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