Module Ir.Winograd

The Winograd F(2x2, 3x3) convolution rewrite (gh-ocannl-505): a member of the algebraic-rewrite tier over lowered code (Rewrites), gated by the config key winograd_conv (default off, on in the approximate profile, pinned off in reproducible).

A 2-D 3x3 stride-1 convolution lowers to one nest d[.., oh, ow, oc] += a[.., oh + kh, ow + kw, ic] * b[oc, kh, kw, ic] (the input physically padded for a padded conv, so its reads are offset-free). The rewrite replaces that nest with four, over three minted nodes (xi, nu in 0..3 index the 4x4 transformed tile):

• the kernel transform U = G b G^T, per output and reduction channel, recomputed every call (so a trained kernel's update is always seen);
• the input transform V = B^T a B, per 4x4 input tile read at (2 th, 2 tw);
• sixteen per-position contractions M[xi, nu, .., th, tw, oc] += V[xi, nu, ..] * U[xi, nu, oc, ic] behind a zeroing of M -- the write loops in d's axis order and the reduction innermost, the lowered shape of a product, which the matmul sketches then schedule;
• the inverse transform A^T M A into each 2x2 output tile, ACCUMULATED into d, so the zeroing ahead of the original nest (or any prior contents) keeps its meaning.

An odd output extent is covered by Winograd on its even prefix and by the original statement, over fresh symbols, on the last row and the last column (disjoint strips). For an even covered extent E the last tile reads input coordinate E + 1, exactly the direct form's maximum, so the rewrite reads nothing the direct form does not.

Recognition, on one perfect serial nest with a single Set of the form d = d + x * y (either order, or fma): exactly two axes of one operand read o + k (unit coefficients, offset 0), with o a plain output iterator of d and k a loop over 0..2 that the other operand reads plainly; each of those four symbols occurs only there (separability); besides them, exactly one output-channel symbol (in d and the kernel) and one reduction-channel symbol (in the input and the kernel), every other loop a batch axis of d and the input, all indices plain iterators or fixed; all loops from 0, both output extents at least 2; three distinct single-precision nodes. Everything else -- other windows, strides, dilations, padding offsets, precisions, channel rows of several axes -- is left alone. Autotune.detect_conv answers the scheduling question for the same nests but lives above this library (and refuses singleton axes, which this rewrite carries through: a batch of one is a fixed index here); test/operations/winograd_conv pins the two to agree on the eligible sites.

Only the forward accumulation is rewritten: a conv's data gradient writes at affine indices and its weight gradient's "window" is the feature map, so both keep the direct form.

The transformed-domain arithmetic changes rounding beyond reassociation (the kernel transform scales by halves and quarters, the tile transforms add and subtract), so results move within the approximate tier's tolerance; on inputs whose every partial sum is exactly representable the two forms agree bitwise, since every coefficient is dyadic. The rewrite is idempotent, as the tier's fixpoint requires: transformed reads index 2 t + i, the contractions have no window, and a remainder strip's output loop does not start at 0.