YES

Problem 1: 

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

Infeasibility Conditions:
f(b) ->* x, g(a) ->* x

Problem 1: 

Obtaining a model using Mace4:

 -> Usable Rules:
a -> b
c -> f(a)
c -> g(b)
f(a) -> c
f(x) -> g(x) | f(x) ->* g(b)

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 3266956 was started by sandbox2 on z020.star.cs.uiowa.edu,
Tue Jul 30 09:05:55 2024
The command was "./mace4 -c -f /tmp/mace43266943-2.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace43266943-2.in

assign(max_seconds,100).

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

formulas(goals).
(exists x3 (->*(f(b),x3) & ->*(g(a),x3))) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause).  [assumption].
2 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence) # label(non_clause).  [assumption].
3 ->*(f(x1),g(b)) -> ->(f(x1),g(x1)) # label(replacement) # label(non_clause).  [assumption].
4 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause).  [assumption].
5 (exists x3 (->*(f(b),x3) & ->*(g(a),x3))) # label(goal) # label(non_clause) # label(goal).  [goal].

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

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

formulas(mace4_clauses).
-->(x,y) | ->(f(x),f(y)) # label(congruence).
-->(x,y) | ->(g(x),g(y)) # label(congruence).
->(a,b) # label(replacement).
->(c,f(a)) # label(replacement).
->(c,g(b)) # label(replacement).
->(f(a),c) # label(replacement).
-->*(f(x),g(b)) | ->(f(x),g(x)) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(f(b),x) | -->*(g(a),x) # label(goal).
end_of_list.

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

% There are no natural numbers in the input.

============================== DOMAIN SIZE 2 =========================

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

For domain size 2.

Current CPU time: 0.00 seconds (total CPU time: 0.01 seconds).
Ground clauses: seen=26, kept=22.
Selections=12, assignments=23, propagations=45, current_models=0.
Rewrite_terms=192, rewrite_bools=130, indexes=64.
Rules_from_neg_clauses=12, cross_offs=12.

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

============================== DOMAIN SIZE 3 =========================

============================== MODEL =================================

interpretation( 3, [number=1, seconds=0], [

        function(a, [ 0 ]),

        function(c, [ 0 ]),

        function(b, [ 1 ]),

        function(f(_), [ 0, 2, 0 ]),

        function(g(_), [ 1, 1, 1 ]),

        relation(->*(_,_), [
			   1, 1, 1,
			   0, 1, 0,
			   0, 0, 1 ]),

        relation(->(_,_), [
			   1, 1, 1,
			   0, 1, 0,
			   0, 0, 1 ])
]).

============================== end of model ==========================

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

For domain size 3.

Current CPU time: 0.00 seconds (total CPU time: 0.01 seconds).
Ground clauses: seen=58, kept=49.
Selections=10, assignments=14, propagations=41, current_models=1.
Rewrite_terms=162, rewrite_bools=139, indexes=55.
Rules_from_neg_clauses=5, cross_offs=11.

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

User_CPU=0.01, System_CPU=0.00, Wall_clock=0.

Exiting with 1 model.

Process 3266956 exit (max_models) Tue Jul 30 09:05:55 2024
The process finished Tue Jul 30 09:05:55 2024


Mace4 cooked interpretation:

% number = 1
% seconds = 0

% Interpretation of size 3

a = 0.

c = 0.

b = 1.

f(0) = 0.
f(1) = 2.
f(2) = 0.

g(0) = 1.
g(1) = 1.
g(2) = 1.

  ->*(0,0).
  ->*(0,1).
  ->*(0,2).
- ->*(1,0).
  ->*(1,1).
- ->*(1,2).
- ->*(2,0).
- ->*(2,1).
  ->*(2,2).

  ->(0,0).
  ->(0,1).
  ->(0,2).
- ->(1,0).
  ->(1,1).
- ->(1,2).
- ->(2,0).
- ->(2,1).
  ->(2,2).


The problem is infeasible.