This is an internal implementation file of the hash map. Users of the hash map should not rely on the contents of this file.

File contents: Verification of associative lists

theorem Std.Internal.List.assoc_induction {α : Type u} {β : α → Type v} {motive : List ((a : α) × β a) → Prop} (nil : motive []) (cons : ∀ (k : α) (v : β k) (tail : List ((a : α) × β a)), motive tail → motive (⟨k, v⟩ :: tail)) (t : List ((a : α) × β a)) : motive t

def Std.Internal.List.getEntry? {α : Type u} {β : α → Type v} [BEq α] (a : α) : List ((a : α) × β a) → Option ((a : α) × β a)

Internal implementation detail of the hash map

def Std.Internal.List.getEntryD {α : Type u} {β : α → Type v} [BEq α] (a : α) (fallback : (a : α) × β a) : List ((a : α) × β a) → (a : α) × β a

Internal implementation detail of the hash map

def Std.Internal.List.getEntry! {α : Type u} {β : α → Type v} [BEq α] (a : α) [Inhabited ((a : α) × β a)] : List ((a : α) × β a) → (a : α) × β a

Internal implementation detail of the hash map

@[simp]

theorem Std.Internal.List.getEntry?_nil {α : Type u} {β : α → Type v} [BEq α] {a : α} : getEntry? a [] = none

@[simp]

theorem Std.Internal.List.getEntryD_nil {α : Type u} {β : α → Type v} [BEq α] {a : α} (fallback : (a : α) × β a) : getEntryD a fallback [] = fallback

@[simp]

theorem Std.Internal.List.getEntry!_nil {α : Type u} {β : α → Type v} [BEq α] {a : α} [Inhabited ((a : α) × β a)] : getEntry! a [] = default

theorem Std.Internal.List.getEntry?_cons {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} : getEntry? a (⟨k, v⟩ :: l) = if (k == a) = true then some ⟨k, v⟩ else getEntry? a l

theorem Std.Internal.List.getEntry?_eq_find {α : Type u} {β : α → Type v} [BEq α] {k : α} {l : List ((a : α) × β a)} : getEntry? k l = List.find? (fun (x : (a : α) × β a) => x.fst == k) l

theorem Std.Internal.List.getEntry?_cons_of_true {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} (h : (k == a) = true) : getEntry? a (⟨k, v⟩ :: l) = some ⟨k, v⟩

theorem Std.Internal.List.getEntry?_cons_of_false {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} (h : (k == a) = false) : getEntry? a (⟨k, v⟩ :: l) = getEntry? a l

@[simp]

theorem Std.Internal.List.getEntry?_cons_self {α : Type u} {β : α → Type v} [BEq α] [ReflBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : getEntry? k (⟨k, v⟩ :: l) = some ⟨k, v⟩

theorem Std.Internal.List.getEntryD_cons {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} {fallback : (a : α) × β a} : getEntryD a fallback (⟨k, v⟩ :: l) = if (k == a) = true then ⟨k, v⟩ else getEntryD a fallback l

theorem Std.Internal.List.getEntryD_cons_of_true {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} {fallback : (a : α) × β a} (h : (k == a) = true) : getEntryD a fallback (⟨k, v⟩ :: l) = ⟨k, v⟩

theorem Std.Internal.List.getEntryD_cons_of_false {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} {fallback : (a : α) × β a} (h : (k == a) = false) : getEntryD a fallback (⟨k, v⟩ :: l) = getEntryD a fallback l

@[simp]

theorem Std.Internal.List.getEntryD_cons_self {α : Type u} {β : α → Type v} [BEq α] [ReflBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} {fallback : (a : α) × β a} : getEntryD k fallback (⟨k, v⟩ :: l) = ⟨k, v⟩

theorem Std.Internal.List.getEntry!_cons {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} [Inhabited ((a : α) × β a)] : getEntry! a (⟨k, v⟩ :: l) = if (k == a) = true then ⟨k, v⟩ else getEntry! a l

theorem Std.Internal.List.getEntry!_cons_of_true {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} [Inhabited ((a : α) × β a)] (h : (k == a) = true) : getEntry! a (⟨k, v⟩ :: l) = ⟨k, v⟩

theorem Std.Internal.List.getEntry!_cons_of_false {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} [Inhabited ((a : α) × β a)] (h : (k == a) = false) : getEntry! a (⟨k, v⟩ :: l) = getEntry! a l

@[simp]

theorem Std.Internal.List.getEntry!_cons_self {α : Type u} {β : α → Type v} [BEq α] [ReflBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} [Inhabited ((a : α) × β a)] : getEntry! k (⟨k, v⟩ :: l) = ⟨k, v⟩

theorem Std.Internal.List.beq_of_getEntry?_eq_some {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {a : α} {p : (a : α) × β a} (h : getEntry? a l = some p) : (p.fst == a) = true

theorem Std.Internal.List.getEntry?_congr {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {a b : α} (h : (a == b) = true) : getEntry? a l = getEntry? b l

theorem Std.Internal.List.getEntryD_eq_getEntry? {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {a : α} {fallback : (a : α) × β a} : getEntryD a fallback l = (getEntry? a l).getD fallback

theorem Std.Internal.List.getEntry!_eq_getEntry? {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {a : α} [Inhabited ((a : α) × β a)] : getEntry! a l = (getEntry? a l).get!

theorem Std.Internal.List.keys_eq_map {α : Type u} {β : α → Type v} {l : List ((a : α) × β a)} : keys l = List.map (fun (x : (a : α) × β a) => x.fst) l

theorem Std.Internal.List.values_eq_map {α : Type u} {β : Type v} {l : List ((_ : α) × β)} : values l = List.map (fun (x : (_ : α) × β) => x.snd) l

theorem Std.Internal.List.getEntry?_eq_some_iff {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {e : (a : α) × β a} {k : α} (hd : DistinctKeys l) : getEntry? k l = some e ↔ (k == e.fst) = true ∧ e ∈ l

theorem Std.Internal.List.mem_iff_getEntry?_eq_some {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {p : (a : α) × β a} (h : DistinctKeys l) : p ∈ l ↔ getEntry? p.fst l = some p

theorem Std.Internal.List.isEmpty_eq_false_iff_exists_isSome_getEntry? {α : Type u} {β : α → Type v} [BEq α] [ReflBEq α] {l : List ((a : α) × β a)} : l.isEmpty = false ↔ ∃ (a : α), (getEntry? a l).isSome = true

theorem Std.Internal.List.isEmpty_iff_forall_isSome_getEntry? {α : Type u} {β : α → Type v} [BEq α] [ReflBEq α] {l : List ((a : α) × β a)} : l.isEmpty = true ↔ ∀ (a : α), (getEntry? a l).isSome = false

def Std.Internal.List.getValue? {α : Type u} {β : Type v} [BEq α] (a : α) : List ((_ : α) × β) → Option β

Internal implementation detail of the hash map

@[simp]

theorem Std.Internal.List.getValue?_nil {α : Type u} {β : Type v} [BEq α] {a : α} : getValue? a [] = none

theorem Std.Internal.List.getValue?_cons {α : Type u} {β : Type v} [BEq α] {l : List ((_ : α) × β)} {k a : α} {v : β} : getValue? a (⟨k, v⟩ :: l) = if (k == a) = true then some v else getValue? a l

theorem Std.Internal.List.getValue?_cons_of_true {α : Type u} {β : Type v} [BEq α] {l : List ((_ : α) × β)} {k a : α} {v : β} (h : (k == a) = true) : getValue? a (⟨k, v⟩ :: l) = some v

theorem Std.Internal.List.getValue?_cons_of_false {α : Type u} {β : Type v} [BEq α] {l : List ((_ : α) × β)} {k a : α} {v : β} (h : (k == a) = false) : getValue? a (⟨k, v⟩ :: l) = getValue? a l

@[simp]

theorem Std.Internal.List.getValue?_cons_self {α : Type u} {β : Type v} [BEq α] [ReflBEq α] {l : List ((_ : α) × β)} {k : α} {v : β} : getValue? k (⟨k, v⟩ :: l) = some v

theorem Std.Internal.List.getValue?_eq_getEntry? {α : Type u} {β : Type v} [BEq α] {l : List ((_ : α) × β)} {a : α} : getValue? a l = Option.map (fun (x : (_ : α) × β) => x.snd) (getEntry? a l)

theorem Std.Internal.List.getValue?_congr {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {a b : α} (h : (a == b) = true) : getValue? a l = getValue? b l

theorem Std.Internal.List.isEmpty_eq_false_iff_exists_isSome_getValue? {α : Type u} {β : Type v} [BEq α] [ReflBEq α] {l : List ((_ : α) × β)} : l.isEmpty = false ↔ ∃ (a : α), (getValue? a l).isSome = true

def Std.Internal.List.getValueCast? {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] (a : α) : List ((a : α) × β a) → Option (β a)

Internal implementation detail of the hash map

def Std.Internal.List.beqModel {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] [(k : α) → BEq (β k)] (l₁ l₂ : List ((a : α) × β a)) : Bool

Internal implementation detail of the hash map

@[simp]

theorem Std.Internal.List.getValueCast?_nil {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {a : α} : getValueCast? a [] = none

theorem Std.Internal.List.getValueCast?_cons {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} : getValueCast? a (⟨k, v⟩ :: l) = if h : (k == a) = true then some (cast ⋯ v) else getValueCast? a l

theorem Std.Internal.List.getValueCast?_cons_of_true {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} (h : (k == a) = true) : getValueCast? a (⟨k, v⟩ :: l) = some (cast ⋯ v)

theorem Std.Internal.List.getValueCast?_cons_of_false {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} (h : (k == a) = false) : getValueCast? a (⟨k, v⟩ :: l) = getValueCast? a l

@[simp]

theorem Std.Internal.List.getValueCast?_cons_self {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : getValueCast? k (⟨k, v⟩ :: l) = some v

theorem Std.Internal.List.getValue?_eq_getValueCast? {α : Type u} [BEq α] [LawfulBEq α] {β : Type v} {l : List ((_ : α) × β)} {a : α} : getValue? a l = getValueCast? a l

def Std.Internal.List.Const.beqModel {α : Type u} {β : Type v} [BEq α] [BEq β] (l₁ l₂ : List ((_ : α) × β)) : Bool

Internal implementation detail of the hash map

theorem Std.Internal.List.isSome_getValueCast?_eq_isSome_getEntry? {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {a : α} : (getValueCast? a l).isSome = (getEntry? a l).isSome

theorem Std.Internal.List.isEmpty_eq_false_iff_exists_isSome_getValueCast? {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} : l.isEmpty = false ↔ ∃ (a : α), (getValueCast? a l).isSome = true

def Std.Internal.List.containsKey {α : Type u} {β : α → Type v} [BEq α] (a : α) : List ((a : α) × β a) → Bool

Internal implementation detail of the hash map

@[simp]

theorem Std.Internal.List.containsKey_nil {α : Type u} {β : α → Type v} [BEq α] {a : α} : containsKey a [] = false

@[simp]

theorem Std.Internal.List.containsKey_cons {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} : containsKey a (⟨k, v⟩ :: l) = (k == a || containsKey a l)

theorem Std.Internal.List.containsKey_cons_eq_false {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} : containsKey a (⟨k, v⟩ :: l) = false ↔ (k == a) = false ∧ containsKey a l = false

theorem Std.Internal.List.containsKey_cons_eq_true {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} : containsKey a (⟨k, v⟩ :: l) = true ↔ (k == a) = true ∨ containsKey a l = true

theorem Std.Internal.List.containsKey_cons_of_beq {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} (h : (k == a) = true) : containsKey a (⟨k, v⟩ :: l) = true

@[simp]

theorem Std.Internal.List.containsKey_cons_self {α : Type u} {β : α → Type v} [BEq α] [ReflBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : containsKey k (⟨k, v⟩ :: l) = true

theorem Std.Internal.List.containsKey_cons_of_containsKey {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} (h : containsKey a l = true) : containsKey a (⟨k, v⟩ :: l) = true

theorem Std.Internal.List.containsKey_of_containsKey_cons {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} (h₁ : containsKey a (⟨k, v⟩ :: l) = true) (h₂ : (k == a) = false) : containsKey a l = true

theorem Std.Internal.List.containsKey_eq_isSome_getEntry? {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {a : α} : containsKey a l = (getEntry? a l).isSome

theorem Std.Internal.List.containsKey_eq_true_iff_exists_mem {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {a : α} : containsKey a l = true ↔ ∃ (p : (a : α) × β a), p ∈ l ∧ (p.fst == a) = true

theorem Std.Internal.List.containsKey_of_mem {α : Type u} {β : α → Type v} [BEq α] [ReflBEq α] {l : List ((a : α) × β a)} {p : (a : α) × β a} (hp : p ∈ l) : containsKey p.fst l = true

theorem Std.Internal.List.containsKey_eq_contains_map_fst {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k : α} : containsKey k l = (List.map Sigma.fst l).contains k

@[simp]

theorem Std.Internal.List.keys_nil {α : Type u} {β : α → Type v} : keys [] = []

@[simp]

theorem Std.Internal.List.values_nil {α : Type u} {β : Type v} : values [] = []

@[simp]

theorem Std.Internal.List.keys_cons {α : Type u} {β : α → Type v} {l : List ((a : α) × β a)} {k : α} {v : β k} : keys (⟨k, v⟩ :: l) = k :: keys l

theorem Std.Internal.List.length_keys_eq_length {α : Type u} {β : α → Type v} (l : List ((a : α) × β a)) : (keys l).length = l.length

theorem Std.Internal.List.isEmpty_keys_eq_isEmpty {α : Type u} {β : α → Type v} (l : List ((a : α) × β a)) : (keys l).isEmpty = l.isEmpty

theorem Std.Internal.List.containsKey_eq_keys_contains {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {a : α} : containsKey a l = (keys l).contains a

theorem Std.Internal.List.containsKey_of_mem_keys {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {a : α} (h : a ∈ keys l) : containsKey a l = true

@[simp]

theorem Std.Internal.List.DistinctKeys.nil {α : Type u} {β : α → Type v} [BEq α] : DistinctKeys []

theorem Std.Internal.List.DistinctKeys.def {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} : DistinctKeys l ↔ List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) l

theorem Std.Internal.List.DistinctKeys.perm_keys {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l l' : List ((a : α) × β a)} (h : (keys l').Perm (keys l)) : DistinctKeys l → DistinctKeys l'

theorem Std.Internal.List.DistinctKeys.perm {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l l' : List ((a : α) × β a)} (h : l'.Perm l) : DistinctKeys l → DistinctKeys l'

theorem Std.Internal.List.DistinctKeys.congr {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l l' : List ((a : α) × β a)} (h : l.Perm l') : DistinctKeys l ↔ DistinctKeys l'

theorem Std.Internal.List.distinctKeys_of_sublist_keys {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] {l : List ((a : α) × β a)} {l' : List ((a : α) × γ a)} (h : (keys l').Sublist (keys l)) : DistinctKeys l → DistinctKeys l'

theorem Std.Internal.List.distinctKeys_of_sublist {α : Type u} {β : α → Type v} [BEq α] {l l' : List ((a : α) × β a)} (h : l'.Sublist l) : DistinctKeys l → DistinctKeys l'

theorem Std.Internal.List.DistinctKeys.of_keys_eq {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] {l : List ((a : α) × β a)} {l' : List ((a : α) × γ a)} (h : keys l = keys l') : DistinctKeys l → DistinctKeys l'

theorem Std.Internal.List.containsKey_iff_exists {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {a : α} : containsKey a l = true ↔ ∃ (a' : α), a' ∈ keys l ∧ (a == a') = true

theorem Std.Internal.List.containsKey_eq_false_iff_forall_mem_keys {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {a : α} : containsKey a l = false ↔ ∀ (a' : α), a' ∈ keys l → (a == a') = false

theorem Std.Internal.List.containsKey_eq_false_iff {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {a : α} : containsKey a l = false ↔ ∀ (b : (a : α) × β a), b ∈ l → (a == b.fst) = false

@[simp]

theorem Std.Internal.List.distinctKeys_cons_iff {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : DistinctKeys (⟨k, v⟩ :: l) ↔ DistinctKeys l ∧ containsKey k l = false

theorem Std.Internal.List.DistinctKeys.tail {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : DistinctKeys (⟨k, v⟩ :: l) → DistinctKeys l

theorem Std.Internal.List.DistinctKeys.containsKey_eq_false {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : DistinctKeys (⟨k, v⟩ :: l) → containsKey k l = false

theorem Std.Internal.List.DistinctKeys.cons {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} (h : containsKey k l = false) : DistinctKeys l → DistinctKeys (⟨k, v⟩ :: l)

theorem Std.Internal.List.isEmpty_eq_false_of_containsKey {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {a : α} (h : containsKey a l = true) : l.isEmpty = false

theorem Std.Internal.List.isEmpty_eq_false_iff_exists_containsKey {α : Type u} {β : α → Type v} [BEq α] [ReflBEq α] {l : List ((a : α) × β a)} : l.isEmpty = false ↔ ∃ (a : α), containsKey a l = true

theorem Std.Internal.List.isEmpty_iff_forall_containsKey {α : Type u} {β : α → Type v} [BEq α] [ReflBEq α] {l : List ((a : α) × β a)} : l.isEmpty = true ↔ ∀ (a : α), containsKey a l = false

@[simp]

theorem Std.Internal.List.getEntry?_eq_none {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {a : α} : getEntry? a l = none ↔ containsKey a l = false

@[simp]

theorem Std.Internal.List.getValue?_eq_none {α : Type u} {β : Type v} [BEq α] {l : List ((_ : α) × β)} {a : α} : getValue? a l = none ↔ containsKey a l = false

theorem Std.Internal.List.containsKey_eq_isSome_getValue? {α : Type u} {β : Type v} [BEq α] {l : List ((_ : α) × β)} {a : α} : containsKey a l = (getValue? a l).isSome

theorem Std.Internal.List.containsKey_eq_isSome_getValueCast? {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {a : α} : containsKey a l = (getValueCast? a l).isSome

theorem Std.Internal.List.containsKey_of_getValueCast?_eq_some {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {a : α} {b : β a} (h : getValueCast? a l = some b) : containsKey a l = true

theorem Std.Internal.List.containsKey_of_getValue_eq_some {α : Type u} [BEq α] [LawfulBEq α] {β : Type v} {l : List ((_ : α) × β)} {a : α} {b : β} (h : getValue? a l = some b) : containsKey a l = true

theorem Std.Internal.List.getValueCast?_eq_none {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {a : α} (h : containsKey a l = false) : getValueCast? a l = none

theorem Std.Internal.List.containsKey_congr {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {a b : α} (h : (a == b) = true) : containsKey a l = containsKey b l

theorem Std.Internal.List.containsKey_of_beq {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {a b : α} (hla : containsKey a l = true) (hab : (a == b) = true) : containsKey b l = true

def Std.Internal.List.getEntry {α : Type u} {β : α → Type v} [BEq α] (a : α) (l : List ((a : α) × β a)) (h : containsKey a l = true) : (a : α) × β a

Internal implementation detail of the hash map

theorem Std.Internal.List.getEntry?_eq_some_getEntry {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {a : α} (h : containsKey a l = true) : getEntry? a l = some (getEntry a l h)

theorem Std.Internal.List.getEntry_eq_of_getEntry?_eq_some {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} (h : getEntry? a l = some ⟨k, v⟩) {h' : containsKey a l = true} : getEntry a l h' = ⟨k, v⟩

theorem Std.Internal.List.getEntry_cons {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} (h : containsKey a (⟨k, v⟩ :: l) = true) : getEntry a (⟨k, v⟩ :: l) h = if h' : (k == a) = true then ⟨k, v⟩ else getEntry a l ⋯

theorem Std.Internal.List.getEntry_congr {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {a b : α} (h : (a == b) = true) {h₁ : containsKey a l = true} {h₂ : containsKey b l = true} : getEntry a l h₁ = getEntry b l h₂

theorem Std.Internal.List.getEntry_cons_of_beq {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} (h : (k == a) = true) : getEntry a (⟨k, v⟩ :: l) ⋯ = ⟨k, v⟩

@[simp]

theorem Std.Internal.List.getEntry_cons_self {α : Type u} {β : α → Type v} [BEq α] [ReflBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : getEntry k (⟨k, v⟩ :: l) ⋯ = ⟨k, v⟩

theorem Std.Internal.List.getEntry_cons_of_false {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} {h₁ : containsKey a (⟨k, v⟩ :: l) = true} (h₂ : (k == a) = false) : getEntry a (⟨k, v⟩ :: l) h₁ = getEntry a l ⋯

theorem Std.Internal.List.getEntry_mem {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} {h : containsKey k l = true} : getEntry k l h ∈ l

theorem Std.Internal.List.getEntry_of_mem {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {x : (a : α) × β a} (h : x ∈ l) (hl : DistinctKeys l) : getEntry x.fst l ⋯ = x

def Std.Internal.List.getValue {α : Type u} {β : Type v} [BEq α] (a : α) (l : List ((_ : α) × β)) (h : containsKey a l = true) : β

Internal implementation detail of the hash map

theorem Std.Internal.List.getValue?_eq_some_getValue {α : Type u} {β : Type v} [BEq α] {l : List ((_ : α) × β)} {a : α} (h : containsKey a l = true) : getValue? a l = some (getValue a l h)

theorem Std.Internal.List.getValue?_eq_some_iff {α : Type u} {β : Type v} [BEq α] {l : List ((_ : α) × β)} {k : α} {v : β} : getValue? k l = some v ↔ ∃ (h : containsKey k l = true), getValue k l h = v

theorem Std.Internal.List.getValue_cons_of_beq {α : Type u} {β : Type v} [BEq α] {l : List ((_ : α) × β)} {k a : α} {v : β} (h : (k == a) = true) : getValue a (⟨k, v⟩ :: l) ⋯ = v

@[simp]

theorem Std.Internal.List.getValue_cons_self {α : Type u} {β : Type v} [BEq α] [ReflBEq α] {l : List ((_ : α) × β)} {k : α} {v : β} : getValue k (⟨k, v⟩ :: l) ⋯ = v

theorem Std.Internal.List.getValue_cons_of_false {α : Type u} {β : Type v} [BEq α] {l : List ((_ : α) × β)} {k a : α} {v : β} {h₁ : containsKey a (⟨k, v⟩ :: l) = true} (h₂ : (k == a) = false) : getValue a (⟨k, v⟩ :: l) h₁ = getValue a l ⋯

theorem Std.Internal.List.getValue_cons {α : Type u} {β : Type v} [BEq α] {l : List ((_ : α) × β)} {k a : α} {v : β} {h : containsKey a (⟨k, v⟩ :: l) = true} : getValue a (⟨k, v⟩ :: l) h = if h' : (k == a) = true then v else getValue a l ⋯

theorem Std.Internal.List.getValue_congr {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {a b : α} (hab : (a == b) = true) {h : containsKey a l = true} : getValue a l h = getValue b l ⋯

theorem Std.Internal.List.getValue_of_mem {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l : List ((_ : α) × β)} {x : (_ : α) × β} (h : x ∈ l) {h' : containsKey x.fst l = true} (distinct : DistinctKeys l) : getValue x.fst l h' = x.snd

theorem Std.Internal.List.getValue_eq_getEntry_snd {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l : List ((_ : α) × β)} {k : α} {h' : containsKey k l = true} : getValue k l h' = (getEntry k l h').snd

def Std.Internal.List.getValueCast {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] (a : α) (l : List ((a : α) × β a)) (h : containsKey a l = true) : β a

Internal implementation detail of the hash map

theorem Std.Internal.List.getValueCast?_eq_some_getValueCast {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {a : α} (h : containsKey a l = true) : getValueCast? a l = some (getValueCast a l h)

theorem Std.Internal.List.getValueCast?_eq_some_iff {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : getValueCast? k l = some v ↔ ∃ (h : containsKey k l = true), getValueCast k l h = v

theorem Std.Internal.List.getValueCast_cons {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} (h : containsKey a (⟨k, v⟩ :: l) = true) : getValueCast a (⟨k, v⟩ :: l) h = if h' : (k == a) = true then cast ⋯ v else getValueCast a l ⋯

theorem Std.Internal.List.getValueCast_mem {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {a : α} (h : containsKey a l = true) : ⟨a, getValueCast a l h⟩ ∈ l

theorem Std.Internal.List.getValueCast_of_mem {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {x : (a : α) × β a} (h : x ∈ l) (distinct : DistinctKeys l) : getValueCast x.fst l ⋯ = x.snd

theorem Std.Internal.List.getValue_eq_getValueCast {α : Type u} {β : Type v} [BEq α] [LawfulBEq α] {l : List ((_ : α) × β)} {a : α} {h : containsKey a l = true} : getValue a l h = getValueCast a l h

def Std.Internal.List.getValueCastD {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] (a : α) (l : List ((a : α) × β a)) (fallback : β a) : β a

Internal implementation detail of the hash map

@[simp]

theorem Std.Internal.List.getValueCastD_nil {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {a : α} {fallback : β a} : getValueCastD a [] fallback = fallback

theorem Std.Internal.List.getValueCastD_eq_getValueCast? {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {a : α} {fallback : β a} : getValueCastD a l fallback = (getValueCast? a l).getD fallback

theorem Std.Internal.List.getValueCastD_eq_fallback {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {a : α} {fallback : β a} (h : containsKey a l = false) : getValueCastD a l fallback = fallback

theorem Std.Internal.List.getValueCast_eq_getValueCastD {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {a : α} {fallback : β a} (h : containsKey a l = true) : getValueCast a l h = getValueCastD a l fallback

theorem Std.Internal.List.getValueCast?_eq_some_getValueCastD {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {a : α} {fallback : β a} (h : containsKey a l = true) : getValueCast? a l = some (getValueCastD a l fallback)

def Std.Internal.List.getValueCast! {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] (a : α) [Inhabited (β a)] (l : List ((a : α) × β a)) : β a

Internal implementation detail of the hash map

@[simp]

theorem Std.Internal.List.getValueCast!_nil {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {a : α} [Inhabited (β a)] : getValueCast! a [] = default

theorem Std.Internal.List.getValueCast!_eq_getValueCast? {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {a : α} [Inhabited (β a)] : getValueCast! a l = (getValueCast? a l).get!

theorem Std.Internal.List.getValueCast!_eq_default {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {a : α} [Inhabited (β a)] (h : containsKey a l = false) : getValueCast! a l = default

theorem Std.Internal.List.getValueCast_eq_getValueCast! {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {a : α} [Inhabited (β a)] (h : containsKey a l = true) : getValueCast a l h = getValueCast! a l

theorem Std.Internal.List.getValueCast?_eq_some_getValueCast! {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {a : α} [Inhabited (β a)] (h : containsKey a l = true) : getValueCast? a l = some (getValueCast! a l)

theorem Std.Internal.List.getValueCast!_eq_getValueCastD_default {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {a : α} [Inhabited (β a)] : getValueCast! a l = getValueCastD a l default

def Std.Internal.List.getValueD {α : Type u} {β : Type v} [BEq α] (a : α) (l : List ((_ : α) × β)) (fallback : β) : β

Internal implementation detail of the hash map

@[simp]

theorem Std.Internal.List.getValueD_nil {α : Type u} {β : Type v} [BEq α] {a : α} {fallback : β} : getValueD a [] fallback = fallback

theorem Std.Internal.List.getValueD_eq_getValue? {α : Type u} {β : Type v} [BEq α] {l : List ((_ : α) × β)} {a : α} {fallback : β} : getValueD a l fallback = (getValue? a l).getD fallback

theorem Std.Internal.List.getValueD_eq_fallback {α : Type u} {β : Type v} [BEq α] {l : List ((_ : α) × β)} {a : α} {fallback : β} (h : containsKey a l = false) : getValueD a l fallback = fallback

theorem Std.Internal.List.getValue_eq_getValueD {α : Type u} {β : Type v} [BEq α] {l : List ((_ : α) × β)} {a : α} {fallback : β} (h : containsKey a l = true) : getValue a l h = getValueD a l fallback

theorem Std.Internal.List.getValue?_eq_some_getValueD {α : Type u} {β : Type v} [BEq α] {l : List ((_ : α) × β)} {a : α} {fallback : β} (h : containsKey a l = true) : getValue? a l = some (getValueD a l fallback)

theorem Std.Internal.List.getValueD_eq_getValueCastD {α : Type u} {β : Type v} [BEq α] [LawfulBEq α] {l : List ((_ : α) × β)} {a : α} {fallback : β} : getValueD a l fallback = getValueCastD a l fallback

theorem Std.Internal.List.getValueD_congr {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {a b : α} {fallback : β} (hab : (a == b) = true) : getValueD a l fallback = getValueD b l fallback

def Std.Internal.List.getValue! {α : Type u} {β : Type v} [BEq α] [Inhabited β] (a : α) (l : List ((_ : α) × β)) : β

Internal implementation detail of the hash map

@[simp]

theorem Std.Internal.List.getValue!_nil {α : Type u} {β : Type v} [BEq α] [Inhabited β] {a : α} : getValue! a [] = default

theorem Std.Internal.List.getValue!_eq_getValue? {α : Type u} {β : Type v} [BEq α] [Inhabited β] {l : List ((_ : α) × β)} {a : α} : getValue! a l = (getValue? a l).get!

theorem Std.Internal.List.getValue!_eq_default {α : Type u} {β : Type v} [BEq α] [Inhabited β] {l : List ((_ : α) × β)} {a : α} (h : containsKey a l = false) : getValue! a l = default

theorem Std.Internal.List.getValue_eq_getValue! {α : Type u} {β : Type v} [BEq α] [Inhabited β] {l : List ((_ : α) × β)} {a : α} (h : containsKey a l = true) : getValue a l h = getValue! a l

theorem Std.Internal.List.getValue?_eq_some_getValue! {α : Type u} {β : Type v} [BEq α] [Inhabited β] {l : List ((_ : α) × β)} {a : α} (h : containsKey a l = true) : getValue? a l = some (getValue! a l)

theorem Std.Internal.List.getValue!_eq_getValueCast! {α : Type u} {β : Type v} [BEq α] [LawfulBEq α] [Inhabited β] {l : List ((_ : α) × β)} {a : α} : getValue! a l = getValueCast! a l

theorem Std.Internal.List.getValue!_congr {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] [Inhabited β] {l : List ((_ : α) × β)} {a b : α} (hab : (a == b) = true) : getValue! a l = getValue! b l

theorem Std.Internal.List.getValue!_eq_getValueD_default {α : Type u} {β : Type v} [BEq α] [Inhabited β] {l : List ((_ : α) × β)} {a : α} : getValue! a l = getValueD a l default

def Std.Internal.List.getKey? {α : Type u} {β : α → Type v} [BEq α] (a : α) : List ((a : α) × β a) → Option α

Internal implementation detail of the hash map

@[simp]

theorem Std.Internal.List.getKey?_nil {α : Type u} {β : α → Type v} [BEq α] {a : α} : getKey? a [] = none

@[simp]

theorem Std.Internal.List.getKey?_cons {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} : getKey? a (⟨k, v⟩ :: l) = if (k == a) = true then some k else getKey? a l

theorem Std.Internal.List.getKey?_cons_of_true {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} (h : (k == a) = true) : getKey? a (⟨k, v⟩ :: l) = some k

theorem Std.Internal.List.getKey?_cons_of_false {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} (h : (k == a) = false) : getKey? a (⟨k, v⟩ :: l) = getKey? a l

theorem Std.Internal.List.getKey?_eq_getEntry? {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {a : α} : getKey? a l = Option.map (fun (x : (a : α) × β a) => x.fst) (getEntry? a l)

theorem Std.Internal.List.fst_mem_keys_of_mem {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {a : (a : α) × β a} {l : List ((a : α) × β a)} (hm : a ∈ l) : a.fst ∈ keys l

theorem Std.Internal.List.getKey?_eq_some_iff {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {k k' : α} (hd : DistinctKeys l) : getKey? k l = some k' ↔ (k == k') = true ∧ k' ∈ keys l

theorem Std.Internal.List.getKey?_beq {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {a : α} : Option.all (fun (x : α) => x == a) (getKey? a l) = true

theorem Std.Internal.List.getKey?_congr {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {k k' : α} (h : (k == k') = true) : getKey? k l = getKey? k' l

theorem Std.Internal.List.containsKey_eq_isSome_getKey? {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {a : α} : containsKey a l = (getKey? a l).isSome

theorem Std.Internal.List.getEntry?_eq_some_iff_getKey?_eq_some_getValue?_eq_some {α : Type u} [BEq α] {β : Type v} {l : List ((_ : α) × β)} {k k' : α} {v : β} : getEntry? k l = some ⟨k', v⟩ ↔ getKey? k l = some k' ∧ getValue? k l = some v

theorem Std.Internal.List.containsKey_of_getKey?_eq_some {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {a a' : α} (h : getKey? a l = some a') : containsKey a' l = true

def Std.Internal.List.getKey {α : Type u} {β : α → Type v} [BEq α] (a : α) (l : List ((a : α) × β a)) (h : containsKey a l = true) : α

Internal implementation detail of the hash map

theorem Std.Internal.List.getKey?_eq_some_getKey {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {a : α} (h : containsKey a l = true) : getKey? a l = some (getKey a l h)

theorem Std.Internal.List.getKey_cons {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} {h : containsKey a (⟨k, v⟩ :: l) = true} : getKey a (⟨k, v⟩ :: l) h = if h' : (k == a) = true then k else getKey a l ⋯

theorem Std.Internal.List.getKey?_eq_some_iff' {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {k k' : α} : getKey? k l = some k' ↔ ∃ (h : containsKey k l = true), getKey k l h = k'

theorem Std.Internal.List.getKey_beq {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {a : α} (h : containsKey a l = true) : (getKey a l h == a) = true

@[simp]

theorem Std.Internal.List.getKey_eq {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {a : α} (h : containsKey a l = true) : getKey a l h = a

theorem Std.Internal.List.getKey?_eq_some {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {a : α} (h : containsKey a l = true) : getKey? a l = some a

theorem Std.Internal.List.getKey_congr {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {k k' : α} (h : (k == k') = true) {h' : containsKey k l = true} {h'' : containsKey k' l = true} : getKey k l h' = getKey k' l h''

theorem Std.Internal.List.getKey_of_mem {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {x : (a : α) × β a} (h : x ∈ l) (distinct : DistinctKeys l) {h' : containsKey x.fst l = true} : getKey x.fst l h' = x.fst

theorem Std.Internal.List.getKey_eq_getEntry_fst {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {a : α} {h : containsKey a l = true} : getKey a l h = (getEntry a l h).fst

theorem Std.Internal.List.forall_mem_keys_iff_forall_containsKey_getKey {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {p : α → Prop} (hl : DistinctKeys l) : (∀ (k : α), k ∈ keys l → p k) ↔ ∀ (k : α) (h : containsKey k l = true), p (getKey k l h)

def Std.Internal.List.getKeyD {α : Type u} {β : α → Type v} [BEq α] (a : α) (l : List ((a : α) × β a)) (fallback : α) : α

Internal implementation detail of the hash map

@[simp]

theorem Std.Internal.List.getKeyD_nil {α : Type u} {β : α → Type v} [BEq α] {a fallback : α} : getKeyD a [] fallback = fallback

theorem Std.Internal.List.getKeyD_eq_getKey? {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {a fallback : α} : getKeyD a l fallback = (getKey? a l).getD fallback

theorem Std.Internal.List.getKeyD_eq_fallback {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {a fallback : α} (h : containsKey a l = false) : getKeyD a l fallback = fallback

theorem Std.Internal.List.getKey_eq_getKeyD {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {a fallback : α} (h : containsKey a l = true) : getKey a l h = getKeyD a l fallback

theorem Std.Internal.List.getKeyD_congr {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {k k' fallback : α} (h : (k == k') = true) : getKeyD k l fallback = getKeyD k' l fallback

theorem Std.Internal.List.getKeyD_eq_of_containsKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k fallback : α} (h : containsKey k l = true) : getKeyD k l fallback = k

theorem Std.Internal.List.getKey?_eq_some_getKeyD {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {a fallback : α} (h : containsKey a l = true) : getKey? a l = some (getKeyD a l fallback)

def Std.Internal.List.getKey! {α : Type u} {β : α → Type v} [BEq α] [Inhabited α] (a : α) (l : List ((a : α) × β a)) : α

Internal implementation detail of the hash map

@[simp]

theorem Std.Internal.List.getKey!_nil {α : Type u} {β : α → Type v} [BEq α] [Inhabited α] {a : α} : getKey! a [] = default

theorem Std.Internal.List.getKey!_eq_getKey? {α : Type u} {β : α → Type v} [BEq α] [Inhabited α] {l : List ((a : α) × β a)} {a : α} : getKey! a l = (getKey? a l).get!

theorem Std.Internal.List.getKey!_eq_default {α : Type u} {β : α → Type v} [BEq α] [Inhabited α] {l : List ((a : α) × β a)} {a : α} (h : containsKey a l = false) : getKey! a l = default

theorem Std.Internal.List.getKey_eq_getKey! {α : Type u} {β : α → Type v} [BEq α] [Inhabited α] {l : List ((a : α) × β a)} {a : α} (h : containsKey a l = true) : getKey a l h = getKey! a l

theorem Std.Internal.List.getKey!_congr {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] [Inhabited α] {l : List ((a : α) × β a)} {k k' : α} (h : (k == k') = true) : getKey! k l = getKey! k' l

theorem Std.Internal.List.getKey!_eq_of_containsKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] [Inhabited α] {l : List ((a : α) × β a)} {k : α} (h : containsKey k l = true) : getKey! k l = k

theorem Std.Internal.List.getKey?_eq_some_getKey! {α : Type u} {β : α → Type v} [BEq α] [Inhabited α] {l : List ((a : α) × β a)} {a : α} (h : containsKey a l = true) : getKey? a l = some (getKey! a l)

theorem Std.Internal.List.getKey!_eq_getKeyD_default {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] [Inhabited α] {l : List ((a : α) × β a)} {a : α} : getKey! a l = getKeyD a l default

theorem Std.Internal.List.getEntry?_eq_getValueCast? {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {a : α} : getEntry? a l = Option.map (fun (v : β a) => ⟨a, v⟩) (getValueCast? a l)

theorem Std.Internal.List.getEntry_eq_getKey_getValue {α : Type u} [BEq α] {β : Type v} {l : List ((_ : α) × β)} {a : α} (h : containsKey a l = true) : getEntry a l h = ⟨getKey a l h, getValue a l h⟩

def Std.Internal.List.replaceEntry {α : Type u} {β : α → Type v} [BEq α] (k : α) (v : β k) : List ((a : α) × β a) → List ((a : α) × β a)

Internal implementation detail of the hash map

@[simp]

theorem Std.Internal.List.replaceEntry_nil {α : Type u} {β : α → Type v} [BEq α] {k : α} {v : β k} : replaceEntry k v [] = []

theorem Std.Internal.List.replaceEntry_cons {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k k' : α} {v : β k} {v' : β k'} : replaceEntry k v (⟨k', v'⟩ :: l) = if (k' == k) = true then ⟨k, v⟩ :: l else ⟨k', v'⟩ :: replaceEntry k v l

theorem Std.Internal.List.replaceEntry_cons_of_true {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k k' : α} {v : β k} {v' : β k'} (h : (k' == k) = true) : replaceEntry k v (⟨k', v'⟩ :: l) = ⟨k, v⟩ :: l

theorem Std.Internal.List.replaceEntry_cons_of_false {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k k' : α} {v : β k} {v' : β k'} (h : (k' == k) = false) : replaceEntry k v (⟨k', v'⟩ :: l) = ⟨k', v'⟩ :: replaceEntry k v l

theorem Std.Internal.List.replaceEntry_of_containsKey_eq_false {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} (h : containsKey k l = false) : replaceEntry k v l = l

@[simp]

theorem Std.Internal.List.isEmpty_replaceEntry {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : (replaceEntry k v l).isEmpty = l.isEmpty

theorem Std.Internal.List.mem_replaceEntry_of_beq_eq_false {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {a : α} {b : β a} {l : List ((a : α) × β a)} (p : (a : α) × β a) (hne : (p.fst == a) = false) : p ∈ replaceEntry a b l ↔ p ∈ l

theorem Std.Internal.List.mem_replaceEntry_of_key_ne {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {a : α} {b : β a} {l : List ((a : α) × β a)} (p : (a : α) × β a) (hne : p.fst ≠ a) : p ∈ replaceEntry a b l ↔ p ∈ l

theorem Std.Internal.List.getEntry?_replaceEntry_of_containsKey_eq_false {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {a k : α} {v : β k} (hl : containsKey k l = false) : getEntry? a (replaceEntry k v l) = getEntry? a l

theorem Std.Internal.List.getEntry?_replaceEntry_of_false {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {a k : α} {v : β k} (h : (k == a) = false) : getEntry? a (replaceEntry k v l) = getEntry? a l

theorem Std.Internal.List.getEntry?_replaceEntry_of_true {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {a k : α} {v : β k} (hl : containsKey k l = true) (h : (k == a) = true) : getEntry? a (replaceEntry k v l) = some ⟨k, v⟩

theorem Std.Internal.List.getEntry?_replaceEntry {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {a k : α} {v : β k} : getEntry? a (replaceEntry k v l) = if containsKey k l = true ∧ (k == a) = true then some ⟨k, v⟩ else getEntry? a l

@[simp]

theorem Std.Internal.List.containsKey_replaceEntry {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {a k : α} {v : β k} : containsKey a (replaceEntry k v l) = containsKey a l

theorem Std.Internal.List.getEntry_replaceEntry_of_true {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {a k : α} {v : β k} (hl : containsKey a (replaceEntry k v l) = true) (h : (k == a) = true) : getEntry a (replaceEntry k v l) hl = ⟨k, v⟩

@[deprecated Std.Internal.List.getEntry_mem (since := "2025-10-29")]

theorem Std.Internal.List.mem_getEntry {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} (hl : containsKey k l = true) : getEntry k l hl ∈ l

theorem Std.Internal.List.mem_replaceEntry_of_true {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} (hl : containsKey k l = true) : ⟨k, v⟩ ∈ replaceEntry k v l

@[simp]

theorem Std.Internal.List.length_replaceEntry {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : (replaceEntry k v l).length = l.length

theorem Std.Internal.List.getValue?_replaceEntry_of_containsKey_eq_false {α : Type u} {β : Type v} [BEq α] {l : List ((_ : α) × β)} {k a : α} {v : β} (hl : containsKey k l = false) : getValue? a (replaceEntry k v l) = getValue? a l

theorem Std.Internal.List.getValue?_replaceEntry_of_false {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {k a : α} {v : β} (h : (k == a) = false) : getValue? a (replaceEntry k v l) = getValue? a l

theorem Std.Internal.List.getValue?_replaceEntry_of_true {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {k a : α} {v : β} (hl : containsKey k l = true) (h : (k == a) = true) : getValue? a (replaceEntry k v l) = some v

theorem Std.Internal.List.getValueCast?_replaceEntry {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {a k : α} {v : β k} : getValueCast? a (replaceEntry k v l) = if h : containsKey k l = true ∧ (k == a) = true then some (cast ⋯ v) else getValueCast? a l

theorem Std.Internal.List.getKey?_replaceEntry {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {a k : α} {v : β k} : getKey? a (replaceEntry k v l) = if containsKey k l = true ∧ (k == a) = true then some k else getKey? a l

def Std.Internal.List.eraseKey {α : Type u} {β : α → Type v} [BEq α] (k : α) : List ((a : α) × β a) → List ((a : α) × β a)

Internal implementation detail of the hash map

@[simp]

theorem Std.Internal.List.eraseKey_nil {α : Type u} {β : α → Type v} [BEq α] {k : α} : eraseKey k [] = []

theorem Std.Internal.List.eraseKey_cons {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k k' : α} {v' : β k'} : eraseKey k (⟨k', v'⟩ :: l) = if (k' == k) = true then l else ⟨k', v'⟩ :: eraseKey k l

theorem Std.Internal.List.eraseKey_cons_of_beq {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k k' : α} {v' : β k'} (h : (k' == k) = true) : eraseKey k (⟨k', v'⟩ :: l) = l

@[simp]

theorem Std.Internal.List.eraseKey_cons_self {α : Type u} {β : α → Type v} [BEq α] [ReflBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : eraseKey k (⟨k, v⟩ :: l) = l

theorem Std.Internal.List.eraseKey_cons_of_false {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k k' : α} {v' : β k'} (h : (k' == k) = false) : eraseKey k (⟨k', v'⟩ :: l) = ⟨k', v'⟩ :: eraseKey k l

theorem Std.Internal.List.eraseKey_of_containsKey_eq_false {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} (h : containsKey k l = false) : eraseKey k l = l

theorem Std.Internal.List.mem_eraseKey_of_key_beq_eq_false {α : Type u} {β : α → Type v} [BEq α] {a : α} {l : List ((a : α) × β a)} (p : (a : α) × β a) (hne : (p.fst == a) = false) : p ∈ eraseKey a l ↔ p ∈ l

theorem Std.Internal.List.mem_eraseKey_of_key_ne {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {a : α} {l : List ((a : α) × β a)} (p : (a : α) × β a) (hne : p.fst ≠ a) : p ∈ eraseKey a l ↔ p ∈ l

theorem Std.Internal.List.sublist_eraseKey {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} : (eraseKey k l).Sublist l

theorem Std.Internal.List.length_eraseKey {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} : (eraseKey k l).length = if containsKey k l = true then l.length - 1 else l.length

theorem Std.Internal.List.length_eraseKey_le {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} : (eraseKey k l).length ≤ l.length

theorem Std.Internal.List.length_le_length_eraseKey {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} : l.length ≤ (eraseKey k l).length + 1

theorem Std.Internal.List.isEmpty_eraseKey {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} : (eraseKey k l).isEmpty = (l.isEmpty || l.length == 1 && containsKey k l)

theorem Std.Internal.List.DistinctKeys.replaceEntry {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} (h : DistinctKeys l) : DistinctKeys (List.replaceEntry k v l)

theorem Std.Internal.List.getEntry?_of_mem {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} (hl : DistinctKeys l) {k k' : α} (hk : (k == k') = true) {v : β k} (hkv : ⟨k, v⟩ ∈ l) : getEntry? k' l = some ⟨k, v⟩

def Std.Internal.List.insertEntry {α : Type u} {β : α → Type v} [BEq α] (k : α) (v : β k) (l : List ((a : α) × β a)) : List ((a : α) × β a)

Internal implementation detail of the hash map

@[simp]

theorem Std.Internal.List.insertEntry_nil {α : Type u} {β : α → Type v} [BEq α] {k : α} {v : β k} : insertEntry k v [] = [⟨k, v⟩]

theorem Std.Internal.List.insertEntry_cons_of_false {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k k' : α} {v : β k} {v' : β k'} (h : (k' == k) = false) : (insertEntry k v (⟨k', v'⟩ :: l)).Perm (⟨k', v'⟩ :: insertEntry k v l)

theorem Std.Internal.List.insertEntry_cons_of_beq {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k k' : α} {v : β k} {v' : β k'} (h : (k' == k) = true) : insertEntry k v (⟨k', v'⟩ :: l) = ⟨k, v⟩ :: l

@[simp]

theorem Std.Internal.List.insertEntry_cons_self {α : Type u} {β : α → Type v} [BEq α] [ReflBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : insertEntry k v (⟨k, v⟩ :: l) = ⟨k, v⟩ :: l

theorem Std.Internal.List.insertEntry_of_containsKey {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} (h : containsKey k l = true) : insertEntry k v l = replaceEntry k v l

theorem Std.Internal.List.insertEntry_of_containsKey_eq_false {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} (h : containsKey k l = false) : insertEntry k v l = ⟨k, v⟩ :: l

theorem Std.Internal.List.mem_insertEntry_of_key_beq_eq_false {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {a : α} {b : β a} {l : List ((a : α) × β a)} (p : (a : α) × β a) (hne : (p.fst == a) = false) : p ∈ insertEntry a b l ↔ p ∈ l

theorem Std.Internal.List.mem_insertEntry_of_key_ne {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {a : α} {b : β a} {l : List ((a : α) × β a)} (p : (a : α) × β a) (hne : p.fst ≠ a) : p ∈ insertEntry a b l ↔ p ∈ l

theorem Std.Internal.List.DistinctKeys.insertEntry {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} (h : DistinctKeys l) : DistinctKeys (List.insertEntry k v l)

@[simp]

theorem Std.Internal.List.isEmpty_insertEntry {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : (insertEntry k v l).isEmpty = false

theorem Std.Internal.List.length_insertEntry {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : (insertEntry k v l).length = if containsKey k l = true then l.length else l.length + 1

theorem Std.Internal.List.length_le_length_insertEntry {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : l.length ≤ (insertEntry k v l).length

theorem Std.Internal.List.length_insertEntry_le {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : (insertEntry k v l).length ≤ l.length + 1

theorem Std.Internal.List.getValue?_insertEntry_of_beq {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {k a : α} {v : β} (h : (k == a) = true) : getValue? a (insertEntry k v l) = some v

theorem Std.Internal.List.getValue?_insertEntry_of_false {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {k a : α} {v : β} (h : (k == a) = false) : getValue? a (insertEntry k v l) = getValue? a l

theorem Std.Internal.List.getValue?_insertEntry {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {k a : α} {v : β} : getValue? a (insertEntry k v l) = if (k == a) = true then some v else getValue? a l

theorem Std.Internal.List.getValue?_insertEntry_self {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l : List ((_ : α) × β)} {k : α} {v : β} : getValue? k (insertEntry k v l) = some v

theorem Std.Internal.List.getEntry?_insertEntry {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} : getEntry? a (insertEntry k v l) = if (k == a) = true then some ⟨k, v⟩ else getEntry? a l

theorem Std.Internal.List.getValueCast?_insertEntry {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} : getValueCast? a (insertEntry k v l) = if h : (k == a) = true then some (cast ⋯ v) else getValueCast? a l

theorem Std.Internal.List.getValueCast?_insertEntry_self {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : getValueCast? k (insertEntry k v l) = some v

theorem Std.Internal.List.getValueCast!_insertEntry {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k a : α} [Inhabited (β a)] {v : β k} : getValueCast! a (insertEntry k v l) = if h : (k == a) = true then cast ⋯ v else getValueCast! a l

theorem Std.Internal.List.getValueCast!_insertEntry_self {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k : α} [Inhabited (β k)] {v : β k} : getValueCast! k (insertEntry k v l) = v

theorem Std.Internal.List.getValueCastD_insertEntry {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k a : α} {fallback : β a} {v : β k} : getValueCastD a (insertEntry k v l) fallback = if h : (k == a) = true then cast ⋯ v else getValueCastD a l fallback

theorem Std.Internal.List.getValueCastD_insertEntry_self {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k : α} {fallback v : β k} : getValueCastD k (insertEntry k v l) fallback = v

theorem Std.Internal.List.getValue!_insertEntry {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] [Inhabited β] {l : List ((_ : α) × β)} {k a : α} {v : β} : getValue! a (insertEntry k v l) = if (k == a) = true then v else getValue! a l

theorem Std.Internal.List.getValue!_insertEntry_self {α : Type u} {β : Type v} [BEq α] [EquivBEq α] [Inhabited β] {l : List ((_ : α) × β)} {k : α} {v : β} : getValue! k (insertEntry k v l) = v

theorem Std.Internal.List.getValueD_insertEntry {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {k a : α} {fallback v : β} : getValueD a (insertEntry k v l) fallback = if (k == a) = true then v else getValueD a l fallback

theorem Std.Internal.List.getValueD_insertEntry_self {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l : List ((_ : α) × β)} {k : α} {fallback v : β} : getValueD k (insertEntry k v l) fallback = v

theorem Std.Internal.List.getKey?_insertEntry {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} : getKey? a (insertEntry k v l) = if (k == a) = true then some k else getKey? a l

theorem Std.Internal.List.getKey?_insertEntry_self {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : getKey? k (insertEntry k v l) = some k

theorem Std.Internal.List.getKey?_eq_none {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {a : α} (h : containsKey a l = false) : getKey? a l = none

theorem Std.Internal.List.getKey!_insertEntry {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] [Inhabited α] {l : List ((a : α) × β a)} {k a : α} {v : β k} : getKey! a (insertEntry k v l) = if (k == a) = true then k else getKey! a l

theorem Std.Internal.List.getKey!_insertEntry_self {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] [Inhabited α] {l : List ((a : α) × β a)} {k : α} {v : β k} : getKey! k (insertEntry k v l) = k

theorem Std.Internal.List.getKeyD_insertEntry {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {k a fallback : α} {v : β k} : getKeyD a (insertEntry k v l) fallback = if (k == a) = true then k else getKeyD a l fallback

theorem Std.Internal.List.getKeyD_insertEntry_self {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {k fallback : α} {v : β k} : getKeyD k (insertEntry k v l) fallback = k

theorem Std.Internal.List.containsKey_insertEntry {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} : containsKey a (insertEntry k v l) = (k == a || containsKey a l)

theorem Std.Internal.List.containsKey_insertEntry_of_beq {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} (h : (k == a) = true) : containsKey a (insertEntry k v l) = true

theorem Std.Internal.List.containsKey_insertEntry_self {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : containsKey k (insertEntry k v l) = true

theorem Std.Internal.List.containsKey_of_containsKey_insertEntry {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} (h₁ : containsKey a (insertEntry k v l) = true) (h₂ : (k == a) = false) : containsKey a l = true

theorem Std.Internal.List.getValueCast_insertEntry {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} {h : containsKey a (insertEntry k v l) = true} : getValueCast a (insertEntry k v l) h = if h' : (k == a) = true then cast ⋯ v else getValueCast a l ⋯

theorem Std.Internal.List.getEntry_insertEntry {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} {h : containsKey a (insertEntry k v l) = true} : getEntry a (insertEntry k v l) h = if h' : (k == a) = true then ⟨k, v⟩ else getEntry a l ⋯

theorem Std.Internal.List.beq_of_getEntry_eq {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {p : (a : α) × β a} {k : α} {h₁ : containsKey k l = true} (h : getEntry k l h₁ = p) : (p.fst == k) = true

theorem Std.Internal.List.getValueCast_insertEntry_self {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : getValueCast k (insertEntry k v l) ⋯ = v

theorem Std.Internal.List.getValue_insertEntry {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {k a : α} {v : β} {h : containsKey a (insertEntry k v l) = true} : getValue a (insertEntry k v l) h = if h' : (k == a) = true then v else getValue a l ⋯

theorem Std.Internal.List.getValue_insertEntry_self {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l : List ((_ : α) × β)} {k : α} {v : β} : getValue k (insertEntry k v l) ⋯ = v

theorem Std.Internal.List.getKey_insertEntry {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} {h : containsKey a (insertEntry k v l) = true} : getKey a (insertEntry k v l) h = if h' : (k == a) = true then k else getKey a l ⋯

theorem Std.Internal.List.getKey_insertEntry_self {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : getKey k (insertEntry k v l) ⋯ = k

def Std.Internal.List.insertEntryIfNew {α : Type u} {β : α → Type v} [BEq α] (k : α) (v : β k) (l : List ((a : α) × β a)) : List ((a : α) × β a)

Internal implementation detail of the hash map

theorem Std.Internal.List.insertEntryIfNew_of_containsKey {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} (h : containsKey k l = true) : insertEntryIfNew k v l = l

theorem Std.Internal.List.insertEntryIfNew_of_containsKey_eq_false {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} (h : containsKey k l = false) : insertEntryIfNew k v l = ⟨k, v⟩ :: l

theorem Std.Internal.List.DistinctKeys.insertEntryIfNew {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {k : α} {v : β k} {l : List ((a : α) × β a)} (h : DistinctKeys l) : DistinctKeys (List.insertEntryIfNew k v l)

@[simp]

theorem Std.Internal.List.isEmpty_insertEntryIfNew {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : (insertEntryIfNew k v l).isEmpty = false

theorem Std.Internal.List.getEntry?_insertEntryIfNew {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} : getEntry? a (insertEntryIfNew k v l) = if (k == a && !containsKey k l) = true then some ⟨k, v⟩ else getEntry? a l

theorem Std.Internal.List.getValueCast?_insertEntryIfNew {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} : getValueCast? a (insertEntryIfNew k v l) = if h : (k == a) = true ∧ containsKey k l = false then some (cast ⋯ v) else getValueCast? a l

theorem Std.Internal.List.getValue?_insertEntryIfNew {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {k a : α} {v : β} : getValue? a (insertEntryIfNew k v l) = if (k == a) = true ∧ containsKey k l = false then some v else getValue? a l

theorem Std.Internal.List.containsKey_insertEntryIfNew {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} : containsKey a (insertEntryIfNew k v l) = (k == a || containsKey a l)

theorem Std.Internal.List.containsKey_insertEntryIfNew_self {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : containsKey k (insertEntryIfNew k v l) = true

theorem Std.Internal.List.containsKey_of_containsKey_insertEntryIfNew {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} (h₁ : containsKey a (insertEntryIfNew k v l) = true) (h₂ : (k == a) = false) : containsKey a l = true

theorem Std.Internal.List.containsKey_of_containsKey_insertEntryIfNew' {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} (h₁ : containsKey a (insertEntryIfNew k v l) = true) (h₂ : ¬((k == a) = true ∧ containsKey k l = false)) : containsKey a l = true

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

theorem Std.Internal.List.getValueCast_insertEntryIfNew {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} {h : containsKey a (insertEntryIfNew k v l) = true} : getValueCast a (insertEntryIfNew k v l) h = if h' : (k == a) = true ∧ containsKey k l = false then cast ⋯ v else getValueCast a l ⋯

theorem Std.Internal.List.getValue_insertEntryIfNew {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {k a : α} {v : β} {h : containsKey a (insertEntryIfNew k v l) = true} : getValue a (insertEntryIfNew k v l) h = if h' : (k == a) = true ∧ containsKey k l = false then v else getValue a l ⋯

theorem Std.Internal.List.getValueCast!_insertEntryIfNew {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} [Inhabited (β a)] : getValueCast! a (insertEntryIfNew k v l) = if h : (k == a) = true ∧ containsKey k l = false then cast ⋯ v else getValueCast! a l

theorem Std.Internal.List.getValue!_insertEntryIfNew {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] [Inhabited β] {l : List ((_ : α) × β)} {k a : α} {v : β} : getValue! a (insertEntryIfNew k v l) = if (k == a) = true ∧ containsKey k l = false then v else getValue! a l

theorem Std.Internal.List.getValueCastD_insertEntryIfNew {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} {fallback : β a} : getValueCastD a (insertEntryIfNew k v l) fallback = if h : (k == a) = true ∧ containsKey k l = false then cast ⋯ v else getValueCastD a l fallback

theorem Std.Internal.List.getValueD_insertEntryIfNew {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {k a : α} {fallback v : β} : getValueD a (insertEntryIfNew k v l) fallback = if (k == a) = true ∧ containsKey k l = false then v else getValueD a l fallback

theorem Std.Internal.List.getKey?_insertEntryIfNew {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} : getKey? a (insertEntryIfNew k v l) = if (k == a) = true ∧ containsKey k l = false then some k else getKey? a l

theorem Std.Internal.List.getKey_insertEntryIfNew {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a : α} {v : β k} {h : containsKey a (insertEntryIfNew k v l) = true} : getKey a (insertEntryIfNew k v l) h = if h' : (k == a) = true ∧ containsKey k l = false then k else getKey a l ⋯

theorem Std.Internal.List.getKey!_insertEntryIfNew {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] [Inhabited α] {l : List ((a : α) × β a)} {k a : α} {v : β k} : getKey! a (insertEntryIfNew k v l) = if (k == a) = true ∧ containsKey k l = false then k else getKey! a l

theorem Std.Internal.List.getKeyD_insertEntryIfNew {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a fallback : α} {v : β k} : getKeyD a (insertEntryIfNew k v l) fallback = if (k == a) = true ∧ containsKey k l = false then k else getKeyD a l fallback

theorem Std.Internal.List.length_insertEntryIfNew {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : (insertEntryIfNew k v l).length = if containsKey k l = true then l.length else l.length + 1

theorem Std.Internal.List.length_le_length_insertEntryIfNew {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : l.length ≤ (insertEntryIfNew k v l).length

theorem Std.Internal.List.length_insertEntryIfNew_le {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} : (insertEntryIfNew k v l).length ≤ l.length + 1

@[simp]

theorem Std.Internal.List.keys_eraseKey {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k : α} : keys (eraseKey k l) = (keys l).erase k

theorem Std.Internal.List.DistinctKeys.eraseKey {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k : α} : DistinctKeys l → DistinctKeys (List.eraseKey k l)

theorem Std.Internal.List.getEntry?_eraseKey_self {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k : α} (h : DistinctKeys l) : getEntry? k (eraseKey k l) = none

theorem Std.Internal.List.getEntry?_eraseKey_of_beq {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a : α} (hl : DistinctKeys l) (hka : (k == a) = true) : getEntry? a (eraseKey k l) = none

theorem Std.Internal.List.getEntry?_eraseKey_of_false {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a : α} (hka : (k == a) = false) : getEntry? a (eraseKey k l) = getEntry? a l

theorem Std.Internal.List.getEntry?_eraseKey {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a : α} (hl : DistinctKeys l) : getEntry? a (eraseKey k l) = if (k == a) = true then none else getEntry? a l

theorem Std.Internal.List.keys_filterMap' {α : Type u} {β : α → Type v} {γ : α → Type w} {l : List ((a : α) × β a)} {f : (a : α) → β a → Option (γ a)} : keys (List.filterMap (fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) = keys (List.filter (fun (p : (a : α) × β a) => (f p.fst p.snd).isSome) l)

theorem Std.Internal.List.mem_keys_iff_contains {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k : α} : k ∈ keys l ↔ containsKey k l = true

theorem Std.Internal.List.keys_filter {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {f : (a : α) → β a → Bool} (hl : DistinctKeys l) : keys (List.filter (fun (x : (a : α) × β a) => f x.fst x.snd) l) = (List.filter (fun (x : { x : α // x ∈ keys l }) => f x.val (getValueCast x.val l ⋯)) (keys l).attach).unattach

theorem Std.Internal.List.Const.keys_filter {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {l : List ((_ : α) × β)} {f : α → β → Bool} (hl : DistinctKeys l) : keys (List.filter (fun (x : (_ : α) × β) => f x.fst x.snd) l) = (List.filter (fun (x : { x : α // x ∈ keys l }) => f x.val (getValue x.val l ⋯)) (keys l).attach).unattach

@[simp]

theorem Std.Internal.List.keys_map {α : Type u} {β : α → Type v} {γ : α → Type w} {l : List ((a : α) × β a)} {f : (a : α) → β a → γ a} : keys (List.map (fun (p : (a : α) × β a) => ⟨p.fst, f p.fst p.snd⟩) l) = keys l

theorem Std.Internal.List.DistinctKeys.filterMap {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {f : (a : α) → β a → Option (γ a)} : DistinctKeys l → DistinctKeys (List.filterMap (fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)) l)

theorem Std.Internal.List.DistinctKeys.map {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] {l : List ((a : α) × β a)} {f : (a : α) → β a → γ a} (h : DistinctKeys l) : DistinctKeys (List.map (fun (p : (a : α) × β a) => ⟨p.fst, f p.fst p.snd⟩) l)

theorem Std.Internal.List.DistinctKeys.filter {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {f : (a : α) → β a → Bool} (h : DistinctKeys l) : DistinctKeys (List.filter (fun (p : (a : α) × β a) => f p.fst p.snd) l)

theorem Std.Internal.List.getValue?_eraseKey_self {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {k : α} (h : DistinctKeys l) : getValue? k (eraseKey k l) = none

theorem Std.Internal.List.getValue?_eraseKey_of_beq {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {k a : α} (hl : DistinctKeys l) (hka : (k == a) = true) : getValue? a (eraseKey k l) = none

theorem Std.Internal.List.getValue?_eraseKey_of_false {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {k a : α} (hka : (k == a) = false) : getValue? a (eraseKey k l) = getValue? a l

theorem Std.Internal.List.getValue?_eraseKey {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {k a : α} (hl : DistinctKeys l) : getValue? a (eraseKey k l) = if (k == a) = true then none else getValue? a l

theorem Std.Internal.List.getKey?_eraseKey {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a : α} (hl : DistinctKeys l) : getKey? a (eraseKey k l) = if (k == a) = true then none else getKey? a l

theorem Std.Internal.List.getKey?_eraseKey_self {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : getKey? k (eraseKey k l) = none

theorem Std.Internal.List.getKey!_eraseKey {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] [Inhabited α] {l : List ((a : α) × β a)} {k a : α} (hl : DistinctKeys l) : getKey! a (eraseKey k l) = if (k == a) = true then default else getKey! a l

theorem Std.Internal.List.getKey!_eraseKey_self {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] [Inhabited α] {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : getKey! k (eraseKey k l) = default

theorem Std.Internal.List.getKeyD_eraseKey {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a fallback : α} (hl : DistinctKeys l) : getKeyD a (eraseKey k l) fallback = if (k == a) = true then fallback else getKeyD a l fallback

theorem Std.Internal.List.getKeyD_eraseKey_self {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k fallback : α} (hl : DistinctKeys l) : getKeyD k (eraseKey k l) fallback = fallback

theorem Std.Internal.List.containsKey_eraseKey_self {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k : α} (h : DistinctKeys l) : containsKey k (eraseKey k l) = false

theorem Std.Internal.List.containsKey_eraseKey_of_beq {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a : α} (hl : DistinctKeys l) (hka : (a == k) = true) : containsKey a (eraseKey k l) = false

theorem Std.Internal.List.containsKey_eraseKey_of_false {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a : α} (hka : (k == a) = false) : containsKey a (eraseKey k l) = containsKey a l

theorem Std.Internal.List.containsKey_eraseKey {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a : α} (hl : DistinctKeys l) : containsKey a (eraseKey k l) = (!k == a && containsKey a l)

theorem Std.Internal.List.isEmpty_eq_isEmpty_eraseKey_and_not_containsKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (k : α) : l.isEmpty = ((eraseKey k l).isEmpty && !containsKey k l)

theorem Std.Internal.List.isEmpty_eq_false_of_isEmpty_eraseKey_eq_false {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} (he : (eraseKey k l).isEmpty = false) : l.isEmpty = false

theorem Std.Internal.List.getValueCast?_eraseKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k a : α} (hl : DistinctKeys l) : getValueCast? a (eraseKey k l) = if (k == a) = true then none else getValueCast? a l

theorem Std.Internal.List.getValueCast?_eraseKey_self {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : getValueCast? k (eraseKey k l) = none

theorem Std.Internal.List.getValueCast!_eraseKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k a : α} [Inhabited (β a)] (hl : DistinctKeys l) : getValueCast! a (eraseKey k l) = if (k == a) = true then default else getValueCast! a l

theorem Std.Internal.List.getValueCast!_eraseKey_self {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k : α} [Inhabited (β k)] (hl : DistinctKeys l) : getValueCast! k (eraseKey k l) = default

theorem Std.Internal.List.getValueCastD_eraseKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k a : α} {fallback : β a} (hl : DistinctKeys l) : getValueCastD a (eraseKey k l) fallback = if (k == a) = true then fallback else getValueCastD a l fallback

theorem Std.Internal.List.getValueCastD_eraseKey_self {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k : α} {fallback : β k} (hl : DistinctKeys l) : getValueCastD k (eraseKey k l) fallback = fallback

theorem Std.Internal.List.getValue!_eraseKey {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] [Inhabited β] {l : List ((_ : α) × β)} {k a : α} (hl : DistinctKeys l) : getValue! a (eraseKey k l) = if (k == a) = true then default else getValue! a l

theorem Std.Internal.List.getValue!_eraseKey_self {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] [Inhabited β] {l : List ((_ : α) × β)} {k : α} (hl : DistinctKeys l) : getValue! k (eraseKey k l) = default

theorem Std.Internal.List.getValueD_eraseKey {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {k a : α} {fallback : β} (hl : DistinctKeys l) : getValueD a (eraseKey k l) fallback = if (k == a) = true then fallback else getValueD a l fallback

theorem Std.Internal.List.getValueD_eraseKey_self {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {k : α} {fallback : β} (hl : DistinctKeys l) : getValueD k (eraseKey k l) fallback = fallback

theorem Std.Internal.List.containsKey_of_containsKey_eraseKey {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a : α} (hl : DistinctKeys l) : containsKey a (eraseKey k l) = true → containsKey a l = true

theorem Std.Internal.List.getValueCast_eraseKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k a : α} {h : containsKey a (eraseKey k l) = true} (hl : DistinctKeys l) : getValueCast a (eraseKey k l) h = getValueCast a l ⋯

theorem Std.Internal.List.getValue_eraseKey {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {k a : α} {h : containsKey a (eraseKey k l) = true} (hl : DistinctKeys l) : getValue a (eraseKey k l) h = getValue a l ⋯

theorem Std.Internal.List.getKey_eraseKey {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} {k a : α} {h : containsKey a (eraseKey k l) = true} (hl : DistinctKeys l) : getKey a (eraseKey k l) h = getKey a l ⋯

theorem Std.Internal.List.getEntry?_of_perm {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l l' : List ((a : α) × β a)} {a : α} (hl : DistinctKeys l) (h : l.Perm l') : getEntry? a l = getEntry? a l'

theorem Std.Internal.List.getEntryD_of_perm {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l l' : List ((a : α) × β a)} {a : α} {fallback : (a : α) × β a} (hl : DistinctKeys l) (h : l.Perm l') : getEntryD a fallback l = getEntryD a fallback l'

theorem Std.Internal.List.getEntry!_of_perm {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l l' : List ((a : α) × β a)} {a : α} [Inhabited ((a : α) × β a)] (hl : DistinctKeys l) (h : l.Perm l') : getEntry! a l = getEntry! a l'

theorem Std.Internal.List.containsKey_of_perm {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l l' : List ((a : α) × β a)} {k : α} (h : l.Perm l') : containsKey k l = containsKey k l'

theorem Std.Internal.List.getValueCast?_of_perm {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l l' : List ((a : α) × β a)} {a : α} (hl : DistinctKeys l) (h : l.Perm l') : getValueCast? a l = getValueCast? a l'

theorem Std.Internal.List.getValueCast_of_perm {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l l' : List ((a : α) × β a)} {a : α} {h' : containsKey a l = true} (hl : DistinctKeys l) (h : l.Perm l') : getValueCast a l h' = getValueCast a l' ⋯

theorem Std.Internal.List.getEntry_of_perm {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l l' : List ((a : α) × β a)} {a : α} {h' : containsKey a l = true} (hl : DistinctKeys l) (h : l.Perm l') : getEntry a l h' = getEntry a l' ⋯

theorem Std.Internal.List.getValueCast!_of_perm {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l l' : List ((a : α) × β a)} {a : α} [Inhabited (β a)] (hl : DistinctKeys l) (h : l.Perm l') : getValueCast! a l = getValueCast! a l'

theorem Std.Internal.List.getValueCastD_of_perm {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l l' : List ((a : α) × β a)} {a : α} {fallback : β a} (hl : DistinctKeys l) (h : l.Perm l') : getValueCastD a l fallback = getValueCastD a l' fallback

theorem Std.Internal.List.getValue?_of_perm {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l l' : List ((_ : α) × β)} {a : α} (hl : DistinctKeys l) (h : l.Perm l') : getValue? a l = getValue? a l'

theorem Std.Internal.List.getValue_of_perm {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l l' : List ((_ : α) × β)} {a : α} {h' : containsKey a l = true} (hl : DistinctKeys l) (h : l.Perm l') : getValue a l h' = getValue a l' ⋯

theorem Std.Internal.List.getValue!_of_perm {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] [Inhabited β] {l l' : List ((_ : α) × β)} {a : α} (hl : DistinctKeys l) (h : l.Perm l') : getValue! a l = getValue! a l'

theorem Std.Internal.List.getValueD_of_perm {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l l' : List ((_ : α) × β)} {a : α} {fallback : β} (hl : DistinctKeys l) (h : l.Perm l') : getValueD a l fallback = getValueD a l' fallback

theorem Std.Internal.List.getKey?_of_perm {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l l' : List ((a : α) × β a)} {a : α} (hl : DistinctKeys l) (h : l.Perm l') : getKey? a l = getKey? a l'

theorem Std.Internal.List.getKey_of_perm {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l l' : List ((a : α) × β a)} {a : α} {h' : containsKey a l = true} (hl : DistinctKeys l) (h : l.Perm l') : getKey a l h' = getKey a l' ⋯

theorem Std.Internal.List.getKey!_of_perm {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] [Inhabited α] {l l' : List ((a : α) × β a)} {a : α} (hl : DistinctKeys l) (h : l.Perm l') : getKey! a l = getKey! a l'

theorem Std.Internal.List.getKeyD_of_perm {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l l' : List ((a : α) × β a)} {a fallback : α} (hl : DistinctKeys l) (h : l.Perm l') : getKeyD a l fallback = getKeyD a l' fallback

theorem Std.Internal.List.perm_cons_getEntry {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} {a : α} (h : containsKey a l = true) : ∃ (l' : List ((a : α) × β a)), l.Perm (getEntry a l h :: l')

theorem Std.Internal.List.getEntry?_ext {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l l' : List ((a : α) × β a)} (hl : DistinctKeys l) (hl' : DistinctKeys l') (h : ∀ (a : α), getEntry? a l = getEntry? a l') : l.Perm l'

theorem Std.Internal.List.getValueCast?_ext {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l l' : List ((a : α) × β a)} (hl : DistinctKeys l) (hl' : DistinctKeys l') (h : ∀ (a : α), getValueCast? a l = getValueCast? a l') : l.Perm l'

theorem Std.Internal.List.getKey_getValue?_ext {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {l l' : List ((_ : α) × β)} (hl : DistinctKeys l) (hl' : DistinctKeys l') (hk : ∀ (a : α) (h : containsKey a l = true) (h' : containsKey a l' = true), getKey a l h = getKey a l' h') (hv : ∀ (a : α), getValue? a l = getValue? a l') : l.Perm l'

theorem Std.Internal.List.getKey?_getValue?_ext {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {l l' : List ((_ : α) × β)} (hl : DistinctKeys l) (hl' : DistinctKeys l') (hk : ∀ (a : α), getKey? a l = getKey? a l') (hv : ∀ (a : α), getValue? a l = getValue? a l') : l.Perm l'

theorem Std.Internal.List.getKey?_ext {α : Type u} [BEq α] [EquivBEq α] {l l' : List ((_ : α) × Unit)} (hl : DistinctKeys l) (hl' : DistinctKeys l') (h : ∀ (a : α), getKey? a l = getKey? a l') : l.Perm l'

theorem Std.Internal.List.containsKey_ext {α : Type u} [BEq α] [LawfulBEq α] {l l' : List ((_ : α) × Unit)} (hl : DistinctKeys l) (hl' : DistinctKeys l') (h : ∀ (a : α), containsKey a l = containsKey a l') : l.Perm l'

theorem Std.Internal.List.replaceEntry_of_perm {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l l' : List ((a : α) × β a)} {k : α} {v : β k} (hl : DistinctKeys l) (h : l.Perm l') : (replaceEntry k v l).Perm (replaceEntry k v l')

theorem Std.Internal.List.insertEntry_of_perm {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l l' : List ((a : α) × β a)} {k : α} {v : β k} (hl : DistinctKeys l) (h : l.Perm l') : (insertEntry k v l).Perm (insertEntry k v l')

theorem Std.Internal.List.insertEntryIfNew_of_perm {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l l' : List ((a : α) × β a)} {k : α} {v : β k} (hl : DistinctKeys l) (h : l.Perm l') : (insertEntryIfNew k v l).Perm (insertEntryIfNew k v l')

theorem Std.Internal.List.eraseKey_of_perm {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l l' : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) (h : l.Perm l') : (eraseKey k l).Perm (eraseKey k l')

@[simp]

theorem Std.Internal.List.getEntry?_append {α : Type u} {β : α → Type v} [BEq α] {l l' : List ((a : α) × β a)} {a : α} : getEntry? a (l ++ l') = (getEntry? a l).or (getEntry? a l')

theorem Std.Internal.List.getEntry?_append_of_containsKey_eq_false {α : Type u} {β : α → Type v} [BEq α] {l l' : List ((a : α) × β a)} {a : α} (h : containsKey a l' = false) : getEntry? a (l ++ l') = getEntry? a l

theorem Std.Internal.List.getEntryD_append_of_containsKey_eq_false {α : Type u} {β : α → Type v} [BEq α] {l l' : List ((a : α) × β a)} {a : α} {fallback : (a : α) × β a} (h : containsKey a l' = false) : getEntryD a fallback (l ++ l') = getEntryD a fallback l

theorem Std.Internal.List.getEntry!_append_of_containsKey_eq_false {α : Type u} {β : α → Type v} [BEq α] {l l' : List ((a : α) × β a)} {a : α} [Inhabited ((a : α) × β a)] (h : containsKey a l' = false) : getEntry! a (l ++ l') = getEntry! a l

@[simp]

theorem Std.Internal.List.containsKey_append {α : Type u} {β : α → Type v} [BEq α] {l l' : List ((a : α) × β a)} {a : α} : containsKey a (l ++ l') = (containsKey a l || containsKey a l')

theorem Std.Internal.List.containsKey_flatMap_eq_false {α : Type u} {β : α → Type v} [BEq α] {γ : Type w} {l : List γ} {f : γ → List ((a : α) × β a)} {a : α} (h : ∀ (i : Nat) (h : i < l.length), containsKey a (f l[i]) = false) : containsKey a (List.flatMap f l) = false

theorem Std.Internal.List.containsKey_append_of_not_contains_right {α : Type u} {β : α → Type v} [BEq α] {l l' : List ((a : α) × β a)} {a : α} (hl' : containsKey a l' = false) : containsKey a (l ++ l') = containsKey a l

