mathlib documentation

data.multiset.dedup

Erasing duplicates in a multiset. #

dedup #

def multiset.dedup {α : Type u_1} [decidable_eq α] (s : multiset α) :

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

Equations

s.dedup = quot.lift_on s (λ (l : list α), ↑(l.dedup)) multiset.dedup._proof_1

@[simp]

theorem multiset.coe_dedup {α : Type u_1} [decidable_eq α] (l : list α) :

↑l.dedup = ↑(l.dedup)

@[simp]

theorem multiset.dedup_zero {α : Type u_1} [decidable_eq α] :

@[simp]

theorem multiset.mem_dedup {α : Type u_1} [decidable_eq α] {a : α} {s : multiset α} :

a ∈ s.dedup ↔ a ∈ s

@[simp]

theorem multiset.dedup_cons_of_mem {α : Type u_1} [decidable_eq α] {a : α} {s : multiset α} :

a ∈ s → (a ::ₘ s).dedup = s.dedup

@[simp]

theorem multiset.dedup_cons_of_not_mem {α : Type u_1} [decidable_eq α] {a : α} {s : multiset α} :

a ∉ s → (a ::ₘ s).dedup = a ::ₘ s.dedup

theorem multiset.dedup_le {α : Type u_1} [decidable_eq α] (s : multiset α) :

theorem multiset.dedup_subset {α : Type u_1} [decidable_eq α] (s : multiset α) :

theorem multiset.subset_dedup {α : Type u_1} [decidable_eq α] (s : multiset α) :

@[simp]

theorem multiset.dedup_subset' {α : Type u_1} [decidable_eq α] {s t : multiset α} :

s.dedup ⊆ t ↔ s ⊆ t

@[simp]

theorem multiset.subset_dedup' {α : Type u_1} [decidable_eq α] {s t : multiset α} :

s ⊆ t.dedup ↔ s ⊆ t

@[simp]

theorem multiset.nodup_dedup {α : Type u_1} [decidable_eq α] (s : multiset α) :

theorem multiset.dedup_eq_self {α : Type u_1} [decidable_eq α] {s : multiset α} :

s.dedup = s ↔ s.nodup

theorem multiset.nodup.dedup {α : Type u_1} [decidable_eq α] {s : multiset α} :

s.nodup → s.dedup = s

Alias of dedup_eq_self.

theorem multiset.dedup_eq_zero {α : Type u_1} [decidable_eq α] {s : multiset α} :

s.dedup = 0 ↔ s = 0

@[simp]

theorem multiset.dedup_singleton {α : Type u_1} [decidable_eq α] {a : α} :

{a}.dedup = {a}

theorem multiset.le_dedup {α : Type u_1} [decidable_eq α] {s t : multiset α} :

s ≤ t.dedup ↔ s ≤ t ∧ s.nodup

theorem multiset.dedup_ext {α : Type u_1} [decidable_eq α] {s t : multiset α} :

s.dedup = t.dedup ↔ ∀ (a : α), a ∈ s ↔ a ∈ t

theorem multiset.dedup_map_dedup_eq {α : Type u_1} {β : Type u_2} [decidable_eq α] [decidable_eq β] (f : α → β) (s : multiset α) :

(multiset.map f s.dedup).dedup = (multiset.map f s).dedup

@[simp]

theorem multiset.dedup_nsmul {α : Type u_1} [decidable_eq α] {s : multiset α} {n : ℕ} (h0 : n ≠ 0) :

(n • s).dedup = s.dedup

theorem multiset.nodup.le_dedup_iff_le {α : Type u_1} [decidable_eq α] {s t : multiset α} (hno : s.nodup) :

s ≤ t.dedup ↔ s ≤ t

theorem multiset.nodup.le_nsmul_iff_le {α : Type u_1} {s t : multiset α} {n : ℕ} (h : s.nodup) (hn : n ≠ 0) :

s ≤ n • t ↔ s ≤ t