NO

Problem 1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x y z)
(REPLACEMENT-MAP
(a)
(f 1, 2)
(b)
(fSNonEmpty)
)
(RULES
a -> b
f(f(x,y),z) -> f(b,b)
f(b,x) -> f(b,b)
f(x,a) -> f(b,b)
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 

Problem 1: 
Not CS-TRS Procedure:
R is not a CS-TRS

Problem 1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x y z)
(REPLACEMENT-MAP
(a)
(f 1, 2)
(b)
(fSNonEmpty)
)
(RULES
a -> b
f(f(x,y),z) -> f(b,b)
f(b,x) -> f(b,b)
f(x,a) -> f(b,b)
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

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

-> Rlps:
 (rule: a -> b, id: 1, possubterms: a->[])
 (rule: f(f(x,y),z) -> f(b,b), id: 2, possubterms: f(f(x,y),z)->[], f(x,y)->[1])
 (rule: f(b,x) -> f(b,b), id: 3, possubterms: f(b,x)->[], b->[1])
 (rule: f(x,a) -> f(b,b), id: 4, possubterms: f(x,a)->[], a->[2])

-> Unifications:
(R2 unifies with R2 at p: [1], l: f(f(x,y),z), lp: f(x,y), sig: {x -> f(x',y'),y -> z'}, l': f(f(x',y'),z'), r: f(b,b), r': f(b,b))
(R2 unifies with R3 at p: [1], l: f(f(x,y),z), lp: f(x,y), sig: {x -> b,y -> x'}, l': f(b,x'), r: f(b,b), r': f(b,b))
(R2 unifies with R4 at p: [1], l: f(f(x,y),z), lp: f(x,y), sig: {x -> x',y -> a}, l': f(x',a), r: f(b,b), r': f(b,b))
(R4 unifies with R2 at p: [], l: f(x,a), lp: f(x,a), sig: {x -> f(x',y),z -> a}, l': f(f(x',y),z), r: f(b,b), r': f(b,b))
(R4 unifies with R3 at p: [], l: f(x,a), lp: f(x,a), sig: {x -> b,x' -> a}, l': f(b,x'), r: f(b,b), r': f(b,b))
(R4 unifies with R1 at p: [2], l: f(x,a), lp: a, sig: {}, l': a, r: f(b,b), r': b)

-> Critical pairs info:
<f(f(b,b),z),f(b,b)>   =>  Not trivial, Not overlay, Proper, NW0, N1
<f(f(b,b),z),f(b,b)>   =>  Not trivial, Not overlay, Proper, NW0, N2
<f(f(b,b),z),f(b,b)>   =>  Not trivial, Not overlay, Proper, NW0, N3
<f(b,b),f(b,b)>   =>  Trivial, Overlay, Proper, NW0, N4
<f(b,b),f(b,b)>   =>  Trivial, Overlay, Proper, NW0, N5
<f(x,b),f(b,b)>   =>  Not trivial, Not overlay, Proper, NW0, N6

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

The problem is not confluent.