YES

Problem 1: 


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


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)
(REPLACEMENT-MAP
(a)
(f 1)
(b)
(fSNonEmpty)
(g 1)
)
(RULES
a -> b
f(a) -> g(a)
f(b) -> g(b)
)
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

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


-> Rlps:
 (rule: a -> b, id: 1, possubterms: a->[])
 (rule: f(a) -> g(a), id: 2, possubterms: f(a)->[], a->[1])
 (rule: f(b) -> g(b), id: 3, possubterms: f(b)->[], b->[1])

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

-> Critical pairs info:
<f(b),g(a)>   =>  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:
(REPLACEMENT-MAP
(a)
(f 1)
(b)
(fSNonEmpty)
(g 1)
)
(RULES
a -> b
f(a) -> g(a)
f(b) -> g(b)
)
Critical Pairs:
<f(b),g(a)>   =>  Not trivial, Not overlay, Proper, NW0, N1
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Strong Confluence Procedure:
-> Rewritings:
s: f(b)
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('f(b)', D0)
ID: 1 => ('g(b)', D1, R3, P[], S{}), NR: 'g(b)'
t: g(a)
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('g(a)', D0)
ID: 1 => ('g(b)', D1, R1, P[1], S{}), NR: 'b'
SNodesPath1: f(b) -> g(b)
TNodesPath1: g(a) -> g(b)
SNodesPath2: f(b) -> g(b)
TNodesPath2: g(a) -> g(b)
f(b) ->= g(b) *<- g(a)
g(a) ->= g(b) *<- f(b)
"Strongly closed critical pair"

The problem is confluent.