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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/temp
1
============================== NewAuto ===============================
2
NewAuto (32) version May-2007, May 2007.
3
Process 21989 was started by mccune on cleo,
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Mon Jun 4 22:53:25 2007
5
The command was "../newauto -f BA.in".
6
============================== end of head ===========================
7
8
============================== INPUT =================================
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10
% Reading from file BA.in
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assign(order,kbo).
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set(lex_order_vars).
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assign(lex_dep_demod_lim,0).
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assign(max_megs,750).
16
assign(sos_limit,20000).
17
assign(demod_step_limit,200).
18
assign(demod_size_limit,100000).
19
clear(print_kept).
20
clear(print_given).
21
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formulas(sos).
23
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
24
end_of_list.
25
26
formulas(goals).
27
f(x,y) = f(y,x) # label("Commutativity").
28
end_of_list.
29
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formulas(extra_assumptions).
31
f(f(f(y,f(x,y)),y),f(x,f(z,y))) = x # label("Sh2/A14_m").
32
f(f(y,f(y,f(y,x))),f(x,f(y,z))) = x # label("C1/A1").
33
f(f(y,f(y,f(x,y))),f(x,f(z,y))) = x # label("C2/A4 (Wald 1)").
34
f(f(y,f(y,f(x,x))),f(x,f(z,y))) = x # label("C3/A6").
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f(f(y,f(y,f(x,z))),f(x,f(z,y))) = x # label("C4/A7").
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f(f(y,f(y,f(z,x))),f(x,f(y,z))) = x # label("C5/A8 (Wald 2))").
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f(f(y,f(f(x,y),y)),f(x,f(y,z))) = x # label("C6/A9").
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f(f(y,f(y,f(y,x))),f(x,f(z,y))) = x # label("C7/A2").
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f(f(f(f(y,x),y),y),f(x,f(z,y))) = x # label("C8/A12_m").
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f(f(f(f(y,x),z),z),f(x,f(y,z))) = x # label("C9/A13_m").
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f(f(f(y,f(y,x)),y),f(x,f(z,y))) = x # label("C10/A16_m").
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f(f(f(y,f(x,x)),y),f(x,f(z,y))) = x # label("C11/A18_m").
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f(f(f(f(y,x),z),z),f(x,f(z,y))) = x # label("C12/A19_m").
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f(f(f(f(y,x),y),y),f(x,f(y,z))) = x # label("C13/A20_m").
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f(f(f(y,f(x,z)),y),f(x,f(y,z))) = x # label("C14/A21_m").
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f(f(f(y,f(z,x)),y),f(x,f(y,z))) = x # label("C15/A23_m").
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f(f(f(y,f(y,x)),y),f(x,f(y,z))) = x # label("C16/A24_m").
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end_of_list.
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============================== end of input ==========================
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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") # label(non_clause) # label(goal). [goal].
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============================== end of process non-clausal formulas ===
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***********************************************
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AUTO SKETCHES ITERATION 1
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***********************************************
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Starting a search with 17 assumptions:
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formulas(assumptions_active).
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f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m"). [assumption].
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f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y # label("C1/A1"). [assumption].
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f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
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f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
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f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
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f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
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f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
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f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
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f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
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f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
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f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
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f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
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f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
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f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
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f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
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f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
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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 21990 started.
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============================== PROCESS INITIAL CLAUSES ===============
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% Clauses before input processing:
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formulas(usable).
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end_of_list.
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formulas(sos).
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f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
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f(c2,c1) != f(c1,c2) # label("Commutativity"). [deny(1)].
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f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m"). [assumption].
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f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y # label("C1/A1"). [assumption].
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f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
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f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
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f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
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f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
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f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
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f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
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f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
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f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
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f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
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f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
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f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
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f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
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f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
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f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
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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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formulas(demodulators).
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end_of_list.
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============================== PREDICATE ELIMINATION =================
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No predicates eliminated.
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============================== end predicate elimination =============
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Auto_denials:
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% copying label "Commutativity" to answer in negative clause
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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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Predicate symbol precedence: predicate_order([ = ]).
131
Function symbol precedence: function_order([ c1, c2, f ]).
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Skipping inverse_order, because term ordering is KBO.
133
Unfolding symbols: (none).
134
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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).
138
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Auto_process settings: (no changes).
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============================== end of process initial clauses ========
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============================== CLAUSES FOR SEARCH ====================
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% Clauses after input processing:
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formulas(usable).
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end_of_list.
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formulas(sos).
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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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4 f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m"). [assumption].
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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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6 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
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7 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
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8 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
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9 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
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10 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
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11 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
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12 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
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13 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
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14 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
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15 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
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16 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
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17 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
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18 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
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19 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
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20 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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formulas(demodulators).
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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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4 f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m"). [assumption].
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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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6 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
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7 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
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8 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
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9 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
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10 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
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11 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
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12 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
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13 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
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14 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
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15 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
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16 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
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17 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
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18 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
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19 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
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20 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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============================== end of clauses for search =============
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============================== SEARCH ================================
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% Starting search at 0.00 seconds.
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% Operation f is commutative; C redundancy checks enabled.
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============================== PROOF =================================
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% Proof 1 at 1.52 (+ 0.01) seconds: "Commutativity".
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% Length of proof is 36.
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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.
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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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6 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
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9 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
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10 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
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12 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
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13 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
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14 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
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15 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
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18 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [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))].
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1022 f(f(f(x,f(x,x)),x),f(x,x)) = x. [para(2(a,1),33(a,1,2,2))].
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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))].
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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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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)].
234
1206 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [para(1121(a,1),15(a,1,1,1))].
235
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)])].
237
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)])].
238
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)])].
239
2567 f(f(x,f(y,y)),f(x,y)) = x. [para(1531(a,1),10(a,1,2,2)),rewrite([1866(6)])].
240
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)])].
241
2657 f(f(x,x),y) = f(y,f(x,x)). [para(2567(a,1),14(a,1,2)),rewrite([1875(5)])].
242
2909 f(x,f(f(x,x),x)) = f(x,x). [para(1121(a,1),2613(a,1,1)),rewrite([1121(4)])].
243
2954 f(x,y) = f(y,x). [para(2(a,1),2657(a,1,1)),rewrite([2909(4),2909(5),1121(4)])].
244
2955 $F # answer("Commutativity"). [resolve(2954,a,3,a)].
245
246
============================== end of proof ==========================
247
248
============================== STATISTICS ============================
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250
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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Weight_deleted=129. Literals_deleted=0.
253
Forward_subsumed=3954. Back_subsumed=152.
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Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
255
New_demodulators=2947 (0 lex), Back_demodulated=205. Back_unit_deleted=0.
256
Demod_attempts=180094. Demod_rewrites=11763.
257
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
258
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
259
Megabytes=4.84.
260
User_CPU=1.52, System_CPU=0.01, Wall_clock=1.
261
262
============================== end of statistics =====================
263
264
============================== end of search =========================
265
266
Exiting with 1 proof.
267
268
Process 21990 exit (max_proofs) Mon Jun 4 22:53:26 2007
269
270
Successful proof using the following 9 assumptions:
271
272
formulas(assumptions_in_proof).
273
f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y # label("C1/A1"). [assumption].
274
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
275
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
276
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
277
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
278
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
279
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
280
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
281
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
282
end_of_list.
283
284
Including 60 new hint clauses:
285
286
formulas(new_hints).
287
$F # label("Commutativity") # label(non_clause) # label(goal).
288
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
289
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
290
f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y # label("C1/A1").
291
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").
292
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").
293
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
294
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").
295
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
296
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
297
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").
298
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
299
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)).
300
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)).
301
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
302
f(f(f(x,f(x,x)),x),f(x,x)) = x.
303
f(f(f(f(x,f(x,f(x,x))),x),f(x,f(x,f(x,x)))),f(f(x,f(x,f(x,x))),f(x,f(x,f(x,x))))) = f(x,f(x,f(x,x))).
304
f(x,f(f(x,f(x,f(x,x))),f(x,f(x,f(x,x))))) = f(x,f(x,f(x,x))).
305
f(x,x) = f(x,f(x,f(x,x))).
306
f(x,f(x,f(x,x))) = f(x,x).
307
f(f(x,x),f(x,f(x,y))) = x.
308
f(f(x,f(x,f(x,x))),f(x,x)) = x.
309
f(f(x,x),f(x,x)) = x.
310
f(f(x,x),f(x,f(y,x))) = x.
311
f(f(f(x,x),f(f(x,x),f(x,f(x,x)))),f(x,x)) = x.
312
f(f(f(x,x),x),f(x,x)) = x.
313
f(f(x,f(x,x)),f(f(x,x),f(x,x))) = f(x,x).
314
f(f(x,f(x,x)),x) = f(x,x).
315
f(f(x,f(f(x,x),x)),f(x,x)) = x.
316
f(f(f(f(f(x,x),x),f(x,x)),f(x,x)),f(x,x)) = x.
317
f(f(x,f(x,x)),f(x,x)) = x.
318
f(f(f(x,f(x,x)),f(x,f(x,f(x,x)))),f(f(x,f(f(x,x),x)),f(x,x))) = f(x,f(f(x,x),x)).
319
f(f(f(x,f(x,x)),f(x,x)),f(f(x,f(f(x,x),x)),f(x,x))) = f(x,f(f(x,x),x)).
320
f(x,f(f(x,f(f(x,x),x)),f(x,x))) = f(x,f(f(x,x),x)).
321
f(x,x) = f(x,f(f(x,x),x)).
322
f(x,f(f(x,x),x)) = f(x,x).
323
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
324
f(f(x,x),f(x,f(y,f(f(x,y),y)))) = x.
325
f(f(x,f(f(f(x,f(x,x)),x),x)),x) = f(x,f(x,x)).
326
f(f(x,f(f(x,x),x)),x) = f(x,f(x,x)).
327
f(f(x,x),x) = f(x,f(x,x)).
328
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)).
329
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
330
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
331
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).
332
f(f(x,f(x,f(y,f(x,x)))),x) = f(x,x).
333
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
334
f(f(x,f(y,y)),f(x,y)) = x.
335
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
336
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
337
f(f(x,y),f(x,f(y,y))) = x.
338
f(f(f(x,f(x,f(y,f(x,x)))),x),y) = f(y,f(x,x)).
339
f(f(x,x),y) = f(y,f(x,x)).
340
f(x,f(f(x,x),f(f(x,x),f(x,x)))) = f(x,x).
341
f(x,f(f(x,x),x)) = f(x,x).
342
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
343
f(x,y) = f(y,f(f(x,x),f(x,f(f(x,x),x)))).
344
f(x,y) = f(y,f(f(x,x),f(x,x))).
345
f(x,y) = f(y,x).
346
$F # answer("Commutativity").
347
end_of_list.
348
349
Attempt to eliminate one assumption:
350
351
f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y # label("C1/A1").
352
353
***********************************************
354
AUTO SKETCHES ITERATION 2
355
***********************************************
356
357
Starting a search with 16 assumptions:
358
359
formulas(assumptions_active).
360
f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m"). [assumption].
361
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
362
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
363
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
364
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
365
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
366
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
367
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
368
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
369
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
370
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
371
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
372
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
373
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
374
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
375
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
376
end_of_list.
377
378
Child search process 21991 started.
379
380
============================== PROCESS INITIAL CLAUSES ===============
381
382
% Clauses before input processing:
383
384
formulas(usable).
385
end_of_list.
386
387
formulas(sos).
388
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
389
f(c2,c1) != f(c1,c2) # label("Commutativity"). [deny(1)].
390
f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m"). [assumption].
391
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
392
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
393
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
394
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
395
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
396
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
397
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
398
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
399
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
400
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
401
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
402
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
403
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
404
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
405
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
406
end_of_list.
407
408
formulas(demodulators).
409
end_of_list.
410
411
% 60 hints input.
412
413
============================== PREDICATE ELIMINATION =================
414
415
No predicates eliminated.
416
417
============================== end predicate elimination =============
418
419
Auto_denials:
420
% copying label "Commutativity" to answer in negative clause
421
422
Term ordering decisions:
423
Function symbol KB weights: c1=1. c2=1. f=1.
424
Predicate symbol precedence: predicate_order([ = ]).
425
Function symbol precedence: function_order([ c1, c2, f ]).
426
Skipping inverse_order, because term ordering is KBO.
427
Unfolding symbols: (none).
428
429
Auto_inference settings:
430
% set(paramodulation). % (positive equality literals)
431
% set(paramodulation) -> set(back_demod).
432
433
Auto_process settings: (no changes).
434
435
============================== end of process initial clauses ========
436
437
============================== CLAUSES FOR SEARCH ====================
438
439
% Clauses after input processing:
440
441
formulas(usable).
442
end_of_list.
443
444
formulas(sos).
445
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
446
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
447
4 f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m"). [assumption].
448
5 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
449
6 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
450
7 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
451
8 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
452
9 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
453
10 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
454
11 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
455
12 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
456
13 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
457
14 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
458
15 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
459
16 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
460
17 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
461
18 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
462
19 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
463
end_of_list.
464
465
formulas(demodulators).
466
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
467
4 f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m"). [assumption].
468
5 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
469
6 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
470
7 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
471
8 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
472
9 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
473
10 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
474
11 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
475
12 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
476
13 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
477
14 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
478
15 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
479
16 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
480
17 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
481
18 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
482
19 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
483
end_of_list.
484
485
% 60 hints processed (13 are redundant).
486
487
============================== end of clauses for search =============
488
489
============================== SEARCH ================================
490
491
% Starting search at 0.00 seconds.
492
493
% Operation f is commutative; C redundancy checks enabled.
494
495
============================== PROOF =================================
496
497
% Proof 1 at 0.68 (+ 0.00) seconds: "Commutativity".
498
% Length of proof is 28.
499
% Level of proof is 11.
500
% Maximum clause weight is 37.
501
% Given clauses 35.
502
503
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
504
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
505
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
506
4 f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m"). [assumption].
507
5 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
508
8 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
509
9 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
510
13 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
511
14 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
512
19 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
513
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))].
514
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))].
515
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))].
516
837 f(f(f(x,f(x,x)),x),f(x,x)) = x. [para(2(a,1),32(a,1,2,2))].
517
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))].
518
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))].
519
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))].
520
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)].
521
1350 f(f(x,f(f(x,x),x)),f(x,x)) = x. [back_rewrite(837),rewrite([1267(3)])].
522
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)].
523
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)])].
524
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)])].
525
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)])].
526
1451 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(1267),rewrite([1358(6)])].
527
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)].
528
1787 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)])].
529
1824 f(x,y) = f(y,x). [para(2(a,1),1787(a,1,1)),rewrite([2(8)])].
530
1825 $F # answer("Commutativity"). [resolve(1824,a,3,a)].
531
532
============================== end of proof ==========================
533
534
============================== STATISTICS ============================
535
536
Given=35. Generated=3495. Kept=1823. proofs=1.
537
Usable=29. Sos=1392. Demods=1426. Limbo=7, Disabled=412. Hints=60.
538
Weight_deleted=53. Literals_deleted=0.
539
Forward_subsumed=1619. Back_subsumed=93.
540
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
541
New_demodulators=1818 (0 lex), Back_demodulated=301. Back_unit_deleted=0.
542
Demod_attempts=80052. Demod_rewrites=5188.
543
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
544
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
545
Megabytes=2.60.
546
User_CPU=0.68, System_CPU=0.00, Wall_clock=2.
547
548
============================== end of statistics =====================
549
550
============================== end of search =========================
551
552
Exiting with 1 proof.
553
554
Process 21991 exit (max_proofs) Mon Jun 4 22:53:27 2007
555
556
Successful proof using the following 7 assumptions:
557
558
formulas(assumptions_in_proof).
559
f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m"). [assumption].
560
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
561
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
562
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
563
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
564
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
565
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
566
end_of_list.
567
568
Including 48 new hint clauses:
569
570
formulas(new_hints).
571
$F # label("Commutativity") # label(non_clause) # label(goal).
572
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
573
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
574
f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m").
575
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").
576
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").
577
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
578
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
579
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").
580
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
581
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)).
582
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).
583
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
584
f(f(f(x,f(x,x)),x),f(x,x)) = x.
585
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
586
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))).
587
f(f(f(x,f(y,x)),f(x,f(x,f(y,x)))),x) = f(x,f(f(x,x),x)).
588
f(f(f(x,f(y,x)),f(x,f(x,f(y,x)))),x) = f(f(x,f(x,x)),x).
589
f(x,f(f(x,x),x)) = f(f(x,f(x,x)),x).
590
f(f(x,f(x,x)),x) = f(x,f(f(x,x),x)).
591
f(f(x,f(f(x,x),x)),f(x,x)) = x.
592
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)).
593
f(x,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
594
f(x,x) = f(x,f(f(x,x),x)).
595
f(x,f(f(x,x),x)) = f(x,x).
596
f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,f(f(x,x),x)),x))) = x.
597
f(f(f(x,x),x),f(x,f(f(x,f(f(x,x),x)),x))) = x.
598
f(f(f(x,x),x),f(x,f(f(x,x),x))) = x.
599
f(f(f(x,x),x),f(x,x)) = x.
600
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).
601
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).
602
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).
603
f(x,f(f(x,x),f(y,f(x,f(f(x,x),x))))) = f(x,x).
604
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x).
605
f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x.
606
f(f(x,f(x,x)),f(x,f(y,f(x,f(f(x,x),x))))) = x.
607
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
608
f(f(x,f(x,x)),x) = f(x,x).
609
f(f(f(x,f(x,x)),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
610
f(f(x,x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
611
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y).
612
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)).
613
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
614
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
615
f(f(x,x),y) = f(y,f(x,x)).
616
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
617
f(x,y) = f(y,x).
618
$F # answer("Commutativity").
619
end_of_list.
620
621
Attempt to eliminate one assumption:
622
623
f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m").
624
625
***********************************************
626
AUTO SKETCHES ITERATION 3
627
***********************************************
628
629
Starting a search with 15 assumptions:
630
631
formulas(assumptions_active).
632
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
633
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
634
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
635
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
636
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
637
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
638
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
639
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
640
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
641
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
642
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
643
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
644
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
645
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
646
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
647
end_of_list.
648
649
Child search process 21994 started.
650
651
============================== PROCESS INITIAL CLAUSES ===============
652
653
% Clauses before input processing:
654
655
formulas(usable).
656
end_of_list.
657
658
formulas(sos).
659
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
660
f(c2,c1) != f(c1,c2) # label("Commutativity"). [deny(1)].
661
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
662
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
663
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
664
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
665
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
666
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
667
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
668
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
669
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
670
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
671
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
672
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
673
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
674
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
675
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
676
end_of_list.
677
678
formulas(demodulators).
679
end_of_list.
680
681
% 108 hints input.
682
683
============================== PREDICATE ELIMINATION =================
684
685
No predicates eliminated.
686
687
============================== end predicate elimination =============
688
689
Auto_denials:
690
% copying label "Commutativity" to answer in negative clause
691
692
Term ordering decisions:
693
Function symbol KB weights: c1=1. c2=1. f=1.
694
Predicate symbol precedence: predicate_order([ = ]).
695
Function symbol precedence: function_order([ c1, c2, f ]).
696
Skipping inverse_order, because term ordering is KBO.
697
Unfolding symbols: (none).
698
699
Auto_inference settings:
700
% set(paramodulation). % (positive equality literals)
701
% set(paramodulation) -> set(back_demod).
702
703
Auto_process settings: (no changes).
704
705
============================== end of process initial clauses ========
706
707
============================== CLAUSES FOR SEARCH ====================
708
709
% Clauses after input processing:
710
711
formulas(usable).
712
end_of_list.
713
714
formulas(sos).
715
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
716
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
717
4 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
718
5 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
719
6 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
720
7 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
721
8 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
722
9 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
723
10 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
724
11 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
725
12 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
726
13 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
727
14 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
728
15 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
729
16 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
730
17 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
731
18 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
732
end_of_list.
733
734
formulas(demodulators).
735
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
736
4 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
737
5 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
738
6 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
739
7 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
740
8 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
741
9 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
742
10 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
743
11 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
744
12 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
745
13 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
746
14 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
747
15 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
748
16 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
749
17 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
750
18 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
751
end_of_list.
752
753
% 108 hints processed (41 are redundant).
754
755
============================== end of clauses for search =============
756
757
============================== SEARCH ================================
758
759
% Starting search at 0.00 seconds.
760
761
% Operation f is commutative; C redundancy checks enabled.
762
763
============================== PROOF =================================
764
765
% Proof 1 at 0.76 (+ 0.00) seconds: "Commutativity".
766
% Length of proof is 44.
767
% Level of proof is 14.
768
% Maximum clause weight is 43.
769
% Given clauses 25.
770
771
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
772
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
773
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
774
4 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
775
8 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
776
9 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
777
12 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
778
14 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
779
16 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
780
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))].
781
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))].
782
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))].
783
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))].
784
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))].
785
758 f(f(f(x,f(x,x)),x),f(x,x)) = x. [para(2(a,1),23(a,1,2,2))].
786
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))].
787
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))].
788
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))].
789
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))].
790
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))].
791
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))].
792
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)])].
793
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)])].
794
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))].
795
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))].
796
1181 f(x,f(f(x,x),x)) = f(x,x). [para(8(a,1),1096(a,1,1)),flip(a)].
797
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)])].
798
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)])].
799
1319 f(f(x,f(x,x)),f(x,x)) = x. [back_rewrite(821),rewrite([1238(7)])].
800
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)])].
801
1381 f(f(x,x),f(x,x)) = x. [para(1181(a,1),2(a,1,2)),rewrite([1181(3)])].
802
1390 f(f(x,x),f(x,f(x,y))) = x. [para(1181(a,1),8(a,1,1))].
803
1404 f(f(f(x,x),x),x) = f(x,x). [para(1181(a,1),16(a,1,1,1)),rewrite([1381(5)])].
804
1417 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(1323),rewrite([1390(4),1404(3)]),flip(a)].
805
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)])].
806
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)])].
807
1552 f(x,f(f(y,x),x)) = f(x,y). [back_rewrite(1240),rewrite([1417(2),1319(4)]),flip(a)].
808
1579 f(f(x,y),x) = f(x,f(x,y)). [back_rewrite(1449),rewrite([1552(5),1552(6),1552(6)])].
809
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)])].
810
1729 f(f(f(x,y),y),y) = f(x,y). [back_rewrite(349),rewrite([1552(4),1642(4),1552(6)])].
811
1762 f(x,f(x,f(y,x))) = f(y,x). [back_rewrite(98),rewrite([1552(3),1729(3),1579(3),1552(6)])].
812
1932 f(f(x,y),y) = f(y,f(x,y)). [back_rewrite(1448),rewrite([1762(4),1552(6),1762(5)])].
813
2067 f(x,y) = f(y,x). [back_rewrite(1552),rewrite([1932(2),1762(3)])].
814
2068 $F # answer("Commutativity"). [resolve(2067,a,3,a)].
815
816
============================== end of proof ==========================
817
818
============================== STATISTICS ============================
819
820
Given=25. Generated=3521. Kept=2066. proofs=1.
821
Usable=6. Sos=368. Demods=505. Limbo=135, Disabled=1573. Hints=108.
822
Weight_deleted=33. Literals_deleted=0.
823
Forward_subsumed=1422. Back_subsumed=83.
824
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
825
New_demodulators=2055 (0 lex), Back_demodulated=1473. Back_unit_deleted=0.
826
Demod_attempts=80861. Demod_rewrites=6184.
827
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
828
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
829
Megabytes=2.55.
830
User_CPU=0.76, System_CPU=0.00, Wall_clock=3.
831
832
============================== end of statistics =====================
833
834
============================== end of search =========================
835
836
Exiting with 1 proof.
837
838
Process 21994 exit (max_proofs) Mon Jun 4 22:53:28 2007
839
840
Successful proof using the following 6 assumptions:
841
842
formulas(assumptions_in_proof).
843
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)"). [assumption].
844
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
845
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
846
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
847
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
848
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
849
end_of_list.
850
851
Including 78 new hint clauses:
852
853
formulas(new_hints).
854
$F # label("Commutativity") # label(non_clause) # label(goal).
855
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
856
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
857
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").
858
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
859
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").
860
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
861
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
862
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
863
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)).
864
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)).
865
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
866
f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)).
867
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
868
f(f(f(x,f(x,x)),x),f(x,x)) = x.
869
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
870
f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x.
871
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)).
872
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)))).
873
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))).
874
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))).
875
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)).
876
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)).
877
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)).
878
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))))).
879
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))))).
880
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))))).
881
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))))).
882
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))))).
883
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))))).
884
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)).
885
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)).
886
f(f(f(x,f(y,x)),f(x,f(x,f(y,x)))),x) = f(x,f(f(x,x),x)).
887
f(x,x) = f(x,f(f(x,x),x)).
888
f(x,f(f(x,x),x)) = f(x,x).
889
f(x,f(f(x,x),f(y,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
890
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,f(f(x,x),x)).
891
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x).
892
f(f(f(f(x,x),x),f(x,f(f(x,x),x))),y) = f(x,f(f(y,x),x)).
893
f(f(f(f(x,x),x),f(x,x)),y) = f(x,f(f(y,x),x)).
894
f(f(x,f(x,x)),f(x,x)) = x.
895
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))).
896
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),x),x)) = f(f(x,x),x).
897
f(f(x,f(f(x,x),x)),f(x,x)) = x.
898
f(f(x,x),f(x,x)) = x.
899
f(f(x,x),f(x,f(x,y))) = x.
900
f(f(f(x,x),x),f(f(x,x),f(x,x))) = f(x,x).
901
f(f(f(x,x),x),x) = f(x,x).
902
f(x,f(f(f(x,x),x),x)) = f(f(x,x),x).
903
f(x,f(x,x)) = f(f(x,x),x).
904
f(f(x,x),x) = f(x,f(x,x)).
905
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)))).
906
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)).
907
f(f(f(x,f(x,x)),f(x,x)),y) = f(x,f(f(y,x),x)).
908
f(x,y) = f(x,f(f(y,x),x)).
909
f(x,f(f(y,x),x)) = f(x,y).
910
f(f(x,y),f(f(x,f(x,y)),f(x,y))) = f(x,f(f(f(x,y),x),x)).
911
f(f(x,y),x) = f(x,f(f(f(x,y),x),x)).
912
f(f(x,y),x) = f(x,f(x,y)).
913
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)).
914
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)).
915
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)).
916
f(f(f(x,y),f(f(f(y,z),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)).
917
f(f(f(x,y),f(y,z)),z) = f(y,f(f(z,y),y)).
