NO

Problem 1: 


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


Problem 1: 

CleanTRS Procedure:
R was updated by simple cleaning of the TRS

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


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)) -> b(c(x))
a(c(x)) -> c(b(x))
b(a(x)) -> a(a(x))
b(b(x)) -> b(a(x))
b(c(x)) -> a(b(x))
c(b(x)) -> a(b(x))
c(c(x)) -> a(a(x))
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

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

-> Rlps:
 (rule: a(a(x)) -> b(c(x)), id: 1, possubterms: a(a(x))->[], a(x)->[1])
 (rule: a(c(x)) -> c(b(x)), id: 2, possubterms: a(c(x))->[], c(x)->[1])
 (rule: b(a(x)) -> a(a(x)), id: 3, possubterms: b(a(x))->[], a(x)->[1])
 (rule: b(b(x)) -> b(a(x)), id: 4, possubterms: b(b(x))->[], b(x)->[1])
 (rule: b(c(x)) -> a(b(x)), id: 5, possubterms: b(c(x))->[], c(x)->[1])
 (rule: c(b(x)) -> a(b(x)), id: 6, possubterms: c(b(x))->[], b(x)->[1])
 (rule: c(c(x)) -> a(a(x)), id: 7, possubterms: c(c(x))->[], c(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: b(c(x)), r': b(c(x')))
(R1 unifies with R2 at p: [1], l: a(a(x)), lp: a(x), sig: {x -> c(x')}, l': a(c(x')), r: b(c(x)), r': c(b(x')))
(R2 unifies with R6 at p: [1], l: a(c(x)), lp: c(x), sig: {x -> b(x')}, l': c(b(x')), r: c(b(x)), r': a(b(x')))
(R2 unifies with R7 at p: [1], l: a(c(x)), lp: c(x), sig: {x -> c(x')}, l': c(c(x')), r: c(b(x)), r': a(a(x')))
(R3 unifies with R1 at p: [1], l: b(a(x)), lp: a(x), sig: {x -> a(x')}, l': a(a(x')), r: a(a(x)), r': b(c(x')))
(R3 unifies with R2 at p: [1], l: b(a(x)), lp: a(x), sig: {x -> c(x')}, l': a(c(x')), r: a(a(x)), r': c(b(x')))
(R4 unifies with R3 at p: [1], l: b(b(x)), lp: b(x), sig: {x -> a(x')}, l': b(a(x')), r: b(a(x)), r': a(a(x')))
(R4 unifies with R4 at p: [1], l: b(b(x)), lp: b(x), sig: {x -> b(x')}, l': b(b(x')), r: b(a(x)), r': b(a(x')))
(R4 unifies with R5 at p: [1], l: b(b(x)), lp: b(x), sig: {x -> c(x')}, l': b(c(x')), r: b(a(x)), r': a(b(x')))
(R5 unifies with R6 at p: [1], l: b(c(x)), lp: c(x), sig: {x -> b(x')}, l': c(b(x')), r: a(b(x)), r': a(b(x')))
(R5 unifies with R7 at p: [1], l: b(c(x)), lp: c(x), sig: {x -> c(x')}, l': c(c(x')), r: a(b(x)), r': a(a(x')))
(R6 unifies with R3 at p: [1], l: c(b(x)), lp: b(x), sig: {x -> a(x')}, l': b(a(x')), r: a(b(x)), r': a(a(x')))
(R6 unifies with R4 at p: [1], l: c(b(x)), lp: b(x), sig: {x -> b(x')}, l': b(b(x')), r: a(b(x)), r': b(a(x')))
(R6 unifies with R5 at p: [1], l: c(b(x)), lp: b(x), sig: {x -> c(x')}, l': b(c(x')), r: a(b(x)), r': a(b(x')))
(R7 unifies with R6 at p: [1], l: c(c(x)), lp: c(x), sig: {x -> b(x')}, l': c(b(x')), r: a(a(x)), r': a(b(x')))
(R7 unifies with R7 at p: [1], l: c(c(x)), lp: c(x), sig: {x -> c(x')}, l': c(c(x')), r: a(a(x)), r': a(a(x')))

-> Critical pairs info:
<c(a(b(x'))),a(a(b(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N1
<b(a(a(x'))),a(b(c(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N2
<a(b(c(x'))),b(c(a(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N3
<b(b(c(x'))),a(a(a(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N4
<b(a(b(x'))),b(a(c(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N5
<c(a(a(x'))),a(a(c(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N6
<b(b(a(x'))),b(a(b(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N7
<b(a(a(x'))),b(a(a(x')))>   =>  Trivial, Not overlay, Proper, NW0, N8
<a(a(b(x'))),c(b(b(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N9
<a(a(a(x'))),c(b(c(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N10
<b(c(b(x'))),a(a(c(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N11
<a(c(b(x'))),b(c(c(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N12
<c(b(a(x'))),a(b(b(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N13
<c(a(a(x'))),a(b(a(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N14
<b(a(b(x'))),a(b(b(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N15
<c(a(b(x'))),a(b(c(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N16

-> 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: c(a(b(x')))
Nodes: [0]
Edges: []
ID: 0 => ('c(a(b(x')))', D0)
t: a(a(b(x')))
Nodes: [0,1,2,3,4,5,6,7]
Edges: [(0,1),(1,2),(1,3),(2,4),(3,0),(4,5),(5,6),(5,7),(6,4),(7,0)]
ID: 0 => ('a(a(b(x')))', D0)
ID: 1 => ('b(c(b(x')))', D1, R1, P[], S{x4 -> b(x')}), NR: 'b(c(b(x')))'
ID: 2 => ('a(b(b(x')))', D2, R5, P[], S{x8 -> b(x')}), NR: 'a(b(b(x')))'
ID: 3 => ('b(a(b(x')))', D2, R6, P[1], S{x9 -> x'}), NR: 'a(b(x'))'
ID: 4 => ('a(b(a(x')))', D3, R4, P[1], S{x7 -> x'}), NR: 'b(a(x'))'
ID: 5 => ('a(a(a(x')))', D4, R3, P[1], S{x6 -> x'}), NR: 'a(a(x'))'
ID: 6 => ('b(c(a(x')))', D5, R1, P[], S{x4 -> a(x')}), NR: 'b(c(a(x')))'
ID: 7 => ('a(b(c(x')))', D5, R1, P[1], S{x4 -> x'}), NR: 'b(c(x'))'
c(a(b(x'))) ->* no union *<- a(a(b(x')))
"Not joinable"

The problem is not confluent.