Finite intervals of finitely supported functions

This file provides the LocallyFiniteOrder instance for Π₀ i, α i when α itself is locally finite and calculates the cardinality of its finite intervals.

def Finset.dfinsupp {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → Zero (α i)] (s : Finset ι) (t : (i : ι) → Finset (α i)) : Finset (Π₀ (i : ι), α i)

Finitely supported product of finsets.

Equations

• Finset.dfinsupp s t = Finset.map { toFun := fun (f : (a : ι) → a ∈ s → α a) => DFinsupp.mk s fun (i : ↑↑s) => f ↑i ⋯, inj' := ⋯ } (Finset.pi s t)

@[simp]

theorem Finset.card_dfinsupp {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → Zero (α i)] (s : Finset ι) (t : (i : ι) → Finset (α i)) : (Finset.dfinsupp s t).card = Finset.prod s fun (i : ι) => (t i).card

theorem Finset.mem_dfinsupp_iff {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → Zero (α i)] {s : Finset ι} {f : Π₀ (i : ι), α i} {t : (i : ι) → Finset (α i)} [(i : ι) → DecidableEq (α i)] : f ∈ Finset.dfinsupp s t ↔ DFinsupp.support f ⊆ s ∧ ∀ i ∈ s, f i ∈ t i

@[simp]

theorem Finset.mem_dfinsupp_iff_of_support_subset {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → Zero (α i)] {s : Finset ι} {f : Π₀ (i : ι), α i} [(i : ι) → DecidableEq (α i)] {t : Π₀ (i : ι), Finset (α i)} (ht : DFinsupp.support t ⊆ s) : f ∈ Finset.dfinsupp s ⇑t ↔ ∀ (i : ι), f i ∈ t i

When t is supported on s, f ∈ s.dfinsupp t precisely means that f is pointwise in t.

def DFinsupp.singleton {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] (f : Π₀ (i : ι), α i) : Π₀ (i : ι), Finset (α i)

Pointwise Finset.singleton bundled as a DFinsupp.

Equations

• One or more equations did not get rendered due to their size.

theorem DFinsupp.mem_singleton_apply_iff {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] {f : Π₀ (i : ι), α i} {i : ι} {a : α i} : a ∈ (DFinsupp.singleton f) i ↔ a = f i

def DFinsupp.rangeIcc {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → PartialOrder (α i)] [(i : ι) → LocallyFiniteOrder (α i)] (f : Π₀ (i : ι), α i) (g : Π₀ (i : ι), α i) : Π₀ (i : ι), Finset (α i)

Pointwise Finset.Icc bundled as a DFinsupp.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem DFinsupp.rangeIcc_apply {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → PartialOrder (α i)] [(i : ι) → LocallyFiniteOrder (α i)] (f : Π₀ (i : ι), α i) (g : Π₀ (i : ι), α i) (i : ι) : (DFinsupp.rangeIcc f g) i = Finset.Icc (f i) (g i)

theorem DFinsupp.mem_rangeIcc_apply_iff {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → PartialOrder (α i)] [(i : ι) → LocallyFiniteOrder (α i)] {f : Π₀ (i : ι), α i} {g : Π₀ (i : ι), α i} {i : ι} {a : α i} : a ∈ (DFinsupp.rangeIcc f g) i ↔ f i ≤ a ∧ a ≤ g i

theorem DFinsupp.support_rangeIcc_subset {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → PartialOrder (α i)] [(i : ι) → LocallyFiniteOrder (α i)] {f : Π₀ (i : ι), α i} {g : Π₀ (i : ι), α i} [DecidableEq ι] [(i : ι) → DecidableEq (α i)] : DFinsupp.support (DFinsupp.rangeIcc f g) ⊆ DFinsupp.support f ∪ DFinsupp.support g

def DFinsupp.pi {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [DecidableEq ι] [(i : ι) → DecidableEq (α i)] (f : Π₀ (i : ι), Finset (α i)) : Finset (Π₀ (i : ι), α i)

Given a finitely supported function f : Π₀ i, Finset (α i), one can define the finset f.pi of all finitely supported functions whose value at i is in f i for all i.

Equations

• DFinsupp.pi f = Finset.dfinsupp (DFinsupp.support f) ⇑f

@[simp]

theorem DFinsupp.mem_pi {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [DecidableEq ι] [(i : ι) → DecidableEq (α i)] {f : Π₀ (i : ι), Finset (α i)} {g : Π₀ (i : ι), α i} : g ∈ DFinsupp.pi f ↔ ∀ (i : ι), g i ∈ f i

@[simp]

theorem DFinsupp.card_pi {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [DecidableEq ι] [(i : ι) → DecidableEq (α i)] (f : Π₀ (i : ι), Finset (α i)) : (DFinsupp.pi f).card = DFinsupp.prod f fun (i : ι) => ↑(f i).card

instance DFinsupp.instLocallyFiniteOrder {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → PartialOrder (α i)] [(i : ι) → Zero (α i)] [(i : ι) → LocallyFiniteOrder (α i)] : LocallyFiniteOrder (Π₀ (i : ι), α i)

Equations

• One or more equations did not get rendered due to their size.

theorem DFinsupp.Icc_eq {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → PartialOrder (α i)] [(i : ι) → Zero (α i)] [(i : ι) → LocallyFiniteOrder (α i)] (f : Π₀ (i : ι), α i) (g : Π₀ (i : ι), α i) : Finset.Icc f g = Finset.dfinsupp (DFinsupp.support f ∪ DFinsupp.support g) ⇑(DFinsupp.rangeIcc f g)

theorem DFinsupp.card_Icc {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → PartialOrder (α i)] [(i : ι) → Zero (α i)] [(i : ι) → LocallyFiniteOrder (α i)] (f : Π₀ (i : ι), α i) (g : Π₀ (i : ι), α i) : (Finset.Icc f g).card = Finset.prod (DFinsupp.support f ∪ DFinsupp.support g) fun (i : ι) => (Finset.Icc (f i) (g i)).card

theorem DFinsupp.card_Ico {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → PartialOrder (α i)] [(i : ι) → Zero (α i)] [(i : ι) → LocallyFiniteOrder (α i)] (f : Π₀ (i : ι), α i) (g : Π₀ (i : ι), α i) : (Finset.Ico f g).card = (Finset.prod (DFinsupp.support f ∪ DFinsupp.support g) fun (i : ι) => (Finset.Icc (f i) (g i)).card) - 1

theorem DFinsupp.card_Ioc {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → PartialOrder (α i)] [(i : ι) → Zero (α i)] [(i : ι) → LocallyFiniteOrder (α i)] (f : Π₀ (i : ι), α i) (g : Π₀ (i : ι), α i) : (Finset.Ioc f g).card = (Finset.prod (DFinsupp.support f ∪ DFinsupp.support g) fun (i : ι) => (Finset.Icc (f i) (g i)).card) - 1

theorem DFinsupp.card_Ioo {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → PartialOrder (α i)] [(i : ι) → Zero (α i)] [(i : ι) → LocallyFiniteOrder (α i)] (f : Π₀ (i : ι), α i) (g : Π₀ (i : ι), α i) : (Finset.Ioo f g).card = (Finset.prod (DFinsupp.support f ∪ DFinsupp.support g) fun (i : ι) => (Finset.Icc (f i) (g i)).card) - 2

theorem DFinsupp.card_uIcc {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → Lattice (α i)] [(i : ι) → Zero (α i)] [(i : ι) → LocallyFiniteOrder (α i)] (f : Π₀ (i : ι), α i) (g : Π₀ (i : ι), α i) : (Finset.uIcc f g).card = Finset.prod (DFinsupp.support f ∪ DFinsupp.support g) fun (i : ι) => (Finset.uIcc (f i) (g i)).card

theorem DFinsupp.card_Iic {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → CanonicallyOrderedAddCommMonoid (α i)] [(i : ι) → LocallyFiniteOrder (α i)] (f : Π₀ (i : ι), α i) : (Finset.Iic f).card = Finset.prod (DFinsupp.support f) fun (i : ι) => (Finset.Iic (f i)).card

theorem DFinsupp.card_Iio {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → CanonicallyOrderedAddCommMonoid (α i)] [(i : ι) → LocallyFiniteOrder (α i)] (f : Π₀ (i : ι), α i) : (Finset.Iio f).card = (Finset.prod (DFinsupp.support f) fun (i : ι) => (Finset.Iic (f i)).card) - 1