============================== Prover9 ===============================
Prover9 (32) version 2009-02A, February 2009.
Process 15830 was started by mccune on cleo,
Wed Feb 25 12:25:50 2009
The command was "/home/mccune/bin/prover9 -f subset_trans_expand.in".
============================== end of head ===========================

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

% Reading from file subset_trans_expand.in

set(expand_relational_defs).

formulas(assumptions).
(all x all y (subset(x,y) <-> (all z (member(z,x) -> member(z,y))))).
end_of_list.

formulas(goals).
(all x all y all z (subset(x,y) & subset(y,z) -> subset(x,z))).
end_of_list.

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

============================== EXPAND RELATIONAL DEFINITIONS =========

% Relational Definitions:
1 (all x all y (subset(x,y) <-> (all z (member(z,x) -> member(z,y))))).  [assumption].

% Formulas Being Expanded:
2 (all x all y all z (subset(x,y) & subset(y,z) -> subset(x,z))).  [goal].

============================== end of expand relational definitions ==

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

% Formulas that are not ordinary clauses:
3 (all x0 all x1 all x2 ((all z (member(z,x0) -> member(z,x1))) & (all z (member(z,x1) -> member(z,x2))) -> (all z (member(z,x0) -> member(z,x2))))) # label(non_clause) # label(goal).  [expand_def(2,1)].

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

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

% Clauses before input processing:

formulas(usable).
end_of_list.

formulas(sos).
-member(x,c1) | member(x,c2).  [deny(3)].
-member(x,c2) | member(x,c3).  [deny(3)].
member(c4,c1).  [deny(3)].
-member(c4,c3).  [deny(3)].
end_of_list.

formulas(demodulators).
end_of_list.

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

No predicates eliminated.

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

Auto_denials:  (no changes).

Term ordering decisions:
Predicate symbol precedence:  predicate_order([ member ]).
Function symbol precedence:  function_order([ c1, c2, c3, c4 ]).
After inverse_order:  (no changes).
Unfolding symbols: (none).

Auto_inference settings:
  % set(neg_binary_resolution).  % (HNE depth_diff=0)
  % clear(ordered_res).  % (HNE depth_diff=0)
  % set(ur_resolution).  % (HNE depth_diff=0)
    % set(ur_resolution) -> set(pos_ur_resolution).
    % set(ur_resolution) -> set(neg_ur_resolution).

Auto_process settings:  (no changes).

kept:      4 -member(x,c1) | member(x,c2).  [deny(3)].
kept:      5 -member(x,c2) | member(x,c3).  [deny(3)].
kept:      6 member(c4,c1).  [deny(3)].
kept:      7 -member(c4,c3).  [deny(3)].

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

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

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
4 -member(x,c1) | member(x,c2).  [deny(3)].
5 -member(x,c2) | member(x,c3).  [deny(3)].
6 member(c4,c1).  [deny(3)].
7 -member(c4,c3).  [deny(3)].
end_of_list.

formulas(demodulators).
end_of_list.

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

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

% Starting search at 0.01 seconds.

given #1 (I,wt=6): 4 -member(x,c1) | member(x,c2).  [deny(3)].

given #2 (I,wt=6): 5 -member(x,c2) | member(x,c3).  [deny(3)].

given #3 (I,wt=3): 6 member(c4,c1).  [deny(3)].

given #4 (I,wt=3): 7 -member(c4,c3).  [deny(3)].

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

% Proof 1 at 0.01 (+ 0.00) seconds.
% Length of proof is 10.
% Level of proof is 4.
% Maximum clause weight is 6.
% Given clauses 4.

1 (all x all y (subset(x,y) <-> (all z (member(z,x) -> member(z,y))))) # label(non_clause).  [assumption].
2 (all x all y all z (subset(x,y) & subset(y,z) -> subset(x,z))) # label(non_clause).  [goal].
3 (all x0 all x1 all x2 ((all z (member(z,x0) -> member(z,x1))) & (all z (member(z,x1) -> member(z,x2))) -> (all z (member(z,x0) -> member(z,x2))))) # label(non_clause) # label(goal).  [expand_def(2,1)].
4 -member(x,c1) | member(x,c2).  [deny(3)].
5 -member(x,c2) | member(x,c3).  [deny(3)].
6 member(c4,c1).  [deny(3)].
7 -member(c4,c3).  [deny(3)].
8 member(c4,c2).  [ur(4,a,6,a)].
9 -member(c4,c2).  [resolve(7,a,5,b)].
10 $F.  [resolve(9,a,8,a)].

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

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

Given=4. Generated=6. Kept=6. proofs=1.
Usable=4. Sos=1. Demods=0. Limbo=0, Disabled=4. Hints=0.
Kept_by_rule=0, Deleted_by_rule=0.
Forward_subsumed=0. Back_subsumed=0.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0.
Demod_attempts=0. Demod_rewrites=0.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=2.
Megabytes=0.03.
User_CPU=0.01, System_CPU=0.00, Wall_clock=0.

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

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

THEOREM PROVED

Exiting with 1 proof.

Process 15830 exit (max_proofs) Wed Feb 25 12:25:50 2009
