English
Dual statement: for any α with a complete lattice structure, the infimum over n of f(n.unpair.1, n.unpair.2) equals the infimum over all i,j of f(i,j), after switching to the dual lattice.
Русский
Двойственное утверждение: для любой.complete lattice α равенство инфimum по n f(n.unpair.1, n.unpair.2) равно инфимину по всем i,j f(i,j) после перехода к двойственности.
LaTeX
$$$ \inf_{n \in \mathbb{N}} f(\operatorname{fst}(\operatorname{unpair}(n)), \operatorname{snd}(\operatorname{unpair}(n))) = \inf_{i\in\mathbb{N}, j\in\mathbb{N}} f(i,j). $$$
Lean4
theorem iInf_unpair {α} [CompleteLattice α] (f : ℕ → ℕ → α) :
⨅ n : ℕ, f n.unpair.1 n.unpair.2 = ⨅ (i : ℕ) (j : ℕ), f i j :=
iSup_unpair (show ℕ → ℕ → αᵒᵈ from f)
