main.bend

bend-ml-tensor-array@0.1.18.0 · 1,867 lines · 0 definitions

1# bend-ml-tensor-array: tensors over a flat Array<F32>, with the shape in the type.
2#
3#   import bend-ml-tensor-array@0.1.18.0/main.bend as TA
4#
5# Mat<r, c> stores r*c numbers in an Array<F32>, row by row (index i*c + j). The
6# dimensions are erased type parameters: the same guarantee as bend-ml-tensor
7# (a product with wrong dimensions does not compile), but ~50x faster than lists:
8# Array.get/set by index, and products in parallel blocks of rows.
9#
10# An Array is affine (a single owner): every operation that reads an operand returns it
11# together with the result (as Array.get does), in records MM, AR, ... Use Mat.clone
12# when you need two copies.
13#
14# What is NOT proved: the invariant "the Array has capacity >= r*c" holds because
15# the constructors (Mat.zeros, Mat.of_list) allocate the right size; and F32 is
16# validated by tests against PyTorch (reference/test_tensor_array.py), not by proof.
17
18import Base
19
20# ---------------------------------------------------------------------
21# helper types
22# ---------------------------------------------------------------------
23
24type R is Type:
25  R{a: Array<F32>, b: Array<F32>, x: F32}
26
27type G is Type:
28  G{a: Array<F32>, b: Array<F32>, c: Array<F32>}
29
30# A dot product with four partial sums. Each step adds its product to a0 and rotates the sums
31# (a1, a2, a3, a0), so each sum gets every fourth product and the processor can work on the next
32# addition before the last one has finished; at the end the sums are added in pairs. The order of
33# the additions is fixed, so the result is the same on every run and every thread count, but it is
34# not the order of a single running sum (NOTES.md, exp. 17).
35def dotr(k: Nat, +ia: U32, +ib: U32, +sa: U32, +sb: U32, a0: F32, a1: F32, a2: F32, a3: F32, ra: Array<F32> & F32, rb: Array<F32> & F32) -> R:
36  match k ra rb:
37    case 0n Tuple{a, x} Tuple{b, y}:
38      R{a, b, ((a0 + a1 : F32) + (a2 + a3 : F32) : F32)}
39    case 1n+p Tuple{a, +x} Tuple{b, +y}:
40      dotr(p, (ia + sa : U32), (ib + sb : U32), sa, sb, a1, a2, a3, (a0 + (x * y : F32) : F32), Array.get(F32, a, (ia + sa : U32)), Array.get(F32, b, (ib + sb : U32)))
41
42def dotg(k: Nat, +ia: U32, +ib: U32, +sa: U32, +sb: U32, acc: F32, ra: Array<F32> & F32, rb: Array<F32> & F32) -> R:
43  dotr(k, ia, ib, sa, sb, acc, 0.0, 0.0, 0.0, ra, rb)
44
45# writes one cell of C: c[idx] = x, or c[idx] = c[idx] + x when ac (an accumulating product, 0.1.10.0). x is computed
46# exactly as without ac and added once, so C + A·B has the bits of adding a separately computed A·B to C.
47
48def cwr_add(+idx: U32, +x: F32, r: Array<F32> & F32) -> Array<F32>:
49  match r:
50    case Tuple{c, +v}:
51      Array.set(F32, c, idx, (v + x : F32))
52
53def cwr(ac: Bool, +idx: U32, +x: F32, c: Array<F32>) -> Array<F32>:
54  match ac:
55    case True{}:
56      cwr_add(idx, x, Array.get(F32, c, idx))
57    case False{}:
58      Array.set(F32, c, idx, x)
59
60# row i of C, columns j.. ; r is the product of the current column (already computed)
61
62def gcols(+ac: Bool, mleft: Nat, +kn: Nat, +ia0: U32, +ib0: U32, +sa: U32, +sb: U32, +bcol: U32, +cidx: U32, c: Array<F32>, r: R) -> G:
63  match mleft r:
64    case 0n R{a, b, +x}:
65      G{a, b, cwr(ac, cidx, x, c)}
66    case 1n+p R{a, b, +x}:
67      gcols(ac, p, kn, ia0, (ib0 + bcol : U32), sa, sb, bcol, (cidx + 1 : U32), cwr(ac, cidx, x, c), dotg(kn, ia0, (ib0 + bcol : U32), sa, sb, 0.0, Array.get(F32, a, ia0), Array.get(F32, b, (ib0 + bcol : U32))))
68
69def grows(+ac: Bool, nleft: Nat, +mm1: Nat, +kn: Nat, +i: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +mu: U32, g: G) -> G:
70  match nleft g:
71    case 0n G{a, b, c}:
72      G{a, b, c}
73    case 1n+p G{a, b, c}:
74      grows(ac, p, mm1, kn, (i + 1 : U32), arow, sa, sb, bcol, mu, gcols(ac, mm1, kn, (i * arow : U32), 0, sa, sb, bcol, (i * mu : U32), c, dotg(kn, (i * arow : U32), 0, sa, sb, 0.0, Array.get(F32, a, (i * arow : U32)), Array.get(F32, b, 0))))
75
76# C (n x m) = A · B with k steps, or C + A · B when ac; c is the output array (with capacity n*m)
77
78def gemm_ac(+ac: Bool, +n: Nat, +m: Nat, +k: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array<F32>, b: Array<F32>, c: Array<F32>) -> G:
79  grows(ac, n, Nat.sub(m, 1n), k, 0, arow, sa, sb, bcol, U32.from_nat(m), G{a, b, c})
80
81def gemm(+n: Nat, +m: Nat, +k: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array<F32>, b: Array<F32>, c: Array<F32>) -> G:
82  gemm_ac(False{}, n, m, k, arow, sa, sb, bcol, a, b, c)
83
84
85# ---------------------------------------------------------------------
86# parallel gemm: 2^d blocks of rows of C, each with its own copy of A and B,
87# returning lists that are written into C at the end
88# ---------------------------------------------------------------------
89
90type S3 is Type:
91  S3{a: Array<F32>, b: Array<F32>, out: List<&2, F32>}
92
93def lcols(mleft: Nat, +kn: Nat, +ia0: U32, +ib0: U32, +sa: U32, +sb: U32, +bcol: U32, out: List<&2, F32>, r: R) -> S3:
94  match mleft r:
95    case 0n R{a, b, +x}:
96      S3{a, b, x <> out}
97    case 1n+p R{a, b, +x}:
98      lcols(p, kn, ia0, (ib0 + bcol : U32), sa, sb, bcol, x <> out, dotg(kn, ia0, (ib0 + bcol : U32), sa, sb, 0.0, Array.get(F32, a, ia0), Array.get(F32, b, (ib0 + bcol : U32))))
99
100def lrows(nleft: Nat, +mm1: Nat, +kn: Nat, +i: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, st: S3) -> S3:
101  match nleft st:
102    case 0n S3{a, b, out}:
103      S3{a, b, out}
104    case 1n+p S3{a, b, out}:
105      lrows(p, mm1, kn, (i + 1 : U32), arow, sa, sb, bcol, lcols(mm1, kn, (i * arow : U32), 0, sa, sb, bcol, out, dotg(kn, (i * arow : U32), 0, sa, sb, 0.0, Array.get(F32, a, (i * arow : U32)), Array.get(F32, b, 0))))
106
107def lblock_fin(st: S3) -> List<&2, F32>:
108  match st:
109    case S3{a, b, out}:
110      List.reverse(&2, F32, out)
111
112def lblock(cnt: Nat, +mm1: Nat, +kn: Nat, +i: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array<F32>, b: Array<F32>) -> List<&2, F32>:
113  lblock_fin(lrows(cnt, mm1, kn, i, arow, sa, sb, bcol, S3{a, b, Nil{}}))
114
115def lpar(d: Nat, +cnt: Nat, +mm1: Nat, +kn: Nat, +i: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, ra: Array<F32> & Array<F32>, rb: Array<F32> & Array<F32>) -> List<&2, F32>:
116  match d ra rb:
117    case 0n Tuple{a, a2} Tuple{b, b2}:
118      lblock(cnt, mm1, kn, i, arow, sa, sb, bcol, a, b)
119    case 1n++q Tuple{a, a2} Tuple{b, b2}:
120      x y = lpar(q, Nat.div(cnt, 2n), mm1, kn, i, arow, sa, sb, bcol, Array.clone(F32, a), Array.clone(F32, b)) lpar(q, Nat.sub(cnt, Nat.div(cnt, 2n)), mm1, kn, (i + U32.from_nat(Nat.div(cnt, 2n)) : U32), arow, sa, sb, bcol, Array.clone(F32, a2), Array.clone(F32, b2))
121      List.append(&2, F32, x, y)
122
123def write_l(xs: List<&2, F32>, +idx: U32, c: Array<F32>) -> Array<F32>:
124  match xs:
125    case Nil{}:
126      c
127    case Con{h, t}:
128      write_l(t, (idx + 1 : U32), Array.set(F32, c, idx, h))
129
130# c[idx..] = list, or c[idx..] += list when ac
131
132def write_la(+ac: Bool, xs: List<&2, F32>, +idx: U32, c: Array<F32>) -> Array<F32>:
133  match xs:
134    case Nil{}:
135      c
136    case Con{+h, t}:
137      write_la(ac, t, (idx + 1 : U32), cwr(ac, idx, h, c))
138
139def gp3(+ac: Bool, a: Array<F32>, b: Array<F32>, c: Array<F32>, xs: List<&2, F32>) -> G:
140  G{a, b, write_la(ac, xs, 0, c)}
141
142def gp2(d: Nat, +n: Nat, +mm1: Nat, +kn: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, c: Array<F32>, pa: Array<F32> & Array<F32>, pb: Array<F32> & Array<F32>) -> G:
143  match pa pb:
144    case Tuple{a1, a2} Tuple{b1, b2}:
145      gp3(False{}, a1, b1, c, lpar(d, n, mm1, kn, 0, arow, sa, sb, bcol, Array.clone(F32, a2), Array.clone(F32, b2)))
146
147# C (n x m) = A · B, with 2^d blocks of rows in parallel; returns A, B (intact) and C
148
149def gemm_par(d: Nat, +n: Nat, +m: Nat, +k: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array<F32>, b: Array<F32>, c: Array<F32>) -> G:
150  gp2(d, n, Nat.sub(m, 1n), k, arow, sa, sb, bcol, c, Array.clone(F32, a), Array.clone(F32, b))
151
152
153# ---------------------------------------------------------------------
154# parallel gemm by COLUMNS, for n = 1 (matrix · vector): each block computes a
155# range of columns of C, with its own copy of the operands
156# ---------------------------------------------------------------------
157
158def cblock_fin(st: S3) -> List<&2, F32>:
159  match st:
160    case S3{a, b, out}:
161      List.reverse(&2, F32, out)
162
163# cnt columns starting at j0 (cnt >= 1); row 0 of A is at ia0
164
165def cblock_go(cnt1: Nat, +kn: Nat, +ia0: U32, +j0: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array<F32>, b: Array<F32>) -> List<&2, F32>:
166  cblock_fin(lcols(cnt1, kn, ia0, (j0 * bcol : U32), sa, sb, bcol, Nil{}, dotg(kn, ia0, (j0 * bcol : U32), sa, sb, 0.0, Array.get(F32, a, ia0), Array.get(F32, b, (j0 * bcol : U32)))))
167
168def cblock(cnt: Nat, +kn: Nat, +ia0: U32, +j0: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array<F32>, b: Array<F32>) -> List<&2, F32>:
169  match cnt:
170    case 0n:
171      Nil{}
172    case 1n+p:
173      cblock_go(p, kn, ia0, j0, sa, sb, bcol, a, b)
174
175def cpar(d: Nat, +cnt: Nat, +kn: Nat, +ia0: U32, +j0: U32, +sa: U32, +sb: U32, +bcol: U32, ra: Array<F32> & Array<F32>, rb: Array<F32> & Array<F32>) -> List<&2, F32>:
176  match d ra rb:
177    case 0n Tuple{a, a2} Tuple{b, b2}:
178      cblock(cnt, kn, ia0, j0, sa, sb, bcol, a, b)
179    case 1n++q Tuple{a, a2} Tuple{b, b2}:
180      x y = cpar(q, Nat.div(cnt, 2n), kn, ia0, j0, sa, sb, bcol, Array.clone(F32, a), Array.clone(F32, b)) cpar(q, Nat.sub(cnt, Nat.div(cnt, 2n)), kn, ia0, (j0 + U32.from_nat(Nat.div(cnt, 2n)) : U32), sa, sb, bcol, Array.clone(F32, a2), Array.clone(F32, b2))
181      List.append(&2, F32, x, y)
182
183def cp2(+ac: Bool, d: Nat, +m: Nat, +kn: Nat, +sa: U32, +sb: U32, +bcol: U32, c: Array<F32>, pa: Array<F32> & Array<F32>, pb: Array<F32> & Array<F32>) -> G:
184  match pa pb:
185    case Tuple{a1, a2} Tuple{b1, b2}:
186      gp3(ac, a1, b1, c, cpar(d, m, kn, 0, 0, sa, sb, bcol, Array.clone(F32, a2), Array.clone(F32, b2)))
187
188# C (1 x m) = A (1 x k) · B, with 2^d blocks of columns in parallel. Since 0.1.12.0 no product calls it (one row runs
189# the scalar loop, see gemm_any3): it clones both operands at every node. Kept for callers of earlier versions.