918
f(f(f(x,y),f(y,z)),z) = f(y,z).
919
f(f(f(f(f(x,y),f(y,z)),z),z),z) = f(y,f(f(z,y),y)).
920
f(f(f(y,z),z),z) = f(y,f(f(z,y),y)).
921
f(f(f(x,y),y),y) = f(x,y).
922
f(f(x,f(f(f(y,x),x),x)),x) = f(y,f(f(x,y),y)).
923
f(f(x,f(y,x)),x) = f(y,f(f(x,y),y)).
924
f(x,f(x,f(y,x))) = f(y,f(f(x,y),y)).
925
f(x,f(x,f(y,x))) = f(y,x).
926
f(f(x,y),f(f(y,f(x,y)),f(x,y))) = f(y,f(y,f(y,f(x,y)))).
927
f(f(x,y),y) = f(y,f(y,f(y,f(x,y)))).
928
f(f(x,y),y) = f(y,f(x,y)).
929
f(x,f(x,f(y,x))) = f(x,y).
930
f(x,y) = f(y,x).
931
$F # answer("Commutativity").
932
end_of_list.
933
934
Attempt to eliminate one assumption:
935
936
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").
937
938
***********************************************
939
AUTO SKETCHES ITERATION 4
940
***********************************************
941
942
Starting a search with 14 assumptions:
943
944
formulas(assumptions_active).
945
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
946
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
947
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
948
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
949
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
950
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
951
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
952
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
953
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
954
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
955
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
956
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
957
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
958
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
959
end_of_list.
960
961
Child search process 21995 started.
962
963
============================== PROCESS INITIAL CLAUSES ===============
964
965
% Clauses before input processing:
966
967
formulas(usable).
968
end_of_list.
969
970
formulas(sos).
971
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
972
f(c2,c1) != f(c1,c2) # label("Commutativity"). [deny(1)].
973
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
974
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
975
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
976
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
977
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
978
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
979
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
980
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
981
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
982
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
983
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
984
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
985
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
986
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
987
end_of_list.
988
989
formulas(demodulators).
990
end_of_list.
991
992
% 186 hints input.
993
994
============================== PREDICATE ELIMINATION =================
995
996
No predicates eliminated.
997
998
============================== end predicate elimination =============
999
1000
Auto_denials:
1001
% copying label "Commutativity" to answer in negative clause
1002
1003
Term ordering decisions:
1004
Function symbol KB weights: c1=1. c2=1. f=1.
1005
Predicate symbol precedence: predicate_order([ = ]).
1006
Function symbol precedence: function_order([ c1, c2, f ]).
1007
Skipping inverse_order, because term ordering is KBO.
1008
Unfolding symbols: (none).
1009
1010
Auto_inference settings:
1011
% set(paramodulation). % (positive equality literals)
1012
% set(paramodulation) -> set(back_demod).
1013
1014
Auto_process settings: (no changes).
1015
1016
============================== end of process initial clauses ========
1017
1018
============================== CLAUSES FOR SEARCH ====================
1019
1020
% Clauses after input processing:
1021
1022
formulas(usable).
1023
end_of_list.
1024
1025
formulas(sos).
1026
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
1027
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
1028
4 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
1029
5 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
1030
6 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
1031
7 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1032
8 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1033
9 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1034
10 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1035
11 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1036
12 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
1037
13 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1038
14 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1039
15 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
1040
16 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
1041
17 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1042
end_of_list.
1043
1044
formulas(demodulators).
1045
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
1046
4 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
1047
5 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
1048
6 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
1049
7 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1050
8 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1051
9 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1052
10 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1053
11 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1054
12 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
1055
13 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1056
14 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1057
15 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
1058
16 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
1059
17 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1060
end_of_list.
1061
1062
% 186 hints processed (71 are redundant).
1063
1064
============================== end of clauses for search =============
1065
1066
============================== SEARCH ================================
1067
1068
% Starting search at 0.01 seconds.
1069
1070
% Operation f is commutative; C redundancy checks enabled.
1071
1072
============================== PROOF =================================
1073
1074
% Proof 1 at 0.87 (+ 0.00) seconds: "Commutativity".
1075
% Length of proof is 52.
1076
% Level of proof is 16.
1077
% Maximum clause weight is 51.
1078
% Given clauses 28.
1079
1080
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
1081
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
1082
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
1083
4 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
1084
5 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
1085
7 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1086
8 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1087
9 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1088
10 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1089
11 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1090
13 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1091
14 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1092
17 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1093
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))].
1094
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))].
1095
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))].
1096
687 f(f(f(x,f(x,x)),x),f(x,x)) = x. [para(2(a,1),22(a,1,2,2))].
1097
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))].
1098
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))].
1099
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))].
1100
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))].
1101
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))].
1102
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))].
1103
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))].
1104
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))].
1105
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)])].
1106
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))].
1107
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))].
1108
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))].
1109
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))].
1110
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)])].
1111
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)].
1112
1273 f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)). [back_rewrite(870),rewrite([1203(4)])].
1113
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)])].
1114
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)])].
1115
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)])].
1116
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)])].
1117
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))].
1118
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)].
1119
1688 f(f(f(f(x,x),x),x),x) = f(f(x,x),x). [para(1612(a,1),1025(a,2,2))].
1120
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)])].
1121
1700 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(1670),rewrite([1687(6)])].
1122
1722 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(1307),rewrite([1700(3),1687(5),1687(6),1688(4)])].
1123
1723 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(1294),rewrite([1687(4),1700(6)])].
1124
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)])].
1125
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)].
1126
1889 f(f(x,y),x) = f(x,f(x,y)). [back_rewrite(1736),rewrite([1750(5),1750(6),1750(6)])].
1127
1893 f(f(x,y),y) = f(y,f(x,y)). [back_rewrite(1694),rewrite([1750(5),1750(6),1750(6)])].
1128
2496 f(x,f(x,f(y,x))) = f(x,y). [back_rewrite(1750),rewrite([1893(2)])].
1129
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)])].
1130
2530 f(x,y) = f(y,x). [back_rewrite(1092),rewrite([1893(2),1889(3),2496(3),2513(5),1889(3),2496(3)])].
1131
2531 $F # answer("Commutativity"). [resolve(2530,a,3,a)].
1132
1133
============================== end of proof ==========================
1134
1135
============================== STATISTICS ============================
1136
1137
Given=28. Generated=4010. Kept=2529. proofs=1.
1138
Usable=7. Sos=265. Demods=891. Limbo=637, Disabled=1635. Hints=186.
1139
Weight_deleted=37. Literals_deleted=0.
1140
Forward_subsumed=1444. Back_subsumed=59.
1141
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
1142
New_demodulators=2478 (0 lex), Back_demodulated=1560. Back_unit_deleted=0.
1143
Demod_attempts=100609. Demod_rewrites=7865.
1144
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
1145
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
1146
Megabytes=3.50.
1147
User_CPU=0.87, System_CPU=0.00, Wall_clock=4.
1148
1149
============================== end of statistics =====================
1150
1151
============================== end of search =========================
1152
1153
Exiting with 1 proof.
1154
1155
Process 21995 exit (max_proofs) Mon Jun 4 22:53:29 2007
1156
1157
Successful proof using the following 10 assumptions:
1158
1159
formulas(assumptions_in_proof).
1160
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6"). [assumption].
1161
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
1162
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1163
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1164
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1165
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1166
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1167
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1168
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1169
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1170
end_of_list.
1171
1172
Including 99 new hint clauses:
1173
1174
formulas(new_hints).
1175
$F # label("Commutativity") # label(non_clause) # label(goal).
1176
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
1177
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
1178
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").
1179
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").
1180
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
1181
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").
1182
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").
1183
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
1184
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
1185
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
1186
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
1187
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
1188
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)).
1189
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
1190
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
1191
f(f(f(x,f(x,x)),x),f(x,x)) = x.
1192
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
1193
f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x.
1194
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)).
1195
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))).
1196
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).
1197
f(f(x,f(f(f(x,f(x,x)),x),x)),x) = f(x,f(x,x)).
1198
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)).
1199
f(f(f(f(x,f(x,f(x,x))),x),x),x) = f(x,f(x,x)).
1200
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)))).
1201
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).
1202
f(f(f(x,y),y),y) = f(x,f(f(y,x),x)).
1203
f(f(x,f(f(f(x,y),y),y)),f(f(y,x),f(x,x))) = f(y,x).
1204
f(f(f(x,f(f(y,x),x)),y),f(y,f(f(x,y),y))) = y.
1205
f(f(f(f(x,x),f(y,x)),f(y,x)),f(f(f(x,y),y),y)) = x.
1206
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)).
1207
f(f(x,f(x,f(x,x))),x) = f(x,f(x,x)).
1208
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).
1209
f(x,f(x,f(f(x,x),x))) = f(f(x,f(f(x,x),x)),x).
1210
f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))).
1211
f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)).
1212
f(f(f(x,x),f(x,f(x,x))),f(x,f(x,x))) = f(f(x,f(x,x)),x).
1213
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)).
1214
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))))).
1215
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))))).
1216
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))))).
1217
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))))).
1218
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))))).
1219
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))))).
1220
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))))).
1221
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)).
1222
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)))).
1223
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)))).
1224
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)))).
1225
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)))).
1226
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)))).
1227
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)))).
1228
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)))).
1229
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)))).
1230
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)))).
1231
f(f(x,f(x,x)),f(x,x)) = x.
1232
f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))) = f(x,x).
1233
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))).
1234
f(f(f(x,f(x,x)),x),f(x,f(x,x))) = f(f(x,x),f(x,f(x,x))).
1235
x = f(f(x,x),f(x,f(x,x))).
1236
f(f(x,x),f(x,f(x,x))) = x.
1237
f(f(f(f(x,x),x),x),x) = f(f(x,x),x).
1238
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)).
1239
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)).
1240
f(f(x,f(x,x)),x) = f(x,x).
1241
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)).
1242
f(f(f(f(x,x),x),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)).
1243
f(f(f(f(x,x),x),x),x) = f(x,f(x,x)).
1244
f(f(x,x),x) = f(x,f(x,x)).
1245
f(x,f(x,f(x,x))) = f(f(x,f(x,x)),x).
1246
f(x,f(x,f(x,x))) = f(x,x).
1247
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)).
1248
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)).
1249
f(f(f(x,f(x,x)),f(x,f(f(x,x),x))),y) = f(x,f(f(y,x),x)).
1250
f(f(f(x,f(x,x)),f(x,f(x,f(x,x)))),y) = f(x,f(f(y,x),x)).
1251
f(f(f(x,f(x,x)),f(x,x)),y) = f(x,f(f(y,x),x)).
1252
f(x,y) = f(x,f(f(y,x),x)).
1253
f(x,f(f(y,x),x)) = f(x,y).
1254
f(f(x,y),f(f(x,f(x,y)),f(x,y))) = f(x,f(f(f(x,y),x),x)).
1255
f(f(x,y),x) = f(x,f(f(f(x,y),x),x)).
1256
f(f(x,y),x) = f(x,f(x,y)).
1257
f(f(x,y),f(f(y,f(x,y)),f(x,y))) = f(y,f(f(f(x,y),y),y)).
1258
f(f(x,y),y) = f(y,f(f(f(x,y),y),y)).
1259
f(f(x,y),y) = f(y,f(x,y)).
1260
f(x,f(x,f(y,x))) = f(x,y).
1261
f(f(f(y,x),f(f(x,x),f(y,x))),f(f(f(x,y),y),y)) = x.
1262
f(f(f(y,x),f(f(x,x),f(y,x))),f(f(y,f(x,y)),y)) = x.
1263
f(f(f(y,x),f(f(x,x),f(y,x))),f(y,f(y,f(x,y)))) = x.
1264
f(f(f(y,x),f(f(x,x),f(y,x))),f(y,x)) = x.
1265
f(f(y,x),f(f(y,x),f(f(x,x),f(y,x)))) = x.
1266
f(f(x,y),f(y,y)) = y.
1267
f(f(x,f(f(y,f(x,y)),y)),f(f(y,x),f(x,x))) = f(y,x).
1268
f(f(x,f(y,f(y,f(x,y)))),f(f(y,x),f(x,x))) = f(y,x).
1269
f(f(x,f(y,x)),f(f(y,x),f(x,x))) = f(y,x).
1270
f(f(x,f(y,x)),x) = f(y,x).
1271
f(x,f(x,f(y,x))) = f(y,x).
1272
f(x,y) = f(y,x).
1273
$F # answer("Commutativity").
1274
end_of_list.
1275
1276
Attempt to eliminate one assumption:
1277
1278
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").
1279
1280
***********************************************
1281
AUTO SKETCHES ITERATION 5
1282
***********************************************
1283
1284
Starting a search with 13 assumptions:
1285
1286
formulas(assumptions_active).
1287
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
1288
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
1289
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1290
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1291
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1292
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1293
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1294
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
1295
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1296
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1297
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
1298
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
1299
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1300
end_of_list.
1301
1302
Child search process 21996 started.
1303
1304
============================== PROCESS INITIAL CLAUSES ===============
1305
1306
% Clauses before input processing:
1307
1308
formulas(usable).
1309
end_of_list.
1310
1311
formulas(sos).
1312
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
1313
f(c2,c1) != f(c1,c2) # label("Commutativity"). [deny(1)].
1314
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
1315
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
1316
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1317
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1318
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1319
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1320
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1321
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
1322
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1323
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1324
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
1325
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
1326
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1327
end_of_list.
1328
1329
formulas(demodulators).
1330
end_of_list.
1331
1332
% 285 hints input.
1333
1334
============================== PREDICATE ELIMINATION =================
1335
1336
No predicates eliminated.
1337
1338
============================== end predicate elimination =============
1339
1340
Auto_denials:
1341
% copying label "Commutativity" to answer in negative clause
1342
1343
Term ordering decisions:
1344
Function symbol KB weights: c1=1. c2=1. f=1.
1345
Predicate symbol precedence: predicate_order([ = ]).
1346
Function symbol precedence: function_order([ c1, c2, f ]).
1347
Skipping inverse_order, because term ordering is KBO.
1348
Unfolding symbols: (none).
1349
1350
Auto_inference settings:
1351
% set(paramodulation). % (positive equality literals)
1352
% set(paramodulation) -> set(back_demod).
1353
1354
Auto_process settings: (no changes).
1355
1356
============================== end of process initial clauses ========
1357
1358
============================== CLAUSES FOR SEARCH ====================
1359
1360
% Clauses after input processing:
1361
1362
formulas(usable).
1363
end_of_list.
1364
1365
formulas(sos).
1366
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
1367
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
1368
4 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
1369
5 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
1370
6 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1371
7 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1372
8 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1373
9 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1374
10 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1375
11 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
1376
12 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1377
13 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1378
14 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
1379
15 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
1380
16 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1381
end_of_list.
1382
1383
formulas(demodulators).
1384
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
1385
4 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
1386
5 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
1387
6 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1388
7 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1389
8 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1390
9 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1391
10 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1392
11 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
1393
12 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1394
13 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1395
14 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
1396
15 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
1397
16 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1398
end_of_list.
1399
1400
% 285 hints processed (128 are redundant).
1401
1402
============================== end of clauses for search =============
1403
1404
============================== SEARCH ================================
1405
1406
% Starting search at 0.02 seconds.
1407
1408
% Operation f is commutative; C redundancy checks enabled.
1409
1410
============================== PROOF =================================
1411
1412
% Proof 1 at 0.74 (+ 0.00) seconds: "Commutativity".
1413
% Length of proof is 59.
1414
% Level of proof is 17.
1415
% Maximum clause weight is 51.
1416
% Given clauses 30.
1417
1418
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
1419
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
1420
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
1421
4 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
1422
6 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1423
7 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1424
8 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1425
9 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1426
10 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1427
12 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1428
13 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1429
15 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
1430
16 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1431
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))].
1432
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))].
1433
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))].
1434
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))].
1435
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))].
1436
611 f(f(f(x,f(x,x)),x),f(x,x)) = x. [para(2(a,1),21(a,1,2,2))].
1437
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))].
1438
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)])].
1439
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))].
1440
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))].
1441
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)])].
1442
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))].
1443
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))].
1444
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)])].
1445
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))].
1446
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)].
1447
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))].
1448
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)].
1449
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))].
1450
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)])].
1451
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)].
1452
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)])].
1453
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)])].
1454
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))].
1455
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)].
1456
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)])].
1457
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)])].
1458
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)])].
1459
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))].
1460
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)])].
1461
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))].
1462
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)])].
1463
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)])].
1464
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))].
1465
1551 f(x,f(f(x,x),x)) = f(x,x). [back_rewrite(1221),rewrite([1548(8)]),flip(a)].
1466
1554 f(f(x,f(x,x)),f(x,x)) = x. [back_rewrite(1548),rewrite([1551(3),1551(5)])].
1467
1586 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(1293),rewrite([1551(3),1551(6)])].
1468
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)].
1469
1602 f(x,f(f(y,x),x)) = f(y,x). [back_rewrite(1343),rewrite([1599(4),1554(5)])].
1470
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)])].
1471
1768 f(f(x,y),y) = f(y,f(y,x)). [back_rewrite(1159),rewrite([1602(4),1602(5)])].
1472
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)].
1473
2065 f(x,f(x,f(x,y))) = f(y,x). [back_rewrite(1602),rewrite([1768(2)])].
1474
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)])].
1475
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)])].
1476
2139 $F # answer("Commutativity"). [resolve(2138,a,3,a)].
1477
1478
============================== end of proof ==========================
1479
1480
============================== STATISTICS ============================
1481
1482
Given=30. Generated=3223. Kept=2137. proofs=1.
1483
Usable=15. Sos=613. Demods=985. Limbo=370, Disabled=1153. Hints=285.
1484
Weight_deleted=0. Literals_deleted=0.
1485
Forward_subsumed=1086. Back_subsumed=44.
1486
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
1487
New_demodulators=2095 (0 lex), Back_demodulated=1094. Back_unit_deleted=0.
1488
Demod_attempts=67957. Demod_rewrites=4889.
1489
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
1490
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
1491
Megabytes=2.83.
1492
User_CPU=0.74, System_CPU=0.00, Wall_clock=5.
1493
1494
============================== end of statistics =====================
1495
1496
============================== end of search =========================
1497
1498
Exiting with 1 proof.
1499
1500
Process 21996 exit (max_proofs) Mon Jun 4 22:53:30 2007
1501
1502
Successful proof using the following 10 assumptions:
1503
1504
formulas(assumptions_in_proof).
1505
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7"). [assumption].
1506
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1507
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1508
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1509
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1510
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1511
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1512
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1513
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
1514
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1515
end_of_list.
1516
1517
Including 101 new hint clauses:
1518
1519
formulas(new_hints).
1520
$F # label("Commutativity") # label(non_clause) # label(goal).
1521
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
1522
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
1523
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").
1524
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
1525
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").
1526
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").
1527
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
1528
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
1529
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
1530
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
1531
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
1532
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
1533
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
1534
f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)).
1535
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
1536
f(f(f(f(f(x,y),f(f(x,f(x,z)),x)),z),z),z) = f(f(x,f(x,z)),x).
1537
f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(y,f(y,x)),y).
1538
f(f(f(x,f(x,x)),x),f(x,x)) = x.
1539
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
1540
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)).
1541
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)).
1542
f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x.
1543
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).
1544
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)).
1545
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)).
1546
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)).
1547
f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)).
1548
f(f(x,f(f(f(x,f(x,x)),x),x)),x) = f(x,f(x,x)).
1549
f(f(f(f(x,f(x,f(x,x))),x),x),x) = f(x,f(x,x)).
1550
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)))).
1551
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).
1552
f(f(f(x,y),y),y) = f(x,f(f(y,x),x)).
1553
f(x,f(x,f(x,x))) = f(x,f(f(f(x,f(x,x)),x),x)).
1554
f(x,f(f(f(x,f(x,x)),x),x)) = f(x,f(x,f(x,x))).
1555
f(f(f(x,f(f(y,x),x)),y),f(y,f(f(x,y),y))) = y.
1556
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))).
1557
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)).
1558
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))).
1559
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)).
1560
f(f(x,f(x,f(x,x))),x) = f(x,f(x,x)).
1561
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)))).
1562
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)))).
1563
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)))).
1564
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))))).
1565
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).
1566
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)).
1567
f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)).
1568
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))).
1569
f(x,f(f(f(y,x),x),x)) = f(x,f(y,f(f(x,y),y))).
1570
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)).
1571
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).
1572
f(x,f(x,f(f(x,x),x))) = f(f(x,f(f(x,x),x)),x).
1573
f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))).
1574
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)).
1575
f(f(f(x,x),f(x,f(x,x))),f(x,f(x,x))) = f(f(x,f(x,x)),x).
1576
f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
1577
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,x).
1578
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)).
1579
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)).
1580
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).
1581
f(f(x,x),f(f(x,f(x,x)),x)) = f(f(x,f(x,x)),f(x,x)).
1582
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))).
1583
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
1584
f(x,x) = f(x,f(f(x,x),x)).
1585
f(x,f(f(x,x),x)) = f(x,x).
1586
f(f(x,f(x,x)),f(x,f(f(x,x),x))) = x.
1587
f(f(x,f(x,x)),f(x,x)) = x.
1588
f(f(x,f(x,x)),x) = f(x,f(f(x,x),x)).
1589
f(f(x,f(x,x)),x) = f(x,x).
1590
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))).
1591
f(f(x,f(x,x)),f(f(x,f(x,x)),x)) = f(f(x,x),f(x,f(x,x))).
1592
f(f(x,f(x,x)),f(x,x)) = f(f(x,x),f(x,f(x,x))).
1593
x = f(f(x,x),f(x,f(x,x))).
1594
f(f(x,x),f(x,f(x,x))) = x.
1595
f(x,f(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x).
1596
f(x,f(f(y,x),x)) = f(y,x).
1597
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).
1598
f(f(y,x),x) = f(x,f(y,f(f(x,y),y))).
1599
f(f(x,y),y) = f(y,f(y,x)).
1600
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).
1601
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).
1602
f(x,f(f(y,x),x)) = f(f(f(y,x),x),x).
1603
f(x,f(x,f(x,y))) = f(f(f(y,x),x),x).
1604
f(x,f(x,f(x,y))) = f(f(x,f(x,y)),x).
1605
f(f(x,f(x,y)),x) = f(x,f(x,f(x,y))).
1606
f(x,f(x,f(x,y))) = f(y,x).
1607
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(f(y,f(y,x)),y).
1608
f(f(f(x,f(f(y,z),f(x,y))),x),x) = f(f(y,f(y,x)),y).
1609
f(x,f(x,f(x,f(f(y,z),f(x,y))))) = f(f(y,f(y,x)),y).
1610
f(f(f(y,z),f(x,y)),x) = f(f(y,f(y,x)),y).
1611
f(f(f(y,z),f(x,y)),x) = f(y,f(y,f(y,x))).
1612
f(f(f(x,y),f(z,x)),z) = f(z,x).
1613
f(f(f(f(f(x,y),f(x,f(x,f(x,z)))),z),z),z) = f(f(x,f(x,z)),x).
1614
f(f(f(f(f(x,y),f(z,x)),z),z),z) = f(f(x,f(x,z)),x).
1615
f(f(f(z,x),z),z) = f(f(x,f(x,z)),x).
1616
f(z,f(z,f(z,x))) = f(f(x,f(x,z)),x).
1617
f(x,z) = f(f(x,f(x,z)),x).
1618
f(x,z) = f(x,f(x,f(x,z))).
1619
f(x,y) = f(y,x).
1620
$F # answer("Commutativity").
1621
end_of_list.
1622
1623
Attempt to eliminate one assumption:
1624
1625
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").
1626
1627
***********************************************
1628
AUTO SKETCHES ITERATION 6
1629
***********************************************
1630
1631
Starting a search with 12 assumptions:
1632
1633
formulas(assumptions_active).
1634
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
1635
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1636
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1637
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1638
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1639
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1640
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
1641
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1642
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1643
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
1644
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
1645
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1646
end_of_list.
1647
1648
Child search process 21997 started.
1649
1650
============================== PROCESS INITIAL CLAUSES ===============
1651
1652
% Clauses before input processing:
1653
1654
formulas(usable).
1655
end_of_list.
1656
1657
formulas(sos).
1658
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
1659
f(c2,c1) != f(c1,c2) # label("Commutativity"). [deny(1)].
1660
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
1661
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1662
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1663
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1664
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1665
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1666
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
1667
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1668
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1669
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
1670
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
1671
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1672
end_of_list.
1673
1674
formulas(demodulators).
1675
end_of_list.
1676
1677
% 386 hints input.
1678
1679
============================== PREDICATE ELIMINATION =================
1680
1681
No predicates eliminated.
1682
1683
============================== end predicate elimination =============
1684
1685
Auto_denials:
1686
% copying label "Commutativity" to answer in negative clause
1687
1688
Term ordering decisions:
1689
Function symbol KB weights: c1=1. c2=1. f=1.
1690
Predicate symbol precedence: predicate_order([ = ]).
1691
Function symbol precedence: function_order([ c1, c2, f ]).
1692
Skipping inverse_order, because term ordering is KBO.
1693
Unfolding symbols: (none).
1694
1695
Auto_inference settings:
1696
% set(paramodulation). % (positive equality literals)
1697
% set(paramodulation) -> set(back_demod).
1698
1699
Auto_process settings: (no changes).
1700
1701
============================== end of process initial clauses ========
1702
1703
============================== CLAUSES FOR SEARCH ====================
1704
1705
% Clauses after input processing:
1706
1707
formulas(usable).
1708
end_of_list.
1709
1710
formulas(sos).
1711
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
1712
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
1713
4 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
1714
5 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1715
6 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1716
7 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1717
8 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1718
9 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1719
10 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
1720
11 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1721
12 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1722
13 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
1723
14 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
1724
15 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1725
end_of_list.
1726
1727
formulas(demodulators).
1728
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
1729
4 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
1730
5 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1731
6 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1732
7 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1733
8 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1734
9 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1735
10 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
1736
11 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1737
12 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1738
13 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
1739
14 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
1740
15 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1741
end_of_list.
1742
1743
% 386 hints processed (172 are redundant).
1744
1745
============================== end of clauses for search =============
1746
1747
============================== SEARCH ================================
1748
1749
% Starting search at 0.02 seconds.
1750
1751
% Operation f is commutative; C redundancy checks enabled.
1752
1753
============================== PROOF =================================
1754
1755
% Proof 1 at 0.36 (+ 0.00) seconds: "Commutativity".
1756
% Length of proof is 37.
