Hash set lemmas

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

@[simp]

theorem Std.HashSet.isEmpty_empty {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {c : Nat}, (Std.HashSet.empty c).isEmpty = true

@[simp]

theorem Std.HashSet.isEmpty_emptyc {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α}, ∅.isEmpty = true

@[simp]

theorem Std.HashSet.isEmpty_insert {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {a : α}, (m.insert a).isEmpty = false

theorem Std.HashSet.mem_iff_contains {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} {a : α}, a ∈ m ↔ m.contains a = true

theorem Std.HashSet.contains_congr {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {a b : α}, (a == b) = true → m.contains a = m.contains b

theorem Std.HashSet.mem_congr {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {a b : α}, (a == b) = true → (a ∈ m ↔ b ∈ m)

@[simp]

theorem Std.HashSet.contains_empty {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {a : α} {c : Nat}, (Std.HashSet.empty c).contains a = false

@[simp]

theorem Std.HashSet.not_mem_empty {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {a : α} {c : Nat}, ¬a ∈ Std.HashSet.empty c

@[simp]

theorem Std.HashSet.contains_emptyc {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {a : α}, ∅.contains a = false

@[simp]

theorem Std.HashSet.not_mem_emptyc {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {a : α}, ¬a ∈ ∅

theorem Std.HashSet.contains_of_isEmpty {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {a : α}, m.isEmpty = true → m.contains a = false

theorem Std.HashSet.not_mem_of_isEmpty {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {a : α}, m.isEmpty = true → ¬a ∈ m

theorem Std.HashSet.isEmpty_eq_false_iff_exists_contains_eq_true {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α], m.isEmpty = false ↔ ∃ (a : α), m.contains a = true

theorem Std.HashSet.isEmpty_eq_false_iff_exists_mem {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α], m.isEmpty = false ↔ ∃ (a : α), a ∈ m

theorem Std.HashSet.isEmpty_iff_forall_contains {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α], m.isEmpty = true ↔ ∀ (a : α), m.contains a = false

theorem Std.HashSet.isEmpty_iff_forall_not_mem {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α], m.isEmpty = true ↔ ∀ (a : α), ¬a ∈ m

@[simp]

theorem Std.HashSet.insert_eq_insert {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} {a : α}, insert a m = m.insert a

@[simp]

theorem Std.HashSet.singleton_eq_insert {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {a : α}, {a} = ∅.insert a

@[simp]

theorem Std.HashSet.contains_insert {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k a : α}, (m.insert k).contains a = (k == a || m.contains a)

@[simp]

theorem Std.HashSet.mem_insert {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k a : α}, a ∈ m.insert k ↔ (k == a) = true ∨ a ∈ m

theorem Std.HashSet.contains_of_contains_insert {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k a : α}, (m.insert k).contains a = true → (k == a) = false → m.contains a = true

theorem Std.HashSet.mem_of_mem_insert {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k a : α}, a ∈ m.insert k → (k == a) = false → a ∈ m

theorem Std.HashSet.contains_of_contains_insert' {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k a : α}, (m.insert k).contains a = true → ¬((k == a) = true ∧ m.contains k = false) → m.contains a = true

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

theorem Std.HashSet.mem_of_mem_insert' {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k a : α}, a ∈ m.insert k → ¬((k == a) = true ∧ ¬k ∈ m) → a ∈ m

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

@[simp]

theorem Std.HashSet.contains_insert_self {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k : α}, (m.insert k).contains k = true

@[simp]

theorem Std.HashSet.mem_insert_self {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k : α}, k ∈ m.insert k

@[simp]

theorem Std.HashSet.size_empty {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {c : Nat}, (Std.HashSet.empty c).size = 0

@[simp]

theorem Std.HashSet.size_emptyc {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α}, ∅.size = 0

theorem Std.HashSet.isEmpty_eq_size_eq_zero {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α}, m.isEmpty = (m.size == 0)

theorem Std.HashSet.size_insert {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k : α}, (m.insert k).size = if k ∈ m then m.size else m.size + 1

theorem Std.HashSet.size_le_size_insert {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k : α}, m.size ≤ (m.insert k).size

theorem Std.HashSet.size_insert_le {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k : α}, (m.insert k).size ≤ m.size + 1

