/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
bits(0) -> 0
bits(s(x)) -> s(bits(half(s(x))))
half(0) -> 0
half(s(0)) -> 0
half(s(s(x))) -> s(half(x))
)

Problem 1: 

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

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 BITS(s(x)) -> BITS(half(s(x)))
 BITS(s(x)) -> HALF(s(x))
 HALF(s(s(x))) -> HALF(x)
-> Rules:
 bits(0) -> 0
 bits(s(x)) -> s(bits(half(s(x))))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))

Problem 1: 

SCC Processor:
-> Pairs:
 BITS(s(x)) -> BITS(half(s(x)))
 BITS(s(x)) -> HALF(s(x))
 HALF(s(s(x))) -> HALF(x)
-> Rules:
 bits(0) -> 0
 bits(s(x)) -> s(bits(half(s(x))))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 HALF(s(s(x))) -> HALF(x)
->->-> Rules:
 bits(0) -> 0
 bits(s(x)) -> s(bits(half(s(x))))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
->->Cycle:
->->-> Pairs:
 BITS(s(x)) -> BITS(half(s(x)))
->->-> Rules:
 bits(0) -> 0
 bits(s(x)) -> s(bits(half(s(x))))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 HALF(s(s(x))) -> HALF(x)
-> Rules:
 bits(0) -> 0
 bits(s(x)) -> s(bits(half(s(x))))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
->Projection:
 pi(HALF) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 bits(0) -> 0
 bits(s(x)) -> s(bits(half(s(x))))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 BITS(s(x)) -> BITS(half(s(x)))
-> Rules:
 bits(0) -> 0
 bits(s(x)) -> s(bits(half(s(x))))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
-> Usable rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[half](X) = 1/2.X
[0] = 0
[s](X) = X + 1/2
[BITS](X) = X

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 bits(0) -> 0
 bits(s(x)) -> s(bits(half(s(x))))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.