YES

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty x y vNonEmpty x)
(STRATEGY CONTEXTSENSITIVE
(+ 1 2)
(0)
(a)
(fSNonEmpty)
(s 1)
)
(RULES
+(0,x) -> x
+(s(x),y) -> s(+(x,y))
)
]
Infeasibility Conditions:
+(x,a) ->* 0

Problem 1: 

Obtaining a model using Mace4:

 -> Usable Rules:
+(0,x) -> x
+(s(x),y) -> s(+(x,y))

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 3368123 was started by sandbox on z018.star.cs.uiowa.edu,
Tue Jul 30 09:35:24 2024
The command was "./mace4 -c -f /tmp/mace43368110-2.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace43368110-2.in

assign(max_seconds,100).

formulas(assumptions).
->(x1,y) -> ->(+(x1,x2),+(y,x2)) # label(congruence).
->(x2,y) -> ->(+(x1,x2),+(x1,y)) # label(congruence).
->(x1,y) -> ->(s(x1),s(y)) # label(congruence).
->(+(0,x1),x1) # label(replacement).
->(+(s(x1),x2),s(+(x1,x2))) # label(replacement).
->*(x,x) # label(reflexivity).
->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity).
end_of_list.

formulas(goals).
(exists x4 ->*(+(x4,a),0)) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 ->(x1,y) -> ->(+(x1,x2),+(y,x2)) # label(congruence) # label(non_clause).  [assumption].
2 ->(x2,y) -> ->(+(x1,x2),+(x1,y)) # label(congruence) # label(non_clause).  [assumption].
3 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause).  [assumption].
4 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause).  [assumption].
5 (exists x4 ->*(+(x4,a),0)) # label(goal) # label(non_clause) # label(goal).  [goal].

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

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

formulas(mace4_clauses).
-->(x,y) | ->(+(x,z),+(y,z)) # label(congruence).
-->(x,y) | ->(+(z,x),+(z,y)) # label(congruence).
-->(x,y) | ->(s(x),s(y)) # label(congruence).
->(+(0,x),x) # label(replacement).
->(+(s(x),y),s(+(x,y))) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(+(x,a),0) # 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(0, [ 0 ]),

        function(a, [ 1 ]),

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

        function(+(_,_), [
			   0, 1,
			   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=38, kept=34.
Selections=5, assignments=6, propagations=14, current_models=1.
Rewrite_terms=79, rewrite_bools=51, indexes=21.
Rules_from_neg_clauses=5, cross_offs=5.

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

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

Exiting with 1 model.

Process 3368123 exit (max_models) Tue Jul 30 09:35:24 2024
The process finished Tue Jul 30 09:35:24 2024


Mace4 cooked interpretation:

% number = 1
% seconds = 0

% Interpretation of size 2

0 = 0.

a = 1.

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

+(0,0) = 0.
+(0,1) = 1.
+(1,0) = 0.
+(1,1) = 1.

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

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


The problem is infeasible.