Prod.Lex.le_iff

theorem Prod.Lex.le_iff : ∀ {α : Type u_1} {β : Type u_2} [inst : LT α] [inst_1 : LE β] (a b : α × β), toLex a ≤ toLex b ↔ a.1 < b.1 ∨ a.1 = b.1 ∧ a.2 ≤ b.2

This theorem, `Prod.Lex.le_iff`, states that for any two pairs of types `α` and `β`, denoted by `a` and `b`, where `α` has a less than (`<`) operation and `β` has a less than or equal to (`≤`) operation, the `toLex` function applied to `a` is less than or equal to the `toLex` function applied to `b` if and only if the first element of `a` is less than the first element of `b` or the first elements of `a` and `b` are equal and the second element of `a` is less than or equal to the second element of `b`. In essence, it's providing a relation between the `toLex` function and the ordering of the elements within the pairs.

More concisely: For types `α` and `β` with less-than relations `<` and `≤`, respectively, the `toLex (a, x)` is less than or equal to `toLex (b, y)` if and only if `x < y` or `x = y` and `a < b`.