333 lines
12 KiB
Text
333 lines
12 KiB
Text
/-
|
||
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
|
||
Module: data.finset
|
||
Author: Leonardo de Moura
|
||
|
||
Finite sets
|
||
-/
|
||
import data.fintype data.nat data.list.perm data.subtype algebra.binary
|
||
open nat quot list subtype binary function
|
||
open [declarations] perm
|
||
|
||
definition nodup_list (A : Type) := {l : list A | nodup l}
|
||
|
||
variable {A : Type}
|
||
|
||
definition to_nodup_list_of_nodup {l : list A} (n : nodup l) : nodup_list A :=
|
||
tag l n
|
||
|
||
definition to_nodup_list [h : decidable_eq A] (l : list A) : nodup_list A :=
|
||
@to_nodup_list_of_nodup A (erase_dup l) (nodup_erase_dup l)
|
||
|
||
namespace finset
|
||
|
||
private definition eqv (l₁ l₂ : nodup_list A) :=
|
||
perm (elt_of l₁) (elt_of l₂)
|
||
|
||
local infix ~ := eqv
|
||
|
||
private definition eqv.refl (l : nodup_list A) : l ~ l :=
|
||
!perm.refl
|
||
|
||
private definition eqv.symm {l₁ l₂ : nodup_list A} : l₁ ~ l₂ → l₂ ~ l₁ :=
|
||
assume p, perm.symm p
|
||
|
||
private definition eqv.trans {l₁ l₂ l₃ : nodup_list A} : l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ :=
|
||
assume p₁ p₂, perm.trans p₁ p₂
|
||
|
||
definition nodup_list_setoid [instance] (A : Type) : setoid (nodup_list A) :=
|
||
setoid.mk (@eqv A) (mk_equivalence (@eqv A) (@eqv.refl A) (@eqv.symm A) (@eqv.trans A))
|
||
|
||
definition finset (A : Type) : Type :=
|
||
quot (nodup_list_setoid A)
|
||
|
||
definition to_finset_of_nodup (l : list A) (n : nodup l) : finset A :=
|
||
⟦to_nodup_list_of_nodup n⟧
|
||
|
||
definition to_finset [h : decidable_eq A] (l : list A) : finset A :=
|
||
⟦to_nodup_list l⟧
|
||
|
||
definition has_decidable_eq [instance] [h : decidable_eq A] : decidable_eq (finset A) :=
|
||
λ s₁ s₂, quot.rec_on_subsingleton₂ s₁ s₂
|
||
(λ l₁ l₂,
|
||
match decidable_perm (elt_of l₁) (elt_of l₂) with
|
||
| decidable.inl e := decidable.inl (quot.sound e)
|
||
| decidable.inr n := decidable.inr (λ e : ⟦l₁⟧ = ⟦l₂⟧, absurd (quot.exact e) n)
|
||
end)
|
||
|
||
definition singleton (a : A) : finset A :=
|
||
to_finset_of_nodup [a] !nodup_singleton
|
||
|
||
definition mem (a : A) (s : finset A) : Prop :=
|
||
quot.lift_on s (λ l, a ∈ elt_of l)
|
||
(λ l₁ l₂ (e : l₁ ~ l₂), propext (iff.intro
|
||
(λ ainl₁, mem_perm e ainl₁)
|
||
(λ ainl₂, mem_perm (perm.symm e) ainl₂)))
|
||
|
||
infix `∈` := mem
|
||
notation a ∉ b := ¬ mem a b
|
||
|
||
theorem mem_of_mem_list {a : A} {l : nodup_list A} : a ∈ elt_of l → a ∈ ⟦l⟧ :=
|
||
λ ainl, ainl
|
||
|
||
