774 lines
27 KiB
Text
774 lines
27 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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Author: Jeremy Avigad, Leonardo de Moura
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-/
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import logic.connectives logic.identities algebra.binary
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open eq.ops binary function
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definition set (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 (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 ext {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 ⊆ := subset
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definition superset (s t : set X) : Prop := t ⊆ s
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infix ⊇ := superset
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theorem subset.refl (a : set X) : a ⊆ a := take x, assume H, H
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theorem subset.trans {a b c : set X} (subab : a ⊆ b) (subbc : b ⊆ c) : a ⊆ c :=
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take x, assume ax, subbc (subab ax)
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theorem subset.antisymm {a b : set X} (h₁ : a ⊆ b) (h₂ : b ⊆ a) : a = b :=
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ext (λ x, iff.intro (λ ina, h₁ ina) (λ inb, h₂ inb))
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-- an alterantive name
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theorem eq_of_subset_of_subset {a b : set X} (h₁ : a ⊆ b) (h₂ : b ⊆ a) : a = b :=
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subset.antisymm h₁ h₂
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theorem mem_of_subset_of_mem {s₁ s₂ : set X} {a : X} : s₁ ⊆ s₂ → a ∈ s₁ → a ∈ s₂ :=
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assume h₁ h₂, h₁ _ h₂
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/- strict subset -/
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definition strict_subset (a b : set X) := a ⊆ b ∧ a ≠ b
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infix ` ⊂ `:50 := strict_subset
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theorem strict_subset.irrefl (a : set X) : ¬ a ⊂ a :=
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assume h, absurd rfl (and.elim_right h)
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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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theorem bounded_exists.intro {P : X → Prop} {s : set X} {x : X} (xs : x ∈ s) (Px : P x) :
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∃₀ x ∈ s, P x :=
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exists.intro x (and.intro xs Px)
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/- empty set -/
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definition empty : 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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theorem eq_empty_of_forall_not_mem {s : set X} (H : ∀ x, x ∉ s) : s = ∅ :=
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ext (take x, iff.intro
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(assume xs, absurd xs (H x))
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(assume xe, absurd xe !not_mem_empty))
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section
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open classical
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theorem exists_mem_of_ne_empty {s : set X} (H : s ≠ ∅) : ∃ x, x ∈ s :=
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by_contradiction (assume H', H (eq_empty_of_forall_not_mem (forall_not_of_not_exists H')))
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end
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theorem empty_subset (s : set X) : ∅ ⊆ s :=
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take x, assume H, false.elim H
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theorem eq_empty_of_subset_empty {s : set X} (H : s ⊆ ∅) : s = ∅ :=
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subset.antisymm H (empty_subset s)
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theorem subset_empty_iff (s : set X) : s ⊆ ∅ ↔ s = ∅ :=
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iff.intro eq_empty_of_subset_empty (take xeq, by rewrite xeq; apply subset.refl ∅)
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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_iff (x : X) : x ∈ univ ↔ true := !iff.refl
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theorem mem_univ_eq (x : X) : x ∈ univ = true := rfl
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theorem empty_ne_univ [h : inhabited X] : (empty : set X) ≠ univ :=
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assume H : empty = univ,
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absurd (mem_univ (inhabited.value h)) (eq.rec_on H (not_mem_empty _))
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theorem subset_univ (s : set X) : s ⊆ univ := λ x H, trivial
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theorem eq_univ_of_univ_subset {s : set X} (H : univ ⊆ s) : s = univ :=
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eq_of_subset_of_subset (subset_univ s) H
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theorem eq_univ_of_forall {s : set X} (H : ∀ x, x ∈ s) : s = univ :=
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ext (take x, iff.intro (assume H', trivial) (assume H', H x))
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/- union -/
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definition union (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_left {x : X} {a : set X} (b : set X) : x ∈ a → x ∈ a ∪ b :=
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assume h, or.inl h
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theorem mem_union_right {x : X} {b : set X} (a : set X) : x ∈ b → x ∈ a ∪ b :=
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assume h, or.inr h
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theorem mem_unionl {x : X} {a b : set X} : x ∈ a → x ∈ a ∪ b :=
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assume h, or.inl h
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theorem mem_unionr {x : X} {a b : set X} : x ∈ b → x ∈ a ∪ b :=
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assume h, or.inr h
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theorem mem_or_mem_of_mem_union {x : X} {a b : set X} (H : x ∈ a ∪ b) : x ∈ a ∨ x ∈ b := H
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theorem mem_union.elim {x : X} {a b : set X} {P : Prop}
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(H₁ : x ∈ a ∪ b) (H₂ : x ∈ a → P) (H₃ : x ∈ b → P) : P :=
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or.elim H₁ H₂ H₃
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theorem mem_union_iff (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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ext (take x, !or_self)
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theorem union_empty (a : set X) : a ∪ ∅ = a :=
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ext (take x, !or_false)
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theorem empty_union (a : set X) : ∅ ∪ a = a :=
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ext (take x, !false_or)
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theorem union_comm (a b : set X) : a ∪ b = b ∪ a :=
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ext (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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ext (take x, or.assoc)
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theorem union_left_comm (s₁ s₂ s₃ : set X) : s₁ ∪ (s₂ ∪ s₃) = s₂ ∪ (s₁ ∪ s₃) :=
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!left_comm union_comm union_assoc s₁ s₂ s₃
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theorem union_right_comm (s₁ s₂ s₃ : set X) : (s₁ ∪ s₂) ∪ s₃ = (s₁ ∪ s₃) ∪ s₂ :=
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!right_comm union_comm union_assoc s₁ s₂ s₃
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theorem subset_union_left (s t : set X) : s ⊆ s ∪ t := λ x H, or.inl H
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theorem subset_union_right (s t : set X) : t ⊆ s ∪ t := λ x H, or.inr H
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theorem union_subset {s t r : set X} (sr : s ⊆ r) (tr : t ⊆ r) : s ∪ t ⊆ r :=
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λ x xst, or.elim xst (λ xs, sr xs) (λ xt, tr xt)
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/- intersection -/
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definition inter (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_iff (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 mem_inter {x : X} {a b : set X} (Ha : x ∈ a) (Hb : x ∈ b) : x ∈ a ∩ b :=
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and.intro Ha Hb
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theorem mem_of_mem_inter_left {x : X} {a b : set X} (H : x ∈ a ∩ b) : x ∈ a :=
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and.left H
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theorem mem_of_mem_inter_right {x : X} {a b : set X} (H : x ∈ a ∩ b) : x ∈ b :=
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and.right H
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theorem inter_self (a : set X) : a ∩ a = a :=
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ext (take x, !and_self)
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theorem inter_empty (a : set X) : a ∩ ∅ = ∅ :=
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ext (take x, !and_false)
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theorem empty_inter (a : set X) : ∅ ∩ a = ∅ :=
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ext (take x, !false_and)
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theorem inter_comm (a b : set X) : a ∩ b = b ∩ a :=
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ext (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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ext (take x, !and.assoc)
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theorem inter_left_comm (s₁ s₂ s₃ : set X) : s₁ ∩ (s₂ ∩ s₃) = s₂ ∩ (s₁ ∩ s₃) :=
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!left_comm inter_comm inter_assoc s₁ s₂ s₃
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theorem inter_right_comm (s₁ s₂ s₃ : set X) : (s₁ ∩ s₂) ∩ s₃ = (s₁ ∩ s₃) ∩ s₂ :=
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!right_comm inter_comm inter_assoc s₁ s₂ s₃
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theorem inter_univ (a : set X) : a ∩ univ = a :=
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ext (take x, !and_true)
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theorem univ_inter (a : set X) : univ ∩ a = a :=
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ext (take x, !true_and)
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theorem inter_subset_left (s t : set X) : s ∩ t ⊆ s := λ x H, and.left H
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theorem inter_subset_right (s t : set X) : s ∩ t ⊆ t := λ x H, and.right H
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theorem subset_inter {s t r : set X} (rs : r ⊆ s) (rt : r ⊆ t) : r ⊆ s ∩ t :=
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λ x xr, and.intro (rs xr) (rt xr)
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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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ext (take x, !and.left_distrib)
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theorem inter_distrib_right (s t u : set X) : (s ∪ t) ∩ u = (s ∩ u) ∪ (t ∩ u) :=
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ext (take x, !and.right_distrib)
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theorem union_distrib_left (s t u : set X) : s ∪ (t ∩ u) = (s ∪ t) ∩ (s ∪ u) :=
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ext (take x, !or.left_distrib)
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theorem union_distrib_right (s t u : set X) : (s ∩ t) ∪ u = (s ∪ u) ∩ (t ∪ u) :=
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ext (take x, !or.right_distrib)
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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 `{` binder ` | ` r:(scoped:1 P, set_of P) `}` := r
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-- {x ∈ s | P}
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definition sep (P : X → Prop) (s : set X) : set X := λx, x ∈ s ∧ P x
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notation `{` binder ` ∈ ` s ` | ` r:(scoped:1 p, sep p s) `}` := r
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/- insert -/
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definition insert (x : X) (a : set X) : set X := {y : X | y = x ∨ y ∈ a}
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-- '{x, y, z}
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notation `'{`:max a:(foldr `, ` (x b, insert x b) ∅) `}`:0 := a
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theorem subset_insert (x : X) (a : set X) : a ⊆ insert x a :=
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take y, assume ys, or.inr ys
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theorem mem_insert (x : X) (s : set X) : x ∈ insert x s :=
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or.inl rfl
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theorem mem_insert_of_mem {x : X} {s : set X} (y : X) : x ∈ s → x ∈ insert y s :=
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assume h, or.inr h
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theorem eq_or_mem_of_mem_insert {x a : X} {s : set X} : x ∈ insert a s → x = a ∨ x ∈ s :=
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assume h, h
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theorem mem_of_mem_insert_of_ne {x a : X} {s : set X} (xin : x ∈ insert a s) : x ≠ a → x ∈ s :=
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or_resolve_right (eq_or_mem_of_mem_insert xin)
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theorem mem_insert_eq (x a : X) (s : set X) : x ∈ insert a s = (x = a ∨ x ∈ s) :=
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propext (iff.intro !eq_or_mem_of_mem_insert
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(or.rec (λH', (eq.substr H' !mem_insert)) !mem_insert_of_mem))
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theorem insert_eq_of_mem {a : X} {s : set X} (H : a ∈ s) : insert a s = s :=
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ext (λ x, eq.substr (mem_insert_eq x a s)
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(or_iff_right_of_imp (λH1, eq.substr H1 H)))
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theorem insert.comm (x y : X) (s : set X) : insert x (insert y s) = insert y (insert x s) :=
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ext (take a, by rewrite [*mem_insert_eq, propext !or.left_comm])
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-- useful in proofs by induction
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theorem forall_of_forall_insert {P : X → Prop} {a : X} {s : set X}
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(H : ∀ x, x ∈ insert a s → P x) :
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∀ x, x ∈ s → P x :=
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λ x xs, H x (!mem_insert_of_mem xs)
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/- singleton -/
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theorem mem_singleton_iff (a b : X) : a ∈ '{b} ↔ a = b :=
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iff.intro
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(assume ainb, or.elim ainb (λ aeqb, aeqb) (λ f, false.elim f))
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(assume aeqb, or.inl aeqb)
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theorem mem_singleton (a : X) : a ∈ '{a} := !mem_insert
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theorem eq_of_mem_singleton {x y : X} (h : x ∈ '{y}) : x = y :=
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or.elim (eq_or_mem_of_mem_insert h)
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(suppose x = y, this)
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(suppose x ∈ ∅, absurd this !not_mem_empty)
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theorem mem_singleton_of_eq {x y : X} (H : x = y) : x ∈ '{y} :=
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eq.symm H ▸ mem_singleton y
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theorem insert_eq (x : X) (s : set X) : insert x s = '{x} ∪ s :=
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ext (take y, iff.intro
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(suppose y ∈ insert x s,
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or.elim this (suppose y = x, or.inl (or.inl this)) (suppose y ∈ s, or.inr this))
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(suppose y ∈ '{x} ∪ s,
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or.elim this
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(suppose y ∈ '{x}, or.inl (eq_of_mem_singleton this))
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(suppose y ∈ s, or.inr this)))
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/- separation -/
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theorem mem_sep {s : set X} {P : X → Prop} {x : X} (xs : x ∈ s) (Px : P x) : x ∈ {x ∈ s | P x} :=
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and.intro xs Px
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theorem eq_sep_of_subset {s t : set X} (ssubt : s ⊆ t) : s = {x ∈ t | x ∈ s} :=
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ext (take x, iff.intro
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(suppose x ∈ s, and.intro (ssubt this) this)
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(suppose x ∈ {x ∈ t | x ∈ s}, and.right this))
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theorem mem_sep_iff {s : set X} {P : X → Prop} {x : X} : x ∈ {x ∈ s | P x} ↔ x ∈ s ∧ P x :=
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!iff.refl
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theorem sep_subset (s : set X) (P : X → Prop) : {x ∈ s | P x} ⊆ s :=