190
191def gemm_par_cols(+ac: Bool, d: Nat, +m: Nat, +k: Nat, +sa: U32, +sb: U32, +bcol: U32, a: Array<F32>, b: Array<F32>, c: Array<F32>) -> G:
192  cp2(ac, d, m, k, sa, sb, bcol, c, Array.clone(F32, a), Array.clone(F32, b))
193
194# ---------------------------------------------------------------------
195# element-wise operations (in place, by index)
196# ---------------------------------------------------------------------
197
198# ---------------------------------------------------------------------
199# gemm: C[i][j] = sum_p A[i*arow + p*sa] * B[p*sb + j*bcol]
200# (nn: arow=K, sa=1, sb=M, bcol=1 | nt: B stored m x k: sb=1, bcol=K | tn: A stored k x n: arow=1, sa=N)
201# ---------------------------------------------------------------------
202
203# acc += A[ia + t*sa] * B[ib + t*sb] for k steps; ra, rb already carry the first pair
204
205type P2 is Type:
206  P2{a: Array<F32>, b: Array<F32>}
207
208type AL is Type:
209  AL{c: Array<F32>, l: List<&2, F32>}
210
211# list of m elements starting at idx (reads with Array.get, returning the array)
212
213def read_l(mleft: Nat, +idx: U32, acc: List<&2, F32>, rc: Array<F32> & F32) -> AL:
214  match mleft rc:
215    case 0n Tuple{c, +v}:
216      AL{c, List.reverse(&2, F32, v <> acc)}
217    case 1n+p Tuple{c, +v}:
218      read_l(p, (idx + 1 : U32), v <> acc, Array.get(F32, c, (idx + 1 : U32)))
219
220# writes the list starting at idx
221
222def addl(xs: List<&2, F32>, +idx: U32, rc: Array<F32> & F32) -> Array<F32>:
223  match xs rc:
224    case Nil{} Tuple{c, v}:
225      c
226    case Con{+h, t} Tuple{c, +v}:
227      addl(t, (idx + 1 : U32), Array.get(F32, Array.set(F32, c, idx, (v + h : F32)), (idx + 1 : U32)))
228
229# adds a bias (a list of m) to each of the n rows of c (n x m)
230
231def bias_rows(nleft: Nat, +i: U32, +mu: U32, +bl: List<&2, F32>, c: Array<F32>) -> Array<F32>:
232  match nleft:
233    case 0n:
234      c
235    case 1n+p:
236      bias_rows(p, (i + 1 : U32), mu, bl, addl(bl, (i * mu : U32), Array.get(F32, c, (i * mu : U32))))
237
238# in-place relu on the first n elements
239
240def relu_l(nleft: Nat, +t: U32, rc: Array<F32> & F32) -> Array<F32>:
241  match nleft rc:
242    case 0n Tuple{c, v}:
243      c
244    case 1n+p Tuple{c, +v}:
245      relu_l(p, (t + 1 : U32), Array.get(F32, Array.set(F32, c, t, F32.max(v, 0.0)), (t + 1 : U32)))
246
247def pos2(b: Bool) -> F32:
248  match b:
249    case True{}:
250      1.0
251    case False{}:
252      0.0
253
254def pos(+y: F32) -> F32:
255  pos2(F32.is_gt(y, 0.0))
256
257# d[t] *= (h[t] > 0): the relu gradient; returns both arrays
258
259def mask_l(nleft: Nat, +t: U32, rd: Array<F32> & F32, rh: Array<F32> & F32) -> P2:
260  match nleft rd rh:
261    case 0n Tuple{d, x} Tuple{h, y}:
262      P2{d, h}
263    case 1n+p Tuple{d, +x} Tuple{h, +y}:
264      mask_l(p, (t + 1 : U32), Array.get(F32, Array.set(F32, d, t, (x * pos(y) : F32)), (t + 1 : U32)), Array.get(F32, h, (t + 1 : U32)))
265
266# w[t] -= lr * dw[t]
267
268def maxl2(xs: List<&2, F32>, cur: F32) -> F32:
269  match xs:
270    case Nil{}:
271      cur
272    case Con{+h, t}:
273      maxl2(t, F32.max(cur, h))
274
275def maxl(xs: List<&2, F32>) -> F32:
276  match xs:
277    case Nil{}:
278      0.0
279    case Con{+h, t}:
280      maxl2(t, h)
281
282def expl(xs: List<&2, F32>, +m: F32) -> List<&2, F32>:
283  match xs:
284    case Nil{}:
285      Nil{}
286    case Con{+h, t}:
287      F32.exp((h - m : F32)) <> expl(t, m)
288
289def suml(xs: List<&2, F32>, acc: F32) -> F32:
290  match xs:
291    case Nil{}:
292      acc
293    case Con{+h, t}:
294      suml(t, (acc + h : F32))
295
296def divl(xs: List<&2, F32>, +s: F32) -> List<&2, F32>:
297  match xs:
298    case Nil{}:
299      Nil{}
300    case Con{+h, t}:
301      (h / s : F32) <> divl(t, s)
302
303def softmax2(+es: List<&2, F32>) -> List<&2, F32>:
304  divl(es, suml(es, 0.0))
305
306def softmax(+xs: List<&2, F32>) -> List<&2, F32>:
307  softmax2(expl(xs, maxl(xs)))
308
309# p[label]
310
311def sgd_l(nleft: Nat, +t: U32, +lr: F32, rw: Array<F32> & F32, rd: Array<F32> & F32) -> P2:
312  match nleft rw rd:
313    case 0n Tuple{w, x} Tuple{d, y}:
314      P2{w, d}
315    case 1n+p Tuple{w, +x} Tuple{d, +y}:
316      sgd_l(p, (t + 1 : U32), lr, Array.get(F32, Array.set(F32, w, t, (x - (lr * y : F32) : F32)), (t + 1 : U32)), Array.get(F32, d, (t + 1 : U32)))
317
318# ---------------------------------------------------------------------
319# softmax + cross-entropy per row (lists of 10 elements: negligible cost)
320# ---------------------------------------------------------------------
321
322def pick(ps: List<&2, F32>, n: Nat) -> F32:
323  match ps n:
324    case Con{+h, t} 0n:
325      h
326    case Con{h, t} 1n+q:
327      pick(t, q)
328    case Nil{} _:
329      0.0
330
331# (p - onehot(label)) / n
332
333def hit(b: Bool) -> F32:
334  match b:
335    case True{}:
336      1.0
337    case False{}:
338      0.0
339
340def grad_l(ps: List<&2, F32>, +label: Nat, +j: Nat, +nf: F32) -> List<&2, F32>:
341  match ps:
342    case Nil{}:
343      Nil{}
344    case Con{+h, t}:
345      (((h - hit(Nat.is_eq(label, j)) : F32) / nf) : F32) <> grad_l(t, label, 1n+j, nf)
346
347type L2 is Type:
348  L2{loss: F32, c: Array<F32>}
349
350def ce_row2(+ps: List<&2, F32>, +label: Nat, +nf: F32, +idx: U32, loss: F32, c: Array<F32>) -> L2:
351  L2{(loss - F32.log(F32.max(pick(ps, label), 0.000000000001)) : F32), write_l(grad_l(ps, label, 0n, nf), idx, c)}
352
353def ce_row(al: AL, +label: Nat, +nf: F32, +idx: U32, loss: F32) -> L2:
354  match al:
355    case AL{c, l}:
356      ce_row2(softmax(l), label, nf, idx, loss, c)
357
358# walks the rows of z2 (n x m): r carries the accumulated loss and the array; at the end the array has become the gradient
359
360type Mat<-r: Nat, -c: Nat> is Type:
361  Mat{d: Array<F32>}
362
363def ce_rows(labels: List<&2, Nat>, +mm1: Nat, +mu: U32, +nf: F32, +i: U32, r: L2) -> L2:
364  match labels r:
365    case Nil{} L2{loss, c}:
366      L2{loss, c}
367    case Con{+y, t} L2{loss, c}:
368      ce_rows(t, mm1, mu, nf, (i + 1 : U32), ce_row(read_l(mm1, (i * mu : U32), Nil{}, Array.get(F32, c, (i * mu : U32))), y, nf, (i * mu : U32), loss))
369
370# ---- the same rows over the array itself: no lists ----
371# The list version above reads each row into a list and walks it five times. This one walks the row in place with tail
372# loops, with the same operations in the same order (the max from the first cell, exp(x - max) summed from +0.0 left to
373# right, p = e / s, the loss term read before the row is overwritten, (p - onehot) / n written back), so the same bits;
374# MetalBend measured 3.4x fewer cycles on 64 rows of 49,152 (2026-10-25). The list version stays as the reference
375# tests/ce_array.bend compares it with.
376
377type AF is Type:
378  AF{c: Array<F32>, x: F32}
379
380# the max of k + 1 cells from idx, from m (the first cell's value, already read in r)
381def cea_max(k: Nat, +idx: U32, +m: F32, r: Array<F32> & F32) -> AF:
382  match k r:
383    case 0n Tuple{c, +x}:
384      AF{c, F32.max(m, x)}
385    case 1n+p Tuple{c, +x}:
386      cea_max(p, (idx + 1 : U32), F32.max(m, x), Array.get(F32, c, (idx + 1 : U32)))
387
388# cells idx..idx + k become exp(cell - m); returns their sum from s, added left to right
389def cea_exp(k: Nat, +idx: U32, +m: F32, +s: F32, r: Array<F32> & F32) -> AF:
390  match k r:
391    case 0n Tuple{c, +x}:
392      +e = F32.exp((x - m : F32))
393      AF{Array.set(F32, c, idx, e), (s + e : F32)}
394    case 1n+p Tuple{c, +x}:
395      +e = F32.exp((x - m : F32))
396      cea_exp(p, (idx + 1 : U32), m, (s + e : F32), Array.get(F32, Array.set(F32, c, idx, e), (idx + 1 : U32)))
397
398# cell idx (column j) becomes ((e / s) - onehot) / nf, for k + 1 cells
399def cea_grad(k: Nat, +idx: U32, +j: Nat, +label: Nat, +s: F32, +nf: F32, r: Array<F32> & F32) -> Array<F32>:
400  match k r:
401    case 0n Tuple{c, +x}:
402      Array.set(F32, c, idx, ((((x / s : F32) - hit(Nat.is_eq(label, j)) : F32) / nf) : F32))
403    case 1n+p Tuple{c, +x}:
404      cea_grad(p, (idx + 1 : U32), 1n+j, label, s, nf, Array.get(F32, Array.set(F32, c, idx, ((((x / s : F32) - hit(Nat.is_eq(label, j)) : F32) / nf) : F32)), (idx + 1 : U32)))
405
406def cea_p2(+s: F32, r: Array<F32> & F32) -> AF:
407  match r:
408    case Tuple{c, +e}:
409      AF{c, (e / s : F32)}
410
411# p[label] = e[label] / s, or 0.0 for a label past the row (pick's Nil case)
412def cea_p(inside: Bool, +o: U32, +label: Nat, +s: F32, c: Array<F32>) -> AF:
413  match inside:
414    case True{}:
415      cea_p2(s, Array.get(F32, c, (o + U32.from_nat(label) : U32)))
416    case False{}:
417      AF{c, 0.0}
418
419def cea_row3(+mm1: Nat, +o: U32, +label: Nat, +s: F32, +nf: F32, +loss: F32, q: AF) -> L2:
420  match q:
421    case AF{c, +p}:
422      L2{(loss - F32.log(F32.max(p, 0.000000000001)) : F32), cea_grad(mm1, o, 0n, label, s, nf, Array.get(F32, c, o))}
423
424def cea_row2(+mm1: Nat, +o: U32, +label: Nat, +nf: F32, +loss: F32, q: AF) -> L2:
425  match q:
426    case AF{c, +s}:
427      cea_row3(mm1, o, label, s, nf, loss, cea_p(Nat.is_le(label, mm1), o, label, s, c))
428
429def cea_row1(+mm1: Nat, +o: U32, +label: Nat, +nf: F32, +loss: F32, q: AF) -> L2:
430  match q:
431    case AF{c, +m}:
432      cea_row2(mm1, o, label, nf, loss, cea_exp(mm1, o, m, 0.0, Array.get(F32, c, o)))
433
434# a row's max starts from its first cell (maxl), then folds over the k + 1 cells from it
435def cea_row0(+mm1: Nat, +o: U32, +label: Nat, +nf: F32, +loss: F32, r: Array<F32> & F32) -> L2:
436  match r:
437    case Tuple{c, +x0}:
438      cea_row1(mm1, o, label, nf, loss, cea_max(mm1, o, x0, Array.get(F32, c, o)))
439
440# one row per label, in order, as ce_rows
441def cea_rows(labels: List<&2, Nat>, +mm1: Nat, +mu: U32, +nf: F32, +i: U32, r: L2) -> L2:
442  match labels r:
443    case Nil{} L2{loss, c}:
444      L2{loss, c}
445    case Con{+y, t} L2{+loss, c}:
446      cea_rows(t, mm1, mu, nf, (i + 1 : U32), cea_row0(mm1, (i * mu : U32), y, nf, loss, Array.get(F32, c, (i * mu : U32))))
447
448# ---- helper: hits per row ----
449
450type CR is Type:
451  CR{c: Array<F32>, n: Nat}
452
453def gt_head(xs: List<&2, F32>, +v: F32) -> Bool:
454  match xs:
455    case Nil{}:
456      False{}
457    case Con{+h, t}:
458      F32.is_gt(h, v)
459
460def argmax2(xs: List<&2, F32>, g: Bool, +i: Nat, +bi: Nat, +bv: F32) -> Nat:
461  match xs g:
462    case Nil{} _:
463      bi
464    case Con{+h, +t} True{}:
465      argmax2(t, gt_head(t, h), 1n+i, i, h)
466    case Con{h, +t} False{}:
467      argmax2(t, gt_head(t, bv), 1n+i, bi, bv)
468
469def argmax(xs: List<&2, F32>) -> Nat:
470  match xs:
471    case Nil{}:
472      0n
473    case Con{+h, +t}:
474      argmax2(t, gt_head(t, h), 1n, 0n, h)
475
476def one(b: Bool) -> Nat:
477  match b:
478    case True{}:
479      1n
480    case False{}:
481      0n
482
483def count_rows3(+y: Nat, n: Nat, al: AL) -> CR:
484  match al:
485    case AL{c, l}:
486      CR{c, Nat.add(n, one(Nat.is_eq(argmax(l), y)))}
487
488def count_rows(labels: List<&2, Nat>, +mm1: Nat, +mu: U32, +i: U32, r: CR) -> CR:
489  match labels r:
490    case Nil{} CR{c, n}:
491      CR{c, n}
492    case Con{+y, t} CR{c, n}:
493      count_rows(t, mm1, mu, (i + 1 : U32), count_rows3(y, n, read_l(mm1, (i * mu : U32), Nil{}, Array.get(F32, c, (i * mu : U32)))))
494
495
496# =====================================================================
497# Typed API: Mat<r, c> (the dimensions enter the type)
498# =====================================================================
499
500# =====================================================================
501# Array capacity: a PROVED depth for the number of slots
502# =====================================================================
503#
504# cap_depth(n) is the d such that an Array with 2^d slots holds n elements. It is computed by
505# structural recursion on n (not with U32.log2) so that the law cap_ok below can prove
506# `n <= 2^cap_depth(n)`. What stays trusted is only that Array.new(T, d, v) allocates 2^d slots.
