The layout-in program, uniform: it computes the pull by relabel initL tgtL

The layout-in program, uniform: it computes the pull by relabel initL tgtL.

The following holds (Lean statement, in the scope of module Solutions.KFV35LayIn):

theorem NLogN.KFV.layIn_ok {eSw : Fin Pp} {dW : Fin HS} {Kc : ℝ} (HSw : SwOK I iprog eSw dW Kc)
    (hd0 : dW ≠ 0) (hd56 : dW.val < 56) (hKc : 0 ≤ Kc) (hdW : dW ∉ r3)
    (hK : 0 < K) (hℓ : 2 ≤ ℓ) (hq1 : 1 ≤ qq ℓ K) {mk p W B : ℕ} (hmk : mk = mkOf (flOf D long))
    (hB : (pLP D ℓ K long).n + ℓ + 5 ≤ B) (hNB : D * qq ℓ K + D + 1 ≤ B) (hWB : W ≤ B) :
    LayerU (r := 2 ^ ℓ) I iprog (fun s => LRg s (pLP D ℓ K long) mk W) (layIn eSw dW)
      (pull (binSh (pLP D ℓ K long).n) (binSh (pLP D ℓ K long).n)
        (relabel (initL D ℓ long) (tgtL D ℓ K long)) 0) (2 ^ (pLP D ℓ K long).n) p W
      (cLayIn D ℓ K long Kc (pLP D ℓ K long).n W B)

Context. The lemma belongs to the part KFV §38: the layout-in program (three phases) as one LayerU: layIn = ph1 ; ph2 ; ph3. The three primitive lists are well formed (ops1_wf, ops2_wf, ops3_wf), so the composed run is the pull by relabel initL tgtL (runArr_eq_pull_init, applyN_lay3). layIn_ok.

Formalization Note. This lemma belongs to layer 5 of the Lean formalization (integer-multiplication-kappa campaign) of the OpenAI preprint Integer multiplication below $n\log n$: 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.