/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 z:S)
(RULES
purge(.(x:S,y:S)) -> .(x:S,purge(remove(x:S,y:S)))
purge(nil) -> nil
remove(x:S,.(y:S,z:S)) -> if(=(x:S,y:S),remove(x:S,z:S),.(y:S,remove(x:S,z:S)))
remove(x:S,nil) -> nil
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 purge(.(x:S,y:S)) -> .(x:S,purge(remove(x:S,y:S)))
 purge(nil) -> nil
 remove(x:S,.(y:S,z:S)) -> if(=(x:S,y:S),remove(x:S,z:S),.(y:S,remove(x:S,z:S)))
 remove(x:S,nil) -> nil
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 PURGE(.(x:S,y:S)) -> PURGE(remove(x:S,y:S))
 PURGE(.(x:S,y:S)) -> REMOVE(x:S,y:S)
 REMOVE(x:S,.(y:S,z:S)) -> REMOVE(x:S,z:S)
-> Rules:
 purge(.(x:S,y:S)) -> .(x:S,purge(remove(x:S,y:S)))
 purge(nil) -> nil
 remove(x:S,.(y:S,z:S)) -> if(=(x:S,y:S),remove(x:S,z:S),.(y:S,remove(x:S,z:S)))
 remove(x:S,nil) -> nil

Problem 1: 

SCC Processor:
-> Pairs:
 PURGE(.(x:S,y:S)) -> PURGE(remove(x:S,y:S))
 PURGE(.(x:S,y:S)) -> REMOVE(x:S,y:S)
 REMOVE(x:S,.(y:S,z:S)) -> REMOVE(x:S,z:S)
-> Rules:
 purge(.(x:S,y:S)) -> .(x:S,purge(remove(x:S,y:S)))
 purge(nil) -> nil
 remove(x:S,.(y:S,z:S)) -> if(=(x:S,y:S),remove(x:S,z:S),.(y:S,remove(x:S,z:S)))
 remove(x:S,nil) -> nil
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 REMOVE(x:S,.(y:S,z:S)) -> REMOVE(x:S,z:S)
->->-> Rules:
 purge(.(x:S,y:S)) -> .(x:S,purge(remove(x:S,y:S)))
 purge(nil) -> nil
 remove(x:S,.(y:S,z:S)) -> if(=(x:S,y:S),remove(x:S,z:S),.(y:S,remove(x:S,z:S)))
 remove(x:S,nil) -> nil
->->Cycle:
->->-> Pairs:
 PURGE(.(x:S,y:S)) -> PURGE(remove(x:S,y:S))
->->-> Rules:
 purge(.(x:S,y:S)) -> .(x:S,purge(remove(x:S,y:S)))
 purge(nil) -> nil
 remove(x:S,.(y:S,z:S)) -> if(=(x:S,y:S),remove(x:S,z:S),.(y:S,remove(x:S,z:S)))
 remove(x:S,nil) -> nil


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 REMOVE(x:S,.(y:S,z:S)) -> REMOVE(x:S,z:S)
-> Rules:
 purge(.(x:S,y:S)) -> .(x:S,purge(remove(x:S,y:S)))
 purge(nil) -> nil
 remove(x:S,.(y:S,z:S)) -> if(=(x:S,y:S),remove(x:S,z:S),.(y:S,remove(x:S,z:S)))
 remove(x:S,nil) -> nil
->Projection:
 pi(REMOVE) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 purge(.(x:S,y:S)) -> .(x:S,purge(remove(x:S,y:S)))
 purge(nil) -> nil
 remove(x:S,.(y:S,z:S)) -> if(=(x:S,y:S),remove(x:S,z:S),.(y:S,remove(x:S,z:S)))
 remove(x:S,nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 PURGE(.(x:S,y:S)) -> PURGE(remove(x:S,y:S))
-> Rules:
 purge(.(x:S,y:S)) -> .(x:S,purge(remove(x:S,y:S)))
 purge(nil) -> nil
 remove(x:S,.(y:S,z:S)) -> if(=(x:S,y:S),remove(x:S,z:S),.(y:S,remove(x:S,z:S)))
 remove(x:S,nil) -> nil
-> Usable rules:
 remove(x:S,.(y:S,z:S)) -> if(=(x:S,y:S),remove(x:S,z:S),.(y:S,remove(x:S,z:S)))
 remove(x:S,nil) -> nil
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[purge](X) = 0
[remove](X1,X2) = X1 + 2.X2 + 1
[.](X1,X2) = 2.X1 + 2.X2 + 2
[=](X1,X2) = 2.X2
[fSNonEmpty] = 0
[if](X1,X2,X3) = 2.X1 + X2 + 1
[nil] = 1
[PURGE](X) = 2.X
[REMOVE](X1,X2) = 0

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 purge(.(x:S,y:S)) -> .(x:S,purge(remove(x:S,y:S)))
 purge(nil) -> nil
 remove(x:S,.(y:S,z:S)) -> if(=(x:S,y:S),remove(x:S,z:S),.(y:S,remove(x:S,z:S)))
 remove(x:S,nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.