NO

Problem 1: 


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

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

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

-> Unifications:
(R2 unifies with R1 at p: [], l: a(x), lp: a(x), sig: {x -> x'}, l': a(x'), r: g(h(b(x))), r': g(c(x')))
(R4 unifies with R1 at p: [1], l: g(a(x)), lp: a(x), sig: {x -> x'}, l': a(x'), r: g(h(a(x))), r': g(c(x')))
(R4 unifies with R2 at p: [1], l: g(a(x)), lp: a(x), sig: {x -> x'}, l': a(x'), r: g(h(a(x))), r': g(h(b(x'))))

-> Critical pairs info:
<g(c(x')),g(h(b(x')))>   =>  Not trivial, Overlay, Proper, NW0, N1
<g(g(c(x'))),g(h(a(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N2
<g(g(h(b(x')))),g(h(a(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N3

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

No Convergence InfChecker Procedure:
Infeasible convergence?
YES

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty x x1 vNonEmpty z0)
(STRATEGY CONTEXTSENSITIVE
(a 1)
(b 1)
(g 1)
(c 1)
(c_x1)
(fSNonEmpty)
(h 1)
)
(RULES
a(x) -> g(c(x))
a(x) -> g(h(b(x)))
b(x) -> g(b(x))
g(a(x)) -> g(h(a(x)))
)]
Infeasibility Conditions:
g(c(c_x1)) ->* z0, g(h(b(c_x1))) ->* z0

Problem 1: 

Obtaining a model using Mace4:

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

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 2775313 was started by shintani on shintani-XPS-13-9310,
Fri Jun  9 16:27:40 2023
The command was "./mace4 -c -f /tmp/mace42775296-2.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace42775296-2.in

assign(max_seconds,10).

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

formulas(goals).
(exists x4 (->*(g(c(c_x1)),x4) & ->*(g(h(b(c_x1))),x4))) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 ->(x1,y) -> ->(a(x1),a(y)) # label(congruence) # label(non_clause).  [assumption].
2 ->(x1,y) -> ->(b(x1),b(y)) # label(congruence) # label(non_clause).  [assumption].
3 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence) # label(non_clause).  [assumption].
4 ->(x1,y) -> ->(c(x1),c(y)) # label(congruence) # label(non_clause).  [assumption].
5 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence) # label(non_clause).  [assumption].
6 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause).  [assumption].
7 (exists x4 (->*(g(c(c_x1)),x4) & ->*(g(h(b(c_x1))),x4))) # label(goal) # label(non_clause) # label(goal).  [goal].

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

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

formulas(mace4_clauses).
-->(x,y) | ->(a(x),a(y)) # label(congruence).
-->(x,y) | ->(b(x),b(y)) # label(congruence).
-->(x,y) | ->(g(x),g(y)) # label(congruence).
-->(x,y) | ->(c(x),c(y)) # label(congruence).
-->(x,y) | ->(h(x),h(y)) # label(congruence).
->(a(x),g(c(x))) # label(replacement).
->(a(x),g(h(b(x)))) # label(replacement).
->(b(x),g(b(x))) # label(replacement).
->(g(a(x)),g(h(a(x)))) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(g(c(c_x1)),x) | -->*(g(h(b(c_x1))),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=40, kept=36.
Selections=615, assignments=1229, propagations=539, current_models=0.
Rewrite_terms=8470, rewrite_bools=6042, indexes=919.
Rules_from_neg_clauses=36, cross_offs=36.

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

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

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

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

        function(c_x1, [ 0 ]),

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

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

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

        function(c(_), [ 1, 1, 1 ]),

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

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

        relation(->(_,_), [
			   1, 1, 1,
			   0, 1, 0,
			   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=90, kept=81.
Selections=30, assignments=67, propagations=76, current_models=1.
Rewrite_terms=675, rewrite_bools=430, indexes=236.
Rules_from_neg_clauses=5, cross_offs=14.

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

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

Exiting with 1 model.

Process 2775313 exit (max_models) Fri Jun  9 16:27:40 2023
The process finished Fri Jun  9 16:27:40 2023


Mace4 cooked interpretation:



The problem is infeasible.


The problem is not confluent.