/- Copyright (c) 2015 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Jeremy Avigad Interactions between finset and set. -/ import data.finset.comb data.set.function open nat eq.ops set namespace finset variable {A : Type} variable [deceq : decidable_eq A] definition to_set [coercion] (s : finset A) : set A := λx, x ∈ s abbreviation ts := @to_set A variables (s t : finset A) (x y : A) theorem mem_eq_mem_to_set : (x ∈ s) = (x ∈ ts s) := rfl definition to_set.inj {s₁ s₂ : finset A} : to_set s₁ = to_set s₂ → s₁ = s₂ := λ h, ext (λ a, iff.of_eq (calc (a ∈ s₁) = (a ∈ ts s₁) : mem_eq_mem_to_set ... = (a ∈ ts s₂) : h ... = (a ∈ s₂) : mem_eq_mem_to_set)) /- operations -/ theorem mem_to_set_empty : (x ∈ ts ∅) = (x ∈ ∅) := rfl theorem to_set_empty : ts ∅ = (∅ : set A) := rfl theorem mem_to_set_univ [h : fintype A] : (x ∈ ts univ) = (x ∈ set.univ) := propext (iff.intro (assume H, trivial) (assume H, !mem_univ)) theorem to_set_univ [h : fintype A] : ts univ = (set.univ : set A) := funext (λ x, !mem_to_set_univ) include deceq theorem mem_to_set_insert : x ∈ insert y s = (x ∈ set.insert y s) := !finset.mem_insert_eq theorem to_set_insert : insert y s = set.insert y s := funext (λ x, !mem_to_set_insert) theorem mem_to_set_union : x ∈ s ∪ t = (x ∈ ts s ∪ ts t) := !finset.mem_union_eq theorem to_set_union : ts (s ∪ t) = ts s ∪ ts t := funext (λ x, !mem_to_set_union) theorem mem_to_set_inter : x ∈ s ∩ t = (x ∈ ts s ∩ ts t) := !finset.mem_inter_eq theorem to_set_inter : ts (s ∩ t) = ts s ∩ ts t := funext (λ x, !mem_to_set_inter) theorem mem_to_set_diff : x ∈ s \ t = (x ∈ ts s \ ts t) := !finset.mem_diff_eq theorem to_set_diff : ts (s \ t) = ts s \ ts t := funext (λ x, !mem_to_set_diff) theorem mem_to_set_filter (p : A → Prop) [h : decidable_pred p] : x ∈ filter p s = (x ∈ set.filter p s) := !finset.mem_filter_eq theorem to_set_filter (p : A → Prop) [h : decidable_pred p] : filter p s = set.filter p s := funext (λ x, !mem_to_set_filter) theorem mem_to_set_image {B : Type} [h : decidable_eq B] (f : A → B) {s : finset A} {y : B} : y ∈ image f s = (y ∈ set.image f s) := !finset.mem_image_eq theorem to_set_image {B : Type} [h : decidable_eq B] (f : A → B) (s : finset A) : image f s = set.image f s := funext (λ x, !mem_to_set_image) /- relations -/ definition decidable_mem_to_set [instance] (x : A) (s : finset A) : decidable (x ∈ ts s) := decidable_of_decidable_of_eq _ !mem_eq_mem_to_set theorem eq_eq_to_set_eq : (s = t) = (ts s = ts t) := propext (iff.intro (assume H, H ▸ rfl) (assume H, ext (take x, by rewrite [mem_eq_mem_to_set s, H]))) definition decidable_to_set_eq [instance] (s t : finset A) : decidable (ts s = ts t) := decidable_of_decidable_of_eq _ !eq_eq_to_set_eq theorem subset_eq_to_set_subset (s t : finset A) : (s ⊆ t) = (ts s ⊆ ts t) := propext (iff.intro (assume H, take x xs, mem_of_subset_of_mem H xs) (assume H, subset_of_forall H)) definition decidable_to_set_subset (s t : finset A) : decidable (ts s ⊆ ts t) := decidable_of_decidable_of_eq _ !subset_eq_to_set_subset /- bounded quantifiers -/ definition decidable_bounded_forall (s : finset A) (p : A → Prop) [h : decidable_pred p] : decidable (∀₀ x ∈ ts s, p x) := decidable_of_decidable_of_iff _ !all_iff_forall definition decidable_bounded_exists (s : finset A) (p : A → Prop) [h : decidable_pred p] : decidable (∃₀ x ∈ ts s, p x) := decidable_of_decidable_of_iff _ !any_iff_exists end finset