of_runsTo (Schönhage–Strassen in the item language: stack layout and Fermat-ring arithmetic)

The following holds (Lean statement, in the scope of module NLogN.SSBase):

theorem NLogN.IRuns.of_runsTo {S P : ℕ} {I : List Sym} {iprog : Fin P → IStmt S P} {p : IStmt S P}
    {s t : IState S} {c : ℕ} (n : ℕ) (h : runsTo (IStmt.run iprog n p s) t c) :
    IRuns I iprog p s t c

Context. The lemma belongs to the part Schönhage–Strassen in the item language: stack layout and Fermat-ring arithmetic: The program uses NS item stacks with fixed roles, so that stack (in)equalities are decided by simp. The current ring ℤ/(2^n+1) is given by the ruler 2^n on top of RN and the modulus 2^n+1 on top of RF.

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.