Intervals in a pi type

This file shows that (dependent) functions to locally finite orders equipped with the pointwise order are locally finite and calculates the cardinality of their intervals.

instance Pi.instLocallyFiniteOrderForAllPreorderToPreorder {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → PartialOrder (α i)] [(i : ι) → LocallyFiniteOrder (α i)] : LocallyFiniteOrder ((i : ι) → α i)

theorem Pi.Icc_eq {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → PartialOrder (α i)] [(i : ι) → LocallyFiniteOrder (α i)] (a : (i : ι) → α i) (b : (i : ι) → α i) : Finset.Icc a b = Fintype.piFinset fun i => Finset.Icc (a i) (b i)

theorem Pi.card_Icc {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → PartialOrder (α i)] [(i : ι) → LocallyFiniteOrder (α i)] (a : (i : ι) → α i) (b : (i : ι) → α i) : Finset.card (Finset.Icc a b) = Finset.prod Finset.univ fun i => Finset.card (Finset.Icc (a i) (b i))

theorem Pi.card_Ico {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → PartialOrder (α i)] [(i : ι) → LocallyFiniteOrder (α i)] (a : (i : ι) → α i) (b : (i : ι) → α i) : Finset.card (Finset.Ico a b) = (Finset.prod Finset.univ fun i => Finset.card (Finset.Icc (a i) (b i))) - 1

theorem Pi.card_Ioc {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → PartialOrder (α i)] [(i : ι) → LocallyFiniteOrder (α i)] (a : (i : ι) → α i) (b : (i : ι) → α i) : Finset.card (Finset.Ioc a b) = (Finset.prod Finset.univ fun i => Finset.card (Finset.Icc (a i) (b i))) - 1

theorem Pi.card_Ioo {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → PartialOrder (α i)] [(i : ι) → LocallyFiniteOrder (α i)] (a : (i : ι) → α i) (b : (i : ι) → α i) : Finset.card (Finset.Ioo a b) = (Finset.prod Finset.univ fun i => Finset.card (Finset.Icc (a i) (b i))) - 2

instance Pi.instLocallyFiniteOrderBotForAllPreorderToPreorder {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → PartialOrder (α i)] [(i : ι) → LocallyFiniteOrderBot (α i)] : LocallyFiniteOrderBot ((i : ι) → α i)

theorem Pi.card_Iic {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → PartialOrder (α i)] [(i : ι) → LocallyFiniteOrderBot (α i)] (b : (i : ι) → α i) : Finset.card (Finset.Iic b) = Finset.prod Finset.univ fun i => Finset.card (Finset.Iic (b i))

theorem Pi.card_Iio {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → PartialOrder (α i)] [(i : ι) → LocallyFiniteOrderBot (α i)] (b : (i : ι) → α i) : Finset.card (Finset.Iio b) = (Finset.prod Finset.univ fun i => Finset.card (Finset.Iic (b i))) - 1

instance Pi.instLocallyFiniteOrderTopForAllPreorderToPreorder {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → PartialOrder (α i)] [(i : ι) → LocallyFiniteOrderTop (α i)] : LocallyFiniteOrderTop ((i : ι) → α i)

theorem Pi.card_Ici {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → PartialOrder (α i)] [(i : ι) → LocallyFiniteOrderTop (α i)] (a : (i : ι) → α i) : Finset.card (Finset.Ici a) = Finset.prod Finset.univ fun i => Finset.card (Finset.Ici (a i))

theorem Pi.card_Ioi {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → PartialOrder (α i)] [(i : ι) → LocallyFiniteOrderTop (α i)] (a : (i : ι) → α i) : Finset.card (Finset.Ioi a) = (Finset.prod Finset.univ fun i => Finset.card (Finset.Ici (a i))) - 1

theorem Pi.uIcc_eq {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → Lattice (α i)] [(i : ι) → LocallyFiniteOrder (α i)] (a : (i : ι) → α i) (b : (i : ι) → α i) : Finset.uIcc a b = Fintype.piFinset fun i => Finset.uIcc (a i) (b i)

theorem Pi.card_uIcc {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → Lattice (α i)] [(i : ι) → LocallyFiniteOrder (α i)] (a : (i : ι) → α i) (b : (i : ι) → α i) : Finset.card (Finset.uIcc a b) = Finset.prod Finset.univ fun i => Finset.card (Finset.uIcc (a i) (b i))