507
508# a type chosen by a Bool: rewriting through it refutes True == False
509def BD(b: Bool, t: Type, f: Type) -> Type:
510  match b:
511    case True{}:
512      t
513    case False{}:
514      f
515
516# a <= b, by direct recursion
517def leb(a: Nat, b: Nat) -> Bool:
518  match a b:
519    case 0n _:
520      True{}
521    case 1n+p 0n:
522      False{}
523    case 1n+p 1n+q:
524      leb(p, q)
525
526def pow2(d: Nat) -> Nat:
527  match d:
528    case 0n:
529      1n
530    case 1n++q:
531      Nat.add(pow2(q), pow2(q))
532
533# smallest power of two >= n, by structural recursion on n: returns (depth, 2^depth)
534def cap_pick(ok: Bool, d: Nat, +p: Nat) -> Nat & Nat:
535  match ok:
536    case True{}:
537      (d, p)
538    case False{}:
539      (1n+d, Nat.add(p, p))
540
541def cap_step(r: Nat & Nat, +n: Nat) -> Nat & Nat:
542  match r:
543    case Tuple{+d, +p}:
544      cap_pick(leb(n, p), d, p)
545
546def cap_pair(n: Nat) -> Nat & Nat:
547  match n:
548    case 0n:
549      (0n, 1n)
550    case 1n++m:
551      cap_step(cap_pair(m), 1n+m)
552
553def fst_nat(r: Nat & Nat) -> Nat:
554  match r:
555    case Tuple{d, p}:
556      d
557
558def snd_nat(r: Nat & Nat) -> Nat:
559  match r:
560    case Tuple{d, p}:
561      p
562
563# leb is Base's Nat.is_le, case for case. leb walks its arguments one step at a time in the compiled program; Base's
564# comparison is one native comparison there (0.1.13.0).
565def le_leb(a: Nat, b: Nat) -> {Nat.is_le(a, b) == leb(a, b) : Bool}:
566  match a b:
567    case 0n 0n:
568      {==}
569    case 0n 1n+q:
570      {==}
571    case 1n+p 0n:
572      {==}
573    case 1n+p 1n+q:
574      le_leb(p, q)
575
576def leb_of(+a: Nat, +b: Nat, e: {True{} == Nat.is_le(a, b) : Bool}) -> {True{} == leb(a, b) : Bool}:
577  %le_leb(a, b) : {True{} == _ : Bool}
578  e
579
580# the same pair by doubling (0.1.10.0): p doubles until n <= p, log2(n) steps in a tail loop, where cap_pair takes n
581# steps of a recursion that is not a tail call, each with a comparison of up to n steps. Its fuel is n, which always
582# suffices (2^n >= n); if it ran out, the answer would be cap_pair's, so the law below needs no arithmetic about how
583# many doublings there are. ok is Nat.is_le(n, p), computed by the caller (match only inspects a parameter); with leb
584# it was n steps in all, 152 ms for n = 786,432 on one GPU lane (0.1.13.0).
585def cap_dbl(f: Nat, ok: Bool, +n: Nat, +d: Nat, +p: Nat) -> Nat & Nat:
586  match f ok:
587    case _ True{}:
588      (d, p)
589    case 0n False{}:
590      cap_pair(n)
591    case 1n+g False{}:
592      cap_dbl(g, Nat.is_le(n, Nat.add(p, p)), n, 1n+d, Nat.add(p, p))
593
594# the depth d such that an Array with 2^d slots holds n elements (the smallest such d)
595def cap_depth(+n: Nat) -> Nat:
596  fst_nat(cap_dbl(n, Nat.is_le(n, 1n), n, 0n, 1n))
597
598# ---- lemmas ----
599
600# a <= b  ->  a <= 1 + b
601def leb_succ(a: Nat, b: Nat, e: {True{} == leb(a, b) : Bool}) -> {True{} == leb(a, 1n+b) : Bool}:
602  match a b:
603    case 0n _:
604      {==}
605    case 1n+p 0n:
606      %e : BD(_, Unit, {True{} == leb(1n+p, 1n) : Bool})
607      Unit{}
608    case 1n+p 1n+q:
609      leb_succ(p, q, e)
610
611# a <= b  ->  a <= c + b
612def leb_up_l(+a: Nat, +b: Nat, c: Nat, +e: {True{} == leb(a, b) : Bool}) -> {True{} == leb(a, Nat.add(c, b)) : Bool}:
613  match c:
614    case 0n:
615      e
616    case 1n++c2:
617      leb_succ(a, Nat.add(c2, b), leb_up_l(a, b, c2, e))
618
619# a <= b  ->  a <= b + c
620def leb_up(a: Nat, b: Nat, c: Nat, e: {True{} == leb(a, b) : Bool}) -> {True{} == leb(a, Nat.add(b, c)) : Bool}:
621  match a b:
622    case 0n _:
623      {==}
624    case 1n+p 0n:
625      %e : BD(_, Unit, {True{} == leb(1n+p, Nat.add(0n, c)) : Bool})
626      Unit{}
627    case 1n+p 1n+q:
628      leb_up(p, q, c, e)
629
630# a <= b  and  c <= d  ->  a + c <= b + d
631def leb_add(a: Nat, b: Nat, c: Nat, d: Nat, e1: {True{} == leb(a, b) : Bool}, e2: {True{} == leb(c, d) : Bool}) -> {True{} == leb(Nat.add(a, c), Nat.add(b, d)) : Bool}:
632  match a b:
633    case 0n _:
634      leb_up_l(c, d, b, e2)
635    case 1n+p 0n:
636      %e1 : BD(_, Unit, {True{} == leb(Nat.add(1n+p, c), Nat.add(0n, d)) : Bool})
637      Unit{}
638    case 1n+p 1n+q:
639      leb_add(p, q, c, d, e1, e2)
640
641# 1 <= 2^d
642def pow2_pos(d: Nat) -> {True{} == leb(1n, pow2(d)) : Bool}:
643  match d:
644    case 0n:
645      {==}
646    case 1n++q:
647      leb_up(1n, pow2(q), pow2(q), pow2_pos(q))
648
649# 1 <= p when p = 2^d
650def pos_p(d: Nat, p: Nat, i1: {p == pow2(d) : Nat}) -> {True{} == leb(1n, p) : Bool}:
651  %Equal.sym(Nat, p, pow2(d), i1) : {True{} == leb(1n, _) : Bool}
652  pow2_pos(d)
653
654# p + p == 2^(1+d) when p == 2^d
655def step_eq(d: Nat, p: Nat, i1: {p == pow2(d) : Nat}) -> {Nat.add(p, p) == pow2(1n+d) : Nat}:
656  %i1 : {Nat.add(p, p) == Nat.add(_, _) : Nat}
657  {==}
658
659# one step: given the invariant for m, the invariant for 1+m holds whichever way the comparison goes
660def step_ok(ok: Bool, +d: Nat, +p: Nat, +m: Nat, +e: {ok == leb(1n+m, p) : Bool}, +i1: {p == pow2(d) : Nat}, +i2: {True{} == leb(m, p) : Bool}) -> {snd_nat(cap_pick(ok, d, p)) == pow2(fst_nat(cap_pick(ok, d, p))) : Nat} & {True{} == leb(1n+m, snd_nat(cap_pick(ok, d, p))) : Bool}:
661  match ok:
662    case True{}:
663      (i1, e)
664    case False{}:
665      (step_eq(d, p, i1), leb_add(1n, p, m, p, pos_p(d, p, i1), i2))
666
667
668# the invariant for cap_step on a pair, opened
669def step_pair_ok(+m: Nat, r: Nat & Nat, i1: {snd_nat(r) == pow2(fst_nat(r)) : Nat}, i2: {True{} == leb(m, snd_nat(r)) : Bool}) -> {snd_nat(cap_step(r, 1n+m)) == pow2(fst_nat(cap_step(r, 1n+m))) : Nat} & {True{} == leb(1n+m, snd_nat(cap_step(r, 1n+m))) : Bool}:
670  match r:
671    case Tuple{+d, +p}:
672      step_ok(leb(1n+m, p), d, p, m, {==}, i1, i2)
673
674def cap_succ(+m: Nat, ih: {snd_nat(cap_pair(m)) == pow2(fst_nat(cap_pair(m))) : Nat} & {True{} == leb(m, snd_nat(cap_pair(m))) : Bool}) -> {snd_nat(cap_pair(1n+m)) == pow2(fst_nat(cap_pair(1n+m))) : Nat} & {True{} == leb(1n+m, snd_nat(cap_pair(1n+m))) : Bool}:
675  match ih:
676    case Tuple{i1, i2}:
677      step_pair_ok(m, cap_pair(m), i1, i2)
678
679# the invariant of cap_pair: the second component is 2^(first) and is >= n
680def cap_inv(n: Nat) -> {snd_nat(cap_pair(n)) == pow2(fst_nat(cap_pair(n))) : Nat} & {True{} == leb(n, snd_nat(cap_pair(n))) : Bool}:
681  match n:
682    case 0n:
683      ({==}, {==})
684    case 1n++m:
685      cap_succ(m, cap_inv(m))
686
687# the invariant of cap_dbl: whichever way it ends (n <= p found, or the fuel out and cap_pair's answer), the second
688# component is 2^(first) and is >= n
689def dbl_inv(f: Nat, ok: Bool, +n: Nat, +d: Nat, +p: Nat, e: {ok == Nat.is_le(n, p) : Bool}, i1: {p == pow2(d) : Nat}) -> {snd_nat(cap_dbl(f, ok, n, d, p)) == pow2(fst_nat(cap_dbl(f, ok, n, d, p))) : Nat} & {True{} == leb(n, snd_nat(cap_dbl(f, ok, n, d, p))) : Bool}:
690  match f ok:
691    case _ True{}:
692      (i1, leb_of(n, p, e))
693    case 0n False{}:
694      cap_inv(n)
695    case 1n+g False{}:
696      dbl_inv(g, Nat.is_le(n, Nat.add(p, p)), n, 1n+d, Nat.add(p, p), {==}, step_eq(d, p, i1))
697
698def dbl_final(n: Nat, pr: {snd_nat(cap_dbl(n, Nat.is_le(n, 1n), n, 0n, 1n)) == pow2(fst_nat(cap_dbl(n, Nat.is_le(n, 1n), n, 0n, 1n))) : Nat} & {True{} == leb(n, snd_nat(cap_dbl(n, Nat.is_le(n, 1n), n, 0n, 1n))) : Bool}) -> {True{} == leb(n, pow2(cap_depth(n))) : Bool}:
699  match pr:
700    case Tuple{i1, i2}:
701      %i1 : {True{} == leb(n, _) : Bool}
702      i2
703
704# LAW (cap_ok): an Array with 2^cap_depth(n) slots always has room for n elements.
705law cap_ok:
706  for +n: Nat
707  {True{} == leb(n, pow2(cap_depth(n))) : Bool}
708
709def cap_ok(n):
710  dbl_final(n, dbl_inv(n, Nat.is_le(n, 1n), n, 0n, 1n, {==}, {==}))
711
712# r x c of zeros, with the exact capacity (smallest power of 2)
713
714def Mat.zeros(+r: Nat, +c: Nat) -> Mat<r, c>:
715  Mat{Array.new(F32, cap_depth(Nat.mul(r, c)), 0.0)}
716
717def Mat.fill(+r: Nat, +c: Nat, +v: F32) -> Mat<r, c>:
718  Mat{Array.new(F32, cap_depth(Nat.mul(r, c)), v)}
719
720def fill_list(xs: List<&2, F32>, +i: U32, a: Array<F32>) -> Array<F32>:
721  match xs:
722    case Nil{}:
723      a
724    case Con{h, t}:
725      fill_list(t, (i + 1 : U32), Array.set(F32, a, i, h))
726
727def len_is(xs: List<&2, F32>, n: Nat) -> Bool:
728  match xs n:
729    case Nil{} 0n:
730      True{}
731    case Con{h, t} 1n+p:
732      len_is(t, p)
733    case _ _:
734      False{}
735
736def Mat.of_list2(-r: Nat, -c: Nat, +rr: Nat, +cc: Nat, ok: Bool, xs: List<&2, F32>) -> Maybe<&1, Mat<r, c>>:
737  match ok:
738    case True{}:
739      Some{Mat{fill_list(xs, 0, Array.new(F32, cap_depth(Nat.mul(rr, cc)), 0.0))}}
740    case False{}:
741      None{}
742
743# Mat<r, c> from r*c numbers in row order; None if the size does not match
744
745def Mat.of_list(+r: Nat, +c: Nat, +xs: List<&2, F32>) -> Maybe<&1, Mat<r, c>>:
746  Mat.of_list2(r, c, r, c, len_is(xs, Nat.mul(r, c)), xs)
747
748# reads the first n numbers, returning the matrix
749
750# Mat<r, c> without checking the list size (a short list leaves the rest at zero, a long one is cut
751# by the capacity): only for when the size is known by construction. Prefer Mat.of_list.
752
753def Mat.from_list(+r: Nat, +c: Nat, xs: List<&2, F32>) -> Mat<r, c>:
754  Mat{fill_list(xs, 0, Array.new(F32, cap_depth(Nat.mul(r, c)), 0.0))}
755
756# writes the numbers of xs into m starting at flat index i (row-major); numbers past the
757# capacity are dropped. For loading a large matrix block by block without building one big list.
758def Mat.fill_at(-r: Nat, -c: Nat, +i: U32, xs: List<&2, F32>, m: Mat<r, c>) -> Mat<r, c>:
759  match m:
760    case Mat{d}:
761      Mat{fill_list(xs, i, d)}
762
763type ML<-r: Nat, -c: Nat> is Type:
764  ML{m: Mat<r, c>, l: List<&2, F32>}
765
766def Mat.to_list2(-r: Nat, -c: Nat, al: AL) -> ML<r, c>:
767  match al:
768    case AL{d, l}:
769      ML{Mat{d}, l}
770
771def ML.len(-r: Nat, -c: Nat, ml: ML<r, c>) -> Nat:
772  match ml:
773    case ML{m, l}:
774      List.length(&2, F32, l)
775
776# n numbers from index 0, in row order; n = 0 gives the empty list (a Mat with no numbers)
777def Mat.to_list_count(-r: Nat, -c: Nat, d: Array<F32>, n: Nat) -> ML<r, c>:
778  match n:
779    case 0n:
780      ML{Mat{d}, Nil{}}
781    case 1n+p:
782      Mat.to_list2(r, c, read_l(p, 0, Nil{}, Array.get(F32, d, 0)))
783
784# two independent copies
785
786def Mat.to_list(+r: Nat, +c: Nat, m: Mat<r, c>) -> ML<r, c>:
787  match m:
788    case Mat{d}:
789      Mat.to_list_count(r, c, d, Nat.mul(r, c))
790
791type MM2<-r: Nat, -c: Nat> is Type:
792  MM2{a: Mat<r, c>, b: Mat<r, c>}
793
794def Mat.clone2(-r: Nat, -c: Nat, p: Array<F32> & Array<F32>) -> MM2<r, c>:
795  match p:
796    case Tuple{x, y}:
797      MM2{Mat{x}, Mat{y}}
798
799# ---- products: C (n x m) = A · B, with 2^par blocks of rows in parallel ----
800
801# C = A(n x k) · B(k x m)
802
803def Mat.clone(-r: Nat, -c: Nat, m: Mat<r, c>) -> MM2<r, c>:
804  match m:
805    case Mat{d}:
806      Mat.clone2(r, c, Array.clone(F32, d))
807
808type MMul<-n: Nat, -k: Nat, -m: Nat> is Type:
809  MMul{a: Mat<n, k>, b: Mat<k, m>, c: Mat<n, m>}
810
811# C = A(n x k) · Bᵀ, with B stored m x k
812
813type MMulNT<-n: Nat, -k: Nat, -m: Nat> is Type:
814  MMulNT{a: Mat<n, k>, b: Mat<m, k>, c: Mat<n, m>}
815
816# C = Aᵀ · B, with A stored k x n
817
818type MMulTN<-n: Nat, -k: Nat, -m: Nat> is Type:
819  MMulTN{a: Mat<k, n>, b: Mat<k, m>, c: Mat<n, m>}
820
821# ---------------------------------------------------------------------
822# gemm in tiles of 8 (0.1.8.0): for products with 2 or more rows, A's rows and B's columns are packed once into cells of
823# 8 F32 (a row of k numbers is ceil(k/8) cells, zeros after the last), so the inner loop reads 8 numbers per array
824# access instead of one. The order of the additions is that of `dotr`: product p of a dot goes to partial sum p mod 4,
825# in order, and the four sums are combined as dotr combines them after k steps (its rotation depends on k mod 4). The
826# padding adds +0 to a sum that starts at +0, which changes nothing, so the results are the same bits as the scalar
827# loop's. Products with one row (matrix · vector) keep the scalar loop: packing B would cost as much as the product.