1757
% Level of proof is 15.
1758
% Maximum clause weight is 35.
1759
% Given clauses 31.
1760
1761
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
1762
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
1763
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
1764
4 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
1765
5 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1766
8 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1767
9 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1768
11 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1769
12 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1770
15 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1771
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))].
1772
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))].
1773
515 f(f(f(x,f(x,x)),x),f(x,x)) = x. [para(2(a,1),20(a,1,2,2))].
1774
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))].
1775
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)])].
1776
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))].
1777
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))].
1778
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))].
1779
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)])].
1780
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)])].
1781
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))].
1782
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)])].
1783
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)])].
1784
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)])].
1785
861 f(x,f(f(x,x),x)) = f(x,x). [back_rewrite(548),rewrite([827(4),685(8)]),flip(a)].
1786
889 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [back_rewrite(834),rewrite([861(3),861(5)])].
1787
896 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(827),rewrite([861(3),861(5)])].
1788
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)])].
1789
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)])].
1790
924 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(861),rewrite([896(2)])].
1791
928 f(f(x,f(x,x)),f(x,x)) = x. [back_rewrite(918),rewrite([924(3),924(5)])].
1792
937 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(905),rewrite([924(3),924(6)])].
1793
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)])].
1794
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)].
1795
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)])].
1796
1203 f(x,y) = f(y,x). [para(2(a,1),1172(a,1,1)),rewrite([2(8)])].
1797
1204 $F # answer("Commutativity"). [resolve(1203,a,3,a)].
1798
1799
============================== end of proof ==========================
1800
1801
============================== STATISTICS ============================
1802
1803
Given=31. Generated=2036. Kept=1202. proofs=1.
1804
Usable=22. Sos=728. Demods=756. Limbo=8, Disabled=457. Hints=386.
1805
Weight_deleted=0. Literals_deleted=0.
1806
Forward_subsumed=834. Back_subsumed=51.
1807
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
1808
New_demodulators=1197 (0 lex), Back_demodulated=392. Back_unit_deleted=0.
1809
Demod_attempts=36008. Demod_rewrites=2734.
1810
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
1811
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
1812
Megabytes=1.74.
1813
User_CPU=0.36, System_CPU=0.00, Wall_clock=5.
1814
1815
============================== end of statistics =====================
1816
1817
============================== end of search =========================
1818
1819
Exiting with 1 proof.
1820
1821
Process 21997 exit (max_proofs) Mon Jun 4 22:53:30 2007
1822
1823
Successful proof using the following 7 assumptions:
1824
1825
formulas(assumptions_in_proof).
1826
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))"). [assumption].
1827
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1828
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1829
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1830
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1831
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1832
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1833
end_of_list.
1834
1835
Including 64 new hint clauses:
1836
1837
formulas(new_hints).
1838
$F # label("Commutativity") # label(non_clause) # label(goal).
1839
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
1840
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
1841
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").
1842
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
1843
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
1844
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
1845
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
1846
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
1847
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
1848
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
1849
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
1850
f(f(f(x,f(x,x)),x),f(x,x)) = x.
1851
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
1852
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)).
1853
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)).
1854
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).
1855
f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x.
1856
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))).
1857
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)).
1858
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)).
1859
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)).
1860
f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)).
1861
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)))).
1862
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)))).
1863
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)))).
1864
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
1865
f(f(f(x,f(f(x,x),x)),x),f(x,f(y,f(x,f(f(x,x),x))))) = x.
1866
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)).
1867
f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
1868
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))).
1869
f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))).
1870
f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x.
1871
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)).
1872
f(x,x) = f(x,f(f(x,x),x)).
1873
f(x,f(f(x,x),x)) = f(x,x).
1874
f(f(x,f(x,x)),f(x,f(y,f(x,f(f(x,x),x))))) = x.
1875
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
1876
f(f(x,x),x) = f(x,f(x,f(f(x,x),x))).
1877
f(f(x,x),x) = f(x,f(x,x)).
1878
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(f(x,x),x)).
1879
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(x,f(x,x))).
1880
f(f(x,f(x,f(x,f(x,x)))),f(x,f(f(x,x),x))) = x.
1881
f(f(x,f(x,f(x,f(x,x)))),f(x,f(x,f(x,x)))) = x.
1882
f(x,f(x,f(x,x))) = f(x,x).
1883
f(f(x,f(x,x)),f(x,f(x,f(x,x)))) = x.
1884
f(f(x,f(x,x)),f(x,x)) = x.
1885
f(f(x,f(x,x)),x) = f(x,f(x,f(x,x))).
1886
f(f(x,f(x,x)),x) = f(x,x).
1887
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).
1888
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).
1889
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).
1890
f(x,f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x).
1891
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x).
1892
f(f(f(x,f(x,x)),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
1893
f(f(x,x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
1894
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y).
1895
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)).
1896
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
1897
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
1898
f(f(x,x),y) = f(y,f(x,x)).
1899
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
1900
f(x,y) = f(y,x).
1901
$F # answer("Commutativity").
1902
end_of_list.
1903
1904
Attempt to eliminate one assumption:
1905
1906
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").
1907
1908
***********************************************
1909
AUTO SKETCHES ITERATION 7
1910
***********************************************
1911
1912
Starting a search with 11 assumptions:
1913
1914
formulas(assumptions_active).
1915
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1916
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1917
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1918
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1919
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1920
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
1921
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1922
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1923
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
1924
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
1925
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1926
end_of_list.
1927
1928
Child search process 21998 started.
1929
1930
============================== PROCESS INITIAL CLAUSES ===============
1931
1932
% Clauses before input processing:
1933
1934
formulas(usable).
1935
end_of_list.
1936
1937
formulas(sos).
1938
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
1939
f(c2,c1) != f(c1,c2) # label("Commutativity"). [deny(1)].
1940
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1941
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1942
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1943
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1944
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1945
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
1946
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1947
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
1948
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
1949
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
1950
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
1951
end_of_list.
1952
1953
formulas(demodulators).
1954
end_of_list.
1955
1956
% 450 hints input.
1957
1958
============================== PREDICATE ELIMINATION =================
1959
1960
No predicates eliminated.
1961
1962
============================== end predicate elimination =============
1963
1964
Auto_denials:
1965
% copying label "Commutativity" to answer in negative clause
1966
1967
Term ordering decisions:
1968
Function symbol KB weights: c1=1. c2=1. f=1.
1969
Predicate symbol precedence: predicate_order([ = ]).
1970
Function symbol precedence: function_order([ c1, c2, f ]).
1971
Skipping inverse_order, because term ordering is KBO.
1972
Unfolding symbols: (none).
1973
1974
Auto_inference settings:
1975
% set(paramodulation). % (positive equality literals)
1976
% set(paramodulation) -> set(back_demod).
1977
1978
Auto_process settings: (no changes).
1979
1980
============================== end of process initial clauses ========
1981
1982
============================== CLAUSES FOR SEARCH ====================
1983
1984
% Clauses after input processing:
1985
1986
formulas(usable).
1987
end_of_list.
1988
1989
formulas(sos).
1990
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
1991
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
1992
4 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
1993
5 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
1994
6 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
1995
7 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
1996
8 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
1997
9 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
1998
10 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
1999
11 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2000
12 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2001
13 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2002
14 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2003
end_of_list.
2004
2005
formulas(demodulators).
2006
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
2007
4 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
2008
5 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
2009
6 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
2010
7 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2011
8 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2012
9 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2013
10 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2014
11 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2015
12 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2016
13 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2017
14 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2018
end_of_list.
2019
2020
% 450 hints processed (218 are redundant).
2021
2022
============================== end of clauses for search =============
2023
2024
============================== SEARCH ================================
2025
2026
% Starting search at 0.03 seconds.
2027
2028
% Operation f is commutative; C redundancy checks enabled.
2029
2030
============================== PROOF =================================
2031
2032
% Proof 1 at 0.33 (+ 0.00) seconds: "Commutativity".
2033
% Length of proof is 42.
2034
% Level of proof is 15.
2035
% Maximum clause weight is 35.
2036
% Given clauses 30.
2037
2038
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
2039
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
2040
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
2041
4 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
2042
5 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
2043
7 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2044
8 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2045
10 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2046
11 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2047
13 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2048
14 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2049
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))].
2050
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))].
2051
433 f(f(f(x,f(x,x)),x),f(x,x)) = x. [para(2(a,1),25(a,1,2,2))].
2052
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))].
2053
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)])].
2054
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))].
2055
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))].
2056
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))].
2057
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)])].
2058
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))].
2059
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)])].
2060
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))].
2061
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))].
2062
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)])].
2063
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)])].
2064
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)])].
2065
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)])].
2066
747 f(x,f(f(x,x),x)) = f(x,x). [back_rewrite(462),rewrite([714(4),586(8)]),flip(a)].
2067
772 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [back_rewrite(723),rewrite([747(3),747(5)])].
2068
774 f(f(x,f(x,f(x,x))),x) = f(x,f(x,x)). [back_rewrite(721),rewrite([747(3),747(7)])].
2069
780 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(714),rewrite([747(3),747(5)])].
2070
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)])].
2071
832 f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)). [back_rewrite(574),rewrite([774(4)])].
2072
836 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(747),rewrite([780(2)])].
2073
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)])].
2074
883 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(789),rewrite([836(3),836(6)])].
2075
899 f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x). [back_rewrite(868),rewrite([883(4)])].
2076
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)].
2077
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)])].
2078
1105 f(x,y) = f(y,x). [para(2(a,1),1094(a,1,1)),rewrite([2(8)])].
2079
1106 $F # answer("Commutativity"). [resolve(1105,a,3,a)].
2080
2081
============================== end of proof ==========================
2082
2083
============================== STATISTICS ============================
2084
2085
Given=30. Generated=1871. Kept=1104. proofs=1.
2086
Usable=21. Sos=622. Demods=649. Limbo=8, Disabled=465. Hints=450.
2087
Weight_deleted=0. Literals_deleted=0.
2088
Forward_subsumed=767. Back_subsumed=51.
2089
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
2090
New_demodulators=1099 (0 lex), Back_demodulated=401. Back_unit_deleted=0.
2091
Demod_attempts=33582. Demod_rewrites=2408.
2092
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
2093
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
2094
Megabytes=1.63.
2095
User_CPU=0.33, System_CPU=0.00, Wall_clock=5.
2096
2097
============================== end of statistics =====================
2098
2099
============================== end of search =========================
2100
2101
Exiting with 1 proof.
2102
2103
Process 21998 exit (max_proofs) Mon Jun 4 22:53:30 2007
2104
2105
Successful proof using the following 8 assumptions:
2106
2107
formulas(assumptions_in_proof).
2108
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [assumption].
2109
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
2110
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2111
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2112
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2113
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2114
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2115
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2116
end_of_list.
2117
2118
Including 66 new hint clauses:
2119
2120
formulas(new_hints).
2121
$F # label("Commutativity") # label(non_clause) # label(goal).
2122
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
2123
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
2124
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
2125
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").
2126
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
2127
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
2128
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
2129
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
2130
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
2131
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
2132
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
2133
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
2134
f(f(f(x,f(x,x)),x),f(x,x)) = x.
2135
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
2136
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)).
2137
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)).
2138
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).
2139
f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x.
2140
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))).
2141
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)).
2142
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)).
2143
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)).
2144
f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)).
2145
f(f(f(f(x,f(x,f(x,x))),x),x),x) = f(x,f(x,x)).
2146
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)))).
2147
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)))).
2148
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)))).
2149
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
2150
f(f(f(x,f(f(x,x),x)),x),f(x,f(y,f(x,f(f(x,x),x))))) = x.
2151
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))).
2152
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)).
2153
f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
2154
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))).
2155
f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))).
2156
f(f(x,f(f(x,f(f(x,x),x)),x)),x) = f(x,f(x,f(f(x,x),x))).
2157
f(f(x,f(x,f(x,f(f(x,x),x)))),x) = f(x,f(x,f(f(x,x),x))).
2158
f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x.
2159
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)).
2160
f(x,x) = f(x,f(f(x,x),x)).
2161
f(x,f(f(x,x),x)) = f(x,x).
2162
f(f(x,f(x,x)),f(x,f(y,f(x,f(f(x,x),x))))) = x.
2163
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
2164
f(f(x,f(x,f(x,x))),x) = f(x,f(x,f(f(x,x),x))).
2165
f(f(x,f(x,f(x,x))),x) = f(x,f(x,x)).
2166
f(f(x,x),x) = f(x,f(x,f(f(x,x),x))).
2167
f(f(x,x),x) = f(x,f(x,x)).
2168
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(f(x,x),x)).
2169
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(x,f(x,x))).
2170
f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)).
2171
f(x,f(x,f(x,x))) = f(x,x).
2172
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).
2173
f(x,f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x).
2174
f(f(x,f(x,x)),x) = f(x,f(x,f(x,x))).
2175
f(f(x,f(x,x)),x) = f(x,x).
2176
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x).
2177
f(f(f(x,f(x,x)),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
2178
f(f(x,x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
2179
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y).
2180
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)).
2181
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
2182
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
2183
f(f(x,x),y) = f(y,f(x,x)).
2184
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
2185
f(x,y) = f(y,x).
2186
$F # answer("Commutativity").
2187
end_of_list.
2188
2189
Attempt to eliminate one assumption:
2190
2191
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
2192
2193
***********************************************
2194
AUTO SKETCHES ITERATION 8
2195
***********************************************
2196
2197
Starting a search with 10 assumptions:
2198
2199
formulas(assumptions_active).
2200
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
2201
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
2202
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2203
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2204
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2205
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2206
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2207
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2208
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2209
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2210
end_of_list.
2211
2212
Child search process 21999 started.
2213
2214
============================== PROCESS INITIAL CLAUSES ===============
2215
2216
% Clauses before input processing:
2217
2218
formulas(usable).
2219
end_of_list.
2220
2221
formulas(sos).
2222
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
2223
f(c2,c1) != f(c1,c2) # label("Commutativity"). [deny(1)].
2224
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
2225
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
2226
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2227
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2228
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2229
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2230
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2231
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2232
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2233
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2234
end_of_list.
2235
2236
formulas(demodulators).
2237
end_of_list.
2238
2239
% 516 hints input.
2240
2241
============================== PREDICATE ELIMINATION =================
2242
2243
No predicates eliminated.
2244
2245
============================== end predicate elimination =============
2246
2247
Auto_denials:
2248
% copying label "Commutativity" to answer in negative clause
2249
2250
Term ordering decisions:
2251
Function symbol KB weights: c1=1. c2=1. f=1.
2252
Predicate symbol precedence: predicate_order([ = ]).
2253
Function symbol precedence: function_order([ c1, c2, f ]).
2254
Skipping inverse_order, because term ordering is KBO.
2255
Unfolding symbols: (none).
2256
2257
Auto_inference settings:
2258
% set(paramodulation). % (positive equality literals)
2259
% set(paramodulation) -> set(back_demod).
2260
2261
Auto_process settings: (no changes).
2262
2263
============================== end of process initial clauses ========
2264
2265
============================== CLAUSES FOR SEARCH ====================
2266
2267
% Clauses after input processing:
2268
2269
formulas(usable).
2270
end_of_list.
2271
2272
formulas(sos).
2273
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
2274
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
2275
4 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
2276
5 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
2277
6 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2278
7 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2279
8 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2280
9 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2281
10 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2282
11 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2283
12 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2284
13 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2285
end_of_list.
2286
2287
formulas(demodulators).
2288
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
2289
4 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
2290
5 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
2291
6 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2292
7 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2293
8 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2294
9 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2295
10 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2296
11 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2297
12 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2298
13 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2299
end_of_list.
2300
2301
% 516 hints processed (261 are redundant).
2302
2303
============================== end of clauses for search =============
2304
2305
============================== SEARCH ================================
2306
2307
% Starting search at 0.02 seconds.
2308
2309
% Operation f is commutative; C redundancy checks enabled.
2310
2311
============================== PROOF =================================
2312
2313
% Proof 1 at 0.27 (+ 0.01) seconds: "Commutativity".
2314
% Length of proof is 38.
2315
% Level of proof is 15.
2316
% Maximum clause weight is 35.
2317
% Given clauses 28.
2318
2319
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
2320
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
2321
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
2322
4 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
2323
6 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2324
7 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2325
9 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2326
10 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2327
12 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2328
13 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2329
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))].
2330
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))].
2331
375 f(f(f(x,f(x,x)),x),f(x,x)) = x. [para(2(a,1),18(a,1,2,2))].
2332
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))].
2333
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)])].
2334
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))].
2335
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))].
2336
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)])].
2337
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))].
2338
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)])].
2339
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)])].
2340
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))].
2341
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)])].
2342
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)])].
2343
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)])].
2344
644 f(x,f(f(x,x),x)) = f(x,x). [back_rewrite(404),rewrite([613(4),523(8)]),flip(a)].
2345
668 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [back_rewrite(622),rewrite([644(3),644(5)])].
2346
676 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(613),rewrite([644(3),644(5)])].
2347
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)])].
2348
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)])].
2349
705 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(644),rewrite([676(2)])].
2350
709 f(f(x,f(x,x)),f(x,x)) = x. [back_rewrite(697),rewrite([705(3),705(5)])].
2351
715 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(686),rewrite([705(3),705(6)])].
2352
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)])].
2353
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)].
2354
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)])].
2355
947 f(x,y) = f(y,x). [para(2(a,1),935(a,1,1)),rewrite([2(8)])].
2356
948 $F # answer("Commutativity"). [resolve(947,a,3,a)].
2357
2358
============================== end of proof ==========================
2359
2360
============================== STATISTICS ============================
2361
2362
Given=28. Generated=1599. Kept=946. proofs=1.
2363
Usable=20. Sos=561. Demods=588. Limbo=9, Disabled=367. Hints=516.
2364
Weight_deleted=0. Literals_deleted=0.
2365
Forward_subsumed=653. Back_subsumed=41.
2366
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
2367
New_demodulators=941 (0 lex), Back_demodulated=314. Back_unit_deleted=0.
2368
Demod_attempts=28728. Demod_rewrites=2185.
2369
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
2370
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
2371
Megabytes=1.47.
2372
User_CPU=0.27, System_CPU=0.01, Wall_clock=6.
2373
2374
============================== end of statistics =====================
2375
2376
============================== end of search =========================
2377
2378
Exiting with 1 proof.
2379
2380
Process 21999 exit (max_proofs) Mon Jun 4 22:53:31 2007
2381
2382
Successful proof using the following 7 assumptions:
2383
2384
formulas(assumptions_in_proof).
2385
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2"). [assumption].
2386
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2387
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2388
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2389
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2390
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2391
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2392
end_of_list.
2393
2394
Including 67 new hint clauses:
2395
2396
formulas(new_hints).
2397
$F # label("Commutativity") # label(non_clause) # label(goal).
2398
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
2399
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
2400
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").
2401
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
2402
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
2403
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
2404
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
2405
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
2406
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
2407
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
2408
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
2409
f(f(f(x,f(x,x)),x),f(x,x)) = x.
2410
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
2411
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)).
2412
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)).
2413
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).
2414
f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x.
2415
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.
2416
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.
2417
f(f(x,f(f(y,x),x)),f(y,f(x,f(f(x,x),x)))) = y.
2418
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))).
2419
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)).
2420
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)).
2421
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)).
2422
f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)).
2423
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)))).
2424
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)))).
2425
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)))).
2426
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
2427
f(f(f(x,f(f(x,x),x)),x),f(x,f(y,f(x,f(f(x,x),x))))) = x.
2428
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)).
2429
f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
2430
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))).
2431
f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))).
2432
f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x.
2433
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)).
2434
f(x,x) = f(x,f(f(x,x),x)).
2435
f(x,f(f(x,x),x)) = f(x,x).
2436
f(f(x,f(x,x)),f(x,f(y,f(x,f(f(x,x),x))))) = x.
2437
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
2438
f(f(x,x),x) = f(x,f(x,f(f(x,x),x))).
2439
f(f(x,x),x) = f(x,f(x,x)).
2440
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(f(x,x),x)).
2441
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(x,f(x,x))).
2442
f(f(x,f(x,f(x,f(x,x)))),f(x,f(f(x,x),x))) = x.
2443
f(f(x,f(x,f(x,f(x,x)))),f(x,f(x,f(x,x)))) = x.
2444
f(x,f(x,f(x,x))) = f(x,x).
2445
f(f(x,f(x,x)),f(x,f(x,f(x,x)))) = x.
2446
f(f(x,f(x,x)),f(x,x)) = x.
2447
f(f(x,f(x,x)),x) = f(x,f(x,f(x,x))).
2448
f(f(x,f(x,x)),x) = f(x,x).
2449
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).
2450
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).
2451
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).
2452
f(x,f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x).
2453
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x).
2454
f(f(f(x,f(x,x)),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
2455
f(f(x,x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
2456
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y).
2457
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)).
2458
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
2459
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
2460
f(f(x,x),y) = f(y,f(x,x)).
2461
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
2462
f(x,y) = f(y,x).
2463
$F # answer("Commutativity").
2464
end_of_list.
2465
2466
Attempt to eliminate one assumption:
2467
2468
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").
2469
2470
***********************************************
2471
AUTO SKETCHES ITERATION 9
2472
***********************************************
2473
2474
Starting a search with 9 assumptions:
2475
2476
formulas(assumptions_active).
2477
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
2478
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2479
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2480
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2481
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2482
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2483
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2484
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2485
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2486
end_of_list.
2487
2488
Child search process 22000 started.
2489
2490
============================== PROCESS INITIAL CLAUSES ===============
2491
2492
% Clauses before input processing:
2493
2494
formulas(usable).
2495
end_of_list.
2496
2497
formulas(sos).
2498
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
2499
f(c2,c1) != f(c1,c2) # label("Commutativity"). [deny(1)].
2500
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
2501
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2502
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2503
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2504
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2505
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2506
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2507
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2508
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2509
end_of_list.
2510
2511
formulas(demodulators).
2512
end_of_list.
2513
2514
% 583 hints input.
2515
2516
============================== PREDICATE ELIMINATION =================
2517
2518
No predicates eliminated.
2519
2520
============================== end predicate elimination =============
2521
2522
Auto_denials:
2523
% copying label "Commutativity" to answer in negative clause
2524
2525
Term ordering decisions:
2526
Function symbol KB weights: c1=1. c2=1. f=1.
2527
Predicate symbol precedence: predicate_order([ = ]).
2528
Function symbol precedence: function_order([ c1, c2, f ]).
2529
Skipping inverse_order, because term ordering is KBO.
2530
Unfolding symbols: (none).
2531
2532
Auto_inference settings:
2533
% set(paramodulation). % (positive equality literals)
2534
% set(paramodulation) -> set(back_demod).
2535
2536
Auto_process settings: (no changes).
2537
2538
============================== end of process initial clauses ========
2539
2540
============================== CLAUSES FOR SEARCH ====================
2541
2542
% Clauses after input processing:
2543
2544
formulas(usable).
2545
end_of_list.
2546
2547
formulas(sos).
2548
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
2549
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
2550
4 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
2551
5 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2552
6 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2553
7 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2554
8 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2555
9 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2556
10 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2557
11 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2558
12 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2559
end_of_list.
2560
2561
formulas(demodulators).
2562
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
2563
4 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
2564
5 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2565
6 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2566
7 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2567
8 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2568
9 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2569
10 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2570
11 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2571
12 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2572
end_of_list.
2573
2574
% 583 hints processed (323 are redundant).
2575
2576
============================== end of clauses for search =============
2577
2578
============================== SEARCH ================================
2579
2580
% Starting search at 0.03 seconds.
2581
2582
% Operation f is commutative; C redundancy checks enabled.
2583
2584
============================== PROOF =================================
2585
2586
% Proof 1 at 1.04 (+ 0.00) seconds: "Commutativity".
2587
% Length of proof is 51.
2588
% Level of proof is 16.
2589
% Maximum clause weight is 51.
2590
% Given clauses 36.
2591
2592
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
2593
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
2594
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
2595
4 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
2596
5 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2597
6 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2598
8 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2599
9 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2600
11 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2601
12 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2602
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))].
2603
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))].
2604
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))].
2605
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))].
2606
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))].
2607
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))].
2608
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))].
2609
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))].
2610
533 f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x. [para(436(a,1),9(a,1,1))].
2611
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)])].
2612
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)].
2613
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))].
2614
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)].
2615
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)].
2616
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)].
2617
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)])].
2618
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)])].
2619
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))].
2620
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)])].
2621
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)])].
2622
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))].
2623
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)])].
2624
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)])].
2625
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)])].
2626
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)])].
2627
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)])].
2628
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)])].
2629
1365 f(x,f(f(x,x),x)) = f(x,x). [para(318(a,1),13(a,1,2)),rewrite([533(8)]),flip(a)].
2630
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)])].
2631
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)])].
2632
1529 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(903),rewrite([1365(3),1365(6)])].
2633
1533 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(897),rewrite([1365(3),1365(6)])].
2634
1577 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(840),rewrite([1365(3),1365(5)])].
2635
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)])].
2636
1610 f(f(x,x),f(x,x)) = x. [back_rewrite(1483),rewrite([1577(5),1529(7)])].
2637
1620 f(f(x,f(y,y)),f(x,y)) = x. [para(1610(a,1),2(a,1,2,2)),rewrite([1513(6)])].
2638
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)])].
2639
1796 f(f(x,x),y) = f(y,f(x,x)). [para(1620(a,1),6(a,1,2)),rewrite([1586(5)])].
2640
1953 f(x,f(f(x,x),x)) = f(x,x). [para(1610(a,1),1781(a,1,1)),rewrite([1610(4)])].
2641
1976 f(x,y) = f(y,x). [para(2(a,1),1796(a,1,1)),rewrite([1953(4),1953(5),1610(4)])].
2642
1977 $F # answer("Commutativity"). [resolve(1976,a,3,a)].
2643
2644
============================== end of proof ==========================
2645
2646
============================== STATISTICS ============================
2647
2648
Given=36. Generated=3828. Kept=1975. proofs=1.
2649
Usable=31. Sos=1569. Demods=1546. Limbo=9, Disabled=376. Hints=583.
2650
Weight_deleted=127. Literals_deleted=0.
2651
Forward_subsumed=1726. Back_subsumed=59.
2652
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
2653
New_demodulators=1901 (0 lex), Back_demodulated=306. Back_unit_deleted=0.
2654
Demod_attempts=112174. Demod_rewrites=6972.
2655
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
2656
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
2657
Megabytes=3.97.
2658
User_CPU=1.04, System_CPU=0.00, Wall_clock=7.
2659
2660
============================== end of statistics =====================
2661
2662
============================== end of search =========================
2663
2664
Exiting with 1 proof.
2665
2666
Process 22000 exit (max_proofs) Mon Jun 4 22:53:32 2007
2667
2668
Successful proof using the following 7 assumptions:
2669
2670
formulas(assumptions_in_proof).
2671
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m"). [assumption].
2672
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2673
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2674
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2675
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2676
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2677
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2678
end_of_list.
2679
2680
Including 95 new hint clauses:
2681
2682
formulas(new_hints).
2683
$F # label("Commutativity") # label(non_clause) # label(goal).
2684
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
2685
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
2686
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").
2687
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
2688
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
2689
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
2690
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
2691
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
2692
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
2693
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)).
2694
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
2695
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
2696
f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y.
2697
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)).
2698
f(f(f(x,y),y),y) = f(x,f(f(y,x),x)).
2699
f(f(f(x,f(f(y,x),x)),y),f(y,f(f(x,y),y))) = y.
2700
f(f(f(f(x,x),f(y,x)),f(y,x)),f(f(f(x,y),y),y)) = x.
2701
f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x.
2702
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))).
2703
f(x,f(f(f(y,x),x),x)) = f(x,f(y,f(f(x,y),y))).
2704
f(x,f(x,f(f(x,x),x))) = f(x,f(f(f(x,f(f(x,x),x)),x),x)).
2705
f(x,f(f(f(x,f(f(x,x),x)),x),x)) = f(x,f(x,f(f(x,x),x))).
2706
f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x.
2707
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)))).
2708
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)))).
2709
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)))).
2710
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)))).
2711
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).
2712
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)))).
2713
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).
2714
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)).
2715
f(f(x,f(x,f(y,f(x,f(f(x,x),x))))),x) = f(x,f(f(x,x),x)).
2716
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))).
2717
f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))).
2718
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))).
2719
f(x,f(f(x,f(x,f(f(x,x),x))),x)) = f(x,f(x,f(f(x,x),x))).
2720
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)).
2721
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)).
2722
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)).
2723
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)).
2724
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
2725
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))))).
2726
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)).
2727
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)))).
2728
f(x,f(f(f(x,f(x,f(f(x,x),x))),x),x)) = f(x,f(f(x,x),x)).
2729
f(x,f(x,f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
2730
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))).
2731
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)).
2732
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)).
2733
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)).
2734
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)).
2735
f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
2736
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))).
2737
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.
2738
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.
2739
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.
2740
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.
2741
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.
2742
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)).
2743
f(x,x) = f(x,f(f(x,x),x)).
2744
f(x,f(f(x,x),x)) = f(x,x).
2745
f(f(f(f(x,x),f(x,x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
2746
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,f(f(x,x),x))) = x.
2747
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,x)) = x.
2748
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))).
2749
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))).
2750
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,f(f(x,x),x))).
2751
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
2752
f(f(x,f(x,x)),x) = f(x,f(f(x,x),x)).
2753
f(f(x,f(x,x)),x) = f(x,x).
2754
f(x,f(x,f(x,x))) = f(x,f(f(x,x),x)).
2755
f(x,f(x,f(x,x))) = f(x,x).
2756
f(f(x,x),x) = f(x,f(x,f(f(x,x),x))).
2757
f(f(x,x),x) = f(x,f(x,x)).
2758
f(f(x,f(x,f(y,f(x,f(x,f(x,x)))))),x) = f(x,f(f(x,x),x)).
2759
f(f(x,f(x,f(y,f(x,x)))),x) = f(x,f(f(x,x),x)).
2760
f(f(x,f(x,f(y,f(x,x)))),x) = f(x,f(x,f(x,x))).
2761
f(f(x,f(x,f(y,f(x,x)))),x) = f(x,x).
2762
f(f(f(x,x),f(f(x,x),f(x,x))),f(x,x)) = x.
2763
f(f(x,x),f(x,x)) = x.
2764
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
2765
f(f(x,f(y,y)),f(x,y)) = x.
2766
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
2767
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
2768
f(f(x,y),f(x,f(y,y))) = x.
2769
f(f(f(x,f(x,f(y,f(x,x)))),x),y) = f(y,f(x,x)).
2770
f(f(x,x),y) = f(y,f(x,x)).
2771
f(x,f(f(x,x),f(f(x,x),f(x,x)))) = f(x,x).
2772
f(x,f(f(x,x),x)) = f(x,x).
2773
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
2774
f(x,y) = f(y,f(f(x,x),f(x,f(f(x,x),x)))).
2775
f(x,y) = f(y,f(f(x,x),f(x,x))).
2776
f(x,y) = f(y,x).
2777
$F # answer("Commutativity").
2778
end_of_list.
2779
2780
Attempt to eliminate one assumption:
2781
2782
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").
2783
2784
***********************************************
2785
AUTO SKETCHES ITERATION 10
2786
***********************************************
2787
2788
Starting a search with 8 assumptions:
2789
2790
formulas(assumptions_active).
2791
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2792
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2793
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2794
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2795
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2796
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2797
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2798
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2799
end_of_list.
2800
2801
Child search process 22004 started.
2802
2803
============================== PROCESS INITIAL CLAUSES ===============
2804
2805
% Clauses before input processing:
2806
2807
formulas(usable).
2808
end_of_list.
2809
2810
formulas(sos).
2811
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
2812
f(c2,c1) != f(c1,c2) # label("Commutativity"). [deny(1)].
2813
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2814
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2815
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2816
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2817
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2818
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2819
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2820
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2821
end_of_list.
2822
2823
formulas(demodulators).
2824
end_of_list.
2825
2826
% 678 hints input.
2827
2828
============================== PREDICATE ELIMINATION =================
2829
2830
No predicates eliminated.
2831
2832
============================== end predicate elimination =============
2833
2834
Auto_denials:
2835
% copying label "Commutativity" to answer in negative clause
2836
2837
Term ordering decisions:
2838
Function symbol KB weights: c1=1. c2=1. f=1.
2839
Predicate symbol precedence: predicate_order([ = ]).
2840
Function symbol precedence: function_order([ c1, c2, f ]).
2841
Skipping inverse_order, because term ordering is KBO.
2842
Unfolding symbols: (none).
2843
2844
Auto_inference settings:
2845
% set(paramodulation). % (positive equality literals)
2846
% set(paramodulation) -> set(back_demod).
2847
2848
Auto_process settings: (no changes).
2849
2850
============================== end of process initial clauses ========
2851
2852
============================== CLAUSES FOR SEARCH ====================
2853
2854
% Clauses after input processing:
2855
2856
formulas(usable).
2857
end_of_list.
2858
2859
formulas(sos).
2860
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
2861
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
2862
4 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2863
5 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2864
6 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2865
7 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2866
8 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2867
9 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2868
10 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2869
11 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2870
end_of_list.
2871
2872
formulas(demodulators).
2873
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
2874
4 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2875
5 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2876
6 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
2877
7 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2878
8 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2879
9 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2880
10 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2881
11 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2882
end_of_list.
2883
2884
% 678 hints processed (374 are redundant).
2885
2886
============================== end of clauses for search =============
2887
2888
============================== SEARCH ================================
2889
2890
% Starting search at 0.03 seconds.
2891
2892
% Operation f is commutative; C redundancy checks enabled.
2893
2894
============================== PROOF =================================
2895
2896
% Proof 1 at 0.87 (+ 0.00) seconds: "Commutativity".
2897
% Length of proof is 54.
2898
% Level of proof is 19.
2899
% Maximum clause weight is 63.
2900
% Given clauses 38.
2901
2902
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
2903
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
2904
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
2905
4 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2906
5 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2907
7 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2908
8 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2909
9 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2910
10 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2911
11 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2912
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))].
2913
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))].
2914
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))].
2915
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))].
2916
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))].
2917
461 f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x. [para(377(a,1),8(a,1,1))].
2918
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)])].
2919
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)].
2920
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)])].
2921
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)].
2922
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)].
2923
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)].
2924
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))].
2925
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)])].
2926
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)].
2927
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)])].
2928
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)])].
2929
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)].
2930
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)])].
2931
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)])].
2932
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)].
2933
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)])].
2934
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)].
2935
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)])].
2936
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)])].
2937
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)])].
2938
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)])].
2939
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))].
2940
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)].
2941
1338 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(1181),rewrite([1316(4),1317(3),1317(6)])].
2942
1403 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [back_rewrite(1251),rewrite([1317(3),1317(5)])].
2943
1424 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(1221),rewrite([1317(3),1317(5)])].
2944
1429 f(f(x,f(x,x)),x) = f(x,x). [back_rewrite(1212),rewrite([1424(2),1338(3),1424(5),1338(6)])].
2945
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)])].
2946
1484 f(f(x,x),f(x,f(x,y))) = x. [back_rewrite(461),rewrite([1424(2),1338(3)])].
2947
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))].
2948
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)])].
2949
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)])].
2950
1613 f(f(x,f(y,y)),f(x,y)) = x. [back_rewrite(1496),rewrite([1583(6)])].
2951
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)])].
2952
1645 f(f(x,x),y) = f(y,f(x,x)). [para(1613(a,1),5(a,1,2)),rewrite([1570(5)])].
2953
1807 f(x,f(f(x,x),x)) = f(x,x). [para(1432(a,1),1634(a,1,1)),rewrite([1432(4)])].
2954
1830 f(x,y) = f(y,x). [para(2(a,1),1645(a,1,1)),rewrite([1807(4),1807(5),1432(4)])].
2955
1831 $F # answer("Commutativity"). [resolve(1830,a,3,a)].
2956
2957
============================== end of proof ==========================
2958
2959
============================== STATISTICS ============================
2960
2961
Given=38. Generated=3590. Kept=1829. proofs=1.
2962
Usable=30. Sos=1349. Demods=1331. Limbo=9, Disabled=450. Hints=678.
2963
Weight_deleted=59. Literals_deleted=0.
2964
Forward_subsumed=1702. Back_subsumed=55.
2965
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
2966
New_demodulators=1752 (0 lex), Back_demodulated=385. Back_unit_deleted=0.
2967
Demod_attempts=97686. Demod_rewrites=6444.
2968
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
2969
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
2970
Megabytes=3.47.
2971
User_CPU=0.87, System_CPU=0.00, Wall_clock=8.
2972
2973
============================== end of statistics =====================
2974
2975
============================== end of search =========================
2976
2977
Exiting with 1 proof.
2978
2979
Process 22004 exit (max_proofs) Mon Jun 4 22:53:33 2007
2980
2981
Successful proof using the following 7 assumptions:
2982
2983
formulas(assumptions_in_proof).
2984
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m"). [assumption].
2985
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
2986
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
2987
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
2988
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
2989
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
2990
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
2991
end_of_list.
2992
2993
Including 125 new hint clauses:
2994
2995
formulas(new_hints).
2996
$F # label("Commutativity") # label(non_clause) # label(goal).
2997
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
2998
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
2999
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
3000
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
3001
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
3002
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
3003
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
3004
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
3005
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
3006
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
3007
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
3008
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)).
3009
f(f(f(x,y),y),y) = f(x,f(f(y,x),x)).
3010
f(f(f(x,f(f(y,x),x)),y),f(y,f(f(x,y),y))) = y.
3011
f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x.
3012
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))).
3013
f(x,f(f(f(y,x),x),x)) = f(x,f(y,f(f(x,y),y))).
3014
f(x,f(x,f(f(x,x),x))) = f(x,f(f(f(x,f(f(x,x),x)),x),x)).
3015
f(x,f(f(f(x,f(f(x,x),x)),x),x)) = f(x,f(x,f(f(x,x),x))).
3016
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)).
3017
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)).
3018
f(x,f(f(x,f(f(x,x),x)),x)) = f(x,f(f(x,x),x)).
3019
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)))).
3020
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)))).
3021
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)))).
3022
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)))).
3023
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).
3024
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)))).
3025
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).
3026
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.
3027
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))))).
3028
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))))).
3029
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))))).
3030
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))))).
3031
f(f(f(f(x,f(f(x,x),x)),x),x),x) = f(f(x,f(f(x,x),x)),x).
3032
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))).
3033
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))).
3034
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))).
3035
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).
3036
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)).
3037
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)).
3038
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)).
3039
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.
3040
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))).
3041
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))).
3042
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))).
3043
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))).
3044
x = f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))).
3045
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
3046
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))).
3047
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))).
3048
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.
3049
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))).
3050
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))).
3051
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))).
3052
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))).
3053
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))).
3054
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))).
3055
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))).
3056
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))).
3057
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))).
3058
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))).
3059
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))).
3060
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))).
3061
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).
3062
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)).
3063
f(x,f(f(f(f(x,x),x),x),x)) = f(x,f(f(x,x),x)).
3064
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))).
3065
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))).
3066
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))).
3067
f(x,f(f(x,x),x)) = f(f(f(x,f(f(x,x),x)),x),x).
3068
f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)).
3069
f(f(f(x,f(f(x,x),x)),x),f(x,f(y,f(x,f(f(x,x),x))))) = x.
3070
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)).
3071
f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
3072
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))).
3073
f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))).
3074
f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x.
3075
f(f(f(f(x,x),x),x),x) = f(x,f(x,f(f(x,x),x))).
3076
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)).
3077
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)).
3078
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)).
3079
f(x,x) = f(x,f(f(x,x),x)).
3080
f(x,f(f(x,x),x)) = f(x,x).
3081
f(x,f(x,f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
3082
f(x,f(x,f(x,x))) = f(x,f(f(x,x),x)).
3083
f(x,f(x,f(x,x))) = f(x,x).
3084
f(f(x,f(x,x)),f(x,f(y,f(x,f(f(x,x),x))))) = x.
3085
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
3086
f(f(x,x),x) = f(x,f(x,f(f(x,x),x))).
3087
f(f(x,x),x) = f(x,f(x,x)).
3088
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(f(x,x),x)).
3089
f(f(x,f(x,x)),x) = f(x,f(f(x,x),x)).
3090
f(f(x,f(x,x)),x) = f(x,f(x,f(x,x))).
3091
f(f(x,f(x,x)),x) = f(x,x).
3092
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.
3093
f(f(f(f(x,x),f(x,x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
3094
f(f(f(f(x,x),f(x,x)),f(x,f(x,f(x,x)))),f(x,f(f(x,x),x))) = x.
3095
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,f(f(x,x),x))) = x.
3096
f(f(f(x,x),f(f(x,x),f(x,x))),f(x,f(f(x,x),x))) = x.
3097
f(f(f(x,x),f(f(x,x),f(x,x))),f(x,f(x,f(x,x)))) = x.
3098
f(f(f(x,x),f(f(x,x),f(x,x))),f(x,x)) = x.
3099
f(f(x,x),f(x,x)) = x.
3100
f(f(x,f(x,f(x,x))),f(x,f(x,y))) = x.
3101
f(f(x,x),f(x,f(x,y))) = x.
3102
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
3103
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).
3104
f(f(x,f(x,f(y,f(x,x)))),x) = f(x,x).
3105
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)).
3106
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
3107
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
3108
f(f(x,f(y,y)),f(x,y)) = x.
3109
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
3110
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
3111
f(f(x,y),f(x,f(y,y))) = x.
3112
f(f(f(x,f(x,f(y,f(x,x)))),x),y) = f(y,f(x,x)).
3113
f(f(x,x),y) = f(y,f(x,x)).
3114
f(x,f(f(x,x),f(f(x,x),f(x,x)))) = f(x,x).
3115
f(x,f(f(x,x),x)) = f(x,x).
3116
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
3117
f(x,y) = f(y,f(f(x,x),f(x,f(f(x,x),x)))).
3118
f(x,y) = f(y,f(f(x,x),f(x,x))).
3119
f(x,y) = f(y,x).
3120
$F # answer("Commutativity").
3121
end_of_list.
3122
3123
Attempt to eliminate one assumption:
3124
3125
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
3126
3127
***********************************************
3128
AUTO SKETCHES ITERATION 11
3129
***********************************************
3130
3131
Starting a search with 7 assumptions:
3132
3133
formulas(assumptions_active).
3134
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
3135
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3136
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3137
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3138
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3139
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3140
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3141
end_of_list.
3142
3143
Child search process 22007 started.
3144
3145
============================== PROCESS INITIAL CLAUSES ===============
3146
3147
% Clauses before input processing:
3148
3149
formulas(usable).
3150
end_of_list.
3151
3152
formulas(sos).
3153
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3154
f(c2,c1) != f(c1,c2) # label("Commutativity"). [deny(1)].
3155
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
3156
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3157
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3158
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3159
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3160
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3161
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3162
end_of_list.
3163
3164
formulas(demodulators).
3165
end_of_list.
3166
3167
% 803 hints input.
3168
3169
============================== PREDICATE ELIMINATION =================
3170
3171
No predicates eliminated.
3172
3173
============================== end predicate elimination =============
3174
3175
Auto_denials:
3176
% copying label "Commutativity" to answer in negative clause
3177
3178
Term ordering decisions:
3179
Function symbol KB weights: c1=1. c2=1. f=1.
3180
Predicate symbol precedence: predicate_order([ = ]).
3181
Function symbol precedence: function_order([ c1, c2, f ]).
3182
Skipping inverse_order, because term ordering is KBO.
3183
Unfolding symbols: (none).
3184
3185
Auto_inference settings:
3186
% set(paramodulation). % (positive equality literals)
3187
% set(paramodulation) -> set(back_demod).
3188
3189
Auto_process settings: (no changes).
3190
3191
============================== end of process initial clauses ========
3192
3193
============================== CLAUSES FOR SEARCH ====================
3194
3195
% Clauses after input processing:
3196
3197
formulas(usable).
3198
end_of_list.
3199
3200
formulas(sos).
3201
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3202
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
3203
4 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
3204
5 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3205
6 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3206
7 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3207
8 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3208
9 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3209
10 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3210
end_of_list.
3211
3212
formulas(demodulators).
3213
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3214
4 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
3215
5 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3216
6 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3217
7 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3218
8 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3219
9 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3220
10 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3221
end_of_list.
3222
3223
% 803 hints processed (449 are redundant).
3224
3225
============================== end of clauses for search =============
3226
3227
============================== SEARCH ================================
3228
3229
% Starting search at 0.04 seconds.
3230
3231
% Operation f is commutative; C redundancy checks enabled.
3232
3233
============================== PROOF =================================
3234
3235
% Proof 1 at 0.47 (+ 0.00) seconds: "Commutativity".
3236
% Length of proof is 32.
3237
% Level of proof is 15.
3238
% Maximum clause weight is 25.
3239
% Given clauses 34.
3240
3241
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
3242
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3243
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
3244
4 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
3245
5 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3246
6 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3247
7 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3248
9 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3249
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))].
3250
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))].
3251
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))].
3252
375 f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x. [para(305(a,1),7(a,1,1))].
3253
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)])].
3254
549 f(x,f(f(x,x),x)) = f(x,x). [para(4(a,1),454(a,1,1)),flip(a)].
3255
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)])].
3256
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)])].
3257
637 f(f(x,x),f(x,f(y,x))) = x. [para(549(a,1),2(a,1,1))].
3258
639 f(f(x,x),f(x,x)) = x. [para(549(a,1),2(a,1,2)),rewrite([549(3)])].
3259
646 f(f(x,f(x,x)),f(x,x)) = x. [para(549(a,1),6(a,1,2)),rewrite([639(3)])].
3260
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))].
3261
667 f(f(x,f(x,x)),x) = f(x,x). [para(639(a,1),2(a,1,2)),rewrite([582(3)])].
3262
669 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [para(639(a,1),5(a,1,1,1))].
3263
737 f(f(x,x),x) = f(x,f(x,x)). [para(646(a,1),2(a,1,2)),rewrite([667(3),549(3)])].
3264
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)])].
3265
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)])].
3266
940 f(f(x,f(y,y)),f(x,y)) = x. [back_rewrite(666),rewrite([920(6)])].
3267
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))].
3268
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)])].
3269
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)])].
3270
1187 f(f(x,x),y) = f(y,f(x,x)). [back_rewrite(961),rewrite([1158(3),667(3)])].
3271
1199 f(x,y) = f(y,x). [para(2(a,1),1187(a,1,1)),rewrite([2(8)])].
3272
1200 $F # answer("Commutativity"). [resolve(1199,a,3,a)].
3273
3274
============================== end of proof ==========================
3275
3276
============================== STATISTICS ============================
3277
3278
Given=34. Generated=2340. Kept=1198. proofs=1.
3279
Usable=26. Sos=890. Demods=883. Limbo=9, Disabled=281. Hints=803.
3280
Weight_deleted=0. Literals_deleted=0.
3281
Forward_subsumed=1142. Back_subsumed=54.
3282
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
3283
New_demodulators=1135 (0 lex), Back_demodulated=218. Back_unit_deleted=0.
3284
Demod_attempts=47705. Demod_rewrites=3591.
3285
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
3286
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
3287
Megabytes=2.22.
3288
User_CPU=0.47, System_CPU=0.00, Wall_clock=8.
3289
3290
============================== end of statistics =====================
3291
3292
============================== end of search =========================
3293
3294
Exiting with 1 proof.
3295
3296
Process 22007 exit (max_proofs) Mon Jun 4 22:53:33 2007
3297
3298
Successful proof using the following 5 assumptions:
3299
3300
formulas(assumptions_in_proof).
3301
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m"). [assumption].
3302
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3303
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3304
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3305
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3306
end_of_list.
3307
3308
Including 52 new hint clauses:
3309
3310
formulas(new_hints).
3311
$F # label("Commutativity") # label(non_clause) # label(goal).
3312
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
3313
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
3314
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
3315
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").
3316
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
3317
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
3318
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
3319
f(f(f(x,y),x),f(f(y,f(f(y,y),y)),f(z,x))) = f(y,f(f(y,y),y)).
3320
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
3321
f(f(f(x,y),y),y) = f(x,f(f(y,x),x)).
3322
f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x.
3323
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)).
3324
f(f(f(f(x,y),x),f(x,y)),x) = f(x,f(f(x,x),x)).
3325
f(x,x) = f(x,f(f(x,x),x)).
3326
f(x,f(f(x,x),x)) = f(x,x).
3327
f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)).
3328
f(f(f(x,x),x),x) = f(x,f(f(x,x),x)).
3329
f(f(f(x,x),x),x) = f(x,x).
3330
f(f(f(x,y),x),f(f(y,y),f(z,x))) = f(y,f(f(y,y),y)).
3331
f(f(f(x,y),x),f(f(y,y),f(z,x))) = f(y,y).
3332
f(f(x,x),f(x,f(y,x))) = x.
3333
f(f(x,f(f(x,x),x)),f(x,x)) = x.
3334
f(f(x,x),f(x,x)) = x.
3335
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,x)) = x.
3336
f(f(x,f(x,x)),f(x,x)) = x.
3337
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
3338
f(f(x,f(f(f(x,x),x),x)),x) = f(x,x).
3339
f(f(x,f(x,x)),x) = f(x,x).
3340
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
3341
f(f(x,f(f(f(x,f(x,x)),x),x)),x) = f(x,f(x,x)).
3342
f(f(x,f(f(x,x),x)),x) = f(x,f(x,x)).
3343
f(f(x,x),x) = f(x,f(x,x)).
3344
f(f(f(x,x),f(x,x)),f(f(x,x),x)) = f(x,x).
3345
f(x,f(f(x,x),x)) = f(x,x).
3346
f(x,f(x,f(x,x))) = f(x,x).
3347
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)).
3348
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
3349
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
3350
f(f(x,f(y,y)),f(x,y)) = x.
3351
f(f(f(x,f(x,f(y,f(x,x)))),x),y) = f(y,f(x,x)).
3352
f(f(f(f(x,x),y),y),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
3353
f(f(f(f(x,x),y),y),y) = f(y,f(x,x)).
3354
f(f(f(f(x,x),x),f(x,x)),f(f(f(f(x,x),y),y),y)) = f(x,x).
3355
f(f(f(x,f(x,x)),f(x,x)),f(f(f(f(x,x),y),y),y)) = f(x,x).
3356
f(x,f(f(f(f(x,x),y),y),y)) = f(x,x).
3357
f(x,f(y,f(x,x))) = f(x,x).
3358
f(f(f(x,f(x,x)),x),y) = f(y,f(x,x)).
3359
f(f(x,x),y) = f(y,f(x,x)).
3360
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
3361
f(x,y) = f(y,x).
3362
$F # answer("Commutativity").
3363
end_of_list.
3364
3365
Attempt to eliminate one assumption:
3366
3367
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
3368
3369
***********************************************
3370
AUTO SKETCHES ITERATION 12
3371
***********************************************
3372
3373
Starting a search with 6 assumptions:
3374
3375
formulas(assumptions_active).
3376
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3377
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3378
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3379
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3380
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3381
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3382
end_of_list.
3383
3384
Child search process 22008 started.
3385
3386
============================== PROCESS INITIAL CLAUSES ===============
3387
3388
% Clauses before input processing:
3389
3390
formulas(usable).
3391
end_of_list.
3392
3393
formulas(sos).
3394
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3395
f(c2,c1) != f(c1,c2) # label("Commutativity"). [deny(1)].
3396
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3397
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3398
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3399
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3400
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3401
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3402
end_of_list.
3403
3404
formulas(demodulators).
3405
end_of_list.
3406
3407
% 855 hints input.
3408
3409
============================== PREDICATE ELIMINATION =================
3410
3411
No predicates eliminated.
3412
3413
============================== end predicate elimination =============
3414
3415
Auto_denials:
3416
% copying label "Commutativity" to answer in negative clause
3417
3418
Term ordering decisions:
3419
Function symbol KB weights: c1=1. c2=1. f=1.
3420
Predicate symbol precedence: predicate_order([ = ]).
3421
Function symbol precedence: function_order([ c1, c2, f ]).
3422
Skipping inverse_order, because term ordering is KBO.
3423
Unfolding symbols: (none).
3424
3425
Auto_inference settings:
3426
% set(paramodulation). % (positive equality literals)
3427
% set(paramodulation) -> set(back_demod).
3428
3429
Auto_process settings: (no changes).
3430
3431
============================== end of process initial clauses ========
3432
3433
============================== CLAUSES FOR SEARCH ====================
3434
3435
% Clauses after input processing:
3436
3437
formulas(usable).
3438
end_of_list.
3439
3440
formulas(sos).
3441
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3442
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
3443
4 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3444
5 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3445
6 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3446
7 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3447
8 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3448
9 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3449
end_of_list.
3450
3451
formulas(demodulators).
