/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR L N V V1 V2 X X1 X2 X3) (RULES U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ) Problem 1: Dependency Pairs Processor: -> Pairs: U11#(active(X)) -> U11#(X) U11#(mark(X)) -> U11#(X) U21#(active(X)) -> U21#(X) U21#(mark(X)) -> U21#(X) U31#(active(X)) -> U31#(X) U31#(mark(X)) -> U31#(X) U41#(active(X1),X2) -> U41#(X1,X2) U41#(mark(X1),X2) -> U41#(X1,X2) U41#(X1,active(X2)) -> U41#(X1,X2) U41#(X1,mark(X2)) -> U41#(X1,X2) U42#(active(X)) -> U42#(X) U42#(mark(X)) -> U42#(X) U51#(active(X1),X2) -> U51#(X1,X2) U51#(mark(X1),X2) -> U51#(X1,X2) U51#(X1,active(X2)) -> U51#(X1,X2) U51#(X1,mark(X2)) -> U51#(X1,X2) U52#(active(X)) -> U52#(X) U52#(mark(X)) -> U52#(X) U61#(active(X1),X2,X3) -> U61#(X1,X2,X3) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) U62#(active(X1),X2) -> U62#(X1,X2) U62#(mark(X1),X2) -> U62#(X1,X2) U62#(X1,active(X2)) -> U62#(X1,X2) U62#(X1,mark(X2)) -> U62#(X1,X2) ACTIVE(U11(tt)) -> MARK(tt) ACTIVE(U21(tt)) -> MARK(tt) ACTIVE(U31(tt)) -> MARK(tt) ACTIVE(U41(tt,V2)) -> U42#(isNatIList(V2)) ACTIVE(U41(tt,V2)) -> ISNATILIST(V2) ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U42(tt)) -> MARK(tt) ACTIVE(U51(tt,V2)) -> U52#(isNatList(V2)) ACTIVE(U51(tt,V2)) -> ISNATLIST(V2) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U52(tt)) -> MARK(tt) ACTIVE(U61(tt,L,N)) -> U62#(isNat(N),L) ACTIVE(U61(tt,L,N)) -> ISNAT(N) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> LENGTH(L) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(U62(tt,L)) -> S(length(L)) ACTIVE(isNat(length(V1))) -> U11#(isNatList(V1)) ACTIVE(isNat(length(V1))) -> ISNATLIST(V1) ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1))) ACTIVE(isNat(s(V1))) -> U21#(isNat(V1)) ACTIVE(isNat(s(V1))) -> ISNAT(V1) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNat(0)) -> MARK(tt) ACTIVE(isNatIList(cons(V1,V2))) -> U41#(isNat(V1),V2) ACTIVE(isNatIList(cons(V1,V2))) -> ISNAT(V1) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatIList(zeros)) -> MARK(tt) ACTIVE(isNatIList(V)) -> U31#(isNatList(V)) ACTIVE(isNatIList(V)) -> ISNATLIST(V) ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V))) ACTIVE(isNatList(cons(V1,V2))) -> U51#(isNat(V1),V2) ACTIVE(isNatList(cons(V1,V2))) -> ISNAT(V1) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(isNatList(nil)) -> MARK(tt) ACTIVE(length(cons(N,L))) -> U61#(isNatList(L),L,N) ACTIVE(length(cons(N,L))) -> ISNATLIST(L) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(length(nil)) -> MARK(0) ACTIVE(zeros) -> MARK(cons(0,zeros)) CONS(active(X1),X2) -> CONS(X1,X2) CONS(mark(X1),X2) -> CONS(X1,X2) CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) ISNAT(active(X)) -> ISNAT(X) ISNAT(mark(X)) -> ISNAT(X) ISNATILIST(active(X)) -> ISNATILIST(X) ISNATILIST(mark(X)) -> ISNATILIST(X) ISNATLIST(active(X)) -> ISNATLIST(X) ISNATLIST(mark(X)) -> ISNATLIST(X) LENGTH(active(X)) -> LENGTH(X) LENGTH(mark(X)) -> LENGTH(X) MARK(U11(X)) -> U11#(mark(X)) MARK(U11(X)) -> ACTIVE(U11(mark(X))) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> U21#(mark(X)) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> U31#(mark(X)) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> U41#(mark(X1),X2) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> U42#(mark(X)) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> U51#(mark(X1),X2) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> U52#(mark(X)) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> U61#(mark(X1),X2,X3) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> U62#(mark(X1),X2) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> CONS(mark(X1),X2) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(s(X)) -> S(mark(X)) MARK(zeros) -> ACTIVE(zeros) S(active(X)) -> S(X) S(mark(X)) -> S(X) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1: SCC Processor: -> Pairs: U11#(active(X)) -> U11#(X) U11#(mark(X)) -> U11#(X) U21#(active(X)) -> U21#(X) U21#(mark(X)) -> U21#(X) U31#(active(X)) -> U31#(X) U31#(mark(X)) -> U31#(X) U41#(active(X1),X2) -> U41#(X1,X2) U41#(mark(X1),X2) -> U41#(X1,X2) U41#(X1,active(X2)) -> U41#(X1,X2) U41#(X1,mark(X2)) -> U41#(X1,X2) U42#(active(X)) -> U42#(X) U42#(mark(X)) -> U42#(X) U51#(active(X1),X2) -> U51#(X1,X2) U51#(mark(X1),X2) -> U51#(X1,X2) U51#(X1,active(X2)) -> U51#(X1,X2) U51#(X1,mark(X2)) -> U51#(X1,X2) U52#(active(X)) -> U52#(X) U52#(mark(X)) -> U52#(X) U61#(active(X1),X2,X3) -> U61#(X1,X2,X3) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) U62#(active(X1),X2) -> U62#(X1,X2) U62#(mark(X1),X2) -> U62#(X1,X2) U62#(X1,active(X2)) -> U62#(X1,X2) U62#(X1,mark(X2)) -> U62#(X1,X2) ACTIVE(U11(tt)) -> MARK(tt) ACTIVE(U21(tt)) -> MARK(tt) ACTIVE(U31(tt)) -> MARK(tt) ACTIVE(U41(tt,V2)) -> U42#(isNatIList(V2)) ACTIVE(U41(tt,V2)) -> ISNATILIST(V2) ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U42(tt)) -> MARK(tt) ACTIVE(U51(tt,V2)) -> U52#(isNatList(V2)) ACTIVE(U51(tt,V2)) -> ISNATLIST(V2) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U52(tt)) -> MARK(tt) ACTIVE(U61(tt,L,N)) -> U62#(isNat(N),L) ACTIVE(U61(tt,L,N)) -> ISNAT(N) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> LENGTH(L) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(U62(tt,L)) -> S(length(L)) ACTIVE(isNat(length(V1))) -> U11#(isNatList(V1)) ACTIVE(isNat(length(V1))) -> ISNATLIST(V1) ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1))) ACTIVE(isNat(s(V1))) -> U21#(isNat(V1)) ACTIVE(isNat(s(V1))) -> ISNAT(V1) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNat(0)) -> MARK(tt) ACTIVE(isNatIList(cons(V1,V2))) -> U41#(isNat(V1),V2) ACTIVE(isNatIList(cons(V1,V2))) -> ISNAT(V1) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatIList(zeros)) -> MARK(tt) ACTIVE(isNatIList(V)) -> U31#(isNatList(V)) ACTIVE(isNatIList(V)) -> ISNATLIST(V) ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V))) ACTIVE(isNatList(cons(V1,V2))) -> U51#(isNat(V1),V2) ACTIVE(isNatList(cons(V1,V2))) -> ISNAT(V1) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(isNatList(nil)) -> MARK(tt) ACTIVE(length(cons(N,L))) -> U61#(isNatList(L),L,N) ACTIVE(length(cons(N,L))) -> ISNATLIST(L) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(length(nil)) -> MARK(0) ACTIVE(zeros) -> MARK(cons(0,zeros)) CONS(active(X1),X2) -> CONS(X1,X2) CONS(mark(X1),X2) -> CONS(X1,X2) CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) ISNAT(active(X)) -> ISNAT(X) ISNAT(mark(X)) -> ISNAT(X) ISNATILIST(active(X)) -> ISNATILIST(X) ISNATILIST(mark(X)) -> ISNATILIST(X) ISNATLIST(active(X)) -> ISNATLIST(X) ISNATLIST(mark(X)) -> ISNATLIST(X) LENGTH(active(X)) -> LENGTH(X) LENGTH(mark(X)) -> LENGTH(X) MARK(U11(X)) -> U11#(mark(X)) MARK(U11(X)) -> ACTIVE(U11(mark(X))) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> U21#(mark(X)) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> U31#(mark(X)) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> U41#(mark(X1),X2) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> U42#(mark(X)) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> U51#(mark(X1),X2) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> U52#(mark(X)) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> U61#(mark(X1),X2,X3) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> U62#(mark(X1),X2) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> CONS(mark(X1),X2) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(s(X)) -> S(mark(X)) MARK(zeros) -> ACTIVE(zeros) S(active(X)) -> S(X) S(mark(X)) -> S(X) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: S(active(X)) -> S(X) S(mark(X)) -> S(X) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: LENGTH(active(X)) -> LENGTH(X) LENGTH(mark(X)) -> LENGTH(X) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: ISNATLIST(active(X)) -> ISNATLIST(X) ISNATLIST(mark(X)) -> ISNATLIST(X) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: ISNATILIST(active(X)) -> ISNATILIST(X) ISNATILIST(mark(X)) -> ISNATILIST(X) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: ISNAT(active(X)) -> ISNAT(X) ISNAT(mark(X)) -> ISNAT(X) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: CONS(active(X1),X2) -> CONS(X1,X2) CONS(mark(X1),X2) -> CONS(X1,X2) CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U62#(active(X1),X2) -> U62#(X1,X2) U62#(mark(X1),X2) -> U62#(X1,X2) U62#(X1,active(X2)) -> U62#(X1,X2) U62#(X1,mark(X2)) -> U62#(X1,X2) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U61#(active(X1),X2,X3) -> U61#(X1,X2,X3) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U52#(active(X)) -> U52#(X) U52#(mark(X)) -> U52#(X) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U51#(active(X1),X2) -> U51#(X1,X2) U51#(mark(X1),X2) -> U51#(X1,X2) U51#(X1,active(X2)) -> U51#(X1,X2) U51#(X1,mark(X2)) -> U51#(X1,X2) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U42#(active(X)) -> U42#(X) U42#(mark(X)) -> U42#(X) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U41#(active(X1),X2) -> U41#(X1,X2) U41#(mark(X1),X2) -> U41#(X1,X2) U41#(X1,active(X2)) -> U41#(X1,X2) U41#(X1,mark(X2)) -> U41#(X1,X2) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U31#(active(X)) -> U31#(X) U31#(mark(X)) -> U31#(X) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U21#(active(X)) -> U21#(X) U21#(mark(X)) -> U21#(X) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U11#(active(X)) -> U11#(X) U11#(mark(X)) -> U11#(X) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V))) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U11(X)) -> ACTIVE(U11(mark(X))) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) The problem is decomposed in 16 subproblems. Problem 1.1: Subterm Processor: -> Pairs: S(active(X)) -> S(X) S(mark(X)) -> S(X) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(S) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: LENGTH(active(X)) -> LENGTH(X) LENGTH(mark(X)) -> LENGTH(X) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(LENGTH) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: ISNATLIST(active(X)) -> ISNATLIST(X) ISNATLIST(mark(X)) -> ISNATLIST(X) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(ISNATLIST) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: ISNATILIST(active(X)) -> ISNATILIST(X) ISNATILIST(mark(X)) -> ISNATILIST(X) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(ISNATILIST) = 1 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Subterm Processor: -> Pairs: ISNAT(active(X)) -> ISNAT(X) ISNAT(mark(X)) -> ISNAT(X) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(ISNAT) = 1 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.6: Subterm Processor: -> Pairs: CONS(active(X1),X2) -> CONS(X1,X2) CONS(mark(X1),X2) -> CONS(X1,X2) CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(CONS) = 1 Problem 1.6: SCC Processor: -> Pairs: CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.6: Subterm Processor: -> Pairs: CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(CONS) = 2 Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.7: Subterm Processor: -> Pairs: U62#(active(X1),X2) -> U62#(X1,X2) U62#(mark(X1),X2) -> U62#(X1,X2) U62#(X1,active(X2)) -> U62#(X1,X2) U62#(X1,mark(X2)) -> U62#(X1,X2) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U62#) = 1 Problem 1.7: SCC Processor: -> Pairs: U62#(X1,active(X2)) -> U62#(X1,X2) U62#(X1,mark(X2)) -> U62#(X1,X2) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U62#(X1,active(X2)) -> U62#(X1,X2) U62#(X1,mark(X2)) -> U62#(X1,X2) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.7: Subterm Processor: -> Pairs: U62#(X1,active(X2)) -> U62#(X1,X2) U62#(X1,mark(X2)) -> U62#(X1,X2) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U62#) = 2 Problem 1.7: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.8: Subterm Processor: -> Pairs: U61#(active(X1),X2,X3) -> U61#(X1,X2,X3) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U61#) = 1 Problem 1.8: SCC Processor: -> Pairs: U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.8: Subterm Processor: -> Pairs: U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U61#) = 2 Problem 1.8: SCC Processor: -> Pairs: U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.8: Subterm Processor: -> Pairs: U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U61#) = 3 Problem 1.8: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.9: Subterm Processor: -> Pairs: U52#(active(X)) -> U52#(X) U52#(mark(X)) -> U52#(X) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U52#) = 1 Problem 1.9: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.10: Subterm Processor: -> Pairs: U51#(active(X1),X2) -> U51#(X1,X2) U51#(mark(X1),X2) -> U51#(X1,X2) U51#(X1,active(X2)) -> U51#(X1,X2) U51#(X1,mark(X2)) -> U51#(X1,X2) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U51#) = 1 Problem 1.10: SCC Processor: -> Pairs: U51#(X1,active(X2)) -> U51#(X1,X2) U51#(X1,mark(X2)) -> U51#(X1,X2) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U51#(X1,active(X2)) -> U51#(X1,X2) U51#(X1,mark(X2)) -> U51#(X1,X2) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.10: Subterm Processor: -> Pairs: U51#(X1,active(X2)) -> U51#(X1,X2) U51#(X1,mark(X2)) -> U51#(X1,X2) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U51#) = 2 Problem 1.10: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.11: Subterm Processor: -> Pairs: U42#(active(X)) -> U42#(X) U42#(mark(X)) -> U42#(X) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U42#) = 1 Problem 1.11: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.12: Subterm Processor: -> Pairs: U41#(active(X1),X2) -> U41#(X1,X2) U41#(mark(X1),X2) -> U41#(X1,X2) U41#(X1,active(X2)) -> U41#(X1,X2) U41#(X1,mark(X2)) -> U41#(X1,X2) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U41#) = 1 Problem 1.12: SCC Processor: -> Pairs: U41#(X1,active(X2)) -> U41#(X1,X2) U41#(X1,mark(X2)) -> U41#(X1,X2) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U41#(X1,active(X2)) -> U41#(X1,X2) U41#(X1,mark(X2)) -> U41#(X1,X2) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.12: Subterm Processor: -> Pairs: U41#(X1,active(X2)) -> U41#(X1,X2) U41#(X1,mark(X2)) -> U41#(X1,X2) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U41#) = 2 Problem 1.12: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.13: Subterm Processor: -> Pairs: U31#(active(X)) -> U31#(X) U31#(mark(X)) -> U31#(X) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U31#) = 1 Problem 1.13: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.14: Subterm Processor: -> Pairs: U21#(active(X)) -> U21#(X) U21#(mark(X)) -> U21#(X) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U21#) = 1 Problem 1.14: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.15: Subterm Processor: -> Pairs: U11#(active(X)) -> U11#(X) U11#(mark(X)) -> U11#(X) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U11#) = 1 Problem 1.15: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V))) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U11(X)) -> ACTIVE(U11(mark(X))) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2.