YES

Problem 1: 


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

New Replacement Map:
(+ 2)
(- 1 2)
(p 1)
(s 1)
(0)
(fSNonEmpty)

New problem:

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


Problem 1: 
Termination Procedure:
Terminating?
YES

Problem 1: 

(VAR vu95NonEmpty vNonEmpty x y)
(STRATEGY CONTEXTSENSITIVE
(u43 2)
(u45 1 2)
(p 1)
(s 1)
(num0)
(fSNonEmpty)
)
(RULES
u43(x,p(y)) -> p(u43(x,y))
u43(x,s(y)) -> s(u43(x,y))
u43(x,num0) -> x
u45(p(x),y) -> p(u45(x,y))
u45(s(x),y) -> s(u45(x,y))
u45(num0,num0) -> num0
u45(x,p(y)) -> s(u45(x,y))
u45(x,s(y)) -> p(u45(x,y))
p(s(x)) -> x
s(p(x)) -> x
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 U43(x,p(y)) -> U43(x,y)
 U43(x,p(y)) -> P(u43(x,y))
 U43(x,s(y)) -> U43(x,y)
 U43(x,s(y)) -> S(u43(x,y))
 U43(x,num0) -> x
 U45(p(x),y) -> U45(x,y)
 U45(p(x),y) -> P(u45(x,y))
 U45(s(x),y) -> U45(x,y)
 U45(s(x),y) -> S(u45(x,y))
 U45(x,p(y)) -> U45(x,y)
 U45(x,p(y)) -> S(u45(x,y))
 U45(x,s(y)) -> U45(x,y)
 U45(x,s(y)) -> P(u45(x,y))
-> Rules:
 u43(x,p(y)) -> p(u43(x,y))
 u43(x,s(y)) -> s(u43(x,y))
 u43(x,num0) -> x
 u45(p(x),y) -> p(u45(x,y))
 u45(s(x),y) -> s(u45(x,y))
 u45(num0,num0) -> num0
 u45(x,p(y)) -> s(u45(x,y))
 u45(x,s(y)) -> p(u45(x,y))
 p(s(x)) -> x
 s(p(x)) -> x
-> Unhiding Rules:
 Empty

Problem 1: 

SCC Processor:
-> Pairs:
 U43(x,p(y)) -> U43(x,y)
 U43(x,p(y)) -> P(u43(x,y))
 U43(x,s(y)) -> U43(x,y)
 U43(x,s(y)) -> S(u43(x,y))
 U43(x,num0) -> x
 U45(p(x),y) -> U45(x,y)
 U45(p(x),y) -> P(u45(x,y))
 U45(s(x),y) -> U45(x,y)
 U45(s(x),y) -> S(u45(x,y))
 U45(x,p(y)) -> U45(x,y)
 U45(x,p(y)) -> S(u45(x,y))
 U45(x,s(y)) -> U45(x,y)
 U45(x,s(y)) -> P(u45(x,y))
-> Rules:
 u43(x,p(y)) -> p(u43(x,y))
 u43(x,s(y)) -> s(u43(x,y))
 u43(x,num0) -> x
 u45(p(x),y) -> p(u45(x,y))
 u45(s(x),y) -> s(u45(x,y))
 u45(num0,num0) -> num0
 u45(x,p(y)) -> s(u45(x,y))
 u45(x,s(y)) -> p(u45(x,y))
 p(s(x)) -> x
 s(p(x)) -> x
-> Unhiding rules:
 Empty
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U45(p(x),y) -> U45(x,y)
 U45(s(x),y) -> U45(x,y)
 U45(x,p(y)) -> U45(x,y)
 U45(x,s(y)) -> U45(x,y)
->->-> Rules:
 u43(x,p(y)) -> p(u43(x,y))
 u43(x,s(y)) -> s(u43(x,y))
 u43(x,num0) -> x
 u45(p(x),y) -> p(u45(x,y))
 u45(s(x),y) -> s(u45(x,y))
 u45(num0,num0) -> num0
 u45(x,p(y)) -> s(u45(x,y))
 u45(x,s(y)) -> p(u45(x,y))
 p(s(x)) -> x
 s(p(x)) -> x
->->-> Unhiding rules:
 Empty
->->Cycle:
->->-> Pairs:
 U43(x,p(y)) -> U43(x,y)
 U43(x,s(y)) -> U43(x,y)
->->-> Rules:
 u43(x,p(y)) -> p(u43(x,y))
 u43(x,s(y)) -> s(u43(x,y))
 u43(x,num0) -> x
 u45(p(x),y) -> p(u45(x,y))
 u45(s(x),y) -> s(u45(x,y))
 u45(num0,num0) -> num0
 u45(x,p(y)) -> s(u45(x,y))
 u45(x,s(y)) -> p(u45(x,y))
 p(s(x)) -> x
 s(p(x)) -> x
->->-> Unhiding rules:
 Empty


