YES

Problem 1: 


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


Problem 1: 

Problem 1: 

Commutativity Transform Procedure:

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


Problem 1: 


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

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

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

-> Unifications:
(R3 unifies with R1 at p: [], l: +(y,s(0)), lp: +(y,s(0)), sig: {y -> s(0),y' -> s(0)}, l': +(s(0),y'), r: s(+(0,y)), r': s(+(0,y')))
(R3 unifies with R2 at p: [], l: +(y,s(0)), lp: +(y,s(0)), sig: {y -> 0,y' -> s(0)}, l': +(0,y'), r: s(+(0,y)), r': y')
(R4 unifies with R1 at p: [], l: +(y,0), lp: +(y,0), sig: {y -> s(0),y' -> 0}, l': +(s(0),y'), r: y, r': s(+(0,y')))
(R4 unifies with R2 at p: [], l: +(y,0), lp: +(y,0), sig: {y -> 0,y' -> 0}, l': +(0,y'), r: y, r': y')
(R5 unifies with R5 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(+(0,0)),s(0)>   =>  Not trivial, Overlay, Proper, NW0, N1
<0,0>   =>  Trivial, Overlay, Proper, NW0, N2
<s(x'),s(x')>   =>  Trivial, Not overlay, Proper, NW0, N3
<s(+(0,s(0))),s(+(0,s(0)))>   =>  Trivial, Overlay, Proper, NW0, N4
<s(0),s(+(0,0))>   =>  Not trivial, Overlay, Proper, NW0, N5

-> 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 1.1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR y x)
(REPLACEMENT-MAP
(+ 1, 2)
(s 1)
(0)
(fSNonEmpty)
)
(RULES
+(s(0),y) -> s(+(0,y))
+(0,y) -> y
+(y,s(0)) -> s(+(0,y))
+(y,0) -> y
s(s(x)) -> x
)
Critical Pairs:
<s(0),s(+(0,0))>   =>  Not trivial, Overlay, Proper, NW0, N5
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Strong Confluence Procedure:
-> Rewritings:
s: s(0)
Nodes: [0]
Edges: []
ID: 0 => ('s(0)', D0)
t: s(+(0,0))
Nodes: [0,1]
Edges: [(0,1),(0,1)]
ID: 0 => ('s(+(0,0))', D0)
ID: 1 => ('s(0)', D1, R2, P[1], S{x4 -> 0}), NR: '0'
SNodesPath1: s(0)
TNodesPath1: s(+(0,0)) -> s(0)
SNodesPath2: s(0)
TNodesPath2: s(+(0,0)) -> s(0)
s(0) ->= s(0) *<- s(+(0,0))
s(+(0,0)) ->= s(0) *<- s(0)
"Strongly closed critical pair"

The problem is confluent.