MAYBE Problem: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(U11(tt(),V1,V2)) -> isNat#(V1) active#(U11(tt(),V1,V2)) -> U12#(isNat(V1),V2) active#(U12(tt(),V2)) -> isNat#(V2) active#(U12(tt(),V2)) -> U13#(isNat(V2)) active#(U21(tt(),V1)) -> isNat#(V1) active#(U21(tt(),V1)) -> U22#(isNat(V1)) active#(U41(tt(),M,N)) -> plus#(N,M) active#(U41(tt(),M,N)) -> s#(plus(N,M)) active#(isNat(plus(V1,V2))) -> isNatKind#(V2) active#(isNat(plus(V1,V2))) -> isNatKind#(V1) active#(isNat(plus(V1,V2))) -> and#(isNatKind(V1),isNatKind(V2)) active#(isNat(plus(V1,V2))) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2) active#(isNat(s(V1))) -> isNatKind#(V1) active#(isNat(s(V1))) -> U21#(isNatKind(V1),V1) active#(isNatKind(plus(V1,V2))) -> isNatKind#(V2) active#(isNatKind(plus(V1,V2))) -> isNatKind#(V1) active#(isNatKind(plus(V1,V2))) -> and#(isNatKind(V1),isNatKind(V2)) active#(isNatKind(s(V1))) -> isNatKind#(V1) active#(plus(N,0())) -> isNatKind#(N) active#(plus(N,0())) -> isNat#(N) active#(plus(N,0())) -> and#(isNat(N),isNatKind(N)) active#(plus(N,0())) -> U31#(and(isNat(N),isNatKind(N)),N) active#(plus(N,s(M))) -> isNatKind#(N) active#(plus(N,s(M))) -> isNat#(N) active#(plus(N,s(M))) -> and#(isNat(N),isNatKind(N)) active#(plus(N,s(M))) -> isNatKind#(M) active#(plus(N,s(M))) -> isNat#(M) active#(plus(N,s(M))) -> and#(isNat(M),isNatKind(M)) active#(plus(N,s(M))) -> and#(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) active#(plus(N,s(M))) -> U41#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) active#(U11(X1,X2,X3)) -> active#(X1) active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) active#(U12(X1,X2)) -> active#(X1) active#(U12(X1,X2)) -> U12#(active(X1),X2) active#(U13(X)) -> active#(X) active#(U13(X)) -> U13#(active(X)) active#(U21(X1,X2)) -> active#(X1) active#(U21(X1,X2)) -> U21#(active(X1),X2) active#(U22(X)) -> active#(X) active#(U22(X)) -> U22#(active(X)) active#(U31(X1,X2)) -> active#(X1) active#(U31(X1,X2)) -> U31#(active(X1),X2) active#(U41(X1,X2,X3)) -> active#(X1) active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(plus(X1,X2)) -> active#(X1) active#(plus(X1,X2)) -> plus#(active(X1),X2) active#(plus(X1,X2)) -> active#(X2) active#(plus(X1,X2)) -> plus#(X1,active(X2)) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U12#(mark(X1),X2) -> U12#(X1,X2) U13#(mark(X)) -> U13#(X) U21#(mark(X1),X2) -> U21#(X1,X2) U22#(mark(X)) -> U22#(X) U31#(mark(X1),X2) -> U31#(X1,X2) U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) s#(mark(X)) -> s#(X) plus#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(U12(X1,X2)) -> proper#(X2) proper#(U12(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) proper#(isNat(X)) -> proper#(X) proper#(isNat(X)) -> isNat#(proper(X)) proper#(U13(X)) -> proper#(X) proper#(U13(X)) -> U13#(proper(X)) proper#(U21(X1,X2)) -> proper#(X2) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2)) proper#(U22(X)) -> proper#(X) proper#(U22(X)) -> U22#(proper(X)) proper#(U31(X1,X2)) -> proper#(X2) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2)) proper#(U41(X1,X2,X3)) -> proper#(X3) proper#(U41(X1,X2,X3)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X1) proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(plus(X1,X2)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X1) proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(isNatKind(X)) -> proper#(X) proper#(isNatKind(X)) -> isNatKind#(proper(X)) U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) U12#(ok(X1),ok(X2)) -> U12#(X1,X2) isNat#(ok(X)) -> isNat#(X) U13#(ok(X)) -> U13#(X) U21#(ok(X1),ok(X2)) -> U21#(X1,X2) U22#(ok(X)) -> U22#(X) U31#(ok(X1),ok(X2)) -> U31#(X1,X2) U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3) s#(ok(X)) -> s#(X) plus#(ok(X1),ok(X2)) -> plus#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) isNatKind#(ok(X)) -> isNatKind#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(U11(tt(),V1,V2)) -> isNat#(V1) active#(U11(tt(),V1,V2)) -> U12#(isNat(V1),V2) active#(U12(tt(),V2)) -> isNat#(V2) active#(U12(tt(),V2)) -> U13#(isNat(V2)) active#(U21(tt(),V1)) -> isNat#(V1) active#(U21(tt(),V1)) -> U22#(isNat(V1)) active#(U41(tt(),M,N)) -> plus#(N,M) active#(U41(tt(),M,N)) -> s#(plus(N,M)) active#(isNat(plus(V1,V2))) -> isNatKind#(V2) active#(isNat(plus(V1,V2))) -> isNatKind#(V1) active#(isNat(plus(V1,V2))) -> and#(isNatKind(V1),isNatKind(V2)) active#(isNat(plus(V1,V2))) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2) active#(isNat(s(V1))) -> isNatKind#(V1) active#(isNat(s(V1))) -> U21#(isNatKind(V1),V1) active#(isNatKind(plus(V1,V2))) -> isNatKind#(V2) active#(isNatKind(plus(V1,V2))) -> isNatKind#(V1) active#(isNatKind(plus(V1,V2))) -> and#(isNatKind(V1),isNatKind(V2)) active#(isNatKind(s(V1))) -> isNatKind#(V1) active#(plus(N,0())) -> isNatKind#(N) active#(plus(N,0())) -> isNat#(N) active#(plus(N,0())) -> and#(isNat(N),isNatKind(N)) active#(plus(N,0())) -> U31#(and(isNat(N),isNatKind(N)),N) active#(plus(N,s(M))) -> isNatKind#(N) active#(plus(N,s(M))) -> isNat#(N) active#(plus(N,s(M))) -> and#(isNat(N),isNatKind(N)) active#(plus(N,s(M))) -> isNatKind#(M) active#(plus(N,s(M))) -> isNat#(M) active#(plus(N,s(M))) -> and#(isNat(M),isNatKind(M)) active#(plus(N,s(M))) -> and#(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) active#(plus(N,s(M))) -> U41#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) active#(U11(X1,X2,X3)) -> active#(X1) active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) active#(U12(X1,X2)) -> active#(X1) active#(U12(X1,X2)) -> U12#(active(X1),X2) active#(U13(X)) -> active#(X) active#(U13(X)) -> U13#(active(X)) active#(U21(X1,X2)) -> active#(X1) active#(U21(X1,X2)) -> U21#(active(X1),X2) active#(U22(X)) -> active#(X) active#(U22(X)) -> U22#(active(X)) active#(U31(X1,X2)) -> active#(X1) active#(U31(X1,X2)) -> U31#(active(X1),X2) active#(U41(X1,X2,X3)) -> active#(X1) active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(plus(X1,X2)) -> active#(X1) active#(plus(X1,X2)) -> plus#(active(X1),X2) active#(plus(X1,X2)) -> active#(X2) active#(plus(X1,X2)) -> plus#(X1,active(X2)) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U12#(mark(X1),X2) -> U12#(X1,X2) U13#(mark(X)) -> U13#(X) U21#(mark(X1),X2) -> U21#(X1,X2) U22#(mark(X)) -> U22#(X) U31#(mark(X1),X2) -> U31#(X1,X2) U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) s#(mark(X)) -> s#(X) plus#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(U12(X1,X2)) -> proper#(X2) proper#(U12(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) proper#(isNat(X)) -> proper#(X) proper#(isNat(X)) -> isNat#(proper(X)) proper#(U13(X)) -> proper#(X) proper#(U13(X)) -> U13#(proper(X)) proper#(U21(X1,X2)) -> proper#(X2) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2)) proper#(U22(X)) -> proper#(X) proper#(U22(X)) -> U22#(proper(X)) proper#(U31(X1,X2)) -> proper#(X2) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2)) proper#(U41(X1,X2,X3)) -> proper#(X3) proper#(U41(X1,X2,X3)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X1) proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(plus(X1,X2)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X1) proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(isNatKind(X)) -> proper#(X) proper#(isNatKind(X)) -> isNatKind#(proper(X)) U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) U12#(ok(X1),ok(X2)) -> U12#(X1,X2) isNat#(ok(X)) -> isNat#(X) U13#(ok(X)) -> U13#(X) U21#(ok(X1),ok(X2)) -> U21#(X1,X2) U22#(ok(X)) -> U22#(X) U31#(ok(X1),ok(X2)) -> U31#(X1,X2) U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3) s#(ok(X)) -> s#(X) plus#(ok(X1),ok(X2)) -> plus#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) isNatKind#(ok(X)) -> isNatKind#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: ... EDG Processor: DPs: active#(U11(tt(),V1,V2)) -> isNat#(V1) active#(U11(tt(),V1,V2)) -> U12#(isNat(V1),V2) active#(U12(tt(),V2)) -> isNat#(V2) active#(U12(tt(),V2)) -> U13#(isNat(V2)) active#(U21(tt(),V1)) -> isNat#(V1) active#(U21(tt(),V1)) -> U22#(isNat(V1)) active#(U41(tt(),M,N)) -> plus#(N,M) active#(U41(tt(),M,N)) -> s#(plus(N,M)) active#(isNat(plus(V1,V2))) -> isNatKind#(V2) active#(isNat(plus(V1,V2))) -> isNatKind#(V1) active#(isNat(plus(V1,V2))) -> and#(isNatKind(V1),isNatKind(V2)) active#(isNat(plus(V1,V2))) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2) active#(isNat(s(V1))) -> isNatKind#(V1) active#(isNat(s(V1))) -> U21#(isNatKind(V1),V1) active#(isNatKind(plus(V1,V2))) -> isNatKind#(V2) active#(isNatKind(plus(V1,V2))) -> isNatKind#(V1) active#(isNatKind(plus(V1,V2))) -> and#(isNatKind(V1),isNatKind(V2)) active#(isNatKind(s(V1))) -> isNatKind#(V1) active#(plus(N,0())) -> isNatKind#(N) active#(plus(N,0())) -> isNat#(N) active#(plus(N,0())) -> and#(isNat(N),isNatKind(N)) active#(plus(N,0())) -> U31#(and(isNat(N),isNatKind(N)),N) active#(plus(N,s(M))) -> isNatKind#(N) active#(plus(N,s(M))) -> isNat#(N) active#(plus(N,s(M))) -> and#(isNat(N),isNatKind(N)) active#(plus(N,s(M))) -> isNatKind#(M) active#(plus(N,s(M))) -> isNat#(M) active#(plus(N,s(M))) -> and#(isNat(M),isNatKind(M)) active#(plus(N,s(M))) -> and#(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) active#(plus(N,s(M))) -> U41#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) active#(U11(X1,X2,X3)) -> active#(X1) active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) active#(U12(X1,X2)) -> active#(X1) active#(U12(X1,X2)) -> U12#(active(X1),X2) active#(U13(X)) -> active#(X) active#(U13(X)) -> U13#(active(X)) active#(U21(X1,X2)) -> active#(X1) active#(U21(X1,X2)) -> U21#(active(X1),X2) active#(U22(X)) -> active#(X) active#(U22(X)) -> U22#(active(X)) active#(U31(X1,X2)) -> active#(X1) active#(U31(X1,X2)) -> U31#(active(X1),X2) active#(U41(X1,X2,X3)) -> active#(X1) active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(plus(X1,X2)) -> active#(X1) active#(plus(X1,X2)) -> plus#(active(X1),X2) active#(plus(X1,X2)) -> active#(X2) active#(plus(X1,X2)) -> plus#(X1,active(X2)) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U12#(mark(X1),X2) -> U12#(X1,X2) U13#(mark(X)) -> U13#(X) U21#(mark(X1),X2) -> U21#(X1,X2) U22#(mark(X)) -> U22#(X) U31#(mark(X1),X2) -> U31#(X1,X2) U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) s#(mark(X)) -> s#(X) plus#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(U12(X1,X2)) -> proper#(X2) proper#(U12(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) proper#(isNat(X)) -> proper#(X) proper#(isNat(X)) -> isNat#(proper(X)) proper#(U13(X)) -> proper#(X) proper#(U13(X)) -> U13#(proper(X)) proper#(U21(X1,X2)) -> proper#(X2) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2)) proper#(U22(X)) -> proper#(X) proper#(U22(X)) -> U22#(proper(X)) proper#(U31(X1,X2)) -> proper#(X2) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2)) proper#(U41(X1,X2,X3)) -> proper#(X3) proper#(U41(X1,X2,X3)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X1) proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(plus(X1,X2)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X1) proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(isNatKind(X)) -> proper#(X) proper#(isNatKind(X)) -> isNatKind#(proper(X)) U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) U12#(ok(X1),ok(X2)) -> U12#(X1,X2) isNat#(ok(X)) -> isNat#(X) U13#(ok(X)) -> U13#(X) U21#(ok(X1),ok(X2)) -> U21#(X1,X2) U22#(ok(X)) -> U22#(X) U31#(ok(X1),ok(X2)) -> U31#(X1,X2) U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3) s#(ok(X)) -> s#(X) plus#(ok(X1),ok(X2)) -> plus#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) isNatKind#(ok(X)) -> isNatKind#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: ... SCC Processor: #sccs: 15 #rules: 57 #arcs: 1489/12544 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: proper#(isNatKind(X)) -> proper#(X) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X1) proper#(plus(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) proper#(U41(X1,X2,X3)) -> proper#(X1) proper#(U41(X1,X2,X3)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X3) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> proper#(X2) proper#(U22(X)) -> proper#(X) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> proper#(X2) proper#(U13(X)) -> proper#(X) proper#(isNat(X)) -> proper#(X) proper#(U12(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X3) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [isNatKind](x0) = x0 + 1, [0] = 1, [and](x0, x1) = x0 + x1, [s](x0) = x0, [plus](x0, x1) = x0 + x1 + 1, [U41](x0, x1, x2) = x0 + x1 + x2, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0 + x1, [U13](x0) = x0, [mark](x0) = 0, [U12](x0, x1) = x0 + x1, [isNat](x0) = x0, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x0 + x1 + x2 + 1, [tt] = 0 orientation: proper#(isNatKind(X)) = X + 2 >= X + 1 = proper#(X) proper#(and(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(and(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(plus(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(plus(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(s(X)) = X + 1 >= X + 1 = proper#(X) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + 1 = proper#(X1) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X2 + 1 = proper#(X2) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X3 + 1 = proper#(X3) proper#(U31(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(U31(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(U22(X)) = X + 1 >= X + 1 = proper#(X) proper#(U21(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(U21(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(U13(X)) = X + 1 >= X + 1 = proper#(X) proper#(isNat(X)) = X + 1 >= X + 1 = proper#(X) proper#(U12(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(U12(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(U11(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = proper#(X1) proper#(U11(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X2 + 1 = proper#(X2) proper#(U11(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X3 + 1 = proper#(X3) active(U11(tt(),V1,V2)) = V1 + V2 + 2 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = V2 + 1 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 + 1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 0 = mark(tt()) active(U31(tt(),N)) = N + 1 >= 0 = mark(N) active(U41(tt(),M,N)) = M + N + 1 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = X + 1 >= 0 = mark(X) active(isNat(0())) = 2 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + V2 + 2 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = V1 + 1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 3 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = V1 + V2 + 3 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = V1 + 2 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N + 3 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N + 2 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U12(active(X1),X2) active(U13(X)) = X + 1 >= X + 1 = U13(active(X)) active(U21(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(active(X1),X2,X3) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(plus(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = plus(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) U11(mark(X1),X2,X3) = X2 + X3 + 1 >= 0 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = X2 >= 0 = mark(U12(X1,X2)) U13(mark(X)) = 0 >= 0 = mark(U13(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = X2 + X3 >= 0 = mark(U41(X1,X2,X3)) s(mark(X)) = 0 >= 0 = mark(s(X)) plus(mark(X1),X2) = X2 + 1 >= 0 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + 1 >= 0 = mark(plus(X1,X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = X1 + X2 >= X1 + X2 = U12(proper(X1),proper(X2)) proper(isNat(X)) = X >= X = isNat(proper(X)) proper(U13(X)) = X >= X = U13(proper(X)) proper(U21(X1,X2)) = X1 + X2 >= X1 + X2 = U21(proper(X1),proper(X2)) proper(U22(X)) = X >= X = U22(proper(X)) proper(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X >= X = s(proper(X)) proper(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(isNatKind(X)) = X + 1 >= X + 1 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U12(X1,X2)) isNat(ok(X)) = X >= X = ok(isNat(X)) U13(ok(X)) = X >= X = ok(U13(X)) U21(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) U31(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U41(X1,X2,X3)) s(ok(X)) = X >= X = ok(s(X)) plus(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) proper#(U41(X1,X2,X3)) -> proper#(X1) proper#(U41(X1,X2,X3)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X3) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> proper#(X2) proper#(U22(X)) -> proper#(X) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> proper#(X2) proper#(U13(X)) -> proper#(X) proper#(isNat(X)) -> proper#(X) proper#(U12(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> proper#(X2) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = x0 + x1 + 1, [s](x0) = x0, [plus](x0, x1) = 0, [U41](x0, x1, x2) = x0 + x1 + x2, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0 + x1, [U13](x0) = x0, [mark](x0) = 0, [U12](x0, x1) = x0 + x1 + 1, [isNat](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: proper#(and(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(and(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(s(X)) = X + 1 >= X + 1 = proper#(X) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + 1 = proper#(X1) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X2 + 1 = proper#(X2) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X3 + 1 = proper#(X3) proper#(U31(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(U31(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(U22(X)) = X + 1 >= X + 1 = proper#(X) proper#(U21(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(U21(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(U13(X)) = X + 1 >= X + 1 = proper#(X) proper#(isNat(X)) = X + 1 >= X + 1 = proper#(X) proper#(U12(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(U12(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = V2 + 1 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = N >= 0 = mark(N) active(U41(tt(),M,N)) = M + N >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = X + 1 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = 0 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = 0 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = 0 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = 0 >= 0 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U12(active(X1),X2) active(U13(X)) = X >= X = U13(active(X)) active(U21(X1,X2)) = X1 + X2 >= X1 + X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(active(X1),X2,X3) active(s(X)) = X >= X = s(active(X)) active(plus(X1,X2)) = 0 >= 0 = plus(active(X1),X2) active(plus(X1,X2)) = 0 >= 0 = plus(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) U11(mark(X1),X2,X3) = 0 >= 0 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = X2 + 1 >= 0 = mark(U12(X1,X2)) U13(mark(X)) = 0 >= 0 = mark(U13(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = X2 + X3 >= 0 = mark(U41(X1,X2,X3)) s(mark(X)) = 0 >= 0 = mark(s(X)) plus(mark(X1),X2) = 0 >= 0 = mark(plus(X1,X2)) plus(X1,mark(X2)) = 0 >= 0 = mark(plus(X1,X2)) and(mark(X1),X2) = X2 + 1 >= 0 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = 0 >= 0 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U12(proper(X1),proper(X2)) proper(isNat(X)) = X >= X = isNat(proper(X)) proper(U13(X)) = X >= X = U13(proper(X)) proper(U21(X1,X2)) = X1 + X2 >= X1 + X2 = U21(proper(X1),proper(X2)) proper(U22(X)) = X >= X = U22(proper(X)) proper(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X >= X = s(proper(X)) proper(plus(X1,X2)) = 0 >= 0 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U12(X1,X2)) isNat(ok(X)) = X >= X = ok(isNat(X)) U13(ok(X)) = X >= X = ok(U13(X)) U21(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) U31(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U41(X1,X2,X3)) s(ok(X)) = X >= X = ok(s(X)) plus(ok(X1),ok(X2)) = 0 >= 0 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(and(X1,X2)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(s(X)) -> proper#(X) proper#(U41(X1,X2,X3)) -> proper#(X1) proper#(U41(X1,X2,X3)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X3) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> proper#(X2) proper#(U22(X)) -> proper#(X) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> proper#(X2) proper#(U13(X)) -> proper#(X) proper#(isNat(X)) -> proper#(X) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = 0, [s](x0) = x0 + 1, [plus](x0, x1) = x0, [U41](x0, x1, x2) = x0 + x1 + x2 + 1, [U31](x0, x1) = x0 + x1 + 1, [U22](x0) = x0 + 1, [U21](x0, x1) = x0 + x1 + 1, [U13](x0) = x0 + 1, [mark](x0) = 0, [U12](x0, x1) = 0, [isNat](x0) = x0 + 1, [active](x0) = x0, [U11](x0, x1, x2) = 0, [tt] = 1 orientation: proper#(s(X)) = X + 2 >= X + 1 = proper#(X) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = proper#(X1) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X2 + 1 = proper#(X2) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X3 + 1 = proper#(X3) proper#(U31(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(U31(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(U22(X)) = X + 2 >= X + 1 = proper#(X) proper#(U21(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(U21(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(U13(X)) = X + 2 >= X + 1 = proper#(X) proper#(isNat(X)) = X + 2 >= X + 1 = proper#(X) active(U11(tt(),V1,V2)) = 0 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 2 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 + 2 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 2 >= 0 = mark(tt()) active(U31(tt(),N)) = N + 2 >= 0 = mark(N) active(U41(tt(),M,N)) = M + N + 2 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + 1 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = V1 + 2 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = N >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = 0 >= 0 = U11(active(X1),X2,X3) active(U12(X1,X2)) = 0 >= 0 = U12(active(X1),X2) active(U13(X)) = X + 1 >= X + 1 = U13(active(X)) active(U21(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(active(X1),X2,X3) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(plus(X1,X2)) = X1 >= X1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 >= X1 = plus(X1,active(X2)) active(and(X1,X2)) = 0 >= 0 = and(active(X1),X2) U11(mark(X1),X2,X3) = 0 >= 0 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = 0 >= 0 = mark(U12(X1,X2)) U13(mark(X)) = 1 >= 0 = mark(U13(X)) U21(mark(X1),X2) = X2 + 1 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 1 >= 0 = mark(U22(X)) U31(mark(X1),X2) = X2 + 1 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = X2 + X3 + 1 >= 0 = mark(U41(X1,X2,X3)) s(mark(X)) = 1 >= 0 = mark(s(X)) plus(mark(X1),X2) = 0 >= 0 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 >= 0 = mark(plus(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = 0 >= 0 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2)) = 0 >= 0 = U12(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U13(X)) = X + 1 >= X + 1 = U13(proper(X)) proper(U21(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X + 1 >= X + 1 = U22(proper(X)) proper(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(plus(X1,X2)) = X1 >= X1 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = 0 >= 0 = ok(U12(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U13(ok(X)) = X + 1 >= X + 1 = ok(U13(X)) U21(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) U31(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = ok(U41(X1,X2,X3)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) plus(ok(X1),ok(X2)) = X1 >= X1 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: active#(and(X1,X2)) -> active#(X1) active#(plus(X1,X2)) -> active#(X2) active#(plus(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) active#(U41(X1,X2,X3)) -> active#(X1) active#(U31(X1,X2)) -> active#(X1) active#(U22(X)) -> active#(X) active#(U21(X1,X2)) -> active#(X1) active#(U13(X)) -> active#(X) active#(U12(X1,X2)) -> active#(X1) active#(U11(X1,X2,X3)) -> active#(X1) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [isNatKind](x0) = 1, [0] = 0, [and](x0, x1) = x0 + 1, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + x1 + 1, [U41](x0, x1, x2) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = 0, [U12](x0, x1) = x0, [isNat](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = x0 + x2, [tt] = 0 orientation: active#(and(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(plus(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = active#(X2) active#(plus(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = active#(X1) active#(s(X)) = X + 2 >= X + 1 = active#(X) active#(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = active#(X1) active#(U31(X1,X2)) = X1 + 1 >= X1 + 1 = active#(X1) active#(U22(X)) = X + 1 >= X + 1 = active#(X) active#(U21(X1,X2)) = X1 + 1 >= X1 + 1 = active#(X1) active#(U13(X)) = X + 1 >= X + 1 = active#(X) active#(U12(X1,X2)) = X1 + 1 >= X1 + 1 = active#(X1) active#(U11(X1,X2,X3)) = X1 + X3 + 1 >= X1 + 1 = active#(X1) active(U11(tt(),V1,V2)) = V2 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = 0 >= 0 = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = 1 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + V2 + 1 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = V1 + 1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = 1 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 1 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N + 2 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X1 + X3 >= X1 + X3 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 >= X1 = U12(active(X1),X2) active(U13(X)) = X >= X = U13(active(X)) active(U21(X1,X2)) = X1 >= X1 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2)) active(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(active(X1),X2) U11(mark(X1),X2,X3) = X3 >= 0 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = 0 >= 0 = mark(U12(X1,X2)) U13(mark(X)) = 0 >= 0 = mark(U13(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) s(mark(X)) = 1 >= 0 = mark(s(X)) plus(mark(X1),X2) = X2 + 1 >= 0 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + 1 >= 0 = mark(plus(X1,X2)) and(mark(X1),X2) = 1 >= 0 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = X1 + X3 >= X1 + X3 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = X1 >= X1 = U12(proper(X1),proper(X2)) proper(isNat(X)) = X >= X = isNat(proper(X)) proper(U13(X)) = X >= X = U13(proper(X)) proper(U21(X1,X2)) = X1 >= X1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X >= X = U22(proper(X)) proper(U31(X1,X2)) = X1 >= X1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = X1 >= X1 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(isNatKind(X)) = 1 >= 1 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = 0 >= 0 = ok(U12(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U13(ok(X)) = 0 >= 0 = ok(U13(X)) U21(ok(X1),ok(X2)) = 0 >= 0 = ok(U21(X1,X2)) U22(ok(X)) = 0 >= 0 = ok(U22(X)) U31(ok(X1),ok(X2)) = 0 >= 0 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U41(X1,X2,X3)) s(ok(X)) = 1 >= 0 = ok(s(X)) plus(ok(X1),ok(X2)) = 1 >= 0 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = 1 >= 0 = ok(and(X1,X2)) isNatKind(ok(X)) = 1 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: active#(U41(X1,X2,X3)) -> active#(X1) active#(U31(X1,X2)) -> active#(X1) active#(U22(X)) -> active#(X) active#(U21(X1,X2)) -> active#(X1) active#(U13(X)) -> active#(X) active#(U12(X1,X2)) -> active#(X1) active#(U11(X1,X2,X3)) -> active#(X1) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [isNatKind](x0) = 1, [0] = 0, [and](x0, x1) = 1, [s](x0) = 1, [plus](x0, x1) = 1, [U41](x0, x1, x2) = x0 + 1, [U31](x0, x1) = x0 + 1, [U22](x0) = x0 + 1, [U21](x0, x1) = x0 + 1, [U13](x0) = x0 + 1, [mark](x0) = 1, [U12](x0, x1) = x0 + 1, [isNat](x0) = 1, [active](x0) = x0, [U11](x0, x1, x2) = x0 + 1, [tt] = 0 orientation: active#(U41(X1,X2,X3)) = X1 + 2 >= X1 + 1 = active#(X1) active#(U31(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(U22(X)) = X + 2 >= X + 1 = active#(X) active#(U21(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(U13(X)) = X + 2 >= X + 1 = active#(X) active#(U12(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(U11(X1,X2,X3)) = X1 + 2 >= X1 + 1 = active#(X1) active(U11(tt(),V1,V2)) = 1 >= 1 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 1 >= 1 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),N)) = 1 >= 1 = mark(N) active(U41(tt(),M,N)) = 1 >= 1 = mark(s(plus(N,M))) active(and(tt(),X)) = 1 >= 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(plus(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(plus(N,0())) = 1 >= 1 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = 1 >= 1 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 + 1 >= X1 + 1 = U12(active(X1),X2) active(U13(X)) = X + 1 >= X + 1 = U13(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3) active(s(X)) = 1 >= 1 = s(active(X)) active(plus(X1,X2)) = 1 >= 1 = plus(active(X1),X2) active(plus(X1,X2)) = 1 >= 1 = plus(X1,active(X2)) active(and(X1,X2)) = 1 >= 1 = and(active(X1),X2) U11(mark(X1),X2,X3) = 2 >= 1 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = 2 >= 1 = mark(U12(X1,X2)) U13(mark(X)) = 2 >= 1 = mark(U13(X)) U21(mark(X1),X2) = 2 >= 1 = mark(U21(X1,X2)) U22(mark(X)) = 2 >= 1 = mark(U22(X)) U31(mark(X1),X2) = 2 >= 1 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = 2 >= 1 = mark(U41(X1,X2,X3)) s(mark(X)) = 1 >= 1 = mark(s(X)) plus(mark(X1),X2) = 1 >= 1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = 1 >= 1 = mark(plus(X1,X2)) and(mark(X1),X2) = 1 >= 1 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = X1 + 1 >= X1 + 1 = U12(proper(X1),proper(X2)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(U13(X)) = X + 1 >= X + 1 = U13(proper(X)) proper(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X + 1 >= X + 1 = U22(proper(X)) proper(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(plus(X1,X2)) = 1 >= 1 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(isNatKind(X)) = 1 >= 1 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U12(X1,X2)) isNat(ok(X)) = 1 >= 1 = ok(isNat(X)) U13(ok(X)) = X + 1 >= X + 1 = ok(U13(X)) U21(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) s(ok(X)) = 1 >= 1 = ok(s(X)) plus(ok(X1),ok(X2)) = 1 >= 1 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = 1 >= 1 = ok(and(X1,X2)) isNatKind(ok(X)) = 1 >= 1 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: isNatKind#(ok(X)) -> isNatKind#(X) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [isNatKind#](x0) = x0, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [isNatKind](x0) = x0, [0] = 0, [and](x0, x1) = x0, [s](x0) = x0, [plus](x0, x1) = x0 + 1, [U41](x0, x1, x2) = x1, [U31](x0, x1) = x0, [U22](x0) = x0, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = 0, [U12](x0, x1) = x0 + 1, [isNat](x0) = x0, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x2, [tt] = 0 orientation: isNatKind#(ok(X)) = X + 1 >= X = isNatKind#(X) active(U11(tt(),V1,V2)) = V2 + 1 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 2 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 0 = mark(tt()) active(U21(tt(),V1)) = 1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 0 = mark(tt()) active(U31(tt(),N)) = 1 >= 0 = mark(N) active(U41(tt(),M,N)) = M + 1 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = 1 >= 0 = mark(X) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + 2 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = V1 + 1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = V1 + 2 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = V1 + 1 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N + 2 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = N + 2 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X3 + 1 >= X3 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 + 2 >= X1 + 2 = U12(active(X1),X2) active(U13(X)) = X + 1 >= X + 1 = U13(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X2 + 1 >= X2 = U41(active(X1),X2,X3) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(plus(X1,X2)) = X1 + 2 >= X1 + 2 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + 2 >= X1 + 1 = plus(X1,active(X2)) active(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(active(X1),X2) U11(mark(X1),X2,X3) = X3 >= 0 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = 1 >= 0 = mark(U12(X1,X2)) U13(mark(X)) = 0 >= 0 = mark(U13(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = X2 >= 0 = mark(U41(X1,X2,X3)) s(mark(X)) = 0 >= 0 = mark(s(X)) plus(mark(X1),X2) = 1 >= 0 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + 1 >= 0 = mark(plus(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2)) = X1 + 2 >= X1 + 2 = U12(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U13(X)) = X + 1 >= X + 1 = U13(proper(X)) proper(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X + 1 >= X + 1 = U22(proper(X)) proper(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = X2 + 1 >= X2 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(plus(X1,X2)) = X1 + 2 >= X1 + 2 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(isNatKind(X)) = X + 1 >= X + 1 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(U12(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U13(ok(X)) = X + 1 >= X + 1 = ok(U13(X)) U21(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = X2 + 1 >= X2 + 1 = ok(U41(X1,X2,X3)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) plus(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: isNat#(ok(X)) -> isNat#(X) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [isNat#](x0) = x0, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [isNatKind](x0) = x0, [0] = 0, [and](x0, x1) = x0, [s](x0) = x0, [plus](x0, x1) = x0 + 1, [U41](x0, x1, x2) = x1, [U31](x0, x1) = x0, [U22](x0) = x0, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = 0, [U12](x0, x1) = x0 + 1, [isNat](x0) = x0, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x2, [tt] = 0 orientation: isNat#(ok(X)) = X + 1 >= X = isNat#(X) active(U11(tt(),V1,V2)) = V2 + 1 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 2 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 0 = mark(tt()) active(U21(tt(),V1)) = 1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 0 = mark(tt()) active(U31(tt(),N)) = 1 >= 0 = mark(N) active(U41(tt(),M,N)) = M + 1 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = 1 >= 0 = mark(X) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + 2 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = V1 + 1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = V1 + 2 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = V1 + 1 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N + 2 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = N + 2 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X3 + 1 >= X3 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 + 2 >= X1 + 2 = U12(active(X1),X2) active(U13(X)) = X + 1 >= X + 1 = U13(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X2 + 1 >= X2 = U41(active(X1),X2,X3) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(plus(X1,X2)) = X1 + 2 >= X1 + 2 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + 2 >= X1 + 1 = plus(X1,active(X2)) active(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(active(X1),X2) U11(mark(X1),X2,X3) = X3 >= 0 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = 1 >= 0 = mark(U12(X1,X2)) U13(mark(X)) = 0 >= 0 = mark(U13(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = X2 >= 0 = mark(U41(X1,X2,X3)) s(mark(X)) = 0 >= 0 = mark(s(X)) plus(mark(X1),X2) = 1 >= 0 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + 1 >= 0 = mark(plus(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2)) = X1 + 2 >= X1 + 2 = U12(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U13(X)) = X + 1 >= X + 1 = U13(proper(X)) proper(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X + 1 >= X + 1 = U22(proper(X)) proper(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = X2 + 1 >= X2 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(plus(X1,X2)) = X1 + 2 >= X1 + 2 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(isNatKind(X)) = X + 1 >= X + 1 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(U12(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U13(ok(X)) = X + 1 >= X + 1 = ok(U13(X)) U21(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = X2 + 1 >= X2 + 1 = ok(U41(X1,X2,X3)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) plus(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U11#](x0, x1, x2) = x2 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [isNatKind](x0) = x0, [0] = 0, [and](x0, x1) = x0, [s](x0) = x0, [plus](x0, x1) = x0 + 1, [U41](x0, x1, x2) = x0, [U31](x0, x1) = x1, [U22](x0) = x0, [U21](x0, x1) = x1, [U13](x0) = x0, [mark](x0) = 0, [U12](x0, x1) = x1 + 1, [isNat](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = x2, [tt] = 1 orientation: U11#(ok(X1),ok(X2),ok(X3)) = X3 + 2 >= X3 + 1 = U11#(X1,X2,X3) U11#(mark(X1),X2,X3) = X3 + 1 >= X3 + 1 = U11#(X1,X2,X3) active(U11(tt(),V1,V2)) = V2 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = V2 + 1 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 0 = mark(tt()) active(U31(tt(),N)) = N >= 0 = mark(N) active(U41(tt(),M,N)) = 1 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = 1 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + 1 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = V1 + 1 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = V1 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = N + 1 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X3 >= X3 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X2 + 1 >= X2 + 1 = U12(active(X1),X2) active(U13(X)) = X >= X = U13(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X2 >= X2 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3) active(s(X)) = X >= X = s(active(X)) active(plus(X1,X2)) = X1 + 1 >= X1 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + 1 >= X1 + 1 = plus(X1,active(X2)) active(and(X1,X2)) = X1 >= X1 = and(active(X1),X2) U11(mark(X1),X2,X3) = X3 >= 0 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = X2 + 1 >= 0 = mark(U12(X1,X2)) U13(mark(X)) = 0 >= 0 = mark(U13(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) s(mark(X)) = 0 >= 0 = mark(s(X)) plus(mark(X1),X2) = 1 >= 0 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + 1 >= 0 = mark(plus(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 2 >= 2 = ok(tt()) proper(U12(X1,X2)) = X2 + 2 >= X2 + 2 = U12(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U13(X)) = X + 1 >= X + 1 = U13(proper(X)) proper(U21(X1,X2)) = X2 + 1 >= X2 + 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X + 1 >= X + 1 = U22(proper(X)) proper(U31(X1,X2)) = X2 + 1 >= X2 + 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(plus(X1,X2)) = X1 + 2 >= X1 + 2 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(isNatKind(X)) = X + 1 >= X + 1 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(U12(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U13(ok(X)) = X + 1 >= X + 1 = ok(U13(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) U31(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) plus(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U11#](x0, x1, x2) = x0, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [plus](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x0 + x1 + x2, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = x0 + 1, [U12](x0, x1) = x0, [isNat](x0) = 0, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x0, [tt] = 0 orientation: U11#(mark(X1),X2,X3) = X1 + 1 >= X1 = U11#(X1,X2,X3) active(U11(tt(),V1,V2)) = 1 >= 1 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 1 >= 1 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),N)) = N + 1 >= N + 1 = mark(N) active(U41(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(plus(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N + 1 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 + 1 >= X1 + 1 = U12(active(X1),X2) active(U13(X)) = X + 1 >= X + 1 = U13(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(active(X1),X2,X3) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) U11(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U12(X1,X2)) U13(mark(X)) = X + 1 >= X + 1 = mark(U13(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U41(X1,X2,X3)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = 0 >= 0 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = 0 >= 0 = U12(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U13(X)) = 0 >= 0 = U13(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(plus(X1,X2)) = 0 >= 0 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = 0 >= 0 = ok(U12(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U13(ok(X)) = 0 >= 0 = ok(U13(X)) U21(ok(X1),ok(X2)) = 0 >= 0 = ok(U21(X1,X2)) U22(ok(X)) = 0 >= 0 = ok(U22(X)) U31(ok(X1),ok(X2)) = 0 >= 0 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U41(X1,X2,X3)) s(ok(X)) = 0 >= 0 = ok(s(X)) plus(ok(X1),ok(X2)) = 0 >= 0 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U12#(ok(X1),ok(X2)) -> U12#(X1,X2) U12#(mark(X1),X2) -> U12#(X1,X2) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U12#](x0, x1) = x1 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [isNatKind](x0) = x0, [0] = 0, [and](x0, x1) = x1, [s](x0) = x0, [plus](x0, x1) = x1, [U41](x0, x1, x2) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [U21](x0, x1) = x1, [U13](x0) = x0, [mark](x0) = 0, [U12](x0, x1) = x0, [isNat](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = x2, [tt] = 0 orientation: U12#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = U12#(X1,X2) U12#(mark(X1),X2) = X2 + 1 >= X2 + 1 = U12#(X1,X2) active(U11(tt(),V1,V2)) = V2 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = 0 >= 0 = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = X >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V2 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = V2 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = V1 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = 0 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X3 >= X3 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 >= X1 = U12(active(X1),X2) active(U13(X)) = X >= X = U13(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3) active(s(X)) = X >= X = s(active(X)) active(plus(X1,X2)) = X2 >= X2 = plus(active(X1),X2) active(plus(X1,X2)) = X2 >= X2 = plus(X1,active(X2)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) U11(mark(X1),X2,X3) = X3 >= 0 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = 0 >= 0 = mark(U12(X1,X2)) U13(mark(X)) = 0 >= 0 = mark(U13(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) s(mark(X)) = 0 >= 0 = mark(s(X)) plus(mark(X1),X2) = X2 >= 0 = mark(plus(X1,X2)) plus(X1,mark(X2)) = 0 >= 0 = mark(plus(X1,X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = 1 >= 1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2)) = 1 >= 1 = U12(proper(X1),proper(X2)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(U13(X)) = 1 >= 1 = U13(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = 1 >= 1 = U22(proper(X)) proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = 1 >= 1 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(plus(X1,X2)) = 1 >= 1 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(isNatKind(X)) = 1 >= 1 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U12(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U13(ok(X)) = X + 1 >= X + 1 = ok(U13(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) plus(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: U12#(mark(X1),X2) -> U12#(X1,X2) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U12#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x0 + x1 + x2 + 1, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = x0 + 1, [U12](x0, x1) = x0, [isNat](x0) = 0, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x0, [tt] = 0 orientation: U12#(mark(X1),X2) = X1 + 2 >= X1 + 1 = U12#(X1,X2) active(U11(tt(),V1,V2)) = 1 >= 1 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 1 >= 1 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),N)) = N + 1 >= N + 1 = mark(N) active(U41(tt(),M,N)) = M + N + 2 >= M + N + 2 = mark(s(plus(N,M))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(plus(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N + 1 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N + 2 >= M + N + 2 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 + 1 >= X1 + 1 = U12(active(X1),X2) active(U13(X)) = X + 1 >= X + 1 = U13(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = U41(active(X1),X2,X3) active(s(X)) = X + 2 >= X + 2 = s(active(X)) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) U11(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U12(X1,X2)) U13(mark(X)) = X + 1 >= X + 1 = mark(U13(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = mark(U41(X1,X2,X3)) s(mark(X)) = X + 2 >= X + 2 = mark(s(X)) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = X1 >= X1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = X1 >= X1 = U12(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U13(X)) = X >= X = U13(proper(X)) proper(U21(X1,X2)) = X1 >= X1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X >= X = U22(proper(X)) proper(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(plus(X1,X2)) = X1 + X2 >= X1 + X2 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = 0 >= 0 = ok(U12(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U13(ok(X)) = 0 >= 0 = ok(U13(X)) U21(ok(X1),ok(X2)) = 0 >= 0 = ok(U21(X1,X2)) U22(ok(X)) = 0 >= 0 = ok(U22(X)) U31(ok(X1),ok(X2)) = 0 >= 0 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = 1 >= 0 = ok(U41(X1,X2,X3)) s(ok(X)) = 1 >= 0 = ok(s(X)) plus(ok(X1),ok(X2)) = 0 >= 0 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U13#(ok(X)) -> U13#(X) U13#(mark(X)) -> U13#(X) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U13#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [plus](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x0 + x1 + x2, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = x0 + 1, [U12](x0, x1) = x0, [isNat](x0) = 0, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x0, [tt] = 0 orientation: U13#(ok(X)) = X + 1 >= X + 1 = U13#(X) U13#(mark(X)) = X + 2 >= X + 1 = U13#(X) active(U11(tt(),V1,V2)) = 1 >= 1 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 1 >= 1 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),N)) = N + 1 >= N + 1 = mark(N) active(U41(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(plus(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N + 1 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 + 1 >= X1 + 1 = U12(active(X1),X2) active(U13(X)) = X + 1 >= X + 1 = U13(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(active(X1),X2,X3) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) U11(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U12(X1,X2)) U13(mark(X)) = X + 1 >= X + 1 = mark(U13(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U41(X1,X2,X3)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = 0 >= 0 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = 0 >= 0 = U12(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U13(X)) = 0 >= 0 = U13(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(plus(X1,X2)) = 0 >= 0 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = X1 >= X1 = ok(U12(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U13(ok(X)) = X >= X = ok(U13(X)) U21(ok(X1),ok(X2)) = X1 >= X1 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) U31(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U41(X1,X2,X3)) s(ok(X)) = X >= X = ok(s(X)) plus(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U13#(ok(X)) -> U13#(X) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U13#](x0) = x0, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [isNatKind](x0) = x0, [0] = 0, [and](x0, x1) = x0, [s](x0) = x0, [plus](x0, x1) = x0 + 1, [U41](x0, x1, x2) = x1, [U31](x0, x1) = x0, [U22](x0) = x0, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = 0, [U12](x0, x1) = x0 + 1, [isNat](x0) = x0, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x2, [tt] = 0 orientation: U13#(ok(X)) = X + 1 >= X = U13#(X) active(U11(tt(),V1,V2)) = V2 + 1 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 2 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 0 = mark(tt()) active(U21(tt(),V1)) = 1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 0 = mark(tt()) active(U31(tt(),N)) = 1 >= 0 = mark(N) active(U41(tt(),M,N)) = M + 1 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = 1 >= 0 = mark(X) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + 2 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = V1 + 1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = V1 + 2 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = V1 + 1 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N + 2 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = N + 2 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X3 + 1 >= X3 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 + 2 >= X1 + 2 = U12(active(X1),X2) active(U13(X)) = X + 1 >= X + 1 = U13(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X2 + 1 >= X2 = U41(active(X1),X2,X3) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(plus(X1,X2)) = X1 + 2 >= X1 + 2 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + 2 >= X1 + 1 = plus(X1,active(X2)) active(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(active(X1),X2) U11(mark(X1),X2,X3) = X3 >= 0 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = 1 >= 0 = mark(U12(X1,X2)) U13(mark(X)) = 0 >= 0 = mark(U13(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = X2 >= 0 = mark(U41(X1,X2,X3)) s(mark(X)) = 0 >= 0 = mark(s(X)) plus(mark(X1),X2) = 1 >= 0 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + 1 >= 0 = mark(plus(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2)) = X1 + 2 >= X1 + 2 = U12(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U13(X)) = X + 1 >= X + 1 = U13(proper(X)) proper(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X + 1 >= X + 1 = U22(proper(X)) proper(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = X2 + 1 >= X2 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(plus(X1,X2)) = X1 + 2 >= X1 + 2 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(isNatKind(X)) = X + 1 >= X + 1 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(U12(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U13(ok(X)) = X + 1 >= X + 1 = ok(U13(X)) U21(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = X2 + 1 >= X2 + 1 = ok(U41(X1,X2,X3)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) plus(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U21#(ok(X1),ok(X2)) -> U21#(X1,X2) U21#(mark(X1),X2) -> U21#(X1,X2) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U21#](x0, x1) = x1 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [isNatKind](x0) = x0, [0] = 0, [and](x0, x1) = x1, [s](x0) = x0, [plus](x0, x1) = x1, [U41](x0, x1, x2) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [U21](x0, x1) = x1, [U13](x0) = x0, [mark](x0) = 0, [U12](x0, x1) = x0, [isNat](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = x2, [tt] = 0 orientation: U21#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = U21#(X1,X2) U21#(mark(X1),X2) = X2 + 1 >= X2 + 1 = U21#(X1,X2) active(U11(tt(),V1,V2)) = V2 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = 0 >= 0 = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = X >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V2 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = V2 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = V1 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = 0 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X3 >= X3 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 >= X1 = U12(active(X1),X2) active(U13(X)) = X >= X = U13(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3) active(s(X)) = X >= X = s(active(X)) active(plus(X1,X2)) = X2 >= X2 = plus(active(X1),X2) active(plus(X1,X2)) = X2 >= X2 = plus(X1,active(X2)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) U11(mark(X1),X2,X3) = X3 >= 0 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = 0 >= 0 = mark(U12(X1,X2)) U13(mark(X)) = 0 >= 0 = mark(U13(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) s(mark(X)) = 0 >= 0 = mark(s(X)) plus(mark(X1),X2) = X2 >= 0 = mark(plus(X1,X2)) plus(X1,mark(X2)) = 0 >= 0 = mark(plus(X1,X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = 1 >= 1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2)) = 1 >= 1 = U12(proper(X1),proper(X2)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(U13(X)) = 1 >= 1 = U13(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = 1 >= 1 = U22(proper(X)) proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = 1 >= 1 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(plus(X1,X2)) = 1 >= 1 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(isNatKind(X)) = 1 >= 1 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U12(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U13(ok(X)) = X + 1 >= X + 1 = ok(U13(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) plus(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: U21#(mark(X1),X2) -> U21#(X1,X2) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U21#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x0 + x1 + x2 + 1, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = x0 + 1, [U12](x0, x1) = x0, [isNat](x0) = 0, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x0, [tt] = 0 orientation: U21#(mark(X1),X2) = X1 + 2 >= X1 + 1 = U21#(X1,X2) active(U11(tt(),V1,V2)) = 1 >= 1 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 1 >= 1 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),N)) = N + 1 >= N + 1 = mark(N) active(U41(tt(),M,N)) = M + N + 2 >= M + N + 2 = mark(s(plus(N,M))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(plus(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N + 1 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N + 2 >= M + N + 2 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 + 1 >= X1 + 1 = U12(active(X1),X2) active(U13(X)) = X + 1 >= X + 1 = U13(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = U41(active(X1),X2,X3) active(s(X)) = X + 2 >= X + 2 = s(active(X)) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) U11(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U12(X1,X2)) U13(mark(X)) = X + 1 >= X + 1 = mark(U13(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = mark(U41(X1,X2,X3)) s(mark(X)) = X + 2 >= X + 2 = mark(s(X)) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = X1 >= X1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = X1 >= X1 = U12(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U13(X)) = X >= X = U13(proper(X)) proper(U21(X1,X2)) = X1 >= X1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X >= X = U22(proper(X)) proper(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(plus(X1,X2)) = X1 + X2 >= X1 + X2 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = 0 >= 0 = ok(U12(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U13(ok(X)) = 0 >= 0 = ok(U13(X)) U21(ok(X1),ok(X2)) = 0 >= 0 = ok(U21(X1,X2)) U22(ok(X)) = 0 >= 0 = ok(U22(X)) U31(ok(X1),ok(X2)) = 0 >= 0 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = 1 >= 0 = ok(U41(X1,X2,X3)) s(ok(X)) = 1 >= 0 = ok(s(X)) plus(ok(X1),ok(X2)) = 0 >= 0 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U22#(ok(X)) -> U22#(X) U22#(mark(X)) -> U22#(X) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U22#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [plus](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x0 + x1 + x2, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = x0 + 1, [U12](x0, x1) = x0, [isNat](x0) = 0, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x0, [tt] = 0 orientation: U22#(ok(X)) = X + 1 >= X + 1 = U22#(X) U22#(mark(X)) = X + 2 >= X + 1 = U22#(X) active(U11(tt(),V1,V2)) = 1 >= 1 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 1 >= 1 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),N)) = N + 1 >= N + 1 = mark(N) active(U41(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(plus(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N + 1 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 + 1 >= X1 + 1 = U12(active(X1),X2) active(U13(X)) = X + 1 >= X + 1 = U13(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(active(X1),X2,X3) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) U11(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U12(X1,X2)) U13(mark(X)) = X + 1 >= X + 1 = mark(U13(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U41(X1,X2,X3)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = 0 >= 0 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = 0 >= 0 = U12(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U13(X)) = 0 >= 0 = U13(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(plus(X1,X2)) = 0 >= 0 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = X1 >= X1 = ok(U12(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U13(ok(X)) = X >= X = ok(U13(X)) U21(ok(X1),ok(X2)) = X1 >= X1 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) U31(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U41(X1,X2,X3)) s(ok(X)) = X >= X = ok(s(X)) plus(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U22#(ok(X)) -> U22#(X) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U22#](x0) = x0, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [isNatKind](x0) = x0, [0] = 0, [and](x0, x1) = x0, [s](x0) = x0, [plus](x0, x1) = x0 + 1, [U41](x0, x1, x2) = x1, [U31](x0, x1) = x0, [U22](x0) = x0, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = 0, [U12](x0, x1) = x0 + 1, [isNat](x0) = x0, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x2, [tt] = 0 orientation: U22#(ok(X)) = X + 1 >= X = U22#(X) active(U11(tt(),V1,V2)) = V2 + 1 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 2 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 0 = mark(tt()) active(U21(tt(),V1)) = 1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 0 = mark(tt()) active(U31(tt(),N)) = 1 >= 0 = mark(N) active(U41(tt(),M,N)) = M + 1 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = 1 >= 0 = mark(X) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + 2 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = V1 + 1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = V1 + 2 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = V1 + 1 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N + 2 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = N + 2 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X3 + 1 >= X3 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 + 2 >= X1 + 2 = U12(active(X1),X2) active(U13(X)) = X + 1 >= X + 1 = U13(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X2 + 1 >= X2 = U41(active(X1),X2,X3) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(plus(X1,X2)) = X1 + 2 >= X1 + 2 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + 2 >= X1 + 1 = plus(X1,active(X2)) active(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(active(X1),X2) U11(mark(X1),X2,X3) = X3 >= 0 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = 1 >= 0 = mark(U12(X1,X2)) U13(mark(X)) = 0 >= 0 = mark(U13(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = X2 >= 0 = mark(U41(X1,X2,X3)) s(mark(X)) = 0 >= 0 = mark(s(X)) plus(mark(X1),X2) = 1 >= 0 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + 1 >= 0 = mark(plus(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2)) = X1 + 2 >= X1 + 2 = U12(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U13(X)) = X + 1 >= X + 1 = U13(proper(X)) proper(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X + 1 >= X + 1 = U22(proper(X)) proper(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = X2 + 1 >= X2 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(plus(X1,X2)) = X1 + 2 >= X1 + 2 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(isNatKind(X)) = X + 1 >= X + 1 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(U12(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U13(ok(X)) = X + 1 >= X + 1 = ok(U13(X)) U21(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = X2 + 1 >= X2 + 1 = ok(U41(X1,X2,X3)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) plus(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U31#(ok(X1),ok(X2)) -> U31#(X1,X2) U31#(mark(X1),X2) -> U31#(X1,X2) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U31#](x0, x1) = x1 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [isNatKind](x0) = x0, [0] = 0, [and](x0, x1) = x1, [s](x0) = x0, [plus](x0, x1) = x1, [U41](x0, x1, x2) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [U21](x0, x1) = x1, [U13](x0) = x0, [mark](x0) = 0, [U12](x0, x1) = x0, [isNat](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = x2, [tt] = 0 orientation: U31#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = U31#(X1,X2) U31#(mark(X1),X2) = X2 + 1 >= X2 + 1 = U31#(X1,X2) active(U11(tt(),V1,V2)) = V2 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = 0 >= 0 = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = X >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V2 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = V2 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = V1 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = 0 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X3 >= X3 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 >= X1 = U12(active(X1),X2) active(U13(X)) = X >= X = U13(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3) active(s(X)) = X >= X = s(active(X)) active(plus(X1,X2)) = X2 >= X2 = plus(active(X1),X2) active(plus(X1,X2)) = X2 >= X2 = plus(X1,active(X2)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) U11(mark(X1),X2,X3) = X3 >= 0 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = 0 >= 0 = mark(U12(X1,X2)) U13(mark(X)) = 0 >= 0 = mark(U13(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) s(mark(X)) = 0 >= 0 = mark(s(X)) plus(mark(X1),X2) = X2 >= 0 = mark(plus(X1,X2)) plus(X1,mark(X2)) = 0 >= 0 = mark(plus(X1,X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = 1 >= 1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2)) = 1 >= 1 = U12(proper(X1),proper(X2)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(U13(X)) = 1 >= 1 = U13(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = 1 >= 1 = U22(proper(X)) proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = 1 >= 1 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(plus(X1,X2)) = 1 >= 1 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(isNatKind(X)) = 1 >= 1 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U12(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U13(ok(X)) = X + 1 >= X + 1 = ok(U13(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) plus(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: U31#(mark(X1),X2) -> U31#(X1,X2) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U31#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x0 + x1 + x2 + 1, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = x0 + 1, [U12](x0, x1) = x0, [isNat](x0) = 0, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x0, [tt] = 0 orientation: U31#(mark(X1),X2) = X1 + 2 >= X1 + 1 = U31#(X1,X2) active(U11(tt(),V1,V2)) = 1 >= 1 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 1 >= 1 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),N)) = N + 1 >= N + 1 = mark(N) active(U41(tt(),M,N)) = M + N + 2 >= M + N + 2 = mark(s(plus(N,M))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(plus(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N + 1 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N + 2 >= M + N + 2 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 + 1 >= X1 + 1 = U12(active(X1),X2) active(U13(X)) = X + 1 >= X + 1 = U13(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = U41(active(X1),X2,X3) active(s(X)) = X + 2 >= X + 2 = s(active(X)) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) U11(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U12(X1,X2)) U13(mark(X)) = X + 1 >= X + 1 = mark(U13(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = mark(U41(X1,X2,X3)) s(mark(X)) = X + 2 >= X + 2 = mark(s(X)) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = X1 >= X1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = X1 >= X1 = U12(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U13(X)) = X >= X = U13(proper(X)) proper(U21(X1,X2)) = X1 >= X1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X >= X = U22(proper(X)) proper(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(plus(X1,X2)) = X1 + X2 >= X1 + X2 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = 0 >= 0 = ok(U12(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U13(ok(X)) = 0 >= 0 = ok(U13(X)) U21(ok(X1),ok(X2)) = 0 >= 0 = ok(U21(X1,X2)) U22(ok(X)) = 0 >= 0 = ok(U22(X)) U31(ok(X1),ok(X2)) = 0 >= 0 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = 1 >= 0 = ok(U41(X1,X2,X3)) s(ok(X)) = 1 >= 0 = ok(s(X)) plus(ok(X1),ok(X2)) = 0 >= 0 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3) U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U41#](x0, x1, x2) = x2 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [isNatKind](x0) = x0, [0] = 0, [and](x0, x1) = x0, [s](x0) = x0, [plus](x0, x1) = x0 + 1, [U41](x0, x1, x2) = x0, [U31](x0, x1) = x1, [U22](x0) = x0, [U21](x0, x1) = x1, [U13](x0) = x0, [mark](x0) = 0, [U12](x0, x1) = x1 + 1, [isNat](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = x2, [tt] = 1 orientation: U41#(ok(X1),ok(X2),ok(X3)) = X3 + 2 >= X3 + 1 = U41#(X1,X2,X3) U41#(mark(X1),X2,X3) = X3 + 1 >= X3 + 1 = U41#(X1,X2,X3) active(U11(tt(),V1,V2)) = V2 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = V2 + 1 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 0 = mark(tt()) active(U31(tt(),N)) = N >= 0 = mark(N) active(U41(tt(),M,N)) = 1 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = 1 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + 1 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = V1 + 1 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = V1 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = N + 1 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X3 >= X3 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X2 + 1 >= X2 + 1 = U12(active(X1),X2) active(U13(X)) = X >= X = U13(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X2 >= X2 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3) active(s(X)) = X >= X = s(active(X)) active(plus(X1,X2)) = X1 + 1 >= X1 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + 1 >= X1 + 1 = plus(X1,active(X2)) active(and(X1,X2)) = X1 >= X1 = and(active(X1),X2) U11(mark(X1),X2,X3) = X3 >= 0 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = X2 + 1 >= 0 = mark(U12(X1,X2)) U13(mark(X)) = 0 >= 0 = mark(U13(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) s(mark(X)) = 0 >= 0 = mark(s(X)) plus(mark(X1),X2) = 1 >= 0 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + 1 >= 0 = mark(plus(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 2 >= 2 = ok(tt()) proper(U12(X1,X2)) = X2 + 2 >= X2 + 2 = U12(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U13(X)) = X + 1 >= X + 1 = U13(proper(X)) proper(U21(X1,X2)) = X2 + 1 >= X2 + 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X + 1 >= X + 1 = U22(proper(X)) proper(U31(X1,X2)) = X2 + 1 >= X2 + 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(plus(X1,X2)) = X1 + 2 >= X1 + 2 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(isNatKind(X)) = X + 1 >= X + 1 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(U12(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U13(ok(X)) = X + 1 >= X + 1 = ok(U13(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) U31(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) plus(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U41#](x0, x1, x2) = x0, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [plus](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x0 + x1 + x2, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = x0 + 1, [U12](x0, x1) = x0, [isNat](x0) = 0, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x0, [tt] = 0 orientation: U41#(mark(X1),X2,X3) = X1 + 1 >= X1 = U41#(X1,X2,X3) active(U11(tt(),V1,V2)) = 1 >= 1 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 1 >= 1 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),N)) = N + 1 >= N + 1 = mark(N) active(U41(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(plus(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N + 1 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 + 1 >= X1 + 1 = U12(active(X1),X2) active(U13(X)) = X + 1 >= X + 1 = U13(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(active(X1),X2,X3) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) U11(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U12(X1,X2)) U13(mark(X)) = X + 1 >= X + 1 = mark(U13(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U41(X1,X2,X3)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = 0 >= 0 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = 0 >= 0 = U12(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U13(X)) = 0 >= 0 = U13(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(plus(X1,X2)) = 0 >= 0 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = 0 >= 0 = ok(U12(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U13(ok(X)) = 0 >= 0 = ok(U13(X)) U21(ok(X1),ok(X2)) = 0 >= 0 = ok(U21(X1,X2)) U22(ok(X)) = 0 >= 0 = ok(U22(X)) U31(ok(X1),ok(X2)) = 0 >= 0 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U41(X1,X2,X3)) s(ok(X)) = 0 >= 0 = ok(s(X)) plus(ok(X1),ok(X2)) = 0 >= 0 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: s#(ok(X)) -> s#(X) s#(mark(X)) -> s#(X) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [plus](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x0 + x1 + x2, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = x0 + 1, [U12](x0, x1) = x0, [isNat](x0) = 0, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x0, [tt] = 0 orientation: s#(ok(X)) = X + 1 >= X + 1 = s#(X) s#(mark(X)) = X + 2 >= X + 1 = s#(X) active(U11(tt(),V1,V2)) = 1 >= 1 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 1 >= 1 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),N)) = N + 1 >= N + 1 = mark(N) active(U41(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(plus(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N + 1 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 + 1 >= X1 + 1 = U12(active(X1),X2) active(U13(X)) = X + 1 >= X + 1 = U13(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(active(X1),X2,X3) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) U11(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U12(X1,X2)) U13(mark(X)) = X + 1 >= X + 1 = mark(U13(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U41(X1,X2,X3)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = 0 >= 0 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = 0 >= 0 = U12(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U13(X)) = 0 >= 0 = U13(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(plus(X1,X2)) = 0 >= 0 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = X1 >= X1 = ok(U12(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U13(ok(X)) = X >= X = ok(U13(X)) U21(ok(X1),ok(X2)) = X1 >= X1 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) U31(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U41(X1,X2,X3)) s(ok(X)) = X >= X = ok(s(X)) plus(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: s#(ok(X)) -> s#(X) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [isNatKind](x0) = x0, [0] = 0, [and](x0, x1) = x0, [s](x0) = x0, [plus](x0, x1) = x0 + 1, [U41](x0, x1, x2) = x1, [U31](x0, x1) = x0, [U22](x0) = x0, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = 0, [U12](x0, x1) = x0 + 1, [isNat](x0) = x0, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x2, [tt] = 0 orientation: s#(ok(X)) = X + 1 >= X = s#(X) active(U11(tt(),V1,V2)) = V2 + 1 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 2 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 0 = mark(tt()) active(U21(tt(),V1)) = 1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 0 = mark(tt()) active(U31(tt(),N)) = 1 >= 0 = mark(N) active(U41(tt(),M,N)) = M + 1 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = 1 >= 0 = mark(X) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V1 + 2 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = V1 + 1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = V1 + 2 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = V1 + 1 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = N + 2 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = N + 2 >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X3 + 1 >= X3 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 + 2 >= X1 + 2 = U12(active(X1),X2) active(U13(X)) = X + 1 >= X + 1 = U13(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X2 + 1 >= X2 = U41(active(X1),X2,X3) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(plus(X1,X2)) = X1 + 2 >= X1 + 2 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + 2 >= X1 + 1 = plus(X1,active(X2)) active(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(active(X1),X2) U11(mark(X1),X2,X3) = X3 >= 0 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = 1 >= 0 = mark(U12(X1,X2)) U13(mark(X)) = 0 >= 0 = mark(U13(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = X2 >= 0 = mark(U41(X1,X2,X3)) s(mark(X)) = 0 >= 0 = mark(s(X)) plus(mark(X1),X2) = 1 >= 0 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + 1 >= 0 = mark(plus(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2)) = X1 + 2 >= X1 + 2 = U12(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U13(X)) = X + 1 >= X + 1 = U13(proper(X)) proper(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X + 1 >= X + 1 = U22(proper(X)) proper(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = X2 + 1 >= X2 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(plus(X1,X2)) = X1 + 2 >= X1 + 2 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(isNatKind(X)) = X + 1 >= X + 1 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(U12(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U13(ok(X)) = X + 1 >= X + 1 = ok(U13(X)) U21(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = X2 + 1 >= X2 + 1 = ok(U41(X1,X2,X3)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) plus(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: plus#(ok(X1),ok(X2)) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) plus#(mark(X1),X2) -> plus#(X1,X2) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [plus#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [plus](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x0 + x1 + x2, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = x0 + 1, [U12](x0, x1) = x0, [isNat](x0) = 0, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x0, [tt] = 0 orientation: plus#(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = plus#(X1,X2) plus#(X1,mark(X2)) = X1 + 1 >= X1 + 1 = plus#(X1,X2) plus#(mark(X1),X2) = X1 + 2 >= X1 + 1 = plus#(X1,X2) active(U11(tt(),V1,V2)) = 1 >= 1 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 1 >= 1 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),N)) = N + 1 >= N + 1 = mark(N) active(U41(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(plus(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N + 1 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 + 1 >= X1 + 1 = U12(active(X1),X2) active(U13(X)) = X + 1 >= X + 1 = U13(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(active(X1),X2,X3) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) U11(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U12(X1,X2)) U13(mark(X)) = X + 1 >= X + 1 = mark(U13(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U41(X1,X2,X3)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = 0 >= 0 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = 0 >= 0 = U12(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U13(X)) = 0 >= 0 = U13(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(plus(X1,X2)) = 0 >= 0 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = X1 >= X1 = ok(U12(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U13(ok(X)) = X >= X = ok(U13(X)) U21(ok(X1),ok(X2)) = X1 >= X1 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) U31(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U41(X1,X2,X3)) s(ok(X)) = X >= X = ok(s(X)) plus(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: plus#(ok(X1),ok(X2)) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [plus#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [plus](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x0 + x1 + x2, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = x0 + 1, [U12](x0, x1) = x0, [isNat](x0) = 0, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x0, [tt] = 0 orientation: plus#(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = plus#(X1,X2) plus#(X1,mark(X2)) = X2 + 2 >= X2 + 1 = plus#(X1,X2) active(U11(tt(),V1,V2)) = 1 >= 1 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 1 >= 1 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),N)) = N + 1 >= N + 1 = mark(N) active(U41(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(plus(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N + 1 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 + 1 >= X1 + 1 = U12(active(X1),X2) active(U13(X)) = X + 1 >= X + 1 = U13(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(active(X1),X2,X3) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) U11(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U12(X1,X2)) U13(mark(X)) = X + 1 >= X + 1 = mark(U13(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U41(X1,X2,X3)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = 0 >= 0 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = 0 >= 0 = U12(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U13(X)) = 0 >= 0 = U13(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(plus(X1,X2)) = 0 >= 0 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = X1 >= X1 = ok(U12(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U13(ok(X)) = X >= X = ok(U13(X)) U21(ok(X1),ok(X2)) = X1 >= X1 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) U31(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U41(X1,X2,X3)) s(ok(X)) = X >= X = ok(s(X)) plus(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: plus#(ok(X1),ok(X2)) -> plus#(X1,X2) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [plus#](x0, x1) = x0 + x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [isNatKind](x0) = x0, [0] = 0, [and](x0, x1) = x1, [s](x0) = x0, [plus](x0, x1) = x1, [U41](x0, x1, x2) = x1, [U31](x0, x1) = x0, [U22](x0) = x0, [U21](x0, x1) = x1 + 1, [U13](x0) = x0, [mark](x0) = 0, [U12](x0, x1) = x0, [isNat](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = x0 + 1, [tt] = 0 orientation: plus#(ok(X1),ok(X2)) = X1 + X2 + 3 >= X1 + X2 + 1 = plus#(X1,X2) active(U11(tt(),V1,V2)) = 1 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 + 1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = 0 >= 0 = mark(N) active(U41(tt(),M,N)) = M >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = X >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V2 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = V2 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = V1 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = 0 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 >= X1 = U12(active(X1),X2) active(U13(X)) = X >= X = U13(active(X)) active(U21(X1,X2)) = X2 + 1 >= X2 + 1 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X2 >= X2 = U41(active(X1),X2,X3) active(s(X)) = X >= X = s(active(X)) active(plus(X1,X2)) = X2 >= X2 = plus(active(X1),X2) active(plus(X1,X2)) = X2 >= X2 = plus(X1,active(X2)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) U11(mark(X1),X2,X3) = 1 >= 0 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = 0 >= 0 = mark(U12(X1,X2)) U13(mark(X)) = 0 >= 0 = mark(U13(X)) U21(mark(X1),X2) = X2 + 1 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = X2 >= 0 = mark(U41(X1,X2,X3)) s(mark(X)) = 0 >= 0 = mark(s(X)) plus(mark(X1),X2) = X2 >= 0 = mark(plus(X1,X2)) plus(X1,mark(X2)) = 0 >= 0 = mark(plus(X1,X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = X1 + 2 >= X1 + 2 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2)) = X1 + 1 >= X1 + 1 = U12(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U13(X)) = X + 1 >= X + 1 = U13(proper(X)) proper(U21(X1,X2)) = X2 + 2 >= X2 + 2 = U21(proper(X1),proper(X2)) proper(U22(X)) = X + 1 >= X + 1 = U22(proper(X)) proper(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = X2 + 1 >= X2 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(plus(X1,X2)) = X2 + 1 >= X2 + 1 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(isNatKind(X)) = X + 1 >= X + 1 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = X1 + 2 >= X1 + 2 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U12(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U13(ok(X)) = X + 1 >= X + 1 = ok(U13(X)) U21(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = X2 + 1 >= X2 + 1 = ok(U41(X1,X2,X3)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) plus(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: and#(ok(X1),ok(X2)) -> and#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [and#](x0, x1) = x1 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [isNatKind](x0) = x0, [0] = 0, [and](x0, x1) = x1, [s](x0) = x0, [plus](x0, x1) = x1, [U41](x0, x1, x2) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [U21](x0, x1) = x1, [U13](x0) = x0, [mark](x0) = 0, [U12](x0, x1) = x0, [isNat](x0) = x0, [active](x0) = x0, [U11](x0, x1, x2) = x2, [tt] = 0 orientation: and#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = and#(X1,X2) and#(mark(X1),X2) = X2 + 1 >= X2 + 1 = and#(X1,X2) active(U11(tt(),V1,V2)) = V2 >= 0 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 0 >= 0 = mark(U13(isNat(V2))) active(U13(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),N)) = 0 >= 0 = mark(N) active(U41(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M))) active(and(tt(),X)) = X >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(plus(V1,V2))) = V2 >= 0 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(plus(V1,V2))) = V2 >= 0 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = V1 >= 0 = mark(isNatKind(V1)) active(plus(N,0())) = 0 >= 0 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M >= 0 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X3 >= X3 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 >= X1 = U12(active(X1),X2) active(U13(X)) = X >= X = U13(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3) active(s(X)) = X >= X = s(active(X)) active(plus(X1,X2)) = X2 >= X2 = plus(active(X1),X2) active(plus(X1,X2)) = X2 >= X2 = plus(X1,active(X2)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) U11(mark(X1),X2,X3) = X3 >= 0 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = 0 >= 0 = mark(U12(X1,X2)) U13(mark(X)) = 0 >= 0 = mark(U13(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) s(mark(X)) = 0 >= 0 = mark(s(X)) plus(mark(X1),X2) = X2 >= 0 = mark(plus(X1,X2)) plus(X1,mark(X2)) = 0 >= 0 = mark(plus(X1,X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = 1 >= 1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2)) = 1 >= 1 = U12(proper(X1),proper(X2)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(U13(X)) = 1 >= 1 = U13(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = 1 >= 1 = U22(proper(X)) proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = 1 >= 1 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(plus(X1,X2)) = 1 >= 1 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(isNatKind(X)) = 1 >= 1 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U12(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U13(ok(X)) = X + 1 >= X + 1 = ok(U13(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) plus(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: and#(mark(X1),X2) -> and#(X1,X2) TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [and#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [isNatKind](x0) = 0, [0] = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x0 + x1 + x2 + 1, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0, [U13](x0) = x0, [mark](x0) = x0 + 1, [U12](x0, x1) = x0, [isNat](x0) = 0, [active](x0) = x0 + 1, [U11](x0, x1, x2) = x0, [tt] = 0 orientation: and#(mark(X1),X2) = X1 + 2 >= X1 + 1 = and#(X1,X2) active(U11(tt(),V1,V2)) = 1 >= 1 = mark(U12(isNat(V1),V2)) active(U12(tt(),V2)) = 1 >= 1 = mark(U13(isNat(V2))) active(U13(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),N)) = N + 1 >= N + 1 = mark(N) active(U41(tt(),M,N)) = M + N + 2 >= M + N + 2 = mark(s(plus(N,M))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(plus(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatKind(V2))) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(plus(N,0())) = N + 1 >= N + 1 = mark(U31(and(isNat(N),isNatKind(N)),N)) active(plus(N,s(M))) = M + N + 2 >= M + N + 2 = mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2,X3) active(U12(X1,X2)) = X1 + 1 >= X1 + 1 = U12(active(X1),X2) active(U13(X)) = X + 1 >= X + 1 = U13(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2) active(U41(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = U41(active(X1),X2,X3) active(s(X)) = X + 2 >= X + 2 = s(active(X)) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) U11(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2,X3)) U12(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U12(X1,X2)) U13(mark(X)) = X + 1 >= X + 1 = mark(U13(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U31(X1,X2)) U41(mark(X1),X2,X3) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = mark(U41(X1,X2,X3)) s(mark(X)) = X + 2 >= X + 2 = mark(s(X)) plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(U11(X1,X2,X3)) = X1 >= X1 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2)) = X1 >= X1 = U12(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U13(X)) = X >= X = U13(proper(X)) proper(U21(X1,X2)) = X1 >= X1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X >= X = U22(proper(X)) proper(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(plus(X1,X2)) = X1 + X2 >= X1 + X2 = plus(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) = 0 >= 0 = ok(U12(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U13(ok(X)) = 0 >= 0 = ok(U13(X)) U21(ok(X1),ok(X2)) = 0 >= 0 = ok(U21(X1,X2)) U22(ok(X)) = 0 >= 0 = ok(U22(X)) U31(ok(X1),ok(X2)) = 0 >= 0 = ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) = 1 >= 0 = ok(U41(X1,X2,X3)) s(ok(X)) = 1 >= 0 = ok(s(X)) plus(ok(X1),ok(X2)) = 0 >= 0 = ok(plus(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: 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(0())) -> mark(tt()) 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(0())) -> mark(tt()) 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)) active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2)) -> U12(active(X1),X2) active(U13(X)) -> U13(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2) -> mark(U12(X1,X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed