feat(library/data/finset/partition): add theory of binary partition

library/data/finset/partition.lean

@@ -73,5 +73,54 @@ ext (take a, iff.intro
     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
+lemma card_binary_Union_disjoint_sets (P : finset A → Prop) [decP : decidable_pred P] {S : finset (finset A)} :
+  disjoint_sets S → Sum S card = Sum {s ∈ S | P s} card + Sum {s ∈ S | ¬P s} card :=
+assume Pds, calc
+  Sum S card = card (Union S id) : card_Union_of_disjoint S id Pds
+         ... = 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