148 lines
3.6 KiB
Plaintext
148 lines
3.6 KiB
Plaintext
import Mathlib.Data.Set.Basic
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import Common.Logic.Basic
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/-! # Common.Set.Basic
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Additional theorems and definitions useful in the context of `Set`s.
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-/
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namespace Set
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/-! ## Minkowski Sum -/
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/-
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The Minkowski sum of two `Set`s `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 minkowskiSum {α : Type u} [Add α] (s t : Set α) :=
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{ x | ∃ a ∈ s, ∃ b ∈ t, x = a + b }
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/--
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The sum of two `Set`s is nonempty **iff** the summands are nonempty.
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-/
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theorem nonempty_minkowski_sum_iff_nonempty_add_nonempty {α : Type u} [Add α]
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{s t : Set α}
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: (minkowskiSum 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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/-! ## Characteristic Function -/
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/--
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The characteristic function of a `Set` `S`.
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It returns `1` if the specified input belongs to `S` and `0` otherwise.
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-/
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def characteristic (S : Set α) (x : α) [Decidable (x ∈ S)] : Nat :=
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if x ∈ S then 1 else 0
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/-! ## Subsets -/
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/--
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Every `Set` is a subset of itself.
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-/
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theorem subset_self (S : Set α) : S ⊆ S := by
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intro _ hs
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exact hs
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/--
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If `Set` `A` is a subset of `Set` `B`, then `A ∪ B = B`.
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-/
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theorem left_subset_union_eq_self {A B : Set α} (h : A ⊆ B)
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: A ∪ B = B := by
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rw [Set.ext_iff]
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intro x
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apply Iff.intro
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· intro hU
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apply Or.elim hU
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· intro hA
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exact h hA
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· simp
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· intro hB
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exact Or.inr hB
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/--
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If `Set` `B` is a subset of `Set` `A`, then `A ∪ B = B`.
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-/
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theorem right_subset_union_eq_self {A B : Set α} (h : B ⊆ A)
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: A ∪ B = A := by
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rw [Set.union_comm]
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exact left_subset_union_eq_self h
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/--
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If `x` and `y` are members of `Set` `A`, it follows `{x, y}` is a subset of `A`.
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-/
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theorem mem_mem_imp_pair_subset {x y : α}
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(hx : x ∈ A) (hy : y ∈ A) : ({x, y} : Set α) ⊆ A := by
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intro a ha
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apply Or.elim ha
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· intro hx'
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rwa [hx']
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· intro hy'
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rwa [hy']
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/-! ## Powerset -/
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/--
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Every `Set` is a member of its own powerset.
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-/
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theorem self_mem_powerset_self {A : Set α}
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: A ∈ 𝒫 A := subset_self A
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/-! ## Symmetric Difference -/
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/--
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`x` is a member of the `symmDiff` of `A` and `B` **iff** `x ∈ A ∧ x ∉ B` or
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`x ∉ A ∧ x ∈ B`.
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-/
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theorem mem_symm_diff_iff_exclusive_mem {A B : Set α}
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: x ∈ (A ∆ B) ↔ (x ∈ A ∧ x ∉ B) ∨ (x ∉ A ∧ x ∈ B) := by
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unfold symmDiff
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apply Iff.intro
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· intro hx
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simp at hx
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conv => arg 2; rw [and_comm]
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exact hx
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· intro hx
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simp
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conv => arg 2; rw [and_comm]
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exact hx
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/--
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`x` is not a member of the `symmDiff` of `A` and `B` **iff** `x ∈ A ∩ B` or
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`x ∉ A ∪ B`.
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This is the contraposition of `mem_symm_diff_iff_exclusive_mem`.
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-/
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theorem not_mem_symm_diff_inter_or_not_union {A B : Set α}
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: x ∉ (A ∆ B) ↔ (x ∈ A ∩ B) ∨ (x ∉ A ∪ B) := by
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show ¬(x ∈ A ∧ ¬x ∈ B ∨ x ∈ B ∧ ¬x ∈ A) ↔ x ∈ A ∧ x ∈ B ∨ ¬(x ∈ A ∨ x ∈ B)
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rw [
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not_or_de_morgan,
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not_and_de_morgan, not_and_de_morgan,
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not_not, not_not,
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not_or_de_morgan
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]
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apply Iff.intro
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· intro nx
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apply Or.elim nx.left
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· intro nA
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exact Or.elim nx.right (Or.inr ⟨nA, ·⟩) (absurd · nA)
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· intro hB
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exact Or.elim nx.right (absurd hB ·) (Or.inl ⟨·, hB⟩)
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· intro hx
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apply Or.elim hx
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· intro hy
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exact ⟨Or.inr hy.right, Or.inr hy.left⟩
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· intro hy
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exact ⟨Or.inl hy.left, Or.inl hy.right⟩
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end Set |