1	# The mathematical model the laws speak about: vectors and matrices as lists of Nat, a matrix as its list of rows.
2	# Nothing here runs in a model; it is the specification that LAWS.bend states facts about. Nat makes the algebra
3	# exact (no rounding), so a law proved here says the formulas are right; whether the F32 code computes them to F32's
4	# precision is tested against PyTorch, never proved (docs/design.md, "Proved, tested, trusted").
5	
6	# bend-lint: disable-file bend-lint-rules/performance/missed_fork -- the exact model the laws speak about not-expected-because: never run for speed
7	# bend-lint: disable-file bend-lint-rules/performance/loop_invariant_call -- the exact model the laws speak about not-expected-because: never run for speed
8	import Base
9	import ./main.bend as T
10	
11	# the dot product of the common prefix
12	def dotn(xs: List<&2, Nat>, ys: List<&2, Nat>) -> Nat:
13	  match xs ys:
14	    case Con{x, xt} Con{y, yt}:
15	      Nat.add(Nat.mul(x, y), dotn(xt, yt))
16	    case _ _:
17	      0n
18	
19	# vector sum; the shorter vector counts as padded with zeros
20	def vadd(xs: List<&2, Nat>, ys: List<&2, Nat>) -> List<&2, Nat>:
21	  match xs ys:
22	    case Nil{} _:
23	      ys
24	    case Con{x, xt} Nil{}:
25	      Con{x, xt}
26	    case Con{x, xt} Con{y, yt}:
27	      Nat.add(x, y) <> vadd(xt, yt)
28	
29	def vscale(+k: Nat, xs: List<&2, Nat>) -> List<&2, Nat>:
30	  match xs:
31	    case Nil{}:
32	      Nil{}
33	    case Con{x, t}:
34	      Nat.mul(k, x) <> vscale(k, t)
35	
36	# M·v: one dot product per row (a linear layer's forward pass, y = W·x)
37	def mv(rows: List<&2, List<&2, Nat>>, +v: List<&2, Nat>) -> List<&2, Nat>:
38	  match rows:
39	    case Nil{}:
40	      Nil{}
41	    case Con{r, t}:
42	      dotn(r, v) <> mv(t, v)
43	
44	# Mᵀ·g: the rows weighted by g and summed (a linear layer's backward pass to its input, dx = Wᵀ·dy)
45	def mtv(rows: List<&2, List<&2, Nat>>, g: List<&2, Nat>) -> List<&2, Nat>:
46	  match rows g:
47	    case Con{r, rt} Con{gi, gt}:
48	      vadd(vscale(gi, r), mtv(rt, gt))
49	    case _ _:
50	      Nil{}
51	
52	# g ⊗ v: row i is g[i]·v (a linear layer's weight gradient, dW = dy ⊗ x)
53	def outer(g: List<&2, Nat>, +v: List<&2, Nat>) -> List<&2, List<&2, Nat>>:
54	  match g:
55	    case Nil{}:
56	      Nil{}
57	    case Con{gi, gt}:
58	      vscale(gi, v) <> outer(gt, v)
59	
60	# the sum of the products of matching entries of two matrices (Frobenius inner product)
61	def frob(a: List<&2, List<&2, Nat>>, b: List<&2, List<&2, Nat>>) -> Nat:
62	  match a b:
63	    case Con{ar, at} Con{br, bt}:
64	      Nat.add(dotn(ar, br), frob(at, bt))
65	    case _ _:
66	      0n
67	
68	# a <= b, by recursion (simpler to reason about than Nat.is_le, which goes through Nat.cmp)
69	def le(a: Nat, b: Nat) -> Bool:
70	  match a b:
71	    case 0n _:
72	      True{}
73	    case 1n+p 0n:
74	      False{}
75	    case 1n+p 1n+q:
76	      le(p, q)
77	
78	# ---- linear layers of the model as BendTorch layers, so the laws can run T.Seq and T.Fork's own code on them ----
79	
80	# row-wise sum of two matrices (the shorter counts as padded)
81	def madd(a: List<&2, List<&2, Nat>>, b: List<&2, List<&2, Nat>>) -> List<&2, List<&2, Nat>>:
82	  match a b:
83	    case Nil{} _:
84	      b
85	    case Con{r, rt} Nil{}:
86	      Con{r, rt}
87	    case Con{r, rt} Con{s, st}:
88	      vadd(r, s) <> madd(rt, st)
89	
90	# y = W·x, saving x
91	def lin_fwd(+w: List<&2, List<&2, Nat>>, +x: List<&2, Nat>) -> T.Fwd<List<&2, List<&2, Nat>>, List<&2, Nat>, List<&2, Nat>>:
92	  T.Fwd{w, mv(w, x), x}
93	
94	# dx = Wᵀ·dy, and dW += dy ⊗ x
95	def lin_bwd(+w: List<&2, List<&2, Nat>>, +x: List<&2, Nat>, +dy: List<&2, Nat>, g: List<&2, List<&2, Nat>>) -> T.Bwd<List<&2, List<&2, Nat>>, List<&2, Nat>>:
96	  T.Bwd{w, mtv(w, dy), madd(g, outer(dy, x))}
97	
98	def LinL() -> T.Layer<List<&2, List<&2, Nat>>, List<&2, Nat>, List<&2, Nat>, List<&2, Nat>>:
99	  T.Layer{p => x => lin_fwd(p, x), p => s => dy => g => lin_bwd(p, s, dy, g), op => p => g => st => T.no_step(List<&2, List<&2, Nat>>, op, p, g, st), u => Nil{}, p => T.PL{p, Nil{}}, p => xs => off => T.PA{p, xs, off}, p => xs => off => T.PA{p, xs, off}, u => 0}
100	
101	# two linear layers in sequence, and side by side on one input, through BendTorch's combinators
102	def SeqL() -> T.Layer<T.Two<List<&2, List<&2, Nat>>, List<&2, List<&2, Nat>>>, List<&2, Nat>, List<&2, Nat>, T.Two<List<&2, Nat>, List<&2, Nat>>>:
103	  T.Seq(~List<&2, List<&2, Nat>>, ~List<&2, List<&2, Nat>>, ~List<&2, Nat>, ~List<&2, Nat>, ~List<&2, Nat>, ~List<&2, Nat>, ~List<&2, Nat>, ~LinL(), ~LinL())
104	
105	def dupv(+x: List<&2, Nat>) -> T.Two<List<&2, Nat>, List<&2, Nat>>:
106	  T.Two{x, x}
107	
108	def ForkL() -> T.Layer<T.Two<List<&2, List<&2, Nat>>, List<&2, List<&2, Nat>>>, List<&2, Nat>, T.Two<List<&2, Nat>, List<&2, Nat>>, T.Two<List<&2, Nat>, List<&2, Nat>>>:
109	  T.Fork(~List<&2, List<&2, Nat>>, ~List<&2, List<&2, Nat>>, ~List<&2, Nat>, ~List<&2, Nat>, ~List<&2, Nat>, ~List<&2, Nat>, ~List<&2, Nat>, ~(x => dupv(x)), ~(a => b => vadd(a, b)), ~LinL(), ~LinL())
110	
111	# two linear layers on the two vectors of a pair, through T.Both
112	def BothL() -> T.Layer<T.Two<List<&2, List<&2, Nat>>, List<&2, List<&2, Nat>>>, T.Two<List<&2, Nat>, List<&2, Nat>>, T.Two<List<&2, Nat>, List<&2, Nat>>, T.Two<List<&2, Nat>, List<&2, Nat>>>:
113	  T.Both(~List<&2, List<&2, Nat>>, ~List<&2, List<&2, Nat>>, ~List<&2, Nat>, ~List<&2, Nat>, ~List<&2, Nat>, ~List<&2, Nat>, ~List<&2, Nat>, ~List<&2, Nat>, ~LinL(), ~LinL())
114	
115	# a + b for a pair of vectors; the backward pass hands dy to both (T.Add2 over vectors)
116	def add_fwd(+p: Unit, x: T.Two<List<&2, Nat>, List<&2, Nat>>) -> T.Fwd<Unit, List<&2, Nat>, Unit>:
117	  match x:
118	    case T.Two{a, b}:
119	      T.Fwd{p, vadd(a, b), Unit{}}
120	
121	def AddL() -> T.Layer<Unit, T.Two<List<&2, Nat>, List<&2, Nat>>, List<&2, Nat>, Unit>:
122	  T.Layer{p => x => add_fwd(p, x), p => s => dy => g => T.Bwd{p, dupv(dy), g}, op => p => g => st => T.no_step(Unit, op, p, g, st), u => Unit{}, p => T.PL{p, Nil{}}, p => xs => off => T.PA{p, xs, off}, p => xs => off => T.PA{p, xs, off}, u => 0}
