NO

Problem 1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x y)
(REPLACEMENT-MAP
(+ 1, 2)
(0)
(fSNonEmpty)
(s 1)
)
(RULES
+(0,0) -> 0
+(s(x),y) -> s(+(x,y))
+(x,s(y)) -> s(+(y,x))
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 

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

Problem 1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x y)
(REPLACEMENT-MAP
(+ 1, 2)
(0)
(fSNonEmpty)
(s 1)
)
(RULES
+(0,0) -> 0
+(s(x),y) -> s(+(x,y))
+(x,s(y)) -> s(+(y,x))
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Levy Procedure:
-> Rules:
 +(0,0) -> 0
 +(s(x),y) -> s(+(x,y))
 +(x,s(y)) -> s(+(y,x))
-> Vars:
 x, y, x, y

-> Rlps:
 (rule: +(0,0) -> 0, id: 1, possubterms: +(0,0)->[], 0->[1], 0->[2])
 (rule: +(s(x),y) -> s(+(x,y)), id: 2, possubterms: +(s(x),y)->[], s(x)->[1])
 (rule: +(x,s(y)) -> s(+(y,x)), id: 3, possubterms: +(x,s(y))->[], s(y)->[2])

-> Unifications:
(R3 unifies with R2 at p: [], l: +(x,s(y)), lp: +(x,s(y)), sig: {x -> s(x'),y' -> s(y)}, l': +(s(x'),y'), r: s(+(y,x)), r': s(+(x',y')))

-> Critical pairs info:
<s(+(x',s(y))),s(+(y,s(x')))>   =>  Not trivial, 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: 
No Convergence Brute Force Procedure:
-> Rewritings:
s: s(+(x',s(y)))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('s(+(x',s(y)))', D0)
ID: 1 => ('s(s(+(y,x')))', D1, R3, P[1], S{x7 -> x', x8 -> y}), NR: 's(+(y,x'))'
t: s(+(y,s(x')))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('s(+(y,s(x')))', D0)
ID: 1 => ('s(s(+(x',y)))', D1, R3, P[1], S{x7 -> y, x8 -> x'}), NR: 's(+(x',y))'
s(+(x',s(y))) ->* no union *<- s(+(y,s(x')))
"Not joinable"

The problem is not confluent.