The test at the start of FM

The test at the start of FM.

The following holds (Lean statement, in the scope of module NLogN.SSFM6):

theorem NLogN.fmTest_runs (s : IState NS) (N : ℕ) (LP : List Item) (hP : s.G PRM = .int N :: LP) :
    IRuns I iprog (.seq (blenS PRM ZZ) (.seq (.op (.const T4 (-(fmThr : ℤ)))) (.op (.add T4 ZZ ZZ))))
      s (s.upd ZZ (.int (-(fmThr : ℤ) + fmL N) :: s.G ZZ))
      (1000 * (fmL N + 1) ^ 2 + 1000 * (fmL N + 15))

Context. The lemma belongs to the part FM by induction on the size.

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.