YES Problem 1: (VAR v_NonEmpty:S fun:S x:S xs:S y:S z:S) (RULES app(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> app(app(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(h,x:S)) app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,0),1) -> 0 app(app(g,0),1) -> 1 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(h,app(app(g,x:S),y:S)) -> app(h,x:S) ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: APP(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> APP(app(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(h,x:S)) APP(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> APP(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)) APP(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> APP(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)) APP(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> APP(f,app(app(g,x:S),y:S)) APP(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> APP(h,x:S) APP(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(cons,x:S),app(app(filter,fun:S),xs:S)) APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(filter2,app(fun:S,x:S)),fun:S),x:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(filter2,app(fun:S,x:S)),fun:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(filter2,app(fun:S,x:S)) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(map,fun:S),xs:S) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(cons,app(fun:S,x:S)) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) APP(h,app(app(g,x:S),y:S)) -> APP(h,x:S) -> Rules: app(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> app(app(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(h,x:S)) app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,0),1) -> 0 app(app(g,0),1) -> 1 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(h,app(app(g,x:S),y:S)) -> app(h,x:S) Problem 1: SCC Processor: -> Pairs: APP(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> APP(app(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(h,x:S)) APP(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> APP(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)) APP(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> APP(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)) APP(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> APP(f,app(app(g,x:S),y:S)) APP(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> APP(h,x:S) APP(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(cons,x:S),app(app(filter,fun:S),xs:S)) APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(filter2,app(fun:S,x:S)),fun:S),x:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(filter2,app(fun:S,x:S)),fun:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(filter2,app(fun:S,x:S)) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(map,fun:S),xs:S) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(cons,app(fun:S,x:S)) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) APP(h,app(app(g,x:S),y:S)) -> APP(h,x:S) -> Rules: app(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> app(app(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(h,x:S)) app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,0),1) -> 0 app(app(g,0),1) -> 1 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(h,app(app(g,x:S),y:S)) -> app(h,x:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: APP(h,app(app(g,x:S),y:S)) -> APP(h,x:S) ->->-> Rules: app(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> app(app(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(h,x:S)) app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,0),1) -> 0 app(app(g,0),1) -> 1 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(h,app(app(g,x:S),y:S)) -> app(h,x:S) ->->Cycle: ->->-> Pairs: APP(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(map,fun:S),xs:S) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) ->->-> Rules: app(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> app(app(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(h,x:S)) app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,0),1) -> 0 app(app(g,0),1) -> 1 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(h,app(app(g,x:S),y:S)) -> app(h,x:S) The problem is decomposed in 2 subproblems. Problem 1.1: Subterm Processor: -> Pairs: APP(h,app(app(g,x:S),y:S)) -> APP(h,x:S) -> Rules: app(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> app(app(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(h,x:S)) app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,0),1) -> 0 app(app(g,0),1) -> 1 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(h,app(app(g,x:S),y:S)) -> app(h,x:S) ->Projection: pi(APP) = 2 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: app(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> app(app(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(h,x:S)) app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,0),1) -> 0 app(app(g,0),1) -> 1 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(h,app(app(g,x:S),y:S)) -> app(h,x:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: APP(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(map,fun:S),xs:S) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) -> Rules: app(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> app(app(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(h,x:S)) app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,0),1) -> 0 app(app(g,0),1) -> 1 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(h,app(app(g,x:S),y:S)) -> app(h,x:S) ->Projection: pi(APP) = 2 Problem 1.2: SCC Processor: -> Pairs: APP(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) -> Rules: app(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> app(app(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(h,x:S)) app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,0),1) -> 0 app(app(g,0),1) -> 1 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(h,app(app(g,x:S),y:S)) -> app(h,x:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite.