Loogle!

Result

Found 45 declarations mentioning WithBot.map.

• WithBot.map 📋 Mathlib.Order.WithBot
  {α : Type u_1} {β : Type u_2} (f : α → β) : WithBot α → WithBot β
• WithBot.map_coe 📋 Mathlib.Order.WithBot
  {α : Type u_1} {β : Type u_2} (f : α → β) (a : α) : WithBot.map f ↑a = ↑(f a)
• WithBot.map_bot 📋 Mathlib.Order.WithBot
  {α : Type u_1} {β : Type u_2} (f : α → β) : WithBot.map f ⊥ = ⊥
• Monotone.withBot_map 📋 Mathlib.Order.WithBot
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} : Monotone f → Monotone (WithBot.map f)
• StrictMono.withBot_map 📋 Mathlib.Order.WithBot
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} : StrictMono f → StrictMono (WithBot.map f)
• WithBot.map_eq_bot_iff 📋 Mathlib.Order.WithBot
  {α : Type u_1} {β : Type u_2} {f : α → β} {a : WithBot α} : WithBot.map f a = ⊥ ↔ a = ⊥
• WithBot.map₂_coe_left 📋 Mathlib.Order.WithBot
  {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 => f a b) b
• WithBot.map₂_coe_right 📋 Mathlib.Order.WithBot
  {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 => f x b) a
• WithBot.monotone_map_iff 📋 Mathlib.Order.WithBot
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} : Monotone (WithBot.map f) ↔ Monotone f
• WithBot.strictMono_map_iff 📋 Mathlib.Order.WithBot
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} : StrictMono (WithBot.map f) ↔ StrictMono f
• WithBot.map_eq_some_iff 📋 Mathlib.Order.WithBot
  {α : Type u_1} {β : Type u_2} {f : α → β} {y : β} {v : WithBot α} : WithBot.map f v = ↑y ↔ ∃ x, v = ↑x ∧ f x = y
• WithBot.some_eq_map_iff 📋 Mathlib.Order.WithBot
  {α : Type u_1} {β : Type u_2} {f : α → β} {y : β} {v : WithBot α} : ↑y = WithBot.map f v ↔ ∃ x, v = ↑x ∧ f x = y
• WithBot.map_comm 📋 Mathlib.Order.WithBot
  {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)
• WithBot.map_le_iff 📋 Mathlib.Order.WithBot
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {x y : WithBot α} (f : α → β) (mono_iff : ∀ {a b : α}, f a ≤ f b ↔ a ≤ b) : WithBot.map f x ≤ WithBot.map f y ↔ x ≤ y
• WithBot.map_ofDual 📋 Mathlib.Order.WithBot
  {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
• WithTop.map_ofDual 📋 Mathlib.Order.WithBot
  {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
• WithBot.map_toDual 📋 Mathlib.Order.WithBot
  {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
• WithTop.map_toDual 📋 Mathlib.Order.WithBot
  {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
• WithBot.toDual_map 📋 Mathlib.Order.WithBot
  {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)
• WithTop.toDual_map 📋 Mathlib.Order.WithBot
  {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)
• WithBot.ofDual_map 📋 Mathlib.Order.WithBot
  {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)
• WithTop.ofDual_map 📋 Mathlib.Order.WithBot
  {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)
• WithBot.map_one 📋 Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
  {α : Type u} [One α] {β : Type u_1} (f : α → β) : WithBot.map f 1 = ↑(f 1)
• WithBot.map_zero 📋 Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
  {α : Type u} [Zero α] {β : Type u_1} (f : α → β) : WithBot.map f 0 = ↑(f 0)
• WithBot.map_natCast 📋 Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
  {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : α → β} (n : ℕ) : WithBot.map f ↑n = ↑(f ↑n)
• WithBot.map_ofNat 📋 Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
  {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : α → β} (n : ℕ) [n.AtLeastTwo] : WithBot.map f (OfNat.ofNat n) = ↑(f (OfNat.ofNat n))
• WithBot.map_eq_one_iff 📋 Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
  {β : Type v} {α : Type u_1} {f : α → β} {v : WithBot α} [One β] : WithBot.map f v = 1 ↔ ∃ x, v = ↑x ∧ f x = 1
• WithBot.map_eq_zero_iff 📋 Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
  {β : Type v} {α : Type u_1} {f : α → β} {v : WithBot α} [Zero β] : WithBot.map f v = 0 ↔ ∃ x, v = ↑x ∧ f x = 0
• WithBot.one_eq_map_iff 📋 Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
  {β : Type v} {α : Type u_1} {f : α → β} {v : WithBot α} [One β] : 1 = WithBot.map f v ↔ ∃ x, v = ↑x ∧ f x = 1
• WithBot.zero_eq_map_iff 📋 Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
  {β : Type v} {α : Type u_1} {f : α → β} {v : WithBot α} [Zero β] : 0 = WithBot.map f v ↔ ∃ x, v = ↑x ∧ f x = 0
• WithBot.map_eq_natCast_iff 📋 Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
  {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : β → α} {n : ℕ} {a : WithBot β} : WithBot.map f a = ↑n ↔ ∃ x, a = ↑x ∧ f x = ↑n
• WithBot.natCast_eq_map_iff 📋 Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
  {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : β → α} {n : ℕ} {a : WithBot β} : ↑n = WithBot.map f a ↔ ∃ x, a = ↑x ∧ f x = ↑n
• WithBot.map_eq_ofNat_iff 📋 Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
  {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : β → α} {n : ℕ} [n.AtLeastTwo] {a : WithBot β} : WithBot.map f a = OfNat.ofNat n ↔ ∃ x, a = ↑x ∧ f x = ↑n
• WithBot.ofNat_eq_map_iff 📋 Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
  {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : β → α} {n : ℕ} [n.AtLeastTwo] {a : WithBot β} : OfNat.ofNat n = WithBot.map f a ↔ ∃ x, a = ↑x ∧ f x = ↑n
• OneHom.withBotMap.eq_1 📋 Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
  {M : Type u_1} {N : Type u_2} [One M] [One N] (f : OneHom M N) : f.withBotMap = { toFun := WithBot.map ⇑f, map_one' := ⋯ }
• ZeroHom.withBotMap.eq_1 📋 Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
  {M : Type u_1} {N : Type u_2} [Zero M] [Zero N] (f : ZeroHom M N) : f.withBotMap = { toFun := WithBot.map ⇑f, map_zero' := ⋯ }
• AddHom.withBotMap_apply 📋 Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
  {M : Type u_1} {N : Type u_2} [Add M] [Add N] (f : M →ₙ+ N) : ⇑f.withBotMap = WithBot.map ⇑f
• AddMonoidHom.withBotMap_apply 📋 Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
  {M : Type u_1} {N : Type u_2} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) : ⇑f.withBotMap = WithBot.map ⇑f
• OneHom.withBotMap_apply 📋 Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
  {M : Type u_1} {N : Type u_2} [One M] [One N] (f : OneHom M N) : ⇑f.withBotMap = WithBot.map ⇑f
• ZeroHom.withBotMap_apply 📋 Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
  {M : Type u_1} {N : Type u_2} [Zero M] [Zero N] (f : ZeroHom M N) : ⇑f.withBotMap = WithBot.map ⇑f
• WithBot.map_add 📋 Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
  {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 : WithBot α) : WithBot.map (⇑f) (a + b) = WithBot.map (⇑f) a + WithBot.map (⇑f) b
• OrderHom.withBotMap_coe 📋 Mathlib.Order.Hom.WithTopBot
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α →o β) : ⇑f.withBotMap = WithBot.map ⇑f
• LatticeHom.withBot_apply 📋 Mathlib.Order.Hom.WithTopBot
  {α : Type u_1} {β : Type u_2} [Lattice α] [Lattice β] (f : LatticeHom α β) (a : WithBot α) : f.withBot a = WithBot.map (⇑f) a
• Finset.map_ofDual_max 📋 Mathlib.Data.Finset.Max
  {α : Type u_2} [LinearOrder α] (s : Finset αᵒᵈ) : WithBot.map (⇑OrderDual.ofDual) s.max = (Finset.image (⇑OrderDual.ofDual) s).min
• Finset.map_toDual_max 📋 Mathlib.Data.Finset.Max
  {α : Type u_2} [LinearOrder α] (s : Finset α) : WithBot.map (⇑OrderDual.toDual) s.max = (Finset.image (⇑OrderDual.toDual) s).min

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using (by default) Ctrl-K Ctrl-S. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO.

This is Loogle revision 19971e9 serving mathlib revision bce1d65