The step runner computes runArr when every delegated interchange is correct

The step runner computes runArr when every delegated interchange is correct.

The following holds (Lean statement, in the scope of module Solutions.KIL2Run):

theorem NLogN.KIL.runCode_runArr (scode : Step n Q → IStmt HS Pp) (jOf : Fin n → Fin n → Fin Pp)
    (child : Fin n → Fin n → List ℤ → List ℤ) (t : IState HS) (R : List Item) (Bb cS : ℕ)
    (cC : Fin n → Fin n → ℕ) (steps : List (Step n Q))
    (hS : ∀ i u c, Step.shift i u c ∈ steps → ∀ ys, DOK sh Bb ys →
      IRuns I iprog (scode (.shift i u c)) (t.upd 0 (ints ys ++ R))
        (t.upd 0 (ints ((Step.shift i u c).arr Λ 0 ys) ++ R)) cS)
    (hC : ∀ a b, Step.swap a b ∈ steps → ∀ 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)) (cC a b) ∧
      child a b ys = (Step.swap a b).arr Λ 0 ys)
    (xs : List ℤ) (hx : DOK sh Bb xs) :
    IRuns I iprog (runCode scode jOf steps) (t.upd 0 (ints xs ++ R))
      (t.upd 0 (ints (AModel.runArr Λ 0 steps xs) ++ R)) (runCostI cS cC steps)

Context. The lemma belongs to the part KIL2: running a list of steps, interchanges delegated to child procedures: runCode scode jOf steps is item-language code (IStmt, so it may call procedures): a shift step runs the code scode s, an interchange swap a b is the call .call (jOf a b).

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.