1	# bend-ml-tensor-array: tensors over a flat Array<F32>, with the shape in the type.
2	#
3	#   import bend-ml-tensor-array@0.1.12.0/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	
18	import Base
19	
20	# ---------------------------------------------------------------------
21	# helper types
22	# ---------------------------------------------------------------------
23	
24	type R is Type:
25	  R{a: Array<F32>, b: Array<F32>, x: F32}
26	
27	type 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).
35	def 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	
42	def 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	
48	def 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	
53	def 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	
62	def 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	
69	def 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	
78	def 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	
81	def 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	
90	type S3 is Type:
91	  S3{a: Array<F32>, b: Array<F32>, out: List<&2, F32>}
92	
93	def 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	
100	def 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	
107	def lblock_fin(st: S3) -> List<&2, F32>:
108	  match st:
109	    case S3{a, b, out}:
110	      List.reverse(&2, F32, out)
111	
112	def 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	
115	def 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	
123	def 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	
132	def 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	
139	def 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	
142	def 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	
149	def 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	
158	def 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	
165	def 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	
168	def 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	
175	def 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	
183	def 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	
191	def 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	
205	type P2 is Type:
206	  P2{a: Array<F32>, b: Array<F32>}
207	
208	type 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	
213	def 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	
222	def 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	
231	def 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	
240	def 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	
247	def pos2(b: Bool) -> F32:
248	  match b:
249	    case True{}:
250	      1.0
251	    case False{}:
252	      0.0
253	
254	def 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	
259	def 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	
268	def 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	
275	def 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	
282	def 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	
289	def 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	
296	def 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	
303	def softmax2(+es: List<&2, F32>) -> List<&2, F32>:
304	  divl(es, suml(es, 0.0))
305	
306	def softmax(+xs: List<&2, F32>) -> List<&2, F32>:
307	  softmax2(expl(xs, maxl(xs)))
308	
309	# p[label]
310	
311	def 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	
322	def 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	
333	def hit(b: Bool) -> F32:
334	  match b:
335	    case True{}:
336	      1.0
337	    case False{}:
338	      0.0
339	
340	def 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	
347	type L2 is Type:
348	  L2{loss: F32, c: Array<F32>}
349	
350	def 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	
353	def 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	
360	type Mat<-r: Nat, -c: Nat> is Type:
361	  Mat{d: Array<F32>}
362	
363	def 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	# ---- helper: hits per row ----
371	
372	type CR is Type:
373	  CR{c: Array<F32>, n: Nat}
374	
375	def gt_head(xs: List<&2, F32>, +v: F32) -> Bool:
376	  match xs:
377	    case Nil{}:
378	      False{}
379	    case Con{+h, t}:
380	      F32.is_gt(h, v)
381	
382	def argmax2(xs: List<&2, F32>, g: Bool, +i: Nat, +bi: Nat, +bv: F32) -> Nat:
383	  match xs g:
384	    case Nil{} _:
385	      bi
386	    case Con{+h, +t} True{}:
387	      argmax2(t, gt_head(t, h), 1n+i, i, h)
388	    case Con{h, +t} False{}:
389	      argmax2(t, gt_head(t, bv), 1n+i, bi, bv)
390	
391	def argmax(xs: List<&2, F32>) -> Nat:
392	  match xs:
393	    case Nil{}:
394	      0n
395	    case Con{+h, +t}:
396	      argmax2(t, gt_head(t, h), 1n, 0n, h)
397	
398	def one(b: Bool) -> Nat:
399	  match b:
400	    case True{}:
401	      1n
402	    case False{}:
403	      0n
404	
405	def count_rows3(+y: Nat, n: Nat, al: AL) -> CR:
406	  match al:
407	    case AL{c, l}:
408	      CR{c, Nat.add(n, one(Nat.is_eq(argmax(l), y)))}
409	
410	def count_rows(labels: List<&2, Nat>, +mm1: Nat, +mu: U32, +i: U32, r: CR) -> CR:
411	  match labels r:
412	    case Nil{} CR{c, n}:
413	      CR{c, n}
414	    case Con{+y, t} CR{c, n}:
415	      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)))))
416	
417	
418	# =====================================================================
419	# Typed API: Mat<r, c> (the dimensions enter the type)
420	# =====================================================================
421	
422	# =====================================================================
423	# Array capacity: a PROVED depth for the number of slots
424	# =====================================================================
425	#
426	# cap_depth(n) is the d such that an Array with 2^d slots holds n elements. It is computed by
427	# structural recursion on n (not with U32.log2) so that the law cap_ok below can prove
428	# `n <= 2^cap_depth(n)`. What stays trusted is only that Array.new(T, d, v) allocates 2^d slots.
429	
430	# a type chosen by a Bool: rewriting through it refutes True == False
431	def BD(b: Bool, t: Type, f: Type) -> Type:
432	  match b:
433	    case True{}:
434	      t
435	    case False{}:
436	      f
437	
438	# a <= b, by direct recursion
439	def leb(a: Nat, b: Nat) -> Bool:
440	  match a b:
441	    case 0n _:
442	      True{}
443	    case 1n+p 0n:
444	      False{}
445	    case 1n+p 1n+q:
446	      leb(p, q)
447	
448	def pow2(d: Nat) -> Nat:
449	  match d:
450	    case 0n:
451	      1n
452	    case 1n++q:
453	      Nat.add(pow2(q), pow2(q))
454	
455	# smallest power of two >= n, by structural recursion on n: returns (depth, 2^depth)
456	def cap_pick(ok: Bool, d: Nat, +p: Nat) -> Nat & Nat:
457	  match ok:
458	    case True{}:
459	      (d, p)
460	    case False{}:
461	      (1n+d, Nat.add(p, p))
462	
463	def cap_step(r: Nat & Nat, +n: Nat) -> Nat & Nat:
464	  match r:
465	    case Tuple{+d, +p}:
466	      cap_pick(leb(n, p), d, p)
467	
468	def cap_pair(n: Nat) -> Nat & Nat:
469	  match n:
470	    case 0n:
471	      (0n, 1n)
472	    case 1n++m:
473	      cap_step(cap_pair(m), 1n+m)
474	
475	def fst_nat(r: Nat & Nat) -> Nat:
476	  match r:
477	    case Tuple{d, p}:
478	      d
479	
480	def snd_nat(r: Nat & Nat) -> Nat:
481	  match r:
482	    case Tuple{d, p}:
483	      p
484	
485	# 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
486	# comparison is one native comparison there (0.1.13.0).
487	def le_leb(a: Nat, b: Nat) -> {Nat.is_le(a, b) == leb(a, b) : Bool}:
488	  match a b:
489	    case 0n 0n:
490	      {==}
491	    case 0n 1n+q:
492	      {==}
493	    case 1n+p 0n:
494	      {==}
495	    case 1n+p 1n+q:
496	      le_leb(p, q)
497	
498	def leb_of(+a: Nat, +b: Nat, e: {True{} == Nat.is_le(a, b) : Bool}) -> {True{} == leb(a, b) : Bool}:
499	  %le_leb(a, b) : {True{} == _ : Bool}
500	  e
501	
502	# 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
503	# 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
504	# suffices (2^n >= n); if it ran out, the answer would be cap_pair's, so the law below needs no arithmetic about how
505	# many doublings there are. ok is Nat.is_le(n, p), computed by the caller (match only inspects a parameter); with leb
506	# it was n steps in all, 152 ms for n = 786,432 on one GPU lane (0.1.13.0).
