encK_encK (KFV §23: uniform layers in a context)
ProvedNLogN.KFV.encK_encKThe following holds (Lean statement, in the scope of module Solutions.KFV19Layer):
theorem NLogN.KFV.encK_encK (s : IState HS) (a b : Fin HS → List ℤ) :
NLogN.KIL.encK (NLogN.KIL.encK s a) b = NLogN.KIL.encK s (fun j => b j ++ a j)
Context. The lemma belongs to the part KFV §23: uniform layers in a context: LayerU Ctx code T N p W cost: on any state encK s (o0 (gflat zs ++ l)) (only stack 0 managed)
whose base s satisfies Ctx, code replaces the array by one representing T xs. The round's
context (layout registers, flags) lives in the base, so a layer may call procedures that need
every other managed stack empty (Proposition 14).
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 AModel
variable {Pp : ℕ} {I : List Sym} {iprog : Fin Pp → IStmt HS Pp}
theorem NLogN.KFV.encK_encK (s : IState HS) (a b : Fin HS → List ℤ) :
NLogN.KIL.encK (NLogN.KIL.encK s a) b = NLogN.KIL.encK s (fun j => b j ++ a j) := by sorry