getElem_pull (Array model: mixed-radix arrays, pull maps, and linear address maps via Bruhat f)

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

theorem AModel.getElem_pull (sh sh' : Shape) (g : List ℕ → List ℕ) (d : α) (xs : List α) (j : ℕ)
    (hj : j < (AModel.pull sh sh' g d xs).length) :
    (AModel.pull sh sh' g d xs)[j] = xs.getD (encode sh (g (decode sh' j))) d

Context. The lemma belongs to the part Array model: mixed-radix arrays, pull maps, and linear address maps via Bruhat factors: Arrays are lists in row-major lexicographic order over fields (first field most significant). All denotations are plain List functions.

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.