YES Problem 1: (VAR v_NonEmpty:S L:S X:S X1:S X2:S XS:S Y:S YS:S) (RULES active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: ACTIVE(app(cons(X:S,XS:S),YS:S)) -> APP(XS:S,YS:S) ACTIVE(app(cons(X:S,XS:S),YS:S)) -> CONS(X:S,app(XS:S,YS:S)) ACTIVE(app(cons(X:S,XS:S),YS:S)) -> MARK(cons(X:S,app(XS:S,YS:S))) ACTIVE(app(nil,YS:S)) -> MARK(YS:S) ACTIVE(from(X:S)) -> CONS(X:S,from(s(X:S))) ACTIVE(from(X:S)) -> FROM(s(X:S)) ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) ACTIVE(from(X:S)) -> S(X:S) ACTIVE(prefix(L:S)) -> CONS(nil,zWadr(L:S,prefix(L:S))) ACTIVE(prefix(L:S)) -> MARK(cons(nil,zWadr(L:S,prefix(L:S)))) ACTIVE(prefix(L:S)) -> ZWADR(L:S,prefix(L:S)) ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> APP(Y:S,cons(X:S,nil)) ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> CONS(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S)) ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> CONS(X:S,nil) ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> ZWADR(XS:S,YS:S) ACTIVE(zWadr(nil,YS:S)) -> MARK(nil) ACTIVE(zWadr(XS:S,nil)) -> MARK(nil) APP(active(X1:S),X2:S) -> APP(X1:S,X2:S) APP(mark(X1:S),X2:S) -> APP(X1:S,X2:S) APP(X1:S,active(X2:S)) -> APP(X1:S,X2:S) APP(X1:S,mark(X2:S)) -> APP(X1:S,X2:S) CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) FROM(active(X:S)) -> FROM(X:S) FROM(mark(X:S)) -> FROM(X:S) MARK(app(X1:S,X2:S)) -> ACTIVE(app(mark(X1:S),mark(X2:S))) MARK(app(X1:S,X2:S)) -> APP(mark(X1:S),mark(X2:S)) MARK(app(X1:S,X2:S)) -> MARK(X1:S) MARK(app(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> CONS(mark(X1:S),X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> FROM(mark(X:S)) MARK(from(X:S)) -> MARK(X:S) MARK(prefix(X:S)) -> ACTIVE(prefix(mark(X:S))) MARK(prefix(X:S)) -> MARK(X:S) MARK(prefix(X:S)) -> PREFIX(mark(X:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(s(X:S)) -> S(mark(X:S)) MARK(zWadr(X1:S,X2:S)) -> ACTIVE(zWadr(mark(X1:S),mark(X2:S))) MARK(zWadr(X1:S,X2:S)) -> MARK(X1:S) MARK(zWadr(X1:S,X2:S)) -> MARK(X2:S) MARK(zWadr(X1:S,X2:S)) -> ZWADR(mark(X1:S),mark(X2:S)) PREFIX(active(X:S)) -> PREFIX(X:S) PREFIX(mark(X:S)) -> PREFIX(X:S) S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) ZWADR(active(X1:S),X2:S) -> ZWADR(X1:S,X2:S) ZWADR(mark(X1:S),X2:S) -> ZWADR(X1:S,X2:S) ZWADR(X1:S,active(X2:S)) -> ZWADR(X1:S,X2:S) ZWADR(X1:S,mark(X2:S)) -> ZWADR(X1:S,X2:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) Problem 1: SCC Processor: -> Pairs: ACTIVE(app(cons(X:S,XS:S),YS:S)) -> APP(XS:S,YS:S) ACTIVE(app(cons(X:S,XS:S),YS:S)) -> CONS(X:S,app(XS:S,YS:S)) ACTIVE(app(cons(X:S,XS:S),YS:S)) -> MARK(cons(X:S,app(XS:S,YS:S))) ACTIVE(app(nil,YS:S)) -> MARK(YS:S) ACTIVE(from(X:S)) -> CONS(X:S,from(s(X:S))) ACTIVE(from(X:S)) -> FROM(s(X:S)) ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) ACTIVE(from(X:S)) -> S(X:S) ACTIVE(prefix(L:S)) -> CONS(nil,zWadr(L:S,prefix(L:S))) ACTIVE(prefix(L:S)) -> MARK(cons(nil,zWadr(L:S,prefix(L:S)))) ACTIVE(prefix(L:S)) -> ZWADR(L:S,prefix(L:S)) ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> APP(Y:S,cons(X:S,nil)) ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> CONS(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S)) ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> CONS(X:S,nil) ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> ZWADR(XS:S,YS:S) ACTIVE(zWadr(nil,YS:S)) -> MARK(nil) ACTIVE(zWadr(XS:S,nil)) -> MARK(nil) APP(active(X1:S),X2:S) -> APP(X1:S,X2:S) APP(mark(X1:S),X2:S) -> APP(X1:S,X2:S) APP(X1:S,active(X2:S)) -> APP(X1:S,X2:S) APP(X1:S,mark(X2:S)) -> APP(X1:S,X2:S) CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) FROM(active(X:S)) -> FROM(X:S) FROM(mark(X:S)) -> FROM(X:S) MARK(app(X1:S,X2:S)) -> ACTIVE(app(mark(X1:S),mark(X2:S))) MARK(app(X1:S,X2:S)) -> APP(mark(X1:S),mark(X2:S)) MARK(app(X1:S,X2:S)) -> MARK(X1:S) MARK(app(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> CONS(mark(X1:S),X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> FROM(mark(X:S)) MARK(from(X:S)) -> MARK(X:S) MARK(prefix(X:S)) -> ACTIVE(prefix(mark(X:S))) MARK(prefix(X:S)) -> MARK(X:S) MARK(prefix(X:S)) -> PREFIX(mark(X:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(s(X:S)) -> S(mark(X:S)) MARK(zWadr(X1:S,X2:S)) -> ACTIVE(zWadr(mark(X1:S),mark(X2:S))) MARK(zWadr(X1:S,X2:S)) -> MARK(X1:S) MARK(zWadr(X1:S,X2:S)) -> MARK(X2:S) MARK(zWadr(X1:S,X2:S)) -> ZWADR(mark(X1:S),mark(X2:S)) PREFIX(active(X:S)) -> PREFIX(X:S) PREFIX(mark(X:S)) -> PREFIX(X:S) S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) ZWADR(active(X1:S),X2:S) -> ZWADR(X1:S,X2:S) ZWADR(mark(X1:S),X2:S) -> ZWADR(X1:S,X2:S) ZWADR(X1:S,active(X2:S)) -> ZWADR(X1:S,X2:S) ZWADR(X1:S,mark(X2:S)) -> ZWADR(X1:S,X2:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ZWADR(active(X1:S),X2:S) -> ZWADR(X1:S,X2:S) ZWADR(mark(X1:S),X2:S) -> ZWADR(X1:S,X2:S) ZWADR(X1:S,active(X2:S)) -> ZWADR(X1:S,X2:S) ZWADR(X1:S,mark(X2:S)) -> ZWADR(X1:S,X2:S) ->->-> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->->Cycle: ->->-> Pairs: S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) ->->-> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->->Cycle: ->->-> Pairs: PREFIX(active(X:S)) -> PREFIX(X:S) PREFIX(mark(X:S)) -> PREFIX(X:S) ->->-> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->->Cycle: ->->-> Pairs: FROM(active(X:S)) -> FROM(X:S) FROM(mark(X:S)) -> FROM(X:S) ->->-> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->->Cycle: ->->-> Pairs: CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) ->->-> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->->Cycle: ->->-> Pairs: APP(active(X1:S),X2:S) -> APP(X1:S,X2:S) APP(mark(X1:S),X2:S) -> APP(X1:S,X2:S) APP(X1:S,active(X2:S)) -> APP(X1:S,X2:S) APP(X1:S,mark(X2:S)) -> APP(X1:S,X2:S) ->->-> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->->Cycle: ->->-> Pairs: ACTIVE(app(cons(X:S,XS:S),YS:S)) -> MARK(cons(X:S,app(XS:S,YS:S))) ACTIVE(app(nil,YS:S)) -> MARK(YS:S) ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) ACTIVE(prefix(L:S)) -> MARK(cons(nil,zWadr(L:S,prefix(L:S)))) ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) MARK(app(X1:S,X2:S)) -> ACTIVE(app(mark(X1:S),mark(X2:S))) MARK(app(X1:S,X2:S)) -> MARK(X1:S) MARK(app(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> MARK(X:S) MARK(prefix(X:S)) -> ACTIVE(prefix(mark(X:S))) MARK(prefix(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zWadr(X1:S,X2:S)) -> ACTIVE(zWadr(mark(X1:S),mark(X2:S))) MARK(zWadr(X1:S,X2:S)) -> MARK(X1:S) MARK(zWadr(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) The problem is decomposed in 7 subproblems. Problem 1.1: Subterm Processor: -> Pairs: ZWADR(active(X1:S),X2:S) -> ZWADR(X1:S,X2:S) ZWADR(mark(X1:S),X2:S) -> ZWADR(X1:S,X2:S) ZWADR(X1:S,active(X2:S)) -> ZWADR(X1:S,X2:S) ZWADR(X1:S,mark(X2:S)) -> ZWADR(X1:S,X2:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Projection: pi(ZWADR) = 1 Problem 1.1: SCC Processor: -> Pairs: ZWADR(X1:S,active(X2:S)) -> ZWADR(X1:S,X2:S) ZWADR(X1:S,mark(X2:S)) -> ZWADR(X1:S,X2:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ZWADR(X1:S,active(X2:S)) -> ZWADR(X1:S,X2:S) ZWADR(X1:S,mark(X2:S)) -> ZWADR(X1:S,X2:S) ->->-> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) Problem 1.1: Subterm Processor: -> Pairs: ZWADR(X1:S,active(X2:S)) -> ZWADR(X1:S,X2:S) ZWADR(X1:S,mark(X2:S)) -> ZWADR(X1:S,X2:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Projection: pi(ZWADR) = 2 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Projection: pi(S) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: PREFIX(active(X:S)) -> PREFIX(X:S) PREFIX(mark(X:S)) -> PREFIX(X:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Projection: pi(PREFIX) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: FROM(active(X:S)) -> FROM(X:S) FROM(mark(X:S)) -> FROM(X:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Projection: pi(FROM) = 1 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Subterm Processor: -> Pairs: CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Projection: pi(CONS) = 1 Problem 1.5: SCC Processor: -> Pairs: CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) ->->-> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) Problem 1.5: Subterm Processor: -> Pairs: CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Projection: pi(CONS) = 2 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.6: Subterm Processor: -> Pairs: APP(active(X1:S),X2:S) -> APP(X1:S,X2:S) APP(mark(X1:S),X2:S) -> APP(X1:S,X2:S) APP(X1:S,active(X2:S)) -> APP(X1:S,X2:S) APP(X1:S,mark(X2:S)) -> APP(X1:S,X2:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Projection: pi(APP) = 1 Problem 1.6: SCC Processor: -> Pairs: APP(X1:S,active(X2:S)) -> APP(X1:S,X2:S) APP(X1:S,mark(X2:S)) -> APP(X1:S,X2:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: APP(X1:S,active(X2:S)) -> APP(X1:S,X2:S) APP(X1:S,mark(X2:S)) -> APP(X1:S,X2:S) ->->-> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) Problem 1.6: Subterm Processor: -> Pairs: APP(X1:S,active(X2:S)) -> APP(X1:S,X2:S) APP(X1:S,mark(X2:S)) -> APP(X1:S,X2:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Projection: pi(APP) = 2 Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(app(cons(X:S,XS:S),YS:S)) -> MARK(cons(X:S,app(XS:S,YS:S))) ACTIVE(app(nil,YS:S)) -> MARK(YS:S) ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) ACTIVE(prefix(L:S)) -> MARK(cons(nil,zWadr(L:S,prefix(L:S)))) ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) MARK(app(X1:S,X2:S)) -> ACTIVE(app(mark(X1:S),mark(X2:S))) MARK(app(X1:S,X2:S)) -> MARK(X1:S) MARK(app(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> MARK(X:S) MARK(prefix(X:S)) -> ACTIVE(prefix(mark(X:S))) MARK(prefix(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zWadr(X1:S,X2:S)) -> ACTIVE(zWadr(mark(X1:S),mark(X2:S))) MARK(zWadr(X1:S,X2:S)) -> MARK(X1:S) MARK(zWadr(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) -> Usable rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [app](X1,X2) = 2.X1 + 2.X2 + 2 [cons](X1,X2) = X1 + 2 [from](X) = 2.X + 2 [mark](X) = X [prefix](X) = 2.X + 2 [s](X) = 2.X [zWadr](X1,X2) = 2.X1 + 2.X2 + 2 [fSNonEmpty] = 0 [nil] = 0 [ACTIVE](X) = 2.X + 2 [APP](X1,X2) = 0 [CONS](X1,X2) = 0 [FROM](X) = 0 [MARK](X) = 2.X + 2 [PREFIX](X) = 0 [S](X) = 0 [ZWADR](X1,X2) = 0 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(app(nil,YS:S)) -> MARK(YS:S) ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) ACTIVE(prefix(L:S)) -> MARK(cons(nil,zWadr(L:S,prefix(L:S)))) ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) MARK(app(X1:S,X2:S)) -> ACTIVE(app(mark(X1:S),mark(X2:S))) MARK(app(X1:S,X2:S)) -> MARK(X1:S) MARK(app(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> MARK(X:S) MARK(prefix(X:S)) -> ACTIVE(prefix(mark(X:S))) MARK(prefix(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zWadr(X1:S,X2:S)) -> ACTIVE(zWadr(mark(X1:S),mark(X2:S))) MARK(zWadr(X1:S,X2:S)) -> MARK(X1:S) MARK(zWadr(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(app(nil,YS:S)) -> MARK(YS:S) ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) ACTIVE(prefix(L:S)) -> MARK(cons(nil,zWadr(L:S,prefix(L:S)))) ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) MARK(app(X1:S,X2:S)) -> ACTIVE(app(mark(X1:S),mark(X2:S))) MARK(app(X1:S,X2:S)) -> MARK(X1:S) MARK(app(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> MARK(X:S) MARK(prefix(X:S)) -> ACTIVE(prefix(mark(X:S))) MARK(prefix(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zWadr(X1:S,X2:S)) -> ACTIVE(zWadr(mark(X1:S),mark(X2:S))) MARK(zWadr(X1:S,X2:S)) -> MARK(X1:S) MARK(zWadr(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(app(nil,YS:S)) -> MARK(YS:S) ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) ACTIVE(prefix(L:S)) -> MARK(cons(nil,zWadr(L:S,prefix(L:S)))) ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) MARK(app(X1:S,X2:S)) -> ACTIVE(app(mark(X1:S),mark(X2:S))) MARK(app(X1:S,X2:S)) -> MARK(X1:S) MARK(app(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> MARK(X:S) MARK(prefix(X:S)) -> ACTIVE(prefix(mark(X:S))) MARK(prefix(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zWadr(X1:S,X2:S)) -> ACTIVE(zWadr(mark(X1:S),mark(X2:S))) MARK(zWadr(X1:S,X2:S)) -> MARK(X1:S) MARK(zWadr(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) -> Usable rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [app](X1,X2) = 2.X1 + 2.X2 + 2 [cons](X1,X2) = X1 + 2 [from](X) = 2.X + 2 [mark](X) = X [prefix](X) = 2.X + 2 [s](X) = 2.X + 1 [zWadr](X1,X2) = 2.X1 + 2.X2 + 1 [fSNonEmpty] = 0 [nil] = 0 [ACTIVE](X) = 2.X + 2 [APP](X1,X2) = 0 [CONS](X1,X2) = 0 [FROM](X) = 0 [MARK](X) = 2.X + 2 [PREFIX](X) = 0 [S](X) = 0 [ZWADR](X1,X2) = 0 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) ACTIVE(prefix(L:S)) -> MARK(cons(nil,zWadr(L:S,prefix(L:S)))) ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) MARK(app(X1:S,X2:S)) -> ACTIVE(app(mark(X1:S),mark(X2:S))) MARK(app(X1:S,X2:S)) -> MARK(X1:S) MARK(app(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> MARK(X:S) MARK(prefix(X:S)) -> ACTIVE(prefix(mark(X:S))) MARK(prefix(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zWadr(X1:S,X2:S)) -> ACTIVE(zWadr(mark(X1:S),mark(X2:S))) MARK(zWadr(X1:S,X2:S)) -> MARK(X1:S) MARK(zWadr(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) ACTIVE(prefix(L:S)) -> MARK(cons(nil,zWadr(L:S,prefix(L:S)))) ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) MARK(app(X1:S,X2:S)) -> ACTIVE(app(mark(X1:S),mark(X2:S))) MARK(app(X1:S,X2:S)) -> MARK(X1:S) MARK(app(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> MARK(X:S) MARK(prefix(X:S)) -> ACTIVE(prefix(mark(X:S))) MARK(prefix(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zWadr(X1:S,X2:S)) -> ACTIVE(zWadr(mark(X1:S),mark(X2:S))) MARK(zWadr(X1:S,X2:S)) -> MARK(X1:S) MARK(zWadr(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) ACTIVE(prefix(L:S)) -> MARK(cons(nil,zWadr(L:S,prefix(L:S)))) ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) MARK(app(X1:S,X2:S)) -> ACTIVE(app(mark(X1:S),mark(X2:S))) MARK(app(X1:S,X2:S)) -> MARK(X1:S) MARK(app(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> MARK(X:S) MARK(prefix(X:S)) -> ACTIVE(prefix(mark(X:S))) MARK(prefix(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zWadr(X1:S,X2:S)) -> ACTIVE(zWadr(mark(X1:S),mark(X2:S))) MARK(zWadr(X1:S,X2:S)) -> MARK(X1:S) MARK(zWadr(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) -> Usable rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [app](X1,X2) = 2.X1 + X2 [cons](X1,X2) = 2.X1 + 1 [from](X) = 2.X + 2 [mark](X) = X [prefix](X) = 2.X + 2 [s](X) = 2.X + 2 [zWadr](X1,X2) = 2.X1 + 2.X2 + 1 [fSNonEmpty] = 0 [nil] = 0 [ACTIVE](X) = 2.X + 2 [APP](X1,X2) = 0 [CONS](X1,X2) = 0 [FROM](X) = 0 [MARK](X) = 2.X + 2 [PREFIX](X) = 0 [S](X) = 0 [ZWADR](X1,X2) = 0 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(prefix(L:S)) -> MARK(cons(nil,zWadr(L:S,prefix(L:S)))) ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) MARK(app(X1:S,X2:S)) -> ACTIVE(app(mark(X1:S),mark(X2:S))) MARK(app(X1:S,X2:S)) -> MARK(X1:S) MARK(app(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> MARK(X:S) MARK(prefix(X:S)) -> ACTIVE(prefix(mark(X:S))) MARK(prefix(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zWadr(X1:S,X2:S)) -> ACTIVE(zWadr(mark(X1:S),mark(X2:S))) MARK(zWadr(X1:S,X2:S)) -> MARK(X1:S) MARK(zWadr(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(prefix(L:S)) -> MARK(cons(nil,zWadr(L:S,prefix(L:S)))) ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) MARK(app(X1:S,X2:S)) -> ACTIVE(app(mark(X1:S),mark(X2:S))) MARK(app(X1:S,X2:S)) -> MARK(X1:S) MARK(app(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> MARK(X:S) MARK(prefix(X:S)) -> ACTIVE(prefix(mark(X:S))) MARK(prefix(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zWadr(X1:S,X2:S)) -> ACTIVE(zWadr(mark(X1:S),mark(X2:S))) MARK(zWadr(X1:S,X2:S)) -> MARK(X1:S) MARK(zWadr(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(prefix(L:S)) -> MARK(cons(nil,zWadr(L:S,prefix(L:S)))) ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) MARK(app(X1:S,X2:S)) -> ACTIVE(app(mark(X1:S),mark(X2:S))) MARK(app(X1:S,X2:S)) -> MARK(X1:S) MARK(app(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> MARK(X:S) MARK(prefix(X:S)) -> ACTIVE(prefix(mark(X:S))) MARK(prefix(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zWadr(X1:S,X2:S)) -> ACTIVE(zWadr(mark(X1:S),mark(X2:S))) MARK(zWadr(X1:S,X2:S)) -> MARK(X1:S) MARK(zWadr(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) -> Usable rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [app](X1,X2) = 2.X1 + 2.X2 [cons](X1,X2) = X1 + 1 [from](X) = X + 1 [mark](X) = X [prefix](X) = 2.X + 2 [s](X) = 2.X + 1 [zWadr](X1,X2) = 2.X1 + 2.X2 + 2 [fSNonEmpty] = 0 [nil] = 0 [ACTIVE](X) = 2.X + 2 [APP](X1,X2) = 0 [CONS](X1,X2) = 0 [FROM](X) = 0 [MARK](X) = 2.X + 2 [PREFIX](X) = 0 [S](X) = 0 [ZWADR](X1,X2) = 0 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) MARK(app(X1:S,X2:S)) -> ACTIVE(app(mark(X1:S),mark(X2:S))) MARK(app(X1:S,X2:S)) -> MARK(X1:S) MARK(app(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> MARK(X:S) MARK(prefix(X:S)) -> ACTIVE(prefix(mark(X:S))) MARK(prefix(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zWadr(X1:S,X2:S)) -> ACTIVE(zWadr(mark(X1:S),mark(X2:S))) MARK(zWadr(X1:S,X2:S)) -> MARK(X1:S) MARK(zWadr(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) MARK(app(X1:S,X2:S)) -> ACTIVE(app(mark(X1:S),mark(X2:S))) MARK(app(X1:S,X2:S)) -> MARK(X1:S) MARK(app(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> MARK(X:S) MARK(prefix(X:S)) -> ACTIVE(prefix(mark(X:S))) MARK(prefix(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zWadr(X1:S,X2:S)) -> ACTIVE(zWadr(mark(X1:S),mark(X2:S))) MARK(zWadr(X1:S,X2:S)) -> MARK(X1:S) MARK(zWadr(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) MARK(app(X1:S,X2:S)) -> ACTIVE(app(mark(X1:S),mark(X2:S))) MARK(app(X1:S,X2:S)) -> MARK(X1:S) MARK(app(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> MARK(X:S) MARK(prefix(X:S)) -> ACTIVE(prefix(mark(X:S))) MARK(prefix(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zWadr(X1:S,X2:S)) -> ACTIVE(zWadr(mark(X1:S),mark(X2:S))) MARK(zWadr(X1:S,X2:S)) -> MARK(X1:S) MARK(zWadr(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) -> Usable rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [app](X1,X2) = 2.X1 + X2 [cons](X1,X2) = 2.X1 [from](X) = 2.X + 2 [mark](X) = X [prefix](X) = 2.X + 1 [s](X) = 2.X + 2 [zWadr](X1,X2) = 2.X1 + 2.X2 + 2 [fSNonEmpty] = 0 [nil] = 0 [ACTIVE](X) = 2.X [APP](X1,X2) = 0 [CONS](X1,X2) = 0 [FROM](X) = 0 [MARK](X) = 2.X [PREFIX](X) = 0 [S](X) = 0 [ZWADR](X1,X2) = 0 Problem 1.7: SCC Processor: -> Pairs: MARK(app(X1:S,X2:S)) -> ACTIVE(app(mark(X1:S),mark(X2:S))) MARK(app(X1:S,X2:S)) -> MARK(X1:S) MARK(app(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> MARK(X:S) MARK(prefix(X:S)) -> ACTIVE(prefix(mark(X:S))) MARK(prefix(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zWadr(X1:S,X2:S)) -> ACTIVE(zWadr(mark(X1:S),mark(X2:S))) MARK(zWadr(X1:S,X2:S)) -> MARK(X1:S) MARK(zWadr(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(app(X1:S,X2:S)) -> MARK(X1:S) MARK(app(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(from(X:S)) -> MARK(X:S) MARK(prefix(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(zWadr(X1:S,X2:S)) -> MARK(X1:S) MARK(zWadr(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) Problem 1.7: Subterm Processor: -> Pairs: MARK(app(X1:S,X2:S)) -> MARK(X1:S) MARK(app(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(from(X:S)) -> MARK(X:S) MARK(prefix(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(zWadr(X1:S,X2:S)) -> MARK(X1:S) MARK(zWadr(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Projection: pi(MARK) = 1 Problem 1.7: SCC Processor: -> Pairs: Empty -> Rules: active(app(cons(X:S,XS:S),YS:S)) -> mark(cons(X:S,app(XS:S,YS:S))) active(app(nil,YS:S)) -> mark(YS:S) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) active(prefix(L:S)) -> mark(cons(nil,zWadr(L:S,prefix(L:S)))) active(zWadr(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(app(Y:S,cons(X:S,nil)),zWadr(XS:S,YS:S))) active(zWadr(nil,YS:S)) -> mark(nil) active(zWadr(XS:S,nil)) -> mark(nil) app(active(X1:S),X2:S) -> app(X1:S,X2:S) app(mark(X1:S),X2:S) -> app(X1:S,X2:S) app(X1:S,active(X2:S)) -> app(X1:S,X2:S) app(X1:S,mark(X2:S)) -> app(X1:S,X2:S) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(app(X1:S,X2:S)) -> active(app(mark(X1:S),mark(X2:S))) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(from(X:S)) -> active(from(mark(X:S))) mark(prefix(X:S)) -> active(prefix(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(zWadr(X1:S,X2:S)) -> active(zWadr(mark(X1:S),mark(X2:S))) mark(nil) -> active(nil) prefix(active(X:S)) -> prefix(X:S) prefix(mark(X:S)) -> prefix(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) zWadr(active(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(mark(X1:S),X2:S) -> zWadr(X1:S,X2:S) zWadr(X1:S,active(X2:S)) -> zWadr(X1:S,X2:S) zWadr(X1:S,mark(X2:S)) -> zWadr(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite.