File: Mathlib/Data/DFinsupp/Multiset.lean

full_name: DFinsupp.addZeroClass'
code: /-- Non-dependent special case of `DFinsupp.addZeroClass` to help typeclass search. -/
instance addZeroClass' {β} [AddZeroClass β] : AddZeroClass (Π₀ _ : α, β) 
goal: AddZeroClass (Π₀ (x : α), β)

full_name: DFinsupp.toMultiset
code: /-- A DFinsupp version of `Finsupp.toMultiset`. -/
def toMultiset : (Π₀ _ : α, ℕ) →+ Multiset α 
goal: (Π₀ (x : α), ℕ) →+ Multiset α

full_name: DFinsupp.toMultiset_single
code: @[simp]
theorem toMultiset_single (a : α) (n : ℕ) :
    toMultiset (DFinsupp.single a n) = Multiset.replicate n a 
goal: DFinsupp.toMultiset (DFinsupp.single a n) = Multiset.replicate n a

full_name: Multiset.toDFinsupp
code: /-- A DFinsupp version of `Multiset.toFinsupp`. -/
def toDFinsupp : Multiset α →+ Π₀ _ : α, ℕ 
goal: Multiset α →+ Π₀ (x : α), ℕ

full_name: Multiset.toDFinsupp_apply
code: @[simp]
theorem toDFinsupp_apply (s : Multiset α) (a : α) : Multiset.toDFinsupp s a = s.count a 
goal: (Multiset.toDFinsupp s) a = Multiset.count a s

full_name: Multiset.toDFinsupp_support
code: @[simp]
theorem toDFinsupp_support (s : Multiset α) : s.toDFinsupp.support = s.toFinset 
goal: (Multiset.toDFinsupp s).support = s.toFinset

full_name: Multiset.toDFinsupp_replicate
code: @[simp]
theorem toDFinsupp_replicate (a : α) (n : ℕ) :
    toDFinsupp (Multiset.replicate n a) = DFinsupp.single a n 
goal: Multiset.toDFinsupp (Multiset.replicate n a) = DFinsupp.single a n

full_name: Multiset.toDFinsupp_singleton
code: @[simp]
theorem toDFinsupp_singleton (a : α) : toDFinsupp {a} = DFinsupp.single a 1 
goal: Multiset.toDFinsupp {a} = DFinsupp.single a 1

full_name: Multiset.equivDFinsupp
code: /-- `Multiset.toDFinsupp` as an `AddEquiv`. -/
@[simps! apply symm_apply]
def equivDFinsupp : Multiset α ≃+ Π₀ _ : α, ℕ 
goal: Multiset α ≃+ Π₀ (x : α), ℕ

full_name: Multiset.equivDFinsupp_symm_apply
code: /-- `Multiset.toDFinsupp` as an `AddEquiv`. -/
@[simps! apply symm_apply]
def equivDFinsupp : Multiset α ≃+ Π₀ _ : α, ℕ 
goal: Multiset.equivDFinsupp.symm a = DFinsupp.toMultiset a

full_name: Multiset.equivDFinsupp_apply
code: /-- `Multiset.toDFinsupp` as an `AddEquiv`. -/
@[simps! apply symm_apply]
def equivDFinsupp : Multiset α ≃+ Π₀ _ : α, ℕ 
goal: Multiset.equivDFinsupp a = Multiset.toDFinsupp a

full_name: Multiset.toDFinsupp_toMultiset
code: @[simp]
theorem toDFinsupp_toMultiset (s : Multiset α) : DFinsupp.toMultiset (Multiset.toDFinsupp s) = s 
goal: DFinsupp.toMultiset (Multiset.toDFinsupp s) = s

full_name: Multiset.toDFinsupp_injective
code: theorem toDFinsupp_injective : Injective (toDFinsupp : Multiset α → Π₀ _a, ℕ) 
goal: Function.Injective ⇑Multiset.toDFinsupp

full_name: Multiset.toDFinsupp_inj
code: @[simp]
theorem toDFinsupp_inj : toDFinsupp s = toDFinsupp t ↔ s = t 
goal: Multiset.toDFinsupp s = Multiset.toDFinsupp t ↔ s = t

full_name: Multiset.toDFinsupp_le_toDFinsupp
code: @[simp]
theorem toDFinsupp_le_toDFinsupp : toDFinsupp s ≤ toDFinsupp t ↔ s ≤ t 
goal: Multiset.toDFinsupp s ≤ Multiset.toDFinsupp t ↔ s ≤ t

full_name: Multiset.toDFinsupp_lt_toDFinsupp
code: @[simp]
theorem toDFinsupp_lt_toDFinsupp : toDFinsupp s < toDFinsupp t ↔ s < t 
goal: Multiset.toDFinsupp s < Multiset.toDFinsupp t ↔ s < t

full_name: Multiset.toDFinsupp_inter
code: @[simp]
theorem toDFinsupp_inter (s t : Multiset α) : toDFinsupp (s ∩ t) = toDFinsupp s ⊓ toDFinsupp t 
goal: Multiset.toDFinsupp (s ∩ t) = Multiset.toDFinsupp s ⊓ Multiset.toDFinsupp t

full_name: Multiset.toDFinsupp_union
code: @[simp]
theorem toDFinsupp_union (s t : Multiset α) : toDFinsupp (s ∪ t) = toDFinsupp s ⊔ toDFinsupp t 
goal: Multiset.toDFinsupp (s ∪ t) = Multiset.toDFinsupp s ⊔ Multiset.toDFinsupp t

full_name: DFinsupp.toMultiset_toDFinsupp
code: @[simp]
theorem toMultiset_toDFinsupp (f : Π₀ _ : α, ℕ) :
    Multiset.toDFinsupp (DFinsupp.toMultiset f) = f 
goal: Multiset.toDFinsupp (DFinsupp.toMultiset f) = f

full_name: DFinsupp.toMultiset_injective
code: theorem toMultiset_injective : Injective (toMultiset : (Π₀ _a, ℕ) → Multiset α) 
goal: Function.Injective ⇑DFinsupp.toMultiset

full_name: DFinsupp.toMultiset_inj
code: @[simp]
theorem toMultiset_inj : toMultiset f = toMultiset g ↔ f = g 
goal: DFinsupp.toMultiset f = DFinsupp.toMultiset g ↔ f = g

full_name: DFinsupp.toMultiset_le_toMultiset
code: @[simp]
theorem toMultiset_le_toMultiset : toMultiset f ≤ toMultiset g ↔ f ≤ g 
goal: DFinsupp.toMultiset f ≤ DFinsupp.toMultiset g ↔ f ≤ g

full_name: DFinsupp.toMultiset_lt_toMultiset
code: @[simp]
theorem toMultiset_lt_toMultiset : toMultiset f < toMultiset g ↔ f < g 
goal: DFinsupp.toMultiset f < DFinsupp.toMultiset g ↔ f < g

full_name: DFinsupp.toMultiset_inf
code: @[simp]
theorem toMultiset_inf : toMultiset (f ⊓ g) = toMultiset f ∩ toMultiset g 
goal: DFinsupp.toMultiset (f ⊓ g) = DFinsupp.toMultiset f ∩ DFinsupp.toMultiset g

full_name: DFinsupp.toMultiset_sup
code: @[simp]
theorem toMultiset_sup : toMultiset (f ⊔ g) = toMultiset f∪ toMultiset g 
goal: DFinsupp.toMultiset (f ⊔ g) = DFinsupp.toMultiset f ∪ DFinsupp.toMultiset g