828# ---------------------------------------------------------------------
829
830type T8 is Data:
831  T8{v0: F32, v1: F32, v2: F32, v3: F32, v4: F32, v5: F32, v6: F32, v7: F32}
832
833def t8_zero() -> T8:
834  T8{0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0}
835
836def pick0(b: Bool, +v: F32) -> F32:
837  match b:
838    case True{}:
839      v
840    case False{}:
841      0.0
842
843type PK is Type:
844  PK{a: Array<F32>, t: Array<T8>}
845
846# one cell: the 8 numbers of a at idx, idx + st, ..., those at positions q0 + q >= lim (past the end of the line) as 0
847def pk_g7(+v0: F32, +v1: F32, +v2: F32, +v3: F32, +v4: F32, +v5: F32, +v6: F32, +q0: U32, +lim: U32, +idx: U32, +st: U32, +o: U32, t: Array<T8>, r: Array<F32> & F32) -> PK:
848  match r:
849    case Tuple{a, +v}:
850      PK{a, Array.set(T8, t, o, T8{v0, v1, v2, v3, v4, v5, v6, pick0(U32.is_lt((q0 + 7 : U32), lim), v)})}
851
852def pk_g6(+v0: F32, +v1: F32, +v2: F32, +v3: F32, +v4: F32, +v5: F32, +q0: U32, +lim: U32, +idx: U32, +st: U32, +o: U32, t: Array<T8>, r: Array<F32> & F32) -> PK:
853  match r:
854    case Tuple{a, +v}:
855      pk_g7(v0, v1, v2, v3, v4, v5, pick0(U32.is_lt((q0 + 6 : U32), lim), v), q0, lim, idx, st, o, t, Array.get(F32, a, (idx + (st * 7 : U32) : U32)))
856
857def pk_g5(+v0: F32, +v1: F32, +v2: F32, +v3: F32, +v4: F32, +q0: U32, +lim: U32, +idx: U32, +st: U32, +o: U32, t: Array<T8>, r: Array<F32> & F32) -> PK:
858  match r:
859    case Tuple{a, +v}:
860      pk_g6(v0, v1, v2, v3, v4, pick0(U32.is_lt((q0 + 5 : U32), lim), v), q0, lim, idx, st, o, t, Array.get(F32, a, (idx + (st * 6 : U32) : U32)))
861
862def pk_g4(+v0: F32, +v1: F32, +v2: F32, +v3: F32, +q0: U32, +lim: U32, +idx: U32, +st: U32, +o: U32, t: Array<T8>, r: Array<F32> & F32) -> PK:
863  match r:
864    case Tuple{a, +v}:
865      pk_g5(v0, v1, v2, v3, pick0(U32.is_lt((q0 + 4 : U32), lim), v), q0, lim, idx, st, o, t, Array.get(F32, a, (idx + (st * 5 : U32) : U32)))
866
867def pk_g3(+v0: F32, +v1: F32, +v2: F32, +q0: U32, +lim: U32, +idx: U32, +st: U32, +o: U32, t: Array<T8>, r: Array<F32> & F32) -> PK:
868  match r:
869    case Tuple{a, +v}:
870      pk_g4(v0, v1, v2, pick0(U32.is_lt((q0 + 3 : U32), lim), v), q0, lim, idx, st, o, t, Array.get(F32, a, (idx + (st * 4 : U32) : U32)))
871
872def pk_g2(+v0: F32, +v1: F32, +q0: U32, +lim: U32, +idx: U32, +st: U32, +o: U32, t: Array<T8>, r: Array<F32> & F32) -> PK:
873  match r:
874    case Tuple{a, +v}:
875      pk_g3(v0, v1, pick0(U32.is_lt((q0 + 2 : U32), lim), v), q0, lim, idx, st, o, t, Array.get(F32, a, (idx + (st * 3 : U32) : U32)))
876
877def pk_g1(+v0: F32, +q0: U32, +lim: U32, +idx: U32, +st: U32, +o: U32, t: Array<T8>, r: Array<F32> & F32) -> PK:
878  match r:
879    case Tuple{a, +v}:
880      pk_g2(v0, pick0(U32.is_lt((q0 + 1 : U32), lim), v), q0, lim, idx, st, o, t, Array.get(F32, a, (idx + (st * 2 : U32) : U32)))
881
882def pk_g0(+q0: U32, +lim: U32, +idx: U32, +st: U32, +o: U32, t: Array<T8>, r: Array<F32> & F32) -> PK:
883  match r:
884    case Tuple{a, +v}:
885      pk_g1(pick0(U32.is_lt((q0 + 0 : U32), lim), v), q0, lim, idx, st, o, t, Array.get(F32, a, (idx + (st * 1 : U32) : U32)))
886
887# the cells of one line, cell o.. of t; q0 counts the line's numbers before the cell
888def pk_line(tl: Nat, +q0: U32, +lim: U32, +base: U32, +st: U32, +o: U32, r: PK) -> PK:
889  match tl r:
890    case 0n r:
891      r
892    case 1n+q PK{a, t}:
893      pk_line(q, (q0 + 8 : U32), lim, (base + (st * 8 : U32) : U32), st, (o + 1 : U32), pk_g0(q0, lim, base, st, o, t, Array.get(F32, a, base)))
894
895# lines count lines; source line i starts at i * lstep (in a), and goes to cell row o (cells o * tku .. in t)
896def pk_lines(cnt: Nat, +tk: Nat, +tku: U32, +ku: U32, +i: U32, +o: U32, +lstep: U32, +st: U32, r: PK) -> PK:
897  match cnt:
898    case 0n:
899      r
900    case 1n+p:
901      pk_lines(p, tk, tku, ku, (i + 1 : U32), (o + 1 : U32), lstep, st, pk_line(tk, 0, ku, (i * lstep : U32), st, (o * tku : U32), r))
902
903def pk_pack(+cnt: Nat, +tk: Nat, +k: Nat, +lstep: U32, +st: U32, a: Array<F32>) -> PK:
904  pk_lines(cnt, tk, U32.from_nat(tk), U32.from_nat(k), 0, 0, lstep, st, PK{a, Array.new(T8, cap_depth(Nat.mul(cnt, tk)), t8_zero())})
905
906# the dot of two lines of cells: tk cells from ia (in a) and ib (in b); ra, rb hold the arrays and the cells at ia, ib,
907# already read. Product q of a cell goes to sum q mod 4, the first four before the last four, as dotr adds them.
908def t8_dot(tk: Nat, +ia: U32, +ib: U32, s0: F32, s1: F32, s2: F32, s3: F32, ra: Array<T8> & T8, rb: Array<T8> & T8) -> TD:
909  match tk ra rb:
910    case 0n Tuple{a, x} Tuple{b, y}:
911      TD{a, b, s0, s1, s2, s3}
912    case 1n+p Tuple{a, T8{x0, x1, x2, x3, x4, x5, x6, x7}} Tuple{b, T8{y0, y1, y2, y3, y4, y5, y6, y7}}:
913      t8_dot(p, (ia + 1 : U32), (ib + 1 : U32), ((s0 + (x0 * y0 : F32) : F32) + (x4 * y4 : F32) : F32), ((s1 + (x1 * y1 : F32) : F32) + (x5 * y5 : F32) : F32), ((s2 + (x2 * y2 : F32) : F32) + (x6 * y6 : F32) : F32), ((s3 + (x3 * y3 : F32) : F32) + (x7 * y7 : F32) : F32), Array.get(T8, a, (ia + 1 : U32)), Array.get(T8, b, (ib + 1 : U32)))
914
915type TD is Type:
916  TD{a: Array<T8>, b: Array<T8>, s0: F32, s1: F32, s2: F32, s3: F32}
917
918# dotr's final sum after k products: the sums sit rotated by k mod 4 when its loop ends
919def t8_comb(+rot: Nat, +s0: F32, +s1: F32, +s2: F32, +s3: F32) -> F32:
920  match rot:
921    case 0n:
922      ((s0 + s1 : F32) + (s2 + s3 : F32) : F32)
923    case 1n:
924      ((s1 + s2 : F32) + (s3 + s0 : F32) : F32)
925    case 2n:
926      ((s2 + s3 : F32) + (s0 + s1 : F32) : F32)
927    case _:
928      ((s3 + s0 : F32) + (s1 + s2 : F32) : F32)
929
930type TR is Type:
931  TR{a: Array<T8>, b: Array<T8>, x: F32}
932
933def t8_fin(+rot: Nat, d: TD) -> TR:
934  match d:
935    case TD{a, b, +s0, +s1, +s2, +s3}:
936      TR{a, b, t8_comb(rot, s0, s1, s2, s3)}
937
938def t8_cell(+tk: Nat, +tku: U32, +rot: Nat, +i: U32, +j: U32, a: Array<T8>, b: Array<T8>) -> TR:
939  t8_fin(rot, t8_dot(tk, (i * tku : U32), (j * tku : U32), 0.0, 0.0, 0.0, 0.0, Array.get(T8, a, (i * tku : U32)), Array.get(T8, b, (j * tku : U32))))
940
941type TG is Type:
942  TG{a: Array<T8>, b: Array<T8>, c: Array<F32>}
943
944def t8_set(+ac: Bool, +idx: U32, c: Array<F32>, r: TR) -> TG:
945  match r:
946    case TR{a, b, +x}:
947      TG{a, b, cwr(ac, idx, x, c)}
948
949def t8_cols(+ac: Bool, mleft: Nat, +tk: Nat, +tku: U32, +rot: Nat, +i: U32, +j: U32, +cidx: U32, g: TG) -> TG:
950  match mleft g:
951    case 0n g:
952      g
953    case 1n+p TG{a, b, c}:
954      t8_cols(ac, p, tk, tku, rot, i, (j + 1 : U32), (cidx + 1 : U32), t8_set(ac, cidx, c, t8_cell(tk, tku, rot, i, j, a, b)))
955
956def t8_rows(+ac: Bool, nleft: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, +i: U32, +mu: U32, g: TG) -> TG:
957  match nleft:
958    case 0n:
959      g
960    case 1n+p:
961      t8_rows(ac, p, m, tk, tku, rot, (i + 1 : U32), mu, t8_cols(ac, m, tk, tku, rot, i, 0, (i * mu : U32), g))
962
963# rows i0 .. i0+cnt-1 of C as a list, in order (the parallel version's leaves)
964type TL is Type:
965  TL{a: Array<T8>, b: Array<T8>, out: List<&2, F32>}
966
967# cells j.. of row i, m - j of them; r is cell j, already computed
968def t8_lcols(mleft: Nat, +tk: Nat, +tku: U32, +rot: Nat, +i: U32, +j: U32, out: List<&2, F32>, r: TR) -> TL:
969  match mleft r:
970    case 0n TR{a, b, +x}:
971      TL{a, b, x <> out}
972    case 1n+p TR{a, b, +x}:
973      t8_lcols(p, tk, tku, rot, i, (j + 1 : U32), x <> out, t8_cell(tk, tku, rot, i, (j + 1 : U32), a, b))
974
975def t8_lrows(nleft: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, +i: U32, l: TL) -> TL:
976  match nleft l:
977    case 0n l:
978      l
979    case 1n+p TL{a, b, out}:
980      t8_lrows(p, m, tk, tku, rot, (i + 1 : U32), t8_lcols(Nat.sub(m, 1n), tk, tku, rot, i, 0, out, t8_cell(tk, tku, rot, i, 0, a, b)))
981
982def t8_lfin(l: TL) -> List<&2, F32>:
983  match l:
984    case TL{a, b, out}:
985      List.reverse(&2, F32, out)
986
987type PKK is Type:
988  PKK{a: Array<F32>, b: Array<F32>, ta: Array<T8>, tb: Array<T8>}
989
990def t8_pk2(pa: PK, pb: PK) -> PKK:
991  match pa pb:
992    case PK{a, ta} PK{b, tb}:
993      PKK{a, b, ta, tb}
994
995def t8_g(a: Array<F32>, b: Array<F32>, g: TG) -> G:
996  match g:
997    case TG{ta, tb, c}:
998      G{a, b, c}
999
1000def t8_out_seq(+ac: Bool, +n: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, c: Array<F32>, p: PKK) -> G:
1001  match p:
1002    case PKK{a, b, ta, tb}:
1003      t8_g(a, b, t8_rows(ac, n, m, tk, tku, rot, 0, U32.from_nat(m), TG{ta, tb, c}))
1004
1005# ---- the parallel path (0.1.9.0): the larger operand is packed into one piece per block, so a block owns its piece;
1006# the other operand is copied once per node of the fork tree (2^d - 1 copies), instead of both at every node ----
1007
1008type PT is Type:
1009  PLeaf{t: Array<T8>}
1010  PNode{l: PT, r: PT}
1011
1012type PKT is Type:
1013  PKT{a: Array<F32>, t: PT}
1014
1015def pkt_leaf(r: PK) -> PKT:
1016  match r:
1017    case PK{a, t}:
1018      PKT{a, PLeaf{t}}
1019
1020def pkt_then(r: PKT, k: Array<F32> -> PT -> PKT) -> PKT:
1021  match r:
1022    case PKT{a, t}:
1023      k(a, t)
1024
1025# lines i .. i+cnt-1 of a, packed into 2^d pieces (halves of the line count, as the fork tree splits them)
1026def pk_tree(+d: Nat, +cnt: Nat, +i: U32, +tk: Nat, +tku: U32, +ku: U32, +lstep: U32, +st: U32, a: Array<F32>) -> PKT:
1027  match d:
1028    case 0n:
1029      pkt_leaf(pk_lines(cnt, tk, tku, ku, i, 0, lstep, st, PK{a, Array.new(T8, cap_depth(Nat.mul(cnt, tk)), t8_zero())}))
1030    case 1n+q:
1031      pkt_then(pk_tree(q, Nat.div(cnt, 2n), i, tk, tku, ku, lstep, st, a), a2 => l => pkt_then(pk_tree(q, Nat.sub(cnt, Nat.div(cnt, 2n)), (i + U32.from_nat(Nat.div(cnt, 2n)) : U32), tk, tku, ku, lstep, st, a2), a3 => r => PKT{a3, PNode{l, r}}))
1032
1033def t8_with2(-R: Type, p: Array<T8> & Array<T8>, k: Array<T8> -> Array<T8> -> R) -> R:
1034  match p:
1035    case Tuple{x, y}:
1036      k(x, y)
1037
1038# split by rows of C: the pieces are A's rows, B is shared; the result is C's rows in order
1039def t8_rpar(t: PT, +cnt: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, b: Array<T8>) -> List<&2, F32>:
1040  match t:
1041    case PLeaf{a}:
1042      t8_lfin(t8_lrows(cnt, m, tk, tku, rot, 0, TL{a, b, Nil{}}))
1043    case PNode{l, r}:
1044      t8_with2(List<&2, F32>, Array.clone(T8, b), b1 => b2 => (
1045        x y = t8_rpar(l, Nat.div(cnt, 2n), m, tk, tku, rot, b1) t8_rpar(r, Nat.sub(cnt, Nat.div(cnt, 2n)), m, tk, tku, rot, b2)
1046        List.append(&2, F32, x, y)))
1047
1048# a block of C's columns j0 .. j0+w-1 for every row, row by row
1049type CB is Data:
1050  CB{j0: U32, w: Nat, xs: List<&2, F32>}
1051