@[simp]

theorem Std.HashSet.erase_empty {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {a : α} {c : Nat}, (Std.HashSet.empty c).erase a = Std.HashSet.empty c

@[simp]

theorem Std.HashSet.erase_emptyc {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {a : α}, ∅.erase a = ∅

@[simp]

theorem Std.HashSet.isEmpty_erase {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k : α}, (m.erase k).isEmpty = (m.isEmpty || m.size == 1 && m.contains k)

@[simp]

theorem Std.HashSet.contains_erase {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k a : α}, (m.erase k).contains a = (!k == a && m.contains a)

@[simp]

theorem Std.HashSet.mem_erase {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k a : α}, a ∈ m.erase k ↔ (k == a) = false ∧ a ∈ m

theorem Std.HashSet.contains_of_contains_erase {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k a : α}, (m.erase k).contains a = true → m.contains a = true

theorem Std.HashSet.mem_of_mem_erase {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k a : α}, a ∈ m.erase k → a ∈ m

theorem Std.HashSet.size_erase {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k : α}, (m.erase k).size = if k ∈ m then m.size - 1 else m.size

theorem Std.HashSet.size_erase_le {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k : α}, (m.erase k).size ≤ m.size

theorem Std.HashSet.size_le_size_erase {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k : α}, m.size ≤ (m.erase k).size + 1

@[simp]

theorem Std.HashSet.get?_empty {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {a : α} {c : Nat}, (Std.HashSet.empty c).get? a = none

@[simp]

theorem Std.HashSet.get?_emptyc {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {a : α}, ∅.get? a = none

theorem Std.HashSet.get?_of_isEmpty {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {a : α}, m.isEmpty = true → m.get? a = none

theorem Std.HashSet.get?_insert {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k a : α}, (m.insert k).get? a = if (k == a) = true ∧ ¬k ∈ m then some k else m.get? a

theorem Std.HashSet.contains_eq_isSome_get? {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {a : α}, m.contains a = (m.get? a).isSome

theorem Std.HashSet.get?_eq_none_of_contains_eq_false {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {a : α}, m.contains a = false → m.get? a = none

theorem Std.HashSet.get?_eq_none {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {a : α}, ¬a ∈ m → m.get? a = none

theorem Std.HashSet.get?_erase {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k a : α}, (m.erase k).get? a = if (k == a) = true then none else m.get? a

@[simp]

theorem Std.HashSet.get?_erase_self {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k : α}, (m.erase k).get? k = none

theorem Std.HashSet.get_insert {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst_1 : LawfulHashable α] {k a : α} {h₁ : a ∈ m.insert k}, (m.insert k).get a h₁ = if h₂ : (k == a) = true ∧ ¬k ∈ m then k else m.get a ⋯

@[simp]

theorem Std.HashSet.get_erase {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst_1 : LawfulHashable α] {k a : α} {h' : a ∈ m.erase k}, (m.erase k).get a h' = m.get a ⋯

theorem Std.HashSet.get?_eq_some_get {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {a : α} {h' : a ∈ m}, m.get? a = some (m.get a h')

@[simp]

theorem Std.HashSet.get!_empty {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} [inst : Inhabited α] {a : α} {c : Nat}, (Std.HashSet.empty c).get! a = default

