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feat(library/data/finset/basic.lean): add lots of theorems, do minor renaming

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Jeremy Avigad 2015-05-08 13:10:17 +10:00
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library/data/finset/basic.lean
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@ -1,11 +1,9 @@
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura, Jeremy Avigad
Module: data.finset
Author: Leonardo de Moura
Finite sets
Finite sets.
-/
import data.fintype data.nat data.list.perm data.subtype algebra.binary
open nat quot list subtype binary function
@ -57,9 +55,6 @@ definition has_decidable_eq [instance] [h : decidable_eq A] : decidable_eq (fins
| 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
@ -75,9 +70,19 @@ theorem mem_of_mem_list {a : A} {l : nodup_list A} : a ∈ elt_of l → a ∈
theorem mem_list_of_mem {a : A} {l : nodup_list A} : a ∈ ⟦l⟧ → a ∈ elt_of l :=
λ ainl, ainl
/- singleton -/
definition singleton (a : A) : finset A :=
to_finset_of_nodup [a] !nodup_singleton
theorem mem_singleton (a : A) : a ∈ singleton a :=
mem_of_mem_list !mem_cons
theorem eq_of_mem_singleton {x a : A} : x ∈ singleton a → x = a :=
list.mem_singleton
theorem mem_singleton_eq (x a : A) : (x ∈ singleton a) = (x = a) :=
propext (iff.intro eq_of_mem_singleton (assume H, eq.subst H !mem_singleton))
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
@ -88,7 +93,7 @@ definition decidable_mem [instance] [h : decidable_eq A] : ∀ (a : A) (s : fins
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 :=
theorem mem_to_finset_of_nodup {a : A} {l : list A} (n : nodup l) : a ∈ l → a ∈ to_finset_of_nodup l n :=
λ ainl, ainl
/- extensionality -/
@ -104,6 +109,9 @@ notation `∅` := !empty
theorem not_mem_empty (a : A) : a ∉ ∅ :=
λ aine : a ∈ ∅, aine
theorem mem_empty_eq (x : A) : x ∈ ∅ = false :=
propext (iff.mp' !iff_false_iff_not !not_mem_empty)
/- universe -/
definition univ [h : fintype A] : finset A :=
to_finset_of_nodup (@fintype.elems A h) (@fintype.unique A h)
@ -111,6 +119,8 @@ 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
theorem mem_univ_eq [h : fintype A] (x : A) : x ∈ univ = true := propext (iff_true_intro !mem_univ)
/- card -/
definition card (s : finset A) : nat :=
quot.lift_on s
@ -133,13 +143,42 @@ 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))
-- set builder notation
notation `{[`:max a:(foldr `,` (x b, insert x b) ∅) `]}`:0 := a
notation `⦃` a:(foldr `,` (x b, insert x b) ∅) `⦄` := a
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)
(λ l : nodup_list A, mem_to_finset_of_nodup _ !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))
(λ (l : nodup_list A) (ainl : a ∈ ⟦l⟧), mem_to_finset_of_nodup _ (list.mem_insert_of_mem _ ainl))
theorem eq_or_mem_of_mem_insert {x a : A} {s : finset A} : x ∈ insert a s → x = a ∨ x ∈ s :=
quot.induction_on s (λ l : nodup_list A, λ H, list.eq_or_mem_of_mem_insert H)
theorem mem_insert_eq (x a : A) (s : finset A) : x ∈ insert a s = (x = a ∨ x ∈ s) :=
propext (iff.intro
(!eq_or_mem_of_mem_insert)
(assume H, or.elim H
(assume H' : x = a, eq.subst (eq.symm H') !mem_insert)
(assume H' : x ∈ s, !mem_insert_of_mem H')))
theorem insert_empty_eq (a : A) : ⦃a⦄ = singleton a := rfl
theorem insert_eq_of_mem {a : A} {s : finset A} (H : a ∈ s) : insert a s = s :=
ext
take x,
begin
rewrite [!mem_insert_eq],
show x = a ∨ x ∈ s ↔ x ∈ s, from
iff.intro
(assume H1, or.elim H1
(assume H2 : x = a, eq.subst (eq.symm H2) H)
(assume H2, H2))
(assume H1, or.inr H1)
end
theorem card_insert_of_mem {a : A} {s : finset A} : a ∈ s → card (insert a s) = card s :=
quot.induction_on s
@ -148,6 +187,32 @@ quot.induction_on s
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)
protected theorem induction {P : finset A → Prop}
(H1 : P empty)
(H2 : ∀⦃s : finset A⦄, ∀{a : A}, a ∉ s → P s → P (insert a s)) :
∀s, P s :=
take s,
quot.induction_on s
take u,
subtype.destruct u
take l,
list.induction_on l
(assume nodup_l, H1)
(take a l',
assume IH nodup_al',
have anl' : a ∉ l', from not_mem_of_nodup_cons nodup_al',
assert H3 : list.insert a l' = a :: l', from insert_eq_of_not_mem anl',
assert nodup_l' : nodup l', from nodup_of_nodup_cons nodup_al',
assert P_l' : P (quot.mk (subtype.tag l' nodup_l')), from IH nodup_l',
assert H4 : P (insert a (quot.mk (subtype.tag l' nodup_l'))), from H2 anl' P_l',
begin rewrite [eq.symm H3], apply H4 end)
protected theorem induction_on {P : finset A → Prop} (s : finset A)
(H1 : P empty)
(H2 : ∀⦃s : finset A⦄, ∀{a : A}, a ∉ s → P s → P (insert a s)) :
P s :=
induction H1 H2 s
end insert
/- erase -/
@ -171,22 +236,6 @@ theorem card_erase_of_not_mem {a : A} {s : finset A} : a ∉ s → card (erase a
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]
@ -204,18 +253,27 @@ 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_l {a : A} {s₁ : finset A} {s₂ : finset A} : a ∈ s₁ → a ∈ s₁ ∪ s₂ :=
mem_union_left s₂
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_union_r {a : A} {s₂ : finset A} {s₁ : finset A} : a ∈ s₂ → a ∈ s₁ ∪ s₂ :=
mem_union_right s₁
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
theorem mem_union_iff (a : A) (s₁ s₂ : finset A) : a ∈ s₁ ∪ s₂ ↔ a ∈ s₁ ∨ a ∈ s₂ :=
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)))
(λ i, mem_union_right _ i))
theorem mem_union_eq (a : A) (s₁ s₂ : finset A) : (a ∈ s₁ ∪ s₂) = (a ∈ s₁ ∨ a ∈ s₂) :=
propext !mem_union_iff
theorem union.comm (s₁ s₂ : finset A) : s₁ ∪ s₂ = s₂ ∪ s₁ :=
ext (λ a, by rewrite [*mem_union_eq]; exact or.comm)
@ -236,6 +294,17 @@ ext (λ a, iff.intro
theorem empty_union (s : finset A) : ∅ ∪ s = s :=
calc ∅ ∪ s = s ∪ ∅ : union.comm
... = s : union_empty
theorem insert_eq (a : A) (s : finset A) : insert a s = singleton a ∪ s :=
ext (take x,
calc
x ∈ insert a s ↔ x ∈ insert a s : iff.refl
... = (x = a ∨ x ∈ s) : mem_insert_eq
... = (x ∈ singleton a ∨ x ∈ s) : mem_singleton_eq
... = (x ∈ ⦃a⦄ ∪ s) : mem_union_eq)
theorem insert_union (a : A) (s t : finset A) : insert a (s ∪ t) = insert a s ∪ t :=
by rewrite [*insert_eq, union.assoc]
end union
/- inter -/
@ -258,13 +327,16 @@ quot.induction_on₂ s₁ s₂ (λ l₁ l₂ ainl₁l₂, list.mem_of_mem_inter_
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 :=
theorem mem_inter {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
theorem mem_inter_iff (a : A) (s₁ s₂ : finset A) : a ∈ s₁ ∩ s₂ ↔ a ∈ s₁ ∧ a ∈ s₂ :=
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)))
(λ h, mem_inter (and.elim_left h) (and.elim_right h))
theorem mem_inter_eq (a : A) (s₁ s₂ : finset A) : (a ∈ s₁ ∩ s₂) = (a ∈ s₁ ∧ a ∈ s₂) :=
propext !mem_inter_iff
theorem inter.comm (s₁ s₂ : finset A) : s₁ ∩ s₂ = s₂ ∩ s₁ :=
ext (λ a, by rewrite [*mem_inter_eq]; exact and.comm)
@ -275,7 +347,7 @@ 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))
(λ h, mem_inter h h))
theorem inter_empty (s : finset A) : s ∩ ∅ = ∅ :=
ext (λ a, iff.intro
@ -285,8 +357,79 @@ ext (λ a, iff.intro
theorem empty_inter (s : finset A) : ∅ ∩ s = ∅ :=
calc ∅ ∩ s = s ∩ ∅ : inter.comm
... = ∅ : inter_empty
theorem singleton_inter_of_mem {a : A} {s : finset A} (H : a ∈ s) :
singleton a ∩ s = singleton a :=
ext (take x,
begin
rewrite [mem_inter_eq, !mem_singleton_eq],
exact iff.intro
(assume H1 : x = a ∧ x ∈ s, and.left H1)
(assume H1 : x = a, and.intro H1 (eq.subst (eq.symm H1) H))
end)
theorem singleton_inter_of_not_mem {a : A} {s : finset A} (H : a ∉ s) :
singleton a ∩ s = ∅ :=
ext (take x,
begin
rewrite [mem_inter_eq, !mem_singleton_eq, mem_empty_eq],
exact iff.intro
(assume H1 : x = a ∧ x ∈ s, H (eq.subst (and.left H1) (and.right H1)))
(false.elim)
end)
end inter
/- distributivity laws -/
section inter
variable [h : decidable_eq A]
include h
theorem inter.distrib_left (s t u : finset A) : s ∩ (t ∪ u) = (s ∩ t) ∪ (s ∩ u) :=
ext (take x, by rewrite [mem_inter_eq, *mem_union_eq, *mem_inter_eq]; apply and.distrib_left)
theorem inter.distrib_right (s t u : finset A) : (s ∪ t) ∩ u = (s ∩ u) ∪ (t ∩ u) :=
ext (take x, by rewrite [mem_inter_eq, *mem_union_eq, *mem_inter_eq]; apply and.distrib_right)
theorem union.distrib_left (s t u : finset A) : s ∪ (t ∩ u) = (s ∪ t) ∩ (s ∪ u) :=
ext (take x, by rewrite [mem_union_eq, *mem_inter_eq, *mem_union_eq]; apply or.distrib_left)
theorem union.distrib_right (s t u : finset A) : (s ∩ t) ∪ u = (s ∪ u) ∩ (t ∪ u) :=
ext (take x, by rewrite [mem_union_eq, *mem_inter_eq, *mem_union_eq]; apply or.distrib_right)
end inter
/- disjoint -/
-- Mainly for internal use; library will use s₁ ∩ s₂ = ∅. Note that it does not require decidable equality.
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.elim {s₁ s₂ : finset A} {x : A} :
disjoint s₁ s₂ → x ∈ s₁ → x ∈ s₂ → false :=
quot.induction_on₂ s₁ s₂ (take u₁ u₂, assume H H1 H2, H x H1 H2)
theorem disjoint.intro {s₁ s₂ : finset A} : (∀{x : A}, x ∈ s₁ → x ∈ s₂ → false) → disjoint s₁ s₂ :=
quot.induction_on₂ s₁ s₂ (take u₁ u₂, assume H, H)
theorem inter_empty_of_disjoint [h : decidable_eq A] {s₁ s₂ : finset A} (H : disjoint s₁ s₂) : s₁ ∩ s₂ = ∅ :=
ext (take x, iff_false_intro (assume H1,
disjoint.elim H (mem_of_mem_inter_left H1) (mem_of_mem_inter_right H1)))
theorem disjoint_of_inter_empty [h : decidable_eq A] {s₁ s₂ : finset A} (H : s₁ ∩ s₂ = ∅) : disjoint s₁ s₂ :=
disjoint.intro (take x H1 H2,
have H3 : x ∈ s₁ ∩ s₂, from mem_inter H1 H2,
!not_mem_empty (eq.subst H H3))
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)
/- subset -/
definition subset (s₁ s₂ : finset A) : Prop :=
quot.lift_on₂ s₁ s₂
@ -297,19 +440,19 @@ quot.lift_on₂ s₁ s₂
infix `⊆`:50 := subset
theorem nil_sub (s : finset A) : ∅ ⊆ s :=
theorem empty_subset (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 :=
theorem subset_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 :=
theorem subset.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 :=
theorem subset.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 :=
theorem mem_of_subset_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 -/
@ -328,6 +471,12 @@ 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
theorem mem_upto_iff (a n : nat) : a ∈ upto n ↔ a < n :=
iff.intro lt_of_mem_upto mem_upto_of_lt
theorem mem_upto_eq (a n : nat) : a ∈ upto n = (a < n) :=
propext !mem_upto_iff
end upto
end finset
abbreviation finset := finset.finset