NO

Problem 1: 


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

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

-> Rlps:
 (rule: a(a(x)) -> b(x), id: 1, possubterms: a(a(x))->[], a(x)->[1])
 (rule: b(a(x)) -> a(b(x)), id: 2, possubterms: b(a(x))->[], a(x)->[1])
 (rule: b(b(c(x))) -> c(a(x)), id: 3, possubterms: b(b(c(x)))->[], b(c(x))->[1], c(x)->[1, 1])
 (rule: b(b(x)) -> a(a(a(x))), id: 4, possubterms: b(b(x))->[], b(x)->[1])
 (rule: c(a(x)) -> b(a(c(x))), id: 5, possubterms: c(a(x))->[], a(x)->[1])

-> Unifications:
(R1 unifies with R1 at p: [1], l: a(a(x)), lp: a(x), sig: {x -> a(x')}, l': a(a(x')), r: b(x), r': b(x'))
(R2 unifies with R1 at p: [1], l: b(a(x)), lp: a(x), sig: {x -> a(x')}, l': a(a(x')), r: a(b(x)), r': b(x'))
(R3 unifies with R5 at p: [1,1], l: b(b(c(x))), lp: c(x), sig: {x -> a(x')}, l': c(a(x')), r: c(a(x)), r': b(a(c(x'))))
(R4 unifies with R3 at p: [], l: b(b(x)), lp: b(b(x)), sig: {x -> c(x')}, l': b(b(c(x'))), r: a(a(a(x))), r': c(a(x')))
(R4 unifies with R2 at p: [1], l: b(b(x)), lp: b(x), sig: {x -> a(x')}, l': b(a(x')), r: a(a(a(x))), r': a(b(x')))
(R4 unifies with R3 at p: [1], l: b(b(x)), lp: b(x), sig: {x -> b(c(x'))}, l': b(b(c(x'))), r: a(a(a(x))), r': c(a(x')))
(R4 unifies with R4 at p: [1], l: b(b(x)), lp: b(x), sig: {x -> b(x')}, l': b(b(x')), r: a(a(a(x))), r': a(a(a(x'))))
(R5 unifies with R1 at p: [1], l: c(a(x)), lp: a(x), sig: {x -> a(x')}, l': a(a(x')), r: b(a(c(x))), r': b(x'))

-> Critical pairs info:
<b(a(b(x'))),a(a(a(a(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N1
<b(b(x')),a(b(a(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N2
<b(a(a(a(x')))),a(a(a(b(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N3
<c(a(x')),a(a(a(c(x'))))>   =>  Not trivial, Overlay, Proper, NW0, N4
<a(b(x')),b(a(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N5
<c(b(x')),b(a(c(a(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N6
<b(b(b(a(c(x'))))),c(a(a(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N7
<b(c(a(x'))),a(a(a(b(c(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N8

-> 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)
(c 1)
(fSNonEmpty)
)
(RULES
a(a(x)) -> b(x)
b(a(x)) -> a(b(x))
b(b(c(x))) -> c(a(x))
b(b(x)) -> a(a(a(x)))
c(a(x)) -> b(a(c(x)))
)
Critical Pairs:
<c(b(x')),b(a(c(a(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N6
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

No Convergence InfChecker Procedure:
Infeasible convergence?
YES

Problem 1: 

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

Problem 1: 

Obtaining a model using AGES:

 -> Theory (Usable Rules):
a(a(x)) -> b(x)
b(a(x)) -> a(b(x))
b(b(c(x))) -> c(a(x))
b(b(x)) -> a(a(a(x)))
c(a(x)) -> b(a(c(x)))

 -> AGES Output:




The problem is infeasible.


The problem is not confluent.