NO

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty x vNonEmpty x1)
(STRATEGY CONTEXTSENSITIVE
(e 1)
(o 1)
(0)
(fSNonEmpty)
(false)
(s 1)
(true)
)
(RULES
e(0) -> true
e(s(x)) -> false | e(x) ->* true
e(s(x)) -> true | o(x) ->* true
o(0) -> true
o(s(x)) -> false | o(x) ->* true
o(s(x)) -> true | e(x) ->* true
)
]

Infeasibility Conditions:
e(x1) ->* true, o(x1) ->* true

Problem 1: 

Obtaining a proof using Prover9:

 -> Prover9 Output:
============================== Prover9 ===============================
Prover9 (64) version 2009-11A, November 2009.
Process 3406812 was started by sandbox2 on z025.star.cs.uiowa.edu,
Tue Jul 30 09:50:40 2024
The command was "./prover9 -f /tmp/prover93406802-0.in".
============================== end of head ===========================

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

% Reading from file /tmp/prover93406802-0.in

assign(max_seconds,20).

formulas(assumptions).
->_s0(x1,y) -> ->_s0(e(x1),e(y)) # label(congruence).
->_s0(x1,y) -> ->_s0(o(x1),o(y)) # label(congruence).
->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence).
->_s0(e(0),true) # label(replacement).
->*_s0(e(x1),true) -> ->_s0(e(s(x1)),false) # label(replacement).
->*_s0(o(x1),true) -> ->_s0(e(s(x1)),true) # label(replacement).
->_s0(o(0),true) # label(replacement).
->*_s0(o(x1),true) -> ->_s0(o(s(x1)),false) # label(replacement).
->*_s0(e(x1),true) -> ->_s0(o(s(x1)),true) # label(replacement).
->*_s0(x,x) # label(reflexivity).
->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity).
end_of_list.

formulas(goals).
(exists x3 (->*_s0(e(x3),true) & ->*_s0(o(x3),true))) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 ->_s0(x1,y) -> ->_s0(e(x1),e(y)) # label(congruence) # label(non_clause).  [assumption].
2 ->_s0(x1,y) -> ->_s0(o(x1),o(y)) # label(congruence) # label(non_clause).  [assumption].
3 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause).  [assumption].
4 ->*_s0(e(x1),true) -> ->_s0(e(s(x1)),false) # label(replacement) # label(non_clause).  [assumption].
5 ->*_s0(o(x1),true) -> ->_s0(e(s(x1)),true) # label(replacement) # label(non_clause).  [assumption].
6 ->*_s0(o(x1),true) -> ->_s0(o(s(x1)),false) # label(replacement) # label(non_clause).  [assumption].
7 ->*_s0(e(x1),true) -> ->_s0(o(s(x1)),true) # label(replacement) # label(non_clause).  [assumption].
8 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
9 (exists x3 (->*_s0(e(x3),true) & ->*_s0(o(x3),true))) # 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(e(x),e(y)) # label(congruence).  [clausify(1)].
-->_s0(x,y) | ->_s0(o(x),o(y)) # label(congruence).  [clausify(2)].
-->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(3)].
->_s0(e(0),true) # label(replacement).  [assumption].
-->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement).  [clausify(4)].
-->*_s0(o(x),true) | ->_s0(e(s(x)),true) # label(replacement).  [clausify(5)].
->_s0(o(0),true) # label(replacement).  [assumption].
-->*_s0(o(x),true) | ->_s0(o(s(x)),false) # label(replacement).  [clausify(6)].
-->*_s0(e(x),true) | ->_s0(o(s(x)),true) # label(replacement).  [clausify(7)].
->*_s0(x,x) # label(reflexivity).  [assumption].
-->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(8)].
-->*_s0(e(x),true) | -->*_s0(o(x),true) # label(goal).  [deny(9)].
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([ true, 0, false, e, o, s ]).
After inverse_order:  (no changes).
Unfolding symbols: (none).

Auto_inference settings:
  % set(neg_binary_resolution).  % (HNE depth_diff=-7)
  % clear(ordered_res).  % (HNE depth_diff=-7)
  % set(ur_resolution).  % (HNE depth_diff=-7)
    % 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:      10 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence).  [clausify(1)].
kept:      11 -->_s0(x,y) | ->_s0(o(x),o(y)) # label(congruence).  [clausify(2)].
kept:      12 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(3)].
kept:      13 ->_s0(e(0),true) # label(replacement).  [assumption].
kept:      14 -->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement).  [clausify(4)].
kept:      15 -->*_s0(o(x),true) | ->_s0(e(s(x)),true) # label(replacement).  [clausify(5)].
kept:      16 ->_s0(o(0),true) # label(replacement).  [assumption].
kept:      17 -->*_s0(o(x),true) | ->_s0(o(s(x)),false) # label(replacement).  [clausify(6)].
kept:      18 -->*_s0(e(x),true) | ->_s0(o(s(x)),true) # label(replacement).  [clausify(7)].
kept:      19 ->*_s0(x,x) # label(reflexivity).  [assumption].
kept:      20 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(8)].
kept:      21 -->*_s0(e(x),true) | -->*_s0(o(x),true) # label(goal) # answer(goal).  [deny(9)].

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

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

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
10 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence).  [clausify(1)].
11 -->_s0(x,y) | ->_s0(o(x),o(y)) # label(congruence).  [clausify(2)].
12 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(3)].
13 ->_s0(e(0),true) # label(replacement).  [assumption].
14 -->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement).  [clausify(4)].
15 -->*_s0(o(x),true) | ->_s0(e(s(x)),true) # label(replacement).  [clausify(5)].
16 ->_s0(o(0),true) # label(replacement).  [assumption].
17 -->*_s0(o(x),true) | ->_s0(o(s(x)),false) # label(replacement).  [clausify(6)].
18 -->*_s0(e(x),true) | ->_s0(o(s(x)),true) # label(replacement).  [clausify(7)].
19 ->*_s0(x,x) # label(reflexivity).  [assumption].
20 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(8)].
21 -->*_s0(e(x),true) | -->*_s0(o(x),true) # label(goal) # answer(goal).  [deny(9)].
end_of_list.

formulas(demodulators).
end_of_list.

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

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

% Starting search at 0.00 seconds.

given #1 (I,wt=8): 10 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence).  [clausify(1)].

given #2 (I,wt=8): 11 -->_s0(x,y) | ->_s0(o(x),o(y)) # label(congruence).  [clausify(2)].

given #3 (I,wt=8): 12 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(3)].

given #4 (I,wt=4): 13 ->_s0(e(0),true) # label(replacement).  [assumption].

given #5 (I,wt=9): 14 -->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement).  [clausify(4)].

given #6 (I,wt=9): 15 -->*_s0(o(x),true) | ->_s0(e(s(x)),true) # label(replacement).  [clausify(5)].

given #7 (I,wt=4): 16 ->_s0(o(0),true) # label(replacement).  [assumption].

given #8 (I,wt=9): 17 -->*_s0(o(x),true) | ->_s0(o(s(x)),false) # label(replacement).  [clausify(6)].

given #9 (I,wt=9): 18 -->*_s0(e(x),true) | ->_s0(o(s(x)),true) # label(replacement).  [clausify(7)].

given #10 (I,wt=3): 19 ->*_s0(x,x) # label(reflexivity).  [assumption].

given #11 (I,wt=9): 20 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(8)].

given #12 (I,wt=8): 21 -->*_s0(e(x),true) | -->*_s0(o(x),true) # label(goal) # answer(goal).  [deny(9)].

given #13 (A,wt=6): 22 ->_s0(s(e(0)),s(true)).  [ur(12,a,13,a)].

given #14 (F,wt=11): 30 -->*_s0(o(x),true) | -->_s0(e(x),y) | -->*_s0(y,true) # answer(goal).  [resolve(21,a,20,c)].

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

% Proof 1 at 0.00 (+ 0.00) seconds: goal.
% Length of proof is 10.
% Level of proof is 3.
% Maximum clause weight is 11.000.
% Given clauses 14.

8 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
9 (exists x3 (->*_s0(e(x3),true) & ->*_s0(o(x3),true))) # label(goal) # label(non_clause) # label(goal).  [goal].
13 ->_s0(e(0),true) # label(replacement).  [assumption].
16 ->_s0(o(0),true) # label(replacement).  [assumption].
19 ->*_s0(x,x) # label(reflexivity).  [assumption].
20 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(8)].
21 -->*_s0(e(x),true) | -->*_s0(o(x),true) # label(goal) # answer(goal).  [deny(9)].
28 ->*_s0(o(0),true).  [ur(20,a,16,a,b,19,a)].
30 -->*_s0(o(x),true) | -->_s0(e(x),y) | -->*_s0(y,true) # answer(goal).  [resolve(21,a,20,c)].
39 $F # answer(goal).  [resolve(30,b,13,a),unit_del(a,28),unit_del(b,19)].

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

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

Given=14. Generated=30. Kept=29. proofs=1.
Usable=14. Sos=12. Demods=0. Limbo=3, Disabled=12. 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=7. Nonunit_bsub_feature_tests=14.
Megabytes=0.09.
User_CPU=0.00, System_CPU=0.00, Wall_clock=0.

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

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

THEOREM PROVED

Exiting with 1 proof.

Process 3406812 exit (max_proofs) Tue Jul 30 09:50:40 2024


The problem is feasible.