NO

Problem 1: 


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

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

-> Rlps:
 (rule: a(b(x)) -> c(d(x)), id: 1, possubterms: a(b(x))->[], b(x)->[1])
 (rule: a(x) -> e(d(x)), id: 2, possubterms: a(x)->[])
 (rule: b(x) -> d(c(x)), id: 3, possubterms: b(x)->[])
 (rule: d(d(x)) -> b(e(x)), id: 4, possubterms: d(d(x))->[], d(x)->[1])
 (rule: d(x) -> x, id: 5, possubterms: d(x)->[])
 (rule: e(c(x)) -> d(a(x)), id: 6, possubterms: e(c(x))->[], c(x)->[1])

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

-> Critical pairs info:
<b(e(x')),d(x')>   =>  Not trivial, Overlay, Proper, NW0, N1
<a(d(c(x'))),c(d(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N2
<c(d(x')),e(d(b(x')))>   =>  Not trivial, Overlay, Proper, NW0, N3
<d(x'),b(e(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N4
<d(b(e(x'))),b(e(d(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(x')
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('d(x')', D0)
ID: 1 => ('x'', D1, R5, P[], S{x8 -> x'}), NR: 'x''
t: b(e(x'))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('b(e(x'))', D0)
ID: 1 => ('d(c(e(x')))', D1, R3, P[], S{x6 -> e(x')}), NR: 'd(c(e(x')))'
ID: 2 => ('c(e(x'))', D2, R5, P[], S{x8 -> c(e(x'))}), NR: 'c(e(x'))'
d(x') ->* no union *<- b(e(x'))
"Not joinable"

The problem is not confluent.