============================== NewAuto ===============================
NewAuto (32) version May-2007, May 2007.
Process 21989 was started by mccune on cleo,
Mon Jun  4 22:53:25 2007
The command was "../newauto -f BA.in".
============================== end of head ===========================

============================== INPUT =================================

% Reading from file BA.in

assign(order,kbo).
set(lex_order_vars).
assign(lex_dep_demod_lim,0).
assign(max_megs,750).
assign(sos_limit,20000).
assign(demod_step_limit,200).
assign(demod_size_limit,100000).
clear(print_kept).
clear(print_given).

formulas(sos).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
end_of_list.

formulas(goals).
f(x,y) = f(y,x) # label("Commutativity").
end_of_list.

formulas(extra_assumptions).
f(f(f(y,f(x,y)),y),f(x,f(z,y))) = x # label("Sh2/A14_m").
f(f(y,f(y,f(y,x))),f(x,f(y,z))) = x # label("C1/A1").
f(f(y,f(y,f(x,y))),f(x,f(z,y))) = x # label("C2/A4 (Wald 1)").
f(f(y,f(y,f(x,x))),f(x,f(z,y))) = x # label("C3/A6").
f(f(y,f(y,f(x,z))),f(x,f(z,y))) = x # label("C4/A7").
f(f(y,f(y,f(z,x))),f(x,f(y,z))) = x # label("C5/A8 (Wald 2))").
f(f(y,f(f(x,y),y)),f(x,f(y,z))) = x # label("C6/A9").
f(f(y,f(y,f(y,x))),f(x,f(z,y))) = x # label("C7/A2").
f(f(f(f(y,x),y),y),f(x,f(z,y))) = x # label("C8/A12_m").
f(f(f(f(y,x),z),z),f(x,f(y,z))) = x # label("C9/A13_m").
f(f(f(y,f(y,x)),y),f(x,f(z,y))) = x # label("C10/A16_m").
f(f(f(y,f(x,x)),y),f(x,f(z,y))) = x # label("C11/A18_m").
f(f(f(f(y,x),z),z),f(x,f(z,y))) = x # label("C12/A19_m").
f(f(f(f(y,x),y),y),f(x,f(y,z))) = x # label("C13/A20_m").
f(f(f(y,f(x,z)),y),f(x,f(y,z))) = x # label("C14/A21_m").
f(f(f(y,f(z,x)),y),f(x,f(y,z))) = x # label("C15/A23_m").
f(f(f(y,f(y,x)),y),f(x,f(y,z))) = x # label("C16/A24_m").
end_of_list.

============================== end of input ==========================

============================== PROCESS NON-CLAUSAL FORMULAS ==========

% Formulas that are not ordinary clauses:
1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal).  [goal].

============================== end of process non-clausal formulas ===

***********************************************
AUTO SKETCHES ITERATION 1
***********************************************

Starting a search with 17 assumptions:

formulas(assumptions_active).
f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m").  [assumption].
f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y # label("C1/A1").  [assumption].
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").  [assumption].
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").  [assumption].
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Child search process 21990 started.

============================== PROCESS INITIAL CLAUSES ===============

% Clauses before input processing:

formulas(usable).
end_of_list.

formulas(sos).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
f(c2,c1) != f(c1,c2) # label("Commutativity").  [deny(1)].
f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m").  [assumption].
f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y # label("C1/A1").  [assumption].
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").  [assumption].
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").  [assumption].
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
end_of_list.

============================== PREDICATE ELIMINATION =================

No predicates eliminated.

============================== end predicate elimination =============

Auto_denials:
  % copying label "Commutativity" to answer in negative clause

Term ordering decisions:
Function symbol KB weights:  c1=1. c2=1. f=1.
Predicate symbol precedence:  predicate_order([ = ]).
Function symbol precedence:  function_order([ c1, c2, f ]).
Skipping inverse_order, because term ordering is KBO.
Unfolding symbols: (none).

Auto_inference settings:
  % set(paramodulation).  % (positive equality literals)
    % set(paramodulation) -> set(back_demod).

Auto_process settings:  (no changes).

============================== end of process initial clauses ========

============================== CLAUSES FOR SEARCH ====================

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m").  [assumption].
5 f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y # label("C1/A1").  [assumption].
6 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").  [assumption].
7 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").  [assumption].
8 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
9 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
10 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
11 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
12 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
13 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
14 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
15 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
16 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
17 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
18 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
19 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
20 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
4 f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m").  [assumption].
5 f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y # label("C1/A1").  [assumption].
6 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").  [assumption].
7 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").  [assumption].
8 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
9 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
10 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
11 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
12 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
13 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
14 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
15 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
16 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
17 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
18 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
19 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
20 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

============================== end of clauses for search =============

============================== SEARCH ================================

% Starting search at 0.00 seconds.

% Operation f is commutative; C redundancy checks enabled.

============================== PROOF =================================

% Proof 1 at 1.52 (+ 0.01) seconds: "Commutativity".
% Length of proof is 36.
% Level of proof is 12.
% Maximum clause weight is 37.
% Given clauses 52.

1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal).  [goal].
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
5 f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y # label("C1/A1").  [assumption].
6 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").  [assumption].
9 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
10 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
12 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
13 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
14 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
15 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
18 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
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))].
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))].
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))].
1022 f(f(f(x,f(x,x)),x),f(x,x)) = x.  [para(2(a,1),33(a,1,2,2))].
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)].
1120 f(f(x,x),f(x,f(x,y))) = x.  [para(1074(a,1),5(a,1,1))].
1121 f(f(x,x),f(x,x)) = x.  [para(1074(a,1),5(a,1,2)),rewrite([1074(3)])].
1123 f(f(x,x),f(x,f(y,x))) = x.  [para(1074(a,1),6(a,1,1))].
1125 f(f(f(x,x),x),f(x,x)) = x.  [para(1074(a,1),6(a,1,2)),rewrite([1120(5)])].
1130 f(f(x,f(x,x)),x) = f(x,x).  [para(1074(a,1),9(a,1,1,2)),rewrite([1121(5)])].
1134 f(f(x,f(f(x,x),x)),f(x,x)) = x.  [para(1074(a,1),10(a,1,2))].
1139 f(f(x,f(x,x)),f(x,x)) = x.  [para(1074(a,1),12(a,1,2)),rewrite([1125(4)])].
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)].
1206 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.  [para(1121(a,1),15(a,1,1,1))].
1531 f(f(x,x),f(x,f(y,f(f(x,y),y)))) = x.  [para(23(a,1),1123(a,1,2,2))].
1567 f(f(x,x),x) = f(x,f(x,x)).  [para(1139(a,1),2(a,1,2)),rewrite([1130(3),1160(3)])].
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)])].
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)])].
2567 f(f(x,f(y,y)),f(x,y)) = x.  [para(1531(a,1),10(a,1,2,2)),rewrite([1866(6)])].
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)])].
2657 f(f(x,x),y) = f(y,f(x,x)).  [para(2567(a,1),14(a,1,2)),rewrite([1875(5)])].
2909 f(x,f(f(x,x),x)) = f(x,x).  [para(1121(a,1),2613(a,1,1)),rewrite([1121(4)])].
2954 f(x,y) = f(y,x).  [para(2(a,1),2657(a,1,1)),rewrite([2909(4),2909(5),1121(4)])].
2955 $F # answer("Commutativity").  [resolve(2954,a,3,a)].

============================== end of proof ==========================

============================== STATISTICS ============================

Given=52. Generated=7036. Kept=2953. proofs=1.
Usable=45. Sos=2543. Demods=2590. Limbo=7, Disabled=376. Hints=0.
Weight_deleted=129. Literals_deleted=0.
Forward_subsumed=3954. Back_subsumed=152.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=2947 (0 lex), Back_demodulated=205. Back_unit_deleted=0.
Demod_attempts=180094. Demod_rewrites=11763.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
Megabytes=4.84.
User_CPU=1.52, System_CPU=0.01, Wall_clock=1.

============================== end of statistics =====================

============================== end of search =========================

Exiting with 1 proof.

Process 21990 exit (max_proofs) Mon Jun  4 22:53:26 2007

Successful proof using the following 9 assumptions:

formulas(assumptions_in_proof).
f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y # label("C1/A1").  [assumption].
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").  [assumption].
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
end_of_list.

Including 60 new hint clauses:

formulas(new_hints).
$F # label("Commutativity") # label(non_clause) # label(goal).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y # label("C1/A1").
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
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)).
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)).
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
f(f(f(x,f(x,x)),x),f(x,x)) = x.
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))).
f(x,f(f(x,f(x,f(x,x))),f(x,f(x,f(x,x))))) = f(x,f(x,f(x,x))).
f(x,x) = f(x,f(x,f(x,x))).
f(x,f(x,f(x,x))) = f(x,x).
f(f(x,x),f(x,f(x,y))) = x.
f(f(x,f(x,f(x,x))),f(x,x)) = x.
f(f(x,x),f(x,x)) = x.
f(f(x,x),f(x,f(y,x))) = x.
f(f(f(x,x),f(f(x,x),f(x,f(x,x)))),f(x,x)) = x.
f(f(f(x,x),x),f(x,x)) = x.
f(f(x,f(x,x)),f(f(x,x),f(x,x))) = f(x,x).
f(f(x,f(x,x)),x) = f(x,x).
f(f(x,f(f(x,x),x)),f(x,x)) = x.
f(f(f(f(f(x,x),x),f(x,x)),f(x,x)),f(x,x)) = x.
f(f(x,f(x,x)),f(x,x)) = x.
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)).
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)).
f(x,f(f(x,f(f(x,x),x)),f(x,x))) = 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(x,x).
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
f(f(x,x),f(x,f(y,f(f(x,y),y)))) = x.
f(f(x,f(f(f(x,f(x,x)),x),x)),x) = f(x,f(x,x)).
f(f(x,f(f(x,x),x)),x) = f(x,f(x,x)).
f(f(x,x),x) = f(x,f(x,x)).
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)).
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
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).
f(f(x,f(x,f(y,f(x,x)))),x) = f(x,x).
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
f(f(x,f(y,y)),f(x,y)) = x.
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
f(f(x,y),f(x,f(y,y))) = x.
f(f(f(x,f(x,f(y,f(x,x)))),x),y) = f(y,f(x,x)).
f(f(x,x),y) = f(y,f(x,x)).
f(x,f(f(x,x),f(f(x,x),f(x,x)))) = f(x,x).
f(x,f(f(x,x),x)) = f(x,x).
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
f(x,y) = f(y,f(f(x,x),f(x,f(f(x,x),x)))).
f(x,y) = f(y,f(f(x,x),f(x,x))).
f(x,y) = f(y,x).
$F # answer("Commutativity").
end_of_list.

Attempt to eliminate one assumption:

   f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y # label("C1/A1").

***********************************************
AUTO SKETCHES ITERATION 2
***********************************************

Starting a search with 16 assumptions:

formulas(assumptions_active).
f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m").  [assumption].
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").  [assumption].
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").  [assumption].
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Child search process 21991 started.

============================== PROCESS INITIAL CLAUSES ===============

% Clauses before input processing:

formulas(usable).
end_of_list.

formulas(sos).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
f(c2,c1) != f(c1,c2) # label("Commutativity").  [deny(1)].
f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m").  [assumption].
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").  [assumption].
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").  [assumption].
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
end_of_list.

% 60 hints input.

============================== PREDICATE ELIMINATION =================

No predicates eliminated.

============================== end predicate elimination =============

Auto_denials:
  % copying label "Commutativity" to answer in negative clause

Term ordering decisions:
Function symbol KB weights:  c1=1. c2=1. f=1.
Predicate symbol precedence:  predicate_order([ = ]).
Function symbol precedence:  function_order([ c1, c2, f ]).
Skipping inverse_order, because term ordering is KBO.
Unfolding symbols: (none).

Auto_inference settings:
  % set(paramodulation).  % (positive equality literals)
    % set(paramodulation) -> set(back_demod).

Auto_process settings:  (no changes).

============================== end of process initial clauses ========

============================== CLAUSES FOR SEARCH ====================

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m").  [assumption].
5 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").  [assumption].
6 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").  [assumption].
7 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
8 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
9 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
10 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
11 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
12 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
13 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
14 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
15 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
16 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
17 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
18 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
19 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
4 f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m").  [assumption].
5 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").  [assumption].
6 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").  [assumption].
7 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
8 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
9 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
10 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
11 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
12 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
13 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
14 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
15 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
16 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
17 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
18 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
19 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

% 60 hints processed (13 are redundant).

============================== end of clauses for search =============

============================== SEARCH ================================

% Starting search at 0.00 seconds.

% Operation f is commutative; C redundancy checks enabled.

============================== PROOF =================================

% Proof 1 at 0.68 (+ 0.00) seconds: "Commutativity".
% Length of proof is 28.
% Level of proof is 11.
% Maximum clause weight is 37.
% Given clauses 35.

1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal).  [goal].
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m").  [assumption].
5 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").  [assumption].
8 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
9 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
13 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
14 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
19 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
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))].
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))].
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))].
837 f(f(f(x,f(x,x)),x),f(x,x)) = x.  [para(2(a,1),32(a,1,2,2))].
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))].
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))].
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))].
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)].
1350 f(f(x,f(f(x,x),x)),f(x,x)) = x.  [back_rewrite(837),rewrite([1267(3)])].
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)].
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)])].
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)])].
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)])].
1451 f(f(x,f(x,x)),x) = f(x,x).  [back_rewrite(1267),rewrite([1358(6)])].
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)].
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)])].
1824 f(x,y) = f(y,x).  [para(2(a,1),1787(a,1,1)),rewrite([2(8)])].
1825 $F # answer("Commutativity").  [resolve(1824,a,3,a)].

============================== end of proof ==========================

============================== STATISTICS ============================

Given=35. Generated=3495. Kept=1823. proofs=1.
Usable=29. Sos=1392. Demods=1426. Limbo=7, Disabled=412. Hints=60.
Weight_deleted=53. Literals_deleted=0.
Forward_subsumed=1619. Back_subsumed=93.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=1818 (0 lex), Back_demodulated=301. Back_unit_deleted=0.
Demod_attempts=80052. Demod_rewrites=5188.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
Megabytes=2.60.
User_CPU=0.68, System_CPU=0.00, Wall_clock=2.

============================== end of statistics =====================

============================== end of search =========================

Exiting with 1 proof.

Process 21991 exit (max_proofs) Mon Jun  4 22:53:27 2007

Successful proof using the following 7 assumptions:

formulas(assumptions_in_proof).
f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m").  [assumption].
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").  [assumption].
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Including 48 new hint clauses:

formulas(new_hints).
$F # label("Commutativity") # label(non_clause) # label(goal).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m").
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
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)).
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).
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
f(f(f(x,f(x,x)),x),f(x,x)) = x.
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
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))).
f(f(f(x,f(y,x)),f(x,f(x,f(y,x)))),x) = f(x,f(f(x,x),x)).
f(f(f(x,f(y,x)),f(x,f(x,f(y,x)))),x) = f(f(x,f(x,x)),x).
f(x,f(f(x,x),x)) = f(f(x,f(x,x)),x).
f(f(x,f(x,x)),x) = f(x,f(f(x,x),x)).
f(f(x,f(f(x,x),x)),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(f(x,x),x)))) = f(x,f(f(x,x),x)).
f(x,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))) = 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(x,x).
f(f(f(x,f(f(x,x),x)),x),f(x,f(f(x,f(f(x,x),x)),x))) = x.
f(f(f(x,x),x),f(x,f(f(x,f(f(x,x),x)),x))) = x.
f(f(f(x,x),x),f(x,f(f(x,x),x))) = x.
f(f(f(x,x),x),f(x,x)) = x.
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).
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).
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).
f(x,f(f(x,x),f(y,f(x,f(f(x,x),x))))) = f(x,x).
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x).
f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x.
f(f(x,f(x,x)),f(x,f(y,f(x,f(f(x,x),x))))) = x.
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
f(f(x,f(x,x)),x) = f(x,x).
f(f(f(x,f(x,x)),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
f(f(x,x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y).
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)).
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
f(f(x,x),y) = f(y,f(x,x)).
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
f(x,y) = f(y,x).
$F # answer("Commutativity").
end_of_list.

Attempt to eliminate one assumption:

   f(f(f(x,f(y,x)),x),f(y,f(z,x))) = y # label("Sh2/A14_m").

***********************************************
AUTO SKETCHES ITERATION 3
***********************************************

Starting a search with 15 assumptions:

formulas(assumptions_active).
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").  [assumption].
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").  [assumption].
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Child search process 21994 started.

============================== PROCESS INITIAL CLAUSES ===============

% Clauses before input processing:

formulas(usable).
end_of_list.

formulas(sos).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
f(c2,c1) != f(c1,c2) # label("Commutativity").  [deny(1)].
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").  [assumption].
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").  [assumption].
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
end_of_list.

% 108 hints input.

============================== PREDICATE ELIMINATION =================

No predicates eliminated.

============================== end predicate elimination =============

Auto_denials:
  % copying label "Commutativity" to answer in negative clause

Term ordering decisions:
Function symbol KB weights:  c1=1. c2=1. f=1.
Predicate symbol precedence:  predicate_order([ = ]).
Function symbol precedence:  function_order([ c1, c2, f ]).
Skipping inverse_order, because term ordering is KBO.
Unfolding symbols: (none).

Auto_inference settings:
  % set(paramodulation).  % (positive equality literals)
    % set(paramodulation) -> set(back_demod).

Auto_process settings:  (no changes).

============================== end of process initial clauses ========

============================== CLAUSES FOR SEARCH ====================

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").  [assumption].
5 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").  [assumption].
6 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
7 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
8 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
9 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
10 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
11 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
12 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
13 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
14 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
15 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
16 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
17 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
18 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
4 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").  [assumption].
5 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").  [assumption].
6 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
7 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
8 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
9 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
10 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
11 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
12 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
13 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
14 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
15 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
16 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
17 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
18 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

% 108 hints processed (41 are redundant).

============================== end of clauses for search =============

============================== SEARCH ================================

% Starting search at 0.00 seconds.

% Operation f is commutative; C redundancy checks enabled.

============================== PROOF =================================

% Proof 1 at 0.76 (+ 0.00) seconds: "Commutativity".
% Length of proof is 44.
% Level of proof is 14.
% Maximum clause weight is 43.
% Given clauses 25.

1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal).  [goal].
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").  [assumption].
8 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
9 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
12 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
14 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
16 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
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))].
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))].
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))].
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))].
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))].
758 f(f(f(x,f(x,x)),x),f(x,x)) = x.  [para(2(a,1),23(a,1,2,2))].
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))].
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))].
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))].
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))].
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))].
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))].
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)])].
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)])].
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))].
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))].
1181 f(x,f(f(x,x),x)) = f(x,x).  [para(8(a,1),1096(a,1,1)),flip(a)].
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)])].
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)])].
1319 f(f(x,f(x,x)),f(x,x)) = x.  [back_rewrite(821),rewrite([1238(7)])].
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)])].
1381 f(f(x,x),f(x,x)) = x.  [para(1181(a,1),2(a,1,2)),rewrite([1181(3)])].
1390 f(f(x,x),f(x,f(x,y))) = x.  [para(1181(a,1),8(a,1,1))].
1404 f(f(f(x,x),x),x) = f(x,x).  [para(1181(a,1),16(a,1,1,1)),rewrite([1381(5)])].
1417 f(f(x,x),x) = f(x,f(x,x)).  [back_rewrite(1323),rewrite([1390(4),1404(3)]),flip(a)].
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)])].
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)])].
1552 f(x,f(f(y,x),x)) = f(x,y).  [back_rewrite(1240),rewrite([1417(2),1319(4)]),flip(a)].
1579 f(f(x,y),x) = f(x,f(x,y)).  [back_rewrite(1449),rewrite([1552(5),1552(6),1552(6)])].
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)])].
1729 f(f(f(x,y),y),y) = f(x,y).  [back_rewrite(349),rewrite([1552(4),1642(4),1552(6)])].
1762 f(x,f(x,f(y,x))) = f(y,x).  [back_rewrite(98),rewrite([1552(3),1729(3),1579(3),1552(6)])].
1932 f(f(x,y),y) = f(y,f(x,y)).  [back_rewrite(1448),rewrite([1762(4),1552(6),1762(5)])].
2067 f(x,y) = f(y,x).  [back_rewrite(1552),rewrite([1932(2),1762(3)])].
2068 $F # answer("Commutativity").  [resolve(2067,a,3,a)].

============================== end of proof ==========================

============================== STATISTICS ============================

Given=25. Generated=3521. Kept=2066. proofs=1.
Usable=6. Sos=368. Demods=505. Limbo=135, Disabled=1573. Hints=108.
Weight_deleted=33. Literals_deleted=0.
Forward_subsumed=1422. Back_subsumed=83.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=2055 (0 lex), Back_demodulated=1473. Back_unit_deleted=0.
Demod_attempts=80861. Demod_rewrites=6184.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
Megabytes=2.55.
User_CPU=0.76, System_CPU=0.00, Wall_clock=3.

============================== end of statistics =====================

============================== end of search =========================

Exiting with 1 proof.

Process 21994 exit (max_proofs) Mon Jun  4 22:53:28 2007

Successful proof using the following 6 assumptions:

formulas(assumptions_in_proof).
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").  [assumption].
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
end_of_list.

Including 78 new hint clauses:

formulas(new_hints).
$F # label("Commutativity") # label(non_clause) # label(goal).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
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)).
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)).
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)).
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
f(f(f(x,f(x,x)),x),f(x,x)) = x.
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x.
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)).
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)))).
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))).
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))).
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)).
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)).
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)).
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))))).
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))))).
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))))).
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))))).
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))))).
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))))).
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)).
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)).
f(f(f(x,f(y,x)),f(x,f(x,f(y,x)))),x) = 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(x,x).
f(x,f(f(x,x),f(y,f(x,f(f(x,x),x))))) = f(x,f(f(x,x),x)).
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,f(f(x,x),x)).
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x).
f(f(f(f(x,x),x),f(x,f(f(x,x),x))),y) = f(x,f(f(y,x),x)).
f(f(f(f(x,x),x),f(x,x)),y) = f(x,f(f(y,x),x)).
f(f(x,f(x,x)),f(x,x)) = x.
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))).
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,x),x),x)) = f(f(x,x),x).
f(f(x,f(f(x,x),x)),f(x,x)) = x.
f(f(x,x),f(x,x)) = x.
f(f(x,x),f(x,f(x,y))) = x.
f(f(f(x,x),x),f(f(x,x),f(x,x))) = f(x,x).
f(f(f(x,x),x),x) = f(x,x).
f(x,f(f(f(x,x),x),x)) = f(f(x,x),x).
f(x,f(x,x)) = f(f(x,x),x).
f(f(x,x),x) = f(x,f(x,x)).
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)))).
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)).
f(f(f(x,f(x,x)),f(x,x)),y) = f(x,f(f(y,x),x)).
f(x,y) = f(x,f(f(y,x),x)).
f(x,f(f(y,x),x)) = f(x,y).
f(f(x,y),f(f(x,f(x,y)),f(x,y))) = f(x,f(f(f(x,y),x),x)).
f(f(x,y),x) = f(x,f(f(f(x,y),x),x)).
f(f(x,y),x) = f(x,f(x,y)).
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)).
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)).
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)).
f(f(f(x,y),f(f(f(y,z),f(x,y)),f(x,y))),z) = f(y,f(f(z,y),y)).
f(f(f(x,y),f(y,z)),z) = f(y,f(f(z,y),y)).
f(f(f(x,y),f(y,z)),z) = f(y,z).
f(f(f(f(f(x,y),f(y,z)),z),z),z) = f(y,f(f(z,y),y)).
f(f(f(y,z),z),z) = f(y,f(f(z,y),y)).
f(f(f(x,y),y),y) = f(x,y).
f(f(x,f(f(f(y,x),x),x)),x) = f(y,f(f(x,y),y)).
f(f(x,f(y,x)),x) = f(y,f(f(x,y),y)).
f(x,f(x,f(y,x))) = f(y,f(f(x,y),y)).
f(x,f(x,f(y,x))) = f(y,x).
f(f(x,y),f(f(y,f(x,y)),f(x,y))) = f(y,f(y,f(y,f(x,y)))).
f(f(x,y),y) = f(y,f(y,f(y,f(x,y)))).
f(f(x,y),y) = f(y,f(x,y)).
f(x,f(x,f(y,x))) = f(x,y).
f(x,y) = f(y,x).
$F # answer("Commutativity").
end_of_list.

Attempt to eliminate one assumption:

   f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y # label("C2/A4 (Wald 1)").

***********************************************
AUTO SKETCHES ITERATION 4
***********************************************

Starting a search with 14 assumptions:

formulas(assumptions_active).
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").  [assumption].
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Child search process 21995 started.

============================== PROCESS INITIAL CLAUSES ===============

% Clauses before input processing:

formulas(usable).
end_of_list.

formulas(sos).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
f(c2,c1) != f(c1,c2) # label("Commutativity").  [deny(1)].
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").  [assumption].
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
end_of_list.

% 186 hints input.

============================== PREDICATE ELIMINATION =================

No predicates eliminated.

============================== end predicate elimination =============

Auto_denials:
  % copying label "Commutativity" to answer in negative clause

Term ordering decisions:
Function symbol KB weights:  c1=1. c2=1. f=1.
Predicate symbol precedence:  predicate_order([ = ]).
Function symbol precedence:  function_order([ c1, c2, f ]).
Skipping inverse_order, because term ordering is KBO.
Unfolding symbols: (none).

Auto_inference settings:
  % set(paramodulation).  % (positive equality literals)
    % set(paramodulation) -> set(back_demod).

Auto_process settings:  (no changes).

============================== end of process initial clauses ========

============================== CLAUSES FOR SEARCH ====================

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").  [assumption].
5 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
6 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
7 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
8 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
9 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
10 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
11 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
12 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
13 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
14 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
15 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
16 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
17 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
4 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").  [assumption].
5 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
6 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
7 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
8 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
9 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
10 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
11 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
12 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
13 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
14 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
15 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
16 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
17 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

% 186 hints processed (71 are redundant).

============================== end of clauses for search =============

============================== SEARCH ================================

% Starting search at 0.01 seconds.

% Operation f is commutative; C redundancy checks enabled.

============================== PROOF =================================

% Proof 1 at 0.87 (+ 0.00) seconds: "Commutativity".
% Length of proof is 52.
% Level of proof is 16.
% Maximum clause weight is 51.
% Given clauses 28.

1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal).  [goal].
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").  [assumption].
5 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
7 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
8 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
9 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
10 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
11 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
13 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
14 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
17 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
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))].
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))].
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))].
687 f(f(f(x,f(x,x)),x),f(x,x)) = x.  [para(2(a,1),22(a,1,2,2))].
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))].
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))].
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))].
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))].
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))].
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))].
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))].
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))].
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)])].
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))].
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))].
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))].
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))].
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)])].
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)].
1273 f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)).  [back_rewrite(870),rewrite([1203(4)])].
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)])].
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)])].
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)])].
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)])].
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))].
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)].
1688 f(f(f(f(x,x),x),x),x) = f(f(x,x),x).  [para(1612(a,1),1025(a,2,2))].
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)])].
1700 f(f(x,f(x,x)),x) = f(x,x).  [back_rewrite(1670),rewrite([1687(6)])].
1722 f(f(x,x),x) = f(x,f(x,x)).  [back_rewrite(1307),rewrite([1700(3),1687(5),1687(6),1688(4)])].
1723 f(x,f(x,f(x,x))) = f(x,x).  [back_rewrite(1294),rewrite([1687(4),1700(6)])].
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)])].
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)].
1889 f(f(x,y),x) = f(x,f(x,y)).  [back_rewrite(1736),rewrite([1750(5),1750(6),1750(6)])].
1893 f(f(x,y),y) = f(y,f(x,y)).  [back_rewrite(1694),rewrite([1750(5),1750(6),1750(6)])].
2496 f(x,f(x,f(y,x))) = f(x,y).  [back_rewrite(1750),rewrite([1893(2)])].
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)])].
2530 f(x,y) = f(y,x).  [back_rewrite(1092),rewrite([1893(2),1889(3),2496(3),2513(5),1889(3),2496(3)])].
2531 $F # answer("Commutativity").  [resolve(2530,a,3,a)].

============================== end of proof ==========================

============================== STATISTICS ============================

Given=28. Generated=4010. Kept=2529. proofs=1.
Usable=7. Sos=265. Demods=891. Limbo=637, Disabled=1635. Hints=186.
Weight_deleted=37. Literals_deleted=0.
Forward_subsumed=1444. Back_subsumed=59.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=2478 (0 lex), Back_demodulated=1560. Back_unit_deleted=0.
Demod_attempts=100609. Demod_rewrites=7865.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
Megabytes=3.50.
User_CPU=0.87, System_CPU=0.00, Wall_clock=4.

============================== end of statistics =====================

============================== end of search =========================

Exiting with 1 proof.

Process 21995 exit (max_proofs) Mon Jun  4 22:53:29 2007

Successful proof using the following 10 assumptions:

formulas(assumptions_in_proof).
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").  [assumption].
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Including 99 new hint clauses:

formulas(new_hints).
$F # label("Commutativity") # label(non_clause) # label(goal).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
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)).
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
f(f(f(x,f(x,x)),x),f(x,x)) = x.
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x.
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)).
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))).
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).
f(f(x,f(f(f(x,f(x,x)),x),x)),x) = f(x,f(x,x)).
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)).
f(f(f(f(x,f(x,f(x,x))),x),x),x) = f(x,f(x,x)).
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)))).
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).
f(f(f(x,y),y),y) = f(x,f(f(y,x),x)).
f(f(x,f(f(f(x,y),y),y)),f(f(y,x),f(x,x))) = f(y,x).
f(f(f(x,f(f(y,x),x)),y),f(y,f(f(x,y),y))) = y.
f(f(f(f(x,x),f(y,x)),f(y,x)),f(f(f(x,y),y),y)) = x.
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)).
f(f(x,f(x,f(x,x))),x) = f(x,f(x,x)).
f(f(f(x,f(f(x,x),x)),f(x,f(x,f(f(x,x),x)))),f(x,f(f(x,x),x))) = f(f(x,f(f(x,x),x)),x).
f(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) = f(x,f(x,f(f(x,x),x))).
f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)).
f(f(f(x,x),f(x,f(x,x))),f(x,f(x,x))) = f(f(x,f(x,x)),x).
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)).
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))))).
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))))).
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))))).
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))))).
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))))).
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))))).
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))))).
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)).
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)))).
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)))).
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)))).
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)))).
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)))).
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)))).
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)))).
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)))).
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)))).
f(f(x,f(x,x)),f(x,x)) = x.
f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))) = f(x,x).
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))).
f(f(f(x,f(x,x)),x),f(x,f(x,x))) = f(f(x,x),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(f(f(f(x,x),x),x),x) = f(f(x,x),x).
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)).
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)).
f(f(x,f(x,x)),x) = f(x,x).
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)).
f(f(f(f(x,x),x),f(f(x,x),f(x,f(x,x)))),x) = f(x,f(x,x)).
f(f(f(f(x,x),x),x),x) = f(x,f(x,x)).
f(f(x,x),x) = f(x,f(x,x)).
f(x,f(x,f(x,x))) = f(f(x,f(x,x)),x).
f(x,f(x,f(x,x))) = f(x,x).
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)).
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)).
f(f(f(x,f(x,x)),f(x,f(f(x,x),x))),y) = f(x,f(f(y,x),x)).
f(f(f(x,f(x,x)),f(x,f(x,f(x,x)))),y) = f(x,f(f(y,x),x)).
f(f(f(x,f(x,x)),f(x,x)),y) = f(x,f(f(y,x),x)).
f(x,y) = f(x,f(f(y,x),x)).
f(x,f(f(y,x),x)) = f(x,y).
f(f(x,y),f(f(x,f(x,y)),f(x,y))) = f(x,f(f(f(x,y),x),x)).
f(f(x,y),x) = f(x,f(f(f(x,y),x),x)).
f(f(x,y),x) = f(x,f(x,y)).
f(f(x,y),f(f(y,f(x,y)),f(x,y))) = f(y,f(f(f(x,y),y),y)).
f(f(x,y),y) = f(y,f(f(f(x,y),y),y)).
f(f(x,y),y) = f(y,f(x,y)).
f(x,f(x,f(y,x))) = f(x,y).
f(f(f(y,x),f(f(x,x),f(y,x))),f(f(f(x,y),y),y)) = x.
f(f(f(y,x),f(f(x,x),f(y,x))),f(f(y,f(x,y)),y)) = x.
f(f(f(y,x),f(f(x,x),f(y,x))),f(y,f(y,f(x,y)))) = x.
f(f(f(y,x),f(f(x,x),f(y,x))),f(y,x)) = x.
f(f(y,x),f(f(y,x),f(f(x,x),f(y,x)))) = x.
f(f(x,y),f(y,y)) = y.
f(f(x,f(f(y,f(x,y)),y)),f(f(y,x),f(x,x))) = f(y,x).
f(f(x,f(y,f(y,f(x,y)))),f(f(y,x),f(x,x))) = f(y,x).
f(f(x,f(y,x)),f(f(y,x),f(x,x))) = f(y,x).
f(f(x,f(y,x)),x) = f(y,x).
f(x,f(x,f(y,x))) = f(y,x).
f(x,y) = f(y,x).
$F # answer("Commutativity").
end_of_list.

Attempt to eliminate one assumption:

   f(f(x,f(x,f(y,y))),f(y,f(z,x))) = y # label("C3/A6").

***********************************************
AUTO SKETCHES ITERATION 5
***********************************************

Starting a search with 13 assumptions:

formulas(assumptions_active).
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Child search process 21996 started.

============================== PROCESS INITIAL CLAUSES ===============

% Clauses before input processing:

formulas(usable).
end_of_list.

formulas(sos).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
f(c2,c1) != f(c1,c2) # label("Commutativity").  [deny(1)].
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
end_of_list.

% 285 hints input.

============================== PREDICATE ELIMINATION =================

No predicates eliminated.

============================== end predicate elimination =============

Auto_denials:
  % copying label "Commutativity" to answer in negative clause

Term ordering decisions:
Function symbol KB weights:  c1=1. c2=1. f=1.
Predicate symbol precedence:  predicate_order([ = ]).
Function symbol precedence:  function_order([ c1, c2, f ]).
Skipping inverse_order, because term ordering is KBO.
Unfolding symbols: (none).

Auto_inference settings:
  % set(paramodulation).  % (positive equality literals)
    % set(paramodulation) -> set(back_demod).

Auto_process settings:  (no changes).

============================== end of process initial clauses ========

============================== CLAUSES FOR SEARCH ====================

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
5 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
6 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
7 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
8 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
9 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
10 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
11 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
12 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
13 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
14 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
15 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
16 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
4 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
5 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
6 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
7 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
8 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
9 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
10 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
11 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
12 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
13 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
14 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
15 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
16 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

% 285 hints processed (128 are redundant).

============================== end of clauses for search =============

============================== SEARCH ================================

% Starting search at 0.02 seconds.

% Operation f is commutative; C redundancy checks enabled.

============================== PROOF =================================

% Proof 1 at 0.74 (+ 0.00) seconds: "Commutativity".
% Length of proof is 59.
% Level of proof is 17.
% Maximum clause weight is 51.
% Given clauses 30.

1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal).  [goal].
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
6 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
7 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
8 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
9 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
10 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
12 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
13 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
15 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
16 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
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))].
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))].
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))].
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))].
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))].
611 f(f(f(x,f(x,x)),x),f(x,x)) = x.  [para(2(a,1),21(a,1,2,2))].
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))].
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)])].
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))].
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))].
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)])].
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))].
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))].
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)])].
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))].
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)].
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))].
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)].
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))].
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)])].
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)].
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)])].
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)])].
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))].
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)].
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)])].
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)])].
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)])].
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))].
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)])].
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))].
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)])].
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)])].
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))].
1551 f(x,f(f(x,x),x)) = f(x,x).  [back_rewrite(1221),rewrite([1548(8)]),flip(a)].
1554 f(f(x,f(x,x)),f(x,x)) = x.  [back_rewrite(1548),rewrite([1551(3),1551(5)])].
1586 f(f(x,f(x,x)),x) = f(x,x).  [back_rewrite(1293),rewrite([1551(3),1551(6)])].
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)].
1602 f(x,f(f(y,x),x)) = f(y,x).  [back_rewrite(1343),rewrite([1599(4),1554(5)])].
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)])].
1768 f(f(x,y),y) = f(y,f(y,x)).  [back_rewrite(1159),rewrite([1602(4),1602(5)])].
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)].
2065 f(x,f(x,f(x,y))) = f(y,x).  [back_rewrite(1602),rewrite([1768(2)])].
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)])].
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)])].
2139 $F # answer("Commutativity").  [resolve(2138,a,3,a)].

============================== end of proof ==========================

============================== STATISTICS ============================

Given=30. Generated=3223. Kept=2137. proofs=1.
Usable=15. Sos=613. Demods=985. Limbo=370, Disabled=1153. Hints=285.
Weight_deleted=0. Literals_deleted=0.
Forward_subsumed=1086. Back_subsumed=44.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=2095 (0 lex), Back_demodulated=1094. Back_unit_deleted=0.
Demod_attempts=67957. Demod_rewrites=4889.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
Megabytes=2.83.
User_CPU=0.74, System_CPU=0.00, Wall_clock=5.

============================== end of statistics =====================

============================== end of search =========================

Exiting with 1 proof.

Process 21996 exit (max_proofs) Mon Jun  4 22:53:30 2007

Successful proof using the following 10 assumptions:

formulas(assumptions_in_proof).
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").  [assumption].
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Including 101 new hint clauses:

formulas(new_hints).
$F # label("Commutativity") # label(non_clause) # label(goal).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y)).
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
f(f(f(f(f(x,y),f(f(x,f(x,z)),x)),z),z),z) = f(f(x,f(x,z)),x).
f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(y,f(y,x)),y).
f(f(f(x,f(x,x)),x),f(x,x)) = x.
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
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)).
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)).
f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x.
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).
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)).
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)).
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)).
f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)).
f(f(x,f(f(f(x,f(x,x)),x),x)),x) = f(x,f(x,x)).
f(f(f(f(x,f(x,f(x,x))),x),x),x) = f(x,f(x,x)).
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)))).
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).
f(f(f(x,y),y),y) = f(x,f(f(y,x),x)).
f(x,f(x,f(x,x))) = f(x,f(f(f(x,f(x,x)),x),x)).
f(x,f(f(f(x,f(x,x)),x),x)) = f(x,f(x,f(x,x))).
f(f(f(x,f(f(y,x),x)),y),f(y,f(f(x,y),y))) = y.
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))).
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)).
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))).
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)).
f(f(x,f(x,f(x,x))),x) = f(x,f(x,x)).
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)))).
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)))).
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)))).
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))))).
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).
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)).
f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)).
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))).
f(x,f(f(f(y,x),x),x)) = f(x,f(y,f(f(x,y),y))).
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)).
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).
f(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) = f(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))),x) = f(x,f(f(x,x),x)).
f(f(f(x,x),f(x,f(x,x))),f(x,f(x,x))) = f(f(x,f(x,x)),x).
f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,x).
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)).
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)).
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).
f(f(x,x),f(f(x,f(x,x)),x)) = f(f(x,f(x,x)),f(x,x)).
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))).
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
f(x,x) = f(x,f(f(x,x),x)).
f(x,f(f(x,x),x)) = f(x,x).
f(f(x,f(x,x)),f(x,f(f(x,x),x))) = x.
f(f(x,f(x,x)),f(x,x)) = x.
f(f(x,f(x,x)),x) = f(x,f(f(x,x),x)).
f(f(x,f(x,x)),x) = f(x,x).
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))).
f(f(x,f(x,x)),f(f(x,f(x,x)),x)) = f(f(x,x),f(x,f(x,x))).
f(f(x,f(x,x)),f(x,x)) = f(f(x,x),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(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x).
f(x,f(f(y,x),x)) = f(y,x).
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).
f(f(y,x),x) = f(x,f(y,f(f(x,y),y))).
f(f(x,y),y) = f(y,f(y,x)).
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).
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).
f(x,f(f(y,x),x)) = f(f(f(y,x),x),x).
f(x,f(x,f(x,y))) = f(f(f(y,x),x),x).
f(x,f(x,f(x,y))) = f(f(x,f(x,y)),x).
f(f(x,f(x,y)),x) = f(x,f(x,f(x,y))).
f(x,f(x,f(x,y))) = f(y,x).
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(f(y,f(y,x)),y).
f(f(f(x,f(f(y,z),f(x,y))),x),x) = f(f(y,f(y,x)),y).
f(x,f(x,f(x,f(f(y,z),f(x,y))))) = f(f(y,f(y,x)),y).
f(f(f(y,z),f(x,y)),x) = f(f(y,f(y,x)),y).
f(f(f(y,z),f(x,y)),x) = f(y,f(y,f(y,x))).
f(f(f(x,y),f(z,x)),z) = f(z,x).
f(f(f(f(f(x,y),f(x,f(x,f(x,z)))),z),z),z) = f(f(x,f(x,z)),x).
f(f(f(f(f(x,y),f(z,x)),z),z),z) = f(f(x,f(x,z)),x).
f(f(f(z,x),z),z) = f(f(x,f(x,z)),x).
f(z,f(z,f(z,x))) = f(f(x,f(x,z)),x).
f(x,z) = f(f(x,f(x,z)),x).
f(x,z) = f(x,f(x,f(x,z))).
f(x,y) = f(y,x).
$F # answer("Commutativity").
end_of_list.

Attempt to eliminate one assumption:

   f(f(x,f(x,f(y,z))),f(y,f(z,x))) = y # label("C4/A7").

***********************************************
AUTO SKETCHES ITERATION 6
***********************************************

Starting a search with 12 assumptions:

formulas(assumptions_active).
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Child search process 21997 started.

============================== PROCESS INITIAL CLAUSES ===============

% Clauses before input processing:

formulas(usable).
end_of_list.

formulas(sos).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
f(c2,c1) != f(c1,c2) # label("Commutativity").  [deny(1)].
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
end_of_list.

% 386 hints input.

============================== PREDICATE ELIMINATION =================

No predicates eliminated.

============================== end predicate elimination =============

Auto_denials:
  % copying label "Commutativity" to answer in negative clause

Term ordering decisions:
Function symbol KB weights:  c1=1. c2=1. f=1.
Predicate symbol precedence:  predicate_order([ = ]).
Function symbol precedence:  function_order([ c1, c2, f ]).
Skipping inverse_order, because term ordering is KBO.
Unfolding symbols: (none).

Auto_inference settings:
  % set(paramodulation).  % (positive equality literals)
    % set(paramodulation) -> set(back_demod).

Auto_process settings:  (no changes).

============================== end of process initial clauses ========

============================== CLAUSES FOR SEARCH ====================

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
5 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
6 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
7 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
8 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
9 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
10 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
11 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
12 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
13 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
14 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
15 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
4 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
5 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
6 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
7 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
8 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
9 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
10 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
11 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
12 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
13 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
14 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
15 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

% 386 hints processed (172 are redundant).

============================== end of clauses for search =============

============================== SEARCH ================================

% Starting search at 0.02 seconds.

% Operation f is commutative; C redundancy checks enabled.

============================== PROOF =================================

% Proof 1 at 0.36 (+ 0.00) seconds: "Commutativity".
% Length of proof is 37.
% Level of proof is 15.
% Maximum clause weight is 35.
% Given clauses 31.

1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal).  [goal].
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
5 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
8 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
9 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
11 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
12 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
15 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
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))].
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))].
515 f(f(f(x,f(x,x)),x),f(x,x)) = x.  [para(2(a,1),20(a,1,2,2))].
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))].
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)])].
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))].
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))].
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))].
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)])].
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)])].
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))].
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)])].
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)])].
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)])].
861 f(x,f(f(x,x),x)) = f(x,x).  [back_rewrite(548),rewrite([827(4),685(8)]),flip(a)].
889 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.  [back_rewrite(834),rewrite([861(3),861(5)])].
896 f(f(x,x),x) = f(x,f(x,x)).  [back_rewrite(827),rewrite([861(3),861(5)])].
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)])].
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)])].
924 f(x,f(x,f(x,x))) = f(x,x).  [back_rewrite(861),rewrite([896(2)])].
928 f(f(x,f(x,x)),f(x,x)) = x.  [back_rewrite(918),rewrite([924(3),924(5)])].
937 f(f(x,f(x,x)),x) = f(x,x).  [back_rewrite(905),rewrite([924(3),924(6)])].
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)])].
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)].
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)])].
1203 f(x,y) = f(y,x).  [para(2(a,1),1172(a,1,1)),rewrite([2(8)])].
1204 $F # answer("Commutativity").  [resolve(1203,a,3,a)].

============================== end of proof ==========================

============================== STATISTICS ============================

Given=31. Generated=2036. Kept=1202. proofs=1.
Usable=22. Sos=728. Demods=756. Limbo=8, Disabled=457. Hints=386.
Weight_deleted=0. Literals_deleted=0.
Forward_subsumed=834. Back_subsumed=51.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=1197 (0 lex), Back_demodulated=392. Back_unit_deleted=0.
Demod_attempts=36008. Demod_rewrites=2734.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
Megabytes=1.74.
User_CPU=0.36, System_CPU=0.00, Wall_clock=5.

============================== end of statistics =====================

============================== end of search =========================

Exiting with 1 proof.

Process 21997 exit (max_proofs) Mon Jun  4 22:53:30 2007

Successful proof using the following 7 assumptions:

formulas(assumptions_in_proof).
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").  [assumption].
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Including 64 new hint clauses:

formulas(new_hints).
$F # label("Commutativity") # label(non_clause) # label(goal).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
f(f(f(x,f(x,x)),x),f(x,x)) = x.
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
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)).
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)).
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).
f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x.
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))).
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)).
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)).
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)).
f(f(f(x,f(f(x,x),x)),x),x) = 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(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(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)))).
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)))).
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
f(f(f(x,f(f(x,x),x)),x),f(x,f(y,f(x,f(f(x,x),x))))) = x.
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)).
f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
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))).
f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))).
f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,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,x) = f(x,f(f(x,x),x)).
f(x,f(f(x,x),x)) = f(x,x).
f(f(x,f(x,x)),f(x,f(y,f(x,f(f(x,x),x))))) = x.
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
f(f(x,x),x) = f(x,f(x,f(f(x,x),x))).
f(f(x,x),x) = f(x,f(x,x)).
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(f(x,x),x)).
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(x,f(x,x))).
f(f(x,f(x,f(x,f(x,x)))),f(x,f(f(x,x),x))) = x.
f(f(x,f(x,f(x,f(x,x)))),f(x,f(x,f(x,x)))) = x.
f(x,f(x,f(x,x))) = f(x,x).
f(f(x,f(x,x)),f(x,f(x,f(x,x)))) = x.
f(f(x,f(x,x)),f(x,x)) = x.
f(f(x,f(x,x)),x) = f(x,f(x,f(x,x))).
f(f(x,f(x,x)),x) = f(x,x).
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).
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).
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).
f(x,f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x).
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x).
f(f(f(x,f(x,x)),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
f(f(x,x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y).
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)).
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
f(f(x,x),y) = f(y,f(x,x)).
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
f(x,y) = f(y,x).
$F # answer("Commutativity").
end_of_list.

Attempt to eliminate one assumption:

   f(f(x,f(x,f(y,z))),f(z,f(x,y))) = z # label("C5/A8 (Wald 2))").

***********************************************
AUTO SKETCHES ITERATION 7
***********************************************

Starting a search with 11 assumptions:

formulas(assumptions_active).
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Child search process 21998 started.

============================== PROCESS INITIAL CLAUSES ===============

% Clauses before input processing:

formulas(usable).
end_of_list.

formulas(sos).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
f(c2,c1) != f(c1,c2) # label("Commutativity").  [deny(1)].
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
end_of_list.

% 450 hints input.

============================== PREDICATE ELIMINATION =================

No predicates eliminated.

============================== end predicate elimination =============

Auto_denials:
  % copying label "Commutativity" to answer in negative clause

Term ordering decisions:
Function symbol KB weights:  c1=1. c2=1. f=1.
Predicate symbol precedence:  predicate_order([ = ]).
Function symbol precedence:  function_order([ c1, c2, f ]).
Skipping inverse_order, because term ordering is KBO.
Unfolding symbols: (none).

Auto_inference settings:
  % set(paramodulation).  % (positive equality literals)
    % set(paramodulation) -> set(back_demod).

Auto_process settings:  (no changes).

============================== end of process initial clauses ========

============================== CLAUSES FOR SEARCH ====================

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
5 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
6 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
7 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
8 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
9 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
10 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
11 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
12 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
13 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
14 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
4 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
5 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
6 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
7 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
8 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
9 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
10 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
11 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
12 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
13 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
14 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

% 450 hints processed (218 are redundant).

============================== end of clauses for search =============

============================== SEARCH ================================

% Starting search at 0.03 seconds.

% Operation f is commutative; C redundancy checks enabled.

============================== PROOF =================================

% Proof 1 at 0.33 (+ 0.00) seconds: "Commutativity".
% Length of proof is 42.
% Level of proof is 15.
% Maximum clause weight is 35.
% Given clauses 30.

1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal).  [goal].
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
5 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
7 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
8 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
10 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
11 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
13 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
14 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
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))].
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))].
433 f(f(f(x,f(x,x)),x),f(x,x)) = x.  [para(2(a,1),25(a,1,2,2))].
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))].
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)])].
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))].
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))].
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))].
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)])].
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))].
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)])].
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))].
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))].
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)])].
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)])].
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)])].
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)])].
747 f(x,f(f(x,x),x)) = f(x,x).  [back_rewrite(462),rewrite([714(4),586(8)]),flip(a)].
772 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.  [back_rewrite(723),rewrite([747(3),747(5)])].
774 f(f(x,f(x,f(x,x))),x) = f(x,f(x,x)).  [back_rewrite(721),rewrite([747(3),747(7)])].
780 f(f(x,x),x) = f(x,f(x,x)).  [back_rewrite(714),rewrite([747(3),747(5)])].
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)])].
832 f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)).  [back_rewrite(574),rewrite([774(4)])].
836 f(x,f(x,f(x,x))) = f(x,x).  [back_rewrite(747),rewrite([780(2)])].
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)])].
883 f(f(x,f(x,x)),x) = f(x,x).  [back_rewrite(789),rewrite([836(3),836(6)])].
899 f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x).  [back_rewrite(868),rewrite([883(4)])].
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)].
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)])].
1105 f(x,y) = f(y,x).  [para(2(a,1),1094(a,1,1)),rewrite([2(8)])].
1106 $F # answer("Commutativity").  [resolve(1105,a,3,a)].

============================== end of proof ==========================

============================== STATISTICS ============================

Given=30. Generated=1871. Kept=1104. proofs=1.
Usable=21. Sos=622. Demods=649. Limbo=8, Disabled=465. Hints=450.
Weight_deleted=0. Literals_deleted=0.
Forward_subsumed=767. Back_subsumed=51.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=1099 (0 lex), Back_demodulated=401. Back_unit_deleted=0.
Demod_attempts=33582. Demod_rewrites=2408.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
Megabytes=1.63.
User_CPU=0.33, System_CPU=0.00, Wall_clock=5.

============================== end of statistics =====================

============================== end of search =========================

Exiting with 1 proof.

Process 21998 exit (max_proofs) Mon Jun  4 22:53:30 2007

Successful proof using the following 8 assumptions:

formulas(assumptions_in_proof).
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").  [assumption].
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Including 66 new hint clauses:

formulas(new_hints).
$F # label("Commutativity") # label(non_clause) # label(goal).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
f(f(f(x,f(x,x)),x),f(x,x)) = x.
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
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)).
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)).
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).
f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x.
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))).
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)).
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)).
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)).
f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)).
f(f(f(f(x,f(x,f(x,x))),x),x),x) = f(x,f(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(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(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)))).
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)))).
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
f(f(f(x,f(f(x,x),x)),x),f(x,f(y,f(x,f(f(x,x),x))))) = x.
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))).
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)).
f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
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))).
f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))).
f(f(x,f(f(x,f(f(x,x),x)),x)),x) = f(x,f(x,f(f(x,x),x))).
f(f(x,f(x,f(x,f(f(x,x),x)))),x) = f(x,f(x,f(f(x,x),x))).
f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,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,x) = f(x,f(f(x,x),x)).
f(x,f(f(x,x),x)) = f(x,x).
f(f(x,f(x,x)),f(x,f(y,f(x,f(f(x,x),x))))) = x.
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
f(f(x,f(x,f(x,x))),x) = f(x,f(x,f(f(x,x),x))).
f(f(x,f(x,f(x,x))),x) = f(x,f(x,x)).
f(f(x,x),x) = f(x,f(x,f(f(x,x),x))).
f(f(x,x),x) = f(x,f(x,x)).
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(f(x,x),x)).
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(x,f(x,x))).
f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)).
f(x,f(x,f(x,x))) = f(x,x).
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).
f(x,f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x).
f(f(x,f(x,x)),x) = f(x,f(x,f(x,x))).
f(f(x,f(x,x)),x) = f(x,x).
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x).
f(f(f(x,f(x,x)),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
f(f(x,x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y).
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)).
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
f(f(x,x),y) = f(y,f(x,x)).
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
f(x,y) = f(y,x).
$F # answer("Commutativity").
end_of_list.

Attempt to eliminate one assumption:

   f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").

***********************************************
AUTO SKETCHES ITERATION 8
***********************************************

Starting a search with 10 assumptions:

formulas(assumptions_active).
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Child search process 21999 started.

============================== PROCESS INITIAL CLAUSES ===============

% Clauses before input processing:

formulas(usable).
end_of_list.

formulas(sos).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
f(c2,c1) != f(c1,c2) # label("Commutativity").  [deny(1)].
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
end_of_list.

% 516 hints input.

============================== PREDICATE ELIMINATION =================

No predicates eliminated.

============================== end predicate elimination =============

Auto_denials:
  % copying label "Commutativity" to answer in negative clause

Term ordering decisions:
Function symbol KB weights:  c1=1. c2=1. f=1.
Predicate symbol precedence:  predicate_order([ = ]).
Function symbol precedence:  function_order([ c1, c2, f ]).
Skipping inverse_order, because term ordering is KBO.
Unfolding symbols: (none).

Auto_inference settings:
  % set(paramodulation).  % (positive equality literals)
    % set(paramodulation) -> set(back_demod).

Auto_process settings:  (no changes).

============================== end of process initial clauses ========

============================== CLAUSES FOR SEARCH ====================

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
5 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
6 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
7 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
8 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
9 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
10 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
11 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
12 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
13 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
4 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
5 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
6 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
7 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
8 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
9 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
10 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
11 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
12 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
13 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

% 516 hints processed (261 are redundant).

============================== end of clauses for search =============

============================== SEARCH ================================

% Starting search at 0.02 seconds.

% Operation f is commutative; C redundancy checks enabled.

============================== PROOF =================================

% Proof 1 at 0.27 (+ 0.01) seconds: "Commutativity".
% Length of proof is 38.
% Level of proof is 15.
% Maximum clause weight is 35.
% Given clauses 28.

1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal).  [goal].
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
6 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
7 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
9 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
10 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
12 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
13 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
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))].
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))].
375 f(f(f(x,f(x,x)),x),f(x,x)) = x.  [para(2(a,1),18(a,1,2,2))].
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))].
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)])].
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))].
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))].
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)])].
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))].
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)])].
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)])].
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))].
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)])].
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)])].
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)])].
644 f(x,f(f(x,x),x)) = f(x,x).  [back_rewrite(404),rewrite([613(4),523(8)]),flip(a)].
668 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.  [back_rewrite(622),rewrite([644(3),644(5)])].
676 f(f(x,x),x) = f(x,f(x,x)).  [back_rewrite(613),rewrite([644(3),644(5)])].
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)])].
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)])].
705 f(x,f(x,f(x,x))) = f(x,x).  [back_rewrite(644),rewrite([676(2)])].
709 f(f(x,f(x,x)),f(x,x)) = x.  [back_rewrite(697),rewrite([705(3),705(5)])].
715 f(f(x,f(x,x)),x) = f(x,x).  [back_rewrite(686),rewrite([705(3),705(6)])].
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)])].
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)].
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)])].
947 f(x,y) = f(y,x).  [para(2(a,1),935(a,1,1)),rewrite([2(8)])].
948 $F # answer("Commutativity").  [resolve(947,a,3,a)].

============================== end of proof ==========================

============================== STATISTICS ============================

Given=28. Generated=1599. Kept=946. proofs=1.
Usable=20. Sos=561. Demods=588. Limbo=9, Disabled=367. Hints=516.
Weight_deleted=0. Literals_deleted=0.
Forward_subsumed=653. Back_subsumed=41.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=941 (0 lex), Back_demodulated=314. Back_unit_deleted=0.
Demod_attempts=28728. Demod_rewrites=2185.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
Megabytes=1.47.
User_CPU=0.27, System_CPU=0.01, Wall_clock=6.

============================== end of statistics =====================

============================== end of search =========================

Exiting with 1 proof.

