The convolution property: pointwise products of transforms are the transform of the cyclic convolution

The convolution property: pointwise products of transforms are the transform of the cyclic convolution.

The following holds (Lean statement, in the scope of module NLogN.SSAlg):

theorem NLogN.pmul_difZ (j E : ℕ) (hjE : j = 0 ∨ 2 ^ j * E = 2 * n) (A B : List ℤ)
    (hA : A.length = 2 ^ j) (hB : B.length = 2 ^ j) :
    pmulZ (2 ^ n + 1) (difZ (2 ^ n + 1) E j A) (difZ (2 ^ n + 1) E j B) =
      difZ (2 ^ n + 1) E j (cycZ (2 ^ n + 1) (2 ^ j) A B)

Context. The lemma belongs to the part Algebra of the Fermat-ring FFT: inversion and the convolution property.

Formalization Note. This lemma belongs to the item-language (IL) layer of the integer-multiplication-kappa campaign: tape-level programs (register programs over 64 item stacks of the NLogN multitape framework, definition bundles IntMul_HvdH_M_Defs1–IntMul_HvdH_M_Defs8) that implement the data-movement primitives of the OpenAI preprint Integer multiplication below $n\log n$ (field moves, row split/merge, shifts, the uniform Proposition 11 interchange, Lemma 12 with fine $O(1)$-per-element costs, Gaussian record primitives, Proposition 14 pieces) with exact list denotations from the array-model bundles IntMul_Kappa_K_Defs* and explicit cost bounds. All names refer to the definition bundles up to IntMul_Kappa_K_Defs18.