Documentation

Mathlib.Order.Interval.Set.Defs

Intervals #

In any preorder α, we define intervals (which on each side can be either infinite, open, or closed) using the following naming conventions:

Each interval has the name I + letter for left side + letter for right side. For instance, Ioc a b denotes the interval (a, b].

We also define a typeclass Set.OrdConnected saying that a set includes Set.Icc a b whenever it contains both a and b.

def Set.Ioo {α : Type u_1} [Preorder α] (a b : α) :
Set α

Ioo a b is the left-open right-open interval $(a, b)$.

Equations
    Instances For
      @[simp]
      theorem Set.mem_Ioo {α : Type u_1} [Preorder α] {a b x : α} :
      x Ioo a b a < x x < b
      theorem Set.Ioo_def {α : Type u_1} [Preorder α] (a b : α) :
      {x : α | a < x x < b} = Ioo a b
      def Set.Ico {α : Type u_1} [Preorder α] (a b : α) :
      Set α

      Ico a b is the left-closed right-open interval $[a, b)$.

      Equations
        Instances For
          @[simp]
          theorem Set.mem_Ico {α : Type u_1} [Preorder α] {a b x : α} :
          x Ico a b a x x < b
          theorem Set.Ico_def {α : Type u_1} [Preorder α] (a b : α) :
          {x : α | a x x < b} = Ico a b
          def Set.Iio {α : Type u_1} [Preorder α] (b : α) :
          Set α

          Iio b is the left-infinite right-open interval $(-∞, b)$.

          Equations
            Instances For
              @[simp]
              theorem Set.mem_Iio {α : Type u_1} [Preorder α] {b x : α} :
              x Iio b x < b
              theorem Set.Iio_def {α : Type u_1} [Preorder α] (a : α) :
              {x : α | x < a} = Iio a
              def Set.Icc {α : Type u_1} [Preorder α] (a b : α) :
              Set α

              Icc a b is the left-closed right-closed interval $[a, b]$.

              Equations
                Instances For
                  @[simp]
                  theorem Set.mem_Icc {α : Type u_1} [Preorder α] {a b x : α} :
                  x Icc a b a x x b
                  theorem Set.Icc_def {α : Type u_1} [Preorder α] (a b : α) :
                  {x : α | a x x b} = Icc a b
                  def Set.Iic {α : Type u_1} [Preorder α] (b : α) :
                  Set α

                  Iic b is the left-infinite right-closed interval $(-∞, b]$.

                  Equations
                    Instances For
                      @[simp]
                      theorem Set.mem_Iic {α : Type u_1} [Preorder α] {b x : α} :
                      x Iic b x b
                      theorem Set.Iic_def {α : Type u_1} [Preorder α] (b : α) :
                      {x : α | x b} = Iic b
                      def Set.Ioc {α : Type u_1} [Preorder α] (a b : α) :
                      Set α

                      Ioc a b is the left-open right-closed interval $(a, b]$.

                      Equations
                        Instances For
                          @[simp]
                          theorem Set.mem_Ioc {α : Type u_1} [Preorder α] {a b x : α} :
                          x Ioc a b a < x x b
                          theorem Set.Ioc_def {α : Type u_1} [Preorder α] (a b : α) :
                          {x : α | a < x x b} = Ioc a b
                          def Set.Ici {α : Type u_1} [Preorder α] (a : α) :
                          Set α

                          Ici a is the left-closed right-infinite interval $[a, ∞)$.

                          Equations
                            Instances For
                              @[simp]
                              theorem Set.mem_Ici {α : Type u_1} [Preorder α] {a x : α} :
                              x Ici a a x
                              theorem Set.Ici_def {α : Type u_1} [Preorder α] (a : α) :
                              {x : α | a x} = Ici a
                              def Set.Ioi {α : Type u_1} [Preorder α] (a : α) :
                              Set α

                              Ioi a is the left-open right-infinite interval $(a, ∞)$.

                              Equations
                                Instances For
                                  @[simp]
                                  theorem Set.mem_Ioi {α : Type u_1} [Preorder α] {a x : α} :
                                  x Ioi a a < x
                                  theorem Set.Ioi_def {α : Type u_1} [Preorder α] (a : α) :
                                  {x : α | a < x} = Ioi a
                                  class Set.OrdConnected {α : Type u_1} [Preorder α] (s : Set α) :

                                  We say that a set s : Set α is OrdConnected if for all x y ∈ s it includes the interval [[x, y]]. If α is a DenselyOrdered ConditionallyCompleteLinearOrder with the OrderTopology, then this condition is equivalent to IsPreconnected s. If α is a LinearOrderedField, then this condition is also equivalent to Convex α s.

                                  • out' x : α (hx : x s) y : α (hy : y s) : Icc x y s

                                    s : Set α is OrdConnected if for all x y ∈ s it includes the interval [[x, y]].

                                  Instances