YES

Problem 1: 


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


Problem 1: 

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

Problem 1: 
Linearity Procedure:
Linear?
YES

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

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

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

-> 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'))))

-> Critical pairs info:
<f(f(g(f(x')))),f(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 1.1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(f 1)
(fSNonEmpty)
(g 1)
)
(RULES
f(f(x)) -> f(g(f(x)))
)
Critical Pairs:
<f(f(g(f(x')))),f(g(f(f(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N1
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Strong Confluence Procedure:
-> Rewritings:
s: f(f(g(f(x'))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('f(f(g(f(x'))))', D0)
ID: 1 => ('f(g(f(g(f(x')))))', D1, R1, P[], S{x4 -> g(f(x'))}), NR: 'f(g(f(g(f(x')))))'
t: f(g(f(f(x'))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('f(g(f(f(x'))))', D0)
ID: 1 => ('f(g(f(g(f(x')))))', D1, R1, P[1, 1], S{x4 -> x'}), NR: 'f(g(f(x')))'
SNodesPath1: f(f(g(f(x')))) -> f(g(f(g(f(x')))))
TNodesPath1: f(g(f(f(x')))) -> f(g(f(g(f(x')))))
SNodesPath2: f(f(g(f(x')))) -> f(g(f(g(f(x')))))
TNodesPath2: f(g(f(f(x')))) -> f(g(f(g(f(x')))))
f(f(g(f(x')))) ->= f(g(f(g(f(x'))))) *<- f(g(f(f(x'))))
f(g(f(f(x')))) ->= f(g(f(g(f(x'))))) *<- f(f(g(f(x'))))
"Strongly closed critical pair"

The problem is confluent.