/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 f:S x:S xs:S y:S)
(RULES
app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
app(app(map,f:S),nil) -> nil
app(app(plus,app(s,x:S)),y:S) -> app(s,app(app(plus,x:S),y:S))
app(app(plus,0),y:S) -> y:S
inc -> app(map,app(plus,app(s,0)))
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(app(plus,app(s,x:S)),y:S) -> app(s,app(app(plus,x:S),y:S))
 app(app(plus,0),y:S) -> y:S
 inc -> app(map,app(plus,app(s,0)))
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(map,f:S),xs:S)
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(cons,app(f:S,x:S))
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x:S)
 APP(app(plus,app(s,x:S)),y:S) -> APP(app(plus,x:S),y:S)
 APP(app(plus,app(s,x:S)),y:S) -> APP(s,app(app(plus,x:S),y:S))
-> Rules:
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(app(plus,app(s,x:S)),y:S) -> app(s,app(app(plus,x:S),y:S))
 app(app(plus,0),y:S) -> y:S
 inc -> app(map,app(plus,app(s,0)))

Problem 1: 

SCC Processor:
-> Pairs:
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(map,f:S),xs:S)
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(cons,app(f:S,x:S))
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x:S)
 APP(app(plus,app(s,x:S)),y:S) -> APP(app(plus,x:S),y:S)
 APP(app(plus,app(s,x:S)),y:S) -> APP(s,app(app(plus,x:S),y:S))
-> Rules:
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(app(plus,app(s,x:S)),y:S) -> app(s,app(app(plus,x:S),y:S))
 app(app(plus,0),y:S) -> y:S
 inc -> app(map,app(plus,app(s,0)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(plus,app(s,x:S)),y:S) -> APP(app(plus,x:S),y:S)
->->-> Rules:
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(app(plus,app(s,x:S)),y:S) -> app(s,app(app(plus,x:S),y:S))
 app(app(plus,0),y:S) -> y:S
 inc -> app(map,app(plus,app(s,0)))
->->Cycle:
->->-> Pairs:
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(map,f:S),xs:S)
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x:S)
->->-> Rules:
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(app(plus,app(s,x:S)),y:S) -> app(s,app(app(plus,x:S),y:S))
 app(app(plus,0),y:S) -> y:S
 inc -> app(map,app(plus,app(s,0)))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Reduction Pairs Processor:
-> Pairs:
 APP(app(plus,app(s,x:S)),y:S) -> APP(app(plus,x:S),y:S)
-> Rules:
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(app(plus,app(s,x:S)),y:S) -> app(s,app(app(plus,x:S),y:S))
 app(app(plus,0),y:S) -> y:S
 inc -> app(map,app(plus,app(s,0)))
-> Usable rules:
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(app(plus,app(s,x:S)),y:S) -> app(s,app(app(plus,x:S),y:S))
 app(app(plus,0),y:S) -> y:S
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[app](X1,X2) = X1.X2 + X1 + 1
[inc] = 0
[0] = 2
[cons] = 0
[fSNonEmpty] = 0
[map] = 1
[nil] = 2
[plus] = 1
[s] = 1
[APP](X1,X2) = X1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(app(plus,app(s,x:S)),y:S) -> app(s,app(app(plus,x:S),y:S))
 app(app(plus,0),y:S) -> y:S
 inc -> app(map,app(plus,app(s,0)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(map,f:S),xs:S)
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x:S)
-> Rules:
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(app(plus,app(s,x:S)),y:S) -> app(s,app(app(plus,x:S),y:S))
 app(app(plus,0),y:S) -> y:S
 inc -> app(map,app(plus,app(s,0)))
->Projection:
 pi(APP) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(map,f:S),xs:S)
-> Rules:
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(app(plus,app(s,x:S)),y:S) -> app(s,app(app(plus,x:S),y:S))
 app(app(plus,0),y:S) -> y:S
 inc -> app(map,app(plus,app(s,0)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(map,f:S),xs:S)
->->-> Rules:
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(app(plus,app(s,x:S)),y:S) -> app(s,app(app(plus,x:S),y:S))
 app(app(plus,0),y:S) -> y:S
 inc -> app(map,app(plus,app(s,0)))

Problem 1.2: 

Subterm Processor:
-> Pairs:
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(map,f:S),xs:S)
-> Rules:
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(app(plus,app(s,x:S)),y:S) -> app(s,app(app(plus,x:S),y:S))
 app(app(plus,0),y:S) -> y:S
 inc -> app(map,app(plus,app(s,0)))
->Projection:
 pi(APP) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(app(plus,app(s,x:S)),y:S) -> app(s,app(app(plus,x:S),y:S))
 app(app(plus,0),y:S) -> y:S
 inc -> app(map,app(plus,app(s,0)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.