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

Equations
    Instances For
      source
      @[simp]
      theorem Multiset.sup_coe {α : Type u_1} [SemilatticeSup α] [OrderBot α] (l : List α) :
      (↑l).sup = List.foldr (fun (x1 x2 : α) => x1 ⊔ x2) ⊥ l
      source
      @[simp]
      theorem Multiset.sup_zero {α : Type u_1} [SemilatticeSup α] [OrderBot α] :
      sup 0 = ⊥
      source
      @[simp]
      theorem Multiset.sup_cons {α : Type u_1} [SemilatticeSup α] [OrderBot α] (a : α) (s : Multiset α) :
      (a ::ₘ s).sup = a ⊔ s.sup
      source
      @[simp]
      theorem Multiset.sup_singleton {α : Type u_1} [SemilatticeSup α] [OrderBot α] {a : α} :
      {a}.sup = a
      source
      @[simp]
      theorem Multiset.sup_add {α : Type u_1} [SemilatticeSup α] [OrderBot α] (s₁ s₂ : Multiset α) :
      (s₁ + s₂).sup = s₁.sup ⊔ s₂.sup
      source
      @[simp]
      theorem Multiset.sup_le {α : Type u_1} [SemilatticeSup α] [OrderBot α] {s : Multiset α} {a : α} :
      s.sup ≤ a ↔ ∀ 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 ≤ s.sup
      source
      theorem Multiset.sup_mono {α : Type u_1} [SemilatticeSup α] [OrderBot α] {s₁ s₂ : Multiset α} (h : s₁ ⊆ s₂) :
      s₁.sup ≤ s₂.sup
      source
      @[simp]
      theorem Multiset.sup_dedup {α : Type u_1} [SemilatticeSup α] [OrderBot α] [DecidableEq α] (s : Multiset α) :
      s.dedup.sup = s.sup
      source
      @[simp]
      theorem Multiset.sup_ndunion {α : Type u_1} [SemilatticeSup α] [OrderBot α] [DecidableEq α] (s₁ s₂ : Multiset α) :
      (s₁.ndunion s₂).sup = s₁.sup ⊔ s₂.sup
      source
      @[simp]
      theorem Multiset.sup_union {α : Type u_1} [SemilatticeSup α] [OrderBot α] [DecidableEq α] (s₁ s₂ : Multiset α) :
      (s₁ ∪ s₂).sup = s₁.sup ⊔ s₂.sup
      source
      @[simp]
      theorem Multiset.sup_ndinsert {α : Type u_1} [SemilatticeSup α] [OrderBot α] [DecidableEq α] (a : α) (s : Multiset α) :
      (ndinsert a s).sup = a ⊔ s.sup
      source
      theorem Multiset.nodup_sup_iff {α : Type u_2} [DecidableEq α] {m : Multiset (Multiset α)} :
      m.sup.Nodup ↔ ∀ a ∈ m, a.Nodup

      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

      Equations
        Instances For
          source
          @[simp]
          theorem Multiset.inf_coe {α : Type u_1} [SemilatticeInf α] [OrderTop α] (l : List α) :
          (↑l).inf = List.foldr (fun (x1 x2 : α) => x1 ⊓ x2) ⊤ l
          source
          @[simp]
          theorem Multiset.inf_zero {α : Type u_1} [SemilatticeInf α] [OrderTop α] :
          inf 0 = ⊤
          source
          @[simp]
          theorem Multiset.inf_cons {α : Type u_1} [SemilatticeInf α] [OrderTop α] (a : α) (s : Multiset α) :
          (a ::ₘ s).inf = a ⊓ s.inf
          source
          @[simp]
          theorem Multiset.inf_singleton {α : Type u_1} [SemilatticeInf α] [OrderTop α] {a : α} :
          {a}.inf = a
          source
          @[simp]
          theorem Multiset.inf_add {α : Type u_1} [SemilatticeInf α] [OrderTop α] (s₁ s₂ : Multiset α) :
          (s₁ + s₂).inf = s₁.inf ⊓ s₂.inf
          source
          @[simp]
          theorem Multiset.le_inf {α : Type u_1} [SemilatticeInf α] [OrderTop α] {s : Multiset α} {a : α} :
          a ≤ s.inf ↔ ∀ 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) :
          s.inf ≤ a
          source
          theorem Multiset.inf_mono {α : Type u_1} [SemilatticeInf α] [OrderTop α] {s₁ s₂ : Multiset α} (h : s₁ ⊆ s₂) :
          s₂.inf ≤ s₁.inf
          source
          @[simp]
          theorem Multiset.inf_dedup {α : Type u_1} [SemilatticeInf α] [OrderTop α] [DecidableEq α] (s : Multiset α) :
          s.dedup.inf = s.inf
          source
          @[simp]
          theorem Multiset.inf_ndunion {α : Type u_1} [SemilatticeInf α] [OrderTop α] [DecidableEq α] (s₁ s₂ : Multiset α) :
          (s₁.ndunion s₂).inf = s₁.inf ⊓ s₂.inf
          source
          @[simp]
          theorem Multiset.inf_union {α : Type u_1} [SemilatticeInf α] [OrderTop α] [DecidableEq α] (s₁ s₂ : Multiset α) :
          (s₁ ∪ s₂).inf = s₁.inf ⊓ s₂.inf
          source
          @[simp]
          theorem Multiset.inf_ndinsert {α : Type u_1} [SemilatticeInf α] [OrderTop α] [DecidableEq α] (a : α) (s : Multiset α) :
          (ndinsert a s).inf = a ⊓ s.inf