NO

Problem 1: 


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


Problem 1: 

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

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


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

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

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


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

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

-> Critical pairs info:
<f(a,b),f(b,f(f(a,b),a))>   =>  Not trivial, Not overlay, Proper, NW0, N1
<f(b,a),f(b,f(f(a,b),a))>   =>  Not trivial, Not overlay, Proper, NW0, N2

-> 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(a,b)
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('f(a,b)', D0)
ID: 1 => ('f(b,b)', D1, R1, P[1], S{}), NR: 'b'
t: f(b,f(f(a,b),a))
Nodes: [0,1,2,3]
Edges: [(0,1),(0,2),(1,3),(2,3)]
ID: 0 => ('f(b,f(f(a,b),a))', D0)
ID: 1 => ('f(b,f(f(b,b),a))', D1, R1, P[2, 1, 1], S{}), NR: 'b'
ID: 2 => ('f(b,f(f(a,b),b))', D1, R1, P[2, 2], S{}), NR: 'b'
ID: 3 => ('f(b,f(f(b,b),b))', D2, R1, P[2, 2], S{}), NR: 'b'
f(a,b) ->* no union *<- f(b,f(f(a,b),a))
"Not joinable"

The problem is not confluent.