YES

Problem 1: 

(VAR x y)
(THEORY 
(AC plus))
(RULES
plus(0,y) -> y
plus(s(x),0) -> s(x)
plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
)

Problem 1: 

Dependency Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0,y),x2) -> PLUS(y,x2)
 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(x),plus(y,0))
 PLUS(plus(s(x),s(y)),x2) -> PLUS(y,0)
 PLUS(s(x),s(y)) -> PLUS(s(x),plus(y,0))
 PLUS(s(x),s(y)) -> PLUS(y,0)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0,y),x2) -> PLUS(y,x2)
 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(x),plus(y,0))
 PLUS(plus(s(x),s(y)),x2) -> PLUS(y,0)
 PLUS(s(x),s(y)) -> PLUS(s(x),plus(y,0))
 PLUS(s(x),s(y)) -> PLUS(y,0)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(0,y),x2) -> PLUS(y,x2)
 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(x),plus(y,0))
 PLUS(plus(s(x),s(y)),x2) -> PLUS(y,0)
 PLUS(s(x),s(y)) -> PLUS(s(x),plus(y,0))
 PLUS(s(x),s(y)) -> PLUS(y,0)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
->->-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0,y),x2) -> PLUS(y,x2)
 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(x),plus(y,0))
 PLUS(plus(s(x),s(y)),x2) -> PLUS(y,0)
 PLUS(s(x),s(y)) -> PLUS(s(x),plus(y,0))
 PLUS(s(x),s(y)) -> PLUS(y,0)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> Usable Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[0] = 0
[s](X) = X + 2
[PLUS](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0,y),x2) -> PLUS(y,x2)
 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(y,0)
 PLUS(s(x),s(y)) -> PLUS(s(x),plus(y,0))
 PLUS(s(x),s(y)) -> PLUS(y,0)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(0,y),x2) -> PLUS(y,x2)
 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(y,0)
 PLUS(s(x),s(y)) -> PLUS(s(x),plus(y,0))
 PLUS(s(x),s(y)) -> PLUS(y,0)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
->->-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0,y),x2) -> PLUS(y,x2)
 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(y,0)
 PLUS(s(x),s(y)) -> PLUS(s(x),plus(y,0))
 PLUS(s(x),s(y)) -> PLUS(y,0)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> Usable Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[0] = 0
[s](X) = X + 2
[PLUS](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0,y),x2) -> PLUS(y,x2)
 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2)
 PLUS(s(x),s(y)) -> PLUS(s(x),plus(y,0))
 PLUS(s(x),s(y)) -> PLUS(y,0)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(0,y),x2) -> PLUS(y,x2)
 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2)
 PLUS(s(x),s(y)) -> PLUS(s(x),plus(y,0))
 PLUS(s(x),s(y)) -> PLUS(y,0)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
->->-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0,y),x2) -> PLUS(y,x2)
 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2)
 PLUS(s(x),s(y)) -> PLUS(s(x),plus(y,0))
 PLUS(s(x),s(y)) -> PLUS(y,0)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> Usable Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[0] = 0
[s](X) = X + 2
[PLUS](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0,y),x2) -> PLUS(y,x2)
 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2)
 PLUS(s(x),s(y)) -> PLUS(y,0)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(0,y),x2) -> PLUS(y,x2)
 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2)
 PLUS(s(x),s(y)) -> PLUS(y,0)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
->->-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0,y),x2) -> PLUS(y,x2)
 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2)
 PLUS(s(x),s(y)) -> PLUS(y,0)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> Usable Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[0] = 0
[s](X) = X + 2
[PLUS](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0,y),x2) -> PLUS(y,x2)
 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(0,y),x2) -> PLUS(y,x2)
 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
->->-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0,y),x2) -> PLUS(y,x2)
 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> Usable Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[0] = 2
[s](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
->->-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2)
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> Usable Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2 + 2
[0] = 0
[s](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
->->-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> Usable Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[0] = 2
[s](X) = 2
[PLUS](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 Empty
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 plus(0,y) -> y
 plus(s(x),0) -> s(x)
 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.