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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(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(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(app(X1:S,X2:S)) -> ACTIVE(X2:S) ACTIVE(app(X1:S,X2:S)) -> APP(active(X1:S),X2:S) ACTIVE(app(X1:S,X2:S)) -> APP(X1:S,active(X2:S)) ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S) ACTIVE(from(X:S)) -> ACTIVE(X:S) ACTIVE(from(X:S)) -> CONS(X:S,from(s(X:S))) ACTIVE(from(X:S)) -> FROM(active(X:S)) ACTIVE(from(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)) -> ZWADR(L:S,prefix(L:S)) ACTIVE(prefix(X:S)) -> ACTIVE(X:S) ACTIVE(prefix(X:S)) -> PREFIX(active(X:S)) ACTIVE(s(X:S)) -> ACTIVE(X:S) ACTIVE(s(X:S)) -> S(active(X: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))) -> ZWADR(XS:S,YS:S) ACTIVE(zWadr(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(zWadr(X1:S,X2:S)) -> ACTIVE(X2:S) ACTIVE(zWadr(X1:S,X2:S)) -> ZWADR(active(X1:S),X2:S) ACTIVE(zWadr(X1:S,X2:S)) -> ZWADR(X1:S,active(X2:S)) APP(mark(X1:S),X2:S) -> APP(X1:S,X2:S) APP(ok(X1:S),ok(X2:S)) -> APP(X1:S,X2:S) APP(X1:S,mark(X2:S)) -> APP(X1:S,X2:S) CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) FROM(mark(X:S)) -> FROM(X:S) FROM(ok(X:S)) -> FROM(X:S) PREFIX(mark(X:S)) -> PREFIX(X:S) PREFIX(ok(X:S)) -> PREFIX(X:S) PROPER(app(X1:S,X2:S)) -> APP(proper(X1:S),proper(X2:S)) PROPER(app(X1:S,X2:S)) -> PROPER(X1:S) PROPER(app(X1:S,X2:S)) -> PROPER(X2:S) PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S)) PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) PROPER(from(X:S)) -> FROM(proper(X:S)) PROPER(from(X:S)) -> PROPER(X:S) PROPER(prefix(X:S)) -> PREFIX(proper(X:S)) PROPER(prefix(X:S)) -> PROPER(X:S) PROPER(s(X:S)) -> PROPER(X:S) PROPER(s(X:S)) -> S(proper(X:S)) PROPER(zWadr(X1:S,X2:S)) -> PROPER(X1:S) PROPER(zWadr(X1:S,X2:S)) -> PROPER(X2:S) PROPER(zWadr(X1:S,X2:S)) -> ZWADR(proper(X1:S),proper(X2:S)) S(mark(X:S)) -> S(X:S) S(ok(X:S)) -> S(X:S) TOP(mark(X:S)) -> PROPER(X:S) TOP(mark(X:S)) -> TOP(proper(X:S)) TOP(ok(X:S)) -> ACTIVE(X:S) TOP(ok(X:S)) -> TOP(active(X:S)) ZWADR(mark(X1:S),X2:S) -> ZWADR(X1:S,X2:S) ZWADR(ok(X1:S),ok(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(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(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(app(X1:S,X2:S)) -> ACTIVE(X2:S) ACTIVE(app(X1:S,X2:S)) -> APP(active(X1:S),X2:S) ACTIVE(app(X1:S,X2:S)) -> APP(X1:S,active(X2:S)) ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S) ACTIVE(from(X:S)) -> ACTIVE(X:S) ACTIVE(from(X:S)) -> CONS(X:S,from(s(X:S))) ACTIVE(from(X:S)) -> FROM(active(X:S)) ACTIVE(from(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)) -> ZWADR(L:S,prefix(L:S)) ACTIVE(prefix(X:S)) -> ACTIVE(X:S) ACTIVE(prefix(X:S)) -> PREFIX(active(X:S)) ACTIVE(s(X:S)) -> ACTIVE(X:S) ACTIVE(s(X:S)) -> S(active(X: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))) -> ZWADR(XS:S,YS:S) ACTIVE(zWadr(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(zWadr(X1:S,X2:S)) -> ACTIVE(X2:S) ACTIVE(zWadr(X1:S,X2:S)) -> ZWADR(active(X1:S),X2:S) ACTIVE(zWadr(X1:S,X2:S)) -> ZWADR(X1:S,active(X2:S)) APP(mark(X1:S),X2:S) -> APP(X1:S,X2:S) APP(ok(X1:S),ok(X2:S)) -> APP(X1:S,X2:S) APP(X1:S,mark(X2:S)) -> APP(X1:S,X2:S) CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) FROM(mark(X:S)) -> FROM(X:S) FROM(ok(X:S)) -> FROM(X:S) PREFIX(mark(X:S)) -> PREFIX(X:S) PREFIX(ok(X:S)) -> PREFIX(X:S) PROPER(app(X1:S,X2:S)) -> APP(proper(X1:S),proper(X2:S)) PROPER(app(X1:S,X2:S)) -> PROPER(X1:S) PROPER(app(X1:S,X2:S)) -> PROPER(X2:S) PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S)) PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) PROPER(from(X:S)) -> FROM(proper(X:S)) PROPER(from(X:S)) -> PROPER(X:S) PROPER(prefix(X:S)) -> PREFIX(proper(X:S)) PROPER(prefix(X:S)) -> PROPER(X:S) PROPER(s(X:S)) -> PROPER(X:S) PROPER(s(X:S)) -> S(proper(X:S)) PROPER(zWadr(X1:S,X2:S)) -> PROPER(X1:S) PROPER(zWadr(X1:S,X2:S)) -> PROPER(X2:S) PROPER(zWadr(X1:S,X2:S)) -> ZWADR(proper(X1:S),proper(X2:S)) S(mark(X:S)) -> S(X:S) S(ok(X:S)) -> S(X:S) TOP(mark(X:S)) -> PROPER(X:S) TOP(mark(X:S)) -> TOP(proper(X:S)) TOP(ok(X:S)) -> ACTIVE(X:S) TOP(ok(X:S)) -> TOP(active(X:S)) ZWADR(mark(X1:S),X2:S) -> ZWADR(X1:S,X2:S) ZWADR(ok(X1:S),ok(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ZWADR(mark(X1:S),X2:S) -> ZWADR(X1:S,X2:S) ZWADR(ok(X1:S),ok(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) ->->Cycle: ->->-> Pairs: S(mark(X:S)) -> S(X:S) S(ok(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) ->->Cycle: ->->-> Pairs: PREFIX(mark(X:S)) -> PREFIX(X:S) PREFIX(ok(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) ->->Cycle: ->->-> Pairs: FROM(mark(X:S)) -> FROM(X:S) FROM(ok(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) ->->Cycle: ->->-> Pairs: CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(ok(X1:S),ok(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) ->->Cycle: ->->-> Pairs: APP(mark(X1:S),X2:S) -> APP(X1:S,X2:S) APP(ok(X1:S),ok(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) ->->Cycle: ->->-> Pairs: PROPER(app(X1:S,X2:S)) -> PROPER(X1:S) PROPER(app(X1:S,X2:S)) -> PROPER(X2:S) PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) PROPER(from(X:S)) -> PROPER(X:S) PROPER(prefix(X:S)) -> PROPER(X:S) PROPER(s(X:S)) -> PROPER(X:S) PROPER(zWadr(X1:S,X2:S)) -> PROPER(X1:S) PROPER(zWadr(X1:S,X2:S)) -> PROPER(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) ->->Cycle: ->->-> Pairs: ACTIVE(app(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(app(X1:S,X2:S)) -> ACTIVE(X2:S) ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(from(X:S)) -> ACTIVE(X:S) ACTIVE(prefix(X:S)) -> ACTIVE(X:S) ACTIVE(s(X:S)) -> ACTIVE(X:S) ACTIVE(zWadr(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(zWadr(X1:S,X2:S)) -> ACTIVE(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) ->->Cycle: ->->-> Pairs: TOP(mark(X:S)) -> TOP(proper(X:S)) TOP(ok(X:S)) -> TOP(active(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) The problem is decomposed in 9 subproblems. Problem 1.1: Subterm Processor: -> Pairs: ZWADR(mark(X1:S),X2:S) -> ZWADR(X1:S,X2:S) ZWADR(ok(X1:S),ok(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) ->Projection: pi(ZWADR) = 1 Problem 1.1: SCC Processor: -> Pairs: 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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) Problem 1.1: Subterm Processor: -> Pairs: 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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(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(mark(X:S)) -> S(X:S) S(ok(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(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(mark(X:S)) -> PREFIX(X:S) PREFIX(ok(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(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(mark(X:S)) -> FROM(X:S) FROM(ok(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(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(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(ok(X1:S),ok(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) ->Projection: pi(CONS) = 1 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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(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(mark(X1:S),X2:S) -> APP(X1:S,X2:S) APP(ok(X1:S),ok(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) ->Projection: pi(APP) = 1 Problem 1.6: SCC Processor: -> Pairs: 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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) Problem 1.6: Subterm Processor: -> Pairs: 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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.7: Subterm Processor: -> Pairs: PROPER(app(X1:S,X2:S)) -> PROPER(X1:S) PROPER(app(X1:S,X2:S)) -> PROPER(X2:S) PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) PROPER(from(X:S)) -> PROPER(X:S) PROPER(prefix(X:S)) -> PROPER(X:S) PROPER(s(X:S)) -> PROPER(X:S) PROPER(zWadr(X1:S,X2:S)) -> PROPER(X1:S) PROPER(zWadr(X1:S,X2:S)) -> PROPER(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) ->Projection: pi(PROPER) = 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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.8: Subterm Processor: -> Pairs: ACTIVE(app(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(app(X1:S,X2:S)) -> ACTIVE(X2:S) ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(from(X:S)) -> ACTIVE(X:S) ACTIVE(prefix(X:S)) -> ACTIVE(X:S) ACTIVE(s(X:S)) -> ACTIVE(X:S) ACTIVE(zWadr(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(zWadr(X1:S,X2:S)) -> ACTIVE(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) ->Projection: pi(ACTIVE) = 1 Problem 1.8: 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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.9: Reduction Pairs Processor: -> Pairs: TOP(mark(X:S)) -> TOP(proper(X:S)) TOP(ok(X:S)) -> TOP(active(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(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 + 1 [from](X) = 2.X + 2 [prefix](X) = 2.X + 2 [proper](X) = X [s](X) = X [top](X) = 0 [zWadr](X1,X2) = 2.X1 + 2.X2 + 2 [fSNonEmpty] = 0 [mark](X) = X + 1 [nil] = 0 [ok](X) = X [ACTIVE](X) = 0 [APP](X1,X2) = 0 [CONS](X1,X2) = 0 [FROM](X) = 0 [PREFIX](X) = 0 [PROPER](X) = 0 [S](X) = 0 [TOP](X) = 2.X [ZWADR](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: TOP(ok(X:S)) -> TOP(active(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TOP(ok(X:S)) -> TOP(active(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) Problem 1.9: Reduction Pairs Processor: -> Pairs: TOP(ok(X:S)) -> TOP(active(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2.X + 1 [app](X1,X2) = 2.X1 + X2 + 2 [cons](X1,X2) = X1 [from](X) = 2.X + 1 [prefix](X) = X [proper](X) = 0 [s](X) = 2.X + 2 [top](X) = 0 [zWadr](X1,X2) = X1 + 2.X2 + 1 [fSNonEmpty] = 0 [mark](X) = X + 1 [nil] = 0 [ok](X) = 2.X + 2 [ACTIVE](X) = 0 [APP](X1,X2) = 0 [CONS](X1,X2) = 0 [FROM](X) = 0 [PREFIX](X) = 0 [PROPER](X) = 0 [S](X) = 0 [TOP](X) = X [ZWADR](X1,X2) = 0 Problem 1.9: 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(app(X1:S,X2:S)) -> app(active(X1:S),X2:S) active(app(X1:S,X2:S)) -> app(X1:S,active(X2:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(from(X:S)) -> from(active(X: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(prefix(X:S)) -> prefix(active(X:S)) active(s(X:S)) -> s(active(X: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(X1:S,X2:S)) -> zWadr(active(X1:S),X2:S) active(zWadr(X1:S,X2:S)) -> zWadr(X1:S,active(X2:S)) active(zWadr(XS:S,nil)) -> mark(nil) app(mark(X1:S),X2:S) -> mark(app(X1:S,X2:S)) app(ok(X1:S),ok(X2:S)) -> ok(app(X1:S,X2:S)) app(X1:S,mark(X2:S)) -> mark(app(X1:S,X2:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) from(mark(X:S)) -> mark(from(X:S)) from(ok(X:S)) -> ok(from(X:S)) prefix(mark(X:S)) -> mark(prefix(X:S)) prefix(ok(X:S)) -> ok(prefix(X:S)) proper(app(X1:S,X2:S)) -> app(proper(X1:S),proper(X2:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(from(X:S)) -> from(proper(X:S)) proper(prefix(X:S)) -> prefix(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(zWadr(X1:S,X2:S)) -> zWadr(proper(X1:S),proper(X2:S)) proper(nil) -> ok(nil) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) zWadr(mark(X1:S),X2:S) -> mark(zWadr(X1:S,X2:S)) zWadr(ok(X1:S),ok(X2:S)) -> ok(zWadr(X1:S,X2:S)) zWadr(X1:S,mark(X2:S)) -> mark(zWadr(X1:S,X2:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite.