170 lines
5.1 KiB
Text
170 lines
5.1 KiB
Text
/-
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Copyright (c) 2014 Jeremy Avigad. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: data.set.basic
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Author: Jeremy Avigad, Leonardo de Moura
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-/
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import logic
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open eq.ops
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definition set [reducible] (X : Type) := X → Prop
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namespace set
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variable {X : Type}
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/- membership and subset -/
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definition mem [reducible] (x : X) (a : set X) := a x
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infix `∈` := mem
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notation a ∉ b := ¬ mem a b
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theorem setext {a b : set X} (H : ∀x, x ∈ a ↔ x ∈ b) : a = b :=
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funext (take x, propext (H x))
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definition subset (a b : set X) := ∀⦃x⦄, x ∈ a → x ∈ b
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infix `⊆`:50 := subset
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/- bounded quantification -/
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abbreviation bounded_forall (a : set X) (P : X → Prop) := ∀⦃x⦄, x ∈ a → P x
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notation `forallb` binders `∈` a `,` r:(scoped:1 P, P) := bounded_forall a r
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notation `∀₀` binders `∈` a `,` r:(scoped:1 P, P) := bounded_forall a r
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abbreviation bounded_exists (a : set X) (P : X → Prop) := ∃⦃x⦄, x ∈ a ∧ P x
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notation `existsb` binders `∈` a `,` r:(scoped:1 P, P) := bounded_exists a r
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notation `∃₀` binders `∈` a `,` r:(scoped:1 P, P) := bounded_exists a r
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/- empty set -/
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definition empty [reducible] : set X := λx, false
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notation `∅` := empty
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theorem not_mem_empty (x : X) : ¬ (x ∈ ∅) :=
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assume H : x ∈ ∅, H
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theorem mem_empty_eq (x : X) : x ∈ ∅ = false := rfl
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/- universal set -/
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definition univ : set X := λx, true
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theorem mem_univ (x : X) : x ∈ univ := trivial
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theorem mem_univ_eq (x : X) : x ∈ univ = true := rfl
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/- union -/
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definition union [reducible] (a b : set X) : set X := λx, x ∈ a ∨ x ∈ b
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notation a ∪ b := union a b
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theorem mem_union (x : X) (a b : set X) : x ∈ a ∪ b ↔ x ∈ a ∨ x ∈ b := !iff.refl
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theorem mem_union_eq (x : X) (a b : set X) : x ∈ a ∪ b = (x ∈ a ∨ x ∈ b) := rfl
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theorem union_self (a : set X) : a ∪ a = a :=
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setext (take x, !or_self)
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theorem union_empty (a : set X) : a ∪ ∅ = a :=
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setext (take x, !or_false)
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theorem empty_union (a : set X) : ∅ ∪ a = a :=
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setext (take x, !false_or)
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theorem union.comm (a b : set X) : a ∪ b = b ∪ a :=
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setext (take x, or.comm)
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theorem union_assoc (a b c : set X) : (a ∪ b) ∪ c = a ∪ (b ∪ c) :=
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setext (take x, or.assoc)
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/- intersection -/
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definition inter [reducible] (a b : set X) : set X := λx, x ∈ a ∧ x ∈ b
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notation a ∩ b := inter a b
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theorem mem_inter (x : X) (a b : set X) : x ∈ a ∩ b ↔ x ∈ a ∧ x ∈ b := !iff.refl
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theorem mem_inter_eq (x : X) (a b : set X) : x ∈ a ∩ b = (x ∈ a ∧ x ∈ b) := rfl
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theorem inter_self (a : set X) : a ∩ a = a :=
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setext (take x, !and_self)
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theorem inter_empty (a : set X) : a ∩ ∅ = ∅ :=
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setext (take x, !and_false)
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theorem empty_inter (a : set X) : ∅ ∩ a = ∅ :=
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setext (take x, !false_and)
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theorem inter.comm (a b : set X) : a ∩ b = b ∩ a :=
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setext (take x, !and.comm)
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theorem inter.assoc (a b c : set X) : (a ∩ b) ∩ c = a ∩ (b ∩ c) :=
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setext (take x, !and.assoc)
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/- distributivity laws -/
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theorem inter.distrib_left (s t u : set X) : s ∩ (t ∪ u) = (s ∩ t) ∪ (s ∩ u) :=
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setext (take x, !and.distrib_left)
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theorem inter.distrib_right (s t u : set X) : (s ∪ t) ∩ u = (s ∩ u) ∪ (t ∩ u) :=
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setext (take x, !and.distrib_right)
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theorem union.distrib_left (s t u : set X) : s ∪ (t ∩ u) = (s ∪ t) ∩ (s ∪ u) :=
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setext (take x, !or.distrib_left)
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theorem union.distrib_right (s t u : set X) : (s ∩ t) ∪ u = (s ∪ u) ∩ (t ∪ u) :=
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setext (take x, !or.distrib_right)
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/- set-builder notation -/
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-- {x : X | P}
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definition set_of (P : X → Prop) : set X := P
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notation `{` binders `|` r:(scoped:1 P, set_of P) `}` := r
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-- {x ∈ s | P}
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definition filter (P : X → Prop) (s : set X) : set X := λx, x ∈ s ∧ P x
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notation `{` binders ∈ s `|` r:(scoped:1 p, filter p s) `}` := r
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-- {[x, y, z]} or ⦃x, y, z⦄
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definition insert (x : X) (a : set X) : set X := {y : X | y = x ∨ y ∈ a}
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notation `{[`:max a:(foldr `,` (x b, insert x b) ∅) `]}`:0 := a
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notation `⦃` a:(foldr `,` (x b, insert x b) ∅) `⦄` := a
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/- set difference -/
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definition diff (s t : set X) : set X := {x ∈ s | x ∉ t}
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infix `\`:70 := diff
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theorem mem_of_mem_diff {s t : set X} {x : X} (H : x ∈ s \ t) : x ∈ s :=
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and.left H
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theorem not_mem_of_mem_diff {s t : set X} {x : X} (H : x ∈ s \ t) : x ∉ t :=
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and.right H
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theorem mem_diff {s t : set X} {x : X} (H1 : x ∈ s) (H2 : x ∉ t) : x ∈ s \ t :=
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and.intro H1 H2
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theorem mem_diff_iff (s t : set X) (x : X) : x ∈ s \ t ↔ x ∈ s ∧ x ∉ t := !iff.refl
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theorem mem_diff_eq (s t : set X) (x : X) : x ∈ s \ t = (x ∈ s ∧ x ∉ t) := rfl
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/- large unions -/
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section
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variables {I : Type}
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variable a : set I
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variable b : I → set X
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variable C : set (set X)
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definition Inter : set X := {x : X | ∀i, x ∈ b i}
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definition bInter : set X := {x : X | ∀₀ i ∈ a, x ∈ b i}
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definition sInter : set X := {x : X | ∀₀ c ∈ C, x ∈ c}
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definition Union : set X := {x : X | ∃i, x ∈ b i}
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definition bUnion : set X := {x : X | ∃₀ i ∈ a, x ∈ b i}
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definition sUnion : set X := {x : X | ∃₀ c ∈ C, x ∈ c}
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-- TODO: need notation for these
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end
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end set
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