YES

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty x y vNonEmpty x2 x)
(STRATEGY CONTEXTSENSITIVE
(add 1 2)
(gcd 1 2)
(0)
(fSNonEmpty)
(false)
(leq 1 2)
(s 1)
)
(RULES
add(0,y) -> y
add(s(x),y) -> s(add(x,y))
gcd(add(x,y),y) -> gcd(x,y)
gcd(0,x) -> x
gcd(x,0) -> x
gcd(x,y) -> gcd(y,x) | leq(y,x) ->* false
gcd(y,add(x,y)) -> gcd(x,y)
)
]

Infeasibility Conditions:
leq(x2,add(x,x2)) ->* false

Problem 1: 

Obtaining a model using Mace4:

 -> Usable Rules:
add(0,y) -> y
add(s(x),y) -> s(add(x,y))
gcd(add(x,y),y) -> gcd(x,y)
gcd(0,x) -> x
gcd(x,0) -> x
gcd(x,y) -> gcd(y,x) | leq(y,x) ->* false
gcd(y,add(x,y)) -> gcd(x,y)

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 3187348 was started by sandbox2 on z029.star.cs.uiowa.edu,
Tue Jul 30 08:40:55 2024
The command was "./mace4 -c -f /tmp/mace43187335-2.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace43187335-2.in

assign(max_seconds,100).

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

formulas(goals).
(exists x4 exists x5 ->*(leq(x4,add(x5,x4)),false)) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 ->(x1,y) -> ->(add(x1,x2),add(y,x2)) # label(congruence) # label(non_clause).  [assumption].
2 ->(x2,y) -> ->(add(x1,x2),add(x1,y)) # label(congruence) # label(non_clause).  [assumption].
3 ->(x1,y) -> ->(gcd(x1,x2),gcd(y,x2)) # label(congruence) # label(non_clause).  [assumption].
4 ->(x2,y) -> ->(gcd(x1,x2),gcd(x1,y)) # label(congruence) # label(non_clause).  [assumption].
5 ->(x1,y) -> ->(leq(x1,x2),leq(y,x2)) # label(congruence) # label(non_clause).  [assumption].
6 ->(x2,y) -> ->(leq(x1,x2),leq(x1,y)) # label(congruence) # label(non_clause).  [assumption].
7 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause).  [assumption].
8 ->*(leq(x2,x1),false) -> ->(gcd(x1,x2),gcd(x2,x1)) # label(replacement) # label(non_clause).  [assumption].
9 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause).  [assumption].
10 (exists x4 exists x5 ->*(leq(x4,add(x5,x4)),false)) # label(goal) # label(non_clause) # label(goal).  [goal].

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

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

formulas(mace4_clauses).
-->(x,y) | ->(add(x,z),add(y,z)) # label(congruence).
-->(x,y) | ->(add(z,x),add(z,y)) # label(congruence).
-->(x,y) | ->(gcd(x,z),gcd(y,z)) # label(congruence).
-->(x,y) | ->(gcd(z,x),gcd(z,y)) # label(congruence).
-->(x,y) | ->(leq(x,z),leq(y,z)) # label(congruence).
-->(x,y) | ->(leq(z,x),leq(z,y)) # label(congruence).
-->(x,y) | ->(s(x),s(y)) # label(congruence).
->(add(0,x),x) # label(replacement).
->(add(s(x),y),s(add(x,y))) # label(replacement).
->(gcd(add(x,y),y),gcd(x,y)) # label(replacement).
->(gcd(0,x),x) # label(replacement).
->(gcd(x,0),x) # label(replacement).
-->*(leq(x,y),false) | ->(gcd(y,x),gcd(x,y)) # label(replacement).
->(gcd(x,add(y,x)),gcd(y,x)) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(leq(x,add(y,x)),false) # 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(false, [ 0 ]),

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

        function(add(_,_), [
			   0, 0,
			   0, 0 ]),

        function(gcd(_,_), [
			   0, 0,
			   0, 0 ]),

        function(leq(_,_), [
			   1, 1,
			   1, 1 ]),

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

        relation(->(_,_), [
			   1, 1,
			   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=88, kept=84.
Selections=11, assignments=11, propagations=13, current_models=1.
Rewrite_terms=184, rewrite_bools=121, indexes=23.
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 3187348 exit (max_models) Tue Jul 30 08:40:55 2024
The process finished Tue Jul 30 08:40:55 2024


Mace4 cooked interpretation:

% number = 1
% seconds = 0

% Interpretation of size 2

0 = 0.

false = 0.

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

add(0,0) = 0.
add(0,1) = 0.
add(1,0) = 0.
add(1,1) = 0.

gcd(0,0) = 0.
gcd(0,1) = 0.
gcd(1,0) = 0.
gcd(1,1) = 0.

leq(0,0) = 1.
leq(0,1) = 1.
leq(1,0) = 1.
leq(1,1) = 1.

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

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


The problem is infeasible.