3452
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3453
4 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3454
5 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3455
6 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3456
7 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3457
8 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3458
9 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3459
end_of_list.
3460
3461
% 855 hints processed (479 are redundant).
3462
3463
============================== end of clauses for search =============
3464
3465
============================== SEARCH ================================
3466
3467
% Starting search at 0.04 seconds.
3468
3469
% Operation f is commutative; C redundancy checks enabled.
3470
3471
============================== PROOF =================================
3472
3473
% Proof 1 at 0.50 (+ 0.01) seconds: "Commutativity".
3474
% Length of proof is 44.
3475
% Level of proof is 18.
3476
% Maximum clause weight is 39.
3477
% Given clauses 34.
3478
3479
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
3480
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3481
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
3482
4 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3483
5 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3484
6 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3485
8 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3486
9 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3487
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))].
3488
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))].
3489
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))].
3490
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))].
3491
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))].
3492
366 f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x. [para(302(a,1),6(a,1,1))].
3493
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))].
3494
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)])].
3495
547 f(x,f(f(x,x),x)) = f(x,x). [para(4(a,1),450(a,1,1)),flip(a)].
3496
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)])].
3497
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)])].
3498
694 f(f(x,x),f(x,f(y,x))) = x. [para(547(a,1),2(a,1,1))].
3499
696 f(f(x,x),f(x,x)) = x. [para(547(a,1),2(a,1,2)),rewrite([547(3)])].
3500
701 f(f(x,f(x,x)),f(x,x)) = x. [para(547(a,1),5(a,1,2)),rewrite([696(3)])].
3501
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))].
3502
720 f(f(x,f(x,x)),x) = f(x,x). [para(696(a,1),2(a,1,2)),rewrite([580(3)])].
3503
721 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [para(696(a,1),4(a,1,1,1))].
3504
783 f(f(x,x),x) = f(x,f(x,x)). [para(701(a,1),2(a,1,2)),rewrite([720(3),547(3)])].
3505
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)])].
3506
835 f(x,f(f(f(f(x,x),y),y),y)) = f(x,x). [back_rewrite(637),rewrite([783(2),701(4)])].
3507
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)])].
3508
966 f(f(x,f(y,y)),f(x,y)) = x. [back_rewrite(719),rewrite([946(6)])].
3509
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)])].
3510
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))].
3511
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))].
3512
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))].
3513
1071 f(f(f(x,f(x,f(x,y))),x),x) = f(x,y). [para(979(a,1),9(a,1,2))].
3514
1148 f(x,f(y,f(x,x))) = f(x,x). [para(302(a,1),835(a,1,2)),rewrite([946(6)])].
3515
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)])].
3516
1183 f(f(f(x,y),x),x) = f(x,y). [back_rewrite(1066),rewrite([1177(3)])].
3517
1184 f(f(f(x,y),y),y) = f(y,x). [back_rewrite(1052),rewrite([1177(3)])].
3518
1189 f(x,f(x,f(x,y))) = f(x,y). [back_rewrite(1071),rewrite([1183(5)])].
3519
1190 f(f(x,y),x) = f(x,f(x,y)). [back_rewrite(1059),rewrite([1183(4),1184(6),1183(6)])].
3520
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)])].
3521
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)])].
3522
1254 $F # answer("Commutativity"). [resolve(1253,a,3,a)].
3523
3524
============================== end of proof ==========================
3525
3526
============================== STATISTICS ============================
3527
3528
Given=34. Generated=2426. Kept=1252. proofs=1.
3529
Usable=23. Sos=727. Demods=786. Limbo=70, Disabled=439. Hints=855.
3530
Weight_deleted=2. Literals_deleted=0.
3531
Forward_subsumed=1172. Back_subsumed=57.
3532
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
3533
New_demodulators=1189 (0 lex), Back_demodulated=374. Back_unit_deleted=0.
3534
Demod_attempts=52607. Demod_rewrites=4432.
3535
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
3536
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
3537
Megabytes=2.23.
3538
User_CPU=0.50, System_CPU=0.01, Wall_clock=9.
3539
3540
============================== end of statistics =====================
3541
3542
============================== end of search =========================
3543
3544
Exiting with 1 proof.
3545
3546
Process 22008 exit (max_proofs) Mon Jun 4 22:53:34 2007
3547
3548
Successful proof using the following 5 assumptions:
3549
3550
formulas(assumptions_in_proof).
3551
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m"). [assumption].
3552
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3553
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_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
Including 86 new hint clauses:
3559
3560
formulas(new_hints).
3561
$F # label("Commutativity") # label(non_clause) # label(goal).
3562
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
3563
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
3564
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").
3565
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
3566
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
3567
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
3568
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
3569
f(f(f(x,y),x),f(f(y,f(f(y,y),y)),f(z,x))) = f(y,f(f(y,y),y)).
3570
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
3571
f(f(f(x,f(f(y,z),f(f(f(y,x),y),y))),x),x) = f(f(f(y,x),y),y).
3572
f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(y,f(y,x)),y).
3573
f(f(f(x,y),y),y) = f(x,f(f(y,x),x)).
3574
f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x.
3575
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)).
3576
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)).
3577
f(f(f(f(x,y),x),f(x,y)),x) = f(x,f(f(x,x),x)).
3578
f(x,x) = f(x,f(f(x,x),x)).
3579
f(x,f(f(x,x),x)) = f(x,x).
3580
f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)).
3581
f(f(f(x,x),x),x) = f(x,f(f(x,x),x)).
3582
f(f(f(x,x),x),x) = f(x,x).
3583
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)).
3584
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)).
3585
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)).
3586
f(f(f(f(x,x),x),f(x,x)),f(f(f(f(x,x),y),y),y)) = f(x,x).
3587
f(f(x,x),f(x,f(y,x))) = x.
3588
f(f(x,f(f(x,x),x)),f(x,x)) = x.
3589
f(f(x,x),f(x,x)) = x.
3590
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,x)) = x.
3591
f(f(x,f(x,x)),f(x,x)) = x.
3592
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
3593
f(f(x,f(f(f(x,x),x),x)),x) = f(x,x).
3594
f(f(x,f(x,x)),x) = f(x,x).
3595
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
3596
f(f(x,f(f(f(x,f(x,x)),x),x)),x) = f(x,f(x,x)).
3597
f(f(x,f(f(x,x),x)),x) = f(x,f(x,x)).
3598
f(f(x,x),x) = f(x,f(x,x)).
3599
f(f(f(x,x),f(x,x)),f(f(x,x),x)) = f(x,x).
3600
f(x,f(f(x,x),x)) = f(x,x).
3601
f(x,f(x,f(x,x))) = f(x,x).
3602
f(f(f(x,f(x,x)),f(x,x)),f(f(f(f(x,x),y),y),y)) = f(x,x).
3603
f(x,f(f(f(f(x,x),y),y),y)) = f(x,x).
3604
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)).
3605
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
3606
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
3607
f(f(x,f(y,y)),f(x,y)) = x.
3608
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
3609
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
3610
f(f(x,y),f(x,f(y,y))) = x.
3611
f(f(f(f(f(x,x),f(y,x)),y),y),y) = f(y,x).
3612
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).
3613
f(f(f(x,f(f(y,y),f(x,y))),x),x) = f(x,y).
3614
f(f(f(x,f(x,f(x,y))),x),x) = f(x,y).
3615
f(x,f(f(x,x),f(f(y,f(x,x)),f(x,x)))) = f(x,x).
3616
f(x,f(y,f(x,x))) = f(x,x).
3617
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))).
3618
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))).
3619
f(f(x,x),f(y,x)) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
3620
f(f(x,x),f(y,x)) = f(f(x,f(x,f(x,x))),f(x,f(f(x,x),x))).
3621
f(f(x,x),f(y,x)) = f(f(x,x),f(x,f(f(x,x),x))).
3622
f(f(x,x),f(y,x)) = f(f(x,x),f(x,f(x,f(x,x)))).
3623
f(f(x,x),f(y,x)) = f(f(x,x),f(x,x)).
3624
f(f(x,x),f(y,x)) = x.
3625
f(f(f(x,y),x),x) = f(x,y).
3626
f(f(f(x,y),y),y) = f(y,x).
3627
f(x,f(x,f(x,y))) = f(x,y).
3628
f(f(f(x,f(x,y)),f(x,y)),f(x,y)) = f(f(f(x,f(x,y)),x),x).
3629
f(f(x,y),x) = f(f(f(x,f(x,y)),x),x).
3630
f(f(x,y),x) = f(x,f(x,y)).
3631
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(f(y,f(y,x)),y).
3632
f(f(f(x,f(f(y,z),f(y,x))),x),x) = f(f(y,f(y,x)),y).
3633
f(f(x,f(x,f(f(y,z),f(y,x)))),x) = f(f(y,f(y,x)),y).
3634
f(x,f(x,f(x,f(f(y,z),f(y,x))))) = f(f(y,f(y,x)),y).
3635
f(x,f(f(y,z),f(y,x))) = f(f(y,f(y,x)),y).
3636
f(x,f(f(y,z),f(y,x))) = f(y,f(y,f(y,x))).
3637
f(x,f(f(y,z),f(y,x))) = f(y,x).
3638
f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(f(y,x),y),y).
3639
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(f(f(y,x),y),y).
3640
f(f(f(x,f(f(y,z),f(y,x))),x),x) = f(f(f(y,x),y),y).
3641
f(f(f(y,x),x),x) = f(f(f(y,x),y),y).
3642
f(x,y) = f(f(f(y,x),y),y).
3643
f(x,y) = f(f(y,f(y,x)),y).
3644
f(x,y) = f(y,f(y,f(y,x))).
3645
f(x,y) = f(y,x).
3646
$F # answer("Commutativity").
3647
end_of_list.
3648
3649
Attempt to eliminate one assumption:
3650
3651
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").
3652
3653
***********************************************
3654
AUTO SKETCHES ITERATION 13
3655
***********************************************
3656
3657
Starting a search with 5 assumptions:
3658
3659
formulas(assumptions_active).
3660
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3661
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3662
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3663
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3664
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3665
end_of_list.
3666
3667
Child search process 22009 started.
3668
3669
============================== PROCESS INITIAL CLAUSES ===============
3670
3671
% Clauses before input processing:
3672
3673
formulas(usable).
3674
end_of_list.
3675
3676
formulas(sos).
3677
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3678
f(c2,c1) != f(c1,c2) # label("Commutativity"). [deny(1)].
3679
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3680
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3681
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3682
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3683
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3684
end_of_list.
3685
3686
formulas(demodulators).
3687
end_of_list.
3688
3689
% 941 hints input.
3690
3691
============================== PREDICATE ELIMINATION =================
3692
3693
No predicates eliminated.
3694
3695
============================== end predicate elimination =============
3696
3697
Auto_denials:
3698
% copying label "Commutativity" to answer in negative clause
3699
3700
Term ordering decisions:
3701
Function symbol KB weights: c1=1. c2=1. f=1.
3702
Predicate symbol precedence: predicate_order([ = ]).
3703
Function symbol precedence: function_order([ c1, c2, f ]).
3704
Skipping inverse_order, because term ordering is KBO.
3705
Unfolding symbols: (none).
3706
3707
Auto_inference settings:
3708
% set(paramodulation). % (positive equality literals)
3709
% set(paramodulation) -> set(back_demod).
3710
3711
Auto_process settings: (no changes).
3712
3713
============================== end of process initial clauses ========
3714
3715
============================== CLAUSES FOR SEARCH ====================
3716
3717
% Clauses after input processing:
3718
3719
formulas(usable).
3720
end_of_list.
3721
3722
formulas(sos).
3723
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3724
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
3725
4 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3726
5 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3727
6 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3728
7 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3729
8 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3730
end_of_list.
3731
3732
formulas(demodulators).
3733
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3734
4 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3735
5 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3736
6 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3737
7 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3738
8 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3739
end_of_list.
3740
3741
% 941 hints processed (525 are redundant).
3742
3743
============================== end of clauses for search =============
3744
3745
============================== SEARCH ================================
3746
3747
% Starting search at 0.05 seconds.
3748
3749
% Operation f is commutative; C redundancy checks enabled.
3750
3751
============================== PROOF =================================
3752
3753
% Proof 1 at 0.67 (+ 0.00) seconds: "Commutativity".
3754
% Length of proof is 39.
3755
% Level of proof is 15.
3756
% Maximum clause weight is 37.
3757
% Given clauses 33.
3758
3759
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
3760
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3761
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
3762
4 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m"). [assumption].
3763
5 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3764
6 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3765
7 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3766
8 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3767
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))].
3768
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))].
3769
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))].
3770
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))].
3771
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))].
3772
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))].
3773
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)])].
3774
460 f(x,f(f(x,x),x)) = f(x,x). [para(127(a,1),9(a,1,2)),rewrite([127(8)]),flip(a)].
3775
472 f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)). [back_rewrite(452),rewrite([460(3),460(7)])].
3776
476 f(f(x,x),f(x,f(y,x))) = x. [para(460(a,1),2(a,1,1))].
3777
478 f(f(x,x),f(x,x)) = x. [para(460(a,1),2(a,1,2)),rewrite([460(3)])].
3778
482 f(f(x,f(x,x)),f(x,x)) = x. [para(460(a,1),4(a,1,2)),rewrite([478(3)])].
3779
486 f(f(f(x,x),x),x) = f(x,x). [para(460(a,1),6(a,1,1,1)),rewrite([478(5)])].
3780
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))].
3781
512 f(f(x,f(x,x)),x) = f(x,x). [para(478(a,1),2(a,1,2)),rewrite([486(3)])].
3782
535 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(472),rewrite([512(3)])].
3783
556 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(460),rewrite([535(2)])].
3784
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))].
3785
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)])].
3786
621 f(f(x,x),f(x,f(x,y))) = x. [para(512(a,1),8(a,1,1))].
3787
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)])].
3788
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)])].
3789
1037 f(f(x,f(y,y)),f(x,y)) = x. [back_rewrite(511),rewrite([1009(6)])].
3790
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)])].
3791
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))].
3792
1263 f(x,f(y,f(x,x))) = f(x,x). [para(574(a,1),970(a,1,2)),rewrite([1009(6)])].
3793
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)])].
3794
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)])].
3795
1311 f(f(f(x,y),x),x) = f(x,y). [back_rewrite(1175),rewrite([1299(3)])].
3796
1320 f(x,y) = f(y,x). [back_rewrite(1305),rewrite([1311(3),1311(4)])].
3797
1321 $F # answer("Commutativity"). [resolve(1320,a,3,a)].
3798
3799
============================== end of proof ==========================
3800
3801
============================== STATISTICS ============================
3802
3803
Given=33. Generated=2831. Kept=1319. proofs=1.
3804
Usable=27. Sos=982. Demods=991. Limbo=9, Disabled=307. Hints=941.
3805
Weight_deleted=93. Literals_deleted=0.
3806
Forward_subsumed=1419. Back_subsumed=64.
3807
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
3808
New_demodulators=1279 (0 lex), Back_demodulated=236. Back_unit_deleted=0.
3809
Demod_attempts=81940. Demod_rewrites=6087.
3810
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
3811
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
3812
Megabytes=2.93.
3813
User_CPU=0.67, System_CPU=0.00, Wall_clock=9.
3814
3815
============================== end of statistics =====================
3816
3817
============================== end of search =========================
3818
3819
Exiting with 1 proof.
3820
3821
Process 22009 exit (max_proofs) Mon Jun 4 22:53:34 2007
3822
3823
Successful proof using the following 5 assumptions:
3824
3825
formulas(assumptions_in_proof).
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
Including 75 new hint clauses:
3834
3835
formulas(new_hints).
3836
$F # label("Commutativity") # label(non_clause) # label(goal).
3837
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
3838
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
3839
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
3840
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
3841
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
3842
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
3843
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
3844
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)).
3845
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
3846
f(f(f(x,f(f(y,z),f(f(f(y,x),y),y))),x),x) = f(f(f(y,x),y),y).
3847
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
3848
f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y.
3849
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))).
3850
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)))).
3851
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)))).
3852
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)))).
3853
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)))).
3854
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)))).
3855
f(f(f(x,f(x,f(f(x,x),x))),x),x) = f(x,f(x,f(f(x,x),x))).
3856
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)).
3857
f(x,x) = f(x,f(f(x,x),x)).
3858
f(x,f(f(x,x),x)) = f(x,x).
3859
f(f(f(x,f(x,x)),x),x) = f(x,f(x,f(f(x,x),x))).
3860
f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)).
3861
f(f(x,x),f(x,f(y,x))) = x.
3862
f(f(x,f(f(x,x),x)),f(x,x)) = x.
3863
f(f(x,x),f(x,x)) = x.
3864
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,x)) = x.
3865
f(f(x,f(x,x)),f(x,x)) = x.
3866
f(f(f(x,x),x),f(f(x,x),f(x,x))) = f(x,x).
3867
f(f(f(x,x),x),x) = f(x,x).
3868
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
3869
f(f(x,f(f(f(x,x),x),x)),x) = f(x,x).
3870
f(f(x,f(x,x)),x) = f(x,x).
3871
f(f(x,x),x) = f(x,f(x,x)).
3872
f(x,f(x,f(x,x))) = f(x,x).
3873
f(f(f(x,y),y),y) = f(x,f(f(y,x),x)).
3874
f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x.
3875
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
3876
f(f(x,x),f(x,f(x,y))) = x.
3877
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)).
3878
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)).
3879
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)).
3880
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)).
3881
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)).
3882
f(x,f(f(f(f(x,f(f(x,x),x)),y),y),y)) = f(x,f(f(x,x),x)).
3883
f(x,f(f(f(f(x,f(x,f(x,x))),y),y),y)) = f(x,f(f(x,x),x)).
3884
f(x,f(f(f(f(x,x),y),y),y)) = f(x,f(f(x,x),x)).
3885
f(x,f(f(f(f(x,x),y),y),y)) = f(x,f(x,f(x,x))).
3886
f(x,f(f(f(f(x,x),y),y),y)) = f(x,x).
3887
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)).
3888
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
3889
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
3890
f(f(x,f(y,y)),f(x,y)) = x.
3891
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
3892
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
3893
f(f(x,y),f(x,f(y,y))) = x.
3894
f(f(f(x,f(f(y,y),f(x,y))),x),x) = f(x,y).
3895
f(x,f(f(x,x),f(f(y,f(x,x)),f(x,x)))) = f(x,x).
3896
f(x,f(y,f(x,x))) = f(x,x).
3897
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))).
3898
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))).
3899
f(f(x,x),f(y,x)) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
3900
f(f(x,x),f(y,x)) = f(f(x,f(x,f(x,x))),f(x,f(f(x,x),x))).
3901
f(f(x,x),f(y,x)) = f(f(x,x),f(x,f(f(x,x),x))).
3902
f(f(x,x),f(y,x)) = f(f(x,x),f(x,f(x,f(x,x)))).
3903
f(f(x,x),f(y,x)) = f(f(x,x),f(x,x)).
3904
f(f(x,x),f(y,x)) = x.
3905
f(f(f(x,f(f(y,y),f(f(f(y,x),y),y))),x),x) = f(f(f(y,x),y),y).
3906
f(f(f(x,y),x),x) = f(f(f(y,x),y),y).
3907
f(f(f(x,y),x),x) = f(x,y).
3908
f(x,y) = f(f(f(y,x),y),y).
3909
f(x,y) = f(y,x).
3910
$F # answer("Commutativity").
3911
end_of_list.
3912
3913
Attempt to eliminate one assumption:
3914
3915
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
3916
3917
***********************************************
3918
AUTO SKETCHES ITERATION 14
3919
***********************************************
3920
3921
Starting a search with 4 assumptions:
3922
3923
formulas(assumptions_active).
3924
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3925
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3926
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3927
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3928
end_of_list.
3929
3930
Child search process 22010 started.
3931
3932
============================== PROCESS INITIAL CLAUSES ===============
3933
3934
% Clauses before input processing:
3935
3936
formulas(usable).
3937
end_of_list.
3938
3939
formulas(sos).
3940
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3941
f(c2,c1) != f(c1,c2) # label("Commutativity"). [deny(1)].
3942
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3943
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3944
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3945
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3946
end_of_list.
3947
3948
formulas(demodulators).
3949
end_of_list.
3950
3951
% 1016 hints input.
3952
3953
============================== PREDICATE ELIMINATION =================
3954
3955
No predicates eliminated.
3956
3957
============================== end predicate elimination =============
3958
3959
Auto_denials:
3960
% copying label "Commutativity" to answer in negative clause
3961
3962
Term ordering decisions:
3963
Function symbol KB weights: c1=1. c2=1. f=1.
3964
Predicate symbol precedence: predicate_order([ = ]).
3965
Function symbol precedence: function_order([ c1, c2, f ]).
3966
Skipping inverse_order, because term ordering is KBO.
3967
Unfolding symbols: (none).
3968
3969
Auto_inference settings:
3970
% set(paramodulation). % (positive equality literals)
3971
% set(paramodulation) -> set(back_demod).
3972
3973
Auto_process settings: (no changes).
3974
3975
============================== end of process initial clauses ========
3976
3977
============================== CLAUSES FOR SEARCH ====================
3978
3979
% Clauses after input processing:
3980
3981
formulas(usable).
3982
end_of_list.
3983
3984
formulas(sos).
3985
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3986
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
3987
4 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3988
5 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3989
6 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3990
7 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3991
end_of_list.
3992
3993
formulas(demodulators).
3994
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
3995
4 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
3996
5 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
3997
6 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
3998
7 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
3999
end_of_list.
4000
4001
% 1016 hints processed (574 are redundant).
4002
4003
============================== end of clauses for search =============
4004
4005
============================== SEARCH ================================
4006
4007
% Starting search at 0.05 seconds.
4008
4009
% Operation f is commutative; C redundancy checks enabled.
4010
4011
============================== PROOF =================================
4012
4013
% Proof 1 at 0.72 (+ 0.00) seconds: "Commutativity".
4014
% Length of proof is 68.
4015
% Level of proof is 22.
4016
% Maximum clause weight is 55.
4017
% Given clauses 43.
4018
4019
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
4020
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
4021
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
4022
4 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
4023
6 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
4024
7 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4025
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))].
4026
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))].
4027
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))].
4028
71 f(f(f(f(f(f(x,y),x),x),f(f(f(f(x,y),x),x),z)),f(f(f(x,y),x),x)),f(z,y)) = z. [para(4(a,1),7(a,1,2,2))].
4029
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))].
4030
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))].
4031
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))].
4032
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))].
4033
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))].
4034
188 f(f(x,f(f(y,x),x)),f(y,f(f(z,f(z,x)),z))) = y. [para(83(a,1),2(a,1,2,2))].
4035
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)])].
4036
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)])].
4037
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))].
4038
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)])].
4039
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)])].
4040
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))].
4041
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)])].
4042
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))].
4043
384 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)])].
4044
387 f(x,f(f(x,x),x)) = f(x,x). [para(90(a,1),8(a,1,2)),rewrite([90(8)]),flip(a)].
4045
426 f(f(x,f(f(y,x),x)),f(y,y)) = y. [back_rewrite(384),rewrite([387(6)])].
4046
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)])].
4047
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)])].
4048
469 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(313),rewrite([387(3),387(5)])].
4049
489 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(387),rewrite([469(2)])].
4050
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)])].
4051
511 f(f(x,x),f(x,f(y,x))) = x. [para(469(a,1),2(a,1,1,2)),rewrite([489(3)])].
4052
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)])].
4053
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)])].
4054
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)])].
4055
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)].
4056
542 f(f(x,f(y,y)),f(x,y)) = x. [para(530(a,1),2(a,1,2,2)),rewrite([526(6)])].
4057
590 f(f(x,x),f(x,f(x,y))) = x. [para(517(a,1),7(a,1,1))].
4058
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)])].
4059
643 f(f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),f(x,x)),f(f(y,f(y,x)),y)) = f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x). [para(83(a,1),542(a,1,2))].
4060
703 f(f(f(x,f(x,f(x,y))),x),x) = f(x,y). [para(604(a,1),7(a,1,2))].
4061
715 f(f(f(f(x,y),x),x),f(f(f(y,f(f(x,z),f(f(f(x,y),x),x))),y),f(y,y))) = f(f(y,f(f(x,z),f(f(f(x,y),x),x))),y). [para(50(a,1),604(a,1,1))].
4062
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))].
4063
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)])].
4064
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))].
4065
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)])].
4066
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)].
4067
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)])].
4068
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)])].
4069
1014 f(f(x,f(f(f(x,y),x),x)),x) = f(x,y). [para(604(a,1),858(a,1,2))].
4070
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)])].
4071
1052 f(f(f(x,y),x),x) = f(x,y). [back_rewrite(1014),rewrite([1029(4)])].
4072
1069 f(f(x,f(y,z)),f(f(f(x,y),u),x)) = x. [back_rewrite(986),rewrite([1029(5),1052(12)])].
4073
1086 f(x,f(x,f(x,y))) = f(x,y). [back_rewrite(963),rewrite([1052(3),1052(5)])].
4074
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)])].
4075
1145 f(f(x,y),x) = f(x,f(x,y)). [back_rewrite(726),rewrite([1029(3),1029(5)])].
4076
1170 f(f(x,y),f(x,x)) = x. [back_rewrite(426),rewrite([1029(3)])].
4077
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)])].
4078
1224 f(f(x,y),f(x,f(z,y))) = x # label("Sh1/A10"). [back_rewrite(188),rewrite([1029(3),1145(4),1086(4)])].
4079
1259 f(x,f(x,f(f(y,z),f(y,x)))) = f(f(y,x),x). [back_rewrite(715),rewrite([1145(2),1145(3),1086(3),1145(4),1145(5),1086(5),1145(6),1170(8),1145(5),1145(6),1086(6),1145(7)]),flip(a)].
4080
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)])].
4081
1302 f(f(f(x,y),z),f(z,y)) = z. [back_rewrite(71),rewrite([1145(2),1145(3),1086(3),1145(3),1145(4),1086(4),1145(6),1145(7),1086(7),1145(6),1086(6)])].
4082
1326 f(f(x,y),y) = f(y,f(x,y)). [back_rewrite(643),rewrite([1145(4),1086(4),1145(5),1259(5),1302(4),1145(3),1086(3),1145(6),1086(6),1145(7),1259(7)]),flip(a)].
4083
1411 f(f(x,y),f(y,f(z,x))) = y # label("C8/A12_m"). [para(1301(a,1),1174(a,1,2,2))].
4084
1427 f(f(f(x,y),f(z,y)),z) = f(z,y). [para(1224(a,1),1411(a,1,2))].