The problem is decomposed in 2 subproblems.

Problem 1.1: 

SubNColl Processor:
-> Pairs:
 U45(p(x),y) -> U45(x,y)
 U45(s(x),y) -> U45(x,y)
 U45(x,p(y)) -> U45(x,y)
 U45(x,s(y)) -> U45(x,y)
-> Rules:
 u43(x,p(y)) -> p(u43(x,y))
 u43(x,s(y)) -> s(u43(x,y))
 u43(x,num0) -> x
 u45(p(x),y) -> p(u45(x,y))
 u45(s(x),y) -> s(u45(x,y))
 u45(num0,num0) -> num0
 u45(x,p(y)) -> s(u45(x,y))
 u45(x,s(y)) -> p(u45(x,y))
 p(s(x)) -> x
 s(p(x)) -> x
-> Unhiding rules:
 Empty
->Projection:
 pi(U45) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 U45(x,p(y)) -> U45(x,y)
 U45(x,s(y)) -> U45(x,y)
-> Rules:
 u43(x,p(y)) -> p(u43(x,y))
 u43(x,s(y)) -> s(u43(x,y))
 u43(x,num0) -> x
 u45(p(x),y) -> p(u45(x,y))
 u45(s(x),y) -> s(u45(x,y))
 u45(num0,num0) -> num0
 u45(x,p(y)) -> s(u45(x,y))
 u45(x,s(y)) -> p(u45(x,y))
 p(s(x)) -> x
 s(p(x)) -> x
-> Unhiding rules:
 Empty
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U45(x,p(y)) -> U45(x,y)
 U45(x,s(y)) -> U45(x,y)
->->-> Rules:
 u43(x,p(y)) -> p(u43(x,y))
 u43(x,s(y)) -> s(u43(x,y))
 u43(x,num0) -> x
 u45(p(x),y) -> p(u45(x,y))
 u45(s(x),y) -> s(u45(x,y))
 u45(num0,num0) -> num0
 u45(x,p(y)) -> s(u45(x,y))
 u45(x,s(y)) -> p(u45(x,y))
 p(s(x)) -> x
 s(p(x)) -> x
->->-> Unhiding rules:
 Empty

Problem 1.1: 

SubNColl Processor:
-> Pairs:
 U45(x,p(y)) -> U45(x,y)
 U45(x,s(y)) -> U45(x,y)
-> Rules:
 u43(x,p(y)) -> p(u43(x,y))
 u43(x,s(y)) -> s(u43(x,y))
 u43(x,num0) -> x
 u45(p(x),y) -> p(u45(x,y))
 u45(s(x),y) -> s(u45(x,y))
 u45(num0,num0) -> num0
 u45(x,p(y)) -> s(u45(x,y))
 u45(x,s(y)) -> p(u45(x,y))
 p(s(x)) -> x
 s(p(x)) -> x
-> Unhiding rules:
 Empty
->Projection:
 pi(U45) = 2

Problem 1.1: 

Basic Processor:
-> Pairs:
 Empty
-> Rules:
 u43(x,p(y)) -> p(u43(x,y))
 u43(x,s(y)) -> s(u43(x,y))
 u43(x,num0) -> x
 u45(p(x),y) -> p(u45(x,y))
 u45(s(x),y) -> s(u45(x,y))
 u45(num0,num0) -> num0
 u45(x,p(y)) -> s(u45(x,y))
 u45(x,s(y)) -> p(u45(x,y))
 p(s(x)) -> x
 s(p(x)) -> x
-> Unhiding rules:
 Empty
-> Result:
 Set P is empty

The problem is finite.

Problem 1.2: 

SubNColl Processor:
-> Pairs:
 U43(x,p(y)) -> U43(x,y)
 U43(x,s(y)) -> U43(x,y)
-> Rules:
 u43(x,p(y)) -> p(u43(x,y))
 u43(x,s(y)) -> s(u43(x,y))
 u43(x,num0) -> x
 u45(p(x),y) -> p(u45(x,y))
 u45(s(x),y) -> s(u45(x,y))
 u45(num0,num0) -> num0
 u45(x,p(y)) -> s(u45(x,y))
 u45(x,s(y)) -> p(u45(x,y))
 p(s(x)) -> x
 s(p(x)) -> x
-> Unhiding rules:
 Empty
->Projection:
 pi(U43) = 2

Problem 1.2: 

Basic Processor:
-> Pairs:
 Empty
-> Rules:
 u43(x,p(y)) -> p(u43(x,y))
 u43(x,s(y)) -> s(u43(x,y))
 u43(x,num0) -> x
 u45(p(x),y) -> p(u45(x,y))
 u45(s(x),y) -> s(u45(x,y))
 u45(num0,num0) -> num0
 u45(x,p(y)) -> s(u45(x,y))
 u45(x,s(y)) -> p(u45(x,y))
 p(s(x)) -> x
 s(p(x)) -> x
-> Unhiding rules:
 Empty
-> Result:
 Set P is empty

The problem is finite.


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

Huet Levy NW Procedure:
-> Rules:
 +(x,p(y)) -> p(+(x,y))
 +(x,s(y)) -> s(+(x,y))
 +(x,0) -> x
 -(p(x),y) -> p(-(x,y))
 -(s(x),y) -> s(-(x,y))
 -(0,0) -> 0
 -(x,p(y)) -> s(-(x,y))
 -(x,s(y)) -> p(-(x,y))
 p(s(x)) -> x
 s(p(x)) -> x
-> Vars:
 x, y, x, y, x, x, y, x, y, x, y, x, y, x, x
-> UVars:
 (UV-RuleId: 1, UV-LActive: [x, y], UV-RActive: [x, y], UV-LFrozen: [], UV-RFrozen: [])
 (UV-RuleId: 2, UV-LActive: [x, y], UV-RActive: [x, y], UV-LFrozen: [], UV-RFrozen: [])
 (UV-RuleId: 3, UV-LActive: [x], UV-RActive: [x], UV-LFrozen: [], UV-RFrozen: [])
 (UV-RuleId: 4, UV-LActive: [x, y], UV-RActive: [x, y], UV-LFrozen: [], UV-RFrozen: [])
 (UV-RuleId: 5, UV-LActive: [x, y], UV-RActive: [x, y], UV-LFrozen: [], UV-RFrozen: [])
 (UV-RuleId: 6, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: [])
 (UV-RuleId: 7, UV-LActive: [x, y], UV-RActive: [x, y], UV-LFrozen: [], UV-RFrozen: [])
 (UV-RuleId: 8, UV-LActive: [x, y], UV-RActive: [x, y], UV-LFrozen: [], UV-RFrozen: [])
 (UV-RuleId: 9, UV-LActive: [x], UV-RActive: [x], UV-LFrozen: [], UV-RFrozen: [])
 (UV-RuleId: 10, UV-LActive: [x], UV-RActive: [x], UV-LFrozen: [], UV-RFrozen: [])

