NO

Problem 1: 


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


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

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

-> Rlps:
 (rule: f(g(f(x))) -> g(f(g(x))), id: 1, possubterms: f(g(f(x)))->[], g(f(x))->[1], f(x)->[1, 1])
 (rule: f(c) -> c, id: 2, possubterms: f(c)->[], c->[1])
 (rule: g(c) -> c, id: 3, possubterms: g(c)->[], c->[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: g(f(g(x))), r': g(f(g(x'))))
(R1 unifies with R2 at p: [1,1], l: f(g(f(x))), lp: f(x), sig: {x -> c}, l': f(c), r: g(f(g(x))), r': c)

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

The problem is not confluent.