Documentation

Mathlib.Data.Multiset.Dedup

Erasing duplicates in a multiset. #

dedup #

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

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

Instances For
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    @[simp]
    theorem Multiset.coe_dedup {α : Type u_1} [DecidableEq α] (l : List α) :
    Multiset.dedup ↑l = ↑(List.dedup l)
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    @[simp]
    theorem Multiset.dedup_zero {α : Type u_1} [DecidableEq α] :
    Multiset.dedup 0 = 0
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    @[simp]
    theorem Multiset.mem_dedup {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} :
    a ∈ Multiset.dedup s ↔ a ∈ s
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    @[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
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    @[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
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    theorem Multiset.dedup_le {α : Type u_1} [DecidableEq α] (s : Multiset α) :
    Multiset.dedup s ≤ s
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    theorem Multiset.dedup_subset {α : Type u_1} [DecidableEq α] (s : Multiset α) :
    Multiset.dedup s ⊆ s
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    theorem Multiset.subset_dedup {α : Type u_1} [DecidableEq α] (s : Multiset α) :
    s ⊆ Multiset.dedup s
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    @[simp]
    theorem Multiset.dedup_subset' {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} :
    Multiset.dedup s ⊆ t ↔ s ⊆ t
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    @[simp]
    theorem Multiset.subset_dedup' {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} :
    s ⊆ Multiset.dedup t ↔ s ⊆ t
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    @[simp]
    theorem Multiset.nodup_dedup {α : Type u_1} [DecidableEq α] (s : Multiset α) :
    Multiset.Nodup (Multiset.dedup s)
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    theorem Multiset.dedup_eq_self {α : Type u_1} [DecidableEq α] {s : Multiset α} :
    Multiset.dedup s = s ↔ Multiset.Nodup s
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    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.

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    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
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    @[simp]
    theorem Multiset.dedup_idem {α : Type u_1} [DecidableEq α] {m : Multiset α} :
    Multiset.dedup (Multiset.dedup m) = Multiset.dedup m
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    @[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)
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    theorem Multiset.dedup_eq_zero {α : Type u_1} [DecidableEq α] {s : Multiset α} :
    Multiset.dedup s = 0 ↔ s = 0
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    @[simp]
    theorem Multiset.dedup_singleton {α : Type u_1} [DecidableEq α] {a : α} :
    Multiset.dedup {a} = {a}
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    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
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    theorem Multiset.le_dedup_self {α : Type u_1} [DecidableEq α] {s : Multiset α} :
    s ≤ Multiset.dedup s ↔ Multiset.Nodup s
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    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
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    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)
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    @[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
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    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
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    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