/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR f x xs y z) (RULES app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs) app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs)) app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs) app(app(filter,f),nil) -> nil app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(plus,x),app(s,y)) -> app(s,app(app(plus,x),y)) app(app(plus,x),0) -> x app(app(times,x),app(app(plus,y),app(s,z))) -> app(app(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))),app(app(times,x),app(s,z))) app(app(times,x),app(s,y)) -> app(app(plus,app(app(times,x),y)),x) app(app(times,x),0) -> 0 ) Problem 1: Dependency Pairs Processor: -> Pairs: APP(app(app(app(filter2,false),f),x),xs) -> APP(app(filter,f),xs) APP(app(app(app(filter2,true),f),x),xs) -> APP(app(cons,x),app(app(filter,f),xs)) APP(app(app(app(filter2,true),f),x),xs) -> APP(app(filter,f),xs) APP(app(filter,f),app(app(cons,x),xs)) -> APP(app(app(app(filter2,app(f,x)),f),x),xs) APP(app(filter,f),app(app(cons,x),xs)) -> APP(app(app(filter2,app(f,x)),f),x) APP(app(filter,f),app(app(cons,x),xs)) -> APP(app(filter2,app(f,x)),f) APP(app(filter,f),app(app(cons,x),xs)) -> APP(filter2,app(f,x)) APP(app(filter,f),app(app(cons,x),xs)) -> APP(f,x) APP(app(map,f),app(app(cons,x),xs)) -> APP(app(cons,app(f,x)),app(app(map,f),xs)) APP(app(map,f),app(app(cons,x),xs)) -> APP(app(map,f),xs) APP(app(map,f),app(app(cons,x),xs)) -> APP(cons,app(f,x)) APP(app(map,f),app(app(cons,x),xs)) -> APP(f,x) APP(app(plus,x),app(s,y)) -> APP(app(plus,x),y) APP(app(plus,x),app(s,y)) -> APP(s,app(app(plus,x),y)) APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(app(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))),app(app(times,x),app(s,z))) APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(app(plus,y),app(app(times,app(s,z)),0)) APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(app(times,app(s,z)),0) APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0))) APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(app(times,x),app(s,z)) APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))) APP(app(times,x),app(s,y)) -> APP(app(plus,app(app(times,x),y)),x) APP(app(times,x),app(s,y)) -> APP(app(times,x),y) APP(app(times,x),app(s,y)) -> APP(plus,app(app(times,x),y)) -> Rules: app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs) app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs)) app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs) app(app(filter,f),nil) -> nil app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(plus,x),app(s,y)) -> app(s,app(app(plus,x),y)) app(app(plus,x),0) -> x app(app(times,x),app(app(plus,y),app(s,z))) -> app(app(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))),app(app(times,x),app(s,z))) app(app(times,x),app(s,y)) -> app(app(plus,app(app(times,x),y)),x) app(app(times,x),0) -> 0 Problem 1: SCC Processor: -> Pairs: APP(app(app(app(filter2,false),f),x),xs) -> APP(app(filter,f),xs) APP(app(app(app(filter2,true),f),x),xs) -> APP(app(cons,x),app(app(filter,f),xs)) APP(app(app(app(filter2,true),f),x),xs) -> APP(app(filter,f),xs) APP(app(filter,f),app(app(cons,x),xs)) -> APP(app(app(app(filter2,app(f,x)),f),x),xs) APP(app(filter,f),app(app(cons,x),xs)) -> APP(app(app(filter2,app(f,x)),f),x) APP(app(filter,f),app(app(cons,x),xs)) -> APP(app(filter2,app(f,x)),f) APP(app(filter,f),app(app(cons,x),xs)) -> APP(filter2,app(f,x)) APP(app(filter,f),app(app(cons,x),xs)) -> APP(f,x) APP(app(map,f),app(app(cons,x),xs)) -> APP(app(cons,app(f,x)),app(app(map,f),xs)) APP(app(map,f),app(app(cons,x),xs)) -> APP(app(map,f),xs) APP(app(map,f),app(app(cons,x),xs)) -> APP(cons,app(f,x)) APP(app(map,f),app(app(cons,x),xs)) -> APP(f,x) APP(app(plus,x),app(s,y)) -> APP(app(plus,x),y) APP(app(plus,x),app(s,y)) -> APP(s,app(app(plus,x),y)) APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(app(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))),app(app(times,x),app(s,z))) APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(app(plus,y),app(app(times,app(s,z)),0)) APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(app(times,app(s,z)),0) APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0))) APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(app(times,x),app(s,z)) APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))) APP(app(times,x),app(s,y)) -> APP(app(plus,app(app(times,x),y)),x) APP(app(times,x),app(s,y)) -> APP(app(times,x),y) APP(app(times,x),app(s,y)) -> APP(plus,app(app(times,x),y)) -> Rules: app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs) app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs)) app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs) app(app(filter,f),nil) -> nil app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(plus,x),app(s,y)) -> app(s,app(app(plus,x),y)) app(app(plus,x),0) -> x app(app(times,x),app(app(plus,y),app(s,z))) -> app(app(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))),app(app(times,x),app(s,z))) app(app(times,x),app(s,y)) -> app(app(plus,app(app(times,x),y)),x) app(app(times,x),0) -> 0 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: APP(app(plus,x),app(s,y)) -> APP(app(plus,x),y) ->->-> Rules: app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs) app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs)) app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs) app(app(filter,f),nil) -> nil app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(plus,x),app(s,y)) -> app(s,app(app(plus,x),y)) app(app(plus,x),0) -> x app(app(times,x),app(app(plus,y),app(s,z))) -> app(app(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))),app(app(times,x),app(s,z))) app(app(times,x),app(s,y)) -> app(app(plus,app(app(times,x),y)),x) app(app(times,x),0) -> 0 ->->Cycle: ->->-> Pairs: APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0))) APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(app(times,x),app(s,z)) APP(app(times,x),app(s,y)) -> APP(app(times,x),y) ->->-> Rules: app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs) app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs)) app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs) app(app(filter,f),nil) -> nil app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(plus,x),app(s,y)) -> app(s,app(app(plus,x),y)) app(app(plus,x),0) -> x app(app(times,x),app(app(plus,y),app(s,z))) -> app(app(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))),app(app(times,x),app(s,z))) app(app(times,x),app(s,y)) -> app(app(plus,app(app(times,x),y)),x) app(app(times,x),0) -> 0 ->->Cycle: ->->-> Pairs: APP(app(app(app(filter2,false),f),x),xs) -> APP(app(filter,f),xs) APP(app(app(app(filter2,true),f),x),xs) -> APP(app(filter,f),xs) APP(app(filter,f),app(app(cons,x),xs)) -> APP(app(app(app(filter2,app(f,x)),f),x),xs) APP(app(filter,f),app(app(cons,x),xs)) -> APP(f,x) APP(app(map,f),app(app(cons,x),xs)) -> APP(app(map,f),xs) APP(app(map,f),app(app(cons,x),xs)) -> APP(f,x) ->->-> Rules: app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs) app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs)) app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs) app(app(filter,f),nil) -> nil app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(plus,x),app(s,y)) -> app(s,app(app(plus,x),y)) app(app(plus,x),0) -> x app(app(times,x),app(app(plus,y),app(s,z))) -> app(app(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))),app(app(times,x),app(s,z))) app(app(times,x),app(s,y)) -> app(app(plus,app(app(times,x),y)),x) app(app(times,x),0) -> 0 The problem is decomposed in 3 subproblems. Problem 1.1: Subterm Processor: -> Pairs: APP(app(plus,x),app(s,y)) -> APP(app(plus,x),y) -> Rules: app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs) app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs)) app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs) app(app(filter,f),nil) -> nil app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(plus,x),app(s,y)) -> app(s,app(app(plus,x),y)) app(app(plus,x),0) -> x app(app(times,x),app(app(plus,y),app(s,z))) -> app(app(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))),app(app(times,x),app(s,z))) app(app(times,x),app(s,y)) -> app(app(plus,app(app(times,x),y)),x) app(app(times,x),0) -> 0 ->Projection: pi(APP) = 2 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs) app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs)) app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs) app(app(filter,f),nil) -> nil app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(plus,x),app(s,y)) -> app(s,app(app(plus,x),y)) app(app(plus,x),0) -> x app(app(times,x),app(app(plus,y),app(s,z))) -> app(app(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))),app(app(times,x),app(s,z))) app(app(times,x),app(s,y)) -> app(app(plus,app(app(times,x),y)),x) app(app(times,x),0) -> 0 ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Narrowing Processor: -> Pairs: APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0))) APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(app(times,x),app(s,z)) APP(app(times,x),app(s,y)) -> APP(app(times,x),y) -> Rules: app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs) app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs)) app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs) app(app(filter,f),nil) -> nil app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(plus,x),app(s,y)) -> app(s,app(app(plus,x),y)) app(app(plus,x),0) -> x app(app(times,x),app(app(plus,y),app(s,z))) -> app(app(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))),app(app(times,x),app(s,z))) app(app(times,x),app(s,y)) -> app(app(plus,app(app(times,x),y)),x) app(app(times,x),0) -> 0 ->Narrowed Pairs: ->->Original Pair: APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0))) ->-> Narrowed pairs: APP(app(times,x5),app(app(plus,x6),app(s,x7))) -> APP(app(times,x5),app(app(plus,x6),0)) Problem 1.2: SCC Processor: -> Pairs: APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(app(times,x),app(s,z)) APP(app(times,x),app(s,y)) -> APP(app(times,x),y) APP(app(times,x5),app(app(plus,x6),app(s,x7))) -> APP(app(times,x5),app(app(plus,x6),0)) -> Rules: app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs) app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs)) app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs) app(app(filter,f),nil) -> nil app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(plus,x),app(s,y)) -> app(s,app(app(plus,x),y)) app(app(plus,x),0) -> x app(app(times,x),app(app(plus,y),app(s,z))) -> app(app(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))),app(app(times,x),app(s,z))) app(app(times,x),app(s,y)) -> app(app(plus,app(app(times,x),y)),x) app(app(times,x),0) -> 0 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(app(times,x),app(s,z)) APP(app(times,x),app(s,y)) -> APP(app(times,x),y) APP(app(times,x5),app(app(plus,x6),app(s,x7))) -> APP(app(times,x5),app(app(plus,x6),0)) ->->-> Rules: app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs) app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs)) app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs) app(app(filter,f),nil) -> nil app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(plus,x),app(s,y)) -> app(s,app(app(plus,x),y)) app(app(plus,x),0) -> x app(app(times,x),app(app(plus,y),app(s,z))) -> app(app(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))),app(app(times,x),app(s,z))) app(app(times,x),app(s,y)) -> app(app(plus,app(app(times,x),y)),x) app(app(times,x),0) -> 0 Problem 1.2: Narrowing Processor: -> Pairs: APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(app(times,x),app(s,z)) APP(app(times,x),app(s,y)) -> APP(app(times,x),y) APP(app(times,x5),app(app(plus,x6),app(s,x7))) -> APP(app(times,x5),app(app(plus,x6),0)) -> Rules: app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs) app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs)) app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs) app(app(filter,f),nil) -> nil app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(plus,x),app(s,y)) -> app(s,app(app(plus,x),y)) app(app(plus,x),0) -> x app(app(times,x),app(app(plus,y),app(s,z))) -> app(app(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))),app(app(times,x),app(s,z))) app(app(times,x),app(s,y)) -> app(app(plus,app(app(times,x),y)),x) app(app(times,x),0) -> 0 ->Narrowed Pairs: ->->Original Pair: APP(app(times,x5),app(app(plus,x6),app(s,x7))) -> APP(app(times,x5),app(app(plus,x6),0)) ->-> Narrowed pairs: APP(app(times,x13),app(app(plus,x),app(s,x15))) -> APP(app(times,x13),x) Problem 1.2: SCC Processor: -> Pairs: APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(app(times,x),app(s,z)) APP(app(times,x),app(s,y)) -> APP(app(times,x),y) APP(app(times,x13),app(app(plus,x),app(s,x15))) -> APP(app(times,x13),x) -> Rules: app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs) app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs)) app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs) app(app(filter,f),nil) -> nil app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(plus,x),app(s,y)) -> app(s,app(app(plus,x),y)) app(app(plus,x),0) -> x