YES

Problem 1: 


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

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

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

-> Unifications:
(R3 unifies with R1 at p: [], l: +(x,0), lp: +(x,0), sig: {x -> 0,y -> 0}, l': +(0,y), r: x, r': y)
(R3 unifies with R2 at p: [], l: +(x,0), lp: +(x,0), sig: {x -> s(x'),y -> 0}, l': +(s(x'),y), r: x, r': s(+(x',y)))
(R4 unifies with R1 at p: [], l: +(x,s(y)), lp: +(x,s(y)), sig: {x -> 0,y' -> s(y)}, l': +(0,y'), r: s(+(x,y)), r': y')
(R4 unifies with R2 at p: [], l: +(x,s(y)), lp: +(x,s(y)), sig: {x -> s(x'),y' -> s(y)}, l': +(s(x'),y'), r: s(+(x,y)), r': s(+(x',y')))
(R5 unifies with R1 at p: [], l: +(x,y), lp: +(x,y), sig: {x -> 0,y -> y'}, l': +(0,y'), r: +(y,x), r': y')
(R5 unifies with R2 at p: [], l: +(x,y), lp: +(x,y), sig: {x -> s(x'),y -> y'}, l': +(s(x'),y'), r: +(y,x), r': s(+(x',y')))
(R5 unifies with R3 at p: [], l: +(x,y), lp: +(x,y), sig: {x -> x',y -> 0}, l': +(x',0), r: +(y,x), r': x')
(R5 unifies with R4 at p: [], l: +(x,y), lp: +(x,y), sig: {x -> x',y -> s(y')}, l': +(x',s(y')), r: +(y,x), r': s(+(x',y')))

-> Critical pairs info:
<0,0>   =>  Trivial, Overlay, Proper, NW0, N1
<y',+(y',0)>   =>  Not trivial, Overlay, Proper, NW0, N2
<s(+(x',y')),+(y',s(x'))>   =>  Not trivial, Overlay, Proper, NW0, N3
<x',+(0,x')>   =>  Not trivial, Overlay, Proper, NW0, N4
<s(+(x',0)),s(x')>   =>  Not trivial, Overlay, Proper, NW0, N5
<s(y),s(+(0,y))>   =>  Not trivial, Overlay, Proper, NW0, N6
<s(+(x',s(y))),s(+(s(x'),y))>   =>  Not trivial, Overlay, Proper, NW0, N7
<s(+(x',y')),+(s(y'),x')>   =>  Not trivial, Overlay, Proper, NW0, N8

-> 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 



The problem is decomposed in 7 subproblems.

Problem 1.1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x y x' y')
(REPLACEMENT-MAP
(+ 1, 2)
(0)
(fSNonEmpty)
(s 1)
)
(RULES
+(0,y) -> y
+(s(x),y) -> s(+(x,y))
+(x,0) -> x
+(x,s(y)) -> s(+(x,y))
+(x,y) -> +(y,x)
)
Critical Pairs:
<y',+(y',0)>   =>  Not trivial, Overlay, Proper, NW0, N2
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Strong Confluence Procedure:
-> Rewritings:
s: y'
Nodes: [0]
Edges: []
ID: 0 => ('y'', D0)
t: +(y',0)
Nodes: [0,1,2]
Edges: [(0,1),(0,2),(2,0),(2,1)]
ID: 0 => ('+(y',0)', D0)
ID: 1 => ('y'', D1, R3, P[], S{x9 -> y'}), NR: 'y''
ID: 2 => ('+(0,y')', D1, R5, P[], S{x12 -> y', x13 -> 0}), NR: '+(0,y')'
SNodesPath1: y'
TNodesPath1: +(y',0) -> y'
SNodesPath2: y'
TNodesPath2: +(y',0) -> y'
y' ->= y' *<- +(y',0)
+(y',0) ->= y' *<- y'
"Strongly closed critical pair"

The problem is confluent.

Problem 1.2: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x y x' y')
(REPLACEMENT-MAP
(+ 1, 2)
(0)
(fSNonEmpty)
(s 1)
)
(RULES
+(0,y) -> y
+(s(x),y) -> s(+(x,y))
+(x,0) -> x
+(x,s(y)) -> s(+(x,y))
+(x,y) -> +(y,x)
)
Critical Pairs:
<s(+(x',y')),+(y',s(x'))>   =>  Not trivial, Overlay, Proper, NW0, N3
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Strong Confluence Procedure:
-> Rewritings:
s: s(+(x',y'))
Nodes: [0,1]
Edges: [(0,1),(1,0)]
ID: 0 => ('s(+(x',y'))', D0)
ID: 1 => ('s(+(y',x'))', D1, R5, P[1], S{x12 -> x', x13 -> y'}), NR: '+(y',x')'
t: +(y',s(x'))
Nodes: [0,1,2,3]
Edges: [(0,1),(0,2),(1,3),(2,0),(2,3),(3,1)]
ID: 0 => ('+(y',s(x'))', D0)
ID: 1 => ('s(+(y',x'))', D1, R4, P[], S{x10 -> y', x11 -> x'}), NR: 's(+(y',x'))'
ID: 2 => ('+(s(x'),y')', D1, R5, P[], S{x12 -> y', x13 -> s(x')}), NR: '+(s(x'),y')'
ID: 3 => ('s(+(x',y'))', D2, R5, P[1], S{x12 -> y', x13 -> x'}), NR: '+(x',y')'
SNodesPath1: s(+(x',y'))
TNodesPath1: +(y',s(x')) -> s(+(y',x')) -> s(+(x',y'))
SNodesPath2: s(+(x',y')) -> s(+(y',x'))
TNodesPath2: +(y',s(x')) -> s(+(y',x'))
s(+(x',y')) ->= s(+(x',y')) *<- +(y',s(x'))
+(y',s(x')) ->= s(+(y',x')) *<- s(+(x',y'))
"Strongly closed critical pair"