theorem mem_list_of_mem {a : A} {l : nodup_list A} : a ∈ ⟦l⟧ → a ∈ elt_of l :=
|
||
λ ainl, ainl
|
||
|
||
theorem mem_singleton (a : A) : a ∈ singleton a :=
|
||
mem_of_mem_list !mem_cons
|
||
|
||
definition decidable_mem [instance] [h : decidable_eq A] : ∀ (a : A) (s : finset A), decidable (a ∈ s) :=
|
||
λ a s, quot.rec_on_subsingleton s
|
||
(λ l, match list.decidable_mem a (elt_of l) with
|
||
| decidable.inl p := decidable.inl (mem_of_mem_list p)
|
||
| decidable.inr n := decidable.inr (λ p, absurd (mem_list_of_mem p) n)
|
||
end)
|
||
|
||
theorem mem_to_finset [h : decidable_eq A] {a : A} {l : list A} : a ∈ l → a ∈ to_finset l :=
|
||
λ ainl, mem_erase_dup ainl
|
||
|
||
theorem mem_to_finset_of_nodub {a : A} {l : list A} (n : nodup l) : a ∈ l → a ∈ to_finset_of_nodup l n :=
|
||
λ ainl, ainl
|
||
|
||
/- extensionality -/
|
||
theorem ext {s₁ s₂ : finset A} : (∀ a, a ∈ s₁ ↔ a ∈ s₂) → s₁ = s₂ :=
|
||
quot.induction_on₂ s₁ s₂ (λ l₁ l₂ e, quot.sound (perm_ext (has_property l₁) (has_property l₂) e))
|
||
|
||
/- empty -/
|
||
definition empty : finset A :=
|
||
to_finset_of_nodup [] nodup_nil
|
||
|
||
notation `∅` := !empty
|
||
|
||
theorem not_mem_empty (a : A) : a ∉ ∅ :=
|
||
λ aine : a ∈ ∅, aine
|
||
|
||
/- universe -/
|
||
definition univ [h : fintype A] : finset A :=
|
||
to_finset_of_nodup (@fintype.elems A h) (@fintype.unique A h)
|
||
|
||
theorem mem_univ [h : fintype A] (x : A) : x ∈ univ :=
|
||
fintype.complete x
|
||
|
||
/- card -/
|
||
definition card (s : finset A) : nat :=
|
||
quot.lift_on s
|
||
(λ l, length (elt_of l))
|
||
(λ l₁ l₂ p, length_eq_length_of_perm p)
|
||
|
||
theorem card_empty : card (@empty A) = 0 :=
|
||
rfl
|
||
|
||
theorem card_singleton (a : A) : card (singleton a) = 1 :=
|
||
rfl
|
||
|
||
/- insert -/
|
||
section insert
|
||
variable [h : decidable_eq A]
|
||
include h
|
||
|
||
definition insert (a : A) (s : finset A) : finset A :=
|
||
quot.lift_on s
|
||
(λ l, to_finset_of_nodup (insert a (elt_of l)) (nodup_insert a (has_property l)))
|
||
(λ (l₁ l₂ : nodup_list A) (p : l₁ ~ l₂), quot.sound (perm_insert a p))
|
||
|
||
theorem mem_insert (a : A) (s : finset A) : a ∈ insert a s :=
|
||
quot.induction_on s
|
||
(λ l : nodup_list A, mem_to_finset_of_nodub _ !list.mem_insert)
|
||
|
||
theorem mem_insert_of_mem {a : A} {s : finset A} (b : A) : a ∈ s → a ∈ insert b s :=
|
||
quot.induction_on s
|
||
(λ (l : nodup_list A) (ainl : a ∈ ⟦l⟧), mem_to_finset_of_nodub _ (list.mem_insert_of_mem _ ainl))
|
||
|
||
theorem card_insert_of_mem {a : A} {s : finset A} : a ∈ s → card (insert a s) = card s :=
