Cost of one uniform level (selection, entry, body, exit): linear part plus the child calls

Cost of one uniform level (selection, entry, body, exit): linear part plus the child calls.

The following holds (Lean statement, in the scope of module Solutions.KIL5Prog):

theorem NLogN.KIL.levelCost_le (N P L' : ℕ) (A B Cs : Shape) (cC Bb : ℕ) :
    cEntry W m Bb + (bodyCI C tpl N P L' A B Cs cC Bb + cExit W m Bb) + 25 ≤
      Klev C tpl * ((vol (P :: W * L' :: hdShape A B Cs m N) + 1) * (Bb + 2) ^ 2) +
        SWn C tpl * cC

Context. The lemma belongs to the part KIL5: the uniform procedure (one fixed program for every level).

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.