19361f0196
see discussion at #604
98 lines
4.2 KiB
Text
98 lines
4.2 KiB
Text
/-
|
||
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
Author: Jeremy Avigad
|
||
|
||
Finite unions and intersections on finsets.
|
||
|
||
Note: for the moment we only do unions. We need to generalize bigops for intersections.
|
||
-/
|
||
import data.finset.comb algebra.group_bigops
|
||
open list
|
||
|
||
namespace finset
|
||
|
||
variables {A B : Type} [deceqA : decidable_eq A] [deceqB : decidable_eq B]
|
||
|
||
/- Unionl and Union -/
|
||
|
||
section union
|
||
|
||
definition to_comm_monoid_Union (B : Type) [deceqB : decidable_eq B] :
|
||
algebra.comm_monoid (finset B) :=
|
||
⦃ algebra.comm_monoid,
|
||
mul := union,
|
||
mul_assoc := union.assoc,
|
||
one := empty,
|
||
mul_one := union_empty,
|
||
one_mul := empty_union,
|
||
mul_comm := union.comm
|
||
⦄
|
||
|
||
open [classes] algebra
|
||
local attribute finset.to_comm_monoid_Union [instance]
|
||
|
||
include deceqB
|
||
|
||
definition Unionl (l : list A) (f : A → finset B) : finset B := algebra.Prodl l f
|
||
notation `⋃` binders `←` l, r:(scoped f, Unionl l f) := r
|
||
definition Union (s : finset A) (f : A → finset B) : finset B := algebra.Prod s f
|
||
notation `⋃` binders `∈` s, r:(scoped f, finset.Union s f) := r
|
||
|
||
theorem Unionl_nil (f : A → finset B) : Unionl [] f = ∅ := algebra.Prodl_nil f
|
||
theorem Unionl_cons (f : A → finset B) (a : A) (l : list A) :
|
||
Unionl (a::l) f = f a ∪ Unionl l f := algebra.Prodl_cons f a l
|
||
theorem Unionl_append (l₁ l₂ : list A) (f : A → finset B) :
|
||
Unionl (l₁++l₂) f = Unionl l₁ f ∪ Unionl l₂ f := algebra.Prodl_append l₁ l₂ f
|
||
theorem Unionl_mul (l : list A) (f g : A → finset B) :
|
||
Unionl l (λx, f x ∪ g x) = Unionl l f ∪ Unionl l g := algebra.Prodl_mul l f g
|
||
section deceqA
|
||
include deceqA
|
||
theorem Unionl_insert_of_mem (f : A → finset B) {a : A} {l : list A} (H : a ∈ l) :
|
||
Unionl (list.insert a l) f = Unionl l f := algebra.Prodl_insert_of_mem f H
|
||
theorem Unionl_insert_of_not_mem (f : A → finset B) {a : A} {l : list A} (H : a ∉ l) :
|
||
Unionl (list.insert a l) f = f a ∪ Unionl l f := algebra.Prodl_insert_of_not_mem f H
|
||
theorem Unionl_union {l₁ l₂ : list A} (f : A → finset B) (d : list.disjoint l₁ l₂) :
|
||
Unionl (list.union l₁ l₂) f = Unionl l₁ f ∪ Unionl l₂ f := algebra.Prodl_union f d
|
||
theorem Unionl_empty (l : list A) : Unionl l (λ x, ∅) = (∅ : finset B) := algebra.Prodl_one l
|
||
end deceqA
|
||
|
||
theorem Union_empty (f : A → finset B) : Union ∅ f = ∅ := algebra.Prod_empty f
|
||
theorem Union_mul (s : finset A) (f g : A → finset B) :
|
||
Union s (λx, f x ∪ g x) = Union s f ∪ Union s g := algebra.Prod_mul s f g
|
||
section deceqA
|
||
include deceqA
|
||
theorem Union_insert_of_mem (f : A → finset B) {a : A} {s : finset A} (H : a ∈ s) :
|
||
Union (insert a s) f = Union s f := algebra.Prod_insert_of_mem f H
|
||
theorem Union_insert_of_not_mem (f : A → finset B) {a : A} {s : finset A} (H : a ∉ s) :
|
||
Union (insert a s) f = f a ∪ Union s f := algebra.Prod_insert_of_not_mem f H
|
||
theorem Union_union (f : A → finset B) {s₁ s₂ : finset A} (disj : s₁ ∩ s₂ = ∅) :
|
||
Union (s₁ ∪ s₂) f = Union s₁ f ∪ Union s₂ f := algebra.Prod_union f disj
|
||
theorem Union_ext {s : finset A} {f g : A → finset B} (H : ∀x, x ∈ s → f x = g x) :
|
||
Union s f = Union s g := algebra.Prod_ext H
|
||
theorem Union_empty' (s : finset A) : Union s (λ x, ∅) = (∅ : finset B) := algebra.Prod_one s
|
||
|
||
-- this will eventually be an instance of something more general
|
||
theorem inter_Union (s : finset B) (t : finset A) (f : A → finset B) :
|
||
s ∩ (⋃ x ∈ t, f x) = (⋃ x ∈ t, s ∩ f x) :=
|
||
finset.induction_on t
|
||
(by rewrite [*Union_empty, inter_empty])
|
||
(take s' x, assume H : x ∉ s',
|
||
assume IH,
|
||
by rewrite [*Union_insert_of_not_mem _ H, inter.distrib_left, IH])
|
||
|
||
theorem mem_Union_iff (s : finset A) (f : A → finset B) (b : B) :
|
||
b ∈ (⋃ x ∈ s, f x) ↔ (∃ x, x ∈ s ∧ b ∈ f x ) :=
|
||
finset.induction_on s
|
||
(by rewrite [exists_mem_empty_eq])
|
||
(take s' a, assume H : a ∉ s', assume IH,
|
||
by rewrite [Union_insert_of_not_mem _ H, mem_union_eq, IH, exists_mem_insert_eq])
|
||
|
||
theorem mem_Union_eq (s : finset A) (f : A → finset B) (b : B) :
|
||
b ∈ (⋃ x ∈ s, f x) = (∃ x, x ∈ s ∧ b ∈ f x ) :=
|
||
propext !mem_Union_iff
|
||
end deceqA
|
||
|
||
end union
|
||
|
||
end finset
|