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

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

• f.singleton = { toFun := fun (i : ι) => {f i}, support' := Trunc.map (fun (s : { s : Multiset ι // ∀ (i : ι), i ∈ s ∨ f.toFun i = 0 }) => ⟨↑s, ⋯⟩) f.support' }

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 ∈ f.singleton 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 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 g : Π₀ (i : ι), α i) (i : ι) : (f.rangeIcc 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 g : Π₀ (i : ι), α i} {i : ι} {a : α i} : a ∈ (f.rangeIcc 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 g : Π₀ (i : ι), α i} [DecidableEq ι] [(i : ι) → DecidableEq (α i)] : (f.rangeIcc g).support ⊆ f.support ∪ g.support

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

• f.pi = f.support.dfinsupp ⇑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 ∈ f.pi ↔ ∀ (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)) : f.pi.card = f.prod 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

• DFinsupp.instLocallyFiniteOrder = LocallyFiniteOrder.ofIcc (Π₀ (i : ι), α i) (fun (f g : Π₀ (i : ι), α i) => (f.support ∪ g.support).dfinsupp ⇑(f.rangeIcc g)) ⋯

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 g : Π₀ (i : ι), α i) : Finset.Icc f g = (f.support ∪ g.support).dfinsupp ⇑(f.rangeIcc 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 g : Π₀ (i : ι), α i) : (Finset.Icc f g).card = ∏ i ∈ f.support ∪ g.support, (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 g : Π₀ (i : ι), α i) : (Finset.Ico f g).card = ∏ i ∈ f.support ∪ g.support, (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 g : Π₀ (i : ι), α i) : (Finset.Ioc f g).card = ∏ i ∈ f.support ∪ g.support, (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 g : Π₀ (i : ι), α i) : (Finset.Ioo f g).card = ∏ i ∈ f.support ∪ g.support, (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 g : Π₀ (i : ι), α i) : (Finset.uIcc f g).card = ∏ i ∈ f.support ∪ g.support, (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) → AddCommMonoid (α i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), CanonicallyOrderedAdd (α i)] [(i : ι) → LocallyFiniteOrder (α i)] (f : Π₀ (i : ι), α i) : (Finset.Iic f).card = ∏ i ∈ f.support, (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) → AddCommMonoid (α i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), CanonicallyOrderedAdd (α i)] [(i : ι) → LocallyFiniteOrder (α i)] (f : Π₀ (i : ι), α i) : (Finset.Iio f).card = ∏ i ∈ f.support, (Finset.Iic (f i)).card - 1