YES

Problem 1: 

(VAR v_NonEmpty:S X:S Y:S)
(RULES
f(0,1,X:S) -> h(X:S,X:S)
g(X:S,Y:S) -> X:S
g(X:S,Y:S) -> Y:S
h(0,X:S) -> f(0,X:S,X:S)
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(0,1,X:S) -> H(X:S,X:S)
 H(0,X:S) -> F(0,X:S,X:S)
-> Rules:
 f(0,1,X:S) -> h(X:S,X:S)
 g(X:S,Y:S) -> X:S
 g(X:S,Y:S) -> Y:S
 h(0,X:S) -> f(0,X:S,X:S)

Problem 1: 

SCC Processor:
-> Pairs:
 F(0,1,X:S) -> H(X:S,X:S)
 H(0,X:S) -> F(0,X:S,X:S)
-> Rules:
 f(0,1,X:S) -> h(X:S,X:S)
 g(X:S,Y:S) -> X:S
 g(X:S,Y:S) -> Y:S
 h(0,X:S) -> f(0,X:S,X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(0,1,X:S) -> H(X:S,X:S)
 H(0,X:S) -> F(0,X:S,X:S)
->->-> Rules:
 f(0,1,X:S) -> h(X:S,X:S)
 g(X:S,Y:S) -> X:S
 g(X:S,Y:S) -> Y:S
 h(0,X:S) -> f(0,X:S,X:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 F(0,1,X:S) -> H(X:S,X:S)
 H(0,X:S) -> F(0,X:S,X:S)
-> Rules:
 f(0,1,X:S) -> h(X:S,X:S)
 g(X:S,Y:S) -> X:S
 g(X:S,Y:S) -> Y:S
 h(0,X:S) -> f(0,X:S,X:S)
-> Usable rules:
 Empty
->Mace4 Output:
 ============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 27553 was started by sandbox2 on n150.star.cs.uiowa.edu,
Mon Jun 22 08:44:38 2020
The command was "./mace4 -c -f /tmp/mace41497983152038664370.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace41497983152038664370.in

assign(max_seconds,20).

formulas(assumptions).
gtrsim_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility).
succeq_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility).
gtrsim_s0(x,y) & succeq_s0(y,z) -> gtrsim_s0(x,z) # label(compatibility).
arrow_s0(x1,y) -> arrow_s0(f2(x1,x2,x3),f2(y,x2,x3)) # label(congruence).
arrow_s0(x2,y) -> arrow_s0(f2(x1,x2,x3),f2(x1,y,x3)) # label(congruence).
arrow_s0(x3,y) -> arrow_s0(f2(x1,x2,x3),f2(x1,x2,y)) # label(congruence).
arrow_s0(x1,y) -> arrow_s0(f3(x1,x2),f3(y,x2)) # label(congruence).
arrow_s0(x2,y) -> arrow_s0(f3(x1,x2),f3(x1,y)) # label(congruence).
arrow_s0(x1,y) -> arrow_s0(f4(x1,x2),f4(y,x2)) # label(congruence).
arrow_s0(x2,y) -> arrow_s0(f4(x1,x2),f4(x1,y)) # label(congruence).
arrow_s0(x1,y) -> arrow_s0(f9(x1,x2,x3),f9(y,x2,x3)) # label(congruence).
arrow_s0(x2,y) -> arrow_s0(f9(x1,x2,x3),f9(x1,y,x3)) # label(congruence).
arrow_s0(x3,y) -> arrow_s0(f9(x1,x2,x3),f9(x1,x2,y)) # label(congruence).
arrow_s0(x1,y) -> arrow_s0(f10(x1,x2),f10(y,x2)) # label(congruence).
arrow_s0(x2,y) -> arrow_s0(f10(x1,x2),f10(x1,y)) # label(congruence).
arrow_s0(x,y) -> gtrsim_s0(x,y) # label(inclusion).
sqsupset_s0(f9(f5,f6,x1),f10(x1,x1)) # label(replacement).
succeq_s0(f10(f5,x1),f9(f5,x1,x1)) # label(replacement).
sqsupset_s0(x,y) -> sqsupsetStar_s0(x,y) # label(inclusion).
sqsupset_s0(x,y) & sqsupsetStar_s0(y,z) -> sqsupsetStar_s0(x,z) # label(compatibility).
end_of_list.

formulas(goals).
(exists x sqsupsetStar_s0(x,x)) # label(wellfoundedness).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 gtrsim_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility) # label(non_clause).  [assumption].
2 succeq_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility) # label(non_clause).  [assumption].
3 gtrsim_s0(x,y) & succeq_s0(y,z) -> gtrsim_s0(x,z) # label(compatibility) # label(non_clause).  [assumption].
4 arrow_s0(x1,y) -> arrow_s0(f2(x1,x2,x3),f2(y,x2,x3)) # label(congruence) # label(non_clause).  [assumption].
5 arrow_s0(x2,y) -> arrow_s0(f2(x1,x2,x3),f2(x1,y,x3)) # label(congruence) # label(non_clause).  [assumption].
6 arrow_s0(x3,y) -> arrow_s0(f2(x1,x2,x3),f2(x1,x2,y)) # label(congruence) # label(non_clause).  [assumption].
7 arrow_s0(x1,y) -> arrow_s0(f3(x1,x2),f3(y,x2)) # label(congruence) # label(non_clause).  [assumption].
8 arrow_s0(x2,y) -> arrow_s0(f3(x1,x2),f3(x1,y)) # label(congruence) # label(non_clause).  [assumption].
9 arrow_s0(x1,y) -> arrow_s0(f4(x1,x2),f4(y,x2)) # label(congruence) # label(non_clause).  [assumption].
10 arrow_s0(x2,y) -> arrow_s0(f4(x1,x2),f4(x1,y)) # label(congruence) # label(non_clause).  [assumption].
11 arrow_s0(x1,y) -> arrow_s0(f9(x1,x2,x3),f9(y,x2,x3)) # label(congruence) # label(non_clause).  [assumption].
12 arrow_s0(x2,y) -> arrow_s0(f9(x1,x2,x3),f9(x1,y,x3)) # label(congruence) # label(non_clause).  [assumption].
13 arrow_s0(x3,y) -> arrow_s0(f9(x1,x2,x3),f9(x1,x2,y)) # label(congruence) # label(non_clause).  [assumption].
14 arrow_s0(x1,y) -> arrow_s0(f10(x1,x2),f10(y,x2)) # label(congruence) # label(non_clause).  [assumption].
15 arrow_s0(x2,y) -> arrow_s0(f10(x1,x2),f10(x1,y)) # label(congruence) # label(non_clause).  [assumption].
16 arrow_s0(x,y) -> gtrsim_s0(x,y) # label(inclusion) # label(non_clause).  [assumption].
17 sqsupset_s0(x,y) -> sqsupsetStar_s0(x,y) # label(inclusion) # label(non_clause).  [assumption].
18 sqsupset_s0(x,y) & sqsupsetStar_s0(y,z) -> sqsupsetStar_s0(x,z) # label(compatibility) # label(non_clause).  [assumption].
19 (exists x sqsupsetStar_s0(x,x)) # label(wellfoundedness) # label(non_clause) # label(goal).  [goal].

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

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

formulas(mace4_clauses).
-gtrsim_s0(x,y) | -sqsupset_s0(y,z) | sqsupset_s0(x,z) # label(compatibility).
-succeq_s0(x,y) | -sqsupset_s0(y,z) | sqsupset_s0(x,z) # label(compatibility).
-gtrsim_s0(x,y) | -succeq_s0(y,z) | gtrsim_s0(x,z) # label(compatibility).
-arrow_s0(x,y) | arrow_s0(f2(x,z,u),f2(y,z,u)) # label(congruence).
-arrow_s0(x,y) | arrow_s0(f2(z,x,u),f2(z,y,u)) # label(congruence).
-arrow_s0(x,y) | arrow_s0(f2(z,u,x),f2(z,u,y)) # label(congruence).
-arrow_s0(x,y) | arrow_s0(f3(x,z),f3(y,z)) # label(congruence).
-arrow_s0(x,y) | arrow_s0(f3(z,x),f3(z,y)) # label(congruence).
-arrow_s0(x,y) | arrow_s0(f4(x,z),f4(y,z)) # label(congruence).
-arrow_s0(x,y) | arrow_s0(f4(z,x),f4(z,y)) # label(congruence).
-arrow_s0(x,y) | arrow_s0(f9(x,z,u),f9(y,z,u)) # label(congruence).
-arrow_s0(x,y) | arrow_s0(f9(z,x,u),f9(z,y,u)) # label(congruence).
-arrow_s0(x,y) | arrow_s0(f9(z,u,x),f9(z,u,y)) # label(congruence).
-arrow_s0(x,y) | arrow_s0(f10(x,z),f10(y,z)) # label(congruence).
-arrow_s0(x,y) | arrow_s0(f10(z,x),f10(z,y)) # label(congruence).
-arrow_s0(x,y) | gtrsim_s0(x,y) # label(inclusion).
sqsupset_s0(f9(f5,f6,x),f10(x,x)) # label(replacement).
succeq_s0(f10(f5,x),f9(f5,x,x)) # label(replacement).
-sqsupset_s0(x,y) | sqsupsetStar_s0(x,y) # label(inclusion).
-sqsupset_s0(x,y) | -sqsupsetStar_s0(y,z) | sqsupsetStar_s0(x,z) # label(compatibility).
-sqsupsetStar_s0(x,x) # label(wellfoundedness).
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(f5, [ 0 ]),

        function(f6, [ 1 ]),

        function(f10(_,_), [
			   0, 1,
			   0, 0 ]),

        function(f3(_,_), [
			   0, 0,
			   0, 0 ]),

        function(f4(_,_), [
			   0, 0,
			   0, 0 ]),

        function(f2(_,_,_), [
			   0, 0,
			   0, 0,
			   0, 0,
			   0, 0 ]),

        function(f9(_,_,_), [
			   0, 0,
			   1, 1,
			   0, 0,
			   0, 0 ]),

        relation(arrow_s0(_,_), [
			   0, 0,
			   0, 0 ]),

        relation(gtrsim_s0(_,_), [
			   0, 0,
			   0, 0 ]),

        relation(sqsupsetStar_s0(_,_), [
			   0, 0,
			   1, 0 ]),

        relation(sqsupset_s0(_,_), [
			   0, 0,
			   1, 0 ]),

        relation(succeq_s0(_,_), [
			   1, 0,
			   0, 1 ])
]).

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

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

For domain size 2.

Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds).
Ground clauses: seen=190, kept=190.
Selections=33, assignments=36, propagations=28, current_models=1.
Rewrite_terms=377, rewrite_bools=327, indexes=44.
Rules_from_neg_clauses=7, cross_offs=7.

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

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

Exiting with 1 model.

Process 27553 exit (max_models) Mon Jun 22 08:44:38 2020
The process finished Mon Jun 22 08:44:38 2020


Mace4 cooked interpretation:

% number = 1
% seconds = 0

% Interpretation of size 2

f5 = 0.

f6 = 1.

f10(0,0) = 0.
f10(0,1) = 1.
f10(1,0) = 0.
f10(1,1) = 0.

f3(0,0) = 0.
f3(0,1) = 0.
f3(1,0) = 0.
f3(1,1) = 0.

f4(0,0) = 0.
f4(0,1) = 0.
f4(1,0) = 0.
f4(1,1) = 0.

f2(0,0,0) = 0.
f2(0,0,1) = 0.
f2(0,1,0) = 0.
f2(0,1,1) = 0.
f2(1,0,0) = 0.
f2(1,0,1) = 0.
f2(1,1,0) = 0.
f2(1,1,1) = 0.

f9(0,0,0) = 0.
f9(0,0,1) = 0.
f9(0,1,0) = 1.
f9(0,1,1) = 1.
f9(1,0,0) = 0.
f9(1,0,1) = 0.
f9(1,1,0) = 0.
f9(1,1,1) = 0.

- arrow_s0(0,0).
- arrow_s0(0,1).
- arrow_s0(1,0).
- arrow_s0(1,1).

- gtrsim_s0(0,0).
- gtrsim_s0(0,1).
- gtrsim_s0(1,0).
- gtrsim_s0(1,1).

- sqsupsetStar_s0(0,0).
- sqsupsetStar_s0(0,1).
  sqsupsetStar_s0(1,0).
- sqsupsetStar_s0(1,1).

- sqsupset_s0(0,0).
- sqsupset_s0(0,1).
  sqsupset_s0(1,0).
- sqsupset_s0(1,1).

  succeq_s0(0,0).
- succeq_s0(0,1).
- succeq_s0(1,0).
  succeq_s0(1,1).


Problem 1: 

SCC Processor:
-> Pairs:
 H(0,X:S) -> F(0,X:S,X:S)
-> Rules:
 f(0,1,X:S) -> h(X:S,X:S)
 g(X:S,Y:S) -> X:S
 g(X:S,Y:S) -> Y:S
 h(0,X:S) -> f(0,X:S,X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.