NO

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty x y vNonEmpty x1)
(STRATEGY CONTEXTSENSITIVE
(f 1 2)
(g 1)
(h 1)
(a)
(c 1)
(fSNonEmpty)
(s 1)
)
(RULES
f(x,y) -> g(s(x)) | c(g(x)) ->* c(a)
f(x,y) -> h(s(x)) | c(h(x)) ->* c(a)
g(s(x)) -> x
h(s(x)) -> x
)
]

Infeasibility Conditions:
c(g(x1)) ->* c(a), c(h(x1)) ->* c(a)

Problem 1: 

Obtaining a proof using Prover9:

 -> Prover9 Output:
============================== Prover9 ===============================
Prover9 (64) version 2009-11A, November 2009.
Process 3317208 was started by sandbox2 on z016.star.cs.uiowa.edu,
Tue Jul 30 09:17:25 2024
The command was "./prover9 -f /tmp/prover93317197-0.in".
============================== end of head ===========================

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

% Reading from file /tmp/prover93317197-0.in

assign(max_seconds,20).

formulas(assumptions).
->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence).
->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence).
->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence).
->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence).
->_s0(x1,y) -> ->_s0(c(x1),c(y)) # label(congruence).
->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence).
->*_s0(c(g(x1)),c(a)) -> ->_s0(f(x1,x2),g(s(x1))) # label(replacement).
->*_s0(c(h(x1)),c(a)) -> ->_s0(f(x1,x2),h(s(x1))) # label(replacement).
->_s0(g(s(x1)),x1) # label(replacement).
->_s0(h(s(x1)),x1) # label(replacement).
->*_s0(x,x) # label(reflexivity).
->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity).
end_of_list.

formulas(goals).
(exists x4 (->*_s0(c(g(x4)),c(a)) & ->*_s0(c(h(x4)),c(a)))) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause).  [assumption].
2 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause).  [assumption].
3 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause).  [assumption].
4 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence) # label(non_clause).  [assumption].
5 ->_s0(x1,y) -> ->_s0(c(x1),c(y)) # label(congruence) # label(non_clause).  [assumption].
6 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause).  [assumption].
7 ->*_s0(c(g(x1)),c(a)) -> ->_s0(f(x1,x2),g(s(x1))) # label(replacement) # label(non_clause).  [assumption].
8 ->*_s0(c(h(x1)),c(a)) -> ->_s0(f(x1,x2),h(s(x1))) # label(replacement) # label(non_clause).  [assumption].
9 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
10 (exists x4 (->*_s0(c(g(x4)),c(a)) & ->*_s0(c(h(x4)),c(a)))) # label(goal) # label(non_clause) # label(goal).  [goal].

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

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

% Clauses before input processing:

formulas(usable).
end_of_list.

formulas(sos).
-->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence).  [clausify(1)].
-->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence).  [clausify(2)].
-->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence).  [clausify(3)].
-->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence).  [clausify(4)].
-->_s0(x,y) | ->_s0(c(x),c(y)) # label(congruence).  [clausify(5)].
-->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(6)].
-->*_s0(c(g(x)),c(a)) | ->_s0(f(x,y),g(s(x))) # label(replacement).  [clausify(7)].
-->*_s0(c(h(x)),c(a)) | ->_s0(f(x,y),h(s(x))) # label(replacement).  [clausify(8)].
->_s0(g(s(x)),x) # label(replacement).  [assumption].
->_s0(h(s(x)),x) # label(replacement).  [assumption].
->*_s0(x,x) # label(reflexivity).  [assumption].
-->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(9)].
-->*_s0(c(g(x)),c(a)) | -->*_s0(c(h(x)),c(a)) # label(goal).  [deny(10)].
end_of_list.

formulas(demodulators).
end_of_list.

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

No predicates eliminated.

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

Auto_denials:
  % copying label goal to answer in negative clause

Term ordering decisions:
Predicate symbol precedence:  predicate_order([ ->_s0, ->*_s0 ]).
Function symbol precedence:  function_order([ a, f, c, s, g, h ]).
After inverse_order:  (no changes).
Unfolding symbols: (none).

Auto_inference settings:
  % set(neg_binary_resolution).  % (HNE depth_diff=-6)
  % clear(ordered_res).  % (HNE depth_diff=-6)
  % set(ur_resolution).  % (HNE depth_diff=-6)
    % set(ur_resolution) -> set(pos_ur_resolution).
    % set(ur_resolution) -> set(neg_ur_resolution).

Auto_process settings:
  % set(unit_deletion).  % (Horn set with negative nonunits)

kept:      11 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence).  [clausify(1)].
kept:      12 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence).  [clausify(2)].
kept:      13 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence).  [clausify(3)].
kept:      14 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence).  [clausify(4)].
kept:      15 -->_s0(x,y) | ->_s0(c(x),c(y)) # label(congruence).  [clausify(5)].
kept:      16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(6)].
kept:      17 -->*_s0(c(g(x)),c(a)) | ->_s0(f(x,y),g(s(x))) # label(replacement).  [clausify(7)].
kept:      18 -->*_s0(c(h(x)),c(a)) | ->_s0(f(x,y),h(s(x))) # label(replacement).  [clausify(8)].
kept:      19 ->_s0(g(s(x)),x) # label(replacement).  [assumption].
kept:      20 ->_s0(h(s(x)),x) # label(replacement).  [assumption].
kept:      21 ->*_s0(x,x) # label(reflexivity).  [assumption].
kept:      22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(9)].
kept:      23 -->*_s0(c(g(x)),c(a)) | -->*_s0(c(h(x)),c(a)) # label(goal) # answer(goal).  [deny(10)].

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

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

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
11 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence).  [clausify(1)].
12 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence).  [clausify(2)].
13 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence).  [clausify(3)].
14 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence).  [clausify(4)].
15 -->_s0(x,y) | ->_s0(c(x),c(y)) # label(congruence).  [clausify(5)].
16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(6)].
17 -->*_s0(c(g(x)),c(a)) | ->_s0(f(x,y),g(s(x))) # label(replacement).  [clausify(7)].
18 -->*_s0(c(h(x)),c(a)) | ->_s0(f(x,y),h(s(x))) # label(replacement).  [clausify(8)].
19 ->_s0(g(s(x)),x) # label(replacement).  [assumption].
20 ->_s0(h(s(x)),x) # label(replacement).  [assumption].
21 ->*_s0(x,x) # label(reflexivity).  [assumption].
22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(9)].
23 -->*_s0(c(g(x)),c(a)) | -->*_s0(c(h(x)),c(a)) # label(goal) # answer(goal).  [deny(10)].
end_of_list.

formulas(demodulators).
end_of_list.

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

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

% Starting search at 0.01 seconds.

given #1 (I,wt=10): 11 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence).  [clausify(1)].

given #2 (I,wt=10): 12 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence).  [clausify(2)].

given #3 (I,wt=8): 13 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence).  [clausify(3)].

given #4 (I,wt=8): 14 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence).  [clausify(4)].

given #5 (I,wt=8): 15 -->_s0(x,y) | ->_s0(c(x),c(y)) # label(congruence).  [clausify(5)].

given #6 (I,wt=8): 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(6)].

given #7 (I,wt=13): 17 -->*_s0(c(g(x)),c(a)) | ->_s0(f(x,y),g(s(x))) # label(replacement).  [clausify(7)].

given #8 (I,wt=13): 18 -->*_s0(c(h(x)),c(a)) | ->_s0(f(x,y),h(s(x))) # label(replacement).  [clausify(8)].

given #9 (I,wt=5): 19 ->_s0(g(s(x)),x) # label(replacement).  [assumption].

given #10 (I,wt=5): 20 ->_s0(h(s(x)),x) # label(replacement).  [assumption].

given #11 (I,wt=3): 21 ->*_s0(x,x) # label(reflexivity).  [assumption].

given #12 (I,wt=9): 22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(9)].

given #13 (I,wt=12): 23 -->*_s0(c(g(x)),c(a)) | -->*_s0(c(h(x)),c(a)) # label(goal) # answer(goal).  [deny(10)].

given #14 (A,wt=7): 24 ->_s0(s(g(s(x))),s(x)).  [ur(16,a,19,a)].

given #15 (F,wt=15): 38 -->*_s0(c(h(x)),c(a)) | -->_s0(c(g(x)),y) | -->*_s0(y,c(a)) # answer(goal).  [resolve(23,a,22,c)].

given #16 (F,wt=12): 49 -->*_s0(c(h(x)),c(a)) | -->_s0(c(g(x)),c(a)) # answer(goal).  [resolve(38,c,21,a)].

given #17 (F,wt=10): 51 -->*_s0(c(h(x)),c(a)) | -->_s0(g(x),a) # answer(goal).  [resolve(49,b,15,b)].

given #18 (F,wt=7): 53 -->*_s0(c(h(s(a))),c(a)) # answer(goal).  [resolve(51,b,19,a)].

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

% Proof 1 at 0.01 (+ 0.00) seconds: goal.
% Length of proof is 15.
% Level of proof is 6.
% Maximum clause weight is 15.000.
% Given clauses 18.

5 ->_s0(x1,y) -> ->_s0(c(x1),c(y)) # label(congruence) # label(non_clause).  [assumption].
9 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
10 (exists x4 (->*_s0(c(g(x4)),c(a)) & ->*_s0(c(h(x4)),c(a)))) # label(goal) # label(non_clause) # label(goal).  [goal].
15 -->_s0(x,y) | ->_s0(c(x),c(y)) # label(congruence).  [clausify(5)].
19 ->_s0(g(s(x)),x) # label(replacement).  [assumption].
20 ->_s0(h(s(x)),x) # label(replacement).  [assumption].
21 ->*_s0(x,x) # label(reflexivity).  [assumption].
22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(9)].
23 -->*_s0(c(g(x)),c(a)) | -->*_s0(c(h(x)),c(a)) # label(goal) # answer(goal).  [deny(10)].
31 ->_s0(c(h(s(x))),c(x)).  [ur(15,a,20,a)].
38 -->*_s0(c(h(x)),c(a)) | -->_s0(c(g(x)),y) | -->*_s0(y,c(a)) # answer(goal).  [resolve(23,a,22,c)].
49 -->*_s0(c(h(x)),c(a)) | -->_s0(c(g(x)),c(a)) # answer(goal).  [resolve(38,c,21,a)].
51 -->*_s0(c(h(x)),c(a)) | -->_s0(g(x),a) # answer(goal).  [resolve(49,b,15,b)].
53 -->*_s0(c(h(s(a))),c(a)) # answer(goal).  [resolve(51,b,19,a)].
55 $F # answer(goal).  [ur(22,b,21,a,c,53,a),unit_del(a,31)].

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

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

Given=18. Generated=47. Kept=44. proofs=1.
Usable=18. Sos=25. Demods=0. Limbo=1, Disabled=13. Hints=0.
Kept_by_rule=0, Deleted_by_rule=0.
Forward_subsumed=2. Back_subsumed=0.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0.
Demod_attempts=0. Demod_rewrites=0.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=14. Nonunit_bsub_feature_tests=48.
Megabytes=0.13.
User_CPU=0.01, System_CPU=0.00, Wall_clock=0.

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

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

THEOREM PROVED

Exiting with 1 proof.

Process 3317208 exit (max_proofs) Tue Jul 30 09:17:25 2024


The problem is feasible.