YES

Problem 1: 


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


Problem 1: 

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

Problem 1: 
CSR Converter From Canonical u-Replacement Map Procedure [CSUR20]:

Original Replacement Map:
(+ 1 2)
(s 1)
(0)
(fSNonEmpty)

New Replacement Map:
(+ 1)
(s 1)
(0)
(fSNonEmpty)

New problem:

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


Problem 1: 
Termination Procedure:
Terminating?
YES

Problem 1: 

(VAR vu95NonEmpty vNonEmpty y x)
(STRATEGY CONTEXTSENSITIVE
(u43 1)
(s 1)
(num0)
(fSNonEmpty)
)
(RULES
u43(s(x),y) -> s(u43(x,y))
u43(num0,y) -> y
s(s(x)) -> x
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 U43(s(x),y) -> U43(x,y)
 U43(s(x),y) -> S(u43(x,y))
 U43(num0,y) -> y
-> Rules:
 u43(s(x),y) -> s(u43(x,y))
 u43(num0,y) -> y
 s(s(x)) -> x
-> Unhiding Rules:
 Empty

Problem 1: 

SCC Processor:
-> Pairs:
 U43(s(x),y) -> U43(x,y)
 U43(s(x),y) -> S(u43(x,y))
 U43(num0,y) -> y
-> Rules:
 u43(s(x),y) -> s(u43(x,y))
 u43(num0,y) -> y
 s(s(x)) -> x
-> Unhiding rules:
 Empty
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U43(s(x),y) -> U43(x,y)
->->-> Rules:
 u43(s(x),y) -> s(u43(x,y))
 u43(num0,y) -> y
 s(s(x)) -> x
->->-> Unhiding rules:
 Empty

Problem 1: 

SubNColl Processor:
-> Pairs:
 U43(s(x),y) -> U43(x,y)
-> Rules:
 u43(s(x),y) -> s(u43(x,y))
 u43(num0,y) -> y
 s(s(x)) -> x
-> Unhiding rules:
 Empty
->Projection:
 pi(U43) = 1

Problem 1: 

Basic Processor:
-> Pairs:
 Empty
-> Rules:
 u43(s(x),y) -> s(u43(x,y))
 u43(num0,y) -> y
 s(s(x)) -> x
-> Unhiding rules:
 Empty
-> Result:
 Set P is empty

The problem is finite.


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

Huet Levy NW Procedure:
-> Rules:
 +(s(x),y) -> s(+(x,y))
 +(0,y) -> y
 s(s(x)) -> x
-> Vars:
 y, x, y, x
-> UVars:
 (UV-RuleId: 1, UV-LActive: [y, x], UV-RActive: [y, x], UV-LFrozen: [], UV-RFrozen: [])
 (UV-RuleId: 2, UV-LActive: [y], UV-RActive: [y], UV-LFrozen: [], UV-RFrozen: [])
 (UV-RuleId: 3, UV-LActive: [x], UV-RActive: [x], UV-LFrozen: [], UV-RFrozen: [])

-> Rlps:
 (rule: +(s(x),y) -> s(+(x,y)), id: 1, possubterms: +(s(x),y)->[], s(x)->[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:
(R1 unifies with R3 at p: [1], l: +(s(x),y), lp: s(x), sig: {x -> s(x')}, l': s(s(x')), r: s(+(x,y)), r': x')
(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:
<+(x',y),s(+(s(x'),y))>   =>  Not trivial, Not overlay, Proper, NW0, N1
<s(x'),s(x')>   =>  Trivial, Not 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 CS-TRS, Left-homogeneous u-replacing variables

Problem 1.1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR y x x')
(REPLACEMENT-MAP
(+ 1)
(s 1)
(0)
(fSNonEmpty)
)
(RULES
+(s(x),y) -> s(+(x,y))
+(0,y) -> y
s(s(x)) -> x
)
Critical Pairs:
<+(x',y),s(+(s(x'),y))>   =>  Not trivial, Not overlay, Proper, NW0, N1
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: +(x',y)
Nodes: [0]
Edges: []
ID: 0 => ('+(x',y)', D0)
t: s(+(s(x'),y))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('s(+(s(x'),y))', D0)
ID: 1 => ('s(s(+(x',y)))', D1, R1, P[1], S{x6 -> y, x7 -> x'}), NR: 's(+(x',y))'
ID: 2 => ('+(x',y)', D2, R3, P[], S{x9 -> +(x',y)}), NR: '+(x',y)'
SNodesPath: +(x',y)
TNodesPath: s(+(s(x'),y)) -> s(s(+(x',y))) -> +(x',y)
+(x',y) ->* +(x',y) *<- s(+(s(x'),y))
"Joinable"

The problem is confluent.