1052# split by columns of C: the pieces are B's columns, A is shared
1053def t8_cpar(t: PT, +cnt: Nat, +j0: U32, +n: Nat, +tk: Nat, +tku: U32, +rot: Nat, a: Array<T8>) -> List<&2, CB>:
1054  match t:
1055    case PLeaf{b}:
1056      [CB{j0, cnt, t8_lfin(t8_lrows(n, cnt, tk, tku, rot, 0, TL{a, b, Nil{}}))}]
1057    case PNode{l, r}:
1058      t8_with2(List<&2, CB>, Array.clone(T8, a), a1 => a2 => (
1059        x y = t8_cpar(l, Nat.div(cnt, 2n), j0, n, tk, tku, rot, a1) t8_cpar(r, Nat.sub(cnt, Nat.div(cnt, 2n)), (j0 + U32.from_nat(Nat.div(cnt, 2n)) : U32), n, tk, tku, rot, a2)
1060        List.append(&2, CB, x, y)))
1061
1062type RW is Type:
1063  RW{c: Array<F32>, xs: List<&2, F32>}
1064
1065# w numbers of xs into c from idx; what is left of xs comes back
1066def cb_row(+ac: Bool, w: Nat, +idx: U32, c: Array<F32>, xs: List<&2, F32>) -> RW:
1067  match w xs:
1068    case 1n+p Con{+h, t}:
1069      cb_row(ac, p, (idx + 1 : U32), cwr(ac, idx, h, c), t)
1070    case _ _:
1071      RW{c, xs}
1072
1073def cb_rows(+ac: Bool, n: Nat, +i: U32, +j0: U32, +w: Nat, +mu: U32, r: RW) -> Array<F32>:
1074  match n r:
1075    case 0n RW{c, xs}:
1076      c
1077    case 1n+p RW{c, xs}:
1078      cb_rows(ac, p, (i + 1 : U32), j0, w, mu, cb_row(ac, w, ((i * mu : U32) + j0 : U32), c, xs))
1079
1080def cb_write(+ac: Bool, bs: List<&2, CB>, +n: Nat, +mu: U32, c: Array<F32>) -> Array<F32>:
1081  match bs:
1082    case Nil{}:
1083      c
1084    case Con{CB{+j0, +w, xs}, t}:
1085      cb_write(ac, t, n, mu, cb_rows(ac, n, 0, j0, w, mu, RW{c, xs}))
1086
1087def t8_pr2(+ac: Bool, +n: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, c: Array<F32>, ra: PKT, rb: PK) -> G:
1088  match ra rb:
1089    case PKT{a, t} PK{b, tb}:
1090      G{a, b, write_la(ac, t8_rpar(t, n, m, tk, tku, rot, tb), 0, c)}
1091
1092def t8_pc2(+ac: Bool, +n: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, c: Array<F32>, ra: PK, rb: PKT) -> G:
1093  match ra rb:
1094    case PK{a, ta} PKT{b, t}:
1095      G{a, b, cb_write(ac, t8_cpar(t, m, 0, n, tk, tku, rot, ta), n, U32.from_nat(m), c)}
1096
1097# 2^d blocks along the larger of n (rows of C, pieces of A) and m (columns of C, pieces of B)
1098def t8_par2(+ac: Bool, byrows: Bool, d: Nat, +n: Nat, +m: Nat, +k: Nat, +tk: Nat, +tku: U32, +rot: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array<F32>, b: Array<F32>, c: Array<F32>) -> G:
1099  match byrows:
1100    case True{}:
1101      t8_pr2(ac, n, m, tk, tku, rot, c, pk_tree(d, n, 0, tk, tku, U32.from_nat(k), arow, sa, a), pk_pack(m, tk, k, bcol, sb, b))
1102    case False{}:
1103      t8_pc2(ac, n, m, tk, tku, rot, c, pk_pack(n, tk, k, arow, sa, a), pk_tree(d, m, 0, tk, tku, U32.from_nat(k), bcol, sb, b))
1104
1105def gemm_t8b(+ac: Bool, d: Nat, +n: Nat, +m: Nat, +k: Nat, +tk: Nat, +tku: U32, +rot: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array<F32>, b: Array<F32>, c: Array<F32>) -> G:
1106  match d:
1107    case 0n:
1108      t8_out_seq(ac, n, m, tk, tku, rot, c, t8_pk2(pk_pack(n, tk, k, arow, sa, a), pk_pack(m, tk, k, bcol, sb, b)))
1109    case 1n+q:
1110      t8_par2(ac, Nat.is_le(m, n), 1n+q, n, m, k, tk, tku, rot, arow, sa, sb, bcol, a, b, c)
1111
1112# C (n x m) = A · B (or C + A · B when ac) in cells of 8, sequential (d = 0) or with 2^d blocks of rows in parallel; A
1113# and B come back intact
1114def gemm_t8_ac(+ac: Bool, d: Nat, +n: Nat, +m: Nat, +k: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array<F32>, b: Array<F32>, c: Array<F32>) -> G:
1115  gemm_t8b(ac, d, n, m, k, Nat.div(Nat.add(k, 7n), 8n), U32.from_nat(Nat.div(Nat.add(k, 7n), 8n)), Nat.mod(k, 4n), arow, sa, sb, bcol, a, b, c)
1116
1117def gemm_t8(d: Nat, +n: Nat, +m: Nat, +k: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array<F32>, b: Array<F32>, c: Array<F32>) -> G:
1118  gemm_t8_ac(False{}, d, n, m, k, arow, sa, sb, bcol, a, b, c)
1119
1120# one row (matrix · vector): the scalar loop at every depth; two or more rows: cells of 8. Up to 0.1.11.0 one row in
1121# parallel ran gemm_par_cols, which clones both operands at every node of the fork tree: a 50,257 x 768 weight matrix at
1122# depth 4 ran out of the default heap (5.1 GB) and 8192 x 2048 took 124 ms against 10 ms sequentially. Packing the
1123# matrix into cells of 8 instead costs more than the product, which reads each weight once (64 ms against 24 at
1124# 50,257 x 768). Matrix · vector in parallel is Bands.matvec. Measured 2026-10-16 (NOTES.md).
1125def gemm_any3(+ac: Bool, one: Bool, d: Nat, +n: Nat, +m: Nat, +k: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array<F32>, b: Array<F32>, c: Array<F32>) -> G:
1126  match one:
1127    case True{}:
1128      gemm_ac(ac, n, m, k, arow, sa, sb, bcol, a, b, c)
1129    case False{}:
1130      gemm_t8_ac(ac, d, n, m, k, arow, sa, sb, bcol, a, b, c)
1131
1132def gemm_any2(+ac: Bool, d: Nat, +n: Nat, +m: Nat, +k: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array<F32>, b: Array<F32>, c: Array<F32>) -> G:
1133  gemm_any3(ac, Nat.is_le(n, 1n), d, n, m, k, arow, sa, sb, bcol, a, b, c)
1134
1135def gemm_any(+ac: Bool, d: Nat, +n: Nat, +m: Nat, +k: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array<F32>, b: Array<F32>, c: Array<F32>) -> G:
1136  gemm_any2(ac, d, n, m, k, arow, sa, sb, bcol, a, b, c)
1137
1138def mmul_fin(-n: Nat, -k: Nat, -m: Nat, g: G) -> MMul<n, k, m>:
1139  match g:
1140    case G{a, b, c}:
1141      MMul{Mat{a}, Mat{b}, Mat{c}}
1142
1143def mmul_nt_fin(-n: Nat, -k: Nat, -m: Nat, g: G) -> MMulNT<n, k, m>:
1144  match g:
1145    case G{a, b, c}:
1146      MMulNT{Mat{a}, Mat{b}, Mat{c}}
1147
1148def mmul_tn_fin(-n: Nat, -k: Nat, -m: Nat, g: G) -> MMulTN<n, k, m>:
1149  match g:
1150    case G{a, b, c}:
1151      MMulTN{Mat{a}, Mat{b}, Mat{c}}
1152
1153# ---- the split depth of a product (0.1.11.0): par is an upper bound, and a product too small to gain splits 0 times ----
1154#
1155# A product of n x k by k x m splits into 2^par blocks only when it does at least 2^23 multiply-adds and each cell of C
1156# sums at least 64 of them; otherwise it runs sequentially. Measured on 16 threads (2026-10-16, 18 shapes): below 2^23 the
1157# split costs more than it saves, and with a short inner dimension (a weight gradient over a few tokens) every split
1158# was 2-3.6x slower, because each block hands its rows of C back as a list that is then written into C, a cost per cell
1159# of C that k multiply-adds per cell do not pay for. The result does not depend on the depth: the same bits at every
1160# depth (tests/t8.bend, tests/acc.bend, tests/split.bend).
1161def split_pick(big: Bool, d: Nat) -> Nat:
1162  match big:
1163    case True{}:
1164      d
1165    case False{}:
1166      0n
1167
1168def split_depth(d: Nat, +n: Nat, +m: Nat, +k: Nat) -> Nat:
1169  split_pick(Bool.and(Nat.is_le(64n, k), Nat.is_le(8388608n, Nat.mul(Nat.mul(n, m), k))), d)
1170
1171def mm_a(+n: Nat, +k: Nat, +m: Nat, d: Nat, a: Array<F32>, b: Array<F32>) -> MMul<n, k, m>:
1172  mmul_fin(n, k, m, gemm_any(False{}, d, n, m, k, U32.from_nat(k), 1, U32.from_nat(m), 1, a, b, Array.new(F32, cap_depth(Nat.mul(n, m)), 0.0)))
1173
1174# C = A · B   (n x k) · (k x m); par = log2 of the most parallel blocks (0 = sequential); split_depth picks 0 or par
1175
1176def Mat.matmul(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: Mat<n, k>, b: Mat<k, m>) -> MMul<n, k, m>:
1177  match a b:
1178    case Mat{x} Mat{y}:
1179      mm_a(n, k, m, split_depth(par, n, m, k), x, y)
1180
1181# C + A · B, into C (n x m); A and B come back with the result, as Mat.matmul returns them. Each cell of A · B is
1182# computed as Mat.matmul computes it and added once to C's cell: the same bits as Mat.add(c, Mat.matmul(a, b)'s c),
1183# without allocating the product or walking it a second time.
1184
1185def Mat.matmul_acc(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: Mat<n, k>, b: Mat<k, m>, c: Mat<n, m>) -> MMul<n, k, m>:
1186  match a b c:
1187    case Mat{x} Mat{y} Mat{z}:
1188      mmul_fin(n, k, m, gemm_any(True{}, split_depth(par, n, m, k), n, m, k, U32.from_nat(k), 1, U32.from_nat(m), 1, x, y, z))
1189
1190def mm_nt(+n: Nat, +k: Nat, +m: Nat, d: Nat, a: Array<F32>, b: Array<F32>) -> MMulNT<n, k, m>:
1191  mmul_nt_fin(n, k, m, gemm_any(False{}, d, n, m, k, U32.from_nat(k), 1, 1, U32.from_nat(k), a, b, Array.new(F32, cap_depth(Nat.mul(n, m)), 0.0)))
1192
1193# C = A · Bᵀ   (n x k) · (m x k)ᵀ: for weights stored with one row per output
1194
1195def Mat.matmul_nt(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: Mat<n, k>, b: Mat<m, k>) -> MMulNT<n, k, m>:
1196  match a b:
1197    case Mat{x} Mat{y}:
1198      mm_nt(n, k, m, split_depth(par, n, m, k), x, y)
1199
1200# C + A · Bᵀ, into C (n x m)
1201
1202def Mat.matmul_nt_acc(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: Mat<n, k>, b: Mat<m, k>, c: Mat<n, m>) -> MMulNT<n, k, m>:
1203  match a b c:
1204    case Mat{x} Mat{y} Mat{z}:
1205      mmul_nt_fin(n, k, m, gemm_any(True{}, split_depth(par, n, m, k), n, m, k, U32.from_nat(k), 1, 1, U32.from_nat(k), x, y, z))
1206
1207def mm_tn(+n: Nat, +k: Nat, +m: Nat, d: Nat, a: Array<F32>, b: Array<F32>) -> MMulTN<n, k, m>:
1208  mmul_tn_fin(n, k, m, gemm_any(False{}, d, n, m, k, 1, U32.from_nat(n), U32.from_nat(m), 1, a, b, Array.new(F32, cap_depth(Nat.mul(n, m)), 0.0)))
1209
1210# C = Aᵀ · B   (k x n)ᵀ · (k x m): weight gradient, X^T · dY
1211
1212def Mat.matmul_tn(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: Mat<k, n>, b: Mat<k, m>) -> MMulTN<n, k, m>:
1213  match a b:
1214    case Mat{x} Mat{y}:
1215      mm_tn(n, k, m, split_depth(par, n, m, k), x, y)
1216
1217# C + Aᵀ · B, into C (n x m): a weight gradient added into the gradient it accumulates, dW += Xᵀ · dY
1218
1219def Mat.matmul_tn_acc(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: Mat<k, n>, b: Mat<k, m>, c: Mat<n, m>) -> MMulTN<n, k, m>:
1220  match a b c:
1221    case Mat{x} Mat{y} Mat{z}:
1222      mmul_tn_fin(n, k, m, gemm_any(True{}, split_depth(par, n, m, k), n, m, k, 1, U32.from_nat(n), U32.from_nat(m), 1, x, y, z))
1223
1224# ---- a precise product for long dots (0.1.14.0) ----
1225#
1226# Mat.matmul_precise is Mat.matmul with k summed in blocks of 1024 (the last one 1..1024): block b's columns of A and
1227# rows of B are copied out, the first block's product written into C, each next one's added to it as matmul_acc adds,
1228# so every cell is ((x0 + x1) + x2) .., x_b the block's dot with the usual four partial sums. A dot of 49,152 products
1229# as four running sums of 12,288 lost the small terms (SmolLM2's head: its gradient 1.8x further from float64 than
1230# PyTorch's float32, 0.03x in blocks; BendTorch NOTES, 2026-10-18). Opt-in: it copies the slices and runs one product
1231# a block, and the default products keep their speed and bits. Up to 1024 steps it is Mat.matmul itself.
