/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 fun x xs)
(RULES
app(app(app(app(filter2,false),fun),x),xs) -> app(app(filter,fun),xs)
app(app(app(app(filter2,true),fun),x),xs) -> app(app(cons,x),app(app(filter,fun),xs))
app(app(filter,fun),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(fun,x)),fun),x),xs)
app(app(filter,fun),nil) -> nil
app(app(map,fun),app(app(cons,x),xs)) -> app(app(cons,app(fun,x)),app(app(map,fun),xs))
app(app(map,fun),nil) -> nil
app(g,app(g,x)) -> app(g,app(h,app(g,x)))
app(g,app(h,app(g,x))) -> app(g,x)
app(h,app(h,x)) -> app(h,app(app(f,app(h,x)),x))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 APP(app(app(app(filter2,false),fun),x),xs) -> APP(app(filter,fun),xs)
 APP(app(app(app(filter2,true),fun),x),xs) -> APP(app(cons,x),app(app(filter,fun),xs))
 APP(app(app(app(filter2,true),fun),x),xs) -> APP(app(filter,fun),xs)
 APP(app(filter,fun),app(app(cons,x),xs)) -> APP(app(app(app(filter2,app(fun,x)),fun),x),xs)
 APP(app(filter,fun),app(app(cons,x),xs)) -> APP(app(app(filter2,app(fun,x)),fun),x)
 APP(app(filter,fun),app(app(cons,x),xs)) -> APP(app(filter2,app(fun,x)),fun)
 APP(app(filter,fun),app(app(cons,x),xs)) -> APP(filter2,app(fun,x))
 APP(app(filter,fun),app(app(cons,x),xs)) -> APP(fun,x)
 APP(app(map,fun),app(app(cons,x),xs)) -> APP(app(cons,app(fun,x)),app(app(map,fun),xs))
 APP(app(map,fun),app(app(cons,x),xs)) -> APP(app(map,fun),xs)
 APP(app(map,fun),app(app(cons,x),xs)) -> APP(cons,app(fun,x))
 APP(app(map,fun),app(app(cons,x),xs)) -> APP(fun,x)
 APP(g,app(g,x)) -> APP(g,app(h,app(g,x)))
 APP(g,app(g,x)) -> APP(h,app(g,x))
 APP(h,app(h,x)) -> APP(app(f,app(h,x)),x)
 APP(h,app(h,x)) -> APP(f,app(h,x))
 APP(h,app(h,x)) -> APP(h,app(app(f,app(h,x)),x))
-> Rules:
 app(app(app(app(filter2,false),fun),x),xs) -> app(app(filter,fun),xs)
 app(app(app(app(filter2,true),fun),x),xs) -> app(app(cons,x),app(app(filter,fun),xs))
 app(app(filter,fun),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(fun,x)),fun),x),xs)
 app(app(filter,fun),nil) -> nil
 app(app(map,fun),app(app(cons,x),xs)) -> app(app(cons,app(fun,x)),app(app(map,fun),xs))
 app(app(map,fun),nil) -> nil
 app(g,app(g,x)) -> app(g,app(h,app(g,x)))
 app(g,app(h,app(g,x))) -> app(g,x)
 app(h,app(h,x)) -> app(h,app(app(f,app(h,x)),x))

Problem 1: 

SCC Processor:
-> Pairs:
 APP(app(app(app(filter2,false),fun),x),xs) -> APP(app(filter,fun),xs)
 APP(app(app(app(filter2,true),fun),x),xs) -> APP(app(cons,x),app(app(filter,fun),xs))
 APP(app(app(app(filter2,true),fun),x),xs) -> APP(app(filter,fun),xs)
 APP(app(filter,fun),app(app(cons,x),xs)) -> APP(app(app(app(filter2,app(fun,x)),fun),x),xs)
 APP(app(filter,fun),app(app(cons,x),xs)) -> APP(app(app(filter2,app(fun,x)),fun),x)
 APP(app(filter,fun),app(app(cons,x),xs)) -> APP(app(filter2,app(fun,x)),fun)
 APP(app(filter,fun),app(app(cons,x),xs)) -> APP(filter2,app(fun,x))
 APP(app(filter,fun),app(app(cons,x),xs)) -> APP(fun,x)
 APP(app(map,fun),app(app(cons,x),xs)) -> APP(app(cons,app(fun,x)),app(app(map,fun),xs))
 APP(app(map,fun),app(app(cons,x),xs)) -> APP(app(map,fun),xs)
 APP(app(map,fun),app(app(cons,x),xs)) -> APP(cons,app(fun,x))