The problem is confluent.

Problem 1.3: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x y x' y')
(REPLACEMENT-MAP
(+ 1, 2)
(0)
(fSNonEmpty)
(s 1)
)
(RULES
+(0,y) -> y
+(s(x),y) -> s(+(x,y))
+(x,0) -> x
+(x,s(y)) -> s(+(x,y))
+(x,y) -> +(y,x)
)
Critical Pairs:
<x',+(0,x')>   =>  Not trivial, Overlay, Proper, NW0, N4
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Strong Confluence Procedure:
-> Rewritings:
s: x'
Nodes: [0]
Edges: []
ID: 0 => ('x'', D0)
t: +(0,x')
Nodes: [0,1,2]
Edges: [(0,1),(0,2),(2,0),(2,1)]
ID: 0 => ('+(0,x')', D0)
ID: 1 => ('x'', D1, R1, P[], S{x6 -> x'}), NR: 'x''
ID: 2 => ('+(x',0)', D1, R5, P[], S{x12 -> 0, x13 -> x'}), NR: '+(x',0)'
SNodesPath1: x'
TNodesPath1: +(0,x') -> x'
SNodesPath2: x'
TNodesPath2: +(0,x') -> x'
x' ->= x' *<- +(0,x')
+(0,x') ->= x' *<- x'
"Strongly closed critical pair"

The problem is confluent.

Problem 1.4: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x y x' y')
(REPLACEMENT-MAP
(+ 1, 2)
(0)
(fSNonEmpty)
(s 1)
)
(RULES
+(0,y) -> y
+(s(x),y) -> s(+(x,y))
+(x,0) -> x
+(x,s(y)) -> s(+(x,y))
+(x,y) -> +(y,x)
)
Critical Pairs:
<s(+(x',0)),s(x')>   =>  Not trivial, Overlay, Proper, NW0, N5
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Strong Confluence Procedure:
-> Rewritings:
s: s(+(x',0))
Nodes: [0,1,2]
Edges: [(0,1),(0,2),(2,0),(2,1)]
ID: 0 => ('s(+(x',0))', D0)
ID: 1 => ('s(x')', D1, R3, P[1], S{x9 -> x'}), NR: 'x''
ID: 2 => ('s(+(0,x'))', D1, R5, P[1], S{x12 -> x', x13 -> 0}), NR: '+(0,x')'
t: s(x')
Nodes: [0]
Edges: []
ID: 0 => ('s(x')', D0)
SNodesPath1: s(+(x',0)) -> s(x')
TNodesPath1: s(x')
SNodesPath2: s(+(x',0)) -> s(x')
TNodesPath2: s(x')
s(+(x',0)) ->= s(x') *<- s(x')
s(x') ->= s(x') *<- s(+(x',0))
"Strongly closed critical pair"

The problem is confluent.

Problem 1.5: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x y x' y')
(REPLACEMENT-MAP
(+ 1, 2)
(0)
(fSNonEmpty)
(s 1)
)
(RULES
+(0,y) -> y
+(s(x),y) -> s(+(x,y))
+(x,0) -> x
+(x,s(y)) -> s(+(x,y))
+(x,y) -> +(y,x)
)
Critical Pairs:
<s(y),s(+(0,y))>   =>  Not trivial, Overlay, Proper, NW0, N6
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Strong Confluence Procedure:
-> Rewritings:
s: s(y)
Nodes: [0]
Edges: []
ID: 0 => ('s(y)', D0)
t: s(+(0,y))
Nodes: [0,1,2]
Edges: [(0,1),(0,2),(2,0),(2,1)]
ID: 0 => ('s(+(0,y))', D0)
ID: 1 => ('s(y)', D1, R1, P[1], S{x6 -> y}), NR: 'y'
ID: 2 => ('s(+(y,0))', D1, R5, P[1], S{x12 -> 0, x13 -> y}), NR: '+(y,0)'
SNodesPath1: s(y)
TNodesPath1: s(+(0,y)) -> s(y)
SNodesPath2: s(y)
TNodesPath2: s(+(0,y)) -> s(y)
s(y) ->= s(y) *<- s(+(0,y))
s(+(0,y)) ->= s(y) *<- s(y)
"Strongly closed critical pair"

The problem is confluent.