@[simp]

theorem Std.Internal.List.getValue?_append {α : Type u} {β : Type v} [BEq α] {l l' : List ((_ : α) × β)} {a : α} : getValue? a (l ++ l') = (getValue? a l).or (getValue? a l')

theorem Std.Internal.List.getValue?_append_of_containsKey_eq_false {α : Type u} {β : Type v} [BEq α] {l l' : List ((_ : α) × β)} {a : α} (h : containsKey a l' = false) : getValue? a (l ++ l') = getValue? a l

theorem Std.Internal.List.getValue_append_of_containsKey_eq_false {α : Type u} {β : Type v} [BEq α] {l l' : List ((_ : α) × β)} {a : α} {h' : containsKey a (l ++ l') = true} (h : containsKey a l' = false) : getValue a (l ++ l') h' = getValue a l ⋯

theorem Std.Internal.List.getValueCast?_append_of_containsKey_eq_false {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l l' : List ((a : α) × β a)} {a : α} (hl' : containsKey a l' = false) : getValueCast? a (l ++ l') = getValueCast? a l

theorem Std.Internal.List.getValueCast_append_of_containsKey_eq_false {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l l' : List ((a : α) × β a)} {a : α} {h : containsKey a (l ++ l') = true} (hl' : containsKey a l' = false) : getValueCast a (l ++ l') h = getValueCast a l ⋯

theorem Std.Internal.List.getEntry_append_of_containsKey_eq_false {α : Type u} {β : α → Type v} [BEq α] {l l' : List ((a : α) × β a)} {a : α} {h : containsKey a (l ++ l') = true} (hl' : containsKey a l' = false) : getEntry a (l ++ l') h = getEntry a l ⋯

theorem Std.Internal.List.getKey?_append_of_containsKey_eq_false {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l l' : List ((a : α) × β a)} {a : α} (hl' : containsKey a l' = false) : getKey? a (l ++ l') = getKey? a l

theorem Std.Internal.List.getKey_append_of_containsKey_eq_false {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l l' : List ((a : α) × β a)} {a : α} {h : containsKey a (l ++ l') = true} (hl' : containsKey a l' = false) : getKey a (l ++ l') h = getKey a l ⋯

theorem Std.Internal.List.replaceEntry_append_of_containsKey_left {α : Type u} {β : α → Type v} [BEq α] {l l' : List ((a : α) × β a)} {k : α} {v : β k} (h : containsKey k l = true) : replaceEntry k v (l ++ l') = replaceEntry k v l ++ l'

theorem Std.Internal.List.replaceEntry_append_of_containsKey_left_eq_false {α : Type u} {β : α → Type v} [BEq α] {l l' : List ((a : α) × β a)} {k : α} {v : β k} (h : containsKey k l = false) : replaceEntry k v (l ++ l') = l ++ replaceEntry k v l'

theorem Std.Internal.List.replaceEntry_append_of_containsKey_right_eq_false {α : Type u} {β : α → Type v} [BEq α] {l l' : List ((a : α) × β a)} {k : α} {v : β k} (h : containsKey k l' = false) : replaceEntry k v (l ++ l') = replaceEntry k v l ++ l'

theorem Std.Internal.List.insertEntry_append_of_not_contains_right {α : Type u} {β : α → Type v} [BEq α] {l l' : List ((a : α) × β a)} {k : α} {v : β k} (h' : containsKey k l' = false) : insertEntry k v (l ++ l') = insertEntry k v l ++ l'

theorem Std.Internal.List.eraseKey_append_of_containsKey_right_eq_false {α : Type u} {β : α → Type v} [BEq α] {l l' : List ((a : α) × β a)} {k : α} (h : containsKey k l' = false) : eraseKey k (l ++ l') = eraseKey k l ++ l'

theorem Std.Internal.List.mem_iff_getValueCast?_eq_some {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k : α} {v : β k} {l : List ((a : α) × β a)} (h : DistinctKeys l) : ⟨k, v⟩ ∈ l ↔ getValueCast? k l = some v

theorem Std.Internal.List.find?_eq_some_iff_getValueCast?_eq_some {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} (h : DistinctKeys l) : List.find? (fun (x : (a : α) × β a) => x.fst == k) l = some ⟨k, v⟩ ↔ getValueCast? k l = some v

theorem Std.Internal.List.find?_eq_none_iff_containsKey_eq_false {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k : α} : List.find? (fun (x : (a : α) × β a) => x.fst == k) l = none ↔ containsKey k l = false

theorem Std.Internal.List.pairwise_fst_eq_false {α : Type u} {β : α → Type v} [BEq α] {l : List ((a : α) × β a)} (h : DistinctKeys l) : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) l

theorem Std.Internal.List.map_fst_map_toProd_eq_keys {α : Type u} {β : Type v} {l : List ((_ : α) × β)} : List.map Prod.fst (List.map (fun (x : (_ : α) × β) => (x.fst, x.snd)) l) = keys l

theorem Std.Internal.List.find?_map_eq_none_iff_containsKey_eq_false {α : Type u} [BEq α] [PartialEquivBEq α] {β : Type v} {l : List ((_ : α) × β)} {k : α} : List.find? (fun (x : α × β) => x.fst == k) (List.map (fun (x : (_ : α) × β) => (x.fst, x.snd)) l) = none ↔ containsKey k l = false

theorem Std.Internal.List.mem_map_toProd_iff_mem {α : Type u} {β : Type v} {k : α} {v : β} {l : List ((_ : α) × β)} : ⟨k, v⟩ ∈ l ↔ (k, v) ∈ List.map (fun (x : (_ : α) × β) => (x.fst, x.snd)) l

theorem Std.Internal.List.mem_iff_getValue?_eq_some {α : Type u} [BEq α] [LawfulBEq α] {β : Type v} {k : α} {v : β} {l : List ((_ : α) × β)} (h : DistinctKeys l) : ⟨k, v⟩ ∈ l ↔ getValue? k l = some v

theorem Std.Internal.List.mem_map_toProd_iff_getValue?_eq_some {α : Type u} [BEq α] [LawfulBEq α] {β : Type v} {k : α} {v : β} {l : List ((_ : α) × β)} (h : DistinctKeys l) : (k, v) ∈ List.map (fun (x : (_ : α) × β) => (x.fst, x.snd)) l ↔ getValue? k l = some v

theorem Std.Internal.List.find?_map_toProd_eq_some_iff_getKey?_eq_some_and_getValue?_eq_some {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {k k' : α} {v : β} {l : List ((_ : α) × β)} : List.find? (fun (a : α × β) => a.fst == k) (List.map (fun (x : (_ : α) × β) => (x.fst, x.snd)) l) = some (k', v) ↔ getKey? k l = some k' ∧ getValue? k l = some v

theorem Std.Internal.List.mem_iff_getKey?_eq_some_and_getValue?_eq_some {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {k : α} {v : β} {l : List ((_ : α) × β)} (h : DistinctKeys l) : ⟨k, v⟩ ∈ l ↔ getKey? k l = some k ∧ getValue? k l = some v

theorem Std.Internal.List.getValue?_eq_some_iff_exists_beq_and_mem_toList {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l : List ((_ : α) × β)} {k : α} {v : β} (h : DistinctKeys l) : getValue? k l = some v ↔ ∃ (k' : α), (k == k') = true ∧ (k', v) ∈ List.map (fun (x : (_ : α) × β) => (x.fst, x.snd)) l

theorem Std.Internal.List.mem_map_toProd_iff_getKey?_eq_some_and_getValue?_eq_some {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {k : α} {v : β} {l : List ((_ : α) × β)} (h : DistinctKeys l) : (k, v) ∈ List.map (fun (x : (_ : α) × β) => (x.fst, x.snd)) l ↔ getKey? k l = some k ∧ getValue? k l = some v

theorem Std.Internal.List.pairwise_fst_eq_false_map_toProd {α : Type u} [BEq α] {β : Type v} {l : List ((_ : α) × β)} (h : DistinctKeys l) : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) (List.map (fun (x : (_ : α) × β) => (x.fst, x.snd)) l)

theorem Std.Internal.List.foldlM_eq_foldlM_toProd {α : Type u} {β : Type v} {δ : Type w} {m' : Type w → Type w'} [Monad m'] [LawfulMonad m'] {l : List ((_ : α) × β)} {f : δ → α → β → m' δ} {init : δ} : List.foldlM (fun (a : δ) (b : (_ : α) × β) => f a b.fst b.snd) init l = List.foldlM (fun (a : δ) (b : α × β) => f a b.fst b.snd) init (List.map (fun (x : (_ : α) × β) => (x.fst, x.snd)) l)

theorem Std.Internal.List.foldl_eq_foldl_toProd {α : Type u} {β : Type v} {δ : Type w} {l : List ((_ : α) × β)} {f : δ → α → β → δ} {init : δ} : List.foldl (fun (a : δ) (b : (_ : α) × β) => f a b.fst b.snd) init l = List.foldl (fun (a : δ) (b : α × β) => f a b.fst b.snd) init (List.map (fun (x : (_ : α) × β) => (x.fst, x.snd)) l)

theorem Std.Internal.List.foldrM_eq_foldrM_toProd {α : Type u} {β : Type v} {δ : Type w} {m' : Type w → Type w'} [Monad m'] [LawfulMonad m'] {l : List ((_ : α) × β)} {f : α → β → δ → m' δ} {init : δ} : List.foldrM (fun (a : (_ : α) × β) (b : δ) => f a.fst a.snd b) init l = List.foldrM (fun (a : α × β) (b : δ) => f a.fst a.snd b) init (List.map (fun (x : (_ : α) × β) => (x.fst, x.snd)) l)

theorem Std.Internal.List.foldrM_eq_foldrM_toProd' {α : Type u} {β : Type v} {δ : Type w} {m' : Type w → Type w'} [Monad m'] [LawfulMonad m'] {l : List ((_ : α) × β)} {f : δ → α → β → m' δ} {init : δ} : List.foldrM (fun (a : (_ : α) × β) (b : δ) => f b a.fst a.snd) init l = List.foldrM (fun (a : α × β) (b : δ) => f b a.fst a.snd) init (List.map (fun (x : (_ : α) × β) => (x.fst, x.snd)) l)

theorem Std.Internal.List.foldr_eq_foldr_toProd {α : Type u} {β : Type v} {δ : Type w} {l : List ((_ : α) × β)} {f : α → β → δ → δ} {init : δ} : List.foldr (fun (a : (_ : α) × β) (b : δ) => f a.fst a.snd b) init l = List.foldr (fun (a : α × β) (b : δ) => f a.fst a.snd b) init (List.map (fun (x : (_ : α) × β) => (x.fst, x.snd)) l)

theorem Std.Internal.List.foldr_eq_foldr_toProd' {α : Type u} {β : Type v} {δ : Type w} {l : List ((_ : α) × β)} {f : δ → α → β → δ} {init : δ} : List.foldr (fun (a : (_ : α) × β) (b : δ) => f b a.fst a.snd) init l = List.foldr (fun (a : α × β) (b : δ) => f b a.fst a.snd) init (List.map (fun (x : (_ : α) × β) => (x.fst, x.snd)) l)

theorem Std.Internal.List.forM_eq_forM_toProd {α : Type u} {β : Type v} {m' : Type w → Type w'} [Monad m'] [LawfulMonad m'] {l : List ((_ : α) × β)} {f : α → β → m' PUnit} : (forM l fun (a : (_ : α) × β) => f a.fst a.snd) = forM (List.map (fun (x : (_ : α) × β) => (x.fst, x.snd)) l) fun (a : α × β) => f a.fst a.snd

theorem Std.Internal.List.forIn_eq_forIn_toProd {α : Type u} {β : Type v} {δ : Type w} {m' : Type w → Type w'} [Monad m'] [LawfulMonad m'] {l : List ((_ : α) × β)} {f : α → β → δ → m' (ForInStep δ)} {init : δ} : (forIn l init fun (a : (_ : α) × β) (d : δ) => f a.fst a.snd d) = forIn (List.map (fun (x : (_ : α) × β) => (x.fst, x.snd)) l) init fun (a : α × β) (d : δ) => f a.fst a.snd d

theorem Std.Internal.List.foldlM_eq_foldlM_keys {α : Type u} {β : α → Type v} {δ : Type w} {m' : Type w → Type w'} [Monad m'] [LawfulMonad m'] {l : List ((a : α) × β a)} {f : δ → α → m' δ} {init : δ} : List.foldlM (fun (a : δ) (b : (a : α) × β a) => f a b.fst) init l = List.foldlM f init (keys l)

theorem Std.Internal.List.foldl_eq_foldl_keys {α : Type u} {β : α → Type v} {δ : Type w} {l : List ((a : α) × β a)} {f : δ → α → δ} {init : δ} : List.foldl (fun (a : δ) (b : (a : α) × β a) => f a b.fst) init l = List.foldl f init (keys l)

theorem Std.Internal.List.foldrM_eq_foldrM_keys {α : Type u} {β : α → Type v} {δ : Type w} {m' : Type w → Type w'} [Monad m'] [LawfulMonad m'] {l : List ((a : α) × β a)} {f : α → δ → m' δ} {init : δ} : List.foldrM (fun (a : (a : α) × β a) (b : δ) => f a.fst b) init l = List.foldrM f init (keys l)

theorem Std.Internal.List.foldrM_eq_foldrM_keys' {α : Type u} {β : α → Type v} {δ : Type w} {m' : Type w → Type w'} [Monad m'] [LawfulMonad m'] {l : List ((a : α) × β a)} {f : δ → α → m' δ} {init : δ} : List.foldrM (fun (a : (a : α) × β a) (b : δ) => f b a.fst) init l = List.foldrM (fun (a : α) (b : δ) => f b a) init (keys l)

theorem Std.Internal.List.foldr_eq_foldr_keys {α : Type u} {β : α → Type v} {δ : Type w} {l : List ((a : α) × β a)} {f : α → δ → δ} {init : δ} : List.foldr (fun (a : (a : α) × β a) (b : δ) => f a.fst b) init l = List.foldr f init (keys l)

theorem Std.Internal.List.foldr_eq_foldr_keys' {α : Type u} {β : α → Type v} {δ : Type w} {l : List ((a : α) × β a)} {f : δ → α → δ} {init : δ} : List.foldr (fun (a : (a : α) × β a) (b : δ) => f b a.fst) init l = List.foldr (fun (a : α) (b : δ) => f b a) init (keys l)

theorem Std.Internal.List.forM_eq_forM_keys {α : Type u} {β : α → Type v} {m' : Type w → Type w'} [Monad m'] [LawfulMonad m'] {l : List ((a : α) × β a)} {f : α → m' PUnit} : (l.forM fun (a : (a : α) × β a) => f a.fst) = (keys l).forM f

theorem Std.Internal.List.forIn_eq_forIn_keys {α : Type u} {β : α → Type v} {δ : Type w} {m' : Type w → Type w'} [Monad m'] [LawfulMonad m'] {f : α → δ → m' (ForInStep δ)} {init : δ} {l : List ((a : α) × β a)} : (forIn l init fun (a : (a : α) × β a) (d : δ) => f a.fst d) = forIn (keys l) init f

def Std.Internal.List.insertList {α : Type u} {β : α → Type v} [BEq α] (l toInsert : List ((a : α) × β a)) : List ((a : α) × β a)

Internal implementation detail of the hash map

theorem Std.Internal.List.DistinctKeys.insertList {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l₁ l₂ : List ((a : α) × β a)} (h : DistinctKeys l₁) : DistinctKeys (List.insertList l₁ l₂)

theorem Std.Internal.List.insertList_perm_of_perm_first {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l1 l2 toInsert : List ((a : α) × β a)} (h : l1.Perm l2) (distinct : DistinctKeys l1) : (insertList l1 toInsert).Perm (insertList l2 toInsert)

theorem Std.Internal.List.insertList_cons_perm {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {p : (a : α) × β a} (hl₁ : DistinctKeys l₁) (hl₂ : DistinctKeys (p :: l₂)) : (insertList l₁ (p :: l₂)).Perm (insertEntry p.fst p.snd (insertList l₁ l₂))

theorem Std.Internal.List.getEntry?_insertList {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} (distinct_l : DistinctKeys l) (distinct_toInsert : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) toInsert) (k : α) : getEntry? k (insertList l toInsert) = (getEntry? k toInsert).or (getEntry? k l)

theorem Std.Internal.List.getEntry?_insertList_of_contains_eq_false {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (not_contains : containsKey k toInsert = false) : getEntry? k (insertList l toInsert) = getEntry? k l

theorem Std.Internal.List.getEntry?_insertList_of_contains_eq_true {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (distinct_l : DistinctKeys l) (distinct_toInsert : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) toInsert) (contains : containsKey k toInsert = true) : getEntry? k (insertList l toInsert) = getEntry? k toInsert

theorem Std.Internal.List.containsKey_insertList {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l toInsert : List ((a : α) × β a)} {k : α} : containsKey k (insertList l toInsert) = (containsKey k l || (List.map Sigma.fst toInsert).contains k)

theorem Std.Internal.List.containsKey_insertList_disj_of_containsKey {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l toInsert : List ((a : α) × β a)} {k : α} : containsKey k (insertList l toInsert) = (containsKey k l || containsKey k toInsert)

theorem Std.Internal.List.containsKey_of_containsKey_insertList {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (h₁ : containsKey k (insertList l toInsert) = true) (h₂ : (List.map Sigma.fst toInsert).contains k = false) : containsKey k l = true

theorem Std.Internal.List.getValueCast?_insertList_of_contains_eq_false {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (not_contains : (List.map Sigma.fst toInsert).contains k = false) : getValueCast? k (insertList l toInsert) = getValueCast? k l

theorem Std.Internal.List.getValueCast?_insertList_of_mem {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l toInsert : List ((a : α) × β a)} (distinct_l : DistinctKeys l) {k k' : α} (k_beq : (k == k') = true) {v : β k} (distinct_toInsert : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) toInsert) (mem : ⟨k, v⟩ ∈ toInsert) : getValueCast? k' (insertList l toInsert) = some (cast ⋯ v)

theorem Std.Internal.List.getValueCast_insertList_of_mem {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l toInsert : List ((a : α) × β a)} (distinct_l : DistinctKeys l) {k k' : α} (k_beq : (k == k') = true) (v : β k) (distinct_toInsert : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) toInsert) (mem : ⟨k, v⟩ ∈ toInsert) {h : containsKey k' (insertList l toInsert) = true} : getValueCast k' (insertList l toInsert) h = cast ⋯ v

theorem Std.Internal.List.getValueCast!_insertList_of_contains_eq_false {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l toInsert : List ((a : α) × β a)} {k : α} [Inhabited (β k)] (not_contains : (List.map Sigma.fst toInsert).contains k = false) : getValueCast! k (insertList l toInsert) = getValueCast! k l

theorem Std.Internal.List.getValueCast!_insertList_of_mem {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l toInsert : List ((a : α) × β a)} (distinct_l : DistinctKeys l) {k k' : α} (k_beq : (k == k') = true) (v : β k) [Inhabited (β k')] (distinct_toInsert : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) toInsert) (mem : ⟨k, v⟩ ∈ toInsert) : getValueCast! k' (insertList l toInsert) = cast ⋯ v

theorem Std.Internal.List.getValueCastD_insertList_of_contains_eq_false {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l toInsert : List ((a : α) × β a)} {k : α} {fallback : β k} (not_mem : (List.map Sigma.fst toInsert).contains k = false) : getValueCastD k (insertList l toInsert) fallback = getValueCastD k l fallback

theorem Std.Internal.List.getValueCastD_insertList_of_mem {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l toInsert : List ((a : α) × β a)} (distinct_l : DistinctKeys l) {k k' : α} (k_beq : (k == k') = true) {v : β k} {fallback : β k'} (distinct_toInsert : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) toInsert) (mem : ⟨k, v⟩ ∈ toInsert) : getValueCastD k' (insertList l toInsert) fallback = cast ⋯ v

theorem Std.Internal.List.getKey?_insertList_of_contains_eq_false {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (not_contains : (List.map Sigma.fst toInsert).contains k = false) : getKey? k (insertList l toInsert) = getKey? k l

theorem Std.Internal.List.getKey?_insertList_of_mem {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} (distinct_l : DistinctKeys l) {k k' : α} (k_beq : (k == k') = true) (distinct_toInsert : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) toInsert) (mem : k ∈ List.map Sigma.fst toInsert) : getKey? k' (insertList l toInsert) = some k

theorem Std.Internal.List.getKey_insertList_of_contains_eq_false {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (not_contains : (List.map Sigma.fst toInsert).contains k = false) {h : containsKey k (insertList l toInsert) = true} : getKey k (insertList l toInsert) h = getKey k l ⋯

theorem Std.Internal.List.getKey_insertList_of_mem {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} (distinct_l : DistinctKeys l) {k k' : α} (k_beq : (k == k') = true) (distinct_toInsert : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) toInsert) (mem : k ∈ List.map Sigma.fst toInsert) {h : containsKey k' (insertList l toInsert) = true} : getKey k' (insertList l toInsert) h = k

theorem Std.Internal.List.getKey!_insertList_of_contains_eq_false {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] [Inhabited α] {l toInsert : List ((a : α) × β a)} {k : α} (contains_false : (List.map Sigma.fst toInsert).contains k = false) : getKey! k (insertList l toInsert) = getKey! k l

theorem Std.Internal.List.getKey!_insertList_of_mem {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] [Inhabited α] {l toInsert : List ((a : α) × β a)} (distinct_l : DistinctKeys l) {k k' : α} (k_beq : (k == k') = true) (distinct_toInsert : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) toInsert) (mem : k ∈ List.map Sigma.fst toInsert) : getKey! k' (insertList l toInsert) = k

theorem Std.Internal.List.getKeyD_insertList_of_contains_eq_false {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} {k fallback : α} (not_contains : (List.map Sigma.fst toInsert).contains k = false) : getKeyD k (insertList l toInsert) fallback = getKeyD k l fallback

theorem Std.Internal.List.getKeyD_insertList_of_mem {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} (distinct_l : DistinctKeys l) {k k' fallback : α} (k_beq : (k == k') = true) (distinct_toInsert : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) toInsert) (mem : k ∈ List.map Sigma.fst toInsert) : getKeyD k' (insertList l toInsert) fallback = k

theorem Std.Internal.List.perm_insertList {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} (distinct_l : DistinctKeys l) (distinct_toInsert : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) toInsert) (distinct_both : ∀ (a : α), containsKey a l = true → (List.map Sigma.fst toInsert).contains a = false) : (insertList l toInsert).Perm (l ++ toInsert)

theorem Std.Internal.List.length_insertList {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} (distinct_l : DistinctKeys l) (distinct_toInsert : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) toInsert) (distinct_both : ∀ (a : α), containsKey a l = true → (List.map Sigma.fst toInsert).contains a = false) : (insertList l toInsert).length = l.length + toInsert.length

theorem Std.Internal.List.length_le_length_insertList {α : Type u} {β : α → Type v} [BEq α] {l toInsert : List ((a : α) × β a)} : l.length ≤ (insertList l toInsert).length

theorem Std.Internal.List.length_insertList_le {α : Type u} {β : α → Type v} [BEq α] {l toInsert : List ((a : α) × β a)} : (insertList l toInsert).length ≤ l.length + toInsert.length

theorem Std.Internal.List.isEmpty_insertList {α : Type u} {β : α → Type v} [BEq α] {l toInsert : List ((a : α) × β a)} : (insertList l toInsert).isEmpty = (l.isEmpty && toInsert.isEmpty)

def Std.Internal.List.insertListIfNew {α : Type u} {β : α → Type v} [BEq α] (l toInsert : List ((a : α) × β a)) : List ((a : α) × β a)

Internal implementation detail of the hash map

theorem Std.Internal.List.DistinctKeys.insertListIfNew {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l₁ l₂ : List ((a : α) × β a)} (h : DistinctKeys l₁) : DistinctKeys (List.insertListIfNew l₁ l₂)

theorem Std.Internal.List.insertListIfNew_perm_of_perm_first {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l1 l2 toInsert : List ((a : α) × β a)} (h : l1.Perm l2) (distinct : DistinctKeys l1) : (insertListIfNew l1 toInsert).Perm (insertListIfNew l2 toInsert)

theorem Std.Internal.List.containsKey_insertListIfNew {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l toInsert : List ((a : α) × β a)} {k : α} : containsKey k (insertListIfNew l toInsert) = (containsKey k l || (List.map Sigma.fst toInsert).contains k)

def Std.Internal.List.insertSmallerList {α : Type u} {β : α → Type v} [BEq α] (l₁ l₂ : List ((a : α) × β a)) : List ((a : α) × β a)

Internal implementation detail of the hash map

theorem Std.Internal.List.contains_insertList_iff {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} {k : α} : containsKey k (insertList l toInsert) = true ↔ containsKey k l = true ∨ containsKey k toInsert = true

theorem Std.Internal.List.contains_of_contains_insertList_of_contains_eq_false_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (h₁ : containsKey k (insertList l toInsert) = true) (h₂ : containsKey k toInsert = false) : containsKey k l = true

theorem Std.Internal.List.contains_of_contains_insertList_of_contains_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (h₁ : containsKey k (insertList l toInsert) = true) (h₂ : containsKey k l = false) : containsKey k toInsert = true

theorem Std.Internal.List.contains_insertList_of_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (contains : containsKey k l = true) : containsKey k (insertList l toInsert) = true

theorem Std.Internal.List.contains_insertList_of_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (contains : containsKey k toInsert = true) : containsKey k (insertList l toInsert) = true

theorem Std.Internal.List.DistinctKeys_impl_Pairwise_distinct {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} (distinct_l : DistinctKeys l) : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) l

theorem Std.Internal.List.insertList_perm_of_perm_second {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l1 l2 l : List ((a : α) × β a)} (h : l1.Perm l2) (distinct_l : DistinctKeys l) (distinct : DistinctKeys l1) : (insertList l l1).Perm (insertList l l2)

theorem Std.Internal.List.insertList_congr {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l1 l2 l3 l4 : List ((a : α) × β a)} (h₁ : l1.Perm l3) (h₂ : l2.Perm l4) (hd₁ : DistinctKeys l1) (hd₂ : DistinctKeys l2) (hd₃ : DistinctKeys l3) : (insertList l1 l2).Perm (insertList l3 l4)

theorem Std.Internal.List.getEntry?_insertList_of_contains_left_eq_false {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (distinct_l : DistinctKeys l) (distinct_toInsert : List.Pairwise (fun (a b : (a : α) × β a) => (a.fst == b.fst) = false) toInsert) (not_contains : containsKey k l = false) : getEntry? k (insertList l toInsert) = getEntry? k toInsert

theorem Std.Internal.List.getKey?_insertList {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) : getKey? k (insertList l toInsert) = (getKey? k toInsert).or (getKey? k l)

theorem Std.Internal.List.getKey_insertList_of_contains_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) {m : containsKey k (insertList l toInsert) = true} (contains : containsKey k toInsert = true) : getKey k (insertList l toInsert) m = getKey k toInsert contains

theorem Std.Internal.List.getKeyD_insertList_of_contains_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} {k fallback : α} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) : containsKey k toInsert = true → getKeyD k (insertList l toInsert) fallback = getKeyD k toInsert fallback

theorem Std.Internal.List.getKey!_insertList_of_contains_right_eq_false {α : Type u} {β : α → Type v} [Inhabited α] [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (not_contains : containsKey k toInsert = false) : getKey! k (insertList l toInsert) = getKey! k l

theorem Std.Internal.List.getKey?_insertList_of_contains_eq_false_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (not_contains : containsKey k toInsert = false) : getKey? k (insertList l toInsert) = getKey? k l

theorem Std.Internal.List.getKey?_insertList_of_contains_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) (not_contains : containsKey k l = false) : getKey? k (insertList l toInsert) = getKey? k toInsert

theorem Std.Internal.List.getKey_insertList_of_contains_left_of_contains_right_eq_false {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} {k : α} {m : containsKey k (insertList l toInsert) = true} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) (mem : containsKey k l = true) : (∃ (mem₂ : containsKey k toInsert = true), getKey k (insertList l toInsert) m = getKey k toInsert mem₂) ∨ getKey k (insertList l toInsert) m = getKey k l mem

theorem Std.Internal.List.getKeyD_insertList {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} {k fallback : α} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) : getKeyD k (insertList l toInsert) fallback = getKeyD k toInsert (getKeyD k l fallback)

theorem Std.Internal.List.getKeyD_insertList_of_contains_eq_false_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} {k fallback : α} (not_contains : containsKey k toInsert = false) : getKeyD k (insertList l toInsert) fallback = getKeyD k l fallback

theorem Std.Internal.List.getKeyD_insertList_of_contains_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} {k fallback : α} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) (not_contains : containsKey k l = false) : getKeyD k (insertList l toInsert) fallback = getKeyD k toInsert fallback

theorem Std.Internal.List.getKey!_insertList_of_contains_left_eq_false {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] [Inhabited α] {l toInsert : List ((a : α) × β a)} {k : α} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) : containsKey k l = false → getKey! k (insertList l toInsert) = getKey! k toInsert

theorem Std.Internal.List.getValueCast?_insertList_of_contains_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) : containsKey k l = false → getValueCast? k (insertList l toInsert) = getValueCast? k toInsert

theorem Std.Internal.List.getValueCast?_of_insertList {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) : getValueCast? k (insertList l toInsert) = (getValueCast? k toInsert).or (getValueCast? k l)

theorem Std.Internal.List.getValueCastD_of_insertList {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l toInsert : List ((a : α) × β a)} {k : α} {fallback : β k} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) : getValueCastD k (insertList l toInsert) fallback = getValueCastD k toInsert (getValueCastD k l fallback)

theorem Std.Internal.List.getValueCast_insertList_of_contains_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) (contains : containsKey k (insertList l toInsert) = true) (not_contains : containsKey k l = false) : getValueCast k (insertList l toInsert) contains = getValueCast k toInsert ⋯

theorem Std.Internal.List.getKey_insertList_of_contains_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) (contains : containsKey k (insertList l toInsert) = true) (not_contains : containsKey k l = false) : getKey k (insertList l toInsert) contains = getKey k toInsert ⋯

theorem Std.Internal.List.getValueCast_insertList_of_contains_right {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) (contains : containsKey k toInsert = true) {h : containsKey k (insertList l toInsert) = true} : getValueCast k (insertList l toInsert) h = getValueCast k toInsert contains

theorem Std.Internal.List.getValueCastD_insertList_of_contains_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l toInsert : List ((a : α) × β a)} {k : α} {fallback : β k} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) (not_contains : containsKey k l = false) : getValueCastD k (insertList l toInsert) fallback = getValueCastD k toInsert fallback

theorem Std.Internal.List.length_insertList_distinct {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) : (∀ (a : α), containsKey a l = true → containsKey a toInsert = false) → (insertList l toInsert).length = l.length + toInsert.length

theorem Std.Internal.List.getKey?_insertList_of_mem_of_not_mem {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (contains : containsKey k l = true) (not_contains : containsKey k toInsert = false) : getKey? k (insertList l toInsert) = some k

theorem Std.Internal.List.insertListIfNew_perm_insertList {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} (hd₁ : DistinctKeys l₁) (hd₂ : DistinctKeys l₂) : (insertListIfNew l₁ l₂).Perm (insertList l₂ l₁)

theorem Std.Internal.List.insertList_perm_insertSmallerList {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) : (insertList l toInsert).Perm (insertSmallerList l toInsert)

theorem Std.Internal.List.length_left_le_length_insertListIfNew {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} : l.length ≤ (insertListIfNew l toInsert).length

theorem Std.Internal.List.length_right_le_length_insertList {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) : toInsert.length ≤ (insertList l toInsert).length

theorem Std.Internal.List.insertList_insertEntry_right_equiv_insertEntry_insertList {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l toInsert : List ((a : α) × β a)} (p : (a : α) × β a) (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) : (insertList l (insertEntry p.fst p.snd toInsert)).Perm (insertEntry p.fst p.snd (insertList l toInsert))

theorem Std.Internal.List.length_le_length_of_containsKey {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} (dl₁ : DistinctKeys l₁) (dl₂ : DistinctKeys l₂) (h : ∀ (a : α), containsKey a l₁ = true → containsKey a l₂ = true) : l₁.length ≤ l₂.length

theorem Std.Internal.List.containsKey_of_length_eq {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} (dl₁ : DistinctKeys l₁) (dl₂ : DistinctKeys l₂) (hl : l₂.length = l₁.length) (hs : ∀ (a : α), containsKey a l₁ = true → containsKey a l₂ = true) (a : α) : containsKey a l₂ = true → containsKey a l₁ = true

def Std.Internal.List.Prod.toSigma {α : Type u} {β : Type v} (p : α × β) : (_ : α) × β

Internal implementation detail of the hash map

@[simp]

theorem Std.Internal.List.Prod.fst_comp_toSigma {α : Type u} {β : Type v} : Sigma.fst ∘ toSigma = Prod.fst

def Std.Internal.List.insertListConst {α : Type u} {β : Type v} [BEq α] (l : List ((_ : α) × β)) (toInsert : List (α × β)) : List ((_ : α) × β)

Internal implementation detail of the hash map

theorem Std.Internal.List.containsKey_insertListConst {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {toInsert : List (α × β)} {k : α} : containsKey k (insertListConst l toInsert) = (containsKey k l || (List.map Prod.fst toInsert).contains k)

theorem Std.Internal.List.containsKey_of_containsKey_insertListConst {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {toInsert : List (α × β)} {k : α} (h₁ : containsKey k (insertListConst l toInsert) = true) (h₂ : (List.map Prod.fst toInsert).contains k = false) : containsKey k l = true

theorem Std.Internal.List.getKey?_insertListConst_of_contains_eq_false {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l : List ((_ : α) × β)} {toInsert : List (α × β)} {k : α} (not_contains : (List.map Prod.fst toInsert).contains k = false) : getKey? k (insertListConst l toInsert) = getKey? k l

theorem Std.Internal.List.getKey?_insertListConst_of_mem {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l : List ((_ : α) × β)} {toInsert : List (α × β)} (distinct_l : DistinctKeys l) {k k' : α} (k_beq : (k == k') = true) (distinct_toInsert : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) toInsert) (mem : k ∈ List.map Prod.fst toInsert) : getKey? k' (insertListConst l toInsert) = some k

theorem Std.Internal.List.getKey_insertListConst_of_contains_eq_false {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l : List ((_ : α) × β)} {toInsert : List (α × β)} {k : α} (not_contains : (List.map Prod.fst toInsert).contains k = false) {h : containsKey k (insertListConst l toInsert) = true} : getKey k (insertListConst l toInsert) h = getKey k l ⋯

theorem Std.Internal.List.getKey_insertListConst_of_mem {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l : List ((_ : α) × β)} {toInsert : List (α × β)} (distinct_l : DistinctKeys l) {k k' : α} (k_beq : (k == k') = true) (distinct_toInsert : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) toInsert) (mem : k ∈ List.map Prod.fst toInsert) {h : containsKey k' (insertListConst l toInsert) = true} : getKey k' (insertListConst l toInsert) h = k

theorem Std.Internal.List.getKey!_insertListConst_of_contains_eq_false {α : Type u} {β : Type v} [BEq α] [EquivBEq α] [Inhabited α] {l : List ((_ : α) × β)} {toInsert : List (α × β)} {k : α} (not_contains : (List.map Prod.fst toInsert).contains k = false) : getKey! k (insertListConst l toInsert) = getKey! k l

theorem Std.Internal.List.getKey!_insertListConst_of_mem {α : Type u} {β : Type v} [BEq α] [EquivBEq α] [Inhabited α] {l : List ((_ : α) × β)} {toInsert : List (α × β)} (distinct_l : DistinctKeys l) {k k' : α} (k_beq : (k == k') = true) (distinct_toInsert : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) toInsert) (mem : k ∈ List.map Prod.fst toInsert) : getKey! k' (insertListConst l toInsert) = k

theorem Std.Internal.List.getKeyD_insertListConst_of_contains_eq_false {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l : List ((_ : α) × β)} {toInsert : List (α × β)} {k fallback : α} (not_contains : (List.map Prod.fst toInsert).contains k = false) : getKeyD k (insertListConst l toInsert) fallback = getKeyD k l fallback

theorem Std.Internal.List.getKeyD_insertListConst_of_mem {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l : List ((_ : α) × β)} {toInsert : List (α × β)} (distinct_l : DistinctKeys l) {k k' fallback : α} (k_beq : (k == k') = true) (distinct_toInsert : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) toInsert) (mem : k ∈ List.map Prod.fst toInsert) : getKeyD k' (insertListConst l toInsert) fallback = k

theorem Std.Internal.List.length_insertListConst {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l : List ((_ : α) × β)} {toInsert : List (α × β)} (distinct_l : DistinctKeys l) (distinct_toInsert : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) toInsert) (distinct_both : ∀ (a : α), containsKey a l = true → (List.map Prod.fst toInsert).contains a = false) : (insertListConst l toInsert).length = l.length + toInsert.length

theorem Std.Internal.List.length_le_length_insertListConst {α : Type u} {β : Type v} [BEq α] {l : List ((_ : α) × β)} {toInsert : List (α × β)} : l.length ≤ (insertListConst l toInsert).length

theorem Std.Internal.List.length_insertListConst_le {α : Type u} {β : Type v} [BEq α] {l : List ((_ : α) × β)} {toInsert : List (α × β)} : (insertListConst l toInsert).length ≤ l.length + toInsert.length

theorem Std.Internal.List.isEmpty_insertListConst {α : Type u} {β : Type v} [BEq α] {l : List ((_ : α) × β)} {toInsert : List (α × β)} : (insertListConst l toInsert).isEmpty = (l.isEmpty && toInsert.isEmpty)

theorem Std.Internal.List.getValue?_insertListConst_of_contains_eq_false {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {toInsert : List (α × β)} {k : α} (not_contains : (List.map Prod.fst toInsert).contains k = false) : getValue? k (insertListConst l toInsert) = getValue? k l

theorem Std.Internal.List.getValue?_insertList_of_contains_eq_false_right {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l toInsert : List ((_ : α) × β)} {k : α} (not_contains : containsKey k toInsert = false) : getValue? k (insertList l toInsert) = getValue? k l

theorem Std.Internal.List.getValueD_insertList_of_contains_eq_false_right {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l toInsert : List ((_ : α) × β)} {k : α} {fallback : β} (not_contains : containsKey k toInsert = false) : getValueD k (insertList l toInsert) fallback = getValueD k l fallback

theorem Std.Internal.List.getValue_insertList_of_contains_eq_false_right {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l toInsert : List ((_ : α) × β)} {k : α} (not_contains : containsKey k toInsert = false) {p1 : containsKey k (insertList l toInsert) = true} {p2 : containsKey k l = true} : getValue k (insertList l toInsert) p1 = getValue k l p2

theorem Std.Internal.List.getValue?_insertList_of_contains_eq_false_left {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l toInsert : List ((_ : α) × β)} {k : α} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) : containsKey k l = false → getValue? k (insertList l toInsert) = getValue? k toInsert

theorem Std.Internal.List.getValueD_insertList_of_contains_eq_false_left {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l toInsert : List ((_ : α) × β)} {k : α} {fallback : β} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) : containsKey k l = false → getValueD k (insertList l toInsert) fallback = getValueD k toInsert fallback

theorem Std.Internal.List.getValue?_insertList {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l toInsert : List ((_ : α) × β)} {k : α} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) : getValue? k (insertList l toInsert) = (getValue? k toInsert).or (getValue? k l)

theorem Std.Internal.List.getValueD_insertList {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l toInsert : List ((_ : α) × β)} {k : α} {fallback : β} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) : getValueD k (insertList l toInsert) fallback = getValueD k toInsert (getValueD k l fallback)

theorem Std.Internal.List.getValue_insertList_of_contains_eq_false_left {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l toInsert : List ((_ : α) × β)} {k : α} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) (contains : containsKey k (insertList l toInsert) = true) (not_contains : containsKey k l = false) : getValue k (insertList l toInsert) contains = getValue k toInsert ⋯

theorem Std.Internal.List.getValue_insertList_of_contains_right {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l toInsert : List ((_ : α) × β)} {k : α} (distinct_l : DistinctKeys l) (distinct_toInsert : DistinctKeys toInsert) (contains : containsKey k toInsert = true) {h : containsKey k (insertList l toInsert) = true} : getValue k (insertList l toInsert) h = getValue k toInsert contains

theorem Std.Internal.List.getValue?_insertListConst_of_mem {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l : List ((_ : α) × β)} {toInsert : List (α × β)} (distinct_l : DistinctKeys l) {k k' : α} (k_beq : (k == k') = true) {v : β} (distinct_toInsert : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) toInsert) (mem : (k, v) ∈ toInsert) : getValue? k' (insertListConst l toInsert) = some v

theorem Std.Internal.List.getValue_insertListConst_of_contains_eq_false {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {toInsert : List (α × β)} {k : α} {not_contains : (List.map Prod.fst toInsert).contains k = false} {h : containsKey k (insertListConst l toInsert) = true} : getValue k (insertListConst l toInsert) h = getValue k l ⋯

theorem Std.Internal.List.getValue_insertListConst_of_mem {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l : List ((_ : α) × β)} {toInsert : List (α × β)} (distinct_l : DistinctKeys l) {k k' : α} (k_beq : (k == k') = true) {v : β} (distinct_toInsert : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) toInsert) (mem : (k, v) ∈ toInsert) {h : containsKey k' (insertListConst l toInsert) = true} : getValue k' (insertListConst l toInsert) h = v

theorem Std.Internal.List.getValue!_insertListConst_of_contains_eq_false {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] [Inhabited β] {l : List ((_ : α) × β)} {toInsert : List (α × β)} {k : α} (not_contains : (List.map Prod.fst toInsert).contains k = false) : getValue! k (insertListConst l toInsert) = getValue! k l

theorem Std.Internal.List.getValue!_insertListConst_of_mem {α : Type u} {β : Type v} [BEq α] [EquivBEq α] [Inhabited β] {l : List ((_ : α) × β)} {toInsert : List (α × β)} {k k' : α} {v : β} (distinct_l : DistinctKeys l) (k_beq : (k == k') = true) (distinct_toInsert : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) toInsert) (mem : (k, v) ∈ toInsert) : getValue! k' (insertListConst l toInsert) = v

theorem Std.Internal.List.getValueD_insertListConst_of_contains_eq_false {α : Type u} {β : Type v} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × β)} {toInsert : List (α × β)} {k : α} {fallback : β} (not_contains : (List.map Prod.fst toInsert).contains k = false) : getValueD k (insertListConst l toInsert) fallback = getValueD k l fallback

theorem Std.Internal.List.getValueD_insertListConst_of_mem {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l : List ((_ : α) × β)} {toInsert : List (α × β)} {k k' : α} {v fallback : β} (distinct_l : DistinctKeys l) (k_beq : (k == k') = true) (distinct_toInsert : List.Pairwise (fun (a b : α × β) => (a.fst == b.fst) = false) toInsert) (mem : (k, v) ∈ toInsert) : getValueD k' (insertListConst l toInsert) fallback = v

def Std.Internal.List.insertListIfNewUnit {α : Type u} [BEq α] (l : List ((_ : α) × Unit)) (toInsert : List α) : List ((_ : α) × Unit)

Internal implementation detail of the hash map

theorem Std.Internal.List.insertListIfNewUnit_perm_of_perm_first {α : Type u} [BEq α] [EquivBEq α] {l1 l2 : List ((_ : α) × Unit)} {toInsert : List α} (h : l1.Perm l2) (distinct : DistinctKeys l1) : (insertListIfNewUnit l1 toInsert).Perm (insertListIfNewUnit l2 toInsert)

theorem Std.Internal.List.DistinctKeys.insertListIfNewUnit {α : Type u} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × Unit)} {toInsert : List α} (distinct : DistinctKeys l) : DistinctKeys (List.insertListIfNewUnit l toInsert)

theorem Std.Internal.List.getEntry?_insertListIfNewUnit {α : Type u} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × Unit)} {toInsert : List α} {k : α} : getEntry? k (insertListIfNewUnit l toInsert) = (getEntry? k l).or (getEntry? k (List.map (fun (x : α) => ⟨x, ()⟩) toInsert))

theorem Std.Internal.List.DistinctKeys.mapUnit {α : Type u} [BEq α] {l : List α} (distinct : List.Pairwise (fun (a b : α) => (a == b) = false) l) : DistinctKeys (List.map (fun (x : α) => ⟨x, ()⟩) l)

theorem Std.Internal.List.getEntry?_insertListIfNewUnit_of_contains_eq_false {α : Type u} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × Unit)} {toInsert : List α} {k : α} (not_contains : toInsert.contains k = false) : getEntry? k (insertListIfNewUnit l toInsert) = getEntry? k l

theorem Std.Internal.List.containsKey_insertListIfNewUnit {α : Type u} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × Unit)} {toInsert : List α} {k : α} : containsKey k (insertListIfNewUnit l toInsert) = (containsKey k l || toInsert.contains k)

theorem Std.Internal.List.containsKey_of_containsKey_insertListIfNewUnit {α : Type u} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × Unit)} {toInsert : List α} {k : α} (h₂ : toInsert.contains k = false) : containsKey k (insertListIfNewUnit l toInsert) = true → containsKey k l = true

theorem Std.Internal.List.getKey?_insertListIfNewUnit_of_contains_eq_false_of_contains_eq_false {α : Type u} [BEq α] [EquivBEq α] {l : List ((_ : α) × Unit)} {toInsert : List α} {k : α} (h' : containsKey k l = false) (h : toInsert.contains k = false) : getKey? k (insertListIfNewUnit l toInsert) = none

theorem Std.Internal.List.getKey?_insertListIfNewUnit_of_contains_eq_false_of_mem {α : Type u} [BEq α] [EquivBEq α] {l : List ((_ : α) × Unit)} {toInsert : List α} {k k' : α} (k_beq : (k == k') = true) (mem' : containsKey k l = false) (distinct : List.Pairwise (fun (a b : α) => (a == b) = false) toInsert) (mem : k ∈ toInsert) : getKey? k' (insertListIfNewUnit l toInsert) = some k

theorem Std.Internal.List.getKey?_insertListIfNewUnit_of_contains {α : Type u} [BEq α] [EquivBEq α] {l : List ((_ : α) × Unit)} {toInsert : List α} {k : α} (h : containsKey k l = true) : getKey? k (insertListIfNewUnit l toInsert) = getKey? k l

theorem Std.Internal.List.getKey_insertListIfNewUnit_of_contains_eq_false_of_mem {α : Type u} [BEq α] [EquivBEq α] {l : List ((_ : α) × Unit)} {toInsert : List α} {k k' : α} (k_beq : (k == k') = true) {h : containsKey k' (insertListIfNewUnit l toInsert) = true} (contains_eq_false : containsKey k l = false) (distinct : List.Pairwise (fun (a b : α) => (a == b) = false) toInsert) (mem : k ∈ toInsert) : getKey k' (insertListIfNewUnit l toInsert) h = k

theorem Std.Internal.List.getKey_insertListIfNewUnit_of_contains {α : Type u} [BEq α] [EquivBEq α] {l : List ((_ : α) × Unit)} {toInsert : List α} {k : α} (contains : containsKey k l = true) {h : containsKey k (insertListIfNewUnit l toInsert) = true} : getKey k (insertListIfNewUnit l toInsert) h = getKey k l contains

theorem Std.Internal.List.getKey!_insertListIfNewUnit_of_contains_eq_false_of_contains_eq_false {α : Type u} [BEq α] [EquivBEq α] [Inhabited α] {l : List ((_ : α) × Unit)} {toInsert : List α} {k : α} (contains_eq_false : containsKey k l = false) (contains_eq_false' : toInsert.contains k = false) : getKey! k (insertListIfNewUnit l toInsert) = default

theorem Std.Internal.List.getKey!_insertListIfNewUnit_of_contains_eq_false_of_mem {α : Type u} [BEq α] [EquivBEq α] [Inhabited α] {l : List ((_ : α) × Unit)} {toInsert : List α} {k k' : α} (k_beq : (k == k') = true) (h : containsKey k l = false) (distinct : List.Pairwise (fun (a b : α) => (a == b) = false) toInsert) (mem : k ∈ toInsert) : getKey! k' (insertListIfNewUnit l toInsert) = k

theorem Std.Internal.List.getKey!_insertListIfNewUnit_of_contains {α : Type u} [BEq α] [EquivBEq α] [Inhabited α] {l : List ((_ : α) × Unit)} {toInsert : List α} {k : α} (h : containsKey k l = true) : getKey! k (insertListIfNewUnit l toInsert) = getKey! k l

theorem Std.Internal.List.getKeyD_insertListIfNewUnit_of_contains_eq_false_of_contains_eq_false {α : Type u} [BEq α] [EquivBEq α] {l : List ((_ : α) × Unit)} {toInsert : List α} {k fallback : α} (contains_eq_false : containsKey k l = false) (contains_eq_false' : toInsert.contains k = false) : getKeyD k (insertListIfNewUnit l toInsert) fallback = fallback

theorem Std.Internal.List.getKeyD_insertListIfNewUnit_of_contains_eq_false_of_mem {α : Type u} [BEq α] [EquivBEq α] {l : List ((_ : α) × Unit)} {toInsert : List α} {k k' fallback : α} (k_beq : (k == k') = true) (h : containsKey k l = false) (distinct : List.Pairwise (fun (a b : α) => (a == b) = false) toInsert) (mem : k ∈ toInsert) : getKeyD k' (insertListIfNewUnit l toInsert) fallback = k

theorem Std.Internal.List.getKeyD_insertListIfNewUnit_of_contains {α : Type u} [BEq α] [EquivBEq α] {l : List ((_ : α) × Unit)} {toInsert : List α} {k fallback : α} (contains : containsKey k l = true) : getKeyD k (insertListIfNewUnit l toInsert) fallback = getKeyD k l fallback

theorem Std.Internal.List.length_insertListIfNewUnit {α : Type u} [BEq α] [EquivBEq α] {l : List ((_ : α) × Unit)} {toInsert : List α} (distinct_l : DistinctKeys l) (distinct_toInsert : List.Pairwise (fun (a b : α) => (a == b) = false) toInsert) (distinct_both : ∀ (a : α), containsKey a l = true → toInsert.contains a = false) : (insertListIfNewUnit l toInsert).length = l.length + toInsert.length

theorem Std.Internal.List.length_le_length_insertListIfNewUnit {α : Type u} [BEq α] [EquivBEq α] {l : List ((_ : α) × Unit)} {toInsert : List α} : l.length ≤ (insertListIfNewUnit l toInsert).length

theorem Std.Internal.List.length_insertListIfNewUnit_le {α : Type u} [BEq α] [EquivBEq α] {l : List ((_ : α) × Unit)} {toInsert : List α} : (insertListIfNewUnit l toInsert).length ≤ l.length + toInsert.length

theorem Std.Internal.List.isEmpty_insertListIfNewUnit {α : Type u} [BEq α] {l : List ((_ : α) × Unit)} {toInsert : List α} : (insertListIfNewUnit l toInsert).isEmpty = (l.isEmpty && toInsert.isEmpty)

theorem Std.Internal.List.getValue?_list_unit {α : Type u} [BEq α] {l : List ((_ : α) × Unit)} {k : α} : getValue? k l = if containsKey k l = true then some () else none

theorem Std.Internal.List.getValue?_insertListIfNewUnit {α : Type u} [BEq α] [PartialEquivBEq α] {l : List ((_ : α) × Unit)} {toInsert : List α} {k : α} : getValue? k (insertListIfNewUnit l toInsert) = if containsKey k l = true ∨ toInsert.contains k = true then some () else none

def Std.Internal.List.alterKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] (k : α) (f : Option (β k) → Option (β k)) (l : List ((a : α) × β a)) : List ((a : α) × β a)

Internal implementation detail of the hash map

theorem Std.Internal.List.length_alterKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k : α} {f : Option (β k) → Option (β k)} {l : List ((a : α) × β a)} : (alterKey k f l).length = if h : containsKey k l = true then if (f (some (getValueCast k l h))).isSome = true then l.length else l.length - 1 else if (f none).isSome = true then l.length + 1 else l.length

theorem Std.Internal.List.length_alterKey' {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k : α} {f : Option (β k) → Option (β k)} {l : List ((a : α) × β a)} : (alterKey k f l).length = if containsKey k l = true ∧ (f (getValueCast? k l)).isNone = true then l.length - 1 else if containsKey k l = false ∧ (f (getValueCast? k l)).isSome = true then l.length + 1 else l.length

theorem Std.Internal.List.alterKey_cons_perm {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k : α} {f : Option (β k) → Option (β k)} {k' : α} {v' : β k'} {l : List ((a : α) × β a)} : (alterKey k f (⟨k', v'⟩ :: l)).Perm (if hk : (k' == k) = true then match f (some (cast ⋯ v')) with | none => l | some v => ⟨k, v⟩ :: l else ⟨k', v'⟩ :: alterKey k f l)

theorem Std.Internal.List.isEmpty_alterKey_eq_isEmpty_eraseKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k : α} {f : Option (β k) → Option (β k)} {l : List ((a : α) × β a)} : (alterKey k f l).isEmpty = ((eraseKey k l).isEmpty && (f (getValueCast? k l)).isNone)

theorem Std.Internal.List.isEmpty_alterKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k : α} {f : Option (β k) → Option (β k)} {l : List ((a : α) × β a)} : (alterKey k f l).isEmpty = ((l.isEmpty || l.length == 1 && containsKey k l) && (f (getValueCast? k l)).isNone)

theorem Std.Internal.List.alterKey_of_perm {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {a : α} {f : Option (β a) → Option (β a)} {l l' : List ((a : α) × β a)} (hl : DistinctKeys l) (hp : l.Perm l') : (alterKey a f l).Perm (alterKey a f l')

theorem Std.Internal.List.alterKey_append_of_containsKey_right_eq_false {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {a : α} {f : Option (β a) → Option (β a)} {l l' : List ((a : α) × β a)} (hc : containsKey a l' = false) : alterKey a f (l ++ l') = alterKey a f l ++ l'

