YES

Problem 1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x)
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 

Problem 1: 
Not CS-TRS Procedure:
R is not a CS-TRS

Problem 1: 
Termination Procedure:
Terminating?
YES

Problem 1: 

(VAR vu95NonEmpty vNonEmpty x)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ASharp(A(A(A(x)))) -> aSharp(a(a(x)))
 ASharp(A(A(A(x)))) -> aSharp(a(x))
 ASharp(A(A(A(x)))) -> aSharp(x)
 ASharp(A(A(b(x)))) -> CSharp(x)
 ASharp(A(A(b(x)))) -> aSharp(C(x))
 ASharp(A(A(b(x)))) -> aSharp(a(C(x)))
 ASharp(A(A(b(x)))) -> aSharp(a(a(C(x))))
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

SCC Processor:
-> Pairs:
 ASharp(A(A(A(x)))) -> aSharp(a(a(x)))
 ASharp(A(A(A(x)))) -> aSharp(a(x))
 ASharp(A(A(A(x)))) -> aSharp(x)
 ASharp(A(A(b(x)))) -> CSharp(x)
 ASharp(A(A(b(x)))) -> aSharp(C(x))
 ASharp(A(A(b(x)))) -> aSharp(a(C(x)))
 ASharp(A(A(b(x)))) -> aSharp(a(a(C(x))))
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ASharp(A(A(A(x)))) -> aSharp(a(a(x)))
 ASharp(A(A(A(x)))) -> aSharp(a(x))
 ASharp(A(A(A(x)))) -> aSharp(x)
 ASharp(A(A(b(x)))) -> CSharp(x)
 ASharp(A(A(b(x)))) -> aSharp(C(x))
 ASharp(A(A(b(x)))) -> aSharp(a(C(x)))
 ASharp(A(A(b(x)))) -> aSharp(a(a(C(x))))
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ASharp(A(A(A(x)))) -> aSharp(a(a(x)))
 ASharp(A(A(A(x)))) -> aSharp(a(x))
 ASharp(A(A(A(x)))) -> aSharp(x)
 ASharp(A(A(b(x)))) -> CSharp(x)
 ASharp(A(A(b(x)))) -> aSharp(C(x))
 ASharp(A(A(b(x)))) -> aSharp(a(C(x)))
 ASharp(A(A(b(x)))) -> aSharp(a(a(C(x))))
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 2.X + 2
[BSharp](X) = 2.X + 2
[CSharp](X) = 2.X + 2
[aSharp](X) = X + 2
[bSharp](X) = 2.X
[cSharp](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ASharp(A(A(A(x)))) -> aSharp(a(x))
 ASharp(A(A(A(x)))) -> aSharp(x)
 ASharp(A(A(b(x)))) -> CSharp(x)
 ASharp(A(A(b(x)))) -> aSharp(C(x))
 ASharp(A(A(b(x)))) -> aSharp(a(C(x)))
 ASharp(A(A(b(x)))) -> aSharp(a(a(C(x))))
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ASharp(A(A(A(x)))) -> aSharp(a(x))
 ASharp(A(A(A(x)))) -> aSharp(x)
 ASharp(A(A(b(x)))) -> CSharp(x)
 ASharp(A(A(b(x)))) -> aSharp(C(x))
 ASharp(A(A(b(x)))) -> aSharp(a(C(x)))
 ASharp(A(A(b(x)))) -> aSharp(a(a(C(x))))
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ASharp(A(A(A(x)))) -> aSharp(a(x))
 ASharp(A(A(A(x)))) -> aSharp(x)
 ASharp(A(A(b(x)))) -> CSharp(x)
 ASharp(A(A(b(x)))) -> aSharp(C(x))
 ASharp(A(A(b(x)))) -> aSharp(a(C(x)))
 ASharp(A(A(b(x)))) -> aSharp(a(a(C(x))))
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 2.X + 2
[BSharp](X) = X + 2
[CSharp](X) = 2.X + 2
[aSharp](X) = 2.X + 2
[bSharp](X) = X + 2
[cSharp](X) = X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ASharp(A(A(A(x)))) -> aSharp(x)
 ASharp(A(A(b(x)))) -> CSharp(x)
 ASharp(A(A(b(x)))) -> aSharp(C(x))
 ASharp(A(A(b(x)))) -> aSharp(a(C(x)))
 ASharp(A(A(b(x)))) -> aSharp(a(a(C(x))))
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ASharp(A(A(A(x)))) -> aSharp(x)
 ASharp(A(A(b(x)))) -> CSharp(x)
 ASharp(A(A(b(x)))) -> aSharp(C(x))
 ASharp(A(A(b(x)))) -> aSharp(a(C(x)))
 ASharp(A(A(b(x)))) -> aSharp(a(a(C(x))))
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ASharp(A(A(A(x)))) -> aSharp(x)
 ASharp(A(A(b(x)))) -> CSharp(x)
 ASharp(A(A(b(x)))) -> aSharp(C(x))
 ASharp(A(A(b(x)))) -> aSharp(a(C(x)))
 ASharp(A(A(b(x)))) -> aSharp(a(a(C(x))))
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 2.X + 2
[BSharp](X) = 2.X + 2
[CSharp](X) = 2.X + 2
[aSharp](X) = X + 2
[bSharp](X) = 2.X + 2
[cSharp](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ASharp(A(A(b(x)))) -> CSharp(x)
 ASharp(A(A(b(x)))) -> aSharp(C(x))
 ASharp(A(A(b(x)))) -> aSharp(a(C(x)))
 ASharp(A(A(b(x)))) -> aSharp(a(a(C(x))))
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ASharp(A(A(b(x)))) -> CSharp(x)
 ASharp(A(A(b(x)))) -> aSharp(C(x))
 ASharp(A(A(b(x)))) -> aSharp(a(C(x)))
 ASharp(A(A(b(x)))) -> aSharp(a(a(C(x))))
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ASharp(A(A(b(x)))) -> CSharp(x)
 ASharp(A(A(b(x)))) -> aSharp(C(x))
 ASharp(A(A(b(x)))) -> aSharp(a(C(x)))
 ASharp(A(A(b(x)))) -> aSharp(a(a(C(x))))
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 2.X + 2
[BSharp](X) = 2.X + 2
[CSharp](X) = 2.X + 2
[aSharp](X) = 2.X + 2
[bSharp](X) = X
[cSharp](X) = 2.X

Problem 1: 

SCC Processor:
-> Pairs:
 ASharp(A(A(b(x)))) -> aSharp(C(x))
 ASharp(A(A(b(x)))) -> aSharp(a(C(x)))
 ASharp(A(A(b(x)))) -> aSharp(a(a(C(x))))
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ASharp(A(A(b(x)))) -> aSharp(C(x))
 ASharp(A(A(b(x)))) -> aSharp(a(C(x)))
 ASharp(A(A(b(x)))) -> aSharp(a(a(C(x))))
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ASharp(A(A(b(x)))) -> aSharp(C(x))
 ASharp(A(A(b(x)))) -> aSharp(a(C(x)))
 ASharp(A(A(b(x)))) -> aSharp(a(a(C(x))))
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 2.X + 2
[BSharp](X) = 2.X + 2
[CSharp](X) = 2.X + 2
[aSharp](X) = 2.X + 2
[bSharp](X) = 2.X + 1
[cSharp](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ASharp(A(A(b(x)))) -> aSharp(a(C(x)))
 ASharp(A(A(b(x)))) -> aSharp(a(a(C(x))))
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ASharp(A(A(b(x)))) -> aSharp(a(C(x)))
 ASharp(A(A(b(x)))) -> aSharp(a(a(C(x))))
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ASharp(A(A(b(x)))) -> aSharp(a(C(x)))
 ASharp(A(A(b(x)))) -> aSharp(a(a(C(x))))
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 2.X + 2
[BSharp](X) = 2.X + 2
[CSharp](X) = 2.X + 2
[aSharp](X) = 2.X + 1
[bSharp](X) = 2.X
[cSharp](X) = X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ASharp(A(A(b(x)))) -> aSharp(a(a(C(x))))
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ASharp(A(A(b(x)))) -> aSharp(a(a(C(x))))
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ASharp(A(A(b(x)))) -> aSharp(a(a(C(x))))
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 2.X + 2
[BSharp](X) = 2.X + 2
[CSharp](X) = 2.X + 1
[aSharp](X) = 2.X + 1
[bSharp](X) = 2.X + 2
[cSharp](X) = 2.X + 1

Problem 1: 

SCC Processor:
-> Pairs:
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ASharp(C(x)) -> bSharp(x)
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 2.X + 2
[BSharp](X) = 2.X + 2
[CSharp](X) = 2.X + 1
[aSharp](X) = 2.X + 1
[bSharp](X) = 2.X + 2
[cSharp](X) = X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ASharp(c(c(c(x)))) -> CSharp(C(C(x)))
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 2.X + 2
[BSharp](X) = 2.X + 2
[CSharp](X) = X + 2
[aSharp](X) = X + 2
[bSharp](X) = 2.X + 2
[cSharp](X) = X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ASharp(c(c(c(x)))) -> CSharp(C(x))
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 1
[B](X) = 2.X + 1
[C](X) = 2.X + 1
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 2.X
[BSharp](X) = 2.X + 1
[CSharp](X) = 2.X + 2
[aSharp](X) = 2.X + 1
[bSharp](X) = 2.X + 2
[cSharp](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ASharp(c(c(c(x)))) -> CSharp(x)
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 1
[B](X) = 2.X + 1
[C](X) = 2.X + 1
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 2.X + 1
[BSharp](X) = 2.X + 2
[CSharp](X) = 2.X + 2
[aSharp](X) = 2.X + 2
[bSharp](X) = 2.X + 2
[cSharp](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ASharp(c(c(c(x)))) -> bSharp(C(C(C(x))))
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 2.X + 2
[BSharp](X) = X + 2
[CSharp](X) = 2.X + 2
[aSharp](X) = X + 2
[bSharp](X) = X + 2
[cSharp](X) = X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> ASharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 BSharp(a(a(a(x)))) -> ASharp(A(x))
 BSharp(a(a(a(x)))) -> ASharp(x)
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(a(a(a(x)))) -> ASharp(A(A(x)))
 aSharp(a(a(a(x)))) -> ASharp(A(x))
 aSharp(a(a(a(x)))) -> ASharp(x)
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 bSharp(c(x)) -> ASharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 BSharp(A(x)) -> cSharp(x)
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 0
[BSharp](X) = 2.X + 2
[CSharp](X) = 2.X + 2
[aSharp](X) = 2.X
[bSharp](X) = 2.X + 2
[cSharp](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 BSharp(B(B(B(x)))) -> bSharp(b(b(x)))
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 1
[B](X) = 2.X + 1
[C](X) = 2.X + 1
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 0
[BSharp](X) = 2.X + 2
[CSharp](X) = 2.X + 2
[aSharp](X) = X + 2
[bSharp](X) = 2.X + 1
[cSharp](X) = X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 BSharp(B(B(B(x)))) -> bSharp(b(x))
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 0
[BSharp](X) = 2.X
[CSharp](X) = 2.X + 2
[aSharp](X) = 2.X + 2
[bSharp](X) = 2.X
[cSharp](X) = 2.X

Problem 1: 

