Equivalence between `Multiset` and `ℕ`-valued finitely supported functions #

This defines Finsupp.toMultiset the equivalence between α →₀ ℕ and Multiset α, along with Multiset.toFinsupp the reverse equivalence and Finsupp.orderIsoMultiset the equivalence promoted to an order isomorphism.

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def Finsupp.toMultiset {α : Type u_1} :

(α →₀ ℕ) →+ Multiset α

Given f : α →₀ ℕ, f.toMultiset is the multiset with multiplicities given by the values of f on the elements of α. We define this function as an AddMonoidHom.

Under the additional assumption of [DecidableEq α], this is available as Multiset.toFinsupp : Multiset α ≃+ (α →₀ ℕ); the two declarations are separate as this assumption is only needed for one direction.

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

Instances For

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theorem Finsupp.toMultiset_zero {α : Type u_1} :

Finsupp.toMultiset 0 = 0

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theorem Finsupp.toMultiset_add {α : Type u_1} (m : α →₀ ℕ) (n : α →₀ ℕ) :

Finsupp.toMultiset (m + n) = Finsupp.toMultiset m + Finsupp.toMultiset n

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theorem Finsupp.toMultiset_apply {α : Type u_1} (f : α →₀ ℕ) :

Finsupp.toMultiset f = Finsupp.sum f fun (a : α) (n : ℕ) => n • {a}

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@[simp]

theorem Finsupp.toMultiset_single {α : Type u_1} (a : α) (n : ℕ) :

Finsupp.toMultiset (Finsupp.single a n) = n • {a}

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theorem Finsupp.toMultiset_sum {α : Type u_1} {ι : Type u_3} {f : ι → α →₀ ℕ} (s : Finset ι) :

Finsupp.toMultiset (Finset.sum s fun (i : ι) => f i) = Finset.sum s fun (i : ι) => Finsupp.toMultiset (f i)

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theorem Finsupp.toMultiset_sum_single {ι : Type u_3} (s : Finset ι) (n : ℕ) :

Finsupp.toMultiset (Finset.sum s fun (i : ι) => Finsupp.single i n) = n • s.val

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theorem Finsupp.card_toMultiset {α : Type u_1} (f : α →₀ ℕ) :

Multiset.card (Finsupp.toMultiset f) = Finsupp.sum f fun (x : α) => id

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theorem Finsupp.toMultiset_map {α : Type u_1} {β : Type u_2} (f : α →₀ ℕ) (g : α → β) :

Multiset.map g (Finsupp.toMultiset f) = Finsupp.toMultiset (Finsupp.mapDomain g f)

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@[simp]

theorem Finsupp.sum_toMultiset {α : Type u_1} [AddCommMonoid α] (f : α →₀ ℕ) :

Multiset.sum (Finsupp.toMultiset f) = Finsupp.sum f fun (a : α) (n : ℕ) => n • a

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@[simp]

theorem Finsupp.prod_toMultiset {α : Type u_1} [CommMonoid α] (f : α →₀ ℕ) :

Multiset.prod (Finsupp.toMultiset f) = Finsupp.prod f fun (a : α) (n : ℕ) => a ^ n

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@[simp]

theorem Finsupp.toFinset_toMultiset {α : Type u_1} [DecidableEq α] (f : α →₀ ℕ) :

Multiset.toFinset (Finsupp.toMultiset f) = f.support

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@[simp]

theorem Finsupp.count_toMultiset {α : Type u_1} [DecidableEq α] (f : α →₀ ℕ) (a : α) :

Multiset.count a (Finsupp.toMultiset f) = f a

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theorem Finsupp.toMultiset_sup {α : Type u_1} [DecidableEq α] (f : α →₀ ℕ) (g : α →₀ ℕ) :

Finsupp.toMultiset (f ⊔ g) = Finsupp.toMultiset f ∪ Finsupp.toMultiset g

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theorem Finsupp.toMultiset_inf {α : Type u_1} [DecidableEq α] (f : α →₀ ℕ) (g : α →₀ ℕ) :

Finsupp.toMultiset (f ⊓ g) = Finsupp.toMultiset f ∩ Finsupp.toMultiset g

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@[simp]

theorem Finsupp.mem_toMultiset {α : Type u_1} (f : α →₀ ℕ) (i : α) :

i ∈ Finsupp.toMultiset f ↔ i ∈ f.support

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@[simp]

theorem Multiset.toFinsupp_symm_apply {α : Type u_1} [DecidableEq α] (f : α →₀ ℕ) :

(AddEquiv.symm Multiset.toFinsupp) f = Finsupp.toMultiset f

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def Multiset.toFinsupp {α : Type u_1} [DecidableEq α] :

Multiset α ≃+ (α →₀ ℕ)

Given a multiset s, s.toFinsupp returns the finitely supported function on ℕ given by the multiplicities of the elements of s.

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@[simp]

theorem Multiset.toFinsupp_support {α : Type u_1} [DecidableEq α] (s : Multiset α) :

(Multiset.toFinsupp s).support = Multiset.toFinset s

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@[simp]

theorem Multiset.toFinsupp_apply {α : Type u_1} [DecidableEq α] (s : Multiset α) (a : α) :

(Multiset.toFinsupp s) a = Multiset.count a s

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theorem Multiset.toFinsupp_zero {α : Type u_1} [DecidableEq α] :

Multiset.toFinsupp 0 = 0

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theorem Multiset.toFinsupp_add {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) :

Multiset.toFinsupp (s + t) = Multiset.toFinsupp s + Multiset.toFinsupp t

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@[simp]

theorem Multiset.toFinsupp_singleton {α : Type u_1} [DecidableEq α] (a : α) :

Multiset.toFinsupp {a} = Finsupp.single a 1

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@[simp]

theorem Multiset.toFinsupp_toMultiset {α : Type u_1} [DecidableEq α] (s : Multiset α) :

Finsupp.toMultiset (Multiset.toFinsupp s) = s

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theorem Multiset.toFinsupp_eq_iff {α : Type u_1} [DecidableEq α] {s : Multiset α} {f : α →₀ ℕ} :

Multiset.toFinsupp s = f ↔ s = Finsupp.toMultiset f

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theorem Multiset.toFinsupp_union {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) :

Multiset.toFinsupp (s ∪ t) = Multiset.toFinsupp s ⊔ Multiset.toFinsupp t

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theorem Multiset.toFinsupp_inter {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) :

Multiset.toFinsupp (s ∩ t) = Multiset.toFinsupp s ⊓ Multiset.toFinsupp t

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@[simp]

theorem Finsupp.toMultiset_toFinsupp {α : Type u_1} [DecidableEq α] (f : α →₀ ℕ) :

Multiset.toFinsupp (Finsupp.toMultiset f) = f

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theorem Finsupp.toMultiset_eq_iff {α : Type u_1} [DecidableEq α] {f : α →₀ ℕ} {s : Multiset α} :

Finsupp.toMultiset f = s ↔ f = Multiset.toFinsupp s

As an order isomorphism #

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def Finsupp.orderIsoMultiset {ι : Type u_3} [DecidableEq ι] :

(ι →₀ ℕ) ≃o Multiset ι

Finsupp.toMultiset as an order isomorphism.

Equations

Finsupp.orderIsoMultiset = { toEquiv := (AddEquiv.symm Multiset.toFinsupp).toEquiv, map_rel_iff' := (_ : ∀ {f g : ι →₀ ℕ}, Finsupp.toMultiset f ≤ Finsupp.toMultiset g ↔ f ≤ g) }