HepLean 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.

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

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    theorem Multiset.count_dedup {α : Type u_1} [DecidableEq α] (m : Multiset α) (a : α) :
    Multiset.count a m.dedup = 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} :
    m.dedup.dedup = m.dedup
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    theorem Multiset.dedup_eq_zero {α : Type u_1} [DecidableEq α] {s : Multiset α} :
    s.dedup = 0 ↔ s = 0
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    @[simp]
    theorem Multiset.dedup_singleton {α : Type u_1} [DecidableEq α] {a : α} :
    {a}.dedup = {a}
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    theorem Multiset.le_dedup {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} :
    s ≤ t.dedup ↔ s ≤ t ∧ s.Nodup
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    theorem Multiset.le_dedup_self {α : Type u_1} [DecidableEq α] {s : Multiset α} :
    s ≤ s.dedup ↔ s.Nodup
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    theorem Multiset.dedup_ext {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} :
    s.dedup = t.dedup ↔ ∀ (a : α), a ∈ s ↔ a ∈ t
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    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) :
    (Multiset.map f s).dedup = Multiset.map f s.dedup
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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.map f s.dedup).dedup = (Multiset.map f s).dedup
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    @[simp]
    theorem Multiset.dedup_nsmul {α : Type u_1} [DecidableEq α] {s : Multiset α} {n : ℕ} (h0 : n ≠ 0) :
    (n • s).dedup = s.dedup
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    theorem Multiset.Nodup.le_dedup_iff_le {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} (hno : s.Nodup) :
    s ≤ t.dedup ↔ s ≤ t
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    theorem Multiset.Subset.dedup_add_right {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} (h : s ⊆ t) :
    (s + t).dedup = t.dedup
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    theorem Multiset.Subset.dedup_add_left {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} (h : t ⊆ s) :
    (s + t).dedup = s.dedup
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    theorem Multiset.Disjoint.dedup_add {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} (h : s.Disjoint t) :
    (s + t).dedup = s.dedup + t.dedup
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    theorem List.Subset.dedup_append_left {α : Type u_1} [DecidableEq α] {s : List α} {t : List α} (h : t ⊆ s) :
    (s ++ t).dedup.Perm s.dedup

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

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