/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES

Problem 1: 

(VAR x1)
(RULES
a(a(b(x1))) -> d(b(a(x1)))
a(a(x1)) -> a(b(a(x1)))
a(d(x1)) -> d(a(x1))
b(a(a(x1))) -> a(b(c(x1)))
b(d(x1)) -> a(b(x1))
c(a(x1)) -> a(c(x1))
c(b(x1)) -> b(a(x1))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A(a(b(x1))) -> A(x1)
 A(a(b(x1))) -> B(a(x1))
 A(a(x1)) -> A(b(a(x1)))
 A(a(x1)) -> B(a(x1))
 A(d(x1)) -> A(x1)
 B(a(a(x1))) -> A(b(c(x1)))
 B(a(a(x1))) -> B(c(x1))
 B(a(a(x1))) -> C(x1)
 B(d(x1)) -> A(b(x1))
 B(d(x1)) -> B(x1)
 C(a(x1)) -> A(c(x1))
 C(a(x1)) -> C(x1)
 C(b(x1)) -> A(x1)
 C(b(x1)) -> B(a(x1))
-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))

Problem 1: 

SCC Processor:
-> Pairs:
 A(a(b(x1))) -> A(x1)
 A(a(b(x1))) -> B(a(x1))
 A(a(x1)) -> A(b(a(x1)))
 A(a(x1)) -> B(a(x1))
 A(d(x1)) -> A(x1)
 B(a(a(x1))) -> A(b(c(x1)))
 B(a(a(x1))) -> B(c(x1))
 B(a(a(x1))) -> C(x1)
 B(d(x1)) -> A(b(x1))
 B(d(x1)) -> B(x1)
 C(a(x1)) -> A(c(x1))
 C(a(x1)) -> C(x1)
 C(b(x1)) -> A(x1)
 C(b(x1)) -> B(a(x1))
-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(a(b(x1))) -> A(x1)
 A(a(b(x1))) -> B(a(x1))
 A(a(x1)) -> A(b(a(x1)))
 A(a(x1)) -> B(a(x1))
 A(d(x1)) -> A(x1)
 B(a(a(x1))) -> A(b(c(x1)))
 B(a(a(x1))) -> B(c(x1))
 B(a(a(x1))) -> C(x1)
 B(d(x1)) -> A(b(x1))
 B(d(x1)) -> B(x1)
 C(a(x1)) -> A(c(x1))
 C(a(x1)) -> C(x1)
 C(b(x1)) -> A(x1)
 C(b(x1)) -> B(a(x1))
->->-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A(a(b(x1))) -> A(x1)
 A(a(b(x1))) -> B(a(x1))
 A(a(x1)) -> A(b(a(x1)))
 A(a(x1)) -> B(a(x1))
 A(d(x1)) -> A(x1)
 B(a(a(x1))) -> A(b(c(x1)))
 B(a(a(x1))) -> B(c(x1))
 B(a(a(x1))) -> C(x1)
 B(d(x1)) -> A(b(x1))
 B(d(x1)) -> B(x1)
 C(a(x1)) -> A(c(x1))
 C(a(x1)) -> C(x1)
 C(b(x1)) -> A(x1)
 C(b(x1)) -> B(a(x1))
-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
-> Usable rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = 2.X + 2
[b](X) = X
[c](X) = 2.X + 2
[d](X) = 2.X + 2
[A](X) = 2.X + 2
[B](X) = X
[C](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 A(a(b(x1))) -> B(a(x1))
 A(a(x1)) -> A(b(a(x1)))
 A(a(x1)) -> B(a(x1))
 A(d(x1)) -> A(x1)
 B(a(a(x1))) -> A(b(c(x1)))
 B(a(a(x1))) -> B(c(x1))
 B(a(a(x1))) -> C(x1)
 B(d(x1)) -> A(b(x1))
 B(d(x1)) -> B(x1)
 C(a(x1)) -> A(c(x1))
 C(a(x1)) -> C(x1)
 C(b(x1)) -> A(x1)
 C(b(x1)) -> B(a(x1))
-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(a(b(x1))) -> B(a(x1))
 A(a(x1)) -> A(b(a(x1)))
 A(a(x1)) -> B(a(x1))
 A(d(x1)) -> A(x1)
 B(a(a(x1))) -> A(b(c(x1)))
 B(a(a(x1))) -> B(c(x1))
 B(a(a(x1))) -> C(x1)
 B(d(x1)) -> A(b(x1))
 B(d(x1)) -> B(x1)
 C(a(x1)) -> A(c(x1))
 C(a(x1)) -> C(x1)
 C(b(x1)) -> A(x1)
 C(b(x1)) -> B(a(x1))
->->-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A(a(b(x1))) -> B(a(x1))
 A(a(x1)) -> A(b(a(x1)))
 A(a(x1)) -> B(a(x1))
 A(d(x1)) -> A(x1)
 B(a(a(x1))) -> A(b(c(x1)))
 B(a(a(x1))) -> B(c(x1))
 B(a(a(x1))) -> C(x1)
 B(d(x1)) -> A(b(x1))
 B(d(x1)) -> B(x1)
 C(a(x1)) -> A(c(x1))
 C(a(x1)) -> C(x1)
 C(b(x1)) -> A(x1)
 C(b(x1)) -> B(a(x1))
-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
-> Usable rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = 2.X + 2
[b](X) = X
[c](X) = 2.X + 2
[d](X) = 2.X + 2
[A](X) = 2.X + 1
[B](X) = X
[C](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 A(a(x1)) -> A(b(a(x1)))
 A(a(x1)) -> B(a(x1))
 A(d(x1)) -> A(x1)
 B(a(a(x1))) -> A(b(c(x1)))
 B(a(a(x1))) -> B(c(x1))
 B(a(a(x1))) -> C(x1)
 B(d(x1)) -> A(b(x1))
 B(d(x1)) -> B(x1)
 C(a(x1)) -> A(c(x1))
 C(a(x1)) -> C(x1)
 C(b(x1)) -> A(x1)
 C(b(x1)) -> B(a(x1))
-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(a(x1)) -> A(b(a(x1)))
 A(a(x1)) -> B(a(x1))
 A(d(x1)) -> A(x1)
 B(a(a(x1))) -> A(b(c(x1)))
 B(a(a(x1))) -> B(c(x1))
 B(a(a(x1))) -> C(x1)
 B(d(x1)) -> A(b(x1))
 B(d(x1)) -> B(x1)
 C(a(x1)) -> A(c(x1))
 C(a(x1)) -> C(x1)
 C(b(x1)) -> A(x1)
 C(b(x1)) -> B(a(x1))
->->-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A(a(x1)) -> A(b(a(x1)))
 A(a(x1)) -> B(a(x1))
 A(d(x1)) -> A(x1)
 B(a(a(x1))) -> A(b(c(x1)))
 B(a(a(x1))) -> B(c(x1))
 B(a(a(x1))) -> C(x1)
 B(d(x1)) -> A(b(x1))
 B(d(x1)) -> B(x1)
 C(a(x1)) -> A(c(x1))
 C(a(x1)) -> C(x1)
 C(b(x1)) -> A(x1)
 C(b(x1)) -> B(a(x1))
-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
-> Usable rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = 2.X + 2
[b](X) = X
[c](X) = 2.X + 2
[d](X) = 2.X + 2
[A](X) = 2.X + 1
[B](X) = X
[C](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 A(a(x1)) -> A(b(a(x1)))
 A(d(x1)) -> A(x1)
 B(a(a(x1))) -> A(b(c(x1)))
 B(a(a(x1))) -> B(c(x1))
 B(a(a(x1))) -> C(x1)
 B(d(x1)) -> A(b(x1))
 B(d(x1)) -> B(x1)
 C(a(x1)) -> A(c(x1))
 C(a(x1)) -> C(x1)
 C(b(x1)) -> A(x1)
 C(b(x1)) -> B(a(x1))
-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(a(x1)) -> A(b(a(x1)))
 A(d(x1)) -> A(x1)
->->-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
->->Cycle:
->->-> Pairs:
 B(a(a(x1))) -> B(c(x1))
 B(a(a(x1))) -> C(x1)
 B(d(x1)) -> B(x1)
 C(a(x1)) -> C(x1)
 C(b(x1)) -> B(a(x1))
->->-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 A(a(x1)) -> A(b(a(x1)))
 A(d(x1)) -> A(x1)
-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
-> Usable rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = 2.X + 2
[b](X) = X
[c](X) = 2.X + 2
[d](X) = 2.X + 2
[A](X) = X

