YES

Problem 1: 

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

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

Problem 1: 

Obtaining a model using Mace4:

 -> Usable Rules:
f(a,x) -> a
f(b,x) -> b

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 3175610 was started by sandbox2 on z006.star.cs.uiowa.edu,
Tue Jul 30 08:31:55 2024
The command was "./mace4 -c -f /tmp/mace43175597-2.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace43175597-2.in

assign(max_seconds,100).

formulas(assumptions).
->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence).
->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence).
->(x1,y) -> ->(g(x1,x2),g(y,x2)) # label(congruence).
->(x2,y) -> ->(g(x1,x2),g(x1,y)) # label(congruence).
->(f(a,x1),a) # label(replacement).
->(f(b,x1),b) # label(replacement).
->*(x,x) # label(reflexivity).
->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity).
end_of_list.

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

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

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

% Formulas that are not ordinary clauses:
1 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause).  [assumption].
2 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause).  [assumption].
3 ->(x1,y) -> ->(g(x1,x2),g(y,x2)) # label(congruence) # label(non_clause).  [assumption].
4 ->(x2,y) -> ->(g(x1,x2),g(x1,y)) # label(congruence) # label(non_clause).  [assumption].
5 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause).  [assumption].
6 ->*(f(a,b),b) & ->*(f(a,a),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,z),f(y,z)) # label(congruence).
-->(x,y) | ->(f(z,x),f(z,y)) # label(congruence).
-->(x,y) | ->(g(x,z),g(y,z)) # label(congruence).
-->(x,y) | ->(g(z,x),g(z,y)) # label(congruence).
->(f(a,x),a) # label(replacement).
->(f(b,x),b) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(f(a,b),b) | -->*(f(a,a),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, [ 1 ]),

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

        function(g(_,_), [
			   0, 0,
			   0, 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=47, kept=43.
Selections=9, assignments=11, propagations=13, current_models=1.
Rewrite_terms=106, rewrite_bools=75, indexes=20.
Rules_from_neg_clauses=3, cross_offs=3.

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

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

Exiting with 1 model.

Process 3175610 exit (max_models) Tue Jul 30 08:31:55 2024
The process finished Tue Jul 30 08:31:55 2024


Mace4 cooked interpretation:

% number = 1
% seconds = 0

% Interpretation of size 2

a = 0.

b = 1.

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

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

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

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


The problem is infeasible.