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) = 2.X + 2 [length](X) = 2.X + 2 [mark](X) = X [s](X) = X [0] = 0 [nil] = 2 [tt] = 2 [zeros] = 0 [ACTIVE](X) = 2.X + 1 [MARK](X) = 2.X + 1 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V))) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U11(X)) -> ACTIVE(U11(mark(X))) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V))) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U11(X)) -> ACTIVE(U11(mark(X))) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V))) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U11(X)) -> ACTIVE(U11(mark(X))) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2.X [U21](X) = 2.X [U31](X) = 2.X + 1 [U41](X1,X2) = X1 + 2.X2 + 2 [U42](X) = X [U51](X1,X2) = 2.X1 [U52](X) = 2.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) = 0 [isNatIList](X) = 2.X + 2 [isNatList](X) = 0 [length](X) = X [mark](X) = X [s](X) = 2.X [0] = 0 [nil] = 1 [tt] = 0 [zeros] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U11(X)) -> ACTIVE(U11(mark(X))) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U11(X)) -> ACTIVE(U11(mark(X))) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U11(X)) -> ACTIVE(U11(mark(X))) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 1 [U21](X) = 2 [U31](X) = 0 [U41](X1,X2) = 2 [U42](X) = 0 [U51](X1,X2) = 2 [U52](X) = 2 [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 [nil] = 0 [tt] = 0 [zeros] = 2 [ACTIVE](X) = X [MARK](X) = 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2.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 + 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) = 2.X + 2 [length](X) = 2.X + 2 [mark](X) = X [s](X) = X [0] = 0 [nil] = 1 [tt] = 2 [zeros] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 1 [U21](X) = 1 [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 + 2 [cons](X1,X2) = 1 [isNat](X) = 2 [isNatIList](X) = 2 [isNatList](X) = 2 [length](X) = 2 [mark](X) = 2.X + 2 [s](X) = 2 [0] = 0 [nil] = 2 [tt] = 1 [zeros] = 2 [ACTIVE](X) = X [MARK](X) = 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2 [U21](X) = 2 [U31](X) = 1 [U41](X1,X2) = 2 [U42](X) = 1 [U51](X1,X2) = 2 [U52](X) = 2 [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] = 2 [nil] = 0 [tt] = 1 [zeros] = 2 [ACTIVE](X) = X [MARK](X) = 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = X + 2 [U21](X) = X [U31](X) = X + 1 [U41](X1,X2) = 2.X1 + X2 [U42](X) = X [U51](X1,X2) = X1 + 2.X2 [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) = X + 1 [isNatIList](X) = X + 2 [isNatList](X) = X + 1 [length](X) = 2.X + 2 [mark](X) = X [s](X) = X [0] = 0 [nil] = 2 [tt] = 1 [zeros] = 0 [ACTIVE](X) = 2.X + 1 [MARK](X) = 2.X + 1 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2.X [U21](X) = 2.X [U31](X) = X + 2 [U41](X1,X2) = 2.X1 + X2 + 2 [U42](X) = X [U51](X1,X2) = 2.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 + X2 [isNat](X) = 0 [isNatIList](X) = X + 2 [isNatList](X) = 0 [length](X) = 2.X [mark](X) = X [s](X) = X [0] = 0 [nil] = 0 [tt] = 0 [zeros] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2 [U21](X) = 2 [U31](X) = 1 [U41](X1,X2) = 2 [U42](X) = 1 [U51](X1,X2) = 2 [U52](X) = 2 [U61](X1,X2,X3) = 2 [U62](X1,X2) = 2 [active](X) = 2.X + 2 [cons](X1,X2) = 0 [isNat](X) = 2 [isNatIList](X) = 2 [isNatList](X) = 2 [length](X) = 2 [mark](X) = 2.X + 2 [s](X) = 2 [0] = 0 [nil] = 2 [tt] = 1 [zeros] = 2 [ACTIVE](X) = X [MARK](X) = 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 0 [U21](X) = 0 [U31](X) = 1 [U41](X1,X2) = 1 [U42](X) = 1 [U51](X1,X2) = 1 [U52](X) = 0 [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) = 1 [0] = 0 [nil] = 2 [tt] = 0 [zeros] = 1 [ACTIVE](X) = 