1232
1233type SL is Type:
1234  SL{a: Array<F32>, s: Array<F32>}
1235
1236# d[di..] = src[si..], k numbers; r holds src[si], already read
1237def sl_row(k: Nat, +si: U32, +di: U32, d: Array<F32>, r: Array<F32> & F32) -> SL:
1238  match k r:
1239    case 0n Tuple{a, x}: SL{a, d}
1240    case 1n+p Tuple{a, +x}: sl_row(p, (si + 1 : U32), (di + 1 : U32), Array.set(F32, d, di, x), Array.get(F32, a, (si + 1 : U32)))
1241
1242# rows i.. of a slice: w numbers a row from src[i*st + o], into d[i*w]
1243def sl_rows(r: Nat, +i: U32, +w: Nat, +w32: U32, +st: U32, +o: U32, sl: SL) -> SL:
1244  match r sl:
1245    case 0n sl: sl
1246    case 1n+p SL{a, d}: sl_rows(p, (i + 1 : U32), w, w32, st, o, sl_row(w, ((i * st : U32) + o : U32), (i * w32 : U32), d, Array.get(F32, a, ((i * st : U32) + o : U32))))
1247
1248def blk_len(+bi: U32, +k: U32) -> Nat:
1249  U32.to_nat(Bool.pick(U32, U32.is_lt((k - (bi * 1024 : U32) : U32), 1024), (k - (bi * 1024 : U32) : U32), 1024))
1250
1251def mp_s4(a: Array<F32>, b: Array<F32>, g: G) -> G:
1252  match g:
1253    case G{x, y, c}: G{a, b, c}
1254
1255def mp_s3(+ac: Bool, +par: Nat, +n: Nat, +m: Nat, +w: Nat, +w32: U32, +m32: U32, a: Array<F32>, as: Array<F32>, c: Array<F32>, sb: SL) -> G:
1256  match sb:
1257    case SL{b, bs}: mp_s4(a, b, gemm_any(ac, split_depth(par, n, m, w), n, m, w, w32, 1, m32, 1, as, bs, c))
1258
1259def mp_s2(+ac: Bool, +par: Nat, +n: Nat, +m: Nat, +w: Nat, +w32: U32, +m32: U32, +o: U32, c: Array<F32>, b: Array<F32>, sa: SL) -> G:
1260  match sa:
1261    case SL{a, as}: mp_s3(ac, par, n, m, w, w32, m32, a, as, c, sl_row(Nat.mul(w, m), (o * m32 : U32), 0, Array.new(F32, cap_depth(Nat.mul(w, m)), 0.0), Array.get(F32, b, (o * m32 : U32))))
1262
1263# block bi: A's columns o..o+w (A is n x k) and B's rows o..o+w (B is k x m), multiplied into C (added when ac)
1264def mp_step(+ac: Bool, +par: Nat, +n: Nat, +m: Nat, +k32: U32, +bi: U32, a: Array<F32>, b: Array<F32>, c: Array<F32>) -> G:
1265  mp_s2(ac, par, n, m, blk_len(bi, k32), U32.from_nat(blk_len(bi, k32)), U32.from_nat(m), (bi * 1024 : U32), c, b, sl_rows(n, 0, blk_len(bi, k32), U32.from_nat(blk_len(bi, k32)), k32, (bi * 1024 : U32), SL{a, Array.new(F32, cap_depth(Nat.mul(n, blk_len(bi, k32))), 0.0)}))
1266
1267def mp_go(nb: Nat, +bi: U32, +par: Nat, +n: Nat, +m: Nat, +k32: U32, g: G) -> G:
1268  match nb g:
1269    case 0n g: g
1270    case 1n++q G{a, b, c}: mp_go(q, (bi + 1 : U32), par, n, m, k32, mp_step(U32.is_lt(0, bi), par, n, m, k32, bi, a, b, c))
1271
1272def mm_precise2(long: Bool, +n: Nat, +k: Nat, +m: Nat, par: Nat, a: Mat<n, k>, b: Mat<k, m>) -> MMul<n, k, m>:
1273  match long a b:
1274    case True{} Mat{x} Mat{y}: mmul_fin(n, k, m, mp_go(U32.to_nat(U32.div((U32.from_nat(k) + 1023 : U32), 1024)), 0, par, n, m, U32.from_nat(k), G{x, y, Array.new(F32, cap_depth(Nat.mul(n, m)), 0.0)}))
1275    case False{} a b: Mat.matmul(n, k, m, par, a, b)
1276
1277# C = A · B (n x k) · (k x m), k summed in blocks of 1024 (above); opt-in
1278def Mat.matmul_precise(+n: Nat, +k: Nat, +m: Nat, +par: Nat, a: Mat<n, k>, b: Mat<k, m>) -> MMul<n, k, m>:
1279  mm_precise2(Nat.is_lt(1024n, k), n, k, m, par, a, b)
1280
1281# ---- element-wise and training operations ----
1282
1283# adds the bias (one row, m numbers) to each row of C (n x m)
1284
1285type MBias<-n: Nat, -m: Nat> is Type:
1286  MBias{c: Mat<n, m>, b: Mat<1n, m>}
1287
1288def add_row2(-n: Nat, -m: Nat, +nn: Nat, +mm: Nat, cd: Array<F32>, al: AL) -> MBias<n, m>:
1289  match al:
1290    case AL{bd, bl}:
1291      MBias{Mat{bias_rows(nn, 0, U32.from_nat(mm), bl, cd)}, Mat{bd}}
1292
1293def Mat.add_row(+n: Nat, +m: Nat, c: Mat<n, m>, b: Mat<1n, m>) -> MBias<n, m>:
1294  match c b:
1295    case Mat{cd} Mat{bd}:
1296      add_row2(n, m, n, m, cd, read_l(Nat.sub(m, 1n), 0, Nil{}, Array.get(F32, bd, 0)))
1297
1298# pair of matrices of the same shape (the result of an operation that reads two operands)
1299
1300type MPair<-r: Nat, -c: Nat> is Type:
1301  MPair{a: Mat<r, c>, b: Mat<r, c>}
1302
1303def mpair_fin(-r: Nat, -c: Nat, p: P2) -> MPair<r, c>:
1304  match p:
1305    case P2{a, b}:
1306      MPair{Mat{a}, Mat{b}}
1307
1308def relu_fin(-r: Nat, -c: Nat, d: Array<F32>) -> Mat<r, c>:
1309  Mat{d}
1310
1311def Mat.relu(+r: Nat, +c: Nat, a: Mat<r, c>) -> Mat<r, c>:
1312  match a:
1313    case Mat{d}:
1314      relu_fin(r, c, relu_l(Nat.mul(r, c), 0, Array.get(F32, d, 0)))
1315
1316# (dy * (h > 0), h): the gradient that passes through the relu
1317
1318def Mat.relu_bwd(+r: Nat, +c: Nat, dy: Mat<r, c>, h: Mat<r, c>) -> MPair<r, c>:
1319  match dy h:
1320    case Mat{d} Mat{x}:
1321      mpair_fin(r, c, mask_l(Nat.mul(r, c), 0, Array.get(F32, d, 0), Array.get(F32, x, 0)))
1322
1323# (w - lr * dw, dw): one SGD step
1324
1325def Mat.sgd(+r: Nat, +c: Nat, +lr: F32, w: Mat<r, c>, dw: Mat<r, c>) -> MPair<r, c>:
1326  match w dw:
1327    case Mat{x} Mat{y}:
1328      mpair_fin(r, c, sgd_l(Nat.mul(r, c), 0, lr, Array.get(F32, x, 0), Array.get(F32, y, 0)))
1329
1330# (a + b, b): element-wise sum (a - (-1)*b is exactly a + b)
1331
1332def Mat.add(+r: Nat, +c: Nat, a: Mat<r, c>, b: Mat<r, c>) -> MPair<r, c>:
1333  match a b:
1334    case Mat{x} Mat{y}:
1335      mpair_fin(r, c, sgd_l(Nat.mul(r, c), 0, (0.0 - 1.0 : F32), Array.get(F32, x, 0), Array.get(F32, y, 0)))
1336
1337# sum of each column (n x m -> 1 x m), returning the matrix
1338
1339type MColSum<-n: Nat, -m: Nat> is Type:
1340  MColSum{a: Mat<n, m>, s: Mat<1n, m>}
1341
1342def colsum_fin(-n: Nat, -m: Nat, g: G) -> MColSum<n, m>:
1343  match g:
1344    case G{ones, a, s}:
1345      MColSum{Mat{a}, Mat{s}}
1346
1347def Mat.col_sums(+n: Nat, +m: Nat, a: Mat<n, m>) -> MColSum<n, m>:
1348  match a:
1349    case Mat{d}:
1350      colsum_fin(n, m, gemm(1n, m, n, 0, 1, U32.from_nat(m), 1, Array.new(F32, cap_depth(n), 1.0), d, Array.new(F32, cap_depth(m), 0.0)))
1351
1352# s + the sum of each column, into s (1 x m): a bias gradient added into its accumulator, with the bits of
1353# Mat.add(s, Mat.col_sums(a)'s s)
1354
1355def Mat.col_sums_acc(+n: Nat, +m: Nat, a: Mat<n, m>, s: Mat<1n, m>) -> MColSum<n, m>:
1356  match a s:
1357    case Mat{d} Mat{z}:
1358      colsum_fin(n, m, gemm_ac(True{}, 1n, m, n, 0, 1, U32.from_nat(m), 1, Array.new(F32, cap_depth(n), 1.0), d, z))
1359
1360# mean cross-entropy and its gradient on the logits: (softmax - one-hot) / n.
1361# labels has one entry per row (n); the gradient replaces the logits.
1362
1363type MCE<-n: Nat, -c: Nat> is Type:
1364  MCE{loss: F32, grad: Mat<n, c>}
1365
1366def ce_fin(-n: Nat, -c: Nat, +nf: F32, r: L2) -> MCE<n, c>:
1367  match r:
1368    case L2{loss, d}:
1369      MCE{(loss / nf : F32), Mat{d}}
1370
1371def Mat.softmax_ce(+n: Nat, +c: Nat, z: Mat<n, c>, labels: List<&2, Nat>) -> MCE<n, c>:
1372  match z:
1373    case Mat{d}:
1374      ce_fin(n, c, F32.from_nat(n), cea_rows(labels, Nat.sub(c, 1n), U32.from_nat(c), F32.from_nat(n), 0, L2{0.0, d}))
1375
1376# argmax hits per row against labels
1377
1378type MHits<-n: Nat, -c: Nat> is Type:
1379  MHits{z: Mat<n, c>, hits: Nat}
1380
1381def hits_fin(-n: Nat, -c: Nat, r: CR) -> MHits<n, c>:
1382  match r:
1383    case CR{d, k}:
1384      MHits{Mat{d}, k}
1385
1386def Mat.count_correct(+n: Nat, +c: Nat, z: Mat<n, c>, labels: List<&2, Nat>) -> MHits<n, c>:
1387  match z:
1388    case Mat{d}:
1389      hits_fin(n, c, count_rows(labels, Nat.sub(c, 1n), U32.from_nat(c), 0, CR{d, 0n}))
1390
1391# a row as a list (returning the matrix) and the other way around
1392
1393type MRow<-r: Nat, -c: Nat> is Type:
1394  MRow{m: Mat<r, c>, row: List<&2, F32>}
1395
1396def row_fin(-r: Nat, -c: Nat, al: AL) -> MRow<r, c>:
1397  match al:
1398    case AL{d, l}:
1399      MRow{Mat{d}, l}
1400
1401def Mat.read_row(+r: Nat, +c: Nat, m: Mat<r, c>, +i: U32) -> MRow<r, c>:
1402  match m:
1403    case Mat{d}:
1404      row_fin(r, c, read_l(Nat.sub(c, 1n), (i * U32.from_nat(c) : U32), Nil{}, Array.get(F32, d, (i * U32.from_nat(c) : U32))))
1405
1406def Mat.write_row(-r: Nat, +c: Nat, m: Mat<r, c>, +i: U32, xs: List<&2, F32>) -> Mat<r, c>:
1407  match m:
1408    case Mat{d}:
1409      Mat{write_l(xs, (i * U32.from_nat(c) : U32), d)}
1410
1411# ---- variants that check the number of labels ----
1412
1413# does the list of labels have exactly n entries (one per row)?
