mathlib3 documentation

data.multiset.dedup

Erasing duplicates in a multiset. #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

dedup #

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

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

Equations
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@[simp]
theorem multiset.coe_dedup {α : Type u_1} [decidable_eq α] (l : list α) :
↑l.dedup = ↑(l.dedup)
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@[simp]
theorem multiset.dedup_zero {α : Type u_1} [decidable_eq α] :
0.dedup = 0
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@[simp]
theorem multiset.mem_dedup {α : Type u_1} [decidable_eq 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} [decidable_eq 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} [decidable_eq 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} [decidable_eq α] (s : multiset α) :
s.dedup ≤ s
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theorem multiset.dedup_subset {α : Type u_1} [decidable_eq α] (s : multiset α) :
s.dedup ⊆ s
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theorem multiset.subset_dedup {α : Type u_1} [decidable_eq α] (s : multiset α) :
s ⊆ s.dedup
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@[simp]
theorem multiset.dedup_subset' {α : Type u_1} [decidable_eq α] {s t : multiset α} :
s.dedup ⊆ t ↔ s ⊆ t
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@[simp]
theorem multiset.subset_dedup' {α : Type u_1} [decidable_eq α] {s t : multiset α} :
s ⊆ t.dedup ↔ s ⊆ t
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@[simp]
theorem multiset.nodup_dedup {α : Type u_1} [decidable_eq α] (s : multiset α) :
s.dedup.nodup
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theorem multiset.dedup_eq_self {α : Type u_1} [decidable_eq α] {s : multiset α} :
s.dedup = s ↔ s.nodup
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theorem multiset.nodup.dedup {α : Type u_1} [decidable_eq 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} [decidable_eq α] (m : multiset α) (a : α) :
multiset.count a m.dedup = ite (a ∈ m) 1 0
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@[simp]
theorem multiset.dedup_idempotent {α : Type u_1} [decidable_eq α] {m : multiset α} :
m.dedup.dedup = m.dedup
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@[simp]
theorem multiset.dedup_bind_dedup {α : Type u_1} {β : Type u_2} [decidable_eq α] [decidable_eq β] (m : multiset α) (f : α → multiset β) :
(m.dedup.bind f).dedup = (m.bind f).dedup
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theorem multiset.dedup_eq_zero {α : Type u_1} [decidable_eq α] {s : multiset α} :
s.dedup = 0 ↔ s = 0
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@[simp]
theorem multiset.dedup_singleton {α : Type u_1} [decidable_eq α] {a : α} :
{a}.dedup = {a}
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theorem multiset.le_dedup {α : Type u_1} [decidable_eq α] {s t : multiset α} :
s ≤ t.dedup ↔ s ≤ t ∧ s.nodup
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theorem multiset.le_dedup_self {α : Type u_1} [decidable_eq α] {s : multiset α} :
s ≤ s.dedup ↔ s.nodup
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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
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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
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@[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
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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
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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