NO

Problem 1: 


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

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

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

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

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

The problem is not confluent.