NO

Problem 1: 


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

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

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

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

-> Critical pairs info:
<x',g(f(f(x')))>   =>  Not trivial, Overlay, Proper, NW0, N1
<f(g(x')),g(f(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N2
<f(g(g(f(x')))),g(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: x'
Nodes: [0]
Edges: []
ID: 0 => ('x'', D0)
t: g(f(f(x')))
Nodes: [0]
Edges: []
ID: 0 => ('g(f(f(x')))', D0)
x' ->* no union *<- g(f(f(x')))
"Not joinable"

The problem is not confluent.