YES

Problem 1: 

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

Problem 1: 

Obtaining a model using Mace4:

 -> Usable Rules:
f(x) -> f(g(x))
h(x) -> h(h(x))

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 3264600 was started by sandbox on z036.star.cs.uiowa.edu,
Tue Jul 30 09:16:10 2024
The command was "./mace4 -c -f /tmp/mace43264587-2.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace43264587-2.in

assign(max_seconds,100).

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

formulas(goals).
(exists x3 (->*(h(c),x3) & ->*(h(f(c)),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) -> ->(h(x1),h(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 (exists x3 (->*(h(c),x3) & ->*(h(f(c)),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) | ->(h(x),h(y)) # label(congruence).
-->(x,y) | ->(g(x),g(y)) # label(congruence).
->(f(x),f(g(x))) # label(replacement).
->(h(x),h(h(x))) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(h(c),x) | -->*(h(f(c)),x) # 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(c, [ 0 ]),

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

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

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

        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=28, kept=24.
Selections=7, assignments=10, propagations=9, current_models=1.
Rewrite_terms=62, rewrite_bools=44, indexes=18.
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 3264600 exit (max_models) Tue Jul 30 09:16:10 2024
The process finished Tue Jul 30 09:16:10 2024


Mace4 cooked interpretation:

% number = 1
% seconds = 0

% Interpretation of size 2

c = 0.

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

h(0) = 0.
h(1) = 1.

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

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

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


The problem is infeasible.