NO

Problem 1: 


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


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)
(fSNonEmpty)
(g 1)
)
(RULES
f(f(x)) -> f(g(f(x)))
f(x) -> x
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

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

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

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

-> Critical pairs info:
<f(g(f(x'))),f(x')>   =>  Not trivial, Overlay, Proper, NW0, N1
<f(f(g(f(x')))),f(g(f(f(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N2
<f(x'),f(g(f(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N3

-> 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')
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('f(x')', D0)
ID: 1 => ('x'', D1, R2, P[], S{x5 -> x'}), NR: 'x''
t: f(g(f(x')))
Nodes: [0,1,2,3]
Edges: [(0,1),(0,2),(1,3),(2,3)]
ID: 0 => ('f(g(f(x')))', D0)
ID: 1 => ('g(f(x'))', D1, R2, P[], S{x5 -> g(f(x'))}), NR: 'g(f(x'))'
ID: 2 => ('f(g(x'))', D1, R2, P[1, 1], S{x5 -> x'}), NR: 'x''
ID: 3 => ('g(x')', D2, R2, P[1], S{x5 -> x'}), NR: 'x''
f(x') ->* no union *<- f(g(f(x')))
"Not joinable"

The problem is not confluent.