73 lines
1.6 KiB
Text
73 lines
1.6 KiB
Text
/-
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Copyright (c) 2016 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Leonardo de Moura
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-/
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prelude
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import init.logic init.collection
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universe variables u v
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definition set (A : Type u) := A → Prop
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namespace set
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variables {A : Type u} {B : Type v}
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definition mem (a : A) (s : set A) :=
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s a
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infix ∈ := mem
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notation a ∉ s := ¬ mem a s
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definition subset (s₁ s₂ : set A) : Prop :=
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∀ ⦃a⦄, a ∈ s₁ → a ∈ s₂
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infix ⊆ := subset
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definition superset (s₁ s₂ : set A) : Prop :=
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s₂ ⊆ s₁
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infix ⊇ := superset
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definition set_of (p : A → Prop) : set A :=
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p
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private definition sep (p : A → Prop) (s : set A) : set A :=
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λ a, a ∈ s ∧ p a
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instance : separable A set :=
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⟨sep⟩
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private definition empty : set A :=
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λ a, false
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private definition insert (a : A) (s : set A) : set A :=
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λ b, b = a ∨ b ∈ s
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instance : insertable A set :=
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⟨empty, insert⟩
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definition union (s₁ s₂ : set A) : set A :=
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λ a, a ∈ s₁ ∨ a ∈ s₂
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notation s₁ ∪ s₂ := union s₁ s₂
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definition inter (s₁ s₂ : set A) : set A :=
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λ a, a ∈ s₁ ∧ a ∈ s₂
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notation s₁ ∩ s₂ := inter s₁ s₂
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definition compl (s : set A) : set A :=
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λ a, a ∉ s
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instance : has_neg (set A) :=
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⟨compl⟩
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definition diff (s t : set A) : set A :=
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{a ∈ s | a ∉ t}
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infix `\`:70 := diff
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definition powerset (s : set A) : set (set A) :=
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λ t : set A, t ⊆ s
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prefix `𝒫`:100 := powerset
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definition image (f : A → B) (s : set A) : set B :=
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λ b, ∃ a, a ∈ s ∧ f a = b
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infix ` ' ` := image
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end set
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