bookshelf/Common/Set/Basic.lean

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import Mathlib.Data.Set.Basic
/-! # Common.Set.Basic
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Additional theorems and definitions useful in the context of sets.
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-/
namespace Set
/-
The Minkowski sum of two sets `s` and `t` is the set
`s + t = { a + b : a ∈ s, b ∈ t }`.
-/
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def minkowskiSum {α : Type u} [Add α] (s t : Set α) :=
{ x | ∃ a ∈ s, ∃ b ∈ t, x = a + b }
/--
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The sum of two sets is nonempty **iff** the summands are nonempty.
-/
theorem nonempty_minkowski_sum_iff_nonempty_add_nonempty {α : Type u} [Add α]
{s t : Set α}
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: (minkowskiSum 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⟩⟩⟩⟩⟩
/--
The characteristic function of a set `S`.
It returns `1` if the specified input belongs to `S` and `0` otherwise.
-/
def characteristic (S : Set α) (x : α) [Decidable (x ∈ S)] : Nat :=
if x ∈ S then 1 else 0
end Set