import ContractExact
namespace Submission
open ExactRuntime FloatLib.Floats.Formats.BinaryInterchange FloatLib.Floats
open USize renaming lt_iff_toNat_lt→γ,le_size→ς,toNat_lt_size→Τ,size_eq_two_pow→Ϟ,toNat_zero→Ϥ,le_refl→Ϭ
open Nat renaming and_two_pow_sub_one_eq_mod→θ,shiftRight_eq_div_pow→ι,mod_eq_of_lt→ξ,pow_le_pow_right→Ε,mul_le_mul_right→Ζ,shiftLeft_eq→Μ,succ_mul→ϋ,two_pow_add_eq_or_of_lt→ϓ,add_comm→ϔ,and_le_right→ϙ,toUSize_eq→ϛ,div_eq_of_lt→ϣ,add_sub_cancel_left→ϥ,le_log2→Ϧ,mul_comm→ϩ,add_zero→Ϫ,one_shiftLeft→ϭ,mul_one→ϱ,div_lt_of_lt_mul→ϲ,mod_lt→ϳ,add_mul_mod_self_right→ϵ,lt_log2_self→Ϸ
open UInt64 renaming le_iff_toNat_le→κ,toNat_add→μ,lt_iff_toNat_lt→ν,toNat_sub_of_le→χ,toNat_and→ψ,toNat_shiftRight→Α,toNat_inj→Β,toNat_shiftLeft→Ι,toNat_sub→Κ,toNat_ofNat_of_lt'→Ξ,toNat_mul→Ϛ,toNat_toUInt32→ϟ,toNat_zero→Ϯ
open Bool renaming false_eq_true→ο,and_eq_true→υ,eq_iff_iff→ϻ
open UInt32 renaming toNat_inj→Η,toNat_and→Λ,toNat_shiftRight→Ρ,toNat_toUInt64→Χ,lt_iff_toNat_lt→ϖ
open Model renaming toModel_ofFloatModel→Θ,isFinite_roundDyadic_of_isIEEE_of_abs_toReal_le_posMaxFinite→Υ,toReal_eq_unpackedToReal_toModel→ό,toDyadic?_ofFields_of_isFinite→ϒ,toReal_roundDyadic_eq_roundAt→ϗ,toDyadic?_isSome_eq_isFinite→Ϝ,abs_signed_mul_bpow_le_toReal_posMaxFinite→Ϡ,ofFloatModel_pack→Ϣ,toDyadic?_ieee_eq_model→ϧ,roundDyadicGeneral_eq_roundDyadic→ϰ,roundedMantissa_shift_quotient→Ϻ
open Finset renaming mem_range→Ν,sum_congr→Ο,sum_le_sum→Ϗ,sum_Ico_succ_top→ϐ,sum_range_succ→ϝ,sum_nonneg→ϡ,sum_add_distrib→ϫ,card_range→ϯ,sum_mul→϶,sum_le_sum_of_subset_of_nonneg→ϸ,abs_sum_le_sum_abs→Ϲ
open Real renaming sqrt_nonneg→ϊ
open Float.Model.UnpackedFloat renaming unpackExponent_packComponents→ύ,unpackMantissa_packComponents→ώ
open Array renaming size_uset→ϑ,getElem_ofFn→Ϩ,size_replicate→ϴ
open Int renaming cast_abs→ϕ
open FloatLib.Numerics.Dyadic renaming cast_toRat→Ϙ
local notation "А" => by omega
local notation "Б" => by decide
local notation "В" => by positivity
local notation "Г" => by norm_num
local notation "Д" => UInt64
local notation "Е" => UInt32 local postfix:max "а"=>UInt64.toNat local postfix:max "б"=>USize.toNat local postfix:max "в"=>Array.size local postfix:max "г"=>Nat.toUSize local postfix:max "д"=>FloatArray.size local postfix:max "е"=>UInt32.toNat local postfix:max "ж"=>Nat.toUInt64 local postfix:max "з"=>Nat.log2 local postfix:max "и"=>Fin.val local postfix:max "й"=>Int.toNat local postfix:max "к"=>UInt32.toUInt64 local postfix:max "л"=>UInt64.toUInt32 local postfix:max "м"=>Int.natAbs local postfix:max "н"=>Int64.toInt open UInt64 renaming reduceToNat→iZ,toNat→jI,size→jT,toNat_or→kp,toUInt32→kr,toNat_one→kW,toNat_toBitVec→ll,ofNat→lP open Finset renaming range→i0,Ico→jr,sum_range_succ'→jY,sum_sub_distrib→j9,sum_const→kg,sum_Ico_eq_sum_range→kz,mul_sum→kJ open FloatLib.Floats.Formats.BinaryInterchange.FloatFormat renaming binary64→i5 open FloatLib.Floats.Formats.BinaryInterchange.Model renaming ofFloatModel→i6,roundDyadic→i8,unpackedToReal→jp,toReal→jJ,toReal_eq→jK,roundDyadic_of_representable→jO,roundAt_mono→jP,unpackedToDyadic?→jU,isFinite→jW,unpackSign_packComponents→kb,roundAt→kh,toModel→ki,toReal_add_eq_roundAt→kx,posMaxFinite→kG,pow2_eq_two_pow→ln,ofUInt32→ls,ofFloat32Model→ly,add→lB,ofModel→lH,modelSignBit→lL open ExactRuntime renaming expOf→i7,Word→i9,idxOf→je,rowVariance→jg,stableNormDomain→ji,sqrtRational→jk,rowMean→jo,signOf→jw,mantOf→jF,toWord→jH,layernormStable→jN,finiteWord→jS,oneW→kf,rmsnormStable→kj,mantOf_lt_24→kw,dotOf→kB,finishWord→kC,isFinite_ofUInt32→kM,ofIntDyadic→kN,finishWord_of_ne→kV,Fmt→lr,layernormSpec→lJ,fracOf→lW open Real renaming sqrt→ja,sq_sqrt→lb,le_sqrt_of_sq_le→lq,sqrt_mul→lI,sqrt_sq→lU open Nat renaming reduceMod→jb,toUInt64_eq→jL,testBit_eq_decide_div_mod_eq→jZ,cast_ofNat→j0,zero_add→j6,add_mul_div_right→kc,le_div_iff_mul_le→kd,zero_le→km,add_mul→ko,cast_natAbs→ku,pos_of_ne_zero→kD,and_one_is_mod→kF,lt_of_lt_of_le→kH,cast_zero→kS,log2_self_le→kU,le_mul_of_pos_right→kX,cast_sub→k9,zero_mul→la,and_two_pow→lc,two_pow_pos→ld,cast_nonneg→lj,div_add_mod→lk,log2_lt→lu,pow_lt_pow_right→lz,log2_eq_log_two→lC,mod_le→lK,cast_one→lQ,abs_cast→lS,one_mul→l0 open Float.Model.UnpackedFloat renaming unpack→jc,abs→jm,isFinite→jn,mul→jq,pack→jA,packedInfinity→jG,accuracyOfFraction→jQ,roundWithAccuracy→kR,unpackExponent→lp,packedZero→lD,round→lG,Sign→lR open USize renaming reduceToNat→jd,size→ke,toNat_add→ks,toNat_one→lf,toNat_mul→li open FloatLib.Numerics renaming Dyadic→jf,binaryRadix→jh,shiftRightJam→jx,roundShiftRightEven_shiftRightJam→jV,roundQuotientEven→kQ open UInt32 renaming reduceToNat→jj,toNat_zero→j5 open Float.Model renaming totalExponent→jl,pack→kY,sqrt→lZ,add→l4,mul→l5,sub→l7 open FloatLib.Numerics.Dyadic renaming ofScaledInt→js,toReal→jz,signedSignificand→jC,cast_signedSignificand→kP,toRat_eq_zero_iff→kZ,ofScaledInt_toRat→k8,toRat→lX open Float.Model.Format renaming binary64→jt,binary32→ju open Array renaming ofFn→jv,replicate→jM,all_eq_true→j8,usize→kk,getElem?_eq_getElem→kL,size_set→lA,setIfInBounds→lE,getElem?_ofFn→lF,getElem?_set→lM,ext_getElem?→lN,uset_eq_set→lT,getElem_set→lV,set!→l8,uset→l9 open FloatLib.Numerics.FixedWord renaming roundShiftRightEven→jy open Float.Model.UnpackedFloat.ExtendedMantissa renaming shiftRightOne→jB open FloatArray renaming size→jD,emptyWithCapacity→jE,set!→kt,uget→lh,usize→lY,uset→l6 open FloatLib.Floats.Formats.Flocq renaming bpow→jR open FloatLib.Numerics.RationalBinary renaming scaleByPowerOfTwo→j7 open FloatLib.Floats.ExecFloat.Binary renaming toModel_ofFloat_int64ToFloat→ka open Int renaming ofNat_eq_natCast→kl,toNat_of_nonneg→kv,log→ky,lt_zpow_succ_log_self→kK,abs_eq_natAbs→kO,zpow_log_le_self→lv open FloatLib.Floats.Formats.BinaryInterchange.Model.Reduction.Semantics renaming finiteDot_toList_eq_slice→kA open BitVec renaming toInt_eq_toNat_cond→kE open Float32 renaming ofBits→kI open FloatLib.Floats.Formats.BinaryInterchange.Model.Reduction.Internal renaming dotState_exact_toRat→kT,dotState→lO open Bool renaming or_eq_true→lg open Max renaming max→lm open FloatLib.Floats.Formats.Flocq.bpow renaming pos→lx open Or renaming inr→ma local notation "о"=>Nat local notation "п"=>Int local notation "р"=>Float local notation "с"=>USize local notation "т"=>Word local notation "у"=>Array local notation "ф"=>Bool
section
open FloatLib.Numerics
lemma aD (s:ф) (m:о) (e:п) (j:о) (h:j+25<= mз):
  i8 lr ⟨s,jx m j,e+j⟩=i8 lr ⟨s,m,e⟩:=by
 have g:m≠0:=by rintro rfl;simp at h
 have o:2^mз<= m:=kU g;have u:2^(j+1)<= m:=le_trans (Ε Г А) o;have d:(jx m j)з=mз-j:=log2_shiftRightJam m j
 have t:jx m j≠0:=
  shiftRightJam_ne_zero m j (le_trans (Ε Г А) u)
 rw[<- ϰ,<- ϰ];unfold Model.roundDyadicGeneral;simp only[t,g,beq_iff_eq,ite_false,d];have q:((mз-j:о):п)+(e+j)=(mз:п)+e:=А;rw[q];have r:Fmt.fracWidth=23:=rfl;have p:Fmt.minSubnormalExponent=-149:=rfl;have l:Fmt.minNormalExponent=-126:=rfl;rw[r,p,l];split_ifs with h1 h2 h3 h4<;>try omega
 .rfl
 .congr 1
  obtain ⟨a,v⟩:∃a:о,e- -149=Int.negSucc a:=⟨(-(e+149)-1)й,by omega⟩
  have w:e+j- -149=Int.negSucc (a-j):=А;rw[v,w];dsimp only;rw[show a-j+1=(a+1)-j by omega]
  exact jV m j (a+1) А
 .congr 1
  rw[show mз-j-23=(mз-23)-j by omega]
  exact jV m j _ А
end
lemma ae (n:о) (f:о-> Е) (k:о) (d:k<n):
  (jv (n:=n) λi=>f iи)[k]! =f k:=by
 rw[getElem!_pos _ k (by simpa using d),Ϩ]
namespace Core
def tm (a b:т):п:=sv a*sv b*2^(idxOf a+idxOf b)
def ah (n:о) (fa fb:о-> т):п:=∑k∈Finset.range n,tm (fa k) (fb k)
def am (n:о) (fa fb:о-> т):Prop:=∀k,k<n-> expOf (fa k)≠255/\expOf (fb k)≠255
lemma aE (n:о) (d e:о-> i9):
  ah (n+1) d e=ah n d e+tm (d n) (e n):=ϝ _ _
lemma aU (n:о) (m o:о-> i9) (l:am n m o):
  kB n m o=kC (ah n m o) (-298) (decide (n≠0)&&
   dallNZ (jv (n:=n) λi=>m iи) (jv (n:=n) λi=>o iи) 0 n):=by
 unfold kB dotSpec;set A:=jv (n:=n) λi=>m iи
 set B:=jv (n:=n) λi=>o iи
 set L:=A.map (ls (fmt:=lr));set R:=B.map (ls (fmt:=lr));have j:Lв=Rв:=by simp[L,R,A,B]
 have e:∀k (h:k<Lв),jW L[k]=true:=λk h=>by
  simp[L,A,kM] at h⊢;exact (l k h).1
 have g:∀k (h:k<Rв),jW R[k]=true:=λk h=>by
  simp[R,B,kM] at h⊢;exact (l k h).2
 have p:∀k,0<= k-> k<0+Lв-> i7 A[k]!≠255/\i7 B[k]!≠255:=λk _ h=>by
  simp[L,A] at h;simp only[A,B,ae n _ k h];exact l k h
 have v:∀i c,dsum A B i c=∑k∈i0 c,tm A[i+k]! B[i+k]!:=λi c=>by
  induction c generalizing i with
  | zero=>simp[dsum]
  | succ c w=>
   rw[dsum,w (i+1),jY,add_comm];exact congrArg₂ _ (Ο rfl λk _=>by rw[show i+1+k=i+(k+1) by omega]) rfl
 have d:=(finiteDotSlice_eq A B 0 Lв (by simp[L]) (by simp[j,R]) p).trans (c:=(ah n m o:ℚ)*2^(-298:ℤ)) (by simp only[v,ah,L,A,B,Array.size_map,Array.size_ofFn];congr 2;exact Ο rfl λk p=>by simp at p;simp[ae n _ k p])
 have r:=kT (destination:=lr) L R j
 rw[<- Model.Reduction.map_value_dot]
 by_cases q:ah n m o=0
 .have hz0:(Model.Reduction.Internal.dotState L R j:Model.Reduction.Internal.State lr).exact.significand=0:=by
   rw[<- kZ,r,d,q];simp
  rw[Model.Reduction.dot_status_eq_zero_of_finite lr L R .nearest j e g hz0,q,finishWord_zero]
  unfold lO
  obtain ⟨t,u⟩:=dotStateLoop_flags A B {} 0 Lв (by simp[L]) (by simp[j,R]) p
  simp only[Except.map];rw[t,u];simp[L,A]
 .have hnz:Model.Reduction.Semantics.finiteDot L.toList R.toList≠.ok 0:=by
   rw[kA L R j];intro h;injection h with h;rw[d] at h;simp at h;exact q h
  rw[Model.Reduction.dot_status_eq_round_of_finite_nonzero lr L R .nearest j e g hnz]
  simp only[Except.map]
  exact (kV _ (ah n m o) (-298) _ q (by rw[r,d])).symm
lemma a9 (n:о) (d e:о-> i9) (g:am n d e)
  (h:ah n d e≠0):
  kB n d e=kC (ah n d e) (-298) false:=by
 simp[aU n d e g,kC,h]
@[inline] def dq (w:Е):Д:=if (w>>>23)&&&0xFF!=0 then 1 else 0
@[inline] def mW (w:Е):Д:=(w&&&0x7FFFFF)к ||| (dq w<<<23)
@[inline] def iW (w:Е):Д:=((w>>>23)&&&0xFF)к-dq w
def bi (x:Д):п:=x.toBitVec.toInt
lemma aN (w:Е):(mW w)а=jF w:=by
 have e:(w&&&0x7FFFFF)е<2^23:=by rw[Λ];exact lt_of_le_of_lt ϙ Б
 unfold mW dq jF i7 lW;by_cases d:(w>>>23)&&&0xFF=0
 .simp[d]
 .simp only[show ((w>>>23)&&&0xFF!=0)=true by simpa using d,d,ite_true,ite_false]
  rw[kp,Χ,show ((1:Д)<<<23)а=2^23*1 from rfl,Nat.or_comm,
   <- ϓ e,ϱ,ϔ]
lemma bf (w:Е):(iW w)а=je w:=by
 unfold iW dq je i7;have g:=expOf_toNat_lt w;unfold i7 at g;by_cases e:(w>>>23)&&&0xFF=0
 .simp[e]
 .have d:1<=((w>>>23)&&&0xFF)е:=
   Nat.one_le_iff_ne_zero.2 λh=>e (Η.mp (by simpa using h))
  simp only[show ((w>>>23)&&&0xFF!=0)=true by simpa using e,e,ite_true,ite_false]
  rw[χ _ _ (κ.mpr (by rw[Χ];simpa using d)),
   Χ]
  rfl
lemma aP (w:Е):jw w=true↔2^31<= wе:=by
 have:=w.toNat_lt
 rw[jw,bne_iff_ne,ne_eq,<- Η,Λ,show (0x80000000:Е)е=2^31 from rfl,
  lc,jZ]
 by_cases h:wе/2^31%2=1<;>simp[h]<;>omega
@[inline] def sx (k x:Д):Д:=bif x>>>k==0 then x else x>>>k
@[inline] def sr (k x r:Д):Д:=bif x>>>k==0 then r else r+k
@[inline] def an (x0:Д):Д:=
 let x1:=sx 32 x0
 let r1:=sr 32 x0 0
 let x2:=sx 16 x1
 let r2:=sr 16 x1 r1
 let x3:=sx 8 x2
 let r3:=sr 8 x2 r2
 let x4:=sx 4 x3
 let r4:=sr 4 x3 r3
 let x5:=sx 2 x4
 let r5:=sr 2 x4 r4
 sr 1 x5 r5
@[inline] def cq (value shift:Д):Д:=
 let quotient:=value>>>shift
 let remainder:=value&&&((1<<<shift)-1)
 let half:=(1:Д)<<<(shift-1)
 quotient+(if remainder+(quotient&&&1)>half then 1 else 0)
@[inline] def aJ (value shift:Д):Д:=
 if shift==0 then value
 else if shift<64 then cq value shift
 else if shift==64 then (if value>0x8000000000000000 then 1 else 0)
 else 0
@[inline] def cd (sign:ф) (e f:Е):Е:=
 (if sign then (0x80000000:Е) else 0) ||| ((e&&&0xff)<<<23) ||| (f&&&0x007fffff)
@[inline] def cj (n:Д):Е:=
 (if n<0x800000 then 0 else n-0x800000)л
@[inline] def aC (sign:ф) (product scale:Д):Е:=
 if product==0 then
  if sign then 0x80000000 else 0
 else
  let leading:=an product
  let position:=leading+scale
  if position<172 then
   let fraction:=
    if scale<149 then aJ product (149-scale) else product<<<(scale-149)
   if fraction==0 then (if sign then 0x80000000 else 0)
   else if fraction==0x800000 then cd sign 1 0
   else cd sign 0 fractionл
  else
   let roundedMantissa:=
    if leading<23 then product<<<(23-leading) else aJ product (leading-23)
   let carry:=roundedMantissa==0x1000000
   let normalizedPosition:=if carry then position+1 else position
   if normalizedPosition>425 then (if sign then 0xff800000 else 0x7f800000)
   else
    let normalizedMantissa:=if carry then 0x800000 else roundedMantissa
    cd sign (normalizedPosition-171)л (cj normalizedMantissa)
open Model.NativeBinary32 in
lemma cn (n k:о) (h:n>>>k≠0):nз=k+(n>>>k)з:=by
 rw[ι] at*
 have:=(Ϧ (by rintro rfl;simp at h)).2 (Nat.div_pos_iff.1 (kD h)).2
 rw[lC,lC,Nat.log_div_base_pow] at*;omega
lemma bs (e r x k:Д) (o:kа<=32) (l:0<kа)
  (d:eаз=rа+xаз) (m:xа<2^(2*kа)):
  eаз=(sr k x r)а+(sx k x)аз/\(sx k x)а<2^kа:=by
 have:eаз<64:=by
  by_cases h:eа=0<;>simp[h,lu,e.toNat_lt]
 have g:kа%64=kа:=ξ А;unfold sx sr;by_cases j:x>>>k=0<;>simp[j,g,d]
 .simpa[g,ι,Nat.div_eq_zero_iff] using congrArg jI j
 have:=cn xа kа (by simpa[<- Β,g] using j)
 refine ⟨by omega,?_⟩
 rw[ι];exact ϲ (by rwa[<- Nat.pow_add,show kа+kа=2*kа by omega])
lemma aG (x:Д):an x=x.log2:=by
 apply Β.mp;obtain ⟨o,e⟩:=bs x 0 x 32 Б Б (by simp) x.toNat_lt
 obtain ⟨p,g⟩:=bs x _ _ 16 Б Б o e;obtain ⟨q,h⟩:=bs x _ _ 8 Б Б p g;obtain ⟨r,j⟩:=bs x _ _ 4 Б Б q h;obtain ⟨t,l⟩:=bs x _ _ 2 Б Б r j;obtain ⟨d,m⟩:=bs x _ _ 1 Б Б t l;rw[Nat.log2_def (sx 1 _)а,if_neg (by simpa using m)] at d
 unfold an;exact d.symm
lemma cF (e d:Д):
  aJ e d=jy e dа:=by
 unfold aJ jy;by_cases g:d=0
 .subst g;rfl
 have:=d.toNat_lt;have h:dа≠0:=by simpa[<- Β] using g
 simp only[g,h,beq_iff_eq,UInt64.ofNat_toNat,ν,↓reduceIte,
  show lP (dа-1)=d-1 from Β.mp (by simp[Κ];omega),
  show d=64↔dа=64 by rw[<- Β];rfl]
 refine ite_congr rfl (λj=>?_) λ_=>rfl
 have:(e&&&((1<<<d)-1))а<2^63:=by
  have l:2^dа<=2^63:=Ε two_pos А
  rw[ψ,χ,Ι]<;>simp[κ,
   ξ j,ϭ,ξ (by omega:2^d.toNat<18446744073709551616),Nat.one_le_two_pow]
  exact (ϳ _ (ld _)).trans_le l
 have:((e>>>d)&&&1)а<=1:=by simp[ψ];omega
 have:=(e>>>d).toNat_lt
 simp only[cq,gt_iff_lt,ν,μ,beq_iff_eq,<- Β,
  apply_ite jI,kW,iZ]
 generalize (e&&&((1<<<d)-1))а=r at*
 generalize ((e>>>d)&&&1)а=b at*
 split_ifs<;>omega
lemma bQ (n:Д):cj n=UInt32.ofNat (nа-0x800000):=by
 unfold cj;split<;>simp_all[kr,Nat.toUInt32,ν,χ,κ,Nat.sub_eq_zero_of_le,le_of_lt]
open Model.NativeBinary32 in
lemma bH (w:Е):jH (ofUInt32 w)=w:=by
 conv_rhs=>rw[<- toUInt32_ofUInt32 w]
 exact Η.mp (ξ (BitVec.isLt _))
open Model.NativeBinary32 in
lemma ax (e:ф) (m d:Д) (g:dа<=506):
  aC e m d=jH (i8 lr ⟨e,mа,(dа:п)-298⟩):=by
 erw[<- ofUInt32_roundDyadic,bH,<- roundProduct_eq_roundDyadic _ _ _ g];unfold aC roundProduct cd mkBits;simp only[aG,cF,bQ,kr,Nat.toUInt32,FloatLib.Numerics.FixedWord.log2Word_eq_log2];rfl
def az (hi lo:Д):п:=bi hi*2^64+loа
@[inline] def bK (hi lo:Д):Д:=if hi>>>63!=0 then (0:Д)-lo else lo
@[inline] def bJ (hi lo:Д):Д:=
 if hi>>>63!=0 then (if lo==0 then (0:Д)-hi else ~~~hi) else hi
lemma ce (d:Д):(d>>>63!=0)=decide (2^63<= dа):=by
 have:=d.toNat_lt;rw[ϻ];simp[<- Β,ι]
lemma a8 (d e:Д):
  az d e=(dа*2^64+eа:п)-(if 2^63<= dа then 2^128 else 0):=by
 unfold az bi;rw[kE,ll];omega
lemma cT (d e:Д):
  (az d e)м=(bJ d e)а*2^64+(bK d e)а /\
  decide (az d e<0)=(d>>>63!=0):=by
 have:=d.toNat_lt;have:=e.toNat_lt;unfold bJ bK;rw[a8,ce];by_cases g:2^63<= dа<;>simp only[g,decide_true,decide_false,ite_true,ite_false,ο]
 .split<;>simp_all[jT,UInt64.toNat_not,<- Β]<;>omega
 .simp;omega
@[inline] def bm (hi lo B:Д):Д:=
 if hi==0&&lo==0 then 0x100000000 else
 if B>442 then 0x100000000 else
 let neg:=hi>>>63!=0
 let mlo:=bK hi lo
 let mhi:=bJ hi lo
 if mhi==0 then (aC neg mlo B)к
 else if mhi>>>32==0 then
  let q:=(mhi<<<32)+(mlo>>>32)
  (aC neg (if mlo&&&0xFFFFFFFF==0 then q else q ||| 1) (B+32))к
 else (aC neg (if mlo==0 then mhi else mhi ||| 1) (B+64))к
lemma bl (n:о) (m o:о-> i9) (e g B:Д)
  (q:am n m o) (j:ah n m o=az e g*2^Bа)
  (h:bm e g B<0x100000000):kB n m o=(bm e g B)л:=by
 obtain ⟨r,t⟩:=cT e g;have:=g.toNat_lt;have:=e.toNat_lt;simp only[bm] at h⊢;have w:¬(e==0&&g==0)=true:=λc=>by simp[c] at h
 have u:¬B>442:=λc=>by simp[c] at h
 simp only[w,u,ο,↓reduceIte];clear h
 have v:ah n m o≠0:=by
  rw[j];refine mul_ne_zero (λp=>w ?_) В
  rw[a8] at p;simp only[υ,beq_iff_eq,<- Β,Ϯ];split at p<;>omega
 rw[a9 n m o q v,kV (kN (az e g) ((Bа:п)-298)) _ _ _ v (by rw[toRat_ofIntDyadic,j];push_cast;rw[zpow_sub₀ two_ne_zero,zpow_natCast];field_simp),kN,t,r]
 symm;simp only[apply_ite kr,UInt32.toUInt32_toUInt64];generalize bJ e g=l at*
 generalize bK e g=d at*
 have:=d.toNat_lt;simp only[ν,gt_iff_lt,iZ,not_lt] at u;split<;>rename_i p<;>simp only[beq_iff_eq,<- Β,Ϯ] at p
 .simp[p,ax _ _ B А]
 split<;>rename_i z<;>simp[<- Β,ι] at z
 .rw[ax _ _ _ (by simp[μ];omega),<- aD _ (lа*2^64+dа) _ 32 ((Ϧ А).2 А)]
  congr 3
  swap;. simp[μ];omega
  have x:(lа*2^64+dа)/2^32=(l<<<32+d>>>32)а:=by
   simp[Μ,ι];omega
  have y:(lа*2^64+dа)%2^32=(d&&&4294967295)а:=by
   rw[ψ,show (4294967295:Д)а=2^32-1 from rfl,θ];omega
  simp only[jx,iZ,x,y,apply_ite jI,kp,beq_iff_eq,
   <- Β]
 .rw[ax _ _ _ (by simp[μ];omega),<- aD _ (lа*2^64+dа) _ 64 ((Ϧ А).2 А)]
  congr 3
  swap;. simp[μ];omega
  simp only[jx,iZ,show (lа*2^64+dа)/2^64=lа by omega,show (lа*2^64+dа)%2^64=dа by omega,apply_ite jI,kp,beq_iff_eq,
   <- Β]
@[inline] def aj (N:Д):Д:=(N+(0x3FFFFFFFFF:Д)+((N>>>39)&&&1))>>>39
lemma dL (n:ф) (e f:Е):
  (cd n e f)е=(if n then 2^31 else 0)+eе%2^8*2^23+fе%2^23:=by
 have d:=ϳ fе (ld 23);have:=ϳ eе (ld 8)
 have g:((e&&&0xff)<<<23 ||| (f&&&0x007fffff))е=2^23*(eе%2^8)+fе%2^23:=by
  rw[UInt32.toNat_or,UInt32.toNat_shiftLeft,Λ,Λ,show (0xff:Е)е=2^8-1 from rfl,
   show (0x007fffff:Е)е=2^23-1 from rfl,θ,
   θ,show (23:Е)е%32=23 from rfl,Μ,ϩ,ξ (by simp;omega),
   <- ϓ d]
 unfold cd;rw[UInt32.or_assoc];cases n<;>simp only[ο,↓reduceIte,UInt32.zero_or,UInt32.toNat_or,g]
 .omega
 .rw[show (0x80000000:Е)е=2^31*1 from rfl,<- ϓ А];omega
lemma bT (n:ф) (N g:Д) (l:2^62<= Nа) (m:Nа<2^63) (t:110<= gа)
  (j:gа<363):
  (aC n N g)е=(if n then 2^31 else 0)+((gа-110)*2^23+(aj N)а) /\
   2^23<=(aj N)а/\(aj N)а<=2^24:=by
 have o:(N>>>39)а=Nа/2^39:=by simp[ι]
 have e:((N>>>39)&&&1)а=Nа/2^39%2:=by simp[o,kF]
 have r:(aj N)а=Nа/2^39+if 2^38<Nа%2^39+Nа/2^39%2 then 1 else 0:=by
  unfold aj;simp (disch:=omega) only[Α,μ,e,iZ,
   ι,Nat.reducePow,ξ]
  generalize Nа=x;clear*-;split<;>omega
 have h:2^23<= Nа/2^39/\Nа/2^39<2^24:=by clear*-l m;omega
 have d:(aJ N (62-23))а=(aj N)а/\2^23<=(aj N)а/\(aj N)а<=2^24:=by
  have p:(N&&&0x7FFFFFFFFF)а=Nа%2^39:=by
   rw[ψ];exact θ _ 39
  rw[r];show (cq N 39)а=_/\_;unfold cq
  simp only[show ((1:Д)<<<39)-1=0x7FFFFFFFFF from rfl,show (1:Д)<<<(39-1)=0x4000000000 from
   rfl,gt_iff_lt,ν,μ,p,e,o,apply_ite jI,iZ]
  clear*-h;split_ifs<;>omega
 refine ⟨?_,d.2⟩;unfold aC
 simp only[show an N=62 by rw[aG,<- Β];exact (Nat.log2_eq_iff (by omega:N.toNat≠0)).2 ⟨l,m⟩,
  show (N==0)=false by simp[<- Β];omega,show ¬62+g<172 by simp[ν];omega,
  show ¬(62:Д)<23 by decide,ite_false,ο]
 obtain ⟨d,e⟩:=d;generalize aJ N (62-23)=R at d⊢;rw[<- d] at e⊢;clear*-e t j;by_cases q:Rа=16777216<;>simp[q,<- Β,bQ,ν]<;>split<;>(try omega)<;>simp[dL,UInt32.toNat_sub,UInt32.toNat_add]<;>omega
lemma cD (n:ф) (M σ:Д) (e:Mа≠0) (d:σа<=506):
  aC n M σ=kC ((if n then-(Mа:п) else Mа)*2^σа) (-298) false:=by
 rw[ax _ _ _ d];refine (kV _ _ _ _ (mul_ne_zero (by split<;>omega) В) ?_).symm
 cases n<;>simp[lX,jC,zpow_sub₀]<;>field_simp
end Core open Core
namespace Dense
lemma al {k n:с} (h:k<n):(k+1)б=kб+1:=by
 have:=γ.mp h;have:=n.toNat_lt_size;have:=Ϟ;rw[ks,lf,ξ А]
lemma ab {a b:с} (h:aб+bб<ke):(a+b)б=aб+bб:=by
 rw[ks,ξ h]
lemma aa {n:о} (h:n<ke):nгб=n:=by
 rw[ϛ,USize.toNat_ofNat_of_lt' h]
lemma aH (x:Е):x.toUSizeб=xе:=by simp[UInt32.toNat_toUSize]
lemma bt (A:у Е) (u:с) (h:uб<Aв):A.uget u h=A[uб]!:=by
 simp_all
lemma au (p:Д):
  bi p=(pа:п)-(if 2^63<= pа then 2^64 else 0):=by
 unfold bi;rw[kE,ll];split<;>split<;>push_cast<;>omega
@[inline] def aS (x:Е):Е:=x&&&0x7FFFFFFF
lemma aR (x:Е):(aS x)е=xе%2^31:=by
 unfold aS;rw[Λ];exact θ _ 31
lemma be (x:Е):(i7 x)е=(aS x)е/2^23:=by
 have:(i7 x)е=xе/2^23%2^8:=by
  unfold i7;rw[Λ,Ρ];simp only[jj,jb,ι];exact θ _ 8
 rw[this,aR,show 2^31=2^23*2^8 by norm_num,Nat.mod_mul_right_div_self]
lemma cM (x:Е):i7 x=0↔(aS x)е<2^23:=by
 rw[<- Η,be];simp;try omega
lemma gI (x:Е):je x=(aS x)е/2^23-1:=by
 have:=be x;unfold je;split<;>rename_i h
 .have:=(cM x).mp h;omega
 .have:=mt (cM x).mpr h;omega
lemma br (x:Е):je x=(i7 x)е-1:=by rw[gI,be]
lemma gT (w:Е):wе/2^23=(i7 w)е+256*(wе/2^31):=by
 rw[be,aR];have:=w.toNat_lt;omega
lemma bO (x:Е):(i7 x)е<256:=by rw[be,aR];omega
lemma gG (w:Е):(w&&&0x7FFFFF)е=wе%2^23:=by
 rw[Λ];exact θ _ 23
lemma bV (w:Е): |sv w| <2^24:=by
 have:=kw w;unfold sv;split<;>simp<;>omega
lemma ay {k:с} {a:у Е} (h:k<a.usize):kб<aв:=by
 have:=γ.mp h;rw[kk,ϛ,USize.toNat_ofNat'] at this;exact lt_of_lt_of_le this (lK _ _)
@[inline] def dy (h l m s:Д):Д×Д:=
 let l':=l+(m<<<s)
 (h+(m>>>1>>>(63-s))+(if l'<l then 1 else 0),l')
lemma cY (h l m s:Д) (j:sа<=60)
  (g:hа*2^64+lа+mа*2^sа<2^128):
  (dy h l m s).1а*2^64+(dy h l m s).2а=hа*2^64+lа+mа*2^sа:=by
 have d:(63-s)а=63-sа:=by
  rw[χ _ _ (by simp[κ];omega)];rfl
 have e:mа/2/2^(63-sа)=mа*2^sа/2^64:=by
  rw[Nat.div_div_eq_div_mul,show (2:о)^64=2^1*2^(63-sа)*2^sа by
   rw[<- pow_add,<- pow_add];congr 1;omega,pow_one,Nat.mul_div_mul_right _ _ (by positivity:0<2^s.toNat)]
 have:=h.toNat_lt;have:=l.toNat_lt
 simp only[dy,ν,μ,Ι,Α,d,
  Μ,ι,ξ (by omega:s.toNat<64),
  ξ (by omega:63-s.toNat<64),iZ,jb,pow_one,e]
 split<;>simp only[iZ]<;>omega
@[specialize] def go (fa fb:о-> т) (n t:о) (B:Д):Д:=
 if t<n then
  go fa fb n (t+1) (if Core.mW (fa t)*Core.mW (fb t)==0 then B else min B (Core.iW (fa t)+Core.iW (fb t)))
 else B
termination_by n-t
@[specialize] def bX (fa fb:о-> т) (n:о) (B:Д) (t:о) (pH pL nH nL:Д):Д:=
 if t<n then
  let a:=fa t
  let b:=fb t
  let m:=Core.mW a*Core.mW b
  let e:=Core.iW a+Core.iW b
  if expOf a==255 || expOf b==255 || m !=0&&(e<B || 60<e-B) then 0x100000000 else
  if m==0 then bX fa fb n B (t+1) pH pL nH nL else
  let g:=signOf a==signOf b
  let q:=dy (if g then pH else nH) (if g then pL else nL) m (e-B)
  bX fa fb n B (t+1) (if g then q.1 else pH) (if g then q.2 else pL) (if g then nH else q.1) (if g then nL else q.2)
 else Core.bm (pH-nH-(if pL<nL then 1 else 0)) (pL-nL) B
termination_by n-t
lemma iY (h e g d:Д) (l:hа*2^64+eа<2^126)
  (j:gа*2^64+dа<2^126):
  az (h-g-(if e<d then 1 else 0)) (e-d)=
   ((hа*2^64+eа:о):п)-((gа*2^64+dа:о):п):=by
 have:=e.toNat_lt;have:=d.toNat_lt;have:=h.toNat_lt;have:=g.toNat_lt;rw[a8];simp only[ν,Κ,apply_ite jI,kW,Ϯ];split<;>split<;>push_cast<;>omega
lemma cZ (d g:о-> i9) (n:о) (B:Д) (hn:n<2^18):
  ∀k t (v w r u:Д),n-t=k-> t<= n-> am t d g ->
   vа*2^64+wа<= t*2^108-> rа*2^64+uа<= t*2^108 ->
   ah t d g=(((vа*2^64+wа:о):п)-((rа*2^64+uа:о):п))*
    2^Bа ->
   bX d g n B t v w r u<0x100000000-> kB n d g=(bX d g n B t v w r u)л:=by
 intro k;induction k<;>intro t v w r u hk ht Z K J M j
 .obtain rfl:t=n:=А
  rw[bX,if_neg А] at j⊢;exact bl _ _ _ _ _ _ Z (by rw[M,iY _ _ _ _ А А]) j
 rename_i k Y;have W:t<n:=А;have y:n-(t+1)=k:=А;have z:t+1<= n:=А;rw[bX,if_pos W] at j⊢;dsimp only at j⊢
 split at j;. exact absurd j Б
 rename_i X;rw[if_neg X]
 simp only[lg,beq_iff_eq,υ,bne_iff_ne,ne_eq,decide_eq_true_eq,not_or,
  not_and,ν,iZ] at X
 obtain ⟨⟨G,I⟩,hw⟩:=X;have x:am (t+1) d g:=λs E=>(Nat.lt_succ_iff_lt_or_eq.mp E).elim (Z s) λe=>e ▸ ⟨G,I⟩
 have o:jF (d t)*jF (g t)<2^48:=
  (Nat.mul_lt_mul'' (kw _) (kw _)).trans_eq Г
 have S:(mW (d t)*mW (g t))а=jF (d t)*jF (g t):=by
  rw[Ϛ,aN,aN,ξ (o.trans Г)]
 have D:=aE t d g;have m:=Ζ (2^108) t.le_succ;split at j
 .rename_i F
  rw[if_pos F];have N:jF (d t)*jF (g t)=0:=by rw[<- S,beq_iff_eq.mp F];rfl
  refine Y (t+1) v w r u y z x (K.trans m) (J.trans m) ?_ j;rw[D,M,add_eq_left];unfold tm sv;rcases Nat.mul_eq_zero.mp N with h | h<;>simp[h]
 rename_i F;rw[if_neg F];obtain ⟨hB,O⟩:=hw λh=>F (by simp[h])
 have a0:=idxOf_le _ G;have a1:=idxOf_le _ I
 have Q:(iW (d t)+iW (g t))а=je (d t)+je (g t):=by
  rw[μ,bf,bf,ξ А]
 rw[Q] at hB
 have E:(iW (d t)+iW (g t)-B)а=je (d t)+je (g t)-Bа:=by
  rw[χ _ _ (by rw[κ,Q];omega),Q]
 rw[E] at O
 have V:(2:п)^(je (d t)+je (g t))=2^(je (d t)+je (g t)-Bа)*2^Bа:=by
  rw[<- pow_add];congr 1;omega
 generalize l:(jF (d t)*jF (g t))*2^(je (d t)+je (g t)-Bа)=T
 have p:T<=2^108:=
  l ▸ (Nat.mul_le_mul o.le (Ε Г (by omega:_<=60))).trans_eq Г
 have A:tm (d t) (g t)=(if jw (d t)==jw (g t) then (T:п) else-T)*2^Bа:=by
  rw[<- l];unfold tm sv;rw[V];cases jw (d t)<;>cases jw (g t)<;>simp<;>ring
 have U:=λH L (as:Hа*2^64+Lа<= t*2^108)=>cY H L _ _ (by rw[E];omega)
  (by rw[S,E,l];omega)
 rw[S,E,l] at U;cases hg:(jw (d t)==jw (g t))<;>simp only[hg,ite_true,ite_false,ο] at A j⊢
 .have q:=U r u J
  refine Y (t+1) v w _ _ y z x (K.trans m) (by rw[q];exact ϋ t _ ▸ Nat.add_le_add J p) ?_ j
  rw[D,M,q,A];push_cast;ring
 .have q:=U v w K
  refine Y (t+1) _ _ r u y z x (by rw[q];exact ϋ t _ ▸ Nat.add_le_add K p) (J.trans m) ?_ j
  rw[D,M,q,A];push_cast;ring
@[inline] def eW (n:о) (fa fb:о-> т):т:=
 let r:=if n<262144 then bX fa fb n (go fa fb n 0 1023) 0 0 0 0 0 else 0x100000000
 if r<0x100000000 then rл else dotOf n fa fb
lemma cJ (n:о) (d e:о-> i9):eW n d e=kB n d e:=by
 unfold eW;dsimp only;by_cases g:n<262144
 .rw[if_pos g];split
  .exact (cZ d e n _ А _ 0 0 0 0 0 rfl А (λ_ h=>absurd h А)
    (by simp) (by simp) (by simp[ah]) ‹_›).symm
  .rfl
 .rw[if_neg g,if_neg Б]
@[noinline] def ap (n:о) (relu:ф) (X W b:@& у т) (xo s j:о):т:=
 act relu (eW (n+1) (λt=>if t<n then X[xo+t*s]! else oneW)
  (λt=>if t<n then W[j*n+t]! else b[j]!))
