YES Problem 1: (VAR v_NonEmpty:S N:S X:S X1:S X2:S XS:S Y:S YS:S) (RULES active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> CONS(s(X:S),incr(XS:S)) ACTIVE(incr(cons(X:S,XS:S))) -> INCR(XS:S) ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(incr(cons(X:S,XS:S))) -> S(X:S) ACTIVE(repItems(cons(X:S,XS:S))) -> CONS(X:S,cons(X:S,repItems(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> CONS(X:S,repItems(XS:S)) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(repItems(cons(X:S,XS:S))) -> REPITEMS(XS:S) ACTIVE(repItems(nil)) -> MARK(nil) ACTIVE(tail(cons(X:S,XS:S))) -> MARK(XS:S) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> CONS(X:S,take(N:S,XS:S)) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> TAKE(N:S,XS:S) ACTIVE(take(0,XS:S)) -> MARK(nil) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> CONS(pair(X:S,Y:S),zip(XS:S,YS:S)) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> PAIR(X:S,Y:S) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> ZIP(XS:S,YS:S) ACTIVE(zip(nil,XS:S)) -> MARK(nil) ACTIVE(zip(X:S,nil)) -> MARK(nil) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) 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) INCR(active(X:S)) -> INCR(X:S) INCR(mark(X:S)) -> INCR(X: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(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> INCR(mark(X:S)) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> ACTIVE(pair(mark(X1:S),mark(X2:S))) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(pair(X1:S,X2:S)) -> PAIR(mark(X1:S),mark(X2:S)) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(repItems(X:S)) -> MARK(X:S) MARK(repItems(X:S)) -> REPITEMS(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(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> TAIL(mark(X:S)) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(take(X1:S,X2:S)) -> TAKE(mark(X1:S),mark(X2:S)) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ZIP(mark(X1:S),mark(X2:S)) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) PAIR(active(X1:S),X2:S) -> PAIR(X1:S,X2:S) PAIR(mark(X1:S),X2:S) -> PAIR(X1:S,X2:S) PAIR(X1:S,active(X2:S)) -> PAIR(X1:S,X2:S) PAIR(X1:S,mark(X2:S)) -> PAIR(X1:S,X2:S) REPITEMS(active(X:S)) -> REPITEMS(X:S) REPITEMS(mark(X:S)) -> REPITEMS(X:S) S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) TAIL(active(X:S)) -> TAIL(X:S) TAIL(mark(X:S)) -> TAIL(X:S) TAKE(active(X1:S),X2:S) -> TAKE(X1:S,X2:S) TAKE(mark(X1:S),X2:S) -> TAKE(X1:S,X2:S) TAKE(X1:S,active(X2:S)) -> TAKE(X1:S,X2:S) TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S) ZIP(active(X1:S),X2:S) -> ZIP(X1:S,X2:S) ZIP(mark(X1:S),X2:S) -> ZIP(X1:S,X2:S) ZIP(X1:S,active(X2:S)) -> ZIP(X1:S,X2:S) ZIP(X1:S,mark(X2:S)) -> ZIP(X1:S,X2:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> CONS(s(X:S),incr(XS:S)) ACTIVE(incr(cons(X:S,XS:S))) -> INCR(XS:S) ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(incr(cons(X:S,XS:S))) -> S(X:S) ACTIVE(repItems(cons(X:S,XS:S))) -> CONS(X:S,cons(X:S,repItems(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> CONS(X:S,repItems(XS:S)) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(repItems(cons(X:S,XS:S))) -> REPITEMS(XS:S) ACTIVE(repItems(nil)) -> MARK(nil) ACTIVE(tail(cons(X:S,XS:S))) -> MARK(XS:S) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> CONS(X:S,take(N:S,XS:S)) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> TAKE(N:S,XS:S) ACTIVE(take(0,XS:S)) -> MARK(nil) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> CONS(pair(X:S,Y:S),zip(XS:S,YS:S)) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> PAIR(X:S,Y:S) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> ZIP(XS:S,YS:S) ACTIVE(zip(nil,XS:S)) -> MARK(nil) ACTIVE(zip(X:S,nil)) -> MARK(nil) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) 