YES Problem 1: (VAR v_NonEmpty:S f:S x:S xs:S y:S) (RULES app(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> app(app(filter,f:S),xs:S) app(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,f:S),xs:S)) app(app(.,app(app(/,y:S),x:S)),x:S) -> y:S app(app(.,e),x:S) -> x:S app(app(.,x:S),app(app(\,x:S),y:S)) -> y:S app(app(.,x:S),e) -> x:S app(app(/,app(app(.,y:S),x:S)),x:S) -> y:S app(app(/,x:S),app(app(\,y:S),x:S)) -> y:S app(app(/,x:S),e) -> x:S app(app(/,x:S),x:S) -> e app(app(\,app(app(/,x:S),y:S)),x:S) -> y:S app(app(\,e),x:S) -> x:S app(app(\,x:S),app(app(.,x:S),y:S)) -> y:S app(app(\,x:S),x:S) -> e app(app(filter,f:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) app(app(filter,f:S),nil) -> nil 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 ) Problem 1: Dependency Pairs Processor: -> Pairs: APP(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> APP(app(filter,f:S),xs:S) APP(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> APP(app(cons,x:S),app(app(filter,f:S),xs:S)) APP(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> APP(app(filter,f:S),xs:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(app(app(filter2,app(f:S,x:S)),f:S),x:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(app(filter2,app(f:S,x:S)),f:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(filter2,app(f:S,x:S)) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x: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) -> Rules: app(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> app(app(filter,f:S),xs:S) app(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,f:S),xs:S)) app(app(.,app(app(/,y:S),x:S)),x:S) -> y:S app(app(.,e),x:S) -> x:S app(app(.,x:S),app(app(\,x:S),y:S)) -> y:S app(app(.,x:S),e) -> x:S app(app(/,app(app(.,y:S),x:S)),x:S) -> y:S app(app(/,x:S),app(app(\,y:S),x:S)) -> y:S app(app(/,x:S),e) -> x:S app(app(/,x:S),x:S) -> e app(app(\,app(app(/,x:S),y:S)),x:S) -> y:S app(app(\,e),x:S) -> x:S app(app(\,x:S),app(app(.,x:S),y:S)) -> y:S app(app(\,x:S),x:S) -> e app(app(filter,f:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) app(app(filter,f:S),nil) -> nil 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 Problem 1: SCC Processor: -> Pairs: APP(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> APP(app(filter,f:S),xs:S) APP(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> APP(app(cons,x:S),app(app(filter,f:S),xs:S)) APP(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> APP(app(filter,f:S),xs:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(app(app(filter2,app(f:S,x:S)),f:S),x:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(app(filter2,app(f:S,x:S)),f:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(filter2,app(f:S,x:S)) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x: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) -> Rules: app(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> app(app(filter,f:S),xs:S) app(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,f:S),xs:S)) app(app(.,app(app(/,y:S),x:S)),x:S) -> y:S app(app(.,e),x:S) -> x:S app(app(.,x:S),app(app(\,x:S),y:S)) -> y:S app(app(.,x:S),e) -> x:S app(app(/,app(app(.,y:S),x:S)),x:S) -> y:S app(app(/,x:S),app(app(\,y:S),x:S)) -> y:S app(app(/,x:S),e) -> x:S app(app(/,x:S),x:S) -> e app(app(\,app(app(/,x:S),y:S)),x:S) -> y:S app(app(\,e),x:S) -> x:S app(app(\,x:S),app(app(.,x:S),y:S)) -> y:S app(app(\,x:S),x:S) -> e app(app(filter,f:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) app(app(filter,f:S),nil) -> nil 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 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: APP(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> APP(app(filter,f:S),xs:S) APP(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> APP(app(filter,f:S),xs:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x: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(f:S,x:S) ->->-> Rules: app(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> app(app(filter,f:S),xs:S) app(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,f:S),xs:S)) app(app(.,app(app(/,y:S),x:S)),x:S) -> y:S app(app(.,e),x:S) -> x:S app(app(.,x:S),app(app(\,x:S),y:S)) -> y:S app(app(.,x:S),e) -> x:S app(app(/,app(app(.,y:S),x:S)),x:S) -> y:S app(app(/,x:S),app(app(\,y:S),x:S)) -> y:S app(app(/,x:S),e) -> x:S app(app(/,x:S),x:S) -> e app(app(\,app(app(/,x:S),y:S)),x:S) -> y:S app(app(\,e),x:S) -> x:S app(app(\,x:S),app(app(.,x:S),y:S)) -> y:S app(app(\,x:S),x:S) -> e app(app(filter,f:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) app(app(filter,f:S),nil) -> nil 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 Problem 1: Subterm Processor: -> Pairs: APP(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> APP(app(filter,f:S),xs:S) APP(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> APP(app(filter,f:S),xs:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x: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(f:S,x:S) -> Rules: app(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> app(app(filter,f:S),xs:S) app(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,f:S),xs:S)) app(app(.,app(app(/,y:S),x:S)),x:S) -> y:S app(app(.,e),x:S) -> x:S app(app(.,x:S),app(app(\,x:S),y:S)) -> y:S app(app(.,x:S),e) -> x:S app(app(/,app(app(.,y:S),x:S)),x:S) -> y:S app(app(/,x:S),app(app(\,y:S),x:S)) -> y:S app(app(/,x:S),e) -> x:S app(app(/,x:S),x:S) -> e app(app(\,app(app(/,x:S),y:S)),x:S) -> y:S app(app(\,e),x:S) -> x:S app(app(\,x:S),app(app(.,x:S),y:S)) -> y:S app(app(\,x:S),x:S) -> e app(app(filter,f:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) app(app(filter,f:S),nil) -> nil 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 ->Projection: pi(APP) = 2 Problem 1: SCC Processor: -> Pairs: APP(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> APP(app(filter,f:S),xs:S) APP(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> APP(app(filter,f:S),xs:S) -> Rules: app(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> app(app(filter,f:S),xs:S) app(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,f:S),xs:S)) app(app(.,app(app(/,y:S),x:S)),x:S) -> y:S app(app(.,e),x:S) -> x:S app(app(.,x:S),app(app(\,x:S),y:S)) -> y:S app(app(.,x:S),e) -> x:S app(app(/,app(app(.,y:S),x:S)),x:S) -> y:S app(app(/,x:S),app(app(\,y:S),x:S)) -> y:S app(app(/,x:S),e) -> x:S app(app(/,x:S),x:S) -> e app(app(\,app(app(/,x:S),y:S)),x:S) -> y:S app(app(\,e),x:S) -> x:S app(app(\,x:S),app(app(.,x:S),y:S)) -> y:S app(app(\,x:S),x:S) -> e app(app(filter,f:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) app(app(filter,f:S),nil) -> nil 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 ->Strongly Connected Components: There is no strongly connected component The problem is finite.