Generic counted loop

Generic counted loop.

The following holds (Lean statement, in the scope of module Solutions.KFMS0):

theorem NLogN.KFM.genLoop_run (i a : Fin HS) (n : Ex HS) (body : RS) (s : IState HS) (N : ℕ)
    (G : ℕ → ℤ → ℤ) (a0 : ℤ) (La : List Item) (hia : i ≠ a) (hi56 : i.val < 56)
    (hn : n.Clean) (hnok : n.ok s) (hN : n.eval s = N) (hw : i ∉ body.writes)
    (ha : s.G a = .int a0 :: La)
    (hb : ∀ k < N, body.ok (lpSt s i a La N k (accSeq G a0 k)) ∧
      body.run (lpSt s i a La N k (accSeq G a0 k)) =
        (lpSt s i a La N k (accSeq G a0 k)).upd a (.int (G k (accSeq G a0 k)) :: La)) :
    (rep i n body).ok s ∧ (rep i n body).run s = s.upd a (.int (accSeq G a0 N) :: La)

Context. The lemma belongs to the part KFM S0: generic counted loops of register programs (RS.rep) for the setup arithmetic.

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_Defs22.