2.X [MARK](X) = 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2.X + 2 [U21](X) = X [U31](X) = 2.X [U41](X1,X2) = 2.X1 + 2.X2 [U42](X) = X [U51](X1,X2) = X1 + X2 [U52](X) = X [U61](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1 [U62](X1,X2) = X1 + 2.X2 + 1 [active](X) = X [cons](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 2.X + 1 [isNatIList](X) = 2.X + 2 [isNatList](X) = X + 1 [length](X) = 2.X + 2 [mark](X) = X [s](X) = X [0] = 0 [nil] = 1 [tt] = 1 [zeros] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 1 [U21](X) = X [U31](X) = 2 [U41](X1,X2) = X1 + X2 + 1 [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) = 2.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 [nil] = 1 [tt] = 1 [zeros] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt,L,N)) -> MARK(U62(isNat(N),L)) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2.X + 2 [U21](X) = X + 2 [U31](X) = 2.X + 2 [U41](X1,X2) = 2.X2 + 2 [U42](X) = X [U51](X1,X2) = X2 [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.X + 2 [isNatIList](X) = 2.X + 2 [isNatList](X) = X [length](X) = 2.X + 2 [mark](X) = X [s](X) = X + 1 [0] = 0 [nil] = 2 [tt] = 2 [zeros] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U62(tt,L)) -> MARK(s(length(L))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2.X + 1 [U21](X) = X + 2 [U31](X) = 2.X [U41](X1,X2) = 2.X2 + 2 [U42](X) = X [U51](X1,X2) = 2.X2 [U52](X) = 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) = 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 [nil] = 2 [tt] = 2 [zeros] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2.X + 2 [U21](X) = X + 1 [U31](X) = 2.X + 1 [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 + 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 [nil] = 2 [tt] = 2 [zeros] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(length(cons(N,L))) -> MARK(U61(isNatList(L),L,N)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2 [U21](X) = X + 1 [U31](X) = 2 [U41](X1,X2) = 2.X2 + 2 [U42](X) = X [U51](X1,X2) = X2 [U52](X) = 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 [nil] = 2 [tt] = 2 [zeros] = 0 [ACTIVE](X) = 2.X + 1 [MARK](X) = 2.X + 1 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U21(X)) -> MARK(X) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2.X + 2 [U21](X) = X + 2 [U31](X) = 2 [U41](X1,X2) = 2 [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) = 2.X1 + 2.X2 [isNat](X) = X + 2 [isNatIList](X) = 2 [isNatList](X) = X [length](X) = 2.X + 2 [mark](X) = X [s](X) = X + 2 [0] = 0 [nil] = 2 [tt] = 2 [zeros] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 0 [U62](X1,X2) = 0 [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 [nil] = 2 [tt] = 1 [zeros] = 1 [ACTIVE](X) = 2.X [MARK](X) = 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U62(X1,X2)) -> ACTIVE(U62(mark(X1),X2)) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 0 [U21](X) = 1 [U31](X) = 2 [U41](X1,X2) = 2 [U42](X) = 2 [U51](X1,X2) = 2 [U52](X) = 1 [U61](X1,X2,X3) = 2 [U62](X1,X2) = 1 [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] = 1 [nil] = 2 [tt] = 0 [zeros] = 2 [ACTIVE](X) = X [MARK](X) = 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(U62(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2.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) = 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 + 1 [0] = 0 [nil] = 2 [tt] = 2 [zeros] = 0 [ACTIVE](X) = X + 2 [MARK](X) = X + 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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 + 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] = 1 [nil] = 2 [tt] = 1 [zeros] = 1 [ACTIVE](X) = 2.X [MARK](X) = 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 1 [U21](X) = 1 [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 + 2 [cons](X1,X2) = 2 [isNat](X) = 1 [isNatIList](X) = 2 [isNatList](X) = 2 [length](X) = 2 [mark](X) = 2.X + 2 [s](X) = 2 [0] = 2 [nil] = 0 [tt] = 0 [zeros] = 2 [ACTIVE](X) = X [MARK](X) = 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 1 [U21](X) = 2 [U31](X) = 1 [U41](X1,X2) = 1 [U42](X) = 1 [U51](X1,X2) = 1 [U52](X) = 1 [U61](X1,X2,X3) = 0 [U62](X1,X2) = 0 [active](X) = 2.X + 2 [cons](X1,X2) = X2 [isNat](X) = 2 [isNatIList](X) = 1 [isNatList](X) = 1 [length](X) = 0 [mark](X) = 2.X + 2 [s](X) = 0 [0] = 0 [nil] = 2 [tt] = 1 [zeros] = 1 [ACTIVE](X) = 2.X [MARK](X) = 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2 [U21](X) = 2 [U31](X) = X [U41](X1,X2) = X1 + X2 [U42](X) = X [U51](X1,X2) = 2.X2 [U52](X) = 2.X [U61](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 [U62](X1,X2) = X1 + 2.X2 + 2 [active](X) = X [cons](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 2 [isNatIList](X) = X + 2 [isNatList](X) = X [length](X) = 2.X + 2 [mark](X) = X [s](X) = X + 2 [0] = 0 [nil] = 2 [tt] = 2 [zeros] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 0 [U21](X) = X [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) = X [cons](X1,X2) = X1 [isNat](X) = 0 [isNatIList](X) = 0 [isNatList](X) = 0 [length](X) = 0 [mark](X) = 2.X [s](X) = 2.X [0] = 0 [nil] = 1 [tt] = 0 [zeros] = 2 [ACTIVE](X) = X + 2 [MARK](X) = 2.X + 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2.X [U21](X) = 1 [U31](X) = 0 [U41](X1,X2) = 0 [U42](X) = X [U51](X1,X2) = 0 [U52](X) = X [U61](X1,X2,X3) = 2.X1 [U62](X1,X2) = 0 [active](X) = X [cons](X1,X2) = 2.X1 + 1 [isNat](X) = 2 [isNatIList](X) = 0 [isNatList](X) = 0 [length](X) = 0 [mark](X) = 2.X [s](X) = 2.X [0] = 0 [nil] = 2 [tt] = 0 [zeros] = 2 [ACTIVE](X) = 2 [MARK](X) = 2.X + 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 1 [isNat](X) = 1 [isNatIList](X) = 1 [isNatList](X) = 1 [length](X) = 1 [mark](X) = 2 [s](X) = 0 [0] = 0 [nil] = 0 [tt] = 1 [zeros] = 1 [ACTIVE](X) = 2.X [MARK](X) = 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2 [U21](X) = X [U31](X) = 2 [U41](X1,X2) = 2 [U42](X) = X [U51](X1,X2) = 2.X2 [U52](X) = 2.X [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2 [U62](X1,X2) = 2.X1 + 2.X2 + 2 [active](X) = X [cons](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 2 [isNatIList](X) = 2 [isNatList](X) = X [length](X) = 2.X + 2 [mark](X) = X [s](X) = X + 2 [0] = 0 [nil] = 2 [tt] = 2 [zeros] = 0 [ACTIVE](X) = X [MARK](X) = X Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U41(tt,V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 2 ->Bound: 1 ->Interpretation: [U11](X) = [0;1] [U21](X) = [1;1] [U31](X) = [1 1;0 0].X [U41](X1,X2) = [0 0;1 1].X1 + [0 0;1 0].X2 + [0;1] [U42](X) = [0 0;1 1].X + [1;0] [U51](X1,X2) = [0 0;1 1].X1 + [0 0;1 0].X2 + [1;0] [U52](X) = [0 0;1 1].X + [1;0] [U61](X1,X2,X3) = [1 1;0 0].X1 + [0 0;1 0].X2 + [0 0;1 1].X3 + [0;1] [U62](X1,X2) = [1 1;0 0].X1 + [1 0;0 0].X2 + [1;0] [active](X) = [1 1;0 0].X [cons](X1,X2) = [1 1;1 1].X1 + [1 0;0 0].X2 + [1;0] [isNat](X) = [1 1;0 0].X + [1;0] [isNatIList](X) = [0 0;1 0].X + [0;1] [isNatList](X) = [0 0;1 0].X + [1;0] [length](X) = [1 1;1 1].X [mark](X) = [1 1;0 0].X [s](X) = [0;1] [0] = 0 [nil] = [1;1] [tt] = [0;1] [zeros] = [1;1] [ACTIVE](X) = [0 1;1 1].X + [1;1] [MARK](X) = [0 1;1 1].X + [1;1] Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U41(X1,X2)) -> ACTIVE(U41(mark(X1),X2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X) = 2 [U21](X) = 1 [U31](X) = 1 [U41](X1,X2) = 0 [U42](X) = 0 [U51](X1,X2) = 1 [U52](X) = 1 [U61](X1,X2,X3) = 2 [U62](X1,X2) = 2 [active](X) = 2.X + 2 [cons](X1,X2) = 0 [isNat](X) = 2 [isNatIList](X) = 1 [isNatList](X) = 1 [length](X) = 