NO

Problem 1: 


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

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

-> Rlps:
 (rule: a(b(a(a(a(a(a(a(a(a(a(x))))))))))) -> a(a(a(a(a(a(a(a(a(a(a(b(a(b(x)))))))))))))), id: 1, possubterms: a(b(a(a(a(a(a(a(a(a(a(x)))))))))))->[], b(a(a(a(a(a(a(a(a(a(x))))))))))->[1], a(a(a(a(a(a(a(a(a(x)))))))))->[1, 1], a(a(a(a(a(a(a(a(x))))))))->[1, 1, 1], a(a(a(a(a(a(a(x)))))))->[1, 1, 1, 1], a(a(a(a(a(a(x))))))->[1, 1, 1, 1, 1], a(a(a(a(a(x)))))->[1, 1, 1, 1, 1, 1], a(a(a(a(x))))->[1, 1, 1, 1, 1, 1, 1], a(a(a(x)))->[1, 1, 1, 1, 1, 1, 1, 1], a(a(x))->[1, 1, 1, 1, 1, 1, 1, 1, 1], a(x)->[1, 1, 1, 1, 1, 1, 1, 1, 1, 1])

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

-> Critical pairs info:
<a(b(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(b(a(b(x')))))))))))))))))))))))),a(a(a(a(a(a(a(a(a(a(a(b(a(b(b(a(a(a(a(a(a(a(a(a(x'))))))))))))))))))))))))>   =>  Not trivial, Not 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: 
No Convergence Brute Force Procedure:
-> Rewritings:
s: a(b(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(b(a(b(x'))))))))))))))))))))))))
Nodes: [0,1,2,3]
Edges: [(0,1),(1,2),(2,3)]
ID: 0 => ('a(b(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(b(a(b(x'))))))))))))))))))))))))', D0)
ID: 1 => ('a(a(a(a(a(a(a(a(a(a(a(b(a(b(a(a(a(a(a(a(a(a(a(a(b(a(b(x')))))))))))))))))))))))))))', D1, R1, P[], S{x4 -> a(a(a(a(a(a(a(a(a(a(b(a(b(x')))))))))))))}), NR: 'a(a(a(a(a(a(a(a(a(a(a(b(a(b(a(a(a(a(a(a(a(a(a(a(b(a(b(x')))))))))))))))))))))))))))'
ID: 2 => ('a(a(a(a(a(a(a(a(a(a(a(b(a(a(a(a(a(a(a(a(a(a(a(b(a(b(a(b(a(b(x'))))))))))))))))))))))))))))))', D2, R1, P[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], S{x4 -> a(b(a(b(x'))))}), NR: 'a(a(a(a(a(a(a(a(a(a(a(b(a(b(a(b(a(b(x'))))))))))))))))))'
ID: 3 => ('a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(b(a(b(a(a(b(a(b(a(b(a(b(x')))))))))))))))))))))))))))))))))', D3, R1, P[1, 1, 1, 1, 1, 1, 1, 1, 1, 1], S{x4 -> a(a(b(a(b(a(b(a(b(x')))))))))}), NR: 'a(a(a(a(a(a(a(a(a(a(a(b(a(b(a(a(b(a(b(a(b(a(b(x')))))))))))))))))))))))'
t: a(a(a(a(a(a(a(a(a(a(a(b(a(b(b(a(a(a(a(a(a(a(a(a(x'))))))))))))))))))))))))
Nodes: [0]
Edges: []
ID: 0 => ('a(a(a(a(a(a(a(a(a(a(a(b(a(b(b(a(a(a(a(a(a(a(a(a(x'))))))))))))))))))))))))', D0)
a(b(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(a(b(a(b(x')))))))))))))))))))))))) ->* no union *<- a(a(a(a(a(a(a(a(a(a(a(b(a(b(b(a(a(a(a(a(a(a(a(a(x'))))))))))))))))))))))))
"Not joinable"

The problem is not confluent.