1414def labels_len_is(xs: List<&2, Nat>, n: Nat) -> Bool:
1415  match xs n:
1416    case Nil{} 0n:
1417      True{}
1418    case Con{h, t} 1n+p:
1419      labels_len_is(t, p)
1420    case _ _:
1421      False{}
1422
1423def ce_checked(ok: Bool, +n: Nat, +c: Nat, z: Mat<n, c>, labels: List<&2, Nat>) -> Maybe<&1, MCE<n, c>>:
1424  match ok:
1425    case True{}:
1426      Some{Mat.softmax_ce(n, c, z, labels)}
1427    case False{}:
1428      None{}
1429
1430# like Mat.softmax_ce, but returns None unless labels has exactly n entries
1431def Mat.softmax_ce_checked(+n: Nat, +c: Nat, z: Mat<n, c>, +labels: List<&2, Nat>) -> Maybe<&1, MCE<n, c>>:
1432  ce_checked(labels_len_is(labels, n), n, c, z, labels)
1433
1434def hits_checked(ok: Bool, +n: Nat, +c: Nat, z: Mat<n, c>, labels: List<&2, Nat>) -> Maybe<&1, MHits<n, c>>:
1435  match ok:
1436    case True{}:
1437      Some{Mat.count_correct(n, c, z, labels)}
1438    case False{}:
1439      None{}
1440
1441# like Mat.count_correct, but returns None unless labels has exactly n entries
1442def Mat.count_correct_checked(+n: Nat, +c: Nat, z: Mat<n, c>, +labels: List<&2, Nat>) -> Maybe<&1, MHits<n, c>>:
1443  hits_checked(labels_len_is(labels, n), n, c, z, labels)
1444
1445# ---------------------------------------------------------------------
1446# Bands<r, c>: a matrix cut into bands of rows, for matrix · vector in parallel
1447# without copying the weights.
1448#
1449# Mat.matmul_nt with n = 1 can only run in parallel by cloning the whole weight
1450# matrix for every task, and in matrix · vector each weight is read once, so the
1451# copies cost more than the products (NOTES.md, exp. 9). Bands keeps each band in
1452# its own Array: a task takes its band and nothing is copied except the input vector.
1453#
1454# A node with r rows has children with half(r) and r - half(r) rows. The split is
1455# part of the type, so the row count of every band follows from r, and the law
1456# half_cover says that the two halves always cover the r rows exactly.
1457# ---------------------------------------------------------------------
1458
1459# n / 2 rounded down. A tail loop with an accumulator: a non-tail recursion here would put a
1460# continuation on every node of the band tree and keep the bands from running in parallel.
1461def half.go(n: Nat, acc: Nat) -> Nat:
1462  match n:
1463    case 0n:
1464      acc
1465    case 1n+p:
1466      match p:
1467        case 0n:
1468          acc
1469        case 1n+q:
1470          half.go(q, 1n+acc)
1471
1472def half(n: Nat) -> Nat:
1473  half.go(n, 0n)
1474
1475# a <= a
1476def leb_refl(+a: Nat) -> {True{} == leb(a, a) : Bool}:
1477  match a:
1478    case 0n:
1479      {==}
1480    case 1n+p:
1481      leb_refl(p)
1482
1483# 1 + (a + b) == a + (1 + b)
1484def add_succ_r(+a: Nat, -b: Nat) -> {1n+Nat.add(a, b) == Nat.add(a, 1n+b) : Nat}:
1485  match a:
1486    case 0n:
1487      {==}
1488    case 1n+p:
1489      %add_succ_r(p, b) : {2n+Nat.add(p, b) == 1n+_ : Nat}
1490      {==}
1491
1492# a + 0 == a
1493def add_zero_r(+a: Nat) -> {Nat.add(a, 0n) == a : Nat}:
1494  match a:
1495    case 0n:
1496      {==}
1497    case 1n+p:
1498      %add_zero_r(p) : {1n+Nat.add(p, 0n) == 1n+_ : Nat}
1499      {==}
1500
1501# half.go(n, acc) <= n + acc
1502def half_go_le(+n: Nat, +acc: Nat) -> {True{} == leb(half.go(n, acc), Nat.add(n, acc)) : Bool}:
1503  match n:
1504    case 0n:
1505      leb_refl(acc)
1506    case 1n+p:
1507      match p:
1508        case 0n:
1509          leb_succ(acc, acc, leb_refl(acc))
1510        case 1n+q:
1511          %Equal.sym(Nat, 1n+Nat.add(q, acc), Nat.add(q, 1n+acc), add_succ_r(q, acc)) : {True{} == leb(half.go(q, 1n+acc), 1n+_) : Bool}
1512          leb_succ(half.go(q, 1n+acc), Nat.add(q, 1n+acc), half_go_le(q, 1n+acc))
1513
1514# half(n) <= n
1515def half_le(+n: Nat) -> {True{} == leb(half(n), n) : Bool}:
1516  %add_zero_r(n) : {True{} == leb(half(n), _) : Bool}
1517  half_go_le(n, 0n)
1518
1519# a <= r  ->  r == a + (r - a)
1520def add_sub(+a: Nat, +r: Nat, e: {True{} == leb(a, r) : Bool}) -> {r == Nat.add(a, Nat.sub(r, a)) : Nat}:
1521  match a r:
1522    case 0n 0n:
1523      {==}
1524    case 0n 1n+q:
1525      {==}
1526    case 1n+p 0n:
1527      %e : BD(_, Unit, {0n == Nat.add(1n+p, Nat.sub(0n, 1n+p)) : Nat})
1528      Unit{}
1529    case 1n+p 1n+q:
1530      %add_sub(p, q, e) : {1n+q == 1n+_ : Nat}
1531      {==}
1532
1533# LAW (half_cover): the two bands of a node, half(r) rows and r - half(r) rows,
1534# add up to exactly r rows: no row is lost and none is counted twice.
1535law half_cover:
1536  for +n: Nat
1537  {n == Nat.add(half(n), Nat.sub(n, half(n))) : Nat}
1538
1539def half_cover(n):
1540  add_sub(half(n), n, half_le(n))
1541
1542type Bands<-r: Nat, -c: Nat> is Type:
1543  BLeaf{m: Mat<r, c>}
1544  BNode{x: Bands<half(r), c>, y: Bands<Nat.sub(r, half(r)), c>}
1545
1546# r x c of zeros in 2^d bands (d = 0: a single band, the same as a Mat)
1547def Bands.zeros(+d: Nat, +r: Nat, +c: Nat) -> Bands<r, c>:
1548  match d:
1549    case 0n:
1550      BLeaf{Mat.zeros(r, c)}
1551    case 1n+q:
1552      BNode{Bands.zeros(q, half(r), c), Bands.zeros(q, Nat.sub(r, half(r)), c)}
1553
1554# the row counts of the two bands follow the row count of the node: if rr == r at run
1555# time, the same holds for both halves (the bands carry r only in their type)
1556def eq_half(-a: Nat, -b: Nat, e: {a == b : Nat}) -> {half(a) == half(b) : Nat}:
1557  %e : {half(a) == half(_) : Nat}
1558  {==}
1559
1560def eq_rest(-a: Nat, -b: Nat, e: {a == b : Nat}) -> {Nat.sub(a, half(a)) == Nat.sub(b, half(b)) : Nat}:
1561  %e : {Nat.sub(a, half(a)) == Nat.sub(_, half(_)) : Nat}
1562  {==}
1563
1564# does a block of n numbers starting at flat index i end before the second band of a
1565# node with r rows? does it start in the second band?
1566def bfill_l(+r: Nat, +c: Nat, +i: Nat, +n: Nat) -> Bool:
1567  Nat.is_le(Nat.add(i, n), Nat.mul(half(r), c))
1568
1569def bfill_r(+r: Nat, +c: Nat, +i: Nat) -> Bool:
1570  Nat.is_le(Nat.mul(half(r), c), i)
1571
1572# writes the first n numbers of xs at index i (and nothing past them)
1573def fill_n(n: Nat, xs: List<&2, F32>, +i: U32, a: Array<F32>) -> Array<F32>:
1574  match n xs:
1575    case 1n+p Con{h, t}:
1576      fill_n(p, t, (i + 1 : U32), Array.set(F32, a, i, h))
1577    case _ _:
1578      a
1579
1580def fill_band(-r: Nat, -c: Nat, n: Nat, xs: List<&2, F32>, +i: U32, m: Mat<r, c>) -> Mat<r, c>:
1581  match m:
1582    case Mat{d}:
1583      Mat{fill_n(n, xs, i, d)}
1584
1585# writes the first n numbers of xs at flat index i; l and rt say which bands they touch
1586# (bfill_l, bfill_r); rr is the row count r at run time. A block that crosses into the second
1587# band is not copied: the first band writes only its part, the second gets the rest by List.drop.
1588def bfill(+c: Nat, -r: Nat, b: Bands<r, c>, l: Bool, rt: Bool, +rr: Nat, -e: {rr == r : Nat}, +i: Nat, +n: Nat, +xs: List<&2, F32>) -> Bands<r, c>:
1589  match b l rt:
1590    case BLeaf{m} _ _:
1591      BLeaf{fill_band(r, c, n, xs, U32.from_nat(i), m)}
1592    case BNode{x, y} True{} _:
1593      BNode{bfill(c, half(r), x, bfill_l(half(rr), c, i, n), bfill_r(half(rr), c, i), half(rr), eq_half(rr, r, e), i, n, xs), y}
1594    case BNode{x, y} False{} True{}:
1595      BNode{x, bfill(c, Nat.sub(r, half(r)), y, bfill_l(Nat.sub(rr, half(rr)), c, Nat.sub(i, Nat.mul(half(rr), c)), n), bfill_r(Nat.sub(rr, half(rr)), c, Nat.sub(i, Nat.mul(half(rr), c))), Nat.sub(rr, half(rr)), eq_rest(rr, r, e), Nat.sub(i, Nat.mul(half(rr), c)), n, xs)}
1596    case BNode{x, y} False{} False{}:
1597      BNode{bfill(c, half(r), x, bfill_l(half(rr), c, i, Nat.sub(Nat.mul(half(rr), c), i)), bfill_r(half(rr), c, i), half(rr), eq_half(rr, r, e), i, Nat.sub(Nat.mul(half(rr), c), i), xs), bfill(c, Nat.sub(r, half(r)), y, bfill_l(Nat.sub(rr, half(rr)), c, 0n, Nat.sub(n, Nat.sub(Nat.mul(half(rr), c), i))), bfill_r(Nat.sub(rr, half(rr)), c, 0n), Nat.sub(rr, half(rr)), eq_rest(rr, r, e), 0n, Nat.sub(n, Nat.sub(Nat.mul(half(rr), c), i)), List.drop(&2, F32, xs, Nat.sub(Nat.mul(half(rr), c), i)))}
1598
1599def bfill_top(ok: Bool, +r: Nat, +c: Nat, b: Bands<r, c>, +i: Nat, +n: Nat, +xs: List<&2, F32>) -> Bands<r, c>:
1600  match ok:
1601    case True{}:
1602      bfill(c, r, b, bfill_l(r, c, i, n), bfill_r(r, c, i), r, {==}, i, n, xs)
1603    case False{}:
1604      bfill(c, r, b, bfill_l(r, c, i, Nat.sub(Nat.mul(r, c), i)), bfill_r(r, c, i), r, {==}, i, Nat.sub(Nat.mul(r, c), i), xs)
1605
1606# writes the n numbers of xs into b starting at flat index i (row-major, as Mat.fill_at);
1607# numbers past r*c are dropped. For loading a large matrix block by block.
1608def Bands.fill_at(+r: Nat, +c: Nat, +i: Nat, +n: Nat, +xs: List<&2, F32>, b: Bands<r, c>) -> Bands<r, c>:
1609  bfill_top(Nat.is_le(Nat.add(i, n), Nat.mul(r, c)), r, c, b, i, n, xs)
1610
1611# ---- y = B · x for B with r rows of c numbers (a weight matrix stored out x in) ----
1612
1613# B (intact) and y = B · x, typed like Mat.matmul_nt: x is 1 x c, y is 1 x r
1614type BV<-r: Nat, -c: Nat> is Type:
1615  BV{b: Bands<r, c>, y: Mat<1n, r>}
1616
1617# the bands and y as a list in row order (the result of Bands.matvec_l)
1618type BL<-r: Nat, -c: Nat> is Type:
1619  BL{b: Bands<r, c>, y: List<&2, F32>}
1620
1621def bv_g2(-r: Nat, -c: Nat, w: Array<F32>, al: AL) -> BL<r, c>:
1622  match al:
1623    case AL{d, l}:
1624      BL{BLeaf{Mat{w}}, l}
1625
1626# the band's 1 + m rows came out in an Array (one number per row); read them back as a list
1627def bv_g(-r: Nat, -c: Nat, +m: Nat, g: G) -> BL<r, c>:
1628  match g:
1629    case G{a, b, y}:
1630      bv_g2(r, c, b, read_l(m, 0, Nil{}, Array.get(F32, y, 0)))
1631
1632# the rows of one band, one dot product each, with the gemm of Mat.matmul_nt (n = 1), so the
1633# results are identical. Writing the results into an Array and reading them back is faster than
1634# building the list row by row (NOTES.md, exp. 14).
1635def bv_gemm(-r: Nat, +c: Nat, rows: Nat, +rr: Nat, x: Array<F32>, w: Array<F32>) -> BL<r, c>:
1636  match rows:
1637    case 0n:
1638      BL{BLeaf{Mat{w}}, Nil{}}
1639    case 1n+p:
1640      bv_g(r, c, p, gemm(1n, rr, c, U32.from_nat(c), 1, 1, U32.from_nat(c), x, w, Array.new(F32, cap_depth(rr), 0.0)))
1641
1642def bv_mat(-r: Nat, +c: Nat, +rr: Nat, x: Array<F32>, m: Mat<r, c>) -> BL<r, c>:
1643  match m:
1644    case Mat{w}:
1645      bv_gemm(r, c, rr, rr, x, w)
1646
1647def bv_join(-r: Nat, -c: Nat, p: BL<half(r), c>, q: BL<Nat.sub(r, half(r)), c>) -> BL<r, c>:
1648  match p q:
1649    case BL{bx, yx} BL{by, yy}:
1650      BL{BNode{bx, by}, List.append(&2, F32, yx, yy)}
1651
1652# the two bands of a node run in parallel, each with its own copy of x (c numbers);
1653# xp is a pair of copies of x (a leaf uses one). xp comes after the erased e: up to Bend 2.0.36 an erased
1654# parameter at the end of the list turned the parallel let into two sequential calls (NOTES.md, exp. 11; fixed in
1655# 2.0.37 by bendlang/bend#1377). The order is kept, and harmless.
1656def bmv(+c: Nat, -r: Nat, b: Bands<r, c>, +rr: Nat, -e: {rr == r : Nat}, xp: Array<F32> & Array<F32>) -> BL<r, c>:
1657  match b xp:
1658    case BLeaf{m} Tuple{x, x2}:
1659      bv_mat(r, c, rr, x, m)
1660    case BNode{bx, by} Tuple{x1, x2}:
1661      p q = bmv(c, half(r), bx, half(rr), eq_half(rr, r, e), Array.clone(F32, x1)) bmv(c, Nat.sub(r, half(r)), by, Nat.sub(rr, half(rr)), eq_rest(rr, r, e), Array.clone(F32, x2))
1662      bv_join(r, c, p, q)
1663
1664def bv_fin(+r: Nat, -c: Nat, bl: BL<r, c>) -> BV<r, c>:
1665  match bl:
1666    case BL{b, y}:
1667      BV{b, Mat.from_list(1n, r, y)}
1668
1669# y = B · x for B (r x c, one row per output) and x (1 x c); returns B intact and y (1 x r).
1670# The same product as Mat.matmul_nt(1n, c, r, ..), and the same numbers, bit for bit.
1671def Bands.matvec(+r: Nat, +c: Nat, b: Bands<r, c>, x: Mat<1n, c>) -> BV<r, c>:
1672  match x:
1673    case Mat{xa}:
1674      bv_fin(r, c, bmv(c, r, b, r, {==}, Array.clone(F32, xa)))
1675
1676# the same product with x and y as lists (x: c numbers, y: r numbers in row order), without the
1677# Mat conversions of Bands.matvec: for callers that already hold the vector as a list
1678def Bands.matvec_l(+r: Nat, +c: Nat, b: Bands<r, c>, +x: List<&2, F32>) -> BL<r, c>:
1679  bmv(c, r, b, r, {==}, Array.clone(F32, fill_list(x, 0, Array.new(F32, cap_depth(c), 0.0))))
1680
1681# ---- reading rows and the whole matrix ----
1682
1683type BRow<-r: Nat, -c: Nat> is Type:
1684  BRow{b: Bands<r, c>, row: List<&2, F32>}
1685
1686def brow_fin(-r: Nat, -c: Nat, al: AL) -> BRow<r, c>:
1687  match al:
1688    case AL{d, l}:
1689      BRow{BLeaf{Mat{d}}, l}
1690
1691# n numbers starting at flat index i of one band (a band may have no rows)
1692def brow_leaf2(-r: Nat, -c: Nat, n: Nat, +i: U32, d: Array<F32>) -> BRow<r, c>:
1693  match n:
1694    case 0n:
1695      BRow{BLeaf{Mat{d}}, Nil{}}
1696    case 1n+p:
1697      brow_fin(r, c, read_l(p, i, Nil{}, Array.get(F32, d, i)))
1698
1699def brow_leaf(-r: Nat, -c: Nat, m: Mat<r, c>, +i: U32, n: Nat) -> BRow<r, c>:
1700  match m:
1701    case Mat{d}:
1702      brow_leaf2(r, c, n, i, d)
1703
1704def brow_l(-r: Nat, -c: Nat, p: BRow<half(r), c>, y: Bands<Nat.sub(r, half(r)), c>) -> BRow<r, c>:
1705  match p:
1706    case BRow{bx, row}:
1707      BRow{BNode{bx, y}, row}
1708
1709def brow_r(-r: Nat, -c: Nat, x: Bands<half(r), c>, q: BRow<Nat.sub(r, half(r)), c>) -> BRow<r, c>:
1710  match q:
1711    case BRow{by, row}:
1712      BRow{BNode{x, by}, row}
1713
1714# row i; left says whether i is in the first band of the node (i < half(rr))
1715def brow(+c: Nat, -r: Nat, b: Bands<r, c>, left: Bool, +rr: Nat, -e: {rr == r : Nat}, +i: Nat) -> BRow<r, c>:
1716  match b left:
1717    case BLeaf{m} _:
1718      brow_leaf(r, c, m, U32.from_nat(Nat.mul(i, c)), c)
1719    case BNode{x, y} True{}:
1720      brow_l(r, c, brow(c, half(r), x, Nat.is_lt(i, half(half(rr))), half(rr), eq_half(rr, r, e), i), y)
1721    case BNode{x, y} False{}:
1722      brow_r(r, c, x, brow(c, Nat.sub(r, half(r)), y, Nat.is_lt(Nat.sub(i, half(rr)), half(Nat.sub(rr, half(rr)))), Nat.sub(rr, half(rr)), eq_rest(rr, r, e), Nat.sub(i, half(rr))))
1723
1724# row i (c numbers), for example an embedding lookup
1725def Bands.read_row(+r: Nat, +c: Nat, b: Bands<r, c>, +i: Nat) -> BRow<r, c>:
1726  brow(c, r, b, Nat.is_lt(i, half(r)), r, {==}, i)
1727
1728def bl_join(-r: Nat, -c: Nat, p: BRow<half(r), c>, q: BRow<Nat.sub(r, half(r)), c>) -> BRow<r, c>:
1729  match p q:
1730    case BRow{bx, xs} BRow{by, ys}:
1731      BRow{BNode{bx, by}, List.append(&2, F32, xs, ys)}
1732
1733def blist(+c: Nat, -r: Nat, b: Bands<r, c>, +rr: Nat, -e: {rr == r : Nat}) -> BRow<r, c>:
1734  match b:
1735    case BLeaf{m}:
1736      brow_leaf(r, c, m, 0, Nat.mul(rr, c))
1737    case BNode{x, y}:
1738      bl_join(r, c, blist(c, half(r), x, half(rr), eq_half(rr, r, e)), blist(c, Nat.sub(r, half(r)), y, Nat.sub(rr, half(rr)), eq_rest(rr, r, e)))
1739
1740# all r*c numbers in row order (in BRow.row)
1741def Bands.to_list(+r: Nat, +c: Nat, b: Bands<r, c>) -> BRow<r, c>:
1742  blist(c, r, b, r, {==})
1743
1744# ---- the list path returns exactly r numbers ----
1745#
1746# The numbers do not matter here, only how many there are: each lemma below counts the list a
1747# function builds, following the function's own recursion.
1748
1749def bl_len(-r: Nat, -c: Nat, bl: BL<r, c>) -> Nat:
1750  match bl:
1751    case BL{b, y}:
1752      List.length(&2, F32, y)
1753
1754# s == a + (1 + b)  ->  s == 1 + (a + b)
1755def succ_out(-s: Nat, +a: Nat, +b: Nat, e: {s == Nat.add(a, 1n+b) : Nat}) -> {s == 1n+Nat.add(a, b) : Nat}:
1756  %Equal.sym(Nat, 1n+Nat.add(a, b), Nat.add(a, 1n+b), add_succ_r(a, b)) : {s == _ : Nat}
1757  e
1758
1759# s == 1 + (p + (1 + l))  ->  s == 2 + (p + l)
1760def len_step(-s: Nat, +p: Nat, +l: Nat, e: {s == 1n+Nat.add(p, 1n+l) : Nat}) -> {s == 2n+Nat.add(p, l) : Nat}:
1761  %Equal.sym(Nat, 1n+Nat.add(p, l), Nat.add(p, 1n+l), add_succ_r(p, l)) : {s == 1n+_ : Nat}
1762  e
1763
1764# s == n + 0  ->  s == n
1765def zero_out(-s: Nat, +n: Nat, e: {s == Nat.add(n, 0n) : Nat}) -> {s == n : Nat}:
1766  %add_zero_r(n) : {s == _ : Nat}
1767  e
1768
1769# reversing onto acc adds the lengths
1770def rev_len(+xs: List<&2, F32>, +acc: List<&2, F32>) -> {List.length(&2, F32, List.reverse.go(&2, F32, xs, acc)) == Nat.add(List.length(&2, F32, xs), List.length(&2, F32, acc)) : Nat}:
1771  match xs:
1772    case Nil{}:
1773      {==}
1774    case Con{h, t}:
1775      succ_out(List.length(&2, F32, List.reverse.go(&2, F32, t, h <> acc)), List.length(&2, F32, t), List.length(&2, F32, acc), rev_len(t, h <> acc))
1776
1777# appending adds the lengths
1778def app_len(+xs: List<&2, F32>, +ys: List<&2, F32>) -> {List.length(&2, F32, List.append(&2, F32, xs, ys)) == Nat.add(List.length(&2, F32, xs), List.length(&2, F32, ys)) : Nat}:
1779  match xs:
1780    case Nil{}:
1781      {==}
1782    case Con{h, t}:
1783      %app_len(t, ys) : {1n+List.length(&2, F32, List.append(&2, F32, t, ys)) == 1n+_ : Nat}
1784      {==}
1785
1786# the same count for the list of a whole Mat: to_list2 wraps read_l's answer, and ML.len counts it
1787def read_ml_len(-r: Nat, -c: Nat, +mleft: Nat, +idx: U32, +acc: List<&2, F32>, rc: Array<F32> & F32) -> {ML.len(r, c, Mat.to_list2(r, c, read_l(mleft, idx, acc, rc))) == Nat.add(1n+mleft, List.length(&2, F32, acc)) : Nat}:
1788  match mleft rc:
1789    case 0n Tuple{d, +v}:
1790      zero_out(ML.len(r, c, Mat.to_list2(r, c, read_l(0n, idx, acc, Tuple{d, v}))), 1n+List.length(&2, F32, acc), rev_len(v <> acc, Nil{}))
1791    case 1n+p Tuple{d, +v}:
1792      len_step(ML.len(r, c, Mat.to_list2(r, c, read_l(p, (idx + 1 : U32), v <> acc, Array.get(F32, d, (idx + 1 : U32))))), p, List.length(&2, F32, acc), read_ml_len(r, c, p, (idx + 1 : U32), v <> acc, Array.get(F32, d, (idx + 1 : U32))))
1793
1794# a leaf's list: read_l reads mleft + 1 numbers onto acc and reverses them. By induction on
1795# mleft: with 0 left it reads one number (rev_len counts the reversed list); with 1 + p left it
1796# reads one and recurses, and len_step moves the 1 out of the sum.
1797law leaf_len:
1798  for -r: Nat
1799  for -c: Nat
1800  for w: Array<F32>
1801  for +mleft: Nat
1802  for +idx: U32
1803  for +acc: List<&2, F32>
1804  for rc: Array<F32> & F32
1805  {bl_len(r, c, bv_g2(r, c, w, read_l(mleft, idx, acc, rc))) == Nat.add(1n+mleft, List.length(&2, F32, acc)) : Nat}
1806
1807def leaf_len(r, c, w, mleft, idx, acc, rc):
1808  match mleft rc:
1809    case 0n Tuple{d, +v}:
1810      zero_out(List.length(&2, F32, List.reverse.go(&2, F32, v <> acc, Nil{})), 1n+List.length(&2, F32, acc), rev_len(v <> acc, Nil{}))
1811    case 1n+p Tuple{d, +v}:
1812      len_step(bl_len(r, c, bv_g2(r, c, w, read_l(p, (idx + 1 : U32), v <> acc, Array.get(F32, d, (idx + 1 : U32))))), p, List.length(&2, F32, acc), leaf_len(r, c, w, p, (idx + 1 : U32), v <> acc, Array.get(F32, d, (idx + 1 : U32))))
1813
1814# whatever the gemm wrote, reading 1 + p rows back gives 1 + p numbers (G has one constructor, so
1815# matching g opens it, and leaf_len counts the list)
1816def g_len(-r: Nat, -c: Nat, +p: Nat, g: G) -> {bl_len(r, c, bv_g(r, c, p, g)) == 1n+p : Nat}:
1817  match g:
1818    case G{a, b, y}:
1819      zero_out(bl_len(r, c, bv_g2(r, c, b, read_l(p, 0, Nil{}, Array.get(F32, y, 0)))), 1n+p, leaf_len(r, c, b, p, 0, Nil{}, Array.get(F32, y, 0)))
1820
1821# LAW (band_len): the product of one band with `rows` rows gives exactly `rows` numbers.
1822# 0 rows give the empty list; 1 + p rows go through the gemm, and g_len counts what is read back.
1823law band_len:
1824  for -r: Nat
1825  for +c: Nat
1826  for +rows: Nat
1827  for x: Array<F32>
1828  for w: Array<F32>
1829  {bl_len(r, c, bv_gemm(r, c, rows, rows, x, w)) == rows : Nat}
1830
1831def band_len(r, c, rows, x, w):
1832  match rows:
1833    case 0n:
1834      {==}
1835    case 1n+p:
1836      g_len(r, c, p, gemm(1n, 1n+p, c, U32.from_nat(c), 1, 1, U32.from_nat(c), x, w, Array.new(F32, cap_depth(1n+p), 0.0)))
1837
1838# joining two halves of h1 and h2 numbers gives h1 + h2 numbers. (The statement for the whole tree,
1839# "Bands.matvec_l gives r numbers", would apply this to the results of the two recursive calls and
1840# to the induction hypotheses about the same calls; that uses each band's Array twice, which Bend's
1841# affine rules refuse. It stays open, covered by tests: NOTES.md, v3.1.)
1842def join_len(-r: Nat, -c: Nat, -h1: Nat, -h2: Nat, p: BL<half(r), c>, q: BL<Nat.sub(r, half(r)), c>, hp: {bl_len(half(r), c, p) == h1 : Nat}, hq: {bl_len(Nat.sub(r, half(r)), c, q) == h2 : Nat}) -> {bl_len(r, c, bv_join(r, c, p, q)) == Nat.add(h1, h2) : Nat}:
1843  match p q:
1844    case BL{bx, +yx} BL{by, +yy}:
1845      %hp : {List.length(&2, F32, List.append(&2, F32, yx, yy)) == Nat.add(_, h2) : Nat}
1846      %hq : {List.length(&2, F32, List.append(&2, F32, yx, yy)) == Nat.add(List.length(&2, F32, yx), _) : Nat}
1847      app_len(yx, yy)
1848
1849# Mat.to_list_count reads exactly n numbers, and n = 0 reads none.
1850def to_list_count_len(-r: Nat, -c: Nat, d: Array<F32>, +n: Nat) -> {ML.len(r, c, Mat.to_list_count(r, c, d, n)) == n : Nat}:
1851  match n:
1852    case 0n:
1853      {==}
1854    case 1n+p:
1855      zero_out(ML.len(r, c, Mat.to_list_count(r, c, d, 1n+p)), 1n+p, read_ml_len(r, c, p, 0, Nil{}, Array.get(F32, d, 0)))
1856
1857# LAW (to_list_len): Mat.to_list reads exactly r*c numbers, no more and no fewer.
1858law to_list_len:
1859  for +r: Nat
1860  for +c: Nat
1861  for m: Mat<r, c>
1862  {ML.len(r, c, Mat.to_list(r, c, m)) == Nat.mul(r, c) : Nat}
1863
1864def to_list_len(r, c, m):
1865  match m:
1866    case Mat{d}:
1867      to_list_count_len(r, c, d, Nat.mul(r, c))