YES

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty x vNonEmpty x1)
(STRATEGY CONTEXTSENSITIVE
(eq 1 2)
(even 1)
(odd 1)
(0)
(T)
(fSNonEmpty)
(false)
(s 1)
(true)
)
(RULES
eq(x,x) -> eq(T,T)
even(0) -> true
even(s(x)) -> false | eq(odd(x),false) ->* eq(T,T)
even(s(x)) -> true | eq(odd(x),true) ->* eq(T,T)
odd(0) -> false
odd(s(x)) -> false | eq(even(x),false) ->* eq(T,T)
odd(s(x)) -> true | eq(even(x),true) ->* eq(T,T)
)
]

Infeasibility Conditions:
eq(even(x1),true) ->* eq(T,T), eq(even(x1),false) ->* eq(T,T)

Problem 1: 

Obtaining a model using Mace4:

 -> Usable Rules:
eq(x,x) -> eq(T,T)
even(0) -> true
even(s(x)) -> false | eq(odd(x),false) ->* eq(T,T)
even(s(x)) -> true | eq(odd(x),true) ->* eq(T,T)
odd(0) -> false
odd(s(x)) -> false | eq(even(x),false) ->* eq(T,T)
odd(s(x)) -> true | eq(even(x),true) ->* eq(T,T)

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 3163031 was started by sandbox on z009.star.cs.uiowa.edu,
Tue Jul 30 08:22:55 2024
The command was "./mace4 -c -f /tmp/mace43163014-2.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace43163014-2.in

assign(max_seconds,100).

formulas(assumptions).
->(x1,y) -> ->(eq(x1,x2),eq(y,x2)) # label(congruence).
->(x2,y) -> ->(eq(x1,x2),eq(x1,y)) # label(congruence).
->(x1,y) -> ->(even(x1),even(y)) # label(congruence).
->(x1,y) -> ->(odd(x1),odd(y)) # label(congruence).
->(x1,y) -> ->(s(x1),s(y)) # label(congruence).
->(eq(x1,x1),eq(T,T)) # label(replacement).
->(even(0),true) # label(replacement).
->*(eq(odd(x1),false),eq(T,T)) -> ->(even(s(x1)),false) # label(replacement).
->*(eq(odd(x1),true),eq(T,T)) -> ->(even(s(x1)),true) # label(replacement).
->(odd(0),false) # label(replacement).
->*(eq(even(x1),false),eq(T,T)) -> ->(odd(s(x1)),false) # label(replacement).
->*(eq(even(x1),true),eq(T,T)) -> ->(odd(s(x1)),true) # label(replacement).
->*(x,x) # label(reflexivity).
->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity).
end_of_list.

formulas(goals).
(exists x3 (->*(eq(even(x3),true),eq(T,T)) & ->*(eq(even(x3),false),eq(T,T)))) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 ->(x1,y) -> ->(eq(x1,x2),eq(y,x2)) # label(congruence) # label(non_clause).  [assumption].
2 ->(x2,y) -> ->(eq(x1,x2),eq(x1,y)) # label(congruence) # label(non_clause).  [assumption].
3 ->(x1,y) -> ->(even(x1),even(y)) # label(congruence) # label(non_clause).  [assumption].
4 ->(x1,y) -> ->(odd(x1),odd(y)) # label(congruence) # label(non_clause).  [assumption].
5 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause).  [assumption].
6 ->*(eq(odd(x1),false),eq(T,T)) -> ->(even(s(x1)),false) # label(replacement) # label(non_clause).  [assumption].
7 ->*(eq(odd(x1),true),eq(T,T)) -> ->(even(s(x1)),true) # label(replacement) # label(non_clause).  [assumption].
8 ->*(eq(even(x1),false),eq(T,T)) -> ->(odd(s(x1)),false) # label(replacement) # label(non_clause).  [assumption].
9 ->*(eq(even(x1),true),eq(T,T)) -> ->(odd(s(x1)),true) # label(replacement) # label(non_clause).  [assumption].
10 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause).  [assumption].
11 (exists x3 (->*(eq(even(x3),true),eq(T,T)) & ->*(eq(even(x3),false),eq(T,T)))) # label(goal) # label(non_clause) # label(goal).  [goal].

============================== end of process non-clausal formulas ===

============================== CLAUSES FOR SEARCH ====================

formulas(mace4_clauses).
-->(x,y) | ->(eq(x,z),eq(y,z)) # label(congruence).
-->(x,y) | ->(eq(z,x),eq(z,y)) # label(congruence).
-->(x,y) | ->(even(x),even(y)) # label(congruence).
-->(x,y) | ->(odd(x),odd(y)) # label(congruence).
-->(x,y) | ->(s(x),s(y)) # label(congruence).
->(eq(x,x),eq(T,T)) # label(replacement).
->(even(0),true) # label(replacement).
-->*(eq(odd(x),false),eq(T,T)) | ->(even(s(x)),false) # label(replacement).
-->*(eq(odd(x),true),eq(T,T)) | ->(even(s(x)),true) # label(replacement).
->(odd(0),false) # label(replacement).
-->*(eq(even(x),false),eq(T,T)) | ->(odd(s(x)),false) # label(replacement).
-->*(eq(even(x),true),eq(T,T)) | ->(odd(s(x)),true) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(eq(even(x),true),eq(T,T)) | -->*(eq(even(x),false),eq(T,T)) # 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(true, [ 0 ]),

        function(0, [ 0 ]),

        function(T, [ 0 ]),

        function(false, [ 1 ]),

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

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

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

        function(eq(_,_), [
			   0, 1,
			   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: 0.00 seconds).
Ground clauses: seen=52, kept=48.
Selections=168, assignments=329, propagations=108, current_models=1.
Rewrite_terms=2918, rewrite_bools=1523, indexes=334.
Rules_from_neg_clauses=16, cross_offs=16.

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

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

Exiting with 1 model.

Process 3163031 exit (max_models) Tue Jul 30 08:22:55 2024
The process finished Tue Jul 30 08:22:55 2024


Mace4 cooked interpretation:

% number = 1
% seconds = 0

% Interpretation of size 2

true = 0.

0 = 0.

T = 0.

false = 1.

even(0) = 0.
even(1) = 1.

odd(0) = 1.
odd(1) = 0.

s(0) = 1.
s(1) = 0.

eq(0,0) = 0.
eq(0,1) = 1.
eq(1,0) = 1.
eq(1,1) = 0.

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

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


The problem is infeasible.