Problem 1.1: 

SCC Processor:
-> Pairs:
 A(a(x1)) -> A(b(a(x1)))
-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(a(x1)) -> A(b(a(x1)))
->->-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 A(a(x1)) -> A(b(a(x1)))
-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
-> Usable rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = 2.X + 1
[b](X) = 1/2.X
[c](X) = 2.X + 1/2
[d](X) = 2.X + 2
[A](X) = 1/2.X

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 B(a(a(x1))) -> B(c(x1))
 B(a(a(x1))) -> C(x1)
 B(d(x1)) -> B(x1)
 C(a(x1)) -> C(x1)
 C(b(x1)) -> B(a(x1))
-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
-> Usable rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = 2.X + 2
[b](X) = X
[c](X) = 2.X + 2
[d](X) = 2.X + 2
[B](X) = X
[C](X) = 2.X + 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 B(a(a(x1))) -> C(x1)
 B(d(x1)) -> B(x1)
 C(a(x1)) -> C(x1)
 C(b(x1)) -> B(a(x1))
-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 B(a(a(x1))) -> C(x1)
 B(d(x1)) -> B(x1)
 C(a(x1)) -> C(x1)
 C(b(x1)) -> B(a(x1))
->->-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 B(a(a(x1))) -> C(x1)
 B(d(x1)) -> B(x1)
 C(a(x1)) -> C(x1)
 C(b(x1)) -> B(a(x1))
-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
-> Usable rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = 2.X + 2
[b](X) = X
[c](X) = 2.X + 2
[d](X) = 2.X + 2
[B](X) = X
[C](X) = 2.X + 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 B(d(x1)) -> B(x1)
 C(a(x1)) -> C(x1)
 C(b(x1)) -> B(a(x1))
-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 B(d(x1)) -> B(x1)
->->-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
->->Cycle:
->->-> Pairs:
 C(a(x1)) -> C(x1)
->->-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))


The problem is decomposed in 2 subproblems.

Problem 1.2.1: 

Subterm Processor:
-> Pairs:
 B(d(x1)) -> B(x1)
-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
->Projection:
 pi(B) = 1

Problem 1.2.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2.2: 

Subterm Processor:
-> Pairs:
 C(a(x1)) -> C(x1)
-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
->Projection:
 pi(C) = 1

Problem 1.2.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(a(b(x1))) -> d(b(a(x1)))
 a(a(x1)) -> a(b(a(x1)))
 a(d(x1)) -> d(a(x1))
 b(a(a(x1))) -> a(b(c(x1)))
 b(d(x1)) -> a(b(x1))
 c(a(x1)) -> a(c(x1))
 c(b(x1)) -> b(a(x1))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.