Problem 1.6: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x y x' y')
(REPLACEMENT-MAP
(+ 1, 2)
(0)
(fSNonEmpty)
(s 1)
)
(RULES
+(0,y) -> y
+(s(x),y) -> s(+(x,y))
+(x,0) -> x
+(x,s(y)) -> s(+(x,y))
+(x,y) -> +(y,x)
)
Critical Pairs:
<s(+(x',s(y))),s(+(s(x'),y))>   =>  Not trivial, Overlay, Proper, NW0, N7
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Strong Confluence Procedure:
-> Rewritings:
s: s(+(x',s(y)))
Nodes: [0,1,2,3]
Edges: [(0,1),(0,2),(1,3),(2,0),(2,3),(3,1)]
ID: 0 => ('s(+(x',s(y)))', D0)
ID: 1 => ('s(s(+(x',y)))', D1, R4, P[1], S{x10 -> x', x11 -> y}), NR: 's(+(x',y))'
ID: 2 => ('s(+(s(y),x'))', D1, R5, P[1], S{x12 -> x', x13 -> s(y)}), NR: '+(s(y),x')'
ID: 3 => ('s(s(+(y,x')))', D2, R5, P[1, 1], S{x12 -> x', x13 -> y}), NR: '+(y,x')'
t: s(+(s(x'),y))
Nodes: [0,1,2,3]
Edges: [(0,1),(0,2),(1,3),(2,0),(2,3),(3,1)]
ID: 0 => ('s(+(s(x'),y))', D0)
ID: 1 => ('s(s(+(x',y)))', D1, R2, P[1], S{x7 -> x', x8 -> y}), NR: 's(+(x',y))'
ID: 2 => ('s(+(y,s(x')))', D1, R5, P[1], S{x12 -> s(x'), x13 -> y}), NR: '+(y,s(x'))'
ID: 3 => ('s(s(+(y,x')))', D2, R5, P[1, 1], S{x12 -> x', x13 -> y}), NR: '+(y,x')'
SNodesPath1: s(+(x',s(y))) -> s(s(+(x',y)))
TNodesPath1: s(+(s(x'),y)) -> s(s(+(x',y)))
SNodesPath2: s(+(x',s(y))) -> s(s(+(x',y)))
TNodesPath2: s(+(s(x'),y)) -> s(s(+(x',y)))
s(+(x',s(y))) ->= s(s(+(x',y))) *<- s(+(s(x'),y))
s(+(s(x'),y)) ->= s(s(+(x',y))) *<- s(+(x',s(y)))
"Strongly closed critical pair"

The problem is confluent.

Problem 1.7: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x y x' y')
(REPLACEMENT-MAP
(+ 1, 2)
(0)
(fSNonEmpty)
(s 1)
)
(RULES
+(0,y) -> y
+(s(x),y) -> s(+(x,y))
+(x,0) -> x
+(x,s(y)) -> s(+(x,y))
+(x,y) -> +(y,x)
)
Critical Pairs:
<s(+(x',y')),+(s(y'),x')>   =>  Not trivial, Overlay, Proper, NW0, N8
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Strong Confluence Procedure:
-> Rewritings:
s: s(+(x',y'))
Nodes: [0,1]
Edges: [(0,1),(1,0)]
ID: 0 => ('s(+(x',y'))', D0)
ID: 1 => ('s(+(y',x'))', D1, R5, P[1], S{x12 -> x', x13 -> y'}), NR: '+(y',x')'
t: +(s(y'),x')
Nodes: [0,1,2,3]
Edges: [(0,1),(0,2),(1,3),(2,0),(2,3),(3,1)]
ID: 0 => ('+(s(y'),x')', D0)
ID: 1 => ('s(+(y',x'))', D1, R2, P[], S{x7 -> y', x8 -> x'}), NR: 's(+(y',x'))'
ID: 2 => ('+(x',s(y'))', D1, R5, P[], S{x12 -> s(y'), x13 -> x'}), NR: '+(x',s(y'))'
ID: 3 => ('s(+(x',y'))', D2, R5, P[1], S{x12 -> y', x13 -> x'}), NR: '+(x',y')'
SNodesPath1: s(+(x',y'))
TNodesPath1: +(s(y'),x') -> s(+(y',x')) -> s(+(x',y'))
SNodesPath2: s(+(x',y')) -> s(+(y',x'))
TNodesPath2: +(s(y'),x') -> s(+(y',x'))
s(+(x',y')) ->= s(+(x',y')) *<- +(s(y'),x')
+(s(y'),x') ->= s(+(y',x')) *<- s(+(x',y'))
"Strongly closed critical pair"

The problem is confluent.