YES

Problem 1: 

(THEORY 
(AC ac1 ac2))
(RULES
ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
)

Problem 1: 

Dependency Pairs Processor:
-> FAxioms:
 AC1(ac1(x0,x1),x2) = AC1(x0,ac1(x1,x2))
 AC1(x0,x1) = AC1(x1,x0)
 AC2(ac2(x0,x1),x2) = AC2(x0,ac2(x1,x2))
 AC2(x0,x1) = AC2(x1,x0)
-> Pairs:
 AC1(ac1(a,ac2(b,c)),x0) -> AC1(ac1(b,f(ac2(a,c))),x0)
 AC1(ac1(a,ac2(b,c)),x0) -> AC1(b,f(ac2(a,c)))
 AC1(ac1(a,ac2(b,c)),x0) -> AC2(a,c)
 AC1(a,ac2(b,c)) -> AC1(b,f(ac2(a,c)))
 AC1(a,ac2(b,c)) -> AC2(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC1(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(ac2(b,f(ac1(a,c))),x0)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC1(a,c)
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC1(ac1(x0,x1),x2) -> AC1(x0,x1)
 AC1(x0,ac1(x1,x2)) -> AC1(x1,x2)
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)

Problem 1: 

SCC Processor:
-> FAxioms:
 AC1(ac1(x0,x1),x2) = AC1(x0,ac1(x1,x2))
 AC1(x0,x1) = AC1(x1,x0)
 AC2(ac2(x0,x1),x2) = AC2(x0,ac2(x1,x2))
 AC2(x0,x1) = AC2(x1,x0)
-> Pairs:
 AC1(ac1(a,ac2(b,c)),x0) -> AC1(ac1(b,f(ac2(a,c))),x0)
 AC1(ac1(a,ac2(b,c)),x0) -> AC1(b,f(ac2(a,c)))
 AC1(ac1(a,ac2(b,c)),x0) -> AC2(a,c)
 AC1(a,ac2(b,c)) -> AC1(b,f(ac2(a,c)))
 AC1(a,ac2(b,c)) -> AC2(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC1(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(ac2(b,f(ac1(a,c))),x0)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC1(a,c)
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC1(ac1(x0,x1),x2) -> AC1(x0,x1)
 AC1(x0,ac1(x1,x2)) -> AC1(x1,x2)
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 AC1(ac1(a,ac2(b,c)),x0) -> AC1(ac1(b,f(ac2(a,c))),x0)
 AC1(ac1(a,ac2(b,c)),x0) -> AC1(b,f(ac2(a,c)))
 AC1(ac1(a,ac2(b,c)),x0) -> AC2(a,c)
 AC1(a,ac2(b,c)) -> AC1(b,f(ac2(a,c)))
 AC1(a,ac2(b,c)) -> AC2(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC1(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(ac2(b,f(ac1(a,c))),x0)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC1(a,c)
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> FAxioms:
 ac1(ac1(x0,x1),x2) -> ac1(x0,ac1(x1,x2))
 ac1(x0,x1) -> ac1(x1,x0)
 ac2(ac2(x0,x1),x2) -> ac2(x0,ac2(x1,x2))
 ac2(x0,x1) -> ac2(x1,x0)
 AC1(ac1(x0,x1),x2) -> AC1(x0,ac1(x1,x2))
 AC1(x0,x1) -> AC1(x1,x0)
 AC2(ac2(x0,x1),x2) -> AC2(x0,ac2(x1,x2))
 AC2(x0,x1) -> AC2(x1,x0)
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
->->-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC1(ac1(x0,x1),x2) -> AC1(x0,x1)
 AC1(x0,ac1(x1,x2)) -> AC1(x1,x2)
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 AC1(ac1(x0,x1),x2) = AC1(x0,ac1(x1,x2))
 AC1(x0,x1) = AC1(x1,x0)
 AC2(ac2(x0,x1),x2) = AC2(x0,ac2(x1,x2))
 AC2(x0,x1) = AC2(x1,x0)
-> Pairs:
 AC1(ac1(a,ac2(b,c)),x0) -> AC1(ac1(b,f(ac2(a,c))),x0)
 AC1(ac1(a,ac2(b,c)),x0) -> AC1(b,f(ac2(a,c)))
 AC1(ac1(a,ac2(b,c)),x0) -> AC2(a,c)
 AC1(a,ac2(b,c)) -> AC1(b,f(ac2(a,c)))
 AC1(a,ac2(b,c)) -> AC2(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC1(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(ac2(b,f(ac1(a,c))),x0)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC1(a,c)
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Usable Equations:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> Usable Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC1(ac1(x0,x1),x2) -> AC1(x0,x1)
 AC1(x0,ac1(x1,x2)) -> AC1(x1,x2)
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[ac1](X1,X2) = X1 + X2 + 1
[ac2](X1,X2) = X1 + X2 + 2
[a] = 0
[b] = 2
[c] = 2
[f](X) = 2
[AC1](X1,X2) = X1 + X2 + 1
[AC2](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1: 

SCC Processor:
-> FAxioms:
 AC1(ac1(x0,x1),x2) = AC1(x0,ac1(x1,x2))
 AC1(x0,x1) = AC1(x1,x0)
 AC2(ac2(x0,x1),x2) = AC2(x0,ac2(x1,x2))
 AC2(x0,x1) = AC2(x1,x0)
-> Pairs:
 AC1(ac1(a,ac2(b,c)),x0) -> AC1(b,f(ac2(a,c)))
 AC1(ac1(a,ac2(b,c)),x0) -> AC2(a,c)
 AC1(a,ac2(b,c)) -> AC1(b,f(ac2(a,c)))
 AC1(a,ac2(b,c)) -> AC2(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC1(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(ac2(b,f(ac1(a,c))),x0)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC1(a,c)
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC1(ac1(x0,x1),x2) -> AC1(x0,x1)
 AC1(x0,ac1(x1,x2)) -> AC1(x1,x2)
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 AC1(ac1(a,ac2(b,c)),x0) -> AC1(b,f(ac2(a,c)))
 AC1(ac1(a,ac2(b,c)),x0) -> AC2(a,c)
 AC1(a,ac2(b,c)) -> AC1(b,f(ac2(a,c)))
 AC1(a,ac2(b,c)) -> AC2(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC1(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(ac2(b,f(ac1(a,c))),x0)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC1(a,c)
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> FAxioms:
 ac1(ac1(x0,x1),x2) -> ac1(x0,ac1(x1,x2))
 ac1(x0,x1) -> ac1(x1,x0)
 ac2(ac2(x0,x1),x2) -> ac2(x0,ac2(x1,x2))
 ac2(x0,x1) -> ac2(x1,x0)
 AC1(ac1(x0,x1),x2) -> AC1(x0,ac1(x1,x2))
 AC1(x0,x1) -> AC1(x1,x0)
 AC2(ac2(x0,x1),x2) -> AC2(x0,ac2(x1,x2))
 AC2(x0,x1) -> AC2(x1,x0)
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
->->-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC1(ac1(x0,x1),x2) -> AC1(x0,x1)
 AC1(x0,ac1(x1,x2)) -> AC1(x1,x2)
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 AC1(ac1(x0,x1),x2) = AC1(x0,ac1(x1,x2))
 AC1(x0,x1) = AC1(x1,x0)
 AC2(ac2(x0,x1),x2) = AC2(x0,ac2(x1,x2))
 AC2(x0,x1) = AC2(x1,x0)
-> Pairs:
 AC1(ac1(a,ac2(b,c)),x0) -> AC1(b,f(ac2(a,c)))
 AC1(ac1(a,ac2(b,c)),x0) -> AC2(a,c)
 AC1(a,ac2(b,c)) -> AC1(b,f(ac2(a,c)))
 AC1(a,ac2(b,c)) -> AC2(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC1(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(ac2(b,f(ac1(a,c))),x0)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC1(a,c)
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Usable Equations:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> Usable Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC1(ac1(x0,x1),x2) -> AC1(x0,x1)
 AC1(x0,ac1(x1,x2)) -> AC1(x1,x2)
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[ac1](X1,X2) = X1 + X2 + 2
[ac2](X1,X2) = X1 + X2 + 2
[a] = 2
[b] = 2
[c] = 1
[f](X) = 2
[AC1](X1,X2) = X1 + X2 + 2
[AC2](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1: 

SCC Processor:
-> FAxioms:
 AC1(ac1(x0,x1),x2) = AC1(x0,ac1(x1,x2))
 AC1(x0,x1) = AC1(x1,x0)
 AC2(ac2(x0,x1),x2) = AC2(x0,ac2(x1,x2))
 AC2(x0,x1) = AC2(x1,x0)
-> Pairs:
 AC1(ac1(a,ac2(b,c)),x0) -> AC2(a,c)
 AC1(a,ac2(b,c)) -> AC1(b,f(ac2(a,c)))
 AC1(a,ac2(b,c)) -> AC2(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC1(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(ac2(b,f(ac1(a,c))),x0)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC1(a,c)
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC1(ac1(x0,x1),x2) -> AC1(x0,x1)
 AC1(x0,ac1(x1,x2)) -> AC1(x1,x2)
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 AC1(ac1(a,ac2(b,c)),x0) -> AC2(a,c)
 AC1(a,ac2(b,c)) -> AC1(b,f(ac2(a,c)))
 AC1(a,ac2(b,c)) -> AC2(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC1(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(ac2(b,f(ac1(a,c))),x0)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC1(a,c)
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> FAxioms:
 ac1(ac1(x0,x1),x2) -> ac1(x0,ac1(x1,x2))
 ac1(x0,x1) -> ac1(x1,x0)
 ac2(ac2(x0,x1),x2) -> ac2(x0,ac2(x1,x2))
 ac2(x0,x1) -> ac2(x1,x0)
 AC1(ac1(x0,x1),x2) -> AC1(x0,ac1(x1,x2))
 AC1(x0,x1) -> AC1(x1,x0)
 AC2(ac2(x0,x1),x2) -> AC2(x0,ac2(x1,x2))
 AC2(x0,x1) -> AC2(x1,x0)
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
->->-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC1(ac1(x0,x1),x2) -> AC1(x0,x1)
 AC1(x0,ac1(x1,x2)) -> AC1(x1,x2)
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 AC1(ac1(x0,x1),x2) = AC1(x0,ac1(x1,x2))
 AC1(x0,x1) = AC1(x1,x0)
 AC2(ac2(x0,x1),x2) = AC2(x0,ac2(x1,x2))
 AC2(x0,x1) = AC2(x1,x0)
-> Pairs:
 AC1(ac1(a,ac2(b,c)),x0) -> AC2(a,c)
 AC1(a,ac2(b,c)) -> AC1(b,f(ac2(a,c)))
 AC1(a,ac2(b,c)) -> AC2(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC1(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(ac2(b,f(ac1(a,c))),x0)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC1(a,c)
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Usable Equations:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> Usable Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC1(ac1(x0,x1),x2) -> AC1(x0,x1)
 AC1(x0,ac1(x1,x2)) -> AC1(x1,x2)
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[ac1](X1,X2) = X1 + X2 + 2
[ac2](X1,X2) = X1 + X2 + 2
[a] = 0
[b] = 2
[c] = 0
[f](X) = 2
[AC1](X1,X2) = 1
[AC2](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> FAxioms:
 AC1(ac1(x0,x1),x2) = AC1(x0,ac1(x1,x2))
 AC1(x0,x1) = AC1(x1,x0)
 AC2(ac2(x0,x1),x2) = AC2(x0,ac2(x1,x2))
 AC2(x0,x1) = AC2(x1,x0)
-> Pairs:
 AC1(a,ac2(b,c)) -> AC1(b,f(ac2(a,c)))
 AC1(a,ac2(b,c)) -> AC2(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC1(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(ac2(b,f(ac1(a,c))),x0)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC1(a,c)
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC1(ac1(x0,x1),x2) -> AC1(x0,x1)
 AC1(x0,ac1(x1,x2)) -> AC1(x1,x2)
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 AC1(a,ac2(b,c)) -> AC1(b,f(ac2(a,c)))
 AC1(a,ac2(b,c)) -> AC2(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC1(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(ac2(b,f(ac1(a,c))),x0)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC1(a,c)
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> FAxioms:
 ac1(ac1(x0,x1),x2) -> ac1(x0,ac1(x1,x2))
 ac1(x0,x1) -> ac1(x1,x0)
 ac2(ac2(x0,x1),x2) -> ac2(x0,ac2(x1,x2))
 ac2(x0,x1) -> ac2(x1,x0)
 AC1(ac1(x0,x1),x2) -> AC1(x0,ac1(x1,x2))
 AC1(x0,x1) -> AC1(x1,x0)
 AC2(ac2(x0,x1),x2) -> AC2(x0,ac2(x1,x2))
 AC2(x0,x1) -> AC2(x1,x0)
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
->->-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC1(ac1(x0,x1),x2) -> AC1(x0,x1)
 AC1(x0,ac1(x1,x2)) -> AC1(x1,x2)
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 AC1(ac1(x0,x1),x2) = AC1(x0,ac1(x1,x2))
 AC1(x0,x1) = AC1(x1,x0)
 AC2(ac2(x0,x1),x2) = AC2(x0,ac2(x1,x2))
 AC2(x0,x1) = AC2(x1,x0)
-> Pairs:
 AC1(a,ac2(b,c)) -> AC1(b,f(ac2(a,c)))
 AC1(a,ac2(b,c)) -> AC2(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC1(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(ac2(b,f(ac1(a,c))),x0)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC1(a,c)
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Usable Equations:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> Usable Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC1(ac1(x0,x1),x2) -> AC1(x0,x1)
 AC1(x0,ac1(x1,x2)) -> AC1(x1,x2)
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[ac1](X1,X2) = X1 + X2 + 2
[ac2](X1,X2) = X1 + X2
[a] = 2
[b] = 2
[c] = 0
[f](X) = 1
[AC1](X1,X2) = 2.X1 + 2.X2 + 1
[AC2](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1: 

