NO

Problem 1: 


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


Problem 1: 

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

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

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

-> Rlps:
 (rule: +(p(x),y) -> +(x,p(y)), id: 1, possubterms: +(p(x),y)->[], p(x)->[1])
 (rule: +(s(x),y) -> s(+(x,y)), id: 2, possubterms: +(s(x),y)->[], s(x)->[1])
 (rule: +(0,y) -> y, id: 3, possubterms: +(0,y)->[], 0->[1])
 (rule: p(s(0)) -> 0, id: 4, possubterms: p(s(0))->[], s(0)->[1], 0->[1, 1])
 (rule: p(s(x)) -> s(p(x)), id: 5, possubterms: p(s(x))->[], s(x)->[1])
 (rule: s(p(x)) -> p(s(x)), id: 6, possubterms: s(p(x))->[], p(x)->[1])

-> Unifications:
(R1 unifies with R4 at p: [1], l: +(p(x),y), lp: p(x), sig: {x -> s(0)}, l': p(s(0)), r: +(x,p(y)), r': 0)
(R1 unifies with R5 at p: [1], l: +(p(x),y), lp: p(x), sig: {x -> s(x')}, l': p(s(x')), r: +(x,p(y)), r': s(p(x')))
(R2 unifies with R6 at p: [1], l: +(s(x),y), lp: s(x), sig: {x -> p(x')}, l': s(p(x')), r: s(+(x,y)), r': p(s(x')))
(R5 unifies with R4 at p: [], l: p(s(x)), lp: p(s(x)), sig: {x -> 0}, l': p(s(0)), r: s(p(x)), r': 0)
(R5 unifies with R6 at p: [1], l: p(s(x)), lp: s(x), sig: {x -> p(x')}, l': s(p(x')), r: s(p(x)), r': p(s(x')))
(R6 unifies with R4 at p: [1], l: s(p(x)), lp: p(x), sig: {x -> s(0)}, l': p(s(0)), r: p(s(x)), r': 0)
(R6 unifies with R5 at p: [1], l: s(p(x)), lp: p(x), sig: {x -> s(x')}, l': p(s(x')), r: p(s(x)), r': s(p(x')))

-> Critical pairs info:
<s(s(p(x'))),p(s(s(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N1
<0,s(p(0))>   =>  Not trivial, Overlay, Proper, NW0, N2
<+(0,y),+(s(0),p(y))>   =>  Not trivial, Not overlay, Proper, NW0, N3
<p(p(s(x'))),s(p(p(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N4
<s(0),p(s(s(0)))>   =>  Not trivial, Not overlay, Proper, NW0, N5
<+(p(s(x')),y),s(+(p(x'),y))>   =>  Not trivial, Not overlay, Proper, NW0, N6
<+(s(p(x')),y),+(s(x'),p(y))>   =>  Not trivial, Not overlay, Proper, NW0, N7

-> 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: +(0,y)
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('+(0,y)', D0)
ID: 1 => ('y', D1, R3, P[], S{x10 -> y}), NR: 'y'
t: +(s(0),p(y))
Nodes: [0,1,2,3]
Edges: [(0,1),(1,2),(2,3),(3,2)]
ID: 0 => ('+(s(0),p(y))', D0)
ID: 1 => ('s(+(0,p(y)))', D1, R2, P[], S{x8 -> p(y), x9 -> 0}), NR: 's(+(0,p(y)))'
ID: 2 => ('s(p(y))', D2, R3, P[1], S{x10 -> p(y)}), NR: 'p(y)'
ID: 3 => ('p(s(y))', D3, R6, P[], S{x12 -> y}), NR: 'p(s(y))'
+(0,y) ->* no union *<- +(s(0),p(y))
"Not joinable"

The problem is not confluent.