Dependent hash map lemmas #

This file contains lemmas about Std.Data.DHashMap. Most of the lemmas require EquivBEq α and LawfulHashable α for the key type α. The easiest way to obtain these instances is to provide an instance of LawfulBEq α.

source

@[simp]

theorem Std.DHashMap.isEmpty_emptyWithCapacity {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {c : Nat} :

(emptyWithCapacity c).isEmpty = true

source

@[simp]

theorem Std.DHashMap.isEmpty_empty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} :

∅.isEmpty = true

source

@[reducible, inline, deprecated Std.DHashMap.isEmpty_empty (since := "2025-03-12")]

abbrev Std.DHashMap.isEmpty_emptyc {α : Type u_1} {β : α → Type u_2} {x✝ : BEq α} {x✝¹ : Hashable α} :

∅.isEmpty = true

Equations

@Std.DHashMap.isEmpty_emptyc = @Std.DHashMap.isEmpty_empty

Instances For

source

@[simp]

theorem Std.DHashMap.isEmpty_insert {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β k} :

(m.insert k v).isEmpty = false

source

theorem Std.DHashMap.mem_iff_contains {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} {a : α} :

a ∈ m ↔ m.contains a = true

source

theorem Std.DHashMap.contains_congr {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {a b : α} (hab : (a == b) = true) :

m.contains a = m.contains b

source

theorem Std.DHashMap.mem_congr {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {a b : α} (hab : (a == b) = true) :

a ∈ m ↔ b ∈ m

source

@[simp]

theorem Std.DHashMap.contains_emptyWithCapacity {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {a : α} {c : Nat} :

(emptyWithCapacity c).contains a = false

source

@[simp]

theorem Std.DHashMap.not_mem_emptyWithCapacity {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {a : α} {c : Nat} :

¬a ∈ emptyWithCapacity c

source

@[simp]

theorem Std.DHashMap.contains_empty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {a : α} :

∅.contains a = false

source

@[reducible, inline, deprecated Std.DHashMap.contains_empty (since := "2025-03-12")]

abbrev Std.DHashMap.contains_emptyc {α : Type u_1} {β : α → Type u_2} {x✝ : BEq α} {x✝¹ : Hashable α} {a : α} :

∅.contains a = false

Equations

@Std.DHashMap.contains_emptyc = @Std.DHashMap.contains_empty

Instances For

source

@[simp]

theorem Std.DHashMap.not_mem_empty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {a : α} :

¬a ∈ ∅

source

@[reducible, inline, deprecated Std.DHashMap.not_mem_empty (since := "2025-03-12")]

abbrev Std.DHashMap.not_mem_emptyc {α : Type u_1} {β : α → Type u_2} {x✝ : BEq α} {x✝¹ : Hashable α} {a : α} :

¬a ∈ ∅

Equations

@Std.DHashMap.not_mem_emptyc = @Std.DHashMap.not_mem_empty

Instances For

source

theorem Std.DHashMap.contains_of_isEmpty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} :

m.isEmpty = true → m.contains a = false

source

theorem Std.DHashMap.not_mem_of_isEmpty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} :

m.isEmpty = true → ¬a ∈ m

source

theorem Std.DHashMap.isEmpty_eq_false_iff_exists_contains_eq_true {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] :

m.isEmpty = false ↔ ∃ (a : α), m.contains a = true

source

theorem Std.DHashMap.isEmpty_eq_false_iff_exists_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] :

m.isEmpty = false ↔ ∃ (a : α), a ∈ m

source

theorem Std.DHashMap.isEmpty_iff_forall_contains {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] :

m.isEmpty = true ↔ ∀ (a : α), m.contains a = false

source

theorem Std.DHashMap.isEmpty_iff_forall_not_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] :

m.isEmpty = true ↔ ∀ (a : α), ¬a ∈ m

source

@[simp]

theorem Std.DHashMap.insert_eq_insert {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} {p : (a : α) × β a} :

Insert.insert p m = m.insert p.fst p.snd

source

@[simp]

theorem Std.DHashMap.singleton_eq_insert {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {p : (a : α) × β a} :

{p} = ∅.insert p.fst p.snd

source

@[simp]

theorem Std.DHashMap.contains_insert {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β k} :

(m.insert k v).contains a = (k == a || m.contains a)

source

@[simp]

theorem Std.DHashMap.mem_insert {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β k} :

a ∈ m.insert k v ↔ (k == a) = true ∨ a ∈ m

source

theorem Std.DHashMap.contains_of_contains_insert {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β k} :

(m.insert k v).contains a = true → (k == a) = false → m.contains a = true

source

theorem Std.DHashMap.mem_of_mem_insert {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β k} :

a ∈ m.insert k v → (k == a) = false → a ∈ m

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theorem Std.DHashMap.contains_insert_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β k} :

(m.insert k v).contains k = true

source

theorem Std.DHashMap.mem_insert_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β k} :

k ∈ m.insert k v

source

@[simp]

theorem Std.DHashMap.size_emptyWithCapacity {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {c : Nat} :

(emptyWithCapacity c).size = 0

source

@[simp]

theorem Std.DHashMap.size_empty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} :

∅.size = 0

source

@[reducible, inline, deprecated Std.DHashMap.size_empty (since := "2025-03-12")]

abbrev Std.DHashMap.size_emptyc {α : Type u_1} {β : α → Type u_2} {x✝ : BEq α} {x✝¹ : Hashable α} :

∅.size = 0

Equations

@Std.DHashMap.size_emptyc = @Std.DHashMap.size_empty

Instances For

source

theorem Std.DHashMap.isEmpty_eq_size_eq_zero {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} :

m.isEmpty = (m.size == 0)

source

theorem Std.DHashMap.size_insert {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β k} :

(m.insert k v).size = if k ∈ m then m.size else m.size + 1

source

theorem Std.DHashMap.size_le_size_insert {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β k} :

m.size ≤ (m.insert k v).size

source

theorem Std.DHashMap.size_insert_le {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β k} :

(m.insert k v).size ≤ m.size + 1

source

@[simp]

theorem Std.DHashMap.erase_emptyWithCapacity {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {k : α} {c : Nat} :

(emptyWithCapacity c).erase k = emptyWithCapacity c

source

@[simp]

theorem Std.DHashMap.erase_empty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {k : α} :

∅.erase k = ∅

source

@[reducible, inline, deprecated Std.DHashMap.erase_empty (since := "2025-03-12")]

abbrev Std.DHashMap.erase_emptyc {α : Type u_1} {β : α → Type u_2} {x✝ : BEq α} {x✝¹ : Hashable α} {k : α} :

∅.erase k = ∅

Equations

@Std.DHashMap.erase_emptyc = @Std.DHashMap.erase_empty

Instances For

source

@[simp]

theorem Std.DHashMap.isEmpty_erase {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} :

(m.erase k).isEmpty = (m.isEmpty || m.size == 1 && m.contains k)

source

@[simp]

theorem Std.DHashMap.contains_erase {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} :

(m.erase k).contains a = (!k == a && m.contains a)

source

@[simp]

theorem Std.DHashMap.mem_erase {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} :

a ∈ m.erase k ↔ (k == a) = false ∧ a ∈ m

source

theorem Std.DHashMap.contains_of_contains_erase {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} :

(m.erase k).contains a = true → m.contains a = true

source

theorem Std.DHashMap.mem_of_mem_erase {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} :

a ∈ m.erase k → a ∈ m

source

theorem Std.DHashMap.size_erase {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} :

(m.erase k).size = if k ∈ m then m.size - 1 else m.size

source

theorem Std.DHashMap.size_erase_le {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} :

(m.erase k).size ≤ m.size

source

theorem Std.DHashMap.size_le_size_erase {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} :

m.size ≤ (m.erase k).size + 1

source

@[simp]

theorem Std.DHashMap.containsThenInsert_fst {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} {k : α} {v : β k} :

(m.containsThenInsert k v).fst = m.contains k

source

@[simp]

theorem Std.DHashMap.containsThenInsert_snd {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} {k : α} {v : β k} :

(m.containsThenInsert k v).snd = m.insert k v

source

@[simp]

theorem Std.DHashMap.containsThenInsertIfNew_fst {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} {k : α} {v : β k} :

(m.containsThenInsertIfNew k v).fst = m.contains k

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@[simp]

theorem Std.DHashMap.containsThenInsertIfNew_snd {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} {k : α} {v : β k} :

(m.containsThenInsertIfNew k v).snd = m.insertIfNew k v

source

@[simp]

theorem Std.DHashMap.get?_emptyWithCapacity {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [LawfulBEq α] {a : α} {c : Nat} :

(emptyWithCapacity c).get? a = none

source

@[simp]

theorem Std.DHashMap.get?_empty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [LawfulBEq α] {a : α} :

∅.get? a = none

source

@[reducible, inline, deprecated Std.DHashMap.get?_empty (since := "2025-03-12")]

abbrev Std.DHashMap.get?_emptyc {α : Type u_1} {β : α → Type u_2} {x✝ : BEq α} {x✝¹ : Hashable α} [LawfulBEq α] {a : α} :

∅.get? a = none

Equations

@Std.DHashMap.get?_emptyc = @Std.DHashMap.get?_empty

Instances For

source

theorem Std.DHashMap.get?_of_isEmpty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {a : α} :

m.isEmpty = true → m.get? a = none

source

theorem Std.DHashMap.get?_insert {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {a k : α} {v : β k} :

(m.insert k v).get? a = if h : (k == a) = true then some (cast ⋯ v) else m.get? a

source

@[simp]

theorem Std.DHashMap.get?_insert_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {v : β k} :

(m.insert k v).get? k = some v

source

theorem Std.DHashMap.contains_eq_isSome_get? {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {a : α} :

m.contains a = (m.get? a).isSome

source

theorem Std.DHashMap.get?_eq_none_of_contains_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {a : α} :

m.contains a = false → m.get? a = none

source

theorem Std.DHashMap.get?_eq_none {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {a : α} :

¬a ∈ m → m.get? a = none

source

theorem Std.DHashMap.get?_erase {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k a : α} :

(m.erase k).get? a = if (k == a) = true then none else m.get? a

source

@[simp]

theorem Std.DHashMap.get?_erase_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} :

(m.erase k).get? k = none

source

@[simp]

theorem Std.DHashMap.Const.get?_emptyWithCapacity {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {a : α} {c : Nat} :

get? (emptyWithCapacity c) a = none

source

@[simp]

theorem Std.DHashMap.Const.get?_empty {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {a : α} :

get? ∅ a = none

source

@[reducible, inline, deprecated Std.DHashMap.Const.get?_empty (since := "2025-03-12")]

abbrev Std.DHashMap.Const.get?_emptyc {α : Type u_1} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type u_2} {a : α} :

get? ∅ a = none

Equations

@Std.DHashMap.Const.get?_emptyc = @Std.DHashMap.Const.get?_empty

Instances For

source

theorem Std.DHashMap.Const.get?_of_isEmpty {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {a : α} :

m.isEmpty = true → get? m a = none

source

theorem Std.DHashMap.Const.get?_insert {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β} :

get? (m.insert k v) a = if (k == a) = true then some v else get? m a

source

@[simp]

theorem Std.DHashMap.Const.get?_insert_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} :

get? (m.insert k v) k = some v

source

theorem Std.DHashMap.Const.contains_eq_isSome_get? {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {a : α} :

m.contains a = (get? m a).isSome

source

theorem Std.DHashMap.Const.get?_eq_none_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {a : α} :

m.contains a = false → get? m a = none

source

theorem Std.DHashMap.Const.get?_eq_none {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {a : α} :

¬a ∈ m → get? m a = none

source

theorem Std.DHashMap.Const.get?_erase {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k a : α} :

get? (m.erase k) a = if (k == a) = true then none else get? m a

source

@[simp]

theorem Std.DHashMap.Const.get?_erase_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} :

get? (m.erase k) k = none

source

theorem Std.DHashMap.Const.get?_eq_get? {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [LawfulBEq α] {a : α} :

get? m a = m.get? a

source

theorem Std.DHashMap.Const.get?_congr {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {a b : α} (hab : (a == b) = true) :

get? m a = get? m b

source

theorem Std.DHashMap.get_insert {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k a : α} {v : β k} {h₁ : a ∈ m.insert k v} :

(m.insert k v).get a h₁ = if h₂ : (k == a) = true then cast ⋯ v else m.get a ⋯

source

@[simp]

theorem Std.DHashMap.get_insert_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {v : β k} :

(m.insert k v).get k ⋯ = v

source

@[simp]

theorem Std.DHashMap.get_erase {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k a : α} {h' : a ∈ m.erase k} :

(m.erase k).get a h' = m.get a ⋯

source

theorem Std.DHashMap.get?_eq_some_get {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {a : α} {h : a ∈ m} :

m.get? a = some (m.get a h)

source

theorem Std.DHashMap.Const.get_insert {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β} {h₁ : a ∈ m.insert k v} :

get (m.insert k v) a h₁ = if h₂ : (k == a) = true then v else get m a ⋯

source

@[simp]

theorem Std.DHashMap.Const.get_insert_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} :

get (m.insert k v) k ⋯ = v

source

@[simp]

theorem Std.DHashMap.Const.get_erase {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k a : α} {h' : a ∈ m.erase k} :

get (m.erase k) a h' = get m a ⋯

source

theorem Std.DHashMap.Const.get?_eq_some_get {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {a : α} {h : a ∈ m} :

get? m a = some (get m a h)

source

theorem Std.DHashMap.Const.get_eq_get {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [LawfulBEq α] {a : α} {h : a ∈ m} :

get m a h = m.get a h

source

theorem Std.DHashMap.Const.get_congr {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {a b : α} (hab : (a == b) = true) {h' : a ∈ m} :

get m a h' = get m b ⋯

source

@[simp]

theorem Std.DHashMap.get!_emptyWithCapacity {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [LawfulBEq α] {a : α} [Inhabited (β a)] {c : Nat} :

(emptyWithCapacity c).get! a = default

source

@[simp]

theorem Std.DHashMap.get!_empty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [LawfulBEq α] {a : α} [Inhabited (β a)] :

∅.get! a = default

source

@[reducible, inline, deprecated Std.DHashMap.get!_empty (since := "2025-03-12")]

abbrev Std.DHashMap.get!_emptyc {α : Type u_1} {β : α → Type u_2} {x✝ : BEq α} {x✝¹ : Hashable α} [LawfulBEq α] {a : α} [Inhabited (β a)] :

∅.get! a = default

Equations

@Std.DHashMap.get!_emptyc = @Std.DHashMap.get!_empty

Instances For

source

theorem Std.DHashMap.get!_of_isEmpty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {a : α} [Inhabited (β a)] :

m.isEmpty = true → m.get! a = default

source

theorem Std.DHashMap.get!_insert {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k a : α} [Inhabited (β a)] {v : β k} :

(m.insert k v).get! a = if h : (k == a) = true then cast ⋯ v else m.get! a

source

@[simp]

theorem Std.DHashMap.get!_insert_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {a : α} [Inhabited (β a)] {b : β a} :

(m.insert a b).get! a = b

source

theorem Std.DHashMap.get!_eq_default_of_contains_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {a : α} [Inhabited (β a)] :

m.contains a = false → m.get! a = default

source

theorem Std.DHashMap.get!_eq_default {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {a : α} [Inhabited (β a)] :

¬a ∈ m → m.get! a = default

source

theorem Std.DHashMap.get!_erase {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k a : α} [Inhabited (β a)] :

(m.erase k).get! a = if (k == a) = true then default else m.get! a

source

@[simp]

theorem Std.DHashMap.get!_erase_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} [Inhabited (β k)] :

(m.erase k).get! k = default

source

theorem Std.DHashMap.get?_eq_some_get!_of_contains {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {a : α} [Inhabited (β a)] :

m.contains a = true → m.get? a = some (m.get! a)

source

theorem Std.DHashMap.get?_eq_some_get! {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {a : α} [Inhabited (β a)] :

a ∈ m → m.get? a = some (m.get! a)

source

theorem Std.DHashMap.get!_eq_get!_get? {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {a : α} [Inhabited (β a)] :

m.get! a = (m.get? a).get!

source

theorem Std.DHashMap.get_eq_get! {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {a : α} [Inhabited (β a)] {h : a ∈ m} :

m.get a h = m.get! a

source

@[simp]

theorem Std.DHashMap.Const.get!_emptyWithCapacity {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} [Inhabited β] {a : α} {c : Nat} :

get! (emptyWithCapacity c) a = default

source

@[simp]

theorem Std.DHashMap.Const.get!_empty {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} [Inhabited β] {a : α} :

get! ∅ a = default

source

@[reducible, inline, deprecated Std.DHashMap.Const.get!_empty (since := "2025-03-12")]

abbrev Std.DHashMap.Const.get!_emptyc {α : Type u_1} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type u_2} [Inhabited β] {a : α} :

get! ∅ a = default

Equations

@Std.DHashMap.Const.get!_emptyc = @Std.DHashMap.Const.get!_empty

Instances For

source

theorem Std.DHashMap.Const.get!_of_isEmpty {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {a : α} :

m.isEmpty = true → get! m a = default

source

theorem Std.DHashMap.Const.get!_insert {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {k a : α} {v : β} :

get! (m.insert k v) a = if (k == a) = true then v else get! m a

source

@[simp]

theorem Std.DHashMap.Const.get!_insert_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {k : α} {v : β} :

get! (m.insert k v) k = v

source

theorem Std.DHashMap.Const.get!_eq_default_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {a : α} :

m.contains a = false → get! m a = default

source

theorem Std.DHashMap.Const.get!_eq_default {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {a : α} :

¬a ∈ m → get! m a = default

source

theorem Std.DHashMap.Const.get!_erase {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {k a : α} :

get! (m.erase k) a = if (k == a) = true then default else get! m a

source

@[simp]

theorem Std.DHashMap.Const.get!_erase_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {k : α} :

get! (m.erase k) k = default

source

theorem Std.DHashMap.Const.get?_eq_some_get!_of_contains {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {a : α} :

m.contains a = true → get? m a = some (get! m a)

source

theorem Std.DHashMap.Const.get?_eq_some_get! {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {a : α} :

a ∈ m → get? m a = some (get! m a)

source

theorem Std.DHashMap.Const.get!_eq_get!_get? {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {a : α} :

get! m a = (get? m a).get!

source

theorem Std.DHashMap.Const.get_eq_get! {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {a : α} {h : a ∈ m} :

get m a h = get! m a

source

theorem Std.DHashMap.Const.get!_eq_get! {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [LawfulBEq α] [Inhabited β] {a : α} :

get! m a = m.get! a

source

theorem Std.DHashMap.Const.get!_congr {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {a b : α} (hab : (a == b) = true) :

get! m a = get! m b

source

@[simp]

theorem Std.DHashMap.getD_emptyWithCapacity {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [LawfulBEq α] {a : α} {fallback : β a} {c : Nat} :

(emptyWithCapacity c).getD a fallback = fallback

source

@[simp]

theorem Std.DHashMap.getD_empty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [LawfulBEq α] {a : α} {fallback : β a} :

∅.getD a fallback = fallback

source

@[reducible, inline, deprecated Std.DHashMap.getD_empty (since := "2025-03-12")]

abbrev Std.DHashMap.getD_emptyc {α : Type u_1} {β : α → Type u_2} {x✝ : BEq α} {x✝¹ : Hashable α} [LawfulBEq α] {a : α} {fallback : β a} :

∅.getD a fallback = fallback

Equations

@Std.DHashMap.getD_emptyc = @Std.DHashMap.getD_empty

Instances For

source

theorem Std.DHashMap.getD_of_isEmpty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {a : α} {fallback : β a} :

m.isEmpty = true → m.getD a fallback = fallback

source

theorem Std.DHashMap.getD_insert {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k a : α} {fallback : β a} {v : β k} :

(m.insert k v).getD a fallback = if h : (k == a) = true then cast ⋯ v else m.getD a fallback

source

@[simp]

theorem Std.DHashMap.getD_insert_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {fallback v : β k} :

(m.insert k v).getD k fallback = v

source

theorem Std.DHashMap.getD_eq_fallback_of_contains_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {a : α} {fallback : β a} :

m.contains a = false → m.getD a fallback = fallback

source

theorem Std.DHashMap.getD_eq_fallback {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {a : α} {fallback : β a} :

¬a ∈ m → m.getD a fallback = fallback

source

theorem Std.DHashMap.getD_erase {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k a : α} {fallback : β a} :

(m.erase k).getD a fallback = if (k == a) = true then fallback else m.getD a fallback

source

@[simp]

theorem Std.DHashMap.getD_erase_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {fallback : β k} :

(m.erase k).getD k fallback = fallback

source

theorem Std.DHashMap.get?_eq_some_getD_of_contains {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {a : α} {fallback : β a} :

m.contains a = true → m.get? a = some (m.getD a fallback)

source

theorem Std.DHashMap.get?_eq_some_getD {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {a : α} {fallback : β a} :

a ∈ m → m.get? a = some (m.getD a fallback)

source

theorem Std.DHashMap.getD_eq_getD_get? {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {a : α} {fallback : β a} :

m.getD a fallback = (m.get? a).getD fallback

source

theorem Std.DHashMap.get_eq_getD {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {a : α} {fallback : β a} {h : a ∈ m} :

m.get a h = m.getD a fallback

source

theorem Std.DHashMap.get!_eq_getD_default {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {a : α} [Inhabited (β a)] :

m.get! a = m.getD a default

source

@[simp]

theorem Std.DHashMap.Const.getD_emptyWithCapacity {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {a : α} {fallback : β} {c : Nat} :

getD (emptyWithCapacity c) a fallback = fallback

source

@[simp]

theorem Std.DHashMap.Const.getD_empty {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {a : α} {fallback : β} :

getD ∅ a fallback = fallback

source

@[reducible, inline, deprecated Std.DHashMap.Const.getD_empty (since := "2025-03-12")]

abbrev Std.DHashMap.Const.getD_emptyc {α : Type u_1} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type u_2} {a : α} {fallback : β} :

getD ∅ a fallback = fallback

Equations

@Std.DHashMap.Const.getD_emptyc = @Std.DHashMap.Const.getD_empty

Instances For

source

theorem Std.DHashMap.Const.getD_of_isEmpty {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {a : α} {fallback : β} :

m.isEmpty = true → getD m a fallback = fallback

source

theorem Std.DHashMap.Const.getD_insert {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k a : α} {fallback v : β} :

getD (m.insert k v) a fallback = if (k == a) = true then v else getD m a fallback

source

@[simp]

theorem Std.DHashMap.Const.getD_insert_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} {fallback v : β} :

getD (m.insert k v) k fallback = v

source

theorem Std.DHashMap.Const.getD_eq_fallback_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {a : α} {fallback : β} :

m.contains a = false → getD m a fallback = fallback

source

theorem Std.DHashMap.Const.getD_eq_fallback {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {a : α} {fallback : β} :

¬a ∈ m → getD m a fallback = fallback

source

theorem Std.DHashMap.Const.getD_erase {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k a : α} {fallback : β} :

getD (m.erase k) a fallback = if (k == a) = true then fallback else getD m a fallback

source

@[simp]

theorem Std.DHashMap.Const.getD_erase_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} {fallback : β} :

getD (m.erase k) k fallback = fallback

source

theorem Std.DHashMap.Const.get?_eq_some_getD_of_contains {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {a : α} {fallback : β} :

m.contains a = true → get? m a = some (getD m a fallback)

source

theorem Std.DHashMap.Const.get?_eq_some_getD {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {a : α} {fallback : β} :

a ∈ m → get? m a = some (getD m a fallback)

source

theorem Std.DHashMap.Const.getD_eq_getD_get? {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {a : α} {fallback : β} :

getD m a fallback = (get? m a).getD fallback

source

theorem Std.DHashMap.Const.get_eq_getD {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {a : α} {fallback : β} {h : a ∈ m} :

get m a h = getD m a fallback

source

theorem Std.DHashMap.Const.get!_eq_getD_default {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {a : α} :

get! m a = getD m a default

source

theorem Std.DHashMap.Const.getD_eq_getD {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [LawfulBEq α] {a : α} {fallback : β} :

getD m a fallback = m.getD a fallback

source

theorem Std.DHashMap.Const.getD_congr {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {a b : α} {fallback : β} (hab : (a == b) = true) :

getD m a fallback = getD m b fallback

source

@[simp]

theorem Std.DHashMap.getKey?_emptyWithCapacity {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {a : α} {c : Nat} :

(emptyWithCapacity c).getKey? a = none

source

@[simp]

theorem Std.DHashMap.getKey?_empty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {a : α} :

∅.getKey? a = none

source

@[reducible, inline, deprecated Std.DHashMap.getKey?_empty (since := "2025-03-12")]

abbrev Std.DHashMap.getKey?_emptyc {α : Type u_1} {β : α → Type u_2} {x✝ : BEq α} {x✝¹ : Hashable α} {a : α} :

∅.getKey? a = none

Equations

@Std.DHashMap.getKey?_emptyc = @Std.DHashMap.getKey?_empty

Instances For

source

theorem Std.DHashMap.getKey?_of_isEmpty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} :

m.isEmpty = true → m.getKey? a = none

source

theorem Std.DHashMap.getKey?_insert {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {a k : α} {v : β k} :

(m.insert k v).getKey? a = if (k == a) = true then some k else m.getKey? a

source

@[simp]

theorem Std.DHashMap.getKey?_insert_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β k} :

(m.insert k v).getKey? k = some k

source

theorem Std.DHashMap.contains_eq_isSome_getKey? {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} :

m.contains a = (m.getKey? a).isSome

source

theorem Std.DHashMap.getKey?_eq_none_of_contains_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} :

m.contains a = false → m.getKey? a = none

source

theorem Std.DHashMap.getKey?_eq_none {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} :

¬a ∈ m → m.getKey? a = none

source

theorem Std.DHashMap.getKey?_erase {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} :

(m.erase k).getKey? a = if (k == a) = true then none else m.getKey? a

source

@[simp]

theorem Std.DHashMap.getKey?_erase_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} :

(m.erase k).getKey? k = none

source

theorem Std.DHashMap.getKey?_beq {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} :

Option.all (fun (x : α) => x == k) (m.getKey? k) = true

source

theorem Std.DHashMap.getKey?_congr {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k k' : α} (h : (k == k') = true) :

m.getKey? k = m.getKey? k'

source

theorem Std.DHashMap.getKey?_eq_some_of_contains {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} (h : m.contains k = true) :

m.getKey? k = some k

source

theorem Std.DHashMap.getKey?_eq_some {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} (h : k ∈ m) :

m.getKey? k = some k

source

theorem Std.DHashMap.getKey_insert {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β k} {h₁ : a ∈ m.insert k v} :

(m.insert k v).getKey a h₁ = if h₂ : (k == a) = true then k else m.getKey a ⋯

source

@[simp]

theorem Std.DHashMap.getKey_insert_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β k} :

(m.insert k v).getKey k ⋯ = k

source

@[simp]

theorem Std.DHashMap.getKey_erase {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {h' : a ∈ m.erase k} :

(m.erase k).getKey a h' = m.getKey a ⋯

source

theorem Std.DHashMap.getKey?_eq_some_getKey {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} {h : a ∈ m} :

m.getKey? a = some (m.getKey a h)

source

theorem Std.DHashMap.getKey_beq {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (h : k ∈ m) :

(m.getKey k h == k) = true

source

theorem Std.DHashMap.getKey_congr {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k₁ k₂ : α} (h : (k₁ == k₂) = true) (h₁ : k₁ ∈ m) :

m.getKey k₁ h₁ = m.getKey k₂ ⋯

source

theorem Std.DHashMap.getKey_eq {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} (h : k ∈ m) :

m.getKey k h = k

source

@[simp]

theorem Std.DHashMap.getKey!_emptyWithCapacity {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [Inhabited α] {a : α} {c : Nat} :

(emptyWithCapacity c).getKey! a = default

source

@[simp]

theorem Std.DHashMap.getKey!_empty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [Inhabited α] {a : α} :

∅.getKey! a = default

source

@[reducible, inline, deprecated Std.DHashMap.getKey!_empty (since := "2025-03-12")]

abbrev Std.DHashMap.getKey!_emptyc {α : Type u_1} {β : α → Type u_2} {x✝ : BEq α} {x✝¹ : Hashable α} [Inhabited α] {a : α} :

∅.getKey! a = default

Equations

@Std.DHashMap.getKey!_emptyc = @Std.DHashMap.getKey!_empty

Instances For

source

theorem Std.DHashMap.getKey!_of_isEmpty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {a : α} :

m.isEmpty = true → m.getKey! a = default

source

theorem Std.DHashMap.getKey!_insert {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k a : α} {v : β k} :

(m.insert k v).getKey! a = if (k == a) = true then k else m.getKey! a

source

@[simp]

theorem Std.DHashMap.getKey!_insert_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {a : α} {b : β a} :

(m.insert a b).getKey! a = a

source

theorem Std.DHashMap.getKey!_eq_default_of_contains_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {a : α} :

m.contains a = false → m.getKey! a = default

source

theorem Std.DHashMap.getKey!_eq_default {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {a : α} :

¬a ∈ m → m.getKey! a = default

source

theorem Std.DHashMap.getKey!_erase {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k a : α} :

(m.erase k).getKey! a = if (k == a) = true then default else m.getKey! a

source

@[simp]

theorem Std.DHashMap.getKey!_erase_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k : α} :

(m.erase k).getKey! k = default

source

theorem Std.DHashMap.getKey?_eq_some_getKey!_of_contains {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {a : α} :

m.contains a = true → m.getKey? a = some (m.getKey! a)

source

theorem Std.DHashMap.getKey?_eq_some_getKey! {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {a : α} :

a ∈ m → m.getKey? a = some (m.getKey! a)

source

theorem Std.DHashMap.getKey!_eq_get!_getKey? {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {a : α} :

m.getKey! a = (m.getKey? a).get!

source

theorem Std.DHashMap.getKey_eq_getKey! {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {a : α} {h : a ∈ m} :

m.getKey a h = m.getKey! a

source

theorem Std.DHashMap.getKey!_congr {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k k' : α} (h : (k == k') = true) :

m.getKey! k = m.getKey! k'

source

theorem Std.DHashMap.getKey!_eq_of_contains {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] [Inhabited α] {k : α} (h : m.contains k = true) :

m.getKey! k = k

source

theorem Std.DHashMap.getKey!_eq_of_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] [Inhabited α] {k : α} (h : k ∈ m) :

m.getKey! k = k

source

@[simp]

theorem Std.DHashMap.getKeyD_emptyWithCapacity {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {a fallback : α} {c : Nat} :

(emptyWithCapacity c).getKeyD a fallback = fallback

source

@[simp]

theorem Std.DHashMap.getKeyD_empty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {a fallback : α} :

∅.getKeyD a fallback = fallback

source

@[reducible, inline, deprecated Std.DHashMap.getKeyD_empty (since := "2025-03-12")]

abbrev Std.DHashMap.getKeyD_emptyc {α : Type u_1} {β : α → Type u_2} {x✝ : BEq α} {x✝¹ : Hashable α} {a fallback : α} :

∅.getKeyD a fallback = fallback

Equations

@Std.DHashMap.getKeyD_emptyc = @Std.DHashMap.getKeyD_empty

Instances For

source

theorem Std.DHashMap.getKeyD_of_isEmpty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {a fallback : α} :

m.isEmpty = true → m.getKeyD a fallback = fallback

source

theorem Std.DHashMap.getKeyD_insert {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k a fallback : α} {v : β k} :

(m.insert k v).getKeyD a fallback = if (k == a) = true then k else m.getKeyD a fallback

source

@[simp]

theorem Std.DHashMap.getKeyD_insert_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k fallback : α} {v : β k} :

(m.insert k v).getKeyD k fallback = k

source

theorem Std.DHashMap.getKeyD_eq_fallback_of_contains_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {a fallback : α} :

m.contains a = false → m.getKeyD a fallback = fallback

source

theorem Std.DHashMap.getKeyD_eq_fallback {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {a fallback : α} :

¬a ∈ m → m.getKeyD a fallback = fallback

source

theorem Std.DHashMap.getKeyD_erase {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k a fallback : α} :

(m.erase k).getKeyD a fallback = if (k == a) = true then fallback else m.getKeyD a fallback

source

@[simp]

theorem Std.DHashMap.getKeyD_erase_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k fallback : α} :

(m.erase k).getKeyD k fallback = fallback

source

theorem Std.DHashMap.getKey?_eq_some_getKeyD_of_contains {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {a fallback : α} :

m.contains a = true → m.getKey? a = some (m.getKeyD a fallback)

source

theorem Std.DHashMap.getKey?_eq_some_getKeyD {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {a fallback : α} :

a ∈ m → m.getKey? a = some (m.getKeyD a fallback)

source

theorem Std.DHashMap.getKeyD_eq_getD_getKey? {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {a fallback : α} :

m.getKeyD a fallback = (m.getKey? a).getD fallback

source

theorem Std.DHashMap.getKey_eq_getKeyD {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {a fallback : α} {h : a ∈ m} :

m.getKey a h = m.getKeyD a fallback

source

theorem Std.DHashMap.getKey!_eq_getKeyD_default {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {a : α} :

m.getKey! a = m.getKeyD a default

source

theorem Std.DHashMap.getKeyD_congr {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k k' fallback : α} (h : (k == k') = true) :

m.getKeyD k fallback = m.getKeyD k' fallback

source

theorem Std.DHashMap.getKeyD_eq_of_contains {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k fallback : α} (h : m.contains k = true) :

m.getKeyD k fallback = k

source

theorem Std.DHashMap.getKeyD_eq_of_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k fallback : α} (h : k ∈ m) :

m.getKeyD k fallback = k

source

@[simp]

theorem Std.DHashMap.isEmpty_insertIfNew {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β k} :

(m.insertIfNew k v).isEmpty = false

source

@[simp]

theorem Std.DHashMap.contains_insertIfNew {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β k} :

(m.insertIfNew k v).contains a = (k == a || m.contains a)

source

@[simp]

theorem Std.DHashMap.mem_insertIfNew {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β k} :

a ∈ m.insertIfNew k v ↔ (k == a) = true ∨ a ∈ m

source

theorem Std.DHashMap.contains_insertIfNew_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β k} :

(m.insertIfNew k v).contains k = true

source

theorem Std.DHashMap.mem_insertIfNew_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β k} :

k ∈ m.insertIfNew k v

source

theorem Std.DHashMap.contains_of_contains_insertIfNew {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β k} :

(m.insertIfNew k v).contains a = true → (k == a) = false → m.contains a = true

source

theorem Std.DHashMap.mem_of_mem_insertIfNew {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β k} :

a ∈ m.insertIfNew k v → (k == a) = false → a ∈ m

source

theorem Std.DHashMap.contains_of_contains_insertIfNew' {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β k} :

(m.insertIfNew k v).contains a = true → ¬((k == a) = true ∧ m.contains k = false) → m.contains a = true

This is a restatement of contains_of_contains_insertIfNew that is written to exactly match the proof obligation in the statement of get_insertIfNew.

source

theorem Std.DHashMap.mem_of_mem_insertIfNew' {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β k} :

a ∈ m.insertIfNew k v → ¬((k == a) = true ∧ ¬k ∈ m) → a ∈ m

This is a restatement of mem_of_mem_insertIfNew that is written to exactly match the proof obligation in the statement of get_insertIfNew.

source

theorem Std.DHashMap.size_insertIfNew {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β k} :

(m.insertIfNew k v).size = if k ∈ m then m.size else m.size + 1

source

theorem Std.DHashMap.size_le_size_insertIfNew {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β k} :

m.size ≤ (m.insertIfNew k v).size

source

theorem Std.DHashMap.size_insertIfNew_le {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β k} :

(m.insertIfNew k v).size ≤ m.size + 1

source

theorem Std.DHashMap.get?_insertIfNew {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k a : α} {v : β k} :

(m.insertIfNew k v).get? a = if h : (k == a) = true ∧ ¬k ∈ m then some (cast ⋯ v) else m.get? a

source

theorem Std.DHashMap.get_insertIfNew {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k a : α} {v : β k} {h₁ : a ∈ m.insertIfNew k v} :

(m.insertIfNew k v).get a h₁ = if h₂ : (k == a) = true ∧ ¬k ∈ m then cast ⋯ v else m.get a ⋯

source

theorem Std.DHashMap.get!_insertIfNew {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k a : α} [Inhabited (β a)] {v : β k} :

(m.insertIfNew k v).get! a = if h : (k == a) = true ∧ ¬k ∈ m then cast ⋯ v else m.get! a

source

theorem Std.DHashMap.getD_insertIfNew {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k a : α} {fallback : β a} {v : β k} :

(m.insertIfNew k v).getD a fallback = if h : (k == a) = true ∧ ¬k ∈ m then cast ⋯ v else m.getD a fallback

source

theorem Std.DHashMap.Const.get?_insertIfNew {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β} :

get? (m.insertIfNew k v) a = if (k == a) = true ∧ ¬k ∈ m then some v else get? m a

source

theorem Std.DHashMap.Const.get_insertIfNew {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β} {h₁ : a ∈ m.insertIfNew k v} :

get (m.insertIfNew k v) a h₁ = if h₂ : (k == a) = true ∧ ¬k ∈ m then v else get m a ⋯

source

theorem Std.DHashMap.Const.get!_insertIfNew {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {k a : α} {v : β} :

get! (m.insertIfNew k v) a = if (k == a) = true ∧ ¬k ∈ m then v else get! m a

source

theorem Std.DHashMap.Const.getD_insertIfNew {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k a : α} {fallback v : β} :

getD (m.insertIfNew k v) a fallback = if (k == a) = true ∧ ¬k ∈ m then v else getD m a fallback

source

theorem Std.DHashMap.getKey?_insertIfNew {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β k} :

(m.insertIfNew k v).getKey? a = if (k == a) = true ∧ ¬k ∈ m then some k else m.getKey? a

source

theorem Std.DHashMap.getKey_insertIfNew {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β k} {h₁ : a ∈ m.insertIfNew k v} :

(m.insertIfNew k v).getKey a h₁ = if h₂ : (k == a) = true ∧ ¬k ∈ m then k else m.getKey a ⋯

source

theorem Std.DHashMap.getKey!_insertIfNew {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k a : α} {v : β k} :

(m.insertIfNew k v).getKey! a = if (k == a) = true ∧ ¬k ∈ m then k else m.getKey! a

source

theorem Std.DHashMap.getKeyD_insertIfNew {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k a fallback : α} {v : β k} :

(m.insertIfNew k v).getKeyD a fallback = if (k == a) = true ∧ ¬k ∈ m then k else m.getKeyD a fallback

source

@[simp]

theorem Std.DHashMap.getThenInsertIfNew?_fst {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {v : β k} :

(m.getThenInsertIfNew? k v).fst = m.get? k

source

@[simp]

theorem Std.DHashMap.getThenInsertIfNew?_snd {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {v : β k} :

(m.getThenInsertIfNew? k v).snd = m.insertIfNew k v

source

@[simp]

theorem Std.DHashMap.Const.getThenInsertIfNew?_fst {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} {k : α} {v : β} :

(getThenInsertIfNew? m k v).fst = get? m k

source

@[simp]

theorem Std.DHashMap.Const.getThenInsertIfNew?_snd {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} {k : α} {v : β} :

(getThenInsertIfNew? m k v).snd = m.insertIfNew k v

source

@[simp]

theorem Std.DHashMap.length_keys {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] :

m.keys.length = m.size

source

@[simp]

theorem Std.DHashMap.isEmpty_keys {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] :

m.keys.isEmpty = m.isEmpty

source

@[simp]

theorem Std.DHashMap.contains_keys {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} :

m.keys.contains k = m.contains k

source

@[simp]

theorem Std.DHashMap.mem_keys {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} :

k ∈ m.keys ↔ k ∈ m

source

theorem Std.DHashMap.distinct_keys {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] :

List.Pairwise (fun (a b : α) => (a == b) = false) m.keys

source

@[simp]

theorem Std.DHashMap.map_fst_toList_eq_keys {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] :

List.map Sigma.fst m.val.toList = m.val.keys

source

@[simp, deprecated Std.DHashMap.map_fst_toList_eq_keys (since := "2025-02-28")]

theorem Std.DHashMap.map_sigma_fst_toList_eq_keys {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] :

List.map Sigma.fst m.val.toList = m.val.keys

source

@[simp]

theorem Std.DHashMap.length_toList {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] :

m.toList.length = m.size

source

@[simp]

theorem Std.DHashMap.isEmpty_toList {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] :

m.toList.isEmpty = m.isEmpty

source

@[simp]

theorem Std.DHashMap.mem_toList_iff_get?_eq_some {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {v : β k} :

⟨k, v⟩ ∈ m.toList ↔ m.get? k = some v

source

theorem Std.DHashMap.find?_toList_eq_some_iff_get?_eq_some {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {v : β k} :

List.find? (fun (x : (a : α) × β a) => x.fst == k) m.toList = some ⟨k, v⟩ ↔ m.get? k = some v

source

theorem Std.DHashMap.find?_toList_eq_none_iff_contains_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} :

List.find? (fun (x : (a : α) × β a) => x.fst == k) m.toList = none ↔ m.contains k = false

source

@[simp]

theorem Std.DHashMap.find?_toList_eq_none_iff_not_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} :

List.find? (fun (x : (a : α) × β a) => x.fst == k) m.toList = none ↔ ¬k ∈ m

source

theorem Std.DHashMap.distinct_keys_toList {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] :

List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) m.toList

source

@[simp]

theorem Std.DHashMap.Const.map_fst_toList_eq_keys {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] :

List.map Prod.fst (toList m) = m.keys

source

@[simp, deprecated Std.DHashMap.Const.map_fst_toList_eq_keys (since := "2025-02-28")]

theorem Std.DHashMap.Const.map_prod_fst_toList_eq_keys {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] :

List.map Prod.fst (toList m) = m.keys

source

@[simp]

theorem Std.DHashMap.Const.length_toList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] :

(toList m).length = m.size

source

@[simp]

theorem Std.DHashMap.Const.isEmpty_toList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] :

(toList m).isEmpty = m.isEmpty

source

@[simp]

theorem Std.DHashMap.Const.mem_toList_iff_get?_eq_some {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [LawfulBEq α] {k : α} {v : β} :

(k, v) ∈ toList m ↔ get? m k = some v

source

@[simp]

theorem Std.DHashMap.Const.mem_toList_iff_getKey?_eq_some_and_get?_eq_some {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} :

(k, v) ∈ toList m ↔ m.getKey? k = some k ∧ get? m k = some v

source

theorem Std.DHashMap.Const.get?_eq_some_iff_exists_beq_and_mem_toList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} :

get? m k = some v ↔ ∃ (k' : α), (k == k') = true ∧ (k', v) ∈ toList m

source

theorem Std.DHashMap.Const.find?_toList_eq_some_iff_getKey?_eq_some_and_get?_eq_some {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k k' : α} {v : β} :

List.find? (fun (a : α × β) => a.fst == k) (toList m) = some (k', v) ↔ m.getKey? k = some k' ∧ get? m k = some v

source

theorem Std.DHashMap.Const.find?_toList_eq_none_iff_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} :

List.find? (fun (x : α × β) => x.fst == k) (toList m) = none ↔ m.contains k = false

source

@[simp]

theorem Std.DHashMap.Const.find?_toList_eq_none_iff_not_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} :

List.find? (fun (x : α × β) => x.fst == k) (toList m) = none ↔ ¬k ∈ m

source

theorem Std.DHashMap.Const.distinct_keys_toList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] :

List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) (toList m)

source

theorem Std.DHashMap.foldM_eq_foldlM_toList {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} {δ : Type w} {m' : Type w → Type w} [Monad m'] [LawfulMonad m'] {f : δ → (a : α) → β a → m' δ} {init : δ} :

foldM f init m = List.foldlM (fun (a : δ) (b : (a : α) × β a) => f a b.fst b.snd) init m.toList

source

theorem Std.DHashMap.fold_eq_foldl_toList {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} {δ : Type w} {f : δ → (a : α) → β a → δ} {init : δ} :

fold f init m = List.foldl (fun (a : δ) (b : (a : α) × β a) => f a b.fst b.snd) init m.toList

source

@[simp]

theorem Std.DHashMap.forM_eq_forM {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} {m' : Type w → Type w} [Monad m'] [LawfulMonad m'] {f : (a : α) → β a → m' PUnit} :

forM f m = ForM.forM m fun (a : (a : α) × β a) => f a.fst a.snd

source

theorem Std.DHashMap.forM_eq_forM_toList {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} {m' : Type w → Type w} [Monad m'] [LawfulMonad m'] {f : (a : α) × β a → m' PUnit} :

ForM.forM m f = ForM.forM m.toList f

source

@[simp]

theorem Std.DHashMap.forIn_eq_forIn {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} {δ : Type w} {m' : Type w → Type w} [Monad m'] [LawfulMonad m'] {f : (a : α) → β a → δ → m' (ForInStep δ)} {init : δ} :

forIn f init m = ForIn.forIn m init fun (a : (a : α) × β a) (b : δ) => f a.fst a.snd b

source

theorem Std.DHashMap.forIn_eq_forIn_toList {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} {δ : Type w} {m' : Type w → Type w} [Monad m'] [LawfulMonad m'] {f : (a : α) × β a → δ → m' (ForInStep δ)} {init : δ} :

ForIn.forIn m init f = ForIn.forIn m.toList init f

source

theorem Std.DHashMap.foldM_eq_foldlM_keys {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} {δ : Type w} {m' : Type w → Type w} [Monad m'] [LawfulMonad m'] {f : δ → α → m' δ} {init : δ} :

foldM (fun (d : δ) (a : α) (x : β a) => f d a) init m = List.foldlM f init m.keys

source

theorem Std.DHashMap.fold_eq_foldl_keys {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} {δ : Type w} {f : δ → α → δ} {init : δ} :

fold (fun (d : δ) (a : α) (x : β a) => f d a) init m = List.foldl f init m.keys

source

theorem Std.DHashMap.forM_eq_forM_keys {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} {m' : Type w → Type w} [Monad m'] [LawfulMonad m'] {f : α → m' PUnit} :

(ForM.forM m fun (a : (a : α) × β a) => f a.fst) = m.keys.forM f

source

theorem Std.DHashMap.forIn_eq_forIn_keys {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} {δ : Type w} {m' : Type w → Type w} [Monad m'] [LawfulMonad m'] {f : α → δ → m' (ForInStep δ)} {init : δ} :

(ForIn.forIn m init fun (a : (a : α) × β a) (d : δ) => f a.fst d) = ForIn.forIn m.keys init f

source

theorem Std.DHashMap.Const.foldM_eq_foldlM_toList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {δ : Type w} {m' : Type w → Type w} {β : Type v} {m : DHashMap α fun (x : α) => β} [Monad m'] [LawfulMonad m'] {f : δ → α → β → m' δ} {init : δ} :

foldM f init m = List.foldlM (fun (a : δ) (b : α × β) => f a b.fst b.snd) init (toList m)

source

theorem Std.DHashMap.Const.fold_eq_foldl_toList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {δ : Type w} {β : Type v} {m : DHashMap α fun (x : α) => β} {f : δ → α → β → δ} {init : δ} :

fold f init m = List.foldl (fun (a : δ) (b : α × β) => f a b.fst b.snd) init (toList m)

source

theorem Std.DHashMap.Const.forM_eq_forMUncurried {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m' : Type w → Type w} {β : Type v} {m : DHashMap α fun (x : α) => β} [Monad m'] [LawfulMonad m'] {f : α → β → m' PUnit} :

forM f m = forMUncurried (fun (a : α × β) => f a.fst a.snd) m

source

theorem Std.DHashMap.Const.forMUncurried_eq_forM_toList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m' : Type w → Type w} {β : Type v} {m : DHashMap α fun (x : α) => β} [Monad m'] [LawfulMonad m'] {f : α × β → m' PUnit} :

forMUncurried f m = (toList m).forM f

source

@[deprecated Std.DHashMap.Const.forMUncurried_eq_forM_toList (since := "2025-03-02")]

theorem Std.DHashMap.Const.forM_eq_forM_toList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m' : Type w → Type w} {β : Type v} {m : DHashMap α fun (x : α) => β} [Monad m'] [LawfulMonad m'] {f : α → β → m' PUnit} :

forM f m = (toList m).forM fun (a : α × β) => f a.fst a.snd

Deprecated, use forMUncurried_eq_forM_toList together with forM_eq_forMUncurried instead.

source

theorem Std.DHashMap.Const.forIn_eq_forInUncurried {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {δ : Type w} {m' : Type w → Type w} {β : Type v} {m : DHashMap α fun (x : α) => β} [Monad m'] [LawfulMonad m'] {f : α → β → δ → m' (ForInStep δ)} {init : δ} :

forIn f init m = forInUncurried (fun (a : α × β) (b : δ) => f a.fst a.snd b) init m

source

theorem Std.DHashMap.Const.forInUncurried_eq_forIn_toList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {δ : Type w} {m' : Type w → Type w} {β : Type v} {m : DHashMap α fun (x : α) => β} [Monad m'] [LawfulMonad m'] {f : α × β → δ → m' (ForInStep δ)} {init : δ} :

forInUncurried f init m = ForIn.forIn (toList m) init f

source

@[deprecated Std.DHashMap.Const.forInUncurried_eq_forIn_toList (since := "2025-03-02")]

theorem Std.DHashMap.Const.forIn_eq_forIn_toList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {δ : Type w} {m' : Type w → Type w} {β : Type v} {m : DHashMap α fun (x : α) => β} [Monad m'] [LawfulMonad m'] {f : α × β → δ → m' (ForInStep δ)} {init : δ} :

forInUncurried f init m = ForIn.forIn (toList m) init f

Deprecated, use forInUncurried_eq_forIn_toList together with forIn_eq_forInUncurried instead.

source

@[deprecated Std.DHashMap.foldM_eq_foldlM_keys (since := "2025-02-28")]

theorem Std.DHashMap.Const.foldM_eq_foldlM_keys {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {δ : Type w} {m' : Type w → Type w} {m : DHashMap α fun (x : α) => Unit} [Monad m'] [LawfulMonad m'] {f : δ → α → m' δ} {init : δ} :

foldM (fun (d : δ) (a : α) (x : Unit) => f d a) init m = List.foldlM f init m.keys

source

@[deprecated Std.DHashMap.fold_eq_foldl_keys (since := "2025-02-28")]

theorem Std.DHashMap.Const.fold_eq_foldl_keys {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {δ : Type w} {m : DHashMap α fun (x : α) => Unit} {f : δ → α → δ} {init : δ} :

fold (fun (d : δ) (a : α) (x : Unit) => f d a) init m = List.foldl f init m.keys

source

@[deprecated Std.DHashMap.forM_eq_forM_keys (since := "2025-02-28")]

theorem Std.DHashMap.Const.forM_eq_forM_keys {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m' : Type w → Type w} {m : DHashMap α fun (x : α) => Unit} [Monad m'] [LawfulMonad m'] {f : α → m' PUnit} :

forM (fun (a : α) (x : Unit) => f a) m = m.keys.forM f

source

@[deprecated Std.DHashMap.forIn_eq_forIn_keys (since := "2025-02-28")]

theorem Std.DHashMap.Const.forIn_eq_forIn_keys {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {δ : Type w} {m' : Type w → Type w} {m : DHashMap α fun (x : α) => Unit} [Monad m'] [LawfulMonad m'] {f : α → δ → m' (ForInStep δ)} {init : δ} :

forIn (fun (a : α) (x : Unit) (d : δ) => f a d) init m = ForIn.forIn m.keys init f

source

@[simp]

theorem Std.DHashMap.insertMany_nil {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} :

m.insertMany [] = m

source

@[simp]

theorem Std.DHashMap.insertMany_list_singleton {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} {k : α} {v : β k} :

m.insertMany [⟨k, v⟩] = m.insert k v

source

theorem Std.DHashMap.insertMany_cons {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} {l : List ((a : α) × β a)} {k : α} {v : β k} :

m.insertMany (⟨k, v⟩ :: l) = (m.insert k v).insertMany l

source

@[simp]

theorem Std.DHashMap.contains_insertMany_list {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} {k : α} :

(m.insertMany l).contains k = (m.contains k || (List.map Sigma.fst l).contains k)

source

@[simp]

theorem Std.DHashMap.mem_insertMany_list {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} {k : α} :

k ∈ m.insertMany l ↔ k ∈ m ∨ (List.map Sigma.fst l).contains k = true

source

theorem Std.DHashMap.mem_of_mem_insertMany_list {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} {k : α} (mem : k ∈ m.insertMany l) (contains_eq_false : (List.map Sigma.fst l).contains k = false) :

k ∈ m

source

theorem Std.DHashMap.get?_insertMany_list_of_contains_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {l : List ((a : α) × β a)} {k : α} (contains_eq_false : (List.map Sigma.fst l).contains k = false) :

(m.insertMany l).get? k = m.get? k

source

theorem Std.DHashMap.get?_insertMany_list_of_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {l : List ((a : α) × β a)} {k k' : α} (k_beq : (k == k') = true) {v : β k} (distinct : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) l) (mem : ⟨k, v⟩ ∈ l) :

(m.insertMany l).get? k' = some (cast ⋯ v)

source

theorem Std.DHashMap.get_insertMany_list_of_contains_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {l : List ((a : α) × β a)} {k : α} (contains_eq_false : (List.map Sigma.fst l).contains k = false) {h : k ∈ m.insertMany l} :

(m.insertMany l).get k h = m.get k ⋯

source

theorem Std.DHashMap.get_insertMany_list_of_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {l : List ((a : α) × β a)} {k k' : α} (k_beq : (k == k') = true) {v : β k} (distinct : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) l) (mem : ⟨k, v⟩ ∈ l) {h : k' ∈ m.insertMany l} :

(m.insertMany l).get k' h = cast ⋯ v

source

theorem Std.DHashMap.get!_insertMany_list_of_contains_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {l : List ((a : α) × β a)} {k : α} [Inhabited (β k)] (contains_eq_false : (List.map Sigma.fst l).contains k = false) :

(m.insertMany l).get! k = m.get! k

source

theorem Std.DHashMap.get!_insertMany_list_of_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {l : List ((a : α) × β a)} {k k' : α} (k_beq : (k == k') = true) {v : β k} [Inhabited (β k')] (distinct : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) l) (mem : ⟨k, v⟩ ∈ l) :

(m.insertMany l).get! k' = cast ⋯ v

source

theorem Std.DHashMap.getD_insertMany_list_of_contains_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {l : List ((a : α) × β a)} {k : α} {fallback : β k} (contains_eq_false : (List.map Sigma.fst l).contains k = false) :

(m.insertMany l).getD k fallback = m.getD k fallback

source

theorem Std.DHashMap.getD_insertMany_list_of_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {l : List ((a : α) × β a)} {k k' : α} (k_beq : (k == k') = true) {v : β k} {fallback : β k'} (distinct : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) l) (mem : ⟨k, v⟩ ∈ l) :

(m.insertMany l).getD k' fallback = cast ⋯ v

source

theorem Std.DHashMap.getKey?_insertMany_list_of_contains_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} {k : α} (contains_eq_false : (List.map Sigma.fst l).contains k = false) :

(m.insertMany l).getKey? k = m.getKey? k

source

theorem Std.DHashMap.getKey?_insertMany_list_of_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} {k k' : α} (k_beq : (k == k') = true) (distinct : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) l) (mem : k ∈ List.map Sigma.fst l) :

(m.insertMany l).getKey? k' = some k

source

theorem Std.DHashMap.getKey_insertMany_list_of_contains_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} {k : α} (contains_eq_false : (List.map Sigma.fst l).contains k = false) {h : k ∈ m.insertMany l} :

(m.insertMany l).getKey k h = m.getKey k ⋯

source

theorem Std.DHashMap.getKey_insertMany_list_of_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} {k k' : α} (k_beq : (k == k') = true) (distinct : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) l) (mem : k ∈ List.map Sigma.fst l) {h : k' ∈ m.insertMany l} :

(m.insertMany l).getKey k' h = k

source

theorem Std.DHashMap.getKey!_insertMany_list_of_contains_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {l : List ((a : α) × β a)} {k : α} (contains_eq_false : (List.map Sigma.fst l).contains k = false) :

(m.insertMany l).getKey! k = m.getKey! k

source

theorem Std.DHashMap.getKey!_insertMany_list_of_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {l : List ((a : α) × β a)} {k k' : α} (k_beq : (k == k') = true) (distinct : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) l) (mem : k ∈ List.map Sigma.fst l) :

(m.insertMany l).getKey! k' = k

source

theorem Std.DHashMap.getKeyD_insertMany_list_of_contains_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} {k fallback : α} (contains_eq_false : (List.map Sigma.fst l).contains k = false) :

(m.insertMany l).getKeyD k fallback = m.getKeyD k fallback

source

theorem Std.DHashMap.getKeyD_insertMany_list_of_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} {k k' fallback : α} (k_beq : (k == k') = true) (distinct : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) l) (mem : k ∈ List.map Sigma.fst l) :

(m.insertMany l).getKeyD k' fallback = k

source

theorem Std.DHashMap.size_insertMany_list {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} (distinct : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) l) :

(∀ (a : α), a ∈ m → (List.map Sigma.fst l).contains a = false) → (m.insertMany l).size = m.size + l.length

source

theorem Std.DHashMap.size_le_size_insertMany_list {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} :

m.size ≤ (m.insertMany l).size

source

theorem Std.DHashMap.size_insertMany_list_le {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} :

(m.insertMany l).size ≤ m.size + l.length

source

@[simp]

theorem Std.DHashMap.isEmpty_insertMany_list {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} :

(m.insertMany l).isEmpty = (m.isEmpty && l.isEmpty)

source

@[simp]

theorem Std.DHashMap.Const.insertMany_nil {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} :

insertMany m [] = m

source

@[simp]

theorem Std.DHashMap.Const.insertMany_list_singleton {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} {k : α} {v : β} :

insertMany m [(k, v)] = m.insert k v

source

theorem Std.DHashMap.Const.insertMany_cons {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} {l : List (α × β)} {k : α} {v : β} :

insertMany m ((k, v) :: l) = insertMany (m.insert k v) l

source

@[simp]

theorem Std.DHashMap.Const.contains_insertMany_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k : α} :

(insertMany m l).contains k = (m.contains k || (List.map Prod.fst l).contains k)

source

@[simp]

theorem Std.DHashMap.Const.mem_insertMany_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k : α} :

k ∈ insertMany m l ↔ k ∈ m ∨ (List.map Prod.fst l).contains k = true

source

theorem Std.DHashMap.Const.mem_of_mem_insertMany_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k : α} (mem : k ∈ insertMany m l) (contains_eq_false : (List.map Prod.fst l).contains k = false) :

k ∈ m

source

theorem Std.DHashMap.Const.getKey?_insertMany_list_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) :

(insertMany m l).getKey? k = m.getKey? k

source

theorem Std.DHashMap.Const.getKey?_insertMany_list_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k k' : α} (k_beq : (k == k') = true) (distinct : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) l) (mem : k ∈ List.map Prod.fst l) :

(insertMany m l).getKey? k' = some k

source

theorem Std.DHashMap.Const.getKey_insertMany_list_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) {h : k ∈ insertMany m l} :

(insertMany m l).getKey k h = m.getKey k ⋯

source

theorem Std.DHashMap.Const.getKey_insertMany_list_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k k' : α} (k_beq : (k == k') = true) (distinct : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) l) (mem : k ∈ List.map Prod.fst l) {h : k' ∈ insertMany m l} :

(insertMany m l).getKey k' h = k

source

theorem Std.DHashMap.Const.getKey!_insertMany_list_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {l : List (α × β)} {k : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) :

(insertMany m l).getKey! k = m.getKey! k

source

theorem Std.DHashMap.Const.getKey!_insertMany_list_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {l : List (α × β)} {k k' : α} (k_beq : (k == k') = true) (distinct : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) l) (mem : k ∈ List.map Prod.fst l) :

(insertMany m l).getKey! k' = k

source

theorem Std.DHashMap.Const.getKeyD_insertMany_list_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k fallback : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) :

(insertMany m l).getKeyD k fallback = m.getKeyD k fallback

source

theorem Std.DHashMap.Const.getKeyD_insertMany_list_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k k' fallback : α} (k_beq : (k == k') = true) (distinct : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) l) (mem : k ∈ List.map Prod.fst l) :

(insertMany m l).getKeyD k' fallback = k

source

theorem Std.DHashMap.Const.size_insertMany_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} (distinct : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) l) :

(∀ (a : α), a ∈ m → (List.map Prod.fst l).contains a = false) → (insertMany m l).size = m.size + l.length

source

theorem Std.DHashMap.Const.size_le_size_insertMany_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} :

m.size ≤ (insertMany m l).size

source

theorem Std.DHashMap.Const.size_insertMany_list_le {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} :

(insertMany m l).size ≤ m.size + l.length

source

@[simp]

theorem Std.DHashMap.Const.isEmpty_insertMany_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} :

(insertMany m l).isEmpty = (m.isEmpty && l.isEmpty)

source

theorem Std.DHashMap.Const.get?_insertMany_list_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) :

get? (insertMany m l) k = get? m k

source

theorem Std.DHashMap.Const.get?_insertMany_list_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k k' : α} (k_beq : (k == k') = true) {v : β} (distinct : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) l) (mem : (k, v) ∈ l) :

get? (insertMany m l) k' = some v

source

theorem Std.DHashMap.Const.get_insertMany_list_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) {h : k ∈ insertMany m l} :

get (insertMany m l) k h = get m k ⋯

source

theorem Std.DHashMap.Const.get_insertMany_list_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k k' : α} (k_beq : (k == k') = true) {v : β} (distinct : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) l) (mem : (k, v) ∈ l) {h : k' ∈ insertMany m l} :

get (insertMany m l) k' h = v

source

theorem Std.DHashMap.Const.get!_insertMany_list_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {l : List (α × β)} {k : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) :

get! (insertMany m l) k = get! m k

source

theorem Std.DHashMap.Const.get!_insertMany_list_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {l : List (α × β)} {k k' : α} (k_beq : (k == k') = true) {v : β} (distinct : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) l) (mem : (k, v) ∈ l) :

get! (insertMany m l) k' = v

source

theorem Std.DHashMap.Const.getD_insertMany_list_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k : α} {fallback : β} (contains_eq_false : (List.map Prod.fst l).contains k = false) :

getD (insertMany m l) k fallback = getD m k fallback

source

theorem Std.DHashMap.Const.getD_insertMany_list_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k k' : α} (k_beq : (k == k') = true) {v fallback : β} (distinct : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) l) (mem : (k, v) ∈ l) :

getD (insertMany m l) k' fallback = v

source

@[simp]

theorem Std.DHashMap.Const.insertManyIfNewUnit_nil {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} :

insertManyIfNewUnit m [] = m

source

@[simp]

theorem Std.DHashMap.Const.insertManyIfNewUnit_list_singleton {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} {k : α} :

insertManyIfNewUnit m [k] = m.insertIfNew k ()

source

theorem Std.DHashMap.Const.insertManyIfNewUnit_cons {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} {l : List α} {k : α} :

insertManyIfNewUnit m (k :: l) = insertManyIfNewUnit (m.insertIfNew k ()) l

source

@[simp]

theorem Std.DHashMap.Const.contains_insertManyIfNewUnit_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} :

(insertManyIfNewUnit m l).contains k = (m.contains k || l.contains k)

source

@[simp]

theorem Std.DHashMap.Const.mem_insertManyIfNewUnit_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} :

k ∈ insertManyIfNewUnit m l ↔ k ∈ m ∨ l.contains k = true

source

theorem Std.DHashMap.Const.mem_of_mem_insertManyIfNewUnit_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} (contains_eq_false : l.contains k = false) :

k ∈ insertManyIfNewUnit m l → k ∈ m

source

theorem Std.DHashMap.Const.getKey?_insertManyIfNewUnit_list_of_not_mem_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} (not_mem : ¬k ∈ m) (contains_eq_false : l.contains k = false) :

(insertManyIfNewUnit m l).getKey? k = none

source

theorem Std.DHashMap.Const.getKey?_insertManyIfNewUnit_list_of_not_mem_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} [EquivBEq α] [LawfulHashable α] {l : List α} {k k' : α} (k_beq : (k == k') = true) (not_mem : ¬k ∈ m) (distinct : List.Pairwise (fun (a b : α) => (a == b) = false) l) (mem : k ∈ l) :

(insertManyIfNewUnit m l).getKey? k' = some k

source

theorem Std.DHashMap.Const.getKey?_insertManyIfNewUnit_list_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} (h' : k ∈ m) :

(insertManyIfNewUnit m l).getKey? k = m.getKey? k

source

theorem Std.DHashMap.Const.getKey_insertManyIfNewUnit_list_of_not_mem_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} [EquivBEq α] [LawfulHashable α] {l : List α} {k k' : α} (k_beq : (k == k') = true) (not_mem : ¬k ∈ m) (distinct : List.Pairwise (fun (a b : α) => (a == b) = false) l) (mem : k ∈ l) {h : k' ∈ insertManyIfNewUnit m l} :

(insertManyIfNewUnit m l).getKey k' h = k

source

theorem Std.DHashMap.Const.getKey_insertManyIfNewUnit_list_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} (mem : k ∈ m) {h : k ∈ insertManyIfNewUnit m l} :

(insertManyIfNewUnit m l).getKey k h = m.getKey k mem

source

theorem Std.DHashMap.Const.getKey!_insertManyIfNewUnit_list_of_not_mem_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} [EquivBEq α] [LawfulHashable α] [Inhabited α] {l : List α} {k : α} (not_mem : ¬k ∈ m) (contains_eq_false : l.contains k = false) :

(insertManyIfNewUnit m l).getKey! k = default

source

theorem Std.DHashMap.Const.getKey!_insertManyIfNewUnit_list_of_not_mem_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} [EquivBEq α] [LawfulHashable α] [Inhabited α] {l : List α} {k k' : α} (k_beq : (k == k') = true) (not_mem : ¬k ∈ m) (distinct : List.Pairwise (fun (a b : α) => (a == b) = false) l) (mem : k ∈ l) :

(insertManyIfNewUnit m l).getKey! k' = k

source

theorem Std.DHashMap.Const.getKey!_insertManyIfNewUnit_list_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} [EquivBEq α] [LawfulHashable α] [Inhabited α] {l : List α} {k : α} (mem : k ∈ m) :

(insertManyIfNewUnit m l).getKey! k = m.getKey! k

source

theorem Std.DHashMap.Const.getKeyD_insertManyIfNewUnit_list_of_not_mem_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} [EquivBEq α] [LawfulHashable α] {l : List α} {k fallback : α} (not_mem : ¬k ∈ m) (contains_eq_false : l.contains k = false) :

(insertManyIfNewUnit m l).getKeyD k fallback = fallback

source

theorem Std.DHashMap.Const.getKeyD_insertManyIfNewUnit_list_of_not_mem_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} [EquivBEq α] [LawfulHashable α] {l : List α} {k k' fallback : α} (k_beq : (k == k') = true) (not_mem : ¬k ∈ m) (distinct : List.Pairwise (fun (a b : α) => (a == b) = false) l) (mem : k ∈ l) :

(insertManyIfNewUnit m l).getKeyD k' fallback = k

source

theorem Std.DHashMap.Const.getKeyD_insertManyIfNewUnit_list_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} [EquivBEq α] [LawfulHashable α] {l : List α} {k fallback : α} (mem : k ∈ m) :

(insertManyIfNewUnit m l).getKeyD k fallback = m.getKeyD k fallback

source

theorem Std.DHashMap.Const.size_insertManyIfNewUnit_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} [EquivBEq α] [LawfulHashable α] {l : List α} (distinct : List.Pairwise (fun (a b : α) => (a == b) = false) l) :

(∀ (a : α), a ∈ m → l.contains a = false) → (insertManyIfNewUnit m l).size = m.size + l.length

source

theorem Std.DHashMap.Const.size_le_size_insertManyIfNewUnit_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} [EquivBEq α] [LawfulHashable α] {l : List α} :

m.size ≤ (insertManyIfNewUnit m l).size

source

theorem Std.DHashMap.Const.size_insertManyIfNewUnit_list_le {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} [EquivBEq α] [LawfulHashable α] {l : List α} :

(insertManyIfNewUnit m l).size ≤ m.size + l.length

source

@[simp]

theorem Std.DHashMap.Const.isEmpty_insertManyIfNewUnit_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} [EquivBEq α] [LawfulHashable α] {l : List α} :

(insertManyIfNewUnit m l).isEmpty = (m.isEmpty && l.isEmpty)

source

theorem Std.DHashMap.Const.get?_insertManyIfNewUnit_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} :

get? (insertManyIfNewUnit m l) k = if k ∈ m ∨ l.contains k = true then some () else none

source

theorem Std.DHashMap.Const.get_insertManyIfNewUnit_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} {l : List α} {k : α} {h : k ∈ insertManyIfNewUnit m l} :

get (insertManyIfNewUnit m l) k h = ()

source

theorem Std.DHashMap.Const.get!_insertManyIfNewUnit_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} {l : List α} {k : α} :

get! (insertManyIfNewUnit m l) k = ()

source

theorem Std.DHashMap.Const.getD_insertManyIfNewUnit_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α fun (x : α) => Unit} {l : List α} {k : α} {fallback : Unit} :

getD (insertManyIfNewUnit m l) k fallback = ()

source

@[simp]

theorem Std.DHashMap.ofList_nil {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} :

ofList [] = ∅

source

@[simp]

theorem Std.DHashMap.ofList_singleton {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {k : α} {v : β k} :

ofList [⟨k, v⟩] = ∅.insert k v

source

theorem Std.DHashMap.ofList_cons {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {k : α} {v : β k} {tl : List ((a : α) × β a)} :

ofList (⟨k, v⟩ :: tl) = (∅.insert k v).insertMany tl

source

@[simp]

theorem Std.DHashMap.contains_ofList {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} {k : α} :

(ofList l).contains k = (List.map Sigma.fst l).contains k

source

@[simp]

theorem Std.DHashMap.mem_ofList {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} {k : α} :

k ∈ ofList l ↔ (List.map Sigma.fst l).contains k = true

source

theorem Std.DHashMap.get?_ofList_of_contains_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [LawfulBEq α] {l : List ((a : α) × β a)} {k : α} (contains_eq_false : (List.map Sigma.fst l).contains k = false) :

(ofList l).get? k = none

source

theorem Std.DHashMap.get?_ofList_of_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [LawfulBEq α] {l : List ((a : α) × β a)} {k k' : α} (k_beq : (k == k') = true) {v : β k} (distinct : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) l) (mem : ⟨k, v⟩ ∈ l) :

(ofList l).get? k' = some (cast ⋯ v)

source

theorem Std.DHashMap.get_ofList_of_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [LawfulBEq α] {l : List ((a : α) × β a)} {k k' : α} (k_beq : (k == k') = true) {v : β k} (distinct : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) l) (mem : ⟨k, v⟩ ∈ l) {h : k' ∈ ofList l} :

(ofList l).get k' h = cast ⋯ v

source

theorem Std.DHashMap.get!_ofList_of_contains_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [LawfulBEq α] {l : List ((a : α) × β a)} {k : α} [Inhabited (β k)] (contains_eq_false : (List.map Sigma.fst l).contains k = false) :

(ofList l).get! k = default

source

theorem Std.DHashMap.get!_ofList_of_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [LawfulBEq α] {l : List ((a : α) × β a)} {k k' : α} (k_beq : (k == k') = true) {v : β k} [Inhabited (β k')] (distinct : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) l) (mem : ⟨k, v⟩ ∈ l) :

(ofList l).get! k' = cast ⋯ v

source

theorem Std.DHashMap.getD_ofList_of_contains_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [LawfulBEq α] {l : List ((a : α) × β a)} {k : α} {fallback : β k} (contains_eq_false : (List.map Sigma.fst l).contains k = false) :

(ofList l).getD k fallback = fallback

source

theorem Std.DHashMap.getD_ofList_of_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [LawfulBEq α] {l : List ((a : α) × β a)} {k k' : α} (k_beq : (k == k') = true) {v : β k} {fallback : β k'} (distinct : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) l) (mem : ⟨k, v⟩ ∈ l) :

(ofList l).getD k' fallback = cast ⋯ v

source

theorem Std.DHashMap.getKey?_ofList_of_contains_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} {k : α} (contains_eq_false : (List.map Sigma.fst l).contains k = false) :

(ofList l).getKey? k = none

source

theorem Std.DHashMap.getKey?_ofList_of_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} {k k' : α} (k_beq : (k == k') = true) (distinct : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) l) (mem : k ∈ List.map Sigma.fst l) :

(ofList l).getKey? k' = some k

source

theorem Std.DHashMap.getKey_ofList_of_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} {k k' : α} (k_beq : (k == k') = true) (distinct : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) l) (mem : k ∈ List.map Sigma.fst l) {h : k' ∈ ofList l} :

(ofList l).getKey k' h = k

source

theorem Std.DHashMap.getKey!_ofList_of_contains_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] [Inhabited α] {l : List ((a : α) × β a)} {k : α} (contains_eq_false : (List.map Sigma.fst l).contains k = false) :

(ofList l).getKey! k = default

source

theorem Std.DHashMap.getKey!_ofList_of_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] [Inhabited α] {l : List ((a : α) × β a)} {k k' : α} (k_beq : (k == k') = true) (distinct : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) l) (mem : k ∈ List.map Sigma.fst l) :

(ofList l).getKey! k' = k

source

theorem Std.DHashMap.getKeyD_ofList_of_contains_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} {k fallback : α} (contains_eq_false : (List.map Sigma.fst l).contains k = false) :

(ofList l).getKeyD k fallback = fallback

source

theorem Std.DHashMap.getKeyD_ofList_of_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} {k k' fallback : α} (k_beq : (k == k') = true) (distinct : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) l) (mem : k ∈ List.map Sigma.fst l) :

(ofList l).getKeyD k' fallback = k

source

theorem Std.DHashMap.size_ofList {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} (distinct : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) l) :

(ofList l).size = l.length

source

theorem Std.DHashMap.size_ofList_le {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} :

(ofList l).size ≤ l.length

source

@[simp]

theorem Std.DHashMap.isEmpty_ofList {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List ((a : α) × β a)} :

(ofList l).isEmpty = l.isEmpty

source

@[simp]

theorem Std.DHashMap.Const.ofList_nil {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} :

ofList [] = ∅

source

@[simp]

theorem Std.DHashMap.Const.ofList_singleton {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {k : α} {v : β} :

ofList [(k, v)] = ∅.insert k v

source

theorem Std.DHashMap.Const.ofList_cons {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {k : α} {v : β} {tl : List (α × β)} :

ofList ((k, v) :: tl) = insertMany (∅.insert k v) tl

source

@[simp]

theorem Std.DHashMap.Const.contains_ofList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k : α} :

(ofList l).contains k = (List.map Prod.fst l).contains k

source

@[simp]

theorem Std.DHashMap.Const.mem_ofList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k : α} :

k ∈ ofList l ↔ (List.map Prod.fst l).contains k = true

source

theorem Std.DHashMap.Const.get?_ofList_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} [LawfulBEq α] {l : List (α × β)} {k : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) :

get? (ofList l) k = none

source

theorem Std.DHashMap.Const.get?_ofList_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} [LawfulBEq α] {l : List (α × β)} {k k' : α} (k_beq : (k == k') = true) {v : β} (distinct : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) l) (mem : (k, v) ∈ l) :

get? (ofList l) k' = some v

source

theorem Std.DHashMap.Const.get_ofList_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} [LawfulBEq α] {l : List (α × β)} {k k' : α} (k_beq : (k == k') = true) {v : β} (distinct : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) l) (mem : (k, v) ∈ l) {h : k' ∈ ofList l} :

get (ofList l) k' h = v

source

theorem Std.DHashMap.Const.get!_ofList_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} [LawfulBEq α] {l : List (α × β)} {k : α} [Inhabited β] (contains_eq_false : (List.map Prod.fst l).contains k = false) :

get! (ofList l) k = default

source

theorem Std.DHashMap.Const.get!_ofList_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} [LawfulBEq α] {l : List (α × β)} {k k' : α} (k_beq : (k == k') = true) {v : β} [Inhabited β] (distinct : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) l) (mem : (k, v) ∈ l) :

get! (ofList l) k' = v

source

theorem Std.DHashMap.Const.getD_ofList_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} [LawfulBEq α] {l : List (α × β)} {k : α} {fallback : β} (contains_eq_false : (List.map Prod.fst l).contains k = false) :

getD (ofList l) k fallback = fallback

source

theorem Std.DHashMap.Const.getD_ofList_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} [LawfulBEq α] {l : List (α × β)} {k k' : α} (k_beq : (k == k') = true) {v fallback : β} (distinct : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) l) (mem : (k, v) ∈ l) :

getD (ofList l) k' fallback = v

source

theorem Std.DHashMap.Const.getKey?_ofList_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) :

(ofList l).getKey? k = none

source

theorem Std.DHashMap.Const.getKey?_ofList_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k k' : α} (k_beq : (k == k') = true) (distinct : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) l) (mem : k ∈ List.map Prod.fst l) :

(ofList l).getKey? k' = some k

source

theorem Std.DHashMap.Const.getKey_ofList_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k k' : α} (k_beq : (k == k') = true) (distinct : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) l) (mem : k ∈ List.map Prod.fst l) {h : k' ∈ ofList l} :

(ofList l).getKey k' h = k

source

theorem Std.DHashMap.Const.getKey!_ofList_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} [EquivBEq α] [LawfulHashable α] [Inhabited α] {l : List (α × β)} {k : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) :

(ofList l).getKey! k = default

source

theorem Std.DHashMap.Const.getKey!_ofList_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} [EquivBEq α] [LawfulHashable α] [Inhabited α] {l : List (α × β)} {k k' : α} (k_beq : (k == k') = true) (distinct : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) l) (mem : k ∈ List.map Prod.fst l) :

(ofList l).getKey! k' = k

source

theorem Std.DHashMap.Const.getKeyD_ofList_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k fallback : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) :

(ofList l).getKeyD k fallback = fallback

source

theorem Std.DHashMap.Const.getKeyD_ofList_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k k' fallback : α} (k_beq : (k == k') = true) (distinct : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) l) (mem : k ∈ List.map Prod.fst l) :

(ofList l).getKeyD k' fallback = k

source

theorem Std.DHashMap.Const.size_ofList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} (distinct : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) l) :

(ofList l).size = l.length

source

theorem Std.DHashMap.Const.size_ofList_le {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} :

(ofList l).size ≤ l.length

source

@[simp]

theorem Std.DHashMap.Const.isEmpty_ofList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} :

(ofList l).isEmpty = l.isEmpty

source

@[simp]

theorem Std.DHashMap.Const.unitOfList_nil {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} :

unitOfList [] = ∅

source

@[simp]

theorem Std.DHashMap.Const.unitOfList_singleton {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {k : α} :

unitOfList [k] = ∅.insertIfNew k ()

source

theorem Std.DHashMap.Const.unitOfList_cons {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {hd : α} {tl : List α} :

unitOfList (hd :: tl) = insertManyIfNewUnit (∅.insertIfNew hd ()) tl

source

@[simp]

theorem Std.DHashMap.Const.contains_unitOfList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} :

(unitOfList l).contains k = l.contains k

source

@[simp]

theorem Std.DHashMap.Const.mem_unitOfList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} :

k ∈ unitOfList l ↔ l.contains k = true

source

theorem Std.DHashMap.Const.getKey?_unitOfList_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} (contains_eq_false : l.contains k = false) :

(unitOfList l).getKey? k = none

source

theorem Std.DHashMap.Const.getKey?_unitOfList_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List α} {k k' : α} (k_beq : (k == k') = true) (distinct : List.Pairwise (fun (a b : α) => (a == b) = false) l) (mem : k ∈ l) :

(unitOfList l).getKey? k' = some k

source

theorem Std.DHashMap.Const.getKey_unitOfList_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List α} {k k' : α} (k_beq : (k == k') = true) (distinct : List.Pairwise (fun (a b : α) => (a == b) = false) l) (mem : k ∈ l) {h : k' ∈ unitOfList l} :

(unitOfList l).getKey k' h = k

source

theorem Std.DHashMap.Const.getKey!_unitOfList_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] [Inhabited α] {l : List α} {k : α} (contains_eq_false : l.contains k = false) :

(unitOfList l).getKey! k = default

source

theorem Std.DHashMap.Const.getKey!_unitOfList_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] [Inhabited α] {l : List α} {k k' : α} (k_beq : (k == k') = true) (distinct : List.Pairwise (fun (a b : α) => (a == b) = false) l) (mem : k ∈ l) :

(unitOfList l).getKey! k' = k

source

theorem Std.DHashMap.Const.getKeyD_unitOfList_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List α} {k fallback : α} (contains_eq_false : l.contains k = false) :

(unitOfList l).getKeyD k fallback = fallback

source

theorem Std.DHashMap.Const.getKeyD_unitOfList_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List α} {k k' fallback : α} (k_beq : (k == k') = true) (distinct : List.Pairwise (fun (a b : α) => (a == b) = false) l) (mem : k ∈ l) :

(unitOfList l).getKeyD k' fallback = k

source

theorem Std.DHashMap.Const.size_unitOfList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List α} (distinct : List.Pairwise (fun (a b : α) => (a == b) = false) l) :

(unitOfList l).size = l.length

source

theorem Std.DHashMap.Const.size_unitOfList_le {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List α} :

(unitOfList l).size ≤ l.length

source

@[simp]

theorem Std.DHashMap.Const.isEmpty_unitOfList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List α} :

(unitOfList l).isEmpty = l.isEmpty

source

@[simp]

theorem Std.DHashMap.Const.get?_unitOfList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} :

get? (unitOfList l) k = if l.contains k = true then some () else none

source

@[simp]

theorem Std.DHashMap.Const.get_unitOfList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {l : List α} {k : α} {h : k ∈ unitOfList l} :

get (unitOfList l) k h = ()

source

@[simp]

theorem Std.DHashMap.Const.get!_unitOfList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {l : List α} {k : α} :

get! (unitOfList l) k = ()

source

@[simp]

theorem Std.DHashMap.Const.getD_unitOfList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {l : List α} {k : α} {fallback : Unit} :

getD (unitOfList l) k fallback = ()

source

theorem Std.DHashMap.isEmpty_alter_eq_isEmpty_erase {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {f : Option (β k) → Option (β k)} :

(m.alter k f).isEmpty = ((m.erase k).isEmpty && (f (m.get? k)).isNone)

source

@[simp]

theorem Std.DHashMap.isEmpty_alter {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {f : Option (β k) → Option (β k)} :

(m.alter k f).isEmpty = ((m.isEmpty || m.size == 1 && m.contains k) && (f (m.get? k)).isNone)

source

theorem Std.DHashMap.contains_alter {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k k' : α} {f : Option (β k) → Option (β k)} :

(m.alter k f).contains k' = if (k == k') = true then (f (m.get? k)).isSome else m.contains k'

source

theorem Std.DHashMap.mem_alter {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k k' : α} {f : Option (β k) → Option (β k)} :

k' ∈ m.alter k f ↔ if (k == k') = true then (f (m.get? k)).isSome = true else k' ∈ m

source

theorem Std.DHashMap.mem_alter_of_beq {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k k' : α} {f : Option (β k) → Option (β k)} (h : (k == k') = true) :

k' ∈ m.alter k f ↔ (f (m.get? k)).isSome = true

source

@[simp]

theorem Std.DHashMap.contains_alter_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {f : Option (β k) → Option (β k)} :

(m.alter k f).contains k = (f (m.get? k)).isSome

source

@[simp]

theorem Std.DHashMap.mem_alter_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {f : Option (β k) → Option (β k)} :

k ∈ m.alter k f ↔ (f (m.get? k)).isSome = true

source

theorem Std.DHashMap.contains_alter_of_beq_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k k' : α} {f : Option (β k) → Option (β k)} (h : (k == k') = false) :

(m.alter k f).contains k' = m.contains k'

source

theorem Std.DHashMap.mem_alter_of_beq_eq_false {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k k' : α} {f : Option (β k) → Option (β k)} (h : (k == k') = false) :

k' ∈ m.alter k f ↔ k' ∈ m

source

theorem Std.DHashMap.size_alter {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {f : Option (β k) → Option (β k)} :

(m.alter k f).size = if k ∈ m ∧ (f (m.get? k)).isNone = true then m.size - 1 else if ¬k ∈ m ∧ (f (m.get? k)).isSome = true then m.size + 1 else m.size

source

theorem Std.DHashMap.size_alter_eq_add_one {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {f : Option (β k) → Option (β k)} (h : ¬k ∈ m) (h' : (f (m.get? k)).isSome = true) :

(m.alter k f).size = m.size + 1

source

theorem Std.DHashMap.size_alter_eq_sub_one {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {f : Option (β k) → Option (β k)} (h : k ∈ m) (h' : (f (m.get? k)).isNone = true) :

(m.alter k f).size = m.size - 1

source

theorem Std.DHashMap.size_alter_eq_self_of_not_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {f : Option (β k) → Option (β k)} (h : ¬k ∈ m) (h' : (f (m.get? k)).isNone = true) :

(m.alter k f).size = m.size

source

theorem Std.DHashMap.size_alter_eq_self_of_mem {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {f : Option (β k) → Option (β k)} (h : k ∈ m) (h' : (f (m.get? k)).isSome = true) :

(m.alter k f).size = m.size

source

theorem Std.DHashMap.size_alter_le_size {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {f : Option (β k) → Option (β k)} :

(m.alter k f).size ≤ m.size + 1

source

theorem Std.DHashMap.size_le_size_alter {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {f : Option (β k) → Option (β k)} :

m.size - 1 ≤ (m.alter k f).size

source

theorem Std.DHashMap.get?_alter {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k k' : α} {f : Option (β k) → Option (β k)} :

(m.alter k f).get? k' = if h : (k == k') = true then cast ⋯ (f (m.get? k)) else m.get? k'

source

@[simp]

theorem Std.DHashMap.get?_alter_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {f : Option (β k) → Option (β k)} :

(m.alter k f).get? k = f (m.get? k)

source

theorem Std.DHashMap.get_alter {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k k' : α} {f : Option (β k) → Option (β k)} {h : k' ∈ m.alter k f} :

(m.alter k f).get k' h = if heq : (k == k') = true then cast ⋯ ((f (m.get? k)).get ⋯) else m.get k' ⋯

source

@[simp]

theorem Std.DHashMap.get_alter_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {f : Option (β k) → Option (β k)} {h : k ∈ m.alter k f} :

(m.alter k f).get k h = (f (m.get? k)).get ⋯

source

theorem Std.DHashMap.get!_alter {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k k' : α} [hi : Inhabited (β k')] {f : Option (β k) → Option (β k)} :

(m.alter k f).get! k' = if heq : (k == k') = true then (Option.map (cast ⋯) (f (m.get? k))).get! else m.get! k'

source

@[simp]

theorem Std.DHashMap.get!_alter_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} [Inhabited (β k)] {f : Option (β k) → Option (β k)} :

(m.alter k f).get! k = (f (m.get? k)).get!

source

theorem Std.DHashMap.getD_alter {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k k' : α} {fallback : β k'} {f : Option (β k) → Option (β k)} :

(m.alter k f).getD k' fallback = if heq : (k == k') = true then (Option.map (cast ⋯) (f (m.get? k))).getD fallback else m.getD k' fallback

source

@[simp]

theorem Std.DHashMap.getD_alter_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {fallback : β k} {f : Option (β k) → Option (β k)} :

(m.alter k f).getD k fallback = (f (m.get? k)).getD fallback

source

theorem Std.DHashMap.getKey?_alter {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k k' : α} {f : Option (β k) → Option (β k)} :

(m.alter k f).getKey? k' = if (k == k') = true then if (f (m.get? k)).isSome = true then some k else none else m.getKey? k'

source

theorem Std.DHashMap.getKey?_alter_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {f : Option (β k) → Option (β k)} :

(m.alter k f).getKey? k = if (f (m.get? k)).isSome = true then some k else none

source

theorem Std.DHashMap.getKey!_alter {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] [Inhabited α] {k k' : α} {f : Option (β k) → Option (β k)} :

(m.alter k f).getKey! k' = if (k == k') = true then if (f (m.get? k)).isSome = true then k else default else m.getKey! k'

source

theorem Std.DHashMap.getKey!_alter_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] [Inhabited α] {k : α} {f : Option (β k) → Option (β k)} :

(m.alter k f).getKey! k = if (f (m.get? k)).isSome = true then k else default

source

theorem Std.DHashMap.getKey_alter {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] [Inhabited α] {k k' : α} {f : Option (β k) → Option (β k)} {h : k' ∈ m.alter k f} :

(m.alter k f).getKey k' h = if heq : (k == k') = true then k else m.getKey k' ⋯

source

@[simp]

theorem Std.DHashMap.getKey_alter_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] [Inhabited α] {k : α} {f : Option (β k) → Option (β k)} {h : k ∈ m.alter k f} :

(m.alter k f).getKey k h = k

source

theorem Std.DHashMap.getKeyD_alter {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k k' fallback : α} {f : Option (β k) → Option (β k)} :

(m.alter k f).getKeyD k' fallback = if (k == k') = true then if (f (m.get? k)).isSome = true then k else fallback else m.getKeyD k' fallback

source

@[simp]

theorem Std.DHashMap.getKeyD_alter_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] [Inhabited α] {k fallback : α} {f : Option (β k) → Option (β k)} :

(m.alter k f).getKeyD k fallback = if (f (m.get? k)).isSome = true then k else fallback

source

theorem Std.DHashMap.Const.isEmpty_alter_eq_isEmpty_erase {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} {f : Option β → Option β} :

(alter m k f).isEmpty = ((m.erase k).isEmpty && (f (get? m k)).isNone)

source

@[simp]

theorem Std.DHashMap.Const.isEmpty_alter {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} {f : Option β → Option β} :

(alter m k f).isEmpty = ((m.isEmpty || m.size == 1 && m.contains k) && (f (get? m k)).isNone)

source

theorem Std.DHashMap.Const.contains_alter {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : Option β → Option β} :

(alter m k f).contains k' = if (k == k') = true then (f (get? m k)).isSome else m.contains k'

source

theorem Std.DHashMap.Const.mem_alter {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : Option β → Option β} :

k' ∈ alter m k f ↔ if (k == k') = true then (f (get? m k)).isSome = true else k' ∈ m

source

theorem Std.DHashMap.Const.mem_alter_of_beq {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : Option β → Option β} (h : (k == k') = true) :

k' ∈ alter m k f ↔ (f (get? m k)).isSome = true

source

@[simp]

theorem Std.DHashMap.Const.contains_alter_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} {f : Option β → Option β} :

(alter m k f).contains k = (f (get? m k)).isSome

source

@[simp]

theorem Std.DHashMap.Const.mem_alter_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} {f : Option β → Option β} :

k ∈ alter m k f ↔ (f (get? m k)).isSome = true

source

theorem Std.DHashMap.Const.contains_alter_of_beq_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : Option β → Option β} (h : (k == k') = false) :

(alter m k f).contains k' = m.contains k'

source

theorem Std.DHashMap.Const.mem_alter_of_beq_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : Option β → Option β} (h : (k == k') = false) :

k' ∈ alter m k f ↔ k' ∈ m

source

theorem Std.DHashMap.Const.size_alter {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [LawfulBEq α] {k : α} {f : Option β → Option β} :

(alter m k f).size = if k ∈ m ∧ (f (get? m k)).isNone = true then m.size - 1 else if ¬k ∈ m ∧ (f (get? m k)).isSome = true then m.size + 1 else m.size

source

theorem Std.DHashMap.Const.size_alter_eq_add_one {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [LawfulBEq α] {k : α} {f : Option β → Option β} (h : ¬k ∈ m) (h' : (f (get? m k)).isSome = true) :

(alter m k f).size = m.size + 1

source

theorem Std.DHashMap.Const.size_alter_eq_sub_one {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [LawfulBEq α] {k : α} {f : Option β → Option β} (h : k ∈ m) (h' : (f (get? m k)).isNone = true) :

(alter m k f).size = m.size - 1

source

theorem Std.DHashMap.Const.size_alter_eq_self_of_not_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [LawfulBEq α] {k : α} {f : Option β → Option β} (h : ¬k ∈ m) (h' : (f (get? m k)).isNone = true) :

(alter m k f).size = m.size

source

theorem Std.DHashMap.Const.size_alter_eq_self_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [LawfulBEq α] {k : α} {f : Option β → Option β} (h : k ∈ m) (h' : (f (get? m k)).isSome = true) :

(alter m k f).size = m.size

source

theorem Std.DHashMap.Const.size_alter_le_size {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [LawfulBEq α] {k : α} {f : Option β → Option β} :

(alter m k f).size ≤ m.size + 1

source

theorem Std.DHashMap.Const.size_le_size_alter {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [LawfulBEq α] {k : α} {f : Option β → Option β} :

m.size - 1 ≤ (alter m k f).size

source

theorem Std.DHashMap.Const.get?_alter {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : Option β → Option β} :

get? (alter m k f) k' = if (k == k') = true then f (get? m k) else get? m k'

source

@[simp]

theorem Std.DHashMap.Const.get?_alter_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} {f : Option β → Option β} :

get? (alter m k f) k = f (get? m k)

source

theorem Std.DHashMap.Const.get_alter {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : Option β → Option β} {h : k' ∈ alter m k f} :

get (alter m k f) k' h = if heq : (k == k') = true then (f (get? m k)).get ⋯ else get m k' ⋯

source

@[simp]

theorem Std.DHashMap.Const.get_alter_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} {f : Option β → Option β} {h : k ∈ alter m k f} :

get (alter m k f) k h = (f (get? m k)).get ⋯

source

theorem Std.DHashMap.Const.get!_alter {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k k' : α} [Inhabited β] {f : Option β → Option β} :

get! (alter m k f) k' = if (k == k') = true then (f (get? m k)).get! else get! m k'

source

@[simp]

theorem Std.DHashMap.Const.get!_alter_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} [Inhabited β] {f : Option β → Option β} :

get! (alter m k f) k = (f (get? m k)).get!

source

theorem Std.DHashMap.Const.getD_alter {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k k' : α} {fallback : β} {f : Option β → Option β} :

getD (alter m k f) k' fallback = if (k == k') = true then (f (get? m k)).getD fallback else getD m k' fallback

source

@[simp]

theorem Std.DHashMap.Const.getD_alter_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} {fallback : β} {f : Option β → Option β} :

getD (alter m k f) k fallback = (f (get? m k)).getD fallback

source

theorem Std.DHashMap.Const.getKey?_alter {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : Option β → Option β} :

(alter m k f).getKey? k' = if (k == k') = true then if (f (get? m k)).isSome = true then some k else none else m.getKey? k'

source

theorem Std.DHashMap.Const.getKey?_alter_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} {f : Option β → Option β} :

(alter m k f).getKey? k = if (f (get? m k)).isSome = true then some k else none

source

theorem Std.DHashMap.Const.getKey!_alter {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k k' : α} {f : Option β → Option β} :

(alter m k f).getKey! k' = if (k == k') = true then if (f (get? m k)).isSome = true then k else default else m.getKey! k'

source

theorem Std.DHashMap.Const.getKey!_alter_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k : α} {f : Option β → Option β} :

(alter m k f).getKey! k = if (f (get? m k)).isSome = true then k else default

source

theorem Std.DHashMap.Const.getKey_alter {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k k' : α} {f : Option β → Option β} {h : k' ∈ alter m k f} :

(alter m k f).getKey k' h = if heq : (k == k') = true then k else m.getKey k' ⋯

source

@[simp]

theorem Std.DHashMap.Const.getKey_alter_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k : α} {f : Option β → Option β} {h : k ∈ alter m k f} :

(alter m k f).getKey k h = k

source

theorem Std.DHashMap.Const.getKeyD_alter {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k k' fallback : α} {f : Option β → Option β} :

(alter m k f).getKeyD k' fallback = if (k == k') = true then if (f (get? m k)).isSome = true then k else fallback else m.getKeyD k' fallback

source

theorem Std.DHashMap.Const.getKeyD_alter_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k fallback : α} {f : Option β → Option β} :

(alter m k f).getKeyD k fallback = if (f (get? m k)).isSome = true then k else fallback

source

@[simp]

theorem Std.DHashMap.isEmpty_modify {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {f : β k → β k} :

(m.modify k f).isEmpty = m.isEmpty

source

@[simp]

theorem Std.DHashMap.contains_modify {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k k' : α} {f : β k → β k} :

(m.modify k f).contains k' = m.contains k'

source

@[simp]

theorem Std.DHashMap.mem_modify {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k k' : α} {f : β k → β k} :

k' ∈ m.modify k f ↔ k' ∈ m

source

@[simp]

theorem Std.DHashMap.size_modify {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {f : β k → β k} :

(m.modify k f).size = m.size

source

theorem Std.DHashMap.get?_modify {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k k' : α} {f : β k → β k} :

(m.modify k f).get? k' = if h : (k == k') = true then cast ⋯ (Option.map f (m.get? k)) else m.get? k'

source

@[simp]

theorem Std.DHashMap.get?_modify_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {f : β k → β k} :

(m.modify k f).get? k = Option.map f (m.get? k)

source

theorem Std.DHashMap.get_modify {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k k' : α} {f : β k → β k} (h : k' ∈ m.modify k f) :

(m.modify k f).get k' h = if heq : (k == k') = true then cast ⋯ (f (m.get k ⋯)) else m.get k' ⋯

source

@[simp]

theorem Std.DHashMap.get_modify_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {f : β k → β k} {h : k ∈ m.modify k f} :

(m.modify k f).get k h = f (m.get k ⋯)

source

theorem Std.DHashMap.get!_modify {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k k' : α} [hi : Inhabited (β k')] {f : β k → β k} :

(m.modify k f).get! k' = if heq : (k == k') = true then (Option.map (cast ⋯) (Option.map f (m.get? k))).get! else m.get! k'

source

@[simp]

theorem Std.DHashMap.get!_modify_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} [Inhabited (β k)] {f : β k → β k} :

(m.modify k f).get! k = (Option.map f (m.get? k)).get!

source

theorem Std.DHashMap.getD_modify {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k k' : α} {fallback : β k'} {f : β k → β k} :

(m.modify k f).getD k' fallback = if heq : (k == k') = true then (Option.map (cast ⋯) (Option.map f (m.get? k))).getD fallback else m.getD k' fallback

source

@[simp]

theorem Std.DHashMap.getD_modify_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {fallback : β k} {f : β k → β k} :

(m.modify k f).getD k fallback = (Option.map f (m.get? k)).getD fallback

source

theorem Std.DHashMap.getKey?_modify {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k k' : α} {f : β k → β k} :

(m.modify k f).getKey? k' = if (k == k') = true then if k ∈ m then some k else none else m.getKey? k'

source

theorem Std.DHashMap.getKey?_modify_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k : α} {f : β k → β k} :

(m.modify k f).getKey? k = if k ∈ m then some k else none

source

theorem Std.DHashMap.getKey!_modify {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] [Inhabited α] {k k' : α} {f : β k → β k} :

(m.modify k f).getKey! k' = if (k == k') = true then if k ∈ m then k else default else m.getKey! k'

source

theorem Std.DHashMap.getKey!_modify_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] [Inhabited α] {k : α} {f : β k → β k} :

(m.modify k f).getKey! k = if k ∈ m then k else default

source

theorem Std.DHashMap.getKey_modify {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] [Inhabited α] {k k' : α} {f : β k → β k} {h : k' ∈ m.modify k f} :

(m.modify k f).getKey k' h = if (k == k') = true then k else m.getKey k' ⋯

source

@[simp]

theorem Std.DHashMap.getKey_modify_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] [Inhabited α] {k : α} {f : β k → β k} {h : k ∈ m.modify k f} :

(m.modify k f).getKey k h = k

source

theorem Std.DHashMap.getKeyD_modify {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] {k k' fallback : α} {f : β k → β k} :

(m.modify k f).getKeyD k' fallback = if (k == k') = true then if k ∈ m then k else fallback else m.getKeyD k' fallback

source

theorem Std.DHashMap.getKeyD_modify_self {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [LawfulBEq α] [Inhabited α] {k fallback : α} {f : β k → β k} :

(m.modify k f).getKeyD k fallback = if k ∈ m then k else fallback

source

@[simp]

theorem Std.DHashMap.Const.isEmpty_modify {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} {f : β → β} :

(modify m k f).isEmpty = m.isEmpty

source

@[simp]

theorem Std.DHashMap.Const.contains_modify {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : β → β} :

(modify m k f).contains k' = m.contains k'

source

@[simp]

theorem Std.DHashMap.Const.mem_modify {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : β → β} :

k' ∈ modify m k f ↔ k' ∈ m

source

@[simp]

theorem Std.DHashMap.Const.size_modify {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} {f : β → β} :

(modify m k f).size = m.size

source

theorem Std.DHashMap.Const.get?_modify {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : β → β} :

get? (modify m k f) k' = if (k == k') = true then Option.map f (get? m k) else get? m k'

source

@[simp]

theorem Std.DHashMap.Const.get?_modify_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} {f : β → β} :

get? (modify m k f) k = Option.map f (get? m k)

source

theorem Std.DHashMap.Const.get_modify {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : β → β} {h : k' ∈ modify m k f} :

get (modify m k f) k' h = if heq : (k == k') = true then f (get m k ⋯) else get m k' ⋯

source

@[simp]

theorem Std.DHashMap.Const.get_modify_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} {f : β → β} {h : k ∈ modify m k f} :

get (modify m k f) k h = f (get m k ⋯)

source

theorem Std.DHashMap.Const.get!_modify {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k k' : α} [Inhabited β] {f : β → β} :

get! (modify m k f) k' = if (k == k') = true then (Option.map f (get? m k)).get! else get! m k'

source

@[simp]

theorem Std.DHashMap.Const.get!_modify_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} [Inhabited β] {f : β → β} :

get! (modify m k f) k = (Option.map f (get? m k)).get!

source

theorem Std.DHashMap.Const.getD_modify {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k k' : α} {fallback : β} {f : β → β} :

getD (modify m k f) k' fallback = if (k == k') = true then (Option.map f (get? m k)).getD fallback else getD m k' fallback

source

@[simp]

theorem Std.DHashMap.Const.getD_modify_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} {fallback : β} {f : β → β} :

getD (modify m k f) k fallback = (Option.map f (get? m k)).getD fallback

source

theorem Std.DHashMap.Const.getKey?_modify {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : β → β} :

(modify m k f).getKey? k' = if (k == k') = true then if k ∈ m then some k else none else m.getKey? k'

source

theorem Std.DHashMap.Const.getKey?_modify_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} {f : β → β} :

(modify m k f).getKey? k = if k ∈ m then some k else none

source

theorem Std.DHashMap.Const.getKey!_modify {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k k' : α} {f : β → β} :

(modify m k f).getKey! k' = if (k == k') = true then if k ∈ m then k else default else m.getKey! k'

source

theorem Std.DHashMap.Const.getKey!_modify_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k : α} {f : β → β} :

(modify m k f).getKey! k = if k ∈ m then k else default

source

theorem Std.DHashMap.Const.getKey_modify {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k k' : α} {f : β → β} {h : k' ∈ modify m k f} :

(modify m k f).getKey k' h = if (k == k') = true then k else m.getKey k' ⋯

source

@[simp]

theorem Std.DHashMap.Const.getKey_modify_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k : α} {f : β → β} {h : k ∈ modify m k f} :

(modify m k f).getKey k h = k

source

theorem Std.DHashMap.Const.getKeyD_modify {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k k' fallback : α} {f : β → β} :

(modify m k f).getKeyD k' fallback = if (k == k') = true then if k ∈ m then k else fallback else m.getKeyD k' fallback

source

theorem Std.DHashMap.Const.getKeyD_modify_self {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k fallback : α} {f : β → β} :

(modify m k f).getKeyD k fallback = if k ∈ m then k else fallback

source

theorem Std.DHashMap.Equiv.refl {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} (m : DHashMap α β) :

m.Equiv m

source

theorem Std.DHashMap.Equiv.rfl {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} :

m.Equiv m

source

theorem Std.DHashMap.Equiv.symm {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} :

m₁.Equiv m₂ → m₂.Equiv m₁

source

theorem Std.DHashMap.Equiv.trans {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ m₃ : DHashMap α β} :

m₁.Equiv m₂ → m₂.Equiv m₃ → m₁.Equiv m₃

source

theorem Std.DHashMap.Equiv.comm {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} :

m₁.Equiv m₂ ↔ m₂.Equiv m₁

source

theorem Std.DHashMap.Equiv.congr_left {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ m₃ : DHashMap α β} (h : m₁.Equiv m₂) :

m₁.Equiv m₃ ↔ m₂.Equiv m₃

source

theorem Std.DHashMap.Equiv.congr_right {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ m₃ : DHashMap α β} (h : m₁.Equiv m₂) :

m₃.Equiv m₁ ↔ m₃.Equiv m₂

source

theorem Std.DHashMap.Equiv.isEmpty_eq {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} [EquivBEq α] [LawfulHashable α] (h : m₁.Equiv m₂) :

m₁.isEmpty = m₂.isEmpty

source

theorem Std.DHashMap.Equiv.size_eq {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} [EquivBEq α] [LawfulHashable α] (h : m₁.Equiv m₂) :

m₁.size = m₂.size

source

theorem Std.DHashMap.Equiv.contains_eq {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (h : m₁.Equiv m₂) :

m₁.contains k = m₂.contains k

source

theorem Std.DHashMap.Equiv.mem_iff {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (h : m₁.Equiv m₂) :

k ∈ m₁ ↔ k ∈ m₂

source

theorem Std.DHashMap.Equiv.toList_perm {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} (h : m₁.Equiv m₂) :

m₁.toList.Perm m₂.toList

source

theorem Std.DHashMap.Equiv.of_toList_perm {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} (h : m₁.toList.Perm m₂.toList) :

m₁.Equiv m₂

source

theorem Std.DHashMap.Equiv.keys_perm {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} (h : m₁.Equiv m₂) :

m₁.keys.Perm m₂.keys

source

theorem Std.DHashMap.Equiv.get?_eq {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} [LawfulBEq α] {k : α} (h : m₁.Equiv m₂) :

m₁.get? k = m₂.get? k

source

theorem Std.DHashMap.Equiv.get_eq {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} [LawfulBEq α] {k : α} (hk : k ∈ m₁) (h : m₁.Equiv m₂) :

m₁.get k hk = m₂.get k ⋯

source

theorem Std.DHashMap.Equiv.get!_eq {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} [LawfulBEq α] {k : α} [Inhabited (β k)] (h : m₁.Equiv m₂) :

m₁.get! k = m₂.get! k

source

theorem Std.DHashMap.Equiv.getD_eq {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} [LawfulBEq α] {k : α} {fallback : β k} (h : m₁.Equiv m₂) :

m₁.getD k fallback = m₂.getD k fallback

source

theorem Std.DHashMap.Equiv.getKey?_eq {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (h : m₁.Equiv m₂) :

m₁.getKey? k = m₂.getKey? k

source

theorem Std.DHashMap.Equiv.getKey_eq {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (hk : k ∈ m₁) (h : m₁.Equiv m₂) :

m₁.getKey k hk = m₂.getKey k ⋯

source

theorem Std.DHashMap.Equiv.getKey!_eq {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k : α} (h : m₁.Equiv m₂) :

m₁.getKey! k = m₂.getKey! k

source

theorem Std.DHashMap.Equiv.getKeyD_eq {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} [EquivBEq α] [LawfulHashable α] {k fallback : α} (h : m₁.Equiv m₂) :

m₁.getKeyD k fallback = m₂.getKeyD k fallback

source

theorem Std.DHashMap.Equiv.insert {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} [EquivBEq α] [LawfulHashable α] (k : α) (v : β k) (h : m₁.Equiv m₂) :

(m₁.insert k v).Equiv (m₂.insert k v)

source

theorem Std.DHashMap.Equiv.erase {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} [EquivBEq α] [LawfulHashable α] (k : α) (h : m₁.Equiv m₂) :

(m₁.erase k).Equiv (m₂.erase k)

source

theorem Std.DHashMap.Equiv.insertIfNew {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} [EquivBEq α] [LawfulHashable α] (k : α) (v : β k) (h : m₁.Equiv m₂) :

(m₁.insertIfNew k v).Equiv (m₂.insertIfNew k v)

source

theorem Std.DHashMap.Equiv.insertMany_list {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} [EquivBEq α] [LawfulHashable α] (l : List ((a : α) × β a)) (h : m₁.Equiv m₂) :

(m₁.insertMany l).Equiv (m₂.insertMany l)

source

theorem Std.DHashMap.Equiv.alter {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} [LawfulBEq α] (k : α) (f : Option (β k) → Option (β k)) (h : m₁.Equiv m₂) :

(m₁.alter k f).Equiv (m₂.alter k f)

source

theorem Std.DHashMap.Equiv.modify {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} [LawfulBEq α] (k : α) (f : β k → β k) (h : m₁.Equiv m₂) :

(m₁.modify k f).Equiv (m₂.modify k f)

source

theorem Std.DHashMap.Equiv.filter {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} (f : (a : α) → β a → Bool) (h : m₁.Equiv m₂) :

(DHashMap.filter f m₁).Equiv (DHashMap.filter f m₂)

source

theorem Std.DHashMap.Equiv.map {α : Type u} {β : α → Type v} {γ : α → Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} (f : (a : α) → β a → γ a) (h : m₁.Equiv m₂) :

(DHashMap.map f m₁).Equiv (DHashMap.map f m₂)

source

theorem Std.DHashMap.Equiv.filterMap {α : Type u} {β : α → Type v} {γ : α → Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} (f : (a : α) → β a → Option (γ a)) (h : m₁.Equiv m₂) :

(DHashMap.filterMap f m₁).Equiv (DHashMap.filterMap f m₂)

source

theorem Std.DHashMap.Equiv.of_forall_get?_eq {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} [LawfulBEq α] (h : ∀ (k : α), m₁.get? k = m₂.get? k) :

m₁.Equiv m₂

source

theorem Std.DHashMap.Equiv.constToList_perm {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m₁ m₂ : DHashMap α fun (x : α) => β} (h : m₁.Equiv m₂) :

(Const.toList m₁).Perm (Const.toList m₂)

source

theorem Std.DHashMap.Equiv.of_constToList_perm {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m₁ m₂ : DHashMap α fun (x : α) => β} (h : (Const.toList m₁).Perm (Const.toList m₂)) :

m₁.Equiv m₂

source

theorem Std.DHashMap.Equiv.of_keys_unit_perm {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α fun (x : α) => Unit} (h : m₁.keys.Perm m₂.keys) :

m₁.Equiv m₂

source

theorem Std.DHashMap.Equiv.constGet?_eq {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m₁ m₂ : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} (h : m₁.Equiv m₂) :

Const.get? m₁ k = Const.get? m₂ k

source

theorem Std.DHashMap.Equiv.constGet_eq {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m₁ m₂ : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} (hk : k ∈ m₁) (h : m₁.Equiv m₂) :

Const.get m₁ k hk = Const.get m₂ k ⋯

source

theorem Std.DHashMap.Equiv.constGet!_eq {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m₁ m₂ : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {k : α} (h : m₁.Equiv m₂) :

Const.get! m₁ k = Const.get! m₂ k

source

theorem Std.DHashMap.Equiv.constGetD_eq {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m₁ m₂ : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] {k : α} {fallback : β} (h : m₁.Equiv m₂) :

Const.getD m₁ k fallback = Const.getD m₂ k fallback

source

theorem Std.DHashMap.Equiv.constInsertMany_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m₁ m₂ : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] (l : List (α × β)) (h : m₁.Equiv m₂) :

(Const.insertMany m₁ l).Equiv (Const.insertMany m₂ l)

source

theorem Std.DHashMap.Equiv.constInsertManyIfNewUnit_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {m₁ m₂ : DHashMap α fun (x : α) => Unit} (l : List α) (h : m₁.Equiv m₂) :

(Const.insertManyIfNewUnit m₁ l).Equiv (Const.insertManyIfNewUnit m₂ l)

source

theorem Std.DHashMap.Equiv.constAlter {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m₁ m₂ : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] (k : α) (f : Option β → Option β) (h : m₁.Equiv m₂) :

(Const.alter m₁ k f).Equiv (Const.alter m₂ k f)

source

theorem Std.DHashMap.Equiv.constModify {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m₁ m₂ : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] (k : α) (f : β → β) (h : m₁.Equiv m₂) :

(Const.modify m₁ k f).Equiv (Const.modify m₂ k f)

source

theorem Std.DHashMap.Equiv.of_forall_getKey?_eq_of_forall_constGet?_eq {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m₁ m₂ : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] (hk : ∀ (k : α), m₁.getKey? k = m₂.getKey? k) (hv : ∀ (k : α), Const.get? m₁ k = Const.get? m₂ k) :

m₁.Equiv m₂

source

theorem Std.DHashMap.Equiv.of_forall_getKey?_unit_eq {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {m₁ m₂ : DHashMap α fun (x : α) => Unit} (h : ∀ (k : α), m₁.getKey? k = m₂.getKey? k) :

m₁.Equiv m₂

source

theorem Std.DHashMap.Equiv.of_forall_contains_unit_eq {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [LawfulBEq α] {m₁ m₂ : DHashMap α fun (x : α) => Unit} (h : ∀ (k : α), m₁.contains k = m₂.contains k) :

m₁.Equiv m₂

source

theorem Std.DHashMap.Equiv.of_forall_mem_unit_iff {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [LawfulBEq α] {m₁ m₂ : DHashMap α fun (x : α) => Unit} (h : ∀ (k : α), k ∈ m₁ ↔ k ∈ m₂) :

m₁.Equiv m₂

source

@[simp]

theorem Std.DHashMap.equiv_emptyWithCapacity_iff_isEmpty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {c : Nat} :

m.Equiv (emptyWithCapacity c) ↔ m.isEmpty = true

source

@[simp]

theorem Std.DHashMap.equiv_empty_iff_isEmpty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] :

m.Equiv ∅ ↔ m.isEmpty = true

source

@[reducible, inline, deprecated Std.DHashMap.equiv_empty_iff_isEmpty (since := "2025-03-11")]

abbrev Std.DHashMap.equiv_emptyc_iff_isEmpty {α : Type u_1} {β : α → Type u_2} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] :

m.Equiv ∅ ↔ m.isEmpty = true

Equations

@Std.DHashMap.equiv_emptyc_iff_isEmpty = @Std.DHashMap.equiv_empty_iff_isEmpty

Instances For

source

theorem Std.DHashMap.empty_equivWithCapacity_iff_isEmpty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] {c : Nat} :

(emptyWithCapacity c).Equiv m ↔ m.isEmpty = true

source

@[simp]

theorem Std.DHashMap.empty_equiv_iff_isEmpty {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] :

∅.Equiv m ↔ m.isEmpty = true

source

@[reducible, inline, deprecated Std.DHashMap.empty_equiv_iff_isEmpty (since := "2025-03-11")]

abbrev Std.DHashMap.emptyc_equiv_iff_isEmpty {α : Type u_1} {β : α → Type u_2} {x✝ : BEq α} {x✝¹ : Hashable α} {m : DHashMap α β} [EquivBEq α] [LawfulHashable α] :

∅.Equiv m ↔ m.isEmpty = true

Equations

@Std.DHashMap.emptyc_equiv_iff_isEmpty = @Std.DHashMap.empty_equiv_iff_isEmpty

Instances For

source

theorem Std.DHashMap.equiv_iff_toList_perm {α : Type u} {β : α → Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : DHashMap α β} [EquivBEq α] [LawfulHashable α] :

m₁.Equiv m₂ ↔ m₁.toList.Perm m₂.toList

source

theorem Std.DHashMap.Const.equiv_iff_toList_perm {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {β : Type v} {m₁ m₂ : DHashMap α fun (x : α) => β} [EquivBEq α] [LawfulHashable α] :

m₁.Equiv m₂ ↔ (toList m₁).Perm (toList m₂)

source

theorem Std.DHashMap.Const.equiv_iff_keys_unit_perm {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {m₁ m₂ : DHashMap α fun (x : α) => Unit} :

m₁.Equiv m₂ ↔ m₁.keys.Perm m₂.keys

Documentation

Std.Data.DHashMap.Lemmas

Dependent hash map lemmas #