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
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
|
; $Id: quant.scm,v 1.4 2008/01/25 13:30:24 logik Exp $
(add-var-name "x" "y" "z" (py "alpha"))
(add-predconst-name "P" (make-arity))
(add-predconst-name "Qpredconst" (make-arity (py "alpha")))
(add-predconst-name "Rpredconst" (make-arity (py "alpha") (py "alpha")))
(add-token
"Q"
'predconst-name
(string-and-arity-to-predconst-parse-function
"Qpredconst" (make-arity (make-tvar -1 "alpha"))))
(add-token
"R"
'pred-infix
(lambda (x y)
((string-and-arity-to-predconst-parse-function
"Rpredconst" (make-arity (make-tvar -1 "alpha") (make-tvar -1 "alpha")))
(- 1) x y)))
(add-predconst-display "Qpredconst" 'predconst-name "Q")
(add-predconst-display "Rpredconst" 'pred-infix "R")
(formula-to-string (pf "Q x"))
(formula-to-string (pf "x R y"))
; (load"~/minlog/lib/nat.scm")
; (formula-to-string (pf "Q nat"))
(formula-to-string (pf "Q17 x"))
(define all-imp (pf "(all x.Q1 x -> Q2 x) -> all x Q1 x -> all x Q2 x"))
(set-goal all-imp)
(assume 1 2 "x")
(use 1)
(use 2)
(dnp)
(define all-imp (pf "(all x.Q1 x -> Q2 x) -> all x Q1 x -> all x Q2 x"))
(set-goal all-imp)
(search)
(dnp)
(define all-and1 (pf "(all x.Q1 x & Q2 x) -> all x Q1 x & all x Q2 x"))
(set-goal all-and1)
(assume 1)
(split)
(assume "x")
(use 1)
(assume "x")
(use 1)
(dnp)
(set-goal all-and1)
(search)
(dnp)
(define all-and2 (pf " all x Q1 x & all x Q2 x -> all x.Q1 x & Q2 x"))
(set-goal all-and2)
(assume 1 "x")
(split)
(use 1)
(use 1)
(dnp)
(set-goal all-and2)
(search)
(dnp)
(define all-exca (pf "all x Q x -> (all x.Q x -> F) -> F"))
(set-goal all-exca)
(assume 1 2)
(use 2 (pt "x"))
(use 1)
(dnp)
(set-goal all-exca)
(search)
(dnp)
(add-global-assumption "Symm" (pf "all x,y.x R y -> y R x"))
(add-global-assumption "Trans" (pf "all x,y,z.x R y -> y R z -> x R z"))
(set-goal (pf "all x,y.x R y -> x R x"))
(assume "x" "y" 1)
(use "Trans" (pt "y"))
(use 1)
(use "Symm")
(use 1)
(set-goal (pf "all x,y.x R y -> x R x"))
(search 1 '("Symm" 1) '("Trans" 1))
(dnp)
; Now we treat somewhat systematically how in classical logic one can
; deal with quantifiers in implications. - Below we shall do the same
; for the constructive existential quantifier.
; qf1m is obtained from the formula (all x Q x -> P) -> ex x.Q x -> P
; by translating `ex' into `not all not' and adding stability of Q:
(define qf1m (pf "(all x.((Q x -> F) -> F) -> Q x)
-> (all x Q x -> P)
-> (all x.(Q x -> P) -> F)
-> F"))
(set-goal qf1m)
(assume 1 2 3)
(use 3 (pt "x"))
(assume 4)
(use 2)
(assume "x1")
(use 1)
(assume 5)
(use 3 (pt "x1"))
(assume 6)
(use 2)
(assume "x2")
(use 1)
(assume 7)
(use 5)
(use 6)
(set-goal qf1m)
(search)
(dnp)
(define qf2 (pf "(P -> all y Q y) -> all y. P -> Q y"))
(set-goal qf2)
(assume 1 "y" 2)
(use 1)
(use 2)
(set-goal qf2)
(search)
(dnp)
; qf3 is obtained from the formula (ex x Q x -> P) -> all x.Q x -> P
; by translating `ex' into `not all not':
(define qf3 (pf "(((all x.Q x -> F) -> F) -> P) -> all x.Q x -> P"))
(set-goal qf3)
(assume 1 "x" 2)
(use 1)
(assume 3)
(use 3 (pt "x"))
(use 2)
(set-goal qf3)
(search)
(dnp)
; qf4m is obtained from the formula (P -> ex y Q y) -> ex y.P -> Q y
; by translating `ex' into `not all not' and adding ef-falso for Q:
(define qf4m (pf "(all y.F -> Q y)
-> (P -> (all y.Q y -> F) -> F)
-> (all y.(P -> Q y) -> F)
-> F"))
(set-goal qf4m)
(assume 1 2 3)
(use 3 (pt "y"))
(assume 4)
(use 1)
(use 2)
(use 4)
(assume "y1" 5)
(use 3 (pt "y1"))
(assume 6)
(use 5)
(set-goal qf4m)
(search)
(dnp)
; qf5m is obtained from the formula (ex x.Q x -> P) -> all x Q x -> P
; by translating `ex' into `not all not' and adding stability of P:
(define qf5m (pf "(((P -> F) -> F) -> P)
-> ((all x.(Q x -> P) -> F) -> F)
-> all x Q x
-> P"))
(set-goal qf5m)
(assume 1 2 3)
(use 1)
(assume 4)
(use 2)
(assume "x" 5)
(use 4)
(use 5)
(use 3)
(set-goal qf5m)
(search)
(dnp)
(define qf6 (pf "(all y. P -> Q y) -> P -> all y Q y"))
(set-goal qf6)
(assume 1 2 "y")
(use 1)
(use 2)
(set-goal qf6)
(search)
(dnp)
; qf7m is obtained from the formula (all x.Q x -> P) -> ex x Q x -> P
; by translating `ex' into `not all not' and adding stability of P:
(define qf7m (pf "(((P -> F) -> F) -> P)
-> (all x.Q x -> P)
-> ((all x.Q x -> F) -> F)
-> P"))
(set-goal qf7m)
(assume 1 2 3)
(use 1)
(assume 4)
(use 3)
(assume "x" 5)
(use 4)
(use 2 (pt "x"))
(use 5)
(set-goal qf7m)
(search)
(dnp)
; qf8 is obtained from the formula (ex y.P -> Q y) -> P -> ex y Q y
; by translating `ex' into `not all not':
(define qf8 (pf "((all y.(P -> Q y) -> F) -> F)
-> P
-> (all y.Q y -> F)
-> F"))
(set-goal qf8)
(assume 1 2 3)
(use 1)
(assume "y" 4)
(use 3 (pt "y"))
(use 4)
(use 2)
(set-goal qf8)
(search)
(dnp)
; Some more examples involving classical existence
(define drinker (pf "(all y.((Q y -> F) -> F) -> Q y) ->
exca x.Q x -> all y Q y"))
(set-goal drinker)
(search)
(dnp)
; Now we treat the constructive existential quantifier
(define ex-and1 (pf "(ex x.Q1 x & Q2 x) -> ex x Q1 x & ex x Q2 x"))
(set-goal ex-and1)
(assume 1)
(split)
(ex-elim 1)
(assume "x" 2)
(ex-intro (pt "x"))
(use 2)
(ex-elim 1)
(assume "x" 2)
(ex-intro (pt "x"))
(use 2)
(set-goal ex-and1)
(search)
(dnp)
; Normalized extracted term:
(dnet) ;"[x0]x0@x0"
(define ex-and2 (pf "ex x Q x & P -> ex x.Q x & P"))
(set-goal ex-and2)
(assume 1)
(inst-with 1 'left)
(ex-elim 2)
(assume "x" 3)
(ex-intro (pt "x"))
(split)
(use 3)
(use 1)
(set-goal ex-and2)
(search)
(dnp)
(dnet) ;"[x0]x0"
(define all-ex (pf "all x Q x -> ex x Q x"))
(set-goal all-ex)
(assume 1)
(ex-intro (pt "x"))
(use 1)
(set-goal all-ex)
(search)
(dnp)
(dnet) ;"x"
; qf1 is the formula (all x Q x -> P) -> ex x.Q x -> P
; It is not provable in minimal logic.
; (define qf2 (pf "(P -> all y Q y) -> all y. P -> Q y")) see above
(define qf3 (pf "(ex x Q x -> P) -> all x.Q x -> P"))
(set-goal qf3)
(assume 1 "x" 2)
(use 1)
(ex-intro (pt "x"))
(use 2)
(set-goal qf3)
(search)
(dnp)
; qf4 is the formula (P -> ex y Q y) -> ex y.P -> Q y
; It is not provable in minimal logic.
(define qf5 (pf "(ex x.Q x -> P) -> all x Q x -> P"))
(set-goal qf5)
(assume 1 2)
(ex-elim 1)
(assume "x" 3)
(use 3)
(use 2)
(set-goal qf5)
(assume 1 2)
(ex-elim 1)
(search)
(dnp)
; (define qf6 (pf "(all y. P -> Q y) -> P -> all y Q y")) see above
(define qf7 (pf "(all x.Q x -> P) -> ex x Q x -> P"))
(set-goal qf7)
(assume 1 2)
(ex-elim 2)
(use 1)
(set-goal qf7)
(assume 1 2)
(ex-elim 2)
(search)
(define qf8 (pf "(ex y.P -> Q y) -> P -> ex y Q y"))
(set-goal qf8)
(assume 1 2)
(ex-elim 1)
(assume "x" 3)
(ex-intro (pt "x"))
(use 3)
(use 2)
(set-goal qf8)
(search)
(dnp)
(dnet) ;"[x0]x0"
;Local Variables:
;mode: scheme
;End:
|
