YES Problem: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Proof: DP Processor: DPs: terms#(N) -> sqr#(N) sqr#(s(X)) -> dbl#(X) sqr#(s(X)) -> sqr#(X) sqr#(s(X)) -> add#(sqr(X),dbl(X)) sqr#(s(X)) -> s#(add(sqr(X),dbl(X))) dbl#(s(X)) -> dbl#(X) dbl#(s(X)) -> s#(dbl(X)) dbl#(s(X)) -> s#(s(dbl(X))) add#(s(X),Y) -> add#(X,Y) add#(s(X),Y) -> s#(add(X,Y)) first#(s(X),cons(Y,Z)) -> activate#(Z) activate#(n__terms(X)) -> activate#(X) activate#(n__terms(X)) -> terms#(activate(X)) activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> s#(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)) TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X TDG Processor: DPs: terms#(N) -> sqr#(N) sqr#(s(X)) -> dbl#(X) sqr#(s(X)) -> sqr#(X) sqr#(s(X)) -> add#(sqr(X),dbl(X)) sqr#(s(X)) -> s#(add(sqr(X),dbl(X))) dbl#(s(X)) -> dbl#(X) dbl#(s(X)) -> s#(dbl(X)) dbl#(s(X)) -> s#(s(dbl(X))) add#(s(X),Y) -> add#(X,Y) add#(s(X),Y) -> s#(add(X,Y)) first#(s(X),cons(Y,Z)) -> activate#(Z) activate#(n__terms(X)) -> activate#(X) activate#(n__terms(X)) -> terms#(activate(X)) activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> s#(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)) TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X graph: 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__s(X)) -> s#(activate(X)) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__s(X)) -> activate#(X) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__terms(X)) -> terms#(activate(X)) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__terms(X)) -> activate#(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__s(X)) -> s#(activate(X)) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> activate#(X) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__terms(X)) -> terms#(activate(X)) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__terms(X)) -> activate#(X) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) -> first#(s(X),cons(Y,Z)) -> activate#(Z) activate#(n__terms(X)) -> activate#(X) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) activate#(n__terms(X)) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__terms(X)) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__terms(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__terms(X)) -> activate#(X) -> activate#(n__s(X)) -> activate#(X) activate#(n__terms(X)) -> activate#(X) -> activate#(n__terms(X)) -> terms#(activate(X)) activate#(n__terms(X)) -> activate#(X) -> activate#(n__terms(X)) -> activate#(X) activate#(n__terms(X)) -> terms#(activate(X)) -> terms#(N) -> sqr#(N) activate#(n__s(X)) -> activate#(X) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) activate#(n__s(X)) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__s(X)) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__s(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__s(X)) -> activate#(X) -> activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> activate#(X) -> activate#(n__terms(X)) -> terms#(activate(X)) activate#(n__s(X)) -> activate#(X) -> activate#(n__terms(X)) -> activate#(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__s(X)) -> s#(activate(X)) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__s(X)) -> activate#(X) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__terms(X)) -> terms#(activate(X)) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__terms(X)) -> activate#(X) add#(s(X),Y) -> add#(X,Y) -> add#(s(X),Y) -> s#(add(X,Y)) add#(s(X),Y) -> add#(X,Y) -> add#(s(X),Y) -> add#(X,Y) dbl#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> s#(s(dbl(X))) dbl#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> s#(dbl(X)) dbl#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> dbl#(X) sqr#(s(X)) -> add#(sqr(X),dbl(X)) -> add#(s(X),Y) -> s#(add(X,Y)) sqr#(s(X)) -