Grover's algorithm finds the marked item

For a natural number $N$ and a chosen index $w_0$ belonging to $\{0,1,\dots,N-1\}$ (the type ascription w0 : Fin N forces $N\ge 1$, since a term of this type can only exist when the index set is nonempty; hence $N=0$ is not a reachable case here), the statement asserts the existence of at least one natural number $k$ (with $k=0$ permitted, corresponding to applying the operator zero times) such that $1-\dfrac{1}{N}\le \big|\langle e_{w_0},\, G^{k}(u_N)\rangle\big|^2$, where: $e_{w_0}\in\mathbb{C}^N$ is the standard basis vector with a $1$ in coordinate $w_0$ and $0$ elsewhere; $u_N\in\mathbb{C}^N$ is the vector all of whose $N$ coordinates equal $1/\sqrt N$ (a well-defined nonzero vector since $N\ge1$ here); $G$ denotes the fixed linear operator on $\mathbb{C}^N$ named groverIterate w0 (its definition is not given in this excerpt); $G^k$ denotes $k$-fold self-composition of $G$; $\langle \cdot,\cdot\rangle$ is the complex inner product on $\mathbb{C}^N$ that is conjugate-linear in its first slot and linear in its second (so $\langle e_{w_0}, v\rangle$ picks out the $w_0$-th coordinate of $v$); $|\cdot|$ is the complex modulus, so $|\langle e_{w_0},G^k(u_N)\rangle|^2$ is a nonnegative real number, specifically the squared modulus of the $w_0$-th coordinate of $G^k(u_N)$; and the right-hand bound $1-1/N$ is a real number in $[0,1)$ for $N\ge1$. The claim is purely existential — it asserts that some such $k$ makes the displayed inequality hold, not that it holds for all $k$, not a rate of convergence, and not an exact value or limit; no upper bound on $k$ is given, and no other property of $k$ is asserted. As written, the proof of this theorem is left as sorry, so the statement is currently unproved (an admitted axiom, not a verified fact) in this file.