t
  let e:=nfOf (inW+H+2)*(dv (xpOf true inW H wih whh bih bhh).2.1*aS V 0 (inW+H+2) 0+xNA)
  let S:=xSum true inW H wih whh bih bhh V (xpOf true inW H wih whh bih bhh).1 (vx_le inW H x h t hk.1) hk.2
  hnX H whh bhh h u (gT (D.loOf e) (D.spOf (D.loOf e)) e (S.get! (4*u+2)) (λ_=>hnSlow H whh bhh h u))=
   hnSlow H whh bhh h u:=by
 intro V e S
 obtain ⟨q,-,-,-⟩:=xq_of true inW H wih whh bih bhh hq
 obtain ⟨hS,ha⟩:=slotF true inW H x wih whh bih bhh h t hk u 2 hu ה (wHN inW H whh bhh u) (λj=>rfl)
 have hwg: ∀j,good (wHN inW H whh bhh u j)=true/\wHN inW H whh bhh u j&&&В<=
   (xpOf true inW H wih whh bih bhh).2.1:= λj=>by
  unfold wHN
  split
  · exact ⟨good_zero,by rw[show (0:Word)&&&В=0 from rfl];exact UInt32.zero_le⟩
  split
  · exact ⟨q.gu _,q.lu _⟩
  split
  · exact ⟨good_zero,by rw[show (0:Word)&&&В=0 from rfl];exact UInt32.zero_le⟩
  · exact ⟨q.gc _,q.lc _⟩
 have hf1: ∀j,j< inW+H+2->expOf (gX inW H x h t j)≠255/\expOf (wHN inW H whh bhh u j)≠255:=
  λj _=>⟨qAe _ (gX_good inW H x h t hx hc j),qAe _ (hwg j).1⟩
 rcases gT_or (inW+H+2) א א (gX inW H x h t) (wHN inW H whh bhh u)
  (λj _=>gX_good inW H x h t hx hc j) (λj _=>(hwg j).1) _ q.gv q.pv (λj _=>(hwg j).2) _ _ hS ha
  (λ_=>hnSlow H whh bhh h u) with h1 | h1
 · rw[h1];unfold hnX;split<;>rfl
 · rw[h1]
   unfold hnX
   split
   · rfl
   · rename_i hnz
     have h0:(∑ j∈Ἣ (inW+H+2),rationalWord (gX inW H x h t j)*
       rationalWord (wHN inW H whh bhh u j))≠0:= λhz=>
      hnz (by rw[dotOf_zero _ _ _ hf1 hz];rfl)
     unfold hnSlow
     rw[Pinned.dotFn_eq_dotOf]
     refine dotOf_sum _ _ _ _ _ _ hf1 (λj hj=>?_) (hn_sum inW H x whh bhh h t u) h0
     by_cases hjH:j<H
     · simp only[hjH,ite_true];exact ⟨qAe _ (all_good h hc j),qAe _ (q.gu _)⟩
     · simp only[hjH,ite_false];exact ⟨qAe _ good_one,qAe _ (q.gc _)⟩
lemma cdX_eq (inW H:Nat) (x wih whh bih bhh h:ё) (t:Nat)
  (hq:(xpOf true inW H wih whh bih bhh).2.2=true) (hx: ∀i:Nat,good x[i]! =true)
  (hc:h.all good=true)
  (hk:inW+H+3<2^20/\(xpOf true inW H wih whh bih bhh).1.size<2^32) (u:Nat) (hu:u<H)
  (hn r:Word):
  let V:=vOf inW H x h t
  let S:=xSum true inW H wih whh bih bhh V (xpOf true inW H wih whh bih bhh).1 (vx_le inW H x h t hk.1) hk.2
  cdX inW H x wih bih t u (S.get! (4*u+3)) (aS V 0 (inW+H+2) 0) (xpOf true inW H wih whh bih bhh).2.1 hn r=
   candSlow inW H x wih bih t u hn r:=by
 intro V S
 obtain ⟨q,-,-,-⟩:=xq_of true inW H wih whh bih bhh hq
 obtain ⟨hS,ha⟩:=slotF true inW H x wih whh bih bhh h t hk u 3 hu ה (wCD inW H wih bih u) (λj=>rfl)
 unfold cdX
 split
 · rename_i hg
   simp only[Bool.and_eq_true,decide_eq_true_eq] at hg
   obtain ⟨⟨ghn,gr⟩,rle⟩:=hg
   let K:=inW+H+2
   let xw:Nat->Word:= λj=>if j<K then gX inW H x h t j else hn
   let ww:Nat->Word:= λj=>if j<K then wCD inW H wih bih u j else r
   have hwg: ∀j,good (wCD inW H wih bih u j)=true/\wCD inW H wih bih u j&&&В<=
     (xpOf true inW H wih whh bih bhh).2.1:= λj=>by
    unfold wCD
    split
    · exact ⟨q.gw _,q.lw _⟩
    split
    · exact ⟨good_zero,by rw[show (0:Word)&&&В=0 from rfl];exact UInt32.zero_le⟩
    split
    · exact ⟨q.gb _,q.lb _⟩
    · exact ⟨good_zero,by rw[show (0:Word)&&&В=0 from rfl];exact UInt32.zero_le⟩
   have hxg: ∀j,j<K+1->good (xw j)=true:= λj _=>by
    show good (if j<K then _ else _)=true
    split
    · exact gX_good inW H x h t hx hc j
    · exact ghn
   have hwg': ∀j,j<K+1->good (ww j)=true/\ww j&&&В<=(xpOf true inW H wih whh bih bhh).2.1:=
    λj _=>by
     show good (if j<K then _ else _)=true/\(if j<K then _ else _)&&&В<=_
     split
     · exact hwg j
     · exact ⟨gr,rle⟩
   have hs0:S.get! (4*u+3)+dv hn*dv r=
     (ἄ (K+1)).foldl (λacc j=>acc+dv (xw j)*dv (ww j)) 0:=by
    rw[List.range_succ,List.foldl_append,hS]
    simp only[ὔ,List.foldl_nil,xw,ww,Nat.lt_irrefl,ite_false]
    congr 1
    exact foldl_ext _ _ _ _ λj hj b=>by rw[Ὦ (List.mem_range.mp hj),Ὦ (List.mem_range.mp hj)]
   have ha0:aS V 0 (inW+H+2) 0+(dv hn).abs=
     (ἄ (K+1)).foldl (λacc j=>acc+(dv (xw j)).abs) 0:=by
    rw[List.range_succ,List.foldl_append,ha]
    simp only[ὔ,List.foldl_nil,xw,Nat.lt_irrefl,ite_false]
    congr 1
    exact foldl_ext _ _ _ _ λj hj b=>by rw[Ὦ (List.mem_range.mp hj)]
   have hf1: ∀j,j<K+1->expOf (xw j)≠255/\expOf (ww j)≠255:=
    λj hj=>⟨qAe _ (hxg j hj),qAe _ (hwg' j hj).1⟩
   dsimp only
   rcases gZ_or (K+1) א xw ww hxg (λj hj=>(hwg' j hj).1) _ q.gv q.pv (λj hj=>(hwg' j hj).2)
    _ _ hs0 ha0 (λ_=>candSlow inW H x wih bih t u hn r) with h1 | h1
   · rw[h1];split<;>rfl
   · rw[h1,C.qI0 (K+1) xw ww א hf1]
     split
     · rfl
     · rename_i hnz
       have h0:(∑ j∈Ἣ (K+1),rationalWord (xw j)*rationalWord (ww j))≠0:= λhz=>
        hnz (by rw[dotOf_zero _ _ _ hf1 hz];rfl)
       unfold candSlow
       rw[Pinned.dotFn_eq_dotOf]
       refine dotOf_sum _ _ _ _ _ _ hf1 (λj hj=>?_) (cd_sum inW H x wih bih h t u hn r) h0
       by_cases j1:j< inW
       · rw[Ὦ j1,Ὦ j1];exact ⟨qAe _ (hx _),qAe _ (q.gw _)⟩
       · by_cases j2:j=inW
         · subst j2;rw[Ὧ (Nat.lt_irrefl _),Ὦ rfl,Ὧ (Nat.lt_irrefl _),Ὦ rfl]
           exact ⟨qAe _ good_one,qAe _ (q.gb _)⟩
         · rw[Ὧ j1,Ὧ j2,Ὧ j1,Ὧ j2];exact ⟨qAe _ ghn,qAe _ gr⟩
 · rfl
lemma gruStepX_eq (sig tanh:Word->Word) (inW H:Nat) (x wih whh bih bhh:ё)
  (hq:(xpOf true inW H wih whh bih bhh).2.2=true) (hx: ∀i:Nat,good x[i]! =true):
  gruStepX sig tanh inW H x wih whh bih bhh (xpOf true inW H wih whh bih bhh).1 (xpOf true inW H wih whh bih bhh).2.1=
   gruStepWith Pinned.dotFn sig tanh inW H x wih whh bih bhh:=by
 funext t st
 obtain ⟨q,-,-,-⟩:=xq_of true inW H wih whh bih bhh hq
 unfold gruStepX
 split
 · rename_i hc
   split
   · rename_i hk
     have hg: ∀g u,u<H->g<2->
       gT (D.loOf (nfOf (inW+H+2)*(dv (xpOf true inW H wih whh bih bhh).2.1*
           aS (vOf inW H x st.1 t) 0 (inW+H+2) 0+xNA)))
        (D.spOf (D.loOf (nfOf (inW+H+2)*(dv (xpOf true inW H wih whh bih bhh).2.1*
          aS (vOf inW H x st.1 t) 0 (inW+H+2) 0+xNA))))
        (nfOf (inW+H+2)*(dv (xpOf true inW H wih whh bih bhh).2.1*
          aS (vOf inW H x st.1 t) 0 (inW+H+2) 0+xNA))
        ((xSum true inW H wih whh bih bhh (vOf inW H x st.1 t) (xpOf true inW H wih whh bih bhh).1
          (vx_le inW H x st.1 t hk.1) hk.2).get! (4*u+g))
        (λ_=>xSlow inW H x wih whh bih bhh st.1 t (g*H+u))=
