-
Notifications
You must be signed in to change notification settings - Fork 1
/
equal.ml
297 lines (246 loc) · 11.9 KB
/
equal.ml
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
(* ========================================================================= *)
(* First order logic with equality. *)
(* *)
(* Copyright (c) 2003-2007, John Harrison. (See "LICENSE.txt" for details.) *)
(* ========================================================================= *)
let is_eq = function (Atom(R("=",_))) -> true | _ -> false;;
let mk_eq s t = Atom(R("=",[s;t]));;
let dest_eq fm =
match fm with
Atom(R("=",[s;t])) -> s,t
| _ -> failwith "dest_eq: not an equation";;
let lhs eq = fst(dest_eq eq) and rhs eq = snd(dest_eq eq);;
(* ------------------------------------------------------------------------- *)
(* The set of predicates in a formula. *)
(* ------------------------------------------------------------------------- *)
let rec predicates fm = atom_union (fun (R(p,a)) -> [p,length a]) fm;;
(* ------------------------------------------------------------------------- *)
(* Code to generate equality axioms for functions. *)
(* ------------------------------------------------------------------------- *)
let function_congruence (f,n) =
if n = 0 then [] else
let argnames_x = map (fun n -> "x"^(string_of_int n)) (1 -- n)
and argnames_y = map (fun n -> "y"^(string_of_int n)) (1 -- n) in
let args_x = map (fun x -> Var x) argnames_x
and args_y = map (fun x -> Var x) argnames_y in
let ant = end_itlist mk_and (map2 mk_eq args_x args_y)
and con = mk_eq (Fn(f,args_x)) (Fn(f,args_y)) in
[itlist mk_forall (argnames_x @ argnames_y) (Imp(ant,con))];;
(* ------------------------------------------------------------------------- *)
(* Example. *)
(* ------------------------------------------------------------------------- *)
START_INTERACTIVE;;
function_congruence ("f",3);;
function_congruence ("+",2);;
END_INTERACTIVE;;
(* ------------------------------------------------------------------------- *)
(* And for predicates. *)
(* ------------------------------------------------------------------------- *)
let predicate_congruence (p,n) =
if n = 0 then [] else
let argnames_x = map (fun n -> "x"^(string_of_int n)) (1 -- n)
and argnames_y = map (fun n -> "y"^(string_of_int n)) (1 -- n) in
let args_x = map (fun x -> Var x) argnames_x
and args_y = map (fun x -> Var x) argnames_y in
let ant = end_itlist mk_and (map2 mk_eq args_x args_y)
and con = Imp(Atom(R(p,args_x)),Atom(R(p,args_y))) in
[itlist mk_forall (argnames_x @ argnames_y) (Imp(ant,con))];;
(* ------------------------------------------------------------------------- *)
(* Hence implement logic with equality just by adding equality "axioms". *)
(* ------------------------------------------------------------------------- *)
let equivalence_axioms =
[<<forall x. x = x>>; <<forall x y z. x = y /\ x = z ==> y = z>>];;
let equalitize fm =
let allpreds = predicates fm in
if not (mem ("=",2) allpreds) then fm else
let preds = subtract allpreds ["=",2] and funcs = functions fm in
let axioms = itlist (union ** function_congruence) funcs
(itlist (union ** predicate_congruence) preds
equivalence_axioms) in
Imp(end_itlist mk_and axioms,fm);;
(* ------------------------------------------------------------------------- *)
(* A simple example (see EWD1266a and the application to Morley's theorem). *)
(* ------------------------------------------------------------------------- *)
START_INTERACTIVE;;
let ewd = equalitize
<<(forall x. f(x) ==> g(x)) /\
(exists x. f(x)) /\
(forall x y. g(x) /\ g(y) ==> x = y)
==> forall y. g(y) ==> f(y)>>;;
meson ewd;;
(* ------------------------------------------------------------------------- *)
(* Wishnu Prasetya's example (even nicer with an "exists unique" primitive). *)
(* ------------------------------------------------------------------------- *)
let wishnu = equalitize
<<(exists x. x = f(g(x)) /\ forall x'. x' = f(g(x')) ==> x = x') <=>
(exists y. y = g(f(y)) /\ forall y'. y' = g(f(y')) ==> y = y')>>;;
time meson wishnu;;
(* ------------------------------------------------------------------------- *)
(* More challenging equational problems. (Size 18, 61814 seconds.) *)
(* ------------------------------------------------------------------------- *)
(*********
(meson ** equalitize)
<<(forall x y z. x * (y * z) = (x * y) * z) /\
(forall x. 1 * x = x) /\
(forall x. i(x) * x = 1)
==> forall x. x * i(x) = 1>>;;
********)
(* ------------------------------------------------------------------------- *)
(* Other variants not mentioned in book. *)
(* ------------------------------------------------------------------------- *)
(*************
(meson ** equalitize)
<<(forall x y z. x * (y * z) = (x * y) * z) /\
(forall x. 1 * x = x) /\
(forall x. x * 1 = x) /\
(forall x. x * x = 1)
==> forall x y. x * y = y * x>>;;
(* ------------------------------------------------------------------------- *)
(* With symmetry at leaves and one-sided congruences (Size = 16, 54659 s). *)
(* ------------------------------------------------------------------------- *)
let fm =
<<(forall x. x = x) /\
(forall x y z. x * (y * z) = (x * y) * z) /\
(forall x y z. =((x * y) * z,x * (y * z))) /\
(forall x. 1 * x = x) /\
(forall x. x = 1 * x) /\
(forall x. i(x) * x = 1) /\
(forall x. 1 = i(x) * x) /\
(forall x y. x = y ==> i(x) = i(y)) /\
(forall x y z. x = y ==> x * z = y * z) /\
(forall x y z. x = y ==> z * x = z * y) /\
(forall x y z. x = y /\ y = z ==> x = z)
==> forall x. x * i(x) = 1>>;;
time meson fm;;
(* ------------------------------------------------------------------------- *)
(* Newer version of stratified equalities. *)
(* ------------------------------------------------------------------------- *)
let fm =
<<(forall x y z. axiom(x * (y * z),(x * y) * z)) /\
(forall x y z. axiom((x * y) * z,x * (y * z)) /\
(forall x. axiom(1 * x,x)) /\
(forall x. axiom(x,1 * x)) /\
(forall x. axiom(i(x) * x,1)) /\
(forall x. axiom(1,i(x) * x)) /\
(forall x x'. x = x' ==> cchain(i(x),i(x'))) /\
(forall x x' y y'. x = x' /\ y = y' ==> cchain(x * y,x' * y'))) /\
(forall s t. axiom(s,t) ==> achain(s,t)) /\
(forall s t u. axiom(s,t) /\ (t = u) ==> achain(s,u)) /\
(forall x x' u. x = x' /\ achain(i(x'),u) ==> cchain(i(x),u)) /\
(forall x x' y y' u.
x = x' /\ y = y' /\ achain(x' * y',u) ==> cchain(x * y,u)) /\
(forall s t. cchain(s,t) ==> s = t) /\
(forall s t. achain(s,t) ==> s = t) /\
(forall t. t = t)
==> forall x. x * i(x) = 1>>;;
time meson fm;;
let fm =
<<(forall x y z. axiom(x * (y * z),(x * y) * z)) /\
(forall x y z. axiom((x * y) * z,x * (y * z)) /\
(forall x. axiom(1 * x,x)) /\
(forall x. axiom(x,1 * x)) /\
(forall x. axiom(i(x) * x,1)) /\
(forall x. axiom(1,i(x) * x)) /\
(forall x x'. x = x' ==> cong(i(x),i(x'))) /\
(forall x x' y y'. x = x' /\ y = y' ==> cong(x * y,x' * y'))) /\
(forall s t. axiom(s,t) ==> achain(s,t)) /\
(forall s t. cong(s,t) ==> cchain(s,t)) /\
(forall s t u. axiom(s,t) /\ (t = u) ==> achain(s,u)) /\
(forall s t u. cong(s,t) /\ achain(t,u) ==> cchain(s,u)) /\
(forall s t. cchain(s,t) ==> s = t) /\
(forall s t. achain(s,t) ==> s = t) /\
(forall t. t = t)
==> forall x. x * i(x) = 1>>;;
time meson fm;;
(* ------------------------------------------------------------------------- *)
(* Showing congruence closure. *)
(* ------------------------------------------------------------------------- *)
let fm = equalitize
<<forall c. f(f(f(f(f(c))))) = c /\ f(f(f(c))) = c ==> f(c) = c>>;;
time meson fm;;
let fm =
<<axiom(f(f(f(f(f(c))))),c) /\
