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