namespace F
noncomputable def rv (v:р):ℝ:=Model.toReal (Model.ofFloatModel v.toModel)
def fv (v:р):Prop:=Model.isFinite (Model.ofFloatModel v.toModel)=true
lemma cf (v:р):rv v=jp v.toModel.unpack:=by
 unfold rv;rw[ό Б,Θ]
lemma bB (v:р):fv v↔v.toModel.unpack.isFinite=true:=by
 unfold fv;rw[<- Ϝ,ϧ _ Б,Θ];cases v.toModel.unpack<;>rfl
lemma dU (x y:р) (d:fv x) (e:fv y):
  i6 (x+y).toModel=lB (i6 x.toModel) (i6 y.toModel):=by
 change i6 (l4 x.toModel y.toModel)=_;rw[l4,Ϣ]
 have h:=Model.add_eq_ofModel_add_of_isFinite (fmt:=i5) Б
  (i6 x.toModel) (i6 y.toModel)
  (by rw[Θ];exact (bB x).1 d) (by rw[Θ];exact (bB y).1 e)
 rw[Θ,Θ] at h;exact h.symm
lemma pm (E:п) (e:E<=971):(2:ℝ)^E<= jJ (kG i5):=by
 have d:FloatFormat.binary64.maxNormalExponent-Int.ofNat FloatFormat.binary64.fracWidth=971:=Б
 have h:=Ϡ i5 false 1 E
  Б (by rw[d];exact e)
 simp only[ο,ite_false,lQ,one_mul] at h;change |(2:ℝ)^E| <= _ at h
 simpa only[abs_of_pos (zpow_pos (by norm_num:(0:ℝ)<2) E)] using h
open FloatLib.Floats.Formats.Flocq in
lemma re (x:ℝ) (E:п) (g:-1021<= E) (e: |x| <(2:ℝ)^E):
  |Model.roundAt i5 x-x| <= (2:ℝ)^(E-53)/2:=by
 by_cases d:x=0
 .subst x;rw[Model.roundAt_zero];simp only[sub_zero,abs_zero];positivity
 have h:magnitude jh x<= E:=magnitude_le_of_abs_lt_bpow _ x E d e;apply (Model.abs_roundAt_sub_le i5 x).trans;apply div_le_div_of_nonneg_right _ Г;change ulp jh (Model.fexpOf i5) x<=(2:ℝ)^(E-53);rw[ulp.of_ne_zero _ _ x d];change (2:ℝ)^(lm (magnitude jh x-53) (-1074))<=(2:ℝ)^(E-53);apply zpow_le_zpow_right₀ Г;omega
lemma db (x y:р) (d:fv x) (e:fv y) (j: |rv x+rv y| <= jJ (kG i5)):
  fv (x+y)/\rv (x+y)=kh i5 (rv x+rv y):=by
 have h:=dU x y d e
 have g:=Model.isFinite_add_of_abs_toReal_add_le_posMaxFinite (i6 x.toModel)
  (i6 y.toModel) Б d e j
 refine ⟨by unfold fv;rw[h];exact g,?_⟩
 unfold rv;rw[h];exact kx _ _ Б g
lemma gF (x y:р) (l:fv x) (m:fv y) (E:п) (j:-1021<= E) (d:E<=971)
  (e: |rv x+rv y| <(2:ℝ)^E):
  fv (x+y) /\ |rv (x+y)-(rv x+rv y)| <= (2:ℝ)^(E-53)/2:=by
 obtain ⟨g,h⟩:=db x y l m (e.le.trans (pm E d));exact ⟨g,by rw[h];exact re _ E j e⟩
lemma rx (d:jf) (l:d.significand<2^53) (g:-1074<= d.exponent)
  (e: |d.toReal| <= (2:ℝ)^(971:п)):
  kh i5 d.toReal=d.toReal:=by
 obtain ⟨h,j⟩:=jO i5 Б d d.significand 0
  (by simp) l g (e.trans (pm 971 le_rfl))
 rw[<- ϗ _ Б d h,j]
lemma gx (x y:р) (o:fv x) (p:fv y) (d:jf) (g:d.toReal=rv x+rv y)
  (m:d.significand<2^53) (l:-1074<= d.exponent) (e: |d.toReal| <= (2:ℝ)^(971:п)):
  fv (x+y)/\rv (x+y)=rv x+rv y:=by
 obtain ⟨h,j⟩:=db x y o p (by rw[<- g];exact e.trans (pm 971 le_rfl));exact ⟨h,by rw[j,<- g];exact rx d m l e⟩
lemma cy (i:Int64) (e:iнм<2^53):fv i.toFloat/\rv i.toFloat=iн:=by
 have g:i6 i.toFloat.toModel=
   i8 i5 (js iн 0):=
  ka i
 have d:(js iн 0).toReal=iн:=by
  rw[<- Ϙ,k8];simp
 obtain ⟨h,j⟩:=jO i5 Б
  (js iн 0) iнм 0 (ϱ _).symm
  e (by change _<=(0:ℤ);decide) (by
   rw[d,<- ϕ,kO];exact le_trans (by exact_mod_cast e.le) (pm 53 Г))
 exact ⟨by unfold fv;rw[g];exact h,by unfold rv;rw[g,j,d]⟩
lemma gO (x y:р):i6 (x*y).toModel=
  lH i5
   (jq jt x.toModel.unpack y.toModel.unpack):=by
 change i6 (l5 x.toModel y.toModel)=_;rw[l5,Ϣ]
lemma eV (b:BitVec 64) (s:lR) (m:о) (e:п) (h:0< m)
  (d:jc jt b=.finite s m e h):
  2^52<= m\/e=-1074:=by
 revert d;unfold jc;dsimp only;split
 .split<;>intro d<;>cases d
 .split
  .split
   .intro d;cases d
   .rename_i l _
    intro d;injection d with _ _ j;right;rw[<- j,l];decide
  .intro d;injection d with _ g _
   left;rw[<- g,BitVec.toNat_append];exact le_trans Б Nat.left_le_or
lemma d4 (x y:р) (w:fv x) (z:fv y) (v: |rv x*rv y| <= (2:ℝ)^(971:п)):
  fv (x*y)/\rv (x*y)=kh i5 (rv x*rv y):=by
 simp only[bB] at w z;rw[cf x,cf y] at v⊢;unfold fv rv;rw[gO];generalize A:x.toModel.unpack=X at w v⊢;generalize B:y.toModel.unpack=Y at z v⊢;cases X<;>cases Y<;>simp only[jn,ο] at w z
 case finite.finite p j e t q l g u=>
  have D:=eV _ p j e t A;have E:=eV _ q l g u B
  have o:e+g<= Float.Model.Format.binary64.targetExponent
    (jl (j*l) (e+g)):=by
   change e+g<= lm ((Nat.log2 (j*l):п)+1+(e+g)-53) (-1074);have:=(Ϧ (n:=j*l) (k:=52) В).2;have:=kX j u;have:=Nat.le_mul_of_pos_left l t;omega
  let r:jf:=⟨xor (lL p) (lL q),j*l,e+g⟩
  have h:lH i5 (jq jt
    (.finite p j e t) (.finite q l g u))=i8 i5 r:=by
   rw[jq,Model.roundWithAccuracy_exact_eq_round_of_le_targetExponent _ _ _ _ o];cases p<;>cases q<;>rfl
  have d:r.toReal=jp (.finite p j e t)*jp (.finite q l g u):=
   Model.Dyadic.toReal_mul ⟨_,j,e⟩ ⟨_,l,g⟩
  have m:=Υ i5 Б r
   (by rw[d];exact v.trans (pm 971 le_rfl))
  rw[h];exact ⟨m,by rw[ϗ _ Б r m,d]⟩
 all_goals simp[jq,Model.isFinite_ofModel_zero i5 Б,
  Model.toReal_ofModel_zero i5 Б]
lemma aw (s:ф) (m:о) (e:п):
  (⟨s,m,e⟩:jf).toReal=(if s then-(m:ℝ) else m)*(2:ℝ)^e:=by
 simp only[jz,kP];split<;>simp
def pw:о-> р
 | 0=>(1:Int64).toFloat
 | k+1=>let p:=pw k;p+p
lemma gh (k:о) (g:k<=900):fv (pw k)/\rv (pw k)=(2:ℝ)^k:=by
 induction k with
 | zero=>have:=cy 1 Б;simp only[pw];exact ⟨this.1,by rw[this.2];simp⟩
 | succ k e=>
  obtain ⟨f,v⟩:=e А
  have d:(⟨false,1,((k+1:о):п)⟩:jf).toReal=(2:ℝ)^(k+1):=by
   rw[aw];simp only[ο,ite_false,lQ,one_mul];exact zpow_natCast 2 (k+1)
  have:=gx (pw k) (pw k) f f _ (by rw[d,v,pow_succ];ring)
   (show (1:о)<2^53 by norm_num) (show (-1074:п)<=((k+1:о):п) by omega) (by
    rw[d,abs_of_pos В,<- zpow_natCast];exact zpow_le_zpow_right₀ Г А)
  simp only[pw];exact ⟨this.1,by rw[this.2,v,pow_succ];ring⟩
def nm:р:=(-1:Int64).toFloat
def P:FloatArray:=⟨Array.ofFn (n:=512) λi=>if iи<256 then pw (iи-1) else nm*pw (iи-256-1)⟩
lemma ag:Pд=512:=by simp[P,jD]
lemma e8 (i:с) (h:iб<Pд):
  P.uget i h=if iб<256 then pw (iб-1) else nm*pw (iб-256-1):=by
 simp[P,lh]
lemma cW (w:Е):(w>>>23)е=wе/2^23:=by
 rw[Ρ];simp[ι]
@[inline] def dw (w:Е):р:=
 (Core.mW w).toInt64.toFloat*P.uget (w>>>23).toUSize (by
  rw[ag,aH,cW];have:=w.toNat_lt;omega)
lemma bn (z e:п):(kN z e).toReal=(z:ℝ)*(2:ℝ)^e:=by
 rw[<- Ϙ,toRat_ofIntDyadic];push_cast;rfl
lemma cp (x y:р) (o:fv x) (p:fv y) (z e:п) (g:rv x*rv y=(z:ℝ)*(2:ℝ)^e)
  (m:zм<2^53) (l:-1074<= e) (d: |(z:ℝ)*(2:ℝ)^e| <= (2:ℝ)^(971:п)):
  fv (x*y)/\rv (x*y)=(z:ℝ)*(2:ℝ)^e:=by
 obtain ⟨h,j⟩:=d4 x y o p (by rw[g];exact d)
 have:=rx (kN z e) m l (by rw[bn];exact d)
 rw[bn] at this;exact ⟨h,by rw[j,g,this]⟩
lemma eT (w:Е):(sv w)м<2^24:=by
 have:=kw w;unfold sv;split<;>simp<;>omega
lemma ez (a b c:о) (e:a+b<= c) (h:c<=971) (z:п) (g:zм<2^a):
  |(z:ℝ)*(2:ℝ)^(b:п)| <= (2:ℝ)^(971:п):=by
 rw[abs_mul,abs_of_pos (by positivity:(0:ℝ)<2^(b:п)),zpow_natCast]
 have d: |(z:ℝ)| <= 2^a:=by
  rw[<- ϕ,kO];exact_mod_cast g.le
 refine (mul_le_mul_of_nonneg_right d В).trans ?_;rw[<- pow_add,<- zpow_natCast];exact zpow_le_zpow_right₀ Г А
lemma mV (u:Д) (n:о) (d:uа=n) (e:n<2^25):
  fv u.toInt64.toFloat/\rv u.toInt64.toFloat=(n:п):=by
 have:u.toInt64н=n:=by rw[show u.toInt64н=bi u from rfl,au,d];split<;>omega
 rw[<- this];exact cy _ (by rw[this];simp;omega)
lemma qV (c:о) (A:FloatArray) (o:∀i h,A.uget i h=if iб<256 then pw (iб-c) else nm*pw (iб-256-c))
  (w:Е) (h):fv (A.uget (w>>>23).toUSize h) /\ ∃σ:п,
  (∀v,jw v=jw w-> σ*jF v=sv v)/\σм=1 /\
   rv (A.uget (w>>>23).toUSize h)=σ*2^((i7 w)е-c):=by
 have:=bO w;have:=w.toNat_lt;have d:=aP w;obtain ⟨j,l⟩:=gh ((i7 w)е-c) А;rw[o,aH,cW,gT];split
 .rw[show (i7 w)е+256*(wе/2^31)-c=(i7 w)е-c by omega]
  exact ⟨j,1,λv e=>by simp only[sv,e,d];split<;>omega,rfl,by rw[l];push_cast;ring⟩
 .rw[show (i7 w)е+256*(wе/2^31)-256-c=(i7 w)е-c by omega]
  have g:=cy (-1) Б
  obtain ⟨m,p⟩:=cp _ _ g.1 j (-1) ((i7 w)е-c:ℕ) (by rw[g.2,l,zpow_natCast];rfl)
   Б А (ez 1 _ 971 А le_rfl _ Б)
  exact ⟨m,-1,λv e=>by simp only[sv,e,d];split<;>omega,rfl,by rw[<- zpow_natCast];exact p⟩
lemma dG (c q:о) (A:FloatArray) (o:∀i h,A.uget i h=if iб<256 then pw (iб-c) else nm*pw (iб-256-c))
  (u:Д) (v w:Е) (h) (r:q<=2) (x:uа=q*jF v)
  (t:jw v=jw w) (p:(q:ℝ)*2^((i7 w)е-c)=2^je v):
  fv (u.toInt64.toFloat*A.uget (w>>>23).toUSize h) /\
   rv (u.toInt64.toFloat*A.uget (w>>>23).toUSize h)=(sv v:ℝ)*2^je v:=by
 have:=kw v;have:=Ζ (jF v) r;have:=bO w;obtain ⟨g,j⟩:=mV u _ x А;obtain ⟨l,σ,d,m,y⟩:=qV c A o w h;have e:(((q*jF v:о):п)*σ)м<2^25:=by rw[Int.natAbs_mul,m];simp;omega
 have:=cp _ _ g l _ ((i7 w)е-c:о) (by rw[j,y,zpow_natCast];push_cast;ring)
  (e.trans Г) А (ez 25 _ 971 А le_rfl _ e)
 exact ⟨this.1,by rw[this.2,<- p,zpow_natCast,<- d v t];push_cast;ring⟩
lemma av (w:Е) (_:i7 w≠255):fv (dw w)/\rv (dw w)=(sv w:ℝ)*2^je w:=
 dG 1 1 P e8 _ w w _ А (by simp[aN]) rfl (by simp[br])
lemma ey (x w:Е) (j:i7 x≠255) (h:i7 w≠255):
  fv (dw x*dw w)/\rv (dw x*dw w)=(tm x w:ℝ):=by
 obtain ⟨e,l⟩:=av x j;obtain ⟨g,m⟩:=av w h
 have d:(sv x*sv w)м<2^48:=by
  rw[Int.natAbs_mul];exact (Nat.mul_lt_mul'' (eT x) (eT w)).trans_eq Г
 have:=cp _ _ e g (sv x*sv w) (je x+je w:о)
  (by rw[l,m,zpow_natCast,pow_add];push_cast;ring) (lt_trans d Г) А
  (ez 48 _ 971 (by rw[br,br];have:=bO x;have:=bO w;omega) le_rfl _ d)
 refine ⟨this.1,?_⟩;rw[this.2,tm,zpow_natCast];push_cast;ring
@[inline] def eF (b:Д):Д:=(b>>>52)&&&2047
@[inline] def mF (b:Д):Д:=(b&&&0xfffffffffffff) ||| 0x10000000000000
lemma gP (u:Д):(u&&&0x8000000000000000!=0)=(u>>>63!=0):=by
 apply ϻ.mpr;simp only[bne_iff_ne,ne_eq,<- Β];rw[ψ,Α];change ¬uа&&&2^63=0↔¬uа>>>63=0;rw[lc,ι,jZ];have d:uа/2^63<2:=by have h:=UInt64.toNat_lt u;omega
 rw[ξ d];by_cases e:uа/2^63=1<;>simp<;>omega
lemma dc (v:р):i6 v.toModel=Model.ofFields i5 (v.toBits>>>63!=0)
  (((v.toBits>>>52)&&&2047)а) ((v.toBits&&&0xfffffffffffff)а):=by
 rw[<- Model.ofFields_signBit_expField_fracField (i6 v.toModel)];congr 1;rw[<- gP];apply ϻ.mpr;simp only[Model.signBit,bne_iff_ne,ne_eq];rw[<- UInt64.toBitVec_inj];exact Iff.rfl
lemma fq (v:р):((v.toBits>>>52)&&&2047)а<2^11/\(v.toBits&&&0xfffffffffffff)а<2^52:=by
 simp only[ψ];exact ⟨lt_of_le_of_lt ϙ Б,lt_of_le_of_lt ϙ Б⟩
lemma aK (v:р) (e:fv v) (g:¬eF v.toBits=0):
  rv v=(⟨v.toBits>>>63!=0,(v.toBits&&&0xfffffffffffff)а+2^52,
   ((eF v.toBits)а:п)-1075⟩:jf).toReal:=by
 unfold eF at g⊢;have j:=(fq v).1;have l:=(fq v).2;have d:((v.toBits>>>52)&&&2047)а≠0:=λh=>g (Β.mp (by simpa using h))
 unfold fv at e;unfold rv;rw[dc] at e⊢;rw[jK,ϒ _ _ _ _ j l e,if_neg d]
 simp only[show FloatFormat.binary64.fracWidth=52 from rfl,show FloatFormat.binary64.exponentBias=1023 from rfl,
  ln]
 congr 2<;>(try simp only[kl,j0])<;>ring
lemma bp (z:Model lr):ofW (jH z)=z:=by
 have h:z.toNatBits<2^32:=Model.toNatBits_lt_two_pow z;simp only[ofW,jH,ls,UInt32.toNat_ofNat',ξ h];exact Model.ofNatBits_toNatBits z
lemma cs (w:i9) (d:i7 w≠255):jJ (ofW w)=(sv w:ℝ)*(2:ℝ)^((je w:п)-149):=by
 rw[ofW,jK,toDyadic_ofUInt32 w d];change (decodeD w).toReal=_;rw[<- Ϙ,decodeD_toRat];push_cast;rfl
lemma ev (w:i9) (j:i7 w≠255) (m:w&&&0x7FFFFFFF≠0):jJ (ofW w)≠0:=by
 rw[cs w j];refine mul_ne_zero ?_ (zpow_ne_zero _ Г)
 have l:jF w≠0:=by
  unfold jF;split
  .rename_i d
   have e:=(cM w).mp d;rw[aR] at e;have g:(aS w)е≠0:=λh=>m (Η.mp (h.trans UInt32.toNat_zero.symm))
   rw[aR] at g;unfold lW;rw[gG];omega
  .omega
 unfold sv;split<;>simp[l]
lemma eM (d g h:jf) (m:d.toReal<= g.toReal) (o:g.toReal<= h.toReal)
  (w:jH (i8 lr d)=jH (i8 lr h))
  (u:i7 (jH (i8 lr d))≠255) (x:jH (i8 lr d)&&&0x7FFFFFFF≠0)
  (v: |g.toReal| <= (2:ℝ)^(127:п)):
  jH (i8 lr g)=jH (i8 lr d):=by
 have e:=bp (i8 lr d);have j:jW (i8 lr d)=true:=by rw[<- e,ofW,kM];simpa using u
 have l:=Υ lr Б g (v.trans
  ((Ϡ lr false (2^23) 104 Б Б).trans' (by
   change _<= |(if false=true then (-1:ℝ) else 1)*((2^23:ℕ):ℝ)*(2:ℝ)^(104:ℤ)|;norm_num)))
 have t:i8 lr h=i8 lr d:=by rw[<- bp (i8 lr h),<- w,e]
 have r:=ϗ lr Б;have p:=jP lr m;have q:=jP lr o;rw[<- r d j,<- r g l] at p;rw[<- r g l,<- r h (by rw[t];exact j),t] at q
 rw[Model.eq_of_toReal_eq_of_nonzero (le_antisymm q p) (by rw[le_antisymm q p,<- e];exact ev _ u x)]
lemma iF (x:о) (e:x<2^53) (d:x%2^29≠2^28):
  (aj (lP (x*2^10)))а=(x+2^28)/2^29:=by
 simp only[aj,Α,μ,ψ,iZ,
  Ξ (by simp only[jT];omega:x*2^10<UInt64.size),jb,
  ι,kF]
 grind
@[inline,irreducible] def ai (S:р) (h:Д):Д:=
 let b:=S.toBits
 let f:=b&&&0xfffffffffffff
 let E:=eF b
 let d:=h+1075-E
 let g:=(1:Д)<<<d
 let u:=(f+0x10000000)&&&0x1FFFFFFF
 let r:=((b&&&0x7FFFFFFFFFFFFFFF)+0x10000000)>>>29
 if d<=26&&E-1195<253&&u-(g+1)<0x1FFFFFFF-(g+g)&&r&&&0x7FFFFF!=0 then
  ((b>>>63)<<<31)+r-0x255000000
 else 0x100000000
lemma rb (b:Д) (h:(((b&&&0x7FFFFFFFFFFFFFFF)+0x10000000)>>>29&&&0x7FFFFF!=0)=true):
  2^28<=(b&&&0xfffffffffffff)а/\(b&&&0xfffffffffffff)а+2^28<2^52:=by
 simp only[bne_iff_ne,ne_eq,<- Β,ψ,Α,μ,
  iZ,jb,ι] at h⊢
 rw[θ _ 63,θ _ 23] at h;rw[θ _ 52];omega
lemma fJ (h E f:Д) (p:hа<2^63) (m:Eа<2048) (o:fа<2^52)
  (d:h+1075-E<=26) (g:E-1195<253)
  (e:((f+0x10000000)&&&0x1FFFFFFF)-(((1:Д)<<<(h+1075-E))+1)<
   0x1FFFFFFF-(((1:Д)<<<(h+1075-E))+((1:Д)<<<(h+1075-E))))
  (l:2^28<= fа/\fа+2^28<2^52):
  Eа<= hа+1075/\hа+1075-Eа<=26/\1195<= Eа/\Eа<=1447 /\
   2^(hа+1075-Eа)+1<=(fа+2^28)%2^29 /\
   (fа+2^28)%2^29+2^(hа+1075-Eа)<2^29 /\
   2^(hа+1075-Eа)<= fа/\fа+2^(hа+1075-Eа)<2^52:=by
 simp only[κ,ν,Κ,μ,ψ,
  Ι,iZ,kW,ϭ] at d g e
 have j:(2^64-Eа+(hа+1075)%2^64)%2^64=hа+1075-Eа:=А;rw[j] at d e;have:=Ε (n:=2) Г d;rw[ξ (by omega:_<64),θ _ 29] at e
 omega
lemma fK (b:Д) (d:1195<=(eF b)а):
  (((b>>>63)<<<31)+(((b&&&0x7FFFFFFFFFFFFFFF)+0x10000000)>>>29)-0x255000000)а=
   bа/2^63*2^31+(((eF b)а-1085-110)*2^23+
    (2^23+((b&&&0xfffffffffffff)а+2^28)/2^29)):=by
 simp only[eF,Κ,μ,Ι,Α,
  ψ,iZ,jb,ι,Μ] at d⊢
 rw[θ _ 11] at d⊢;rw[θ _ 63,θ _ 52];have:=b.toNat_lt;omega
lemma ch (S:р) (hS:fv S) (h:Д) (hh:hа<2^63) (T:п)
  (x: |rv S-(T:ℝ)| <= (2:ℝ)^(hа:п)) (as: |(T:ℝ)| <= (2:ℝ)^(425:п))
  (F:ai S h<0x100000000):T≠0/\(ai S h)л=kC T (-298) false:=by
 unfold ai at F⊢;dsimp only at F⊢
 split;swap;. simp[*] at F
 rename_i u;simp only[υ,decide_eq_true_eq] at u;obtain ⟨⟨⟨X,Y⟩,Z⟩,c4⟩:=u
 have I:(if (S.toBits>>>63!=0)=true then 2^31 else 0)=S.toBitsа/2^63*2^31:=by
  simp only[ce,decide_eq_true_eq];have:=S.toBits.toNat_lt;split<;>omega
 obtain ⟨d1,d2,A,e2,u1,u2,f1,f2⟩:=fJ h (eF S.toBits) _ hh (fq S).1 (fq S).2 X Y Z (rb _ c4);have Q:=fK S.toBits A;have O:=aK S hS (λh0=>by rw[h0] at A;simp at A)
 rw[<- I] at Q
 have H:(eF S.toBits-1085)а=(eF S.toBits)а-1085:=by
  rw[Κ];simp only[iZ];omega
 generalize hW:kr _=W;generalize (S.toBits>>>63!=0)=y at*
 generalize (eF S.toBits)а=d at*
 generalize hG:2^(hа+1075-d)=G at*
 generalize (S.toBits&&&0xfffffffffffff)а=g at*
 have j:Wе=(if y then 2^31 else 0)+((d-1085-110)*2^23+(2^23+(g+2^28)/2^29)):=by
  rw[<- hW,ϟ,Q];split<;>omega
 have v:∀N,g+2^52<= N+G-> N<= g+2^52+G ->
   jH (i8 lr ⟨y,N*2^10,((d-1085:ℕ):ℤ)-298⟩)=W:=λN l r=>by
  have n:=Ξ (show N*2^10<jT by simp only[jT];omega)
  have w:=(bT y (lP (N*2^10)) (eF S.toBits-1085) А А А
   А).1
  rw[<- n,<- H,<- ax _ _ _ А];exact Η.mp (by rw[w,iF N А А,j];omega)
 have U:i7 W≠255:=λJ=>by
  have:=be W;simp only[J,aR,j,jj] at this;split at this<;>omega
 have V:W&&&0x7FFFFFFF≠0:=λJ=>by
  have:=aR W;simp only[aS,J,j,j5] at this;split at this<;>omega
 have o:=v (g+2^52-G) А А;have p:=v (g+2^52+G) А А;have D:G<g+2^52:=А;rw[aw] at O
 have hU:(2:ℝ)^(hа:п)=G*(2:ℝ)^((d:п)-1075):=by
  rw[<- hG,Nat.cast_pow,j0,<- zpow_natCast,<- zpow_add₀ two_ne_zero];congr 1;omega
 have B:∀m:ℕ,((m*2^10:ℕ):ℝ)*2^(((d-1085:ℕ):ℤ)-298)=
   m*2^((d:ℤ)-1075)*2^(-298:ℤ):=λm=>by
  rw[Nat.cast_mul,mul_assoc,mul_assoc,Nat.cast_pow,j0,<- zpow_natCast (2:ℝ) 10,<- zpow_add₀ two_ne_zero,
   <- zpow_add₀ two_ne_zero,show ((10:ℕ):ℤ)+(((d-1085:ℕ):ℤ)-298)=(d:ℤ)-1075+-298 by omega]
 have hu:(0:ℝ)<(2:ℝ)^((d:п)-1075):=В;have e:(0:ℝ)<(2:ℝ)^(-298:п):=В
 have t: |(kN T (-298)).toReal| <= (2:ℝ)^(127:п):=by
  rw[bn,abs_mul,abs_of_pos e,show (127:ℤ)=425+-298 by norm_num,zpow_add₀ two_ne_zero];gcongr
 rw[hU,O] at x;generalize g+2^52=M at x o p D;have z:((M-G:ℕ):ℝ)=M-G:=k9 D.le;have E:(G:ℝ)<M:=by exact_mod_cast D
 cases y<;>simp only[ο,ite_false,ite_true] at x<;>obtain ⟨K,L⟩:=abs_le.mp x<;>
  have q:T≠0:=by rintro rfl;push_cast at K L;nlinarith only[K,L,E,hu]
 all_goals refine ⟨q,?_⟩;simp only[kC,q,↓reduceIte]
 on_goal 2=>refine ((eM _ _ _ ?_ ?_ (p.trans o.symm) (p ▸ U) (p ▸ V) t).trans p).symm
 refine ((eM _ _ _ ?_ ?_ (o.trans p.symm) (o ▸ U) (o ▸ V) t).trans o).symm;all_goals simp only[bn,aw,ο,ite_false,ite_true,neg_mul,B,z];push_cast;nlinarith only[K,L,e]
@[specialize] def ac (p:о-> р) (n t:о) (a:р):р:=
 if t<n then ac p n (t+1) (a+p t) else a
termination_by n-t
structure A8 where
 a0:р
 a1:р
 a2:р
 a3:р
 a4:р
 a5:р
 a6:р
 a7:р
local instance:Inhabited A8:=⟨⟨0,0,0,0,0,0,0,0⟩⟩
@[inline] def A8.f (s:A8):о-> р
 | 0=>s.a0 | 1=>s.a1 | 2=>s.a2 | 3=>s.a3 | 4=>s.a4 | 5=>s.a5 | 6=>s.a6 | _=>s.a7
lemma uk {i k:с} {S:о} (d:kб<8) (h:iб+8<= S) (e:S<2^32):
  (i+k)б<S:=by
 have:=ς;rw[ab А];omega
lemma ug (X:FloatArray) (i:с) (h:iб<Xд):X.uget i h=X.data[iб]!:=by
 rcases X with ⟨d⟩;simp only[lh];rw[getElem!_pos d iб h];rfl
lemma gf (Y:FloatArray) (v:р):(Y.push v)д=Yд+1:=by
 rcases Y with ⟨d⟩;simp[FloatArray.push,jD]
lemma bD (Y:FloatArray) (i:о) (v:р):(Y.set! i v)д=Yд:=by
 rcases Y with ⟨d⟩;simp[kt,jD]
lemma sg (Y:FloatArray) (i q:о) (v:р) (e:q<Yд):
  (Y.set! i v).data[q]! =if i=q then v else Y.data[q]!:=by
 rcases Y with ⟨d⟩;change q<dв at e
 simp only[kt,l8,getElem!_pos d q e,getElem!_pos (d.setIfInBounds i v) q (by simpa using e),
  Array.getElem_setIfInBounds e]
lemma pg (Y:FloatArray) (v:р) (q:о) (e:q<Yд+1):
  (Y.push v).data[q]! =if q<Yд then Y.data[q]! else v:=by
 rcases Y with ⟨d⟩;change q<dв+1 at e;change (d.push v)[q]! =if q<dв then d[q]! else v;rw[getElem!_pos _ _ (by simp;omega),Array.getElem_push];split<;>simp[*,getElem!_pos]
@[specialize] def bk (f:о-> р) (N:о) (Y:FloatArray):FloatArray:=
 if Yд<N then bk f N (Y.push (f Yд)) else Y
termination_by N-Yд
decreasing_by rw[gf];omega
lemma bM (f:о-> р) (N:о):∀k (Y:FloatArray),N-Yд=k-> Yд<= N ->
  (bk f N Y)д=N/\(∀q<Yд,(bk f N Y).data[q]! =Y.data[q]!) /\
   ∀q,Yд<= q-> q<N->(bk f N Y).data[q]! =f q:=by
 intro k;induction k<;>intro Y r p
 .rw[bk,if_neg А];exact ⟨by omega,λq _=>rfl,λq h d=>by omega⟩
 rename_i o;rw[bk,if_pos А];have t:=gf Y (f Yд);obtain ⟨l,g,m⟩:=o (Y.push (f Yд)) А А;refine ⟨l,λq j=>by rw[g q А,pg _ _ _ А,if_pos j],λq h d=>?_⟩
 rcases eq_or_ne q Yд with rfl | e
 .rw[g _ А,pg _ _ _ А,if_neg (lt_irrefl _)]
 exact m q А d
lemma f6 (v:р) (h:(eF v.toBits)а<2047):fv v:=by
 unfold fv;rw[dc];have e:=(fq v).1;have g:=(fq v).2;rw[Model.isFinite_ofFields _ _ _ _ e g];have d:FloatFormat.binary64.expAllOnesNat=2047:=Б;unfold eF at h;simp only[show FloatFormat.binary64.encoding=.ieee from rfl,d,bne_iff_ne,ne_eq];omega
lemma bj (v:р) (g:fv v): |rv v| <(2:ℝ)^(((eF v.toBits)а:п)-1022):=by
 have d:∀(m k:ℕ) (e f:ℤ),m<2^k-> k+e<= f->(m:ℝ)*2^e<2^f:=λm k e f h j=>
  (mul_lt_mul_of_pos_right (by exact_mod_cast h:(m:ℝ)<2^k) (zpow_pos two_pos e)).trans_le
   (by rw[<- zpow_natCast,<- zpow_add₀ two_ne_zero];exact zpow_le_zpow_right₀ Г j)
 unfold rv eF;unfold fv at g;rw[dc] at g⊢;rw[jK,ϒ _ _ _ _ (fq v).1 (fq v).2 g];have:=(fq v).2;split_ifs with l<;>simp only[Model.Dyadic.abs_toReal]
 .simp;positivity
 .exact d _ 52 _ _ this (by rw[l];decide)
 .exact d _ 53 _ _ (by simp only[i5,ln];omega) (by simp[i5];omega)
lemma iL (x y:р) (l:fv x) (m:fv y) (r:fv (x+y)) (E:п) (e:-1074<= E) (g:E<=971)
  (h: |rv (x+y)| <(2:ℝ)^E): |rv x+rv y| <(2:ℝ)^E:=by
 have j:=dU x y l m;have q:=kx (i6 x.toModel) (i6 y.toModel) Б (j ▸ r);unfold rv at h⊢;rw[j,q,abs_lt] at h;have p:=rx ⟨false,1,E⟩ Г e (by simp[aw];exact zpow_le_zpow_right₀ Г g)
 have o:=rx ⟨true,1,E⟩ Г e (by simp[aw];exact zpow_le_zpow_right₀ Г g)
 simp only[aw,ο,ite_false,ite_true,lQ,one_mul,neg_mul] at p o
 exact abs_lt.2 ⟨lt_of_not_ge λd=>by linarith[jP i5 d],
  lt_of_not_ge λd=>by linarith[jP i5 d]⟩
@[specialize] def sT (p:о-> р) (n t:о) (a:р) (m:Д):р×Д:=
 if t<n then
  let a':=a+p t
  let e:=eF a'.toBits
  sT p n (t+1) a' (if m<e then e else m)
 else (a,m)
termination_by n-t
lemma aA (m e:Д):(if m<e then e else m)а=lm mа eа:=by
 split<;>rename_i h<;>simp only[ν] at h<;>omega
lemma fn (p:о-> р) (n:о):∀k t a m,n-t=k-> mа<=(sT p n t a m).2а:=by
 intro k;induction k with
 | zero=>intro t a m h;rw[sT,if_neg А]
 | succ k d=>
  intro t a m h;rw[sT,if_pos А];dsimp only;have:=d (t+1) (a+p t) (if m<eF (a+p t).toBits then eF (a+p t).toBits else m) А;rw[aA] at this;omega
lemma eK (p:о-> р) (z:о-> п) (q:п) (n K:о) (u:1<= K) (v:K<=1993)
  (x:∀t<n,fv (p t)/\rv (p t)=z t):
  ∀m t a M,n-t=m-> t<= n-> fv a ->
   |rv a-((q:ℝ)+∑s∈i0 t,(z s:ℝ))| <= t*(2:ℝ)^((K:п)-1076) ->
   (sT p n t a M).2а<= K ->
   fv (sT p n t a M).1 /\
    |rv (sT p n t a M).1-((q:ℝ)+∑s∈i0 n,(z s:ℝ))| <= n*(2:ℝ)^((K:п)-1076):=by
 intro m;induction m<;>intro t a M B D h g j
 .obtain rfl:t=n:=А
  rw[sT,if_neg (lt_irrefl _)];exact ⟨h,g⟩
 rename_i y;have d:t<n:=А;rw[sT,if_pos d] at j⊢;dsimp only at j⊢;obtain ⟨l,A⟩:=x t d;have w:=fn p n _ (t+1) (a+p t) (if M<eF (a+p t).toBits then eF (a+p t).toBits else M) rfl;rw[aA] at w;have o:=f6 (a+p t) А
 obtain ⟨r,e⟩:=gF a (p t) h l _ А А (iL a (p t) h l o ((K:п)-1022) А А
  ((bj _ o).trans_le (zpow_le_zpow_right₀ Г А)))
 refine y (t+1) (a+p t) _ А А r ?_ j;rw[show (K:п)-1022-53=((K:п)-1076)+1 by ring,zpow_add_one₀ two_ne_zero,mul_div_cancel_right₀ _ two_ne_zero] at e
 push_cast[ϝ,abs_le] at e g⊢;constructor<;>linarith only[e.1,e.2,g.1,g.2,A]
lemma cw (n:о) (d e:о-> i9) (g:am n d e) (S:р) (m:fv S) (h:Д)
  (q:hа<2^63) (o: |rv S-(ah n d e:ℝ)| <= (2:ℝ)^(hа:п))
  (l: |(ah n d e:ℝ)| <= (2:ℝ)^(425:п)) (p:ai S h<0x100000000):
  (ai S h)л=kB n d e:=by
 obtain ⟨j,r⟩:=ch S m h q _ o l p;rw[r,a9 n d e g j]
def aI (A:у т) (i:с) (m:Е):Е:=
 if h:i<A.usize then
  let e:=(A.uget i (ay h)>>>23)&&&255
  aI A (i+1) (if m<e then e else m)
 else m
termination_by Aв-iб
decreasing_by have:=ay h;rw[al h];omega
lemma bN (A:у i9) (l:Aв<ke):∀k (i:с) (m:Е),Aв-iб=k ->
  mе<=(aI A i m)е /\ ∀q,iб<= q-> q<Aв->(i7 A[q]!)е<=(aI A i m)е:=by
 intro k;induction k<;>intro i m u
 .have h:¬i<A.usize:=λh=>by have:=ay h;omega
  rw[aI,dif_neg h];exact ⟨le_rfl,λq g d=>by omega⟩
 rename_i t;have h:i<A.usize:=by rw[γ,kk,aa l];omega
 rw[aI,dif_pos h];dsimp only
 have o:∀e:Е,(if m<e then e else m)е=lm mе eе:=λe=>by
  split<;>rename_i j<;>simp only[ϖ] at j<;>omega
 obtain ⟨p,r⟩:=t (i+1) (if m<(A.uget i (ay h)>>>23)&&&255 then (A.uget i (ay h)>>>23)&&&255 else m)
  (by rw[al h];omega)
 rw[o] at p;refine ⟨by omega,λq g d=>?_⟩
 rcases eq_or_ne q iб with rfl | e
 .rw[<- bt A i (ay h)];unfold i7;omega
 exact r q (by rw[al h];omega) d
lemma da (A:у i9) (j:Aв<ke) (h:aI A 0 0<255) (q:о):
  i7 A[q]!≠255/\je A[q]! <= (aI A 0 0)е-1:=by
 have g:=ϖ.mp h
 have d:(i7 A[q]!)е<=(aI A 0 0)е:=by
  by_cases d:q<Aв
  .exact (bN A j _ 0 0 rfl).2 q (by simp) d
  .rw[getElem!_neg A q d];exact km _
 refine ⟨λe=>by rw[e] at d;simp at d g;omega,by rw[br];omega⟩
@[inline] def s8 (Y:FloatArray) (o:с) (e0 e1 e2 e3 e4 e5 e6 e7:р):FloatArray:=
 (((((((Y.set! oб e0).set! (o+1)б e1).set! (o+2)б e2).set! (o+3)б e3).set! (o+4)б e4).set!
  (o+5)б e5).set! (o+6)б e6).set! (o+7)б e7
lemma dQ (Y:FloatArray) (o:с) (e:о-> р) (h:oб+8<ke):
  (s8 Y o (e 0) (e 1) (e 2) (e 3) (e 4) (e 5) (e 6) (e 7))д=Yд /\ ∀q<Yд,
   (s8 Y o (e 0) (e 1) (e 2) (e 3) (e 4) (e 5) (e 6) (e 7)).data[q]! =if oб<= q/\q<oб+8 then e (q-oб) else Y.data[q]!:=by
 refine ⟨by simp only[s8,bD],λq d=>?_⟩
 simp (disch:=(try simp only[bD,USize.reduceToNat]);omega) only[s8,sg,ab,jd]
 split_ifs<;>first | rfl | omega | (congr 1;omega)
lemma gD (X:у i9) (k:о):X.getD k 0=X[k]!:=by
 unfold Array.getD;split<;>rename_i h
 .rw[getElem!_pos X k h];rfl
 .rw[getElem!_neg X k h];rfl
@[inline] def xr (X:у т) (q st:с) (r:о):р:=dw (X.getD (q+rг*st)б 0)
def bx (X:у т) (st s:с):о-> с-> с-> FloatArray-> FloatArray
 | 0,_,_,Y=>Y
 | k+1,q,o,Y=>bx X st s k (q+s) (o+8) (s8 Y o (xr X q st 0) (xr X q st 1) (xr X q st 2) (xr X q st 3)
   (xr X q st 4) (xr X q st 5) (xr X q st 6) (xr X q st 7))
lemma dD (X:у i9) (d s:с):∀k (q o:с) (Y:FloatArray),
  qб+k*sб+8*dб+oб+8*k<ke-> oб+8*k<= Yд ->
  (bx X d s k q o Y)д=Yд/\(∀p<oб,(bx X d s k q o Y).data[p]! =Y.data[p]!) /\
   ∀τ r,τ<k-> r<8->(bx X d s k q o Y).data[oб+8*τ+r]! =dw X[qб+τ*sб+r*dб]!:=by
 intro k;induction k with
 | zero=>intro q o Y _ _;exact ⟨rfl,λ_ _=>rfl,λτ r h=>absurd h А⟩
 | succ k y=>
  intro q o Y v A;rw[ϋ] at v;have:=ς;have:=Ϟ;obtain ⟨l,t⟩:=dQ Y o (xr X q d) А;have u:(q+s)б=qб+sб:=ab А;have g:(o+8)б=oб+8:=by rw[ab (by simp;omega)];simp
  obtain ⟨w,e,x⟩:=y (q+s) (o+8) _ А (by rw[l];omega)
  rw[g] at e;rw[u,g] at x;refine ⟨by rw[bx,w,l],λp B=>by rw[bx,e p А,t p А,if_neg А],λτ r z j=>?_⟩
  rw[bx];rcases τ with _ | τ
  .have m:(rг*d)б=r*dб:=by
    rw[li,aa А,ξ (by have:=Ζ dб j.le;omega)]
   rw[e _ А,t _ А,if_pos А,show oб+8*0+r-oб=r by omega,xr,
    ab (by rw[m];have:=Ζ dб j.le;omega),m,gD];simp
  .grind
def ZT:у FloatArray:=Array.ofFn (n:=10) λk=>⟨Array.replicate (2^(kи+7)) 0⟩
def zT (n:о):FloatArray:=
 let k:=Nat.log2 (8*n)+1-7
 if k<10 then ZT[k]! else ⟨Array.replicate (8*n) 0⟩
lemma gu (n:о):8*n<=(zT n)д/\((zT n)д=8*n\/(zT n)д<=2^16):=by
 unfold zT;dsimp only;split
 .rw[getElem!_pos ZT _ (by simp[ZT];omega)]
  simp only[ZT,Ϩ,jD,ϴ]
  exact ⟨(Ϸ (n:=8*n)).le.trans (Ε Б А),
   .inr (Ε Б А)⟩
 .simp[jD]
@[inline] def rX (X:у т) (i:с):р:=if h:i<X.usize then dw (X.uget i (ay h)) else dw 0
lemma de {Y:FloatArray} {i:с} (h:i<Y.usize):iб<Yд:=by
 have:=γ.mp h;simp only[lY,ϛ,USize.toNat_ofNat] at this;exact lt_of_lt_of_le this (lK _ _)
@[inline] def wF (Y:FloatArray) (i:с) (v:р):FloatArray:=if h:i<Y.usize then Y.uset i v (de h) else Y
@[inline] def cE (F a:р):Д:=let e:=eF (F*a).toBits;if 1078<e then e-1078 else 0
@[inline] def tS (relu:ф) (L n:о) (p:о-> р) (bpf:Unit-> р) (fbk:Unit-> т):т:=
 let r:=sT p n 0 (bpf ()) 0
 let c:=if 1<= r.2а/\r.2а<=1993/\1076<= r.2а+L then
  ai r.1 (r.2а+L-1076)ж else 0x100000000
 if c<0x100000000 then act relu cл else fbk ()
@[inline] def tf (relu:ф) (S:р) (hh:Д) (sl:Unit-> т):т:=
 let c:=ai S hh
 if c<0x100000000 then act relu cл else sl ()
lemma gg (X:FloatArray) (i:о):X.get! i=X.data[i]!:=by rcases X with ⟨d⟩;rfl
lemma gk (z:ℝ): |Model.roundAt i5 z-z| <= (2:ℝ)^(-53:п)* |z| +2^(-1075:п):=by
 by_cases h: |z| <(2:ℝ)^(-1021:п)
 .refine (re z (-1021) le_rfl h).trans ?_
  rw[div_eq_mul_inv,<- zpow_sub_one₀ two_ne_zero];norm_num
 .have j:0< |z| :=lt_of_lt_of_le В (not_lt.1 h)
  have d:=lv (b:=2) Г j;have e:=kK (b:=2) Г |z|
  simp only[j0] at d e
  have g:(-1021:п)<= ky 2 |z| +1:=by
   by_contra l;linarith[zpow_le_zpow_right₀ (by norm_num:(1:ℝ)<=2) (by omega:Int.log 2 |z| +1<=-1021)]
  refine (re z _ g e).trans ?_
  rw[div_eq_mul_inv,<- zpow_sub_one₀ two_ne_zero,show ky 2 |z| +1-53-1=-53+ky 2 |z| by ring,
   zpow_add₀ two_ne_zero]
  linarith[mul_le_mul_of_nonneg_left d (by positivity:(0:ℝ)<=2^(-53:п)),
   show (0:ℝ)<=2^(-1075:п) by positivity]
lemma iN (x y:р) (g:fv x) (h:fv y) (d: |rv x+rv y| <= (2:ℝ)^(970:п)):
  fv (x+y) /\ |rv (x+y)-(rv x+rv y)| <= (2:ℝ)^(-53:п)* |rv x+rv y| +2^(-1075:п):=by
 obtain ⟨f,e⟩:=db x y g h (d.trans ((zpow_le_zpow_right₀ Г Г).trans (pm 971 le_rfl)));exact ⟨f,by rw[e];exact gk _⟩
lemma iO (x y:р) (g:fv x) (h:fv y) (d: |rv x*rv y| <= (2:ℝ)^(971:п)):
  fv (x*y) /\ |rv (x*y)-rv x*rv y| <= (2:ℝ)^(-53:п)* |rv x*rv y| +2^(-1075:п):=by
 obtain ⟨f,e⟩:=d4 x y g h d;exact ⟨f,by rw[e];exact gk _⟩
lemma fe (v:р) (d:fv v) (e:eF v.toBits=0):
  rv v=(⟨v.toBits>>>63!=0,(v.toBits&&&0xfffffffffffff)а,-1074⟩:jf).toReal:=by
 unfold eF at e;unfold fv at d;unfold rv;rw[dc] at d⊢;rw[jK,ϒ _ _ _ _ (fq v).1 (fq v).2 d,ite_eq_left (by rw[e];rfl)]
 split_ifs with g
 .dsimp only;rw[aw,aw,g];simp
 .rfl
lemma bu (v:р) (e:fv v):
  |rv v| <(2:ℝ)^(((eF v.toBits)а:п)-1022):=by
 have g:=(fq v).2;by_cases d:eF v.toBits=0
 .rw[fe v e d,aw,show (eF v.toBits)а=0 by rw[d];rfl,
   show (2:ℝ)^(((0:о):п)-1022)=2^52*2^(-1074:п) by
    rw[show (2:ℝ)^52=2^(52:п) by norm_num,<- zpow_add₀ two_ne_zero];norm_num]
  have:((v.toBits&&&0xfffffffffffff)а:ℝ)<2^52:=by exact_mod_cast g
  split<;>simp only[neg_mul,abs_neg,abs_mul,abs_zpow,abs_two,lS]<;>
   exact mul_lt_mul_of_pos_right this В
 .rw[aK v e d,aw,show (2:ℝ)^(((eF v.toBits)а:п)-1022)=
   2^53*2^(((eF v.toBits)а:п)-1075) by
    rw[show (2:ℝ)^53=2^(53:п) by norm_num,<- zpow_add₀ two_ne_zero];congr 1;ring]
  have:(((v.toBits&&&0xfffffffffffff)а+2^52:о):ℝ)<2^53:=by
   exact_mod_cast (by omega:(v.toBits&&&0xfffffffffffff).toNat+2^52<2^53)
  split<;>simp only[neg_mul,abs_neg,abs_mul,abs_zpow,abs_two,lS]<;>
   exact mul_lt_mul_of_pos_right this В
lemma gm (p:о-> р) (z:о-> ℝ) (d:ℝ) (n:о) (I:∀t<n,fv (p t)/\rv (p t)=z t)
  (U:n<=2^32) (F: |d| +∑s∈i0 n,|z s| <= (2:ℝ)^(968:п)):
  ∀m t a,n-t=m-> t<= n-> fv a ->
   |rv a-(d+∑s∈i0 t,z s)| <=
    t*((2:ℝ)^(-53:п)*(1+2^(-20:п))*(|d| +∑s∈i0 t,|z s|)+2^(-1074:п)) ->
   fv (ac p n t a) /\ |rv (ac p n t a)-(d+∑s∈i0 n,z s)| <=
    n*((2:ℝ)^(-53:п)*(1+2^(-20:п))*(|d| +∑s∈i0 n,|z s|)+2^(-1074:п)):=by
 have j:(2:ℝ)^(970:п)=2^(968:п)*4:=by
  rw[show (970:п)=968+2 by norm_num,zpow_add₀ two_ne_zero];norm_num
 have K:(2:ℝ)^(-1075:п)=2^(-1074:п)/2:=by
  rw[show (-1074:п)=-1075+1 by norm_num,zpow_add_one₀ two_ne_zero];ring
 have r:(4294967296:ℝ)*2^(-1074:п)<=2^(968:п):=by
  rw[show (4294967296:ℝ)=2^(32:п) by norm_num,<- zpow_add₀ two_ne_zero]
  exact zpow_le_zpow_right₀ Г Г
 have l:(0:ℝ)<2^(-1074:п):=В;intro m;induction m with
 | zero=>
  intro t a G J o h;obtain rfl:t=n:=А;rw[ac,if_neg (lt_irrefl _)];exact ⟨o,h⟩
 | succ m L=>
  intro t a G J o h;have g:t<n:=А;rw[ac,if_pos g];obtain ⟨D,y⟩:=I t g;have v:=ϸ (Finset.range_mono g) (λi _ _=>abs_nonneg (z i))
  have w:=(abs_add_le d _).trans (add_le_add le_rfl (Ϲ z (i0 (t+1))));simp only[ϝ] at v w;have x:=abs_add_le (rv a-(d+∑s∈i0 t,z s)) (d+(∑s∈i0 t,z s+z t))
  rw[show rv a-(d+∑s∈i0 t,z s)+(d+(∑s∈i0 t,z s+z t))=rv a+rv (p t) by
   rw[y];ring] at x
  have e:(t:ℝ)<=4294967296:=by exact_mod_cast (by omega:t<=2^32)
  have M:=mul_le_mul_of_nonneg_right e (abs_nonneg d);have N:=mul_le_mul_of_nonneg_right e (ϡ λi (_:i∈i0 t)=>abs_nonneg (z i))
  have V:=mul_le_mul_of_nonneg_right e (abs_nonneg (z t));have O:=mul_le_mul_of_nonneg_right e l.le;have q:0<= ∑s∈i0 t,|z s| :=ϡ λi _=>abs_nonneg (z i)
  have W:=mul_nonneg (lj t:(0:ℝ)<= t) (abs_nonneg (z t));have Q:=abs_nonneg d;have S:=abs_nonneg (z t)
  have E: |rv a-(d+∑s∈i0 t,z s)| <= 1/9007199254740992*(1+1/1048576)*(t* |d|)+
    1/9007199254740992*(1+1/1048576)*(t*∑s∈i0 t,|z s|)+t*2^(-1074:п):=h.trans_eq (by ring)
  obtain ⟨B,A⟩:=iN a (p t) o D (by
   rw[j];generalize (2:ℝ)^(968:п)=H at*;generalize (2:ℝ)^(-1074:п)=η at*
   linarith only[x,E,w,v,F,M,N,O,r,q,Q,S,l])
  refine L (t+1) (a+p t) А А B ?_;rw[K,show (2:ℝ)^(-53:п)=1/9007199254740992 by norm_num] at A
  have u:=abs_add_le (rv (a+p t)-(rv a+rv (p t))) (rv a-(d+∑s∈i0 t,z s))
  rw[show rv (a+p t)-(rv a+rv (p t))+(rv a-(d+∑s∈i0 t,z s))=
   rv (a+p t)-(d+(∑s∈i0 t,z s+z t)) by rw[y];ring] at u
  simp only[ϝ];generalize (2:ℝ)^(-1074:п)=η at*
  calc
   _<=1/9007199254740992*(1+1/1048576)*(t* |d| +t*∑s∈i0 t,|z s| +t* |z t| +
     (|d| +(∑s∈i0 t,|z s| + |z t|)))+t*η+η:=by linarith
   _=_:=by push_cast;ring
lemma fx (m:Float.Model):(kY m.unpack.abs).unpack=m.unpack.abs:=by
 have d:(kY m.unpack).unpack=m.unpack:=by
  have:=Model.toFloatModel_ofFloatModel m;conv_rhs=>rw[<- this]
  rfl
 generalize m.unpack=u at d⊢;simp only[kY,Float.Model.unpack,UInt64.toBitVec_ofBitVec] at d⊢;cases u with
 | notANumber=>exact d
 | finite s h e g=>
  simp only[jm,jA,jG] at d⊢;split_ifs at d⊢<;>
  simp only[jc,ύ,
   ώ,kb] at d⊢<;>
  split_ifs at d⊢<;>simp_all
 | _=>
  simp only[jm,jA,jG,
   lD,jc,
   ύ,ώ,
   kb] at d⊢
  split_ifs at d⊢<;>simp_all
lemma gE (v:р) (d:fv v):fv v.abs/\rv v.abs= |rv v| :=by
 have e:v.abs.toModel.unpack=v.toModel.unpack.abs:=fx v.toModel;rw[bB] at d⊢;rw[cf,cf,e];generalize v.toModel.unpack=u at d⊢;cases u with
 | notANumber | infinity _=>simp[jn] at d
 | zero _=>simp[jm,jn,jp,jU]
 | finite s j e h=>
  refine ⟨by simp[jm,jn],?_⟩
  simp only[jm,Model.unpackedToReal_finite,abs_mul];have g:=lx jh e;split<;>split<;>simp_all[abs_of_pos g]
lemma gX (S:р) (l:fv S) (h:Д) (B:hа<2^63) (T:ℝ)
  (A: |rv S-T| <= (2:ℝ)^(hа:п)) (d:ai S h<0x100000000): |T| <= (2:ℝ)^(425:п):=by
 unfold ai at d;dsimp only at d;split at d
 swap;. exact absurd d Б
 rename_i g;simp only[υ,decide_eq_true_eq] at g;obtain ⟨⟨⟨q,r⟩,t⟩,u⟩:=g;obtain ⟨v,w,j,x,-,-,-,m⟩:=fJ h (eF S.toBits) (S.toBits&&&0xfffffffffffff) B (fq S).1 (fq S).2 q r t (rb _ u);have e:=aK S l (λy=>by rw[y] at j;simp at j)
 rw[aw] at e;generalize (eF S.toBits)а=E at v w j x m e;generalize (S.toBits&&&0xfffffffffffff)а=f at m e;have p:(0:ℝ)<2^((E:п)-1075):=В
 have z:((f+2^52:о):ℝ)+((2^(hа+1075-E):о):ℝ)<=2^53:=by
  exact_mod_cast (by omega:f+2^52+2^(h.toNat+1075-E)<=2^53)
 have l: |rv S| =((f+2^52:о):ℝ)*2^((E:п)-1075):=by
  rw[e,abs_mul,abs_of_pos p];split<;>simp only[abs_neg,lS]
 have:=abs_sub (rv S) (rv S-T);rw[sub_sub_cancel,l] at this
 rw[show (2:ℝ)^(hа:п)=((2^(hа+1075-E):о):ℝ)*2^((E:п)-1075) by
  push_cast;rw[<- zpow_natCast,<- zpow_add₀ two_ne_zero];congr 1;omega] at A
 have o:(2:ℝ)^(53:о)*2^((E:п)-1075)<=2^(425:п):=by
  rw[<- zpow_natCast,<- zpow_add₀ two_ne_zero];exact zpow_le_zpow_right₀ Г А
 linarith[mul_le_mul_of_nonneg_right z p.le]
lemma fO (n:о) (e g:о-> i9) (d:am n e g) (S:р) (j:fv S) (h:Д)
  (o:hа<2^63) (l: |rv S-(ah n e g:ℝ)| <= (2:ℝ)^(hа:п))
  (m:ai S h<0x100000000):(ai S h)л=kB n e g:=
 cw n e g d S j h o l (gX S j h o _ l m) m
lemma dx (Y:FloatArray) (g:Yд<2^32) (i:с) (v:р):wF Y i v=Y.set! iб v:=by
 rcases Y with ⟨d⟩;unfold wF;split
 next h=>simp only[l6,kt,l9,l8,lE];rw[dite_eq_left (show iб<dв from de h)]
 next h=>
  simp only[kt,l8,lE]
  rw[dite_eq_right λe=>h (by
   rw[γ,lY,aa (lt_of_lt_of_le g ς)];exact e)]
lemma dr (X:у i9) (e:Xв<2^32) (i:с):rX X i=dw X[iб]!:=by
 unfold rX;split
 .rw[bt]
 next h=>
  rw[getElem!_neg X _ λd=>h (by
   rw[γ,kk,aa (lt_of_lt_of_le e ς)];exact d)];rfl
lemma b7 (p:о-> р) (n:о):∀m t a,n-t=m ->
  ac p (n+1) (t+1) a=ac (λτ=>p (τ+1)) n t a:=by
 intro m;induction m with
 | zero=>intro t a h;unfold ac;rw[if_neg А,if_neg А]
 | succ m d=>intro t a h;unfold ac;rw[if_pos А,if_pos А];exact d (t+1) _ А
lemma gW (n:ℕ) (σ j E A C H M:ℝ) (u:0<= σ) (d:0<A) (m:1<= C) (r:(n:ℝ)<=2^(32:п))
  (o:E<= n*((2:ℝ)^(-53:п)*(1+2^(-20:п))*(|j| +A*σ)+2^(-1074:п))+2^(26:п)*σ)
  (g:(2:ℝ)^(26:п)<= A*2^(-53:п))
  (h:9*((n:ℝ)+1)*σ*(1-2^(-20:п))<= C*2^(56:п)+2^(-900:п))
  (t:(n:ℝ)* |j| <= H*2^(49:п))
  (p:A*C<= M) (q:H<= M) (e:(2:ℝ)^(79:п)<= M):E<= M:=by
 have v:(n:ℝ)*2^(-1074:п)<=1:=(mul_le_mul_of_nonneg_right r В).trans
  (by rw[<- zpow_add₀ two_ne_zero];exact zpow_le_one_of_nonpos₀ Г Г)
 have l:(2:ℝ)^(-900:п)<=1:=zpow_le_one_of_nonpos₀ Г Г;generalize (2:ℝ)^(-1074:п)=η at*
 generalize (2:ℝ)^(-900:п)=ε at*
 norm_num at o g h t e
 linarith[mul_le_mul_of_nonneg_right g u,mul_le_mul_of_nonneg_left h d.le,mul_le_mul_of_nonneg_left l d.le,
  mul_le_mul_of_nonneg_left m d.le]
lemma dE (n:о) (H:n<2^29) (a:р) (σ:ℝ) (D:fv a) (u:0<= σ) (J:σ<=2^(400:п))
  (d: |rv a-σ| <= n*((2:ℝ)^(-53:п)*(1+2^(-20:п))*σ+2^(-1074:п))):
  9*((n:ℝ)+1)*σ*(1-2^(-20:п)) <=
   2^(((cE (9*(n+1))ж.toInt64.toFloat a)а:п)+56)+2^(-900:п):=by
 have e:(9*(n+1))ж.toInt64н=((9*(n+1):о):п):=by
  rw[show (9*(n+1))ж.toInt64н=bi (9*(n+1))ж from rfl,au,
   jL,Ξ (by simp only[jT];omega)]
  split<;>omega
 obtain ⟨w,h⟩:=cy (9*(n+1))ж.toInt64 (by rw[e];simp;omega)
 rw[e] at h;push_cast at h
 have r:(2:ℝ)^(((eF ((9*(n+1))ж.toInt64.toFloat*a).toBits)а:п)-1022) <=
   2^(((cE (9*(n+1))ж.toInt64.toFloat a)а:п)+56):=by
  apply zpow_le_zpow_right₀ Г;unfold cE;dsimp only;split
  .rw[χ _ _ (UInt64.le_of_lt ‹_›)];have:=ν.mp ‹_›
   simp only[iZ] at this⊢;omega
  .have:=κ.mp (UInt64.not_lt.mp ‹_›);simp only[iZ] at this⊢;omega
 have t:(n:ℝ)<=536870912:=by exact_mod_cast H.le
 have I:(2:ℝ)^(-1074:п)=2^(-1075:п)*2:=by rw[<- zpow_add_one₀ two_ne_zero];norm_num
 have V:(2:ℝ)^(-1075:п)*2^64<=2^(-900:п):=by
  rw[<- zpow_natCast,<- zpow_add₀ two_ne_zero];exact zpow_le_zpow_right₀ Г Г
 have S:(2:ℝ)^40*2^(400:п)<=2^(971:п):=by
  rw[<- zpow_natCast,<- zpow_add₀ two_ne_zero];exact zpow_le_zpow_right₀ Г Г
 have U:(1:ℝ)<=2^(400:п):=one_le_zpow₀ Г Г;have g:(0:ℝ)<2^(-1075:п):=В;have F:(2:ℝ)^(-900:п)<=1:=zpow_le_one_of_nonpos₀ Г Г
 rw[show (2:ℝ)^(-53:п)=1/9007199254740992 by norm_num,show (2:ℝ)^(-20:п)=1/1048576 by norm_num,
  I,mul_add,<- mul_assoc] at d
 have W:=mul_le_mul_of_nonneg_right t g.le;generalize G:(n:ℝ)*(1/9007199254740992*(1+1/1048576))=K at d;have m:K<=1/8388608:=by rw[<- G];nlinarith
 have X:=mul_le_mul_of_nonneg_right m u
 have z: |rv a| <= 3*2^(400:п):=by
  have:=abs_sub_abs_le_abs_sub (rv a) σ;rw[abs_of_nonneg u] at this;generalize (2:ℝ)^(-1075:п)=y at*;generalize (2:ℝ)^(-900:п)=ε at*
  generalize (2:ℝ)^(400:п)=Z at*;linarith
 have E: |rv (9*(n+1))ж.toInt64.toFloat*rv a| <= 2^(971:п):=by
  rw[h,abs_mul,abs_of_pos В];have:=mul_le_mul_of_nonneg_left z (by positivity:(0:ℝ)<=9*(n+1))
  have:=mul_le_mul_of_nonneg_right t (by positivity:(0:ℝ)<=2^(400:п))
  generalize (2:ℝ)^(400:п)=Z at*;generalize (2:ℝ)^(971:п)=L at*;linarith
 obtain ⟨o,j⟩:=iO _ a w D E;have O:=lt_of_le_of_lt' r (lt_of_le_of_lt (abs_nonneg _) (bu _ o));have M:=lt_of_lt_of_le (lt_of_le_of_lt (le_abs_self _) (bu _ o)) r;rw[h,show (2:ℝ)^(-53:п)=1/9007199254740992 by norm_num] at j
 rw[show (2:ℝ)^(-20:п)=1/1048576 by norm_num]
 generalize (2:ℝ)^(((cE (9*(n+1))ж.toInt64.toFloat a)а:п)+56)=Q at*
 generalize rv ((9*(n+1))ж.toInt64.toFloat*a)=P at*
 generalize rv a=A at*
 generalize (2:ℝ)^(-1075:п)=y at*
 generalize (2:ℝ)^(-900:п)=ε at*
 have l:(n:ℝ)*(y*2)<=1073741824*y:=by linarith
 generalize (n:ℝ)*(y*2)=v at d l;have q:(0:ℝ)<=9*(n+1):=В;have B:=mul_le_mul_of_nonneg_right (by linarith:9*((n:ℝ)+1)<=2^33) g.le
 generalize 9*((n:ℝ)+1)=N at*
 obtain ⟨x,-⟩:=abs_le.mp d;rcases le_or_gt 0 A with p | p
 .rw[abs_of_nonneg (mul_nonneg q p)] at j
  have:=mul_le_mul_of_nonneg_left x (by linarith:0<= N*(1-1/9007199254740992))
  nlinarith[(abs_le.mp j).1]
 .have:=mul_le_mul_of_nonneg_left (by linarith:σ<=2*v) q
  nlinarith
lemma fW (x:i9) (d:i7 x≠255): |rv (dw x)| <= 2^(277:п):=by
 rw[(av x d).2,abs_mul,abs_pow,abs_two];have:(i7 x)е≠255:=λh=>d (Η.mp (by simpa using h))
 have:=bO x
 calc
  _<=(2:ℝ)^24*2^253:=mul_le_mul (by rw[<- ϕ];exact_mod_cast (bV x).le)
    (pow_le_pow_right₀ Г (by rw[br];omega)) В В
  _=2^(277:п):=by rw[<- pow_add,<- zpow_natCast];rfl
lemma f0:fv (0:р)/\rv (0:р)=0:=by
 have e:(0:р).toModel.unpack=.zero .positive:=rfl;refine ⟨by rw[bB,e];rfl,by rw[cf,e];simp[jp,jU]⟩
lemma io (b s d τ r:о) (o:s<2^32) (g:d<2^32) (p:τ<2^32) (l:r<2^32)
  (h:b+τ*s+r*d<2^32):
  (bг+τг*sг+rг*dг)б=b+τ*s+r*d:=by
 have:=ς;have:=Ϟ
 have m:∀x y:о,x<2^32-> y<2^32-> x*y<2^32->(xг*yг)б=x*y:=
  λx y q t h=>by rw[li,aa А,aa А,ξ А]
 have e:=m τ s p o А;have j:=m r d l g А
 rw[ab (by rw[ab (by rw[aa А,e];omega),aa А,e,j];omega),
  ab (by rw[aa А,e];omega),aa А,e,j]
lemma fw (b:i9) (d:о) (g:je b<= d): |(tm kf b:ℝ)| <= 2^(173+d):=by
 unfold tm
 rw[show sv kf=2^23 by decide,show je kf=126 by decide,Int.cast_mul,Int.cast_mul,abs_mul,abs_mul,
  Int.cast_pow,Int.cast_pow,Int.cast_ofNat,abs_pow,abs_pow,abs_two]
 have e: |(sv b:ℝ)| <= 2^24:=by rw[<- ϕ];exact_mod_cast (bV _).le
 calc
  (2:ℝ)^23* |(sv b:ℝ)| *2^(126+je b)<=2^23*2^24*2^(126+d):=by gcongr;norm_num
  _=2^(173+d):=by rw[<- pow_add,<- pow_add];congr 1;omega
lemma fh (b:Д):(eF b)а<=2047:=by
 unfold eF;simp only[ψ,iZ];exact ϙ
lemma ff (F a:р):(cE F a)а<=969:=by
 unfold cE;dsimp only;split
 .rename_i h;have:=fh (F*a).toBits;have d:=ν.mp h
  rw[χ _ _ (UInt64.le_of_lt h)];simp only[iZ] at d⊢;omega
 .simp
abbrev Iv (F:о-> т) (a:у т) (P:о-> Prop) (A:у т):Prop:=
 Aв=aв /\ ∀q,(A[q]?=a[q]? \/ A[q]?=some (F q))/\(P q-> q<aв-> A[q]?=some (F q))
lemma cz (F:о-> i9) (e g:у i9) (S U:о-> Prop) (j:∀q,U q-> S q)
  (h:Iv F e S g):Iv F e U g:=
 ⟨h.1,λq=>⟨(h.2 q).1,λd l=>(h.2 q).2 (j q d) l⟩⟩
def aV (A:у т) (e:с) (he:e<= A.usize) (i:с) (m:Е):Е:=
 if h:i<e then aV A e he (i+1) (max m ((A.uget i (ay (γ.mpr
  (Nat.lt_of_lt_of_le (γ.mp h) (USize.le_iff_toNat_le.mp he))))>>>23)&&&255)) else m
termination_by eб-iб
decreasing_by have:=γ.mp h;rw[al h];omega
lemma cL (A:у i9) (e:с) (d:e<= A.usize):∀k (i:с) (m:Е),eб-iб=k ->
  mе<=(aV A e d i m)е /\ ∀q,iб<= q-> q<eб->(i7 A[q]!)е<=(aV A e d i m)е:=by
 intro k;induction k with
 | zero=>
  intro i m o;rw[aV,dif_neg λh=>by have:=γ.mp h;omega];exact ⟨le_rfl,λq v g=>by omega⟩
 | succ k u=>
  intro i m o;have h:i<e:=γ.mpr А;have j:i<A.usize:=γ.mpr (kH (γ.mp h) (USize.le_iff_toNat_le.mp d));rw[aV,dif_pos h];obtain ⟨r,t⟩:=u (i+1) (lm m ((A.uget i (ay j)>>>23)&&&255)) (by rw[al h];omega)
  rw[UInt32.toNat_max] at r;refine ⟨by omega,λq l g=>?_⟩
  by_cases p:q=iб
  .subst p;rw[<- bt A i (ay j)];unfold i7;omega
  .exact t q (by rw[al h];omega) g
lemma dN (n:о) (e d:о-> i9):∀c i,i+c=n ->
  dallNZ (jv (n:=n+1) λt=>if tи<n then d tи else kf)
   (jv (n:=n+1) λt=>if tи<n then e tи else 0x80000000) i (c+1)=
  dallNZ (jv (n:=n) λt=>e tи) (jv (n:=n) λt=>d tи) i c:=by
 intro c;induction c with
 | zero=>
  intro i h;obtain rfl:i=n:=А
  simp only[dallNZ,ae (i+1) (λt=>if t< i then d t else kf) i А,
   ae (i+1) (λt=>if t< i then e t else 0x80000000) i А,lt_irrefl,if_false];rfl
 | succ c g=>
  intro i h
  rw[dallNZ,g (i+1) А,dallNZ,ae (n+1) (λt=>if t<n then d t else kf) i А,
   ae (n+1) (λt=>if t<n then e t else 0x80000000) i А,ae n e i А,
   ae n d i А]
  simp only[show i<n by omega,↓reduceIte,ϩ (jF (d i)),Bool.xor_comm (jw (d i))]
lemma dz (n:о) (d e:о-> i9) (h:0<n) (g:am n d e):
  kB (n+1) (λt=>if t<n then e t else kf) (λt=>if t<n then d t else 0x80000000)=
   kB n d e:=by
 rw[aU _ _ _ λt _=>by split;exacts [(g t ‹_›).symm,by decide],aU _ _ _ g,aE,
  dN n d e n 0 А]
 congr 1
 .simp only[lt_irrefl,if_false,show tm kf 0x80000000=0 by decide,add_zero]
  exact Ο rfl λt j=>by simp only[if_pos (Ν.mp j)];unfold tm;rw[ϔ];ring
 .simp[h.ne']
end F
lemma co {a:о} (h:a<=4294967295):a<2^32:=А
end Dense
namespace Bn
namespace NativeAdd
lemma cu (x:Float32):ly x.toModel=ofW x.toBits:=
 congrArg Model.mk (ofUInt32_bits x.toBits).symm
lemma fk (x:Float32):jH (ly x.toModel)=x.toBits:=
 UInt32.ofNat_toNat
lemma aZ (a:i9) (h:i7 a≠255):
  (kI a).toBits=a/\(kI a).isFinite=true:=by
 have g:lp (spec:=ju) a.toBitVec≠-1#8:=λd=>
  h (Η.mp ((show (i7 a)е=(a.toBitVec.extractLsb' 23 8).toNat by
   simp only[i7,Λ,Ρ,BitVec.extractLsb'_toNat];exact θ _ 8).trans (congrArg BitVec.toNat d:)))
 have e:(kI a).toBits=a:=by
  change UInt32.ofBitVec (jA ju (jc ju a.toBitVec))=a
  rw[Model.pack_unpack_eq,if_neg (g ·.1)]
 refine ⟨e,?_⟩;change (jc ju (kI a).toBits.toBitVec).isFinite=true;simp only[e,jc,g,ite_false];split_ifs<;>rfl
@[noinline] def cN (relu:ф) (a b:т):т:=
 act relu (toWord (Model.add (ofW a) (ofW b)))
@[inline] def cK (relu:ф) (w:т):т:=
 w&&&~~~((0:т)-((if relu then w>>>31 else 0)&&&(if w&&&0x7FFFFFFF<=0x7F800000 then 1 else 0)))
lemma gQ (w:i9):(w&&&0x80000000!=0)=decide (2^31<= wе):=by
 have e:=w.toNat_lt
 rw[ϻ,bne_iff_ne,ne_eq,<- Η,Λ,decide_eq_true_iff,
  show (0x80000000:Е)е=2^31 from rfl,lc]
 simp only[j5,Nat.testBit,ι,Nat.one_and_eq_mod_two];constructor
 .intro h;by_contra d;apply h;rw[ϣ А];simp
 .intro h;rw[show wе/2^31=1 by omega];simp
lemma dA (d:ф) (w:i9):cK d w=act d w:=by
 have v:=w.toNat_lt;have t:(w>>>31)е=wе/2^31:=by rw[Ρ];simp[ι]
 have m:=Dense.aR w;have q:=Dense.be w
 have r:(lW w)е=wе%2^23:=by
  unfold lW;rw[Λ];exact θ _ 23
 unfold Dense.aS at m q
 have o:∀x:i9,x&&&~~~((0:i9)-0)=x:=λx=>by
  apply Η.mp;rw[Λ];rw[show (~~~((0:i9)-0))е=2^32-1 from rfl,θ,ξ x.toNat_lt]
 have u:∀x:i9,x&&&~~~((0:i9)-1)=0:=λx=>by
  apply Η.mp;rw[Λ];simp
 have j:w>>>31=if 2^31<= wе then 1 else 0:=by
  apply Η.mp;rw[t];split<;>simp<;>omega
 have p:(w&&&0x7FFFFFFF<=0x7F800000)↔wе%2^31<=0x7F800000:=by
  rw[UInt32.le_iff_toNat_le,m];rfl
 have l:(w&&&0x80000000!=0&&!(i7 w=255&&lW w !=0))=true↔
   (2^31<= wе/\wе%2^31<=0x7F800000):=by
  rw[gQ]
  simp only[υ,decide_eq_true_eq,Bool.not_eq_true',Bool.and_eq_false_iff,decide_eq_false_iff_not,
   bne_eq_false_iff_eq,<- Η,q,r,m,jj,j5]
  omega
 unfold cK act;cases d
 .simp only[ο,ite_false,UInt32.zero_and,o]
 simp only[ite_true];unfold reluW;rw[j];by_cases e:2^31<= wе
 .rw[if_pos e]
  by_cases g:wе%2^31<=0x7F800000
  .rw[if_pos (p.mpr g),show (1:i9)&&&1=1 from rfl,u,if_pos (l.mpr ⟨e,g⟩)]
  .rw[if_neg (mt p.mp g),show (1:i9)&&&0=0 from rfl,o,if_neg (λh=>g (l.mp h).2)]
 .rw[if_neg e,UInt32.zero_and,o,if_neg (λh=>e (l.mp h).1)]
@[inline] def add (relu:ф) (a b:т):т:=
 if expOf a !=255&&expOf b !=255 then
  cK relu (Float32.ofBits a+Float32.ofBits b).toBits
 else cN relu a b
lemma dV (d:ф) (a b:i9):
  add d a b=act d (jH (lB (ofW a) (ofW b))):=by
 unfold add;split
 .rename_i h
  rw[dA];simp only[υ,bne_iff_ne] at h;have e:=aZ a h.1;have g:=aZ b h.2;have j:=ExecFloat.Binary.toModel_ofFloat32_add_of_isFinite _ _ e.2 g.2;simp only[ExecFloat.Binary.toModel_ofFloat32,cu,e.1,g.1] at j;rw[show lB (ofW a) (ofW b)=ofW (Float32.ofBits a+Float32.ofBits b).toBits from j.symm,<- cu,fk]
 .rfl
end NativeAdd
@[inline] def ad (rb:Е) (a b:т) (_slow:Unit-> т):т:=NativeAdd.add (rb !=0) a b
@[specialize] def bW (stop:с) (f:(i:с)-> iб<stopб-> т-> т) (i:с)
  (x:у т) (h:stopб<= xв):у т:=
 if hi:i<stop then
  have hs:=γ.mp hi
  bW stop f (i+1) (x.uset i (f i hs (x.uget i А)) А) (by rw[ϑ];exact h)
 else x
termination_by stopб-iб
decreasing_by rw[Dense.al hi];omega
lemma cI (d:с) (f:(i:с)-> iб<dб-> i9-> i9) (F:о-> i9-> i9)
  (l:∀i e w,f i e w=F iб w) (i:с) (x:у i9) (h:dб<= xв):
  (bW d f i x h)в=xв /\
  ∀j,(bW d f i x h)[j]?=if iб<= j/\j<dб then x[j]?.map (F j) else x[j]?:=by
 fun_induction bW
 .next i x h e m g=>
  refine ⟨by rw[g.1,ϑ],λj=>?_⟩
  rw[g.2,Dense.al e,lT,lM А];split_ifs<;>first | rfl | omega | (subst_vars;rw[kL А,Option.map_some,l];rfl)
 .next i x h e=>exact ⟨rfl,λj=>by rw[if_neg (by rw[γ] at e;omega)]⟩
lemma fd (d:ф) (a b:i9) (e:Unit-> i9)
  (g:e ()=act d (jH (lB (ofW a) (ofW b)))):
  ad (if d then 1 else 0) a b e=act d (jH (lB (ofW a) (ofW b))):=by
 cases d<;>simp[ad,NativeAdd.dV]
@[noinline] def gH (relu:ф) (x y:т):т:=act relu (toWord (Model.add (ofW x) (ofW y)))
def bq (relu:ф) (a b:у т):у т:=
 if h:aв<= bв/\aв<USize.size then
  let rb:Е:=if relu then 1 else 0
  bW aвг (λi hi x=>
   let y:=b.uget i (by rw[Dense.aa h.2] at hi;omega)
   ad rb x y λ_=>gH relu x y) 0 a (by rw[Dense.aa h.2])
 else addReluSpec relu a b
lemma cg:bq=addReluSpec:=by
 funext d a b;unfold bq;dsimp only
 split;swap;. rfl
 rename_i h;have g:=Dense.aa h.2;refine lN λj=>?_
 rw[(cI _ _ (λj x=>act d (jH (lB (ofW x) (ofW b[j]!)))) (λi l x=>?_) 0 a _).2 j,
  addReluSpec,lF]
 .by_cases e:j<aв
  .rw[if_pos ⟨by simp,by rwa[g]⟩,dif_pos e,kL e];rfl
  .rw[if_neg (by rw[g];omega),dif_neg e,Array.getElem?_eq_none А]
 .rw[fd d _ _ _ rfl,getElem!_pos b iб А];rfl
end Bn open Bn
namespace Dense
namespace F
@[inline] def aF (relu:ф) (w:т):т:=if relu then w&&&((w>>>31)-1) else w
lemma fc (d:ф) (w:i9) (m:i7 w≠255):aF d w=act d w:=by
 unfold aF act reluW;cases d
 .simp
 have l:=aP w
 have j:(w>>>31)е=wе/2^31:=by
  rw[Ρ];simp[ι]
 have o:=w.toNat_lt;by_cases h:2^31<= wе
 .have g:w>>>31=1:=Η.mp (by rw[j];simp only[UInt32.toNat_one];omega)
  have e:jw w=true:=l.mpr h;unfold jw at e;simp[g,e,m]
 .have g:w>>>31=0:=Η.mp (by rw[j];simp only[j5];omega)
  have e:jw w≠true:=λq=>h (l.mp q)
  unfold jw at e;simp only[Bool.not_eq_true] at e;simp[g,e]
lemma fg (S:р) (h:Д) (e:ai S h<0x100000000):i7 (ai S h)л≠255:=by
 unfold ai at e⊢;dsimp only at e⊢;split at e
 swap;. exact absurd e Б
 rename_i d;rw[if_pos d];simp only[υ,decide_eq_true_eq] at d;obtain ⟨⟨⟨-,l⟩,-⟩,-⟩:=d
 have m:(eF S.toBits)а<2048:=by
  unfold eF;rw[ψ];exact lt_of_le_of_lt ϙ Б
 have o:(S.toBits&&&0xfffffffffffff)а<2^52:=by
  rw[ψ];exact lt_of_le_of_lt ϙ Б
 rw[ν,Κ] at l;simp only[iZ] at l;have p:1195<=(eF S.toBits)а:=А;have r:=fK S.toBits p;have q:S.toBitsа/2^63<2:=by have:=S.toBits.toNat_lt;omega
 rw[ν] at e;simp only[iZ] at e;intro g;have j:=congrArg UInt32.toNat g;rw[i7,Λ,Ρ,ϟ] at j;simp only[jj,jb,ι] at j;rw[θ _ 8,ξ e,r] at j;omega
@[inline] def cH (relu:ф) (S:р) (hh:Д) (sl:Unit-> т):т:=
 let c:=ai S hh
 if c<0x100000000 then aF relu cл else sl ()
lemma eU (d:ф) (S:р) (e:Д) (g:Unit-> i9):cH d S e g=tf d S e g:=by
 unfold cH tf;dsimp only;split
 .rename_i h;rw[fc _ _ (fg _ _ h)]
 .rfl
lemma gS (x w:i9) (e:о) (d:(i7 w)е<= e):
  |(tm x w:ℝ)| <= |(sv x:ℝ)*2^je x| *2^(lm e 56+23):=by
 unfold tm;push_cast
 rw[show (sv x:ℝ)*sv w*2^(je x+je w)=(sv x*2^je x)*(sv w*2^je w) by
  rw[pow_add];ring,abs_mul]
 gcongr;rw[abs_mul,abs_pow,abs_two]
 refine (mul_le_mul (b:=2^24) (by rw[<- ϕ];exact_mod_cast (bV _).le)
  (pow_le_pow_right₀ Г (by rw[br];omega:_<= max e 56-1)) В В).trans_eq ?_
 rw[<- pow_add];congr 1;omega
lemma iD (l:ф) (n:о) (j g:о-> i9) (d:i9) (p:о-> р) (F v L:о) (e:i9) (c r:Д)
  (S:р) (o u:Unit-> i9)
  (E:∀s<n,i7 (j s)≠255) (t:i7 d≠255/\je d<= v) (a2:174+v<= F)
  (hL:n+2<=2^L) (M:L<=27) (O:F+L<=425) (hn:8*n<2^32)
  (hp:∀s<n,i7 (g s)≠255-> fv (p s)/\rv (p s)=tm (j s) (g s))
  (Z:S=ac p n 0 (dw kf*dw d))
  (N:∀s<n,(i7 (g s))е<= eе)
  (w:9*((n:ℝ)+1)*(∑s∈i0 n,|rv (dw (j s))|)*(1-2^(-20:п)) <=
   2^((cа:п)+56)+2^(-900:п))
  (as:cа<=1000) (a0:rа=F+L-50)
  (B:o ()=act l (kB (n+1) (λs=>if s<n then j s else kf) (λs=>if s<n then g s else d)))
  (hk:(∀s<n,i7 (g s)≠255) ->
   u ()=act l (kB (n+1) (λs=>if s<n then j s else kf) (λs=>if s<n then g s else d))):
  (if e<255&&(if eк<56 then 56 else eк)+23+c<=400 then
    cH l S (if (if eк<56 then 56 else eк)+23+c<r then r
     else (if eк<56 then 56 else eк)+23+c) u
   else o ())=
  act l (kB (n+1) (λs=>if s<n then j s else kf) (λs=>if s<n then g s else d)):=by
 have P:∀{a b:ℕ},a<= b->(2:ℝ)^a<=2^b:=pow_le_pow_right₀ Г;generalize Y:(if eк<56 then 56 else eк)+23+c=q
 have hq:qа=lm eе 56+23+cа:=by
  have:=UInt32.toNat_lt e
  rw[<- Y];split<;>rename_i h<;>simp only[ν,Χ,μ,
   iZ] at h⊢<;>omega
 split;swap;. exact B
 rename_i hg;simp only[υ,decide_eq_true_eq,ϖ,κ,
  jj,iZ] at hg
 generalize W:(if q<r then r else q)=A
 have a1:Aа=lm qа rа:=by
  rw[<- W];split<;>rename_i h<;>simp only[ν] at h<;>omega
 have D:∀s<n,i7 (g s)≠255:=λs m h=>by have:=N s m;simp[h] at this;omega
 have K:=ey kf d Б t.1;generalize y:(∑s∈i0 n,|rv (dw (j s))|)=z at w
 have I:0<= z:=y ▸ ϡ λ_ _=>abs_nonneg _
 rw[zpow_add₀ two_ne_zero,zpow_natCast] at w;have Q:=fw d v t.2;have J:(n:ℝ)<=2^32:=by exact_mod_cast (show n<=2^32 by omega)
 have X:z<= n*2^277:=
  y ▸ (Ϗ λs (m:s∈i0 n)=>fW _ (E s (Ν.1 m))).trans_eq (by simp)
 have hZ:∑s∈i0 n,|(tm (j s) (g s):ℝ)| <= 2^(lm eе 56+23)*z:=by
  rw[<- y,kJ]
  exact Ϗ λs m=>by
   rw[(av _ (E s (Ν.1 m))).2];exact (gS _ _ _ (N s (Ν.1 m))).trans_eq (mul_comm _ _)
 have U: |(tm kf d:ℝ)| +∑s∈i0 n,|(tm (j s) (g s):ℝ)| <= 2^(968:п):=by
  rw[show (2:ℝ)^(968:п)=2^967+2^967 by rw[<- two_mul,<- pow_succ'];norm_num]
  refine add_le_add (Q.trans (P А)) (hZ.trans ((mul_le_mul (P (by omega:_<=400))
   (X.trans (mul_le_mul_of_nonneg_right J В)) I В).trans ?_))
  rw[<- pow_add,<- pow_add];exact P Г
 obtain ⟨H,G⟩:=gm p (λs=>(tm (j s) (g s):ℝ)) (tm kf d:ℝ) n
  (λs m=>hp s m (D s m)) А U n 0 _ А (km _) K.1 (by rw[K.2];simp)
 rw[<- Z] at H G
 have hT:(ah (n+1) (λs=>if s<n then j s else kf) (λs=>if s<n then g s else d):ℝ)=
   tm kf d+∑s∈i0 n,(tm (j s) (g s):ℝ):=by
  rw[aE,ah];push_cast;simp only[lt_irrefl,ite_false];rw[add_comm];congr 1;exact Ο rfl λs m=>by simp[Ν.1 m]
 have V: |rv S-(ah (n+1) (λs=>if s<n then j s else kf)
   (λs=>if s<n then g s else d):ℝ)| <= (2:ℝ)^(Aа:п):=by
  rw[hT]
  refine gW n z (tm kf d:ℝ) _ (2^(lm eе 56+23)) _ (2^rа) _ I В
   (one_le_pow₀ Г) (by simpa using J) ?_ ?_ w ?_ ?_ ?_ ?_
  .exact (G.trans (by gcongr)).trans (le_add_of_nonneg_right В)
  .rw[<- zpow_natCast,<- zpow_add₀ two_ne_zero];exact zpow_le_zpow_right₀ Г А
  .rw[show (2:ℝ)^(49:п)=2^49 by norm_num,<- pow_add]
   refine (mul_le_mul (by exact_mod_cast (show n<=2^L by omega):(n:ℝ)<=2^L) Q (abs_nonneg _)
    В).trans ?_
   rw[<- pow_add];exact P А
  .rw[<- pow_add,zpow_natCast];exact P А
  .rw[zpow_natCast];exact P А
  .exact zpow_le_zpow_right₀ Г А
 rw[eU];unfold tf;dsimp only;split
 .rename_i x
  have hf:am (n+1) (λs=>if s<n then j s else kf) (λs=>if s<n then g s else d):=λk _=>by
   by_cases h:k<n<;>simp only[h,↓reduceIte];exacts [⟨E k h,D k h⟩,⟨by decide,t.1⟩]
  rw[fO _ _ _ hf S H A А V x]
 .exact hk D
lemma fu (d:ф) (n:о) (g h:о-> i9) (p:о-> р) (e:Unit-> р) (L:о)
  (m:Unit-> i9) (z:∀s<n,fv (p s)/\rv (p s)=tm (g s) (h s))
  (o:fv (e ())/\rv (e ())=tm (g n) (h n)) (q:am (n+1) g h)
  (A:n+2<=2^L) (u:L<=27) (v:m ()=act d (kB (n+1) g h)):
  tS d L n p e m=act d (kB (n+1) g h):=by
 unfold tS;dsimp only;generalize w:sT p n 0 (e ()) 0=r;split
 next t=>
  split
  next y=>
   obtain ⟨l,j⟩:=eK _ _ (tm (g n) (h n)) n r.2а t.1 t.2.1 z n 0 _ 0 rfl А
    o.1 (by simp[o.2]) (by rw[w])
   rw[w] at j l
   have x:((r.2а+L-1076)ж)а=r.2а+L-1076:=by
    rw[jL,Ξ (by simp only[jT];omega)]
   refine congrArg _ (fO _ _ _ q _ l _ (by rw[x];omega) ?_ y)
   rw[aE,x];unfold ah;push_cast;rw[add_comm]
   exact j.trans ((mul_le_mul_of_nonneg_right (by exact_mod_cast (by omega:n<=2^L):(n:ℝ)<=2^L) В).trans_eq
    (by rw[<- zpow_natCast,<- zpow_add₀ two_ne_zero];exact congrArg (2^·) А))
  .exact v
 .rw[if_neg Б];exact v
end F
end Dense
namespace Sr
structure Ent {α:Type} (f:у т-> у т-> о-> α) where
 w:у т
 b:у т
 n:о
 v:α
 ok:v=f w b n
 a:с
structure St {α:Type} (f:у т-> у т-> о-> α) where
 e:у (Ent f)
 x:ByteArray
 hd:о
 tl:о
 u:о
structure Tab {α:Type} (f:у т-> у т-> о-> α) where
 r:ST.Ref Unit (St f)
 z:Ent f
 zx:ByteArray
 cap:о
variable {α:Type} {f:у i9-> у i9-> о-> α}
def Tab.gM (f:у т-> у т-> о-> α) (N cap:о):Tab f:=
 let z:Ent f:=⟨#[],#[],0,f #[] #[] 0,rfl,0⟩
 let zx:ByteArray:=⟨Array.replicate (4*N) 0⟩
 ⟨(ST.Prim.mkRef ⟨Array.replicate N z,zx,0,0,0⟩ (Void.mk ())).val,z,zx,cap⟩
@[inline] def Ent.dh (e:Ent f) (w b:у т) (n:о):ф:=
 e.n==n&&withPtrEq e.w w (λ_=>e.w==w) (λh=>by rw[h];simp)&&
  withPtrEq e.b b (λ_=>e.b==b) (λh=>by rw[h];simp)
lemma Ent.cx {e:Ent f} {w b:у i9} {n:о} (h:e.dh w b n=true):e.v=f w b n:=by
 simp only[Ent.dh,withPtrEq,υ,beq_iff_eq] at h;obtain ⟨⟨d,g⟩,j⟩:=h;rw[e.ok,d,g,j]
@[inline] def Ent.sz (e:Ent f):о:=e.wв+e.bв+64
@[inline] def hs (a:с):с:=(((a>>>4).toUInt64*0x9E3779B97F4A7C15)>>>48).toUSize
lemma ul {β:Type} {a:у β} {i:с} (h:i<a.usize):iб<aв:=by
 have d:=γ.mp h;simp only[kk,ϛ] at d;exact kH d (by simp[lK])
lemma ub {a:ByteArray} {i:с} (h:i<a.usize):iб<a.size:=by
 have d:=γ.mp h;simp only[ByteArray.usize,ϛ] at d;exact kH d (by simp[lK])
def eY (x:ByteArray) (m:с) (c:UInt8):о-> с-> ByteArray
 | 0,_=>x
 | p+1,i=>
  let k:=i&&&m
  if h:k<x.usize then
   if x.uget k (ub h)==0 then x.uset k c (ub h) else eY x m c p (i+1)
  else x
def eX (e:у (Ent f)) (m:с) (i:о) (x:ByteArray):ByteArray:=
 if h:i<eв then
  let q:=e[i]
  eX e m (i+1) (if q.w.isEmpty then x else eY x m (i+1).toUInt8 x.size (hs q.a))
 else x
termination_by eв-i
@[inline] def St.dX (s:St f) (z:Ent f):St f:=
 let o:=s.e.getD s.hd z
 ⟨s.e.setIfInBounds s.hd z,s.x,(s.hd+1)%s.eв,s.tl,if o.w.isEmpty then s.u else s.u-o.sz⟩
def St.eH (t:Tab f) (sz:о):о-> St f-> St f
 | 0,s=>s
 | p+1,s=>if s.u+sz<= t.cap then s else St.eH t sz p (s.dX t.z)
@[noinline] def Tab.cU (t:Tab f) (w b:у т) (n:о) (a:с):{y // y=f w b n}:=
 let y:=f w b n
 let q:Ent f:=⟨w,b,n,y,rfl,a⟩
 let sz:=q.sz
 if 4*sz<= t.cap then
  let s:=(ST.Prim.Ref.get t.r (Void.mk ())).val
  let s:=St.eH t sz s.eв s
  let s:=if (s.e.getD s.tl t.z).w.isEmpty then s else s.dX t.z
  let e:=s.e.setIfInBounds s.tl q
  let x:=eX e (t.zx.usize-1) 0 t.zx
  let o:=(ST.Prim.Ref.swap t.r ⟨e,x,s.hd,(s.tl+1)%eв,s.u+sz⟩ (Void.mk ())).val
  if o.u=0xFFFFFFFFFFFF then
   if h:(o.e.getD 0 t.z).dh w b n=true then ⟨_,Ent.cx h⟩ else ⟨y,rfl⟩
  else ⟨y,rfl⟩
 else ⟨y,rfl⟩
def St.cr (s:St f) (t:Tab f) (w b:у т) (n:о) (a m:с):о-> с-> {y // y=f w b n}
 | 0,_=>t.cU w b n a
 | p+1,i=>
  let k:=i&&&m
  if h:k<s.x.usize then
   let c:=s.x.uget k (ub h)
   if c==0 then t.cU w b n a else
    let j:=(c-1).toUSize
    if hj:j<s.e.usize then
     let e:=s.e.uget j (ul hj)
     if e.a==a then
      if hk:e.dh w b n=true then ⟨e.v,Ent.cx hk⟩ else s.cr t w b n a m p (i+1)
     else s.cr t w b n a m p (i+1)
    else s.cr t w b n a m p (i+1)
  else t.cU w b n a
local instance {β:Type} (y:β):Subsingleton {z // z=y} :=
 ⟨λp q=>Subtype.ext (p.2.trans q.2.symm)⟩
@[inline] def Tab.get (t:Tab f) (w b:у т) (n:о):{y // y=f w b n}:=
 withPtrAddr w (λa=>
  let s:=(ST.Prim.Ref.get t.r (Void.mk ())).val
  s.cr t w b n a (s.x.usize-1) s.x.size (hs a)) (λ_ _=>Subsingleton.elim _ _)
end Sr open Sr
namespace Dense
namespace F
namespace Pn
def dR (X:у т) (st:с):о-> с-> с-> FloatArray-> р-> FloatArray
 | 0,_,o,Y,a=>wF Y o a
 | k+1,q,o,Y,a=>let v:=rX X q;dR X st k (q+st) (o+1) (wF Y o v) (a+v.abs)
def dO (X:у т) (st s m:с) (n:о):о-> с-> с-> FloatArray-> FloatArray
 | 0,_,_,Y=>Y
 | k+1,q,o,Y=>dO X st s m n k (q+s) (o+m) (dR X st n q o Y 0)
def qL (W:у т) (su m:с) (n:о):о-> с-> FloatArray-> FloatArray
 | 0,_,Y=>Y
 | k+1,q,Y=>qL W su m n k (q+m) (bx W su 1 n q (q+1) Y)
lemma dv (q d:с) (τ:о):q+d+τг*d=q+(τ+1)г*d:=by
 rw[ϛ,ϛ,USize.ofNat_add,USize.add_mul,show USize.ofNat 1=1 from rfl,USize.one_mul,
  USize.add_assoc,USize.add_comm d]
lemma uz (q d:с):q+(0:о)г*d=q:=by
 simp[ϛ,show USize.ofNat 0=0 from rfl]
lemma dH (X:у i9) (d:с):∀k (q o:с) (Y:FloatArray) (a:р),
  Yд<2^32-> oб+k<Yд ->
  (dR X d k q o Y a)д=Yд /\
  (∀i,i<Yд-> i<oб\/oб+k< i->(dR X d k q o Y a).data[i]! =Y.data[i]!) /\
  (∀τ<k,(dR X d k q o Y a).data[oб+τ]! =rX X (q+τг*d)) /\
  (dR X d k q o Y a).data[oб+k]! =ac (λτ=>(rX X (q+τг*d)).abs) k 0 a:=by
 intro k;induction k with
 | zero=>
  intro q o Y a g l;rw[dR,dx Y g,ac,if_neg А];refine ⟨bD ..,λi e v=>?_,λτ h=>absurd h А,?_⟩
  .rw[sg _ _ _ _ e,if_neg А]
  .rw[sg _ _ _ _ l,if_pos А]
 | succ k u=>
  intro q o Y a g l;have:=ς
  have e:(o+1)б=oб+1:=by
   rw[ab (by simp only[jd];omega)];simp only[jd]
  rw[dR,dx Y g]
  obtain ⟨m,p,r,t⟩:=u (q+d) (o+1) (Y.set! oб (rX X q)) (a+(rX X q).abs)
   (by rw[bD];exact g) (by rw[bD,e];omega)
  rw[bD] at m;refine ⟨m,λi j w=>?_,λτ h=>?_,?_⟩
  .rw[p i (by rw[bD];exact j) А,sg _ _ _ _ j,if_neg А]
  .rcases τ with _ | τ
   .rw[p _ (by rw[bD];omega) А,sg _ _ _ _ А,if_pos А,uz]
   .have:=r τ А
    rw[e,show oб+1+τ=oб+(τ+1) by omega] at this
    rw[this,dv]
  .have:=t
   rw[e,show oб+1+k=oб+(k+1) by omega] at this
   rw[this];conv_rhs=>rw[ac,if_pos А,b7 _ k _ 0 _ rfl]
   simp only[dv,uz]
lemma dF (X:у i9) (d s m:с) (n:о) (v:mб=n+1) (F:n+1<2^32):
  ∀k (q o:с) (Y:FloatArray),Yд<2^32-> oб+k*(n+1)<= Yд ->
  (dO X d s m n k q o Y)д=Yд /\
  (∀i,i<Yд-> i<oб->(dO X d s m n k q o Y).data[i]! =Y.data[i]!) /\
  (∀j<k,∀τ<n,(dO X d s m n k q o Y).data[oб+j*(n+1)+τ]! =
   rX X (q+jг*s+τг*d)) /\
  (∀j<k,(dO X d s m n k q o Y).data[oб+j*(n+1)+n]! =
   ac (λτ=>(rX X (q+jг*s+τг*d)).abs) n 0 0):=by
 intro k;induction k with
 | zero=>intro q o Y _ _;exact ⟨rfl,λ_ _ _=>rfl,λj h=>absurd h А,λj h=>absurd h А⟩
 | succ k z=>
  intro q o Y p w;rw[ϋ] at w;have:=ς;have g:(o+m)б=oб+(n+1):=by rw[ab А,v]
  obtain ⟨e,A,B,D⟩:=dH X d n q o Y 0 p А
  obtain ⟨u,l,x,y⟩:=z (q+s) (o+m) (dR X d n q o Y 0) (by rw[e];exact p)
   (by rw[e,g];omega)
  rw[e] at u;rw[dO];refine ⟨u,λi r E=>?_,λj h τ t=>?_,λj h=>?_⟩
  .rw[l i (by rw[e];exact r) А,A i r А]
  .rcases j with _ | j
   .rw[la,Ϫ,l _ (by rw[e];omega) А,B τ t,uz]
   .have:=x j А τ t
    rw[g,show oб+(n+1)+j*(n+1)=oб+(j+1)*(n+1) by rw[ϋ];omega] at this
    rw[this,dv]
  .rcases j with _ | j
   .rw[la,Ϫ,l _ (by rw[e];omega) А,D]
    simp only[uz]
   .have:=y j А
    rw[g,show oб+(n+1)+j*(n+1)=oб+(j+1)*(n+1) by rw[ϋ];omega] at this
    rw[this];simp only[dv]
lemma dP (W:у i9) (d m:с) (n:о) (A:mб=8*n) (B:dб=n):
  ∀k (q:с) (Y:FloatArray),2*(qб+k*(8*n))+32*n<2^32-> qб+k*(8*n)+1<= Yд ->
  (qL W d m n k q Y)д=Yд /\
  (∀i,i<qб+1->(qL W d m n k q Y).data[i]! =Y.data[i]!) /\
  ∀g<k,∀τ<n,∀r<8,(qL W d m n k q Y).data[qб+g*(8*n)+1+8*τ+r]! =
   dw W[qб+g*(8*n)+τ+r*n]!:=by
 intro k;induction k with
 | zero=>intro q Y _ _;exact ⟨rfl,λ_ _=>rfl,λg h=>absurd h А⟩
 | succ k E=>
  intro q Y x y;rw[ko,l0] at x y;have:=ς;have e:(q+m)б=qб+8*n:=by rw[ab А,A]
  have j:(q+1)б=qб+1:=by rw[ab (by simp only[jd];omega)];simp only[jd]
  obtain ⟨u,v,w⟩:=dD W d 1 n q (q+1) Y (by simp only[jd,j];omega)
   (by rw[j];omega)
  simp only[jd,ϱ,B,j] at v w;obtain ⟨p,t,D⟩:=E (q+m) (bx W d 1 n q (q+1) Y) (by rw[e];omega) (by rw[u,e];omega)
  rw[u] at p;rw[qL];refine ⟨p,λi z=>?_,λg h τ o r l=>?_⟩
  .rw[t i А,v i z]
  .rcases g with _ | g
   .rw[la,Ϫ,t _ (by rw[e];omega),w τ r o l,Ϫ]
   .have:=D g А τ o r l
    rw[e,show qб+8*n+g*(8*n)=qб+(g+1)*(8*n) by rw[ko,l0];omega] at this
    exact this
@[inline] def pp (V P:FloatArray) (vo xb r s:о):р:=V.get! (vo+s)*P.get! (xb+8*s+r)
@[noinline] def sQ (n:о) (relu:ф) (X W b:@& у т) (V P:@& FloatArray) (vo xb r xo st j:о):т:=
 tS relu (Nat.log2 (n+1)+1) n (pp V P vo xb r) (λ_=>dw oneW*dw b[j]!) λ_=>ap n relu X W b xo st j
@[inline] def pO (relu ok:ф) (hh:Д) (n:о) (X W b:у т) (xo st o r:о) (V P:FloatArray) (vo:с)
  (q:с) (on:ф) (acc:у т) (S:р):у т:=
 if h:on=true/\q<acc.usize then
  acc.uset q (if ok then cH relu S hh (λ_=>sQ n relu X W b V P voб (o*n+1) r xo st (o+r))
   else ap n relu X W b xo st (o+r)) (ay h.2)
 else acc
@[noinline] def fP (relu ok:ф) (hh:Д) (n:о) (X W b:у т) (xo st o:о) (q0 so mu:с)
  (V P:FloatArray) (vo:с) (a0 a1 a2 a3 a4 a5 a6 a7:р) (acc:у т):у т:=
 pO relu ok hh n X W b xo st o 7 V P vo (q0+7*so) (decide (7< mu))
 (pO relu ok hh n X W b xo st o 6 V P vo (q0+6*so) (decide (6< mu))
 (pO relu ok hh n X W b xo st o 5 V P vo (q0+5*so) (decide (5< mu))
 (pO relu ok hh n X W b xo st o 4 V P vo (q0+4*so) (decide (4< mu))
 (pO relu ok hh n X W b xo st o 3 V P vo (q0+3*so) (decide (3< mu))
 (pO relu ok hh n X W b xo st o 2 V P vo (q0+2*so) (decide (2< mu))
 (pO relu ok hh n X W b xo st o 1 V P vo (q0+1*so) (decide (1< mu))
 (pO relu ok hh n X W b xo st o 0 V P vo (q0+0*so) (decide (0< mu)) acc a0) a1) a2) a3) a4) a5) a6) a7
lemma dI (d Z:ф) (N:Д) (n:о) (X W b:у i9) (B A o:о) (r l p:с)
  (V P:FloatArray) (vo:с) (D E G H J K L M:р) (m:у i9) (F:о-> i9)
  (as:rб+8*lб<=4294967295) (hz:mв<=4294967295)
  (hv:∀k<8,k<pб->(if Z then cH d (A8.f ⟨D,E,G,H,J,K,L,M⟩ k) N
    (λ_=>sQ n d X W b V P voб (o*n+1) k B A (o+k)) else ap n d X W b B A (o+k))=
   F (rб+k*lб)):
  Iv F m (λq=>∃k,k<8/\k<pб/\q=rб+k*lб)
   (fP d Z N n X W b B A o r l p V P vo D E G H J K L M m):=by
 let I j:=rб+j*lб;let Q k:=Iv F m λq=>∃j,j<k/\j<pб/\q=I j
 have ul:=ς
 have g:∀k {a z on},Q k a-> k<8->(k<pб-> USize.toNat z=I k)-> on=decide (k<pб) ->
   Q (k+1) (pO d Z N n X W b B A o k V P vo z on a (A8.f ⟨D,E,G,H,J,K,L,M⟩ k)):=by
  intro k a z on ⟨h1,y⟩ hk S x;unfold pO;split
  next v=>
   have w:k<pб:=by simpa[x] using v.1
   refine ⟨by simp[h1],λq=>?_⟩
   rw[lT,lM,hv k hk w];by_cases h:zб=q
   .subst h;simp[S w,I]
   .rw[if_neg h]
    exact ⟨(y q).1,λ⟨j,O,t,u⟩=>(y q).2 ⟨j,Nat.lt_of_le_of_ne (Nat.le_of_lt_succ O) λe=>by
     subst e;exact h ((S t).trans u.symm),t,u⟩⟩
  next v=>
   refine ⟨h1,λq=>⟨(y q).1,λ⟨j,O,t,u⟩ U=>(y q).2 ⟨j,Nat.lt_of_le_of_ne (Nat.le_of_lt_succ O)
    λe=>v ⟨by simp[x];omega,?_⟩,t,u⟩ U⟩⟩
   subst e;rw[γ,kk,aa А,S t,<- u];omega
 have Y:∀c:с,cб<8->(r+c*l)б=rб+cб*lб:=λc v=>by
  have m1:cб*lб<=8*lб:=Ζ _ v.le
  have e:(c*l)б=cб*lб:=by
   rw[li,<- Ϟ,ξ А]
  rw[ab А,e]
 unfold fP
 refine g 7 (g 6 (g 5 (g 4 (g 3 (g 2 (g 1 (g 0 ?_ ?_ ?_ ?_) ?_ ?_ ?_) ?_ ?_ ?_) ?_ ?_ ?_) ?_ ?_ ?_)
  ?_ ?_ ?_) ?_ ?_ ?_) ?_ ?_ ?_
 all_goals first
  | decide
  | (intro _;rw[Y _ (by simp only[jd];omega)];simp only[jd,I])
  | simp only[γ,jd]
  | simp[Q,Iv]
def pk (V P:FloatArray) (vo nu:с) (hs:Vд<2^32/\Pд<2^32) (hV:voб+nuб<= Vд)
  (t i:с) (hi:iб+8*(nuб-tб)<= Pд) (a0 a1 a2 a3 a4 a5 a6 a7:р)
  (relu ok:ф) (hh:Д) (n:о) (X W b:у т) (xo st o:о) (q0 so mu:с) (acc:у т):
  у т:=
 if h:t<nu then
  have ht:=γ.mp h
  have hI:iб+8<= Pд:=А
  let x:=V.uget (vo+t) (by have:=ς;rw[ab А];omega)
  pk V P vo nu hs hV (t+1) (i+8)
   (by rw[al h,ab (by have:=ς;simp only[USize.reduceToNat];omega)];simp only[USize.reduceToNat];omega)
   (a0+x*P.uget (i+0) (uk (by simp) hI hs.2))
   (a1+x*P.uget (i+1) (uk (by simp) hI hs.2))
   (a2+x*P.uget (i+2) (uk (by simp) hI hs.2))
   (a3+x*P.uget (i+3) (uk (by simp) hI hs.2))
   (a4+x*P.uget (i+4) (uk (by simp) hI hs.2))
   (a5+x*P.uget (i+5) (uk (by simp) hI hs.2))
   (a6+x*P.uget (i+6) (uk (by simp) hI hs.2))
   (a7+x*P.uget (i+7) (uk (by simp) hI hs.2)) relu ok hh n X W b xo st o q0 so mu acc
 else fP relu ok hh n X W b xo st o q0 so mu V P vo a0 a1 a2 a3 a4 a5 a6 a7 acc
termination_by nuб-tб
decreasing_by have:=γ.mp h;rw[al h];omega
lemma fl (V P:FloatArray) (d g:с) (hs:Vд<2^32/\Pд<2^32) (Y:dб+gб<= Vд)
  (l M:ф) (H:Д) (n:о) (X W b:у i9) (S Q o:о) (N O L:с) (u:у i9)
  (j:о):∀m (t i:с) (I:iб+8*(gб-tб)<= Pд) (v w x y z A B D:р),
   gб-tб=m-> iб=j+8*tб ->
   pk V P d g hs Y t i I v w x y z A B D l M H n X W b S Q o N O L u=
    fP l M H n X W b S Q o N O L V P d (ac (pp V P dб j 0) gб tб v)
     (ac (pp V P dб j 1) gб tб w) (ac (pp V P dб j 2) gб tб x)
     (ac (pp V P dб j 3) gб tб y) (ac (pp V P dб j 4) gб tб z)
     (ac (pp V P dб j 5) gб tб A) (ac (pp V P dб j 6) gб tб B)
     (ac (pp V P dб j 7) gб tб D) u:=by
 intro m;induction m<;>intro t i I v w x y z A B D hm K
 .rw[pk,dif_neg (by rw[γ];omega)]
  have e:∀p a,ac p gб tб a=a:=λp a=>by rw[ac,if_neg А]
  simp only[e]
 rename_i ih;have:=ς;have J:tб<gб:=А;have h:t<g:=γ.mpr J;have U:(i+8)б=iб+8:=by rw[ab (by simp only[jd];omega)];simp only[jd]
 have hw:(d+t)б=dб+tб:=ab А
 have q:∀r a,ac (pp V P dб j r) gб tб a=
   ac (pp V P dб j r) gб (tб+1) (a+pp V P dб j r tб):=λr a=>by
  rw[ac,if_pos J]
 have e:∀(c:с) (r:о) (G:cб=r) (Z:r<8) (E:(d+t)б<Vд) (F:(i+c)б<Pд),
   V.uget (d+t) E*P.uget (i+c) F=pp V P dб j r tб:=λc r G Z E F=>by
  rw[ug,ug,pp,gg,gg,hw,ab А,K,G]
 rw[pk,dif_pos h]
 rw[ih (t+1) (i+8) _ _ _ _ _ _ _ _ _ (by rw[al h];omega) (by rw[al h,U,K];ring),al h,
  q 0 v,q 1 w,q 2 x,q 3 y,q 4 z,q 5 A,q 6 B,q 7 D]
 simp only[e 0 0 (by simp) А,e 1 1 (by simp) А,e 2 2 (by simp) А,e 3 3 (by simp) А,
  e 4 4 (by simp) А,e 5 5 (by simp) А,e 6 6 (by simp) А,e 7 7 (by simp) А]
@[specialize] def l2 (U V:о) (f:о-> о-> у т-> у т) (u v:о) (a:у т):у т:=
 if u<U then if v<V then l2 U V f u (v+1) (f u v a) else l2 U V f (u+1) 0 a else a
termination_by (U-u,V-v)
lemma dJ (U V N:о) (f:о-> о-> у i9-> у i9) (F:о-> i9) (C:о-> о-> о-> Prop)
  (D:∀u v a,u<U-> v<V-> aв=N-> Iv F a (C u v) (f u v a)) (u v:о) (a:у i9):
  aв=N->(∀q,(∃d g,d<U/\g<V/\(d<u\/d=u/\g<v)/\C d g q)-> q<aв-> a[q]?=some (F q)) ->
  (l2 U V f u v a)в=aв /\
   ∀q,(∃d g,d<U/\g<V/\C d g q)-> q<aв->(l2 U V f u v a)[q]?=some (F q):=by
 induction u,v,a using l2.induct U V f with
 | case1 u v a m x y=>
  intro w r;rw[l2,if_pos m,if_pos x];obtain ⟨E,z⟩:=D u v a m x w
  obtain ⟨G,H⟩:=y А λq ⟨d,g,o,p,A,l⟩ j=>by
   have t:q<aв:=А;by_cases e:d=u/\g=v
   .obtain ⟨rfl,rfl⟩:=e;exact (z q).2 l t
   .have:=r q ⟨d,g,o,p,by omega,l⟩ t
    rcases (z q).1 with h | h
    .rw[h,this]
    .exact h
  exact ⟨G.trans E,λq B j=>H q B А⟩
 | case2 u v a m x y=>
  intro w r;rw[l2,if_pos m,if_neg x];exact y w λq ⟨d,g,o,p,A,l⟩ j=>r q ⟨d,g,o,p,by omega,l⟩ j
 | case3 u v a m=>
  intro w r;rw[l2,if_neg m];exact ⟨rfl,λq ⟨d,g,o,p,l⟩ j=>r q ⟨d,g,o,p,by omega,l⟩ j⟩
lemma d6 (U V N:о) (f:о-> о-> у i9-> у i9) (F:о-> i9) (C:о-> о-> о-> Prop)
  (l:∀u v a,u<U-> v<V-> aв=N-> Iv F a (C u v) (f u v a))
  (j:∀q,q<N-> ∃u v,u<U/\v<V/\C u v q):
  l2 U V f 0 0 (jM N 0)=jv (n:=N) λq=>F qи:=by
 obtain ⟨p,e⟩:=dJ U V N f F C l 0 0 (jM N 0) (by simp)
  λq ⟨r,t,_,_,h,_⟩ _=>А
 apply lN;intro q;by_cases d:q<N
 .obtain ⟨u,v,m,o,g⟩:=j q d
  rw[e q ⟨u,v,m,o,g⟩ (by simpa using d),lF];simp[d]
 .simp_all
structure PD where
 Q:FloatArray
 B:FloatArray
 bw:у т
 e:т
 eb:т
def pd (W b:у т) (n:о):PD:=
 let ow:=Wв/n
 let G:=(ow+7)/8
 let bw:=if b.isEmpty then Array.replicate ow 0x80000000 else b
 ⟨qL W nг (8*n)г n G 0 (zT (n*G+1)),
  bk (λo=>dw oneW*rX bw oг) (8*G) (FloatArray.emptyWithCapacity (8*G)),bw,aI W 0 0,aI bw 0 0⟩
def oF (t:ф) (R n ow:о) (relu:ф) (X W b:у т) (q:о):т:=
 act relu (dotOf (n+1) (λs=>if s<n then X[if t then q%R+s*R else q/ow*n+s]! else oneW)
  (λs=>if s<n then W[(if t then q/R else q%ow)*n+s]! else b[if t then q/R else q%ow]!))
lemma ts {a S:о} (h:a<= S) (d:S<2^32):aгб=a:=
 aa (by have:=ς;omega)
@[inline] def eO (t:ф) (R n ow:о) (relu:ф) (X W bw:у т) (Y Q B:FloatArray) (F:р) (hb e6:Д)
  (eok:ф) (hs:Yд<2^32/\Qд<2^32) (hB:Bд<2^32) (g r:о) (a:у т):у т:=
 let vo:=r*(n+1)
 let o:=8*g
 if h:vo+n<Yд/\o*n+1+8*n<= Qд/\o+8<= Bд then
  have hn:nгб=n:=ts А hs.1
  have hv:voгб=vo:=ts А hs.1
  have hb8:oгб+8<= Bд:=by rw[ts А hB];exact h.2.2
  let bo:=oг
  let p:=e6+23+cE F (Y.uget (vo+n)г (by rw[ts А hs.1];exact h.1))
  pk Y Q voг nг hs (by rw[hv,hn];omega) 0 (o*n+1)г
   (by rw[ts А hs.2,hn];simp only[Ϥ,Nat.sub_zero];omega)
   (B.uget (bo+0) (uk (by simp) hb8 hB)) (B.uget (bo+1) (uk (by simp) hb8 hB)) (B.uget (bo+2) (uk (by simp) hb8 hB))
   (B.uget (bo+3) (uk (by simp) hb8 hB)) (B.uget (bo+4) (uk (by simp) hb8 hB)) (B.uget (bo+5) (uk (by simp) hb8 hB))
   (B.uget (bo+6) (uk (by simp) hb8 hB)) (B.uget (bo+7) (uk (by simp) hb8 hB))
   relu (eok&&p<=400) (if p<hb then hb else p) n X W bw (if t then r else r*n) (if t then R else 1) o
   (if t then o*R+r else r*ow+o)г (if t then R else 1)г (min 8 (ow-o))г a
 else a
lemma cR (X:у i9) (n:о) (h:n<2^29) (j:∀q:о,i7 X[q]!≠255) (f:о-> о) (g:о-> р)
  (o:∀τ<n,g τ=dw X[f τ]!):
  9*((n:ℝ)+1)*(∑τ∈i0 n,|rv (dw X[f τ]!)|)*(1-2^(-20:п)) <=
   2^(((cE (9*(n+1))ж.toInt64.toFloat (ac (λτ=>(g τ).abs) n 0 0))а:п)+56)+
    2^(-900:п):=by
 have m:0<= ∑τ∈i0 n,|rv (dw X[f τ]!)| :=ϡ λ_ _=>abs_nonneg _
 have e:∑τ∈i0 n,|rv (dw X[f τ]!)| <= 2^(400:п):=by
  refine (Ϗ λτ _=>fW _ (j _)).trans ?_
  have:(n:ℝ)<=2^29:=by exact_mod_cast h.le
  rw[kg,ϯ,nsmul_eq_mul]
  calc
   (n:ℝ)*2^(277:п)<=2^29*2^(277:п):=by gcongr
   _<=2^(400:п):=by
    rw[<- zpow_natCast,<- zpow_add₀ two_ne_zero];exact zpow_le_zpow_right₀ Г Г
 obtain ⟨l,d⟩:=gm (λτ=>(g τ).abs) (λτ=> |rv (dw X[f τ]!)|) 0 n
  (λτ p=>by rw[o τ p];exact gE _ (av _ (j _)).1) А
  (by simpa using e.trans (zpow_le_zpow_right₀ Г (by norm_num:(400:ℤ)<=968))) n 0 0 rfl (km _) f0.1
  (by simp[f0.2])
 simp only[abs_zero,zero_add,abs_abs] at d;exact dE n h _ _ l m e d
lemma cX (t:ф) (R n d:о) (l:ф) (X W u:у i9) (Y Q B:FloatArray) (F:р) (hb e6:Д)
  (G:ф) (N:Yд<2^32/\Qд<2^32) (x:Bд<2^32) (e:i9) (S U L:о)
  (a1:L=Nat.log2 (n+1)+1) (a6:e6=if eк<56 then 56 else eк) (w:G=decide (e<255))
  (hF:F=(9*(n+1))ж.toInt64.toFloat) (a7:hb=(S+L-50)ж)
  (hx:∀q:о,i7 X[q]!≠255) (A:∀q:о,i7 u[q]!≠255/\je u[q]! <= U)
  (ba:174+U<= S) (hL:n+2<=2^L) (z:L<=27) (H:S+L<=425) (hn:8*n<2^32)
  (he:∀q:о,(i7 W[q]!)е<= eе) (b1:R*(n+1)<= Yд)
  (a4:∀j<R,∀τ<n,Y.data[j*(n+1)+τ]! =dw X[(if t then j else j*n)+τ*(if t then R else 1)]!)
  (a5:∀j<R,9*((n:ℝ)+1)*(∑τ∈i0 n,
    |rv (dw X[(if t then j else j*n)+τ*(if t then R else 1)]!)|)*(1-2^(-20:п)) <=
   2^(((cE F Y.data[j*(n+1)+n]!)а:п)+56)+2^(-900:п))
  (a2:∀g<(d+7)/8,8*g*n+1+8*n<= Qд)
  (hQ:∀g<(d+7)/8,∀τ<n,∀r<8,Q.data[8*g*n+1+8*τ+r]! =dw W[(8*g+r)*n+τ]!)
  (b0:8*((d+7)/8)<= Bд) (as:∀o<d,B.data[o]! =dw kf*dw u[o]!)
  (b2:R*d+9*R+d+8<=4294967295) (g r:о) (hg:g<(d+7)/8) (M:r<R) (a:у i9)
  (ha:aв=R*d):
  Iv (oF t R n d l X W u) a
   (λq=>∃k,k<8/\8*g+k<d/\q=if t then (8*g+k)*R+r else r*d+(8*g+k))
   (eO t R n d l X W u Y Q B F hb e6 G N x g r a):=by
 subst a1 a6 w hF a7;have ul:=ς;have bz:8*g<d:=А
 have h1:r*(n+1)+n<Yд:=by
  have:=Ζ (n+1) M;rw[ϋ] at this;omega
 have h2:=a2 g hg;have y:8*g+8<= Bд:=А;have bo:nгб=n:=ts А N.1;have O:(r*(n+1))гб=r*(n+1):=ts А N.1;have p:(8*g)гб=8*g:=ts А x;have bv:(8*g*n+1)гб=8*g*n+1:=ts А N.2
 have a0:∀(c:с) (m:о) (hc:cб=m) (hm:m<8) (hh:((8*g)г+c)б<Bд) (ho:8*g+m<d),
   B.uget ((8*g)г+c) hh=dw kf*dw u[8*g+m]!:=λc m hc hm hh ho=>by
  rw[ug,ab А,p,hc,as _ ho]
 have a3:∀h,Y.uget (r*(n+1)+n)г h=Y.data[r*(n+1)+n]!:=λh=>by rw[ug,ts h1.le N.1]
 have m1:8*g*R<= d*R:=Ζ R bz.le;have m2:r*d<= R*d:=Ζ d M.le;have cm:d*R=R*d:=ϩ _ _
 have D:(if t then 8*g*R+r else r*d+8*g)гб=if t then 8*g*R+r else r*d+8*g:=
  aa (by cases t<;>simp only[ο,↓reduceIte]<;>omega)
 have E:(if t then R else 1)гб=if t then R else 1:=
  aa (by cases t<;>simp only[ο,↓reduceIte]<;>omega)
 have J:(min 8 (d-8*g))гб=min 8 (d-8*g):=aa А
 have hI:∀k,(if t then 8*g*R+r else r*d+8*g)+k*(if t then R else 1)=
   if t then (8*g+k)*R+r else r*d+(8*g+k):=λk=>by
  cases t<;>simp only[ο,↓reduceIte,ϱ,ko]<;>omega
 have dm:∀a c m:о,c< m->(a*m+c)/m=a/\(a*m+c)%m=c:=λa c m h=>
  ⟨by rw[ϔ,kc _ _ А,ϣ h,j6],
   by rw[ϔ,ϵ,ξ h]⟩
 unfold eO;dsimp only;rw[dif_pos ⟨h1,h2,y⟩,fl _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (8*g*n+1) _ _ _ _ _ _ _ _ _ _ _ _ rfl (by rw[bv];simp)]
 simp only[a3,bo,O,Ϥ];refine cz _ _ _ _ _ (λq ⟨k,hk,w,hq⟩=>⟨k,hk,?_,?_⟩) (dI _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ a _ ?_ А λk hk bd=>?_)
 .rw[D,E];cases t<;>simp only[ο,↓reduceIte]<;>omega
 rotate_left
 .rw[J];omega
 .rw[D,E,hI];exact hq
 rw[J] at bd;have w:8*g+k<d:=А;rw[D,E,hI]
 have hA:A8.f ⟨ac (pp Y Q (r*(n+1)) (8*g*n+1) 0) n 0 (B.uget ((8*g)г+0) (uk (by simp) (by rw[p];exact y) x)),
   ac (pp Y Q (r*(n+1)) (8*g*n+1) 1) n 0 (B.uget ((8*g)г+1) (uk (by simp) (by rw[p];exact y) x)),
   ac (pp Y Q (r*(n+1)) (8*g*n+1) 2) n 0 (B.uget ((8*g)г+2) (uk (by simp) (by rw[p];exact y) x)),
   ac (pp Y Q (r*(n+1)) (8*g*n+1) 3) n 0 (B.uget ((8*g)г+3) (uk (by simp) (by rw[p];exact y) x)),
   ac (pp Y Q (r*(n+1)) (8*g*n+1) 4) n 0 (B.uget ((8*g)г+4) (uk (by simp) (by rw[p];exact y) x)),
   ac (pp Y Q (r*(n+1)) (8*g*n+1) 5) n 0 (B.uget ((8*g)г+5) (uk (by simp) (by rw[p];exact y) x)),
   ac (pp Y Q (r*(n+1)) (8*g*n+1) 6) n 0 (B.uget ((8*g)г+6) (uk (by simp) (by rw[p];exact y) x)),
   ac (pp Y Q (r*(n+1)) (8*g*n+1) 7) n 0 (B.uget ((8*g)г+7) (uk (by simp) (by rw[p];exact y) x))⟩ k=
   ac (pp Y Q (r*(n+1)) (8*g*n+1) k) n 0 (dw kf*dw u[8*g+k]!):=by
  interval_cases k<;>(simp only[A8.f];exact congrArg (ac _ n 0) (a0 _ _ (by simp only[jd]) А _ w))
 rw[hA];simp only[O]
 have K:∀s<n,i7 W[(8*g+k)*n+s]!≠255-> fv (pp Y Q (r*(n+1)) (8*g*n+1) k s) /\
   rv (pp Y Q (r*(n+1)) (8*g*n+1) k s)=tm X[(if t then r else r*n)+s*(if t then R else 1)]! W[(8*g+k)*n+s]!:=
  λs v h=>by rw[pp,gg,gg,a4 r M s v,hQ g hg s v k hk];exact ey _ _ (hx _) h
 have I:ap n l X W u (if t then r else r*n) (if t then R else 1) (8*g+k)=
   act l (kB (n+1) (λs=>if s<n then X[(if t then r else r*n)+s*(if t then R else 1)]! else kf)
    (λs=>if s<n then W[(8*g+k)*n+s]! else u[8*g+k]!)):=by simp only[ap,cJ]
 refine (iD l n (λs=>X[(if t then r else r*n)+s*(if t then R else 1)]!) (λs=>W[(8*g+k)*n+s]!)
  u[8*g+k]! (pp Y Q (r*(n+1)) (8*g*n+1) k) S U _ e _ _ _ (λ_=>ap n l X W u (if t then r else r*n)
  (if t then R else 1) (8*g+k)) (λ_=>sQ n l X W u Y Q (r*(n+1)) (8*g*n+1) k (if t then r else r*n)
  (if t then R else 1) (8*g+k)) (λs _=>hx _) (A _) ba hL z H hn K rfl (λs _=>he _)
  (a5 r M) (Nat.le_trans (ff _ _) (by decide:969<=1000)) ?_ I (λhw=>?_)).trans ?_
 .rw[jL,Ξ (by simp only[jT];omega)]
 .refine fu _ _ _ _ _ _ _ _ (λs v=>?_) ?_ (λs _=>?_) hL z I
  .simpa only[if_pos v] using K s v (hw s v)
  .simpa only[lt_irrefl,↓reduceIte] using ey kf u[8*g+k]! Б (A _).1
  .by_cases h:s<n<;>simp only[h,↓reduceIte]
   exacts [⟨hx _,hw s h⟩,⟨by decide,(A _).1⟩]
 .unfold oF
  cases t
  .obtain ⟨V,Z⟩:=dm r (8*g+k) d w
   simp only[V,Z,ο,↓reduceIte,ϱ]
  .obtain ⟨V,Z⟩:=dm (8*g+k) r R M
   simp only[V,Z,↓reduceIte]
@[inline] def eI (t:ф) (R n ow:о) (relu:ф) (X W:у т) (d:PD) (ex:т) (Y:FloatArray):у т:=
 let L:=Nat.log2 (n+1)+1
 let ib:=174+(d.ebе-1)
 if h:ex<255/\d.eb<255/\ib+L<=425/\L<=27/\0<n/\8*n<=4294967295/\Xв<=4294967295 /\
   ow*n<= Wв/\2*Wв+64*n<=4294967295/\d.bwв<=4294967295/\R*ow+9*R+ow+8<=4294967295 /\
   R*(n+1)+8<=4294967295/\R*(if t then 1 else n)+(n+1)*(if t then R else 1)<=4294967295 /\
   Yд<=4294967295/\d.Qд<=4294967295/\d.Bд<=4294967295 then
  let e6:=d.eк
  l2 ((ow+7)/8) R (eO t R n ow relu X W d.bw Y d.Q d.B (9*(n+1))ж.toInt64.toFloat (ib+L-50)ж
   (if e6<56 then 56 else e6) (decide (d.e<255)) ⟨co h.2.2.2.2.2.2.2.2.2.2.2.2.2.1,co h.2.2.2.2.2.2.2.2.2.2.2.2.2.2.1⟩
   (co h.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2)) 0 0 (Array.replicate (R*ow) 0)
 else Array.ofFn (n:=R*ow) λq=>
  ap n relu X W d.bw (if t then qи%R else qи/ow*n) (if t then R else 1) (if t then qи/R else qи%ow)
@[inline] def dt (t:ф) (R n ow:о) (relu:ф) (X W:у т) (d:PD) (ex:т):у т:=
 eI t R n ow relu X W d ex (dO X (if t then R else 1)г (if t then 1 else n)г (n+1)г n R (0:о)г 0
  (zT ((R*(n+1)+7)/8)))
lemma dM (W b:у i9) (n d:о) (w:0<n) (x:d*n<= Wв) (H:2*Wв+64*n<=4294967295)
  (l:(pd W b n).bwв<2^32):
  (∀g<(d+7)/8,8*g*n+1+8*n<=(pd W b n).Qд) /\
  (∀g<(d+7)/8,∀τ<n,∀r<8,(pd W b n).Q.data[8*g*n+1+8*τ+r]! =dw W[(8*g+r)*n+τ]!) /\
  8*((d+7)/8)<=(pd W b n).Bд /\ ∀o<d,(pd W b n).B.data[o]! =dw kf*dw (pd W b n).bw[o]!:=by
 have I:=ς;have t:d<= Wв/n:=(kd w).2 x;have u:Wв/n*n<= Wв:=Nat.div_mul_le_self _ _;have j:Wв/n<= Wв:=Nat.div_le_self _ _;generalize q:Wв/n=O at t u j
 have E:(O+7)/8*(8*n)<= Wв+7*n:=by
  have:8*((O+7)/8)*n<=(O+7)*n:=Ζ n А;rw[ko] at this;rw[show (O+7)/8*(8*n)=8*((O+7)/8)*n by ac_rfl];omega
 have F:(d+7)/8<=(O+7)/8:=А;obtain ⟨J,-⟩:=gu (n*((O+7)/8)+1);have G:8*(n*((O+7)/8)+1)=(O+7)/8*(8*n)+8:=by ring
 have h:(0:о)гб=0:=aa А
 obtain ⟨B,-,D⟩:=dP W nг (8*n)г n (aa А) (aa А) ((O+7)/8)
  (0:о)г (zT (n*((O+7)/8)+1)) (by rw[h];omega) (by rw[h];omega)
 have e:=bM (λo=>dw kf*rX (pd W b n).bw oг) (8*((O+7)/8)) _
  (jE (8*((O+7)/8))) rfl (km _)
 have p:(pd W b n).Q=qL W nг (8*n)г n ((O+7)/8) (0:о)г (zT (n*((O+7)/8)+1)):=by
  simp only[pd,q];rfl
 have m:(pd W b n).B=bk (λo=>dw kf*rX (pd W b n).bw oг) (8*((O+7)/8))
   (jE (8*((O+7)/8))):=by
  simp only[pd,q]
 refine ⟨λg v=>?_,λg v τ A r z=>?_,?_,λo y=>?_⟩
 .have:=Ζ (8*n) (show g+1<=(O+7)/8 by omega)
  rw[ko,l0] at this;rw[p,B,show 8*g*n=g*(8*n) by ac_rfl];omega
 .have:=D g А τ A r z
  rw[h,j6] at this;rw[p,show 8*g*n=g*(8*n) by ac_rfl,this]
  congr 2;rw[ko,show 8*g*n=g*(8*n) by ac_rfl];omega
 .rw[m,e.1];omega
 .rw[m,e.2.2 o (km _) А]
  rw[dr _ l,aa А]
lemma dp {c:Prop} [Decidable c] {α:Type} {a:c-> α} {b:¬c-> α} {y:α} (d:∀h:c,a h=y)
  (e:∀h,b h=y):dite c a b=y:=by
 by_cases h:c
 .rw[dif_pos h];exact d h
 .rw[dif_neg h];exact e h
lemma cG (t:ф) (R n d:о) (e:ф) (X W b:у i9) (A:i9) (x:A=aI X 0 0):
  dt t R n d e X W (pd W b n) A=jv (n:=R*d) λq=>oF t R n d e X W (pd W b n).bw qи:=by
 have ul:=ς;unfold dt eI;dsimp only;refine dp (λh=>?_) λ_=>?_
 swap
 .congr 1;funext q;unfold oF;cases t<;>simp only[ap,cJ,ϱ,↓reduceIte,ο]
 obtain ⟨L,U,V,Z,g5,g6,g7,g8,g9,u,v,M,N,O,Q,S⟩:=h;obtain ⟨G,H,I,J⟩:=dM W b n d g5 g8 g9 (co u);have D:=da X А (x ▸ L);have w:=da (pd W b n).bw А U
 have he:∀q:о,(i7 W[q]!)е<=(pd W b n).eе:=λq=>by
  by_cases m:q<Wв
  .exact (bN W А _ 0 0 rfl).2 q (by simp) m
  .rw[getElem!_neg W q m];exact km _
 have hL:n+2<=2^(Nat.log2 (n+1)+1):=@Ϸ (n+1);have as:R<= R*(n+1):=kX _ А;obtain ⟨z1,z2⟩:=gu ((R*(n+1)+7)/8);have hZ:(zT ((R*(n+1)+7)/8))д<2^32:=by rcases z2 with h | h<;>omega
 have z:(if t then 1 else n)<2^32:=by cases t<;>simp only[ο,↓reduceIte]<;>omega
 have y:(if t then R else 1)<2^32:=by cases t<;>simp only[ο,↓reduceIte]<;>omega
 have K:=dF X (if t then R else 1)г (if t then 1 else n)г (n+1)г n (aa А)
  А R (0:о)г 0 _ hZ (by simp only[Ϥ,j6];omega)
 simp only[Ϥ,j6] at K;obtain ⟨K1,-,E,F⟩:=K
 have p:∀j<R,∀τ<n,rX X ((0:о)г+jг*(if t then 1 else n)г+τг*(if t then R else 1)г)=
   dw X[(if t then j else j*n)+τ*(if t then R else 1)]!:=λj l τ o=>by
  have m1:=Ζ (if t then 1 else n) l.le;have m2:=Ζ (if t then R else 1) (show τ<= n+1 by omega)
  rw[dr X А,io 0 _ _ j τ z y А А А];cases t<;>simp only[ο,↓reduceIte,j6,ϱ]
 generalize dO X (if t then R else 1)г (if t then 1 else n)г (n+1)г n R (0:о)г 0 _=Y at*
 refine d6 _ _ _ _ _ (λg r q=>∃k,k<8/\8*g+k<d/\q=if t then (8*g+k)*R+r else r*d+(8*g+k))
  (λg r a hg hr ha=>cX t R n d e X W _ Y _ _ _ _ _ _ _ _ (pd W b n).e _ ((pd W b n).ebе-1) _ rfl rfl rfl rfl rfl
   (λq=>(D q).1) w le_rfl hL Z V А he А (λj l τ o=>by rw[E j l τ o,p j l τ o])
   (λj l=>by
    rw[F j l]
    exact cR X n А (λq=>(D q).1) (λτ=>(if t then j else j*n)+τ*(if t then R else 1)) _
     (λτ o=>p j l τ o)) G H I J v g r hg hr a ha) λq m=>?_
 have hw:0<d:=kD λh=>by subst h;simp at m
 have hR:0<R:=kD λh=>by subst h;simp at m
 cases t
 .have h1:=lk q d
  have B:=ϳ q hw;refine ⟨q%d/8,q/d,by omega,ϲ (by rwa[ϩ]),q%d%8,by omega,by omega,?_⟩
  simp only[ο,↓reduceIte];rw[ϩ (q/d)];omega
 .have h1:=lk q R
  have B:=ϳ q hR;have h3:q/R<d:=ϲ m;refine ⟨q/R/8,q%R,by omega,B,q/R%8,by omega,by omega,?_⟩
  simp only[↓reduceIte];rw[show 8*(q/R/8)+q/R%8=q/R by omega,ϩ (q/R)];omega
@[noinline] def TP:Sr.Tab pd:=Sr.Tab.gM pd 64 (2^22)
@[inline] def ci (W b:у т) (n:о):PD:=if Wв<16 then pd W b n else (TP.get W b n).1
lemma b8 (W b:у i9) (n:о):ci W b n=pd W b n:=by
 unfold ci;split
 .rfl
 .exact (TP.get W b n).2
def cP (rows n ow:о) (relu:ф) (X W b:у т):у т:=
 if 0<ow/\bв=ow then dt false rows n ow relu X W (ci W b n) (aI X 0 0) else linearSpec rows n ow relu X W b
lemma dK:cP=linearSpec:=by
 funext e n j d X W b
 unfold cP;split;swap;. rfl
 rename_i h
 have g:(pd W b n).bw=b:=by
  have:b.isEmpty=false:=by simp[Array.isEmpty,h.2];omega
  simp[pd,this]
 rw[b8,cG _ _ _ _ _ _ _ _ _ rfl,g];unfold linearSpec linearWith oF;simp only[ο,↓reduceIte]
def ck (batch m k n:о) (A B:у т):у т:=
 let d:=ci A #[] k
 let ex:=aI B 0 0
 if batch=1/\0<k/\m*k<= Aв/\d.e<255/\ex<255/\Aв<=4294967295/\Bв<=4294967295 then
  dt true n k m false B A d ex
 else gemmSpec batch m k n A B
lemma cO:ck=gemmSpec:=by
 funext d m k n A B
 unfold ck;dsimp only;split;swap;. rfl
 rename_i h;obtain ⟨rfl,j,l,g,o,x,y⟩:=h;rw[b8] at g⊢;have D:=ς;have v:=F.da B А o;have u:=F.da A А g;rw[cG _ _ _ _ _ _ _ _ _ rfl];have p:(pd A #[] k).bw=jM (Aв/k) 0x80000000:=by simp[pd]
 have z:m<= Aв/k:=(kd j).2 l;unfold gemmSpec gemmWith;refine Array.ext (by simp[ϩ]) λq e _=>?_
 simp only[Array.size_ofFn] at e;have r:q/n< m:=ϲ e
 have w:(pd A #[] k).bw[q/n]! =0x80000000:=by
  rw[p,getElem!_pos _ _ (by rw[ϴ];omega),Array.getElem_replicate]
 simp only[Ϩ,oF,↓reduceIte,w,act,ϣ (show q< m*n by rwa[ϩ]),
  ξ r,la,j6,ο]
 exact (F.dz k (λt=>A[q/n*k+t]!) (λt=>B[q%n+t*n]!) j λt _=>⟨(u _).1,(v _).1⟩).trans
  (by simp only[ϔ (q%n)])
end Pn
end F
end Dense
section
open Float.Model.UnpackedFloat
local notation "ZA" => rationalWord
local notation "ZB" => Array Word
local notation "ZC" => Core.mW
local notation "ZD" => Core.iW
local notation "ZE" => FloatFormat.toModel Fmt
local notation "ZF" => zpow_natCast
local notation "ZG" => UInt64.toNat_ofNat'
local notation "ZH" => decide_eq_true_eq
local notation "ZI" => Core.aN
local notation "ZJ" => Nat.div_eq_zero_iff_lt
local notation "ZK" => sqrtLower
local notation "ZL" => zpow_add₀
local notation "ZM" => Finset.range
local notation "ZO" => mantOf_lt_24
local notation "ZQ" => two_ne_zero
local notation "ZR" => Core.an
local notation "ZS" => Core.bf
local notation "ZT" => roundRational
local notation "ZU" => Nat.div_add_mod'
local notation "ZV" => Nat.div_le_self
local notation "ZW" => mantOf
local notation "ZX" => signOf
local notation "ZY" => beq_iff_eq
local notation "ZZ" => idxOf
local notation "XA" => ite_false
local notation "XB" => zpow_sub₀
local notation "XC" => two_ne_zero'
local notation "XD" => mul_assoc
local notation "XE" => ExtendedMantissa.ofMantissaAndAccuracy
local notation "XF" => UInt64.ofNat
local notation "XG" => Bool.xor
local notation "XH" => rowVariance
local notation "XI" => (А)
local notation "XJ" => (В)
local notation "XK" => (by simp)
namespace Norm
def Ya:Accuracy->о
 | .exact=>0
 | .inexact _=>1
lemma Yc {α:Type} (f:α->α) (n:о) (a:α):
  Nat.repeat f (n+1) a=Nat.repeat f n (f a):=by
 induction n generalizing a with
 | zero=>rfl
 | succ n d=>exact congrArg f (d a)
lemma QA (R k:о) (d:Accuracy):
  XE (2*R+Ya d) .exact>>>(k+2)=
   XE R d>>>(k+1):=by
 show Nat.repeat jB (k+2) _=
  Nat.repeat jB (k+1) _
 rw[Yc,Yc,Yc];congr 1;rcases d with _ | ⟨_ | _ | _⟩<;>
 simp[ExtendedMantissa.ofMantissaAndAccuracy,jB,Ya]<;>omega
open FloatLib.Numerics in
lemma VY (q r D t:о) (e:0<D) (d:r<D) (g:1<=t):
  kQ (q*D+r) (D*2^t)=
   kQ (2*q+(if r=0 then 0 else 1)) (2^(t+1)):=by
 obtain ⟨k,rfl⟩:∃k,t=k+1:=⟨t-1,XI⟩;have h:=Ϻ (2*q+Ya (jQ r D)) 1 (k+1+1)
  one_ne_zero
 rw[one_mul,Nat.div_one,Nat.mod_one,show jQ 0 1=.exact from rfl,QA] at h;rw[<-Ϻ _ _ _ XI,
  show (if r=0 then 0 else 1)=Ya (jQ r D) by
   unfold jQ;split<;>rfl,<-h,
  show (q*D+r)/D=q by rw[ϔ,kc _ _ e,ϣ d,zero_add],
  show (q*D+r)%D=r by rw[ϔ,ϵ,ξ d]]
lemma Y5 (e:п):Model.bpow e=(2:ℝ)^e:=by rfl
lemma Vi (q b:о) (d:q≠0) (h:b<=1):(2*q+b)з=qз+1:=by
 have e:=kU d;have g:=Ϸ (n:=q);rw[Nat.log2_eq_iff XI];omega
open FloatLib.Numerics in
lemma V0 (s:ф) (P d:о) (o:P≠0) (l:d≠0) (q r D:о)
  (x:0<D) (t:r<D) (e:п) (h:(P:ℝ)/d=(q+r/D)*(2:ℝ)^e)
  (z:2^25<=q) (A:-126<=(qз:п)+e) (B:(qз:п)+e<=126):
  Model.roundRat lr s P d=
   i8 lr ⟨s,2*q+(if r=0 then 0 else 1),e-1⟩:=by
 have j:q≠0:=XI;have m:=(Ϧ j).2 z;have u:((2^qз:ℕ):ℝ)<=q:=mod_cast kU j;have v:(q:ℝ)+1<=((2^(qз+1):ℕ):ℝ):=mod_cast Ϸ (n:=q);have y:(if r=0 then 0 else 1)<=1:=by split<;>omega
 have g:=Vi q _ j y;have G:(r:ℝ)/D<1:=by rw[div_lt_one XJ];exact_mod_cast t
 have E:(0:ℝ)<=r/D:=XJ;push_cast at u v;have w:RationalBinary.floorLog2 P d=(qз:п)+e:=by
  apply Model.floorLog2_eq_of_bounds P d _
  .rw[h,Y5,ZL ZQ,ZF];gcongr;grind
  .rw[h,Y5,show (qз:п)+e+1=((qз+1:о):п)+e by
    push_cast;ring,ZL ZQ,ZF];gcongr;linarith
 have p:kQ (j7 P d (23-((qз:п)+e))).1
   (j7 P d (23-((qз:п)+e))).2=
   roundShiftRightEven (2*q+if r=0 then 0 else 1) ((2*q+if r=0 then 0 else 1)з-23):=by
  rw[roundShiftRightEven_eq_roundQuotientEven,g,show qз+1-23=(qз-23)+1 by
   clear*-m g;omega,<-VY q r D _ x t XI]
  apply Model.roundQuotientEven_eq_of_rat_eq _ _ _ _
   (RationalBinary.scaleByPowerOfTwo_snd_ne_zero _ _ _ l) XJ
  rw[Model.scaleByPowerOfTwo_real];unfold Model.scaledRatToReal;rw[h,Y5,XD,<-ZL ZQ,
   show e+(23-((qз:п)+e))=-((qз-23:о):п) by clear*-m g;omega,
   zpow_neg,ZF]
  push_cast;field_simp
 generalize F:2*q+(if r=0 then 0 else 1)=J at g p⊢;rw[i8,ite_eq_left Б,Model.ieeeRoundDyadic_eq_internal];simp only[Model.roundRat,Model.roundRatScaled,show Fmt.isIEEE=true by decide,↓reduceDIte,
  Model.ieeeRoundRatScaled,Model.Internal.ieeeRoundDyadic,ZY,o,l,XA,w,
  Int.add_zero,kl,g,show (FloatFormat.ieeeMaxNormalExponent lr:п)=127 by decide,
  show FloatFormat.ieeeMinNormalExponent lr=-126 by decide,
  show (FloatFormat.normalMantissaExpOffset lr:п)=150 by decide,
  show Fmt.fracWidth=23 by decide,show J≠0 by omega,
  show ((qз+1:о):п)+(e-1)=(qз:п)+e by push_cast;ring,
  show¬((qз:п)+e>127) by omega,show¬((qз:п)+e< -150) by omega,
  show¬((qз:п)+e< -126) by omega,show 23<=qз+1 by omega,ite_true,
  j0,p]
lemma Y8 (v:ℚ) (w:v≠0) (s:ф) (u:decide (v<0)=s)
  (q:о) (ρ:ℚ) (o:0<=ρ) (g:ρ<1) (e:п)
  (j:|v|=((q:ℚ)+ρ)*(2:ℚ)^e)
  (p:2^25<=q) (l:-126<=(qз:п)+e) (m:(qз:п)+e<=126):
  ZT v=
   jH (i8 lr ⟨s,2*q+(if ρ=0 then 0 else 1),e-1⟩):=by
 have h:ρ=(ρ.numй:ℚ)/ρ.den:=by
  conv_lhs=>rw[<-Rat.num_div_den ρ]
  congr 1;exact_mod_cast (kv (Rat.num_nonneg.mpr o)).symm
 have d:=ρ.den_pos;generalize ρ.den=D,ρ.numй=r at h d;subst h;have t:r<D:=by rw[div_lt_one (mod_cast d)] at g;exact_mod_cast g
 unfold roundRational;rw[u,V0 s _ _ XJ v.den_nz q r D d t e ?_ p l m]
 .simp[d.ne']
 .have:=congrArg (λz:ℚ=>(z:ℝ)) j;rw[Rat.abs_def,Rat.divInt_eq_div] at this;push_cast at this;rwa[ku,ϕ]
lemma Q7 (R:о) (s:п) (h:2^25<=R):
  -s+1<=(ZE).targetExponent (jl R (-s)):=by
 have:=(Ϧ XI).2 h;have:(ZE).mantissaBits<=25:=Б;unfold Float.Model.Format.targetExponent jl;omega
lemma Vd (v:ℚ) (q:0<v) (e d:о) (t:0<d) (s:п)
  (p:(e:ℝ)/d=(v:ℝ)*(2:ℝ)^(2*s))
  (o:2^25<= ZK e d):
  jH (lH lr (kR (ZE) .positive
   (ZK e d) (-s) (sqrtAccuracy e d)))=jk v:=by
 have g:=sqrtAccuracy_correct e d t;have h:ZK e d≠0:=XI;have j:=le_trans (show-s<=-s+1 by omega) (Q7 _ s o);have r:ja ((e:ℝ)/d)*jR
   jh (-s)=ja (v:ℝ):=by
  rw[show jR jh (-s)=(2:ℝ)^(-s)
   from Y5 _,p,show (2:ℝ)^(2*s)=((2:ℝ)^s)^2 by
   rw[<-ZF,<-zpow_mul];ring_nf,lI XJ,
   lU XJ,XD,<-ZL ZQ]
  simp
 have l:=Model.cexp_accuracy_mul_bpow_eq_targetExponent lr _ (-s) _ _ Б h g;have m:=Model.scaledMantissa_accuracy_mul_bpow lr _ (-s) _ _ Б h g;rw[r] at l m;rw[sqrtRational_eq_math,Model.roundWithAccuracy_eq_finishRoundedMantissa];simp only[sqrtMathWord,not_le.mpr q,XA,l,m,
  <-Model.roundedMantissa_shiftToTargetExponent_eq_nearestEven lr _ (-s) _ _ g j]
 rw[Model.shiftToTargetExponent_eq_of_le_targetExponent lr _ _ _ j];rfl
lemma Y1 (w:i9):
  ZA w=(sv w:ℚ)*2^((ZZ w:п)-149):=by
 unfold rationalWord;rw[decodeD_toRat]
lemma VT (w:i9):|(sv w:ℚ)|=ZW w:=by
 unfold sv;split_ifs<;>simp
lemma YN (w:i9):(sv w:ℚ)=0↔ZW w=0:=by
 unfold sv;split_ifs<;>simp
lemma Vr (w:i9):(sv w:ℚ)=(if ZX w then-1 else 1)*ZW w:=by
 unfold sv;simp
lemma Y7 (x:Д):(ZR x)а=xаз:=by
 rw[aG];rfl
lemma YK (w:i9):(ZC w==0)=decide (ZW w=0):=by
 by_cases h:ZW w=0
 .have:ZC w=0:=Β.mp (by rw[ZI,h];rfl);grind
 .have:ZC w≠0:=λd=>h (by rw[<-ZI,d];rfl)
  grind
lemma Yk (n:о) (f:о->ℚ) (a:ℚ):
  (List.range n).foldl (λt j=>t+f j) a=a+∑j∈ZM n,f j:=by
 induction n generalizing a with
 | zero=>simp
 | succ n d=>
  rw[List.range_succ,List.foldl_append,d,ϝ];simp;ring
def Yf (num den:о) (s:п):т:=
 toWord (Model.ofModel Fmt (roundWithAccuracy (ZE) .positive
  (ZK num den) (-s) (sqrtAccuracy num den)))
open Float.Model in
lemma Q4 (spec:Format) (g:lR) (R:о) (e:п) (d:Accuracy) (l:R≠0)
  (p:e+1<= spec.targetExponent (jl R e)):
  kR spec g R e d=lG spec g (2*R+Ya d) (e-1):=by
 have m:Ya d<=1:=by cases d<;>simp[Ya]
 have h:jl (2*R+Ya d) (e-1)=jl R e:=by
  unfold jl;rw[Vi R _ l m];omega
 unfold lG decreaseExponent;simp only[h];rw[show (e-1-spec.targetExponent (jl R e))й=0 by omega];simp;unfold kR shiftToTargetExponent shiftToExponent;rw[h];generalize o:spec.targetExponent (jl R e)=T at*
 obtain ⟨k,j⟩:∃k,(T-e)й=k+1:=⟨(T-e)й-1,XI⟩;dsimp only;rw[j,show (T-(e-1))й=k+2 by omega,QA,show e-1+((k+2:о):п)=e+((k+1:о):п) by push_cast;ring]
lemma Vh (e d:о):
  Ya (sqrtAccuracy e d)=if e=ZK e d^2*d then 0 else 1:=by
 unfold sqrtAccuracy;dsimp only;split_ifs<;>rfl
lemma V9 (e d:о) (s:п) (p:33554432<= ZK e d)
  (l:ZK e d>>>31=0) (m:-209<= s) (o:s<=297):
  Yf e d s=aC false (XF (2*ZK e d+
   (if e=ZK e d*ZK e d*d then 0 else 1)))
   (XF (297-s)й):=by
 set R:=ZK e d with g
 have h:(XF (297-s)й)а=(297-s)й:=by
  rw[ZG,ξ XI]
 have q:(if e=R*R*d then 0 else 1)<=1:=by split_ifs<;>omega
 have j:(XF (2*R+(if e=R*R*d then 0 else 1)))а=
   2*R+(if e=R*R*d then 0 else 1):=by
  rw[ZG,ξ XI]
 rw[ax _ _ _ XI,h,j];unfold Yf;rw[<-g,Q4 _ _ R (-s) _ XI (Q7 R s XI),Vh,<-g,sq,
  show (((297-s)й:о):п)-298=-s-1 by omega]
 rfl
def V8 (B:о) (c:п):о:=if 0<=c then B<<<cй else B
def YF (Dn:о) (c:п):о:=if 0<=c then Dn else Dn<<<(-c)й
lemma VE (B d:о) (e:0<d) (c:п):
  0<YF d c/\((V8 B c:ℚ)/YF d c=(B:ℚ)/d*2^c):=by
 unfold V8 YF;split_ifs with hc<;>rw[Μ]
 .refine ⟨e,?_⟩;push_cast;rw[<-ZF (2:ℚ) cй,show ((cй:о):п)=c by omega];ring
 .refine ⟨XJ,?_⟩;push_cast;rw[<-ZF (2:ℚ) (-c)й,show (((-c)й:о):п)=-c by omega];have:(2:ℚ)^(-c)≠0:=XJ;rw[zpow_neg];grind
@[inline] def V2 (r:о) (st:ф):о:=if r !=0 || st then 1 else 0
lemma VI (N M:о) (e:0<M) (δ:ℚ) (g:0<=δ) (j:δ<1):
  (N:ℚ)+δ=((N/M:о)+((N%M:о)+δ)/M)*M/\0<=((N%M:о)+δ)/M/\
   ((N%M:о)+δ)/M<1/\(((N%M:о)+δ)/M=0↔N%M=0/\δ=0):=by
 have d:(0:ℚ)<M:=mod_cast e;have l:((N%M:о):ℚ)+1<=M:=mod_cast ϳ N e;refine ⟨by rw[show (N:ℚ)=(N/M:о)*M+(N%M:о) from mod_cast (ZU N M).symm];field_simp;ring,XJ,by rw[div_lt_one d];linarith,?_⟩;rw[div_eq_zero_iff,or_iff_left d.ne'];constructor
 .intro h;have:(0:ℚ)<=(N%M:о):=XJ;exact ⟨mod_cast (by linarith:((N%M:о):ℚ)=0),by linarith⟩
 .rintro ⟨h,rfl⟩;simp[h]
lemma Qx (s:ф) (W:о) (l:ф) (d:о) (e:п) (v δ:ℚ) (M σ:Д)
  (A:v≠0->decide (v<0)=s) (p:0<=δ) (q:δ<1) (w:l=true↔δ≠0) (x:0<d)
  (r:|v|=((W:ℚ)+δ)*2^e/d) (z:2^25<=W/d) (t:-126<=((W/d)з:п)+e)
  (u:((W/d)з:п)+e<=126) (y:Mа=2*(W/d)+V2 (W%d) l)
  (B:(σа:п)=e+297):
  aC s M σ=ZT v:=by
 obtain ⟨E,m,o,g⟩:=VI W d x δ p q;have h:|v|=(((W/d:о):ℚ)+((W%d:о)+δ)/d)*2^e:=by
  rw[r,E];field_simp
 have j:v≠0:=abs_pos.1 (h▸XJ);rw[ax _ _ _ XI,Y8 v j s (A j) _ _ m o _ h z t u,y];congr 3
 .clear*-g w;simp only[V2,g];grind
 omega
def sg (s:ф) (M:ℚ):ℚ:=if s then-M else M
lemma Q6 (w:i9):(ZC w !=0)=!decide (ZW w=0):=by
 rw[show (ZC w !=0)=!(ZC w==0) from rfl,YK]
lemma Y2 (w:i9):ZA w=sg (ZX w) (ZW w)*2^((ZZ w:п)-149):=by
 rw[Y1,Vr];unfold sg;simp
lemma aL (w:i9):je w<= 254:=by
 have:=expOf_toNat_lt w;unfold je;grind
namespace NF
open Dense.F
open Dense (au)
open FloatLib.Floats.Formats.Flocq in
lemma ds (z:ℝ): |Model.roundAt i5 z-z| <= |z| /2^53+1/2^1075:=by
 by_cases g:z=0
 .subst g;simp
 have l:0< |z| :=abs_pos.mpr g
 have d: |z| <(2:ℝ)^(ky 2 |z| +1):=by
  have:=kK (R:=ℝ) (b:=2) Г |z|;omega
 have m:(2:ℝ)^(ky 2 |z|)<= |z| :=by
  have:=lv (R:=ℝ) (b:=2) Г l;omega
 have o:(0:ℝ)<=1/2^1075:=XJ;by_cases j:-1021<= ky 2 |z| +1
 .have:=re z _ j d
  have e:(2:ℝ)^(ky 2 |z| +1-53)/2=(2:ℝ)^(ky 2 |z|)/2^53:=by
   rw[show ky 2 |z| +1-53=ky 2 |z| +(-52) by ring,zpow_add₀ Г]
   ring
  grind
 .have h: |z| <(2:ℝ)^(-1021:ℤ):=lt_of_lt_of_le d (zpow_le_zpow_right₀ Г А)
  have:=re z (-1021) le_rfl h
  have e:(2:ℝ)^((-1021:ℤ)-53)/2=1/2^1075:=by
   rw[show ((-1021:ℤ)-53)=-((1074:ℕ):ℤ) by norm_num,zpow_neg,zpow_natCast];grind
  grind
lemma fA (x y:р) (h:fv x) (j:fv y) (g: |rv x+rv y| <= (2:ℝ)^(971:п)):
  fv (x+y) /\ |rv (x+y)-(rv x+rv y)| <= |rv x+rv y| /2^53+1/2^1075:=by
 obtain ⟨d,e⟩:=db x y h j (g.trans (pm 971 le_rfl));exact ⟨d,by rw[e];exact ds _⟩
lemma fM (x y:р) (h:fv x) (j:fv y) (g: |rv x*rv y| <= (2:ℝ)^(971:п)):
  fv (x*y) /\ |rv (x*y)-rv x*rv y| <= |rv x*rv y| /2^53+1/2^1075:=by
 obtain ⟨d,e⟩:=d4 x y h j g;exact ⟨d,by rw[e];exact ds _⟩
lemma f8 (b:Д):(mF b)а=(b&&&0xfffffffffffff)а+2^52:=by
 have d:(b&&&0xfffffffffffff)а<2^52:=by
  rw[ψ];exact lt_of_le_of_lt ϙ Б
 unfold mF;rw[kp,show (0x10000000000000:Д)а=2^52*1 from rfl,Nat.or_comm,
  <- ϓ d,ϱ,ϔ]
lemma fo (v Y U:ℝ) (d:ф) (M L c:ℕ) (o:0<U) (j:v=(if d then-(M:ℝ) else M)*U)
  (u:L=M%2^27) (g:2^c<L) (h:L+2^c<2^27) (e: |v-Y| <= 2^c*U):
  ((M/2^27:ℕ):ℝ)*2^27*U<(if d then-Y else Y) /\
   (if d then-Y else Y)<(((M/2^27:ℕ):ℝ)+1)*2^27*U:=by
 have w:(M:ℝ)=((M/2^27:ℕ):ℝ)*2^27+L:=by
  rw[u];exact_mod_cast (Nat.div_add_mod' M (2^27)).symm
 have l:((2^c:ℕ):ℝ)<L:=by exact_mod_cast g
 have m:(L:ℝ)+((2^c:ℕ):ℝ)<2^27:=by exact_mod_cast h
 push_cast at l m;have p:(2:ℝ)^c*U<L*U:=mul_lt_mul_of_pos_right l o;have q:(L:ℝ)*U+2^c*U<2^27*U:=by linarith only[mul_lt_mul_of_pos_right m o]
 obtain ⟨r,t⟩:=abs_le.1 e
 cases d<;>simp only[ite_true,ο,ite_false] at j⊢<;>rw[w] at j<;>
  constructor<;>linarith only[j,r,t,p,q]
end NF
namespace NE
open NF Dense.F
open Dense (au)
@[inline] def mx (a b:Д):Д:=if a<b then b else a
lemma aA (m e:Д):(mx m e)а=lm mа eа:=by
 unfold mx;split<;>rename_i h<;>simp only[ν] at h<;>omega
@[inline] def pk (s:ф) (q σ:Д):т:=
 ((if s then (0x80000000:Д) else 0)+((σ-146)<<<23+Core.aj (((2:Д)*q+1)<<<36)))л
lemma gb (s:ф) (q σ:Д) (j:2^25<= qа) (l:qа<2^26) (m:146<= σа)
  (o:σа<=398):pk s q σ=aC s ((2:Д)*q+1) σ:=by
 have d:((2:Д)*q+1)а=2*qа+1:=by
  rw[μ,Ϛ];simp only[iZ];omega
 have g:(((2:Д)*q+1)<<<36)а=(2*qа+1)*2^36:=by
  rw[Ι,d];simp only[iZ,jb,Μ];omega
 have h:(σ-36)а=σа-36:=by
  rw[χ σ 36 (κ.2 (by simp only[iZ];omega))];rfl
 have e:aC s ((2:Д)*q+1) σ=aC s (((2:Д)*q+1)<<<36) (σ-36):=by
  rw[cD s ((2:Д)*q+1) σ А А,
   cD s (((2:Д)*q+1)<<<36) (σ-36) А А,d,g,h,
   show (2:п)^σа=2^36*2^(σа-36) by rw[<- pow_add,Nat.add_sub_cancel' (show 36<= σа by omega)]]
  cases s<;>simp only[ite_true,ite_false,ο]<;>push_cast<;>ring_nf
 have p:(pk s q σ)е=(if s then 2^31 else 0)+((σа-146)*2^23+
   (aj (((2:Д)*q+1)<<<36))а):=by
  obtain ⟨-,-,t⟩:=bT s (((2:Д)*q+1)<<<36) (σ-36) А А А А;unfold pk
  rw[ϟ,μ,μ,Ι,
   χ σ 146 (κ.2 (by simp only[iZ];omega))]
  cases s<;>simp only[ite_true,ite_false,ο,iZ,jb,Μ]<;>omega
 obtain ⟨r,-,-⟩:=bT s (((2:Д)*q+1)<<<36) (σ-36) А А А А;rw[e];apply Η.mp;omega
lemma mo (a:ℝ) (p q:ℕ) (h: |a| <= 2^p) (d:p<= q): |a| <= 2^q:=
 h.trans (pow_le_pow_right₀ Г d)
lemma z9 (a:ℝ) (p:ℕ) (h: |a| <= 2^p) (d:p<=971): |a| <= (2:ℝ)^(971:ℤ):=by
 rw[show (971:ℤ)=((971:ℕ):ℤ) from rfl,zpow_natCast];exact mo a p 971 h d
lemma wB (w:i9): |(sv w:ℝ)*2^je w| <= 2^(24+je w):=by
 rw[abs_mul,abs_of_pos (by positivity:(0:ℝ)<2^idxOf w),pow_add]
 have: |(sv w:ℝ)| <= 2^24:=by
  rw[<- ϕ,kO];exact_mod_cast (eT w).le
 exact mul_le_mul_of_nonneg_right this В
end NE
namespace NR
open NF Dense.F
open Dense (au)
@[inline] def pt (j:о):р:=P.get! (j+1)
lemma iC (j:о) (g:j<=254):fv (pt j)/\rv (pt j)=(2:ℝ)^j:=by
 have h:pt j=pw j:=by
  have d:P.get! (j+1)=P.data[j+1]!:=by rcases P with ⟨e⟩;rfl
  unfold pt;rw[d,getElem!_pos P.data (j+1) (by simp[P];omega)]
  simp only[P,Ϩ];rw[if_pos А];rfl
 rw[h];exact gh j А
lemma iy (n:о) (g:n<2^29):fv nж.toInt64.toFloat/\rv nж.toInt64.toFloat=n:=by
 have d:nжа=n:=Ξ (show n<2^64 by omega)
 have e:nж.toInt64н=n:=by show bi nж=n;rw[au,d];split<;>omega
 have:=cy _ (by rw[e];simp;omega);rw[e] at this;omega
lemma d0 (a:ZB) (h:∀i (d:i<aв),jS a[i]=true) (i:о) (d:i<aв):i7 a[i]!≠255:=by
 have:=h i d;rw[getElem!_pos a i d];unfold jS at this;grind
end NR
lemma uS {i:с} (h:iб+1<ke):(i+1)б=iб+1:=by
 rw[ks,lf];exact ξ h
lemma Qg {α:Type} [Inhabited α] {m k:о} (f:Fin m->α) (h:k< m):
  (jv f)[k]! =f ⟨k,h⟩:=by
 rw[getElem!_pos _ k (by simpa using h),Ϩ]
@[inline] def SN (rows n:о) (eps:т) (x g:ZB):ф:=
 rows>0&&n>0&&xв==rows*n&&gв==n&&x.all finiteWord&&g.all finiteWord&&finiteWord eps&&(!ZX eps&&ZC eps !=0)
lemma e9 (e n:о) (d:i9) (x g:ZB):SN e n d x g=ji e n d x g:=by
 unfold SN ji;congr 1;rw[Q6,Y2];have:(0:ℚ)<2^((ZZ d:п)-149):=XJ;unfold sg;cases ZX d<;>simp[this,Nat.pos_iff_ne_zero]
lemma fI {d n:о} {e:i9} {x g:ZB} (h:ji d n e x g=true):
  0<n/\0<ZA e/\(∀i,i<d*n-> i7 x[i]!≠255)/\∀j,j<n-> i7 g[j]!≠255:=by
 simp only[ji,υ,ZH,gt_iff_lt] at h;have p:xв=d*n:=by simpa using h.1.1.1.1.1.2
 have l:gв=n:=by simpa using h.1.1.1.1.2
 exact ⟨h.1.1.1.1.1.1.2,h.2,λi m=>NR.d0 x (by simpa[j8] using h.1.1.1.2) i А,
  λj o=>NR.d0 g (by simpa[j8] using h.1.1.2) j А⟩
namespace NF
open Dense.F
lemma gR (x y:р) (d:fv x) (e:fv y):
  i6 (x-y).toModel=Model.sub (i6 x.toModel) (i6 y.toModel):=by
 change i6 (l7 x.toModel y.toModel)=_;rw[l7,Ϣ]
 have h:=Model.sub_eq_ofModel_sub_of_isFinite (fmt:=i5) Б
  (i6 x.toModel) (i6 y.toModel)
  (by rw[Θ];exact (bB x).1 d) (by rw[Θ];exact (bB y).1 e)
 rw[Θ,Θ] at h;exact h.symm
lemma il (x y:р) (d:fv x) (e:fv y) (j: |rv x-rv y| <= jJ (kG i5)):
  fv (x-y)/\rv (x-y)=kh i5 (rv x-rv y):=by
 have h:=gR x y d e
 have g:=Model.isFinite_sub_of_abs_toReal_sub_le_posMaxFinite (i6 x.toModel)
  (i6 y.toModel) Б d e j
 refine ⟨by unfold fv;rw[h];exact g,?_⟩
 unfold rv;rw[h];exact Model.toReal_sub_eq_roundAt _ _ Б g
lemma iT (x:р):ki (i6 x.sqrt.toModel)=ki (Model.sqrt (i6 x.toModel)):=by
 change ki (i6 (lZ x.toModel))=_;rw[lZ,Ϣ,show jt=FloatFormat.binary64.toModel from rfl];have h:=Model.canonicalizeModel_sqrt_eq_model (fmt:=i5) Б (i6 x.toModel);rw[Θ] at h;rw[<- h,Model.toModel_canonicalizeModel]
lemma ir {y z:Model i5} (h:ki y=ki z):jW y=jW z/\jJ y=jJ z:=by
 constructor
 .rw[<- Ϝ,ϧ _ Б,<- Ϝ,ϧ _ Б,h]
 .rw[ό Б,ό Б,h]
lemma iS (x:р) (l:fv x) (g:0<rv x):Model.signBit (i6 x.toModel)=false:=by
 obtain ⟨d,h⟩:=Model.exists_toDyadic?_of_isFinite l;rw[<- Model.sign_eq_signBit_of_toDyadic?_some h];have e:rv x=d.toReal:=by unfold rv;simp[jK,h]
 cases j:d.negative with
 | false=>rfl
 | true=>
  exfalso;rw[e] at g;simp only[jz,j] at g;have:(0:ℝ)<=(d.significand:ℝ)*(2:ℝ)^d.exponent:=В;simp only[jC,j] at g;rw[kl] at g;push_cast at g;linarith
lemma ik (x:р) (e:fv x) (h:0<rv x):fv x.sqrt/\rv x.sqrt=kh i5 (ja (rv x)):=by
 have g:=iS x e h;have d:=Model.isFinite_sqrt_of_isFinite _ Б e (ma g);have j:=Model.toReal_sqrt_eq_roundAt _ Б e (ma g);obtain ⟨a,b⟩:=ir (iT x);unfold fv rv;rw[a,b];exact ⟨d,j⟩
lemma fS (x y:р) (h:fv x) (j:fv y) (g:|rv x-rv y|<=(2:ℝ)^(971:п)):
  fv (x-y)/\|rv (x-y)-(rv x-rv y)|<=|rv x-rv y|/2^53+1/2^1075:=by
 obtain ⟨d,e⟩:=il x y h j (g.trans (pm 971 le_rfl));exact ⟨d,by rw[e];exact ds _⟩
lemma fQ (x:р) (h:fv x) (d:0<rv x):fv x.sqrt/\|rv x.sqrt-ja (rv x)|<= ja (rv x)/2^53+1/2^1075:=by
 obtain ⟨e,g⟩:=ik x h d;refine ⟨e,?_⟩;rw[g];have:=ds (ja (rv x));rwa[abs_of_nonneg (ϊ _)] at this
lemma rc (y r:р) (p:fv y) (q:1<= rv y) (o:rv y<=2^400) (v:(eF r.toBits)а<1100)
  (h:(r*y).toBits-0x3FEFFFFFFFFFFFFF<2):fv r/\|rv r-1/rv y|<=3/2^53/rv y:=by
 have g:=f6 r А;have d:(0:ℝ)<rv y:=by linarith
 have e:|rv r|<=2^(78:п):=(bj r g).le.trans (zpow_le_zpow_right₀ Г А)
 obtain ⟨m,l⟩:=fM r y g p (by
  rw[abs_mul,abs_of_pos d];simp only[zpow_ofNat] at e⊢;exact (mul_le_mul e o d.le В).trans (by rw[<-pow_add];exact pow_le_pow_right₀ Г Г))
 have u:|rv (r*y)-1|<=1/2^53:=by
  have:=ν.mp h;have:=(r*y).toBits.toNat_lt;simp only[Κ,iZ] at*
  obtain j|j:(r*y).toBits=0x3FEFFFFFFFFFFFFF\/(r*y).toBits=0x3FF0000000000000:=by simp only[<-Β,iZ];omega
  all_goals rw[aK _ m (by rw[j];decide),aw,j];norm_num+decide [eF]
 have w:=abs_sub_le (rv r*rv y) (rv (r*y)) 1;have x:=abs_sub_abs_le_abs_sub (rv r*rv y) 1;have t:(1:ℝ)/2^1075<=1/2^60:=by gcongr<;>norm_num
 rw[abs_sub_comm] at l;generalize (1:ℝ)/2^1075=τ at*
 refine ⟨g,?_⟩;rw[show rv r-1/rv y=(rv r*rv y-1)/rv y by field_simp,abs_div,abs_of_pos d];gcongr
 norm_num at*;linarith
end NF
open NF Dense.F
@[inline] def pq (B:Д):т:=(((B>>>63)<<<31)+(((B&&&0x7FFFFFFFFFFFFFFF)+0x10000000)>>>29)-((1045:Д)<<<23))л
lemma iM (q:Д) (h:qа<2^26):(aj ((2*q+1)<<<36))а=(qа+2)/4:=by
 have e:((2*q+1)<<<36)а=(2*qа+1)*2^36:=by
  simp only[Ι,μ,Ϛ,iZ,jb,Μ];omega
 unfold aj;simp only[Α,μ,ψ,e,iZ,jb,ι,kF];generalize qа=a at*
 obtain ⟨k,j,d,rfl⟩:∃k j,j<8/\a=8*k+j:=⟨a/8,a%8,ϳ _ Г,(lk a 8).symm⟩;interval_cases j<;>omega
lemma g0 (B:Д):(eF B)а=Bа/2^52%2048:=by
 unfold eF;rw[ψ,Α];simp only[iZ,jb,ι];exact θ _ 11
lemma iu (B:Д):(mF B)а=Bа%2^52+2^52:=by
 rw[f8,ψ];congr 1;exact θ _ 52
lemma iA (B:Д) (d:146<=(eF B-900)а) (e:(eF B-900)а<=398):pq B=NE.pk (B>>>63!=0) (mF B>>>27) (eF B-900):=by
 have g:(eF B)а=Bа/2^52%2048:=by
  unfold eF;rw[ψ,Α];simp only[iZ,jb,ι];exact θ _ 11
 have h:=f8 B;have q:(B&&&0xfffffffffffff)а=Bа%2^52:=by rw[ψ];exact θ _ 52
 have j:(B>>>63)а=Bа/2^63:=by rw[Α];simp only[iZ,jb,ι]
 have p:(B&&&0x7FFFFFFFFFFFFFFF)а=Bа%2^63:=by rw[ψ];exact θ _ 63
 have t:=B.toNat_lt;rw[q] at h;have l:(mF B>>>27)а=(Bа%2^52+2^52)/2^27:=by rw[Α,h];simp only[iZ,jb,ι]
 have o:(eF B-900)а=Bа/2^52%2048-900:=by
  rw[Κ,g];simp only[iZ] at d e⊢;rw[Κ,g] at d e;simp only[iZ] at d e;omega
 have m:=iM (mF B>>>27) (by rw[l];omega)
 rw[l] at m
 have r:(B>>>63!=0)=decide (Bа/2^63=1):=by
  rw[ϻ];simp only[bne_iff_ne,ne_eq,decide_eq_true_eq,<-Β,j,iZ];omega
 apply Η.mp;unfold pq NE.pk;rw[r]
 simp only[ϟ,μ,Κ,Ι,Α,
  j,p,o,m,iZ,jb,Μ,ι]
 rw[o] at d e;generalize Bа=N at*
 split<;>simp only[iZ,decide_eq_true_eq] at*<;>omega
lemma gc (B:Д) (d:1046<=(eF B)а) (e:(eF B)а<=1298):pq B=NE.pk (B>>>63!=0) (mF B>>>27) (eF B-900):=by
 have:(eF B-900)а=(eF B)а-900:=χ _ _ (κ.2 (by simp only[iZ];omega))
 exact iA B А А
lemma tc (B c:Д) (l:c<=25) (h:(B&&&0x7FFFFFF)-((1:Д)<<<c)-1<0x7FFFFFF-2*((1:Д)<<<c)):
  2^cа<Bа%2^27/\Bа%2^27+2^cа<2^27:=by
 have d:cа<=25:=κ.1 l
 have e:((1:Д)<<<c)а=2^cа:=by
  rw[Ι,show (1:Д)а=1 from rfl,ξ (show cа<64 by omega),ϭ]
  exact ξ (lz Г (show cа<64 by omega))
 have g:(B&&&0x7FFFFFF)а=Bа%2^27:=by rw[ψ];exact θ _ 27
 have:2^cа<=2^25:=Ε Г d;have j:=ν.1 h;simp only[Κ,Ϛ,e,g,iZ] at j;have:=ϳ Bа (show 2^27>0 by norm_num)
 omega
lemma te (E:Д) (h:E-1046<=252):1046<=Eа/\Eа<=1298:=by
 have d:=κ.1 h;rw[Κ] at d;have:=E.toNat_lt;simp only[iZ] at d;omega
lemma cl (V:р) (w:fv V) (c:ℕ) (Z:ℝ) (g:1046<=(eF V.toBits)а) (q:(eF V.toBits)а<=1298)
  (j:2^c<V.toBitsа%2^27) (l:V.toBitsа%2^27+2^c<2^27)
  (u:|rv V-Z|<=2^c*(2:ℝ)^(((eF V.toBits)а:п)-1075)):
  2^25<=(mF V.toBits>>>27)а/\(mF V.toBits>>>27)а<2^26/\
  ((mF V.toBits>>>27)а:ℝ)*2^(((eF V.toBits)а:п)-1048)<(if V.toBits>>>63!=0 then-Z else Z)/\
  (if V.toBits>>>63!=0 then-Z else Z)<(((mF V.toBits>>>27)а:ℝ)+1)*2^(((eF V.toBits)а:п)-1048):=by
 have o:¬eF V.toBits=0:=λh=>by rw[h] at g;simp at g
 have d:=aK V w o;rw[aw] at d;have p:=iu V.toBits;have r:(V.toBits&&&0xfffffffffffff)а=V.toBitsа%2^52:=by rw[ψ];exact θ _ 52
 rw[r,show V.toBitsа%2^52+2^52=(mF V.toBits)а by rw[p]] at d
 have t:V.toBitsа%2^27=(mF V.toBits)а%2^27:=by rw[p];omega
 rw[t] at j l;have m:=fo _ Z _ _ _ _ c В d rfl j l u;have v:(mF V.toBits>>>27)а=(mF V.toBits)а/2^27:=by rw[Α];simp only[iZ,jb,ι]
 have x:=V.toBits.toNat_lt
 have e:(2:ℝ)^27*2^(((eF V.toBits)а:п)-1075)=2^(((eF V.toBits)а:п)-1048):=by
  rw[<-zpow_natCast,<-zpow_add₀ Г];congr 1;push_cast;ring
 rw[v];refine ⟨by omega,by omega,?_,?_⟩
 .rw[<-e,<-mul_assoc];exact m.1
 .rw[<-e,<-mul_assoc];exact m.2
lemma eC (V:р) (K:fv V) (c:ℕ) (Y:ℚ) (l:1046<=(eF V.toBits)а) (m:(eF V.toBits)а<=1298)
  (r:2^c<V.toBitsа%2^27) (t:V.toBitsа%2^27+2^c<2^27)
  (I:|rv V-(Y:ℝ)*2^149|<=2^c*(2:ℝ)^(((eF V.toBits)а:п)-1075)):pq V.toBits=ZT Y:=by
 obtain ⟨B,H,G,D⟩:=cl V K c _ l m r t I;have L:(eF V.toBits-900)а=(eF V.toBits)а-900:=by rw[χ _ _ (κ.2 (by simp only[iZ];omega))];rfl
 rw[gc _ l m,NE.gb _ _ _ B H А А];set q:=(mF V.toBits>>>27)а with J;set s:=V.toBits>>>63!=0 with N;set E:=(eF V.toBits)а with M;set e:ℤ:=(E:ℤ)-1197 with u;have k:∀a:ℝ,a*2^((E:ℤ)-1048)=a*2^e*2^149:=λa=>by rw[mul_assoc,<-zpow_natCast (2:ℝ) 149,<-zpow_add₀ Г];congr 2;rw[u];push_cast;ring
 have F:(if s then-((Y:ℝ)*2^149) else (Y:ℝ)*2^149)=((if s then-Y else Y:ℚ):ℝ)*2^149:=by split<;>push_cast<;>ring
 rw[k,F] at G;rw[k,F] at D;have A:=lt_of_mul_lt_mul_right G В;have v:=lt_of_mul_lt_mul_right D В;have p:(q:ℚ)*2^e<(if s then-Y else Y):=by exact_mod_cast (by push_cast;ring:(((q:ℚ)*2^e:ℚ):ℝ)=(q:ℝ)*2^e).trans_lt A
 have w:(if s then-Y else Y)<((q:ℚ)+1)*2^e:=by exact_mod_cast v.trans_eq (by push_cast;ring:((q:ℝ)+1)*2^e=((((q:ℚ)+1)*2^e:ℚ):ℝ))
 have o:(0:ℚ)<2^e:=В;have j:0<(if s then-Y else Y):=lt_of_le_of_lt В p;have d:|Y|=(if s then-Y else Y):=by grind
 set δ:ℚ:=(if s then-Y else Y)/2^e-q with x;have y:0<δ:=by rw[x,sub_pos,lt_div_iff₀ o];exact p
 have z:δ<1:=by rw[x,sub_lt_iff_lt_add,div_lt_iff₀ o];linarith only[w]
 have g:qз=25:=le_antisymm (Nat.lt_succ_iff.1 ((lu А).2 H)) ((Ϧ А).2 B)
 rw[<-Qx s q true 1 e Y δ ((2:Д)*(mF V.toBits>>>27)+1) (eF V.toBits-900)
  (by intro _;split at d<;>rename_i h<;>simp only[h,ite_true,ο,ite_false] at j⊢<;>
   simp only[decide_eq_true_eq,decide_eq_false_iff_not,not_lt]<;>linarith only[j])
  y.le z (by grind) one_pos (by rw[d,x];field_simp;ring) (by simpa using B) (by simp[g];omega) (by simp[g];omega)
  (by rw[μ,Ϛ];simp[V2,<-J];omega) (by rw[L];omega)]
lemma dC (t:р) (N:fv t) (w:ℕ) (v:ℚ) (p:0<v) (e:1046<=(eF t.toBits)а) (g:(eF t.toBits)а<=1298)
  (x:2^w<t.toBitsа%2^27) (y:t.toBitsа%2^27+2^w<2^27)
  (K:|rv t-ja (v:ℝ)*2^149|<=2^w*(2:ℝ)^(((eF t.toBits)а:п)-1075)):pq t.toBits=jk v:=by
 obtain ⟨U,V,l,o⟩:=cl t N w _ e g x y K;set q:=(mF t.toBits>>>27)а with Y;set E:=(eF t.toBits)а with W;have X:0<ja (v:ℝ)*2^149:=mul_pos (Real.sqrt_pos.2 (by exact_mod_cast p)) В
 have I:(t.toBits>>>63!=0)=false:=by
  by_contra h;simp only[Bool.not_eq_false] at h;rw[if_pos h] at l;linarith[show (0:ℝ)<=(q:ℝ)*2^((E:ℤ)-1048) by positivity]
 rw[I] at l o;simp only[ο,ite_false] at l o;set s:ℤ:=1197-E with M;have k:(2:ℝ)^149=2^s*2^((E:ℤ)-1048):=by rw[<-zpow_natCast,<-zpow_add₀ two_ne_zero];congr 1;omega
 rw[k,<-mul_assoc] at l o;have O:=lt_of_mul_lt_mul_right l В;have Q:=lt_of_mul_lt_mul_right o В
 have j:(ja v*2^s)^2=(v:ℝ)*2^(2*s):=by
  rw[mul_pow,lb (by exact_mod_cast p.le),<-zpow_natCast,<-zpow_mul];ring_nf
 have B:(q:ℝ)^2<(v:ℝ)*2^(2*s):=by rw[<-j];exact pow_lt_pow_left₀ O В Г
 have D:(v:ℝ)*2^(2*s)<((q:ℝ)+1)^2:=by rw[<-j];exact pow_lt_pow_left₀ Q В Г
 have L:0<v.num:=Rat.num_pos.2 p;set a:=v.numм with J
 have u:(v:ℝ)=a/v.den:=by
  rw[J,ku,ϕ,abs_of_pos (by exact_mod_cast L)];exact_mod_cast (Rat.num_div_den v).symm
 obtain ⟨d,r⟩:=VE a v.den v.den_pos (2*s);set num:=V8 a (2*s);set den:=YF v.den (2*s);have G:(num:ℝ)/den=(v:ℝ)*2^(2*s):=by rw[u];have:=congrArg (λz:ℚ=>(z:ℝ)) r;push_cast at this;exact this
 rw[<-G] at B D;have H:(0:ℝ)<den:=by exact_mod_cast d
 rw[sq] at B;rw[sq] at D;have F:q*q*den<num:=by exact_mod_cast (lt_div_iff₀ H).1 B
 have S:num<(q+1)*(q+1)*den:=by exact_mod_cast (div_lt_iff₀ H).1 D
 have m:sqrtLower num den=q:=(Nat.eq_sqrt.2 ⟨(kd d).2 F.le,(Nat.div_lt_iff_lt_mul d).2 S⟩).symm;have A:(eF t.toBits-900)а=E-900:=by rw[χ _ _ (κ.2 (by simp only[iZ];omega))];rfl
 rw[<-Vd v p num den d s G (by rw[m];omega)]
 change pq t.toBits=Yf num den s
 rw[V9 num den s (by rw[m];omega) (by rw[m];exact Nat.shiftRight_eq_zero _ _ А) А А,m,
  if_neg (by intro h;rw[h] at F;exact lt_irrefl _ F),gc _ e g,I,NE.gb _ _ _ U V (by rw[A];omega) (by rw[A];omega)]
 congr 1
 .apply Β.mp;rw[μ,Ϛ,ZG];simp only[iZ];omega
 .apply Β.mp;rw[ZG,A];omega
lemma fY (w:i9):(i7 w)е=wе/2^23%256:=by
 unfold i7;rw[Λ,Ρ];simp only[jj,jb,ι];exact θ _ 8
lemma f5 (w:i9):(lW w)е=wе%2^23:=by
 unfold lW;rw[Λ];simp only[jj];exact θ _ 23
lemma wV (w:i9) (K R:ℕ) (h:wе=K*2^23+R) (p:2^23<= R) (u:R<=2^24) (v:K<=252):
  i7 w≠255/\jw w=false/\(sv w:ℝ)*2^je w=R*2^K:=by
 have g:=fY w;have t:=f5 w
 have o:jw w=false:=by
  have:¬2^31<= wе:=А;exact Bool.eq_false_iff.2 (λd=>this ((aP w).1 d))
 have q:(i7 w)е≠255:=by rw[g,h];omega
 refine ⟨λd=>q (by rw[d];rfl),o,?_⟩
 have e:i7 w≠0:=λd=>by rw[d] at g;simp at g;omega
 unfold sv jF je;rw[o,if_neg e,if_neg e];simp only[ο,ite_false];rw[t,g,h];rcases Nat.lt_or_ge R (2^24) with r|r
 .have l:(K*2^23+R)/2^23%256=K+1:=А
  have m:(K*2^23+R)%2^23=R-2^23:=А;rw[l,m,show K+1-1=K by omega,Nat.cast_add,k9 p];push_cast;ring
 .have j:R=2^24:=А
  subst j;have l:(K*2^23+2^24)/2^23%256=K+2:=А;have m:(K*2^23+2^24)%2^23=0:=А;rw[l,m,show K+2-1=K+1 by omega];push_cast;ring
lemma iB (B:Д) (r:1046<=(eF B)а) (t:(eF B)а<=1298) (h:Bа<2^63):
  (pq B)е=((eF B)а-1046)*2^23+(2^23+(Bа%2^52+2^28)/2^29):=by
 have q:=g0 B;have d:(eF B)а=Bа/2^52:=by rw[q];exact ξ А
 have j:((B>>>63)<<<31)а=0:=by
  rw[Ι,Α];simp only[iZ,jb,ι,Μ];rw[ϣ h]
 have l:(B&&&0x7FFFFFFFFFFFFFFF)а=Bа:=by
  rw[ψ];simp only[iZ];rw[show 9223372036854775807=2^63-1 by norm_num,θ];exact ξ h
 have m:(((B&&&0x7FFFFFFFFFFFFFFF)+0x10000000)>>>29)а=(Bа+2^28)/2^29:=by
  rw[Α,μ,l];simp only[iZ,jb,ι];rw[ξ А];norm_num
 have o:(Bа+2^28)/2^29=Bа/2^52*2^23+(Bа%2^52+2^28)/2^29:=А
 have e:(((B>>>63)<<<31)+(((B&&&0x7FFFFFFFFFFFFFFF)+0x10000000)>>>29))а=(Bа+2^28)/2^29:=by
  rw[μ,j,m,zero_add];exact ξ А
 have g:((1045:Д)<<<23)а=1045*2^23:=Б
 have p:(((B>>>63)<<<31)+(((B&&&0x7FFFFFFFFFFFFFFF)+0x10000000)>>>29)-((1045:Д)<<<23))а=(Bа+2^28)/2^29-1045*2^23:=by
  rw[χ _ _ (κ.2 (by rw[e,g];omega)),e,g]
 unfold pq;rw[ϟ,p,o,d];omega
lemma bP (b:р) (g:fv b) (l:0<rv b) (j:¬eF b.toBits=0):b.toBitsа<2^63/\rv b=((b.toBitsа%2^52+2^52:ℕ):ℝ)*(2:ℝ)^(((eF b.toBits)а:п)-1075):=by
 have h:=aK b g j;rw[aw] at h
 have e:(b.toBits>>>63!=0)=false:=by
  by_contra d;simp only[Bool.not_eq_false] at d;rw[d,if_pos rfl] at h;have:(0:ℝ)<=((b.toBits&&&0xfffffffffffff)а+2^52:ℕ)*(2:ℝ)^(((eF b.toBits)а:п)-1075):=В;push_cast at this h;linarith
 rw[e] at h;simp only[ο,ite_false] at h;refine ⟨?_,?_⟩
 .have:b.toBits>>>63=0:=by simpa using e
  have:=congrArg jI this;rw[Α] at this;simp only[iZ,jb,ι] at this;have:=UInt64.toNat_lt b.toBits;omega
 .rw[h];congr 2;rw[ψ];simp only[iZ];rw[show (4503599627370495:ℕ)=2^52-1 by norm_num,θ]
lemma iz (b:р) (m:fv b) (j:0<rv b) (d:1046<=(eF b.toBits)а) (l:(eF b.toBits)а<=1298):
  i7 (pq b.toBits)≠255/\jw (pq b.toBits)=false/\|(sv (pq b.toBits):ℝ)*2^je (pq b.toBits)-rv b|<= rv b/2^24:=by
 obtain ⟨o,p⟩:=bP b m j (λh=>by rw[h] at d;simp at d)
 have q:=iB b.toBits d l o;obtain ⟨w,x,y⟩:=wV _ _ _ q А А А;refine ⟨w,x,?_⟩;rw[y,p];set E:=(eF b.toBits)а;set F:=b.toBitsа%2^52;set R:=2^23+(F+2^28)/2^29;have r:R*2^29<= F+2^52+2^28:=А;have t:F+2^52+2^28<R*2^29+2^29:=А;have e:(2:ℝ)^(E-1046)=2^(29:ℤ)*2^((E:ℤ)-1075):=by rw[<-zpow_natCast,<-zpow_add₀ Г];congr 1;omega
 have g:(R:ℝ)*2^(E-1046)=((R*2^29:ℕ):ℝ)*(2:ℝ)^((E:п)-1075):=by rw[e];push_cast;norm_num;ring
 rw[g,<-sub_mul,abs_mul,abs_of_pos (by positivity:(0:ℝ)<2^((E:п)-1075))]
 have u:|((R*2^29:ℕ):ℝ)-((F+2^52:ℕ):ℝ)|<=2^28:=by
  rw[abs_le];constructor
  .have:((F+2^52:ℕ):ℝ)<((R*2^29:ℕ):ℝ)+2^29-2^28:=by
    have:F+2^52+2^28<R*2^29+2^29:=t;have:=(Nat.cast_lt (α:=ℝ)).2 this;push_cast at this⊢;linarith
   linarith
  .have:=(Nat.cast_le (α:=ℝ)).2 r;push_cast at this⊢;linarith
 have v:(2:ℝ)^28<=((F+2^52:ℕ):ℝ)/2^24:=by
  have:(0:ℝ)<= F:=lj _;push_cast;rw[le_div_iff₀ В];norm_num;try linarith
 calc
  (|((R*2^29:ℕ):ℝ)-((F+2^52:ℕ):ℝ)|)*2^((E:п)-1075)<=((F+2^52:ℕ):ℝ)/2^24*2^((E:п)-1075):=
    mul_le_mul_of_nonneg_right (u.trans v) В
  _=_:=by ring
@[inline] def aq (a:ZB):FloatArray:=bk (λj=>dw a[j]!) aв (FloatArray.emptyWithCapacity aв)
lemma af (a:ZB):(aq a)д=aв/\∀j,j<aв->(aq a).data[j]! =dw a[j]!:=by
 have d:(jE aв)д=0:=rfl;obtain ⟨e,-,g⟩:=bM (λj=>dw a[j]!) aв aв _ (by rw[d];omega) (by rw[d];omega)
 exact ⟨e,λj h=>g j (by rw[d];omega) h⟩
lemma ao (X:FloatArray) (i:с) (h:iб<Xд):X.uget i h=X.data[iб]!:=by
 rcases X with ⟨d⟩;simp only[lh];rw[getElem!_pos d iб h];rfl
lemma u1 {i:с} (h:iб+1<2^32):(i+1)б=iб+1:=by
 rw[ks,lf];exact ξ (by have:=Ϟ;have:=ς;omega)
@[specialize] def PP (f:т->р) (x:ZB) (e:с) (he:eб<= xв) (hb:eб+1<2^32) (i:с) (hi:iб<= eб) (s t:р):р:=
 have:=ς
 if h:i+1<e then
  have h1:=u1 (i:=i) А
  have:=γ.mp h
  have h2:(i+1+1)б=iб+2:=by rw[u1 А,h1]
  PP f x e he hb (i+1+1) А (s+f (x.uget i А)) (t+f (x.uget (i+1) А))
 else if h:i<e then s+f (x.uget i (by have:=γ.mp h;omega))+t else s+t
termination_by eб-iб
decreasing_by omega
lemma df:fv (0:р)/\rv (0:р)=0:=by
 have h:(0:р).toModel.unpack=.zero .positive:=by rfl
 refine ⟨(bB 0).2 (by rw[h];rfl),?_⟩;rw[cf,h];simp[jp,jU,jz]
lemma dg (a b:р) (A B X Y d e:ℝ) (j:fv a) (l:fv b) (m:|rv a-A|<= X) (o:|rv b-B|<= Y) (p:|A|<= d) (q:|B|<= e)
  (r:d+e<=2^960) (t:X+Y<=2^30*((1/2^53+1/2^73)*(d+e)+1/2^1074)):
  fv (a+b)/\|rv (a+b)-(A+B)|<= X+Y+((1/2^53+1/2^73)*(d+e)+1/2^1074):=by
 have k:=abs_add_le (rv a) (rv b);have u:=abs_sub_abs_le_abs_sub (rv a) A;have v:=abs_sub_abs_le_abs_sub (rv b) B;have w:=abs_nonneg A;have x:=abs_nonneg B;have y:(1:ℝ)/2^1074<=1:=by rw[div_le_one В];exact one_le_pow₀ Г
 have z:(0:ℝ)<=1/2^1074:=В
 obtain ⟨g,h⟩:=fA a b j l (by
  rw[zpow_ofNat,show (2:ℝ)^971=2048*2^960 by rw[show (2048:ℝ)=2^11 by norm_num,<-pow_add]]
  have:(2:ℝ)^30<=2^960:=pow_le_pow_right₀ Г Г;generalize (2:ℝ)^960=G at*;generalize (1:ℝ)/2^1074=T at*;linarith)
 refine ⟨g,?_⟩;rw[show (1:ℝ)/2^1075=1/2^1074/2 by rw[pow_succ];ring] at h
 have:|rv a+rv b-(A+B)|<=|rv a-A|+|rv b-B|:=by rw[show rv a+rv b-(A+B)=rv a-A+(rv b-B) by ring];exact abs_add_le _ _
 have:=abs_sub_le (rv (a+b)) (rv a+rv b) (A+B);generalize (1:ℝ)/2^1074=T at*;linarith
lemma ak (x:ZB) (j:с) (d:jб<xв):x.uget j d=x[jб]!:=by rw[getElem!_pos x jб d];rfl
lemma ew (s F:р) (e d g:ℝ) (c:ℕ) (h:c<=2^30) (p:fv s) (o:fv F) (m:|rv s-e|<= c*((1/2^53+1/2^73)*d+1/2^1074))
  (l:|e|<= d) (j:|rv F|<= g) (t:d+g<=2^960):
  fv (s+F)/\|rv (s+F)-(e+rv F)|<=((c+1:ℕ):ℝ)*((1/2^53+1/2^73)*(d+g)+1/2^1074)/\|e+rv F|<= d+g:=by
 have h:(c:ℝ)<=2^30:=by exact_mod_cast h
 have q:=(abs_nonneg _).trans l;have r:=(abs_nonneg _).trans j;have:=mul_le_mul_of_nonneg_right h q;have:=mul_nonneg c.cast_nonneg r;have:=mul_le_mul_of_nonneg_right h (show (0:ℝ)<=1/2^1074 by positivity)
 obtain ⟨a,b⟩:=dg s F e (rv F) _ 0 d g p o m (by simp) l j t (by linarith)
 exact ⟨a,by push_cast;linarith,(abs_add_le _ _).trans (add_le_add l j)⟩
lemma ga (a b:р) (A B g h:ℝ) (d e N:ℕ) (l:d+e=N) (o:2*d<= N+1) (p:2*e<= N+1) (j:N<=2^30) (y:fv a) (z:fv b)
  (w:|rv a-A|<= d*((1/2^53+1/2^73)*g+1/2^1074)) (x:|rv b-B|<= e*((1/2^53+1/2^73)*h+1/2^1074))
  (r:|A|<= g) (t:|B|<= h) (D:g+h<=2^960):
  fv (a+b)/\|rv (a+b)-(A+B)|<=((N:ℝ)+3)/2*((1/2^53+1/2^73)*(g+h))+((N:ℝ)+1)*(1/2^1074):=by
 have j:(N:ℝ)<=2^30:=by exact_mod_cast j
 have l:(d:ℝ)+e=N:=by exact_mod_cast l
 have o:2*(d:ℝ)<=N+1:=by exact_mod_cast o
 have p:2*(e:ℝ)<=N+1:=by exact_mod_cast p
 have m:=(abs_nonneg _).trans r;have q:=(abs_nonneg _).trans t;have:=mul_le_mul_of_nonneg_right o m;have:=mul_le_mul_of_nonneg_right p q;have:=mul_le_mul_of_nonneg_right j m;have:=mul_le_mul_of_nonneg_right j q;have:=mul_nonneg d.cast_nonneg q;have:=mul_nonneg e.cast_nonneg m;have:=mul_le_mul_of_nonneg_right j (show (0:ℝ)<=1/2^1074 by positivity)
 obtain ⟨u,v⟩:=dg a b A B _ _ g h y z w x r t D (by rw[<-l] at*;linarith)
 exact ⟨u,by rw[<-l] at*;linarith⟩
lemma fy (f:i9->р) (x:ZB) (e:с) (O:eб<= xв) (N:eб+1<2^32) (d:о) (w:о->ℝ)
  (v:∀k,d<= k->k<eб->fv (f x[k]!)/\|rv (f x[k]!)|<= w k)
  (V:∑k∈jr d eб,w k<=2^960) (hn:eб<= d+2^29)
  (i:с) (W:iб<= eб) (s t:р) (c:о) (m q o r:ℝ) (hc:iб=d+2*c) (G:fv s) (H:fv t)
  (E:|rv s-m|<= c*((1/2^53+1/2^73)*o+1/2^1074)) (F:|rv t-q|<= c*((1/2^53+1/2^73)*r+1/2^1074)) (y:|m|<= o) (z:|q|<= r)
  (M:m+q=∑k∈jr d iб,rv (f x[k]!)) (l:o+r=∑k∈jr d iб,w k):
  fv (PP f x e O N i W s t)/\|rv (PP f x e O N i W s t)-∑k∈jr d eб,rv (f x[k]!)|<=
   (((eб-d:ℕ):ℝ)+3)/2*((1/2^53+1/2^73)*∑k∈jr d eб,w k)+(((eб-d:ℕ):ℝ)+1)*(1/2^1074):=by
 have S:∀k,d<= k->k<eб->0<= w k:=λk u L=>(abs_nonneg _).trans (v k u L).2
 have Q:∀j≤eб,∑k∈jr d j,w k<=2^960:=λj hj=>(ϸ
  (Finset.Ico_subset_Ico_right hj) (λk hk _=>(Finset.mem_Ico.1 hk).elim (S k))).trans V
 have:=(abs_nonneg _).trans y;have:=(abs_nonneg _).trans z;have:=ς;rw[PP];dsimp only;split
 next h=>
  have u:=u1 (i:=i) А;have h0:=γ.mp h;have L:(i+1+1)б=iб+1+1:=by rw[u1 А,u]
  rw[ak x i А,ak x (i+1) А,u];obtain ⟨I,J⟩:=v iб А А;obtain ⟨g3,g4⟩:=v (iб+1) А А
  have U:∀g:ℕ->ℝ,∑k∈jr d (i+1+1)б,g k=∑k∈jr d iб,g k+g iб+g (iб+1):=λg=>by
   rw[L,ϐ А,ϐ А]
  have:=Q _ (by omega:(i+1+1).toNat<=_);rw[U,<-l] at this
  have:=S iб А А;have:=S (iб+1) А А;obtain ⟨A,B,D⟩:=ew s _ m o (w iб) c А G I E y J (by linarith)
  obtain ⟨X,Y,Z⟩:=ew t _ q r (w (iб+1)) c А H g3 F z g4 (by linarith)
  exact fy f x e O N d w v V hn (i+1+1) А _ _ (c+1) _ _ _ _ А A X B Y D Z (by rw[U,<-M];ring) (by rw[U,<-l];ring)
 next h=>
  split
  next K=>
   have h0:=γ.mp K;have u:=u1 (i:=i) А;have p:eб=iб+1:=by have:=mt γ.mpr h;omega
   rw[ak x i А];obtain ⟨I,J⟩:=v iб А А;have:=Q (iб+1) А;rw[ϐ А,<-l] at this;obtain ⟨A,B,D⟩:=ew s _ m o (w iб) c А G I E y J (by linarith)
   rw[p,ϐ А,ϐ А,<-M,<-l,add_right_comm m,add_right_comm o];exact ga _ t _ q _ r (c+1) c _ А А А А A H B F D z (by linarith)
  next K=>
   have p:eб=iб:=by have:=mt γ.mpr K;omega
   rw[p,<-M,<-l];exact ga s t m q o r c c _ А А А А G H E F y z (by rw[l];exact Q _ А)
termination_by eб-iб
decreasing_by all_goals omega
lemma fz (f:i9->р) (x:ZB) (e:с) (h:eб<= xв) (g:eб+1<2^32) (d:с) (j:dб<= eб) (w:о->ℝ)
  (m:∀k,dб<= k->k<eб->fv (f x[k]!)/\|rv (f x[k]!)|<= w k)
  (l:∑k∈jr dб eб,w k<=2^960) (o:eб<= dб+2^29):
  fv (PP f x e h g d j 0 0)/\|rv (PP f x e h g d j 0 0)-∑k∈jr dб eб,rv (f x[k]!)|<=
   (((eб-dб:ℕ):ℝ)+3)/2*((1/2^53+1/2^73)*∑k∈jr dб eб,w k)+(((eб-dб:ℕ):ℝ)+1)*(1/2^1074):=
 fy f x e h g dб w m l o d j 0 0 0 0 0 0 0 А df.1 df.1 (by rw[df.2];simp) (by rw[df.2];simp)
  (by simp) (by simp) (by simp) (by simp)
structure RS where
 ok:ф
 m:р
 ri:р
 d:т
 A:Д
@[inline] def rq (y r:р):ф:=eF r.toBits<1100&&(r*y).toBits-0x3FEFFFFFFFFFFFFF<2
@[inline] def dn (V:р) (w:Д) (k:Д->ф):ф×т:=
 let B:=(Float.sqrt V).toBits
 let t:=(1:Д)<<<w
 (eF B-1046<=252&&((B&&&0x7FFFFFF)-t-1<0x7FFFFFF-2*t||k B),pq B)
@[inline] def rL (Ln:Д) (k:Д->ф) (Ep rn:р) (x:ZB) (lo hi:с) (h:hiб<= xв) (hb:hiб+1<2^32) (hl:loб<= hiб):RS:=
 let m:=PP dw x hi h hb lo hl 0 0*rn
 let V:=PP (λw=>let d:=dw w-m;d*d) x hi h hb lo hl 0 0*rn+Ep
 let t:=dn V Ln k
 let D:=dw t.2
 let ri:=1/D
 let a:=eF m.toBits
 let b:=eF D.toBits
 let mD:=if b<= a then a-b+1 else 0
 ⟨t.1&&Ln+mD<=25&&rq D ri,m,ri,t.2,Ln+mD+1047⟩
@[inline] def rR (Ln:Д) (k:Д->ф) (Ep rn:р) (x:ZB) (lo hi:с) (h:hiб<= xв) (hb:hiб+1<2^32) (hl:loб<= hiб):RS:=
 let V:=PP (λw=>let d:=dw w;d*d) x hi h hb lo hl 0 0*rn+Ep
 let t:=dn V Ln k
 let D:=dw t.2
 let ri:=1/D
 ⟨t.1&&rq D ri,0,ri,t.2,0⟩
@[noinline] def sL (n:о) (x:ZB) (r:о) (xw gw bw d:т):т:=ZT ((ZA xw-rowMean n x r)*ZA gw/ZA d+ZA bw)
def mL (x:ZB) (b:о):о->Д->Д
 | 0,u=>u
 | k+1,u=>let w:=x[b]!;let i:=Core.iW w;mL x (b+1) k (if Core.mW w==0||u< i then u else i)
@[inline] def zS (w:т) (lo:Д):о:=let M:=Core.mW w;let h:=Core.iW w-lo;if h<=38 then (M<<<h)а else Mа<<<hа
def sS (x:ZB) (lo:Д) (b:о):о->о->о->о×о
 | 0,p,q=>(p,q)
 | k+1,p,q=>let w:=x[b]!
  if Core.mW w==0 then sS x lo (b+1) k p q else if ZX w then sS x lo (b+1) k p (q+zS w lo) else sS x lo (b+1) k (p+zS w lo) q
@[noinline] def sI (n:о) (x:ZB) (r:о) (xw gw bw d:т):т:=
 let lo:=mL x (r*n) n 255
 let S:=sS x lo (r*n) n 0 0
 let X:=if Core.mW xw==0 then 0 else n*zS xw lo
 let T:=(if ZX xw then-(X:п) else X)-((S.1:п)-S.2)
 let s1:п:=(loа:п)+idxOf gw-idxOf d
 let s2:п:=idxOf bw
 let f:=min s1 s2
 let md:=ZW d
 let P:=T*sv gw*((1<<<(s1-f)й:о):п)+sv bw*(n*md:о)*((1<<<(s2-f)й:о):п)
 let Q:=n*md
 let a:=Pм
 let k1:п:=26+(Qз:п)-(aз:п)
 let k:=if V8 a k1/YF Q k1<67108864 then k1 else k1-1
 let W:=V8 a k
 let Dd:=YF Q k
 let q:=W/Dd
 let e:=f-149-k
 if 0<Q&&a==0 then ZT 0 else
 if 33554432<= q&&q<67108864&&-151<= e&&e<=101&&0<Q then Core.aC (decide (P<0)) (2*q+V2 (W%Dd) false)ж (e+297)йж
 else sL n x r xw gw bw d
lemma el (x:ZB):∀k b u,(mL x b k u)а<= uа/\∀j<k,ZW x[b+j]!≠0->(mL x b k u)а<= je x[b+j]!:=by
 intro k;induction k with
 | zero=>intro b u;exact ⟨le_rfl,λj h=>absurd h (Nat.not_lt_zero _)⟩
 | succ k m=>
  intro b u;simp only[mL];generalize l:(if mW x[b]! ==0||u< iW x[b]! then u else iW x[b]!)=v
  have d:vа<= uа/\(ZW x[b]!≠0->vа<= je x[b]!):=by
   rw[<-l];have:=ZS x[b]!;split<;>rename_i h<;>simp only[lg,YK,decide_eq_true_eq,ν,not_or,not_lt] at h<;>omega
  obtain ⟨o,e⟩:=m (b+1) v;refine ⟨by omega,λj p g=>?_⟩
  rcases j with _|j
  .have:=d.2 g;simp only[Ϫ] at*;omega
  .have:=e j А (by rwa[show b+1+j=b+(j+1) by omega]);rw[show b+1+j=b+(j+1) by omega] at this;exact this
lemma eZ (w:i9) (d:Д) (h:dа<= je w):zS w d=ZW w*2^(je w-dа):=by
 have o:=ZS w;have j:=ZI w;have m:=ZO w;have g:(iW w-d)а=je w-dа:=by rw[χ _ _ (κ.2 А),o]
 unfold zS;dsimp only;split<;>rename_i l
 .have e:je w-dа<=38:=by have:=κ.1 l;rw[g] at this;simpa using this
  rw[Ι,j,g,ξ (show je w-dа<64 by omega),Μ]
  apply ξ;calc ZW w*2^(je w-dа)<2^24*2^38:=Nat.mul_lt_mul_of_lt_of_le m (Ε Г e) В
   _=2^62:=Г
   _<2^64:=Г
 .rw[Μ,j,g]
lemma iR (x:ZB) (d:Д):∀k b p q,(∀j<k,ZW x[b+j]!≠0->dа<= je x[b+j]!)->
  ((sS x d b k p q).1:ℤ)-(sS x d b k p q).2=(p:ℤ)-q+∑j∈ZM k,sv x[b+j]!*2^(je x[b+j]!-dа):=by
 intro k;induction k with
 | zero=>intro b p q _;simp[sS]
 | succ k m=>
  intro b p q h;have e:∀j<k,ZW x[b+1+j]!≠0->dа<= je x[b+1+j]!:=λj r=>by rw[show b+1+j=b+(j+1) by omega];exact h _ А
  rw[jY,Ο rfl (λt _=>by rw[show b+(t+1)=b+1+t by omega]),Ϫ]
  simp only[sS];split<;>rename_i l
  .rw[m _ _ _ e];have:ZW x[b]! =0:=by simpa[YK] using l
   simp[sv,this]
  .have g:=h 0 А (by simpa[YK] using l);simp only[Ϫ] at g
   split<;>rename_i o<;>rw[m _ _ _ e,eZ _ _ g]<;>simp[sv,o]<;>ring
lemma c7 (w:i9) (d:ℕ) (h:ZW w=0\/d<= je w):((ZA w:ℚ))=(sv w*2^(je w-d):ℤ)*(2:ℚ)^((d:ℤ)-149):=by
 rw[Y1];rcases h with h|h
 .have:(sv w:ℚ)=0:=(YN w).2 h;rw[this];simp[show sv w=0 by exact_mod_cast this]
 .push_cast;rw[mul_assoc,<-zpow_natCast,<-zpow_add₀ Г];congr 2;push_cast[h];ring
lemma gj (n:о) (x:ZB) (r:о):jo n x r=(((sS x (mL x (r*n) n 255) (r*n) n 0 0).1:ℤ)-(sS x (mL x (r*n) n 255) (r*n) n 0 0).2:ℤ)*(2:ℚ)^(((mL x (r*n) n 255)а:ℤ)-149)/n:=by
 obtain ⟨-,d⟩:=el x n (r*n) 255;generalize mL x (r*n) n 255=e at d⊢;have g:=iR x e n (r*n) 0 0 (λj l h=>d j l h)
 unfold jo;rw[Yk,zero_add];congr 1;rw[g];simp only[kS,sub_zero,zero_add];push_cast;rw[϶];refine Ο rfl λt m=>?_
 rw[c7 _ eа (by by_cases h:ZW x[r*n+t]! =0;exact Or.inl h;exact ma (d t (Ν.1 m) h))];push_cast;ring
lemma iP (n:о) (x:ZB) (r j:о) (z y d:i9) (O:j<n) (hd:ZX d=false):
  sI n x r x[r*n+j]! z y d=ZT ((ZA x[r*n+j]!-jo n x r)*ZA z/ZA d+ZA y):=by
 have hn:0<n:=А;obtain ⟨-,p⟩:=el x n (r*n) 255;unfold sI sL;lift_lets;intro u S X T A B f v P Q a k1 k W w q e;have t:jo n x r=((S.1:ℤ)-S.2:ℤ)*(2:ℚ)^((uа:ℤ)-149)/n:=gj n x r
 have hX:(ZA x[r*n+j]!:ℚ)=((if ZX x[r*n+j]! then-(X:ℤ) else X:ℤ):ℚ)/n*(2:ℚ)^((uа:ℤ)-149):=by
  by_cases m:ZW x[r*n+j]! =0
  .have:X=0:=by simp only[X,YK,m,decide_true,ite_true]
   rw[c7 _ uа (Or.inl m),this,show sv x[r*n+j]! =0 by exact_mod_cast (YN _).2 m];simp
  .have:X=n*(ZW x[r*n+j]!*2^(je x[r*n+j]!-uа)):=by
    simp only[X,YK,m,decide_false,ο,ite_false];rw[eZ _ _ (p j O m)]
   rw[c7 _ uа (ma (p j O m)),this];unfold sv;split<;>push_cast<;>field_simp
 have J:=Y1 z;have F:=Y1 y;have G:=Y1 d;have K:(sv d:ℚ)=v:=by unfold sv;simp[hd,v]
 have l:(ZA x[r*n+j]!-jo n x r)*ZA z/ZA d+ZA y=(P:ℚ)*2^f/(Q*2^149)\/Q=0:=by
  by_cases o:Q=0;exact ma o;left;have as:(v:ℚ)≠0:=by intro h;apply o;simp only[Q];exact_mod_cast (show n*v=0 by rw[show v=0 by exact_mod_cast h];simp)
  rw[hX,t,J,F,G,K];have L:(2:ℚ)^(A-f)=2^A/2^f:=by rw[zpow_sub₀ Г]
  have M:(2:ℚ)^(B-f)=2^B/2^f:=by rw[zpow_sub₀ Г]
  have H:0<= A-f:=by simp only[f];omega
  have I:0<= B-f:=by simp only[f];omega
  simp only[P,Q,T,ϭ];push_cast;rw[<-zpow_natCast,<-zpow_natCast,kv H,kv I,L,M];simp only[A,B,zpow_sub₀ (show (2:ℚ)≠0 by norm_num),zpow_add₀ (show (2:ℚ)≠0 by norm_num)]
  field_simp
 split_ifs with D N
 .simp only[υ,decide_eq_true_eq,beq_iff_eq] at D
  obtain ⟨o,ha⟩:=D;rcases l with l|l
  .rw[l,show P=0 from Int.natAbs_eq_zero.1 ha];simp
  .omega
 .simp only[υ,decide_eq_true_eq] at N
  obtain ⟨⟨⟨⟨U,q2⟩,L⟩,M⟩,o⟩:=N;rcases l with l|l;swap;omega
  rw[l];have E:a≠0:=by intro h;apply D;simp[o,h]
  obtain ⟨Y,Z⟩:=VE a Q o k;have hP:(P:ℚ)≠0:=by intro h;apply E;simp only[a];exact_mod_cast Int.natAbs_eq_zero.2 (by exact_mod_cast h)
  have g:qз=25:=le_antisymm (Nat.lt_succ_iff.1 ((lu А).2 q2)) ((Ϧ А).2 U);refine Qx _ W false w e _ 0 _ _ ?_ le_rfl one_pos (by simp) Y ?_ U (by rw[g];omega) (by rw[g];omega) ?_ ?_
  .intro _;congr 1;apply propext;constructor<;>intro h
   .by_contra V;push Not at V;have:(0:ℚ)<= P*2^f/(Q*2^149):=В
    linarith
   .apply div_neg_of_neg_of_pos (mul_neg_of_neg_of_pos (by exact_mod_cast h) В) В
  .rw[add_zero,abs_div,abs_mul,abs_of_pos (show (0:ℚ)<2^f by positivity),abs_of_pos (show (0:ℚ)<Q*2^149 by positivity),
    show |(P:ℚ)|=a by simp only[a,ku,ϕ]]
   rw[mul_div_assoc,show (W:ℚ)*2^e/w=(W:ℚ)/w*2^e by ring,Z,show e=f-149-k from rfl]
   rw[zpow_sub₀ Г,zpow_sub₀ Г];field_simp
  .rw[jL,ZG];have:V2 (W%w) false<=1:=by unfold V2;split<;>omega
   exact ξ А
  .rw[jL,ZG,ξ А];omega
 .rfl
lemma zQ (w:i9) (d:Д) (h:ZW w=0\/dа<= je w):(ZA w)^2=((zS w d:ℕ):ℚ)^2*((2:ℚ)^((dа:ℤ)-149))^2:=by
 rw[c7 _ _ h,mul_pow];congr 1;rcases h with h|h
 .have:mW w=0:=Β.mp (by rw[ZI,h];rfl)
  rw[show sv w=0 by exact_mod_cast (YN _).2 h];unfold zS;simp[this]
 .rw[eZ _ _ h];unfold sv;split<;>push_cast<;>ring
def qZ (x:ZB) (lo:Д) (b:о):о->о->о
 | 0,a=>a
 | k+1,a=>let z:=zS x[b]! lo;qZ x lo (b+1) k (a+z*z)
lemma iE (x:ZB) (d:Д):∀k b a,(qZ x d b k a:ℚ)=a+∑j∈ZM k,((zS x[b+j]! d:ℕ):ℚ)^2:=by
 intro k;induction k with
 | zero=>intro b a;simp[qZ]
 | succ k e=>
  intro b a;rw[jY,Ο rfl (λt _=>by rw[show b+(t+1)=b+1+t by omega]),Ϫ,qZ,e];push_cast;ring
def V5 (A Dn u:о) (eps:т):т:=
 let E:=ZW eps
 let ie:=ZZ eps
 let w0:=min (2*u) (ie+149)
 let B:=(A<<<(2*u-w0))+((Dn*E)<<<(ie+149-w0))
 let s:п:=(350-(w0:п)-((Bз:п)-(Dnз:п)))/2
 let c:п:=(w0:п)-298+2*s
 let num:=V8 B c
 let den:=YF Dn c
 let R:=ZK num den
 if 33554432<=R then Yf num den s
 else sqrtRational ((A:ℚ)*2^(2*(u:п)-298)/Dn+ZA eps)
lemma YW (A e u:о) (d:i9) (t:0<e) (o:0<ZA d):
  V5 A e u d=
   jk ((A:ℚ)*2^(2*(u:п)-298)/e+ZA d):=by
 unfold V5;dsimp only;split_ifs with hr
 .have r:(sv d:ℚ)=ZW d:=by
   rw[<-VT,abs_of_pos (pos_of_mul_pos_left (by rwa[Y1] at o) XJ)]
  have p:=add_pos_of_nonneg_of_pos (XJ:(0:ℚ)<=A*2^(2*(u:п)-298)/e) o;rw[Y1,r] at p⊢;generalize ZW d=E at*
  generalize ZZ d=l at*
  generalize x:min (2*u) (l+149)=g at*
  generalize v:(A<<<(2*u-g))+((e*E)<<<(l+149-g))=B at*
  generalize (350-(g:п)-((Bз:п)-(eз:п)))/2=s at*
  obtain ⟨q,w⟩:=VE B e t ((g:п)-298+2*s);refine Vd _ p _ _ q s ?_ XI;have m:∀(k:о) (j:п),j=k+(g:п)-298->
    (2:ℚ)^j=2^k*2^((g:п)-298):=by
   intro k j h;rw[h,<-ZF,<-ZL ZQ];ring_nf
  have:=congrArg (λz:ℚ=>(z:ℝ)) (w.trans (by
   rw[<-v,ZL ZQ,m (2*u-g) (2*(u:п)-298) XI,
    m (l+149-g) ((l:п)-149) XI,Μ,Μ]
   push_cast;field_simp:
   _=((A:ℚ)*2^(2*(u:п)-298)/e+E*2^((l:п)-149))*2^(2*s)))
  push_cast at this⊢;omega
 .rfl
@[noinline] def xD (n:о) (x:ZB) (r:о) (eps:т) (c:ф) (B:Д):ф:=
 let lo:=mL x (r*n) n 255
 let S:=sS x lo (r*n) n 0 0
 let s:=if c then ((S.1:п)-S.2)м else 0
 V5 (n*n*qZ x lo (r*n) n 0-n*s*s) (n*n*n) loа eps==pq B
@[noinline] def sQ (xw gw d:т):т:=ZT (ZA xw*ZA gw/ZA d)
@[inline] def eL (n:о) (x:ZB) (r:о) (e:RS) (xw gw bw:т) (gv bv:р):т:=
 let v:=(dw xw-e.m)*(gv*e.ri)+bv
 let B:=v.toBits
 let E:=eF B
 let c:=NE.mx (NE.mx (E+5) (Core.iW bw+1051)) (Core.iW gw+e.A)-E
 let t:=(1:Д)<<<c
 if E-1046<=252&&c<=25&&(B&&&0x7FFFFFF)-t-1<0x7FFFFFF-2*t then pq B
 else sI n x r xw gw bw e.d
@[inline] def c9 (e:RS) (xw gw:т) (gv:р):т:=
 let v:=dw xw*(gv*e.ri)
 let B:=v.toBits
 let L:=B&&&0x7FFFFFF
 if eF B-1046<=252&&L-17<0x7FFFFDF then pq B
 else if xw&&&0x7FFFFFFF==0 then 0 else sQ xw gw e.d
@[inline] def eR (e:RS) (xw gw:т) (gv:р):т:=
 let v:=dw xw*(gv*e.ri)
 let B:=v.toBits
 let z:Д:=((xw&&&0x7FFFFFFF)к-1)>>>(63:Д)
 let m:Д:=z-1
 if ((eF B-1046)&&&m:Д)<=252&&((B&&&0x7FFFFFF)|||(z<<<(8:Д)))-17<0x7FFFFDF then pq B&&&mл
 else if xw&&&0x7FFFFFFF==0 then 0 else sQ xw gw e.d
lemma uB {r d n:о} (j:r<d) (l:d*n+1<2^32):r*n+n<= d*n/\(r*n)гб=r*n/\nгб=n/\((r*n)г+nг)б=r*n+n:=by
 have e:=Ζ n (Nat.succ_le_of_lt j);rw[ϋ] at e;have:=ς;have g:=Dense.aa (n:=r*n) А;have h:=Dense.aa (n:=n) А;exact ⟨e,g,h,by rw[Dense.ab (by have:=ς;omega),g,h]⟩
@[specialize] def QE (lo N:с) (f:(j:с)->j<N->т) (hN:loб+Nб<2^32) (i:с) (out:ZB) (ho:loб+Nб<= outв):ZB:=
 if h:i<N then
  have:=γ.mp h
  QE lo N f hN (i+1) (out.uset (lo+i) (f i h) (by rw[Dense.ab (by have:=ς;omega)];omega)) (by rw[ϑ];exact ho)
 else out
termination_by Nб-iб
decreasing_by have:=Τ N;rw[uS А];omega
lemma bL (e N:с) (f:(j:с)->j<N->i9) (g:eб+Nб<2^32) (i:с) (d:ZB) (h:eб+Nб<= dв):
  (QE e N f g i d h)в=dв:=by
 unfold QE;split
 .rw[bL];exact ϑ ..
 .rfl
termination_by Nб-iб
decreasing_by have:=γ.mp ‹_›;have:=Τ N;rw[uS А];omega
lemma im (a:ZB) (i:о) (v:i9) (h:i<aв) (q:о):(a.set i v h)[q]! =if i=q then v else a[q]!:=by
 by_cases d:q<aв
 .rw[getElem!_pos _ q (by simpa using d),getElem!_pos a q d,lV]
 .rw[getElem!_neg _ q (by simpa using d),getElem!_neg a q d];split<;>first|rfl|omega
lemma ct (e N:с) (f:(j:с)->j<N->i9) (F:о->i9) (l:eб+Nб<2^32)
  (p:∀j h,f j h=F (eб+jб)) (i:с) (d:ZB) (o:eб+Nб<= dв):
   (QE e N f l i d o)в=dв/\∀q,(QE e N f l i d o)[q]! =if eб+iб<= q/\q<eб+Nб then F q else d[q]!:=by
 rw[QE];split
 .rename_i h
  have r:=γ.mp h;have w:=ς;have m:(i+1)б=iб+1:=uS А;have g:(e+i)б=eб+iб:=Dense.ab А;obtain ⟨u,v⟩:=ct e N f F l p (i+1) (d.uset (e+i) (f i h) (by rw[g];omega)) (by rw[ϑ];exact o)
  refine ⟨by rw[u,ϑ],λq=>?_⟩
  rw[v q,m];simp only[l9];rw[im,g,p];split_ifs<;>first|rfl|omega|(subst_vars;rfl)
 .rename_i h
  have r:¬iб<Nб:=λt=>h (γ.mpr t)
  exact ⟨rfl,λq=>by rw[if_neg А]⟩
termination_by Nб-iб
decreasing_by have:=ς;rw[uS А];omega
@[noinline] def fL (n:о) (eps:т) (x g b:ZB) (r:о) (lo N:с) (hN:loб+Nб<2^32) (out:ZB) (ho:loб+Nб<= outв):ZB:=
 let M:=rowMean n x r
 let d:=sqrtRational (rowVariance n x r M+ZA eps)
 QE lo N (λj _=>ZT ((ZA x[r*n+jб]!-M)*ZA g[jб]!/ZA d+ZA b[jб]!)) hN 0 out ho
@[noinline] def fR (n:о) (eps:т) (x g:ZB) (r:о) (lo N:с) (hN:loб+Nб<2^32) (out:ZB) (ho:loб+Nб<= outв):ZB:=
 let d:=sqrtRational (rowVariance n x r 0+ZA eps)
 QE lo N (λj _=>ZT (ZA x[r*n+jб]!*ZA g[jб]!/ZA d)) hN 0 out ho
def RL (rows n:о) (Ln:Д) (Ep rn:р) (eps:т) (x g b:ZB) (gd bd:FloatArray) (hn:rows*n+1<2^32)
  (hx:rows*n<= xв) (hg:n<= gв) (hb:n<= bв) (hgd:n<= gdд) (hbd:n<= bdд) (r:о) (out:ZB) (ho:rows*n<= outв):ZB:=
 if hr:r<rows then
  have h:=uB hr hn;have h1:=h.1;have h2:=h.2.1;have h3:=h.2.2.1;have h4:=h.2.2.2
  let e:=rL Ln (xD n x r eps true) Ep rn x (r*n)г ((r*n)г+nг) (by rw[h4];omega) (by rw[h4];omega) (by rw[h4,h2];omega)
  RL rows n Ln Ep rn eps x g b gd bd hn hx hg hb hgd hbd (r+1)
   (if e.ok then QE (r*n)г nг (λj hj=>
    have:=γ.mp hj
    eL n x r e (x.uget ((r*n)г+j) (by rw[Dense.ab (by have:=ς;omega)];omega)) (g.uget j А)
     (b.uget j А) (gd.uget j А) (bd.uget j А)) (by rw[h2,h3];omega) 0 out (by rw[h2,h3];omega)
    else fL n eps x g b r (r*n)г nг (by rw[h2,h3];omega) out (by rw[h2,h3];omega))
   (by split<;>simp only[bL,fL]<;>exact ho)
 else out
termination_by rows-r
def RR (rows n:о) (Ln:Д) (Ep rn:р) (eps:т) (x g:ZB) (gd:FloatArray) (hn:rows*n+1<2^32)
  (hx:rows*n<= xв) (hg:n<= gв) (hgd:n<= gdд) (r:о) (out:ZB) (ho:rows*n<= outв):ZB:=
 if hr:r<rows then
  have h:=uB hr hn;have h1:=h.1;have h2:=h.2.1;have h3:=h.2.2.1;have h4:=h.2.2.2
  let e:=rR Ln (xD n x r eps false) Ep rn x (r*n)г ((r*n)г+nг) (by rw[h4];omega) (by rw[h4];omega) (by rw[h4,h2];omega)
  RR rows n Ln Ep rn eps x g gd hn hx hg hgd (r+1)
   (if e.ok then QE (r*n)г nг (λj hj=>
    have:=γ.mp hj
    eR e (x.uget ((r*n)г+j) (by rw[Dense.ab (by have:=ς;omega)];omega)) (g.uget j А)
     (gd.uget j А)) (by rw[h2,h3];omega) 0 out (by rw[h2,h3];omega)
    else fR n eps x g r (r*n)г nг (by rw[h2,h3];omega) out (by rw[h2,h3];omega))
   (by split<;>simp only[bL,fR]<;>exact ho)
 else out
termination_by rows-r
@[inline] def lw (n:о):Д:=Core.an (n+16)ж+1
@[inline] def aQ (rows n:о) (eps:т) (x g:ZB):ф:=
 SN rows n eps x g&&rows*n<=4294967294&&n<=1048576&&rq nж.toInt64.toFloat (1/nж.toInt64.toFloat)
lemma dZ (w:i9) (g:i7 w≠255):fv (dw w)/\rv (dw w)=(ZA w:ℝ)*2^149:=by
 obtain ⟨d,e⟩:=av w g;refine ⟨d,?_⟩;rw[e,Y1];push_cast;rw[mul_assoc,<-zpow_natCast,<-zpow_natCast,<-zpow_add₀ Г];congr 2;push_cast;ring
lemma d7 (n:о) (l:n<=2^20):(lw n)а<=21/\(n:ℝ)+17<=2^(lw n)а/\(32:ℝ)<=2^(lw n)а:=by
 have d:((n+16)ж)а=n+16:=Ξ (show n+16<2^64 by omega)
 have g:(an (n+16)ж)а=(n+16)з:=by rw[Y7,d]
 have h:(n+16)з<=20:=Nat.lt_succ_iff.1 ((lu А).2 А);have j:4<=(n+16)з:=(Ϧ А).2 А;have e:(lw n)а=(n+16)з+1:=by unfold lw;rw[μ,g];simp only[iZ];omega
 rw[e];refine ⟨by omega,?_,?_⟩
 .have:=Ϸ (n:=n+16);have:(n+17:ℕ)<=2^((n+16)з+1):=А
  exact_mod_cast this
 .have:2^5<=2^((n+16)з+1):=Ε Г А
  exact_mod_cast this
lemma eJ {e n:о} {d:i9} {x g:ZB} (h:ji e n d x g=true):i7 d≠255:=by
 simp only[ji,υ,ZH,gt_iff_lt] at h;have:=h.1.2;unfold jS at this;simpa using this
lemma dY (d:i9) (h:i7 d≠255) (j:0<ZA d):
  fv (dw d*NR.pt 149)/\|rv (dw d*NR.pt 149)-(ZA d:ℝ)*2^298|<=(ZA d:ℝ)*2^298/2^53+1/2^1075/\(2:ℝ)^149<=(ZA d:ℝ)*2^298:=by
 obtain ⟨p,q⟩:=dZ d h;obtain ⟨v,o⟩:=NR.iC 149 Г;have l:(ZA d:ℝ)*2^298=rv (dw d)*rv (NR.pt 149):=by rw[q,o,mul_assoc,<-pow_add]
 have t:(1:ℤ)<= sv d:=by
  have:(0:ℚ)<(sv d:ℚ)*2^((ZZ d:п)-149):=by rw[<-Y1];exact j
  have:(0:ℚ)<(sv d:ℚ):=pos_of_mul_pos_left this В;exact_mod_cast this
 have x:(2:ℝ)^(-149:ℤ)<=(ZA d:ℝ):=by
  have e:(ZA d:ℝ)=(sv d:ℝ)*2^((ZZ d:п)-149):=by rw[Y1];push_cast;rfl
  rw[e];have:(1:ℝ)<=(sv d:ℝ):=by exact_mod_cast t
  calc
   (2:ℝ)^(-149:ℤ)=1*2^(-149:ℤ):=by ring
   _<=(sv d:ℝ)*2^((ZZ d:п)-149):=mul_le_mul this (zpow_le_zpow_right₀ Г А) В (by linarith)
 have r:|rv (dw d)*rv (NR.pt 149)|<=(2:ℝ)^(971:п):=by
  rw[(av d h).2,o,abs_mul,abs_of_pos (by positivity:(0:ℝ)<2^149)];have:=NE.wB d;have w:=aL d
  calc
   (|(sv d:ℝ)*2^ZZ d|)*2^149<=2^(24+ZZ d)*2^149:=by gcongr
   _=2^(24+ZZ d+149):=by ring
   _<=(2:ℝ)^(971:ℕ):=pow_le_pow_right₀ Г А
   _=_:=Г
 obtain ⟨u,m⟩:=NF.fM _ _ p v r;have g:0<= rv (dw d)*rv (NR.pt 149):=by rw[<-l];exact mul_nonneg (by exact_mod_cast j.le) В
 rw[abs_of_nonneg g] at m;refine ⟨u,by rw[l];exact m,?_⟩
 calc
  (2:ℝ)^149=2^(-149:ℤ)*2^298:=by rw[<-zpow_natCast,<-zpow_natCast,<-zpow_add₀ Г];norm_num
  _<=_:=by gcongr
lemma eG (n:о) (g:1<= n) (j:n<=2^20) (h:rq nж.toInt64.toFloat (1/nж.toInt64.toFloat)=true):
  fv (1/nж.toInt64.toFloat)/\|rv (1/nж.toInt64.toFloat)-1/n|<=3/2^53/n:=by
 obtain ⟨e,d⟩:=NR.iy n А;simp only[rq,υ,decide_eq_true_eq] at h;have l:(n:ℝ)<=2^20:=by exact_mod_cast j
 have m:(2:ℝ)^20<=2^400:=pow_le_pow_right₀ Г Г
 have:=rc _ _ e (by rw[d];exact_mod_cast g) (by rw[d];exact l.trans m)
  (by have:=ν.mp h.1;simpa using this) h.2
 rw[d] at this;exact ⟨this.1,this.2⟩
lemma tS:(1:ℝ)/2^1075<=1/\(1:ℝ)/2^1074<=1/\(0:ℝ)<=1/2^1075/\(0:ℝ)<=1/2^1074:=by
 refine ⟨?_,?_,by positivity,by positivity⟩<;>rw[div_le_one В]<;>exact one_le_pow₀ Г
lemma gn (V T ρ:ℝ) (d:0<T) (g:0<= V) (h:|V-T|<=ρ*T):|Real.sqrt V-ja T|<=ρ*ja T:=by
 have a:=ϊ V;have b:=Real.sqrt_pos.2 d;have p:0<√V+√T:=by linarith
 have e:(√V+√T)*|√V-√T|=|V-T|:=by
  rw[<-abs_of_pos p,<-abs_mul,show (√V+√T)*(√V-√T)=V-T by linear_combination lb g-lb d.le]
 refine le_of_mul_le_mul_left ?_ p;rw[e];linarith[show ρ*T=ρ*√T*√T by rw[mul_assoc,Real.mul_self_sqrt d.le],mul_nonneg (mul_nonneg (show 0<=ρ by nlinarith[abs_nonneg (V-T)]) b.le) a]
lemma dR (n g:ℕ) (V T e U ρ:ℝ) (y:n<=2^20) (v:0<T) (m:0<V) (x:|V-T|<=ρ*T) (q:ρ<=((n:ℝ)+32)/2^54)
  (t:|e-ja V|<= ja V/2^53+1/2^1075) (u:e<2^53*U) (w:1/2^29<= U) (h:(n:ℝ)+17<=2^g) (j:(32:ℝ)<=2^g):
  |e-ja T|<=2^g*U:=by
 have r:=gn V T ρ v m.le x;obtain ⟨z,A⟩:=abs_le.1 r;obtain ⟨B,D⟩:=abs_le.1 t;have p:=ϊ T;have o:(n:ℝ)<=2^20:=by exact_mod_cast y
 have d:0<= U:=le_trans В w;have E:=mul_le_mul_of_nonneg_right (q.trans (by rw[div_le_div_iff₀ В В];norm_num;linarith):ρ<=1/2^33) p
 have F:=mul_le_mul_of_nonneg_right q p;have:(1:ℝ)/2^1075<=1/2^59:=by gcongr<;>norm_num
 have l:ja T<=2^53*U*(1+1/2^30):=by linarith
 have:=mul_le_mul_of_nonneg_left l (by positivity:(0:ℝ)<=((n:ℝ)+32)/2^54)
 linarith[abs_sub_le e (ja V) (ja T),mul_le_mul_of_nonneg_right h d,mul_le_mul_of_nonneg_right j d,mul_le_mul_of_nonneg_right o d]
lemma vP (n:ℕ) (q g h:р) (A S d j:ℝ) (y:1<= n) (Q:n<=2^20) (w:0<= A) (H:A<=2^700) (x:0<= j) (O:j<=1)
  (G:fv q) (v:fv g) (F:fv h)
  (hS:|S-A|<=4/2^53*A+j) (hq:|rv q-S|<=((n:ℝ)+3)/2*((1/2^53+1/2^73)*((1+4/2^53)*A+j))+((n:ℝ)+1)*(1/2^1074))
  (B:|rv g-1/n|<=3/2^53/n) (L:|rv h-d|<= d/2^53+1/2^1075) (I:2^149<= d) (K:d<=2^700):
  fv (q*g+h)/\|rv (q*g+h)-(A/n+d)|<=((n:ℝ)+30)/2^54*(A/n+d):=by
 obtain ⟨t1,t2,t3,t4⟩:=tS;have o:(1:ℝ)<= n:=by exact_mod_cast y
 have p:(n:ℝ)<=2^20:=by exact_mod_cast Q
 have m:0<d:=lt_of_lt_of_le В I;have h2:((n:ℝ)+3)/2*((1/2^53+1/2^73)*((1+4/2^53)*A+j))<=(2^20+3)/2*((1/2^53+1/2^73)*((1+4/2^53)*2^700+1)):=by gcongr
 have h3:((n:ℝ)+1)*(1/2^1074)<=(2^20+1)*1:=by gcongr
 have k1:=abs_sub_abs_le_abs_sub (rv q) S;have W:=abs_sub_abs_le_abs_sub S A;have X:=abs_sub_abs_le_abs_sub (rv g) (1/n);have Y:=abs_sub_abs_le_abs_sub (rv h) d;have:(1:ℝ)/n<=1:=div_le_one_of_le₀ o В;have:3/2^53/(n:ℝ)<=3/2^53:=div_le_self В o;rw[abs_of_nonneg w] at W;rw[abs_of_pos m] at Y;rw[abs_of_pos (by positivity:(0:ℝ)<1/n)] at X
 have h8:(8:ℝ)*2^700<=2^971:=by rw[show (8:ℝ)=2^3 by norm_num,<-pow_add];exact pow_le_pow_right₀ Г Г
 have h6:(2:ℝ)^60<=2^700:=pow_le_pow_right₀ Г Г;have z:=mul_le_mul (show |rv q|<=2*2^700 by generalize (2:ℝ)^700=T at*;linarith) (show |rv g|<=2 by linarith) (abs_nonneg _) В
 rw[<-abs_mul] at z;obtain ⟨Z,m2⟩:=NF.fM q g G v (by rw[zpow_ofNat];generalize (2:ℝ)^700=T at*;generalize (2:ℝ)^971=U at*;linarith)
 have:=abs_add_le (rv (q*g)) (rv h);have:=abs_sub_abs_le_abs_sub (rv (q*g)) (rv q*rv g);obtain ⟨E,a2⟩:=NF.fA _ _ Z F (by rw[zpow_ofNat];generalize (2:ℝ)^700=T at*;generalize (2:ℝ)^971=U at*;linarith)
 refine ⟨E,?_⟩;generalize rv (q*g+h)=V at*;generalize rv (q*g)=l at*;generalize rv q=q at*;generalize rv g=g at*;generalize rv h=h at*
 generalize (1:ℝ)/2^1075=τ at*
 generalize (1:ℝ)/2^1074=τ' at*
 have r:(0:ℝ)<n:=by linarith
 have t:|q-A|<=(((n:ℝ)+3)/2*((1/2^53+1/2^73)*(1+4/2^53))+4/2^53)*A+(((n:ℝ)+3)/2*((1/2^53+1/2^73)*j)+((n:ℝ)+1)*τ'+j):=by
  linarith[abs_sub_le q S A]
 have M:((n:ℝ)+3)/2*((1/2^53+1/2^73)*j)+((n:ℝ)+1)*τ'+j<=2^22:=by
  linarith[mul_le_mul p O x В,mul_le_mul p t2 t4 В]
 have D:((n:ℝ)+3)/2*((1/2^53+1/2^73)*(1+4/2^53))+4/2^53<=((n:ℝ)+13)/2^54:=by linarith
 generalize ((n:ℝ)+3)/2*((1/2^53+1/2^73)*(1+4/2^53))+4/2^53=c at t D;generalize ((n:ℝ)+3)/2*((1/2^53+1/2^73)*j)+((n:ℝ)+1)*τ'+j=J at t M
 have N:|n*g-1|<=3/2^53:=by
  rw[show n*g-1=n*(g-1/n) by field_simp,abs_mul,abs_of_pos r];exact (mul_le_mul_of_nonneg_left B r.le).trans_eq (by field_simp)
 have e:=abs_add_le (q-A) (q*(n*g-1));rw[show q-A+q*(n*g-1)=n*(q*g-A/n) by field_simp;ring,abs_mul,abs_mul,abs_of_pos r] at e
 have hD:|q*g-A/n|<=(c*(1+3/2^53)+3/2^53)*(A/n)+(1+3/2^53)*(J/n):=by
  rw[show (c*(1+3/2^53)+3/2^53)*(A/n)+(1+3/2^53)*(J/n)=((c*(1+3/2^53)+3/2^53)*A+(1+3/2^53)*J)/n by ring,le_div_iff₀ r];linarith[mul_le_mul_of_nonneg_left N (abs_nonneg q),abs_sub_abs_le_abs_sub q A,abs_of_nonneg w]
 have u:0<= A/n:=div_nonneg w r.le;have:|l+h-(A/n+d)|<=|l-A/n|+|h-d|:=by rw[show l+h-(A/n+d)=l-A/n+(h-d) by ring];exact abs_add_le _ _
 have:=abs_sub_abs_le_abs_sub (l+h) (A/n+d);rw[abs_of_pos (add_pos_of_nonneg_of_pos u m)] at this;have:=abs_sub_abs_le_abs_sub (q*g) (A/n);rw[abs_of_nonneg u] at this
 linarith[abs_sub_le V (l+h) (A/n+d),abs_sub_le l (q*g) (A/n),mul_le_mul_of_nonneg_right D u,mul_le_mul_of_nonneg_right p u,
  (div_le_iff₀ r).2 (by linarith:J<=2^22*n),le_mul_of_one_le_left m.le o]
lemma c8 (v:р) (e:fv v) (d:¬eF v.toBits=0):(2:ℝ)^(((eF v.toBits)а:п)-1023)<=|rv v|:=by
 rw[aK v e d,aw,abs_mul,abs_of_pos (by positivity:(0:ℝ)<2^(((eF v.toBits).toNat:п)-1075))]
 have:|(if v.toBits>>>63!=0 then-(((v.toBits&&&0xfffffffffffff)а+2^52:ℕ):ℝ) else ((v.toBits&&&0xfffffffffffff)а+2^52:ℕ))|=
   (((v.toBits&&&0xfffffffffffff)а+2^52:ℕ):ℝ):=by split<;>(try simp only[abs_neg])<;>exact abs_of_nonneg В
 rw[this]
 calc
  (2:ℝ)^(((eF v.toBits)а:п)-1023)=2^52*2^(((eF v.toBits)а:п)-1075):=by
    rw[<-zpow_natCast,<-zpow_add₀ Г];congr 1;push_cast;ring
  _<=_:=by gcongr;push_cast;linarith[show (0:ℝ)<=((v.toBits&&&0xfffffffffffff)а:ℝ) by positivity]
lemma bU (v:р) (d:fv v):|rv v|<2^53*(2:ℝ)^(((eF v.toBits)а:п)-1075):=by
 have:=bj v d;rwa[show (2:ℝ)^(((eF v.toBits)а:п)-1022)=2^53*2^(((eF v.toBits)а:п)-1075) by
  rw[<-zpow_natCast,<-zpow_add₀ Г];congr 1;push_cast;ring] at this
lemma dW (V:р) (o:Д) (k:Д->ф) (u:fv V) (m:0<rv V) (d:(dn V o k).1=true):
  fv V.sqrt/\1046<=(eF V.sqrt.toBits)а/\(eF V.sqrt.toBits)а<=1298/\
  |rv V.sqrt-ja (rv V)|<= ja (rv V)/2^53+1/2^1075/\
  rv V.sqrt<2^53*(2:ℝ)^(((eF V.sqrt.toBits)а:п)-1075)/\(2:ℝ)^(((eF V.sqrt.toBits)а:п)-1023)<= rv V.sqrt/\
  (dn V o k).2=pq V.sqrt.toBits/\i7 (pq V.sqrt.toBits)≠255/\jw (pq V.sqrt.toBits)=false/\
  |(sv (pq V.sqrt.toBits):ℝ)*2^je (pq V.sqrt.toBits)-rv V.sqrt|<= rv V.sqrt/2^24:=by
 unfold dn at d;simp only[υ,decide_eq_true_eq] at d;obtain ⟨j,p⟩:=te _ d.1;clear d;obtain ⟨l,r⟩:=fQ V u m;have q:=bU _ l;have g:=c8 _ l λh=>by rw[h] at j;simp at j
 have e:0<rv V.sqrt:=by
  have:(2:ℝ)^23<=(2:ℝ)^(((eF V.sqrt.toBits)а:п)-1023):=by
   rw[<-zpow_natCast];exact zpow_le_zpow_right₀ Г (by push_cast;omega)
  have k:=(abs_le.1 r).1;have:0<= ja (rv V):=ϊ _;have t:(1:ℝ)/2^1075<=1:=tS.1;rcases abs_cases (rv V.sqrt) with ⟨h,_⟩|⟨h,_⟩
  .linarith[show (0:ℝ)<2^23 by positivity]
  .rw[h] at g;have:(1:ℝ)<=2^23:=one_le_pow₀ Г;nlinarith
 rw[abs_of_pos e] at q g;obtain ⟨v,w,x⟩:=iz _ l e j p;exact ⟨l,j,p,r,q,g,rfl,v,w,x⟩
lemma fU (V:р) (n:ℕ) (k:Д->ф) (v:ℚ) (ρ:ℝ) (j:n<=2^20) (u:fv V) (l:0<v) (m:|rv V-(v:ℝ)*2^298|<=ρ*((v:ℝ)*2^298))
  (o:ρ<=((n:ℝ)+32)/2^54) (D:∀B,k B=true->pq B=jk v)
  (d:(dn V (lw n) k).1=true):(dn V (lw n) k).2=jk v:=by
 obtain ⟨K,G,H⟩:=d7 n j;have y:(0:ℝ)<(v:ℝ)*2^298:=mul_pos (by exact_mod_cast l) В
 have E:(n:ℝ)<=2^20:=by exact_mod_cast j
 have F:ρ<=1/2:=o.trans (by rw[div_le_div_iff₀ В В];nlinarith)
 have e:0<rv V:=by have:=(abs_le.1 m).1;nlinarith
 obtain ⟨I,r,t,J,w,x,A,-⟩:=dW V (lw n) k u e d;rw[A];unfold dn at d;simp only[υ,lg,decide_eq_true_eq] at d;rcases d.2 with h|h;swap;exact D _ h
 obtain ⟨p,q⟩:=tc _ (lw n) (κ.2 (by simp only[iZ];omega)) h
 have z:(1:ℝ)/2^29<=(2:ℝ)^(((eF V.sqrt.toBits)а:п)-1075):=by
  rw[one_div,<-zpow_neg_one,<-zpow_natCast,<-zpow_mul];exact zpow_le_zpow_right₀ Г (by push_cast;omega)
 have g:=dR n (lw n)а (rv V) _ (rv V.sqrt) _ ρ j y e m o J w z G H;apply dC _ I (lw n)а v l r t p q
 have:ja ((v:ℝ)*2^298)=ja (v:ℝ)*2^149:=by
  rw[lI (by exact_mod_cast l.le),show (2:ℝ)^298=(2^149)^2 by rw[<-pow_mul]]
  rw[lU В]
 rw[<-this];exact g
lemma pS (f:i9->р) (x:ZB) (m l:с) (h:lб<= xв) (r:lб+1<2^32) (v:mб<= lб)
  (n e:ℕ) (p:mб=e) (o:lб=e+n) (A:n<=2^20) (d:ℕ->ℝ)
  (t:∀j<n,fv (f x[e+j]!)/\|rv (f x[e+j]!)-d j^2|<=4/2^53*d j^2+1/2^1000/\d j^2<=2^600):
  fv (PP f x l h r m v 0 0)/\
  |rv (PP f x l h r m v 0 0)-∑j∈i0 n,rv (f x[e+j]!)|<=((n:ℝ)+3)/2*((1/2^53+1/2^73)*((1+4/2^53)*∑j∈i0 n,d j^2+n/2^1000))+((n:ℝ)+1)*(1/2^1074)/\
  |∑j∈i0 n,rv (f x[e+j]!)-∑j∈i0 n,d j^2|<=4/2^53*∑j∈i0 n,d j^2+n/2^1000:=by
 have q:∀g:ℕ->ℝ,∑k∈jr mб lб,g k=∑j∈i0 n,g (e+j):=λg=>by
  rw[kz,p,o,ϥ]
 have z:∀a:ℝ,∑j∈i0 n,(a*d j^2+1/2^1000)=a*∑j∈i0 n,d j^2+n/2^1000:=λa=>by
  rw[ϫ,<-kJ];simp;ring
 have D:∑j∈i0 n,d j^2<= n*2^600:=by simpa using Finset.sum_le_card_nsmul _ _ _ λj u=>(t j (Ν.1 u)).2.2
 have w:(n:ℝ)<=2^20:=by exact_mod_cast A
 have:(2:ℝ)^22*2^600<=2^960:=by rw[<-pow_add];exact pow_le_pow_right₀ Г Г
 have:=mul_le_mul_of_nonneg_right w (by positivity:(0:ℝ)<=2^600)
 have:(1:ℝ)<=2^600:=one_le_pow₀ Г;have:(n:ℝ)/2^1000<= n:=div_le_self В (one_le_pow₀ Г)
 obtain ⟨B,y⟩:=fz f x l h r m v (λk=>(1+4/2^53)*d (k-e)^2+1/2^1000)
  (λk E F=>by
   obtain ⟨j,rfl⟩:∃j,k=e+j:=⟨k-e,by omega⟩
   obtain ⟨a,b,-⟩:=t j А;simp only[ϥ];refine ⟨a,?_⟩;have:=abs_sub_abs_le_abs_sub (rv (f x[e+j]!)) (d j^2);rw[abs_of_nonneg (sq_nonneg (d j))] at this;linarith)
  (by simp only[q,ϥ,z];generalize (2:ℝ)^600=T at*;generalize (2:ℝ)^960=U at*;linarith) А
 rw[q,q] at y;simp only[o,p,ϥ,z] at y;refine ⟨B,y,?_⟩;rw[<-j9,<-z];exact (Ϲ _ _).trans (Ϗ λj u=>(t j (Ν.1 u)).2.1)
lemma dB (w:i9) (d:i7 w≠255):|rv (dw w)|<=2^278:=by
 rw[(av w d).2];exact (NE.wB w).trans (pow_le_pow_right₀ Г (by have:=aL w;omega))
lemma tR (w:i9) (h:i7 w≠255):fv (dw w*dw w)/\|rv (dw w*dw w)-rv (dw w)^2|<=4/2^53*rv (dw w)^2+1/2^1000/\rv (dw w)^2<=2^600:=by
 have g:=(av _ h).1;have d:=dB _ h
 have e:rv (dw w)^2<=2^556:=by
  have:=sq_le_sq' (abs_le.1 d).1 (abs_le.1 d).2;rw[<-pow_mul] at this;exact this
 have j:|rv (dw w)*rv (dw w)|<=(2:ℝ)^(971:п):=by
  rw[abs_mul_self,<-sq];exact e.trans (by rw[show (971:ℤ)=((971:ℕ):ℤ) from rfl,zpow_natCast];exact pow_le_pow_right₀ Г Г)
 obtain ⟨l,m⟩:=NF.fM _ _ g g j;refine ⟨l,?_,e.trans (pow_le_pow_right₀ Г Г)⟩;rw[abs_mul_self,<-sq] at m;have:(1:ℝ)/2^1075<=1/2^1000:=one_div_le_one_div_of_le В (pow_le_pow_right₀ Г Г);have:rv (dw w)^2/2^53<=4/2^53*rv (dw w)^2:=by have:=sq_nonneg (rv (dw w));rw[div_le_iff₀ В];nlinarith
 linarith
lemma sA (n:ℕ) (g:n<=2^20) (d:ℕ->ℝ) (h:∀j<n,d j^2<=2^600):∑j∈i0 n,d j^2<=2^700:=by
 calc
  _<=∑j∈i0 n,(2:ℝ)^600:=Ϗ λj e=>h j (Ν.1 e)
  _=n*2^600:=by simp
  _<=2^20*2^600:=by gcongr;exact_mod_cast g
  _<=2^700:=by rw[<-pow_add];exact pow_le_pow_right₀ Г Г
lemma fX (d:i9) (g:i7 d≠255):(ZA d:ℝ)*2^298<=2^700:=by
 have:=dB d g;rw[(dZ d g).2] at this;have e:(ZA d:ℝ)*2^298=(ZA d:ℝ)*2^149*2^149:=by rw[mul_assoc,<-pow_add]
 rw[e];calc (ZA d:ℝ)*2^149*2^149<=2^278*2^149:=by gcongr;exact le_trans (le_abs_self _) this
  _<=2^700:=by rw[<-pow_add];exact pow_le_pow_right₀ Г Г
lemma nJ (n:ℕ) (e:n<=2^20):(n:ℝ)/2^1000<=1:=by
 have d:(n:ℝ)<=2^20:=by exact_mod_cast e
 rw[div_le_one В];exact d.trans (pow_le_pow_right₀ Г Г)
lemma iK (n:о) (e:i9) (x:ZB) (r:о) (t g:с) (h:gб<= xв) (v:gб+1<2^32) (z:tб<= gб)
  (A:tб=r*n) (y:gб=r*n+n) (q:1<= n) (p:n<=2^20) (l:∀j<n,i7 x[r*n+j]!≠255)
  (m:i7 e≠255) (u:0<ZA e) (B:rq nж.toInt64.toFloat (1/nж.toInt64.toFloat)=true):
  fv (PP (λw=>let d:=dw w;d*d) x g h v t z 0 0*(1/nж.toInt64.toFloat)+dw e*NR.pt 149)/\
  |rv (PP (λw=>let d:=dw w;d*d) x g h v t z 0 0*(1/nж.toInt64.toFloat)+dw e*NR.pt 149)-
   ((jg n x r 0+ZA e:ℚ):ℝ)*2^298|<=((n:ℝ)+30)/2^54*(((jg n x r 0+ZA e:ℚ):ℝ)*2^298):=by
 obtain ⟨J,K⟩:=eG n q p B;obtain ⟨D,E,F⟩:=dY e m u;obtain ⟨G,H,I⟩:=pS (λw=>let d:=dw w;d*d) x t g h v z n (r*n) A y p (λj=>rv (dw x[r*n+j]!)) (λj o=>tR _ (l j o))
 rw[show ((jg n x r 0+ZA e:ℚ):ℝ)*2^298=(∑j∈i0 n,rv (dw x[r*n+j]!)^2)/n+(ZA e:ℝ)*2^298 by
  unfold jg;rw[Yk];push_cast;rw[add_mul,div_mul_eq_mul_div,zero_add,϶];congr 2;refine Ο rfl λj o=>?_
  rw[sub_zero,(dZ _ (l j (Ν.1 o))).2,mul_pow,show (2:ℝ)^298=(2^149)^2 by rw[<-pow_mul]]]
 exact vP n _ _ _ _ _ _ _ q p (ϡ λj _=>sq_nonneg _) (sA n p _ λj o=>(tR _ (l j o)).2.2) В (nJ n p) G J D I H K E F (fX e m)
lemma fT (V:р) (d:Д) (k:Д->ф) (q:fv V) (j:0<rv V) (l:(dn V d k).1=true)
  (g:rq (dw (dn V d k).2) (1/dw (dn V d k).2)=true):
  i7 (dn V d k).2≠255/\jw (dn V d k).2=false/\1<= rv (dw (dn V d k).2)/\fv (1/dw (dn V d k).2)/\
   |rv (1/dw (dn V d k).2)-1/rv (dw (dn V d k).2)|<=3/2^53/rv (dw (dn V d k).2):=by
 obtain ⟨z,w,x,A,y,r,t,m,u,v⟩:=dW V d k q j l;rw[t] at g⊢;obtain ⟨o,p⟩:=av _ m
 have e:1<= rv (dw (pq V.sqrt.toBits)):=by
  rw[p];have:(2:ℝ)^23<=(2:ℝ)^(((eF V.sqrt.toBits)а:п)-1023):=by
   rw[<-zpow_natCast];exact zpow_le_zpow_right₀ Г (by push_cast;omega)
  linarith[(abs_le.1 v).1]
 have h:rv (dw (pq V.sqrt.toBits))<=2^400:=(le_abs_self _).trans ((dB _ m).trans (pow_le_pow_right₀ Г Г));simp only[rq,υ,decide_eq_true_eq] at g;have:=rc _ _ o e h (by have:=ν.mp g.1;simpa using this) g.2
 exact ⟨m,u,e,this.1,this.2⟩
lemma eD (n:ℕ) (x:ZB) (r:ℕ) (m:ℚ):0<= jg n x r m:=by unfold jg;rw[Yk];positivity
lemma gq (n:о) (x:ZB) (r:о) (g:i9) (c:ф) (p:0<n) (w:0<ZA g) (B:Д) (h:xD n x r g c B=true):
  pq B=jk (jg n x r (if c then jo n x r else 0)+ZA g):=by
 obtain ⟨-,m⟩:=el x n (r*n) 255;have o:=gj n x r;have z:jo n x r*n=∑j∈ZM n,ZA x[r*n+j]!:=by unfold jo;rw[Yk,zero_add];field_simp
 have l:=eD n x r (if c then jo n x r else 0);unfold xD at h;rw[<-beq_iff_eq.1 h,YW _ _ _ _ (Nat.mul_pos (Nat.mul_pos p p) p) w];congr 2;generalize mL x (r*n) n 255=d at m o⊢;generalize sS x d (r*n) n 0 0=S at o⊢
 have v:jg n x r (if c then jo n x r else 0)=((qZ x d (r*n) n 0:ℚ)*((2:ℚ)^((dа:ℤ)-149))^2-n*((if c then ((S.1:ℤ)-S.2)м else 0:ℕ):ℚ)^2*((2:ℚ)^((dа:ℤ)-149))^2/n^2)/n:=by
  rw[iE,kS,zero_add,϶,Ο rfl λj y=>(zQ _ _ (by by_cases h:ZW x[r*n+j]! =0;exact Or.inl h;exact ma (m j (Ν.1 y) h))).symm]
  unfold jg;rw[Yk,zero_add];congr 1;simp only[sub_sq,ϫ,j9,<-϶,<-kJ,kg,ϯ,nsmul_eq_mul];cases c<;>simp only[ο,ite_false,ite_true,<-z,o,ku,ϕ,sq_abs,kS]<;>ring
 have e:((2:ℚ)^((dа:ℤ)-149))^2=2^(2*(dа:ℤ)-298):=by rw[<-zpow_natCast,<-zpow_mul];congr 1;push_cast;ring
 have t:(0:ℚ)<2^(2*(dа:ℤ)-298):=В;rw[v,e] at l⊢;generalize (2:ℚ)^(2*(dа:ℤ)-298)=K at t l⊢;generalize qZ x d (r*n) n 0=Q at l⊢;generalize (if c then ((S.1:ℤ)-S.2)м else 0)=s at l⊢;have u:((Q:ℚ)*K-n*(s:ℚ)^2*K/n^2)/n=(n*n*Q-n*s*s)*K/(n*n*n):=by field_simp
 rw[u] at l⊢;have:(0:ℚ)<=n*n*Q-n*s*s:=by by_contra q;push_neg at q;exact absurd l (not_le.2 (div_neg_of_neg_of_pos (mul_neg_of_neg_of_pos q t) В))
 rw[k9 (by exact_mod_cast sub_nonneg.1 this)];push_cast;ring
lemma iI (l:Д) (k:Д->ф) (j p:р) (x:ZB) (g e:с) (h:eб<= xв) (m:eб+1<2^32) (o:gб<= eб):
  rR l k j p x g e h m o=⟨(dn (PP (λw=>let d:=dw w;d*d) x e h m g o 0 0*p+j) l k).1&&
   rq (dw (dn (PP (λw=>let d:=dw w;d*d) x e h m g o 0 0*p+j) l k).2) (1/dw (dn (PP (λw=>let d:=dw w;d*d) x e h m g o 0 0*p+j) l k).2),
   0,1/dw (dn (PP (λw=>let d:=dw w;d*d) x e h m g o 0 0*p+j) l k).2,(dn (PP (λw=>let d:=dw w;d*d) x e h m g o 0 0*p+j) l k).2,0⟩:=rfl
lemma iG (n:ℕ) (V:р) (k:Д->ф) (v:ℚ) (d:n<=2^20) (g:fv V) (j:0<v) (h:|rv V-(v:ℝ)*2^298|<=((n:ℝ)+30)/2^54*((v:ℝ)*2^298))
  (m:∀B,k B=true->pq B=jk v)
  (e:((dn V (lw n) k).1&&rq (dw (dn V (lw n) k).2) (1/dw (dn V (lw n) k).2))=true):
  (dn V (lw n) k).2=jk v/\i7 (dn V (lw n) k).2≠255/\jw (dn V (lw n) k).2=false/\1<= rv (dw (dn V (lw n) k).2)/\
   fv (1/dw (dn V (lw n) k).2)/\|rv (1/dw (dn V (lw n) k).2)-1/rv (dw (dn V (lw n) k).2)|<=3/2^53/rv (dw (dn V (lw n) k).2):=by
 obtain ⟨p,-,-⟩:=d7 n d;have l:(0:ℝ)<(v:ℝ)*2^298:=mul_pos (by exact_mod_cast j) В
 simp only[υ] at e;have o:0<rv V:=by linarith[(abs_le.1 h).1,mul_le_mul_of_nonneg_right (show (n:ℝ)<=2^20 by exact_mod_cast d) l.le]
 exact ⟨fU V n k v _ d g j h (by gcongr;norm_num) m e.1,fT V (lw n) k g o e.1 e.2⟩
lemma pB (a b:ℝ) (p q:ℕ) (d:|a|<=2^p) (e:|b|<=2^q):|a*b|<=2^(p+q):=by
 rw[abs_mul,pow_add];exact mul_le_mul d e (abs_nonneg _) В
lemma rB (y z:ℝ) (p:ℕ) (d:|z|<=2^p) (h:|y-z|<=|z|/2^53+1/2^1075):|y|<=2^(p+1):=by
 have t:(1:ℝ)/2^1075<=1/4:=one_div_le_one_div_of_le Г (le_trans (by norm_num:(4:ℝ)<=2^2) (pow_le_pow_right₀ Г Г))
 have:(1:ℝ)<=2^p:=one_le_pow₀ Г;rw[pow_succ];generalize (1:ℝ)/2^1075=τ at*;linarith[abs_sub_abs_le_abs_sub y z,div_le_div_of_nonneg_left (abs_nonneg z) (by norm_num:(0:ℝ)<2) (by norm_num:(2:ℝ)<=2^53)]
lemma aB (a b:ℝ) (p:ℕ) (d:|a|<=2^p) (e:|b|<=2^p):|a+b|<=2^(p+1):=by
 rw[pow_succ];linarith[abs_add_le a b]
lemma eE (D e:ℝ) (g:1<= D) (h:|e-1/D|<=3/2^53/D):|e|<=2^1:=by
 have d:0<D:=by linarith
 have:=abs_sub_abs_le_abs_sub e (1/D);rw[abs_of_pos (one_div_pos.2 d)] at this;have:1/D<=1:=by rw[div_le_one d];exact g
 have:3/2^53/D<=3/2^53:=by rw[div_le_iff₀ d];nlinarith
 rw[pow_one];linarith
lemma gp (d:i9) (e:р) (h:i7 d≠255) (g:fv e) (l:|rv e|<=2^1):
  fv (dw d*e)/\|rv (dw d*e)-rv (dw d)*rv e|<=|rv (dw d)*rv e|/2^53+1/2^1075/\|rv (dw d*e)|<=2^280:=by
 have k:=pB _ _ 278 1 (dB d h) l;obtain ⟨m,j⟩:=NF.fM _ _ (dZ d h).1 g (NE.z9 _ _ k Г);exact ⟨m,j,rB _ _ 279 k j⟩
lemma zE (a b:ℕ) (c:ℤ) (h:(a:ℤ)-b<= c):(2:ℝ)^a/2^b<=(2:ℝ)^c:=by
 rw[<-zpow_natCast,<-zpow_natCast,<-zpow_sub₀ Г];exact zpow_le_zpow_right₀ Г h
lemma wb (w:i9) (d:i7 w≠255) (a M:ℕ) (h:a+je w+1046<=M):2^a*|rv (dw w)|/2^53<=(2:ℝ)^((M:ℤ)-1075):=by
 calc
  (2:ℝ)^a*|rv (dw w)|/2^53<=2^a*2^(24+je w)/2^53:=by gcongr;rw[(av _ d).2];exact NE.wB w
  _=2^(a+(24+je w))/2^53:=by rw[<-pow_add]
  _<=_:=zE _ _ _ (by push_cast;omega)
lemma gZ (v Z Y e G U d K:ℝ) (k:ℕ) (p:|v-Z|<=71/10/2^53*|v|+61/10/2^53*|e|+K) (m:|Z-Y|<=2^k*|G|/2^53) (o:|v|<2^53*U)
  (h:32*U<=d) (j:2^5*|e|/2^53<=d) (g:2^(k+1)*|G|/2^53<=d) (l:K*64<=d):|v-Y|<=d:=by
 rw[pow_succ] at g;have:=abs_sub_le v Z Y;have:=abs_nonneg v;have:=abs_nonneg e;linarith
lemma jX (X:ℝ) (d:|X|<=2^281):(|X|)*(1/2^1075)<=1/2^794/\(1:ℝ)/2^1075<=1/2^794/\8*((1:ℝ)/2^794)<=1/2^700/\(0:ℝ)<=1/2^794:=by
 refine ⟨?_,?_,?_,by positivity⟩
 .calc (|X|)*(1/2^1075)<=2^281*(1/2^1075):=by gcongr
   _=1/2^794:=by rw[mul_one_div,div_eq_div_iff В В,one_mul,<-pow_add]
 .exact one_div_le_one_div_of_le В (pow_le_pow_right₀ Г Г)
 .rw[mul_one_div,div_le_div_iff₀ В В,one_mul,show (8:ℝ)=2^3 by norm_num,<-pow_add]
  exact pow_le_pow_right₀ Г Г
lemma fZ (X G D d e v:ℝ) (p:1<= D) (j:|d-1/D|<=3/2^53/D) (h:|e-G*d|<=|G*d|/2^53+1/2^1075)
  (q:|v-X*e|<=|X*e|/2^53+1/2^1075) (l:|X|<=2^281):|v-X*G/D|<=501/100/2^53*|v|+1/2^700:=by
 have g:0<D:=by linarith
 obtain ⟨r,t,u,w⟩:=jX X l;have:(0:ℝ)<=1/2^1075:=В;generalize (1:ℝ)/2^1075=τ at*
 generalize (1:ℝ)/2^794=σ at*
 generalize (1:ℝ)/2^700=K at*
 have m:|X*G*d-X*G/D|<=3/2^53*|X*G/D|:=by
  rw[show X*G*d-X*G/D=X*G*(d-1/D) by field_simp,abs_mul,abs_div,abs_of_pos g,show 3/2^53*(|X*G|/D)=(|X*G|)*(3/2^53/D) by ring]
  exact mul_le_mul_of_nonneg_left j (abs_nonneg _)
 have o:|X*e-X*G*d|<=|X*G*d|/2^53+σ:=by
  rw[show X*e-X*G*d=X*(e-G*d) by ring,abs_mul,mul_assoc,abs_mul X]
  linarith[mul_le_mul_of_nonneg_left h (abs_nonneg X)]
 generalize X*G/D=Y at*
 linarith[abs_sub_le v (X*e) Y,abs_sub_le (X*e) (X*G*d) Y,abs_sub_abs_le_abs_sub (X*G*d) Y,abs_sub_abs_le_abs_sub (X*e) Y,
  abs_sub_abs_le_abs_sub Y v,abs_sub_comm Y v,abs_nonneg v]
lemma ia (j G D u t s q l v:ℝ) (y:1<= D) (p:|u-1/D|<=3/2^53/D)
  (x:|s-j|<=|j|/2^53+1/2^1075) (o:|t-G*u|<=|G*u|/2^53+1/2^1075) (B:|q-s*t|<=|s*t|/2^53+1/2^1075)
  (V:|v-(q+l)|<=|q+l|/2^53+1/2^1075) (A:|j|<=2^280) (z:|G|<=2^278):
  |v-(j*G/D+l)|<=71/10/2^53*|v|+61/10/2^53*|l|+1/2^699:=by
 have m:0<D:=by linarith
 have r:|s|<=2^281:=rB _ _ 280 A x
 have Q:=fZ s G D u t q y p o B r
 have F:|s*G/D-j*G/D|<=|j*G/D|/2^53+1/2^1075*2^278:=by
  have e:s*G/D-j*G/D=(s-j)*(|G|/D)*(if 0<= G then 1 else-1):=by
   split<;>rename_i h
   .rw[abs_of_nonneg h];field_simp
   .rw[abs_of_neg (by linarith)];field_simp;try ring
  have H:|G|/D<=2^278:=(div_le_self (abs_nonneg _) y).trans z
  rw[e,abs_mul,abs_mul,abs_of_nonneg (div_nonneg (abs_nonneg _) m.le)];have:|(if 0<= G then (1:ℝ) else-1)|=1:=by split<;>simp
  rw[this,mul_one]
  calc
   (|s-j|)*(|G|/D)<=(|j|/2^53+1/2^1075)*(|G|/D):=mul_le_mul_of_nonneg_right x (div_nonneg (abs_nonneg _) m.le)
   _=|j*G/D|/2^53+1/2^1075*(|G|/D):=by rw[abs_div,abs_mul,abs_of_pos m];ring
   _<=_:=by gcongr
 have I:(1:ℝ)/2^1075*2^278<=1/2^797:=by rw[div_mul_eq_mul_div,one_mul,div_le_div_iff₀ В В,one_mul,<-pow_add]
 have W:(1:ℝ)/2^1075<=1/2^797:=one_div_le_one_div_of_le В (pow_le_pow_right₀ Г Г)
 have X:(1:ℝ)/2^797*16<=1/2^700:=by
  rw[div_mul_eq_mul_div,one_mul,div_le_div_iff₀ В В,one_mul,show (16:ℝ)=2^4 by norm_num,<-pow_add]
  exact pow_le_pow_right₀ Г Г
 have Y:(1:ℝ)/2^700*2=1/2^699:=by rw[div_mul_eq_mul_div,one_mul,div_eq_div_iff В В,one_mul,<-pow_succ']
 have Z:(0:ℝ)<=1/2^797:=В;have tτ:(0:ℝ)<=1/2^1075:=В;have J:=abs_sub_le v (q+l) (j*G/D+l)
 have E:|q+l-(j*G/D+l)|<=|q-s*G/D|+|s*G/D-j*G/D|:=by
  rw[show q+l-(j*G/D+l)=(q-s*G/D)+(s*G/D-j*G/D) by ring];exact abs_add_le _ _
 have L:|q+l|<=|v|+|v-(q+l)|:=by have:=abs_sub_abs_le_abs_sub (q+l) v;rw[abs_sub_comm] at this;linarith
 have M:|q|<=|q+l|+|l|:=by have:=abs_sub_abs_le_abs_sub q (q+l);rw[show q-(q+l)=-l by ring,abs_neg] at this;linarith
 have N:|j*G/D|<=|q|+|q-s*G/D|+|s*G/D-j*G/D|:=by
  have:=abs_sub_abs_le_abs_sub (j*G/D) q
  have:|j*G/D-q|<=|q-s*G/D|+|s*G/D-j*G/D|:=by
   rw[abs_sub_comm,show q-j*G/D=(q-s*G/D)+(s*G/D-j*G/D) by ring];exact abs_add_le _ _
  linarith
 clear r A z p o B x;generalize (2:ℝ)^278=P at F I;generalize (1:ℝ)/2^1075=τ at*;generalize (1:ℝ)/2^797=σ at*;generalize (1:ℝ)/2^700=K at*;generalize (1:ℝ)/2^699=O at*
 have S:=abs_nonneg v;have U:=abs_nonneg l;generalize |v-(j*G/D+l)|=w at*;generalize |q+l-(j*G/D+l)|=w2 at*;generalize |v-(q+l)|=a at*
 generalize |q-s*G/D|=b at*;generalize |s*G/D-j*G/D|=c at*;generalize |q+l|=d at*;generalize |q|=e at*
 generalize |j*G/D|=f at*;generalize |v|=g at*;generalize |l|=h at*
 linarith
lemma zW (w:i9) (h:(w&&&0x7FFFFFFF==0)=true):ZA w=0:=by
 have d:(w&&&0x7FFFFFFF)е=0:=by simp only[beq_iff_eq] at h;rw[h];rfl
 rw[Λ] at d;simp only[jj] at d;rw[show (2147483647:ℕ)=2^31-1 by norm_num,θ] at d
 have e:=fY w;have g:=f5 w
 have j:ZW w=0:=by
  unfold jF;have:(i7 w)е=0:=by rw[e];omega
  have:i7 w=0:=Η.mp (by rw[this];rfl)
  rw[if_pos this,g];omega
 rw[Y1,show (sv w:ℚ)=0 from (YN w).2 j,zero_mul]
lemma gB (j g d:i9) (l:р) (b:ф) (B:i7 j≠255) (A:i7 g≠255) (x:i7 d≠255) (o:1<= rv (dw d))
  (t:fv l) (p:|rv l-1/rv (dw d)|<=3/2^53/rv (dw d)):c9 ⟨b,0,l,d,0⟩ j g (dw g)=ZT (ZA j*ZA g/ZA d):=by
 unfold c9;dsimp only;split
 .rename_i w
  simp only[υ,decide_eq_true_eq] at w;obtain ⟨M,N⟩:=dZ j B;obtain ⟨I,J,L⟩:=gp g l A t (eE _ _ o p);have y:=dB j B;obtain ⟨D,H⟩:=NF.fM _ _ M I (NE.z9 _ _ (pB _ _ 278 280 y L) Г);obtain ⟨q,F⟩:=te _ w.1
  have z:16<(dw j*(dw g*l)).toBitsа%2^27/\(dw j*(dw g*l)).toBitsа%2^27+16<2^27:=by
   have:=ν.mp w.2;rw[Κ,ψ] at this;simp only[iZ] at this;rw[show (134217727:ℕ)=2^27-1 by norm_num,θ] at this
   have:=ϳ (dw j*(dw g*l)).toBitsа (show 0<2^27 by norm_num);omega
  apply eC _ D 4 _ q F (by simpa using z.1) (by simpa using z.2)
  have m:0<rv (dw d):=by linarith
  obtain ⟨-,r⟩:=dZ d x;have:(ZA d:ℝ)≠0:=λh=>by rw[h] at r;rw[r] at m;simp at m
