sF_le_Pv (KFUU §7d: the uniform table set-up of one shift)

The following holds (Lean statement, in the scope of module Solutions.KFUURot4):

theorem NLogN.KFUU.sF_le_Pv (sF : ℕ → ℕ) {i : ℕ} (hs : ∀ j < i, 1 ≤ sF j) {j : ℕ} (hj : j < i) :
    sF j ≤ Pv sF i

Context. The lemma belongs to the part KFUU §7d: the uniform table set-up of one shift: tabU neg (register code): from the registers oI = i, oQ = Q = s_i, oMu = μ, oPi = P_i, oBv = B and the sizes s_{i-1}, …, s_0 on oSP, it pushes the offset table tbl Q (acL neg sF μ Q i) on stack 14, P_i, Q, B on stacks 1, 2, 4 (the parameters of KFC rotF) and -Q on sNQ; oSP is restored. tabU_sp, cost cTab.

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.