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Mathlib.Data.Multiset.FinsetOps

Preparations for defining operations on Finset. #

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

finset insert #

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

Equations
Instances For
    @[simp]
    theorem Multiset.coe_ndinsert {α : Type u_1} [DecidableEq α] (a : α) (l : List α) :
    Multiset.ndinsert a l = (insert a l)
    @[simp]
    theorem Multiset.ndinsert_zero {α : Type u_1} [DecidableEq α] (a : α) :
    @[simp]
    theorem Multiset.ndinsert_of_mem {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} :
    a sMultiset.ndinsert a s = s
    @[simp]
    theorem Multiset.ndinsert_of_not_mem {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} :
    asMultiset.ndinsert a s = a ::ₘ s
    @[simp]
    theorem Multiset.mem_ndinsert {α : Type u_1} [DecidableEq α] {a : α} {b : α} {s : Multiset α} :
    @[simp]
    theorem Multiset.le_ndinsert_self {α : Type u_1} [DecidableEq α] (a : α) (s : Multiset α) :
    theorem Multiset.mem_ndinsert_self {α : Type u_1} [DecidableEq α] (a : α) (s : Multiset α) :
    theorem Multiset.mem_ndinsert_of_mem {α : Type u_1} [DecidableEq α] {a : α} {b : α} {s : Multiset α} (h : a s) :
    @[simp]
    theorem Multiset.length_ndinsert_of_mem {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} (h : a s) :
    Multiset.card (Multiset.ndinsert a s) = Multiset.card s
    @[simp]
    theorem Multiset.length_ndinsert_of_not_mem {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} (h : as) :
    Multiset.card (Multiset.ndinsert a s) = Multiset.card s + 1
    theorem Multiset.dedup_cons {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} :
    (a ::ₘ s).dedup = Multiset.ndinsert a s.dedup
    theorem Multiset.Nodup.ndinsert {α : Type u_1} [DecidableEq α] {s : Multiset α} (a : α) :
    s.Nodup(Multiset.ndinsert a s).Nodup
    theorem Multiset.ndinsert_le {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} {t : Multiset α} :
    theorem Multiset.attach_ndinsert {α : Type u_1} [DecidableEq α] (a : α) (s : Multiset α) :
    (Multiset.ndinsert a s).attach = Multiset.ndinsert a, (Multiset.map (fun (p : { x : α // x s }) => p, ) s.attach)
    @[simp]
    theorem Multiset.disjoint_ndinsert_left {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} {t : Multiset α} :
    (Multiset.ndinsert a s).Disjoint t at s.Disjoint t
    @[simp]
    theorem Multiset.disjoint_ndinsert_right {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} {t : Multiset α} :
    s.Disjoint (Multiset.ndinsert a t) as s.Disjoint t

    finset union #

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

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

      finset inter #

      def Multiset.ndinter {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : 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 a union operation on finset, but this is more efficient.)

      Equations
      Instances For
        @[simp]
        theorem Multiset.coe_ndinter {α : Type u_1} [DecidableEq α] (l₁ : List α) (l₂ : List α) :
        (↑l₁).ndinter l₂ = (l₁ l₂)
        @[simp]
        theorem Multiset.zero_ndinter {α : Type u_1} [DecidableEq α] (s : Multiset α) :
        @[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
        @[simp]
        theorem Multiset.ndinter_cons_of_not_mem {α : Type u_1} [DecidableEq α] {a : α} (s : Multiset α) {t : Multiset α} (h : at) :
        (a ::ₘ s).ndinter t = s.ndinter t
        @[simp]
        theorem Multiset.mem_ndinter {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} {a : α} :
        a s.ndinter t a s a t
        @[simp]
        theorem Multiset.Nodup.ndinter {α : Type u_1} [DecidableEq α] {s : Multiset α} (t : Multiset α) :
        s.Nodup(s.ndinter t).Nodup
        theorem Multiset.le_ndinter {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} {u : Multiset α} :
        s t.ndinter u s t s u
        theorem Multiset.ndinter_le_left {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) :
        s.ndinter t s
        theorem Multiset.ndinter_subset_left {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) :
        s.ndinter t s
        theorem Multiset.ndinter_subset_right {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) :
        s.ndinter t t
        theorem Multiset.ndinter_le_right {α : Type u_1} [DecidableEq α] {s : Multiset α} (t : Multiset α) (d : s.Nodup) :
        s.ndinter t t
        theorem Multiset.inter_le_ndinter {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) :
        s t s.ndinter t
        @[simp]
        theorem Multiset.ndinter_eq_inter {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} (d : s.Nodup) :
        s.ndinter t = s t
        theorem Multiset.ndinter_eq_zero_iff_disjoint {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} :
        s.ndinter t = 0 s.Disjoint t
        theorem Multiset.Disjoint.ndinter_eq_zero {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} :
        s.Disjoint ts.ndinter t = 0

        Alias of the reverse direction of Multiset.ndinter_eq_zero_iff_disjoint.

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