lean2/library/data/set/basic.lean

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/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Jeremy Avigad, Leonardo de Moura
-/
import logic
open eq.ops
definition set [reducible] (X : Type) := X → Prop
namespace set
variable {X : Type}
/- membership and subset -/
definition mem [reducible] (x : X) (a : set X) := a x
infix `∈` := mem
notation a ∉ b := ¬ mem a b
theorem setext {a b : set X} (H : ∀x, x ∈ a ↔ x ∈ b) : a = b :=
funext (take x, propext (H x))
definition subset (a b : set X) := ∀⦃x⦄, x ∈ a → x ∈ b
infix `⊆` := subset
theorem subset.refl (a : set X) : a ⊆ a := take x, assume H, H
theorem subset.trans (a b c : set X) (subab : a ⊆ b) (subbc : b ⊆ c) : a ⊆ c :=
take x, assume ax, subbc (subab ax)
theorem subset.antisymm (a b : set X) (h₁ : a ⊆ b) (h₂ : b ⊆ a) : a = b :=
setext (λ x, iff.intro (λ ina, h₁ ina) (λ inb, h₂ inb))
definition strict_subset (a b : set X) := a ⊆ b ∧ a ≠ b
infix `⊂`:50 := strict_subset
theorem strict_subset.irrefl (a : set X) : ¬ a ⊂ a :=
assume h, absurd rfl (and.elim_right h)
/- bounded quantification -/
abbreviation bounded_forall (a : set X) (P : X → Prop) := ∀⦃x⦄, x ∈ a → P x
notation `forallb` binders `∈` a `,` r:(scoped:1 P, P) := bounded_forall a r
notation `∀₀` binders `∈` a `,` r:(scoped:1 P, P) := bounded_forall a r
abbreviation bounded_exists (a : set X) (P : X → Prop) := ∃⦃x⦄, x ∈ a ∧ P x
notation `existsb` binders `∈` a `,` r:(scoped:1 P, P) := bounded_exists a r
notation `∃₀` binders `∈` a `,` r:(scoped:1 P, P) := bounded_exists a r
/- empty set -/
definition empty [reducible] : set X := λx, false
notation `∅` := empty
theorem not_mem_empty (x : X) : ¬ (x ∈ ∅) :=
assume H : x ∈ ∅, H
theorem mem_empty_eq (x : X) : x ∈ ∅ = false := rfl
/- universal set -/
definition univ : set X := λx, true
theorem mem_univ (x : X) : x ∈ univ := trivial
theorem mem_univ_eq (x : X) : x ∈ univ = true := rfl
theorem empty_ne_univ [h : inhabited X] : (empty : set X) ≠ univ :=
assume H : empty = univ,
absurd (mem_univ (inhabited.value h)) (eq.rec_on H (not_mem_empty _))
/- union -/
definition union [reducible] (a b : set X) : set X := λx, x ∈ a x ∈ b
notation a b := union a b
theorem mem_union (x : X) (a b : set X) : x ∈ a b ↔ x ∈ a x ∈ b := !iff.refl
theorem mem_union_eq (x : X) (a b : set X) : x ∈ a b = (x ∈ a x ∈ b) := rfl
theorem union_self (a : set X) : a a = a :=
setext (take x, !or_self)
theorem union_empty (a : set X) : a ∅ = a :=
setext (take x, !or_false)
theorem empty_union (a : set X) : ∅ a = a :=
setext (take x, !false_or)
theorem union.comm (a b : set X) : a b = b a :=
setext (take x, or.comm)
theorem union.assoc (a b c : set X) : (a b) c = a (b c) :=
setext (take x, or.assoc)
/- intersection -/
definition inter [reducible] (a b : set X) : set X := λx, x ∈ a ∧ x ∈ b
notation a ∩ b := inter a b
theorem mem_inter (x : X) (a b : set X) : x ∈ a ∩ b ↔ x ∈ a ∧ x ∈ b := !iff.refl
theorem mem_inter_eq (x : X) (a b : set X) : x ∈ a ∩ b = (x ∈ a ∧ x ∈ b) := rfl
theorem inter_self (a : set X) : a ∩ a = a :=
setext (take x, !and_self)
theorem inter_empty (a : set X) : a ∩ ∅ = ∅ :=
setext (take x, !and_false)
theorem empty_inter (a : set X) : ∅ ∩ a = ∅ :=
setext (take x, !false_and)
theorem inter.comm (a b : set X) : a ∩ b = b ∩ a :=
setext (take x, !and.comm)
theorem inter.assoc (a b c : set X) : (a ∩ b) ∩ c = a ∩ (b ∩ c) :=
setext (take x, !and.assoc)
theorem inter_univ (a : set X) : a ∩ univ = a :=
setext (take x, !and_true)
theorem univ_inter (a : set X) : univ ∩ a = a :=
setext (take x, !true_and)
/- distributivity laws -/
theorem inter.distrib_left (s t u : set X) : s ∩ (t u) = (s ∩ t) (s ∩ u) :=
setext (take x, !and.distrib_left)
theorem inter.distrib_right (s t u : set X) : (s t) ∩ u = (s ∩ u) (t ∩ u) :=
setext (take x, !and.distrib_right)
theorem union.distrib_left (s t u : set X) : s (t ∩ u) = (s t) ∩ (s u) :=
setext (take x, !or.distrib_left)
theorem union.distrib_right (s t u : set X) : (s ∩ t) u = (s u) ∩ (t u) :=
setext (take x, !or.distrib_right)
/- set-builder notation -/
-- {x : X | P}
definition set_of (P : X → Prop) : set X := P
notation `{` binders `|` r:(scoped:1 P, set_of P) `}` := r
-- {x ∈ s | P}
definition filter (P : X → Prop) (s : set X) : set X := λx, x ∈ s ∧ P x
notation `{` binders ∈ s `|` r:(scoped:1 p, filter p s) `}` := r
-- {[x, y, z]}
definition insert (x : X) (a : set X) : set X := {y : X | y = x y ∈ a}
notation `{[`:max a:(foldr `,` (x b, insert x b) ∅) `]}`:0 := a
/- set difference -/
definition diff (s t : set X) : set X := {x ∈ s | x ∉ t}
infix `\`:70 := diff
theorem mem_of_mem_diff {s t : set X} {x : X} (H : x ∈ s \ t) : x ∈ s :=
and.left H
theorem not_mem_of_mem_diff {s t : set X} {x : X} (H : x ∈ s \ t) : x ∉ t :=
and.right H
theorem mem_diff {s t : set X} {x : X} (H1 : x ∈ s) (H2 : x ∉ t) : x ∈ s \ t :=
and.intro H1 H2
theorem mem_diff_iff (s t : set X) (x : X) : x ∈ s \ t ↔ x ∈ s ∧ x ∉ t := !iff.refl
theorem mem_diff_eq (s t : set X) (x : X) : x ∈ s \ t = (x ∈ s ∧ x ∉ t) := rfl
/- large unions -/
section
variables {I : Type}
variable a : set I
variable b : I → set X
variable C : set (set X)
definition Inter : set X := {x : X | ∀i, x ∈ b i}
definition bInter : set X := {x : X | ∀₀ i ∈ a, x ∈ b i}
definition sInter : set X := {x : X | ∀₀ c ∈ C, x ∈ c}
definition Union : set X := {x : X | ∃i, x ∈ b i}
definition bUnion : set X := {x : X | ∃₀ i ∈ a, x ∈ b i}
definition sUnion : set X := {x : X | ∃₀ c ∈ C, x ∈ c}
-- TODO: need notation for these
end
end set