 APP(app(map,fun),app(app(cons,x),xs)) -> APP(fun,x)
 APP(g,app(g,x)) -> APP(g,app(h,app(g,x)))
 APP(g,app(g,x)) -> APP(h,app(g,x))
 APP(h,app(h,x)) -> APP(app(f,app(h,x)),x)
 APP(h,app(h,x)) -> APP(f,app(h,x))
 APP(h,app(h,x)) -> APP(h,app(app(f,app(h,x)),x))
-> Rules:
 app(app(app(app(filter2,false),fun),x),xs) -> app(app(filter,fun),xs)
 app(app(app(app(filter2,true),fun),x),xs) -> app(app(cons,x),app(app(filter,fun),xs))
 app(app(filter,fun),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(fun,x)),fun),x),xs)
 app(app(filter,fun),nil) -> nil
 app(app(map,fun),app(app(cons,x),xs)) -> app(app(cons,app(fun,x)),app(app(map,fun),xs))
 app(app(map,fun),nil) -> nil
 app(g,app(g,x)) -> app(g,app(h,app(g,x)))
 app(g,app(h,app(g,x))) -> app(g,x)
 app(h,app(h,x)) -> app(h,app(app(f,app(h,x)),x))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(g,app(g,x)) -> APP(g,app(h,app(g,x)))
->->-> Rules:
 app(app(app(app(filter2,false),fun),x),xs) -> app(app(filter,fun),xs)
 app(app(app(app(filter2,true),fun),x),xs) -> app(app(cons,x),app(app(filter,fun),xs))
 app(app(filter,fun),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(fun,x)),fun),x),xs)
 app(app(filter,fun),nil) -> nil
 app(app(map,fun),app(app(cons,x),xs)) -> app(app(cons,app(fun,x)),app(app(map,fun),xs))
 app(app(map,fun),nil) -> nil
 app(g,app(g,x)) -> app(g,app(h,app(g,x)))
 app(g,app(h,app(g,x))) -> app(g,x)
 app(h,app(h,x)) -> app(h,app(app(f,app(h,x)),x))
->->Cycle:
->->-> Pairs:
 APP(app(app(app(filter2,false),fun),x),xs) -> APP(app(filter,fun),xs)
 APP(app(app(app(filter2,true),fun),x),xs) -> APP(app(filter,fun),xs)
 APP(app(filter,fun),app(app(cons,x),xs)) -> APP(app(app(app(filter2,app(fun,x)),fun),x),xs)
 APP(app(filter,fun),app(app(cons,x),xs)) -> APP(fun,x)
 APP(app(map,fun),app(app(cons,x),xs)) -> APP(app(map,fun),xs)
 APP(app(map,fun),app(app(cons,x),xs)) -> APP(fun,x)
->->-> Rules:
 app(app(app(app(filter2,false),fun),x),xs) -> app(app(filter,fun),xs)
 app(app(app(app(filter2,true),fun),x),xs) -> app(app(cons,x),app(app(filter,fun),xs))
 app(app(filter,fun),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(fun,x)),fun),x),xs)
 app(app(filter,fun),nil) -> nil
 app(app(map,fun),app(app(cons,x),xs)) -> app(app(cons,app(fun,x)),app(app(map,fun),xs))
 app(app(map,fun),nil) -> nil
 app(g,app(g,x)) -> app(g,app(h,app(g,x)))
 app(g,app(h,app(g,x))) -> app(g,x)
 app(h,app(h,x)) -> app(h,app(app(f,app(h,x)),x))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 APP(g,app(g,x)) -> APP(g,app(h,app(g,x)))
-> Rules:
 app(app(app(app(filter2,false),fun),x),xs) -> app(app(filter,fun),xs)
 app(app(app(app(filter2,true),fun),x),xs) -> app(app(cons,x),app(app(filter,fun),xs))
 app(app(filter,fun),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(fun,x)),fun),x),xs)
 app(app(filter,fun),nil) -> nil
 app(app(map,fun),app(app(cons,x),xs)) -> app(app(cons,app(fun,x)),app(app(map,fun),xs))
 app(app(map,fun),nil) -> nil
 app(g,app(g,x)) -> app(g,app(h,app(g,x)))
 app(g,app(h,app(g,x))) -> app(g,x)
 app(h,app(h,x)) -> app(h,app(app(f,app(h,x)),x))
-> Usable rules:
