Every level
ProvedNLogN.KFW.lvQ_allEvery level.
The following holds (Lean statement, in the scope of module Solutions.KFWLev2):
theorem NLogN.KFW.lvQ_all {Kc : ℝ} (hRow : RowU Isym tbl rowC Kc) (hKc : 0 ≤ Kc)
(hj : tbl jL = levC jL rowC)
(hK6 : 6 ≤ K) (hr : 1 ≤ r) (hKZ : 80 * Z * (Z + 2) ≤ 2 ^ K) (hZZ : Z ≤ Zr) (hZr : Zr + 2 ≤ r)
(hHZ : H + 1 < 2 ^ Z) : ∀ k, LvQ Isym tbl jL rowC K r Z Zr H k (cbW Kc K k)
Context. The lemma belongs to the part KFW: the levels of the uniform level procedure.
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.KFW
open IntMul KIL KFD KFB KFG KFH KFN Complex Butterfly
variable {Pp : ℕ} {Isym : List Sym} {tbl : Fin Pp → IStmt HS Pp}
variable {jL : Fin Pp} {rowC : IStmt HS Pp} {K r Z Zr H : ℕ}
theorem NLogN.KFW.lvQ_all {Kc : ℝ} (hRow : RowU Isym tbl rowC Kc) (hKc : 0 ≤ Kc)
(hj : tbl jL = levC jL rowC)
(hK6 : 6 ≤ K) (hr : 1 ≤ r) (hKZ : 80 * Z * (Z + 2) ≤ 2 ^ K) (hZZ : Z ≤ Zr) (hZr : Zr + 2 ≤ r)
(hHZ : H + 1 < 2 ^ Z) : ∀ k, LvQ Isym tbl jL rowC K r Z Zr H k (cbW Kc K k) := by sorry