runArrR_cons (KFV §39: running a layout schedule backwards)

The following holds (Lean statement, in the scope of module Solutions.KFV36LayRev):

theorem NLogN.KFV.runArrR_cons {α : Type*} (n : ℕ) (d : α) (o : LOp) (ops : List LOp) (xs : List α) :
    runArrR n d (o :: ops) xs = pull (binSh n) (binSh n) (permIdx o.idx) d (runArrR n d ops xs)

Context. The lemma belongs to the part KFV §39: running a layout schedule backwards: runArrR n d ops applies the primitives of ops in reverse order, each by its forward position map (pull by permIdx o.idx): it undoes runArr (runArrR_runArr), hence it is the pull by the reverse relabelling (runArrR_eq_pull). runArrR_flatMap_range: a schedule given as a flatMap over range N runs backwards as an iterL loop over the reversed index.

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.