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.

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

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

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

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

@[simp]

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

@[simp]

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

@[simp]

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

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

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

@[simp]

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

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

@[simp]

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

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

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

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

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

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

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

theorem Multiset.Subset.dedup_add_right {α : Type u_1} [DecidableEq α] {s t : Multiset α} (h : s ⊆ t) : (s + t).dedup = t.dedup

theorem Multiset.Subset.dedup_add_left {α : Type u_1} [DecidableEq α] {s t : Multiset α} (h : t ⊆ s) : (s + t).dedup = s.dedup

theorem Multiset.Disjoint.dedup_add {α : Type u_1} [DecidableEq α] {s t : Multiset α} (h : Disjoint s t) : (s + t).dedup = s.dedup + t.dedup

theorem List.Subset.dedup_append_left {α : Type u_1} [DecidableEq α] {s t : List α} (h : t ⊆ s) : (s ++ t).dedup.Perm s.dedup

Note that the stronger List.Subset.dedup_append_right is proved earlier.