main.bend
bend-ml-tensor-array@0.1.11.0 · 1,671 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@0.1.11.0/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
189
190def 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:
191 cp2(ac, d, m, k, sa, sb, bcol, c, Array.clone(F32, a), Array.clone(F32, b))
192
193# ---------------------------------------------------------------------
194# element-wise operations (in place, by index)
195# ---------------------------------------------------------------------
196
197# ---------------------------------------------------------------------
198# gemm: C[i][j] = sum_p A[i*arow + p*sa] * B[p*sb + j*bcol]
199# (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)
200# ---------------------------------------------------------------------
201
202# acc += A[ia + t*sa] * B[ib + t*sb] for k steps; ra, rb already carry the first pair
203
204type P2 is Type:
205 P2{a: Array<F32>, b: Array<F32>}
206
207type AL is Type:
208 AL{c: Array<F32>, l: List<&2, F32>}
209
210# list of m elements starting at idx (reads with Array.get, returning the array)
211
212def read_l(mleft: Nat, +idx: U32, acc: List<&2, F32>, rc: Array<F32> & F32) -> AL:
213 match mleft rc:
214 case 0n Tuple{c, +v}:
215 AL{c, List.reverse(&2, F32, v <> acc)}
216 case 1n+p Tuple{c, +v}:
217 read_l(p, (idx + 1 : U32), v <> acc, Array.get(F32, c, (idx + 1 : U32)))
218
219# writes the list starting at idx
220
221def addl(xs: List<&2, F32>, +idx: U32, rc: Array<F32> & F32) -> Array<F32>:
222 match xs rc:
223 case Nil{} Tuple{c, v}:
224 c
225 case Con{+h, t} Tuple{c, +v}:
226 addl(t, (idx + 1 : U32), Array.get(F32, Array.set(F32, c, idx, (v + h : F32)), (idx + 1 : U32)))
227
228# adds a bias (a list of m) to each of the n rows of c (n x m)
229
230def bias_rows(nleft: Nat, +i: U32, +mu: U32, +bl: List<&2, F32>, c: Array<F32>) -> Array<F32>:
231 match nleft:
232 case 0n:
233 c
234 case 1n+p:
235 bias_rows(p, (i + 1 : U32), mu, bl, addl(bl, (i * mu : U32), Array.get(F32, c, (i * mu : U32))))
236
237# in-place relu on the first n elements
238
239def relu_l(nleft: Nat, +t: U32, rc: Array<F32> & F32) -> Array<F32>:
240 match nleft rc:
241 case 0n Tuple{c, v}:
242 c
243 case 1n+p Tuple{c, +v}:
244 relu_l(p, (t + 1 : U32), Array.get(F32, Array.set(F32, c, t, F32.max(v, 0.0)), (t + 1 : U32)))
245
246def pos2(b: Bool) -> F32:
247 match b:
248 case True{}:
249 1.0
250 case False{}:
251 0.0
252
253def pos(+y: F32) -> F32:
254 pos2(F32.is_gt(y, 0.0))
255
256# d[t] *= (h[t] > 0): the relu gradient; returns both arrays
257
258def mask_l(nleft: Nat, +t: U32, rd: Array<F32> & F32, rh: Array<F32> & F32) -> P2:
259 match nleft rd rh:
260 case 0n Tuple{d, x} Tuple{h, y}:
261 P2{d, h}
262 case 1n+p Tuple{d, +x} Tuple{h, +y}:
263 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)))
264
265# w[t] -= lr * dw[t]
266
267def maxl2(xs: List<&2, F32>, cur: F32) -> F32:
268 match xs:
269 case Nil{}:
270 cur
271 case Con{+h, t}:
272 maxl2(t, F32.max(cur, h))
273
274def maxl(xs: List<&2, F32>) -> F32:
275 match xs:
276 case Nil{}:
277 0.0
278 case Con{+h, t}:
279 maxl2(t, h)
280
281def expl(xs: List<&2, F32>, +m: F32) -> List<&2, F32>:
282 match xs:
283 case Nil{}:
284 Nil{}
285 case Con{+h, t}:
286 F32.exp((h - m : F32)) <> expl(t, m)
287
288def suml(xs: List<&2, F32>, acc: F32) -> F32:
289 match xs:
290 case Nil{}:
291 acc
292 case Con{+h, t}:
293 suml(t, (acc + h : F32))
294
295def divl(xs: List<&2, F32>, +s: F32) -> List<&2, F32>:
296 match xs:
297 case Nil{}:
298 Nil{}
299 case Con{+h, t}:
300 (h / s : F32) <> divl(t, s)
301
302def softmax2(+es: List<&2, F32>) -> List<&2, F32>:
303 divl(es, suml(es, 0.0))
304
305def softmax(+xs: List<&2, F32>) -> List<&2, F32>:
306 softmax2(expl(xs, maxl(xs)))
307
308# p[label]
309
310def sgd_l(nleft: Nat, +t: U32, +lr: F32, rw: Array<F32> & F32, rd: Array<F32> & F32) -> P2:
311 match nleft rw rd:
312 case 0n Tuple{w, x} Tuple{d, y}:
313 P2{w, d}
314 case 1n+p Tuple{w, +x} Tuple{d, +y}:
315 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)))
316
317# ---------------------------------------------------------------------
318# softmax + cross-entropy per row (lists of 10 elements: negligible cost)
319# ---------------------------------------------------------------------
320
321def pick(ps: List<&2, F32>, n: Nat) -> F32:
322 match ps n:
323 case Con{+h, t} 0n:
324 h
325 case Con{h, t} 1n+q:
326 pick(t, q)
327 case Nil{} _:
328 0.0
329
330# (p - onehot(label)) / n
331
332def hit(b: Bool) -> F32:
333 match b:
334 case True{}:
335 1.0
336 case False{}:
337 0.0
338
339def grad_l(ps: List<&2, F32>, +label: Nat, +j: Nat, +nf: F32) -> List<&2, F32>:
340 match ps:
341 case Nil{}:
342 Nil{}
343 case Con{+h, t}:
344 (((h - hit(Nat.is_eq(label, j)) : F32) / nf) : F32) <> grad_l(t, label, 1n+j, nf)
345
346type L2 is Type:
347 L2{loss: F32, c: Array<F32>}
348
349def ce_row2(+ps: List<&2, F32>, +label: Nat, +nf: F32, +idx: U32, loss: F32, c: Array<F32>) -> L2:
350 L2{(loss - F32.log(F32.max(pick(ps, label), 0.000000000001)) : F32), write_l(grad_l(ps, label, 0n, nf), idx, c)}
351
352def ce_row(al: AL, +label: Nat, +nf: F32, +idx: U32, loss: F32) -> L2:
353 match al:
354 case AL{c, l}:
355 ce_row2(softmax(l), label, nf, idx, loss, c)
356
357# 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
358
359type Mat<-r: Nat, -c: Nat> is Type:
360 Mat{d: Array<F32>}
361
362def ce_rows(labels: List<&2, Nat>, +mm1: Nat, +mu: U32, +nf: F32, +i: U32, r: L2) -> L2:
363 match labels r:
364 case Nil{} L2{loss, c}:
365 L2{loss, c}
366 case Con{+y, t} L2{loss, c}:
367 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))
368
369# ---- helper: hits per row ----
370
371type CR is Type:
372 CR{c: Array<F32>, n: Nat}
373
374def gt_head(xs: List<&2, F32>, +v: F32) -> Bool:
375 match xs:
376 case Nil{}:
377 False{}
378 case Con{+h, t}:
379 F32.is_gt(h, v)
380
381def argmax2(xs: List<&2, F32>, g: Bool, +i: Nat, +bi: Nat, +bv: F32) -> Nat:
382 match xs g:
383 case Nil{} _:
384 bi
385 case Con{+h, +t} True{}:
386 argmax2(t, gt_head(t, h), 1n+i, i, h)
387 case Con{h, +t} False{}:
388 argmax2(t, gt_head(t, bv), 1n+i, bi, bv)
389
390def argmax(xs: List<&2, F32>) -> Nat:
391 match xs:
392 case Nil{}:
393 0n
394 case Con{+h, +t}:
395 argmax2(t, gt_head(t, h), 1n, 0n, h)
396
397def one(b: Bool) -> Nat:
398 match b:
399 case True{}:
400 1n
401 case False{}:
402 0n
403
404def count_rows3(+y: Nat, n: Nat, al: AL) -> CR:
405 match al:
406 case AL{c, l}:
407 CR{c, Nat.add(n, one(Nat.is_eq(argmax(l), y)))}
408
409def count_rows(labels: List<&2, Nat>, +mm1: Nat, +mu: U32, +i: U32, r: CR) -> CR:
410 match labels r:
411 case Nil{} CR{c, n}:
412 CR{c, n}
413 case Con{+y, t} CR{c, n}:
414 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)))))
415
416
417# =====================================================================
418# Typed API: Mat<r, c> (the dimensions enter the type)
419# =====================================================================
420
421# =====================================================================
422# Array capacity: a PROVED depth for the number of slots
423# =====================================================================
424#
425# cap_depth(n) is the d such that an Array with 2^d slots holds n elements. It is computed by
426# structural recursion on n (not with U32.log2) so that the law cap_ok below can prove
427# `n <= 2^cap_depth(n)`. What stays trusted is only that Array.new(T, d, v) allocates 2^d slots.
