MAYBE Problem: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(U21(tt(),M,N)) -> plus#(N,M) active#(U21(tt(),M,N)) -> s#(plus(N,M)) active#(U41(tt(),M,N)) -> x#(N,M) active#(U41(tt(),M,N)) -> plus#(x(N,M),N) active#(isNat(plus(V1,V2))) -> isNat#(V2) active#(isNat(plus(V1,V2))) -> isNat#(V1) active#(isNat(plus(V1,V2))) -> and#(isNat(V1),isNat(V2)) active#(isNat(s(V1))) -> isNat#(V1) active#(isNat(x(V1,V2))) -> isNat#(V2) active#(isNat(x(V1,V2))) -> isNat#(V1) active#(isNat(x(V1,V2))) -> and#(isNat(V1),isNat(V2)) active#(plus(N,0())) -> isNat#(N) active#(plus(N,0())) -> U11#(isNat(N),N) active#(plus(N,s(M))) -> isNat#(N) active#(plus(N,s(M))) -> isNat#(M) active#(plus(N,s(M))) -> and#(isNat(M),isNat(N)) active#(plus(N,s(M))) -> U21#(and(isNat(M),isNat(N)),M,N) active#(x(N,0())) -> isNat#(N) active#(x(N,0())) -> U31#(isNat(N)) active#(x(N,s(M))) -> isNat#(N) active#(x(N,s(M))) -> isNat#(M) active#(x(N,s(M))) -> and#(isNat(M),isNat(N)) active#(x(N,s(M))) -> U41#(and(isNat(M),isNat(N)),M,N) active#(U11(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> U11#(active(X1),X2) active#(U21(X1,X2,X3)) -> active#(X1) active#(U21(X1,X2,X3)) -> U21#(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#(U31(X)) -> active#(X) active#(U31(X)) -> U31#(active(X)) active#(U41(X1,X2,X3)) -> active#(X1) active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3) active#(x(X1,X2)) -> active#(X1) active#(x(X1,X2)) -> x#(active(X1),X2) active#(x(X1,X2)) -> active#(X2) active#(x(X1,X2)) -> x#(X1,active(X2)) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) U11#(mark(X1),X2) -> U11#(X1,X2) U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) s#(mark(X)) -> s#(X) plus#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) U31#(mark(X)) -> U31#(X) U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) x#(mark(X1),X2) -> x#(X1,X2) x#(X1,mark(X2)) -> x#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) proper#(U11(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(U21(X1,X2,X3)) -> proper#(X3) proper#(U21(X1,X2,X3)) -> proper#(X2) proper#(U21(X1,X2,X3)) -> proper#(X1) proper#(U21(X1,X2,X3)) -> U21#(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#(U31(X)) -> proper#(X) proper#(U31(X)) -> U31#(proper(X)) 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#(x(X1,X2)) -> proper#(X2) proper#(x(X1,X2)) -> proper#(X1) proper#(x(X1,X2)) -> x#(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#(isNat(X)) -> proper#(X) proper#(isNat(X)) -> isNat#(proper(X)) U11#(ok(X1),ok(X2)) -> U11#(X1,X2) U21#(ok(X1),ok(X2),ok(X3)) -> U21#(X1,X2,X3) s#(ok(X)) -> s#(X) plus#(ok(X1),ok(X2)) -> plus#(X1,X2) U31#(ok(X)) -> U31#(X) U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3) x#(ok(X1),ok(X2)) -> x#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) isNat#(ok(X)) -> isNat#(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(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(U21(tt(),M,N)) -> plus#(N,M) active#(U21(tt(),M,N)) -> s#(plus(N,M)) active#(U41(tt(),M,N)) -> x#(N,M) active#(U41(tt(),M,N)) -> plus#(x(N,M),N) active#(isNat(plus(V1,V2))) -> isNat#(V2) active#(isNat(plus(V1,V2))) -> isNat#(V1) active#(isNat(plus(V1,V2))) -> and#(isNat(V1),isNat(V2)) active#(isNat(s(V1))) -> isNat#(V1) active#(isNat(x(V1,V2))) -> isNat#(V2) active#(isNat(x(V1,V2))) -> isNat#(V1) active#(isNat(x(V1,V2))) -> and#(isNat(V1),isNat(V2)) active#(plus(N,0())) -> isNat#(N) active#(plus(N,0())) -> U11#(isNat(N),N) active#(plus(N,s(M))) -> isNat#(N) active#(plus(N,s(M))) -> isNat#(M) active#(plus(N,s(M))) -> and#(isNat(M),isNat(N)) active#(plus(N,s(M))) -> U21#(and(isNat(M),isNat(N)),M,N) active#(x(N,0())) -> isNat#(N) active#(x(N,0())) -> U31#(isNat(N)) active#(x(N,s(M))) -> isNat#(N) active#(x(N,s(M))) -> isNat#(M) active#(x(N,s(M))) -> and#(isNat(M),isNat(N)) active#(x(N,s(M))) -> U41#(and(isNat(M),isNat(N)),M,N) active#(U11(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> U11#(active(X1),X2) active#(U21(X1,X2,X3)) -> active#(X1) active#(U21(X1,X2,X3)) -> U21#(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#(U31(X)) -> active#(X) active#(U31(X)) -> U31#(active(X)) active#(U41(X1,X2,X3)) -> active#(X1) active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3) active#(x(X1,X2)) -> active#(X1) active#(x(X1,X2)) -> x#(active(X1),X2) active#(x(X1,X2)) -> active#(X2) active#(x(X1,X2)) -> x#(X1,active(X2)) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) U11#(mark(X1),X2) -> U11#(X1,X2) U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) s#(mark(X)) -> s#(X) plus#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) U31#(mark(X)) -> U31#(X) U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) x#(mark(X1),X2) -> x#(X1,X2) x#(X1,mark(X2)) -> x#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) proper#(U11(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(U21(X1,X2,X3)) -> proper#(X3) proper#(U21(X1,X2,X3)) -> proper#(X2) proper#(U21(X1,X2,X3)) -> proper#(X1) proper#(U21(X1,X2,X3)) -> U21#(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#(U31(X)) -> proper#(X) proper#(U31(X)) -> U31#(proper(X)) 