Documentation

Mathlib.Data.Multiset.Lattice

Lattice operations on multisets #

sup #

source
def Multiset.sup {α : Type u_1} [SemilatticeSup α] [OrderBot α] (s : Multiset α) :
α

Supremum of a multiset: sup {a, b, c} = a ⊔ b ⊔ c

Instances For
    source
    @[simp]
    theorem Multiset.sup_coe {α : Type u_1} [SemilatticeSup α] [OrderBot α] (l : List α) :
    Multiset.sup ↑l = List.foldr (fun x x_1 => x ⊔ x_1) ⊥ l
    source
    @[simp]
    theorem Multiset.sup_zero {α : Type u_1} [SemilatticeSup α] [OrderBot α] :
    Multiset.sup 0 = ⊥
    source
    @[simp]
    theorem Multiset.sup_cons {α : Type u_1} [SemilatticeSup α] [OrderBot α] (a : α) (s : Multiset α) :
    Multiset.sup (a ::ₘ s) = a ⊔ Multiset.sup s
    source
    @[simp]
    theorem Multiset.sup_singleton {α : Type u_1} [SemilatticeSup α] [OrderBot α] {a : α} :
    Multiset.sup {a} = a
    source
    @[simp]
    theorem Multiset.sup_add {α : Type u_1} [SemilatticeSup α] [OrderBot α] (s₁ : Multiset α) (s₂ : Multiset α) :
    Multiset.sup (s₁ + s₂) = Multiset.sup s₁ ⊔ Multiset.sup s₂
    source
    theorem Multiset.sup_le {α : Type u_1} [SemilatticeSup α] [OrderBot α] {s : Multiset α} {a : α} :
    Multiset.sup s ≤ a ↔ ∀ (b : α), b ∈ s → b ≤ a
    source
    theorem Multiset.le_sup {α : Type u_1} [SemilatticeSup α] [OrderBot α] {s : Multiset α} {a : α} (h : a ∈ s) :
    a ≤ Multiset.sup s
    source
    theorem Multiset.sup_mono {α : Type u_1} [SemilatticeSup α] [OrderBot α] {s₁ : Multiset α} {s₂ : Multiset α} (h : s₁ ⊆ s₂) :
    Multiset.sup s₁ ≤ Multiset.sup s₂
    source
    @[simp]
    theorem Multiset.sup_dedup {α : Type u_1} [SemilatticeSup α] [OrderBot α] [DecidableEq α] (s : Multiset α) :
    Multiset.sup (Multiset.dedup s) = Multiset.sup s
    source
    @[simp]
    theorem Multiset.sup_ndunion {α : Type u_1} [SemilatticeSup α] [OrderBot α] [DecidableEq α] (s₁ : Multiset α) (s₂ : Multiset α) :
    Multiset.sup (Multiset.ndunion s₁ s₂) = Multiset.sup s₁ ⊔ Multiset.sup s₂
    source
    @[simp]
    theorem Multiset.sup_union {α : Type u_1} [SemilatticeSup α] [OrderBot α] [DecidableEq α] (s₁ : Multiset α) (s₂ : Multiset α) :
    Multiset.sup (s₁ ∪ s₂) = Multiset.sup s₁ ⊔ Multiset.sup s₂
    source
    @[simp]
    theorem Multiset.sup_ndinsert {α : Type u_1} [SemilatticeSup α] [OrderBot α] [DecidableEq α] (a : α) (s : Multiset α) :
    Multiset.sup (Multiset.ndinsert a s) = a ⊔ Multiset.sup s
    source
    theorem Multiset.nodup_sup_iff {α : Type u_2} [DecidableEq α] {m : Multiset (Multiset α)} :
    Multiset.Nodup (Multiset.sup m) ↔ ∀ (a : Multiset α), a ∈ m → Multiset.Nodup a

    inf #

    source
    def Multiset.inf {α : Type u_1} [SemilatticeInf α] [OrderTop α] (s : Multiset α) :
    α

    Infimum of a multiset: inf {a, b, c} = a ⊓ b ⊓ c

    Instances For
      source
      @[simp]
      theorem Multiset.inf_coe {α : Type u_1} [SemilatticeInf α] [OrderTop α] (l : List α) :
      Multiset.inf ↑l = List.foldr (fun x x_1 => x ⊓ x_1) ⊤ l
      source
      @[simp]
      theorem Multiset.inf_zero {α : Type u_1} [SemilatticeInf α] [OrderTop α] :
      Multiset.inf 0 = ⊤
      source
      @[simp]
      theorem Multiset.inf_cons {α : Type u_1} [SemilatticeInf α] [OrderTop α] (a : α) (s : Multiset α) :
      Multiset.inf (a ::ₘ s) = a ⊓ Multiset.inf s
      source
      @[simp]
      theorem Multiset.inf_singleton {α : Type u_1} [SemilatticeInf α] [OrderTop α] {a : α} :
      Multiset.inf {a} = a
      source
      @[simp]
      theorem Multiset.inf_add {α : Type u_1} [SemilatticeInf α] [OrderTop α] (s₁ : Multiset α) (s₂ : Multiset α) :
      Multiset.inf (s₁ + s₂) = Multiset.inf s₁ ⊓ Multiset.inf s₂
      source
      theorem Multiset.le_inf {α : Type u_1} [SemilatticeInf α] [OrderTop α] {s : Multiset α} {a : α} :
      a ≤ Multiset.inf s ↔ ∀ (b : α), b ∈ s → a ≤ b
      source
      theorem Multiset.inf_le {α : Type u_1} [SemilatticeInf α] [OrderTop α] {s : Multiset α} {a : α} (h : a ∈ s) :
      Multiset.inf s ≤ a
      source
      theorem Multiset.inf_mono {α : Type u_1} [SemilatticeInf α] [OrderTop α] {s₁ : Multiset α} {s₂ : Multiset α} (h : s₁ ⊆ s₂) :
      Multiset.inf s₂ ≤ Multiset.inf s₁
      source
      @[simp]
      theorem Multiset.inf_dedup {α : Type u_1} [SemilatticeInf α] [OrderTop α] [DecidableEq α] (s : Multiset α) :
      Multiset.inf (Multiset.dedup s) = Multiset.inf s
      source
      @[simp]
      theorem Multiset.inf_ndunion {α : Type u_1} [SemilatticeInf α] [OrderTop α] [DecidableEq α] (s₁ : Multiset α) (s₂ : Multiset α) :
      Multiset.inf (Multiset.ndunion s₁ s₂) = Multiset.inf s₁ ⊓ Multiset.inf s₂
      source
      @[simp]
      theorem Multiset.inf_union {α : Type u_1} [SemilatticeInf α] [OrderTop α] [DecidableEq α] (s₁ : Multiset α) (s₂ : Multiset α) :
      Multiset.inf (s₁ ∪ s₂) = Multiset.inf s₁ ⊓ Multiset.inf s₂
      source
      @[simp]
      theorem Multiset.inf_ndinsert {α : Type u_1} [SemilatticeInf α] [OrderTop α] [DecidableEq α] (a : α) (s : Multiset α) :
      Multiset.inf (Multiset.ndinsert a s) = a ⊓ Multiset.inf s