Documentation

Mathlib.Order.Bounds.Defs

Definitions about upper/lower bounds #

In this file we define:

def upperBounds {α : Type u_1} [LE α] (s : Set α) :
Set α

The set of upper bounds of a set.

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      def lowerBounds {α : Type u_1} [LE α] (s : Set α) :
      Set α

      The set of lower bounds of a set.

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          def BddAbove {α : Type u_1} [LE α] (s : Set α) :

          A set is bounded above if there exists an upper bound.

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              def BddBelow {α : Type u_1} [LE α] (s : Set α) :

              A set is bounded below if there exists a lower bound.

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                  def IsLeast {α : Type u_1} [LE α] (s : Set α) (a : α) :

                  a is a least element of a set s; for a partial order, it is unique if exists.

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                      def IsGreatest {α : Type u_1} [LE α] (s : Set α) (a : α) :

                      a is a greatest element of a set s; for a partial order, it is unique if exists.

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                          def IsLUB {α : Type u_1} [LE α] (s : Set α) :
                          αProp

                          a is a least upper bound of a set s; for a partial order, it is unique if exists.

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                              def IsGLB {α : Type u_1} [LE α] (s : Set α) :
                              αProp

                              a is a greatest lower bound of a set s; for a partial order, it is unique if exists.

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                                  def IsCofinalFor {α : Type u_1} [LE α] (s t : Set α) :

                                  A set s is said to be cofinal for a set t if, for all a ∈ s there exists b ∈ t such that a ≤ b.

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                                      def IsCoinitialFor {α : Type u_1} [LE α] (s t : Set α) :

                                      A set s is said to be coinitial for a set t if, for all a ∈ s there exists b ∈ t such that b ≤ a.

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                                          def IsCofinal {α : Type u_1} [LE α] (s : Set α) :

                                          A set is cofinal when for every x : α there exists y ∈ s with x ≤ y.

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