428
429# a type chosen by a Bool: rewriting through it refutes True == False
430def BD(b: Bool, t: Type, f: Type) -> Type:
431 match b:
432 case True{}:
433 t
434 case False{}:
435 f
436
437# a <= b, by direct recursion
438def leb(a: Nat, b: Nat) -> Bool:
439 match a b:
440 case 0n _:
441 True{}
442 case 1n+p 0n:
443 False{}
444 case 1n+p 1n+q:
445 leb(p, q)
446
447def pow2(d: Nat) -> Nat:
448 match d:
449 case 0n:
450 1n
451 case 1n++q:
452 Nat.add(pow2(q), pow2(q))
453
454# smallest power of two >= n, by structural recursion on n: returns (depth, 2^depth)
455def cap_pick(ok: Bool, d: Nat, +p: Nat) -> Nat & Nat:
456 match ok:
457 case True{}:
458 (d, p)
459 case False{}:
460 (1n+d, Nat.add(p, p))
461
462def cap_step(r: Nat & Nat, +n: Nat) -> Nat & Nat:
463 match r:
464 case Tuple{+d, +p}:
465 cap_pick(leb(n, p), d, p)
466
467def cap_pair(n: Nat) -> Nat & Nat:
468 match n:
469 case 0n:
470 (0n, 1n)
471 case 1n++m:
472 cap_step(cap_pair(m), 1n+m)
473
474def fst_nat(r: Nat & Nat) -> Nat:
475 match r:
476 case Tuple{d, p}:
477 d
478
479def snd_nat(r: Nat & Nat) -> Nat:
480 match r:
481 case Tuple{d, p}:
482 p
483
484# 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
485# 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
486# suffices (2^n >= n); if it ran out, the answer would be cap_pair's, so the law below needs no arithmetic about how
487# many doublings there are. ok is leb(n, p), computed by the caller (match only inspects a parameter).
488def cap_dbl(f: Nat, ok: Bool, +n: Nat, +d: Nat, +p: Nat) -> Nat & Nat:
489 match f ok:
490 case _ True{}:
491 (d, p)
492 case 0n False{}:
493 cap_pair(n)
494 case 1n+g False{}:
495 cap_dbl(g, leb(n, Nat.add(p, p)), n, 1n+d, Nat.add(p, p))
496
497# the depth d such that an Array with 2^d slots holds n elements (the smallest such d)
498def cap_depth(+n: Nat) -> Nat:
499 fst_nat(cap_dbl(n, leb(n, 1n), n, 0n, 1n))
500
501# ---- lemmas ----
502
503# a <= b -> a <= 1 + b
504def leb_succ(a: Nat, b: Nat, e: {True{} == leb(a, b) : Bool}) -> {True{} == leb(a, 1n+b) : Bool}:
505 match a b:
506 case 0n _:
507 {==}
508 case 1n+p 0n:
509 %e : BD(_, Unit, {True{} == leb(1n+p, 1n) : Bool})
510 Unit{}
511 case 1n+p 1n+q:
512 leb_succ(p, q, e)
513
514# a <= b -> a <= c + b
515def leb_up_l(+a: Nat, +b: Nat, c: Nat, +e: {True{} == leb(a, b) : Bool}) -> {True{} == leb(a, Nat.add(c, b)) : Bool}:
516 match c:
517 case 0n:
518 e
519 case 1n++c2:
520 leb_succ(a, Nat.add(c2, b), leb_up_l(a, b, c2, e))
521
522# a <= b -> a <= b + c
523def leb_up(a: Nat, b: Nat, c: Nat, e: {True{} == leb(a, b) : Bool}) -> {True{} == leb(a, Nat.add(b, c)) : Bool}:
524 match a b:
525 case 0n _:
526 {==}
527 case 1n+p 0n:
528 %e : BD(_, Unit, {True{} == leb(1n+p, Nat.add(0n, c)) : Bool})
529 Unit{}
530 case 1n+p 1n+q:
531 leb_up(p, q, c, e)
532
533# a <= b and c <= d -> a + c <= b + d
534def 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}:
535 match a b:
536 case 0n _:
537 leb_up_l(c, d, b, e2)
538 case 1n+p 0n:
539 %e1 : BD(_, Unit, {True{} == leb(Nat.add(1n+p, c), Nat.add(0n, d)) : Bool})
540 Unit{}
541 case 1n+p 1n+q:
542 leb_add(p, q, c, d, e1, e2)
543
544# 1 <= 2^d
545def pow2_pos(d: Nat) -> {True{} == leb(1n, pow2(d)) : Bool}:
546 match d:
547 case 0n:
548 {==}
549 case 1n++q:
550 leb_up(1n, pow2(q), pow2(q), pow2_pos(q))
551
552# 1 <= p when p = 2^d
553def pos_p(d: Nat, p: Nat, i1: {p == pow2(d) : Nat}) -> {True{} == leb(1n, p) : Bool}:
554 %Equal.sym(Nat, p, pow2(d), i1) : {True{} == leb(1n, _) : Bool}
555 pow2_pos(d)
556
557# p + p == 2^(1+d) when p == 2^d
558def step_eq(d: Nat, p: Nat, i1: {p == pow2(d) : Nat}) -> {Nat.add(p, p) == pow2(1n+d) : Nat}:
559 %i1 : {Nat.add(p, p) == Nat.add(_, _) : Nat}
560 {==}
561
562# one step: given the invariant for m, the invariant for 1+m holds whichever way the comparison goes
563def 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}:
564 match ok:
565 case True{}:
566 (i1, e)
567 case False{}:
568 (step_eq(d, p, i1), leb_add(1n, p, m, p, pos_p(d, p, i1), i2))
569
570
571# the invariant for cap_step on a pair, opened
572def 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}:
573 match r:
574 case Tuple{+d, +p}:
575 step_ok(leb(1n+m, p), d, p, m, {==}, i1, i2)
576
577def 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}:
578 match ih:
579 case Tuple{i1, i2}:
580 step_pair_ok(m, cap_pair(m), i1, i2)
581
582# the invariant of cap_pair: the second component is 2^(first) and is >= n
583def 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}:
584 match n:
585 case 0n:
586 ({==}, {==})
587 case 1n++m:
588 cap_succ(m, cap_inv(m))
589
590# the invariant of cap_dbl: whichever way it ends (n <= p found, or the fuel out and cap_pair's answer), the second
591# component is 2^(first) and is >= n
592def dbl_inv(f: Nat, ok: Bool, +n: Nat, +d: Nat, +p: Nat, e: {ok == leb(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}:
593 match f ok:
594 case _ True{}:
595 (i1, e)
596 case 0n False{}:
597 cap_inv(n)
598 case 1n+g False{}:
599 dbl_inv(g, leb(n, Nat.add(p, p)), n, 1n+d, Nat.add(p, p), {==}, step_eq(d, p, i1))
600
601def dbl_final(n: Nat, pr: {snd_nat(cap_dbl(n, leb(n, 1n), n, 0n, 1n)) == pow2(fst_nat(cap_dbl(n, leb(n, 1n), n, 0n, 1n))) : Nat} & {True{} == leb(n, snd_nat(cap_dbl(n, leb(n, 1n), n, 0n, 1n))) : Bool}) -> {True{} == leb(n, pow2(cap_depth(n))) : Bool}:
602 match pr:
603 case Tuple{i1, i2}:
604 %i1 : {True{} == leb(n, _) : Bool}
605 i2
606
607# LAW (cap_ok): an Array with 2^cap_depth(n) slots always has room for n elements.
