bookshelf/Bookshelf/Real/Set/Basic.lean

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import Mathlib.Data.Real.Basic
/-! # Bookshelf.Real.Set.Basic
A collection of useful definitions and theorems regarding sets.
-/
namespace Real
/--
The Minkowski sum of two sets `s` and `t` is the set
`s + t = { a + b : a ∈ s, b ∈ t }`.
-/
def minkowski_sum (s t : Set ) :=
{ x | ∃ a ∈ s, ∃ b ∈ t, x = a + b }
/--
The sum of two sets is nonempty **iff** the summands are nonempty.
-/
def nonempty_minkowski_sum_iff_nonempty_add_nonempty {s t : Set }
: (minkowski_sum s t).Nonempty ↔ s.Nonempty ∧ t.Nonempty := by
apply Iff.intro
· intro h
have ⟨x, hx⟩ := h
have ⟨a, ⟨ha, ⟨b, ⟨hb, _⟩⟩⟩⟩ := hx
apply And.intro
· exact ⟨a, ha⟩
· exact ⟨b, hb⟩
· intro ⟨⟨a, ha⟩, ⟨b, hb⟩⟩
exact ⟨a + b, ⟨a, ⟨ha, ⟨b, ⟨hb, rfl⟩⟩⟩⟩⟩
end Real