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[Merged by Bors] - feat: Unions and intersections indexed by Finset.sigma #10851

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[Merged by Bors] - feat: Unions and intersections indexed by Finset.sigma #10851

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212d06e
feat: Unions and intersections indexed by `Finset.sigma`
YaelDillies Feb 22, 2024
124b3c1
dedup namespace
YaelDillies Feb 22, 2024
e94fd91
use `Π`
YaelDillies Feb 28, 2024
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2 changes: 2 additions & 0 deletions Mathlib/Data/Fintype/Card.lean
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Expand Up @@ -538,6 +538,8 @@ theorem card_eq_zero_iff : card α = 0 ↔ IsEmpty α := by
card_eq_zero_iff.2 ‹_›
#align fintype.card_eq_zero Fintype.card_eq_zero

alias card_of_isEmpty := card_eq_zero

theorem card_eq_one_iff_nonempty_unique : card α = 1 ↔ Nonempty (Unique α) :=
⟨fun h =>
let ⟨d, h⟩ := Fintype.card_eq_one_iff.mp h
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31 changes: 19 additions & 12 deletions Mathlib/Data/Fintype/Sigma.lean
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Expand Up @@ -19,20 +19,27 @@ open Nat

universe u v

variable {α β γ : Type*}
variable {ι α β γ : Type*} {κ : ι → Type*} [Π i, Fintype (κ i)]

open Finset Function

instance {α : Type*} (β : α → Type*) [Fintype α] [∀ a, Fintype (β a)] : Fintype (Sigma β) :=
⟨univ.sigma fun _ => univ, fun ⟨a, b⟩ => by simp⟩
lemma Set.biUnion_finsetSigma_univ (s : Finset ι) (f : Sigma κ → Set α) :
⋃ ij ∈ s.sigma fun _ ↦ Finset.univ, f ij = ⋃ i ∈ s, ⋃ j, f ⟨i, j⟩ := by aesop

@[simp]
theorem Finset.univ_sigma_univ {α : Type*} {β : α → Type*} [Fintype α] [∀ a, Fintype (β a)] :
((univ : Finset α).sigma fun a => (univ : Finset (β a))) = univ :=
rfl
#align finset.univ_sigma_univ Finset.univ_sigma_univ
lemma Set.biUnion_finsetSigma_univ' (s : Finset ι) (f : Π i, κ i → Set α) :
⋃ i ∈ s, ⋃ j, f i j = ⋃ ij ∈ s.sigma fun _ ↦ Finset.univ, f ij.1 ij.2 := by aesop

lemma Set.biInter_finsetSigma_univ (s : Finset ι) (f : Sigma κ → Set α) :
⋂ ij ∈ s.sigma fun _ ↦ Finset.univ, f ij = ⋂ i ∈ s, ⋂ j, f ⟨i, j⟩ := by aesop

lemma Set.biInter_finsetSigma_univ' (s : Finset ι) (f : Π i, κ i → Set α) :
⋂ i ∈ s, ⋂ j, f i j = ⋂ ij ∈ s.sigma fun _ ↦ Finset.univ, f ij.1 ij.2 := by aesop

instance PSigma.fintype {α : Type*} {β : α → Type*} [Fintype α] [∀ a, Fintype (β a)] :
Fintype (Σ'a, β a) :=
Fintype.ofEquiv _ (Equiv.psigmaEquivSigma _).symm
#align psigma.fintype PSigma.fintype
variable [Fintype ι] [Π i, Fintype (κ i)]

instance Sigma.instFintype : Fintype (Σ i, κ i) := ⟨univ.sigma fun _ ↦ univ, by simp⟩
instance PSigma.instFintype : Fintype (Σ' i, κ i) := .ofEquiv _ (Equiv.psigmaEquivSigma _).symm
#align psigma.fintype PSigma.instFintype

@[simp] lemma Finset.univ_sigma_univ : univ.sigma (fun _ ↦ univ) = (univ : Finset (Σ i, κ i)) := rfl
#align finset.univ_sigma_univ Finset.univ_sigma_univ
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