YES Problem: and(true(),X) -> activate(X) and(false(),Y) -> false() if(true(),X,Y) -> activate(X) if(false(),X,Y) -> activate(Y) add(0(),X) -> activate(X) add(s(X),Y) -> s(n__add(activate(X),activate(Y))) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z))) from(X) -> cons(activate(X),n__from(n__s(activate(X)))) add(X1,X2) -> n__add(X1,X2) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__add(X1,X2)) -> add(activate(X1),X2) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(X) activate(n__s(X)) -> s(X) activate(X) -> X Proof: DP Processor: DPs: and#(true(),X) -> activate#(X) if#(true(),X,Y) -> activate#(X) if#(false(),X,Y) -> activate#(Y) add#(0(),X) -> activate#(X) add#(s(X),Y) -> activate#(Y) add#(s(X),Y) -> activate#(X) add#(s(X),Y) -> s#(n__add(activate(X),activate(Y))) first#(s(X),cons(Y,Z)) -> activate#(Z) first#(s(X),cons(Y,Z)) -> activate#(X) first#(s(X),cons(Y,Z)) -> activate#(Y) from#(X) -> activate#(X) activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__add(X1,X2)) -> add#(activate(X1),X2) activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) activate#(n__from(X)) -> from#(X) activate#(n__s(X)) -> s#(X) TRS: and(true(),X) -> activate(X) and(false(),Y) -> false() if(true(),X,Y) -> activate(X) if(false(),X,Y) -> activate(Y) add(0(),X) -> activate(X) add(s(X),Y) -> s(n__add(activate(X),activate(Y))) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z))) from(X) -> cons(activate(X),n__from(n__s(activate(X)))) add(X1,X2) -> n__add(X1,X2) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__add(X1,X2)) -> add(activate(X1),X2) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(X) activate(n__s(X)) -> s(X) activate(X) -> X TDG Processor: DPs: and#(true(),X) -> activate#(X) if#(true(),X,Y) -> activate#(X) if#(false(),X,Y) -> activate#(Y) add#(0(),X) -> activate#(X) add#(s(X),Y) -> activate#(Y) add#(s(X),Y) -> activate#(X) add#(s(X),Y) -> s#(n__add(activate(X),activate(Y))) first#(s(X),cons(Y,Z)) -> activate#(Z) first#(s(X),cons(Y,Z)) -> activate#(X) first#(s(X),cons(Y,Z)) -> activate#(Y) from#(X) -> activate#(X) activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__add(X1,X2)) -> add#(activate(X1),X2) activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) activate#(n__from(X)) -> from#(X) activate#(n__s(X)) -> s#(X) TRS: and(true(),X) -> activate(X) and(false(),Y) -> false() if(true(),X,Y) -> activate(X) if(false(),X,Y) -> activate(Y) add(0(),X) -> activate(X) add(s(X),Y) -> s(n__add(activate(X),activate(Y))) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z))) from(X) -> cons(activate(X),n__from(n__s(activate(X)))) add(X1,X2) -> n__add(X1,X2) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__add(X1,X2)) -> add(activate(X1),X2) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(X) activate(n__s(X)) -> s(X) activate(X) -> X graph: from#(X) -> activate#(X) -> activate#(n__s(X)) -> s#(X) from#(X) -> activate#(X) -> activate#(n__from(X)) -> from#(X) from#(X) -> activate#(X) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) from#(X) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X1) from#(X) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X2) from#(X) -> activate#(X) -> activate#(n__add(X1,X2)) -> add#(activate(X1),X2) from#(X) -> activate#(X) -> activate#(n__add(X1,X2)) -> activate#(X1) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__s(X)) -> s#(X) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__from(X)) -> from#(X) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__first(X1,X2)) -> activate#(X1) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__first(X1,X2)) -> activate#(X2) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__add(X1,X2)) -> add#(activate(X1),X2) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__add(X1,X2)) -> activate#(X1) first#(s(X),cons(Y,Z)) -> activate#(Y) -> activate#(n__s(X)) -> s#(X) first#(s(X),cons(Y,Z)) -> activate#(Y) -> activate#(n__from(X)) -> from#(X) first#(s(X),cons(Y,Z)) -> activate#(Y) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) -> activate#(Y) -> activate#(n__first(X1,X2)) -> activate#(X1) first#(s(X),cons(Y,Z)) -> activate#(Y) -> activate#(n__first(X1,X2)) -> activate#(X2) first#(s(X),cons(Y,Z)) -> activate#(Y) -> activate#(n__add(X1,X2)) -> add#(activate(X1),X2) first#(s(X),cons(Y,Z)) -> activate#(Y) -> activate#(n__add(X1,X2)) -> activate#(X1) first#(s(X),cons(Y,Z)) -> activate#(X) -> activate#(n__s(X)) -> s#(X) first#(s(X),cons(Y,Z)) -> activate#(X) -> activate#(n__from(X)) -> from#(X) first#(s(X),cons(Y,Z)) -> activate#(X) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X1) first#(s(X),cons(Y,Z)) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X2) first#(s(X),cons(Y,Z)) -> activate#(X) -> activate#(n__add(X1,X2)) -> add#(activate(X1),X2) first#(s(X),cons(Y,Z)) -> activate#(X) -> activate#(n__add(X1,X2)) -> activate#(X1) add#(s(X),Y) -> activate#(Y) -> activate#(n__s(X)) -> s#(X) add#(s(X),Y) -> activate#(Y) -> activate#(n__from(X)) -> from#(X) add#(s(X),Y) -> activate#(Y) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) add#(s(X),Y) -> activate#(Y) -> activate#(n__first(X1,X2)) -> activate#(X1) add#(s(X),Y) -> activate#(Y) -> activate#(n__first(X1,X2)) -> activate#(X2) add#(s(X),Y) -> activate#(Y) -> activate#(n__add(X1,X2)) -> add#(activate(X1),X2) add#(s(X),Y) -> activate#(Y) -> activate#(n__add(X1,X2)) -> activate#(X1) add#(s(X),Y) -> activate#(X) -> activate#(n__s(X)) -> s#(X) add#(s(X),Y) -> activate#(X) -> activate#(n__from(X)) -> from#(X) add#(s(X),Y) -> activate#(X) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) add#(s(X),Y) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X1) add#(s(X),Y) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X2) add#(s(X),Y) -> activate#(X) -> activate#(n__add(X1,X2)) -> add#(activate(X1),X2) add#(s(X),Y) -> activate#(X) -> activate#(n__add(X1,X2)) -> activate#(X1) add#(0(),X) -> activate#(X) -> activate#(n__s(X)) -> s#(X) add#(0(),X) -> activate#(X) -> activate#(n__from(X)) -> from#(X) add#(0(),X) -> activate#(X) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) add#(0(),X) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X1) add#(0(),X) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X2) add#(0(),X) -> activate#(X) -> activate#(n__add(X1,X2)) -> add#(activate(X1),X2) add#(0(),X) -> activate#(X) -> activate#(n__add(X1,X2)) -> activate#(X1) if#(false(),X,Y) -> activate#(Y) -> activate#(n__s(X)) -> s#(X) if#(false(),X,Y) -> activate#(Y) -> activate#(n__from(X)) -> from#(X) if#(false(),X,Y) -> activate#(Y) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) if#(false(),X,Y) -> activate#(Y) -> activate#(n__first(X1,X2)) -> activate#(X1) if#(false(),X,Y) -> activate#(Y) -> activate#(n__first(X1,X2)) -> activate#(X2) if#(false(),X,Y) -> activate#(Y) -> activate#(n__add(X1,X2)) -> add#(activate(X1),X2) if#(false(),X,Y) -> activate#(Y) -> activate#(n__add(X1,X2)) -> activate#(X1) if#(true(),X,Y) -> activate#(X) -> activate#(n__s(X)) -> s#(X) if#(true(),X,Y) -> activate#(X) -> activate#(n__from(X)) -> from#(X) if#(true(),X,Y) -> activate#(X) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) if#(true(),X,Y) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X1) if#(true(),X,Y) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X2) if#(true(),X,Y) -> activate#(X) -> activate#(n__add(X1,X2)) -> add#(activate(X1),X2) if#(true(),X,Y) -> activate#(X) -> activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__from(X)) -> from#(X) -> from#(X) -> activate#(X) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) -> first#(s(X),cons(Y,Z)) -> activate#(Y) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) -> first#(s(X),cons(Y,Z)) -> activate#(X) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) -> first#(s(X),cons(Y,Z)) -> activate#(Z) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__s(X)) -> s#(X) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__from(X)) -> from#(X) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__add(X1,X2)) -> add#(activate(X1),X2) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> s#(X) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__from(X)) -> from#(X) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__add(X1,X2)) -> add#(activate(X1),X2) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__add(X1,X2)) -> add#(activate(X1),X2) -> add#(s(X),Y) -> s#(n__add(activate(X),activate(Y))) activate#(n__add(X1,X2)) -> add#(activate(X1),X2) -> add#(s(X),Y) -> activate#(X) activate#(n__add(X1,X2)) -> add#(activate(X1),X2) -> add#(s(X),Y) -> activate#(Y) activate#(n__add(X1,X2)) -> add#(activate(X1),X2) -> add#(0(),X) -> activate#(X) activate#(n__add(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> s#(X) activate#(n__add(X1,X2)) -> activate#(X1) -> activate#(n__from(X)) -> from#(X) activate#(n__add(X1,X2)) -> activate#(X1) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) activate#(n__add(X1,X2)) -> activate#(X1) -> activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__add(X1,X2)) -> activate#(X1) -> activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__add(X1,X2)) -> activate#(X1) -> activate#(n__add(X1,X2)) -> add#(activate(X1),X2) activate#(n__add(X1,X2)) -> activate#(X1) -> activate#(n__add(X1,X2)) -> activate#(X1) and#(true(),X) -> activate#(X) -> activate#(n__s(X)) -> s#(X) and#(true(),X) -> activate#(X) -> activate#(n__from(X)) -> from#(X) and#(true(),X) -> activate#(X) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) and#(true(),X) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X1) and#(true(),X) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X2) and#(true(),X) -> activate#(X) -> activate#(n__add(X1,X2)) -> add#(activate(X1),X2) and#(true(),X) -> activate#(X) -> activate#(n__add(X1,X2)) -> activate#(X1) SCC Processor: #sccs: 1 #rules: 13 #arcs: 99/324 DPs: from#(X) -> activate#(X) activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__add(X1,X2)) -> add#(activate(X1),X2) add#(0(),X) -> activate#(X) activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) -> activate#(Z) activate#(n__from(X)) -> from#(X) first#(s(X),cons(Y,Z)) -> activate#(X) first#(s(X),cons(Y,Z)) -> activate#(Y) add#(s(X),Y) -> activate#(Y) add#(s(X),Y) -> activate#(X) TRS: and(true(),X) -> activate(X) and(false(),Y) -> false() if(true(),X,Y) -> activate(X) if(false(),X,Y) -> activate(Y) add(0(),X) -> activate(X) add(s(X),Y) -> s(n__add(activate(X),activate(Y))) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z))) from(X) -> cons(activate(X),n__from(n__s(activate(X)))) add(X1,X2) -> n__add(X1,X2) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__add(X1,X2)) -> add(activate(X1),X2) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(X) activate(n__s(X)) -> s(X) activate(X) -> X Matrix Interpretation Processor: dim=1 interpretation: [from#](x0) = 1/2x0 + 1/2, [first#](x0, x1) = 1/2x0 + x1, [add#](x0, x1) = 1/2x0 + 1/2x1, [activate#](x0) = 1/2x0, [n__from](x0) = x0 + 2, [n__s](x0) = x0 + 1, [from](x0) = x0 + 2, [n__first](x0, x1) = 2x0 + 2x1 + 1, [cons](x0, x1) = 1/2x0 + 1/2x1 + 1/2, [nil] = 0, [first](x0, x1) = 2x0 + 2x1 + 1, [n__add](x0, x1) = x0 + x1 + 1, [s](x0) = x0 + 1, [add](x0, x1) = x0 + x1 + 1, [0] = 1, [if](x0, x1, x2) = 2x0 + x1 + 2x2 + 1, [false] = 1, [activate](x0) = x0, [and](x0, x1) = 2x1 + 3, [true] = 0 orientation: from#(X) = 1/2X + 1/2 >= 1/2X = activate#(X) activate#(n__add(X1,X2)) = 1/2X1 + 1/2X2 + 1/2 >= 1/2X1 = activate#(X1) activate#(n__add(X1,X2)) = 1/2X1 + 1/2X2 + 1/2 >= 1/2X1 + 1/2X2 = add#(activate(X1),X2) add#(0(),X) = 1/2X + 1/2 >= 1/2X = activate#(X) activate#(n__first(X1,X2)) = X1 + X2 + 1/2 >= 1/2X2 = activate#(X2) activate#(n__first(X1,X2)) = X1 + X2 + 1/2 >= 1/2X1 = activate#(X1) activate#(n__first(X1,X2)) = X1 + X2 + 1/2 >= 1/2X1 + X2 = first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) = 1/2X + 1/2Y + 1/2Z + 1 >= 1/2Z = activate#(Z) activate#(n__from(X)) = 1/2X + 1 >= 1/2X + 1/2 = from#(X) first#(s(X),cons(Y,Z)) = 1/2X + 1/2Y + 1/2Z + 1 >= 1/2X = activate#(X) first#(s(X),cons(Y,Z)) = 1/2X + 1/2Y + 1/2Z + 1 >= 1/2Y = activate#(Y) add#(s(X),Y) = 1/2X + 1/2Y + 1/2 >= 1/2Y = activate#(Y) add#(s(X),Y) = 1/2X + 1/2Y + 1/2 >= 1/2X = activate#(X) and(true(),X) = 2X + 3 >= X = activate(X) and(false(),Y) = 2Y + 3 >= 1 = false() if(true(),X,Y) = X + 2Y + 1 >= X = activate(X) if(false(),X,Y) = X + 2Y + 3 >= Y = activate(Y) add(0(),X) = X + 2 >= X = activate(X) add(s(X),Y) = X + Y + 2 >= X + Y + 2 = s(n__add(activate(X),activate(Y))) first(0(),X) = 2X + 3 >= 0 = nil() first(s(X),cons(Y,Z)) = 2X + Y + Z + 4 >= X + 1/2Y + Z + 1 = cons(activate(Y),n__first(activate(X),activate(Z))) from(X) = X + 2 >= X + 2 = cons(activate(X),n__from(n__s(activate(X)))) add(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = n__add(X1,X2) first(X1,X2) = 2X1 + 2X2 + 1 >= 2X1 + 2X2 + 1 = n__first(X1,X2) from(X) = X + 2 >= X + 2 = n__from(X) s(X) = X + 1 >= X + 1 = n__s(X) activate(n__add(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = add(activate(X1),X2) activate(n__first(X1,X2)) = 2X1 + 2X2 + 1 >= 2X1 + 2X2 + 1 = first(activate(X1),activate(X2)) activate(n__from(X)) = X + 2 >= X + 2 = from(X) activate(n__s(X)) = X + 1 >= X + 1 = s(X) activate(X) = X >= X = X problem: DPs: TRS: and(true(),X) -> activate(X) and(false(),Y) -> false() if(true(),X,Y) -> activate(X) if(false(),X,Y) -> activate(Y) add(0(),X) -> activate(X) add(s(X),Y) -> s(n__add(activate(X),activate(Y))) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z))) from(X) -> cons(activate(X),n__from(n__s(activate(X)))) add(X1,X2) -> n__add(X1,X2) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__add(X1,X2)) -> add(activate(X1),X2) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(X) activate(n__s(X)) -> s(X) activate(X) -> X Qed