valid_revH_KR (KFU §9: the runs of `Φ` and `Φ⁻¹`)

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

theorem NLogN.KFU.valid_revH_KR (d : ℕ) (f : ℕ → ℕ) {a : List ℕ} (ha : Valid (shK d f) a) :
    Valid (NLogN.KFU.shR d f) (revH d a)

Context. The lemma belongs to the part KFU §9: the runs of Φ and Φ⁻¹: phiC (stack 0: ints xs ++ R, xs on S_k = shK (sP)):

1. pad S_k → T_k (d+1 scans),
2. reverse the axes on T_k → T_r (revC, Lemma 12 swaps + bit moves),
3. delete the padding T_r → S_r,
4. the shifts i = d-1, …, 0 (forward coefficients csF),
5. pad S_r → T_r.

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.