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#(x(X1,X2)) -> proper#(X2) proper#(x(X1,X2)) -> proper#(X1) proper#(x(X1,X2)) -> x#(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#(isNat(X)) -> proper#(X) proper#(isNat(X)) -> isNat#(proper(X)) U11#(ok(X1),ok(X2)) -> U11#(X1,X2) U21#(ok(X1),ok(X2),ok(X3)) -> U21#(X1,X2,X3) s#(ok(X)) -> s#(X) plus#(ok(X1),ok(X2)) -> plus#(X1,X2) U31#(ok(X)) -> U31#(X) U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3) x#(ok(X1),ok(X2)) -> x#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) isNat#(ok(X)) -> isNat#(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(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: ... SCC Processor: #sccs: 12 #rules: 48 #arcs: 1073/8464 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: active#(U11(X1,X2)) -> active#(X1) active#(U21(X1,X2,X3)) -> active#(X1) active#(s(X)) -> active#(X) active#(plus(X1,X2)) -> active#(X1) active#(plus(X1,X2)) -> active#(X2) active#(U31(X)) -> active#(X) active#(U41(X1,X2,X3)) -> active#(X1) active#(x(X1,X2)) -> active#(X1) active#(x(X1,X2)) -> active#(X2) active#(and(X1,X2)) -> active#(X1) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(active#) = 0 problem: DPs: TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: proper#(U11(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) proper#(U21(X1,X2,X3)) -> proper#(X3) proper#(U21(X1,X2,X3)) -> proper#(X2) proper#(U21(X1,X2,X3)) -> proper#(X1) proper#(s(X)) -> proper#(X) proper#(plus(X1,X2)) -> proper#(X2) proper#(plus(X1,X2)) -> proper#(X1) proper#(U31(X)) -> proper#(X) proper#(U41(X1,X2,X3)) -> proper#(X3) proper#(U41(X1,X2,X3)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X1) proper#(x(X1,X2)) -> proper#(X2) proper#(x(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(isNat(X)) -> proper#(X) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(proper#) = 0 problem: DPs: TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: isNat#(ok(X)) -> isNat#(X) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(isNat#) = 0 problem: DPs: TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: and#(mark(X1),X2) -> and#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(and#) = 1 problem: DPs: and#(mark(X1),X2) -> and#(X1,X2) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(and#) = 0 problem: DPs: TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: x#(mark(X1),X2) -> x#(X1,X2) x#(X1,mark(X2)) -> x#(X1,X2) x#(ok(X1),ok(X2)) -> x#(X1,X2) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(x#) = 1 problem: DPs: x#(mark(X1),X2) -> x#(X1,X2) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(x#) = 0 problem: DPs: TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U41#) = 2 problem: DPs: U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U41#) = 0 problem: DPs: TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U31#(mark(X)) -> U31#(X) U31#(ok(X)) -> U31#(X) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U31#) = 0 problem: DPs: TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: plus#(mark(X1),X2) -> plus#(X1,X2) plus#(X1,mark(X2)) -> plus#(X1,X2) plus#(ok(X1),ok(X2)) -> plus#(X1,X2) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(plus#) = 1 problem: DPs: plus#(mark(X1),X2) -> plus#(X1,X2) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(plus#) = 0 problem: DPs: TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: s#(mark(X)) -> s#(X) s#(ok(X)) -> s#(X) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(s#) = 0 problem: DPs: TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) U21#(ok(X1),ok(X2),ok(X3)) -> U21#(X1,X2,X3) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U21#) = 2 problem: DPs: U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U21#) = 0 problem: DPs: TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U11#(mark(X1),X2) -> U11#(X1,X2) U11#(ok(X1),ok(X2)) -> U11#(X1,X2) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U11#) = 1 problem: DPs: U11#(mark(X1),X2) -> U11#(X1,X2) TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U11#) = 0 problem: DPs: TRS: active(U11(tt(),N)) -> mark(N) active(U21(tt(),M,N)) -> mark(s(plus(N,M))) active(U31(tt())) -> mark(0()) active(U41(tt(),M,N)) -> mark(plus(x(N,M),N)) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2))) active(plus(N,0())) -> mark(U11(isNat(N),N)) active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N)) active(x(N,0())) -> mark(U31(isNat(N))) active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U21(X1,X2,X3)) -> U21(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(U31(X)) -> U31(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(x(X1,X2)) -> x(active(X1),X2) active(x(X1,X2)) -> x(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U11(mark(X1),X2) -> mark(U11(X1,X2)) U21(mark(X1),X2,X3) -> mark(U21(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)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) x(mark(X1),X2) -> mark(x(X1,X2)) x(X1,mark(X2)) -> mark(x(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(U31(X)) -> U31(proper(X)) proper(0()) -> ok(0()) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(x(X1,X2)) -> x(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) x(ok(X1),ok(X2)) -> ok(x(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed