NO

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty x vNonEmpty)
(STRATEGY CONTEXTSENSITIVE
(a)
(c)
(f 1)
(h 1)
(b)
(fSNonEmpty)
(g 1)
(k 1)
)
(RULES
a -> b
c -> k(f(a))
c -> k(g(b))
f(x) -> g(x) | h(f(x)) ->* k(g(b))
h(f(a)) -> c
h(x) -> k(x)
)
]

Infeasibility Conditions:
h(f(a)) ->* k(g(b))

Problem 1: 

Obtaining a proof using Prover9:

 -> Prover9 Output:
============================== Prover9 ===============================
Prover9 (64) version 2009-11A, November 2009.
Process 3256793 was started by sandbox on z006.star.cs.uiowa.edu,
Tue Jul 30 08:59:55 2024
The command was "./prover9 -f /tmp/prover93256782-0.in".
============================== end of head ===========================

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

% Reading from file /tmp/prover93256782-0.in

assign(max_seconds,20).

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

formulas(goals).
->*_s0(h(f(a)),k(g(b))) # 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),f(y)) # label(congruence) # label(non_clause).  [assumption].
2 ->_s0(x1,y) -> ->_s0(h(x1),h(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(k(x1),k(y)) # label(congruence) # label(non_clause).  [assumption].
5 ->*_s0(h(f(x1)),k(g(b))) -> ->_s0(f(x1),g(x1)) # label(replacement) # label(non_clause).  [assumption].
6 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
7 ->*_s0(h(f(a)),k(g(b))) # 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),f(y)) # label(congruence).  [clausify(1)].
-->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence).  [clausify(2)].
-->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence).  [clausify(3)].
-->_s0(x,y) | ->_s0(k(x),k(y)) # label(congruence).  [clausify(4)].
->_s0(a,b) # label(replacement).  [assumption].
->_s0(c,k(f(a))) # label(replacement).  [assumption].
->_s0(c,k(g(b))) # label(replacement).  [assumption].
-->*_s0(h(f(x)),k(g(b))) | ->_s0(f(x),g(x)) # label(replacement).  [clausify(5)].
->_s0(h(f(a)),c) # label(replacement).  [assumption].
->_s0(h(x),k(x)) # label(replacement).  [assumption].
->*_s0(x,x) # label(reflexivity).  [assumption].
-->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(6)].
-->*_s0(h(f(a)),k(g(b))) # 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([ a, c, b, f, k, h, g ]).
After inverse_order:  (no changes).
Unfolding symbols: (none).

Auto_inference settings:
  % set(neg_binary_resolution).  % (HNE depth_diff=-3)
  % clear(ordered_res).  % (HNE depth_diff=-3)
  % set(ur_resolution).  % (HNE depth_diff=-3)
    % set(ur_resolution) -> set(pos_ur_resolution).
    % set(ur_resolution) -> set(neg_ur_resolution).

Auto_process settings:  (no changes).

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

given #2 (I,wt=8): 9 -->_s0(x,y) | ->_s0(h(x),h(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(k(x),k(y)) # label(congruence).  [clausify(4)].

given #5 (I,wt=3): 12 ->_s0(a,b) # label(replacement).  [assumption].

given #6 (I,wt=5): 13 ->_s0(c,k(f(a))) # label(replacement).  [assumption].

given #7 (I,wt=5): 14 ->_s0(c,k(g(b))) # label(replacement).  [assumption].

given #8 (I,wt=12): 15 -->*_s0(h(f(x)),k(g(b))) | ->_s0(f(x),g(x)) # label(replacement).  [clausify(5)].

given #9 (I,wt=5): 16 ->_s0(h(f(a)),c) # label(replacement).  [assumption].

given #10 (I,wt=5): 17 ->_s0(h(x),k(x)) # label(replacement).  [assumption].

given #11 (I,wt=3): 18 ->*_s0(x,x) # label(reflexivity).  [assumption].

given #12 (I,wt=9): 19 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(6)].

given #13 (I,wt=7): 20 -->*_s0(h(f(a)),k(g(b))) # label(goal) # answer(goal).  [deny(7)].

============================== 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 13.

6 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
7 ->*_s0(h(f(a)),k(g(b))) # label(goal) # label(non_clause) # label(goal).  [goal].
14 ->_s0(c,k(g(b))) # label(replacement).  [assumption].
16 ->_s0(h(f(a)),c) # label(replacement).  [assumption].
18 ->*_s0(x,x) # label(reflexivity).  [assumption].
19 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(6)].
20 -->*_s0(h(f(a)),k(g(b))) # label(goal) # answer(goal).  [deny(7)].
43 ->*_s0(c,k(g(b))).  [ur(19,a,14,a,b,18,a)].
49 -->*_s0(c,k(g(b))) # answer(goal).  [ur(19,a,16,a,c,20,a)].
50 $F # answer(goal).  [resolve(49,a,43,a)].

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

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

Given=13. Generated=42. Kept=42. proofs=1.
Usable=13. Sos=25. Demods=0. Limbo=3, Disabled=13. 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=7.
Megabytes=0.10.
User_CPU=0.01, System_CPU=0.00, Wall_clock=0.

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

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

THEOREM PROVED

Exiting with 1 proof.

Process 3256793 exit (max_proofs) Tue Jul 30 08:59:55 2024


The problem is feasible.