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 Arctic Interpretation Processor: dimension: 1 interpretation: [from#](x0) = x0 + 0, [first#](x0, x1) = 4x0 + 4x1 + 0, [add#](x0, x1) = x0 + 3x1 + 0, [activate#](x0) = x0 + 0, [n__from](x0) = x0, [n__s](x0) = x0, [from](x0) = x0, [n__first](x0, x1) = 4x0 + 4x1, [cons](x0, x1) = x0 + x1, [nil] = 0, [first](x0, x1) = 4x0 + 4x1, [n__add](x0, x1) = x0 + 3x1, [s](x0) = x0, [add](x0, x1) = x0 + 3x1, [0] = 1, [if](x0, x1, x2) = 2x1 + x2 + 4, [false] = 3, [activate](x0) = x0, [and](x0, x1) = x0 + 4x1 + 2, [true] = 2 orientation: from#(X) = X + 0 >= X + 0 = activate#(X) activate#(n__add(X1,X2)) = X1 + 3X2 + 0 >= X1 + 0 = activate#(X1) activate#(n__add(X1,X2)) = X1 + 3X2 + 0 >= X1 + 3X2 + 0 = add#(activate(X1),X2) add#(0(),X) = 3X + 1 >= X + 0 = activate#(X) activate#(n__first(X1,X2)) = 4X1 + 4X2 + 0 >= X2 + 0 = activate#(X2) activate#(n__first(X1,X2)) = 4X1 + 4X2 + 0 >= X1 + 0 = activate#(X1) activate#(n__first(X1,X2)) = 4X1 + 4X2 + 0 >= 4X1 + 4X2 + 0 = first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) = 4X + 4Y + 4Z + 0 >= Z + 0 = activate#(Z) activate#(n__from(X)) = X + 0 >= X + 0 = from#(X) first#(s(X),cons(Y,Z)) = 4X + 4Y + 4Z + 0 >= X + 0 = activate#(X) first#(s(X),cons(Y,Z)) = 4X + 4Y + 4Z + 0 >= Y + 0 = activate#(Y) add#(s(X),Y) = X + 3Y + 0 >= Y + 0 = activate#(Y) add#(s(X),Y) = X + 3Y + 0 >= X + 0 = activate#(X) and(true(),X) = 4X + 2 >= X = activate(X) and(false(),Y) = 4Y + 3 >= 3 = false() if(true(),X,Y) = 2X + Y + 4 >= X = activate(X) if(false(),X,Y) = 2X + Y + 4 >= Y = activate(Y) add(0(),X) = 3X + 1 >= X = activate(X) add(s(X),Y) = X + 3Y >= X + 3Y = s(n__add(activate(X),activate(Y))) first(0(),X) = 4X + 5 >= 0 = nil() first(s(X),cons(Y,Z)) = 4X + 4Y + 4Z >= 4X + Y + 4Z = cons(activate(Y),n__first(activate(X),activate(Z))) from(X) = X >= X = cons(activate(X),n__from(n__s(activate(X)))) add(X1,X2) = X1 + 3X2 >= X1 + 3X2 = n__add(X1,X2) first(X1,X2) = 4X1 + 4X2 >= 4X1 + 4X2 = n__first(X1,X2) from(X) = X >= X = n__from(X) s(X) = X >= X = n__s(X) activate(n__add(X1,X2)) = X1 + 3X2 >= X1 + 3X2 = add(activate(X1),X2) activate(n__first(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = first(activate(X1),activate(X2)) activate(n__from(X)) = X >= X = from(X) activate(n__s(X)) = X >= X = s(X) activate(X) = X >= X = 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#(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 Arctic Interpretation Processor: dimension: 1 interpretation: [from#](x0) = x0, [first#](x0, x1) = x0 + x1, [add#](x0, x1) = x0 + x1, [activate#](x0) = x0, [n__from](x0) = x0, [n__s](x0) = x0, [from](x0) = x0, [n__first](x0, x1) = x0 + x1, [cons](x0, x1) = x0 + x1, [nil] = 1, [first](x0, x1) = x0 + x1, [n__add](x0, x1) = 6x0 + 4x1, [s](x0) = x0, [add](x0, x1) = 6x0 + 4x1, [0] = 1, [if](x0, x1, x2) = 4x0 + 4x1 + x2 + 0, [false] = 0, [activate](x0) = x0, [and](x0, x1) = x1 + 0, [true] = 1 orientation: from#(X) = X >= X = activate#(X) activate#(n__add(X1,X2)) = 6X1 + 4X2 >= X1 = activate#(X1) activate#(n__add(X1,X2)) = 6X1 + 4X2 >= X1 + X2 = add#(activate(X1),X2) activate#(n__first(X1,X2)) = X1 + X2 >= X2 = activate#(X2) activate#(n__first(X1,X2)) = X1 + X2 >= X1 = activate#(X1) activate#(n__first(X1,X2)) = X1 + X2 >= X1 + X2 = first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) = X + Y + Z >= Z = activate#(Z) activate#(n__from(X)) = X >= X = from#(X) first#(s(X),cons(Y,Z)) = X + Y + Z >= X = activate#(X) first#(s(X),cons(Y,Z)) = X + Y + Z >= Y = activate#(Y) add#(s(X),Y) = X + Y >= Y = activate#(Y) add#(s(X),Y) = X + Y >= X = activate#(X) and(true(),X) = X + 0 >= X = activate(X) and(false(),Y) = Y + 0 >= 0 = false() if(true(),X,Y) = 4X + Y + 5 >= X = activate(X) if(false(),X,Y) = 4X + Y + 4 >= Y = activate(Y) add(0(),X) = 4X + 7 >= X = activate(X) add(s(X),Y) = 6X + 4Y >= 6X + 4Y = s(n__add(activate(X),activate(Y))) first(0(),X) = X + 1 >= 1 = nil() first(s(X),cons(Y,Z)) = X + Y + Z >= X + Y + Z = cons(activate(Y),n__first(activate(X),activate(Z))) from(X) = X >= X = cons(activate(X),n__from(n__s(activate(X)))) add(X1,X2) = 6X1 + 4X2 >= 6X1 + 4X2 = n__add(X1,X2) first(X1,X2) = X1 + X2 >= X1 + X2 = n__first(X1,X2) from(X) = X >= X = n__from(X) s(X) = X >= X = n__s(X) activate(n__add(X1,X2)) = 6X1 + 4X2 >= 6X1 + 4X2 = add(activate(X1),X2) activate(n__first(X1,X2)) = X1 + X2 >= X1 + X2 = first(activate(X1),activate(X2)) activate(n__from(X)) = X >= X = from(X) activate(n__s(X)) = X >= X = s(X) activate(X) = X >= X = X problem: DPs: from#(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 SCC Processor: #sccs: 1 #rules: 8 #arcs: 67/100 DPs: from#(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) 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 Arctic Interpretation Processor: dimension: 1 interpretation: [from#](x0) = 6x0 + 0, [first#](x0, x1) = 4x0 + 3x1 + 2, [activate#](x0) = 2x0 + 0, [n__from](x0) = 4x0, [n__s](x0) = x0, [from](x0) = 4x0, [n__first](x0, x1) = 4x0 + 1x1 + 0, [cons](x0, x1) = 4x0 + x1, [nil] = 6, [first](x0, x1) = 4x0 + 1x1 + 0, [n__add](x0, x1) = x0 + 1x1, [s](x0) = x0, [add](x0, x1) = x0 + 1x1, [0] = 2, [if](x0, x1, x2) = x0 + 4x1 + 4x2 + 10, [false] = 6, [activate](x0) = x0, [and](x0, x1) = 5x1 + 11, [true] = 3 orientation: from#(X) = 6X + 0 >= 2X + 0 = activate#(X) activate#(n__first(X1,X2)) = 6X1 + 3X2 + 2 >= 2X2 + 0 = activate#(X2) activate#(n__first(X1,X2)) = 6X1 + 3X2 + 2 >= 2X1 + 0 = activate#(X1) activate#(n__first(X1,X2)) = 6X1 + 3X2 + 2 >= 4X1 + 3X2 + 2 = first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) = 4X + 7Y + 3Z + 2 >= 2Z + 0 = activate#(Z) activate#(n__from(X)) = 6X + 0 >= 6X + 0 = from#(X) first#(s(X),cons(Y,Z)) = 4X + 7Y + 3Z + 2 >= 2X + 0 = activate#(X) first#(s(X),cons(Y,Z)) = 4X + 7Y + 3Z + 2 >= 2Y + 0 = activate#(Y) and(true(),X) = 5X + 11 >= X = activate(X) and(false(),Y) = 5Y + 11 >= 6 = false() if(true(),X,Y) = 4X + 4Y + 10 >= X = activate(X) if(false(),X,Y) = 4X + 4Y + 10 >= Y = activate(Y) add(0(),X) = 1X + 2 >= X = activate(X) add(s(X),Y) = X + 1Y >= X + 1Y = s(n__add(activate(X),activate(Y))) first(0(),X) = 1X + 6 >= 6 = nil() first(s(X),cons(Y,Z)) = 4X + 5Y + 1Z + 0 >= 4X + 4Y + 1Z + 0 = cons(activate(Y),n__first(activate(X),activate(Z))) from(X) = 4X >= 4X = cons(activate(X),n__from(n__s(activate(X)))) add(X1,X2) = X1 + 1X2 >= X1 + 1X2 = n__add(X1,X2) first(X1,X2) = 4X1 + 1X2 + 0 >= 4X1 + 1X2 + 0 = n__first(X1,X2) from(X) = 4X >= 4X = n__from(X) s(X) = X >= X = n__s(X) activate(n__add(X1,X2)) = X1 + 1X2 >= X1 + 1X2 = add(activate(X1),X2) activate(n__first(X1,X2)) = 4X1 + 1X2 + 0 >= 4X1 + 1X2 + 0 = first(activate(X1),activate(X2)) activate(n__from(X)) = 4X >= 4X = from(X) activate(n__s(X)) = X >= X = s(X) activate(X) = X >= X = X problem: DPs: from#(X) -> activate#(X) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) activate#(n__from(X)) -> from#(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: 2 #arcs: 28/9 DPs: from#(X) -> activate#(X) activate#(n__from(X)) -> from#(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 Subterm Criterion Processor: simple projection: pi(activate#) = 0 pi(from#) = 0 problem: DPs: from#(X) -> 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: 2/1