dsafe_of (KFV §37: phase 3 of the layout (the chunk exchanges) as a fixed IL loop)

The following holds (Lean statement, in the scope of module Solutions.KFV34LayLoop3):

theorem NLogN.KFV.dsafe_of {dW : Fin HS} (hd : dW ∉ r3) (e : Ex HS) (h : ∀ r ∈ e.regs, r ∈ r3) :
    DSafe dW e

Context. The lemma belongs to the part KFV §37: phase 3 of the layout (the chunk exchanges) as a fixed IL loop: Iteration t < N = D q: the chase x ← src t; N times (x ← src x while x < t) runs in the register gV (an RS counter loop on gT2), then, unless x = t, the chunks t and x (after the prefix Lst) are exchanged by the Lemma 12 entry (xchC). ph3_ok: the loop computes runArr n 0 (ops3 …).

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.