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
|
% --------------------------------------------------------------------- %
% Copyright (c) Jim Grundy 1992 %
% All rights reserved %
% %
% Jim Grundy, hereafter referred to as `the Author', retains the %
% copyright and all other legal rights to the Software contained in %
% this file, hereafter referred to as `the Software'. %
% %
% The Software is made available free of charge on an `as is' basis. %
% No guarantee, either express or implied, of maintenance, reliability, %
% merchantability or suitability for any purpose is made by the Author. %
% %
% The user is granted the right to make personal or internal use %
% of the Software provided that both: %
% 1. The Software is not used for commercial gain. %
% 2. The user shall not hold the Author liable for any consequences %
% arising from use of the Software. %
% %
% The user is granted the right to further distribute the Software %
% provided that both: %
% 1. The Software and this statement of rights is not modified. %
% 2. The Software does not form part or the whole of a system %
% distributed for commercial gain. %
% %
% The user is granted the right to modify the Software for personal or %
% internal use provided that all of the following conditions are %
% observed: %
% 1. The user does not distribute the modified software. %
% 2. The modified software is not used for commercial gain. %
% 3. The Author retains all rights to the modified software. %
% %
% Anyone seeking a licence to use this software for commercial purposes %
% is invited to contact the Author. %
% --------------------------------------------------------------------- %
%============================================================================%
% CONTENTS: generic and basic window inference rules %
%============================================================================%
%$Id: basic_close.ml,v 3.1 1993/12/07 14:15:19 jg Exp $%
begin_section basic_close;;
% (|- f = g) %
% -------------------- RATOR_CLOSE "g x" %
% (|- (f x) = (g x)) %
let RATOR_CLOSE tm th = (AP_THM th (rand tm)) ? failwith `RATOR_CLOSE`;;
% (|- y = x) %
% --------------------- RAND_CLOSE "f x" %
% (|- (f y) = (f x)) %
let RAND_CLOSE tm th = (AP_TERM (rator tm) th) ? failwith `RAND_CLOSE`;;
% (|- t = u) %
% ---------------------- BODY_CLOSE "\x.u" %
% (|- (\x.t) = (\x.u)) %
let BODY_CLOSE tm th = (ABS (bndvar tm) th) ? failwith `BODY_CONV`;;
let COND1_THM =
prove
(
"!R A B C D. (!x:*. R x x) ==>
(A ==> R D B) ==> (R (A => D | C) (A => B | C))"
,
(REPEAT GEN_TAC) THEN
DISCH_TAC THEN
(BOOL_CASES_TAC "A:bool") THEN
(ASM_REWRITE_TAC [])
) ;;
% (A |- D R B) %
% --------------------------------- COND1_CLOSE "A => B | C" %
% (|- (A => D | C) R (A => B | C) %
let COND1_CLOSE tm th =
let (a,b,c) = dest_cond tm in
let d = rand (rator (concl th)) in
let r = rator (rator (concl th)) in
let x = genvar (type_of b) in
let rref = GEN x (reflexive (mk_comb ((mk_comb (r, x)), x))) in
MP (MP (ISPECL [r; a; b; c; d] COND1_THM) rref) (DISCH a th) ;;
let COND2_THM =
prove
(
"!R A B C D. (!x:*. R x x) ==>
((~A) ==> R D C) ==> (R (A => B | D) (A => B | C))"
,
(REPEAT GEN_TAC) THEN
DISCH_TAC THEN
(BOOL_CASES_TAC "A:bool") THEN
(ASM_REWRITE_TAC [])
) ;;
% (~A |- D R C) %
% --------------------------------- COND2_CLOSE "A => B | C" %
% (|- (A => B | D) R (A => B | C) %
let COND2_CLOSE tm th =
let (a,b,c) = dest_cond tm in
let d = rand (rator (concl th)) in
let r = rator (rator (concl th)) in
let x = genvar (type_of c) in
let rref = GEN x (reflexive (mk_comb ((mk_comb (r, x)), x))) in
MP (MP (ISPECL [r; a; b; c; d] COND2_THM) rref) (DISCH (mk_neg a) th) ;;
let BODY2_THM =
prove
(
"!(c:*) (f:*->**) (g:*->**) (r:**->**->bool).
