101 lines
2.3 KiB
Text
101 lines
2.3 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.meta.interactive
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import init.category.lawful
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universes u v
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def set (α : Type u) := α → Prop
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def set_of {α : Type u} (p : α → Prop) : set α :=
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p
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namespace set
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variables {α : Type u} {β : Type v}
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protected def mem (a : α) (s : set α) :=
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s a
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instance : has_mem α (set α) :=
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⟨set.mem⟩
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protected def subset (s₁ s₂ : set α) :=
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∀ ⦃a⦄, a ∈ s₁ → a ∈ s₂
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instance : has_subset (set α) :=
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⟨set.subset⟩
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protected def sep (p : α → Prop) (s : set α) : set α :=
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{a | a ∈ s ∧ p a}
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instance : has_sep α (set α) :=
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⟨set.sep⟩
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instance : has_emptyc (set α) :=
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⟨λ a, false⟩
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def univ : set α :=
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λ a, true
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protected def insert (a : α) (s : set α) : set α :=
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{b | b = a ∨ b ∈ s}
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instance : has_insert α (set α) :=
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⟨set.insert⟩
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protected def union (s₁ s₂ : set α) : set α :=
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{a | a ∈ s₁ ∨ a ∈ s₂}
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instance : has_union (set α) :=
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⟨set.union⟩
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protected def inter (s₁ s₂ : set α) : set α :=
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{a | a ∈ s₁ ∧ a ∈ s₂}
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instance : has_inter (set α) :=
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⟨set.inter⟩
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def compl (s : set α) : set α :=
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{a | a ∉ s}
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instance : has_neg (set α) :=
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⟨compl⟩
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protected def diff (s t : set α) : set α :=
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{a ∈ s | a ∉ t}
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instance : has_sdiff (set α) :=
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⟨set.diff⟩
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def powerset (s : set α) : set (set α) :=
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{t | t ⊆ s}
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prefix `𝒫`:100 := powerset
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@[reducible]
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def sUnion (s : set (set α)) : set α := {t | ∃ a ∈ s, t ∈ a}
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prefix `⋃₀`:110 := sUnion
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def image (f : α → β) (s : set α) : set β :=
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{b | ∃ a, a ∈ s ∧ f a = b}
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instance : functor set :=
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{ map := @set.image }
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instance : is_lawful_functor set :=
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{ id_map := begin
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intros _ s, funext b,
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dsimp [image, set_of],
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exact propext ⟨λ ⟨b', ⟨_, _⟩⟩, ‹b' = b› ▸ ‹s b'›,
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λ _, ⟨b, ⟨‹s b›, rfl⟩⟩⟩,
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end,
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comp_map := begin
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intros, funext c,
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dsimp [image, set_of, function.comp],
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exact propext ⟨λ ⟨a, ⟨h₁, h₂⟩⟩, ⟨g a, ⟨⟨a, ⟨h₁, rfl⟩⟩, h₂⟩⟩,
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λ ⟨b, ⟨⟨a, ⟨h₁, h₂⟩⟩, h₃⟩⟩, ⟨a, ⟨h₁, h₂.symm ▸ h₃⟩⟩⟩
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end }
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end set
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