/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR v_NonEmpty:S f:S x:S xs:S y:S) (RULES app(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> app(app(filter,f:S),xs:S) app(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,f:S),xs:S)) app(app(filter,f:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) app(app(filter,f:S),nil) -> nil app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S)) app(app(map,f:S),nil) -> nil app(D,app(app(*,x:S),y:S)) -> app(app(+,app(app(*,y:S),app(D,x:S))),app(app(*,x:S),app(D,y:S))) app(D,app(app(+,x:S),y:S)) -> app(app(+,app(D,x:S)),app(D,y:S)) app(D,app(app(-,x:S),y:S)) -> app(app(-,app(D,x:S)),app(D,y:S)) app(D,app(app(div,x:S),y:S)) -> app(app(-,app(app(div,app(D,x:S)),y:S)),app(app(div,app(app(*,x:S),app(D,y:S))),app(app(pow,y:S),2))) app(D,app(app(pow,x:S),y:S)) -> app(app(+,app(app(*,app(app(*,y:S),app(app(pow,x:S),app(app(-,y:S),1)))),app(D,x:S))),app(app(*,app(app(*,app(app(pow,x:S),y:S)),app(ln,x:S))),app(D,y:S))) app(D,app(ln,x:S)) -> app(app(div,app(D,x:S)),x:S) app(D,app(minus,x:S)) -> app(minus,app(D,x:S)) app(D,constant) -> 0 app(D,t) -> 1 ) Problem 1: Innermost Equivalent Processor: -> Rules: app(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> app(app(filter,f:S),xs:S) app(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,f:S),xs:S)) app(app(filter,f:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) app(app(filter,f:S),nil) -> nil app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S)) app(app(map,f:S),nil) -> nil app(D,app(app(*,x:S),y:S)) -> app(app(+,app(app(*,y:S),app(D,x:S))),app(app(*,x:S),app(D,y:S))) app(D,app(app(+,x:S),y:S)) -> app(app(+,app(D,x:S)),app(D,y:S)) app(D,app(app(-,x:S),y:S)) -> app(app(-,app(D,x:S)),app(D,y:S)) app(D,app(app(div,x:S),y:S)) -> app(app(-,app(app(div,app(D,x:S)),y:S)),app(app(div,app(app(*,x:S),app(D,y:S))),app(app(pow,y:S),2))) app(D,app(app(pow,x:S),y:S)) -> app(app(+,app(app(*,app(app(*,y:S),app(app(pow,x:S),app(app(-,y:S),1)))),app(D,x:S))),app(app(*,app(app(*,app(app(pow,x:S),y:S)),app(ln,x:S))),app(D,y:S))) app(D,app(ln,x:S)) -> app(app(div,app(D,x:S)),x:S) app(D,app(minus,x:S)) -> app(minus,app(D,x:S)) app(D,constant) -> 0 app(D,t) -> 1 -> 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),f:S),x:S),xs:S) -> APP(app(filter,f:S),xs:S) APP(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> APP(app(cons,x:S),app(app(filter,f:S),xs:S)) APP(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> APP(app(filter,f:S),xs:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(app(app(filter2,app(f:S,x:S)),f:S),x:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(app(filter2,app(f:S,x:S)),f:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(filter2,app(f:S,x:S)) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x:S) APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S)) APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(map,f:S),xs:S) APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(cons,app(f:S,x:S)) APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x:S) APP(D,app(app(*,x:S),y:S)) -> APP(app(*,x:S),app(D,y:S)) APP(D,app(app(*,x:S),y:S)) -> APP(app(*,y:S),app(D,x:S)) APP(D,app(app(*,x:S),y:S)) -> APP(app(+,app(app(*,y:S),app(D,x:S))),app(app(*,x:S),app(D,y:S))) APP(D,app(app(*,x:S),y:S)) -> APP(+,app(app(*,y:S),app(D,x:S))) APP(D,app(app(*,x:S),y:S)) -> APP(D,x:S) APP(D,app(app(*,x:S),y:S)) -> APP(D,y:S) APP(D,app(app(+,x:S),y:S)) -> APP(app(+,app(D,x:S)),app(D,y:S)) APP(D,app(app(+,x:S),y:S)) -> APP(+,app(D,x:S)) APP(D,app(app(+,x:S),y:S)) -> APP(D,x:S) APP(D,app(app(+,x:S),y:S)) -> APP(D,y:S) APP(D,app(app(-,x:S),y:S)) -> APP(app(-,app(D,x:S)),app(D,y:S)) APP(D,app(app(-,x:S),y:S)) -> APP(-,app(D,x:S)) APP(D,app(app(-,x:S),y:S)) -> APP(D,x:S) APP(D,app(app(-,x:S),y:S)) -> APP(D,y:S) APP(D,app(app(div,x:S),y:S)) -> APP(app(*,x:S),app(D,y:S)) APP(D,app(app(div,x:S),y:S)) -> APP(app(-,app(app(div,app(D,x:S)),y:S)),app(app(div,app(app(*,x:S),app(D,y:S))),app(app(pow,y:S),2))) APP(D,app(app(div,x:S),y:S)) -> APP(app(div,app(app(*,x:S),app(D,y:S))),app(app(pow,y:S),2)) APP(D,app(app(div,x:S),y:S)) -> APP(app(div,app(D,x:S)),y:S) APP(D,app(app(div,x:S),y:S)) -> APP(-,app(app(div,app(D,x:S)),y:S)) APP(D,app(app(div,x:S),y:S)) -> APP(D,x:S) APP(D,app(app(div,x:S),y:S)) -> APP(D,y:S) APP(D,app(app(div,x:S),y:S)) -> APP(div,app(app(*,x:S),app(D,y:S))) APP(D,app(app(div,x:S),y:S)) -> APP(div,app(D,x:S)) APP(D,app(app(pow,x:S),y:S)) -> APP(app(*,app(app(*,app(app(pow,x:S),y:S)),app(ln,x:S))),app(D,y:S)) APP(D,app(app(pow,x:S),y:S)) -> APP(app(*,app(app(*,y:S),app(app(pow,x:S),app(app(-,y:S),1)))),app(D,x:S)) APP(D,app(app(pow,x:S),y:S)) -> APP(app(+,app(app(*,app(app(*,y:S),app(app(pow,x:S),app(app(-,y:S),1)))),app(D,x:S))),app(app(*,app(app(*,app(app(pow,x:S),y:S)),app(ln,x:S))),app(D,y:S))) APP(D,app(app(pow,x:S),y:S)) -> APP(+,app(app(*,app(app(*,y:S),app(app(pow,x:S),app(app(-,y:S),1)))),app(D,x:S))) APP(D,app(app(pow,x:S),y:S)) -> APP(D,x:S) APP(D,app(app(pow,x:S),y:S)) -> APP(D,y:S) APP(D,app(ln,x:S)) -> APP(app(div,app(D,x:S)),x:S) APP(D,app(ln,x:S)) -> APP(D,x:S) APP(D,app(ln,x:S)) -> APP(div,app(D,x:S)) APP(D,app(minus,x:S)) -> APP(D,x:S) APP(D,app(minus,x:S)) -> APP(minus,app(D,x:S)) -> Rules: app(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> app(app(filter,f:S),xs:S) app(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,f:S),xs:S)) app(app(filter,f:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) app(app(filter,f:S),nil) -> nil app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S)) app(app(map,f:S),nil) -> nil app(D,app(app(*,x:S),y:S)) -> app(app(+,app(app(*,y:S),app(D,x:S))),app(app(*,x:S),app(D,y:S))) app(D,app(app(+,x:S),y:S)) -> app(app(+,app(D,x:S)),app(D,y:S)) app(D,app(app(-,x:S),y:S)) -> app(app(-,app(D,x:S)),app(D,y:S)) app(D,app(app(div,x:S),y:S)) -> app(app(-,app(app(div,app(D,x:S)),y:S)),app(app(div,app(app(*,x:S),app(D,y:S))),app(app(pow,y:S),2))) app(D,app(app(pow,x:S),y:S)) -> app(app(+,app(app(*,app(app(*,y:S),app(app(pow,x:S),app(app(-,y:S),1)))),app(D,x:S))),app(app(*,app(app(*,app(app(pow,x:S),y:S)),app(ln,x:S))),app(D,y:S))) app(D,app(ln,x:S)) -> app(app(div,app(D,x:S)),x:S) app(D,app(minus,x:S)) -> app(minus,app(D,x:S)) app(D,constant) -> 0 app(D,t) -> 1 Problem 1: SCC Processor: -> Pairs: APP(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> APP(app(filter,f:S),xs:S) APP(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> APP(app(cons,x:S),app(app(filter,f:S),xs:S)) APP(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> APP(app(filter,f:S),xs:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(app(app(filter2,app(f:S,x:S)),f:S),x:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(app(filter2,app(f:S,x:S)),f:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(filter2,app(f:S,x:S)) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x:S) APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S)) APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(map,f:S),xs:S) APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(cons,app(f:S,x:S)) APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x:S) APP(D,app(app(*,x:S),y:S)) -> APP(app(*,x:S),app(D,y:S)) APP(D,app(app(*,x:S),y:S)) -> APP(app(*,y:S),app(D,x:S)) APP(D,app(app(*,x:S),y:S)) -> APP(app(+,app(app(*,y:S),app(D,x:S))),app(app(*,x:S),app(D,y:S))) APP(D,app(app(*,x:S),y:S)) -> APP(+,app(app(*,y:S),app(D,x:S))) APP(D,app(app(*,x:S),y:S)) -> APP(D,x:S) APP(D,app(app(*,x:S),y:S)) -> APP(D,y:S) APP(D,app(app(+,x:S),y:S)) -> APP(app(+,app(D,x:S)),app(D,y:S)) APP(D,app(app(+,x:S),y:S)) -> APP(+,app(D,x:S)) APP(D,app(app(+,x:S),y:S)) -> APP(D,x:S) APP(D,app(app(+,x:S),y:S)) -> APP(D,y:S) APP(D,app(app(-,x:S),y:S)) -> APP(app(-,app(D,x:S)),app(D,y:S)) APP(D,app(app(-,x:S),y:S)) -> APP(-,app(D,x:S)) APP(D,app(app(-,x:S),y:S)) -> APP(D,x:S) APP(D,app(app(-,x:S),y:S)) -> APP(D,y:S) APP(D,app(app(div,x:S),y:S)) -> APP(app(*,x:S),app(D,y:S)) APP(D,app(app(div,x:S),y:S)) -> APP(app(-,app(app(div,app(D,x:S)),y:S)),app(app(div,app(app(*,x:S),app(D,y:S))),app(app(pow,y:S),2))) APP(D,app(app(div,x:S),y:S)) -> APP(app(div,app(app(*,x:S),app(D,y:S))),app(app(pow,y:S),2)) APP(D,app(app(div,x:S),y:S)) -> APP(app(div,app(D,x:S)),y:S) APP(D,app(app(div,x:S),y:S)) -> APP(-,app(app(div,app(D,x:S)),y:S)) APP(D,app(app(div,x:S),y:S)) -> APP(D,x:S) APP(D,app(app(div,x:S),y:S)) -> APP(D,y:S) APP(D,app(app(div,x:S),y:S)) -> APP(div,app(app(*,x:S),app(D,y:S))) APP(D,app(app(div,x:S),y:S)) -> APP(div,app(D,x:S)) APP(D,app(app(pow,x:S),y:S)) -> APP(app(*,app(app(*,app(app(pow,x:S),y:S)),app(ln,x:S))),app(D,y:S)) APP(D,app(app(pow,x:S),y:S)) -> APP(app(*,app(app(*,y:S),app(app(pow,x:S),app(app(-,y:S),1)))),app(D,x:S)) APP(D,app(app(pow,x:S),y:S)) -> APP(app(+,app(app(*,app(app(*,y:S),app(app(pow,x:S),app(app(-,y:S),1)))),app(D,x:S))),app(app(*,app(app(*,app(app(pow,x:S),y:S)),app(ln,x:S))),app(D,y:S))) APP(D,app(app(pow,x:S),y:S)) -> APP(+,app(app(*,app(app(*,y:S),app(app(pow,x:S),app(app(-,y:S),1)))),app(D,x:S))) APP(D,app(app(pow,x:S),y:S)) -> APP(D,x:S) APP(D,app(app(pow,x:S),y:S)) -> APP(D,y:S) APP(D,app(ln,x:S)) -> APP(app(div,app(D,x:S)),x:S) APP(D,app(ln,x:S)) -> APP(D,x:S) APP(D,app(ln,x:S)) -> APP(div,app(D,x:S)) APP(D,app(minus,x:S)) -> APP(D,x:S) APP(D,app(minus,x:S)) -> APP(minus,app(D,x:S)) -> Rules: app(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> app(app(filter,f:S),xs:S) app(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,f:S),xs:S)) app(app(filter,f:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) app(app(filter,f:S),nil) -> nil app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S)) app(app(map,f:S),nil) -> nil app(D,app(app(*,x:S),y:S)) -> app(app(+,app(app(*,y:S),app(D,x:S))),app(app(*,x:S),app(D,y:S))) app(D,app(app(+,x:S),y:S)) -> app(app(+,app(D,x:S)),app(D,y:S)) app(D,app(app(-,x:S),y:S)) -> app(app(-,app(D,x:S)),app(D,y:S)) app(D,app(app(div,x:S),y:S)) -> app(app(-,app(app(div,app(D,x:S)),y:S)),app(app(div,app(app(*,x:S),app(D,y:S))),app(app(pow,y:S),2))) app(D,app(app(pow,x:S),y:S)) -> app(app(+,app(app(*,app(app(*,y:S),app(app(pow,x:S),app(app(-,y:S),1)))),app(D,x:S))),app(app(*,app(app(*,app(app(pow,x:S),y:S)),app(ln,x:S))),app(D,y:S))) app(D,app(ln,x:S)) -> app(app(div,app(D,x:S)),x:S) app(D,app(minus,x:S)) -> app(minus,app(D,x:S)) app(D,constant) -> 0 app(D,t) -> 1 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: APP(D,app(app(*,x:S),y:S)) -> APP(D,x:S) APP(D,app(app(*,x:S),y:S)) -> APP(D,y:S) APP(D,app(app(+,x:S),y:S)) -> APP(D,x:S) APP(D,app(app(+,x:S),y:S)) -> APP(D,y:S) APP(D,app(app(-,x:S),y:S)) -> APP(D,x:S) APP(D,app(app(-,x:S),y:S)) -> APP(D,y:S) APP(D,app(app(div,x:S),y:S)) -> APP(D,x:S) APP(D,app(app(div,x:S),y:S)) -> APP(D,y:S) APP(D,app(app(pow,x:S),y:S)) -> APP(D,x:S) APP(D,app(app(pow,x:S),y:S)) -> APP(D,y:S) APP(D,app(ln,x:S)) -> APP(D,x:S) APP(D,app(minus,x:S)) -> APP(D,x:S) ->->-> Rules: app(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> app(app(filter,f:S),xs:S) app(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,f:S),xs:S)) app(app(filter,f:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) app(app(filter,f:S),nil) -> nil app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S)) app(app(map,f:S),nil) -> nil app(D,app(app(*,x:S),y:S)) -> app(app(+,app(app(*,y:S),app(D,x:S))),app(app(*,x:S),app(D,y:S))) app(D,app(app(+,x:S),y:S)) -> app(app(+,app(D,x:S)),app(D,y:S)) app(D,app(app(-,x:S),y:S)) -> app(app(-,app(D,x:S)),app(D,y:S)) app(D,app(app(div,x:S),y:S)) -> app(app(-,app(app(div,app(D,x:S)),y:S)),app(app(div,app(app(*,x:S),app(D,y:S))),app(app(pow,y:S),2))) app(D,app(app(pow,x:S),y:S)) -> app(app(+,app(app(*,app(app(*,y:S),app(app(pow,x:S),app(app(-,y:S),1)))),app(D,x:S))),app(app(*,app(app(*,app(app(pow,x:S),y:S)),app(ln,x:S))),app(D,y:S))) app(D,app(ln,x:S)) -> app(app(div,app(D,x:S)),x:S) app(D,app(minus,x:S)) -> app(minus,app(D,x:S)) app(D,constant) -> 0 app(D,t) -> 1 ->->Cycle: ->->-> Pairs: APP(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> APP(app(filter,f:S),xs:S) APP(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> APP(app(filter,f:S),xs:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x:S) APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(map,f:S),xs:S) APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x:S) ->->-> Rules: app(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> app(app(filter,f:S),xs:S) app(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,f:S),xs:S)) app(app(filter,f:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) app(app(filter,f:S),nil) -> nil app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S)) app(app(map,f:S),nil) -> nil app(D,app(app(*,x:S),y:S)) -> app(app(+,app(app(*,y:S),app(D,x:S))),app(app(*,x:S),app(D,y:S))) app(D,app(app(+,x:S),y:S)) -> app(app(+,app(D,x:S)),app(D,y:S)) app(D,app(app(-,x:S),y:S)) -> app(app(-,app(D,x:S)),app(D,y:S)) app(D,app(app(div,x:S),y:S)) -> app(app(-,app(app(div,app(D,x:S)),y:S)),app(app(div,app(app(*,x:S),app(D,y:S))),app(app(pow,y:S),2))) app(D,app(app(pow,x:S),y:S)) -> app(app(+,app(app(*,app(app(*,y:S),app(app(pow,x:S),app(app(-,y:S),1)))),app(D,x:S))),app(app(*,app(app(*,app(app(pow,x:S),y:S)),app(ln,x:S))),app(D,y:S))) app(D,app(ln,x:S)) -> app(app(div,app(D,x:S)),x:S) app(D,app(minus,x:S)) -> app(minus,app(D,x:S)) app(D,constant) -> 0 app(D,t) -> 1 The problem is decomposed in 2 subproblems. Problem 1.1: Subterm Processor: -> Pairs: APP(D,app(app(*,x:S),y:S)) -> APP(D,x:S) APP(D,app(app(*,x:S),y:S)) -> APP(D,y:S) APP(D,app(app(+,x:S),y:S)) -> APP(D,x:S) APP(D,app(app(+,x:S),y:S)) -> APP(D,y:S) APP(D,app(app(-,x:S),y:S)) -> APP(D,x:S) APP(D,app(app(-,x:S),y:S)) -> APP(D,y:S) APP(D,app(app(div,x:S),y:S)) -> APP(D,x:S) APP(D,app(app(div,x:S),y:S)) -> APP(D,y:S) APP(D,app(app(pow,x:S),y:S)) -> APP(D,x:S) APP(D,app(app(pow,x:S),y:S)) -> APP(D,y:S) APP(D,app(ln,x:S)) -> APP(D,x:S) APP(D,app(minus,x:S)) -> APP(D,x:S) -> Rules: app(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> app(app(filter,f:S),xs:S) app(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,f:S),xs:S)) app(app(filter,f:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) app(app(filter,f:S),nil) -> nil app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S)) app(app(map,f:S),nil) -> nil app(D,app(app(*,x:S),y:S)) -> app(app(+,app(app(*,y:S),app(D,x:S))),app(app(*,x:S),app(D,y:S))) app(D,app(app(+,x:S),y:S)) -> app(app(+,app(D,x:S)),app(D,y:S)) app(D,app(app(-,x:S),y:S)) -> app(app(-,app(D,x:S)),app(D,y:S)) app(D,app(app(div,x:S),y:S)) -> app(app(-,app(app(div,app(D,x:S)),y:S)),app(app(div,app(app(*,x:S),app(D,y:S))),app(app(pow,y:S),2))) app(D,app(app(pow,x:S),y:S)) -> app(app(+,app(app(*,app(app(*,y:S),app(app(pow,x:S),app(app(-,y:S),1)))),app(D,x:S))),app(app(*,app(app(*,app(app(pow,x:S),y:S)),app(ln,x:S))),app(D,y:S))) app(D,app(ln,x:S)) -> app(app(div,app(D,x:S)),x:S) app(D,app(minus,x:S)) -> app(minus,app(D,x:S)) app(D,constant) -> 0 app(D,t) -> 1 ->Projection: pi(APP) = 2 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: app(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> app(app(filter,f:S),xs:S) app(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,f:S),xs:S)) app(app(filter,f:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) app(app(filter,f:S),nil) -> nil app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S)) app(app(map,f:S),nil) -> nil app(D,app(app(*,x:S),y:S)) -> app(app(+,app(app(*,y:S),app(D,x:S))),app(app(*,x:S),app(D,y:S))) app(D,app(app(+,x:S),y:S)) -> app(app(+,app(D,x:S)),app(D,y:S)) app(D,app(app(-,x:S),y:S)) -> app(app(-,app(D,x:S)),app(D,y:S)) app(D,app(app(div,x:S),y:S)) -> app(app(-,app(app(div,app(D,x:S)),y:S)),app(app(div,app(app(*,x:S),app(D,y:S))),app(app(pow,y:S),2))) app(D,app(app(pow,x:S),y:S)) -> app(app(+,app(app(*,app(app(*,y:S),app(app(pow,x:S),app(app(-,y:S),1)))),app(D,x:S))),app(app(*,app(app(*,app(app(pow,x:S),y:S)),app(ln,x:S))),app(D,y:S))) app(D,app(ln,x:S)) -> app(app(div,app(D,x:S)),x:S) app(D,app(minus,x:S)) -> app(minus,app(D,x:S)) app(D,constant) -> 0 app(D,t) -> 1 ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: APP(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> APP(app(filter,f:S),xs:S) APP(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> APP(app(filter,f:S),xs:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) APP(app(filter,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x:S) APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(map,f:S),xs:S) APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x:S) -> Rules: app(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> app(app(filter,f:S),xs:S) app(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,f:S),xs:S)) app(app(filter,f:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) app(app(filter,f:S),nil) -> nil app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S)) app(app(map,f:S),nil) -> nil app(D,app(app(*,x:S),y:S)) -> app(app(+,app(app(*,y:S),app(D,x:S))),app(app(*,x:S),app(D,y:S))) app(D,app(app(+,x:S),y:S)) -> app(app(+,app(D,x:S)),app(D,y:S)) app(D,app(app(-,x:S),y:S)) -> app(app(-,app(D,x:S)),app(D,y:S)) app(D,app(app(div,x:S),y:S)) -> app(app(-,app(app(div,app(D,x:S)),y:S)),app(app(div,app(app(*,x:S),app(D,y:S))),app(app(pow,y:S),2))) app(D,app(app(pow,x:S),y:S)) -> app(app(+,app(app(*,app(app(*,y:S),app(app(pow,x:S),app(app(-,y:S),1)))),app(D,x:S))),app(app(*,app(app(*,app(app(pow,x:S),y:S)),app(ln,x:S))),app(D,y:S))) app(D,app(ln,x:S)) -> app(app(div,app(D,x:S)),x:S) app(D,app(minus,x:S)) -> app(minus,app(D,x:S)) app(D,constant) -> 0 app(D,t) -> 1 ->Projection: pi(APP) = 2 Problem 1.2: SCC Processor: -> Pairs: APP(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> APP(app(filter,f:S),xs:S) APP(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> APP(app(filter,f:S),xs:S) -> Rules: app(app(app(app(filter2,ffalse),f:S),x:S),xs:S) -> app(app(filter,f:S),xs:S) app(app(app(app(filter2,ttrue),f:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,f:S),xs:S)) app(app(filter,f:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(f:S,x:S)),f:S),x:S),xs:S) app(app(filter,f:S),nil) -> nil app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S)) app(app(map,f:S),nil) -> nil app(D,app(app(*,x:S),y:S)) -> app(app(+,app(app(*,y:S),app(D,x:S))),app(app(*,x:S),app(D,y:S))) app(D,app(app(+,x:S),y:S)) -> app(app(+,app(D,x:S)),app(D,y:S)) app(D,app(app(-,x:S),y:S)) -> app(app(-,app(D,x:S)),app(D,y:S)) app(D,app(app(div,x:S),y:S)) -> app(app(-,app(app(div,app(D,x:S)),y:S)),app(app(div,app(app(*,x:S),app(D,y:S))),app(app(pow,y:S),2))) app(D,app(app(pow,x:S),y:S)) -> app(app(+,app(app(*,app(app(*,y:S),app(app(pow,x:S),app(app(-,y:S),1)))),app(D,x:S))),app(app(*,app(app(*,app(app(pow,x:S),y:S)),app(ln,x:S))),app(D,y:S))) app(D,app(ln,x:S)) -> app(app(div,app(D,x:S)),x:S) app(D,app(minus,x:S)) -> app(minus,app(D,x:S)) app(D,constant) -> 0 app(D,t) -> 1 ->Strongly Connected Components: There is no strongly connected component The problem is finite.