Extensional hash map lemmas

This module contains lemmas about Std.ExtHashMap.

@[simp]

theorem Std.ExtHashMap.isEmpty_iff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : m.isEmpty = true ↔ m = ∅

@[simp]

theorem Std.ExtHashMap.isEmpty_eq_false_iff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : m.isEmpty = false ↔ ¬m = ∅

@[simp]

theorem Std.ExtHashMap.empty_eq {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} : ∅ = m ↔ m = ∅

@[simp]

theorem Std.ExtHashMap.emptyWithCapacity_eq {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {c : Nat} : emptyWithCapacity c = ∅

@[simp]

theorem Std.ExtHashMap.not_insert_eq_empty {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : ¬m.insert k v = ∅

theorem Std.ExtHashMap.mem_iff_contains {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} : a ∈ m ↔ m.contains a = true

@[simp]

theorem Std.ExtHashMap.contains_iff_mem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} : m.contains a = true ↔ a ∈ m

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

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

@[simp]

theorem Std.ExtHashMap.contains_empty {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {a : α} : ∅.contains a = false

@[simp]

theorem Std.ExtHashMap.not_mem_empty {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {a : α} : ¬a ∈ ∅

theorem Std.ExtHashMap.eq_empty_iff_forall_contains {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : m = ∅ ↔ ∀ (a : α), m.contains a = false

theorem Std.ExtHashMap.eq_empty_iff_forall_not_mem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : m = ∅ ↔ ∀ (a : α), ¬a ∈ m

@[simp]

theorem Std.ExtHashMap.insert_eq_insert {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {p : α × β} : Insert.insert p m = m.insert p.fst p.snd

@[simp]

theorem Std.ExtHashMap.singleton_eq_insert {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {p : α × β} : {p} = ∅.insert p.fst p.snd

@[simp]

theorem Std.ExtHashMap.contains_insert {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β} : (m.insert k v).contains a = (k == a || m.contains a)

@[simp]

theorem Std.ExtHashMap.mem_insert {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β} : a ∈ m.insert k v ↔ (k == a) = true ∨ a ∈ m

theorem Std.ExtHashMap.contains_of_contains_insert {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β} : (m.insert k v).contains a = true → (k == a) = false → m.contains a = true

theorem Std.ExtHashMap.mem_of_mem_insert {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β} : a ∈ m.insert k v → (k == a) = false → a ∈ m

theorem Std.ExtHashMap.contains_insert_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : (m.insert k v).contains k = true

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

@[simp]

theorem Std.ExtHashMap.size_empty {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] : ∅.size = 0

theorem Std.ExtHashMap.eq_empty_iff_size_eq_zero {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : m = ∅ ↔ m.size = 0

theorem Std.ExtHashMap.size_insert {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : (m.insert k v).size = if k ∈ m then m.size else m.size + 1

theorem Std.ExtHashMap.size_le_size_insert {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : m.size ≤ (m.insert k v).size

theorem Std.ExtHashMap.size_insert_le {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : (m.insert k v).size ≤ m.size + 1

@[simp]

theorem Std.ExtHashMap.erase_empty {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {a : α} : ∅.erase a = ∅

@[simp]

theorem Std.ExtHashMap.erase_eq_empty_iff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : m.erase k = ∅ ↔ m = ∅ ∨ m.size = 1 ∧ k ∈ m

@[simp]

theorem Std.ExtHashMap.contains_erase {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} : (m.erase k).contains a = (!k == a && m.contains a)

@[simp]

theorem Std.ExtHashMap.mem_erase {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} : a ∈ m.erase k ↔ (k == a) = false ∧ a ∈ m

theorem Std.ExtHashMap.contains_of_contains_erase {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} : (m.erase k).contains a = true → m.contains a = true

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

theorem Std.ExtHashMap.size_erase {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : (m.erase k).size = if k ∈ m then m.size - 1 else m.size

theorem Std.ExtHashMap.size_erase_le {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : (m.erase k).size ≤ m.size

theorem Std.ExtHashMap.size_le_size_erase {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : m.size ≤ (m.erase k).size + 1

@[simp]

theorem Std.ExtHashMap.containsThenInsert_fst {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : (m.containsThenInsert k v).fst = m.contains k

@[simp]

theorem Std.ExtHashMap.containsThenInsert_snd {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : (m.containsThenInsert k v).snd = m.insert k v

@[simp]

theorem Std.ExtHashMap.containsThenInsertIfNew_fst {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : (m.containsThenInsertIfNew k v).fst = m.contains k

@[simp]

theorem Std.ExtHashMap.containsThenInsertIfNew_snd {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : (m.containsThenInsertIfNew k v).snd = m.insertIfNew k v

@[simp]

theorem Std.ExtHashMap.get_eq_getElem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} {h : a ∈ m} : m.get a h = m[a]

@[simp]

theorem Std.ExtHashMap.get?_eq_getElem? {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} : m.get? a = m[a]?

@[simp]

theorem Std.ExtHashMap.get!_eq_getElem! {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {a : α} : m.get! a = m[a]!

@[simp]

theorem Std.ExtHashMap.getElem?_empty {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {a : α} : ∅[a]? = none

theorem Std.ExtHashMap.getElem?_insert {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β} : (m.insert k v)[a]? = if (k == a) = true then some v else m[a]?

@[simp]

theorem Std.ExtHashMap.getElem?_insert_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : (m.insert k v)[k]? = some v

theorem Std.ExtHashMap.contains_eq_isSome_getElem? {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} : m.contains a = m[a]?.isSome

@[simp]

theorem Std.ExtHashMap.isSome_getElem?_eq_contains {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} : m[a]?.isSome = m.contains a

theorem Std.ExtHashMap.mem_iff_isSome_getElem? {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} : a ∈ m ↔ m[a]?.isSome = true

@[simp]

theorem Std.ExtHashMap.isSome_getElem?_iff_mem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} : m[a]?.isSome = true ↔ a ∈ m

theorem Std.ExtHashMap.getElem?_eq_some_iff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : m[k]? = some v ↔ ∃ (h : k ∈ m), m[k] = v

theorem Std.ExtHashMap.getElem?_eq_none_of_contains_eq_false {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} : m.contains a = false → m[a]? = none

theorem Std.ExtHashMap.getElem?_eq_none {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} : ¬a ∈ m → m[a]? = none

theorem Std.ExtHashMap.getElem?_erase {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} : (m.erase k)[a]? = if (k == a) = true then none else m[a]?

@[simp]

theorem Std.ExtHashMap.getElem?_erase_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : (m.erase k)[k]? = none

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

theorem Std.ExtHashMap.getElem_insert {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β} {h₁ : a ∈ m.insert k v} : (m.insert k v)[a] = if h₂ : (k == a) = true then v else m[a]

@[simp]

theorem Std.ExtHashMap.getElem_insert_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : (m.insert k v)[k] = v

@[simp]

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

theorem Std.ExtHashMap.getElem?_eq_some_getElem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} (h' : a ∈ m) : m[a]? = some m[a]

theorem Std.ExtHashMap.getElem_eq_get_getElem? {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} {h : a ∈ m} : m[a] = m[a]?.get ⋯

theorem Std.ExtHashMap.get_getElem? {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} {h : m[a]?.isSome = true} : m[a]?.get h = m[a]

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

@[simp]

theorem Std.ExtHashMap.getElem!_empty {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] [Inhabited β] {a : α} : ∅[a]! = default

theorem Std.ExtHashMap.getElem!_insert {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {k a : α} {v : β} : (m.insert k v)[a]! = if (k == a) = true then v else m[a]!

@[simp]

theorem Std.ExtHashMap.getElem!_insert_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {k : α} {v : β} : (m.insert k v)[k]! = v

theorem Std.ExtHashMap.getElem!_eq_default_of_contains_eq_false {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {a : α} : m.contains a = false → m[a]! = default

theorem Std.ExtHashMap.getElem!_eq_default {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {a : α} : ¬a ∈ m → m[a]! = default

theorem Std.ExtHashMap.getElem!_erase {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {k a : α} : (m.erase k)[a]! = if (k == a) = true then default else m[a]!

@[simp]

theorem Std.ExtHashMap.getElem!_erase_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {k : α} : (m.erase k)[k]! = default

theorem Std.ExtHashMap.getElem?_eq_some_getElem!_of_contains {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {a : α} : m.contains a = true → m[a]? = some m[a]!

theorem Std.ExtHashMap.getElem?_eq_some_getElem! {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {a : α} : a ∈ m → m[a]? = some m[a]!

theorem Std.ExtHashMap.getElem!_eq_get!_getElem? {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {a : α} : m[a]! = m[a]?.get!

theorem Std.ExtHashMap.getElem_eq_getElem! {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {a : α} {h' : a ∈ m} : m[a] = m[a]!

