NO

Problem 1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty)
(REPLACEMENT-MAP
(b)
(c)
(h 1, 2)
(a)
(f 1)
(fSNonEmpty)
)
(RULES
b -> f(b)
c -> a
h(a,c) -> b
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 

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

Problem 1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty)
(REPLACEMENT-MAP
(b)
(c)
(h 1, 2)
(a)
(f 1)
(fSNonEmpty)
)
(RULES
b -> f(b)
c -> a
h(a,c) -> b
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

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


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

-> Unifications:
(R3 unifies with R2 at p: [2], l: h(a,c), lp: c, sig: {}, l': c, r: b, r': a)

-> Critical pairs info:
<h(a,a),b>   =>  Not trivial, Not 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:
(REPLACEMENT-MAP
(b)
(c)
(h 1, 2)
(a)
(f 1)
(fSNonEmpty)
)
(RULES
b -> f(b)
c -> a
h(a,c) -> b
)
Critical Pairs:
<h(a,a),b>   =>  Not trivial, Not overlay, Proper, NW0, N1
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

No Convergence InfChecker Procedure:
Infeasible convergence?
YES

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty vNonEmpty z0)
(STRATEGY CONTEXTSENSITIVE
(b)
(c)
(h 1 2)
(a)
(f 1)
(fSNonEmpty)
)
(RULES
b -> f(b)
c -> a
h(a,c) -> b
)]
Infeasibility Conditions:
h(a,a) ->* z0, b ->* z0

Problem 1: 

Obtaining a model using Mace4:

 -> Usable Rules:
b -> f(b)
h(a,c) -> b

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 2791176 was started by shintani on shintani-XPS-13-9310,
Fri Jun  9 17:31:02 2023
The command was "./mace4 -c -f /tmp/mace42791158-2.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace42791158-2.in

assign(max_seconds,10).

formulas(assumptions).
->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence).
->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence).
->(x1,y) -> ->(f(x1),f(y)) # label(congruence).
->(b,f(b)) # label(replacement).
->(h(a,c),b) # label(replacement).
->*(x,x) # label(reflexivity).
->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity).
end_of_list.

formulas(goals).
(exists x2 (->*(h(a,a),x2) & ->*(b,x2))) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause).  [assumption].
2 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause).  [assumption].
3 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause).  [assumption].
4 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause).  [assumption].
5 (exists x2 (->*(h(a,a),x2) & ->*(b,x2))) # label(goal) # label(non_clause) # label(goal).  [goal].

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

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

formulas(mace4_clauses).
-->(x,y) | ->(h(x,z),h(y,z)) # label(congruence).
-->(x,y) | ->(h(z,x),h(z,y)) # label(congruence).
-->(x,y) | ->(f(x),f(y)) # label(congruence).
->(b,f(b)) # label(replacement).
->(h(a,c),b) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(h(a,a),x) | -->*(b,x) # 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(c, [ 0 ]),

        function(a, [ 1 ]),

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

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

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

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

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

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

For domain size 2.

Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds).
Ground clauses: seen=34, kept=30.
Selections=7, assignments=8, propagations=18, current_models=1.
Rewrite_terms=76, rewrite_bools=71, indexes=17.
Rules_from_neg_clauses=5, cross_offs=5.

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

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

Exiting with 1 model.

Process 2791176 exit (max_models) Fri Jun  9 17:31:02 2023
The process finished Fri Jun  9 17:31:02 2023


Mace4 cooked interpretation:



The problem is infeasible.


The problem is not confluent.