WithBot, WithTop

Adding a bot or a top to an order.

Main declarations

• With<Top/Bot> α: Equips Option α with the order on α plus none as the top/bottom element.

theorem WithBot.nontrivial {α : Type u_1} [Nonempty α] : Nontrivial (WithBot α)

theorem WithBot.coe_injective {α : Type u_1} : Function.Injective WithBot.some

@[simp]

theorem WithBot.coe_inj {α : Type u_1} {a b : α} : ↑a = ↑b ↔ a = b

theorem WithBot.forall {α : Type u_1} {p : WithBot α → Prop} : (∀ (x : WithBot α), p x) ↔ p ⊥ ∧ ∀ (x : α), p ↑x

theorem WithBot.exists {α : Type u_1} {p : WithBot α → Prop} : (∃ (x : WithBot α), p x) ↔ p ⊥ ∨ ∃ (x : α), p ↑x

theorem WithBot.none_eq_bot {α : Type u_1} : none = ⊥

theorem WithBot.some_eq_coe {α : Type u_1} (a : α) : some a = ↑a

@[simp]

theorem WithBot.bot_ne_coe {α : Type u_1} {a : α} : ⊥ ≠ ↑a

@[simp]

theorem WithBot.coe_ne_bot {α : Type u_1} {a : α} : ↑a ≠ ⊥

def WithBot.unbot' {α : Type u_1} (d : α) (x : WithBot α) : α

Specialization of Option.getD to values in WithBot α that respects API boundaries.

Equations

• WithBot.unbot' d x = WithBot.recBotCoe d id x

@[simp]

theorem WithBot.unbot'_bot {α : Type u_5} (d : α) : WithBot.unbot' d ⊥ = d

@[simp]

theorem WithBot.unbot'_coe {α : Type u_5} (d x : α) : WithBot.unbot' d ↑x = x

theorem WithBot.coe_eq_coe {α : Type u_1} {a b : α} : ↑a = ↑b ↔ a = b

theorem WithBot.unbot'_eq_iff {α : Type u_1} {d y : α} {x : WithBot α} : WithBot.unbot' d x = y ↔ x = ↑y ∨ x = ⊥ ∧ y = d

@[simp]

theorem WithBot.unbot'_eq_self_iff {α : Type u_1} {d : α} {x : WithBot α} : WithBot.unbot' d x = d ↔ x = ↑d ∨ x = ⊥

theorem WithBot.unbot'_eq_unbot'_iff {α : Type u_1} {d : α} {x y : WithBot α} : WithBot.unbot' d x = WithBot.unbot' d y ↔ x = y ∨ x = ↑d ∧ y = ⊥ ∨ x = ⊥ ∧ y = ↑d

def WithBot.map {α : Type u_1} {β : Type u_2} (f : α → β) : WithBot α → WithBot β

Lift a map f : α → β to WithBot α → WithBot β. Implemented using Option.map.

Equations

• WithBot.map f = Option.map f

@[simp]

theorem WithBot.map_bot {α : Type u_1} {β : Type u_2} (f : α → β) : WithBot.map f ⊥ = ⊥

@[simp]

theorem WithBot.map_coe {α : Type u_1} {β : Type u_2} (f : α → β) (a : α) : WithBot.map f ↑a = ↑(f a)

@[simp]

theorem WithBot.map_eq_bot_iff {α : Type u_1} {β : Type u_2} {f : α → β} {a : WithBot α} : WithBot.map f a = ⊥ ↔ a = ⊥

theorem WithBot.map_comm {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {f₁ : α → β} {f₂ : α → γ} {g₁ : β → δ} {g₂ : γ → δ} (h : g₁ ∘ f₁ = g₂ ∘ f₂) (a : α) : WithBot.map g₁ (WithBot.map f₁ ↑a) = WithBot.map g₂ (WithBot.map f₂ ↑a)

def WithBot.map₂ {α : Type u_1} {β : Type u_2} {γ : Type u_3} : (α → β → γ) → WithBot α → WithBot β → WithBot γ

The image of a binary function f : α → β → γ as a function WithBot α → WithBot β → WithBot γ.

Mathematically this should be thought of as the image of the corresponding function α × β → γ.

Equations

• WithBot.map₂ = Option.map₂

theorem WithBot.map₂_coe_coe {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β → γ) (a : α) (b : β) : WithBot.map₂ f ↑a ↑b = ↑(f a b)

@[simp]

theorem WithBot.map₂_bot_left {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β → γ) (b : WithBot β) : WithBot.map₂ f ⊥ b = ⊥

@[simp]

theorem WithBot.map₂_bot_right {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β → γ) (a : WithBot α) : WithBot.map₂ f a ⊥ = ⊥

@[simp]

theorem WithBot.map₂_coe_left {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β → γ) (a : α) (b : WithBot β) : WithBot.map₂ f (↑a) b = WithBot.map (fun (b : β) => f a b) b

@[simp]

theorem WithBot.map₂_coe_right {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β → γ) (a : WithBot α) (b : β) : WithBot.map₂ f a ↑b = WithBot.map (fun (x : α) => f x b) a

@[simp]

theorem WithBot.map₂_eq_bot_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β → γ} {a : WithBot α} {b : WithBot β} : WithBot.map₂ f a b = ⊥ ↔ a = ⊥ ∨ b = ⊥

theorem WithBot.ne_bot_iff_exists {α : Type u_1} {x : WithBot α} : x ≠ ⊥ ↔ ∃ (a : α), ↑a = x

theorem WithBot.forall_ne_iff_eq_bot {α : Type u_1} {x : WithBot α} : (∀ (a : α), ↑a ≠ x) ↔ x = ⊥

def WithBot.unbot {α : Type u_1} (x : WithBot α) : x ≠ ⊥ → α

Deconstruct a x : WithBot α to the underlying value in α, given a proof that x ≠ ⊥.

Equations

• WithBot.unbot (some x_2) x_3 = x_2

@[simp]

theorem WithBot.coe_unbot {α : Type u_1} (x : WithBot α) (hx : x ≠ ⊥) : ↑(x.unbot hx) = x

@[simp]

theorem WithBot.unbot_coe {α : Type u_1} (x : α) (h : ↑x ≠ ⊥ := ⋯) : (↑x).unbot h = x

theorem WithBot.canLift {α : Type u_1} : CanLift (WithBot α) α WithBot.some fun (r : WithBot α) => r ≠ ⊥

instance WithBot.instTop {α : Type u_1} [Top α] : Top (WithBot α)

Equations

• WithBot.instTop = { top := ↑⊤ }

@[simp]

theorem WithBot.coe_top {α : Type u_1} [Top α] : ↑⊤ = ⊤

@[simp]

theorem WithBot.coe_eq_top {α : Type u_1} [Top α] {a : α} : ↑a = ⊤ ↔ a = ⊤

@[simp]

theorem WithBot.top_eq_coe {α : Type u_1} [Top α] {a : α} : ⊤ = ↑a ↔ ⊤ = a

theorem WithBot.unbot_eq_iff {α : Type u_1} {a : WithBot α} {b : α} (h : a ≠ ⊥) : a.unbot h = b ↔ a = ↑b

theorem WithBot.eq_unbot_iff {α : Type u_1} {a : α} {b : WithBot α} (h : b ≠ ⊥) : a = b.unbot h ↔ ↑a = b

def Equiv.withBotSubtypeNe {α : Type u_1} : { y : WithBot α // y ≠ ⊥ } ≃ α

The equivalence between the non-bottom elements of WithBot α and α.

Equations

• Equiv.withBotSubtypeNe = { toFun := fun (x : { y : WithBot α // y ≠ ⊥ }) => match x with | ⟨x, h⟩ => x.unbot h, invFun := fun (x : α) => ⟨↑x, ⋯⟩, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem Equiv.withBotSubtypeNe_symm_apply_coe {α : Type u_1} (x : α) : ↑(Equiv.withBotSubtypeNe.symm x) = ↑x

@[simp]

theorem Equiv.withBotSubtypeNe_apply {α : Type u_1} (x✝ : { y : WithBot α // y ≠ ⊥ }) : Equiv.withBotSubtypeNe x✝ = match x✝ with | ⟨x, h⟩ => x.unbot h

@[instance 10]

instance WithBot.le {α : Type u_1} [LE α] : LE (WithBot α)

Equations

• WithBot.le = { le := fun (o₁ o₂ : WithBot α) => ∀ (a : α), o₁ = ↑a → ∃ (b : α), o₂ = ↑b ∧ a ≤ b }

@[simp]

theorem WithBot.coe_le_coe {α : Type u_1} {a b : α} [LE α] : ↑a ≤ ↑b ↔ a ≤ b

instance WithBot.orderBot {α : Type u_1} [LE α] : OrderBot (WithBot α)

Equations

• WithBot.orderBot = OrderBot.mk ⋯

@[simp, deprecated WithBot.coe_le_coe]

theorem WithBot.some_le_some {α : Type u_1} {a b : α} [LE α] : some a ≤ some b ↔ a ≤ b

@[simp, deprecated bot_le]

theorem WithBot.none_le {α : Type u_1} [LE α] {a : WithBot α} : none ≤ a

instance WithBot.orderTop {α : Type u_1} [LE α] [OrderTop α] : OrderTop (WithBot α)

Equations

• WithBot.orderTop = OrderTop.mk ⋯

instance WithBot.instBoundedOrder {α : Type u_1} [LE α] [OrderTop α] : BoundedOrder (WithBot α)

Equations

• WithBot.instBoundedOrder = BoundedOrder.mk

theorem WithBot.not_coe_le_bot {α : Type u_1} [LE α] (a : α) : ¬↑a ≤ ⊥

@[simp]

theorem WithBot.le_bot_iff {α : Type u_1} [LE α] {a : WithBot α} : a ≤ ⊥ ↔ a = ⊥

There is a general version le_bot_iff, but this lemma does not require a PartialOrder.

theorem WithBot.coe_le {α : Type u_1} {a b : α} [LE α] {o : Option α} : b ∈ o → (↑a ≤ o ↔ a ≤ b)

theorem WithBot.coe_le_iff {α : Type u_1} {a : α} [LE α] {x : WithBot α} : ↑a ≤ x ↔ ∃ (b : α), x = ↑b ∧ a ≤ b

theorem WithBot.le_coe_iff {α : Type u_1} {b : α} [LE α] {x : WithBot α} : x ≤ ↑b ↔ ∀ (a : α), x = ↑a → a ≤ b

theorem IsMax.withBot {α : Type u_1} {a : α} [LE α] (h : IsMax a) : IsMax ↑a

theorem WithBot.le_unbot_iff {α : Type u_1} [LE α] {a : α} {b : WithBot α} (h : b ≠ ⊥) : a ≤ b.unbot h ↔ ↑a ≤ b

theorem WithBot.unbot_le_iff {α : Type u_1} [LE α] {a : WithBot α} (h : a ≠ ⊥) {b : α} : a.unbot h ≤ b ↔ a ≤ ↑b

theorem WithBot.unbot'_le_iff {α : Type u_1} [LE α] {a : WithBot α} {b c : α} (h : a = ⊥ → b ≤ c) : WithBot.unbot' b a ≤ c ↔ a ≤ ↑c

@[instance 10]

instance WithBot.lt {α : Type u_1} [LT α] : LT (WithBot α)

Equations

• WithBot.lt = { lt := fun (o₁ o₂ : WithBot α) => ∃ (b : α), o₂ = ↑b ∧ ∀ (a : α), o₁ = ↑a → a < b }

@[simp]

theorem WithBot.coe_lt_coe {α : Type u_1} {a b : α} [LT α] : ↑a < ↑b ↔ a < b

@[simp]

theorem WithBot.bot_lt_coe {α : Type u_1} [LT α] (a : α) : ⊥ < ↑a

@[simp]

theorem WithBot.not_lt_bot {α : Type u_1} [LT α] (a : WithBot α) : ¬a < ⊥

@[simp, deprecated WithBot.coe_lt_coe]

theorem WithBot.some_lt_some {α : Type u_1} {a b : α} [LT α] : some a < some b ↔ a < b

@[simp, deprecated WithBot.bot_lt_coe]

theorem WithBot.none_lt_some {α : Type u_1} [LT α] (a : α) : none < ↑a

@[simp, deprecated not_lt_bot]

theorem WithBot.not_lt_none {α : Type u_1} [LT α] (a : WithBot α) : ¬a < none

theorem WithBot.lt_iff_exists_coe {α : Type u_1} [LT α] {a b : WithBot α} : a < b ↔ ∃ (p : α), b = ↑p ∧ a < ↑p

theorem WithBot.lt_coe_iff {α : Type u_1} {b : α} [LT α] {x : WithBot α} : x < ↑b ↔ ∀ (a : α), x = ↑a → a < b

theorem WithBot.bot_lt_iff_ne_bot {α : Type u_1} [LT α] {x : WithBot α} : ⊥ < x ↔ x ≠ ⊥

A version of bot_lt_iff_ne_bot for WithBot that only requires LT α, not PartialOrder α.

theorem WithBot.lt_unbot_iff {α : Type u_1} [LT α] {a : α} {b : WithBot α} (h : b ≠ ⊥) : a < b.unbot h ↔ ↑a < b

theorem WithBot.unbot_lt_iff {α : Type u_1} [LT α] {a : WithBot α} (h : a ≠ ⊥) {b : α} : a.unbot h < b ↔ a < ↑b

theorem WithBot.unbot'_lt_iff {α : Type u_1} [LT α] {a : WithBot α} {b c : α} (h : a = ⊥ → b < c) : WithBot.unbot' b a < c ↔ a < ↑c

instance WithBot.preorder {α : Type u_1} [Preorder α] : Preorder (WithBot α)