123	
124	# x + W·x through T.ResidualOf, the code of T.Residual: Seq(Fork(Id, linear), AddL)
125	def ResL() -> T.Layer<T.Two<T.Two<Unit, List<&2, List<&2, Nat>>>, Unit>, List<&2, Nat>, List<&2, Nat>, T.Two<T.Two<Unit, List<&2, Nat>>, Unit>>:
126	  T.ResidualOf(~List<&2, Nat>, ~List<&2, List<&2, Nat>>, ~List<&2, Nat>, ~(x => dupv(x)), ~(a => b => vadd(a, b)), ~AddL(), ~LinL())
127	
128	def fy(-P: Type, -Y: Type, -S: Type, r: T.Fwd<P, Y, S>) -> Y:
129	  match r:
130	    case T.Fwd{p, y, s}:
131	      y
132	
133	def fsv(-P: Type, -Y: Type, -S: Type, r: T.Fwd<P, Y, S>) -> S:
134	  match r:
135	    case T.Fwd{p, y, s}:
136	      s
137	
138	def bdx(-P: Type, -X: Type, r: T.Bwd<P, X>) -> X:
139	  match r:
140	    case T.Bwd{p, dx, g}:
141	      dx
142	
143	# the dot product of two pairs of vectors
144	def pdot(a: T.Two<List<&2, Nat>, List<&2, Nat>>, b: T.Two<List<&2, Nat>, List<&2, Nat>>) -> Nat:
145	  match a b:
146	    case T.Two{a1, a2} T.Two{b1, b2}:
147	      Nat.add(dotn(a1, b1), dotn(a2, b2))
148	
149	# ---- The order of a row's additions (the law block_sum). A sum here is the tree of additions it makes over the cells'
150	# positions, not a number: two ways of summing that build the same tree add the same pairs in the same order, so they
151	# give the same bits in F32, or in any arithmetic, with nothing assumed about rounding. ----
152	
153	type Sum is Data:
154	  SCell{i: Nat}
155	  SAdd{a: Sum, b: Sum}
156	
157	# 2^p
158	def pw(p: Nat) -> Nat:
159	  match p:
160	    case 0n:
161	      1n
162	    case 1n+q:
163	      +h = pw(q)
164	      Nat.add(h, h)
165	
166	# the 2^p cells from cell i summed as a balanced tree of pairs: how a GPU leaf sums its block (the device's MSE and
167	# cross-entropy block sums)
168	def bal(+p: Nat, +i: Nat) -> Sum:
169	  match p:
170	    case 0n:
171	      SCell{i}
172	    case 1n+q:
173	      SAdd{bal(q, i), bal(q, Nat.add(i, pw(q)))}
174	
175	# k blocks of 2^p cells, the first from cell lo, each summed by bal; with p = 0, a row of k cells
176	def blocks(+p: Nat, k: Nat, +lo: Nat) -> List<&2, Sum>:
177	  match k:
178	    case 0n:
179	      Nil{}
180	    case 1n+j:
181	      bal(p, lo) <> blocks(p, j, Nat.add(lo, pw(p)))
182	
183	# one pass of main.bend's tree_sum (ts_pass): neighbours added in pairs, an odd last cell carried up as it is
184	def spass(xs: List<&2, Sum>) -> List<&2, Sum>:
185	  match xs:
186	    case Nil{}:
187	      Nil{}
188	    case Con{a, Nil{}}:
189	      a <> Nil{}
190	    case Con{a, Con{b, t}}:
191	      SAdd{a, b} <> spass(t)
192	
193	# r passes of tree_sum; tree_sum (ts_go) stops once one cell is left, and a pass leaves one cell as it is, so the sum of
194	# m cells is passes(r, row) for any r at least the number of passes it makes
195	def passes(r: Nat, xs: List<&2, Sum>) -> List<&2, Sum>:
196	  match r:
197	    case 0n:
198	      xs
199	    case 1n+q:
200	      spass(passes(q, xs))