YES

Problem 1: 


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

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

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

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

-> Critical pairs info:
<s(x'),s(x')>   =>  Trivial, Not overlay, Proper, NW0, N1

-> Problem conclusions:
 Left linear, Right linear, Linear
 Weakly orthogonal, Not almost orthogonal, Not orthogonal
 Huet-Levy confluent, Not Newman confluent
 R is a TRS 


The problem is confluent.