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

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 +(0,y:S) -> y:S
 +(s(x:S),y:S) -> s(+(x:S,y:S))
 -(0,y:S) -> 0
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> 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:
 +#(s(x:S),y:S) -> +#(x:S,y:S)
 -#(s(x:S),s(y:S)) -> -#(x:S,y:S)
-> Rules:
 +(0,y:S) -> y:S
 +(s(x:S),y:S) -> s(+(x:S,y:S))
 -(0,y:S) -> 0
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S

Problem 1: 

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.