[[!redirects Sandbox > history]] [[!redirects Sandbox]] < [[nlab:Sandbox]] $M:(\mathbb{N} \to \mathbb{Q}) \times \mathbb{Q} \to (\mathbb{N} \to \mathbb{Q})$ $$M(a, x)(n) \coloneqq \sum_{i = 0}^{n} \frac{a(i) x^i}{i!}$$ Sequence operations on rational numbers. * Left shift $$\frac{\Gamma, n:\mathbb{N} \vdash a(n):\mathbb{Q}}{\Gamma, n:\mathbb{N} \vdash S_\lambda a(n):\mathbb{Q}}$$ $$\frac{\Gamma, n:\mathbb{N} \vdash a(n):\mathbb{Q}}{\Gamma, n:\mathbb{N} \vdash \delta_{\partial(a)}(n):S_\lambda a(n) =_\mathbb{Q} a(n + 1)}$$ * Right shift $$\frac{\Gamma, n:\mathbb{N} \vdash a(n):\mathbb{Q}}{\Gamma, n:\mathbb{N} \vdash S_\rho a(n):\mathbb{Q}}$$ $$\frac{\Gamma, n:\mathbb{N} \vdash a(n):\mathbb{Q}}{\Gamma, n:\mathbb{N} \vdash \delta_{\partial(a)}(n):S_\rho a(0) =_\mathbb{Q} 0}$$ $$\frac{\Gamma, n:\mathbb{N} \vdash a(n):\mathbb{Q}}{\Gamma, n:\mathbb{N} \vdash \delta_{\partial(a)}(n):S_\rho a(n + 1) =_\mathbb{Q} a(n)}$$ * Derivative $$\frac{\Gamma, n:\mathbb{N} \vdash a(n):\mathbb{Q}}{\Gamma, n:\mathbb{N} \vdash \partial a(n):\mathbb{Q}}$$ $$\frac{\Gamma, n:\mathbb{N} \vdash a(n):\mathbb{Q}}{\Gamma, n:\mathbb{N} \vdash \delta_{\partial(a)}(n):\partial a(n) =_\mathbb{Q} (n + 1) a(n + 1)}$$ $\epsilon$-tolerance and uniformities. category: redirected to nlab