1	import Base
2	import ../../../src/crypto/subtle.bend as Subtle
3	import ../../../spec/crypto/subtle.bend as Spec
4	import ../../lib/logic.bend as L
5	import ./word.bend as W
6	import ./laws.bend as Laws
7	
8	# Gate for src/crypto/subtle.bend: `bend proofs/crypto/subtle/proof.bend`.
9	#
10	# The accumulator invariant (the shape of HACL*'s lbytes_eq proof): after
11	# the common prefix, "lengths agree and the accumulator is zero" is "the
12	# starting accumulator is zero and the lists are equal".
13	law step:
14	  for +acc: U32
15	  for +x: U32
16	  for +y: U32
17	  for +e: Bool
18	  {Bool.and(U32.is_eq(U32.or(acc, U32.xor(x, y)), 0), e) == Bool.and(U32.is_eq(acc, 0), Bool.and(U32.is_eq(x, y), e)) : Bool}
19	
20	def step(acc, x, y, e):
21	  %Equal.sym(Bool, U32.is_eq(U32.or(acc, U32.xor(x, y)), 0),
22	    Bool.and(U32.is_eq(acc, 0), U32.is_eq(U32.xor(x, y), 0)), W.u32_or_zero(acc, U32.xor(x, y))) :
23	    {Bool.and(_, e) == Bool.and(U32.is_eq(acc, 0), Bool.and(U32.is_eq(x, y), e)) : Bool}
24	  %Equal.sym(Bool, U32.is_eq(U32.xor(x, y), 0), U32.is_eq(x, y), W.u32_xor_zero(x, y)) :
25	    {Bool.and(Bool.and(U32.is_eq(acc, 0), _), e) == Bool.and(U32.is_eq(acc, 0), Bool.and(U32.is_eq(x, y), e)) : Bool}
26	  W.and_assoc(U32.is_eq(acc, 0), U32.is_eq(x, y), e)
27	
28	law fold:
29	  for +a: List<&2, U32>
30	  for +b: List<&2, U32>
31	  for +acc: U32
32	  {Bool.and(Nat.is_eq(List.length(&2, U32, a), List.length(&2, U32, b)), U32.is_eq(Subtle.diff(a, b, acc), 0)) ==
33	    Bool.and(U32.is_eq(acc, 0), Spec.equal(a, b)) : Bool}
34	
35	def fold(a, b, acc):
36	  match a b:
37	    case Nil{} Nil{}:
38	      Equal.sym(Bool, Bool.and(U32.is_eq(acc, 0), True{}), U32.is_eq(acc, 0), W.and_true(U32.is_eq(acc, 0)))
39	    case Nil{} y <> ys:
40	      Equal.sym(Bool, Bool.and(U32.is_eq(acc, 0), False{}), False{}, W.and_false(U32.is_eq(acc, 0)))
41	    case x <> xs Nil{}:
42	      Equal.sym(Bool, Bool.and(U32.is_eq(acc, 0), False{}), False{}, W.and_false(U32.is_eq(acc, 0)))
43	    case x <> xs y <> ys:
44	      +acc2 = U32.or(acc, U32.xor(x, y))
45	      Equal.trans(Bool,
46	        Bool.and(Nat.is_eq(List.length(&2, U32, xs), List.length(&2, U32, ys)), U32.is_eq(Subtle.diff(xs, ys, acc2), 0)),
47	        Bool.and(U32.is_eq(acc2, 0), Spec.equal(xs, ys)),
48	        Bool.and(U32.is_eq(acc, 0), Bool.and(U32.is_eq(x, y), Spec.equal(xs, ys))),
49	        fold(xs, ys, acc2),
50	        step(acc, x, y, Spec.equal(xs, ys)))
51	
52	def Laws.Eq.value(a, b):
53	  fold(a, b, 0)
54	
55	law equal_sound:
56	  for +a: List<&2, U32>
57	  for +b: List<&2, U32>
58	  for +h: {Spec.equal(a, b) == True{} : Bool}
59	  {a == b : List<&2, U32>}
60	
61	def equal_sound(a, b, h):
62	  match a b:
63	    case Nil{} Nil{}:
64	      {==}
65	    case Nil{} y <> ys:
66	      Empty.absurd({Nil{} == y <> ys : List<&2, U32>}, L.false_true(h))
67	    case x <> xs Nil{}:
68	      Empty.absurd({x <> xs == Nil{} : List<&2, U32>}, L.false_true(h))
69	    case x <> xs y <> ys:
70	      +exy = W.u32_sound(x, y, L.and_left(U32.is_eq(x, y), Spec.equal(xs, ys), h))
71	      +et = equal_sound(xs, ys, L.and_right(U32.is_eq(x, y), Spec.equal(xs, ys), h))
72	      %et : {x <> xs == y <> _ : List<&2, U32>}
73	      %exy : {x <> xs == _ <> xs : List<&2, U32>}
74	      {==}
75	
76	def Laws.Eq.sound(a, b, h):
77	  equal_sound(a, b, Equal.trans(Bool, Spec.equal(a, b), Subtle.eq(a, b), True{},
78	    Equal.sym(Bool, Subtle.eq(a, b), Spec.equal(a, b), Laws.Eq.value(a, b)), h))
79	
80	law equal_refl:
81	  for +a: List<&2, U32>
82	  {Spec.equal(a, a) == True{} : Bool}
83	
84	def equal_refl(a):
85	  match a:
86	    case Nil{}:
87	      {==}
88	    case x <> xs:
89	      %Equal.sym(Bool, U32.is_eq(x, x), True{}, W.u32_refl(x)) : {Bool.and(_, Spec.equal(xs, xs)) == True{} : Bool}
90	      equal_refl(xs)
91	
92	def Laws.Eq.refl(a):
93	  %Equal.sym(Bool, Subtle.eq(a, a), Spec.equal(a, a), Laws.Eq.value(a, a)) : {_ == True{} : Bool}
94	  equal_refl(a)