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 Usable Rule Processor: 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: 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 add(0(),X) -> activate(X) add(s(X),Y) -> s(n__add(activate(X),activate(Y))) add(X1,X2) -> n__add(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z))) first(X1,X2) -> n__first(X1,X2) from(X) -> cons(activate(X),n__from(n__s(activate(X)))) from(X) -> n__from(X) s(X) -> n__s(X) Arctic Interpretation Processor: dimension: 1 usable rules: 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 add(0(),X) -> activate(X) add(s(X),Y) -> s(n__add(activate(X),activate(Y))) add(X1,X2) -> n__add(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z))) first(X1,X2) -> n__first(X1,X2) from(X) -> cons(activate(X),n__from(n__s(activate(X)))) from(X) -> n__from(X) s(X) -> n__s(X) interpretation: [from#](x0) = 5x0 + 0, [first#](x0, x1) = x0 + x1 + 0, [add#](x0, x1) = x0 + 5x1 + 2, [activate#](x0) = x0 + 0, [n__from](x0) = 7x0, [n__s](x0) = x0, [from](x0) = 7x0, [n__first](x0, x1) = x0 + x1 + 0, [cons](x0, x1) = x0 + x1, [nil] = 0, [first](x0, x1) = x0 + x1 + 0, [n__add](x0, x1) = x0 + 5x1 + 2, [s](x0) = x0, [add](x0, x1) = x0 + 5x1 + 2, [0] = 0, [activate](x0) = x0 orientation: from#(X) = 5X + 0 >= X + 0 = activate#(X) activate#(n__add(X1,X2)) = X1 + 5X2 + 2 >= X1 + 0 = activate#(X1) activate#(n__add(X1,X2)) = X1 + 5X2 + 2 >= X1 + 5X2 + 2 = add#(activate(X1),X2) add#(0(),X) = 5X + 2 >= X + 0 = activate#(X) activate#(n__first(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = activate#(X2) activate#(n__first(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = activate#(X1) activate#(n__first(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) = X + Y + Z + 0 >= Z + 0 = activate#(Z) activate#(n__from(X)) = 7X + 0 >= 5X + 0 = from#(X) first#(s(X),cons(Y,Z)) = X + Y + Z + 0 >= X + 0 = activate#(X) first#(s(X),cons(Y,Z)) = X + Y + Z + 0 >= Y + 0 = activate#(Y) add#(s(X),Y) = X + 5Y + 2 >= Y + 0 = activate#(Y) add#(s(X),Y) = X + 5Y + 2 >= X + 0 = activate#(X) activate(n__add(X1,X2)) = X1 + 5X2 + 2 >= X1 + 5X2 + 2 = add(activate(X1),X2) activate(n__first(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = first(activate(X1),activate(X2)) activate(n__from(X)) = 7X >= 7X = from(X) activate(n__s(X)) = X >= X = s(X) activate(X) = X >= X = X add(0(),X) = 5X + 2 >= X = activate(X) add(s(X),Y) = X + 5Y + 2 >= X + 5Y + 2 = s(n__add(activate(X),activate(Y))) add(X1,X2) = X1 + 5X2 + 2 >= X1 + 5X2 + 2 = n__add(X1,X2) first(0(),X) = X + 0 >= 0 = nil() first(s(X),cons(Y,Z)) = X + Y + Z + 0 >= X + Y + Z + 0 = cons(activate(Y),n__first(activate(X),activate(Z))) first(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = n__first(X1,X2) from(X) = 7X >= 7X = cons(activate(X),n__from(n__s(activate(X)))) from(X) = 7X >= 7X = n__from(X) s(X) = X >= X = n__s(X) problem: DPs: 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)) 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#(X) TRS: 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 add(0(),X) -> activate(X) add(s(X),Y) -> s(n__add(activate(X),activate(Y))) add(X1,X2) -> n__add(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z))) first(X1,X2) -> n__first(X1,X2) from(X) -> cons(activate(X),n__from(n__s(activate(X)))) from(X) -> n__from(X) s(X) -> n__s(X) Restore Modifier: DPs: 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)) 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#(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 Usable Rule