valid_binSh_permIdx (§6.2 Lemma 16: the positional layout permutation `L`)

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

theorem Synth.valid_binSh_permIdx {n : ℕ} {f : ℕ → ℕ} (hf : ∀ k < n, f k < n) {a : List ℕ}
    (ha : AModel.Valid (binSh n) a) : AModel.Valid (binSh n) (permIdx f a)

Context. The lemma belongs to the part §6.2 Lemma 16: the positional layout permutation L.

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.