/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 f x xs y z)
(RULES
app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs)
app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs))
app(app(*,x),app(app(+,y),z)) -> app(app(+,app(app(*,x),y)),app(app(*,x),z))
app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs)
app(app(filter,f),nil) -> nil
app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs))
app(app(map,f),nil) -> nil
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs)
 app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs))
 app(app(*,x),app(app(+,y),z)) -> app(app(+,app(app(*,x),y)),app(app(*,x),z))
 app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs)
 app(app(filter,f),nil) -> nil
 app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs))
 app(app(map,f),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:
 APP(app(app(app(filter2,false),f),x),xs) -> APP(app(filter,f),xs)
 APP(app(app(app(filter2,true),f),x),xs) -> APP(app(cons,x),app(app(filter,f),xs))
 APP(app(app(app(filter2,true),f),x),xs) -> APP(app(filter,f),xs)
 APP(app(*,x),app(app(+,y),z)) -> APP(app(*,x),y)
 APP(app(*,x),app(app(+,y),z)) -> APP(app(*,x),z)
 APP(app(*,x),app(app(+,y),z)) -> APP(app(+,app(app(*,x),y)),app(app(*,x),z))
 APP(app(*,x),app(app(+,y),z)) -> APP(+,app(app(*,x),y))
 APP(app(filter,f),app(app(cons,x),xs)) -> APP(app(app(app(filter2,app(f,x)),f),x),xs)
 APP(app(filter,f),app(app(cons,x),xs)) -> APP(app(app(filter2,app(f,x)),f),x)
 APP(app(filter,f),app(app(cons,x),xs)) -> APP(app(filter2,app(f,x)),f)
 APP(app(filter,f),app(app(cons,x),xs)) -> APP(filter2,app(f,x))
 APP(app(filter,f),app(app(cons,x),xs)) -> APP(f,x)
 APP(app(map,f),app(app(cons,x),xs)) -> APP(app(cons,app(f,x)),app(app(map,f),xs))
 APP(app(map,f),app(app(cons,x),xs)) -> APP(app(map,f),xs)
 APP(app(map,f),app(app(cons,x),xs)) -> APP(cons,app(f,x))
 APP(app(map,f),app(app(cons,x),xs)) -> APP(f,x)
-> Rules:
 app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs)
 app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs))
 app(app(*,x),app(app(+,y),z)) -> app(app(+,app(app(*,x),y)),app(app(*,x),z))
 app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs)
 app(app(filter,f),nil) -> nil
 app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs))
 app(app(map,f),nil) -> nil

Problem 1: 

SCC Processor:
-> Pairs:
 APP(app(app(app(filter2,false),f),x),xs) -> APP(app(filter,f),xs)
 APP(app(app(app(filter2,true),f),x),xs) -> APP(app(cons,x),app(app(filter,f),xs))
 APP(app(app(app(filter2,true),f),x),xs) -> APP(app(filter,f),xs)
 APP(app(*,x),app(app(+,y),z)) -> APP(app(*,x),y)
 APP(app(*,x),app(app(+,y),z)) -> APP(app(*,x),z)
 APP(app(*,x),app(app(+,y),z)) -> APP(app(+,app(app(*,x),y)),app(app(*,x),z))
 APP(app(*,x),app(app(+,y),z)) -> APP(+,app(app(*,x),y))
 APP(app(filter,f),app(app(cons,x),xs)) -> APP(app(app(app(filter2,app(f,x)),f),x),xs)
 APP(app(filter,f),app(app(cons,x),xs)) -> APP(app(app(filter2,app(f,x)),f),x)
 APP(app(filter,f),app(app(cons,x),xs)) -> APP(app(filter2,app(f,x)),f)
 APP(app(filter,f),app(app(cons,x),xs)) -> APP(filter2,app(f,x))
 APP(app(filter,f),app(app(cons,x),xs)) -> APP(f,x)
 APP(app(map,f),app(app(cons,x),xs)) -> APP(app(cons,app(f,x)),app(app(map,f),xs))
 APP(app(map,f),app(app(cons,x),xs)) -> APP(app(map,f),xs)
 APP(app(map,f),app(app(cons,x),xs)) -> APP(cons,app(f,x))
 APP(app(map,f),app(app(cons,x),xs)) -> APP(f,x)
-> Rules:
 app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs)
 app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs))
 app(app(*,x),app(app(+,y),z)) -> app(app(+,app(app(*,x),y)),app(app(*,x),z))
 app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs)
 app(app(filter,f),nil) -> nil
 app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs))
 app(app(map,f),nil) -> nil
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(*,x),app(app(+,y),z)) -> APP(app(*,x),y)
 APP(app(*,x),app(app(+,y),z)) -> APP(app(*,x),z)
->->-> Rules:
 app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs)
 app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs))
 app(app(*,x),app(app(+,y),z)) -> app(app(+,app(app(*,x),y)),app(app(*,x),z))
 app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs)
 app(app(filter,f),nil) -> nil
 app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs))
 app(app(map,f),nil) -> nil
->->Cycle:
->->-> Pairs:
 APP(app(app(app(filter2,false),f),x),xs) -> APP(app(filter,f),xs)
 APP(app(app(app(filter2,true),f),x),xs) -> APP(app(filter,f),xs)
 APP(app(filter,f),app(app(cons,x),xs)) -> APP(app(app(app(filter2,app(f,x)),f),x),xs)
 APP(app(filter,f),app(app(cons,x),xs)) -> APP(f,x)
 APP(app(map,f),app(app(cons,x),xs)) -> APP(app(map,f),xs)
 APP(app(map,f),app(app(cons,x),xs)) -> APP(f,x)
->->-> Rules:
 app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs)
 app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs))
 app(app(*,x),app(app(+,y),z)) -> app(app(+,app(app(*,x),y)),app(app(*,x),z))
 app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs)
 app(app(filter,f),nil) -> nil
 app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs))
 app(app(map,f),nil) -> nil


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 APP(app(*,x),app(app(+,y),z)) -> APP(app(*,x),y)
 APP(app(*,x),app(app(+,y),z)) -> APP(app(*,x),z)
-> Rules:
 app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs)
 app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs))
 app(app(*,x),app(app(+,y),z)) -> app(app(+,app(app(*,x),y)),app(app(*,x),z))
 app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs)
 app(app(filter,f),nil) -> nil
 app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs))
 app(app(map,f),nil) -> nil
->Projection:
 pi(APP) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs)
 app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs))
 app(app(*,x),app(app(+,y),z)) -> app(app(+,app(app(*,x),y)),app(app(*,x),z))
 app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs)
 app(app(filter,f),nil) -> nil
 app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs))
 app(app(map,f),nil) -> nil
->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),f),x),xs) -> APP(app(filter,f),xs)
 APP(app(app(app(filter2,true),f),x),xs) -> APP(app(filter,f),xs)
 APP(app(filter,f),app(app(cons,x),xs)) -> APP(app(app(app(filter2,app(f,x)),f),x),xs)
 APP(app(filter,f),app(app(cons,x),xs)) -> APP(f,x)
 APP(app(map,f),app(app(cons,x),xs)) -> APP(app(map,f),xs)
 APP(app(map,f),app(app(cons,x),xs)) -> APP(f,x)
-> Rules:
 app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs)
 app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs))
 app(app(*,x),app(app(+,y),z)) -> app(app(+,app(app(*,x),y)),app(app(*,x),z))
 app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs)
 app(app(filter,f),nil) -> nil
 app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs))
 app(app(map,f),nil) -> nil
->Projection:
 pi(APP) = 2

Problem 1.2: 

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

The problem is finite.