@[simp]

theorem Std.Internal.List.alterKey_nil {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {a : α} {f : Option (β a) → Option (β a)} : alterKey a f [] = match f none with | none => [] | some b => [⟨a, b⟩]

theorem Std.Internal.List.containsKey_alterKey_self {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {a : α} {f : Option (β a) → Option (β a)} {l : List ((a : α) × β a)} (hl : DistinctKeys l) : containsKey a (alterKey a f l) = (f (getValueCast? a l)).isSome

theorem Std.Internal.List.containsKey_alterKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k k' : α} {f : Option (β k) → Option (β k)} {l : List ((a : α) × β a)} (hl : DistinctKeys l) : containsKey k' (alterKey k f l) = if (k == k') = true then (f (getValueCast? k l)).isSome else containsKey k' l

theorem Std.Internal.List.DistinctKeys.alterKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {a : α} {f : Option (β a) → Option (β a)} {l : List ((a : α) × β a)} (hl : DistinctKeys l) : DistinctKeys (List.alterKey a f l)

theorem Std.Internal.List.mem_alterKey_of_key_ne {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {a : α} {f : Option (β a) → Option (β a)} {l : List ((a : α) × β a)} (p : (a : α) × β a) (hne : p.fst ≠ a) : p ∈ alterKey a f l ↔ p ∈ l

theorem Std.Internal.List.getValueCast?_alterKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] (k k' : α) (f : Option (β k) → Option (β k)) (l : List ((a : α) × β a)) (hl : DistinctKeys l) : getValueCast? k' (alterKey k f l) = if h : (k == k') = true then cast ⋯ (f (getValueCast? k l)) else getValueCast? k' l

theorem Std.Internal.List.getValueCast_alterKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] (k k' : α) (f : Option (β k) → Option (β k)) (l : List ((a : α) × β a)) (hl : DistinctKeys l) (hc : containsKey k' (alterKey k f l) = true) : getValueCast k' (alterKey k f l) hc = if h : (k == k') = true then cast ⋯ ((f (getValueCast? k l)).get ⋯) else getValueCast k' l ⋯

theorem Std.Internal.List.getValueCast_alterKey_self {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] (k : α) (f : Option (β k) → Option (β k)) (l : List ((a : α) × β a)) (hl : DistinctKeys l) (hc : containsKey k (alterKey k f l) = true) : getValueCast k (alterKey k f l) hc = (f (getValueCast? k l)).get ⋯

theorem Std.Internal.List.getValueCast!_alterKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k k' : α} [Inhabited (β k')] {f : Option (β k) → Option (β k)} (l : List ((a : α) × β a)) (hl : DistinctKeys l) : getValueCast! k' (alterKey k f l) = if heq : (k == k') = true then (Option.map (cast ⋯) (f (getValueCast? k l))).get! else getValueCast! k' l

theorem Std.Internal.List.getValueCastD_alterKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k k' : α} {fallback : β k'} {f : Option (β k) → Option (β k)} (l : List ((a : α) × β a)) (hl : DistinctKeys l) : getValueCastD k' (alterKey k f l) fallback = if heq : (k == k') = true then (Option.map (cast ⋯) (f (getValueCast? k l))).getD fallback else getValueCastD k' l fallback

theorem Std.Internal.List.getKey?_alterKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k k' : α} {f : Option (β k) → Option (β k)} (l : List ((a : α) × β a)) (hl : DistinctKeys l) : getKey? k' (alterKey k f l) = if (k == k') = true then if (f (getValueCast? k l)).isSome = true then some k else none else getKey? k' l

theorem Std.Internal.List.getKey!_alterKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] [Inhabited α] {k k' : α} {f : Option (β k) → Option (β k)} (l : List ((a : α) × β a)) (hl : DistinctKeys l) : getKey! k' (alterKey k f l) = if (k == k') = true then if (f (getValueCast? k l)).isSome = true then k else default else getKey! k' l

theorem Std.Internal.List.getKey_alterKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k k' : α} {f : Option (β k) → Option (β k)} (l : List ((a : α) × β a)) (hl : DistinctKeys l) (hc : containsKey k' (alterKey k f l) = true) : getKey k' (alterKey k f l) hc = if heq : (k == k') = true then k else getKey k' l ⋯

theorem Std.Internal.List.getKeyD_alterKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k k' fallback : α} {f : Option (β k) → Option (β k)} (l : List ((a : α) × β a)) (hl : DistinctKeys l) : getKeyD k' (alterKey k f l) fallback = if (k == k') = true then if (f (getValueCast? k l)).isSome = true then k else fallback else getKeyD k' l fallback

def Std.Internal.List.Const.alterKey {α : Type u} {β : Type v} [BEq α] (k : α) (f : Option β → Option β) (l : List ((_ : α) × β)) : List ((_ : α) × β)

Internal implementation detail of the hash map

theorem Std.Internal.List.Const.length_alterKey {α : Type u} [BEq α] {β : Type v} {k : α} {f : Option β → Option β} {l : List ((_ : α) × β)} : (alterKey k f l).length = if h : containsKey k l = true then if (f (some (getValue k l h))).isSome = true then l.length else l.length - 1 else if (f none).isSome = true then l.length + 1 else l.length

theorem Std.Internal.List.Const.length_alterKey' {α : Type u} [BEq α] {β : Type v} {k : α} {f : Option β → Option β} {l : List ((_ : α) × β)} : (alterKey k f l).length = if containsKey k l = true ∧ (f (getValue? k l)).isNone = true then l.length - 1 else if containsKey k l = false ∧ (f (getValue? k l)).isSome = true then l.length + 1 else l.length

theorem Std.Internal.List.Const.length_alterKey_eq_add_one {α : Type u} [BEq α] {β : Type v} {k : α} {f : Option β → Option β} {l : List ((_ : α) × β)} (h : containsKey k l = false) (h' : (f (getValue? k l)).isSome = true) : (alterKey k f l).length = l.length + 1

theorem Std.Internal.List.Const.length_alterKey_eq_sub_one {α : Type u} [BEq α] {β : Type v} {k : α} {f : Option β → Option β} {l : List ((_ : α) × β)} (h : containsKey k l = true) (h' : (f (getValue? k l)).isNone = true) : (alterKey k f l).length = l.length - 1

theorem Std.Internal.List.Const.length_alterKey_eq_self {α : Type u} [BEq α] {β : Type v} {k : α} {f : Option β → Option β} {l : List ((_ : α) × β)} (h : containsKey k l = true) (h' : (f (getValue? k l)).isSome = true) : (alterKey k f l).length = l.length

theorem Std.Internal.List.Const.length_alterKey_eq_self' {α : Type u} [BEq α] {β : Type v} {k : α} {f : Option β → Option β} {l : List ((_ : α) × β)} (h : containsKey k l = false) (h' : (f (getValue? k l)).isNone = true) : (alterKey k f l).length = l.length

theorem Std.Internal.List.Const.alterKey_cons_perm {α : Type u} [BEq α] {β : Type v} {k : α} {f : Option β → Option β} {k' : α} {v' : β} {l : List ((_ : α) × β)} : (alterKey k f (⟨k', v'⟩ :: l)).Perm (if (k' == k) = true then match f (some v') with | none => l | some v => ⟨k, v⟩ :: l else ⟨k', v'⟩ :: alterKey k f l)

theorem Std.Internal.List.Const.isEmpty_alterKey_eq_isEmpty_eraseKey {α : Type u} [BEq α] {β : Type v} {k : α} {f : Option β → Option β} {l : List ((_ : α) × β)} : (alterKey k f l).isEmpty = ((eraseKey k l).isEmpty && (f (getValue? k l)).isNone)

theorem Std.Internal.List.Const.isEmpty_alterKey {α : Type u} [BEq α] {β : Type v} {k : α} {f : Option β → Option β} {l : List ((_ : α) × β)} : (alterKey k f l).isEmpty = ((l.isEmpty || l.length == 1 && containsKey k l) && (f (getValue? k l)).isNone)

theorem Std.Internal.List.Const.alterKey_of_perm {α : Type u} [BEq α] {β : Type v} [EquivBEq α] {a : α} {f : Option β → Option β} {l l' : List ((_ : α) × β)} (hl : DistinctKeys l) (hp : l.Perm l') : (alterKey a f l).Perm (alterKey a f l')

theorem Std.Internal.List.Const.alterKey_append_of_containsKey_right_eq_false {α : Type u} [BEq α] {β : Type v} {a : α} {f : Option β → Option β} {l l' : List ((_ : α) × β)} (hc : containsKey a l' = false) : alterKey a f (l ++ l') = alterKey a f l ++ l'

@[simp]

theorem Std.Internal.List.Const.alterKey_nil {α : Type u} [BEq α] {β : Type v} {a : α} {f : Option β → Option β} : alterKey a f [] = match f none with | none => [] | some b => [⟨a, b⟩]

theorem Std.Internal.List.Const.containsKey_alterKey_self {α : Type u} [BEq α] {β : Type v} [EquivBEq α] {a : α} {f : Option β → Option β} {l : List ((_ : α) × β)} (hl : DistinctKeys l) : containsKey a (alterKey a f l) = (f (getValue? a l)).isSome

theorem Std.Internal.List.Const.mem_alterKey_of_key_not_beq {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {a : α} {f : Option β → Option β} {l : List ((_ : α) × β)} (p : (_ : α) × β) (hne : (p.fst == a) = false) : p ∈ alterKey a f l ↔ p ∈ l

theorem Std.Internal.List.Const.containsKey_alterKey {α : Type u} [BEq α] {β : Type v} [EquivBEq α] {k k' : α} {f : Option β → Option β} {l : List ((_ : α) × β)} (hl : DistinctKeys l) : containsKey k' (alterKey k f l) = if (k == k') = true then (f (getValue? k l)).isSome else containsKey k' l

theorem Std.Internal.List.Const.getValue?_alterKey {α : Type u} [BEq α] {β : Type v} [EquivBEq α] (k k' : α) (f : Option β → Option β) (l : List ((_ : α) × β)) (hl : DistinctKeys l) : getValue? k' (alterKey k f l) = if (k == k') = true then f (getValue? k l) else getValue? k' l

theorem Std.Internal.List.Const.getValue_alterKey {α : Type u} [BEq α] {β : Type v} [EquivBEq α] (k k' : α) (f : Option β → Option β) (l : List ((_ : α) × β)) (hl : DistinctKeys l) (hc : containsKey k' (alterKey k f l) = true) : getValue k' (alterKey k f l) hc = if h : (k == k') = true then (f (getValue? k l)).get ⋯ else getValue k' l ⋯

theorem Std.Internal.List.Const.getValue_alterKey_self {α : Type u} [BEq α] {β : Type v} [EquivBEq α] (k : α) (f : Option β → Option β) (l : List ((_ : α) × β)) (hl : DistinctKeys l) (hc : containsKey k (alterKey k f l) = true) : getValue k (alterKey k f l) hc = (f (getValue? k l)).get ⋯

theorem Std.Internal.List.Const.getValue!_alterKey {α : Type u} [BEq α] {β : Type v} [EquivBEq α] {k k' : α} [Inhabited β] {f : Option β → Option β} (l : List ((_ : α) × β)) (hl : DistinctKeys l) : getValue! k' (alterKey k f l) = if (k == k') = true then (f (getValue? k l)).get! else getValue! k' l

theorem Std.Internal.List.Const.getValueD_alterKey {α : Type u} [BEq α] {β : Type v} [EquivBEq α] {k k' : α} {fallback : β} {f : Option β → Option β} (l : List ((_ : α) × β)) (hl : DistinctKeys l) : getValueD k' (alterKey k f l) fallback = if (k == k') = true then (f (getValue? k l)).getD fallback else getValueD k' l fallback

theorem Std.Internal.List.Const.getKey?_alterKey {α : Type u} [BEq α] {β : Type v} [EquivBEq α] {k k' : α} {f : Option β → Option β} (l : List ((_ : α) × β)) (hl : DistinctKeys l) : getKey? k' (alterKey k f l) = if (k == k') = true then if (f (getValue? k l)).isSome = true then some k else none else getKey? k' l

theorem Std.Internal.List.Const.getKey!_alterKey {α : Type u} [BEq α] {β : Type v} [EquivBEq α] [Inhabited α] {k k' : α} {f : Option β → Option β} (l : List ((_ : α) × β)) (hl : DistinctKeys l) : getKey! k' (alterKey k f l) = if (k == k') = true then if (f (getValue? k l)).isSome = true then k else default else getKey! k' l

theorem Std.Internal.List.Const.getKey_alterKey {α : Type u} [BEq α] {β : Type v} [EquivBEq α] {k k' : α} {f : Option β → Option β} (l : List ((_ : α) × β)) (hl : DistinctKeys l) (hc : containsKey k' (alterKey k f l) = true) : getKey k' (alterKey k f l) hc = if heq : (k == k') = true then k else getKey k' l ⋯

theorem Std.Internal.List.Const.getKeyD_alterKey {α : Type u} [BEq α] {β : Type v} [EquivBEq α] {k k' fallback : α} {f : Option β → Option β} (l : List ((_ : α) × β)) (hl : DistinctKeys l) : getKeyD k' (alterKey k f l) fallback = if (k == k') = true then if (f (getValue? k l)).isSome = true then k else fallback else getKeyD k' l fallback

theorem Std.Internal.List.constAlterKey_eq_alterKey {α : Type u} [BEq α] [LawfulBEq α] {β : Type v} {k : α} {f : Option β → Option β} {l : List ((_ : α) × β)} : Const.alterKey k f l = alterKey k f l

theorem Std.Internal.List.DistinctKeys.constAlterKey {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {a : α} {f : Option β → Option β} {l : List ((_ : α) × β)} (hl : DistinctKeys l) : DistinctKeys (Const.alterKey a f l)

def Std.Internal.List.modifyKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] (k : α) (f : β k → β k) (l : List ((a : α) × β a)) : List ((a : α) × β a)

Internal implementation detail of the hash map

theorem Std.Internal.List.isEmpty_modifyKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] (k : α) (f : β k → β k) (l : List ((a : α) × β a)) : (modifyKey k f l).isEmpty = l.isEmpty

theorem Std.Internal.List.length_modifyKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] (k : α) (f : β k → β k) (l : List ((a : α) × β a)) : (modifyKey k f l).length = l.length

theorem Std.Internal.List.containsKey_modifyKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] (k k' : α) (f : β k → β k) (l : List ((a : α) × β a)) : containsKey k' (modifyKey k f l) = containsKey k' l

theorem Std.Internal.List.modifyKey_eq_alterKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] (k : α) (f : β k → β k) (l : List ((a : α) × β a)) : modifyKey k f l = alterKey k (fun (x : Option (β k)) => Option.map f x) l

theorem Std.Internal.List.DistinctKeys.modifyKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {a : α} {f : β a → β a} {l : List ((a : α) × β a)} (hd : DistinctKeys l) : DistinctKeys (List.modifyKey a f l)

theorem Std.Internal.List.modifyKey_of_perm {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l l' : List ((a : α) × β a)} {k : α} {f : β k → β k} (hl : DistinctKeys l) (h : l.Perm l') : (modifyKey k f l).Perm (modifyKey k f l')

theorem Std.Internal.List.getValueCast?_modifyKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k k' : α} {f : β k → β k} (l : List ((a : α) × β a)) (hl : DistinctKeys l) : getValueCast? k' (modifyKey k f l) = if h : (k == k') = true then cast ⋯ (Option.map f (getValueCast? k l)) else getValueCast? k' l

@[simp]

theorem Std.Internal.List.getValueCast?_modifyKey_self {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k : α} {f : β k → β k} (l : List ((a : α) × β a)) (hl : DistinctKeys l) : getValueCast? k (modifyKey k f l) = Option.map f (getValueCast? k l)

theorem Std.Internal.List.getValueCast_modifyKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k k' : α} {f : β k → β k} (l : List ((a : α) × β a)) (hl : DistinctKeys l) (h : containsKey k' (modifyKey k f l) = true) : getValueCast k' (modifyKey k f l) h = if heq : (k == k') = true then cast ⋯ (f (getValueCast k l ⋯)) else getValueCast k' l ⋯

@[simp]

theorem Std.Internal.List.getValueCast_modifyKey_self {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k : α} {f : β k → β k} (l : List ((a : α) × β a)) (hl : DistinctKeys l) {h : containsKey k (modifyKey k f l) = true} : getValueCast k (modifyKey k f l) h = f (getValueCast k l ⋯)

theorem Std.Internal.List.getValueCast!_modifyKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k k' : α} [hi : Inhabited (β k')] {f : β k → β k} (l : List ((a : α) × β a)) (hl : DistinctKeys l) : getValueCast! k' (modifyKey k f l) = if heq : (k == k') = true then (Option.map (cast ⋯) (Option.map f (getValueCast? k l))).get! else getValueCast! k' l

@[simp]

theorem Std.Internal.List.getValueCast!_modifyKey_self {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k : α} [Inhabited (β k)] {f : β k → β k} (l : List ((a : α) × β a)) (hl : DistinctKeys l) : getValueCast! k (modifyKey k f l) = (Option.map f (getValueCast? k l)).get!

theorem Std.Internal.List.getValueCastD_modifyKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k k' : α} {fallback : β k'} {f : β k → β k} (l : List ((a : α) × β a)) (hl : DistinctKeys l) : getValueCastD k' (modifyKey k f l) fallback = if heq : (k == k') = true then (Option.map (cast ⋯) (Option.map f (getValueCast? k l))).getD fallback else getValueCastD k' l fallback

@[simp]

theorem Std.Internal.List.getValueCastD_modifyKey_self {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k : α} {fallback : β k} {f : β k → β k} (l : List ((a : α) × β a)) (hl : DistinctKeys l) : getValueCastD k (modifyKey k f l) fallback = (Option.map f (getValueCast? k l)).getD fallback

theorem Std.Internal.List.getKey?_modifyKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k k' : α} {f : β k → β k} (l : List ((a : α) × β a)) (hl : DistinctKeys l) : getKey? k' (modifyKey k f l) = if (k == k') = true then if containsKey k l = true then some k else none else getKey? k' l

theorem Std.Internal.List.getKey?_modifyKey_self {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k : α} {f : β k → β k} (l : List ((a : α) × β a)) (hl : DistinctKeys l) : getKey? k (modifyKey k f l) = if containsKey k l = true then some k else none

theorem Std.Internal.List.getKey!_modifyKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] [Inhabited α] {k k' : α} {f : β k → β k} (l : List ((a : α) × β a)) (hl : DistinctKeys l) : getKey! k' (modifyKey k f l) = if (k == k') = true then if containsKey k l = true then k else default else getKey! k' l

theorem Std.Internal.List.getKey!_modifyKey_self {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] [Inhabited α] {k : α} {f : β k → β k} (l : List ((a : α) × β a)) (hl : DistinctKeys l) : getKey! k (modifyKey k f l) = if containsKey k l = true then k else default

theorem Std.Internal.List.getKey_modifyKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k k' : α} {f : β k → β k} (l : List ((a : α) × β a)) (hl : DistinctKeys l) (h : containsKey k' (modifyKey k f l) = true) : getKey k' (modifyKey k f l) h = if (k == k') = true then k else getKey k' l ⋯

@[simp]

theorem Std.Internal.List.getKey_modifyKey_self {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k : α} {f : β k → β k} (l : List ((a : α) × β a)) : DistinctKeys l → ∀ (h : containsKey k (modifyKey k f l) = true), getKey k (modifyKey k f l) h = k

theorem Std.Internal.List.getKeyD_modifyKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k k' fallback : α} {f : β k → β k} (l : List ((a : α) × β a)) (hl : DistinctKeys l) : getKeyD k' (modifyKey k f l) fallback = if (k == k') = true then if containsKey k l = true then k else fallback else getKeyD k' l fallback

theorem Std.Internal.List.getKeyD_modifyKey_self {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {k fallback : α} {f : β k → β k} (l : List ((a : α) × β a)) (hl : DistinctKeys l) : getKeyD k (modifyKey k f l) fallback = if containsKey k l = true then k else fallback

def Std.Internal.List.Const.modifyKey {α : Type u} {β : Type v} [BEq α] (k : α) (f : β → β) (l : List ((_ : α) × β)) : List ((_ : α) × β)

Internal implementation detail of the hash map

theorem Std.Internal.List.Const.isEmpty_modifyKey {α : Type u} {β : Type v} [BEq α] (k : α) (f : β → β) (l : List ((_ : α) × β)) : (modifyKey k f l).isEmpty = l.isEmpty

theorem Std.Internal.List.Const.modifyKey_eq_alterKey {α : Type u} {β : Type v} [BEq α] (k : α) (f : β → β) (l : List ((_ : α) × β)) : modifyKey k f l = alterKey k (fun (x : Option β) => Option.map f x) l

theorem Std.Internal.List.Const.modifyKey_of_perm {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l l' : List ((_ : α) × β)} {k : α} {f : β → β} (hl : DistinctKeys l) (h : l.Perm l') : (modifyKey k f l).Perm (modifyKey k f l')

theorem Std.Internal.List.Const.length_modifyKey {α : Type u} {β : Type v} [BEq α] (k : α) (f : β → β) (l : List ((_ : α) × β)) : (modifyKey k f l).length = l.length

theorem Std.Internal.List.Const.containsKey_modifyKey {α : Type u} {β : Type v} [BEq α] [EquivBEq α] (k k' : α) (f : β → β) (l : List ((_ : α) × β)) : containsKey k' (modifyKey k f l) = containsKey k' l

theorem Std.Internal.List.Const.getValue?_modifyKey {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {k k' : α} {f : β → β} (l : List ((_ : α) × β)) (hl : DistinctKeys l) : getValue? k' (modifyKey k f l) = if (k == k') = true then Option.map f (getValue? k l) else getValue? k' l

@[simp]

theorem Std.Internal.List.Const.getValue?_modifyKey_self {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {k : α} {f : β → β} (l : List ((_ : α) × β)) (hl : DistinctKeys l) : getValue? k (modifyKey k f l) = Option.map f (getValue? k l)

theorem Std.Internal.List.Const.getValue_modifyKey {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {k k' : α} {f : β → β} (l : List ((_ : α) × β)) (hl : DistinctKeys l) (h : containsKey k' (modifyKey k f l) = true) : getValue k' (modifyKey k f l) h = if heq : (k == k') = true then f (getValue k l ⋯) else getValue k' l ⋯

@[simp]

theorem Std.Internal.List.Const.getValue_modifyKey_self {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {k : α} {f : β → β} (l : List ((_ : α) × β)) (hl : DistinctKeys l) {h : containsKey k (modifyKey k f l) = true} : getValue k (modifyKey k f l) h = f (getValue k l ⋯)

theorem Std.Internal.List.Const.getValue!_modifyKey {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {k k' : α} [hi : Inhabited β] {f : β → β} (l : List ((_ : α) × β)) (hl : DistinctKeys l) : getValue! k' (modifyKey k f l) = if (k == k') = true then (Option.map f (getValue? k l)).get! else getValue! k' l

@[simp]

theorem Std.Internal.List.Const.getValue!_modifyKey_self {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {k : α} [Inhabited β] {f : β → β} (l : List ((_ : α) × β)) (hl : DistinctKeys l) : getValue! k (modifyKey k f l) = (Option.map f (getValue? k l)).get!

theorem Std.Internal.List.Const.getValueD_modifyKey {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {k k' : α} {fallback : β} {f : β → β} (l : List ((_ : α) × β)) (hl : DistinctKeys l) : getValueD k' (modifyKey k f l) fallback = if (k == k') = true then (Option.map f (getValue? k l)).getD fallback else getValueD k' l fallback

@[simp]

theorem Std.Internal.List.Const.getValueD_modifyKey_self {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {k : α} {fallback : β} {f : β → β} (l : List ((_ : α) × β)) (hl : DistinctKeys l) : getValueD k (modifyKey k f l) fallback = (Option.map f (getValue? k l)).getD fallback

theorem Std.Internal.List.Const.getKey?_modifyKey {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {k k' : α} {f : β → β} (l : List ((_ : α) × β)) (hl : DistinctKeys l) : getKey? k' (modifyKey k f l) = if (k == k') = true then if containsKey k l = true then some k else none else getKey? k' l

theorem Std.Internal.List.Const.getKey?_modifyKey_self {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {k : α} {f : β → β} (l : List ((_ : α) × β)) (hl : DistinctKeys l) : getKey? k (modifyKey k f l) = if containsKey k l = true then some k else none

theorem Std.Internal.List.Const.getKey!_modifyKey {α : Type u} {β : Type v} [BEq α] [EquivBEq α] [Inhabited α] {k k' : α} {f : β → β} (l : List ((_ : α) × β)) (hl : DistinctKeys l) : getKey! k' (modifyKey k f l) = if (k == k') = true then if containsKey k l = true then k else default else getKey! k' l

theorem Std.Internal.List.Const.getKey!_modifyKey_self {α : Type u} {β : Type v} [BEq α] [EquivBEq α] [Inhabited α] {k : α} {f : β → β} (l : List ((_ : α) × β)) (hl : DistinctKeys l) : getKey! k (modifyKey k f l) = if containsKey k l = true then k else default

theorem Std.Internal.List.Const.getKey_modifyKey {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {k k' : α} {f : β → β} (l : List ((_ : α) × β)) (hl : DistinctKeys l) (h : containsKey k' (modifyKey k f l) = true) : getKey k' (modifyKey k f l) h = if (k == k') = true then k else getKey k' l ⋯

@[simp]

theorem Std.Internal.List.Const.getKey_modifyKey_self {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {k : α} {f : β → β} (l : List ((_ : α) × β)) (hl : DistinctKeys l) (h : containsKey k (modifyKey k f l) = true) : getKey k (modifyKey k f l) h = k

theorem Std.Internal.List.Const.getKeyD_modifyKey {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {k k' fallback : α} {f : β → β} (l : List ((_ : α) × β)) (hl : DistinctKeys l) : getKeyD k' (modifyKey k f l) fallback = if (k == k') = true then if containsKey k l = true then k else fallback else getKeyD k' l fallback

theorem Std.Internal.List.Const.getKeyD_modifyKey_self {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {k fallback : α} {f : β → β} (l : List ((_ : α) × β)) (hl : DistinctKeys l) : getKeyD k (modifyKey k f l) fallback = if containsKey k l = true then k else fallback

theorem Std.Internal.List.constModifyKey_eq_modifyKey {α : Type u} {β : Type v} [BEq α] [LawfulBEq α] {k : α} {f : β → β} {l : List ((_ : α) × β)} : Const.modifyKey k f l = modifyKey k f l

theorem Std.Internal.List.DistinctKeys.constModifyKey {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {a : α} {f : β → β} {l : List ((_ : α) × β)} (hd : DistinctKeys l) : DistinctKeys (Const.modifyKey a f l)

theorem Std.Internal.List.guard_eq_map {α : Type u} {β : α → Type v} (p : (a : α) × β a → Prop) [DecidablePred p] : (Option.guard fun (b : (a : α) × β a) => decide (p b)) = fun (x : (a : α) × β a) => Option.map (fun (y : β x.fst) => ⟨x.fst, y⟩) (if p x then some x.snd else none)

theorem Std.Internal.List.Sigma.snd_congr {α : Type u} {β : α → Type v} {x x' : (a : α) × β a} (h : x = x') : x.snd = cast ⋯ x'.snd

theorem Std.Internal.List.getEntry?_filterMap' {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [EquivBEq α] {f : (a : α) × β a → Option ((a : α) × γ a)} (hf : ∀ (p : (a : α) × β a), Option.all (fun (x : (a : α) × γ a) => x.fst == p.fst) (f p) = true) {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : getEntry? k (List.filterMap f l) = (getEntry? k l).bind f

theorem Std.Internal.List.getEntry?_map' {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [EquivBEq α] {f : (a : α) × β a → (a : α) × γ a} (hf : ∀ (p : (a : α) × β a), ((f p).fst == p.fst) = true) {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : getEntry? k (List.map f l) = Option.map f (getEntry? k l)

theorem Std.Internal.List.getEntry?_filter {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {f : (a : α) × β a → Bool} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : getEntry? k (List.filter f l) = Option.filter f (getEntry? k l)

theorem Std.Internal.List.getValueCast?_filter_containsKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) : getValueCast? k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) = if containsKey k l₂ = true then getValueCast? k l₁ else none

theorem Std.Internal.List.getValueCast?_filter_not_contains {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} {k : α} (dl₁ : DistinctKeys l₁) : getValueCast? k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) = if l₂.contains k = true then none else getValueCast? k l₁

theorem Std.Internal.List.getValueCast?_filter_not_contains_map_fst {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) : getValueCast? k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) = if containsKey k l₂ = true then none else getValueCast? k l₁

theorem Std.Internal.List.getValueCast?_filter_not_contains_of_contains_eq_false_right {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} {k : α} (dl₁ : DistinctKeys l₁) : l₂.contains k = false → getValueCast? k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) = getValueCast? k l₁

theorem Std.Internal.List.getValueCast?_filter_not_contains_map_fst_of_containsKey_eq_false_right {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = false → getValueCast? k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) = getValueCast? k l₁

theorem Std.Internal.List.getValueCast?_filter_not_contains_of_contains_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₁ = false → getValueCast? k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) = none

theorem Std.Internal.List.getValueCast?_filter_not_contains_map_fst_of_containsKey_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₁ = false → getValueCast? k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) = none

theorem Std.Internal.List.getValueCast?_filter_not_contains_of_contains_right {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} {k : α} (dl₁ : DistinctKeys l₁) : l₂.contains k = true → getValueCast? k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) = none

theorem Std.Internal.List.getValueCast?_filter_not_contains_map_fst_of_containsKey_right {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = true → getValueCast? k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) = none

theorem Std.Internal.List.getKey?_filter_containsKey {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) : getKey? k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) = if containsKey k l₂ = true then getKey? k l₁ else none

theorem Std.Internal.List.getKey?_filter_containsKey_of_containsKey_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = true → getKey? k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) = getKey? k l₁

def Std.Internal.List.eraseList {α : Type u} {β : α → Type v} [BEq α] (l : List ((a : α) × β a)) (toErase : List α) : List ((a : α) × β a)

Internal implementation detail of the hash map

theorem Std.Internal.List.eraseList_perm_filter_not_contains {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] (l toErase : List ((a : α) × β a)) (hl : DistinctKeys l) : (eraseList l (List.map Sigma.fst toErase)).Perm (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst toErase).contains p.fst) l)

theorem Std.Internal.List.eraseList_perm_of_perm_first {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l1 l2 : List ((a : α) × β a)} {toErase : List α} (h : l1.Perm l2) (distinct : DistinctKeys l1) : (eraseList l1 toErase).Perm (eraseList l2 toErase)

theorem Std.Internal.List.getKey?_filter_containsKey_of_containsKey_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₁ = false → getKey? k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) = none

theorem Std.Internal.List.getKey?_filter_containsKey_of_containsKey_eq_false_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = false → getKey? k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) = none

theorem Std.Internal.List.getValueCast?_filter_containsKey_of_containsKey_right {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = true → getValueCast? k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) = getValueCast? k l₁

theorem Std.Internal.List.getValueCast?_filter_containsKey_of_containsKey_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₁ = false → getValueCast? k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) = none

theorem Std.Internal.List.getValueCast?_filter_containsKey_of_containsKey_eq_false_right {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = false → getValueCast? k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) = none

theorem Std.Internal.List.getEntry?_filter_containsKey {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) : getEntry? k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) = if containsKey k l₂ = true then getEntry? k l₁ else none

theorem Std.Internal.List.getEntry?_filter_containsKey_of_containsKey_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = true → getEntry? k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) = getEntry? k l₁

theorem Std.Internal.List.getEntry?_filter_containsKey_of_containsKey_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₁ = false → getEntry? k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) = none

theorem Std.Internal.List.getEntry?_filter_containsKey_of_containsKey_eq_false_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = false → getEntry? k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) = none

theorem Std.Internal.List.getEntry?_filterMap {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [EquivBEq α] {f : (a : α) → β a → Option (γ a)} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : getEntry? k (List.filterMap (fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) = (getEntry? k l).bind fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)

theorem Std.Internal.List.getEntry?_map {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [EquivBEq α] {f : (a : α) → β a → γ a} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : getEntry? k (List.map (fun (p : (a : α) × β a) => ⟨p.fst, f p.fst p.snd⟩) l) = Option.map (fun (p : (a : α) × β a) => ⟨p.fst, f p.fst p.snd⟩) (getEntry? k l)

theorem Std.Internal.List.containsKey_filter_containsKey {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {hl₁ : DistinctKeys l₁} {k : α} : containsKey k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) = (containsKey k l₁ && containsKey k l₂)

theorem Std.Internal.List.containsKey_filter_containsKey_iff {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {hl₁ : DistinctKeys l₁} {k : α} : containsKey k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) = true ↔ containsKey k l₁ = true ∧ containsKey k l₂ = true

theorem Std.Internal.List.containsKey_filter_not_contains {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} {hl₁ : DistinctKeys l₁} {k : α} : containsKey k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) = (containsKey k l₁ && !l₂.contains k)

theorem Std.Internal.List.containsKey_filter_not_contains_map_fst {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {hl₁ : DistinctKeys l₁} {k : α} : containsKey k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) = (containsKey k l₁ && !containsKey k l₂)

theorem Std.Internal.List.containsKey_filter_not_contains_iff {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} {hl₁ : DistinctKeys l₁} {k : α} : containsKey k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) = true ↔ containsKey k l₁ = true ∧ ¬l₂.contains k = true

theorem Std.Internal.List.containsKey_filter_not_contains_map_fst_iff {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {hl₁ : DistinctKeys l₁} {k : α} : containsKey k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) = true ↔ containsKey k l₁ = true ∧ ¬containsKey k l₂ = true

theorem Std.Internal.List.getValueCast_filter_not_contains {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} {k : α} (dl₁ : DistinctKeys l₁) {h₁ : containsKey k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) = true} {h₂ : containsKey k l₁ = true} : getValueCast k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) h₁ = getValueCast k l₁ h₂

theorem Std.Internal.List.getValueCast_filter_not_contains_map_fst {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) {h₁ : containsKey k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) = true} {h₂ : containsKey k l₁ = true} : getValueCast k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) h₁ = getValueCast k l₁ h₂

theorem Std.Internal.List.getValueCast_filter_containsKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) {h₁ : containsKey k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) = true} {h₂ : containsKey k l₁ = true} : getValueCast k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) h₁ = getValueCast k l₁ h₂

theorem Std.Internal.List.getEntry_filter_containsKey {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) {h₁ : containsKey k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) = true} {h₂ : containsKey k l₁ = true} : getEntry k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) h₁ = getEntry k l₁ h₂

theorem Std.Internal.List.getKey_filter_containsKey {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) {h₁ : containsKey k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) = true} {h₂ : containsKey k l₁ = true} : getKey k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) h₁ = getKey k l₁ h₂

theorem Std.Internal.List.containsKey_filter_containsKey_eq_false_of_containsKey_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {hl₁ : DistinctKeys l₁} {k : α} : containsKey k l₁ = false → containsKey k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) = false

theorem Std.Internal.List.containsKey_filter_not_contains_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} {hl₁ : DistinctKeys l₁} {k : α} : containsKey k l₁ = false → containsKey k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) = false

theorem Std.Internal.List.containsKey_filter_not_contains_map_fst_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {hl₁ : DistinctKeys l₁} {k : α} : containsKey k l₁ = false → containsKey k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) = false

theorem Std.Internal.List.containsKey_filter_containsKey_eq_false_of_containsKey_eq_false_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {hl₁ : DistinctKeys l₁} {k : α} : containsKey k l₂ = false → containsKey k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) = false

theorem Std.Internal.List.perm_filter_containsKey_of_perm {α : Type u} {β : α → Type v} {l₁ l₂ l₃ : List ((a : α) × β a)} [BEq α] [EquivBEq α] (h : l₂.Perm l₃) (wf₁ : DistinctKeys l₁) : (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁).Perm (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₃) l₁)

theorem Std.Internal.List.perm_filter_not_contains_map_fst_of_perm {α : Type u} {β : α → Type v} {l₁ l₂ l₃ : List ((a : α) × β a)} [BEq α] [EquivBEq α] (h : l₂.Perm l₃) (wf₁ : DistinctKeys l₁) : (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁).Perm (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₃).contains p.fst) l₁)

theorem Std.Internal.List.congr_filter_containsKey_of_perm {α : Type u} {β : α → Type v} {l₁ l₂ l₃ l₄ : List ((a : α) × β a)} [BEq α] [EquivBEq α] (h₁ : l₁.Perm l₃) (h₂ : l₂.Perm l₄) (hd : DistinctKeys l₃) : (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁).Perm (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₄) l₃)

theorem Std.Internal.List.congr_filter_not_contains_map_fst_of_perm {α : Type u} {β : α → Type v} {l₁ l₂ l₃ l₄ : List ((a : α) × β a)} [BEq α] [EquivBEq α] (h₁ : l₁.Perm l₃) (h₂ : l₂.Perm l₄) (hd : DistinctKeys l₃) : (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁).Perm (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₄).contains p.fst) l₃)

theorem Std.Internal.List.containsKey_filter_not_contains_of_contains_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} {hl₁ : DistinctKeys l₁} {k : α} : l₂.contains k = true → containsKey k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) = false

theorem Std.Internal.List.containsKey_filter_not_contains_map_fst_of_contains_map_fst_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {hl₁ : DistinctKeys l₁} {k : α} : containsKey k l₂ = true → containsKey k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) = false

theorem Std.Internal.List.getValueCastD_filter_not_contains {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} {k : α} {fallback : β k} (dl₁ : DistinctKeys l₁) : getValueCastD k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) fallback = if l₂.contains k = true then fallback else getValueCastD k l₁ fallback

theorem Std.Internal.List.getValueCastD_filter_not_contains_map_fst {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} {fallback : β k} (dl₁ : DistinctKeys l₁) : getValueCastD k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) fallback = if containsKey k l₂ = true then fallback else getValueCastD k l₁ fallback

theorem Std.Internal.List.getKey?_filter_not_contains {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} {k : α} (dl₁ : DistinctKeys l₁) : getKey? k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) = if l₂.contains k = true then none else getKey? k l₁

theorem Std.Internal.List.getKey?_filter_not_contains_map_fst {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) : getKey? k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) = if containsKey k l₂ = true then none else getKey? k l₁

theorem Std.Internal.List.getKey?_filter_not_contains_of_contains_eq_false_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} {k : α} (dl₁ : DistinctKeys l₁) : l₂.contains k = false → getKey? k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) = getKey? k l₁

theorem Std.Internal.List.getKey?_filter_not_contains_map_fst_of_containsKey_eq_false_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = false → getKey? k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) = getKey? k l₁

theorem Std.Internal.List.getKey?_filter_not_contains_of_containsKey_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₁ = false → getKey? k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) = none

theorem Std.Internal.List.getKey?_filter_not_contains_map_fst_of_containsKey_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₁ = false → getKey? k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) = none

theorem Std.Internal.List.getKey?_filter_not_contains_of_containsKey_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} {k : α} (dl₁ : DistinctKeys l₁) : l₂.contains k = true → getKey? k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) = none

theorem Std.Internal.List.getKey?_filter_not_contains_map_fst_of_containsKey_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = true → getKey? k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) = none

theorem Std.Internal.List.getKey_filter_not_contains {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} {k : α} (dl₁ : DistinctKeys l₁) {h₁ : containsKey k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) = true} {h₂ : containsKey k l₁ = true} : getKey k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) h₁ = getKey k l₁ h₂

theorem Std.Internal.List.getKey_filter_not_contains_map_fst {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} (dl₁ : DistinctKeys l₁) {h₁ : containsKey k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) = true} {h₂ : containsKey k l₁ = true} : getKey k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) h₁ = getKey k l₁ h₂

theorem Std.Internal.List.getKeyD_filter_not_contains {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} {k fallback : α} (dl₁ : DistinctKeys l₁) : getKeyD k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) fallback = if l₂.contains k = true then fallback else getKeyD k l₁ fallback

theorem Std.Internal.List.getKeyD_filter_not_contains_map_fst {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k fallback : α} (dl₁ : DistinctKeys l₁) : getKeyD k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) fallback = if containsKey k l₂ = true then fallback else getKeyD k l₁ fallback

theorem Std.Internal.List.getKeyD_filter_not_contains_of_contains_eq_false_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} {k fallback : α} (dl₁ : DistinctKeys l₁) : l₂.contains k = false → getKeyD k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) fallback = getKeyD k l₁ fallback

theorem Std.Internal.List.getKeyD_filter_not_contains_map_fst_of_contains_eq_false_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k fallback : α} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = false → getKeyD k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) fallback = getKeyD k l₁ fallback

theorem Std.Internal.List.getKeyD_filter_not_contains_of_contains_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} {k fallback : α} (dl₁ : DistinctKeys l₁) : l₂.contains k = true → getKeyD k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) fallback = fallback

theorem Std.Internal.List.getKeyD_filter_not_contains_map_fst_of_contains_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k fallback : α} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = true → getKeyD k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) fallback = fallback

theorem Std.Internal.List.getKeyD_filter_not_contains_of_contains_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} {k fallback : α} (dl₁ : DistinctKeys l₁) : containsKey k l₁ = false → getKeyD k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) fallback = fallback

theorem Std.Internal.List.getKeyD_filter_not_contains_map_fst_of_contains_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k fallback : α} (dl₁ : DistinctKeys l₁) : containsKey k l₁ = false → getKeyD k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) fallback = fallback

theorem Std.Internal.List.getValueCastD_filter_not_contains_of_containsKey_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} {k : α} {fallback : β k} (dl₁ : DistinctKeys l₁) : containsKey k l₁ = false → getValueCastD k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) fallback = fallback

theorem Std.Internal.List.getValueCastD_filter_not_contains_map_fst_of_containsKey_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} {fallback : β k} (dl₁ : DistinctKeys l₁) : containsKey k l₁ = false → getValueCastD k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) fallback = fallback

theorem Std.Internal.List.length_filter_not_contains_eq_length_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} (hl₁ : DistinctKeys l₁) (w : ∀ (a : α), containsKey a l₁ = true → l₂.contains a = false) : (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁).length = l₁.length

theorem Std.Internal.List.length_filter_not_contains_map_fst_eq_length_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} (hl₁ : DistinctKeys l₁) (w : ∀ (a : α), containsKey a l₁ = true → containsKey a l₂ = false) : (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁).length = l₁.length

theorem Std.Internal.List.size_filter_not_contains_add_size_inter_eq_size_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} (dl₁ : DistinctKeys l₁) : (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁).length + (List.filter (fun (p : (a : α) × β a) => l₂.contains p.fst) l₁).length = l₁.length

theorem Std.Internal.List.size_filter_not_contains_map_fst_add_size_inter_eq_size_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} (dl₁ : DistinctKeys l₁) : (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁).length + (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁).length = l₁.length

theorem Std.Internal.List.isEmpty_filter_not_contains_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} : l₁.isEmpty = true → (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁).isEmpty = true

theorem Std.Internal.List.isEmpty_filter_not_contains_map_fst_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} : l₁.isEmpty = true → (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁).isEmpty = true

theorem Std.Internal.List.isEmpty_filter_not_contains_iff {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} (dl₁ : DistinctKeys l₁) : (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁).isEmpty = true ↔ ∀ (k : α), containsKey k l₁ = true → l₂.contains k = true

theorem Std.Internal.List.isEmpty_filter_not_contains_map_fst_iff {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} (dl₁ : DistinctKeys l₁) : (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁).isEmpty = true ↔ ∀ (k : α), containsKey k l₁ = true → containsKey k l₂ = true

theorem Std.Internal.List.getValue?_filter_not_contains {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ : List ((_ : α) × β)} {l₂ : List α} {k : α} (dl₁ : DistinctKeys l₁) : getValue? k (List.filter (fun (p : (_ : α) × β) => !l₂.contains p.fst) l₁) = if l₂.contains k = true then none else getValue? k l₁

theorem Std.Internal.List.getValue?_filter_not_contains_map_fst {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((_ : α) × β)} {k : α} (dl₁ : DistinctKeys l₁) : getValue? k (List.filter (fun (p : (_ : α) × β) => !(List.map Sigma.fst l₂).contains p.fst) l₁) = if containsKey k l₂ = true then none else getValue? k l₁

theorem Std.Internal.List.getValue?_filter_not_contains_of_contains_eq_false_right {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ : List ((_ : α) × β)} {l₂ : List α} {k : α} (dl₁ : DistinctKeys l₁) : l₂.contains k = false → getValue? k (List.filter (fun (p : (_ : α) × β) => !l₂.contains p.fst) l₁) = getValue? k l₁

theorem Std.Internal.List.getValue?_filter_not_contains_map_fst_of_contains_eq_false_right {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((_ : α) × β)} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = false → getValue? k (List.filter (fun (p : (_ : α) × β) => !(List.map Sigma.fst l₂).contains p.fst) l₁) = getValue? k l₁

theorem Std.Internal.List.getValue?_filter_not_contains_of_containsKey_eq_false_left {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ : List ((_ : α) × β)} {l₂ : List α} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₁ = false → getValue? k (List.filter (fun (p : (_ : α) × β) => !l₂.contains p.fst) l₁) = none

theorem Std.Internal.List.getValue?_filter_not_contains_map_fst_of_containsKey_eq_false_left {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((_ : α) × β)} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₁ = false → getValue? k (List.filter (fun (p : (_ : α) × β) => !(List.map Sigma.fst l₂).contains p.fst) l₁) = none

theorem Std.Internal.List.getValue?_filter_not_contains_of_contains_right {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ : List ((_ : α) × β)} {l₂ : List α} {k : α} (dl₁ : DistinctKeys l₁) : l₂.contains k = true → getValue? k (List.filter (fun (p : (_ : α) × β) => !l₂.contains p.fst) l₁) = none

theorem Std.Internal.List.getValue?_filter_not_contains_map_fst_of_contains_right {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((_ : α) × β)} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = true → getValue? k (List.filter (fun (p : (_ : α) × β) => !(List.map Sigma.fst l₂).contains p.fst) l₁) = none

theorem Std.Internal.List.getValue_filter_not_contains {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ : List ((_ : α) × β)} {l₂ : List α} {k : α} (dl₁ : DistinctKeys l₁) {h₁ : containsKey k (List.filter (fun (p : (_ : α) × β) => !l₂.contains p.fst) l₁) = true} {h₂ : containsKey k l₁ = true} : getValue k (List.filter (fun (p : (_ : α) × β) => !l₂.contains p.fst) l₁) h₁ = getValue k l₁ h₂

theorem Std.Internal.List.getValue_filter_not_contains_map_fst {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((_ : α) × β)} {k : α} (dl₁ : DistinctKeys l₁) {h₁ : containsKey k (List.filter (fun (p : (_ : α) × β) => !(List.map Sigma.fst l₂).contains p.fst) l₁) = true} {h₂ : containsKey k l₁ = true} : getValue k (List.filter (fun (p : (_ : α) × β) => !(List.map Sigma.fst l₂).contains p.fst) l₁) h₁ = getValue k l₁ h₂

theorem Std.Internal.List.getValueD_filter_not_contains {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ : List ((_ : α) × β)} {l₂ : List α} {k : α} {fallback : β} (dl₁ : DistinctKeys l₁) : getValueD k (List.filter (fun (p : (_ : α) × β) => !l₂.contains p.fst) l₁) fallback = if l₂.contains k = true then fallback else getValueD k l₁ fallback

theorem Std.Internal.List.getValueD_filter_not_contains_map_fst {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((_ : α) × β)} {k : α} {fallback : β} (dl₁ : DistinctKeys l₁) : getValueD k (List.filter (fun (p : (_ : α) × β) => !(List.map Sigma.fst l₂).contains p.fst) l₁) fallback = if containsKey k l₂ = true then fallback else getValueD k l₁ fallback

theorem Std.Internal.List.getValueD_filter_not_contains_of_containsKey_eq_false_right {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ : List ((_ : α) × β)} {l₂ : List α} {k : α} {fallback : β} (dl₁ : DistinctKeys l₁) : l₂.contains k = false → getValueD k (List.filter (fun (p : (_ : α) × β) => !l₂.contains p.fst) l₁) fallback = getValueD k l₁ fallback

theorem Std.Internal.List.getValueD_filter_not_contains_map_fst_of_containsKey_eq_false_right {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((_ : α) × β)} {k : α} {fallback : β} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = false → getValueD k (List.filter (fun (p : (_ : α) × β) => !(List.map Sigma.fst l₂).contains p.fst) l₁) fallback = getValueD k l₁ fallback

theorem Std.Internal.List.getValueD_filter_not_contains_of_contains_right {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ : List ((_ : α) × β)} {l₂ : List α} {k : α} {fallback : β} (dl₁ : DistinctKeys l₁) : l₂.contains k = true → getValueD k (List.filter (fun (p : (_ : α) × β) => !l₂.contains p.fst) l₁) fallback = fallback

theorem Std.Internal.List.getValueD_filter_not_contains_map_fst_of_contains_map_fst_right {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((_ : α) × β)} {k : α} {fallback : β} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = true → getValueD k (List.filter (fun (p : (_ : α) × β) => !(List.map Sigma.fst l₂).contains p.fst) l₁) fallback = fallback

theorem Std.Internal.List.getValueD_filter_not_contains_of_containsKey_eq_false_left {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ : List ((_ : α) × β)} {l₂ : List α} {k : α} {fallback : β} (dl₁ : DistinctKeys l₁) : containsKey k l₁ = false → getValueD k (List.filter (fun (p : (_ : α) × β) => !l₂.contains p.fst) l₁) fallback = fallback

theorem Std.Internal.List.getValueD_filter_not_contains_map_fst_of_containsKey_eq_false_left {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((_ : α) × β)} {k : α} {fallback : β} (dl₁ : DistinctKeys l₁) : containsKey k l₁ = false → getValueD k (List.filter (fun (p : (_ : α) × β) => !(List.map Sigma.fst l₂).contains p.fst) l₁) fallback = fallback

theorem Std.Internal.List.getValueCastD_filter_not_contains_of_contains_right {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} {k : α} {fallback : β k} (dl₁ : DistinctKeys l₁) : l₂.contains k = true → getValueCastD k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) fallback = fallback

theorem Std.Internal.List.getValueCastD_filter_not_contains_map_fst_of_contains_map_fst_right {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} {fallback : β k} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = true → getValueCastD k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) fallback = fallback

theorem Std.Internal.List.List.getValue?_filter_containsKey {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((_ : α) × β)} {k : α} (dl₁ : DistinctKeys l₁) : getValue? k (List.filter (fun (p : (_ : α) × β) => containsKey p.fst l₂) l₁) = if containsKey k l₂ = true then getValue? k l₁ else none

theorem Std.Internal.List.List.getValue?_filter_containsKey_of_containsKey_right {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((_ : α) × β)} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = true → getValue? k (List.filter (fun (p : (_ : α) × β) => containsKey p.fst l₂) l₁) = getValue? k l₁

theorem Std.Internal.List.List.getValue?_filter_containsKey_of_containsKey_eq_false_left {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((_ : α) × β)} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₁ = false → getValue? k (List.filter (fun (p : (_ : α) × β) => containsKey p.fst l₂) l₁) = none

theorem Std.Internal.List.getValue?_filter_containsKey_of_containsKey_eq_false_right {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((_ : α) × β)} {k : α} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = false → getValue? k (List.filter (fun (p : (_ : α) × β) => containsKey p.fst l₂) l₁) = none

theorem Std.Internal.List.getValueCastD_filter_containsKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} {fallback : β k} (dl₁ : DistinctKeys l₁) : getValueCastD k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) fallback = if containsKey k l₂ = true then getValueCastD k l₁ fallback else fallback

theorem Std.Internal.List.getValueCastD_filter_containsKey_of_containsKey_right {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} {fallback : β k} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = true → getValueCastD k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) fallback = getValueCastD k l₁ fallback

theorem Std.Internal.List.getValueCastD_filter_containsKey_of_containsKey_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} {fallback : β k} (dl₁ : DistinctKeys l₁) : containsKey k l₁ = false → getValueCastD k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) fallback = fallback

theorem Std.Internal.List.getValueCastD_filter_containsKey_of_containsKey_eq_false_right {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} {fallback : β k} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = false → getValueCastD k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) fallback = fallback

theorem Std.Internal.List.getValueCastD_filter_not_contains_of_contains_eq_false_right {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ : List ((a : α) × β a)} {l₂ : List α} {k : α} {fallback : β k} (dl₁ : DistinctKeys l₁) : l₂.contains k = false → getValueCastD k (List.filter (fun (p : (a : α) × β a) => !l₂.contains p.fst) l₁) fallback = getValueCastD k l₁ fallback

theorem Std.Internal.List.getValueCastD_filter_not_contains_map_fst_of_containsKey_eq_false_right {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l₁ l₂ : List ((a : α) × β a)} {k : α} {fallback : β k} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = false → getValueCastD k (List.filter (fun (p : (a : α) × β a) => !(List.map Sigma.fst l₂).contains p.fst) l₁) fallback = getValueCastD k l₁ fallback

theorem Std.Internal.List.getKeyD_filter_containsKey {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k fallback : α} (dl₁ : DistinctKeys l₁) : getKeyD k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) fallback = if containsKey k l₂ = true then getKeyD k l₁ fallback else fallback

theorem Std.Internal.List.getKeyD_filter_containsKey_of_containsKey_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k fallback : α} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = true → getKeyD k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) fallback = getKeyD k l₁ fallback

theorem Std.Internal.List.getKeyD_filter_containsKey_of_containsKey_eq_false_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k fallback : α} (dl₁ : DistinctKeys l₁) : containsKey k l₁ = false → getKeyD k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) fallback = fallback

theorem Std.Internal.List.getKeyD_filter_containsKey_of_containsKey_eq_false_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} {k fallback : α} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = false → getKeyD k (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁) fallback = fallback

theorem Std.Internal.List.List.getValue_filter_containsKey {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((_ : α) × β)} {k : α} (dl₁ : DistinctKeys l₁) {h₁ : containsKey k (List.filter (fun (p : (_ : α) × β) => containsKey p.fst l₂) l₁) = true} {h₂ : containsKey k l₁ = true} : getValue k (List.filter (fun (p : (_ : α) × β) => containsKey p.fst l₂) l₁) h₁ = getValue k l₁ h₂

theorem Std.Internal.List.List.getValueD_filter_containsKey {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((_ : α) × β)} {k : α} {fallback : β} (dl₁ : DistinctKeys l₁) : getValueD k (List.filter (fun (p : (_ : α) × β) => containsKey p.fst l₂) l₁) fallback = if containsKey k l₂ = true then getValueD k l₁ fallback else fallback

