NO

Problem 1: 


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


Problem 1: 

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

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


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

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

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

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

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

-> Critical pairs info:
<f(x,b),f(a,b)>   =>  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: f(x,b)
Nodes: [0]
Edges: []
ID: 0 => ('f(x,b)', D0)
t: f(a,b)
Nodes: [0,1]
Edges: [(0,1),(1,0)]
ID: 0 => ('f(a,b)', D0)
ID: 1 => ('f(b,b)', D1, R1, P[1], S{}), NR: 'b'
f(x,b) ->* no union *<- f(a,b)
"Not joinable"

The problem is not confluent.