SCC Processor:
-> Pairs:
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 BSharp(B(B(B(x)))) -> bSharp(x)
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 0
[BSharp](X) = 2.X + 2
[CSharp](X) = 2.X + 2
[aSharp](X) = 2.X + 2
[bSharp](X) = 2.X + 2
[cSharp](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 BSharp(B(B(c(x)))) -> bSharp(A(x))
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 0
[BSharp](X) = 2.X + 2
[CSharp](X) = 2.X + 2
[aSharp](X) = 2.X + 2
[bSharp](X) = X + 2
[cSharp](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 BSharp(B(B(c(x)))) -> bSharp(b(A(x)))
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 0
[BSharp](X) = 2.X + 2
[CSharp](X) = 2.X + 2
[aSharp](X) = 2.X + 2
[bSharp](X) = 2.X + 2
[cSharp](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 BSharp(B(B(c(x)))) -> bSharp(b(b(A(x))))
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = X + 2
[B](X) = X + 2
[C](X) = X + 2
[a](X) = X + 2
[b](X) = X + 2
[c](X) = X + 2
[fSNonEmpty] = 0
[ASharp](X) = 0
[BSharp](X) = 2.X + 2
[CSharp](X) = 2.X + 2
[aSharp](X) = 2.X + 1
[bSharp](X) = 2.X + 1
[cSharp](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 bSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 bSharp(b(b(b(x)))) -> BSharp(B(x))
 bSharp(b(b(b(x)))) -> BSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 BSharp(a(a(a(x)))) -> cSharp(A(A(A(x))))
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 0
[BSharp](X) = 2.X + 1
[CSharp](X) = 2.X + 2
[aSharp](X) = 2.X
[bSharp](X) = 0
[cSharp](X) = X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> BSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> BSharp(B(B(x)))
 CSharp(b(b(b(x)))) -> BSharp(B(x))
 CSharp(b(b(b(x)))) -> BSharp(x)
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(a(x)) -> BSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 CSharp(B(x)) -> aSharp(x)
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 0
[BSharp](X) = 0
[CSharp](X) = 2.X + 2
[aSharp](X) = 2.X + 2
[bSharp](X) = 0
[cSharp](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 CSharp(C(C(C(x)))) -> cSharp(c(c(x)))
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = X + 2
[B](X) = X + 2
[C](X) = X + 2
[a](X) = X + 2
[b](X) = X + 2
[c](X) = X + 2
[fSNonEmpty] = 0
[ASharp](X) = 0
[BSharp](X) = 0
[CSharp](X) = 2.X + 2
[aSharp](X) = 2.X + 2
[bSharp](X) = 0
[cSharp](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 CSharp(C(C(C(x)))) -> cSharp(c(x))
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 0
[BSharp](X) = 0
[CSharp](X) = 2.X + 2
[aSharp](X) = 2.X
[bSharp](X) = 0
[cSharp](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 CSharp(C(C(C(x)))) -> cSharp(x)
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 0
[BSharp](X) = 0
[CSharp](X) = 2.X + 2
[aSharp](X) = 2.X + 2
[bSharp](X) = 0
[cSharp](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 CSharp(C(C(a(x)))) -> cSharp(B(x))
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 1
[B](X) = 2.X + 1
[C](X) = 2.X + 1
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 0
[BSharp](X) = 0
[CSharp](X) = 2.X
[aSharp](X) = 2.X + 2
[bSharp](X) = 0
[cSharp](X) = 2.X

Problem 1: 

SCC Processor:
-> Pairs:
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 CSharp(C(C(a(x)))) -> cSharp(c(B(x)))
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 0
[BSharp](X) = 0
[CSharp](X) = 2.X + 2
[aSharp](X) = X
[bSharp](X) = 0
[cSharp](X) = 2.X

Problem 1: 

SCC Processor:
-> Pairs:
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 CSharp(C(C(a(x)))) -> cSharp(c(c(B(x))))
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 0
[BSharp](X) = 0
[CSharp](X) = 2.X + 2
[aSharp](X) = X + 2
[bSharp](X) = 0
[cSharp](X) = 2.X + 1

Problem 1: 

SCC Processor:
-> Pairs:
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
 cSharp(c(c(c(x)))) -> CSharp(C(C(x)))
 cSharp(c(c(c(x)))) -> CSharp(C(x))
 cSharp(c(c(c(x)))) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
->->-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 CSharp(b(b(b(x)))) -> aSharp(B(B(B(x))))
 aSharp(b(x)) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Usable rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[fSNonEmpty] = 0
[ASharp](X) = 0
[BSharp](X) = 0
[CSharp](X) = 2.X
[aSharp](X) = X + 2
[bSharp](X) = 0
[cSharp](X) = 0

Problem 1: 

SCC Processor:
-> Pairs:
 aSharp(b(x)) -> CSharp(x)
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.


Problem 1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x)
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Levy NW Procedure:
-> Rules:
 A(A(A(A(x)))) -> a(a(a(x)))
 A(A(A(b(x)))) -> a(a(a(C(x))))
 A(C(x)) -> b(x)
 A(a(x)) -> x
 A(c(c(c(x)))) -> b(C(C(C(x))))
 B(A(x)) -> c(x)
 B(B(B(B(x)))) -> b(b(b(x)))
 B(B(B(c(x)))) -> b(b(b(A(x))))
 B(a(a(a(x)))) -> c(A(A(A(x))))
 B(b(x)) -> x
 C(B(x)) -> a(x)
 C(C(C(C(x)))) -> c(c(c(x)))
 C(C(C(a(x)))) -> c(c(c(B(x))))
 C(b(b(b(x)))) -> a(B(B(B(x))))
 C(c(x)) -> x
 a(A(x)) -> x
 a(a(a(a(x)))) -> A(A(A(x)))
 a(b(x)) -> C(x)
 b(B(x)) -> x
 b(b(b(b(x)))) -> B(B(B(x)))
 b(c(x)) -> A(x)
 c(C(x)) -> x
 c(a(x)) -> B(x)
 c(c(c(c(x)))) -> C(C(C(x)))
-> Vars:
 x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x

-> Rlps:
 (rule: A(A(A(A(x)))) -> a(a(a(x))), id: 1, possubterms: A(A(A(A(x))))->[], A(A(A(x)))->[1], A(A(x))->[1, 1], A(x)->[1, 1, 1])
 (rule: A(A(A(b(x)))) -> a(a(a(C(x)))), id: 2, possubterms: A(A(A(b(x))))->[], A(A(b(x)))->[1], A(b(x))->[1, 1], b(x)->[1, 1, 1])
 (rule: A(C(x)) -> b(x), id: 3, possubterms: A(C(x))->[], C(x)->[1])
 (rule: A(a(x)) -> x, id: 4, possubterms: A(a(x))->[], a(x)->[1])
 (rule: A(c(c(c(x)))) -> b(C(C(C(x)))), id: 5, possubterms: A(c(c(c(x))))->[], c(c(c(x)))->[1], c(c(x))->[1, 1], c(x)->[1, 1, 1])
 (rule: B(A(x)) -> c(x), id: 6, possubterms: B(A(x))->[], A(x)->[1])
 (rule: B(B(B(B(x)))) -> b(b(b(x))), id: 7, possubterms: B(B(B(B(x))))->[], B(B(B(x)))->[1], B(B(x))->[1, 1], B(x)->[1, 1, 1])
 (rule: B(B(B(c(x)))) -> b(b(b(A(x)))), id: 8, possubterms: B(B(B(c(x))))->[], B(B(c(x)))->[1], B(c(x))->[1, 1], c(x)->[1, 1, 1])
 (rule: B(a(a(a(x)))) -> c(A(A(A(x)))), id: 9, possubterms: B(a(a(a(x))))->[], a(a(a(x)))->[1], a(a(x))->[1, 1], a(x)->[1, 1, 1])
 (rule: B(b(x)) -> x, id: 10, possubterms: B(b(x))->[], b(x)->[1])
 (rule: C(B(x)) -> a(x), id: 11, possubterms: C(B(x))->[], B(x)->[1])
 (rule: C(C(C(C(x)))) -> c(c(c(x))), id: 12, possubterms: C(C(C(C(x))))->[], C(C(C(x)))->[1], C(C(x))->[1, 1], C(x)->[1, 1, 1])
 (rule: C(C(C(a(x)))) -> c(c(c(B(x)))), id: 13, possubterms: C(C(C(a(x))))->[], C(C(a(x)))->[1], C(a(x))->[1, 1], a(x)->[1, 1, 1])
 (rule: C(b(b(b(x)))) -> a(B(B(B(x)))), id: 14, possubterms: C(b(b(b(x))))->[], b(b(b(x)))->[1], b(b(x))->[1, 1], b(x)->[1, 1, 1])
 (rule: C(c(x)) -> x, id: 15, possubterms: C(c(x))->[], c(x)->[1])
 (rule: a(A(x)) -> x, id: 16, possubterms: a(A(x))->[], A(x)->[1])
 (rule: a(a(a(a(x)))) -> A(A(A(x))), id: 17, possubterms: a(a(a(a(x))))->[], a(a(a(x)))->[1], a(a(x))->[1, 1], a(x)->[1, 1, 1])
 (rule: a(b(x)) -> C(x), id: 18, possubterms: a(b(x))->[], b(x)->[1])
 (rule: b(B(x)) -> x, id: 19, possubterms: b(B(x))->[], B(x)->[1])
 (rule: b(b(b(b(x)))) -> B(B(B(x))), id: 20, possubterms: b(b(b(b(x))))->[], b(b(b(x)))->[1], b(b(x))->[1, 1], b(x)->[1, 1, 1])
 (rule: b(c(x)) -> A(x), id: 21, possubterms: b(c(x))->[], c(x)->[1])
 (rule: c(C(x)) -> x, id: 22, possubterms: c(C(x))->[], C(x)->[1])
 (rule: c(a(x)) -> B(x), id: 23, possubterms: c(a(x))->[], a(x)->[1])
 (rule: c(c(c(c(x)))) -> C(C(C(x))), id: 24, possubterms: c(c(c(c(x))))->[], c(c(c(x)))->[1], c(c(x))->[1, 1], c(x)->[1, 1, 1])