Process 21999 exit (max_proofs) Mon Jun  4 22:53:31 2007

Successful proof using the following 7 assumptions:

formulas(assumptions_in_proof).
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Including 67 new hint clauses:

formulas(new_hints).
$F # label("Commutativity") # label(non_clause) # label(goal).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
f(f(f(x,f(x,x)),x),f(x,f(y,f(x,f(x,x))))) = x.
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
f(f(f(x,f(x,x)),x),f(x,x)) = x.
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
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)).
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)).
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).
f(f(x,f(x,x)),f(x,f(f(x,x),f(x,f(x,x))))) = x.
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.
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.
f(f(x,f(f(y,x),x)),f(y,f(x,f(f(x,x),x)))) = y.
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))).
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)).
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)).
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)).
f(f(f(x,f(f(x,x),x)),x),x) = 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(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(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)))).
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)))).
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
f(f(f(x,f(f(x,x),x)),x),f(x,f(y,f(x,f(f(x,x),x))))) = x.
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)).
f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
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))).
f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))).
f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,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,x) = f(x,f(f(x,x),x)).
f(x,f(f(x,x),x)) = f(x,x).
f(f(x,f(x,x)),f(x,f(y,f(x,f(f(x,x),x))))) = x.
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
f(f(x,x),x) = f(x,f(x,f(f(x,x),x))).
f(f(x,x),x) = f(x,f(x,x)).
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(f(x,x),x)).
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(x,f(x,x))).
f(f(x,f(x,f(x,f(x,x)))),f(x,f(f(x,x),x))) = x.
f(f(x,f(x,f(x,f(x,x)))),f(x,f(x,f(x,x)))) = x.
f(x,f(x,f(x,x))) = f(x,x).
f(f(x,f(x,x)),f(x,f(x,f(x,x)))) = x.
f(f(x,f(x,x)),f(x,x)) = x.
f(f(x,f(x,x)),x) = f(x,f(x,f(x,x))).
f(f(x,f(x,x)),x) = f(x,x).
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).
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).
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).
f(x,f(f(x,x),f(y,f(f(x,f(x,x)),x)))) = f(x,x).
f(x,f(f(x,x),f(y,f(x,x)))) = f(x,x).
f(f(f(x,f(x,x)),x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
f(f(x,x),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(f(x,x),y).
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)).
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
f(f(x,x),y) = f(y,f(x,x)).
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
f(x,y) = f(y,x).
$F # answer("Commutativity").
end_of_list.

Attempt to eliminate one assumption:

   f(f(x,f(x,f(x,y))),f(y,f(z,x))) = y # label("C7/A2").

***********************************************
AUTO SKETCHES ITERATION 9
***********************************************

Starting a search with 9 assumptions:

formulas(assumptions_active).
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Child search process 22000 started.

============================== PROCESS INITIAL CLAUSES ===============

% Clauses before input processing:

formulas(usable).
end_of_list.

formulas(sos).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
f(c2,c1) != f(c1,c2) # label("Commutativity").  [deny(1)].
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
end_of_list.

% 583 hints input.

============================== PREDICATE ELIMINATION =================

No predicates eliminated.

============================== end predicate elimination =============

Auto_denials:
  % copying label "Commutativity" to answer in negative clause

Term ordering decisions:
Function symbol KB weights:  c1=1. c2=1. f=1.
Predicate symbol precedence:  predicate_order([ = ]).
Function symbol precedence:  function_order([ c1, c2, f ]).
Skipping inverse_order, because term ordering is KBO.
Unfolding symbols: (none).

Auto_inference settings:
  % set(paramodulation).  % (positive equality literals)
    % set(paramodulation) -> set(back_demod).

Auto_process settings:  (no changes).

============================== end of process initial clauses ========

============================== CLAUSES FOR SEARCH ====================

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
5 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
6 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
7 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
8 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
9 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
10 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
11 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
12 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
4 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
5 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
6 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
7 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
8 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
9 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
10 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
11 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
12 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

% 583 hints processed (323 are redundant).

============================== end of clauses for search =============

============================== SEARCH ================================

% Starting search at 0.03 seconds.

% Operation f is commutative; C redundancy checks enabled.

============================== PROOF =================================

% Proof 1 at 1.04 (+ 0.00) seconds: "Commutativity".
% Length of proof is 51.
% Level of proof is 16.
% Maximum clause weight is 51.
% Given clauses 36.

1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal).  [goal].
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
5 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
6 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
8 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
9 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
11 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
12 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
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))].
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))].
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))].
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))].
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))].
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))].
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))].
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))].
533 f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x.  [para(436(a,1),9(a,1,1))].
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)])].
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)].
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))].
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)].
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)].
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)].
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)])].
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)])].
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))].
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)])].
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)])].
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))].
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)])].
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)])].
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)])].
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)])].
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)])].
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)])].
1365 f(x,f(f(x,x),x)) = f(x,x).  [para(318(a,1),13(a,1,2)),rewrite([533(8)]),flip(a)].
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)])].
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)])].
1529 f(f(x,f(x,x)),x) = f(x,x).  [back_rewrite(903),rewrite([1365(3),1365(6)])].
1533 f(x,f(x,f(x,x))) = f(x,x).  [back_rewrite(897),rewrite([1365(3),1365(6)])].
1577 f(f(x,x),x) = f(x,f(x,x)).  [back_rewrite(840),rewrite([1365(3),1365(5)])].
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)])].
1610 f(f(x,x),f(x,x)) = x.  [back_rewrite(1483),rewrite([1577(5),1529(7)])].
1620 f(f(x,f(y,y)),f(x,y)) = x.  [para(1610(a,1),2(a,1,2,2)),rewrite([1513(6)])].
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)])].
1796 f(f(x,x),y) = f(y,f(x,x)).  [para(1620(a,1),6(a,1,2)),rewrite([1586(5)])].
1953 f(x,f(f(x,x),x)) = f(x,x).  [para(1610(a,1),1781(a,1,1)),rewrite([1610(4)])].
1976 f(x,y) = f(y,x).  [para(2(a,1),1796(a,1,1)),rewrite([1953(4),1953(5),1610(4)])].
1977 $F # answer("Commutativity").  [resolve(1976,a,3,a)].

============================== end of proof ==========================

============================== STATISTICS ============================

Given=36. Generated=3828. Kept=1975. proofs=1.
Usable=31. Sos=1569. Demods=1546. Limbo=9, Disabled=376. Hints=583.
Weight_deleted=127. Literals_deleted=0.
Forward_subsumed=1726. Back_subsumed=59.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=1901 (0 lex), Back_demodulated=306. Back_unit_deleted=0.
Demod_attempts=112174. Demod_rewrites=6972.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
Megabytes=3.97.
User_CPU=1.04, System_CPU=0.00, Wall_clock=7.

============================== end of statistics =====================

============================== end of search =========================

Exiting with 1 proof.

Process 22000 exit (max_proofs) Mon Jun  4 22:53:32 2007

Successful proof using the following 7 assumptions:

formulas(assumptions_in_proof).
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Including 95 new hint clauses:

formulas(new_hints).
$F # label("Commutativity") # label(non_clause) # label(goal).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
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)).
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y.
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)).
f(f(f(x,y),y),y) = f(x,f(f(y,x),x)).
f(f(f(x,f(f(y,x),x)),y),f(y,f(f(x,y),y))) = y.
f(f(f(f(x,x),f(y,x)),f(y,x)),f(f(f(x,y),y),y)) = x.
f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x.
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))).
f(x,f(f(f(y,x),x),x)) = f(x,f(y,f(f(x,y),y))).
f(x,f(x,f(f(x,x),x))) = f(x,f(f(f(x,f(f(x,x),x)),x),x)).
f(x,f(f(f(x,f(f(x,x),x)),x),x)) = f(x,f(x,f(f(x,x),x))).
f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x.
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)))).
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)))).
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)))).
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)))).
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).
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)))).
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).
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)).
f(f(x,f(x,f(y,f(x,f(f(x,x),x))))),x) = f(x,f(f(x,x),x)).
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))).
f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))).
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))).
f(x,f(f(x,f(x,f(f(x,x),x))),x)) = f(x,f(x,f(f(x,x),x))).
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)).
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)).
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)).
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)).
f(f(x,f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
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))))).
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)).
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)))).
f(x,f(f(f(x,f(x,f(f(x,x),x))),x),x)) = f(x,f(f(x,x),x)).
f(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(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))).
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)).
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)).
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)).
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)).
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(y,f(x,f(f(x,x),x))),f(x,f(f(x,x),x)))) = f(y,f(x,f(f(x,x),x))).
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.
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.
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.
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.
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.
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)).
f(x,x) = f(x,f(f(x,x),x)).
f(x,f(f(x,x),x)) = f(x,x).
f(f(f(f(x,x),f(x,x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,f(f(x,x),x))) = x.
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,x)) = x.
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))).
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))).
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,f(f(x,x),x))).
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
f(f(x,f(x,x)),x) = f(x,f(f(x,x),x)).
f(f(x,f(x,x)),x) = f(x,x).
f(x,f(x,f(x,x))) = f(x,f(f(x,x),x)).
f(x,f(x,f(x,x))) = f(x,x).
f(f(x,x),x) = f(x,f(x,f(f(x,x),x))).
f(f(x,x),x) = f(x,f(x,x)).
f(f(x,f(x,f(y,f(x,f(x,f(x,x)))))),x) = f(x,f(f(x,x),x)).
f(f(x,f(x,f(y,f(x,x)))),x) = f(x,f(f(x,x),x)).
f(f(x,f(x,f(y,f(x,x)))),x) = f(x,f(x,f(x,x))).
f(f(x,f(x,f(y,f(x,x)))),x) = f(x,x).
f(f(f(x,x),f(f(x,x),f(x,x))),f(x,x)) = x.
f(f(x,x),f(x,x)) = x.
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
f(f(x,f(y,y)),f(x,y)) = x.
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
f(f(x,y),f(x,f(y,y))) = x.
f(f(f(x,f(x,f(y,f(x,x)))),x),y) = f(y,f(x,x)).
f(f(x,x),y) = f(y,f(x,x)).
f(x,f(f(x,x),f(f(x,x),f(x,x)))) = f(x,x).
f(x,f(f(x,x),x)) = f(x,x).
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
f(x,y) = f(y,f(f(x,x),f(x,f(f(x,x),x)))).
f(x,y) = f(y,f(f(x,x),f(x,x))).
f(x,y) = f(y,x).
$F # answer("Commutativity").
end_of_list.

Attempt to eliminate one assumption:

   f(f(f(f(x,y),x),x),f(y,f(z,x))) = y # label("C8/A12_m").

***********************************************
AUTO SKETCHES ITERATION 10
***********************************************

Starting a search with 8 assumptions:

formulas(assumptions_active).
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Child search process 22004 started.

============================== PROCESS INITIAL CLAUSES ===============

% Clauses before input processing:

formulas(usable).
end_of_list.

formulas(sos).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
f(c2,c1) != f(c1,c2) # label("Commutativity").  [deny(1)].
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
end_of_list.

% 678 hints input.

============================== PREDICATE ELIMINATION =================

No predicates eliminated.

============================== end predicate elimination =============

Auto_denials:
  % copying label "Commutativity" to answer in negative clause

Term ordering decisions:
Function symbol KB weights:  c1=1. c2=1. f=1.
Predicate symbol precedence:  predicate_order([ = ]).
Function symbol precedence:  function_order([ c1, c2, f ]).
Skipping inverse_order, because term ordering is KBO.
Unfolding symbols: (none).

Auto_inference settings:
  % set(paramodulation).  % (positive equality literals)
    % set(paramodulation) -> set(back_demod).

Auto_process settings:  (no changes).

============================== end of process initial clauses ========

============================== CLAUSES FOR SEARCH ====================

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
5 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
6 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
7 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
8 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
9 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
10 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
11 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
4 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
5 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
6 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
7 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
8 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
9 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
10 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
11 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

% 678 hints processed (374 are redundant).

============================== end of clauses for search =============

============================== SEARCH ================================

% Starting search at 0.03 seconds.

% Operation f is commutative; C redundancy checks enabled.

============================== PROOF =================================

% Proof 1 at 0.87 (+ 0.00) seconds: "Commutativity".
% Length of proof is 54.
% Level of proof is 19.
% Maximum clause weight is 63.
% Given clauses 38.

1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal).  [goal].
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
5 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
7 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
8 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
9 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
10 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
11 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
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))].
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))].
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))].
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))].
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))].
461 f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x.  [para(377(a,1),8(a,1,1))].
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)])].
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)].
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)])].
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)].
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)].
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)].
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))].
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)])].
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)].
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)])].
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)])].
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)].
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)])].
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)])].
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)].
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)])].
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)].
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)])].
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)])].
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)])].
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)])].
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))].
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)].
1338 f(x,f(x,f(x,x))) = f(x,x).  [back_rewrite(1181),rewrite([1316(4),1317(3),1317(6)])].
1403 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.  [back_rewrite(1251),rewrite([1317(3),1317(5)])].
1424 f(f(x,x),x) = f(x,f(x,x)).  [back_rewrite(1221),rewrite([1317(3),1317(5)])].
1429 f(f(x,f(x,x)),x) = f(x,x).  [back_rewrite(1212),rewrite([1424(2),1338(3),1424(5),1338(6)])].
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)])].
1484 f(f(x,x),f(x,f(x,y))) = x.  [back_rewrite(461),rewrite([1424(2),1338(3)])].
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))].
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)])].
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)])].
1613 f(f(x,f(y,y)),f(x,y)) = x.  [back_rewrite(1496),rewrite([1583(6)])].
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)])].
1645 f(f(x,x),y) = f(y,f(x,x)).  [para(1613(a,1),5(a,1,2)),rewrite([1570(5)])].
1807 f(x,f(f(x,x),x)) = f(x,x).  [para(1432(a,1),1634(a,1,1)),rewrite([1432(4)])].
1830 f(x,y) = f(y,x).  [para(2(a,1),1645(a,1,1)),rewrite([1807(4),1807(5),1432(4)])].
1831 $F # answer("Commutativity").  [resolve(1830,a,3,a)].

============================== end of proof ==========================

============================== STATISTICS ============================

Given=38. Generated=3590. Kept=1829. proofs=1.
Usable=30. Sos=1349. Demods=1331. Limbo=9, Disabled=450. Hints=678.
Weight_deleted=59. Literals_deleted=0.
Forward_subsumed=1702. Back_subsumed=55.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=1752 (0 lex), Back_demodulated=385. Back_unit_deleted=0.
Demod_attempts=97686. Demod_rewrites=6444.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
Megabytes=3.47.
User_CPU=0.87, System_CPU=0.00, Wall_clock=8.

============================== end of statistics =====================

============================== end of search =========================

Exiting with 1 proof.

Process 22004 exit (max_proofs) Mon Jun  4 22:53:33 2007

Successful proof using the following 7 assumptions:

formulas(assumptions_in_proof).
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Including 125 new hint clauses:

formulas(new_hints).
$F # label("Commutativity") # label(non_clause) # label(goal).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
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)).
f(f(f(x,y),y),y) = f(x,f(f(y,x),x)).
f(f(f(x,f(f(y,x),x)),y),f(y,f(f(x,y),y))) = y.
f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x.
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))).
f(x,f(f(f(y,x),x),x)) = f(x,f(y,f(f(x,y),y))).
f(x,f(x,f(f(x,x),x))) = f(x,f(f(f(x,f(f(x,x),x)),x),x)).
f(x,f(f(f(x,f(f(x,x),x)),x),x)) = f(x,f(x,f(f(x,x),x))).
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)).
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)).
f(x,f(f(x,f(f(x,x),x)),x)) = f(x,f(f(x,x),x)).
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)))).
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)))).
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)))).
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)))).
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).
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)))).
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).
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.
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))))).
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))))).
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))))).
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))))).
f(f(f(f(x,f(f(x,x),x)),x),x),x) = f(f(x,f(f(x,x),x)),x).
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))).
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))).
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))).
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).
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)).
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)).
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)).
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.
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))).
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))).
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))).
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))).
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))) = x.
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))).
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))).
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.
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))).
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))).
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))).
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))).
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))).
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))).
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))).
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))).
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))).
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))).
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))).
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))).
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).
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)).
f(x,f(f(f(f(x,x),x),x),x)) = f(x,f(f(x,x),x)).
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))).
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))).
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(x,f(f(x,x),x)) = f(f(f(x,f(f(x,x),x)),x),x).
f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)).
f(f(f(x,f(f(x,x),x)),x),f(x,f(y,f(x,f(f(x,x),x))))) = x.
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)).
f(f(x,f(x,f(f(x,x),x))),x) = f(x,f(f(x,x),x)).
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))).
f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))).
f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x.
f(f(f(f(x,x),x),x),x) = f(x,f(x,f(f(x,x),x))).
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)).
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)).
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)).
f(x,x) = f(x,f(f(x,x),x)).
f(x,f(f(x,x),x)) = f(x,x).
f(x,f(x,f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
f(x,f(x,f(x,x))) = f(x,f(f(x,x),x)).
f(x,f(x,f(x,x))) = f(x,x).
f(f(x,f(x,x)),f(x,f(y,f(x,f(f(x,x),x))))) = x.
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
f(f(x,x),x) = f(x,f(x,f(f(x,x),x))).
f(f(x,x),x) = f(x,f(x,x)).
f(f(x,f(x,f(x,f(x,x)))),x) = f(x,f(f(x,x),x)).
f(f(x,f(x,x)),x) = f(x,f(f(x,x),x)).
f(f(x,f(x,x)),x) = f(x,f(x,f(x,x))).
f(f(x,f(x,x)),x) = f(x,x).
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.
f(f(f(f(x,x),f(x,x)),f(x,f(f(x,x),x))),f(x,f(f(x,x),x))) = x.
f(f(f(f(x,x),f(x,x)),f(x,f(x,f(x,x)))),f(x,f(f(x,x),x))) = x.
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,f(f(x,x),x))) = x.
f(f(f(x,x),f(f(x,x),f(x,x))),f(x,f(f(x,x),x))) = x.
f(f(f(x,x),f(f(x,x),f(x,x))),f(x,f(x,f(x,x)))) = x.
f(f(f(x,x),f(f(x,x),f(x,x))),f(x,x)) = x.
f(f(x,x),f(x,x)) = x.
f(f(x,f(x,f(x,x))),f(x,f(x,y))) = x.
f(f(x,x),f(x,f(x,y))) = x.
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
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).
f(f(x,f(x,f(y,f(x,x)))),x) = f(x,x).
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)).
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
f(f(x,f(y,y)),f(x,y)) = x.
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
f(f(x,y),f(x,f(y,y))) = x.
f(f(f(x,f(x,f(y,f(x,x)))),x),y) = f(y,f(x,x)).
f(f(x,x),y) = f(y,f(x,x)).
f(x,f(f(x,x),f(f(x,x),f(x,x)))) = f(x,x).
f(x,f(f(x,x),x)) = f(x,x).
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
f(x,y) = f(y,f(f(x,x),f(x,f(f(x,x),x)))).
f(x,y) = f(y,f(f(x,x),f(x,x))).
f(x,y) = f(y,x).
$F # answer("Commutativity").
end_of_list.

Attempt to eliminate one assumption:

   f(f(f(f(x,y),z),z),f(y,f(x,z))) = y # label("C9/A13_m").

***********************************************
AUTO SKETCHES ITERATION 11
***********************************************

Starting a search with 7 assumptions:

formulas(assumptions_active).
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Child search process 22007 started.

============================== PROCESS INITIAL CLAUSES ===============

% Clauses before input processing:

formulas(usable).
end_of_list.

formulas(sos).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
f(c2,c1) != f(c1,c2) # label("Commutativity").  [deny(1)].
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
end_of_list.

% 803 hints input.

============================== PREDICATE ELIMINATION =================

No predicates eliminated.

============================== end predicate elimination =============

Auto_denials:
  % copying label "Commutativity" to answer in negative clause

Term ordering decisions:
Function symbol KB weights:  c1=1. c2=1. f=1.
Predicate symbol precedence:  predicate_order([ = ]).
Function symbol precedence:  function_order([ c1, c2, f ]).
Skipping inverse_order, because term ordering is KBO.
Unfolding symbols: (none).

Auto_inference settings:
  % set(paramodulation).  % (positive equality literals)
    % set(paramodulation) -> set(back_demod).

Auto_process settings:  (no changes).

============================== end of process initial clauses ========

============================== CLAUSES FOR SEARCH ====================

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
5 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
6 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
7 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
8 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
9 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
10 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
4 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
5 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
6 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
7 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
8 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
9 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
10 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

% 803 hints processed (449 are redundant).

============================== end of clauses for search =============

============================== SEARCH ================================

% Starting search at 0.04 seconds.

