YES

Problem 1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x)
(REPLACEMENT-MAP
(f 1)
(fSNonEmpty)
(g 1, 2)
)
(RULES
f(f(x)) -> f(g(f(x),f(x)))
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 

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

Problem 1: 
CSR Converter From Canonical u-Replacement Map Procedure [CSUR20]:

Original Replacement Map:
(f 1)
(fSNonEmpty)
(g 1 2)

New Replacement Map:
(f 1)
(fSNonEmpty)
(g)

New problem:

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


Problem 1: 
Termination Procedure:
Terminating?
YES

Problem 1: 

(VAR vu95NonEmpty vNonEmpty x)
(STRATEGY CONTEXTSENSITIVE
(f 1)
(fSNonEmpty)
(g)
)
(RULES
f(f(x)) -> f(g(f(x),f(x)))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 Empty
-> Rules:
 f(f(x)) -> f(g(f(x),f(x)))
-> Unhiding Rules:
 Empty

Problem 1: 

Basic Processor:
-> Pairs:
 Empty
-> Rules:
 f(f(x)) -> f(g(f(x),f(x)))
-> Unhiding rules:
 Empty
-> Result:
 Set P is empty

The problem is finite.


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

Huet Levy NW Procedure:
-> Rules:
 f(f(x)) -> f(g(f(x),f(x)))
-> Vars:
 x
-> UVars:
 (UV-RuleId: 1, UV-LActive: [x], UV-RActive: [x], UV-LFrozen: [], UV-RFrozen: [])

-> Rlps:
 (rule: f(f(x)) -> f(g(f(x),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),f(x))), r': f(g(f(x'),f(x'))))

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

-> Problem conclusions:
 Left linear, Not right linear, Not linear
 Not weakly orthogonal, Not almost orthogonal, Not orthogonal
 Not Huet-Levy confluent, Not Newman confluent
 R is a CS-TRS, Left-homogeneous u-replacing variables

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

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: f(f(g(f(x'),f(x'))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('f(f(g(f(x'),f(x'))))', D0)
ID: 1 => ('f(g(f(g(f(x'),f(x'))),f(g(f(x'),f(x')))))', D1, R1, P[], S{x4 -> g(f(x'),f(x'))}), NR: 'f(g(f(g(f(x'),f(x'))),f(g(f(x'),f(x')))))'
t: f(g(f(f(x')),f(f(x'))))
Nodes: [0,1,2,3]
Edges: [(0,1),(0,2),(1,3),(2,3)]
ID: 0 => ('f(g(f(f(x')),f(f(x'))))', D0)
ID: 1 => ('f(g(f(g(f(x'),f(x'))),f(f(x'))))', D1, R1, P[1, 1], S{x4 -> x'}), NR: 'f(g(f(x'),f(x')))'
ID: 2 => ('f(g(f(f(x')),f(g(f(x'),f(x')))))', D1, R1, P[1, 2], S{x4 -> x'}), NR: 'f(g(f(x'),f(x')))'
ID: 3 => ('f(g(f(g(f(x'),f(x'))),f(g(f(x'),f(x')))))', D2, R1, P[1, 2], S{x4 -> x'}), NR: 'f(g(f(x'),f(x')))'
SNodesPath: f(f(g(f(x'),f(x')))) -> f(g(f(g(f(x'),f(x'))),f(g(f(x'),f(x')))))
TNodesPath: f(g(f(f(x')),f(f(x')))) -> f(g(f(g(f(x'),f(x'))),f(f(x')))) -> f(g(f(g(f(x'),f(x'))),f(g(f(x'),f(x')))))
f(f(g(f(x'),f(x')))) ->* f(g(f(g(f(x'),f(x'))),f(g(f(x'),f(x'))))) *<- f(g(f(f(x')),f(f(x'))))
"Joinable"

The problem is confluent.