(!v:*. (v=c) ==> (r (f v) (g v))) ==> (r (f c) (g c))"
,
(REPEAT STRIP_TAC) THEN
(REWRITE_TAC
[REWRITE_RULE
[]
(SPEC
"c:*"
(ASSUME "!v:*. (v = c) ==> (r:**->**->bool)(f v)(g v)"))])
);;
% (v = c |- u R t) %
% ----------------------------- BODY2_CLOSE "((\v.t) c)" %
% (|- ((\v.u) c) R ((\v.t) c) %
let BODY2_CLOSE tm th =
let ((v,t),c) = (dest_abs # I) (dest_comb tm) in
let u = rand (rator (concl th)) in
let r = rator (rator (concl th)) in
if mem v (flat (map frees (hyp th))) then
let t1 = GEN v (DISCH (mk_eq(v,c)) th) in
let t2 = ISPECL [c; mk_abs(v,u); mk_abs(v,t); r] BODY2_THM in
let t3 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (ABS_CONV (RAND_CONV
(RATOR_CONV (RAND_CONV BETA_CONV)))))))
t2 in
let t4 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (ABS_CONV (RAND_CONV
(RAND_CONV BETA_CONV))))))
t3 in
MP t4 t1
else
let t1 = BETA_CONV (mk_comb(mk_abs(v,u),v)) in
let t2 = SYM (BETA_CONV (mk_comb(mk_abs(v,t),v))) in
INST [(c,v)] (transitive (CONJ (transitive (CONJ t1 th)) t2)) ;;
let LET_THM =
prove
(
"!(c:*) (f:*->**) (g:*->**) (r:**->**->bool).
(!v:*. (v=c) ==> (r (f v) (g v))) ==>
(r (LET f c) (LET g c))"
,
(REPEAT STRIP_TAC) THEN
(PURE_ONCE_REWRITE_TAC [LET_DEF]) THEN
BETA_TAC THEN
(REWRITE_TAC
[REWRITE_RULE
[]
(SPEC
"c:*"
(ASSUME "!v:*. (v = c) ==> (r:**->**->bool)(f v)(g v)"))])
);;
% (v = c |- u R t) %
% ----------------------------------------- LET_CLOSE "let v = c in t" %
% (|- (let v = c in u) R (let v = c in t) %
let LET_CLOSE tm th =
let ((v,t),c) = (dest_abs # I) (dest_let tm) in
let u = rand (rator (concl th)) in
let r = rator (rator (concl th)) in
let t1 = GEN v (DISCH (mk_eq(v,c)) th) in
let t2 = ISPECL [c; mk_abs(v,u); mk_abs(v,t); r] LET_THM in
let t3 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (ABS_CONV (RAND_CONV
(RATOR_CONV (RAND_CONV BETA_CONV)))))))
t2 in
let t4 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (ABS_CONV (RAND_CONV
(RAND_CONV BETA_CONV))))))
t3 in
MP t4 t1;;
% Put all those rules in the data base. %
store_rule
(
[RATOR],
is_comb,
(\targ.\rel.
let ty = type_of (rator targ) in
mk_const
(
`=`
,
mk_type (`fun`, [ty; mk_type(`fun`, [ty; bool_ty])])
)),
(\targ.\rel. mk_const(`=`, type_of rel)),
KI,
K [],
RATOR_CLOSE
);;
store_rule
(
[RAND],
is_comb,
(
\targ.\rel.
let ty = type_of (rand targ) in
mk_const
(
`=`
,
mk_type (`fun`, [ty; mk_type(`fun`, [ty; bool_ty])])
)
),
(\targ.\rel. mk_const(`=`, type_of rel)),
KI,
K [],
RAND_CLOSE
);;
store_rule
(
[BODY],
is_abs,
(
\targ.\rel.
let ty = ran targ in
mk_const
(
`=`
,
mk_type (`fun`, [ty; mk_type(`fun`, [ty; bool_ty])])
)
),
(\targ.\rel. mk_const(`=`, type_of rel)),
(\tm.\tl. (filter (\th. not (mem (bndvar tm) (thm_frees th))) tl)),
(\tm. [bndvar tm]),
BODY_CLOSE
);;
store_rule
(
[RATOR; RAND],
is_cond,
K I,
K I,
(\tm.\tl. (SMASH (ASSUME (rand (rator (rator tm))))) @ tl),
K [],
COND1_CLOSE
);;
store_rule
(
[RAND],
is_cond,
K I,
K I,
(\tm.\tl.
(SMASH (ASSUME (mk_neg (rand (rator (rator tm)))))) @ tl),
K [],
COND2_CLOSE
);;
store_rule
(
[RATOR; BODY],
(\tm. (is_comb tm) & (is_abs (rator tm))),
K I,
K I,
(\tm.\tl.
let v = bndvar (rator tm) in
filter
(\th. not (mem v (thm_frees th)))
((ASSUME (mk_eq (v, rand tm))).tl)),
(\tm. [bndvar (rator tm)]),
BODY2_CLOSE
);;
store_rule
(
[RATOR; RAND; BODY],
(\tm. is_let tm),
K I,
K I,
(\tm.\tl.
let ((v,_),c) = (dest_abs # I) (dest_let tm) in
(ASSUME (mk_eq (v,c))).
(filter (\th. not (mem v (thm_frees th))) tl)),
(\foc. [bndvar (rand (rator foc))]),
LET_CLOSE
);;
|
