YES

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty x y vNonEmpty x)
(STRATEGY CONTEXTSENSITIVE
(c)
(f 1 2)
(h 1 2)
(d)
(fSNonEmpty)
(g 1)
)
(RULES
c -> f(c,d)
c -> h(c,d)
f(x,y) -> h(g(y),x)
h(x,y) -> f(g(y),x)
)
]
Infeasibility Conditions:
h(f(c,d),d) ->* x, f(c,d) ->* x

Problem 1: 

Obtaining a model using Mace4:

 -> Usable Rules:
c -> f(c,d)
c -> h(c,d)
f(x,y) -> h(g(y),x)
h(x,y) -> f(g(y),x)

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 3287550 was started by sandbox on z009.star.cs.uiowa.edu,
Tue Jul 30 09:05:10 2024
The command was "./mace4 -c -f /tmp/mace43287536-2.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace43287536-2.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) -> ->(h(x1,x2),h(y,x2)) # label(congruence).
->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence).
->(x1,y) -> ->(g(x1),g(y)) # label(congruence).
->(c,f(c,d)) # label(replacement).
->(c,h(c,d)) # label(replacement).
->(f(x1,x2),h(g(x2),x1)) # label(replacement).
->(h(x1,x2),f(g(x2),x1)) # label(replacement).
->*(x,x) # label(reflexivity).
->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity).
end_of_list.

formulas(goals).
(exists x4 (->*(h(f(c,d),d),x4) & ->*(f(c,d),x4))) # 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) -> ->(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause).  [assumption].
4 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause).  [assumption].
5 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence) # label(non_clause).  [assumption].
6 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause).  [assumption].
7 (exists x4 (->*(h(f(c,d),d),x4) & ->*(f(c,d),x4))) # 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) | ->(h(x,z),h(y,z)) # label(congruence).
-->(x,y) | ->(h(z,x),h(z,y)) # label(congruence).
-->(x,y) | ->(g(x),g(y)) # label(congruence).
->(c,f(c,d)) # label(replacement).
->(c,h(c,d)) # label(replacement).
->(f(x,y),h(g(y),x)) # label(replacement).
->(h(x,y),f(g(y),x)) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(h(f(c,d),d),x) | -->*(f(c,d),x) # 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: 0.00 seconds).
Ground clauses: seen=58, kept=54.
Selections=43, assignments=85, propagations=227, current_models=0.
Rewrite_terms=1350, rewrite_bools=1062, indexes=378.
Rules_from_neg_clauses=84, cross_offs=84.

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

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

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

interpretation( 3, [number=1, seconds=0], [

        function(c, [ 0 ]),

        function(d, [ 1 ]),

        function(g(_), [ 0, 1, 2 ]),

        function(f(_,_), [
			   0, 2, 0,
			   0, 2, 1,
			   0, 2, 0 ]),

        function(h(_,_), [
			   0, 0, 0,
			   2, 2, 2,
			   0, 1, 0 ]),

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

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

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

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

For domain size 3.

Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds).
Ground clauses: seen=170, kept=161.
Selections=24, assignments=45, propagations=134, current_models=1.
Rewrite_terms=999, rewrite_bools=915, indexes=225.
Rules_from_neg_clauses=28, cross_offs=73.

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

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

Exiting with 1 model.

Process 3287550 exit (max_models) Tue Jul 30 09:05:10 2024
The process finished Tue Jul 30 09:05:10 2024


Mace4 cooked interpretation:

% number = 1
% seconds = 0

% Interpretation of size 3

c = 0.

d = 1.

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

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

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

  ->*(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.