125 lines
3.5 KiB
Text
125 lines
3.5 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 (T : Type) := T → Prop
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namespace set
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variable {T : Type}
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/- membership and subset -/
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definition mem [reducible] (x : T) (a : set T) := a x
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notation e ∈ a := mem e a
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theorem setext {a b : set T} (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 T) := ∀ 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 T) (P : T → 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 T) (P : T → 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 T := λx, false
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notation `∅` := empty
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theorem mem_empty (x : T) : ¬ (x ∈ ∅) :=
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assume H : x ∈ ∅, H
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/- universal set -/
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definition univ : set T := λx, true
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theorem mem_univ (x : T) : x ∈ univ := trivial
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/- intersection -/
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definition inter [reducible] (a b : set T) : set T := λx, x ∈ a ∧ x ∈ b
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notation a ∩ b := inter a b
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theorem mem_inter (x : T) (a b : set T) : x ∈ a ∩ b ↔ (x ∈ a ∧ x ∈ b) := !iff.refl
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theorem inter_self (a : set T) : a ∩ a = a :=
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setext (take x, !and_self)
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theorem inter_empty (a : set T) : a ∩ ∅ = ∅ :=
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setext (take x, !and_false)
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theorem empty_inter (a : set T) : ∅ ∩ a = ∅ :=
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setext (take x, !false_and)
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theorem inter.comm (a b : set T) : a ∩ b = b ∩ a :=
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setext (take x, !and.comm)
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theorem inter.assoc (a b c : set T) : (a ∩ b) ∩ c = a ∩ (b ∩ c) :=
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setext (take x, !and.assoc)
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/- union -/
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definition union [reducible] (a b : set T) : set T := λx, x ∈ a ∨ x ∈ b
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notation a ∪ b := union a b
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theorem mem_union (x : T) (a b : set T) : x ∈ a ∪ b ↔ (x ∈ a ∨ x ∈ b) := !iff.refl
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theorem union_self (a : set T) : a ∪ a = a :=
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setext (take x, !or_self)
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theorem union_empty (a : set T) : a ∪ ∅ = a :=
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setext (take x, !or_false)
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theorem empty_union (a : set T) : ∅ ∪ a = a :=
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setext (take x, !false_or)
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theorem union.comm (a b : set T) : a ∪ b = b ∪ a :=
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setext (take x, or.comm)
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theorem union_assoc (a b c : set T) : (a ∪ b) ∪ c = a ∪ (b ∪ c) :=
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setext (take x, or.assoc)
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/- set-builder notation -/
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-- {x : T | P}
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definition set_of (P : T → Prop) : set T := P
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notation `{` binders `|` r:(scoped:1 P, set_of P) `}` := r
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-- {[x, y, z]} or ⦃x, y, z⦄
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definition insert (x : T) (a : set T) : set T := {y : T | 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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/- 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 T
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variable C : set (set T)
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definition Inter : set T := {x : T | ∀i, x ∈ b i}
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definition bInter : set T := {x : T | ∀₀ i ∈ a, x ∈ b i}
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definition sInter : set T := {x : T | ∀₀ c ∈ C, x ∈ c}
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definition Union : set T := {x : T | ∃i, x ∈ b i}
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definition bUnion : set T := {x : T | ∃₀ i ∈ a, x ∈ b i}
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definition sUnion : set T := {x : T | ∃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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