YES Problem 1: (VAR v_NonEmpty:S fun:S x:S xs:S y:S) (RULES 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(if,ffalse),x:S),y:S) -> y:S app(app(app(if,ttrue),x:S),y:S) -> x: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,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y: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),nil) -> nil app(f,app(s,x:S)) -> app(f,x:S) app(f,0) -> ttrue app(f,1) -> ffalse ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs 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(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(g,app(s,x:S)),app(s,y:S)) -> APP(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) APP(app(g,app(s,x:S)),app(s,y:S)) -> APP(app(if,app(f,x:S)),app(s,x:S)) APP(app(g,app(s,x:S)),app(s,y:S)) -> APP(f,x:S) APP(app(g,app(s,x:S)),app(s,y:S)) -> APP(if,app(f,x:S)) APP(app(g,x:S),app(c,y:S)) -> APP(app(g,app(s,app(c,y:S))),y:S) APP(app(g,x:S),app(c,y:S)) -> APP(app(g,x:S),app(app(g,app(s,app(c,y:S))),y: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(f,app(s,x:S)) -> APP(f,x:S) -> Rules: 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(if,ffalse),x:S),y:S) -> y:S app(app(app(if,ttrue),x:S),y:S) -> x: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,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y: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),nil) -> nil app(f,app(s,x:S)) -> app(f,x:S) app(f,0) -> ttrue app(f,1) -> ffalse Problem 1: 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(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(g,app(s,x:S)),app(s,y:S)) -> APP(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) APP(app(g,app(s,x:S)),app(s,y:S)) -> APP(app(if,app(f,x:S)),app(s,x:S)) APP(app(g,app(s,x:S)),app(s,y:S)) -> APP(f,x:S) APP(app(g,app(s,x:S)),app(s,y:S)) -> APP(if,app(f,x:S)) APP(app(g,x:S),app(c,y:S)) -> APP(app(g,app(s,app(c,y:S))),y:S) APP(app(g,x:S),app(c,y:S)) -> APP(app(g,x:S),app(app(g,app(s,app(c,y:S))),y: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(f,app(s,x:S)) -> APP(f,x:S) -> Rules: 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(if,ffalse),x:S),y:S) -> y:S app(app(app(if,ttrue),x:S),y:S) -> x: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,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y: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),nil) -> nil app(f,app(s,x:S)) -> app(f,x:S) app(f,0) -> ttrue app(f,1) -> ffalse ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: APP(f,app(s,x:S)) -> APP(f,x:S) ->->-> Rules: 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(if,ffalse),x:S),y:S) -> y:S app(app(app(if,ttrue),x:S),y:S) -> x: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,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y: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),nil) -> nil app(f,app(s,x:S)) -> app(f,x:S) app(f,0) -> ttrue app(f,1) -> ffalse ->->Cycle: ->->-> Pairs: APP(app(g,x:S),app(c,y:S)) -> APP(app(g,app(s,app(c,y:S))),y:S) APP(app(g,x:S),app(c,y:S)) -> APP(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) ->->-> Rules: 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(if,ffalse),x:S),y:S) -> y:S app(app(app(if,ttrue),x:S),y:S) -> x: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,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y: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),nil) -> nil app(f,app(s,x:S)) -> app(f,x:S) app(f,0) -> ttrue app(f,1) -> ffalse ->->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(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(if,ffalse),x:S),y:S) -> y:S app(app(app(if,ttrue),x:S),y:S) -> x: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,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y: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),nil) -> nil app(f,app(s,x:S)) -> app(f,x:S) app(f,0) -> ttrue app(f,1) -> ffalse The problem is decomposed in 3 subproblems. Problem 1.1: Subterm Processor: -> Pairs: APP(f,app(s,x:S)) -> APP(f,x:S) -> Rules: 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(if,ffalse),x:S),y:S) -> y:S app(app(app(if,ttrue),x:S),y:S) -> x: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,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y: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),nil) -> nil app(f,app(s,x:S)) -> app(f,x:S) app(f,0) -> ttrue app(f,1) -> ffalse ->Projection: pi(APP) = 2 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: 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(if,ffalse),x:S),y:S) -> y:S app(app(app(if,ttrue),x:S),y:S) -> x: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,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y: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),nil) -> nil app(f,app(s,x:S)) -> app(f,x:S) app(f,0) -> ttrue app(f,1) -> ffalse ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Reduction