2 [mark](X) = 2.X + 2 [s](X) = 0 [0] = 2 [nil] = 0 [tt] = 0 [zeros] = 0 [ACTIVE](X) = X + 1 [MARK](X) = 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(U51(tt,V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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 1;0 0].X + [0;1] [U41](X1,X2) = [1 0;1 0].X2 + [1;1] [U42](X) = [0 0;1 1].X [U51](X1,X2) = [0 0;1 1].X1 + [0 0;1 0].X2 [U52](X) = [0 0;1 1].X [U61](X1,X2,X3) = [1 1;1 1].X1 + [0 0;1 1].X3 + [1;1] [U62](X1,X2) = [1 1;0 0].X1 + [0;1] [active](X) = [1 1;0 0].X [cons](X1,X2) = [1 1;0 0].X1 + [1 0;0 0].X2 + [1;0] [isNat](X) = [1 0;0 0].X + [1;1] [isNatIList](X) = [1 0;1 0].X + [1;1] [isNatList](X) = [0 0;1 0].X + [0;1] [length](X) = [1 1;1 1].X + [1;1] [mark](X) = [1 1;0 0].X [s](X) = [1;1] [0] = 0 [nil] = [1;1] [tt] = [1;1] [zeros] = [1;1] [ACTIVE](X) = [0 1;0 0].X + [1;1] [MARK](X) = [0 1;0 0].X + [1;1] Problem 1.16: SCC Processor: -> Pairs: ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 0 [U52](X) = 2 [U61](X1,X2,X3) = 2 [U62](X1,X2) = 2 [active](X) = 2 [cons](X1,X2) = 0 [isNat](X) = 2 [isNatIList](X) = 1 [isNatList](X) = 1 [length](X) = 1 [mark](X) = 2 [s](X) = 2 [0] = 2 [nil] = 0 [tt] = 0 [zeros] = 2 [ACTIVE](X) = X + 1 [MARK](X) = 2 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U42(X)) -> MARK(X) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U42(X)) -> MARK(X) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(isNatIList(cons(V1,V2))) -> MARK(U41(isNat(V1),V2)) ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U42(X)) -> MARK(X) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U41](X1,X2) = 2.X1 + 2.X2 [U42](X) = 2.X [U51](X1,X2) = 2.X1 + 2.X2 [U52](X) = X [active](X) = 2.X [cons](X1,X2) = 2.X1 + 2.X2 + 2 [isNat](X) = 1 [isNatIList](X) = X + 1 [isNatList](X) = 2.X + 1 [mark](X) = X [ACTIVE](X) = 2.X [MARK](X) = 2.X + 1 Problem 1.16: SCC Processor: -> Pairs: ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U42(X)) -> MARK(X) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U42(X)) -> MARK(X) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Reduction Pair Processor: -> Pairs: ACTIVE(isNatList(cons(V1,V2))) -> MARK(U51(isNat(V1),V2)) MARK(U42(X)) -> MARK(X) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U42](X) = X [U51](X1,X2) = 2.X2 [U52](X) = 2.X [active](X) = 2.X [cons](X1,X2) = 2.X1 + 2.X2 + 2 [isNat](X) = 2.X + 2 [isNatIList](X) = 2.X [isNatList](X) = X + 2 [mark](X) = 2.X [ACTIVE](X) = 2.X [MARK](X) = 2.X + 2 Problem 1.16: SCC Processor: -> Pairs: MARK(U42(X)) -> MARK(X) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(U42(X)) -> MARK(X) MARK(U52(X)) -> MARK(X) ->->-> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Subterm Processor: -> Pairs: MARK(U42(X)) -> MARK(X) MARK(U52(X)) -> MARK(X) -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(MARK) = 1 Problem 1.16: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(active(X1),X2) -> U41(X1,X2) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2) -> U62(X1,X2) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt,V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt,V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt,L,N)) -> mark(U62(isNat(N),L)) active(U62(tt,L)) -> mark(s(length(L))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(0)) -> mark(tt) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(isNatList(nil)) -> mark(tt) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(length(nil)) -> mark(0) active(zeros) -> mark(cons(0,zeros)) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(U11(X)) -> active(U11(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(nil) -> active(nil) mark(tt) -> active(tt) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite.