YES Problem 1: (VAR v_NonEmpty:S L:S N:S V:S V1:S V2:S X:S X1:S X2:S X3:S) (RULES U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: U11#(active(X:S)) -> U11#(X:S) U11#(mark(X:S)) -> U11#(X:S) U21#(active(X:S)) -> U21#(X:S) U21#(mark(X:S)) -> U21#(X:S) U31#(active(X:S)) -> U31#(X:S) U31#(mark(X:S)) -> U31#(X:S) U41#(active(X1:S),X2:S) -> U41#(X1:S,X2:S) U41#(mark(X1:S),X2:S) -> U41#(X1:S,X2:S) U41#(X1:S,active(X2:S)) -> U41#(X1:S,X2:S) U41#(X1:S,mark(X2:S)) -> U41#(X1:S,X2:S) U42#(active(X:S)) -> U42#(X:S) U42#(mark(X:S)) -> U42#(X:S) U51#(active(X1:S),X2:S) -> U51#(X1:S,X2:S) U51#(mark(X1:S),X2:S) -> U51#(X1:S,X2:S) U51#(X1:S,active(X2:S)) -> U51#(X1:S,X2:S) U51#(X1:S,mark(X2:S)) -> U51#(X1:S,X2:S) U52#(active(X:S)) -> U52#(X:S) U52#(mark(X:S)) -> U52#(X:S) U61#(active(X1:S),X2:S,X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(mark(X1:S),X2:S,X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,active(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,mark(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,active(X3:S)) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,mark(X3:S)) -> U61#(X1:S,X2:S,X3:S) U62#(active(X1:S),X2:S) -> U62#(X1:S,X2:S) U62#(mark(X1:S),X2:S) -> U62#(X1:S,X2:S) U62#(X1:S,active(X2:S)) -> U62#(X1:S,X2:S) U62#(X1:S,mark(X2:S)) -> U62#(X1:S,X2:S) ACTIVE(U11(tt)) -> MARK(tt) ACTIVE(U21(tt)) -> MARK(tt) ACTIVE(U31(tt)) -> MARK(tt) ACTIVE(U41(tt,V2:S)) -> U42#(isNatIList(V2:S)) ACTIVE(U41(tt,V2:S)) -> ISNATILIST(V2:S) ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U42(tt)) -> MARK(tt) ACTIVE(U51(tt,V2:S)) -> U52#(isNatList(V2:S)) ACTIVE(U51(tt,V2:S)) -> ISNATLIST(V2:S) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U52(tt)) -> MARK(tt) ACTIVE(U61(tt,L:S,N:S)) -> U62#(isNat(N:S),L:S) ACTIVE(U61(tt,L:S,N:S)) -> ISNAT(N:S) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> LENGTH(L:S) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(U62(tt,L:S)) -> S(length(L:S)) ACTIVE(isNat(length(V1:S))) -> U11#(isNatList(V1:S)) ACTIVE(isNat(length(V1:S))) -> ISNATLIST(V1:S) ACTIVE(isNat(length(V1:S))) -> MARK(U11(isNatList(V1:S))) ACTIVE(isNat(s(V1:S))) -> U21#(isNat(V1:S)) ACTIVE(isNat(s(V1:S))) -> ISNAT(V1:S) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNat(0)) -> MARK(tt) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> U41#(isNat(V1:S),V2:S) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> ISNAT(V1:S) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatIList(zeros)) -> MARK(tt) ACTIVE(isNatIList(V:S)) -> U31#(isNatList(V:S)) ACTIVE(isNatIList(V:S)) -> ISNATLIST(V:S) ACTIVE(isNatIList(V:S)) -> MARK(U31(isNatList(V:S))) ACTIVE(isNatList(cons(V1:S,V2:S))) -> U51#(isNat(V1:S),V2:S) ACTIVE(isNatList(cons(V1:S,V2:S))) -> ISNAT(V1:S) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(isNatList(nil)) -> MARK(tt) ACTIVE(length(cons(N:S,L:S))) -> U61#(isNatList(L:S),L:S,N:S) ACTIVE(length(cons(N:S,L:S))) -> ISNATLIST(L:S) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(length(nil)) -> MARK(0) ACTIVE(zeros) -> MARK(cons(0,zeros)) 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) ISNAT(active(X:S)) -> ISNAT(X:S) ISNAT(mark(X:S)) -> ISNAT(X:S) ISNATILIST(active(X:S)) -> ISNATILIST(X:S) ISNATILIST(mark(X:S)) -> ISNATILIST(X:S) ISNATLIST(active(X:S)) -> ISNATLIST(X:S) ISNATLIST(mark(X:S)) -> ISNATLIST(X:S) LENGTH(active(X:S)) -> LENGTH(X:S) LENGTH(mark(X:S)) -> LENGTH(X:S) MARK(U11(X:S)) -> U11#(mark(X:S)) MARK(U11(X:S)) -> ACTIVE(U11(mark(X:S))) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> U21#(mark(X:S)) MARK(U21(X:S)) -> ACTIVE(U21(mark(X:S))) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> U31#(mark(X:S)) MARK(U31(X:S)) -> ACTIVE(U31(mark(X:S))) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> U41#(mark(X1:S),X2:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> U42#(mark(X:S)) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> U51#(mark(X1:S),X2:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> U52#(mark(X:S)) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> U61#(mark(X1:S),X2:S,X3:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> U62#(mark(X1:S),X2:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1: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(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> LENGTH(mark(X:S)) MARK(length(X:S)) -> 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(zeros) -> ACTIVE(zeros) S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1: SCC Processor: -> Pairs: U11#(active(X:S)) -> U11#(X:S) U11#(mark(X:S)) -> U11#(X:S) U21#(active(X:S)) -> U21#(X:S) U21#(mark(X:S)) -> U21#(X:S) U31#(active(X:S)) -> U31#(X:S) U31#(mark(X:S)) -> U31#(X:S) U41#(active(X1:S),X2:S) -> U41#(X1:S,X2:S) U41#(mark(X1:S),X2:S) -> U41#(X1:S,X2:S) U41#(X1:S,active(X2:S)) -> U41#(X1:S,X2:S) U41#(X1:S,mark(X2:S)) -> U41#(X1:S,X2:S) U42#(active(X:S)) -> U42#(X:S) U42#(mark(X:S)) -> U42#(X:S) U51#(active(X1:S),X2:S) -> U51#(X1:S,X2:S) U51#(mark(X1:S),X2:S) -> U51#(X1:S,X2:S) U51#(X1:S,active(X2:S)) -> U51#(X1:S,X2:S) U51#(X1:S,mark(X2:S)) -> U51#(X1:S,X2:S) U52#(active(X:S)) -> U52#(X:S) U52#(mark(X:S)) -> U52#(X:S) U61#(active(X1:S),X2:S,X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(mark(X1:S),X2:S,X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,active(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,mark(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,active(X3:S)) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,mark(X3:S)) -> U61#(X1:S,X2:S,X3:S) U62#(active(X1:S),X2:S) -> U62#(X1:S,X2:S) U62#(mark(X1:S),X2:S) -> U62#(X1:S,X2:S) U62#(X1:S,active(X2:S)) -> U62#(X1:S,X2:S) U62#(X1:S,mark(X2:S)) -> U62#(X1:S,X2:S) ACTIVE(U11(tt)) -> MARK(tt) ACTIVE(U21(tt)) -> MARK(tt) ACTIVE(U31(tt)) -> MARK(tt) ACTIVE(U41(tt,V2:S)) -> U42#(isNatIList(V2:S)) ACTIVE(U41(tt,V2:S)) -> ISNATILIST(V2:S) ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U42(tt)) -> MARK(tt) ACTIVE(U51(tt,V2:S)) -> U52#(isNatList(V2:S)) ACTIVE(U51(tt,V2:S)) -> ISNATLIST(V2:S) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U52(tt)) -> MARK(tt) ACTIVE(U61(tt,L:S,N:S)) -> U62#(isNat(N:S),L:S) ACTIVE(U61(tt,L:S,N:S)) -> ISNAT(N:S) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> LENGTH(L:S) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(U62(tt,L:S)) -> S(length(L:S)) ACTIVE(isNat(length(V1:S))) -> U11#(isNatList(V1:S)) ACTIVE(isNat(length(V1:S))) -> ISNATLIST(V1:S) ACTIVE(isNat(length(V1:S))) -> MARK(U11(isNatList(V1:S))) ACTIVE(isNat(s(V1:S))) -> U21#(isNat(V1:S)) ACTIVE(isNat(s(V1:S))) -> ISNAT(V1:S) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNat(0)) -> MARK(tt) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> U41#(isNat(V1:S),V2:S) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> ISNAT(V1:S) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatIList(zeros)) -> MARK(tt) ACTIVE(isNatIList(V:S)) -> U31#(isNatList(V:S)) ACTIVE(isNatIList(V:S)) -> ISNATLIST(V:S) ACTIVE(isNatIList(V:S)) -> MARK(U31(isNatList(V:S))) ACTIVE(isNatList(cons(V1:S,V2:S))) -> U51#(isNat(V1:S),V2:S) ACTIVE(isNatList(cons(V1:S,V2:S))) -> ISNAT(V1:S) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(isNatList(nil)) -> MARK(tt) ACTIVE(length(cons(N:S,L:S))) -> U61#(isNatList(L:S),L:S,N:S) ACTIVE(length(cons(N:S,L:S))) -> ISNATLIST(L:S) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(length(nil)) -> MARK(0) ACTIVE(zeros) -> MARK(cons(0,zeros)) 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) ISNAT(active(X:S)) -> ISNAT(X:S) ISNAT(mark(X:S)) -> ISNAT(X:S) ISNATILIST(active(X:S)) -> ISNATILIST(X:S) ISNATILIST(mark(X:S)) -> ISNATILIST(X:S) ISNATLIST(active(X:S)) -> ISNATLIST(X:S) ISNATLIST(mark(X:S)) -> ISNATLIST(X:S) LENGTH(active(X:S)) -> LENGTH(X:S) LENGTH(mark(X:S)) -> LENGTH(X:S) MARK(U11(X:S)) -> U11#(mark(X:S)) MARK(U11(X:S)) -> ACTIVE(U11(mark(X:S))) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> U21#(mark(X:S)) MARK(U21(X:S)) -> ACTIVE(U21(mark(X:S))) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> U31#(mark(X:S)) MARK(U31(X:S)) -> ACTIVE(U31(mark(X:S))) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> U41#(mark(X1:S),X2:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> U42#(mark(X:S)) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> U51#(mark(X1:S),X2:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> U52#(mark(X:S)) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> U61#(mark(X1:S),X2:S,X3:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> U62#(mark(X1:S),X2:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1: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(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> LENGTH(mark(X:S)) MARK(length(X:S)) -> 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(zeros) -> ACTIVE(zeros) S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: LENGTH(active(X:S)) -> LENGTH(X:S) LENGTH(mark(X:S)) -> LENGTH(X:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: ISNATLIST(active(X:S)) -> ISNATLIST(X:S) ISNATLIST(mark(X:S)) -> ISNATLIST(X:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: ISNATILIST(active(X:S)) -> ISNATILIST(X:S) ISNATILIST(mark(X:S)) -> ISNATILIST(X:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: ISNAT(active(X:S)) -> ISNAT(X:S) ISNAT(mark(X:S)) -> ISNAT(X:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X: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: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U62#(active(X1:S),X2:S) -> U62#(X1:S,X2:S) U62#(mark(X1:S),X2:S) -> U62#(X1:S,X2:S) U62#(X1:S,active(X2:S)) -> U62#(X1:S,X2:S) U62#(X1:S,mark(X2:S)) -> U62#(X1:S,X2:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U61#(active(X1:S),X2:S,X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(mark(X1:S),X2:S,X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,active(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,mark(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,active(X3:S)) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,mark(X3:S)) -> U61#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U52#(active(X:S)) -> U52#(X:S) U52#(mark(X:S)) -> U52#(X:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U51#(active(X1:S),X2:S) -> U51#(X1:S,X2:S) U51#(mark(X1:S),X2:S) -> U51#(X1:S,X2:S) U51#(X1:S,active(X2:S)) -> U51#(X1:S,X2:S) U51#(X1:S,mark(X2:S)) -> U51#(X1:S,X2:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U42#(active(X:S)) -> U42#(X:S) U42#(mark(X:S)) -> U42#(X:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