Documentation

Mathlib.Data.DFinsupp.Interval

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.

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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.

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    @[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
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    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
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    @[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.

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    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.

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      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
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      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.

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        @[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)
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        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
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        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
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        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.

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          @[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
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          @[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
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          instance DFinsupp.instLocallyFiniteOrderDFinsuppInstPreorderDFinsuppToPreorder {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)
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          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)
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          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
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          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
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          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
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          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
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          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
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          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
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          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