-> Rlps:
 (rule: +(x,p(y)) -> p(+(x,y)), id: 1, possubterms: +(x,p(y))->[], p(y)->[2])
 (rule: +(x,s(y)) -> s(+(x,y)), id: 2, possubterms: +(x,s(y))->[], s(y)->[2])
 (rule: +(x,0) -> x, id: 3, possubterms: +(x,0)->[], 0->[2])
 (rule: -(p(x),y) -> p(-(x,y)), id: 4, possubterms: -(p(x),y)->[], p(x)->[1])
 (rule: -(s(x),y) -> s(-(x,y)), id: 5, possubterms: -(s(x),y)->[], s(x)->[1])
 (rule: -(0,0) -> 0, id: 6, possubterms: -(0,0)->[], 0->[1], 0->[2])
 (rule: -(x,p(y)) -> s(-(x,y)), id: 7, possubterms: -(x,p(y))->[], p(y)->[2])
 (rule: -(x,s(y)) -> p(-(x,y)), id: 8, possubterms: -(x,s(y))->[], s(y)->[2])
 (rule: p(s(x)) -> x, id: 9, possubterms: p(s(x))->[], s(x)->[1])
 (rule: s(p(x)) -> x, id: 10, possubterms: s(p(x))->[], p(x)->[1])

-> Unifications:
(R1 unifies with R9 at p: [2], l: +(x,p(y)), lp: p(y), sig: {y -> s(x')}, l': p(s(x')), r: p(+(x,y)), r': x')
(R2 unifies with R10 at p: [2], l: +(x,s(y)), lp: s(y), sig: {y -> p(x')}, l': s(p(x')), r: s(+(x,y)), r': x')
(R4 unifies with R9 at p: [1], l: -(p(x),y), lp: p(x), sig: {x -> s(x')}, l': p(s(x')), r: p(-(x,y)), r': x')
(R5 unifies with R10 at p: [1], l: -(s(x),y), lp: s(x), sig: {x -> p(x')}, l': s(p(x')), r: s(-(x,y)), r': x')
(R7 unifies with R4 at p: [], l: -(x,p(y)), lp: -(x,p(y)), sig: {x -> p(x'),y' -> p(y)}, l': -(p(x'),y'), r: s(-(x,y)), r': p(-(x',y')))
(R7 unifies with R5 at p: [], l: -(x,p(y)), lp: -(x,p(y)), sig: {x -> s(x'),y' -> p(y)}, l': -(s(x'),y'), r: s(-(x,y)), r': s(-(x',y')))
(R7 unifies with R9 at p: [2], l: -(x,p(y)), lp: p(y), sig: {y -> s(x')}, l': p(s(x')), r: s(-(x,y)), r': x')
(R8 unifies with R4 at p: [], l: -(x,s(y)), lp: -(x,s(y)), sig: {x -> p(x'),y' -> s(y)}, l': -(p(x'),y'), r: p(-(x,y)), r': p(-(x',y')))
(R8 unifies with R5 at p: [], l: -(x,s(y)), lp: -(x,s(y)), sig: {x -> s(x'),y' -> s(y)}, l': -(s(x'),y'), r: p(-(x,y)), r': s(-(x',y')))
(R8 unifies with R10 at p: [2], l: -(x,s(y)), lp: s(y), sig: {y -> p(x')}, l': s(p(x')), r: p(-(x,y)), r': x')
(R9 unifies with R10 at p: [1], l: p(s(x)), lp: s(x), sig: {x -> p(x')}, l': s(p(x')), r: x, r': x')
(R10 unifies with R9 at p: [1], l: s(p(x)), lp: p(x), sig: {x -> s(x')}, l': p(s(x')), r: x, r': x')

