NO

Problem 1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x)
(REPLACEMENT-MAP
(a)
(f 1)
(h 1)
(c)
(fSNonEmpty)
(g 1)
)
(RULES
a -> f(c)
a -> h(a)
f(x) -> h(g(x))
h(x) -> f(g(x))
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 

Problem 1: 
Not CS-TRS Procedure:
R is not a CS-TRS

Problem 1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x)
(REPLACEMENT-MAP
(a)
(f 1)
(h 1)
(c)
(fSNonEmpty)
(g 1)
)
(RULES
a -> f(c)
a -> h(a)
f(x) -> h(g(x))
h(x) -> f(g(x))
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Levy Ordered by Num of Vars and Symbs Procedure:
-> Rules:
 a -> f(c)
 a -> h(a)
 f(x) -> h(g(x))
 h(x) -> f(g(x))
-> Vars:
 x, x

-> Rlps:
 (rule: a -> f(c), id: 1, possubterms: a->[])
 (rule: a -> h(a), id: 2, possubterms: a->[])
 (rule: f(x) -> h(g(x)), id: 3, possubterms: f(x)->[])
 (rule: h(x) -> f(g(x)), id: 4, possubterms: h(x)->[])

-> Unifications:
(R2 unifies with R1 at p: [], l: a, lp: a, sig: {}, l': a, r: h(a), r': f(c))

-> Critical pairs info:
<f(c),h(a)>   =>  Not trivial, Overlay, Proper, NW0, N1

-> Problem conclusions:
 Left linear, Right linear, Linear
 Not weakly orthogonal, Not almost orthogonal, Not orthogonal
 Not Huet-Levy confluent, Not Newman confluent
 R is a TRS 


Problem 1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x)
(REPLACEMENT-MAP
(a)
(f 1)
(h 1)
(c)
(fSNonEmpty)
(g 1)
)
(RULES
a -> f(c)
a -> h(a)
f(x) -> h(g(x))
h(x) -> f(g(x))
)
Critical Pairs:
<f(c),h(a)>   =>  Not trivial, Overlay, Proper, NW0, N1
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

No Convergence InfChecker Procedure:
Infeasible convergence?
YES

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty x vNonEmpty z0)
(STRATEGY CONTEXTSENSITIVE
(a)
(f 1)
(h 1)
(c)
(fSNonEmpty)
(g 1)
)
(RULES
a -> f(c)
a -> h(a)
f(x) -> h(g(x))
h(x) -> f(g(x))
)]
Infeasibility Conditions:
f(c) ->* z0, h(a) ->* z0

Problem 1: 

Obtaining a model using Mace4:

 -> Usable Rules:
a -> f(c)
a -> h(a)
f(x) -> h(g(x))
h(x) -> f(g(x))

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 2776412 was started by shintani on shintani-XPS-13-9310,
Fri Jun  9 16:31:58 2023
The command was "./mace4 -c -f /tmp/mace42776395-2.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace42776395-2.in

assign(max_seconds,10).

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

formulas(goals).
(exists x3 (->*(f(c),x3) & ->*(h(a),x3))) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause).  [assumption].
2 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence) # label(non_clause).  [assumption].
3 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence) # label(non_clause).  [assumption].
4 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause).  [assumption].
5 (exists x3 (->*(f(c),x3) & ->*(h(a),x3))) # label(goal) # label(non_clause) # label(goal).  [goal].

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

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

formulas(mace4_clauses).
-->(x,y) | ->(f(x),f(y)) # label(congruence).
-->(x,y) | ->(h(x),h(y)) # label(congruence).
-->(x,y) | ->(g(x),g(y)) # label(congruence).
->(a,f(c)) # label(replacement).
->(a,h(a)) # label(replacement).
->(f(x),h(g(x))) # label(replacement).
->(h(x),f(g(x))) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(f(c),x) | -->*(h(a),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=30, kept=26.
Selections=6, assignments=11, propagations=32, current_models=0.
Rewrite_terms=117, rewrite_bools=36, indexes=45.
Rules_from_neg_clauses=6, cross_offs=6.

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

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

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

For domain size 3.

Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds).
Ground clauses: seen=68, kept=59.
Selections=16, assignments=43, propagations=128, current_models=0.
Rewrite_terms=466, rewrite_bools=495, indexes=152.
Rules_from_neg_clauses=23, cross_offs=50.

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

============================== DOMAIN SIZE 4 =========================

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

interpretation( 4, [number=1, seconds=0], [

        function(a, [ 0 ]),

        function(c, [ 0 ]),

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

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

        function(g(_), [ 3, 3, 3, 1 ]),

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

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

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

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

For domain size 4.

Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds).
Ground clauses: seen=130, kept=114.
Selections=19, assignments=35, propagations=38, current_models=1.
Rewrite_terms=246, rewrite_bools=215, indexes=43.
Rules_from_neg_clauses=3, cross_offs=22.

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

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

Exiting with 1 model.

Process 2776412 exit (max_models) Fri Jun  9 16:31:58 2023
The process finished Fri Jun  9 16:31:58 2023


Mace4 cooked interpretation:



The problem is infeasible.


The problem is not confluent.