1	import Base
2	import ../../../src/crypto/subtle.bend as Subtle
3	import ../../../spec/crypto/subtle.bend as SS
4	import ../../lib/logic.bend as L
5	import ../subtle/word.bend as SW
6	
7	# subtle.eq is list equality (the lemmas of proofs/crypto/subtle/proof.bend,
8	# restated here so that file stays the subtle package's root).
9	
10	law step:
11	  for +acc: U32
12	  for +x: U32
13	  for +y: U32
14	  for +e: Bool
15	  {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}
16	
17	def step(acc, x, y, e):
18	  %Equal.sym(Bool, U32.is_eq(U32.or(acc, U32.xor(x, y)), 0),
19	    Bool.and(U32.is_eq(acc, 0), U32.is_eq(U32.xor(x, y), 0)), SW.u32_or_zero(acc, U32.xor(x, y))) :
20	    {Bool.and(_, e) == Bool.and(U32.is_eq(acc, 0), Bool.and(U32.is_eq(x, y), e)) : Bool}
21	  %Equal.sym(Bool, U32.is_eq(U32.xor(x, y), 0), U32.is_eq(x, y), SW.u32_xor_zero(x, y)) :
22	    {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}
23	  SW.and_assoc(U32.is_eq(acc, 0), U32.is_eq(x, y), e)
24	
25	law fold:
26	  for +a: List<&2, U32>
27	  for +b: List<&2, U32>
28	  for +acc: U32
29	  {Bool.and(Nat.is_eq(List.length(&2, U32, a), List.length(&2, U32, b)), U32.is_eq(Subtle.diff(a, b, acc), 0)) ==
30	    Bool.and(U32.is_eq(acc, 0), SS.equal(a, b)) : Bool}
31	
32	def fold(a, b, acc):
33	  match a b:
34	    case Nil{} Nil{}:
35	      Equal.sym(Bool, Bool.and(U32.is_eq(acc, 0), True{}), U32.is_eq(acc, 0), SW.and_true(U32.is_eq(acc, 0)))
36	    case Nil{} y <> ys:
37	      Equal.sym(Bool, Bool.and(U32.is_eq(acc, 0), False{}), False{}, SW.and_false(U32.is_eq(acc, 0)))
38	    case x <> xs Nil{}:
39	      Equal.sym(Bool, Bool.and(U32.is_eq(acc, 0), False{}), False{}, SW.and_false(U32.is_eq(acc, 0)))
40	    case x <> xs y <> ys:
41	      +acc2 = U32.or(acc, U32.xor(x, y))
42	      Equal.trans(Bool,
43	        Bool.and(Nat.is_eq(List.length(&2, U32, xs), List.length(&2, U32, ys)), U32.is_eq(Subtle.diff(xs, ys, acc2), 0)),
44	        Bool.and(U32.is_eq(acc2, 0), SS.equal(xs, ys)),
45	        Bool.and(U32.is_eq(acc, 0), Bool.and(U32.is_eq(x, y), SS.equal(xs, ys))),
46	        fold(xs, ys, acc2),
47	        step(acc, x, y, SS.equal(xs, ys)))
48	
49	law equal_sound:
50	  for +a: List<&2, U32>
51	  for +b: List<&2, U32>
52	  for +h: {SS.equal(a, b) == True{} : Bool}
53	  {a == b : List<&2, U32>}
54	
55	def equal_sound(a, b, h):
56	  match a b:
57	    case Nil{} Nil{}: {==}
58	    case Nil{} y <> ys: Empty.absurd({Nil{} == y <> ys : List<&2, U32>}, L.false_true(h))
59	    case x <> xs Nil{}: Empty.absurd({x <> xs == Nil{} : List<&2, U32>}, L.false_true(h))
60	    case x <> xs y <> ys:
61	      +exy = SW.u32_sound(x, y, L.and_left(U32.is_eq(x, y), SS.equal(xs, ys), h))
62	      +et = equal_sound(xs, ys, L.and_right(U32.is_eq(x, y), SS.equal(xs, ys), h))
63	      %et : {x <> xs == y <> _ : List<&2, U32>}
64	      %exy : {x <> xs == _ <> xs : List<&2, U32>}
65	      {==}
66	
67	law equal_refl:
68	  for +a: List<&2, U32>
69	  {SS.equal(a, a) == True{} : Bool}
70	
71	def equal_refl(a):
72	  match a:
73	    case Nil{}: {==}
74	    case x <> xs:
75	      %Equal.sym(Bool, U32.is_eq(x, x), True{}, SW.u32_refl(x)) : {Bool.and(_, SS.equal(xs, xs)) == True{} : Bool}
76	      equal_refl(xs)
77	
78	law eq_value:
79	  for +a: List<&2, U32>
80	  for +b: List<&2, U32>
81	  {Subtle.eq(a, b) == SS.equal(a, b) : Bool}
82	
83	def eq_value(a, b):
84	  fold(a, b, 0)