507	def cap_dbl(f: Nat, ok: Bool, +n: Nat, +d: Nat, +p: Nat) -> Nat & Nat:
508	  match f ok:
509	    case _ True{}:
510	      (d, p)
511	    case 0n False{}:
512	      cap_pair(n)
513	    case 1n+g False{}:
514	      cap_dbl(g, Nat.is_le(n, Nat.add(p, p)), n, 1n+d, Nat.add(p, p))
515	
516	# the depth d such that an Array with 2^d slots holds n elements (the smallest such d)
517	def cap_depth(+n: Nat) -> Nat:
518	  fst_nat(cap_dbl(n, Nat.is_le(n, 1n), n, 0n, 1n))
519	
520	# ---- lemmas ----
521	
522	# a <= b  ->  a <= 1 + b
523	def leb_succ(a: Nat, b: Nat, e: {True{} == leb(a, b) : Bool}) -> {True{} == leb(a, 1n+b) : Bool}:
524	  match a b:
525	    case 0n _:
526	      {==}
527	    case 1n+p 0n:
528	      %e : BD(_, Unit, {True{} == leb(1n+p, 1n) : Bool})
529	      Unit{}
530	    case 1n+p 1n+q:
531	      leb_succ(p, q, e)
532	
533	# a <= b  ->  a <= c + b
534	def leb_up_l(+a: Nat, +b: Nat, c: Nat, +e: {True{} == leb(a, b) : Bool}) -> {True{} == leb(a, Nat.add(c, b)) : Bool}:
535	  match c:
536	    case 0n:
537	      e
538	    case 1n++c2:
539	      leb_succ(a, Nat.add(c2, b), leb_up_l(a, b, c2, e))
540	
541	# a <= b  ->  a <= b + c
542	def leb_up(a: Nat, b: Nat, c: Nat, e: {True{} == leb(a, b) : Bool}) -> {True{} == leb(a, Nat.add(b, c)) : Bool}:
543	  match a b:
544	    case 0n _:
545	      {==}
546	    case 1n+p 0n:
547	      %e : BD(_, Unit, {True{} == leb(1n+p, Nat.add(0n, c)) : Bool})
548	      Unit{}
549	    case 1n+p 1n+q:
550	      leb_up(p, q, c, e)
551	
552	# a <= b  and  c <= d  ->  a + c <= b + d
553	def 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}:
554	  match a b:
555	    case 0n _:
556	      leb_up_l(c, d, b, e2)
557	    case 1n+p 0n:
558	      %e1 : BD(_, Unit, {True{} == leb(Nat.add(1n+p, c), Nat.add(0n, d)) : Bool})
559	      Unit{}
560	    case 1n+p 1n+q:
561	      leb_add(p, q, c, d, e1, e2)
562	
563	# 1 <= 2^d
564	def pow2_pos(d: Nat) -> {True{} == leb(1n, pow2(d)) : Bool}:
565	  match d:
566	    case 0n:
567	      {==}
568	    case 1n++q:
569	      leb_up(1n, pow2(q), pow2(q), pow2_pos(q))
570	
571	# 1 <= p when p = 2^d
572	def pos_p(d: Nat, p: Nat, i1: {p == pow2(d) : Nat}) -> {True{} == leb(1n, p) : Bool}:
573	  %Equal.sym(Nat, p, pow2(d), i1) : {True{} == leb(1n, _) : Bool}
574	  pow2_pos(d)
575	
576	# p + p == 2^(1+d) when p == 2^d
577	def step_eq(d: Nat, p: Nat, i1: {p == pow2(d) : Nat}) -> {Nat.add(p, p) == pow2(1n+d) : Nat}:
578	  %i1 : {Nat.add(p, p) == Nat.add(_, _) : Nat}
579	  {==}
580	
581	# one step: given the invariant for m, the invariant for 1+m holds whichever way the comparison goes
582	def 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}:
583	  match ok:
584	    case True{}:
585	      (i1, e)
586	    case False{}:
587	      (step_eq(d, p, i1), leb_add(1n, p, m, p, pos_p(d, p, i1), i2))
588	
589	
590	# the invariant for cap_step on a pair, opened
591	def 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}:
592	  match r:
593	    case Tuple{+d, +p}:
594	      step_ok(leb(1n+m, p), d, p, m, {==}, i1, i2)
595	
596	def 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}:
597	  match ih:
598	    case Tuple{i1, i2}:
599	      step_pair_ok(m, cap_pair(m), i1, i2)
600	
601	# the invariant of cap_pair: the second component is 2^(first) and is >= n
602	def 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}:
603	  match n:
604	    case 0n:
605	      ({==}, {==})
606	    case 1n++m:
607	      cap_succ(m, cap_inv(m))
608	
609	# the invariant of cap_dbl: whichever way it ends (n <= p found, or the fuel out and cap_pair's answer), the second
610	# component is 2^(first) and is >= n
611	def 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}:
612	  match f ok:
613	    case _ True{}:
614	      (i1, leb_of(n, p, e))
615	    case 0n False{}:
616	      cap_inv(n)
617	    case 1n+g False{}:
618	      dbl_inv(g, Nat.is_le(n, Nat.add(p, p)), n, 1n+d, Nat.add(p, p), {==}, step_eq(d, p, i1))
619	
620	def 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}:
621	  match pr:
622	    case Tuple{i1, i2}:
623	      %i1 : {True{} == leb(n, _) : Bool}
624	      i2
625	
626	# LAW (cap_ok): an Array with 2^cap_depth(n) slots always has room for n elements.
627	law cap_ok:
628	  for +n: Nat
629	  {True{} == leb(n, pow2(cap_depth(n))) : Bool}
630	
631	def cap_ok(n):
632	  dbl_final(n, dbl_inv(n, Nat.is_le(n, 1n), n, 0n, 1n, {==}, {==}))
633	
634	# r x c of zeros, with the exact capacity (smallest power of 2)
635	
636	def Mat.zeros(+r: Nat, +c: Nat) -> Mat<r, c>:
637	  Mat{Array.new(F32, cap_depth(Nat.mul(r, c)), 0.0)}
638	
639	def Mat.fill(+r: Nat, +c: Nat, +v: F32) -> Mat<r, c>:
640	  Mat{Array.new(F32, cap_depth(Nat.mul(r, c)), v)}
641	
642	def fill_list(xs: List<&2, F32>, +i: U32, a: Array<F32>) -> Array<F32>:
643	  match xs:
644	    case Nil{}:
645	      a
646	    case Con{h, t}:
647	      fill_list(t, (i + 1 : U32), Array.set(F32, a, i, h))
648	
649	def len_is(xs: List<&2, F32>, n: Nat) -> Bool:
650	  match xs n:
651	    case Nil{} 0n:
652	      True{}
653	    case Con{h, t} 1n+p:
654	      len_is(t, p)
655	    case _ _:
656	      False{}
657	
658	def Mat.of_list2(-r: Nat, -c: Nat, +rr: Nat, +cc: Nat, ok: Bool, xs: List<&2, F32>) -> Maybe<&1, Mat<r, c>>:
659	  match ok:
660	    case True{}:
661	      Some{Mat{fill_list(xs, 0, Array.new(F32, cap_depth(Nat.mul(rr, cc)), 0.0))}}
662	    case False{}:
663	      None{}
664	
665	# Mat<r, c> from r*c numbers in row order; None if the size does not match
666	
667	def Mat.of_list(+r: Nat, +c: Nat, +xs: List<&2, F32>) -> Maybe<&1, Mat<r, c>>:
668	  Mat.of_list2(r, c, r, c, len_is(xs, Nat.mul(r, c)), xs)
669	
670	# reads the first n numbers, returning the matrix
671	
672	# Mat<r, c> without checking the list size (a short list leaves the rest at zero, a long one is cut
673	# by the capacity): only for when the size is known by construction. Prefer Mat.of_list.
674	
675	def Mat.from_list(+r: Nat, +c: Nat, xs: List<&2, F32>) -> Mat<r, c>:
676	  Mat{fill_list(xs, 0, Array.new(F32, cap_depth(Nat.mul(r, c)), 0.0))}
677	
678	# writes the numbers of xs into m starting at flat index i (row-major); numbers past the
679	# capacity are dropped. For loading a large matrix block by block without building one big list.
