/export/starexec/sandbox/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- 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) = x0 + 0, [first#](x0, x1) = x0 + x1 + 0, [add#](x0, x1) = x0 + x1 + 0, [activate#](x0) = x0 + 0, [n__from](x0) = x0 + 0, [n__s](x0) = x0, [from](x0) = x0 + 0, [n__first](x0, x1) = 2x0 + 2x1 + 2, [cons](x0, x1) = x0 + x1 + 0, [nil] = 2, [first](x0, x1) = 2x0 + 2x1 + 2, [n__add](x0, x1) = x0 + x1 + 0, [s](x0) = x0 + 0, [add](x0, x1) = x0 + x1 + 0, [0] = 0, [activate](x0) = x0 + 0 orientation: from#(X) = X + 0 >= X + 0 = activate#(X) 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#(0(),X) = X + 0 >= X + 0 = activate#(X) activate#(n__first(X1,X2)) = 2X1 + 2X2 + 2 >= X2 + 0 = activate#(X2) activate#(n__first(X1,X2)) = 2X1 + 2X2 + 2 >= X1 + 0 = activate#(X1) activate#(n__first(X1,X2)) = 2X1 + 2X2 + 2 >= 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)) = X + 0 >= X + 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 + Y + 0 >= Y + 0 = activate#(Y) add#(s(X),Y) = X + Y + 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)) = 2X1 + 2X2 + 2 >= 2X1 + 2X2 + 2 = 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 + 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 + 0 = n__add(X1,X2) first(0(),X) = 2X + 2 >= 2 = nil() first(s(X),cons(Y,Z)) = 2X + 2Y + 2Z + 2 >= 2X + Y + 2Z + 2 = cons(activate(Y),n__first(activate(X),activate(Z))) first(X1,X2) = 2X1 + 2X2 + 2 >= 2X1 + 2X2 + 2 = n__first(X1,X2) from(X) = X + 0 >= X + 0 = cons(activate(X),n__from(n__s(activate(X)))) from(X) = X + 0 >= X + 0 = n__from(X) s(X) = X + 0 >= 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) add#(0(),X) -> activate#(X) 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) Restore Modifier: 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) 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: 7 #arcs: 67/100 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__from(X)) -> from#(X) 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__from(X)) -> from#(X) 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) = x0 + 0, [add#](x0, x1) = 5x0 + x1 + 0, [activate#](x0) = x0, [n__from](x0) = x0 + 0, [n__s](x0) = x0 + 0, [from](x0) = x0 + 0, [n__first](x0, x1) = 2x0 + x1 + 4, [cons](x0, x1) = x0 + x1 + 0, [nil] = 0, [first](x0, x1) = 2x0 + x1 + 4, [n__add](x0, x1) = 6x0 + x1 + 0, [s](x0) = x0 + 0, [add](x0, x1) = 6x0 + x1 + 0, [0] = 0, [activate](x0) = x0 orientation: from#(X) = X + 0 >= X = activate#(X) activate#(n__add(X1,X2)) = 6X1 + X2 + 0 >= X1 = activate#(X1) activate#(n__add(X1,X2)) = 6X1 + X2 + 0 >= 5X1 + X2 + 0 = add#(activate(X1),X2) add#(0(),X) = X + 5 >= X = activate#(X) activate#(n__from(X)) = X + 0 >= X + 0 = from#(X) add#(s(X),Y) = 5X + Y + 5 >= Y = activate#(Y) add#(s(X),Y) = 5X + Y + 5 >= X = activate#(X) activate(n__add(X1,X2)) = 6X1 + X2 + 0 >= 6X1 + X2 + 0 = add(activate(X1),X2) activate(n__first(X1,X2)) = 2X1 + X2 + 4 >= 2X1 + X2 + 4 = 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 >= X = X add(0(),X) = X + 6 >= X = activate(X) add(s(X),Y) = 6X + Y + 6 >= 6X + Y + 0 = s(n__add(activate(X),activate(Y))) add(X1,X2) = 6X1 + X2 + 0 >= 6X1 + X2 + 0 = n__add(X1,X2) first(0(),X) = X + 4 >= 0 = nil() first(s(X),cons(Y,Z)) = 2X + Y + Z + 4 >= 2X + Y + Z + 4 = cons(activate(Y),n__first(activate(X),activate(Z))) first(X1,X2) = 2X1 + X2 + 4 >= 2X1 + X2 + 4 = n__first(X1,X2) from(X) = X + 0 >= X + 0 = cons(activate(X),n__from(n__s(activate(X)))) from(X) = X + 0 >= X + 0 = n__from(X) s(X) = X + 0 >= X + 0 = n__s(X) problem: DPs: from#(X) -> activate#(X) activate#(n__add(X1,X2)) -> add#(activate(X1),X2) add#(0(),X) -> activate#(X) activate#(n__from(X)) -> from#(X) add#(s(X),Y) -> 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) Restore Modifier: DPs: from#(X) -> activate#(X) activate#(n__add(X1,X2)) -> add#(activate(X1),X2) add#(0(),X) -> activate#(X) activate#(n__from(X)) -> from#(X) add#(s(X),Y) -> 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 Size-Change Termination Processor: 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 The DP: from#(X) -> activate#(X) has the edges: 0 >= 0 The DP: activate#(n__add(X1,X2)) -> add#(activate(X1),X2) has the edges: 0 > 1 The DP: add#(0(),X) -> activate#(X) has the edges: 1 >= 0 The DP: activate#(n__from(X)) -> from#(X) has the edges: 0 > 0 The DP: add#(s(X),Y) -> activate#(Y) has the edges: 1 >= 0 Qed