LS_ne (KFUU §1: uniform padding and deletion scans)
ProvedNLogN.KFUU.LS_neThe following holds (Lean statement, in the scope of module Solutions.KFUUPad):
theorem NLogN.KFUU.LS_ne {i : Fin HS} (h : LS i) : i ≠ 0 ∧ i ≠ NLogN.KFUU.qX ∧ i ≠ NLogN.KFUU.qB ∧ i ≠ NLogN.KFUU.qZ ∧ i ≠ tS ∧ i ≠ NLogN.KFUU.tT ∧
i ≠ pV ∧ i ≠ pXr ∧ i ≠ tq ∧ i ≠ NLogN.KFUU.lA ∧ i ≠ lC ∧ i ≠ lD ∧ i ≠ nR ∧ i ≠ ARGM
Context. The lemma belongs to the part KFUU §1: uniform padding and deletion scans: padIU / unpadIU are KFU's padI X B Z / unpadI X B C with the three literal counts read
from the parameter stacks qX, qB, qZ = 50, 51, 52 (for unpadIU, qZ holds C - B).
padIU_runs / unpadIU_runs follow from padI_spec / unpadI_spec by KFV.ISim.runs.
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
set_option linter.unnecessarySeqFocus false
set_option maxRecDepth 20000
open NLogN
open NLogN.KFUU
open IntMul KIL KFC KFT KFU AModel AModelL12
variable {P : ℕ} {I : List Sym} {iprog : Fin P → IStmt HS P}
theorem NLogN.KFUU.LS_ne {i : Fin HS} (h : LS i) : i ≠ 0 ∧ i ≠ NLogN.KFUU.qX ∧ i ≠ NLogN.KFUU.qB ∧ i ≠ NLogN.KFUU.qZ ∧ i ≠ tS ∧ i ≠ NLogN.KFUU.tT ∧
i ≠ pV ∧ i ≠ pXr ∧ i ≠ tq ∧ i ≠ NLogN.KFUU.lA ∧ i ≠ lC ∧ i ≠ lD ∧ i ≠ nR ∧ i ≠ ARGM := by sorry