Equations

• WithBot.preorder = Preorder.mk ⋯ ⋯ ⋯

instance WithBot.partialOrder {α : Type u_1} [PartialOrder α] : PartialOrder (WithBot α)

Equations

• WithBot.partialOrder = PartialOrder.mk ⋯

theorem WithBot.coe_strictMono {α : Type u_1} [Preorder α] : StrictMono fun (a : α) => ↑a

theorem WithBot.coe_mono {α : Type u_1} [Preorder α] : Monotone fun (a : α) => ↑a

theorem WithBot.monotone_iff {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : WithBot α → β} : Monotone f ↔ (Monotone fun (a : α) => f ↑a) ∧ ∀ (x : α), f ⊥ ≤ f ↑x

@[simp]

theorem WithBot.monotone_map_iff {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} : Monotone (WithBot.map f) ↔ Monotone f

theorem Monotone.withBot_map {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} : Monotone f → Monotone (WithBot.map f)

Alias of the reverse direction of WithBot.monotone_map_iff.

theorem WithBot.strictMono_iff {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : WithBot α → β} : StrictMono f ↔ (StrictMono fun (a : α) => f ↑a) ∧ ∀ (x : α), f ⊥ < f ↑x

theorem WithBot.strictAnti_iff {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : WithBot α → β} : StrictAnti f ↔ (StrictAnti fun (a : α) => f ↑a) ∧ ∀ (x : α), f ↑x < f ⊥

@[simp]

theorem WithBot.strictMono_map_iff {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} : StrictMono (WithBot.map f) ↔ StrictMono f

theorem StrictMono.withBot_map {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} : StrictMono f → StrictMono (WithBot.map f)

Alias of the reverse direction of WithBot.strictMono_map_iff.

theorem WithBot.map_le_iff {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α → β) (mono_iff : ∀ {a b : α}, f a ≤ f b ↔ a ≤ b) (a b : WithBot α) : WithBot.map f a ≤ WithBot.map f b ↔ a ≤ b

theorem WithBot.le_coe_unbot' {α : Type u_1} [Preorder α] (a : WithBot α) (b : α) : a ≤ ↑(WithBot.unbot' b a)

@[simp]

theorem WithBot.lt_coe_bot {α : Type u_1} [Preorder α] [OrderBot α] {x : WithBot α} : x < ↑⊥ ↔ x = ⊥

instance WithBot.semilatticeSup {α : Type u_1} [SemilatticeSup α] : SemilatticeSup (WithBot α)

Equations

• One or more equations did not get rendered due to their size.

theorem WithBot.coe_sup {α : Type u_1} [SemilatticeSup α] (a b : α) : ↑(a ⊔ b) = ↑a ⊔ ↑b

instance WithBot.semilatticeInf {α : Type u_1} [SemilatticeInf α] : SemilatticeInf (WithBot α)

Equations

• WithBot.semilatticeInf = SemilatticeInf.mk (WithBot.map₂ fun (x1 x2 : α) => x1 ⊓ x2) ⋯ ⋯ ⋯

theorem WithBot.coe_inf {α : Type u_1} [SemilatticeInf α] (a b : α) : ↑(a ⊓ b) = ↑a ⊓ ↑b

instance WithBot.lattice {α : Type u_1} [Lattice α] : Lattice (WithBot α)

Equations

• WithBot.lattice = Lattice.mk SemilatticeInf.inf ⋯ ⋯ ⋯

instance WithBot.distribLattice {α : Type u_1} [DistribLattice α] : DistribLattice (WithBot α)

Equations

• WithBot.distribLattice = DistribLattice.mk ⋯

instance WithBot.decidableEq {α : Type u_1} [DecidableEq α] : DecidableEq (WithBot α)

Equations

• WithBot.decidableEq = inferInstanceAs (DecidableEq (Option α))

instance WithBot.decidableLE {α : Type u_1} [LE α] [DecidableRel fun (x1 x2 : α) => x1 ≤ x2] : DecidableRel fun (x1 x2 : WithBot α) => x1 ≤ x2

Equations

• WithBot.decidableLE none x = isTrue ⋯
• WithBot.decidableLE (some x_2) (some y) = if h : x_2 ≤ y then isTrue ⋯ else isFalse ⋯
• WithBot.decidableLE (some x_2) none = isFalse ⋯

instance WithBot.decidableLT {α : Type u_1} [LT α] [DecidableRel fun (x1 x2 : α) => x1 < x2] : DecidableRel fun (x1 x2 : WithBot α) => x1 < x2

Equations

• WithBot.decidableLT none (some x_2) = isTrue ⋯
• WithBot.decidableLT (some x_2) (some y) = if h : x_2 < y then isTrue ⋯ else isFalse ⋯
• x.decidableLT none = isFalse ⋯

theorem WithBot.isTotal_le {α : Type u_1} [LE α] [IsTotal α fun (x1 x2 : α) => x1 ≤ x2] : IsTotal (WithBot α) fun (x1 x2 : WithBot α) => x1 ≤ x2

instance WithBot.linearOrder {α : Type u_1} [LinearOrder α] : LinearOrder (WithBot α)

Equations

• WithBot.linearOrder = Lattice.toLinearOrder (WithBot α)

@[simp]

theorem WithBot.coe_min {α : Type u_1} [LinearOrder α] (x y : α) : ↑(x ⊓ y) = ↑x ⊓ ↑y

@[simp]

theorem WithBot.coe_max {α : Type u_1} [LinearOrder α] (x y : α) : ↑(x ⊔ y) = ↑x ⊔ ↑y

theorem WithBot.instWellFoundedLT {α : Type u_1} [LT α] [WellFoundedLT α] : WellFoundedLT (WithBot α)

theorem WithBot.instWellFoundedGT {α : Type u_1} [LT α] [WellFoundedGT α] : WellFoundedGT (WithBot α)

theorem WithBot.denselyOrdered {α : Type u_1} [LT α] [DenselyOrdered α] [NoMinOrder α] : DenselyOrdered (WithBot α)

theorem WithBot.lt_iff_exists_coe_btwn {α : Type u_1} [Preorder α] [DenselyOrdered α] [NoMinOrder α] {a b : WithBot α} : a < b ↔ ∃ (x : α), a < ↑x ∧ ↑x < b

theorem WithBot.noTopOrder {α : Type u_1} [LE α] [NoTopOrder α] [Nonempty α] : NoTopOrder (WithBot α)

theorem WithBot.noMaxOrder {α : Type u_1} [LT α] [NoMaxOrder α] [Nonempty α] : NoMaxOrder (WithBot α)

theorem WithTop.nontrivial {α : Type u_1} [Nonempty α] : Nontrivial (WithTop α)

theorem WithTop.coe_injective {α : Type u_1} : Function.Injective WithTop.some

theorem WithTop.coe_inj {α : Type u_1} {a b : α} : ↑a = ↑b ↔ a = b

theorem WithTop.forall {α : Type u_1} {p : WithTop α → Prop} : (∀ (x : WithTop α), p x) ↔ p ⊤ ∧ ∀ (x : α), p ↑x

theorem WithTop.exists {α : Type u_1} {p : WithTop α → Prop} : (∃ (x : WithTop α), p x) ↔ p ⊤ ∨ ∃ (x : α), p ↑x

theorem WithTop.none_eq_top {α : Type u_1} : none = ⊤

theorem WithTop.some_eq_coe {α : Type u_1} (a : α) : some a = ↑a

@[simp]

theorem WithTop.top_ne_coe {α : Type u_1} {a : α} : ⊤ ≠ ↑a

@[simp]

theorem WithTop.coe_ne_top {α : Type u_1} {a : α} : ↑a ≠ ⊤

def WithTop.toDual {α : Type u_1} : WithTop α ≃ WithBot αᵒᵈ

WithTop.toDual is the equivalence sending ⊤ to ⊥ and any a : α to toDual a : αᵒᵈ. See WithTop.toDualBotEquiv for the related order-iso.

Equations

• WithTop.toDual = Equiv.refl (WithTop α)

def WithTop.ofDual {α : Type u_1} : WithTop αᵒᵈ ≃ WithBot α

WithTop.ofDual is the equivalence sending ⊤ to ⊥ and any a : αᵒᵈ to ofDual a : α. See WithTop.toDualBotEquiv for the related order-iso.

Equations

• WithTop.ofDual = Equiv.refl (WithTop αᵒᵈ)

def WithBot.toDual {α : Type u_1} : WithBot α ≃ WithTop αᵒᵈ

WithBot.toDual is the equivalence sending ⊥ to ⊤ and any a : α to toDual a : αᵒᵈ. See WithBot.toDual_top_equiv for the related order-iso.

Equations

• WithBot.toDual = Equiv.refl (WithBot α)

def WithBot.ofDual {α : Type u_1} : WithBot αᵒᵈ ≃ WithTop α

WithBot.ofDual is the equivalence sending ⊥ to ⊤ and any a : αᵒᵈ to ofDual a : α. See WithBot.ofDual_top_equiv for the related order-iso.

Equations

• WithBot.ofDual = Equiv.refl (WithBot αᵒᵈ)

@[simp]

theorem WithTop.toDual_symm_apply {α : Type u_1} (a : WithBot αᵒᵈ) : WithTop.toDual.symm a = WithBot.ofDual a

@[simp]

theorem WithTop.ofDual_symm_apply {α : Type u_1} (a : WithBot α) : WithTop.ofDual.symm a = WithBot.toDual a

@[simp]

theorem WithTop.toDual_apply_top {α : Type u_1} : WithTop.toDual ⊤ = ⊥

@[simp]

theorem WithTop.ofDual_apply_top {α : Type u_1} : WithTop.ofDual ⊤ = ⊥

@[simp]

theorem WithTop.toDual_apply_coe {α : Type u_1} (a : α) : WithTop.toDual ↑a = ↑(OrderDual.toDual a)

@[simp]

theorem WithTop.ofDual_apply_coe {α : Type u_1} (a : αᵒᵈ) : WithTop.ofDual ↑a = ↑(OrderDual.ofDual a)

def WithTop.untop' {α : Type u_1} (d : α) (x : WithTop α) : α

Specialization of Option.getD to values in WithTop α that respects API boundaries.

Equations

• WithTop.untop' d x = WithTop.recTopCoe d id x

@[simp]

theorem WithTop.untop'_top {α : Type u_5} (d : α) : WithTop.untop' d ⊤ = d

@[simp]

theorem WithTop.untop'_coe {α : Type u_5} (d x : α) : WithTop.untop' d ↑x = x

@[simp]

theorem WithTop.coe_eq_coe {α : Type u_1} {a b : α} : ↑a = ↑b ↔ a = b

theorem WithTop.untop'_eq_iff {α : Type u_1} {d y : α} {x : WithTop α} : WithTop.untop' d x = y ↔ x = ↑y ∨ x = ⊤ ∧ y = d

@[simp]

theorem WithTop.untop'_eq_self_iff {α : Type u_1} {d : α} {x : WithTop α} : WithTop.untop' d x = d ↔ x = ↑d ∨ x = ⊤

theorem WithTop.untop'_eq_untop'_iff {α : Type u_1} {d : α} {x y : WithTop α} : WithTop.untop' d x = WithTop.untop' d y ↔ x = y ∨ x = ↑d ∧ y = ⊤ ∨ x = ⊤ ∧ y = ↑d

def WithTop.map {α : Type u_1} {β : Type u_2} (f : α → β) : WithTop α → WithTop β

Lift a map f : α → β to WithTop α → WithTop β. Implemented using Option.map.

Equations

• WithTop.map f = Option.map f

@[simp]

theorem WithTop.map_top {α : Type u_1} {β : Type u_2} (f : α → β) : WithTop.map f ⊤ = ⊤

@[simp]

theorem WithTop.map_coe {α : Type u_1} {β : Type u_2} (f : α → β) (a : α) : WithTop.map f ↑a = ↑(f a)

@[simp]

theorem WithTop.map_eq_top_iff {α : Type u_1} {β : Type u_2} {f : α → β} {a : WithTop α} : WithTop.map f a = ⊤ ↔ a = ⊤

theorem WithTop.map_comm {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {f₁ : α → β} {f₂ : α → γ} {g₁ : β → δ} {g₂ : γ → δ} (h : g₁ ∘ f₁ = g₂ ∘ f₂) (a : α) : WithTop.map g₁ (WithTop.map f₁ ↑a) = WithTop.map g₂ (WithTop.map f₂ ↑a)

def WithTop.map₂ {α : Type u_1} {β : Type u_2} {γ : Type u_3} : (α → β → γ) → WithTop α → WithTop β → WithTop γ

The image of a binary function f : α → β → γ as a function WithTop α → WithTop β → WithTop γ.

Mathematically this should be thought of as the image of the corresponding function α × β → γ.

Equations

• WithTop.map₂ = Option.map₂

theorem WithTop.map₂_coe_coe {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β → γ) (a : α) (b : β) : WithTop.map₂ f ↑a ↑b = ↑(f a b)

@[simp]

theorem WithTop.map₂_top_left {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β → γ) (b : WithTop β) : WithTop.map₂ f ⊤ b = ⊤

@[simp]

theorem WithTop.map₂_top_right {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β → γ) (a : WithTop α) : WithTop.map₂ f a ⊤ = ⊤

@[simp]

theorem WithTop.map₂_coe_left {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β → γ) (a : α) (b : WithTop β) : WithTop.map₂ f (↑a) b = WithTop.map (fun (b : β) => f a b) b

@[simp]

theorem WithTop.map₂_coe_right {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β → γ) (a : WithTop α) (b : β) : WithTop.map₂ f a ↑b = WithTop.map (fun (x : α) => f x b) a

@[simp]

theorem WithTop.map₂_eq_top_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β → γ} {a : WithTop α} {b : WithTop β} : WithTop.map₂ f a b = ⊤ ↔ a = ⊤ ∨ b = ⊤

theorem WithTop.map_toDual {α : Type u_1} {β : Type u_2} (f : αᵒᵈ → βᵒᵈ) (a : WithBot α) : WithTop.map f (WithBot.toDual a) = WithBot.map (⇑OrderDual.toDual ∘ f) a

theorem WithTop.map_ofDual {α : Type u_1} {β : Type u_2} (f : α → β) (a : WithBot αᵒᵈ) : WithTop.map f (WithBot.ofDual a) = WithBot.map (⇑OrderDual.ofDual ∘ f) a

theorem WithTop.toDual_map {α : Type u_1} {β : Type u_2} (f : α → β) (a : WithTop α) : WithTop.toDual (WithTop.map f a) = WithBot.map (⇑OrderDual.toDual ∘ f ∘ ⇑OrderDual.ofDual) (WithTop.toDual a)

theorem WithTop.ofDual_map {α : Type u_1} {β : Type u_2} (f : αᵒᵈ → βᵒᵈ) (a : WithTop αᵒᵈ) : WithTop.ofDual (WithTop.map f a) = WithBot.map (⇑OrderDual.ofDual ∘ f ∘ ⇑OrderDual.toDual) (WithTop.ofDual a)

theorem WithTop.ne_top_iff_exists {α : Type u_1} {x : WithTop α} : x ≠ ⊤ ↔ ∃ (a : α), ↑a = x

theorem WithTop.forall_ne_iff_eq_top {α : Type u_1} {x : WithTop α} : (∀ (a : α), ↑a ≠ x) ↔ x = ⊤

def WithTop.untop {α : Type u_1} (x : WithTop α) : x ≠ ⊤ → α

Deconstruct a x : WithTop α to the underlying value in α, given a proof that x ≠ ⊤.

Equations

• WithTop.untop (some x_2) x_3 = x_2

@[simp]

theorem WithTop.coe_untop {α : Type u_1} (x : WithTop α) (hx : x ≠ ⊤) : ↑(x.untop hx) = x

@[simp]

theorem WithTop.untop_coe {α : Type u_1} (x : α) (h : ↑x ≠ ⊤ := ⋯) : (↑x).untop h = x

theorem WithTop.canLift {α : Type u_1} : CanLift (WithTop α) α WithTop.some fun (r : WithTop α) => r ≠ ⊤

instance WithTop.instBot {α : Type u_1} [Bot α] : Bot (WithTop α)

Equations

• WithTop.instBot = { bot := ↑⊥ }

@[simp]

theorem WithTop.coe_bot {α : Type u_1} [Bot α] : ↑⊥ = ⊥

@[simp]

theorem WithTop.coe_eq_bot {α : Type u_1} [Bot α] {a : α} : ↑a = ⊥ ↔ a = ⊥

@[simp]

theorem WithTop.bot_eq_coe {α : Type u_1} [Bot α] {a : α} : ⊥ = ↑a ↔ ⊥ = a

theorem WithTop.untop_eq_iff {α : Type u_1} {a : WithTop α} {b : α} (h : a ≠ ⊤) : a.untop h = b ↔ a = ↑b

theorem WithTop.eq_untop_iff {α : Type u_1} {a : α} {b : WithTop α} (h : b ≠ ⊤) : a = b.untop h ↔ ↑a = b

def Equiv.withTopSubtypeNe {α : Type u_1} : { y : WithTop α // y ≠ ⊤ } ≃ α

The equivalence between the non-top elements of WithTop α and α.

Equations

• Equiv.withTopSubtypeNe = { toFun := fun (x : { y : WithTop α // y ≠ ⊤ }) => match x with | ⟨x, h⟩ => x.untop h, invFun := fun (x : α) => ⟨↑x, ⋯⟩, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem Equiv.withTopSubtypeNe_symm_apply_coe {α : Type u_1} (x : α) : ↑(Equiv.withTopSubtypeNe.symm x) = ↑x

@[simp]

theorem Equiv.withTopSubtypeNe_apply {α : Type u_1} (x✝ : { y : WithTop α // y ≠ ⊤ }) : Equiv.withTopSubtypeNe x✝ = match x✝ with | ⟨x, h⟩ => x.untop h

@[instance 10]

instance WithTop.le {α : Type u_1} [LE α] : LE (WithTop α)

Equations

• WithTop.le = { le := fun (o₁ o₂ : WithTop α) => ∀ (a : α), o₂ = ↑a → ∃ (b : α), o₁ = ↑b ∧ b ≤ a }

theorem WithTop.toDual_le_iff {α : Type u_1} [LE α] {a : WithTop α} {b : WithBot αᵒᵈ} : WithTop.toDual a ≤ b ↔ WithBot.ofDual b ≤ a

theorem WithTop.le_toDual_iff {α : Type u_1} [LE α] {a : WithBot αᵒᵈ} {b : WithTop α} : a ≤ WithTop.toDual b ↔ b ≤ WithBot.ofDual a

@[simp]

theorem WithTop.toDual_le_toDual_iff {α : Type u_1} [LE α] {a b : WithTop α} : WithTop.toDual a ≤ WithTop.toDual b ↔ b ≤ a

theorem WithTop.ofDual_le_iff {α : Type u_1} [LE α] {a : WithTop αᵒᵈ} {b : WithBot α} : WithTop.ofDual a ≤ b ↔ WithBot.toDual b ≤ a

theorem WithTop.le_ofDual_iff {α : Type u_1} [LE α] {a : WithBot α} {b : WithTop αᵒᵈ} : a ≤ WithTop.ofDual b ↔ b ≤ WithBot.toDual a

@[simp]

theorem WithTop.ofDual_le_ofDual_iff {α : Type u_1} [LE α] {a b : WithTop αᵒᵈ} : WithTop.ofDual a ≤ WithTop.ofDual b ↔ b ≤ a

@[simp]

theorem WithTop.coe_le_coe {α : Type u_1} {a b : α} [LE α] : ↑a ≤ ↑b ↔ a ≤ b

@[simp, deprecated WithTop.coe_le_coe]

theorem WithTop.some_le_some {α : Type u_1} {a b : α} [LE α] : some a ≤ some b ↔ a ≤ b

instance WithTop.orderTop {α : Type u_1} [LE α] : OrderTop (WithTop α)

Equations

• WithTop.orderTop = OrderTop.mk ⋯

@[simp, deprecated le_top]

theorem WithTop.le_none {α : Type u_1} [LE α] {a : WithTop α} : a ≤ none

instance WithTop.orderBot {α : Type u_1} [LE α] [OrderBot α] : OrderBot (WithTop α)

Equations

• WithTop.orderBot = OrderBot.mk ⋯

instance WithTop.boundedOrder {α : Type u_1} [LE α] [OrderBot α] : BoundedOrder (WithTop α)

Equations

• WithTop.boundedOrder = BoundedOrder.mk

theorem WithTop.not_top_le_coe {α : Type u_1} [LE α] (a : α) : ¬⊤ ≤ ↑a

@[simp]

theorem WithTop.top_le_iff {α : Type u_1} [LE α] {a : WithTop α} : ⊤ ≤ a ↔ a = ⊤

There is a general version top_le_iff, but this lemma does not require a PartialOrder.

theorem WithTop.le_coe {α : Type u_1} {a b : α} [LE α] {o : Option α} : a ∈ o → (o ≤ ↑b ↔ a ≤ b)

theorem WithTop.le_coe_iff {α : Type u_1} {b : α} [LE α] {x : WithTop α} : x ≤ ↑b ↔ ∃ (a : α), x = ↑a ∧ a ≤ b

theorem WithTop.coe_le_iff {α : Type u_1} {a : α} [LE α] {x : WithTop α} : ↑a ≤ x ↔ ∀ (b : α), x = ↑b → a ≤ b

theorem IsMin.withTop {α : Type u_1} {a : α} [LE α] (h : IsMin a) : IsMin ↑a

theorem WithTop.untop_le_iff {α : Type u_1} [LE α] {a : WithTop α} {b : α} (h : a ≠ ⊤) : a.untop h ≤ b ↔ a ≤ ↑b

theorem WithTop.le_untop_iff {α : Type u_1} [LE α] {a : α} {b : WithTop α} (h : b ≠ ⊤) : a ≤ b.untop h ↔ ↑a ≤ b

theorem WithTop.le_untop'_iff {α : Type u_1} [LE α] {a : WithTop α} {b c : α} (h : a = ⊤ → c ≤ b) : c ≤ WithTop.untop' b a ↔ ↑c ≤ a

@[instance 10]

instance WithTop.lt {α : Type u_1} [LT α] : LT (WithTop α)

Equations

• WithTop.lt = { lt := fun (o₁ o₂ : Option α) => ∃ (b : α), b ∈ o₁ ∧ ∀ (a : α), a ∈ o₂ → b < a }

theorem WithTop.toDual_lt_iff {α : Type u_1} [LT α] {a : WithTop α} {b : WithBot αᵒᵈ} : WithTop.toDual a < b ↔ WithBot.ofDual b < a

theorem WithTop.lt_toDual_iff {α : Type u_1} [LT α] {a : WithBot αᵒᵈ} {b : WithTop α} : a < WithTop.toDual b ↔ b < WithBot.ofDual a

@[simp]

theorem WithTop.toDual_lt_toDual_iff {α : Type u_1} [LT α] {a b : WithTop α} : WithTop.toDual a < WithTop.toDual b ↔ b < a

theorem WithTop.ofDual_lt_iff {α : Type u_1} [LT α] {a : WithTop αᵒᵈ} {b : WithBot α} : WithTop.ofDual a < b ↔ WithBot.toDual b < a

theorem WithTop.lt_ofDual_iff {α : Type u_1} [LT α] {a : WithBot α} {b : WithTop αᵒᵈ} : a < WithTop.ofDual b ↔ b < WithBot.toDual a

@[simp]

theorem WithTop.ofDual_lt_ofDual_iff {α : Type u_1} [LT α] {a b : WithTop αᵒᵈ} : WithTop.ofDual a < WithTop.ofDual b ↔ b < a

theorem WithTop.lt_untop_iff {α : Type u_1} [LT α] {a : α} {b : WithTop α} (h : b ≠ ⊤) : a < b.untop h ↔ ↑a < b

theorem WithTop.untop_lt_iff {α : Type u_1} [LT α] {a : WithTop α} {b : α} (h : a ≠ ⊤) : a.untop h < b ↔ a < ↑b

theorem WithTop.lt_untop'_iff {α : Type u_1} [LT α] {a : WithTop α} {b c : α} (h : a = ⊤ → c < b) : c < WithTop.untop' b a ↔ ↑c < a

@[simp]

theorem WithBot.toDual_symm_apply {α : Type u_1} (a : WithTop αᵒᵈ) : WithBot.toDual.symm a = WithTop.ofDual a

@[simp]

theorem WithBot.ofDual_symm_apply {α : Type u_1} (a : WithTop α) : WithBot.ofDual.symm a = WithTop.toDual a

@[simp]

theorem WithBot.toDual_apply_bot {α : Type u_1} : WithBot.toDual ⊥ = ⊤

@[simp]

theorem WithBot.ofDual_apply_bot {α : Type u_1} : WithBot.ofDual ⊥ = ⊤

@[simp]

theorem WithBot.toDual_apply_coe {α : Type u_1} (a : α) : WithBot.toDual ↑a = ↑(OrderDual.toDual a)

@[simp]

theorem WithBot.ofDual_apply_coe {α : Type u_1} (a : αᵒᵈ) : WithBot.ofDual ↑a = ↑(OrderDual.ofDual a)

theorem WithBot.map_toDual {α : Type u_1} {β : Type u_2} (f : αᵒᵈ → βᵒᵈ) (a : WithTop α) : WithBot.map f (WithTop.toDual a) = WithTop.map (⇑OrderDual.toDual ∘ f) a

theorem WithBot.map_ofDual {α : Type u_1} {β : Type u_2} (f : α → β) (a : WithTop αᵒᵈ) : WithBot.map f (WithTop.ofDual a) = WithTop.map (⇑OrderDual.ofDual ∘ f) a

theorem WithBot.toDual_map {α : Type u_1} {β : Type u_2} (f : α → β) (a : WithBot α) : WithBot.toDual (WithBot.map f a) = WithBot.map (⇑OrderDual.toDual ∘ f ∘ ⇑OrderDual.ofDual) (WithBot.toDual a)

theorem WithBot.ofDual_map {α : Type u_1} {β : Type u_2} (f : αᵒᵈ → βᵒᵈ) (a : WithBot αᵒᵈ) : WithBot.ofDual (WithBot.map f a) = WithBot.map (⇑OrderDual.ofDual ∘ f ∘ ⇑OrderDual.toDual) (WithBot.ofDual a)

theorem WithBot.forall_lt_iff_eq_bot {α : Type u_1} [Preorder α] {x : WithBot α} : (∀ (y : α), x < ↑y) ↔ x = ⊥

theorem WithBot.forall_le_iff_eq_bot {α : Type u_1} [Preorder α] [NoMinOrder α] {x : WithBot α} : (∀ (y : α), x ≤ ↑y) ↔ x = ⊥

theorem WithBot.le_of_forall_lt_iff_le {α : Type u_1} [LinearOrder α] [DenselyOrdered α] [NoMinOrder α] {x y : WithBot α} : (∀ (z : α), x < ↑z → y ≤ ↑z) ↔ y ≤ x

theorem WithBot.ge_of_forall_gt_iff_ge {α : Type u_1} [LinearOrder α] [DenselyOrdered α] [NoMinOrder α] {x y : WithBot α} : (∀ (z : α), ↑z < x → ↑z ≤ y) ↔ x ≤ y

theorem WithBot.toDual_le_iff {α : Type u_1} [LE α] {a : WithBot α} {b : WithTop αᵒᵈ} : WithBot.toDual a ≤ b ↔ WithTop.ofDual b ≤ a

theorem WithBot.le_toDual_iff {α : Type u_1} [LE α] {a : WithTop αᵒᵈ} {b : WithBot α} : a ≤ WithBot.toDual b ↔ b ≤ WithTop.ofDual a

@[simp]

theorem WithBot.toDual_le_toDual_iff {α : Type u_1} [LE α] {a b : WithBot α} : WithBot.toDual a ≤ WithBot.toDual b ↔ b ≤ a

theorem WithBot.ofDual_le_iff {α : Type u_1} [LE α] {a : WithBot αᵒᵈ} {b : WithTop α} : WithBot.ofDual a ≤ b ↔ WithTop.toDual b ≤ a

theorem WithBot.le_ofDual_iff {α : Type u_1} [LE α] {a : WithTop α} {b : WithBot αᵒᵈ} : a ≤ WithBot.ofDual b ↔ b ≤ WithTop.toDual a

@[simp]

theorem WithBot.ofDual_le_ofDual_iff {α : Type u_1} [LE α] {a b : WithBot αᵒᵈ} : WithBot.ofDual a ≤ WithBot.ofDual b ↔ b ≤ a

theorem WithBot.toDual_lt_iff {α : Type u_1} [LT α] {a : WithBot α} {b : WithTop αᵒᵈ} : WithBot.toDual a < b ↔ WithTop.ofDual b < a

theorem WithBot.lt_toDual_iff {α : Type u_1} [LT α] {a : WithTop αᵒᵈ} {b : WithBot α} : a < WithBot.toDual b ↔ b < WithTop.ofDual a

@[simp]

theorem WithBot.toDual_lt_toDual_iff {α : Type u_1} [LT α] {a b : WithBot α} : WithBot.toDual a < WithBot.toDual b ↔ b < a

theorem WithBot.ofDual_lt_iff {α : Type u_1} [LT α] {a : WithBot αᵒᵈ} {b : WithTop α} : WithBot.ofDual a < b ↔ WithTop.toDual b < a

theorem WithBot.lt_ofDual_iff {α : Type u_1} [LT α] {a : WithTop α} {b : WithBot αᵒᵈ} : a < WithBot.ofDual b ↔ b < WithTop.toDual a

@[simp]

theorem WithBot.ofDual_lt_ofDual_iff {α : Type u_1} [LT α] {a b : WithBot αᵒᵈ} : WithBot.ofDual a < WithBot.ofDual b ↔ b < a

@[simp]

theorem WithTop.coe_lt_coe {α : Type u_1} [LT α] {a b : α} : ↑a < ↑b ↔ a < b

@[simp]

theorem WithTop.coe_lt_top {α : Type u_1} [LT α] (a : α) : ↑a < ⊤

@[simp]

theorem WithTop.not_top_lt {α : Type u_1} [LT α] (a : WithTop α) : ¬⊤ < a

@[simp, deprecated WithTop.coe_lt_coe]

theorem WithTop.some_lt_some {α : Type u_1} [LT α] {a b : α} : some a < some b ↔ a < b

@[simp, deprecated WithTop.coe_lt_top]

theorem WithTop.some_lt_none {α : Type u_1} [LT α] (a : α) : some a < none

@[simp, deprecated not_top_lt]

theorem WithTop.not_none_lt {α : Type u_1} [LT α] (a : WithTop α) : ¬none < a

theorem WithTop.lt_iff_exists_coe {α : Type u_1} [LT α] {a b : WithTop α} : a < b ↔ ∃ (p : α), a = ↑p ∧ ↑p < b

theorem WithTop.coe_lt_iff {α : Type u_1} [LT α] {a : α} {x : WithTop α} : ↑a < x ↔ ∀ (b : α), x = ↑b → a < b

theorem WithTop.lt_top_iff_ne_top {α : Type u_1} [LT α] {x : WithTop α} : x < ⊤ ↔ x ≠ ⊤

A version of lt_top_iff_ne_top for WithTop that only requires LT α, not PartialOrder α.

instance WithTop.preorder {α : Type u_1} [Preorder α] : Preorder (WithTop α)

Equations

• WithTop.preorder = Preorder.mk ⋯ ⋯ ⋯

instance WithTop.partialOrder {α : Type u_1} [PartialOrder α] : PartialOrder (WithTop α)

Equations

• WithTop.partialOrder = PartialOrder.mk ⋯

theorem WithTop.coe_strictMono {α : Type u_1} [Preorder α] : StrictMono fun (a : α) => ↑a

theorem WithTop.coe_mono {α : Type u_1} [Preorder α] : Monotone fun (a : α) => ↑a

theorem WithTop.monotone_iff {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : WithTop α → β} : Monotone f ↔ (Monotone fun (a : α) => f ↑a) ∧ ∀ (x : α), f ↑x ≤ f ⊤

@[simp]

theorem WithTop.monotone_map_iff {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} : Monotone (WithTop.map f) ↔ Monotone f

theorem Monotone.withTop_map {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} : Monotone f → Monotone (WithTop.map f)

Alias of the reverse direction of WithTop.monotone_map_iff.

theorem WithTop.strictMono_iff {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : WithTop α → β} : StrictMono f ↔ (StrictMono fun (a : α) => f ↑a) ∧ ∀ (x : α), f ↑x < f ⊤

theorem WithTop.strictAnti_iff {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : WithTop α → β} : StrictAnti f ↔ (StrictAnti fun (a : α) => f ↑a) ∧ ∀ (x : α), f ⊤ < f ↑x

@[simp]

theorem WithTop.strictMono_map_iff {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} : StrictMono (WithTop.map f) ↔ StrictMono f

theorem StrictMono.withTop_map {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} : StrictMono f → StrictMono (WithTop.map f)

Alias of the reverse direction of WithTop.strictMono_map_iff.

theorem WithTop.map_le_iff {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α → β) (a b : WithTop α) (mono_iff : ∀ {a b : α}, f a ≤ f b ↔ a ≤ b) : WithTop.map f a ≤ WithTop.map f b ↔ a ≤ b

theorem WithTop.coe_untop'_le {α : Type u_1} [Preorder α] (a : WithTop α) (b : α) : ↑(WithTop.untop' b a) ≤ a

@[simp]

theorem WithTop.coe_top_lt {α : Type u_1} [Preorder α] [OrderTop α] {x : WithTop α} : ↑⊤ < x ↔ x = ⊤

theorem WithTop.forall_gt_iff_eq_top {α : Type u_1} [Preorder α] {x : WithTop α} : (∀ (y : α), ↑y < x) ↔ x = ⊤

theorem WithTop.forall_ge_iff_eq_top {α : Type u_1} [Preorder α] [NoMaxOrder α] {x : WithTop α} : (∀ (y : α), ↑y ≤ x) ↔ x = ⊤

theorem WithTop.le_of_forall_lt_iff_le {α : Type u_1} [LinearOrder α] [DenselyOrdered α] [NoMaxOrder α] {x y : WithTop α} : (∀ (z : α), x < ↑z → y ≤ ↑z) ↔ y ≤ x

theorem WithTop.ge_of_forall_gt_iff_ge {α : Type u_1} [LinearOrder α] [DenselyOrdered α] [NoMaxOrder α] {x y : WithTop α} : (∀ (z : α), ↑z < x → ↑z ≤ y) ↔ x ≤ y

instance WithTop.semilatticeInf {α : Type u_1} [SemilatticeInf α] : SemilatticeInf (WithTop α)

Equations

• WithTop.semilatticeInf = SemilatticeInf.mk (Option.liftOrGet fun (x1 x2 : α) => x1 ⊓ x2) ⋯ ⋯ ⋯

theorem WithTop.coe_inf {α : Type u_1} [SemilatticeInf α] (a b : α) : ↑(a ⊓ b) = ↑a ⊓ ↑b

instance WithTop.semilatticeSup {α : Type u_1} [SemilatticeSup α] : SemilatticeSup (WithTop α)

Equations

• WithTop.semilatticeSup = SemilatticeSup.mk (WithTop.map₂ fun (x1 x2 : α) => x1 ⊔ x2) ⋯ ⋯ ⋯

theorem WithTop.coe_sup {α : Type u_1} [SemilatticeSup α] (a b : α) : ↑(a ⊔ b) = ↑a ⊔ ↑b

instance WithTop.lattice {α : Type u_1} [Lattice α] : Lattice (WithTop α)

Equations

• WithTop.lattice = Lattice.mk SemilatticeInf.inf ⋯ ⋯ ⋯

instance WithTop.distribLattice {α : Type u_1} [DistribLattice α] : DistribLattice (WithTop α)

Equations

• WithTop.distribLattice = DistribLattice.mk ⋯

instance WithTop.decidableEq {α : Type u_1} [DecidableEq α] : DecidableEq (WithTop α)

Equations

• WithTop.decidableEq = inferInstanceAs (DecidableEq (Option α))

instance WithTop.decidableLE {α : Type u_1} [LE α] [DecidableRel fun (x1 x2 : α) => x1 ≤ x2] : DecidableRel fun (x1 x2 : WithTop α) => x1 ≤ x2

Equations

• x✝.decidableLE x = decidable_of_decidable_of_iff ⋯

instance WithTop.decidableLT {α : Type u_1} [LT α] [DecidableRel fun (x1 x2 : α) => x1 < x2] : DecidableRel fun (x1 x2 : WithTop α) => x1 < x2

Equations

• x✝.decidableLT x = decidable_of_decidable_of_iff ⋯

theorem WithTop.isTotal_le {α : Type u_1} [LE α] [IsTotal α fun (x1 x2 : α) => x1 ≤ x2] : IsTotal (WithTop α) fun (x1 x2 : WithTop α) => x1 ≤ x2

instance WithTop.linearOrder {α : Type u_1} [LinearOrder α] : LinearOrder (WithTop α)

Equations

• WithTop.linearOrder = Lattice.toLinearOrder (WithTop α)

@[simp]

theorem WithTop.coe_min {α : Type u_1} [LinearOrder α] (x y : α) : ↑(x ⊓ y) = ↑x ⊓ ↑y

@[simp]

theorem WithTop.coe_max {α : Type u_1} [LinearOrder α] (x y : α) : ↑(x ⊔ y) = ↑x ⊔ ↑y

theorem WithTop.instWellFoundedLT {α : Type u_1} [LT α] [WellFoundedLT α] : WellFoundedLT (WithTop α)

theorem WithTop.instWellFoundedGT {α : Type u_1} [LT α] [WellFoundedGT α] : WellFoundedGT (WithTop α)

theorem WithTop.trichotomous.lt {α : Type u_1} [Preorder α] [IsTrichotomous α fun (x1 x2 : α) => x1 < x2] : IsTrichotomous (WithTop α) fun (x1 x2 : WithTop α) => x1 < x2

theorem WithTop.IsWellOrder.lt {α : Type u_1} [Preorder α] [IsWellOrder α fun (x1 x2 : α) => x1 < x2] : IsWellOrder (WithTop α) fun (x1 x2 : WithTop α) => x1 < x2

theorem WithTop.trichotomous.gt {α : Type u_1} [Preorder α] [IsTrichotomous α fun (x1 x2 : α) => x1 > x2] : IsTrichotomous (WithTop α) fun (x1 x2 : WithTop α) => x1 > x2

theorem WithTop.IsWellOrder.gt {α : Type u_1} [Preorder α] [IsWellOrder α fun (x1 x2 : α) => x1 > x2] : IsWellOrder (WithTop α) fun (x1 x2 : WithTop α) => x1 > x2

theorem WithBot.trichotomous.lt {α : Type u_1} [Preorder α] [h : IsTrichotomous α fun (x1 x2 : α) => x1 < x2] : IsTrichotomous (WithBot α) fun (x1 x2 : WithBot α) => x1 < x2

theorem WithBot.isWellOrder.lt {α : Type u_1} [Preorder α] [IsWellOrder α fun (x1 x2 : α) => x1 < x2] : IsWellOrder (WithBot α) fun (x1 x2 : WithBot α) => x1 < x2

theorem WithBot.trichotomous.gt {α : Type u_1} [Preorder α] [h : IsTrichotomous α fun (x1 x2 : α) => x1 > x2] : IsTrichotomous (WithBot α) fun (x1 x2 : WithBot α) => x1 > x2

theorem WithBot.isWellOrder.gt {α : Type u_1} [Preorder α] [h : IsWellOrder α fun (x1 x2 : α) => x1 > x2] : IsWellOrder (WithBot α) fun (x1 x2 : WithBot α) => x1 > x2

theorem WithTop.instDenselyOrderedOfNoMaxOrder {α : Type u_1} [LT α] [DenselyOrdered α] [NoMaxOrder α] : DenselyOrdered (WithTop α)

theorem WithTop.lt_iff_exists_coe_btwn {α : Type u_1} [Preorder α] [DenselyOrdered α] [NoMaxOrder α] {a b : WithTop α} : a < b ↔ ∃ (x : α), a < ↑x ∧ ↑x < b

theorem WithTop.noBotOrder {α : Type u_1} [LE α] [NoBotOrder α] [Nonempty α] : NoBotOrder (WithTop α)

theorem WithTop.noMinOrder {α : Type u_1} [LT α] [NoMinOrder α] [Nonempty α] : NoMinOrder (WithTop α)