> add#(sqr(X),dbl(X)) -> add#(s(X),Y) -> add#(X,Y) sqr#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> s#(s(dbl(X))) sqr#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> s#(dbl(X)) sqr#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> dbl#(X) sqr#(s(X)) -> sqr#(X) -> sqr#(s(X)) -> s#(add(sqr(X),dbl(X))) sqr#(s(X)) -> sqr#(X) -> sqr#(s(X)) -> add#(sqr(X),dbl(X)) sqr#(s(X)) -> sqr#(X) -> sqr#(s(X)) -> sqr#(X) sqr#(s(X)) -> sqr#(X) -> sqr#(s(X)) -> dbl#(X) terms#(N) -> sqr#(N) -> sqr#(s(X)) -> s#(add(sqr(X),dbl(X))) terms#(N) -> sqr#(N) -> sqr#(s(X)) -> add#(sqr(X),dbl(X)) terms#(N) -> sqr#(N) -> sqr#(s(X)) -> sqr#(X) terms#(N) -> sqr#(N) -> sqr#(s(X)) -> dbl#(X) SCC Processor: #sccs: 4 #rules: 9 #arcs: 55/324 DPs: activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__terms(X)) -> activate#(X) activate#(n__s(X)) -> 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) TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 usable rules: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X interpretation: [n__terms](x0) = 7x0, [dbl](x0) = 2, [sqr](x0) = x0 + 0, [n__s](x0) = x0, [first#](x0, x1) = x0 + 4x1 + 0, [cons](x0, x1) = x1, [first](x0, x1) = x0 + 4x1 + 0, [terms](x0) = 7x0, [s](x0) = x0, [activate#](x0) = x0, [activate](x0) = x0, [nil] = 0, [add](x0, x1) = x0 + 4x1 + 2, [n__first](x0, x1) = x0 + 4x1 + 0, [0] = 6, [recip](x0) = x0 + 2 orientation: activate#(n__first(X1,X2)) = X1 + 4X2 + 0 >= X2 = activate#(X2) activate#(n__terms(X)) = 7X >= X = activate#(X) activate#(n__s(X)) = X >= X = activate#(X) activate#(n__first(X1,X2)) = X1 + 4X2 + 0 >= X1 = activate#(X1) activate#(n__first(X1,X2)) = X1 + 4X2 + 0 >= X1 + 4X2 + 0 = first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) = X + 4Z + 0 >= Z = activate#(Z) terms(N) = 7N >= 7N = cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) = 6 >= 6 = 0() sqr(s(X)) = X + 0 >= X + 6 = s(add(sqr(X),dbl(X))) dbl(0()) = 2 >= 6 = 0() dbl(s(X)) = 2 >= 2 = s(s(dbl(X))) add(0(),X) = 4X + 6 >= X = X add(s(X),Y) = X + 4Y + 2 >= X + 4Y + 2 = s(add(X,Y)) first(0(),X) = 4X + 6 >= 0 = nil() first(s(X),cons(Y,Z)) = X + 4Z + 0 >= X + 4Z + 0 = cons(Y,n__first(X,activate(Z))) terms(X) = 7X >= 7X = n__terms(X) s(X) = X >= X = n__s(X) first(X1,X2) = X1 + 4X2 + 0 >= X1 + 4X2 + 0 = n__first(X1,X2) activate(n__terms(X)) = 7X >= 7X = terms(activate(X)) activate(n__s(X)) = X >= X = s(activate(X)) activate(n__first(X1,X2)) = X1 + 4X2 + 0 >= X1 + 4X2 + 0 = first(activate(X1),activate(X2)) activate(X) = X >= X = X problem: DPs: activate#(n__s(X)) -> activate#(X) activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Restore Modifier: DPs: activate#(n__s(X)) -> activate#(X) activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X SCC Processor: #sccs: 1 #rules: 2 #arcs: 26/9 DPs: activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__s(X)) -> activate#(X) TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Size-Change Termination Processor: DPs: TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X The DP: activate#(n__first(X1,X2)) -> activate#(X1) has the edges: 0 > 0 The DP: activate#(n__s(X)) -> activate#(X) has the edges: 0 > 0 Qed DPs: sqr#(s(X)) -> sqr#(X) TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Subterm Criterion Processor: simple projection: pi(sqr#) = 0 problem: DPs: TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Qed DPs: add#(s(X),Y) -> add#(X,Y) TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Subterm Criterion Processor: simple projection: pi(add#) = 0 problem: DPs: TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Qed DPs: dbl#(s(X)) -> dbl#(X) TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Subterm Criterion Processor: simple projection: pi(dbl#) = 0 problem: DPs: TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Qed