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) INCR(active(X:S)) -> INCR(X:S) INCR(mark(X:S)) -> INCR(X: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(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> INCR(mark(X:S)) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> ACTIVE(pair(mark(X1:S),mark(X2:S))) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(pair(X1:S,X2:S)) -> PAIR(mark(X1:S),mark(X2:S)) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(repItems(X:S)) -> MARK(X:S) MARK(repItems(X:S)) -> REPITEMS(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(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> TAIL(mark(X:S)) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(take(X1:S,X2:S)) -> TAKE(mark(X1:S),mark(X2:S)) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ZIP(mark(X1:S),mark(X2:S)) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) PAIR(active(X1:S),X2:S) -> PAIR(X1:S,X2:S) PAIR(mark(X1:S),X2:S) -> PAIR(X1:S,X2:S) PAIR(X1:S,active(X2:S)) -> PAIR(X1:S,X2:S) PAIR(X1:S,mark(X2:S)) -> PAIR(X1:S,X2:S) REPITEMS(active(X:S)) -> REPITEMS(X:S) REPITEMS(mark(X:S)) -> REPITEMS(X:S) S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) TAIL(active(X:S)) -> TAIL(X:S) TAIL(mark(X:S)) -> TAIL(X:S) TAKE(active(X1:S),X2:S) -> TAKE(X1:S,X2:S) TAKE(mark(X1:S),X2:S) -> TAKE(X1:S,X2:S) TAKE(X1:S,active(X2:S)) -> TAKE(X1:S,X2:S) TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S) ZIP(active(X1:S),X2:S) -> ZIP(X1:S,X2:S) ZIP(mark(X1:S),X2:S) -> ZIP(X1:S,X2:S) ZIP(X1:S,active(X2:S)) -> ZIP(X1:S,X2:S) ZIP(X1:S,mark(X2:S)) -> ZIP(X1:S,X2:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ZIP(active(X1:S),X2:S) -> ZIP(X1:S,X2:S) ZIP(mark(X1:S),X2:S) -> ZIP(X1:S,X2:S) ZIP(X1:S,active(X2:S)) -> ZIP(X1:S,X2:S) ZIP(X1:S,mark(X2:S)) -> ZIP(X1:S,X2:S) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->->Cycle: ->->-> Pairs: TAKE(active(X1:S),X2:S) -> TAKE(X1:S,X2:S) TAKE(mark(X1:S),X2:S) -> TAKE(X1:S,X2:S) TAKE(X1:S,active(X2:S)) -> TAKE(X1:S,X2:S) TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->->Cycle: ->->-> Pairs: TAIL(active(X:S)) -> TAIL(X:S) TAIL(mark(X:S)) -> TAIL(X:S) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->->Cycle: ->->-> Pairs: S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->->Cycle: ->->-> Pairs: REPITEMS(active(X:S)) -> REPITEMS(X:S) REPITEMS(mark(X:S)) -> REPITEMS(X:S) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->->Cycle: ->->-> Pairs: PAIR(active(X1:S),X2:S) -> PAIR(X1:S,X2:S) PAIR(mark(X1:S),X2:S) -> PAIR(X1:S,X2:S) PAIR(X1:S,active(X2:S)) -> PAIR(X1:S,X2:S) PAIR(X1:S,mark(X2:S)) -> PAIR(X1:S,X2:S) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->->Cycle: ->->-> Pairs: INCR(active(X:S)) -> INCR(X:S) INCR(mark(X:S)) -> INCR(X:S) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(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(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(tail(cons(X:S,XS:S))) -> MARK(XS:S) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> ACTIVE(pair(mark(X1:S),mark(X2:S))) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(repItems(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) The problem is decomposed in 9 subproblems. Problem 1.1: Subterm Processor: -> Pairs: ZIP(active(X1:S),X2:S) -> ZIP(X1:S,X2:S) ZIP(mark(X1:S),X2:S) -> ZIP(X1:S,X2:S) ZIP(X1:S,active(X2:S)) -> ZIP(X1:S,X2:S) ZIP(X1:S,mark(X2:S)) -> ZIP(X1:S,X2:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Projection: pi(ZIP) = 1 Problem 1.1: SCC Processor: -> Pairs: ZIP(X1:S,active(X2:S)) -> ZIP(X1:S,X2:S) ZIP(X1:S,mark(X2:S)) -> ZIP(X1:S,X2:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ZIP(X1:S,active(X2:S)) -> ZIP(X1:S,X2:S) ZIP(X1:S,mark(X2:S)) -> ZIP(X1:S,X2:S) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.1: Subterm Processor: -> Pairs: ZIP(X1:S,active(X2:S)) -> ZIP(X1:S,X2:S) ZIP(X1:S,mark(X2:S)) -> ZIP(X1:S,X2:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Projection: pi(ZIP) = 2 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: TAKE(active(X1:S),X2:S) -> TAKE(X1:S,X2:S) TAKE(mark(X1:S),X2:S) -> TAKE(X1:S,X2:S) TAKE(X1:S,active(X2:S)) -> TAKE(X1:S,X2:S) TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Projection: pi(TAKE) = 1 Problem 1.2: SCC Processor: -> Pairs: TAKE(X1:S,active(X2:S)) -> TAKE(X1:S,X2:S) TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TAKE(X1:S,active(X2:S)) -> TAKE(X1:S,X2:S) TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.2: Subterm Processor: -> Pairs: TAKE(X1:S,active(X2:S)) -> TAKE(X1:S,X2:S) TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Projection: pi(TAKE) = 2 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: TAIL(active(X:S)) -> TAIL(X:S) TAIL(mark(X:S)) -> TAIL(X:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Projection: pi(TAIL) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Projection: pi(S) = 1 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Subterm Processor: -> Pairs: REPITEMS(active(X:S)) -> REPITEMS(X:S) REPITEMS(mark(X:S)) -> REPITEMS(X:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Projection: pi(REPITEMS) = 1 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.6: Subterm Processor: -> Pairs: PAIR(active(X1:S),X2:S) -> PAIR(X1:S,X2:S) PAIR(mark(X1:S),X2:S) -> PAIR(X1:S,X2:S) PAIR(X1:S,active(X2:S)) -> PAIR(X1:S,X2:S) PAIR(X1:S,mark(X2:S)) -> PAIR(X1:S,X2:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Projection: pi(PAIR) = 1 Problem 1.6: SCC Processor: -> Pairs: PAIR(X1:S,active(X2:S)) -> PAIR(X1:S,X2:S) PAIR(X1:S,mark(X2:S)) -> PAIR(X1:S,X2:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: PAIR(X1:S,active(X2:S)) -> PAIR(X1:S,X2:S) PAIR(X1:S,mark(X2:S)) -> PAIR(X1:S,X2:S) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.6: Subterm Processor: -> Pairs: PAIR(X1:S,active(X2:S)) -> PAIR(X1:S,X2:S) PAIR(X1:S,mark(X2:S)) -> PAIR(X1:S,X2:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Projection: pi(PAIR) = 2 Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.7: Subterm Processor: -> Pairs: INCR(active(X:S)) -> INCR(X:S) INCR(mark(X:S)) -> INCR(X:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Projection: pi(INCR) = 1 Problem 1.7: SCC Processor: -> Pairs: Empty -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.8: 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(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Projection: pi(CONS) = 1 Problem 1.8: 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(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(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(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.8: 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(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Projection: pi(CONS) = 2 Problem 1.8: SCC Processor: -> Pairs: Empty -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(tail(cons(X:S,XS:S))) -> MARK(XS:S) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> ACTIVE(pair(mark(X1:S),mark(X2:S))) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(repItems(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + X2 [incr](X) = X [mark](X) = X [pair](X1,X2) = 2.X1 + 2.X2 [repItems](X) = 2.X + 2 [s](X) = X [tail](X) = 2.X + 2 [take](X1,X2) = 2.X1 + 2.X2 + 2 [zip](X1,X2) = 2.X1 + 2.X2 + 2 [0] = 0 [fSNonEmpty] = 0 [nil] = 2 [oddNs] = 2 [pairNs] = 2 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 2.X + 2 [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> ACTIVE(pair(mark(X1:S),mark(X2:S))) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(repItems(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> ACTIVE(pair(mark(X1:S),mark(X2:S))) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(repItems(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> ACTIVE(pair(mark(X1:S),mark(X2:S))) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(repItems(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2 [cons](X1,X2) = 0 [incr](X) = 1 [mark](X) = 2 [pair](X1,X2) = 0 [repItems](X) = 1 [s](X) = 1 [tail](X) = 1 [take](X1,X2) = 1 [zip](X1,X2) = 1 [0] = 0 [fSNonEmpty] = 0 [nil] = 0 [oddNs] = 1 [pairNs] = 1 [ACTIVE](X) = X + 1 [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 2 [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> ACTIVE(pair(mark(X1:S),mark(X2:S))) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(repItems(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> ACTIVE(pair(mark(X1:S),mark(X2:S))) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(repItems(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> ACTIVE(pair(mark(X1:S),mark(X2:S))) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(repItems(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2.X [cons](X1,X2) = 0 [incr](X) = 1 [mark](X) = 2 [pair](X1,X2) = 0 [repItems](X) = 1 [s](X) = 1 [tail](X) = 1 [take](X1,X2) = 1 [zip](X1,X2) = 1 [0] = 0 [fSNonEmpty] = 0 [nil] = 1 [oddNs] = 1 [pairNs] = 1 [ACTIVE](X) = 2.X [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 2 [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(repItems(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(repItems(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(repItems(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + X2 [incr](X) = X [mark](X) = X [pair](X1,X2) = X1 + X2 [repItems](X) = 2.X + 2 [s](X) = X [tail](X) = 2.X + 2 [take](X1,X2) = 2.X1 + 2.X2 + 2 [zip](X1,X2) = X1 + 2.X2 + 2 [0] = 0 [fSNonEmpty] = 0 [nil] = 2 [oddNs] = 2 [pairNs] = 2 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 2.X + 2 [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2.X [cons](X1,X2) = 0 [incr](X) = 1 [mark](X) = 2 [pair](X1,X2) = 1 [repItems](X) = 1 [s](X) = 0 [tail](X) = 1 [take](X1,X2) = 1 [zip](X1,X2) = 1 [0] = 0 [fSNonEmpty] = 0 [nil] = 0 [oddNs] = 1 [pairNs] = 1 [ACTIVE](X) = 2.X [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 2 [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2 [cons](X1,X2) = 0 [incr](X) = 1 [mark](X) = 2 [pair](X1,X2) = 2 [repItems](X) = 1 [s](X) = 0 [tail](X) = 0 [take](X1,X2) = 1 [zip](X1,X2) = 1 [0] = 1 [fSNonEmpty] = 0 [nil] = 2 [oddNs] = 1 [pairNs] = 1 [ACTIVE](X) = 2.X [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 2 [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + X2 [incr](X) = X [mark](X) = X [pair](X1,X2) = 2.X1 + 2.X2 [repItems](X) = 2.X + 2 [s](X) = X [tail](X) = 2.X + 2 [take](X1,X2) = 2.X1 + 2.X2 + 2 [zip](X1,X2) = 2.X1 + 2.X2 + 2 [0] = 0 [fSNonEmpty] = 0 [nil] = 2 [oddNs] = 2 [pairNs] = 2 [ACTIVE](X) = 2.X + 1 [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 2.X + 1 [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = X1 + X2 [incr](X) = X [mark](X) = X [pair](X1,X2) = X1 + 2.X2 [repItems](X) = 2.X + 2 [s](X) = X [tail](X) = X + 1 [take](X1,X2) = 2.X1 + 2.X2 + 2 [zip](X1,X2) = 2.X1 + 2.X2 + 2 [0] = 0 [fSNonEmpty] = 0 [nil] = 2 [oddNs] = 2 [pairNs] = 2 [ACTIVE](X) = 2.X + 1 [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 2.X + 1 [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + X2 [incr](X) = 2.X [mark](X) = X [pair](X1,X2) = 2.X1 + 2.X2 [repItems](X) = 2.X + 2 [s](X) = 2.X [tail](X) = 2.X + 2 [take](X1,X2) = X1 + X2 + 2 [zip](X1,X2) = 2.X1 + 2.X2 + 1 [0] = 0 [fSNonEmpty] = 0 [nil] = 2 [oddNs] = 0 [pairNs] = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 2.X + 2 [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X1:S) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = X1 + X2 [incr](X) = X [mark](X) = X [pair](X1,X2) = 2.X1 + X2 [repItems](X) = 2.X + 2 [s](X) = X [tail](X) = 2.X + 2 [take](X1,X2) = 2.X1 + 2.X2 + 2 [zip](X1,X2) = 2.X1 + X2 + 2 [0] = 0 [fSNonEmpty] = 0 [nil] = 2 [oddNs] = 2 [pairNs] = 2 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 2.X + 2 [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> MARK(X2:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + X2 [incr](X) = X [mark](X) = X [pair](X1,X2) = 2.X1 + X2 [repItems](X) = 2.X + 2 [s](X) = X [tail](X) = 2.X + 2 [take](X1,X2) = X1 + 2.X2 + 2 [zip](X1,X2) = 2.X1 + 2.X2 + 1 [0] = 0 [fSNonEmpty] = 0 [nil] = 2 [oddNs] = 2 [pairNs] = 2 [ACTIVE](X) = 2.X + 1 [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 2.X + 1 [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(repItems(cons(X:S,XS:S))) -> MARK(cons(X:S,cons(X:S,repItems(XS:S)))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + 1/2.X2 [incr](X) = X [mark](X) = X [pair](X1,X2) = X1 + X2 [repItems](X) = 2.X + 2 [s](X) = X [tail](X) = 2.X [take](X1,X2) = 1/2.X1 + X2 + 2 [zip](X1,X2) = 2.X1 + X2 + 1 [0] = 1/2 [fSNonEmpty] = 0 [nil] = 2 [oddNs] = 2 [pairNs] = 2 [ACTIVE](X) = 1/2.X [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 1/2.X [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(repItems(X:S)) -> ACTIVE(repItems(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = X2 [incr](X) = 2 [mark](X) = X [pair](X1,X2) = X1 + 2 [repItems](X) = 0 [s](X) = 2 [tail](X) = X + 2 [take](X1,X2) = 2 [zip](X1,X2) = 2 [0] = 1 [fSNonEmpty] = 0 [nil] = 0 [oddNs] = 2 [pairNs] = 2 [ACTIVE](X) = X [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 2 [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> MARK(cons(X:S,take(N:S,XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + 1/2.X2 + 1/2 [incr](X) = X [mark](X) = X [pair](X1,X2) = X1 + X2 + 1 [repItems](X) = 2.X + 2 [s](X) = X [tail](X) = 2.X + 1/2 [take](X1,X2) = X2 + 1 [zip](X1,X2) = X1 + 2.X2 + 2 [0] = 0 [fSNonEmpty] = 0 [nil] = 1 [oddNs] = 1 [pairNs] = 1 [ACTIVE](X) = 2.X + 1 [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 2.X + 1 [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2 [cons](X1,X2) = 1 [incr](X) = 2 [mark](X) = 2 [pair](X1,X2) = 0 [repItems](X) = 2 [s](X) = 0 [tail](X) = 2 [take](X1,X2) = 0 [zip](X1,X2) = 2 [0] = 2 [fSNonEmpty] = 0 [nil] = 2 [oddNs] = 2 [pairNs] = 2 [ACTIVE](X) = X [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 2 [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> MARK(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = X1 + 1/2.X2 [incr](X) = X [mark](X) = X [pair](X1,X2) = X1 + 2.X2 [repItems](X) = 2.X [s](X) = X [tail](X) = 2.X + 1 [take](X1,X2) = X1 + X2 + 2 [zip](X1,X2) = X1 + 2.X2 + 1 [0] = 0 [fSNonEmpty] = 0 [nil] = 2 [oddNs] = 2 [pairNs] = 1 [ACTIVE](X) = X [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = X [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(zip(X1:S,X2:S)) -> ACTIVE(zip(mark(X1:S),mark(X2:S))) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 1 [cons](X1,X2) = 0 [incr](X) = 1 [mark](X) = 1 [pair](X1,X2) = 0 [repItems](X) = 1 [s](X) = 2 [tail](X) = 1 [take](X1,X2) = 2 [zip](X1,X2) = 0 [0] = 2 [fSNonEmpty] = 0 [nil] = 2 [oddNs] = 1 [pairNs] = 1 [ACTIVE](X) = X + 1 [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 2 [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(oddNs) -> MARK(incr(pairNs)) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + 1/2.X2 [incr](X) = X [mark](X) = X [pair](X1,X2) = X1 + X2 [repItems](X) = 2.X + 2 [s](X) = X [tail](X) = 2.X + 1/2 [take](X1,X2) = 1/2.X1 + 2.X2 + 1 [zip](X1,X2) = X1 + X2 + 1/2 [0] = 0 [fSNonEmpty] = 0 [nil] = 1/2 [oddNs] = 2 [pairNs] = 1 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 2.X + 2 [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(oddNs) -> ACTIVE(oddNs) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(pairNs) -> ACTIVE(pairNs) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(pairNs) -> MARK(cons(0,incr(oddNs))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + 1/2.X2 [incr](X) = X [mark](X) = X [pair](X1,X2) = 2.X1 + 2.X2 [repItems](X) = 2.X + 1 [s](X) = X [tail](X) = 2.X [take](X1,X2) = 2.X1 + X2 + 2 [zip](X1,X2) = 2.X1 + 2.X2 + 1/2 [0] = 0 [fSNonEmpty] = 0 [nil] = 1/2 [oddNs] = 1/2 [pairNs] = 1/2 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 2.X + 2 [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(pairNs) -> ACTIVE(pairNs) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = X1 + 1/2.X2 + 1/2 [incr](X) = X [mark](X) = X [pair](X1,X2) = X1 + X2 + 1 [repItems](X) = 2.X + 2 [s](X) = X [tail](X) = 2.X + 2 [take](X1,X2) = 2.X1 + X2 + 1 [zip](X1,X2) = X1 + 2.X2 [0] = 0 [fSNonEmpty] = 0 [nil] = 1 [oddNs] = 1 [pairNs] = 1 [ACTIVE](X) = 1/2.X + 1 [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 1/2.X + 1 [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X1:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X2 [incr](X) = 2.X [mark](X) = X [pair](X1,X2) = 2.X1 + X2 + 2 [repItems](X) = 0 [s](X) = 2.X + 2 [tail](X) = 2.X + 1 [take](X1,X2) = 2.X2 [zip](X1,X2) = 2.X1 [0] = 1 [fSNonEmpty] = 0 [nil] = 0 [oddNs] = 0 [pairNs] = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 2.X + 2 [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(pair(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X2 [incr](X) = 2.X [mark](X) = X [pair](X1,X2) = X1 + 2.X2 + 2 [repItems](X) = 0 [s](X) = 2.X + 1 [tail](X) = 2.X + 2 [take](X1,X2) = 2.X1 + 2.X2 + 1 [zip](X1,X2) = 2.X1 + 2.X2 [0] = 2 [fSNonEmpty] = 0 [nil] = 0 [oddNs] = 0 [pairNs] = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 2.X + 2 [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = X2 [incr](X) = X [mark](X) = X [pair](X1,X2) = 2.X1 + 2 [repItems](X) = 2.X + 1 [s](X) = X + 2 [tail](X) = 2.X + 2 [take](X1,X2) = 2.X2 + 2 [zip](X1,X2) = 2.X1 + 2.X2 + 2 [0] = 1 [fSNonEmpty] = 0 [nil] = 2 [oddNs] = 2 [pairNs] = 2 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 2.X + 2 [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) -> Usable rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 4 ->Interpretation: [active](X) = X [cons](X1,X2) = 1/2.X2 + 1/3 [incr](X) = X + 1/2 [mark](X) = X [pair](X1,X2) = 3/2.X1 + 3/2.X2 [repItems](X) = 4/3 [s](X) = 1/4.X + 1/3 [tail](X) = 4.X + 2 [take](X1,X2) = 2.X2 + 3 [zip](X1,X2) = 4.X1 + 3 [0] = 1/3 [fSNonEmpty] = 0 [nil] = 1/3 [oddNs] = 4 [pairNs] = 3 [ACTIVE](X) = 3/2.X + 4/3 [CONS](X1,X2) = 0 [INCR](X) = 0 [MARK](X) = 3/2.X + 4/3 [PAIR](X1,X2) = 0 [REPITEMS](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TAKE](X1,X2) = 0 [ZIP](X1,X2) = 0 Problem 1.9: SCC Processor: -> Pairs: MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(incr(X:S)) -> MARK(X:S) ->->-> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) Problem 1.9: Subterm Processor: -> Pairs: MARK(incr(X:S)) -> MARK(X:S) -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Projection: pi(MARK) = 1 Problem 1.9: SCC Processor: -> Pairs: Empty -> Rules: active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) active(repItems(nil)) -> mark(nil) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) active(take(0,XS:S)) -> mark(nil) active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) active(zip(nil,XS:S)) -> mark(nil) active(zip(X:S,nil)) -> mark(nil) active(oddNs) -> mark(incr(pairNs)) active(pairNs) -> mark(cons(0,incr(oddNs))) 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) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(pair(X1:S,X2:S)) -> active(pair(mark(X1:S),mark(X2:S))) mark(repItems(X:S)) -> active(repItems(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(zip(X1:S,X2:S)) -> active(zip(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(oddNs) -> active(oddNs) mark(pairNs) -> active(pairNs) pair(active(X1:S),X2:S) -> pair(X1:S,X2:S) pair(mark(X1:S),X2:S) -> pair(X1:S,X2:S) pair(X1:S,active(X2:S)) -> pair(X1:S,X2:S) pair(X1:S,mark(X2:S)) -> pair(X1:S,X2:S) repItems(active(X:S)) -> repItems(X:S) repItems(mark(X:S)) -> repItems(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) zip(active(X1:S),X2:S) -> zip(X1:S,X2:S) zip(mark(X1:S),X2:S) -> zip(X1:S,X2:S) zip(X1:S,active(X2:S)) -> zip(X1:S,X2:S) zip(X1:S,mark(X2:S)) -> zip(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite.