volU_runs (KFUU §2h: the set-up of phase 2 (volume, powers))

The following holds (Lean statement, in the scope of module Solutions.KFUURev8):

theorem NLogN.KFUU.volU_runs {pM : Fin P} (hM : MulOK I iprog pM NLogN.KFUU.MS) {pS : Fin HS} (hpS : LS pS)
    (S : Shape) (Z : ℕ) (hS1 : ∀ a ∈ S, 1 ≤ a) (hVZ : zsz ((vol S : ℕ) : ℤ) ≤ Z)
    (t : IState HS) (RS LN : List Item) (hpS0 : t.G pS = ints (NLogN.KFUU.zl S) ++ RS)
    (hN : t.G nR = .int (S.length : ℤ) :: LN) :
    IRuns I iprog (volU pM pS) t (t.upd NLogN.KFUU.rB (.int (vol S : ℤ) :: t.G NLogN.KFUU.rB))
      ((S.length + 1) * (2 * cU Z + 200 * Z + 9000))

Context. The lemma belongs to the part KFUU §2h: the set-up of phase 2 (volume, powers).

Formalization Note. This lemma belongs to layer 5 of the Lean formalization (integer-multiplication-kappa campaign) of the OpenAI preprint Integer multiplication below $n\log n$: the uniform item-language machine (register programs over the NLogN multitape framework) assembling the Proposition 14 / Lemma 13 / Proposition 18 pieces, the resampling and chirp stages and the cost recurrences into one fixed machine, culminating in NLogN.KFTop.kappa_bound : IntMul.KappaBound (1 / 2 ^ 182) (the bound of Def_IntMul_MultitapeModel). All names refer to the definition bundles up to IntMul_Kappa_K_Defs33.