% Operation f is commutative; C redundancy checks enabled.

============================== PROOF =================================

% Proof 1 at 0.47 (+ 0.00) seconds: "Commutativity".
% Length of proof is 32.
% Level of proof is 15.
% Maximum clause weight is 25.
% Given clauses 34.

1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal).  [goal].
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
5 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
6 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
7 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
9 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
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))].
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))].
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))].
375 f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x.  [para(305(a,1),7(a,1,1))].
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)])].
549 f(x,f(f(x,x),x)) = f(x,x).  [para(4(a,1),454(a,1,1)),flip(a)].
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)])].
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)])].
637 f(f(x,x),f(x,f(y,x))) = x.  [para(549(a,1),2(a,1,1))].
639 f(f(x,x),f(x,x)) = x.  [para(549(a,1),2(a,1,2)),rewrite([549(3)])].
646 f(f(x,f(x,x)),f(x,x)) = x.  [para(549(a,1),6(a,1,2)),rewrite([639(3)])].
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))].
667 f(f(x,f(x,x)),x) = f(x,x).  [para(639(a,1),2(a,1,2)),rewrite([582(3)])].
669 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.  [para(639(a,1),5(a,1,1,1))].
737 f(f(x,x),x) = f(x,f(x,x)).  [para(646(a,1),2(a,1,2)),rewrite([667(3),549(3)])].
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)])].
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)])].
940 f(f(x,f(y,y)),f(x,y)) = x.  [back_rewrite(666),rewrite([920(6)])].
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))].
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)])].
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)])].
1187 f(f(x,x),y) = f(y,f(x,x)).  [back_rewrite(961),rewrite([1158(3),667(3)])].
1199 f(x,y) = f(y,x).  [para(2(a,1),1187(a,1,1)),rewrite([2(8)])].
1200 $F # answer("Commutativity").  [resolve(1199,a,3,a)].

============================== end of proof ==========================

============================== STATISTICS ============================

Given=34. Generated=2340. Kept=1198. proofs=1.
Usable=26. Sos=890. Demods=883. Limbo=9, Disabled=281. Hints=803.
Weight_deleted=0. Literals_deleted=0.
Forward_subsumed=1142. Back_subsumed=54.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=1135 (0 lex), Back_demodulated=218. Back_unit_deleted=0.
Demod_attempts=47705. Demod_rewrites=3591.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
Megabytes=2.22.
User_CPU=0.47, System_CPU=0.00, Wall_clock=8.

============================== end of statistics =====================

============================== end of search =========================

Exiting with 1 proof.

Process 22007 exit (max_proofs) Mon Jun  4 22:53:33 2007

Successful proof using the following 5 assumptions:

formulas(assumptions_in_proof).
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").  [assumption].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
end_of_list.

Including 52 new hint clauses:

formulas(new_hints).
$F # label("Commutativity") # label(non_clause) # label(goal).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
f(f(f(x,y),x),f(f(y,f(f(y,y),y)),f(z,x))) = f(y,f(f(y,y),y)).
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
f(f(f(x,y),y),y) = f(x,f(f(y,x),x)).
f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x.
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)).
f(f(f(f(x,y),x),f(x,y)),x) = 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(x,x).
f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)).
f(f(f(x,x),x),x) = f(x,f(f(x,x),x)).
f(f(f(x,x),x),x) = f(x,x).
f(f(f(x,y),x),f(f(y,y),f(z,x))) = f(y,f(f(y,y),y)).
f(f(f(x,y),x),f(f(y,y),f(z,x))) = f(y,y).
f(f(x,x),f(x,f(y,x))) = x.
f(f(x,f(f(x,x),x)),f(x,x)) = x.
f(f(x,x),f(x,x)) = x.
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,x)) = x.
f(f(x,f(x,x)),f(x,x)) = x.
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
f(f(x,f(f(f(x,x),x),x)),x) = f(x,x).
f(f(x,f(x,x)),x) = f(x,x).
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
f(f(x,f(f(f(x,f(x,x)),x),x)),x) = f(x,f(x,x)).
f(f(x,f(f(x,x),x)),x) = f(x,f(x,x)).
f(f(x,x),x) = f(x,f(x,x)).
f(f(f(x,x),f(x,x)),f(f(x,x),x)) = f(x,x).
f(x,f(f(x,x),x)) = f(x,x).
f(x,f(x,f(x,x))) = f(x,x).
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)).
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
f(f(x,f(y,y)),f(x,y)) = x.
f(f(f(x,f(x,f(y,f(x,x)))),x),y) = f(y,f(x,x)).
f(f(f(f(x,x),y),y),y) = f(f(x,x),f(f(y,f(x,x)),f(x,x))).
f(f(f(f(x,x),y),y),y) = f(y,f(x,x)).
f(f(f(f(x,x),x),f(x,x)),f(f(f(f(x,x),y),y),y)) = f(x,x).
f(f(f(x,f(x,x)),f(x,x)),f(f(f(f(x,x),y),y),y)) = f(x,x).
f(x,f(f(f(f(x,x),y),y),y)) = f(x,x).
f(x,f(y,f(x,x))) = f(x,x).
f(f(f(x,f(x,x)),x),y) = f(y,f(x,x)).
f(f(x,x),y) = f(y,f(x,x)).
f(x,y) = f(y,f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x)))).
f(x,y) = f(y,x).
$F # answer("Commutativity").
end_of_list.

Attempt to eliminate one assumption:

   f(f(f(x,f(x,y)),x),f(y,f(z,x))) = y # label("C10/A16_m").

***********************************************
AUTO SKETCHES ITERATION 12
***********************************************

Starting a search with 6 assumptions:

formulas(assumptions_active).
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Child search process 22008 started.

============================== PROCESS INITIAL CLAUSES ===============

% Clauses before input processing:

formulas(usable).
end_of_list.

formulas(sos).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
f(c2,c1) != f(c1,c2) # label("Commutativity").  [deny(1)].
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
end_of_list.

% 855 hints input.

============================== PREDICATE ELIMINATION =================

No predicates eliminated.

============================== end predicate elimination =============

Auto_denials:
  % copying label "Commutativity" to answer in negative clause

Term ordering decisions:
Function symbol KB weights:  c1=1. c2=1. f=1.
Predicate symbol precedence:  predicate_order([ = ]).
Function symbol precedence:  function_order([ c1, c2, f ]).
Skipping inverse_order, because term ordering is KBO.
Unfolding symbols: (none).

Auto_inference settings:
  % set(paramodulation).  % (positive equality literals)
    % set(paramodulation) -> set(back_demod).

Auto_process settings:  (no changes).

============================== end of process initial clauses ========

============================== CLAUSES FOR SEARCH ====================

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
5 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
6 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
7 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
8 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
9 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
4 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
5 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
6 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
7 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
8 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
9 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

% 855 hints processed (479 are redundant).

============================== end of clauses for search =============

============================== SEARCH ================================

% Starting search at 0.04 seconds.

% Operation f is commutative; C redundancy checks enabled.

============================== PROOF =================================

% Proof 1 at 0.50 (+ 0.01) seconds: "Commutativity".
% Length of proof is 44.
% Level of proof is 18.
% Maximum clause weight is 39.
% Given clauses 34.

1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal).  [goal].
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
5 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
6 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
8 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
9 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
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))].
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))].
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))].
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))].
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))].
366 f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x.  [para(302(a,1),6(a,1,1))].
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))].
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)])].
547 f(x,f(f(x,x),x)) = f(x,x).  [para(4(a,1),450(a,1,1)),flip(a)].
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)])].
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)])].
694 f(f(x,x),f(x,f(y,x))) = x.  [para(547(a,1),2(a,1,1))].
696 f(f(x,x),f(x,x)) = x.  [para(547(a,1),2(a,1,2)),rewrite([547(3)])].
701 f(f(x,f(x,x)),f(x,x)) = x.  [para(547(a,1),5(a,1,2)),rewrite([696(3)])].
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))].
720 f(f(x,f(x,x)),x) = f(x,x).  [para(696(a,1),2(a,1,2)),rewrite([580(3)])].
721 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.  [para(696(a,1),4(a,1,1,1))].
783 f(f(x,x),x) = f(x,f(x,x)).  [para(701(a,1),2(a,1,2)),rewrite([720(3),547(3)])].
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)])].
835 f(x,f(f(f(f(x,x),y),y),y)) = f(x,x).  [back_rewrite(637),rewrite([783(2),701(4)])].
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)])].
966 f(f(x,f(y,y)),f(x,y)) = x.  [back_rewrite(719),rewrite([946(6)])].
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)])].
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))].
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))].
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))].
1071 f(f(f(x,f(x,f(x,y))),x),x) = f(x,y).  [para(979(a,1),9(a,1,2))].
1148 f(x,f(y,f(x,x))) = f(x,x).  [para(302(a,1),835(a,1,2)),rewrite([946(6)])].
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)])].
1183 f(f(f(x,y),x),x) = f(x,y).  [back_rewrite(1066),rewrite([1177(3)])].
1184 f(f(f(x,y),y),y) = f(y,x).  [back_rewrite(1052),rewrite([1177(3)])].
1189 f(x,f(x,f(x,y))) = f(x,y).  [back_rewrite(1071),rewrite([1183(5)])].
1190 f(f(x,y),x) = f(x,f(x,y)).  [back_rewrite(1059),rewrite([1183(4),1184(6),1183(6)])].
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)])].
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)])].
1254 $F # answer("Commutativity").  [resolve(1253,a,3,a)].

============================== end of proof ==========================

============================== STATISTICS ============================

Given=34. Generated=2426. Kept=1252. proofs=1.
Usable=23. Sos=727. Demods=786. Limbo=70, Disabled=439. Hints=855.
Weight_deleted=2. Literals_deleted=0.
Forward_subsumed=1172. Back_subsumed=57.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=1189 (0 lex), Back_demodulated=374. Back_unit_deleted=0.
Demod_attempts=52607. Demod_rewrites=4432.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
Megabytes=2.23.
User_CPU=0.50, System_CPU=0.01, Wall_clock=9.

============================== end of statistics =====================

============================== end of search =========================

Exiting with 1 proof.

Process 22008 exit (max_proofs) Mon Jun  4 22:53:34 2007

Successful proof using the following 5 assumptions:

formulas(assumptions_in_proof).
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").  [assumption].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Including 86 new hint clauses:

formulas(new_hints).
$F # label("Commutativity") # label(non_clause) # label(goal).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
f(f(f(x,y),x),f(f(y,f(f(y,y),y)),f(z,x))) = f(y,f(f(y,y),y)).
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
f(f(f(x,f(f(y,z),f(f(f(y,x),y),y))),x),x) = f(f(f(y,x),y),y).
f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(y,f(y,x)),y).
f(f(f(x,y),y),y) = f(x,f(f(y,x),x)).
f(f(x,f(f(x,x),x)),f(x,f(x,y))) = x.
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)).
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)).
f(f(f(f(x,y),x),f(x,y)),x) = 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(x,x).
f(f(f(x,f(f(x,x),x)),x),x) = f(x,f(f(x,x),x)).
f(f(f(x,x),x),x) = f(x,f(f(x,x),x)).
f(f(f(x,x),x),x) = f(x,x).
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)).
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)).
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)).
f(f(f(f(x,x),x),f(x,x)),f(f(f(f(x,x),y),y),y)) = f(x,x).
f(f(x,x),f(x,f(y,x))) = x.
f(f(x,f(f(x,x),x)),f(x,x)) = x.
f(f(x,x),f(x,x)) = x.
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,x)) = x.
f(f(x,f(x,x)),f(x,x)) = x.
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
f(f(x,f(f(f(x,x),x),x)),x) = f(x,x).
f(f(x,f(x,x)),x) = f(x,x).
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
f(f(x,f(f(f(x,f(x,x)),x),x)),x) = f(x,f(x,x)).
f(f(x,f(f(x,x),x)),x) = f(x,f(x,x)).
f(f(x,x),x) = f(x,f(x,x)).
f(f(f(x,x),f(x,x)),f(f(x,x),x)) = f(x,x).
f(x,f(f(x,x),x)) = f(x,x).
f(x,f(x,f(x,x))) = f(x,x).
f(f(f(x,f(x,x)),f(x,x)),f(f(f(f(x,x),y),y),y)) = f(x,x).
f(x,f(f(f(f(x,x),y),y),y)) = f(x,x).
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)).
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
f(f(x,f(y,y)),f(x,y)) = x.
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
f(f(x,y),f(x,f(y,y))) = x.
f(f(f(f(f(x,x),f(y,x)),y),y),y) = f(y,x).
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).
f(f(f(x,f(f(y,y),f(x,y))),x),x) = f(x,y).
f(f(f(x,f(x,f(x,y))),x),x) = f(x,y).
f(x,f(f(x,x),f(f(y,f(x,x)),f(x,x)))) = f(x,x).
f(x,f(y,f(x,x))) = f(x,x).
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))).
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))).
f(f(x,x),f(y,x)) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
f(f(x,x),f(y,x)) = f(f(x,f(x,f(x,x))),f(x,f(f(x,x),x))).
f(f(x,x),f(y,x)) = f(f(x,x),f(x,f(f(x,x),x))).
f(f(x,x),f(y,x)) = f(f(x,x),f(x,f(x,f(x,x)))).
f(f(x,x),f(y,x)) = f(f(x,x),f(x,x)).
f(f(x,x),f(y,x)) = x.
f(f(f(x,y),x),x) = f(x,y).
f(f(f(x,y),y),y) = f(y,x).
f(x,f(x,f(x,y))) = f(x,y).
f(f(f(x,f(x,y)),f(x,y)),f(x,y)) = f(f(f(x,f(x,y)),x),x).
f(f(x,y),x) = f(f(f(x,f(x,y)),x),x).
f(f(x,y),x) = f(x,f(x,y)).
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(f(y,f(y,x)),y).
f(f(f(x,f(f(y,z),f(y,x))),x),x) = f(f(y,f(y,x)),y).
f(f(x,f(x,f(f(y,z),f(y,x)))),x) = f(f(y,f(y,x)),y).
f(x,f(x,f(x,f(f(y,z),f(y,x))))) = f(f(y,f(y,x)),y).
f(x,f(f(y,z),f(y,x))) = f(f(y,f(y,x)),y).
f(x,f(f(y,z),f(y,x))) = f(y,f(y,f(y,x))).
f(x,f(f(y,z),f(y,x))) = f(y,x).
f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(f(y,x),y),y).
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(f(f(y,x),y),y).
f(f(f(x,f(f(y,z),f(y,x))),x),x) = f(f(f(y,x),y),y).
f(f(f(y,x),x),x) = f(f(f(y,x),y),y).
f(x,y) = f(f(f(y,x),y),y).
f(x,y) = f(f(y,f(y,x)),y).
f(x,y) = f(y,f(y,f(y,x))).
f(x,y) = f(y,x).
$F # answer("Commutativity").
end_of_list.

Attempt to eliminate one assumption:

   f(f(f(x,f(y,y)),x),f(y,f(z,x))) = y # label("C11/A18_m").

***********************************************
AUTO SKETCHES ITERATION 13
***********************************************

Starting a search with 5 assumptions:

formulas(assumptions_active).
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Child search process 22009 started.

============================== PROCESS INITIAL CLAUSES ===============

% Clauses before input processing:

formulas(usable).
end_of_list.

formulas(sos).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
f(c2,c1) != f(c1,c2) # label("Commutativity").  [deny(1)].
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
end_of_list.

% 941 hints input.

============================== PREDICATE ELIMINATION =================

No predicates eliminated.

============================== end predicate elimination =============

Auto_denials:
  % copying label "Commutativity" to answer in negative clause

Term ordering decisions:
Function symbol KB weights:  c1=1. c2=1. f=1.
Predicate symbol precedence:  predicate_order([ = ]).
Function symbol precedence:  function_order([ c1, c2, f ]).
Skipping inverse_order, because term ordering is KBO.
Unfolding symbols: (none).

Auto_inference settings:
  % set(paramodulation).  % (positive equality literals)
    % set(paramodulation) -> set(back_demod).

Auto_process settings:  (no changes).

============================== end of process initial clauses ========

============================== CLAUSES FOR SEARCH ====================

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
5 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
6 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
7 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
8 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
4 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
5 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
6 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
7 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
8 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

% 941 hints processed (525 are redundant).

============================== end of clauses for search =============

============================== SEARCH ================================

% Starting search at 0.05 seconds.

% Operation f is commutative; C redundancy checks enabled.

============================== PROOF =================================

% Proof 1 at 0.67 (+ 0.00) seconds: "Commutativity".
% Length of proof is 39.
% Level of proof is 15.
% Maximum clause weight is 37.
% Given clauses 33.

1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal).  [goal].
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
5 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
6 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
7 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
8 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
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))].
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))].
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))].
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))].
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))].
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))].
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)])].
460 f(x,f(f(x,x),x)) = f(x,x).  [para(127(a,1),9(a,1,2)),rewrite([127(8)]),flip(a)].
472 f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)).  [back_rewrite(452),rewrite([460(3),460(7)])].
476 f(f(x,x),f(x,f(y,x))) = x.  [para(460(a,1),2(a,1,1))].
478 f(f(x,x),f(x,x)) = x.  [para(460(a,1),2(a,1,2)),rewrite([460(3)])].
482 f(f(x,f(x,x)),f(x,x)) = x.  [para(460(a,1),4(a,1,2)),rewrite([478(3)])].
486 f(f(f(x,x),x),x) = f(x,x).  [para(460(a,1),6(a,1,1,1)),rewrite([478(5)])].
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))].
512 f(f(x,f(x,x)),x) = f(x,x).  [para(478(a,1),2(a,1,2)),rewrite([486(3)])].
535 f(f(x,x),x) = f(x,f(x,x)).  [back_rewrite(472),rewrite([512(3)])].
556 f(x,f(x,f(x,x))) = f(x,x).  [back_rewrite(460),rewrite([535(2)])].
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))].
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)])].
621 f(f(x,x),f(x,f(x,y))) = x.  [para(512(a,1),8(a,1,1))].
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)])].
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)])].
1037 f(f(x,f(y,y)),f(x,y)) = x.  [back_rewrite(511),rewrite([1009(6)])].
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)])].
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))].
1263 f(x,f(y,f(x,x))) = f(x,x).  [para(574(a,1),970(a,1,2)),rewrite([1009(6)])].
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)])].
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)])].
1311 f(f(f(x,y),x),x) = f(x,y).  [back_rewrite(1175),rewrite([1299(3)])].
1320 f(x,y) = f(y,x).  [back_rewrite(1305),rewrite([1311(3),1311(4)])].
1321 $F # answer("Commutativity").  [resolve(1320,a,3,a)].

============================== end of proof ==========================

============================== STATISTICS ============================

Given=33. Generated=2831. Kept=1319. proofs=1.
Usable=27. Sos=982. Demods=991. Limbo=9, Disabled=307. Hints=941.
Weight_deleted=93. Literals_deleted=0.
Forward_subsumed=1419. Back_subsumed=64.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=1279 (0 lex), Back_demodulated=236. Back_unit_deleted=0.
Demod_attempts=81940. Demod_rewrites=6087.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
Megabytes=2.93.
User_CPU=0.67, System_CPU=0.00, Wall_clock=9.

============================== end of statistics =====================

============================== end of search =========================

Exiting with 1 proof.

Process 22009 exit (max_proofs) Mon Jun  4 22:53:34 2007

Successful proof using the following 5 assumptions:

formulas(assumptions_in_proof).
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").  [assumption].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Including 75 new hint clauses:

formulas(new_hints).
$F # label("Commutativity") # label(non_clause) # label(goal).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
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)).
f(f(f(f(f(x,y),f(y,f(f(z,y),y))),z),z),z) = f(y,f(f(z,y),y)).
f(f(f(x,f(f(y,z),f(f(f(y,x),y),y))),x),x) = f(f(f(y,x),y),y).
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y.
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))).
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)))).
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)))).
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)))).
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)))).
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)))).
f(f(f(x,f(x,f(f(x,x),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)))),x) = 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(x,x).
f(f(f(x,f(x,x)),x),x) = f(x,f(x,f(f(x,x),x))).
f(f(f(x,f(x,x)),x),x) = f(x,f(x,x)).
f(f(x,x),f(x,f(y,x))) = x.
f(f(x,f(f(x,x),x)),f(x,x)) = x.
f(f(x,x),f(x,x)) = x.
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,x)) = x.
f(f(x,f(x,x)),f(x,x)) = x.
f(f(f(x,x),x),f(f(x,x),f(x,x))) = f(x,x).
f(f(f(x,x),x),x) = f(x,x).
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
f(f(x,f(f(f(x,x),x),x)),x) = f(x,x).
f(f(x,f(x,x)),x) = f(x,x).
f(f(x,x),x) = f(x,f(x,x)).
f(x,f(x,f(x,x))) = f(x,x).
f(f(f(x,y),y),y) = f(x,f(f(y,x),x)).
f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x.
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
f(f(x,x),f(x,f(x,y))) = x.
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)).
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)).
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)).
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)).
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)).
f(x,f(f(f(f(x,f(f(x,x),x)),y),y),y)) = f(x,f(f(x,x),x)).
f(x,f(f(f(f(x,f(x,f(x,x))),y),y),y)) = f(x,f(f(x,x),x)).
f(x,f(f(f(f(x,x),y),y),y)) = f(x,f(f(x,x),x)).
f(x,f(f(f(f(x,x),y),y),y)) = f(x,f(x,f(x,x))).
f(x,f(f(f(f(x,x),y),y),y)) = f(x,x).
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)).
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
f(f(x,f(y,y)),f(x,y)) = x.
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
f(f(x,y),f(x,f(y,y))) = x.
f(f(f(x,f(f(y,y),f(x,y))),x),x) = f(x,y).
f(x,f(f(x,x),f(f(y,f(x,x)),f(x,x)))) = f(x,x).
f(x,f(y,f(x,x))) = f(x,x).
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))).
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))).
f(f(x,x),f(y,x)) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
f(f(x,x),f(y,x)) = f(f(x,f(x,f(x,x))),f(x,f(f(x,x),x))).
f(f(x,x),f(y,x)) = f(f(x,x),f(x,f(f(x,x),x))).
f(f(x,x),f(y,x)) = f(f(x,x),f(x,f(x,f(x,x)))).
f(f(x,x),f(y,x)) = f(f(x,x),f(x,x)).
f(f(x,x),f(y,x)) = x.
f(f(f(x,f(f(y,y),f(f(f(y,x),y),y))),x),x) = f(f(f(y,x),y),y).
f(f(f(x,y),x),x) = f(f(f(y,x),y),y).
f(f(f(x,y),x),x) = f(x,y).
f(x,y) = f(f(f(y,x),y),y).
f(x,y) = f(y,x).
$F # answer("Commutativity").
end_of_list.

Attempt to eliminate one assumption:

   f(f(f(f(x,y),z),z),f(y,f(z,x))) = y # label("C12/A19_m").

***********************************************
AUTO SKETCHES ITERATION 14
***********************************************

Starting a search with 4 assumptions:

formulas(assumptions_active).
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Child search process 22010 started.

============================== PROCESS INITIAL CLAUSES ===============

% Clauses before input processing:

formulas(usable).
end_of_list.

formulas(sos).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
f(c2,c1) != f(c1,c2) # label("Commutativity").  [deny(1)].
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
end_of_list.

% 1016 hints input.

============================== PREDICATE ELIMINATION =================

No predicates eliminated.

============================== end predicate elimination =============

Auto_denials:
  % copying label "Commutativity" to answer in negative clause

Term ordering decisions:
Function symbol KB weights:  c1=1. c2=1. f=1.
Predicate symbol precedence:  predicate_order([ = ]).
Function symbol precedence:  function_order([ c1, c2, f ]).
Skipping inverse_order, because term ordering is KBO.
Unfolding symbols: (none).

Auto_inference settings:
  % set(paramodulation).  % (positive equality literals)
    % set(paramodulation) -> set(back_demod).

Auto_process settings:  (no changes).

============================== end of process initial clauses ========

============================== CLAUSES FOR SEARCH ====================

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
5 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
6 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
7 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
4 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
5 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
6 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
7 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

% 1016 hints processed (574 are redundant).

============================== end of clauses for search =============

============================== SEARCH ================================

% Starting search at 0.05 seconds.

% Operation f is commutative; C redundancy checks enabled.

============================== PROOF =================================

% Proof 1 at 0.72 (+ 0.00) seconds: "Commutativity".
% Length of proof is 68.
% Level of proof is 22.
% Maximum clause weight is 55.
% Given clauses 43.

1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal).  [goal].
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
6 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
7 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
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))].
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))].
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))].
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))].
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))].
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))].
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))].
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))].
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))].
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))].
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)])].
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)])].
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))].
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)])].
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)])].
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))].
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)])].
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))].
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)])].
387 f(x,f(f(x,x),x)) = f(x,x).  [para(90(a,1),8(a,1,2)),rewrite([90(8)]),flip(a)].
426 f(f(x,f(f(y,x),x)),f(y,y)) = y.  [back_rewrite(384),rewrite([387(6)])].
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)])].
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)])].
469 f(f(x,x),x) = f(x,f(x,x)).  [back_rewrite(313),rewrite([387(3),387(5)])].
489 f(x,f(x,f(x,x))) = f(x,x).  [back_rewrite(387),rewrite([469(2)])].
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)])].
511 f(f(x,x),f(x,f(y,x))) = x.  [para(469(a,1),2(a,1,1,2)),rewrite([489(3)])].
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)])].
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)])].
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)])].
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)].
542 f(f(x,f(y,y)),f(x,y)) = x.  [para(530(a,1),2(a,1,2,2)),rewrite([526(6)])].
590 f(f(x,x),f(x,f(x,y))) = x.  [para(517(a,1),7(a,1,1))].
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)])].
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))].
703 f(f(f(x,f(x,f(x,y))),x),x) = f(x,y).  [para(604(a,1),7(a,1,2))].
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))].
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))].
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)])].
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))].
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)])].
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)].
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)])].
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)])].
1014 f(f(x,f(f(f(x,y),x),x)),x) = f(x,y).  [para(604(a,1),858(a,1,2))].
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)])].
1052 f(f(f(x,y),x),x) = f(x,y).  [back_rewrite(1014),rewrite([1029(4)])].
1069 f(f(x,f(y,z)),f(f(f(x,y),u),x)) = x.  [back_rewrite(986),rewrite([1029(5),1052(12)])].
1086 f(x,f(x,f(x,y))) = f(x,y).  [back_rewrite(963),rewrite([1052(3),1052(5)])].
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)])].
1145 f(f(x,y),x) = f(x,f(x,y)).  [back_rewrite(726),rewrite([1029(3),1029(5)])].
1170 f(f(x,y),f(x,x)) = x.  [back_rewrite(426),rewrite([1029(3)])].
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)])].
1224 f(f(x,y),f(x,f(z,y))) = x # label("Sh1/A10").  [back_rewrite(188),rewrite([1029(3),1145(4),1086(4)])].
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)].
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)])].
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)])].
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)].
1411 f(f(x,y),f(y,f(z,x))) = y # label("C8/A12_m").  [para(1301(a,1),1174(a,1,2,2))].
1427 f(f(f(x,y),f(z,y)),z) = f(z,y).  [para(1224(a,1),1411(a,1,2))].
1455 f(x,y) = f(y,x).  [para(1145(a,1),1427(a,1,1)),rewrite([1326(3),1174(4)])].
1456 $F # answer("Commutativity").  [resolve(1455,a,3,a)].

============================== end of proof ==========================

============================== STATISTICS ============================

Given=43. Generated=3497. Kept=1454. proofs=1.
Usable=14. Sos=244. Demods=260. Limbo=3, Disabled=1198. Hints=1016.
Weight_deleted=84. Literals_deleted=0.
Forward_subsumed=1959. Back_subsumed=204.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=1451 (0 lex), Back_demodulated=988. Back_unit_deleted=0.
Demod_attempts=103191. Demod_rewrites=10948.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
Megabytes=2.57.
User_CPU=0.72, System_CPU=0.00, Wall_clock=10.

============================== end of statistics =====================

============================== end of search =========================

Exiting with 1 proof.

Process 22010 exit (max_proofs) Mon Jun  4 22:53:35 2007

Successful proof using the following 3 assumptions:

formulas(assumptions_in_proof).
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Including 173 new hint clauses:

formulas(new_hints).
$F # label("Commutativity") # label(non_clause) # label(goal).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
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)).
f(f(f(x,f(f(y,z),f(f(f(y,x),y),y))),x),x) = f(f(f(y,x),y),y).
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
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.
f(f(f(x,f(x,f(f(f(y,x),y),y))),x),x) = f(f(f(y,x),y),y).
f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(y,f(y,x)),y).
f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y.
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))).
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).
f(f(x,f(f(y,x),x)),f(y,f(f(z,f(z,x)),z))) = y.
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).
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).
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).
f(f(x,f(x,f(f(x,x),x))),f(x,f(y,f(x,f(f(x,x),x))))) = x.
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)).
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)).
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)).
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)).
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))).
f(f(x,f(f(x,x),x)),x) = f(x,f(x,f(f(x,x),x))).
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))).
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)).
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.
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)))).
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)))).
f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y.
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)).
f(x,x) = f(x,f(f(x,x),x)).
f(x,f(f(x,x),x)) = f(x,x).
f(f(x,f(f(y,x),x)),f(y,y)) = y.
f(f(x,f(x,x)),f(f(x,f(f(x,x),x)),f(y,x))) = f(x,f(f(x,x),x)).
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,f(f(x,x),x)).
f(f(x,f(x,x)),f(f(x,x),f(y,x))) = f(x,x).
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))).
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))).
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))).
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)).
f(f(x,x),x) = f(x,f(x,f(f(x,x),x))).
f(f(x,x),x) = f(x,f(x,x)).
f(x,f(x,f(x,x))) = f(x,x).
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
f(f(x,f(x,f(x,x))),f(x,f(y,x))) = x.
f(f(x,x),f(x,f(y,x))) = x.
f(f(f(x,f(x,f(x,f(x,f(x,x))))),x),x) = f(x,f(f(x,x),x)).
f(f(f(x,f(x,f(x,x))),x),x) = f(x,f(f(x,x),x)).
f(f(f(x,x),x),x) = f(x,f(f(x,x),x)).
f(f(x,f(x,x)),x) = f(x,f(f(x,x),x)).
f(f(x,f(x,x)),x) = f(x,f(x,f(x,x))).
f(f(x,f(x,x)),x) = f(x,x).
f(f(x,f(x,f(x,x))),f(x,f(y,f(f(x,y),y)))) = x.
f(f(x,x),f(x,f(y,f(f(x,y),y)))) = x.
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
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))).
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))).
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))).
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))).
f(f(x,x),f(f(x,x),x)) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
f(f(x,x),f(x,f(x,x))) = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
x = f(f(x,f(f(x,x),x)),f(x,f(f(x,x),x))).
x = f(f(x,f(x,f(x,x))),f(x,f(f(x,x),x))).
x = f(f(x,x),f(x,f(f(x,x),x))).
x = f(f(x,x),f(x,f(x,f(x,x)))).
x = f(f(x,x),f(x,x)).
f(f(x,x),f(x,x)) = x.
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
f(f(x,f(y,y)),f(x,y)) = x.
f(f(x,x),f(x,f(x,y))) = x.
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
f(f(x,y),f(x,f(y,y))) = x.
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).
f(f(f(x,f(x,f(x,y))),x),x) = f(x,y).
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).
f(f(x,f(f(y,x),x)),y) = f(y,f(x,f(f(y,x),x))).
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)).
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)).
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)).
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.
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
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))).
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))).
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))).
f(f(x,f(y,f(f(x,y),y))),f(y,f(f(x,y),y))) = x.
f(x,y) = f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x).
f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x) = f(x,y).
f(f(f(x,f(x,f(f(f(x,y),x),x))),x),x) = f(x,y).
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))).
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))).
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))).
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.
f(f(x,f(f(f(x,y),x),x)),x) = f(x,y).
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).
f(x,f(f(y,x),f(f(x,f(x,x)),f(x,x)))) = f(y,x).
f(x,f(f(y,x),x)) = f(y,x).
f(f(f(x,y),x),x) = f(x,y).
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.
f(f(x,f(y,z)),f(f(f(x,y),u),x)) = x.
f(f(f(x,f(x,f(x,y))),x),x) = f(x,y).
f(x,f(x,f(x,y))) = f(x,y).
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.
f(f(x,f(x,f(y,f(f(x,y),y)))),f(x,f(y,f(f(x,y),y)))) = x.
f(f(x,f(x,f(x,y))),f(x,f(y,f(f(x,y),y)))) = x.
f(f(x,y),f(x,f(y,f(f(x,y),y)))) = x.
f(f(x,y),f(x,f(x,y))) = x.
f(f(y,x),y) = f(y,f(x,f(f(y,x),x))).
f(f(x,y),x) = f(x,f(x,y)).
f(f(x,y),f(x,x)) = x.
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.
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.
f(f(f(x,f(x,u)),x),f(u,f(z,f(f(x,z),z)))) = u.
f(f(x,f(x,f(x,u))),f(u,f(z,f(f(x,z),z)))) = u.
f(f(x,u),f(u,f(z,f(f(x,z),z)))) = u.
f(f(x,y),f(y,f(x,z))) = y.
f(f(y,x),f(y,f(f(z,f(z,x)),z))) = y.
f(f(y,x),f(y,f(z,f(z,f(z,x))))) = y.
f(f(x,y),f(x,f(z,y))) = x.
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).
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).
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).
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).
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).
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).
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).
f(f(x,y),y) = f(f(y,f(f(x,z),f(f(f(x,y),x),x))),y).
f(f(x,y),y) = f(f(y,f(f(x,z),f(f(x,f(x,y)),x))),y).
f(f(x,y),y) = f(f(y,f(f(x,z),f(x,f(x,f(x,y))))),y).
f(f(x,y),y) = f(f(y,f(f(x,z),f(x,y))),y).
f(f(x,y),y) = f(y,f(y,f(f(x,z),f(x,y)))).
f(x,f(x,f(f(y,z),f(y,x)))) = f(f(y,x),x).
f(f(f(x,f(x,f(f(y,f(y,x)),y))),x),x) = f(f(f(y,x),y),y).
f(f(f(x,f(x,f(y,f(y,f(y,x))))),x),x) = f(f(f(y,x),y),y).
f(f(f(x,f(x,f(y,x))),x),x) = f(f(f(y,x),y),y).
f(f(x,f(x,f(x,f(y,x)))),x) = f(f(f(y,x),y),y).
f(f(x,f(y,x)),x) = f(f(f(y,x),y),y).
f(x,f(x,f(y,x))) = f(f(f(y,x),y),y).
f(x,f(x,f(y,x))) = f(f(y,f(y,x)),y).
f(x,f(x,f(y,x))) = f(y,f(y,f(y,x))).
f(x,f(x,f(y,x))) = f(y,x).
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.
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.
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.
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.
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.
f(f(f(f(x,y),f(f(x,y),z)),f(f(f(x,y),x),x)),f(z,y)) = z.
f(f(f(f(x,y),f(f(x,y),z)),f(f(x,f(x,y)),x)),f(z,y)) = z.
f(f(f(f(x,y),f(f(x,y),z)),f(x,f(x,f(x,y)))),f(z,y)) = z.
f(f(f(f(x,y),f(f(x,y),z)),f(x,y)),f(z,y)) = z.
f(f(f(x,y),f(f(x,y),f(f(x,y),z))),f(z,y)) = z.
f(f(f(x,y),z),f(z,y)) = z.
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).
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).
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).
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).
f(x,f(f(y,f(y,x)),y)) = f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x).
f(x,f(y,f(y,f(y,x)))) = f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x).
f(x,f(y,x)) = f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x).
f(x,f(y,x)) = f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x).
f(x,f(y,x)) = f(f(x,f(f(y,z),f(y,x))),x).
f(x,f(y,x)) = f(x,f(x,f(f(y,z),f(y,x)))).
f(x,f(y,x)) = f(f(y,x),x).
f(f(x,y),y) = f(y,f(x,y)).
f(f(x,y),f(y,f(z,x))) = y # label("C8/A12_m").
f(f(f(x,y),f(z,y)),z) = f(z,y).
f(f(f(x,y),f(f(x,y),y)),x) = f(x,y).
f(f(f(x,y),f(y,f(x,y))),x) = f(x,y).
f(x,y) = f(y,x).
$F # answer("Commutativity").
end_of_list.

Attempt to eliminate one assumption:

   f(f(f(f(x,y),x),x),f(y,f(x,z))) = y # label("C13/A20_m").

***********************************************
AUTO SKETCHES ITERATION 15
***********************************************

Starting a search with 3 assumptions:

formulas(assumptions_active).
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Child search process 22011 started.

============================== PROCESS INITIAL CLAUSES ===============

% Clauses before input processing:

formulas(usable).
end_of_list.

formulas(sos).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
f(c2,c1) != f(c1,c2) # label("Commutativity").  [deny(1)].
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
end_of_list.

% 1189 hints input.

============================== PREDICATE ELIMINATION =================

No predicates eliminated.

============================== end predicate elimination =============

Auto_denials:
  % copying label "Commutativity" to answer in negative clause

Term ordering decisions:
Function symbol KB weights:  c1=1. c2=1. f=1.
Predicate symbol precedence:  predicate_order([ = ]).
Function symbol precedence:  function_order([ c1, c2, f ]).
Skipping inverse_order, because term ordering is KBO.
Unfolding symbols: (none).

Auto_inference settings:
  % set(paramodulation).  % (positive equality literals)
    % set(paramodulation) -> set(back_demod).

Auto_process settings:  (no changes).

============================== end of process initial clauses ========

============================== CLAUSES FOR SEARCH ====================

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
5 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
6 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
4 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
5 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
6 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

% 1189 hints processed (624 are redundant).

============================== end of clauses for search =============

============================== SEARCH ================================

% Starting search at 0.06 seconds.

% Operation f is commutative; C redundancy checks enabled.

============================== PROOF =================================

% Proof 1 at 0.86 (+ 0.00) seconds: "Commutativity".
% Length of proof is 60.
% Level of proof is 25.
% Maximum clause weight is 75.
% Given clauses 46.

1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal).  [goal].
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
5 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
6 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
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))].
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))].
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))].
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))].
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))].
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)])].
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)])].
244 f(x,f(f(x,x),x)) = f(x,x).  [para(62(a,1),7(a,1,2)),rewrite([62(8)]),flip(a)].
252 f(f(x,f(f(y,x),x)),f(y,y)) = y.  [back_rewrite(241),rewrite([244(6)])].
257 f(f(x,x),f(x,f(y,x))) = x.  [para(244(a,1),2(a,1,1))].
259 f(f(x,x),f(x,x)) = x.  [para(244(a,1),2(a,1,2)),rewrite([244(3)])].
261 f(f(f(x,x),x),x) = f(x,x).  [para(244(a,1),4(a,1,1,1)),rewrite([259(5)])].
263 f(f(x,f(x,x)),f(x,x)) = x.  [para(244(a,1),4(a,1,2)),rewrite([259(3)])].
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))].
286 f(f(x,f(x,x)),x) = f(x,x).  [para(259(a,1),2(a,1,2)),rewrite([261(3)])].
353 f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x.  [para(263(a,1),2(a,1,1,2))].
354 f(f(x,x),x) = f(x,f(x,x)).  [para(263(a,1),2(a,1,2)),rewrite([286(3),244(3)])].
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)])].
378 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.  [back_rewrite(353),rewrite([354(2)])].
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))].
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)])].
589 f(f(x,f(y,y)),f(x,y)) = x.  [back_rewrite(285),rewrite([567(6)])].
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)])].
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))].
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)])].
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)].
679 f(f(f(x,f(x,f(x,y))),x),x) = f(x,y).  [para(591(a,1),6(a,1,2))].
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))].
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)])].
839 f(f(x,f(f(f(x,y),x),x)),x) = f(x,y).  [para(591(a,1),736(a,1,2))].
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))].
886 f(f(x,f(x,y)),x) = f(x,f(x,f(x,y))).  [para(679(a,1),839(a,1,1,2))].
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)])].
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)])].
996 f(x,f(x,f(x,f(x,f(x,y))))) = f(x,y).  [back_rewrite(679),rewrite([886(4),886(5)])].
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)])].
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)])].
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)])].
1131 f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y # label("C1/A1").  [back_rewrite(6),rewrite([886(3)])].
1137 f(f(x,f(x,f(x,y))),f(x,f(x,y))) = x.  [para(886(a,1),4(a,1,1))].
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)])].
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)])].
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)])].
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)])].
1239 f(x,f(x,f(x,y))) = f(x,y).  [back_rewrite(998),rewrite([1161(2),1161(3),1165(4),1165(4)])].
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)].
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)])].
1472 f(x,f(f(y,z),f(y,x))) = f(y,x).  [back_rewrite(1196),rewrite([1239(4),1239(6),1239(7)])].
1483 f(f(x,y),f(y,f(x,z))) = y.  [back_rewrite(1131),rewrite([1239(3)])].
1586 f(f(x,y),y) = f(y,f(x,y)).  [back_rewrite(1438),rewrite([1472(4),1472(5),1483(5)])].
1587 f(x,f(x,f(y,x))) = f(y,x).  [back_rewrite(1258),rewrite([1472(4)]),flip(a)].
1723 f(f(x,y),f(x,f(z,y))) = x.  [back_rewrite(2),rewrite([1586(2),1587(3)])].
1781 f(x,y) = f(y,x).  [para(1586(a,1),1472(a,1,2)),rewrite([1723(4)])].
1782 $F # answer("Commutativity").  [resolve(1781,a,3,a)].

============================== end of proof ==========================

============================== STATISTICS ============================

Given=46. Generated=3828. Kept=1780. proofs=1.
Usable=15. Sos=242. Demods=262. Limbo=6, Disabled=1521. Hints=1189.
Weight_deleted=42. Literals_deleted=0.
Forward_subsumed=2006. Back_subsumed=227.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=1774 (0 lex), Back_demodulated=1289. Back_unit_deleted=0.
Demod_attempts=100302. Demod_rewrites=10522.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
Megabytes=2.77.
User_CPU=0.86, System_CPU=0.00, Wall_clock=11.

============================== end of statistics =====================

============================== end of search =========================

Exiting with 1 proof.

Process 22011 exit (max_proofs) Mon Jun  4 22:53:36 2007

Successful proof using the following 3 assumptions:

formulas(assumptions_in_proof).
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").  [assumption].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Including 117 new hint clauses:

formulas(new_hints).
$F # label("Commutativity") # label(non_clause) # label(goal).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
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)).
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(y,f(y,x)),y).
f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y.
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).
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)).
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)).
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)))).
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)))).
f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y.
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)).
f(x,x) = f(x,f(f(x,x),x)).
f(x,f(f(x,x),x)) = f(x,x).
f(f(x,f(f(y,x),x)),f(y,y)) = y.
f(f(x,x),f(x,f(y,x))) = x.
f(f(x,f(f(x,x),x)),f(x,x)) = x.
f(f(x,x),f(x,x)) = x.
f(f(f(x,x),x),f(f(x,x),f(x,x))) = f(x,x).
f(f(f(x,x),x),x) = f(x,x).
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,x)) = x.
f(f(x,f(x,x)),f(x,x)) = x.
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
f(f(x,f(f(f(x,x),x),x)),x) = f(x,x).
f(f(x,f(x,x)),x) = f(x,x).
f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x.
f(f(x,f(f(f(x,f(x,x)),x),x)),x) = f(x,f(x,x)).
f(f(x,f(f(x,x),x)),x) = f(x,f(x,x)).
f(f(x,x),x) = f(x,f(x,x)).
f(f(f(x,x),f(x,x)),f(f(x,x),x)) = f(x,x).
f(x,f(f(x,x),x)) = f(x,x).
f(x,f(x,f(x,x))) = f(x,x).
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
f(f(x,f(y,f(f(x,y),y))),f(x,f(f(y,f(f(x,y),y)),x))) = x.
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)).
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
f(f(x,f(y,y)),f(x,y)) = x.
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
f(f(x,y),f(x,f(y,y))) = x.
f(x,f(f(y,f(f(x,y),y)),x)) = 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.
f(x,y) = f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x).
f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x) = f(x,y).
f(f(f(x,f(x,f(x,y))),x),x) = f(x,y).
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
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)).
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)).
f(f(x,f(f(f(x,y),x),x)),x) = f(x,y).
f(f(x,f(f(y,f(y,x)),y)),x) = f(x,f(f(y,z),f(f(y,f(y,x)),y))).
f(f(x,f(x,y)),x) = f(x,f(x,f(x,y))).
f(f(x,f(y,f(y,f(y,x)))),x) = f(x,f(f(y,z),f(f(y,f(y,x)),y))).
f(f(x,f(y,f(y,f(y,x)))),x) = f(x,f(f(y,z),f(y,f(y,f(y,x))))).
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)).
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)).
f(f(x,f(x,f(x,f(x,y)))),x) = f(x,y).
f(x,f(x,f(x,f(x,f(x,y))))) = f(x,y).
f(x,f(x,f(x,f(x,f(f(f(x,y),x),x))))) = f(x,y).
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)).
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)).
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)).
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)).
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)).
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)).
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)).
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)))).
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(f(y,f(y,x)),y).
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(y,f(y,f(y,x))).
f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y.
f(f(x,f(x,f(x,y))),f(x,f(x,y))) = x.
f(f(x,f(f(x,f(x,f(x,y))),x)),x) = f(x,f(x,y)).
f(f(x,f(x,f(x,f(x,f(x,y))))),x) = f(x,f(x,y)).
f(f(x,y),x) = f(x,f(x,y)).
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))).
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))).
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))).
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))).
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)).
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)).
f(x,f(x,f(x,f(x,y)))) = f(x,f(x,y)).
f(f(x,f(x,f(f(y,z),f(y,f(y,f(y,x)))))),x) = f(y,f(y,f(y,x))).
f(x,f(x,f(x,f(f(y,z),f(y,f(y,f(y,x))))))) = f(y,f(y,f(y,x))).
f(x,f(x,f(x,f(x,f(f(x,f(x,y)),x))))) = f(x,y).
f(x,f(x,f(x,f(x,f(x,f(x,f(x,y))))))) = f(x,y).
f(x,f(x,f(x,f(x,f(x,y))))) = f(x,y).
f(x,f(x,f(x,y))) = f(x,y).
f(f(x,f(y,x)),x) = f(x,f(f(y,z),f(y,f(y,f(y,x))))).
f(x,f(x,f(y,x))) = f(x,f(f(y,z),f(y,f(y,f(y,x))))).
f(x,f(x,f(y,x))) = f(x,f(f(y,z),f(y,x))).
f(x,f(f(y,z),f(y,x))) = f(x,f(x,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(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)))).
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)))).
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)))).
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)))).
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)))).
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)))).
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)).
f(x,f(x,f(x,f(f(y,z),f(y,x))))) = f(y,f(y,f(y,x))).
f(x,f(f(y,z),f(y,x))) = f(y,f(y,f(y,x))).
f(x,f(f(y,z),f(y,x))) = f(y,x).
f(f(x,y),f(y,f(x,z))) = y.
f(f(y,x),f(f(x,f(f(y,z),f(y,x))),f(x,f(y,x)))) = f(x,f(y,x)).
f(f(y,x),f(f(y,x),f(x,f(y,x)))) = f(x,f(y,x)).
f(f(x,y),y) = f(y,f(x,y)).
f(y,x) = f(x,f(x,f(y,x))).
f(x,f(x,f(y,x))) = f(y,x).
f(f(x,f(x,f(y,x))),f(y,f(z,x))) = y.
f(f(x,y),f(x,f(z,y))) = x.
f(x,f(f(y,x),f(y,f(y,x)))) = f(y,x).
f(x,y) = f(y,x).
$F # answer("Commutativity").
end_of_list.

Attempt to eliminate one assumption:

   f(f(f(x,f(y,z)),x),f(y,f(x,z))) = y # label("C14/A21_m").

***********************************************
AUTO SKETCHES ITERATION 16
***********************************************

Starting a search with 2 assumptions:

formulas(assumptions_active).
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Child search process 22012 started.

============================== PROCESS INITIAL CLAUSES ===============

% Clauses before input processing:

formulas(usable).
end_of_list.

formulas(sos).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
f(c2,c1) != f(c1,c2) # label("Commutativity").  [deny(1)].
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
end_of_list.

% 1306 hints input.

============================== PREDICATE ELIMINATION =================

No predicates eliminated.

============================== end predicate elimination =============

Auto_denials:
  % copying label "Commutativity" to answer in negative clause

Term ordering decisions:
Function symbol KB weights:  c1=1. c2=1. f=1.
Predicate symbol precedence:  predicate_order([ = ]).
Function symbol precedence:  function_order([ c1, c2, f ]).
Skipping inverse_order, because term ordering is KBO.
Unfolding symbols: (none).

Auto_inference settings:
  % set(paramodulation).  % (positive equality literals)
    % set(paramodulation) -> set(back_demod).

Auto_process settings:  (no changes).

============================== end of process initial clauses ========

============================== CLAUSES FOR SEARCH ====================

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
5 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
4 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
5 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

% 1306 hints processed (676 are redundant).

============================== end of clauses for search =============

============================== SEARCH ================================

% Starting search at 0.07 seconds.

% Operation f is commutative; C redundancy checks enabled.

============================== PROOF =================================

% Proof 1 at 0.75 (+ 0.00) seconds: "Commutativity".
% Length of proof is 66.
% Level of proof is 24.
% Maximum clause weight is 75.
% Given clauses 44.

1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal).  [goal].
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
5 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
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))].
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))].
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))].
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))].
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)])].
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))].
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))].
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)])].
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))].
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)].
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)])].
181 f(x,f(f(x,x),x)) = f(x,x).  [para(35(a,1),6(a,1,2)),rewrite([35(8)]),flip(a)].
189 f(f(x,f(f(y,x),x)),f(y,y)) = y.  [back_rewrite(178),rewrite([181(6)])].
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)])].
194 f(f(x,x),f(x,f(y,x))) = x.  [para(181(a,1),2(a,1,1))].
196 f(f(x,x),f(x,x)) = x.  [para(181(a,1),2(a,1,2)),rewrite([181(3)])].
200 f(f(x,f(x,x)),f(x,x)) = x.  [para(181(a,1),4(a,1,2)),rewrite([196(3)])].
217 f(f(f(x,f(x,x)),x),x) = f(f(x,x),x).  [back_rewrite(190),rewrite([200(5)]),flip(a)].
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))].
251 f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x.  [para(200(a,1),2(a,1,1,2))].
252 f(f(x,x),x) = f(x,f(x,x)).  [para(200(a,1),2(a,1,2)),rewrite([217(4),181(3)])].
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)])].
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)])].
278 f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.  [back_rewrite(251),rewrite([252(2)])].
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))].
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)])].
468 f(f(x,f(y,y)),f(x,y)) = x.  [back_rewrite(218),rewrite([446(6)])].
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)])].
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))].
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)])].
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)].
548 f(f(f(x,f(x,f(x,y))),x),x) = f(x,y).  [para(470(a,1),5(a,1,2))].
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))].
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)])].
694 f(f(x,f(f(f(x,y),x),x)),x) = f(x,y).  [para(470(a,1),603(a,1,2))].
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))].
734 f(f(x,f(x,y)),x) = f(x,f(x,f(x,y))).  [para(548(a,1),694(a,1,1,2))].
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)])].
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)])].
834 f(x,f(x,f(x,f(x,f(x,y))))) = f(x,y).  [back_rewrite(548),rewrite([734(4),734(5)])].
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)])].
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)])].
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)])].
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)])].
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)])].
952 f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y.  [back_rewrite(5),rewrite([734(3)])].
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)])].
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)])].
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)])].
1037 f(x,f(x,f(x,y))) = f(x,y).  [back_rewrite(836),rewrite([980(2),980(3),834(5)])].
1048 f(f(x,y),f(x,f(x,y))) = x.  [back_rewrite(795),rewrite([980(5),1037(8)])].
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)].
1187 f(x,f(f(y,z),f(y,x))) = f(y,x).  [back_rewrite(1004),rewrite([1037(4),1037(6),1037(7)])].
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)])].
1194 f(f(x,y),f(y,f(x,z))) = y.  [back_rewrite(952),rewrite([1037(3)])].
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)])].
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)])].
1326 f(x,f(x,f(y,x))) = f(y,x).  [back_rewrite(1052),rewrite([1187(4)]),flip(a)].
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)])].
1503 f(x,y) = f(y,x).  [para(1204(a,1),1187(a,1,2)),rewrite([1331(4)])].
1504 $F # answer("Commutativity").  [resolve(1503,a,3,a)].

============================== end of proof ==========================

============================== STATISTICS ============================

Given=44. Generated=3393. Kept=1502. proofs=1.
Usable=15. Sos=189. Demods=209. Limbo=6, Disabled=1295. Hints=1306.
Weight_deleted=42. Literals_deleted=0.
Forward_subsumed=1849. Back_subsumed=199.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=1496 (0 lex), Back_demodulated=1092. Back_unit_deleted=0.
Demod_attempts=93146. Demod_rewrites=10101.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
Megabytes=2.55.
User_CPU=0.76, System_CPU=0.00, Wall_clock=12.

============================== end of statistics =====================

============================== end of search =========================

Exiting with 1 proof.

Process 22012 exit (max_proofs) Mon Jun  4 22:53:37 2007

Successful proof using the following 2 assumptions:

formulas(assumptions_in_proof).
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").  [assumption].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Including 154 new hint clauses:

formulas(new_hints).
$F # label("Commutativity") # label(non_clause) # label(goal).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
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)).
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
f(f(f(x,f(f(y,z),f(f(y,f(y,x)),y))),x),x) = f(f(y,f(y,x)),y).
f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y.
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)))).
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)))).
f(f(x,f(f(y,x),x)),f(y,f(f(z,f(z,x)),z))) = y.
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).
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)).
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)).
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.
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)))).
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)))).
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)))).
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)))).
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)))).
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).
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)))).
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)))).
f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y.
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)).
f(x,x) = f(x,f(f(x,x),x)).
f(x,f(f(x,x),x)) = f(x,x).
f(f(x,f(f(y,x),x)),f(y,y)) = y.
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).
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).
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).
f(f(x,x),f(f(x,f(x,x)),f(x,x))) = f(f(f(x,f(x,x)),x),x).
f(f(x,x),f(x,f(y,x))) = x.
f(f(x,f(f(x,x),x)),f(x,x)) = x.
f(f(x,x),f(x,x)) = x.
f(f(f(f(x,x),f(x,x)),f(x,x)),f(x,x)) = x.
f(f(x,f(x,x)),f(x,x)) = x.
f(f(x,x),x) = f(f(f(x,f(x,x)),x),x).
f(f(f(x,f(x,x)),x),x) = f(f(x,x),x).
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x.
f(f(x,f(f(f(x,f(x,x)),x),x)),x) = f(x,f(x,x)).
f(f(x,f(f(x,x),x)),x) = f(x,f(x,x)).
f(f(x,x),x) = f(x,f(x,x)).
f(f(f(x,x),x),f(f(x,x),f(x,x))) = f(x,x).
f(f(x,f(x,x)),f(f(x,x),f(x,x))) = f(x,x).
f(f(x,f(x,x)),x) = f(x,x).
f(f(f(x,x),f(x,x)),f(f(x,x),x)) = f(x,x).
f(x,f(f(x,x),x)) = f(x,x).
f(x,f(x,f(x,x))) = f(x,x).
f(f(x,f(x,x)),f(x,f(y,f(x,x)))) = x.
f(f(x,f(y,f(f(x,y),y))),f(x,f(f(y,f(f(x,y),y)),x))) = x.
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)).
f(f(x,x),f(f(y,f(x,x)),f(f(x,f(x,x)),x))) = f(y,f(x,x)).
f(f(x,x),f(f(y,f(x,x)),f(x,x))) = f(y,f(x,x)).
f(f(x,f(y,y)),f(x,y)) = x.
f(f(x,y),f(x,f(y,f(f(y,y),y)))) = x.
f(f(x,y),f(x,f(y,f(y,f(y,y))))) = x.
f(f(x,y),f(x,f(y,y))) = x.
f(x,f(f(y,f(f(x,y),y)),x)) = 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.
f(x,y) = f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x).
f(f(x,f(x,f(x,f(f(f(x,y),x),x)))),x) = f(x,y).
f(f(f(x,f(x,f(x,y))),x),x) = f(x,y).
f(f(x,f(f(y,x),x)),f(y,f(x,z))) = y # label("C6/A9").
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))).
f(f(f(f(x,y),f(f(x,y),x)),f(x,y)),f(x,y)) = x.
f(f(x,f(f(f(x,y),x),x)),x) = f(x,y).
f(f(x,f(f(y,f(y,x)),y)),x) = f(x,f(f(y,z),f(f(y,f(y,x)),y))).
f(f(x,f(x,y)),x) = f(x,f(x,f(x,y))).
f(f(x,f(y,f(y,f(y,x)))),x) = f(x,f(f(y,z),f(f(y,f(y,x)),y))).
f(f(x,f(y,f(y,f(y,x)))),x) = f(x,f(f(y,z),f(y,f(y,f(y,x))))).
f(f(f(x,y),f(f(x,y),f(f(x,y),x))),f(x,y)) = x.
f(f(x,y),f(f(x,y),f(f(x,y),f(f(x,y),x)))) = x.
f(f(x,f(x,f(x,f(x,y)))),x) = f(x,y).
f(x,f(x,f(x,f(x,f(x,y))))) = f(x,y).
f(x,f(x,f(x,f(x,f(f(f(x,y),x),x))))) = f(x,y).
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.
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.
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.
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)).
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)).
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)).
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)).
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)).
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)).
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)).
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)))).
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)))).
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)))).
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)))).
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)))).
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)))).
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)))).
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(f(y,f(y,x)),y).
f(f(f(x,f(f(y,z),f(y,f(y,f(y,x))))),x),x) = f(y,f(y,f(y,x))).
f(f(x,f(x,f(x,y))),f(y,f(x,z))) = y.
f(f(x,f(f(x,f(x,f(x,y))),x)),x) = f(x,f(x,y)).
f(f(x,f(x,f(x,f(x,f(x,y))))),x) = f(x,f(x,y)).
f(f(x,y),x) = f(x,f(x,y)).
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.
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.
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.
f(f(x,f(x,f(f(y,z),f(y,f(y,f(y,x)))))),x) = f(y,f(y,f(y,x))).
f(x,f(x,f(x,f(f(y,z),f(y,f(y,f(y,x))))))) = f(y,f(y,f(y,x))).
f(x,f(x,f(x,f(x,f(f(x,f(x,y)),x))))) = f(x,y).
f(x,f(x,f(x,f(x,f(x,f(x,f(x,y))))))) = f(x,y).
f(x,f(x,f(x,y))) = f(x,y).
f(f(x,y),f(f(x,y),f(f(x,y),f(x,f(x,y))))) = x.
f(f(x,y),f(x,f(x,y))) = x.
f(f(x,f(y,x)),x) = f(x,f(f(y,z),f(y,f(y,f(y,x))))).
f(x,f(x,f(y,x))) = f(x,f(f(y,z),f(y,f(y,f(y,x))))).
f(x,f(x,f(y,x))) = f(x,f(f(y,z),f(y,x))).
f(x,f(f(y,z),f(y,x))) = f(x,f(x,f(y,x))).
f(x,f(x,f(x,f(f(y,z),f(y,x))))) = f(y,f(y,f(y,x))).
f(x,f(f(y,z),f(y,x))) = f(y,f(y,f(y,x))).
f(x,f(f(y,z),f(y,x))) = f(y,x).
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.
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.
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.
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.
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.
f(f(x,y),f(y,f(x,z))) = y.
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)))).
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)))).
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)))).
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)))).
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)))).
f(f(x,y),f(f(x,y),y)) = f(f(y,f(x,z)),f(y,f(y,f(x,z)))).
f(f(x,y),f(f(x,y),y)) = y.
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)))).
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)))).
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)))).
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)))).
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)))).
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)))).
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)))).
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)))).
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)))).
f(f(y,x),f(f(y,x),f(f(y,x),x))) = f(x,f(y,f(y,f(y,x)))).
f(f(y,x),x) = f(x,f(y,f(y,f(y,x)))).
f(f(x,y),y) = f(y,f(x,y)).
f(y,x) = f(x,f(x,f(y,x))).
f(x,f(x,f(y,x))) = f(y,x).
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.
f(f(y,x),f(y,f(f(x,f(z,x)),f(f(x,f(z,x)),f(z,x))))) = y.
f(f(y,x),f(y,f(f(x,f(z,x)),f(f(z,x),f(x,f(z,x)))))) = y.
f(f(y,x),f(y,f(f(x,f(z,x)),x))) = y.
f(f(y,x),f(y,f(x,f(x,f(z,x))))) = y.
f(f(x,y),f(x,f(z,y))) = x.
f(x,f(f(y,x),f(y,f(y,x)))) = f(y,x).
f(x,y) = f(y,x).
$F # answer("Commutativity").
end_of_list.

Attempt to eliminate one assumption:

   f(f(f(x,f(y,z)),x),f(z,f(x,y))) = z # label("C15/A23_m").

***********************************************
AUTO SKETCHES ITERATION 17
***********************************************

Starting a search with 1 assumptions:

formulas(assumptions_active).
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Child search process 22013 started.

============================== PROCESS INITIAL CLAUSES ===============

% Clauses before input processing:

formulas(usable).
end_of_list.

formulas(sos).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
f(c2,c1) != f(c1,c2) # label("Commutativity").  [deny(1)].
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
end_of_list.

% 1460 hints input.

============================== PREDICATE ELIMINATION =================

No predicates eliminated.

============================== end predicate elimination =============

Auto_denials:
  % copying label "Commutativity" to answer in negative clause

Term ordering decisions:
Function symbol KB weights:  c1=1. c2=1. f=1.
Predicate symbol precedence:  predicate_order([ = ]).
Function symbol precedence:  function_order([ c1, c2, f ]).
Skipping inverse_order, because term ordering is KBO.
Unfolding symbols: (none).

Auto_inference settings:
  % set(paramodulation).  % (positive equality literals)
    % set(paramodulation) -> set(back_demod).

Auto_process settings:  (no changes).

============================== end of process initial clauses ========

============================== CLAUSES FOR SEARCH ====================

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

formulas(demodulators).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
4 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

% 1460 hints processed (771 are redundant).

============================== end of clauses for search =============

============================== SEARCH ================================

% Starting search at 0.08 seconds.

% Operation f is commutative; C redundancy checks enabled.

============================== PROOF =================================

% Proof 1 at 1.68 (+ 0.00) seconds: "Commutativity".
% Length of proof is 57.
% Level of proof is 18.
% Maximum clause weight is 59.
% Given clauses 58.

1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal).  [goal].
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
4 f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
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))].
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))].
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))].
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))].
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))].
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))].
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))].
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)].
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)])].
74 f(x,f(f(x,x),x)) = f(x,x).  [para(18(a,1),5(a,1,2)),rewrite([18(8)]),flip(a)].
77 f(f(x,f(f(y,x),x)),f(y,y)) = y.  [back_rewrite(71),rewrite([74(6)])].
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)])].
79 f(f(x,f(f(f(y,y),x),x)),y) = f(y,y).  [back_rewrite(31),rewrite([74(3),74(8)])].
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)])].
82 f(f(x,x),f(x,f(y,x))) = x.  [para(74(a,1),2(a,1,1))].
84 f(f(x,x),f(x,x)) = x.  [para(74(a,1),2(a,1,2)),rewrite([74(3)])].
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))].
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)])].
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)].
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)])].
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))].
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)])].
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)])].
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)])].
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)])].
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)])].
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)])].
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)])].
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))].
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)])].
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)])].
430 f(f(x,x),x) = f(x,f(x,x)).  [back_rewrite(78),rewrite([405(6),426(5)]),flip(a)].
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)])].
449 f(f(x,f(x,x)),f(x,x)) = x.  [back_rewrite(387),rewrite([426(3),430(2)])].
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)])].
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)])].
1231 f(f(x,x),f(y,x)) = x.  [para(149(a,1),2(a,1,2)),rewrite([430(2),431(3)])].
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)])].
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))].
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)])].
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)])].
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)])].
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))].
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)])].
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)])].
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)])].
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)])].
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)])].
1518 f(f(x,f(y,y)),y) = f(y,y).  [back_rewrite(1249),rewrite([1496(7),1496(13),1502(12)])].
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)])].
1535 f(f(x,x),f(x,y)) = x.  [back_rewrite(1297),rewrite([1499(7),1516(4)])].
1795 f(x,y) = f(y,x).  [para(1231(a,1),1531(a,1,1)),rewrite([1535(4)])].
1796 $F # answer("Commutativity").  [resolve(1795,a,3,a)].

============================== end of proof ==========================

============================== STATISTICS ============================

Given=58. Generated=8110. Kept=1794. proofs=1.
Usable=7. Sos=164. Demods=169. Limbo=0, Disabled=1625. Hints=1460.
Weight_deleted=896. Literals_deleted=0.
Forward_subsumed=5420. Back_subsumed=196.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=1785 (0 lex), Back_demodulated=1426. Back_unit_deleted=0.
Demod_attempts=407851. Demod_rewrites=33848.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
Megabytes=4.27.
User_CPU=1.68, System_CPU=0.00, Wall_clock=13.

============================== end of statistics =====================

============================== end of search =========================

Exiting with 1 proof.

Process 22013 exit (max_proofs) Mon Jun  4 22:53:38 2007

Successful proof using the following 1 assumptions:

formulas(assumptions_in_proof).
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").  [assumption].
end_of_list.

Including 170 new hint clauses:

formulas(new_hints).
$F # label("Commutativity") # label(non_clause) # label(goal).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").
f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").
f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").
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)).
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)).
f(f(f(x,f(x,f(y,f(f(x,y),y)))),x),x) = f(y,f(f(x,y),y)).
f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y.
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)).
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)).
f(f(x,f(f(f(y,f(f(y,y),y)),x),x)),y) = f(y,f(f(y,y),y)).
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)))).
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)))).
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)))).
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)))).
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)))).
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).
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)))).
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)))).
f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y.
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)).
f(x,x) = f(x,f(f(x,x),x)).
f(x,f(f(x,x),x)) = f(x,x).
f(f(x,f(f(y,x),x)),f(y,y)) = y.
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).
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).
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).
f(f(x,x),f(f(x,f(x,x)),f(x,x))) = f(f(f(x,f(x,x)),x),x).
f(f(x,f(f(f(y,y),x),x)),y) = f(y,f(f(y,y),y)).
f(f(x,f(f(f(y,y),x),x)),y) = f(y,y).
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)).
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)).
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)).
f(f(f(f(x,x),x),f(x,x)),f(x,x)) = f(x,f(f(f(x,x),x),x)).
f(f(x,x),f(x,f(y,x))) = x.
f(f(x,f(f(x,x),x)),f(x,x)) = x.
f(f(x,x),f(x,x)) = x.
f(f(f(x,x),f(f(y,f(x,x)),f(x,x))),f(y,x)) = y.
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))).
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))).
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))).
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.
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)).
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)).
f(x,y) = f(y,f(f(f(f(x,y),f(x,y)),y),y)).
f(x,f(f(f(f(y,x),f(y,x)),x),x)) = f(y,x).
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)).
f(x,f(f(y,f(f(x,y),y)),x)) = f(y,f(f(x,y),y)).
f(f(f(x,f(x,y)),x),f(y,f(z,f(f(x,z),z)))) = y.
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)).
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)).
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)).
f(x,f(y,f(f(f(x,f(f(x,x),x)),y),y))) = f(x,f(f(x,x),x)).
f(x,f(y,f(f(f(x,x),y),y))) = f(x,f(f(x,x),x)).
f(x,f(y,f(f(f(x,x),y),y))) = f(x,x).
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))).
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))).
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))).
f(f(x,f(f(y,x),x)),f(y,f(x,f(f(y,x),x)))) = y.
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)))).
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)))).
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)))).
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)))).
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)))).
f(f(f(x,f(x,y)),x),f(y,y)) = f(f(y,y),f(y,f(y,f(f(y,y),y)))).
f(f(f(x,f(x,y)),x),f(y,y)) = f(f(y,y),f(y,f(y,y))).
f(f(f(x,f(x,y)),x),f(y,y)) = y.
f(f(f(f(x,x),x),f(x,x)),f(f(x,x),f(x,x))) = f(x,x).
f(f(f(f(x,x),x),f(x,x)),x) = f(x,x).
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.
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.
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).
f(f(f(x,x),f(x,f(x,x))),f(f(f(x,f(x,x)),x),x)) = f(x,x).
f(x,f(f(f(x,f(x,x)),x),x)) = f(x,x).
f(f(f(f(x,f(x,x)),x),x),f(x,f(y,f(f(f(x,x),y),y)))) = x.
f(f(f(f(x,f(x,x)),x),x),f(x,x)) = x.
f(f(x,x),f(f(x,f(x,x)),f(y,x))) = f(x,f(x,x)).
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))).
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))).
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)).
f(f(x,f(x,x)),x) = f(x,f(f(f(x,f(x,x)),x),x)).
f(f(x,f(x,x)),x) = f(x,x).
f(x,f(x,x)) = f(f(f(x,f(x,x)),x),x).
f(x,f(x,x)) = f(f(x,x),x).
f(f(x,x),x) = f(x,f(x,x)).
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))).
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))).
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))).
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))).
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))).
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))).
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))).
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))).
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))).
f(x,f(x,f(x,x))) = f(f(x,f(x,x)),f(f(x,x),f(f(x,x),x))).
f(x,f(x,f(x,x))) = f(f(x,f(x,x)),f(f(x,x),f(x,f(x,x)))).
f(x,f(x,f(x,x))) = f(f(x,f(x,x)),x).
f(x,f(x,f(x,x))) = f(x,x).
f(f(f(x,x),x),f(x,x)) = x.
f(f(x,f(x,x)),f(x,x)) = x.
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.
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.
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.
f(f(f(x,f(x,x)),x),f(x,f(f(f(f(x,x),x),f(x,x)),y))) = x.
f(f(x,x),f(x,f(f(f(f(x,x),x),f(x,x)),y))) = x.
f(f(x,x),f(x,f(f(f(x,f(x,x)),f(x,x)),y))) = x.
f(f(x,x),f(x,f(x,y))) = x.
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))).
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))).
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))).
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))).
f(f(y,y),f(f(z,f(z,y)),z)) = f(f(y,y),f(y,f(y,y))).
f(f(x,x),f(f(y,f(y,x)),y)) = x.
f(f(x,f(f(x,x),x)),f(y,x)) = x.
f(f(x,f(x,f(x,x))),f(y,x)) = x.
f(f(x,x),f(y,x)) = x.
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)).
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)).
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)).
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)).
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)).
f(x,f(y,f(x,f(f(x,x),x)))) = f(x,f(f(x,x),x)).
f(x,f(y,f(x,f(x,f(x,x))))) = f(x,f(f(x,x),x)).
f(x,f(y,f(x,x))) = f(x,f(f(x,x),x)).
f(x,f(y,f(x,x))) = f(x,f(x,f(x,x))).
f(x,f(y,f(x,x))) = f(x,x).
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).
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))).
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))).
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)).
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)).
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).
f(f(f(x,f(y,y)),f(f(y,y),f(x,f(y,y)))),f(z,f(f(y,z),z))) = y.
f(f(f(x,y),f(x,y)),f(f(x,y),y)) = f(x,y).
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))).
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))).
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))).
f(f(f(x,y),f(x,y)),f(f(f(y,y),y),f(y,y))) = f(x,y).
f(f(f(x,y),f(x,y)),f(f(y,f(y,y)),f(y,y))) = f(x,y).
f(f(f(x,y),f(x,y)),y) = f(x,y).
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)).
f(f(x,y),f(f(f(y,y),f(x,y)),f(x,y))) = f(f(y,y),f(x,y)).
f(f(x,y),f(y,f(x,y))) = f(f(y,y),f(x,y)).
f(f(x,y),f(y,f(x,y))) = y.
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).
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).
f(f(f(x,y),y),f(f(x,y),f(f(f(f(x,y),f(x,y)),y),y))) = f(x,y).
f(f(f(x,y),y),f(f(x,y),f(f(x,y),y))) = f(x,y).
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))).
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))).
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))).
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))).
f(x,f(f(y,x),x)) = f(f(f(y,x),x),f(f(y,x),f(f(y,x),x))).
f(x,f(f(y,x),x)) = f(y,x).
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).
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).
f(f(x,f(y,y)),y) = f(y,y).
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))).
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))).
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))).
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))).
f(f(y,f(f(y,y),y)),z) = f(f(y,y),f(f(z,f(y,y)),f(y,y))).
f(f(y,f(y,f(y,y))),z) = f(f(y,y),f(f(z,f(y,y)),f(y,y))).
f(f(y,y),z) = f(f(y,y),f(f(z,f(y,y)),f(y,y))).
f(f(x,x),y) = f(y,f(x,x)).
f(f(y,y),f(z,f(f(y,z),z))) = y.
f(f(x,x),f(x,y)) = x.
f(x,y) = f(y,f(f(x,x),f(x,x))).
f(x,y) = f(y,x).
$F # answer("Commutativity").
end_of_list.

Attempt to eliminate one assumption:

   f(f(f(x,f(x,y)),x),f(y,f(x,z))) = y # label("C16/A24_m").

***********************************************
AUTO SKETCHES ITERATION 18
***********************************************

Starting a search with 0 assumptions:

formulas(assumptions_active).
end_of_list.

Note: Attempt with no assumptions!


Child search process 22016 started.

============================== PROCESS INITIAL CLAUSES ===============

% Clauses before input processing:

formulas(usable).
end_of_list.

formulas(sos).
f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
f(c2,c1) != f(c1,c2) # label("Commutativity").  [deny(1)].
end_of_list.

formulas(demodulators).
end_of_list.

% 1630 hints input.

============================== PREDICATE ELIMINATION =================

No predicates eliminated.

============================== end predicate elimination =============

Auto_denials:
  % copying label "Commutativity" to answer in negative clause

Term ordering decisions:
Function symbol KB weights:  c1=1. c2=1. f=1.
Predicate symbol precedence:  predicate_order([ = ]).
Function symbol precedence:  function_order([ c1, c2, f ]).
Skipping inverse_order, because term ordering is KBO.
Unfolding symbols: (none).

Auto_inference settings:
  % set(paramodulation).  % (positive equality literals)
    % set(paramodulation) -> set(back_demod).

Auto_process settings:  (no changes).

============================== end of process initial clauses ========

============================== CLAUSES FOR SEARCH ====================

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
end_of_list.

formulas(demodulators).
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
end_of_list.

% 1630 hints processed (814 are redundant).

============================== end of clauses for search =============

============================== SEARCH ================================

% Starting search at 0.10 seconds.

% Operation f is commutative; C redundancy checks enabled.

============================== PROOF =================================

% Proof 1 at 0.31 (+ 0.00) seconds: "Commutativity".
% Length of proof is 25.
% Level of proof is 12.
% Maximum clause weight is 61.
% Given clauses 31.

1 f(x,y) = f(y,x) # label("Commutativity") # label(non_clause) # label(goal).  [goal].
2 f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y # label("Sh1/A10").  [assumption].
3 f(c2,c1) != f(c1,c2) # label("Commutativity") # answer("Commutativity").  [deny(1)].
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))].
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))].
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))].
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)])].
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))].
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)])].
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)].
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)].
56 f(x,f(f(f(f(y,x),f(y,x)),x),x)) = f(y,x).  [back_rewrite(35),rewrite([45(7)])].
62 f(f(x,f(f(y,x),x)),f(y,y)) = y.  [back_rewrite(27),rewrite([45(6)])].
73 f(f(x,x),f(x,f(y,x))) = x.  [para(45(a,1),2(a,1,1))].
104 f(f(x,x),f(y,x)) = x.  [para(56(a,1),2(a,1,2)),rewrite([45(3)])].
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))].
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)])].
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))].
143 f(f(f(x,y),f(x,y)),y) = f(x,y).  [para(104(a,1),104(a,1,2))].
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)])].
150 f(f(x,y),f(y,y)) = y.  [back_rewrite(115),rewrite([143(4),143(7),148(7)])].
157 f(x,f(f(y,x),x)) = f(y,x).  [back_rewrite(56),rewrite([143(4)])].
197 f(f(f(x,y),f(z,y)),x) = f(x,y).  [back_rewrite(6),rewrite([157(4),157(7),157(7)])].
257 f(x,y) = f(y,x).  [para(150(a,1),197(a,1,1))].
258 $F # answer("Commutativity").  [resolve(257,a,3,a)].

============================== end of proof ==========================

============================== STATISTICS ============================

Given=31. Generated=1114. Kept=256. proofs=1.
Usable=17. Sos=63. Demods=79. Limbo=1, Disabled=176. Hints=1630.
Weight_deleted=2. Literals_deleted=0.
Forward_subsumed=856. Back_subsumed=34.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=253 (0 lex), Back_demodulated=140. Back_unit_deleted=0.
Demod_attempts=33613. Demod_rewrites=3985.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0.
Megabytes=1.46.
User_CPU=0.31, System_CPU=0.00, Wall_clock=14.

============================== end of statistics =====================

============================== end of search =========================

Exiting with 1 proof.

Process 22016 exit (max_proofs) Mon Jun  4 22:53:39 2007

Proof is complete (no extra assumptions used).

Autosketches completes after 18 iterations.

THEOREM PROVED

Exiting.

Process 21989 exit (max_proofs) Mon Jun  4 22:53:39 2007
