fintype instance for the product of two fintypes. #

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theorem Set.toFinset_prod {α : Type u_1} {β : Type u_2} (s : Set α) (t : Set β) [Fintype ↑s] [Fintype ↑t] [Fintype ↑(s ×ˢ t)] :

Set.toFinset (s ×ˢ t) = Set.toFinset s ×ˢ Set.toFinset t

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theorem Set.toFinset_off_diag {α : Type u_1} {s : Set α} [DecidableEq α] [Fintype ↑s] [Fintype ↑(Set.offDiag s)] :

Set.toFinset (Set.offDiag s) = Finset.offDiag (Set.toFinset s)

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instance instFintypeProd (α : Type u_4) (β : Type u_5) [Fintype α] [Fintype β] :

Fintype (α × β)

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instFintypeProd α β = { elems := Finset.univ ×ˢ Finset.univ, complete := (_ : ∀ (x : α × β), x ∈ Finset.univ ×ˢ Finset.univ) }

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@[simp]

theorem Finset.univ_product_univ {α : Type u_4} {β : Type u_5} [Fintype α] [Fintype β] :

Finset.univ ×ˢ Finset.univ = Finset.univ

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@[simp]

theorem Fintype.card_prod (α : Type u_4) (β : Type u_5) [Fintype α] [Fintype β] :

Fintype.card (α × β) = Fintype.card α * Fintype.card β

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@[simp]

theorem infinite_prod {α : Type u_1} {β : Type u_2} :

Infinite (α × β) ↔ Infinite α ∧ Nonempty β ∨ Nonempty α ∧ Infinite β

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instance Pi.infinite_of_left {ι : Sort u_4} {π : ι → Type u_5} [∀ (i : ι), Nontrivial (π i)] [Infinite ι] :

Infinite ((i : ι) → π i)

Equations

@Pi.infinite_of_left = @Pi.infinite_of_left.proof_1

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theorem Pi.infinite_of_exists_right {ι : Type u_4} {π : ι → Type u_5} (i : ι) [Infinite (π i)] [∀ (i : ι), Nonempty (π i)] :

Infinite ((i : ι) → π i)

If at least one π i is infinite and the rest nonempty, the pi type of all π is infinite.

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instance Pi.infinite_of_right {ι : Type u_4} {π : ι → Type u_5} [∀ (i : ι), Infinite (π i)] [Nonempty ι] :

Infinite ((i : ι) → π i)

See Pi.infinite_of_exists_right for the case that only one π i is infinite.

Equations

@Pi.infinite_of_right = @Pi.infinite_of_right.proof_1

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instance Function.infinite_of_left {ι : Sort u_4} {π : Type u_5} [Nontrivial π] [Infinite ι] :

Infinite (ι → π)

Non-dependent version of Pi.infinite_of_left.

Equations

@Function.infinite_of_left = @Function.infinite_of_left.proof_1

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instance Function.infinite_of_right {ι : Type u_4} {π : Type u_5} [Infinite π] [Nonempty ι] :

Infinite (ι → π)

Non-dependent version of Pi.infinite_of_exists_right and Pi.infinite_of_right.

Equations

@Function.infinite_of_right = @Function.infinite_of_right.proof_1