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take x, assume H, and.left H
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/- complement -/
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definition complement (s : set X) : set X := {x | x ∉ s}
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prefix `-` := complement
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theorem mem_comp {s : set X} {x : X} (H : x ∉ s) : x ∈ -s := H
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theorem not_mem_of_mem_comp {s : set X} {x : X} (H : x ∈ -s) : x ∉ s := H
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theorem mem_comp_iff (s : set X) (x : X) : x ∈ -s ↔ x ∉ s := !iff.refl
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theorem inter_comp_self (s : set X) : s ∩ -s = ∅ :=
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ext (take x, !and_not_self_iff)
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theorem comp_inter_self (s : set X) : -s ∩ s = ∅ :=
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ext (take x, !not_and_self_iff)
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/- some classical identities -/
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section
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open classical
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theorem comp_empty : -(∅ : set X) = univ :=
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ext (take x, iff.intro (assume H, trivial) (assume H, not_false))
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theorem comp_union (s t : set X) : -(s ∪ t) = -s ∩ -t :=
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ext (take x, !not_or_iff_not_and_not)
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theorem comp_comp (s : set X) : -(-s) = s :=
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ext (take x, !not_not_iff)
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theorem comp_inter (s t : set X) : -(s ∩ t) = -s ∪ -t :=
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ext (take x, !not_and_iff_not_or_not)
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theorem comp_univ : -(univ : set X) = ∅ :=
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by rewrite [-comp_empty, comp_comp]
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theorem union_eq_comp_comp_inter_comp (s t : set X) : s ∪ t = -(-s ∩ -t) :=
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ext (take x, !or_iff_not_and_not)
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theorem inter_eq_comp_comp_union_comp (s t : set X) : s ∩ t = -(-s ∪ -t) :=
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ext (take x, !and_iff_not_or_not)
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theorem union_comp_self (s : set X) : s ∪ -s = univ :=
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ext (take x, !or_not_self_iff)
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theorem comp_union_self (s : set X) : -s ∪ s = univ :=
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ext (take x, !not_or_self_iff)
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theorem complement_compose_complement :
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#function complement ∘ complement = @id (set X) :=
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funext (λ s, comp_comp s)
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end
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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_diff {s t : set X} {x : X} (H1 : x ∈ s) (H2 : x ∉ t) : x ∈ s \ t :=
|
||
and.intro H1 H2
|
||
|
||
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_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
|
||
|
||
theorem diff_eq (s t : set X) : s \ t = s ∩ -t := rfl
|
||
|
||
theorem union_diff_cancel {s t : set X} [dec : Π x, decidable (x ∈ s)] (H : s ⊆ t) : s ∪ (t \ s) = t :=
|
||
ext (take x, iff.intro
|
||
(assume H1 : x ∈ s ∪ (t \ s), or.elim H1 (assume H2, !H H2) (assume H2, and.left H2))
|
||
(assume H1 : x ∈ t,
|
||
decidable.by_cases
|
||
(suppose x ∈ s, or.inl this)
|
||
(suppose x ∉ s, or.inr (and.intro H1 this))))
|
||
|
||
theorem diff_subset (s t : set X) : s \ t ⊆ s := inter_subset_left s _
|
||
|
||
theorem comp_eq_univ_diff (s : set X) : -s = univ \ s :=
|
||
ext (take x, iff.intro (assume H, and.intro trivial H) (assume H, and.right H))
|
||
|
||
/- powerset -/
|
||
|
||
definition powerset (s : set X) : set (set X) := {x : set X | x ⊆ s}
|
||
prefix `𝒫`:100 := powerset
|
||
|
||
theorem mem_powerset {x s : set X} (H : x ⊆ s) : x ∈ 𝒫 s := H
|
||
|
||
theorem subset_of_mem_powerset {x s : set X} (H : x ∈ 𝒫 s) : x ⊆ s := H
