/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- 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(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(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(g,app(app(cons,app(s,x:S)),y:S)) -> app(s,x:S) app(g,app(app(cons,0),y:S)) -> app(g,y:S) app(h,app(app(cons,x:S),y:S)) -> app(h,app(g,app(app(cons,x:S),y:S))) ) Problem 1: Innermost Equivalent Processor: -> 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(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(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(g,app(app(cons,app(s,x:S)),y:S)) -> app(s,x:S) app(g,app(app(cons,0),y:S)) -> app(g,y:S) app(h,app(app(cons,x:S),y:S)) -> app(h,app(g,app(app(cons,x:S),y:S))) -> 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,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(f,app(s,x:S)) -> APP(f,x:S) APP(g,app(app(cons,0),y:S)) -> APP(g,y:S) APP(h,app(app(cons,x:S),y:S)) -> APP(g,app(app(cons,x:S),y:S)) APP(h,app(app(cons,x:S),y:S)) -> APP(h,app(g,app(app(cons,x: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(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(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(g,app(app(cons,app(s,x:S)),y:S)) -> app(s,x:S) app(g,app(app(cons,0),y:S)) -> app(g,y:S) app(h,app(app(cons,x:S),y:S)) -> app(h,app(g,app(app(cons,x:S),y:S))) 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(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) APP(g,app(app(cons,0),y:S)) -> APP(g,y:S) APP(h,app(app(cons,x:S),y:S)) -> APP(g,app(app(cons,x:S),y:S)) APP(h,app(app(cons,x:S),y:S)) -> APP(h,app(g,app(app(cons,x: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(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(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(g,app(app(cons,app(s,x:S)),y:S)) -> app(s,x:S) app(g,app(app(cons,0),y:S)) -> app(g,y:S) app(h,app(app(cons,x:S),y:S)) -> app(h,app(g,app(app(cons,x:S),y:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: APP(g,app(app(cons,0),y:S)) -> APP(g,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(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(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(g,app(app(cons,app(s,x:S)),y:S)) -> app(s,x:S) app(g,app(app(cons,0),y:S)) -> app(g,y:S) app(h,app(app(cons,x:S),y:S)) -> app(h,app(g,app(app(cons,x:S),y:S))) ->->Cycle: ->->-> Pairs: APP(h,app(app(cons,x:S),y:S)) -> APP(h,app(g,app(app(cons,x: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(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(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(g,app(app(cons,app(s,x:S)),y:S)) -> app(s,x:S) app(g,app(app(cons,0),y:S)) -> app(g,y:S) app(h,app(app(cons,x:S),y:S)) -> app(h,app(g,app(app(cons,x:S),y:S))) ->->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(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(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(g,app(app(cons,app(s,x:S)),y:S)) -> app(s,x:S) app(g,app(app(cons,0),y:S)) -> app(g,y:S) app(h,app(app(cons,x:S),y:S)) -> app(h,app(g,app(app(cons,x:S),y: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(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(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(g,app(app(cons,app(s,x:S)),y:S)) -> app(s,x:S) app(g,app(app(cons,0),y:S)) -> app(g,y:S) app(h,app(app(cons,x:S),y:S)) -> app(h,app(g,app(app(cons,x:S),y:S))) The problem is decomposed in 4 subproblems. Problem 1.1: Subterm Processor: -> Pairs: APP(g,app(app(cons,0),y:S)) -> APP(g,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(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(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(g,app(app(cons,app(s,x:S)),y:S)) -> app(s,x:S) app(g,app(app(cons,0),y:S)) -> app(g,y:S) app(h,app(app(cons,x:S),y:S)) -> app(h,app(g,app(app(cons,x:S),y:S))) ->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(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(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(g,app(app(cons,app(s,x:S)),y:S)) -> app(s,x:S) app(g,app(app(cons,0),y:S)) -> app(g,y:S) app(h,app(app(cons,x:S),y:S)) -> app(h,app(g,app(app(cons,x:S),y:S))) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Reduction Pairs Processor: -> Pairs: APP(h,app(app(cons,x:S),y:S)) -> APP(h,app(g,app(app(cons,x: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(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(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(g,app(app(cons,app(s,x:S)),y:S)) -> app(s,x:S) app(g,app(app(cons,0),y:S)) -> app(g,y:S) app(h,app(app(cons,x:S),y:S)) -> app(h,app(g,app(app(cons,x:S),y:S))) -> 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(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(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(g,app(app(cons,app(s,x:S)),y:S)) -> app(s,x:S) app(g,app(app(cons,0),y:S)) -> app(g,y:S) app(h,app(app(cons,x:S),y:S)) -> app(h,app(g,app(app(cons,x:S),y:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [app](X1,X2) = X1 + 1 [0] = 0 [cons] = 2 [f] = 0 [fSNonEmpty] = 0 [false] = 2 [filter] = 2 [filter2] = 0 [g] = 2 [h] = 0 [map] = 2 [nil] = 2 [s] = 2 [true] = 1 [APP](X1,X2) = 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(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(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(g,app(app(cons,app(s,x:S)),y:S)) -> app(s,x:S) app(g,app(app(cons,0),y:S)) -> app(g,y:S) app(h,app(app(cons,x:S),y:S)) -> app(h,app(g,app(app(cons,x:S),y:S))) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: 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(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(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(g,app(app(cons,app(s,x:S)),y:S)) -> app(s,x:S) app(g,app(app(cons,0),y:S)) -> app(g,y:S) app(h,app(app(cons,x:S),y:S)) -> app(h,app(g,app(app(cons,x:S),y:S))) ->Projection: pi(APP) = 2 Problem 1.3: 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(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(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(g,app(app(cons,app(s,x:S)),y:S)) -> app(s,x:S) app(g,app(app(cons,0),y:S)) -> app(g,y:S) app(h,app(app(cons,x:S),y:S)) -> app(h,app(g,app(app(cons,x:S),y:S))) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: 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(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(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(g,app(app(cons,app(s,x:S)),y:S)) -> app(s,x:S) app(g,app(app(cons,0),y:S)) -> app(g,y:S) app(h,app(app(cons,x:S),y:S)) -> app(h,app(g,app(app(cons,x:S),y:S))) ->Projection: pi(APP) = 2 Problem 1.4: 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(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(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(g,app(app(cons,app(s,x:S)),y:S)) -> app(s,x:S) app(g,app(app(cons,0),y:S)) -> app(g,y:S) app(h,app(app(cons,x:S),y:S)) -> app(h,app(g,app(app(cons,x:S),y:S))) ->Strongly Connected Components: There is no strongly connected component The problem is finite.