Fintype instances for pi types #

def Fintype.piFinset {α : Type u_1} [inst : DecidableEq α] [inst : Fintype α] {δ : α → Type u_2} (t : (a : α) → Finset (δ a)) :

Finset ((a : α) → δ a)

Given for all a : α a finset t a of δ a, then one can define the finset Fintype.piFinset t of all functions taking values in t a for all a. This is the analogue of Finset.pi where the base finset is univ (but formally they are not the same, as there is an additional condition i ∈ Finset.univ in the Finset.pi definition).

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theorem Fintype.mem_piFinset {α : Type u_2} [inst : DecidableEq α] [inst : Fintype α] {δ : α → Type u_1} {t : (a : α) → Finset (δ a)} {f : (a : α) → δ a} :

f ∈ Fintype.piFinset t ↔ ∀ (a : α), f a ∈ t a

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theorem Fintype.coe_piFinset {α : Type u_2} [inst : DecidableEq α] [inst : Fintype α] {δ : α → Type u_1} (t : (a : α) → Finset (δ a)) :

↑(Fintype.piFinset t) = Set.pi Set.univ fun a => ↑(t a)

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theorem Fintype.piFinset_subset {α : Type u_2} [inst : DecidableEq α] [inst : Fintype α] {δ : α → Type u_1} (t₁ : (a : α) → Finset (δ a)) (t₂ : (a : α) → Finset (δ a)) (h : ∀ (a : α), t₁ a ⊆ t₂ a) :

Fintype.piFinset t₁ ⊆ Fintype.piFinset t₂

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theorem Fintype.piFinset_empty {α : Type u_1} [inst : DecidableEq α] [inst : Fintype α] {δ : α → Type u_2} [inst : Nonempty α] :

(Fintype.piFinset fun x => ∅) = ∅

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theorem Fintype.piFinset_singleton {α : Type u_1} [inst : DecidableEq α] [inst : Fintype α] {δ : α → Type u_2} (f : (i : α) → δ i) :

(Fintype.piFinset fun i => {f i}) = {f}

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theorem Fintype.piFinset_subsingleton {α : Type u_2} [inst : DecidableEq α] [inst : Fintype α] {δ : α → Type u_1} {f : (i : α) → Finset (δ i)} (hf : ∀ (i : α), Set.Subsingleton ↑(f i)) :

Set.Subsingleton ↑(Fintype.piFinset f)

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theorem Fintype.piFinset_disjoint_of_disjoint {α : Type u_2} [inst : DecidableEq α] [inst : Fintype α] {δ : α → Type u_1} (t₁ : (a : α) → Finset (δ a)) (t₂ : (a : α) → Finset (δ a)) {a : α} (h : Disjoint (t₁ a) (t₂ a)) :

Disjoint (Fintype.piFinset t₁) (Fintype.piFinset t₂)

pi #

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instance Pi.fintype {α : Type u_1} {β : α → Type u_2} [inst : DecidableEq α] [inst : Fintype α] [inst : (a : α) → Fintype (β a)] :

Fintype ((a : α) → β a)

A dependent product of fintypes, indexed by a fintype, is a fintype.

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Pi.fintype = { elems := Fintype.piFinset fun x => Finset.univ, complete := (_ : ∀ (a : (a : α) → β a), a ∈ Fintype.piFinset fun x => Finset.univ) }

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theorem Fintype.piFinset_univ {α : Type u_1} {β : α → Type u_2} [inst : DecidableEq α] [inst : Fintype α] [inst : (a : α) → Fintype (β a)] :

(Fintype.piFinset fun a => Finset.univ) = Finset.univ

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noncomputable instance Function.Embedding.fintype {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst : Fintype β] :

Fintype (α ↪ β)

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Function.Embedding.fintype = Fintype.ofEquiv { f // Function.Injective f } (Equiv.subtypeInjectiveEquivEmbedding α β)

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theorem Finset.univ_pi_univ {α : Type u_1} {β : α → Type u_2} [inst : DecidableEq α] [inst : Fintype α] [inst : (a : α) → Fintype (β a)] :

(Finset.pi Finset.univ fun a => Finset.univ) = Finset.univ