-> Critical pairs info:
<+(x,x'),p(+(x,s(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N1
<s(-(x',p(y))),s(-(s(x'),y))>   =>  Not trivial, Overlay, Proper, NW0, N2
<-(x,x'),s(-(x,s(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N3
<p(-(x',p(y))),s(-(p(x'),y))>   =>  Not trivial, Overlay, Proper, NW0, N4
<-(x',y),s(-(p(x'),y))>   =>  Not trivial, Not overlay, Proper, NW0, N5
<+(x,x'),s(+(x,p(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N6
<-(x,x'),p(-(x,p(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N7
<-(x',y),p(-(s(x'),y))>   =>  Not trivial, Not overlay, Proper, NW0, N8
<p(-(x',s(y))),p(-(p(x'),y))>   =>  Not trivial, Overlay, Proper, NW0, N9
<p(x'),p(x')>   =>  Trivial, Not overlay, Proper, NW0, N10
<s(x'),s(x')>   =>  Trivial, Not overlay, Proper, NW0, N11
<s(-(x',s(y))),p(-(s(x'),y))>   =>  Not trivial, Overlay, Proper, NW0, N12

-> 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



The problem is decomposed in 10 subproblems.

Problem 1.1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x y x')
(REPLACEMENT-MAP
(+ 2)
(- 1, 2)
(p 1)
(s 1)
(0)
(fSNonEmpty)
)
(RULES
+(x,p(y)) -> p(+(x,y))
+(x,s(y)) -> s(+(x,y))
+(x,0) -> x
-(p(x),y) -> p(-(x,y))
-(s(x),y) -> s(-(x,y))
-(0,0) -> 0
-(x,p(y)) -> s(-(x,y))
-(x,s(y)) -> p(-(x,y))
p(s(x)) -> x
s(p(x)) -> x
)
Critical Pairs:
<+(x,x'),p(+(x,s(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N1
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: +(x,x')
Nodes: [0]
Edges: []
ID: 0 => ('+(x,x')', D0)
t: p(+(x,s(x')))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('p(+(x,s(x')))', D0)
ID: 1 => ('p(s(+(x,x')))', D1, R2, P[1], S{x7 -> x, x8 -> x'}), NR: 's(+(x,x'))'
ID: 2 => ('+(x,x')', D2, R9, P[], S{x18 -> +(x,x')}), NR: '+(x,x')'
SNodesPath: +(x,x')
TNodesPath: p(+(x,s(x'))) -> p(s(+(x,x'))) -> +(x,x')
+(x,x') ->* +(x,x') *<- p(+(x,s(x')))
"Joinable"

The problem is confluent.

Problem 1.2: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x y x')
(REPLACEMENT-MAP
(+ 2)
(- 1, 2)
(p 1)
(s 1)
(0)
(fSNonEmpty)
)
(RULES
+(x,p(y)) -> p(+(x,y))
+(x,s(y)) -> s(+(x,y))
+(x,0) -> x
-(p(x),y) -> p(-(x,y))
-(s(x),y) -> s(-(x,y))
-(0,0) -> 0
-(x,p(y)) -> s(-(x,y))
-(x,s(y)) -> p(-(x,y))
p(s(x)) -> x
s(p(x)) -> x
)
Critical Pairs:
<s(-(x',p(y))),s(-(s(x'),y))>   =>  Not trivial, Overlay, Proper, NW0, N2
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: s(-(x',p(y)))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('s(-(x',p(y)))', D0)
ID: 1 => ('s(s(-(x',y)))', D1, R7, P[1], S{x14 -> x', x15 -> y}), NR: 's(-(x',y))'
t: s(-(s(x'),y))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('s(-(s(x'),y))', D0)
ID: 1 => ('s(s(-(x',y)))', D1, R5, P[1], S{x12 -> x', x13 -> y}), NR: 's(-(x',y))'
SNodesPath: s(-(x',p(y))) -> s(s(-(x',y)))
TNodesPath: s(-(s(x'),y)) -> s(s(-(x',y)))
s(-(x',p(y))) ->* s(s(-(x',y))) *<- s(-(s(x'),y))
"Joinable"

The problem is confluent.

