YES

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty x y vNonEmpty x1 y)
(STRATEGY CONTEXTSENSITIVE
(f 1 2)
(h 1)
(a 1)
(b)
(c 1)
(fSNonEmpty)
)
(RULES
f(c(c(c(x))),y) -> a(y) | c(f(c(x),c(c(y)))) ->* c(a(a(b)))
f(c(x),c(c(y))) -> a(a(x)) | c(f(x,y)) ->* c(a(b))
h(a(a(x))) -> a(b) | h(x) ->* b
h(b) -> b
)
]

Infeasibility Conditions:
c(f(c(c(x1)),y)) ->* c(a(b)), c(f(c(x1),c(c(c(c(y)))))) ->* c(a(a(b)))

Problem 1: 

Obtaining a model using Mace4:

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

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 3425417 was started by sandbox on z018.star.cs.uiowa.edu,
Tue Jul 30 09:54:25 2024
The command was "./mace4 -c -f /tmp/mace43425406-2.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace43425406-2.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),h(y)) # label(congruence).
->(x1,y) -> ->(a(x1),a(y)) # label(congruence).
->(x1,y) -> ->(c(x1),c(y)) # label(congruence).
->*(c(f(c(x1),c(c(x2)))),c(a(a(b)))) -> ->(f(c(c(c(x1))),x2),a(x2)) # label(replacement).
->*(c(f(x1,x2)),c(a(b))) -> ->(f(c(x1),c(c(x2))),a(a(x1))) # label(replacement).
->*(h(x1),b) -> ->(h(a(a(x1))),a(b)) # label(replacement).
->(h(b),b) # label(replacement).
->*(x,x) # label(reflexivity).
->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity).
end_of_list.

formulas(goals).
(exists x4 exists x5 (->*(c(f(c(c(x4)),x5)),c(a(b))) & ->*(c(f(c(x4),c(c(c(c(x5)))))),c(a(a(b)))))) # 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),h(y)) # label(congruence) # label(non_clause).  [assumption].
4 ->(x1,y) -> ->(a(x1),a(y)) # label(congruence) # label(non_clause).  [assumption].
5 ->(x1,y) -> ->(c(x1),c(y)) # label(congruence) # label(non_clause).  [assumption].
6 ->*(c(f(c(x1),c(c(x2)))),c(a(a(b)))) -> ->(f(c(c(c(x1))),x2),a(x2)) # label(replacement) # label(non_clause).  [assumption].
7 ->*(c(f(x1,x2)),c(a(b))) -> ->(f(c(x1),c(c(x2))),a(a(x1))) # label(replacement) # label(non_clause).  [assumption].
8 ->*(h(x1),b) -> ->(h(a(a(x1))),a(b)) # label(replacement) # label(non_clause).  [assumption].
9 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause).  [assumption].
10 (exists x4 exists x5 (->*(c(f(c(c(x4)),x5)),c(a(b))) & ->*(c(f(c(x4),c(c(c(c(x5)))))),c(a(a(b)))))) # 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),h(y)) # label(congruence).
-->(x,y) | ->(a(x),a(y)) # label(congruence).
-->(x,y) | ->(c(x),c(y)) # label(congruence).
-->*(c(f(c(x),c(c(y)))),c(a(a(b)))) | ->(f(c(c(c(x))),y),a(y)) # label(replacement).
-->*(c(f(x,y)),c(a(b))) | ->(f(c(x),c(c(y))),a(a(x))) # label(replacement).
-->*(h(x),b) | ->(h(a(a(x))),a(b)) # label(replacement).
->(h(b),b) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(c(f(c(c(x)),y)),c(a(b))) | -->*(c(f(c(x),c(c(c(c(y)))))),c(a(a(b)))) # 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(b, [ 0 ]),

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

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

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

        function(f(_,_), [
			   1, 1,
			   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: 0.01 seconds).
Ground clauses: seen=53, kept=49.
Selections=13, assignments=18, propagations=7, current_models=1.
Rewrite_terms=280, rewrite_bools=95, indexes=87.
Rules_from_neg_clauses=0, cross_offs=0.

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

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

Exiting with 1 model.

Process 3425417 exit (max_models) Tue Jul 30 09:54:25 2024
The process finished Tue Jul 30 09:54:25 2024


Mace4 cooked interpretation:

% number = 1
% seconds = 0

% Interpretation of size 2

b = 0.

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

a(0) = 0.
a(1) = 0.

c(0) = 0.
c(1) = 1.

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

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

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


The problem is infeasible.