103 lines
3.9 KiB
Text
103 lines
3.9 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: Jeremy Avigad
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Interactions between finset and set.
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-/
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import data.finset.comb data.set.function
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open nat eq.ops set
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namespace finset
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variable {A : Type}
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variable [deceq : decidable_eq A]
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definition to_set [coercion] (s : finset A) : set A := λx, x ∈ s
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abbreviation ts := @to_set A
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variables (s t : finset A) (x y : A)
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theorem mem_eq_mem_to_set : x ∈ s = (x ∈ ts s) := rfl
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definition to_set.inj {s₁ s₂ : finset A} : to_set s₁ = to_set s₂ → s₁ = s₂ :=
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λ h, ext (λ a, iff.of_eq (calc
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(a ∈ s₁) = (a ∈ ts s₁) : mem_eq_mem_to_set
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... = (a ∈ ts s₂) : h
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... = (a ∈ s₂) : mem_eq_mem_to_set))
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/- operations -/
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theorem mem_to_set_empty : (x ∈ ts ∅) = (x ∈ ∅) := rfl
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theorem to_set_empty : ts ∅ = (@set.empty A) := rfl
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theorem mem_to_set_univ [h : fintype A] : (x ∈ ts univ) = (x ∈ set.univ) :=
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propext (iff.intro (assume H, trivial) (assume H, !mem_univ))
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theorem to_set_univ [h : fintype A] : ts univ = (set.univ : set A) := funext (λ x, !mem_to_set_univ)
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theorem mem_to_set_upto (x n : ℕ) : x ∈ ts (upto n) = (x ∈ {a | a < n}) := !mem_upto_eq
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theorem to_set_upto (n : ℕ) : ts (upto n) = {a | a < n} := funext (λ x, !mem_to_set_upto)
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include deceq
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theorem mem_to_set_insert : x ∈ insert y s = (x ∈ set.insert y s) := !mem_insert_eq
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theorem to_set_insert : insert y s = set.insert y s := funext (λ x, !mem_to_set_insert)
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theorem mem_to_set_union : x ∈ s ∪ t = (x ∈ ts s ∪ ts t) := !mem_union_eq
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theorem to_set_union : ts (s ∪ t) = ts s ∪ ts t := funext (λ x, !mem_to_set_union)
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theorem mem_to_set_inter : x ∈ s ∩ t = (x ∈ ts s ∩ ts t) := !mem_inter_eq
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theorem to_set_inter : ts (s ∩ t) = ts s ∩ ts t := funext (λ x, !mem_to_set_inter)
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theorem mem_to_set_diff : x ∈ s \ t = (x ∈ ts s \ ts t) := !mem_diff_eq
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theorem to_set_diff : ts (s \ t) = ts s \ ts t := funext (λ x, !mem_to_set_diff)
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theorem mem_to_set_sep (p : A → Prop) [h : decidable_pred p] : x ∈ sep p s = (x ∈ set.sep p s) :=
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!finset.mem_sep_eq
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theorem to_set_sep (p : A → Prop) [h : decidable_pred p] : sep p s = set.sep p s :=
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funext (λ x, !mem_to_set_sep)
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theorem mem_to_set_image {B : Type} [h : decidable_eq B] (f : A → B) {s : finset A} {y : B} :
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y ∈ image f s = (y ∈ set.image f s) := !mem_image_eq
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theorem to_set_image {B : Type} [h : decidable_eq B] (f : A → B) (s : finset A) :
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image f s = set.image f s := funext (λ x, !mem_to_set_image)
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/- relations -/
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definition decidable_mem_to_set [instance] (x : A) (s : finset A) : decidable (x ∈ ts s) :=
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decidable_of_decidable_of_eq _ !mem_eq_mem_to_set
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theorem eq_of_to_set_eq_to_set {s t : finset A} (H : to_set s = to_set t) : s = t :=
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ext (take x, by rewrite [mem_eq_mem_to_set s, H])
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theorem eq_eq_to_set_eq : (s = t) = (ts s = ts t) :=
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propext (iff.intro (assume H, H ▸ rfl) !eq_of_to_set_eq_to_set)
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definition decidable_to_set_eq [instance] (s t : finset A) : decidable (ts s = ts t) :=
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decidable_of_decidable_of_eq _ !eq_eq_to_set_eq
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theorem subset_eq_to_set_subset (s t : finset A) : (s ⊆ t) = (ts s ⊆ ts t) :=
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propext (iff.intro
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(assume H, take x xs, mem_of_subset_of_mem H xs)
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(assume H, subset_of_forall H))
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definition decidable_to_set_subset (s t : finset A) : decidable (ts s ⊆ ts t) :=
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decidable_of_decidable_of_eq _ !subset_eq_to_set_subset
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/- bounded quantifiers -/
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definition decidable_bounded_forall (s : finset A) (p : A → Prop) [h : decidable_pred p] :
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decidable (∀₀ x ∈ ts s, p x) :=
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decidable_of_decidable_of_iff _ !all_iff_forall
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definition decidable_bounded_exists (s : finset A) (p : A → Prop) [h : decidable_pred p] :
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decidable (∃₀ x ∈ ts s, p x) :=
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decidable_of_decidable_of_iff _ !any_iff_exists
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/- properties -/
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theorem inj_on_to_set {B : Type} [h : decidable_eq B] (f : A → B) (s : finset A) :
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inj_on f s = inj_on f (ts s) :=
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rfl
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end finset
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