/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR M N V1 V2 X X1 X2 X3) (RULES U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ) Problem 1: Dependency Pairs Processor: -> Pairs: U11#(active(X1),X2,X3) -> U11#(X1,X2,X3) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) U12#(active(X1),X2) -> U12#(X1,X2) U12#(mark(X1),X2) -> U12#(X1,X2) U12#(X1,active(X2)) -> U12#(X1,X2) U12#(X1,mark(X2)) -> U12#(X1,X2) U13#(active(X)) -> U13#(X) U13#(mark(X)) -> U13#(X) U21#(active(X1),X2) -> U21#(X1,X2) U21#(mark(X1),X2) -> U21#(X1,X2) U21#(X1,active(X2)) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) U22#(active(X)) -> U22#(X) U22#(mark(X)) -> U22#(X) U31#(active(X1),X2) -> U31#(X1,X2) U31#(mark(X1),X2) -> U31#(X1,X2) U31#(X1,active(X2)) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) U41#(active(X1),X2,X3) -> U41#(X1,X2,X3) U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) U41#(X1,active(X2),X3) -> U41#(X1,X2,X3) U41#(X1,mark(X2),X3) -> U41#(X1,X2,X3) U41#(X1,X2,active(X3)) -> U41#(X1,X2,X3) U41#(X1,X2,mark(X3)) -> U41#(X1,X2,X3) ACTIVE(U11(tt,V1,V2)) -> U12#(isNat(V1),V2) ACTIVE(U11(tt,V1,V2)) -> ISNAT(V1) ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> U13#(isNat(V2)) ACTIVE(U12(tt,V2)) -> ISNAT(V2) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U13(tt)) -> MARK(tt) ACTIVE(U21(tt,V1)) -> U22#(isNat(V1)) ACTIVE(U21(tt,V1)) -> ISNAT(V1) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(U22(tt)) -> MARK(tt) ACTIVE(U31(tt,N)) -> MARK(N) ACTIVE(U41(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(U41(tt,M,N)) -> PLUS(N,M) ACTIVE(U41(tt,M,N)) -> S(plus(N,M)) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2) ACTIVE(isNat(plus(V1,V2))) -> AND(isNatKind(V1),isNatKind(V2)) ACTIVE(isNat(plus(V1,V2))) -> ISNATKIND(V1) ACTIVE(isNat(plus(V1,V2))) -> ISNATKIND(V2) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> U21#(isNatKind(V1),V1) ACTIVE(isNat(s(V1))) -> ISNATKIND(V1) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNat(0)) -> MARK(tt) ACTIVE(isNatKind(plus(V1,V2))) -> AND(isNatKind(V1),isNatKind(V2)) ACTIVE(isNatKind(plus(V1,V2))) -> ISNATKIND(V1) ACTIVE(isNatKind(plus(V1,V2))) -> ISNATKIND(V2) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> ISNATKIND(V1) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) ACTIVE(isNatKind(0)) -> MARK(tt) ACTIVE(plus(N,s(M))) -> U41#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) ACTIVE(plus(N,s(M))) -> AND(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) ACTIVE(plus(N,s(M))) -> AND(isNat(M),isNatKind(M)) ACTIVE(plus(N,s(M))) -> AND(isNat(N),isNatKind(N)) ACTIVE(plus(N,s(M))) -> ISNAT(M) ACTIVE(plus(N,s(M))) -> ISNAT(N) ACTIVE(plus(N,s(M))) -> ISNATKIND(M) ACTIVE(plus(N,s(M))) -> ISNATKIND(N) ACTIVE(plus(N,s(M))) -> MARK(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) ACTIVE(plus(N,0)) -> U31#(and(isNat(N),isNatKind(N)),N) ACTIVE(plus(N,0)) -> AND(isNat(N),isNatKind(N)) ACTIVE(plus(N,0)) -> ISNAT(N) ACTIVE(plus(N,0)) -> ISNATKIND(N) ACTIVE(plus(N,0)) -> MARK(U31(and(isNat(N),isNatKind(N)),N)) AND(active(X1),X2) -> AND(X1,X2) AND(mark(X1),X2) -> AND(X1,X2) AND(X1,active(X2)) -> AND(X1,X2) AND(X1,mark(X2)) -> AND(X1,X2) ISNAT(active(X)) -> ISNAT(X) ISNAT(mark(X)) -> ISNAT(X) ISNATKIND(active(X)) -> ISNATKIND(X) ISNATKIND(mark(X)) -> ISNATKIND(X) MARK(U11(X1,X2,X3)) -> U11#(mark(X1),X2,X3) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> U12#(mark(X1),X2) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> U13#(mark(X)) MARK(U13(X)) -> ACTIVE(U13(mark(X))) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> U21#(mark(X1),X2) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> U22#(mark(X)) MARK(U22(X)) -> ACTIVE(U22(mark(X))) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> U31#(mark(X1),X2) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> U41#(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(plus(X1,X2)) -> PLUS(mark(X1),mark(X2)) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(s(X)) -> S(mark(X)) PLUS(active(X1),X2) -> PLUS(X1,X2) PLUS(mark(X1),X2) -> PLUS(X1,X2) PLUS(X1,active(X2)) -> PLUS(X1,X2) PLUS(X1,mark(X2)) -> PLUS(X1,X2) S(active(X)) -> S(X) S(mark(X)) -> S(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1: SCC Processor: -> Pairs: U11#(active(X1),X2,X3) -> U11#(X1,X2,X3) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) U12#(active(X1),X2) -> U12#(X1,X2) U12#(mark(X1),X2) -> U12#(X1,X2) U12#(X1,active(X2)) -> U12#(X1,X2) U12#(X1,mark(X2)) -> U12#(X1,X2) U13#(active(X)) -> U13#(X) U13#(mark(X)) -> U13#(X) U21#(active(X1),X2) -> U21#(X1,X2) U21#(mark(X1),X2) -> U21#(X1,X2) U21#(X1,active(X2)) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) U22#(active(X)) -> U22#(X) U22#(mark(X)) -> U22#(X) U31#(active(X1),X2) -> U31#(X1,X2) U31#(mark(X1),X2) -> U31#(X1,X2) U31#(X1,active(X2)) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) U41#(active(X1),X2,X3) -> U41#(X1,X2,X3) U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) U41#(X1,active(X2),X3) -> U41#(X1,X2,X3) U41#(X1,mark(X2),X3) -> U41#(X1,X2,X3) U41#(X1,X2,active(X3)) -> U41#(X1,X2,X3) U41#(X1,X2,mark(X3)) -> U41#(X1,X2,X3) ACTIVE(U11(tt,V1,V2)) -> U12#(isNat(V1),V2) ACTIVE(U11(tt,V1,V2)) -> ISNAT(V1) ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> U13#(isNat(V2)) ACTIVE(U12(tt,V2)) -> ISNAT(V2) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U13(tt)) -> MARK(tt) ACTIVE(U21(tt,V1)) -> U22#(isNat(V1)) ACTIVE(U21(tt,V1)) -> ISNAT(V1) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(U22(tt)) -> MARK(tt) ACTIVE(U31(tt,N)) -> MARK(N) ACTIVE(U41(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(U41(tt,M,N)) -> PLUS(N,M) ACTIVE(U41(tt,M,N)) -> S(plus(N,M)) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2) ACTIVE(isNat(plus(V1,V2))) -> AND(isNatKind(V1),isNatKind(V2)) ACTIVE(isNat(plus(V1,V2))) -> ISNATKIND(V1) ACTIVE(isNat(plus(V1,V2))) -> ISNATKIND(V2) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> U21#(isNatKind(V1),V1) ACTIVE(isNat(s(V1))) -> ISNATKIND(V1) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNat(0)) -> MARK(tt) ACTIVE(isNatKind(plus(V1,V2))) -> AND(isNatKind(V1),isNatKind(V2)) ACTIVE(isNatKind(plus(V1,V2))) -> ISNATKIND(V1) ACTIVE(isNatKind(plus(V1,V2))) -> ISNATKIND(V2) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> ISNATKIND(V1) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) ACTIVE(isNatKind(0)) -> MARK(tt) ACTIVE(plus(N,s(M))) -> U41#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) ACTIVE(plus(N,s(M))) -> AND(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) ACTIVE(plus(N,s(M))) -> AND(isNat(M),isNatKind(M)) ACTIVE(plus(N,s(M))) -> AND(isNat(N),isNatKind(N)) ACTIVE(plus(N,s(M))) -> ISNAT(M) ACTIVE(plus(N,s(M))) -> ISNAT(N) ACTIVE(plus(N,s(M))) -> ISNATKIND(M) ACTIVE(plus(N,s(M))) -> ISNATKIND(N) ACTIVE(plus(N,s(M))) -> MARK(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) ACTIVE(plus(N,0)) -> U31#(and(isNat(N),isNatKind(N)),N) ACTIVE(plus(N,0)) -> AND(isNat(N),isNatKind(N)) ACTIVE(plus(N,0)) -> ISNAT(N) ACTIVE(plus(N,0)) -> ISNATKIND(N) ACTIVE(plus(N,0)) -> MARK(U31(and(isNat(N),isNatKind(N)),N)) AND(active(X1),X2) -> AND(X1,X2) AND(mark(X1),X2) -> AND(X1,X2) AND(X1,active(X2)) -> AND(X1,X2) AND(X1,mark(X2)) -> AND(X1,X2) ISNAT(active(X)) -> ISNAT(X) ISNAT(mark(X)) -> ISNAT(X) ISNATKIND(active(X)) -> ISNATKIND(X) ISNATKIND(mark(X)) -> ISNATKIND(X) MARK(U11(X1,X2,X3)) -> U11#(mark(X1),X2,X3) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> U12#(mark(X1),X2) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> U13#(mark(X)) MARK(U13(X)) -> ACTIVE(U13(mark(X))) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> U21#(mark(X1),X2) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> U22#(mark(X)) MARK(U22(X)) -> ACTIVE(U22(mark(X))) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> U31#(mark(X1),X2) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> U41#(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(plus(X1,X2)) -> PLUS(mark(X1),mark(X2)) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(s(X)) -> S(mark(X)) PLUS(active(X1),X2) -> PLUS(X1,X2) PLUS(mark(X1),X2) -> PLUS(X1,X2) PLUS(X1,active(X2)) -> PLUS(X1,X2) PLUS(X1,mark(X2)) -> PLUS(X1,X2) S(active(X)) -> S(X) S(mark(X)) -> S(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) 