F2_G (KFV §36: phase 2 of the layout (the low bits) as a fixed IL loop)
ProvedNLogN.KFV.F2_GThe following holds (Lean statement, in the scope of module Solutions.KFV33LayLoop2):
theorem NLogN.KFV.F2_G (N : ℕ) (s : IState HS) (k : ℕ) (v : List ℤ) :
encK (encK s (F2 N k)) (o0 v) = encK s (Function.update (F2 N k) 0 v)
Context. The lemma belongs to the part KFV §36: phase 2 of the layout (the low bits) as a fixed IL loop: Iteration k < b D (u = k / D, i = k mod D) moves the bit at
nL + k + i (qK + b - 1 - u) + qK to nL + k (mvB). The count b D is pushed on gT, the
iteration index is gI. ph2_ok: the loop computes runArr n 0 (ops2 …).
Formalization Note. This lemma belongs to layer 5 of the Lean formalization (integer-multiplication-kappa campaign) of the OpenAI preprint Integer multiplication below : the uniform item-language machine (register programs over the NLogN multitape framework) assembling the Proposition 14 / Lemma 13 / Proposition 18 pieces, the resampling and chirp stages and the cost recurrences into one fixed machine, culminating in NLogN.KFTop.kappa_bound : IntMul.KappaBound (1 / 2 ^ 182) (the bound of Def_IntMul_MultitapeModel). All names refer to the definition bundles up to IntMul_Kappa_K_Defs33.
-- OpenAI-preprint / κ-improvement development (integer-multiplication-kappa), generated by skeleton subtraction.
import Definitions.Def_IntMul_Kappa_K_Defs33
set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000
set_option linter.unusedSectionVars false
set_option linter.unusedSimpArgs false
set_option linter.unusedVariables false
open NLogN
open NLogN.KFV
open IntMul RS KIL KFD KFB KFH KFC KFT Synth AModel AModelL12
variable {Pp : ℕ} {I : List Sym} {iprog : Fin Pp → IStmt HS Pp}
variable (D ℓ K : ℕ) [NeZero D] (long : Fin D → Bool)
theorem NLogN.KFV.F2_G (N : ℕ) (s : IState HS) (k : ℕ) (v : List ℤ) :
encK (encK s (F2 N k)) (o0 v) = encK s (Function.update (F2 N k) 0 v) := by sorry