YES Problem: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X Proof: DP Processor: DPs: U11#(tt(),V2) -> activate#(V2) U11#(tt(),V2) -> isNat#(activate(V2)) U11#(tt(),V2) -> U12#(isNat(activate(V2))) U31#(tt(),N) -> activate#(N) U41#(tt(),M,N) -> activate#(M) U41#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> isNat#(activate(N)) U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) U42#(tt(),M,N) -> activate#(M) U42#(tt(),M,N) -> activate#(N) U42#(tt(),M,N) -> plus#(activate(N),activate(M)) U42#(tt(),M,N) -> s#(plus(activate(N),activate(M))) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNat(activate(V1))) plus#(N,0()) -> isNat#(N) plus#(N,0()) -> U31#(isNat(N),N) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> U41#(isNat(M),M,N) activate#(n__0()) -> 0#() activate#(n__plus(X1,X2)) -> plus#(X1,X2) activate#(n__s(X)) -> s#(X) TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X Matrix Interpretation Processor: dim=1 interpretation: [0#] = 0, [s#](x0) = 0, [plus#](x0, x1) = 4x0 + 4x1, [U42#](x0, x1, x2) = 4x1 + 4x2 + 1, [U41#](x0, x1, x2) = 4x1 + 4x2 + 2, [U31#](x0, x1) = 3x1 + 1, [U21#](x0) = 3, [U12#](x0) = 0, [isNat#](x0) = 4x0, [activate#](x0) = 2x0, [U11#](x0, x1) = 4x1 + 3, [0] = 2, [n__s](x0) = x0 + 2, [n__plus](x0, x1) = 2x0 + 2x1 + 2, [n__0] = 2, [s](x0) = x0 + 2, [plus](x0, x1) = 2x0 + 2x1 + 2, [U42](x0, x1, x2) = 2x1 + 2x2 + 4, [U41](x0, x1, x2) = 2x1 + 2x2 + 4, [U31](x0, x1) = 2x1 + 6, [U21](x0) = 0, [U12](x0) = 0, [isNat](x0) = 0, [activate](x0) = x0, [U11](x0, x1) = 0, [tt] = 0 orientation: U11#(tt(),V2) = 4V2 + 3 >= 2V2 = activate#(V2) U11#(tt(),V2) = 4V2 + 3 >= 4V2 = isNat#(activate(V2)) U11#(tt(),V2) = 4V2 + 3 >= 0 = U12#(isNat(activate(V2))) U31#(tt(),N) = 3N + 1 >= 2N = activate#(N) U41#(tt(),M,N) = 4M + 4N + 2 >= 2M = activate#(M) U41#(tt(),M,N) = 4M + 4N + 2 >= 2N = activate#(N) U41#(tt(),M,N) = 4M + 4N + 2 >= 4N = isNat#(activate(N)) U41#(tt(),M,N) = 4M + 4N + 2 >= 4M + 4N + 1 = U42#(isNat(activate(N)),activate(M),activate(N)) U42#(tt(),M,N) = 4M + 4N + 1 >= 2M = activate#(M) U42#(tt(),M,N) = 4M + 4N + 1 >= 2N = activate#(N) U42#(tt(),M,N) = 4M + 4N + 1 >= 4M + 4N = plus#(activate(N),activate(M)) U42#(tt(),M,N) = 4M + 4N + 1 >= 0 = s#(plus(activate(N),activate(M))) isNat#(n__plus(V1,V2)) = 8V1 + 8V2 + 8 >= 2V2 = activate#(V2) isNat#(n__plus(V1,V2)) = 8V1 + 8V2 + 8 >= 2V1 = activate#(V1) isNat#(n__plus(V1,V2)) = 8V1 + 8V2 + 8 >= 4V1 = isNat#(activate(V1)) isNat#(n__plus(V1,V2)) = 8V1 + 8V2 + 8 >= 4V2 + 3 = U11#(isNat(activate(V1)),activate(V2)) isNat#(n__s(V1)) = 4V1 + 8 >= 2V1 = activate#(V1) isNat#(n__s(V1)) = 4V1 + 8 >= 4V1 = isNat#(activate(V1)) isNat#(n__s(V1)) = 4V1 + 8 >= 3 = U21#(isNat(activate(V1))) plus#(N,0()) = 4N + 8 >= 4N = isNat#(N) plus#(N,0()) = 4N + 8 >= 3N + 1 = U31#(isNat(N),N) plus#(N,s(M)) = 4M + 4N + 8 >= 4M = isNat#(M) plus#(N,s(M)) = 4M + 4N + 8 >= 4M + 4N + 2 = U41#(isNat(M),M,N) activate#(n__0()) = 4 >= 0 = 0#() activate#(n__plus(X1,X2)) = 4X1 + 4X2 + 4 >= 4X1 + 4X2 = plus#(X1,X2) activate#(n__s(X)) = 2X + 4 >= 0 = s#(X) U11(tt(),V2) = 0 >= 0 = U12(isNat(activate(V2))) U12(tt()) = 0 >= 0 = tt() U21(tt()) = 0 >= 0 = tt() U31(tt(),N) = 2N + 6 >= N = activate(N) U41(tt(),M,N) = 2M + 2N + 4 >= 2M + 2N + 4 = U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) = 2M + 2N + 4 >= 2M + 2N + 4 = s(plus(activate(N),activate(M))) isNat(n__0()) = 0 >= 0 = tt() isNat(n__plus(V1,V2)) = 0 >= 0 = U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) = 0 >= 0 = U21(isNat(activate(V1))) plus(N,0()) = 2N + 6 >= 2N + 6 = U31(isNat(N),N) plus(N,s(M)) = 2M + 2N + 6 >= 2M + 2N + 4 = U41(isNat(M),M,N) 0() = 2 >= 2 = n__0() plus(X1,X2) = 2X1 + 2X2 + 2 >= 2X1 + 2X2 + 2 = n__plus(X1,X2) s(X) = X + 2 >= X + 2 = n__s(X) activate(n__0()) = 2 >= 2 = 0() activate(n__plus(X1,X2)) = 2X1 + 2X2 + 2 >= 2X1 + 2X2 + 2 = plus(X1,X2) activate(n__s(X)) = X + 2 >= X + 2 = s(X) activate(X) = X >= X = X problem: DPs: TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X Qed