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(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(int,app(s,x:S)),app(s,y:S)) -> app(intlist,app(app(int,x:S),y:S)) app(app(int,app(s,x:S)),0) -> nil app(app(int,0),app(s,y:S)) -> app(app(cons,0),app(app(int,app(s,0)),app(s,y:S))) app(app(int,0),0) -> app(app(cons,0),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 app(intlist,app(app(cons,x:S),y:S)) -> app(app(cons,app(s,x:S)),app(intlist,y:S)) app(intlist,nil) -> nil ) (STRATEGY INNERMOST) 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(int,app(s,x:S)),app(s,y:S)) -> APP(app(int,x:S),y:S) APP(app(int,app(s,x:S)),app(s,y:S)) -> APP(intlist,app(app(int,x:S),y:S)) APP(app(int,0),app(s,y:S)) -> APP(app(cons,0),app(app(int,app(s,0)),app(s,y:S))) APP(app(int,0),app(s,y:S)) -> APP(app(int,app(s,0)),app(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(intlist,app(app(cons,x:S),y:S)) -> APP(app(cons,app(s,x:S)),app(intlist,y:S)) APP(intlist,app(app(cons,x:S),y:S)) -> APP(intlist,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(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(int,app(s,x:S)),app(s,y:S)) -> app(intlist,app(app(int,x:S),y:S)) app(app(int,app(s,x:S)),0) -> nil app(app(int,0),app(s,y:S)) -> app(app(cons,0),app(app(int,app(s,0)),app(s,y:S))) app(app(int,0),0) -> app(app(cons,0),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 app(intlist,app(app(cons,x:S),y:S)) -> app(app(cons,app(s,x:S)),app(intlist,y:S)) app(intlist,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(int,app(s,x:S)),app(s,y:S)) -> APP(app(int,x:S),y:S) APP(app(int,app(s,x:S)),app(s,y:S)) -> APP(intlist,app(app(int,x:S),y:S)) APP(app(int,0),app(s,y:S)) -> APP(app(cons,0),app(app(int,app(s,0)),app(s,y:S))) APP(app(int,0),app(s,y:S)) -> APP(app(int,app(s,0)),app(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(intlist,app(app(cons,x:S),y:S)) -> APP(app(cons,app(s,x:S)),app(intlist,y:S)) APP(intlist,app(app(cons,x:S),y:S)) -> APP(intlist,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(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(int,app(s,x:S)),app(s,y:S)) -> app(intlist,app(app(int,x:S),y:S)) app(app(int,app(s,x:S)),0) -> nil app(app(int,0),app(s,y:S)) -> app(app(cons,0),app(app(int,app(s,0)),app(s,y:S))) app(app(int,0),0) -> app(app(cons,0),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 app(intlist,app(app(cons,x:S),y:S)) -> app(app(cons,app(s,x:S)),app(intlist,y:S)) app(intlist,nil) -> nil ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: APP(intlist,app(app(cons,x:S),y:S)) -> APP(intlist,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(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(int,app(s,x:S)),app(s,y:S)) -> app(intlist,app(app(int,x:S),y:S)) app(app(int,app(s,x:S)),0) -> nil app(app(int,0),app(s,y:S)) -> app(app(cons,0),app(app(int,app(s,0)),app(s,y:S))) app(app(int,0),0) -> app(app(cons,0),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 app(intlist,app(app(cons,x:S),y:S)) -> app(app(cons,app(s,x:S)),app(intlist,y:S)) app(intlist,nil) -> nil ->->Cycle: ->->-> Pairs: APP(app(int,app(s,x:S)),app(s,y:S)) -> APP(app(int,x:S),y:S) APP(app(int,0),app(s,y:S)) -> APP(app(int,app(s,0)),app(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(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(int,app(s,x:S)),app(s,y:S)) -> app(intlist,app(app(int,x:S),y:S)) app(app(int,app(s,x:S)),0) -> nil app(app(int,0),app(s,y:S)) -> app(app(cons,0),app(app(int,app(s,0)),app(s,y:S))) app(app(int,0),0) -> app(app(cons,0),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 app(intlist,app(app(cons,x:S),y:S)) -> app(app(cons,app(s,x:S)),app(intlist,y:S)) app(intlist,nil) -> nil ->->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(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(int,app(s,x:S)),app(s,y:S)) -> app(intlist,app(app(int,x:S),y:S)) app(app(int,app(s,x:S)),0) -> nil app(app(int,0),app(s,y:S)) -> app(app(cons,0),app(app(int,app(s,0)),app(s,y:S))) app(app(int,0),0) -> app(app(cons,0),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 app(intlist,app(app(cons,x:S),y:S)) -> app(app(cons,app(s,x:S)),app(intlist,y:S)) app(intlist,nil) -> nil The problem is decomposed in 3 subproblems. Problem 1.1: Subterm Processor: -> Pairs: APP(intlist,app(app(cons,x:S),y:S)) -> APP(intlist,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(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(int,app(s,x:S)),app(s,y:S)) -> app(intlist,app(app(int,x:S),y:S)) app(app(int,app(s,x:S)),0) -> nil app(app(int,0),app(s,y:S)) -> app(app(cons,0),app(app(int,app(s,0)),app(s,y:S))) app(app(int,0),0) -> app(app(cons,0),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 app(intlist,app(app(cons,x:S),y:S)) -> app(app(cons,app(s,x:S)),app(intlist,y:S)) app(intlist,nil) -> nil ->Projection: pi(APP) = 2 Problem 1.1: SCC Processor: -> Pairs: Empty -> 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(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(int,app(s,x:S)),app(s,y:S)) -> app(intlist,app(app(int,x:S),y:S)) app(app(int,app(s,x:S)),0) -> nil app(app(int,0),app(s,y:S)) -> app(app(cons,0),app(app(int,app(s,0)),app(s,y:S))) app(app(int,0),0) -> app(app(cons,0),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 app(intlist,app(app(cons,x:S),y:S)) -> app(app(cons,app(s,x:S)),app(intlist,y:S)) app(intlist,nil) -> nil ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: APP(app(int,app(s,x:S)),app(s,y:S)) -> APP(app(int,x:S),y:S) APP(app(int,0),app(s,y:S)) -> APP(app(int,app(s,0)),app(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(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(int,app(s,x:S)),app(s,y:S)) -> app(intlist,app(app(int,x:S),y:S)) app(app(int,app(s,x:S)),0) -> nil app(app(int,0),app(s,y:S)) -> app(app(cons,0),app(app(int,app(s,0)),app(s,y:S))) app(app(int,0),0) -> app(app(cons,0),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 app(intlist,app(app(cons,x:S),y:S)) -> app(app(cons,app(s,x:S)),app(intlist,y:S)) app(intlist,nil) -> nil ->Projection: pi(APP) = 2 Problem 1.2: SCC Processor: -> Pairs: APP(app(int,0),app(s,y:S)) -> APP(app(int,app(s,0)),app(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(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(int,app(s,x:S)),app(s,y:S)) -> app(intlist,app(app(int,x:S),y:S)) app(app(int,app(s,x:S)),0) -> nil app(app(int,0),app(s,y:S)) -> app(app(cons,0),app(app(int,app(s,0)),app(s,y:S))) app(app(int,0),0) -> app(app(cons,0),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 app(intlist,app(app(cons,x:S),y:S)) -> app(app(cons,app(s,x:S)),app(intlist,y:S)) app(intlist,nil) -> nil ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: 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(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(int,app(s,x:S)),app(s,y:S)) -> app(intlist,app(app(int,x:S),y:S)) app(app(int,app(s,x:S)),0) -> nil app(app(int,0),app(s,y:S)) -> app(app(cons,0),app(app(int,app(s,0)),app(s,y:S))) app(app(int,0),0) -> app(app(cons,0),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 app(intlist,app(app(cons,x:S),y:S)) -> app(app(cons,app(s,x:S)),app(intlist,y:S)) app(intlist,nil) -> nil ->Projection: pi(APP) = 2 Problem 1.3: 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(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(int,app(s,x:S)),app(s,y:S)) -> app(intlist,app(app(int,x:S),y:S)) app(app(int,app(s,x:S)),0) -> nil app(app(int,0),app(s,y:S)) -> app(app(cons,0),app(app(int,app(s,0)),app(s,y:S))) app(app(int,0),0) -> app(app(cons,0),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 app(intlist,app(app(cons,x:S),y:S)) -> app(app(cons,app(s,x:S)),app(intlist,y:S)) app(intlist,nil) -> nil ->Strongly Connected Components: There is no strongly connected component The problem is finite.