NO

Problem 1: 


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


Problem 1: 

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

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

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

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

-> Unifications:
(R3 unifies with R1 at p: [], l: plus(x,y), lp: plus(x,y), sig: {y -> y',x -> 0}, l': plus(0,y'), r: qplus(x,y), r': y')
(R3 unifies with R2 at p: [], l: plus(x,y), lp: plus(x,y), sig: {y -> y',x -> s(x')}, l': plus(s(x'),y'), r: qplus(x,y), r': s(plus(x',y')))

-> Critical pairs info:
<s(plus(x',y')),qplus(s(x'),y')>   =>  Not trivial, Overlay, Proper, NW0, N1
<y',qplus(0,y')>   =>  Not trivial, Overlay, Proper, NW0, N2

-> 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(plus(x',y'))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('s(plus(x',y'))', D0)
ID: 1 => ('s(qplus(x',y'))', D1, R3, P[1], S{x9 -> y', x10 -> x'}), NR: 'qplus(x',y')'
t: qplus(s(x'),y')
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('qplus(s(x'),y')', D0)
ID: 1 => ('qplus(x',s(y'))', D1, R5, P[], S{x12 -> y', x13 -> x'}), NR: 'qplus(x',s(y'))'
s(plus(x',y')) ->* no union *<- qplus(s(x'),y')
"Not joinable"

The problem is not confluent.