|
||
|
||
theorem mem_powerset_iff (x s : set X) : x ∈ 𝒫 s ↔ x ⊆ s := !iff.refl
|
||
|
||
/- function image -/
|
||
|
||
section image
|
||
|
||
variables {Y Z : Type}
|
||
|
||
abbreviation eq_on (f1 f2 : X → Y) (a : set X) : Prop :=
|
||
∀₀ x ∈ a, f1 x = f2 x
|
||
|
||
definition image (f : X → Y) (a : set X) : set Y := {y : Y | ∃x, x ∈ a ∧ f x = y}
|
||
notation f `'[`:max a `]` := image f a
|
||
|
||
theorem image_eq_image_of_eq_on {f1 f2 : X → Y} {a : set X} (H1 : eq_on f1 f2 a) :
|
||
f1 '[a] = f2 '[a] :=
|
||
ext (take y, iff.intro
|
||
(assume H2,
|
||
obtain x (H3 : x ∈ a ∧ f1 x = y), from H2,
|
||
have H4 : x ∈ a, from and.left H3,
|
||
have H5 : f2 x = y, from (H1 H4)⁻¹ ⬝ and.right H3,
|
||
exists.intro x (and.intro H4 H5))
|
||
(assume H2,
|
||
obtain x (H3 : x ∈ a ∧ f2 x = y), from H2,
|
||
have H4 : x ∈ a, from and.left H3,
|
||
have H5 : f1 x = y, from (H1 H4) ⬝ and.right H3,
|
||
exists.intro x (and.intro H4 H5)))
|
||
|
||
theorem mem_image {f : X → Y} {a : set X} {x : X} {y : Y}
|
||
(H1 : x ∈ a) (H2 : f x = y) : y ∈ f '[a] :=
|
||
exists.intro x (and.intro H1 H2)
|
||
|
||
theorem mem_image_of_mem (f : X → Y) {x : X} {a : set X} (H : x ∈ a) : f x ∈ image f a :=
|
||
mem_image H rfl
|
||
|
||
lemma image_compose (f : Y → Z) (g : X → Y) (a : set X) : (f ∘ g) '[a] = f '[g '[a]] :=
|
||
ext (take z,
|
||
iff.intro
|
||
(assume Hz : z ∈ (f ∘ g) '[a],
|
||
obtain x (Hx₁ : x ∈ a) (Hx₂ : f (g x) = z), from Hz,
|
||
have Hgx : g x ∈ g '[a], from mem_image Hx₁ rfl,
|
||
show z ∈ f '[g '[a]], from mem_image Hgx Hx₂)
|
||
(assume Hz : z ∈ f '[g '[a]],
|
||
obtain y (Hy₁ : y ∈ g '[a]) (Hy₂ : f y = z), from Hz,
|
||
obtain x (Hz₁ : x ∈ a) (Hz₂ : g x = y), from Hy₁,
|
||
show z ∈ (f ∘ g) '[a], from mem_image Hz₁ (Hz₂⁻¹ ▸ Hy₂)))
|
||
|
||
lemma image_subset {a b : set X} (f : X → Y) (H : a ⊆ b) : f '[a] ⊆ f '[b] :=
|
||
take y, assume Hy : y ∈ f '[a],
|
||
obtain x (Hx₁ : x ∈ a) (Hx₂ : f x = y), from Hy,
|
||
mem_image (H Hx₁) Hx₂
|
||
|
||
theorem image_union (f : X → Y) (s t : set X) :
|
||
image f (s ∪ t) = image f s ∪ image f t :=
|
||
ext (take y, iff.intro
|
||
(assume H : y ∈ image f (s ∪ t),
|
||
obtain x [(xst : x ∈ s ∪ t) (fxy : f x = y)], from H,
|
||
or.elim xst
|
||
(assume xs, or.inl (mem_image xs fxy))
|
||
(assume xt, or.inr (mem_image xt fxy)))
|
||
(assume H : y ∈ image f s ∪ image f t,
|
||
or.elim H
|
||
(assume yifs : y ∈ image f s,
|
||
obtain x [(xs : x ∈ s) (fxy : f x = y)], from yifs,
|
||
mem_image (or.inl xs) fxy)
|
||
(assume yift : y ∈ image f t,
|
||
obtain x [(xt : x ∈ t) (fxy : f x = y)], from yift,
|
||
mem_image (or.inr xt) fxy)))
|
||
|
||
theorem image_empty (f : X → Y) : image f ∅ = ∅ :=
|
||
eq_empty_of_forall_not_mem
|
||
(take y, suppose y ∈ image f ∅,
|
||
obtain x [(H : x ∈ empty) H'], from this,
|
||
H)
|
||
|
||
theorem mem_image_complement (t : set X) (S : set (set X)) :
|
||
t ∈ complement '[S] ↔ -t ∈ S :=
|
||
iff.intro
|
||
(suppose t ∈ complement '[S],
|
||
obtain t' [(Ht' : t' ∈ S) (Ht : -t' = t)], from this,
|
||
show -t ∈ S, by rewrite [-Ht, comp_comp]; exact Ht')
|
||
(suppose -t ∈ S,
|
||
have -(-t) ∈ complement '[S], from mem_image_of_mem complement this,
|
||
show t ∈ complement '[S], from comp_comp t ▸ this)
|
||
|
||
theorem image_id (s : set X) : id '[s] = s :=
|
||
ext (take x, iff.intro
|
||
(suppose x ∈ id '[s],
|
||
obtain x' [(Hx' : x' ∈ s) (x'eq : x' = x)], from this,
|
||
show x ∈ s, by rewrite [-x'eq]; apply Hx')
|
||
(suppose x ∈ s, mem_image_of_mem id this))
|
||
|
||
theorem complement_complement_image (S : set (set X)) :
|
||
complement '[complement '[S]] = S :=
|
||
by rewrite [-image_compose, complement_compose_complement, image_id]
|
||
|
||
end image
|
||
|
||
/- collections of disjoint sets -/
|
||
|
||
definition disjoint_sets (S : set (set X)) : Prop := ∀ a b, a ∈ S → b ∈ S → a ≠ b → a ∩ b = ∅
|
||
|
||
theorem disjoint_sets_empty : disjoint_sets (∅ : set (set X)) :=
|
||
take a b, assume H, !not.elim !not_mem_empty H
|
||
|
||
theorem disjoint_sets_union {s t : set (set X)} (Hs : disjoint_sets s) (Ht : disjoint_sets t)
|
||
(H : ∀ x y, x ∈ s ∧ y ∈ t → x ∩ y = ∅) :
|
||
disjoint_sets (s ∪ t) :=
|
||
take a b, assume Ha Hb Hneq, or.elim Ha
|
||
(assume H1, or.elim Hb
|
||
(suppose b ∈ s, (Hs a b) H1 this Hneq)
|
||
(suppose b ∈ t, (H a b) (and.intro H1 this)))
|
||
(assume H2, or.elim Hb
|
||
(suppose b ∈ s, !inter_comm ▸ ((H b a) (and.intro this H2)))
|
||
(suppose b ∈ t, (Ht a b) H2 this Hneq))
|
||
|
||
theorem disjoint_sets_singleton (s : set (set X)) : disjoint_sets '{s} :=
|
||
take a b, assume Ha Hb Hneq,
|
||
absurd (eq.trans ((iff.elim_left !mem_singleton_iff) Ha) ((iff.elim_left !mem_singleton_iff) Hb)⁻¹)
