Documentation

Mathlib.Algebra.Order.IsBotOne

Typeclasses expressing IsBot 1 and IsBot 0 #

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class IsBotZeroClass (α : Type u_1) [LE α] [Zero α] :
Prop

A typeclass expressing that the 0 of a type is a bottom element. In a partial OrderBot, this is equivalent to ⊥ = 0.

Instances
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    class IsBotOneClass (α : Type u_1) [LE α] [One α] :
    Prop

    A typeclass expressing that the 1 of a type is a bottom element. In a partial OrderBot, this is equivalent to ⊥ = 1.

    Instances
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      theorem isBot_one {α : Type u_1} [LE α] [One α] [IsBotOneClass α] :
      IsBot 1
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      theorem isBot_zero {α : Type u_1} [LE α] [Zero α] [IsBotZeroClass α] :
      IsBot 0
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      @[simp]
      theorem one_le {α : Type u_1} [LE α] [One α] [IsBotOneClass α] {a : α} :
      1 ≤ a
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      @[simp]
      theorem zero_le {α : Type u_1} [LE α] [Zero α] [IsBotZeroClass α] {a : α} :
      0 ≤ a
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      theorem zero_le' {α : Type u_1} [LE α] [Zero α] [IsBotZeroClass α] {a : α} :
      0 ≤ a

      Alias of zero_le.

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      @[implicit_reducible]
      def IsBotOneClass.toOrderBot (α : Type u_1) [LE α] [One α] [IsBotOneClass α] :
      OrderBot α

      Create an OrderBot instance, setting 1 as the bottom element.

      Equations
      Instances For
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        @[implicit_reducible]
        def IsBotZeroClass.toOrderBot (α : Type u_1) [LE α] [Zero α] [IsBotZeroClass α] :
        OrderBot α

        Create an OrderBot instance, setting 0 as the bottom element.

        Equations
        Instances For
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          @[instance 100]
          instance instZeroLEOneClassOfIsBotZeroClass {α : Type u_1} [LE α] [Zero α] [One α] [IsBotZeroClass α] :
          ZeroLEOneClass α
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          @[simp]
          theorem not_lt_one {α : Type u_1} {a : α} [Preorder α] [One α] [IsBotOneClass α] :
          ¬a < 1
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          @[simp]
          theorem not_lt_zero {α : Type u_1} {a : α} [Preorder α] [Zero α] [IsBotZeroClass α] :
          ¬a < 0
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          @[deprecated not_lt_zero (since := "2025-12-03")]
          theorem not_neg {α : Type u_1} {a : α} [Preorder α] [Zero α] [IsBotZeroClass α] :
          ¬a < 0

          Alias of not_lt_zero.

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          theorem not_lt_zero' {α : Type u_1} {a : α} [Preorder α] [Zero α] [IsBotZeroClass α] :
          ¬a < 0

          Alias of not_lt_zero.

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          theorem one_lt_of_gt {α : Type u_1} {a b : α} [Preorder α] [One α] [IsBotOneClass α] (h : a < b) :
          1 < b
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          theorem pos_of_gt {α : Type u_1} {a b : α} [Preorder α] [Zero α] [IsBotZeroClass α] (h : a < b) :
          0 < b
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          theorem LT.lt.one_lt {α : Type u_1} {a b : α} [Preorder α] [One α] [IsBotOneClass α] (h : a < b) :
          1 < b

          Alias of one_lt_of_gt.

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          theorem LT.lt.pos {α : Type u_1} {a b : α} [Preorder α] [Zero α] [IsBotZeroClass α] (h : a < b) :
          0 < b
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          theorem ne_one_of_lt {α : Type u_1} {a b : α} [Preorder α] [One α] [IsBotOneClass α] (h : a < b) :
          b ≠ 1
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          theorem ne_zero_of_lt {α : Type u_1} {a b : α} [Preorder α] [Zero α] [IsBotZeroClass α] (h : a < b) :
          b ≠ 0
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          theorem LT.lt.ne_one {α : Type u_1} {a b : α} [Preorder α] [One α] [IsBotOneClass α] (h : a < b) :
          b ≠ 1

          Alias of ne_one_of_lt.

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          theorem LT.lt.ne_zero {α : Type u_1} {a b : α} [Preorder α] [Zero α] [IsBotZeroClass α] (h : a < b) :
          b ≠ 0
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          theorem bot_eq_one {α : Type u_1} [PartialOrder α] [One α] [IsBotOneClass α] [OrderBot α] :
          ⊥ = 1
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          theorem bot_eq_zero {α : Type u_1} [PartialOrder α] [Zero α] [IsBotZeroClass α] [OrderBot α] :
          ⊥ = 0
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          theorem bot_eq_zero'' {α : Type u_1} [PartialOrder α] [Zero α] [IsBotZeroClass α] [OrderBot α] :
          ⊥ = 0

          Alias of bot_eq_zero.

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          @[simp]
          theorem le_one_iff_eq_one {α : Type u_1} {a : α} [PartialOrder α] [One α] [IsBotOneClass α] :
          a ≤ 1 ↔ a = 1
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          @[simp]
          theorem nonpos_iff_eq_zero {α : Type u_1} {a : α} [PartialOrder α] [Zero α] [IsBotZeroClass α] :
          a ≤ 0 ↔ a = 0
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          theorem le_zero_iff {α : Type u_1} {a : α} [PartialOrder α] [Zero α] [IsBotZeroClass α] :
          a ≤ 0 ↔ a = 0

          Alias of nonpos_iff_eq_zero.

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          theorem eq_one_of_le_one {α : Type u_1} {a : α} [PartialOrder α] [One α] [IsBotOneClass α] :
          a ≤ 1 → a = 1

          Alias of the forward direction of le_one_iff_eq_one.

          source
          theorem eq_zero_of_nonpos {α : Type u_1} {a : α} [PartialOrder α] [Zero α] [IsBotZeroClass α] :
          a ≤ 0 → a = 0
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          theorem LE.le.eq_one {α : Type u_1} {a : α} [PartialOrder α] [One α] [IsBotOneClass α] :
          a ≤ 1 → a = 1

          Alias of the forward direction of le_one_iff_eq_one.

          Alias of the forward direction of le_one_iff_eq_one.

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          theorem LE.le.eq_zero {α : Type u_1} {a : α} [PartialOrder α] [Zero α] [IsBotZeroClass α] :
          a ≤ 0 → a = 0
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          theorem one_lt_iff_ne_one {α : Type u_1} {a : α} [PartialOrder α] [One α] [IsBotOneClass α] :
          1 < a ↔ a ≠ 1
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          theorem pos_iff_ne_zero {α : Type u_1} {a : α} [PartialOrder α] [Zero α] [IsBotZeroClass α] :
          0 < a ↔ a ≠ 0
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          theorem zero_lt_iff {α : Type u_1} {a : α} [PartialOrder α] [Zero α] [IsBotZeroClass α] :
          0 < a ↔ a ≠ 0

          Alias of pos_iff_ne_zero.

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          theorem one_lt_of_ne_one {α : Type u_1} {a : α} [PartialOrder α] [One α] [IsBotOneClass α] :
          a ≠ 1 → 1 < a

          Alias of the reverse direction of one_lt_iff_ne_one.

          source
          theorem pos_of_ne_zero {α : Type u_1} {a : α} [PartialOrder α] [Zero α] [IsBotZeroClass α] :
          a ≠ 0 → 0 < a
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          theorem Ne.one_lt {α : Type u_1} {a : α} [PartialOrder α] [One α] [IsBotOneClass α] :
          a ≠ 1 → 1 < a

          Alias of the reverse direction of one_lt_iff_ne_one.

          Alias of the reverse direction of one_lt_iff_ne_one.

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          theorem Ne.pos {α : Type u_1} {a : α} [PartialOrder α] [Zero α] [IsBotZeroClass α] :
          a ≠ 0 → 0 < a
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          theorem eq_one_or_one_lt {α : Type u_1} [PartialOrder α] [One α] [IsBotOneClass α] (a : α) :
          a = 1 ∨ 1 < a
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          theorem eq_zero_or_pos {α : Type u_1} [PartialOrder α] [Zero α] [IsBotZeroClass α] (a : α) :
          a = 0 ∨ 0 < a
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          theorem one_notMem_iff {α : Type u_1} [PartialOrder α] [One α] [IsBotOneClass α] {s : Set α} :
          ¬1 ∈ s ↔ ∀ (x : α), x ∈ s → 1 < x
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          theorem zero_notMem_iff {α : Type u_1} [PartialOrder α] [Zero α] [IsBotZeroClass α] {s : Set α} :
          ¬0 ∈ s ↔ ∀ (x : α), x ∈ s → 0 < x
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          @[deprecated Ne.pos (since := "2026-02-17")]
          theorem NE.ne.pos {α : Type u_1} {a : α} [PartialOrder α] [Zero α] [IsBotZeroClass α] :
          a ≠ 0 → 0 < a

          Alias of Ne.pos.

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          @[deprecated Ne.one_lt (since := "2026-02-17")]
          theorem NE.ne.one_lt {α : Type u_1} {a : α} [PartialOrder α] [One α] [IsBotOneClass α] :
          a ≠ 1 → 1 < a

          Alias of the reverse direction of one_lt_iff_ne_one.

          Alias of the reverse direction of one_lt_iff_ne_one.

          Alias of the reverse direction of one_lt_iff_ne_one.

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          theorem one_min {α : Type u_1} [LinearOrder α] [One α] [IsBotOneClass α] (a : α) :
          min 1 a = 1
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          theorem zero_min {α : Type u_1} [LinearOrder α] [Zero α] [IsBotZeroClass α] (a : α) :
          min 0 a = 0
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          theorem min_one {α : Type u_1} [LinearOrder α] [One α] [IsBotOneClass α] (a : α) :
          min a 1 = 1
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          theorem min_zero {α : Type u_1} [LinearOrder α] [Zero α] [IsBotZeroClass α] (a : α) :
          min a 0 = 0
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          theorem NeZero.of_gt {α : Type u_1} {a b : α} [Zero α] [Preorder α] [IsBotZeroClass α] (h : a < b) :
          NeZero b
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          theorem NeZero.pos {α : Type u_1} [Zero α] [PartialOrder α] [IsBotZeroClass α] (a : α) [NeZero a] :
          0 < a
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          @[instance 10]
          instance NeZero.of_gt' {α : Type u_1} {a : α} [Zero α] [Preorder α] [IsBotZeroClass α] [One α] [Fact (1 < a)] :
          NeZero a
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          theorem NeZero.of_ge {α : Type u_1} {a b : α} [Zero α] [PartialOrder α] [IsBotZeroClass α] [NeZero a] (h : a ≤ b) :
          NeZero b