/
codi_down_sum.all.p9
390 lines (317 loc) · 16.2 KB
/
codi_down_sum.all.p9
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
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
%axioms from module codi/p9/codi_down_sum.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_codi/codi_down_sum.clif
%imports("http://colore.oor.net/multidim_mereotopology_codi/codi_down.clif").
%imports("http://colore.oor.net/multidim_mereotopology_codi/theorems/codi_down_theoremspo-e1.clif").
% 'codi with downwards closure and sums'
% 'sum-a1: sum is commutative'
all x all y ((sum(x, y) = sum(y, x))).
% 'sum-a2: sum of entities of different dimensions'
all x all y ((less(x, y) -> (y = sum(x, y)))).
% 'sum-a3: every part of y is a part of x+y if x<=y'
all x all y all z ((leq(x, y) & cont(z, y) -> cont(z, sum(x, y)))).
% 'sum-a4: everything contained in the sum has a part contained in x or contained in y'
all x all y all z ((cont(z, sum(x, y)) & -(cont(z, x)) -> cont(difference(z, x), y))).
end_of_list.
%axioms from module codi/p9/codi_down.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_codi/codi_down.clif
%imports("http://colore.oor.net/multidim_mereotopology_codi/codi_int.clif").
%imports("http://colore.oor.net/multidim_mereotopology_codi/ep_ext.clif").
%imports("http://colore.oor.net/multidim_mereotopology_codi/defs/ep.clif").
%imports("http://colore.oor.net/multidim_mereotopology_codi/defs/epp.clif").
%imports("http://colore.oor.net/multidim_mereotopology_codi/defs/po.clif").
% 'closure under differences'
% 'dif-a1: difference is of same dimension'
all x all y ((-(zex(difference(x, y))) -> eqdim(x, difference(x, y)))).
% 'dif-a2: difference with a entity of lower dimension'
all x all y ((less(y, x) -> (x = difference(x, y)))).
% 'dif-a3: constitution of the difference with an entity of greater or equal dimension'
all x all y all z ((leq(x, y) -> (cont(z, x) & less(intersection(z, y), z) <-> cont(z, difference(x, y))))).
% 'dif-a4: zero difference only for contained entities or for zero entity'
all x all y ((zex(difference(x, y)) <-> zex(x) | cont(x, y))).
end_of_list.
%axioms from module codi/theorems/p9/codi_down_theoremsPO-E1.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_codi/theorems/codi_down_theoremspo-e1.clif
%imports("http://colore.oor.net/multidim_mereotopology_codi/theorems/codi_down_theoremsep-e3.clif").
%imports("http://colore.oor.net/multidim_mereotopology_codi/theorems/codi_down_theoremsT11.clif").
% 'parthood defined by po'
all x all y ((-(zex(x)) & all z ((po(x, z) -> po(y, z))) -> p(x, y))).
% 'po-e1: extensionality of po'
all x all y (( all z ((po(x, z) <-> po(y, z))) -> (x = y))).
end_of_list.
%axioms from module codi/p9/codi_int.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_codi/codi_int.clif
%imports("http://colore.oor.net/multidim_mereotopology_codi/codi_linear.clif").
%imports("http://colore.oor.net/multidim_mereotopology_zex/zex.clif").
% 'int-a1: disconnected entities have empty intersection'
all x all y ((-(c(x, y)) -> zex(intersection(x, y)))).
% 'int-a2: the intersection is contained in the intersecting entities (also ensures the intersection is of no greater dimension than necessary)'
all x all y ((-(zex(intersection(x, y))) -> cont(intersection(x, y), x))).
% 'int-a3: the intersection is of greatest possible dimension (determines the dimension of the intersection)'
all x all y all z ((cont(z, x) & cont(z, y) -> leq(z, intersection(x, y)))).
% 'int-a4: the intersection contains everything of the greatest possible dimension (and whatever those things contain)'
all x all y all z ((cont(z, x) & cont(z, y) & eqdim(z, intersection(x, y)) <-> p(z, intersection(x, y)))).
end_of_list.
%axioms from module codi/p9/ep_ext.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_codi/ep_ext.clif
% 'theorems of parthood, not required as axiom!'
% 'ep-t1: parthood reflexive'
all x ((-(zex(x)) -> p(x, x))).
% 'ep-t2: parthood antisymmetric'
all x all y ((p(x, y) & p(y, x) -> (x = y))).
% 'ep-t3: parthood transitive'
all x all y all z ((p(x, y) & p(y, z) -> p(x, z))).
% 'ep-t4'
all x all y all z ((p(x, y) & less(z, x) -> less(z, y))).
% 'ep-t5'
all x all y all z ((p(x, y) & less(y, z) -> less(x, z))).
% 'ep-t6'
all x all y all z ((p(x, y) & eqdim(z, x) -> eqdim(z, y))).
% 'ep-t7'
all x all y all z ((p(x, y) & eqdim(z, y) -> eqdim(z, x))).
% 'ep-t8: parthood requires contact'
all x all y all z ((p(x, y) -> c(x, y))).
% 'ep-t9: extensionality of parthood'
all x all y (( all z ((p(z, x) <-> p(z, y))) -> (x = y))).
end_of_list.
%axioms from module codi/defs/p9/ep.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_codi/defs/ep.clif
% 'basic axioms of traditional parthood (parthood of equidimensional entities) which is a non-strict partial order'
% 'parthood holds between two entities of the same spatial dimension'
% 'ep-d: definition of parthood'
all x all y ((p(x, y) <-> cont(x, y) & eqdim(x, y))).
end_of_list.
%axioms from module codi/defs/p9/epp.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_codi/defs/epp.clif
%imports("http://colore.oor.net/multidim_mereotopology_codi/defs/ep.clif").
% 'epp-d: definition of proper parthood'
all x all y ((pp(x, y) <-> p(x, y) & -((x = y)))).
end_of_list.
%axioms from module codi/defs/p9/po.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_codi/defs/po.clif
%imports("http://colore.oor.net/multidim_mereotopology_codi/defs/ep.clif").
% 'po-d: partial overlap (strong contact)'
all x all y ((po(x, y) <-> exists z (p(z, x) & p(z, y)))).
end_of_list.
%axioms from module codi/theorems/p9/codi_down_theoremsEP-E3.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_codi/theorems/codi_down_theoremsep-e3.clif
%imports("http://colore.oor.net/multidim_mereotopology_codi/theorems/codi_down_theoremsep-e2.clif").
% 'ep-e3: strong supplementation for containment'
all x all y ((-(zex(x)) & -(zex(y)) & -(cont(y, x)) -> exists z (p(z, y) & less(intersection(z, x), z)))).
end_of_list.
%axioms from module codi/theorems/p9/codi_down_theoremsT11.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_codi/theorems/codi_down_theoremst5.clif
%imports("http://colore.oor.net/multidim_mereotopology_codi/codi_down.clif").
%imports("http://colore.oor.net/multidim_mereotopology_codi/defs/min_max_in_dim.clif").
% 'dif-a3: any minimal entity contained in x is contained in either y or x-y. if it were contained in neither but in contact to both, by int-a1 to int-a3, intersections would exist.'
all x all y all z ((p(y, x) & min(z) & cont(z, x) -> cont(z, y) | cont(z, difference(x, y)))).
end_of_list.
%axioms from module codi/p9/codi_linear.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_codi/codi.clif
%imports("http://colore.oor.net/multidim_mereotopology_dim/dim_prime_linear.clif").
%imports("http://colore.oor.net/multidim_mereotopology_cont/cont_c_ext.clif").
%imports("http://colore.oor.net/multidim_mereotopology_codi/defs/ep.clif").
%imports("http://colore.oor.net/multidim_mereotopology_codi/defs/epp.clif").
%imports("http://colore.oor.net/multidim_mereotopology_codi/defs/po.clif").
%imports("http://colore.oor.net/multidim_mereotopology_codi/defs/inc.clif").
%imports("http://colore.oor.net/multidim_mereotopology_codi/defs/sc.clif").
%imports("http://colore.oor.net/multidim_mereotopology_codi/defs/min_max_in_dim.clif").
% 'cd-a1: containment requires that the contained entity has a dimension that is lower or equal to the entity containing it'
all x all y ((cont(x, y) -> leq(x, y))).
end_of_list.
%axioms from module zex/p9/zex.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_dim/zex.clif
% 'forces a special null region (zex)'
% 'z-a1: there exists a zex'
exists x (zex(x)).
end_of_list.
%axioms from module codi/theorems/p9/codi_down_theoremsEP-E2.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_codi/theorems/codi_down_theoremsep-e2.clif
%imports("http://colore.oor.net/multidim_mereotopology_codi/theorems/codi_down_theoremsep-e1.clif").
% 'ep-e2.ii: strong supplementation'
all x all y ((-(zex(x)) & -(zex(y)) & -(p(y, x)) & eqdim(x, y) & po(x, y) & -(pp(y, x)) -> exists z (p(z, y) & -(po(z, x))))).
% 'ep-e2.iii: strong supplementation'
all x all y ((-(zex(x)) & -(zex(y)) & -(p(y, x)) & eqdim(x, y) & -(po(x, y)) -> exists z (p(z, y) & -(po(z, x))))).
% 'ep-e2.iv: strong supplementation'
all x all y ((-(zex(x)) & -(zex(y)) & -(p(y, x)) & -(eqdim(x, y)) -> exists z (p(z, y) & -(po(z, x))))).
end_of_list.
%axioms from module codi/defs/p9/min_max_in_dim.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_codi/defs/min_max_in_dim.clif
%imports("http://colore.oor.net/multidim_mereotopology_codi/defs/epp.clif").
% 'definitional extension'
% 'me-d1: defining maximal entities of a dimension'
all x ((max(x) <-> -(zex(x)) & all y (-(pp(x, y))))).
% 'me-d2: defining minimal entities of a dimension'
all x ((min(x) <-> -(zex(x)) & all y (-(pp(y, x))))).
end_of_list.
%axioms from module dim/p9/dim_prime_linear.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_dim/dim_prime_linear.clif
% 'compact axioms for linear relative dimension'
% 'd-a1: irreflexive'
all x (-(less(x, x))).
% 'd-a2: asymmetric (antisymmetric & irreflexive)'
all x all y ((less(x, y) -> -(less(y, x)))).
% 'd-a3: transitive (corrected from ijcai version: need <= in the second condition to ensure that eqdim is transitive)'
all x all y all z ((less(x, y) & leq(y, z) -> less(x, z))).
% 'd-a4: unique zex'
all x all y ((zex(x) & zex(y) -> (x = y))).
% 'd-a5: zex has dimension lower than anything else'
all x all y ((zex(x) & -(zex(y)) -> less(x, y))).
% 'd-a6: a lowest dimension must exist'
exists x (mindim(x)).
% 'd-d2: define eqdim in terms of <'
all x all y ((eqdim(x, y) <-> -(less(x, y)) & -(less(y, x)))).
% 'd-d3: <='
all x all y ((leq(x, y) <-> less(x, y) | eqdim(x, y))).
% 'd-d4: maximal dimension (codim=0)'
all x ((maxdim(x) <-> all y (leq(y, x)))).
% 'd-d5: minimal dimension (dim=0)'
all x ((mindim(x) <-> -(zex(x)) & all y ((less(y, x) -> zex(y))))).
% 'd-d6: next highest dimension (covers), x covers y if x>y and no z s.t. x>z>y'
all x all y ((covers(x, y) <-> less(y, x) & all z (-(less(y, z) & less(z, x))))).
end_of_list.
%axioms from module cont/p9/cont_c_ext.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_cont/cont_c-ext.clif
%imports("http://colore.oor.net/multidim_mereotopology_cont/cont_c.clif").
%imports("http://colore.oor.net/multidim_mereotopology_cont/cont_ext.clif").
end_of_list.
%axioms from module codi/defs/p9/inc.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_codi/defs/inc.clif
% 'basic axioms of incidence (two entities occupy some same space. it is reserved for entities of differing dimensions. it is a variant of contact.'
%imports("http://colore.oor.net/multidim_mereotopology_codi/defs/ep.clif").
% 'inc-d: incidence holds if and only if some entity of the dimension as the lower of the two incident entities is shared'
all x all y ((inc(x, y) <-> exists z (less(z, x) & cont(z, x) & p(z, y)) | exists z (less(z, y) & cont(z, y) & p(z, x)))).
end_of_list.
%axioms from module codi/defs/p9/sc.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_codi/defs/sc.clif
% 'sc-d: superficial (weak) contact (contact is of a lesser dimension than either involved entity) - prover9 file needs extra parentheses'
all x all y ((sc(x, y) <-> exists z (cont(z, x) & cont(z, y)) & all z ((cont(z, x) & cont(z, y) -> less(z, x) & less(z, y))))).
end_of_list.
%axioms from module codi/theorems/p9/codi_down_theoremsEP-E1.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_codi/theorems/codi_down_theoremsep-e1.clif
%imports("http://colore.oor.net/multidim_mereotopology_codi/codi_down.clif").
%imports("http://colore.oor.net/multidim_mereotopology_codi/theorems/codi_down_theoremst1-t2.clif").
%imports("http://colore.oor.net/multidim_mereotopology_codi/theorems/codi_down_theoremst3-t5.clif").
% 'ep-e1: weak supplementation - original'
all x all y ((pp(x, y) -> exists z (p(z, y) & -(po(z, x))))).
end_of_list.
%axioms from module cont/p9/cont_c.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_cont/cont_c.clif
%imports("http://colore.oor.net/multidim_mereotopology_cont/cont_basic.clif").
% 'c-d: definition of contact in terms of containment'
all x all y ((c(x, y) <-> exists z (cont(z, x) & cont(z, y)))).
end_of_list.
%axioms from module cont/p9/cont_ext.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_cont/cont_ext.clif
%imports("http://colore.oor.net/multidim_mereotopology_cont/cont_basic.clif").
% 'c-e1: cont is extensional'
all x all y (( all z ((cont(z, x) <-> cont(z, y))) -> (x = y))).
end_of_list.
%axioms from module codi/theorems/p9/codi_down_theoremsT1-T2.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_codi/theorems/codi_down_theoremst1-t2.clif
%imports("http://colore.oor.net/multidim_mereotopology_codi/codi_down.clif").
% 'dif-t1'
all x all y ((-(zex(difference(x, y))) -> p(difference(x, y), x))).
% 'dif-t2 - easier'
all x all y ((pp(y, x) & -(zex(difference(x, y))) & p(difference(x, y), x) -> pp(difference(x, y), x))).
% 'dif-t2 - full'
all x all y ((pp(y, x) -> pp(difference(x, y), x))).
end_of_list.
%axioms from module codi/theorems/p9/codi_down_theoremsT3-T5.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_codi/theorems/codi_down_theoremst3-t5.clif
%imports("http://colore.oor.net/multidim_mereotopology_codi/codi_down.clif").
% 'dif-t3'
all x all y ((-(zex(y)) & -(zex(difference(x, y))) -> -(po(y, difference(x, y))))).
% 'dif-t4'
all x all y (-(po(intersection(x, y), difference(x, y)))).
% 'dif-t5a'
all x all y all z ((p(z, difference(x, y)) -> p(z, x))).
% 'dif-t5b'
all x all y all z ((p(z, difference(x, y)) -> -(po(z, intersection(x, y))))).
% 'dif-t5c'
all x all y all z ((p(z, x) & -(po(z, intersection(x, y))) -> p(z, difference(x, y)))).
end_of_list.
%axioms from module cont/p9/cont_basic.p9
%----------------------------------
% automatically generated from cl/kif
formulas(sos).
% Module http://colore.oor.net/multidim_mereotopology_cont/cont_basic.clif
% 'basic axioms of containment (parthood irrespective of dimension) which is a non-strict partial order'
% 'a possibly existing zero extent zex is included'
% 'c-a1: reflexive'
all x ((-(zex(x)) <-> cont(x, x))).
% 'c-a2: antisymmetric'
all x all y ((cont(x, y) & cont(y, x) -> (x = y))).
% 'c-a3: transitive'
all x all y all z ((cont(x, y) & cont(y, z) -> cont(x, z))).
% 'c-a4: zexs are not contained in anything and contain nothing'
all x all y ((zex(x) -> -(cont(y, x)) & -(cont(x, y)))).
end_of_list.