sapL_take (KFY 1: the padded line passes (math side))
ProvedNLogN.KFY.sapL_takeThe following holds (Lean statement, in the scope of module Solutions.KFY1Pad):
theorem NLogN.KFY.sapL_take (p s t α : ℕ) (hs : 0 < s) (X : List GI) :
sapL p s t α (X.take s) = sapL p s t α X
Context. The lemma belongs to the part KFY 1: the padded line passes (math side): The machine keeps the array in the padded box ∏ t_i on every axis (so that the axis rotations of
TransNeed always see the shapes shPad h bs). Axis by axis it applies
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.unusedSimpArgs false
set_option linter.unusedVariables false
set_option linter.unusedSectionVars false
open NLogN
open NLogN.KFY
open IntMul IntMul.HvdH IntMul.HvdH.Ref IntMul.HvdH.RefPf NLogN.KFM IntMul.OpenAI.RT Complex
open KFB (GI gflat)
variable {d : ℕ}
theorem NLogN.KFY.sapL_take (p s t α : ℕ) (hs : 0 < s) (X : List GI) :
sapL p s t α (X.take s) = sapL p s t α X := by sorry