U41#(active(X1:S),X2:S) -> U41#(X1:S,X2:S) U41#(mark(X1:S),X2:S) -> U41#(X1:S,X2:S) U41#(X1:S,active(X2:S)) -> U41#(X1:S,X2:S) U41#(X1:S,mark(X2:S)) -> U41#(X1:S,X2:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U31#(active(X:S)) -> U31#(X:S) U31#(mark(X:S)) -> U31#(X:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U21#(active(X:S)) -> U21#(X:S) U21#(mark(X:S)) -> U21#(X:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U11#(active(X:S)) -> U11#(X:S) U11#(mark(X:S)) -> U11#(X:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(length(V1:S))) -> MARK(U11(isNatList(V1:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatIList(V:S)) -> MARK(U31(isNatList(V:S))) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U11(X:S)) -> ACTIVE(U11(mark(X:S))) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> ACTIVE(U21(mark(X:S))) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> ACTIVE(U31(mark(X:S))) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) The problem is decomposed in 16 subproblems. Problem 1.1: Subterm Processor: -> Pairs: S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(S) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: LENGTH(active(X:S)) -> LENGTH(X:S) LENGTH(mark(X:S)) -> LENGTH(X:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(LENGTH) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: ISNATLIST(active(X:S)) -> ISNATLIST(X:S) ISNATLIST(mark(X:S)) -> ISNATLIST(X:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(ISNATLIST) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: ISNATILIST(active(X:S)) -> ISNATILIST(X:S) ISNATILIST(mark(X:S)) -> ISNATILIST(X:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(ISNATILIST) = 1 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Subterm Processor: -> Pairs: ISNAT(active(X:S)) -> ISNAT(X:S) ISNAT(mark(X:S)) -> ISNAT(X:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(ISNAT) = 1 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.6: 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: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(CONS) = 1 Problem 1.6: 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: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X: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: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.6: 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: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(CONS) = 2 Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.7: Subterm Processor: -> Pairs: U62#(active(X1:S),X2:S) -> U62#(X1:S,X2:S) U62#(mark(X1:S),X2:S) -> U62#(X1:S,X2:S) U62#(X1:S,active(X2:S)) -> U62#(X1:S,X2:S) U62#(X1:S,mark(X2:S)) -> U62#(X1:S,X2:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U62#) = 1 Problem 1.7: SCC Processor: -> Pairs: U62#(X1:S,active(X2:S)) -> U62#(X1:S,X2:S) U62#(X1:S,mark(X2:S)) -> U62#(X1:S,X2:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U62#(X1:S,active(X2:S)) -> U62#(X1:S,X2:S) U62#(X1:S,mark(X2:S)) -> U62#(X1:S,X2:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.7: Subterm Processor: -> Pairs: U62#(X1:S,active(X2:S)) -> U62#(X1:S,X2:S) U62#(X1:S,mark(X2:S)) -> U62#(X1:S,X2:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U62#) = 2 Problem 1.7: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.8: Subterm Processor: -> Pairs: U61#(active(X1:S),X2:S,X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(mark(X1:S),X2:S,X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,active(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,mark(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,active(X3:S)) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,mark(X3:S)) -> U61#(X1:S,X2:S,X3:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U61#) = 1 Problem 1.8: SCC Processor: -> Pairs: U61#(X1:S,active(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,mark(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,active(X3:S)) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,mark(X3:S)) -> U61#(X1:S,X2:S,X3:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U61#(X1:S,active(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,mark(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,active(X3:S)) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,mark(X3:S)) -> U61#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.8: Subterm Processor: -> Pairs: U61#(X1:S,active(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,mark(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,active(X3:S)) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,mark(X3:S)) -> U61#(X1:S,X2:S,X3:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U61#) = 2 Problem 1.8: SCC Processor: -> Pairs: U61#(X1:S,X2:S,active(X3:S)) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,mark(X3:S)) -> U61#(X1:S,X2:S,X3:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U61#(X1:S,X2:S,active(X3:S)) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,mark(X3:S)) -> U61#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.8: Subterm Processor: -> Pairs: U61#(X1:S,X2:S,active(X3:S)) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,mark(X3:S)) -> U61#(X1:S,X2:S,X3:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U61#) = 3 Problem 1.8: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.9: Subterm Processor: -> Pairs: U52#(active(X:S)) -> U52#(X:S) U52#(mark(X:S)) -> U52#(X:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U52#) = 1 Problem 1.9: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.10: Subterm Processor: -> Pairs: U51#(active(X1:S),X2:S) -> U51#(X1:S,X2:S) U51#(mark(X1:S),X2:S) -> U51#(X1:S,X2:S) U51#(X1:S,active(X2:S)) -> U51#(X1:S,X2:S) U51#(X1:S,mark(X2:S)) -> U51#(X1:S,X2:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U51#) = 1 Problem 1.10: SCC Processor: -> Pairs: U51#(X1:S,active(X2:S)) -> U51#(X1:S,X2:S) U51#(X1:S,mark(X2:S)) -> U51#(X1:S,X2:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U51#(X1:S,active(X2:S)) -> U51#(X1:S,X2:S) U51#(X1:S,mark(X2:S)) -> U51#(X1:S,X2:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.10: Subterm Processor: -> Pairs: U51#(X1:S,active(X2:S)) -> U51#(X1:S,X2:S) U51#(X1:S,mark(X2:S)) -> U51#(X1:S,X2:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U51#) = 2 Problem 1.10: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.11: Subterm Processor: -> Pairs: U42#(active(X:S)) -> U42#(X:S) U42#(mark(X:S)) -> U42#(X:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U42#) = 1 Problem 1.11: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.12: Subterm Processor: -> Pairs: U41#(active(X1:S),X2:S) -> U41#(X1:S,X2:S) U41#(mark(X1:S),X2:S) -> U41#(X1:S,X2:S) U41#(X1:S,active(X2:S)) -> U41#(X1:S,X2:S) U41#(X1:S,mark(X2:S)) -> U41#(X1:S,X2:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U41#) = 1 Problem 1.12: SCC Processor: -> Pairs: U41#(X1:S,active(X2:S)) -> U41#(X1:S,X2:S) U41#(X1:S,mark(X2:S)) -> U41#(X1:S,X2:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U41#(X1:S,active(X2:S)) -> U41#(X1:S,X2:S) U41#(X1:S,mark(X2:S)) -> U41#(X1:S,X2:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.12: Subterm Processor: -> Pairs: U41#(X1:S,active(X2:S)) -> U41#(X1:S,X2:S) U41#(X1:S,mark(X2:S)) -> U41#(X1:S,X2:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U41#) = 2 Problem 1.12: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.13: Subterm Processor: -> Pairs: U31#(active(X:S)) -> U31#(X:S) U31#(mark(X:S)) -> U31#(X:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U31#) = 1 Problem 1.13: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.14: Subterm Processor: -> Pairs: U21#(active(X:S)) -> U21#(X:S) U21#(mark(X:S)) -> U21#(X:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U21#) = 1 Problem 1.14: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.15: Subterm Processor: -> Pairs: U11#(active(X:S)) -> U11#(X:S) U11#(mark(X:S)) -> U11#(X:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U11#) = 1 Problem 1.15: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(length(V1:S))) -> MARK(U11(isNatList(V1:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatIList(V:S)) -> MARK(U31(isNatList(V:S))) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U11(X:S)) -> ACTIVE(U11(mark(X:S))) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> ACTIVE(U21(mark(X:S))) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> ACTIVE(U31(mark(X:S))) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = X + 2 [U21](X) = X [U31](X) = X [U41](X1,X2) = X1 + 2.X2 [U42](X) = X [U51](X1,X2) = X1 + 2.X2 [U52](X) = X [U61](X1,X2,X3) = X1 + X2 + 2.X3 [U62](X1,X2) = X1 + X2 [active](X) = X [cons](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 2.X + 2 [isNatIList](X) = 2.X + 2 [isNatList](X) = X + 2 [length](X) = X + 2 [mark](X) = X [s](X) = X [0] = 0 [fSNonEmpty] = 0 [nil] = 2 [tt] = 2 [zeros] = 0 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatIList(V:S)) -> MARK(U31(isNatList(V:S))) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U11(X:S)) -> ACTIVE(U11(mark(X:S))) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> ACTIVE(U21(mark(X:S))) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> ACTIVE(U31(mark(X:S))) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatIList(V:S)) -> MARK(U31(isNatList(V:S))) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U11(X:S)) -> ACTIVE(U11(mark(X:S))) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> ACTIVE(U21(mark(X:S))) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> ACTIVE(U31(mark(X:S))) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatIList(V:S)) -> MARK(U31(isNatList(V:S))) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U11(X:S)) -> ACTIVE(U11(mark(X:S))) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> ACTIVE(U21(mark(X:S))) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> ACTIVE(U31(mark(X:S))) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2.X + 2 [U21](X) = X [U31](X) = X + 1 [U41](X1,X2) = X1 + 2.X2 + 2 [U42](X) = X [U51](X1,X2) = 2.X1 + 2.X2 [U52](X) = X [U61](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U62](X1,X2) = X1 + 2.X2 + 2 [active](X) = X [cons](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 2.X [isNatIList](X) = 2.X + 2 [isNatList](X) = 2.X [length](X) = 2.X + 2 [mark](X) = X [s](X) = X [0] = 0 [fSNonEmpty] = 0 [nil] = 2 [tt] = 0 [zeros] = 0 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U11(X:S)) -> ACTIVE(U11(mark(X:S))) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> ACTIVE(U21(mark(X:S))) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> ACTIVE(U31(mark(X:S))) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U11(X:S)) -> ACTIVE(U11(mark(X:S))) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> ACTIVE(U21(mark(X:S))) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> ACTIVE(U31(mark(X:S))) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U11(X:S)) -> ACTIVE(U11(mark(X:S))) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> ACTIVE(U21(mark(X:S))) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> ACTIVE(U31(mark(X:S))) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 0 [U21](X) = 1 [U31](X) = 1 [U41](X1,X2) = 1 [U42](X) = 1 [U51](X1,X2) = 1 [U52](X) = 1 [U61](X1,X2,X3) = 1 [U62](X1,X2) = 1 [active](X) = 2.X + 2 [cons](X1,X2) = 1 [isNat](X) = 1 [isNatIList](X) = 1 [isNatList](X) = 1 [length](X) = 1 [mark](X) = 2.X + 2 [s](X) = 0 [0] = 1 [fSNonEmpty] = 0 [nil] = 0 [tt] = 0 [zeros] = 1 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = 2.X [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> ACTIVE(U21(mark(X:S))) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> ACTIVE(U31(mark(X:S))) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> ACTIVE(U21(mark(X:S))) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> ACTIVE(U31(mark(X:S))) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> ACTIVE(U21(mark(X:S))) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> ACTIVE(U31(mark(X:S))) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = X + 1 [U21](X) = X [U31](X) = X [U41](X1,X2) = X1 + 2.X2 [U42](X) = X [U51](X1,X2) = X1 + 2.X2 [U52](X) = X [U61](X1,X2,X3) = X1 + 2.X2 + 2.X3 [U62](X1,X2) = X1 + 2.X2 [active](X) = X [cons](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 2.X + 2 [isNatIList](X) = 2.X + 2 [isNatList](X) = X + 2 [length](X) = 2.X + 2 [mark](X) = X [s](X) = X [0] = 0 [fSNonEmpty] = 0 [nil] = 1 [tt] = 2 [zeros] = 0 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> ACTIVE(U21(mark(X:S))) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> ACTIVE(U31(mark(X:S))) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> ACTIVE(U21(mark(X:S))) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> ACTIVE(U31(mark(X:S))) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> ACTIVE(U21(mark(X:S))) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> ACTIVE(U31(mark(X:S))) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2 [U21](X) = 1 [U31](X) = 1 [U41](X1,X2) = 2 [U42](X) = 2 [U51](X1,X2) = 2 [U52](X) = 1 [U61](X1,X2,X3) = 2 [U62](X1,X2) = 2 [active](X) = 2.X + 2 [cons](X1,X2) = 1 [isNat](X) = 2 [isNatIList](X) = 2 [isNatList](X) = 2 [length](X) = 2 [mark](X) = 2.X + 2 [s](X) = 1 [0] = 0 [fSNonEmpty] = 0 [nil] = 2 [tt] = 1 [zeros] = 2 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = X [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> ACTIVE(U31(mark(X:S))) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> ACTIVE(U31(mark(X:S))) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> ACTIVE(U31(mark(X:S))) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2 [U21](X) = 2 [U31](X) = 0 [U41](X1,X2) = 2 [U42](X) = 2 [U51](X1,X2) = 2 [U52](X) = 1 [U61](X1,X2,X3) = 2 [U62](X1,X2) = 2 [active](X) = 2 [cons](X1,X2) = 2 [isNat](X) = 2 [isNatIList](X) = 2 [isNatList](X) = 2 [length](X) = 2 [mark](X) = 2 [s](X) = 2 [0] = 0 [fSNonEmpty] = 0 [nil] = 2 [tt] = 2 [zeros] = 2 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = X [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 1 [U21](X) = X [U31](X) = X + 1 [U41](X1,X2) = 2.X1 + 2.X2 [U42](X) = X [U51](X1,X2) = X1 [U52](X) = X [U61](X1,X2,X3) = 2.X1 + 2.X2 [U62](X1,X2) = 2.X1 + 2.X2 [active](X) = X [cons](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 1 [isNatIList](X) = 2.X + 2 [isNatList](X) = 1 [length](X) = 2.X + 2 [mark](X) = X [s](X) = X [0] = 0 [fSNonEmpty] = 0 [nil] = 2 [tt] = 1 [zeros] = 0 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = X [U21](X) = X [U31](X) = 2 [U41](X1,X2) = X1 + 2.X2 + 1 [U42](X) = X [U51](X1,X2) = X1 [U52](X) = X [U61](X1,X2,X3) = 2.X1 + 2.X2 [U62](X1,X2) = X1 + 2.X2 + 1 [active](X) = X [cons](X1,X2) = X1 + 2.X2 [isNat](X) = 1 [isNatIList](X) = 2.X + 2 [isNatList](X) = 1 [length](X) = 2.X + 2 [mark](X) = X [s](X) = X [0] = 0 [fSNonEmpty] = 0 [nil] = 0 [tt] = 1 [zeros] = 0 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> ACTIVE(U42(mark(X:S))) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 1 [U21](X) = 1 [U31](X) = 1 [U41](X1,X2) = 1 [U42](X) = 0 [U51](X1,X2) = 1 [U52](X) = 1 [U61](X1,X2,X3) = 1 [U62](X1,X2) = 1 [active](X) = 2.X + 2 [cons](X1,X2) = 0 [isNat](X) = 1 [isNatIList](X) = 1 [isNatList](X) = 1 [length](X) = 1 [mark](X) = 2.X + 2 [s](X) = 1 [0] = 0 [fSNonEmpty] = 0 [nil] = 1 [tt] = 0 [zeros] = 1 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = 2.X [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> ACTIVE(U52(mark(X:S))) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2 [U21](X) = 2 [U31](X) = 2 [U41](X1,X2) = 1 [U42](X) = 2 [U51](X1,X2) = 1 [U52](X) = 0 [U61](X1,X2,X3) = 1 [U62](X1,X2) = 1 [active](X) = 2 [cons](X1,X2) = 1 [isNat](X) = 1 [isNatIList](X) = 1 [isNatList](X) = 1 [length](X) = 1 [mark](X) = 2 [s](X) = 0 [0] = 0 [fSNonEmpty] = 0 [nil] = 1 [tt] = 2 [zeros] = 1 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = 2.X [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 1 [U21](X) = X [U31](X) = 2.X [U41](X1,X2) = 2.X2 + 2 [U42](X) = X [U51](X1,X2) = X1 [U52](X) = X [U61](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1 [U62](X1,X2) = 2.X1 + 2.X2 [active](X) = X [cons](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 1 [isNatIList](X) = 2.X + 2 [isNatList](X) = 1 [length](X) = 2.X + 2 [mark](X) = X [s](X) = X [0] = 0 [fSNonEmpty] = 0 [nil] = 0 [tt] = 1 [zeros] = 0 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 0 [U21](X) = X [U31](X) = 2 [U41](X1,X2) = 2.X1 + 2.X2 + 2 [U42](X) = X [U51](X1,X2) = X1 + 1 [U52](X) = X [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2) = 2.X1 + 2.X2 + 2 [active](X) = X [cons](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 0 [isNatIList](X) = 2.X + 2 [isNatList](X) = 1 [length](X) = 2.X + 2 [mark](X) = X [s](X) = X [0] = 0 [fSNonEmpty] = 0 [nil] = 2 [tt] = 0 [zeros] = 0 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U61(tt,L:S,N:S)) -> MARK(U62(isNat(N:S),L:S)) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2 [U21](X) = X [U31](X) = 2.X [U41](X1,X2) = X1 + 2.X2 [U42](X) = X [U51](X1,X2) = X2 + 1 [U52](X) = X [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U62](X1,X2) = X1 + 2.X2 + 1 [active](X) = X [cons](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 2 [isNatIList](X) = 2.X + 2 [isNatList](X) = X + 1 [length](X) = 2.X + 2 [mark](X) = X [s](X) = X [0] = 0 [fSNonEmpty] = 0 [nil] = 1 [tt] = 2 [zeros] = 0 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(U62(tt,L:S)) -> MARK(s(length(L:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2.X + 2 [U21](X) = X [U31](X) = 2.X + 2 [U41](X1,X2) = X1 + 2.X2 [U42](X) = X [U51](X1,X2) = 2.X2 [U52](X) = X [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 [U62](X1,X2) = X1 + 2.X2 [active](X) = X [cons](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 2.X + 2 [isNatIList](X) = 2.X + 2 [isNatList](X) = X [length](X) = 2.X + 1 [mark](X) = X [s](X) = X [0] = 0 [fSNonEmpty] = 0 [nil] = 2 [tt] = 2 [zeros] = 0 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNat(V1:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2.X + 2 [U21](X) = X + 2 [U31](X) = X + 1 [U41](X1,X2) = 2.X2 + 2 [U42](X) = X [U51](X1,X2) = 2.X2 + 1 [U52](X) = X [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U62](X1,X2) = X1 + 2.X2 + 2 [active](X) = X [cons](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 2.X + 2 [isNatIList](X) = 2.X + 2 [isNatList](X) = X + 1 [length](X) = 2.X + 2 [mark](X) = X [s](X) = X + 2 [0] = 0 [fSNonEmpty] = 0 [nil] = 2 [tt] = 2 [zeros] = 0 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(length(cons(N:S,L:S))) -> MARK(U61(isNatList(L:S),L:S,N:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2 [U21](X) = X + 2 [U31](X) = 2.X + 2 [U41](X1,X2) = 2.X2 + 2 [U42](X) = X [U51](X1,X2) = 2.X2 [U52](X) = 2.X [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 [U62](X1,X2) = 2.X1 + 2.X2 + 1 [active](X) = X [cons](X1,X2) = 2.X1 + 2.X2 [isNat](X) = X + 2 [isNatIList](X) = 2.X + 2 [isNatList](X) = X [length](X) = 2.X + 2 [mark](X) = X [s](X) = X + 2 [0] = 0 [fSNonEmpty] = 0 [nil] = 2 [tt] = 2 [zeros] = 0 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2 [U21](X) = X + 1 [U31](X) = 2.X + 1 [U41](X1,X2) = X1 + 2.X2 [U42](X) = X [U51](X1,X2) = 2.X2 [U52](X) = 2.X [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U62](X1,X2) = 2.X1 + 2.X2 + 2 [active](X) = X [cons](X1,X2) = X1 + 2.X2 [isNat](X) = X + 2 [isNatIList](X) = 2.X + 2 [isNatList](X) = X [length](X) = 2.X + 2 [mark](X) = X [s](X) = X + 2 [0] = 0 [fSNonEmpty] = 0 [nil] = 2 [tt] = 2 [zeros] = 0 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2.X [U21](X) = X [U31](X) = 0 [U41](X1,X2) = 2.X1 [U42](X) = 2.X [U51](X1,X2) = 2.X1 [U52](X) = 2.X [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2) = 2.X1 + 2.X2 + 2 [active](X) = X [cons](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 0 [isNatIList](X) = 0 [isNatList](X) = 0 [length](X) = 2.X + 2 [mark](X) = X [s](X) = X [0] = 0 [fSNonEmpty] = 0 [nil] = 2 [tt] = 0 [zeros] = 0 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = 2 [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U62(X1:S,X2:S)) -> ACTIVE(U62(mark(X1:S),X2:S)) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2 [U21](X) = 2 [U31](X) = 2 [U41](X1,X2) = 2 [U42](X) = 2 [U51](X1,X2) = 2 [U52](X) = 2 [U61](X1,X2,X3) = 2 [U62](X1,X2) = 1 [active](X) = 2.X + 2 [cons](X1,X2) = 2 [isNat](X) = 2.X + 2 [isNatIList](X) = 2 [isNatList](X) = 2 [length](X) = 2 [mark](X) = 2.X + 2 [s](X) = 1 [0] = 2 [fSNonEmpty] = 0 [nil] = 2 [tt] = 2 [zeros] = 2 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = X [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U62(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2 [U21](X) = X [U31](X) = 2 [U41](X1,X2) = 2.X2 + 2 [U42](X) = X [U51](X1,X2) = X1 + 2.X2 + 1 [U52](X) = X [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2) = X1 + 2.X2 + 1 [active](X) = X [cons](X1,X2) = X1 + 2.X2 [isNat](X) = 2.X + 1 [isNatIList](X) = X + 2 [isNatList](X) = 2.X + 2 [length](X) = 2.X + 2 [mark](X) = X [s](X) = X [0] = 0 [fSNonEmpty] = 0 [nil] = 2 [tt] = 1 [zeros] = 0 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2 [U21](X) = 2 [U31](X) = 2 [U41](X1,X2) = 2 [U42](X) = 2 [U51](X1,X2) = 2 [U52](X) = 2 [U61](X1,X2,X3) = 2 [U62](X1,X2) = 2 [active](X) = 2.X [cons](X1,X2) = 1 [isNat](X) = 2 [isNatIList](X) = 2 [isNatList](X) = 2 [length](X) = 2 [mark](X) = X + 2 [s](X) = 2 [0] = 2 [fSNonEmpty] = 0 [nil] = 0 [tt] = 2 [zeros] = 2 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = X [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2 [U21](X) = 2 [U31](X) = 0 [U41](X1,X2) = 1 [U42](X) = 2 [U51](X1,X2) = 1 [U52](X) = 1 [U61](X1,X2,X3) = 0 [U62](X1,X2) = 2 [active](X) = 2 [cons](X1,X2) = 0 [isNat](X) = 2 [isNatIList](X) = 1 [isNatList](X) = 1 [length](X) = 0 [mark](X) = 2 [s](X) = 1 [0] = 0 [fSNonEmpty] = 0 [nil] = 2 [tt] = 2 [zeros] = 1 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = 2.X [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2 [U21](X) = 0 [U31](X) = X + 2 [U41](X1,X2) = 2.X1 + 2.X2 + 2 [U42](X) = X [U51](X1,X2) = X1 + 2.X2 [U52](X) = X [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2) = X1 + 2.X2 + 2 [active](X) = X [cons](X1,X2) = 2.X1 + X2 [isNat](X) = 2.X [isNatIList](X) = 2.X + 2 [isNatList](X) = 2.X [length](X) = 2.X + 2 [mark](X) = X [s](X) = X [0] = 0 [fSNonEmpty] = 0 [nil] = 2 [tt] = 0 [zeros] = 1 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 1 [U21](X) = 1 [U31](X) = 2.X [U41](X1,X2) = 0 [U42](X) = 2.X [U51](X1,X2) = 0 [U52](X) = 2.X [U61](X1,X2,X3) = 2.X1 [U62](X1,X2) = 0 [active](X) = X [cons](X1,X2) = X1 [isNat](X) = 2 [isNatIList](X) = 0 [isNatList](X) = 0 [length](X) = 0 [mark](X) = 2.X [s](X) = 2.X [0] = 0 [fSNonEmpty] = 0 [nil] = 1 [tt] = 0 [zeros] = 2 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = X + 2 [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2.X [U21](X) = 1 [U31](X) = 2.X [U41](X1,X2) = 0 [U42](X) = 2.X [U51](X1,X2) = 0 [U52](X) = 2.X [U61](X1,X2,X3) = X1 [U62](X1,X2) = 0 [active](X) = X [cons](X1,X2) = X1 + 1 [isNat](X) = 2 [isNatIList](X) = 0 [isNatList](X) = 0 [length](X) = 0 [mark](X) = 2.X [s](X) = 2.X [0] = 0 [fSNonEmpty] = 0 [nil] = 0 [tt] = 0 [zeros] = 2 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = X + 2 [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 1 [U21](X) = 1 [U31](X) = 1 [U41](X1,X2) = 1 [U42](X) = 1 [U51](X1,X2) = 1 [U52](X) = 1 [U61](X1,X2,X3) = 1 [U62](X1,X2) = 1 [active](X) = 2.X [cons](X1,X2) = 0 [isNat](X) = 1 [isNatIList](X) = 1 [isNatList](X) = 1 [length](X) = 1 [mark](X) = 2 [s](X) = 0 [0] = 0 [fSNonEmpty] = 0 [nil] = 0 [tt] = 1 [zeros] = 1 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = 2.X [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2 [U21](X) = 2 [U31](X) = 2 [U41](X1,X2) = 2.X2 + 2 [U42](X) = X [U51](X1,X2) = 2.X2 + 1 [U52](X) = X [U61](X1,X2,X3) = X1 + 2.X2 [U62](X1,X2) = 2.X2 + 2 [active](X) = X [cons](X1,X2) = 2.X2 [isNat](X) = 2.X + 2 [isNatIList](X) = 2.X + 2 [isNatList](X) = 2.X + 1 [length](X) = 2.X + 1 [mark](X) = X [s](X) = X + 1 [0] = 2 [fSNonEmpty] = 0 [nil] = 1 [tt] = 2 [zeros] = 0 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = X [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = X [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U41(tt,V2:S)) -> MARK(U42(isNatIList(V2:S))) ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 2 ->Bound: 1 ->Interpretation: [U11](X) = [1;1] [U21](X) = [1;1] [U31](X) = [0 0;1 1].X [U41](X1,X2) = [0 0;1 1].X1 + [0 0;0 1].X2 [U42](X) = [0 0;1 1].X + [1;0] [U51](X1,X2) = [0 0;0 1].X2 + [0;1] [U52](X) = [0 0;1 1].X [U61](X1,X2,X3) = [0;1] [U62](X1,X2) = [1;0] [active](X) = [0 0;1 1].X [cons](X1,X2) = [0 0;1 1].X1 + [0 0;0 1].X2 + [0;1] [isNat](X) = [1 1;0 0].X + [1;1] [isNatIList](X) = [0 0;0 1].X + [0;1] [isNatList](X) = [0 0;0 1].X + [1;0] [length](X) = [1;1] [mark](X) = [0 0;1 1].X [s](X) = [1;0] [0] = 0 [fSNonEmpty] = 0 [nil] = [1;1] [tt] = [1;1] [zeros] = [1;1] [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = [0 1;0 1].X + [1;1] [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = [0 1;1 1].X + [1;1] [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U41(X1:S,X2:S)) -> ACTIVE(U41(mark(X1:S),X2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2 [U21](X) = 2 [U31](X) = 2 [U41](X1,X2) = 1 [U42](X) = 1 [U51](X1,X2) = 2 [U52](X) = 2 [U61](X1,X2,X3) = 2 [U62](X1,X2) = 0 [active](X) = 2.X + 2 [cons](X1,X2) = 2.X2 [isNat](X) = 2.X + 2 [isNatIList](X) = 2 [isNatList](X) = 2 [length](X) = 2 [mark](X) = 2.X + 2 [s](X) = 0 [0] = 2 [fSNonEmpty] = 0 [nil] = 2 [tt] = 1 [zeros] = 0 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = X [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(U51(tt,V2:S)) -> MARK(U52(isNatList(V2:S))) ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 2 ->Bound: 1 ->Interpretation: [U11](X) = [0 0;1 1].X + [1;1] [U21](X) = [1;1] [U31](X) = [1;0] [U41](X1,X2) = [1 1;0 0].X1 + [0 1;0 1].X2 + [1;1] [U42](X) = [0 0;1 1].X + [0;1] [U51](X1,X2) = [1 1;0 0].X1 + [0 0;0 1].X2 + [0;1] [U52](X) = [0 0;1 1].X + [1;0] [U61](X1,X2,X3) = [1 1;0 0].X1 + [0 0;0 1].X2 + [1 1;0 1].X3 + [1;1] [U62](X1,X2) = [1 1;0 0].X1 + [0 1;0 0].X2 + [1;1] [active](X) = [0 0;1 1].X [cons](X1,X2) = [1 1;1 1].X1 + [0 0;0 1].X2 + [0;1] [isNat](X) = [1 1;0 0].X [isNatIList](X) = [0 1;0 1].X + [0;1] [isNatList](X) = [0 0;0 1].X [length](X) = [1 1;1 1].X + [1;1] [mark](X) = [0 0;1 1].X [s](X) = [1;1] [0] = 0 [fSNonEmpty] = 0 [nil] = [1;1] [tt] = 0 [zeros] = [1;0] [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = [0 1;1 1].X + [1;1] [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = [0 1;1 1].X + [1;1] [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 0 [U21](X) = 2 [U31](X) = 2 [U41](X1,X2) = 2 [U42](X) = 0 [U51](X1,X2) = 1 [U52](X) = 1 [U61](X1,X2,X3) = 2 [U62](X1,X2) = 2 [active](X) = 2 [cons](X1,X2) = 0 [isNat](X) = 2 [isNatIList](X) = 2 [isNatList](X) = 2 [length](X) = 2 [mark](X) = 2 [s](X) = 2 [0] = 2 [fSNonEmpty] = 0 [nil] = 1 [tt] = 2 [zeros] = 2 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = X [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(isNatIList(cons(V1:S,V2:S))) -> MARK(U41(isNat(V1:S),V2:S)) ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 0 [U21](X) = 0 [U31](X) = 0 [U41](X1,X2) = 2.X1 + 1 [U42](X) = 2.X [U51](X1,X2) = 2.X2 + 2 [U52](X) = 2.X [U61](X1,X2,X3) = 0 [U62](X1,X2) = 0 [active](X) = 2.X [cons](X1,X2) = X1 + 2.X2 + 2 [isNat](X) = 0 [isNatIList](X) = 2 [isNatList](X) = 2.X + 2 [length](X) = 0 [mark](X) = X [s](X) = 0 [0] = 0 [fSNonEmpty] = 0 [nil] = 0 [tt] = 0 [zeros] = 0 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = X + 2 [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = X + 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatIList(X:S)) -> ACTIVE(isNatIList(X:S)) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Reduction Pairs Processor: -> Pairs: ACTIVE(isNatList(cons(V1:S,V2:S))) -> MARK(U51(isNat(V1:S),V2:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 0 [U21](X) = 0 [U31](X) = 0 [U41](X1,X2) = 0 [U42](X) = 2.X [U51](X1,X2) = 0 [U52](X) = 2.X [U61](X1,X2,X3) = 0 [U62](X1,X2) = 0 [active](X) = 0 [cons](X1,X2) = 2.X1 + 1 [isNat](X) = 0 [isNatIList](X) = 0 [isNatList](X) = 2 [length](X) = 0 [mark](X) = 0 [s](X) = 0 [0] = 0 [fSNonEmpty] = 0 [nil] = 0 [tt] = 0 [zeros] = 0 [U11#](X) = 0 [U21#](X) = 0 [U31#](X) = 0 [U41#](X1,X2) = 0 [U42#](X) = 0 [U51#](X1,X2) = 0 [U52#](X) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2) = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [ISNAT](X) = 0 [ISNATILIST](X) = 0 [ISNATLIST](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 Problem 1.16: SCC Processor: -> Pairs: MARK(U42(X:S)) -> MARK(X:S) MARK(U52(X:S)) -> MARK(X:S) MARK(isNatList(X:S)) -> ACTIVE(isNatList(X:S)) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(U42(X:S)) -> MARK(X:S) MARK(U52(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Subterm Processor: -> Pairs: MARK(U42(X:S)) -> MARK(X:S) MARK(U52(X:S)) -> MARK(X:S) -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(MARK) = 1 Problem 1.16: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X:S)) -> U11(X:S) U11(mark(X:S)) -> U11(X:S) U21(active(X:S)) -> U21(X:S) U21(mark(X:S)) -> U21(X:S) U31(active(X:S)) -> U31(X:S) U31(mark(X:S)) -> U31(X:S) U41(active(X1:S),X2:S) -> U41(X1:S,X2:S) U41(mark(X1:S),X2:S) -> U41(X1:S,X2:S) U41(X1:S,active(X2:S)) -> U41(X1:S,X2:S) U41(X1:S,mark(X2:S)) -> U41(X1:S,X2:S) U42(active(X:S)) -> U42(X:S) U42(mark(X:S)) -> U42(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X:S)) -> U52(X:S) U52(mark(X:S)) -> U52(X:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S) -> U62(X1:S,X2:S) U62(mark(X1:S),X2:S) -> U62(X1:S,X2:S) U62(X1:S,active(X2:S)) -> U62(X1:S,X2:S) U62(X1:S,mark(X2:S)) -> U62(X1:S,X2:S) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2:S)) -> mark(U42(isNatIList(V2:S))) active(U42(tt)) -> mark(tt) active(U51(tt,V2:S)) -> mark(U52(isNatList(V2:S))) active(U52(tt)) -> mark(tt) active(U61(tt,L:S,N:S)) -> mark(U62(isNat(N:S),L:S)) active(U62(tt,L:S)) -> mark(s(length(L:S))) active(isNat(length(V1:S))) -> mark(U11(isNatList(V1:S))) active(isNat(s(V1:S))) -> mark(U21(isNat(V1:S))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1:S,V2:S))) -> mark(U41(isNat(V1:S),V2:S)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V:S)) -> mark(U31(isNatList(V:S))) active(isNatList(cons(V1:S,V2:S))) -> mark(U51(isNat(V1:S),V2:S)) active(isNatList(nil)) -> mark(tt) active(length(cons(N:S,L:S))) -> mark(U61(isNatList(L:S),L:S,N:S)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) 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) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatIList(active(X:S)) -> isNatIList(X:S) isNatIList(mark(X:S)) -> isNatIList(X:S) isNatList(active(X:S)) -> isNatList(X:S) isNatList(mark(X:S)) -> isNatList(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(U11(X:S)) -> active(U11(mark(X:S))) mark(U21(X:S)) -> active(U21(mark(X:S))) mark(U31(X:S)) -> active(U31(mark(X:S))) mark(U41(X1:S,X2:S)) -> active(U41(mark(X1:S),X2:S)) mark(U42(X:S)) -> active(U42(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X:S)) -> active(U52(mark(X:S))) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S)) -> active(U62(mark(X1:S),X2:S)) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatIList(X:S)) -> active(isNatIList(X:S)) mark(isNatList(X:S)) -> active(isNatList(X:S)) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite.