Adjoining top/bottom elements to ordered monoids. #

theorem WithTop.addLECancellable_of_ne_top {α : Type u} [Add α] {x : WithTop α} [LE α] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] (hx : x ≠ ⊤) :

AddLECancellable x

source

theorem WithTop.addLECancellable_of_lt_top {α : Type u} [Add α] {x : WithTop α} [Preorder α] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] (hx : x < ⊤) :

AddLECancellable x

source

theorem WithTop.addLECancellable_coe {α : Type u} [Add α] [LE α] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] (a : α) :

AddLECancellable ↑a

source

theorem WithTop.addLECancellable_iff_ne_top {α : Type u} [Add α] {x : WithTop α} [Nonempty α] [Preorder α] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] :

AddLECancellable x ↔ x ≠ ⊤

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@[simp]

theorem WithTop.map_add {α : Type u} {β : Type v} [Add α] {F : Type u_1} [Add β] [FunLike F α β] [AddHomClass F α β] (f : F) (a b : WithTop α) :

map (⇑f) (a + b) = map (⇑f) a + map (⇑f) b

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instance WithTop.addSemigroup {α : Type u} [AddSemigroup α] :

AddSemigroup (WithTop α)

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instance WithTop.addCommSemigroup {α : Type u} [AddCommSemigroup α] :

AddCommSemigroup (WithTop α)

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instance WithTop.addZeroClass {α : Type u} [AddZeroClass α] :

AddZeroClass (WithTop α)

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instance WithTop.addMonoid {α : Type u} [AddMonoid α] :

AddMonoid (WithTop α)

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@[simp]

theorem WithTop.coe_nsmul {α : Type u} [AddMonoid α] (a : α) (n : ℕ) :

↑(n • a) = n • ↑a

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def WithTop.addHom {α : Type u} [AddMonoid α] :

α →+ WithTop α

Coercion from α to WithTop α as an AddMonoidHom.

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@[simp]

theorem WithTop.coe_addHom {α : Type u} [AddMonoid α] :

⇑addHom = some

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instance WithTop.addCommMonoid {α : Type u} [AddCommMonoid α] :

AddCommMonoid (WithTop α)

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instance WithTop.addMonoidWithOne {α : Type u} [AddMonoidWithOne α] :

AddMonoidWithOne (WithTop α)

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@[simp]

theorem WithTop.coe_natCast {α : Type u} [AddMonoidWithOne α] (n : ℕ) :

↑↑n = ↑n

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@[simp]

theorem WithTop.top_ne_natCast {α : Type u} [AddMonoidWithOne α] (n : ℕ) :

⊤ ≠ ↑n

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@[simp]

theorem WithTop.natCast_ne_top {α : Type u} [AddMonoidWithOne α] (n : ℕ) :

↑n ≠ ⊤

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@[simp]

theorem WithTop.natCast_lt_top {α : Type u} [AddMonoidWithOne α] [LT α] (n : ℕ) :

↑n < ⊤

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@[simp]

theorem WithTop.coe_ofNat {α : Type u} [AddMonoidWithOne α] (n : ℕ) [n.AtLeastTwo] :

↑(OfNat.ofNat n) = OfNat.ofNat n

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@[simp]

theorem WithTop.coe_eq_ofNat {α : Type u} [AddMonoidWithOne α] (n : ℕ) [n.AtLeastTwo] (m : α) :

↑m = OfNat.ofNat n ↔ m = OfNat.ofNat n

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@[simp]

theorem WithTop.ofNat_eq_coe {α : Type u} [AddMonoidWithOne α] (n : ℕ) [n.AtLeastTwo] (m : α) :

OfNat.ofNat n = ↑m ↔ OfNat.ofNat n = m

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@[simp]

theorem WithTop.ofNat_ne_top {α : Type u} [AddMonoidWithOne α] (n : ℕ) [n.AtLeastTwo] :

OfNat.ofNat n ≠ ⊤

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@[simp]

theorem WithTop.top_ne_ofNat {α : Type u} [AddMonoidWithOne α] (n : ℕ) [n.AtLeastTwo] :

