/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR f x xs y) (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(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 app(D,app(app(*,x),y)) -> app(app(+,app(app(*,y),app(D,x))),app(app(*,x),app(D,y))) app(D,app(app(+,x),y)) -> app(app(+,app(D,x)),app(D,y)) app(D,app(app(-,x),y)) -> app(app(-,app(D,x)),app(D,y)) app(D,constant) -> 0 app(D,t) -> 1 ) 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(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 app(D,app(app(*,x),y)) -> app(app(+,app(app(*,y),app(D,x))),app(app(*,x),app(D,y))) app(D,app(app(+,x),y)) -> app(app(+,app(D,x)),app(D,y)) app(D,app(app(-,x),y)) -> app(app(-,app(D,x)),app(D,y)) app(D,constant) -> 0 app(D,t) -> 1 -> 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(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) APP(D,app(app(*,x),y)) -> APP(app(*,x),app(D,y)) APP(D,app(app(*,x),y)) -> APP(app(*,y),app(D,x)) APP(D,app(app(*,x),y)) -> APP(app(+,app(app(*,y),app(D,x))),app(app(*,x),app(D,y))) APP(D,app(app(*,x),y)) -> APP(+,app(app(*,y),app(D,x))) APP(D,app(app(*,x),y)) -> APP(D,x) APP(D,app(app(*,x),y)) -> APP(D,y) APP(D,app(app(+,x),y)) -> APP(app(+,app(D,x)),app(D,y)) APP(D,app(app(+,x),y)) -> APP(+,app(D,x)) APP(D,app(app(+,x),y)) -> APP(D,x) APP(D,app(app(+,x),y)) -> APP(D,y) APP(D,app(app(-,x),y)) -> APP(app(-,app(D,x)),app(D,y)) APP(D,app(app(-,x),y)) -> APP(-,app(D,x)) APP(D,app(app(-,x),y)) -> APP(D,x) APP(D,app(app(-,x),y)) -> APP(D,y) -> 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(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 app(D,app(app(*,x),y)) -> app(app(+,app(app(*,y),app(D,x))),app(app(*,x),app(D,y))) app(D,app(app(+,x),y)) -> app(app(+,app(D,x)),app(D,y)) app(D,app(app(-,x),y)) -> app(app(-,app(D,x)),app(D,y)) app(D,constant) -> 0 app(D,t) -> 1 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(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) APP(D,app(app(*,x),y)) -> APP(app(*,x),app(D,y)) APP(D,app(app(*,x),y)) -> APP(app(*,y),app(D,x)) APP(D,app(app(*,x),y)) -> APP(app(+,app(app(*,y),app(D,x))),app(app(*,x),app(D,y))) APP(D,app(app(*,x),y)) -> APP(+,app(app(*,y),app(D,x))) APP(D,app(app(*,x),y)) -> APP(D,x) APP(D,app(app(*,x),y)) -> APP(D,y) APP(D,app(app(+,x),y)) -> APP(app(+,app(D,x)),app(D,y)) APP(D,app(app(+,x),y)) -> APP(+,app(D,x)) APP(D,app(app(+,x),y)) -> APP(D,x) APP(D,app(app(+,x),y)) -> APP(D,y) APP(D,app(app(-,x),y)) -> APP(app(-,app(D,x)),app(D,y)) APP(D,app(app(-,x),y)) -> APP(-,app(D,x)) APP(D,app(app(-,x),y)) -> APP(D,x) APP(D,app(app(-,x),y)) -> APP(D,y) -> 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(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 app(D,app(app(*,x),y)) -> app(app(+,app(app(*,y),app(D,x))),app(app(*,x),app(D,y))) app(D,app(app(+,x),y)) -> app(app(+,app(D,x)),app(D,y)) app(D,app(app(-,x),y)) -> app(app(-,app(D,x)),app(D,y)) app(D,constant) -> 0 app(D,t) -> 1 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: APP(D,app(app(*,x),y)) -> APP(D,x) APP(D,app(app(*,x),y)) -> APP(D,y) APP(D,app(app(+,x),y)) -> APP(D,x) APP(D,app(app(+,x),y)) -> APP(D,y) APP(D,app(app(-,x),y)) -> APP(D,x) APP(D,app(app(-,x),y)) -> APP(D,y) ->->-> 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(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 app(D,app(app(*,x),y)) -> app(app(+,app(app(*,y),app(D,x))),app(app(*,x),app(D,y))) app(D,app(app(+,x),y)) -> app(app(+,app(D,x)),app(D,y)) app(D,app(app(-,x),y)) -> app(app(-,app(D,x)),app(D,y)) app(D,constant) -> 0 app(D,t) -> 1 ->->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(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 app(D,app(app(*,x),y)) -> app(app(+,app(app(*,y),app(D,x))),app(app(*,x),app(D,y))) app(D,app(app(+,x),y)) -> app(app(+,app(D,x)),app(D,y)) app(D,app(app(-,x),y)) -> app(app(-,app(D,x)),app(D,y)) app(D,constant) -> 0 app(D,t) -> 1 The problem is decomposed in 2 subproblems. Problem 1.1: Subterm Processor: -> Pairs: APP(D,app(app(*,x),y)) -> APP(D,x) APP(D,app(app(*,x),y)) -> APP(D,y) APP(D,app(app(+,x),y)) -> APP(D,x) APP(D,app(app(+,x),y)) -> APP(D,y) APP(D,app(app(-,x),y)) -> APP(D,x) APP(D,app(app(-,x),y)) -> APP(D,y) -> 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(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 app(D,app(app(*,x),y)) -> app(app(+,app(app(*,y),app(D,x))),app(app(*,x),app(D,y))) app(D,app(app(+,x),y)) -> app(app(+,app(D,x)),app(D,y)) app(D,app(app(-,x),y)) -> app(app(-,app(D,x)),app(D,y)) app(D,constant) -> 0 app(D,t) -> 1 ->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(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 app(D,app(app(*,x),y)) -> app(app(+,app(app(*,y),app(D,x))),app(app(*,x),app(D,y))) app(D,app(app(+,x),y)) -> app(app(+,app(D,x)),app(D,y)) app(D,app(app(-,x),y)) -> app(app(-,app(D,x)),app(D,y)) app(D,constant) -> 0 app(D,t) -> 1 ->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(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 app(D,app(app(*,x),y)) -> app(app(+,app(app(*,y),app(D,x))),app(app(*,x),app(D,y))) app(D,app(app(+,x),y)) -> app(app(+,app(D,x)),app(D,y)) app(D,app(app(-,x),y)) -> app(app(-,app(D,x)),app(D,y)) app(D,constant) -> 0 app(D,t) -> 1 ->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(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 app(D,app(app(*,x),y)) -> app(app(+,app(app(*,y),app(D,x))),app(app(*,x),app(D,y))) app(D,app(app(+,x),y)) -> app(app(+,app(D,x)),app(D,y)) app(D,app(app(-,x),y)) -> app(app(-,app(D,x)),app(D,y)) app(D,constant) -> 0 app(D,t) -> 1 ->Strongly Connected Components: There is no strongly connected component The problem is finite.