@[simp]

theorem Std.HashSet.get!_emptyc {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} [inst : Inhabited α] {a : α}, ∅.get! a = default

theorem Std.HashSet.get!_of_isEmpty {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : Inhabited α] [inst_1 : EquivBEq α] [inst_2 : LawfulHashable α] {a : α}, m.isEmpty = true → m.get! a = default

theorem Std.HashSet.get!_insert {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : Inhabited α] [inst_1 : EquivBEq α] [inst_2 : LawfulHashable α] {k a : α}, (m.insert k).get! a = if (k == a) = true ∧ ¬k ∈ m then k else m.get! a

theorem Std.HashSet.get!_eq_default_of_contains_eq_false {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : Inhabited α] [inst_1 : EquivBEq α] [inst_2 : LawfulHashable α] {a : α}, m.contains a = false → m.get! a = default

theorem Std.HashSet.get!_eq_default {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : Inhabited α] [inst_1 : EquivBEq α] [inst_2 : LawfulHashable α] {a : α}, ¬a ∈ m → m.get! a = default

theorem Std.HashSet.get!_erase {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : Inhabited α] [inst_1 : EquivBEq α] [inst_2 : LawfulHashable α] {k a : α}, (m.erase k).get! a = if (k == a) = true then default else m.get! a

@[simp]

theorem Std.HashSet.get!_erase_self {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : Inhabited α] [inst_1 : EquivBEq α] [inst_2 : LawfulHashable α] {k : α}, (m.erase k).get! k = default

theorem Std.HashSet.get?_eq_some_get!_of_contains {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] [inst : Inhabited α] {a : α}, m.contains a = true → m.get? a = some (m.get! a)

theorem Std.HashSet.get?_eq_some_get! {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] [inst : Inhabited α] {a : α}, a ∈ m → m.get? a = some (m.get! a)

theorem Std.HashSet.get!_eq_get!_get? {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] [inst : Inhabited α] {a : α}, m.get! a = (m.get? a).get!

theorem Std.HashSet.get_eq_get! {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] [inst : Inhabited α] {a : α} {h' : a ∈ m}, m.get a h' = m.get! a

@[simp]

theorem Std.HashSet.getD_empty {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {a fallback : α} {c : Nat}, (Std.HashSet.empty c).getD a fallback = fallback

@[simp]

theorem Std.HashSet.getD_emptyc {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {a fallback : α}, ∅.getD a fallback = fallback

theorem Std.HashSet.getD_of_isEmpty {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {a fallback : α}, m.isEmpty = true → m.getD a fallback = fallback

theorem Std.HashSet.getD_insert {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k a fallback : α}, (m.insert k).getD a fallback = if (k == a) = true ∧ ¬k ∈ m then k else m.getD a fallback

theorem Std.HashSet.getD_eq_fallback_of_contains_eq_false {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {a fallback : α}, m.contains a = false → m.getD a fallback = fallback

theorem Std.HashSet.getD_eq_fallback {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {a fallback : α}, ¬a ∈ m → m.getD a fallback = fallback

theorem Std.HashSet.getD_erase {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k a fallback : α}, (m.erase k).getD a fallback = if (k == a) = true then fallback else m.getD a fallback

@[simp]

theorem Std.HashSet.getD_erase_self {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {k fallback : α}, (m.erase k).getD k fallback = fallback

theorem Std.HashSet.get?_eq_some_getD_of_contains {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {a fallback : α}, m.contains a = true → m.get? a = some (m.getD a fallback)

theorem Std.HashSet.get?_eq_some_getD {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {a fallback : α}, a ∈ m → m.get? a = some (m.getD a fallback)

theorem Std.HashSet.getD_eq_getD_get? {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {a fallback : α}, m.getD a fallback = (m.get? a).getD fallback

theorem Std.HashSet.get_eq_getD {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] {a fallback : α} {h' : a ∈ m}, m.get a h' = m.getD a fallback

theorem Std.HashSet.get!_eq_getD_default {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} [inst : EquivBEq α] [inst : LawfulHashable α] [inst : Inhabited α] {a : α}, m.get! a = m.getD a default

@[simp]

theorem Std.HashSet.containsThenInsert_fst {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} {k : α}, (m.containsThenInsert k).fst = m.contains k

@[simp]

theorem Std.HashSet.containsThenInsert_snd {α : Type u} : ∀ {x : BEq α} {x_1 : Hashable α} {m : Std.HashSet α} {k : α}, (m.containsThenInsert k).snd = m.insert k