680	def Mat.fill_at(-r: Nat, -c: Nat, +i: U32, xs: List<&2, F32>, m: Mat<r, c>) -> Mat<r, c>:
681	  match m:
682	    case Mat{d}:
683	      Mat{fill_list(xs, i, d)}
684	
685	type ML<-r: Nat, -c: Nat> is Type:
686	  ML{m: Mat<r, c>, l: List<&2, F32>}
687	
688	def Mat.to_list2(-r: Nat, -c: Nat, al: AL) -> ML<r, c>:
689	  match al:
690	    case AL{d, l}:
691	      ML{Mat{d}, l}
692	
693	# two independent copies
694	
695	def Mat.to_list(+r: Nat, +c: Nat, m: Mat<r, c>) -> ML<r, c>:
696	  match m:
697	    case Mat{d}:
698	      Mat.to_list2(r, c, read_l(Nat.sub(Nat.mul(r, c), 1n), 0, Nil{}, Array.get(F32, d, 0)))
699	
700	type MM2<-r: Nat, -c: Nat> is Type:
701	  MM2{a: Mat<r, c>, b: Mat<r, c>}
702	
703	def Mat.clone2(-r: Nat, -c: Nat, p: Array<F32> & Array<F32>) -> MM2<r, c>:
704	  match p:
705	    case Tuple{x, y}:
706	      MM2{Mat{x}, Mat{y}}
707	
708	# ---- products: C (n x m) = A · B, with 2^par blocks of rows in parallel ----
709	
710	# C = A(n x k) · B(k x m)
711	
712	def Mat.clone(-r: Nat, -c: Nat, m: Mat<r, c>) -> MM2<r, c>:
713	  match m:
714	    case Mat{d}:
715	      Mat.clone2(r, c, Array.clone(F32, d))
716	
717	type MMul<-n: Nat, -k: Nat, -m: Nat> is Type:
718	  MMul{a: Mat<n, k>, b: Mat<k, m>, c: Mat<n, m>}
719	
720	# C = A(n x k) · Bᵀ, with B stored m x k
721	
722	type MMulNT<-n: Nat, -k: Nat, -m: Nat> is Type:
723	  MMulNT{a: Mat<n, k>, b: Mat<m, k>, c: Mat<n, m>}
724	
725	# C = Aᵀ · B, with A stored k x n
726	
727	type MMulTN<-n: Nat, -k: Nat, -m: Nat> is Type:
728	  MMulTN{a: Mat<k, n>, b: Mat<k, m>, c: Mat<n, m>}
729	
730	# ---------------------------------------------------------------------
731	# 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
732	# 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
733	# 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,
734	# in order, and the four sums are combined as dotr combines them after k steps (its rotation depends on k mod 4). The
735	# padding adds +0 to a sum that starts at +0, which changes nothing, so the results are the same bits as the scalar
736	# loop's. Products with one row (matrix · vector) keep the scalar loop: packing B would cost as much as the product.
737	# ---------------------------------------------------------------------
738	
739	type T8 is Data:
740	  T8{v0: F32, v1: F32, v2: F32, v3: F32, v4: F32, v5: F32, v6: F32, v7: F32}
741	
742	def t8_zero() -> T8:
743	  T8{0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0}
744	
745	def pick0(b: Bool, +v: F32) -> F32:
746	  match b:
747	    case True{}:
748	      v
749	    case False{}:
750	      0.0
751	
752	type PK is Type:
753	  PK{a: Array<F32>, t: Array<T8>}
754	
755	# 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
756	def 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:
757	  match r:
758	    case Tuple{a, +v}:
759	      PK{a, Array.set(T8, t, o, T8{v0, v1, v2, v3, v4, v5, v6, pick0(U32.is_lt((q0 + 7 : U32), lim), v)})}
760	
761	def 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:
762	  match r:
763	    case Tuple{a, +v}:
764	      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)))
765	
766	def 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:
767	  match r:
768	    case Tuple{a, +v}:
769	      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)))
770	
771	def 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:
772	  match r:
773	    case Tuple{a, +v}:
774	      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)))
775	
776	def 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:
777	  match r:
778	    case Tuple{a, +v}:
779	      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)))
780	
781	def pk_g2(+v0: F32, +v1: F32, +q0: U32, +lim: U32, +idx: U32, +st: U32, +o: U32, t: Array<T8>, r: Array<F32> & F32) -> PK:
782	  match r:
783	    case Tuple{a, +v}:
784	      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)))
785	
786	def pk_g1(+v0: F32, +q0: U32, +lim: U32, +idx: U32, +st: U32, +o: U32, t: Array<T8>, r: Array<F32> & F32) -> PK:
787	  match r:
788	    case Tuple{a, +v}:
789	      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)))
790	
791	def pk_g0(+q0: U32, +lim: U32, +idx: U32, +st: U32, +o: U32, t: Array<T8>, r: Array<F32> & F32) -> PK:
792	  match r:
793	    case Tuple{a, +v}:
794	      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)))
795	
796	# the cells of one line, cell o.. of t; q0 counts the line's numbers before the cell
797	def pk_line(tl: Nat, +q0: U32, +lim: U32, +base: U32, +st: U32, +o: U32, r: PK) -> PK:
798	  match tl r:
799	    case 0n r:
800	      r
801	    case 1n+q PK{a, t}:
802	      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)))
803	
804	# lines count lines; source line i starts at i * lstep (in a), and goes to cell row o (cells o * tku .. in t)
805	def pk_lines(cnt: Nat, +tk: Nat, +tku: U32, +ku: U32, +i: U32, +o: U32, +lstep: U32, +st: U32, r: PK) -> PK:
806	  match cnt:
807	    case 0n:
808	      r
809	    case 1n+p:
810	      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))
811	
812	def pk_pack(+cnt: Nat, +tk: Nat, +k: Nat, +lstep: U32, +st: U32, a: Array<F32>) -> PK:
813	  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())})
814	
815	# 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,
816	# already read. Product q of a cell goes to sum q mod 4, the first four before the last four, as dotr adds them.
817	def 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:
818	  match tk ra rb:
819	    case 0n Tuple{a, x} Tuple{b, y}:
820	      TD{a, b, s0, s1, s2, s3}
821	    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}}:
822	      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)))
823	
824	type TD is Type:
825	  TD{a: Array<T8>, b: Array<T8>, s0: F32, s1: F32, s2: F32, s3: F32}
826	
827	# dotr's final sum after k products: the sums sit rotated by k mod 4 when its loop ends
828	def t8_comb(+rot: Nat, +s0: F32, +s1: F32, +s2: F32, +s3: F32) -> F32:
829	  match rot:
830	    case 0n:
831	      ((s0 + s1 : F32) + (s2 + s3 : F32) : F32)
832	    case 1n:
833	      ((s1 + s2 : F32) + (s3 + s0 : F32) : F32)
834	    case 2n:
835	      ((s2 + s3 : F32) + (s0 + s1 : F32) : F32)
836	    case _:
837	      ((s3 + s0 : F32) + (s1 + s2 : F32) : F32)
838	
839	type TR is Type:
840	  TR{a: Array<T8>, b: Array<T8>, x: F32}
841	
842	def t8_fin(+rot: Nat, d: TD) -> TR:
843	  match d:
844	    case TD{a, b, +s0, +s1, +s2, +s3}:
845	      TR{a, b, t8_comb(rot, s0, s1, s2, s3)}
846	
847	def t8_cell(+tk: Nat, +tku: U32, +rot: Nat, +i: U32, +j: U32, a: Array<T8>, b: Array<T8>) -> TR:
848	  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))))
849	
850	type TG is Type:
851	  TG{a: Array<T8>, b: Array<T8>, c: Array<F32>}
852	
853	def t8_set(+ac: Bool, +idx: U32, c: Array<F32>, r: TR) -> TG:
854	  match r:
855	    case TR{a, b, +x}:
856	      TG{a, b, cwr(ac, idx, x, c)}
857	
858	def t8_cols(+ac: Bool, mleft: Nat, +tk: Nat, +tku: U32, +rot: Nat, +i: U32, +j: U32, +cidx: U32, g: TG) -> TG:
859	  match mleft g:
860	    case 0n g:
861	      g
862	    case 1n+p TG{a, b, c}:
863	      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)))
864	
865	def t8_rows(+ac: Bool, nleft: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, +i: U32, +mu: U32, g: TG) -> TG:
866	  match nleft:
867	    case 0n:
868	      g
869	    case 1n+p:
870	      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))
871	
872	# rows i0 .. i0+cnt-1 of C as a list, in order (the parallel version's leaves)
873	type TL is Type:
874	  TL{a: Array<T8>, b: Array<T8>, out: List<&2, F32>}
875	
876	# cells j.. of row i, m - j of them; r is cell j, already computed
877	def t8_lcols(mleft: Nat, +tk: Nat, +tku: U32, +rot: Nat, +i: U32, +j: U32, out: List<&2, F32>, r: TR) -> TL:
878	  match mleft r:
879	    case 0n TR{a, b, +x}:
880	      TL{a, b, x <> out}
881	    case 1n+p TR{a, b, +x}:
882	      t8_lcols(p, tk, tku, rot, i, (j + 1 : U32), x <> out, t8_cell(tk, tku, rot, i, (j + 1 : U32), a, b))
883	
884	def t8_lrows(nleft: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, +i: U32, l: TL) -> TL:
885	  match nleft l:
886	    case 0n l:
887	      l
888	    case 1n+p TL{a, b, out}:
889	      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)))
890	
891	def t8_lfin(l: TL) -> List<&2, F32>:
892	  match l:
893	    case TL{a, b, out}:
894	      List.reverse(&2, F32, out)
895	
896	type PKK is Type:
897	  PKK{a: Array<F32>, b: Array<F32>, ta: Array<T8>, tb: Array<T8>}
898	
899	def t8_pk2(pa: PK, pb: PK) -> PKK:
900	  match pa pb:
901	    case PK{a, ta} PK{b, tb}:
902	      PKK{a, b, ta, tb}
903	
904	def t8_g(a: Array<F32>, b: Array<F32>, g: TG) -> G:
905	  match g:
906	    case TG{ta, tb, c}:
907	      G{a, b, c}
908	
909	def t8_out_seq(+ac: Bool, +n: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, c: Array<F32>, p: PKK) -> G:
910	  match p:
911	    case PKK{a, b, ta, tb}:
912	      t8_g(a, b, t8_rows(ac, n, m, tk, tku, rot, 0, U32.from_nat(m), TG{ta, tb, c}))
913	
914	# ---- the parallel path (0.1.9.0): the larger operand is packed into one piece per block, so a block owns its piece;
915	# the other operand is copied once per node of the fork tree (2^d - 1 copies), instead of both at every node ----
916	
917	type PT is Type:
918	  PLeaf{t: Array<T8>}
919	  PNode{l: PT, r: PT}
920	
921	type PKT is Type:
922	  PKT{a: Array<F32>, t: PT}
923	
924	def pkt_leaf(r: PK) -> PKT:
925	  match r:
926	    case PK{a, t}:
927	      PKT{a, PLeaf{t}}
928	
929	def pkt_then(r: PKT, k: Array<F32> -> PT -> PKT) -> PKT:
930	  match r:
931	    case PKT{a, t}:
932	      k(a, t)
933	
934	# lines i .. i+cnt-1 of a, packed into 2^d pieces (halves of the line count, as the fork tree splits them)
935	def pk_tree(+d: Nat, +cnt: Nat, +i: U32, +tk: Nat, +tku: U32, +ku: U32, +lstep: U32, +st: U32, a: Array<F32>) -> PKT:
936	  match d:
937	    case 0n:
938	      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())}))
939	    case 1n+q:
940	      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}}))
941	
942	def t8_with2(-R: Type, p: Array<T8> & Array<T8>, k: Array<T8> -> Array<T8> -> R) -> R:
943	  match p:
944	    case Tuple{x, y}:
945	      k(x, y)
946	
947	# split by rows of C: the pieces are A's rows, B is shared; the result is C's rows in order
948	def t8_rpar(t: PT, +cnt: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, b: Array<T8>) -> List<&2, F32>:
949	  match t:
950	    case PLeaf{a}:
951	      t8_lfin(t8_lrows(cnt, m, tk, tku, rot, 0, TL{a, b, Nil{}}))
952	    case PNode{l, r}:
953	      t8_with2(List<&2, F32>, Array.clone(T8, b), b1 => b2 => (
954	        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)
955	        List.append(&2, F32, x, y)))
956	
957	# a block of C's columns j0 .. j0+w-1 for every row, row by row
958	type CB is Data:
959	  CB{j0: U32, w: Nat, xs: List<&2, F32>}
960	
961	# split by columns of C: the pieces are B's columns, A is shared
962	def t8_cpar(t: PT, +cnt: Nat, +j0: U32, +n: Nat, +tk: Nat, +tku: U32, +rot: Nat, a: Array<T8>) -> List<&2, CB>:
963	  match t:
964	    case PLeaf{b}:
965	      [CB{j0, cnt, t8_lfin(t8_lrows(n, cnt, tk, tku, rot, 0, TL{a, b, Nil{}}))}]
966	    case PNode{l, r}:
967	      t8_with2(List<&2, CB>, Array.clone(T8, a), a1 => a2 => (
968	        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)
969	        List.append(&2, CB, x, y)))
970	
971	type RW is Type:
972	  RW{c: Array<F32>, xs: List<&2, F32>}
973	
974	# w numbers of xs into c from idx; what is left of xs comes back
975	def cb_row(+ac: Bool, w: Nat, +idx: U32, c: Array<F32>, xs: List<&2, F32>) -> RW:
976	  match w xs:
977	    case 1n+p Con{+h, t}:
978	      cb_row(ac, p, (idx + 1 : U32), cwr(ac, idx, h, c), t)
979	    case _ _:
980	      RW{c, xs}
981	
982	def cb_rows(+ac: Bool, n: Nat, +i: U32, +j0: U32, +w: Nat, +mu: U32, r: RW) -> Array<F32>:
983	  match n r:
984	    case 0n RW{c, xs}:
985	      c
986	    case 1n+p RW{c, xs}:
987	      cb_rows(ac, p, (i + 1 : U32), j0, w, mu, cb_row(ac, w, ((i * mu : U32) + j0 : U32), c, xs))
988	
989	def cb_write(+ac: Bool, bs: List<&2, CB>, +n: Nat, +mu: U32, c: Array<F32>) -> Array<F32>:
990	  match bs:
991	    case Nil{}:
992	      c
993	    case Con{CB{+j0, +w, xs}, t}:
994	      cb_write(ac, t, n, mu, cb_rows(ac, n, 0, j0, w, mu, RW{c, xs}))
995	
996	def t8_pr2(+ac: Bool, +n: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, c: Array<F32>, ra: PKT, rb: PK) -> G:
997	  match ra rb:
998	    case PKT{a, t} PK{b, tb}:
999	      G{a, b, write_la(ac, t8_rpar(t, n, m, tk, tku, rot, tb), 0, c)}
1000	
1001	def t8_pc2(+ac: Bool, +n: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, c: Array<F32>, ra: PK, rb: PKT) -> G:
1002	  match ra rb:
1003	    case PK{a, ta} PKT{b, t}:
1004	      G{a, b, cb_write(ac, t8_cpar(t, m, 0, n, tk, tku, rot, ta), n, U32.from_nat(m), c)}
1005	
1006	# 2^d blocks along the larger of n (rows of C, pieces of A) and m (columns of C, pieces of B)
1007	def 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:
1008	  match byrows:
1009	    case True{}:
1010	      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))
1011	    case False{}:
1012	      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))
1013	
1014	def 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:
1015	  match d:
1016	    case 0n:
1017	      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)))
1018	    case 1n+q:
1019	      t8_par2(ac, Nat.is_le(m, n), 1n+q, n, m, k, tk, tku, rot, arow, sa, sb, bcol, a, b, c)
1020	
1021	# 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
1022	# and B come back intact
1023	def 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:
1024	  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)
1025	
1026	def 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:
1027	  gemm_t8_ac(False{}, d, n, m, k, arow, sa, sb, bcol, a, b, c)
1028	
1029	# 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
1030	# parallel ran gemm_par_cols, which clones both operands at every node of the fork tree: a 50,257 x 768 weight matrix at
1031	# depth 4 ran out of the default heap (5.1 GB) and 8192 x 2048 took 124 ms against 10 ms sequentially. Packing the
1032	# matrix into cells of 8 instead costs more than the product, which reads each weight once (64 ms against 24 at
1033	# 50,257 x 768). Matrix · vector in parallel is Bands.matvec. Measured 2026-10-16 (NOTES.md).
1034	def 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:
1035	  match one:
1036	    case True{}:
1037	      gemm_ac(ac, n, m, k, arow, sa, sb, bcol, a, b, c)
1038	    case False{}:
1039	      gemm_t8_ac(ac, d, n, m, k, arow, sa, sb, bcol, a, b, c)
1040	
1041	def 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:
1042	  gemm_any3(ac, Nat.is_le(n, 1n), d, n, m, k, arow, sa, sb, bcol, a, b, c)
1043	
1044	def 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:
1045	  gemm_any2(ac, d, n, m, k, arow, sa, sb, bcol, a, b, c)
1046	
1047	def mmul_fin(-n: Nat, -k: Nat, -m: Nat, g: G) -> MMul<n, k, m>:
1048	  match g:
1049	    case G{a, b, c}:
1050	      MMul{Mat{a}, Mat{b}, Mat{c}}
1051	
1052	def mmul_nt_fin(-n: Nat, -k: Nat, -m: Nat, g: G) -> MMulNT<n, k, m>:
1053	  match g:
1054	    case G{a, b, c}:
1055	      MMulNT{Mat{a}, Mat{b}, Mat{c}}
1056	
1057	def mmul_tn_fin(-n: Nat, -k: Nat, -m: Nat, g: G) -> MMulTN<n, k, m>:
1058	  match g:
1059	    case G{a, b, c}:
1060	      MMulTN{Mat{a}, Mat{b}, Mat{c}}
1061	
1062	# ---- 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 ----
1063	#
1064	# 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
1065	# sums at least 64 of them; otherwise it runs sequentially. Measured on 16 threads (2026-10-16, 18 shapes): below 2^23 the
1066	# split costs more than it saves, and with a short inner dimension (a weight gradient over a few tokens) every split
1067	# 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
1068	# 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
1069	# depth (tests/t8.bend, tests/acc.bend, tests/split.bend).
1070	def split_pick(big: Bool, d: Nat) -> Nat:
1071	  match big:
1072	    case True{}:
1073	      d
1074	    case False{}:
1075	      0n
1076	
1077	def split_depth(d: Nat, +n: Nat, +m: Nat, +k: Nat) -> Nat:
1078	  split_pick(Bool.and(Nat.is_le(64n, k), Nat.is_le(8388608n, Nat.mul(Nat.mul(n, m), k))), d)
1079	
1080	def mm_a(+n: Nat, +k: Nat, +m: Nat, d: Nat, a: Array<F32>, b: Array<F32>) -> MMul<n, k, m>:
1081	  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)))
1082	
1083	# C = A · B   (n x k) · (k x m); par = log2 of the most parallel blocks (0 = sequential); split_depth picks 0 or par
1084	
1085	def Mat.matmul(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: Mat<n, k>, b: Mat<k, m>) -> MMul<n, k, m>:
1086	  match a b:
1087	    case Mat{x} Mat{y}:
1088	      mm_a(n, k, m, split_depth(par, n, m, k), x, y)
1089	
1090	# 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
1091	# 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),
1092	# without allocating the product or walking it a second time.
1093	
1094	def 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>:
1095	  match a b c:
1096	    case Mat{x} Mat{y} Mat{z}:
1097	      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))
1098	
1099	def mm_nt(+n: Nat, +k: Nat, +m: Nat, d: Nat, a: Array<F32>, b: Array<F32>) -> MMulNT<n, k, m>:
1100	  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)))
1101	
1102	# C = A · Bᵀ   (n x k) · (m x k)ᵀ: for weights stored with one row per output
1103	
1104	def Mat.matmul_nt(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: Mat<n, k>, b: Mat<m, k>) -> MMulNT<n, k, m>:
1105	  match a b:
1106	    case Mat{x} Mat{y}:
1107	      mm_nt(n, k, m, split_depth(par, n, m, k), x, y)
1108	
1109	# C + A · Bᵀ, into C (n x m)
1110	
1111	def 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>:
1112	  match a b c:
1113	    case Mat{x} Mat{y} Mat{z}:
1114	      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))
1115	
1116	def mm_tn(+n: Nat, +k: Nat, +m: Nat, d: Nat, a: Array<F32>, b: Array<F32>) -> MMulTN<n, k, m>:
1117	  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)))
1118	
1119	# C = Aᵀ · B   (k x n)ᵀ · (k x m): weight gradient, X^T · dY
1120	
1121	def Mat.matmul_tn(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: Mat<k, n>, b: Mat<k, m>) -> MMulTN<n, k, m>:
1122	  match a b:
1123	    case Mat{x} Mat{y}:
1124	      mm_tn(n, k, m, split_depth(par, n, m, k), x, y)
1125	
1126	# C + Aᵀ · B, into C (n x m): a weight gradient added into the gradient it accumulates, dW += Xᵀ · dY
1127	
1128	def 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>:
1129	  match a b c:
1130	    case Mat{x} Mat{y} Mat{z}:
1131	      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))
1132	
1133	# ---- element-wise and training operations ----
1134	
1135	# adds the bias (one row, m numbers) to each row of C (n x m)
1136	
1137	type MBias<-n: Nat, -m: Nat> is Type:
1138	  MBias{c: Mat<n, m>, b: Mat<1n, m>}
1139	
1140	def add_row2(-n: Nat, -m: Nat, +nn: Nat, +mm: Nat, cd: Array<F32>, al: AL) -> MBias<n, m>:
1141	  match al:
1142	    case AL{bd, bl}:
1143	      MBias{Mat{bias_rows(nn, 0, U32.from_nat(mm), bl, cd)}, Mat{bd}}
1144	
1145	def Mat.add_row(+n: Nat, +m: Nat, c: Mat<n, m>, b: Mat<1n, m>) -> MBias<n, m>:
1146	  match c b:
1147	    case Mat{cd} Mat{bd}:
1148	      add_row2(n, m, n, m, cd, read_l(Nat.sub(m, 1n), 0, Nil{}, Array.get(F32, bd, 0)))
1149	
1150	# pair of matrices of the same shape (the result of an operation that reads two operands)
1151	
1152	type MPair<-r: Nat, -c: Nat> is Type:
1153	  MPair{a: Mat<r, c>, b: Mat<r, c>}
1154	
1155	def mpair_fin(-r: Nat, -c: Nat, p: P2) -> MPair<r, c>:
1156	  match p:
1157	    case P2{a, b}:
1158	      MPair{Mat{a}, Mat{b}}
1159	
1160	def relu_fin(-r: Nat, -c: Nat, d: Array<F32>) -> Mat<r, c>:
1161	  Mat{d}
1162	
1163	def Mat.relu(+r: Nat, +c: Nat, a: Mat<r, c>) -> Mat<r, c>:
1164	  match a:
1165	    case Mat{d}:
1166	      relu_fin(r, c, relu_l(Nat.mul(r, c), 0, Array.get(F32, d, 0)))
1167	
1168	# (dy * (h > 0), h): the gradient that passes through the relu
1169	
1170	def Mat.relu_bwd(+r: Nat, +c: Nat, dy: Mat<r, c>, h: Mat<r, c>) -> MPair<r, c>:
1171	  match dy h:
1172	    case Mat{d} Mat{x}:
1173	      mpair_fin(r, c, mask_l(Nat.mul(r, c), 0, Array.get(F32, d, 0), Array.get(F32, x, 0)))
1174	
1175	# (w - lr * dw, dw): one SGD step
1176	
1177	def Mat.sgd(+r: Nat, +c: Nat, +lr: F32, w: Mat<r, c>, dw: Mat<r, c>) -> MPair<r, c>:
1178	  match w dw:
1179	    case Mat{x} Mat{y}:
1180	      mpair_fin(r, c, sgd_l(Nat.mul(r, c), 0, lr, Array.get(F32, x, 0), Array.get(F32, y, 0)))
1181	
1182	# (a + b, b): element-wise sum (a - (-1)*b is exactly a + b)
1183	
1184	def Mat.add(+r: Nat, +c: Nat, a: Mat<r, c>, b: Mat<r, c>) -> MPair<r, c>:
1185	  match a b:
1186	    case Mat{x} Mat{y}:
1187	      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)))
1188	
1189	# sum of each column (n x m -> 1 x m), returning the matrix
1190	
1191	type MColSum<-n: Nat, -m: Nat> is Type:
1192	  MColSum{a: Mat<n, m>, s: Mat<1n, m>}
1193	
1194	def colsum_fin(-n: Nat, -m: Nat, g: G) -> MColSum<n, m>:
1195	  match g:
1196	    case G{ones, a, s}:
1197	      MColSum{Mat{a}, Mat{s}}
1198	
1199	def Mat.col_sums(+n: Nat, +m: Nat, a: Mat<n, m>) -> MColSum<n, m>:
1200	  match a:
1201	    case Mat{d}:
1202	      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)))
1203	
1204	# s + the sum of each column, into s (1 x m): a bias gradient added into its accumulator, with the bits of
1205	# Mat.add(s, Mat.col_sums(a)'s s)
1206	
1207	def Mat.col_sums_acc(+n: Nat, +m: Nat, a: Mat<n, m>, s: Mat<1n, m>) -> MColSum<n, m>:
1208	  match a s:
1209	    case Mat{d} Mat{z}:
1210	      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))
1211	
1212	# mean cross-entropy and its gradient on the logits: (softmax - one-hot) / n.
1213	# labels has one entry per row (n); the gradient replaces the logits.
