/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 f:S x:S xs:S y:S ys:S)
(RULES
app(app(le,app(s,x:S)),app(s,y:S)) -> app(app(le,x:S),y:S)
app(app(le,app(s,x:S)),0) -> ffalse
app(app(le,0),y:S) -> ttrue
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(maxlist,x:S),app(app(cons,y:S),ys:S)) -> app(app(if,app(app(le,x:S),y:S)),app(app(maxlist,y:S),ys:S))
app(app(maxlist,x:S),nil) -> x:S
app(height,app(app(node,x:S),xs:S)) -> app(s,app(app(maxlist,0),app(app(map,height),xs:S)))
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 app(app(le,app(s,x:S)),app(s,y:S)) -> app(app(le,x:S),y:S)
 app(app(le,app(s,x:S)),0) -> ffalse
 app(app(le,0),y:S) -> ttrue
 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(maxlist,x:S),app(app(cons,y:S),ys:S)) -> app(app(if,app(app(le,x:S),y:S)),app(app(maxlist,y:S),ys:S))
 app(app(maxlist,x:S),nil) -> x:S
 app(height,app(app(node,x:S),xs:S)) -> app(s,app(app(maxlist,0),app(app(map,height),xs:S)))
-> 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(le,app(s,x:S)),app(s,y:S)) -> APP(app(le,x:S),y:S)
 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(maxlist,x:S),app(app(cons,y:S),ys:S)) -> APP(app(if,app(app(le,x:S),y:S)),app(app(maxlist,y:S),ys:S))
 APP(app(maxlist,x:S),app(app(cons,y:S),ys:S)) -> APP(app(le,x:S),y:S)
 APP(app(maxlist,x:S),app(app(cons,y:S),ys:S)) -> APP(app(maxlist,y:S),ys:S)
 APP(app(maxlist,x:S),app(app(cons,y:S),ys:S)) -> APP(if,app(app(le,x:S),y:S))
 APP(height,app(app(node,x:S),xs:S)) -> APP(app(map,height),xs:S)
 APP(height,app(app(node,x:S),xs:S)) -> APP(app(maxlist,0),app(app(map,height),xs:S))
 APP(height,app(app(node,x:S),xs:S)) -> APP(s,app(app(maxlist,0),app(app(map,height),xs:S)))
-> Rules:
 app(app(le,app(s,x:S)),app(s,y:S)) -> app(app(le,x:S),y:S)
 app(app(le,app(s,x:S)),0) -> ffalse
 app(app(le,0),y:S) -> ttrue
 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(maxlist,x:S),app(app(cons,y:S),ys:S)) -> app(app(if,app(app(le,x:S),y:S)),app(app(maxlist,y:S),ys:S))
 app(app(maxlist,x:S),nil) -> x:S
 app(height,app(app(node,x:S),xs:S)) -> app(s,app(app(maxlist,0),app(app(map,height),xs:S)))

Problem 1: 

SCC Processor:
-> Pairs:
 APP(app(le,app(s,x:S)),app(s,y:S)) -> APP(app(le,x:S),y:S)
 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(maxlist,x:S),app(app(cons,y:S),ys:S)) -> APP(app(if,app(app(le,x:S),y:S)),app(app(maxlist,y:S),ys:S))
 APP(app(maxlist,x:S),app(app(cons,y:S),ys:S)) -> APP(app(le,x:S),y:S)
 APP(app(maxlist,x:S),app(app(cons,y:S),ys:S)) -> APP(app(maxlist,y:S),ys:S)
 APP(app(maxlist,x:S),app(app(cons,y:S),ys:S)) -> APP(if,app(app(le,x:S),y:S))
 APP(height,app(app(node,x:S),xs:S)) -> APP(app(map,height),xs:S)
 APP(height,app(app(node,x:S),xs:S)) -> APP(app(maxlist,0),app(app(map,height),xs:S))
 APP(height,app(app(node,x:S),xs:S)) -> APP(s,app(app(maxlist,0),app(app(map,height),xs:S)))
-> Rules:
 app(app(le,app(s,x:S)),app(s,y:S)) -> app(app(le,x:S),y:S)
 app(app(le,app(s,x:S)),0) -> ffalse
 app(app(le,0),y:S) -> ttrue
 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(maxlist,x:S),app(app(cons,y:S),ys:S)) -> app(app(if,app(app(le,x:S),y:S)),app(app(maxlist,y:S),ys:S))
 app(app(maxlist,x:S),nil) -> x:S
 app(height,app(app(node,x:S),xs:S)) -> app(s,app(app(maxlist,0),app(app(map,height),xs:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(le,app(s,x:S)),app(s,y:S)) -> APP(app(le,x:S),y:S)
->->-> Rules:
 app(app(le,app(s,x:S)),app(s,y:S)) -> app(app(le,x:S),y:S)
 app(app(le,app(s,x:S)),0) -> ffalse
 app(app(le,0),y:S) -> ttrue
 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(maxlist,x:S),app(app(cons,y:S),ys:S)) -> app(app(if,app(app(le,x:S),y:S)),app(app(maxlist,y:S),ys:S))
 app(app(maxlist,x:S),nil) -> x:S
 app(height,app(app(node,x:S),xs:S)) -> app(s,app(app(maxlist,0),app(app(map,height),xs:S)))
->->Cycle:
->->-> Pairs:
 APP(app(maxlist,x:S),app(app(cons,y:S),ys:S)) -> APP(app(maxlist,y:S),ys:S)
->->-> Rules:
 app(app(le,app(s,x:S)),app(s,y:S)) -> app(app(le,x:S),y:S)
 app(app(le,app(s,x:S)),0) -> ffalse
 app(app(le,0),y:S) -> ttrue
 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(maxlist,x:S),app(app(cons,y:S),ys:S)) -> app(app(if,app(app(le,x:S),y:S)),app(app(maxlist,y:S),ys:S))
 app(app(maxlist,x:S),nil) -> x:S
 app(height,app(app(node,x:S),xs:S)) -> app(s,app(app(maxlist,0),app(app(map,height),xs:S)))
->->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)
 APP(height,app(app(node,x:S),xs:S)) -> APP(app(map,height),xs:S)
->->-> Rules:
 app(app(le,app(s,x:S)),app(s,y:S)) -> app(app(le,x:S),y:S)
 app(app(le,app(s,x:S)),0) -> ffalse
 app(app(le,0),y:S) -> ttrue
 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(maxlist,x:S),app(app(cons,y:S),ys:S)) -> app(app(if,app(app(le,x:S),y:S)),app(app(maxlist,y:S),ys:S))
 app(app(maxlist,x:S),nil) -> x:S
 app(height,app(app(node,x:S),xs:S)) -> app(s,app(app(maxlist,0),app(app(map,height),xs:S)))


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 APP(app(le,app(s,x:S)),app(s,y:S)) -> APP(app(le,x:S),y:S)
-> Rules:
 app(app(le,app(s,x:S)),app(s,y:S)) -> app(app(le,x:S),y:S)
 app(app(le,app(s,x:S)),0) -> ffalse
 app(app(le,0),y:S) -> ttrue
 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(maxlist,x:S),app(app(cons,y:S),ys:S)) -> app(app(if,app(app(le,x:S),y:S)),app(app(maxlist,y:S),ys:S))
 app(app(maxlist,x:S),nil) -> x:S
 app(height,app(app(node,x:S),xs:S)) -> app(s,app(app(maxlist,0),app(app(map,height),xs:S)))
->Projection:
 pi(APP) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(app(le,app(s,x:S)),app(s,y:S)) -> app(app(le,x:S),y:S)
 app(app(le,app(s,x:S)),0) -> ffalse
 app(app(le,0),y:S) -> ttrue
 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(maxlist,x:S),app(app(cons,y:S),ys:S)) -> app(app(if,app(app(le,x:S),y:S)),app(app(maxlist,y:S),ys:S))
 app(app(maxlist,x:S),nil) -> x:S
 app(height,app(app(node,x:S),xs:S)) -> app(s,app(app(maxlist,0),app(app(map,height),xs:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 APP(app(maxlist,x:S),app(app(cons,y:S),ys:S)) -> APP(app(maxlist,y:S),ys:S)
-> Rules:
 app(app(le,app(s,x:S)),app(s,y:S)) -> app(app(le,x:S),y:S)
 app(app(le,app(s,x:S)),0) -> ffalse
 app(app(le,0),y:S) -> ttrue
 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(maxlist,x:S),app(app(cons,y:S),ys:S)) -> app(app(if,app(app(le,x:S),y:S)),app(app(maxlist,y:S),ys:S))
 app(app(maxlist,x:S),nil) -> x:S
 app(height,app(app(node,x:S),xs:S)) -> app(s,app(app(maxlist,0),app(app(map,height),xs:S)))
->Projection:
 pi(APP) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(app(le,app(s,x:S)),app(s,y:S)) -> app(app(le,x:S),y:S)
 app(app(le,app(s,x:S)),0) -> ffalse
 app(app(le,0),y:S) -> ttrue
 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(maxlist,x:S),app(app(cons,y:S),ys:S)) -> app(app(if,app(app(le,x:S),y:S)),app(app(maxlist,y:S),ys:S))
 app(app(maxlist,x:S),nil) -> x:S
 app(height,app(app(node,x:S),xs:S)) -> app(s,app(app(maxlist,0),app(app(map,height),xs:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

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)
 APP(height,app(app(node,x:S),xs:S)) -> APP(app(map,height),xs:S)
-> Rules:
 app(app(le,app(s,x:S)),app(s,y:S)) -> app(app(le,x:S),y:S)
 app(app(le,app(s,x:S)),0) -> ffalse
 app(app(le,0),y:S) -> ttrue
 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(maxlist,x:S),app(app(cons,y:S),ys:S)) -> app(app(if,app(app(le,x:S),y:S)),app(app(maxlist,y:S),ys:S))
 app(app(maxlist,x:S),nil) -> x:S
 app(height,app(app(node,x:S),xs:S)) -> app(s,app(app(maxlist,0),app(app(map,height),xs:S)))
->Projection:
 pi(APP) = 2

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(app(le,app(s,x:S)),app(s,y:S)) -> app(app(le,x:S),y:S)
 app(app(le,app(s,x:S)),0) -> ffalse
 app(app(le,0),y:S) -> ttrue
 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(maxlist,x:S),app(app(cons,y:S),ys:S)) -> app(app(if,app(app(le,x:S),y:S)),app(app(maxlist,y:S),ys:S))
 app(app(maxlist,x:S),nil) -> x:S
 app(height,app(app(node,x:S),xs:S)) -> app(s,app(app(maxlist,0),app(app(map,height),xs:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.