|
||
quot.induction_on s
|
||
(λ (l : nodup_list A) (ainl : a ∈ ⟦l⟧), list.length_insert_of_mem ainl)
|
||
|
||
theorem card_insert_of_not_mem {a : A} {s : finset A} : a ∉ s → card (insert a s) = card s + 1 :=
|
||
quot.induction_on s
|
||
(λ (l : nodup_list A) (nainl : a ∉ ⟦l⟧), list.length_insert_of_not_mem nainl)
|
||
end insert
|
||
|
||
/- erase -/
|
||
section erase
|
||
variable [h : decidable_eq A]
|
||
include h
|
||
|
||
definition erase (a : A) (s : finset A) : finset A :=
|
||
quot.lift_on s
|
||
(λ l, to_finset_of_nodup (erase a (elt_of l)) (nodup_erase_of_nodup a (has_property l)))
|
||
(λ (l₁ l₂ : nodup_list A) (p : l₁ ~ l₂), quot.sound (erase_perm_erase_of_perm a p))
|
||
|
||
theorem mem_erase (a : A) (s : finset A) : a ∉ erase a s :=
|
||
quot.induction_on s
|
||
(λ l, list.mem_erase_of_nodup _ (has_property l))
|
||
|
||
theorem card_erase_of_mem {a : A} {s : finset A} : a ∈ s → card (erase a s) = pred (card s) :=
|
||
quot.induction_on s (λ l ainl, list.length_erase_of_mem ainl)
|
||
|
||
theorem card_erase_of_not_mem {a : A} {s : finset A} : a ∉ s → card (erase a s) = card s :=
|
||
quot.induction_on s (λ l nainl, list.length_erase_of_not_mem nainl)
|
||
end erase
|
||
|
||
/- disjoint -/
|
||
definition disjoint (s₁ s₂ : finset A) : Prop :=
|
||
quot.lift_on₂ s₁ s₂ (λ l₁ l₂, disjoint (elt_of l₁) (elt_of l₂))
|
||
(λ v₁ v₂ w₁ w₂ p₁ p₂, propext (iff.intro
|
||
(λ d₁ a (ainw₁ : a ∈ elt_of w₁),
|
||
have ainv₁ : a ∈ elt_of v₁, from mem_perm (perm.symm p₁) ainw₁,
|
||
have nainv₂ : a ∉ elt_of v₂, from disjoint_left d₁ ainv₁,
|
||
not_mem_perm p₂ nainv₂)
|
||
(λ d₂ a (ainv₁ : a ∈ elt_of v₁),
|
||
have ainw₁ : a ∈ elt_of w₁, from mem_perm p₁ ainv₁,
|
||
have nainw₂ : a ∉ elt_of w₂, from disjoint_left d₂ ainw₁,
|
||
not_mem_perm (perm.symm p₂) nainw₂)))
|
||
|
||
theorem disjoint.comm {s₁ s₂ : finset A} : disjoint s₁ s₂ → disjoint s₂ s₁ :=
|
||
quot.induction_on₂ s₁ s₂ (λ l₁ l₂ d, list.disjoint.comm d)
|
||
|
||
/- union -/
|
||
section union
|
||
variable [h : decidable_eq A]
|
||
include h
|
||
|
||
definition union (s₁ s₂ : finset A) : finset A :=
|
||
quot.lift_on₂ s₁ s₂
|
||
(λ l₁ l₂,
|
||
to_finset_of_nodup (list.union (elt_of l₁) (elt_of l₂))
|
||
(nodup_union_of_nodup_of_nodup (has_property l₁) (has_property l₂)))
|
||
(λ v₁ v₂ w₁ w₂ p₁ p₂, quot.sound (perm_union p₁ p₂))
|
||
|
||
notation s₁ ∪ s₂ := union s₁ s₂
|
||
|
||
theorem mem_union_left {a : A} {s₁ : finset A} (s₂ : finset A) : a ∈ s₁ → a ∈ s₁ ∪ s₂ :=
|
||
quot.induction_on₂ s₁ s₂ (λ l₁ l₂ ainl₁, list.mem_union_left _ ainl₁)
|
||
|
||
theorem mem_union_right {a : A} {s₂ : finset A} (s₁ : finset A) : a ∈ s₂ → a ∈ s₁ ∪ s₂ :=
|
||
quot.induction_on₂ s₁ s₂ (λ l₁ l₂ ainl₂, list.mem_union_right _ ainl₂)
|
||
|
||
theorem mem_or_mem_of_mem_union {a : A} {s₁ s₂ : finset A} : a ∈ s₁ ∪ s₂ → a ∈ s₁ ∨ a ∈ s₂ :=
|
||
quot.induction_on₂ s₁ s₂ (λ l₁ l₂ ainl₁l₂, list.mem_or_mem_of_mem_union ainl₁l₂)
|
||
|
||
theorem mem_union_eq (a : A) (s₁ s₂ : finset A) : (a ∈ s₁ ∪ s₂) = (a ∈ s₁ ∨ a ∈ s₂) :=
|
||
propext (iff.intro
|
||
(λ h, mem_or_mem_of_mem_union h)
|
||
(λ d, or.elim d
|
||
(λ i, mem_union_left _ i)
|
||
(λ i, mem_union_right _ i)))
|
||
|
||
theorem union.comm (s₁ s₂ : finset A) : s₁ ∪ s₂ = s₂ ∪ s₁ :=
|
||
ext (λ a, by rewrite [*mem_union_eq]; exact or.comm)
|
||
|
||
theorem union.assoc (s₁ s₂ s₃ : finset A) : (s₁ ∪ s₂) ∪ s₃ = s₁ ∪ (s₂ ∪ s₃) :=
|
||
ext (λ a, by rewrite [*mem_union_eq]; exact or.assoc)
|
||
|
||
theorem union_self (s : finset A) : s ∪ s = s :=
|
||
ext (λ a, iff.intro
|
||
(λ ain, or.elim (mem_or_mem_of_mem_union ain) (λ i, i) (λ i, i))
|
||
(λ i, mem_union_left _ i))
|
||
|
||
theorem union_empty (s : finset A) : s ∪ ∅ = s :=
|
||
ext (λ a, iff.intro
|
||
(λ ain : a ∈ s ∪ ∅, or.elim (mem_or_mem_of_mem_union ain) (λ i, i) (λ i, absurd i !not_mem_empty))
|
||
(λ i : a ∈ s, mem_union_left _ i))
|
||
|
||
theorem empty_union (s : finset A) : ∅ ∪ s = s :=
|
||
calc ∅ ∪ s = s ∪ ∅ : union.comm
|
||
... = s : union_empty
|
||
end union
|
||
|
||
/- inter -/
|
||
section inter
|
||
variable [h : decidable_eq A]
|
||
include h
|
||
|
||
definition inter (s₁ s₂ : finset A) : finset A :=
|
||
quot.lift_on₂ s₁ s₂
|
||
(λ l₁ l₂,
|
||
to_finset_of_nodup (list.inter (elt_of l₁) (elt_of l₂))
|
||
(nodup_inter_of_nodup _ (has_property l₁)))
|
||
(λ v₁ v₂ w₁ w₂ p₁ p₂, quot.sound (perm_inter p₁ p₂))
|
||
|
||
notation s₁ ∩ s₂ := inter s₁ s₂
|
||
|
||
theorem mem_of_mem_inter_left {a : A} {s₁ s₂ : finset A} : a ∈ s₁ ∩ s₂ → a ∈ s₁ :=
|
||
quot.induction_on₂ s₁ s₂ (λ l₁ l₂ ainl₁l₂, list.mem_of_mem_inter_left ainl₁l₂)
|
||
|
||
theorem mem_of_mem_inter_right {a : A} {s₁ s₂ : finset A} : a ∈ s₁ ∩ s₂ → a ∈ s₂ :=
|
||
quot.induction_on₂ s₁ s₂ (λ l₁ l₂ ainl₁l₂, list.mem_of_mem_inter_right ainl₁l₂)
|
||
|
||
theorem mem_inter_of_mem_of_mem {a : A} {s₁ s₂ : finset A} : a ∈ s₁ → a ∈ s₂ → a ∈ s₁ ∩ s₂ :=
|
||
quot.induction_on₂ s₁ s₂ (λ l₁ l₂ ainl₁ ainl₂, list.mem_inter_of_mem_of_mem ainl₁ ainl₂)
|
||
|
||
theorem mem_inter_eq (a : A) (s₁ s₂ : finset A) : (a ∈ s₁ ∩ s₂) = (a ∈ s₁ ∧ a ∈ s₂) :=
|
||
propext (iff.intro
|
||
(λ h, and.intro (mem_of_mem_inter_left h) (mem_of_mem_inter_right h))
|
||
(λ h, mem_inter_of_mem_of_mem (and.elim_left h) (and.elim_right h)))
|
||
|
||
theorem inter.comm (s₁ s₂ : finset A) : s₁ ∩ s₂ = s₂ ∩ s₁ :=
|
||
ext (λ a, by rewrite [*mem_inter_eq]; exact and.comm)
|
||
|
||
theorem inter.assoc (s₁ s₂ s₃ : finset A) : (s₁ ∩ s₂) ∩ s₃ = s₁ ∩ (s₂ ∩ s₃) :=
|
||
ext (λ a, by rewrite [*mem_inter_eq]; exact and.assoc)
|
||
|
||
theorem inter_self (s : finset A) : s ∩ s = s :=
|
||
ext (λ a, iff.intro
|
||
(λ h, mem_of_mem_inter_right h)
|
||
(λ h, mem_inter_of_mem_of_mem h h))
|
||
|
||
theorem inter_empty (s : finset A) : s ∩ ∅ = ∅ :=
|
||
ext (λ a, iff.intro
|
||
(λ h : a ∈ s ∩ ∅, absurd (mem_of_mem_inter_right h) !not_mem_empty)
|
||
(λ h : a ∈ ∅, absurd h !not_mem_empty))
|
||
|
||
theorem empty_inter (s : finset A) : ∅ ∩ s = ∅ :=
|
||
calc ∅ ∩ s = s ∩ ∅ : inter.comm
|
||
... = ∅ : inter_empty
|
||
end inter
|
||
|
||
/- subset -/
|
||
definition subset (s₁ s₂ : finset A) : Prop :=
|
||
quot.lift_on₂ s₁ s₂
|
||
(λ l₁ l₂, sublist (elt_of l₁) (elt_of l₂))
|
||
(λ v₁ v₂ w₁ w₂ p₁ p₂, propext (iff.intro
|
||
(λ s₁ a i, mem_perm p₂ (s₁ a (mem_perm (perm.symm p₁) i)))
|
||
(λ s₂ a i, mem_perm (perm.symm p₂) (s₂ a (mem_perm p₁ i)))))
|
||
|
||
infix `⊆`:50 := subset
|
||
|
||
theorem nil_sub (s : finset A) : ∅ ⊆ s :=
|
||
quot.induction_on s (λ l, list.nil_sub (elt_of l))
|
||
|
||
theorem sub_univ [h : fintype A] (s : finset A) : s ⊆ univ :=
|
||
quot.induction_on s (λ l a i, fintype.complete a)
|
||
|
||
theorem sub.refl (s : finset A) : s ⊆ s :=
|
||
quot.induction_on s (λ l, list.sub.refl (elt_of l))
|
||
|
||
theorem sub.trans {s₁ s₂ s₃ : finset A} : s₁ ⊆ s₂ → s₂ ⊆ s₃ → s₁ ⊆ s₃ :=
|
||
quot.induction_on₃ s₁ s₂ s₃ (λ l₁ l₂ l₃ h₁ h₂, list.sub.trans h₁ h₂)
|
||
|
||
theorem mem_of_sub_of_mem {s₁ s₂ : finset A} {a : A} : s₁ ⊆ s₂ → a ∈ s₁ → a ∈ s₂ :=
|
||
quot.induction_on₂ s₁ s₂ (λ l₁ l₂ h₁ h₂, h₁ a h₂)
|
||
|
||
/- upto -/
|
||
section upto
|
||
definition upto (n : nat) : finset nat :=
|
||
to_finset_of_nodup (list.upto n) (nodup_upto n)
|
||
|
||
theorem card_upto : ∀ n, card (upto n) = n :=
|
||
list.length_upto
|
||
|
||
theorem lt_of_mem_upto {n a : nat} : a ∈ upto n → a < n :=
|
||
list.lt_of_mem_upto
|
||
|
||
theorem mem_upto_succ_of_mem_upto {n a : nat} : a ∈ upto n → a ∈ upto (succ n) :=
|
||
list.mem_upto_succ_of_mem_upto
|
||
|
||
theorem mem_upto_of_lt {n a : nat} : a < n → a ∈ upto n :=
|
||
list.mem_upto_of_lt
|
||
end upto
|
||
end finset
|
||
abbreviation finset := finset.finset
|