79 lines
1.6 KiB
Text
79 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.category.functor
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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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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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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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⟨@set.image⟩
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end set
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