recRep_pull (KFV §31: one layout primitive on the record array)

The following holds (Lean statement, in the scope of module Solutions.KFV28LayStep):

theorem NLogN.KFV.recRep_pull {zs : List GI} {xs : List (Fin r → ℂ)} (h : RecRep r F zs xs) (sh : Shape)
    (hxs : xs.length = vol sh) {g : List ℕ → List ℕ} (hg : ∀ a, Valid sh a → Valid sh (g a)) :
    RecRep r F (pull (sh ++ [r]) (sh ++ [r]) (extA sh.length g) 0 zs) (pull sh sh g 0 xs)

Context. The lemma belongs to the part KFV §31: one layout primitive on the record array: The records xs (shape binSh n, each r coefficients) are stored as zs (shape binSh n ++ [r]) and on stack 0 as gflat zs (shape binSh n ++ [r, 2]). A layout primitive o acts on the records by pull (binSh n) (permIdx o.inv).

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.