140 lines
5.6 KiB
Text
140 lines
5.6 KiB
Text
/-
|
||
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
Author: Haitao Zhang
|
||
|
||
Partitions of a type A into finite subsets of A. Such a partition is represented by
|
||
a function f : A → finset A which maps every element a : A to its equivalence class.
|
||
-/
|
||
import .card
|
||
open function eq.ops
|
||
|
||
variable {A : Type}
|
||
variable [deceqA : decidable_eq A]
|
||
include deceqA
|
||
|
||
namespace finset
|
||
|
||
definition is_partition (f : A → finset A) := ∀ a b, a ∈ f b = (f a = f b)
|
||
|
||
structure partition : Type :=
|
||
(set : finset A) (part : A → finset A) (is_part : is_partition part)
|
||
(complete : set = Union set part)
|
||
|
||
attribute partition.part [coercion]
|
||
|
||
namespace partition
|
||
|
||
definition equiv_classes (f : partition) : finset (finset A) :=
|
||
image (partition.part f) (partition.set f)
|
||
|
||
lemma equiv_class_disjoint (f : partition) (a1 a2 : finset A) (Pa1 : a1 ∈ equiv_classes f)
|
||
(Pa2 : a2 ∈ equiv_classes f) :
|
||
a1 ≠ a2 → a1 ∩ a2 = ∅ :=
|
||
assume Pne,
|
||
assert Pe1 : _, from exists_of_mem_image Pa1, obtain g1 Pg1, from Pe1,
|
||
assert Pe2 : _, from exists_of_mem_image Pa2, obtain g2 Pg2, from Pe2,
|
||
begin
|
||
apply inter_eq_empty_of_disjoint,
|
||
apply disjoint.intro,
|
||
rewrite [eq.symm (and.right Pg1), eq.symm (and.right Pg2)],
|
||
intro x,
|
||
rewrite [*partition.is_part f],
|
||
intro Pxg1, rewrite [Pxg1, and.right Pg1, and.right Pg2],
|
||
intro Pe, exact absurd Pe Pne
|
||
end
|
||
|
||
theorem class_equation (f : @partition A _) :
|
||
card (partition.set f) = nat.Sum (equiv_classes f) card :=
|
||
let s := (partition.set f), p := (partition.part f), img := image p s in
|
||
calc
|
||
card s = card (Union s p) : partition.complete f
|
||
... = card (Union img id) : image_eq_Union_index_image s p
|
||
... = card (Union (equiv_classes f) id) : rfl
|
||
... = nat.Sum (equiv_classes f) card : card_Union_of_disjoint _ id (equiv_class_disjoint f)
|
||
|
||
lemma equiv_class_refl {f : A → finset A} (Pequiv : is_partition f) : ∀ a, a ∈ f a :=
|
||
take a, by rewrite [Pequiv a a]
|
||
|
||
-- make it a little easier to prove union from restriction
|
||
lemma restriction_imp_union {s : finset A} (f : A → finset A) (Pequiv : is_partition f)
|
||
(Psub : ∀{a}, a ∈ s → f a ⊆ s) :
|
||
s = Union s f :=
|
||
ext (take a, iff.intro
|
||
(assume Pains,
|
||
begin
|
||
rewrite [(Union_insert_of_mem f Pains)⁻¹, Union_insert],
|
||
apply mem_union_l, exact equiv_class_refl Pequiv a
|
||
end)
|
||
(assume Painu,
|
||
have Pclass : ∃ x, x ∈ s ∧ a ∈ f x,
|
||
from iff.elim_left (mem_Union_iff s f _) Painu,
|
||
obtain x Px, from Pclass,
|
||
have Pfx : f x ⊆ s, from Psub (and.left Px),
|
||
mem_of_subset_of_mem Pfx (and.right Px)))
|
||
|
||
lemma binary_union (P : A → Prop) [decP : decidable_pred P] {S : finset A} :
|
||
S = {a ∈ S | P a} ∪ {a ∈ S | ¬(P a)} :=
|
||
ext take a, iff.intro
|
||
(assume Pin, decidable.by_cases
|
||
(λ Pa : P a, mem_union_l (mem_filter_of_mem Pin Pa))
|
||
(λ nPa, mem_union_r (mem_filter_of_mem Pin nPa)))
|
||
(assume Pinu, or.elim (mem_or_mem_of_mem_union Pinu)
|
||
(assume Pin, mem_of_mem_filter Pin)
|
||
(assume Pin, mem_of_mem_filter Pin))
|
||
|
||
lemma binary_inter_empty {P : A → Prop} [decP : decidable_pred P] {S : finset A} :
|
||
{a ∈ S | P a} ∩ {a ∈ S | ¬(P a)} = ∅ :=
|
||
inter_eq_empty (take a, assume Pa nPa, absurd (of_mem_filter Pa) (of_mem_filter nPa))
|
||
|
||
definition disjoint_sets (S : finset (finset A)) : Prop :=
|
||
∀ s₁ s₂ (P₁ : s₁ ∈ S) (P₂ : s₂ ∈ S), s₁ ≠ s₂ → s₁ ∩ s₂ = ∅
|
||
|
||
lemma disjoint_sets_filter_of_disjoint_sets {P : finset A → Prop} [decP : decidable_pred P] {S : finset (finset A)} :
|
||
disjoint_sets S → disjoint_sets {s ∈ S | P s} :=
|
||
assume Pds, take s₁ s₂, assume P₁ P₂, Pds s₁ s₂ (mem_of_mem_filter P₁) (mem_of_mem_filter P₂)
|
||
|
||
lemma binary_inter_empty_Union_disjoint_sets {P : finset A → Prop} [decP : decidable_pred P] {S : finset (finset A)} :
|
||
disjoint_sets S → Union {s ∈ S | P s} id ∩ Union {s ∈ S | ¬P s} id = ∅ :=
|
||
assume Pds, inter_eq_empty (take a, assume Pa nPa,
|
||
obtain s Psin Pains, from iff.elim_left !mem_Union_iff Pa,
|
||
obtain t Ptin Paint, from iff.elim_left !mem_Union_iff nPa,
|
||
assert Pneq : s ≠ t,
|
||
from assume Peq, absurd (Peq ▸ of_mem_filter Psin) (of_mem_filter Ptin),
|
||
Pds s t (mem_of_mem_filter Psin) (mem_of_mem_filter Ptin) Pneq ▸ mem_inter Pains Paint)
|
||
|
||
section
|
||
variables {B: Type} [deceqB : decidable_eq B]
|
||
include deceqB
|
||
|
||
lemma binary_Union (f : A → finset B) {P : A → Prop} [decP : decidable_pred P] {s : finset A} :
|
||
Union s f = Union {a ∈ s | P a} f ∪ Union {a ∈ s | ¬P a} f :=
|
||
begin rewrite [binary_union P at {1}], apply Union_union, exact binary_inter_empty end
|
||
|
||
end
|
||
|
||
open nat
|
||
section
|
||
open algebra
|
||
|
||
variables {B : Type} [acmB : add_comm_monoid B]
|
||
include acmB
|
||
|
||
lemma Sum_binary_union (f : A → B) (P : A → Prop) [decP : decidable_pred P] {S : finset A} :
|
||
Sum S f = Sum {s ∈ S | P s} f + Sum {s ∈ S | ¬P s} f :=
|
||
calc
|
||
Sum S f = Sum ({s ∈ S | P s} ∪ {s ∈ S | ¬(P s)}) f : binary_union
|
||
... = Sum {s ∈ S | P s} f + Sum {s ∈ S | ¬P s} f : Sum_union f binary_inter_empty
|
||
|
||
end
|
||
|
||
lemma card_binary_Union_disjoint_sets (P : finset A → Prop) [decP : decidable_pred P] {S : finset (finset A)} :
|
||
disjoint_sets S → card (Union S id) = Sum {s ∈ S | P s} card + Sum {s ∈ S | ¬P s} card :=
|
||
assume Pds, calc
|
||
card (Union S id)
|
||
= card (Union {s ∈ S | P s} id ∪ Union {s ∈ S | ¬P s} id) : binary_Union
|
||
... = card (Union {s ∈ S | P s} id) + card (Union {s ∈ S | ¬P s} id) : card_union_of_disjoint (binary_inter_empty_Union_disjoint_sets Pds)
|
||
... = Sum {s ∈ S | P s} card + Sum {s ∈ S | ¬P s} card : by rewrite [*(card_Union_of_disjoint _ id (disjoint_sets_filter_of_disjoint_sets Pds))]
|
||
|
||
end partition
|
||
end finset
|