srcT_inj (KFV §31: the chunk phase of the layout, a selection sort by Lemma 12 exchanges)

The following holds (Lean statement, in the scope of module Solutions.KFV27Chunks):

theorem NLogN.KFV.srcT_inj {q x y : ℕ} (hx : x < D * q) (hy : y < D * q) (h : NLogN.KFV.srcT D q x = NLogN.KFV.srcT D q y) :
    x = y

Context. The lemma belongs to the part KFV §31: the chunk phase of the layout, a selection sort by Lemma 12 exchanges: After the single-bit moves (KFV25Moves) the N = D q chunks of K bits are in axis-major order; Lemma 16's layout wants them chunk-major. Position t of the target holds the chunk originally at src t = (t mod D) q + ⌊t / D⌋. Selection sort: at step t the wanted chunk sits at the chase position x_t: start at src t and apply src while the value is < t (chF, fuel N); exchange positions t and x_t (nothing if equal). At most N exchanges, and the positions are computed by a fixed arithmetic loop, so a fixed program can do it.

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.