The implemented frame change Φ_X as IL code

The implemented frame change Φ_X as IL code. The Lemma 10 steps of X (edgeSteps), interchanges delegated to child procedures of cost c₀, compute runArr with cost (4m + 2·pivRank X) · cS + pivRank X · (c₀ + 2) + 1 (cS the cost of one shift).

The following holds (Lean statement, in the scope of module Solutions.KIL2Level):

theorem NLogN.KIL.edgeSteps_il {Rg : Type*} [CommRing Rg] {W : ℕ} (C : Circuit Rg m W) (X : IMat m)
    [NeZero Q] (hQ : 1 < Q) (hX : FacAt Q C.fac X)
    (jOf : Fin (m + m) → Fin (m + m) → Fin Pp)
    (child : Fin (m + m) → Fin (m + m) → List ℤ → List ℤ) (hpos : ∀ a ∈ sh, 1 ≤ a)
    (t : IState HS) (R : List Item) (Bb : ℕ) (hV : zsz ((AModel.vol sh : ℕ) : ℤ) ≤ Bb)
    (hQ2 : zsz ((2 * Q : ℕ) : ℤ) ≤ Bb) (c0 : ℕ)
    (hU : ∀ i u c, Step.shift i u c ∈ AModel.edgeSteps C Q X → SmallUnit2 Q u)
    (hC : ∀ a b, Step.swap a b ∈ AModel.edgeSteps C Q X → ∀ ys, DOK sh Bb ys →
      IRuns I iprog (iprog (jOf a b)) (t.upd 0 (ints ys ++ R))
        (t.upd 0 (ints (child a b ys) ++ R)) c0 ∧
      child a b ys = (Step.swap a b).arr Λ 0 ys)
    (xs : List ℤ) (hx : DOK sh Bb xs) :
    IRuns I iprog (runCode (scodeU2 Λ) jOf (AModel.edgeSteps C Q X)) (t.upd 0 (ints xs ++ R))
      (t.upd 0 (ints (AModel.runArr Λ 0 (AModel.edgeSteps C Q X) xs) ++ R))
      ((4 * m + 2 * pivRank C X) * cShiftU sh.length (AModel.vol sh) Bb + pivRank C X * (c0 + 2) + 1)

Context. The lemma belongs to the part KIL2: the pieces of one Prop. 11 level as IL code.

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 $n\log n$ (field moves, row split/merge, shifts, the uniform Proposition 11 interchange, Lemma 12 with fine $O(1)$-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_Defs18.