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

@[simp]

theorem Std.ExtHashMap.getD_empty {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {a : α} {fallback : β} : ∅.getD a fallback = fallback

theorem Std.ExtHashMap.getD_insert {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {fallback v : β} : (m.insert k v).getD a fallback = if (k == a) = true then v else m.getD a fallback

@[simp]

theorem Std.ExtHashMap.getD_insert_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {fallback v : β} : (m.insert k v).getD k fallback = v

theorem Std.ExtHashMap.getD_eq_fallback_of_contains_eq_false {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} {fallback : β} : m.contains a = false → m.getD a fallback = fallback

theorem Std.ExtHashMap.getD_eq_fallback {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} {fallback : β} : ¬a ∈ m → m.getD a fallback = fallback

theorem Std.ExtHashMap.getD_erase {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {fallback : β} : (m.erase k).getD a fallback = if (k == a) = true then fallback else m.getD a fallback

@[simp]

theorem Std.ExtHashMap.getD_erase_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {fallback : β} : (m.erase k).getD k fallback = fallback

theorem Std.ExtHashMap.getElem?_eq_some_getD_of_contains {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} {fallback : β} : m.contains a = true → m[a]? = some (m.getD a fallback)

theorem Std.ExtHashMap.getElem?_eq_some_getD {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} {fallback : β} : a ∈ m → m[a]? = some (m.getD a fallback)

theorem Std.ExtHashMap.getD_eq_getD_getElem? {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} {fallback : β} : m.getD a fallback = m[a]?.getD fallback

theorem Std.ExtHashMap.getElem_eq_getD {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} {fallback : β} {h' : a ∈ m} : m[a] = m.getD a fallback

theorem Std.ExtHashMap.getElem!_eq_getD_default {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {a : α} : m[a]! = m.getD a default

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

@[simp]

theorem Std.ExtHashMap.getKey?_empty {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {a : α} : ∅.getKey? a = none

theorem Std.ExtHashMap.getKey?_insert {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β} : (m.insert k v).getKey? a = if (k == a) = true then some k else m.getKey? a

@[simp]

theorem Std.ExtHashMap.getKey?_insert_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : (m.insert k v).getKey? k = some k

theorem Std.ExtHashMap.contains_eq_isSome_getKey? {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} : m.contains a = (m.getKey? a).isSome

@[simp]

theorem Std.ExtHashMap.isSome_getKey?_eq_contains {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} : (m.getKey? a).isSome = m.contains a

theorem Std.ExtHashMap.mem_iff_isSome_getKey? {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} : a ∈ m ↔ (m.getKey? a).isSome = true

@[simp]

theorem Std.ExtHashMap.isSome_getKey?_iff_mem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} : (m.getKey? a).isSome = true ↔ a ∈ m

theorem Std.ExtHashMap.mem_of_getKey?_eq_some {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k k' : α} (h : m.getKey? k = some k') : k' ∈ m

theorem Std.ExtHashMap.getKey?_eq_some_iff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k k' : α} : m.getKey? k = some k' ↔ ∃ (h : k ∈ m), m.getKey k h = k'

theorem Std.ExtHashMap.getKey?_eq_none_of_contains_eq_false {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} : m.contains a = false → m.getKey? a = none

theorem Std.ExtHashMap.getKey?_eq_none {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} : ¬a ∈ m → m.getKey? a = none

theorem Std.ExtHashMap.getKey?_erase {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} : (m.erase k).getKey? a = if (k == a) = true then none else m.getKey? a

@[simp]

theorem Std.ExtHashMap.getKey?_erase_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : (m.erase k).getKey? k = none

theorem Std.ExtHashMap.getKey?_beq {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : Option.all (fun (x : α) => x == k) (m.getKey? k) = true

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

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

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

theorem Std.ExtHashMap.getKey_insert {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β} {h₁ : a ∈ m.insert k v} : (m.insert k v).getKey a h₁ = if h₂ : (k == a) = true then k else m.getKey a ⋯

@[simp]

theorem Std.ExtHashMap.getKey_insert_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : (m.insert k v).getKey k ⋯ = k

@[simp]

theorem Std.ExtHashMap.getKey_erase {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {h' : a ∈ m.erase k} : (m.erase k).getKey a h' = m.getKey a ⋯

theorem Std.ExtHashMap.getKey?_eq_some_getKey {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} (h : a ∈ m) : m.getKey? a = some (m.getKey a h)

theorem Std.ExtHashMap.getKey_eq_get_getKey? {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} {h : a ∈ m} : m.getKey a h = (m.getKey? a).get ⋯

@[simp]

theorem Std.ExtHashMap.get_getKey? {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a : α} {h : (m.getKey? a).isSome = true} : (m.getKey? a).get h = m.getKey a ⋯

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

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

@[simp]

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

@[simp]

theorem Std.ExtHashMap.getKey!_empty {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] [Inhabited α] {a : α} : ∅.getKey! a = default

theorem Std.ExtHashMap.getKey!_insert {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k a : α} {v : β} : (m.insert k v).getKey! a = if (k == a) = true then k else m.getKey! a

@[simp]

theorem Std.ExtHashMap.getKey!_insert_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k : α} {v : β} : (m.insert k v).getKey! k = k

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

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

theorem Std.ExtHashMap.getKey!_erase {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap 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

@[simp]

theorem Std.ExtHashMap.getKey!_erase_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k : α} : (m.erase k).getKey! k = default

theorem Std.ExtHashMap.getKey?_eq_some_getKey!_of_contains {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {a : α} : m.contains a = true → m.getKey? a = some (m.getKey! a)

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

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

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

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

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

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

@[simp]

theorem Std.ExtHashMap.getKeyD_empty {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {a fallback : α} : ∅.getKeyD a fallback = fallback

theorem Std.ExtHashMap.getKeyD_insert {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a fallback : α} {v : β} : (m.insert k v).getKeyD a fallback = if (k == a) = true then k else m.getKeyD a fallback

@[simp]

theorem Std.ExtHashMap.getKeyD_insert_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k fallback : α} {v : β} : (m.insert k v).getKeyD k fallback = k

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

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

theorem Std.ExtHashMap.getKeyD_erase {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap 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

@[simp]

theorem Std.ExtHashMap.getKeyD_erase_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k fallback : α} : (m.erase k).getKeyD k fallback = fallback

theorem Std.ExtHashMap.getKey?_eq_some_getKeyD_of_contains {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a fallback : α} : m.contains a = true → m.getKey? a = some (m.getKeyD a fallback)

theorem Std.ExtHashMap.getKey?_eq_some_getKeyD {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a fallback : α} : a ∈ m → m.getKey? a = some (m.getKeyD a fallback)

theorem Std.ExtHashMap.getKeyD_eq_getD_getKey? {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a fallback : α} : m.getKeyD a fallback = (m.getKey? a).getD fallback

theorem Std.ExtHashMap.getKey_eq_getKeyD {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {a fallback : α} {h' : a ∈ m} : m.getKey a h' = m.getKeyD a fallback

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

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

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

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

@[simp]

theorem Std.ExtHashMap.not_insertIfNew_eq_empty {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : ¬m.insertIfNew k v = ∅

@[simp]

theorem Std.ExtHashMap.contains_insertIfNew {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β} : (m.insertIfNew k v).contains a = (k == a || m.contains a)

@[simp]

theorem Std.ExtHashMap.mem_insertIfNew {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β} : a ∈ m.insertIfNew k v ↔ (k == a) = true ∨ a ∈ m

theorem Std.ExtHashMap.contains_insertIfNew_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : (m.insertIfNew k v).contains k = true

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

theorem Std.ExtHashMap.contains_of_contains_insertIfNew {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β} : (m.insertIfNew k v).contains a = true → (k == a) = false → m.contains a = true

theorem Std.ExtHashMap.mem_of_mem_insertIfNew {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β} : a ∈ m.insertIfNew k v → (k == a) = false → a ∈ m

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

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

theorem Std.ExtHashMap.size_insertIfNew {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : (m.insertIfNew k v).size = if k ∈ m then m.size else m.size + 1

theorem Std.ExtHashMap.size_le_size_insertIfNew {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : m.size ≤ (m.insertIfNew k v).size

theorem Std.ExtHashMap.size_insertIfNew_le {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : (m.insertIfNew k v).size ≤ m.size + 1

theorem Std.ExtHashMap.getElem?_insertIfNew {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β} : (m.insertIfNew k v)[a]? = if (k == a) = true ∧ ¬k ∈ m then some v else m[a]?

theorem Std.ExtHashMap.getElem_insertIfNew {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β} {h₁ : a ∈ m.insertIfNew k v} : (m.insertIfNew k v)[a] = if h₂ : (k == a) = true ∧ ¬k ∈ m then v else m[a]

theorem Std.ExtHashMap.getElem!_insertIfNew {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {k a : α} {v : β} : (m.insertIfNew k v)[a]! = if (k == a) = true ∧ ¬k ∈ m then v else m[a]!

theorem Std.ExtHashMap.getD_insertIfNew {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {fallback v : β} : (m.insertIfNew k v).getD a fallback = if (k == a) = true ∧ ¬k ∈ m then v else m.getD a fallback

theorem Std.ExtHashMap.getKey?_insertIfNew {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β} : (m.insertIfNew k v).getKey? a = if (k == a) = true ∧ ¬k ∈ m then some k else m.getKey? a

theorem Std.ExtHashMap.getKey_insertIfNew {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a : α} {v : β} {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 ⋯

theorem Std.ExtHashMap.getKey!_insertIfNew {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k a : α} {v : β} : (m.insertIfNew k v).getKey! a = if (k == a) = true ∧ ¬k ∈ m then k else m.getKey! a

theorem Std.ExtHashMap.getKeyD_insertIfNew {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k a fallback : α} {v : β} : (m.insertIfNew k v).getKeyD a fallback = if (k == a) = true ∧ ¬k ∈ m then k else m.getKeyD a fallback

@[simp]

theorem Std.ExtHashMap.getThenInsertIfNew?_fst {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : (m.getThenInsertIfNew? k v).fst = m.get? k

@[simp]

theorem Std.ExtHashMap.getThenInsertIfNew?_snd {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : (m.getThenInsertIfNew? k v).snd = m.insertIfNew k v

instance Std.ExtHashMap.instLawfulGetElemMem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] : LawfulGetElem (ExtHashMap α β) α β fun (m : ExtHashMap α β) (a : α) => a ∈ m

@[simp]

theorem Std.ExtHashMap.insertMany_nil {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : m.insertMany [] = m

@[simp]

theorem Std.ExtHashMap.insertMany_list_singleton {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : m.insertMany [(k, v)] = m.insert k v

theorem Std.ExtHashMap.insertMany_cons {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k : α} {v : β} : m.insertMany ((k, v) :: l) = (m.insert k v).insertMany l

theorem Std.ExtHashMap.insertMany_append {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {l₁ l₂ : List (α × β)} : m.insertMany (l₁ ++ l₂) = (m.insertMany l₁).insertMany l₂

theorem Std.ExtHashMap.insertMany_ind {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {ρ : Type w} [ForIn Id ρ (α × β)] [EquivBEq α] [LawfulHashable α] {motive : ExtHashMap α β → Prop} (m : ExtHashMap α β) {l : ρ} (init : motive m) (insert : ∀ (m : ExtHashMap α β) (a : α) (b : β), motive m → motive (m.insert a b)) : motive (m.insertMany l)

