YES

Problem 1: 

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

Problem 1: 

Obtaining a model using Mace4:

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

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 3200175 was started by sandbox2 on z019.star.cs.uiowa.edu,
Tue Jul 30 08:41:10 2024
The command was "./mace4 -c -f /tmp/mace43200161-2.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace43200161-2.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(x2,x3)),x2) # label(replacement).
->(f(f(x1,x2),x3),f(f(x1,x3),f(x2,x3))) # label(replacement).
->(f(x1,f(x2,x3)),f(f(x1,x2),f(x1,x3))) # label(replacement).
->*(x,x) # label(reflexivity).
->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity).
end_of_list.

formulas(goals).
(exists x5 (->*(f(a,a),x5) & ->*(a,x5))) # 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,z) -> ->*(x,z) # label(transitivity) # label(non_clause).  [assumption].
4 (exists x5 (->*(f(a,a),x5) & ->*(a,x5))) # 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(y,z)),y) # label(replacement).
->(f(f(x,y),z),f(f(x,z),f(y,z))) # label(replacement).
->(f(x,f(y,z)),f(f(x,y),f(x,z))) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(f(a,a),x) | -->*(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=52, kept=48.
Selections=1, assignments=1, propagations=11, current_models=0.
Rewrite_terms=68, rewrite_bools=31, indexes=17.
Rules_from_neg_clauses=3, cross_offs=3.

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

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

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

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

        function(a, [ 0 ]),

        function(f(_,_), [
			   1, 2, 2,
			   2, 2, 2,
			   2, 2, 2 ]),

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

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

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

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

For domain size 3.

Current CPU time: 0.00 seconds (total CPU time: 0.01 seconds).
Ground clauses: seen=168, kept=159.
Selections=9, assignments=16, propagations=36, current_models=1.
Rewrite_terms=708, rewrite_bools=240, indexes=136.
Rules_from_neg_clauses=8, cross_offs=24.

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

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

Exiting with 1 model.

Process 3200175 exit (max_models) Tue Jul 30 08:41:10 2024
The process finished Tue Jul 30 08:41:10 2024


Mace4 cooked interpretation:

% number = 1
% seconds = 0

% Interpretation of size 3

a = 0.

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

  ->*(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.