YES

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty x y vNonEmpty x1 x2)
(STRATEGY CONTEXTSENSITIVE
(pin 1)
(tc 1)
(a)
(b)
(c)
(fSNonEmpty)
(pout 1)
(z)
)
(RULES
pin(a) -> pout(b)
pin(b) -> pout(c)
tc(x) -> x
tc(x) -> y | pin(x) ->* pout(z), tc(z) ->* y
)
]

Infeasibility Conditions:
pin(x1) ->* pout(z), tc(z) ->* x2

Problem 1: 

Obtaining a model using Mace4:

 -> Usable Rules:
pin(a) -> pout(b)
pin(b) -> pout(c)
tc(x) -> x
tc(x) -> y | pin(x) ->* pout(z), tc(z) ->* y

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 3319418 was started by sandbox on z001.star.cs.uiowa.edu,
Tue Jul 30 09:28:25 2024
The command was "./mace4 -c -f /tmp/mace43319404-2.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace43319404-2.in

assign(max_seconds,100).

formulas(assumptions).
->(x1,y) -> ->(pin(x1),pin(y)) # label(congruence).
->(x1,y) -> ->(tc(x1),tc(y)) # label(congruence).
->(x1,y) -> ->(pout(x1),pout(y)) # label(congruence).
->(pin(a),pout(b)) # label(replacement).
->(pin(b),pout(c)) # label(replacement).
->(tc(x1),x1) # label(replacement).
->*(pin(x1),pout(z)) & ->*(tc(z),x2) -> ->(tc(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 (->*(pin(x4),pout(z)) & ->*(tc(z),x5))) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 ->(x1,y) -> ->(pin(x1),pin(y)) # label(congruence) # label(non_clause).  [assumption].
2 ->(x1,y) -> ->(tc(x1),tc(y)) # label(congruence) # label(non_clause).  [assumption].
3 ->(x1,y) -> ->(pout(x1),pout(y)) # label(congruence) # label(non_clause).  [assumption].
4 ->*(pin(x1),pout(z)) & ->*(tc(z),x2) -> ->(tc(x1),x2) # label(replacement) # label(non_clause).  [assumption].
5 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause).  [assumption].
6 (exists x4 exists x5 (->*(pin(x4),pout(z)) & ->*(tc(z),x5))) # label(goal) # label(non_clause) # label(goal).  [goal].

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

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

formulas(mace4_clauses).
-->(x,y) | ->(pin(x),pin(y)) # label(congruence).
-->(x,y) | ->(tc(x),tc(y)) # label(congruence).
-->(x,y) | ->(pout(x),pout(y)) # label(congruence).
->(pin(a),pout(b)) # label(replacement).
->(pin(b),pout(c)) # label(replacement).
->(tc(x),x) # label(replacement).
-->*(pin(x),pout(z)) | -->*(tc(z),y) | ->(tc(x),y) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(pin(x),pout(z)) | -->*(tc(z),y) # 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(a, [ 0 ]),

        function(b, [ 0 ]),

        function(c, [ 0 ]),

        function(z, [ 1 ]),

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

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

        function(pout(_), [ 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=34, kept=30.
Selections=13, assignments=18, propagations=30, current_models=1.
Rewrite_terms=241, rewrite_bools=139, indexes=109.
Rules_from_neg_clauses=6, cross_offs=6.

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

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

Exiting with 1 model.

Process 3319418 exit (max_models) Tue Jul 30 09:28:25 2024
The process finished Tue Jul 30 09:28:25 2024


Mace4 cooked interpretation:

% number = 1
% seconds = 0

% Interpretation of size 2

a = 0.

b = 0.

c = 0.

z = 1.

pin(0) = 0.
pin(1) = 0.

tc(0) = 0.
tc(1) = 1.

pout(0) = 0.
pout(1) = 1.

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

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


The problem is infeasible.