SCC Processor:
-> FAxioms:
 AC1(ac1(x0,x1),x2) = AC1(x0,ac1(x1,x2))
 AC1(x0,x1) = AC1(x1,x0)
 AC2(ac2(x0,x1),x2) = AC2(x0,ac2(x1,x2))
 AC2(x0,x1) = AC2(x1,x0)
-> Pairs:
 AC1(a,ac2(b,c)) -> AC1(b,f(ac2(a,c)))
 AC1(a,ac2(b,c)) -> AC2(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC1(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(ac2(b,f(ac1(a,c))),x0)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC1(a,c)
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC1(x0,ac1(x1,x2)) -> AC1(x1,x2)
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 AC1(a,ac2(b,c)) -> AC1(b,f(ac2(a,c)))
 AC1(a,ac2(b,c)) -> AC2(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC1(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(ac2(b,f(ac1(a,c))),x0)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC1(a,c)
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> FAxioms:
 ac1(ac1(x0,x1),x2) -> ac1(x0,ac1(x1,x2))
 ac1(x0,x1) -> ac1(x1,x0)
 ac2(ac2(x0,x1),x2) -> ac2(x0,ac2(x1,x2))
 ac2(x0,x1) -> ac2(x1,x0)
 AC1(ac1(x0,x1),x2) -> AC1(x0,ac1(x1,x2))
 AC1(x0,x1) -> AC1(x1,x0)
 AC2(ac2(x0,x1),x2) -> AC2(x0,ac2(x1,x2))
 AC2(x0,x1) -> AC2(x1,x0)
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
->->-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC1(x0,ac1(x1,x2)) -> AC1(x1,x2)
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 AC1(ac1(x0,x1),x2) = AC1(x0,ac1(x1,x2))
 AC1(x0,x1) = AC1(x1,x0)
 AC2(ac2(x0,x1),x2) = AC2(x0,ac2(x1,x2))
 AC2(x0,x1) = AC2(x1,x0)
-> Pairs:
 AC1(a,ac2(b,c)) -> AC1(b,f(ac2(a,c)))
 AC1(a,ac2(b,c)) -> AC2(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC1(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(ac2(b,f(ac1(a,c))),x0)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC1(a,c)
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Usable Equations:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> Usable Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC1(x0,ac1(x1,x2)) -> AC1(x1,x2)
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[ac1](X1,X2) = X1 + X2
[ac2](X1,X2) = X1 + X2 + 1
[a] = 2
[b] = 2
[c] = 2
[f](X) = 2
[AC1](X1,X2) = 2.X1 + 2.X2 + 1
[AC2](X1,X2) = 2.X1 + 2.X2 + 1

Problem 1: 

SCC Processor:
-> FAxioms:
 AC1(ac1(x0,x1),x2) = AC1(x0,ac1(x1,x2))
 AC1(x0,x1) = AC1(x1,x0)
 AC2(ac2(x0,x1),x2) = AC2(x0,ac2(x1,x2))
 AC2(x0,x1) = AC2(x1,x0)
-> Pairs:
 AC1(a,ac2(b,c)) -> AC2(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC1(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(ac2(b,f(ac1(a,c))),x0)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC1(a,c)
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC1(x0,ac1(x1,x2)) -> AC1(x1,x2)
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 AC1(a,ac2(b,c)) -> AC2(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC1(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(ac2(b,f(ac1(a,c))),x0)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC1(a,c)
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> FAxioms:
 ac1(ac1(x0,x1),x2) -> ac1(x0,ac1(x1,x2))
 ac1(x0,x1) -> ac1(x1,x0)
 ac2(ac2(x0,x1),x2) -> ac2(x0,ac2(x1,x2))
 ac2(x0,x1) -> ac2(x1,x0)
 AC1(ac1(x0,x1),x2) -> AC1(x0,ac1(x1,x2))
 AC1(x0,x1) -> AC1(x1,x0)
 AC2(ac2(x0,x1),x2) -> AC2(x0,ac2(x1,x2))
 AC2(x0,x1) -> AC2(x1,x0)
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
->->-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC1(x0,ac1(x1,x2)) -> AC1(x1,x2)
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 AC1(ac1(x0,x1),x2) = AC1(x0,ac1(x1,x2))
 AC1(x0,x1) = AC1(x1,x0)
 AC2(ac2(x0,x1),x2) = AC2(x0,ac2(x1,x2))
 AC2(x0,x1) = AC2(x1,x0)
-> Pairs:
 AC1(a,ac2(b,c)) -> AC2(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC1(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(ac2(b,f(ac1(a,c))),x0)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC1(a,c)
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Usable Equations:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> Usable Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC1(x0,ac1(x1,x2)) -> AC1(x1,x2)
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[ac1](X1,X2) = X1 + X2 + 2
[ac2](X1,X2) = X1 + X2 + 1
[a] = 2
[b] = 2
[c] = 1
[f](X) = 2
[AC1](X1,X2) = 2.X1 + 2.X2 + 1
[AC2](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1: 

SCC Processor:
-> FAxioms:
 AC1(ac1(x0,x1),x2) = AC1(x0,ac1(x1,x2))
 AC1(x0,x1) = AC1(x1,x0)
 AC2(ac2(x0,x1),x2) = AC2(x0,ac2(x1,x2))
 AC2(x0,x1) = AC2(x1,x0)
-> Pairs:
 AC2(ac2(a,ac1(b,c)),x0) -> AC1(a,c)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(ac2(b,f(ac1(a,c))),x0)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC1(a,c)
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC1(x0,ac1(x1,x2)) -> AC1(x1,x2)
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(ac2(b,f(ac1(a,c))),x0)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> FAxioms:
 ac1(ac1(x0,x1),x2) -> ac1(x0,ac1(x1,x2))
 ac1(x0,x1) -> ac1(x1,x0)
 ac2(ac2(x0,x1),x2) -> ac2(x0,ac2(x1,x2))
 ac2(x0,x1) -> ac2(x1,x0)
 AC2(ac2(x0,x1),x2) -> AC2(x0,ac2(x1,x2))
 AC2(x0,x1) -> AC2(x1,x0)
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
->->-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 AC2(ac2(x0,x1),x2) = AC2(x0,ac2(x1,x2))
 AC2(x0,x1) = AC2(x1,x0)
-> Pairs:
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(ac2(b,f(ac1(a,c))),x0)
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Usable Equations:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> Usable Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[ac1](X1,X2) = 0
[ac2](X1,X2) = X1 + X2 + 1
[a] = 2
[b] = 0
[c] = 0
[f](X) = 0
[AC1](X1,X2) = 0
[AC2](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> FAxioms:
 AC2(ac2(x0,x1),x2) = AC2(x0,ac2(x1,x2))
 AC2(x0,x1) = AC2(x1,x0)
-> Pairs:
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> FAxioms:
 ac1(ac1(x0,x1),x2) -> ac1(x0,ac1(x1,x2))
 ac1(x0,x1) -> ac1(x1,x0)
 ac2(ac2(x0,x1),x2) -> ac2(x0,ac2(x1,x2))
 ac2(x0,x1) -> ac2(x1,x0)
 AC2(ac2(x0,x1),x2) -> AC2(x0,ac2(x1,x2))
 AC2(x0,x1) -> AC2(x1,x0)
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
->->-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 AC2(ac2(x0,x1),x2) = AC2(x0,ac2(x1,x2))
 AC2(x0,x1) = AC2(x1,x0)
-> Pairs:
 AC2(ac2(a,ac1(b,c)),x0) -> AC2(b,f(ac1(a,c)))
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Usable Equations:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> Usable Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[ac1](X1,X2) = X1 + X2 + 2
[ac2](X1,X2) = X1 + X2 + 1
[a] = 0
[b] = 1
[c] = 2
[f](X) = 2
[AC1](X1,X2) = 0
[AC2](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> FAxioms:
 AC2(ac2(x0,x1),x2) = AC2(x0,ac2(x1,x2))
 AC2(x0,x1) = AC2(x1,x0)
-> Pairs:
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> FAxioms:
 ac1(ac1(x0,x1),x2) -> ac1(x0,ac1(x1,x2))
 ac1(x0,x1) -> ac1(x1,x0)
 ac2(ac2(x0,x1),x2) -> ac2(x0,ac2(x1,x2))
 ac2(x0,x1) -> ac2(x1,x0)
 AC2(ac2(x0,x1),x2) -> AC2(x0,ac2(x1,x2))
 AC2(x0,x1) -> AC2(x1,x0)
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
->->-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 AC2(ac2(x0,x1),x2) = AC2(x0,ac2(x1,x2))
 AC2(x0,x1) = AC2(x1,x0)
-> Pairs:
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Usable Equations:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> Usable Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC2(ac2(x0,x1),x2) -> AC2(x0,x1)
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[ac1](X1,X2) = 1
[ac2](X1,X2) = X1 + X2 + 2
[a] = 1
[b] = 0
[c] = 0
[f](X) = 2.X
[AC1](X1,X2) = 0
[AC2](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> FAxioms:
 AC2(ac2(x0,x1),x2) = AC2(x0,ac2(x1,x2))
 AC2(x0,x1) = AC2(x1,x0)
-> Pairs:
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> FAxioms:
 ac1(ac1(x0,x1),x2) -> ac1(x0,ac1(x1,x2))
 ac1(x0,x1) -> ac1(x1,x0)
 ac2(ac2(x0,x1),x2) -> ac2(x0,ac2(x1,x2))
 ac2(x0,x1) -> ac2(x1,x0)
 AC2(ac2(x0,x1),x2) -> AC2(x0,ac2(x1,x2))
 AC2(x0,x1) -> AC2(x1,x0)
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
->->-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 AC2(ac2(x0,x1),x2) = AC2(x0,ac2(x1,x2))
 AC2(x0,x1) = AC2(x1,x0)
-> Pairs:
 AC2(a,ac1(b,c)) -> AC2(b,f(ac1(a,c)))
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Usable Equations:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> Usable Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[ac1](X1,X2) = 0
[ac2](X1,X2) = X1 + X2 + 1
[a] = 2
[b] = 0
[c] = 0
[f](X) = 2.X + 1
[AC1](X1,X2) = 0
[AC2](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> FAxioms:
 AC2(ac2(x0,x1),x2) = AC2(x0,ac2(x1,x2))
 AC2(x0,x1) = AC2(x1,x0)
-> Pairs:
 Empty
-> EAxioms:
 ac1(ac1(x0,x1),x2) = ac1(x0,ac1(x1,x2))
 ac1(x0,x1) = ac1(x1,x0)
 ac2(ac2(x0,x1),x2) = ac2(x0,ac2(x1,x2))
 ac2(x0,x1) = ac2(x1,x0)
-> Rules:
 ac1(a,ac2(b,c)) -> ac1(b,f(ac2(a,c)))
 ac2(a,ac1(b,c)) -> ac2(b,f(ac1(a,c)))
-> SRules:
 AC2(x0,ac2(x1,x2)) -> AC2(x1,x2)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.