/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 x y)
(RULES
+(x,0) -> x
+(x,s(y)) -> s(+(x,y))
sum(0) -> 0
sum(s(x)) -> +(sum(x),s(x))
)

Problem 1: 

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

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 +#(x,s(y)) -> +#(x,y)
 SUM(s(x)) -> +#(sum(x),s(x))
 SUM(s(x)) -> SUM(x)
-> Rules:
 +(x,0) -> x
 +(x,s(y)) -> s(+(x,y))
 sum(0) -> 0
 sum(s(x)) -> +(sum(x),s(x))

Problem 1: 

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 +#(x,s(y)) -> +#(x,y)
-> Rules:
 +(x,0) -> x
 +(x,s(y)) -> s(+(x,y))
 sum(0) -> 0
 sum(s(x)) -> +(sum(x),s(x))
->Projection:
 pi(+#) = 2

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.