/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 v_NonEmpty:S x:S y:S)
(RULES
f(s(x:S),s(y:S)) -> s(f(x:S,y:S))
f(x:S,0) -> s(0)
g(0,x:S) -> g(f(x:S,x:S),x:S)
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 f(s(x:S),s(y:S)) -> s(f(x:S,y:S))
 f(x:S,0) -> s(0)
 g(0,x:S) -> g(f(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:
 F(s(x:S),s(y:S)) -> F(x:S,y:S)
 G(0,x:S) -> F(x:S,x:S)
 G(0,x:S) -> G(f(x:S,x:S),x:S)
-> Rules:
 f(s(x:S),s(y:S)) -> s(f(x:S,y:S))
 f(x:S,0) -> s(0)
 g(0,x:S) -> g(f(x:S,x:S),x:S)

Problem 1: 

SCC Processor:
-> Pairs:
 F(s(x:S),s(y:S)) -> F(x:S,y:S)
 G(0,x:S) -> F(x:S,x:S)
 G(0,x:S) -> G(f(x:S,x:S),x:S)
-> Rules:
 f(s(x:S),s(y:S)) -> s(f(x:S,y:S))
 f(x:S,0) -> s(0)
 g(0,x:S) -> g(f(x:S,x:S),x:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(s(x:S),s(y:S)) -> F(x:S,y:S)
->->-> Rules:
 f(s(x:S),s(y:S)) -> s(f(x:S,y:S))
 f(x:S,0) -> s(0)
 g(0,x:S) -> g(f(x:S,x:S),x:S)
->->Cycle:
->->-> Pairs:
 G(0,x:S) -> G(f(x:S,x:S),x:S)
->->-> Rules:
 f(s(x:S),s(y:S)) -> s(f(x:S,y:S))
 f(x:S,0) -> s(0)
 g(0,x:S) -> g(f(x:S,x:S),x:S)


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 F(s(x:S),s(y:S)) -> F(x:S,y:S)
-> Rules:
 f(s(x:S),s(y:S)) -> s(f(x:S,y:S))
 f(x:S,0) -> s(0)
 g(0,x:S) -> g(f(x:S,x:S),x:S)
->Projection:
 pi(F) = 1

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 G(0,x:S) -> G(f(x:S,x:S),x:S)
-> Rules:
 f(s(x:S),s(y:S)) -> s(f(x:S,y:S))
 f(x:S,0) -> s(0)
 g(0,x:S) -> g(f(x:S,x:S),x:S)
-> Usable rules:
 f(s(x:S),s(y:S)) -> s(f(x:S,y:S))
 f(x:S,0) -> s(0)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X1,X2) = 1
[g](X1,X2) = 0
[0] = 2
[fSNonEmpty] = 0
[s](X) = 1
[F](X1,X2) = 0
[G](X1,X2) = X1

Problem 1.2: 

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

The problem is finite.