Processor: DPs: 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)) 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#(X) TRS: 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 add(0(),X) -> activate(X) add(s(X),Y) -> s(n__add(activate(X),activate(Y))) add(X1,X2) -> n__add(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z))) first(X1,X2) -> n__first(X1,X2) from(X) -> cons(activate(X),n__from(n__s(activate(X)))) from(X) -> n__from(X) s(X) -> n__s(X) Arctic Interpretation Processor: dimension: 1 usable rules: 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 add(0(),X) -> activate(X) add(s(X),Y) -> s(n__add(activate(X),activate(Y))) add(X1,X2) -> n__add(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z))) first(X1,X2) -> n__first(X1,X2) from(X) -> cons(activate(X),n__from(n__s(activate(X)))) from(X) -> n__from(X) s(X) -> n__s(X) interpretation: [from#](x0) = x0, [first#](x0, x1) = x0 + x1 + 0, [add#](x0, x1) = x0 + 0, [activate#](x0) = x0, [n__from](x0) = 6x0 + 0, [n__s](x0) = x0, [from](x0) = 6x0 + 0, [n__first](x0, x1) = x0 + 4x1 + 0, [cons](x0, x1) = x0 + x1, [nil] = 1, [first](x0, x1) = x0 + 4x1 + 0, [n__add](x0, x1) = x0 + x1 + 0, [s](x0) = x0, [add](x0, x1) = x0 + x1 + 0, [0] = 2, [activate](x0) = x0 orientation: from#(X) = X >= X = activate#(X) activate#(n__add(X1,X2)) = X1 + X2 + 0 >= X1 = activate#(X1) activate#(n__add(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = add#(activate(X1),X2) activate#(n__first(X1,X2)) = X1 + 4X2 + 0 >= X2 = activate#(X2) activate#(n__first(X1,X2)) = X1 + 4X2 + 0 >= X1 = activate#(X1) activate#(n__first(X1,X2)) = X1 + 4X2 + 0 >= X1 + X2 + 0 = first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) = X + Y + Z + 0 >= Z = activate#(Z) activate#(n__from(X)) = 6X + 0 >= X = from#(X) first#(s(X),cons(Y,Z)) = X + Y + Z + 0 >= X = activate#(X) first#(s(X),cons(Y,Z)) = X + Y + Z + 0 >= Y = activate#(Y) add#(s(X),Y) = X + 0 >= X = activate#(X) activate(n__add(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = add(activate(X1),X2) activate(n__first(X1,X2)) = X1 + 4X2 + 0 >= X1 + 4X2 + 0 = first(activate(X1),activate(X2)) activate(n__from(X)) = 6X + 0 >= 6X + 0 = from(X) activate(n__s(X)) = X >= X = s(X) activate(X) = X >= X = X add(0(),X) = X + 2 >= X = activate(X) add(s(X),Y) = X + Y + 0 >= X + Y + 0 = s(n__add(activate(X),activate(Y))) add(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = n__add(X1,X2) first(0(),X) = 4X + 2 >= 1 = nil() first(s(X),cons(Y,Z)) = X + 4Y + 4Z + 0 >= X + Y + 4Z + 0 = cons(activate(Y),n__first(activate(X),activate(Z))) first(X1,X2) = X1 + 4X2 + 0 >= X1 + 4X2 + 0 = n__first(X1,X2) from(X) = 6X + 0 >= 6X + 0 = cons(activate(X),n__from(n__s(activate(X)))) from(X) = 6X + 0 >= 6X + 0 = n__from(X) s(X) = X >= X = n__s(X) problem: DPs: 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#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) -> activate#(Z) first#(s(X),cons(Y,Z)) -> activate#(X) first#(s(X),cons(Y,Z)) -> activate#(Y) add#(s(X),Y) -> activate#(X) TRS: 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 add(0(),X) -> activate(X) add(s(X),Y) -> s(n__add(activate(X),activate(Y))) add(X1,X2) -> n__add(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z))) first(X1,X2) -> n__first(X1,X2) from(X) -> cons(activate(X),n__from(n__s(activate(X)))) from(X) -> n__from(X) s(X) -> n__s(X) Restore Modifier: DPs: 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#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) -> activate#(Z) first#(s(X),cons(Y,Z)) -> activate#(X) first#(s(X),cons(Y,Z)) -> 