NO

Problem 1: 


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


Problem 1: 

CleanTRS Procedure:
R was updated by simple cleaning of the TRS

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


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

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

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


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

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

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

No Convergence InfChecker Procedure:
Infeasible convergence?
YES

Problem 1: 

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

Problem 1: 

Obtaining a model using Mace4:

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

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 2789711 was started by shintani on shintani-XPS-13-9310,
Fri Jun  9 17:19:15 2023
The command was "./mace4 -c -f /tmp/mace42789693-2.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace42789693-2.in

assign(max_seconds,10).

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

formulas(goals).
(exists x2 (->*(c,x2) & ->*(h(b,c),x2))) # 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,x2),h(y,x2)) # label(congruence) # label(non_clause).  [assumption].
3 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause).  [assumption].
4 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause).  [assumption].
5 (exists x2 (->*(c,x2) & ->*(h(b,c),x2))) # 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,z),h(y,z)) # label(congruence).
-->(x,y) | ->(h(z,x),h(z,y)) # label(congruence).
->(a,f(a)) # label(replacement).
->(b,c) # label(replacement).
->(b,h(b,c)) # label(replacement).
->(c,a) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(c,x) | -->*(h(b,c),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=36, kept=32.
Selections=4, assignments=7, propagations=20, current_models=0.
Rewrite_terms=59, rewrite_bools=66, indexes=28.
Rules_from_neg_clauses=4, cross_offs=4.

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

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

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

interpretation( 3, [number=1, seconds=0], [

        function(a, [ 0 ]),

        function(b, [ 1 ]),

        function(c, [ 0 ]),

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

        function(h(_,_), [
			   2, 1, 0,
			   2, 1, 0,
			   2, 1, 0 ]),

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

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

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

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

For domain size 3.

Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds).
Ground clauses: seen=100, kept=91.
Selections=12, assignments=17, propagations=39, current_models=1.
Rewrite_terms=217, rewrite_bools=215, indexes=39.
Rules_from_neg_clauses=8, cross_offs=21.

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

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

Exiting with 1 model.

Process 2789711 exit (max_models) Fri Jun  9 17:19:15 2023
The process finished Fri Jun  9 17:19:15 2023


Mace4 cooked interpretation:



The problem is infeasible.


The problem is not confluent.