4085
1455 f(x,y) = f(y,x). [para(1145(a,1),1427(a,1,1)),rewrite([1326(3),1174(4)])].
4086
1456 $F # answer("Commutativity"). [resolve(1455,a,3,a)].
4087
4088
============================== end of proof ==========================
4089
4090
============================== STATISTICS ============================
4091
4092
Given=43. Generated=3497. Kept=1454. proofs=1.
4093
Usable=14. Sos=244. Demods=260. Limbo=3, Disabled=1198. Hints=1016.
4094
Weight_deleted=84. Literals_deleted=0.
4095
Forward_subsumed=1959. Back_subsumed=204.
4096
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
4097
New_demodulators=1451 (0 lex), Back_demodulated=988. Back_unit_deleted=0.
4098
Demod_attempts=103191. Demod_rewrites=10948.
4099
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
4100
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
4101
Megabytes=2.57.
4102
User_CPU=0.72, System_CPU=0.00, Wall_clock=10.
4103
4104
============================== end of statistics =====================
4105
4106
============================== end of search =========================
4107
4108
Exiting with 1 proof.
4109
4110
Process 22010 exit (max_proofs) Mon Jun 4 22:53:35 2007
4111
4112
Successful proof using the following 3 assumptions:
4113
4114
formulas(assumptions_in_proof).
4115
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m"). [assumption].
4116
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
4117
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4118
end_of_list.
4119
4120
Including 173 new hint clauses:
4121
4122
formulas(new_hints).
4123
$F # label("Commutativity") # label(non_clause) # label(goal).
4124
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
4125
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
4126
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
4127
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
4128
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
4129
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)).
4130
f(f(f(x,f(f(y,z),f(f(f(y,x),y),y))),x),x) = f(f(f(y,x),y),y).
4131
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
4132
f(f(f(f(f(f(x,y),x),x),f(f(f(f(x,y),x),x),z)),f(f(f(x,y),x),x)),f(z,y)) = z.
4133
f(f(f(x,f(x,f(f(f(y,x),y),y))),x),x) = f(f(f(y,x),y),y).
4134
f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(y,f(y,x)),y).
4135
f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y.
4136
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))).
4137
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).
4138
f(f(x,f(f(y,x),x)),f(y,f(f(z,f(z,x)),z))) = y.
4139
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).
4140
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).
4141
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).
4142
f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x.
4143
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)).
4144
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)).
4145
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)).
4146
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)).
4147
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))).
4148
f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))).
4149
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))).
4150
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)).
4151
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.
4152
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)))).
4153
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)))).
4154
f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y.
4155
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)).
4156
f(x,x) = f(x,f(f(x,x),x)).
4157
f(x,f(f(x,x),x)) = f(x,x).
4158
f(f(x,f(f(y,x),x)),f(y,y)) = y.
4159
f(f(x,f(x,x)),f(f(x,f(f(x,x),x)),f(y,x))) = f(x,f(f(x,x),x)).
4160
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,f(f(x,x),x)).
4161
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,x).
4162
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))).
4163
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))).
4164
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))).
4165
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)).
4166
f(f(x,x),x) = f(x,f(x,f(f(x,x),x))).
4167
f(f(x,x),x) = f(x,f(x,x)).
4168
f(x,f(x,f(x,x))) = f(x,x).
4169
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
4170
f(f(x,f(x,f(x,x))),f(x,f(y,x))) = x.
4171
f(f(x,x),f(x,f(y,x))) = x.
4172
f(f(f(x,f(x,f(x,f(x,f(x,x))))),x),x) = f(x,f(f(x,x),x)).
4173
f(f(f(x,f(x,f(x,x))),x),x) = f(x,f(f(x,x),x)).
4174
f(f(f(x,x),x),x) = f(x,f(f(x,x),x)).
4175
f(f(x,f(x,x)),x) = f(x,f(f(x,x),x)).
4176
f(f(x,f(x,x)),x) = f(x,f(x,f(x,x))).
4177
f(f(x,f(x,x)),x) = f(x,x).
4178
f(f(x,f(x,f(x,x))),f(x,f(y,f(f(x,y),y)))) = x.
4179
f(f(x,x),f(x,f(y,f(f(x,y),y)))) = x.
4180
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
4181
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))).
4182
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))).
4183
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))).
4184
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))).
4185
f(f(x,x),f(f(x,x),x)) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
4186
f(f(x,x),f(x,f(x,x))) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
4187
x = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
4188
x = f(f(x,f(x,f(x,x))),f(x,f(f(x,x),x))).
4189
x = f(f(x,x),f(x,f(f(x,x),x))).
4190
x = f(f(x,x),f(x,f(x,f(x,x)))).
4191
x = f(f(x,x),f(x,x)).
4192
f(f(x,x),f(x,x)) = x.
4193
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
4194
f(f(x,f(y,y)),f(x,y)) = x.
4195
f(f(x,x),f(x,f(x,y))) = x.
4196
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
4197
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
4198
f(f(x,y),f(x,f(y,y))) = x.
4199
f(f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),f(x,x)),f(f(y,f(y,x)),y)) = f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x).
4200
f(f(f(x,f(x,f(x,y))),x),x) = f(x,y).
4201
f(f(f(f(x,y),x),x),f(f(f(y,f(f(x,z),f(f(f(x,y),x),x))),y),f(y,y))) = f(f(y,f(f(x,z),f(f(f(x,y),x),x))),y).
4202
f(f(x,f(f(y,x),x)),y) = f(y,f(x,f(f(y,x),x))).
4203
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)).
4204
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)).
4205
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)).
4206
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.
4207
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
4208
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))).
4209
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))).
4210
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))).
4211
f(f(x,f(y,f(f(x,y),y))),f(y,f(f(x,y),y))) = x.
4212
f(x,y) = f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x).
4213
f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x) = f(x,y).
4214
f(f(f(x,f(x,f(f(f(x,y),x),x))),x),x) = f(x,y).
4215
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))).
4216
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))).
4217
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))).
4218
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.
4219
f(f(x,f(f(f(x,y),x),x)),x) = f(x,y).
4220
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).
4221
f(x,f(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x).
4222
f(x,f(f(y,x),x)) = f(y,x).
4223
f(f(f(x,y),x),x) = f(x,y).
4224
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.
4225
f(f(x,f(y,z)),f(f(f(x,y),u),x)) = x.
4226
f(f(f(x,f(x,f(x,y))),x),x) = f(x,y).
4227
f(x,f(x,f(x,y))) = f(x,y).
4228
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.
4229
f(f(x,f(x,f(y,f(f(x,y),y)))),f(x,f(y,f(f(x,y),y)))) = x.
4230
f(f(x,f(x,f(x,y))),f(x,f(y,f(f(x,y),y)))) = x.
4231
f(f(x,y),f(x,f(y,f(f(x,y),y)))) = x.
4232
f(f(x,y),f(x,f(x,y))) = x.
4233
f(f(y,x),y) = f(y,f(x,f(f(y,x),x))).
4234
f(f(x,y),x) = f(x,f(x,y)).
4235
f(f(x,y),f(x,x)) = x.
4236
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.
4237
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.
4238
f(f(f(x,f(x,u)),x),f(u,f(z,f(f(x,z),z)))) = u.
4239
f(f(x,f(x,f(x,u))),f(u,f(z,f(f(x,z),z)))) = u.
4240
f(f(x,u),f(u,f(z,f(f(x,z),z)))) = u.
4241
f(f(x,y),f(y,f(x,z))) = y.
4242
f(f(y,x),f(y,f(f(z,f(z,x)),z))) = y.
4243
f(f(y,x),f(y,f(z,f(z,f(z,x))))) = y.
4244
f(f(x,y),f(x,f(z,y))) = x.
4245
f(f(f(x,f(x,y)),x),f(f(f(y,f(f(x,z),f(f(f(x,y),x),x))),y),f(y,y))) = f(f(y,f(f(x,z),f(f(f(x,y),x),x))),y).
4246
f(f(x,f(x,f(x,y))),f(f(f(y,f(f(x,z),f(f(f(x,y),x),x))),y),f(y,y))) = f(f(y,f(f(x,z),f(f(f(x,y),x),x))),y).
4247
f(f(x,y),f(f(f(y,f(f(x,z),f(f(f(x,y),x),x))),y),f(y,y))) = f(f(y,f(f(x,z),f(f(f(x,y),x),x))),y).
4248
f(f(x,y),f(f(f(y,f(f(x,z),f(f(x,f(x,y)),x))),y),f(y,y))) = f(f(y,f(f(x,z),f(f(f(x,y),x),x))),y).
4249
f(f(x,y),f(f(f(y,f(f(x,z),f(x,f(x,f(x,y))))),y),f(y,y))) = f(f(y,f(f(x,z),f(f(f(x,y),x),x))),y).
4250
f(f(x,y),f(f(f(y,f(f(x,z),f(x,y))),y),f(y,y))) = f(f(y,f(f(x,z),f(f(f(x,y),x),x))),y).
4251
f(f(x,y),f(f(y,f(y,f(f(x,z),f(x,y)))),f(y,y))) = f(f(y,f(f(x,z),f(f(f(x,y),x),x))),y).
4252
f(f(x,y),y) = f(f(y,f(f(x,z),f(f(f(x,y),x),x))),y).
4253
f(f(x,y),y) = f(f(y,f(f(x,z),f(f(x,f(x,y)),x))),y).
4254
f(f(x,y),y) = f(f(y,f(f(x,z),f(x,f(x,f(x,y))))),y).
4255
f(f(x,y),y) = f(f(y,f(f(x,z),f(x,y))),y).
4256
f(f(x,y),y) = f(y,f(y,f(f(x,z),f(x,y)))).
4257
f(x,f(x,f(f(y,z),f(y,x)))) = f(f(y,x),x).
4258
f(f(f(x,f(x,f(f(y,f(y,x)),y))),x),x) = f(f(f(y,x),y),y).
4259
f(f(f(x,f(x,f(y,f(y,f(y,x))))),x),x) = f(f(f(y,x),y),y).
4260
f(f(f(x,f(x,f(y,x))),x),x) = f(f(f(y,x),y),y).
4261
f(f(x,f(x,f(x,f(y,x)))),x) = f(f(f(y,x),y),y).
4262
f(f(x,f(y,x)),x) = f(f(f(y,x),y),y).
4263
f(x,f(x,f(y,x))) = f(f(f(y,x),y),y).
4264
f(x,f(x,f(y,x))) = f(f(y,f(y,x)),y).
4265
f(x,f(x,f(y,x))) = f(y,f(y,f(y,x))).
4266
f(x,f(x,f(y,x))) = f(y,x).
4267
f(f(f(f(f(x,f(x,y)),x),f(f(f(f(x,y),x),x),z)),f(f(f(x,y),x),x)),f(z,y)) = z.
4268
f(f(f(f(x,f(x,f(x,y))),f(f(f(f(x,y),x),x),z)),f(f(f(x,y),x),x)),f(z,y)) = z.
4269
f(f(f(f(x,y),f(f(f(f(x,y),x),x),z)),f(f(f(x,y),x),x)),f(z,y)) = z.
4270
f(f(f(f(x,y),f(f(f(x,f(x,y)),x),z)),f(f(f(x,y),x),x)),f(z,y)) = z.
4271
f(f(f(f(x,y),f(f(x,f(x,f(x,y))),z)),f(f(f(x,y),x),x)),f(z,y)) = z.
4272
f(f(f(f(x,y),f(f(x,y),z)),f(f(f(x,y),x),x)),f(z,y)) = z.
4273
f(f(f(f(x,y),f(f(x,y),z)),f(f(x,f(x,y)),x)),f(z,y)) = z.
4274
f(f(f(f(x,y),f(f(x,y),z)),f(x,f(x,f(x,y)))),f(z,y)) = z.
4275
f(f(f(f(x,y),f(f(x,y),z)),f(x,y)),f(z,y)) = z.
4276
f(f(f(x,y),f(f(x,y),f(f(x,y),z))),f(z,y)) = z.
4277
f(f(f(x,y),z),f(z,y)) = z.
4278
f(f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),f(x,x)),f(f(y,f(y,x)),y)) = f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x).
4279
f(f(f(f(x,f(f(y,z),f(y,x))),x),f(x,x)),f(f(y,f(y,x)),y)) = f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x).
4280
f(f(f(x,f(x,f(f(y,z),f(y,x)))),f(x,x)),f(f(y,f(y,x)),y)) = f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x).
4281
f(f(f(f(y,x),x),f(x,x)),f(f(y,f(y,x)),y)) = f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x).
4282
f(x,f(f(y,f(y,x)),y)) = f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x).
4283
f(x,f(y,f(y,f(y,x)))) = f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x).
4284
f(x,f(y,x)) = f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x).
4285
f(x,f(y,x)) = f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x).
4286
f(x,f(y,x)) = f(f(x,f(f(y,z),f(y,x))),x).
4287
f(x,f(y,x)) = f(x,f(x,f(f(y,z),f(y,x)))).
4288
f(x,f(y,x)) = f(f(y,x),x).
4289
f(f(x,y),y) = f(y,f(x,y)).
4290
f(f(x,y),f(y,f(z,x))) = y # label("C8/A12_m").
4291
f(f(f(x,y),f(z,y)),z) = f(z,y).
4292
f(f(f(x,y),f(f(x,y),y)),x) = f(x,y).
4293
f(f(f(x,y),f(y,f(x,y))),x) = f(x,y).
4294
f(x,y) = f(y,x).
4295
$F # answer("Commutativity").
4296
end_of_list.
4297
4298
Attempt to eliminate one assumption:
4299
4300
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
4301
4302
***********************************************
4303
AUTO SKETCHES ITERATION 15
4304
***********************************************
4305
4306
Starting a search with 3 assumptions:
4307
4308
formulas(assumptions_active).
4309
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
4310
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
4311
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4312
end_of_list.
4313
4314
Child search process 22011 started.
4315
4316
============================== PROCESS INITIAL CLAUSES ===============
4317
4318
% Clauses before input processing:
4319
4320
formulas(usable).
4321
end_of_list.
4322
4323
formulas(sos).
4324
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
4325
f(c2,c1) != f(c1,c2) # label("Commutativity"). [deny(1)].
4326
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
4327
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
4328
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4329
end_of_list.
4330
4331
formulas(demodulators).
4332
end_of_list.
4333
4334
% 1189 hints input.
4335
4336
============================== PREDICATE ELIMINATION =================
4337
4338
No predicates eliminated.
4339
4340
============================== end predicate elimination =============
4341
4342
Auto_denials:
4343
% copying label "Commutativity" to answer in negative clause
4344
4345
Term ordering decisions:
4346
Function symbol KB weights: c1=1. c2=1. f=1.
4347
Predicate symbol precedence: predicate_order([ = ]).
4348
Function symbol precedence: function_order([ c1, c2, f ]).
4349
Skipping inverse_order, because term ordering is KBO.
4350
Unfolding symbols: (none).
4351
4352
Auto_inference settings:
4353
% set(paramodulation). % (positive equality literals)
4354
% set(paramodulation) -> set(back_demod).
4355
4356
Auto_process settings: (no changes).
4357
4358
============================== end of process initial clauses ========
4359
4360
============================== CLAUSES FOR SEARCH ====================
4361
4362
% Clauses after input processing:
4363
4364
formulas(usable).
4365
end_of_list.
4366
4367
formulas(sos).
4368
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
4369
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
4370
4 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
4371
5 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
4372
6 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4373
end_of_list.
4374
4375
formulas(demodulators).
4376
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
4377
4 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
4378
5 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
4379
6 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4380
end_of_list.
4381
4382
% 1189 hints processed (624 are redundant).
4383
4384
============================== end of clauses for search =============
4385
4386
============================== SEARCH ================================
4387
4388
% Starting search at 0.06 seconds.
4389
4390
% Operation f is commutative; C redundancy checks enabled.
4391
4392
============================== PROOF =================================
4393
4394
% Proof 1 at 0.86 (+ 0.00) seconds: "Commutativity".
4395
% Length of proof is 60.
4396
% Level of proof is 25.
4397
% Maximum clause weight is 75.
4398
% Given clauses 46.
4399
4400
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
4401
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
4402
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
4403
4 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
4404
5 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
4405
6 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4406
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))].
4407
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))].
4408
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))].
4409
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))].
4410
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))].
4411
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)])].
4412
241 f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y. [para(7(a,1),62(a,1,2)),rewrite([62(8),62(15)])].
4413
244 f(x,f(f(x,x),x)) = f(x,x). [para(62(a,1),7(a,1,2)),rewrite([62(8)]),flip(a)].
4414
252 f(f(x,f(f(y,x),x)),f(y,y)) = y. [back_rewrite(241),rewrite([244(6)])].
4415
257 f(f(x,x),f(x,f(y,x))) = x. [para(244(a,1),2(a,1,1))].
4416
259 f(f(x,x),f(x,x)) = x. [para(244(a,1),2(a,1,2)),rewrite([244(3)])].
4417
261 f(f(f(x,x),x),x) = f(x,x). [para(244(a,1),4(a,1,1,1)),rewrite([259(5)])].
4418
263 f(f(x,f(x,x)),f(x,x)) = x. [para(244(a,1),4(a,1,2)),rewrite([259(3)])].
4419
285 f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y. [para(259(a,1),2(a,1,2,2))].
4420
286 f(f(x,f(x,x)),x) = f(x,x). [para(259(a,1),2(a,1,2)),rewrite([261(3)])].
4421
353 f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x. [para(263(a,1),2(a,1,1,2))].
4422
354 f(f(x,x),x) = f(x,f(x,x)). [para(263(a,1),2(a,1,2)),rewrite([286(3),244(3)])].
4423
377 f(x,f(x,f(x,x))) = f(x,x). [para(263(a,1),257(a,1,2,2)),rewrite([259(3),354(2)])].
4424
378 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [back_rewrite(353),rewrite([354(2)])].
4425
466 f(f(x,f(y,f(f(x,y),y))),f(x,f(f(y,f(f(x,y),y)),x))) = x. [para(252(a,1),4(a,1,1,1))].
4426
567 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)). [para(378(a,1),5(a,1,1,1)),rewrite([377(3),286(6)])].
4427
589 f(f(x,f(y,y)),f(x,y)) = x. [back_rewrite(285),rewrite([567(6)])].
4428
591 f(f(x,y),f(x,f(y,y))) = x. [para(2(a,1),589(a,1,1,2)),rewrite([354(3),377(4)])].
4429
639 f(x,f(f(y,f(f(x,y),y)),x)) = f(y,f(f(x,y),y)). [para(252(a,1),589(a,1,1))].
4430
647 f(f(x,f(y,f(f(x,y),y))),f(y,f(f(x,y),y))) = x. [back_rewrite(466),rewrite([639(9)])].
4431
663 f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x) = f(x,y). [back_rewrite(117),rewrite([647(11)]),flip(a)].
4432
679 f(f(f(x,f(x,f(x,y))),x),x) = f(x,y). [para(591(a,1),6(a,1,2))].
4433
736 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [para(679(a,1),2(a,1,2,2))].
4434
751 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(679(a,1),44(a,1,1,1,2,2,2)),rewrite([679(23)])].
4435
839 f(f(x,f(f(f(x,y),x),x)),x) = f(x,y). [para(591(a,1),736(a,1,2))].
4436
870 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(55(a,1),839(a,1,1,2))].
4437
886 f(f(x,f(x,y)),x) = f(x,f(x,f(x,y))). [para(679(a,1),839(a,1,1,2))].
4438
939 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(870),rewrite([886(3),886(9)])].
4439
970 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(751),rewrite([886(14),886(18)])].
4440
996 f(x,f(x,f(x,f(x,f(x,y))))) = f(x,y). [back_rewrite(679),rewrite([886(4),886(5)])].
4441
998 f(x,f(x,f(x,f(x,f(f(f(x,y),x),x))))) = f(x,y). [back_rewrite(663),rewrite([886(7)])].
4442
1087 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([886(4),886(10),886(15),886(22),886(25),886(29),886(32),886(35)])].
4443
1118 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([886(4),886(11)])].
4444
1131 f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y # label("C1/A1"). [back_rewrite(6),rewrite([886(3)])].
4445
1137 f(f(x,f(x,f(x,y))),f(x,f(x,y))) = x. [para(886(a,1),4(a,1,1))].
4446
1161 f(f(x,y),x) = f(x,f(x,y)). [para(886(a,1),839(a,1,1,2,1)),rewrite([886(4),996(5)])].
4447
1162 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(886(a,1),839(a,2)),rewrite([1161(5),1137(8),1161(9)])].
4448
1165 f(x,f(x,f(x,f(x,y)))) = f(x,f(x,y)). [back_rewrite(970),rewrite([1137(15),1161(10),1162(12)])].
4449
1196 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(1118),rewrite([1161(7),1161(8)])].
4450
1239 f(x,f(x,f(x,y))) = f(x,y). [back_rewrite(998),rewrite([1161(2),1161(3),1165(4),1165(4)])].
4451
1258 f(x,f(f(y,z),f(y,x))) = f(x,f(x,f(y,x))). [back_rewrite(939),rewrite([1239(3),1161(3),1239(7)]),flip(a)].
4452
1438 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(1087),rewrite([1239(4),1239(8),1239(12),1239(16),1239(19),1239(21),1239(15)])].
4453
1472 f(x,f(f(y,z),f(y,x))) = f(y,x). [back_rewrite(1196),rewrite([1239(4),1239(6),1239(7)])].
4454
1483 f(f(x,y),f(y,f(x,z))) = y. [back_rewrite(1131),rewrite([1239(3)])].
4455
1586 f(f(x,y),y) = f(y,f(x,y)). [back_rewrite(1438),rewrite([1472(4),1472(5),1483(5)])].
4456
1587 f(x,f(x,f(y,x))) = f(y,x). [back_rewrite(1258),rewrite([1472(4)]),flip(a)].
4457
1723 f(f(x,y),f(x,f(z,y))) = x. [back_rewrite(2),rewrite([1586(2),1587(3)])].
4458
1781 f(x,y) = f(y,x). [para(1586(a,1),1472(a,1,2)),rewrite([1723(4)])].
4459
1782 $F # answer("Commutativity"). [resolve(1781,a,3,a)].
4460
4461
============================== end of proof ==========================
4462
4463
============================== STATISTICS ============================
4464
4465
Given=46. Generated=3828. Kept=1780. proofs=1.
4466
Usable=15. Sos=242. Demods=262. Limbo=6, Disabled=1521. Hints=1189.
4467
Weight_deleted=42. Literals_deleted=0.
4468
Forward_subsumed=2006. Back_subsumed=227.
4469
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
4470
New_demodulators=1774 (0 lex), Back_demodulated=1289. Back_unit_deleted=0.
4471
Demod_attempts=100302. Demod_rewrites=10522.
4472
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
4473
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
4474
Megabytes=2.77.
4475
User_CPU=0.86, System_CPU=0.00, Wall_clock=11.
4476
4477
============================== end of statistics =====================
4478
4479
============================== end of search =========================
4480
4481
Exiting with 1 proof.
4482
4483
Process 22011 exit (max_proofs) Mon Jun 4 22:53:36 2007
4484
4485
Successful proof using the following 3 assumptions:
4486
4487
formulas(assumptions_in_proof).
4488
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m"). [assumption].
4489
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
4490
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4491
end_of_list.
4492
4493
Including 117 new hint clauses:
4494
4495
formulas(new_hints).
4496
$F # label("Commutativity") # label(non_clause) # label(goal).
4497
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
4498
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
4499
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
4500
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
4501
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
4502
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)).
4503
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
4504
f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(y,f(y,x)),y).
4505
f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y.
4506
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).
4507
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)).
4508
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)).
4509
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)))).
4510
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)))).
4511
f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y.
4512
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)).
4513
f(x,x) = f(x,f(f(x,x),x)).
4514
f(x,f(f(x,x),x)) = f(x,x).
4515
f(f(x,f(f(y,x),x)),f(y,y)) = y.
4516
f(f(x,x),f(x,f(y,x))) = x.
4517
f(f(x,f(f(x,x),x)),f(x,x)) = x.
4518
f(f(x,x),f(x,x)) = x.
4519
f(f(f(x,x),x),f(f(x,x),f(x,x))) = f(x,x).
4520
f(f(f(x,x),x),x) = f(x,x).
4521
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,x)) = x.
4522
f(f(x,f(x,x)),f(x,x)) = x.
4523
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
4524
f(f(x,f(f(f(x,x),x),x)),x) = f(x,x).
4525
f(f(x,f(x,x)),x) = f(x,x).
4526
f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x.
4527
f(f(x,f(f(f(x,f(x,x)),x),x)),x) = f(x,f(x,x)).
4528
f(f(x,f(f(x,x),x)),x) = f(x,f(x,x)).
4529
f(f(x,x),x) = f(x,f(x,x)).
4530
f(f(f(x,x),f(x,x)),f(f(x,x),x)) = f(x,x).
4531
f(x,f(f(x,x),x)) = f(x,x).
4532
f(x,f(x,f(x,x))) = f(x,x).
4533
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
4534
f(f(x,f(y,f(f(x,y),y))),f(x,f(f(y,f(f(x,y),y)),x))) = x.
4535
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)).
4536
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
4537
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
4538
f(f(x,f(y,y)),f(x,y)) = x.
4539
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
4540
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
4541
f(f(x,y),f(x,f(y,y))) = x.
4542
f(x,f(f(y,f(f(x,y),y)),x)) = f(y,f(f(x,y),y)).
4543
f(f(x,f(y,f(f(x,y),y))),f(y,f(f(x,y),y))) = x.
4544
f(x,y) = f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x).
4545
f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x) = f(x,y).
4546
f(f(f(x,f(x,f(x,y))),x),x) = f(x,y).
4547
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
4548
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)).
4549
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)).
4550
f(f(x,f(f(f(x,y),x),x)),x) = f(x,y).
4551
f(f(x,f(f(y,f(y,x)),y)),x) = f(x,f(f(y,z),f(f(y,f(y,x)),y))).
4552
f(f(x,f(x,y)),x) = f(x,f(x,f(x,y))).
4553
f(f(x,f(y,f(y,f(y,x)))),x) = f(x,f(f(y,z),f(f(y,f(y,x)),y))).
4554
f(f(x,f(y,f(y,f(y,x)))),x) = f(x,f(f(y,z),f(y,f(y,f(y,x))))).
4555
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)).
4556
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)).
4557
f(f(x,f(x,f(x,f(x,y)))),x) = f(x,y).
4558
f(x,f(x,f(x,f(x,f(x,y))))) = f(x,y).
4559
f(x,f(x,f(x,f(x,f(f(f(x,y),x),x))))) = f(x,y).
4560
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)).
4561
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)).
4562
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)).
4563
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)).
4564
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)).
4565
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)).
4566
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)).
4567
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)))).
4568
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(f(y,f(y,x)),y).
4569
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(y,f(y,f(y,x))).
4570
f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y.
4571
f(f(x,f(x,f(x,y))),f(x,f(x,y))) = x.
4572
f(f(x,f(f(x,f(x,f(x,y))),x)),x) = f(x,f(x,y)).
4573
f(f(x,f(x,f(x,f(x,f(x,y))))),x) = f(x,f(x,y)).
4574
f(f(x,y),x) = f(x,f(x,y)).
4575
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))).
4576
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))).
4577
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))).
4578
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))).
4579
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)).
4580
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)).
4581
f(x,f(x,f(x,f(x,y)))) = f(x,f(x,y)).
4582
f(f(x,f(x,f(f(y,z),f(y,f(y,f(y,x)))))),x) = f(y,f(y,f(y,x))).
4583
f(x,f(x,f(x,f(f(y,z),f(y,f(y,f(y,x))))))) = f(y,f(y,f(y,x))).
4584
f(x,f(x,f(x,f(x,f(f(x,f(x,y)),x))))) = f(x,y).
4585
f(x,f(x,f(x,f(x,f(x,f(x,f(x,y))))))) = f(x,y).
4586
f(x,f(x,f(x,f(x,f(x,y))))) = f(x,y).
4587
f(x,f(x,f(x,y))) = f(x,y).
4588
f(f(x,f(y,x)),x) = f(x,f(f(y,z),f(y,f(y,f(y,x))))).
4589
f(x,f(x,f(y,x))) = f(x,f(f(y,z),f(y,f(y,f(y,x))))).
4590
f(x,f(x,f(y,x))) = f(x,f(f(y,z),f(y,x))).
4591
f(x,f(f(y,z),f(y,x))) = f(x,f(x,f(y,x))).
4592
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)))).
4593
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)))).
4594
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)))).
4595
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)))).
4596
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)))).
4597
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)))).
4598
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)).
4599
f(x,f(x,f(x,f(f(y,z),f(y,x))))) = f(y,f(y,f(y,x))).
4600
f(x,f(f(y,z),f(y,x))) = f(y,f(y,f(y,x))).
4601
f(x,f(f(y,z),f(y,x))) = f(y,x).
4602
f(f(x,y),f(y,f(x,z))) = y.
4603
f(f(y,x),f(f(x,f(f(y,z),f(y,x))),f(x,f(y,x)))) = f(x,f(y,x)).
4604
f(f(y,x),f(f(y,x),f(x,f(y,x)))) = f(x,f(y,x)).
4605
f(f(x,y),y) = f(y,f(x,y)).
4606
f(y,x) = f(x,f(x,f(y,x))).
4607
f(x,f(x,f(y,x))) = f(y,x).
4608
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y.
4609
f(f(x,y),f(x,f(z,y))) = x.
4610
f(x,f(f(y,x),f(y,f(y,x)))) = f(y,x).
4611
f(x,y) = f(y,x).
4612
$F # answer("Commutativity").
4613
end_of_list.
4614
4615
Attempt to eliminate one assumption:
4616
4617
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
4618
4619
***********************************************
4620
AUTO SKETCHES ITERATION 16
4621
***********************************************
4622
4623
Starting a search with 2 assumptions:
4624
4625
formulas(assumptions_active).
4626
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
4627
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4628
end_of_list.
4629
4630
Child search process 22012 started.
4631
4632
============================== PROCESS INITIAL CLAUSES ===============
4633
4634
% Clauses before input processing:
4635
4636
formulas(usable).
4637
end_of_list.
4638
4639
formulas(sos).
4640
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
4641
f(c2,c1) != f(c1,c2) # label("Commutativity"). [deny(1)].
4642
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
4643
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4644
end_of_list.
4645
4646
formulas(demodulators).
4647
end_of_list.
4648
4649
% 1306 hints input.
4650
4651
============================== PREDICATE ELIMINATION =================
4652
4653
No predicates eliminated.
4654
4655
============================== end predicate elimination =============
4656
4657
Auto_denials:
4658
% copying label "Commutativity" to answer in negative clause
4659
4660
Term ordering decisions:
4661
Function symbol KB weights: c1=1. c2=1. f=1.
4662
Predicate symbol precedence: predicate_order([ = ]).
4663
Function symbol precedence: function_order([ c1, c2, f ]).
4664
Skipping inverse_order, because term ordering is KBO.
4665
Unfolding symbols: (none).
4666
4667
Auto_inference settings:
4668
% set(paramodulation). % (positive equality literals)
4669
% set(paramodulation) -> set(back_demod).
4670
4671
Auto_process settings: (no changes).
4672
4673
============================== end of process initial clauses ========
4674
4675
============================== CLAUSES FOR SEARCH ====================
4676
4677
% Clauses after input processing:
4678
4679
formulas(usable).
4680
end_of_list.
4681
4682
formulas(sos).
4683
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
4684
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
4685
4 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
4686
5 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4687
end_of_list.
4688
4689
formulas(demodulators).
4690
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
4691
4 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
4692
5 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4693
end_of_list.
4694
4695
% 1306 hints processed (676 are redundant).
4696
4697
============================== end of clauses for search =============
4698
4699
============================== SEARCH ================================
4700
4701
% Starting search at 0.07 seconds.
4702
4703
% Operation f is commutative; C redundancy checks enabled.
4704
4705
============================== PROOF =================================
4706
4707
% Proof 1 at 0.75 (+ 0.00) seconds: "Commutativity".
4708
% Length of proof is 66.
4709
% Level of proof is 24.
4710
% Maximum clause weight is 75.
4711
% Given clauses 44.
4712
4713
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
4714
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
4715
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
4716
4 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
4717
5 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4718
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))].
4719
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))].
4720
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))].
4721
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))].
4722
46 f(f(f(f(f(x,f(x,y)),x),f(f(f(x,f(x,y)),x),f(f(y,f(x,z)),f(y,f(y,f(x,z)))))),f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x)) = f(f(y,f(x,z)),f(y,f(y,f(x,z)))). [para(5(a,1),24(a,1,1,1,2,2,2,1)),rewrite([5(30)])].
4723
75 f(f(x,f(f(y,x),x)),f(y,f(f(z,f(z,x)),z))) = y. [para(28(a,1),2(a,1,2,2))].
4724
79 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),28(a,1,1,1))].
4725
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)])].
4726
127 f(f(x,f(f(y,x),x)),f(y,f(f(f(f(x,f(f(z,u),f(f(z,f(z,x)),z))),x),f(f(z,f(z,x)),z)),f(f(x,f(f(z,u),f(f(z,f(z,x)),z))),x)))) = y. [para(28(a,1),75(a,1,2,2,1,2))].
4727
173 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)].
4728
178 f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y. [para(6(a,1),35(a,1,2)),rewrite([35(8),35(15)])].
4729
181 f(x,f(f(x,x),x)) = f(x,x). [para(35(a,1),6(a,1,2)),rewrite([35(8)]),flip(a)].
4730
189 f(f(x,f(f(y,x),x)),f(y,y)) = y. [back_rewrite(178),rewrite([181(6)])].
4731
190 f(f(x,x),f(f(x,f(x,x)),f(x,x))) = f(f(f(x,f(x,x)),x),x). [back_rewrite(173),rewrite([181(3),181(4),181(6),181(9)])].
4732
194 f(f(x,x),f(x,f(y,x))) = x. [para(181(a,1),2(a,1,1))].
4733
196 f(f(x,x),f(x,x)) = x. [para(181(a,1),2(a,1,2)),rewrite([181(3)])].
4734
200 f(f(x,f(x,x)),f(x,x)) = x. [para(181(a,1),4(a,1,2)),rewrite([196(3)])].
4735
217 f(f(f(x,f(x,x)),x),x) = f(f(x,x),x). [back_rewrite(190),rewrite([200(5)]),flip(a)].
4736
218 f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y. [para(196(a,1),2(a,1,2,2))].
4737
251 f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x. [para(200(a,1),2(a,1,1,2))].
4738
252 f(f(x,x),x) = f(x,f(x,x)). [para(200(a,1),2(a,1,2)),rewrite([217(4),181(3)])].
4739
276 f(f(x,f(x,x)),x) = f(x,x). [para(196(a,1),200(a,1,1,2)),rewrite([252(2),196(5)])].
4740
277 f(x,f(x,f(x,x))) = f(x,x). [para(200(a,1),194(a,1,2,2)),rewrite([196(3),252(2)])].
4741
278 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x. [back_rewrite(251),rewrite([252(2)])].
4742
362 f(f(x,f(y,f(f(x,y),y))),f(x,f(f(y,f(f(x,y),y)),x))) = x. [para(189(a,1),4(a,1,1,1))].
4743
446 f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)). [para(278(a,1),4(a,1,1,1)),rewrite([277(3),276(6)])].
4744
468 f(f(x,f(y,y)),f(x,y)) = x. [back_rewrite(218),rewrite([446(6)])].
4745
470 f(f(x,y),f(x,f(y,y))) = x. [para(2(a,1),468(a,1,1,2)),rewrite([252(3),277(4)])].
4746
512 f(x,f(f(y,f(f(x,y),y)),x)) = f(y,f(f(x,y),y)). [para(189(a,1),468(a,1,1))].
4747
520 f(f(x,f(y,f(f(x,y),y))),f(y,f(f(x,y),y))) = x. [back_rewrite(362),rewrite([512(9)])].
4748
536 f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x) = f(x,y). [back_rewrite(79),rewrite([520(11)]),flip(a)].
4749
548 f(f(f(x,f(x,f(x,y))),x),x) = f(x,y). [para(470(a,1),5(a,1,2))].
4750
603 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9"). [para(548(a,1),2(a,1,2,2))].
4751
641 f(f(f(f(x,y),f(f(x,y),x)),f(x,y)),f(x,y)) = x. [para(470(a,1),548(a,1,1,1,2,2)),rewrite([470(12)])].
4752
694 f(f(x,f(f(f(x,y),x),x)),x) = f(x,y). [para(470(a,1),603(a,1,2))].
4753
718 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),694(a,1,1,2))].
4754
734 f(f(x,f(x,y)),x) = f(x,f(x,f(x,y))). [para(548(a,1),694(a,1,1,2))].
4755
783 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(718),rewrite([734(3),734(9)])].
4756
795 f(f(x,y),f(f(x,y),f(f(x,y),f(f(x,y),x)))) = x. [back_rewrite(641),rewrite([734(6),734(8)])].
4757
834 f(x,f(x,f(x,f(x,f(x,y))))) = f(x,y). [back_rewrite(548),rewrite([734(4),734(5)])].
4758
836 f(x,f(x,f(x,f(x,f(f(f(x,y),x),x))))) = f(x,y). [back_rewrite(536),rewrite([734(7)])].
4759
897 f(f(x,f(f(y,x),x)),f(y,f(f(f(f(x,f(f(z,u),f(z,f(z,f(z,x))))),x),f(z,f(z,f(z,x)))),f(f(x,f(f(z,u),f(z,f(z,f(z,x))))),x)))) = y. [back_rewrite(127),rewrite([734(7),734(13),734(18)])].
4760
920 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([734(4),734(10),734(15),734(22),734(25),734(29),734(32),734(35)])].
4761
939 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(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(46),rewrite([734(3),734(6),734(17),734(18),734(21),734(22)])].
4762
945 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([734(4),734(11)])].
4763
952 f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y. [back_rewrite(5),rewrite([734(3)])].
4764
980 f(f(x,y),x) = f(x,f(x,y)). [para(734(a,1),694(a,1,1,2,1)),rewrite([734(4),834(5)])].
4765
984 f(f(x,f(f(y,x),x)),f(y,f(f(x,f(x,f(f(z,u),f(z,f(z,f(z,x)))))),f(f(x,f(x,f(f(z,u),f(z,f(z,f(z,x)))))),f(z,f(z,f(z,x))))))) = y. [back_rewrite(897),rewrite([980(10),980(21),980(22)])].
4766
1004 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(945),rewrite([980(7),980(8)])].
4767
1037 f(x,f(x,f(x,y))) = f(x,y). [back_rewrite(836),rewrite([980(2),980(3),834(5)])].
4768
1048 f(f(x,y),f(x,f(x,y))) = x. [back_rewrite(795),rewrite([980(5),1037(8)])].
4769
1052 f(x,f(f(y,z),f(y,x))) = f(x,f(x,f(y,x))). [back_rewrite(783),rewrite([1037(3),980(3),1037(7)]),flip(a)].
4770
1187 f(x,f(f(y,z),f(y,x))) = f(y,x). [back_rewrite(1004),rewrite([1037(4),1037(6),1037(7)])].
4771
1191 f(f(x,f(f(y,x),x)),f(y,f(f(x,f(z,x)),f(f(x,f(z,x)),f(z,x))))) = y. [back_rewrite(984),rewrite([1037(7),1187(7),1037(9),1187(9),1037(10)])].
4772
1194 f(f(x,y),f(y,f(x,z))) = y. [back_rewrite(952),rewrite([1037(3)])].
4773
1199 f(f(x,y),f(f(x,y),y)) = y. [back_rewrite(939),rewrite([1037(3),1037(4),1037(5),1037(6),1048(10),1037(7),1048(10)])].
4774
1204 f(f(x,y),y) = f(y,f(x,y)). [back_rewrite(920),rewrite([1037(4),1187(4),1037(5),1187(5),1037(6),1187(6),1037(7),1187(7),1037(7),1194(7),1199(5),1037(5)])].
4775
1326 f(x,f(x,f(y,x))) = f(y,x). [back_rewrite(1052),rewrite([1187(4)]),flip(a)].
4776
1331 f(f(x,y),f(x,f(z,y))) = x. [back_rewrite(1191),rewrite([1204(2),1326(3),1204(7),1194(7),980(4),1326(4)])].
4777
1503 f(x,y) = f(y,x). [para(1204(a,1),1187(a,1,2)),rewrite([1331(4)])].
4778
1504 $F # answer("Commutativity"). [resolve(1503,a,3,a)].
4779
4780
============================== end of proof ==========================
4781
4782
============================== STATISTICS ============================
4783
4784
Given=44. Generated=3393. Kept=1502. proofs=1.
4785
Usable=15. Sos=189. Demods=209. Limbo=6, Disabled=1295. Hints=1306.
4786
Weight_deleted=42. Literals_deleted=0.
4787
Forward_subsumed=1849. Back_subsumed=199.
4788
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
4789
New_demodulators=1496 (0 lex), Back_demodulated=1092. Back_unit_deleted=0.
4790
Demod_attempts=93146. Demod_rewrites=10101.
4791
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
4792
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
4793
Megabytes=2.55.
4794
User_CPU=0.76, System_CPU=0.00, Wall_clock=12.
4795
4796
============================== end of statistics =====================
4797
4798
============================== end of search =========================
4799
4800
Exiting with 1 proof.
4801
4802
Process 22012 exit (max_proofs) Mon Jun 4 22:53:37 2007
4803
4804
Successful proof using the following 2 assumptions:
4805
4806
formulas(assumptions_in_proof).
4807
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m"). [assumption].
4808
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4809
end_of_list.
4810
4811
Including 154 new hint clauses:
4812
4813
formulas(new_hints).
4814
$F # label("Commutativity") # label(non_clause) # label(goal).
4815
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
4816
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
4817
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
4818
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
4819
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)).
4820
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
4821
f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(y,f(y,x)),y).
4822
f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y.
4823
f(f(f(f(f(x,f(x,y)),x),f(f(f(x,f(x,y)),x),f(f(y,f(x,z)),f(y,f(y,f(x,z)))))),f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x)) = 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)))).
4824
f(f(f(f(f(x,f(x,y)),x),f(f(f(x,f(x,y)),x),f(f(y,f(x,z)),f(y,f(y,f(x,z)))))),f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x)) = f(f(y,f(x,z)),f(y,f(y,f(x,z)))).
4825
f(f(x,f(f(y,x),x)),f(y,f(f(z,f(z,x)),z))) = y.
4826
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).
4827
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)).
4828
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)).
4829
f(f(x,f(f(y,x),x)),f(y,f(f(f(f(x,f(f(z,u),f(f(z,f(z,x)),z))),x),f(f(z,f(z,x)),z)),f(f(x,f(f(z,u),f(f(z,f(z,x)),z))),x)))) = y.
4830
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)))).
4831
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)))).
4832
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)))).
4833
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)))).
4834
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)))).
4835
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).
4836
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)))).
4837
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)))).
4838
f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y.
4839
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)).
4840
f(x,x) = f(x,f(f(x,x),x)).
4841
f(x,f(f(x,x),x)) = f(x,x).
4842
f(f(x,f(f(y,x),x)),f(y,y)) = y.
4843
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).
4844
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).
4845
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).
4846
f(f(x,x),f(f(x,f(x,x)),f(x,x))) = f(f(f(x,f(x,x)),x),x).
4847
f(f(x,x),f(x,f(y,x))) = x.
4848
f(f(x,f(f(x,x),x)),f(x,x)) = x.
4849
f(f(x,x),f(x,x)) = x.
4850
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,x)) = x.
4851
f(f(x,f(x,x)),f(x,x)) = x.
4852
f(f(x,x),x) = f(f(f(x,f(x,x)),x),x).
4853
f(f(f(x,f(x,x)),x),x) = f(f(x,x),x).
4854
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
4855
f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x.
4856
f(f(x,f(f(f(x,f(x,x)),x),x)),x) = f(x,f(x,x)).
4857
f(f(x,f(f(x,x),x)),x) = f(x,f(x,x)).
4858
f(f(x,x),x) = f(x,f(x,x)).
4859
f(f(f(x,x),x),f(f(x,x),f(x,x))) = f(x,x).
4860
f(f(x,f(x,x)),f(f(x,x),f(x,x))) = f(x,x).
4861
f(f(x,f(x,x)),x) = f(x,x).
4862
f(f(f(x,x),f(x,x)),f(f(x,x),x)) = f(x,x).
4863
f(x,f(f(x,x),x)) = f(x,x).
4864
f(x,f(x,f(x,x))) = f(x,x).
4865
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
4866
f(f(x,f(y,f(f(x,y),y))),f(x,f(f(y,f(f(x,y),y)),x))) = x.
4867
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)).
4868
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
4869
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
4870
f(f(x,f(y,y)),f(x,y)) = x.
4871
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
4872
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
4873
f(f(x,y),f(x,f(y,y))) = x.
4874
f(x,f(f(y,f(f(x,y),y)),x)) = f(y,f(f(x,y),y)).
4875
f(f(x,f(y,f(f(x,y),y))),f(y,f(f(x,y),y))) = x.
4876
f(x,y) = f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x).
4877
f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x) = f(x,y).
4878
f(f(f(x,f(x,f(x,y))),x),x) = f(x,y).
4879
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
4880
f(f(f(f(x,y),f(f(x,y),x)),f(x,y)),f(x,y)) = f(f(x,y),f(x,f(y,y))).
4881
f(f(f(f(x,y),f(f(x,y),x)),f(x,y)),f(x,y)) = x.
4882
f(f(x,f(f(f(x,y),x),x)),x) = f(x,y).
4883
f(f(x,f(f(y,f(y,x)),y)),x) = f(x,f(f(y,z),f(f(y,f(y,x)),y))).
4884
f(f(x,f(x,y)),x) = f(x,f(x,f(x,y))).
4885
f(f(x,f(y,f(y,f(y,x)))),x) = f(x,f(f(y,z),f(f(y,f(y,x)),y))).
4886
f(f(x,f(y,f(y,f(y,x)))),x) = f(x,f(f(y,z),f(y,f(y,f(y,x))))).
4887
f(f(f(x,y),f(f(x,y),f(f(x,y),x))),f(x,y)) = x.
4888
f(f(x,y),f(f(x,y),f(f(x,y),f(f(x,y),x)))) = x.
4889
f(f(x,f(x,f(x,f(x,y)))),x) = f(x,y).
4890
f(x,f(x,f(x,f(x,f(x,y))))) = f(x,y).
4891
f(x,f(x,f(x,f(x,f(f(f(x,y),x),x))))) = f(x,y).
4892
f(f(x,f(f(y,x),x)),f(y,f(f(f(f(x,f(f(z,u),f(z,f(z,f(z,x))))),x),f(f(z,f(z,x)),z)),f(f(x,f(f(z,u),f(f(z,f(z,x)),z))),x)))) = y.
4893
f(f(x,f(f(y,x),x)),f(y,f(f(f(f(x,f(f(z,u),f(z,f(z,f(z,x))))),x),f(z,f(z,f(z,x)))),f(f(x,f(f(z,u),f(f(z,f(z,x)),z))),x)))) = y.
4894
f(f(x,f(f(y,x),x)),f(y,f(f(f(f(x,f(f(z,u),f(z,f(z,f(z,x))))),x),f(z,f(z,f(z,x)))),f(f(x,f(f(z,u),f(z,f(z,f(z,x))))),x)))) = y.
4895
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)).
4896
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)).
4897
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)).
4898
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)).
4899
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)).
4900
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)).
4901
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)).
4902
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)))).
4903
f(f(f(f(x,f(x,f(x,y))),f(f(f(x,f(x,y)),x),f(f(y,f(x,z)),f(y,f(y,f(x,z)))))),f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x)) = f(f(y,f(x,z)),f(y,f(y,f(x,z)))).
4904
f(f(f(f(x,f(x,f(x,y))),f(f(x,f(x,f(x,y))),f(f(y,f(x,z)),f(y,f(y,f(x,z)))))),f(f(x,f(x,y)),x)),f(f(x,f(x,y)),x)) = f(f(y,f(x,z)),f(y,f(y,f(x,z)))).
4905
f(f(f(f(x,f(x,f(x,y))),f(f(x,f(x,f(x,y))),f(f(y,f(x,z)),f(y,f(y,f(x,z)))))),f(x,f(x,f(x,y)))),f(f(x,f(x,y)),x)) = f(f(y,f(x,z)),f(y,f(y,f(x,z)))).
4906
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(f(y,f(x,z)),f(y,f(y,f(x,z))))))),f(f(x,f(x,y)),x)) = f(f(y,f(x,z)),f(y,f(y,f(x,z)))).
4907
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(f(y,f(x,z)),f(y,f(y,f(x,z))))))),f(x,f(x,f(x,y)))) = f(f(y,f(x,z)),f(y,f(y,f(x,z)))).
4908
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(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)))).
4909
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(f(y,f(y,x)),y).
4910
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(y,f(y,f(y,x))).
4911
f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y.
4912
f(f(x,f(f(x,f(x,f(x,y))),x)),x) = f(x,f(x,y)).
4913
f(f(x,f(x,f(x,f(x,f(x,y))))),x) = f(x,f(x,y)).
4914
f(f(x,y),x) = f(x,f(x,y)).
4915
f(f(x,f(f(y,x),x)),f(y,f(f(f(x,f(x,f(f(z,u),f(z,f(z,f(z,x)))))),f(z,f(z,f(z,x)))),f(f(x,f(f(z,u),f(z,f(z,f(z,x))))),x)))) = y.
4916
f(f(x,f(f(y,x),x)),f(y,f(f(f(x,f(x,f(f(z,u),f(z,f(z,f(z,x)))))),f(z,f(z,f(z,x)))),f(x,f(x,f(f(z,u),f(z,f(z,f(z,x))))))))) = y.
4917
f(f(x,f(f(y,x),x)),f(y,f(f(x,f(x,f(f(z,u),f(z,f(z,f(z,x)))))),f(f(x,f(x,f(f(z,u),f(z,f(z,f(z,x)))))),f(z,f(z,f(z,x))))))) = y.
4918
f(f(x,f(x,f(f(y,z),f(y,f(y,f(y,x)))))),x) = f(y,f(y,f(y,x))).
4919
f(x,f(x,f(x,f(f(y,z),f(y,f(y,f(y,x))))))) = f(y,f(y,f(y,x))).
4920
f(x,f(x,f(x,f(x,f(f(x,f(x,y)),x))))) = f(x,y).
4921
f(x,f(x,f(x,f(x,f(x,f(x,f(x,y))))))) = f(x,y).
4922
f(x,f(x,f(x,y))) = f(x,y).
4923
f(f(x,y),f(f(x,y),f(f(x,y),f(x,f(x,y))))) = x.
4924
f(f(x,y),f(x,f(x,y))) = x.
4925
f(f(x,f(y,x)),x) = f(x,f(f(y,z),f(y,f(y,f(y,x))))).
4926
f(x,f(x,f(y,x))) = f(x,f(f(y,z),f(y,f(y,f(y,x))))).
4927
f(x,f(x,f(y,x))) = f(x,f(f(y,z),f(y,x))).
4928
f(x,f(f(y,z),f(y,x))) = f(x,f(x,f(y,x))).
4929
f(x,f(x,f(x,f(f(y,z),f(y,x))))) = f(y,f(y,f(y,x))).
4930
f(x,f(f(y,z),f(y,x))) = f(y,f(y,f(y,x))).
4931
f(x,f(f(y,z),f(y,x))) = f(y,x).
4932
f(f(x,f(f(y,x),x)),f(y,f(f(x,f(x,f(f(z,u),f(z,x)))),f(f(x,f(x,f(f(z,u),f(z,f(z,f(z,x)))))),f(z,f(z,f(z,x))))))) = y.
4933
f(f(x,f(f(y,x),x)),f(y,f(f(x,f(z,x)),f(f(x,f(x,f(f(z,u),f(z,f(z,f(z,x)))))),f(z,f(z,f(z,x))))))) = y.
4934
f(f(x,f(f(y,x),x)),f(y,f(f(x,f(z,x)),f(f(x,f(x,f(f(z,u),f(z,x)))),f(z,f(z,f(z,x))))))) = y.
4935
f(f(x,f(f(y,x),x)),f(y,f(f(x,f(z,x)),f(f(x,f(z,x)),f(z,f(z,f(z,x))))))) = y.
4936
f(f(x,f(f(y,x),x)),f(y,f(f(x,f(z,x)),f(f(x,f(z,x)),f(z,x))))) = y.
4937
f(f(x,y),f(y,f(x,z))) = y.
4938
f(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(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)))).
4939
f(f(x,y),f(f(x,y),f(f(x,f(x,f(x,y))),f(f(x,f(x,f(x,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)))).
4940
f(f(x,y),f(f(x,y),f(f(x,y),f(f(x,f(x,f(x,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)))).