⊤ ≠ OfNat.ofNat n

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@[simp]

theorem WithTop.map_ofNat {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : α → β} (n : ℕ) [n.AtLeastTwo] :

map f (OfNat.ofNat n) = ↑(f (OfNat.ofNat n))

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@[simp]

theorem WithTop.map_natCast {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : α → β} (n : ℕ) :

map f ↑n = ↑(f ↑n)

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theorem WithTop.map_eq_ofNat_iff {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : β → α} {n : ℕ} [n.AtLeastTwo] {a : WithTop β} :

map f a = OfNat.ofNat n ↔ ∃ (x : β), a = ↑x ∧ f x = ↑n

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theorem WithTop.ofNat_eq_map_iff {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : β → α} {n : ℕ} [n.AtLeastTwo] {a : WithTop β} :

OfNat.ofNat n = map f a ↔ ∃ (x : β), a = ↑x ∧ f x = ↑n

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theorem WithTop.map_eq_natCast_iff {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : β → α} {n : ℕ} {a : WithTop β} :

map f a = ↑n ↔ ∃ (x : β), a = ↑x ∧ f x = ↑n

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theorem WithTop.natCast_eq_map_iff {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : β → α} {n : ℕ} {a : WithTop β} :

↑n = map f a ↔ ∃ (x : β), a = ↑x ∧ f x = ↑n

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instance WithTop.charZero {α : Type u} [AddMonoidWithOne α] [CharZero α] :

CharZero (WithTop α)

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instance WithTop.addCommMonoidWithOne {α : Type u} [AddCommMonoidWithOne α] :

AddCommMonoidWithOne (WithTop α)

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instance WithTop.existsAddOfLE {α : Type u} [LE α] [Add α] [ExistsAddOfLE α] :

ExistsAddOfLE (WithTop α)

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@[simp]

theorem WithTop.one_lt_top {α : Type u} [One α] [LT α] :

1 < ⊤

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@[simp]

theorem WithTop.top_pos {α : Type u} [Zero α] [LT α] :

0 < ⊤

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@[deprecated WithTop.top_pos (since := "2024-10-22")]

theorem WithTop.zero_lt_top {α : Type u} [Zero α] [LT α] :

0 < ⊤

Alias of WithTop.top_pos.

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@[deprecated WithTop.coe_pos (since := "2024-10-22")]

theorem WithTop.zero_lt_coe {α : Type u} [Zero α] [LT α] (a : α) :

0 < ↑a ↔ 0 < a

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def OneHom.withTopMap {M : Type u_1} {N : Type u_2} [One M] [One N] (f : OneHom M N) :

OneHom (WithTop M) (WithTop N)

A version of WithTop.map for OneHoms.

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def ZeroHom.withTopMap {M : Type u_1} {N : Type u_2} [Zero M] [Zero N] (f : ZeroHom M N) :

ZeroHom (WithTop M) (WithTop N)

A version of WithTop.map for ZeroHoms

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@[simp]

theorem OneHom.withTopMap_apply {M : Type u_1} {N : Type u_2} [One M] [One N] (f : OneHom M N) :

⇑f.withTopMap = WithTop.map ⇑f

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@[simp]

theorem ZeroHom.withTopMap_apply {M : Type u_1} {N : Type u_2} [Zero M] [Zero N] (f : ZeroHom M N) :

⇑f.withTopMap = WithTop.map ⇑f

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def AddHom.withTopMap {M : Type u_1} {N : Type u_2} [Add M] [Add N] (f : M →ₙ+ N) :

WithTop M →ₙ+ WithTop N

A version of WithTop.map for AddHoms.

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@[simp]

theorem AddHom.withTopMap_apply {M : Type u_1} {N : Type u_2} [Add M] [Add N] (f : M →ₙ+ N) :

⇑f.withTopMap = WithTop.map ⇑f

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def AddMonoidHom.withTopMap {M : Type u_1} {N : Type u_2} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) :

WithTop M →+ WithTop N

A version of WithTop.map for AddMonoidHoms.

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@[simp]

theorem AddMonoidHom.withTopMap_apply {M : Type u_1} {N : Type u_2} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) :

⇑f.withTopMap = WithTop.map ⇑f

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instance WithBot.one {α : Type u} [One α] :

One (WithBot α)

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instance WithBot.zero {α : Type u} [Zero α] :

Zero (WithBot α)

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@[simp]

theorem WithBot.coe_one {α : Type u} [One α] :

↑1 = 1

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@[simp]

theorem WithBot.coe_zero {α : Type u} [Zero α] :

↑0 = 0

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@[simp]

theorem WithBot.coe_eq_one {α : Type u} [One α] {a : α} :

↑a = 1 ↔ a = 1

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@[simp]

theorem WithBot.coe_eq_zero {α : Type u} [Zero α] {a : α} :

↑a = 0 ↔ a = 0

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@[simp]

theorem WithBot.one_eq_coe {α : Type u} [One α] {a : α} :

1 = ↑a ↔ a = 1

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@[simp]

theorem WithBot.zero_eq_coe {α : Type u} [Zero α] {a : α} :

0 = ↑a ↔ a = 0

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@[simp]

theorem WithBot.bot_ne_one {α : Type u} [One α] :

⊥ ≠ 1

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@[simp]

theorem WithBot.bot_ne_zero {α : Type u} [Zero α] :

⊥ ≠ 0

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@[simp]

theorem WithBot.one_ne_bot {α : Type u} [One α] :

1 ≠ ⊥

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@[simp]

theorem WithBot.zero_ne_bot {α : Type u} [Zero α] :

0 ≠ ⊥

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@[simp]

theorem WithBot.unbot_one {α : Type u} [One α] :

unbot 1 ⋯ = 1

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@[simp]

theorem WithBot.unbot_zero {α : Type u} [Zero α] :

unbot 0 ⋯ = 0

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@[simp]

theorem WithBot.unbotD_one {α : Type u} [One α] (d : α) :

unbotD d 1 = 1

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@[simp]

theorem WithBot.unbotD_zero {α : Type u} [Zero α] (d : α) :

unbotD d 0 = 0

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@[deprecated WithBot.unbotD_zero (since := "2025-02-06")]

theorem WithBot.unbot_zero' {α : Type u} [Zero α] (d : α) :

unbotD d 0 = 0

Alias of WithBot.unbotD_zero.

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@[deprecated WithBot.unbotD_one (since := "2025-02-06")]

theorem WithBot.unbot_one' {α : Type u} [One α] (d : α) :

unbotD d 1 = 1

Alias of WithBot.unbotD_one.

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@[simp]

theorem WithBot.one_le_coe {α : Type u} [One α] {a : α} [LE α] :

1 ≤ ↑a ↔ 1 ≤ a

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@[simp]

theorem WithBot.coe_nonneg {α : Type u} [Zero α] {a : α} [LE α] :

0 ≤ ↑a ↔ 0 ≤ a

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@[simp]

theorem WithBot.coe_le_one {α : Type u} [One α] {a : α} [LE α] :

↑a ≤ 1 ↔ a ≤ 1

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@[simp]

theorem WithBot.coe_le_zero {α : Type u} [Zero α] {a : α} [LE α] :

↑a ≤ 0 ↔ a ≤ 0

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@[simp]

theorem WithBot.one_lt_coe {α : Type u} [One α] {a : α} [LT α] :

1 < ↑a ↔ 1 < a

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@[simp]

theorem WithBot.coe_pos {α : Type u} [Zero α] {a : α} [LT α] :

0 < ↑a ↔ 0 < a

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@[simp]

theorem WithBot.coe_lt_one {α : Type u} [One α] {a : α} [LT α] :

↑a < 1 ↔ a < 1

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@[simp]

theorem WithBot.coe_lt_zero {α : Type u} [Zero α] {a : α} [LT α] :

↑a < 0 ↔ a < 0

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@[simp]

theorem WithBot.map_one {α : Type u} [One α] {β : Type u_1} (f : α → β) :

map f 1 = ↑(f 1)

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@[simp]

theorem WithBot.map_zero {α : Type u} [Zero α] {β : Type u_1} (f : α → β) :

map f 0 = ↑(f 0)

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theorem WithBot.map_eq_one_iff {β : Type v} {α : Type u_1} {f : α → β} {v : WithBot α} [One β] :

map f v = 1 ↔ ∃ (x : α), v = ↑x ∧ f x = 1

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theorem WithBot.map_eq_zero_iff {β : Type v} {α : Type u_1} {f : α → β} {v : WithBot α} [Zero β] :

map f v = 0 ↔ ∃ (x : α), v = ↑x ∧ f x = 0

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theorem WithBot.one_eq_map_iff {β : Type v} {α : Type u_1} {f : α → β} {v : WithBot α} [One β] :

1 = map f v ↔ ∃ (x : α), v = ↑x ∧ f x = 1

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theorem WithBot.zero_eq_map_iff {β : Type v} {α : Type u_1} {f : α → β} {v : WithBot α} [Zero β] :

0 = map f v ↔ ∃ (x : α), v = ↑x ∧ f x = 0

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instance WithBot.zeroLEOneClass {α : Type u} [One α] [Zero α] [LE α] [ZeroLEOneClass α] :

ZeroLEOneClass (WithBot α)

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instance WithBot.add {α : Type u} [Add α] :

Add (WithBot α)

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@[simp]

theorem WithBot.coe_add {α : Type u} [Add α] (a b : α) :

↑(a + b) = ↑a + ↑b

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@[simp]

theorem WithBot.bot_add {α : Type u} [Add α] (x : WithBot α) :

⊥ + x = ⊥

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@[simp]

theorem WithBot.add_bot {α : Type u} [Add α] (x : WithBot α) :

x + ⊥ = ⊥

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@[simp]

theorem WithBot.add_eq_bot {α : Type u} [Add α] {x y : WithBot α} :

x + y = ⊥ ↔ x = ⊥ ∨ y = ⊥

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theorem WithBot.add_ne_bot {α : Type u} [Add α] {x y : WithBot α} :

x + y ≠ ⊥ ↔ x ≠ ⊥ ∧ y ≠ ⊥

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@[simp]

theorem WithBot.bot_lt_add {α : Type u} [Add α] {x y : WithBot α} [LT α] :

⊥ < x + y ↔ ⊥ < x ∧ ⊥ < y

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theorem WithBot.add_eq_coe {α : Type u} [Add α] {a b : WithBot α} {c : α} :

a + b = ↑c ↔ ∃ (a' : α), ∃ (b' : α), ↑a' = a ∧ ↑b' = b ∧ a' + b' = c

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theorem WithBot.add_coe_eq_bot_iff {α : Type u} [Add α] {x : WithBot α} {b : α} :

x + ↑b = ⊥ ↔ x = ⊥

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theorem WithBot.coe_add_eq_bot_iff {α : Type u} [Add α] {y : WithBot α} {a : α} :

↑a + y = ⊥ ↔ y = ⊥

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theorem WithBot.add_right_inj {α : Type u} [Add α] {x y z : WithBot α} [IsRightCancelAdd α] (hz : z ≠ ⊥) :

x + z = y + z ↔ x = y

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theorem WithBot.add_right_cancel {α : Type u} [Add α] {x y z : WithBot α} [IsRightCancelAdd α] (hz : z ≠ ⊥) (h : x + z = y + z) :

x = y

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theorem WithBot.add_left_inj {α : Type u} [Add α] {x y z : WithBot α} [IsLeftCancelAdd α] (hx : x ≠ ⊥) :

x + y = x + z ↔ y = z

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theorem WithBot.add_left_cancel {α : Type u} [Add α] {x y z : WithBot α} [IsLeftCancelAdd α] (hx : x ≠ ⊥) (h : x + y = x + z) :

y = z

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@[deprecated WithBot.add_left_inj (since := "2025-02-19")]

theorem WithBot.add_left_cancel_iff {α : Type u} [Add α] {x y z : WithBot α} [IsLeftCancelAdd α] (hx : x ≠ ⊥) :

x + y = x + z ↔ y = z

Alias of WithBot.add_left_inj.

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@[deprecated WithBot.add_right_inj (since := "2025-02-19")]

theorem WithBot.add_right_cancel_iff {α : Type u} [Add α] {x y z : WithBot α} [IsRightCancelAdd α] (hz : z ≠ ⊥) :

x + z = y + z ↔ x = y

Alias of WithBot.add_right_inj.

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instance WithBot.addLeftMono {α : Type u} [Add α] [LE α] [AddLeftMono α] :

AddLeftMono (WithBot α)

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instance WithBot.addRightMono {α : Type u} [Add α] [LE α] [AddRightMono α] :

AddRightMono (WithBot α)

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instance WithBot.addLeftReflectLT {α : Type u} [Add α] [LT α] [AddLeftReflectLT α] :

AddLeftReflectLT (WithBot α)

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instance WithBot.addRightReflectLT {α : Type u} [Add α] [LT α] [AddRightReflectLT α] :

AddRightReflectLT (WithBot α)

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theorem WithBot.le_of_add_le_add_left {α : Type u} [Add α] {x y z : WithBot α} [LE α] [AddLeftReflectLE α] (hx : x ≠ ⊥) :

x + y ≤ x + z → y ≤ z

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theorem WithBot.le_of_add_le_add_right {α : Type u} [Add α] {x y z : WithBot α} [LE α] [AddRightReflectLE α] (hz : z ≠ ⊥) :

x + z ≤ y + z → x ≤ y

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theorem WithBot.add_lt_add_left {α : Type u} [Add α] {x y z : WithBot α} [LT α] [AddLeftStrictMono α] (hx : x ≠ ⊥) :

y < z → x + y < x + z

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theorem WithBot.add_lt_add_right {α : Type u} [Add α] {x y z : WithBot α} [LT α] [AddRightStrictMono α] (hz : z ≠ ⊥) :

x < y → x + z < y + z

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theorem WithBot.add_le_add_iff_left {α : Type u} [Add α] {x y z : WithBot α} [LE α] [AddLeftMono α] [AddLeftReflectLE α] (hx : x ≠ ⊥) :

x + y ≤ x + z ↔ y ≤ z

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theorem WithBot.add_le_add_iff_right {α : Type u} [Add α] {x y z : WithBot α} [LE α] [AddRightMono α] [AddRightReflectLE α] (hz : z ≠ ⊥) :

x + z ≤ y + z ↔ x ≤ y

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theorem WithBot.add_lt_add_iff_left {α : Type u} [Add α] {x y z : WithBot α} [LT α] [AddLeftStrictMono α] [AddLeftReflectLT α] (hx : x ≠ ⊥) :

x + y < x + z ↔ y < z

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theorem WithBot.add_lt_add_iff_right {α : Type u} [Add α] {x y z : WithBot α} [LT α] [AddRightStrictMono α] [AddRightReflectLT α] (hz : z ≠ ⊥) :

x + z < y + z ↔ x < y

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theorem WithBot.add_lt_add_of_le_of_lt {α : Type u} [Add α] {w x y z : WithBot α} [Preorder α] [AddLeftStrictMono α] [AddRightMono α] (hw : w ≠ ⊥) (hwy : w ≤ y) (hxz : x < z) :

w + x < y + z

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theorem WithBot.add_lt_add_of_lt_of_le {α : Type u} [Add α] {w x y z : WithBot α} [Preorder α] [AddLeftMono α] [AddRightStrictMono α] (hx : x ≠ ⊥) (hwy : w < y) (hxz : x ≤ z) :

w + x < y + z

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theorem WithBot.addLECancellable_of_ne_bot {α : Type u} [Add α] {x : WithBot α} [LE α] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] (hx : x ≠ ⊥) :

AddLECancellable x

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theorem WithBot.addLECancellable_of_lt_bot {α : Type u} [Add α] {x : WithBot α} [Preorder α] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] (hx : x < ⊥) :

AddLECancellable x

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theorem WithBot.addLECancellable_coe {α : Type u} [Add α] [LE α] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] (a : α) :

AddLECancellable ↑a

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