1214	
1215	type MCE<-n: Nat, -c: Nat> is Type:
1216	  MCE{loss: F32, grad: Mat<n, c>}
1217	
1218	def ce_fin(-n: Nat, -c: Nat, +nf: F32, r: L2) -> MCE<n, c>:
1219	  match r:
1220	    case L2{loss, d}:
1221	      MCE{(loss / nf : F32), Mat{d}}
1222	
1223	def Mat.softmax_ce(+n: Nat, +c: Nat, z: Mat<n, c>, labels: List<&2, Nat>) -> MCE<n, c>:
1224	  match z:
1225	    case Mat{d}:
1226	      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}))
1227	
1228	# argmax hits per row against labels
1229	
1230	type MHits<-n: Nat, -c: Nat> is Type:
1231	  MHits{z: Mat<n, c>, hits: Nat}
1232	
1233	def hits_fin(-n: Nat, -c: Nat, r: CR) -> MHits<n, c>:
1234	  match r:
1235	    case CR{d, k}:
1236	      MHits{Mat{d}, k}
1237	
1238	def Mat.count_correct(+n: Nat, +c: Nat, z: Mat<n, c>, labels: List<&2, Nat>) -> MHits<n, c>:
1239	  match z:
1240	    case Mat{d}:
1241	      hits_fin(n, c, count_rows(labels, Nat.sub(c, 1n), U32.from_nat(c), 0, CR{d, 0n}))
1242	
1243	# a row as a list (returning the matrix) and the other way around
1244	
1245	type MRow<-r: Nat, -c: Nat> is Type:
1246	  MRow{m: Mat<r, c>, row: List<&2, F32>}
1247	
1248	def row_fin(-r: Nat, -c: Nat, al: AL) -> MRow<r, c>:
1249	  match al:
1250	    case AL{d, l}:
1251	      MRow{Mat{d}, l}
1252	
1253	def Mat.read_row(+r: Nat, +c: Nat, m: Mat<r, c>, +i: U32) -> MRow<r, c>:
1254	  match m:
1255	    case Mat{d}:
1256	      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))))
1257	
1258	def Mat.write_row(-r: Nat, +c: Nat, m: Mat<r, c>, +i: U32, xs: List<&2, F32>) -> Mat<r, c>:
1259	  match m:
1260	    case Mat{d}:
1261	      Mat{write_l(xs, (i * U32.from_nat(c) : U32), d)}
1262	
1263	# ---- variants that check the number of labels ----
1264	
1265	# does the list of labels have exactly n entries (one per row)?
1266	def labels_len_is(xs: List<&2, Nat>, n: Nat) -> Bool:
1267	  match xs n:
1268	    case Nil{} 0n:
1269	      True{}
1270	    case Con{h, t} 1n+p:
1271	      labels_len_is(t, p)
1272	    case _ _:
1273	      False{}
1274	
1275	def ce_checked(ok: Bool, +n: Nat, +c: Nat, z: Mat<n, c>, labels: List<&2, Nat>) -> Maybe<&1, MCE<n, c>>:
1276	  match ok:
1277	    case True{}:
1278	      Some{Mat.softmax_ce(n, c, z, labels)}
1279	    case False{}:
1280	      None{}
1281	
1282	# like Mat.softmax_ce, but returns None unless labels has exactly n entries
1283	def Mat.softmax_ce_checked(+n: Nat, +c: Nat, z: Mat<n, c>, +labels: List<&2, Nat>) -> Maybe<&1, MCE<n, c>>:
1284	  ce_checked(labels_len_is(labels, n), n, c, z, labels)
1285	
1286	def hits_checked(ok: Bool, +n: Nat, +c: Nat, z: Mat<n, c>, labels: List<&2, Nat>) -> Maybe<&1, MHits<n, c>>:
1287	  match ok:
1288	    case True{}:
1289	      Some{Mat.count_correct(n, c, z, labels)}
1290	    case False{}:
1291	      None{}
1292	
1293	# like Mat.count_correct, but returns None unless labels has exactly n entries
1294	def Mat.count_correct_checked(+n: Nat, +c: Nat, z: Mat<n, c>, +labels: List<&2, Nat>) -> Maybe<&1, MHits<n, c>>:
1295	  hits_checked(labels_len_is(labels, n), n, c, z, labels)
1296	
1297	# ---------------------------------------------------------------------
1298	# Bands<r, c>: a matrix cut into bands of rows, for matrix · vector in parallel
1299	# without copying the weights.
1300	#
1301	# Mat.matmul_nt with n = 1 can only run in parallel by cloning the whole weight
1302	# matrix for every task, and in matrix · vector each weight is read once, so the
1303	# copies cost more than the products (NOTES.md, exp. 9). Bands keeps each band in
1304	# its own Array: a task takes its band and nothing is copied except the input vector.
1305	#
1306	# A node with r rows has children with half(r) and r - half(r) rows. The split is
1307	# part of the type, so the row count of every band follows from r, and the law
1308	# half_cover says that the two halves always cover the r rows exactly.
1309	# ---------------------------------------------------------------------
1310	
1311	# n / 2 rounded down. A tail loop with an accumulator: a non-tail recursion here would put a
1312	# continuation on every node of the band tree and keep the bands from running in parallel.
1313	def half.go(n: Nat, acc: Nat) -> Nat:
1314	  match n:
1315	    case 0n:
1316	      acc
1317	    case 1n+p:
1318	      match p:
1319	        case 0n:
1320	          acc
1321	        case 1n+q:
1322	          half.go(q, 1n+acc)
1323	
1324	def half(n: Nat) -> Nat:
1325	  half.go(n, 0n)
1326	
1327	# a <= a
1328	def leb_refl(+a: Nat) -> {True{} == leb(a, a) : Bool}:
1329	  match a:
1330	    case 0n:
1331	      {==}
1332	    case 1n+p:
1333	      leb_refl(p)
1334	
1335	# 1 + (a + b) == a + (1 + b)
1336	def add_succ_r(+a: Nat, -b: Nat) -> {1n+Nat.add(a, b) == Nat.add(a, 1n+b) : Nat}:
1337	  match a:
1338	    case 0n:
1339	      {==}
1340	    case 1n+p:
1341	      %add_succ_r(p, b) : {2n+Nat.add(p, b) == 1n+_ : Nat}
1342	      {==}
1343	
1344	# a + 0 == a
1345	def add_zero_r(+a: Nat) -> {Nat.add(a, 0n) == a : Nat}:
1346	  match a:
1347	    case 0n:
1348	      {==}
1349	    case 1n+p:
1350	      %add_zero_r(p) : {1n+Nat.add(p, 0n) == 1n+_ : Nat}
1351	      {==}
1352	
1353	# half.go(n, acc) <= n + acc
1354	def half_go_le(+n: Nat, +acc: Nat) -> {True{} == leb(half.go(n, acc), Nat.add(n, acc)) : Bool}:
1355	  match n:
1356	    case 0n:
1357	      leb_refl(acc)
1358	    case 1n+p:
1359	      match p:
1360	        case 0n:
1361	          leb_succ(acc, acc, leb_refl(acc))
1362	        case 1n+q:
1363	          %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}
1364	          leb_succ(half.go(q, 1n+acc), Nat.add(q, 1n+acc), half_go_le(q, 1n+acc))
1365	
1366	# half(n) <= n
1367	def half_le(+n: Nat) -> {True{} == leb(half(n), n) : Bool}:
1368	  %add_zero_r(n) : {True{} == leb(half(n), _) : Bool}
1369	  half_go_le(n, 0n)
1370	
1371	# a <= r  ->  r == a + (r - a)
1372	def add_sub(+a: Nat, +r: Nat, e: {True{} == leb(a, r) : Bool}) -> {r == Nat.add(a, Nat.sub(r, a)) : Nat}:
1373	  match a r:
1374	    case 0n 0n:
1375	      {==}
1376	    case 0n 1n+q:
1377	      {==}
1378	    case 1n+p 0n:
1379	      %e : BD(_, Unit, {0n == Nat.add(1n+p, Nat.sub(0n, 1n+p)) : Nat})
1380	      Unit{}
1381	    case 1n+p 1n+q:
1382	      %add_sub(p, q, e) : {1n+q == 1n+_ : Nat}
1383	      {==}
1384	
1385	# LAW (half_cover): the two bands of a node, half(r) rows and r - half(r) rows,
1386	# add up to exactly r rows: no row is lost and none is counted twice.
1387	law half_cover:
1388	  for +n: Nat
1389	  {n == Nat.add(half(n), Nat.sub(n, half(n))) : Nat}
1390	
1391	def half_cover(n):
1392	  add_sub(half(n), n, half_le(n))
1393	
1394	type Bands<-r: Nat, -c: Nat> is Type:
1395	  BLeaf{m: Mat<r, c>}
1396	  BNode{x: Bands<half(r), c>, y: Bands<Nat.sub(r, half(r)), c>}
1397	
1398	# r x c of zeros in 2^d bands (d = 0: a single band, the same as a Mat)
1399	def Bands.zeros(+d: Nat, +r: Nat, +c: Nat) -> Bands<r, c>:
1400	  match d:
1401	    case 0n:
1402	      BLeaf{Mat.zeros(r, c)}
1403	    case 1n+q:
1404	      BNode{Bands.zeros(q, half(r), c), Bands.zeros(q, Nat.sub(r, half(r)), c)}
1405	
1406	# the row counts of the two bands follow the row count of the node: if rr == r at run
1407	# time, the same holds for both halves (the bands carry r only in their type)
1408	def eq_half(-a: Nat, -b: Nat, e: {a == b : Nat}) -> {half(a) == half(b) : Nat}:
1409	  %e : {half(a) == half(_) : Nat}
1410	  {==}
1411	
1412	def eq_rest(-a: Nat, -b: Nat, e: {a == b : Nat}) -> {Nat.sub(a, half(a)) == Nat.sub(b, half(b)) : Nat}:
1413	  %e : {Nat.sub(a, half(a)) == Nat.sub(_, half(_)) : Nat}
1414	  {==}
1415	
1416	# does a block of n numbers starting at flat index i end before the second band of a
1417	# node with r rows? does it start in the second band?
1418	def bfill_l(+r: Nat, +c: Nat, +i: Nat, +n: Nat) -> Bool:
1419	  Nat.is_le(Nat.add(i, n), Nat.mul(half(r), c))
1420	
1421	def bfill_r(+r: Nat, +c: Nat, +i: Nat) -> Bool:
1422	  Nat.is_le(Nat.mul(half(r), c), i)
1423	
1424	# writes the first n numbers of xs at index i (and nothing past them)
1425	def fill_n(n: Nat, xs: List<&2, F32>, +i: U32, a: Array<F32>) -> Array<F32>:
1426	  match n xs:
1427	    case 1n+p Con{h, t}:
1428	      fill_n(p, t, (i + 1 : U32), Array.set(F32, a, i, h))
1429	    case _ _:
1430	      a
1431	
1432	def fill_band(-r: Nat, -c: Nat, n: Nat, xs: List<&2, F32>, +i: U32, m: Mat<r, c>) -> Mat<r, c>:
1433	  match m:
1434	    case Mat{d}:
1435	      Mat{fill_n(n, xs, i, d)}
1436	
1437	# writes the first n numbers of xs at flat index i; l and rt say which bands they touch
1438	# (bfill_l, bfill_r); rr is the row count r at run time. A block that crosses into the second
1439	# band is not copied: the first band writes only its part, the second gets the rest by List.drop.
1440	def 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>:
1441	  match b l rt:
1442	    case BLeaf{m} _ _:
1443	      BLeaf{fill_band(r, c, n, xs, U32.from_nat(i), m)}
1444	    case BNode{x, y} True{} _:
1445	      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}
1446	    case BNode{x, y} False{} True{}:
1447	      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)}
1448	    case BNode{x, y} False{} False{}:
1449	      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)))}
1450	
1451	def bfill_top(ok: Bool, +r: Nat, +c: Nat, b: Bands<r, c>, +i: Nat, +n: Nat, +xs: List<&2, F32>) -> Bands<r, c>:
1452	  match ok:
1453	    case True{}:
1454	      bfill(c, r, b, bfill_l(r, c, i, n), bfill_r(r, c, i), r, {==}, i, n, xs)
1455	    case False{}:
1456	      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)
1457	
1458	# writes the n numbers of xs into b starting at flat index i (row-major, as Mat.fill_at);
1459	# numbers past r*c are dropped. For loading a large matrix block by block.
1460	def Bands.fill_at(+r: Nat, +c: Nat, +i: Nat, +n: Nat, +xs: List<&2, F32>, b: Bands<r, c>) -> Bands<r, c>:
1461	  bfill_top(Nat.is_le(Nat.add(i, n), Nat.mul(r, c)), r, c, b, i, n, xs)
1462	
1463	# ---- y = B · x for B with r rows of c numbers (a weight matrix stored out x in) ----
1464	
1465	# B (intact) and y = B · x, typed like Mat.matmul_nt: x is 1 x c, y is 1 x r
1466	type BV<-r: Nat, -c: Nat> is Type:
1467	  BV{b: Bands<r, c>, y: Mat<1n, r>}
1468	
1469	# the bands and y as a list in row order (the result of Bands.matvec_l)
1470	type BL<-r: Nat, -c: Nat> is Type:
1471	  BL{b: Bands<r, c>, y: List<&2, F32>}
1472	
1473	def bv_g2(-r: Nat, -c: Nat, w: Array<F32>, al: AL) -> BL<r, c>:
1474	  match al:
1475	    case AL{d, l}:
1476	      BL{BLeaf{Mat{w}}, l}
1477	
1478	# the band's 1 + m rows came out in an Array (one number per row); read them back as a list
1479	def bv_g(-r: Nat, -c: Nat, +m: Nat, g: G) -> BL<r, c>:
1480	  match g:
1481	    case G{a, b, y}:
1482	      bv_g2(r, c, b, read_l(m, 0, Nil{}, Array.get(F32, y, 0)))
1483	
1484	# the rows of one band, one dot product each, with the gemm of Mat.matmul_nt (n = 1), so the
1485	# results are identical. Writing the results into an Array and reading them back is faster than
1486	# building the list row by row (NOTES.md, exp. 14).
1487	def bv_gemm(-r: Nat, +c: Nat, rows: Nat, +rr: Nat, x: Array<F32>, w: Array<F32>) -> BL<r, c>:
1488	  match rows:
1489	    case 0n:
1490	      BL{BLeaf{Mat{w}}, Nil{}}
1491	    case 1n+p:
1492	      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)))
1493	
1494	def bv_mat(-r: Nat, +c: Nat, +rr: Nat, x: Array<F32>, m: Mat<r, c>) -> BL<r, c>:
1495	  match m:
1496	    case Mat{w}:
1497	      bv_gemm(r, c, rr, rr, x, w)
1498	
1499	def bv_join(-r: Nat, -c: Nat, p: BL<half(r), c>, q: BL<Nat.sub(r, half(r)), c>) -> BL<r, c>:
1500	  match p q:
1501	    case BL{bx, yx} BL{by, yy}:
1502	      BL{BNode{bx, by}, List.append(&2, F32, yx, yy)}
1503	
1504	# the two bands of a node run in parallel, each with its own copy of x (c numbers);
1505	# 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
1506	# parameter at the end of the list turned the parallel let into two sequential calls (NOTES.md, exp. 11; fixed in
1507	# 2.0.37 by bendlang/bend#1377). The order is kept, and harmless.
1508	def bmv(+c: Nat, -r: Nat, b: Bands<r, c>, +rr: Nat, -e: {rr == r : Nat}, xp: Array<F32> & Array<F32>) -> BL<r, c>:
1509	  match b xp:
1510	    case BLeaf{m} Tuple{x, x2}:
1511	      bv_mat(r, c, rr, x, m)
1512	    case BNode{bx, by} Tuple{x1, x2}:
1513	      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))
1514	      bv_join(r, c, p, q)
1515	
1516	def bv_fin(+r: Nat, -c: Nat, bl: BL<r, c>) -> BV<r, c>:
1517	  match bl:
1518	    case BL{b, y}:
1519	      BV{b, Mat.from_list(1n, r, y)}
1520	
1521	# 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).
1522	# The same product as Mat.matmul_nt(1n, c, r, ..), and the same numbers, bit for bit.
1523	def Bands.matvec(+r: Nat, +c: Nat, b: Bands<r, c>, x: Mat<1n, c>) -> BV<r, c>:
1524	  match x:
1525	    case Mat{xa}:
1526	      bv_fin(r, c, bmv(c, r, b, r, {==}, Array.clone(F32, xa)))
1527	
1528	# the same product with x and y as lists (x: c numbers, y: r numbers in row order), without the
1529	# Mat conversions of Bands.matvec: for callers that already hold the vector as a list
1530	def Bands.matvec_l(+r: Nat, +c: Nat, b: Bands<r, c>, +x: List<&2, F32>) -> BL<r, c>:
1531	  bmv(c, r, b, r, {==}, Array.clone(F32, fill_list(x, 0, Array.new(F32, cap_depth(c), 0.0))))
1532	
1533	# ---- reading rows and the whole matrix ----
1534	
1535	type BRow<-r: Nat, -c: Nat> is Type:
1536	  BRow{b: Bands<r, c>, row: List<&2, F32>}
1537	
1538	def brow_fin(-r: Nat, -c: Nat, al: AL) -> BRow<r, c>:
1539	  match al:
1540	    case AL{d, l}:
1541	      BRow{BLeaf{Mat{d}}, l}
1542	
1543	# n numbers starting at flat index i of one band (a band may have no rows)
1544	def brow_leaf2(-r: Nat, -c: Nat, n: Nat, +i: U32, d: Array<F32>) -> BRow<r, c>:
1545	  match n:
1546	    case 0n:
1547	      BRow{BLeaf{Mat{d}}, Nil{}}
1548	    case 1n+p:
1549	      brow_fin(r, c, read_l(p, i, Nil{}, Array.get(F32, d, i)))
1550	
1551	def brow_leaf(-r: Nat, -c: Nat, m: Mat<r, c>, +i: U32, n: Nat) -> BRow<r, c>:
1552	  match m:
1553	    case Mat{d}:
1554	      brow_leaf2(r, c, n, i, d)
1555	
1556	def brow_l(-r: Nat, -c: Nat, p: BRow<half(r), c>, y: Bands<Nat.sub(r, half(r)), c>) -> BRow<r, c>:
1557	  match p:
1558	    case BRow{bx, row}:
1559	      BRow{BNode{bx, y}, row}
1560	
1561	def brow_r(-r: Nat, -c: Nat, x: Bands<half(r), c>, q: BRow<Nat.sub(r, half(r)), c>) -> BRow<r, c>:
1562	  match q:
1563	    case BRow{by, row}:
1564	      BRow{BNode{x, by}, row}
1565	
1566	# row i; left says whether i is in the first band of the node (i < half(rr))
1567	def brow(+c: Nat, -r: Nat, b: Bands<r, c>, left: Bool, +rr: Nat, -e: {rr == r : Nat}, +i: Nat) -> BRow<r, c>:
1568	  match b left:
1569	    case BLeaf{m} _:
1570	      brow_leaf(r, c, m, U32.from_nat(Nat.mul(i, c)), c)
1571	    case BNode{x, y} True{}:
1572	      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)
1573	    case BNode{x, y} False{}:
1574	      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))))
1575	
1576	# row i (c numbers), for example an embedding lookup
1577	def Bands.read_row(+r: Nat, +c: Nat, b: Bands<r, c>, +i: Nat) -> BRow<r, c>:
1578	  brow(c, r, b, Nat.is_lt(i, half(r)), r, {==}, i)
1579	
1580	def bl_join(-r: Nat, -c: Nat, p: BRow<half(r), c>, q: BRow<Nat.sub(r, half(r)), c>) -> BRow<r, c>:
1581	  match p q:
1582	    case BRow{bx, xs} BRow{by, ys}:
1583	      BRow{BNode{bx, by}, List.append(&2, F32, xs, ys)}
1584	
1585	def blist(+c: Nat, -r: Nat, b: Bands<r, c>, +rr: Nat, -e: {rr == r : Nat}) -> BRow<r, c>:
1586	  match b:
1587	    case BLeaf{m}:
1588	      brow_leaf(r, c, m, 0, Nat.mul(rr, c))
1589	    case BNode{x, y}:
1590	      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)))
1591	
1592	# all r*c numbers in row order (in BRow.row)
1593	def Bands.to_list(+r: Nat, +c: Nat, b: Bands<r, c>) -> BRow<r, c>:
1594	  blist(c, r, b, r, {==})
1595	
1596	# ---- the list path returns exactly r numbers ----
1597	#
1598	# The numbers do not matter here, only how many there are: each lemma below counts the list a
1599	# function builds, following the function's own recursion.
1600	
1601	def bl_len(-r: Nat, -c: Nat, bl: BL<r, c>) -> Nat:
1602	  match bl:
1603	    case BL{b, y}:
1604	      List.length(&2, F32, y)
1605	
1606	# s == a + (1 + b)  ->  s == 1 + (a + b)
1607	def succ_out(-s: Nat, +a: Nat, +b: Nat, e: {s == Nat.add(a, 1n+b) : Nat}) -> {s == 1n+Nat.add(a, b) : Nat}:
1608	  %Equal.sym(Nat, 1n+Nat.add(a, b), Nat.add(a, 1n+b), add_succ_r(a, b)) : {s == _ : Nat}
1609	  e
1610	
1611	# s == 1 + (p + (1 + l))  ->  s == 2 + (p + l)
1612	def len_step(-s: Nat, +p: Nat, +l: Nat, e: {s == 1n+Nat.add(p, 1n+l) : Nat}) -> {s == 2n+Nat.add(p, l) : Nat}:
1613	  %Equal.sym(Nat, 1n+Nat.add(p, l), Nat.add(p, 1n+l), add_succ_r(p, l)) : {s == 1n+_ : Nat}
1614	  e
1615	
1616	# s == n + 0  ->  s == n
1617	def zero_out(-s: Nat, +n: Nat, e: {s == Nat.add(n, 0n) : Nat}) -> {s == n : Nat}:
1618	  %add_zero_r(n) : {s == _ : Nat}
1619	  e
1620	
1621	# reversing onto acc adds the lengths
1622	def 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}:
1623	  match xs:
1624	    case Nil{}:
1625	      {==}
1626	    case Con{h, t}:
1627	      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))
1628	
1629	# appending adds the lengths
1630	def 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}:
1631	  match xs:
1632	    case Nil{}:
1633	      {==}
1634	    case Con{h, t}:
1635	      %app_len(t, ys) : {1n+List.length(&2, F32, List.append(&2, F32, t, ys)) == 1n+_ : Nat}
1636	      {==}
1637	
1638	# a leaf's list: read_l reads mleft + 1 numbers onto acc and reverses them. By induction on
1639	# mleft: with 0 left it reads one number (rev_len counts the reversed list); with 1 + p left it
1640	# reads one and recurses, and len_step moves the 1 out of the sum.
1641	law leaf_len:
1642	  for -r: Nat
1643	  for -c: Nat
1644	  for w: Array<F32>
1645	  for +mleft: Nat
1646	  for +idx: U32
1647	  for +acc: List<&2, F32>
1648	  for rc: Array<F32> & F32
1649	  {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}
1650	
1651	def leaf_len(r, c, w, mleft, idx, acc, rc):
1652	  match mleft rc:
1653	    case 0n Tuple{d, +v}:
1654	      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{}))
1655	    case 1n+p Tuple{d, +v}:
1656	      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))))
1657	
1658	# whatever the gemm wrote, reading 1 + p rows back gives 1 + p numbers (G has one constructor, so
1659	# matching g opens it, and leaf_len counts the list)
1660	def g_len(-r: Nat, -c: Nat, +p: Nat, g: G) -> {bl_len(r, c, bv_g(r, c, p, g)) == 1n+p : Nat}:
1661	  match g:
1662	    case G{a, b, y}:
1663	      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)))
1664	
1665	# LAW (band_len): the product of one band with `rows` rows gives exactly `rows` numbers.
1666	# 0 rows give the empty list; 1 + p rows go through the gemm, and g_len counts what is read back.
1667	law band_len:
1668	  for -r: Nat
1669	  for +c: Nat
1670	  for +rows: Nat
1671	  for x: Array<F32>
1672	  for w: Array<F32>
1673	  {bl_len(r, c, bv_gemm(r, c, rows, rows, x, w)) == rows : Nat}
1674	
1675	def band_len(r, c, rows, x, w):
1676	  match rows:
1677	    case 0n:
1678	      {==}
1679	    case 1n+p:
1680	      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)))
1681	
1682	# joining two halves of h1 and h2 numbers gives h1 + h2 numbers. (The statement for the whole tree,
1683	# "Bands.matvec_l gives r numbers", would apply this to the results of the two recursive calls and
1684	# to the induction hypotheses about the same calls; that uses each band's Array twice, which Bend's
1685	# affine rules refuse. It stays open, covered by tests: NOTES.md, v3.1.)
1686	def 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}:
1687	  match p q:
1688	    case BL{bx, +yx} BL{by, +yy}:
1689	      %hp : {List.length(&2, F32, List.append(&2, F32, yx, yy)) == Nat.add(_, h2) : Nat}
1690	      %hq : {List.length(&2, F32, List.append(&2, F32, yx, yy)) == Nat.add(List.length(&2, F32, yx), _) : Nat}
1691	      app_len(yx, yy)
1692