Problem 1.3: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x y x')
(REPLACEMENT-MAP
(+ 2)
(- 1, 2)
(p 1)
(s 1)
(0)
(fSNonEmpty)
)
(RULES
+(x,p(y)) -> p(+(x,y))
+(x,s(y)) -> s(+(x,y))
+(x,0) -> x
-(p(x),y) -> p(-(x,y))
-(s(x),y) -> s(-(x,y))
-(0,0) -> 0
-(x,p(y)) -> s(-(x,y))
-(x,s(y)) -> p(-(x,y))
p(s(x)) -> x
s(p(x)) -> x
)
Critical Pairs:
<-(x,x'),s(-(x,s(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N3
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: -(x,x')
Nodes: [0]
Edges: []
ID: 0 => ('-(x,x')', D0)
t: s(-(x,s(x')))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('s(-(x,s(x')))', D0)
ID: 1 => ('s(p(-(x,x')))', D1, R8, P[1], S{x16 -> x, x17 -> x'}), NR: 'p(-(x,x'))'
ID: 2 => ('-(x,x')', D2, R10, P[], S{x19 -> -(x,x')}), NR: '-(x,x')'
SNodesPath: -(x,x')
TNodesPath: s(-(x,s(x'))) -> s(p(-(x,x'))) -> -(x,x')
-(x,x') ->* -(x,x') *<- s(-(x,s(x')))
"Joinable"

The problem is confluent.

Problem 1.4: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x y x')
(REPLACEMENT-MAP
(+ 2)
(- 1, 2)
(p 1)
(s 1)
(0)
(fSNonEmpty)
)
(RULES
+(x,p(y)) -> p(+(x,y))
+(x,s(y)) -> s(+(x,y))
+(x,0) -> x
-(p(x),y) -> p(-(x,y))
-(s(x),y) -> s(-(x,y))
-(0,0) -> 0
-(x,p(y)) -> s(-(x,y))
-(x,s(y)) -> p(-(x,y))
p(s(x)) -> x
s(p(x)) -> x
)
Critical Pairs:
<p(-(x',p(y))),s(-(p(x'),y))>   =>  Not trivial, Overlay, Proper, NW0, N4
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: p(-(x',p(y)))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('p(-(x',p(y)))', D0)
ID: 1 => ('p(s(-(x',y)))', D1, R7, P[1], S{x14 -> x', x15 -> y}), NR: 's(-(x',y))'
ID: 2 => ('-(x',y)', D2, R9, P[], S{x18 -> -(x',y)}), NR: '-(x',y)'
t: s(-(p(x'),y))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('s(-(p(x'),y))', D0)
ID: 1 => ('s(p(-(x',y)))', D1, R4, P[1], S{x10 -> x', x11 -> y}), NR: 'p(-(x',y))'
ID: 2 => ('-(x',y)', D2, R10, P[], S{x19 -> -(x',y)}), NR: '-(x',y)'
SNodesPath: p(-(x',p(y))) -> p(s(-(x',y))) -> -(x',y)
TNodesPath: s(-(p(x'),y)) -> s(p(-(x',y))) -> -(x',y)
p(-(x',p(y))) ->* -(x',y) *<- s(-(p(x'),y))
"Joinable"

The problem is confluent.

Problem 1.5: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x y x')
(REPLACEMENT-MAP
(+ 2)
(- 1, 2)
(p 1)
(s 1)
(0)
(fSNonEmpty)
)
(RULES
+(x,p(y)) -> p(+(x,y))
+(x,s(y)) -> s(+(x,y))
+(x,0) -> x
-(p(x),y) -> p(-(x,y))
-(s(x),y) -> s(-(x,y))
-(0,0) -> 0
-(x,p(y)) -> s(-(x,y))
-(x,s(y)) -> p(-(x,y))
p(s(x)) -> x
s(p(x)) -> x
)
Critical Pairs:
<-(x',y),s(-(p(x'),y))>   =>  Not trivial, Not overlay, Proper, NW0, N5
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: -(x',y)
Nodes: [0]
Edges: []
ID: 0 => ('-(x',y)', D0)
t: s(-(p(x'),y))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('s(-(p(x'),y))', D0)
ID: 1 => ('s(p(-(x',y)))', D1, R4, P[1], S{x10 -> x', x11 -> y}), NR: 'p(-(x',y))'
ID: 2 => ('-(x',y)', D2, R10, P[], S{x19 -> -(x',y)}), NR: '-(x',y)'
SNodesPath: -(x',y)
TNodesPath: s(-(p(x'),y)) -> s(p(-(x',y))) -> -(x',y)
-(x',y) ->* -(x',y) *<- s(-(p(x'),y))
"Joinable"

The problem is confluent.

Problem 1.6: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x y x')
(REPLACEMENT-MAP
(+ 2)
(- 1, 2)
(p 1)
(s 1)
(0)
(fSNonEmpty)
)
(RULES
+(x,p(y)) -> p(+(x,y))
+(x,s(y)) -> s(+(x,y))
+(x,0) -> x
-(p(x),y) -> p(-(x,y))
-(s(x),y) -> s(-(x,y))
-(0,0) -> 0
-(x,p(y)) -> s(-(x,y))
-(x,s(y)) -> p(-(x,y))
p(s(x)) -> x
s(p(x)) -> x
)
Critical Pairs:
<+(x,x'),s(+(x,p(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N6
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: +(x,x')
Nodes: [0]
Edges: []
ID: 0 => ('+(x,x')', D0)
t: s(+(x,p(x')))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('s(+(x,p(x')))', D0)
ID: 1 => ('s(p(+(x,x')))', D1, R1, P[1], S{x5 -> x, x6 -> x'}), NR: 'p(+(x,x'))'
ID: 2 => ('+(x,x')', D2, R10, P[], S{x19 -> +(x,x')}), NR: '+(x,x')'
SNodesPath: +(x,x')
TNodesPath: s(+(x,p(x'))) -> s(p(+(x,x'))) -> +(x,x')
+(x,x') ->* +(x,x') *<- s(+(x,p(x')))
"Joinable"

The problem is confluent.

Problem 1.7: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x y x')
(REPLACEMENT-MAP
(+ 2)
(- 1, 2)
(p 1)
(s 1)
(0)
(fSNonEmpty)
)
(RULES
+(x,p(y)) -> p(+(x,y))
+(x,s(y)) -> s(+(x,y))
+(x,0) -> x
-(p(x),y) -> p(-(x,y))
-(s(x),y) -> s(-(x,y))
-(0,0) -> 0
-(x,p(y)) -> s(-(x,y))
-(x,s(y)) -> p(-(x,y))
p(s(x)) -> x
s(p(x)) -> x
)
Critical Pairs:
<-(x,x'),p(-(x,p(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N7
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: -(x,x')
Nodes: [0]
Edges: []
ID: 0 => ('-(x,x')', D0)
t: p(-(x,p(x')))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('p(-(x,p(x')))', D0)
ID: 1 => ('p(s(-(x,x')))', D1, R7, P[1], S{x14 -> x, x15 -> x'}), NR: 's(-(x,x'))'
ID: 2 => ('-(x,x')', D2, R9, P[], S{x18 -> -(x,x')}), NR: '-(x,x')'
SNodesPath: -(x,x')
TNodesPath: p(-(x,p(x'))) -> p(s(-(x,x'))) -> -(x,x')
-(x,x') ->* -(x,x') *<- p(-(x,p(x')))
"Joinable"

The problem is confluent.

Problem 1.8: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x y x')
(REPLACEMENT-MAP
(+ 2)
(- 1, 2)
(p 1)
(s 1)
(0)
(fSNonEmpty)
)
(RULES
+(x,p(y)) -> p(+(x,y))
+(x,s(y)) -> s(+(x,y))
+(x,0) -> x
-(p(x),y) -> p(-(x,y))
-(s(x),y) -> s(-(x,y))
-(0,0) -> 0
-(x,p(y)) -> s(-(x,y))
-(x,s(y)) -> p(-(x,y))
p(s(x)) -> x
s(p(x)) -> x
)
Critical Pairs:
<-(x',y),p(-(s(x'),y))>   =>  Not trivial, Not overlay, Proper, NW0, N8
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: -(x',y)
Nodes: [0]
Edges: []
ID: 0 => ('-(x',y)', D0)
t: p(-(s(x'),y))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('p(-(s(x'),y))', D0)
ID: 1 => ('p(s(-(x',y)))', D1, R5, P[1], S{x12 -> x', x13 -> y}), NR: 's(-(x',y))'
ID: 2 => ('-(x',y)', D2, R9, P[], S{x18 -> -(x',y)}), NR: '-(x',y)'
SNodesPath: -(x',y)
TNodesPath: p(-(s(x'),y)) -> p(s(-(x',y))) -> -(x',y)
-(x',y) ->* -(x',y) *<- p(-(s(x'),y))
"Joinable"