       gateWith Pinned.dotFn inW H x wih whh bih bhh st.1 t (g*H+u):= λg u hu hg2=>by
      rw[Pinned.dotFn_eq_dotOf]
      exact gateX_eq true inW H x wih whh bih bhh st.1 t hq hx hc hk u g hu א
       (gWt inW H wih whh bih bhh (g*H+u)) (λj=>by unfold wfX;rw[Ὦ rfl,Ὦ hg2])
       _ (by unfold xSlow;rw[Pinned.dotFn_eq_dotOf]) (λj=>gWt_good q _ j)
     dsimp only
     rw[dite_grU,grU_eq _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ H 0 _ (by rfl),Ἇ]
     unfold gruStepWith
     simp only[]
     congr 1
     congr 1;funext u
     unfold gruUX
     rw[updF_eq]
     repeat (rw[ofFn_get!F _ _ u.isLt];try dsimp only)
     have h0:=hg 0 u.val u.isLt ה
     have h1:=hg 1 u.val u.isLt ה
     simp only[ὒ,Ἇ,Ὑ,Ἵ] at h0 h1
     have hn:=hnX_eq inW H x wih whh bih bhh st.1 t hq hx hc hk u.val u.isLt
     simp only[] at hn
     rw[h0,h1,hn,cdX_eq inW H x wih whh bih bhh st.1 t hq hx hc hk u.val u.isLt]
     unfold candSlow hnSlow
     rfl
   · rfl
 · rfl
@[specialize] def gruLX (sig tanh:Word->Word) (T inW H:Nat) (x wih whh bih bhh:ё):ё:=
 let p:=(xpGet true inW H wih whh bih bhh).1
 if p.2.2&&x.all good then
  recFlat (gruStepX sig tanh inW H x wih whh bih bhh p.1 p.2.1) 0 T (ἇ H 0, #[])
   (Array.emptyWithCapacity (T*H))
 else gruWith Pinned.dotFn sig tanh T inW H x wih whh bih bhh
lemma gruLX_eq (sig tanh:Word->Word):gruLX sig tanh=gruWith Pinned.dotFn sig tanh:=by
 funext T inW H x wih whh bih bhh
 unfold gruLX
 rw[(xpGet true inW H wih whh bih bhh).2]
 dsimp only
 split
 · rename_i hq
   simp only[Bool.and_eq_true] at hq
   rw[gruStepX_eq sig tanh inW H x wih whh bih bhh hq.1 (all_good x hq.2),
    recFlat_eq T H _ _ (λt st=>by unfold gruStepWith;exact Array.size_ofFn)]
   rfl
 · rfl
end RnnX6
end RnnFast
def kernels:Kernels where
 core:=specCore
 dense:=⟨C.linG2,C.gemmC2⟩
 conv:=⟨ConvFast.conv2dF,ConvFast.depthwiseF,Pinned.kernels.conv.convTranspose⟩
 elem:=⟨ConvFast.avgpoolF,ConvFast.maxpoolF,C.addF,C.scaleF⟩
 norm:=⟨D.rmsF,D.lnF⟩
 attn:=specAttn
 rnn:=⟨RnnFast.lstmLX RnnFast.sigmoidF2 RnnFast.tanhF2,RnnFast.gruLX RnnFast.sigmoidF2 RnnFast.tanhF2⟩
lemma correct:Obligation kernels:=by
 have hP:=Pinned.correct
 have hc:Pinned.kernels.conv=specConv:=congrArg Kernels.conv hP
 refine obligation_of_families kernels rfl ?_ ?_ ?_ ?_ ?_ ?_
 · show (⟨C.linG2,C.gemmC2⟩:DenseKernels)=specDense
   rw[C.qHb2,C.qGm2];rfl
 · show (⟨ConvFast.conv2dF,ConvFast.depthwiseF,Pinned.kernels.conv.convTranspose⟩:ConvKernels)=specConv
   rw[ConvFast.conv2dF_eq,ConvFast.depthwiseF_eq,
    show Pinned.kernels.conv.convTranspose=convTransposeSpec from congrArg ConvKernels.convTranspose hc];rfl
 · show (⟨ConvFast.avgpoolF,ConvFast.maxpoolF,C.addF,C.scaleF⟩:ElementKernels)=specElem
   rw[ConvFast.avgpoolF_eq,ConvFast.maxpoolF_eq,C.qFu,C.qFf];rfl
 · show (⟨D.rmsF,D.lnF⟩:NormKernels)=specNorm
   rw[D.qFf,D.qGc];rfl
 · exact rfl
 · show (⟨RnnFast.lstmLX RnnFast.sigmoidF2 RnnFast.tanhF2,RnnFast.gruLX RnnFast.sigmoidF2 RnnFast.tanhF2⟩:RecurrentKernels)=specRnn
   rw[RnnFast.lstmLX_eq,RnnFast.gruLX_eq,RnnFast.sigmoidF2_eq,RnnFast.tanhF2_eq,FastAct.sigmoidF_eq,FastAct.tanhF_eq,Pinned.dotFn_eq_dotOf];rfl
end Submission