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