YES

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty x y z w vNonEmpty)
(STRATEGY CONTEXTSENSITIVE
(f 1 2)
(a)
(b)
(fSNonEmpty)
)
(RULES
f(f(x,y),f(z,w)) -> f(f(x,z),f(y,w))
f(f(x,y),x) -> x
)
]
Infeasibility Conditions:
f(b,a) <-->* f(a,b)

Problem 1: 

Obtaining a model using Mace4:

 -> Usable Rules:
f(f(x,y),f(z,w)) -> f(f(x,z),f(y,w))
f(f(x,y),x) -> x

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 3365057 was started by sandbox on z010.star.cs.uiowa.edu,
Tue Jul 30 09:26:40 2024
The command was "./mace4 -c -f /tmp/mace43365043-2.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace43365043-2.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).
->(f(f(x1,x2),f(x3,x4)),f(f(x1,x3),f(x2,x4))) # label(replacement).
->(f(f(x1,x2),x1),x1) # label(replacement).
->(x,y) | ->(y,x) -> <-->(x,y) # label(convertibility).
<-->*(x,x) # label(covertibility).
<-->(x,y) & <-->*(y,z) -> <-->*(x,z) # label(convertibility).
end_of_list.

formulas(goals).
<-->*(f(b,a),f(a,b)) # 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 ->(x,y) | ->(y,x) -> <-->(x,y) # label(convertibility) # label(non_clause).  [assumption].
4 <-->(x,y) & <-->*(y,z) -> <-->*(x,z) # label(convertibility) # label(non_clause).  [assumption].
5 <-->*(f(b,a),f(a,b)) # 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).
->(f(f(x,y),f(z,u)),f(f(x,z),f(y,u))) # label(replacement).
->(f(f(x,y),x),x) # label(replacement).
-->(x,y) | <-->(x,y) # label(convertibility).
-->(x,y) | <-->(y,x) # label(convertibility).
<-->*(x,x) # label(covertibility).
-<-->(x,y) | -<-->*(y,z) | <-->*(x,z) # label(convertibility).
-<-->*(f(b,a),f(a,b)) # 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 ]),

        relation(->(_,_), [
			   1, 0,
			   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: 0.00 seconds).
Ground clauses: seen=55, kept=51.
Selections=6, assignments=8, propagations=13, current_models=1.
Rewrite_terms=148, rewrite_bools=71, indexes=12.
Rules_from_neg_clauses=2, cross_offs=2.

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

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

Exiting with 1 model.

Process 3365057 exit (max_models) Tue Jul 30 09:26:40 2024
The process finished Tue Jul 30 09:26:40 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.

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

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

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


The problem is infeasible.