vol_shK (KFU §8: the programs `Φ` and `Φ⁻¹`)
ProvedNLogN.KFU.vol_shKThe following holds (Lean statement, in the scope of module Solutions.KFUPhi):
theorem NLogN.KFU.vol_shK (d : ℕ) (f : ℕ → ℕ) : vol (shK d f) = vol (NLogN.KFU.shR d f)
Context. The lemma belongs to the part KFU §8: the programs Φ and Φ⁻¹: Parameters: d axes, sources s_j = sF j (1 ≤ s_j ≤ t_j), padded sizes
t_j = tP d h bL j = 2^h · (2 if bL j), multipliers μ j. Arrays are integer lists with a
trailing component axis of size 2 (Gaussian records in gflat form); the component axis has
index d.
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.unnecessarySeqFocus false
open NLogN
open NLogN.KFU
open IntMul KFC AModel AModelL12
variable {Pp : ℕ} {I : List Sym} {iprog : Fin Pp → IStmt HS Pp}
theorem NLogN.KFU.vol_shK (d : ℕ) (f : ℕ → ℕ) : vol (shK d f) = vol (NLogN.KFU.shR d f) := by sorry