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- Committer: Package Import Robot
- Author(s): Frank Lichtenheld
- Date: 2013-05-25 11:43:32 UTC
- mfrom: (5.1.5 sid)
- Revision ID: package-import@ubuntu.com-20130525114332-lkzco1dti2hwrf7v
Tags: 0.0.200911a-2
* QA upload.
* Upload to unstable.
* Change maintainer to QA group.
* Upload to unstable.
* Change maintainer to QA group.
- files added:
- .pc
- .pc/01-libtoolise.patch
- .pc/01-libtoolise.patch/apps.src
- .pc/01-libtoolise.patch/ladr
- .pc/01-libtoolise.patch/mace4.src
- .pc/01-libtoolise.patch/provers.src
- .pc/01-libtoolise.patch/test.src
- .pc/03-no-2.5isms.patch
- .pc/03-no-2.5isms.patch/utilities
- debian/source
- files removed:
- Changelog.save
- files modified:
- Changelog
added
removed
provers.src/autosketches.examples/BA.out
1
1
============================== NewAuto ===============================
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NewAuto (32) version August-2006A, August 2006.
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Process 13961 was started by mccune on cleo.thornwood,
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Sun Aug 13 21:04:57 2006
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NewAuto (32) version 2009-02A, February 2009.
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Process 10934 was started by mccune on cleo,
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Mon Sep 28 11:57:09 2009
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The command was "newauto -f BA.in".
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============================== end of head ===========================
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assign(max_megs,750).
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assign(sos_limit,20000).
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assign(demod_step_limit,200).
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assign(demod_size_limit,100000).
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assign(demod_increase_limit,100000).
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clear(print_kept).
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clear(print_given).
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============================== PROCESS NON-CLAUSAL FORMULAS ==========
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% Formulas that are not ordinary clauses:
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1 f(x,y) = f(y,x) # label("Commutativity"). [goal].
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1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
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============================== end of process non-clausal formulas ===
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f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
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end_of_list.
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Child search process 13962 started.
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Child search process 10935 started.
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============================== PROCESS INITIAL CLAUSES ===============
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Term ordering decisions:
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Function symbol KB weights: c1=1. c2=1. f=1.
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Relation symbol precedence: lex([ = ]).
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Function symbol precedence: lex([ c1, c2, f ]).
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Predicate symbol precedence: predicate_order([ = ]).
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Function symbol precedence: function_order([ c1, c2, f ]).
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Skipping inverse_order, because term ordering is KBO.
133
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Unfolding symbols: (none).
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Auto_inference settings:
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% set(paramodulation). % (positive equality literals)
137
% set(paramodulation) -> set(back_demod).
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Auto_process settings: (no changes).
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kept: 2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
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kept: 3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
142
kept: 4 f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m"). [assumption].
143
kept: 5 f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y # label("C1/A1"). [assumption].
144
kept: 6 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
145
kept: 7 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
146
kept: 8 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
147
kept: 9 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
148
kept: 10 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
149
kept: 11 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
150
kept: 12 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
151
kept: 13 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
152
kept: 14 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
153
kept: 15 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
154
kept: 16 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
155
kept: 17 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
156
kept: 18 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
157
kept: 19 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
158
kept: 20 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
159
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============================== end of process initial clauses ========
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============================== CLAUSES FOR SEARCH ====================
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============================== PROOF =================================
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% Proof 1 at 1.03 (+ 0.00) seconds: "Commutativity".
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% Proof 1 at 1.02 (+ 0.00) seconds: "Commutativity".
204
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% Length of proof is 36.
205
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% Level of proof is 12.
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% Maximum clause weight is 37.
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% Given clauses 52.
208
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1 f(x,y) = f(y,x) # label("Commutativity"). [goal].
228
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
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2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
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3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
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5 f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y # label("C1/A1"). [assumption].
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21 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)). [para(2(a,1),2(a,1,1,2,1))].
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23 f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)). [para(2(a,1),2(a,1,2))].
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33 f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x. [para(5(a,1),2(a,1,1,2))].
224
929 f(f(f(x,f(x,x)),x),f(x,x)) = x. [para(2(a,1),33(a,1,2,2))].
225
1074 f(x,f(x,f(x,x))) = f(x,x). [para(5(a,1),929(a,1,1,1,2)),rewrite(18(8),5(7)),flip(a)].
243
1022 f(f(f(x,f(x,x)),x),f(x,x)) = x. [para(2(a,1),33(a,1,2,2))].
244
1074 f(x,f(x,f(x,x))) = f(x,x). [para(5(a,1),1022(a,1,1,1,2)),rewrite([18(8),5(7)]),flip(a)].
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1120 f(f(x,x),f(x,f(x,y))) = x. [para(1074(a,1),5(a,1,1))].
227
1121 f(f(x,x),f(x,x)) = x. [para(1074(a,1),5(a,1,2)),rewrite(1074(3))].
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1121 f(f(x,x),f(x,x)) = x. [para(1074(a,1),5(a,1,2)),rewrite([1074(3)])].
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1123 f(f(x,x),f(x,f(y,x))) = x. [para(1074(a,1),6(a,1,1))].
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1125 f(f(f(x,x),x),f(x,x)) = x. [para(1074(a,1),6(a,1,2)),rewrite(1120(5))].
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1130 f(f(x,f(x,x)),x) = f(x,x). [para(1074(a,1),9(a,1,1,2)),rewrite(1121(5))].
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1125 f(f(f(x,x),x),f(x,x)) = x. [para(1074(a,1),6(a,1,2)),rewrite([1120(5)])].
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1130 f(f(x,f(x,x)),x) = f(x,x). [para(1074(a,1),9(a,1,1,2)),rewrite([1121(5)])].
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1134 f(f(x,f(f(x,x),x)),f(x,x)) = x. [para(1074(a,1),10(a,1,2))].
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1139 f(f(x,f(x,x)),f(x,x)) = x. [para(1074(a,1),12(a,1,2)),rewrite(1125(4))].
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1160 f(x,f(f(x,x),x)) = f(x,x). [para(1074(a,1),21(a,1,2,2)),rewrite(1074(5),1139(4),1134(5)),flip(a)].
251
1139 f(f(x,f(x,x)),f(x,x)) = x. [para(1074(a,1),12(a,1,2)),rewrite([1125(4)])].
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1160 f(x,f(f(x,x),x)) = f(x,x). [para(1074(a,1),21(a,1,2,2)),rewrite([1074(5),1139(4),1134(5)]),flip(a)].
234
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1206 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [para(1121(a,1),15(a,1,1,1))].
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1561 f(f(x,x),f(x,f(y,f(f(x,y),y)))) = x. [para(23(a,1),1123(a,1,2,2))].
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1568 f(f(x,x),x) = f(x,f(x,x)). [para(1139(a,1),2(a,1,2)),rewrite(1130(3),1160(3))].
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1868 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)). [para(1206(a,1),9(a,1,1,2)),rewrite(1130(3),1130(6))].
238
1877 f(f(x,f(x,f(y,f(x,x)))),x) = f(x,x). [para(1206(a,1),13(a,1,1,1)),rewrite(1120(10))].
239
2569 f(f(x,f(y,y)),f(x,y)) = x. [para(1561(a,1),10(a,1,2,2)),rewrite(1868(6))].
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2615 f(f(x,y),f(x,f(y,y))) = x. [para(2(a,1),2569(a,1,1,2)),rewrite(1568(3),1074(4))].
241
2659 f(f(x,x),y) = f(y,f(x,x)). [para(2569(a,1),14(a,1,2)),rewrite(1877(5))].
242
2802 f(x,f(f(x,x),x)) = f(x,x). [para(1121(a,1),2615(a,1,1)),rewrite(1121(4))].
243
2949 f(x,y) = f(y,x). [para(2(a,1),2659(a,1,1)),rewrite(2802(4),2802(5),1121(4))].
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2950 $F # answer("Commutativity"). [resolve(2949,a,3,a)].
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1531 f(f(x,x),f(x,f(y,f(f(x,y),y)))) = x. [para(23(a,1),1123(a,1,2,2))].
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1567 f(f(x,x),x) = f(x,f(x,x)). [para(1139(a,1),2(a,1,2)),rewrite([1130(3),1160(3)])].
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1866 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)). [para(1206(a,1),9(a,1,1,2)),rewrite([1130(3),1130(6)])].
257
1875 f(f(x,f(x,f(y,f(x,x)))),x) = f(x,x). [para(1206(a,1),13(a,1,1,1)),rewrite([1120(10)])].
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2567 f(f(x,f(y,y)),f(x,y)) = x. [para(1531(a,1),10(a,1,2,2)),rewrite([1866(6)])].
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2613 f(f(x,y),f(x,f(y,y))) = x. [para(2(a,1),2567(a,1,1,2)),rewrite([1567(3),1074(4)])].
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2657 f(f(x,x),y) = f(y,f(x,x)). [para(2567(a,1),14(a,1,2)),rewrite([1875(5)])].
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2909 f(x,f(f(x,x),x)) = f(x,x). [para(1121(a,1),2613(a,1,1)),rewrite([1121(4)])].
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2954 f(x,y) = f(y,x). [para(2(a,1),2657(a,1,1)),rewrite([2909(4),2909(5),1121(4)])].
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2955 $F # answer("Commutativity"). [resolve(2954,a,3,a)].
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============================== end of proof ==========================
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============================== STATISTICS ============================
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Given=52. Generated=7039. Kept=2948. proofs=1.
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Usable=45. Sos=2545. Demods=2592. Limbo=7, Disabled=369. Hints=0.
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Weight_deleted=129. Literals_deleted=0.
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Forward_subsumed=3962. Back_subsumed=151.
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Given=52. Generated=7036. Kept=2953. proofs=1.
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Usable=45. Sos=2543. Demods=2590. Limbo=7, Disabled=376. Hints=0.
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Kept_by_rule=0, Deleted_by_rule=129.
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Forward_subsumed=3954. Back_subsumed=152.
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Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
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New_demodulators=2942 (0 lex), Back_demodulated=199. Back_unit_deleted=0.
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Demod_attempts=180112. Demod_rewrites=11794.
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New_demodulators=2947 (0 lex), Back_demodulated=205. Back_unit_deleted=0.
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Demod_attempts=180094. Demod_rewrites=11763.
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Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
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Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
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Megabytes=4.65.
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User_CPU=1.03, System_CPU=0.00, Wall_clock=1.
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Megabytes=4.88.
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User_CPU=1.02, System_CPU=0.00, Wall_clock=1.
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============================== end of statistics =====================
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Exiting with 1 proof.
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Process 13962 exit (max_proofs) Sun Aug 13 21:04:58 2006
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Process 10935 exit (max_proofs) Mon Sep 28 11:57:10 2009
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Successful proof using the following 9 assumptions:
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Including 60 new hint clauses:
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formulas(new_hints).
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$F # label("Commutativity").
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$F # label("Commutativity") # label(non_clause) # label(goal).
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f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
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f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
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f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y # label("C1/A1").
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f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
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end_of_list.
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Child search process 13963 started.
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Child search process 10936 started.
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============================== PROCESS INITIAL CLAUSES ===============
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Term ordering decisions:
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Function symbol KB weights: c1=1. c2=1. f=1.
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Relation symbol precedence: lex([ = ]).
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Function symbol precedence: lex([ c1, c2, f ]).
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Predicate symbol precedence: predicate_order([ = ]).
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Function symbol precedence: function_order([ c1, c2, f ]).
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Skipping inverse_order, because term ordering is KBO.
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Unfolding symbols: (none).
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447
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Auto_inference settings:
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% set(paramodulation). % (positive equality literals)
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% set(paramodulation) -> set(back_demod).
432
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Auto_process settings: (no changes).
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kept: 2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
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kept: 3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
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kept: 4 f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m"). [assumption].
456
kept: 5 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
457
kept: 6 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
458
kept: 7 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
459
kept: 8 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
460
kept: 9 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
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kept: 10 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
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kept: 11 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
463
kept: 12 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
464
kept: 13 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
465
kept: 14 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
466
kept: 15 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
467
kept: 16 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
468
kept: 17 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
469
kept: 18 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
470
kept: 19 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
471
435
472
============================== end of process initial clauses ========
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473
437
474
============================== CLAUSES FOR SEARCH ====================
482
519
19 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
483
520
end_of_list.
484
521
485
% 60 hints processed (3 are redundant).
522
% 47 hints (60 processed, 13 redundant).
486
523
487
524
============================== end of clauses for search =============
488
525
494
531
495
532
============================== PROOF =================================
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533
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% Proof 1 at 0.88 (+ 0.00) seconds: "Commutativity".
498
% Length of proof is 45.
499
% Level of proof is 12.
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% Maximum clause weight is 81.
501
% Given clauses 45.
534
% Proof 1 at 0.40 (+ 0.01) seconds: "Commutativity".
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% Length of proof is 28.
536
% Level of proof is 11.
537
% Maximum clause weight is 37.
538
% Given clauses 34.
502
539
503
1 f(x,y) = f(y,x) # label("Commutativity"). [goal].
540
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
504
541
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
505
542
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
506
543
4 f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m"). [assumption].
507
544
5 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
508
545
8 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
509
546
9 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
510
12 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
511
547
13 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
512
548
14 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
513
549
19 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
514
550
20 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)). [para(2(a,1),2(a,1,1,2,1))].
515
22 f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)). [para(2(a,1),2(a,1,2))].
516
551
23 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(f(z,f(x,z)),z),f(u,f(x,f(y,z))))) = f(f(z,f(x,z)),z). [para(4(a,1),2(a,1,1,2,1))].
517
552
32 f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x. [para(5(a,1),2(a,1,1,2))].
518
553
837 f(f(f(x,f(x,x)),x),f(x,x)) = x. [para(2(a,1),32(a,1,2,2))].
519
554
877 f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y. [para(837(a,1),2(a,1,2,2))].
520
948 f(f(f(x,f(y,x)),f(x,f(x,f(y,x)))),x) = f(x,f(f(x,x),x)). [para(9(a,1),20(a,1,2))].
521
1008 f(f(f(x,y),f(f(z,f(x,y)),f(x,y))),f(z,x)) = z. [para(877(a,1),2(a,1,2,2))].
522
1018 f(f(f(x,y),f(f(x,y),f(z,f(x,y)))),f(z,x)) = z. [para(877(a,1),5(a,1,2,2))].
523
1479 f(f(x,f(x,x)),x) = f(x,f(f(x,x),x)). [para(19(a,1),23(a,1,2)),rewrite(948(7)),flip(a)].
524
1485 f(f(f(f(x,f(f(x,x),x)),f(y,f(x,x))),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(y,f(x,x))))),f(f(f(f(x,x),x),f(x,x)),f(z,f(f(x,f(f(x,x),x)),f(y,f(x,x)))))) = f(f(f(x,x),f(f(x,f(f(x,x),x)),f(x,x))),f(x,x)). [para(837(a,1),23(a,1,2,1,1,2)),rewrite(1479(3),1479(9),1479(12),1479(24),1479(34))].
525
1577 f(f(x,f(f(x,x),x)),f(x,x)) = x. [back_rewrite(837),rewrite(1479(3))].
526
1582 f(f(f(f(x,f(f(x,x),x)),f(y,f(x,x))),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(y,f(x,x))))),f(f(f(f(x,x),x),f(x,x)),f(z,f(f(x,f(f(x,x),x)),f(y,f(x,x)))))) = f(f(f(x,x),x),f(x,x)). [back_rewrite(1485),rewrite(1577(36))].
527
1584 f(x,f(f(x,x),x)) = f(x,x). [para(2(a,1),1577(a,1,1,2,1)),rewrite(9(8),2(7)),flip(a)].
528
1587 f(f(f(x,x),x),x) = f(x,x). [para(1577(a,1),4(a,1,2)),rewrite(1584(3),1584(3),1584(6))].
529
1593 f(f(f(x,x),x),f(x,x)) = x. [para(1577(a,1),8(a,1,1,2)),rewrite(1584(3),1584(5),1584(5))].
530
1601 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [para(1577(a,1),14(a,1,1,1)),rewrite(1584(3),1584(5))].
531
1633 f(f(f(x,x),f(f(x,x),x)),f(f(x,x),x)) = f(x,x). [para(1577(a,1),1018(a,1,1,2,2)),rewrite(1584(7),1584(10))].
532
1639 f(f(f(f(x,x),f(y,f(x,x))),f(f(x,x),f(f(x,x),f(y,f(x,x))))),f(x,f(z,f(f(x,x),f(y,f(x,x)))))) = f(f(x,x),f(x,x)). [para(1577(a,1),23(a,1,2,1,1,2)),rewrite(1584(3),1584(7),1584(8),1593(15),1584(14),1584(22),1479(25),1584(25))].
533
1644 f(f(x,x),f(x,x)) = x. [back_rewrite(1582),rewrite(1584(3),1584(7),1584(8),1593(15),1584(14),1639(18),1593(7))].
534
1689 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(1479),rewrite(1584(6))].
535
1700 f(f(x,x),f(x,f(y,x))) = x. [para(1584(a,1),2(a,1,1))].
536
1707 f(f(x,x),f(x,f(x,y))) = x. [para(1584(a,1),9(a,1,1))].
537
1757 f(f(x,x),x) = f(x,f(x,x)). [para(1593(a,1),1008(a,1,1,2,1)),rewrite(1707(4),1587(3)),flip(a)].
538
1797 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(1633),rewrite(1757(3),1707(4),1757(2))].
539
1978 f(f(x,x),f(x,f(y,f(f(x,y),y)))) = x. [para(22(a,1),1700(a,1,2,2))].
540
2259 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)). [para(1601(a,1),8(a,1,1,2)),rewrite(1689(3),1689(6))].
541
2267 f(f(x,f(x,f(y,f(x,x)))),x) = f(x,x). [para(1601(a,1),12(a,1,1,1)),rewrite(1707(10))].
542
2415 f(f(x,f(y,y)),f(x,y)) = x. [para(1978(a,1),9(a,1,2,2)),rewrite(2259(6))].
543
2467 f(f(x,y),f(x,f(y,y))) = x. [para(2(a,1),2415(a,1,1,2)),rewrite(1757(3),1797(4))].
544
2507 f(f(x,x),y) = f(y,f(x,x)). [para(2415(a,1),13(a,1,2)),rewrite(2267(5))].
545
2658 f(x,f(f(x,x),x)) = f(x,x). [para(1644(a,1),2467(a,1,1)),rewrite(1644(4))].
546
2688 f(x,y) = f(y,x). [para(2(a,1),2507(a,1,1)),rewrite(2658(4),2658(5),1644(4))].
547
2689 $F # answer("Commutativity"). [resolve(2688,a,3,a)].
555
966 f(f(f(x,f(x,f(f(x,x),f(f(y,f(x,x)),f(x,x))))),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))). [para(877(a,1),13(a,1,2))].
556
1119 f(f(f(x,f(y,x)),f(x,f(x,f(y,x)))),x) = f(x,f(f(x,x),x)). [para(9(a,1),20(a,1,2))].
557
1267 f(f(x,f(x,x)),x) = f(x,f(f(x,x),x)). [para(19(a,1),23(a,1,2)),rewrite([1119(7)]),flip(a)].
558
1350 f(f(x,f(f(x,x),x)),f(x,x)) = x. [back_rewrite(837),rewrite([1267(3)])].
559
1358 f(x,f(f(x,x),x)) = f(x,x). [para(2(a,1),1350(a,1,1,2,1)),rewrite([9(8),2(7)]),flip(a)].
560
1367 f(f(f(x,x),x),f(x,x)) = x. [para(1350(a,1),8(a,1,1,2)),rewrite([1358(3),1358(5),1358(5)])].
561
1373 f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x). [para(1350(a,1),13(a,1,1,1,2)),rewrite([1358(3),1358(5),1367(4),1358(4)])].
562
1375 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [para(1350(a,1),14(a,1,1,1)),rewrite([1358(3),1358(5)])].
563
1451 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(1267),rewrite([1358(6)])].
564
1458 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y). [back_rewrite(966),rewrite([1373(7),1451(3)]),flip(a)].
565
1776 f(f(x,x),y) = f(y,f(x,x)). [para(1375(a,1),8(a,1,1,2)),rewrite([1451(3),1451(6),1458(6)])].
566
1813 f(x,y) = f(y,x). [para(2(a,1),1776(a,1,1)),rewrite([2(8)])].
567
1814 $F # answer("Commutativity"). [resolve(1813,a,3,a)].
548
568
549
569
============================== end of proof ==========================
550
570
551
571
============================== STATISTICS ============================
552
572
553
Given=45. Generated=5918. Kept=2687. proofs=1.
554
Usable=37. Sos=2126. Demods=2166. Limbo=7, Disabled=534. Hints=60.
555
Weight_deleted=135. Literals_deleted=0.
556
Forward_subsumed=3096. Back_subsumed=198.
573
Given=34. Generated=3404. Kept=1812. proofs=1.
574
Usable=29. Sos=1389. Demods=1423. Limbo=7, Disabled=404. Hints=60.
575
Kept_by_rule=0, Deleted_by_rule=53.
576
Forward_subsumed=1539. Back_subsumed=85.
557
577
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
558
New_demodulators=2682 (0 lex), Back_demodulated=318. Back_unit_deleted=0.
559
Demod_attempts=152649. Demod_rewrites=9595.
578
New_demodulators=1807 (0 lex), Back_demodulated=301. Back_unit_deleted=0.
579
Demod_attempts=78219. Demod_rewrites=4851.
560
580
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
561
581
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
562
Megabytes=3.94.
563
User_CPU=0.88, System_CPU=0.00, Wall_clock=2.
582
Megabytes=2.61.
583
User_CPU=0.40, System_CPU=0.01, Wall_clock=2.
564
584
565
585
============================== end of statistics =====================
566
586
568
588
569
589
Exiting with 1 proof.
570
590
571
Process 13963 exit (max_proofs) Sun Aug 13 21:04:59 2006
591
Process 10936 exit (max_proofs) Mon Sep 28 11:57:11 2009
572
592
573
Successful proof using the following 8 assumptions:
593
Successful proof using the following 7 assumptions:
574
594
575
595
formulas(assumptions_in_proof).
576
596
f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m"). [assumption].
577
597
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
578
598
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
579
599
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
580
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
581
600
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
582
601
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
583
602
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
584
603
end_of_list.
585
604
586
Including 95 new hint clauses:
605
Including 48 new hint clauses:
587
606
588
607
formulas(new_hints).
589
$F # label("Commutativity").
608
$F # label("Commutativity") # label(non_clause) # label(goal).
590
609
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
591
610
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
592
611
f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m").
593
612
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").
594
613
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").
595
614
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
596
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
597
615
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
598
616
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").
599
617
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
600
618
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)).
601
f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)).
602
619
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(f(z,f(x,z)),z),f(u,f(x,f(y,z))))) = f(f(z,f(x,z)),z).
603
620
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
604
621
f(f(f(x,f(x,x)),x),f(x,x)) = x.
605
622
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
623
f(f(f(x,f(x,f(f(x,x),f(f(y,f(x,x)),f(x,x))))),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
606
624
f(f(f(x,f(y,x)),f(x,f(x,f(y,x)))),x) = f(x,f(f(x,x),x)).
607
f(f(f(x,y),f(f(z,f(x,y)),f(x,y))),f(z,x)) = z.
608
f(f(f(x,y),f(f(x,y),f(z,f(x,y)))),f(z,x)) = z.
609
625
f(f(f(x,f(y,x)),f(x,f(x,f(y,x)))),x) = f(f(x,f(x,x)),x).
610
626
f(x,f(f(x,x),x)) = f(f(x,f(x,x)),x).
611
627
f(f(x,f(x,x)),x) = f(x,f(f(x,x),x)).
612
f(f(f(f(f(x,f(x,x)),x),f(y,f(x,x))),f(f(f(x,f(x,x)),x),f(f(f(x,f(x,x)),x),f(y,f(x,x))))),f(f(f(f(x,x),x),f(x,x)),f(z,f(f(f(x,f(x,x)),x),f(y,f(x,x)))))) = f(f(f(x,x),f(f(f(x,f(x,x)),x),f(x,x))),f(x,x)).
613
f(f(f(f(x,f(f(x,x),x)),f(y,f(x,x))),f(f(f(x,f(x,x)),x),f(f(f(x,f(x,x)),x),f(y,f(x,x))))),f(f(f(f(x,x),x),f(x,x)),f(z,f(f(f(x,f(x,x)),x),f(y,f(x,x)))))) = f(f(f(x,x),f(f(f(x,f(x,x)),x),f(x,x))),f(x,x)).
614
f(f(f(f(x,f(f(x,x),x)),f(y,f(x,x))),f(f(x,f(f(x,x),x)),f(f(f(x,f(x,x)),x),f(y,f(x,x))))),f(f(f(f(x,x),x),f(x,x)),f(z,f(f(f(x,f(x,x)),x),f(y,f(x,x)))))) = f(f(f(x,x),f(f(f(x,f(x,x)),x),f(x,x))),f(x,x)).
615
f(f(f(f(x,f(f(x,x),x)),f(y,f(x,x))),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(y,f(x,x))))),f(f(f(f(x,x),x),f(x,x)),f(z,f(f(f(x,f(x,x)),x),f(y,f(x,x)))))) = f(f(f(x,x),f(f(f(x,f(x,x)),x),f(x,x))),f(x,x)).
616
f(f(f(f(x,f(f(x,x),x)),f(y,f(x,x))),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(y,f(x,x))))),f(f(f(f(x,x),x),f(x,x)),f(z,f(f(x,f(f(x,x),x)),f(y,f(x,x)))))) = f(f(f(x,x),f(f(f(x,f(x,x)),x),f(x,x))),f(x,x)).
617
f(f(f(f(x,f(f(x,x),x)),f(y,f(x,x))),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(y,f(x,x))))),f(f(f(f(x,x),x),f(x,x)),f(z,f(f(x,f(f(x,x),x)),f(y,f(x,x)))))) = f(f(f(x,x),f(f(x,f(f(x,x),x)),f(x,x))),f(x,x)).
618
628
f(f(x,f(f(x,x),x)),f(x,x)) = x.
619
f(f(f(f(x,f(f(x,x),x)),f(y,f(x,x))),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(y,f(x,x))))),f(f(f(f(x,x),x),f(x,x)),f(z,f(f(x,f(f(x,x),x)),f(y,f(x,x)))))) = f(f(f(x,x),x),f(x,x)).
620
629
f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
621
630
f(x,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
622
631
f(x,x) = f(x,f(f(x,x),x)).
623
632
f(x,f(f(x,x),x)) = f(x,x).
624
f(f(f(x,f(f(x,f(f(x,x),x)),x)),x),x) = f(x,f(f(x,x),x)).
625
f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)).
626
f(f(f(x,x),x),x) = f(x,f(f(x,x),x)).
627
f(f(f(x,x),x),x) = f(x,x).
628
633
f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,f(f(x,x),x)),x))) = x.
629
634
f(f(f(x,x),x),f(x,f(f(x,f(f(x,x),x)),x))) = x.
630
635
f(f(f(x,x),x),f(x,f(f(x,x),x))) = x.
631
636
f(f(f(x,x),x),f(x,x)) = x.
637
f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),f(f(x,x),f(y,f(x,f(f(x,x),x))))) = f(x,x).
638
f(f(f(f(x,x),x),f(x,f(f(x,x),x))),f(f(x,x),f(y,f(x,f(f(x,x),x))))) = f(x,x).
639
f(f(f(f(x,x),x),f(x,x)),f(f(x,x),f(y,f(x,f(f(x,x),x))))) = f(x,x).
640
f(x,f(f(x,x),f(y,f(x,f(f(x,x),x))))) = f(x,x).
641
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x).
632
642
f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x.
633
643
f(f(x,f(x,x)),f(x,f(y,f(x,f(f(x,x),x))))) = x.
634
644
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
635
f(f(f(x,x),f(f(x,x),x)),f(f(x,f(f(x,x),x)),x)) = f(x,f(f(x,x),x)).
636
f(f(f(x,x),f(f(x,x),x)),f(f(x,x),x)) = f(x,f(f(x,x),x)).
637
f(f(f(x,x),f(f(x,x),x)),f(f(x,x),x)) = f(x,x).
638
f(f(f(f(x,f(f(x,x),x)),f(y,f(x,x))),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(y,f(x,x))))),f(f(f(f(x,x),x),f(x,x)),f(z,f(f(x,f(f(x,x),x)),f(y,f(x,x)))))) = f(f(f(x,x),f(f(x,f(f(x,x),x)),f(x,x))),f(x,x)).
639
f(f(f(f(x,x),f(y,f(x,x))),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(y,f(x,x))))),f(f(f(f(x,x),x),f(x,x)),f(z,f(f(x,f(f(x,x),x)),f(y,f(x,x)))))) = f(f(f(x,x),f(f(x,f(f(x,x),x)),f(x,x))),f(x,x)).
640
f(f(f(f(x,x),f(y,f(x,x))),f(f(x,x),f(f(x,f(f(x,x),x)),f(y,f(x,x))))),f(f(f(f(x,x),x),f(x,x)),f(z,f(f(x,f(f(x,x),x)),f(y,f(x,x)))))) = f(f(f(x,x),f(f(x,f(f(x,x),x)),f(x,x))),f(x,x)).
641
f(f(f(f(x,x),f(y,f(x,x))),f(f(x,x),f(f(x,x),f(y,f(x,x))))),f(f(f(f(x,x),x),f(x,x)),f(z,f(f(x,f(f(x,x),x)),f(y,f(x,x)))))) = f(f(f(x,x),f(f(x,f(f(x,x),x)),f(x,x))),f(x,x)).
642
f(f(f(f(x,x),f(y,f(x,x))),f(f(x,x),f(f(x,x),f(y,f(x,x))))),f(x,f(z,f(f(x,f(f(x,x),x)),f(y,f(x,x)))))) = f(f(f(x,x),f(f(x,f(f(x,x),x)),f(x,x))),f(x,x)).
643
f(f(f(f(x,x),f(y,f(x,x))),f(f(x,x),f(f(x,x),f(y,f(x,x))))),f(x,f(z,f(f(x,x),f(y,f(x,x)))))) = f(f(f(x,x),f(f(x,f(f(x,x),x)),f(x,x))),f(x,x)).
644
f(f(f(f(x,x),f(y,f(x,x))),f(f(x,x),f(f(x,x),f(y,f(x,x))))),f(x,f(z,f(f(x,x),f(y,f(x,x)))))) = f(f(f(x,x),f(f(x,x),f(x,x))),f(x,x)).
645
f(f(f(f(x,x),f(y,f(x,x))),f(f(x,x),f(f(x,x),f(y,f(x,x))))),f(x,f(z,f(f(x,x),f(y,f(x,x)))))) = f(f(x,x),f(f(f(x,x),f(x,x)),f(x,x))).
646
f(f(f(f(x,x),f(y,f(x,x))),f(f(x,x),f(f(x,x),f(y,f(x,x))))),f(x,f(z,f(f(x,x),f(y,f(x,x)))))) = f(f(x,x),f(x,x)).
647
f(f(f(f(x,x),f(y,f(x,x))),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(y,f(x,x))))),f(f(f(f(x,x),x),f(x,x)),f(z,f(f(x,f(f(x,x),x)),f(y,f(x,x)))))) = f(f(f(x,x),x),f(x,x)).
648
f(f(f(f(x,x),f(y,f(x,x))),f(f(x,x),f(f(x,f(f(x,x),x)),f(y,f(x,x))))),f(f(f(f(x,x),x),f(x,x)),f(z,f(f(x,f(f(x,x),x)),f(y,f(x,x)))))) = f(f(f(x,x),x),f(x,x)).
649
f(f(f(f(x,x),f(y,f(x,x))),f(f(x,x),f(f(x,x),f(y,f(x,x))))),f(f(f(f(x,x),x),f(x,x)),f(z,f(f(x,f(f(x,x),x)),f(y,f(x,x)))))) = f(f(f(x,x),x),f(x,x)).
650
f(f(f(f(x,x),f(y,f(x,x))),f(f(x,x),f(f(x,x),f(y,f(x,x))))),f(x,f(z,f(f(x,f(f(x,x),x)),f(y,f(x,x)))))) = f(f(f(x,x),x),f(x,x)).
651
f(f(f(f(x,x),f(y,f(x,x))),f(f(x,x),f(f(x,x),f(y,f(x,x))))),f(x,f(z,f(f(x,x),f(y,f(x,x)))))) = f(f(f(x,x),x),f(x,x)).
652
f(f(x,x),f(x,x)) = f(f(f(x,x),x),f(x,x)).
653
f(f(x,x),f(x,x)) = x.
654
645
f(f(x,f(x,x)),x) = f(x,x).
655
f(f(x,x),f(x,f(y,x))) = x.
656
f(f(x,x),f(x,f(x,y))) = x.
657
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),x),x)) = f(f(x,x),x).
658
f(x,f(f(f(x,x),x),x)) = f(f(x,x),x).
659
f(x,f(x,x)) = f(f(x,x),x).
660
f(f(x,x),x) = f(x,f(x,x)).
661
f(f(f(x,x),f(x,f(x,x))),f(f(x,x),x)) = f(x,x).
662
f(x,f(f(x,x),x)) = f(x,x).
663
f(x,f(x,f(x,x))) = f(x,x).
664
f(f(x,x),f(x,f(y,f(f(x,y),y)))) = x.
646
f(f(f(x,f(x,x)),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
647
f(f(x,x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
648
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y).
665
649
f(f(f(x,f(x,x)),x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
666
650
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
667
651
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
668
f(f(x,f(x,f(y,f(x,x)))),f(f(x,x),f(x,f(x,f(y,f(x,x)))))) = f(x,x).
669
f(f(x,f(x,f(y,f(x,x)))),x) = f(x,x).
670
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
671
f(f(x,f(y,y)),f(x,y)) = x.
672
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
673
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
674
f(f(x,y),f(x,f(y,y))) = x.
675
f(f(f(x,f(x,f(y,f(x,x)))),x),y) = f(y,f(x,x)).
676
652
f(f(x,x),y) = f(y,f(x,x)).
677
f(x,f(f(x,x),f(f(x,x),f(x,x)))) = f(x,x).
678
f(x,f(f(x,x),x)) = f(x,x).
679
653
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
680
f(x,y) = f(y,f(f(x,x),f(x,f(f(x,x),x)))).
681
f(x,y) = f(y,f(f(x,x),f(x,x))).
682
654
f(x,y) = f(y,x).
683
655
$F # answer("Commutativity").
684
656
end_of_list.
711
683
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
712
684
end_of_list.
713
685
714
Child search process 13964 started.
686
Child search process 10937 started.
715
687
716
688
============================== PROCESS INITIAL CLAUSES ===============
717
689
743
715
formulas(demodulators).
744
716
end_of_list.
745
717
746
% 155 hints input.
718
% 108 hints input.
747
719
748
720
============================== PREDICATE ELIMINATION =================
749
721
756
728
757
729
Term ordering decisions:
758
730
Function symbol KB weights: c1=1. c2=1. f=1.
759
Relation symbol precedence: lex([ = ]).
760
Function symbol precedence: lex([ c1, c2, f ]).
731
Predicate symbol precedence: predicate_order([ = ]).
732
Function symbol precedence: function_order([ c1, c2, f ]).
761
733
Skipping inverse_order, because term ordering is KBO.
762
734
Unfolding symbols: (none).
763
735
764
736
Auto_inference settings:
765
737
% set(paramodulation). % (positive equality literals)
766
% set(paramodulation) -> set(back_demod).
767
738
768
739
Auto_process settings: (no changes).
769
740
741
kept: 2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
742
kept: 3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
743
kept: 4 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
744
kept: 5 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
745
kept: 6 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
746
kept: 7 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
747
kept: 8 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
748
kept: 9 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
749
kept: 10 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
750
kept: 11 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
751
kept: 12 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
752
kept: 13 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
753
kept: 14 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
754
kept: 15 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
755
kept: 16 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
756
kept: 17 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
757
kept: 18 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
758
770
759
============================== end of process initial clauses ========
771
760
772
761
============================== CLAUSES FOR SEARCH ====================
815
804
18 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
816
805
end_of_list.
817
806
818
% 155 hints processed (49 are redundant).
807
% 67 hints (108 processed, 41 redundant).
819
808
820
809
============================== end of clauses for search =============
821
810
827
816
828
817
============================== PROOF =================================
829
818
830
% Proof 1 at 0.73 (+ 0.00) seconds: "Commutativity".
831
% Length of proof is 39.
832
% Level of proof is 11.
833
% Maximum clause weight is 37.
834
% Given clauses 43.
819
% Proof 1 at 0.45 (+ 0.01) seconds: "Commutativity".
820
% Length of proof is 44.
821
% Level of proof is 14.
822
% Maximum clause weight is 43.
823
% Given clauses 25.
835
824
836
1 f(x,y) = f(y,x) # label("Commutativity"). [goal].
825
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
837
826
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
838
827
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
839
828
4 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
840
5 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
841
6 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
842
7 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
843
829
8 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
844
10 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
845
11 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
830
9 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
846
831
12 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
847
832
14 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
833
16 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
848
834
19 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)). [para(2(a,1),2(a,1,1,2,1))].
849
835
21 f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)). [para(2(a,1),2(a,1,2))].
850
836
23 f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x. [para(4(a,1),2(a,1,1,2))].
851
25 f(f(f(x,f(y,z)),f(f(x,f(y,z)),x)),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)). [para(2(a,1),4(a,1,1,2,2))].
837
98 f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)). [para(2(a,1),8(a,1,2))].
838
349 f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)). [para(2(a,1),14(a,1,2))].
852
839
758 f(f(f(x,f(x,x)),x),f(x,x)) = x. [para(2(a,1),23(a,1,2,2))].
853
840
796 f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y. [para(758(a,1),2(a,1,2,2))].
854
841
821 f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x. [para(758(a,1),14(a,1,1,1))].
855
865 f(f(x,f(x,x)),f(f(f(x,f(x,x)),f(x,x)),f(y,f(x,x)))) = f(f(x,f(x,x)),f(x,x)). [para(796(a,1),10(a,1,1,1))].
856
947 f(x,f(f(x,f(f(x,x),x)),f(y,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)). [para(8(a,1),19(a,1,1))].
857
949 f(f(f(x,f(y,x)),f(x,f(x,f(y,x)))),x) = f(x,f(f(x,x),x)). [para(8(a,1),19(a,1,2))].
858
1223 f(x,f(f(x,x),x)) = f(x,x). [para(8(a,1),949(a,1,1)),flip(a)].
859
1283 f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x). [back_rewrite(947),rewrite(1223(3),1223(4),1223(8))].
860
1288 f(f(x,f(x,x)),f(x,x)) = x. [back_rewrite(821),rewrite(1283(7))].
861
1290 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [back_rewrite(865),rewrite(1288(6),1288(10))].
862
1292 f(f(x,x),f(x,f(y,x))) = x. [para(1223(a,1),2(a,1,1))].
863
1293 f(f(x,x),f(x,x)) = x. [para(1223(a,1),2(a,1,2)),rewrite(1223(3))].
864
1295 f(f(x,f(x,x)),x) = f(x,x). [para(1223(a,1),6(a,1,1,2)),rewrite(1293(5))].
865
1302 f(f(x,x),f(x,f(x,y))) = x. [para(1223(a,1),8(a,1,1))].
866
1558 f(f(x,x),f(x,f(y,f(f(x,y),y)))) = x. [para(21(a,1),1292(a,1,2,2))].
867
1697 f(f(f(x,y),f(f(x,y),x)),x) = f(x,x). [para(1302(a,1),5(a,1,2)),rewrite(1293(5))].
868
1927 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)). [para(1290(a,1),7(a,1,1,2)),rewrite(1295(3),1295(6))].
869
1936 f(f(x,f(x,f(y,f(x,x)))),x) = f(x,x). [para(1290(a,1),11(a,1,1,1)),rewrite(1302(10))].
870
2065 f(f(x,f(y,y)),f(x,y)) = x. [para(1558(a,1),8(a,1,2,2)),rewrite(1927(6))].
871
2159 f(f(x,x),y) = f(y,f(x,x)). [para(2065(a,1),12(a,1,2)),rewrite(1936(5))].
872
2241 f(x,f(f(x,x),x)) = f(x,x). [para(8(a,1),25(a,1,2)),rewrite(1697(7)),flip(a)].
873
2454 f(x,y) = f(y,x). [para(2(a,1),2159(a,1,1)),rewrite(2241(4),2241(5),1293(4))].
874
2455 $F # answer("Commutativity"). [resolve(2454,a,3,a)].
842
857 f(f(f(f(x,y),f(x,y)),f(f(x,y),f(f(x,y),f(x,y)))),f(f(x,f(f(f(x,y),x),x)),f(x,y))) = f(x,f(f(f(x,y),x),x)). [para(8(a,1),796(a,1,1,2,1))].
843
862 f(f(f(f(x,y),f(x,y)),f(f(x,y),f(f(x,y),f(x,y)))),f(f(y,f(y,f(y,f(x,y)))),f(x,y))) = f(y,f(y,f(y,f(x,y)))). [para(9(a,1),796(a,1,1,2,1))].
844
878 f(f(f(x,f(x,f(f(x,x),f(f(y,f(x,x)),f(x,x))))),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))). [para(796(a,1),12(a,1,2))].
845
916 f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),f(f(x,f(x,x)),f(x,x))),x)) = f(f(x,x),f(f(x,f(x,x)),f(x,x))). [para(796(a,1),796(a,1,1,2,1))].
846
993 f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)). [para(21(a,1),21(a,1,1,1)),rewrite([21(16),21(20)])].
847
999 f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),y) = f(x,f(f(y,x),x)). [para(2(a,1),878(a,1,1,1,2,2,1)),rewrite([2(13),2(14),2(10),2(16),2(16),2(17)])].
848
1094 f(x,f(f(x,f(f(x,x),x)),f(y,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)). [para(8(a,1),19(a,1,1))].
849
1096 f(f(f(x,f(y,x)),f(x,f(x,f(y,x)))),x) = f(x,f(f(x,x),x)). [para(8(a,1),19(a,1,2))].
850
1181 f(x,f(f(x,x),x)) = f(x,x). [para(8(a,1),1096(a,1,1)),flip(a)].
851
1238 f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x). [back_rewrite(1094),rewrite([1181(3),1181(4),1181(8)])].
852
1240 f(f(f(f(x,x),x),f(x,x)),y) = f(x,f(f(y,x),x)). [back_rewrite(999),rewrite([1181(3),1181(5)])].
853
1319 f(f(x,f(x,x)),f(x,x)) = x. [back_rewrite(821),rewrite([1238(7)])].
854
1323 f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),x),x)) = f(f(x,x),x). [back_rewrite(916),rewrite([1319(9),1319(13)])].
855
1381 f(f(x,x),f(x,x)) = x. [para(1181(a,1),2(a,1,2)),rewrite([1181(3)])].
856
1390 f(f(x,x),f(x,f(x,y))) = x. [para(1181(a,1),8(a,1,1))].
857
1404 f(f(f(x,x),x),x) = f(x,x). [para(1181(a,1),16(a,1,1,1)),rewrite([1381(5)])].
858
1417 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(1323),rewrite([1390(4),1404(3)]),flip(a)].
859
1448 f(f(x,y),f(f(y,f(y,f(y,f(x,y)))),f(x,y))) = f(y,f(y,f(y,f(x,y)))). [back_rewrite(862),rewrite([1390(9)])].
860
1449 f(f(x,y),f(f(x,f(f(f(x,y),x),x)),f(x,y))) = f(x,f(f(f(x,y),x),x)). [back_rewrite(857),rewrite([1390(9)])].
861
1552 f(x,f(f(y,x),x)) = f(x,y). [back_rewrite(1240),rewrite([1417(2),1319(4)]),flip(a)].
862
1579 f(f(x,y),x) = f(x,f(x,y)). [back_rewrite(1449),rewrite([1552(5),1552(6),1552(6)])].
863
1642 f(f(f(x,y),f(y,z)),z) = f(y,z). [back_rewrite(993),rewrite([1552(3),1552(4),1552(5),1552(7),1552(7),1552(7)])].
864
1729 f(f(f(x,y),y),y) = f(x,y). [back_rewrite(349),rewrite([1552(4),1642(4),1552(6)])].
865
1762 f(x,f(x,f(y,x))) = f(y,x). [back_rewrite(98),rewrite([1552(3),1729(3),1579(3),1552(6)])].
866
1932 f(f(x,y),y) = f(y,f(x,y)). [back_rewrite(1448),rewrite([1762(4),1552(6),1762(5)])].
867
2067 f(x,y) = f(y,x). [back_rewrite(1552),rewrite([1932(2),1762(3)])].
868
2068 $F # answer("Commutativity"). [resolve(2067,a,3,a)].
875
869
876
870
============================== end of proof ==========================
877
871
878
872
============================== STATISTICS ============================
879
873
880
Given=43. Generated=5183. Kept=2453. proofs=1.
881
Usable=36. Sos=2049. Demods=2089. Limbo=7, Disabled=377. Hints=155.
882
Weight_deleted=107. Literals_deleted=0.
883
Forward_subsumed=2623. Back_subsumed=177.
874
Given=25. Generated=3521. Kept=2066. proofs=1.
875
Usable=6. Sos=368. Demods=505. Limbo=135, Disabled=1573. Hints=108.
876
Kept_by_rule=0, Deleted_by_rule=33.
877
Forward_subsumed=1422. Back_subsumed=83.
884
878
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
885
New_demodulators=2449 (0 lex), Back_demodulated=183. Back_unit_deleted=0.
886
Demod_attempts=130318. Demod_rewrites=7832.
879
New_demodulators=2055 (0 lex), Back_demodulated=1473. Back_unit_deleted=0.
880
Demod_attempts=80861. Demod_rewrites=6184.
887
881
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
888
882
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
889
Megabytes=3.78.
890
User_CPU=0.73, System_CPU=0.00, Wall_clock=3.
883
Megabytes=2.58.
884
User_CPU=0.45, System_CPU=0.02, Wall_clock=2.
891
885
892
886
============================== end of statistics =====================
893
887
895
889
896
890
Exiting with 1 proof.
897
891
898
Process 13964 exit (max_proofs) Sun Aug 13 21:05:00 2006
892
Process 10937 exit (max_proofs) Mon Sep 28 11:57:11 2009
899
893
900
Successful proof using the following 9 assumptions:
894
Successful proof using the following 6 assumptions:
901
895
902
896
formulas(assumptions_in_proof).
903
897
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
904
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
905
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
906
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
907
898
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
908
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
909
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
899
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
910
900
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
911
901
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
902
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
912
903
end_of_list.
913
904
914
Including 56 new hint clauses:
905
Including 78 new hint clauses:
915
906
916
907
formulas(new_hints).
917
$F # label("Commutativity").
908
$F # label("Commutativity") # label(non_clause) # label(goal).
918
909
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
919
910
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
920
911
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").
921
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").
922
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").
923
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").
924
912
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
925
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").
926
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
913
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").
927
914
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
928
915
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
916
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
929
917
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)).
930
918
f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)).
931
919
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
932
f(f(f(x,f(y,z)),f(f(x,f(y,z)),x)),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)).
920
f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)).
921
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
933
922
f(f(f(x,f(x,x)),x),f(x,x)) = x.
934
923
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
935
924
f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x.
936
f(f(x,f(x,x)),f(f(f(x,f(x,x)),f(x,x)),f(y,f(x,x)))) = f(f(x,f(x,x)),f(x,x)).
925
f(f(f(f(x,y),f(x,y)),f(f(x,y),f(f(x,y),f(x,y)))),f(f(x,f(f(f(x,y),x),x)),f(x,y))) = f(x,f(f(f(x,y),x),x)).
926
f(f(f(f(x,y),f(x,y)),f(f(x,y),f(f(x,y),f(x,y)))),f(f(y,f(y,f(y,f(x,y)))),f(x,y))) = f(y,f(y,f(y,f(x,y)))).
927
f(f(f(x,f(x,f(f(x,x),f(f(y,f(x,x)),f(x,x))))),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
928
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),f(f(x,f(x,x)),f(x,x))),x)) = f(f(x,x),f(f(x,f(x,x)),f(x,x))).
929
f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(f(f(u,x),f(f(f(x,f(f(y,x),x)),f(u,x)),f(u,x))),y)),f(f(f(u,x),f(f(f(x,f(f(y,x),x)),f(u,x)),f(u,x))),y))),z) = f(y,f(f(z,y),y)).
930
f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(f(f(u,x),f(f(f(x,f(f(y,x),x)),f(u,x)),f(u,x))),y))),z) = f(y,f(f(z,y),y)).
931
f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
932
f(f(f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))))),f(x,f(f(x,x),x))),y) = f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))).
933
f(f(f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(f(y,x),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))))),f(x,f(f(x,x),x))),y) = f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))).
934
f(f(f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(f(y,x),x)))),f(x,f(f(x,x),x))),y) = f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))).
935
f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),y) = f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))).
936
f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),y) = f(x,f(f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))).
937
f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),y) = f(x,f(f(y,x),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))).
938
f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),y) = f(x,f(f(y,x),x)).
937
939
f(x,f(f(x,f(f(x,x),x)),f(y,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
938
940
f(f(f(x,f(y,x)),f(x,f(x,f(y,x)))),x) = f(x,f(f(x,x),x)).
939
941
f(x,x) = f(x,f(f(x,x),x)).
941
943
f(x,f(f(x,x),f(y,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
942
944
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,f(f(x,x),x)).
943
945
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x).
946
f(f(f(f(x,x),x),f(x,f(f(x,x),x))),y) = f(x,f(f(y,x),x)).
947
f(f(f(f(x,x),x),f(x,x)),y) = f(x,f(f(y,x),x)).
944
948
f(f(x,f(x,x)),f(x,x)) = x.
945
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = f(f(x,f(x,x)),f(x,x)).
946
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
947
f(f(x,x),f(x,f(y,x))) = x.
949
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),x),x)) = f(f(x,x),f(f(x,f(x,x)),f(x,x))).
950
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),x),x)) = f(f(x,x),x).
948
951
f(f(x,f(f(x,x),x)),f(x,x)) = x.
949
952
f(f(x,x),f(x,x)) = x.
950
f(f(x,f(x,x)),f(f(x,x),f(x,x))) = f(x,x).
951
f(f(x,f(x,x)),x) = f(x,x).
952
953
f(f(x,x),f(x,f(x,y))) = x.
953
f(f(x,x),f(x,f(y,f(f(x,y),y)))) = x.
954
f(f(f(x,y),f(f(x,y),f(f(x,x),f(x,x)))),x) = f(x,x).
955
f(f(f(x,y),f(f(x,y),x)),x) = f(x,x).
956
f(f(f(x,f(x,x)),x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
957
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
958
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
959
f(f(x,f(x,f(y,f(x,x)))),f(f(x,x),f(x,f(x,f(y,f(x,x)))))) = f(x,x).
960
f(f(x,f(x,f(y,f(x,x)))),x) = f(x,x).
961
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
962
f(f(x,f(y,y)),f(x,y)) = x.
963
f(f(f(x,f(x,f(y,f(x,x)))),x),y) = f(y,f(x,x)).
964
f(f(x,x),y) = f(y,f(x,x)).
965
f(f(f(x,f(y,x)),f(f(x,f(y,x)),x)),x) = f(x,f(f(x,x),x)).
966
f(x,x) = f(x,f(f(x,x),x)).
967
f(x,f(f(x,x),x)) = f(x,x).
968
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
969
f(x,y) = f(y,f(f(x,x),f(x,f(f(x,x),x)))).
970
f(x,y) = f(y,f(f(x,x),f(x,x))).
954
f(f(f(x,x),x),f(f(x,x),f(x,x))) = f(x,x).
955
f(f(f(x,x),x),x) = f(x,x).
956
f(x,f(f(f(x,x),x),x)) = f(f(x,x),x).
957
f(x,f(x,x)) = f(f(x,x),x).
958
f(f(x,x),x) = f(x,f(x,x)).
959
f(f(x,y),f(f(y,f(y,f(y,f(x,y)))),f(x,y))) = f(y,f(y,f(y,f(x,y)))).
960
f(f(x,y),f(f(x,f(f(f(x,y),x),x)),f(x,y))) = f(x,f(f(f(x,y),x),x)).
961
f(f(f(x,f(x,x)),f(x,x)),y) = f(x,f(f(y,x),x)).
962
f(x,y) = f(x,f(f(y,x),x)).
963
f(x,f(f(y,x),x)) = f(x,y).
964
f(f(x,y),f(f(x,f(x,y)),f(x,y))) = f(x,f(f(f(x,y),x),x)).
965
f(f(x,y),x) = f(x,f(f(f(x,y),x),x)).
966
f(f(x,y),x) = f(x,f(x,y)).
967
f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
968
f(f(f(x,y),f(f(f(y,z),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
969
f(f(f(x,y),f(f(f(y,z),f(x,y)),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
970
f(f(f(x,y),f(f(f(y,z),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)).
971
f(f(f(x,y),f(y,z)),z) = f(y,f(f(z,y),y)).
972
f(f(f(x,y),f(y,z)),z) = f(y,z).
973
f(f(f(f(f(x,y),f(y,z)),z),z),z) = f(y,f(f(z,y),y)).
974
f(f(f(y,z),z),z) = f(y,f(f(z,y),y)).
975
f(f(f(x,y),y),y) = f(x,y).
976
f(f(x,f(f(f(y,x),x),x)),x) = f(y,f(f(x,y),y)).
977
f(f(x,f(y,x)),x) = f(y,f(f(x,y),y)).
978
f(x,f(x,f(y,x))) = f(y,f(f(x,y),y)).
979
f(x,f(x,f(y,x))) = f(y,x).
980
f(f(x,y),f(f(y,f(x,y)),f(x,y))) = f(y,f(y,f(y,f(x,y)))).
981
f(f(x,y),y) = f(y,f(y,f(y,f(x,y)))).
982
f(f(x,y),y) = f(y,f(x,y)).
983
f(x,f(x,f(y,x))) = f(x,y).
971
984
f(x,y) = f(y,x).
972
985
$F # answer("Commutativity").
973
986
end_of_list.
999
1012
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1000
1013
end_of_list.
1001
1014
1002
Child search process 13965 started.
1015
Child search process 10938 started.
1003
1016
1004
1017
============================== PROCESS INITIAL CLAUSES ===============
1005
1018
1030
1043
formulas(demodulators).
1031
1044
end_of_list.
1032
1045
1033
% 211 hints input.
1046
% 186 hints input.
1034
1047
1035
1048
============================== PREDICATE ELIMINATION =================
1036
1049
1043
1056
1044
1057
Term ordering decisions:
1045
1058
Function symbol KB weights: c1=1. c2=1. f=1.
1046
Relation symbol precedence: lex([ = ]).
1047
Function symbol precedence: lex([ c1, c2, f ]).
1059
Predicate symbol precedence: predicate_order([ = ]).
1060
Function symbol precedence: function_order([ c1, c2, f ]).
1048
1061
Skipping inverse_order, because term ordering is KBO.
1049
1062
Unfolding symbols: (none).
1050
1063
1051
1064
Auto_inference settings:
1052
1065
% set(paramodulation). % (positive equality literals)
1053
% set(paramodulation) -> set(back_demod).
1054
1066
1055
1067
Auto_process settings: (no changes).
1056
1068
1069
kept: 2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
1070
kept: 3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
1071
kept: 4 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
1072
kept: 5 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
1073
kept: 6 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
1074
kept: 7 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1075
kept: 8 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1076
kept: 9 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1077
kept: 10 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1078
kept: 11 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1079
kept: 12 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
1080
kept: 13 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1081
kept: 14 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1082
kept: 15 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
1083
kept: 16 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
1084
kept: 17 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1085
1057
1086
============================== end of process initial clauses ========
1058
1087
1059
1088
============================== CLAUSES FOR SEARCH ====================
1100
1129
17 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1101
1130
end_of_list.
1102
1131
1103
% 211 hints processed (89 are redundant).
1132
% 114 hints (186 processed, 72 redundant).
1104
1133
1105
1134
============================== end of clauses for search =============
1106
1135
1107
1136
============================== SEARCH ================================
1108
1137
1109
% Starting search at 0.00 seconds.
1138
% Starting search at 0.01 seconds.
1110
1139
1111
1140
% Operation f is commutative; C redundancy checks enabled.
1112
1141
1113
1142
============================== PROOF =================================
1114
1143
1115
% Proof 1 at 0.43 (+ 0.00) seconds: "Commutativity".
1116
% Length of proof is 56.
1144
% Proof 1 at 0.54 (+ 0.02) seconds: "Commutativity".
1145
% Length of proof is 52.
1117
1146
% Level of proof is 16.
1118
% Maximum clause weight is 39.
1119
% Given clauses 27.
1147
% Maximum clause weight is 51.
1148
% Given clauses 28.
1120
1149
1121
1 f(x,y) = f(y,x) # label("Commutativity"). [goal].
1150
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
1122
1151
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
1123
1152
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
1124
1153
4 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
1125
6 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
1154
5 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
1126
1155
7 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1156
8 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1157
9 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1127
1158
10 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1128
1159
11 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1129
12 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
1130
1160
13 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1131
15 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
1161
14 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1132
1162
17 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1133
1163
18 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)). [para(2(a,1),2(a,1,1,2,1))].
1134
20 f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)). [para(2(a,1),2(a,1,2))].
1135
1164
22 f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x. [para(4(a,1),2(a,1,1,2))].
1136
71 f(f(f(x,y),f(f(f(x,f(f(z,x),x)),f(x,y)),f(x,y))),z) = f(x,f(f(z,x),x)). [para(7(a,1),2(a,1,2))].
1165
302 f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)). [para(2(a,1),13(a,1,2))].
1137
1166
687 f(f(f(x,f(x,x)),x),f(x,x)) = x. [para(2(a,1),22(a,1,2,2))].
1138
1167
724 f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y. [para(687(a,1),2(a,1,2,2))].
1139
725 f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)). [para(2(a,1),687(a,1,1,1,2)),rewrite(2(15))].
1140
1168
748 f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x. [para(687(a,1),13(a,1,1,1))].
1169
781 f(f(f(f(x,y),f(x,y)),f(f(x,y),f(f(x,y),f(x,y)))),f(f(x,f(f(f(x,y),x),x)),f(x,y))) = f(x,f(f(f(x,y),x),x)). [para(7(a,1),724(a,1,1,2,1))].
1141
1170
802 f(f(f(x,f(x,f(f(x,x),f(f(y,f(x,x)),f(x,x))))),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))). [para(724(a,1),11(a,1,2))].
1142
840 f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),f(f(x,f(x,x)),f(x,x))),x)) = f(f(x,x),f(f(x,f(x,x)),f(x,x))). [para(724(a,1),724(a,1,1,2,1))].
1143
844 f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)). [para(2(a,1),748(a,1,1,2)),rewrite(2(14),2(17),2(13))].
1144
850 f(f(f(x,f(x,x)),x),f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),x))) = f(f(x,x),f(x,f(x,x))). [para(748(a,1),6(a,1,1,2))].
1145
864 f(f(f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),f(x,f(x,x)))),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)). [para(748(a,1),12(a,1,2))].
1146
883 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),x)) = f(z,f(f(x,z),z)). [para(2(a,1),18(a,1,2,2))].
1147
904 f(x,f(f(x,f(f(x,x),x)),f(y,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)). [para(7(a,1),18(a,1,1))].
1148
961 f(f(f(x,f(x,f(f(x,x),x))),x),f(x,x)) = x. [para(18(a,1),687(a,1,1,1,2)),rewrite(7(8),7(12),7(13),7(13),7(15))].
1149
965 f(f(x,f(x,x)),f(x,x)) = x. [para(18(a,1),748(a,1,2,2,1)),rewrite(7(8),7(8),7(8),7(10),7(13),7(13),7(13),7(8),7(12))].
1150
969 f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),x),x)) = f(f(x,x),x). [back_rewrite(840),rewrite(965(9),965(13))].
1151
974 f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),f(y,f(x,x)))) = f(x,f(x,x)). [para(965(a,1),2(a,1,1,2,1))].
1152
988 f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))) = f(x,x). [para(965(a,1),10(a,1,1,1))].
1153
1006 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(864),rewrite(974(10),988(7))].
1154
1013 f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),x)) = f(x,f(x,x)). [back_rewrite(969),rewrite(1006(6),1006(10))].
1155
1015 f(f(f(x,f(x,f(x,f(x,x)))),x),f(x,x)) = x. [back_rewrite(961),rewrite(1006(2))].
1156
1020 f(x,f(f(x,f(x,f(x,x))),f(y,f(x,f(x,f(x,x)))))) = f(x,f(x,f(x,x))). [back_rewrite(904),rewrite(1006(2),1006(5),1006(11))].
1157
1022 f(f(f(x,f(x,f(x,x))),x),x) = f(x,f(x,f(x,x))). [back_rewrite(844),rewrite(1006(2),1006(7))].
1158
1023 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(725),rewrite(1006(2),1006(6),15(8),1006(3)),flip(a)].
1159
1042 f(f(x,x),f(x,f(x,x))) = x. [back_rewrite(850),rewrite(1013(11),17(6)),flip(a)].
1160
1043 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(1022),rewrite(1023(3),1006(2),1023(6))].
1161
1044 f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x). [back_rewrite(1020),rewrite(1023(3),1023(4),1023(8))].
1162
1047 f(f(x,x),f(x,x)) = x. [back_rewrite(1015),rewrite(1023(3),1043(3))].
1163
1106 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y). [back_rewrite(802),rewrite(1044(7),1043(3)),flip(a)].
1164
1161 f(f(x,y),f(f(y,f(f(f(x,y),y),y)),f(x,y))) = f(y,f(f(f(x,y),y),y)). [para(1047(a,1),18(a,1,2,2)),rewrite(1042(9))].
1165
1162 f(f(x,x),f(x,f(y,x))) = x. [para(1006(a,1),2(a,1,1,2)),rewrite(1023(3))].
1166
1189 f(f(x,y),f(x,f(f(f(x,y),f(x,y)),y))) = x. [para(1006(a,1),15(a,1,1,1)),rewrite(965(9))].
1167
1321 f(x,f(f(y,x),x)) = f(x,y). [para(1043(a,1),20(a,1,1,1)),rewrite(1043(7),1043(9),1106(9),1162(5)),flip(a)].
1168
1398 f(f(x,y),y) = f(y,f(x,y)). [back_rewrite(1161),rewrite(1321(5),1321(6),1321(6))].
1169
1462 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(z,f(x,z))),x)) = f(z,f(z,f(x,z))). [back_rewrite(883),rewrite(1398(8),1398(13))].
1170
1549 f(f(f(x,y),f(f(x,y),f(f(x,f(x,f(z,x))),f(x,y)))),z) = f(x,f(x,f(z,x))). [back_rewrite(71),rewrite(1398(3),1398(8),1398(12))].
1171
1585 f(x,f(x,f(y,x))) = f(x,y). [back_rewrite(1321),rewrite(1398(2))].
1172
1920 f(f(f(x,y),f(x,z)),z) = f(x,z). [back_rewrite(1549),rewrite(1585(5),1585(7),1585(7))].
1173
1991 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(x,f(z,x))) = f(z,x). [back_rewrite(1462),rewrite(1585(9),1398(8),1585(12))].
1174
2068 f(f(x,y),f(x,f(x,y))) = x. [back_rewrite(1189),rewrite(1920(5))].
1175
2186 f(x,y) = f(y,x). [back_rewrite(1991),rewrite(2068(6),1585(3))].
1176
2187 $F # answer("Commutativity"). [resolve(2186,a,3,a)].
1171
838 f(f(f(x,x),f(x,f(x,x))),f(f(f(x,f(x,x)),x),x)) = f(f(x,f(x,x)),x). [para(687(a,1),724(a,1,1,2,1))].
1172
854 f(f(x,f(f(f(x,f(x,x)),x),x)),x) = f(x,f(x,x)). [para(748(a,1),7(a,1,2))].
1173
858 f(f(f(f(f(f(x,x),f(x,f(x,x))),f(x,f(x,x))),f(f(x,x),f(x,f(x,x)))),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)). [para(748(a,1),9(a,1,2))].
1174
870 f(f(f(f(x,f(x,f(x,x))),x),x),x) = f(x,f(x,x)). [para(748(a,1),14(a,1,2))].
1175
881 f(f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))) = f(f(x,f(f(x,x),x)),x). [para(2(a,1),854(a,1,1,2,1,1,2)),rewrite([2(30)])].
1176
1025 f(f(f(x,y),y),y) = f(x,f(f(y,x),x)). [para(9(a,1),302(a,1,1,1,1))].
1177
1092 f(f(x,f(f(f(x,y),y),y)),f(f(y,x),f(x,x))) = f(y,x). [para(1025(a,2),5(a,1,1,2))].
1178
1131 f(f(f(x,f(f(y,x),x)),y),f(y,f(f(x,y),y))) = y. [para(1025(a,1),10(a,1,1,1))].
1179
1159 f(f(f(f(x,x),f(y,x)),f(y,x)),f(f(f(x,y),y),y)) = x. [para(1025(a,2),13(a,1,2))].
1180
1203 f(f(x,f(x,f(x,x))),x) = f(x,f(x,x)). [para(1025(a,2),854(a,1,1)),rewrite([8(6)])].
1181
1270 f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))). [back_rewrite(881),rewrite([1131(11),7(8)]),flip(a)].
1182
1273 f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)). [back_rewrite(870),rewrite([1203(4)])].
1183
1294 f(f(f(x,x),f(x,f(x,x))),f(x,f(x,x))) = f(f(x,f(x,x)),x). [back_rewrite(838),rewrite([1273(8)])].
1184
1307 f(f(f(f(f(x,f(x,x)),x),f(f(x,x),f(x,f(x,x)))),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)). [back_rewrite(858),rewrite([1294(7)])].
1185
1397 f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),y) = f(x,f(f(y,x),x)). [para(2(a,1),802(a,1,1,1,2,2,1)),rewrite([2(13),2(14),2(10),1270(4),2(16),2(16),2(17)])].
1186
1612 f(f(x,f(x,x)),f(x,x)) = x. [para(18(a,1),748(a,1,2,2,1)),rewrite([7(8),7(8),7(8),7(10),7(13),7(13),7(13),7(8),7(12)])].
1187
1670 f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))) = f(x,x). [para(1612(a,1),10(a,1,1,1))].
1188
1687 f(f(x,x),f(x,f(x,x))) = x. [para(1612(a,1),1025(a,2,2,1)),rewrite([1294(7),17(6)]),flip(a)].
1189
1688 f(f(f(f(x,x),x),x),x) = f(f(x,x),x). [para(1612(a,1),1025(a,2,2))].
1190
1694 f(f(x,y),f(f(y,f(f(f(x,y),y),y)),f(x,y))) = f(y,f(f(f(x,y),y),y)). [para(1612(a,1),18(a,1,2,2)),rewrite([1687(9)])].
1191
1700 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(1670),rewrite([1687(6)])].
1192
1722 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(1307),rewrite([1700(3),1687(5),1687(6),1688(4)])].
1193
1723 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(1294),rewrite([1687(4),1700(6)])].
1194
1736 f(f(x,y),f(f(x,f(f(f(x,y),x),x)),f(x,y))) = f(x,f(f(f(x,y),x),x)). [back_rewrite(781),rewrite([1687(9)])].
1195
1750 f(x,f(f(y,x),x)) = f(x,y). [back_rewrite(1397),rewrite([1722(2),1723(3),1722(4),1723(5),1612(4)]),flip(a)].
1196
1889 f(f(x,y),x) = f(x,f(x,y)). [back_rewrite(1736),rewrite([1750(5),1750(6),1750(6)])].
1197
1893 f(f(x,y),y) = f(y,f(x,y)). [back_rewrite(1694),rewrite([1750(5),1750(6),1750(6)])].
1198
2496 f(x,f(x,f(y,x))) = f(x,y). [back_rewrite(1750),rewrite([1893(2)])].
1199
2513 f(f(x,y),f(y,y)) = y. [back_rewrite(1159),rewrite([1893(5),1893(7),1889(8),2496(8),1889(7),2496(7)])].
1200
2530 f(x,y) = f(y,x). [back_rewrite(1092),rewrite([1893(2),1889(3),2496(3),2513(5),1889(3),2496(3)])].
1201
2531 $F # answer("Commutativity"). [resolve(2530,a,3,a)].
1177
1202
1178
1203
============================== end of proof ==========================
1179
1204
1180
1205
============================== STATISTICS ============================
1181
1206
1182
Given=27. Generated=3244. Kept=2185. proofs=1.
1183
Usable=11. Sos=709. Demods=836. Limbo=118, Disabled=1362. Hints=211.
1184
Weight_deleted=21. Literals_deleted=0.
1185
Forward_subsumed=1038. Back_subsumed=106.
1207
Given=28. Generated=4010. Kept=2529. proofs=1.
1208
Usable=7. Sos=265. Demods=891. Limbo=637, Disabled=1635. Hints=186.
1209
Kept_by_rule=0, Deleted_by_rule=37.
1210
Forward_subsumed=1444. Back_subsumed=59.
1186
1211
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
1187
New_demodulators=2180 (0 lex), Back_demodulated=1240. Back_unit_deleted=0.
1188
Demod_attempts=66317. Demod_rewrites=4839.
1212
New_demodulators=2478 (0 lex), Back_demodulated=1560. Back_unit_deleted=0.
1213
Demod_attempts=100609. Demod_rewrites=7865.
1189
1214
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
1190
1215
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
1191
Megabytes=1.83.
1192
User_CPU=0.43, System_CPU=0.00, Wall_clock=4.
1216
Megabytes=3.53.
1217
User_CPU=0.54, System_CPU=0.02, Wall_clock=3.
1193
1218
1194
1219
============================== end of statistics =====================
1195
1220
1197
1222
1198
1223
Exiting with 1 proof.
1199
1224
1200
Process 13965 exit (max_proofs) Sun Aug 13 21:05:01 2006
1225
Process 10938 exit (max_proofs) Mon Sep 28 11:57:12 2009
1201
1226
1202
Successful proof using the following 9 assumptions:
1227
Successful proof using the following 10 assumptions:
1203
1228
1204
1229
formulas(assumptions_in_proof).
1205
1230
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
1206
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
1231
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
1207
1232
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1233
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1234
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1208
1235
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1209
1236
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1210
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
1211
1237
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1212
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
1238
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1213
1239
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1214
1240
end_of_list.
1215
1241
1216
Including 111 new hint clauses:
1242
Including 99 new hint clauses:
1217
1243
1218
1244
formulas(new_hints).
1219
$F # label("Commutativity").
1245
$F # label("Commutativity") # label(non_clause) # label(goal).
1220
1246
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
1221
1247
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
1222
1248
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").
1223
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").
1249
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").
1224
1250
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
1251
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").
1252
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").
1225
1253
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
1226
1254
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
1227
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").
1228
1255
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
1229
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
1256
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
1230
1257
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
1231
1258
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)).
1232
f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)).
1233
1259
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
1234
f(f(f(x,y),f(f(f(x,f(f(z,x),x)),f(x,y)),f(x,y))),z) = f(x,f(f(z,x),x)).
1260
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
1235
1261
f(f(f(x,f(x,x)),x),f(x,x)) = x.
1236
1262
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
1237
f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
1238
f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
1239
1263
f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x.
1264
f(f(f(f(x,y),f(x,y)),f(f(x,y),f(f(x,y),f(x,y)))),f(f(x,f(f(f(x,y),x),x)),f(x,y))) = f(x,f(f(f(x,y),x),x)).
1240
1265
f(f(f(x,f(x,f(f(x,x),f(f(y,f(x,x)),f(x,x))))),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
1241
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),f(f(x,f(x,x)),f(x,x))),x)) = f(f(x,x),f(f(x,f(x,x)),f(x,x))).
1242
f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))))) = f(x,f(f(x,x),x)).
1243
f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(x,f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))))) = f(x,f(f(x,x),x)).
1244
f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(x,f(f(x,f(f(x,x),x)),x)))) = f(x,f(f(x,x),x)).
1245
f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)).
1246
f(f(f(x,f(x,x)),x),f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),x))) = f(f(x,x),f(x,f(x,x))).
1247
f(f(f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),f(x,f(x,x)))),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)).
1248
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),x)) = f(z,f(f(x,z),z)).
1249
f(x,f(f(x,f(f(x,x),x)),f(y,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
1250
f(f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1251
f(f(f(x,f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1252
f(f(f(x,f(x,f(f(x,x),x))),x),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1253
f(f(f(x,f(x,f(f(x,x),x))),x),f(x,f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1254
f(f(f(x,f(x,f(f(x,x),x))),x),f(x,x)) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1255
f(f(f(x,f(x,f(f(x,x),x))),x),f(x,x)) = x.
1266
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,f(x,x)),x),x)) = f(f(x,f(x,x)),x).
1267
f(f(x,f(f(f(x,f(x,x)),x),x)),x) = f(x,f(x,x)).
1268
f(f(f(f(f(f(x,x),f(x,f(x,x))),f(x,f(x,x))),f(f(x,x),f(x,f(x,x)))),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)).
1269
f(f(f(f(x,f(x,f(x,x))),x),x),x) = f(x,f(x,x)).
1270
f(f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))) = f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
1271
f(f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))) = f(f(x,f(f(x,x),x)),x).
1272
f(f(f(x,y),y),y) = f(x,f(f(y,x),x)).
1273
f(f(x,f(f(f(x,y),y),y)),f(f(y,x),f(x,x))) = f(y,x).
1274
f(f(f(x,f(f(y,x),x)),y),f(y,f(f(x,y),y))) = y.
1275
f(f(f(f(x,x),f(y,x)),f(y,x)),f(f(f(x,y),y),y)) = x.
1276
f(f(f(f(x,f(x,f(x,x))),f(x,f(x,x))),f(x,f(x,x))),x) = f(x,f(x,x)).
1277
f(f(x,f(x,f(x,x))),x) = f(x,f(x,x)).
1278
f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))) = f(f(x,f(f(x,x),x)),x).
1279
f(x,f(x,f(f(x,x),x))) = f(f(x,f(f(x,x),x)),x).
1280
f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))).
1281
f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)).
1282
f(f(f(x,x),f(x,f(x,x))),f(x,f(x,x))) = f(f(x,f(x,x)),x).
1283
f(f(f(f(f(x,f(x,x)),x),f(f(x,x),f(x,f(x,x)))),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)).
1284
f(f(f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))))),f(x,f(f(x,x),x))),y) = f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))).
1285
f(f(f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(f(y,x),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))))),f(x,f(f(x,x),x))),y) = f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))).
1286
f(f(f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(f(y,x),x)))),f(x,f(f(x,x),x))),y) = f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))).
1287
f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),y) = f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))).
1288
f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),y) = f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))).
1289
f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),y) = f(x,f(f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))).
1290
f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),y) = f(x,f(f(y,x),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))).
1291
f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),y) = f(x,f(f(y,x),x)).
1256
1292
f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1257
1293
f(f(x,f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1258
1294
f(f(x,f(x,f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1263
1299
f(f(x,f(x,x)),f(x,f(f(x,f(f(x,x),x)),f(x,f(x,x))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1264
1300
f(f(x,f(x,x)),f(x,x)) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1265
1301
f(f(x,f(x,x)),f(x,x)) = x.
1266
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),x),x)) = f(f(x,x),f(f(x,f(x,x)),f(x,x))).
1267
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),x),x)) = f(f(x,x),x).
1268
f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),f(y,f(x,x)))) = f(x,f(x,x)).
1269
1302
f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))) = f(x,x).
1270
f(f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)).
1271
f(f(x,x),x) = f(x,f(x,x)).
1272
f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),x)) = f(f(x,x),x).
1273
f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),x)) = f(x,f(x,x)).
1274
f(f(f(x,f(x,f(x,f(x,x)))),x),f(x,x)) = x.
1275
f(x,f(f(x,f(x,f(x,x))),f(y,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
1276
f(x,f(f(x,f(x,f(x,x))),f(y,f(x,f(x,f(x,x)))))) = f(x,f(f(x,x),x)).
1277
f(x,f(f(x,f(x,f(x,x))),f(y,f(x,f(x,f(x,x)))))) = f(x,f(x,f(x,x))).
1278
f(f(f(x,f(x,f(x,x))),x),x) = f(x,f(f(x,x),x)).
1279
f(f(f(x,f(x,f(x,x))),x),x) = f(x,f(x,f(x,x))).
1280
f(f(f(f(x,f(x,f(x,x))),x),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
1281
f(f(f(f(x,f(x,f(x,x))),x),f(x,f(x,f(x,x)))),x) = f(x,f(f(x,x),x)).
1282
f(x,x) = f(x,f(f(x,x),x)).
1283
f(x,x) = f(x,f(x,f(x,x))).
1284
f(x,f(x,f(x,x))) = f(x,x).
1303
f(f(f(f(x,x),f(x,f(x,x))),f(x,f(x,x))),f(x,f(x,x))) = f(f(x,x),f(x,f(x,x))).
1285
1304
f(f(f(x,f(x,x)),x),f(x,f(x,x))) = f(f(x,x),f(x,f(x,x))).
1286
1305
x = f(f(x,x),f(x,f(x,x))).
1287
1306
f(f(x,x),f(x,f(x,x))) = x.
1288
f(f(f(x,x),x),x) = f(x,f(x,f(x,x))).
1289
f(f(x,f(x,x)),x) = f(x,f(x,f(x,x))).
1290
f(f(x,f(x,x)),x) = f(x,x).
1291
f(x,f(f(x,x),f(y,f(x,f(x,f(x,x)))))) = f(x,f(x,f(x,x))).
1292
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,f(x,f(x,x))).
1293
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x).
1294
f(f(f(x,f(x,x)),x),f(x,x)) = x.
1295
f(f(x,x),f(x,x)) = x.
1296
f(f(f(x,f(x,x)),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
1297
f(f(x,x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
1298
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y).
1307
f(f(f(f(x,x),x),x),x) = f(f(x,x),x).
1299
1308
f(f(f(f(x,y),f(x,y)),f(f(x,y),f(f(x,y),f(x,y)))),f(f(y,f(f(f(x,y),y),y)),f(x,y))) = f(y,f(f(f(x,y),y),y)).
1300
1309
f(f(x,y),f(f(y,f(f(f(x,y),y),y)),f(x,y))) = f(y,f(f(f(x,y),y),y)).
1301
f(f(x,f(x,f(x,x))),f(x,f(y,x))) = x.
1302
f(f(x,x),f(x,f(y,x))) = x.
1303
f(f(f(f(x,y),f(f(x,y),f(x,y))),f(f(x,y),f(x,y))),f(x,f(f(f(x,y),f(x,y)),y))) = x.
1304
f(f(x,y),f(x,f(f(f(x,y),f(x,y)),y))) = x.
1305
f(f(f(x,x),f(f(f(x,f(f(y,x),x)),f(f(x,f(x,x)),x)),f(f(x,f(x,x)),x))),y) = f(x,f(f(y,x),x)).
1306
f(f(f(x,x),f(f(f(x,f(f(y,x),x)),f(x,x)),f(f(x,f(x,x)),x))),y) = f(x,f(f(y,x),x)).
1307
f(f(f(x,x),f(f(f(x,f(f(y,x),x)),f(x,x)),f(x,x))),y) = f(x,f(f(y,x),x)).
1308
f(f(f(x,x),f(x,f(f(y,x),x))),y) = f(x,f(f(y,x),x)).
1310
f(f(x,f(x,x)),x) = f(x,x).
1311
f(f(f(f(x,x),f(f(x,x),f(x,f(x,x)))),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)).
1312
f(f(f(f(x,x),x),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)).
1313
f(f(f(f(x,x),x),x),x) = f(x,f(x,x)).
1314
f(f(x,x),x) = f(x,f(x,x)).
1315
f(x,f(x,f(x,x))) = f(f(x,f(x,x)),x).
1316
f(x,f(x,f(x,x))) = f(x,x).
1317
f(f(x,y),f(f(x,f(f(f(x,y),x),x)),f(x,y))) = f(x,f(f(f(x,y),x),x)).
1318
f(f(f(x,f(x,f(x,f(x,x)))),f(x,f(f(x,x),x))),y) = f(x,f(f(y,x),x)).
1319
f(f(f(x,f(x,x)),f(x,f(f(x,x),x))),y) = f(x,f(f(y,x),x)).
1320
f(f(f(x,f(x,x)),f(x,f(x,f(x,x)))),y) = f(x,f(f(y,x),x)).
1321
f(f(f(x,f(x,x)),f(x,x)),y) = f(x,f(f(y,x),x)).
1309
1322
f(x,y) = f(x,f(f(y,x),x)).
1310
1323
f(x,f(f(y,x),x)) = f(x,y).
1324
f(f(x,y),f(f(x,f(x,y)),f(x,y))) = f(x,f(f(f(x,y),x),x)).
1325
f(f(x,y),x) = f(x,f(f(f(x,y),x),x)).
1326
f(f(x,y),x) = f(x,f(x,y)).
1311
1327
f(f(x,y),f(f(y,f(x,y)),f(x,y))) = f(y,f(f(f(x,y),y),y)).
1312
1328
f(f(x,y),y) = f(y,f(f(f(x,y),y),y)).
1313
1329
f(f(x,y),y) = f(y,f(x,y)).
1314
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(z,f(x,z))),x)) = f(z,f(f(x,z),z)).
1315
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(z,f(x,z))),x)) = f(z,f(z,f(x,z))).
1316
f(f(f(x,y),f(f(f(x,f(x,f(z,x))),f(x,y)),f(x,y))),z) = f(x,f(f(z,x),x)).
1317
f(f(f(x,y),f(f(x,y),f(f(x,f(x,f(z,x))),f(x,y)))),z) = f(x,f(f(z,x),x)).
1318
f(f(f(x,y),f(f(x,y),f(f(x,f(x,f(z,x))),f(x,y)))),z) = f(x,f(x,f(z,x))).
1319
1330
f(x,f(x,f(y,x))) = f(x,y).
1320
f(f(f(x,y),f(f(x,y),f(f(x,z),f(x,y)))),z) = f(x,f(x,f(z,x))).
1321
f(f(f(x,y),f(x,z)),z) = f(x,f(x,f(z,x))).
1322
f(f(f(x,y),f(x,z)),z) = f(x,z).
1323
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,x),x)) = f(z,f(z,f(x,z))).
1324
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(x,f(z,x))) = f(z,f(z,f(x,z))).
1325
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(x,f(z,x))) = f(z,x).
1326
f(f(x,y),f(x,f(x,y))) = x.
1327
f(x,f(x,f(z,x))) = f(z,x).
1331
f(f(f(y,x),f(f(x,x),f(y,x))),f(f(f(x,y),y),y)) = x.
1332
f(f(f(y,x),f(f(x,x),f(y,x))),f(f(y,f(x,y)),y)) = x.
1333
f(f(f(y,x),f(f(x,x),f(y,x))),f(y,f(y,f(x,y)))) = x.
1334
f(f(f(y,x),f(f(x,x),f(y,x))),f(y,x)) = x.
1335
f(f(y,x),f(f(y,x),f(f(x,x),f(y,x)))) = x.
1336
f(f(x,y),f(y,y)) = y.
1337
f(f(x,f(f(y,f(x,y)),y)),f(f(y,x),f(x,x))) = f(y,x).
1338
f(f(x,f(y,f(y,f(x,y)))),f(f(y,x),f(x,x))) = f(y,x).
1339
f(f(x,f(y,x)),f(f(y,x),f(x,x))) = f(y,x).
1340
f(f(x,f(y,x)),x) = f(y,x).
1341
f(x,f(x,f(y,x))) = f(y,x).
1328
1342
f(x,y) = f(y,x).
1329
1343
$F # answer("Commutativity").
1330
1344
end_of_list.
1355
1369
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1356
1370
end_of_list.
1357
1371
1358
Child search process 13966 started.
1372
Child search process 10939 started.
1359
1373
1360
1374
============================== PROCESS INITIAL CLAUSES ===============
1361
1375
1385
1399
formulas(demodulators).
1386
1400
end_of_list.
1387
1401
1388
% 322 hints input.
1402
% 285 hints input.
1389
1403
1390
1404
============================== PREDICATE ELIMINATION =================
1391
1405
1398
1412
1399
1413
Term ordering decisions:
1400
1414
Function symbol KB weights: c1=1. c2=1. f=1.
1401
Relation symbol precedence: lex([ = ]).
1402
Function symbol precedence: lex([ c1, c2, f ]).
1415
Predicate symbol precedence: predicate_order([ = ]).
1416
Function symbol precedence: function_order([ c1, c2, f ]).
1403
1417
Skipping inverse_order, because term ordering is KBO.
1404
1418
Unfolding symbols: (none).
1405
1419
1406
1420
Auto_inference settings:
1407
1421
% set(paramodulation). % (positive equality literals)
1408
% set(paramodulation) -> set(back_demod).
1409
1422
1410
1423
Auto_process settings: (no changes).
1411
1424
1425
kept: 2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
1426
kept: 3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
1427
kept: 4 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
1428
kept: 5 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
1429
kept: 6 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1430
kept: 7 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1431
kept: 8 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1432
kept: 9 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1433
kept: 10 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1434
kept: 11 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
1435
kept: 12 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1436
kept: 13 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1437
kept: 14 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
1438
kept: 15 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
1439
kept: 16 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1440
1412
1441
============================== end of process initial clauses ========
1413
1442
1414
1443
============================== CLAUSES FOR SEARCH ====================
1453
1482
16 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1454
1483
end_of_list.
1455
1484
1456
% 322 hints processed (121 are redundant).
1485
% 154 hints (285 processed, 131 redundant).
1457
1486
1458
1487
============================== end of clauses for search =============
1459
1488
1460
1489
============================== SEARCH ================================
1461
1490
1462
% Starting search at 0.00 seconds.
1491
% Starting search at 0.02 seconds.
1463
1492
1464
1493
% Operation f is commutative; C redundancy checks enabled.
1465
1494
1466
1495
============================== PROOF =================================
1467
1496
1468
% Proof 1 at 0.53 (+ 0.00) seconds: "Commutativity".
1469
% Length of proof is 57.
1497
% Proof 1 at 0.45 (+ 0.00) seconds: "Commutativity".
1498
% Length of proof is 59.
1470
1499
% Level of proof is 17.
1471
% Maximum clause weight is 35.
1472
% Given clauses 28.
1500
% Maximum clause weight is 51.
1501
% Given clauses 30.
1473
1502
1474
1 f(x,y) = f(y,x) # label("Commutativity"). [goal].
1503
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
1475
1504
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
1476
1505
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
1477
1506
4 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
1478
5 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
1479
1507
6 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1508
7 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1480
1509
8 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1481
1510
9 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1482
1511
10 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1483
11 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
1484
1512
12 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1485
14 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
1513
13 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1514
15 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
1486
1515
16 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1487
17 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)). [para(2(a,1),2(a,1,1,2,1))].
1488
19 f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)). [para(2(a,1),2(a,1,2))].
1489
1516
21 f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x. [para(4(a,1),2(a,1,1,2))].
1490
544 f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)). [para(2(a,1),16(a,1,2))].
1491
612 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),x)) = f(z,f(f(x,z),z)). [para(2(a,1),17(a,1,2,2))].
1492
630 f(x,f(f(x,f(f(x,x),x)),f(y,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)). [para(6(a,1),17(a,1,1))].
1493
686 f(f(f(x,f(x,x)),x),f(x,x)) = x. [para(2(a,1),21(a,1,2,2))].
1494
725 f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)). [para(17(a,1),21(a,1,2,2)),rewrite(2(10),2(15))].
1495
727 f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y. [para(686(a,1),2(a,1,2,2))].
1496
749 f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x. [para(686(a,1),12(a,1,1,1))].
1497
762 f(f(f(x,f(x,f(f(x,x),x))),x),f(x,x)) = x. [para(17(a,1),686(a,1,1,1,2)),rewrite(6(8),6(12),6(13),6(13),6(15))].
1498
830 f(f(f(x,f(x,f(f(x,x),f(f(y,f(x,x)),f(x,x))))),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))). [para(727(a,1),10(a,1,2))].
1499
874 f(f(f(x,x),f(x,f(x,x))),f(f(f(x,f(x,f(f(x,x),x))),x),x)) = f(f(x,f(x,f(f(x,x),x))),x). [para(762(a,1),727(a,1,1,2,1))].
1500
876 f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),f(f(x,f(x,x)),f(x,x))),x)) = f(f(x,x),f(f(x,f(x,x)),f(x,x))). [para(727(a,1),727(a,1,1,2,1))].
1501
900 f(f(f(f(x,y),x),x),f(y,f(z,f(f(x,z),z)))) = y. [para(19(a,1),8(a,1,2,2))].
1502
956 f(f(f(x,f(x,x)),x),f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),x))) = f(f(x,x),f(x,f(x,x))). [para(749(a,1),5(a,1,1,2))].
1503
970 f(f(f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),f(x,f(x,x)))),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)). [para(749(a,1),11(a,1,2))].
1504
984 f(f(f(x,f(x,f(x,f(x,x)))),x),x) = f(x,f(x,x)). [para(749(a,1),16(a,1,2))].
1505
987 f(f(x,f(x,x)),f(x,x)) = x. [para(17(a,1),749(a,1,2,2,1)),rewrite(6(8),6(8),6(8),6(10),6(13),6(13),6(13),6(8),6(12))].
1506
989 f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),x),x)) = f(f(x,x),x). [back_rewrite(876),rewrite(987(9),987(13))].
1507
993 f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),f(y,f(x,x)))) = f(x,f(x,x)). [para(987(a,1),2(a,1,1,2,1))].
1508
1006 f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))) = f(x,x). [para(987(a,1),9(a,1,1,1))].
1509
1026 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(970),rewrite(993(10),1006(7))].
1510
1033 f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),x)) = f(x,f(x,x)). [back_rewrite(989),rewrite(1026(6),1026(10))].
1511
1038 f(f(f(x,x),f(x,f(x,x))),f(x,f(x,x))) = f(f(x,f(x,f(x,f(x,x)))),x). [back_rewrite(874),rewrite(1026(6),984(10),1026(9))].
1512
1069 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(725),rewrite(1026(2),1026(6),14(8),1026(3)),flip(a)].
1513
1071 f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x). [back_rewrite(630),rewrite(1026(2),1069(3),1026(3),1069(4),1026(7),1069(8))].
1514
1089 f(f(x,x),f(x,f(x,x))) = x. [back_rewrite(956),rewrite(1033(11),16(6)),flip(a)].
1515
1093 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(1038),rewrite(1089(4),1069(3),1069(4)),flip(a)].
1516
1101 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y). [back_rewrite(830),rewrite(1071(7),1093(3)),flip(a)].
1517
1160 f(f(x,x),f(x,x)) = x. [back_rewrite(686),rewrite(1093(3))].
1518
1215 f(f(x,y),f(f(y,f(f(f(x,y),y),y)),f(x,y))) = f(y,f(f(f(x,y),y),y)). [para(1160(a,1),17(a,1,2,2)),rewrite(1089(9))].
1519
1216 f(f(x,x),f(x,f(y,x))) = x. [para(1026(a,1),2(a,1,1,2)),rewrite(1069(3))].
1520
1409 f(x,f(f(y,x),x)) = f(x,y). [para(1093(a,1),19(a,1,1,1)),rewrite(1093(7),1093(9),1101(9),1216(5)),flip(a)].
1521
1436 f(f(x,y),y) = f(y,f(x,y)). [back_rewrite(1215),rewrite(1409(5),1409(6),1409(6))].
1522
1493 f(f(x,f(f(x,y),x)),f(y,f(z,f(z,f(x,z))))) = y. [back_rewrite(900),rewrite(1436(3),1436(5))].
1523
1573 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(z,f(x,z))),x)) = f(z,f(z,f(x,z))). [back_rewrite(612),rewrite(1436(8),1436(13))].
1524
1578 f(x,f(f(x,f(x,f(y,f(y,f(x,y))))),x)) = f(y,f(y,f(x,y))). [back_rewrite(544),rewrite(1436(2),1436(7),1436(9))].
1525
1679 f(x,f(x,f(y,x))) = f(x,y). [back_rewrite(1409),rewrite(1436(2))].
1526
2106 f(x,f(f(x,y),x)) = f(y,x). [back_rewrite(1578),rewrite(1679(3),1679(3),1679(6))].
1527
2111 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(x,f(z,x))) = f(z,x). [back_rewrite(1573),rewrite(1679(9),1436(8),1679(12))].
1528
2179 f(f(x,y),f(x,f(z,y))) = x. [back_rewrite(1493),rewrite(2106(3),1679(4))].
1529
2360 f(x,y) = f(y,x). [back_rewrite(2111),rewrite(2179(6),1679(3))].
1530
2361 $F # answer("Commutativity"). [resolve(2360,a,3,a)].
1517
53 f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)). [para(2(a,1),6(a,1,2))].
1518
254 f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)). [para(2(a,1),12(a,1,2))].
1519
589 f(f(f(f(f(x,y),f(f(x,f(x,z)),x)),z),z),z) = f(f(x,f(x,z)),x). [para(16(a,1),12(a,1,2))].
1520
603 f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(y,f(y,x)),y). [para(16(a,1),15(a,1,2))].
1521
611 f(f(f(x,f(x,x)),x),f(x,x)) = x. [para(2(a,1),21(a,1,2,2))].
1522
647 f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y. [para(611(a,1),2(a,1,2,2))].
1523
648 f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)). [para(2(a,1),611(a,1,1,1,2)),rewrite([2(15)])].
1524
670 f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x. [para(611(a,1),12(a,1,1,1))].
1525
755 f(f(f(x,x),f(x,f(x,x))),f(f(f(x,f(x,x)),x),x)) = f(f(x,f(x,x)),x). [para(611(a,1),647(a,1,1,2,1))].
1526
761 f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)). [para(2(a,1),670(a,1,1,2)),rewrite([2(14),2(17),2(13)])].
1527
770 f(f(x,f(f(f(x,f(x,x)),x),x)),x) = f(x,f(x,x)). [para(670(a,1),6(a,1,2))].
1528
786 f(f(f(f(x,f(x,f(x,x))),x),x),x) = f(x,f(x,x)). [para(670(a,1),13(a,1,2))].
1529
797 f(f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))) = f(f(x,f(f(x,x),x)),x). [para(2(a,1),770(a,1,1,2,1,1,2)),rewrite([2(30)])].
1530
975 f(f(f(x,y),y),y) = f(x,f(f(y,x),x)). [para(8(a,1),254(a,1,1,1,1))].
1531
1040 f(x,f(f(f(x,f(x,x)),x),x)) = f(x,f(x,f(x,x))). [para(4(a,1),975(a,1,1)),flip(a)].
1532
1078 f(f(f(x,f(f(y,x),x)),y),f(y,f(f(x,y),y))) = y. [para(975(a,1),9(a,1,1,1))].
1533
1146 f(f(f(x,f(x,x)),x),f(f(f(x,x),f(f(x,f(x,x)),x)),f(f(x,f(x,x)),x))) = f(f(x,f(x,x)),f(x,x)). [para(611(a,1),975(a,1,1,1)),flip(a)].
1534
1147 f(f(f(f(x,x),f(f(x,f(x,x)),x)),f(f(x,f(x,x)),x)),f(f(x,f(x,x)),x)) = f(f(x,x),f(x,f(x,x))). [para(611(a,1),975(a,2,2,1))].
1535
1150 f(f(x,f(x,f(x,x))),x) = f(x,f(x,x)). [para(975(a,2),770(a,1,1)),rewrite([7(6)])].
1536
1151 f(f(x,f(x,f(x,x))),f(f(x,f(x,f(x,f(x,x)))),f(x,f(x,f(x,x))))) = f(f(f(x,f(x,x)),x),x). [para(770(a,1),975(a,1,1,1)),rewrite([1040(9),1040(12),1040(16)]),flip(a)].
1537
1153 f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)). [para(975(a,1),786(a,1)),rewrite([1151(12)])].
1538
1159 f(x,f(f(f(y,x),x),x)) = f(x,f(y,f(f(x,y),y))). [para(975(a,1),53(a,2,2)),rewrite([53(12)])].
1539
1178 f(f(f(f(f(x,y),y),y),f(f(y,x),x)),f(f(y,x),x)) = f(x,f(f(f(f(y,x),x),x),x)). [para(975(a,2),975(a,1,1,1))].
1540
1220 f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))). [back_rewrite(797),rewrite([1078(11),6(8)]),flip(a)].
1541
1221 f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)). [back_rewrite(648),rewrite([1220(4)])].
1542
1268 f(f(f(x,x),f(x,f(x,x))),f(x,f(x,x))) = f(f(x,f(x,x)),x). [back_rewrite(755),rewrite([1153(8)])].
1543
1293 f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)). [back_rewrite(761),rewrite([1220(4)])].
1544
1301 f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,x). [para(1150(a,1),10(a,1,1))].
1545
1336 f(f(f(x,f(x,x)),x),f(f(f(x,x),f(y,x)),f(f(x,f(x,x)),z))) = f(f(x,x),f(y,x)). [para(1301(a,1),13(a,1,1,1,1)),rewrite([1268(7)])].
1546
1343 f(f(f(x,x),f(x,f(x,x))),f(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x). [para(1301(a,1),15(a,1,1,1))].
1547
1356 f(f(x,x),f(f(x,f(x,x)),x)) = f(f(x,f(x,x)),f(x,x)). [back_rewrite(1146),rewrite([1336(13)])].
1548
1357 f(f(f(f(x,f(x,x)),f(x,x)),f(f(x,f(x,x)),x)),f(f(x,f(x,x)),x)) = f(f(x,x),f(x,f(x,x))). [back_rewrite(1147),rewrite([1356(5)])].
1549
1548 f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x. [para(1220(a,1),1078(a,1,1))].
1550
1551 f(x,f(f(x,x),x)) = f(x,x). [back_rewrite(1221),rewrite([1548(8)]),flip(a)].
1551
1554 f(f(x,f(x,x)),f(x,x)) = x. [back_rewrite(1548),rewrite([1551(3),1551(5)])].
1552
1586 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(1293),rewrite([1551(3),1551(6)])].
1553
1599 f(f(x,x),f(x,f(x,x))) = x. [back_rewrite(1357),rewrite([1554(4),1586(3),1586(5),1554(4)]),flip(a)].
1554
1602 f(x,f(f(y,x),x)) = f(y,x). [back_rewrite(1343),rewrite([1599(4),1554(5)])].
1555
1759 f(f(f(f(f(x,y),y),y),f(f(y,x),x)),f(f(y,x),x)) = f(f(f(y,x),x),x). [back_rewrite(1178),rewrite([1602(14)])].
1556
1768 f(f(x,y),y) = f(y,f(y,x)). [back_rewrite(1159),rewrite([1602(4),1602(5)])].
1557
2023 f(f(x,f(x,y)),x) = f(x,f(x,f(x,y))). [back_rewrite(1759),rewrite([1768(2),1768(5),10(6),1768(2),1768(5)]),flip(a)].
1558
2065 f(x,f(x,f(x,y))) = f(y,x). [back_rewrite(1602),rewrite([1768(2)])].
1559
2133 f(f(f(x,y),f(z,x)),z) = f(z,x). [back_rewrite(603),rewrite([2023(4),2065(4),1768(6),2065(6),2023(7),2065(7)])].
1560
2138 f(x,y) = f(y,x). [back_rewrite(589),rewrite([2023(4),2065(4),2133(4),1768(3),2065(3),2023(4),2065(4)])].
1561
2139 $F # answer("Commutativity"). [resolve(2138,a,3,a)].
1531
1562
1532
1563
============================== end of proof ==========================
1533
1564
1534
1565
============================== STATISTICS ============================
1535
1566
1536
Given=28. Generated=3642. Kept=2359. proofs=1.
1537
Usable=11. Sos=674. Demods=864. Limbo=181, Disabled=1507. Hints=322.
1538
Weight_deleted=66. Literals_deleted=0.
1539
Forward_subsumed=1217. Back_subsumed=113.
1567
Given=30. Generated=3223. Kept=2137. proofs=1.
1568
Usable=15. Sos=613. Demods=985. Limbo=370, Disabled=1153. Hints=285.
1569
Kept_by_rule=0, Deleted_by_rule=0.
1570
Forward_subsumed=1086. Back_subsumed=44.
1540
1571
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
1541
New_demodulators=2354 (0 lex), Back_demodulated=1379. Back_unit_deleted=0.
1542
Demod_attempts=85839. Demod_rewrites=6134.
1572
New_demodulators=2095 (0 lex), Back_demodulated=1094. Back_unit_deleted=0.
1573
Demod_attempts=67952. Demod_rewrites=4888.
1543
1574
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
1544
1575
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
1545
Megabytes=2.15.
1546
User_CPU=0.53, System_CPU=0.00, Wall_clock=5.
1576
Megabytes=2.87.
1577
User_CPU=0.45, System_CPU=0.00, Wall_clock=3.
1547
1578
1548
1579
============================== end of statistics =====================
1549
1580
1551
1582
1552
1583
Exiting with 1 proof.
1553
1584
1554
Process 13966 exit (max_proofs) Sun Aug 13 21:05:02 2006
1585
Process 10939 exit (max_proofs) Mon Sep 28 11:57:12 2009
1555
1586
1556
1587
Successful proof using the following 10 assumptions:
1557
1588
1558
1589
formulas(assumptions_in_proof).
1559
1590
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
1560
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
1561
1591
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1592
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1562
1593
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1563
1594
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1564
1595
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1565
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
1566
1596
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1567
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
1597
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1598
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
1568
1599
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1569
1600
end_of_list.
1570
1601
1571
Including 113 new hint clauses:
1602
Including 101 new hint clauses:
1572
1603
1573
1604
formulas(new_hints).
1574
$F # label("Commutativity").
1605
$F # label("Commutativity") # label(non_clause) # label(goal).
1575
1606
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
1576
1607
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
1577
1608
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").
1578
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").
1579
1609
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
1610
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").
1580
1611
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").
1581
1612
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
1582
1613
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
1583
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").
1584
1614
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
1585
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
1615
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
1616
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
1586
1617
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
1587
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)).
1588
f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)).
1589
1618
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
1590
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
1591
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),x)) = f(z,f(f(x,z),z)).
1592
f(x,f(f(x,f(f(x,x),x)),f(y,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
1619
f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)).
1620
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
1621
f(f(f(f(f(x,y),f(f(x,f(x,z)),x)),z),z),z) = f(f(x,f(x,z)),x).
1622
f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(y,f(y,x)),y).
1593
1623
f(f(f(x,f(x,x)),x),f(x,x)) = x.
1594
f(f(f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
1624
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
1595
1625
f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
1596
1626
f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
1597
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
1598
1627
f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x.
1599
f(f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1600
f(f(f(x,f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1601
f(f(f(x,f(x,f(f(x,x),x))),x),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1602
f(f(f(x,f(x,f(f(x,x),x))),x),f(x,f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1603
f(f(f(x,f(x,f(f(x,x),x))),x),f(x,x)) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1604
f(f(f(x,f(x,f(f(x,x),x))),x),f(x,x)) = x.
1605
f(f(f(x,f(x,f(f(x,x),f(f(y,f(x,x)),f(x,x))))),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
1606
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,f(x,f(f(x,x),x))),x),x)) = f(f(x,f(x,f(f(x,x),x))),x).
1607
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),f(f(x,f(x,x)),f(x,x))),x)) = f(f(x,x),f(f(x,f(x,x)),f(x,x))).
1608
f(f(f(f(x,y),x),x),f(y,f(z,f(f(x,z),z)))) = y.
1609
f(f(f(x,f(x,x)),x),f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),x))) = f(f(x,x),f(x,f(x,x))).
1610
f(f(f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),f(x,f(x,x)))),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)).
1611
f(f(f(x,f(x,f(x,f(x,x)))),x),x) = f(x,f(x,x)).
1612
f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1613
f(f(x,f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1614
f(f(x,f(x,f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1615
f(f(x,f(x,x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1616
f(f(x,f(x,x)),f(x,f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1617
f(f(x,f(x,x)),f(x,f(f(x,f(f(x,x),x)),f(x,f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1618
f(f(x,f(x,x)),f(x,f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1619
f(f(x,f(x,x)),f(x,f(f(x,f(f(x,x),x)),f(x,f(x,x))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1620
f(f(x,f(x,x)),f(x,x)) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1628
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,f(x,x)),x),x)) = f(f(x,f(x,x)),x).
1629
f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))))) = f(x,f(f(x,x),x)).
1630
f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(x,f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))))) = f(x,f(f(x,x),x)).
1631
f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(x,f(f(x,f(f(x,x),x)),x)))) = f(x,f(f(x,x),x)).
1632
f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)).
1633
f(f(x,f(f(f(x,f(x,x)),x),x)),x) = f(x,f(x,x)).
1634
f(f(f(f(x,f(x,f(x,x))),x),x),x) = f(x,f(x,x)).
1635
f(f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))) = f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
1636
f(f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))) = f(f(x,f(f(x,x),x)),x).
1637
f(f(f(x,y),y),y) = f(x,f(f(y,x),x)).
1638
f(x,f(x,f(x,x))) = f(x,f(f(f(x,f(x,x)),x),x)).
1639
f(x,f(f(f(x,f(x,x)),x),x)) = f(x,f(x,f(x,x))).
1640
f(f(f(x,f(f(y,x),x)),y),f(y,f(f(x,y),y))) = y.
1641
f(f(x,f(x,x)),f(x,x)) = f(f(f(x,f(x,x)),x),f(f(f(x,x),f(f(x,f(x,x)),x)),f(f(x,f(x,x)),x))).
1642
f(f(f(x,f(x,x)),x),f(f(f(x,x),f(f(x,f(x,x)),x)),f(f(x,f(x,x)),x))) = f(f(x,f(x,x)),f(x,x)).
1643
f(f(f(f(x,x),f(f(x,f(x,x)),x)),f(f(x,f(x,x)),x)),f(f(x,f(x,x)),x)) = f(f(x,x),f(x,f(x,x))).
1644
f(f(f(f(x,f(x,f(x,x))),f(x,f(x,x))),f(x,f(x,x))),x) = f(x,f(x,x)).
1645
f(f(x,f(x,f(x,x))),x) = f(x,f(x,x)).
1646
f(f(f(x,f(x,x)),x),x) = f(f(x,f(f(f(x,f(x,x)),x),x)),f(f(x,f(x,f(f(f(x,f(x,x)),x),x))),f(x,f(f(f(x,f(x,x)),x),x)))).
1647
f(f(f(x,f(x,x)),x),x) = f(f(x,f(x,f(x,x))),f(f(x,f(x,f(f(f(x,f(x,x)),x),x))),f(x,f(f(f(x,f(x,x)),x),x)))).
1648
f(f(f(x,f(x,x)),x),x) = f(f(x,f(x,f(x,x))),f(f(x,f(x,f(x,f(x,x)))),f(x,f(f(f(x,f(x,x)),x),x)))).
1649
f(f(f(x,f(x,x)),x),x) = f(f(x,f(x,f(x,x))),f(f(x,f(x,f(x,f(x,x)))),f(x,f(x,f(x,x))))).
1650
f(f(x,f(x,f(x,x))),f(f(x,f(x,f(x,f(x,x)))),f(x,f(x,f(x,x))))) = f(f(f(x,f(x,x)),x),x).
1651
f(f(x,f(x,f(x,x))),f(f(x,f(x,f(x,f(x,x)))),f(x,f(x,f(x,x))))) = f(x,f(x,x)).
1652
f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)).
1653
f(f(f(x,y),f(f(f(y,f(f(f(x,y),y),y)),f(x,y)),f(x,y))),f(x,y)) = f(y,f(x,f(f(y,x),x))).
1654
f(x,f(f(f(y,x),x),x)) = f(x,f(y,f(f(x,y),y))).
1655
f(f(f(f(f(x,y),y),y),f(f(y,x),x)),f(f(y,x),x)) = f(x,f(f(f(f(y,x),x),x),x)).
1656
f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))) = f(f(x,f(f(x,x),x)),x).
1657
f(x,f(x,f(f(x,x),x))) = f(f(x,f(f(x,x),x)),x).
1658
f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))).
1659
f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
1660
f(f(f(x,x),f(x,f(x,x))),f(x,f(x,x))) = f(f(x,f(x,x)),x).
1661
f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
1662
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,x).
1663
f(f(f(f(x,x),f(x,f(x,x))),f(x,f(x,x))),f(f(f(x,x),f(y,x)),f(f(x,f(x,x)),z))) = f(f(x,x),f(y,x)).
1664
f(f(f(x,f(x,x)),x),f(f(f(x,x),f(y,x)),f(f(x,f(x,x)),z))) = f(f(x,x),f(y,x)).
1665
f(f(f(x,x),f(x,f(x,x))),f(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x).
1666
f(f(x,x),f(f(x,f(x,x)),x)) = f(f(x,f(x,x)),f(x,x)).
1667
f(f(f(f(x,f(x,x)),f(x,x)),f(f(x,f(x,x)),x)),f(f(x,f(x,x)),x)) = f(f(x,x),f(x,f(x,x))).
1668
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
1669
f(x,x) = f(x,f(f(x,x),x)).
1670
f(x,f(f(x,x),x)) = f(x,x).
1671
f(f(x,f(x,x)),f(x,f(f(x,x),x))) = x.
1621
1672
f(f(x,f(x,x)),f(x,x)) = x.
1622
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),x),x)) = f(f(x,x),f(f(x,f(x,x)),f(x,x))).
1623
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),x),x)) = f(f(x,x),x).
1624
f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),f(y,f(x,x)))) = f(x,f(x,x)).
1625
f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))) = f(x,x).
1626
f(f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)).
1627
f(f(x,x),x) = f(x,f(x,x)).
1628
f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),x)) = f(f(x,x),x).
1629
f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),x)) = f(x,f(x,x)).
1630
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,f(x,f(x,f(x,x)))),x),x)) = f(f(x,f(x,f(f(x,x),x))),x).
1631
f(f(f(x,x),f(x,f(x,x))),f(x,f(x,x))) = f(f(x,f(x,f(f(x,x),x))),x).
1632
f(f(f(x,x),f(x,f(x,x))),f(x,f(x,x))) = f(f(x,f(x,f(x,f(x,x)))),x).
1633
f(f(f(f(x,f(x,f(x,x))),x),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
1634
f(f(f(f(x,f(x,f(x,x))),x),f(x,f(x,f(x,x)))),x) = f(x,f(f(x,x),x)).
1635
f(x,x) = f(x,f(f(x,x),x)).
1636
f(x,x) = f(x,f(x,f(x,x))).
1637
f(x,f(x,f(x,x))) = f(x,x).
1638
f(x,f(f(x,f(x,f(x,x))),f(y,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
1639
f(x,f(f(x,x),f(y,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
1640
f(x,f(f(x,x),f(y,f(x,f(x,f(x,x)))))) = f(x,f(f(x,x),x)).
1641
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,f(f(x,x),x)).
1642
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,f(x,f(x,x))).
1643
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x).
1644
f(f(f(x,f(x,x)),x),f(x,f(x,x))) = f(f(x,x),f(x,f(x,x))).
1673
f(f(x,f(x,x)),x) = f(x,f(f(x,x),x)).
1674
f(f(x,f(x,x)),x) = f(x,x).
1675
f(f(x,f(f(x,f(x,x)),x)),f(f(x,f(x,x)),x)) = f(f(x,x),f(x,f(x,x))).
1676
f(f(x,f(x,x)),f(f(x,f(x,x)),x)) = f(f(x,x),f(x,f(x,x))).
1677
f(f(x,f(x,x)),f(x,x)) = f(f(x,x),f(x,f(x,x))).
1645
1678
x = f(f(x,x),f(x,f(x,x))).
1646
1679
f(f(x,x),f(x,f(x,x))) = x.
1647
f(x,f(x,f(x,x))) = f(f(x,f(x,f(x,f(x,x)))),x).
1648
f(x,x) = f(f(x,f(x,f(x,f(x,x)))),x).
1649
f(x,x) = f(f(x,f(x,x)),x).
1650
f(f(x,f(x,x)),x) = f(x,x).
1651
f(f(f(x,f(x,x)),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
1652
f(f(x,x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
1653
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y).
1654
f(f(x,x),f(x,x)) = x.
1655
f(f(f(f(x,y),f(x,y)),f(f(x,y),f(f(x,y),f(x,y)))),f(f(y,f(f(f(x,y),y),y)),f(x,y))) = f(y,f(f(f(x,y),y),y)).
1656
f(f(x,y),f(f(y,f(f(f(x,y),y),y)),f(x,y))) = f(y,f(f(f(x,y),y),y)).
1657
f(f(x,f(x,f(x,x))),f(x,f(y,x))) = x.
1658
f(f(x,x),f(x,f(y,x))) = x.
1659
f(f(f(x,x),f(f(f(x,f(f(y,x),x)),f(f(x,f(x,x)),x)),f(f(x,f(x,x)),x))),y) = f(x,f(f(y,x),x)).
1660
f(f(f(x,x),f(f(f(x,f(f(y,x),x)),f(x,x)),f(f(x,f(x,x)),x))),y) = f(x,f(f(y,x),x)).
1661
f(f(f(x,x),f(f(f(x,f(f(y,x),x)),f(x,x)),f(x,x))),y) = f(x,f(f(y,x),x)).
1662
f(f(f(x,x),f(x,f(f(y,x),x))),y) = f(x,f(f(y,x),x)).
1663
f(x,y) = f(x,f(f(y,x),x)).
1664
f(x,f(f(y,x),x)) = f(x,y).
1665
f(f(x,y),f(f(y,f(x,y)),f(x,y))) = f(y,f(f(f(x,y),y),y)).
1666
f(f(x,y),y) = f(y,f(f(f(x,y),y),y)).
1667
f(f(x,y),y) = f(y,f(x,y)).
1668
f(f(x,f(f(x,y),x)),f(y,f(z,f(f(x,z),z)))) = y.
1669
f(f(x,f(f(x,y),x)),f(y,f(z,f(z,f(x,z))))) = y.
1670
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(z,f(x,z))),x)) = f(z,f(f(x,z),z)).
1671
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(z,f(x,z))),x)) = f(z,f(z,f(x,z))).
1672
f(f(f(x,f(x,f(y,f(y,f(x,y))))),x),x) = f(y,f(f(x,y),y)).
1673
f(x,f(f(x,f(x,f(y,f(y,f(x,y))))),x)) = f(y,f(f(x,y),y)).
1674
f(x,f(f(x,f(x,f(y,f(y,f(x,y))))),x)) = f(y,f(y,f(x,y))).
1675
f(x,f(x,f(y,x))) = f(x,y).
1676
f(x,f(f(x,f(x,f(y,x))),x)) = f(y,f(y,f(x,y))).
1677
f(x,f(f(x,y),x)) = f(y,f(y,f(x,y))).
1678
f(x,f(f(x,y),x)) = f(y,x).
1679
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,x),x)) = f(z,f(z,f(x,z))).
1680
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(x,f(z,x))) = f(z,f(z,f(x,z))).
1681
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(x,f(z,x))) = f(z,x).
1682
f(f(y,x),f(y,f(z,f(z,f(x,z))))) = y.
1683
f(f(x,y),f(x,f(z,y))) = x.
1684
f(x,f(x,f(z,x))) = f(z,x).
1680
f(x,f(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x).
1681
f(x,f(f(y,x),x)) = f(y,x).
1682
f(f(f(f(f(x,y),y),y),f(f(y,x),x)),f(f(y,x),x)) = f(f(f(y,x),x),x).
1683
f(f(y,x),x) = f(x,f(y,f(f(x,y),y))).
1684
f(f(x,y),y) = f(y,f(y,x)).
1685
f(f(f(f(y,f(y,x)),y),f(f(y,x),x)),f(f(y,x),x)) = f(f(f(y,x),x),x).
1686
f(f(f(f(y,f(y,x)),y),f(x,f(x,y))),f(f(y,x),x)) = f(f(f(y,x),x),x).
1687
f(x,f(f(y,x),x)) = f(f(f(y,x),x),x).
1688
f(x,f(x,f(x,y))) = f(f(f(y,x),x),x).
1689
f(x,f(x,f(x,y))) = f(f(x,f(x,y)),x).
1690
f(f(x,f(x,y)),x) = f(x,f(x,f(x,y))).
1691
f(x,f(x,f(x,y))) = f(y,x).
1692
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(f(y,f(y,x)),y).
1693
f(f(f(x,f(f(y,z),f(x,y))),x),x) = f(f(y,f(y,x)),y).
1694
f(x,f(x,f(x,f(f(y,z),f(x,y))))) = f(f(y,f(y,x)),y).
1695
f(f(f(y,z),f(x,y)),x) = f(f(y,f(y,x)),y).
1696
f(f(f(y,z),f(x,y)),x) = f(y,f(y,f(y,x))).
1697
f(f(f(x,y),f(z,x)),z) = f(z,x).
1698
f(f(f(f(f(x,y),f(x,f(x,f(x,z)))),z),z),z) = f(f(x,f(x,z)),x).
1699
f(f(f(f(f(x,y),f(z,x)),z),z),z) = f(f(x,f(x,z)),x).
1700
f(f(f(z,x),z),z) = f(f(x,f(x,z)),x).
1701
f(z,f(z,f(z,x))) = f(f(x,f(x,z)),x).
1702
f(x,z) = f(f(x,f(x,z)),x).
1703
f(x,z) = f(x,f(x,f(x,z))).
1685
1704
f(x,y) = f(y,x).
1686
1705
$F # answer("Commutativity").
1687
1706
end_of_list.
1711
1730
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1712
1731
end_of_list.
1713
1732
1714
Child search process 13967 started.
1733
Child search process 10940 started.
1715
1734
1716
1735
============================== PROCESS INITIAL CLAUSES ===============
1717
1736
1740
1759
formulas(demodulators).
1741
1760
end_of_list.
1742
1761
1743
% 435 hints input.
1762
% 386 hints input.
1744
1763
1745
1764
============================== PREDICATE ELIMINATION =================
1746
1765
1753
1772
1754
1773
Term ordering decisions:
1755
1774
Function symbol KB weights: c1=1. c2=1. f=1.
1756
Relation symbol precedence: lex([ = ]).
1757
Function symbol precedence: lex([ c1, c2, f ]).
1775
Predicate symbol precedence: predicate_order([ = ]).
1776
Function symbol precedence: function_order([ c1, c2, f ]).
1758
1777
Skipping inverse_order, because term ordering is KBO.
1759
1778
Unfolding symbols: (none).
1760
1779
1761
1780
Auto_inference settings:
1762
1781
% set(paramodulation). % (positive equality literals)
1763
% set(paramodulation) -> set(back_demod).
1764
1782
1765
1783
Auto_process settings: (no changes).
1766
1784
1785
kept: 2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
1786
kept: 3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
1787
kept: 4 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
1788
kept: 5 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1789
kept: 6 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1790
kept: 7 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1791
kept: 8 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1792
kept: 9 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1793
kept: 10 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
1794
kept: 11 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1795
kept: 12 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1796
kept: 13 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
1797
kept: 14 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
1798
kept: 15 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1799
1767
1800
============================== end of process initial clauses ========
1768
1801
1769
1802
============================== CLAUSES FOR SEARCH ====================
1806
1839
15 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1807
1840
end_of_list.
1808
1841
1809
% 435 hints processed (209 are redundant).
1842
% 208 hints (386 processed, 178 redundant).
1810
1843
1811
1844
============================== end of clauses for search =============
1812
1845
1813
1846
============================== SEARCH ================================
1814
1847
1815
% Starting search at 0.00 seconds.
1848
% Starting search at 0.02 seconds.
1816
1849
1817
1850
% Operation f is commutative; C redundancy checks enabled.
1818
1851
1819
1852
============================== PROOF =================================
1820
1853
1821
% Proof 1 at 0.41 (+ 0.00) seconds: "Commutativity".
1822
% Length of proof is 54.
1823
% Level of proof is 17.
1824
% Maximum clause weight is 29.
1825
% Given clauses 27.
1854
% Proof 1 at 0.21 (+ 0.00) seconds: "Commutativity".
1855
% Length of proof is 37.
1856
% Level of proof is 15.
1857
% Maximum clause weight is 35.
1858
% Given clauses 31.
1826
1859
1827
1 f(x,y) = f(y,x) # label("Commutativity"). [goal].
1860
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
1828
1861
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
1829
1862
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
1830
1863
4 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
1831
1864
5 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1832
1865
8 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1833
1866
9 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1834
10 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
1835
1867
11 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1836
13 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
1868
12 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1837
1869
15 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1838
16 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)). [para(2(a,1),2(a,1,1,2,1))].
1839
18 f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)). [para(2(a,1),2(a,1,2))].
1840
1870
20 f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x. [para(4(a,1),2(a,1,1,2))].
1841
34 f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)). [para(2(a,1),5(a,1,2))].
1842
121 f(f(f(x,y),f(f(f(f(y,f(y,z)),y),f(x,y)),f(x,y))),z) = f(f(y,f(y,z)),y). [para(9(a,1),2(a,1,2))].
1843
236 f(f(f(f(f(x,y),f(f(y,f(y,z)),y)),z),z),z) = f(f(y,f(y,z)),y). [para(9(a,1),11(a,1,2))].
1871
455 f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)). [para(2(a,1),15(a,1,2))].
1844
1872
515 f(f(f(x,f(x,x)),x),f(x,x)) = x. [para(2(a,1),20(a,1,2,2))].
1845
561 f(x,f(f(x,f(f(x,x),x)),f(y,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)). [para(5(a,1),16(a,1,1))].
1846
616 f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)). [para(16(a,1),20(a,1,2,2)),rewrite(2(10),2(15))].
1847
621 f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y. [para(515(a,1),2(a,1,2,2))].
1848
639 f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x. [para(515(a,1),11(a,1,1,1))].
1849
652 f(f(f(x,f(x,f(f(x,x),x))),x),f(x,x)) = x. [para(16(a,1),515(a,1,1,1,2)),rewrite(5(8),5(12),5(13),5(13),5(15))].
1850
710 f(f(f(x,f(x,f(f(x,x),f(f(y,f(x,x)),f(x,x))))),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))). [para(621(a,1),9(a,1,2))].
1851
754 f(f(f(x,x),f(x,f(x,x))),f(f(f(x,f(x,f(f(x,x),x))),x),x)) = f(f(x,f(x,f(f(x,x),x))),x). [para(652(a,1),621(a,1,1,2,1))].
1852
756 f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),f(f(x,f(x,x)),f(x,x))),x)) = f(f(x,x),f(f(x,f(x,x)),f(x,x))). [para(621(a,1),621(a,1,1,2,1))].
1853
762 f(f(f(x,f(x,x)),x),f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),x))) = f(f(x,x),f(x,f(x,x))). [para(639(a,1),4(a,1,1,2))].
1854
776 f(f(f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),f(x,f(x,x)))),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)). [para(639(a,1),10(a,1,2))].
1855
790 f(f(f(x,f(x,f(x,f(x,x)))),x),x) = f(x,f(x,x)). [para(639(a,1),15(a,1,2))].
1856
793 f(f(x,f(x,x)),f(x,x)) = x. [para(16(a,1),639(a,1,2,2,1)),rewrite(5(8),5(8),5(8),5(10),5(13),5(13),5(13),5(8),5(12))].
1857
794 f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),x),x)) = f(f(x,x),x). [back_rewrite(756),rewrite(793(9),793(13))].
1858
864 f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),f(y,f(x,x)))) = f(x,f(x,x)). [para(793(a,1),2(a,1,1,2,1))].
1859
874 f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))) = f(x,x). [para(793(a,1),8(a,1,1,1))].
1860
894 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(776),rewrite(864(10),874(7))].
1861
902 f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),x)) = f(x,f(x,x)). [back_rewrite(794),rewrite(894(6),894(10))].
1862
906 f(f(f(x,x),f(x,f(x,x))),f(x,f(x,x))) = f(f(x,f(x,f(x,f(x,x)))),x). [back_rewrite(754),rewrite(894(6),790(10),894(9))].
1863
934 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(616),rewrite(894(2),894(6),13(8),894(3)),flip(a)].
1864
936 f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x). [back_rewrite(561),rewrite(894(2),934(3),894(3),934(4),894(7),934(8))].
1865
951 f(f(x,x),f(x,f(x,x))) = x. [back_rewrite(762),rewrite(902(11),15(6)),flip(a)].
1866
955 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(906),rewrite(951(4),934(3),934(4)),flip(a)].
1867
962 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y). [back_rewrite(710),rewrite(936(7),955(3)),flip(a)].
1868
1015 f(f(x,x),f(x,x)) = x. [back_rewrite(515),rewrite(955(3))].
1869
1066 f(f(x,y),f(f(y,f(f(f(x,y),y),y)),f(x,y))) = f(y,f(f(f(x,y),y),y)). [para(1015(a,1),16(a,1,2,2)),rewrite(951(9))].
1870
1067 f(f(x,x),f(x,f(y,x))) = x. [para(894(a,1),2(a,1,1,2)),rewrite(934(3))].
1871
1242 f(x,f(f(y,x),x)) = f(x,y). [para(955(a,1),18(a,1,1,1)),rewrite(955(7),955(9),962(9),1067(5)),flip(a)].
1872
1269 f(f(x,y),y) = f(y,f(x,y)). [back_rewrite(1066),rewrite(1242(5),1242(6),1242(6))].
1873
1448 f(f(f(x,y),f(f(x,y),f(f(f(y,f(y,z)),y),f(x,y)))),z) = f(f(y,f(y,z)),y). [back_rewrite(121),rewrite(1269(8))].
1874
1475 f(f(x,f(x,f(f(y,f(y,f(x,y))),x))),x) = f(y,f(y,f(x,y))). [back_rewrite(34),rewrite(1269(2),1269(5),1269(9))].
1875
1492 f(x,f(x,f(y,x))) = f(x,y). [back_rewrite(1242),rewrite(1269(2))].
1876
1705 f(f(x,f(f(f(y,z),f(f(z,f(z,x)),z)),x)),x) = f(f(z,f(z,x)),z). [back_rewrite(236),rewrite(1269(7))].
1877
1817 f(f(x,f(x,y)),x) = f(y,x). [back_rewrite(1475),rewrite(1492(3),1269(2),1492(3),1492(6))].
1878
1841 f(f(f(x,y),f(z,y)),z) = f(z,y). [back_rewrite(1448),rewrite(1817(5),1492(7),1817(7))].
1879
2006 f(x,y) = f(y,x). [back_rewrite(1705),rewrite(1817(4),1841(4),1817(3),1817(4))].
1880
2007 $F # answer("Commutativity"). [resolve(2006,a,3,a)].
1873
547 f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y. [para(515(a,1),2(a,1,2,2))].
1874
548 f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)). [para(2(a,1),515(a,1,1,1,2)),rewrite([2(15)])].
1875
561 f(f(f(f(f(x,f(x,x)),x),x),f(f(x,f(x,x)),x)),f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x). [para(515(a,1),9(a,1,1,1,2))].
1876
566 f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x. [para(515(a,1),11(a,1,1,1))].
1877
609 f(f(f(x,f(x,f(f(x,x),f(f(y,f(x,x)),f(x,x))))),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))). [para(547(a,1),9(a,1,2))].
1878
651 f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)). [para(2(a,1),566(a,1,1,2)),rewrite([2(14),2(17),2(13)])].
1879
685 f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x. [para(2(a,1),651(a,1,1,1,2,1)),rewrite([5(8),2(18),5(16)])].
1880
805 f(f(f(x,f(f(x,x),x)),x),f(x,f(y,f(x,f(f(x,x),x))))) = x. [para(685(a,1),2(a,1,1,2))].
1881
817 f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)). [para(685(a,1),8(a,1,1,1)),rewrite([5(12)])].
1882
827 f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))). [para(685(a,1),12(a,1,2)),rewrite([455(7)])].
1883
834 f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x. [back_rewrite(805),rewrite([827(4)])].
1884
861 f(x,f(f(x,x),x)) = f(x,x). [back_rewrite(548),rewrite([827(4),685(8)]),flip(a)].
1885
889 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [back_rewrite(834),rewrite([861(3),861(5)])].
1886
896 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(827),rewrite([861(3),861(5)])].
1887
905 f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(x,f(x,x))). [back_rewrite(817),rewrite([896(2),896(7)])].
1888
918 f(f(x,f(x,f(x,f(x,x)))),f(x,f(x,f(x,x)))) = x. [back_rewrite(685),rewrite([896(2),896(6)])].
1889
924 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(861),rewrite([896(2)])].
1890
928 f(f(x,f(x,x)),f(x,x)) = x. [back_rewrite(918),rewrite([924(3),924(5)])].
1891
937 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(905),rewrite([924(3),924(6)])].
1892
975 f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x). [back_rewrite(561),rewrite([937(3),896(2),937(5),928(4),937(4)])].
1893
1021 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y). [back_rewrite(609),rewrite([975(7),937(3)]),flip(a)].
1894
1172 f(f(x,x),y) = f(y,f(x,x)). [para(889(a,1),4(a,1,1,2)),rewrite([937(3),937(6),1021(6)])].
1895
1203 f(x,y) = f(y,x). [para(2(a,1),1172(a,1,1)),rewrite([2(8)])].
1896
1204 $F # answer("Commutativity"). [resolve(1203,a,3,a)].
1881
1897
1882
1898
============================== end of proof ==========================
1883
1899
1884
1900
============================== STATISTICS ============================
1885
1901
1886
Given=27. Generated=3158. Kept=2005. proofs=1.
1887
Usable=10. Sos=646. Demods=843. Limbo=189, Disabled=1173. Hints=435.
1888
Weight_deleted=67. Literals_deleted=0.
1889
Forward_subsumed=1086. Back_subsumed=62.
1902
Given=31. Generated=2036. Kept=1202. proofs=1.
1903
Usable=22. Sos=728. Demods=756. Limbo=8, Disabled=457. Hints=386.
1904
Kept_by_rule=0, Deleted_by_rule=0.
1905
Forward_subsumed=834. Back_subsumed=51.
1890
1906
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
1891
New_demodulators=2000 (0 lex), Back_demodulated=1097. Back_unit_deleted=0.
1892
Demod_attempts=76540. Demod_rewrites=5442.
1907
New_demodulators=1197 (0 lex), Back_demodulated=392. Back_unit_deleted=0.
1908
Demod_attempts=35978. Demod_rewrites=2731.
1893
1909
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
1894
1910
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
1895
Megabytes=2.02.
1896
User_CPU=0.41, System_CPU=0.00, Wall_clock=5.
1911
Megabytes=1.76.
1912
User_CPU=0.21, System_CPU=0.00, Wall_clock=4.
1897
1913
1898
1914
============================== end of statistics =====================
1899
1915
1901
1917
1902
1918
Exiting with 1 proof.
1903
1919
1904
Process 13967 exit (max_proofs) Sun Aug 13 21:05:02 2006
1920
Process 10940 exit (max_proofs) Mon Sep 28 11:57:13 2009
1905
1921
1906
Successful proof using the following 8 assumptions:
1922
Successful proof using the following 7 assumptions:
1907
1923
1908
1924
formulas(assumptions_in_proof).
1909
1925
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
1910
1926
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1911
1927
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1912
1928
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1913
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
1914
1929
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1915
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
1930
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1916
1931
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1917
1932
end_of_list.
1918
1933
1919
Including 110 new hint clauses:
1934
Including 64 new hint clauses:
1920
1935
1921
1936
formulas(new_hints).
1922
$F # label("Commutativity").
1937
$F # label("Commutativity") # label(non_clause) # label(goal).
1923
1938
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
1924
1939
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
1925
1940
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").
1926
1941
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
1927
1942
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
1928
1943
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
1929
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").
1930
1944
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
1931
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
1945
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
1932
1946
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
1933
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)).
1934
f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)).
1935
1947
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
1936
f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)).
1937
f(f(f(x,y),f(f(f(f(y,f(y,z)),y),f(x,y)),f(x,y))),z) = f(f(y,f(y,z)),y).
1938
f(f(f(f(f(x,y),f(f(y,f(y,z)),y)),z),z),z) = f(f(y,f(y,z)),y).
1948
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
1939
1949
f(f(f(x,f(x,x)),x),f(x,x)) = x.
1940
f(x,f(f(x,f(f(x,x),x)),f(y,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
1941
f(f(f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
1950
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
1942
1951
f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
1943
1952
f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
1944
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
1953
f(f(f(f(f(x,f(x,x)),x),x),f(f(x,f(x,x)),x)),f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x).
1945
1954
f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x.
1946
f(f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1947
f(f(f(x,f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1948
f(f(f(x,f(x,f(f(x,x),x))),x),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1949
f(f(f(x,f(x,f(f(x,x),x))),x),f(x,f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1950
f(f(f(x,f(x,f(f(x,x),x))),x),f(x,x)) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1951
f(f(f(x,f(x,f(f(x,x),x))),x),f(x,x)) = x.
1952
1955
f(f(f(x,f(x,f(f(x,x),f(f(y,f(x,x)),f(x,x))))),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
1953
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,f(x,f(f(x,x),x))),x),x)) = f(f(x,f(x,f(f(x,x),x))),x).
1954
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),f(f(x,f(x,x)),f(x,x))),x)) = f(f(x,x),f(f(x,f(x,x)),f(x,x))).
1955
f(f(f(x,f(x,x)),x),f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),x))) = f(f(x,x),f(x,f(x,x))).
1956
f(f(f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),f(x,f(x,x)))),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)).
1957
f(f(f(x,f(x,f(x,f(x,x)))),x),x) = f(x,f(x,x)).
1958
f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1959
f(f(x,f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1960
f(f(x,f(x,f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1961
f(f(x,f(x,x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1962
f(f(x,f(x,x)),f(x,f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1963
f(f(x,f(x,x)),f(x,f(f(x,f(f(x,x),x)),f(x,f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1964
f(f(x,f(x,x)),f(x,f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1965
f(f(x,f(x,x)),f(x,f(f(x,f(f(x,x),x)),f(x,f(x,x))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1966
f(f(x,f(x,x)),f(x,x)) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1967
f(f(x,f(x,x)),f(x,x)) = x.
1968
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),x),x)) = f(f(x,x),f(f(x,f(x,x)),f(x,x))).
1969
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),x),x)) = f(f(x,x),x).
1970
f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),f(y,f(x,x)))) = f(x,f(x,x)).
1971
f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))) = f(x,x).
1972
f(f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)).
1956
f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))))) = f(x,f(f(x,x),x)).
1957
f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(x,f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))))) = f(x,f(f(x,x),x)).
1958
f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(x,f(f(x,f(f(x,x),x)),x)))) = f(x,f(f(x,x),x)).
1959
f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)).
1960
f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
1961
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
1962
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
1963
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
1964
f(f(f(x,f(f(x,x),x)),x),f(x,f(y,f(x,f(f(x,x),x))))) = x.
1965
f(f(x,f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
1966
f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
1967
f(f(f(f(x,f(x,f(x,f(f(x,x),x)))),x),x),x) = f(x,f(x,f(f(x,x),x))).
1968
f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))).
1969
f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x.
1970
f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
1971
f(x,x) = f(x,f(f(x,x),x)).
1972
f(x,f(f(x,x),x)) = f(x,x).
1973
f(f(x,f(x,x)),f(x,f(y,f(x,f(f(x,x),x))))) = x.
1974
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
1975
f(f(x,x),x) = f(x,f(x,f(f(x,x),x))).
1973
1976
f(f(x,x),x) = f(x,f(x,x)).
1974
f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),x)) = f(f(x,x),x).
1975
f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),x)) = f(x,f(x,x)).
1976
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,f(x,f(x,f(x,x)))),x),x)) = f(f(x,f(x,f(f(x,x),x))),x).
1977
f(f(f(x,x),f(x,f(x,x))),f(x,f(x,x))) = f(f(x,f(x,f(f(x,x),x))),x).
1978
f(f(f(x,x),f(x,f(x,x))),f(x,f(x,x))) = f(f(x,f(x,f(x,f(x,x)))),x).
1979
f(f(f(f(x,f(x,f(x,x))),x),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
1980
f(f(f(f(x,f(x,f(x,x))),x),f(x,f(x,f(x,x)))),x) = f(x,f(f(x,x),x)).
1981
f(x,x) = f(x,f(f(x,x),x)).
1982
f(x,x) = f(x,f(x,f(x,x))).
1977
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(f(x,x),x)).
1978
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(x,f(x,x))).
1979
f(f(x,f(x,f(x,f(x,x)))),f(x,f(f(x,x),x))) = x.
1980
f(f(x,f(x,f(x,f(x,x)))),f(x,f(x,f(x,x)))) = x.
1983
1981
f(x,f(x,f(x,x))) = f(x,x).
1984
f(x,f(f(x,f(x,f(x,x))),f(y,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
1985
f(x,f(f(x,x),f(y,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
1986
f(x,f(f(x,x),f(y,f(x,f(x,f(x,x)))))) = f(x,f(f(x,x),x)).
1987
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,f(f(x,x),x)).
1988
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,f(x,f(x,x))).
1982
f(f(x,f(x,x)),f(x,f(x,f(x,x)))) = x.
1983
f(f(x,f(x,x)),f(x,x)) = x.
1984
f(f(x,f(x,x)),x) = f(x,f(x,f(x,x))).
1985
f(f(x,f(x,x)),x) = f(x,x).
1986
f(f(f(f(x,x),x),f(f(x,f(x,x)),x)),f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x).
1987
f(f(f(x,f(x,x)),f(f(x,f(x,x)),x)),f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x).
1988
f(f(f(x,f(x,x)),f(x,x)),f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x).
1989
f(x,f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x).
1989
1990
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x).
1990
f(f(f(x,f(x,x)),x),f(x,f(x,x))) = f(f(x,x),f(x,f(x,x))).
1991
x = f(f(x,x),f(x,f(x,x))).
1992
f(f(x,x),f(x,f(x,x))) = x.
1993
f(x,f(x,f(x,x))) = f(f(x,f(x,f(x,f(x,x)))),x).
1994
f(x,x) = f(f(x,f(x,f(x,f(x,x)))),x).
1995
f(x,x) = f(f(x,f(x,x)),x).
1996
f(f(x,f(x,x)),x) = f(x,x).
1997
1991
f(f(f(x,f(x,x)),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
1998
1992
f(f(x,x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
1999
1993
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y).
2000
f(f(x,x),f(x,x)) = x.
2001
f(f(f(f(x,y),f(x,y)),f(f(x,y),f(f(x,y),f(x,y)))),f(f(y,f(f(f(x,y),y),y)),f(x,y))) = f(y,f(f(f(x,y),y),y)).
2002
f(f(x,y),f(f(y,f(f(f(x,y),y),y)),f(x,y))) = f(y,f(f(f(x,y),y),y)).
2003
f(f(x,f(x,f(x,x))),f(x,f(y,x))) = x.
2004
f(f(x,x),f(x,f(y,x))) = x.
2005
f(f(f(x,x),f(f(f(x,f(f(y,x),x)),f(f(x,f(x,x)),x)),f(f(x,f(x,x)),x))),y) = f(x,f(f(y,x),x)).
2006
f(f(f(x,x),f(f(f(x,f(f(y,x),x)),f(x,x)),f(f(x,f(x,x)),x))),y) = f(x,f(f(y,x),x)).
2007
f(f(f(x,x),f(f(f(x,f(f(y,x),x)),f(x,x)),f(x,x))),y) = f(x,f(f(y,x),x)).
2008
f(f(f(x,x),f(x,f(f(y,x),x))),y) = f(x,f(f(y,x),x)).
2009
f(x,y) = f(x,f(f(y,x),x)).
2010
f(x,f(f(y,x),x)) = f(x,y).
2011
f(f(x,y),f(f(y,f(x,y)),f(x,y))) = f(y,f(f(f(x,y),y),y)).
2012
f(f(x,y),y) = f(y,f(f(f(x,y),y),y)).
2013
f(f(x,y),y) = f(y,f(x,y)).
2014
f(f(f(x,y),f(f(x,y),f(f(f(y,f(y,z)),y),f(x,y)))),z) = f(f(y,f(y,z)),y).
2015
f(f(x,f(f(f(y,f(y,f(x,y))),x),x)),x) = f(y,f(f(x,y),y)).
2016
f(f(x,f(x,f(f(y,f(y,f(x,y))),x))),x) = f(y,f(f(x,y),y)).
2017
f(f(x,f(x,f(f(y,f(y,f(x,y))),x))),x) = f(y,f(y,f(x,y))).
2018
f(x,f(x,f(y,x))) = f(x,y).
2019
f(f(x,f(f(f(y,z),f(f(z,f(z,x)),z)),x)),x) = f(f(z,f(z,x)),z).
2020
f(f(x,f(x,f(f(y,x),x))),x) = f(y,f(y,f(x,y))).
2021
f(f(x,f(x,f(x,f(y,x)))),x) = f(y,f(y,f(x,y))).
2022
f(f(x,f(x,y)),x) = f(y,f(y,f(x,y))).
2023
f(f(x,f(x,y)),x) = f(y,x).
2024
f(f(f(x,y),f(f(x,y),f(f(z,y),f(x,y)))),z) = f(f(y,f(y,z)),y).
2025
f(f(f(x,y),f(z,y)),z) = f(f(y,f(y,z)),y).
2026
f(f(f(x,y),f(z,y)),z) = f(z,y).
2027
f(f(x,f(f(f(y,z),f(x,z)),x)),x) = f(f(z,f(z,x)),z).
2028
f(f(x,f(x,z)),x) = f(f(z,f(z,x)),z).
2029
f(z,x) = f(f(z,f(z,x)),z).
1994
f(f(f(x,f(x,x)),x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
1995
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
1996
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
1997
f(f(x,x),y) = f(y,f(x,x)).
1998
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
2030
1999
f(x,y) = f(y,x).
2031
2000
$F # answer("Commutativity").
2032
2001
end_of_list.
2055
2024
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2056
2025
end_of_list.
2057
2026
2058
Child search process 13968 started.
2027
Child search process 10941 started.
2059
2028
2060
2029
============================== PROCESS INITIAL CLAUSES ===============
2061
2030
2083
2052
formulas(demodulators).
2084
2053
end_of_list.
2085
2054
2086
% 545 hints input.
2055
% 450 hints input.
2087
2056
2088
2057
============================== PREDICATE ELIMINATION =================
2089
2058
2096
2065
2097
2066
Term ordering decisions:
2098
2067
Function symbol KB weights: c1=1. c2=1. f=1.
2099
Relation symbol precedence: lex([ = ]).
2100
Function symbol precedence: lex([ c1, c2, f ]).
2068
Predicate symbol precedence: predicate_order([ = ]).
2069
Function symbol precedence: function_order([ c1, c2, f ]).
2101
2070
Skipping inverse_order, because term ordering is KBO.
2102
2071
Unfolding symbols: (none).
2103
2072
2104
2073
Auto_inference settings:
2105
2074
% set(paramodulation). % (positive equality literals)
2106
% set(paramodulation) -> set(back_demod).
2107
2075
2108
2076
Auto_process settings: (no changes).
2109
2077
2078
kept: 2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
2079
kept: 3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
2080
kept: 4 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
2081
kept: 5 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
2082
kept: 6 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
2083
kept: 7 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2084
kept: 8 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2085
kept: 9 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2086
kept: 10 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2087
kept: 11 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2088
kept: 12 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2089
kept: 13 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2090
kept: 14 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2091
2110
2092
============================== end of process initial clauses ========
2111
2093
2112
2094
============================== CLAUSES FOR SEARCH ====================
2147
2129
14 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2148
2130
end_of_list.
2149
2131
2150
% 545 hints processed (298 are redundant).
2132
% 225 hints (450 processed, 225 redundant).
2151
2133
2152
2134
============================== end of clauses for search =============
2153
2135
2154
2136
============================== SEARCH ================================
2155
2137
2156
% Starting search at 0.01 seconds.
2138
% Starting search at 0.02 seconds.
2157
2139
2158
2140
% Operation f is commutative; C redundancy checks enabled.
2159
2141
2160
2142
============================== PROOF =================================
2161
2143
2162
% Proof 1 at 0.28 (+ 0.00) seconds: "Commutativity".
2163
% Length of proof is 60.
2164
% Level of proof is 18.
2165
% Maximum clause weight is 43.
2166
% Given clauses 25.
2144
% Proof 1 at 0.20 (+ 0.00) seconds: "Commutativity".
2145
% Length of proof is 42.
2146
% Level of proof is 15.
2147
% Maximum clause weight is 35.
2148
% Given clauses 30.
2167
2149
2168
1 f(x,y) = f(y,x) # label("Commutativity"). [goal].
2150
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
2169
2151
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
2170
2152
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
2171
2153
4 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
2172
2154
5 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
2173
2155
7 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2174
2156
8 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2175
9 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2176
2157
10 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2177
12 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2158
11 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2159
13 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2178
2160
14 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2179
15 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)). [para(2(a,1),2(a,1,1,2,1))].
2180
17 f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)). [para(2(a,1),2(a,1,2))].
2181
22 f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)). [para(2(a,1),4(a,1,2))].
2182
2161
25 f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x. [para(5(a,1),2(a,1,1,2))].
2183
148 f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)). [para(2(a,1),10(a,1,2))].
2162
378 f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)). [para(2(a,1),14(a,1,2))].
2184
2163
433 f(f(f(x,f(x,x)),x),f(x,x)) = x. [para(2(a,1),25(a,1,2,2))].
2185
2164
461 f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y. [para(433(a,1),2(a,1,2,2))].
2186
462 f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)). [para(2(a,1),433(a,1,1,1,2)),rewrite(2(15))].
2165
462 f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)). [para(2(a,1),433(a,1,1,1,2)),rewrite([2(15)])].
2166
464 f(f(f(f(x,f(x,x)),x),f(f(f(x,f(x,x)),x),x)),f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x). [para(433(a,1),5(a,1,1,2,2))].
2187
2167
477 f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x. [para(433(a,1),10(a,1,1,1))].
2188
496 f(x,f(f(x,f(f(x,x),x)),f(y,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)). [para(4(a,1),15(a,1,1))].
2189
553 f(f(f(x,f(x,f(f(x,x),x))),x),f(x,x)) = x. [para(15(a,1),433(a,1,1,1,2)),rewrite(4(8),4(12),4(13),4(13),4(15))].
2190
558 f(f(f(x,x),f(x,f(x,x))),f(f(f(x,f(x,f(f(x,x),x))),x),f(y,f(x,x)))) = f(f(x,f(x,f(f(x,x),x))),x). [para(553(a,1),2(a,1,1,2,1))].
2191
561 f(f(x,f(x,f(x,f(f(x,f(x,f(f(x,x),x))),x)))),x) = f(f(x,f(x,f(f(x,x),x))),x). [para(553(a,1),5(a,1,2))].
2192
576 f(f(f(x,f(f(f(x,f(x,f(f(x,x),x))),x),x)),x),x) = f(f(x,f(x,f(f(x,x),x))),x). [para(553(a,1),12(a,1,2))].
2193
586 f(f(f(f(x,y),f(x,y)),f(f(x,y),f(f(x,y),f(x,y)))),f(f(x,f(f(f(x,y),x),x)),f(x,y))) = f(x,f(f(f(x,y),x),x)). [para(4(a,1),461(a,1,1,2,1))].
2194
591 f(f(f(f(x,y),f(x,y)),f(f(x,y),f(f(x,y),f(x,y)))),f(f(y,f(y,f(y,f(x,y)))),f(x,y))) = f(y,f(y,f(y,f(x,y)))). [para(5(a,1),461(a,1,1,2,1))].
2195
607 f(f(f(x,f(x,f(f(x,x),f(f(y,f(x,x)),f(x,x))))),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))). [para(461(a,1),8(a,1,2))].
2196
657 f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)). [para(2(a,1),477(a,1,1,2)),rewrite(2(14),2(17),2(13))].
2197
669 f(f(f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),f(x,f(x,x)))),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)). [para(477(a,1),9(a,1,2))].
2198
683 f(f(f(x,f(x,f(x,f(x,x)))),x),x) = f(x,f(x,x)). [para(477(a,1),14(a,1,2))].
2199
686 f(f(x,f(x,x)),f(x,x)) = x. [para(15(a,1),477(a,1,2,2,1)),rewrite(4(8),4(8),4(8),4(10),4(13),4(13),4(13),4(8),4(12))].
2200
691 f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),f(y,f(x,x)))) = f(x,f(x,x)). [para(686(a,1),2(a,1,1,2,1))].
2201
699 f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))) = f(x,x). [para(686(a,1),7(a,1,1,1))].
2202
717 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(669),rewrite(691(10),699(7))].
2203
725 f(f(f(x,f(x,f(x,x))),x),x) = f(x,f(x,f(x,x))). [back_rewrite(657),rewrite(717(2),717(7))].
2204
736 f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(x,f(x,x))). [back_rewrite(576),rewrite(717(2),683(6),725(5),717(5)),flip(a)].
2205
747 f(f(x,f(x,f(x,f(x,f(x,f(x,x)))))),x) = f(x,f(x,f(x,x))). [back_rewrite(561),rewrite(717(2),736(5),717(9),736(12))].
2206
750 f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,f(x,x))),f(y,f(x,x)))) = f(x,f(x,f(x,x))). [back_rewrite(558),rewrite(717(6),736(9),717(13),736(16))].
2207
751 f(f(x,f(x,f(x,x))),f(x,x)) = x. [back_rewrite(553),rewrite(717(2),736(5))].
2208
755 f(x,f(f(x,f(x,f(x,x))),f(y,f(x,f(x,f(x,x)))))) = f(x,f(x,f(x,x))). [back_rewrite(496),rewrite(717(2),717(5),717(11))].
2209
756 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(462),rewrite(717(2),717(6),12(8),717(3)),flip(a)].
2210
772 f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x). [back_rewrite(755),rewrite(756(3),756(4),756(8))].
2211
774 f(f(x,x),f(x,x)) = x. [back_rewrite(751),rewrite(756(3))].
2212
775 f(f(f(x,x),f(x,f(x,x))),f(f(x,x),f(y,f(x,x)))) = f(x,x). [back_rewrite(750),rewrite(756(7),756(12))].
2213
777 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(747),rewrite(756(3),756(3),756(6))].
2214
786 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y). [back_rewrite(607),rewrite(772(7),777(3)),flip(a)].
2215
916 f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)). [para(17(a,1),17(a,1,1,1)),rewrite(17(16),17(20))].
2216
921 f(f(x,x),f(x,f(x,x))) = x. [para(15(a,1),774(a,1,1)),rewrite(717(2),756(3),717(3),756(4),717(4),756(5),717(7),756(8),717(8),756(9),775(10),774(3),717(2),756(3),717(3),756(4)),flip(a)].
2217
937 f(f(x,y),f(f(y,f(y,f(y,f(x,y)))),f(x,y))) = f(y,f(y,f(y,f(x,y)))). [back_rewrite(591),rewrite(921(9))].
2218
938 f(f(x,y),f(f(x,f(f(f(x,y),x),x)),f(x,y))) = f(x,f(f(f(x,y),x),x)). [back_rewrite(586),rewrite(921(9))].
2219
939 f(f(x,x),f(x,f(y,x))) = x. [para(717(a,1),2(a,1,1,2)),rewrite(756(3))].
2220
1005 f(x,f(f(y,x),x)) = f(x,y). [para(777(a,1),17(a,1,1,1)),rewrite(777(7),777(9),786(9),939(5)),flip(a)].
2221
1017 f(f(x,y),x) = f(x,f(x,y)). [back_rewrite(938),rewrite(1005(5),1005(6),1005(6))].
2222
1022 f(f(f(x,y),f(y,z)),z) = f(y,z). [back_rewrite(916),rewrite(1005(3),1005(4),1005(5),1005(7),1005(7),1005(7))].
2223
1149 f(f(f(x,y),y),y) = f(x,y). [back_rewrite(148),rewrite(1005(4),1022(4),1005(6))].
2224
1175 f(x,f(x,f(y,x))) = f(y,x). [back_rewrite(22),rewrite(1005(3),1149(3),1017(3),1005(6))].
2225
1354 f(f(x,y),y) = f(y,f(x,y)). [back_rewrite(937),rewrite(1175(4),1005(6),1175(5))].
2226
1448 f(x,y) = f(y,x). [back_rewrite(1005),rewrite(1354(2),1175(3))].
2227
1449 $F # answer("Commutativity"). [resolve(1448,a,3,a)].
2168
514 f(f(f(x,f(x,f(f(x,x),f(f(y,f(x,x)),f(x,x))))),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))). [para(461(a,1),8(a,1,2))].
2169
556 f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)). [para(2(a,1),477(a,1,1,2)),rewrite([2(14),2(17),2(13)])].
2170
574 f(f(f(f(x,f(x,f(x,x))),x),x),x) = f(x,f(x,x)). [para(477(a,1),11(a,1,2))].
2171
586 f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x. [para(2(a,1),556(a,1,1,1,2,1)),rewrite([4(8),2(18),4(16)])].
2172
694 f(f(f(x,f(f(x,x),x)),x),f(x,f(y,f(x,f(f(x,x),x))))) = x. [para(586(a,1),2(a,1,1,2))].
2173
698 f(f(x,f(f(f(x,f(x,f(f(x,x),x))),x),x)),x) = f(x,f(x,f(f(x,x),x))). [para(586(a,1),4(a,1,2))].
2174
704 f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)). [para(586(a,1),7(a,1,1,1)),rewrite([4(12)])].
2175
714 f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))). [para(586(a,1),11(a,1,2)),rewrite([378(7)])].
2176
721 f(f(x,f(x,f(x,f(f(x,x),x)))),x) = f(x,f(x,f(f(x,x),x))). [back_rewrite(698),rewrite([704(5),714(4)])].
2177
723 f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x. [back_rewrite(694),rewrite([714(4)])].
2178
747 f(x,f(f(x,x),x)) = f(x,x). [back_rewrite(462),rewrite([714(4),586(8)]),flip(a)].
2179
772 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [back_rewrite(723),rewrite([747(3),747(5)])].
2180
774 f(f(x,f(x,f(x,x))),x) = f(x,f(x,x)). [back_rewrite(721),rewrite([747(3),747(7)])].
2181
780 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(714),rewrite([747(3),747(5)])].
2182
789 f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(x,f(x,x))). [back_rewrite(704),rewrite([780(2),780(7)])].
2183
832 f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)). [back_rewrite(574),rewrite([774(4)])].
2184
836 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(747),rewrite([780(2)])].
2185
868 f(x,f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x). [back_rewrite(464),rewrite([832(7),14(6)])].
2186
883 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(789),rewrite([836(3),836(6)])].
2187
899 f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x). [back_rewrite(868),rewrite([883(4)])].
2188
940 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y). [back_rewrite(514),rewrite([899(7),883(3)]),flip(a)].
2189
1094 f(f(x,x),y) = f(y,f(x,x)). [para(772(a,1),13(a,1,1,1)),rewrite([836(3),883(6),940(6)])].
2190
1105 f(x,y) = f(y,x). [para(2(a,1),1094(a,1,1)),rewrite([2(8)])].
2191
1106 $F # answer("Commutativity"). [resolve(1105,a,3,a)].
2228
2192
2229
2193
============================== end of proof ==========================
2230
2194
2231
2195
============================== STATISTICS ============================
2232
2196
2233
Given=25. Generated=2474. Kept=1447. proofs=1.
2234
Usable=5. Sos=371. Demods=468. Limbo=94, Disabled=989. Hints=545.
2235
Weight_deleted=22. Literals_deleted=0.
2236
Forward_subsumed=1005. Back_subsumed=69.
2197
Given=30. Generated=1871. Kept=1104. proofs=1.
2198
Usable=21. Sos=622. Demods=649. Limbo=8, Disabled=465. Hints=450.
2199
Kept_by_rule=0, Deleted_by_rule=0.
2200
Forward_subsumed=767. Back_subsumed=51.
2237
2201
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
2238
New_demodulators=1442 (0 lex), Back_demodulated=907. Back_unit_deleted=0.
2239
Demod_attempts=53887. Demod_rewrites=4439.
2202
New_demodulators=1099 (0 lex), Back_demodulated=401. Back_unit_deleted=0.
2203
Demod_attempts=33545. Demod_rewrites=2404.
2240
2204
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
2241
2205
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
2242
Megabytes=1.59.
2243
User_CPU=0.28, System_CPU=0.00, Wall_clock=6.
2206
Megabytes=1.65.
2207
User_CPU=0.20, System_CPU=0.00, Wall_clock=4.
2244
2208
2245
2209
============================== end of statistics =====================
2246
2210
2248
2212
2249
2213
Exiting with 1 proof.
2250
2214
2251
Process 13968 exit (max_proofs) Sun Aug 13 21:05:03 2006
2215
Process 10941 exit (max_proofs) Mon Sep 28 11:57:13 2009
2252
2216
2253
2217
Successful proof using the following 8 assumptions:
2254
2218
2257
2221
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
2258
2222
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2259
2223
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2260
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2261
2224
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2262
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2225
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2226
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2263
2227
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2264
2228
end_of_list.
2265
2229
2266
Including 144 new hint clauses:
2230
Including 66 new hint clauses:
2267
2231
2268
2232
formulas(new_hints).
2269
$F # label("Commutativity").
2233
$F # label("Commutativity") # label(non_clause) # label(goal).
2270
2234
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
2271
2235
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
2272
2236
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
2273
2237
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").
2274
2238
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
2275
2239
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
2276
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").
2277
2240
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
2278
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
2241
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
2242
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
2279
2243
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
2280
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)).
2281
f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)).
2282
f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)).
2283
2244
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
2284
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
2245
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
2285
2246
f(f(f(x,f(x,x)),x),f(x,x)) = x.
2286
2247
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
2287
2248
f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
2288
2249
f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
2250
f(f(f(f(x,f(x,x)),x),f(f(f(x,f(x,x)),x),x)),f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x).
2289
2251
f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x.
2290
f(x,f(f(x,f(f(x,x),x)),f(y,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
2291
f(f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2292
f(f(f(x,f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2293
f(f(f(x,f(x,f(f(x,x),x))),x),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2294
f(f(f(x,f(x,f(f(x,x),x))),x),f(x,f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2295
f(f(f(x,f(x,f(f(x,x),x))),x),f(x,x)) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2296
f(f(f(x,f(x,f(f(x,x),x))),x),f(x,x)) = x.
2297
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,f(x,f(f(x,x),x))),x),f(y,f(x,x)))) = f(f(x,f(x,f(f(x,x),x))),x).
2298
f(f(x,f(x,f(x,f(f(x,f(x,f(f(x,x),x))),x)))),x) = f(f(x,f(x,f(f(x,x),x))),x).
2299
f(f(f(x,f(f(f(x,f(x,f(f(x,x),x))),x),x)),x),x) = f(f(x,f(x,f(f(x,x),x))),x).
2300
f(f(f(f(x,y),f(x,y)),f(f(x,y),f(f(x,y),f(x,y)))),f(f(x,f(f(f(x,y),x),x)),f(x,y))) = f(x,f(f(f(x,y),x),x)).
2301
f(f(f(f(x,y),f(x,y)),f(f(x,y),f(f(x,y),f(x,y)))),f(f(y,f(y,f(y,f(x,y)))),f(x,y))) = f(y,f(y,f(y,f(x,y)))).
2302
2252
f(f(f(x,f(x,f(f(x,x),f(f(y,f(x,x)),f(x,x))))),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
2303
2253
f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))))) = f(x,f(f(x,x),x)).
2304
2254
f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(x,f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))))) = f(x,f(f(x,x),x)).
2305
2255
f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(x,f(f(x,f(f(x,x),x)),x)))) = f(x,f(f(x,x),x)).
2306
2256
f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)).
2307
f(f(f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),f(x,f(x,x)))),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)).
2308
f(f(f(x,f(x,f(x,f(x,x)))),x),x) = f(x,f(x,x)).
2309
f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2310
f(f(x,f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2311
f(f(x,f(x,f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2312
f(f(x,f(x,x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2313
f(f(x,f(x,x)),f(x,f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2314
f(f(x,f(x,x)),f(x,f(f(x,f(f(x,x),x)),f(x,f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2315
f(f(x,f(x,x)),f(x,f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2316
f(f(x,f(x,x)),f(x,f(f(x,f(f(x,x),x)),f(x,f(x,x))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2317
f(f(x,f(x,x)),f(x,x)) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2318
f(f(x,f(x,x)),f(x,x)) = x.
2319
f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),f(y,f(x,x)))) = f(x,f(x,x)).
2320
f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))) = f(x,x).
2321
f(f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)).
2257
f(f(f(f(x,f(x,f(x,x))),x),x),x) = f(x,f(x,x)).
2258
f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
2259
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
2260
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2261
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
2262
f(f(f(x,f(f(x,x),x)),x),f(x,f(y,f(x,f(f(x,x),x))))) = x.
2263
f(f(x,f(f(f(x,f(x,f(f(x,x),x))),x),x)),x) = f(x,f(x,f(f(x,x),x))).
2264
f(f(x,f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
2265
f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
2266
f(f(f(f(x,f(x,f(x,f(f(x,x),x)))),x),x),x) = f(x,f(x,f(f(x,x),x))).
2267
f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))).
2268
f(f(x,f(f(x,f(f(x,x),x)),x)),x) = f(x,f(x,f(f(x,x),x))).
2269
f(f(x,f(x,f(x,f(f(x,x),x)))),x) = f(x,f(x,f(f(x,x),x))).
2270
f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x.
2271
f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
2272
f(x,x) = f(x,f(f(x,x),x)).
2273
f(x,f(f(x,x),x)) = f(x,x).
2274
f(f(x,f(x,x)),f(x,f(y,f(x,f(f(x,x),x))))) = x.
2275
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
2276
f(f(x,f(x,f(x,x))),x) = f(x,f(x,f(f(x,x),x))).
2277
f(f(x,f(x,f(x,x))),x) = f(x,f(x,x)).
2278
f(f(x,x),x) = f(x,f(x,f(f(x,x),x))).
2322
2279
f(f(x,x),x) = f(x,f(x,x)).
2323
f(f(f(x,f(x,f(x,x))),x),x) = f(x,f(f(x,x),x)).
2324
f(f(f(x,f(x,f(x,x))),x),x) = f(x,f(x,f(x,x))).
2325
f(f(f(x,f(f(f(x,f(x,f(x,f(x,x)))),x),x)),x),x) = f(f(x,f(x,f(f(x,x),x))),x).
2326
f(f(f(x,f(x,f(x,x))),x),x) = f(f(x,f(x,f(f(x,x),x))),x).
2327
f(x,f(x,f(x,x))) = f(f(x,f(x,f(f(x,x),x))),x).
2328
f(x,f(x,f(x,x))) = f(f(x,f(x,f(x,f(x,x)))),x).
2280
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(f(x,x),x)).
2329
2281
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(x,f(x,x))).
2330
f(f(x,f(x,f(x,f(f(x,f(x,f(x,f(x,x)))),x)))),x) = f(f(x,f(x,f(f(x,x),x))),x).
2331
f(f(x,f(x,f(x,f(x,f(x,f(x,x)))))),x) = f(f(x,f(x,f(f(x,x),x))),x).
2332
f(f(x,f(x,f(x,f(x,f(x,f(x,x)))))),x) = f(f(x,f(x,f(x,f(x,x)))),x).
2333
f(f(x,f(x,f(x,f(x,f(x,f(x,x)))))),x) = f(x,f(x,f(x,x))).
2334
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,f(x,f(x,f(x,x)))),x),f(y,f(x,x)))) = f(f(x,f(x,f(f(x,x),x))),x).
2335
f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,f(x,x))),f(y,f(x,x)))) = f(f(x,f(x,f(f(x,x),x))),x).
2336
f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,f(x,x))),f(y,f(x,x)))) = f(f(x,f(x,f(x,f(x,x)))),x).
2337
f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,f(x,x))),f(y,f(x,x)))) = f(x,f(x,f(x,x))).
2338
f(f(f(x,f(x,f(x,f(x,x)))),x),f(x,x)) = x.
2339
f(f(x,f(x,f(x,x))),f(x,x)) = x.
2340
f(x,f(f(x,f(x,f(x,x))),f(y,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
2341
f(x,f(f(x,f(x,f(x,x))),f(y,f(x,f(x,f(x,x)))))) = f(x,f(f(x,x),x)).
2342
f(x,f(f(x,f(x,f(x,x))),f(y,f(x,f(x,f(x,x)))))) = f(x,f(x,f(x,x))).
2343
f(f(f(f(x,f(x,f(x,x))),x),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
2344
f(f(f(f(x,f(x,f(x,x))),x),f(x,f(x,f(x,x)))),x) = f(x,f(f(x,x),x)).
2345
f(x,x) = f(x,f(f(x,x),x)).
2346
f(x,x) = f(x,f(x,f(x,x))).
2282
f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)).
2347
2283
f(x,f(x,f(x,x))) = f(x,x).
2348
f(x,f(f(x,x),f(y,f(x,f(x,f(x,x)))))) = f(x,f(x,f(x,x))).
2349
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,f(x,f(x,x))).
2350
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x).
2351
f(f(x,x),f(x,x)) = x.
2352
f(f(f(x,x),f(x,f(x,x))),f(f(x,x),f(y,f(x,x)))) = f(x,f(x,f(x,x))).
2353
f(f(f(x,x),f(x,f(x,x))),f(f(x,x),f(y,f(x,x)))) = f(x,x).
2354
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(x,f(x,x))).
2284
f(f(f(f(x,f(x,x)),x),f(x,f(x,x))),f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x).
2285
f(x,f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x).
2355
2286
f(f(x,f(x,x)),x) = f(x,f(x,f(x,x))).
2356
2287
f(f(x,f(x,x)),x) = f(x,x).
2288
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x).
2357
2289
f(f(f(x,f(x,x)),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
2358
2290
f(f(x,x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
2359
2291
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y).
2360
f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(f(f(u,x),f(f(f(x,f(f(y,x),x)),f(u,x)),f(u,x))),y)),f(f(f(u,x),f(f(f(x,f(f(y,x),x)),f(u,x)),f(u,x))),y))),z) = f(y,f(f(z,y),y)).
2361
f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(f(f(u,x),f(f(f(x,f(f(y,x),x)),f(u,x)),f(u,x))),y))),z) = f(y,f(f(z,y),y)).
2362
f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
2363
f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2364
f(f(x,f(x,f(x,x))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2365
f(f(x,x),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2366
f(f(x,x),f(f(f(x,f(x,f(x,x))),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2367
f(f(x,x),f(f(f(x,x),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2368
f(f(x,x),f(f(f(x,x),f(x,f(x,f(x,f(x,x))))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2369
f(f(x,x),f(f(f(x,x),f(x,f(x,x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2370
f(f(x,x),f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,f(x,x))),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2371
f(f(x,x),f(f(f(x,x),f(x,f(x,x))),f(f(x,x),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2372
f(f(x,x),f(f(f(x,x),f(x,f(x,x))),f(f(x,x),f(x,f(x,f(x,f(x,x))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2373
f(f(x,x),f(f(f(x,x),f(x,f(x,x))),f(f(x,x),f(x,f(x,x))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2374
f(f(x,x),f(x,x)) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2375
x = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2376
x = f(f(x,f(x,f(x,x))),f(x,f(x,f(f(x,x),x)))).
2377
x = f(f(x,x),f(x,f(x,f(f(x,x),x)))).
2378
x = f(f(x,x),f(x,f(x,f(x,f(x,x))))).
2379
x = f(f(x,x),f(x,f(x,x))).
2380
f(f(x,x),f(x,f(x,x))) = x.
2381
f(f(x,y),f(f(y,f(y,f(y,f(x,y)))),f(x,y))) = f(y,f(y,f(y,f(x,y)))).
2382
f(f(x,y),f(f(x,f(f(f(x,y),x),x)),f(x,y))) = f(x,f(f(f(x,y),x),x)).
2383
f(f(x,f(x,f(x,x))),f(x,f(y,x))) = x.
2384
f(f(x,x),f(x,f(y,x))) = x.
2385
f(f(f(x,x),f(f(f(x,f(f(y,x),x)),f(f(x,f(x,x)),x)),f(f(x,f(x,x)),x))),y) = f(x,f(f(y,x),x)).
2386
f(f(f(x,x),f(f(f(x,f(f(y,x),x)),f(x,x)),f(f(x,f(x,x)),x))),y) = f(x,f(f(y,x),x)).
2387
f(f(f(x,x),f(f(f(x,f(f(y,x),x)),f(x,x)),f(x,x))),y) = f(x,f(f(y,x),x)).
2388
f(f(f(x,x),f(x,f(f(y,x),x))),y) = f(x,f(f(y,x),x)).
2389
f(x,y) = f(x,f(f(y,x),x)).
2390
f(x,f(f(y,x),x)) = f(x,y).
2391
f(f(x,y),f(f(x,f(x,y)),f(x,y))) = f(x,f(f(f(x,y),x),x)).
2392
f(f(x,y),x) = f(x,f(f(f(x,y),x),x)).
2393
f(f(x,y),x) = f(x,f(x,y)).
2394
f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
2395
f(f(f(x,y),f(f(f(y,z),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
2396
f(f(f(x,y),f(f(f(y,z),f(x,y)),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
2397
f(f(f(x,y),f(f(f(y,z),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)).
2398
f(f(f(x,y),f(y,z)),z) = f(y,f(f(z,y),y)).
2399
f(f(f(x,y),f(y,z)),z) = f(y,z).
2400
f(f(f(f(f(x,y),f(y,z)),z),z),z) = f(y,f(f(z,y),y)).
2401
f(f(f(y,z),z),z) = f(y,f(f(z,y),y)).
2402
f(f(f(x,y),y),y) = f(x,y).
2403
f(f(x,f(f(f(y,x),x),x)),x) = f(y,f(f(x,y),y)).
2404
f(f(x,f(y,x)),x) = f(y,f(f(x,y),y)).
2405
f(x,f(x,f(y,x))) = f(y,f(f(x,y),y)).
2406
f(x,f(x,f(y,x))) = f(y,x).
2407
f(f(x,y),f(f(y,f(x,y)),f(x,y))) = f(y,f(y,f(y,f(x,y)))).
2408
f(f(x,y),y) = f(y,f(y,f(y,f(x,y)))).
2409
f(f(x,y),y) = f(y,f(x,y)).
2410
f(x,f(x,f(y,x))) = f(x,y).
2292
f(f(x,f(x,f(x,x))),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
2293
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
2294
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
2295
f(f(x,x),y) = f(y,f(x,x)).
2296
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
2411
2297
f(x,y) = f(y,x).
2412
2298
$F # answer("Commutativity").
2413
2299
end_of_list.
2435
2321
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2436
2322
end_of_list.
2437
2323
2438
Child search process 13969 started.
2324
Child search process 10942 started.
2439
2325
2440
2326
============================== PROCESS INITIAL CLAUSES ===============
2441
2327
2462
2348
formulas(demodulators).
2463
2349
end_of_list.
2464
2350
2465
% 689 hints input.
2351
% 516 hints input.
2466
2352
2467
2353
============================== PREDICATE ELIMINATION =================
2468
2354
2475
2361
2476
2362
Term ordering decisions:
2477
2363
Function symbol KB weights: c1=1. c2=1. f=1.
2478
Relation symbol precedence: lex([ = ]).
2479
Function symbol precedence: lex([ c1, c2, f ]).
2364
Predicate symbol precedence: predicate_order([ = ]).
2365
Function symbol precedence: function_order([ c1, c2, f ]).
2480
2366
Skipping inverse_order, because term ordering is KBO.
2481
2367
Unfolding symbols: (none).
2482
2368
2483
2369
Auto_inference settings:
2484
2370
% set(paramodulation). % (positive equality literals)
2485
% set(paramodulation) -> set(back_demod).
2486
2371
2487
2372
Auto_process settings: (no changes).
2488
2373
2374
kept: 2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
2375
kept: 3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
2376
kept: 4 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
2377
kept: 5 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
2378
kept: 6 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2379
kept: 7 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2380
kept: 8 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2381
kept: 9 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2382
kept: 10 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2383
kept: 11 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2384
kept: 12 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2385
kept: 13 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2386
2489
2387
============================== end of process initial clauses ========
2490
2388
2491
2389
============================== CLAUSES FOR SEARCH ====================
2524
2422
13 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2525
2423
end_of_list.
2526
2424
2527
% 689 hints processed (381 are redundant).
2425
% 247 hints (516 processed, 269 redundant).
2528
2426
2529
2427
============================== end of clauses for search =============
2530
2428
2531
2429
============================== SEARCH ================================
2532
2430
2533
% Starting search at 0.01 seconds.
2431
% Starting search at 0.02 seconds.
2534
2432
2535
2433
% Operation f is commutative; C redundancy checks enabled.
2536
2434
2537
2435
============================== PROOF =================================
2538
2436
2539
% Proof 1 at 0.29 (+ 0.01) seconds: "Commutativity".
2540
% Length of proof is 53.
2541
% Level of proof is 19.
2542
% Maximum clause weight is 47.
2543
% Given clauses 29.
2437
% Proof 1 at 0.17 (+ 0.00) seconds: "Commutativity".
2438
% Length of proof is 38.
2439
% Level of proof is 15.
2440
% Maximum clause weight is 35.
2441
% Given clauses 28.
2544
2442
2545
1 f(x,y) = f(y,x) # label("Commutativity"). [goal].
2443
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
2546
2444
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
2547
2445
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
2548
2446
4 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
2549
5 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
2550
2447
6 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2551
2448
7 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2552
8 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2553
2449
9 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2554
11 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2450
10 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2555
2451
12 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2556
2452
13 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2557
14 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)). [para(2(a,1),2(a,1,1,2,1))].
2558
16 f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)). [para(2(a,1),2(a,1,2))].
2559
2453
18 f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x. [para(4(a,1),2(a,1,1,2))].
2560
257 f(f(f(x,f(f(y,z),f(z,f(f(x,z),z)))),x),x) = f(z,f(f(x,z),z)). [para(2(a,1),12(a,1,2))].
2454
322 f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)). [para(2(a,1),13(a,1,2))].
2561
2455
375 f(f(f(x,f(x,x)),x),f(x,x)) = x. [para(2(a,1),18(a,1,2,2))].
2562
2456
402 f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y. [para(375(a,1),2(a,1,2,2))].
2563
404 f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)). [para(2(a,1),375(a,1,1,1,2)),rewrite(2(15))].
2457
404 f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)). [para(2(a,1),375(a,1,1,1,2)),rewrite([2(15)])].
2458
413 f(f(f(f(f(x,f(x,x)),x),x),f(f(x,f(x,x)),x)),f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x). [para(375(a,1),7(a,1,1,1,2))].
2564
2459
418 f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x. [para(375(a,1),9(a,1,1,1))].
2565
431 f(f(x,f(f(y,x),x)),f(y,f(x,f(f(x,x),x)))) = y. [para(2(a,1),402(a,1,1,1)),rewrite(2(7),2(8))].
2460
431 f(f(x,f(f(y,x),x)),f(y,f(x,f(f(x,x),x)))) = y. [para(2(a,1),402(a,1,1,1)),rewrite([2(7),2(8)])].
2566
2461
453 f(f(f(x,f(x,f(f(x,x),f(f(y,f(x,x)),f(x,x))))),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))). [para(402(a,1),7(a,1,2))].
2567
497 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),x)) = f(z,f(f(x,z),z)). [para(2(a,1),14(a,1,2,2))].
2568
552 f(f(f(x,f(x,f(f(x,x),x))),x),f(x,x)) = x. [para(14(a,1),375(a,1,1,1,2)),rewrite(431(8),431(12),431(13),431(13),431(15))].
2569
558 f(f(x,f(x,f(x,f(f(x,f(x,f(f(x,x),x))),x)))),x) = f(f(x,f(x,f(f(x,x),x))),x). [para(552(a,1),4(a,1,2))].
2570
559 f(f(f(x,f(f(x,f(x,f(f(x,x),x))),x)),f(f(x,f(x,f(f(x,x),x))),x)),f(f(x,x),f(y,f(f(x,f(x,f(f(x,x),x))),x)))) = f(x,x). [para(552(a,1),5(a,1,1,1,1))].
2571
573 f(f(f(x,f(f(f(x,f(x,f(f(x,x),x))),x),x)),x),x) = f(f(x,f(x,f(f(x,x),x))),x). [para(552(a,1),11(a,1,2))].
2572
581 f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)). [para(2(a,1),418(a,1,1,2)),rewrite(2(14),2(17),2(13))].
2573
591 f(f(f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),f(x,f(x,x)))),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)). [para(418(a,1),8(a,1,2))].
2574
605 f(f(f(x,f(x,f(x,f(x,x)))),x),x) = f(x,f(x,x)). [para(418(a,1),13(a,1,2))].
2575
608 f(f(x,f(x,x)),f(x,x)) = x. [para(14(a,1),418(a,1,2,2,1)),rewrite(431(8),431(8),431(8),431(10),431(13),431(13),431(13),2(8),431(12))].
2576
613 f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),f(y,f(x,x)))) = f(x,f(x,x)). [para(608(a,1),2(a,1,1,2,1))].
2577
620 f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))) = f(x,x). [para(608(a,1),6(a,1,1,1))].
2578
638 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(591),rewrite(613(10),620(7))].
2579
646 f(f(f(x,f(x,f(x,x))),x),x) = f(x,f(x,f(x,x))). [back_rewrite(581),rewrite(638(2),638(7))].
2580
652 f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(x,f(x,x))). [back_rewrite(573),rewrite(638(2),605(6),646(5),638(5)),flip(a)].
2581
662 f(f(f(x,f(x,f(x,f(x,x)))),f(x,f(x,f(x,x)))),f(f(x,x),f(y,f(x,f(x,f(x,x)))))) = f(x,x). [back_rewrite(559),rewrite(638(2),652(5),638(6),652(9),638(11),652(14))].
2582
663 f(f(x,f(x,f(x,f(x,f(x,f(x,x)))))),x) = f(x,f(x,f(x,x))). [back_rewrite(558),rewrite(638(2),652(5),638(9),652(12))].
2583
670 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(404),rewrite(638(2),638(6),11(8),638(3)),flip(a)].
2584
688 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(663),rewrite(670(3),670(3),670(6))].
2585
689 f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x). [back_rewrite(662),rewrite(670(3),670(5),608(4),670(4))].
2586
730 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y). [back_rewrite(453),rewrite(689(7),688(3)),flip(a)].
2587
840 f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)). [para(16(a,1),16(a,1,1,1)),rewrite(16(16),16(20))].
2588
843 f(f(x,x),f(x,f(y,x))) = x. [para(638(a,1),2(a,1,1,2)),rewrite(670(3))].
2589
910 f(x,f(f(y,x),x)) = f(x,y). [para(688(a,1),16(a,1,1,1)),rewrite(688(7),688(9),730(9),843(5)),flip(a)].
2590
925 f(f(f(x,y),f(y,z)),z) = f(y,z). [back_rewrite(840),rewrite(910(3),910(4),910(5),910(7),910(7),910(7))].
2591
1020 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,x),x)) = f(z,x). [back_rewrite(497),rewrite(910(9),910(12))].
2592
1030 f(f(f(x,f(f(y,z),f(z,x))),x),x) = f(z,x). [back_rewrite(257),rewrite(910(4),910(9))].
2593
1274 f(x,f(f(y,z),f(z,x))) = f(x,z). [para(925(a,1),910(a,1,2,1)),rewrite(910(3)),flip(a)].
2594
1287 f(f(f(x,y),x),x) = f(y,x). [back_rewrite(1030),rewrite(1274(4))].
2595
1384 f(f(x,y),f(x,f(z,y))) = x. [back_rewrite(5),rewrite(1287(3))].
2596
1441 f(x,y) = f(y,x). [back_rewrite(1020),rewrite(1384(6),910(3))].
2597
1442 $F # answer("Commutativity"). [resolve(1441,a,3,a)].
2462
495 f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)). [para(2(a,1),418(a,1,1,2)),rewrite([2(14),2(17),2(13)])].
2463
523 f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x. [para(2(a,1),495(a,1,1,1,2,1)),rewrite([431(8),2(18),431(16)])].
2464
594 f(f(f(x,f(f(x,x),x)),x),f(x,f(y,f(x,f(f(x,x),x))))) = x. [para(523(a,1),2(a,1,1,2))].
2465
601 f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)). [para(523(a,1),6(a,1,1,1)),rewrite([431(12)])].
2466
613 f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))). [para(523(a,1),10(a,1,2)),rewrite([322(7)])].
2467
622 f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x. [back_rewrite(594),rewrite([613(4)])].
2468
644 f(x,f(f(x,x),x)) = f(x,x). [back_rewrite(404),rewrite([613(4),523(8)]),flip(a)].
2469
668 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [back_rewrite(622),rewrite([644(3),644(5)])].
2470
676 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(613),rewrite([644(3),644(5)])].
2471
686 f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(x,f(x,x))). [back_rewrite(601),rewrite([676(2),676(7)])].
2472
697 f(f(x,f(x,f(x,f(x,x)))),f(x,f(x,f(x,x)))) = x. [back_rewrite(523),rewrite([676(2),676(6)])].
2473
705 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(644),rewrite([676(2)])].
2474
709 f(f(x,f(x,x)),f(x,x)) = x. [back_rewrite(697),rewrite([705(3),705(5)])].
2475
715 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(686),rewrite([705(3),705(6)])].
2476
750 f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x). [back_rewrite(413),rewrite([715(3),676(2),715(5),709(4),715(4)])].
2477
787 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y). [back_rewrite(453),rewrite([750(7),715(3)]),flip(a)].
2478
935 f(f(x,x),y) = f(y,f(x,x)). [para(668(a,1),12(a,1,1,1)),rewrite([705(3),715(6),787(6)])].
2479
947 f(x,y) = f(y,x). [para(2(a,1),935(a,1,1)),rewrite([2(8)])].
2480
948 $F # answer("Commutativity"). [resolve(947,a,3,a)].
2598
2481
2599
2482
============================== end of proof ==========================
2600
2483
2601
2484
============================== STATISTICS ============================
2602
2485
2603
Given=29. Generated=2570. Kept=1440. proofs=1.
2604
Usable=16. Sos=585. Demods=737. Limbo=140, Disabled=710. Hints=689.
2605
Weight_deleted=31. Literals_deleted=0.
2606
Forward_subsumed=1099. Back_subsumed=76.
2486
Given=28. Generated=1599. Kept=946. proofs=1.
2487
Usable=20. Sos=561. Demods=588. Limbo=9, Disabled=367. Hints=516.
2488
Kept_by_rule=0, Deleted_by_rule=0.
2489
Forward_subsumed=653. Back_subsumed=41.
2607
2490
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
2608
New_demodulators=1422 (0 lex), Back_demodulated=622. Back_unit_deleted=0.
2609
Demod_attempts=57687. Demod_rewrites=3953.
2491
New_demodulators=941 (0 lex), Back_demodulated=314. Back_unit_deleted=0.
2492
Demod_attempts=28691. Demod_rewrites=2181.
2610
2493
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
2611
2494
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
2612
Megabytes=2.08.
2613
User_CPU=0.29, System_CPU=0.01, Wall_clock=6.
2495
Megabytes=1.49.
2496
User_CPU=0.17, System_CPU=0.00, Wall_clock=4.
2614
2497
2615
2498
============================== end of statistics =====================
2616
2499
2618
2501
2619
2502
Exiting with 1 proof.
2620
2503
2621
Process 13969 exit (max_proofs) Sun Aug 13 21:05:03 2006
2504
Process 10942 exit (max_proofs) Mon Sep 28 11:57:13 2009
2622
2505
2623
Successful proof using the following 9 assumptions:
2506
Successful proof using the following 7 assumptions:
2624
2507
2625
2508
formulas(assumptions_in_proof).
2626
2509
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
2627
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
2628
2510
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2629
2511
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2630
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2631
2512
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2632
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2513
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2633
2514
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2634
2515
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2635
2516
end_of_list.
2636
2517
2637
Including 116 new hint clauses:
2518
Including 67 new hint clauses:
2638
2519
2639
2520
formulas(new_hints).
2640
$F # label("Commutativity").
2521
$F # label("Commutativity") # label(non_clause) # label(goal).
2641
2522
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
2642
2523
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
2643
2524
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").
2644
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").
2645
2525
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
2646
2526
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
2647
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").
2648
2527
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
2649
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
2528
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
2650
2529
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
2651
2530
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
2652
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)).
2653
f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)).
2654
2531
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
2655
f(f(f(x,f(f(y,z),f(z,f(f(x,z),z)))),x),x) = f(z,f(f(x,z),z)).
2532
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
2656
2533
f(f(f(x,f(x,x)),x),f(x,x)) = x.
2657
2534
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
2658
2535
f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
2659
2536
f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
2537
f(f(f(f(f(x,f(x,x)),x),x),f(f(x,f(x,x)),x)),f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x).
2660
2538
f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x.
2661
2539
f(f(x,f(f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))),f(y,f(x,f(f(x,x),x)))) = y.
2662
2540
f(f(x,f(f(y,x),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))),f(y,f(x,f(f(x,x),x)))) = y.
2663
2541
f(f(x,f(f(y,x),x)),f(y,f(x,f(f(x,x),x)))) = y.
2664
2542
f(f(f(x,f(x,f(f(x,x),f(f(y,f(x,x)),f(x,x))))),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
2665
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),x)) = f(z,f(f(x,z),z)).
2666
f(f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2667
f(f(f(x,f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2668
f(f(f(x,f(x,f(f(x,x),x))),x),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2669
f(f(f(x,f(x,f(f(x,x),x))),x),f(x,f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2670
f(f(f(x,f(x,f(f(x,x),x))),x),f(x,x)) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2671
f(f(f(x,f(x,f(f(x,x),x))),x),f(x,x)) = x.
2672
f(f(x,f(x,f(x,f(f(x,f(x,f(f(x,x),x))),x)))),x) = f(f(x,f(x,f(f(x,x),x))),x).
2673
f(f(f(x,f(f(x,f(x,f(f(x,x),x))),x)),f(f(x,f(x,f(f(x,x),x))),x)),f(f(x,x),f(y,f(f(x,f(x,f(f(x,x),x))),x)))) = f(x,x).
2674
f(f(f(x,f(f(f(x,f(x,f(f(x,x),x))),x),x)),x),x) = f(f(x,f(x,f(f(x,x),x))),x).
2675
2543
f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))))) = f(x,f(f(x,x),x)).
2676
2544
f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(x,f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))))))) = f(x,f(f(x,x),x)).
2677
2545
f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(x,f(f(x,f(f(x,x),x)),x)))) = f(x,f(f(x,x),x)).
2678
2546
f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)).
2679
f(f(f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),f(x,f(x,x)))),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)).
2680
f(f(f(x,f(x,f(x,f(x,x)))),x),x) = f(x,f(x,x)).
2681
f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2682
f(f(x,f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2683
f(f(x,f(x,f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2684
f(f(x,f(x,x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2685
f(f(x,f(x,x)),f(x,f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2686
f(f(x,f(x,x)),f(x,f(f(x,f(f(x,x),x)),f(x,f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2687
f(f(x,f(x,x)),f(x,f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2688
f(f(x,f(x,x)),f(x,f(f(x,f(f(x,x),x)),f(x,f(x,x))))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2689
f(f(x,f(x,x)),f(x,x)) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2690
f(f(x,f(x,x)),f(x,x)) = x.
2691
f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,x)),f(y,f(x,x)))) = f(x,f(x,x)).
2692
f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))) = f(x,x).
2693
f(f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)).
2547
f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
2548
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
2549
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))).
2550
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
2551
f(f(f(x,f(f(x,x),x)),x),f(x,f(y,f(x,f(f(x,x),x))))) = x.
2552
f(f(x,f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
2553
f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
2554
f(f(f(f(x,f(x,f(x,f(f(x,x),x)))),x),x),x) = f(x,f(x,f(f(x,x),x))).
2555
f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))).
2556
f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x.
2557
f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
2558
f(x,x) = f(x,f(f(x,x),x)).
2559
f(x,f(f(x,x),x)) = f(x,x).
2560
f(f(x,f(x,x)),f(x,f(y,f(x,f(f(x,x),x))))) = x.
2561
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
2562
f(f(x,x),x) = f(x,f(x,f(f(x,x),x))).
2694
2563
f(f(x,x),x) = f(x,f(x,x)).
2695
f(f(f(x,f(x,f(x,x))),x),x) = f(x,f(f(x,x),x)).
2696
f(f(f(x,f(x,f(x,x))),x),x) = f(x,f(x,f(x,x))).
2697
f(f(f(x,f(f(f(x,f(x,f(x,f(x,x)))),x),x)),x),x) = f(f(x,f(x,f(f(x,x),x))),x).
2698
f(f(f(x,f(x,f(x,x))),x),x) = f(f(x,f(x,f(f(x,x),x))),x).
2699
f(x,f(x,f(x,x))) = f(f(x,f(x,f(f(x,x),x))),x).
2700
f(x,f(x,f(x,x))) = f(f(x,f(x,f(x,f(x,x)))),x).
2564
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(f(x,x),x)).
2701
2565
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(x,f(x,x))).
2702
f(f(f(x,f(f(x,f(x,f(x,f(x,x)))),x)),f(f(x,f(x,f(f(x,x),x))),x)),f(f(x,x),f(y,f(f(x,f(x,f(f(x,x),x))),x)))) = f(x,x).
2703
f(f(f(x,f(x,f(x,f(x,x)))),f(f(x,f(x,f(f(x,x),x))),x)),f(f(x,x),f(y,f(f(x,f(x,f(f(x,x),x))),x)))) = f(x,x).
2704
f(f(f(x,f(x,f(x,f(x,x)))),f(f(x,f(x,f(x,f(x,x)))),x)),f(f(x,x),f(y,f(f(x,f(x,f(f(x,x),x))),x)))) = f(x,x).
2705
f(f(f(x,f(x,f(x,f(x,x)))),f(x,f(x,f(x,x)))),f(f(x,x),f(y,f(f(x,f(x,f(f(x,x),x))),x)))) = f(x,x).
2706
f(f(f(x,f(x,f(x,f(x,x)))),f(x,f(x,f(x,x)))),f(f(x,x),f(y,f(f(x,f(x,f(x,f(x,x)))),x)))) = f(x,x).
2707
f(f(f(x,f(x,f(x,f(x,x)))),f(x,f(x,f(x,x)))),f(f(x,x),f(y,f(x,f(x,f(x,x)))))) = f(x,x).
2708
f(f(x,f(x,f(x,f(f(x,f(x,f(x,f(x,x)))),x)))),x) = f(f(x,f(x,f(f(x,x),x))),x).
2709
f(f(x,f(x,f(x,f(x,f(x,f(x,x)))))),x) = f(f(x,f(x,f(f(x,x),x))),x).
2710
f(f(x,f(x,f(x,f(x,f(x,f(x,x)))))),x) = f(f(x,f(x,f(x,f(x,x)))),x).
2711
f(f(x,f(x,f(x,f(x,f(x,f(x,x)))))),x) = f(x,f(x,f(x,x))).
2712
f(f(f(f(x,f(x,f(x,x))),x),f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
2713
f(f(f(f(x,f(x,f(x,x))),x),f(x,f(x,f(x,x)))),x) = f(x,f(f(x,x),x)).
2714
f(x,x) = f(x,f(f(x,x),x)).
2715
f(x,x) = f(x,f(x,f(x,x))).
2566
f(f(x,f(x,f(x,f(x,x)))),f(x,f(f(x,x),x))) = x.
2567
f(f(x,f(x,f(x,f(x,x)))),f(x,f(x,f(x,x)))) = x.
2716
2568
f(x,f(x,f(x,x))) = f(x,x).
2717
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(x,f(x,x))).
2569
f(f(x,f(x,x)),f(x,f(x,f(x,x)))) = x.
2570
f(f(x,f(x,x)),f(x,x)) = x.
2718
2571
f(f(x,f(x,x)),x) = f(x,f(x,f(x,x))).
2719
2572
f(f(x,f(x,x)),x) = f(x,x).
2720
f(f(f(x,f(x,x)),f(x,f(x,f(x,x)))),f(f(x,x),f(y,f(x,f(x,f(x,x)))))) = f(x,x).
2721
f(f(f(x,f(x,x)),f(x,x)),f(f(x,x),f(y,f(x,f(x,f(x,x)))))) = f(x,x).
2722
f(x,f(f(x,x),f(y,f(x,f(x,f(x,x)))))) = f(x,x).
2573
f(f(f(f(x,x),x),f(f(x,f(x,x)),x)),f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x).
2574
f(f(f(x,f(x,x)),f(f(x,f(x,x)),x)),f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x).
2575
f(f(f(x,f(x,x)),f(x,x)),f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x).
2576
f(x,f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x).
2723
2577
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x).
2724
2578
f(f(f(x,f(x,x)),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
2725
2579
f(f(x,x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
2726
2580
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y).
2727
f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(f(f(u,x),f(f(f(x,f(f(y,x),x)),f(u,x)),f(u,x))),y)),f(f(f(u,x),f(f(f(x,f(f(y,x),x)),f(u,x)),f(u,x))),y))),z) = f(y,f(f(z,y),y)).
2728
f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(f(f(u,x),f(f(f(x,f(f(y,x),x)),f(u,x)),f(u,x))),y))),z) = f(y,f(f(z,y),y)).
2729
f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
2730
f(f(x,f(x,f(x,x))),f(x,f(y,x))) = x.
2731
f(f(x,x),f(x,f(y,x))) = x.
2732
f(f(f(x,x),f(f(f(x,f(f(y,x),x)),f(f(x,f(x,x)),x)),f(f(x,f(x,x)),x))),y) = f(x,f(f(y,x),x)).
2733
f(f(f(x,x),f(f(f(x,f(f(y,x),x)),f(x,x)),f(f(x,f(x,x)),x))),y) = f(x,f(f(y,x),x)).
2734
f(f(f(x,x),f(f(f(x,f(f(y,x),x)),f(x,x)),f(x,x))),y) = f(x,f(f(y,x),x)).
2735
f(f(f(x,x),f(x,f(f(y,x),x))),y) = f(x,f(f(y,x),x)).
2736
f(x,y) = f(x,f(f(y,x),x)).
2737
f(x,f(f(y,x),x)) = f(x,y).
2738
f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
2739
f(f(f(x,y),f(f(f(y,z),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
2740
f(f(f(x,y),f(f(f(y,z),f(x,y)),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
2741
f(f(f(x,y),f(f(f(y,z),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)).
2742
f(f(f(x,y),f(y,z)),z) = f(y,f(f(z,y),y)).
2743
f(f(f(x,y),f(y,z)),z) = f(y,z).
2744
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,x),x)) = f(z,f(f(x,z),z)).
2745
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,x),x)) = f(z,x).
2746
f(f(f(x,f(f(y,z),f(z,x))),x),x) = f(z,f(f(x,z),z)).
2747
f(f(f(x,f(f(y,z),f(z,x))),x),x) = f(z,x).
2748
f(x,f(f(y,x),x)) = f(x,f(f(z,y),f(y,x))).
2749
f(x,y) = f(x,f(f(z,y),f(y,x))).
2750
f(x,f(f(y,z),f(z,x))) = f(x,z).
2751
f(f(f(x,y),x),x) = f(y,x).
2752
f(f(x,y),f(x,f(z,y))) = x.
2753
f(x,f(f(z,x),x)) = f(z,x).
2581
f(f(x,f(x,f(x,x))),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
2582
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
2583
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
2584
f(f(x,x),y) = f(y,f(x,x)).
2585
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
2754
2586
f(x,y) = f(y,x).
2755
2587
$F # answer("Commutativity").
2756
2588
end_of_list.
2777
2609
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2778
2610
end_of_list.
2779
2611
2780
Child search process 13970 started.
2612
Child search process 10943 started.
2781
2613
2782
2614
============================== PROCESS INITIAL CLAUSES ===============
2783
2615
2803
2635
formulas(demodulators).
2804
2636
end_of_list.
2805
2637
2806
% 805 hints input.
2638
% 583 hints input.
2807
2639
2808
2640
============================== PREDICATE ELIMINATION =================
2809
2641
2816
2648
2817
2649
Term ordering decisions:
2818
2650
Function symbol KB weights: c1=1. c2=1. f=1.
2819
Relation symbol precedence: lex([ = ]).
2820
Function symbol precedence: lex([ c1, c2, f ]).
2651
Predicate symbol precedence: predicate_order([ = ]).
2652
Function symbol precedence: function_order([ c1, c2, f ]).
2821
2653
Skipping inverse_order, because term ordering is KBO.
2822
2654
Unfolding symbols: (none).
2823
2655
2824
2656
Auto_inference settings:
2825
2657
% set(paramodulation). % (positive equality literals)
2826
% set(paramodulation) -> set(back_demod).
2827
2658
2828
2659
Auto_process settings: (no changes).
2829
2660
2661
kept: 2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
2662
kept: 3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
2663
kept: 4 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
2664
kept: 5 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2665
kept: 6 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2666
kept: 7 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2667
kept: 8 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2668
kept: 9 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2669
kept: 10 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2670
kept: 11 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2671
kept: 12 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2672
2830
2673
============================== end of process initial clauses ========
2831
2674
2832
2675
============================== CLAUSES FOR SEARCH ====================
2863
2706
12 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2864
2707
end_of_list.
2865
2708
2866
% 805 hints processed (471 are redundant).
2709
% 252 hints (583 processed, 331 redundant).
2867
2710
2868
2711
============================== end of clauses for search =============
2869
2712
2870
2713
============================== SEARCH ================================
2871
2714
2872
% Starting search at 0.01 seconds.
2715
% Starting search at 0.02 seconds.
2873
2716
2874
2717
% Operation f is commutative; C redundancy checks enabled.
2875
2718
2876
2719
============================== PROOF =================================
2877
2720
2878
% Proof 1 at 0.62 (+ 0.00) seconds: "Commutativity".
2879
% Length of proof is 60.
2880
% Level of proof is 17.
2881
% Maximum clause weight is 59.
2882
% Given clauses 39.
2721
% Proof 1 at 0.69 (+ 0.00) seconds: "Commutativity".
2722
% Length of proof is 51.
2723
% Level of proof is 16.
2724
% Maximum clause weight is 51.
2725
% Given clauses 36.
2883
2726
2884
1 f(x,y) = f(y,x) # label("Commutativity"). [goal].
2727
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
2885
2728
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
2886
2729
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
2887
2730
4 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
2888
7 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2731
5 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2732
6 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2889
2733
8 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2890
2734
9 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2891
10 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2892
2735
11 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2893
2736
12 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2894
2737
13 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)). [para(2(a,1),2(a,1,1,2,1))].
2895
16 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(f(f(z,x),z),z),f(u,f(x,f(y,z))))) = f(f(f(z,x),z),z). [para(4(a,1),2(a,1,1,2,1))].
2896
17 f(f(f(x,y),f(f(f(f(f(y,z),y),y),f(x,y)),f(x,y))),z) = f(f(f(y,z),y),y). [para(4(a,1),2(a,1,2))].
2897
2738
85 f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)). [para(2(a,1),8(a,1,2))].
2898
2739
271 f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)). [para(2(a,1),12(a,1,2))].
2899
2740
318 f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y. [para(271(a,1),2(a,1,2,2))].
2900
324 f(f(f(f(x,y),x),x),f(y,f(z,f(f(x,z),z)))) = y. [para(271(a,1),4(a,1,2,2))].
2901
348 f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(y,z))) = f(y,f(x,f(f(y,x),x))). [para(271(a,1),9(a,1,1))].
2902
504 f(f(f(x,y),y),y) = f(x,f(f(y,x),x)). [para(4(a,1),85(a,1,1,1,1))].
2903
605 f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(x,f(x,f(f(x,x),x))). [para(13(a,1),271(a,1,1,1,2)),rewrite(318(8),318(12),318(13),318(17),348(18))].
2904
610 f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y. [para(13(a,1),318(a,1,2)),rewrite(318(8),318(15))].
2905
613 f(x,f(f(x,x),x)) = f(x,x). [para(318(a,1),13(a,1,2)),rewrite(318(8)),flip(a)].
2906
616 f(f(f(f(x,f(f(y,y),f(y,f(y,y)))),x),x),f(y,y)) = f(f(y,y),f(y,f(y,y))). [para(13(a,1),324(a,1,2)),rewrite(613(3),613(4),613(10),613(12),613(13))].
2907
621 f(f(x,f(f(y,x),x)),f(y,y)) = y. [back_rewrite(610),rewrite(613(6))].
2908
622 f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)). [back_rewrite(605),rewrite(613(3),613(7))].
2909
629 f(f(x,x),f(x,f(y,x))) = x. [para(613(a,1),2(a,1,1))].
2910
631 f(f(x,x),f(x,x)) = x. [para(613(a,1),2(a,1,2)),rewrite(613(3))].
2911
646 f(f(x,f(x,x)),f(x,x)) = x. [para(613(a,1),8(a,1,2)),rewrite(631(3))].
2912
650 f(f(f(x,x),x),x) = f(x,x). [para(613(a,1),10(a,1,1,1)),rewrite(631(5))].
2913
668 f(f(f(f(x,y),x),x),f(y,y)) = y. [back_rewrite(616),rewrite(629(4),629(9))].
2914
673 f(f(x,f(x,x)),x) = f(x,x). [para(631(a,1),2(a,1,2)),rewrite(650(3))].
2915
678 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [para(631(a,1),7(a,1,1,1))].
2916
690 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(622),rewrite(673(3))].
2917
712 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(613),rewrite(690(2))].
2918
759 f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,x). [para(646(a,1),13(a,1,1,1)),rewrite(646(6),673(3),673(5),690(4),712(5),646(7),673(9),690(8),712(9))].
2919
856 f(f(x,x),f(x,f(x,y))) = x. [para(673(a,1),12(a,1,1))].
2920
1029 f(f(x,f(x,f(x,y))),f(x,x)) = x. [para(10(a,1),668(a,1,1,1))].
2921
1058 f(f(f(f(x,f(x,f(x,f(x,y)))),x),x),x) = f(x,f(x,f(x,y))). [para(1029(a,1),4(a,1,2))].
2922
1067 f(f(f(x,f(f(x,f(x,f(x,y))),f(x,f(x,f(x,y))))),x),x) = f(x,f(x,f(x,y))). [para(1029(a,1),7(a,1,2))].
2923
1087 f(x,f(f(x,f(x,f(x,y))),f(x,f(x,f(x,y))))) = f(x,f(x,f(x,y))). [para(1029(a,1),621(a,1,1,2,1)),rewrite(856(4))].
2924
1091 f(f(f(x,f(x,f(x,y))),x),x) = f(x,f(x,f(x,y))). [back_rewrite(1067),rewrite(1087(8))].
2925
1094 f(f(x,f(x,f(x,f(x,y)))),x) = f(x,f(x,f(x,y))). [back_rewrite(1058),rewrite(1091(6))].
2926
1181 f(x,f(f(f(y,x),x),x)) = f(x,f(y,f(f(x,y),y))). [para(504(a,1),271(a,2,2)),rewrite(271(12))].
2927
1240 f(f(x,f(f(y,x),x)),f(f(f(y,y),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))) = y. [para(621(a,1),504(a,1,1,1)),rewrite(646(4)),flip(a)].
2928
1257 f(f(f(f(x,y),x),x),f(f(f(y,y),f(f(f(x,y),x),x)),f(f(f(x,y),x),x))) = y. [para(668(a,1),504(a,1,1,1)),rewrite(646(4)),flip(a)].
2929
1263 f(f(x,y),f(f(y,f(x,y)),f(x,y))) = f(f(x,f(f(y,x),x)),y). [para(504(a,1),504(a,1,1)),flip(a)].
2930
1331 f(f(f(f(f(x,y),x),x),f(f(y,f(z,x)),f(y,f(y,f(z,x))))),f(f(f(f(x,y),x),x),f(f(f(y,f(z,x)),f(y,f(y,f(z,x)))),f(u,f(y,f(z,x)))))) = f(f(f(x,y),x),x). [para(16(a,1),10(a,1,1,1))].
2931
1333 f(f(f(f(f(f(x,y),x),x),f(f(f(y,f(z,x)),f(y,f(y,f(z,x)))),f(u,f(y,f(z,x))))),f(f(f(x,y),x),x)),f(f(f(x,y),x),x)) = f(f(y,f(z,x)),f(y,f(y,f(z,x)))). [para(16(a,1),10(a,1,2))].
2932
1406 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)). [para(678(a,1),11(a,1,1,1)),rewrite(712(3),673(6))].
2933
1605 f(f(f(f(f(x,y),x),x),f(x,x)),y) = f(f(f(x,y),x),x). [para(673(a,1),17(a,1,1,1)),rewrite(673(7),673(9),1406(9))].
2934
1799 f(x,f(f(y,x),x)) = f(y,x). [para(759(a,1),11(a,1,1,1)),rewrite(856(4),646(5))].
2935
1969 f(f(x,y),x) = f(x,f(y,x)). [back_rewrite(1263),rewrite(1799(6),1799(5)),flip(a)].
2936
1971 f(f(x,x),f(x,y)) = x. [back_rewrite(1257),rewrite(1969(2),1969(3),1799(3),1969(4),1969(5),1799(5),1969(6),1969(7),1799(7),1799(7))].
2937
1986 f(f(x,y),f(x,f(x,y))) = x. [back_rewrite(1240),rewrite(1799(3),1799(5),1971(4),1799(4))].
2938
2018 f(f(x,y),y) = f(y,f(y,x)). [back_rewrite(1181),rewrite(1799(4),1799(5))].
2939
2565 f(x,f(x,f(x,y))) = f(x,y). [back_rewrite(1605),rewrite(1969(2),1969(3),2018(2),1029(5),1969(3),1969(4),2018(3)),flip(a)].
2940
2627 f(f(x,y),f(y,f(z,f(y,f(u,x))))) = y. [back_rewrite(1333),rewrite(1969(2),1969(3),2018(2),2565(3),1986(7),1969(8),1969(9),2018(8),2565(9),1969(8),1969(10),1969(11),2018(10),2565(11),1969(10),2018(9),2565(10),1986(12))].
2941
2629 f(x,f(x,f(y,x))) = f(y,x). [back_rewrite(1331),rewrite(1969(2),1969(3),2018(2),2565(3),1986(7),2018(2),1969(4),1969(5),2018(4),2565(5),1986(9),2627(8),1969(3),1969(2),1969(5),1969(6),2018(5),2565(6))].
2942
2676 f(x,y) = f(y,x). [back_rewrite(1094),rewrite(2565(3),1969(3),1969(2),2629(3),2565(4))].
2943
2677 $F # answer("Commutativity"). [resolve(2676,a,3,a)].
2741
321 f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = f(x,f(f(f(x,f(f(x,x),x)),x),x)). [para(2(a,1),271(a,1,1,1,2))].
2742
436 f(f(f(x,y),y),y) = f(x,f(f(y,x),x)). [para(4(a,1),85(a,1,1,1,1))].
2743
500 f(f(f(x,f(f(y,x),x)),y),f(y,f(f(x,y),y))) = y. [para(436(a,1),5(a,1,1,1))].
2744
529 f(f(f(f(x,x),f(y,x)),f(y,x)),f(f(f(x,y),y),y)) = x. [para(436(a,2),8(a,1,2))].
2745
533 f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x. [para(436(a,1),9(a,1,1))].
2746
571 f(x,f(f(f(y,x),x),x)) = f(x,f(y,f(f(x,y),y))). [para(436(a,1),271(a,2,2)),rewrite([271(12)])].
2747
604 f(x,f(f(f(x,f(f(x,x),x)),x),x)) = f(x,f(x,f(f(x,x),x))). [back_rewrite(321),rewrite([500(8)]),flip(a)].
2748
613 f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x. [para(500(a,1),4(a,1,1,1))].
2749
729 f(f(x,f(f(y,x),x)),f(f(y,f(f(f(x,y),y),y)),f(x,f(f(y,x),x)))) = f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))). [para(571(a,2),271(a,2,2,1)),rewrite([271(16)]),flip(a)].
2750
742 f(f(x,f(f(y,x),x)),f(f(y,f(f(f(x,y),y),y)),f(x,f(f(y,x),x)))) = f(f(f(f(x,f(f(y,x),x)),y),y),y). [para(571(a,2),436(a,2,2,1)),flip(a)].
2751
747 f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))) = f(f(f(f(x,f(f(y,x),x)),y),y),y). [back_rewrite(729),rewrite([742(12)]),flip(a)].
2752
829 f(f(x,f(x,f(y,f(x,f(f(x,x),x))))),x) = f(x,f(f(x,x),x)). [para(613(a,1),5(a,1,1,1)),rewrite([533(16)])].
2753
840 f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))). [para(613(a,1),9(a,1,2)),rewrite([271(7)])].
2754
845 f(f(x,f(x,f(x,f(f(x,x),x)))),f(f(y,f(x,f(f(x,x),x))),f(f(x,f(x,f(f(x,x),x))),x))) = f(y,f(x,f(f(x,x),x))). [para(613(a,1),11(a,1,1,1))].
2755
865 f(x,f(f(x,f(x,f(f(x,x),x))),x)) = f(x,f(x,f(f(x,x),x))). [back_rewrite(604),rewrite([840(4)])].
2756
892 f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),f(f(f(x,f(x,f(f(x,x),x))),x),x)) = f(x,f(f(x,x),x)). [para(840(a,1),500(a,1,1,1,2,1)),rewrite([865(6),747(20),840(12)])].
2757
893 f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x. [para(840(a,1),500(a,1,1))].
2758
894 f(x,f(f(f(x,f(x,f(f(x,x),x))),x),x)) = f(x,f(x,f(x,f(f(x,x),x)))). [para(840(a,1),571(a,1,2,1,1)),rewrite([893(18),840(11)])].
2759
897 f(x,f(x,f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)). [back_rewrite(892),rewrite([893(8),894(7)])].
2760
899 f(f(x,f(f(x,x),x)),f(f(y,f(x,f(f(x,x),x))),f(f(x,f(x,f(f(x,x),x))),x))) = f(y,f(x,f(f(x,x),x))). [back_rewrite(845),rewrite([897(5)])].
2761
903 f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)). [para(2(a,1),893(a,1,1,2,2,1)),rewrite([533(11),840(4),2(14),533(12)])].
2762
923 f(f(x,f(f(x,x),x)),f(f(y,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))) = f(y,f(x,f(f(x,x),x))). [back_rewrite(899),rewrite([903(12)])].
2763
953 f(f(f(f(x,x),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x. [para(903(a,1),529(a,1,1,1,2)),rewrite([903(10),897(14),533(17),897(14)])].
2764
1365 f(x,f(f(x,x),x)) = f(x,x). [para(318(a,1),13(a,1,2)),rewrite([533(8)]),flip(a)].
2765
1483 f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,x)) = x. [back_rewrite(953),rewrite([1365(4),1365(6),1365(8)])].
2766
1513 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)). [back_rewrite(923),rewrite([1365(3),1365(4),1365(6),1365(9)])].
2767
1529 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(903),rewrite([1365(3),1365(6)])].
2768
1533 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(897),rewrite([1365(3),1365(6)])].
2769
1577 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(840),rewrite([1365(3),1365(5)])].
2770
1586 f(f(x,f(x,f(y,f(x,x)))),x) = f(x,x). [back_rewrite(829),rewrite([1577(2),1533(3),1577(7),1533(8)])].
2771
1610 f(f(x,x),f(x,x)) = x. [back_rewrite(1483),rewrite([1577(5),1529(7)])].
2772
1620 f(f(x,f(y,y)),f(x,y)) = x. [para(1610(a,1),2(a,1,2,2)),rewrite([1513(6)])].
2773
1781 f(f(x,y),f(x,f(y,y))) = x. [para(2(a,1),1620(a,1,1,2)),rewrite([1577(3),1533(4)])].
2774
1796 f(f(x,x),y) = f(y,f(x,x)). [para(1620(a,1),6(a,1,2)),rewrite([1586(5)])].
2775
1953 f(x,f(f(x,x),x)) = f(x,x). [para(1610(a,1),1781(a,1,1)),rewrite([1610(4)])].
2776
1976 f(x,y) = f(y,x). [para(2(a,1),1796(a,1,1)),rewrite([1953(4),1953(5),1610(4)])].
2777
1977 $F # answer("Commutativity"). [resolve(1976,a,3,a)].
2944
2778
2945
2779
============================== end of proof ==========================
2946
2780
2947
2781
============================== STATISTICS ============================
2948
2782
2949
Given=39. Generated=5177. Kept=2675. proofs=1.
2950
Usable=25. Sos=676. Demods=1394. Limbo=707, Disabled=1277. Hints=805.
2951
Weight_deleted=110. Literals_deleted=0.
2952
Forward_subsumed=2392. Back_subsumed=75.
2783
Given=36. Generated=3828. Kept=1975. proofs=1.
2784
Usable=31. Sos=1569. Demods=1546. Limbo=9, Disabled=376. Hints=583.
2785
Kept_by_rule=0, Deleted_by_rule=127.
2786
Forward_subsumed=1726. Back_subsumed=59.
2953
2787
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
2954
New_demodulators=2625 (0 lex), Back_demodulated=1191. Back_unit_deleted=0.
2955
Demod_attempts=146841. Demod_rewrites=14242.
2788
New_demodulators=1901 (0 lex), Back_demodulated=306. Back_unit_deleted=0.
2789
Demod_attempts=112136. Demod_rewrites=6968.
2956
2790
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
2957
2791
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
2958
Megabytes=4.41.
2959
User_CPU=0.62, System_CPU=0.00, Wall_clock=7.
2792
Megabytes=4.01.
2793
User_CPU=0.69, System_CPU=0.00, Wall_clock=5.
2960
2794
2961
2795
============================== end of statistics =====================
2962
2796
2964
2798
2965
2799
Exiting with 1 proof.
2966
2800
2967
Process 13970 exit (max_proofs) Sun Aug 13 21:05:04 2006
2801
Process 10943 exit (max_proofs) Mon Sep 28 11:57:14 2009
2968
2802
2969
2803
Successful proof using the following 7 assumptions:
2970
2804
2971
2805
formulas(assumptions_in_proof).
2972
2806
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
2973
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2807
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2808
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2974
2809
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2975
2810
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2976
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2977
2811
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2978
2812
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2979
2813
end_of_list.
2980
2814
2981
Including 163 new hint clauses:
2815
Including 95 new hint clauses:
2982
2816
2983
2817
formulas(new_hints).
2984
$F # label("Commutativity").
2818
$F # label("Commutativity") # label(non_clause) # label(goal).
2985
2819
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
2986
2820
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
2987
2821
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").
2988
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").
2822
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
2823
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
2989
2824
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
2990
2825
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
2991
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
2992
2826
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
2993
2827
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
2994
2828
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)).
2995
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(f(f(z,x),z),z),f(u,f(x,f(y,z))))) = f(f(f(z,x),z),z).
2996
f(f(f(x,y),f(f(f(f(f(y,z),y),y),f(x,y)),f(x,y))),z) = f(f(f(y,z),y),y).
2997
2829
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
2998
2830
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
2999
2831
f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y.
3000
f(f(f(f(x,y),x),x),f(y,f(z,f(f(x,z),z)))) = y.
3001
f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(y,z))) = f(y,f(x,f(f(y,x),x))).
2832
f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = f(x,f(f(f(x,f(f(x,x),x)),x),x)).
3002
2833
f(f(f(x,y),y),y) = f(x,f(f(y,x),x)).
3003
f(f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
3004
f(f(f(x,f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
3005
f(f(f(x,f(x,f(f(x,x),x))),x),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
3006
f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
3007
f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(f(x,f(f(x,x),x)),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
3008
f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(x,f(x,f(f(x,x),x))).
3009
f(f(x,f(f(f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))),x),x)),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
3010
f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
3011
f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y.
2834
f(f(f(x,f(f(y,x),x)),y),f(y,f(f(x,y),y))) = y.
2835
f(f(f(f(x,x),f(y,x)),f(y,x)),f(f(f(x,y),y),y)) = x.
2836
f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x.
2837
f(f(f(f(x,y),f(f(x,y),f(y,f(f(f(x,y),y),y)))),f(x,y)),f(x,y)) = f(y,f(x,f(f(y,x),x))).
2838
f(x,f(f(f(y,x),x),x)) = f(x,f(y,f(f(x,y),y))).
2839
f(x,f(x,f(f(x,x),x))) = f(x,f(f(f(x,f(f(x,x),x)),x),x)).
2840
f(x,f(f(f(x,f(f(x,x),x)),x),x)) = f(x,f(x,f(f(x,x),x))).
2841
f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x.
2842
f(f(f(x,f(x,f(f(y,f(f(x,y),y)),f(f(x,f(y,f(f(x,y),y))),f(y,f(f(x,y),y)))))),x),x) = f(f(y,f(f(x,y),y)),f(f(x,f(f(f(y,x),x),x)),f(y,f(f(x,y),y)))).
2843
f(f(y,f(f(x,y),y)),f(f(x,f(y,f(f(x,y),y))),f(y,f(f(x,y),y)))) = f(f(y,f(f(x,y),y)),f(f(x,f(f(f(y,x),x),x)),f(y,f(f(x,y),y)))).
2844
f(f(x,f(f(y,x),x)),f(f(y,f(f(f(x,y),y),y)),f(x,f(f(y,x),x)))) = f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))).
2845
f(f(f(f(x,f(f(y,x),x)),y),y),y) = f(f(x,f(f(y,x),x)),f(f(y,f(f(f(x,y),y),y)),f(x,f(f(y,x),x)))).
2846
f(f(x,f(f(y,x),x)),f(f(y,f(f(f(x,y),y),y)),f(x,f(f(y,x),x)))) = f(f(f(f(x,f(f(y,x),x)),y),y),y).
2847
f(f(f(f(x,f(f(y,x),x)),y),y),y) = f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))).
2848
f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))) = f(f(f(f(x,f(f(y,x),x)),y),y),y).
2849
f(f(x,f(x,f(y,f(x,f(f(x,x),x))))),f(f(x,f(f(x,x),x)),f(x,f(x,f(y,f(x,f(f(x,x),x))))))) = f(x,f(f(x,x),x)).
2850
f(f(x,f(x,f(y,f(x,f(f(x,x),x))))),x) = f(x,f(f(x,x),x)).
2851
f(f(f(f(x,f(x,f(x,f(f(x,x),x)))),x),x),x) = f(x,f(x,f(f(x,x),x))).
2852
f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))).
2853
f(f(x,f(x,f(x,f(f(x,x),x)))),f(f(y,f(x,f(f(x,x),x))),f(f(x,f(x,f(f(x,x),x))),x))) = f(y,f(x,f(f(x,x),x))).
2854
f(x,f(f(x,f(x,f(f(x,x),x))),x)) = f(x,f(x,f(f(x,x),x))).
2855
f(f(f(x,f(f(x,f(x,f(f(x,x),x))),x)),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
2856
f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
2857
f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),f(f(f(f(x,f(f(x,x),x)),x),x),x)) = f(x,f(f(x,x),x)).
2858
f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),f(f(f(x,f(x,f(f(x,x),x))),x),x)) = f(x,f(f(x,x),x)).
2859
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
2860
f(x,f(f(f(x,f(x,f(f(x,x),x))),x),x)) = f(x,f(f(x,f(f(x,x),x)),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))))).
2861
f(x,f(f(f(x,f(x,f(f(x,x),x))),x),x)) = f(x,f(f(x,f(f(x,x),x)),x)).
2862
f(x,f(f(f(x,f(x,f(f(x,x),x))),x),x)) = f(x,f(x,f(x,f(f(x,x),x)))).
2863
f(x,f(f(f(x,f(x,f(f(x,x),x))),x),x)) = f(x,f(f(x,x),x)).
2864
f(x,f(x,f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
2865
f(f(x,f(f(x,x),x)),f(f(y,f(x,f(f(x,x),x))),f(f(x,f(x,f(f(x,x),x))),x))) = f(y,f(x,f(f(x,x),x))).
2866
f(f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
2867
f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
2868
f(f(x,f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
2869
f(f(x,f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
2870
f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
2871
f(f(x,f(f(x,x),x)),f(f(y,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))) = f(y,f(x,f(f(x,x),x))).
2872
f(f(f(f(x,x),f(x,f(f(x,x),x))),f(f(x,f(x,f(f(x,x),x))),x)),f(f(f(x,f(x,f(x,f(f(x,x),x)))),f(x,f(x,f(f(x,x),x)))),f(x,f(x,f(f(x,x),x))))) = x.
2873
f(f(f(f(x,x),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),f(f(f(x,f(x,f(x,f(f(x,x),x)))),f(x,f(x,f(f(x,x),x)))),f(x,f(x,f(f(x,x),x))))) = x.
2874
f(f(f(f(x,x),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(x,f(f(x,x),x))))) = x.
2875
f(f(f(f(x,x),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),f(x,f(x,f(x,f(f(x,x),x))))) = x.
2876
f(f(f(f(x,x),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
3012
2877
f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),x) = f(x,f(f(x,x),x)).
3013
2878
f(x,x) = f(x,f(f(x,x),x)).
3014
2879
f(x,f(f(x,x),x)) = f(x,x).
3015
f(f(f(f(x,f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y))))),x),x),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
3016
f(f(f(f(x,f(f(y,y),f(y,f(y,f(f(y,y),y))))),x),x),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
3017
f(f(f(f(x,f(f(y,y),f(y,f(y,y)))),x),x),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
3018
f(f(f(f(x,f(f(y,y),f(y,f(y,y)))),x),x),f(y,y)) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
3019
f(f(f(f(x,f(f(y,y),f(y,f(y,y)))),x),x),f(y,y)) = f(f(y,y),f(y,f(y,f(f(y,y),y)))).
3020
f(f(f(f(x,f(f(y,y),f(y,f(y,y)))),x),x),f(y,y)) = f(f(y,y),f(y,f(y,y))).
3021
f(f(x,f(f(y,x),x)),f(y,y)) = y.
3022
f(f(f(x,f(x,x)),x),x) = f(x,f(x,f(f(x,x),x))).
3023
f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)).
3024
f(f(x,x),f(x,f(y,x))) = x.
3025
f(f(x,f(f(x,x),x)),f(x,x)) = x.
3026
f(f(x,x),f(x,x)) = x.
2880
f(f(f(f(x,x),f(x,x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
2881
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,f(f(x,x),x))) = x.
3027
2882
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,x)) = x.
3028
f(f(x,f(x,x)),f(x,x)) = x.
3029
f(f(f(x,x),x),f(f(x,x),f(x,x))) = f(x,x).
3030
f(f(f(x,x),x),x) = f(x,x).
3031
f(f(f(f(x,y),x),x),f(y,y)) = f(f(y,y),f(y,f(y,y))).
3032
f(f(f(f(x,y),x),x),f(y,y)) = y.
3033
f(f(x,f(f(f(x,x),x),x)),x) = f(x,x).
2883
f(f(x,x),f(f(y,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))) = f(y,f(x,f(f(x,x),x))).
2884
f(f(x,x),f(f(y,f(x,x)),f(x,f(f(x,x),x)))) = f(y,f(x,f(f(x,x),x))).
2885
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,f(f(x,x),x))).
2886
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
2887
f(f(x,f(x,x)),x) = f(x,f(f(x,x),x)).
3034
2888
f(f(x,f(x,x)),x) = f(x,x).
3035
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
2889
f(x,f(x,f(x,x))) = f(x,f(f(x,x),x)).
2890
f(x,f(x,f(x,x))) = f(x,x).
2891
f(f(x,x),x) = f(x,f(x,f(f(x,x),x))).
3036
2892
f(f(x,x),x) = f(x,f(x,x)).
3037
f(x,f(x,f(x,x))) = f(x,x).
3038
f(f(x,f(f(x,f(x,x)),f(f(x,f(x,x)),f(x,x)))),f(f(x,f(f(f(x,f(x,x)),x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
3039
f(f(x,f(f(x,f(x,x)),x)),f(f(x,f(f(f(x,f(x,x)),x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
3040
f(f(x,f(x,x)),f(f(x,f(f(f(x,f(x,x)),x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
3041
f(f(x,f(x,x)),f(f(x,f(f(x,x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
3042
f(f(x,f(x,x)),f(f(x,f(x,f(x,x))),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
3043
f(f(x,f(x,x)),f(f(x,x),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
3044
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,f(f(f(x,f(x,x)),x),x)).
3045
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,f(f(x,x),x)).
3046
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,f(x,f(x,x))).
3047
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,x).
3048
f(f(x,x),f(x,f(x,y))) = x.
3049
f(f(x,f(x,f(x,y))),f(x,x)) = x.
3050
f(f(f(f(x,f(x,f(x,f(x,y)))),x),x),x) = f(x,f(x,f(x,y))).
3051
f(f(f(x,f(f(x,f(x,f(x,y))),f(x,f(x,f(x,y))))),x),x) = f(x,f(x,f(x,y))).
3052
f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,f(x,y))),f(x,f(x,f(x,y))))) = f(x,f(x,f(x,y))).
3053
f(x,f(f(x,f(x,f(x,y))),f(x,f(x,f(x,y))))) = f(x,f(x,f(x,y))).
3054
f(f(f(x,f(x,f(x,y))),x),x) = f(x,f(x,f(x,y))).
3055
f(f(x,f(x,f(x,f(x,y)))),x) = f(x,f(x,f(x,y))).
3056
f(f(f(f(x,y),f(f(x,y),f(y,f(f(f(x,y),y),y)))),f(x,y)),f(x,y)) = f(y,f(x,f(f(y,x),x))).
3057
f(x,f(f(f(y,x),x),x)) = f(x,f(y,f(f(x,y),y))).
3058
f(f(x,f(x,x)),f(x,x)) = f(f(y,f(f(x,y),y)),f(f(f(x,x),f(y,f(f(x,y),y))),f(y,f(f(x,y),y)))).
3059
x = f(f(y,f(f(x,y),y)),f(f(f(x,x),f(y,f(f(x,y),y))),f(y,f(f(x,y),y)))).
3060
f(f(x,f(f(y,x),x)),f(f(f(y,y),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))) = y.
3061
f(f(x,f(x,x)),f(x,x)) = f(f(f(f(y,x),y),y),f(f(f(x,x),f(f(f(y,x),y),y)),f(f(f(y,x),y),y))).
3062
x = f(f(f(f(y,x),y),y),f(f(f(x,x),f(f(f(y,x),y),y)),f(f(f(y,x),y),y))).
3063
f(f(f(f(x,y),x),x),f(f(f(y,y),f(f(f(x,y),x),x)),f(f(f(x,y),x),x))) = y.
3064
f(f(x,f(f(y,x),x)),y) = f(f(x,y),f(f(y,f(x,y)),f(x,y))).
3065
f(f(x,y),f(f(y,f(x,y)),f(x,y))) = f(f(x,f(f(y,x),x)),y).
3066
f(f(f(f(f(x,y),x),x),f(f(y,f(z,x)),f(y,f(y,f(z,x))))),f(f(f(f(x,y),x),x),f(f(f(y,f(z,x)),f(y,f(y,f(z,x)))),f(u,f(y,f(z,x)))))) = f(f(f(x,y),x),x).
3067
f(f(f(f(f(f(x,y),x),x),f(f(f(y,f(z,x)),f(y,f(y,f(z,x)))),f(u,f(y,f(z,x))))),f(f(f(x,y),x),x)),f(f(f(x,y),x),x)) = f(f(y,f(z,x)),f(y,f(y,f(z,x)))).
3068
f(f(x,f(x,f(x,x))),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
3069
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
3070
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
3071
f(f(f(x,x),f(f(f(f(f(x,y),x),x),f(f(x,f(x,x)),x)),f(f(x,f(x,x)),x))),y) = f(f(f(x,y),x),x).
3072
f(f(f(x,x),f(f(f(f(f(x,y),x),x),f(x,x)),f(f(x,f(x,x)),x))),y) = f(f(f(x,y),x),x).
3073
f(f(f(x,x),f(f(f(f(f(x,y),x),x),f(x,x)),f(x,x))),y) = f(f(f(x,y),x),x).
3074
f(f(f(f(f(x,y),x),x),f(x,x)),y) = f(f(f(x,y),x),x).
3075
f(f(f(x,x),f(x,f(x,x))),f(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x).
3076
f(x,f(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x).
3077
f(x,f(f(y,x),x)) = f(y,x).
3078
f(y,f(x,y)) = f(f(x,f(f(y,x),x)),y).
3079
f(y,f(x,y)) = f(f(y,x),y).
3080
f(f(x,y),x) = f(x,f(y,x)).
3081
f(f(f(x,f(y,x)),x),f(f(f(y,y),f(f(f(x,y),x),x)),f(f(f(x,y),x),x))) = y.
3082
f(f(x,f(f(y,x),x)),f(f(f(y,y),f(f(f(x,y),x),x)),f(f(f(x,y),x),x))) = y.
3083
f(f(y,x),f(f(f(y,y),f(f(f(x,y),x),x)),f(f(f(x,y),x),x))) = y.
3084
f(f(y,x),f(f(f(y,y),f(f(x,f(y,x)),x)),f(f(f(x,y),x),x))) = y.
3085
f(f(y,x),f(f(f(y,y),f(x,f(f(y,x),x))),f(f(f(x,y),x),x))) = y.
3086
f(f(y,x),f(f(f(y,y),f(y,x)),f(f(f(x,y),x),x))) = y.
3087
f(f(y,x),f(f(f(y,y),f(y,x)),f(f(x,f(y,x)),x))) = y.
3088
f(f(y,x),f(f(f(y,y),f(y,x)),f(x,f(f(y,x),x)))) = y.
3089
f(f(y,x),f(f(f(y,y),f(y,x)),f(y,x))) = y.
3090
f(f(x,x),f(x,y)) = x.
3091
f(f(y,x),f(f(f(y,y),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))) = y.
3092
f(f(y,x),f(f(f(y,y),f(y,x)),f(x,f(f(y,x),x)))) = y.
3093
f(f(y,x),f(y,f(x,f(f(y,x),x)))) = y.
3094
f(f(x,y),f(x,f(x,y))) = x.
3095
f(f(y,x),x) = f(x,f(y,f(f(x,y),y))).
3096
f(f(x,y),y) = f(y,f(y,x)).
3097
f(f(f(f(x,f(y,x)),x),f(x,x)),y) = f(f(f(x,y),x),x).
3098
f(f(f(x,f(f(y,x),x)),f(x,x)),y) = f(f(f(x,y),x),x).
3099
f(f(f(x,f(x,f(x,y))),f(x,x)),y) = f(f(f(x,y),x),x).
3100
f(x,y) = f(f(f(x,y),x),x).
3101
f(x,y) = f(f(x,f(y,x)),x).
3102
f(x,y) = f(x,f(f(y,x),x)).
3103
f(x,y) = f(x,f(x,f(x,y))).
3104
f(x,f(x,f(x,y))) = f(x,y).
3105
f(f(f(f(f(x,f(y,x)),x),f(f(f(y,f(z,x)),f(y,f(y,f(z,x)))),f(u,f(y,f(z,x))))),f(f(f(x,y),x),x)),f(f(f(x,y),x),x)) = f(f(y,f(z,x)),f(y,f(y,f(z,x)))).
3106
f(f(f(f(x,f(f(y,x),x)),f(f(f(y,f(z,x)),f(y,f(y,f(z,x)))),f(u,f(y,f(z,x))))),f(f(f(x,y),x),x)),f(f(f(x,y),x),x)) = f(f(y,f(z,x)),f(y,f(y,f(z,x)))).
3107
f(f(f(f(x,f(x,f(x,y))),f(f(f(y,f(z,x)),f(y,f(y,f(z,x)))),f(u,f(y,f(z,x))))),f(f(f(x,y),x),x)),f(f(f(x,y),x),x)) = f(f(y,f(z,x)),f(y,f(y,f(z,x)))).
3108
f(f(f(f(x,y),f(f(f(y,f(z,x)),f(y,f(y,f(z,x)))),f(u,f(y,f(z,x))))),f(f(f(x,y),x),x)),f(f(f(x,y),x),x)) = f(f(y,f(z,x)),f(y,f(y,f(z,x)))).
3109
f(f(f(f(x,y),f(y,f(u,f(y,f(z,x))))),f(f(f(x,y),x),x)),f(f(f(x,y),x),x)) = f(f(y,f(z,x)),f(y,f(y,f(z,x)))).
3110
f(f(f(f(x,y),f(y,f(u,f(y,f(z,x))))),f(f(x,f(y,x)),x)),f(f(f(x,y),x),x)) = f(f(y,f(z,x)),f(y,f(y,f(z,x)))).
3111
f(f(f(f(x,y),f(y,f(u,f(y,f(z,x))))),f(x,f(f(y,x),x))),f(f(f(x,y),x),x)) = f(f(y,f(z,x)),f(y,f(y,f(z,x)))).
3112
f(f(f(f(x,y),f(y,f(u,f(y,f(z,x))))),f(x,f(x,f(x,y)))),f(f(f(x,y),x),x)) = f(f(y,f(z,x)),f(y,f(y,f(z,x)))).
3113
f(f(f(f(x,y),f(y,f(u,f(y,f(z,x))))),f(x,y)),f(f(f(x,y),x),x)) = f(f(y,f(z,x)),f(y,f(y,f(z,x)))).
3114
f(f(f(x,y),f(f(y,f(u,f(y,f(z,x)))),f(x,y))),f(f(f(x,y),x),x)) = f(f(y,f(z,x)),f(y,f(y,f(z,x)))).
3115
f(f(f(x,y),f(f(y,f(u,f(y,f(z,x)))),f(x,y))),f(f(x,f(y,x)),x)) = f(f(y,f(z,x)),f(y,f(y,f(z,x)))).
3116
f(f(f(x,y),f(f(y,f(u,f(y,f(z,x)))),f(x,y))),f(x,f(f(y,x),x))) = f(f(y,f(z,x)),f(y,f(y,f(z,x)))).
3117
f(f(f(x,y),f(f(y,f(u,f(y,f(z,x)))),f(x,y))),f(x,f(x,f(x,y)))) = f(f(y,f(z,x)),f(y,f(y,f(z,x)))).
3118
f(f(f(x,y),f(f(y,f(u,f(y,f(z,x)))),f(x,y))),f(x,y)) = f(f(y,f(z,x)),f(y,f(y,f(z,x)))).
3119
f(f(x,y),f(f(f(y,f(u,f(y,f(z,x)))),f(x,y)),f(x,y))) = f(f(y,f(z,x)),f(y,f(y,f(z,x)))).
3120
f(f(x,y),f(f(x,y),f(f(x,y),f(y,f(u,f(y,f(z,x))))))) = f(f(y,f(z,x)),f(y,f(y,f(z,x)))).
3121
f(f(x,y),f(y,f(u,f(y,f(z,x))))) = f(f(y,f(z,x)),f(y,f(y,f(z,x)))).
3122
f(f(x,y),f(y,f(z,f(y,f(u,x))))) = y.
3123
f(f(f(f(x,f(y,x)),x),f(f(y,f(z,x)),f(y,f(y,f(z,x))))),f(f(f(f(x,y),x),x),f(f(f(y,f(z,x)),f(y,f(y,f(z,x)))),f(u,f(y,f(z,x)))))) = f(f(f(x,y),x),x).
3124
f(f(f(x,f(f(y,x),x)),f(f(y,f(z,x)),f(y,f(y,f(z,x))))),f(f(f(f(x,y),x),x),f(f(f(y,f(z,x)),f(y,f(y,f(z,x)))),f(u,f(y,f(z,x)))))) = f(f(f(x,y),x),x).
3125
f(f(f(x,f(x,f(x,y))),f(f(y,f(z,x)),f(y,f(y,f(z,x))))),f(f(f(f(x,y),x),x),f(f(f(y,f(z,x)),f(y,f(y,f(z,x)))),f(u,f(y,f(z,x)))))) = f(f(f(x,y),x),x).
3126
f(f(f(x,y),f(f(y,f(z,x)),f(y,f(y,f(z,x))))),f(f(f(f(x,y),x),x),f(f(f(y,f(z,x)),f(y,f(y,f(z,x)))),f(u,f(y,f(z,x)))))) = f(f(f(x,y),x),x).
3127
f(f(f(x,y),y),f(f(f(f(x,y),x),x),f(f(f(y,f(z,x)),f(y,f(y,f(z,x)))),f(u,f(y,f(z,x)))))) = f(f(f(x,y),x),x).
3128
f(f(y,f(y,x)),f(f(f(f(x,y),x),x),f(f(f(y,f(z,x)),f(y,f(y,f(z,x)))),f(u,f(y,f(z,x)))))) = f(f(f(x,y),x),x).
3129
f(f(y,f(y,x)),f(f(f(x,f(y,x)),x),f(f(f(y,f(z,x)),f(y,f(y,f(z,x)))),f(u,f(y,f(z,x)))))) = f(f(f(x,y),x),x).
3130
f(f(y,f(y,x)),f(f(x,f(f(y,x),x)),f(f(f(y,f(z,x)),f(y,f(y,f(z,x)))),f(u,f(y,f(z,x)))))) = f(f(f(x,y),x),x).
3131
f(f(y,f(y,x)),f(f(x,f(x,f(x,y))),f(f(f(y,f(z,x)),f(y,f(y,f(z,x)))),f(u,f(y,f(z,x)))))) = f(f(f(x,y),x),x).
3132
f(f(y,f(y,x)),f(f(x,y),f(f(f(y,f(z,x)),f(y,f(y,f(z,x)))),f(u,f(y,f(z,x)))))) = f(f(f(x,y),x),x).
3133
f(f(y,f(y,x)),f(f(x,y),f(y,f(u,f(y,f(z,x)))))) = f(f(f(x,y),x),x).
3134
f(f(y,f(y,x)),y) = f(f(f(x,y),x),x).
3135
f(y,f(f(y,x),y)) = f(f(f(x,y),x),x).
3136
f(y,f(y,f(x,y))) = f(f(f(x,y),x),x).
3137
f(y,f(y,f(x,y))) = f(f(x,f(y,x)),x).
3138
f(y,f(y,f(x,y))) = f(x,f(f(y,x),x)).
3139
f(y,f(y,f(x,y))) = f(x,f(x,f(x,y))).
3140
f(x,f(x,f(y,x))) = f(y,x).
3141
f(f(x,f(x,y)),x) = f(x,f(x,f(x,y))).
3142
f(x,f(f(x,y),x)) = f(x,f(x,f(x,y))).
3143
f(x,f(x,f(y,x))) = f(x,f(x,f(x,y))).
3144
f(y,x) = f(x,f(x,f(x,y))).
2893
f(f(x,f(x,f(y,f(x,f(x,f(x,x)))))),x) = f(x,f(f(x,x),x)).
2894
f(f(x,f(x,f(y,f(x,x)))),x) = f(x,f(f(x,x),x)).
2895
f(f(x,f(x,f(y,f(x,x)))),x) = f(x,f(x,f(x,x))).
2896
f(f(x,f(x,f(y,f(x,x)))),x) = f(x,x).
2897
f(f(f(x,x),f(f(x,x),f(x,x))),f(x,x)) = x.
2898
f(f(x,x),f(x,x)) = x.
2899
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
2900
f(f(x,f(y,y)),f(x,y)) = x.
2901
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
2902
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
2903
f(f(x,y),f(x,f(y,y))) = x.
2904
f(f(f(x,f(x,f(y,f(x,x)))),x),y) = f(y,f(x,x)).
2905
f(f(x,x),y) = f(y,f(x,x)).
2906
f(x,f(f(x,x),f(f(x,x),f(x,x)))) = f(x,x).
2907
f(x,f(f(x,x),x)) = f(x,x).
2908
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
2909
f(x,y) = f(y,f(f(x,x),f(x,f(f(x,x),x)))).
2910
f(x,y) = f(y,f(f(x,x),f(x,x))).
3145
2911
f(x,y) = f(y,x).
3146
2912
$F # answer("Commutativity").
3147
2913
end_of_list.
3167
2933
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3168
2934
end_of_list.
3169
2935
3170
Child search process 13971 started.
2936
Child search process 10944 started.
3171
2937
3172
2938
============================== PROCESS INITIAL CLAUSES ===============
3173
2939
3192
2958
formulas(demodulators).
3193
2959
end_of_list.
3194
2960
3195
% 968 hints input.
2961
% 678 hints input.
3196
2962
3197
2963
============================== PREDICATE ELIMINATION =================
3198
2964
3205
2971
3206
2972
Term ordering decisions:
3207
2973
Function symbol KB weights: c1=1. c2=1. f=1.
3208
Relation symbol precedence: lex([ = ]).
3209
Function symbol precedence: lex([ c1, c2, f ]).
2974
Predicate symbol precedence: predicate_order([ = ]).
2975
Function symbol precedence: function_order([ c1, c2, f ]).
3210
2976
Skipping inverse_order, because term ordering is KBO.
3211
2977
Unfolding symbols: (none).
3212
2978
3213
2979
Auto_inference settings:
3214
2980
% set(paramodulation). % (positive equality literals)
3215
% set(paramodulation) -> set(back_demod).
3216
2981
3217
2982
Auto_process settings: (no changes).
3218
2983
2984
kept: 2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
2985
kept: 3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
2986
kept: 4 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2987
kept: 5 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2988
kept: 6 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2989
kept: 7 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2990
kept: 8 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2991
kept: 9 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2992
kept: 10 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2993
kept: 11 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2994
3219
2995
============================== end of process initial clauses ========
3220
2996
3221
2997
============================== CLAUSES FOR SEARCH ====================
3250
3026
11 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3251
3027
end_of_list.
3252
3028
3253
% 968 hints processed (507 are redundant).
3029
% 295 hints (678 processed, 383 redundant).
3254
3030
3255
3031
============================== end of clauses for search =============
3256
3032
3257
3033
============================== SEARCH ================================
3258
3034
3259
% Starting search at 0.01 seconds.
3035
% Starting search at 0.02 seconds.
3260
3036
3261
3037
% Operation f is commutative; C redundancy checks enabled.
3262
3038
3263
3039
============================== PROOF =================================
3264
3040
3265
% Proof 1 at 0.82 (+ 0.01) seconds: "Commutativity".
3266
% Length of proof is 72.
3267
% Level of proof is 17.
3268
% Maximum clause weight is 67.
3269
% Given clauses 50.
3041
% Proof 1 at 0.57 (+ 0.00) seconds: "Commutativity".
3042
% Length of proof is 54.
3043
% Level of proof is 19.
3044
% Maximum clause weight is 63.
3045
% Given clauses 38.
3270
3046
3271
1 f(x,y) = f(y,x) # label("Commutativity"). [goal].
3047
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
3272
3048
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3273
3049
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
3274
3050
4 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
3275
3051
5 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
3276
6 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3277
3052
7 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3278
3053
8 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3279
3054
9 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3280
3055
10 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3281
3056
11 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3282
12 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)). [para(2(a,1),2(a,1,1,2,1))].
3283
14 f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)). [para(2(a,1),2(a,1,2))].
3284
16 f(f(f(x,y),f(f(f(f(f(x,z),y),y),f(x,y)),f(x,y))),z) = f(f(f(x,z),y),y). [para(4(a,1),2(a,1,2))].
3285
29 f(f(f(f(x,y),f(f(x,y),f(y,f(f(z,y),y)))),f(x,y)),z) = f(y,f(f(z,y),y)). [para(2(a,1),5(a,1,2))].
3286
40 f(f(f(x,y),x),f(f(y,f(f(y,y),y)),f(z,x))) = f(y,f(f(y,y),y)). [para(2(a,1),6(a,1,1,1,2))].
3287
206 f(f(f(x,f(f(y,z),f(f(f(y,x),y),y))),x),x) = f(f(f(y,x),y),y). [para(8(a,1),10(a,1,2))].
3057
62 f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)). [para(2(a,1),7(a,1,2))].
3288
3058
224 f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)). [para(2(a,1),11(a,1,2))].
3289
259 f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(y,f(y,x)),y). [para(11(a,1),10(a,1,2))].
3290
314 f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y. [para(224(a,1),2(a,1,2,2))].
3291
324 f(f(f(x,f(x,y)),x),f(y,f(z,f(f(x,z),z)))) = y. [para(224(a,1),5(a,1,2,2))].
3292
338 f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(y,z))) = f(y,f(x,f(f(y,x),x))). [para(224(a,1),8(a,1,1))].
3293
358 f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(x,f(x,f(f(x,x),x))). [para(12(a,1),224(a,1,1,1,2)),rewrite(314(8),314(12),314(13),314(17),338(18))].
3294
360 f(f(f(f(x,f(x,f(y,f(f(x,y),y)))),f(f(x,f(x,f(y,f(f(x,y),y)))),f(x,f(y,f(f(x,y),y))))),f(x,f(x,f(y,f(f(x,y),y))))),f(x,f(x,f(y,f(f(x,y),y))))) = f(x,f(y,f(f(x,y),y))). [para(224(a,1),224(a,1,1,1,2,2,2)),rewrite(224(35))].
3295
411 f(x,f(f(x,x),x)) = f(x,x). [para(314(a,1),12(a,1,2)),rewrite(314(8)),flip(a)].
3296
414 f(f(x,f(f(f(f(y,y),f(y,f(y,y))),x),x)),f(y,y)) = f(f(y,y),f(y,f(y,y))). [para(12(a,1),314(a,1,2)),rewrite(411(3),411(4),411(10),411(12),411(13))].
3297
425 f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)). [back_rewrite(358),rewrite(411(3),411(7))].
3298
427 f(f(f(x,y),x),f(f(y,y),f(z,x))) = f(y,y). [back_rewrite(40),rewrite(411(5),411(9))].
3299
429 f(f(x,x),f(x,f(y,f(f(x,x),x)))) = x. [para(411(a,1),2(a,1,1,2,1)),rewrite(427(7))].
3300
430 f(f(x,x),f(x,f(y,x))) = x. [para(411(a,1),2(a,1,1))].
3301
432 f(f(x,x),f(x,x)) = x. [para(411(a,1),2(a,1,2)),rewrite(411(3))].
3302
444 f(f(x,f(x,x)),f(x,x)) = x. [para(411(a,1),7(a,1,2)),rewrite(432(3))].
3303
448 f(f(f(x,x),x),x) = f(x,x). [para(411(a,1),9(a,1,1,1)),rewrite(432(5))].
3304
462 f(f(x,f(f(y,x),x)),f(y,y)) = y. [back_rewrite(414),rewrite(430(4),430(9))].
3305
466 f(f(x,f(x,x)),x) = f(x,x). [para(432(a,1),2(a,1,2)),rewrite(448(3))].
3306
471 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [para(432(a,1),6(a,1,1,1))].
3307
483 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(425),rewrite(466(3))].
3308
500 f(f(x,x),f(x,f(y,f(x,f(x,x))))) = x. [back_rewrite(429),rewrite(483(3))].
3309
501 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(411),rewrite(483(2))].
3310
603 f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,x). [para(444(a,1),12(a,1,1,1)),rewrite(444(6),466(3),466(5),483(4),501(5),444(7),466(9),483(8),501(9))].
3311
624 f(f(x,x),f(x,f(x,y))) = x. [para(466(a,1),11(a,1,1))].
3312
730 f(f(x,f(x,x)),f(f(x,y),f(x,f(x,x)))) = f(x,y). [para(624(a,1),10(a,1,1,1)),rewrite(483(5))].
3313
743 f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)). [para(462(a,1),2(a,1,2))].
3314
751 f(f(f(x,f(f(y,f(f(x,y),y)),f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)). [para(462(a,1),6(a,1,2))].
3315
781 f(x,f(f(y,f(f(x,y),y)),f(y,f(f(x,y),y)))) = f(y,f(f(x,y),y)). [para(462(a,1),462(a,1,1,2,1)),rewrite(624(4))].
3316
782 f(f(f(x,f(f(y,x),x)),y),y) = f(x,f(f(y,x),x)). [back_rewrite(751),rewrite(781(8))].
3317
784 f(f(x,f(y,f(f(x,y),y))),x) = f(y,f(f(x,y),y)). [back_rewrite(743),rewrite(782(5))].
3318
952 f(f(x,f(x,f(y,f(x,x)))),x) = f(x,x). [para(471(a,1),4(a,1,1,1)),rewrite(624(10))].
3319
972 f(f(f(f(x,f(x,x)),f(f(x,f(x,x)),f(f(x,f(y,f(x,x))),f(x,f(x,f(y,f(x,x))))))),f(x,f(x,x))),f(x,f(x,x))) = f(f(x,f(y,f(x,x))),f(x,f(x,f(y,f(x,x))))). [para(471(a,1),224(a,1,1,1,2,2,2,1)),rewrite(471(29))].
3320
974 f(f(f(x,f(x,x)),f(f(y,f(x,f(x,x))),f(x,f(x,x)))),f(y,f(f(x,f(z,f(x,x))),f(x,f(x,f(z,f(x,x))))))) = y. [para(471(a,1),314(a,1,2,2,2,1))].
3321
1029 f(f(x,f(x,x)),f(f(y,f(x,f(x,x))),f(x,f(x,x)))) = f(y,f(x,f(x,x))). [para(500(a,1),10(a,1,1,1)),rewrite(483(7))].
3322
1044 f(f(x,f(y,f(y,y))),f(x,f(f(y,f(z,f(y,y))),f(y,f(y,f(z,f(y,y))))))) = x. [back_rewrite(974),rewrite(1029(9))].
3323
1111 f(x,f(x,f(y,f(f(x,y),y)))) = f(y,f(f(x,y),y)). [para(224(a,1),16(a,1,1,2,1,1)),rewrite(462(6),624(4),224(12))].
3324
1185 f(f(f(f(x,f(f(y,x),x)),f(f(x,f(f(y,x),x)),f(y,f(x,f(f(y,x),x))))),f(x,f(f(y,x),x))),f(x,f(f(y,x),x))) = f(y,f(x,f(f(y,x),x))). [back_rewrite(360),rewrite(1111(5),1111(8),1111(17),1111(21))].
3325
1305 f(x,f(x,f(y,f(x,x)))) = f(x,f(x,x)). [para(952(a,1),16(a,1,1,2,1,1,1)),rewrite(483(3),444(5),624(4),952(9),483(6))].
3326
1307 f(f(x,f(y,f(x,x))),f(x,f(x,x))) = x. [back_rewrite(972),rewrite(1305(11),730(11),471(6),501(3),624(4),1305(7)),flip(a)].
3327
1308 f(f(x,f(y,f(y,y))),f(x,y)) = x. [back_rewrite(1044),rewrite(1305(10),1307(9))].
3328
1357 f(f(x,f(f(f(f(x,y),f(f(z,f(x,y)),f(x,y))),x),x)),z) = f(f(x,y),f(f(z,f(x,y)),f(x,y))). [para(1308(a,1),14(a,1,1,1)),rewrite(1308(11),1308(12))].
3329
1602 f(x,f(f(y,x),x)) = f(y,x). [para(603(a,1),10(a,1,1,1)),rewrite(624(4),444(5))].
3330
1749 f(f(f(x,f(y,z)),y),x) = f(x,f(y,z)). [back_rewrite(1357),rewrite(1602(6),1602(5),1602(10))].
3331
1806 f(f(x,y),f(f(x,y),f(x,f(x,y)))) = f(x,f(x,y)). [back_rewrite(1185),rewrite(1602(3),1602(4),1602(5),1602(9),1602(11),1749(10),1602(9))].
3332
1967 f(f(x,f(x,y)),x) = f(x,y). [back_rewrite(784),rewrite(1602(3),1602(6))].
3333
1969 f(f(f(x,y),x),x) = f(x,y). [back_rewrite(782),rewrite(1602(3),1602(6))].
3334
2122 f(f(x,y),f(y,f(x,z))) = y. [back_rewrite(324),rewrite(1967(3),1602(4))].
3335
2177 f(f(f(x,y),f(z,y)),z) = f(z,y). [back_rewrite(29),rewrite(1602(5),1967(7),1602(7))].
3336
2185 f(f(x,y),f(x,f(z,y))) = x. [back_rewrite(2),rewrite(1602(3))].
3337
2257 f(x,f(f(y,z),f(y,x))) = f(y,x). [back_rewrite(259),rewrite(1967(4),1969(6),1967(7))].
3338
2315 f(f(f(x,y),y),y) = f(x,y). [back_rewrite(206),rewrite(1969(4),2257(4),1969(6))].
3339
2401 f(f(x,y),x) = f(x,f(x,y)). [back_rewrite(1806),rewrite(2185(5))].
3340
2843 f(f(x,y),y) = f(y,f(x,y)). [para(2315(a,1),1602(a,1,2)),flip(a)].
3341
3315 f(x,y) = f(y,x). [para(2401(a,1),2177(a,1,1)),rewrite(2843(3),2122(4))].
3342
3316 $F # answer("Commutativity"). [resolve(3315,a,3,a)].
3059
269 f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = f(x,f(f(f(x,f(f(x,x),x)),x),x)). [para(2(a,1),224(a,1,1,1,2))].
3060
377 f(f(f(x,y),y),y) = f(x,f(f(y,x),x)). [para(4(a,1),62(a,1,1,1,1))].
3061
428 f(f(f(x,f(f(y,x),x)),y),f(y,f(f(x,y),y))) = y. [para(377(a,1),4(a,1,1,1))].
3062
461 f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x. [para(377(a,1),8(a,1,1))].
3063
499 f(x,f(f(f(y,x),x),x)) = f(x,f(y,f(f(x,y),y))). [para(377(a,1),224(a,2,2)),rewrite([224(12)])].
3064
532 f(x,f(f(f(x,f(f(x,x),x)),x),x)) = f(x,f(x,f(f(x,x),x))). [back_rewrite(269),rewrite([428(8)]),flip(a)].
3065
572 f(x,f(f(x,f(f(x,x),x)),x)) = f(x,f(f(x,x),x)). [para(461(a,1),461(a,1,2,2)),rewrite([2(10),461(8)])].
3066
686 f(f(x,f(f(y,x),x)),f(f(y,f(f(f(x,y),y),y)),f(x,f(f(y,x),x)))) = f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))). [para(499(a,2),224(a,2,2,1)),rewrite([224(16)]),flip(a)].
3067
699 f(f(x,f(f(y,x),x)),f(f(y,f(f(f(x,y),y),y)),f(x,f(f(y,x),x)))) = f(f(f(f(x,f(f(y,x),x)),y),y),y). [para(499(a,2),377(a,2,2,1)),flip(a)].
3068
706 f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))) = f(f(f(f(x,f(f(y,x),x)),y),y),y). [back_rewrite(686),rewrite([699(12)]),flip(a)].
3069
778 f(f(f(f(f(x,f(f(x,x),x)),x),x),f(f(x,f(x,f(f(x,x),x))),f(f(f(x,f(f(x,x),x)),x),x))),f(x,f(y,f(f(f(x,f(f(x,x),x)),x),x)))) = x. [para(532(a,1),2(a,1,1,2,1))].
3070
781 f(f(f(f(x,f(f(x,x),x)),x),x),x) = f(f(x,f(f(x,x),x)),x). [para(2(a,1),532(a,1,2,1,1,2,1)),rewrite([461(11),706(12),2(19),461(17)])].
3071
810 f(f(f(f(x,f(f(x,x),x)),x),x),f(f(x,f(x,f(f(x,x),x))),f(f(f(x,f(f(x,x),x)),x),x))) = f(f(x,f(f(x,x),x)),x). [para(532(a,1),377(a,2,2,1)),rewrite([781(6),781(6)]),flip(a)].
3072
813 f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),x)) = f(x,f(f(x,x),x)). [para(532(a,1),428(a,1,1,1)),rewrite([706(20),781(14)])].
3073
819 f(f(f(x,f(f(x,x),x)),x),f(x,f(y,f(f(f(x,f(f(x,x),x)),x),x)))) = x. [back_rewrite(778),rewrite([810(16)])].
3074
1154 f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x. [para(813(a,1),4(a,1,2)),rewrite([706(12),781(6),781(6),428(8)]),flip(a)].
3075
1166 f(f(f(f(x,f(f(x,x),x)),f(x,x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x. [para(813(a,1),9(a,1,2)),rewrite([1154(11),1154(21)])].
3076
1172 f(f(f(x,f(x,f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),x))))),x),x) = f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),x))). [para(813(a,1),224(a,1,1,1,2,2,2,1)),rewrite([1154(8),1154(8),1154(23),1154(24),1154(29),572(26)])].
3077
1178 f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),x))) = f(f(f(x,f(f(x,x),x)),x),x). [para(813(a,1),377(a,2,2,1)),rewrite([1154(12),1154(13),781(6),1154(12)]),flip(a)].
3078
1181 f(x,f(f(f(f(x,x),x),x),x)) = f(x,f(f(x,x),x)). [para(377(a,2),813(a,1,2,1)),rewrite([1154(8)])].
3079
1194 f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)). [back_rewrite(1172),rewrite([1178(13),532(6),224(7),1178(16)]),flip(a)].
3080
1205 f(f(f(x,f(f(x,x),x)),x),f(x,f(y,f(x,f(f(x,x),x))))) = x. [back_rewrite(819),rewrite([1194(9)])].
3081
1212 f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)). [para(1154(a,1),4(a,1,1,1)),rewrite([461(12)])].
3082
1221 f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))). [para(1154(a,1),8(a,1,2)),rewrite([224(7)])].
3083
1251 f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x. [back_rewrite(1205),rewrite([1221(4)])].
3084
1316 f(f(f(f(x,x),x),x),x) = f(x,f(x,f(f(x,x),x))). [para(377(a,2),1221(a,1,1))].
3085
1317 f(x,f(f(x,x),x)) = f(x,x). [para(1221(a,1),461(a,1,2,2)),rewrite([2(10),461(8),461(8)]),flip(a)].
3086
1338 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(1181),rewrite([1316(4),1317(3),1317(6)])].
3087
1403 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [back_rewrite(1251),rewrite([1317(3),1317(5)])].
3088
1424 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(1221),rewrite([1317(3),1317(5)])].
3089
1429 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(1212),rewrite([1424(2),1338(3),1424(5),1338(6)])].
3090
1432 f(f(x,x),f(x,x)) = x. [back_rewrite(1166),rewrite([1424(2),1338(3),1424(5),1338(6),1424(5),1424(7),1338(8),1429(7)])].
3091
1484 f(f(x,x),f(x,f(x,y))) = x. [back_rewrite(461),rewrite([1424(2),1338(3)])].
3092
1496 f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y. [para(1432(a,1),2(a,1,2,2))].
3093
1570 f(f(x,f(x,f(y,f(x,x)))),x) = f(x,x). [para(1403(a,1),4(a,1,1,1)),rewrite([1484(10)])].
3094
1583 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)). [para(1403(a,1),10(a,1,1,1)),rewrite([1338(3),1429(6)])].
3095
1613 f(f(x,f(y,y)),f(x,y)) = x. [back_rewrite(1496),rewrite([1583(6)])].
3096
1634 f(f(x,y),f(x,f(y,y))) = x. [para(2(a,1),1613(a,1,1,2)),rewrite([1424(3),1338(4)])].
3097
1645 f(f(x,x),y) = f(y,f(x,x)). [para(1613(a,1),5(a,1,2)),rewrite([1570(5)])].
3098
1807 f(x,f(f(x,x),x)) = f(x,x). [para(1432(a,1),1634(a,1,1)),rewrite([1432(4)])].
3099
1830 f(x,y) = f(y,x). [para(2(a,1),1645(a,1,1)),rewrite([1807(4),1807(5),1432(4)])].
3100
1831 $F # answer("Commutativity"). [resolve(1830,a,3,a)].
3343
3101
3344
3102
============================== end of proof ==========================
3345
3103
3346
3104
============================== STATISTICS ============================
3347
3105
3348
Given=50. Generated=6194. Kept=3314. proofs=1.
3349
Usable=11. Sos=795. Demods=819. Limbo=14, Disabled=2503. Hints=968.
3350
Weight_deleted=126. Literals_deleted=0.
3351
Forward_subsumed=2754. Back_subsumed=388.
3106
Given=38. Generated=3590. Kept=1829. proofs=1.
3107
Usable=30. Sos=1349. Demods=1331. Limbo=9, Disabled=450. Hints=678.
3108
Kept_by_rule=0, Deleted_by_rule=59.
3109
Forward_subsumed=1702. Back_subsumed=55.
3352
3110
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
3353
New_demodulators=3309 (0 lex), Back_demodulated=2105. Back_unit_deleted=0.
3354
Demod_attempts=153354. Demod_rewrites=12777.
3111
New_demodulators=1752 (0 lex), Back_demodulated=385. Back_unit_deleted=0.
3112
Demod_attempts=97640. Demod_rewrites=6439.
3355
3113
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
3356
3114
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
3357
Megabytes=4.49.
3358
User_CPU=0.82, System_CPU=0.01, Wall_clock=8.
3115
Megabytes=3.50.
3116
User_CPU=0.57, System_CPU=0.00, Wall_clock=5.
3359
3117
3360
3118
============================== end of statistics =====================
3361
3119
3363
3121
3364
3122
Exiting with 1 proof.
3365
3123
3366
Process 13971 exit (max_proofs) Sun Aug 13 21:05:05 2006
3124
Process 10944 exit (max_proofs) Mon Sep 28 11:57:14 2009
3367
3125
3368
Successful proof using the following 8 assumptions:
3126
Successful proof using the following 7 assumptions:
3369
3127
3370
3128
formulas(assumptions_in_proof).
3371
3129
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
3372
3130
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
3373
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3374
3131
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3375
3132
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3376
3133
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3378
3135
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3379
3136
end_of_list.
3380
3137
3381
Including 149 new hint clauses:
3138
Including 125 new hint clauses:
3382
3139
3383
3140
formulas(new_hints).
3384
$F # label("Commutativity").
3141
$F # label("Commutativity") # label(non_clause) # label(goal).
3385
3142
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
3386
3143
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
3387
3144
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
3388
3145
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
3389
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").
3390
3146
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
3391
3147
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
3392
3148
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
3393
3149
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
3394
3150
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
3395
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)).
3396
f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)).
3397
f(f(f(x,y),f(f(f(f(f(x,z),y),y),f(x,y)),f(x,y))),z) = f(f(f(x,z),y),y).
3398
f(f(f(f(x,y),f(f(x,y),f(y,f(f(z,y),y)))),f(x,y)),z) = f(y,f(f(z,y),y)).
3151
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
3152
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
3153
f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = f(x,f(f(f(x,f(f(x,x),x)),x),x)).
3154
f(f(f(x,y),y),y) = f(x,f(f(y,x),x)).
3155
f(f(f(x,f(f(y,x),x)),y),f(y,f(f(x,y),y))) = y.
3156
f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x.
3157
f(f(f(f(x,y),f(f(x,y),f(y,f(f(f(x,y),y),y)))),f(x,y)),f(x,y)) = f(y,f(x,f(f(y,x),x))).
3158
f(x,f(f(f(y,x),x),x)) = f(x,f(y,f(f(x,y),y))).
3159
f(x,f(x,f(f(x,x),x))) = f(x,f(f(f(x,f(f(x,x),x)),x),x)).
3160
f(x,f(f(f(x,f(f(x,x),x)),x),x)) = f(x,f(x,f(f(x,x),x))).
3161
f(f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),x)) = f(x,f(f(x,x),x)).
3162
f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),x)) = f(x,f(f(x,x),x)).
3163
f(x,f(f(x,f(f(x,x),x)),x)) = f(x,f(f(x,x),x)).
3164
f(f(f(x,f(x,f(f(y,f(f(x,y),y)),f(f(x,f(y,f(f(x,y),y))),f(y,f(f(x,y),y)))))),x),x) = f(f(y,f(f(x,y),y)),f(f(x,f(f(f(y,x),x),x)),f(y,f(f(x,y),y)))).
3165
f(f(y,f(f(x,y),y)),f(f(x,f(y,f(f(x,y),y))),f(y,f(f(x,y),y)))) = f(f(y,f(f(x,y),y)),f(f(x,f(f(f(y,x),x),x)),f(y,f(f(x,y),y)))).
3166
f(f(x,f(f(y,x),x)),f(f(y,f(f(f(x,y),y),y)),f(x,f(f(y,x),x)))) = f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))).
3167
f(f(f(f(x,f(f(y,x),x)),y),y),y) = f(f(x,f(f(y,x),x)),f(f(y,f(f(f(x,y),y),y)),f(x,f(f(y,x),x)))).
3168
f(f(x,f(f(y,x),x)),f(f(y,f(f(f(x,y),y),y)),f(x,f(f(y,x),x)))) = f(f(f(f(x,f(f(y,x),x)),y),y),y).
3169
f(f(f(f(x,f(f(y,x),x)),y),y),y) = f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))).
3170
f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))) = f(f(f(f(x,f(f(y,x),x)),y),y),y).
3171
f(f(f(f(f(x,f(f(x,x),x)),x),x),f(f(x,f(x,f(f(x,x),x))),f(f(f(x,f(f(x,x),x)),x),x))),f(x,f(y,f(f(f(x,f(f(x,x),x)),x),x)))) = x.
3172
f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))) = f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))))).
3173
f(f(x,f(f(x,x),x)),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))) = f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))))).
3174
f(f(f(f(x,f(f(x,x),x)),x),x),x) = f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))))).
3175
f(f(f(f(x,f(f(x,x),x)),x),x),x) = f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))).
3176
f(f(f(f(x,f(f(x,x),x)),x),x),x) = f(f(x,f(f(x,x),x)),x).
3177
f(f(f(f(f(f(x,f(f(x,x),x)),x),x),x),x),x) = f(f(f(f(x,f(f(x,x),x)),x),x),f(f(x,f(x,f(f(x,x),x))),f(f(f(x,f(f(x,x),x)),x),x))).
3178
f(f(f(f(x,f(f(x,x),x)),x),x),x) = f(f(f(f(x,f(f(x,x),x)),x),x),f(f(x,f(x,f(f(x,x),x))),f(f(f(x,f(f(x,x),x)),x),x))).
3179
f(f(x,f(f(x,x),x)),x) = f(f(f(f(x,f(f(x,x),x)),x),x),f(f(x,f(x,f(f(x,x),x))),f(f(f(x,f(f(x,x),x)),x),x))).
3180
f(f(f(f(x,f(f(x,x),x)),x),x),f(f(x,f(x,f(f(x,x),x))),f(f(f(x,f(f(x,x),x)),x),x))) = f(f(x,f(f(x,x),x)),x).
3181
f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
3182
f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),f(f(f(f(x,f(f(x,x),x)),x),x),x)) = f(x,f(f(x,x),x)).
3183
f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),x)) = f(x,f(f(x,x),x)).
3184
f(f(f(x,f(f(x,x),x)),x),f(x,f(y,f(f(f(x,f(f(x,x),x)),x),x)))) = x.
3185
f(f(f(f(f(x,f(f(x,x),x)),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))),x),x),f(x,f(f(x,x),x))) = f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))).
3186
f(f(f(f(f(f(f(x,f(f(x,x),x)),x),x),x),x),x),f(x,f(f(x,x),x))) = f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))).
3187
f(f(f(f(f(x,f(f(x,x),x)),x),x),x),f(x,f(f(x,x),x))) = f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))).
3188
f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))) = f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))).
3189
x = f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))).
3190
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
3191
f(f(f(f(x,f(f(x,x),x)),f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))).
3192
f(f(f(f(x,f(f(x,x),x)),f(x,x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))).
3193
f(f(f(f(x,f(f(x,x),x)),f(x,x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
3194
f(f(f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),x))))),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))) = f(f(f(x,f(f(x,x),x)),x),f(f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),x)),f(f(x,f(f(x,x),x)),x))).
3195
f(f(f(x,f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),x))))),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))) = f(f(f(x,f(f(x,x),x)),x),f(f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),x)),f(f(x,f(f(x,x),x)),x))).
3196
f(f(f(x,f(x,f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),x))))),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))) = f(f(f(x,f(f(x,x),x)),x),f(f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),x)),f(f(x,f(f(x,x),x)),x))).
3197
f(f(f(x,f(x,f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),x))))),x),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))) = f(f(f(x,f(f(x,x),x)),x),f(f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),x)),f(f(x,f(f(x,x),x)),x))).
3198
f(f(f(x,f(x,f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),x))))),x),x) = f(f(f(x,f(f(x,x),x)),x),f(f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),x)),f(f(x,f(f(x,x),x)),x))).
3199
f(f(f(x,f(x,f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),x))))),x),x) = f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,f(f(x,x),x)),x)),f(f(x,f(f(x,x),x)),x))).
3200
f(f(f(x,f(x,f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),x))))),x),x) = f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),x))).
3201
f(f(f(f(f(x,f(f(x,x),x)),x),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))) = f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),x))).
3202
f(f(f(f(f(x,f(f(x,x),x)),x),x),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))) = f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),x))).
3203
f(f(f(f(f(x,f(f(x,x),x)),x),x),x),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))) = f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),x))).
3204
f(f(f(x,f(f(x,x),x)),x),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))) = f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),x))).
3205
f(f(f(x,f(f(x,x),x)),x),x) = f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),x))).
3206
f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),x))) = f(f(f(x,f(f(x,x),x)),x),x).
3207
f(f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))),f(f(f(f(x,x),x),x),x)) = f(x,f(f(x,x),x)).
3208
f(x,f(f(f(f(x,x),x),x),x)) = f(x,f(f(x,x),x)).
3209
f(f(f(x,f(x,f(f(f(x,f(f(x,x),x)),x),x))),x),x) = f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),x))).
3210
f(f(f(x,f(x,f(x,f(f(x,x),x)))),x),x) = f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),x))).
3211
f(x,f(f(x,x),x)) = f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),x))).
3212
f(x,f(f(x,x),x)) = f(f(f(x,f(f(x,x),x)),x),x).
3213
f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)).
3214
f(f(f(x,f(f(x,x),x)),x),f(x,f(y,f(x,f(f(x,x),x))))) = x.
3215
f(f(x,f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
3216
f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
3217
f(f(f(f(x,f(x,f(x,f(f(x,x),x)))),x),x),x) = f(x,f(x,f(f(x,x),x))).
3218
f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))).
3219
f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x.
3220
f(f(f(f(x,x),x),x),x) = f(x,f(x,f(f(x,x),x))).
3221
f(f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
3222
f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
3223
f(x,f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
3224
f(x,x) = f(x,f(f(x,x),x)).
3225
f(x,f(f(x,x),x)) = f(x,x).
3226
f(x,f(x,f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
3227
f(x,f(x,f(x,x))) = f(x,f(f(x,x),x)).
3228
f(x,f(x,f(x,x))) = f(x,x).
3229
f(f(x,f(x,x)),f(x,f(y,f(x,f(f(x,x),x))))) = x.
3230
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
3231
f(f(x,x),x) = f(x,f(x,f(f(x,x),x))).
3232
f(f(x,x),x) = f(x,f(x,x)).
3233
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(f(x,x),x)).
3234
f(f(x,f(x,x)),x) = f(x,f(f(x,x),x)).
3235
f(f(x,f(x,x)),x) = f(x,f(x,f(x,x))).
3236
f(f(x,f(x,x)),x) = f(x,x).
3237
f(f(f(f(x,f(x,f(x,x))),f(x,x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
3238
f(f(f(f(x,x),f(x,x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
3239
f(f(f(f(x,x),f(x,x)),f(x,f(x,f(x,x)))),f(x,f(f(x,x),x))) = x.
3240
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,f(f(x,x),x))) = x.
3241
f(f(f(x,x),f(f(x,x),f(x,x))),f(x,f(f(x,x),x))) = x.
3242
f(f(f(x,x),f(f(x,x),f(x,x))),f(x,f(x,f(x,x)))) = x.
3243
f(f(f(x,x),f(f(x,x),f(x,x))),f(x,x)) = x.
3244
f(f(x,x),f(x,x)) = x.
3245
f(f(x,f(x,f(x,x))),f(x,f(x,y))) = x.
3246
f(f(x,x),f(x,f(x,y))) = x.
3247
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
3248
f(f(x,f(x,f(y,f(x,x)))),f(f(x,x),f(x,f(x,f(y,f(x,x)))))) = f(x,x).
3249
f(f(x,f(x,f(y,f(x,x)))),x) = f(x,x).
3250
f(f(x,f(x,f(x,x))),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
3251
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
3252
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
3253
f(f(x,f(y,y)),f(x,y)) = x.
3254
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
3255
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
3256
f(f(x,y),f(x,f(y,y))) = x.
3257
f(f(f(x,f(x,f(y,f(x,x)))),x),y) = f(y,f(x,x)).
3258
f(f(x,x),y) = f(y,f(x,x)).
3259
f(x,f(f(x,x),f(f(x,x),f(x,x)))) = f(x,x).
3260
f(x,f(f(x,x),x)) = f(x,x).
3261
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
3262
f(x,y) = f(y,f(f(x,x),f(x,f(f(x,x),x)))).
3263
f(x,y) = f(y,f(f(x,x),f(x,x))).
3264
f(x,y) = f(y,x).
3265
$F # answer("Commutativity").
3266
end_of_list.
3267
3268
Attempt to eliminate one assumption:
3269
3270
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
3271
3272
***********************************************
3273
AUTO SKETCHES ITERATION 11
3274
***********************************************
3275
3276
Starting a search with 7 assumptions:
3277
3278
formulas(assumptions_active).
3279
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
3280
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3281
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3282
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3283
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3284
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3285
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3286
end_of_list.
3287
3288
Child search process 10945 started.
3289
3290
============================== PROCESS INITIAL CLAUSES ===============
3291
3292
% Clauses before input processing:
3293
3294
formulas(usable).
3295
end_of_list.
3296
3297
formulas(sos).
3298
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3299
f(c2,c1) != f(c1,c2) # label("Commutativity"). [deny(1)].
3300
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
3301
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3302
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3303
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3304
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3305
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3306
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3307
end_of_list.
3308
3309
formulas(demodulators).
3310
end_of_list.
3311
3312
% 803 hints input.
3313
3314
============================== PREDICATE ELIMINATION =================
3315
3316
No predicates eliminated.
3317
3318
============================== end predicate elimination =============
3319
3320
Auto_denials:
3321
% copying label "Commutativity" to answer in negative clause
3322
3323
Term ordering decisions:
3324
Function symbol KB weights: c1=1. c2=1. f=1.
3325
Predicate symbol precedence: predicate_order([ = ]).
3326
Function symbol precedence: function_order([ c1, c2, f ]).
3327
Skipping inverse_order, because term ordering is KBO.
3328
Unfolding symbols: (none).
3329
3330
Auto_inference settings:
3331
% set(paramodulation). % (positive equality literals)
3332
3333
Auto_process settings: (no changes).
3334
3335
kept: 2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3336
kept: 3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
3337
kept: 4 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
3338
kept: 5 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3339
kept: 6 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3340
kept: 7 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3341
kept: 8 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3342
kept: 9 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3343
kept: 10 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3344
3345
============================== end of process initial clauses ========
3346
3347
============================== CLAUSES FOR SEARCH ====================
3348
3349
% Clauses after input processing:
3350
3351
formulas(usable).
3352
end_of_list.
3353
3354
formulas(sos).
3355
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3356
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
3357
4 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
3358
5 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3359
6 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3360
7 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3361
8 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3362
9 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3363
10 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3364
end_of_list.
3365
3366
formulas(demodulators).
3367
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3368
4 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
3369
5 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3370
6 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3371
7 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3372
8 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3373
9 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3374
10 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3375
end_of_list.
3376
3377
% 343 hints (803 processed, 460 redundant).
3378
3379
============================== end of clauses for search =============
3380
3381
============================== SEARCH ================================
3382
3383
% Starting search at 0.03 seconds.
3384
3385
% Operation f is commutative; C redundancy checks enabled.
3386
3387
============================== PROOF =================================
3388
3389
% Proof 1 at 0.30 (+ 0.00) seconds: "Commutativity".
3390
% Length of proof is 32.
3391
% Level of proof is 15.
3392
% Maximum clause weight is 25.
3393
% Given clauses 34.
3394
3395
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
3396
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3397
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
3398
4 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
3399
5 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3400
6 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3401
7 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3402
9 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3403
23 f(f(f(x,y),x),f(f(y,f(f(y,y),y)),f(z,x))) = f(y,f(f(y,y),y)). [para(2(a,1),5(a,1,1,1,2))].
3404
39 f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)). [para(2(a,1),6(a,1,2))].
3405
305 f(f(f(x,y),y),y) = f(x,f(f(y,x),x)). [para(4(a,1),39(a,1,1,1,1))].
3406
375 f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x. [para(305(a,1),7(a,1,1))].
3407
454 f(f(f(f(x,y),x),f(x,y)),x) = f(x,f(f(x,x),x)). [para(375(a,1),5(a,1,2)),rewrite([2(8)])].
3408
549 f(x,f(f(x,x),x)) = f(x,x). [para(4(a,1),454(a,1,1)),flip(a)].
3409
582 f(f(f(x,x),x),x) = f(x,x). [para(454(a,1),39(a,1,1,1)),rewrite([549(3),549(6)])].
3410
634 f(f(f(x,y),x),f(f(y,y),f(z,x))) = f(y,y). [back_rewrite(23),rewrite([549(5),549(9)])].
3411
637 f(f(x,x),f(x,f(y,x))) = x. [para(549(a,1),2(a,1,1))].
3412
639 f(f(x,x),f(x,x)) = x. [para(549(a,1),2(a,1,2)),rewrite([549(3)])].
3413
646 f(f(x,f(x,x)),f(x,x)) = x. [para(549(a,1),6(a,1,2)),rewrite([639(3)])].
3414
666 f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y. [para(639(a,1),2(a,1,2,2))].
3415
667 f(f(x,f(x,x)),x) = f(x,x). [para(639(a,1),2(a,1,2)),rewrite([582(3)])].
3416
669 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [para(639(a,1),5(a,1,1,1))].
3417
737 f(f(x,x),x) = f(x,f(x,x)). [para(646(a,1),2(a,1,2)),rewrite([667(3),549(3)])].
3418
757 f(x,f(x,f(x,x))) = f(x,x). [para(646(a,1),637(a,1,2,2)),rewrite([639(3),737(2)])].
3419
920 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)). [para(669(a,1),9(a,1,1,1)),rewrite([757(3),667(6)])].
3420
940 f(f(x,f(y,y)),f(x,y)) = x. [back_rewrite(666),rewrite([920(6)])].
3421
961 f(f(f(x,f(x,f(y,f(x,x)))),x),y) = f(y,f(x,x)). [para(940(a,1),4(a,1,2))].
3422
1151 f(f(f(f(x,x),y),y),y) = f(y,f(x,x)). [para(634(a,1),39(a,1,1,1,1)),rewrite([920(10)])].
3423
1158 f(x,f(y,f(x,x))) = f(x,x). [para(305(a,2),634(a,1,2)),rewrite([737(2),940(4),1151(4)])].
3424
1187 f(f(x,x),y) = f(y,f(x,x)). [back_rewrite(961),rewrite([1158(3),667(3)])].
3425
1199 f(x,y) = f(y,x). [para(2(a,1),1187(a,1,1)),rewrite([2(8)])].
3426
1200 $F # answer("Commutativity"). [resolve(1199,a,3,a)].
3427
3428
============================== end of proof ==========================
3429
3430
============================== STATISTICS ============================
3431
3432
Given=34. Generated=2340. Kept=1198. proofs=1.
3433
Usable=26. Sos=890. Demods=883. Limbo=9, Disabled=281. Hints=803.
3434
Kept_by_rule=0, Deleted_by_rule=0.
3435
Forward_subsumed=1142. Back_subsumed=54.
3436
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
3437
New_demodulators=1135 (0 lex), Back_demodulated=218. Back_unit_deleted=0.
3438
Demod_attempts=47628. Demod_rewrites=3579.
3439
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
3440
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
3441
Megabytes=2.24.
3442
User_CPU=0.30, System_CPU=0.00, Wall_clock=6.
3443
3444
============================== end of statistics =====================
3445
3446
============================== end of search =========================
3447
3448
Exiting with 1 proof.
3449
3450
Process 10945 exit (max_proofs) Mon Sep 28 11:57:15 2009
3451
3452
Successful proof using the following 5 assumptions:
3453
3454
formulas(assumptions_in_proof).
3455
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
3456
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3457
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3458
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3459
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3460
end_of_list.
3461
3462
Including 52 new hint clauses:
3463
3464
formulas(new_hints).
3465
$F # label("Commutativity") # label(non_clause) # label(goal).
3466
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
3467
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
3468
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
3469
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").
3470
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
3471
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
3472
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
3399
3473
f(f(f(x,y),x),f(f(y,f(f(y,y),y)),f(z,x))) = f(y,f(f(y,y),y)).
3400
f(f(f(x,f(f(y,z),f(f(f(y,x),y),y))),x),x) = f(f(f(y,x),y),y).
3401
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
3402
f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(y,f(y,x)),y).
3403
f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y.
3404
f(f(f(x,f(x,y)),x),f(y,f(z,f(f(x,z),z)))) = y.
3405
f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(y,z))) = f(y,f(x,f(f(y,x),x))).
3406
f(f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
3407
f(f(f(x,f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
3408
f(f(f(x,f(x,f(f(x,x),x))),x),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
3409
f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
3410
f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(f(x,f(f(x,x),x)),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
3411
f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(x,f(x,f(f(x,x),x))).
3412
f(f(f(f(x,f(x,f(y,f(f(x,y),y)))),f(f(x,f(x,f(y,f(f(x,y),y)))),f(x,f(y,f(f(x,y),y))))),f(x,f(x,f(y,f(f(x,y),y))))),f(x,f(x,f(y,f(f(x,y),y))))) = f(x,f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x)).
3413
f(f(f(f(x,f(x,f(y,f(f(x,y),y)))),f(f(x,f(x,f(y,f(f(x,y),y)))),f(x,f(y,f(f(x,y),y))))),f(x,f(x,f(y,f(f(x,y),y))))),f(x,f(x,f(y,f(f(x,y),y))))) = f(x,f(y,f(f(x,y),y))).
3414
f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),x) = f(x,f(f(x,x),x)).
3474
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
3475
f(f(f(x,y),y),y) = f(x,f(f(y,x),x)).
3476
f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x.
3477
f(f(f(f(x,y),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))),f(x,y)),x) = f(x,f(f(x,x),x)).
3478
f(f(f(f(x,y),x),f(x,y)),x) = f(x,f(f(x,x),x)).
3415
3479
f(x,x) = f(x,f(f(x,x),x)).
3416
3480
f(x,f(f(x,x),x)) = f(x,x).
3417
f(f(x,f(f(f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))),x),x)),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
3418
f(f(x,f(f(f(f(y,y),f(y,f(y,f(f(y,y),y)))),x),x)),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
3419
f(f(x,f(f(f(f(y,y),f(y,f(y,y))),x),x)),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
3420
f(f(x,f(f(f(f(y,y),f(y,f(y,y))),x),x)),f(y,y)) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
3421
f(f(x,f(f(f(f(y,y),f(y,f(y,y))),x),x)),f(y,y)) = f(f(y,y),f(y,f(y,f(f(y,y),y)))).
3422
f(f(x,f(f(f(f(y,y),f(y,f(y,y))),x),x)),f(y,y)) = f(f(y,y),f(y,f(y,y))).
3423
f(f(f(x,f(x,x)),x),x) = f(x,f(x,f(f(x,x),x))).
3424
f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)).
3481
f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)).
3482
f(f(f(x,x),x),x) = f(x,f(f(x,x),x)).
3483
f(f(f(x,x),x),x) = f(x,x).
3425
3484
f(f(f(x,y),x),f(f(y,y),f(z,x))) = f(y,f(f(y,y),y)).
3426
3485
f(f(f(x,y),x),f(f(y,y),f(z,x))) = f(y,y).
3427
f(f(f(f(x,x),x),f(f(x,x),f(f(x,x),x))),f(x,f(y,f(f(x,x),x)))) = x.
3428
f(f(x,x),f(x,f(y,f(f(x,x),x)))) = x.
3429
3486
f(f(x,x),f(x,f(y,x))) = x.
3430
3487
f(f(x,f(f(x,x),x)),f(x,x)) = x.
3431
3488
f(f(x,x),f(x,x)) = x.
3432
3489
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,x)) = x.
3433
3490
f(f(x,f(x,x)),f(x,x)) = x.
3434
f(f(f(x,x),x),f(f(x,x),f(x,x))) = f(x,x).
3491
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
3492
f(f(x,f(f(f(x,x),x),x)),x) = f(x,x).
3493
f(f(x,f(x,x)),x) = f(x,x).
3494
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
3495
f(f(x,f(f(f(x,f(x,x)),x),x)),x) = f(x,f(x,x)).
3496
f(f(x,f(f(x,x),x)),x) = f(x,f(x,x)).
3497
f(f(x,x),x) = f(x,f(x,x)).
3498
f(f(f(x,x),f(x,x)),f(f(x,x),x)) = f(x,x).
3499
f(x,f(f(x,x),x)) = f(x,x).
3500
f(x,f(x,f(x,x))) = f(x,x).
3501
f(f(x,f(x,f(x,x))),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
3502
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
3503
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
3504
f(f(x,f(y,y)),f(x,y)) = x.
3505
f(f(f(x,f(x,f(y,f(x,x)))),x),y) = f(y,f(x,x)).
3506
f(f(f(f(x,x),y),y),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
3507
f(f(f(f(x,x),y),y),y) = f(y,f(x,x)).
3508
f(f(f(f(x,x),x),f(x,x)),f(f(f(f(x,x),y),y),y)) = f(x,x).
3509
f(f(f(x,f(x,x)),f(x,x)),f(f(f(f(x,x),y),y),y)) = f(x,x).
3510
f(x,f(f(f(f(x,x),y),y),y)) = f(x,x).
3511
f(x,f(y,f(x,x))) = f(x,x).
3512
f(f(f(x,f(x,x)),x),y) = f(y,f(x,x)).
3513
f(f(x,x),y) = f(y,f(x,x)).
3514
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
3515
f(x,y) = f(y,x).
3516
$F # answer("Commutativity").
3517
end_of_list.
3518
3519
Attempt to eliminate one assumption:
3520
3521
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
3522
3523
***********************************************
3524
AUTO SKETCHES ITERATION 12
3525
***********************************************
3526
3527
Starting a search with 6 assumptions:
3528
3529
formulas(assumptions_active).
3530
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3531
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3532
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3533
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3534
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3535
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3536
end_of_list.
3537
3538
Child search process 10946 started.
3539
3540
============================== PROCESS INITIAL CLAUSES ===============
3541
3542
% Clauses before input processing:
3543
3544
formulas(usable).
3545
end_of_list.
3546
3547
formulas(sos).
3548
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3549
f(c2,c1) != f(c1,c2) # label("Commutativity"). [deny(1)].
3550
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3551
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3552
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3553
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3554
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3555
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3556
end_of_list.
3557
3558
formulas(demodulators).
3559
end_of_list.
3560
3561
% 855 hints input.
3562
3563
============================== PREDICATE ELIMINATION =================
3564
3565
No predicates eliminated.
3566
3567
============================== end predicate elimination =============
3568
3569
Auto_denials:
3570
% copying label "Commutativity" to answer in negative clause
3571
3572
Term ordering decisions:
3573
Function symbol KB weights: c1=1. c2=1. f=1.
3574
Predicate symbol precedence: predicate_order([ = ]).
3575
Function symbol precedence: function_order([ c1, c2, f ]).
3576
Skipping inverse_order, because term ordering is KBO.
3577
Unfolding symbols: (none).
3578
3579
Auto_inference settings:
3580
% set(paramodulation). % (positive equality literals)
3581
3582
Auto_process settings: (no changes).
3583
3584
kept: 2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3585
kept: 3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
3586
kept: 4 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3587
kept: 5 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3588
kept: 6 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3589
kept: 7 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3590
kept: 8 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3591
kept: 9 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3592
3593
============================== end of process initial clauses ========
3594
3595
============================== CLAUSES FOR SEARCH ====================
3596
3597
% Clauses after input processing:
3598
3599
formulas(usable).
3600
end_of_list.
3601
3602
formulas(sos).
3603
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3604
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
3605
4 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3606
5 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3607
6 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3608
7 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3609
8 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3610
9 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3611
end_of_list.
3612
3613
formulas(demodulators).
3614
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3615
4 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3616
5 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3617
6 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3618
7 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3619
8 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3620
9 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3621
end_of_list.
3622
3623
% 361 hints (855 processed, 494 redundant).
3624
3625
============================== end of clauses for search =============
3626
3627
============================== SEARCH ================================
3628
3629
% Starting search at 0.03 seconds.
3630
3631
% Operation f is commutative; C redundancy checks enabled.
3632
3633
============================== PROOF =================================
3634
3635
% Proof 1 at 0.32 (+ 0.00) seconds: "Commutativity".
3636
% Length of proof is 44.
3637
% Level of proof is 18.
3638
% Maximum clause weight is 39.
3639
% Given clauses 34.
3640
3641
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
3642
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3643
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
3644
4 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3645
5 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3646
6 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3647
8 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3648
9 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3649
15 f(f(f(x,y),x),f(f(y,f(f(y,y),y)),f(z,x))) = f(y,f(f(y,y),y)). [para(2(a,1),4(a,1,1,1,2))].
3650
27 f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)). [para(2(a,1),5(a,1,2))].
3651
119 f(f(f(x,f(f(y,z),f(f(f(y,x),y),y))),x),x) = f(f(f(y,x),y),y). [para(6(a,1),8(a,1,2))].
3652
162 f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(y,f(y,x)),y). [para(9(a,1),8(a,1,2))].
3653
302 f(f(f(x,y),y),y) = f(x,f(f(y,x),x)). [para(4(a,1),27(a,1,1,1,1))].
3654
366 f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x. [para(302(a,1),6(a,1,1))].
3655
425 f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),f(f(f(f(x,f(f(x,x),x)),y),y),y)) = f(x,f(f(x,x),x)). [para(302(a,2),15(a,1,2))].
3656
450 f(f(f(f(x,y),x),f(x,y)),x) = f(x,f(f(x,x),x)). [para(366(a,1),4(a,1,2)),rewrite([2(8)])].
3657
547 f(x,f(f(x,x),x)) = f(x,x). [para(4(a,1),450(a,1,1)),flip(a)].
3658
580 f(f(f(x,x),x),x) = f(x,x). [para(450(a,1),27(a,1,1,1)),rewrite([547(3),547(6)])].
3659
637 f(f(f(f(x,x),x),f(x,x)),f(f(f(f(x,x),y),y),y)) = f(x,x). [back_rewrite(425),rewrite([547(3),547(5),547(7),547(12)])].
3660
694 f(f(x,x),f(x,f(y,x))) = x. [para(547(a,1),2(a,1,1))].
3661
696 f(f(x,x),f(x,x)) = x. [para(547(a,1),2(a,1,2)),rewrite([547(3)])].
3662
701 f(f(x,f(x,x)),f(x,x)) = x. [para(547(a,1),5(a,1,2)),rewrite([696(3)])].
3663
719 f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y. [para(696(a,1),2(a,1,2,2))].
3664
720 f(f(x,f(x,x)),x) = f(x,x). [para(696(a,1),2(a,1,2)),rewrite([580(3)])].
3665
721 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [para(696(a,1),4(a,1,1,1))].
3666
783 f(f(x,x),x) = f(x,f(x,x)). [para(701(a,1),2(a,1,2)),rewrite([720(3),547(3)])].
3667
803 f(x,f(x,f(x,x))) = f(x,x). [para(701(a,1),694(a,1,2,2)),rewrite([696(3),783(2)])].
3668
835 f(x,f(f(f(f(x,x),y),y),y)) = f(x,x). [back_rewrite(637),rewrite([783(2),701(4)])].
3669
946 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)). [para(721(a,1),8(a,1,1,1)),rewrite([803(3),720(6)])].
3670
966 f(f(x,f(y,y)),f(x,y)) = x. [back_rewrite(719),rewrite([946(6)])].
3671
979 f(f(x,y),f(x,f(y,y))) = x. [para(2(a,1),966(a,1,1,2)),rewrite([783(3),803(4)])].
3672
1052 f(f(f(f(f(x,x),f(y,x)),y),y),y) = f(y,x). [para(979(a,1),5(a,1,2))].
3673
1059 f(f(f(f(f(x,f(x,y)),x),x),f(x,y)),f(x,y)) = f(f(f(x,f(x,y)),x),x). [para(6(a,1),979(a,1,2))].
3674
1066 f(f(f(x,f(f(y,y),f(x,y))),x),x) = f(x,y). [para(979(a,1),8(a,1,2))].
3675
1071 f(f(f(x,f(x,f(x,y))),x),x) = f(x,y). [para(979(a,1),9(a,1,2))].
3676
1148 f(x,f(y,f(x,x))) = f(x,x). [para(302(a,1),835(a,1,2)),rewrite([946(6)])].
3677
1177 f(f(x,x),f(y,x)) = x. [para(2(a,1),1148(a,1,2,2)),rewrite([783(2),1148(3),783(5),1148(6),783(6),1148(7),696(6)])].
3678
1183 f(f(f(x,y),x),x) = f(x,y). [back_rewrite(1066),rewrite([1177(3)])].
3679
1184 f(f(f(x,y),y),y) = f(y,x). [back_rewrite(1052),rewrite([1177(3)])].
3680
1189 f(x,f(x,f(x,y))) = f(x,y). [back_rewrite(1071),rewrite([1183(5)])].
3681
1190 f(f(x,y),x) = f(x,f(x,y)). [back_rewrite(1059),rewrite([1183(4),1184(6),1183(6)])].
3682
1244 f(x,f(f(y,z),f(y,x))) = f(y,x). [back_rewrite(162),rewrite([1190(4),1189(4),1190(5),1190(6),1189(6),1190(7),1189(7)])].
3683
1253 f(x,y) = f(y,x). [back_rewrite(119),rewrite([1190(3),1190(4),1189(4),1244(4),1184(3),1190(3),1190(4),1189(4)])].
3684
1254 $F # answer("Commutativity"). [resolve(1253,a,3,a)].
3685
3686
============================== end of proof ==========================
3687
3688
============================== STATISTICS ============================
3689
3690
Given=34. Generated=2426. Kept=1252. proofs=1.
3691
Usable=23. Sos=727. Demods=786. Limbo=70, Disabled=439. Hints=855.
3692
Kept_by_rule=0, Deleted_by_rule=2.
3693
Forward_subsumed=1172. Back_subsumed=57.
3694
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
3695
New_demodulators=1189 (0 lex), Back_demodulated=374. Back_unit_deleted=0.
3696
Demod_attempts=52484. Demod_rewrites=4414.
3697
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
3698
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
3699
Megabytes=2.26.
3700
User_CPU=0.32, System_CPU=0.00, Wall_clock=6.
3701
3702
============================== end of statistics =====================
3703
3704
============================== end of search =========================
3705
3706
Exiting with 1 proof.
3707
3708
Process 10946 exit (max_proofs) Mon Sep 28 11:57:15 2009
3709
3710
Successful proof using the following 5 assumptions:
3711
3712
formulas(assumptions_in_proof).
3713
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3714
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3715
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3716
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3717
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3718
end_of_list.
3719
3720
Including 86 new hint clauses:
3721
3722
formulas(new_hints).
3723
$F # label("Commutativity") # label(non_clause) # label(goal).
3724
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
3725
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
3726
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").
3727
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
3728
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
3729
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
3730
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
3731
f(f(f(x,y),x),f(f(y,f(f(y,y),y)),f(z,x))) = f(y,f(f(y,y),y)).
3732
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
3733
f(f(f(x,f(f(y,z),f(f(f(y,x),y),y))),x),x) = f(f(f(y,x),y),y).
3734
f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(y,f(y,x)),y).
3735
f(f(f(x,y),y),y) = f(x,f(f(y,x),x)).
3736
f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x.
3737
f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),f(f(f(f(x,f(f(x,x),x)),y),y),y)) = f(x,f(f(x,x),x)).
3738
f(f(f(f(x,y),f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))),f(x,y)),x) = f(x,f(f(x,x),x)).
3739
f(f(f(f(x,y),x),f(x,y)),x) = f(x,f(f(x,x),x)).
3740
f(x,x) = f(x,f(f(x,x),x)).
3741
f(x,f(f(x,x),x)) = f(x,x).
3742
f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)).
3743
f(f(f(x,x),x),x) = f(x,f(f(x,x),x)).
3435
3744
f(f(f(x,x),x),x) = f(x,x).
3436
f(f(x,f(f(y,x),x)),f(y,y)) = f(f(y,y),f(y,f(y,y))).
3437
f(f(x,f(f(y,x),x)),f(y,y)) = y.
3745
f(f(f(f(x,x),x),f(x,f(f(x,x),x))),f(f(f(f(x,f(f(x,x),x)),y),y),y)) = f(x,f(f(x,x),x)).
3746
f(f(f(f(x,x),x),f(x,x)),f(f(f(f(x,f(f(x,x),x)),y),y),y)) = f(x,f(f(x,x),x)).
3747
f(f(f(f(x,x),x),f(x,x)),f(f(f(f(x,x),y),y),y)) = f(x,f(f(x,x),x)).
3748
f(f(f(f(x,x),x),f(x,x)),f(f(f(f(x,x),y),y),y)) = f(x,x).
3749
f(f(x,x),f(x,f(y,x))) = x.
3750
f(f(x,f(f(x,x),x)),f(x,x)) = x.
3751
f(f(x,x),f(x,x)) = x.
3752
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,x)) = x.
3753
f(f(x,f(x,x)),f(x,x)) = x.
3754
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
3438
3755
f(f(x,f(f(f(x,x),x),x)),x) = f(x,x).
3439
3756
f(f(x,f(x,x)),x) = f(x,x).
3440
3757
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
3758
f(f(x,f(f(f(x,f(x,x)),x),x)),x) = f(x,f(x,x)).
3759
f(f(x,f(f(x,x),x)),x) = f(x,f(x,x)).
3441
3760
f(f(x,x),x) = f(x,f(x,x)).
3442
f(f(x,x),f(x,f(y,f(x,f(x,x))))) = x.
3761
f(f(f(x,x),f(x,x)),f(f(x,x),x)) = f(x,x).
3762
f(x,f(f(x,x),x)) = f(x,x).
3443
3763
f(x,f(x,f(x,x))) = f(x,x).
3444
f(f(x,f(f(x,f(x,x)),f(f(x,f(x,x)),f(x,x)))),f(f(x,f(f(f(x,f(x,x)),x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
3445
f(f(x,f(f(x,f(x,x)),x)),f(f(x,f(f(f(x,f(x,x)),x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
3446
f(f(x,f(x,x)),f(f(x,f(f(f(x,f(x,x)),x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
3447
f(f(x,f(x,x)),f(f(x,f(f(x,x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
3448
f(f(x,f(x,x)),f(f(x,f(x,f(x,x))),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
3449
f(f(x,f(x,x)),f(f(x,x),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
3450
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,f(f(f(x,f(x,x)),x),x)).
3451
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,f(f(x,x),x)).
3452
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,f(x,f(x,x))).
3453
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,x).
3454
f(f(x,x),f(x,f(x,y))) = x.
3455
f(f(x,f(x,x)),f(f(x,y),f(f(x,x),x))) = f(x,y).
3456
f(f(x,f(x,x)),f(f(x,y),f(x,f(x,x)))) = f(x,y).
3457
f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)).
3458
f(f(f(x,f(f(y,f(f(x,y),y)),f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
3459
f(f(f(x,x),f(x,f(x,x))),f(f(y,f(f(x,y),y)),f(y,f(f(x,y),y)))) = f(y,f(f(x,y),y)).
3460
f(x,f(f(y,f(f(x,y),y)),f(y,f(f(x,y),y)))) = f(y,f(f(x,y),y)).
3461
f(f(f(x,f(f(y,x),x)),y),y) = f(x,f(f(y,x),x)).
3462
f(f(x,f(y,f(f(x,y),y))),x) = f(y,f(f(x,y),y)).
3463
f(f(x,f(x,f(y,f(x,x)))),f(f(x,x),f(x,f(x,f(y,f(x,x)))))) = f(x,x).
3464
f(f(x,f(x,f(y,f(x,x)))),x) = f(x,x).
3465
f(f(f(f(x,f(x,x)),f(f(x,f(x,x)),f(f(x,f(y,f(x,x))),f(x,f(x,f(y,f(x,x))))))),f(x,f(x,x))),f(x,f(x,x))) = f(f(x,f(y,f(x,x))),f(f(f(x,f(x,x)),f(x,f(y,f(x,x)))),f(x,f(y,f(x,x))))).
3466
f(f(f(f(x,f(x,x)),f(f(x,f(x,x)),f(f(x,f(y,f(x,x))),f(x,f(x,f(y,f(x,x))))))),f(x,f(x,x))),f(x,f(x,x))) = f(f(x,f(y,f(x,x))),f(x,f(x,f(y,f(x,x))))).
3467
f(f(f(x,f(x,x)),f(f(y,f(x,f(x,x))),f(x,f(x,x)))),f(y,f(f(x,f(z,f(x,x))),f(x,f(x,f(z,f(x,x))))))) = y.
3468
f(f(x,f(x,x)),f(f(y,f(x,f(x,x))),f(f(x,x),x))) = f(y,f(x,f(x,x))).
3469
f(f(x,f(x,x)),f(f(y,f(x,f(x,x))),f(x,f(x,x)))) = f(y,f(x,f(x,x))).
3470
f(f(x,f(y,f(y,y))),f(x,f(f(y,f(z,f(y,y))),f(y,f(y,f(z,f(y,y))))))) = x.
3471
f(f(f(x,x),f(f(f(y,f(f(x,y),y)),f(x,x)),f(x,x))),f(x,f(y,f(f(x,y),y)))) = f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x).
3472
f(f(f(x,x),f(x,f(x,x))),f(x,f(y,f(f(x,y),y)))) = f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x).
3473
f(x,f(x,f(y,f(f(x,y),y)))) = f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x).
3474
f(x,f(x,f(y,f(f(x,y),y)))) = f(y,f(f(x,y),y)).
3475
f(f(f(f(y,f(f(x,y),y)),f(f(x,f(x,f(y,f(f(x,y),y)))),f(x,f(y,f(f(x,y),y))))),f(x,f(x,f(y,f(f(x,y),y))))),f(x,f(x,f(y,f(f(x,y),y))))) = f(x,f(y,f(f(x,y),y))).
3476
f(f(f(f(y,f(f(x,y),y)),f(f(y,f(f(x,y),y)),f(x,f(y,f(f(x,y),y))))),f(x,f(x,f(y,f(f(x,y),y))))),f(x,f(x,f(y,f(f(x,y),y))))) = f(x,f(y,f(f(x,y),y))).
3477
f(f(f(f(y,f(f(x,y),y)),f(f(y,f(f(x,y),y)),f(x,f(y,f(f(x,y),y))))),f(y,f(f(x,y),y))),f(x,f(x,f(y,f(f(x,y),y))))) = f(x,f(y,f(f(x,y),y))).
3478
f(f(f(f(x,f(f(y,x),x)),f(f(x,f(f(y,x),x)),f(y,f(x,f(f(y,x),x))))),f(x,f(f(y,x),x))),f(x,f(f(y,x),x))) = f(y,f(x,f(f(y,x),x))).
3479
f(f(f(x,x),f(f(f(f(x,x),x),f(x,x)),f(x,x))),f(x,f(y,f(x,x)))) = f(f(f(x,f(x,f(y,f(x,x)))),x),x).
3480
f(f(f(x,x),f(f(f(x,f(x,x)),f(x,x)),f(x,x))),f(x,f(y,f(x,x)))) = f(f(f(x,f(x,f(y,f(x,x)))),x),x).
3481
f(f(f(x,x),f(x,f(x,x))),f(x,f(y,f(x,x)))) = f(f(f(x,f(x,f(y,f(x,x)))),x),x).
3482
f(x,f(x,f(y,f(x,x)))) = f(f(f(x,f(x,f(y,f(x,x)))),x),x).
3483
f(x,f(x,f(y,f(x,x)))) = f(f(x,x),x).
3484
f(x,f(x,f(y,f(x,x)))) = f(x,f(x,x)).
3485
f(f(f(f(x,f(x,x)),f(f(x,f(x,x)),f(f(x,f(y,f(x,x))),f(x,f(x,x))))),f(x,f(x,x))),f(x,f(x,x))) = f(f(x,f(y,f(x,x))),f(x,f(x,f(y,f(x,x))))).
3486
f(f(f(f(x,f(x,x)),f(x,f(y,f(x,x)))),f(x,f(x,x))),f(x,f(x,x))) = f(f(x,f(y,f(x,x))),f(x,f(x,f(y,f(x,x))))).
3487
f(f(x,f(x,f(x,x))),f(x,f(x,x))) = f(f(x,f(y,f(x,x))),f(x,f(x,f(y,f(x,x))))).
3488
f(f(x,x),f(x,f(x,x))) = f(f(x,f(y,f(x,x))),f(x,f(x,f(y,f(x,x))))).
3489
x = f(f(x,f(y,f(x,x))),f(x,f(x,f(y,f(x,x))))).
3490
x = f(f(x,f(y,f(x,x))),f(x,f(x,x))).
3491
f(f(x,f(y,f(x,x))),f(x,f(x,x))) = x.
3492
f(f(x,f(y,f(y,y))),f(x,f(f(y,f(z,f(y,y))),f(y,f(y,y))))) = x.
3493
f(f(x,f(y,f(y,y))),f(x,y)) = x.
3494
f(f(x,f(f(f(f(x,y),f(f(z,f(x,y)),f(x,y))),f(f(x,f(y,f(y,y))),f(x,y))),f(f(x,f(y,f(y,y))),f(x,y)))),z) = f(f(x,y),f(f(z,f(x,y)),f(x,y))).
3495
f(f(x,f(f(f(f(x,y),f(f(z,f(x,y)),f(x,y))),x),f(f(x,f(y,f(y,y))),f(x,y)))),z) = f(f(x,y),f(f(z,f(x,y)),f(x,y))).
3496
f(f(x,f(f(f(f(x,y),f(f(z,f(x,y)),f(x,y))),x),x)),z) = f(f(x,y),f(f(z,f(x,y)),f(x,y))).
3497
f(f(f(x,x),f(x,f(x,x))),f(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x).
3498
f(x,f(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x).
3499
f(x,f(f(y,x),x)) = f(y,x).
3500
f(f(x,f(f(f(z,f(x,y)),x),x)),z) = f(f(x,y),f(f(z,f(x,y)),f(x,y))).
3501
f(f(f(z,f(x,y)),x),z) = f(f(x,y),f(f(z,f(x,y)),f(x,y))).
3502
f(f(f(x,f(y,z)),y),x) = f(x,f(y,z)).
3503
f(f(f(f(y,x),f(f(x,f(f(y,x),x)),f(y,f(x,f(f(y,x),x))))),f(x,f(f(y,x),x))),f(x,f(f(y,x),x))) = f(y,f(x,f(f(y,x),x))).
3504
f(f(f(f(y,x),f(f(y,x),f(y,f(x,f(f(y,x),x))))),f(x,f(f(y,x),x))),f(x,f(f(y,x),x))) = f(y,f(x,f(f(y,x),x))).
3505
f(f(f(f(y,x),f(f(y,x),f(y,f(y,x)))),f(x,f(f(y,x),x))),f(x,f(f(y,x),x))) = f(y,f(x,f(f(y,x),x))).
3506
f(f(f(f(y,x),f(f(y,x),f(y,f(y,x)))),f(y,x)),f(x,f(f(y,x),x))) = f(y,f(x,f(f(y,x),x))).
3507
f(f(f(f(y,x),f(f(y,x),f(y,f(y,x)))),f(y,x)),f(y,x)) = f(y,f(x,f(f(y,x),x))).
3508
f(f(y,x),f(f(y,x),f(y,f(y,x)))) = f(y,f(x,f(f(y,x),x))).
3509
f(f(x,y),f(f(x,y),f(x,f(x,y)))) = f(x,f(x,y)).
3510
f(f(x,f(x,y)),x) = f(y,f(f(x,y),y)).
3511
f(f(x,f(x,y)),x) = f(x,y).
3512
f(f(f(y,x),y),y) = f(x,f(f(y,x),x)).
3764
f(f(f(x,f(x,x)),f(x,x)),f(f(f(f(x,x),y),y),y)) = f(x,x).
3765
f(x,f(f(f(f(x,x),y),y),y)) = f(x,x).
3766
f(f(x,f(x,f(x,x))),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
3767
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
3768
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
3769
f(f(x,f(y,y)),f(x,y)) = x.
3770
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
3771
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
3772
f(f(x,y),f(x,f(y,y))) = x.
3773
f(f(f(f(f(x,x),f(y,x)),y),y),y) = f(y,x).
3774
f(f(f(f(f(x,f(x,y)),x),x),f(x,y)),f(x,y)) = f(f(f(x,f(x,y)),x),x).
3775
f(f(f(x,f(f(y,y),f(x,y))),x),x) = f(x,y).
3776
f(f(f(x,f(x,f(x,y))),x),x) = f(x,y).
3777
f(x,f(f(x,x),f(f(y,f(x,x)),f(x,x)))) = f(x,x).
3778
f(x,f(y,f(x,x))) = f(x,x).
3779
f(f(x,f(f(x,x),x)),f(y,x)) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
3780
f(f(x,f(x,f(x,x))),f(y,x)) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
3781
f(f(x,x),f(y,x)) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
3782
f(f(x,x),f(y,x)) = f(f(x,f(x,f(x,x))),f(x,f(f(x,x),x))).
3783
f(f(x,x),f(y,x)) = f(f(x,x),f(x,f(f(x,x),x))).
3784
f(f(x,x),f(y,x)) = f(f(x,x),f(x,f(x,f(x,x)))).
3785
f(f(x,x),f(y,x)) = f(f(x,x),f(x,x)).
3786
f(f(x,x),f(y,x)) = x.
3513
3787
f(f(f(x,y),x),x) = f(x,y).
3514
f(f(x,y),f(y,f(z,f(f(x,z),z)))) = y.
3515
f(f(x,y),f(y,f(x,z))) = y.
3516
f(f(f(f(x,y),f(f(x,y),f(z,y))),f(x,y)),z) = f(y,f(f(z,y),y)).
3517
f(f(f(x,y),f(z,y)),z) = f(y,f(f(z,y),y)).
3518
f(f(f(x,y),f(z,y)),z) = f(z,y).
3519
f(f(x,y),f(x,f(z,y))) = x.
3788
f(f(f(x,y),y),y) = f(y,x).
3789
f(x,f(x,f(x,y))) = f(x,y).
3790
f(f(f(x,f(x,y)),f(x,y)),f(x,y)) = f(f(f(x,f(x,y)),x),x).
3791
f(f(x,y),x) = f(f(f(x,f(x,y)),x),x).
3792
f(f(x,y),x) = f(x,f(x,y)).
3793
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(f(y,f(y,x)),y).
3520
3794
f(f(f(x,f(f(y,z),f(y,x))),x),x) = f(f(y,f(y,x)),y).
3795
f(f(x,f(x,f(f(y,z),f(y,x)))),x) = f(f(y,f(y,x)),y).
3796
f(x,f(x,f(x,f(f(y,z),f(y,x))))) = f(f(y,f(y,x)),y).
3521
3797
f(x,f(f(y,z),f(y,x))) = f(f(y,f(y,x)),y).
3798
f(x,f(f(y,z),f(y,x))) = f(y,f(y,f(y,x))).
3522
3799
f(x,f(f(y,z),f(y,x))) = f(y,x).
3800
f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(f(y,x),y),y).
3801
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(f(f(y,x),y),y).
3523
3802
f(f(f(x,f(f(y,z),f(y,x))),x),x) = f(f(f(y,x),y),y).
3524
3803
f(f(f(y,x),x),x) = f(f(f(y,x),y),y).
3525
f(f(f(x,y),y),y) = f(x,y).
3526
f(f(x,y),x) = f(x,f(x,y)).
3527
f(x,f(y,x)) = f(f(y,x),x).
3528
f(f(x,y),y) = f(y,f(x,y)).
3529
f(f(f(x,y),f(f(x,y),y)),x) = f(x,y).
3530
f(f(f(x,y),f(y,f(x,y))),x) = f(x,y).
3531
f(x,y) = f(y,x).
3532
$F # answer("Commutativity").
3533
end_of_list.
3534
3535
Attempt to eliminate one assumption:
3536
3537
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
3538
3539
***********************************************
3540
AUTO SKETCHES ITERATION 11
3541
***********************************************
3542
3543
Starting a search with 7 assumptions:
3544
3545
formulas(assumptions_active).
3546
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
3547
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3548
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3549
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3550
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3551
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3552
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3553
end_of_list.
3554
3555
Child search process 13972 started.
3556
3557
============================== PROCESS INITIAL CLAUSES ===============
3558
3559
% Clauses before input processing:
3560
3561
formulas(usable).
3562
end_of_list.
3563
3564
formulas(sos).
3565
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3566
f(c2,c1) != f(c1,c2) # label("Commutativity"). [deny(1)].
3567
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
3568
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3569
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3570
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3571
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3572
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3573
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3574
end_of_list.
3575
3576
formulas(demodulators).
3577
end_of_list.
3578
3579
% 1117 hints input.
3580
3581
============================== PREDICATE ELIMINATION =================
3582
3583
No predicates eliminated.
3584
3585
============================== end predicate elimination =============
3586
3587
Auto_denials:
3588
% copying label "Commutativity" to answer in negative clause
3589
3590
Term ordering decisions:
3591
Function symbol KB weights: c1=1. c2=1. f=1.
3592
Relation symbol precedence: lex([ = ]).
3593
Function symbol precedence: lex([ c1, c2, f ]).
3594
Skipping inverse_order, because term ordering is KBO.
3595
Unfolding symbols: (none).
3596
3597
Auto_inference settings:
3598
% set(paramodulation). % (positive equality literals)
3599
% set(paramodulation) -> set(back_demod).
3600
3601
Auto_process settings: (no changes).
3602
3603
============================== end of process initial clauses ========
3604
3605
============================== CLAUSES FOR SEARCH ====================
3606
3607
% Clauses after input processing:
3608
3609
formulas(usable).
3610
end_of_list.
3611
3612
formulas(sos).
3613
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3614
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
3615
4 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
3616
5 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3617
6 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3618
7 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3619
8 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3620
9 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3621
10 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3622
end_of_list.
3623
3624
formulas(demodulators).
3625
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3626
4 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
3627
5 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3628
6 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3629
7 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3630
8 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3631
9 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3632
10 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3633
end_of_list.
3634
3635
% 1117 hints processed (567 are redundant).
3636
3637
============================== end of clauses for search =============
3638
3639
============================== SEARCH ================================
3640
3641
% Starting search at 0.02 seconds.
3642
3643
% Operation f is commutative; C redundancy checks enabled.
3644
3645
============================== PROOF =================================
3646
3647
% Proof 1 at 0.38 (+ 0.01) seconds: "Commutativity".
3648
% Length of proof is 41.
3649
% Level of proof is 14.
3650
% Maximum clause weight is 0.
3651
% Given clauses 54.
3652
3653
1 f(x,y) = f(y,x) # label("Commutativity"). [goal].
3654
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3655
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
3656
4 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
3657
5 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3658
6 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3659
7 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3660
8 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3661
9 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3662
10 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3663
11 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)). [para(2(a,1),2(a,1,1,2,1))].
3664
172 f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)). [para(2(a,1),10(a,1,2))].
3665
209 f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y. [para(172(a,1),2(a,1,2,2))].
3666
214 f(f(f(x,f(x,y)),x),f(y,f(z,f(f(x,z),z)))) = y. [para(172(a,1),4(a,1,2,2))].
3667
229 f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(y,z))) = f(y,f(x,f(f(y,x),x))). [para(172(a,1),7(a,1,1))].
3668
287 f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(x,f(x,f(f(x,x),x))). [para(11(a,1),172(a,1,1,1,2)),rewrite(209(8),209(12),209(13),209(17),229(18))].
3669
293 f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)). [para(209(a,1),2(a,1,2))].
3670
341 f(x,f(f(x,x),x)) = f(x,x). [para(209(a,1),11(a,1,2)),rewrite(209(8)),flip(a)].
3671
344 f(f(x,f(f(f(f(y,y),f(y,f(y,y))),x),x)),f(y,y)) = f(f(y,y),f(y,f(y,y))). [para(11(a,1),209(a,1,2)),rewrite(341(3),341(4),341(10),341(12),341(13))].
3672
349 f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)). [back_rewrite(287),rewrite(341(3),341(7))].
3673
354 f(f(x,x),f(x,f(y,x))) = x. [para(341(a,1),2(a,1,1))].
3674
356 f(f(x,x),f(x,x)) = x. [para(341(a,1),2(a,1,2)),rewrite(341(3))].
3675
364 f(f(x,f(x,x)),f(x,x)) = x. [para(341(a,1),6(a,1,2)),rewrite(356(3))].
3676
368 f(f(f(x,x),x),x) = f(x,x). [para(341(a,1),8(a,1,1,1)),rewrite(356(5))].
3677
382 f(f(x,f(f(y,x),x)),f(y,y)) = y. [back_rewrite(344),rewrite(354(4),354(9))].
3678
386 f(f(x,f(x,x)),x) = f(x,x). [para(356(a,1),2(a,1,2)),rewrite(368(3))].
3679
402 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(349),rewrite(386(3))].
3680
417 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(341),rewrite(402(2))].
3681
509 f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,x). [para(364(a,1),11(a,1,1,1)),rewrite(364(6),386(3),386(5),402(4),417(5),364(7),386(9),402(8),417(9))].
3682
529 f(f(x,x),f(x,f(x,y))) = x. [para(386(a,1),10(a,1,1))].
3683
637 f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)). [para(382(a,1),2(a,1,2))].
3684
641 f(f(f(x,f(f(y,f(f(x,y),y)),f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)). [para(382(a,1),5(a,1,2))].
3685
671 f(x,f(f(y,f(f(x,y),y)),f(y,f(f(x,y),y)))) = f(y,f(f(x,y),y)). [para(382(a,1),382(a,1,1,2,1)),rewrite(529(4))].
3686
672 f(f(f(x,f(f(y,x),x)),y),y) = f(x,f(f(y,x),x)). [back_rewrite(641),rewrite(671(8))].
3687
674 f(f(x,f(y,f(f(x,y),y))),x) = f(y,f(f(x,y),y)). [back_rewrite(637),rewrite(672(5))].
3688
1075 f(x,f(f(y,x),x)) = f(y,x). [para(509(a,1),9(a,1,1,1)),rewrite(529(4),364(5))].
3689
1220 f(f(x,f(x,y)),x) = f(x,y). [back_rewrite(674),rewrite(1075(3),1075(6))].
3690
1373 f(f(f(x,y),f(y,z)),x) = f(x,y). [back_rewrite(293),rewrite(1075(3),1075(4),1075(5),1075(7),1075(7),1075(7))].
3691
1426 f(f(x,y),f(y,f(x,z))) = y. [back_rewrite(214),rewrite(1220(3),1075(4))].
3692
2274 f(x,y) = f(y,x). [para(1426(a,1),1373(a,1,1))].
3693
2275 $F # answer("Commutativity"). [resolve(2274,a,3,a)].
3694
3695
============================== end of proof ==========================
3696
3697
============================== STATISTICS ============================
3698
3699
Given=54. Generated=5148. Kept=2273. proofs=1.
3700
Usable=15. Sos=524. Demods=533. Limbo=7, Disabled=1735. Hints=1117.
3701
Weight_deleted=95. Literals_deleted=0.
3702
Forward_subsumed=2780. Back_subsumed=320.
3703
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
3704
New_demodulators=2259 (0 lex), Back_demodulated=1406. Back_unit_deleted=0.
3705
Demod_attempts=125447. Demod_rewrites=11854.
3706
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
3707
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
3708
Megabytes=3.28.
3709
User_CPU=0.38, System_CPU=0.01, Wall_clock=9.
3710
3711
============================== end of statistics =====================
3712
3713
============================== end of search =========================
3714
3715
Exiting with 1 proof.
3716
3717
Process 13972 exit (max_proofs) Sun Aug 13 21:05:06 2006
3718
3719
Successful proof using the following 7 assumptions:
3720
3721
formulas(assumptions_in_proof).
3722
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
3723
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3724
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3725
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3726
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3727
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3728
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3729
end_of_list.
3730
3731
Including 78 new hint clauses:
3732
3733
formulas(new_hints).
3734
$F # label("Commutativity").
3735
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
3736
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
3737
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
3738
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").
3739
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
3740
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
3741
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
3742
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
3743
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
3744
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)).
3745
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
3746
f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y.
3747
f(f(f(x,f(x,y)),x),f(y,f(z,f(f(x,z),z)))) = y.
3748
f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(y,z))) = f(y,f(x,f(f(y,x),x))).
3749
f(f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
3750
f(f(f(x,f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
3751
f(f(f(x,f(x,f(f(x,x),x))),x),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
3752
f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
3753
f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(f(x,f(f(x,x),x)),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
3754
f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(x,f(x,f(f(x,x),x))).
3755
f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
3756
f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),x) = f(x,f(f(x,x),x)).
3757
f(x,x) = f(x,f(f(x,x),x)).
3758
f(x,f(f(x,x),x)) = f(x,x).
3759
f(f(x,f(f(f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))),x),x)),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
3760
f(f(x,f(f(f(f(y,y),f(y,f(y,f(f(y,y),y)))),x),x)),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
3761
f(f(x,f(f(f(f(y,y),f(y,f(y,y))),x),x)),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
3762
f(f(x,f(f(f(f(y,y),f(y,f(y,y))),x),x)),f(y,y)) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
3763
f(f(x,f(f(f(f(y,y),f(y,f(y,y))),x),x)),f(y,y)) = f(f(y,y),f(y,f(y,f(f(y,y),y)))).
3764
f(f(x,f(f(f(f(y,y),f(y,f(y,y))),x),x)),f(y,y)) = f(f(y,y),f(y,f(y,y))).
3765
f(f(f(x,f(x,x)),x),x) = f(x,f(x,f(f(x,x),x))).
3766
f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)).
3767
f(f(x,x),f(x,f(y,x))) = x.
3768
f(f(x,f(f(x,x),x)),f(x,x)) = x.
3769
f(f(x,x),f(x,x)) = x.
3770
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,x)) = x.
3771
f(f(x,f(x,x)),f(x,x)) = x.
3772
f(f(f(x,x),x),f(f(x,x),f(x,x))) = f(x,x).
3773
f(f(f(x,x),x),x) = f(x,x).
3774
f(f(x,f(f(y,x),x)),f(y,y)) = f(f(y,y),f(y,f(y,y))).
3775
f(f(x,f(f(y,x),x)),f(y,y)) = y.
3776
f(f(x,f(f(f(x,x),x),x)),x) = f(x,x).
3777
f(f(x,f(x,x)),x) = f(x,x).
3778
f(f(x,x),x) = f(x,f(x,x)).
3779
f(x,f(x,f(x,x))) = f(x,x).
3780
f(f(x,f(f(x,f(x,x)),f(f(x,f(x,x)),f(x,x)))),f(f(x,f(f(f(x,f(x,x)),x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
3781
f(f(x,f(f(x,f(x,x)),x)),f(f(x,f(f(f(x,f(x,x)),x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
3782
f(f(x,f(x,x)),f(f(x,f(f(f(x,f(x,x)),x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
3783
f(f(x,f(x,x)),f(f(x,f(f(x,x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
3784
f(f(x,f(x,x)),f(f(x,f(x,f(x,x))),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
3785
f(f(x,f(x,x)),f(f(x,x),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
3786
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,f(f(f(x,f(x,x)),x),x)).
3787
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,f(f(x,x),x)).
3788
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,f(x,f(x,x))).
3789
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,x).
3790
f(f(x,x),f(x,f(x,y))) = x.
3791
f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)).
3792
f(f(f(x,f(f(y,f(f(x,y),y)),f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
3793
f(f(f(x,x),f(x,f(x,x))),f(f(y,f(f(x,y),y)),f(y,f(f(x,y),y)))) = f(y,f(f(x,y),y)).
3794
f(x,f(f(y,f(f(x,y),y)),f(y,f(f(x,y),y)))) = f(y,f(f(x,y),y)).
3795
f(f(f(x,f(f(y,x),x)),y),y) = f(x,f(f(y,x),x)).
3796
f(f(x,f(y,f(f(x,y),y))),x) = f(y,f(f(x,y),y)).
3797
f(f(f(x,x),f(x,f(x,x))),f(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x).
3798
f(x,f(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x).
3799
f(x,f(f(y,x),x)) = f(y,x).
3800
f(f(x,f(x,y)),x) = f(y,f(f(x,y),y)).
3801
f(f(x,f(x,y)),x) = f(x,y).
3802
f(f(f(y,x),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
3803
f(f(f(y,x),f(f(f(z,y),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
3804
f(f(f(y,x),f(f(f(z,y),f(y,x)),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
3805
f(f(f(y,x),f(f(f(z,y),f(y,x)),f(y,x))),z) = f(y,f(f(z,y),y)).
3806
f(f(f(z,y),f(y,x)),z) = f(y,f(f(z,y),y)).
3807
f(f(f(x,y),f(y,z)),x) = f(x,y).
3808
f(f(x,y),f(y,f(z,f(f(x,z),z)))) = y.
3809
f(f(x,y),f(y,f(x,z))) = y.
3810
f(x,y) = f(y,x).
3811
$F # answer("Commutativity").
3812
end_of_list.
3813
3814
Attempt to eliminate one assumption:
3815
3816
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
3817
3818
***********************************************
3819
AUTO SKETCHES ITERATION 12
3820
***********************************************
3821
3822
Starting a search with 6 assumptions:
3823
3824
formulas(assumptions_active).
3825
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3826
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3827
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3828
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3829
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3830
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3831
end_of_list.
3832
3833
Child search process 13974 started.
3834
3835
============================== PROCESS INITIAL CLAUSES ===============
3836
3837
% Clauses before input processing:
3838
3839
formulas(usable).
3840
end_of_list.
3841
3842
formulas(sos).
3843
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3844
f(c2,c1) != f(c1,c2) # label("Commutativity"). [deny(1)].
3845
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3846
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3847
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3848
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3849
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3850
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3851
end_of_list.
3852
3853
formulas(demodulators).
3854
end_of_list.
3855
3856
% 1195 hints input.
3857
3858
============================== PREDICATE ELIMINATION =================
3859
3860
No predicates eliminated.
3861
3862
============================== end predicate elimination =============
3863
3864
Auto_denials:
3865
% copying label "Commutativity" to answer in negative clause
3866
3867
Term ordering decisions:
3868
Function symbol KB weights: c1=1. c2=1. f=1.
3869
Relation symbol precedence: lex([ = ]).
3870
Function symbol precedence: lex([ c1, c2, f ]).
3871
Skipping inverse_order, because term ordering is KBO.
3872
Unfolding symbols: (none).
3873
3874
Auto_inference settings:
3875
% set(paramodulation). % (positive equality literals)
3876
% set(paramodulation) -> set(back_demod).
3877
3878
Auto_process settings: (no changes).
3879
3880
============================== end of process initial clauses ========
3881
3882
============================== CLAUSES FOR SEARCH ====================
3883
3884
% Clauses after input processing:
3885
3886
formulas(usable).
3887
end_of_list.
3888
3889
formulas(sos).
3890
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3891
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
3892
4 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3893
5 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3894
6 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3895
7 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3896
8 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3897
9 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3898
end_of_list.
3899
3900
formulas(demodulators).
3901
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3902
4 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3903
5 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3904
6 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3905
7 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3906
8 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3907
9 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3908
end_of_list.
3909
3910
% 1195 hints processed (637 are redundant).
3911
3912
============================== end of clauses for search =============
3913
3914
============================== SEARCH ================================
3915
3916
% Starting search at 0.02 seconds.
3917
3918
% Operation f is commutative; C redundancy checks enabled.
3919
3920
============================== PROOF =================================
3921
3922
% Proof 1 at 0.40 (+ 0.00) seconds: "Commutativity".
3923
% Length of proof is 38.
3924
% Level of proof is 14.
3925
% Maximum clause weight is 0.
3926
% Given clauses 53.
3927
3928
1 f(x,y) = f(y,x) # label("Commutativity"). [goal].
3929
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3930
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
3931
4 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3932
5 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3933
6 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3934
7 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3935
8 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3936
9 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3937
10 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)). [para(2(a,1),2(a,1,1,2,1))].
3938
137 f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)). [para(2(a,1),9(a,1,2))].
3939
169 f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y. [para(137(a,1),2(a,1,2,2))].
3940
185 f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(y,z))) = f(y,f(x,f(f(y,x),x))). [para(137(a,1),6(a,1,1))].
3941
204 f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)). [para(169(a,1),2(a,1,2))].
3942
285 f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(x,f(x,f(f(x,x),x))). [para(10(a,1),137(a,1,1,1,2)),rewrite(169(8),169(12),169(13),169(17),185(18))].
3943
290 f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y. [para(10(a,1),169(a,1,2)),rewrite(169(8),169(15))].
3944
293 f(x,f(f(x,x),x)) = f(x,x). [para(169(a,1),10(a,1,2)),rewrite(169(8)),flip(a)].
3945
296 f(f(x,f(f(y,x),x)),f(y,y)) = y. [back_rewrite(290),rewrite(293(6))].
3946
297 f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)). [back_rewrite(285),rewrite(293(3),293(7))].
3947
305 f(f(x,x),f(x,x)) = x. [para(293(a,1),2(a,1,2)),rewrite(293(3))].
3948
311 f(f(x,f(x,x)),f(x,x)) = x. [para(293(a,1),5(a,1,2)),rewrite(305(3))].
3949
315 f(f(f(x,x),x),x) = f(x,x). [para(293(a,1),7(a,1,1,1)),rewrite(305(5))].
3950
332 f(f(x,f(x,x)),x) = f(x,x). [para(305(a,1),2(a,1,2)),rewrite(315(3))].
3951
347 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(297),rewrite(332(3))].
3952
361 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(293),rewrite(347(2))].
3953
397 f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,x). [para(311(a,1),10(a,1,1,1)),rewrite(311(6),332(3),332(5),347(4),361(5),311(7),332(9),347(8),361(9))].
3954
465 f(f(x,x),f(x,f(x,y))) = x. [para(332(a,1),9(a,1,1))].
3955
569 f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)). [para(296(a,1),2(a,1,2))].
3956
572 f(f(f(x,f(f(y,f(f(x,y),y)),f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)). [para(296(a,1),4(a,1,2))].
3957
602 f(x,f(f(y,f(f(x,y),y)),f(y,f(f(x,y),y)))) = f(y,f(f(x,y),y)). [para(296(a,1),296(a,1,1,2,1)),rewrite(465(4))].
3958
603 f(f(f(x,f(f(y,x),x)),y),y) = f(x,f(f(y,x),x)). [back_rewrite(572),rewrite(602(8))].
3959
605 f(f(x,f(y,f(f(x,y),y))),x) = f(y,f(f(x,y),y)). [back_rewrite(569),rewrite(603(5))].
3960
967 f(x,f(f(y,x),x)) = f(y,x). [para(397(a,1),8(a,1,1,1)),rewrite(465(4),311(5))].
3961
1099 f(f(x,f(x,y)),x) = f(x,y). [back_rewrite(605),rewrite(967(3),967(6))].
3962
1271 f(f(f(x,y),f(y,z)),x) = f(x,y). [back_rewrite(204),rewrite(967(3),967(4),967(5),967(7),967(7),967(7))].
3963
1360 f(f(x,y),f(y,f(x,z))) = y. [back_rewrite(9),rewrite(1099(3))].
3964
2173 f(x,y) = f(y,x). [para(1360(a,1),1271(a,1,1))].
3965
2174 $F # answer("Commutativity"). [resolve(2173,a,3,a)].
3966
3967
============================== end of proof ==========================
3968
3969
============================== STATISTICS ============================
3970
3971
Given=53. Generated=4904. Kept=2172. proofs=1.
3972
Usable=15. Sos=513. Demods=527. Limbo=12, Disabled=1639. Hints=1195.
3973
Weight_deleted=84. Literals_deleted=0.
3974
Forward_subsumed=2648. Back_subsumed=308.
3975
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
3976
New_demodulators=2158 (0 lex), Back_demodulated=1323. Back_unit_deleted=0.
3977
Demod_attempts=118413. Demod_rewrites=11678.
3978
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
3979
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
3980
Megabytes=3.17.
3981
User_CPU=0.40, System_CPU=0.00, Wall_clock=10.
3982
3983
============================== end of statistics =====================
3984
3985
============================== end of search =========================
3986
3987
Exiting with 1 proof.
3988
3989
Process 13974 exit (max_proofs) Sun Aug 13 21:05:07 2006
3990
3991
Successful proof using the following 6 assumptions:
3992
3993
formulas(assumptions_in_proof).
3994
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3995
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3996
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3997
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3998
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3999
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4000
end_of_list.
4001
4002
Including 70 new hint clauses:
4003
4004
formulas(new_hints).
4005
$F # label("Commutativity").
4006
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
4007
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
4008
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").
4009
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
4010
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
4011
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
4012
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
4013
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
4014
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)).
4015
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
4016
f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y.
4017
f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(y,z))) = f(y,f(x,f(f(y,x),x))).
4018
f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
4019
f(f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
4020
f(f(f(x,f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
4021
f(f(f(x,f(x,f(f(x,x),x))),x),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
4022
f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
4023
f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(f(x,f(f(x,x),x)),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
4024
f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(x,f(x,f(f(x,x),x))).
4025
f(f(x,f(f(f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))),x),x)),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
4026
f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
4027
f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y.
4028
f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),x) = f(x,f(f(x,x),x)).
4029
f(x,x) = f(x,f(f(x,x),x)).
4030
f(x,f(f(x,x),x)) = f(x,x).
4031
f(f(x,f(f(y,x),x)),f(y,y)) = y.
4032
f(f(f(x,f(x,x)),x),x) = f(x,f(x,f(f(x,x),x))).
4033
f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)).
4034
f(f(x,f(f(x,x),x)),f(x,x)) = x.
4035
f(f(x,x),f(x,x)) = x.
4036
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,x)) = x.
4037
f(f(x,f(x,x)),f(x,x)) = x.
4038
f(f(f(x,x),x),f(f(x,x),f(x,x))) = f(x,x).
4039
f(f(f(x,x),x),x) = f(x,x).
4040
f(f(x,f(f(f(x,x),x),x)),x) = f(x,x).
4041
f(f(x,f(x,x)),x) = f(x,x).
4042
f(f(x,x),x) = f(x,f(x,x)).
4043
f(x,f(x,f(x,x))) = f(x,x).
4044
f(f(x,f(f(x,f(x,x)),f(f(x,f(x,x)),f(x,x)))),f(f(x,f(f(f(x,f(x,x)),x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
4045
f(f(x,f(f(x,f(x,x)),x)),f(f(x,f(f(f(x,f(x,x)),x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
4046
f(f(x,f(x,x)),f(f(x,f(f(f(x,f(x,x)),x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
4047
f(f(x,f(x,x)),f(f(x,f(f(x,x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
4048
f(f(x,f(x,x)),f(f(x,f(x,f(x,x))),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
4049
f(f(x,f(x,x)),f(f(x,x),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
4050
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,f(f(f(x,f(x,x)),x),x)).
4051
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,f(f(x,x),x)).
4052
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,f(x,f(x,x))).
4053
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,x).
4054
f(f(x,x),f(x,f(x,y))) = x.
4055
f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)).
4056
f(f(f(x,f(f(y,f(f(x,y),y)),f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
4057
f(f(f(x,x),f(x,f(x,x))),f(f(y,f(f(x,y),y)),f(y,f(f(x,y),y)))) = f(y,f(f(x,y),y)).
4058
f(x,f(f(y,f(f(x,y),y)),f(y,f(f(x,y),y)))) = f(y,f(f(x,y),y)).
4059
f(f(f(x,f(f(y,x),x)),y),y) = f(x,f(f(y,x),x)).
4060
f(f(x,f(y,f(f(x,y),y))),x) = f(y,f(f(x,y),y)).
4061
f(f(f(x,x),f(x,f(x,x))),f(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x).
4062
f(x,f(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x).
4063
f(x,f(f(y,x),x)) = f(y,x).
4064
f(f(x,f(x,y)),x) = f(y,f(f(x,y),y)).
4065
f(f(x,f(x,y)),x) = f(x,y).
4066
f(f(f(y,x),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
4067
f(f(f(y,x),f(f(f(z,y),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
4068
f(f(f(y,x),f(f(f(z,y),f(y,x)),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
4069
f(f(f(y,x),f(f(f(z,y),f(y,x)),f(y,x))),z) = f(y,f(f(z,y),y)).
4070
f(f(f(z,y),f(y,x)),z) = f(y,f(f(z,y),y)).
4071
f(f(f(x,y),f(y,z)),x) = f(x,y).
4072
f(f(x,y),f(y,f(x,z))) = y.
3804
f(x,y) = f(f(f(y,x),y),y).
3805
f(x,y) = f(f(y,f(y,x)),y).
3806
f(x,y) = f(y,f(y,f(y,x))).
4073
3807
f(x,y) = f(y,x).
4074
3808
$F # answer("Commutativity").
4075
3809
end_of_list.
4092
3826
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4093
3827
end_of_list.
4094
3828
4095
Child search process 13975 started.
3829
Child search process 10947 started.
4096
3830
4097
3831
============================== PROCESS INITIAL CLAUSES ===============
4098
3832
4114
3848
formulas(demodulators).
4115
3849
end_of_list.
4116
3850
4117
% 1265 hints input.
3851
% 941 hints input.
4118
3852
4119
3853
============================== PREDICATE ELIMINATION =================
4120
3854
4127
3861
4128
3862
Term ordering decisions:
4129
3863
Function symbol KB weights: c1=1. c2=1. f=1.
4130
Relation symbol precedence: lex([ = ]).
4131
Function symbol precedence: lex([ c1, c2, f ]).
3864
Predicate symbol precedence: predicate_order([ = ]).
3865
Function symbol precedence: function_order([ c1, c2, f ]).
4132
3866
Skipping inverse_order, because term ordering is KBO.
4133
3867
Unfolding symbols: (none).
4134
3868
4135
3869
Auto_inference settings:
4136
3870
% set(paramodulation). % (positive equality literals)
4137
% set(paramodulation) -> set(back_demod).
4138
3871
4139
3872
Auto_process settings: (no changes).
4140
3873
3874
kept: 2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3875
kept: 3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
3876
kept: 4 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3877
kept: 5 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3878
kept: 6 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3879
kept: 7 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3880
kept: 8 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3881
4141
3882
============================== end of process initial clauses ========
4142
3883
4143
3884
============================== CLAUSES FOR SEARCH ====================
4166
3907
8 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4167
3908
end_of_list.
4168
3909
4169
% 1265 hints processed (705 are redundant).
3910
% 397 hints (941 processed, 544 redundant).
4170
3911
4171
3912
============================== end of clauses for search =============
4172
3913
4173
3914
============================== SEARCH ================================
4174
3915
4175
% Starting search at 0.02 seconds.
3916
% Starting search at 0.04 seconds.
4176
3917
4177
3918
% Operation f is commutative; C redundancy checks enabled.
4178
3919
4179
3920
============================== PROOF =================================
4180
3921
4181
% Proof 1 at 0.25 (+ 0.00) seconds: "Commutativity".
4182
% Length of proof is 52.
4183
% Level of proof is 17.
4184
% Maximum clause weight is 61.
3922
% Proof 1 at 0.43 (+ 0.01) seconds: "Commutativity".
3923
% Length of proof is 39.
3924
% Level of proof is 15.
3925
% Maximum clause weight is 37.
4185
3926
% Given clauses 33.
4186
3927
4187
1 f(x,y) = f(y,x) # label("Commutativity"). [goal].
3928
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
4188
3929
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
4189
3930
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
4190
3931
4 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
4193
3934
7 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
4194
3935
8 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4195
3936
9 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)). [para(2(a,1),2(a,1,1,2,1))].
4196
11 f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)). [para(2(a,1),2(a,1,2))].
4197
12 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(f(f(z,x),y),y),f(u,f(x,f(y,z))))) = f(f(f(z,x),y),y). [para(4(a,1),2(a,1,1,2,1))].
4198
13 f(f(f(x,y),f(f(f(f(f(y,z),x),x),f(x,y)),f(x,y))),z) = f(f(f(y,z),x),x). [para(4(a,1),2(a,1,2))].
4199
3937
17 f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)). [para(2(a,1),4(a,1,2))].
3938
82 f(f(f(x,f(f(y,z),f(f(f(y,x),y),y))),x),x) = f(f(f(y,x),y),y). [para(5(a,1),7(a,1,2))].
4200
3939
100 f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)). [para(2(a,1),8(a,1,2))].
4201
3940
127 f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y. [para(100(a,1),2(a,1,2,2))].
4202
3941
138 f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(y,z))) = f(y,f(x,f(f(y,x),x))). [para(100(a,1),5(a,1,1))].
4203
267 f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(x,f(x,f(f(x,x),x))). [para(9(a,1),100(a,1,1,1,2)),rewrite(127(8),127(12),127(13),127(17),138(18))].
4204
272 f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y. [para(9(a,1),127(a,1,2)),rewrite(127(8),127(15))].
4205
275 f(x,f(f(x,x),x)) = f(x,x). [para(127(a,1),9(a,1,2)),rewrite(127(8)),flip(a)].
4206
279 f(f(x,f(f(y,x),x)),f(y,y)) = y. [back_rewrite(272),rewrite(275(6))].
4207
280 f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)). [back_rewrite(267),rewrite(275(3),275(7))].
4208
284 f(f(x,x),f(x,f(y,x))) = x. [para(275(a,1),2(a,1,1))].
4209
286 f(f(x,x),f(x,x)) = x. [para(275(a,1),2(a,1,2)),rewrite(275(3))].
4210
290 f(f(x,f(x,x)),f(x,x)) = x. [para(275(a,1),4(a,1,2)),rewrite(286(3))].
4211
294 f(f(f(x,x),x),x) = f(x,x). [para(275(a,1),6(a,1,1,1)),rewrite(286(5))].
4212
315 f(f(x,f(x,x)),x) = f(x,x). [para(286(a,1),2(a,1,2)),rewrite(294(3))].
4213
330 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(280),rewrite(315(3))].
4214
346 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(275),rewrite(330(2))].
4215
364 f(f(f(x,y),y),y) = f(x,f(f(y,x),x)). [para(284(a,1),17(a,1,1,1,1))].
4216
384 f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,x). [para(290(a,1),9(a,1,1,1)),rewrite(290(6),315(3),315(5),330(4),346(5),290(7),315(9),330(8),346(9))].
4217
449 f(f(x,x),f(x,f(x,y))) = x. [para(315(a,1),8(a,1,1))].
4218
564 f(f(f(x,f(f(y,f(f(x,y),y)),x)),x),x) = f(y,f(f(x,y),y)). [para(279(a,1),6(a,1,2))].
4219
868 f(f(f(f(x,y),f(f(x,y),f(y,f(x,f(f(y,x),x))))),f(x,y)),f(x,y)) = f(y,f(f(f(x,y),y),y)). [para(364(a,1),100(a,1,1,1,2,2,2))].
4220
901 f(f(f(f(f(x,y),y),f(z,y)),f(f(f(x,y),y),f(f(f(x,y),y),f(z,y)))),f(f(y,f(f(x,f(f(y,x),x)),y)),f(u,f(f(f(x,y),y),f(z,y))))) = f(y,f(f(f(f(x,y),y),y),y)). [para(364(a,1),9(a,1,2,1,2,1))].
4221
902 f(f(f(f(x,y),f(z,y)),f(f(x,y),f(f(x,y),f(z,y)))),f(f(y,f(x,f(f(y,x),x))),f(u,f(f(x,y),f(z,y))))) = f(y,f(f(f(x,y),y),y)). [para(364(a,1),9(a,1,2,1,2))].
4222
910 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(f(f(z,x),x),x),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)). [para(364(a,2),9(a,1,2,1))].
4223
925 f(f(f(x,y),f(f(f(y,f(f(z,f(f(y,z),z)),y)),f(x,y)),f(x,y))),f(f(z,y),y)) = f(y,f(f(f(f(z,y),y),y),y)). [para(364(a,1),11(a,1,1,2,1,1,2,1))].
4224
926 f(f(f(x,y),f(f(f(y,f(z,f(f(y,z),z))),f(x,y)),f(x,y))),f(z,y)) = f(y,f(f(f(z,y),y),y)). [para(364(a,1),11(a,1,1,2,1,1,2))].
4225
971 f(f(x,y),f(f(y,f(x,y)),f(x,y))) = f(f(x,f(f(y,x),x)),y). [para(364(a,1),12(a,2,1)),rewrite(910(18))].
4226
1072 f(x,f(x,f(y,f(f(x,y),y)))) = f(y,f(f(x,y),y)). [para(100(a,1),13(a,1,1,2,1,1)),rewrite(279(6),449(4),100(12))].
4227
1153 f(f(f(f(x,y),y),y),y) = f(f(x,f(f(y,x),x)),y). [para(364(a,1),13(a,2,1)),rewrite(13(14))].
4228
1266 f(x,f(f(y,x),x)) = f(y,x). [para(384(a,1),7(a,1,1,1)),rewrite(449(4),290(5))].
4229
1367 f(f(f(f(x,y),y),y),y) = f(f(y,x),y). [back_rewrite(1153),rewrite(1266(7))].
4230
1428 f(x,f(x,f(x,y))) = f(x,y). [back_rewrite(1072),rewrite(1266(3),1266(6))].
4231
1467 f(f(x,y),x) = f(x,f(y,x)). [back_rewrite(971),rewrite(1266(6),1266(5)),flip(a)].
4232
1499 f(f(f(x,f(x,y)),f(z,x)),f(y,x)) = f(f(y,x),x). [back_rewrite(926),rewrite(1266(4),1266(8),1266(10))].
4233
1500 f(f(f(x,y),y),y) = f(y,f(y,f(x,y))). [back_rewrite(925),rewrite(1266(4),1467(3),1266(9),1499(8),1367(7),1467(5))].
4234
1514 f(f(f(f(x,y),f(z,y)),f(f(x,y),f(f(x,y),f(z,y)))),f(f(y,f(y,x)),f(u,f(f(x,y),f(z,y))))) = f(y,f(x,y)). [back_rewrite(902),rewrite(1266(12),1500(20),1428(21))].
4235
1515 f(x,f(x,f(y,x))) = f(y,x). [back_rewrite(901),rewrite(1266(15),1467(14),1514(22),1266(3),1500(4),1467(5),1467(4),1266(4)),flip(a)].
4236
1534 f(f(x,y),y) = f(y,f(x,y)). [back_rewrite(868),rewrite(1266(5),1467(8),1467(7),1515(8),1499(6),1500(5),1515(5))].
4237
1607 f(x,y) = f(y,x). [back_rewrite(564),rewrite(1534(2),1515(3),1467(2),1515(3),1534(2),1467(3),1534(2),1515(3),1534(3),1515(4))].
4238
1608 $F # answer("Commutativity"). [resolve(1607,a,3,a)].
3942
452 f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(x,f(x,f(f(x,x),x))). [para(9(a,1),100(a,1,1,1,2)),rewrite([127(8),127(12),127(13),127(17),138(18)])].
3943
460 f(x,f(f(x,x),x)) = f(x,x). [para(127(a,1),9(a,1,2)),rewrite([127(8)]),flip(a)].
3944
472 f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)). [back_rewrite(452),rewrite([460(3),460(7)])].
3945
476 f(f(x,x),f(x,f(y,x))) = x. [para(460(a,1),2(a,1,1))].
3946
478 f(f(x,x),f(x,x)) = x. [para(460(a,1),2(a,1,2)),rewrite([460(3)])].
3947
482 f(f(x,f(x,x)),f(x,x)) = x. [para(460(a,1),4(a,1,2)),rewrite([478(3)])].
3948
486 f(f(f(x,x),x),x) = f(x,x). [para(460(a,1),6(a,1,1,1)),rewrite([478(5)])].
3949
511 f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y. [para(478(a,1),2(a,1,2,2))].
3950
512 f(f(x,f(x,x)),x) = f(x,x). [para(478(a,1),2(a,1,2)),rewrite([486(3)])].
3951
535 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(472),rewrite([512(3)])].
3952
556 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(460),rewrite([535(2)])].
3953
574 f(f(f(x,y),y),y) = f(x,f(f(y,x),x)). [para(476(a,1),17(a,1,1,1,1))].
3954
585 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [para(482(a,1),2(a,1,1,2)),rewrite([535(2)])].
3955
621 f(f(x,x),f(x,f(x,y))) = x. [para(512(a,1),8(a,1,1))].
3956
970 f(x,f(f(f(f(x,x),y),y),y)) = f(x,x). [para(574(a,2),9(a,1,2)),rewrite([535(2),556(3),535(3),556(4),621(4),535(2),556(3),535(7),556(8)])].
3957
1009 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)). [para(585(a,1),7(a,1,1,1)),rewrite([556(3),512(6)])].
3958
1037 f(f(x,f(y,y)),f(x,y)) = x. [back_rewrite(511),rewrite([1009(6)])].
3959
1044 f(f(x,y),f(x,f(y,y))) = x. [para(2(a,1),1037(a,1,1,2)),rewrite([535(3),556(4)])].
3960
1175 f(f(f(x,f(f(y,y),f(x,y))),x),x) = f(x,y). [para(1044(a,1),7(a,1,2))].
3961
1263 f(x,f(y,f(x,x))) = f(x,x). [para(574(a,1),970(a,1,2)),rewrite([1009(6)])].
3962
1299 f(f(x,x),f(y,x)) = x. [para(2(a,1),1263(a,1,2,2)),rewrite([535(2),1263(3),535(5),1263(6),535(6),1263(7),478(6)])].
3963
1305 f(f(f(x,y),x),x) = f(f(f(y,x),y),y). [para(1263(a,1),82(a,1,1,1,2,1)),rewrite([1299(5)])].
3964
1311 f(f(f(x,y),x),x) = f(x,y). [back_rewrite(1175),rewrite([1299(3)])].
3965
1320 f(x,y) = f(y,x). [back_rewrite(1305),rewrite([1311(3),1311(4)])].
3966
1321 $F # answer("Commutativity"). [resolve(1320,a,3,a)].
4239
3967
4240
3968
============================== end of proof ==========================
4241
3969
4242
3970
============================== STATISTICS ============================
4243
3971
4244
Given=33. Generated=3357. Kept=1606. proofs=1.
4245
Usable=30. Sos=678. Demods=1031. Limbo=341, Disabled=563. Hints=1265.
4246
Weight_deleted=135. Literals_deleted=0.
4247
Forward_subsumed=1616. Back_subsumed=20.
3972
Given=33. Generated=2831. Kept=1319. proofs=1.
3973
Usable=27. Sos=982. Demods=991. Limbo=9, Disabled=307. Hints=941.
3974
Kept_by_rule=0, Deleted_by_rule=93.
3975
Forward_subsumed=1419. Back_subsumed=64.
4248
3976
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
4249
New_demodulators=1561 (0 lex), Back_demodulated=536. Back_unit_deleted=0.
4250
Demod_attempts=100045. Demod_rewrites=7790.
3977
New_demodulators=1279 (0 lex), Back_demodulated=236. Back_unit_deleted=0.
3978
Demod_attempts=81810. Demod_rewrites=6068.
4251
3979
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
4252
3980
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
4253
Megabytes=3.48.
4254
User_CPU=0.25, System_CPU=0.00, Wall_clock=10.
3981
Megabytes=2.96.
3982
User_CPU=0.43, System_CPU=0.01, Wall_clock=7.
4255
3983
4256
3984
============================== end of statistics =====================
4257
3985
4259
3987
4260
3988
Exiting with 1 proof.
4261
3989
4262
Process 13975 exit (max_proofs) Sun Aug 13 21:05:07 2006
3990
Process 10947 exit (max_proofs) Mon Sep 28 11:57:16 2009
4263
3991
4264
3992
Successful proof using the following 5 assumptions:
4265
3993
4271
3999
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4272
4000
end_of_list.
4273
4001
4274
Including 117 new hint clauses:
4002
Including 75 new hint clauses:
4275
4003
4276
4004
formulas(new_hints).
4277
$F # label("Commutativity").
4005
$F # label("Commutativity") # label(non_clause) # label(goal).
4278
4006
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
4279
4007
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
4280
4008
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
4283
4011
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
4284
4012
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
4285
4013
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)).
4286
f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)).
4287
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(f(f(z,x),y),y),f(u,f(x,f(y,z))))) = f(f(f(z,x),y),y).
4288
f(f(f(x,y),f(f(f(f(f(y,z),x),x),f(x,y)),f(x,y))),z) = f(f(f(y,z),x),x).
4289
4014
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
4015
f(f(f(x,f(f(y,z),f(f(f(y,x),y),y))),x),x) = f(f(f(y,x),y),y).
4290
4016
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
4291
4017
f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y.
4292
4018
f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(y,z))) = f(y,f(x,f(f(y,x),x))).
4296
4022
f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
4297
4023
f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(f(x,f(f(x,x),x)),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
4298
4024
f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(x,f(x,f(f(x,x),x))).
4299
f(f(x,f(f(f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))),x),x)),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
4300
f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
4301
f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y.
4302
4025
f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),x) = f(x,f(f(x,x),x)).
4303
4026
f(x,x) = f(x,f(f(x,x),x)).
4304
4027
f(x,f(f(x,x),x)) = f(x,x).
4305
f(f(x,f(f(y,x),x)),f(y,y)) = y.
4306
4028
f(f(f(x,f(x,x)),x),x) = f(x,f(x,f(f(x,x),x))).
4307
4029
f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)).
4308
4030
f(f(x,x),f(x,f(y,x))) = x.
4312
4034
f(f(x,f(x,x)),f(x,x)) = x.
4313
4035
f(f(f(x,x),x),f(f(x,x),f(x,x))) = f(x,x).
4314
4036
f(f(f(x,x),x),x) = f(x,x).
4037
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
4315
4038
f(f(x,f(f(f(x,x),x),x)),x) = f(x,x).
4316
4039
f(f(x,f(x,x)),x) = f(x,x).
4317
4040
f(f(x,x),x) = f(x,f(x,x)).
4318
4041
f(x,f(x,f(x,x))) = f(x,x).
4319
4042
f(f(f(x,y),y),y) = f(x,f(f(y,x),x)).
4320
f(f(x,f(f(x,f(x,x)),f(f(x,f(x,x)),f(x,x)))),f(f(x,f(f(f(x,f(x,x)),x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
4321
f(f(x,f(f(x,f(x,x)),x)),f(f(x,f(f(f(x,f(x,x)),x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
4322
f(f(x,f(x,x)),f(f(x,f(f(f(x,f(x,x)),x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
4323
f(f(x,f(x,x)),f(f(x,f(f(x,x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
4324
f(f(x,f(x,x)),f(f(x,f(x,f(x,x))),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
4325
f(f(x,f(x,x)),f(f(x,x),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
4326
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,f(f(f(x,f(x,x)),x),x)).
4327
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,f(f(x,x),x)).
4328
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,f(x,f(x,x))).
4329
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,x).
4043
f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x.
4044
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
4330
4045
f(f(x,x),f(x,f(x,y))) = x.
4331
f(f(f(x,f(f(y,f(f(x,y),y)),x)),x),x) = f(y,f(f(x,y),y)).
4332
f(f(f(f(x,y),f(f(x,y),f(y,f(x,f(f(y,x),x))))),f(x,y)),f(x,y)) = f(y,f(f(f(x,y),y),y)).
4333
f(f(f(f(f(x,y),y),f(z,y)),f(f(f(x,y),y),f(f(f(x,y),y),f(z,y)))),f(f(y,f(f(x,f(f(y,x),x)),y)),f(u,f(f(f(x,y),y),f(z,y))))) = f(y,f(f(f(f(x,y),y),y),y)).
4334
f(f(f(f(x,y),f(z,y)),f(f(x,y),f(f(x,y),f(z,y)))),f(f(y,f(x,f(f(y,x),x))),f(u,f(f(x,y),f(z,y))))) = f(y,f(f(f(x,y),y),y)).
4335
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(f(f(z,x),x),x),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)).
4336
f(f(f(x,y),f(f(f(y,f(f(z,f(f(y,z),z)),y)),f(x,y)),f(x,y))),f(f(z,y),y)) = f(y,f(f(f(f(z,y),y),y),y)).
4337
f(f(f(x,y),f(f(f(y,f(z,f(f(y,z),z))),f(x,y)),f(x,y))),f(z,y)) = f(y,f(f(f(z,y),y),y)).
4338
f(f(f(x,f(x,f(y,x))),f(x,f(x,f(x,f(y,x))))),f(f(f(f(f(y,x),x),x),x),f(z,f(x,f(x,f(y,x)))))) = f(f(y,f(f(x,y),y)),x).
4339
f(f(x,y),f(f(y,f(x,y)),f(x,y))) = f(f(x,f(f(y,x),x)),y).
4340
f(f(f(x,x),f(f(f(y,f(f(x,y),y)),f(x,x)),f(x,x))),f(x,f(y,f(f(x,y),y)))) = f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x).
4341
f(f(f(x,x),f(x,f(x,x))),f(x,f(y,f(f(x,y),y)))) = f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x).
4342
f(x,f(x,f(y,f(f(x,y),y)))) = f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x).
4343
f(x,f(x,f(y,f(f(x,y),y)))) = f(y,f(f(x,y),y)).
4344
f(f(f(x,f(y,x)),f(f(f(f(f(f(y,x),x),x),x),f(x,f(y,x))),f(x,f(y,x)))),x) = f(f(y,f(f(x,y),y)),x).
4345
f(f(f(f(x,y),y),y),y) = f(f(x,f(f(y,x),x)),y).
4346
f(f(f(x,x),f(x,f(x,x))),f(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x).
4347
f(x,f(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x).
4348
f(x,f(f(y,x),x)) = f(y,x).
4349
f(f(f(f(x,y),y),y),y) = f(f(y,x),y).
4350
f(x,f(x,f(x,y))) = f(y,f(f(x,y),y)).
4351
f(x,f(x,f(x,y))) = f(x,y).
4352
f(y,f(x,y)) = f(f(x,f(f(y,x),x)),y).
4353
f(y,f(x,y)) = f(f(y,x),y).
4354
f(f(x,y),x) = f(x,f(y,x)).
4355
f(f(f(x,y),f(f(f(y,f(y,z)),f(x,y)),f(x,y))),f(z,y)) = f(y,f(f(f(z,y),y),y)).
4356
f(f(f(y,f(y,z)),f(x,y)),f(z,y)) = f(y,f(f(f(z,y),y),y)).
4357
f(f(f(x,f(x,y)),f(z,x)),f(y,x)) = f(f(y,x),x).
4358
f(f(f(x,y),f(f(f(y,f(f(y,z),y)),f(x,y)),f(x,y))),f(f(z,y),y)) = f(y,f(f(f(f(z,y),y),y),y)).
4359
f(f(f(x,y),f(f(f(y,f(y,f(z,y))),f(x,y)),f(x,y))),f(f(z,y),y)) = f(y,f(f(f(f(z,y),y),y),y)).
4360
f(f(f(y,f(y,f(z,y))),f(x,y)),f(f(z,y),y)) = f(y,f(f(f(f(z,y),y),y),y)).
4361
f(f(f(z,y),y),y) = f(y,f(f(f(f(z,y),y),y),y)).
4362
f(f(f(z,y),y),y) = f(y,f(f(y,z),y)).
4363
f(f(f(x,y),y),y) = f(y,f(y,f(x,y))).
4364
f(f(f(f(x,y),f(z,y)),f(f(x,y),f(f(x,y),f(z,y)))),f(f(y,f(y,x)),f(u,f(f(x,y),f(z,y))))) = f(y,f(f(f(x,y),y),y)).
4365
f(f(f(f(x,y),f(z,y)),f(f(x,y),f(f(x,y),f(z,y)))),f(f(y,f(y,x)),f(u,f(f(x,y),f(z,y))))) = f(y,f(y,f(y,f(x,y)))).
4366
f(f(f(f(x,y),f(z,y)),f(f(x,y),f(f(x,y),f(z,y)))),f(f(y,f(y,x)),f(u,f(f(x,y),f(z,y))))) = f(y,f(x,y)).
4367
f(f(f(f(f(x,y),y),f(z,y)),f(f(f(x,y),y),f(f(f(x,y),y),f(z,y)))),f(f(y,f(f(y,x),y)),f(u,f(f(f(x,y),y),f(z,y))))) = f(y,f(f(f(f(x,y),y),y),y)).
4368
f(f(f(f(f(x,y),y),f(z,y)),f(f(f(x,y),y),f(f(f(x,y),y),f(z,y)))),f(f(y,f(y,f(x,y))),f(u,f(f(f(x,y),y),f(z,y))))) = f(y,f(f(f(f(x,y),y),y),y)).
4369
f(y,f(f(x,y),y)) = f(y,f(f(f(f(x,y),y),y),y)).
4370
f(x,y) = f(y,f(f(f(f(x,y),y),y),y)).
4371
f(x,y) = f(y,f(f(y,f(y,f(x,y))),y)).
4372
f(x,y) = f(y,f(y,f(f(y,f(x,y)),y))).
4373
f(x,y) = f(y,f(y,f(y,f(f(x,y),y)))).
4374
f(x,y) = f(y,f(y,f(x,y))).
4375
f(x,f(x,f(y,x))) = f(y,x).
4376
f(f(f(f(x,y),f(f(x,y),f(y,f(y,x)))),f(x,y)),f(x,y)) = f(y,f(f(f(x,y),y),y)).
4377
f(f(f(x,y),f(f(f(x,y),f(y,f(y,x))),f(x,y))),f(x,y)) = f(y,f(f(f(x,y),y),y)).
4378
f(f(f(x,y),f(f(x,y),f(f(y,f(y,x)),f(x,y)))),f(x,y)) = f(y,f(f(f(x,y),y),y)).
4379
f(f(f(y,f(y,x)),f(x,y)),f(x,y)) = f(y,f(f(f(x,y),y),y)).
4380
f(f(x,y),y) = f(y,f(f(f(x,y),y),y)).
4381
f(f(x,y),y) = f(y,f(y,f(y,f(x,y)))).
4382
f(f(x,y),y) = f(y,f(x,y)).
4383
f(f(f(x,f(f(y,f(y,f(x,y))),x)),x),x) = f(y,f(f(x,y),y)).
4384
f(f(f(x,f(f(x,y),x)),x),x) = f(y,f(f(x,y),y)).
4385
f(f(f(x,f(x,f(y,x))),x),x) = f(y,f(f(x,y),y)).
4386
f(f(f(y,x),x),x) = f(y,f(f(x,y),y)).
4387
f(f(x,f(y,x)),x) = f(y,f(f(x,y),y)).
4388
f(x,f(f(y,x),x)) = f(y,f(f(x,y),y)).
4389
f(x,f(x,f(y,x))) = f(y,f(f(x,y),y)).
4390
f(y,x) = f(y,f(f(x,y),y)).
4391
f(y,x) = f(y,f(y,f(x,y))).
4046
f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(f(f(f(x,f(f(x,x),x)),y),y),y)) = f(x,f(f(x,x),x)).
4047
f(f(f(x,f(x,f(x,x))),f(x,f(x,f(f(x,x),x)))),f(f(f(f(x,f(f(x,x),x)),y),y),y)) = f(x,f(f(x,x),x)).
4048
f(f(f(x,x),f(x,f(x,f(f(x,x),x)))),f(f(f(f(x,f(f(x,x),x)),y),y),y)) = f(x,f(f(x,x),x)).
4049
f(f(f(x,x),f(x,f(x,f(x,f(x,x))))),f(f(f(f(x,f(f(x,x),x)),y),y),y)) = f(x,f(f(x,x),x)).
4050
f(f(f(x,x),f(x,f(x,x))),f(f(f(f(x,f(f(x,x),x)),y),y),y)) = f(x,f(f(x,x),x)).
4051
f(x,f(f(f(f(x,f(f(x,x),x)),y),y),y)) = f(x,f(f(x,x),x)).
4052
f(x,f(f(f(f(x,f(x,f(x,x))),y),y),y)) = f(x,f(f(x,x),x)).
4053
f(x,f(f(f(f(x,x),y),y),y)) = f(x,f(f(x,x),x)).
4054
f(x,f(f(f(f(x,x),y),y),y)) = f(x,f(x,f(x,x))).
4055
f(x,f(f(f(f(x,x),y),y),y)) = f(x,x).
4056
f(f(x,f(x,f(x,x))),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
4057
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
4058
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
4059
f(f(x,f(y,y)),f(x,y)) = x.
4060
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
4061
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
4062
f(f(x,y),f(x,f(y,y))) = x.
4063
f(f(f(x,f(f(y,y),f(x,y))),x),x) = f(x,y).
4064
f(x,f(f(x,x),f(f(y,f(x,x)),f(x,x)))) = f(x,x).
4065
f(x,f(y,f(x,x))) = f(x,x).
4066
f(f(x,f(f(x,x),x)),f(y,x)) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
4067
f(f(x,f(x,f(x,x))),f(y,x)) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
4068
f(f(x,x),f(y,x)) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
4069
f(f(x,x),f(y,x)) = f(f(x,f(x,f(x,x))),f(x,f(f(x,x),x))).
4070
f(f(x,x),f(y,x)) = f(f(x,x),f(x,f(f(x,x),x))).
4071
f(f(x,x),f(y,x)) = f(f(x,x),f(x,f(x,f(x,x)))).
4072
f(f(x,x),f(y,x)) = f(f(x,x),f(x,x)).
4073
f(f(x,x),f(y,x)) = x.
4074
f(f(f(x,f(f(y,y),f(f(f(y,x),y),y))),x),x) = f(f(f(y,x),y),y).
4075
f(f(f(x,y),x),x) = f(f(f(y,x),y),y).
4076
f(f(f(x,y),x),x) = f(x,y).
4077
f(x,y) = f(f(f(y,x),y),y).
4392
4078
f(x,y) = f(y,x).
4393
4079
$F # answer("Commutativity").
4394
4080
end_of_list.
4410
4096
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4411
4097
end_of_list.
4412
4098
4413
Child search process 13976 started.
4099
Child search process 10948 started.
4414
4100
4415
4101
============================== PROCESS INITIAL CLAUSES ===============
4416
4102
4431
4117
formulas(demodulators).
4432
4118
end_of_list.
4433
4119
4434
% 1382 hints input.
4120
% 1016 hints input.
4435
4121
4436
4122
============================== PREDICATE ELIMINATION =================
4437
4123
4444
4130
4445
4131
Term ordering decisions:
4446
4132
Function symbol KB weights: c1=1. c2=1. f=1.
4447
Relation symbol precedence: lex([ = ]).
4448
Function symbol precedence: lex([ c1, c2, f ]).
4133
Predicate symbol precedence: predicate_order([ = ]).
4134
Function symbol precedence: function_order([ c1, c2, f ]).
4449
4135
Skipping inverse_order, because term ordering is KBO.
4450
4136
Unfolding symbols: (none).
4451
4137
4452
4138
Auto_inference settings:
4453
4139
% set(paramodulation). % (positive equality literals)
4454
% set(paramodulation) -> set(back_demod).
4455
4140
4456
4141
Auto_process settings: (no changes).
4457
4142
4143
kept: 2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
4144
kept: 3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
4145
kept: 4 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
4146
kept: 5 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
4147
kept: 6 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
4148
kept: 7 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4149
4458
4150
============================== end of process initial clauses ========
4459
4151
4460
4152
============================== CLAUSES FOR SEARCH ====================
4481
4173
7 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4482
4174
end_of_list.
4483
4175
4484
% 1382 hints processed (776 are redundant).
4176
% 420 hints (1016 processed, 596 redundant).
4485
4177
4486
4178
============================== end of clauses for search =============
4487
4179
4488
4180
============================== SEARCH ================================
4489
4181
4490
% Starting search at 0.02 seconds.
4182
% Starting search at 0.03 seconds.
4491
4183
4492
4184
% Operation f is commutative; C redundancy checks enabled.
4493
4185
4494
4186
============================== PROOF =================================
4495
4187
4496
% Proof 1 at 0.48 (+ 0.01) seconds: "Commutativity".
4497
% Length of proof is 39.
4498
% Level of proof is 14.
4188
% Proof 1 at 0.48 (+ 0.00) seconds: "Commutativity".
4189
% Length of proof is 59.
4190
% Level of proof is 22.
4499
4191
% Maximum clause weight is 55.
4500
% Given clauses 50.
4192
% Given clauses 51.
4501
4193
4502
1 f(x,y) = f(y,x) # label("Commutativity"). [goal].
4194
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
4503
4195
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
4504
4196
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
4505
4197
4 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
4506
5 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
4507
4198
6 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
4508
4199
7 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4509
4200
8 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)). [para(2(a,1),2(a,1,1,2,1))].
4510
10 f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)). [para(2(a,1),2(a,1,2))].
4511
43 f(f(f(x,f(f(y,z),f(z,f(f(x,z),z)))),x),x) = f(z,f(f(x,z),z)). [para(2(a,1),6(a,1,2))].
4512
4201
50 f(f(f(x,f(f(y,z),f(f(f(y,x),y),y))),x),x) = f(f(f(y,x),y),y). [para(4(a,1),6(a,1,2))].
4513
4202
68 f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)). [para(2(a,1),7(a,1,2))].
4203
72 f(f(f(x,f(x,f(f(f(y,x),y),y))),x),x) = f(f(f(y,x),y),y). [para(4(a,1),7(a,1,2))].
4204
83 f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(y,f(y,x)),y). [para(7(a,1),6(a,1,2))].
4514
4205
90 f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y. [para(68(a,1),2(a,1,2,2))].
4515
189 f(f(f(x,y),f(x,f(f(f(x,y),x),x))),f(x,f(f(f(x,y),x),x))) = f(f(f(x,f(x,f(f(f(x,y),x),x))),x),x). [para(2(a,1),50(a,1,1,1))].
4516
191 f(f(f(f(x,y),x),x),f(f(f(x,z),f(f(f(x,y),x),x)),f(y,u))) = f(f(x,z),f(f(f(x,y),x),x)). [para(50(a,1),4(a,1,1))].
4517
232 f(f(f(f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),u),f(f(x,f(y,z)),f(x,f(x,f(y,z))))),f(f(x,f(y,z)),f(x,f(x,f(y,z))))),f(u,f(z,f(f(x,z),z)))) = u. [para(8(a,1),4(a,1,2,2))].
4518
256 f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y. [para(8(a,1),90(a,1,2)),rewrite(90(8),90(15))].
4519
259 f(x,f(f(x,x),x)) = f(x,x). [para(90(a,1),8(a,1,2)),rewrite(90(8)),flip(a)].
4520
266 f(f(x,f(f(y,x),x)),f(y,y)) = y. [back_rewrite(256),rewrite(259(6))].
4521
271 f(f(x,x),f(x,f(y,x))) = x. [para(259(a,1),2(a,1,1))].
4522
273 f(f(x,x),f(x,x)) = x. [para(259(a,1),2(a,1,2)),rewrite(259(3))].
4523
277 f(f(f(x,x),x),x) = f(x,x). [para(259(a,1),5(a,1,1,1)),rewrite(273(5))].
4524
300 f(f(x,f(x,x)),x) = f(x,x). [para(273(a,1),2(a,1,2)),rewrite(277(3))].
4525
345 f(f(f(x,y),x),x) = f(y,f(f(x,y),y)). [para(271(a,1),43(a,1,1,1,2))].
4526
425 f(f(x,x),f(x,f(x,y))) = x. [para(300(a,1),7(a,1,1))].
4527
530 f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)). [para(266(a,1),2(a,1,2))].
4528
740 f(f(x,f(y,f(f(x,y),y))),f(y,f(f(x,y),y))) = f(f(x,f(z,y)),f(x,f(x,f(z,y)))). [para(2(a,1),345(a,1,1,1)),rewrite(2(16))].
4529
778 f(f(f(x,f(x,f(f(f(x,y),x),x))),x),x) = f(y,f(f(x,y),y)). [para(345(a,2),68(a,1,1,1,2,2))].
4530
835 f(f(f(x,f(f(y,x),x)),f(f(f(x,z),x),x)),y) = f(x,f(f(y,x),x)). [para(345(a,1),10(a,1,1,2,2)),rewrite(191(15))].
4531
847 f(f(x,f(y,f(f(x,y),y))),f(y,f(f(x,y),y))) = x. [para(266(a,1),345(a,1,1,1)),rewrite(266(14),425(12))].
4532
911 f(f(x,f(y,z)),f(x,f(x,f(y,z)))) = x. [back_rewrite(740),rewrite(847(8)),flip(a)].
4533
921 f(f(f(x,f(x,f(f(f(x,y),x),x))),x),x) = f(x,y). [back_rewrite(189),rewrite(847(11)),flip(a)].
4534
1080 f(f(f(f(x,y),x),x),f(y,f(z,f(f(x,z),z)))) = y. [back_rewrite(232),rewrite(911(6),911(7),911(8))].
4535
1107 f(x,f(f(y,x),x)) = f(y,x). [back_rewrite(778),rewrite(921(7)),flip(a)].
4536
1117 f(f(f(x,y),x),x) = f(x,y). [back_rewrite(530),rewrite(1107(3),1107(4),1107(6))].
4537
1153 f(f(x,y),f(y,f(x,z))) = y. [back_rewrite(1080),rewrite(1117(3),1107(4))].
4538
1240 f(f(f(x,y),f(y,z)),x) = f(x,y). [back_rewrite(835),rewrite(1107(3),1117(4),1107(7))].
4539
1960 f(x,y) = f(y,x). [para(1153(a,1),1240(a,1,1))].
4540
1961 $F # answer("Commutativity"). [resolve(1960,a,3,a)].
4206
95 f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(y,z))) = f(y,f(x,f(f(y,x),x))). [para(68(a,1),4(a,1,1))].
4207
153 f(f(f(x,y),f(x,f(f(f(x,y),x),x))),f(x,f(f(f(x,y),x),x))) = f(f(f(x,f(x,f(f(f(x,y),x),x))),x),x). [para(2(a,1),50(a,1,1,1))].
4208
192 f(f(f(x,f(x,f(f(f(x,y),x),x))),x),x) = f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x). [para(2(a,1),83(a,1,1,1)),rewrite([153(11)])].
4209
223 f(f(f(x,y),f(x,f(f(f(x,y),x),x))),f(x,f(f(f(x,y),x),x))) = f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x). [back_rewrite(153),rewrite([192(18)])].
4210
265 f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x. [para(95(a,1),2(a,1,1))].
4211
309 f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(y,x))) = f(x,f(f(x,x),x)). [para(2(a,1),265(a,1,1,2,2,1)),rewrite([90(11),2(17),90(15)])].
4212
313 f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))). [para(265(a,1),4(a,1,2)),rewrite([68(7)])].
4213
318 f(f(x,f(x,f(x,f(f(x,x),x)))),f(f(y,f(x,f(f(x,x),x))),f(f(x,f(x,f(f(x,x),x))),x))) = f(y,f(x,f(f(x,x),x))). [para(265(a,1),6(a,1,1,1))].
4214
333 f(f(x,f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(y,x))) = f(x,f(f(x,x),x)). [back_rewrite(309),rewrite([313(4)])].
4215
376 f(f(f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),u)),f(f(x,f(y,z)),f(x,f(x,f(y,z))))),f(u,f(z,f(f(x,z),z)))) = u. [para(8(a,1),7(a,1,2,2))].
4216
387 f(x,f(f(x,x),x)) = f(x,x). [para(90(a,1),8(a,1,2)),rewrite([90(8)]),flip(a)].
4217
449 f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,x). [back_rewrite(333),rewrite([387(3),387(5),387(9)])].
4218
464 f(f(x,f(x,f(x,x))),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)). [back_rewrite(318),rewrite([387(3),387(6),387(8),387(13)])].
4219
469 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(313),rewrite([387(3),387(5)])].
4220
489 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(387),rewrite([469(2)])].
4221
499 f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)). [back_rewrite(464),rewrite([489(3)])].
4222
511 f(f(x,x),f(x,f(y,x))) = x. [para(469(a,1),2(a,1,1,2)),rewrite([489(3)])].
4223
517 f(f(x,f(x,x)),x) = f(x,x). [para(469(a,1),68(a,1,1,1,2,2,2)),rewrite([489(3),489(3),469(2),469(5),489(6)])].
4224
518 f(f(x,x),f(x,f(y,f(f(x,y),y)))) = x. [para(469(a,1),90(a,1,1,2)),rewrite([489(3)])].
4225
526 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)). [back_rewrite(499),rewrite([517(6)])].
4226
530 f(f(x,x),f(x,x)) = x. [para(2(a,1),489(a,1,2,2)),rewrite([469(2),489(3),469(3),489(4),469(3),511(4),469(2),489(3),469(3),489(4)]),flip(a)].
4227
542 f(f(x,f(y,y)),f(x,y)) = x. [para(530(a,1),2(a,1,2,2)),rewrite([526(6)])].
4228
590 f(f(x,x),f(x,f(x,y))) = x. [para(517(a,1),7(a,1,1))].
4229
604 f(f(x,y),f(x,f(y,y))) = x. [para(2(a,1),542(a,1,1,2)),rewrite([469(3),489(4)])].
4230
703 f(f(f(x,f(x,f(x,y))),x),x) = f(x,y). [para(604(a,1),7(a,1,2))].
4231
726 f(f(x,f(f(y,x),x)),y) = f(y,f(x,f(f(y,x),x))). [para(604(a,1),95(a,1,2))].
4232
809 f(f(f(f(x,f(y,f(f(x,y),y))),f(f(f(x,x),z),x)),f(x,f(y,f(f(x,y),y)))),f(x,f(y,f(f(x,y),y)))) = x. [para(518(a,1),50(a,1,1,1,2,2,1,1)),rewrite([542(10),518(24),542(22)])].
4233
858 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [para(703(a,1),2(a,1,2,2))].
4234
860 f(f(x,f(y,f(f(x,y),y))),f(y,f(f(x,y),y))) = x. [para(2(a,1),703(a,1,1,1,2,2)),rewrite([726(7),858(8),2(14)])].
4235
912 f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x) = f(x,y). [back_rewrite(223),rewrite([860(11)]),flip(a)].
4236
963 f(f(f(x,f(x,f(f(f(x,y),x),x))),x),x) = f(x,y). [back_rewrite(192),rewrite([912(14)])].
4237
986 f(f(f(f(x,f(y,z)),f(f(f(y,f(f(x,y),y)),u),x)),f(x,f(y,z))),f(x,f(y,z))) = x. [para(858(a,1),50(a,1,1,1,2,2,1,1)),rewrite([860(14),858(20),860(22)])].
4238
1014 f(f(x,f(f(f(x,y),x),x)),x) = f(x,y). [para(604(a,1),858(a,1,2))].
4239
1029 f(x,f(f(y,x),x)) = f(y,x). [para(449(a,1),6(a,1,1,1)),rewrite([590(4),542(5)])].
4240
1052 f(f(f(x,y),x),x) = f(x,y). [back_rewrite(1014),rewrite([1029(4)])].
4241
1069 f(f(x,f(y,z)),f(f(f(x,y),u),x)) = x. [back_rewrite(986),rewrite([1029(5),1052(12)])].
4242
1086 f(x,f(x,f(x,y))) = f(x,y). [back_rewrite(963),rewrite([1052(3),1052(5)])].
4243
1128 f(f(x,y),f(x,f(y,z))) = x. [back_rewrite(858),rewrite([1029(3)])].
4244
1133 f(f(x,y),f(x,f(x,y))) = x. [back_rewrite(809),rewrite([1029(3),1069(6),1029(3),1086(3),1029(4)])].
4245
1145 f(f(x,y),x) = f(x,f(x,y)). [back_rewrite(726),rewrite([1029(3),1029(5)])].
4246
1174 f(f(x,y),f(y,f(x,z))) = y # label("C1/A1"). [back_rewrite(376),rewrite([1133(6),1133(6),1133(8),1145(3),1086(3),1029(4)])].
4247
1301 f(x,f(x,f(y,x))) = f(y,x). [back_rewrite(72),rewrite([1145(2),1145(3),1086(3),1145(4),1086(4),1145(3),1145(5),1145(6),1086(6)])].
4248
1410 f(f(x,y),f(y,f(z,x))) = y # label("C8/A12_m"). [para(1301(a,1),1174(a,1,2,2))].
4249
1411 f(f(x,x),f(y,x)) = x. [para(1301(a,1),1174(a,1,2))].
4250
1580 f(f(f(x,y),f(z,x)),z) = f(z,x). [para(1128(a,1),1410(a,1,2))].
4251
1599 f(x,y) = f(y,x). [para(1411(a,1),1580(a,1,1))].
4252
1600 $F # answer("Commutativity"). [resolve(1599,a,3,a)].
4541
4253
4542
4254
============================== end of proof ==========================
4543
4255
4544
4256
============================== STATISTICS ============================
4545
4257
4546
Given=50. Generated=4622. Kept=1959. proofs=1.
4547
Usable=15. Sos=446. Demods=466. Limbo=6, Disabled=1497. Hints=1382.
4548
Weight_deleted=118. Literals_deleted=0.
4549
Forward_subsumed=2545. Back_subsumed=199.
4258
Given=51. Generated=4232. Kept=1598. proofs=1.
4259
Usable=21. Sos=356. Demods=381. Limbo=8, Disabled=1218. Hints=1016.
4260
Kept_by_rule=0, Deleted_by_rule=84.
4261
Forward_subsumed=2550. Back_subsumed=221.
4550
4262
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
4551
New_demodulators=1933 (0 lex), Back_demodulated=1292. Back_unit_deleted=0.
4552
Demod_attempts=113300. Demod_rewrites=10237.
4263
New_demodulators=1592 (0 lex), Back_demodulated=991. Back_unit_deleted=0.
4264
Demod_attempts=111852. Demod_rewrites=12132.
4553
4265
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
4554
4266
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
4555
Megabytes=3.25.
4556
User_CPU=0.48, System_CPU=0.01, Wall_clock=11.
4267
Megabytes=2.67.
4268
User_CPU=0.48, System_CPU=0.00, Wall_clock=7.
4557
4269
4558
4270
============================== end of statistics =====================
4559
4271
4561
4273
4562
4274
Exiting with 1 proof.
4563
4275
4564
Process 13976 exit (max_proofs) Sun Aug 13 21:05:08 2006
4276
Process 10948 exit (max_proofs) Mon Sep 28 11:57:16 2009
4565
4277
4566
Successful proof using the following 4 assumptions:
4278
Successful proof using the following 3 assumptions:
4567
4279
4568
4280
formulas(assumptions_in_proof).
4569
4281
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
4570
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
4571
4282
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
4572
4283
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4573
4284
end_of_list.
4574
4285
4575
Including 60 new hint clauses:
4286
Including 125 new hint clauses:
4576
4287
4577
4288
formulas(new_hints).
4578
$F # label("Commutativity").
4289
$F # label("Commutativity") # label(non_clause) # label(goal).
4579
4290
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
4580
4291
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
4581
4292
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
4582
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
4583
4293
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
4584
4294
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
4585
4295
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)).
4586
f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)).
4587
f(f(f(x,f(f(y,z),f(z,f(f(x,z),z)))),x),x) = f(z,f(f(x,z),z)).
4588
4296
f(f(f(x,f(f(y,z),f(f(f(y,x),y),y))),x),x) = f(f(f(y,x),y),y).
4589
4297
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
4298
f(f(f(x,f(x,f(f(f(y,x),y),y))),x),x) = f(f(f(y,x),y),y).
4299
f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(y,f(y,x)),y).
4590
4300
f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y.
4301
f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(y,z))) = f(y,f(x,f(f(y,x),x))).
4591
4302
f(f(f(x,y),f(x,f(f(f(x,y),x),x))),f(x,f(f(f(x,y),x),x))) = f(f(f(x,f(x,f(f(f(x,y),x),x))),x),x).
4592
f(f(f(f(x,y),x),x),f(f(f(x,z),f(f(f(x,y),x),x)),f(y,u))) = f(f(x,z),f(f(f(x,y),x),x)).
4593
f(f(f(f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),u),f(f(x,f(y,z)),f(x,f(x,f(y,z))))),f(f(x,f(y,z)),f(x,f(x,f(y,z))))),f(u,f(z,f(f(x,z),z)))) = u.
4594
f(f(x,f(f(f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))),x),x)),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
4595
f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
4596
f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y.
4303
f(f(f(x,y),f(x,f(f(f(x,y),x),x))),f(x,f(f(f(x,y),x),x))) = f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x).
4304
f(f(f(x,f(x,f(f(f(x,y),x),x))),x),x) = f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x).
4305
f(f(f(x,y),f(x,f(f(f(x,y),x),x))),f(x,f(f(f(x,y),x),x))) = f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x).
4306
f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x.
4307
f(f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(x,f(f(x,x),x)),f(y,f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))))))) = f(x,f(f(x,x),x)).
4308
f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(y,f(f(x,f(f(x,x),x)),f(f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))))))) = f(x,f(f(x,x),x)).
4309
f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(y,f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))))) = f(x,f(f(x,x),x)).
4310
f(f(f(x,f(f(x,x),x)),x),f(f(x,f(f(x,x),x)),f(y,x))) = f(x,f(f(x,x),x)).
4311
f(f(f(f(x,f(x,f(x,f(f(x,x),x)))),x),x),x) = f(x,f(x,f(f(x,x),x))).
4312
f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))).
4313
f(f(x,f(x,f(x,f(f(x,x),x)))),f(f(y,f(x,f(f(x,x),x))),f(f(x,f(x,f(f(x,x),x))),x))) = f(y,f(x,f(f(x,x),x))).
4314
f(f(x,f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(y,x))) = f(x,f(f(x,x),x)).
4315
f(f(f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),u)),f(f(x,f(y,z)),f(x,f(x,f(y,z))))),f(u,f(z,f(f(x,z),z)))) = u.
4597
4316
f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),x) = f(x,f(f(x,x),x)).
4598
4317
f(x,x) = f(x,f(f(x,x),x)).
4599
4318
f(x,f(f(x,x),x)) = f(x,x).
4600
f(f(x,f(f(y,x),x)),f(y,y)) = y.
4319
f(f(x,f(x,x)),f(f(x,f(f(x,x),x)),f(y,x))) = f(x,f(f(x,x),x)).
4320
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,f(f(x,x),x)).
4321
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,x).
4322
f(f(x,f(x,f(x,x))),f(f(y,f(x,f(f(x,x),x))),f(f(x,f(x,f(f(x,x),x))),x))) = f(y,f(x,f(f(x,x),x))).
4323
f(f(x,f(x,f(x,x))),f(f(y,f(x,x)),f(f(x,f(x,f(f(x,x),x))),x))) = f(y,f(x,f(f(x,x),x))).
4324
f(f(x,f(x,f(x,x))),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,f(f(x,x),x))).
4325
f(f(x,f(x,f(x,x))),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
4326
f(f(x,x),x) = f(x,f(x,f(f(x,x),x))).
4327
f(f(x,x),x) = f(x,f(x,x)).
4328
f(x,f(x,f(x,x))) = f(x,x).
4329
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
4330
f(f(x,f(x,f(x,x))),f(x,f(y,x))) = x.
4601
4331
f(f(x,x),f(x,f(y,x))) = x.
4602
f(f(x,f(f(x,x),x)),f(x,x)) = x.
4332
f(f(f(x,f(x,f(x,f(x,f(x,x))))),x),x) = f(x,f(f(x,x),x)).
4333
f(f(f(x,f(x,f(x,x))),x),x) = f(x,f(f(x,x),x)).
4334
f(f(f(x,x),x),x) = f(x,f(f(x,x),x)).
4335
f(f(x,f(x,x)),x) = f(x,f(f(x,x),x)).
4336
f(f(x,f(x,x)),x) = f(x,f(x,f(x,x))).
4337
f(f(x,f(x,x)),x) = f(x,x).
4338
f(f(x,f(x,f(x,x))),f(x,f(y,f(f(x,y),y)))) = x.
4339
f(f(x,x),f(x,f(y,f(f(x,y),y)))) = x.
4340
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
4341
f(f(x,f(f(x,x),x)),f(f(x,f(f(x,x),x)),x)) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
4342
f(f(x,f(x,f(x,x))),f(f(x,f(f(x,x),x)),x)) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
4343
f(f(x,x),f(f(x,f(f(x,x),x)),x)) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
4344
f(f(x,x),f(f(x,f(x,f(x,x))),x)) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
4345
f(f(x,x),f(f(x,x),x)) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
4346
f(f(x,x),f(x,f(x,x))) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
4347
x = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
4348
x = f(f(x,f(x,f(x,x))),f(x,f(f(x,x),x))).
4349
x = f(f(x,x),f(x,f(f(x,x),x))).
4350
x = f(f(x,x),f(x,f(x,f(x,x)))).
4351
x = f(f(x,x),f(x,x)).
4603
4352
f(f(x,x),f(x,x)) = x.
4604
f(f(f(x,x),x),f(f(x,x),f(x,x))) = f(x,x).
4605
f(f(f(x,x),x),x) = f(x,x).
4606
f(f(x,f(f(f(x,x),x),x)),x) = f(x,x).
4607
f(f(x,f(x,x)),x) = f(x,x).
4608
f(f(f(x,y),x),x) = f(y,f(f(x,y),y)).
4353
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
4354
f(f(x,f(y,y)),f(x,y)) = x.
4609
4355
f(f(x,x),f(x,f(x,y))) = x.
4610
f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)).
4611
f(f(x,f(y,f(f(x,y),y))),f(y,f(f(x,y),y))) = f(f(x,f(z,y)),f(f(f(y,f(f(x,y),y)),f(x,f(z,y))),f(x,f(z,y)))).
4612
f(f(x,f(y,f(f(x,y),y))),f(y,f(f(x,y),y))) = f(f(x,f(z,y)),f(x,f(x,f(z,y)))).
4613
f(f(f(x,f(x,f(f(f(x,y),x),x))),x),x) = f(y,f(f(x,y),y)).
4614
f(f(f(f(f(x,y),x),x),f(f(f(x,f(f(z,x),x)),f(f(f(x,y),x),x)),f(y,f(f(x,y),y)))),z) = f(x,f(f(z,x),x)).
4615
f(f(f(x,f(f(y,x),x)),f(f(f(x,z),x),x)),y) = f(x,f(f(y,x),x)).
4616
f(f(x,f(y,f(f(x,y),y))),f(y,f(f(x,y),y))) = f(f(x,x),f(f(f(y,f(f(x,y),y)),f(x,x)),f(x,x))).
4617
f(f(x,f(y,f(f(x,y),y))),f(y,f(f(x,y),y))) = f(f(x,x),f(x,f(x,x))).
4356
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
4357
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
4358
f(f(x,y),f(x,f(y,y))) = x.
4359
f(f(f(x,f(x,f(x,y))),x),x) = f(x,y).
4360
f(f(x,f(f(y,x),x)),y) = f(y,f(x,f(f(y,x),x))).
4361
f(f(f(f(x,f(y,f(f(x,y),y))),f(f(f(x,x),z),f(f(x,f(x,x)),f(x,x)))),f(x,f(y,f(f(x,y),y)))),f(x,f(y,f(f(x,y),y)))) = f(f(f(f(x,x),f(x,f(y,f(f(x,y),y)))),f(x,x)),f(x,x)).
4362
f(f(f(f(x,f(y,f(f(x,y),y))),f(f(f(x,x),z),x)),f(x,f(y,f(f(x,y),y)))),f(x,f(y,f(f(x,y),y)))) = f(f(f(f(x,x),f(x,f(y,f(f(x,y),y)))),f(x,x)),f(x,x)).
4363
f(f(f(f(x,f(y,f(f(x,y),y))),f(f(f(x,x),z),x)),f(x,f(y,f(f(x,y),y)))),f(x,f(y,f(f(x,y),y)))) = f(f(x,f(x,x)),f(x,x)).
4364
f(f(f(f(x,f(y,f(f(x,y),y))),f(f(f(x,x),z),x)),f(x,f(y,f(f(x,y),y)))),f(x,f(y,f(f(x,y),y)))) = x.
4365
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
4366
f(f(f(f(x,f(f(y,x),x)),f(f(x,f(f(y,x),x)),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x))) = f(f(x,f(f(y,x),x)),f(y,f(z,x))).
4367
f(f(f(f(x,f(f(y,x),x)),f(y,f(x,f(f(y,x),x)))),f(x,f(f(y,x),x))),f(x,f(f(y,x),x))) = f(f(x,f(f(y,x),x)),f(y,f(z,x))).
4368
f(f(y,f(x,f(f(y,x),x))),f(x,f(f(y,x),x))) = f(f(x,f(f(y,x),x)),f(y,f(z,x))).
4618
4369
f(f(x,f(y,f(f(x,y),y))),f(y,f(f(x,y),y))) = x.
4619
x = f(f(x,f(z,y)),f(x,f(x,f(z,y)))).
4620
f(f(x,f(y,z)),f(x,f(x,f(y,z)))) = x.
4621
f(x,y) = f(f(f(x,f(x,f(f(f(x,y),x),x))),x),x).
4370
f(x,y) = f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x).
4371
f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x) = f(x,y).
4622
4372
f(f(f(x,f(x,f(f(f(x,y),x),x))),x),x) = f(x,y).
4623
f(f(f(f(x,u),f(f(x,f(y,z)),f(x,f(x,f(y,z))))),f(f(x,f(y,z)),f(x,f(x,f(y,z))))),f(u,f(z,f(f(x,z),z)))) = u.
4624
f(f(f(f(x,u),x),f(f(x,f(y,z)),f(x,f(x,f(y,z))))),f(u,f(z,f(f(x,z),z)))) = u.
4625
f(f(f(f(x,y),x),x),f(y,f(z,f(f(x,z),z)))) = y.
4626
f(x,y) = f(y,f(f(x,y),y)).
4373
f(f(f(f(x,f(y,z)),f(f(f(y,f(f(x,y),y)),u),f(f(x,f(y,f(f(x,y),y))),f(y,f(f(x,y),y))))),f(x,f(y,z))),f(x,f(y,z))) = f(f(f(f(y,f(f(x,y),y)),f(x,f(y,z))),f(y,f(f(x,y),y))),f(y,f(f(x,y),y))).
4374
f(f(f(f(x,f(y,z)),f(f(f(y,f(f(x,y),y)),u),x)),f(x,f(y,z))),f(x,f(y,z))) = f(f(f(f(y,f(f(x,y),y)),f(x,f(y,z))),f(y,f(f(x,y),y))),f(y,f(f(x,y),y))).
4375
f(f(f(f(x,f(y,z)),f(f(f(y,f(f(x,y),y)),u),x)),f(x,f(y,z))),f(x,f(y,z))) = f(f(x,f(y,f(f(x,y),y))),f(y,f(f(x,y),y))).
4376
f(f(f(f(x,f(y,z)),f(f(f(y,f(f(x,y),y)),u),x)),f(x,f(y,z))),f(x,f(y,z))) = x.
4377
f(f(x,f(f(f(x,y),x),x)),x) = f(x,y).
4378
f(f(f(x,x),f(x,f(x,x))),f(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x).
4379
f(x,f(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x).
4627
4380
f(x,f(f(y,x),x)) = f(y,x).
4628
f(f(x,f(f(f(x,y),x),x)),x) = f(y,f(f(x,y),y)).
4629
f(f(f(x,y),x),x) = f(y,f(f(x,y),y)).
4630
4381
f(f(f(x,y),x),x) = f(x,y).
4631
f(f(x,y),f(y,f(z,f(f(x,z),z)))) = y.
4382
f(f(f(f(x,f(y,z)),f(f(f(x,y),u),x)),f(x,f(y,z))),f(x,f(y,z))) = x.
4383
f(f(x,f(y,z)),f(f(f(x,y),u),x)) = x.
4384
f(f(f(x,f(x,f(x,y))),x),x) = f(x,y).
4385
f(x,f(x,f(x,y))) = f(x,y).
4386
f(f(x,y),f(x,f(y,z))) = x.
4387
f(f(f(f(x,f(x,y)),f(f(f(x,x),z),x)),f(x,f(y,f(f(x,y),y)))),f(x,f(y,f(f(x,y),y)))) = x.
4388
f(f(x,f(x,f(y,f(f(x,y),y)))),f(x,f(y,f(f(x,y),y)))) = x.
4389
f(f(x,f(x,f(x,y))),f(x,f(y,f(f(x,y),y)))) = x.
4390
f(f(x,y),f(x,f(y,f(f(x,y),y)))) = x.
4391
f(f(x,y),f(x,f(x,y))) = x.
4392
f(f(y,x),y) = f(y,f(x,f(f(y,x),x))).
4393
f(f(x,y),x) = f(x,f(x,y)).
4394
f(f(f(x,f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),u)),f(f(x,f(y,z)),f(x,f(x,f(y,z))))),f(u,f(z,f(f(x,z),z)))) = u.
4395
f(f(f(x,f(x,u)),f(f(x,f(y,z)),f(x,f(x,f(y,z))))),f(u,f(z,f(f(x,z),z)))) = u.
4396
f(f(f(x,f(x,u)),x),f(u,f(z,f(f(x,z),z)))) = u.
4397
f(f(x,f(x,f(x,u))),f(u,f(z,f(f(x,z),z)))) = u.
4398
f(f(x,u),f(u,f(z,f(f(x,z),z)))) = u.
4632
4399
f(f(x,y),f(y,f(x,z))) = y.
4633
f(f(f(y,x),f(f(f(x,z),x),x)),y) = f(x,f(f(y,x),x)).
4634
f(f(f(y,x),f(x,z)),y) = f(x,f(f(y,x),x)).
4635
f(f(f(x,y),f(y,z)),x) = f(x,y).
4400
f(f(f(x,f(x,f(f(y,f(y,x)),y))),x),x) = f(f(f(y,x),y),y).
4401
f(f(f(x,f(x,f(y,f(y,f(y,x))))),x),x) = f(f(f(y,x),y),y).
4402
f(f(f(x,f(x,f(y,x))),x),x) = f(f(f(y,x),y),y).
4403
f(f(x,f(x,f(x,f(y,x)))),x) = f(f(f(y,x),y),y).
4404
f(f(x,f(y,x)),x) = f(f(f(y,x),y),y).
4405
f(x,f(x,f(y,x))) = f(f(f(y,x),y),y).
4406
f(x,f(x,f(y,x))) = f(f(y,f(y,x)),y).
4407
f(x,f(x,f(y,x))) = f(y,f(y,f(y,x))).
4408
f(x,f(x,f(y,x))) = f(y,x).
4409
f(f(x,y),f(y,f(z,x))) = y # label("C8/A12_m").
4410
f(f(x,x),f(y,x)) = x.
4411
f(f(f(x,y),f(z,x)),z) = f(z,x).
4636
4412
f(x,y) = f(y,x).
4637
4413
$F # answer("Commutativity").
4638
4414
end_of_list.
4653
4429
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4654
4430
end_of_list.
4655
4431
4656
Child search process 13977 started.
4432
Child search process 10949 started.
4657
4433
4658
4434
============================== PROCESS INITIAL CLAUSES ===============
4659
4435
4673
4449
formulas(demodulators).
4674
4450
end_of_list.
4675
4451
4676
% 1442 hints input.
4452
% 1141 hints input.
4677
4453
4678
4454
============================== PREDICATE ELIMINATION =================
4679
4455
4686
4462
4687
4463
Term ordering decisions:
4688
4464
Function symbol KB weights: c1=1. c2=1. f=1.
4689
Relation symbol precedence: lex([ = ]).
4690
Function symbol precedence: lex([ c1, c2, f ]).
4465
Predicate symbol precedence: predicate_order([ = ]).
4466
Function symbol precedence: function_order([ c1, c2, f ]).
4691
4467
Skipping inverse_order, because term ordering is KBO.
4692
4468
Unfolding symbols: (none).
4693
4469
4694
4470
Auto_inference settings:
4695
4471
% set(paramodulation). % (positive equality literals)
4696
% set(paramodulation) -> set(back_demod).
4697
4472
4698
4473
Auto_process settings: (no changes).
4699
4474
4475
kept: 2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
4476
kept: 3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
4477
kept: 4 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
4478
kept: 5 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
4479
kept: 6 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4480
4700
4481
============================== end of process initial clauses ========
4701
4482
4702
4483
============================== CLAUSES FOR SEARCH ====================
4721
4502
6 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4722
4503
end_of_list.
4723
4504
4724
% 1442 hints processed (816 are redundant).
4505
% 488 hints (1141 processed, 653 redundant).
4725
4506
4726
4507
============================== end of clauses for search =============
4727
4508
4728
4509
============================== SEARCH ================================
4729
4510
4730
% Starting search at 0.03 seconds.
4511
% Starting search at 0.04 seconds.
4731
4512
4732
4513
% Operation f is commutative; C redundancy checks enabled.
4733
4514
4734
4515
============================== PROOF =================================
4735
4516
4736
% Proof 1 at 0.38 (+ 0.00) seconds: "Commutativity".
4737
% Length of proof is 38.
4738
% Level of proof is 15.
4739
% Maximum clause weight is 55.
4740
% Given clauses 49.
4517
% Proof 1 at 0.49 (+ 0.00) seconds: "Commutativity".
4518
% Length of proof is 54.
4519
% Level of proof is 26.
4520
% Maximum clause weight is 75.
4521
% Given clauses 42.
4741
4522
4742
1 f(x,y) = f(y,x) # label("Commutativity"). [goal].
4523
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
4743
4524
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
4744
4525
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
4745
4526
4 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
4746
4527
5 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
4747
4528
6 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4748
4529
7 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)). [para(2(a,1),2(a,1,1,2,1))].
4749
10 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(f(y,f(x,z)),y),f(u,f(x,f(y,z))))) = f(f(y,f(x,z)),y). [para(4(a,1),2(a,1,1,2,1))].
4750
4530
44 f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)). [para(2(a,1),6(a,1,2))].
4751
78 f(f(f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),u)),f(f(x,f(y,z)),f(x,f(x,f(y,z))))),f(u,f(z,f(f(x,z),z)))) = u. [para(7(a,1),6(a,1,2,2))].
4752
85 f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y. [para(44(a,1),2(a,1,2,2))].
4753
108 f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)). [para(85(a,1),2(a,1,2))].
4754
112 f(f(x,f(y,f(f(x,y),y))),f(x,f(f(y,f(f(x,y),y)),f(z,f(f(y,z),z))))) = x. [para(85(a,1),4(a,1,1,1))].
4755
114 f(f(f(x,f(f(y,f(f(x,y),y)),f(z,f(f(y,z),z)))),x),x) = f(y,f(f(x,y),y)). [para(85(a,1),4(a,1,2))].
4756
130 f(x,f(f(x,x),x)) = f(x,x). [para(85(a,1),7(a,1,2)),rewrite(85(8)),flip(a)].
4757
133 f(f(x,f(f(f(f(y,y),f(y,f(y,y))),x),x)),f(y,y)) = f(f(y,y),f(y,f(y,y))). [para(7(a,1),85(a,1,2)),rewrite(130(3),130(4),130(10),130(12),130(13))].
4758
147 f(f(x,x),f(x,f(y,x))) = x. [para(130(a,1),2(a,1,1))].
4759
149 f(f(x,x),f(x,x)) = x. [para(130(a,1),2(a,1,2)),rewrite(130(3))].
4760
151 f(f(f(x,x),x),x) = f(x,x). [para(130(a,1),4(a,1,1,1)),rewrite(149(5))].
4761
153 f(f(x,f(x,x)),f(x,x)) = x. [para(130(a,1),4(a,1,2)),rewrite(149(3))].
4762
167 f(f(x,f(f(y,x),x)),f(y,y)) = y. [back_rewrite(133),rewrite(147(4),147(9))].
4763
171 f(f(x,f(x,x)),x) = f(x,x). [para(149(a,1),2(a,1,2)),rewrite(151(3))].
4764
235 f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,x). [para(151(a,1),2(a,1,1,2))].
4765
299 f(f(x,x),f(x,f(x,y))) = x. [para(171(a,1),6(a,1,1))].
4766
373 f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)). [para(167(a,1),2(a,1,2))].
4767
425 f(f(f(x,f(x,y)),f(x,f(x,f(x,y)))),y) = f(f(x,f(x,y)),x). [para(6(a,1),10(a,1,2))].
4768
716 f(x,f(f(y,x),x)) = f(y,x). [para(235(a,1),5(a,1,1,1)),rewrite(299(4),153(5))].
4769
883 f(f(f(x,y),x),x) = f(x,y). [back_rewrite(373),rewrite(716(3),716(4),716(6))].
4770
978 f(x,f(f(x,y),f(y,z))) = f(x,y). [back_rewrite(114),rewrite(716(3),716(4),883(6),716(7))].
4771
980 f(f(x,f(x,y)),f(x,y)) = x. [back_rewrite(112),rewrite(716(3),716(5),716(6),978(6))].
4772
983 f(f(f(x,y),f(y,z)),x) = f(x,y). [back_rewrite(108),rewrite(716(3),716(4),716(5),716(7),716(7),716(7))].
4773
1007 f(f(f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),u)),f(f(x,f(y,z)),f(x,f(x,f(y,z))))),f(u,f(x,z))) = u. [back_rewrite(78),rewrite(716(24))].
4774
1023 f(x,f(x,f(x,y))) = f(x,y). [back_rewrite(44),rewrite(716(3),883(5),716(6))].
4775
1038 f(f(x,y),f(x,f(z,y))) = x. [back_rewrite(2),rewrite(716(3))].
4776
1095 f(f(x,f(x,y)),x) = f(x,y). [back_rewrite(425),rewrite(1023(5),980(4)),flip(a)].
4777
1109 f(f(x,y),f(y,f(x,z))) = y. [back_rewrite(1007),rewrite(1038(6),1038(6),1038(8),1095(3))].
4778
1681 f(x,y) = f(y,x). [para(1109(a,1),983(a,1,1))].
4779
1682 $F # answer("Commutativity"). [resolve(1681,a,3,a)].
4531
55 f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(y,f(y,x)),y). [para(6(a,1),5(a,1,2))].
4532
62 f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y. [para(44(a,1),2(a,1,2,2))].
4533
117 f(f(f(x,y),f(x,f(f(f(x,y),x),x))),f(x,f(f(f(x,y),x),x))) = f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x). [para(2(a,1),55(a,1,1,1))].
4534
134 f(f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),f(x,f(f(y,f(y,x)),y)))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))) = f(x,f(f(y,f(y,x)),y)). [para(55(a,1),44(a,1,1,1,2,2,2)),rewrite([55(40)])].
4535
200 f(x,f(f(x,x),x)) = f(x,x). [para(62(a,1),7(a,1,2)),rewrite([62(8)]),flip(a)].
4536
211 f(f(x,x),f(x,f(y,x))) = x. [para(200(a,1),2(a,1,1))].
4537
213 f(f(x,x),f(x,x)) = x. [para(200(a,1),2(a,1,2)),rewrite([200(3)])].
4538
215 f(f(f(x,x),x),x) = f(x,x). [para(200(a,1),4(a,1,1,1)),rewrite([213(5)])].
4539
217 f(f(x,f(x,x)),f(x,x)) = x. [para(200(a,1),4(a,1,2)),rewrite([213(3)])].
4540
236 f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y. [para(213(a,1),2(a,1,2,2))].
4541
237 f(f(x,f(x,x)),x) = f(x,x). [para(213(a,1),2(a,1,2)),rewrite([215(3)])].
4542
298 f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x. [para(217(a,1),2(a,1,1,2))].
4543
299 f(f(x,x),x) = f(x,f(x,x)). [para(217(a,1),2(a,1,2)),rewrite([237(3),200(3)])].
4544
320 f(x,f(x,f(x,x))) = f(x,x). [para(217(a,1),211(a,1,2,2)),rewrite([213(3),299(2)])].
4545
321 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [back_rewrite(298),rewrite([299(2)])].
4546
474 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)). [para(321(a,1),5(a,1,1,1)),rewrite([320(3),237(6)])].
4547
494 f(f(x,f(y,y)),f(x,y)) = x. [back_rewrite(236),rewrite([474(6)])].
4548
496 f(f(x,y),f(x,f(y,y))) = x. [para(2(a,1),494(a,1,1,2)),rewrite([299(3),320(4)])].
4549
573 f(f(f(x,f(x,f(x,y))),x),x) = f(x,y). [para(496(a,1),6(a,1,2))].
4550
627 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [para(573(a,1),2(a,1,2,2))].
4551
643 f(f(f(f(x,f(x,f(x,y))),f(f(x,f(x,f(x,y))),f(x,f(x,y)))),f(x,f(x,f(x,y)))),f(x,f(x,f(x,y)))) = f(x,f(x,y)). [para(573(a,1),44(a,1,1,1,2,2,2)),rewrite([573(23)])].
4552
722 f(f(x,f(f(f(x,y),x),x)),x) = f(x,y). [para(496(a,1),627(a,1,2))].
4553
740 f(f(x,y),f(x,f(x,f(f(x,y),x)))) = x. [para(722(a,1),5(a,1,1))].
4554
766 f(f(x,f(x,y)),x) = f(x,f(x,f(x,y))). [para(573(a,1),722(a,1,1,2))].
4555
769 f(f(f(x,f(f(f(x,y),x),x)),f(x,f(x,f(x,f(x,f(f(f(x,y),x),x)))))),f(x,f(f(f(x,y),x),x))) = f(x,y). [para(722(a,1),722(a,1,1,2,1,1)),rewrite([117(15),766(11),722(22)])].
4556
833 f(f(x,f(x,f(x,y))),f(f(x,f(x,f(x,y))),f(f(x,f(x,f(x,y))),f(f(x,f(x,f(x,y))),f(x,f(x,y)))))) = f(x,f(x,y)). [back_rewrite(643),rewrite([766(14),766(18)])].
4557
858 f(x,f(x,f(x,f(x,f(x,y))))) = f(x,y). [back_rewrite(573),rewrite([766(4),766(5)])].
4558
900 f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(x,f(y,f(y,f(y,x)))))))) = f(x,f(y,f(y,f(y,x)))). [back_rewrite(134),rewrite([766(4),766(10),766(15),766(22),766(25),766(29),766(32),766(35)])].
4559
932 f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(y,f(y,f(y,x))). [back_rewrite(55),rewrite([766(4),766(11)])].
4560
945 f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y # label("C1/A1"). [back_rewrite(6),rewrite([766(3)])].
4561
951 f(f(x,f(x,f(x,y))),f(x,f(x,y))) = x. [para(766(a,1),4(a,1,1))].
4562
974 f(f(x,y),x) = f(x,f(x,y)). [para(766(a,1),722(a,1,1,2,1)),rewrite([766(4),858(5)])].
4563
975 f(f(x,f(x,y)),f(f(x,f(x,y)),f(x,f(x,f(x,y))))) = f(x,f(x,f(x,y))). [para(766(a,1),722(a,2)),rewrite([974(5),951(8),974(9)])].
4564
990 f(x,f(x,f(x,f(x,y)))) = f(x,f(x,y)). [back_rewrite(833),rewrite([951(15),974(10),975(12)])].
4565
1015 f(x,f(x,f(x,f(f(y,z),f(y,f(y,f(y,x))))))) = f(y,f(y,f(y,x))). [back_rewrite(932),rewrite([974(7),974(8)])].
4566
1077 f(x,f(x,f(x,y))) = f(x,y). [back_rewrite(769),rewrite([974(2),974(3),990(4),974(4),974(5),990(6),990(6),974(8),974(9),990(10),974(9),975(9)])].
4567
1086 f(f(x,y),f(x,f(x,y))) = x. [back_rewrite(740),rewrite([974(3),1077(4)])].
4568
1244 f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(x,f(y,x)))) = f(x,f(y,x)). [back_rewrite(900),rewrite([1077(4),1077(8),1077(12),1077(16),1077(19),1077(21),1077(15)])].
4569
1284 f(x,f(f(y,z),f(y,x))) = f(y,x). [back_rewrite(1015),rewrite([1077(4),1077(6),1077(7)])].
4570
1311 f(f(x,y),f(y,f(x,z))) = y. [back_rewrite(945),rewrite([1077(3)])].
4571
1416 f(f(x,y),y) = f(y,f(x,y)). [back_rewrite(1244),rewrite([1284(4),1284(5),1311(5)])].
4572
1629 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [back_rewrite(2),rewrite([1416(2)])].
4573
1660 f(x,f(x,f(y,x))) = f(y,x). [para(1311(a,1),1086(a,1,1)),rewrite([1311(5),1416(2)])].
4574
1669 f(f(x,y),f(x,f(z,y))) = x. [back_rewrite(1629),rewrite([1660(3)])].
4575
1807 f(x,y) = f(y,x). [para(1416(a,1),1284(a,1,2)),rewrite([1669(4)])].
4576
1808 $F # answer("Commutativity"). [resolve(1807,a,3,a)].
4780
4577
4781
4578
============================== end of proof ==========================
4782
4579
4783
4580
============================== STATISTICS ============================
4784
4581
4785
Given=49. Generated=3930. Kept=1680. proofs=1.
4786
Usable=15. Sos=376. Demods=402. Limbo=12, Disabled=1281. Hints=1442.
4787
Weight_deleted=132. Literals_deleted=0.
4788
Forward_subsumed=2118. Back_subsumed=257.
4582
Given=42. Generated=3543. Kept=1806. proofs=1.
4583
Usable=12. Sos=222. Demods=238. Limbo=6, Disabled=1570. Hints=1141.
4584
Kept_by_rule=0, Deleted_by_rule=40.
4585
Forward_subsumed=1697. Back_subsumed=229.
4789
4586
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
4790
New_demodulators=1678 (0 lex), Back_demodulated=1019. Back_unit_deleted=0.
4791
Demod_attempts=99651. Demod_rewrites=9251.
4587
New_demodulators=1799 (0 lex), Back_demodulated=1336. Back_unit_deleted=0.
4588
Demod_attempts=92312. Demod_rewrites=9388.
4792
4589
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
4793
4590
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
4794
Megabytes=2.83.
4795
User_CPU=0.38, System_CPU=0.00, Wall_clock=11.
4591
Megabytes=2.62.
4592
User_CPU=0.49, System_CPU=0.00, Wall_clock=8.
4796
4593
4797
4594
============================== end of statistics =====================
4798
4595
4800
4597
4801
4598
Exiting with 1 proof.
4802
4599
4803
Process 13977 exit (max_proofs) Sun Aug 13 21:05:08 2006
4600
Process 10949 exit (max_proofs) Mon Sep 28 11:57:17 2009
4804
4601
4805
4602
Successful proof using the following 3 assumptions:
4806
4603
4810
4607
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4811
4608
end_of_list.
4812
4609
4813
Including 72 new hint clauses:
4610
Including 116 new hint clauses:
4814
4611
4815
4612
formulas(new_hints).
4816
$F # label("Commutativity").
4613
$F # label("Commutativity") # label(non_clause) # label(goal).
4817
4614
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
4818
4615
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
4819
4616
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
4820
4617
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
4821
4618
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
4822
4619
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)).
4823
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(f(y,f(x,z)),y),f(u,f(x,f(y,z))))) = f(f(y,f(x,z)),y).
4824
4620
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
4825
f(f(f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),u)),f(f(x,f(y,z)),f(x,f(x,f(y,z))))),f(u,f(z,f(f(x,z),z)))) = u.
4621
f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(y,f(y,x)),y).
4826
4622
f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y.
4827
f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
4828
f(f(x,f(y,f(f(x,y),y))),f(x,f(f(y,f(f(x,y),y)),f(z,f(f(y,z),z))))) = x.
4829
f(f(f(x,f(f(y,f(f(x,y),y)),f(z,f(f(y,z),z)))),x),x) = f(y,f(f(x,y),y)).
4623
f(f(f(x,y),f(x,f(f(f(x,y),x),x))),f(x,f(f(f(x,y),x),x))) = f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x).
4624
f(f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),f(x,f(f(y,f(y,x)),y)))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))) = f(x,f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x)).
4625
f(f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),f(x,f(f(y,f(y,x)),y)))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))) = f(x,f(f(y,f(y,x)),y)).
4830
4626
f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),x) = f(x,f(f(x,x),x)).
4831
4627
f(x,x) = f(x,f(f(x,x),x)).
4832
4628
f(x,f(f(x,x),x)) = f(x,x).
4833
f(f(x,f(f(f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))),x),x)),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
4834
f(f(x,f(f(f(f(y,y),f(y,f(y,f(f(y,y),y)))),x),x)),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
4835
f(f(x,f(f(f(f(y,y),f(y,f(y,y))),x),x)),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
4836
f(f(x,f(f(f(f(y,y),f(y,f(y,y))),x),x)),f(y,y)) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
4837
f(f(x,f(f(f(f(y,y),f(y,f(y,y))),x),x)),f(y,y)) = f(f(y,y),f(y,f(y,f(f(y,y),y)))).
4838
f(f(x,f(f(f(f(y,y),f(y,f(y,y))),x),x)),f(y,y)) = f(f(y,y),f(y,f(y,y))).
4839
4629
f(f(x,x),f(x,f(y,x))) = x.
4840
4630
f(f(x,f(f(x,x),x)),f(x,x)) = x.
4841
4631
f(f(x,x),f(x,x)) = x.
4843
4633
f(f(f(x,x),x),x) = f(x,x).
4844
4634
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,x)) = x.
4845
4635
f(f(x,f(x,x)),f(x,x)) = x.
4846
f(f(x,f(f(y,x),x)),f(y,y)) = f(f(y,y),f(y,f(y,y))).
4847
f(f(x,f(f(y,x),x)),f(y,y)) = y.
4636
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
4848
4637
f(f(x,f(f(f(x,x),x),x)),x) = f(x,x).
4849
4638
f(f(x,f(x,x)),x) = f(x,x).
4850
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,x).
4851
f(f(x,x),f(x,f(x,y))) = x.
4852
f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)).
4853
f(f(f(x,f(x,y)),f(x,f(x,f(x,y)))),y) = f(f(x,f(x,y)),x).
4854
f(f(f(x,x),f(x,f(x,x))),f(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x).
4855
f(x,f(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x).
4639
f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x.
4640
f(f(x,f(f(f(x,f(x,x)),x),x)),x) = f(x,f(x,x)).
4641
f(f(x,f(f(x,x),x)),x) = f(x,f(x,x)).
4642
f(f(x,x),x) = f(x,f(x,x)).
4643
f(f(f(x,x),f(x,x)),f(f(x,x),x)) = f(x,x).
4644
f(x,f(f(x,x),x)) = f(x,x).
4645
f(x,f(x,f(x,x))) = f(x,x).
4646
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
4647
f(f(x,f(x,f(x,x))),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
4648
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
4649
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
4650
f(f(x,f(y,y)),f(x,y)) = x.
4651
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
4652
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
4653
f(f(x,y),f(x,f(y,y))) = x.
4654
f(f(f(x,f(x,f(x,y))),x),x) = f(x,y).
4655
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
4656
f(f(f(f(x,f(x,f(x,y))),f(f(x,f(x,f(x,y))),f(x,f(x,y)))),f(x,f(x,f(x,y)))),f(x,f(x,f(x,y)))) = f(x,f(f(f(x,f(x,f(x,y))),x),x)).
4657
f(f(f(f(x,f(x,f(x,y))),f(f(x,f(x,f(x,y))),f(x,f(x,y)))),f(x,f(x,f(x,y)))),f(x,f(x,f(x,y)))) = f(x,f(x,y)).
4658
f(f(x,f(f(f(x,y),x),x)),x) = f(x,y).
4659
f(f(x,y),f(x,f(x,f(f(x,y),x)))) = x.
4660
f(f(x,f(x,y)),x) = f(x,f(x,f(x,y))).
4661
f(f(f(x,f(f(f(x,y),x),x)),f(f(f(x,y),f(x,f(f(f(x,y),x),x))),f(x,f(f(f(x,y),x),x)))),f(x,f(f(f(x,y),x),x))) = f(f(x,f(f(f(x,y),x),x)),x).
4662
f(f(f(x,f(f(f(x,y),x),x)),f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x)),f(x,f(f(f(x,y),x),x))) = f(f(x,f(f(f(x,y),x),x)),x).
4663
f(f(f(x,f(f(f(x,y),x),x)),f(x,f(x,f(x,f(x,f(f(f(x,y),x),x)))))),f(x,f(f(f(x,y),x),x))) = f(f(x,f(f(f(x,y),x),x)),x).
4664
f(f(f(x,f(f(f(x,y),x),x)),f(x,f(x,f(x,f(x,f(f(f(x,y),x),x)))))),f(x,f(f(f(x,y),x),x))) = f(x,y).
4665
f(f(f(x,f(x,f(x,y))),f(f(x,f(x,f(x,y))),f(f(x,f(x,f(x,y))),f(x,f(x,y))))),f(x,f(x,f(x,y)))) = f(x,f(x,y)).
4666
f(f(x,f(x,f(x,y))),f(f(x,f(x,f(x,y))),f(f(x,f(x,f(x,y))),f(f(x,f(x,f(x,y))),f(x,f(x,y)))))) = f(x,f(x,y)).
4667
f(f(x,f(x,f(x,f(x,y)))),x) = f(x,y).
4668
f(x,f(x,f(x,f(x,f(x,y))))) = f(x,y).
4669
f(f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),f(x,f(f(y,f(y,x)),y)))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))) = f(x,f(f(y,f(y,x)),y)).
4670
f(f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(x,f(f(y,f(y,x)),y)))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))) = f(x,f(f(y,f(y,x)),y)).
4671
f(f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(x,f(y,f(y,f(y,x)))))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))) = f(x,f(f(y,f(y,x)),y)).
4672
f(f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(x,f(y,f(y,f(y,x)))))),f(x,f(f(y,z),f(y,f(y,f(y,x)))))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))) = f(x,f(f(y,f(y,x)),y)).
4673
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(x,f(y,f(y,f(y,x))))))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))) = f(x,f(f(y,f(y,x)),y)).
4674
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(x,f(y,f(y,f(y,x))))))),f(x,f(f(y,z),f(y,f(y,f(y,x)))))) = f(x,f(f(y,f(y,x)),y)).
4675
f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(x,f(y,f(y,f(y,x)))))))) = f(x,f(f(y,f(y,x)),y)).
4676
f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(x,f(y,f(y,f(y,x)))))))) = f(x,f(y,f(y,f(y,x)))).
4677
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(f(y,f(y,x)),y).
4678
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(y,f(y,f(y,x))).
4679
f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y.
4680
f(f(x,f(x,f(x,y))),f(x,f(x,y))) = x.
4681
f(f(x,f(f(x,f(x,f(x,y))),x)),x) = f(x,f(x,y)).
4682
f(f(x,f(x,f(x,f(x,f(x,y))))),x) = f(x,f(x,y)).
4683
f(f(x,y),x) = f(x,f(x,y)).
4684
f(f(f(x,f(x,y)),f(f(f(f(x,f(x,y)),x),f(x,f(x,y))),f(x,f(x,y)))),f(x,f(x,y))) = f(x,f(x,f(x,y))).
4685
f(f(f(x,f(x,y)),f(f(f(x,f(x,f(x,y))),f(x,f(x,y))),f(x,f(x,y)))),f(x,f(x,y))) = f(x,f(x,f(x,y))).
4686
f(f(f(x,f(x,y)),f(x,f(x,f(x,y)))),f(x,f(x,y))) = f(x,f(x,f(x,y))).
4687
f(f(x,f(x,y)),f(f(x,f(x,y)),f(x,f(x,f(x,y))))) = f(x,f(x,f(x,y))).
4688
f(f(x,f(x,f(x,y))),f(f(x,f(x,f(x,y))),f(f(x,f(x,f(x,y))),x))) = f(x,f(x,y)).
4689
f(f(x,f(x,f(x,y))),f(f(x,f(x,f(x,y))),f(x,f(x,f(x,f(x,y)))))) = f(x,f(x,y)).
4690
f(x,f(x,f(x,f(x,y)))) = f(x,f(x,y)).
4691
f(f(x,f(x,f(f(y,z),f(y,f(y,f(y,x)))))),x) = f(y,f(y,f(y,x))).
4692
f(x,f(x,f(x,f(f(y,z),f(y,f(y,f(y,x))))))) = f(y,f(y,f(y,x))).
4693
f(f(f(x,f(f(x,f(x,y)),x)),f(x,f(x,f(x,f(x,f(f(f(x,y),x),x)))))),f(x,f(f(f(x,y),x),x))) = f(x,y).
4694
f(f(f(x,f(x,f(x,f(x,y)))),f(x,f(x,f(x,f(x,f(f(f(x,y),x),x)))))),f(x,f(f(f(x,y),x),x))) = f(x,y).
4695
f(f(f(x,f(x,y)),f(x,f(x,f(x,f(x,f(f(f(x,y),x),x)))))),f(x,f(f(f(x,y),x),x))) = f(x,y).
4696
f(f(f(x,f(x,y)),f(x,f(x,f(x,f(x,f(f(x,f(x,y)),x)))))),f(x,f(f(f(x,y),x),x))) = f(x,y).
4697
f(f(f(x,f(x,y)),f(x,f(x,f(x,f(x,f(x,f(x,f(x,y)))))))),f(x,f(f(f(x,y),x),x))) = f(x,y).
4698
f(f(f(x,f(x,y)),f(x,f(x,f(x,f(x,f(x,y)))))),f(x,f(f(f(x,y),x),x))) = f(x,y).
4699
f(f(f(x,f(x,y)),f(x,f(x,f(x,y)))),f(x,f(f(f(x,y),x),x))) = f(x,y).
4700
f(f(f(x,f(x,y)),f(x,f(x,f(x,y)))),f(x,f(f(x,f(x,y)),x))) = f(x,y).
4701
f(f(f(x,f(x,y)),f(x,f(x,f(x,y)))),f(x,f(x,f(x,f(x,y))))) = f(x,y).
4702
f(f(f(x,f(x,y)),f(x,f(x,f(x,y)))),f(x,f(x,y))) = f(x,y).
4703
f(f(x,f(x,y)),f(f(x,f(x,y)),f(x,f(x,f(x,y))))) = f(x,y).
4704
f(x,f(x,f(x,y))) = f(x,y).
4705
f(f(x,y),f(x,f(x,f(x,f(x,y))))) = x.
4706
f(f(x,y),f(x,f(x,y))) = x.
4707
f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(x,f(y,f(y,f(y,x)))))))) = f(x,f(y,f(y,f(y,x)))).
4708
f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(x,f(y,f(y,f(y,x)))))))) = f(x,f(y,f(y,f(y,x)))).
4709
f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(x,f(y,f(y,f(y,x)))))))) = f(x,f(y,f(y,f(y,x)))).
4710
f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(x,f(y,f(y,f(y,x)))))))) = f(x,f(y,f(y,f(y,x)))).
4711
f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(x,f(y,x)))))) = f(x,f(y,f(y,f(y,x)))).
4712
f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(x,f(y,x)))) = f(x,f(y,f(y,f(y,x)))).
4713
f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(x,f(y,x)))) = f(x,f(y,x)).
4714
f(x,f(x,f(x,f(f(y,z),f(y,x))))) = f(y,f(y,f(y,x))).
4715
f(x,f(f(y,z),f(y,x))) = f(y,f(y,f(y,x))).
4716
f(x,f(f(y,z),f(y,x))) = f(y,x).
4717
f(f(x,y),f(y,f(x,z))) = y.
4718
f(f(y,x),f(f(x,f(f(y,z),f(y,x))),f(x,f(y,x)))) = f(x,f(y,x)).
4719
f(f(y,x),f(f(y,x),f(x,f(y,x)))) = f(x,f(y,x)).
4720
f(f(x,y),y) = f(y,f(x,y)).
4721
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y.
4722
f(x,f(f(y,x),f(f(y,x),f(x,f(y,z))))) = f(y,x).
4856
4723
f(x,f(f(y,x),x)) = f(y,x).
4857
f(f(x,f(f(f(x,y),x),x)),x) = f(y,f(f(x,y),y)).
4858
f(f(f(x,y),x),x) = f(y,f(f(x,y),y)).
4859
f(f(f(x,y),x),x) = f(x,y).
4860
f(f(f(x,f(f(x,y),f(z,f(f(y,z),z)))),x),x) = f(y,f(f(x,y),y)).
4861
f(f(f(x,f(f(x,y),f(y,z))),x),x) = f(y,f(f(x,y),y)).
4862
f(x,f(f(x,y),f(y,z))) = f(y,f(f(x,y),y)).
4863
f(x,f(f(x,y),f(y,z))) = f(x,y).
4864
f(f(x,f(x,y)),f(x,f(f(y,f(f(x,y),y)),f(z,f(f(y,z),z))))) = x.
4865
f(f(x,f(x,y)),f(x,f(f(x,y),f(z,f(f(y,z),z))))) = x.
4866
f(f(x,f(x,y)),f(x,f(f(x,y),f(y,z)))) = x.
4867
f(f(x,f(x,y)),f(x,y)) = x.
4868
f(f(f(y,x),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
4869
f(f(f(y,x),f(f(f(z,y),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
4870
f(f(f(y,x),f(f(f(z,y),f(y,x)),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
4871
f(f(f(y,x),f(f(f(z,y),f(y,x)),f(y,x))),z) = f(y,f(f(z,y),y)).
4872
f(f(f(z,y),f(y,x)),z) = f(y,f(f(z,y),y)).
4873
f(f(f(x,y),f(y,z)),x) = f(x,y).
4874
f(f(f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),u)),f(f(x,f(y,z)),f(x,f(x,f(y,z))))),f(u,f(x,z))) = u.
4875
f(f(f(x,f(x,f(x,y))),x),x) = f(y,f(f(x,y),y)).
4876
f(x,f(x,f(x,y))) = f(y,f(f(x,y),y)).
4877
f(x,f(x,f(x,y))) = f(x,y).
4724
f(x,f(x,f(y,x))) = f(y,x).
4878
4725
f(f(x,y),f(x,f(z,y))) = x.
4879
f(f(f(x,f(x,y)),f(x,y)),y) = f(f(x,f(x,y)),x).
4880
f(x,y) = f(f(x,f(x,y)),x).
4881
f(f(x,f(x,y)),x) = f(x,y).
4882
f(f(f(x,f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),u)),f(f(x,f(y,z)),f(x,f(x,f(y,z))))),f(u,f(x,z))) = u.
4883
f(f(f(x,f(x,u)),f(f(x,f(y,z)),f(x,f(x,f(y,z))))),f(u,f(x,z))) = u.
4884
f(f(f(x,f(x,u)),x),f(u,f(x,z))) = u.
4885
f(f(x,y),f(y,f(x,z))) = y.
4726
f(x,f(f(y,x),f(y,f(y,x)))) = f(y,x).
4886
4727
f(x,y) = f(y,x).
4887
4728
$F # answer("Commutativity").
4888
4729
end_of_list.
4902
4743
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4903
4744
end_of_list.
4904
4745
4905
Child search process 13978 started.
4746
Child search process 10950 started.
4906
4747
4907
4748
============================== PROCESS INITIAL CLAUSES ===============
4908
4749
4921
4762
formulas(demodulators).
4922
4763
end_of_list.
4923
4764
4924
% 1514 hints input.
4765
% 1257 hints input.
4925
4766
4926
4767
============================== PREDICATE ELIMINATION =================
4927
4768
4934
4775
4935
4776
Term ordering decisions:
4936
4777
Function symbol KB weights: c1=1. c2=1. f=1.
4937
Relation symbol precedence: lex([ = ]).
4938
Function symbol precedence: lex([ c1, c2, f ]).
4778
Predicate symbol precedence: predicate_order([ = ]).
4779
Function symbol precedence: function_order([ c1, c2, f ]).
4939
4780
Skipping inverse_order, because term ordering is KBO.
4940
4781
Unfolding symbols: (none).
4941
4782
4942
4783
Auto_inference settings:
4943
4784
% set(paramodulation). % (positive equality literals)
4944
% set(paramodulation) -> set(back_demod).
4945
4785
4946
4786
Auto_process settings: (no changes).
4947
4787
4788
kept: 2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
4789
kept: 3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
4790
kept: 4 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
4791
kept: 5 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4792
4948
4793
============================== end of process initial clauses ========
4949
4794
4950
4795
============================== CLAUSES FOR SEARCH ====================
4967
4812
5 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4968
4813
end_of_list.
4969
4814
4970
% 1514 hints processed (867 are redundant).
4815
% 551 hints (1257 processed, 706 redundant).
4971
4816
4972
4817
============================== end of clauses for search =============
4973
4818
4974
4819
============================== SEARCH ================================
4975
4820
4976
% Starting search at 0.02 seconds.
4821
% Starting search at 0.06 seconds.
4977
4822
4978
4823
% Operation f is commutative; C redundancy checks enabled.
4979
4824
4980
4825
============================== PROOF =================================
4981
4826
4982
% Proof 1 at 0.28 (+ 0.00) seconds: "Commutativity".
4983
% Length of proof is 37.
4984
% Level of proof is 17.
4985
% Maximum clause weight is 29.
4986
% Given clauses 48.
4827
% Proof 1 at 0.43 (+ 0.00) seconds: "Commutativity".
4828
% Length of proof is 54.
4829
% Level of proof is 25.
4830
% Maximum clause weight is 75.
4831
% Given clauses 42.
4987
4832
4988
1 f(x,y) = f(y,x) # label("Commutativity"). [goal].
4833
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
4989
4834
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
4990
4835
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
4991
4836
4 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
4992
4837
5 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4993
4838
6 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)). [para(2(a,1),2(a,1,1,2,1))].
4994
10 f(f(f(x,y),f(f(f(f(x,f(y,z)),x),f(x,y)),f(x,y))),z) = f(f(x,f(y,z)),x). [para(4(a,1),2(a,1,2))].
4995
4839
24 f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)). [para(2(a,1),5(a,1,2))].
4840
28 f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(y,f(y,x)),y). [para(5(a,1),4(a,1,2))].
4996
4841
35 f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y. [para(24(a,1),2(a,1,2,2))].
4997
64 f(f(x,f(f(x,x),x)),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))) = f(f(f(x,f(x,f(f(x,x),x))),x),x). [para(6(a,1),24(a,1,1,1,2)),rewrite(35(8),35(12),35(13),35(17)),flip(a)].
4998
70 f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)). [para(35(a,1),2(a,1,2))].
4999
92 f(x,f(f(x,x),x)) = f(x,x). [para(35(a,1),6(a,1,2)),rewrite(35(8)),flip(a)].
5000
95 f(f(x,f(f(f(f(y,y),f(y,f(y,y))),x),x)),f(y,y)) = f(f(y,y),f(y,f(y,y))). [para(6(a,1),35(a,1,2)),rewrite(92(3),92(4),92(10),92(12),92(13))].
5001
100 f(f(x,x),f(f(x,f(x,x)),f(x,x))) = f(f(f(x,f(x,x)),x),x). [back_rewrite(64),rewrite(92(3),92(4),92(6),92(9))].
5002
103 f(f(x,x),f(x,f(y,x))) = x. [para(92(a,1),2(a,1,1))].
5003
105 f(f(x,x),f(x,x)) = x. [para(92(a,1),2(a,1,2)),rewrite(92(3))].
5004
109 f(f(x,f(x,x)),f(x,x)) = x. [para(92(a,1),4(a,1,2)),rewrite(105(3))].
5005
120 f(f(x,f(f(y,x),x)),f(y,y)) = y. [back_rewrite(95),rewrite(103(4),103(9))].
5006
121 f(f(f(x,f(x,x)),x),x) = f(f(x,x),x). [back_rewrite(100),rewrite(109(5)),flip(a)].
5007
174 f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x. [para(109(a,1),2(a,1,1,2))].
5008
175 f(f(x,x),x) = f(x,f(x,x)). [para(109(a,1),2(a,1,2)),rewrite(121(4),92(3))].
5009
185 f(f(x,f(f(x,f(x,x)),x)),f(f(x,f(x,f(x,x))),f(y,x))) = f(x,f(x,f(x,x))). [para(109(a,1),6(a,1,1,1)),rewrite(109(6),121(8),175(6),109(11),121(14),175(12))].
5010
193 f(f(x,f(x,x)),x) = f(x,x). [para(105(a,1),109(a,1,1,2)),rewrite(175(2),105(5))].
5011
194 f(x,f(x,f(x,x))) = f(x,x). [para(109(a,1),103(a,1,2,2)),rewrite(105(3),175(2))].
5012
195 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [back_rewrite(174),rewrite(175(2))].
5013
207 f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,x). [back_rewrite(185),rewrite(193(3),194(5),194(9))].
5014
214 f(f(x,f(x,x)),f(x,f(f(x,x),y))) = x. [para(175(a,1),5(a,1,1,1,2)),rewrite(103(4))].
5015
228 f(f(x,x),f(x,f(x,y))) = x. [para(193(a,1),5(a,1,1))].
5016
563 f(x,f(y,f(x,x))) = f(x,x). [para(195(a,1),10(a,1,1,2,1,1,1)),rewrite(193(3),194(4),193(5),105(4),193(4),228(4),195(9),194(6))].
5017
647 f(x,f(f(x,x),y)) = f(x,x). [para(214(a,1),10(a,1,1,2,1,1,1)),rewrite(193(3),563(4),193(5),105(4),193(4),228(4),214(9),563(6))].
5018
650 f(f(x,x),f(x,y)) = x. [para(2(a,1),647(a,1,2,1)),rewrite(175(2),563(3),175(5),563(6),175(6),563(7),105(6))].
5019
679 f(x,f(f(y,x),x)) = f(y,x). [para(207(a,1),4(a,1,1,1)),rewrite(650(4),109(5))].
5020
894 f(f(x,y),f(x,x)) = x. [back_rewrite(120),rewrite(679(3))].
5021
919 f(f(f(x,y),f(y,z)),x) = f(x,y). [back_rewrite(70),rewrite(679(3),679(4),679(5),679(7),679(7),679(7))].
5022
1210 f(f(x,y),f(y,y)) = y. [para(679(a,1),894(a,1,1))].
5023
1307 f(x,y) = f(y,x). [para(647(a,1),919(a,1,1,2)),rewrite(1210(3))].
5024
1308 $F # answer("Commutativity"). [resolve(1307,a,3,a)].
4842
90 f(f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),f(x,f(f(y,f(y,x)),y)))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))) = f(x,f(f(y,f(y,x)),y)). [para(28(a,1),24(a,1,1,1,2,2,2)),rewrite([28(40)])].
4843
136 f(f(x,f(f(x,x),x)),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))) = f(f(f(x,f(x,f(f(x,x),x))),x),x). [para(6(a,1),24(a,1,1,1,2)),rewrite([35(8),35(12),35(13),35(17)]),flip(a)].
4844
144 f(x,f(f(x,x),x)) = f(x,x). [para(35(a,1),6(a,1,2)),rewrite([35(8)]),flip(a)].
4845
151 f(f(x,x),f(f(x,f(x,x)),f(x,x))) = f(f(f(x,f(x,x)),x),x). [back_rewrite(136),rewrite([144(3),144(4),144(6),144(9)])].
4846
155 f(f(x,x),f(x,f(y,x))) = x. [para(144(a,1),2(a,1,1))].
4847
157 f(f(x,x),f(x,x)) = x. [para(144(a,1),2(a,1,2)),rewrite([144(3)])].
4848
161 f(f(x,f(x,x)),f(x,x)) = x. [para(144(a,1),4(a,1,2)),rewrite([157(3)])].
4849
175 f(f(f(x,f(x,x)),x),x) = f(f(x,x),x). [back_rewrite(151),rewrite([161(5)]),flip(a)].
4850
176 f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y. [para(157(a,1),2(a,1,2,2))].
4851
207 f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x. [para(161(a,1),2(a,1,1,2))].
4852
208 f(f(x,x),x) = f(x,f(x,x)). [para(161(a,1),2(a,1,2)),rewrite([175(4),144(3)])].
4853
230 f(f(x,f(x,x)),x) = f(x,x). [para(157(a,1),161(a,1,1,2)),rewrite([208(2),157(5)])].
4854
231 f(x,f(x,f(x,x))) = f(x,x). [para(161(a,1),155(a,1,2,2)),rewrite([157(3),208(2)])].
4855
232 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [back_rewrite(207),rewrite([208(2)])].
4856
367 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)). [para(232(a,1),4(a,1,1,1)),rewrite([231(3),230(6)])].
4857
387 f(f(x,f(y,y)),f(x,y)) = x. [back_rewrite(176),rewrite([367(6)])].
4858
389 f(f(x,y),f(x,f(y,y))) = x. [para(2(a,1),387(a,1,1,2)),rewrite([208(3),231(4)])].
4859
456 f(f(f(x,f(x,f(x,y))),x),x) = f(x,y). [para(389(a,1),5(a,1,2))].
4860
508 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [para(456(a,1),2(a,1,2,2))].
4861
591 f(f(x,f(f(f(x,y),x),x)),x) = f(x,y). [para(389(a,1),508(a,1,2))].
4862
602 f(f(x,y),f(x,f(x,f(f(x,y),x)))) = x. [para(591(a,1),4(a,1,1))].
4863
615 f(f(x,f(f(y,f(y,x)),y)),x) = f(x,f(f(y,z),f(f(y,f(y,x)),y))). [para(28(a,1),591(a,1,1,2))].
4864
628 f(f(x,f(x,y)),x) = f(x,f(x,f(x,y))). [para(456(a,1),591(a,1,1,2))].
4865
665 f(f(x,f(y,f(y,f(y,x)))),x) = f(x,f(f(y,z),f(y,f(y,f(y,x))))). [back_rewrite(615),rewrite([628(3),628(9)])].
4866
709 f(x,f(x,f(x,f(x,f(x,y))))) = f(x,y). [back_rewrite(456),rewrite([628(4),628(5)])].
4867
751 f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(x,f(y,f(y,f(y,x)))))))) = f(x,f(y,f(y,f(y,x)))). [back_rewrite(90),rewrite([628(4),628(10),628(15),628(22),628(25),628(29),628(32),628(35)])].
4868
777 f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(y,f(y,f(y,x))). [back_rewrite(28),rewrite([628(4),628(11)])].
4869
784 f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y # label("C1/A1"). [back_rewrite(5),rewrite([628(3)])].
4870
805 f(x,f(f(x,y),f(f(x,y),f(x,f(x,y))))) = f(x,y). [para(389(a,1),602(a,1,1)),rewrite([389(6)])].
4871
833 f(f(x,y),x) = f(x,f(x,y)). [para(628(a,1),591(a,1,1,2,1)),rewrite([628(4),709(5)])].
4872
836 f(f(x,f(x,y)),f(x,y)) = x. [para(628(a,1),602(a,1,2,2,2)),rewrite([709(7)])].
4873
837 f(f(x,y),f(f(x,y),f(f(x,y),f(x,f(x,f(x,f(x,y))))))) = x. [para(602(a,1),628(a,1,1,2)),rewrite([833(2),836(4),833(5)]),flip(a)].
4874
838 f(f(x,y),f(x,f(x,y))) = x. [para(602(a,1),628(a,2,2,2)),rewrite([833(4),833(10),837(10),833(3)]),flip(a)].
4875
882 f(x,f(x,f(x,f(f(y,z),f(y,f(y,f(y,x))))))) = f(y,f(y,f(y,x))). [back_rewrite(777),rewrite([833(7),833(8)])].
4876
921 f(x,f(f(y,z),f(y,f(y,f(y,x))))) = f(x,f(x,f(y,f(y,f(y,x))))). [back_rewrite(665),rewrite([833(5)]),flip(a)].
4877
1022 f(x,f(x,f(x,y))) = f(x,y). [back_rewrite(805),rewrite([838(5),833(2)])].
4878
1101 f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(x,f(y,x)))) = f(x,f(y,x)). [back_rewrite(751),rewrite([1022(4),1022(8),1022(12),1022(16),1022(19),1022(21),1022(15)])].
4879
1132 f(x,f(f(y,z),f(y,x))) = f(x,f(x,f(y,x))). [back_rewrite(921),rewrite([1022(4),1022(7)])].
4880
1160 f(x,f(f(y,z),f(y,x))) = f(y,x). [back_rewrite(882),rewrite([1022(4),1022(6),1022(7)])].
4881
1176 f(f(x,y),f(y,f(x,z))) = y. [back_rewrite(784),rewrite([1022(3)])].
4882
1217 f(x,f(x,f(y,x))) = f(y,x). [back_rewrite(1132),rewrite([1160(4)]),flip(a)].
4883
1218 f(f(x,y),y) = f(y,f(x,y)). [back_rewrite(1101),rewrite([1160(4),1160(5),1176(5)])].
4884
1319 f(f(x,y),f(x,f(z,y))) = x. [back_rewrite(2),rewrite([1218(2),1217(3)])].
4885
1364 f(x,y) = f(y,x). [para(1218(a,1),1160(a,1,2)),rewrite([1319(4)])].
4886
1365 $F # answer("Commutativity"). [resolve(1364,a,3,a)].
5025
4887
5026
4888
============================== end of proof ==========================
5027
4889
5028
4890
============================== STATISTICS ============================
5029
4891
5030
Given=48. Generated=3471. Kept=1306. proofs=1.
5031
Usable=18. Sos=303. Demods=319. Limbo=0, Disabled=988. Hints=1514.
5032
Weight_deleted=124. Literals_deleted=0.
5033
Forward_subsumed=2041. Back_subsumed=231.
4892
Given=42. Generated=3082. Kept=1363. proofs=1.
4893
Usable=14. Sos=170. Demods=190. Limbo=8, Disabled=1174. Hints=1257.
4894
Kept_by_rule=0, Deleted_by_rule=40.
4895
Forward_subsumed=1679. Back_subsumed=207.
5034
4896
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
5035
New_demodulators=1303 (0 lex), Back_demodulated=753. Back_unit_deleted=0.
5036
Demod_attempts=89826. Demod_rewrites=8825.
4897
New_demodulators=1355 (0 lex), Back_demodulated=963. Back_unit_deleted=0.
4898
Demod_attempts=84601. Demod_rewrites=9084.
5037
4899
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
5038
4900
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
5039
Megabytes=2.64.
5040
User_CPU=0.28, System_CPU=0.00, Wall_clock=11.
4901
Megabytes=2.37.
4902
User_CPU=0.43, System_CPU=0.00, Wall_clock=8.
5041
4903
5042
4904
============================== end of statistics =====================
5043
4905
5045
4907
5046
4908
Exiting with 1 proof.
5047
4909
5048
Process 13978 exit (max_proofs) Sun Aug 13 21:05:08 2006
4910
Process 10950 exit (max_proofs) Mon Sep 28 11:57:17 2009
5049
4911
5050
4912
Successful proof using the following 2 assumptions:
5051
4913
5054
4916
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
5055
4917
end_of_list.
5056
4918
5057
Including 102 new hint clauses:
4919
Including 118 new hint clauses:
5058
4920
5059
4921
formulas(new_hints).
5060
$F # label("Commutativity").
4922
$F # label("Commutativity") # label(non_clause) # label(goal).
5061
4923
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
5062
4924
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
5063
4925
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
5064
4926
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
5065
4927
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)).
5066
f(f(f(x,y),f(f(f(f(x,f(y,z)),x),f(x,y)),f(x,y))),z) = f(f(x,f(y,z)),x).
5067
4928
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
4929
f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(y,f(y,x)),y).
5068
4930
f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y.
4931
f(f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),f(x,f(f(y,f(y,x)),y)))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))) = f(x,f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x)).
4932
f(f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),f(x,f(f(y,f(y,x)),y)))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))) = f(x,f(f(y,f(y,x)),y)).
5069
4933
f(f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
5070
4934
f(f(f(x,f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
5071
4935
f(f(f(x,f(x,f(f(x,x),x))),x),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
5072
4936
f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
5073
4937
f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(f(x,f(f(x,x),x)),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
5074
4938
f(f(x,f(f(x,x),x)),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))) = f(f(f(x,f(x,f(f(x,x),x))),x),x).
5075
f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
5076
4939
f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),x) = f(x,f(f(x,x),x)).
5077
4940
f(x,x) = f(x,f(f(x,x),x)).
5078
4941
f(x,f(f(x,x),x)) = f(x,x).
5079
f(f(x,f(f(f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))),x),x)),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
5080
f(f(x,f(f(f(f(y,y),f(y,f(y,f(f(y,y),y)))),x),x)),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
5081
f(f(x,f(f(f(f(y,y),f(y,f(y,y))),x),x)),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
5082
f(f(x,f(f(f(f(y,y),f(y,f(y,y))),x),x)),f(y,y)) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
5083
f(f(x,f(f(f(f(y,y),f(y,f(y,y))),x),x)),f(y,y)) = f(f(y,y),f(y,f(y,f(f(y,y),y)))).
5084
f(f(x,f(f(f(f(y,y),f(y,f(y,y))),x),x)),f(y,y)) = f(f(y,y),f(y,f(y,y))).
5085
4942
f(f(x,x),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))) = f(f(f(x,f(x,f(f(x,x),x))),x),x).
5086
4943
f(f(x,x),f(f(x,f(x,x)),f(x,f(f(x,x),x)))) = f(f(f(x,f(x,f(f(x,x),x))),x),x).
5087
4944
f(f(x,x),f(f(x,f(x,x)),f(x,x))) = f(f(f(x,f(x,f(f(x,x),x))),x),x).
5091
4948
f(f(x,x),f(x,x)) = x.
5092
4949
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,x)) = x.
5093
4950
f(f(x,f(x,x)),f(x,x)) = x.
5094
f(f(x,f(f(y,x),x)),f(y,y)) = f(f(y,y),f(y,f(y,y))).
5095
f(f(x,f(f(y,x),x)),f(y,y)) = y.
5096
4951
f(f(x,x),x) = f(f(f(x,f(x,x)),x),x).
5097
4952
f(f(f(x,f(x,x)),x),x) = f(f(x,x),x).
4953
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
5098
4954
f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x.
5099
4955
f(f(x,f(f(f(x,f(x,x)),x),x)),x) = f(x,f(x,x)).
5100
4956
f(f(x,f(f(x,x),x)),x) = f(x,f(x,x)).
5101
4957
f(f(x,x),x) = f(x,f(x,x)).
5102
f(f(x,f(f(x,f(x,x)),f(f(x,f(x,x)),f(x,x)))),f(f(x,f(f(f(x,f(x,x)),x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
5103
f(f(x,f(f(x,f(x,x)),x)),f(f(x,f(f(f(x,f(x,x)),x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
5104
f(f(x,f(f(x,f(x,x)),x)),f(f(x,f(f(x,x),x)),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
5105
f(f(x,f(f(x,f(x,x)),x)),f(f(x,f(x,f(x,x))),f(y,f(f(x,f(x,x)),f(x,x))))) = f(x,f(f(f(x,f(x,x)),x),x)).
5106
f(f(x,f(f(x,f(x,x)),x)),f(f(x,f(x,f(x,x))),f(y,x))) = f(x,f(f(f(x,f(x,x)),x),x)).
5107
f(f(x,f(f(x,f(x,x)),x)),f(f(x,f(x,f(x,x))),f(y,x))) = f(x,f(f(x,x),x)).
5108
f(f(x,f(f(x,f(x,x)),x)),f(f(x,f(x,f(x,x))),f(y,x))) = f(x,f(x,f(x,x))).
5109
4958
f(f(f(x,x),x),f(f(x,x),f(x,x))) = f(x,x).
5110
4959
f(f(x,f(x,x)),f(f(x,x),f(x,x))) = f(x,x).
5111
4960
f(f(x,f(x,x)),x) = f(x,x).
5113
4962
f(x,f(f(x,x),x)) = f(x,x).
5114
4963
f(x,f(x,f(x,x))) = f(x,x).
5115
4964
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
5116
f(f(x,f(x,x)),f(f(x,f(x,f(x,x))),f(y,x))) = f(x,f(x,f(x,x))).
5117
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,f(x,f(x,x))).
5118
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,x).
5119
f(f(f(f(x,x),f(x,f(x,x))),f(x,x)),f(x,f(f(x,x),y))) = x.
5120
f(f(x,f(x,x)),f(x,f(f(x,x),y))) = x.
5121
f(f(x,x),f(x,f(x,y))) = x.
5122
f(f(f(f(x,f(x,x)),x),f(f(f(x,f(x,f(x,x))),f(f(x,f(x,x)),x)),f(f(x,f(x,x)),x))),f(y,f(x,x))) = f(f(f(x,f(x,x)),f(x,f(y,f(x,x)))),f(x,f(x,x))).
5123
f(f(f(x,x),f(f(f(x,f(x,f(x,x))),f(f(x,f(x,x)),x)),f(f(x,f(x,x)),x))),f(y,f(x,x))) = f(f(f(x,f(x,x)),f(x,f(y,f(x,x)))),f(x,f(x,x))).
5124
f(f(f(x,x),f(f(f(x,x),f(f(x,f(x,x)),x)),f(f(x,f(x,x)),x))),f(y,f(x,x))) = f(f(f(x,f(x,x)),f(x,f(y,f(x,x)))),f(x,f(x,x))).
5125
f(f(f(x,x),f(f(f(x,x),f(x,x)),f(f(x,f(x,x)),x))),f(y,f(x,x))) = f(f(f(x,f(x,x)),f(x,f(y,f(x,x)))),f(x,f(x,x))).
5126
f(f(f(x,x),f(x,f(f(x,f(x,x)),x))),f(y,f(x,x))) = f(f(f(x,f(x,x)),f(x,f(y,f(x,x)))),f(x,f(x,x))).
5127
f(f(f(x,x),f(x,f(x,x))),f(y,f(x,x))) = f(f(f(x,f(x,x)),f(x,f(y,f(x,x)))),f(x,f(x,x))).
5128
f(x,f(y,f(x,x))) = f(f(f(x,f(x,x)),f(x,f(y,f(x,x)))),f(x,f(x,x))).
5129
f(x,f(y,f(x,x))) = f(x,f(x,f(x,x))).
5130
f(x,f(y,f(x,x))) = f(x,x).
5131
f(f(f(f(x,f(x,x)),x),f(f(f(x,f(x,f(x,x))),f(f(x,f(x,x)),x)),f(f(x,f(x,x)),x))),f(f(x,x),y)) = f(f(f(x,f(x,x)),f(x,f(f(x,x),y))),f(x,f(x,x))).
5132
f(f(f(x,x),f(f(f(x,f(x,f(x,x))),f(f(x,f(x,x)),x)),f(f(x,f(x,x)),x))),f(f(x,x),y)) = f(f(f(x,f(x,x)),f(x,f(f(x,x),y))),f(x,f(x,x))).
5133
f(f(f(x,x),f(f(f(x,x),f(f(x,f(x,x)),x)),f(f(x,f(x,x)),x))),f(f(x,x),y)) = f(f(f(x,f(x,x)),f(x,f(f(x,x),y))),f(x,f(x,x))).
5134
f(f(f(x,x),f(f(f(x,x),f(x,x)),f(f(x,f(x,x)),x))),f(f(x,x),y)) = f(f(f(x,f(x,x)),f(x,f(f(x,x),y))),f(x,f(x,x))).
5135
f(f(f(x,x),f(x,f(f(x,f(x,x)),x))),f(f(x,x),y)) = f(f(f(x,f(x,x)),f(x,f(f(x,x),y))),f(x,f(x,x))).
5136
f(f(f(x,x),f(x,f(x,x))),f(f(x,x),y)) = f(f(f(x,f(x,x)),f(x,f(f(x,x),y))),f(x,f(x,x))).
5137
f(x,f(f(x,x),y)) = f(f(f(x,f(x,x)),f(x,f(f(x,x),y))),f(x,f(x,x))).
5138
f(x,f(f(x,x),y)) = f(x,f(x,f(x,x))).
5139
f(x,f(f(x,x),y)) = f(x,x).
5140
f(f(x,f(f(x,x),x)),f(x,y)) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
5141
f(f(x,f(x,f(x,x))),f(x,y)) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
5142
f(f(x,x),f(x,y)) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
5143
f(f(x,x),f(x,y)) = f(f(x,f(x,f(x,x))),f(x,f(f(x,x),x))).
5144
f(f(x,x),f(x,y)) = f(f(x,x),f(x,f(f(x,x),x))).
5145
f(f(x,x),f(x,y)) = f(f(x,x),f(x,f(x,f(x,x)))).
5146
f(f(x,x),f(x,y)) = f(f(x,x),f(x,x)).
5147
f(f(x,x),f(x,y)) = x.
5148
f(f(f(x,x),f(x,f(x,x))),f(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x).
5149
f(x,f(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x).
5150
f(x,f(f(y,x),x)) = f(y,x).
5151
f(f(x,y),f(x,x)) = x.
5152
f(f(f(y,x),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
5153
f(f(f(y,x),f(f(f(z,y),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
5154
f(f(f(y,x),f(f(f(z,y),f(y,x)),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
5155
f(f(f(y,x),f(f(f(z,y),f(y,x)),f(y,x))),z) = f(y,f(f(z,y),y)).
5156
f(f(f(z,y),f(y,x)),z) = f(y,f(f(z,y),y)).
5157
f(f(f(x,y),f(y,z)),x) = f(x,y).
5158
f(f(x,y),f(y,y)) = y.
5159
f(f(f(x,y),f(y,y)),x) = f(x,y).
4965
f(f(x,f(x,f(x,x))),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
4966
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
4967
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
4968
f(f(x,f(y,y)),f(x,y)) = x.
4969
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
4970
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
4971
f(f(x,y),f(x,f(y,y))) = x.
4972
f(f(f(x,f(x,f(x,y))),x),x) = f(x,y).
4973
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
4974
f(f(x,f(f(f(x,y),x),x)),x) = f(x,y).
4975
f(f(x,y),f(x,f(x,f(f(x,y),x)))) = x.
4976
f(f(x,f(f(y,f(y,x)),y)),x) = f(x,f(f(y,z),f(f(y,f(y,x)),y))).
4977
f(f(x,f(x,y)),x) = f(x,f(x,f(x,y))).
4978
f(f(x,f(y,f(y,f(y,x)))),x) = f(x,f(f(y,z),f(f(y,f(y,x)),y))).
4979
f(f(x,f(y,f(y,f(y,x)))),x) = f(x,f(f(y,z),f(y,f(y,f(y,x))))).
4980
f(f(x,f(x,f(x,f(x,y)))),x) = f(x,y).
4981
f(x,f(x,f(x,f(x,f(x,y))))) = f(x,y).
4982
f(f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),f(x,f(f(y,f(y,x)),y)))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))) = f(x,f(f(y,f(y,x)),y)).
4983
f(f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(x,f(f(y,f(y,x)),y)))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))) = f(x,f(f(y,f(y,x)),y)).
4984
f(f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(x,f(y,f(y,f(y,x)))))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))) = f(x,f(f(y,f(y,x)),y)).
4985
f(f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(x,f(y,f(y,f(y,x)))))),f(x,f(f(y,z),f(y,f(y,f(y,x)))))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))) = f(x,f(f(y,f(y,x)),y)).
4986
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(x,f(y,f(y,f(y,x))))))),f(x,f(f(y,z),f(f(y,f(y,x)),y)))) = f(x,f(f(y,f(y,x)),y)).
4987
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(x,f(y,f(y,f(y,x))))))),f(x,f(f(y,z),f(y,f(y,f(y,x)))))) = f(x,f(f(y,f(y,x)),y)).
4988
f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(x,f(y,f(y,f(y,x)))))))) = f(x,f(f(y,f(y,x)),y)).
4989
f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(x,f(y,f(y,f(y,x)))))))) = f(x,f(y,f(y,f(y,x)))).
4990
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(f(y,f(y,x)),y).
4991
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(y,f(y,f(y,x))).
4992
f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y.
4993
f(x,f(f(x,y),f(f(x,y),f(f(f(x,y),f(x,f(y,y))),f(x,y))))) = f(x,y).
4994
f(x,f(f(x,y),f(f(x,y),f(x,f(x,y))))) = f(x,y).
4995
f(f(x,f(f(x,f(x,f(x,y))),x)),x) = f(x,f(x,y)).
4996
f(f(x,f(x,f(x,f(x,f(x,y))))),x) = f(x,f(x,y)).
4997
f(f(x,y),x) = f(x,f(x,y)).
4998
f(f(x,f(x,y)),f(x,f(x,f(x,f(x,f(x,y)))))) = x.
4999
f(f(x,f(x,y)),f(x,y)) = x.
5000
f(f(f(x,y),x),f(x,y)) = f(f(x,y),f(f(x,y),f(f(x,y),f(x,f(x,f(f(x,y),x)))))).
5001
f(f(x,f(x,y)),f(x,y)) = f(f(x,y),f(f(x,y),f(f(x,y),f(x,f(x,f(f(x,y),x)))))).
5002
x = f(f(x,y),f(f(x,y),f(f(x,y),f(x,f(x,f(f(x,y),x)))))).
5003
x = f(f(x,y),f(f(x,y),f(f(x,y),f(x,f(x,f(x,f(x,y))))))).
5004
f(f(x,y),f(f(x,y),f(f(x,y),f(x,f(x,f(x,f(x,y))))))) = x.
5005
f(f(f(x,y),f(f(x,y),f(x,f(x,f(f(x,y),x))))),f(x,y)) = f(f(x,y),f(f(x,y),x)).
5006
f(f(f(x,y),f(f(x,y),f(x,f(x,f(x,f(x,y)))))),f(x,y)) = f(f(x,y),f(f(x,y),x)).
5007
f(f(x,y),f(f(x,y),f(f(x,y),f(x,f(x,f(x,f(x,y))))))) = f(f(x,y),f(f(x,y),x)).
5008
x = f(f(x,y),f(f(x,y),x)).
5009
x = f(f(x,y),f(x,f(x,y))).
5010
f(f(x,y),f(x,f(x,y))) = x.
5011
f(f(x,f(x,f(f(y,z),f(y,f(y,f(y,x)))))),x) = f(y,f(y,f(y,x))).
5012
f(x,f(x,f(x,f(f(y,z),f(y,f(y,f(y,x))))))) = f(y,f(y,f(y,x))).
5013
f(x,f(x,f(y,f(y,f(y,x))))) = f(x,f(f(y,z),f(y,f(y,f(y,x))))).
5014
f(x,f(f(y,z),f(y,f(y,f(y,x))))) = f(x,f(x,f(y,f(y,f(y,x))))).
5015
f(x,f(f(x,y),x)) = f(x,y).
5016
f(x,f(x,f(x,y))) = f(x,y).
5017
f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(x,f(y,f(y,f(y,x)))))))) = f(x,f(y,f(y,f(y,x)))).
5018
f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(x,f(y,f(y,f(y,x)))))))) = f(x,f(y,f(y,f(y,x)))).
5019
f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),f(x,f(y,f(y,f(y,x)))))))) = f(x,f(y,f(y,f(y,x)))).
5020
f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(x,f(y,f(y,f(y,x)))))))) = f(x,f(y,f(y,f(y,x)))).
5021
f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(x,f(y,x)))))) = f(x,f(y,f(y,f(y,x)))).
5022
f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(x,f(y,x)))) = f(x,f(y,f(y,f(y,x)))).
5023
f(f(x,f(f(y,z),f(y,x))),f(f(x,f(f(y,z),f(y,x))),f(x,f(y,x)))) = f(x,f(y,x)).
5024
f(x,f(f(y,z),f(y,x))) = f(x,f(x,f(y,f(y,f(y,x))))).
5025
f(x,f(f(y,z),f(y,x))) = f(x,f(x,f(y,x))).
5026
f(x,f(x,f(x,f(f(y,z),f(y,x))))) = f(y,f(y,f(y,x))).
5027
f(x,f(f(y,z),f(y,x))) = f(y,f(y,f(y,x))).
5028
f(x,f(f(y,z),f(y,x))) = f(y,x).
5029
f(f(x,y),f(y,f(x,z))) = y.
5030
f(y,x) = f(x,f(x,f(y,x))).
5031
f(x,f(x,f(y,x))) = f(y,x).
5032
f(f(y,x),f(f(x,f(f(y,z),f(y,x))),f(x,f(y,x)))) = f(x,f(y,x)).
5033
f(f(y,x),f(f(y,x),f(x,f(y,x)))) = f(x,f(y,x)).
5034
f(f(x,y),y) = f(y,f(x,y)).
5035
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y.
5036
f(f(x,y),f(x,f(z,y))) = x.
5037
f(x,f(f(y,x),f(y,f(y,x)))) = f(y,x).
5160
5038
f(x,y) = f(y,x).
5161
5039
$F # answer("Commutativity").
5162
5040
end_of_list.
5175
5053
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
5176
5054
end_of_list.
5177
5055
5178
Child search process 13979 started.
5056
Child search process 10951 started.
5179
5057
5180
5058
============================== PROCESS INITIAL CLAUSES ===============
5181
5059
5193
5071
formulas(demodulators).
5194
5072
end_of_list.
5195
5073
5196
% 1616 hints input.
5074
% 1375 hints input.
5197
5075
5198
5076
============================== PREDICATE ELIMINATION =================
5199
5077
5206
5084
5207
5085
Term ordering decisions:
5208
5086
Function symbol KB weights: c1=1. c2=1. f=1.
5209
Relation symbol precedence: lex([ = ]).
5210
Function symbol precedence: lex([ c1, c2, f ]).
5087
Predicate symbol precedence: predicate_order([ = ]).
5088
Function symbol precedence: function_order([ c1, c2, f ]).
5211
5089
Skipping inverse_order, because term ordering is KBO.
5212
5090
Unfolding symbols: (none).
5213
5091
5214
5092
Auto_inference settings:
5215
5093
% set(paramodulation). % (positive equality literals)
5216
% set(paramodulation) -> set(back_demod).
5217
5094
5218
5095
Auto_process settings: (no changes).
5219
5096
5097
kept: 2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
5098
kept: 3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
5099
kept: 4 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
5100
5220
5101
============================== end of process initial clauses ========
5221
5102
5222
5103
============================== CLAUSES FOR SEARCH ====================
5237
5118
4 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
5238
5119
end_of_list.
5239
5120
5240
% 1616 hints processed (922 are redundant).
5121
% 579 hints (1375 processed, 796 redundant).
5241
5122
5242
5123
============================== end of clauses for search =============
5243
5124
5244
5125
============================== SEARCH ================================
5245
5126
5246
% Starting search at 0.03 seconds.
5127
% Starting search at 0.06 seconds.
5247
5128
5248
5129
% Operation f is commutative; C redundancy checks enabled.
5249
5130
5250
5131
============================== PROOF =================================
5251
5132
5252
5133
% Proof 1 at 0.88 (+ 0.01) seconds: "Commutativity".
5253
% Length of proof is 81.
5254
% Level of proof is 27.
5134
% Length of proof is 57.
5135
% Level of proof is 18.
5255
5136
% Maximum clause weight is 59.
5256
% Given clauses 63.
5137
% Given clauses 55.
5257
5138
5258
1 f(x,y) = f(y,x) # label("Commutativity"). [goal].
5139
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
5259
5140
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
5260
5141
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
5261
5142
4 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
5262
5143
5 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)). [para(2(a,1),2(a,1,1,2,1))].
5263
6 f(f(f(x,f(y,z)),f(f(u,f(x,f(y,z))),f(x,f(y,z)))),f(u,x)) = u. [para(2(a,1),2(a,1,2,2))].
5264
9 f(f(f(x,y),f(f(f(f(x,f(x,z)),x),f(x,y)),f(x,y))),z) = f(f(x,f(x,z)),x). [para(4(a,1),2(a,1,2))].
5265
11 f(f(f(f(x,f(f(y,x),x)),f(f(x,f(f(y,x),x)),z)),f(x,f(f(y,x),x))),f(z,y)) = z. [para(2(a,1),4(a,1,2,2))].
5144
7 f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)). [para(2(a,1),2(a,1,2))].
5266
5145
12 f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)). [para(2(a,1),4(a,1,2))].
5267
5146
18 f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y. [para(12(a,1),2(a,1,2,2))].
5147
21 f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = f(x,f(f(f(x,f(f(x,x),x)),x),x)). [para(2(a,1),12(a,1,1,1,2))].
5268
5148
28 f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)). [para(18(a,1),2(a,1,2))].
5269
37 f(f(x,f(f(y,f(f(x,y),y)),x)),f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),f(z,f(f(x,z),z)))) = f(f(x,f(x,f(y,f(f(x,y),y)))),x). [para(12(a,1),18(a,1,1,2,1))].
5270
53 f(f(x,f(f(x,x),x)),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))) = f(f(f(x,f(x,f(f(x,x),x))),x),x). [para(5(a,1),12(a,1,1,1,2)),rewrite(18(8),18(12),18(13),18(17)),flip(a)].
5271
58 f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y. [para(5(a,1),18(a,1,2)),rewrite(18(8),18(15))].
5272
61 f(x,f(f(x,x),x)) = f(x,x). [para(18(a,1),5(a,1,2)),rewrite(18(8)),flip(a)].
5273
64 f(f(x,f(f(y,x),x)),f(y,y)) = y. [back_rewrite(58),rewrite(61(6))].
5274
65 f(f(x,x),f(f(x,f(x,x)),f(x,x))) = f(f(f(x,f(x,x)),x),x). [back_rewrite(53),rewrite(61(3),61(4),61(6),61(9))].
5275
69 f(f(x,x),f(x,f(y,x))) = x. [para(61(a,1),2(a,1,1))].
5276
71 f(f(x,x),f(x,x)) = x. [para(61(a,1),2(a,1,2)),rewrite(61(3))].
5277
101 f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)). [para(64(a,1),2(a,1,2))].
5278
115 f(x,f(f(y,f(f(x,y),y)),f(y,f(f(x,y),y)))) = f(y,f(f(x,y),y)). [para(64(a,1),64(a,1,1,2,1)),rewrite(69(4))].
5279
116 f(f(f(x,y),f(f(z,f(x,y)),f(x,y))),f(z,x)) = z. [para(6(a,1),2(a,1,2,2))].
5280
229 f(x,f(f(y,f(f(x,y),y)),x)) = f(y,f(f(x,y),y)). [para(64(a,1),116(a,1,1,2,1)),rewrite(69(4))].
5281
251 f(f(x,f(f(y,x),x)),f(f(f(y,f(y,f(x,f(f(y,x),x)))),y),f(z,f(f(y,z),z)))) = f(f(y,f(y,f(x,f(f(y,x),x)))),y). [back_rewrite(37),rewrite(229(5))].
5282
257 f(x,f(f(f(x,f(x,x)),x),x)) = f(x,x). [para(65(a,1),2(a,1,2)),rewrite(71(4),69(4))].
5283
270 f(f(f(f(x,f(x,x)),x),x),f(x,x)) = x. [para(65(a,1),64(a,1,1))].
5284
290 f(f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),f(x,f(f(y,x),x))) = y. [para(64(a,1),101(a,1,1,2,1,1,2,1)),rewrite(69(7),64(22),69(20))].
5285
304 f(f(f(x,f(x,y)),x),f(y,f(z,f(f(x,z),z)))) = y. [para(115(a,1),4(a,1,2,2))].
5286
305 f(f(f(x,f(x,y)),x),f(f(f(y,f(x,z)),f(y,f(y,f(x,z)))),f(f(y,f(x,z)),f(y,f(y,f(x,z)))))) = f(f(y,f(x,z)),f(y,f(y,f(x,z)))). [para(4(a,1),115(a,1,2,1,2,1)),rewrite(4(17),4(25))].
5287
380 f(f(f(x,f(x,y)),x),f(y,y)) = y. [para(5(a,1),304(a,1,2)),rewrite(61(3),61(4),69(4),61(6),61(8),61(9),69(9))].
5288
423 f(f(f(f(f(x,f(x,y)),x),f(f(f(x,f(x,y)),x),y)),f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x)) = y. [para(380(a,1),12(a,1,1,1,2,2,2,1)),rewrite(69(10),380(22),69(20))].
5289
449 f(x,f(f(f(y,f(y,x)),y),x)) = f(f(y,f(y,x)),y). [para(380(a,1),116(a,1,1,2,1)),rewrite(69(4))].
5290
454 f(f(f(f(x,f(x,y)),x),f(f(y,f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x))),f(f(x,f(x,y)),x)) = y. [para(380(a,1),101(a,1,1,2,1,1,2,1)),rewrite(69(7),380(22),69(20))].
5291
468 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(257),rewrite(449(5))].
5292
493 f(f(f(x,x),x),f(x,x)) = x. [back_rewrite(270),rewrite(468(3))].
5293
510 f(f(x,x),f(f(x,f(x,x)),f(x,x))) = f(f(x,x),x). [back_rewrite(65),rewrite(468(9))].
5294
528 f(f(x,x),f(f(x,f(x,x)),f(y,x))) = f(x,f(x,x)). [para(468(a,1),2(a,1,1,2,1)),rewrite(61(3))].
5295
531 f(f(x,x),f(x,f(x,y))) = x. [para(468(a,1),4(a,1,1))].
5296
552 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(510),rewrite(528(6)),flip(a)].
5297
565 f(f(x,f(x,x)),f(x,x)) = x. [back_rewrite(493),rewrite(552(2))].
5298
602 f(x,f(f(x,f(x,y)),f(x,f(x,y)))) = f(x,f(x,y)). [para(531(a,1),380(a,1,1,1,2)),rewrite(552(2),565(4))].
5299
830 f(f(f(f(f(x,y),f(f(z,f(x,y)),f(x,y))),z),f(f(x,y),f(f(z,f(x,y)),f(x,y)))),f(f(z,x),z)) = f(z,x). [para(116(a,1),11(a,1,1,1,2))].
5300
848 f(f(f(f(f(x,y),f(f(f(f(x,f(x,z)),x),f(x,y)),f(x,y))),f(f(x,f(x,z)),x)),f(f(x,y),f(f(f(f(x,f(x,z)),x),f(x,y)),f(x,y)))),f(z,f(f(x,f(x,z)),x))) = z. [para(9(a,1),11(a,1,1,1,2))].
5301
940 f(f(x,f(x,y)),f(x,x)) = x. [para(602(a,1),64(a,1,1,2,1)),rewrite(531(14))].
5302
943 f(f(x,f(x,y)),f(x,f(x,f(x,y)))) = x. [para(602(a,1),116(a,1,1,2,1)),rewrite(531(14))].
5303
945 f(f(x,f(f(f(x,f(x,y)),x),x)),x) = f(x,f(x,y)). [para(602(a,1),101(a,1,1,2,1,1,2,1)),rewrite(531(14),602(17),531(20))].
5304
973 f(x,f(f(x,f(x,y)),x)) = f(x,f(x,y)). [para(940(a,1),116(a,1,1,2,1)),rewrite(531(4))].
5305
1069 f(f(f(f(x,f(x,y)),x),f(f(f(f(f(x,f(x,y)),x),y),f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x))),f(f(x,f(x,y)),x)) = f(f(f(x,f(x,y)),x),y). [para(4(a,1),945(a,1,1,2,1,1,2)),rewrite(4(29))].
5306
1070 f(f(x,f(y,f(f(x,y),y))),x) = f(x,f(x,f(y,f(f(x,y),y)))). [para(12(a,1),945(a,1,1,2))].
5307
1081 f(f(x,f(x,f(y,f(f(x,y),y)))),x) = f(x,f(y,f(f(x,y),y))). [para(115(a,1),945(a,1,1,2,1,1,2)),rewrite(1070(5),973(7),115(14))].
5308
1114 f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(z,f(f(y,z),z)))) = f(y,f(x,f(f(y,x),x))). [back_rewrite(251),rewrite(1081(9),1081(18))].
5309
1121 f(x,f(x,f(y,f(f(x,y),y)))) = f(y,f(f(x,y),y)). [back_rewrite(12),rewrite(1081(6),1070(5))].
5310
1124 f(f(x,f(y,f(f(x,y),y))),f(y,f(f(x,y),y))) = x. [back_rewrite(290),rewrite(1114(12))].
5311
1136 f(f(x,f(f(y,x),x)),y) = f(y,f(x,f(f(y,x),x))). [back_rewrite(1081),rewrite(1121(5))].
5312
1165 f(f(f(x,f(x,y)),x),f(y,f(f(x,f(x,y)),x))) = y. [back_rewrite(848),rewrite(1136(13),1124(23))].
5313
1166 f(x,f(f(x,y),x)) = f(x,y). [back_rewrite(830),rewrite(1136(7),1124(14))].
5314
1242 f(f(f(x,f(x,y)),x),f(f(f(x,f(x,y)),x),y)) = y. [para(380(a,1),1121(a,1,2,2,2,1)),rewrite(531(10),380(14),531(12))].
5315
1268 f(f(x,f(f(y,f(y,x)),y)),f(f(y,f(y,x)),y)) = x. [back_rewrite(423),rewrite(1242(8))].
5316
1285 f(f(f(f(x,f(x,y)),x),y),f(f(x,f(x,y)),x)) = y. [back_rewrite(454),rewrite(1268(11))].
5317
1292 f(f(f(x,f(x,y)),x),y) = f(y,f(f(x,f(x,y)),x)). [back_rewrite(1069),rewrite(1285(11),1165(8)),flip(a)].
5318
1329 f(x,f(x,f(f(y,f(y,x)),y))) = f(f(y,f(y,x)),y). [back_rewrite(449),rewrite(1292(4))].
5319
1367 f(f(f(f(x,f(x,y)),x),f(x,f(f(f(f(x,f(x,y)),x),x),x))),y) = f(f(x,f(x,y)),x). [para(1124(a,1),9(a,1,1,2)),rewrite(1136(10))].
5320
1375 f(f(f(x,f(x,y)),x),x) = f(x,f(x,y)). [para(940(a,1),1124(a,1,1,2,2,1)),rewrite(531(6),940(8),531(7))].
5321
1379 f(f(f(f(x,f(x,y)),x),f(x,f(x,y))),y) = f(f(x,f(x,y)),x). [back_rewrite(1367),rewrite(1375(7),1166(7))].
5322
1395 f(f(x,f(x,f(x,y))),x) = f(x,f(x,y)). [back_rewrite(945),rewrite(1375(4))].
5323
1491 f(f(x,f(x,y)),x) = f(x,f(x,f(x,y))). [para(1375(a,1),1121(a,1,2,2,2)),rewrite(943(8),1375(7))].
5324
1503 f(f(x,f(x,f(x,f(x,y)))),f(x,f(x,f(x,y)))) = f(f(x,f(x,f(x,y))),f(x,f(x,y))). [para(1375(a,1),1375(a,2,2)),rewrite(1491(3),1491(6),1491(7),943(8),1491(3),1491(7),1491(11))].
5325
1574 f(x,f(x,f(x,f(x,y)))) = f(x,f(x,y)). [back_rewrite(1395),rewrite(1491(4))].
5326
1575 f(f(f(x,f(x,f(x,y))),f(x,f(x,y))),y) = f(x,f(x,f(x,y))). [back_rewrite(1379),rewrite(1491(3),1491(10))].
5327
1591 f(x,f(x,f(y,f(y,f(y,x))))) = f(y,f(y,f(y,x))). [back_rewrite(1329),rewrite(1491(3),1491(8))].
5328
1626 f(f(x,f(x,f(x,y))),y) = f(y,f(x,f(x,f(x,y)))). [back_rewrite(1292),rewrite(1491(3),1491(7))].
5329
1816 f(f(x,f(x,f(x,y))),f(f(f(y,f(x,z)),f(y,f(y,f(x,z)))),f(f(y,f(x,z)),f(y,f(y,f(x,z)))))) = f(f(y,f(x,z)),f(y,f(y,f(x,z)))). [back_rewrite(305),rewrite(1491(3))].
5330
1838 f(f(x,f(x,f(x,y))),f(x,f(x,y))) = x. [back_rewrite(1503),rewrite(1574(4),943(6)),flip(a)].
5331
1852 f(x,f(x,f(x,y))) = f(x,y). [back_rewrite(1575),rewrite(1838(6)),flip(a)].
5332
1867 f(f(x,y),f(x,f(x,y))) = x. [back_rewrite(1838),rewrite(1852(3))].
5333
1883 f(f(x,y),f(y,y)) = y. [back_rewrite(1816),rewrite(1852(3),1867(7),1867(7),1867(9))].
5334
1981 f(f(x,y),y) = f(y,f(x,y)). [back_rewrite(1626),rewrite(1852(3),1852(5))].
5335
2010 f(x,f(x,f(y,x))) = f(y,x). [back_rewrite(1591),rewrite(1852(3),1852(6))].
5336
2272 f(f(f(x,y),f(y,z)),x) = f(x,y). [back_rewrite(28),rewrite(1981(2),2010(3),1981(3),2010(4),1981(4),2010(5),1981(6),2010(7),1981(6),2010(7),1981(6),2010(7))].
5337
2309 f(x,y) = f(y,x). [para(1883(a,1),2272(a,1,1))].
5338
2310 $F # answer("Commutativity"). [resolve(2309,a,3,a)].
5149
31 f(f(x,f(f(f(y,f(f(y,y),y)),x),x)),y) = f(y,f(f(y,y),y)). [para(2(a,1),18(a,1,2))].
5150
66 f(f(x,f(f(x,x),x)),f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))) = f(f(f(x,f(x,f(f(x,x),x))),x),x). [para(5(a,1),12(a,1,1,1,2)),rewrite([18(8),18(12),18(13),18(17)]),flip(a)].
5151
71 f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y. [para(5(a,1),18(a,1,2)),rewrite([18(8),18(15)])].
5152
74 f(x,f(f(x,x),x)) = f(x,x). [para(18(a,1),5(a,1,2)),rewrite([18(8)]),flip(a)].
5153
77 f(f(x,f(f(y,x),x)),f(y,y)) = y. [back_rewrite(71),rewrite([74(6)])].
5154
78 f(f(x,x),f(f(x,f(x,x)),f(x,x))) = f(f(f(x,f(x,x)),x),x). [back_rewrite(66),rewrite([74(3),74(4),74(6),74(9)])].
5155
79 f(f(x,f(f(f(y,y),x),x)),y) = f(y,y). [back_rewrite(31),rewrite([74(3),74(8)])].
5156
80 f(f(f(f(x,x),x),f(x,x)),f(x,x)) = f(x,f(f(f(x,x),x),x)). [back_rewrite(21),rewrite([74(3),74(5),74(7),74(9)])].
5157
82 f(f(x,x),f(x,f(y,x))) = x. [para(74(a,1),2(a,1,1))].
5158
84 f(f(x,x),f(x,x)) = x. [para(74(a,1),2(a,1,2)),rewrite([74(3)])].
5159
97 f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y. [para(84(a,1),2(a,1,2,2))].
5160
134 f(x,f(f(f(f(y,x),f(y,x)),x),x)) = f(y,x). [para(79(a,1),7(a,1,1,2,1)),rewrite([74(7),84(7)]),flip(a)].
5161
198 f(x,f(f(y,f(f(x,y),y)),x)) = f(y,f(f(x,y),y)). [para(79(a,1),97(a,1,1,2,1)),rewrite([84(4),82(4),84(3),84(8)])].
5162
239 f(f(f(x,f(x,y)),x),f(y,f(z,f(f(x,z),z)))) = y. [para(198(a,1),4(a,1,2,2))].
5163
251 f(x,f(y,f(f(f(x,x),y),y))) = f(x,x). [para(198(a,1),5(a,1,2)),rewrite([74(3),74(4),82(4),74(3),74(8)])].
5164
255 f(f(x,f(f(f(y,y),x),x)),f(f(y,y),f(x,f(f(f(y,y),x),x)))) = f(y,y). [para(79(a,1),198(a,1,2,1,2,1)),rewrite([74(7),79(16),74(14)])].
5165
287 f(f(f(x,f(x,y)),x),f(y,y)) = y. [para(5(a,1),239(a,1,2)),rewrite([74(3),74(4),82(4),74(6),74(8),74(9),82(9)])].
5166
330 f(f(f(f(x,x),x),f(x,x)),x) = f(x,x). [para(84(a,1),287(a,1,1,1,2)),rewrite([84(7)])].
5167
347 f(f(f(x,f(f(f(x,x),x),x)),f(f(f(x,x),x),f(x,x))),f(x,f(f(f(f(x,x),x),f(x,x)),y))) = x. [para(330(a,1),4(a,1,1,1,2)),rewrite([80(6)])].
5168
363 f(x,f(f(f(x,f(x,x)),x),x)) = f(x,x). [para(78(a,1),2(a,1,2)),rewrite([84(4),82(4)])].
5169
369 f(f(f(f(x,f(x,x)),x),x),f(x,x)) = x. [para(78(a,1),18(a,1,1)),rewrite([251(9)])].
5170
387 f(f(x,x),f(f(x,f(x,x)),f(y,x))) = f(x,f(x,x)). [para(363(a,1),2(a,1,1))].
5171
406 f(x,f(f(f(f(f(x,f(x,x)),x),x),f(f(x,x),f(f(f(x,f(x,x)),x),x))),x)) = f(f(f(f(x,f(x,x)),x),x),f(f(x,x),f(f(f(x,f(x,x)),x),x))). [para(363(a,1),198(a,1,2,1,2,1)),rewrite([363(22)])].
5172
407 f(f(x,f(x,x)),x) = f(x,x). [para(363(a,1),198(a,1,2,1)),rewrite([82(6),363(8)])].
5173
411 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(78),rewrite([387(6),407(5)]),flip(a)].
5174
412 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(406),rewrite([407(3),411(2),407(6),411(5),82(6),407(3),411(2),407(6),411(5),407(9),411(8),82(9),407(6)])].
5175
430 f(f(x,f(x,x)),f(x,x)) = x. [back_rewrite(369),rewrite([407(3),411(2)])].
5176
439 f(f(x,x),f(x,f(x,y))) = x. [back_rewrite(347),rewrite([411(2),407(3),411(4),430(6),407(3),411(3),430(5)])].
5177
648 f(f(x,x),f(f(y,f(y,x)),y)) = x. [para(287(a,1),28(a,1,1,2,1,1,2,1)),rewrite([439(8),251(9),255(11),287(11),439(9)])].
5178
743 f(f(f(x,f(y,y)),f(f(y,f(x,f(y,y))),f(x,f(y,y)))),f(z,f(f(y,z),z))) = y. [para(77(a,1),7(a,1,1,2,1,1,2,1)),rewrite([439(6),77(19),439(17)])].
5179
1127 f(f(x,x),f(y,x)) = x. [para(134(a,1),2(a,1,2)),rewrite([411(2),412(3)])].
5180
1141 f(x,f(y,f(x,x))) = f(x,x). [para(134(a,1),5(a,1,2)),rewrite([411(2),412(3),411(3),412(4),439(4),411(2),412(3),411(5),412(6)])].
5181
1144 f(f(f(f(f(f(x,f(y,y)),f(x,f(y,y))),f(y,y)),f(y,y)),f(f(x,f(y,y)),f(f(f(f(x,f(y,y)),f(x,f(y,y))),f(y,y)),f(y,y)))),y) = f(y,y). [para(134(a,1),79(a,1,1,2,1))].
5182
1149 f(x,f(f(f(f(f(f(y,x),f(y,x)),x),x),f(f(y,x),f(f(f(f(y,x),f(y,x)),x),x))),x)) = f(f(f(f(f(y,x),f(y,x)),x),x),f(f(y,x),f(f(f(f(y,x),f(y,x)),x),x))). [para(134(a,1),198(a,1,2,1,2,1)),rewrite([134(26)])].
5183
1180 f(f(f(f(f(x,y),f(x,y)),y),y),f(f(f(f(x,y),f(x,y)),f(f(f(f(x,y),f(x,y)),y),y)),f(f(f(f(x,y),f(x,y)),y),y))) = f(x,y). [para(134(a,1),134(a,1,2,1,1,1)),rewrite([134(12),134(27)])].
5184
1185 f(f(f(x,f(y,y)),f(f(y,y),f(x,f(y,y)))),f(z,f(f(y,z),z))) = y. [back_rewrite(743),rewrite([1141(5)])].
5185
1352 f(f(f(x,y),f(x,y)),f(f(x,y),y)) = f(x,y). [para(1127(a,1),82(a,1,2,2))].
5186
1359 f(f(f(f(x,y),f(y,f(x,y))),f(f(f(f(y,y),f(f(z,f(y,y)),f(y,y))),f(f(x,y),f(y,f(x,y)))),f(f(x,y),f(y,f(x,y))))),z) = f(f(y,y),f(f(z,f(y,y)),f(y,y))). [para(1127(a,1),28(a,1,1,1,2,1)),rewrite([1127(14),1127(19)])].
5187
1367 f(f(f(x,y),f(x,y)),y) = f(x,y). [para(1127(a,1),648(a,1,2,1,2)),rewrite([411(5),430(7)])].
5188
1369 f(f(x,y),f(y,f(x,y))) = y. [para(1127(a,1),134(a,1,2,1,1,1)),rewrite([1127(4),1127(4),1127(7)])].
5189
1372 f(f(f(x,y),y),f(f(x,y),f(f(x,y),y))) = f(x,y). [back_rewrite(1180),rewrite([1367(4),1367(9),1352(8),1367(7)])].
5190
1385 f(x,f(f(y,x),x)) = f(y,x). [back_rewrite(1149),rewrite([1367(4),1367(7),1372(7),1367(7),1367(10),1372(10)])].
5191
1387 f(f(x,f(y,y)),y) = f(y,y). [back_rewrite(1144),rewrite([1367(7),1367(13),1372(12)])].
5192
1399 f(f(x,x),y) = f(y,f(x,x)). [back_rewrite(1359),rewrite([1369(4),1385(6),1369(6),1387(3),1369(5),411(2),1141(3),1385(8)])].
5193
1403 f(f(x,x),f(x,y)) = x. [back_rewrite(1185),rewrite([1369(7),1385(4)])].
5194
1635 f(x,y) = f(y,x). [para(1127(a,1),1399(a,1,1)),rewrite([1403(4)])].
5195
1636 $F # answer("Commutativity"). [resolve(1635,a,3,a)].
5339
5196
5340
5197
============================== end of proof ==========================
5341
5198
5342
5199
============================== STATISTICS ============================
5343
5200
5344
Given=63. Generated=7622. Kept=2308. proofs=1.
5345
Usable=16. Sos=238. Demods=253. Limbo=1, Disabled=2055. Hints=1616.
5346
Weight_deleted=400. Literals_deleted=0.
5347
Forward_subsumed=4914. Back_subsumed=267.
5201
Given=55. Generated=6946. Kept=1634. proofs=1.
5202
Usable=6. Sos=171. Demods=175. Limbo=0, Disabled=1459. Hints=1375.
5203
Kept_by_rule=0, Deleted_by_rule=586.
5204
Forward_subsumed=4726. Back_subsumed=194.
5348
5205
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
5349
New_demodulators=2301 (0 lex), Back_demodulated=1785. Back_unit_deleted=0.
5350
Demod_attempts=272126. Demod_rewrites=25458.
5206
New_demodulators=1625 (0 lex), Back_demodulated=1262. Back_unit_deleted=0.
5207
Demod_attempts=308933. Demod_rewrites=26939.
5351
5208
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
5352
5209
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
5353
Megabytes=4.30.
5354
User_CPU=0.88, System_CPU=0.01, Wall_clock=12.
5210
Megabytes=3.79.
5211
User_CPU=0.88, System_CPU=0.01, Wall_clock=9.
5355
5212
5356
5213
============================== end of statistics =====================
5357
5214
5359
5216
5360
5217
Exiting with 1 proof.
5361
5218
5362
Process 13979 exit (max_proofs) Sun Aug 13 21:05:09 2006
5219
Process 10951 exit (max_proofs) Mon Sep 28 11:57:18 2009
5363
5220
5364
5221
Successful proof using the following 1 assumptions:
5365
5222
5367
5224
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
5368
5225
end_of_list.
5369
5226
5370
Including 176 new hint clauses:
5227
Including 173 new hint clauses:
5371
5228
5372
5229
formulas(new_hints).
5373
$F # label("Commutativity").
5230
$F # label("Commutativity") # label(non_clause) # label(goal).
5374
5231
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
5375
5232
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
5376
5233
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
5377
5234
f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)).
5378
f(f(f(x,f(y,z)),f(f(u,f(x,f(y,z))),f(x,f(y,z)))),f(u,x)) = u.
5379
f(f(f(x,y),f(f(f(f(x,f(x,z)),x),f(x,y)),f(x,y))),z) = f(f(x,f(x,z)),x).
5380
f(f(f(f(x,f(f(y,x),x)),f(f(x,f(f(y,x),x)),z)),f(x,f(f(y,x),x))),f(z,y)) = z.
5235
f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)).
5381
5236
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
5382
5237
f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y.
5238
f(f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = f(x,f(f(f(x,f(f(x,x),x)),x),x)).
5383
5239
f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
5384
f(f(x,f(f(y,f(f(x,y),y)),x)),f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),f(z,f(f(x,z),z)))) = f(f(x,f(x,f(y,f(f(x,y),y)))),x).
5240
f(f(x,f(f(f(y,f(f(y,y),y)),x),x)),y) = f(y,f(f(y,y),y)).
5385
5241
f(f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
5386
5242
f(f(f(x,f(x,f(f(x,x),x))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
5387
5243
f(f(f(x,f(x,f(f(x,x),x))),x),f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x))))) = f(f(x,f(f(x,x),x)),f(f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))).
5399
5255
f(f(x,x),f(f(x,f(x,x)),f(x,f(f(x,x),x)))) = f(f(f(x,f(x,f(f(x,x),x))),x),x).
5400
5256
f(f(x,x),f(f(x,f(x,x)),f(x,x))) = f(f(f(x,f(x,f(f(x,x),x))),x),x).
5401
5257
f(f(x,x),f(f(x,f(x,x)),f(x,x))) = f(f(f(x,f(x,x)),x),x).
5258
f(f(x,f(f(f(y,y),x),x)),y) = f(y,f(f(y,y),y)).
5259
f(f(x,f(f(f(y,y),x),x)),y) = f(y,y).
5260
f(f(f(f(x,x),x),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = f(x,f(f(f(x,f(f(x,x),x)),x),x)).
5261
f(f(f(f(x,x),x),f(x,x)),f(x,f(f(x,x),x))) = f(x,f(f(f(x,f(f(x,x),x)),x),x)).
5262
f(f(f(f(x,x),x),f(x,x)),f(x,x)) = f(x,f(f(f(x,f(f(x,x),x)),x),x)).
5263
f(f(f(f(x,x),x),f(x,x)),f(x,x)) = f(x,f(f(f(x,x),x),x)).
5402
5264
f(f(x,x),f(x,f(y,x))) = x.
5403
5265
f(f(x,f(f(x,x),x)),f(x,x)) = x.
5404
5266
f(f(x,x),f(x,x)) = x.
5405
f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)).
5406
f(f(f(x,x),f(x,f(x,x))),f(f(y,f(f(x,y),y)),f(y,f(f(x,y),y)))) = f(y,f(f(x,y),y)).
5407
f(x,f(f(y,f(f(x,y),y)),f(y,f(f(x,y),y)))) = f(y,f(f(x,y),y)).
5408
f(f(f(x,y),f(f(z,f(x,y)),f(x,y))),f(z,x)) = z.
5409
f(f(f(x,x),f(x,f(x,x))),f(f(y,f(f(x,y),y)),x)) = f(y,f(f(x,y),y)).
5267
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
5268
f(f(f(x,y),f(f(f(x,y),f(x,y)),f(x,y))),f(f(x,y),f(x,y))) = f(y,f(f(f(f(x,y),f(x,y)),y),y)).
5269
f(f(f(x,y),f(x,y)),f(f(x,y),f(x,y))) = f(y,f(f(f(f(x,y),f(x,y)),y),y)).
5270
f(x,y) = f(y,f(f(f(f(x,y),f(x,y)),y),y)).
5271
f(x,f(f(f(f(y,x),f(y,x)),x),x)) = f(y,x).
5272
f(f(f(x,x),f(f(f(x,x),f(x,x)),f(x,x))),f(f(y,f(f(f(f(x,x),f(x,x)),y),y)),x)) = f(y,f(f(f(f(x,x),f(x,x)),y),y)).
5273
f(f(f(x,x),f(x,f(x,x))),f(f(y,f(f(f(f(x,x),f(x,x)),y),y)),x)) = f(y,f(f(f(f(x,x),f(x,x)),y),y)).
5274
f(x,f(f(y,f(f(f(f(x,x),f(x,x)),y),y)),x)) = f(y,f(f(f(f(x,x),f(x,x)),y),y)).
5275
f(x,f(f(y,f(f(x,y),y)),x)) = f(y,f(f(f(f(x,x),f(x,x)),y),y)).
5410
5276
f(x,f(f(y,f(f(x,y),y)),x)) = f(y,f(f(x,y),y)).
5411
f(f(x,f(f(y,x),x)),f(f(f(y,f(y,f(x,f(f(y,x),x)))),y),f(z,f(f(y,z),z)))) = f(f(y,f(y,f(x,f(f(y,x),x)))),y).
5412
f(f(f(x,x),f(f(f(x,x),f(x,x)),f(x,x))),f(f(f(x,f(x,x)),x),x)) = f(x,x).
5413
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,f(x,x)),x),x)) = f(x,x).
5414
f(x,f(f(f(x,f(x,x)),x),x)) = f(x,x).
5415
f(f(f(f(x,f(x,x)),x),x),f(x,x)) = x.
5416
f(f(f(x,f(f(y,x),x)),f(f(f(f(y,y),f(y,f(y,y))),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),f(x,f(f(y,x),x))) = f(f(y,y),f(f(f(x,f(f(y,x),x)),f(y,y)),f(y,y))).
5417
f(f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),f(x,f(f(y,x),x))) = f(f(y,y),f(f(f(x,f(f(y,x),x)),f(y,y)),f(y,y))).
5418
f(f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),f(x,f(f(y,x),x))) = f(f(y,y),f(y,f(y,y))).
5419
f(f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),f(x,f(f(y,x),x))) = y.
5420
5277
f(f(f(x,f(x,y)),x),f(y,f(z,f(f(x,z),z)))) = y.
5421
f(f(f(x,f(x,y)),x),f(f(f(y,f(x,z)),f(y,f(y,f(x,z)))),f(f(y,f(x,z)),f(f(f(f(x,f(x,y)),x),f(y,f(x,z))),f(y,f(x,z)))))) = f(f(y,f(x,z)),f(f(f(f(x,f(x,y)),x),f(y,f(x,z))),f(y,f(x,z)))).
5422
f(f(f(x,f(x,y)),x),f(f(f(y,f(x,z)),f(y,f(y,f(x,z)))),f(f(y,f(x,z)),f(y,f(y,f(x,z)))))) = f(f(y,f(x,z)),f(f(f(f(x,f(x,y)),x),f(y,f(x,z))),f(y,f(x,z)))).
5423
f(f(f(x,f(x,y)),x),f(f(f(y,f(x,z)),f(y,f(y,f(x,z)))),f(f(y,f(x,z)),f(y,f(y,f(x,z)))))) = f(f(y,f(x,z)),f(y,f(y,f(x,z)))).
5278
f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(y,f(f(f(x,f(f(x,x),x)),y),y))) = f(x,f(f(x,x),x)).
5279
f(f(f(x,x),f(x,f(x,f(f(x,x),x)))),f(y,f(f(f(x,f(f(x,x),x)),y),y))) = f(x,f(f(x,x),x)).
5280
f(f(f(x,x),f(x,f(x,x))),f(y,f(f(f(x,f(f(x,x),x)),y),y))) = f(x,f(f(x,x),x)).
5281
f(x,f(y,f(f(f(x,f(f(x,x),x)),y),y))) = f(x,f(f(x,x),x)).
5282
f(x,f(y,f(f(f(x,x),y),y))) = f(x,f(f(x,x),x)).
5283
f(x,f(y,f(f(f(x,x),y),y))) = f(x,x).
5284
f(f(x,f(f(f(y,y),x),x)),f(f(y,f(f(y,y),y)),f(x,f(f(f(y,y),x),x)))) = f(y,f(f(f(x,f(f(f(y,y),x),x)),y),y)).
5285
f(f(x,f(f(f(y,y),x),x)),f(f(y,y),f(x,f(f(f(y,y),x),x)))) = f(y,f(f(f(x,f(f(f(y,y),x),x)),y),y)).
5286
f(f(x,f(f(f(y,y),x),x)),f(f(y,y),f(x,f(f(f(y,y),x),x)))) = f(y,f(f(y,y),y)).
5287
f(f(x,f(f(f(y,y),x),x)),f(f(y,y),f(x,f(f(f(y,y),x),x)))) = f(y,y).
5424
5288
f(f(f(x,f(x,f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))))),x),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
5425
5289
f(f(f(x,f(x,f(f(y,y),f(y,f(y,f(f(y,y),y)))))),x),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
5426
5290
f(f(f(x,f(x,f(f(y,y),f(y,f(y,y))))),x),f(y,f(f(y,y),y))) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
5429
5293
f(f(f(x,f(x,y)),x),f(y,y)) = f(f(y,y),f(y,f(y,f(f(y,y),y)))).
5430
5294
f(f(f(x,f(x,y)),x),f(y,y)) = f(f(y,y),f(y,f(y,y))).
5431
5295
f(f(f(x,f(x,y)),x),f(y,y)) = y.
5432
f(f(f(f(f(x,f(x,y)),x),f(f(f(x,f(x,y)),x),f(f(y,y),f(y,f(y,y))))),f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x)) = f(f(y,y),f(f(f(f(x,f(x,y)),x),f(y,y)),f(y,y))).
5433
f(f(f(f(f(x,f(x,y)),x),f(f(f(x,f(x,y)),x),y)),f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x)) = f(f(y,y),f(f(f(f(x,f(x,y)),x),f(y,y)),f(y,y))).
5434
f(f(f(f(f(x,f(x,y)),x),f(f(f(x,f(x,y)),x),y)),f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x)) = f(f(y,y),f(y,f(y,y))).
5435
f(f(f(f(f(x,f(x,y)),x),f(f(f(x,f(x,y)),x),y)),f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x)) = y.
5436
f(f(f(x,x),f(x,f(x,x))),f(f(f(y,f(y,x)),y),x)) = f(f(y,f(y,x)),y).
5437
f(x,f(f(f(y,f(y,x)),y),x)) = f(f(y,f(y,x)),y).
5438
f(f(f(f(x,f(x,y)),x),f(f(f(f(y,y),f(y,f(y,y))),f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x))),f(f(x,f(x,y)),x)) = f(f(y,y),f(f(f(f(x,f(x,y)),x),f(y,y)),f(y,y))).
5439
f(f(f(f(x,f(x,y)),x),f(f(y,f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x))),f(f(x,f(x,y)),x)) = f(f(y,y),f(f(f(f(x,f(x,y)),x),f(y,y)),f(y,y))).
5440
f(f(f(f(x,f(x,y)),x),f(f(y,f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x))),f(f(x,f(x,y)),x)) = f(f(y,y),f(y,f(y,y))).
5441
f(f(f(f(x,f(x,y)),x),f(f(y,f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x))),f(f(x,f(x,y)),x)) = y.
5442
f(f(x,f(x,x)),x) = f(x,x).
5443
f(f(f(x,x),x),f(x,x)) = x.
5444
f(f(x,x),f(f(x,f(x,x)),f(x,x))) = f(f(x,x),x).
5445
f(f(x,f(f(x,x),x)),f(f(x,f(x,x)),f(y,x))) = f(x,f(x,x)).
5296
f(f(f(f(x,x),x),f(x,x)),f(f(x,x),f(x,x))) = f(x,x).
5297
f(f(f(f(x,x),x),f(x,x)),x) = f(x,x).
5298
f(f(f(f(f(f(x,x),x),f(x,x)),f(x,x)),f(f(f(x,x),x),f(x,x))),f(x,f(f(f(f(x,x),x),f(x,x)),y))) = x.
5299
f(f(f(x,f(f(f(x,x),x),x)),f(f(f(x,x),x),f(x,x))),f(x,f(f(f(f(x,x),x),f(x,x)),y))) = x.
5300
f(f(f(x,x),f(f(f(x,x),f(x,x)),f(x,x))),f(f(f(x,f(x,x)),x),x)) = f(x,x).
5301
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,f(x,x)),x),x)) = f(x,x).
5302
f(x,f(f(f(x,f(x,x)),x),x)) = f(x,x).
5303
f(f(f(f(x,f(x,x)),x),x),f(x,f(y,f(f(f(x,x),y),y)))) = x.
5304
f(f(f(f(x,f(x,x)),x),x),f(x,x)) = x.
5446
5305
f(f(x,x),f(f(x,f(x,x)),f(y,x))) = f(x,f(x,x)).
5447
f(f(x,x),f(x,f(x,y))) = x.
5306
f(x,f(f(f(f(f(x,f(x,x)),x),x),f(f(x,x),f(f(f(x,f(x,x)),x),x))),x)) = f(f(f(f(x,f(x,x)),x),x),f(f(x,f(f(f(x,f(x,x)),x),x)),f(f(f(x,f(x,x)),x),x))).
5307
f(x,f(f(f(f(f(x,f(x,x)),x),x),f(f(x,x),f(f(f(x,f(x,x)),x),x))),x)) = f(f(f(f(x,f(x,x)),x),x),f(f(x,x),f(f(f(x,f(x,x)),x),x))).
5308
f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))) = f(x,f(f(f(x,f(x,x)),x),x)).
5309
f(f(x,f(x,x)),x) = f(x,f(f(f(x,f(x,x)),x),x)).
5310
f(f(x,f(x,x)),x) = f(x,x).
5311
f(x,f(x,x)) = f(f(f(x,f(x,x)),x),x).
5448
5312
f(x,f(x,x)) = f(f(x,x),x).
5449
5313
f(f(x,x),x) = f(x,f(x,x)).
5314
f(x,f(f(f(f(x,x),x),f(f(x,x),f(f(f(x,f(x,x)),x),x))),x)) = f(f(f(f(x,f(x,x)),x),x),f(f(x,x),f(f(f(x,f(x,x)),x),x))).
5315
f(x,f(f(f(x,f(x,x)),f(f(x,x),f(f(f(x,f(x,x)),x),x))),x)) = f(f(f(f(x,f(x,x)),x),x),f(f(x,x),f(f(f(x,f(x,x)),x),x))).
5316
f(x,f(f(f(x,f(x,x)),f(f(x,x),f(f(x,x),x))),x)) = f(f(f(f(x,f(x,x)),x),x),f(f(x,x),f(f(f(x,f(x,x)),x),x))).
5317
f(x,f(f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))),x)) = f(f(f(f(x,f(x,x)),x),x),f(f(x,x),f(f(f(x,f(x,x)),x),x))).
5318
f(x,f(f(f(x,f(x,x)),x),x)) = f(f(f(f(x,f(x,x)),x),x),f(f(x,x),f(f(f(x,f(x,x)),x),x))).
5319
f(x,f(f(x,x),x)) = f(f(f(f(x,f(x,x)),x),x),f(f(x,x),f(f(f(x,f(x,x)),x),x))).
5320
f(x,f(x,f(x,x))) = f(f(f(f(x,f(x,x)),x),x),f(f(x,x),f(f(f(x,f(x,x)),x),x))).
5321
f(x,f(x,f(x,x))) = f(f(f(x,x),x),f(f(x,x),f(f(f(x,f(x,x)),x),x))).
5322
f(x,f(x,f(x,x))) = f(f(x,f(x,x)),f(f(x,x),f(f(f(x,f(x,x)),x),x))).
5323
f(x,f(x,f(x,x))) = f(f(x,f(x,x)),f(f(x,x),f(f(x,x),x))).
5324
f(x,f(x,f(x,x))) = f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))).
5325
f(x,f(x,f(x,x))) = f(f(x,f(x,x)),x).
5326
f(x,f(x,f(x,x))) = f(x,x).
5327
f(f(f(x,x),x),f(x,x)) = x.
5450
5328
f(f(x,f(x,x)),f(x,x)) = x.
5451
f(f(f(f(x,x),x),f(x,x)),f(f(x,f(x,y)),f(x,f(x,y)))) = f(x,f(x,y)).
5452
f(f(f(x,f(x,x)),f(x,x)),f(f(x,f(x,y)),f(x,f(x,y)))) = f(x,f(x,y)).
5453
f(x,f(f(x,f(x,y)),f(x,f(x,y)))) = f(x,f(x,y)).
5454
f(f(f(f(f(x,y),f(f(z,f(x,y)),f(x,y))),z),f(f(x,y),f(f(z,f(x,y)),f(x,y)))),f(f(z,x),z)) = f(z,x).
5455
f(f(f(f(f(x,y),f(f(f(f(x,f(x,z)),x),f(x,y)),f(x,y))),f(f(x,f(x,z)),x)),f(f(x,y),f(f(f(f(x,f(x,z)),x),f(x,y)),f(x,y)))),f(z,f(f(x,f(x,z)),x))) = z.
5456
f(f(f(f(x,f(x,y)),f(x,f(x,y))),f(f(x,f(x,y)),f(f(x,f(x,y)),f(x,f(x,y))))),f(x,x)) = x.
5457
f(f(x,f(x,y)),f(x,x)) = x.
5458
f(f(f(f(x,f(x,y)),f(x,f(x,y))),f(f(x,f(x,y)),f(f(x,f(x,y)),f(x,f(x,y))))),f(x,f(x,f(x,y)))) = x.
5459
f(f(x,f(x,y)),f(x,f(x,f(x,y)))) = x.
5460
f(f(x,f(f(f(f(f(x,f(x,y)),f(x,f(x,y))),f(f(x,f(x,y)),f(f(x,f(x,y)),f(x,f(x,y))))),x),x)),x) = f(f(f(x,f(x,y)),f(x,f(x,y))),f(f(x,f(f(x,f(x,y)),f(x,f(x,y)))),f(f(x,f(x,y)),f(x,f(x,y))))).
5461
f(f(x,f(f(f(x,f(x,y)),x),x)),x) = f(f(f(x,f(x,y)),f(x,f(x,y))),f(f(x,f(f(x,f(x,y)),f(x,f(x,y)))),f(f(x,f(x,y)),f(x,f(x,y))))).
5462
f(f(x,f(f(f(x,f(x,y)),x),x)),x) = f(f(f(x,f(x,y)),f(x,f(x,y))),f(f(x,f(x,y)),f(f(x,f(x,y)),f(x,f(x,y))))).
5463
f(f(x,f(f(f(x,f(x,y)),x),x)),x) = f(x,f(x,y)).
5464
f(f(f(x,x),f(x,f(x,x))),f(f(x,f(x,y)),x)) = f(x,f(x,y)).
5465
f(x,f(f(x,f(x,y)),x)) = f(x,f(x,y)).
5466
f(f(f(f(x,f(x,y)),x),f(f(f(f(f(x,f(x,y)),x),y),f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x))),f(f(x,f(x,y)),x)) = f(f(f(x,f(x,y)),x),f(f(f(x,f(x,y)),x),f(y,f(x,z)))).
5467
f(f(f(f(x,f(x,y)),x),f(f(f(f(f(x,f(x,y)),x),y),f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x))),f(f(x,f(x,y)),x)) = f(f(f(x,f(x,y)),x),y).
5468
f(f(x,f(y,f(f(x,y),y))),x) = f(x,f(x,f(y,f(f(x,y),y)))).
5469
f(f(x,f(f(f(x,f(y,f(f(x,y),y))),x),x)),x) = f(x,f(x,f(f(y,f(f(x,y),y)),f(y,f(f(x,y),y))))).
5470
f(f(x,f(f(x,f(x,f(y,f(f(x,y),y)))),x)),x) = f(x,f(x,f(f(y,f(f(x,y),y)),f(y,f(f(x,y),y))))).
5471
f(f(x,f(x,f(y,f(f(x,y),y)))),x) = f(x,f(x,f(f(y,f(f(x,y),y)),f(y,f(f(x,y),y))))).
5472
f(f(x,f(x,f(y,f(f(x,y),y)))),x) = f(x,f(y,f(f(x,y),y))).
5473
f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(z,f(f(y,z),z)))) = f(f(y,f(y,f(x,f(f(y,x),x)))),y).
5474
f(f(x,f(f(y,x),x)),f(f(y,f(x,f(f(y,x),x))),f(z,f(f(y,z),z)))) = f(y,f(x,f(f(y,x),x))).
5475
f(f(x,f(y,f(f(x,y),y))),x) = f(y,f(f(x,y),y)).
5476
f(x,f(x,f(y,f(f(x,y),y)))) = f(y,f(f(x,y),y)).
5477
f(f(x,f(y,f(f(x,y),y))),f(y,f(f(x,y),y))) = x.
5478
f(f(x,f(f(y,x),x)),y) = f(y,f(x,f(f(y,x),x))).
5479
f(f(f(f(f(x,f(x,z)),x),f(f(x,y),f(f(f(f(x,f(x,z)),x),f(x,y)),f(x,y)))),f(f(x,y),f(f(f(f(x,f(x,z)),x),f(x,y)),f(x,y)))),f(z,f(f(x,f(x,z)),x))) = z.
5480
f(f(f(x,f(x,y)),x),f(y,f(f(x,f(x,y)),x))) = y.
5481
f(f(f(z,f(f(x,y),f(f(z,f(x,y)),f(x,y)))),f(f(x,y),f(f(z,f(x,y)),f(x,y)))),f(f(z,x),z)) = f(z,x).
5482
f(x,f(f(x,y),x)) = f(x,y).
5483
f(f(f(x,f(x,y)),x),f(f(f(x,f(x,y)),x),f(f(y,y),f(y,f(y,y))))) = f(f(y,y),f(f(f(f(x,f(x,y)),x),f(y,y)),f(y,y))).
5484
f(f(f(x,f(x,y)),x),f(f(f(x,f(x,y)),x),y)) = f(f(y,y),f(f(f(f(x,f(x,y)),x),f(y,y)),f(y,y))).
5485
f(f(f(x,f(x,y)),x),f(f(f(x,f(x,y)),x),y)) = f(f(y,y),f(y,f(y,y))).
5486
f(f(f(x,f(x,y)),x),f(f(f(x,f(x,y)),x),y)) = y.
5487
f(f(x,f(f(y,f(y,x)),y)),f(f(y,f(y,x)),y)) = x.
5488
f(f(f(f(x,f(x,y)),x),y),f(f(x,f(x,y)),x)) = y.
5489
f(f(f(f(x,f(x,y)),x),f(y,f(f(x,f(x,y)),x))),f(f(x,f(x,y)),x)) = f(f(f(x,f(x,y)),x),y).
5490
f(y,f(f(x,f(x,y)),x)) = f(f(f(x,f(x,y)),x),y).
5491
f(f(f(x,f(x,y)),x),y) = f(y,f(f(x,f(x,y)),x)).
5492
f(x,f(x,f(f(y,f(y,x)),y))) = f(f(y,f(y,x)),y).
5493
f(f(f(x,f(f(f(f(x,f(x,y)),x),x),x)),f(f(x,f(x,y)),x)),y) = f(f(x,f(x,y)),x).
5494
f(f(f(f(x,f(x,y)),x),f(x,f(f(f(f(x,f(x,y)),x),x),x))),y) = f(f(x,f(x,y)),x).
5495
f(f(f(x,f(x,y)),f(f(x,x),f(x,f(x,x)))),f(f(x,x),f(f(f(x,f(x,y)),f(x,x)),f(x,x)))) = f(x,f(x,y)).
5496
f(f(f(x,f(x,y)),x),f(f(x,x),f(f(f(x,f(x,y)),f(x,x)),f(x,x)))) = f(x,f(x,y)).
5497
f(f(f(x,f(x,y)),x),f(f(x,x),f(x,f(x,x)))) = f(x,f(x,y)).
5498
f(f(f(x,f(x,y)),x),x) = f(x,f(x,y)).
5499
f(f(f(f(x,f(x,y)),x),f(x,f(f(x,f(x,y)),x))),y) = f(f(x,f(x,y)),x).
5500
f(f(f(f(x,f(x,y)),x),f(x,f(x,y))),y) = f(f(x,f(x,y)),x).
5501
f(f(x,f(x,f(x,y))),x) = f(x,f(x,y)).
5502
f(f(x,f(x,y)),f(f(x,f(x,y)),f(x,f(x,f(x,y))))) = f(x,f(f(f(x,f(x,y)),x),x)).
5503
f(f(x,f(x,y)),x) = f(x,f(f(f(x,f(x,y)),x),x)).
5504
f(f(x,f(x,y)),x) = f(x,f(x,f(x,y))).
5505
f(f(f(f(f(x,f(x,y)),x),f(f(f(x,f(x,y)),x),x)),f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x)) = f(f(f(x,f(x,y)),x),f(x,f(x,y))).
5506
f(f(f(f(x,f(x,f(x,y))),f(f(f(x,f(x,y)),x),x)),f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x)) = f(f(f(x,f(x,y)),x),f(x,f(x,y))).
5507
f(f(f(f(x,f(x,f(x,y))),f(f(x,f(x,f(x,y))),x)),f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x)) = f(f(f(x,f(x,y)),x),f(x,f(x,y))).
5508
f(f(f(f(x,f(x,f(x,y))),f(x,f(x,f(x,f(x,y))))),f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x)) = f(f(f(x,f(x,y)),x),f(x,f(x,y))).
5509
f(f(x,f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x)) = f(f(f(x,f(x,y)),x),f(x,f(x,y))).
5510
f(f(x,f(x,f(x,f(x,y)))),f(f(x,f(x,y)),x)) = f(f(f(x,f(x,y)),x),f(x,f(x,y))).
5511
f(f(x,f(x,f(x,f(x,y)))),f(x,f(x,f(x,y)))) = f(f(f(x,f(x,y)),x),f(x,f(x,y))).
5512
f(f(x,f(x,f(x,f(x,y)))),f(x,f(x,f(x,y)))) = f(f(x,f(x,f(x,y))),f(x,f(x,y))).
5513
f(x,f(x,f(x,f(x,y)))) = f(x,f(x,y)).
5514
f(f(f(x,f(x,f(x,y))),f(x,f(x,y))),y) = f(f(x,f(x,y)),x).
5515
f(f(f(x,f(x,f(x,y))),f(x,f(x,y))),y) = f(x,f(x,f(x,y))).
5516
f(x,f(x,f(y,f(y,f(y,x))))) = f(f(y,f(y,x)),y).
5517
f(x,f(x,f(y,f(y,f(y,x))))) = f(y,f(y,f(y,x))).
5518
f(f(x,f(x,f(x,y))),y) = f(y,f(f(x,f(x,y)),x)).
5519
f(f(x,f(x,f(x,y))),y) = f(y,f(x,f(x,f(x,y)))).
5520
f(f(x,f(x,f(x,y))),f(f(f(y,f(x,z)),f(y,f(y,f(x,z)))),f(f(y,f(x,z)),f(y,f(y,f(x,z)))))) = f(f(y,f(x,z)),f(y,f(y,f(x,z)))).
5521
f(f(x,f(x,y)),f(x,f(x,f(x,y)))) = f(f(x,f(x,f(x,y))),f(x,f(x,y))).
5522
x = f(f(x,f(x,f(x,y))),f(x,f(x,y))).
5523
f(f(x,f(x,f(x,y))),f(x,f(x,y))) = x.
5524
f(x,y) = f(x,f(x,f(x,y))).
5525
f(x,f(x,f(x,y))) = f(x,y).
5526
f(f(x,y),f(x,f(x,y))) = x.
5527
f(f(x,y),f(f(f(y,f(x,z)),f(y,f(y,f(x,z)))),f(f(y,f(x,z)),f(y,f(y,f(x,z)))))) = f(f(y,f(x,z)),f(y,f(y,f(x,z)))).
5528
f(f(x,y),f(y,f(f(y,f(x,z)),f(y,f(y,f(x,z)))))) = f(f(y,f(x,z)),f(y,f(y,f(x,z)))).
5529
f(f(x,y),f(y,y)) = f(f(y,f(x,z)),f(y,f(y,f(x,z)))).
5530
f(f(x,y),f(y,y)) = y.
5531
f(f(x,y),y) = f(y,f(x,f(x,f(x,y)))).
5532
f(f(x,y),y) = f(y,f(x,y)).
5533
f(x,f(x,f(y,x))) = f(y,f(y,f(y,x))).
5534
f(x,f(x,f(y,x))) = f(y,x).
5535
f(f(f(x,f(x,f(y,x))),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
5536
f(f(f(y,x),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
5537
f(f(f(y,x),f(f(f(y,f(y,f(z,y))),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
5538
f(f(f(y,x),f(f(f(z,y),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
5539
f(f(f(y,x),f(f(f(z,y),f(x,f(x,f(y,x)))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
5540
f(f(f(y,x),f(f(f(z,y),f(y,x)),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y)).
5541
f(f(f(y,x),f(f(f(z,y),f(y,x)),f(x,f(x,f(y,x))))),z) = f(y,f(f(z,y),y)).
5542
f(f(f(y,x),f(f(f(z,y),f(y,x)),f(y,x))),z) = f(y,f(f(z,y),y)).
5543
f(f(f(y,x),f(f(y,x),f(f(z,y),f(y,x)))),z) = f(y,f(f(z,y),y)).
5544
f(f(f(z,y),f(y,x)),z) = f(y,f(f(z,y),y)).
5545
f(f(f(z,y),f(y,x)),z) = f(y,f(y,f(z,y))).
5546
f(f(f(x,y),f(y,z)),x) = f(x,y).
5329
f(f(f(x,f(f(x,f(x,x)),x)),f(f(f(x,x),x),f(x,x))),f(x,f(f(f(f(x,x),x),f(x,x)),y))) = x.
5330
f(f(f(x,f(x,x)),f(f(f(x,x),x),f(x,x))),f(x,f(f(f(f(x,x),x),f(x,x)),y))) = x.
5331
f(f(f(x,f(x,x)),f(f(x,f(x,x)),f(x,x))),f(x,f(f(f(f(x,x),x),f(x,x)),y))) = x.
5332
f(f(f(x,f(x,x)),x),f(x,f(f(f(f(x,x),x),f(x,x)),y))) = x.
5333
f(f(x,x),f(x,f(f(f(f(x,x),x),f(x,x)),y))) = x.
5334
f(f(x,x),f(x,f(f(f(x,f(x,x)),f(x,x)),y))) = x.
5335
f(f(x,x),f(x,f(x,y))) = x.
5336
f(f(f(x,f(f(f(y,y),x),x)),f(f(f(f(y,y),f(y,f(y,y))),f(x,f(f(f(y,y),x),x))),f(x,f(f(f(y,y),x),x)))),f(f(z,f(z,y)),z)) = f(f(y,y),f(f(f(f(z,f(z,y)),z),f(y,y)),f(y,y))).
5337
f(f(f(x,f(f(f(y,y),x),x)),f(f(y,f(x,f(f(f(y,y),x),x))),f(x,f(f(f(y,y),x),x)))),f(f(z,f(z,y)),z)) = f(f(y,y),f(f(f(f(z,f(z,y)),z),f(y,y)),f(y,y))).
5338
f(f(f(x,f(f(f(y,y),x),x)),f(f(y,y),f(x,f(f(f(y,y),x),x)))),f(f(z,f(z,y)),z)) = f(f(y,y),f(f(f(f(z,f(z,y)),z),f(y,y)),f(y,y))).
5339
f(f(y,y),f(f(z,f(z,y)),z)) = f(f(y,y),f(f(f(f(z,f(z,y)),z),f(y,y)),f(y,y))).
5340
f(f(y,y),f(f(z,f(z,y)),z)) = f(f(y,y),f(y,f(y,y))).
5341
f(f(x,x),f(f(y,f(y,x)),y)) = x.
5342
f(f(f(x,f(y,y)),f(f(f(f(y,y),f(y,f(y,y))),f(x,f(y,y))),f(x,f(y,y)))),f(z,f(f(y,z),z))) = f(f(y,y),f(f(f(z,f(f(y,z),z)),f(y,y)),f(y,y))).
5343
f(f(f(x,f(y,y)),f(f(y,f(x,f(y,y))),f(x,f(y,y)))),f(z,f(f(y,z),z))) = f(f(y,y),f(f(f(z,f(f(y,z),z)),f(y,y)),f(y,y))).
5344
f(f(f(x,f(y,y)),f(f(y,f(x,f(y,y))),f(x,f(y,y)))),f(z,f(f(y,z),z))) = f(f(y,y),f(y,f(y,y))).
5345
f(f(f(x,f(y,y)),f(f(y,f(x,f(y,y))),f(x,f(y,y)))),f(z,f(f(y,z),z))) = y.
5346
f(f(x,f(f(x,x),x)),f(y,x)) = x.
5347
f(f(x,f(x,f(x,x))),f(y,x)) = x.
5348
f(f(x,x),f(y,x)) = x.
5349
f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(y,f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
5350
f(f(f(x,f(x,f(x,x))),f(x,f(x,f(f(x,x),x)))),f(y,f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
5351
f(f(f(x,x),f(x,f(x,f(f(x,x),x)))),f(y,f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
5352
f(f(f(x,x),f(x,f(x,f(x,f(x,x))))),f(y,f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
5353
f(f(f(x,x),f(x,f(x,x))),f(y,f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
5354
f(x,f(y,f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
5355
f(x,f(y,f(x,f(x,f(x,x))))) = f(x,f(f(x,x),x)).
5356
f(x,f(y,f(x,x))) = f(x,f(f(x,x),x)).
5357
f(x,f(y,f(x,x))) = f(x,f(x,f(x,x))).
5358
f(x,f(y,f(x,x))) = f(x,x).
5359
f(f(f(f(f(f(x,f(y,y)),f(x,f(y,y))),f(y,y)),f(y,y)),f(f(x,f(y,y)),f(f(f(f(x,f(y,y)),f(x,f(y,y))),f(y,y)),f(y,y)))),y) = f(y,y).
5360
f(x,f(f(f(f(f(f(y,x),f(y,x)),x),x),f(f(y,x),f(f(f(f(y,x),f(y,x)),x),x))),x)) = f(f(f(f(f(y,x),f(y,x)),x),x),f(f(x,f(f(f(f(y,x),f(y,x)),x),x)),f(f(f(f(y,x),f(y,x)),x),x))).
5361
f(x,f(f(f(f(f(f(y,x),f(y,x)),x),x),f(f(y,x),f(f(f(f(y,x),f(y,x)),x),x))),x)) = f(f(f(f(f(y,x),f(y,x)),x),x),f(f(y,x),f(f(f(f(y,x),f(y,x)),x),x))).
5362
f(f(f(f(f(x,y),f(x,y)),y),y),f(f(f(f(x,y),f(y,f(f(f(f(x,y),f(x,y)),y),y))),f(f(f(f(x,y),f(x,y)),y),y)),f(f(f(f(x,y),f(x,y)),y),y))) = f(y,f(f(f(f(x,y),f(x,y)),y),y)).
5363
f(f(f(f(f(x,y),f(x,y)),y),y),f(f(f(f(x,y),f(x,y)),f(f(f(f(x,y),f(x,y)),y),y)),f(f(f(f(x,y),f(x,y)),y),y))) = f(y,f(f(f(f(x,y),f(x,y)),y),y)).
5364
f(f(f(f(f(x,y),f(x,y)),y),y),f(f(f(f(x,y),f(x,y)),f(f(f(f(x,y),f(x,y)),y),y)),f(f(f(f(x,y),f(x,y)),y),y))) = f(x,y).
5365
f(f(f(x,f(y,y)),f(f(y,y),f(x,f(y,y)))),f(z,f(f(y,z),z))) = y.
5366
f(f(f(x,y),f(x,y)),f(f(x,y),y)) = f(x,y).
5367
f(f(f(f(x,y),f(y,f(x,y))),f(f(f(f(y,y),f(f(z,f(y,y)),f(y,y))),f(f(x,y),f(f(f(y,y),f(x,y)),f(x,y)))),f(f(x,y),f(f(f(y,y),f(x,y)),f(x,y))))),z) = f(f(y,y),f(f(z,f(y,y)),f(y,y))).
5368
f(f(f(f(x,y),f(y,f(x,y))),f(f(f(f(y,y),f(f(z,f(y,y)),f(y,y))),f(f(x,y),f(y,f(x,y)))),f(f(x,y),f(f(f(y,y),f(x,y)),f(x,y))))),z) = f(f(y,y),f(f(z,f(y,y)),f(y,y))).
5369
f(f(f(f(x,y),f(y,f(x,y))),f(f(f(f(y,y),f(f(z,f(y,y)),f(y,y))),f(f(x,y),f(y,f(x,y)))),f(f(x,y),f(y,f(x,y))))),z) = f(f(y,y),f(f(z,f(y,y)),f(y,y))).
5370
f(f(f(x,y),f(x,y)),f(f(f(y,y),y),f(y,y))) = f(x,y).
5371
f(f(f(x,y),f(x,y)),f(f(y,f(y,y)),f(y,y))) = f(x,y).
5372
f(f(f(x,y),f(x,y)),y) = f(x,y).
5373
f(f(x,y),f(f(f(y,f(f(y,y),f(x,y))),f(x,y)),f(x,y))) = f(f(y,y),f(x,y)).
5374
f(f(x,y),f(f(f(y,y),f(x,y)),f(x,y))) = f(f(y,y),f(x,y)).
5375
f(f(x,y),f(y,f(x,y))) = f(f(y,y),f(x,y)).
5376
f(f(x,y),f(y,f(x,y))) = y.
5377
f(f(f(x,y),y),f(f(f(f(x,y),f(x,y)),f(f(f(f(x,y),f(x,y)),y),y)),f(f(f(f(x,y),f(x,y)),y),y))) = f(x,y).
5378
f(f(f(x,y),y),f(f(f(f(x,y),f(x,y)),f(f(x,y),y)),f(f(f(f(x,y),f(x,y)),y),y))) = f(x,y).
5379
f(f(f(x,y),y),f(f(x,y),f(f(f(f(x,y),f(x,y)),y),y))) = f(x,y).
5380
f(f(f(x,y),y),f(f(x,y),f(f(x,y),y))) = f(x,y).
5381
f(x,f(f(f(f(y,x),x),f(f(y,x),f(f(f(f(y,x),f(y,x)),x),x))),x)) = f(f(f(f(f(y,x),f(y,x)),x),x),f(f(y,x),f(f(f(f(y,x),f(y,x)),x),x))).
5382
f(x,f(f(f(f(y,x),x),f(f(y,x),f(f(y,x),x))),x)) = f(f(f(f(f(y,x),f(y,x)),x),x),f(f(y,x),f(f(f(f(y,x),f(y,x)),x),x))).
5383
f(x,f(f(y,x),x)) = f(f(f(f(f(y,x),f(y,x)),x),x),f(f(y,x),f(f(f(f(y,x),f(y,x)),x),x))).
5384
f(x,f(f(y,x),x)) = f(f(f(y,x),x),f(f(y,x),f(f(f(f(y,x),f(y,x)),x),x))).
5385
f(x,f(f(y,x),x)) = f(f(f(y,x),x),f(f(y,x),f(f(y,x),x))).
5386
f(x,f(f(y,x),x)) = f(y,x).
5387
f(f(f(f(x,f(y,y)),f(y,y)),f(f(x,f(y,y)),f(f(f(f(x,f(y,y)),f(x,f(y,y))),f(y,y)),f(y,y)))),y) = f(y,y).
5388
f(f(f(f(x,f(y,y)),f(y,y)),f(f(x,f(y,y)),f(f(x,f(y,y)),f(y,y)))),y) = f(y,y).
5389
f(f(x,f(y,y)),y) = f(y,y).
5390
f(f(y,f(f(f(f(y,y),f(f(z,f(y,y)),f(y,y))),f(f(x,y),f(y,f(x,y)))),f(f(x,y),f(y,f(x,y))))),z) = f(f(y,y),f(f(z,f(y,y)),f(y,y))).
5391
f(f(y,f(f(f(z,f(y,y)),f(f(x,y),f(y,f(x,y)))),f(f(x,y),f(y,f(x,y))))),z) = f(f(y,y),f(f(z,f(y,y)),f(y,y))).
5392
f(f(y,f(f(f(z,f(y,y)),y),f(f(x,y),f(y,f(x,y))))),z) = f(f(y,y),f(f(z,f(y,y)),f(y,y))).
5393
f(f(y,f(f(y,y),f(f(x,y),f(y,f(x,y))))),z) = f(f(y,y),f(f(z,f(y,y)),f(y,y))).
5394
f(f(y,f(f(y,y),y)),z) = f(f(y,y),f(f(z,f(y,y)),f(y,y))).
5395
f(f(y,f(y,f(y,y))),z) = f(f(y,y),f(f(z,f(y,y)),f(y,y))).
5396
f(f(y,y),z) = f(f(y,y),f(f(z,f(y,y)),f(y,y))).
5397
f(f(x,x),y) = f(y,f(x,x)).
5398
f(f(y,y),f(z,f(f(y,z),z))) = y.
5399
f(f(x,x),f(x,y)) = x.
5400
f(x,y) = f(y,f(f(x,x),f(x,x))).
5547
5401
f(x,y) = f(y,x).
5548
5402
$F # answer("Commutativity").
5549
5403
end_of_list.
5564
5418
Note: Attempt with no assumptions!
5565
5419
5566
5420
5567
Child search process 13980 started.
5421
Child search process 10952 started.
5568
5422
5569
5423
============================== PROCESS INITIAL CLAUSES ===============
5570
5424
5581
5435
formulas(demodulators).
5582
5436
end_of_list.
5583
5437
5584
% 1792 hints input.
5438
% 1548 hints input.
5585
5439
5586
5440
============================== PREDICATE ELIMINATION =================
5587
5441
5594
5448
5595
5449
Term ordering decisions:
5596
5450
Function symbol KB weights: c1=1. c2=1. f=1.
5597
Relation symbol precedence: lex([ = ]).
5598
Function symbol precedence: lex([ c1, c2, f ]).
5451
Predicate symbol precedence: predicate_order([ = ]).
5452
Function symbol precedence: function_order([ c1, c2, f ]).
5599
5453
Skipping inverse_order, because term ordering is KBO.
5600
5454
Unfolding symbols: (none).
5601
5455
5602
5456
Auto_inference settings:
5603
5457
% set(paramodulation). % (positive equality literals)
5604
% set(paramodulation) -> set(back_demod).
5605
5458
5606
5459
Auto_process settings: (no changes).
5607
5460
5461
kept: 2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
5462
kept: 3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
5463
5608
5464
============================== end of process initial clauses ========
5609
5465
5610
5466
============================== CLAUSES FOR SEARCH ====================
5623
5479
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
5624
5480
end_of_list.
5625
5481
5626
% 1792 hints processed (973 are redundant).
5482
% 709 hints (1548 processed, 839 redundant).
5627
5483
5628
5484
============================== end of clauses for search =============
5629
5485
5630
5486
============================== SEARCH ================================
5631
5487
5632
% Starting search at 0.04 seconds.
5488
% Starting search at 0.07 seconds.
5633
5489
5634
5490
% Operation f is commutative; C redundancy checks enabled.
5635
5491
5636
5492
============================== PROOF =================================
5637
5493
5638
% Proof 1 at 0.24 (+ 0.00) seconds: "Commutativity".
5639
% Length of proof is 30.
5640
% Level of proof is 13.
5641
% Maximum clause weight is 27.
5642
% Given clauses 29.
5494
% Proof 1 at 0.19 (+ 0.00) seconds: "Commutativity".
5495
% Length of proof is 31.
5496
% Level of proof is 14.
5497
% Maximum clause weight is 61.
5498
% Given clauses 32.
5643
5499
5644
1 f(x,y) = f(y,x) # label("Commutativity"). [goal].
5500
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
5645
5501
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
5646
5502
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
5647
4 f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(u,f(x,f(y,z))))) = f(z,f(f(x,z),z)). [para(2(a,1),2(a,1,1,2,1))].
5648
5 f(f(f(x,f(y,z)),f(f(u,f(x,f(y,z))),f(x,f(y,z)))),f(u,x)) = u. [para(2(a,1),2(a,1,2,2))].
5649
5503
6 f(f(f(x,y),f(f(f(y,f(f(z,y),y)),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)). [para(2(a,1),2(a,1,2))].
5650
8 f(f(f(x,y),f(f(z,f(x,y)),f(x,y))),f(z,x)) = z. [para(5(a,1),2(a,1,2,2))].
5651
10 f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y. [para(2(a,1),5(a,1,1,1)),rewrite(2(6),2(7))].
5652
16 f(f(x,f(f(f(f(x,y),f(f(z,f(x,y)),f(x,y))),x),x)),z) = f(f(x,y),f(f(z,f(x,y)),f(x,y))). [para(8(a,1),2(a,1,2))].
5653
24 f(f(x,f(f(f(y,f(f(y,y),y)),x),x)),y) = f(y,f(f(y,y),y)). [para(2(a,1),10(a,1,2))].
5654
41 f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y. [para(4(a,1),10(a,1,2)),rewrite(10(8),10(15))].
5655
44 f(x,f(f(x,x),x)) = f(x,x). [para(10(a,1),4(a,1,2)),rewrite(10(8)),flip(a)].
5656
46 f(f(x,f(f(y,x),x)),f(y,y)) = y. [back_rewrite(41),rewrite(44(6))].
5657
47 f(f(x,f(f(f(y,y),x),x)),y) = f(y,y). [back_rewrite(24),rewrite(44(3),44(8))].
5658
50 f(f(x,x),f(x,x)) = x. [para(44(a,1),2(a,1,2)),rewrite(44(3))].
5659
52 f(f(f(x,y),f(x,y)),f(f(x,y),x)) = f(x,y). [para(44(a,1),8(a,1,1))].
5660
113 f(x,f(f(f(f(y,x),f(y,x)),x),x)) = f(y,x). [para(47(a,1),6(a,1,1,2,1)),rewrite(44(7),50(7)),flip(a)].
5661
136 f(f(f(f(x,y),x),f(f(x,y),f(f(x,y),x))),f(x,y)) = f(f(x,y),f(x,y)). [para(52(a,1),46(a,1,1,2,1)),rewrite(50(14))].
5662
302 f(f(x,y),f(f(f(x,x),f(x,y)),f(x,y))) = x. [para(47(a,1),16(a,1,1,2,1)),rewrite(44(3),50(3)),flip(a)].
5663
304 f(f(f(x,y),x),f(f(x,y),f(f(x,y),x))) = f(x,y). [para(52(a,1),16(a,1,1,2,1,1,2,1)),rewrite(136(10),44(7),50(7),52(9)),flip(a)].
5664
353 f(f(f(x,y),f(x,y)),x) = f(x,y). [para(302(a,1),2(a,1,2)),rewrite(44(7))].
5665
383 f(f(f(x,y),f(f(f(x,f(f(z,x),x)),f(x,y)),f(x,y))),z) = f(x,f(f(z,x),x)). [para(302(a,1),16(a,1,1,2,1,1,1)),rewrite(302(8),302(9),302(17),302(17),302(18))].
5666
385 f(x,f(f(x,y),x)) = f(x,y). [para(302(a,1),302(a,1,1)),rewrite(302(10),353(4),302(8))].
5667
419 f(f(x,x),f(x,y)) = x. [para(385(a,1),2(a,1,2)),rewrite(385(3))].
5668
420 f(f(x,y),f(x,f(x,y))) = x. [para(385(a,1),8(a,1,1,2,1)),rewrite(304(7))].
5669
428 f(f(x,y),f(x,x)) = x. [para(385(a,1),46(a,1,1,2,1)),rewrite(420(7))].
5670
481 f(x,f(f(y,x),x)) = f(x,y). [para(419(a,1),16(a,1,1,2,1,1,1)),rewrite(419(4),419(5),428(6),419(4),419(4),419(4),419(5)),flip(a)].
5671
537 f(f(f(x,y),f(x,z)),z) = f(x,z). [back_rewrite(383),rewrite(481(4),481(7),481(7))].
5672
566 f(x,y) = f(y,x). [back_rewrite(113),rewrite(537(4),481(3))].
5673
567 $F # answer("Commutativity"). [resolve(566,a,3,a)].
5504
8 f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y. [para(6(a,1),2(a,1,2,2))].
5505
18 f(f(x,f(f(f(y,f(f(y,y),y)),x),x)),y) = f(y,f(f(y,y),y)). [para(2(a,1),8(a,1,2))].
5506
27 f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y. [para(2(a,1),18(a,1,1,2,1,1,2,1)),rewrite([8(8),2(17),8(15)])].
5507
30 f(f(f(x,y),f(f(f(x,y),f(f(f(x,y),f(x,y)),f(x,y))),f(x,y))),f(f(x,y),f(f(f(x,y),f(x,y)),f(x,y)))) = f(y,f(f(f(f(x,y),f(f(f(x,y),f(x,y)),f(x,y))),y),y)). [para(18(a,1),6(a,1,1,2,1))].
5508
34 f(f(x,f(f(x,f(f(x,x),x)),x)),f(x,f(f(x,x),x))) = x. [para(18(a,1),18(a,1,1,2,1)),rewrite([2(19),8(17)])].
5509
35 f(x,f(f(f(f(y,x),f(f(f(y,x),f(y,x)),f(y,x))),x),x)) = f(y,x). [back_rewrite(30),rewrite([34(19)]),flip(a)].
5510
45 f(x,f(f(x,x),x)) = f(x,x). [para(27(a,1),18(a,1,1,2,1)),rewrite([8(8)]),flip(a)].
5511
56 f(x,f(f(f(f(y,x),f(y,x)),x),x)) = f(y,x). [back_rewrite(35),rewrite([45(7)])].
5512
65 f(f(x,f(f(f(y,y),x),x)),y) = f(y,y). [back_rewrite(18),rewrite([45(3),45(8)])].
5513
73 f(f(x,x),f(x,f(y,x))) = x. [para(45(a,1),2(a,1,1))].
5514
104 f(f(x,x),f(y,x)) = x. [para(56(a,1),2(a,1,2)),rewrite([45(3)])].
5515
116 f(f(f(f(f(f(x,f(y,y)),f(x,f(y,y))),f(y,y)),f(y,y)),f(f(x,f(y,y)),f(f(f(f(x,f(y,y)),f(x,f(y,y))),f(y,y)),f(y,y)))),y) = f(y,y). [para(56(a,1),65(a,1,1,2,1))].
5516
120 f(f(f(f(f(x,y),f(x,y)),y),y),f(f(f(f(x,y),f(x,y)),f(f(f(f(x,y),f(x,y)),y),y)),f(f(f(f(x,y),f(x,y)),y),y))) = f(x,y). [para(56(a,1),56(a,1,2,1,1,1)),rewrite([56(12),56(27)])].
5517
130 f(f(f(x,y),f(f(z,f(x,y)),f(x,y))),f(z,y)) = z. [para(104(a,1),2(a,1,2,2))].
5518
135 f(f(x,x),f(y,f(f(x,y),y))) = x. [para(6(a,1),104(a,1,2))].
5519
139 f(f(f(x,y),f(x,y)),f(f(x,y),y)) = f(x,y). [para(104(a,1),73(a,1,2,2))].
5520
143 f(f(f(x,y),f(x,y)),y) = f(x,y). [para(104(a,1),104(a,1,2))].
5521
148 f(f(f(x,y),y),f(f(x,y),f(f(x,y),y))) = f(x,y). [back_rewrite(120),rewrite([143(4),143(9),139(8),143(7)])].
5522
149 f(f(x,f(y,y)),y) = f(y,y). [back_rewrite(116),rewrite([143(7),143(13),148(12)])].
5523
157 f(x,f(f(y,x),x)) = f(y,x). [back_rewrite(56),rewrite([143(4)])].
5524
163 f(f(x,x),f(x,y)) = x. [back_rewrite(135),rewrite([157(4)])].
5525
165 f(f(x,f(y,z)),f(x,z)) = x. [back_rewrite(130),rewrite([157(6)])].
5526
199 f(f(x,y),f(x,f(z,y))) = x. [back_rewrite(2),rewrite([157(3)])].
5527
231 f(f(x,x),f(f(y,f(x,x)),f(z,x))) = f(y,f(x,x)). [para(149(a,1),199(a,1,1))].
5528
278 f(f(x,x),y) = f(y,f(x,x)). [para(165(a,1),231(a,1,2))].
5529
281 f(x,y) = f(y,x). [para(104(a,1),278(a,1,1)),rewrite([163(4)])].
5530
282 $F # answer("Commutativity"). [resolve(281,a,3,a)].
5674
5531
5675
5532
============================== end of proof ==========================
5676
5533
5677
5534
============================== STATISTICS ============================
5678
5535
5679
Given=29. Generated=2931. Kept=565. proofs=1.
5680
Usable=13. Sos=39. Demods=134. Limbo=85, Disabled=429. Hints=1792.
5681
Weight_deleted=197. Literals_deleted=0.
5682
Forward_subsumed=2169. Back_subsumed=75.
5536
Given=32. Generated=1239. Kept=280. proofs=1.
5537
Usable=18. Sos=88. Demods=104. Limbo=0, Disabled=175. Hints=1548.
5538
Kept_by_rule=0, Deleted_by_rule=2.
5539
Forward_subsumed=957. Back_subsumed=33.
5683
5540
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
5684
New_demodulators=558 (0 lex), Back_demodulated=352. Back_unit_deleted=0.
5685
Demod_attempts=127373. Demod_rewrites=9340.
5541
New_demodulators=277 (0 lex), Back_demodulated=140. Back_unit_deleted=0.
5542
Demod_attempts=33835. Demod_rewrites=4003.
5686
5543
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
5687
5544
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
5688
Megabytes=2.18.
5689
User_CPU=0.24, System_CPU=0.00, Wall_clock=13.
5545
Megabytes=1.40.
5546
User_CPU=0.19, System_CPU=0.00, Wall_clock=9.
5690
5547
5691
5548
============================== end of statistics =====================
5692
5549
5694
5551
5695
5552
Exiting with 1 proof.
5696
5553
5697
Process 13980 exit (max_proofs) Sun Aug 13 21:05:10 2006
5554
Process 10952 exit (max_proofs) Mon Sep 28 11:57:18 2009
5698
5555
5699
5556
Proof is complete (no extra assumptions used).
5700
5557
5704
5561
5705
5562
Exiting.
5706
5563
5707
Process 13961 exit (max_proofs) Sun Aug 13 21:05:10 2006
5564
Process 10934 exit (max_proofs) Mon Sep 28 11:57:18 2009
Loggerhead is a web-based interface for Breezy
Version: 2.0.1