The problem is confluent.

Problem 1.9: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x y x')
(REPLACEMENT-MAP
(+ 2)
(- 1, 2)
(p 1)
(s 1)
(0)
(fSNonEmpty)
)
(RULES
+(x,p(y)) -> p(+(x,y))
+(x,s(y)) -> s(+(x,y))
+(x,0) -> x
-(p(x),y) -> p(-(x,y))
-(s(x),y) -> s(-(x,y))
-(0,0) -> 0
-(x,p(y)) -> s(-(x,y))
-(x,s(y)) -> p(-(x,y))
p(s(x)) -> x
s(p(x)) -> x
)
Critical Pairs:
<p(-(x',s(y))),p(-(p(x'),y))>   =>  Not trivial, Overlay, Proper, NW0, N9
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: p(-(x',s(y)))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('p(-(x',s(y)))', D0)
ID: 1 => ('p(p(-(x',y)))', D1, R8, P[1], S{x16 -> x', x17 -> y}), NR: 'p(-(x',y))'
t: p(-(p(x'),y))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('p(-(p(x'),y))', D0)
ID: 1 => ('p(p(-(x',y)))', D1, R4, P[1], S{x10 -> x', x11 -> y}), NR: 'p(-(x',y))'
SNodesPath: p(-(x',s(y))) -> p(p(-(x',y)))
TNodesPath: p(-(p(x'),y)) -> p(p(-(x',y)))
p(-(x',s(y))) ->* p(p(-(x',y))) *<- p(-(p(x'),y))
"Joinable"

The problem is confluent.

Problem 1.10: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x y x')
(REPLACEMENT-MAP
(+ 2)
(- 1, 2)
(p 1)
(s 1)
(0)
(fSNonEmpty)
)
(RULES
+(x,p(y)) -> p(+(x,y))
+(x,s(y)) -> s(+(x,y))
+(x,0) -> x
-(p(x),y) -> p(-(x,y))
-(s(x),y) -> s(-(x,y))
-(0,0) -> 0
-(x,p(y)) -> s(-(x,y))
-(x,s(y)) -> p(-(x,y))
p(s(x)) -> x
s(p(x)) -> x
)
Critical Pairs:
<s(-(x',s(y))),p(-(s(x'),y))>   =>  Not trivial, Overlay, Proper, NW0, N12
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: s(-(x',s(y)))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('s(-(x',s(y)))', D0)
ID: 1 => ('s(p(-(x',y)))', D1, R8, P[1], S{x16 -> x', x17 -> y}), NR: 'p(-(x',y))'
ID: 2 => ('-(x',y)', D2, R10, P[], S{x19 -> -(x',y)}), NR: '-(x',y)'
t: p(-(s(x'),y))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('p(-(s(x'),y))', D0)
ID: 1 => ('p(s(-(x',y)))', D1, R5, P[1], S{x12 -> x', x13 -> y}), NR: 's(-(x',y))'
ID: 2 => ('-(x',y)', D2, R9, P[], S{x18 -> -(x',y)}), NR: '-(x',y)'
SNodesPath: s(-(x',s(y))) -> s(p(-(x',y))) -> -(x',y)
TNodesPath: p(-(s(x'),y)) -> p(s(-(x',y))) -> -(x',y)
s(-(x',s(y))) ->* -(x',y) *<- p(-(s(x'),y))
"Joinable"

The problem is confluent.