 app(app(app(app(filter2,false),fun),x),xs) -> app(app(filter,fun),xs)
 app(app(app(app(filter2,true),fun),x),xs) -> app(app(cons,x),app(app(filter,fun),xs))
 app(app(filter,fun),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(fun,x)),fun),x),xs)
 app(app(filter,fun),nil) -> nil
 app(app(map,fun),app(app(cons,x),xs)) -> app(app(cons,app(fun,x)),app(app(map,fun),xs))
 app(app(map,fun),nil) -> nil
 app(g,app(g,x)) -> app(g,app(h,app(g,x)))
 app(g,app(h,app(g,x))) -> app(g,x)
 app(h,app(h,x)) -> app(h,app(app(f,app(h,x)),x))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[app](X1,X2) = X1
[cons] = 0
[f] = 2
[false] = 2
[filter] = 0
[filter2] = 0
[g] = 2
[h] = 1
[map] = 0
[nil] = 0
[true] = 2
[APP](X1,X2) = X2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(app(app(app(filter2,false),fun),x),xs) -> app(app(filter,fun),xs)
 app(app(app(app(filter2,true),fun),x),xs) -> app(app(cons,x),app(app(filter,fun),xs))
 app(app(filter,fun),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(fun,x)),fun),x),xs)
 app(app(filter,fun),nil) -> nil
 app(app(map,fun),app(app(cons,x),xs)) -> app(app(cons,app(fun,x)),app(app(map,fun),xs))
 app(app(map,fun),nil) -> nil
 app(g,app(g,x)) -> app(g,app(h,app(g,x)))
 app(g,app(h,app(g,x))) -> app(g,x)
 app(h,app(h,x)) -> app(h,app(app(f,app(h,x)),x))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 APP(app(app(app(filter2,false),fun),x),xs) -> APP(app(filter,fun),xs)
 APP(app(app(app(filter2,true),fun),x),xs) -> APP(app(filter,fun),xs)
 APP(app(filter,fun),app(app(cons,x),xs)) -> APP(app(app(app(filter2,app(fun,x)),fun),x),xs)
 APP(app(filter,fun),app(app(cons,x),xs)) -> APP(fun,x)
 APP(app(map,fun),app(app(cons,x),xs)) -> APP(app(map,fun),xs)
 APP(app(map,fun),app(app(cons,x),xs)) -> APP(fun,x)
-> Rules:
 app(app(app(app(filter2,false),fun),x),xs) -> app(app(filter,fun),xs)
 app(app(app(app(filter2,true),fun),x),xs) -> app(app(cons,x),app(app(filter,fun),xs))
 app(app(filter,fun),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(fun,x)),fun),x),xs)
 app(app(filter,fun),nil) -> nil
 app(app(map,fun),app(app(cons,x),xs)) -> app(app(cons,app(fun,x)),app(app(map,fun),xs))
 app(app(map,fun),nil) -> nil
 app(g,app(g,x)) -> app(g,app(h,app(g,x)))
 app(g,app(h,app(g,x))) -> app(g,x)
 app(h,app(h,x)) -> app(h,app(app(f,app(h,x)),x))
->Projection:
 pi(APP) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 APP(app(app(app(filter2,false),fun),x),xs) -> APP(app(filter,fun),xs)
 APP(app(app(app(filter2,true),fun),x),xs) -> APP(app(filter,fun),xs)
-> Rules:
 app(app(app(app(filter2,false),fun),x),xs) -> app(app(filter,fun),xs)
 app(app(app(app(filter2,true),fun),x),xs) -> app(app(cons,x),app(app(filter,fun),xs))
 app(app(filter,fun),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(fun,x)),fun),x),xs)
 app(app(filter,fun),nil) -> nil
 app(app(map,fun),app(app(cons,x),xs)) -> app(app(cons,app(fun,x)),app(app(map,fun),xs))
 app(app(map,fun),nil) -> nil
 app(g,app(g,x)) -> app(g,app(h,app(g,x)))
 app(g,app(h,app(g,x))) -> app(g,x)
 app(h,app(h,x)) -> app(h,app(app(f,app(h,x)),x))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.