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
|
============================== Prover9 ===============================
Prover9 (32) version 2008-06A, June 2008.
Process 18728 was started by mccune on cleo,
Mon Jun 23 10:45:04 2008
The command was "/home/mccune/LADR/bin/prover9".
============================== end of head ===========================
============================== INPUT =================================
formulas(sos).
(all x all y (subset(x,y) <-> (all z (member(z,x) -> member(z,y))))).
end_of_list.
formulas(goals).
(all x all y all z (subset(x,y) & subset(y,z) -> subset(x,z))).
end_of_list.
============================== end of input ==========================
============================== PROCESS NON-CLAUSAL FORMULAS ==========
% Formulas that are not ordinary clauses:
1 (all x all y (subset(x,y) <-> (all z (member(z,x) -> member(z,y))))) # label(non_clause). [assumption].
2 (all x all y all z (subset(x,y) & subset(y,z) -> subset(x,z))) # label(non_clause) # label(goal). [goal].
============================== end of process non-clausal formulas ===
============================== PROCESS INITIAL CLAUSES ===============
% Clauses before input processing:
formulas(usable).
end_of_list.
formulas(sos).
-subset(x,y) | -member(z,x) | member(z,y). [clausify(1)].
subset(x,y) | member(f1(x,y),x). [clausify(1)].
subset(x,y) | -member(f1(x,y),y). [clausify(1)].
subset(c1,c2). [deny(2)].
subset(c2,c3). [deny(2)].
-subset(c1,c3). [deny(2)].
end_of_list.
formulas(demodulators).
end_of_list.
============================== PREDICATE ELIMINATION =================
Eliminating subset/2
3 subset(x,y) | member(f1(x,y),x). [clausify(1)].
4 -subset(x,y) | -member(z,x) | member(z,y). [clausify(1)].
Derived: member(f1(x,y),x) | -member(z,x) | member(z,y). [resolve(3,a,4,a)].
5 subset(x,y) | -member(f1(x,y),y). [clausify(1)].
Derived: -member(f1(x,y),y) | -member(z,x) | member(z,y). [resolve(5,a,4,a)].
6 subset(c1,c2). [deny(2)].
Derived: -member(x,c1) | member(x,c2). [resolve(6,a,4,a)].
7 subset(c2,c3). [deny(2)].
Derived: -member(x,c2) | member(x,c3). [resolve(7,a,4,a)].
8 -subset(c1,c3). [deny(2)].
Derived: member(f1(c1,c3),c1). [resolve(8,a,3,a)].
Derived: -member(f1(c1,c3),c3). [resolve(8,a,5,a)].
============================== end predicate elimination =============
Auto_denials: (non-Horn, no changes).
Term ordering decisions:
Predicate symbol precedence: predicate_order([ member ]).
Function symbol precedence: function_order([ c1, c2, c3, f1 ]).
After inverse_order: (no changes).
Unfolding symbols: (none).
Auto_inference settings:
% set(binary_resolution). % (non-Horn)
% set(neg_ur_resolution). % (non-Horn, less than 100 clauses)
Auto_process settings:
% set(factor). % (non-Horn)
% set(unit_deletion). % (non-Horn)
kept: 9 member(f1(x,y),x) | -member(z,x) | member(z,y). [resolve(3,a,4,a)].
kept: 10 -member(f1(x,y),y) | -member(z,x) | member(z,y). [resolve(5,a,4,a)].
kept: 11 -member(x,c1) | member(x,c2). [resolve(6,a,4,a)].
kept: 12 -member(x,c2) | member(x,c3). [resolve(7,a,4,a)].
kept: 13 member(f1(c1,c3),c1). [resolve(8,a,3,a)].
kept: 14 -member(f1(c1,c3),c3). [resolve(8,a,5,a)].
============================== end of process initial clauses ========
============================== CLAUSES FOR SEARCH ====================
% Clauses after input processing:
formulas(usable).
end_of_list.
formulas(sos).
9 member(f1(x,y),x) | -member(z,x) | member(z,y). [resolve(3,a,4,a)].
10 -member(f1(x,y),y) | -member(z,x) | member(z,y). [resolve(5,a,4,a)].
11 -member(x,c1) | member(x,c2). [resolve(6,a,4,a)].
12 -member(x,c2) | member(x,c3). [resolve(7,a,4,a)].
13 member(f1(c1,c3),c1). [resolve(8,a,3,a)].
14 -member(f1(c1,c3),c3). [resolve(8,a,5,a)].
end_of_list.
formulas(demodulators).
end_of_list.
============================== end of clauses for search =============
============================== SEARCH ================================
% Starting search at 0.00 seconds.
given #1 (I,wt=11): 9 member(f1(x,y),x) | -member(z,x) | member(z,y). [resolve(3,a,4,a)].
given #2 (I,wt=11): 10 -member(f1(x,y),y) | -member(z,x) | member(z,y). [resolve(5,a,4,a)].
given #3 (I,wt=6): 11 -member(x,c1) | member(x,c2). [resolve(6,a,4,a)].
given #4 (I,wt=6): 12 -member(x,c2) | member(x,c3). [resolve(7,a,4,a)].
given #5 (I,wt=5): 13 member(f1(c1,c3),c1). [resolve(8,a,3,a)].
given #6 (I,wt=5): 14 -member(f1(c1,c3),c3). [resolve(8,a,5,a)].
============================== PROOF =================================
% Proof 1 at 0.00 (+ 0.00) seconds.
% Length of proof is 14.
% Level of proof is 4.
% Maximum clause weight is 6.
% Given clauses 6.
1 (all x all y (subset(x,y) <-> (all z (member(z,x) -> member(z,y))))) # label(non_clause). [assumption].
2 (all x all y all z (subset(x,y) & subset(y,z) -> subset(x,z))) # label(non_clause) # label(goal). [goal].
3 subset(x,y) | member(f1(x,y),x). [clausify(1)].
4 -subset(x,y) | -member(z,x) | member(z,y). [clausify(1)].
5 subset(x,y) | -member(f1(x,y),y). [clausify(1)].
6 subset(c1,c2). [deny(2)].
7 subset(c2,c3). [deny(2)].
8 -subset(c1,c3). [deny(2)].
11 -member(x,c1) | member(x,c2). [resolve(6,a,4,a)].
12 -member(x,c2) | member(x,c3). [resolve(7,a,4,a)].
13 member(f1(c1,c3),c1). [resolve(8,a,3,a)].
14 -member(f1(c1,c3),c3). [resolve(8,a,5,a)].
15 member(f1(c1,c3),c2). [resolve(13,a,11,a)].
18 $F. [ur(12,b,14,a),unit_del(a,15)].
============================== end of proof ==========================
============================== STATISTICS ============================
Given=6. Generated=12. Kept=9. proofs=1.
Usable=6. Sos=3. Demods=0. Limbo=0, Disabled=12. Hints=0.
Kept_by_rule=0, Deleted_by_rule=0.
Forward_subsumed=2. Back_subsumed=0.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0.
Demod_attempts=0. Demod_rewrites=0.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=6.
Megabytes=0.03.
User_CPU=0.00, System_CPU=0.00, Wall_clock=0.
============================== end of statistics =====================
============================== end of search =========================
THEOREM PROVED
Exiting with 1 proof.
Process 18728 exit (max_proofs) Mon Jun 23 10:45:04 2008
|