/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(b(x1)) -> d(x1)
a(f(x1)) -> a(x1)
b(a(x1)) -> a(b(x1))
d(c(x1)) -> f(a(b(b(c(x1)))))
d(f(x1)) -> f(a(b(x1)))
)

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.