NO

Problem 1: 


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

Huet Levy Procedure:
-> Rules:
 a(a(x)) -> a(b(b(c(x))))
 b(a(x)) -> x
 c(b(x)) -> a(c(x))
-> Vars:
 x, x, x

-> Rlps:
 (rule: a(a(x)) -> a(b(b(c(x)))), id: 1, possubterms: a(a(x))->[], a(x)->[1])
 (rule: b(a(x)) -> x, id: 2, possubterms: b(a(x))->[], a(x)->[1])
 (rule: c(b(x)) -> a(c(x)), id: 3, possubterms: c(b(x))->[], b(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: a(b(b(c(x)))), r': a(b(b(c(x')))))
(R2 unifies with R1 at p: [1], l: b(a(x)), lp: a(x), sig: {x -> a(x')}, l': a(a(x')), r: x, r': a(b(b(c(x')))))
(R3 unifies with R2 at p: [1], l: c(b(x)), lp: b(x), sig: {x -> a(x')}, l': b(a(x')), r: a(c(x)), r': x')

-> Critical pairs info:
<b(a(b(b(c(x'))))),a(x')>   =>  Not trivial, Not overlay, Proper, NW0, N1
<a(a(b(b(c(x'))))),a(b(b(c(a(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N2
<c(x'),a(c(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: 
No Convergence Brute Force Procedure:
-> Rewritings:
s: b(a(b(b(c(x')))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('b(a(b(b(c(x')))))', D0)
ID: 1 => ('b(b(c(x')))', D1, R2, P[], S{x5 -> b(b(c(x')))}), NR: 'b(b(c(x')))'
t: a(x')
Nodes: [0]
Edges: []
ID: 0 => ('a(x')', D0)
b(a(b(b(c(x'))))) ->* no union *<- a(x')
"Not joinable"

The problem is not confluent.