The FM multiplier in the merged table
ProvedNLogN.KFM.kTable_mulOKThe FM multiplier in the merged table.
The following holds (Lean statement, in the scope of module Solutions.KFMTable):
theorem NLogN.KFM.kTable_mulOK {I : List Sym} (ext : Fin E → IStmt HS (7 + E)) :
MulOK I (kTable ext) (pMul E) ⟨fun a b => mulCostH (zsz a + zsz b), fun _ _ => True⟩
Context. The lemma belongs to the part KFM 7: one procedure table for Lemma 12 / Prop 14 and the FM multiplier: Procedure tables are merged by renaming procedure indices (IStmt.ren id g, nlogn HRen).
Formalization Note. This lemma belongs to the item-language (IL) layer of the integer-multiplication-kappa campaign: tape-level programs (register programs over 64 item stacks of the NLogN multitape framework, definition bundles IntMul_HvdH_M_Defs1–IntMul_HvdH_M_Defs8) that implement the data-movement primitives of the OpenAI preprint Integer multiplication below (field moves, row split/merge, shifts, the uniform Proposition 11 interchange, Lemma 12 with fine -per-element costs, Gaussian record primitives, Proposition 14 pieces) with exact list denotations from the array-model bundles IntMul_Kappa_K_Defs* and explicit cost bounds. All names refer to the definition bundles up to IntMul_Kappa_K_Defs22.
-- OpenAI-preprint / κ-improvement development (integer-multiplication-kappa), generated by skeleton subtraction.
import Definitions.Def_IntMul_Kappa_K_Defs22
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.KFM
open IntMul RS AModel KIL KFD KFB KFG KFH KFN Complex P8Lift Butterfly
variable {E : ℕ}
theorem NLogN.KFM.kTable_mulOK {I : List Sym} (ext : Fin E → IStmt HS (7 + E)) :
MulOK I (kTable ext) (pMul E) ⟨fun a b => mulCostH (zsz a + zsz b), fun _ _ => True⟩ := by sorry