  rw[show ((ZA j*ZA g/ZA d:ℚ):ℝ)*2^149=rv (dw j)*rv (dw g)/rv (dw d) by rw[N,(dZ g A).2,r];push_cast;field_simp]
  have e:=fZ _ _ _ _ _ _ o p J H (y.trans (pow_le_pow_right₀ Г Г));have u:=bU _ D;generalize dw j*(dw g*l)=v at e u q⊢;generalize (eF v.toBits)а=E at q u⊢;have:(1:ℝ)/2^700<=2^((E:п)-1075):=by rw[one_div,<-zpow_natCast,<-zpow_neg];exact zpow_le_zpow_right₀ Г А
  generalize (2:ℝ)^((E:п)-1075)=U at*;generalize (1:ℝ)/2^700=K at*
  have:=abs_nonneg (rv v);norm_num at e u⊢;nlinarith
 .split
  .rename_i _ G;rw[zW j G,zero_mul,zero_div];rfl
  .rfl
lemma gz (e:RS) (d j:i9) (g:р) (h:(d&&&0x7FFFFFFF==0)=true):eR e d j g=0:=by
 have l:d&&&0x7FFFFFFF=0:=by simpa using h
 have m:((d&&&0x7FFFFFFF)к-1)>>>(63:Д)=1:=by rw[l];decide
 unfold eR;dsimp only;rw[m,show (1:Д)-1=0 from rfl];split
 .simp
 .rfl
lemma gA (e:RS) (d j:i9) (g:р) (h:(d&&&0x7FFFFFFF==0)=false):eR e d j g=c9 e d j g:=by
 have l:(d&&&0x7FFFFFFF)≠0:=by simpa using h
 have m:(((d&&&0x7FFFFFFF)к-1)>>>(63:Д))=0:=by
  have p:0<(d&&&0x7FFFFFFF)е:=by
   rcases Nat.eq_zero_or_pos (d&&&0x7FFFFFFF)е with q|q
   .exact absurd (Η.mp (by rw[q];rfl)) l
   .exact q
  apply Β.mp;rw[Α,χ _ _ (κ.2 (by simp only[Χ,iZ];omega))]
  simp only[Χ,iZ,jb,ι];have:=(d&&&0x7FFFFFFF).toNat_lt;omega
 unfold eR c9;dsimp only;rw[m];simp
lemma ij (e g d:i9) (h:р) (b:ф) (r:i7 e≠255) (q:i7 g≠255) (m:i7 d≠255) (l:1<= rv (dw d))
  (j:fv h) (o:|rv h-1/rv (dw d)|<=3/2^53/rv (dw d)):eR ⟨b,0,h,d,0⟩ e g (dw g)=ZT (ZA e*ZA g/ZA d):=by
 cases p:e&&&0x7FFFFFFF==0
 .rw[gA _ _ _ _ p];exact gB e g d h b r q m l j o
 .rw[gz _ _ _ _ p,zW e p,zero_mul,zero_div];rfl
lemma gC (h n:о) (e:i9) (x g:ZB) (A:ji h n e x g=true)
  (I:h*n+1<2^32) (m:n<=2^20) (y:rq nж.toInt64.toFloat (1/nж.toInt64.toFloat)=true)
  (r:о) (J:r<h) (h1 h2 h3) (l:(rR (lw n) (xD n x r e false) (dw e*NR.pt 149) (1/nж.toInt64.toFloat) x (r*n)г ((r*n)г+nг) h1 h2 h3).ok=true)
  (j:о) (B:j<n):
  eR (rR (lw n) (xD n x r e false) (dw e*NR.pt 149) (1/nж.toInt64.toFloat) x (r*n)г ((r*n)г+nг) h1 h2 h3) x[r*n+j]! g[j]! (dw g[j]!)=
   ZT (ZA x[r*n+j]!*ZA g[j]!/ZA (jk (jg n x r 0+ZA e))):=by
 obtain ⟨p,o,q,v⟩:=fI A;have u:=eJ A;obtain ⟨O,L,Q,M⟩:=uB J I;have t:∀j<n,i7 x[r*n+j]!≠255:=λj B=>q _ А
 obtain ⟨D,z⟩:=iK n e x r _ _ h1 h2 h3 L M p m t u o y;have K:0<jg n x r 0+ZA e:=by have:=eD n x r 0;linarith
 rw[iI] at l⊢;dsimp only at l⊢;generalize PP (λw=>let d:=dw w;d*d) x ((r*n)г+nг) h1 h2 (r*n)г h3 0 0*(1/nж.toInt64.toFloat)+dw e*NR.pt 149=V at D z l⊢
 obtain ⟨c,E,N,F,G,H⟩:=iG n V _ _ m D K (by push_cast at z⊢;exact z) (gq n x r e false p o) l
 generalize (dn V (lw n) (xD n x r e false)).2=d at c E N F G H⊢;rw[<-c];exact ij _ _ d _ _ (q _ А) (v j B) E F G H
lemma bu (b:Д):(eF b)а<=2047:=by
 unfold eF;rw[ψ];exact Nat.and_le_right.trans (by simp)
lemma gl (a b:ℝ) (g:|a|<=2^278) (j:|b|<=2^279):|a-b|<=2^280:=by
 have h:=abs_sub_le a 0 b;rw[sub_zero,zero_sub,abs_neg] at h;have d:(2:ℝ)^279=2*2^278:=by rw[<-pow_succ']
 have e:(2:ℝ)^280=2*2^279:=by rw[<-pow_succ']
 rw[d] at j;rw[e,d];clear d e;generalize (2:ℝ)^278=T at*;have:0<= T:=le_trans (abs_nonneg _) g;linarith
lemma cS (n:ℕ) (l:1<= n) (d:ℕ->ℝ):∑j∈i0 n,|d j|<= n*ja ((∑j∈i0 n,d j^2)/n):=by
 have g:(0:ℝ)<n:=by exact_mod_cast l
 have h:=sq_sum_le_card_mul_sum_sq (s:=i0 n) (f:=λj=>|d j|)
 simp only[ϯ,sq_abs] at h;have e:(n:ℝ)*∑j∈i0 n,d j^2=(n:ℝ)^2*((∑j∈i0 n,d j^2)/n):=by field_simp
 calc
  _<= ja ((n:ℝ)*∑j∈i0 n,d j^2):=lq h
  _=_:=by rw[e,lI (sq_nonneg _),lU g.le]
lemma mR (n:ℕ) (o:1<= n) (v:n<=2^20) (S g W d m:ℝ) (q:|g|<= W)
  (x:|S-g|<=((n:ℝ)+3)/2*((1/2^53+1/2^73)*W)+((n:ℝ)+1)*(1/2^1074))
  (p:|d-1/n|<=3/2^53/n) (u:|m-S*d|<=|S*d|/2^53+1/2^1075):
  |m-g/n|<=((n:ℝ)+13)/2^54*(W/n)+1/2^1040:=by
 have l:(1:ℝ)<= n:=by exact_mod_cast o
 have h:(n:ℝ)<=2^20:=by exact_mod_cast v
 have j:(0:ℝ)<n:=by linarith
 have w:=le_trans (abs_nonneg _) q
 have r:|n*d-1|<=3/2^53:=by
  rw[show n*d-1=n*(d-1/n) by field_simp,abs_mul,abs_of_pos j];exact (mul_le_mul_of_nonneg_left p j.le).trans_eq (by field_simp)
 have e:=abs_add_three (n*(m-S*d)) (S*(n*d-1)) (S-g);rw[show n*(m-S*d)+S*(n*d-1)+(S-g)=n*(m-g/n) by field_simp;ring,abs_mul,abs_mul,abs_mul,abs_of_pos j] at e
 have:n*|S*d|=(|S|)*|n*d|:=by rw[abs_mul,abs_mul,abs_of_pos j];ring
 have:=abs_sub_abs_le_abs_sub (n*d) 1;have:=abs_sub_abs_le_abs_sub S g;have:=mul_le_mul_of_nonneg_left u j.le;have:=mul_le_mul_of_nonneg_left r (abs_nonneg S);have:=mul_le_mul_of_nonneg_left (show |n*d|<=1+3/2^53 by rw[abs_one] at*;linarith) (abs_nonneg S)
 have:=mul_le_mul_of_nonneg_right h w;have:=mul_le_mul_of_nonneg_right h (by positivity:(0:ℝ)<=1/2^1075)
 have t:(0:ℝ)<=1/2^1074:=В;have:=mul_le_mul_of_nonneg_right h t;have:=mul_le_mul_of_nonneg_right l (by positivity:(0:ℝ)<=1/2^1040)
 have:(1:ℝ)/2^1075<=1/2^1074:=by gcongr<;>norm_num
 have:(1:ℝ)/2^1074*2^25<=1/2^1040:=by rw[div_mul_eq_mul_div,one_mul,div_le_div_iff₀ В В,one_mul,<-pow_add];exact pow_le_pow_right₀ Г Г
 refine le_of_mul_le_mul_left (e.trans ?_) j;rw[show n*(((n:ℝ)+13)/2^54*(W/n)+1/2^1040)=((n:ℝ)+13)/2^54*W+1/2^1040*n by field_simp]
 linarith
lemma mE (n:ℕ) (P Q D R d δ W:ℝ) (v:(n:ℝ)+17<= P) (j:P<=2^21) (w:1<= Q) (u:1<= D)
  (A:δ<=((n:ℝ)+13)/2^54*(W/n)+1/2^1040) (y:W/n<= R+|d|) (x:R<= D*(1+1/2^22)) (z:|d|<= Q*D):δ<= P*Q/2^53*D:=by
 have g:W/n<= D*(1+1/2^22+Q):=by linarith
 have l:(0:ℝ)<= n:=lj n;have q:((n:ℝ)+13)/2^54*(W/n)<=((n:ℝ)+13)/2^54*(D*(1+1/2^22+Q)):=mul_le_mul_of_nonneg_left g В
 have h:((n:ℝ)+13)*(1+1/2^22+Q)<=2*P*Q-15/2:=by
  have m:((n:ℝ)+13)*(1+1/2^22+Q)<=(P-4)*(1+1/2^22+Q):=mul_le_mul_of_nonneg_right (by linarith) (by linarith)
  have o:0<= P*(Q-1):=mul_nonneg (by linarith) (by linarith)
  nlinarith
 have r:((n:ℝ)+13)*(1+1/2^22+Q)*D<=(2*P*Q-15/2)*D:=mul_le_mul_of_nonneg_right h (by linarith)
 have t:(1:ℝ)/2^1040<=15/2*D/2^54:=by
  have a:(1:ℝ)/2^1040<=1/2^54:=one_div_le_one_div_of_le В (pow_le_pow_right₀ Г Г);have b:(1:ℝ)/2^54<=15/2*D/2^54:=by rw[div_le_div_iff_of_pos_right В];linarith
  linarith
 have e:((n:ℝ)+13)/2^54*(D*(1+1/2^22+Q))=((n:ℝ)+13)*(1+1/2^22+Q)*D/2^54:=by ring
 have p:(2*P*Q-15/2)*D/2^54=P*Q/2^53*D-15/2*D/2^54:=by ring
 have:((n:ℝ)+13)*(1+1/2^22+Q)*D/2^54<=(2*P*Q-15/2)*D/2^54:=by gcongr
 linarith
lemma gY (n:ℕ) (v:n<=2^20) (V d g D:ℝ) (e:2^149<= d) (u:|V-d|<=((n:ℝ)+30)/2^54*d) (j:0<V)
  (p:|g-ja V|<= ja V/2^53+1/2^1075) (r:|D-g|<= g/2^24):|D-ja d|<= ja d/2^23:=by
 have h:0<d:=lt_of_lt_of_le В e;have l:=gn V d _ h j.le u;have w:(n:ℝ)<=2^20:=by exact_mod_cast v
 have x:((n:ℝ)+30)/2^54<=1/2^33:=by rw[div_le_div_iff₀ В В];norm_num;linarith
 have y:(2:ℝ)^74<= ja d:=lq (le_trans (by rw[<-pow_mul];exact pow_le_pow_right₀ Г Г) e)
 have z:(1:ℝ)/2^1075<= ja d/2^60:=by
  have a:(1:ℝ)/2^1075<=1/2^60:=one_div_le_one_div_of_le В (pow_le_pow_right₀ Г Г);have o:(1:ℝ)<=2^74:=one_le_pow₀ Г;have b:(1:ℝ)/2^60<= ja d/2^60:=by rw[div_le_div_iff_of_pos_right В];linarith
  linarith
 have A:((n:ℝ)+30)/2^54*ja d<=1/2^33*ja d:=mul_le_mul_of_nonneg_right x (ϊ _);have t:=abs_le.1 r;have q:=abs_le.1 p;have m:=abs_le.1 l;rw[abs_le];constructor<;>nlinarith[ϊ V,ϊ d]
lemma bI (n:ℕ) (r:n<=2^20) (V T δ D:ℝ) (x:0<T) (y:|V-(T+δ^2)|<=((n:ℝ)+30)/2^54*(T+δ^2))
  (j:|δ|<= D/2^28) (p:|D-ja (T+δ^2)|<= ja (T+δ^2)/2^23):|V-T|<=((n:ℝ)+31)/2^54*T:=by
 have t:(n:ℝ)<=2^20:=by exact_mod_cast r
 have o:0<= T+δ^2:=В;have u:=lb o;have d:0<= D:=by have h:=le_trans (abs_nonneg δ) j;rwa[le_div_iff₀ В,zero_mul] at h
 have g:D<= ja (T+δ^2)*(1+1/2^23):=by have:=(abs_le.1 p).2;linarith
 have q:D^2<=(T+δ^2)*(1+1/2^23)^2:=by
  rw[<-u,<-mul_pow];exact pow_le_pow_left₀ d g 2
 have v:δ^2<= D^2/2^56:=by
  have:=pow_le_pow_left₀ (abs_nonneg δ) j 2;rw[sq_abs,div_pow] at this;norm_num at this⊢;linarith
 have w:δ^2<= T/2^55:=by nlinarith
 have z:((n:ℝ)+30)/2^54*δ^2<=1/2^33*δ^2:=mul_le_mul_of_nonneg_right
  (by rw[div_le_div_iff₀ В В];norm_num;linarith) (sq_nonneg δ)
 have e:((n:ℝ)+30)/2^54*(T+δ^2)=((n:ℝ)+30)/2^54*T+((n:ℝ)+30)/2^54*δ^2:=by ring
 have l:((n:ℝ)+31)/2^54*T=((n:ℝ)+30)/2^54*T+T/2^54:=by ring
 have:=abs_sub_le V (T+δ^2) T;have m:|T+δ^2-T|=δ^2:=by rw[add_sub_cancel_left,abs_of_nonneg (sq_nonneg δ)]
 rw[m] at this;rw[l];nlinarith
lemma iQ (n:ℕ) (l:1<= n) (X:ℕ->ℝ) (a:ℝ):∑j∈i0 n,(X j-a)^2=
  ∑j∈i0 n,(X j-(∑k∈i0 n,X k)/n)^2+n*((∑k∈i0 n,X k)/n-a)^2:=by
 have h:(n:ℝ)≠0:=Nat.cast_ne_zero.2 А;generalize g:(∑k∈i0 n,X k)/n=M
 have d:∑j∈i0 n,(X j-M)=0:=by
  rw[j9,kg,ϯ,nsmul_eq_mul,<-g];field_simp;ring
 have e:∀j,(X j-a)^2=(X j-M)^2+2*(M-a)*(X j-M)+(M-a)^2:=λj=>by ring
 simp only[e,ϫ,<-kJ,d,kg,ϯ,nsmul_eq_mul];ring
lemma iU (d s p τ:ℝ) (e:0<=τ) (o:τ+2^58*τ^2<=1/2^1000) (m:|s-d|<=|d|/2^53+τ) (l:|p-s^2|<= s^2/2^53+τ):
  |p-d^2|<=4/2^53*d^2+1/2^1000:=by
 have g:=abs_add_le (2*d*(s-d)) ((s-d)^2);rw[show 2*d*(s-d)+(s-d)^2=s^2-d^2 by ring,abs_mul,abs_mul,abs_two,abs_pow] at g
 have j:=abs_nonneg (s-d);have h:=abs_nonneg d;have q:(|s-d|)^2<=2*(|d|/2^53)^2+2*τ^2:=by nlinarith[sq_nonneg (|d|/2^53-τ)]
 have r:2*(|d|)*(|s-d|)<=2*(|d|)*(|d|/2^53+τ):=by nlinarith
 have t:2*(|d|)*τ<=2^57*τ^2+(|d|)^2/2^57:=by nlinarith[sq_nonneg (2^57*τ-|d|)]
 have u:s^2<= d^2+|s^2-d^2|:=by have:=le_abs_self (s^2-d^2);linarith
 have v:=abs_sub_le p (s^2) (d^2);have w:=sq_abs d;generalize (1:ℝ)/2^1000=ζ at*
 nlinarith
lemma tL (w:i9) (m:р) (g:i7 w≠255) (q:fv m) (r:|rv m|<=2^279):
  fv ((dw w-m)*(dw w-m))/\|rv ((dw w-m)*(dw w-m))-(rv (dw w)-rv m)^2|<=4/2^53*(rv (dw w)-rv m)^2+1/2^1000/\(rv (dw w)-rv m)^2<=2^600:=by
 have p:=(av _ g).1;have d:=gl _ _ (dB _ g) r;obtain ⟨l,o⟩:=NF.fS _ _ p q (NE.z9 _ 280 d Г);have h:|rv (dw w-m)|<=2^281:=rB _ _ 280 d o
 have e:=pB _ _ 281 281 h h;obtain ⟨t,j⟩:=NF.fM _ _ l l (NE.z9 _ _ e Г);rw[abs_mul_self,<-sq] at j;refine ⟨t,iU _ _ _ (1/2^1075) В ?_ o j,?_⟩
 .have:(2:ℝ)^58*(1/2^1075)<=1:=by rw[mul_one_div,div_le_one В];exact pow_le_pow_right₀ Г Г
  have:(1:ℝ)/2^1075<=1/2^1000/2:=by rw[div_div,<-pow_succ];gcongr<;>norm_num
  have:(0:ℝ)<=1/2^1075:=В;generalize (1:ℝ)/2^1075=τ at*;nlinarith
 .have:=sq_le_sq' (abs_le.1 d).1 (abs_le.1 d).2;rw[<-pow_mul] at this
  exact this.trans (pow_le_pow_right₀ Г Г)
lemma it (n:о) (x:ZB) (r o:с) (h:oб<= xв) (v:oб+1<2^32) (z:rб<= oб)
  (d:ℕ) (A:rб=d) (y:oб=d+n) (t:1<= n) (J:n<=2^20) (E:∀j<n,i7 x[d+j]!≠255)
  (L:rq nж.toInt64.toFloat (1/nж.toInt64.toFloat)=true):
  fv (PP dw x o h v r z 0 0*(1/nж.toInt64.toFloat))/\
  |rv (PP dw x o h v r z 0 0*(1/nж.toInt64.toFloat))-(∑j∈i0 n,rv (dw x[d+j]!))/n|<=
   ((n:ℝ)+13)/2^54*((∑j∈i0 n,|rv (dw x[d+j]!)|)/n)+1/2^1040/\
  |rv (PP dw x o h v r z 0 0*(1/nж.toInt64.toFloat))|<=2^279:=by
 obtain ⟨r1,V⟩:=eG n t J L;have K:(1:ℝ)<= n:=by exact_mod_cast t
 have as:(n:ℝ)<=2^20:=by exact_mod_cast J
 have p:(0:ℝ)<n:=by linarith
 have H:∀g:ℕ->ℝ,∑k∈jr rб oб,g k=∑j∈i0 n,g (d+j):=by
  intro g;rw[kz,A,y,show d+n-d=n by omega]
 have m:∑j∈i0 n,|rv (dw x[d+j]!)|<= n*2^278:=by
  calc
   _<=∑j∈i0 n,(2:ℝ)^278:=Ϗ λj Z=>dB _ (E j (Ν.1 Z))
   _=n*2^278:=by simp
 have G:∑j∈i0 n,|rv (dw x[d+j]!)|<=2^298:=m.trans (by
  calc
   (n:ℝ)*2^278<=2^20*2^278:=by gcongr
   _=2^298:=by rw[<-pow_add])
 obtain ⟨U,u⟩:=fz dw x o h v r z (λk=>|rv (dw x[k]!)|)
  (λk Q S=>by
   obtain ⟨j,rfl⟩:∃j,k=d+j:=⟨k-d,by omega⟩
   exact ⟨(av _ (E j А)).1,le_rfl⟩)
  (by rw[H];exact G.trans (pow_le_pow_right₀ Г Г)) А
 rw[H,H,y,A,show d+n-d=n by omega] at u
 have l:=Ϲ (λj=>rv (dw x[d+j]!)) (i0 n)
 generalize ∑j∈i0 n,|rv (dw x[d+j]!)|=W at G m u l⊢
 generalize ∑j∈i0 n,rv (dw x[d+j]!)=F at u l⊢;generalize PP dw x o h v r z 0 0=q at U u⊢;have M:0<= W:=le_trans (abs_nonneg _) l;have D:=eE _ _ K V
 have O:|rv q|<=2^299:=by
  have X:((n:ℝ)+3)/2*((1/2^53+1/2^73)*W)<= W/2:=by
   have:((n:ℝ)+3)/2*(1/2^53+1/2^73)<=1/2:=by rw[div_mul_eq_mul_div,div_le_div_iff₀ Г Г];nlinarith
   nlinarith
  have Y:((n:ℝ)+1)*(1/2^1074)<=1:=by
   have:(1:ℝ)/2^1074<=1/2^21:=one_div_le_one_div_of_le В (pow_le_pow_right₀ Г Г);have:((n:ℝ)+1)*(1/2^1074)<=(2^20+1)*(1/2^21):=by gcongr
   linarith
  have:=abs_sub_abs_le_abs_sub (rv q) F;clear m;have e:(2:ℝ)^299=2*2^298:=by rw[<-pow_succ']
  have I:(2:ℝ)<=2^298:=le_self_pow₀ Г Г;rw[e];clear e;generalize (2:ℝ)^298=T at*;generalize ((n:ℝ)+3)/2*((1/2^53+1/2^73)*W)=c at*
  generalize ((n:ℝ)+1)*(1/2^1074)=j at*;linarith only[this,u,X,Y,l,G,I]
 have B:=pB _ _ 299 1 O D;obtain ⟨m1,m2⟩:=NF.fM _ _ U r1 (NE.z9 _ _ B Г);have N:=mR n t J (rv q) F W _ _ l u V m2;refine ⟨m1,N,?_⟩
 have Q:W/n<=2^278:=by rw[div_le_iff₀ p];calc W<= n*2^278:=m
   _=_:=mul_comm _ _
 have S:|F/n|<=W/n:=by rw[abs_div,abs_of_pos p];exact div_le_div_of_nonneg_right l p.le
 have k3:((n:ℝ)+13)/2^54*(W/n)<= W/n/2:=by
  have:((n:ℝ)+13)/2^54<=1/2:=by rw[div_le_div_iff₀ В В];nlinarith
  have:=mul_le_mul_of_nonneg_right this (div_nonneg M p.le);linarith
 have k4:(1:ℝ)/2^1040<=1:=by rw[div_le_one В];exact one_le_pow₀ Г
 have:=abs_sub_abs_le_abs_sub (rv (q*(1/nж.toInt64.toFloat))) (F/n);have e:(2:ℝ)^279=2*2^278:=by rw[<-pow_succ']
 have I:(2:ℝ)<=2^278:=le_self_pow₀ Г Г;clear G O B m;rw[e];clear e;generalize (2:ℝ)^278=T at*;generalize W/n=w at*;generalize (1:ℝ)/2^1040=ζ at*
 generalize ((n:ℝ)+13)/2^54*w=c at*;linarith only[this,N,Q,S,k3,k4,I]
lemma iJ (n:о) (g:i9) (x:ZB) (t l:с) (h:lб<= xв) (y:lб+1<2^32) (B:tб<= lб)
  (e:ℕ) (D:tб=e) (z:lб=e+n) (q:1<= n) (p:n<=2^20) (r:∀j<n,i7 x[e+j]!≠255)
  (o:i7 g≠255) (v:0<ZA g) (E:rq nж.toInt64.toFloat (1/nж.toInt64.toFloat)=true)
  (m:р) (F:fv m) (G:|rv m|<=2^279):
  fv (PP (λw=>let d:=dw w-m;d*d) x l h y t B 0 0*(1/nж.toInt64.toFloat)+dw g*NR.pt 149)/\
  |rv (PP (λw=>let d:=dw w-m;d*d) x l h y t B 0 0*(1/nж.toInt64.toFloat)+dw g*NR.pt 149)-
   ((∑j∈i0 n,(rv (dw x[e+j]!)-rv m)^2)/n+(ZA g:ℝ)*2^298)|<=
   ((n:ℝ)+30)/2^54*((∑j∈i0 n,(rv (dw x[e+j]!)-rv m)^2)/n+(ZA g:ℝ)*2^298):=by
 obtain ⟨N,O⟩:=eG n q p E;obtain ⟨H,I,J⟩:=dY g o v
 obtain ⟨K,L,M⟩:=pS (λw=>let d:=dw w-m;d*d) x t l h y B n e D z p (λj=>rv (dw x[e+j]!)-rv m)
  (λj A=>tL _ m (r j A) F G)
 have u:0<=∑j∈i0 n,(rv (dw x[e+j]!)-rv m)^2:=ϡ λj _=>sq_nonneg _
 exact vP n _ _ _ _ _ ((ZA g:ℝ)*2^298) _ q p u (sA n p _ λj A=>(tL _ m (r j A) F G).2.2) В
  (nJ n p) K N H M L O I J (fX g o)
lemma iq (m D:р) (o:Д) (u:fv m) (p:fv D) (x:1<= rv D) (h:o=if eF D.toBits<=eF m.toBits then eF m.toBits-eF D.toBits+1 else 0):
  |rv m|<=2^oа*rv D/\oа<=2048:=by
 subst h;have r:=bj m u;have y:=bu m.toBits;have w:=bu D.toBits
 have l:¬eF D.toBits=0:=by
  intro v;have k:=bj D p;rw[v] at k;simp only[Ϯ,kS,zero_sub] at k;have:(2:ℝ)^(-1022:ℤ)<=1:=zpow_le_one_of_nonpos₀ Г Г;have:=le_abs_self (rv D);linarith
 have q:=c8 D p l;rw[abs_of_pos (by linarith)] at q
 split
 .rename_i d
  have g:=κ.1 d
  have e:(eF m.toBits-eF D.toBits+1)а=(eF m.toBits)а-(eF D.toBits)а+1:=by
   rw[μ,χ _ _ d];simp only[iZ];omega
  rw[e];refine ⟨?_,by omega⟩
  calc
   |rv m|<=(2:ℝ)^(((eF m.toBits)а:п)-1022):=r.le
   _=2^((eF m.toBits)а-(eF D.toBits)а+1)*2^(((eF D.toBits)а:п)-1023):=by
    rw[<-zpow_natCast,<-zpow_add₀ Г];congr 1;rw[Nat.cast_add,k9 g];push_cast;ring
   _<=_:=by gcongr
 .rename_i j
  have t:(eF m.toBits)а<(eF D.toBits)а:=by
   have:=UInt64.le_iff_toNat_le.not.1 j;omega
  refine ⟨?_,by simp⟩
  simp only[Ϯ,pow_zero,one_mul]
  calc
   |rv m|<=(2:ℝ)^(((eF m.toBits)а:п)-1022):=r.le
   _<=(2:ℝ)^(((eF D.toBits)а:п)-1023):=zpow_le_zpow_right₀ Г А
   _<=_:=q
lemma gi (e:Д) (k:Д->ф) (r q:р) (x:ZB) (j g:с) (h:gб<= xв) (l:gб+1<2^32) (o:jб<= gб):
  ∃m V p,m=PP dw x g h l j o 0 0*q/\V=PP (λw=>let d:=dw w-m;d*d) x g h l j o 0 0*q+r/\
  p=(if eF (dw (dn V e k).2).toBits<=eF m.toBits then eF m.toBits-eF (dw (dn V e k).2).toBits+1 else 0)/\
  rL e k r q x j g h l o=⟨(dn V e k).1&&e+p<=25&&rq (dw (dn V e k).2) (1/dw (dn V e k).2),m,1/dw (dn V e k).2,(dn V e k).2,e+p+1047⟩:=
 ⟨_,_,_,rfl,rfl,rfl,rfl⟩
lemma f9 (n:о) (x:ZB) (r:о) (d:∀j<n,i7 x[r*n+j]!≠255):
  ((jo n x r:ℚ):ℝ)*2^149=(∑j∈i0 n,rv (dw x[r*n+j]!))/n:=by
 unfold jo;rw[Yk];push_cast;simp only[zero_add,div_mul_eq_mul_div,϶];congr 1;refine Ο rfl λj e=>?_
 rw[(dZ _ (d j (Ν.1 e))).2]
lemma iX (n:о) (x:ZB) (r:о) (d:i9) (e:∀j<n,i7 x[r*n+j]!≠255):
  ((jg n x r (jo n x r)+ZA d:ℚ):ℝ)*2^298=
   (∑j∈i0 n,(rv (dw x[r*n+j]!)-(∑k∈i0 n,rv (dw x[r*n+k]!))/n)^2)/n+(ZA d:ℝ)*2^298:=by
 rw[<-f9 n x r e];unfold jg;rw[Yk];push_cast;simp only[zero_add,add_mul,div_mul_eq_mul_div,϶];congr 2;refine Ο rfl λj g=>?_
 rw[(dZ _ (e j (Ν.1 g))).2,show (2:ℝ)^298=(2^149)^2 by rw[<-pow_mul]]
 ring
lemma cE (E d e A:Д) (k:ℕ) (q:Eа<=2047) (o:dа<=254) (p:eа<=254) (m:Aа=k+1047) (r:k<=25):
  (NE.mx (NE.mx (E+5) (d+1051)) (e+A)-E)а=lm (lm (Eа+5) (dа+1051)) (eа+k+1047)-Eа:=by
 have h:(E+5)а=Eа+5:=by rw[μ];simp only[iZ];omega
 have j:(d+1051)а=dа+1051:=by rw[μ];simp only[iZ];omega
 have l:(e+A)а=eа+k+1047:=by rw[μ,m];omega
 have g:(NE.mx (NE.mx (E+5) (d+1051)) (e+A))а=lm (lm (Eа+5) (dа+1051)) (eа+k+1047):=by
  rw[NE.aA,NE.aA,h,j,l]
 rw[χ _ _ (κ.2 (by rw[g];omega)),g]
lemma ii (n:о) (x:ZB) (r j:о) (hj:j<n) (g l d:i9) (m w:р) (ok:ф) (k:ℕ) (A:Д)
  (L:i7 x[r*n+j]!≠255) (D:i7 g≠255) (q:i7 l≠255) (J:i7 d≠255) (K:ZX d=false) (t:1<= rv (dw d))
  (H:fv w) (u:|rv w-1/rv (dw d)|<=3/2^53/rv (dw d)) (fm:fv m) (hm:|rv m|<=2^279)
  (hA:Aа=k+1047) (hk:k<=25) (hδ:|rv m-((jo n x r:ℚ):ℝ)*2^149|<=2^k/2^53*rv (dw d)):
  eL n x r ⟨ok,m,w,d,A⟩ x[r*n+j]! g l (dw g) (dw l)=ZT ((ZA x[r*n+j]!-jo n x r)*ZA g/ZA d+ZA l):=by
 unfold eL;dsimp only;split
 .rename_i U
  simp only[υ,decide_eq_true_eq] at U;obtain ⟨⟨F,z⟩,B⟩:=U;generalize x[r*n+j]! =y at L F z B⊢;obtain ⟨x1,x2⟩:=dZ y L;obtain ⟨Y,Z⟩:=dZ l q;obtain ⟨-,G⟩:=dZ d J;obtain ⟨p1,p2,p3⟩:=gp g w D H (eE _ _ t u);have N:=gl _ _ (dB y L) hm;obtain ⟨s1,X⟩:=NF.fS _ _ x1 fm (NE.z9 _ 280 N Г);have V:=pB _ _ 281 280 (rB _ _ 280 N X) p3;obtain ⟨q1,W⟩:=NF.fM _ _ s1 p1 (NE.z9 _ _ V Г);obtain ⟨O,v2⟩:=NF.fA _ _ q1 Y (NE.z9 _ _ (aB _ _ 562 (rB _ _ 561 V W) ((dB l q).trans (pow_le_pow_right₀ Г Г))) Г);have e:=ia _ _ _ _ _ _ _ _ _ t u X p2 W v2 N (dB g D);have I:=bU _ O;generalize (dw y-m)*(dw g*w)+dw l=v at O e I F z B⊢;obtain ⟨Q,S⟩:=te _ F;have p:=cE (eF v.toBits) (iW l) (iW g) A k (bu _) (by rw[ZS];exact aL l) (by rw[ZS];exact aL g) hA hk
  rw[ZS,ZS] at p;generalize NE.mx (NE.mx (eF v.toBits+5) (iW l+1051)) (iW g+A)-eF v.toBits=c at z B p;obtain ⟨t1,t2⟩:=tc _ c z B;apply eC _ O cа _ Q S t1 t2;generalize (eF v.toBits)а=E at Q S I p⊢;obtain ⟨M,hM⟩:∃M,M=lm (lm (E+5) (je l+1051)) (je g+k+1047):=⟨_,rfl⟩;rw[<-hM] at p;rw[<-zpow_natCast (2:ℝ) cа,<-zpow_add₀ Г,show (cа:ℤ)+((E:ℤ)-1075)=(M:ℤ)-1075 by omega]
  have o:0<rv (dw d):=by linarith
  refine gZ _ _ _ _ _ _ _ _ k e ?_ I (by rw[show (32:ℝ)=2^(5:ℤ) by norm_num,<-zpow_add₀ Г];exact zpow_le_zpow_right₀ Г А)
   (wb l q 5 _ А) (wb g D (k+1) _ А) (by rw[show (1:ℝ)/2^699*64=2^6/2^699 by ring];exact zE _ _ _ А)
  have:(ZA d:ℝ)≠0:=λh=>by rw[h] at G;rw[G] at o;simp at o
  rw[show (rv (dw y)-rv m)*rv (dw g)/rv (dw d)+rv (dw l)-(((ZA y-jo n x r)*ZA g/ZA d+ZA l:ℚ):ℝ)*2^149=
   (((jo n x r:ℚ):ℝ)*2^149-rv m)*(rv (dw g)/rv (dw d)) by rw[x2,(dZ g D).2,G,Z];push_cast;field_simp;ring,
   abs_mul,abs_sub_comm,abs_div,abs_of_pos o]
  calc
   _<=2^k/2^53*rv (dw d)*(|rv (dw g)|/rv (dw d)):=mul_le_mul_of_nonneg_right hδ (div_nonneg (abs_nonneg _) o.le)
   _=_:=by have:=o.ne';field_simp
 .exact iP n x r j g l d hj K
lemma vp (n:ℕ) (d:n<=2^20) (V A E:ℝ) (g:0<= A) (j:2^149<= E) (h:|V-(A+E)|<=((n:ℝ)+30)/2^54*(A+E)):0<V:=by
 have e:(n:ℝ)<=2^20:=by exact_mod_cast d
 have:((n:ℝ)+30)/2^54*(A+E)<=1/2*(A+E):=mul_le_mul_of_nonneg_right (by rw[div_le_div_iff₀ В В];nlinarith) (by linarith[show (0:ℝ)<2^149 by positivity])
 linarith[(abs_le.1 h).1,show (0:ℝ)<2^149 by positivity]
lemma ip (n:ℕ) (o:1<= n) (A:n<=2^20) (X:ℕ->ℝ) (m d V q D P Q:ℝ) (L:2^149<= d)
  (r:|V-((∑j∈i0 n,(X j-m)^2)/n+d)|<=((n:ℝ)+30)/2^54*((∑j∈i0 n,(X j-m)^2)/n+d)) (w:0<V)
  (B:|q-ja V|<= ja V/2^53+1/2^1075) (u:|D-q|<= q/2^24) (t:1<= D) (I:|m|<= Q*D)
  (z:|m-(∑j∈i0 n,X j)/n|<=((n:ℝ)+13)/2^54*((∑j∈i0 n,|X j|)/n)+1/2^1040)
  (E:(n:ℝ)+17<= P) (v:P<=2^21) (G:1<= Q) (F:P*Q<=2^25):|m-(∑j∈i0 n,X j)/n|<= P*Q/2^53*D/\
  |V-((∑j∈i0 n,(X j-(∑k∈i0 n,X k)/n)^2)/n+d)|<=((n:ℝ)+31)/2^54*((∑j∈i0 n,(X j-(∑k∈i0 n,X k)/n)^2)/n+d):=by
 have l:(0:ℝ)<n:=by exact_mod_cast o
 have K:0<=(∑j∈i0 n,(X j-m)^2)/n:=В;generalize H:(∑j∈i0 n,(X j-m)^2)/n+d=p at r;have g:=gY n A V p q D (by linarith) r w B u
 have x:=cS n o (λj=>X j-m);beta_reduce at x
 have:∑j∈i0 n,|X j|<=∑j∈i0 n,|X j-m|+n*|m|:=by
  simpa[ϫ] using Ϗ λj (_:j∈i0 n)=>abs_add_le (X j-m) m
 have J:=mul_le_mul_of_nonneg_left (Real.sqrt_le_sqrt (by linarith:(∑j∈Finset.range n,(X j-m)^2)/n<=p)) l.le
 have h:=mE n P Q D (ja p) m _ _ E v G t z (by rw[div_le_iff₀ l];nlinarith) (by nlinarith[(abs_le.1 g).1,ϊ p]) I
 refine ⟨h,?_⟩;have e:=iQ n o X m;generalize ∑k∈i0 n,X k=y at e h z⊢;rw[show p=((∑j∈i0 n,(X j-y/n)^2)/n+d)+(m-y/n)^2 by rw[<-H,e,add_div,mul_div_cancel_left₀ _ l.ne'];ring] at r g
 exact bI n A V _ _ D В r (h.trans (by nlinarith)) g
lemma aT {d n:о} {e:i9} {x g:ZB} (h:aQ d n e x g=true):
  ji d n e x g=true/\d*n+1<2^32/\n<=2^20/\rq nж.toInt64.toFloat (1/nж.toInt64.toFloat)=true:=by
 simp only[aQ,υ,decide_eq_true_eq] at h;obtain ⟨⟨⟨j,m⟩,o⟩,l⟩:=h;exact ⟨by rw[<-e9];exact j,by omega,by omega,l⟩
lemma bE {d n:о} {e:i9} {x g:ZB} (h:ji d n e x g=true):xв=d*n/\gв=n:=by
 simp only[ji,υ,ZH,gt_iff_lt] at h;exact ⟨by simpa using h.1.1.1.1.1.2,by simpa using h.1.1.1.1.2⟩
lemma gy (o n:о) (h:i9) (x g b:ZB) (K:ji o n h x g=true) (L:∀j<n,i7 b[j]!≠255)
  (hn:o*n+1<2^32) (l:n<=2^20) (D:rq nж.toInt64.toFloat (1/nж.toInt64.toFloat)=true)
  (r:о) (hr:r<o) (h1 h2 h3) (v:(rL (lw n) (xD n x r h true) (dw h*NR.pt 149) (1/nж.toInt64.toFloat) x (r*n)г ((r*n)г+nг) h1 h2 h3).ok=true)
  (j:о) (z:j<n):
  eL n x r (rL (lw n) (xD n x r h true) (dw h*NR.pt 149) (1/nж.toInt64.toFloat) x (r*n)г ((r*n)г+nг) h1 h2 h3) x[r*n+j]! g[j]! b[j]! (dw g[j]!) (dw b[j]!)=
   ZT ((ZA x[r*n+j]!-jo n x r)*ZA g[j]!/ZA (jk (jg n x r (jo n x r)+ZA h))+ZA b[j]!):=by
 obtain ⟨q,t,Q,N⟩:=fI K;obtain ⟨u1,Z,u3,u4⟩:=uB hr hn;have m:∀j<n,i7 x[r*n+j]!≠255:=λj z=>Q _ А
 obtain ⟨l1,l2,-⟩:=d7 n l;obtain ⟨G,V,F,O,hV,A,e⟩:=gi (lw n) _ (dw h*NR.pt 149) (1/nж.toInt64.toFloat) x _ _ h1 h2 h3;rw[e] at v⊢;simp only[υ,decide_eq_true_eq] at v;obtain ⟨⟨S,o2⟩,o3⟩:=v;obtain ⟨E,W,J⟩:=it n x _ _ h1 h2 h3 (r*n) Z u4 q l m D;rw[<-O] at E W J;obtain ⟨H,U⟩:=iJ n h x _ _ h1 h2 h3 (r*n) Z u4 q l m (eJ K) t D G E J;rw[<-hV] at H U;have X:=(dY h (eJ K) t).2.2;have u:=vp n l (rv V) _ _ В X U;obtain ⟨-,-,-,q2,-,-,Y,-,-,w3⟩:=dW V (lw n) _ H u S;obtain ⟨I,w2,p,w,B⟩:=fT V (lw n) _ H u S o3;generalize M:(dn V (lw n) (xD n x r h true)).2=d at Y I w2 p w B A⊢;rw[<-Y] at w3;obtain ⟨hm,as⟩:=iq G (dw d) F E (av d I).1 p A;have y:(lw n)а+Fа<=25:=by have:=κ.1 o2;rw[μ] at this;simp only[iZ] at this;omega
 obtain ⟨k1,k2⟩:=ip n q l (λj=>rv (dw x[r*n+j]!)) (rv G) _ (rv V) _ (rv (dw d)) (2^(lw n)а) (2^Fа) X U u q2
  (by rw[(av d I).2];exact w3) p hm W l2 (pow_le_pow_right₀ Г l1) (one_le_pow₀ Г) (by rw[<-pow_add];exact pow_le_pow_right₀ Г y)
 have c:=fU V n _ (jg n x r (jo n x r)+ZA h) (((n:ℝ)+31)/2^54) l H (by have:=eD n x r (jo n x r);linarith) (by rw[iX n x r h m];exact k2) (by gcongr;norm_num) (gq n x r h true q t) S
 rw[M] at c;rw[<-c]
 exact ii n x r j z g[j]! b[j]! d G (1/dw d) _ ((lw n)а+Fа) _ (m j z) (N j z) (L j z) I w2 p w B E J
  (by rw[μ,μ];simp only[iZ];omega) y (by rw[f9 n x r m,pow_add];exact k1)
lemma fC (d n:о) (p:Д) (o v:р) (h:i9) (x g b:ZB) (u t:FloatArray) (z:d*n+1<2^32)
  (J:d*n<= xв) (G:n<= gв) (B:n<= bв) (m:n<= uд) (l:n<= tд) (F:о->i9)
  (H:∀j<n,u.data[j]! =dw g[j]!) (D:∀j<n,t.data[j]! =dw b[j]!)
  (Q:∀r<d,∀j<n,F (r*n+j)=ZT ((ZA x[r*n+j]!-jo n x r)*ZA g[j]!/ZA (jk (jg n x r (jo n x r)+ZA h))+ZA b[j]!))
  (E:∀r,r<d->∀M A y,(rL p (xD n x r h true) o v x (r*n)г ((r*n)г+nг) M A y).ok=true->∀j<n,
   eL n x r (rL p (xD n x r h true) o v x (r*n)г ((r*n)г+nг) M A y) x[r*n+j]! g[j]! b[j]! (dw g[j]!) (dw b[j]!)=F (r*n+j))
  (r:о) (e:ZB) (S:d*n<= eв):
  (RL d n p o v h x g b u t z J G B m l r e S)в=eв/\
   ∀q,(RL d n p o v h x g b u t z J G B m l r e S)[q]! =if r*n<= q/\q<d*n then F q else e[q]!:=by
 rw[RL];split
 .rename_i I
  obtain ⟨M,A,y,Y⟩:=uB I z;dsimp only
  have K:∀w L,(wв=eв/\∀q,w[q]! =if (r*n)гб+(0:с)б<= q/\q<(r*n)гб+nгб then F q else e[q]!)->
    (RL d n p o v h x g b u t z J G B m l (r+1) w L)в=eв/\
    ∀q,(RL d n p o v h x g b u t z J G B m l (r+1) w L)[q]! =if r*n<= q/\q<d*n then F q else e[q]!:=by
   intro w L ⟨W,X⟩;obtain ⟨U,V⟩:=fC d n p o v h x g b u t z J G B m l F H D Q E (r+1) w L;refine ⟨U.trans W,λq=>?_⟩;rw[V,X,A,y,Ϥ,ϋ];split_ifs<;>first|rfl|omega
  split
  .rename_i O;exact K _ _ (ct _ _ _ F _ (λj N=>by
    have:=γ.mp N;rw[y] at this;rw[ak,ak,ak,ao,ao,Dense.ab (a:=(r*n)г) (b:=j) (by have:=ς;omega),show (r*n)гб+jб=r*n+jб by omega,H _ this,D _ this]
    exact E r I _ _ _ O _ this) _ _ _)
  .exact K _ _ (ct _ _ _ F _ (λj N=>by have:=γ.mp N;rw[y] at this;rw[A,Q r I _ this]) _ _ _)
 .rename_i I
  refine ⟨rfl,λq=>?_⟩;rw[if_neg]
  intro ⟨Z,k2⟩;have:=Ζ n (Nat.le_of_not_lt I);omega
termination_by d-r
lemma fH (d n:о) (o:Д) (m t:р) (h:i9) (x g:ZB) (p:FloatArray) (v:d*n+1<2^32)
  (E:d*n<= xв) (A:n<= gв) (l:n<= pд) (F:о->i9)
  (B:∀j<n,p.data[j]! =dw g[j]!)
  (L:∀r<d,∀j<n,F (r*n+j)=ZT (ZA x[r*n+j]!*ZA g[j]!/ZA (jk (jg n x r 0+ZA h))))
  (z:∀r,r<d->∀H y u,(rR o (xD n x r h false) m t x (r*n)г ((r*n)г+nг) H y u).ok=true->∀j<n,
   eR (rR o (xD n x r h false) m t x (r*n)г ((r*n)г+nг) H y u) x[r*n+j]! g[j]! (dw g[j]!)=F (r*n+j))
  (r:о) (e:ZB) (M:d*n<= eв):
  (RR d n o m t h x g p v E A l r e M)в=eв/\
   ∀q,(RR d n o m t h x g p v E A l r e M)[q]! =if r*n<= q/\q<d*n then F q else e[q]!:=by
 rw[RR];split
 .rename_i D
  obtain ⟨H,y,u,U⟩:=uB D v;dsimp only
  have K:∀w G,(wв=eв/\∀q,w[q]! =if (r*n)гб+(0:с)б<= q/\q<(r*n)гб+nгб then F q else e[q]!)->
    (RR d n o m t h x g p v E A l (r+1) w G)в=eв/\
    ∀q,(RR d n o m t h x g p v E A l (r+1) w G)[q]! =if r*n<= q/\q<d*n then F q else e[q]!:=by
   intro w G ⟨Q,S⟩;obtain ⟨N,O⟩:=fH d n o m t h x g p v E A l F B L z (r+1) w G;refine ⟨N.trans Q,λq=>?_⟩;rw[O,S,y,u,Ϥ,ϋ];split_ifs<;>first|rfl|omega
  split
  .rename_i J;exact K _ _ (ct _ _ _ F _ (λj I=>by
    have:=γ.mp I;rw[u] at this;rw[ak,ak,ao,Dense.ab (a:=(r*n)г) (b:=j) (by have:=ς;omega),show (r*n)гб+jб=r*n+jб by omega,B _ this]
    exact z r D _ _ _ J _ this) _ _ _)
  .exact K _ _ (ct _ _ _ F _ (λj I=>by have:=γ.mp I;rw[u] at this;rw[y,L r D _ this]) _ _ _)
 .rename_i D
  refine ⟨rfl,λq=>?_⟩;rw[if_neg]
  intro ⟨V,W⟩;have:=Ζ n (Nat.le_of_not_lt D);omega
termination_by d-r
def aO (rows n:о) (eps:т) (x g:ZB):ZB:=
 if h:aQ rows n eps x g then
  have d:=aT h;have z:=bE d.1
  let gd:=aq g
  RR rows n (lw n) (dw eps*NR.pt 149) (1/nж.toInt64.toFloat) eps x g gd d.2.1 А А (by rw[(af g).1];omega) 0 (Array.replicate (rows*n) 0) (by simp)
 else rmsnormSpec rows n eps x g
lemma aX (a b:ZB) (e:aв=bв) (g:∀q<bв,a[q]! =b[q]!):a=b:=by
 apply Array.ext e;intro q h d;have:=g q d;rwa[getElem!_pos a q h,getElem!_pos b q d] at this
lemma f7 {d n r j:ℕ} (g:r<d) (e:j<n):r*n+j<d*n/\(r*n+j)/n=r/\(r*n+j)%n=j:=by
 refine ⟨by have:=Ζ n (Nat.succ_le_of_lt g);rw[ϋ] at this;omega,?_,?_⟩
 .rw[ϔ,kc _ _ А,ϣ e,zero_add]
 .rw[ϔ,ϵ,ξ e]
lemma bY:aO=rmsnormSpec:=by
 funext e n l x g;by_cases h:aQ e n l x g=true
 .rw[aO,dif_pos h]
  have d:=aT h;have z:=bE d.1;unfold rmsnormSpec rmsnormWith;rw[if_pos d.1];have v:(kj e n l x g)в=e*n:=by simp[kj]
  have u:∀r<e,∀j<n,(λq=>(kj e n l x g)[q]!) (r*n+j)=ZT (ZA x[r*n+j]!*ZA g[j]!/ZA (jk (jg n x r 0+ZA l))):=
   (λr o j m=>by
    obtain ⟨p,w,y⟩:=f7 o m;unfold kj;dsimp only;rw[Qg _ p];simp only[w,y];rw[Qg _ o])
  obtain ⟨E,F⟩:=fH e n (lw n) (dw l*NR.pt 149) (1/nж.toInt64.toFloat) l x g (aq g) d.2.1 А А
   (by rw[(af g).1];omega) (λq=>(kj e n l x g)[q]!) (λj m=>(af g).2 j А) u
   (λr o A B D t j m=>(gC e n l x g d.1 d.2.1 d.2.2.1 d.2.2.2 r o A B D t j m).trans (u r o j m).symm) 0 (jM (e*n) 0) (by simp)
  apply aX _ _ (by rw[E,v,ϴ]) (λq p=>?_)
  rw[F q,if_pos ⟨by omega,by rw[v] at p;exact p⟩]
 .rw[aO,dif_neg h]
lemma di {d n:о} {e:i9} {x g b:ZB} (h:(aQ d n e x g&&bв==n&&b.all jS)=true):
  aQ d n e x g=true/\bв=n/\∀j<n,i7 b[j]!≠255:=by
 simp only[υ,beq_iff_eq] at h;obtain ⟨⟨l,m⟩,o⟩:=h;exact ⟨l,m,λj p=>NR.d0 b (by simpa[j8] using o) j А⟩
def ar (rows n:о) (eps:т) (x g b:ZB):ZB:=
 if h:aQ rows n eps x g&&bв==n&&b.all finiteWord then
  have l:=di h;have d:=aT l.1;have z:=bE d.1
  RL rows n (lw n) (dw eps*NR.pt 149) (1/nж.toInt64.toFloat) eps x g b (aq g) (aq b) d.2.1 А А А
   (by rw[(af g).1];omega) (by rw[(af b).1];omega) 0 (Array.replicate (rows*n) 0) (by simp)
 else layernormSpec rows n eps x g b
lemma bF:ar=lJ:=by
 funext e n m x g b;by_cases h:(aQ e n m x g&&bв==n&&b.all jS)=true
 .rw[ar,dif_pos h]
  have l:=di h;have d:=aT l.1;have z:=bE d.1;unfold lJ layernormWith;rw[if_pos (by have v:=h;simp only[υ,beq_iff_eq] at v⊢;exact ⟨⟨d.1,l.2.1⟩,v.2⟩)]
  have y:(jN e n m x g b)в=e*n:=by simp[jN]
  have w:∀r<e,∀j<n,(λq=>(jN e n m x g b)[q]!) (r*n+j)=ZT ((ZA x[r*n+j]!-jo n x r)*ZA g[j]!/ZA (jk (jg n x r (jo n x r)+ZA m))+ZA b[j]!):=
   (λr p j o=>by
    obtain ⟨t,A,B⟩:=f7 p o;unfold jN;dsimp only;rw[Qg _ t];simp only[A,B];simp only[Qg _ p,Fin.val_mk])
  obtain ⟨G,H⟩:=fC e n (lw n) (dw m*NR.pt 149) (1/nж.toInt64.toFloat) m x g b (aq g) (aq b) d.2.1 А А
   А (by rw[(af g).1];omega) (by rw[(af b).1];omega) (λq=>(jN e n m x g b)[q]!)