|
||
Hneq
|
||
|
||
/- large unions -/
|
||
|
||
section large_unions
|
||
variables {I : Type}
|
||
variable a : set I
|
||
variable b : I → set X
|
||
variable C : set (set X)
|
||
|
||
definition sUnion : set X := {x : X | ∃₀ c ∈ C, x ∈ c}
|
||
definition sInter : set X := {x : X | ∀₀ c ∈ C, x ∈ c}
|
||
|
||
prefix `⋃₀`:110 := sUnion
|
||
prefix `⋂₀`:110 := sInter
|
||
|
||
definition Union : set X := {x : X | ∃i, x ∈ b i}
|
||
definition Inter : set X := {x : X | ∀i, x ∈ b i}
|
||
|
||
notation `⋃` binders, r:(scoped f, Union f) := r
|
||
notation `⋂` binders, r:(scoped f, Inter f) := r
|
||
|
||
definition bUnion : set X := {x : X | ∃₀ i ∈ a, x ∈ b i}
|
||
definition bInter : set X := {x : X | ∀₀ i ∈ a, x ∈ b i}
|
||
|
||
notation `⋃` binders `∈` s, r:(scoped f, bUnion s f) := r
|
||
notation `⋂` binders `∈` s, r:(scoped f, bInter s f) := r
|
||
|
||
end large_unions
|
||
|
||
-- sUnion and sInter: a collection (set) of sets
|
||
|
||
theorem mem_sUnion {x : X} {t : set X} {S : set (set X)} (Hx : x ∈ t) (Ht : t ∈ S) :
|
||
x ∈ ⋃₀ S :=
|
||
exists.intro t (and.intro Ht Hx)
|
||
|
||
theorem mem_sInter {x : X} {t : set X} {S : set (set X)} (H : ∀₀ t ∈ S, x ∈ t) :
|
||
x ∈ ⋂₀ S :=
|
||
H
|
||
|
||
theorem sInter_subset_of_mem {S : set (set X)} {t : set X} (tS : t ∈ S) :
|
||
(⋂₀ S) ⊆ t :=
|
||
take x, assume H, H t tS
|
||
|
||
theorem subset_sUnion_of_mem {S : set (set X)} {t : set X} (tS : t ∈ S) :
|
||
t ⊆ (⋃₀ S) :=
|
||
take x, assume H, exists.intro t (and.intro tS H)
|
||
|
||
theorem sUnion_empty : ⋃₀ ∅ = (∅ : set X) :=
|
||
eq_empty_of_forall_not_mem
|
||
(take x, suppose x ∈ sUnion ∅,
|
||
obtain t [(Ht : t ∈ ∅) Ht'], from this,
|
||
show false, from Ht)
|
||
|
||
theorem sInter_empty : ⋂₀ ∅ = (univ : set X) :=
|
||
eq_univ_of_forall (λ x s H, false.elim H)
|
||
|
||
theorem sUnion_singleton (s : set X) : ⋃₀ '{s} = s :=
|
||
ext (take x, iff.intro
|
||
(suppose x ∈ sUnion '{s},
|
||
obtain u [(Hu : u ∈ '{s}) (xu : x ∈ u)], from this,
|
||
have u = s, from eq_of_mem_singleton Hu,
|
||
show x ∈ s, using this, by rewrite -this; apply xu)
|
||
(suppose x ∈ s,
|
||
mem_sUnion this (mem_singleton s)))
|
||
|
||
theorem sInter_singleton (s : set X) : ⋂₀ '{s} = s :=
|
||
ext (take x, iff.intro
|
||
(suppose x ∈ ⋂₀ '{s}, show x ∈ s, from this (mem_singleton s))
|
||
(suppose x ∈ s, take u, suppose u ∈ '{s},
|
||
show x ∈ u, by+ rewrite [eq_of_mem_singleton this]; assumption))
|
||
|
||
theorem sUnion_union (S T : set (set X)) : ⋃₀ (S ∪ T) = ⋃₀ S ∪ ⋃₀ T :=
|
||
ext (take x, iff.intro
|
||
(suppose x ∈ sUnion (S ∪ T),
|
||
obtain u [(Hu : u ∈ S ∪ T) (xu : x ∈ u)], from this,
|
||
or.elim Hu
|
||
(assume uS, or.inl (mem_sUnion xu uS))
|
||
(assume uT, or.inr (mem_sUnion xu uT)))
|
||
(suppose x ∈ sUnion S ∪ sUnion T,
|
||
or.elim this
|
||
(suppose x ∈ sUnion S,
|
||
obtain u [(uS : u ∈ S) (xu : x ∈ u)], from this,
|
||
mem_sUnion xu (or.inl uS))
|
||
(suppose x ∈ sUnion T,
|
||
obtain u [(uT : u ∈ T) (xu : x ∈ u)], from this,
|
||
mem_sUnion xu (or.inr uT))))
|
||
|
||
theorem sInter_union (S T : set (set X)) : ⋂₀ (S ∪ T) = ⋂₀ S ∩ ⋂₀ T :=
|
||
ext (take x, iff.intro
|
||
(assume H : x ∈ ⋂₀ (S ∪ T),
|
||
and.intro (λ u uS, H (or.inl uS)) (λ u uT, H (or.inr uT)))
|
||
(assume H : x ∈ ⋂₀ S ∩ ⋂₀ T,
|
||
take u, suppose u ∈ S ∪ T, or.elim this (λ uS, and.left H u uS) (λ uT, and.right H u uT)))
|
||
|
||
theorem sUnion_insert (s : set X) (T : set (set X)) :
|
||
⋃₀ (insert s T) = s ∪ ⋃₀ T :=
|
||
by rewrite [insert_eq, sUnion_union, sUnion_singleton]
|
||
|
||
theorem sInter_insert (s : set X) (T : set (set X)) :
|
||
⋂₀ (insert s T) = s ∩ ⋂₀ T :=
|
||
by rewrite [insert_eq, sInter_union, sInter_singleton]
|
||
|
||
theorem comp_sUnion (S : set (set X)) :
|
||
- ⋃₀ S = ⋂₀ (complement '[S]) :=
|
||
ext (take x, iff.intro
|
||
(assume H : x ∈ -(⋃₀ S),
|
||
take t, suppose t ∈ complement '[S],
|
||
obtain t' [(Ht' : t' ∈ S) (Ht : -t' = t)], from this,
|
||
have x ∈ -t', from suppose x ∈ t', H (mem_sUnion this Ht'),
|
||
show x ∈ t, using this, by rewrite -Ht; apply this)
|
||
(assume H : x ∈ ⋂₀ (complement '[S]),
|
||
suppose x ∈ ⋃₀ S,
|
||
obtain t [(tS : t ∈ S) (xt : x ∈ t)], from this,
|
||
have -t ∈ complement '[S], from mem_image_of_mem complement tS,
|
||
have x ∈ -t, from H this,
|
||
show false, proof this xt qed))
|
||
|
||
theorem sUnion_eq_comp_sInter_comp (S : set (set X)) :
|
||
⋃₀ S = - ⋂₀ (complement '[S]) :=
|
||
by rewrite [-comp_comp, comp_sUnion]
|
||
|
||
theorem comp_sInter (S : set (set X)) :
|
||
- ⋂₀ S = ⋃₀ (complement '[S]) :=
|
||
by rewrite [sUnion_eq_comp_sInter_comp, complement_complement_image]
|
||
|
||
theorem sInter_eq_comp_sUnion_comp (S : set (set X)) :
|
||
⋂₀ S = -(⋃₀ (complement '[S])) :=
|
||
by rewrite [-comp_comp, comp_sInter]
|
||
|
||
-- Union and Inter: a family of sets indexed by a type
|
||
|
||
theorem Union_subset {I : Type} {b : I → set X} {c : set X} (H : ∀ i, b i ⊆ c) : (⋃ i, b i) ⊆ c :=
|
||
take x,
|
||
suppose x ∈ Union b,
|
||
obtain i (Hi : x ∈ b i), from this,
|
||
show x ∈ c, from H i Hi
|
||
|
||
theorem subset_Inter {I : Type} {b : I → set X} {c : set X} (H : ∀ i, c ⊆ b i) : c ⊆ ⋂ i, b i :=
|
||
λ x cx i, H i cx
|
||
|
||
theorem Union_eq_sUnion_image {X I : Type} (s : I → set X) : (⋃ i, s i) = ⋃₀ (s '[univ]) :=
|
||
ext (take x, iff.intro
|
||
(suppose x ∈ Union s,
|
||
obtain i (Hi : x ∈ s i), from this,
|
||
mem_sUnion Hi (mem_image_of_mem s trivial))
|
||
(suppose x ∈ sUnion (s '[univ]),
|
||
obtain t [(Ht : t ∈ s '[univ]) (Hx : x ∈ t)], from this,
|
||
obtain i [univi (Hi : s i = t)], from Ht,
|
||
exists.intro i (show x ∈ s i, by rewrite Hi; apply Hx)))
|
||
|
||
theorem Inter_eq_sInter_image {X I : Type} (s : I → set X) : (⋂ i, s i) = ⋂₀ (s '[univ]) :=
|
||
ext (take x, iff.intro
|
||
(assume H : x ∈ Inter s,
|
||
take t,
|
||
suppose t ∈ s '[univ],
|
||
obtain i [univi (Hi : s i = t)], from this,
|
||
show x ∈ t, by rewrite -Hi; exact H i)
|
||
(assume H : x ∈ ⋂₀ (s '[univ]),
|
||
take i,
|
||
have s i ∈ s '[univ], from mem_image_of_mem s trivial,
|
||
show x ∈ s i, from H this))
|
||
|
||
theorem comp_Union {X I : Type} (s : I → set X) : - (⋃ i, s i) = (⋂ i, - s i) :=
|
||
by rewrite [Union_eq_sUnion_image, comp_sUnion, -image_compose, -Inter_eq_sInter_image]
|
||
|
||
theorem comp_Inter {X I : Type} (s : I → set X) : -(⋂ i, s i) = (⋃ i, - s i) :=
|
||
by rewrite [Inter_eq_sInter_image, comp_sInter, -image_compose, -Union_eq_sUnion_image]
|
||
|
||
theorem Union_eq_comp_Inter_comp {X I : Type} (s : I → set X) : (⋃ i, s i) = - (⋂ i, - s i) :=
|
||
by rewrite [-comp_comp, comp_Union]
|
||
|
||
theorem Inter_eq_comp_Union_comp {X I : Type} (s : I → set X) : (⋂ i, s i) = - (⋃ i, -s i) :=
|
||
by rewrite [-comp_comp, comp_Inter]
|
||
|
||
-- these are useful for turning binary union / intersection into countable ones
|
||
|
||
definition bin_ext (s t : set X) (n : ℕ) : set X :=
|
||
nat.cases_on n s (λ m, t)
|
||
|
||
lemma Union_bin_ext (s t : set X) : (⋃ i, bin_ext s t i) = s ∪ t :=
|
||
ext (take x, iff.intro
|
||
(assume H,
|
||
obtain i (Hi : x ∈ (bin_ext s t) i), from H,
|
||
by cases i; apply or.inl Hi; apply or.inr Hi)
|
||
(assume H,
|
||
or.elim H
|
||
(suppose x ∈ s, exists.intro 0 this)
|
||
(suppose x ∈ t, exists.intro 1 this)))
|
||
|
||
lemma Inter_bin_ext (s t : set X) : (⋂ i, bin_ext s t i) = s ∩ t :=
|
||
ext (take x, iff.intro
|
||
(assume H, and.intro (H 0) (H 1))
|
||
(assume H, by intro i; cases i;
|
||
apply and.elim_left H; apply and.elim_right H))
|
||
|
||
-- bUnion and bInter: a family of sets indexed by a set ("b" is for bounded)
|
||
|
||
variable {Y : Type}
|
||
|
||
theorem mem_bUnion {s : set X} {f : X → set Y} {x : X} {y : Y}
|
||
(xs : x ∈ s) (yfx : y ∈ f x) :
|
||
y ∈ ⋃ x ∈ s, f x :=
|
||
exists.intro x (and.intro xs yfx)
|
||
|
||
theorem mem_bInter {s : set X} {f : X → set Y} {y : Y} (H : ∀₀ x ∈ s, y ∈ f x) :
|
||
y ∈ ⋂ x ∈ s, f x :=
|
||
H
|
||
|
||
theorem bUnion_subset {s : set X} {t : set Y} {f : X → set Y} (H : ∀₀ x ∈ s, f x ⊆ t) :
|
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(⋃ x ∈ s, f x) ⊆ t :=
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take y, assume Hy,
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obtain x [xs yfx], from Hy,
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show y ∈ t, from H xs yfx
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theorem subset_bInter {s : set X} {t : set Y} {f : X → set Y} (H : ∀₀ x ∈ s, t ⊆ f x) :
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t ⊆ ⋂ x ∈ s, f x :=
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take y, assume yt, take x, assume xs, H xs yt
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theorem subset_bUnion_of_mem {s : set X} {f : X → set Y} {x : X} (xs : x ∈ s) :
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f x ⊆ ⋃ x ∈ s, f x :=
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take y, assume Hy, mem_bUnion xs Hy
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theorem bInter_subset_of_mem {s : set X} {f : X → set Y} {x : X} (xs : x ∈ s) :
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(⋂ x ∈ s, f x) ⊆ f x :=
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take y, assume Hy, Hy x xs
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end set
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