Documentation

Mathlib.Data.DFinsupp.Multiset

Equivalence between Multiset and ℕ-valued finitely supported functions #

This defines DFinsupp.toMultiset the equivalence between Π₀ a : α, ℕ and Multiset α, along with Multiset.toDFinsupp the reverse equivalence.

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instance DFinsupp.addZeroClass' {α : Type u_1} {β : Type u_2} [AddZeroClass β] :
AddZeroClass (Π₀ (x : α), β)

Non-dependent special case of DFinsupp.addZeroClass to help typeclass search.

Equations
  • DFinsupp.addZeroClass' = DFinsupp.addZeroClass
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def DFinsupp.toMultiset {α : Type u_1} [DecidableEq α] :
(Π₀ (x : α), ℕ) →+ Multiset α

A DFinsupp version of Finsupp.toMultiset.

Equations
Instances For
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    @[simp]
    theorem DFinsupp.toMultiset_single {α : Type u_1} [DecidableEq α] (a : α) (n : ℕ) :
    DFinsupp.toMultiset (DFinsupp.single a n) = Multiset.replicate n a
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    def Multiset.toDFinsupp {α : Type u_1} [DecidableEq α] :
    Multiset α →+ Π₀ (x : α), ℕ

    A DFinsupp version of Multiset.toFinsupp.

    Equations
    • Multiset.toDFinsupp = { toFun := fun (s : Multiset α) => { toFun := fun (n : α) => Multiset.count n s, support' := Trunc.mk ⟨s, ⋯⟩ }, map_zero' := ⋯, map_add' := ⋯ }
    Instances For
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      @[simp]
      theorem Multiset.toDFinsupp_apply {α : Type u_1} [DecidableEq α] (s : Multiset α) (a : α) :
      (Multiset.toDFinsupp s) a = Multiset.count a s
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      @[simp]
      theorem Multiset.toDFinsupp_support {α : Type u_1} [DecidableEq α] (s : Multiset α) :
      (Multiset.toDFinsupp s).support = s.toFinset
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      @[simp]
      theorem Multiset.toDFinsupp_replicate {α : Type u_1} [DecidableEq α] (a : α) (n : ℕ) :
      Multiset.toDFinsupp (Multiset.replicate n a) = DFinsupp.single a n
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      @[simp]
      theorem Multiset.toDFinsupp_singleton {α : Type u_1} [DecidableEq α] (a : α) :
      Multiset.toDFinsupp {a} = DFinsupp.single a 1
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      def Multiset.equivDFinsupp {α : Type u_1} [DecidableEq α] :
      Multiset α ≃+ Π₀ (x : α), ℕ

      Multiset.toDFinsupp as an AddEquiv.

      Equations
      • Multiset.equivDFinsupp = Multiset.toDFinsupp.toAddEquiv DFinsupp.toMultiset ⋯ ⋯
      Instances For
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        @[simp]
        theorem Multiset.equivDFinsupp_symm_apply {α : Type u_1} [DecidableEq α] (a : Π₀ (x : α), ℕ) :
        Multiset.equivDFinsupp.symm a = DFinsupp.toMultiset a
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        @[simp]
        theorem Multiset.equivDFinsupp_apply {α : Type u_1} [DecidableEq α] (a : Multiset α) :
        Multiset.equivDFinsupp a = Multiset.toDFinsupp a
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        @[simp]
        theorem Multiset.toDFinsupp_toMultiset {α : Type u_1} [DecidableEq α] (s : Multiset α) :
        DFinsupp.toMultiset (Multiset.toDFinsupp s) = s
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        theorem Multiset.toDFinsupp_injective {α : Type u_1} [DecidableEq α] :
        Function.Injective ⇑Multiset.toDFinsupp
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        @[simp]
        theorem Multiset.toDFinsupp_inj {α : Type u_1} [DecidableEq α] {s t : Multiset α} :
        Multiset.toDFinsupp s = Multiset.toDFinsupp t ↔ s = t
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        @[simp]
        theorem Multiset.toDFinsupp_le_toDFinsupp {α : Type u_1} [DecidableEq α] {s t : Multiset α} :
        Multiset.toDFinsupp s ≤ Multiset.toDFinsupp t ↔ s ≤ t
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        @[simp]
        theorem Multiset.toDFinsupp_lt_toDFinsupp {α : Type u_1} [DecidableEq α] {s t : Multiset α} :
        Multiset.toDFinsupp s < Multiset.toDFinsupp t ↔ s < t
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        @[simp]
        theorem Multiset.toDFinsupp_inter {α : Type u_1} [DecidableEq α] (s t : Multiset α) :
        Multiset.toDFinsupp (s ∩ t) = Multiset.toDFinsupp s ⊓ Multiset.toDFinsupp t
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        @[simp]
        theorem Multiset.toDFinsupp_union {α : Type u_1} [DecidableEq α] (s t : Multiset α) :
        Multiset.toDFinsupp (s ∪ t) = Multiset.toDFinsupp s ⊔ Multiset.toDFinsupp t
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        @[simp]
        theorem DFinsupp.toMultiset_toDFinsupp {α : Type u_1} [DecidableEq α] (f : Π₀ (x : α), ℕ) :
        Multiset.toDFinsupp (DFinsupp.toMultiset f) = f
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        theorem DFinsupp.toMultiset_injective {α : Type u_1} [DecidableEq α] :
        Function.Injective ⇑DFinsupp.toMultiset
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        @[simp]
        theorem DFinsupp.toMultiset_inj {α : Type u_1} [DecidableEq α] {f g : Π₀ (_a : α), ℕ} :
        DFinsupp.toMultiset f = DFinsupp.toMultiset g ↔ f = g
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        @[simp]
        theorem DFinsupp.toMultiset_le_toMultiset {α : Type u_1} [DecidableEq α] {f g : Π₀ (_a : α), ℕ} :
        DFinsupp.toMultiset f ≤ DFinsupp.toMultiset g ↔ f ≤ g
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        @[simp]
        theorem DFinsupp.toMultiset_lt_toMultiset {α : Type u_1} [DecidableEq α] {f g : Π₀ (_a : α), ℕ} :
        DFinsupp.toMultiset f < DFinsupp.toMultiset g ↔ f < g
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        @[simp]
        theorem DFinsupp.toMultiset_inf {α : Type u_1} [DecidableEq α] (f g : Π₀ (_a : α), ℕ) :
        DFinsupp.toMultiset (f ⊓ g) = DFinsupp.toMultiset f ∩ DFinsupp.toMultiset g
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        @[simp]
        theorem DFinsupp.toMultiset_sup {α : Type u_1} [DecidableEq α] (f g : Π₀ (_a : α), ℕ) :
        DFinsupp.toMultiset (f ⊔ g) = DFinsupp.toMultiset f ∪ DFinsupp.toMultiset g