NO

Problem 1: 


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

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

-> Rlps:
 (rule: a(a(x)) -> b(c(x)), id: 1, possubterms: a(a(x))->[], a(x)->[1])
 (rule: b(b(x)) -> c(d(x)), id: 2, possubterms: b(b(x))->[], b(x)->[1])
 (rule: c(c(x)) -> d(d(d(x))), id: 3, possubterms: c(c(x))->[], c(x)->[1])
 (rule: d(d(d(x))) -> a(c(x)), id: 4, possubterms: d(d(d(x)))->[], d(d(x))->[1], d(x)->[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(c(x)), r': b(c(x')))
(R2 unifies with R2 at p: [1], l: b(b(x)), lp: b(x), sig: {x -> b(x')}, l': b(b(x')), r: c(d(x)), r': c(d(x')))
(R3 unifies with R3 at p: [1], l: c(c(x)), lp: c(x), sig: {x -> c(x')}, l': c(c(x')), r: d(d(d(x))), r': d(d(d(x'))))
(R4 unifies with R4 at p: [1], l: d(d(d(x))), lp: d(d(x)), sig: {x -> d(x')}, l': d(d(d(x'))), r: a(c(x)), r': a(c(x')))
(R4 unifies with R4 at p: [1,1], l: d(d(d(x))), lp: d(x), sig: {x -> d(d(x'))}, l': d(d(d(x'))), r: a(c(x)), r': a(c(x')))

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

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

The problem is not confluent.