/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 x)
(RULES
a(d,a(s,x)) -> a(s,a(s,a(d,a(p,a(s,x)))))
a(d,0) -> 0
a(f,a(s,x)) -> a(d,a(f,a(p,a(s,x))))
a(f,0) -> a(s,0)
a(p,a(s,x)) -> x
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 a(d,a(s,x)) -> a(s,a(s,a(d,a(p,a(s,x)))))
 a(d,0) -> 0
 a(f,a(s,x)) -> a(d,a(f,a(p,a(s,x))))
 a(f,0) -> a(s,0)
 a(p,a(s,x)) -> x
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A(d,a(s,x)) -> A(d,a(p,a(s,x)))
 A(d,a(s,x)) -> A(p,a(s,x))
 A(d,a(s,x)) -> A(s,a(d,a(p,a(s,x))))
 A(d,a(s,x)) -> A(s,a(s,a(d,a(p,a(s,x)))))
 A(f,a(s,x)) -> A(d,a(f,a(p,a(s,x))))
 A(f,a(s,x)) -> A(f,a(p,a(s,x)))
 A(f,a(s,x)) -> A(p,a(s,x))
-> Rules:
 a(d,a(s,x)) -> a(s,a(s,a(d,a(p,a(s,x)))))
 a(d,0) -> 0
 a(f,a(s,x)) -> a(d,a(f,a(p,a(s,x))))
 a(f,0) -> a(s,0)
 a(p,a(s,x)) -> x

Problem 1: 

SCC Processor:
-> Pairs:
 A(d,a(s,x)) -> A(d,a(p,a(s,x)))
 A(d,a(s,x)) -> A(p,a(s,x))
 A(d,a(s,x)) -> A(s,a(d,a(p,a(s,x))))
 A(d,a(s,x)) -> A(s,a(s,a(d,a(p,a(s,x)))))
 A(f,a(s,x)) -> A(d,a(f,a(p,a(s,x))))
 A(f,a(s,x)) -> A(f,a(p,a(s,x)))
 A(f,a(s,x)) -> A(p,a(s,x))
-> Rules:
 a(d,a(s,x)) -> a(s,a(s,a(d,a(p,a(s,x)))))
 a(d,0) -> 0
 a(f,a(s,x)) -> a(d,a(f,a(p,a(s,x))))
 a(f,0) -> a(s,0)
 a(p,a(s,x)) -> x
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(d,a(s,x)) -> A(d,a(p,a(s,x)))
->->-> Rules:
 a(d,a(s,x)) -> a(s,a(s,a(d,a(p,a(s,x)))))
 a(d,0) -> 0
 a(f,a(s,x)) -> a(d,a(f,a(p,a(s,x))))
 a(f,0) -> a(s,0)
 a(p,a(s,x)) -> x
->->Cycle:
->->-> Pairs:
 A(f,a(s,x)) -> A(f,a(p,a(s,x)))
->->-> Rules:
 a(d,a(s,x)) -> a(s,a(s,a(d,a(p,a(s,x)))))
 a(d,0) -> 0
 a(f,a(s,x)) -> a(d,a(f,a(p,a(s,x))))
 a(f,0) -> a(s,0)
 a(p,a(s,x)) -> x


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Narrowing Processor:
-> Pairs:
 A(d,a(s,x)) -> A(d,a(p,a(s,x)))
-> Rules:
 a(d,a(s,x)) -> a(s,a(s,a(d,a(p,a(s,x)))))
 a(d,0) -> 0
 a(f,a(s,x)) -> a(d,a(f,a(p,a(s,x))))
 a(f,0) -> a(s,0)
 a(p,a(s,x)) -> x
->Narrowed Pairs:
->->Original Pair:
 A(d,a(s,x)) -> A(d,a(p,a(s,x)))
->-> Narrowed pairs:
 A(d,a(s,x)) -> A(d,x)

Problem 1.1: 

SCC Processor:
-> Pairs:
 A(d,a(s,x)) -> A(d,x)
-> Rules:
 a(d,a(s,x)) -> a(s,a(s,a(d,a(p,a(s,x)))))
 a(d,0) -> 0
 a(f,a(s,x)) -> a(d,a(f,a(p,a(s,x))))
 a(f,0) -> a(s,0)
 a(p,a(s,x)) -> x
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(d,a(s,x)) -> A(d,x)
->->-> Rules:
 a(d,a(s,x)) -> a(s,a(s,a(d,a(p,a(s,x)))))
 a(d,0) -> 0
 a(f,a(s,x)) -> a(d,a(f,a(p,a(s,x))))
 a(f,0) -> a(s,0)
 a(p,a(s,x)) -> x

Problem 1.1: 

Subterm Processor:
-> Pairs:
 A(d,a(s,x)) -> A(d,x)
-> Rules:
 a(d,a(s,x)) -> a(s,a(s,a(d,a(p,a(s,x)))))
 a(d,0) -> 0
 a(f,a(s,x)) -> a(d,a(f,a(p,a(s,x))))
 a(f,0) -> a(s,0)
 a(p,a(s,x)) -> x
->Projection:
 pi(A) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(d,a(s,x)) -> a(s,a(s,a(d,a(p,a(s,x)))))
 a(d,0) -> 0
 a(f,a(s,x)) -> a(d,a(f,a(p,a(s,x))))
 a(f,0) -> a(s,0)
 a(p,a(s,x)) -> x
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Narrowing Processor:
-> Pairs:
 A(f,a(s,x)) -> A(f,a(p,a(s,x)))
-> Rules:
 a(d,a(s,x)) -> a(s,a(s,a(d,a(p,a(s,x)))))
 a(d,0) -> 0
 a(f,a(s,x)) -> a(d,a(f,a(p,a(s,x))))
 a(f,0) -> a(s,0)
 a(p,a(s,x)) -> x
->Narrowed Pairs:
->->Original Pair:
 A(f,a(s,x)) -> A(f,a(p,a(s,x)))
->-> Narrowed pairs:
 A(f,a(s,x)) -> A(f,x)

Problem 1.2: 

SCC Processor:
-> Pairs:
 A(f,a(s,x)) -> A(f,x)
-> Rules:
 a(d,a(s,x)) -> a(s,a(s,a(d,a(p,a(s,x)))))
 a(d,0) -> 0
 a(f,a(s,x)) -> a(d,a(f,a(p,a(s,x))))
 a(f,0) -> a(s,0)
 a(p,a(s,x)) -> x
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(f,a(s,x)) -> A(f,x)
->->-> Rules:
 a(d,a(s,x)) -> a(s,a(s,a(d,a(p,a(s,x)))))
 a(d,0) -> 0
 a(f,a(s,x)) -> a(d,a(f,a(p,a(s,x))))
 a(f,0) -> a(s,0)
 a(p,a(s,x)) -> x

Problem 1.2: 

Subterm Processor:
-> Pairs:
 A(f,a(s,x)) -> A(f,x)
-> Rules:
 a(d,a(s,x)) -> a(s,a(s,a(d,a(p,a(s,x)))))
 a(d,0) -> 0
 a(f,a(s,x)) -> a(d,a(f,a(p,a(s,x))))
 a(f,0) -> a(s,0)
 a(p,a(s,x)) -> x
->Projection:
 pi(A) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(d,a(s,x)) -> a(s,a(s,a(d,a(p,a(s,x)))))
 a(d,0) -> 0
 a(f,a(s,x)) -> a(d,a(f,a(p,a(s,x))))
 a(f,0) -> a(s,0)
 a(p,a(s,x)) -> x
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.