/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 v_NonEmpty:S x:S)
(RULES
conv(0) -> cons(nil,0)
conv(s(x:S)) -> cons(conv(half(s(x:S))),lastbit(s(x:S)))
half(0) -> 0
half(s(0)) -> 0
half(s(s(x:S))) -> s(half(x:S))
lastbit(0) -> 0
lastbit(s(0)) -> s(0)
lastbit(s(s(x:S))) -> lastbit(x:S)
)

Problem 1: 

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

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 CONV(s(x:S)) -> CONV(half(s(x:S)))
 CONV(s(x:S)) -> HALF(s(x:S))
 CONV(s(x:S)) -> LASTBIT(s(x:S))
 HALF(s(s(x:S))) -> HALF(x:S)
 LASTBIT(s(s(x:S))) -> LASTBIT(x:S)
-> Rules:
 conv(0) -> cons(nil,0)
 conv(s(x:S)) -> cons(conv(half(s(x:S))),lastbit(s(x:S)))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x:S))) -> lastbit(x:S)

Problem 1: 

SCC Processor:
-> Pairs:
 CONV(s(x:S)) -> CONV(half(s(x:S)))
 CONV(s(x:S)) -> HALF(s(x:S))
 CONV(s(x:S)) -> LASTBIT(s(x:S))
 HALF(s(s(x:S))) -> HALF(x:S)
 LASTBIT(s(s(x:S))) -> LASTBIT(x:S)
-> Rules:
 conv(0) -> cons(nil,0)
 conv(s(x:S)) -> cons(conv(half(s(x:S))),lastbit(s(x:S)))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x:S))) -> lastbit(x:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 LASTBIT(s(s(x:S))) -> LASTBIT(x:S)
->->-> Rules:
 conv(0) -> cons(nil,0)
 conv(s(x:S)) -> cons(conv(half(s(x:S))),lastbit(s(x:S)))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x:S))) -> lastbit(x:S)
->->Cycle:
->->-> Pairs:
 HALF(s(s(x:S))) -> HALF(x:S)
->->-> Rules:
 conv(0) -> cons(nil,0)
 conv(s(x:S)) -> cons(conv(half(s(x:S))),lastbit(s(x:S)))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x:S))) -> lastbit(x:S)
->->Cycle:
->->-> Pairs:
 CONV(s(x:S)) -> CONV(half(s(x:S)))
->->-> Rules:
 conv(0) -> cons(nil,0)
 conv(s(x:S)) -> cons(conv(half(s(x:S))),lastbit(s(x:S)))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x:S))) -> lastbit(x:S)


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 LASTBIT(s(s(x:S))) -> LASTBIT(x:S)
-> Rules:
 conv(0) -> cons(nil,0)
 conv(s(x:S)) -> cons(conv(half(s(x:S))),lastbit(s(x:S)))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x:S))) -> lastbit(x:S)
->Projection:
 pi(LASTBIT) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 conv(0) -> cons(nil,0)
 conv(s(x:S)) -> cons(conv(half(s(x:S))),lastbit(s(x:S)))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x:S))) -> lastbit(x:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 HALF(s(s(x:S))) -> HALF(x:S)
-> Rules:
 conv(0) -> cons(nil,0)
 conv(s(x:S)) -> cons(conv(half(s(x:S))),lastbit(s(x:S)))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x:S))) -> lastbit(x:S)
->Projection:
 pi(HALF) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 conv(0) -> cons(nil,0)
 conv(s(x:S)) -> cons(conv(half(s(x:S))),lastbit(s(x:S)))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x:S))) -> lastbit(x:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Reduction Pairs Processor:
-> Pairs:
 CONV(s(x:S)) -> CONV(half(s(x:S)))
-> Rules:
 conv(0) -> cons(nil,0)
 conv(s(x:S)) -> cons(conv(half(s(x:S))),lastbit(s(x:S)))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x:S))) -> lastbit(x:S)
-> Usable rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[conv](X) = 0
[half](X) = 1/2.X
[lastbit](X) = 0
[0] = 0
[cons](X1,X2) = 0
[fSNonEmpty] = 0
[nil] = 0
[s](X) = X + 1/2
[CONV](X) = X
[HALF](X) = 0
[LASTBIT](X) = 0

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 conv(0) -> cons(nil,0)
 conv(s(x:S)) -> cons(conv(half(s(x:S))),lastbit(s(x:S)))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x:S))) -> lastbit(x:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.