4941
f(f(x,y),f(f(x,y),f(f(x,y),f(f(x,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)))).
4942
f(f(x,y),f(f(x,y),f(f(x,y),f(f(x,y),y)))) = f(f(y,f(x,z)),f(y,f(y,f(x,z)))).
4943
f(f(x,y),f(f(x,y),y)) = f(f(y,f(x,z)),f(y,f(y,f(x,z)))).
4944
f(f(x,y),f(f(x,y),y)) = y.
4945
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)))).
4946
f(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)))).
4947
f(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)))).
4948
f(f(y,x),f(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)))).
4949
f(f(y,x),f(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)))).
4950
f(f(y,x),f(f(y,x),f(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)))).
4951
f(f(y,x),f(f(y,x),f(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)))).
4952
f(f(y,x),f(f(y,x),f(f(y,x),f(f(y,x),f(x,f(y,f(y,f(y,x)))))))) = f(x,f(y,f(y,f(y,x)))).
4953
f(f(y,x),f(f(y,x),f(f(y,x),f(f(y,x),f(x,f(y,x)))))) = f(x,f(y,f(y,f(y,x)))).
4954
f(f(y,x),f(f(y,x),f(f(y,x),x))) = f(x,f(y,f(y,f(y,x)))).
4955
f(f(y,x),x) = f(x,f(y,f(y,f(y,x)))).
4956
f(f(x,y),y) = f(y,f(x,y)).
4957
f(y,x) = f(x,f(x,f(y,x))).
4958
f(x,f(x,f(y,x))) = f(y,x).
4959
f(f(x,f(x,f(y,x))),f(y,f(f(x,f(z,x)),f(f(x,f(z,x)),f(z,x))))) = y.
4960
f(f(y,x),f(y,f(f(x,f(z,x)),f(f(x,f(z,x)),f(z,x))))) = y.
4961
f(f(y,x),f(y,f(f(x,f(z,x)),f(f(z,x),f(x,f(z,x)))))) = y.
4962
f(f(y,x),f(y,f(f(x,f(z,x)),x))) = y.
4963
f(f(y,x),f(y,f(x,f(x,f(z,x))))) = y.
4964
f(f(x,y),f(x,f(z,y))) = x.
4965
f(x,f(f(y,x),f(y,f(y,x)))) = f(y,x).
4966
f(x,y) = f(y,x).
4967
$F # answer("Commutativity").
4968
end_of_list.
4969
4970
Attempt to eliminate one assumption:
4971
4972
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
4973
4974
***********************************************
4975
AUTO SKETCHES ITERATION 17
4976
***********************************************
4977
4978
Starting a search with 1 assumptions:
4979
4980
formulas(assumptions_active).
4981
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4982
end_of_list.
4983
4984
Child search process 22013 started.
4985
4986
============================== PROCESS INITIAL CLAUSES ===============
4987
4988
% Clauses before input processing:
4989
4990
formulas(usable).
4991
end_of_list.
4992
4993
formulas(sos).
4994
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
4995
f(c2,c1) != f(c1,c2) # label("Commutativity"). [deny(1)].
4996
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
4997
end_of_list.
4998
4999
formulas(demodulators).
5000
end_of_list.
5001
5002
% 1460 hints input.
5003
5004
============================== PREDICATE ELIMINATION =================
5005
5006
No predicates eliminated.
5007
5008
============================== end predicate elimination =============
5009
5010
Auto_denials:
5011
% copying label "Commutativity" to answer in negative clause
5012
5013
Term ordering decisions:
5014
Function symbol KB weights: c1=1. c2=1. f=1.
5015
Predicate symbol precedence: predicate_order([ = ]).
5016
Function symbol precedence: function_order([ c1, c2, f ]).
5017
Skipping inverse_order, because term ordering is KBO.
5018
Unfolding symbols: (none).
5019
5020
Auto_inference settings:
5021
% set(paramodulation). % (positive equality literals)
5022
% set(paramodulation) -> set(back_demod).
5023
5024
Auto_process settings: (no changes).
5025
5026
============================== end of process initial clauses ========
5027
5028
============================== CLAUSES FOR SEARCH ====================
5029
5030
% Clauses after input processing:
5031
5032
formulas(usable).
5033
end_of_list.
5034
5035
formulas(sos).
5036
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
5037
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
5038
4 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
5039
end_of_list.
5040
5041
formulas(demodulators).
5042
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
5043
4 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
5044
end_of_list.
5045
5046
% 1460 hints processed (771 are redundant).
5047
5048
============================== end of clauses for search =============
5049
5050
============================== SEARCH ================================
5051
5052
% Starting search at 0.08 seconds.
5053
5054
% Operation f is commutative; C redundancy checks enabled.
5055
5056
============================== PROOF =================================
5057
5058
% Proof 1 at 1.68 (+ 0.00) seconds: "Commutativity".
5059
% Length of proof is 57.
5060
% Level of proof is 18.
5061
% Maximum clause weight is 59.
5062
% Given clauses 58.
5063
5064
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
5065
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
5066
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
5067
4 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
5068
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))].
5069
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))].
5070
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))].
5071
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))].
5072
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))].
5073
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))].
5074
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))].
5075
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)].
5076
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)])].
5077
74 f(x,f(f(x,x),x)) = f(x,x). [para(18(a,1),5(a,1,2)),rewrite([18(8)]),flip(a)].
5078
77 f(f(x,f(f(y,x),x)),f(y,y)) = y. [back_rewrite(71),rewrite([74(6)])].
5079
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)])].
5080
79 f(f(x,f(f(f(y,y),x),x)),y) = f(y,y). [back_rewrite(31),rewrite([74(3),74(8)])].
5081
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)])].
5082
82 f(f(x,x),f(x,f(y,x))) = x. [para(74(a,1),2(a,1,1))].
5083
84 f(f(x,x),f(x,x)) = x. [para(74(a,1),2(a,1,2)),rewrite([74(3)])].
5084
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))].
5085
127 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([82(6),77(19),82(17)])].
5086
149 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)].
5087
213 f(x,f(f(y,f(f(x,y),y)),x)) = f(y,f(f(x,y),y)). [para(77(a,1),97(a,1,1,2,1)),rewrite([82(4)])].
5088
252 f(f(f(x,f(x,y)),x),f(y,f(z,f(f(x,z),z)))) = y. [para(213(a,1),4(a,1,2,2))].
5089
263 f(x,f(y,f(f(f(x,x),y),y))) = f(x,x). [para(213(a,1),5(a,1,2)),rewrite([74(3),74(4),82(4),74(3),74(8)])].
5090
266 f(f(x,f(f(y,x),x)),f(y,f(x,f(f(y,x),x)))) = y. [para(77(a,1),213(a,1,2,1,2,1)),rewrite([82(7),77(14),82(12)])].
5091
300 f(f(f(x,f(x,y)),x),f(y,y)) = y. [para(5(a,1),252(a,1,2)),rewrite([74(3),74(4),82(4),74(6),74(8),74(9),82(9)])].
5092
345 f(f(f(f(x,x),x),f(x,x)),x) = f(x,x). [para(84(a,1),300(a,1,1,1,2)),rewrite([84(7)])].
5093
364 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(345(a,1),4(a,1,1,1,2)),rewrite([80(6)])].
5094
381 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)])].
5095
387 f(f(f(f(x,f(x,x)),x),x),f(x,x)) = x. [para(78(a,1),18(a,1,1)),rewrite([263(9)])].
5096
405 f(f(x,x),f(f(x,f(x,x)),f(y,x))) = f(x,f(x,x)). [para(381(a,1),2(a,1,1))].
5097
425 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(381(a,1),213(a,1,2,1,2,1)),rewrite([381(22)])].
5098
426 f(f(x,f(x,x)),x) = f(x,x). [para(381(a,1),213(a,1,2,1)),rewrite([82(6),381(8)])].
5099
430 f(f(x,x),x) = f(x,f(x,x)). [back_rewrite(78),rewrite([405(6),426(5)]),flip(a)].
5100
431 f(x,f(x,f(x,x))) = f(x,x). [back_rewrite(425),rewrite([426(3),430(2),426(6),430(5),82(6),426(3),430(2),426(6),430(5),426(9),430(8),82(9),426(6)])].
5101
449 f(f(x,f(x,x)),f(x,x)) = x. [back_rewrite(387),rewrite([426(3),430(2)])].
5102
458 f(f(x,x),f(x,f(x,y))) = x. [back_rewrite(364),rewrite([430(2),426(3),430(4),449(6),426(3),430(3),449(5)])].
5103
696 f(f(x,x),f(f(y,f(y,x)),y)) = x. [para(300(a,1),28(a,1,1,2,1,1,2,1)),rewrite([458(8),263(9),266(11),300(11),458(9)])].
5104
1231 f(f(x,x),f(y,x)) = x. [para(149(a,1),2(a,1,2)),rewrite([430(2),431(3)])].
5105
1245 f(x,f(y,f(x,x))) = f(x,x). [para(149(a,1),5(a,1,2)),rewrite([430(2),431(3),430(3),431(4),458(4),430(2),431(3),430(5),431(6)])].
5106
1249 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(149(a,1),79(a,1,1,2,1))].
5107
1254 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(149(a,1),213(a,1,2,1,2,1)),rewrite([149(26)])].
5108
1289 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(149(a,1),149(a,1,2,1,1,1)),rewrite([149(12),149(27)])].
5109
1297 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(127),rewrite([1245(5)])].
5110
1480 f(f(f(x,y),f(x,y)),f(f(x,y),y)) = f(x,y). [para(1231(a,1),82(a,1,2,2))].
5111
1488 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(1231(a,1),28(a,1,1,1,2,1)),rewrite([1231(14),1231(19)])].
5112
1496 f(f(f(x,y),f(x,y)),y) = f(x,y). [para(1231(a,1),696(a,1,2,1,2)),rewrite([430(5),449(7)])].
5113
1499 f(f(x,y),f(y,f(x,y))) = y. [para(1231(a,1),149(a,1,2,1,1,1)),rewrite([1231(4),1231(4),1231(7)])].
5114
1502 f(f(f(x,y),y),f(f(x,y),f(f(x,y),y))) = f(x,y). [back_rewrite(1289),rewrite([1496(4),1496(9),1480(8),1496(7)])].
5115
1516 f(x,f(f(y,x),x)) = f(y,x). [back_rewrite(1254),rewrite([1496(4),1496(7),1502(7),1496(7),1496(10),1502(10)])].
5116
1518 f(f(x,f(y,y)),y) = f(y,y). [back_rewrite(1249),rewrite([1496(7),1496(13),1502(12)])].
5117
1531 f(f(x,x),y) = f(y,f(x,x)). [back_rewrite(1488),rewrite([1499(4),1516(6),1499(6),1518(3),1499(5),430(2),1245(3),1516(8)])].
5118
1535 f(f(x,x),f(x,y)) = x. [back_rewrite(1297),rewrite([1499(7),1516(4)])].
5119
1795 f(x,y) = f(y,x). [para(1231(a,1),1531(a,1,1)),rewrite([1535(4)])].
5120
1796 $F # answer("Commutativity"). [resolve(1795,a,3,a)].
5121
5122
============================== end of proof ==========================
5123
5124
============================== STATISTICS ============================
5125
5126
Given=58. Generated=8110. Kept=1794. proofs=1.
5127
Usable=7. Sos=164. Demods=169. Limbo=0, Disabled=1625. Hints=1460.
5128
Weight_deleted=896. Literals_deleted=0.
5129
Forward_subsumed=5420. Back_subsumed=196.
5130
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
5131
New_demodulators=1785 (0 lex), Back_demodulated=1426. Back_unit_deleted=0.
5132
Demod_attempts=407851. Demod_rewrites=33848.
5133
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
5134
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
5135
Megabytes=4.27.
5136
User_CPU=1.68, System_CPU=0.00, Wall_clock=13.
5137
5138
============================== end of statistics =====================
5139
5140
============================== end of search =========================
5141
5142
Exiting with 1 proof.
5143
5144
Process 22013 exit (max_proofs) Mon Jun 4 22:53:38 2007
5145
5146
Successful proof using the following 1 assumptions:
5147
5148
formulas(assumptions_in_proof).
5149
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m"). [assumption].
5150
end_of_list.
5151
5152
Including 170 new hint clauses:
5153
5154
formulas(new_hints).
5155
$F # label("Commutativity") # label(non_clause) # label(goal).
5156
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
5157
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
5158
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
5159
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)).
5160
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)).
5161
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
5162
f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y.
5163
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)).
5164
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)).
5165
f(f(x,f(f(f(y,f(f(y,y),y)),x),x)),y) = f(y,f(f(y,y),y)).
5166
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)))).
5167
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)))).
5168
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)))).
5169
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)))).
5170
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)))).
5171
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).
5172
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)))).
5173
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)))).
5174
f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y.
5175
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)).
5176
f(x,x) = f(x,f(f(x,x),x)).
5177
f(x,f(f(x,x),x)) = f(x,x).
5178
f(f(x,f(f(y,x),x)),f(y,y)) = y.
5179
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).
5180
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).
5181
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).
5182
f(f(x,x),f(f(x,f(x,x)),f(x,x))) = f(f(f(x,f(x,x)),x),x).
5183
f(f(x,f(f(f(y,y),x),x)),y) = f(y,f(f(y,y),y)).
5184
f(f(x,f(f(f(y,y),x),x)),y) = f(y,y).
5185
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)).
5186
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)).
5187
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)).
5188
f(f(f(f(x,x),x),f(x,x)),f(x,x)) = f(x,f(f(f(x,x),x),x)).
5189
f(f(x,x),f(x,f(y,x))) = x.
5190
f(f(x,f(f(x,x),x)),f(x,x)) = x.
5191
f(f(x,x),f(x,x)) = x.
5192
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
5193
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))).
5194
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))).
5195
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))).
5196
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.
5197
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)).
5198
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)).
5199
f(x,y) = f(y,f(f(f(f(x,y),f(x,y)),y),y)).
5200
f(x,f(f(f(f(y,x),f(y,x)),x),x)) = f(y,x).
5201
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)).
5202
f(x,f(f(y,f(f(x,y),y)),x)) = f(y,f(f(x,y),y)).
5203
f(f(f(x,f(x,y)),x),f(y,f(z,f(f(x,z),z)))) = y.
5204
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)).
5205
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)).
5206
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)).
5207
f(x,f(y,f(f(f(x,f(f(x,x),x)),y),y))) = f(x,f(f(x,x),x)).
5208
f(x,f(y,f(f(f(x,x),y),y))) = f(x,f(f(x,x),x)).
5209
f(x,f(y,f(f(f(x,x),y),y))) = f(x,x).
5210
f(f(x,f(f(y,x),x)),f(f(f(y,y),f(y,f(y,y))),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))).
5211
f(f(x,f(f(y,x),x)),f(y,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))).
5212
f(f(x,f(f(y,x),x)),f(y,f(x,f(f(y,x),x)))) = f(f(y,y),f(y,f(y,y))).
5213
f(f(x,f(f(y,x),x)),f(y,f(x,f(f(y,x),x)))) = y.
5214
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)))).
5215
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)))).
5216
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)))).
5217
f(f(f(x,f(x,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)))).
5218
f(f(f(x,f(x,y)),x),f(y,y)) = f(f(y,f(f(y,y),y)),f(y,f(y,f(f(y,y),y)))).
5219
f(f(f(x,f(x,y)),x),f(y,y)) = f(f(y,y),f(y,f(y,f(f(y,y),y)))).
5220
f(f(f(x,f(x,y)),x),f(y,y)) = f(f(y,y),f(y,f(y,y))).
5221
f(f(f(x,f(x,y)),x),f(y,y)) = y.
5222
f(f(f(f(x,x),x),f(x,x)),f(f(x,x),f(x,x))) = f(x,x).
5223
f(f(f(f(x,x),x),f(x,x)),x) = f(x,x).
5224
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.
5225
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.
5226
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).
5227
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,f(x,x)),x),x)) = f(x,x).
5228
f(x,f(f(f(x,f(x,x)),x),x)) = f(x,x).
5229
f(f(f(f(x,f(x,x)),x),x),f(x,f(y,f(f(f(x,x),y),y)))) = x.
5230
f(f(f(f(x,f(x,x)),x),x),f(x,x)) = x.
5231
f(f(x,x),f(f(x,f(x,x)),f(y,x))) = f(x,f(x,x)).
5232
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))).
5233
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))).
5234
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)).
5235
f(f(x,f(x,x)),x) = f(x,f(f(f(x,f(x,x)),x),x)).
5236
f(f(x,f(x,x)),x) = f(x,x).
5237
f(x,f(x,x)) = f(f(f(x,f(x,x)),x),x).
5238
f(x,f(x,x)) = f(f(x,x),x).
5239
f(f(x,x),x) = f(x,f(x,x)).
5240
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))).
5241
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))).
5242
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))).
5243
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))).
5244
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))).
5245
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))).
5246
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))).
5247
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))).
5248
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))).
5249
f(x,f(x,f(x,x))) = f(f(x,f(x,x)),f(f(x,x),f(f(x,x),x))).
5250
f(x,f(x,f(x,x))) = f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))).
5251
f(x,f(x,f(x,x))) = f(f(x,f(x,x)),x).
5252
f(x,f(x,f(x,x))) = f(x,x).
5253
f(f(f(x,x),x),f(x,x)) = x.
5254
f(f(x,f(x,x)),f(x,x)) = x.
5255
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.
5256
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.
5257
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.
5258
f(f(f(x,f(x,x)),x),f(x,f(f(f(f(x,x),x),f(x,x)),y))) = x.
5259
f(f(x,x),f(x,f(f(f(f(x,x),x),f(x,x)),y))) = x.
5260
f(f(x,x),f(x,f(f(f(x,f(x,x)),f(x,x)),y))) = x.
5261
f(f(x,x),f(x,f(x,y))) = x.
5262
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))).
5263
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))).
5264
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))).
5265
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))).
5266
f(f(y,y),f(f(z,f(z,y)),z)) = f(f(y,y),f(y,f(y,y))).
5267
f(f(x,x),f(f(y,f(y,x)),y)) = x.
5268
f(f(x,f(f(x,x),x)),f(y,x)) = x.
5269
f(f(x,f(x,f(x,x))),f(y,x)) = x.
5270
f(f(x,x),f(y,x)) = x.
5271
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)).
5272
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)).
5273
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)).
5274
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)).
5275
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)).
5276
f(x,f(y,f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
5277
f(x,f(y,f(x,f(x,f(x,x))))) = f(x,f(f(x,x),x)).
5278
f(x,f(y,f(x,x))) = f(x,f(f(x,x),x)).
5279
f(x,f(y,f(x,x))) = f(x,f(x,f(x,x))).
5280
f(x,f(y,f(x,x))) = f(x,x).
5281
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).
5282
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))).
5283
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))).
5284
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)).
5285
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)).
5286
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).
5287
f(f(f(x,f(y,y)),f(f(y,y),f(x,f(y,y)))),f(z,f(f(y,z),z))) = y.
5288
f(f(f(x,y),f(x,y)),f(f(x,y),y)) = f(x,y).
5289
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))).
5290
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))).
5291
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))).
5292
f(f(f(x,y),f(x,y)),f(f(f(y,y),y),f(y,y))) = f(x,y).
5293
f(f(f(x,y),f(x,y)),f(f(y,f(y,y)),f(y,y))) = f(x,y).
5294
f(f(f(x,y),f(x,y)),y) = f(x,y).
5295
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)).
5296
f(f(x,y),f(f(f(y,y),f(x,y)),f(x,y))) = f(f(y,y),f(x,y)).
5297
f(f(x,y),f(y,f(x,y))) = f(f(y,y),f(x,y)).
5298
f(f(x,y),f(y,f(x,y))) = y.
5299
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).
5300
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).
5301
f(f(f(x,y),y),f(f(x,y),f(f(f(f(x,y),f(x,y)),y),y))) = f(x,y).
5302
f(f(f(x,y),y),f(f(x,y),f(f(x,y),y))) = f(x,y).
5303
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))).
5304
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))).
5305
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))).
5306
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))).
5307
f(x,f(f(y,x),x)) = f(f(f(y,x),x),f(f(y,x),f(f(y,x),x))).
5308
f(x,f(f(y,x),x)) = f(y,x).
5309
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).
5310
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).
5311
f(f(x,f(y,y)),y) = f(y,y).
5312
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))).
5313
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))).
5314
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))).
5315
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))).
5316
f(f(y,f(f(y,y),y)),z) = f(f(y,y),f(f(z,f(y,y)),f(y,y))).
5317
f(f(y,f(y,f(y,y))),z) = f(f(y,y),f(f(z,f(y,y)),f(y,y))).
5318
f(f(y,y),z) = f(f(y,y),f(f(z,f(y,y)),f(y,y))).
5319
f(f(x,x),y) = f(y,f(x,x)).
5320
f(f(y,y),f(z,f(f(y,z),z))) = y.
5321
f(f(x,x),f(x,y)) = x.
5322
f(x,y) = f(y,f(f(x,x),f(x,x))).
5323
f(x,y) = f(y,x).
5324
$F # answer("Commutativity").
5325
end_of_list.
5326
5327
Attempt to eliminate one assumption:
5328
5329
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
5330
5331
***********************************************
5332
AUTO SKETCHES ITERATION 18
5333
***********************************************
5334
5335
Starting a search with 0 assumptions:
5336
5337
formulas(assumptions_active).
5338
end_of_list.
5339
5340
Note: Attempt with no assumptions!
5341
5342
5343
Child search process 22016 started.
5344
5345
============================== PROCESS INITIAL CLAUSES ===============
5346
5347
% Clauses before input processing:
5348
5349
formulas(usable).
5350
end_of_list.
5351
5352
formulas(sos).
5353
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
5354
f(c2,c1) != f(c1,c2) # label("Commutativity"). [deny(1)].
5355
end_of_list.
5356
5357
formulas(demodulators).
5358
end_of_list.
5359
5360
% 1630 hints input.
5361
5362
============================== PREDICATE ELIMINATION =================
5363
5364
No predicates eliminated.
5365
5366
============================== end predicate elimination =============
5367
5368
Auto_denials:
5369
% copying label "Commutativity" to answer in negative clause
5370
5371
Term ordering decisions:
5372
Function symbol KB weights: c1=1. c2=1. f=1.
5373
Predicate symbol precedence: predicate_order([ = ]).
5374
Function symbol precedence: function_order([ c1, c2, f ]).
5375
Skipping inverse_order, because term ordering is KBO.
5376
Unfolding symbols: (none).
5377
5378
Auto_inference settings:
5379
% set(paramodulation). % (positive equality literals)
5380
% set(paramodulation) -> set(back_demod).
5381
5382
Auto_process settings: (no changes).
5383
5384
============================== end of process initial clauses ========
5385
5386
============================== CLAUSES FOR SEARCH ====================
5387
5388
% Clauses after input processing:
5389
5390
formulas(usable).
5391
end_of_list.
5392
5393
formulas(sos).
5394
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
5395
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
5396
end_of_list.
5397
5398
formulas(demodulators).
5399
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
5400
end_of_list.
5401
5402
% 1630 hints processed (814 are redundant).
5403
5404
============================== end of clauses for search =============
5405
5406
============================== SEARCH ================================
5407
5408
% Starting search at 0.10 seconds.
5409
5410
% Operation f is commutative; C redundancy checks enabled.
5411
5412
============================== PROOF =================================
5413
5414
% Proof 1 at 0.31 (+ 0.00) seconds: "Commutativity".
5415
% Length of proof is 25.
5416
% Level of proof is 12.
5417
% Maximum clause weight is 61.
5418
% Given clauses 31.
5419
5420
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal). [goal].
5421
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10"). [assumption].
5422
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity"). [deny(1)].
5423
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))].
5424
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))].
5425
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))].
5426
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)])].
5427
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))].
5428
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)])].
5429
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)].
5430
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)].
5431
56 f(x,f(f(f(f(y,x),f(y,x)),x),x)) = f(y,x). [back_rewrite(35),rewrite([45(7)])].
5432
62 f(f(x,f(f(y,x),x)),f(y,y)) = y. [back_rewrite(27),rewrite([45(6)])].
5433
73 f(f(x,x),f(x,f(y,x))) = x. [para(45(a,1),2(a,1,1))].
5434
104 f(f(x,x),f(y,x)) = x. [para(56(a,1),2(a,1,2)),rewrite([45(3)])].
5435
115 f(f(f(f(f(f(x,y),f(x,y)),y),y),f(f(x,y),f(f(f(f(x,y),f(x,y)),y),y))),f(y,y)) = y. [para(56(a,1),62(a,1,1,2,1))].
5436
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)])].
5437
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))].
5438
143 f(f(f(x,y),f(x,y)),y) = f(x,y). [para(104(a,1),104(a,1,2))].
5439
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)])].
5440
150 f(f(x,y),f(y,y)) = y. [back_rewrite(115),rewrite([143(4),143(7),148(7)])].
5441
157 f(x,f(f(y,x),x)) = f(y,x). [back_rewrite(56),rewrite([143(4)])].
5442
197 f(f(f(x,y),f(z,y)),x) = f(x,y). [back_rewrite(6),rewrite([157(4),157(7),157(7)])].
5443
257 f(x,y) = f(y,x). [para(150(a,1),197(a,1,1))].
5444
258 $F # answer("Commutativity"). [resolve(257,a,3,a)].
5445
5446
============================== end of proof ==========================
5447
5448
============================== STATISTICS ============================
5449
5450
Given=31. Generated=1114. Kept=256. proofs=1.
5451
Usable=17. Sos=63. Demods=79. Limbo=1, Disabled=176. Hints=1630.
5452
Weight_deleted=2. Literals_deleted=0.
5453
Forward_subsumed=856. Back_subsumed=34.
5454
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
5455
New_demodulators=253 (0 lex), Back_demodulated=140. Back_unit_deleted=0.
5456
Demod_attempts=33613. Demod_rewrites=3985.
5457
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
5458
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
5459
Megabytes=1.46.
5460
User_CPU=0.31, System_CPU=0.00, Wall_clock=14.
5461
5462
============================== end of statistics =====================
5463
5464
============================== end of search =========================
5465
5466
Exiting with 1 proof.
5467
5468
Process 22016 exit (max_proofs) Mon Jun 4 22:53:39 2007
5469
5470
Proof is complete (no extra assumptions used).
5471
5472
Autosketches completes after 18 iterations.
5473
5474
THEOREM PROVED
5475
5476
Exiting.
5477
5478
Process 21989 exit (max_proofs) Mon Jun 4 22:53:39 2007
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