axiom(c,f(f(f(f(f(c)))))) /\
axiom(f(f(f(c))),c) /\
axiom(c,f(f(f(c)))) /\
(forall s t. axiom(s,t) ==> achain(s,t)) /\
(forall s t. cong(s,t) ==> cchain(s,t)) /\
(forall s t u. axiom(s,t) /\ (t = u) ==> achain(s,u)) /\
(forall s t u. cong(s,t) /\ achain(t,u) ==> cchain(s,u)) /\
(forall s t. cchain(s,t) ==> s = t) /\
(forall s t. achain(s,t) ==> s = t) /\
(forall t. t = t) /\
(forall x y. x = y ==> cong(f(x),f(y)))
==> f(c) = c>>;;
time meson fm;;
(* ------------------------------------------------------------------------- *)
(* With stratified equalities. *)
(* ------------------------------------------------------------------------- *)
let fm =
<<(forall x y z. eqA (x * (y * z),(x * y) * z)) /\
(forall x y z. eqA ((x * y) * z)) /\
(forall x. eqA (1 * x,x)) /\
(forall x. eqA (x,1 * x)) /\
(forall x. eqA (i(x) * x,1)) /\
(forall x. eqA (1,i(x) * x)) /\
(forall x. eqA (x,x)) /\
(forall x y. eqA (x,y) ==> eqC (i(x),i(y))) /\
(forall x y. eqC (x,y) ==> eqC (i(x),i(y))) /\
(forall x y. eqT (x,y) ==> eqC (i(x),i(y))) /\
(forall w x y z. eqA (w,x) /\ eqA (y,z) ==> eqC (w * y,x * z)) /\
(forall w x y z. eqA (w,x) /\ eqC (y,z) ==> eqC (w * y,x * z)) /\
(forall w x y z. eqA (w,x) /\ eqT (y,z) ==> eqC (w * y,x * z)) /\
(forall w x y z. eqC (w,x) /\ eqA (y,z) ==> eqC (w * y,x * z)) /\
(forall w x y z. eqC (w,x) /\ eqC (y,z) ==> eqC (w * y,x * z)) /\
(forall w x y z. eqC (w,x) /\ eqT (y,z) ==> eqC (w * y,x * z)) /\
(forall w x y z. eqT (w,x) /\ eqA (y,z) ==> eqC (w * y,x * z)) /\
(forall w x y z. eqT (w,x) /\ eqC (y,z) ==> eqC (w * y,x * z)) /\
(forall w x y z. eqT (w,x) /\ eqT (y,z) ==> eqC (w * y,x * z)) /\
(forall x y z. eqA (x,y) /\ eqA (y,z) ==> eqT (x,z)) /\
(forall x y z. eqC (x,y) /\ eqA (y,z) ==> eqT (x,z)) /\
(forall x y z. eqA (x,y) /\ eqC (y,z) ==> eqT (x,z)) /\
(forall x y z. eqA (x,y) /\ eqT (y,z) ==> eqT (x,z)) /\
(forall x y z. eqC (x,y) /\ eqT (y,z) ==> eqT (x,z))
==> forall x. eqT (x * i(x),1)>>;;
(* ------------------------------------------------------------------------- *)
(* With transitivity chains... *)
(* ------------------------------------------------------------------------- *)
let fm =
<<(forall x y z. eqA (x * (y * z),(x * y) * z)) /\
(forall x y z. eqA ((x * y) * z)) /\
(forall x. eqA (1 * x,x)) /\
(forall x. eqA (x,1 * x)) /\
(forall x. eqA (i(x) * x,1)) /\
(forall x. eqA (1,i(x) * x)) /\
(forall x y. eqA (x,y) ==> eqC (i(x),i(y))) /\
(forall x y. eqC (x,y) ==> eqC (i(x),i(y))) /\
(forall w x y. eqA (w,x) ==> eqC (w * y,x * y)) /\
(forall w x y. eqC (w,x) ==> eqC (w * y,x * y)) /\
(forall x y z. eqA (y,z) ==> eqC (x * y,x * z)) /\
(forall x y z. eqC (y,z) ==> eqC (x * y,x * z)) /\
(forall x y z. eqA (x,y) /\ eqA (y,z) ==> eqT (x,z)) /\
(forall x y z. eqC (x,y) /\ eqA (y,z) ==> eqT (x,z)) /\
(forall x y z. eqA (x,y) /\ eqC (y,z) ==> eqT (x,z)) /\
(forall x y z. eqC (x,y) /\ eqC (y,z) ==> eqT (x,z)) /\
(forall x y z. eqA (x,y) /\ eqT (y,z) ==> eqT (x,z)) /\
(forall x y z. eqC (x,y) /\ eqT (y,z) ==> eqT (x,z))
==> forall x. eqT (x * i(x),1) \/ eqC (x * i(x),1)>>;;
time meson fm;;
(* ------------------------------------------------------------------------- *)
(* Enforce canonicity (proof size = 20). *)
(* ------------------------------------------------------------------------- *)
let fm =
<<(forall x y z. eq1(x * (y * z),(x * y) * z)) /\
(forall x y z. eq1((x * y) * z,x * (y * z))) /\
(forall x. eq1(1 * x,x)) /\
(forall x. eq1(x,1 * x)) /\
(forall x. eq1(i(x) * x,1)) /\
(forall x. eq1(1,i(x) * x)) /\
(forall x y z. eq1(x,y) ==> eq1(x * z,y * z)) /\
(forall x y z. eq1(x,y) ==> eq1(z * x,z * y)) /\
(forall x y z. eq1(x,y) /\ eq2(y,z) ==> eq2(x,z)) /\
(forall x y. eq1(x,y) ==> eq2(x,y))
==> forall x. eq2(x,i(x))>>;;
time meson fm;;
******************)
END_INTERACTIVE;;