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(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: PLUS(active(X1),X2) -> PLUS(X1,X2) PLUS(mark(X1),X2) -> PLUS(X1,X2) PLUS(X1,active(X2)) -> PLUS(X1,X2) PLUS(X1,mark(X2)) -> PLUS(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: ISNATKIND(active(X)) -> ISNATKIND(X) ISNATKIND(mark(X)) -> ISNATKIND(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) 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(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: AND(active(X1),X2) -> AND(X1,X2) AND(mark(X1),X2) -> AND(X1,X2) AND(X1,active(X2)) -> AND(X1,X2) AND(X1,mark(X2)) -> AND(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U41#(active(X1),X2,X3) -> U41#(X1,X2,X3) U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) U41#(X1,active(X2),X3) -> U41#(X1,X2,X3) U41#(X1,mark(X2),X3) -> U41#(X1,X2,X3) U41#(X1,X2,active(X3)) -> U41#(X1,X2,X3) U41#(X1,X2,mark(X3)) -> U41#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U31#(active(X1),X2) -> U31#(X1,X2) U31#(mark(X1),X2) -> U31#(X1,X2) U31#(X1,active(X2)) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U22#(active(X)) -> U22#(X) U22#(mark(X)) -> U22#(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U21#(active(X1),X2) -> U21#(X1,X2) U21#(mark(X1),X2) -> U21#(X1,X2) U21#(X1,active(X2)) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U13#(active(X)) -> U13#(X) U13#(mark(X)) -> U13#(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U12#(active(X1),X2) -> U12#(X1,X2) U12#(mark(X1),X2) -> U12#(X1,X2) U12#(X1,active(X2)) -> U12#(X1,X2) U12#(X1,mark(X2)) -> U12#(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U11#(active(X1),X2,X3) -> U11#(X1,X2,X3) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(U31(tt,N)) -> MARK(N) ACTIVE(U41(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) ACTIVE(plus(N,s(M))) -> MARK(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) ACTIVE(plus(N,0)) -> MARK(U31(and(isNat(N),isNatKind(N)),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> ACTIVE(U13(mark(X))) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> ACTIVE(U22(mark(X))) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) The problem is decomposed in 13 subproblems. Problem 1.1: Subterm Processor: -> Pairs: S(active(X)) -> S(X) S(mark(X)) -> S(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(S) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) 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: PLUS(active(X1),X2) -> PLUS(X1,X2) PLUS(mark(X1),X2) -> PLUS(X1,X2) PLUS(X1,active(X2)) -> PLUS(X1,X2) PLUS(X1,mark(X2)) -> PLUS(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(PLUS) = 1 Problem 1.2: SCC Processor: -> Pairs: PLUS(X1,active(X2)) -> PLUS(X1,X2) PLUS(X1,mark(X2)) -> PLUS(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: PLUS(X1,active(X2)) -> PLUS(X1,X2) PLUS(X1,mark(X2)) -> PLUS(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.2: Subterm Processor: -> Pairs: PLUS(X1,active(X2)) -> PLUS(X1,X2) PLUS(X1,mark(X2)) -> PLUS(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(PLUS) = 2 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) 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: ISNATKIND(active(X)) -> ISNATKIND(X) ISNATKIND(mark(X)) -> ISNATKIND(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(ISNATKIND) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) 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: ISNAT(active(X)) -> ISNAT(X) ISNAT(mark(X)) -> ISNAT(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(ISNAT) = 1 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) 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: AND(active(X1),X2) -> AND(X1,X2) AND(mark(X1),X2) -> AND(X1,X2) AND(X1,active(X2)) -> AND(X1,X2) AND(X1,mark(X2)) -> AND(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(AND) = 1 Problem 1.5: SCC Processor: -> Pairs: AND(X1,active(X2)) -> AND(X1,X2) AND(X1,mark(X2)) -> AND(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: AND(X1,active(X2)) -> AND(X1,X2) AND(X1,mark(X2)) -> AND(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.5: Subterm Processor: -> Pairs: AND(X1,active(X2)) -> AND(X1,X2) AND(X1,mark(X2)) -> AND(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(AND) = 2 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) 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: U41#(active(X1),X2,X3) -> U41#(X1,X2,X3) U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) U41#(X1,active(X2),X3) -> U41#(X1,X2,X3) U41#(X1,mark(X2),X3) -> U41#(X1,X2,X3) U41#(X1,X2,active(X3)) -> U41#(X1,X2,X3) U41#(X1,X2,mark(X3)) -> U41#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U41#) = 1 Problem 1.6: SCC Processor: -> Pairs: U41#(X1,active(X2),X3) -> U41#(X1,X2,X3) U41#(X1,mark(X2),X3) -> U41#(X1,X2,X3) U41#(X1,X2,active(X3)) -> U41#(X1,X2,X3) U41#(X1,X2,mark(X3)) -> U41#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U41#(X1,active(X2),X3) -> U41#(X1,X2,X3) U41#(X1,mark(X2),X3) -> U41#(X1,X2,X3) U41#(X1,X2,active(X3)) -> U41#(X1,X2,X3) U41#(X1,X2,mark(X3)) -> U41#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.6: Subterm Processor: -> Pairs: U41#(X1,active(X2),X3) -> U41#(X1,X2,X3) U41#(X1,mark(X2),X3) -> U41#(X1,X2,X3) U41#(X1,X2,active(X3)) -> U41#(X1,X2,X3) U41#(X1,X2,mark(X3)) -> U41#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U41#) = 2 Problem 1.6: SCC Processor: -> Pairs: U41#(X1,X2,active(X3)) -> U41#(X1,X2,X3) U41#(X1,X2,mark(X3)) -> U41#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U41#(X1,X2,active(X3)) -> U41#(X1,X2,X3) U41#(X1,X2,mark(X3)) -> U41#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.6: Subterm Processor: -> Pairs: U41#(X1,X2,active(X3)) -> U41#(X1,X2,X3) U41#(X1,X2,mark(X3)) -> U41#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U41#) = 3 Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) 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: U31#(active(X1),X2) -> U31#(X1,X2) U31#(mark(X1),X2) -> U31#(X1,X2) U31#(X1,active(X2)) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U31#) = 1 Problem 1.7: SCC Processor: -> Pairs: U31#(X1,active(X2)) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U31#(X1,active(X2)) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.7: Subterm Processor: -> Pairs: U31#(X1,active(X2)) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U31#) = 2 Problem 1.7: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) 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: U22#(active(X)) -> U22#(X) U22#(mark(X)) -> U22#(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U22#) = 1 Problem 1.8: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) 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: U21#(active(X1),X2) -> U21#(X1,X2) U21#(mark(X1),X2) -> U21#(X1,X2) U21#(X1,active(X2)) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U21#) = 1 Problem 1.9: SCC Processor: -> Pairs: U21#(X1,active(X2)) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U21#(X1,active(X2)) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.9: Subterm Processor: -> Pairs: U21#(X1,active(X2)) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U21#) = 2 Problem 1.9: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) 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: U13#(active(X)) -> U13#(X) U13#(mark(X)) -> U13#(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U13#) = 1 Problem 1.10: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) 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: U12#(active(X1),X2) -> U12#(X1,X2) U12#(mark(X1),X2) -> U12#(X1,X2) U12#(X1,active(X2)) -> U12#(X1,X2) U12#(X1,mark(X2)) -> U12#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U12#) = 1 Problem 1.11: SCC Processor: -> Pairs: U12#(X1,active(X2)) -> U12#(X1,X2) U12#(X1,mark(X2)) -> U12#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U12#(X1,active(X2)) -> U12#(X1,X2) U12#(X1,mark(X2)) -> U12#(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.11: Subterm Processor: -> Pairs: U12#(X1,active(X2)) -> U12#(X1,X2) U12#(X1,mark(X2)) -> U12#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U12#) = 2 Problem 1.11: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) 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: U11#(active(X1),X2,X3) -> U11#(X1,X2,X3) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U11#) = 1 Problem 1.12: SCC Processor: -> Pairs: U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.12: Subterm Processor: -> Pairs: U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U11#) = 2 Problem 1.12: SCC Processor: -> Pairs: U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.12: Subterm Processor: -> Pairs: U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U11#) = 3 Problem 1.12: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) 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: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(U31(tt,N)) -> MARK(N) ACTIVE(U41(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) ACTIVE(plus(N,s(M))) -> MARK(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) ACTIVE(plus(N,0)) -> MARK(U31(and(isNat(N),isNatKind(N)),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> ACTIVE(U13(mark(X))) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> ACTIVE(U22(mark(X))) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 [U12](X1,X2) = 2.X1 [U13](X) = 2.X [U21](X1,X2) = 2.X1 [U22](X) = 2.X [U31](X1,X2) = X1 + 2.X2 + 2 [U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 0 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [tt] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(U41(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) ACTIVE(plus(N,s(M))) -> MARK(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) ACTIVE(plus(N,0)) -> MARK(U31(and(isNat(N),isNatKind(N)),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> ACTIVE(U13(mark(X))) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> ACTIVE(U22(mark(X))) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(U41(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) ACTIVE(plus(N,s(M))) -> MARK(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) ACTIVE(plus(N,0)) -> MARK(U31(and(isNat(N),isNatKind(N)),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> ACTIVE(U13(mark(X))) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> ACTIVE(U22(mark(X))) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(U41(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) ACTIVE(plus(N,s(M))) -> MARK(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) ACTIVE(plus(N,0)) -> MARK(U31(and(isNat(N),isNatKind(N)),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> ACTIVE(U13(mark(X))) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> ACTIVE(U22(mark(X))) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 [U12](X1,X2) = 2.X1 [U13](X) = X [U21](X1,X2) = 2.X1 [U22](X) = X [U31](X1,X2) = 2.X1 + 2.X2 [U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = X1 + X2 [isNat](X) = 0 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 [s](X) = X + 1 [0] = 2 [tt] = 0 [ACTIVE](X) = 2.X + 1 [MARK](X) = 2.X + 1 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) ACTIVE(plus(N,s(M))) -> MARK(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) ACTIVE(plus(N,0)) -> MARK(U31(and(isNat(N),isNatKind(N)),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> ACTIVE(U13(mark(X))) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> ACTIVE(U22(mark(X))) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) ACTIVE(plus(N,s(M))) -> MARK(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) ACTIVE(plus(N,0)) -> MARK(U31(and(isNat(N),isNatKind(N)),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> ACTIVE(U13(mark(X))) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> ACTIVE(U22(mark(X))) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) ACTIVE(plus(N,s(M))) -> MARK(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) ACTIVE(plus(N,0)) -> MARK(U31(and(isNat(N),isNatKind(N)),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> ACTIVE(U13(mark(X))) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> ACTIVE(U22(mark(X))) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 [U12](X1,X2) = 2.X1 [U13](X) = 2.X [U21](X1,X2) = 2.X1 [U22](X) = 2.X [U31](X1,X2) = 2.X1 + 2.X2 + 2 [U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 0 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [tt] = 0 [ACTIVE](X) = 2.X [MARK](X) = 2.X Problem 1.13: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) ACTIVE(plus(N,0)) -> MARK(U31(and(isNat(N),isNatKind(N)),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> ACTIVE(U13(mark(X))) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> ACTIVE(U22(mark(X))) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) ACTIVE(plus(N,0)) -> MARK(U31(and(isNat(N),isNatKind(N)),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> ACTIVE(U13(mark(X))) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> ACTIVE(U22(mark(X))) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) ACTIVE(plus(N,0)) -> MARK(U31(and(isNat(N),isNatKind(N)),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> ACTIVE(U13(mark(X))) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> ACTIVE(U22(mark(X))) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 [U12](X1,X2) = X1 [U13](X) = 2.X [U21](X1,X2) = 2.X1 [U22](X) = 2.X [U31](X1,X2) = 2.X1 + 2.X2 [U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 0 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 2 [tt] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> ACTIVE(U13(mark(X))) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> ACTIVE(U22(mark(X))) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> ACTIVE(U13(mark(X))) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> ACTIVE(U22(mark(X))) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> ACTIVE(U13(mark(X))) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> ACTIVE(U22(mark(X))) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 1 [U12](X1,X2) = 1 [U13](X) = 0 [U21](X1,X2) = 1 [U22](X) = 1 [U31](X1,X2) = 0 [U41](X1,X2,X3) = 1 [active](X) = 2 [and](X1,X2) = 1 [isNat](X) = 1 [isNatKind](X) = 1 [mark](X) = 2 [plus](X1,X2) = 1 [s](X) = 1 [0] = 2 [tt] = 0 [ACTIVE](X) = 2.X [MARK](X) = 2 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> ACTIVE(U22(mark(X))) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> ACTIVE(U22(mark(X))) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> ACTIVE(U22(mark(X))) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 1 [U12](X1,X2) = 1 [U13](X) = 2 [U21](X1,X2) = 1 [U22](X) = 0 [U31](X1,X2) = 1 [U41](X1,X2,X3) = 0 [active](X) = 2 [and](X1,X2) = 1 [isNat](X) = 1 [isNatKind](X) = 1 [mark](X) = 2 [plus](X1,X2) = 1 [s](X) = 1 [0] = 2 [tt] = 2 [ACTIVE](X) = 2.X [MARK](X) = 2 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2 [U12](X1,X2) = 2 [U13](X) = 2 [U21](X1,X2) = 2 [U22](X) = 1 [U31](X1,X2) = 1 [U41](X1,X2,X3) = 2 [active](X) = 0 [and](X1,X2) = 2 [isNat](X) = 2 [isNatKind](X) = 2 [mark](X) = 0 [plus](X1,X2) = 2 [s](X) = 2 [0] = 2 [tt] = 0 [ACTIVE](X) = X [MARK](X) = 2 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 [U12](X1,X2) = 2.X1 [U13](X) = 2.X [U21](X1,X2) = X1 [U22](X) = 2.X [U31](X1,X2) = 2.X1 + 2.X2 + 2 [U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 0 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 [s](X) = X + 2 [0] = 2 [tt] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> ACTIVE(U41(mark(X1),X2,X3)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 [U12](X1,X2) = 2.X1 [U13](X) = X [U21](X1,X2) = 2.X1 [U22](X) = 2.X [U31](X1,X2) = 2.X1 + X2 + 2 [U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = X1 + 2.X2 [isNat](X) = 0 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [tt] = 0 [ACTIVE](X) = X + 1 [MARK](X) = 2.X + 1 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 [U12](X1,X2) = 2.X1 [U13](X) = 2.X [U21](X1,X2) = X1 [U22](X) = 2.X [U31](X1,X2) = 2.X1 + 2.X2 + 2 [U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = 2.X1 + X2 [isNat](X) = 0 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 [s](X) = X + 2 [0] = 2 [tt] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNat(V1),V2)) ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2) = 2.X1 + X2 + 1 [U13](X) = X + 1 [U21](X1,X2) = 2.X1 + X2 [U22](X) = X [U31](X1,X2) = X2 + 2 [U41](X1,X2,X3) = 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = X [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 0 [tt] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(U12(tt,V2)) -> MARK(U13(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2) = X1 + 2.X2 + 2 [U13](X) = X + 1 [U21](X1,X2) = X1 + 2.X2 + 2 [U22](X) = X + 1 [U31](X1,X2) = X2 [U41](X1,X2,X3) = 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 2.X [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [tt] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(U21(tt,V1)) -> MARK(U22(isNat(V1))) ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2) = X1 + 2.X2 + 2 [U13](X) = X + 2 [U21](X1,X2) = 2.X1 + 2.X2 + 2 [U22](X) = X + 1 [U31](X1,X2) = X2 + 1 [U41](X1,X2,X3) = 2.X2 + X3 + 2 [active](X) = X [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 2.X [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [tt] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2) = X1 + 2.X2 + 2 [U13](X) = X + 2 [U21](X1,X2) = X1 + 2.X2 [U22](X) = X [U31](X1,X2) = X2 [U41](X1,X2,X3) = 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 2.X [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 2 [tt] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2) = X1 + 2.X2 + 1 [U13](X) = X [U21](X1,X2) = 2.X1 + 2.X2 + 2 [U22](X) = X + 1 [U31](X1,X2) = 2.X2 + 2 [U41](X1,X2,X3) = 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 2.X + 1 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [tt] = 0 [ACTIVE](X) = 2.X + 1 [MARK](X) = 2.X + 1 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 0 [U12](X1,X2) = 2 [U13](X) = 2 [U21](X1,X2) = 2 [U22](X) = 0 [U31](X1,X2) = 2 [U41](X1,X2,X3) = 0 [active](X) = 2 [and](X1,X2) = 2 [isNat](X) = 2 [isNatKind](X) = 2 [mark](X) = 2 [plus](X1,X2) = 0 [s](X) = 2 [0] = 2 [tt] = 2 [ACTIVE](X) = X [MARK](X) = 2 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2) = X1 + 2.X2 + 2 [U13](X) = X + 2 [U21](X1,X2) = 2.X1 + 2.X2 + 2 [U22](X) = X + 2 [U31](X1,X2) = 2.X2 + 2 [U41](X1,X2,X3) = X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 2.X [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + X2 + 1 [s](X) = X + 1 [0] = 2 [tt] = 0 [ACTIVE](X) = X + 2 [MARK](X) = 2.X + 2 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U12(X1,X2)) -> ACTIVE(U12(mark(X1),X2)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 0 [U12](X1,X2) = 0 [U13](X) = 2 [U21](X1,X2) = 1 [U22](X) = 2 [U31](X1,X2) = 2 [U41](X1,X2,X3) = 2 [active](X) = 2 [and](X1,X2) = 1 [isNat](X) = 1 [isNatKind](X) = 1 [mark](X) = 2 [plus](X1,X2) = 1 [s](X) = 1 [0] = 2 [tt] = 0 [ACTIVE](X) = 2.X [MARK](X) = 2 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2) = X1 + 2.X2 + 1 [U13](X) = X [U21](X1,X2) = 2.X1 + 2.X2 + 1 [U22](X) = X [U31](X1,X2) = 2.X2 + 2 [U41](X1,X2,X3) = 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = 2.X1 + X2 [isNat](X) = 2.X + 1 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [tt] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [U12](X1,X2) = 2.X2 + 2 [U13](X) = X + 1 [U21](X1,X2) = 2.X1 + 2.X2 + 1 [U22](X) = X [U31](X1,X2) = 2.X2 + 2 [U41](X1,X2,X3) = 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 2.X + 1 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 0 [tt] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2) = 2.X2 + 2 [U13](X) = 0 [U21](X1,X2) = 2.X1 + 2.X2 + 2 [U22](X) = X [U31](X1,X2) = 2.X2 + 2 [U41](X1,X2,X3) = 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 2.X + 2 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [tt] = 0 [ACTIVE](X) = X + 2 [MARK](X) = 2.X + 2 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2 [U12](X1,X2) = X1 [U13](X) = 0 [U21](X1,X2) = 2.X1 + 2 [U22](X) = X [U31](X1,X2) = 2.X1 + X2 [U41](X1,X2,X3) = 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = X1 + 2.X2 [isNat](X) = 2 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 [s](X) = X + 2 [0] = 2 [tt] = 0 [ACTIVE](X) = 2.X + 1 [MARK](X) = 2.X + 1 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(and(tt,X)) -> MARK(X) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X2 + X3 + 1 [U12](X1,X2) = X1 + X2 [U13](X) = 2 [U21](X1,X2) = 2.X2 + 1 [U22](X) = X [U31](X1,X2) = X2 + 2 [U41](X1,X2,X3) = X2 + 2.X3 [active](X) = X [and](X1,X2) = 2.X1 + X2 [isNat](X) = 2.X + 1 [isNatKind](X) = 2.X [mark](X) = X [plus](X1,X2) = 2.X1 + X2 [s](X) = X [0] = 2 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(isNatKind(plus(V1,V2))) -> MARK(and(isNatKind(V1),isNatKind(V2))) ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 0 [U12](X1,X2) = 2.X1 [U13](X) = X [U21](X1,X2) = 0 [U22](X) = 2.X [U31](X1,X2) = X2 + 2 [U41](X1,X2,X3) = 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 0 [isNatKind](X) = 2.X + 1 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 2 [tt] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.13: SCC Processor: -> Pairs: ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Reduction Pair Processor: -> Pairs: ACTIVE(isNatKind(s(V1))) -> MARK(isNatKind(V1)) MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 + 2.X2 + 2.X3 [U12](X1,X2) = 2.X2 + 2 [U13](X) = 2 [U21](X1,X2) = 2.X2 + 2 [U22](X) = X [U31](X1,X2) = X2 + 1 [U41](X1,X2,X3) = 2.X2 + 2.X3 + 2 [active](X) = X [and](X1,X2) = X1 + X2 [isNat](X) = 2.X + 2 [isNatKind](X) = 2.X + 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [tt] = 2 [ACTIVE](X) = X + 2 [MARK](X) = X + 2 Problem 1.13: SCC Processor: -> Pairs: MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Subterm Processor: -> Pairs: MARK(U22(X)) -> MARK(X) MARK(and(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(MARK) = 1 Problem 1.13: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2) -> U12(X1,X2) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U13(active(X)) -> U13(X) U13(mark(X)) -> U13(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X)) -> U22(X) U22(mark(X)) -> U22(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNat(V1),V2)) active(U12(tt,V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt,N)) -> mark(N) active(U41(tt,M,N)) -> mark(s(plus(N,M))) active(and(tt,X)) -> mark(X) active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(plus(N,0)) -> mark(U31(and(isNat(N),isNatKind(N)),N)) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(U13(X)) -> active(U13(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite.