77 lines
2.6 KiB
Text
77 lines
2.6 KiB
Text
/-
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Copyright (c) 2015 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Haitao Zhang
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Partitions of a type A into finite subsets of A. Such a partition is represented by
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a function f : A → finset A which maps every element a : A to its equivalence class.
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-/
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import .card
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open function eq.ops
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variable {A : Type}
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variable [deceqA : decidable_eq A]
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include deceqA
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namespace finset
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definition is_partition (f : A → finset A) := ∀ a b, a ∈ f b = (f a = f b)
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structure partition : Type :=
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(set : finset A) (part : A → finset A) (is_part : is_partition part)
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(complete : set = Union set part)
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attribute partition.part [coercion]
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namespace partition
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definition equiv_classes (f : partition) : finset (finset A) :=
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image (partition.part f) (partition.set f)
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lemma equiv_class_disjoint (f : partition) (a1 a2 : finset A) (Pa1 : a1 ∈ equiv_classes f)
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(Pa2 : a2 ∈ equiv_classes f) :
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a1 ≠ a2 → a1 ∩ a2 = ∅ :=
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assume Pne,
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assert Pe1 : _, from exists_of_mem_image Pa1, obtain g1 Pg1, from Pe1,
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assert Pe2 : _, from exists_of_mem_image Pa2, obtain g2 Pg2, from Pe2,
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begin
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apply inter_eq_empty_of_disjoint,
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apply disjoint.intro,
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rewrite [eq.symm (and.right Pg1), eq.symm (and.right Pg2)],
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intro x,
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rewrite [*partition.is_part f],
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intro Pxg1, rewrite [Pxg1, and.right Pg1, and.right Pg2],
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intro Pe, exact absurd Pe Pne
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end
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theorem class_equation (f : @partition A _) :
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card (partition.set f) = nat.Sum (equiv_classes f) card :=
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let s := (partition.set f), p := (partition.part f), img := image p s in
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calc
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card s = card (Union s p) : partition.complete f
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... = card (Union img id) : image_eq_Union_index_image s p
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... = card (Union (equiv_classes f) id) : rfl
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... = nat.Sum (equiv_classes f) card : card_Union_of_disjoint _ id (equiv_class_disjoint f)
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lemma equiv_class_refl {f : A → finset A} (Pequiv : is_partition f) : ∀ a, a ∈ f a :=
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take a, by rewrite [Pequiv a a]
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-- make it a little easier to prove union from restriction
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lemma restriction_imp_union {s : finset A} (f : A → finset A) (Pequiv : is_partition f)
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(Psub : ∀{a}, a ∈ s → f a ⊆ s) :
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s = Union s f :=
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ext (take a, iff.intro
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(assume Pains,
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begin
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rewrite [(Union_insert_of_mem f Pains)⁻¹, Union_insert],
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apply mem_union_l, exact equiv_class_refl Pequiv a
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end)
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(assume Painu,
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have Pclass : ∃ x, x ∈ s ∧ a ∈ f x,
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from iff.elim_left (mem_Union_iff s f _) Painu,
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obtain x Px, from Pclass,
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have Pfx : f x ⊆ s, from Psub (and.left Px),
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mem_of_subset_of_mem Pfx (and.right Px)))
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end partition
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end finset
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