NO

Problem 1: 


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


Problem 1: 

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

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

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

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

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

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

The problem is not confluent.