Documentation

Mathlib.Data.Multiset.Lattice

Lattice operations on multisets #

sup #

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

Equations
Instances For
    @[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
    @[simp]
    @[simp]
    theorem Multiset.sup_cons {α : Type u_1} [SemilatticeSup α] [OrderBot α] (a : α) (s : Multiset α) :
    @[simp]
    theorem Multiset.sup_singleton {α : Type u_1} [SemilatticeSup α] [OrderBot α] {a : α} :
    @[simp]
    theorem Multiset.sup_add {α : Type u_1} [SemilatticeSup α] [OrderBot α] (s₁ : Multiset α) (s₂ : Multiset α) :
    theorem Multiset.sup_le {α : Type u_1} [SemilatticeSup α] [OrderBot α] {s : Multiset α} {a : α} :
    Multiset.sup s a ∀ (b : α), b sb a
    theorem Multiset.le_sup {α : Type u_1} [SemilatticeSup α] [OrderBot α] {s : Multiset α} {a : α} (h : a s) :
    theorem Multiset.sup_mono {α : Type u_1} [SemilatticeSup α] [OrderBot α] {s₁ : Multiset α} {s₂ : Multiset α} (h : s₁ s₂) :
    @[simp]
    theorem Multiset.sup_ndunion {α : Type u_1} [SemilatticeSup α] [OrderBot α] [DecidableEq α] (s₁ : Multiset α) (s₂ : Multiset α) :
    @[simp]
    theorem Multiset.sup_union {α : Type u_1} [SemilatticeSup α] [OrderBot α] [DecidableEq α] (s₁ : Multiset α) (s₂ : Multiset α) :
    @[simp]

    inf #

    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

    Equations
    Instances For
      @[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
      @[simp]
      @[simp]
      theorem Multiset.inf_cons {α : Type u_1} [SemilatticeInf α] [OrderTop α] (a : α) (s : Multiset α) :
      @[simp]
      theorem Multiset.inf_singleton {α : Type u_1} [SemilatticeInf α] [OrderTop α] {a : α} :
      @[simp]
      theorem Multiset.inf_add {α : Type u_1} [SemilatticeInf α] [OrderTop α] (s₁ : Multiset α) (s₂ : Multiset α) :
      theorem Multiset.le_inf {α : Type u_1} [SemilatticeInf α] [OrderTop α] {s : Multiset α} {a : α} :
      a Multiset.inf s ∀ (b : α), b sa b
      theorem Multiset.inf_le {α : Type u_1} [SemilatticeInf α] [OrderTop α] {s : Multiset α} {a : α} (h : a s) :
      theorem Multiset.inf_mono {α : Type u_1} [SemilatticeInf α] [OrderTop α] {s₁ : Multiset α} {s₂ : Multiset α} (h : s₁ s₂) :
      @[simp]
      theorem Multiset.inf_ndunion {α : Type u_1} [SemilatticeInf α] [OrderTop α] [DecidableEq α] (s₁ : Multiset α) (s₂ : Multiset α) :
      @[simp]
      theorem Multiset.inf_union {α : Type u_1} [SemilatticeInf α] [OrderTop α] [DecidableEq α] (s₁ : Multiset α) (s₂ : Multiset α) :
      @[simp]