NO

Problem 1: 


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

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

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

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

-> 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: a(b(b(b(x'))))
Nodes: [0]
Edges: []
ID: 0 => ('a(b(b(b(x'))))', D0)
t: b(b(b(a(x'))))
Nodes: [0]
Edges: []
ID: 0 => ('b(b(b(a(x'))))', D0)
a(b(b(b(x')))) ->* no union *<- b(b(b(a(x'))))
"Not joinable"

The problem is not confluent.