-> Unifications:
(R1 unifies with R1 at p: [1], l: A(A(A(A(x)))), lp: A(A(A(x))), sig: {x -> A(x')}, l': A(A(A(A(x')))), r: a(a(a(x))), r': a(a(a(x'))))
(R1 unifies with R2 at p: [1], l: A(A(A(A(x)))), lp: A(A(A(x))), sig: {x -> b(x')}, l': A(A(A(b(x')))), r: a(a(a(x))), r': a(a(a(C(x')))))
(R1 unifies with R1 at p: [1,1], l: A(A(A(A(x)))), lp: A(A(x)), sig: {x -> A(A(x'))}, l': A(A(A(A(x')))), r: a(a(a(x))), r': a(a(a(x'))))
(R1 unifies with R2 at p: [1,1], l: A(A(A(A(x)))), lp: A(A(x)), sig: {x -> A(b(x'))}, l': A(A(A(b(x')))), r: a(a(a(x))), r': a(a(a(C(x')))))
(R1 unifies with R1 at p: [1,1,1], l: A(A(A(A(x)))), lp: A(x), sig: {x -> A(A(A(x')))}, l': A(A(A(A(x')))), r: a(a(a(x))), r': a(a(a(x'))))
(R1 unifies with R2 at p: [1,1,1], l: A(A(A(A(x)))), lp: A(x), sig: {x -> A(A(b(x')))}, l': A(A(A(b(x')))), r: a(a(a(x))), r': a(a(a(C(x')))))
(R1 unifies with R3 at p: [1,1,1], l: A(A(A(A(x)))), lp: A(x), sig: {x -> C(x')}, l': A(C(x')), r: a(a(a(x))), r': b(x'))
(R1 unifies with R4 at p: [1,1,1], l: A(A(A(A(x)))), lp: A(x), sig: {x -> a(x')}, l': A(a(x')), r: a(a(a(x))), r': x')
(R1 unifies with R5 at p: [1,1,1], l: A(A(A(A(x)))), lp: A(x), sig: {x -> c(c(c(x')))}, l': A(c(c(c(x')))), r: a(a(a(x))), r': b(C(C(C(x')))))
(R2 unifies with R19 at p: [1,1,1], l: A(A(A(b(x)))), lp: b(x), sig: {x -> B(x')}, l': b(B(x')), r: a(a(a(C(x)))), r': x')
(R2 unifies with R20 at p: [1,1,1], l: A(A(A(b(x)))), lp: b(x), sig: {x -> b(b(b(x')))}, l': b(b(b(b(x')))), r: a(a(a(C(x)))), r': B(B(B(x'))))
(R2 unifies with R21 at p: [1,1,1], l: A(A(A(b(x)))), lp: b(x), sig: {x -> c(x')}, l': b(c(x')), r: a(a(a(C(x)))), r': A(x'))
(R3 unifies with R11 at p: [1], l: A(C(x)), lp: C(x), sig: {x -> B(x')}, l': C(B(x')), r: b(x), r': a(x'))
(R3 unifies with R12 at p: [1], l: A(C(x)), lp: C(x), sig: {x -> C(C(C(x')))}, l': C(C(C(C(x')))), r: b(x), r': c(c(c(x'))))
(R3 unifies with R13 at p: [1], l: A(C(x)), lp: C(x), sig: {x -> C(C(a(x')))}, l': C(C(C(a(x')))), r: b(x), r': c(c(c(B(x')))))
(R3 unifies with R14 at p: [1], l: A(C(x)), lp: C(x), sig: {x -> b(b(b(x')))}, l': C(b(b(b(x')))), r: b(x), r': a(B(B(B(x')))))
(R3 unifies with R15 at p: [1], l: A(C(x)), lp: C(x), sig: {x -> c(x')}, l': C(c(x')), r: b(x), r': x')
(R4 unifies with R16 at p: [1], l: A(a(x)), lp: a(x), sig: {x -> A(x')}, l': a(A(x')), r: x, r': x')
(R4 unifies with R17 at p: [1], l: A(a(x)), lp: a(x), sig: {x -> a(a(a(x')))}, l': a(a(a(a(x')))), r: x, r': A(A(A(x'))))
(R4 unifies with R18 at p: [1], l: A(a(x)), lp: a(x), sig: {x -> b(x')}, l': a(b(x')), r: x, r': C(x'))
(R5 unifies with R24 at p: [1], l: A(c(c(c(x)))), lp: c(c(c(x))), sig: {x -> c(x')}, l': c(c(c(c(x')))), r: b(C(C(C(x)))), r': C(C(C(x'))))
(R5 unifies with R24 at p: [1,1], l: A(c(c(c(x)))), lp: c(c(x)), sig: {x -> c(c(x'))}, l': c(c(c(c(x')))), r: b(C(C(C(x)))), r': C(C(C(x'))))
(R5 unifies with R22 at p: [1,1,1], l: A(c(c(c(x)))), lp: c(x), sig: {x -> C(x')}, l': c(C(x')), r: b(C(C(C(x)))), r': x')
(R5 unifies with R23 at p: [1,1,1], l: A(c(c(c(x)))), lp: c(x), sig: {x -> a(x')}, l': c(a(x')), r: b(C(C(C(x)))), r': B(x'))
(R5 unifies with R24 at p: [1,1,1], l: A(c(c(c(x)))), lp: c(x), sig: {x -> c(c(c(x')))}, l': c(c(c(c(x')))), r: b(C(C(C(x)))), r': C(C(C(x'))))
(R6 unifies with R1 at p: [1], l: B(A(x)), lp: A(x), sig: {x -> A(A(A(x')))}, l': A(A(A(A(x')))), r: c(x), r': a(a(a(x'))))
(R6 unifies with R2 at p: [1], l: B(A(x)), lp: A(x), sig: {x -> A(A(b(x')))}, l': A(A(A(b(x')))), r: c(x), r': a(a(a(C(x')))))
(R6 unifies with R3 at p: [1], l: B(A(x)), lp: A(x), sig: {x -> C(x')}, l': A(C(x')), r: c(x), r': b(x'))
(R6 unifies with R4 at p: [1], l: B(A(x)), lp: A(x), sig: {x -> a(x')}, l': A(a(x')), r: c(x), r': x')
(R6 unifies with R5 at p: [1], l: B(A(x)), lp: A(x), sig: {x -> c(c(c(x')))}, l': A(c(c(c(x')))), r: c(x), r': b(C(C(C(x')))))
(R7 unifies with R7 at p: [1], l: B(B(B(B(x)))), lp: B(B(B(x))), sig: {x -> B(x')}, l': B(B(B(B(x')))), r: b(b(b(x))), r': b(b(b(x'))))
(R7 unifies with R8 at p: [1], l: B(B(B(B(x)))), lp: B(B(B(x))), sig: {x -> c(x')}, l': B(B(B(c(x')))), r: b(b(b(x))), r': b(b(b(A(x')))))
(R7 unifies with R7 at p: [1,1], l: B(B(B(B(x)))), lp: B(B(x)), sig: {x -> B(B(x'))}, l': B(B(B(B(x')))), r: b(b(b(x))), r': b(b(b(x'))))
(R7 unifies with R8 at p: [1,1], l: B(B(B(B(x)))), lp: B(B(x)), sig: {x -> B(c(x'))}, l': B(B(B(c(x')))), r: b(b(b(x))), r': b(b(b(A(x')))))
(R7 unifies with R6 at p: [1,1,1], l: B(B(B(B(x)))), lp: B(x), sig: {x -> A(x')}, l': B(A(x')), r: b(b(b(x))), r': c(x'))
(R7 unifies with R7 at p: [1,1,1], l: B(B(B(B(x)))), lp: B(x), sig: {x -> B(B(B(x')))}, l': B(B(B(B(x')))), r: b(b(b(x))), r': b(b(b(x'))))
(R7 unifies with R8 at p: [1,1,1], l: B(B(B(B(x)))), lp: B(x), sig: {x -> B(B(c(x')))}, l': B(B(B(c(x')))), r: b(b(b(x))), r': b(b(b(A(x')))))
(R7 unifies with R9 at p: [1,1,1], l: B(B(B(B(x)))), lp: B(x), sig: {x -> a(a(a(x')))}, l': B(a(a(a(x')))), r: b(b(b(x))), r': c(A(A(A(x')))))
(R7 unifies with R10 at p: [1,1,1], l: B(B(B(B(x)))), lp: B(x), sig: {x -> b(x')}, l': B(b(x')), r: b(b(b(x))), r': x')
(R8 unifies with R22 at p: [1,1,1], l: B(B(B(c(x)))), lp: c(x), sig: {x -> C(x')}, l': c(C(x')), r: b(b(b(A(x)))), r': x')
(R8 unifies with R23 at p: [1,1,1], l: B(B(B(c(x)))), lp: c(x), sig: {x -> a(x')}, l': c(a(x')), r: b(b(b(A(x)))), r': B(x'))
(R8 unifies with R24 at p: [1,1,1], l: B(B(B(c(x)))), lp: c(x), sig: {x -> c(c(c(x')))}, l': c(c(c(c(x')))), r: b(b(b(A(x)))), r': C(C(C(x'))))
(R9 unifies with R17 at p: [1], l: B(a(a(a(x)))), lp: a(a(a(x))), sig: {x -> a(x')}, l': a(a(a(a(x')))), r: c(A(A(A(x)))), r': A(A(A(x'))))
(R9 unifies with R17 at p: [1,1], l: B(a(a(a(x)))), lp: a(a(x)), sig: {x -> a(a(x'))}, l': a(a(a(a(x')))), r: c(A(A(A(x)))), r': A(A(A(x'))))
(R9 unifies with R16 at p: [1,1,1], l: B(a(a(a(x)))), lp: a(x), sig: {x -> A(x')}, l': a(A(x')), r: c(A(A(A(x)))), r': x')
(R9 unifies with R17 at p: [1,1,1], l: B(a(a(a(x)))), lp: a(x), sig: {x -> a(a(a(x')))}, l': a(a(a(a(x')))), r: c(A(A(A(x)))), r': A(A(A(x'))))
(R9 unifies with R18 at p: [1,1,1], l: B(a(a(a(x)))), lp: a(x), sig: {x -> b(x')}, l': a(b(x')), r: c(A(A(A(x)))), r': C(x'))
(R10 unifies with R19 at p: [1], l: B(b(x)), lp: b(x), sig: {x -> B(x')}, l': b(B(x')), r: x, r': x')
(R10 unifies with R20 at p: [1], l: B(b(x)), lp: b(x), sig: {x -> b(b(b(x')))}, l': b(b(b(b(x')))), r: x, r': B(B(B(x'))))
(R10 unifies with R21 at p: [1], l: B(b(x)), lp: b(x), sig: {x -> c(x')}, l': b(c(x')), r: x, r': A(x'))
(R11 unifies with R6 at p: [1], l: C(B(x)), lp: B(x), sig: {x -> A(x')}, l': B(A(x')), r: a(x), r': c(x'))
(R11 unifies with R7 at p: [1], l: C(B(x)), lp: B(x), sig: {x -> B(B(B(x')))}, l': B(B(B(B(x')))), r: a(x), r': b(b(b(x'))))
(R11 unifies with R8 at p: [1], l: C(B(x)), lp: B(x), sig: {x -> B(B(c(x')))}, l': B(B(B(c(x')))), r: a(x), r': b(b(b(A(x')))))
(R11 unifies with R9 at p: [1], l: C(B(x)), lp: B(x), sig: {x -> a(a(a(x')))}, l': B(a(a(a(x')))), r: a(x), r': c(A(A(A(x')))))
(R11 unifies with R10 at p: [1], l: C(B(x)), lp: B(x), sig: {x -> b(x')}, l': B(b(x')), r: a(x), r': x')
(R12 unifies with R12 at p: [1], l: C(C(C(C(x)))), lp: C(C(C(x))), sig: {x -> C(x')}, l': C(C(C(C(x')))), r: c(c(c(x))), r': c(c(c(x'))))
(R12 unifies with R13 at p: [1], l: C(C(C(C(x)))), lp: C(C(C(x))), sig: {x -> a(x')}, l': C(C(C(a(x')))), r: c(c(c(x))), r': c(c(c(B(x')))))
(R12 unifies with R12 at p: [1,1], l: C(C(C(C(x)))), lp: C(C(x)), sig: {x -> C(C(x'))}, l': C(C(C(C(x')))), r: c(c(c(x))), r': c(c(c(x'))))
(R12 unifies with R13 at p: [1,1], l: C(C(C(C(x)))), lp: C(C(x)), sig: {x -> C(a(x'))}, l': C(C(C(a(x')))), r: c(c(c(x))), r': c(c(c(B(x')))))
(R12 unifies with R11 at p: [1,1,1], l: C(C(C(C(x)))), lp: C(x), sig: {x -> B(x')}, l': C(B(x')), r: c(c(c(x))), r': a(x'))
(R12 unifies with R12 at p: [1,1,1], l: C(C(C(C(x)))), lp: C(x), sig: {x -> C(C(C(x')))}, l': C(C(C(C(x')))), r: c(c(c(x))), r': c(c(c(x'))))
(R12 unifies with R13 at p: [1,1,1], l: C(C(C(C(x)))), lp: C(x), sig: {x -> C(C(a(x')))}, l': C(C(C(a(x')))), r: c(c(c(x))), r': c(c(c(B(x')))))
(R12 unifies with R14 at p: [1,1,1], l: C(C(C(C(x)))), lp: C(x), sig: {x -> b(b(b(x')))}, l': C(b(b(b(x')))), r: c(c(c(x))), r': a(B(B(B(x')))))
(R12 unifies with R15 at p: [1,1,1], l: C(C(C(C(x)))), lp: C(x), sig: {x -> c(x')}, l': C(c(x')), r: c(c(c(x))), r': x')
(R13 unifies with R16 at p: [1,1,1], l: C(C(C(a(x)))), lp: a(x), sig: {x -> A(x')}, l': a(A(x')), r: c(c(c(B(x)))), r': x')
(R13 unifies with R17 at p: [1,1,1], l: C(C(C(a(x)))), lp: a(x), sig: {x -> a(a(a(x')))}, l': a(a(a(a(x')))), r: c(c(c(B(x)))), r': A(A(A(x'))))
(R13 unifies with R18 at p: [1,1,1], l: C(C(C(a(x)))), lp: a(x), sig: {x -> b(x')}, l': a(b(x')), r: c(c(c(B(x)))), r': C(x'))
(R14 unifies with R20 at p: [1], l: C(b(b(b(x)))), lp: b(b(b(x))), sig: {x -> b(x')}, l': b(b(b(b(x')))), r: a(B(B(B(x)))), r': B(B(B(x'))))
(R14 unifies with R20 at p: [1,1], l: C(b(b(b(x)))), lp: b(b(x)), sig: {x -> b(b(x'))}, l': b(b(b(b(x')))), r: a(B(B(B(x)))), r': B(B(B(x'))))
(R14 unifies with R19 at p: [1,1,1], l: C(b(b(b(x)))), lp: b(x), sig: {x -> B(x')}, l': b(B(x')), r: a(B(B(B(x)))), r': x')
(R14 unifies with R20 at p: [1,1,1], l: C(b(b(b(x)))), lp: b(x), sig: {x -> b(b(b(x')))}, l': b(b(b(b(x')))), r: a(B(B(B(x)))), r': B(B(B(x'))))
(R14 unifies with R21 at p: [1,1,1], l: C(b(b(b(x)))), lp: b(x), sig: {x -> c(x')}, l': b(c(x')), r: a(B(B(B(x)))), r': A(x'))
(R15 unifies with R22 at p: [1], l: C(c(x)), lp: c(x), sig: {x -> C(x')}, l': c(C(x')), r: x, r': x')
(R15 unifies with R23 at p: [1], l: C(c(x)), lp: c(x), sig: {x -> a(x')}, l': c(a(x')), r: x, r': B(x'))
(R15 unifies with R24 at p: [1], l: C(c(x)), lp: c(x), sig: {x -> c(c(c(x')))}, l': c(c(c(c(x')))), r: x, r': C(C(C(x'))))
(R16 unifies with R1 at p: [1], l: a(A(x)), lp: A(x), sig: {x -> A(A(A(x')))}, l': A(A(A(A(x')))), r: x, r': a(a(a(x'))))
(R16 unifies with R2 at p: [1], l: a(A(x)), lp: A(x), sig: {x -> A(A(b(x')))}, l': A(A(A(b(x')))), r: x, r': a(a(a(C(x')))))
(R16 unifies with R3 at p: [1], l: a(A(x)), lp: A(x), sig: {x -> C(x')}, l': A(C(x')), r: x, r': b(x'))
(R16 unifies with R4 at p: [1], l: a(A(x)), lp: A(x), sig: {x -> a(x')}, l': A(a(x')), r: x, r': x')
(R16 unifies with R5 at p: [1], l: a(A(x)), lp: A(x), sig: {x -> c(c(c(x')))}, l': A(c(c(c(x')))), r: x, r': b(C(C(C(x')))))
(R17 unifies with R17 at p: [1], l: a(a(a(a(x)))), lp: a(a(a(x))), sig: {x -> a(x')}, l': a(a(a(a(x')))), r: A(A(A(x))), r': A(A(A(x'))))
(R17 unifies with R17 at p: [1,1], l: a(a(a(a(x)))), lp: a(a(x)), sig: {x -> a(a(x'))}, l': a(a(a(a(x')))), r: A(A(A(x))), r': A(A(A(x'))))
(R17 unifies with R16 at p: [1,1,1], l: a(a(a(a(x)))), lp: a(x), sig: {x -> A(x')}, l': a(A(x')), r: A(A(A(x))), r': x')
(R17 unifies with R17 at p: [1,1,1], l: a(a(a(a(x)))), lp: a(x), sig: {x -> a(a(a(x')))}, l': a(a(a(a(x')))), r: A(A(A(x))), r': A(A(A(x'))))
(R17 unifies with R18 at p: [1,1,1], l: a(a(a(a(x)))), lp: a(x), sig: {x -> b(x')}, l': a(b(x')), r: A(A(A(x))), r': C(x'))
(R18 unifies with R19 at p: [1], l: a(b(x)), lp: b(x), sig: {x -> B(x')}, l': b(B(x')), r: C(x), r': x')
(R18 unifies with R20 at p: [1], l: a(b(x)), lp: b(x), sig: {x -> b(b(b(x')))}, l': b(b(b(b(x')))), r: C(x), r': B(B(B(x'))))
(R18 unifies with R21 at p: [1], l: a(b(x)), lp: b(x), sig: {x -> c(x')}, l': b(c(x')), r: C(x), r': A(x'))
(R19 unifies with R6 at p: [1], l: b(B(x)), lp: B(x), sig: {x -> A(x')}, l': B(A(x')), r: x, r': c(x'))
(R19 unifies with R7 at p: [1], l: b(B(x)), lp: B(x), sig: {x -> B(B(B(x')))}, l': B(B(B(B(x')))), r: x, r': b(b(b(x'))))
(R19 unifies with R8 at p: [1], l: b(B(x)), lp: B(x), sig: {x -> B(B(c(x')))}, l': B(B(B(c(x')))), r: x, r': b(b(b(A(x')))))
(R19 unifies with R9 at p: [1], l: b(B(x)), lp: B(x), sig: {x -> a(a(a(x')))}, l': B(a(a(a(x')))), r: x, r': c(A(A(A(x')))))
(R19 unifies with R10 at p: [1], l: b(B(x)), lp: B(x), sig: {x -> b(x')}, l': B(b(x')), r: x, r': x')
(R20 unifies with R20 at p: [1], l: b(b(b(b(x)))), lp: b(b(b(x))), sig: {x -> b(x')}, l': b(b(b(b(x')))), r: B(B(B(x))), r': B(B(B(x'))))
(R20 unifies with R20 at p: [1,1], l: b(b(b(b(x)))), lp: b(b(x)), sig: {x -> b(b(x'))}, l': b(b(b(b(x')))), r: B(B(B(x))), r': B(B(B(x'))))
(R20 unifies with R19 at p: [1,1,1], l: b(b(b(b(x)))), lp: b(x), sig: {x -> B(x')}, l': b(B(x')), r: B(B(B(x))), r': x')
(R20 unifies with R20 at p: [1,1,1], l: b(b(b(b(x)))), lp: b(x), sig: {x -> b(b(b(x')))}, l': b(b(b(b(x')))), r: B(B(B(x))), r': B(B(B(x'))))
(R20 unifies with R21 at p: [1,1,1], l: b(b(b(b(x)))), lp: b(x), sig: {x -> c(x')}, l': b(c(x')), r: B(B(B(x))), r': A(x'))
(R21 unifies with R22 at p: [1], l: b(c(x)), lp: c(x), sig: {x -> C(x')}, l': c(C(x')), r: A(x), r': x')
(R21 unifies with R23 at p: [1], l: b(c(x)), lp: c(x), sig: {x -> a(x')}, l': c(a(x')), r: A(x), r': B(x'))
(R21 unifies with R24 at p: [1], l: b(c(x)), lp: c(x), sig: {x -> c(c(c(x')))}, l': c(c(c(c(x')))), r: A(x), r': C(C(C(x'))))
(R22 unifies with R11 at p: [1], l: c(C(x)), lp: C(x), sig: {x -> B(x')}, l': C(B(x')), r: x, r': a(x'))
(R22 unifies with R12 at p: [1], l: c(C(x)), lp: C(x), sig: {x -> C(C(C(x')))}, l': C(C(C(C(x')))), r: x, r': c(c(c(x'))))
(R22 unifies with R13 at p: [1], l: c(C(x)), lp: C(x), sig: {x -> C(C(a(x')))}, l': C(C(C(a(x')))), r: x, r': c(c(c(B(x')))))
(R22 unifies with R14 at p: [1], l: c(C(x)), lp: C(x), sig: {x -> b(b(b(x')))}, l': C(b(b(b(x')))), r: x, r': a(B(B(B(x')))))
(R22 unifies with R15 at p: [1], l: c(C(x)), lp: C(x), sig: {x -> c(x')}, l': C(c(x')), r: x, r': x')
(R23 unifies with R16 at p: [1], l: c(a(x)), lp: a(x), sig: {x -> A(x')}, l': a(A(x')), r: B(x), r': x')
(R23 unifies with R17 at p: [1], l: c(a(x)), lp: a(x), sig: {x -> a(a(a(x')))}, l': a(a(a(a(x')))), r: B(x), r': A(A(A(x'))))
(R23 unifies with R18 at p: [1], l: c(a(x)), lp: a(x), sig: {x -> b(x')}, l': a(b(x')), r: B(x), r': C(x'))
(R24 unifies with R24 at p: [1], l: c(c(c(c(x)))), lp: c(c(c(x))), sig: {x -> c(x')}, l': c(c(c(c(x')))), r: C(C(C(x))), r': C(C(C(x'))))
(R24 unifies with R24 at p: [1,1], l: c(c(c(c(x)))), lp: c(c(x)), sig: {x -> c(c(x'))}, l': c(c(c(c(x')))), r: C(C(C(x))), r': C(C(C(x'))))
(R24 unifies with R22 at p: [1,1,1], l: c(c(c(c(x)))), lp: c(x), sig: {x -> C(x')}, l': c(C(x')), r: C(C(C(x))), r': x')
(R24 unifies with R23 at p: [1,1,1], l: c(c(c(c(x)))), lp: c(x), sig: {x -> a(x')}, l': c(a(x')), r: C(C(C(x))), r': B(x'))
(R24 unifies with R24 at p: [1,1,1], l: c(c(c(c(x)))), lp: c(x), sig: {x -> c(c(c(x')))}, l': c(c(c(c(x')))), r: C(C(C(x))), r': C(C(C(x'))))

-> Critical pairs info:
<b(b(b(B(B(B(x')))))),B(B(B(b(b(b(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N1
<A(x'),b(c(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N2
<C(x'),a(b(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N3
<c(C(C(C(x')))),C(C(C(c(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N4
<c(x'),B(A(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N5
<C(c(A(A(A(x'))))),a(a(a(a(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N6
<B(B(B(b(b(b(A(x'))))))),b(b(b(B(B(c(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N7
<B(b(x')),c(C(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N8
<B(b(b(b(x')))),b(b(b(B(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N9
<C(C(C(a(x')))),c(c(c(B(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N10
<a(B(B(B(x')))),C(b(b(b(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N11
<c(c(c(C(C(C(x')))))),C(C(C(c(c(c(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N12
<a(A(A(A(x')))),A(A(A(a(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N13
<a(x'),a(x')>   =>  Trivial, Not overlay, Proper, NW0, N14
<C(C(C(a(B(B(B(x'))))))),c(c(c(b(b(b(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N15
<a(a(a(a(C(x'))))),A(A(b(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N16
<A(A(A(x'))),a(a(a(a(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N17
<C(c(c(c(B(x'))))),c(c(c(a(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N18
<b(b(B(B(B(x'))))),B(B(B(b(b(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N19
<C(B(x')),a(x')>   =>  Not trivial, Not overlay, Proper, NW0, N20
<c(A(A(A(x')))),B(a(a(a(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N21
<B(B(B(c(x')))),b(b(b(A(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N22
<B(a(a(A(A(A(x')))))),c(A(A(A(a(a(a(x')))))))>   =>  Not trivial, Not overlay, Proper, NW0, N23
<A(c(c(c(x')))),b(C(C(C(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N24
<c(a(B(B(B(x'))))),b(b(b(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N25
<C(c(x')),a(A(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N26
<c(x'),c(x')>   =>  Trivial, Not overlay, Proper, NW0, N27
<b(c(A(A(A(x'))))),a(a(a(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N28
<C(C(C(A(A(A(x')))))),c(c(c(B(a(a(a(x')))))))>   =>  Not trivial, Not overlay, Proper, NW0, N29
<A(A(A(x'))),a(a(a(C(B(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N30
<C(B(B(B(x')))),a(B(B(B(b(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N31
<B(a(a(a(C(x'))))),c(A(A(b(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N32
<a(x'),C(B(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N33
<B(B(B(B(x')))),b(b(b(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N34
<C(c(c(c(x')))),c(c(c(C(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N35
<B(A(x')),c(x')>   =>  Not trivial, Not overlay, Proper, NW0, N36
<b(B(B(B(x')))),B(B(B(b(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N37
<b(c(x')),A(x')>   =>  Not trivial, Not overlay, Proper, NW0, N38
<b(B(x')),A(a(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N39
<C(C(C(c(c(c(B(x'))))))),c(c(c(C(C(a(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N40
<C(b(B(B(B(x'))))),a(B(B(B(b(b(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N41
<A(C(C(C(x')))),b(C(C(C(c(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N42
<B(B(B(x'))),b(b(b(A(C(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N43
<A(a(x')),b(B(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N44
<b(b(b(A(x')))),B(B(B(c(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N45
<A(c(c(x'))),b(C(C(C(C(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N46
<C(C(C(x'))),c(c(c(c(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N47
<c(a(x')),B(x')>   =>  Not trivial, Not overlay, Proper, NW0, N48
<A(A(A(B(B(B(x')))))),a(a(a(C(b(b(b(x')))))))>   =>  Not trivial, Not overlay, Proper, NW0, N49
<c(c(c(x'))),C(C(C(C(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N50
<a(b(C(C(C(x'))))),c(c(c(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N51
<B(B(b(b(b(x'))))),b(b(b(B(B(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N52
<B(B(B(b(b(b(x')))))),b(b(b(B(B(B(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N53
<a(a(a(x'))),A(A(A(A(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N54
<a(a(A(A(A(x'))))),A(A(A(a(a(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N55
<c(C(x')),B(b(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N56
<C(C(C(C(x')))),c(c(c(B(b(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N57
<B(B(B(C(C(C(x')))))),b(b(b(A(c(c(c(x')))))))>   =>  Not trivial, Not overlay, Proper, NW0, N58
<B(B(B(B(x')))),b(b(b(A(a(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N59
<A(A(A(A(x')))),a(a(a(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N60
<b(C(C(C(x')))),A(c(c(c(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N61
<B(a(a(x'))),c(A(A(A(A(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N62
<B(B(B(c(A(A(A(x'))))))),b(b(b(a(a(a(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N63
<c(c(c(c(x')))),C(C(C(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N64
<A(A(A(a(a(a(x')))))),a(a(a(A(A(A(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N65
<c(c(c(B(x')))),C(C(C(a(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N66
<C(C(c(c(c(x'))))),c(c(c(C(C(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N67
<C(b(b(b(A(x'))))),a(B(B(c(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N68
<c(c(C(C(C(x'))))),C(C(C(c(c(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N69
<a(a(a(a(x')))),A(A(A(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N70
<B(B(b(b(b(A(x')))))),b(b(b(B(c(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N71
<b(b(b(b(x')))),B(B(B(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N72
<B(a(a(C(x')))),c(A(A(A(b(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N73
<A(C(x')),b(x')>   =>  Not trivial, Not overlay, Proper, NW0, N74
<B(A(A(A(x')))),c(A(A(A(a(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N75
<C(b(b(A(x')))),a(B(B(B(c(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N76
<A(c(c(c(B(x'))))),b(C(C(a(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N77
<C(C(C(C(x')))),c(c(c(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N78
<B(x'),B(x')>   =>  Trivial, Not overlay, Proper, NW0, N79
<C(C(c(c(c(B(x')))))),c(c(c(C(a(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N80
<A(c(c(B(x')))),b(C(C(C(a(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N81
<A(A(a(a(a(C(x')))))),a(a(a(A(b(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N82
<c(c(c(c(B(x'))))),C(C(a(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N83
<b(x'),A(C(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N84
<B(b(b(b(A(x'))))),b(b(b(c(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N85
<A(A(A(b(C(C(C(x'))))))),a(a(a(c(c(c(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N86
<a(a(a(A(A(A(x')))))),A(A(A(a(a(a(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N87
<A(c(C(C(C(x'))))),b(C(C(C(c(c(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N88
<A(A(a(a(a(x'))))),a(a(a(A(A(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N89
<C(b(b(x'))),a(B(B(B(B(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N90
<C(C(C(x'))),c(c(c(B(A(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N91
<A(a(a(a(C(x'))))),a(a(a(b(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N92
<C(b(b(B(B(B(x')))))),a(B(B(B(b(b(b(x')))))))>   =>  Not trivial, Not overlay, Proper, NW0, N93
<A(A(A(b(x')))),a(a(a(C(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N94
<A(x'),A(x')>   =>  Trivial, Not overlay, Proper, NW0, N95
<B(x'),c(a(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N96
<B(b(C(C(C(x'))))),c(c(c(c(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N97
<b(b(b(b(A(x'))))),B(B(c(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N98
<B(a(a(a(x')))),c(A(A(A(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N99
<a(b(x')),C(x')>   =>  Not trivial, Not overlay, Proper, NW0, N100
<A(a(a(a(x')))),a(a(a(A(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N101
<A(a(B(B(B(x'))))),b(b(b(b(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N102
<C(b(b(b(x')))),a(B(B(B(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N103
<a(A(x')),C(c(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N104
<A(A(A(a(a(a(C(x'))))))),a(a(a(A(A(b(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N105
<b(x'),b(x')>   =>  Trivial, Not overlay, Proper, NW0, N106
<b(b(b(x'))),B(B(B(B(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N107
<a(a(a(C(x')))),A(A(A(b(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N108
<A(A(A(A(x')))),a(a(a(C(c(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N109
<B(B(B(x'))),b(b(b(b(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N110
<C(x'),C(x')>   =>  Trivial, Not overlay, Proper, NW0, N111
<B(a(A(A(A(x'))))),c(A(A(A(a(a(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N112
<A(c(c(C(C(C(x')))))),b(C(C(C(c(c(c(x')))))))>   =>  Not trivial, Not overlay, Proper, NW0, N113
<C(C(C(c(c(c(x')))))),c(c(c(C(C(C(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N114

-> 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 84 subproblems.

Problem 1.1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<c(C(C(C(x')))),C(C(C(c(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N4
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: c(C(C(C(x'))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('c(C(C(C(x'))))', D0)
ID: 1 => ('C(C(x'))', D1, R22, P[], S{x25 -> C(C(x'))}), NR: 'C(C(x'))'
t: C(C(C(c(x'))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('C(C(C(c(x'))))', D0)
ID: 1 => ('C(C(x'))', D1, R15, P[1, 1], S{x18 -> x'}), NR: 'x''
SNodesPath: c(C(C(C(x')))) -> C(C(x'))
TNodesPath: C(C(C(c(x')))) -> C(C(x'))
c(C(C(C(x')))) ->* C(C(x')) *<- C(C(C(c(x'))))
"Joinable"

The problem is confluent.

Problem 1.2: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<C(c(A(A(A(x'))))),a(a(a(a(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N6
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: C(c(A(A(A(x')))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('C(c(A(A(A(x')))))', D0)
ID: 1 => ('A(A(A(x')))', D1, R15, P[], S{x18 -> A(A(A(x')))}), NR: 'A(A(A(x')))'
t: a(a(a(a(x'))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('a(a(a(a(x'))))', D0)
ID: 1 => ('A(A(A(x')))', D1, R17, P[], S{x20 -> x'}), NR: 'A(A(A(x')))'
SNodesPath: C(c(A(A(A(x'))))) -> A(A(A(x')))
TNodesPath: a(a(a(a(x')))) -> A(A(A(x')))
C(c(A(A(A(x'))))) ->* A(A(A(x'))) *<- a(a(a(a(x'))))
"Joinable"

The problem is confluent.

Problem 1.3: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<B(B(B(b(b(b(A(x'))))))),b(b(b(B(B(c(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N7
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: B(B(B(b(b(b(A(x')))))))
Nodes: [0,1,2,3]
Edges: [(0,1),(1,2),(2,3)]
ID: 0 => ('B(B(B(b(b(b(A(x')))))))', D0)
ID: 1 => ('B(B(b(b(A(x')))))', D1, R10, P[1, 1], S{x13 -> b(b(A(x')))}), NR: 'b(b(A(x')))'
ID: 2 => ('B(b(A(x')))', D2, R10, P[1], S{x13 -> b(A(x'))}), NR: 'b(A(x'))'
ID: 3 => ('A(x')', D3, R10, P[], S{x13 -> A(x')}), NR: 'A(x')'
t: b(b(b(B(B(c(x'))))))
Nodes: [0,1,2,3]
Edges: [(0,1),(1,2),(2,3)]
ID: 0 => ('b(b(b(B(B(c(x'))))))', D0)
ID: 1 => ('b(b(B(c(x'))))', D1, R19, P[1, 1], S{x22 -> B(c(x'))}), NR: 'B(c(x'))'
ID: 2 => ('b(c(x'))', D2, R19, P[1], S{x22 -> c(x')}), NR: 'c(x')'
ID: 3 => ('A(x')', D3, R21, P[], S{x24 -> x'}), NR: 'A(x')'
SNodesPath: B(B(B(b(b(b(A(x'))))))) -> B(B(b(b(A(x'))))) -> B(b(A(x'))) -> A(x')
TNodesPath: b(b(b(B(B(c(x')))))) -> b(b(B(c(x')))) -> b(c(x')) -> A(x')
B(B(B(b(b(b(A(x'))))))) ->* A(x') *<- b(b(b(B(B(c(x'))))))
"Joinable"

The problem is confluent.

Problem 1.4: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<B(b(b(b(x')))),b(b(b(B(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N9
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: B(b(b(b(x'))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('B(b(b(b(x'))))', D0)
ID: 1 => ('b(b(x'))', D1, R10, P[], S{x13 -> b(b(x'))}), NR: 'b(b(x'))'
t: b(b(b(B(x'))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('b(b(b(B(x'))))', D0)
ID: 1 => ('b(b(x'))', D1, R19, P[1, 1], S{x22 -> x'}), NR: 'x''
SNodesPath: B(b(b(b(x')))) -> b(b(x'))
TNodesPath: b(b(b(B(x')))) -> b(b(x'))
B(b(b(b(x')))) ->* b(b(x')) *<- b(b(b(B(x'))))
"Joinable"

The problem is confluent.

Problem 1.5: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<a(A(A(A(x')))),A(A(A(a(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N13
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: a(A(A(A(x'))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('a(A(A(A(x'))))', D0)
ID: 1 => ('A(A(x'))', D1, R16, P[], S{x19 -> A(A(x'))}), NR: 'A(A(x'))'
t: A(A(A(a(x'))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('A(A(A(a(x'))))', D0)
ID: 1 => ('A(A(x'))', D1, R4, P[1, 1], S{x7 -> x'}), NR: 'x''
SNodesPath: a(A(A(A(x')))) -> A(A(x'))
TNodesPath: A(A(A(a(x')))) -> A(A(x'))
a(A(A(A(x')))) ->* A(A(x')) *<- A(A(A(a(x'))))
"Joinable"

The problem is confluent.

Problem 1.6: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<C(C(C(a(B(B(B(x'))))))),c(c(c(b(b(b(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N15
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: C(C(C(a(B(B(B(x')))))))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('C(C(C(a(B(B(B(x')))))))', D0)
ID: 1 => ('c(c(c(B(B(B(B(x')))))))', D1, R13, P[], S{x16 -> B(B(B(x')))}), NR: 'c(c(c(B(B(B(B(x')))))))'
ID: 2 => ('c(c(c(b(b(b(x'))))))', D2, R7, P[1, 1, 1], S{x10 -> x'}), NR: 'b(b(b(x')))'
t: c(c(c(b(b(b(x'))))))
Nodes: [0]
Edges: []
ID: 0 => ('c(c(c(b(b(b(x'))))))', D0)
SNodesPath: C(C(C(a(B(B(B(x'))))))) -> c(c(c(B(B(B(B(x'))))))) -> c(c(c(b(b(b(x'))))))
TNodesPath: c(c(c(b(b(b(x'))))))
C(C(C(a(B(B(B(x'))))))) ->* c(c(c(b(b(b(x')))))) *<- c(c(c(b(b(b(x'))))))
"Joinable"

The problem is confluent.

Problem 1.7: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<a(a(a(a(C(x'))))),A(A(b(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N16
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: a(a(a(a(C(x')))))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('a(a(a(a(C(x')))))', D0)
ID: 1 => ('A(A(A(C(x'))))', D1, R17, P[], S{x20 -> C(x')}), NR: 'A(A(A(C(x'))))'
ID: 2 => ('A(A(b(x')))', D2, R3, P[1, 1], S{x6 -> x'}), NR: 'b(x')'
t: A(A(b(x')))
Nodes: [0]
Edges: []
ID: 0 => ('A(A(b(x')))', D0)
SNodesPath: a(a(a(a(C(x'))))) -> A(A(A(C(x')))) -> A(A(b(x')))
TNodesPath: A(A(b(x')))
a(a(a(a(C(x'))))) ->* A(A(b(x'))) *<- A(A(b(x')))
"Joinable"

The problem is confluent.

Problem 1.8: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<C(c(c(c(B(x'))))),c(c(c(a(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N18
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: C(c(c(c(B(x')))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('C(c(c(c(B(x')))))', D0)
ID: 1 => ('c(c(B(x')))', D1, R15, P[], S{x18 -> c(c(B(x')))}), NR: 'c(c(B(x')))'
t: c(c(c(a(x'))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('c(c(c(a(x'))))', D0)
ID: 1 => ('c(c(B(x')))', D1, R23, P[1, 1], S{x26 -> x'}), NR: 'B(x')'
SNodesPath: C(c(c(c(B(x'))))) -> c(c(B(x')))
TNodesPath: c(c(c(a(x')))) -> c(c(B(x')))
C(c(c(c(B(x'))))) ->* c(c(B(x'))) *<- c(c(c(a(x'))))
"Joinable"

The problem is confluent.

Problem 1.9: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<b(b(B(B(B(x'))))),B(B(B(b(b(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N19
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: b(b(B(B(B(x')))))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('b(b(B(B(B(x')))))', D0)
ID: 1 => ('b(B(B(x')))', D1, R19, P[1], S{x22 -> B(B(x'))}), NR: 'B(B(x'))'
ID: 2 => ('B(x')', D2, R19, P[], S{x22 -> B(x')}), NR: 'B(x')'
t: B(B(B(b(b(x')))))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('B(B(B(b(b(x')))))', D0)
ID: 1 => ('B(B(b(x')))', D1, R10, P[1, 1], S{x13 -> b(x')}), NR: 'b(x')'
ID: 2 => ('B(x')', D2, R10, P[1], S{x13 -> x'}), NR: 'x''
SNodesPath: b(b(B(B(B(x'))))) -> b(B(B(x'))) -> B(x')
TNodesPath: B(B(B(b(b(x'))))) -> B(B(b(x'))) -> B(x')
b(b(B(B(B(x'))))) ->* B(x') *<- B(B(B(b(b(x')))))
"Joinable"

The problem is confluent.

Problem 1.10: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<B(a(a(A(A(A(x')))))),c(A(A(A(a(a(a(x')))))))>   =>  Not trivial, Not overlay, Proper, NW0, N23
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: B(a(a(A(A(A(x'))))))
Nodes: [0,1,2,3]
Edges: [(0,1),(1,2),(2,3)]
ID: 0 => ('B(a(a(A(A(A(x'))))))', D0)
ID: 1 => ('B(a(A(A(x'))))', D1, R16, P[1, 1], S{x19 -> A(A(x'))}), NR: 'A(A(x'))'
ID: 2 => ('B(A(x'))', D2, R16, P[1], S{x19 -> A(x')}), NR: 'A(x')'
ID: 3 => ('c(x')', D3, R6, P[], S{x9 -> x'}), NR: 'c(x')'
t: c(A(A(A(a(a(a(x')))))))
Nodes: [0,1,2,3]
Edges: [(0,1),(1,2),(2,3)]
ID: 0 => ('c(A(A(A(a(a(a(x')))))))', D0)
ID: 1 => ('c(A(A(a(a(x')))))', D1, R4, P[1, 1, 1], S{x7 -> a(a(x'))}), NR: 'a(a(x'))'
ID: 2 => ('c(A(a(x')))', D2, R4, P[1, 1], S{x7 -> a(x')}), NR: 'a(x')'
ID: 3 => ('c(x')', D3, R4, P[1], S{x7 -> x'}), NR: 'x''
SNodesPath: B(a(a(A(A(A(x')))))) -> B(a(A(A(x')))) -> B(A(x')) -> c(x')
TNodesPath: c(A(A(A(a(a(a(x'))))))) -> c(A(A(a(a(x'))))) -> c(A(a(x'))) -> c(x')
B(a(a(A(A(A(x')))))) ->* c(x') *<- c(A(A(A(a(a(a(x')))))))
"Joinable"

The problem is confluent.

Problem 1.11: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<c(a(B(B(B(x'))))),b(b(b(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N25
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: c(a(B(B(B(x')))))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('c(a(B(B(B(x')))))', D0)
ID: 1 => ('B(B(B(B(x'))))', D1, R23, P[], S{x26 -> B(B(B(x')))}), NR: 'B(B(B(B(x'))))'
ID: 2 => ('b(b(b(x')))', D2, R7, P[], S{x10 -> x'}), NR: 'b(b(b(x')))'
t: b(b(b(x')))
Nodes: [0]
Edges: []
ID: 0 => ('b(b(b(x')))', D0)
SNodesPath: c(a(B(B(B(x'))))) -> B(B(B(B(x')))) -> b(b(b(x')))
TNodesPath: b(b(b(x')))
c(a(B(B(B(x'))))) ->* b(b(b(x'))) *<- b(b(b(x')))
"Joinable"

The problem is confluent.

Problem 1.12: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<b(c(A(A(A(x'))))),a(a(a(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N28
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: b(c(A(A(A(x')))))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('b(c(A(A(A(x')))))', D0)
ID: 1 => ('A(A(A(A(x'))))', D1, R21, P[], S{x24 -> A(A(A(x')))}), NR: 'A(A(A(A(x'))))'
ID: 2 => ('a(a(a(x')))', D2, R1, P[], S{x4 -> x'}), NR: 'a(a(a(x')))'
t: a(a(a(x')))
Nodes: [0]
Edges: []
ID: 0 => ('a(a(a(x')))', D0)
SNodesPath: b(c(A(A(A(x'))))) -> A(A(A(A(x')))) -> a(a(a(x')))
TNodesPath: a(a(a(x')))
b(c(A(A(A(x'))))) ->* a(a(a(x'))) *<- a(a(a(x')))
"Joinable"

The problem is confluent.

Problem 1.13: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<C(C(C(A(A(A(x')))))),c(c(c(B(a(a(a(x')))))))>   =>  Not trivial, Not overlay, Proper, NW0, N29
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: C(C(C(A(A(A(x'))))))
Nodes: [0]
Edges: []
ID: 0 => ('C(C(C(A(A(A(x'))))))', D0)
t: c(c(c(B(a(a(a(x')))))))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('c(c(c(B(a(a(a(x')))))))', D0)
ID: 1 => ('c(c(c(c(A(A(A(x')))))))', D1, R9, P[1, 1, 1], S{x12 -> x'}), NR: 'c(A(A(A(x'))))'
ID: 2 => ('C(C(C(A(A(A(x'))))))', D2, R24, P[], S{x27 -> A(A(A(x')))}), NR: 'C(C(C(A(A(A(x'))))))'
SNodesPath: C(C(C(A(A(A(x'))))))
TNodesPath: c(c(c(B(a(a(a(x'))))))) -> c(c(c(c(A(A(A(x'))))))) -> C(C(C(A(A(A(x'))))))
C(C(C(A(A(A(x')))))) ->* C(C(C(A(A(A(x')))))) *<- c(c(c(B(a(a(a(x')))))))
"Joinable"

The problem is confluent.

Problem 1.14: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<A(A(A(x'))),a(a(a(C(B(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N30
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: A(A(A(x')))
Nodes: [0]
Edges: []
ID: 0 => ('A(A(A(x')))', D0)
t: a(a(a(C(B(x')))))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('a(a(a(C(B(x')))))', D0)
ID: 1 => ('a(a(a(a(x'))))', D1, R11, P[1, 1, 1], S{x14 -> x'}), NR: 'a(x')'
ID: 2 => ('A(A(A(x')))', D2, R17, P[], S{x20 -> x'}), NR: 'A(A(A(x')))'
SNodesPath: A(A(A(x')))
TNodesPath: a(a(a(C(B(x'))))) -> a(a(a(a(x')))) -> A(A(A(x')))
A(A(A(x'))) ->* A(A(A(x'))) *<- a(a(a(C(B(x')))))
"Joinable"

The problem is confluent.

Problem 1.15: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<C(B(B(B(x')))),a(B(B(B(b(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N31
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: C(B(B(B(x'))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('C(B(B(B(x'))))', D0)
ID: 1 => ('a(B(B(x')))', D1, R11, P[], S{x14 -> B(B(x'))}), NR: 'a(B(B(x')))'
t: a(B(B(B(b(x')))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('a(B(B(B(b(x')))))', D0)
ID: 1 => ('a(B(B(x')))', D1, R10, P[1, 1, 1], S{x13 -> x'}), NR: 'x''
SNodesPath: C(B(B(B(x')))) -> a(B(B(x')))
TNodesPath: a(B(B(B(b(x'))))) -> a(B(B(x')))
C(B(B(B(x')))) ->* a(B(B(x'))) *<- a(B(B(B(b(x')))))
"Joinable"

The problem is confluent.

Problem 1.16: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<B(a(a(a(C(x'))))),c(A(A(b(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N32
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: B(a(a(a(C(x')))))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('B(a(a(a(C(x')))))', D0)
ID: 1 => ('c(A(A(A(C(x')))))', D1, R9, P[], S{x12 -> C(x')}), NR: 'c(A(A(A(C(x')))))'
ID: 2 => ('c(A(A(b(x'))))', D2, R3, P[1, 1, 1], S{x6 -> x'}), NR: 'b(x')'
t: c(A(A(b(x'))))
Nodes: [0]
Edges: []
ID: 0 => ('c(A(A(b(x'))))', D0)
SNodesPath: B(a(a(a(C(x'))))) -> c(A(A(A(C(x'))))) -> c(A(A(b(x'))))
TNodesPath: c(A(A(b(x'))))
B(a(a(a(C(x'))))) ->* c(A(A(b(x')))) *<- c(A(A(b(x'))))
"Joinable"

The problem is confluent.

Problem 1.17: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<a(x'),C(B(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N33
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: a(x')
Nodes: [0]
Edges: []
ID: 0 => ('a(x')', D0)
t: C(B(x'))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('C(B(x'))', D0)
ID: 1 => ('a(x')', D1, R11, P[], S{x14 -> x'}), NR: 'a(x')'
SNodesPath: a(x')
TNodesPath: C(B(x')) -> a(x')
a(x') ->* a(x') *<- C(B(x'))
"Joinable"

The problem is confluent.

Problem 1.18: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<C(c(c(c(x')))),c(c(c(C(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N35
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: C(c(c(c(x'))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('C(c(c(c(x'))))', D0)
ID: 1 => ('c(c(x'))', D1, R15, P[], S{x18 -> c(c(x'))}), NR: 'c(c(x'))'
t: c(c(c(C(x'))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('c(c(c(C(x'))))', D0)
ID: 1 => ('c(c(x'))', D1, R22, P[1, 1], S{x25 -> x'}), NR: 'x''
SNodesPath: C(c(c(c(x')))) -> c(c(x'))
TNodesPath: c(c(c(C(x')))) -> c(c(x'))
C(c(c(c(x')))) ->* c(c(x')) *<- c(c(c(C(x'))))
"Joinable"

The problem is confluent.

Problem 1.19: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<B(A(x')),c(x')>   =>  Not trivial, Not overlay, Proper, NW0, N36
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: B(A(x'))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('B(A(x'))', D0)
ID: 1 => ('c(x')', D1, R6, P[], S{x9 -> x'}), NR: 'c(x')'
t: c(x')
Nodes: [0]
Edges: []
ID: 0 => ('c(x')', D0)
SNodesPath: B(A(x')) -> c(x')
TNodesPath: c(x')
B(A(x')) ->* c(x') *<- c(x')
"Joinable"

The problem is confluent.

Problem 1.20: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<b(B(B(B(x')))),B(B(B(b(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N37
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: b(B(B(B(x'))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('b(B(B(B(x'))))', D0)
ID: 1 => ('B(B(x'))', D1, R19, P[], S{x22 -> B(B(x'))}), NR: 'B(B(x'))'
t: B(B(B(b(x'))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('B(B(B(b(x'))))', D0)
ID: 1 => ('B(B(x'))', D1, R10, P[1, 1], S{x13 -> x'}), NR: 'x''
SNodesPath: b(B(B(B(x')))) -> B(B(x'))
TNodesPath: B(B(B(b(x')))) -> B(B(x'))
b(B(B(B(x')))) ->* B(B(x')) *<- B(B(B(b(x'))))
"Joinable"

The problem is confluent.

Problem 1.21: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<b(c(x')),A(x')>   =>  Not trivial, Not overlay, Proper, NW0, N38
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: b(c(x'))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('b(c(x'))', D0)
ID: 1 => ('A(x')', D1, R21, P[], S{x24 -> x'}), NR: 'A(x')'
t: A(x')
Nodes: [0]
Edges: []
ID: 0 => ('A(x')', D0)
SNodesPath: b(c(x')) -> A(x')
TNodesPath: A(x')
b(c(x')) ->* A(x') *<- A(x')
"Joinable"

The problem is confluent.

Problem 1.22: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<C(C(C(c(c(c(B(x'))))))),c(c(c(C(C(a(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N40
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: C(C(C(c(c(c(B(x')))))))
Nodes: [0,1,2,3]
Edges: [(0,1),(1,2),(2,3)]
ID: 0 => ('C(C(C(c(c(c(B(x')))))))', D0)
ID: 1 => ('C(C(c(c(B(x')))))', D1, R15, P[1, 1], S{x18 -> c(c(B(x')))}), NR: 'c(c(B(x')))'
ID: 2 => ('C(c(B(x')))', D2, R15, P[1], S{x18 -> c(B(x'))}), NR: 'c(B(x'))'
ID: 3 => ('B(x')', D3, R15, P[], S{x18 -> B(x')}), NR: 'B(x')'
t: c(c(c(C(C(a(x'))))))
Nodes: [0,1,2,3]
Edges: [(0,1),(1,2),(2,3)]
ID: 0 => ('c(c(c(C(C(a(x'))))))', D0)
ID: 1 => ('c(c(C(a(x'))))', D1, R22, P[1, 1], S{x25 -> C(a(x'))}), NR: 'C(a(x'))'
ID: 2 => ('c(a(x'))', D2, R22, P[1], S{x25 -> a(x')}), NR: 'a(x')'
ID: 3 => ('B(x')', D3, R23, P[], S{x26 -> x'}), NR: 'B(x')'
SNodesPath: C(C(C(c(c(c(B(x'))))))) -> C(C(c(c(B(x'))))) -> C(c(B(x'))) -> B(x')
TNodesPath: c(c(c(C(C(a(x')))))) -> c(c(C(a(x')))) -> c(a(x')) -> B(x')
C(C(C(c(c(c(B(x'))))))) ->* B(x') *<- c(c(c(C(C(a(x'))))))
"Joinable"

The problem is confluent.

Problem 1.23: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<C(b(B(B(B(x'))))),a(B(B(B(b(b(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N41
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: C(b(B(B(B(x')))))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('C(b(B(B(B(x')))))', D0)
ID: 1 => ('C(B(B(x')))', D1, R19, P[1], S{x22 -> B(B(x'))}), NR: 'B(B(x'))'
ID: 2 => ('a(B(x'))', D2, R11, P[], S{x14 -> B(x')}), NR: 'a(B(x'))'
t: a(B(B(B(b(b(x'))))))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('a(B(B(B(b(b(x'))))))', D0)
ID: 1 => ('a(B(B(b(x'))))', D1, R10, P[1, 1, 1], S{x13 -> b(x')}), NR: 'b(x')'
ID: 2 => ('a(B(x'))', D2, R10, P[1, 1], S{x13 -> x'}), NR: 'x''
SNodesPath: C(b(B(B(B(x'))))) -> C(B(B(x'))) -> a(B(x'))
TNodesPath: a(B(B(B(b(b(x')))))) -> a(B(B(b(x')))) -> a(B(x'))
C(b(B(B(B(x'))))) ->* a(B(x')) *<- a(B(B(B(b(b(x'))))))
"Joinable"

The problem is confluent.

Problem 1.24: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<A(C(C(C(x')))),b(C(C(C(c(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N42
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: A(C(C(C(x'))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('A(C(C(C(x'))))', D0)
ID: 1 => ('b(C(C(x')))', D1, R3, P[], S{x6 -> C(C(x'))}), NR: 'b(C(C(x')))'
t: b(C(C(C(c(x')))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('b(C(C(C(c(x')))))', D0)
ID: 1 => ('b(C(C(x')))', D1, R15, P[1, 1, 1], S{x18 -> x'}), NR: 'x''
SNodesPath: A(C(C(C(x')))) -> b(C(C(x')))
TNodesPath: b(C(C(C(c(x'))))) -> b(C(C(x')))
A(C(C(C(x')))) ->* b(C(C(x'))) *<- b(C(C(C(c(x')))))
"Joinable"

The problem is confluent.

Problem 1.25: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<B(B(B(x'))),b(b(b(A(C(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N43
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: B(B(B(x')))
Nodes: [0]
Edges: []
ID: 0 => ('B(B(B(x')))', D0)
t: b(b(b(A(C(x')))))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('b(b(b(A(C(x')))))', D0)
ID: 1 => ('b(b(b(b(x'))))', D1, R3, P[1, 1, 1], S{x6 -> x'}), NR: 'b(x')'
ID: 2 => ('B(B(B(x')))', D2, R20, P[], S{x23 -> x'}), NR: 'B(B(B(x')))'
SNodesPath: B(B(B(x')))
TNodesPath: b(b(b(A(C(x'))))) -> b(b(b(b(x')))) -> B(B(B(x')))
B(B(B(x'))) ->* B(B(B(x'))) *<- b(b(b(A(C(x')))))
"Joinable"

The problem is confluent.

Problem 1.26: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<A(a(x')),b(B(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N44
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: A(a(x'))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('A(a(x'))', D0)
ID: 1 => ('x'', D1, R4, P[], S{x7 -> x'}), NR: 'x''
t: b(B(x'))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('b(B(x'))', D0)
ID: 1 => ('x'', D1, R19, P[], S{x22 -> x'}), NR: 'x''
SNodesPath: A(a(x')) -> x'
TNodesPath: b(B(x')) -> x'
A(a(x')) ->* x' *<- b(B(x'))
"Joinable"

The problem is confluent.

Problem 1.27: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<b(b(b(A(x')))),B(B(B(c(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N45
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: b(b(b(A(x'))))
Nodes: [0]
Edges: []
ID: 0 => ('b(b(b(A(x'))))', D0)
t: B(B(B(c(x'))))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('B(B(B(c(x'))))', D0)
ID: 1 => ('b(b(b(A(x'))))', D1, R8, P[], S{x11 -> x'}), NR: 'b(b(b(A(x'))))'
SNodesPath: b(b(b(A(x'))))
TNodesPath: B(B(B(c(x')))) -> b(b(b(A(x'))))
b(b(b(A(x')))) ->* b(b(b(A(x')))) *<- B(B(B(c(x'))))
"Joinable"

The problem is confluent.

Problem 1.28: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<A(c(c(x'))),b(C(C(C(C(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N46
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: A(c(c(x')))
Nodes: [0]
Edges: []
ID: 0 => ('A(c(c(x')))', D0)
t: b(C(C(C(C(x')))))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('b(C(C(C(C(x')))))', D0)
ID: 1 => ('b(c(c(c(x'))))', D1, R12, P[1], S{x15 -> x'}), NR: 'c(c(c(x')))'
ID: 2 => ('A(c(c(x')))', D2, R21, P[], S{x24 -> c(c(x'))}), NR: 'A(c(c(x')))'
SNodesPath: A(c(c(x')))
TNodesPath: b(C(C(C(C(x'))))) -> b(c(c(c(x')))) -> A(c(c(x')))
A(c(c(x'))) ->* A(c(c(x'))) *<- b(C(C(C(C(x')))))
"Joinable"

The problem is confluent.

Problem 1.29: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<A(A(A(B(B(B(x')))))),a(a(a(C(b(b(b(x')))))))>   =>  Not trivial, Not overlay, Proper, NW0, N49
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: A(A(A(B(B(B(x'))))))
Nodes: [0]
Edges: []
ID: 0 => ('A(A(A(B(B(B(x'))))))', D0)
t: a(a(a(C(b(b(b(x')))))))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('a(a(a(C(b(b(b(x')))))))', D0)
ID: 1 => ('a(a(a(a(B(B(B(x')))))))', D1, R14, P[1, 1, 1], S{x17 -> x'}), NR: 'a(B(B(B(x'))))'
ID: 2 => ('A(A(A(B(B(B(x'))))))', D2, R17, P[], S{x20 -> B(B(B(x')))}), NR: 'A(A(A(B(B(B(x'))))))'
SNodesPath: A(A(A(B(B(B(x'))))))
TNodesPath: a(a(a(C(b(b(b(x'))))))) -> a(a(a(a(B(B(B(x'))))))) -> A(A(A(B(B(B(x'))))))
A(A(A(B(B(B(x')))))) ->* A(A(A(B(B(B(x')))))) *<- a(a(a(C(b(b(b(x')))))))
"Joinable"

The problem is confluent.

Problem 1.30: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<a(b(C(C(C(x'))))),c(c(c(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N51
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: a(b(C(C(C(x')))))
Nodes: []
Edges: []

t: c(c(c(x')))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
a(b(C(C(C(x'))))) ->* c(c(c(x'))) *<- c(c(c(x')))
"Joinable"

The problem is confluent.

Problem 1.31: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<B(B(b(b(b(x'))))),b(b(b(B(B(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N52
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: B(B(b(b(b(x')))))
Nodes: []
Edges: []

t: b(b(b(B(B(x')))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
B(B(b(b(b(x'))))) ->* b(x') *<- b(b(b(B(B(x')))))
"Joinable"

The problem is confluent.

Problem 1.32: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<B(B(B(b(b(b(x')))))),b(b(b(B(B(B(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N53
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: B(B(B(b(b(b(x'))))))
Nodes: []
Edges: []

t: b(b(b(B(B(B(x'))))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
B(B(B(b(b(b(x')))))) ->* x' *<- b(b(b(B(B(B(x'))))))
"Joinable"

The problem is confluent.

Problem 1.33: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<a(a(A(A(A(x'))))),A(A(A(a(a(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N55
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: a(a(A(A(A(x')))))
Nodes: []
Edges: []

t: A(A(A(a(a(x')))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
a(a(A(A(A(x'))))) ->* A(x') *<- A(A(A(a(a(x')))))
"Joinable"

The problem is confluent.

Problem 1.34: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<c(C(x')),B(b(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N56
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: c(C(x'))
Nodes: []
Edges: []

t: B(b(x'))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
c(C(x')) ->* x' *<- B(b(x'))
"Joinable"

The problem is confluent.

Problem 1.35: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<C(C(C(C(x')))),c(c(c(B(b(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N57
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: C(C(C(C(x'))))
Nodes: []
Edges: []

t: c(c(c(B(b(x')))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
C(C(C(C(x')))) ->* c(c(c(x'))) *<- c(c(c(B(b(x')))))
"Joinable"

The problem is confluent.

Problem 1.36: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<B(B(B(C(C(C(x')))))),b(b(b(A(c(c(c(x')))))))>   =>  Not trivial, Not overlay, Proper, NW0, N58
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: B(B(B(C(C(C(x'))))))
Nodes: []
Edges: []

t: b(b(b(A(c(c(c(x')))))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
B(B(B(C(C(C(x')))))) ->* B(B(B(C(C(C(x')))))) *<- b(b(b(A(c(c(c(x')))))))
"Joinable"

The problem is confluent.

Problem 1.37: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<B(B(B(B(x')))),b(b(b(A(a(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N59
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: B(B(B(B(x'))))
Nodes: []
Edges: []

t: b(b(b(A(a(x')))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
B(B(B(B(x')))) ->* b(b(b(x'))) *<- b(b(b(A(a(x')))))
"Joinable"

The problem is confluent.

Problem 1.38: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<A(A(A(A(x')))),a(a(a(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N60
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: A(A(A(A(x'))))
Nodes: []
Edges: []

t: a(a(a(x')))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
A(A(A(A(x')))) ->* a(a(a(x'))) *<- a(a(a(x')))
"Joinable"

The problem is confluent.

Problem 1.39: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<b(C(C(C(x')))),A(c(c(c(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N61
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: b(C(C(C(x'))))
Nodes: []
Edges: []

t: A(c(c(c(x'))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
b(C(C(C(x')))) ->* b(C(C(C(x')))) *<- A(c(c(c(x'))))
"Joinable"

The problem is confluent.

Problem 1.40: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<B(a(a(x'))),c(A(A(A(A(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N62
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: B(a(a(x')))
Nodes: []
Edges: []

t: c(A(A(A(A(x')))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
B(a(a(x'))) ->* B(a(a(x'))) *<- c(A(A(A(A(x')))))
"Joinable"

The problem is confluent.

Problem 1.41: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<B(B(B(c(A(A(A(x'))))))),b(b(b(a(a(a(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N63
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: B(B(B(c(A(A(A(x')))))))
Nodes: []
Edges: []

t: b(b(b(a(a(a(x'))))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
B(B(B(c(A(A(A(x'))))))) ->* b(b(b(a(a(a(x')))))) *<- b(b(b(a(a(a(x'))))))
"Joinable"

The problem is confluent.

Problem 1.42: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<c(c(c(c(x')))),C(C(C(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N64
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: c(c(c(c(x'))))
Nodes: []
Edges: []

t: C(C(C(x')))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
c(c(c(c(x')))) ->* C(C(C(x'))) *<- C(C(C(x')))
"Joinable"

The problem is confluent.

Problem 1.43: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<c(c(c(B(x')))),C(C(C(a(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N66
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: c(c(c(B(x'))))
Nodes: []
Edges: []

t: C(C(C(a(x'))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
c(c(c(B(x')))) ->* c(c(c(B(x')))) *<- C(C(C(a(x'))))
"Joinable"

The problem is confluent.

Problem 1.44: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<C(C(c(c(c(x'))))),c(c(c(C(C(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N67
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: C(C(c(c(c(x')))))
Nodes: []
Edges: []

t: c(c(c(C(C(x')))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
C(C(c(c(c(x'))))) ->* c(x') *<- c(c(c(C(C(x')))))
"Joinable"

The problem is confluent.

Problem 1.45: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<C(b(b(b(A(x'))))),a(B(B(c(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N68
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: C(b(b(b(A(x')))))
Nodes: []
Edges: []

t: a(B(B(c(x'))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
C(b(b(b(A(x'))))) ->* a(B(B(c(x')))) *<- a(B(B(c(x'))))
"Joinable"

The problem is confluent.

Problem 1.46: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<c(c(C(C(C(x'))))),C(C(C(c(c(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N69
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: c(c(C(C(C(x')))))
Nodes: []
Edges: []

t: C(C(C(c(c(x')))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
c(c(C(C(C(x'))))) ->* C(x') *<- C(C(C(c(c(x')))))
"Joinable"

The problem is confluent.

Problem 1.47: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<a(a(a(a(x')))),A(A(A(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N70
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: a(a(a(a(x'))))
Nodes: []
Edges: []

t: A(A(A(x')))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
a(a(a(a(x')))) ->* A(A(A(x'))) *<- A(A(A(x')))
"Joinable"

The problem is confluent.

Problem 1.48: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<B(B(b(b(b(A(x')))))),b(b(b(B(c(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N71
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: B(B(b(b(b(A(x'))))))
Nodes: []
Edges: []

t: b(b(b(B(c(x')))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
B(B(b(b(b(A(x')))))) ->* b(A(x')) *<- b(b(b(B(c(x')))))
"Joinable"

The problem is confluent.

Problem 1.49: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<B(a(a(C(x')))),c(A(A(A(b(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N73
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: B(a(a(C(x'))))
Nodes: []
Edges: []

t: c(A(A(A(b(x')))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
B(a(a(C(x')))) ->* B(a(a(C(x')))) *<- c(A(A(A(b(x')))))
"Joinable"

The problem is confluent.

Problem 1.50: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<B(A(A(A(x')))),c(A(A(A(a(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N75
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: B(A(A(A(x'))))
Nodes: []
Edges: []

t: c(A(A(A(a(x')))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
B(A(A(A(x')))) ->* c(A(A(x'))) *<- c(A(A(A(a(x')))))
"Joinable"

The problem is confluent.

Problem 1.51: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<C(b(b(A(x')))),a(B(B(B(c(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N76
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: C(b(b(A(x'))))
Nodes: []
Edges: []

t: a(B(B(B(c(x')))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
C(b(b(A(x')))) ->* C(b(b(A(x')))) *<- a(B(B(B(c(x')))))
"Joinable"

The problem is confluent.

Problem 1.52: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<A(c(c(c(B(x'))))),b(C(C(a(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N77
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: A(c(c(c(B(x')))))
Nodes: []
Edges: []

t: b(C(C(a(x'))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
A(c(c(c(B(x'))))) ->* b(C(C(a(x')))) *<- b(C(C(a(x'))))
"Joinable"

The problem is confluent.

Problem 1.53: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<C(C(C(C(x')))),c(c(c(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N78
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: C(C(C(C(x'))))
Nodes: []
Edges: []

t: c(c(c(x')))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
C(C(C(C(x')))) ->* c(c(c(x'))) *<- c(c(c(x')))
"Joinable"

The problem is confluent.

Problem 1.54: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<C(C(c(c(c(B(x')))))),c(c(c(C(a(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N80
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: C(C(c(c(c(B(x'))))))
Nodes: []
Edges: []

t: c(c(c(C(a(x')))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
C(C(c(c(c(B(x')))))) ->* c(B(x')) *<- c(c(c(C(a(x')))))
"Joinable"

The problem is confluent.

Problem 1.55: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<A(c(c(B(x')))),b(C(C(C(a(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N81
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: A(c(c(B(x'))))
Nodes: []
Edges: []

t: b(C(C(C(a(x')))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
A(c(c(B(x')))) ->* A(c(c(B(x')))) *<- b(C(C(C(a(x')))))
"Joinable"

The problem is confluent.

Problem 1.56: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<A(A(a(a(a(C(x')))))),a(a(a(A(b(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N82
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: A(A(a(a(a(C(x'))))))
Nodes: []
Edges: []

t: a(a(a(A(b(x')))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
A(A(a(a(a(C(x')))))) ->* a(C(x')) *<- a(a(a(A(b(x')))))
"Joinable"

The problem is confluent.

Problem 1.57: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<c(c(c(c(B(x'))))),C(C(a(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N83
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: c(c(c(c(B(x')))))
Nodes: []
Edges: []

t: C(C(a(x')))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
c(c(c(c(B(x'))))) ->* C(C(a(x'))) *<- C(C(a(x')))
"Joinable"

The problem is confluent.

Problem 1.58: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<b(x'),A(C(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N84
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: b(x')
Nodes: []
Edges: []

t: A(C(x'))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
b(x') ->* b(x') *<- A(C(x'))
"Joinable"

The problem is confluent.

Problem 1.59: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<B(b(b(b(A(x'))))),b(b(b(c(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N85
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: B(b(b(b(A(x')))))
Nodes: []
Edges: []

t: b(b(b(c(x'))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
B(b(b(b(A(x'))))) ->* b(b(A(x'))) *<- b(b(b(c(x'))))
"Joinable"

The problem is confluent.

Problem 1.60: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<A(A(A(b(C(C(C(x'))))))),a(a(a(c(c(c(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N86
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: A(A(A(b(C(C(C(x')))))))
Nodes: []
Edges: []

t: a(a(a(c(c(c(x'))))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
A(A(A(b(C(C(C(x'))))))) ->* a(a(a(c(c(c(x')))))) *<- a(a(a(c(c(c(x'))))))
"Joinable"

The problem is confluent.

Problem 1.61: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<a(a(a(A(A(A(x')))))),A(A(A(a(a(a(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N87
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: a(a(a(A(A(A(x'))))))
Nodes: []
Edges: []

t: A(A(A(a(a(a(x'))))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
a(a(a(A(A(A(x')))))) ->* x' *<- A(A(A(a(a(a(x'))))))
"Joinable"

The problem is confluent.

Problem 1.62: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<A(c(C(C(C(x'))))),b(C(C(C(c(c(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N88
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: A(c(C(C(C(x')))))
Nodes: []
Edges: []

t: b(C(C(C(c(c(x'))))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
A(c(C(C(C(x'))))) ->* b(C(x')) *<- b(C(C(C(c(c(x'))))))
"Joinable"

The problem is confluent.

Problem 1.63: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<A(A(a(a(a(x'))))),a(a(a(A(A(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N89
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: A(A(a(a(a(x')))))
Nodes: []
Edges: []

t: a(a(a(A(A(x')))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
A(A(a(a(a(x'))))) ->* a(x') *<- a(a(a(A(A(x')))))
"Joinable"

The problem is confluent.

Problem 1.64: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<C(b(b(x'))),a(B(B(B(B(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N90
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: C(b(b(x')))
Nodes: []
Edges: []

t: a(B(B(B(B(x')))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
C(b(b(x'))) ->* C(b(b(x'))) *<- a(B(B(B(B(x')))))
"Joinable"

The problem is confluent.

Problem 1.65: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<C(C(C(x'))),c(c(c(B(A(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N91
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: C(C(C(x')))
Nodes: []
Edges: []

t: c(c(c(B(A(x')))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
C(C(C(x'))) ->* C(C(C(x'))) *<- c(c(c(B(A(x')))))
"Joinable"

The problem is confluent.

Problem 1.66: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<A(a(a(a(C(x'))))),a(a(a(b(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N92
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: A(a(a(a(C(x')))))
Nodes: []
Edges: []

t: a(a(a(b(x'))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
A(a(a(a(C(x'))))) ->* a(a(C(x'))) *<- a(a(a(b(x'))))
"Joinable"

The problem is confluent.

Problem 1.67: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<C(b(b(B(B(B(x')))))),a(B(B(B(b(b(b(x')))))))>   =>  Not trivial, Not overlay, Proper, NW0, N93
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: C(b(b(B(B(B(x'))))))
Nodes: []
Edges: []

t: a(B(B(B(b(b(b(x')))))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
C(b(b(B(B(B(x')))))) ->* a(x') *<- a(B(B(B(b(b(b(x')))))))
"Joinable"

The problem is confluent.

Problem 1.68: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<B(x'),c(a(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N96
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: B(x')
Nodes: []
Edges: []

t: c(a(x'))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
B(x') ->* B(x') *<- c(a(x'))
"Joinable"

The problem is confluent.

Problem 1.69: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<B(b(C(C(C(x'))))),c(c(c(c(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N97
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: B(b(C(C(C(x')))))
Nodes: []
Edges: []

t: c(c(c(c(x'))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
B(b(C(C(C(x'))))) ->* C(C(C(x'))) *<- c(c(c(c(x'))))
"Joinable"

The problem is confluent.

Problem 1.70: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<b(b(b(b(A(x'))))),B(B(c(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N98
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: b(b(b(b(A(x')))))
Nodes: []
Edges: []

t: B(B(c(x')))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
b(b(b(b(A(x'))))) ->* B(B(c(x'))) *<- B(B(c(x')))
"Joinable"

The problem is confluent.

Problem 1.71: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<B(a(a(a(x')))),c(A(A(A(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N99
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: B(a(a(a(x'))))
Nodes: []
Edges: []

t: c(A(A(A(x'))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
B(a(a(a(x')))) ->* c(A(A(A(x')))) *<- c(A(A(A(x'))))
"Joinable"

The problem is confluent.

Problem 1.72: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<a(b(x')),C(x')>   =>  Not trivial, Not overlay, Proper, NW0, N100
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: a(b(x'))
Nodes: []
Edges: []

t: C(x')
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
a(b(x')) ->* C(x') *<- C(x')
"Joinable"

The problem is confluent.

Problem 1.73: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<A(a(a(a(x')))),a(a(a(A(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N101
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: A(a(a(a(x'))))
Nodes: []
Edges: []

t: a(a(a(A(x'))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
A(a(a(a(x')))) ->* a(a(x')) *<- a(a(a(A(x'))))
"Joinable"

The problem is confluent.

Problem 1.74: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<A(a(B(B(B(x'))))),b(b(b(b(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N102
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: A(a(B(B(B(x')))))
Nodes: []
Edges: []

t: b(b(b(b(x'))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
A(a(B(B(B(x'))))) ->* B(B(B(x'))) *<- b(b(b(b(x'))))
"Joinable"

The problem is confluent.

Problem 1.75: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<C(b(b(b(x')))),a(B(B(B(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N103
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: C(b(b(b(x'))))
Nodes: []
Edges: []

t: a(B(B(B(x'))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
C(b(b(b(x')))) ->* a(B(B(B(x')))) *<- a(B(B(B(x'))))
"Joinable"

The problem is confluent.

Problem 1.76: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<a(A(x')),C(c(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N104
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: a(A(x'))
Nodes: []
Edges: []

t: C(c(x'))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
a(A(x')) ->* x' *<- C(c(x'))
"Joinable"

The problem is confluent.

Problem 1.77: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<A(A(A(a(a(a(C(x'))))))),a(a(a(A(A(b(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N105
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: A(A(A(a(a(a(C(x')))))))
Nodes: []
Edges: []

t: a(a(a(A(A(b(x'))))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
A(A(A(a(a(a(C(x'))))))) ->* C(x') *<- a(a(a(A(A(b(x'))))))
"Joinable"

The problem is confluent.

Problem 1.78: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<b(b(b(x'))),B(B(B(B(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N107
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: b(b(b(x')))
Nodes: []
Edges: []

t: B(B(B(B(x'))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
b(b(b(x'))) ->* b(b(b(x'))) *<- B(B(B(B(x'))))
"Joinable"

The problem is confluent.

Problem 1.79: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<a(a(a(C(x')))),A(A(A(b(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N108
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: a(a(a(C(x'))))
Nodes: []
Edges: []

t: A(A(A(b(x'))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
a(a(a(C(x')))) ->* a(a(a(C(x')))) *<- A(A(A(b(x'))))
"Joinable"

The problem is confluent.

Problem 1.80: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<A(A(A(A(x')))),a(a(a(C(c(x')))))>   =>  Not trivial, Not overlay, Proper, NW0, N109
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: A(A(A(A(x'))))
Nodes: []
Edges: []

t: a(a(a(C(c(x')))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
A(A(A(A(x')))) ->* a(a(a(x'))) *<- a(a(a(C(c(x')))))
"Joinable"

The problem is confluent.

Problem 1.81: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<B(B(B(x'))),b(b(b(b(x'))))>   =>  Not trivial, Not overlay, Proper, NW0, N110
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: B(B(B(x')))
Nodes: []
Edges: []

t: b(b(b(b(x'))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
B(B(B(x'))) ->* B(B(B(x'))) *<- b(b(b(b(x'))))
"Joinable"

The problem is confluent.

Problem 1.82: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<B(a(A(A(A(x'))))),c(A(A(A(a(a(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N112
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: B(a(A(A(A(x')))))
Nodes: []
Edges: []

t: c(A(A(A(a(a(x'))))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
B(a(A(A(A(x'))))) ->* c(A(x')) *<- c(A(A(A(a(a(x'))))))
"Joinable"

The problem is confluent.

Problem 1.83: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<A(c(c(C(C(C(x')))))),b(C(C(C(c(c(c(x')))))))>   =>  Not trivial, Not overlay, Proper, NW0, N113
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: A(c(c(C(C(C(x'))))))
Nodes: []
Edges: []

t: b(C(C(C(c(c(c(x')))))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
A(c(c(C(C(C(x')))))) ->* b(x') *<- b(C(C(C(c(c(c(x')))))))
"Joinable"

The problem is confluent.

Problem 1.84: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(A 1)
(B 1)
(C 1)
(a 1)
(b 1)
(c 1)
(fSNonEmpty)
)
(RULES
A(A(A(A(x)))) -> a(a(a(x)))
A(A(A(b(x)))) -> a(a(a(C(x))))
A(C(x)) -> b(x)
A(a(x)) -> x
A(c(c(c(x)))) -> b(C(C(C(x))))
B(A(x)) -> c(x)
B(B(B(B(x)))) -> b(b(b(x)))
B(B(B(c(x)))) -> b(b(b(A(x))))
B(a(a(a(x)))) -> c(A(A(A(x))))
B(b(x)) -> x
C(B(x)) -> a(x)
C(C(C(C(x)))) -> c(c(c(x)))
C(C(C(a(x)))) -> c(c(c(B(x))))
C(b(b(b(x)))) -> a(B(B(B(x))))
C(c(x)) -> x
a(A(x)) -> x
a(a(a(a(x)))) -> A(A(A(x)))
a(b(x)) -> C(x)
b(B(x)) -> x
b(b(b(b(x)))) -> B(B(B(x)))
b(c(x)) -> A(x)
c(C(x)) -> x
c(a(x)) -> B(x)
c(c(c(c(x)))) -> C(C(C(x)))
)
Critical Pairs:
<C(C(C(c(c(c(x')))))),c(c(c(C(C(C(x'))))))>   =>  Not trivial, Not overlay, Proper, NW0, N114
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: C(C(C(c(c(c(x'))))))
Nodes: []
Edges: []

t: c(c(c(C(C(C(x'))))))
Nodes: []
Edges: []

SNodesPath: 
TNodesPath: 
C(C(C(c(c(c(x')))))) ->* x' *<- c(c(c(C(C(C(x'))))))
"Joinable"

The problem is confluent.