NO

Problem 1: 

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

Infeasibility Conditions:
a ->* d, b ->* d

Problem 1: 

Obtaining a proof using Prover9:

 -> Prover9 Output:
============================== Prover9 ===============================
Prover9 (64) version 2009-11A, November 2009.
Process 2732147 was started by sandbox on z033.star.cs.uiowa.edu,
Thu Jun 27 11:10:59 2024
The command was "./prover9 -f /tmp/prover92732138-0.in".
============================== end of head ===========================

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

% Reading from file /tmp/prover92732138-0.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(s(x1),s(y)) # label(congruence).
->_s0(a,d) # label(replacement).
->_s0(b,d) # label(replacement).
->*_s0(x,x) # label(reflexivity).
->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity).
end_of_list.

formulas(goals).
->*_s0(a,d) & ->*_s0(b,d) # 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(s(x1),s(y)) # label(congruence) # label(non_clause).  [assumption].
6 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
7 ->*_s0(a,d) & ->*_s0(b,d) # 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(s(x),s(y)) # label(congruence).  [clausify(5)].
->_s0(a,d) # label(replacement).  [assumption].
->_s0(b,d) # label(replacement).  [assumption].
->*_s0(x,x) # label(reflexivity).  [assumption].
-->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(6)].
-->*_s0(a,d) | -->*_s0(b,d) # label(goal).  [deny(7)].
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([ d, a, b, f, s, g, h ]).
After inverse_order:  (no changes).
Unfolding symbols: (none).

Auto_inference settings:
  % set(neg_binary_resolution).  % (HNE depth_diff=-5)
  % clear(ordered_res).  % (HNE depth_diff=-5)
  % set(ur_resolution).  % (HNE depth_diff=-5)
    % 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:      8 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence).  [clausify(1)].
kept:      9 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence).  [clausify(2)].
kept:      10 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence).  [clausify(3)].
kept:      11 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence).  [clausify(4)].
kept:      12 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(5)].
kept:      13 ->_s0(a,d) # label(replacement).  [assumption].
kept:      14 ->_s0(b,d) # label(replacement).  [assumption].
kept:      15 ->*_s0(x,x) # label(reflexivity).  [assumption].
kept:      16 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(6)].
kept:      17 -->*_s0(a,d) | -->*_s0(b,d) # label(goal) # answer(goal).  [deny(7)].

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

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

% Clauses after input processing:

formulas(usable).
end_of_list.

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

given #2 (I,wt=10): 9 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence).  [clausify(2)].

given #3 (I,wt=8): 10 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence).  [clausify(3)].

given #4 (I,wt=8): 11 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence).  [clausify(4)].

given #5 (I,wt=8): 12 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(5)].

given #6 (I,wt=3): 13 ->_s0(a,d) # label(replacement).  [assumption].

given #7 (I,wt=3): 14 ->_s0(b,d) # label(replacement).  [assumption].

given #8 (I,wt=3): 15 ->*_s0(x,x) # label(reflexivity).  [assumption].

given #9 (I,wt=9): 16 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(6)].

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

% Proof 1 at 0.01 (+ 0.00) seconds: goal.
% Length of proof is 10.
% Level of proof is 3.
% Maximum clause weight is 9.000.
% Given clauses 9.

6 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
7 ->*_s0(a,d) & ->*_s0(b,d) # label(goal) # label(non_clause) # label(goal).  [goal].
13 ->_s0(a,d) # label(replacement).  [assumption].
14 ->_s0(b,d) # label(replacement).  [assumption].
15 ->*_s0(x,x) # label(reflexivity).  [assumption].
16 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(6)].
17 -->*_s0(a,d) | -->*_s0(b,d) # label(goal) # answer(goal).  [deny(7)].
28 ->*_s0(b,d).  [ur(16,a,14,a,b,15,a)].
29 ->*_s0(a,d).  [ur(16,a,13,a,b,15,a)].
30 $F # answer(goal).  [back_unit_del(17),unit_del(a,29),unit_del(b,28)].

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

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

Given=9. Generated=23. Kept=22. proofs=1.
Usable=9. Sos=10. Demods=0. Limbo=2, Disabled=11. 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=1.
Demod_attempts=0. Demod_rewrites=0.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=1. Nonunit_bsub_feature_tests=10.
Megabytes=0.07.
User_CPU=0.01, System_CPU=0.00, Wall_clock=0.

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

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

THEOREM PROVED

Exiting with 1 proof.

Process 2732147 exit (max_proofs) Thu Jun 27 11:10:59 2024


The problem is feasible.