YES

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty x vNonEmpty)
(STRATEGY CONTEXTSENSITIVE
(p 1)
(pos 1)
(s 1)
(0)
(fSNonEmpty)
(false)
(true)
)
(RULES
p(s(x)) -> x
pos(p(x)) -> false | pos(x) ->* false
pos(s(0)) -> true
pos(s(x)) -> true | pos(x) ->* true
pos(0) -> false
s(p(x)) -> x
)
]

Infeasibility Conditions:
pos(0) ->* true

Problem 1: 

Obtaining a model using Mace4:

 -> Usable Rules:
pos(p(x)) -> false | pos(x) ->* false
pos(s(0)) -> true
pos(s(x)) -> true | pos(x) ->* true
pos(0) -> false

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 3189351 was started by sandbox on z022.star.cs.uiowa.edu,
Tue Jul 30 08:37:40 2024
The command was "./mace4 -c -f /tmp/mace43189338-2.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace43189338-2.in

assign(max_seconds,100).

formulas(assumptions).
->(x1,y) -> ->(p(x1),p(y)) # label(congruence).
->(x1,y) -> ->(pos(x1),pos(y)) # label(congruence).
->(x1,y) -> ->(s(x1),s(y)) # label(congruence).
->*(pos(x1),false) -> ->(pos(p(x1)),false) # label(replacement).
->(pos(s(0)),true) # label(replacement).
->*(pos(x1),true) -> ->(pos(s(x1)),true) # label(replacement).
->(pos(0),false) # label(replacement).
->*(x,x) # label(reflexivity).
->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity).
end_of_list.

formulas(goals).
->*(pos(0),true) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 ->(x1,y) -> ->(p(x1),p(y)) # label(congruence) # label(non_clause).  [assumption].
2 ->(x1,y) -> ->(pos(x1),pos(y)) # label(congruence) # label(non_clause).  [assumption].
3 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause).  [assumption].
4 ->*(pos(x1),false) -> ->(pos(p(x1)),false) # label(replacement) # label(non_clause).  [assumption].
5 ->*(pos(x1),true) -> ->(pos(s(x1)),true) # label(replacement) # label(non_clause).  [assumption].
6 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause).  [assumption].
7 ->*(pos(0),true) # label(goal) # label(non_clause) # label(goal).  [goal].

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

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

formulas(mace4_clauses).
-->(x,y) | ->(p(x),p(y)) # label(congruence).
-->(x,y) | ->(pos(x),pos(y)) # label(congruence).
-->(x,y) | ->(s(x),s(y)) # label(congruence).
-->*(pos(x),false) | ->(pos(p(x)),false) # label(replacement).
->(pos(s(0)),true) # label(replacement).
-->*(pos(x),true) | ->(pos(s(x)),true) # label(replacement).
->(pos(0),false) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(pos(0),true) # 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(true, [ 1 ]),

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

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

        function(s(_), [ 1, 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=29, kept=25.
Selections=8, assignments=11, propagations=12, current_models=1.
Rewrite_terms=72, rewrite_bools=46, indexes=13.
Rules_from_neg_clauses=3, cross_offs=3.

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

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

Exiting with 1 model.

Process 3189351 exit (max_models) Tue Jul 30 08:37:40 2024
The process finished Tue Jul 30 08:37:40 2024


Mace4 cooked interpretation:

% number = 1
% seconds = 0

% Interpretation of size 2

0 = 0.

false = 0.

true = 1.

p(0) = 0.
p(1) = 0.

pos(0) = 0.
pos(1) = 1.

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

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

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


The problem is infeasible.