YES

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty x' x'' x y' y'' y vNonEmpty x1 x x2 x3)
(STRATEGY CONTEXTSENSITIVE
(a)
(f 1 2)
(b)
(fSNonEmpty)
(g 1)
(h 1 2)
)
(RULES
a -> b
f(g(y'),y'') -> h(y,f(g(y),a)) | y' ->* y, y'' ->* y
f(x',x'') -> h(x,f(x,b)) | x' ->* x, x'' ->* x
)
]

Infeasibility Conditions:
g(x1) ->* x, x2 ->* x, x1 ->* x3, x2 ->* x3

Problem 1: 

Obtaining a model using Mace4:

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

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 3297569 was started by sandbox on z020.star.cs.uiowa.edu,
Tue Jul 30 09:16:40 2024
The command was "./mace4 -c -f /tmp/mace43297556-2.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace43297556-2.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) -> ->(g(x1),g(y)) # label(congruence).
->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence).
->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence).
->(a,b) # label(replacement).
->*(x4,x6) & ->*(x5,x6) -> ->(f(g(x4),x5),h(x6,f(g(x6),a))) # label(replacement).
->*(x1,x3) & ->*(x2,x3) -> ->(f(x1,x2),h(x3,f(x3,b))) # label(replacement).
->*(x,x) # label(reflexivity).
->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity).
end_of_list.

formulas(goals).
(exists x8 exists x9 exists x10 exists x11 (->*(g(x8),x9) & ->*(x10,x9) & ->*(x8,x11) & ->*(x10,x11))) # 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) -> ->(g(x1),g(y)) # label(congruence) # label(non_clause).  [assumption].
4 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause).  [assumption].
5 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause).  [assumption].
6 ->*(x4,x6) & ->*(x5,x6) -> ->(f(g(x4),x5),h(x6,f(g(x6),a))) # label(replacement) # label(non_clause).  [assumption].
7 ->*(x1,x3) & ->*(x2,x3) -> ->(f(x1,x2),h(x3,f(x3,b))) # label(replacement) # label(non_clause).  [assumption].
8 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause).  [assumption].
9 (exists x8 exists x9 exists x10 exists x11 (->*(g(x8),x9) & ->*(x10,x9) & ->*(x8,x11) & ->*(x10,x11))) # 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) | ->(g(x),g(y)) # label(congruence).
-->(x,y) | ->(h(x,z),h(y,z)) # label(congruence).
-->(x,y) | ->(h(z,x),h(z,y)) # label(congruence).
->(a,b) # label(replacement).
-->*(x,y) | -->*(z,y) | ->(f(g(x),z),h(y,f(g(y),a))) # label(replacement).
-->*(x,y) | -->*(z,y) | ->(f(x,z),h(y,f(y,b))) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(g(x),y) | -->*(z,y) | -->*(x,u) | -->*(z,u) # 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(g(_), [ 1, 0 ]),

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

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

        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=79, kept=62.
Selections=6, assignments=6, propagations=14, current_models=1.
Rewrite_terms=156, rewrite_bools=95, indexes=36.
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 3297569 exit (max_models) Tue Jul 30 09:16:40 2024
The process finished Tue Jul 30 09:16:40 2024


Mace4 cooked interpretation:

% number = 1
% seconds = 0

% Interpretation of size 2

a = 0.

b = 0.

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

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

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

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

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


The problem is infeasible.