bap0Sl_all (Item 12 (machine side, part 1): the permutation-free `B̃₀` slice program)

The following holds (Lean statement, in the scope of module Solutions.KFMBap0):

theorem NLogN.KFM.bap0Sl_all (M : MulSpec) (p s t α kb Bi Bo mm Φ Bacc Q : ℕ) (hM : MulBnd M mm Φ)
    (H : BapHyp p s t α kb Bi Bo mm Bacc Q) (st : IState HS) (x : List (ℤ × ℤ)) (R : List Item)
    (hC : CtxB p s t α kb st) (hx : x.length = t ∧ NLogN.GL.GB Bi x) (hI : st.G bI = gints x ++ R) :
    bap0SlP.run st = (st.upd bI R).upd bO (gints (b0L p s t α x) ++ st.G bO) ∧
      bap0SlP.ok M st ∧ bap0SlP.cost M st ≤ bap0SlC p s t α mm Φ Bacc Q

Context. The lemma belongs to the part Item 12 (machine side, part 1): the permutation-free B̃₀ slice program.

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.