Erasing duplicates in a multiset.

dedup

def Multiset.dedup {α : Type u_1} [DecidableEq α] (s : Multiset α) : Multiset α

dedup s removes duplicates from s, yielding a nodup multiset.

Equations

• Multiset.dedup s = Quot.liftOn s (fun (l : List α) => ↑(List.dedup l)) ⋯

@[simp]

theorem Multiset.coe_dedup {α : Type u_1} [DecidableEq α] (l : List α) : Multiset.dedup ↑l = ↑(List.dedup l)

@[simp]

theorem Multiset.dedup_zero {α : Type u_1} [DecidableEq α] : Multiset.dedup 0 = 0

@[simp]

theorem Multiset.mem_dedup {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} : a ∈ Multiset.dedup s ↔ a ∈ s

@[simp]

theorem Multiset.dedup_cons_of_mem {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} : a ∈ s → Multiset.dedup (a ::ₘ s) = Multiset.dedup s

@[simp]

theorem Multiset.dedup_cons_of_not_mem {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} : a ∉ s → Multiset.dedup (a ::ₘ s) = a ::ₘ Multiset.dedup s

theorem Multiset.dedup_le {α : Type u_1} [DecidableEq α] (s : Multiset α) : Multiset.dedup s ≤ s

theorem Multiset.dedup_subset {α : Type u_1} [DecidableEq α] (s : Multiset α) : Multiset.dedup s ⊆ s

theorem Multiset.subset_dedup {α : Type u_1} [DecidableEq α] (s : Multiset α) : s ⊆ Multiset.dedup s

@[simp]

theorem Multiset.dedup_subset' {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} : Multiset.dedup s ⊆ t ↔ s ⊆ t

@[simp]

theorem Multiset.subset_dedup' {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} : s ⊆ Multiset.dedup t ↔ s ⊆ t

@[simp]

theorem Multiset.nodup_dedup {α : Type u_1} [DecidableEq α] (s : Multiset α) : Multiset.Nodup (Multiset.dedup s)

theorem Multiset.dedup_eq_self {α : Type u_1} [DecidableEq α] {s : Multiset α} : Multiset.dedup s = s ↔ Multiset.Nodup s

theorem Multiset.Nodup.dedup {α : Type u_1} [DecidableEq α] {s : Multiset α} : Multiset.Nodup s → Multiset.dedup s = s

Alias of the reverse direction of Multiset.dedup_eq_self.

theorem Multiset.count_dedup {α : Type u_1} [DecidableEq α] (m : Multiset α) (a : α) : Multiset.count a (Multiset.dedup m) = if a ∈ m then 1 else 0

@[simp]

theorem Multiset.dedup_idem {α : Type u_1} [DecidableEq α] {m : Multiset α} : Multiset.dedup (Multiset.dedup m) = Multiset.dedup m

@[simp]

theorem Multiset.dedup_bind_dedup {α : Type u_1} {β : Type u_2} [DecidableEq α] [DecidableEq β] (m : Multiset α) (f : α → Multiset β) : Multiset.dedup (Multiset.bind (Multiset.dedup m) f) = Multiset.dedup (Multiset.bind m f)

theorem Multiset.dedup_eq_zero {α : Type u_1} [DecidableEq α] {s : Multiset α} : Multiset.dedup s = 0 ↔ s = 0

@[simp]

theorem Multiset.dedup_singleton {α : Type u_1} [DecidableEq α] {a : α} : Multiset.dedup {a} = {a}

theorem Multiset.le_dedup {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} : s ≤ Multiset.dedup t ↔ s ≤ t ∧ Multiset.Nodup s

theorem Multiset.le_dedup_self {α : Type u_1} [DecidableEq α] {s : Multiset α} : s ≤ Multiset.dedup s ↔ Multiset.Nodup s

theorem Multiset.dedup_ext {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} : Multiset.dedup s = Multiset.dedup t ↔ ∀ (a : α), a ∈ s ↔ a ∈ t

theorem Multiset.dedup_map_dedup_eq {α : Type u_1} {β : Type u_2} [DecidableEq α] [DecidableEq β] (f : α → β) (s : Multiset α) : Multiset.dedup (Multiset.map f (Multiset.dedup s)) = Multiset.dedup (Multiset.map f s)

@[simp]

theorem Multiset.dedup_nsmul {α : Type u_1} [DecidableEq α] {s : Multiset α} {n : ℕ} (h0 : n ≠ 0) : Multiset.dedup (n • s) = Multiset.dedup s

theorem Multiset.Nodup.le_dedup_iff_le {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} (hno : Multiset.Nodup s) : s ≤ Multiset.dedup t ↔ s ≤ t

theorem Multiset.Nodup.le_nsmul_iff_le {α : Type u_1} {s : Multiset α} {t : Multiset α} {n : ℕ} (h : Multiset.Nodup s) (hn : n ≠ 0) : s ≤ n • t ↔ s ≤ t