xT_ge (KFV §31: the chunk phase of the layout, a selection sort by Lemma 12 exchanges)
ProvedNLogN.KFV.xT_geThe following holds (Lean statement, in the scope of module Solutions.KFV27Chunks):
theorem NLogN.KFV.xT_ge {t : ℕ} (ht : t < N) : t ≤ xT src N t
Context. The lemma belongs to the part KFV §31: the chunk phase of the layout, a selection sort by Lemma 12 exchanges: After the single-bit moves (KFV25Moves) the N = D q chunks of K bits are in axis-major
order; Lemma 16's layout wants them chunk-major. Position t of the target holds the chunk
originally at src t = (t mod D) q + ⌊t / D⌋. Selection sort: at step t the wanted chunk sits
at the chase position x_t: start at src t and apply src while the value is < t
(chF, fuel N); exchange positions t and x_t (nothing if equal). At most N exchanges,
and the positions are computed by a fixed arithmetic loop, so a fixed program can do it.
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.unusedSectionVars false
set_option linter.unusedSimpArgs false
open NLogN
open NLogN.KFV
open Synth AModel AModelL12
variable (src : ℕ → ℕ) (N : ℕ)
variable {src N} (hsrc : ∀ x < N, src x < N)
(hinj : ∀ x < N, ∀ y < N, src x = src y → x = y)
include hsrc hinj
theorem NLogN.KFV.xT_ge {t : ℕ} (ht : t < N) : t ≤ xT src N t := by sorry