YES

Problem 1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x)
(REPLACEMENT-MAP
(F 1)
(G 1)
(c 1)
(fSNonEmpty)
)
(RULES
F(c(x)) -> G(x)
G(x) -> F(x)
c(x) -> 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)
(G 1)
(c 1)
(fSNonEmpty)
)
(RULES
F(c(x)) -> G(x)
G(x) -> F(x)
c(x) -> x
)
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Levy Procedure:
-> Rules:
 F(c(x)) -> G(x)
 G(x) -> F(x)
 c(x) -> x
-> Vars:
 x, x, x

-> Rlps:
 (rule: F(c(x)) -> G(x), id: 1, possubterms: F(c(x))->[], c(x)->[1])
 (rule: G(x) -> F(x), id: 2, possubterms: G(x)->[])
 (rule: c(x) -> x, id: 3, possubterms: c(x)->[])

-> Unifications:
(R1 unifies with R3 at p: [1], l: F(c(x)), lp: c(x), sig: {x -> x'}, l': c(x'), r: G(x), r': x')

-> Critical pairs info:
<F(x'),G(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)
(G 1)
(c 1)
(fSNonEmpty)
)
(RULES
F(c(x)) -> G(x)
G(x) -> F(x)
c(x) -> x
)
Critical Pairs:
<F(x'),G(x')>   =>  Not trivial, Not overlay, Proper, NW0, N1
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Strong Confluence Procedure:
-> Rewritings:
s: F(x')
Nodes: [0]
Edges: []
ID: 0 => ('F(x')', D0)
t: G(x')
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('G(x')', D0)
ID: 1 => ('F(x')', D1, R2, P[], S{x5 -> x'}), NR: 'F(x')'
SNodesPath1: F(x')
TNodesPath1: G(x') -> F(x')
SNodesPath2: F(x')
TNodesPath2: G(x') -> F(x')
F(x') ->= F(x') *<- G(x')
G(x') ->= F(x') *<- F(x')
"Strongly closed critical pair"

The problem is confluent.