YES

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty y x z vNonEmpty x y)
(STRATEGY CONTEXTSENSITIVE
(a)
(f 1 2)
(k 1)
(b)
(fSNonEmpty)
(g 1 2)
(h 1)
)
(RULES
a -> b
f(y,a) -> g(x,x) | h(z) ->* x, y ->* k(z)
k(x) -> a
k(x) -> b
k(x) -> h(x)
)
]

Infeasibility Conditions:
f(k(b),x) ->* g(h(a),y)

Problem 1: 

Obtaining a model using Mace4:

 -> Usable Rules:
a -> b
f(y,a) -> g(x,x) | h(z) ->* x, y ->* k(z)
k(x) -> a
k(x) -> b
k(x) -> h(x)

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 3390698 was started by sandbox2 on z024.star.cs.uiowa.edu,
Tue Jul 30 09:35:10 2024
The command was "./mace4 -c -f /tmp/mace43390685-2.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace43390685-2.in

assign(max_seconds,100).

formulas(assumptions).
->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence).
->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence).
->(x1,y) -> ->(k(x1),k(y)) # label(congruence).
->(x1,y) -> ->(g(x1,x2),g(y,x2)) # label(congruence).
->(x2,y) -> ->(g(x1,x2),g(x1,y)) # label(congruence).
->(x1,y) -> ->(h(x1),h(y)) # label(congruence).
->(a,b) # label(replacement).
->*(h(x3),x2) & ->*(x1,k(x3)) -> ->(f(x1,a),g(x2,x2)) # label(replacement).
->(k(x2),a) # label(replacement).
->(k(x2),b) # label(replacement).
->(k(x2),h(x2)) # label(replacement).
->*(x,x) # label(reflexivity).
->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity).
end_of_list.

formulas(goals).
(exists x5 exists x6 ->*(f(k(b),x5),g(h(a),x6))) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause).  [assumption].
2 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause).  [assumption].
3 ->(x1,y) -> ->(k(x1),k(y)) # label(congruence) # label(non_clause).  [assumption].
4 ->(x1,y) -> ->(g(x1,x2),g(y,x2)) # label(congruence) # label(non_clause).  [assumption].
5 ->(x2,y) -> ->(g(x1,x2),g(x1,y)) # label(congruence) # label(non_clause).  [assumption].
6 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence) # label(non_clause).  [assumption].
7 ->*(h(x3),x2) & ->*(x1,k(x3)) -> ->(f(x1,a),g(x2,x2)) # label(replacement) # label(non_clause).  [assumption].
8 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause).  [assumption].
9 (exists x5 exists x6 ->*(f(k(b),x5),g(h(a),x6))) # label(goal) # label(non_clause) # label(goal).  [goal].

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

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

formulas(mace4_clauses).
-->(x,y) | ->(f(x,z),f(y,z)) # label(congruence).
-->(x,y) | ->(f(z,x),f(z,y)) # label(congruence).
-->(x,y) | ->(k(x),k(y)) # label(congruence).
-->(x,y) | ->(g(x,z),g(y,z)) # label(congruence).
-->(x,y) | ->(g(z,x),g(z,y)) # label(congruence).
-->(x,y) | ->(h(x),h(y)) # label(congruence).
->(a,b) # label(replacement).
-->*(h(x),y) | -->*(z,k(x)) | ->(f(z,a),g(y,y)) # label(replacement).
->(k(x),a) # label(replacement).
->(k(x),b) # label(replacement).
->(k(x),h(x)) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(f(k(b),x),g(h(a),y)) # label(goal).
end_of_list.

============================== end of clauses for search =============

% There are no natural numbers in the input.

============================== DOMAIN SIZE 2 =========================

============================== STATISTICS ============================

For domain size 2.

Current CPU time: 0.00 seconds (total CPU time: 0.01 seconds).
Ground clauses: seen=69, kept=65.
Selections=175, assignments=349, propagations=268, current_models=0.
Rewrite_terms=4170, rewrite_bools=2455, indexes=272.
Rules_from_neg_clauses=183, cross_offs=183.

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

============================== DOMAIN SIZE 3 =========================

============================== MODEL =================================

interpretation( 3, [number=1, seconds=7], [

        function(a, [ 0 ]),

        function(b, [ 1 ]),

        function(k(_), [ 2, 0, 2 ]),

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

        function(f(_,_), [
			   0, 0, 0,
			   0, 0, 0,
			   2, 0, 2 ]),

        function(g(_,_), [
			   2, 2, 2,
			   0, 0, 0,
			   2, 2, 2 ]),

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

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

============================== end of model ==========================

============================== STATISTICS ============================

For domain size 3.

Current CPU time: 0.00 seconds (total CPU time: 7.53 seconds).
Ground clauses: seen=202, kept=193.
Selections=1738116, assignments=5202230, propagations=3477432, current_models=1.
Rewrite_terms=104365032, rewrite_bools=54540923, indexes=167788.
Rules_from_neg_clauses=3348023, cross_offs=6752079.

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

User_CPU=7.53, System_CPU=0.29, Wall_clock=8.

Exiting with 1 model.

Process 3390698 exit (max_models) Tue Jul 30 09:35:18 2024
The process finished Tue Jul 30 09:35:18 2024


Mace4 cooked interpretation:

% number = 1
% seconds = 7

% Interpretation of size 3

a = 0.

b = 1.

k(0) = 2.
k(1) = 0.
k(2) = 2.

h(0) = 0.
h(1) = 1.
h(2) = 0.

f(0,0) = 0.
f(0,1) = 0.
f(0,2) = 0.
f(1,0) = 0.
f(1,1) = 0.
f(1,2) = 0.
f(2,0) = 2.
f(2,1) = 0.
f(2,2) = 2.

g(0,0) = 2.
g(0,1) = 2.
g(0,2) = 2.
g(1,0) = 0.
g(1,1) = 0.
g(1,2) = 0.
g(2,0) = 2.
g(2,1) = 2.
g(2,2) = 2.

  ->*(0,0).
  ->*(0,1).
- ->*(0,2).
- ->*(1,0).
  ->*(1,1).
- ->*(1,2).
  ->*(2,0).
  ->*(2,1).
  ->*(2,2).

  ->(0,0).
  ->(0,1).
- ->(0,2).
- ->(1,0).
- ->(1,1).
- ->(1,2).
  ->(2,0).
  ->(2,1).
  ->(2,2).


The problem is infeasible.