Pairs Processor: -> Pairs: APP(app(g,x:S),app(c,y:S)) -> APP(app(g,app(s,app(c,y:S))),y:S) APP(app(g,x:S),app(c,y:S)) -> APP(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) -> Rules: 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(if,ffalse),x:S),y:S) -> y:S app(app(app(if,ttrue),x:S),y:S) -> x: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,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y: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),nil) -> nil app(f,app(s,x:S)) -> app(f,x:S) app(f,0) -> ttrue app(f,1) -> ffalse -> Usable rules: 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(if,ffalse),x:S),y:S) -> y:S app(app(app(if,ttrue),x:S),y:S) -> x: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,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y: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),nil) -> nil app(f,app(s,x:S)) -> app(f,x:S) app(f,0) -> ttrue app(f,1) -> ffalse ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [app](X1,X2) = 2.X1.X2 + X1 [0] = 2 [1] = 0 [c] = 2 [cons] = 0 [f] = 0 [fSNonEmpty] = 0 [false] = 0 [filter] = 0 [filter2] = 0 [g] = 2 [if] = 2 [map] = 2 [nil] = 0 [s] = 0 [true] = 0 [APP](X1,X2) = X1.X2 + X1 + 2.X2 Problem 1.2: SCC Processor: -> Pairs: APP(app(g,x:S),app(c,y:S)) -> APP(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) -> Rules: 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(if,ffalse),x:S),y:S) -> y:S app(app(app(if,ttrue),x:S),y:S) -> x: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,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y: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),nil) -> nil app(f,app(s,x:S)) -> app(f,x:S) app(f,0) -> ttrue app(f,1) -> ffalse ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: APP(app(g,x:S),app(c,y:S)) -> APP(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) ->->-> Rules: 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(if,ffalse),x:S),y:S) -> y:S app(app(app(if,ttrue),x:S),y:S) -> x: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,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y: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),nil) -> nil app(f,app(s,x:S)) -> app(f,x:S) app(f,0) -> ttrue app(f,1) -> ffalse Problem 1.2: Reduction Pairs Processor: -> Pairs: APP(app(g,x:S),app(c,y:S)) -> APP(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) -> Rules: 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(if,ffalse),x:S),y:S) -> y:S app(app(app(if,ttrue),x:S),y:S) -> x: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,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y: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),nil) -> nil app(f,app(s,x:S)) -> app(f,x:S) app(f,0) -> ttrue app(f,1) -> ffalse -> Usable rules: 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(if,ffalse),x:S),y:S) -> y:S app(app(app(if,ttrue),x:S),y:S) -> x: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,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y: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),nil) -> nil app(f,app(s,x:S)) -> app(f,x:S) app(f,0) -> ttrue app(f,1) -> ffalse ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [app](X1,X2) = 2.X1.X2 + X1 [0] = 2 [1] = 0 [c] = 2 [cons] = 0 [f] = 0 [fSNonEmpty] = 0 [false] = 0 [filter] = 0 [filter2] = 0 [g] = 1 [if] = 1 [map] = 1 [nil] = 0 [s] = 0 [true] = 0 [APP](X1,X2) = X1.X2 + 2.X2 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: 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(if,ffalse),x:S),y:S) -> y:S app(app(app(if,ttrue),x:S),y:S) -> x: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,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y: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),nil) -> nil app(f,app(s,x:S)) -> app(f,x:S) app(f,0) -> ttrue app(f,1) -> ffalse ->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),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(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(if,ffalse),x:S),y:S) -> y:S app(app(app(if,ttrue),x:S),y:S) -> x: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,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y: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),nil) -> nil app(f,app(s,x:S)) -> app(f,x:S) app(f,0) -> ttrue app(f,1) -> ffalse ->Projection: pi(APP) = 2 Problem 1.3: 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(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(if,ffalse),x:S),y:S) -> y:S app(app(app(if,ttrue),x:S),y:S) -> x: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,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y: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),nil) -> nil app(f,app(s,x:S)) -> app(f,x:S) app(f,0) -> ttrue app(f,1) -> ffalse ->Strongly Connected Components: There is no strongly connected component The problem is finite.