Documentation

Mathlib.Data.Multiset.Dedup

Erasing duplicates in a multiset. #

dedup #

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

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

Equations
@[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_not_mem {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} :
as(a ::ₘ s).dedup = a ::ₘ s.dedup
theorem Multiset.dedup_le {α : Type u_1} [DecidableEq α] (s : Multiset α) :
theorem Multiset.dedup_subset {α : Type u_1} [DecidableEq α] (s : Multiset α) :
theorem Multiset.subset_dedup {α : Type u_1} [DecidableEq α] (s : Multiset α) :
@[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) :
theorem Multiset.dedup_eq_self {α : Type u_1} [DecidableEq α] {s : Multiset α} :
theorem Multiset.Nodup.dedup {α : Type u_1} [DecidableEq α] {s : Multiset α} :
s.Nodups.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} :
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} :
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} :
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 : αβ} (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 : αβ) (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.