YES Problem: from(X) -> cons(X,n__from(s(X))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,n__cons(Y,Z))) -> rcons(posrecip(activate(Y)),2ndsneg(N,activate(Z))) 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,n__cons(Y,Z))) -> rcons(negrecip(activate(Y)),2ndspos(N,activate(Z))) pi(X) -> 2ndspos(X,from(0())) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) square(X) -> times(X,X) from(X) -> n__from(X) cons(X1,X2) -> n__cons(X1,X2) activate(n__from(X)) -> from(X) activate(n__cons(X1,X2)) -> cons(X1,X2) activate(X) -> X Proof: DP Processor: DPs: from#(X) -> cons#(X,n__from(s(X))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> activate#(Z) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> 2ndsneg#(N,activate(Z)) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> activate#(Y) 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> activate#(Z) 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> 2ndspos#(N,activate(Z)) 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> activate#(Y) pi#(X) -> from#(0()) pi#(X) -> 2ndspos#(X,from(0())) plus#(s(X),Y) -> plus#(X,Y) times#(s(X),Y) -> times#(X,Y) times#(s(X),Y) -> plus#(Y,times(X,Y)) square#(X) -> times#(X,X) activate#(n__from(X)) -> from#(X) activate#(n__cons(X1,X2)) -> cons#(X1,X2) TRS: from(X) -> cons(X,n__from(s(X))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,n__cons(Y,Z))) -> rcons(posrecip(activate(Y)),2ndsneg(N,activate(Z))) 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,n__cons(Y,Z))) -> rcons(negrecip(activate(Y)),2ndspos(N,activate(Z))) pi(X) -> 2ndspos(X,from(0())) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) square(X) -> times(X,X) from(X) -> n__from(X) cons(X1,X2) -> n__cons(X1,X2) activate(n__from(X)) -> from(X) activate(n__cons(X1,X2)) -> cons(X1,X2) activate(X) -> X Usable Rule Processor: DPs: from#(X) -> cons#(X,n__from(s(X))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> activate#(Z) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> 2ndsneg#(N,activate(Z)) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> activate#(Y) 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> activate#(Z) 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> 2ndspos#(N,activate(Z)) 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> activate#(Y) pi#(X) -> from#(0()) pi#(X) -> 2ndspos#(X,from(0())) plus#(s(X),Y) -> plus#(X,Y) times#(s(X),Y) -> times#(X,Y) times#(s(X),Y) -> plus#(Y,times(X,Y)) square#(X) -> times#(X,X) activate#(n__from(X)) -> from#(X) activate#(n__cons(X1,X2)) -> cons#(X1,X2) TRS: activate(n__from(X)) -> from(X) activate(n__cons(X1,X2)) -> cons(X1,X2) activate(X) -> X from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) cons(X1,X2) -> n__cons(X1,X2) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) Matrix Interpretation Processor: dim=1 usable rules: activate(n__from(X)) -> from(X) activate(n__cons(X1,X2)) -> cons(X1,X2) activate(X) -> X from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) cons(X1,X2) -> n__cons(X1,X2) interpretation: [square#](x0) = 6x0 + 4, [times#](x0, x1) = x0 + 5x1, [plus#](x0, x1) = 4x0, [pi#](x0) = 4x0 + 4, [2ndsneg#](x0, x1) = 4x0 + x1 + 1, [activate#](x0) = 6, [2ndspos#](x0, x1) = 2x0 + x1 + 1, [cons#](x0, x1) = 0, [from#](x0) = 2, [times](x0, x1) = 0, [plus](x0, x1) = 5x0, [activate](x0) = x0 + 4, [n__cons](x0, x1) = x1, [0] = 6, [cons](x0, x1) = x1, [n__from](x0) = 1, [s](x0) = 4x0 + 4, [from](x0) = 1 orientation: from#(X) = 2 >= 0 = cons#(X,n__from(s(X))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) = 8N + Z + 9 >= 6 = activate#(Z) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) = 8N + Z + 9 >= 4N + Z + 5 = 2ndsneg#(N,activate(Z)) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) = 8N + Z + 9 >= 6 = activate#(Y) 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) = 16N + Z + 17 >= 6 = activate#(Z) 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) = 16N + Z + 17 >= 2N + Z + 5 = 2ndspos#(N,activate(Z)) 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) = 16N + Z + 17 >= 6 = activate#(Y) pi#(X) = 4X + 4 >= 2 = from#(0()) pi#(X) = 4X + 4 >= 2X + 2 = 2ndspos#(X,from(0())) plus#(s(X),Y) = 16X + 16 >= 4X = plus#(X,Y) times#(s(X),Y) = 4X + 5Y + 4 >= X + 5Y = times#(X,Y) times#(s(X),Y) = 4X + 5Y + 4 >= 4Y = plus#(Y,times(X,Y)) square#(X) = 6X + 4 >= 6X = times#(X,X) activate#(n__from(X)) = 6 >= 2 = from#(X) activate#(n__cons(X1,X2)) = 6 >= 0 = cons#(X1,X2) activate(n__from(X)) = 5 >= 1 = from(X) activate(n__cons(X1,X2)) = X2 + 4 >= X2 = cons(X1,X2) activate(X) = X + 4 >= X = X from(X) = 1 >= 1 = cons(X,n__from(s(X))) from(X) = 1 >= 1 = n__from(X) cons(X1,X2) = X2 >= X2 = n__cons(X1,X2) times(0(),Y) = 0 >= 6 = 0() times(s(X),Y) = 0 >= 5Y = plus(Y,times(X,Y)) plus(0(),Y) = 30 >= Y = Y plus(s(X),Y) = 20X + 20 >= 20X + 4 = s(plus(X,Y)) problem: DPs: TRS: activate(n__from(X)) -> from(X) activate(n__cons(X1,X2)) -> cons(X1,X2) activate(X) -> X from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) cons(X1,X2) -> n__cons(X1,X2) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) Qed