/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
sum(0) -> 0
sum(s(x:S)) -> +(sum(x:S),s(x:S))
sum1(0) -> 0
sum1(s(x:S)) -> s(+(sum1(x:S),+(x:S,x:S)))
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 sum(0) -> 0
 sum(s(x:S)) -> +(sum(x:S),s(x:S))
 sum1(0) -> 0
 sum1(s(x:S)) -> s(+(sum1(x:S),+(x:S,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:
 SUM(s(x:S)) -> SUM(x:S)
 SUM1(s(x:S)) -> SUM1(x:S)
-> Rules:
 sum(0) -> 0
 sum(s(x:S)) -> +(sum(x:S),s(x:S))
 sum1(0) -> 0
 sum1(s(x:S)) -> s(+(sum1(x:S),+(x:S,x:S)))

Problem 1: 

SCC Processor:
-> Pairs:
 SUM(s(x:S)) -> SUM(x:S)
 SUM1(s(x:S)) -> SUM1(x:S)
-> Rules:
 sum(0) -> 0
 sum(s(x:S)) -> +(sum(x:S),s(x:S))
 sum1(0) -> 0
 sum1(s(x:S)) -> s(+(sum1(x:S),+(x:S,x:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 SUM1(s(x:S)) -> SUM1(x:S)
->->-> Rules:
 sum(0) -> 0
 sum(s(x:S)) -> +(sum(x:S),s(x:S))
 sum1(0) -> 0
 sum1(s(x:S)) -> s(+(sum1(x:S),+(x:S,x:S)))
->->Cycle:
->->-> Pairs:
 SUM(s(x:S)) -> SUM(x:S)
->->-> Rules:
 sum(0) -> 0
 sum(s(x:S)) -> +(sum(x:S),s(x:S))
 sum1(0) -> 0
 sum1(s(x:S)) -> s(+(sum1(x:S),+(x:S,x:S)))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 SUM1(s(x:S)) -> SUM1(x:S)
-> Rules:
 sum(0) -> 0
 sum(s(x:S)) -> +(sum(x:S),s(x:S))
 sum1(0) -> 0
 sum1(s(x:S)) -> s(+(sum1(x:S),+(x:S,x:S)))
->Projection:
 pi(SUM1) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 sum(0) -> 0
 sum(s(x:S)) -> +(sum(x:S),s(x:S))
 sum1(0) -> 0
 sum1(s(x:S)) -> s(+(sum1(x:S),+(x:S,x:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 SUM(s(x:S)) -> SUM(x:S)
-> Rules:
 sum(0) -> 0
 sum(s(x:S)) -> +(sum(x:S),s(x:S))
 sum1(0) -> 0
 sum1(s(x:S)) -> s(+(sum1(x:S),+(x:S,x:S)))
->Projection:
 pi(SUM) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 sum(0) -> 0
 sum(s(x:S)) -> +(sum(x:S),s(x:S))
 sum1(0) -> 0
 sum1(s(x:S)) -> s(+(sum1(x:S),+(x:S,x:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.