608law cap_ok:
609 for +n: Nat
610 {True{} == leb(n, pow2(cap_depth(n))) : Bool}
611
612def cap_ok(n):
613 dbl_final(n, dbl_inv(n, leb(n, 1n), n, 0n, 1n, {==}, {==}))
614
615# r x c of zeros, with the exact capacity (smallest power of 2)
616
617def Mat.zeros(+r: Nat, +c: Nat) -> Mat<r, c>:
618 Mat{Array.new(F32, cap_depth(Nat.mul(r, c)), 0.0)}
619
620def Mat.fill(+r: Nat, +c: Nat, +v: F32) -> Mat<r, c>:
621 Mat{Array.new(F32, cap_depth(Nat.mul(r, c)), v)}
622
623def fill_list(xs: List<&2, F32>, +i: U32, a: Array<F32>) -> Array<F32>:
624 match xs:
625 case Nil{}:
626 a
627 case Con{h, t}:
628 fill_list(t, (i + 1 : U32), Array.set(F32, a, i, h))
629
630def len_is(xs: List<&2, F32>, n: Nat) -> Bool:
631 match xs n:
632 case Nil{} 0n:
633 True{}
634 case Con{h, t} 1n+p:
635 len_is(t, p)
636 case _ _:
637 False{}
638
639def Mat.of_list2(-r: Nat, -c: Nat, +rr: Nat, +cc: Nat, ok: Bool, xs: List<&2, F32>) -> Maybe<&1, Mat<r, c>>:
640 match ok:
641 case True{}:
642 Some{Mat{fill_list(xs, 0, Array.new(F32, cap_depth(Nat.mul(rr, cc)), 0.0))}}
643 case False{}:
644 None{}
645
646# Mat<r, c> from r*c numbers in row order; None if the size does not match
647
648def Mat.of_list(+r: Nat, +c: Nat, +xs: List<&2, F32>) -> Maybe<&1, Mat<r, c>>:
649 Mat.of_list2(r, c, r, c, len_is(xs, Nat.mul(r, c)), xs)
650
651# reads the first n numbers, returning the matrix
652
653# Mat<r, c> without checking the list size (a short list leaves the rest at zero, a long one is cut
654# by the capacity): only for when the size is known by construction. Prefer Mat.of_list.
655
656def Mat.from_list(+r: Nat, +c: Nat, xs: List<&2, F32>) -> Mat<r, c>:
657 Mat{fill_list(xs, 0, Array.new(F32, cap_depth(Nat.mul(r, c)), 0.0))}
658
659# writes the numbers of xs into m starting at flat index i (row-major); numbers past the
660# capacity are dropped. For loading a large matrix block by block without building one big list.
661def Mat.fill_at(-r: Nat, -c: Nat, +i: U32, xs: List<&2, F32>, m: Mat<r, c>) -> Mat<r, c>:
662 match m:
663 case Mat{d}:
664 Mat{fill_list(xs, i, d)}
665
666type ML<-r: Nat, -c: Nat> is Type:
667 ML{m: Mat<r, c>, l: List<&2, F32>}
668
669def Mat.to_list2(-r: Nat, -c: Nat, al: AL) -> ML<r, c>:
670 match al:
671 case AL{d, l}:
672 ML{Mat{d}, l}
673
674# two independent copies
675
676def Mat.to_list(+r: Nat, +c: Nat, m: Mat<r, c>) -> ML<r, c>:
677 match m:
678 case Mat{d}:
679 Mat.to_list2(r, c, read_l(Nat.sub(Nat.mul(r, c), 1n), 0, Nil{}, Array.get(F32, d, 0)))
680
681type MM2<-r: Nat, -c: Nat> is Type:
682 MM2{a: Mat<r, c>, b: Mat<r, c>}
683
684def Mat.clone2(-r: Nat, -c: Nat, p: Array<F32> & Array<F32>) -> MM2<r, c>:
685 match p:
686 case Tuple{x, y}:
687 MM2{Mat{x}, Mat{y}}
688
689# ---- products: C (n x m) = A · B, with 2^par blocks of rows in parallel ----
690
691# C = A(n x k) · B(k x m)
692
693def Mat.clone(-r: Nat, -c: Nat, m: Mat<r, c>) -> MM2<r, c>:
694 match m:
695 case Mat{d}:
696 Mat.clone2(r, c, Array.clone(F32, d))
697
698type MMul<-n: Nat, -k: Nat, -m: Nat> is Type:
699 MMul{a: Mat<n, k>, b: Mat<k, m>, c: Mat<n, m>}
700
701# C = A(n x k) · Bᵀ, with B stored m x k
702
703type MMulNT<-n: Nat, -k: Nat, -m: Nat> is Type:
704 MMulNT{a: Mat<n, k>, b: Mat<m, k>, c: Mat<n, m>}
705
706# C = Aᵀ · B, with A stored k x n
707
708type MMulTN<-n: Nat, -k: Nat, -m: Nat> is Type:
709 MMulTN{a: Mat<k, n>, b: Mat<k, m>, c: Mat<n, m>}
710
711# ---------------------------------------------------------------------
712# 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
713# 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
714# 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,
715# in order, and the four sums are combined as dotr combines them after k steps (its rotation depends on k mod 4). The
716# padding adds +0 to a sum that starts at +0, which changes nothing, so the results are the same bits as the scalar
717# loop's. Products with one row (matrix · vector) keep the scalar loop: packing B would cost as much as the product.
718# ---------------------------------------------------------------------
719
720type T8 is Data:
721 T8{v0: F32, v1: F32, v2: F32, v3: F32, v4: F32, v5: F32, v6: F32, v7: F32}
722
723def t8_zero() -> T8:
724 T8{0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0}
725
726def pick0(b: Bool, +v: F32) -> F32:
727 match b:
728 case True{}:
729 v
730 case False{}:
731 0.0
732
733type PK is Type:
734 PK{a: Array<F32>, t: Array<T8>}
735
736# 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
737def 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:
738 match r:
739 case Tuple{a, +v}:
740 PK{a, Array.set(T8, t, o, T8{v0, v1, v2, v3, v4, v5, v6, pick0(U32.is_lt((q0 + 7 : U32), lim), v)})}
741
742def 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:
743 match r:
744 case Tuple{a, +v}:
745 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)))
746
747def 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:
748 match r:
749 case Tuple{a, +v}:
750 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)))
751
752def 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:
753 match r:
754 case Tuple{a, +v}:
755 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)))
756
757def 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:
758 match r:
759 case Tuple{a, +v}:
760 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)))
761
762def pk_g2(+v0: F32, +v1: F32, +q0: U32, +lim: U32, +idx: U32, +st: U32, +o: U32, t: Array<T8>, r: Array<F32> & F32) -> PK:
763 match r:
764 case Tuple{a, +v}:
765 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)))
766
767def pk_g1(+v0: F32, +q0: U32, +lim: U32, +idx: U32, +st: U32, +o: U32, t: Array<T8>, r: Array<F32> & F32) -> PK:
768 match r:
769 case Tuple{a, +v}:
770 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)))
771
772def pk_g0(+q0: U32, +lim: U32, +idx: U32, +st: U32, +o: U32, t: Array<T8>, r: Array<F32> & F32) -> PK:
773 match r:
774 case Tuple{a, +v}:
775 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)))
776
777# the cells of one line, cell o.. of t; q0 counts the line's numbers before the cell
778def pk_line(tl: Nat, +q0: U32, +lim: U32, +base: U32, +st: U32, +o: U32, r: PK) -> PK:
779 match tl r:
780 case 0n r:
781 r
782 case 1n+q PK{a, t}:
783 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)))
784
785# lines count lines; source line i starts at i * lstep (in a), and goes to cell row o (cells o * tku .. in t)
786def pk_lines(cnt: Nat, +tk: Nat, +tku: U32, +ku: U32, +i: U32, +o: U32, +lstep: U32, +st: U32, r: PK) -> PK:
787 match cnt:
788 case 0n:
789 r
790 case 1n+p:
791 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))
792
793def pk_pack(+cnt: Nat, +tk: Nat, +k: Nat, +lstep: U32, +st: U32, a: Array<F32>) -> PK:
794 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())})
795
796# 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,
797# already read. Product q of a cell goes to sum q mod 4, the first four before the last four, as dotr adds them.
798def 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:
799 match tk ra rb:
800 case 0n Tuple{a, x} Tuple{b, y}:
801 TD{a, b, s0, s1, s2, s3}
802 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}}:
803 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)))
804
805type TD is Type:
806 TD{a: Array<T8>, b: Array<T8>, s0: F32, s1: F32, s2: F32, s3: F32}
807
808# dotr's final sum after k products: the sums sit rotated by k mod 4 when its loop ends
809def t8_comb(+rot: Nat, +s0: F32, +s1: F32, +s2: F32, +s3: F32) -> F32:
810 match rot:
811 case 0n:
812 ((s0 + s1 : F32) + (s2 + s3 : F32) : F32)
813 case 1n:
814 ((s1 + s2 : F32) + (s3 + s0 : F32) : F32)
815 case 2n:
816 ((s2 + s3 : F32) + (s0 + s1 : F32) : F32)
817 case _:
818 ((s3 + s0 : F32) + (s1 + s2 : F32) : F32)
819
820type TR is Type:
821 TR{a: Array<T8>, b: Array<T8>, x: F32}
822
823def t8_fin(+rot: Nat, d: TD) -> TR:
824 match d:
825 case TD{a, b, +s0, +s1, +s2, +s3}:
826 TR{a, b, t8_comb(rot, s0, s1, s2, s3)}
827
828def t8_cell(+tk: Nat, +tku: U32, +rot: Nat, +i: U32, +j: U32, a: Array<T8>, b: Array<T8>) -> TR:
829 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))))
830
831type TG is Type:
832 TG{a: Array<T8>, b: Array<T8>, c: Array<F32>}
833
834def t8_set(+ac: Bool, +idx: U32, c: Array<F32>, r: TR) -> TG:
835 match r:
836 case TR{a, b, +x}:
837 TG{a, b, cwr(ac, idx, x, c)}
838
839def t8_cols(+ac: Bool, mleft: Nat, +tk: Nat, +tku: U32, +rot: Nat, +i: U32, +j: U32, +cidx: U32, g: TG) -> TG:
840 match mleft g:
841 case 0n g:
842 g
843 case 1n+p TG{a, b, c}:
844 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)))
845
846def t8_rows(+ac: Bool, nleft: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, +i: U32, +mu: U32, g: TG) -> TG:
847 match nleft:
848 case 0n:
849 g
850 case 1n+p:
851 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))
852
853# rows i0 .. i0+cnt-1 of C as a list, in order (the parallel version's leaves)
854type TL is Type:
855 TL{a: Array<T8>, b: Array<T8>, out: List<&2, F32>}
856
857# cells j.. of row i, m - j of them; r is cell j, already computed
858def t8_lcols(mleft: Nat, +tk: Nat, +tku: U32, +rot: Nat, +i: U32, +j: U32, out: List<&2, F32>, r: TR) -> TL:
859 match mleft r:
860 case 0n TR{a, b, +x}:
861 TL{a, b, x <> out}
862 case 1n+p TR{a, b, +x}:
863 t8_lcols(p, tk, tku, rot, i, (j + 1 : U32), x <> out, t8_cell(tk, tku, rot, i, (j + 1 : U32), a, b))
864
865def t8_lrows(nleft: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, +i: U32, l: TL) -> TL:
866 match nleft l:
867 case 0n l:
868 l
869 case 1n+p TL{a, b, out}:
870 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)))
871
872def t8_lfin(l: TL) -> List<&2, F32>:
873 match l:
874 case TL{a, b, out}:
875 List.reverse(&2, F32, out)
876
877type PKK is Type:
878 PKK{a: Array<F32>, b: Array<F32>, ta: Array<T8>, tb: Array<T8>}
879
880def t8_pk2(pa: PK, pb: PK) -> PKK:
881 match pa pb:
882 case PK{a, ta} PK{b, tb}:
883 PKK{a, b, ta, tb}
884
885def t8_g(a: Array<F32>, b: Array<F32>, g: TG) -> G:
886 match g:
887 case TG{ta, tb, c}:
888 G{a, b, c}
889
890def t8_out_seq(+ac: Bool, +n: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, c: Array<F32>, p: PKK) -> G:
891 match p:
892 case PKK{a, b, ta, tb}:
893 t8_g(a, b, t8_rows(ac, n, m, tk, tku, rot, 0, U32.from_nat(m), TG{ta, tb, c}))
894
895# ---- the parallel path (0.1.9.0): the larger operand is packed into one piece per block, so a block owns its piece;
896# the other operand is copied once per node of the fork tree (2^d - 1 copies), instead of both at every node ----
897
898type PT is Type:
899 PLeaf{t: Array<T8>}
900 PNode{l: PT, r: PT}
901
902type PKT is Type:
903 PKT{a: Array<F32>, t: PT}
904
905def pkt_leaf(r: PK) -> PKT:
906 match r:
907 case PK{a, t}:
908 PKT{a, PLeaf{t}}
909
910def pkt_then(r: PKT, k: Array<F32> -> PT -> PKT) -> PKT:
911 match r:
912 case PKT{a, t}:
913 k(a, t)
914
915# lines i .. i+cnt-1 of a, packed into 2^d pieces (halves of the line count, as the fork tree splits them)
916def pk_tree(+d: Nat, +cnt: Nat, +i: U32, +tk: Nat, +tku: U32, +ku: U32, +lstep: U32, +st: U32, a: Array<F32>) -> PKT:
917 match d:
918 case 0n:
919 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())}))
920 case 1n+q:
921 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}}))
922
923def t8_with2(-R: Type, p: Array<T8> & Array<T8>, k: Array<T8> -> Array<T8> -> R) -> R:
924 match p:
925 case Tuple{x, y}:
926 k(x, y)
927
928# split by rows of C: the pieces are A's rows, B is shared; the result is C's rows in order
929def t8_rpar(t: PT, +cnt: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, b: Array<T8>) -> List<&2, F32>:
930 match t:
931 case PLeaf{a}:
932 t8_lfin(t8_lrows(cnt, m, tk, tku, rot, 0, TL{a, b, Nil{}}))
933 case PNode{l, r}:
934 t8_with2(List<&2, F32>, Array.clone(T8, b), b1 => b2 => (
935 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)
936 List.append(&2, F32, x, y)))
937
938# a block of C's columns j0 .. j0+w-1 for every row, row by row
939type CB is Data:
940 CB{j0: U32, w: Nat, xs: List<&2, F32>}
941
942# split by columns of C: the pieces are B's columns, A is shared
943def t8_cpar(t: PT, +cnt: Nat, +j0: U32, +n: Nat, +tk: Nat, +tku: U32, +rot: Nat, a: Array<T8>) -> List<&2, CB>:
944 match t:
945 case PLeaf{b}:
946 [CB{j0, cnt, t8_lfin(t8_lrows(n, cnt, tk, tku, rot, 0, TL{a, b, Nil{}}))}]
947 case PNode{l, r}:
948 t8_with2(List<&2, CB>, Array.clone(T8, a), a1 => a2 => (
949 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)
950 List.append(&2, CB, x, y)))
951
952type RW is Type:
953 RW{c: Array<F32>, xs: List<&2, F32>}
954
955# w numbers of xs into c from idx; what is left of xs comes back
956def cb_row(+ac: Bool, w: Nat, +idx: U32, c: Array<F32>, xs: List<&2, F32>) -> RW:
957 match w xs:
958 case 1n+p Con{+h, t}:
959 cb_row(ac, p, (idx + 1 : U32), cwr(ac, idx, h, c), t)
960 case _ _:
961 RW{c, xs}
962
963def cb_rows(+ac: Bool, n: Nat, +i: U32, +j0: U32, +w: Nat, +mu: U32, r: RW) -> Array<F32>:
964 match n r:
965 case 0n RW{c, xs}:
966 c
967 case 1n+p RW{c, xs}:
968 cb_rows(ac, p, (i + 1 : U32), j0, w, mu, cb_row(ac, w, ((i * mu : U32) + j0 : U32), c, xs))
969
970def cb_write(+ac: Bool, bs: List<&2, CB>, +n: Nat, +mu: U32, c: Array<F32>) -> Array<F32>:
971 match bs:
972 case Nil{}:
973 c
974 case Con{CB{+j0, +w, xs}, t}:
975 cb_write(ac, t, n, mu, cb_rows(ac, n, 0, j0, w, mu, RW{c, xs}))
976
977def t8_pr2(+ac: Bool, +n: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, c: Array<F32>, ra: PKT, rb: PK) -> G:
978 match ra rb:
979 case PKT{a, t} PK{b, tb}:
980 G{a, b, write_la(ac, t8_rpar(t, n, m, tk, tku, rot, tb), 0, c)}
981
982def t8_pc2(+ac: Bool, +n: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, c: Array<F32>, ra: PK, rb: PKT) -> G:
983 match ra rb:
984 case PK{a, ta} PKT{b, t}:
985 G{a, b, cb_write(ac, t8_cpar(t, m, 0, n, tk, tku, rot, ta), n, U32.from_nat(m), c)}
986
987# 2^d blocks along the larger of n (rows of C, pieces of A) and m (columns of C, pieces of B)
988def 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:
989 match byrows:
990 case True{}:
991 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))
992 case False{}:
993 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))
994
995def 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:
996 match d:
997 case 0n:
998 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)))
999 case 1n+q:
1000 t8_par2(ac, Nat.is_le(m, n), 1n+q, n, m, k, tk, tku, rot, arow, sa, sb, bcol, a, b, c)
1001
1002# 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
1003# and B come back intact
1004def 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:
1005 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)
1006
1007def 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:
1008 gemm_t8_ac(False{}, d, n, m, k, arow, sa, sb, bcol, a, b, c)
1009
1010# one row (matrix · vector): the scalar loop, by columns in parallel when d > 0; two or more rows: cells of 8
1011def 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:
1012 match one d:
1013 case True{} 0n:
1014 gemm_ac(ac, n, m, k, arow, sa, sb, bcol, a, b, c)
1015 case True{} 1n+q:
1016 gemm_par_cols(ac, 1n+q, m, k, sa, sb, bcol, a, b, c)
1017 case False{} d:
1018 gemm_t8_ac(ac, d, n, m, k, arow, sa, sb, bcol, a, b, c)
1019
1020def 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:
1021 gemm_any3(ac, Nat.is_le(n, 1n), d, n, m, k, arow, sa, sb, bcol, a, b, c)
1022
1023def 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:
1024 gemm_any2(ac, d, n, m, k, arow, sa, sb, bcol, a, b, c)
1025
1026def mmul_fin(-n: Nat, -k: Nat, -m: Nat, g: G) -> MMul<n, k, m>:
1027 match g:
1028 case G{a, b, c}:
1029 MMul{Mat{a}, Mat{b}, Mat{c}}
1030
1031def mmul_nt_fin(-n: Nat, -k: Nat, -m: Nat, g: G) -> MMulNT<n, k, m>:
1032 match g:
1033 case G{a, b, c}:
1034 MMulNT{Mat{a}, Mat{b}, Mat{c}}
1035
1036def mmul_tn_fin(-n: Nat, -k: Nat, -m: Nat, g: G) -> MMulTN<n, k, m>:
1037 match g:
1038 case G{a, b, c}:
1039 MMulTN{Mat{a}, Mat{b}, Mat{c}}
1040
1041# ---- 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 ----
1042#
1043# 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
1044# sums at least 64 of them; otherwise it runs sequentially. Measured on 16 threads (bend2-lab, B8 sweep): below 2^23 the
1045# split costs more than it saves, and with a short inner dimension (a weight gradient over a few tokens) every split
1046# 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
1047# 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
1048# depth (tests/t8.bend, tests/acc.bend, tests/split.bend).
1049def split_pick(big: Bool, d: Nat) -> Nat:
1050 match big:
1051 case True{}:
1052 d
1053 case False{}:
1054 0n
1055
1056def split_depth(d: Nat, +n: Nat, +m: Nat, +k: Nat) -> Nat:
1057 split_pick(Bool.and(Nat.is_le(64n, k), Nat.is_le(8388608n, Nat.mul(Nat.mul(n, m), k))), d)
1058
1059def mm_a(+n: Nat, +k: Nat, +m: Nat, d: Nat, a: Array<F32>, b: Array<F32>) -> MMul<n, k, m>:
1060 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)))
1061
1062# C = A · B (n x k) · (k x m); par = log2 of the most parallel blocks (0 = sequential); split_depth picks 0 or par
1063
1064def Mat.matmul(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: Mat<n, k>, b: Mat<k, m>) -> MMul<n, k, m>:
1065 match a b:
1066 case Mat{x} Mat{y}:
1067 mm_a(n, k, m, split_depth(par, n, m, k), x, y)
1068
1069# 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
1070# 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),
1071# without allocating the product or walking it a second time.
1072
1073def 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>:
1074 match a b c:
1075 case Mat{x} Mat{y} Mat{z}:
1076 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))
1077
1078def mm_nt(+n: Nat, +k: Nat, +m: Nat, d: Nat, a: Array<F32>, b: Array<F32>) -> MMulNT<n, k, m>:
1079 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)))
1080
1081# C = A · Bᵀ (n x k) · (m x k)ᵀ: for weights stored with one row per output
1082
1083def Mat.matmul_nt(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: Mat<n, k>, b: Mat<m, k>) -> MMulNT<n, k, m>:
1084 match a b:
1085 case Mat{x} Mat{y}:
1086 mm_nt(n, k, m, split_depth(par, n, m, k), x, y)
1087
1088# C + A · Bᵀ, into C (n x m)
1089
1090def 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>:
1091 match a b c:
1092 case Mat{x} Mat{y} Mat{z}:
1093 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))
1094
1095def mm_tn(+n: Nat, +k: Nat, +m: Nat, d: Nat, a: Array<F32>, b: Array<F32>) -> MMulTN<n, k, m>:
1096 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)))
1097
1098# C = Aᵀ · B (k x n)ᵀ · (k x m): weight gradient, X^T · dY
1099
1100def Mat.matmul_tn(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: Mat<k, n>, b: Mat<k, m>) -> MMulTN<n, k, m>:
1101 match a b:
1102 case Mat{x} Mat{y}:
1103 mm_tn(n, k, m, split_depth(par, n, m, k), x, y)
1104
1105# C + Aᵀ · B, into C (n x m): a weight gradient added into the gradient it accumulates, dW += Xᵀ · dY
1106
1107def 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>:
1108 match a b c:
1109 case Mat{x} Mat{y} Mat{z}:
1110 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))
1111
1112# ---- element-wise and training operations ----
1113
1114# adds the bias (one row, m numbers) to each row of C (n x m)
1115
1116type MBias<-n: Nat, -m: Nat> is Type:
1117 MBias{c: Mat<n, m>, b: Mat<1n, m>}
1118
1119def add_row2(-n: Nat, -m: Nat, +nn: Nat, +mm: Nat, cd: Array<F32>, al: AL) -> MBias<n, m>:
1120 match al:
1121 case AL{bd, bl}:
1122 MBias{Mat{bias_rows(nn, 0, U32.from_nat(mm), bl, cd)}, Mat{bd}}
1123
1124def Mat.add_row(+n: Nat, +m: Nat, c: Mat<n, m>, b: Mat<1n, m>) -> MBias<n, m>:
1125 match c b:
1126 case Mat{cd} Mat{bd}:
1127 add_row2(n, m, n, m, cd, read_l(Nat.sub(m, 1n), 0, Nil{}, Array.get(F32, bd, 0)))
1128
1129# pair of matrices of the same shape (the result of an operation that reads two operands)
1130
1131type MPair<-r: Nat, -c: Nat> is Type:
1132 MPair{a: Mat<r, c>, b: Mat<r, c>}
1133
1134def mpair_fin(-r: Nat, -c: Nat, p: P2) -> MPair<r, c>:
1135 match p:
1136 case P2{a, b}:
1137 MPair{Mat{a}, Mat{b}}
1138
1139def relu_fin(-r: Nat, -c: Nat, d: Array<F32>) -> Mat<r, c>:
1140 Mat{d}
1141
1142def Mat.relu(+r: Nat, +c: Nat, a: Mat<r, c>) -> Mat<r, c>:
1143 match a:
1144 case Mat{d}:
1145 relu_fin(r, c, relu_l(Nat.mul(r, c), 0, Array.get(F32, d, 0)))
1146
1147# (dy * (h > 0), h): the gradient that passes through the relu
1148
1149def Mat.relu_bwd(+r: Nat, +c: Nat, dy: Mat<r, c>, h: Mat<r, c>) -> MPair<r, c>:
1150 match dy h:
1151 case Mat{d} Mat{x}:
1152 mpair_fin(r, c, mask_l(Nat.mul(r, c), 0, Array.get(F32, d, 0), Array.get(F32, x, 0)))
1153
1154# (w - lr * dw, dw): one SGD step
1155
1156def Mat.sgd(+r: Nat, +c: Nat, +lr: F32, w: Mat<r, c>, dw: Mat<r, c>) -> MPair<r, c>:
1157 match w dw:
1158 case Mat{x} Mat{y}:
1159 mpair_fin(r, c, sgd_l(Nat.mul(r, c), 0, lr, Array.get(F32, x, 0), Array.get(F32, y, 0)))
1160
1161# (a + b, b): element-wise sum (a - (-1)*b is exactly a + b)
1162
1163def Mat.add(+r: Nat, +c: Nat, a: Mat<r, c>, b: Mat<r, c>) -> MPair<r, c>:
1164 match a b:
1165 case Mat{x} Mat{y}:
1166 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)))
1167
1168# sum of each column (n x m -> 1 x m), returning the matrix
1169
1170type MColSum<-n: Nat, -m: Nat> is Type:
1171 MColSum{a: Mat<n, m>, s: Mat<1n, m>}
1172
1173def colsum_fin(-n: Nat, -m: Nat, g: G) -> MColSum<n, m>:
1174 match g:
1175 case G{ones, a, s}:
1176 MColSum{Mat{a}, Mat{s}}
1177
1178def Mat.col_sums(+n: Nat, +m: Nat, a: Mat<n, m>) -> MColSum<n, m>:
1179 match a:
1180 case Mat{d}:
1181 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)))
1182
1183# s + the sum of each column, into s (1 x m): a bias gradient added into its accumulator, with the bits of
1184# Mat.add(s, Mat.col_sums(a)'s s)
1185
1186def Mat.col_sums_acc(+n: Nat, +m: Nat, a: Mat<n, m>, s: Mat<1n, m>) -> MColSum<n, m>:
1187 match a s:
1188 case Mat{d} Mat{z}:
1189 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))
1190
1191# mean cross-entropy and its gradient on the logits: (softmax - one-hot) / n.
1192# labels has one entry per row (n); the gradient replaces the logits.
1193
1194type MCE<-n: Nat, -c: Nat> is Type:
1195 MCE{loss: F32, grad: Mat<n, c>}
1196
1197def ce_fin(-n: Nat, -c: Nat, +nf: F32, r: L2) -> MCE<n, c>:
1198 match r:
1199 case L2{loss, d}:
1200 MCE{(loss / nf : F32), Mat{d}}
1201
1202def Mat.softmax_ce(+n: Nat, +c: Nat, z: Mat<n, c>, labels: List<&2, Nat>) -> MCE<n, c>:
1203 match z:
1204 case Mat{d}:
1205 ce_fin(n, c, F32.from_nat(n), ce_rows(labels, Nat.sub(c, 1n), U32.from_nat(c), F32.from_nat(n), 0, L2{0.0, d}))
1206
1207# argmax hits per row against labels
1208
1209type MHits<-n: Nat, -c: Nat> is Type:
1210 MHits{z: Mat<n, c>, hits: Nat}
1211
1212def hits_fin(-n: Nat, -c: Nat, r: CR) -> MHits<n, c>:
1213 match r:
1214 case CR{d, k}:
1215 MHits{Mat{d}, k}
1216
1217def Mat.count_correct(+n: Nat, +c: Nat, z: Mat<n, c>, labels: List<&2, Nat>) -> MHits<n, c>:
1218 match z:
1219 case Mat{d}:
1220 hits_fin(n, c, count_rows(labels, Nat.sub(c, 1n), U32.from_nat(c), 0, CR{d, 0n}))
1221
1222# a row as a list (returning the matrix) and the other way around
1223
1224type MRow<-r: Nat, -c: Nat> is Type:
1225 MRow{m: Mat<r, c>, row: List<&2, F32>}
1226
1227def row_fin(-r: Nat, -c: Nat, al: AL) -> MRow<r, c>:
1228 match al:
1229 case AL{d, l}:
1230 MRow{Mat{d}, l}
1231
1232def Mat.read_row(+r: Nat, +c: Nat, m: Mat<r, c>, +i: U32) -> MRow<r, c>:
1233 match m:
1234 case Mat{d}:
1235 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))))
1236
1237def Mat.write_row(-r: Nat, +c: Nat, m: Mat<r, c>, +i: U32, xs: List<&2, F32>) -> Mat<r, c>:
1238 match m:
1239 case Mat{d}:
1240 Mat{write_l(xs, (i * U32.from_nat(c) : U32), d)}
1241
1242# ---- variants that check the number of labels ----
1243
1244# does the list of labels have exactly n entries (one per row)?
1245def labels_len_is(xs: List<&2, Nat>, n: Nat) -> Bool:
1246 match xs n:
1247 case Nil{} 0n:
1248 True{}
1249 case Con{h, t} 1n+p:
1250 labels_len_is(t, p)
1251 case _ _:
1252 False{}
1253
1254def ce_checked(ok: Bool, +n: Nat, +c: Nat, z: Mat<n, c>, labels: List<&2, Nat>) -> Maybe<&1, MCE<n, c>>:
1255 match ok:
1256 case True{}:
1257 Some{Mat.softmax_ce(n, c, z, labels)}
1258 case False{}:
1259 None{}
1260
1261# like Mat.softmax_ce, but returns None unless labels has exactly n entries
1262def Mat.softmax_ce_checked(+n: Nat, +c: Nat, z: Mat<n, c>, +labels: List<&2, Nat>) -> Maybe<&1, MCE<n, c>>:
1263 ce_checked(labels_len_is(labels, n), n, c, z, labels)
1264
1265def hits_checked(ok: Bool, +n: Nat, +c: Nat, z: Mat<n, c>, labels: List<&2, Nat>) -> Maybe<&1, MHits<n, c>>:
1266 match ok:
1267 case True{}:
1268 Some{Mat.count_correct(n, c, z, labels)}
1269 case False{}:
1270 None{}
1271
1272# like Mat.count_correct, but returns None unless labels has exactly n entries
1273def Mat.count_correct_checked(+n: Nat, +c: Nat, z: Mat<n, c>, +labels: List<&2, Nat>) -> Maybe<&1, MHits<n, c>>:
1274 hits_checked(labels_len_is(labels, n), n, c, z, labels)
1275
1276# ---------------------------------------------------------------------
1277# Bands<r, c>: a matrix cut into bands of rows, for matrix · vector in parallel
1278# without copying the weights.
1279#
1280# Mat.matmul_nt with n = 1 can only run in parallel by cloning the whole weight
1281# matrix for every task, and in matrix · vector each weight is read once, so the
1282# copies cost more than the products (NOTES.md, exp. 9). Bands keeps each band in
1283# its own Array: a task takes its band and nothing is copied except the input vector.
1284#
1285# A node with r rows has children with half(r) and r - half(r) rows. The split is
1286# part of the type, so the row count of every band follows from r, and the law
1287# half_cover says that the two halves always cover the r rows exactly.
1288# ---------------------------------------------------------------------
1289
1290# n / 2 rounded down. A tail loop with an accumulator: a non-tail recursion here would put a
1291# continuation on every node of the band tree and keep the bands from running in parallel.
1292def half.go(n: Nat, acc: Nat) -> Nat:
1293 match n:
1294 case 0n:
1295 acc
1296 case 1n+p:
1297 match p:
1298 case 0n:
1299 acc
1300 case 1n+q:
1301 half.go(q, 1n+acc)
1302
1303def half(n: Nat) -> Nat:
1304 half.go(n, 0n)
1305
1306# a <= a
1307def leb_refl(+a: Nat) -> {True{} == leb(a, a) : Bool}:
1308 match a:
1309 case 0n:
1310 {==}
1311 case 1n+p:
1312 leb_refl(p)
1313
1314# 1 + (a + b) == a + (1 + b)
1315def add_succ_r(+a: Nat, -b: Nat) -> {1n+Nat.add(a, b) == Nat.add(a, 1n+b) : Nat}:
1316 match a:
1317 case 0n:
1318 {==}
1319 case 1n+p:
1320 %add_succ_r(p, b) : {2n+Nat.add(p, b) == 1n+_ : Nat}
1321 {==}
1322
1323# a + 0 == a
1324def add_zero_r(+a: Nat) -> {Nat.add(a, 0n) == a : Nat}:
1325 match a:
1326 case 0n:
1327 {==}
1328 case 1n+p:
1329 %add_zero_r(p) : {1n+Nat.add(p, 0n) == 1n+_ : Nat}
1330 {==}
1331
1332# half.go(n, acc) <= n + acc
1333def half_go_le(+n: Nat, +acc: Nat) -> {True{} == leb(half.go(n, acc), Nat.add(n, acc)) : Bool}:
1334 match n:
1335 case 0n:
1336 leb_refl(acc)
1337 case 1n+p:
1338 match p:
1339 case 0n:
1340 leb_succ(acc, acc, leb_refl(acc))
1341 case 1n+q:
1342 %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}
1343 leb_succ(half.go(q, 1n+acc), Nat.add(q, 1n+acc), half_go_le(q, 1n+acc))
1344
1345# half(n) <= n
1346def half_le(+n: Nat) -> {True{} == leb(half(n), n) : Bool}:
1347 %add_zero_r(n) : {True{} == leb(half(n), _) : Bool}
1348 half_go_le(n, 0n)
1349
1350# a <= r -> r == a + (r - a)
1351def add_sub(+a: Nat, +r: Nat, e: {True{} == leb(a, r) : Bool}) -> {r == Nat.add(a, Nat.sub(r, a)) : Nat}:
1352 match a r:
1353 case 0n 0n:
1354 {==}
1355 case 0n 1n+q:
1356 {==}
1357 case 1n+p 0n:
1358 %e : BD(_, Unit, {0n == Nat.add(1n+p, Nat.sub(0n, 1n+p)) : Nat})
1359 Unit{}
1360 case 1n+p 1n+q:
1361 %add_sub(p, q, e) : {1n+q == 1n+_ : Nat}
1362 {==}
1363
1364# LAW (half_cover): the two bands of a node, half(r) rows and r - half(r) rows,
1365# add up to exactly r rows: no row is lost and none is counted twice.
1366law half_cover:
1367 for +n: Nat
1368 {n == Nat.add(half(n), Nat.sub(n, half(n))) : Nat}
1369
1370def half_cover(n):
1371 add_sub(half(n), n, half_le(n))
1372
1373type Bands<-r: Nat, -c: Nat> is Type:
1374 BLeaf{m: Mat<r, c>}
1375 BNode{x: Bands<half(r), c>, y: Bands<Nat.sub(r, half(r)), c>}
1376
1377# r x c of zeros in 2^d bands (d = 0: a single band, the same as a Mat)
1378def Bands.zeros(+d: Nat, +r: Nat, +c: Nat) -> Bands<r, c>:
1379 match d:
1380 case 0n:
1381 BLeaf{Mat.zeros(r, c)}
1382 case 1n+q:
1383 BNode{Bands.zeros(q, half(r), c), Bands.zeros(q, Nat.sub(r, half(r)), c)}
1384
1385# the row counts of the two bands follow the row count of the node: if rr == r at run
1386# time, the same holds for both halves (the bands carry r only in their type)
1387def eq_half(-a: Nat, -b: Nat, e: {a == b : Nat}) -> {half(a) == half(b) : Nat}:
1388 %e : {half(a) == half(_) : Nat}
1389 {==}
1390
1391def eq_rest(-a: Nat, -b: Nat, e: {a == b : Nat}) -> {Nat.sub(a, half(a)) == Nat.sub(b, half(b)) : Nat}:
1392 %e : {Nat.sub(a, half(a)) == Nat.sub(_, half(_)) : Nat}
1393 {==}
1394
1395# does a block of n numbers starting at flat index i end before the second band of a
1396# node with r rows? does it start in the second band?
1397def bfill_l(+r: Nat, +c: Nat, +i: Nat, +n: Nat) -> Bool:
1398 Nat.is_le(Nat.add(i, n), Nat.mul(half(r), c))
1399
1400def bfill_r(+r: Nat, +c: Nat, +i: Nat) -> Bool:
1401 Nat.is_le(Nat.mul(half(r), c), i)
1402
1403# writes the first n numbers of xs at index i (and nothing past them)
1404def fill_n(n: Nat, xs: List<&2, F32>, +i: U32, a: Array<F32>) -> Array<F32>:
1405 match n xs:
1406 case 1n+p Con{h, t}:
1407 fill_n(p, t, (i + 1 : U32), Array.set(F32, a, i, h))
1408 case _ _:
1409 a
1410
1411def fill_band(-r: Nat, -c: Nat, n: Nat, xs: List<&2, F32>, +i: U32, m: Mat<r, c>) -> Mat<r, c>:
1412 match m:
1413 case Mat{d}:
1414 Mat{fill_n(n, xs, i, d)}
1415
1416# writes the first n numbers of xs at flat index i; l and rt say which bands they touch
1417# (bfill_l, bfill_r); rr is the row count r at run time. A block that crosses into the second
1418# band is not copied: the first band writes only its part, the second gets the rest by List.drop.
1419def 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>:
1420 match b l rt:
1421 case BLeaf{m} _ _:
1422 BLeaf{fill_band(r, c, n, xs, U32.from_nat(i), m)}
1423 case BNode{x, y} True{} _:
1424 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}
1425 case BNode{x, y} False{} True{}:
1426 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)}
1427 case BNode{x, y} False{} False{}:
1428 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)))}
1429
1430def bfill_top(ok: Bool, +r: Nat, +c: Nat, b: Bands<r, c>, +i: Nat, +n: Nat, +xs: List<&2, F32>) -> Bands<r, c>:
1431 match ok:
1432 case True{}:
1433 bfill(c, r, b, bfill_l(r, c, i, n), bfill_r(r, c, i), r, {==}, i, n, xs)
1434 case False{}:
1435 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)
1436
1437# writes the n numbers of xs into b starting at flat index i (row-major, as Mat.fill_at);
1438# numbers past r*c are dropped. For loading a large matrix block by block.
1439def Bands.fill_at(+r: Nat, +c: Nat, +i: Nat, +n: Nat, +xs: List<&2, F32>, b: Bands<r, c>) -> Bands<r, c>:
1440 bfill_top(Nat.is_le(Nat.add(i, n), Nat.mul(r, c)), r, c, b, i, n, xs)
1441
1442# ---- y = B · x for B with r rows of c numbers (a weight matrix stored out x in) ----
1443
1444# B (intact) and y = B · x, typed like Mat.matmul_nt: x is 1 x c, y is 1 x r
1445type BV<-r: Nat, -c: Nat> is Type:
1446 BV{b: Bands<r, c>, y: Mat<1n, r>}
1447
1448# the bands and y as a list in row order (the result of Bands.matvec_l)
1449type BL<-r: Nat, -c: Nat> is Type:
1450 BL{b: Bands<r, c>, y: List<&2, F32>}
1451
1452def bv_g2(-r: Nat, -c: Nat, w: Array<F32>, al: AL) -> BL<r, c>:
1453 match al:
1454 case AL{d, l}:
1455 BL{BLeaf{Mat{w}}, l}
1456
1457# the band's 1 + m rows came out in an Array (one number per row); read them back as a list
1458def bv_g(-r: Nat, -c: Nat, +m: Nat, g: G) -> BL<r, c>:
1459 match g:
1460 case G{a, b, y}:
1461 bv_g2(r, c, b, read_l(m, 0, Nil{}, Array.get(F32, y, 0)))
1462
1463# the rows of one band, one dot product each, with the gemm of Mat.matmul_nt (n = 1), so the
1464# results are identical. Writing the results into an Array and reading them back is faster than
1465# building the list row by row (NOTES.md, exp. 14).
1466def bv_gemm(-r: Nat, +c: Nat, rows: Nat, +rr: Nat, x: Array<F32>, w: Array<F32>) -> BL<r, c>:
1467 match rows:
1468 case 0n:
1469 BL{BLeaf{Mat{w}}, Nil{}}
1470 case 1n+p:
1471 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)))
1472
1473def bv_mat(-r: Nat, +c: Nat, +rr: Nat, x: Array<F32>, m: Mat<r, c>) -> BL<r, c>:
1474 match m:
1475 case Mat{w}:
1476 bv_gemm(r, c, rr, rr, x, w)
1477
1478def bv_join(-r: Nat, -c: Nat, p: BL<half(r), c>, q: BL<Nat.sub(r, half(r)), c>) -> BL<r, c>:
1479 match p q:
1480 case BL{bx, yx} BL{by, yy}:
1481 BL{BNode{bx, by}, List.append(&2, F32, yx, yy)}
1482
1483# the two bands of a node run in parallel, each with its own copy of x (c numbers);
1484# 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
1485# parameter at the end of the list turned the parallel let into two sequential calls (NOTES.md, exp. 11; fixed in
1486# 2.0.37 by bendlang/bend#1377). The order is kept, and harmless.
1487def bmv(+c: Nat, -r: Nat, b: Bands<r, c>, +rr: Nat, -e: {rr == r : Nat}, xp: Array<F32> & Array<F32>) -> BL<r, c>:
1488 match b xp:
1489 case BLeaf{m} Tuple{x, x2}:
1490 bv_mat(r, c, rr, x, m)
1491 case BNode{bx, by} Tuple{x1, x2}:
1492 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))
1493 bv_join(r, c, p, q)
1494
1495def bv_fin(+r: Nat, -c: Nat, bl: BL<r, c>) -> BV<r, c>:
1496 match bl:
1497 case BL{b, y}:
1498 BV{b, Mat.from_list(1n, r, y)}
1499
1500# 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).
1501# The same product as Mat.matmul_nt(1n, c, r, ..), and the same numbers, bit for bit.
1502def Bands.matvec(+r: Nat, +c: Nat, b: Bands<r, c>, x: Mat<1n, c>) -> BV<r, c>:
1503 match x:
1504 case Mat{xa}:
1505 bv_fin(r, c, bmv(c, r, b, r, {==}, Array.clone(F32, xa)))
1506
1507# the same product with x and y as lists (x: c numbers, y: r numbers in row order), without the
1508# Mat conversions of Bands.matvec: for callers that already hold the vector as a list
1509def Bands.matvec_l(+r: Nat, +c: Nat, b: Bands<r, c>, +x: List<&2, F32>) -> BL<r, c>:
1510 bmv(c, r, b, r, {==}, Array.clone(F32, fill_list(x, 0, Array.new(F32, cap_depth(c), 0.0))))
1511
1512# ---- reading rows and the whole matrix ----
1513
1514type BRow<-r: Nat, -c: Nat> is Type:
1515 BRow{b: Bands<r, c>, row: List<&2, F32>}
1516
1517def brow_fin(-r: Nat, -c: Nat, al: AL) -> BRow<r, c>:
1518 match al:
1519 case AL{d, l}:
1520 BRow{BLeaf{Mat{d}}, l}
1521
1522# n numbers starting at flat index i of one band (a band may have no rows)
1523def brow_leaf2(-r: Nat, -c: Nat, n: Nat, +i: U32, d: Array<F32>) -> BRow<r, c>:
1524 match n:
1525 case 0n:
1526 BRow{BLeaf{Mat{d}}, Nil{}}
1527 case 1n+p:
1528 brow_fin(r, c, read_l(p, i, Nil{}, Array.get(F32, d, i)))
1529
1530def brow_leaf(-r: Nat, -c: Nat, m: Mat<r, c>, +i: U32, n: Nat) -> BRow<r, c>:
1531 match m:
1532 case Mat{d}:
1533 brow_leaf2(r, c, n, i, d)
1534
1535def brow_l(-r: Nat, -c: Nat, p: BRow<half(r), c>, y: Bands<Nat.sub(r, half(r)), c>) -> BRow<r, c>:
1536 match p:
1537 case BRow{bx, row}:
1538 BRow{BNode{bx, y}, row}
1539
1540def brow_r(-r: Nat, -c: Nat, x: Bands<half(r), c>, q: BRow<Nat.sub(r, half(r)), c>) -> BRow<r, c>:
1541 match q:
1542 case BRow{by, row}:
1543 BRow{BNode{x, by}, row}
1544
1545# row i; left says whether i is in the first band of the node (i < half(rr))
1546def brow(+c: Nat, -r: Nat, b: Bands<r, c>, left: Bool, +rr: Nat, -e: {rr == r : Nat}, +i: Nat) -> BRow<r, c>:
1547 match b left:
1548 case BLeaf{m} _:
1549 brow_leaf(r, c, m, U32.from_nat(Nat.mul(i, c)), c)
1550 case BNode{x, y} True{}:
1551 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)
1552 case BNode{x, y} False{}:
1553 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))))
1554
1555# row i (c numbers), for example an embedding lookup
1556def Bands.read_row(+r: Nat, +c: Nat, b: Bands<r, c>, +i: Nat) -> BRow<r, c>:
1557 brow(c, r, b, Nat.is_lt(i, half(r)), r, {==}, i)
1558
1559def bl_join(-r: Nat, -c: Nat, p: BRow<half(r), c>, q: BRow<Nat.sub(r, half(r)), c>) -> BRow<r, c>:
1560 match p q:
1561 case BRow{bx, xs} BRow{by, ys}:
1562 BRow{BNode{bx, by}, List.append(&2, F32, xs, ys)}
1563
1564def blist(+c: Nat, -r: Nat, b: Bands<r, c>, +rr: Nat, -e: {rr == r : Nat}) -> BRow<r, c>:
1565 match b:
1566 case BLeaf{m}:
1567 brow_leaf(r, c, m, 0, Nat.mul(rr, c))
1568 case BNode{x, y}:
1569 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)))
1570
1571# all r*c numbers in row order (in BRow.row)
1572def Bands.to_list(+r: Nat, +c: Nat, b: Bands<r, c>) -> BRow<r, c>:
1573 blist(c, r, b, r, {==})
1574
1575# ---- the list path returns exactly r numbers ----
1576#
1577# The numbers do not matter here, only how many there are: each lemma below counts the list a
1578# function builds, following the function's own recursion.
1579
1580def bl_len(-r: Nat, -c: Nat, bl: BL<r, c>) -> Nat:
1581 match bl:
1582 case BL{b, y}:
1583 List.length(&2, F32, y)
1584
1585# s == a + (1 + b) -> s == 1 + (a + b)
1586def succ_out(-s: Nat, +a: Nat, +b: Nat, e: {s == Nat.add(a, 1n+b) : Nat}) -> {s == 1n+Nat.add(a, b) : Nat}:
1587 %Equal.sym(Nat, 1n+Nat.add(a, b), Nat.add(a, 1n+b), add_succ_r(a, b)) : {s == _ : Nat}
1588 e
1589
1590# s == 1 + (p + (1 + l)) -> s == 2 + (p + l)
1591def len_step(-s: Nat, +p: Nat, +l: Nat, e: {s == 1n+Nat.add(p, 1n+l) : Nat}) -> {s == 2n+Nat.add(p, l) : Nat}:
1592 %Equal.sym(Nat, 1n+Nat.add(p, l), Nat.add(p, 1n+l), add_succ_r(p, l)) : {s == 1n+_ : Nat}
1593 e
1594
1595# s == n + 0 -> s == n
1596def zero_out(-s: Nat, +n: Nat, e: {s == Nat.add(n, 0n) : Nat}) -> {s == n : Nat}:
1597 %add_zero_r(n) : {s == _ : Nat}
1598 e
1599
1600# reversing onto acc adds the lengths
1601def 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}:
1602 match xs:
1603 case Nil{}:
1604 {==}
1605 case Con{h, t}:
1606 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))
1607
1608# appending adds the lengths
1609def 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}:
1610 match xs:
1611 case Nil{}:
1612 {==}
1613 case Con{h, t}:
1614 %app_len(t, ys) : {1n+List.length(&2, F32, List.append(&2, F32, t, ys)) == 1n+_ : Nat}
1615 {==}
1616
1617# a leaf's list: read_l reads mleft + 1 numbers onto acc and reverses them. By induction on
1618# mleft: with 0 left it reads one number (rev_len counts the reversed list); with 1 + p left it
1619# reads one and recurses, and len_step moves the 1 out of the sum.
1620law leaf_len:
1621 for -r: Nat
1622 for -c: Nat
1623 for w: Array<F32>
1624 for +mleft: Nat
1625 for +idx: U32
1626 for +acc: List<&2, F32>
1627 for rc: Array<F32> & F32
1628 {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}
1629
1630def leaf_len(r, c, w, mleft, idx, acc, rc):
1631 match mleft rc:
1632 case 0n Tuple{d, +v}:
1633 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{}))
1634 case 1n+p Tuple{d, +v}:
1635 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))))
1636
1637# whatever the gemm wrote, reading 1 + p rows back gives 1 + p numbers (G has one constructor, so
1638# matching g opens it, and leaf_len counts the list)
1639def g_len(-r: Nat, -c: Nat, +p: Nat, g: G) -> {bl_len(r, c, bv_g(r, c, p, g)) == 1n+p : Nat}:
1640 match g:
1641 case G{a, b, y}:
1642 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)))
1643
1644# LAW (band_len): the product of one band with `rows` rows gives exactly `rows` numbers.
1645# 0 rows give the empty list; 1 + p rows go through the gemm, and g_len counts what is read back.
1646law band_len:
1647 for -r: Nat
1648 for +c: Nat
1649 for +rows: Nat
1650 for x: Array<F32>
1651 for w: Array<F32>
1652 {bl_len(r, c, bv_gemm(r, c, rows, rows, x, w)) == rows : Nat}
1653
1654def band_len(r, c, rows, x, w):
1655 match rows:
1656 case 0n:
1657 {==}
1658 case 1n+p:
1659 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)))
1660
1661# joining two halves of h1 and h2 numbers gives h1 + h2 numbers. (The statement for the whole tree,
1662# "Bands.matvec_l gives r numbers", would apply this to the results of the two recursive calls and
1663# to the induction hypotheses about the same calls; that uses each band's Array twice, which Bend's
1664# affine rules refuse. It stays open, covered by tests: NOTES.md, v3.1.)
1665def 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}:
1666 match p q:
1667 case BL{bx, +yx} BL{by, +yy}:
1668 %hp : {List.length(&2, F32, List.append(&2, F32, yx, yy)) == Nat.add(_, h2) : Nat}
1669 %hq : {List.length(&2, F32, List.append(&2, F32, yx, yy)) == Nat.add(List.length(&2, F32, yx), _) : Nat}
1670 app_len(yx, yy)
1671