@[simp]

theorem Std.ExtHashMap.contains_insertMany_list {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k : α} : (m.insertMany l).contains k = (m.contains k || (List.map Prod.fst l).contains k)

@[simp]

theorem Std.ExtHashMap.mem_insertMany_list {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k : α} : k ∈ m.insertMany l ↔ k ∈ m ∨ (List.map Prod.fst l).contains k = true

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

theorem Std.ExtHashMap.mem_insertMany_of_mem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} {ρ : Type w} [ForIn Id ρ (α × β)] [EquivBEq α] [LawfulHashable α] {l : ρ} {k : α} : k ∈ m → k ∈ m.insertMany l

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

theorem Std.ExtHashMap.getElem?_insertMany_list_of_mem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap 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) : (m.insertMany l)[k']? = some v

theorem Std.ExtHashMap.getElem?_insertMany_list {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k : α} : (m.insertMany l)[k]? = (List.findSomeRev? (fun (x : α × β) => match x with | (a, b) => if (a == k) = true then some b else none) l).or m[k]?

theorem Std.ExtHashMap.getElem_insertMany_list_of_contains_eq_false {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap 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 ∈ m.insertMany l} : (m.insertMany l)[k] = m[k]

theorem Std.ExtHashMap.getElem_insertMany_list_of_mem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap 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' ∈ m.insertMany l} : (m.insertMany l)[k'] = v

theorem Std.ExtHashMap.getElem!_insertMany_list_of_contains_eq_false {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap 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) : (m.insertMany l)[k]! = m[k]!

theorem Std.ExtHashMap.getElem!_insertMany_list_of_mem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap 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) : (m.insertMany l)[k']! = v

theorem Std.ExtHashMap.getD_insertMany_list_of_contains_eq_false {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap 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) : (m.insertMany l).getD k fallback = m.getD k fallback

theorem Std.ExtHashMap.getD_insertMany_list_of_mem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap 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) : (m.insertMany l).getD k' fallback = v

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

theorem Std.ExtHashMap.getKey?_insertMany_list_of_mem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap 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) : (m.insertMany l).getKey? k' = some k

theorem Std.ExtHashMap.getKey_insertMany_list_of_contains_eq_false {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap 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 ∈ m.insertMany l} : (m.insertMany l).getKey k h = m.getKey k ⋯

theorem Std.ExtHashMap.getKey_insertMany_list_of_mem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap 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' ∈ m.insertMany l} : (m.insertMany l).getKey k' h = k

theorem Std.ExtHashMap.getKey!_insertMany_list_of_contains_eq_false {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap 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) : (m.insertMany l).getKey! k = m.getKey! k

theorem Std.ExtHashMap.getKey!_insertMany_list_of_mem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap 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) : (m.insertMany l).getKey! k' = k

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

theorem Std.ExtHashMap.getKeyD_insertMany_list_of_mem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap 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) : (m.insertMany l).getKeyD k' fallback = k

theorem Std.ExtHashMap.size_insertMany_list {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap 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) → (m.insertMany l).size = m.size + l.length

theorem Std.ExtHashMap.size_le_size_insertMany_list {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} : m.size ≤ (m.insertMany l).size

theorem Std.ExtHashMap.size_le_size_insertMany {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} {ρ : Type w} [ForIn Id ρ (α × β)] [EquivBEq α] [LawfulHashable α] {l : ρ} : m.size ≤ (m.insertMany l).size

theorem Std.ExtHashMap.size_insertMany_list_le {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} : (m.insertMany l).size ≤ m.size + l.length

@[simp]

theorem Std.ExtHashMap.insertMany_list_eq_empty_iff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} : m.insertMany l = ∅ ↔ m = ∅ ∧ l = []

theorem Std.ExtHashMap.eq_empty_of_insertMany_eq_empty {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} {ρ : Type w} [ForIn Id ρ (α × β)] [EquivBEq α] [LawfulHashable α] {l : ρ} : m.insertMany l = ∅ → m = ∅

theorem Std.ExtHashMap.insertMany_list_eq_foldl {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} : m.insertMany l = List.foldl (fun (acc : ExtHashMap α β) (p : α × β) => acc.insert p.fst p.snd) m l

@[simp]

theorem Std.ExtHashMap.insertManyIfNewUnit_nil {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α Unit} [EquivBEq α] [LawfulHashable α] : m.insertManyIfNewUnit [] = m

@[simp]

theorem Std.ExtHashMap.insertManyIfNewUnit_list_singleton {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α Unit} [EquivBEq α] [LawfulHashable α] {k : α} : m.insertManyIfNewUnit [k] = m.insertIfNew k ()

theorem Std.ExtHashMap.insertManyIfNewUnit_cons {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α Unit} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} : m.insertManyIfNewUnit (k :: l) = (m.insertIfNew k ()).insertManyIfNewUnit l

theorem Std.ExtHashMap.insertManyIfNewUnit_ind {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {ρ : Type w} [ForIn Id ρ α] [EquivBEq α] [LawfulHashable α] {motive : ExtHashMap α Unit → Prop} (m : ExtHashMap α Unit) (l : ρ) (init : motive m) (insert : ∀ (m : ExtHashMap α Unit) (a : α), motive m → motive (m.insertIfNew a ())) : motive (m.insertManyIfNewUnit l)

@[simp]

theorem Std.ExtHashMap.contains_insertManyIfNewUnit_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α Unit} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} : (m.insertManyIfNewUnit l).contains k = (m.contains k || l.contains k)

@[simp]

theorem Std.ExtHashMap.mem_insertManyIfNewUnit_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α Unit} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} : k ∈ m.insertManyIfNewUnit l ↔ k ∈ m ∨ l.contains k = true

theorem Std.ExtHashMap.mem_of_mem_insertManyIfNewUnit_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α Unit} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} (contains_eq_false : l.contains k = false) : k ∈ m.insertManyIfNewUnit l → k ∈ m

theorem Std.ExtHashMap.mem_insertManyIfNewUnit_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α Unit} {ρ : Type w} [ForIn Id ρ α] [EquivBEq α] [LawfulHashable α] {l : ρ} {k : α} : k ∈ m → k ∈ m.insertManyIfNewUnit l

theorem Std.ExtHashMap.getElem?_insertManyIfNewUnit_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α Unit} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} : (m.insertManyIfNewUnit l)[k]? = if k ∈ m ∨ l.contains k = true then some () else none

theorem Std.ExtHashMap.getElem_insertManyIfNewUnit_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α Unit} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} {h : k ∈ m.insertManyIfNewUnit l} : (m.insertManyIfNewUnit l)[k] = ()

theorem Std.ExtHashMap.getElem!_insertManyIfNewUnit_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α Unit} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} : (m.insertManyIfNewUnit l)[k]! = ()

theorem Std.ExtHashMap.getD_insertManyIfNewUnit_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α Unit} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} {fallback : Unit} : (m.insertManyIfNewUnit l).getD k fallback = ()

theorem Std.ExtHashMap.getKey?_insertManyIfNewUnit_list_of_not_mem_of_contains_eq_false {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α Unit} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} (not_mem : ¬k ∈ m) (contains_eq_false : l.contains k = false) : (m.insertManyIfNewUnit l).getKey? k = none

theorem Std.ExtHashMap.getKey?_insertManyIfNewUnit_list_of_not_mem_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap 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) : (m.insertManyIfNewUnit l).getKey? k' = some k

theorem Std.ExtHashMap.getKey?_insertManyIfNewUnit_list_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α Unit} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} (mem : k ∈ m) : (m.insertManyIfNewUnit l).getKey? k = m.getKey? k

theorem Std.ExtHashMap.getKey_insertManyIfNewUnit_list_of_not_mem_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap 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' ∈ m.insertManyIfNewUnit l} : (m.insertManyIfNewUnit l).getKey k' h = k

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

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

theorem Std.ExtHashMap.getKey!_insertManyIfNewUnit_list_of_not_mem_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap 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) : (m.insertManyIfNewUnit l).getKey! k' = k

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

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

theorem Std.ExtHashMap.getKeyD_insertManyIfNewUnit_list_of_not_mem_of_mem {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap 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) : (m.insertManyIfNewUnit l).getKeyD k' fallback = k

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

theorem Std.ExtHashMap.size_insertManyIfNewUnit_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap 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) → (m.insertManyIfNewUnit l).size = m.size + l.length

theorem Std.ExtHashMap.size_le_size_insertManyIfNewUnit_list {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α Unit} [EquivBEq α] [LawfulHashable α] {l : List α} : m.size ≤ (m.insertManyIfNewUnit l).size

theorem Std.ExtHashMap.size_le_size_insertManyIfNewUnit {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α Unit} {ρ : Type w} [ForIn Id ρ α] [EquivBEq α] [LawfulHashable α] {l : ρ} : m.size ≤ (m.insertManyIfNewUnit l).size

theorem Std.ExtHashMap.size_insertManyIfNewUnit_list_le {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α Unit} [EquivBEq α] [LawfulHashable α] {l : List α} : (m.insertManyIfNewUnit l).size ≤ m.size + l.length

@[simp]

theorem Std.ExtHashMap.insertManyIfNewUnit_list_eq_empty_iff {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α Unit} [EquivBEq α] [LawfulHashable α] {l : List α} : m.insertManyIfNewUnit l = ∅ ↔ m = ∅ ∧ l = []

theorem Std.ExtHashMap.eq_empty_of_insertManyIfNewUnit_eq_empty {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α Unit} {ρ : Type w} [ForIn Id ρ α] [EquivBEq α] [LawfulHashable α] {l : ρ} : m.insertManyIfNewUnit l = ∅ → m = ∅

theorem Std.ExtHashMap.insertManyIfNewUnit_list_eq_foldl {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α Unit} [EquivBEq α] [LawfulHashable α] {l : List α} : m.insertManyIfNewUnit l = List.foldl (fun (acc : ExtHashMap α Unit) (a : α) => acc.insertIfNew a ()) m l

@[simp]

theorem Std.ExtHashMap.ofList_nil {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] : ofList [] = ∅

@[simp]

theorem Std.ExtHashMap.ofList_singleton {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} : ofList [(k, v)] = ∅.insert k v

theorem Std.ExtHashMap.ofList_cons {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {k : α} {v : β} {tl : List (α × β)} : ofList ((k, v) :: tl) = (∅.insert k v).insertMany tl

theorem Std.ExtHashMap.ofList_eq_insertMany_empty {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} : ofList l = ∅.insertMany l

@[simp]

theorem Std.ExtHashMap.contains_ofList {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k : α} : (ofList l).contains k = (List.map Prod.fst l).contains k

@[simp]

theorem Std.ExtHashMap.mem_ofList {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k : α} : k ∈ ofList l ↔ (List.map Prod.fst l).contains k = true

theorem Std.ExtHashMap.getElem?_ofList_of_contains_eq_false {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) : (ofList l)[k]? = none

theorem Std.ExtHashMap.getElem?_ofList_of_mem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [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) : (ofList l)[k']? = some v

theorem Std.ExtHashMap.getElem_ofList_of_mem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [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' ∈ ofList l} : (ofList l)[k'] = v

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

theorem Std.ExtHashMap.getElem!_ofList_of_mem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable 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) : (ofList l)[k']! = v

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

theorem Std.ExtHashMap.getD_ofList_of_mem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [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) : (ofList l).getD k' fallback = v

theorem Std.ExtHashMap.getKey?_ofList_of_contains_eq_false {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [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

theorem Std.ExtHashMap.getKey?_ofList_of_mem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [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

theorem Std.ExtHashMap.getKey_ofList_of_mem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [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

theorem Std.ExtHashMap.getKey!_ofList_of_contains_eq_false {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [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

theorem Std.ExtHashMap.getKey!_ofList_of_mem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [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

theorem Std.ExtHashMap.getKeyD_ofList_of_contains_eq_false {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [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

theorem Std.ExtHashMap.getKeyD_ofList_of_mem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [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

theorem Std.ExtHashMap.size_ofList {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [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

theorem Std.ExtHashMap.size_ofList_le {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} : (ofList l).size ≤ l.length

@[simp]

theorem Std.ExtHashMap.ofList_eq_empty_iff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} : ofList l = ∅ ↔ l = []

theorem Std.ExtHashMap.ofList_eq_foldl {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} : ofList l = List.foldl (fun (acc : ExtHashMap α β) (p : α × β) => acc.insert p.fst p.snd) ∅ l

@[simp]

theorem Std.ExtHashMap.unitOfList_nil {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] : unitOfList [] = ∅

@[simp]

theorem Std.ExtHashMap.unitOfList_singleton {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {k : α} : unitOfList [k] = ∅.insertIfNew k ()

theorem Std.ExtHashMap.unitOfList_cons {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {hd : α} {tl : List α} : unitOfList (hd :: tl) = (∅.insertIfNew hd ()).insertManyIfNewUnit tl

theorem Std.ExtHashMap.unitOfList_eq_insertManyIfNewUnit_empty {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List α} : unitOfList l = ∅.insertManyIfNewUnit l

@[simp]

theorem Std.ExtHashMap.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

@[simp]

theorem Std.ExtHashMap.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

@[simp]

theorem Std.ExtHashMap.getElem?_unitOfList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} : (unitOfList l)[k]? = if l.contains k = true then some () else none

@[simp]

theorem Std.ExtHashMap.getElem_unitOfList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} {h : k ∈ unitOfList l} : (unitOfList l)[k] = ()

@[simp]

theorem Std.ExtHashMap.getElem!_unitOfList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} : (unitOfList l)[k]! = ()

@[simp]

theorem Std.ExtHashMap.getD_unitOfList {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List α} {k : α} {fallback : Unit} : (unitOfList l).getD k fallback = ()

theorem Std.ExtHashMap.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

theorem Std.ExtHashMap.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

theorem Std.ExtHashMap.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

theorem Std.ExtHashMap.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

theorem Std.ExtHashMap.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

theorem Std.ExtHashMap.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

theorem Std.ExtHashMap.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

theorem Std.ExtHashMap.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

theorem Std.ExtHashMap.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

@[simp]

theorem Std.ExtHashMap.unitOfList_eq_empty_iff {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List α} : unitOfList l = ∅ ↔ l = []

theorem Std.ExtHashMap.unitOfList_eq_foldl {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {l : List α} : unitOfList l = List.foldl (fun (acc : ExtHashMap α Unit) (a : α) => acc.insertIfNew a ()) ∅ l

@[simp]

theorem Std.ExtHashMap.union_eq {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : m₁.union m₂ = m₁ ∪ m₂

@[simp]

theorem Std.ExtHashMap.contains_union {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : (m₁ ∪ m₂).contains k = (m₁.contains k || m₂.contains k)

theorem Std.ExtHashMap.mem_union_of_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : k ∈ m₁ → k ∈ m₁ ∪ m₂

theorem Std.ExtHashMap.mem_union_of_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : k ∈ m₂ → k ∈ m₁ ∪ m₂

@[simp]

theorem Std.ExtHashMap.mem_union_iff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : k ∈ m₁ ∪ m₂ ↔ k ∈ m₁ ∨ k ∈ m₂

theorem Std.ExtHashMap.mem_of_mem_union_of_not_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : k ∈ m₁ ∪ m₂ → ¬k ∈ m₂ → k ∈ m₁

theorem Std.ExtHashMap.mem_of_mem_union_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : k ∈ m₁ ∪ m₂ → ¬k ∈ m₁ → k ∈ m₂

theorem Std.ExtHashMap.union_insert_right_eq_insert_union {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {p : (_ : α) × β} : m₁ ∪ m₂.insert p.fst p.snd = (m₁ ∪ m₂).insert p.fst p.snd

theorem Std.ExtHashMap.getElem?_union {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : (m₁ ∪ m₂)[k]? = m₂[k]?.or m₁[k]?

theorem Std.ExtHashMap.getElem?_union_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (not_mem : ¬k ∈ m₁) : (m₁ ∪ m₂)[k]? = m₂[k]?

theorem Std.ExtHashMap.getElem?_union_of_not_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (not_mem : ¬k ∈ m₂) : (m₁ ∪ m₂)[k]? = m₁[k]?

theorem Std.ExtHashMap.getElem_union_of_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (mem : k ∈ m₂) : (m₁ ∪ m₂)[k] = m₂[k]

theorem Std.ExtHashMap.getElem_union_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (not_mem : ¬k ∈ m₁) {h' : k ∈ m₁ ∪ m₂} : (m₁ ∪ m₂)[k] = m₂[k]

theorem Std.ExtHashMap.getD_union {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {fallback : β} : (m₁ ∪ m₂).getD k fallback = m₂.getD k (m₁.getD k fallback)

theorem Std.ExtHashMap.getD_union_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {fallback : β} (not_mem : ¬k ∈ m₁) : (m₁ ∪ m₂).getD k fallback = m₂.getD k fallback

theorem Std.ExtHashMap.getD_union_of_not_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {fallback : β} (not_mem : ¬k ∈ m₂) : (m₁ ∪ m₂).getD k fallback = m₁.getD k fallback

theorem Std.ExtHashMap.getElem!_union {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} [Inhabited β] : (m₁ ∪ m₂)[k]! = m₂.getD k m₁[k]!

theorem Std.ExtHashMap.getElem!_union_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} [Inhabited β] (not_mem : ¬k ∈ m₁) : (m₁ ∪ m₂)[k]! = m₂[k]!

theorem Std.ExtHashMap.getElem!_union_of_not_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} [Inhabited β] (not_mem : ¬k ∈ m₂) : (m₁ ∪ m₂)[k]! = m₁[k]!

theorem Std.ExtHashMap.getKey?_union {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : (m₁ ∪ m₂).getKey? k = (m₂.getKey? k).or (m₁.getKey? k)

theorem Std.ExtHashMap.getKey?_union_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (not_mem : ¬k ∈ m₁) : (m₁ ∪ m₂).getKey? k = m₂.getKey? k

theorem Std.ExtHashMap.getKey_union_of_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (mem : k ∈ m₂) : (m₁ ∪ m₂).getKey k ⋯ = m₂.getKey k mem

theorem Std.ExtHashMap.getKey_union_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (not_mem : ¬k ∈ m₁) {h' : k ∈ m₁ ∪ m₂} : (m₁ ∪ m₂).getKey k h' = m₂.getKey k ⋯

theorem Std.ExtHashMap.getKey_union_of_not_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (not_mem : ¬k ∈ m₂) {h' : k ∈ m₁ ∪ m₂} : (m₁ ∪ m₂).getKey k h' = m₁.getKey k ⋯

theorem Std.ExtHashMap.getKeyD_union {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k fallback : α} : (m₁ ∪ m₂).getKeyD k fallback = m₂.getKeyD k (m₁.getKeyD k fallback)

theorem Std.ExtHashMap.getKeyD_union_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k fallback : α} (not_mem : ¬k ∈ m₁) : (m₁ ∪ m₂).getKeyD k fallback = m₂.getKeyD k fallback

theorem Std.ExtHashMap.getKeyD_union_of_not_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k fallback : α} (not_mem : ¬k ∈ m₂) : (m₁ ∪ m₂).getKeyD k fallback = m₁.getKeyD k fallback

theorem Std.ExtHashMap.getKey!_union {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k : α} : (m₁ ∪ m₂).getKey! k = m₂.getKeyD k (m₁.getKey! k)

theorem Std.ExtHashMap.getKey!_union_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [Inhabited α] [EquivBEq α] [LawfulHashable α] {k : α} (not_mem : ¬k ∈ m₁) : (m₁ ∪ m₂).getKey! k = m₂.getKey! k

theorem Std.ExtHashMap.getKey!_union_of_not_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [Inhabited α] [EquivBEq α] [LawfulHashable α] {k : α} (not_mem : ¬k ∈ m₂) : (m₁ ∪ m₂).getKey! k = m₁.getKey! k

theorem Std.ExtHashMap.size_union_of_not_mem {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : (∀ (a : α), a ∈ m₁ → ¬a ∈ m₂) → (m₁ ∪ m₂).size = m₁.size + m₂.size

theorem Std.ExtHashMap.size_left_le_size_union {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : m₁.size ≤ (m₁ ∪ m₂).size

theorem Std.ExtHashMap.size_right_le_size_union {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : m₂.size ≤ (m₁ ∪ m₂).size

theorem Std.ExtHashMap.size_union_le_size_add_size {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : (m₁ ∪ m₂).size ≤ m₁.size + m₂.size