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 SCC Processor: #sccs: 1 #rules: 8 #arcs: 67/81 DPs: activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__add(X1,X2)) -> add#(activate(X1),X2) add#(s(X),Y) -> activate#(X) 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) first#(s(X),cons(Y,Z)) -> activate#(X) first#(s(X),cons(Y,Z)) -> activate#(Y) 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 Usable Rule Processor: DPs: activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__add(X1,X2)) -> add#(activate(X1),X2) add#(s(X),Y) -> activate#(X) 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) first#(s(X),cons(Y,Z)) -> activate#(X) first#(s(X),cons(Y,Z)) -> activate#(Y) TRS: 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 add(0(),X) -> activate(X) add(s(X),Y) -> s(n__add(activate(X),activate(Y))) add(X1,X2) -> n__add(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z))) first(X1,X2) -> n__first(X1,X2) from(X) -> cons(activate(X),n__from(n__s(activate(X)))) from(X) -> n__from(X) s(X) -> n__s(X) Arctic Interpretation Processor: dimension: 1 usable rules: 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 add(0(),X) -> activate(X) add(s(X),Y) -> s(n__add(activate(X),activate(Y))) add(X1,X2) -> n__add(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z))) first(X1,X2) -> n__first(X1,X2) from(X) -> cons(activate(X),n__from(n__s(activate(X)))) from(X) -> n__from(X) s(X) -> n__s(X) interpretation: [first#](x0, x1) = x0 + x1 + 0, [add#](x0, x1) = x0 + x1 + 0, [activate#](x0) = x0 + 0, [n__from](x0) = 4x0 + 4, [n__s](x0) = x0 + 0, [from](x0) = 4x0 + 4, [n__first](x0, x1) = x0 + x1 + 0, [cons](x0, x1) = 1x0 + x1 + 2, [nil] = 0, [first](x0, x1) = x0 + x1 + 0, [n__add](x0, x1) = x0 + x1, [s](x0) = x0 + 0, [add](x0, x1) = x0 + x1 + 0, [0] = 0, [activate](x0) = x0 + 0 orientation: activate#(n__add(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = activate#(X1) activate#(n__add(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = add#(activate(X1),X2) add#(s(X),Y) = X + Y + 0 >= X + 0 = activate#(X) activate#(n__first(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = activate#(X1) activate#(n__first(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) = X + 1Y + Z + 2 >= Z + 0 = activate#(Z) first#(s(X),cons(Y,Z)) = X + 1Y + Z + 2 >= X + 0 = activate#(X) first#(s(X),cons(Y,Z)) = X + 1Y + Z + 2 >= Y + 0 = activate#(Y) activate(n__add(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = add(activate(X1),X2) activate(n__first(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = first(activate(X1),activate(X2)) activate(n__from(X)) = 4X + 4 >= 4X + 4 = from(X) activate(n__s(X)) = X + 0 >= X + 0 = s(X) activate(X) = X + 0 >= X = X add(0(),X) = X + 0 >= X + 0 = activate(X) add(s(X),Y) = X + Y + 0 >= X + Y + 0 = s(n__add(activate(X),activate(Y))) add(X1,X2) = X1 + X2 + 0 >= X1 + X2 = n__add(X1,X2) first(0(),X) = X + 0 >= 0 = nil() first(s(X),cons(Y,Z)) = X + 1Y + Z + 2 >= X + 1Y + Z + 2 = cons(activate(Y),n__first(activate(X),activate(Z))) first(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = n__first(X1,X2) from(X) = 4X + 4 >= 4X + 4 = cons(activate(X),n__from(n__s(activate(X)))) from(X) = 4X + 4 >= 4X + 4 = n__from(X) s(X) = X + 0 >= X + 0 = n__s(X) problem: DPs: activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__add(X1,X2)) -> add#(activate(X1),X2) add#(s(X),Y) -> activate#(X) 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) first#(s(X),cons(Y,Z)) -> activate#(X) TRS: 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 add(0(),X) -> activate(X) add(s(X),Y) -> s(n__add(activate(X),activate(Y))) add(X1,X2) -> n__add(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z))) first(X1,X2) -> n__first(X1,X2) from(X) -> cons(activate(X),n__from(n__s(activate(X)))) from(X) -> n__from(X) s(X) -> n__s(X) Restore Modifier: DPs: activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__add(X1,X2)) -> add#(activate(X1),X2) add#(s(X),Y) -> activate#(X) 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) first#(s(X),cons(Y,Z)) -> 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 Usable Rule Processor: DPs: activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__add(X1,X2)) -> add#(activate(X1),X2) add#(s(X),Y) -> activate#(X) 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) first#(s(X),cons(Y,Z)) -> activate#(X) TRS: 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 add(0(),X) -> activate(X) add(s(X),Y) -> s(n__add(activate(X),activate(Y))) add(X1,X2) -> n__add(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z))) first(X1,X2) -> n__first(X1,X2) from(X) -> cons(activate(X),n__from(n__s(activate(X)))) from(X) -> n__from(X) s(X) -> n__s(X) Arctic Interpretation Processor: dimension: 1 usable rules: 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 add(0(),X) -> activate(X) add(s(X),Y) -> s(n__add(activate(X),activate(Y))) add(X1,X2) -> n__add(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z))) first(X1,X2) -> n__first(X1,X2) from(X) -> cons(activate(X),n__from(n__s(activate(X)))) from(X) -> n__from(X) s(X) -> n__s(X) interpretation: [first#](x0, x1) = x0 + x1 + 0, [add#](x0, x1) = x0 + 0, [activate#](x0) = x0 + 0, [n__from](x0) = x0, [n__s](x0) = x0 + -8, [from](x0) = x0 + 0, [n__first](x0, x1) = 1x0 + 1x1 + 1, [cons](x0, x1) = -7x0 + x1 + -8, [nil] = 1, [first](x0, x1) = 1x0 + 1x1 + 1, [n__add](x0, x1) = x0 + x1 + 0, [s](x0) = x0 + 0, [add](x0, x1) = x0 + x1 + 0, [0] = 5, [activate](x0) = x0 + 0 orientation: activate#(n__add(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = activate#(X1) activate#(n__add(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = add#(activate(X1),X2) add#(s(X),Y) = X + 0 >= X + 0 = activate#(X) activate#(n__first(X1,X2)) = 1X1 + 1X2 + 1 >= X1 + 0 = activate#(X1) activate#(n__first(X1,X2)) = 1X1 + 1X2 + 1 >= X1 + X2 + 0 = first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) = X + -7Y + Z + 0 >= Z + 0 = activate#(Z) first#(s(X),cons(Y,Z)) = X + -7Y + Z + 0 >= X + 0 = activate#(X) activate(n__add(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = add(activate(X1),X2) activate(n__first(X1,X2)) = 1X1 + 1X2 + 1 >= 1X1 + 1X2 + 1 = first(activate(X1),activate(X2)) activate(n__from(X)) = X + 0 >= X + 0 = from(X) activate(n__s(X)) = X + 0 >= X + 0 = s(X) activate(X) = X + 0 >= X = X add(0(),X) = X + 5 >= X + 0 = activate(X) add(s(X),Y) = X + Y + 0 >= X + Y + 0 = s(n__add(activate(X),activate(Y))) add(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = n__add(X1,X2) first(0(),X) = 1X + 6 >= 1 = nil() first(s(X),cons(Y,Z)) = 1X + -6Y + 1Z + 1 >= 1X + -7Y + 1Z + 1 = cons(activate(Y),n__first(activate(X),activate(Z))) first(X1,X2) = 1X1 + 1X2 + 1 >= 1X1 + 1X2 + 1 = n__first(X1,X2) from(X) = X + 0 >= X + 0 = cons(activate(X),n__from(n__s(activate(X)))) from(X) = X + 0 >= X = n__from(X) s(X) = X + 0 >= X + -8 = n__s(X) problem: DPs: activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__add(X1,X2)) -> add#(activate(X1),X2) add#(s(X),Y) -> activate#(X) first#(s(X),cons(Y,Z)) -> activate#(Z) first#(s(X),cons(Y,Z)) -> activate#(X) TRS: 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 add(0(),X) -> activate(X) add(s(X),Y) -> s(n__add(activate(X),activate(Y))) add(X1,X2) -> n__add(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z))) first(X1,X2) -> n__first(X1,X2) from(X) -> cons(activate(X),n__from(n__s(activate(X)))) from(X) -> n__from(X) s(X) -> n__s(X) Restore Modifier: DPs: activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__add(X1,X2)) -> add#(activate(X1),X2) add#(s(X),Y) -> activate#(X) first#(s(X),cons(Y,Z)) -> activate#(Z) first#(s(X),cons(Y,Z)) -> 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 SCC Processor: #sccs: 1 #rules: 3 #arcs: 28/25 DPs: activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__add(X1,X2)) -> add#(activate(X1),X2) 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 Usable Rule Processor: DPs: activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__add(X1,X2)) -> add#(activate(X1),X2) add#(s(X),Y) -> activate#(X) TRS: 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 add(0(),X) -> activate(X) add(s(X),Y) -> s(n__add(activate(X),activate(Y))) add(X1,X2) -> n__add(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z))) first(X1,X2) -> n__first(X1,X2) from(X) -> cons(activate(X),n__from(n__s(activate(X)))) from(X) -> n__from(X) s(X) -> n__s(X) Arctic Interpretation Processor: dimension: 1 usable rules: 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 add(0(),X) -> activate(X) add(s(X),Y) -> s(n__add(activate(X),activate(Y))) add(X1,X2) -> n__add(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z))) first(X1,X2) -> n__first(X1,X2) from(X) -> cons(activate(X),n__from(n__s(activate(X)))) from(X) -> n__from(X) s(X) -> n__s(X) interpretation: [add#](x0, x1) = x0 + 0, [activate#](x0) = x0 + 1, [n__from](x0) = 0, [n__s](x0) = x0, [from](x0) = 5, [n__first](x0, x1) = x0 + x1 + 0, [cons](x0, x1) = 4, [nil] = 2, [first](x0, x1) = x0 + x1 + 1, [n__add](x0, x1) = 1x0 + x1 + 7, [s](x0) = x0 + 1, [add](x0, x1) = 1x0 + x1 + 7, [0] = 2, [activate](x0) = x0 + 6 orientation: activate#(n__add(X1,X2)) = 1X1 + X2 + 7 >= X1 + 1 = activate#(X1) activate#(n__add(X1,X2)) = 1X1 + X2 + 7 >= X1 + 6 = add#(activate(X1),X2) add#(s(X),Y) = X + 1 >= X + 1 = activate#(X) activate(n__add(X1,X2)) = 1X1 + X2 + 7 >= 1X1 + X2 + 7 = add(activate(X1),X2) activate(n__first(X1,X2)) = X1 + X2 + 6 >= X1 + X2 + 6 = first(activate(X1),activate(X2)) activate(n__from(X)) = 6 >= 5 = from(X) activate(n__s(X)) = X + 6 >= X + 1 = s(X) activate(X) = X + 6 >= X = X add(0(),X) = X + 7 >= X + 6 = activate(X) add(s(X),Y) = 1X + Y + 7 >= 1X + Y + 7 = s(n__add(activate(X),activate(Y))) add(X1,X2) = 1X1 + X2 + 7 >= 1X1 + X2 + 7 = n__add(X1,X2) first(0(),X) = X + 2 >= 2 = nil() first(s(X),cons(Y,Z)) = X + 4 >= 4 = cons(activate(Y),n__first(activate(X),activate(Z))) first(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 0 = n__first(X1,X2) from(X) = 5 >= 4 = cons(activate(X),n__from(n__s(activate(X)))) from(X) = 5 >= 0 = n__from(X) s(X) = X + 1 >= X = n__s(X) problem: DPs: add#(s(X),Y) -> activate#(X) TRS: 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 add(0(),X) -> activate(X) add(s(X),Y) -> s(n__add(activate(X),activate(Y))) add(X1,X2) -> n__add(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z))) first(X1,X2) -> n__first(X1,X2) from(X) -> cons(activate(X),n__from(n__s(activate(X)))) from(X) -> n__from(X) s(X) -> n__s(X) Restore Modifier: DPs: 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 SCC Processor: #sccs: 0 #rules: 0 #arcs: 5/1