one_le_ctB (KFUU §2g: phase 2 in IL (one iteration))
ProvedNLogN.KFUU.one_le_ctBThe following holds (Lean statement, in the scope of module Solutions.KFUURev7):
theorem NLogN.KFUU.one_le_ctB (u cz : ℕ) (bs : List Bool) (hcz : 1 ≤ cz) : 1 ≤ vol (ctB u cz bs)
Context. The lemma belongs to the part KFUU §2g: phase 2 in IL (one iteration): p2B divides M by U², calls the swap entry with the descriptor
[w, u, 1, 3, 5, U^k, U, U^{n-2-2k}, U, U^k·Bh], and multiplies A = U^k and C = U^k·Bh by U.
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 KFB KFC KFT KFU AModel AModelL12
variable {P : ℕ} {I : List Sym} {iprog : Fin P → IStmt HS P}
theorem NLogN.KFUU.one_le_ctB (u cz : ℕ) (bs : List Bool) (hcz : 1 ≤ cz) : 1 ≤ vol (ctB u cz bs) := by sorry