@[simp]

theorem Std.ExtHashMap.isEmpty_union {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : (m₁ ∪ m₂).isEmpty = (m₁.isEmpty && m₂.isEmpty)

@[simp]

theorem Std.ExtHashMap.inter_eq {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : m₁.inter m₂ = m₁ ∩ m₂

@[simp]

theorem Std.ExtHashMap.contains_inter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : (m₁ ∩ m₂).contains k = (m₁.contains k && m₂.contains k)

@[simp]

theorem Std.ExtHashMap.mem_inter_iff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : k ∈ m₁ ∩ m₂ ↔ k ∈ m₁ ∧ k ∈ m₂

theorem Std.ExtHashMap.not_mem_inter_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (not_mem : ¬k ∈ m₁) : ¬k ∈ m₁ ∩ m₂

theorem Std.ExtHashMap.not_mem_inter_of_not_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (not_mem : ¬k ∈ m₂) : ¬k ∈ m₁ ∩ m₂

theorem Std.ExtHashMap.getElem?_inter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : (m₁ ∩ m₂)[k]? = if k ∈ m₂ then m₁[k]? else none

theorem Std.ExtHashMap.getElem?_inter_of_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (mem : k ∈ m₂) : (m₁ ∩ m₂)[k]? = m₁[k]?

theorem Std.ExtHashMap.getElem?_inter_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (not_mem : ¬k ∈ m₁) : (m₁ ∩ m₂)[k]? = none

theorem Std.ExtHashMap.getElem?_inter_of_not_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (not_mem : ¬k ∈ m₂) : (m₁ ∩ m₂)[k]? = none

@[simp]

theorem Std.ExtHashMap.getElem_inter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {h_mem : k ∈ m₁ ∩ m₂} : (m₁ ∩ m₂)[k] = m₁[k]

theorem Std.ExtHashMap.getD_inter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {fallback : β} : (m₁ ∩ m₂).getD k fallback = if k ∈ m₂ then m₁.getD k fallback else fallback

theorem Std.ExtHashMap.getD_inter_of_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {fallback : β} (mem : k ∈ m₂) : (m₁ ∩ m₂).getD k fallback = m₁.getD k fallback

theorem Std.ExtHashMap.getD_inter_of_not_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {fallback : β} (not_mem : ¬k ∈ m₂) : (m₁ ∩ m₂).getD k fallback = fallback

theorem Std.ExtHashMap.getD_inter_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {fallback : β} (not_mem : ¬k ∈ m₁) : (m₁ ∩ m₂).getD k fallback = fallback

theorem Std.ExtHashMap.getElem!_inter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} [Inhabited β] : (m₁ ∩ m₂)[k]! = if k ∈ m₂ then m₁[k]! else default

theorem Std.ExtHashMap.getElem!_inter_of_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} [Inhabited β] (mem : k ∈ m₂) : (m₁ ∩ m₂)[k]! = m₁[k]!

theorem Std.ExtHashMap.getElem!_inter_of_not_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} [Inhabited β] (not_mem : ¬k ∈ m₂) : (m₁ ∩ m₂)[k]! = default

theorem Std.ExtHashMap.getElem!_inter_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} [Inhabited β] (not_mem : ¬k ∈ m₁) : (m₁ ∩ m₂)[k]! = default

theorem Std.ExtHashMap.getKey?_inter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : (m₁ ∩ m₂).getKey? k = if k ∈ m₂ then m₁.getKey? k else none

theorem Std.ExtHashMap.getKey?_inter_of_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (mem : k ∈ m₂) : (m₁ ∩ m₂).getKey? k = m₁.getKey? k

theorem Std.ExtHashMap.getKey?_inter_of_not_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (not_mem : ¬k ∈ m₂) : (m₁ ∩ m₂).getKey? k = none

theorem Std.ExtHashMap.getKey?_inter_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (not_mem : ¬k ∈ m₁) : (m₁ ∩ m₂).getKey? k = none

@[simp]

theorem Std.ExtHashMap.getKey_inter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {h_mem : k ∈ m₁ ∩ m₂} : (m₁ ∩ m₂).getKey k h_mem = m₁.getKey k ⋯

theorem Std.ExtHashMap.getKeyD_inter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k fallback : α} : (m₁ ∩ m₂).getKeyD k fallback = if k ∈ m₂ then m₁.getKeyD k fallback else fallback

theorem Std.ExtHashMap.getKeyD_inter_of_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k fallback : α} (mem : k ∈ m₂) : (m₁ ∩ m₂).getKeyD k fallback = m₁.getKeyD k fallback

theorem Std.ExtHashMap.getKeyD_inter_of_not_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k fallback : α} (not_mem : ¬k ∈ m₂) : (m₁ ∩ m₂).getKeyD k fallback = fallback

theorem Std.ExtHashMap.getKeyD_inter_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k fallback : α} (not_mem : ¬k ∈ m₁) : (m₁ ∩ m₂).getKeyD k fallback = fallback

theorem Std.ExtHashMap.getKey!_inter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k : α} : (m₁ ∩ m₂).getKey! k = if k ∈ m₂ then m₁.getKey! k else default

theorem Std.ExtHashMap.getKey!_inter_of_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k : α} (mem : k ∈ m₂) : (m₁ ∩ m₂).getKey! k = m₁.getKey! k

theorem Std.ExtHashMap.getKey!_inter_of_not_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k : α} (not_mem : ¬k ∈ m₂) : (m₁ ∩ m₂).getKey! k = default

theorem Std.ExtHashMap.getKey!_inter_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k : α} (not_mem : ¬k ∈ m₁) : (m₁ ∩ m₂).getKey! k = default

theorem Std.ExtHashMap.size_inter_le_size_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : (m₁ ∩ m₂).size ≤ m₁.size

theorem Std.ExtHashMap.size_inter_le_size_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : (m₁ ∩ m₂).size ≤ m₂.size

theorem Std.ExtHashMap.size_inter_eq_size_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] (h : ∀ (a : α), a ∈ m₁ → a ∈ m₂) : (m₁ ∩ m₂).size = m₁.size

theorem Std.ExtHashMap.size_inter_eq_size_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] (h : ∀ (a : α), a ∈ m₂ → a ∈ m₁) : (m₁ ∩ m₂).size = m₂.size

theorem Std.ExtHashMap.size_add_size_eq_size_union_add_size_inter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : m₁.size + m₂.size = (m₁ ∪ m₂).size + (m₁ ∩ m₂).size

@[simp]

theorem Std.ExtHashMap.isEmpty_inter_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] (h : m₁.isEmpty = true) : (m₁ ∩ m₂).isEmpty = true

@[simp]

theorem Std.ExtHashMap.isEmpty_inter_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] (h : m₂.isEmpty = true) : (m₁ ∩ m₂).isEmpty = true

theorem Std.ExtHashMap.isEmpty_inter_iff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : (m₁ ∩ m₂).isEmpty = true ↔ ∀ (k : α), k ∈ m₁ → ¬k ∈ m₂

theorem Std.ExtHashMap.isEmpty_inter_comm {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : (m₁ ∩ m₂).isEmpty = (m₂ ∩ m₁).isEmpty

theorem Std.ExtHashMap.inter_eq_empty_comm {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : m₁ ∩ m₂ = ∅ ↔ m₂ ∩ m₁ = ∅

@[simp]

theorem Std.ExtHashMap.diff_eq {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : m₁.diff m₂ = m₁ \ m₂

@[simp]

theorem Std.ExtHashMap.contains_diff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : (m₁ \ m₂).contains k = (m₁.contains k && !m₂.contains k)

@[simp]

theorem Std.ExtHashMap.mem_diff_iff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : k ∈ m₁ \ m₂ ↔ k ∈ m₁ ∧ ¬k ∈ m₂

theorem Std.ExtHashMap.not_mem_diff_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (not_mem : ¬k ∈ m₁) : ¬k ∈ m₁ \ m₂

theorem Std.ExtHashMap.not_mem_diff_of_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (mem : k ∈ m₂) : ¬k ∈ m₁ \ m₂

theorem Std.ExtHashMap.get?_diff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : (m₁ \ m₂).get? k = if k ∈ m₂ then none else m₁.get? k

theorem Std.ExtHashMap.get?_diff_of_not_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (not_mem : ¬k ∈ m₂) : (m₁ \ m₂).get? k = m₁.get? k

theorem Std.ExtHashMap.get?_diff_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (not_mem : ¬k ∈ m₁) : (m₁ \ m₂).get? k = none

theorem Std.ExtHashMap.get?_diff_of_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (mem : k ∈ m₂) : (m₁ \ m₂).get? k = none

@[simp]

theorem Std.ExtHashMap.get_diff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {h_mem : k ∈ m₁ \ m₂} : (m₁ \ m₂).get k h_mem = m₁.get k ⋯

theorem Std.ExtHashMap.getD_diff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {fallback : β} : (m₁ \ m₂).getD k fallback = if k ∈ m₂ then fallback else m₁.getD k fallback

theorem Std.ExtHashMap.getD_diff_of_not_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {fallback : β} (not_mem : ¬k ∈ m₂) : (m₁ \ m₂).getD k fallback = m₁.getD k fallback

theorem Std.ExtHashMap.getD_diff_of_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {fallback : β} (mem : k ∈ m₂) : (m₁ \ m₂).getD k fallback = fallback

theorem Std.ExtHashMap.getD_diff_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {fallback : β} (not_mem : ¬k ∈ m₁) : (m₁ \ m₂).getD k fallback = fallback

theorem Std.ExtHashMap.get!_diff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} [Inhabited β] : (m₁ \ m₂).get! k = if k ∈ m₂ then default else m₁.get! k

theorem Std.ExtHashMap.get!_diff_of_not_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} [Inhabited β] (not_mem : ¬k ∈ m₂) : (m₁ \ m₂).get! k = m₁.get! k

theorem Std.ExtHashMap.get!_diff_of_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} [Inhabited β] (mem : k ∈ m₂) : (m₁ \ m₂).get! k = default

theorem Std.ExtHashMap.get!_diff_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} [Inhabited β] (not_mem : ¬k ∈ m₁) : (m₁ \ m₂).get! k = default

theorem Std.ExtHashMap.getKey?_diff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} : (m₁ \ m₂).getKey? k = if k ∈ m₂ then none else m₁.getKey? k

theorem Std.ExtHashMap.getKey?_diff_of_not_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (not_mem : ¬k ∈ m₂) : (m₁ \ m₂).getKey? k = m₁.getKey? k

theorem Std.ExtHashMap.getKey?_diff_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (not_mem : ¬k ∈ m₁) : (m₁ \ m₂).getKey? k = none

theorem Std.ExtHashMap.getKey?_diff_of_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} (mem : k ∈ m₂) : (m₁ \ m₂).getKey? k = none

@[simp]

theorem Std.ExtHashMap.getKey_diff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {h_mem : k ∈ m₁ \ m₂} : (m₁ \ m₂).getKey k h_mem = m₁.getKey k ⋯

theorem Std.ExtHashMap.getKeyD_diff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k fallback : α} : (m₁ \ m₂).getKeyD k fallback = if k ∈ m₂ then fallback else m₁.getKeyD k fallback

theorem Std.ExtHashMap.getKeyD_diff_of_not_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k fallback : α} (not_mem : ¬k ∈ m₂) : (m₁ \ m₂).getKeyD k fallback = m₁.getKeyD k fallback

theorem Std.ExtHashMap.getKeyD_diff_of_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k fallback : α} (mem : k ∈ m₂) : (m₁ \ m₂).getKeyD k fallback = fallback

theorem Std.ExtHashMap.getKeyD_diff_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k fallback : α} (not_mem : ¬k ∈ m₁) : (m₁ \ m₂).getKeyD k fallback = fallback

theorem Std.ExtHashMap.getKey!_diff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k : α} : (m₁ \ m₂).getKey! k = if k ∈ m₂ then default else m₁.getKey! k

theorem Std.ExtHashMap.getKey!_diff_of_not_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k : α} (not_mem : ¬k ∈ m₂) : (m₁ \ m₂).getKey! k = m₁.getKey! k

theorem Std.ExtHashMap.getKey!_diff_of_mem_right {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k : α} (mem : k ∈ m₂) : (m₁ \ m₂).getKey! k = default

theorem Std.ExtHashMap.getKey!_diff_of_not_mem_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k : α} (not_mem : ¬k ∈ m₁) : (m₁ \ m₂).getKey! k = default

theorem Std.ExtHashMap.size_diff_le_size_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : (m₁ \ m₂).size ≤ m₁.size

theorem Std.ExtHashMap.size_diff_eq_size_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] (h : ∀ (a : α), a ∈ m₁ → ¬a ∈ m₂) : (m₁ \ m₂).size = m₁.size

theorem Std.ExtHashMap.size_diff_add_size_inter_eq_size_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : (m₁ \ m₂).size + (m₁ ∩ m₂).size = m₁.size

@[simp]

theorem Std.ExtHashMap.isEmpty_diff_left {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] (h : m₁.isEmpty = true) : (m₁ \ m₂).isEmpty = true

theorem Std.ExtHashMap.isEmpty_diff_iff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m₁ m₂ : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] : (m₁ \ m₂).isEmpty = true ↔ ∀ (k : α), k ∈ m₁ → k ∈ m₂

theorem Std.ExtHashMap.alter_eq_empty_iff_erase_eq_empty {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {f : Option β → Option β} : m.alter k f = ∅ ↔ m.erase k = ∅ ∧ f m[k]? = none

@[simp]

theorem Std.ExtHashMap.alter_eq_empty_iff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {f : Option β → Option β} : m.alter k f = ∅ ↔ (m = ∅ ∨ m.size = 1 ∧ k ∈ m) ∧ f m[k]? = none

theorem Std.ExtHashMap.contains_alter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : Option β → Option β} : (m.alter k f).contains k' = if (k == k') = true then (f m[k]?).isSome else m.contains k'

theorem Std.ExtHashMap.mem_alter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : Option β → Option β} : k' ∈ m.alter k f ↔ if (k == k') = true then (f m[k]?).isSome = true else k' ∈ m

theorem Std.ExtHashMap.mem_alter_of_beq {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : Option β → Option β} (h : (k == k') = true) : k' ∈ m.alter k f ↔ (f m[k]?).isSome = true

@[simp]

theorem Std.ExtHashMap.contains_alter_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {f : Option β → Option β} : (m.alter k f).contains k = (f m[k]?).isSome

@[simp]

theorem Std.ExtHashMap.mem_alter_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {f : Option β → Option β} : k ∈ m.alter k f ↔ (f m[k]?).isSome = true

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

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

theorem Std.ExtHashMap.size_alter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {f : Option β → Option β} : (m.alter k f).size = if k ∈ m ∧ (f m[k]?).isNone = true then m.size - 1 else if ¬k ∈ m ∧ (f m[k]?).isSome = true then m.size + 1 else m.size

theorem Std.ExtHashMap.size_alter_eq_add_one {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {f : Option β → Option β} (h : ¬k ∈ m) (h' : (f m[k]?).isSome = true) : (m.alter k f).size = m.size + 1

theorem Std.ExtHashMap.size_alter_eq_sub_one {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {f : Option β → Option β} (h : k ∈ m) (h' : (f m[k]?).isNone = true) : (m.alter k f).size = m.size - 1

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

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

theorem Std.ExtHashMap.size_alter_le_size {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {f : Option β → Option β} : (m.alter k f).size ≤ m.size + 1

theorem Std.ExtHashMap.size_le_size_alter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {f : Option β → Option β} : m.size - 1 ≤ (m.alter k f).size

theorem Std.ExtHashMap.getElem?_alter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : Option β → Option β} : (m.alter k f)[k']? = if (k == k') = true then f m[k]? else m[k']?

@[simp]

theorem Std.ExtHashMap.getElem?_alter_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {f : Option β → Option β} : (m.alter k f)[k]? = f m[k]?

theorem Std.ExtHashMap.getElem_alter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : Option β → Option β} {h : k' ∈ m.alter k f} : (m.alter k f)[k'] = if heq : (k == k') = true then (f m[k]?).get ⋯ else m[k']

@[simp]

theorem Std.ExtHashMap.getElem_alter_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {f : Option β → Option β} {h : k ∈ m.alter k f} : (m.alter k f)[k] = (f m[k]?).get ⋯

theorem Std.ExtHashMap.getElem!_alter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k k' : α} [Inhabited β] {f : Option β → Option β} : (m.alter k f)[k']! = if (k == k') = true then (f m[k]?).get! else m[k']!

@[simp]

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

theorem Std.ExtHashMap.getD_alter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k k' : α} {fallback : β} {f : Option β → Option β} : (m.alter k f).getD k' fallback = if (k == k') = true then (f m[k]?).getD fallback else m.getD k' fallback

@[simp]

theorem Std.ExtHashMap.getD_alter_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {fallback : β} {f : Option β → Option β} : (m.alter k f).getD k fallback = (f m[k]?).getD fallback

theorem Std.ExtHashMap.getKey?_alter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : Option β → Option β} : (m.alter k f).getKey? k' = if (k == k') = true then if (f m[k]?).isSome = true then some k else none else m.getKey? k'

theorem Std.ExtHashMap.getKey?_alter_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {f : Option β → Option β} : (m.alter k f).getKey? k = if (f m[k]?).isSome = true then some k else none

theorem Std.ExtHashMap.getKey!_alter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k k' : α} {f : Option β → Option β} : (m.alter k f).getKey! k' = if (k == k') = true then if (f m[k]?).isSome = true then k else default else m.getKey! k'

theorem Std.ExtHashMap.getKey!_alter_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k : α} {f : Option β → Option β} : (m.alter k f).getKey! k = if (f m[k]?).isSome = true then k else default

theorem Std.ExtHashMap.getKey_alter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k k' : α} {f : Option β → Option β} {h : k' ∈ m.alter k f} : (m.alter k f).getKey k' h = if heq : (k == k') = true then k else m.getKey k' ⋯

@[simp]

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

theorem Std.ExtHashMap.getKeyD_alter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k k' fallback : α} {f : Option β → Option β} : (m.alter k f).getKeyD k' fallback = if (k == k') = true then if (f m[k]?).isSome = true then k else fallback else m.getKeyD k' fallback

theorem Std.ExtHashMap.getKeyD_alter_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k fallback : α} {f : Option β → Option β} : (m.alter k f).getKeyD k fallback = if (f m[k]?).isSome = true then k else fallback

@[simp]

theorem Std.ExtHashMap.modify_eq_empty_iff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {f : β → β} : m.modify k f = ∅ ↔ m = ∅

@[simp]

theorem Std.ExtHashMap.contains_modify {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : β → β} : (m.modify k f).contains k' = m.contains k'

@[simp]

theorem Std.ExtHashMap.mem_modify {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : β → β} : k' ∈ m.modify k f ↔ k' ∈ m

@[simp]

theorem Std.ExtHashMap.size_modify {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {f : β → β} : (m.modify k f).size = m.size

theorem Std.ExtHashMap.getElem?_modify {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : β → β} : (m.modify k f)[k']? = if (k == k') = true then Option.map f m[k]? else m[k']?

@[simp]

theorem Std.ExtHashMap.getElem?_modify_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {f : β → β} : (m.modify k f)[k]? = Option.map f m[k]?

theorem Std.ExtHashMap.getElem_modify {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : β → β} {h : k' ∈ m.modify k f} : (m.modify k f)[k'] = if heq : (k == k') = true then f m[k] else m[k']

@[simp]

theorem Std.ExtHashMap.getElem_modify_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {f : β → β} {h : k ∈ m.modify k f} : (m.modify k f)[k] = f m[k]

theorem Std.ExtHashMap.getElem!_modify {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k k' : α} [Inhabited β] {f : β → β} : (m.modify k f)[k']! = if (k == k') = true then (Option.map f m[k]?).get! else m[k']!

@[simp]

theorem Std.ExtHashMap.getElem!_modify_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} [Inhabited β] {f : β → β} : (m.modify k f)[k]! = (Option.map f m[k]?).get!

theorem Std.ExtHashMap.getD_modify {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k k' : α} {fallback : β} {f : β → β} : (m.modify k f).getD k' fallback = if (k == k') = true then (Option.map f m[k]?).getD fallback else m.getD k' fallback

@[simp]

theorem Std.ExtHashMap.getD_modify_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {fallback : β} {f : β → β} : (m.modify k f).getD k fallback = (Option.map f m[k]?).getD fallback

theorem Std.ExtHashMap.getKey?_modify {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k k' : α} {f : β → β} : (m.modify k f).getKey? k' = if (k == k') = true then if k ∈ m then some k else none else m.getKey? k'

theorem Std.ExtHashMap.getKey?_modify_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k : α} {f : β → β} : (m.modify k f).getKey? k = if k ∈ m then some k else none

theorem Std.ExtHashMap.getKey!_modify {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k k' : α} {f : β → β} : (m.modify k f).getKey! k' = if (k == k') = true then if k ∈ m then k else default else m.getKey! k'

theorem Std.ExtHashMap.getKey!_modify_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k : α} {f : β → β} : (m.modify k f).getKey! k = if k ∈ m then k else default

theorem Std.ExtHashMap.getKey_modify {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k k' : α} {f : β → β} {h : k' ∈ m.modify k f} : (m.modify k f).getKey k' h = if (k == k') = true then k else m.getKey k' ⋯

@[simp]

theorem Std.ExtHashMap.getKey_modify_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k : α} {f : β → β} {h : k ∈ m.modify k f} : (m.modify k f).getKey k h = k

theorem Std.ExtHashMap.getKeyD_modify {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {k k' fallback : α} {f : β → β} : (m.modify k f).getKeyD k' fallback = if (k == k') = true then if k ∈ m then k else fallback else m.getKeyD k' fallback

theorem Std.ExtHashMap.getKeyD_modify_self {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {k fallback : α} {f : β → β} : (m.modify k f).getKeyD k fallback = if k ∈ m then k else fallback

theorem Std.ExtHashMap.ext_getKey_getElem? {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {m₁ m₂ : ExtHashMap α β} (hk : ∀ (k : α) (hk : k ∈ m₁) (hk' : k ∈ m₂), m₁.getKey k hk = m₂.getKey k hk') (hv : ∀ (k : α), m₁[k]? = m₂[k]?) : m₁ = m₂

theorem Std.ExtHashMap.ext_getKey_getElem?_iff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {m₁ m₂ : ExtHashMap α β} : m₁ = m₂ ↔ (∀ (k : α) (hk : k ∈ m₁) (hk' : k ∈ m₂), m₁.getKey k hk = m₂.getKey k hk') ∧ ∀ (k : α), m₁[k]? = m₂[k]?

theorem Std.ExtHashMap.ext_getElem? {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [LawfulBEq α] {m₁ m₂ : ExtHashMap α β} (h : ∀ (k : α), m₁[k]? = m₂[k]?) : m₁ = m₂

theorem Std.ExtHashMap.ext_getElem?_iff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} [LawfulBEq α] {m₁ m₂ : ExtHashMap α β} : m₁ = m₂ ↔ ∀ (k : α), m₁[k]? = m₂[k]?

theorem Std.ExtHashMap.ext_getKey?_unit {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [EquivBEq α] [LawfulHashable α] {m₁ m₂ : ExtHashMap α Unit} (h : ∀ (k : α), m₁.getKey? k = m₂.getKey? k) : m₁ = m₂

theorem Std.ExtHashMap.ext_contains_unit {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [LawfulBEq α] {m₁ m₂ : ExtHashMap α Unit} (h : ∀ (k : α), m₁.contains k = m₂.contains k) : m₁ = m₂

theorem Std.ExtHashMap.ext_mem_unit {α : Type u} {x✝ : BEq α} {x✝¹ : Hashable α} [LawfulBEq α] {m₁ m₂ : ExtHashMap α Unit} (h : ∀ (k : α), k ∈ m₁ ↔ k ∈ m₂) : m₁ = m₂

theorem Std.ExtHashMap.filterMap_eq_empty_iff {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Option γ} : filterMap f m = ∅ ↔ ∀ (k : α) (h : k ∈ m), f (m.getKey k h) m[k] = none

theorem Std.ExtHashMap.mem_filterMap {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Option γ} {k : α} : k ∈ filterMap f m ↔ ∃ (h : k ∈ m), (f (m.getKey k h) m[k]).isSome = true

theorem Std.ExtHashMap.contains_of_contains_filterMap {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Option γ} {k : α} : (filterMap f m).contains k = true → m.contains k = true

theorem Std.ExtHashMap.mem_of_mem_filterMap {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Option γ} {k : α} : k ∈ filterMap f m → k ∈ m

theorem Std.ExtHashMap.size_filterMap_le_size {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Option γ} : (filterMap f m).size ≤ m.size

theorem Std.ExtHashMap.size_filterMap_eq_size_iff {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Option γ} : (filterMap f m).size = m.size ↔ ∀ (k : α) (h : k ∈ m), (f (m.getKey k h) m[k]).isSome = true

@[simp]

theorem Std.ExtHashMap.getElem?_filterMap {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Option γ} {k : α} : (filterMap f m)[k]? = m[k]?.pbind fun (x : β) (h' : m[k]? = some x) => f (m.getKey k ⋯) x

theorem Std.ExtHashMap.getElem?_filterMap' {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [LawfulBEq α] {f : α → β → Option γ} {k : α} : (filterMap f m)[k]? = m[k]?.bind fun (x : β) => f k x

Simpler variant of getElem?_filterMap when LawfulBEq is available.

theorem Std.ExtHashMap.getElem?_filterMap_of_getKey?_eq_some {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Option γ} {k k' : α} (h : m.getKey? k = some k') : (filterMap f m)[k]? = m[k]?.bind (f k')

theorem Std.ExtHashMap.isSome_apply_of_mem_filterMap {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Option γ} {k : α} (h : k ∈ filterMap f m) : (f (m.getKey k ⋯) m[k]).isSome = true

@[simp]

theorem Std.ExtHashMap.getElem_filterMap {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Option γ} {k : α} {h : k ∈ filterMap f m} : (filterMap f m)[k] = (f (m.getKey k ⋯) m[k]).get ⋯

theorem Std.ExtHashMap.getElem_filterMap' {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [LawfulBEq α] {f : α → β → Option γ} {k : α} {h : k ∈ filterMap f m} : (filterMap f m)[k] = (f k m[k]).get ⋯

Simpler variant of getElem_filterMap when LawfulBEq is available.

theorem Std.ExtHashMap.getElem!_filterMap {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited γ] {f : α → β → Option γ} {k : α} : (filterMap f m)[k]! = (m[k]?.pbind fun (x : β) (h' : m[k]? = some x) => f (m.getKey k ⋯) x).get!

theorem Std.ExtHashMap.getElem!_filterMap_of_getKey?_eq_some {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited γ] {f : α → β → Option γ} {k k' : α} (h : m.getKey? k = some k') : (filterMap f m)[k]! = (m[k]?.bind (f k')).get!

theorem Std.ExtHashMap.getD_filterMap {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Option γ} {k : α} {fallback : γ} : (filterMap f m).getD k fallback = (m[k]?.pbind fun (x : β) (h' : m[k]? = some x) => f (m.getKey k ⋯) x).getD fallback

theorem Std.ExtHashMap.getD_filterMap_of_getKey?_eq_some {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Option γ} {k k' : α} {fallback : γ} (h : m.getKey? k = some k') : (filterMap f m).getD k fallback = (m[k]?.bind (f k')).getD fallback

theorem Std.ExtHashMap.getKey?_filterMap {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Option γ} {k : α} : (filterMap f m).getKey? k = (m.getKey? k).pfilter fun (x : α) (h' : m.getKey? k = some x) => (f x m[x]).isSome

@[simp]

theorem Std.ExtHashMap.getKey_filterMap {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Option γ} {k : α} {h' : k ∈ filterMap f m} : (filterMap f m).getKey k h' = m.getKey k ⋯

theorem Std.ExtHashMap.getKey!_filterMap {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {f : α → β → Option γ} {k : α} : (filterMap f m).getKey! k = ((m.getKey? k).pfilter fun (x : α) (h' : m.getKey? k = some x) => (f x m[x]).isSome).get!

theorem Std.ExtHashMap.getKeyD_filterMap {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Option γ} {k fallback : α} : (filterMap f m).getKeyD k fallback = ((m.getKey? k).pfilter fun (x : α) (h' : m.getKey? k = some x) => (f x m[x]).isSome).getD fallback

theorem Std.ExtHashMap.filterMap_eq_filter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Bool} : filterMap (fun (k : α) => Option.guard fun (v : β) => f k v) m = filter f m

theorem Std.ExtHashMap.filter_eq_empty_iff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Bool} : filter f m = ∅ ↔ ∀ (k : α) (h : k ∈ m), f (m.getKey k h) m[k] = false

theorem Std.ExtHashMap.mem_filter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Bool} {k : α} : k ∈ filter f m ↔ ∃ (h' : k ∈ m), f (m.getKey k h') m[k] = true

theorem Std.ExtHashMap.contains_of_contains_filter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Bool} {k : α} : (filter f m).contains k = true → m.contains k = true

theorem Std.ExtHashMap.mem_of_mem_filter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Bool} {k : α} : k ∈ filter f m → k ∈ m

theorem Std.ExtHashMap.size_filter_le_size {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Bool} : (filter f m).size ≤ m.size

theorem Std.ExtHashMap.size_filter_eq_size_iff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Bool} : (filter f m).size = m.size ↔ ∀ (k : α) (h : k ∈ m), f (m.getKey k h) m[k] = true

theorem Std.ExtHashMap.filter_eq_self_iff {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Bool} : filter f m = m ↔ ∀ (k : α) (h : k ∈ m), f (m.getKey k h) m[k] = true

theorem Std.ExtHashMap.getElem?_filter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Bool} {k : α} : (filter f m)[k]? = m[k]?.pfilter fun (x : β) (h' : m[k]? = some x) => f (m.getKey k ⋯) x

@[simp]

theorem Std.ExtHashMap.getElem?_filter' {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [LawfulBEq α] {f : α → β → Bool} {k : α} : (filter f m)[k]? = Option.filter (f k) m[k]?

Simpler variant of getElem?_filter when LawfulBEq is available.

theorem Std.ExtHashMap.getElem?_filter_of_getKey?_eq_some {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Bool} {k k' : α} : m.getKey? k = some k' → (filter f m)[k]? = Option.filter (fun (x : β) => f k' x) m[k]?

@[simp]

theorem Std.ExtHashMap.getElem_filter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Bool} {k : α} {h' : k ∈ filter f m} : (filter f m)[k] = m[k]

theorem Std.ExtHashMap.getElem!_filter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {f : α → β → Bool} {k : α} : (filter f m)[k]! = (m[k]?.pfilter fun (x : β) (h' : m[k]? = some x) => f (m.getKey k ⋯) x).get!

theorem Std.ExtHashMap.getElem!_filter' {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [LawfulBEq α] [Inhabited β] {f : α → β → Bool} {k : α} : (filter f m)[k]! = (Option.filter (f k) m[k]?).get!

Simpler variant of getElem!_filter when LawfulBEq is available.

theorem Std.ExtHashMap.getElem!_filter_of_getKey?_eq_some {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited β] {f : α → β → Bool} {k k' : α} : m.getKey? k = some k' → (filter f m)[k]! = (Option.filter (f k') m[k]?).get!

theorem Std.ExtHashMap.getD_filter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Bool} {k : α} {fallback : β} : (filter f m).getD k fallback = (m[k]?.pfilter fun (x : β) (h' : m[k]? = some x) => f (m.getKey k ⋯) x).getD fallback

theorem Std.ExtHashMap.getD_filter' {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [LawfulBEq α] {f : α → β → Bool} {k : α} {fallback : β} : (filter f m).getD k fallback = (Option.filter (f k) m[k]?).getD fallback

Simpler variant of getD_filter when LawfulBEq is available.

theorem Std.ExtHashMap.getD_filter_of_getKey?_eq_some {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Bool} {k k' : α} {fallback : β} : m.getKey? k = some k' → (filter f m).getD k fallback = (Option.filter (fun (x : β) => f k' x) m[k]?).getD fallback

theorem Std.ExtHashMap.getKey?_filter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Bool} {k : α} : (filter f m).getKey? k = (m.getKey? k).pfilter fun (x : α) (h' : m.getKey? k = some x) => f x m[x]

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

@[simp]

theorem Std.ExtHashMap.getKey_filter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Bool} {k : α} {h' : k ∈ filter f m} : (filter f m).getKey k h' = m.getKey k ⋯

theorem Std.ExtHashMap.getKey!_filter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {f : α → β → Bool} {k : α} : (filter f m).getKey! k = ((m.getKey? k).pfilter fun (x : α) (h' : m.getKey? k = some x) => f x m[x]).get!

theorem Std.ExtHashMap.getKey!_filter_key {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {f : α → Bool} {k : α} : (filter (fun (k : α) (x : β) => f k) m).getKey! k = (Option.filter f (m.getKey? k)).get!

theorem Std.ExtHashMap.getKeyD_filter {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → Bool} {k fallback : α} : (filter f m).getKeyD k fallback = ((m.getKey? k).pfilter fun (x : α) (h' : m.getKey? k = some x) => f x m[x]).getD fallback

theorem Std.ExtHashMap.getKeyD_filter_key {α : Type u} {β : Type v} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → Bool} {k fallback : α} : (filter (fun (k : α) (x : β) => f k) m).getKeyD k fallback = (Option.filter f (m.getKey? k)).getD fallback

@[simp]

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

@[simp]

theorem Std.ExtHashMap.map_map {α : Type u} {β : Type v} {γ : Type w} {δ : Type w'} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → γ} {g : α → γ → δ} : map g (map f m) = map (fun (k : α) (v : β) => g k (f k v)) m

theorem Std.ExtHashMap.filterMap_equiv_map {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → γ} : filterMap (fun (k : α) (v : β) => some (f k v)) m = map f m

@[simp]

theorem Std.ExtHashMap.map_eq_empty_iff {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → γ} : map f m = ∅ ↔ m = ∅

@[simp]

theorem Std.ExtHashMap.contains_map {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → γ} {k : α} : (map f m).contains k = m.contains k

theorem Std.ExtHashMap.contains_of_contains_map {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → γ} {k : α} : (map f m).contains k = true → m.contains k = true

@[simp]

theorem Std.ExtHashMap.mem_map {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → γ} {k : α} : k ∈ map f m ↔ k ∈ m

theorem Std.ExtHashMap.mem_of_mem_map {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → γ} {k : α} : k ∈ map f m → k ∈ m

@[simp]

theorem Std.ExtHashMap.size_map {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → γ} : (map f m).size = m.size

@[simp]

theorem Std.ExtHashMap.getElem?_map {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [LawfulBEq α] {f : α → β → γ} {k : α} : (map f m)[k]? = Option.map (f k) m[k]?

@[simp]

theorem Std.ExtHashMap.getElem?_map' {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → γ} {k : α} : (map f m)[k]? = Option.pmap (fun (v : β) (h' : k ∈ m) => f (m.getKey k h') v) m[k]? ⋯

Variant of getElem?_map that holds with EquivBEq (i.e. without LawfulBEq).

theorem Std.ExtHashMap.getElem?_map_of_getKey?_eq_some {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → γ} {k k' : α} (h : m.getKey? k = some k') : (map f m)[k]? = Option.map (f k') m[k]?

@[simp]

theorem Std.ExtHashMap.getElem_map {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [LawfulBEq α] {f : α → β → γ} {k : α} {h' : k ∈ map f m} : (map f m)[k] = f k m[k]

@[simp]

theorem Std.ExtHashMap.getElem_map' {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → γ} {k : α} {h' : k ∈ map f m} : (map f m)[k] = f (m.getKey k ⋯) m[k]

Variant of getElem_map that holds with EquivBEq (i.e. without LawfulBEq).

theorem Std.ExtHashMap.getElem!_map {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [LawfulBEq α] [Inhabited γ] {f : α → β → γ} {k : α} : (map f m)[k]! = (Option.map (f k) m[k]?).get!

theorem Std.ExtHashMap.getElem!_map' {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited γ] {f : α → β → γ} {k : α} : (map f m)[k]! = (Option.pmap (fun (v : β) (h : k ∈ m) => f (m.getKey k h) v) m[k]? ⋯).get!

Variant of getElem!_map that holds with EquivBEq (i.e. without LawfulBEq).

theorem Std.ExtHashMap.getElem!_map_of_getKey?_eq_some {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited γ] {f : α → β → γ} {k k' : α} (h : m.getKey? k = some k') : (map f m)[k]! = (Option.map (f k') m[k]?).get!

theorem Std.ExtHashMap.getD_map {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [LawfulBEq α] {f : α → β → γ} {k : α} {fallback : γ} : (map f m).getD k fallback = (Option.map (f k) m[k]?).getD fallback

theorem Std.ExtHashMap.getD_map' {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → γ} {k : α} {fallback : γ} : (map f m).getD k fallback = (Option.pmap (fun (v : β) (h : k ∈ m) => f (m.getKey k h) v) m[k]? ⋯).getD fallback

Variant of getD_map that holds with EquivBEq (i.e. without LawfulBEq).

theorem Std.ExtHashMap.getD_map_of_getKey?_eq_some {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited γ] {f : α → β → γ} {k k' : α} {fallback : γ} (h : m.getKey? k = some k') : (map f m).getD k fallback = (Option.map (f k') m[k]?).getD fallback

@[simp]

theorem Std.ExtHashMap.getKey?_map {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → γ} {k : α} : (map f m).getKey? k = m.getKey? k

@[simp]

theorem Std.ExtHashMap.getKey_map {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → γ} {k : α} {h' : k ∈ map f m} : (map f m).getKey k h' = m.getKey k ⋯

@[simp]

theorem Std.ExtHashMap.getKey!_map {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] [Inhabited α] {f : α → β → γ} {k : α} : (map f m).getKey! k = m.getKey! k

@[simp]

theorem Std.ExtHashMap.getKeyD_map {α : Type u} {β : Type v} {γ : Type w} {x✝ : BEq α} {x✝¹ : Hashable α} {m : ExtHashMap α β} [EquivBEq α] [LawfulHashable α] {f : α → β → γ} {k fallback : α} : (map f m).getKeyD k fallback = m.getKeyD k fallback