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