app(app(times,x),app(app(plus,y),app(s,z))) -> app(app(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))),app(app(times,x),app(s,z))) app(app(times,x),app(s,y)) -> app(app(plus,app(app(times,x),y)),x) app(app(times,x),0) -> 0 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(app(times,x),app(s,z)) APP(app(times,x),app(s,y)) -> APP(app(times,x),y) APP(app(times,x13),app(app(plus,x),app(s,x15))) -> APP(app(times,x13),x) ->->-> Rules: app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs) app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs)) app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs) app(app(filter,f),nil) -> nil app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(plus,x),app(s,y)) -> app(s,app(app(plus,x),y)) app(app(plus,x),0) -> x app(app(times,x),app(app(plus,y),app(s,z))) -> app(app(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))),app(app(times,x),app(s,z))) app(app(times,x),app(s,y)) -> app(app(plus,app(app(times,x),y)),x) app(app(times,x),0) -> 0 Problem 1.2: Subterm Processor: -> Pairs: APP(app(times,x),app(app(plus,y),app(s,z))) -> APP(app(times,x),app(s,z)) APP(app(times,x),app(s,y)) -> APP(app(times,x),y) APP(app(times,x13),app(app(plus,x),app(s,x15))) -> APP(app(times,x13),x) -> Rules: app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs) app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs)) app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs) app(app(filter,f),nil) -> nil app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(plus,x),app(s,y)) -> app(s,app(app(plus,x),y)) app(app(plus,x),0) -> x app(app(times,x),app(app(plus,y),app(s,z))) -> app(app(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))),app(app(times,x),app(s,z))) app(app(times,x),app(s,y)) -> app(app(plus,app(app(times,x),y)),x) app(app(times,x),0) -> 0 ->Projection: pi(APP) = 2 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs) app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs)) app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs) app(app(filter,f),nil) -> nil app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(plus,x),app(s,y)) -> app(s,app(app(plus,x),y)) app(app(plus,x),0) -> x app(app(times,x),app(app(plus,y),app(s,z))) -> app(app(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))),app(app(times,x),app(s,z))) app(app(times,x),app(s,y)) -> app(app(plus,app(app(times,x),y)),x) app(app(times,x),0) -> 0 ->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),f),x),xs) -> APP(app(filter,f),xs) APP(app(app(app(filter2,true),f),x),xs) -> APP(app(filter,f),xs) APP(app(filter,f),app(app(cons,x),xs)) -> APP(app(app(app(filter2,app(f,x)),f),x),xs) APP(app(filter,f),app(app(cons,x),xs)) -> APP(f,x) APP(app(map,f),app(app(cons,x),xs)) -> APP(app(map,f),xs) APP(app(map,f),app(app(cons,x),xs)) -> APP(f,x) -> Rules: app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs) app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs)) app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs) app(app(filter,f),nil) -> nil app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(plus,x),app(s,y)) -> app(s,app(app(plus,x),y)) app(app(plus,x),0) -> x app(app(times,x),app(app(plus,y),app(s,z))) -> app(app(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))),app(app(times,x),app(s,z))) app(app(times,x),app(s,y)) -> app(app(plus,app(app(times,x),y)),x) app(app(times,x),0) -> 0 ->Projection: pi(APP) = 2 Problem 1.3: SCC Processor: -> Pairs: APP(app(app(app(filter2,false),f),x),xs) -> APP(app(filter,f),xs) APP(app(app(app(filter2,true),f),x),xs) -> APP(app(filter,f),xs) -> Rules: app(app(app(app(filter2,false),f),x),xs) -> app(app(filter,f),xs) app(app(app(app(filter2,true),f),x),xs) -> app(app(cons,x),app(app(filter,f),xs)) app(app(filter,f),app(app(cons,x),xs)) -> app(app(app(app(filter2,app(f,x)),f),x),xs) app(app(filter,f),nil) -> nil app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(plus,x),app(s,y)) -> app(s,app(app(plus,x),y)) app(app(plus,x),0) -> x app(app(times,x),app(app(plus,y),app(s,z))) -> app(app(plus,app(app(times,x),app(app(plus,y),app(app(times,app(s,z)),0)))),app(app(times,x),app(s,z))) app(app(times,x),app(s,y)) -> app(app(plus,app(app(times,x),y)),x) app(app(times,x),0) -> 0 ->Strongly Connected Components: There is no strongly connected component The problem is finite.