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


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YES

Problem 1: 

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

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A(b(x1)) -> R(x1)
 D(a(x1)) -> A(a(d(x1)))
 D(a(x1)) -> A(d(x1))
 D(a(x1)) -> D(x1)
 D(x1) -> A(x1)
 R(a(x1)) -> D(r(x1))
 R(a(x1)) -> R(x1)
 R(x1) -> D(x1)
-> Rules:
 a(b(x1)) -> b(r(x1))
 d(a(x1)) -> a(a(d(x1)))
 d(x1) -> a(x1)
 r(a(x1)) -> d(r(x1))
 r(x1) -> d(x1)

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.