Documentation

Mathlib.Data.Multiset.FinsetOps

Preparations for defining operations on Finset. #

The operations here ignore multiplicities, and prepare for defining the corresponding operations on Finset.

finset insert #

source
def Multiset.ndinsert {α : Type u_1} [DecidableEq α] (a : α) (s : Multiset α) :
Multiset α

ndinsert a s is the lift of the list insert operation. This operation does not respect multiplicities, unlike cons, but it is suitable as an insert operation on Finset.

Instances For
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    @[simp]
    theorem Multiset.coe_ndinsert {α : Type u_1} [DecidableEq α] (a : α) (l : List α) :
    ndinsert a ↑l = ↑(insert a l)
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    @[simp]
    theorem Multiset.ndinsert_zero {α : Type u_1} [DecidableEq α] (a : α) :
    ndinsert a 0 = {a}
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    @[simp]
    theorem Multiset.ndinsert_of_mem {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} :
    a ∈ s → ndinsert a s = s
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    @[simp]
    theorem Multiset.ndinsert_of_notMem {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} :
    ¬a ∈ s → ndinsert a s = a ::ₘ s
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    @[simp]
    theorem Multiset.mem_ndinsert {α : Type u_1} [DecidableEq α] {a b : α} {s : Multiset α} :
    a ∈ ndinsert b s ↔ a = b ∨ a ∈ s
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    @[simp]
    theorem Multiset.le_ndinsert_self {α : Type u_1} [DecidableEq α] (a : α) (s : Multiset α) :
    s ≤ ndinsert a s
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    theorem Multiset.mem_ndinsert_self {α : Type u_1} [DecidableEq α] (a : α) (s : Multiset α) :
    a ∈ ndinsert a s
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    theorem Multiset.mem_ndinsert_of_mem {α : Type u_1} [DecidableEq α] {a b : α} {s : Multiset α} (h : a ∈ s) :
    a ∈ ndinsert b s
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    theorem Multiset.length_ndinsert_of_mem {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} (h : a ∈ s) :
    (ndinsert a s).card = s.card
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    theorem Multiset.length_ndinsert_of_notMem {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} (h : ¬a ∈ s) :
    (ndinsert a s).card = s.card + 1
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    theorem Multiset.dedup_cons {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} :
    (a ::ₘ s).dedup = ndinsert a s.dedup
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    theorem Multiset.Nodup.ndinsert {α : Type u_1} [DecidableEq α] {s : Multiset α} (a : α) :
    s.Nodup → (Multiset.ndinsert a s).Nodup
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    theorem Multiset.ndinsert_le {α : Type u_1} [DecidableEq α] {a : α} {s t : Multiset α} :
    ndinsert a s ≤ t ↔ s ≤ t ∧ a ∈ t
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    theorem Multiset.attach_ndinsert {α : Type u_1} [DecidableEq α] (a : α) (s : Multiset α) :
    (ndinsert a s).attach = ndinsert ⟨a, ⋯⟩ (map (fun (p : { x : α // x ∈ s }) => ⟨↑p, ⋯⟩) s.attach)
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    @[simp]
    theorem Multiset.disjoint_ndinsert_left {α : Type u_1} [DecidableEq α] {a : α} {s t : Multiset α} :
    Disjoint (ndinsert a s) t ↔ ¬a ∈ t ∧ Disjoint s t
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    @[simp]
    theorem Multiset.disjoint_ndinsert_right {α : Type u_1} [DecidableEq α] {a : α} {s t : Multiset α} :
    Disjoint s (ndinsert a t) ↔ ¬a ∈ s ∧ Disjoint s t

    finset union #

    source
    def Multiset.ndunion {α : Type u_1} [DecidableEq α] (s t : Multiset α) :
    Multiset α

    ndunion s t is the lift of the list union operation. This operation does not respect multiplicities, unlike s ∪ t, but it is suitable as a union operation on Finset. (s ∪ t would also work as a union operation on finset, but this is more efficient.)

    Instances For
      source
      @[simp]
      theorem Multiset.coe_ndunion {α : Type u_1} [DecidableEq α] (l₁ l₂ : List α) :
      (↑l₁).ndunion ↑l₂ = ↑(l₁ ∪ l₂)
      source
      theorem Multiset.zero_ndunion {α : Type u_1} [DecidableEq α] (s : Multiset α) :
      ndunion 0 s = s
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      @[simp]
      theorem Multiset.cons_ndunion {α : Type u_1} [DecidableEq α] (s t : Multiset α) (a : α) :
      (a ::ₘ s).ndunion t = ndinsert a (s.ndunion t)
      source
      @[simp]
      theorem Multiset.mem_ndunion {α : Type u_1} [DecidableEq α] {s t : Multiset α} {a : α} :
      a ∈ s.ndunion t ↔ a ∈ s ∨ a ∈ t
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      theorem Multiset.le_ndunion_right {α : Type u_1} [DecidableEq α] (s t : Multiset α) :
      t ≤ s.ndunion t
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      theorem Multiset.subset_ndunion_right {α : Type u_1} [DecidableEq α] (s t : Multiset α) :
      t ⊆ s.ndunion t
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      theorem Multiset.ndunion_le_add {α : Type u_1} [DecidableEq α] (s t : Multiset α) :
      s.ndunion t ≤ s + t
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      theorem Multiset.ndunion_le {α : Type u_1} [DecidableEq α] {s t u : Multiset α} :
      s.ndunion t ≤ u ↔ s ⊆ u ∧ t ≤ u
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      theorem Multiset.subset_ndunion_left {α : Type u_1} [DecidableEq α] (s t : Multiset α) :
      s ⊆ s.ndunion t
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      theorem Multiset.le_ndunion_left {α : Type u_1} [DecidableEq α] {s : Multiset α} (t : Multiset α) (d : s.Nodup) :
      s ≤ s.ndunion t
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      theorem Multiset.ndunion_le_union {α : Type u_1} [DecidableEq α] (s t : Multiset α) :
      s.ndunion t ≤ s ∪ t
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      theorem Multiset.Nodup.ndunion {α : Type u_1} [DecidableEq α] (s : Multiset α) {t : Multiset α} :
      t.Nodup → (s.ndunion t).Nodup
      source
      @[simp]
      theorem Multiset.ndunion_eq_union {α : Type u_1} [DecidableEq α] {s t : Multiset α} (d : s.Nodup) :
      s.ndunion t = s ∪ t
      source
      theorem Multiset.dedup_add {α : Type u_1} [DecidableEq α] (s t : Multiset α) :
      (s + t).dedup = s.ndunion t.dedup
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      theorem Multiset.Disjoint.ndunion_eq {α : Type u_1} [DecidableEq α] {s t : Multiset α} (h : Disjoint s t) :
      s.ndunion t = s.dedup + t
      source
      theorem Multiset.Subset.ndunion_eq_right {α : Type u_1} [DecidableEq α] {s t : Multiset α} (h : s ⊆ t) :
      s.ndunion t = t

      finset inter #

      source
      def Multiset.ndinter {α : Type u_1} [DecidableEq α] (s t : Multiset α) :
      Multiset α

      ndinter s t is the lift of the list ∩ operation. This operation does not respect multiplicities, unlike s ∩ t, but it is suitable as an intersection operation on Finset. (s ∩ t would also work as an intersection operation on finset, but this is more efficient.)

      Instances For
        source
        @[simp]
        theorem Multiset.coe_ndinter {α : Type u_1} [DecidableEq α] (l₁ l₂ : List α) :
        (↑l₁).ndinter ↑l₂ = ↑(l₁ ∩ l₂)
        source
        @[simp]
        theorem Multiset.zero_ndinter {α : Type u_1} [DecidableEq α] (s : Multiset α) :
        ndinter 0 s = 0
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        @[simp]
        theorem Multiset.cons_ndinter_of_mem {α : Type u_1} [DecidableEq α] {a : α} (s : Multiset α) {t : Multiset α} (h : a ∈ t) :
        (a ::ₘ s).ndinter t = a ::ₘ s.ndinter t
        source
        @[simp]
        theorem Multiset.ndinter_cons_of_notMem {α : Type u_1} [DecidableEq α] {a : α} (s : Multiset α) {t : Multiset α} (h : ¬a ∈ t) :
        (a ::ₘ s).ndinter t = s.ndinter t
        source
        @[simp]
        theorem Multiset.mem_ndinter {α : Type u_1} [DecidableEq α] {s t : Multiset α} {a : α} :
        a ∈ s.ndinter t ↔ a ∈ s ∧ a ∈ t
        source
        theorem Multiset.Nodup.ndinter {α : Type u_1} [DecidableEq α] {s : Multiset α} (t : Multiset α) :
        s.Nodup → (s.ndinter t).Nodup
        source
        theorem Multiset.le_ndinter {α : Type u_1} [DecidableEq α] {s t u : Multiset α} :
        s ≤ t.ndinter u ↔ s ≤ t ∧ s ⊆ u
        source
        theorem Multiset.ndinter_le_left {α : Type u_1} [DecidableEq α] (s t : Multiset α) :
        s.ndinter t ≤ s
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        theorem Multiset.ndinter_subset_left {α : Type u_1} [DecidableEq α] (s t : Multiset α) :
        s.ndinter t ⊆ s
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        theorem Multiset.ndinter_subset_right {α : Type u_1} [DecidableEq α] (s t : Multiset α) :
        s.ndinter t ⊆ t
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        theorem Multiset.ndinter_le_right {α : Type u_1} [DecidableEq α] {s : Multiset α} (t : Multiset α) (d : s.Nodup) :
        s.ndinter t ≤ t
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        theorem Multiset.inter_le_ndinter {α : Type u_1} [DecidableEq α] (s t : Multiset α) :
        s ∩ t ≤ s.ndinter t
        source
        @[simp]
        theorem Multiset.ndinter_eq_inter {α : Type u_1} [DecidableEq α] {s t : Multiset α} (d : s.Nodup) :
        s.ndinter t = s ∩ t
        source
        @[simp]
        theorem Multiset.Nodup.inter {α : Type u_1} [DecidableEq α] {s : Multiset α} (t : Multiset α) (d : s.Nodup) :
        (s ∩ t).Nodup
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        theorem Multiset.ndinter_eq_zero_iff_disjoint {α : Type u_1} [DecidableEq α] {s t : Multiset α} :
        s.ndinter t = 0 ↔ Disjoint s t
        source
        theorem Multiset.Disjoint.ndinter_eq_zero {α : Type u_1} [DecidableEq α] {s t : Multiset α} :
        Disjoint s t → s.ndinter t = 0

        Alias of the reverse direction of Multiset.ndinter_eq_zero_iff_disjoint.

        source
        theorem Multiset.Subset.ndinter_eq_left {α : Type u_1} [DecidableEq α] {s t : Multiset α} (h : s ⊆ t) :
        s.ndinter t = s