⊤ and ⊥, bounded lattices and variants #

This file defines top and bottom elements (greatest and least elements) of a type, the bounded variants of different kinds of lattices, sets up the typeclass hierarchy between them and provides instances for Prop and fun.

Main declarations #

<Top/Bot> α: Typeclasses to declare the ⊤/⊥ notation.
Order<Top/Bot> α: Order with a top/bottom element.
BoundedOrder α: Order with a top and bottom element.

Top, bottom element #

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class OrderTop (α : Type u) [LE α] extends Top α :

Type u

An order is an OrderTop if it has a greatest element. We state this using a data mixin, holding the value of ⊤ and the greatest element constraint.

top : α
le_top : ∀ (a : α), a ≤ ⊤
⊤ is the greatest element

Instances

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noncomputable def topOrderOrNoTopOrder (α : Type u_1) [LE α] :

OrderTop α ⊕' NoTopOrder α

An order is (noncomputably) either an OrderTop or a NoTopOrder. Use as casesI topOrderOrNoTopOrder α.

Equations

topOrderOrNoTopOrder α = if H : ∀ (a : α), ∃ (b : α), ¬b ≤ a then PSum.inr ⋯ else PSum.inl (OrderTop.mk ⋯)

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@[simp]

theorem le_top {α : Type u} [LE α] [OrderTop α] {a : α} :

a ≤ ⊤

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@[simp]

theorem isTop_top {α : Type u} [LE α] [OrderTop α] :

IsTop ⊤

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@[simp]

theorem isMax_top {α : Type u} [Preorder α] [OrderTop α] :

IsMax ⊤

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@[simp]

theorem not_top_lt {α : Type u} [Preorder α] [OrderTop α] {a : α} :

¬⊤ < a

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theorem ne_top_of_lt {α : Type u} [Preorder α] [OrderTop α] {a b : α} (h : a < b) :

a ≠ ⊤

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theorem LT.lt.ne_top {α : Type u} [Preorder α] [OrderTop α] {a b : α} (h : a < b) :

a ≠ ⊤

Alias of ne_top_of_lt.

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theorem lt_top_of_lt {α : Type u} [Preorder α] [OrderTop α] {a b : α} (h : a < b) :

a < ⊤

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theorem LT.lt.lt_top {α : Type u} [Preorder α] [OrderTop α] {a b : α} (h : a < b) :

a < ⊤

Alias of lt_top_of_lt.

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@[simp]

theorem isMax_iff_eq_top {α : Type u} [PartialOrder α] [OrderTop α] {a : α} :

IsMax a ↔ a = ⊤

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@[simp]

theorem isTop_iff_eq_top {α : Type u} [PartialOrder α] [OrderTop α] {a : α} :

IsTop a ↔ a = ⊤

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theorem not_isMax_iff_ne_top {α : Type u} [PartialOrder α] [OrderTop α] {a : α} :

¬IsMax a ↔ a ≠ ⊤

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theorem not_isTop_iff_ne_top {α : Type u} [PartialOrder α] [OrderTop α] {a : α} :

¬IsTop a ↔ a ≠ ⊤

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theorem IsMax.eq_top {α : Type u} [PartialOrder α] [OrderTop α] {a : α} :

IsMax a → a = ⊤

Alias of the forward direction of isMax_iff_eq_top.

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theorem IsTop.eq_top {α : Type u} [PartialOrder α] [OrderTop α] {a : α} :

IsTop a → a = ⊤

Alias of the forward direction of isTop_iff_eq_top.

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@[simp]

theorem top_le_iff {α : Type u} [PartialOrder α] [OrderTop α] {a : α} :

⊤ ≤ a ↔ a = ⊤

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theorem top_unique {α : Type u} [PartialOrder α] [OrderTop α] {a : α} (h : ⊤ ≤ a) :

a = ⊤

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theorem eq_top_iff {α : Type u} [PartialOrder α] [OrderTop α] {a : α} :

a = ⊤ ↔ ⊤ ≤ a

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theorem eq_top_mono {α : Type u} [PartialOrder α] [OrderTop α] {a b : α} (h : a ≤ b) (h₂ : a = ⊤) :

b = ⊤

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theorem lt_top_iff_ne_top {α : Type u} [PartialOrder α] [OrderTop α] {a : α} :

a < ⊤ ↔ a ≠ ⊤

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@[simp]

theorem not_lt_top_iff {α : Type u} [PartialOrder α] [OrderTop α] {a : α} :

¬a < ⊤ ↔ a = ⊤

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theorem eq_top_or_lt_top {α : Type u} [PartialOrder α] [OrderTop α] (a : α) :

a = ⊤ ∨ a < ⊤

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theorem Ne.lt_top {α : Type u} [PartialOrder α] [OrderTop α] {a : α} (h : a ≠ ⊤) :

a < ⊤

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theorem Ne.lt_top' {α : Type u} [PartialOrder α] [OrderTop α] {a : α} (h : ⊤ ≠ a) :

a < ⊤

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theorem ne_top_of_le_ne_top {α : Type u} [PartialOrder α] [OrderTop α] {a b : α} (hb : b ≠ ⊤) (hab : a ≤ b) :

a ≠ ⊤

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theorem top_not_mem_iff {α : Type u} [PartialOrder α] [OrderTop α] {s : Set α} :

¬⊤ ∈ s ↔ ∀ (x : α), x ∈ s → x < ⊤

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theorem not_isMin_top {α : Type u} [PartialOrder α] [OrderTop α] [Nontrivial α] :

¬IsMin ⊤

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theorem OrderTop.ext_top {α : Type u_1} {hA : PartialOrder α} (A : OrderTop α) {hB : PartialOrder α} (B : OrderTop α) (H : ∀ (x y : α), x ≤ y ↔ x ≤ y) :

⊤ = ⊤

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class OrderBot (α : Type u) [LE α] extends Bot α :

Type u

An order is an OrderBot if it has a least element. We state this using a data mixin, holding the value of ⊥ and the least element constraint.

bot : α
bot_le : ∀ (a : α), ⊥ ≤ a
⊥ is the least element

Instances

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noncomputable def botOrderOrNoBotOrder (α : Type u_1) [LE α] :

OrderBot α ⊕' NoBotOrder α

An order is (noncomputably) either an OrderBot or a NoBotOrder. Use as casesI botOrderOrNoBotOrder α.

Equations

botOrderOrNoBotOrder α = if H : ∀ (a : α), ∃ (b : α), ¬a ≤ b then PSum.inr ⋯ else PSum.inl (OrderBot.mk ⋯)

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@[simp]

theorem bot_le {α : Type u} [LE α] [OrderBot α] {a : α} :

⊥ ≤ a

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@[simp]

theorem isBot_bot {α : Type u} [LE α] [OrderBot α] :

IsBot ⊥

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instance OrderDual.instTop (α : Type u) [Bot α] :

Top αᵒᵈ

Equations

OrderDual.instTop α = { top := ⊥ }

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instance OrderDual.instBot (α : Type u) [Top α] :

Bot αᵒᵈ

Equations

OrderDual.instBot α = { bot := ⊤ }

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instance OrderDual.instOrderTop (α : Type u) [LE α] [OrderBot α] :

OrderTop αᵒᵈ

Equations

OrderDual.instOrderTop α = OrderTop.mk ⋯

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instance OrderDual.instOrderBot (α : Type u) [LE α] [OrderTop α] :

OrderBot αᵒᵈ

Equations

OrderDual.instOrderBot α = OrderBot.mk ⋯

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@[simp]

theorem OrderDual.ofDual_bot (α : Type u) [Top α] :

OrderDual.ofDual ⊥ = ⊤

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@[simp]

theorem OrderDual.ofDual_top (α : Type u) [Bot α] :

OrderDual.ofDual ⊤ = ⊥

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@[simp]

theorem OrderDual.toDual_bot (α : Type u) [Bot α] :

OrderDual.toDual ⊥ = ⊤

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@[simp]

theorem OrderDual.toDual_top (α : Type u) [Top α] :

OrderDual.toDual ⊤ = ⊥

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@[simp]

theorem isMin_bot {α : Type u} [Preorder α] [OrderBot α] :

IsMin ⊥

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@[simp]

theorem not_lt_bot {α : Type u} [Preorder α] [OrderBot α] {a : α} :

¬a < ⊥

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theorem ne_bot_of_gt {α : Type u} [Preorder α] [OrderBot α] {a b : α} (h : a < b) :

b ≠ ⊥

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theorem LT.lt.ne_bot {α : Type u} [Preorder α] [OrderBot α] {a b : α} (h : a < b) :

b ≠ ⊥

Alias of ne_bot_of_gt.

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theorem bot_lt_of_lt {α : Type u} [Preorder α] [OrderBot α] {a b : α} (h : a < b) :

⊥ < b

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theorem LT.lt.bot_lt {α : Type u} [Preorder α] [OrderBot α] {a b : α} (h : a < b) :

⊥ < b

Alias of bot_lt_of_lt.

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@[simp]

theorem isMin_iff_eq_bot {α : Type u} [PartialOrder α] [OrderBot α] {a : α} :

IsMin a ↔ a = ⊥

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@[simp]

theorem isBot_iff_eq_bot {α : Type u} [PartialOrder α] [OrderBot α] {a : α} :

IsBot a ↔ a = ⊥

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theorem not_isMin_iff_ne_bot {α : Type u} [PartialOrder α] [OrderBot α] {a : α} :

¬IsMin a ↔ a ≠ ⊥

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theorem not_isBot_iff_ne_bot {α : Type u} [PartialOrder α] [OrderBot α] {a : α} :

¬IsBot a ↔ a ≠ ⊥

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theorem IsMin.eq_bot {α : Type u} [PartialOrder α] [OrderBot α] {a : α} :

IsMin a → a = ⊥

Alias of the forward direction of isMin_iff_eq_bot.

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theorem IsBot.eq_bot {α : Type u} [PartialOrder α] [OrderBot α] {a : α} :

IsBot a → a = ⊥

Alias of the forward direction of isBot_iff_eq_bot.

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@[simp]

theorem le_bot_iff {α : Type u} [PartialOrder α] [OrderBot α] {a : α} :

a ≤ ⊥ ↔ a = ⊥

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theorem bot_unique {α : Type u} [PartialOrder α] [OrderBot α] {a : α} (h : a ≤ ⊥) :

a = ⊥

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theorem eq_bot_iff {α : Type u} [PartialOrder α] [OrderBot α] {a : α} :

a = ⊥ ↔ a ≤ ⊥

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theorem eq_bot_mono {α : Type u} [PartialOrder α] [OrderBot α] {a b : α} (h : a ≤ b) (h₂ : b = ⊥) :

a = ⊥

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theorem bot_lt_iff_ne_bot {α : Type u} [PartialOrder α] [OrderBot α] {a : α} :

⊥ < a ↔ a ≠ ⊥

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@[simp]

theorem not_bot_lt_iff {α : Type u} [PartialOrder α] [OrderBot α] {a : α} :

¬⊥ < a ↔ a = ⊥

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theorem eq_bot_or_bot_lt {α : Type u} [PartialOrder α] [OrderBot α] (a : α) :

a = ⊥ ∨ ⊥ < a

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theorem eq_bot_of_minimal {α : Type u} [PartialOrder α] [OrderBot α] {a : α} (h : ∀ (b : α), ¬b < a) :

a = ⊥

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theorem Ne.bot_lt {α : Type u} [PartialOrder α] [OrderBot α] {a : α} (h : a ≠ ⊥) :

⊥ < a

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theorem Ne.bot_lt' {α : Type u} [PartialOrder α] [OrderBot α] {a : α} (h : ⊥ ≠ a) :

⊥ < a

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theorem ne_bot_of_le_ne_bot {α : Type u} [PartialOrder α] [OrderBot α] {a b : α} (hb : b ≠ ⊥) (hab : b ≤ a) :

a ≠ ⊥

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theorem bot_not_mem_iff {α : Type u} [PartialOrder α] [OrderBot α] {s : Set α} :

¬⊥ ∈ s ↔ ∀ (x : α), x ∈ s → ⊥ < x

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theorem not_isMax_bot {α : Type u} [PartialOrder α] [OrderBot α] [Nontrivial α] :

¬IsMax ⊥

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theorem OrderBot.ext_bot {α : Type u_1} {hA : PartialOrder α} (A : OrderBot α) {hB : PartialOrder α} (B : OrderBot α) (H : ∀ (x y : α), x ≤ y ↔ x ≤ y) :

⊥ = ⊥

Bounded order #

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class BoundedOrder (α : Type u) [LE α] extends OrderTop α, OrderBot α :

Type u

A bounded order describes an order (≤) with a top and bottom element, denoted ⊤ and ⊥ respectively.

top : α
le_top : ∀ (a : α), a ≤ ⊤
bot : α
bot_le : ∀ (a : α), ⊥ ≤ a

Instances

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instance OrderDual.instBoundedOrder (α : Type u) [LE α] [BoundedOrder α] :

BoundedOrder αᵒᵈ

Equations

OrderDual.instBoundedOrder α = BoundedOrder.mk

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theorem OrderBot.instSubsingleton {α : Type u} [PartialOrder α] :

Subsingleton (OrderBot α)

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theorem OrderTop.instSubsingleton {α : Type u} [PartialOrder α] :

Subsingleton (OrderTop α)

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theorem BoundedOrder.instSubsingleton {α : Type u} [PartialOrder α] :

Subsingleton (BoundedOrder α)

Function lattices #

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instance Pi.instBotForall {ι : Type u_1} {α' : ι → Type u_2} [(i : ι) → Bot (α' i)] :

Bot ((i : ι) → α' i)

Equations

Pi.instBotForall = { bot := fun (x : ι) => ⊥ }

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@[simp]

theorem Pi.bot_apply {ι : Type u_1} {α' : ι → Type u_2} [(i : ι) → Bot (α' i)] (i : ι) :

⊥ i = ⊥

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theorem Pi.bot_def {ι : Type u_1} {α' : ι → Type u_2} [(i : ι) → Bot (α' i)] :

⊥ = fun (x : ι) => ⊥

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instance Pi.instTopForall {ι : Type u_1} {α' : ι → Type u_2} [(i : ι) → Top (α' i)] :

Top ((i : ι) → α' i)

Equations

Pi.instTopForall = { top := fun (x : ι) => ⊤ }

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@[simp]

theorem Pi.top_apply {ι : Type u_1} {α' : ι → Type u_2} [(i : ι) → Top (α' i)] (i : ι) :

⊤ i = ⊤

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theorem Pi.top_def {ι : Type u_1} {α' : ι → Type u_2} [(i : ι) → Top (α' i)] :

⊤ = fun (x : ι) => ⊤

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instance Pi.instOrderTop {ι : Type u_1} {α' : ι → Type u_2} [(i : ι) → LE (α' i)] [(i : ι) → OrderTop (α' i)] :

OrderTop ((i : ι) → α' i)

Equations

Pi.instOrderTop = OrderTop.mk ⋯

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instance Pi.instOrderBot {ι : Type u_1} {α' : ι → Type u_2} [(i : ι) → LE (α' i)] [(i : ι) → OrderBot (α' i)] :

OrderBot ((i : ι) → α' i)

Equations

Pi.instOrderBot = OrderBot.mk ⋯

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instance Pi.instBoundedOrder {ι : Type u_1} {α' : ι → Type u_2} [(i : ι) → LE (α' i)] [(i : ι) → BoundedOrder (α' i)] :

BoundedOrder ((i : ι) → α' i)

Equations

Pi.instBoundedOrder = BoundedOrder.mk

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theorem eq_bot_of_bot_eq_top {α : Type u} [PartialOrder α] [BoundedOrder α] (hα : ⊥ = ⊤) (x : α) :

x = ⊥

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theorem eq_top_of_bot_eq_top {α : Type u} [PartialOrder α] [BoundedOrder α] (hα : ⊥ = ⊤) (x : α) :

x = ⊤

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theorem subsingleton_of_top_le_bot {α : Type u} [PartialOrder α] [BoundedOrder α] (h : ⊤ ≤ ⊥) :

Subsingleton α

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theorem subsingleton_of_bot_eq_top {α : Type u} [PartialOrder α] [BoundedOrder α] (hα : ⊥ = ⊤) :

Subsingleton α

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theorem subsingleton_iff_bot_eq_top {α : Type u} [PartialOrder α] [BoundedOrder α] :

⊥ = ⊤ ↔ Subsingleton α

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@[reducible, inline]

abbrev OrderTop.lift {α : Type u} {β : Type v} [LE α] [Top α] [LE β] [OrderTop β] (f : α → β) (map_le : ∀ (a b : α), f a ≤ f b → a ≤ b) (map_top : f ⊤ = ⊤) :

OrderTop α

Pullback an OrderTop.

Equations

OrderTop.lift f map_le map_top = OrderTop.mk ⋯

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@[reducible, inline]

abbrev OrderBot.lift {α : Type u} {β : Type v} [LE α] [Bot α] [LE β] [OrderBot β] (f : α → β) (map_le : ∀ (a b : α), f a ≤ f b → a ≤ b) (map_bot : f ⊥ = ⊥) :

OrderBot α

Pullback an OrderBot.

Equations

OrderBot.lift f map_le map_bot = OrderBot.mk ⋯

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@[reducible, inline]

abbrev BoundedOrder.lift {α : Type u} {β : Type v} [LE α] [Top α] [Bot α] [LE β] [BoundedOrder β] (f : α → β) (map_le : ∀ (a b : α), f a ≤ f b → a ≤ b) (map_top : f ⊤ = ⊤) (map_bot : f ⊥ = ⊥) :

BoundedOrder α

Pullback a BoundedOrder.

Equations