YES

Problem 1: 

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

Problem 1: 

Obtaining a model using Mace4:

 -> Usable Rules:
a -> b

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 2727398 was started by sandbox on z036.star.cs.uiowa.edu,
Thu Jun 27 11:10:54 2024
The command was "./mace4 -c -f /tmp/mace42727381-2.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace42727381-2.in

assign(max_seconds,100).

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

formulas(goals).
->*(F(a),F(g(a))) # 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) -> ->(F(x1),F(y)) # label(congruence) # label(non_clause).  [assumption].
3 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence) # label(non_clause).  [assumption].
4 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause).  [assumption].
5 ->*(F(a),F(g(a))) # 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) | ->(F(x),F(y)) # label(congruence).
-->(x,y) | ->(g(x),g(y)) # label(congruence).
->(a,b) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(F(a),F(g(a))) # label(goal).
end_of_list.

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

% There are no natural numbers in the input.

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

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

interpretation( 2, [number=1, seconds=0], [

        function(a, [ 0 ]),

        function(b, [ 0 ]),

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

        function(F(_), [ 0, 1 ]),

        function(g(_), [ 1, 0 ]),

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

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

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

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

For domain size 2.

Current CPU time: 0.00 seconds (total CPU time: 0.01 seconds).
Ground clauses: seen=24, kept=20.
Selections=8, assignments=9, propagations=8, current_models=1.
Rewrite_terms=33, rewrite_bools=35, indexes=8.
Rules_from_neg_clauses=1, cross_offs=1.

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

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

Exiting with 1 model.

Process 2727398 exit (max_models) Thu Jun 27 11:10:54 2024
The process finished Thu Jun 27 11:10:54 2024


Mace4 cooked interpretation:

% number = 1
% seconds = 0

% Interpretation of size 2

a = 0.

b = 0.

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

F(0) = 0.
F(1) = 1.

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

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

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


The problem is infeasible.