   (λj o=>(af g).2 j А) (λj o=>(af b).2 j А) w
   (λr p D E F u j o=>(gy e n m x g b d.1 l.2.2 d.2.1 d.2.2.1 d.2.2.2 r p D E F u j o).trans (w r p j o).symm) 0 (jM (e*n) 0) (by simp)
  apply aX _ _ (by rw[G,y,ϴ]) (λq t=>?_)
  rw[H q,if_pos ⟨by omega,by rw[y] at t;exact t⟩]
 .rw[ar,dif_neg h]
@[inline] def bC (T:@& FloatArray) (hT:Tд=512) (w:т):р:=
 (Core.mW w).toInt64.toFloat*T.uget (w>>>23).toUSize (by rw[hT,Dense.aH,cW];have:=w.toNat_lt;omega)
lemma bg (w:i9):bC P ag w=dw w:=rfl
lemma aW (Y:FloatArray) (i:с) (v:р) (h:iб<Yд):(Y.uset i v h)д=Yд:=by
 rcases Y with ⟨d⟩;simp only[l6,jD,l9];apply lA
lemma bA (Y:FloatArray) (i:с) (v:р) (h:iб<Yд) (q:о):
  (Y.uset i v h).data[q]! =if iб=q then v else Y.data[q]!:=by
 have g:iб<Y.dataв:=h;have e:(Y.uset i v h).data=Y.data.set iб v g:=rfl;rw[e];by_cases d:q<Y.dataв
 .rw[getElem!_pos _ q (by rw[lA];exact d),getElem!_pos _ q d,lV]
 .rw[getElem!_neg _ q (by rw[lA];exact d),getElem!_neg _ q d];split<;>first|rfl|omega
@[specialize] def bc (f:р->р) (X:@& FloatArray) (e:с) (he:eб<= Xд) (hb:eб+1<2^32) (i:с) (hi:iб<= eб) (s t:р):р:=
 have:=ς
 if h:i+1<e then
  have h1:=u1 (i:=i) А
  have:=γ.mp h
  have h2:(i+1+1)б=iб+2:=by rw[u1 А,h1]
  bc f X e he hb (i+1+1) А (s+f (X.uget i А)) (t+f (X.uget (i+1) А))
 else if h:i<e then s+f (X.uget i (by have:=γ.mp h;omega))+t else s+t
termination_by eб-iб
decreasing_by omega
lemma eo (f:р->р) (X:FloatArray) (x:ZB) (d N:с) (A:Nб<= Xд) (z:Nб+1<2^32)
  (B:(d+N)б<= xв) (m:(d+N)б+1<2^32) (y:dб+Nб<2^32)
  (o:∀q<Nб,X.data[q]! =dw x[dб+q]!) (i:с) (G:iб<= Nб) (v:(d+i)б<=(d+N)б) (s t:р):
  bc f X N A z i G s t=PP (λw=>f (dw w)) x (d+N) B m (d+i) v s t:=by
 have H:=ς;have l:(d+N)б=dб+Nб:=Dense.ab А;have e:(d+i)б=dб+iб:=Dense.ab А;rw[bc,PP];dsimp only;by_cases h:i+1<N
 .have F:=γ.mp h;have r:=u1 (i:=i) А
  have p:d+i+1<d+N:=γ.mpr (by rw[u1 А,e,l];omega)
  rw[dif_pos h,dif_pos p];have g:(i+1+1)б=iб+2:=by rw[u1 А,r]
  have E:d+i+1+1=d+(i+1+1):=by
   apply USize.toNat_inj.mp;rw[u1 (by rw[u1 А,e];omega),u1 А,e,Dense.ab (by rw[g];omega),g];omega
  have D:d+i+1=d+(i+1):=by apply USize.toNat_inj.mp;rw[u1 А,e,Dense.ab А,r];omega
  rw[eo f X x d N A z B m y o (i+1+1) А (by rw[Dense.ab (by rw[g];omega),g,l];omega)]
  congr 1
  .exact E.symm
  .rw[ao,ak,o _ А,e]
  .rw[ao,ak,o _ А,D,Dense.ab (a:=d) (b:=i+1) А,r]
 .have p:¬d+i+1<d+N:=λu=>h (γ.mpr (by have:=γ.mp u;rw[u1 А,e,l] at this;rw[u1 А];omega))
  rw[dif_neg h,dif_neg p];by_cases g:i<N
  .have F:=γ.mp g
   have j:d+i<d+N:=γ.mpr (by rw[e,l];omega)
   rw[dif_pos g,dif_pos j,ao,ak,o _ А,e]
  .have j:¬d+i<d+N:=λu=>g (γ.mpr (by have:=γ.mp u;rw[e,l] at this;omega))
   rw[dif_neg g,dif_neg j]
termination_by Nб-iб
decreasing_by all_goals (have:=γ.mp h;rw[u1 А,u1 А];omega)
@[specialize] def aY (T:@& FloatArray) (hT:Tд=512) (x:ZB) (lo N:с) (hN:loб+Nб<2^32) (hx:loб+Nб<= xв) (hb:Nб+1<2^32)
  (i:с) (hi:iб<= Nб) (s t:р) (Y:FloatArray) (hY:Nб+1<= Yд):FloatArray:=
 have:=ς
 if h:i+1<N then
  have h1:=u1 (i:=i) А
  have:=γ.mp h
  have h2:(i+1+1)б=iб+2:=by rw[u1 А,h1]
  let u:=bC T hT (x.uget (lo+i) (by rw[Dense.ab А];omega))
  let v:=bC T hT (x.uget (lo+(i+1)) (by rw[Dense.ab А];omega))
  aY T hT x lo N hN hx hb (i+1+1) А (s+u) (t+v) ((Y.uset i u А).uset (i+1) v (by rw[aW];omega)) (by rw[aW,aW];exact hY)
 else if h:i<N then
  have:=γ.mp h
  let u:=bC T hT (x.uget (lo+i) (by rw[Dense.ab А];omega))
  (Y.uset i u А).uset N (s+u+t) (by rw[aW];omega)
 else Y.uset N (s+t) А
termination_by Nб-iб
decreasing_by omega
lemma bS (T:FloatArray) (e:Tд=512) (x:ZB) (g N:с) (hN hx hb) (i:с) (hi) (s t:р) (Y:FloatArray) (hY):
  (aY T e x g N hN hx hb i hi s t Y hY)д=Yд:=by
 have m:=ς;rw[aY];dsimp only;split
 .rename_i h
  have j:=γ.mp h;have d:=u1 (i:=i) А;have l:(i+1+1)б=iб+2:=by rw[u1 А,d]
  rw[bS,aW,aW]
 .split
  .rw[aW,aW]
  .rw[aW]
termination_by Nб-iб
decreasing_by omega
lemma ed (x:ZB) (d N:с) (hN hx hb) (i:с) (hi) (s t:р) (Y:FloatArray) (hY):
  ∀q<Nб,(aY P ag x d N hN hx hb i hi s t Y hY).data[q]! =if iб<= q then dw x[dб+q]! else Y.data[q]!:=by
 have v:=ς;rw[aY];dsimp only;split
 .rename_i h
  have o:=γ.mp h;have j:=u1 (i:=i) А;have u:(i+1+1)б=iб+2:=by rw[u1 А,j]
  have e:(d+i)б=dб+iб:=Dense.ab А;have p:(d+(i+1))б=dб+(iб+1):=by rw[Dense.ab А,j]
  intro q m;rw[ed,u,bA,bA,j,bg,bg,ak,ak,e,p]
  .split_ifs<;>first|rfl|omega|(subst_vars;rfl)
  .exact m
 .split
  .rename_i h g
   have o:=γ.mp g;have r:¬iб+1<Nб:=λl=>h (γ.mpr (by rw[u1 А];exact l))
   have e:(d+i)б=dб+iб:=Dense.ab А;intro q m;rw[bA,bA,bg,ak,e];split_ifs<;>first|rfl|omega|(subst_vars;rfl)
  .rename_i h g
   have o:¬iб<Nб:=λl=>g (γ.mpr l)
   intro q m;rw[bA,if_neg А,if_neg А]
termination_by Nб-iб
decreasing_by omega
lemma ec (x:ZB) (d N:с) (hN hx hb) (X:FloatArray) (p:Nб<= Xд) (i:с) (hi) (s t:р) (Y:FloatArray) (hY)
  (j:∀q,iб<= q->q<Nб->X.data[q]! =dw x[dб+q]!):
  (aY P ag x d N hN hx hb i hi s t Y hY).data[Nб]! =bc id X N p hb i hi s t:=by
 have z:=ς;rw[aY,bc];dsimp only;split
 .rename_i h
  have o:=γ.mp h;have g:=u1 (i:=i) А;have v:(i+1+1)б=iб+2:=by rw[u1 А,g]
  rw[ec x d N hN hx hb X p (i+1+1) _ _ _ _ _ (λq w y=>j q (by rw[v] at w;omega) y)]
  have e:(d+i)б=dб+iб:=Dense.ab А;have l:(d+(i+1))б=dб+(iб+1):=by rw[Dense.ab А,g]
  have r:bC P ag (x.uget (d+i) (by rw[e];omega))=id (X.uget i А):=by
   rw[bg,ak,e,ao,j _ le_rfl А];rfl
  have u:bC P ag (x.uget (d+(i+1)) (by rw[l];omega))=id (X.uget (i+1) (by rw[g];omega)):=by
   rw[bg,ak,l,ao,g,j _ А А];rfl
  rw[r,u]
 .rename_i h
  split
  .rename_i m
   have o:=γ.mp m;have e:(d+i)б=dб+iб:=Dense.ab А;rw[bA,if_pos rfl,bg,ak,e,ao,j _ le_rfl o];rfl
  .rename_i m
   rw[bA,if_pos rfl]
termination_by Nб-iб
decreasing_by omega
@[inline] def dl (T:@& FloatArray) (hT:Tд=512) (Ln:Д) (k:Д->ф) (Ep rn:р) (X:@& FloatArray) (N:с) (he:Nб<= Xд) (hb:Nб+1<2^32):RS:=
 let m:=bc id X N he hb 0 (by simp) 0 0*rn
 let V:=bc (λv=>let d:=v-m;d*d) X N he hb 0 (by simp) 0 0*rn+Ep
 let t:=dn V Ln k
 let D:=bC T hT t.2
 let ri:=1/D
 let a:=eF m.toBits
 let b:=eF D.toBits
 let mD:=if b<= a then a-b+1 else 0
 ⟨t.1&&Ln+mD<=25&&rq D ri,m,ri,t.2,Ln+mD+1047⟩
lemma gN (p:Д) (k:Д->ф) (o r:р) (X:FloatArray) (x:ZB) (d N:с) (j:Nб<= Xд) (g:Nб+1<2^32)
  (h:(d+N)б<= xв) (e:(d+N)б+1<2^32) (l:dб<=(d+N)б) (m:dб+Nб<2^32)
  (u:∀q<Nб,X.data[q]! =dw x[dб+q]!):
  dl P ag p k o r X N j g=rL p k o r x d (d+N) h e l:=by
 have v:∀(f:р->р) s t,bc f X N j g 0 (by simp) s t=PP (λw=>f (dw w)) x (d+N) h e d l s t:=by
  intro f s t;rw[eo f X x d N j g h e m u 0 (by simp) (by simp only[USize.add_zero];exact l) s t]
  simp only[USize.add_zero]
 simp only[dl,rL,v,bg,id]
@[specialize] def bR (m ri:р) (gd bd:@& FloatArray) (N:с) (hg:Nб<= gdд) (hb:Nб<= bdд) (i:с) (Y:FloatArray) (hY:Nб<= Yд):FloatArray:=
 if h:i<N then
  have:=γ.mp h
  bR m ri gd bd N hg hb (i+1) (Y.uset i ((Y.uget i А-m)*(gd.uget i А*ri)+bd.uget i А) А) (by rw[aW];exact hY)
 else Y
termination_by Nб-iб
decreasing_by have:=Τ N;rw[uS А];omega
lemma d9 (m g:р) (e d:FloatArray) (N:с) (hg hb) (i:с) (Y:FloatArray) (hY):(bR m g e d N hg hb i Y hY)д=Yд:=by
 unfold bR;split
 .rw[d9];exact aW ..
 .rfl
termination_by Nб-iб
decreasing_by have:=γ.mp ‹_›;have:=Τ N;rw[uS А];omega
lemma d8 (m j:р) (g e:FloatArray) (N:с) (hg hb) (i:с) (Y:FloatArray) (hY):
  ∀q,(bR m j g e N hg hb i Y hY).data[q]! =if iб<= q/\q<Nб then (Y.data[q]!-m)*(g.data[q]!*j)+e.data[q]! else Y.data[q]!:=by
 rw[bR];split
 .rename_i h
  have l:=γ.mp h;have d:(i+1)б=iб+1:=uS (by have:=Τ N;omega)
  intro q;rw[d8,d,bA,ao,ao,ao];split_ifs<;>first|rfl|omega|(subst_vars;rfl)
 .rename_i h
  have l:¬iб<Nб:=λo=>h (γ.mpr o)
  exact λq=>by rw[if_neg А]
termination_by Nб-iб
decreasing_by have:=Τ N;rw[uS А];omega
@[noinline] def dk (n:о) (x:ZB) (r:о) (e:RS) (xw gw bw:т) (B:Д):т:=
 let E:=eF B
 let c:=NE.mx (NE.mx (E+5) (Core.iW bw+1051)) (Core.iW gw+e.A)-E
 let t:=(1:Д)<<<c
 if E-1046<=252&&c<=25&&(B&&&0x7FFFFFF)-t-1<0x7FFFFFF-2*t then pq B
 else sI n x r xw gw bw e.d
lemma ic (n:о) (x:ZB) (r:о) (e:RS) (d l h:i9) (j g:р):
  eL n x r e d l h j g=dk n x r e d l h ((dw d-e.m)*(j*e.ri)+g).toBits:=rfl
lemma du (c:Д) (h:cа<64):((1:Д)<<<c)а=2^cа:=by
 rw[Ι,show (1:Д)а=1 from rfl,ξ h,ϭ];exact ξ (lz Г h)
lemma gL (n:о) (x:ZB) (r:о) (e:RS) (xw j h:i9) (B g d:Д)
  (U:(iW h)а+1051<= gа) (V:(iW j)а+e.Aа<= gа) (W:gа<=4000)
  (O:dа=lm 1046 (gа-25)) (Q:dа<=1298)
  (G:(eF B-d<=1298-d&&((B+(1:Д)<<<(NE.mx (eF B+5) g-eF B))&&&0x7FFFFFF)>(1:Д)<<<(NE.mx (eF B+5) g-eF B)+(1:Д)<<<(NE.mx (eF B+5) g-eF B))=true):
  dk n x r e xw j h B=pq B:=by
 simp only[υ,decide_eq_true_eq] at G;obtain ⟨F,t⟩:=G;unfold dk;dsimp only;rw[if_pos];have N:=bu B;generalize eF B=E at F t N⊢;have X:=aL h;have Y:=aL j;rw[<-ZS] at X Y;have q:=κ.1 F;simp only[Κ,iZ] at q;have S:dа<= Eа/\Eа<=1298:=by have:=d.toNat_lt;omega
 have D:(E+5)а=Eа+5:=by rw[μ];simp only[iZ];omega
 have H:(NE.mx (E+5) g)а=lm (Eа+5) gа:=by rw[NE.aA,D]
 have p:(NE.mx (E+5) g-E)а=lm (Eа+5) gа-Eа:=by
  rw[χ _ _ (κ.2 (by rw[H];omega)),H]
 have m:(NE.mx (E+5) g-E)а<=25:=by rw[p];omega
 generalize NE.mx (E+5) g-E=l at t p m;have J:=du l А;have v:(iW h+1051)а=(iW h)а+1051:=by rw[μ];simp only[iZ];omega
 have w:(iW j+e.A)а=(iW j)а+e.Aа:=by rw[μ];omega
 have o:(NE.mx (NE.mx (E+5) (iW h+1051)) (iW j+e.A))а=lm (lm (Eа+5) ((iW h)а+1051)) ((iW j)а+e.Aа):=by
  rw[NE.aA,NE.aA,D,v,w]
 have z:(NE.mx (NE.mx (E+5) (iW h+1051)) (iW j+e.A)-E)а=lm (lm (Eа+5) ((iW h)а+1051)) ((iW j)а+e.Aа)-Eа:=by
  rw[χ _ _ (κ.2 (by rw[o];omega)),o]
 generalize NE.mx (NE.mx (E+5) (iW h+1051)) (iW j+e.A)-E=c at z;have L:cа<= lа:=by rw[p,z];omega
 have K:=du c А;have I:2^cа<=2^lа:=Ε Г L;have y:2^lа<=2^25:=Ε Г m;have Z:1<=2^cа:=Nat.one_le_two_pow;generalize 2^cа=T at K I Z;generalize 2^lа=T' at J I y;have u:=ν.1 t
 rw[ψ,μ,μ,J,show (0x7FFFFFF:Д)а=2^27-1 from rfl,
  θ] at u
 have hB:=B.toNat_lt;simp only[υ,decide_eq_true_eq];refine ⟨⟨κ.2 ?_,κ.2 (by simp only[iZ];omega)⟩,ν.2 ?_⟩
 .simp only[Κ,iZ];omega
 .have M:(B&&&0x7FFFFFF)а=Bа%2^27:=by rw[ψ];exact θ _ 27
  simp only[Κ,Ϛ,M,K,iZ];omega
@[inline] def cV (T:@& FloatArray) (hT:Tд=512) (Ln:Д) (k:Д->ф) (Ep rn:р) (X:@& FloatArray) (N:с) (he:Nб+1<= Xд) (hb:Nб+1<2^32):RS:=
 let m:=X.uget N А*rn
 let V:=bc (λv=>let d:=v-m;d*d) X N А hb 0 (by simp) 0 0*rn+Ep
 let t:=dn V Ln k
 let D:=bC T hT t.2
 let ri:=1/D
 let a:=eF m.toBits
 let b:=eF D.toBits
 let mD:=if b<= a then a-b+1 else 0
 ⟨t.1&&Ln+mD<=25&&rq D ri,m,ri,t.2,Ln+mD+1047⟩
lemma fj (T:FloatArray) (h:Tд=512) (g:Д) (k:Д->ф) (e j:р) (X:FloatArray) (N:с) (l:Nб+1<= Xд) (d:Nб+1<2^32)
  (m:X.data[Nб]! =bc id X N А d 0 (by simp) 0 0):cV T h g k e j X N l d=dl T h g k e j X N А d:=by
 simp only[cV,dl,ao,m]
lemma iv (B t d:Д) (m:tа<= dа) (g:dа<=2^25) (h:((B+d)&&&0x7FFFFFF)>d+d):((B+t)&&&0x7FFFFFF)>t+t:=by
 have e:∀u:Д,uа<=2^25->((B+u)&&&0x7FFFFFF)а=(Bа%2^27+uа)%2^27:=λu l=>by
  rw[ψ,μ,show (0x7FFFFFF:Д)а=2^27-1 from rfl,θ];omega
 have f:∀u:Д,uа<=2^25->(u+u)а=2*uа:=λu l=>by rw[μ];omega
 have j:=ν.1 h;rw[e _ g,f _ g] at j;apply ν.2;rw[e _ А,f _ А];have:=Bа%2^27;omega
lemma iw (B e d S:Д) (r:Sа=2^(eа-dа)) (H:dа<= eа) (A:eа<= dа+25)
  (t:(eF B-d<=1298-d&&((B+NE.mx 32 (S>>>(eF B-d)))&&&0x7FFFFFF)>NE.mx 32 (S>>>(eF B-d))+NE.mx 32 (S>>>(eF B-d)))=true) (z:dа<=1298):
  (eF B-d<=1298-d&&((B+(1:Д)<<<(NE.mx (eF B+5) e-eF B))&&&0x7FFFFFF)>(1:Д)<<<(NE.mx (eF B+5) e-eF B)+(1:Д)<<<(NE.mx (eF B+5) e-eF B))=true:=by
 simp only[υ,decide_eq_true_eq] at t⊢;obtain ⟨h,q⟩:=t;refine ⟨h,?_⟩;have y:=bu B;generalize eF B=E at h q y⊢;have j:=κ.1 h;simp only[Κ,iZ] at j;have l:dа<= Eа/\Eа<=1298:=by have:=d.toNat_lt;omega
 have D:(E-d)а=Eа-dа:=χ _ _ (κ.2 l.1);have x:(E+5)а=Eа+5:=by rw[μ];simp only[iZ];omega
 have w:(NE.mx (E+5) e)а=lm (Eа+5) eа:=by rw[NE.aA,x]
 have p:(NE.mx (E+5) e-E)а=lm (Eа+5) eа-Eа:=by
  rw[χ _ _ (κ.2 (by rw[w];omega)),w]
 have G:=du (NE.mx (E+5) e-E) (by rw[p];omega)
 have o:(S>>>(E-d))а=2^(eа-dа)/2^(Eа-dа)\/2^(Eа-dа)>=2^26:=by
  by_cases m:Eа-dа<64
  .left;rw[Α,r,D,ξ m,ι]
  .right;exact le_trans (Ε Г (show 26<=64 by norm_num)) (Ε Г А)
 have g:Sа<=2^25:=by rw[r];exact Ε Г А
 have F:(S>>>(E-d))а<=2^25:=by
  rw[Α,ι];exact le_trans (Nat.div_le_self _ _) g
 have v:(NE.mx 32 (S>>>(E-d)))а=lm 32 (S>>>(E-d))а:=NE.aA _ _;apply iv _ _ _ ?_ (by rw[v];omega) q
 rw[G,v,p];by_cases I:eа<= Eа+5
 .rw[show lm (Eа+5) eа-Eа=5 by omega];norm_num
 .rw[show lm (Eа+5) eа-Eа=eа-Eа by omega]
  rcases o with u|u
  .rw[u,Nat.pow_div А Г,show eа-dа-(Eа-dа)=eа-Eа by omega];omega
  .exfalso;have:2^(Eа-dа)<=2^25:=Ε Г А;omega
@[specialize] def b9 (n:о) (x g b:ZB) (r:о) (e:RS) (S Elo Wd:Д) (Y:@& FloatArray) (lo N:с) (hN:loб+Nб<2^32)
  (hx:loб+Nб<= xв) (hg:Nб<= gв) (hb:Nб<= bв) (hY:Nб<= Yд) (i:с) (out:ZB) (ho:loб+Nб<= outв):ZB:=
 if h:i<N then
  have:=γ.mp h
  let B:=(Y.uget i А).toBits
  let d:=eF B-Elo
  let t:=NE.mx 32 (S>>>d)
  let w:=if d<=Wd&&((B+t)&&&0x7FFFFFF)>t+t then pq B
   else dk n x r e (x.uget (lo+i) (by rw[Dense.ab (by have:=ς;omega)];omega)) (g.uget i А) (b.uget i А) B
  b9 n x g b r e S Elo Wd Y lo N hN hx hg hb hY (i+1) (out.uset (lo+i) w (by rw[Dense.ab (by have:=ς;omega)];omega))
   (by rw[ϑ];exact ho)
 else out
termination_by Nб-iб
decreasing_by have:=Τ N;rw[uS А];omega
lemma ek (n:о) (x g b:ZB) (r:о) (e:RS) (S h j:Д) (Y:FloatArray) (l N:с) (hN hx hg hb hY) (i:с) (d:ZB) (ho):
  (b9 n x g b r e S h j Y l N hN hx hg hb hY i d ho)в=dв:=by
 unfold b9;split
 .rw[ek];exact ϑ ..
 .rfl
termination_by Nб-iб
decreasing_by have:=γ.mp ‹_›;have:=Τ N;rw[uS А];omega
lemma gU (d N:с) (f:(j:с)->j<N->i9) (l:dб+Nб<2^32) (i:с) (a e:ZB) (m:dб+Nб<= aв) (g:dб+Nб<= eв)
  (h:a=e):QE d N f l i a m=QE d N f l i e g:=by subst h;rfl
lemma iV (a:ZB) (k:с) (v d:i9) (p:kб<aв) (e:kб<aв) (h:v=d):a.uset k v p=a.uset k d e:=by subst h;rfl
lemma ej (n:о) (x g b:ZB) (r:о) (e:RS) (l S d:Д) (u t Y:FloatArray) (p N:с) (hN hx hg hb hY)
  (H:Nб<= uд) (G:Nб<= tд)
  (B:∀q<Nб,Y.data[q]! =(dw x[pб+q]!-e.m)*(u.data[q]!*e.ri)+t.data[q]!)
  (w:∀q<Nб,(iW b[q]!)а+1051<= lа) (y:∀q<Nб,(iW g[q]!)а+e.Aа<= lа) (z:lа<=4000)
  (v:dа=lm 1046 (lа-25)) (o:dа<=1298) (I:Sа=2^(lа-dа)) (F:1051<= lа) (i:с) (D:ZB) (ho):
  b9 n x g b r e S d (1298-d) Y p N hN hx hg hb hY i D ho=
   QE p N (λj K=>
    have:=γ.mp K
    eL n x r e (x.uget (p+j) (by rw[Dense.ab (by have:=ς;omega)];omega)) (g.uget j А)
     (b.uget j А) (u.uget j А) (t.uget j А)) hN i D ho:=by
 rw[b9,QE];split
 .rename_i h
  have E:=γ.mp h;have M:=ς;dsimp only;rw[ej n x g b r e l S d u t Y p N hN hx hg hb hY H G B w y z v o I F (i+1)];apply gU;apply iV
  have L:Y.uget i А=(dw (x.uget (p+i) (by rw[Dense.ab А];omega))-e.m)*(u.uget i А*e.ri)+t.uget i А:=by
   rw[ao,ao,ao,ak,B _ E,Dense.ab А]
  rw[L,ic];split
  .rename_i J
   exact (gL n x r e _ _ _ _ l d (by rw[ak];exact w _ E) (by rw[ak];exact y _ E) z v o (iw _ l d S I А А J o)).symm
  .rfl
 .rfl
termination_by Nб-iб
decreasing_by have:=ς;rw[uS А];omega
def bb (a:ZB) (i:о) (k:Д):Д:=
 if h:i<aв then bb a (i+1) (NE.mx k (Core.iW a[i])) else k
termination_by aв-i
lemma em (a:ZB) (i:о) (k:Д):kа<=(bb a i k)а:=by
 rw[bb];split
 .exact le_trans (by rw[NE.aA];omega) (em a (i+1) _)
 .exact le_rfl
termination_by aв-i
lemma cQ (a:ZB) (i:о) (k:Д) (m:kа<=254):(bb a i k)а<=254/\∀j,i<= j->j<aв->(iW a[j]!)а<=(bb a i k)а:=by
 rw[bb];split
 .rename_i h
  have q:(iW a[i])а<=254:=by rw[ZS];exact aL _
  have l:(NE.mx k (iW a[i]))а=lm kа (iW a[i])а:=NE.aA _ _;obtain ⟨o,p⟩:=cQ a (i+1) _ (by rw[l];omega)
  refine ⟨o,λj e d=>?_⟩
  by_cases g:j=i
  .subst g;rw[getElem!_pos a j d];exact le_trans (by rw[l];omega) (em a (j+1) _)
  .exact p j А d
 .exact ⟨m,λj e d=>by omega⟩
termination_by aв-i
lemma iH (j:Д) (k:Д->ф) (g p:р) (x:ZB) (o l:с) (h hb hl) (d:(rL j k g p x o l h hb hl).ok=true):
  (rL j k g p x o l h hb hl).Aа<=1072:=by
 obtain ⟨m,V,q,-,-,-,e⟩:=gi j k g p x o l h hb hl;rw[e] at d⊢;simp only[υ,decide_eq_true_eq] at d;have:=κ.1 d.1.2;rw[μ];simp only[iZ] at this⊢;omega
@[inline] def eu (e:RS) (KB KG:Д):ф:=e.ok&&NE.mx 1046 (NE.mx KB (KG+e.A)-25)<=1298
@[inline] def mZ (e:RS) (KB KG:Д) (gd bd:@& FloatArray) (N:с) (hg:Nб<= gdд) (hb:Nб<= bdд) (X:FloatArray) (hX:Nб<= Xд):FloatArray:=
 if eu e KB KG then bR e.m e.ri gd bd N hg hb 0 X hX else X
lemma ix (e:RS) (d g:Д) (j h:FloatArray) (N:с) (hg hb) (X:FloatArray) (hX):(mZ e d g j h N hg hb X hX)д=Xд:=by
 unfold mZ;split
 .exact d9 ..
 .rfl
@[inline] def cC (n:о) (x g b:ZB) (r:о) (eps:т) (e:RS) (KB KG:Д) (gd bd Z:FloatArray) (lo N:с) (hN:loб+Nб<2^32)
  (hx:loб+Nб<= xв) (hg:Nб<= gв) (hb:Nб<= bв) (hgd:Nб<= gdд) (hbd:Nб<= bdд) (hZ:Nб<= Zд)
  (out:ZB) (ho:loб+Nб<= outв):ZB:=
 if eu e KB KG then
  b9 n x g b r e ((1:Д)<<<(NE.mx KB (KG+e.A)-NE.mx 1046 (NE.mx KB (KG+e.A)-25))) (NE.mx 1046 (NE.mx KB (KG+e.A)-25)) (1298-NE.mx 1046 (NE.mx KB (KG+e.A)-25)) Z lo N hN hx hg hb hZ 0 out ho
 else if e.ok then QE lo N (λj hj=>
  have:=γ.mp hj
  eL n x r e (x.uget (lo+j) (by rw[Dense.ab (by have:=ς;omega)];omega)) (g.uget j А)
   (b.uget j А) (gd.uget j А) (bd.uget j А)) hN 0 out ho
 else fL n eps x g b r lo N hN out ho
lemma gK (n:о) (x g b:ZB) (r:о) (h:i9) (e:RS) (j l:Д) (o m Z:FloatArray) (p N:с) (hN hx hg hb hgd hbd hZ) (d:ZB) (ho):
  (cC n x g b r h e j l o m Z p N hN hx hg hb hgd hbd hZ d ho)в=dв:=by
 unfold cC;split
 .exact ek ..
 .split
  .exact bL ..
  .simp only[fL];exact bL ..
lemma gJ (n:о) (x g b:ZB) (r:о) (z:i9) (e E:RS) (Y:e=E) (V:e.ok=true->e.Aа<=1072) (l d:Д) (y N:с)
  (G:∀j<Nб,(iW b[j]!)а+1051<= lа) (L:1051<= lа) (M:lа<=1305)
  (H:∀j<Nб,(iW g[j]!)а<= dа) (O:dа<=254)
  (v u X:FloatArray) (hN hx hg hb hgd hbd) (I:Nб<= Xд) (hZ)
  (J:∀q<Nб,X.data[q]! =dw x[yб+q]!) (t:ZB) (ho):
  cC n x g b r z e l d v u (mZ e l d v u N hgd hbd X I) y N hN hx hg hb hgd hbd hZ t ho=
   if E.ok then QE y N (λj Z=>
    have:=γ.mp Z
    eL n x r E (x.uget (y+j) (by rw[Dense.ab (by have:=ς;omega)];omega)) (g.uget j А)
     (b.uget j А) (v.uget j А) (u.uget j А)) hN 0 t ho
   else fL n z x g b r y N hN t ho:=by
 subst Y;unfold cC;split
 .rename_i S
  have D:=S;simp only[eu,υ,decide_eq_true_eq] at D;obtain ⟨K,U⟩:=D;rw[if_pos K];have W:=V K;have F:(d+e.A)а=dа+e.Aа:=by rw[μ];omega
  have o:(NE.mx l (d+e.A))а=lm lа (dа+e.Aа):=by rw[NE.aA,F]
  have B:(NE.mx l (d+e.A)-25)а=lm lа (dа+e.Aа)-25:=by
   rw[χ _ _ (κ.2 (by rw[o];simp only[iZ];omega)),o];rfl
  have h:(NE.mx 1046 (NE.mx l (d+e.A)-25))а=lm 1046 (lm lа (dа+e.Aа)-25):=by rw[NE.aA,B];rfl
  have p:=κ.1 U;simp only[iZ] at p;have hz:mZ e l d v u N hgd hbd X I=bR e.m e.ri v u N hgd hbd 0 X I:=by unfold mZ;rw[if_pos S]
  apply ej n x g b r e _ _ _ v u _ y N hN hx hg hb hZ hgd hbd
  .intro q w;rw[hz,d8,if_pos ⟨by simp,w⟩,J q w]
  .intro q w;rw[o];have:=G q w;omega
  .intro q w;rw[o];have:=H q w;omega
  .rw[o];omega
  .rw[h,o]
  .rw[h] at p⊢;exact p
  .have Q:(NE.mx l (d+e.A)-NE.mx 1046 (NE.mx l (d+e.A)-25))а=lm lа (dа+e.Aа)-lm 1046 (lm lа (dа+e.Aа)-25):=by
    rw[χ _ _ (κ.2 (by rw[h,o];omega)),h,o]
   rw[du _ (by rw[Q];omega),Q,h,o]
  .rw[o];omega
 .rfl
def dj (T:@& FloatArray) (hT:Tд=512) (rows n:о) (Ln:Д) (Ep rn:р) (eps:т) (KB KG:Д) (x g b:ZB) (gd bd:FloatArray) (hn:rows*n+1<2^32)
  (hx:rows*n<= xв) (hg:n<= gв) (hb:n<= bв) (hgd:n<= gdд) (hbd:n<= bdд) (r:о) (Y:FloatArray) (hY:n+1<= Yд) (out:ZB) (ho:rows*n<= outв):ZB:=
 if hr:r<rows then
  have h:=uB hr hn;have h1:=h.1;have h2:=h.2.1;have h3:=h.2.2.1;have h4:=h.2.2.2
  let X:=aY T hT x (r*n)г nг (by rw[h2,h3];omega) (by rw[h2,h3];omega) (by rw[h3];omega) 0 (by simp) 0 0 Y (by rw[h3];omega)
  have hX:n+1<= Xд:=by rw[show Xд=Yд from bS ..];exact hY
  let e:=cV T hT Ln (xD n x r eps true) Ep rn X nг (by rw[h3];omega) (by rw[h3];omega)
  let Z:=mZ e KB KG gd bd nг (by rw[h3];omega) (by rw[h3];omega) X (by rw[h3];omega)
  have hZ:n+1<= Zд:=by rw[show Zд=Xд from ix ..];exact hX
  dj T hT rows n Ln Ep rn eps KB KG x g b gd bd hn hx hg hb hgd hbd (r+1) Z hZ
   (cC n x g b r eps e KB KG gd bd Z (r*n)г nг (by rw[h2,h3];omega) (by rw[h2,h3];omega) (by rw[h3];omega) (by rw[h3];omega)
    (by rw[h3];omega) (by rw[h3];omega) (by rw[h3];omega) out (by rw[h2,h3];omega))
   (by rw[gK];exact ho)
 else out
termination_by rows-r
lemma fG (d n:о) (l:Д) (j q:р) (e:i9) (x g b:ZB) (p o:FloatArray) (hn hx hg hb hgd hbd) (r:о) (a m:ZB) (ha ha') (h:a=m):
  RL d n l j q e x g b p o hn hx hg hb hgd hbd r a ha=RL d n l j q e x g b p o hn hx hg hb hgd hbd r m ha':=by subst h;rfl
lemma er (e n:о) (y:Д) (v F:р) (h:i9) (u w:Д) (x g b:ZB) (H G:FloatArray) (hn hx hg hb hgd hbd)
  (z:∀j<n,(iW b[j]!)а+1051<= uа) (o:1051<= uа) (p:uа<=1305)
  (A:∀j<n,(iW g[j]!)а<= wа) (t:wа<=254) (r:о) (Y:FloatArray) (hY) (E:ZB) (ho):
  dj P ag e n y v F h u w x g b H G hn hx hg hb hgd hbd r Y hY E ho=RL e n y v F h x g b H G hn hx hg hb hgd hbd r E ho:=by
 rw[dj,RL];split
 .rename_i N
  obtain ⟨O,l,d,I⟩:=uB N hn;have Q:=ς;have J:=ed x (r*n)г nг (by rw[l,d];omega) (by rw[l,d];omega) (by rw[d];omega) 0 (by simp) 0 0 Y (by rw[d];omega)
  have m:∀q<nгб,(aY P ag x (r*n)г nг (by rw[l,d];omega) (by rw[l,d];omega) (by rw[d];omega) 0 (by simp) 0 0 Y (by rw[d];omega)).data[q]! =dw x[(r*n)гб+q]!:=
   λq K=>by rw[J q K,if_pos (by simp)]
  have M:=ec x (r*n)г nг (by rw[l,d];omega) (by rw[l,d];omega) (by rw[d];omega)
   (aY P ag x (r*n)г nг (by rw[l,d];omega) (by rw[l,d];omega) (by rw[d];omega) 0 (by simp) 0 0 Y (by rw[d];omega)) (by rw[bS,d];omega)
   0 (by simp) 0 0 Y (by rw[d];omega) (λq _ K=>m q K)
  have L:=(fj P ag y (xD n x r h true) v F _ nг (by rw[bS,d];omega) (by rw[d];omega) M).trans
   (gN y (xD n x r h true) v F (aY P ag x (r*n)г nг (by rw[l,d];omega) (by rw[l,d];omega) (by rw[d];omega) 0 (by simp) 0 0 Y (by rw[d];omega)) x
   (r*n)г nг (by rw[bS,d];omega) (by rw[d];omega) (by rw[I];omega) (by rw[I];omega) (by rw[I,l];omega) (by rw[l,d];omega) m)
  exact (er e n y v F h u w x g b H G hn hx hg hb hgd hbd z o p A t (r+1) _ _ _ _).trans
   (fG _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (gJ n x g b r h _ _ L (λD=>by rw[L] at D⊢;exact iH _ _ _ _ _ _ _ _ _ _ D) u w _ _
    (λj B=>z j (by rw[d] at B;exact B)) o p (λj B=>A j (by rw[d] at B;exact B)) t H G _ _ _ _ _ _ _ _ _
    m _ _))
 .rfl
termination_by e-r
lemma fB (e N:с) (f g:(j:с)->j<N->i9) (l:eб+Nб<2^32) (i:с) (a:ZB) (m:eб+Nб<= aв)
  (h:∀j d,f j d=g j d):QE e N f l i a m=QE e N g l i a m:=by
 have:f=g:=funext λj=>funext λd=>h j d
 subst this;rfl
lemma d5 {α:Type} {c:Prop} [Decidable c] (a d b:α) (h:c->a=d):(if c then a else b)=(if c then d else b):=by
 split
 .exact h ‹_›
 .rfl
def c5 (rows n:о) (Ln:Д) (Ep rn:р) (eps:т) (x g b:ZB) (hn:rows*n+1<2^32)
  (hx:rows*n<= xв) (hg:n<= gв) (hb:n<= bв) (r:о) (out:ZB) (ho:rows*n<= outв):ZB:=
 if hr:r<rows then
  have h:=uB hr hn;have h1:=h.1;have h2:=h.2.1;have h3:=h.2.2.1;have h4:=h.2.2.2
  let e:=rL Ln (xD n x r eps true) Ep rn x (r*n)г ((r*n)г+nг) (by rw[h4];omega) (by rw[h4];omega) (by rw[h4,h2];omega)
  c5 rows n Ln Ep rn eps x g b hn hx hg hb (r+1)
   (if e.ok then QE (r*n)г nг (λj hj=>
    have:=γ.mp hj
    let gw:=g.uget j А;let bw:=b.uget j А
    eL n x r e (x.uget ((r*n)г+j) (by rw[Dense.ab (by have:=ς;omega)];omega)) gw bw (dw gw) (dw bw)) (by rw[h2,h3];omega) 0 out (by rw[h2,h3];omega)
    else fL n eps x g b r (r*n)г nг (by rw[h2,h3];omega) out (by rw[h2,h3];omega))
   (by split<;>simp only[bL,fL]<;>exact ho)
 else out
termination_by rows-r
lemma dS (d n:о) (q:Д) (p v:р) (e:i9) (x g b:ZB) (u t:FloatArray) (hn hx hg hb hgd hbd)
  (m:∀j<n,u.data[j]! =dw g[j]!) (l:∀j<n,t.data[j]! =dw b[j]!) (r:о) (o:ZB) (ho):
  c5 d n q p v e x g b hn hx hg hb r o ho=RL d n q p v e x g b u t hn hx hg hb hgd hbd r o ho:=by
 rw[c5,RL];split
 .rename_i z
  obtain ⟨A,B,w,D⟩:=uB z hn;refine (dS d n q p v e x g b u t hn hx hg hb hgd hbd m l (r+1) _ _).trans (fG _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ?_);apply d5;intro _;apply fB;intro j y;have h:=γ.mp y;rw[w] at h;dsimp only;rw[ao,ao,m _ h,l _ h,ak g,ak b]
 .rfl
termination_by d-r
lemma cB (a:ZB) (m:Е) (o:aв<2^32) (h:(aV a a.usize (Ϭ _) 0 m)е<255):mе<255/\∀q<aв,i7 a[q]!≠255:=by
 have j:a.usizeб=aв:=by rw[kk,Dense.aa (by have:=ς;omega)]
 obtain ⟨l,d⟩:=cL a a.usize (Ϭ _) _ 0 m rfl;exact ⟨by omega,λq g e=>by have:=d q (by simp) (by rw[j];exact g);rw[e] at this;simp only[jj] at this;omega⟩
lemma cA (a:ZB) (e:∀q<aв,i7 a[q]!≠255):a.all jS=true:=by
 rw[j8];intro i d;have:=e i d;rw[getElem!_pos a i d] at this;unfold jS;simpa using this
@[inline] def ee (rows n:о) (eps:т) (x g b:ZB):ф:=
 rows>0&&n>0&&xв==rows*n&&gв==n&&bв==n&&rows*n<=4294967294&&n<=1048576&&
 aV x x.usize (Ϭ _) 0 (aV g g.usize (Ϭ _) 0 (aV b b.usize (Ϭ _) 0 0))<255&&
 finiteWord eps&&(!ZX eps&&ZC eps !=0)&&rq nж.toInt64.toFloat (1/nж.toInt64.toFloat)
lemma ef {d n:о} {e:i9} {x g b:ZB} (h:ee d n e x g b=true):(aQ d n e x g&&bв==n&&b.all jS)=true:=by
 simp only[ee,υ,decide_eq_true_eq,beq_iff_eq] at h;obtain ⟨⟨⟨⟨⟨⟨⟨⟨⟨⟨q,r⟩,t⟩,u⟩,v⟩,w⟩,y⟩,z⟩,A⟩,j⟩,l⟩:=h;have m:=cB x _ А (ϖ.1 z);have o:=cB g _ А m.1;have p:=cB b _ А o.1;simp only[aQ,SN,υ,decide_eq_true_eq,beq_iff_eq,cA x m.2,cA g o.2,cA b p.2];exact ⟨⟨⟨⟨⟨⟨⟨⟨⟨⟨⟨⟨q,r⟩,t⟩,u⟩,trivial⟩,trivial⟩,A⟩,j⟩,w⟩,y⟩,l⟩,v⟩,trivial⟩
def aM (rows n:о) (eps:т) (x g b:ZB):ZB:=
 if h:ee rows n eps x g b then
  have l:=di (ef h);have d:=aT l.1;have z:=bE d.1
  if rows<4 then c5 rows n (lw n) (dw eps*NR.pt 149) (1/nж.toInt64.toFloat) eps x g b d.2.1 А А А 0 (Array.replicate (rows*n) 0) (by simp)
  else dj P ag rows n (lw n) (dw eps*NR.pt 149) (1/nж.toInt64.toFloat) eps (bb b 0 0+1051) (bb g 0 0) x g b (aq g) (aq b) d.2.1 А А А
   (by rw[(af g).1];omega) (by rw[(af b).1];omega) 0 (aq (Array.replicate (n+1) 0)) (by rw[(af _).1];simp) (Array.replicate (rows*n) 0) (by simp)
 else layernormSpec rows n eps x g b
lemma bG:aM=lJ:=by
 funext q n r x g b;unfold aM;split
 .rename_i h
  have o:=ef h;rw[<-bF];unfold ar;rw[dif_pos o];have l:=di o;have d:=aT l.1;have z:=bE d.1;dsimp only;split
  .exact dS _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (λj m=>(af g).2 j А) (λj m=>(af b).2 j А) _ _ _
  .have t:=cQ b 0 0 (by simp);have p:=cQ g 0 0 (by simp)
   have e:(bb b 0 0+1051)а=(bb b 0 0)а+1051:=by rw[μ];simp only[iZ];omega
   exact er _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (λj m=>by rw[e];have:=t.2 j А А;omega) (by rw[e];omega)
    (by rw[e];omega) (λj m=>p.2 j А А) p.1 _ _ _ _ _
 .rfl
def c6 (rows n:о) (Ln:Д) (Ep rn:р) (eps:т) (x g:ZB) (hn:rows*n+1<2^32) (hx:rows*n<= xв) (hg:n<= gв) (r:о) (out:ZB) (ho:rows*n<= outв):ZB:=
 if hr:r<rows then
  have h:=uB hr hn;have h1:=h.1;have h2:=h.2.1;have h3:=h.2.2.1;have h4:=h.2.2.2
  let e:=rR Ln (xD n x r eps false) Ep rn x (r*n)г ((r*n)г+nг) (by rw[h4];omega) (by rw[h4];omega) (by rw[h4,h2];omega)
  c6 rows n Ln Ep rn eps x g hn hx hg (r+1)
   (if e.ok then QE (r*n)г nг (λj hj=>
    have:=γ.mp hj
    let gw:=g.uget j А
    eR e (x.uget ((r*n)г+j) (by rw[Dense.ab (by have:=ς;omega)];omega)) gw (dw gw)) (by rw[h2,h3];omega) 0 out (by rw[h2,h3];omega)
    else fR n eps x g r (r*n)г nг (by rw[h2,h3];omega) out (by rw[h2,h3];omega))
   (by split<;>simp only[bL,fR]<;>exact ho)
 else out
termination_by rows-r
lemma gV (d n:о) (l:Д) (j p:р) (e:i9) (x g:ZB) (o:FloatArray) (hn hx hg hgd) (r:о) (a m:ZB) (ha ha') (h:a=m):
  RR d n l j p e x g o hn hx hg hgd r a ha=RR d n l j p e x g o hn hx hg hgd r m ha':=by subst h;rfl
lemma dT (d n:о) (p:Д) (o t:р) (e:i9) (x g:ZB) (q:FloatArray) (hn hx hg hgd)
  (h:∀j<n,q.data[j]! =dw g[j]!) (r:о) (m:ZB) (ho):
  c6 d n p o t e x g hn hx hg r m ho=RR d n p o t e x g q hn hx hg hgd r m ho:=by
 rw[c6,RR];split
 .rename_i w
  obtain ⟨y,z,u,A⟩:=uB w hn;refine (dT d n p o t e x g q hn hx hg hgd h (r+1) _ _).trans (gV _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ?_);apply d5;intro _;apply fB;intro j v;have l:=γ.mp v;rw[u] at l;dsimp only;rw[ao,h _ l,ak g]
 .rfl
termination_by d-r
@[inline] def eg (rows n:о) (eps:т) (x g:ZB):ф:=
 rows>0&&n>0&&xв==rows*n&&gв==n&&rows*n<=4294967294&&n<=1048576&&
 aV x x.usize (Ϭ _) 0 (aV g g.usize (Ϭ _) 0 0)<255&&
 finiteWord eps&&(!ZX eps&&ZC eps !=0)&&rq nж.toInt64.toFloat (1/nж.toInt64.toFloat)
lemma eh {d n:о} {e:i9} {x g:ZB} (h:eg d n e x g=true):aQ d n e x g=true:=by
 simp only[eg,υ,decide_eq_true_eq,beq_iff_eq] at h;obtain ⟨⟨⟨⟨⟨⟨⟨⟨⟨p,q⟩,r⟩,t⟩,u⟩,v⟩,w⟩,y⟩,j⟩,l⟩:=h;have m:=cB x _ А (ϖ.1 w);have o:=cB g _ А m.1;simp only[aQ,SN,υ,decide_eq_true_eq,beq_iff_eq,cA x m.2,cA g o.2];exact ⟨⟨⟨⟨⟨⟨⟨⟨⟨⟨p,q⟩,r⟩,t⟩,trivial⟩,trivial⟩,y⟩,j⟩,u⟩,v⟩,l⟩
def bh (rows n:о) (eps:т) (x g:ZB):ZB:=
 if h:eg rows n eps x g then
  have d:=aT (eh h);have z:=bE d.1
  if rows<4 then c6 rows n (lw n) (dw eps*NR.pt 149) (1/nж.toInt64.toFloat) eps x g d.2.1 А А 0 (Array.replicate (rows*n) 0) (by simp)
  else
   let gd:=aq g
   RR rows n (lw n) (dw eps*NR.pt 149) (1/nж.toInt64.toFloat) eps x g gd d.2.1 А А (by rw[(af g).1];omega) 0 (Array.replicate (rows*n) 0) (by simp)
 else rmsnormSpec rows n eps x g
lemma bZ:bh=rmsnormSpec:=by
 funext l n m x g;unfold bh;split
 .rename_i h
  have e:=eh h;rw[<-bY];unfold aO;rw[dif_pos e];have d:=aT e;have z:=bE d.1;dsimp only;split
  .exact dT _ _ _ _ _ _ _ _ _ _ _ _ _ (λj o=>(af g).2 j А) _ _ _
  .rfl
 .rfl
end Norm
end
namespace CV
theorem dc{n}{a b c d:Nat→Word}(h:∀t,t<n→a t=c t)(g:∀t,t<n→b t=d t):dotOf n a b=dotOf n c d:=by
 unfold dotOf;congr 1<;>apply Array.ext (by simp)<;>intro i j _<;>simp only[Array.size_ofFn] at j<;>simp[h i j,g i j]
theorem ge{n}(f:Fin n→Word)(k)(h:k<n):(Array.ofFn f)[k]! =f ⟨k,h⟩:=by rw[getElem!_pos _ k (by simpa using h),Array.getElem_ofFn]
@[inline] def im(s:ConvShape)(x:Array Word):Array Word:=Array.ofFn (n:=(s.ho*s.wo)*s.taps) λr=>windowAt s x 0 (r.1/s.taps/s.wo%s.ho) (r.1/s.taps%s.wo) (r.1%s.taps)
@[inline] def cl(s:ConvShape)(u:Bool)(x w b:Array Word):Array Word:=
 if s.groups=1∧0<s.taps then let L:=Dense.F.Pn.cP (s.ho*s.wo) s.taps s.cout u (im s x) w b;Array.ofFn (n:=s.cout*(s.ho*s.wo)) λq=>L[(q.1%(s.ho*s.wo))*s.cout+q.1/(s.ho*s.wo)]!
 else conv2dWith Pinned.dotFn s u x w b
theorem cl_eq:cl=conv2dWith dotOf:=by
 funext s u x w b;unfold cl;split_ifs with h
 · obtain ⟨hg,hk⟩:=h;rw[Dense.F.Pn.dK];generalize hP:s.ho*s.wo=P
   have p0:0<P:=hP▸Nat.mul_pos (by simp[ConvShape.ho]) (by simp[ConvShape.wo])
   apply Array.ext (by simp[conv2dWith,hP]);intro q h1 _;simp only[Array.size_ofFn] at h1
   have o:q/P<s.cout:=(Nat.div_lt_iff_lt_mul p0).2 h1;have p:q%P<P:=Nat.mod_lt _ p0
   have r:q%P*s.cout+q/P<P*s.cout:=calc q%P*s.cout+q/P<q%P*s.cout+s.cout:=by omega
     _=(q%P+1)*s.cout:=by ring
     _≤P*s.cout:=Nat.mul_le_mul_right _ p
   have c:0<s.cout:=Nat.lt_of_le_of_lt (Nat.zero_le _) o
   have e:s.opg=s.cout:=by simp[ConvShape.opg,hg]
   have y:q%P/s.wo%s.ho=q/s.wo%s.ho:=by rw[<-hP,Nat.mul_comm s.ho s.wo,Nat.mod_mul_right_div_self,Nat.mod_mod]
   have z:q%P%s.wo=q%s.wo:=by rw[<-hP];exact Nat.mod_mul_left_mod _ _ _
   simp only[Array.getElem_ofFn];unfold linearSpec linearWith;rw[ge _ _ r]
   simp only[conv2dWith,Array.getElem_ofFn,hP,Nat.add_comm _ (q/P),Nat.add_mul_div_right _ _ c,Nat.div_eq_of_lt o,Nat.zero_add,Nat.add_mul_mod_self_right,Nat.mod_eq_of_lt o,e]
   congr 1;apply dc
   · intro t _;split_ifs with k
     · have i:q%P*s.taps+t<P*s.taps:=calc q%P*s.taps+t<q%P*s.taps+s.taps:=by omega
         _=(q%P+1)*s.taps:=by ring
         _≤P*s.taps:=Nat.mul_le_mul_right _ p
       unfold im;rw[ge _ _ (hP▸i)]
       simp only[hP,Nat.add_comm _ t,Nat.add_mul_div_right _ _ hk,Nat.div_eq_of_lt k,Nat.zero_add,Nat.add_mul_mod_self_right,Nat.mod_eq_of_lt k,y,z]
     · rfl
   · intro t _;rfl
 · rw[Pinned.dotFn_eq_dotOf]
end CV
def kernels:Kernels:=
 {Pinned.kernels with
  dense:=⟨Dense.F.Pn.cP,Dense.F.Pn.ck⟩
  conv:={Pinned.kernels.conv with conv2d:=CV.cl}
  elem:={Pinned.kernels.elem with addRelu:=Bn.bq}
  norm:=⟨Norm.bh,Norm.aM⟩}
theorem correct:Obligation kernels:=by
 have h:=Pinned.correct
 unfold Obligation at h ⊢
 unfold kernels
 rw[h,CV.cl_eq,Dense.F.Pn.dK,Dense.F.Pn.cO,cg,Norm.bZ,Norm.bG]
 rfl
end Submission