theorem Std.Internal.List.List.getValueD_filter_containsKey_of_containsKey_right {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((_ : α) × β)} {k : α} {fallback : β} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = true → getValueD k (List.filter (fun (p : (_ : α) × β) => containsKey p.fst l₂) l₁) fallback = getValueD k l₁ fallback

theorem Std.Internal.List.getValueD_filter_containsKey_of_containsKey_eq_false_left {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((_ : α) × β)} {k : α} {fallback : β} (dl₁ : DistinctKeys l₁) : containsKey k l₁ = false → getValueD k (List.filter (fun (p : (_ : α) × β) => containsKey p.fst l₂) l₁) fallback = fallback

theorem Std.Internal.List.getValueD_filter_containsKey_of_containsKey_eq_false_right {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((_ : α) × β)} {k : α} {fallback : β} (dl₁ : DistinctKeys l₁) : containsKey k l₂ = false → getValueD k (List.filter (fun (p : (_ : α) × β) => containsKey p.fst l₂) l₁) fallback = fallback

theorem Std.Internal.List.containsKey_of_containsKey_filterMap' {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [EquivBEq α] {f : (a : α) × β a → Option ((a : α) × γ a)} (hf : ∀ (p : (a : α) × β a), Option.all (fun (x : (a : α) × γ a) => x.fst == p.fst) (f p) = true) {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) (h : containsKey k (List.filterMap f l) = true) : containsKey k l = true

theorem Std.Internal.List.containsKey_of_containsKey_filterMap {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [EquivBEq α] {f : (a : α) → β a → Option (γ a)} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) (h : containsKey k (List.filterMap (fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) = true) : containsKey k l = true

theorem Std.Internal.List.snd_eq_getValueCast_of_getEntry?_eq_some {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k : α} {y : (a : α) × β a} (h : getEntry? k l = some y) : y.snd = cast ⋯ (getValueCast k l ⋯)

theorem Std.Internal.List.eq_getKey_getValue_of_getEntry?_eq_some {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {l : List ((_ : α) × β)} {k : α} {y : (_ : α) × β} (h : getEntry? k l = some y) : y = ⟨getKey k l ⋯, getValue k l ⋯⟩

theorem Std.Internal.List.isSome_apply_of_containsKey_filterMap {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Option (γ a)} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) (h : containsKey k (List.filterMap (fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) = true) : (f k (getValueCast k l ⋯)).isSome = true

theorem Std.Internal.List.Const.isSome_apply_of_containsKey_filterMap {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {γ : Type w} {f : α → β → Option γ} {l : List ((_ : α) × β)} {k : α} (hl : DistinctKeys l) (h : containsKey k (List.filterMap (fun (p : (_ : α) × β) => Option.map (fun (x : γ) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) = true) : (f (getKey k l ⋯) (getValue k l ⋯)).isSome = true

theorem Std.Internal.List.containsKey_of_containsKey_filter {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {f : (a : α) → β a → Bool} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) (h : containsKey k (List.filter (fun (p : (a : α) × β a) => f p.fst p.snd) l) = true) : containsKey k l = true

theorem Std.Internal.List.Const.containsKey_filterMap {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {γ : Type w} {f : α → β → Option γ} {l : List ((_ : α) × β)} {k : α} (hl : DistinctKeys l) : containsKey k (List.filterMap (fun (p : (_ : α) × β) => Option.map (fun (x : γ) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) = if h : containsKey k l = true then (f (getKey k l h) (getValue k l h)).isSome else false

theorem Std.Internal.List.Const.containsKey_filter {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {f : α → β → Bool} {l : List ((_ : α) × β)} {k : α} (hl : DistinctKeys l) : containsKey k (List.filter (fun (p : (_ : α) × β) => f p.fst p.snd) l) = if h : containsKey k l = true then f (getKey k l h) (getValue k l h) else false

theorem Std.Internal.List.containsKey_map {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] {f : (a : α) → β a → γ a} {l : List ((a : α) × β a)} {k : α} : containsKey k (List.map (fun (p : (a : α) × β a) => ⟨p.fst, f p.fst p.snd⟩) l) = containsKey k l

theorem Std.Internal.List.apply_eq_true_of_containsKey_filter {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Bool} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) (h : containsKey k (List.filter (fun (p : (a : α) × β a) => f p.fst p.snd) l) = true) : f k (getValueCast k l ⋯) = true

theorem Std.Internal.List.getValueCast?_filterMap {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Option (γ a)} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : getValueCast? k (List.filterMap (fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) = (getValueCast? k l).bind (f k)

theorem Std.Internal.List.getValueCast!_filterMap {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Option (γ a)} {l : List ((a : α) × β a)} {k : α} [Inhabited (γ k)] (hl : DistinctKeys l) : getValueCast! k (List.filterMap (fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) = ((getValueCast? k l).bind (f k)).get!

theorem Std.Internal.List.getValueCastD_filterMap {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Option (γ a)} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) {fallback : γ k} : getValueCastD k (List.filterMap (fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) fallback = ((getValueCast? k l).bind (f k)).getD fallback

theorem Std.Internal.List.getValueCast?_filter {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Bool} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : getValueCast? k (List.filter (fun (p : (a : α) × β a) => f p.fst p.snd) l) = Option.filter (f k) (getValueCast? k l)

theorem Std.Internal.List.getValueCast!_filter {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Bool} {l : List ((a : α) × β a)} {k : α} [Inhabited (β k)] (hl : DistinctKeys l) : getValueCast! k (List.filter (fun (p : (a : α) × β a) => f p.fst p.snd) l) = (Option.filter (f k) (getValueCast? k l)).get!

theorem Std.Internal.List.getValueCastD_filter {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Bool} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) {fallback : β k} : getValueCastD k (List.filter (fun (p : (a : α) × β a) => f p.fst p.snd) l) fallback = (Option.filter (f k) (getValueCast? k l)).getD fallback

theorem Std.Internal.List.getValueCast?_map {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [LawfulBEq α] {f : (a : α) → β a → γ a} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : getValueCast? k (List.map (fun (p : (a : α) × β a) => ⟨p.fst, f p.fst p.snd⟩) l) = Option.map (f k) (getValueCast? k l)

theorem Std.Internal.List.getValueCast!_map {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [LawfulBEq α] {f : (a : α) → β a → γ a} {l : List ((a : α) × β a)} {k : α} [Inhabited (γ k)] (hl : DistinctKeys l) : getValueCast! k (List.map (fun (p : (a : α) × β a) => ⟨p.fst, f p.fst p.snd⟩) l) = (Option.map (f k) (getValueCast? k l)).get!

theorem Std.Internal.List.getValueCastD_map {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [LawfulBEq α] {f : (a : α) → β a → γ a} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) {fallback : γ k} : getValueCastD k (List.map (fun (p : (a : α) × β a) => ⟨p.fst, f p.fst p.snd⟩) l) fallback = (Option.map (f k) (getValueCast? k l)).getD fallback

theorem Std.Internal.List.getValueCast_filterMap {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Option (γ a)} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) (h : containsKey k (List.filterMap (fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) = true) : getValueCast k (List.filterMap (fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) h = (f k (getValueCast k l ⋯)).get ⋯

theorem Std.Internal.List.getValueCast_filter {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Bool} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) (h : containsKey k (List.filter (fun (p : (a : α) × β a) => f p.fst p.snd) l) = true) : getValueCast k (List.filter (fun (p : (a : α) × β a) => f p.fst p.snd) l) h = getValueCast k l ⋯

theorem Std.Internal.List.getValueCast_map {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [LawfulBEq α] {f : (a : α) → β a → γ a} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) (h : containsKey k (List.map (fun (p : (a : α) × β a) => ⟨p.fst, f p.fst p.snd⟩) l) = true) : getValueCast k (List.map (fun (p : (a : α) × β a) => ⟨p.fst, f p.fst p.snd⟩) l) h = f k (getValueCast k l ⋯)

theorem Std.Internal.List.containsKey_filterMap {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Option (γ a)} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : containsKey k (List.filterMap (fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) = Option.any (fun (x : β k) => (f k x).isSome) (getValueCast? k l)

theorem Std.Internal.List.containsKey_filter {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Bool} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : containsKey k (List.filter (fun (p : (a : α) × β a) => f p.fst p.snd) l) = Option.any (f k) (getValueCast? k l)

theorem Std.Internal.List.containsKey_filter_key {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {f : α → Bool} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : containsKey k (List.filter (fun (p : (a : α) × β a) => f p.fst) l) = true ↔ ∃ (h : containsKey k l = true), f (getKey k l h) = true

theorem Std.Internal.List.Option.exists_eq_some_and_iff {α : Type u} (o : Option α) (p : α → Prop) : (∃ (x : α), o = some x ∧ p x) ↔ ∃ (h : o.isSome = true), p (o.get h)

theorem Std.Internal.List.containsKey_filterMap_iff {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Option (γ a)} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : containsKey k (List.filterMap (fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) = true ↔ ∃ (h : containsKey k l = true), (f k (getValueCast k l h)).isSome = true

theorem Std.Internal.List.containsKey_filter_iff {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Bool} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : containsKey k (List.filter (fun (p : (a : α) × β a) => f p.fst p.snd) l) = true ↔ ∃ (h : containsKey k l = true), f k (getValueCast k l h) = true

theorem Std.Internal.List.insertEntry_perm_filter {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] (k : α) (v : β k) {l : List ((a : α) × β a)} (hl : DistinctKeys l) : (insertEntry k v l).Perm (⟨k, v⟩ :: List.filter (fun (x : (a : α) × β a) => decide ¬(k == x.fst) = true) l)

theorem Std.Internal.List.Const.map_insertEntry_perm_filter_map {α : Type u} {β : Type v} [BEq α] [EquivBEq α] (k : α) (v : β) {l : List ((_ : α) × β)} (hl : DistinctKeys l) : (List.map (fun (x : (_ : α) × β) => (x.fst, x.snd)) (insertEntry k v l)).Perm ((k, v) :: List.filter (fun (x : α × β) => !k == x.fst) (List.map (fun (x : (_ : α) × β) => (x.fst, x.snd)) l))

theorem Std.Internal.List.keys_insertEntryIfNew_perm {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] (k : α) (v : β k) {l : List ((a : α) × β a)} : (keys (insertEntryIfNew k v l)).Perm (if containsKey k l = true then keys l else k :: keys l)

theorem Std.Internal.List.Const.containsKey_filterMap_iff {α : Type u} {β : Type v} {γ : Type w} [BEq α] [EquivBEq α] {f : α → β → Option γ} {l : List ((_ : α) × β)} {k : α} (hl : DistinctKeys l) : containsKey k (List.filterMap (fun (p : (_ : α) × β) => Option.map (fun (x : γ) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) = true ↔ ∃ (h : containsKey k l = true), (f (getKey k l h) (getValue k l h)).isSome = true

theorem Std.Internal.List.Const.containsKey_filter_iff {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {f : α → β → Bool} {l : List ((_ : α) × β)} {k : α} (hl : DistinctKeys l) : containsKey k (List.filter (fun (p : (_ : α) × β) => f p.fst p.snd) l) = true ↔ ∃ (h : containsKey k l = true), f (getKey k l h) (getValue k l h) = true

theorem Std.Internal.List.getKey?_filterMap {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Option (γ a)} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : getKey? k (List.filterMap (fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) = (getKey? k l).pfilter fun (x : α) (h : getKey? k l = some x) => (f x (getValueCast x l ⋯)).isSome

theorem Std.Internal.List.getKey!_filterMap {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [LawfulBEq α] [Inhabited α] {f : (a : α) → β a → Option (γ a)} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : getKey! k (List.filterMap (fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) = ((getKey? k l).pfilter fun (x : α) (h : getKey? k l = some x) => (f x (getValueCast x l ⋯)).isSome).get!

theorem Std.Internal.List.getKeyD_filterMap {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Option (γ a)} {l : List ((a : α) × β a)} {k fallback : α} (hl : DistinctKeys l) : getKeyD k (List.filterMap (fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) fallback = ((getKey? k l).pfilter fun (x : α) (h : getKey? k l = some x) => (f x (getValueCast x l ⋯)).isSome).getD fallback

theorem Std.Internal.List.getValueCast_eq_get_getValueCast? {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {a : α} {l : List ((a : α) × β a)} {h : containsKey a l = true} : getValueCast a l h = (getValueCast? a l).get ⋯

theorem Std.Internal.List.getValueCast_insertList_of_contains_eq_false {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l toInsert : List ((a : α) × β a)} {k : α} (not_contains : (List.map Sigma.fst toInsert).contains k = false) {h : containsKey k (insertList l toInsert) = true} : getValueCast k (insertList l toInsert) h = getValueCast k l ⋯

theorem Std.Internal.List.getKey?_filter {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Bool} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : getKey? k (List.filter (fun (p : (a : α) × β a) => f p.fst p.snd) l) = (getKey? k l).pfilter fun (x : α) (h : getKey? k l = some x) => f x (getValueCast x l ⋯)

theorem Std.Internal.List.getKey?_filter_key {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {f : α → Bool} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : getKey? k (List.filter (fun (p : (a : α) × β a) => f p.fst) l) = Option.filter f (getKey? k l)

theorem Std.Internal.List.getKey!_filter {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] [Inhabited α] {f : (a : α) → β a → Bool} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : getKey! k (List.filter (fun (p : (a : α) × β a) => f p.fst p.snd) l) = ((getKey? k l).pfilter fun (x : α) (h : getKey? k l = some x) => f x (getValueCast x l ⋯)).get!

theorem Std.Internal.List.getKey!_filter_key {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] [Inhabited α] {f : α → Bool} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : getKey! k (List.filter (fun (p : (a : α) × β a) => f p.fst) l) = (Option.filter f (getKey? k l)).get!

theorem Std.Internal.List.getKeyD_filter {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Bool} {l : List ((a : α) × β a)} {k fallback : α} (hl : DistinctKeys l) : getKeyD k (List.filter (fun (p : (a : α) × β a) => f p.fst p.snd) l) fallback = ((getKey? k l).pfilter fun (x : α) (h : getKey? k l = some x) => f x (getValueCast x l ⋯)).getD fallback

theorem Std.Internal.List.getKeyD_filter_key {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {f : α → Bool} {l : List ((a : α) × β a)} {k fallback : α} (hl : DistinctKeys l) : getKeyD k (List.filter (fun (p : (a : α) × β a) => f p.fst) l) fallback = (Option.filter f (getKey? k l)).getD fallback

theorem Std.Internal.List.getKey?_map {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [EquivBEq α] {f : (a : α) → β a → γ a} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : getKey? k (List.map (fun (p : (a : α) × β a) => ⟨p.fst, f p.fst p.snd⟩) l) = getKey? k l

theorem Std.Internal.List.getKey_map {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [EquivBEq α] {f : (a : α) → β a → γ a} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) {h : containsKey k (List.map (fun (p : (a : α) × β a) => ⟨p.fst, f p.fst p.snd⟩) l) = true} : getKey k (List.map (fun (p : (a : α) × β a) => ⟨p.fst, f p.fst p.snd⟩) l) h = getKey k l ⋯

theorem Std.Internal.List.getKey!_map {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [EquivBEq α] [Inhabited α] {f : (a : α) → β a → γ a} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) : getKey! k (List.map (fun (p : (a : α) × β a) => ⟨p.fst, f p.fst p.snd⟩) l) = getKey! k l

theorem Std.Internal.List.getKeyD_map {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [EquivBEq α] {f : (a : α) → β a → γ a} {l : List ((a : α) × β a)} {k fallback : α} (hl : DistinctKeys l) : getKeyD k (List.map (fun (p : (a : α) × β a) => ⟨p.fst, f p.fst p.snd⟩) l) fallback = getKeyD k l fallback

theorem Std.Internal.List.getKey_filterMap {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [EquivBEq α] {f : (a : α) → β a → Option (γ a)} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) {h : containsKey k (List.filterMap (fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) = true} : getKey k (List.filterMap (fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) h = getKey k l ⋯

theorem Std.Internal.List.getKey_filter {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {f : (a : α) → β a → Bool} {l : List ((a : α) × β a)} {k : α} (hl : DistinctKeys l) {h : containsKey k (List.filter (fun (p : (a : α) × β a) => f p.fst p.snd) l) = true} : getKey k (List.filter (fun (p : (a : α) × β a) => f p.fst p.snd) l) h = getKey k l ⋯

theorem Std.Internal.List.getValue_map {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {γ : Type w} {f : α → β → γ} {l : List ((_ : α) × β)} {k : α} (hl : DistinctKeys l) {h : containsKey k (List.map (fun (p : (_ : α) × β) => ⟨p.fst, f p.fst p.snd⟩) l) = true} : getValue k (List.map (fun (p : (_ : α) × β) => ⟨p.fst, f p.fst p.snd⟩) l) h = f (getKey k l ⋯) (getValue k l ⋯)

theorem Std.Internal.List.getValue_filterMap {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {γ : Type w} {f : α → β → Option γ} {l : List ((_ : α) × β)} {k : α} (hl : DistinctKeys l) {h : containsKey k (List.filterMap (fun (p : (_ : α) × β) => Option.map (fun (x : γ) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) = true} : getValue k (List.filterMap (fun (p : (_ : α) × β) => Option.map (fun (x : γ) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) h = (f (getKey k l ⋯) (getValue k l ⋯)).get ⋯

theorem Std.Internal.List.getValue_filter {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {f : α → β → Bool} {l : List ((_ : α) × β)} {k : α} (hl : DistinctKeys l) {h : containsKey k (List.filter (fun (p : (_ : α) × β) => f p.fst p.snd) l) = true} : getValue k (List.filter (fun (p : (_ : α) × β) => f p.fst p.snd) l) h = getValue k l ⋯

theorem Std.Internal.List.length_filterMap_eq_length_iff {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Option (γ a)} {l : List ((a : α) × β a)} (distinct : DistinctKeys l) : (List.filterMap (fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)) l).length = l.length ↔ ∀ (a : α) (h : containsKey a l = true), (f a (getValueCast a l h)).isSome = true

theorem Std.Internal.List.forall_mem_iff_forall_contains_getValueCast {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {p : (a : α) → β a → Prop} (distinct : DistinctKeys l) : (∀ (x : (a : α) × β a), x ∈ l → p x.fst x.snd) ↔ ∀ (a : α) (h : containsKey a l = true), p a (getValueCast a l h)

theorem Std.Internal.List.length_filter_eq_length_iff {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Bool} {l : List ((a : α) × β a)} (distinct : DistinctKeys l) : (List.filter (fun (p : (a : α) × β a) => f p.fst p.snd) l).length = l.length ↔ ∀ (a : α) (h : containsKey a l = true), f a (getValueCast a l h) = true

theorem Std.Internal.List.length_filter_key_eq_length_iff {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {f : α → Bool} {l : List ((a : α) × β a)} (hl : DistinctKeys l) : (List.filter (fun (p : (a : α) × β a) => f p.fst) l).length = l.length ↔ ∀ (a : α) (h : containsKey a l = true), f (getKey a l h) = true

theorem Std.Internal.List.length_filter_containsKey_eq_length_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} (hl₁ : DistinctKeys l₁) (w : ∀ (a : α), containsKey a l₁ = true → containsKey a l₂ = true) : (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁).length = l₁.length

theorem Std.Internal.List.length_filter_containsKey_of_length_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} (hl₁ : DistinctKeys l₁) (hl₂ : DistinctKeys l₂) (w : ∀ (a : α), containsKey a l₂ = true → containsKey a l₁ = true) : (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁).length = l₂.length

theorem Std.Internal.List.size_add_size_eq_size_insertList_add_size_filter_containsKey {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} (hl₁ : DistinctKeys l₁) (hl₂ : DistinctKeys l₂) : l₁.length + l₂.length = (insertList l₁ l₂).length + (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁).length

theorem Std.Internal.List.length_filter_containsKey_le {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} (dl₁ : DistinctKeys l₂) (dl₂ : DistinctKeys l₁) : (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁).length ≤ l₂.length

theorem Std.Internal.List.isEmpty_filter_containsKey_iff {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} (dl₁ : DistinctKeys l₁) : (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁).isEmpty = true ↔ ∀ (k : α), containsKey k l₁ = true → containsKey k l₂ = false

theorem Std.Internal.List.isEmpty_filter_containsKey_comm {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} (dl₁ : DistinctKeys l₁) (dl₂ : DistinctKeys l₂) : (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁).isEmpty = (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₁) l₂).isEmpty

theorem Std.Internal.List.nil_of_containsKey_eq_false {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l : List ((a : α) × β a)} : (∀ (k : α), containsKey k l = false) ↔ l = []

theorem Std.Internal.List.isEmpty_filter_containsKey_left {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} : l₁.isEmpty = true → (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁).isEmpty = true

theorem Std.Internal.List.isEmpty_filter_containsKey_right {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {l₁ l₂ : List ((a : α) × β a)} : l₂.isEmpty = true → (List.filter (fun (p : (a : α) × β a) => containsKey p.fst l₂) l₁).isEmpty = true

theorem Std.Internal.List.perm_filter_self_iff {α : Type u} {f : α → Bool} {l : List α} : (List.filter f l).Perm l ↔ ∀ (a : α), a ∈ l → f a = true

theorem Std.Internal.List.perm_filter_self_iff_forall_containsKey {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Bool} {l : List ((a : α) × β a)} (hl : DistinctKeys l) : (List.filter (fun (p : (a : α) × β a) => f p.fst p.snd) l).Perm l ↔ ∀ (a : α) (h : containsKey a l = true), f a (getValueCast a l h) = true

theorem Std.Internal.List.perm_filter_key_self_iff_forall_containsKey {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {f : α → Bool} {l : List ((a : α) × β a)} (hl : DistinctKeys l) : (List.filter (fun (p : (a : α) × β a) => f p.fst) l).Perm l ↔ ∀ (a : α) (h : containsKey a l = true), f (getKey a l h) = true

theorem Std.Internal.List.Const.perm_filter_self_iff_forall_containsKey {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {f : α → β → Bool} {l : List ((_ : α) × β)} (hl : DistinctKeys l) : (List.filter (fun (p : (_ : α) × β) => f p.fst p.snd) l).Perm l ↔ ∀ (a : α) (h : containsKey a l = true), f (getKey a l h) (getValue a l h) = true

theorem Std.Internal.List.isEmpty_filterMap_eq_true {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Option (γ a)} {l : List ((a : α) × β a)} (distinct : DistinctKeys l) : (List.filterMap (fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)) l).isEmpty = true ↔ ∀ (k : α) (h : containsKey k l = true), f k (getValueCast k l h) = none

theorem Std.Internal.List.isEmpty_filterMap_eq_false {α : Type u} {β : α → Type v} {γ : α → Type w} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Option (γ a)} {l : List ((a : α) × β a)} (distinct : DistinctKeys l) : (List.filterMap (fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)) l).isEmpty = false ↔ ∃ (k : α), ∃ (h : containsKey k l = true), (f k (getValueCast k l h)).isSome = true

theorem Std.Internal.List.isEmpty_filter_eq_true {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Bool} {l : List ((a : α) × β a)} (distinct : DistinctKeys l) : (List.filter (fun (p : (a : α) × β a) => f p.fst p.snd) l).isEmpty = true ↔ ∀ (k : α) (h : containsKey k l = true), f k (getValueCast k l h) = false

theorem Std.Internal.List.isEmpty_filter_eq_false {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] {f : (a : α) → β a → Bool} {l : List ((a : α) × β a)} (distinct : DistinctKeys l) : (List.filter (fun (p : (a : α) × β a) => f p.fst p.snd) l).isEmpty = false ↔ ∃ (k : α), ∃ (h : containsKey k l = true), f k (getValueCast k l h) = true

theorem Std.Internal.List.isEmpty_filter_key_iff {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {f : α → Bool} {l : List ((a : α) × β a)} (hl : DistinctKeys l) : (List.filter (fun (p : (a : α) × β a) => f p.fst) l).isEmpty = true ↔ ∀ (k : α) (h : containsKey k l = true), f (getKey k l h) = false

theorem Std.Internal.List.beqModel_eq_true_of_perm {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] [(k : α) → BEq (β k)] [∀ (k : α), ReflBEq (β k)] {l₁ l₂ : List ((a : α) × β a)} (hl₁ : DistinctKeys l₁) : l₁.Perm l₂ → beqModel l₁ l₂ = true

theorem Std.Internal.List.Const.beqModel_eq_true_of_perm {α : Type u} {β : Type v} [BEq α] [EquivBEq α] [BEq β] [ReflBEq β] {l₁ l₂ : List ((_ : α) × β)} (hl₁ : DistinctKeys l₁) : l₁.Perm l₂ → beqModel l₁ l₂ = true

theorem Std.Internal.List.perm_of_beqModel {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] [(k : α) → BEq (β k)] [∀ (k : α), LawfulBEq (β k)] {l₁ l₂ : List ((a : α) × β a)} (hl₁ : DistinctKeys l₁) (hl₂ : DistinctKeys l₂) : beqModel l₁ l₂ = true → l₁.Perm l₂

theorem Std.Internal.List.beqModel_congr {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] [(k : α) → BEq (β k)] {l₁ l₂ l₃ l₄ : List ((a : α) × β a)} (hl : DistinctKeys l₂) (p₁ : l₁.Perm l₃) (p₂ : l₂.Perm l₄) : beqModel l₁ l₂ = beqModel l₃ l₄

theorem Std.Internal.List.Const.beqModel_congr {α : Type u} {β : Type v} [BEq α] [EquivBEq α] [BEq β] {l₁ l₂ l₃ l₄ : List ((_ : α) × β)} (hl : DistinctKeys l₂) (p₁ : l₁.Perm l₃) (p₂ : l₂.Perm l₄) : beqModel l₁ l₂ = beqModel l₃ l₄

theorem Std.Internal.List.beqModel_eq_constBeqModel {α : Type u} {β : Type v} [BEq α] [LawfulBEq α] [BEq β] {l₁ l₂ : List ((_ : α) × β)} : beqModel l₁ l₂ = Const.beqModel l₁ l₂

theorem Std.Internal.List.Const.perm_of_beqModel {α : Type u} {β : Type v} [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β] {l₁ l₂ : List ((_ : α) × β)} (hl₁ : DistinctKeys l₁) (hl₂ : DistinctKeys l₂) : beqModel l₁ l₂ = true → l₁.Perm l₂

theorem Std.Internal.List.beqModel_iff_perm {α : Type u} {β : α → Type v} [BEq α] [LawfulBEq α] [(k : α) → BEq (β k)] [∀ (k : α), LawfulBEq (β k)] {l₁ l₂ : List ((a : α) × β a)} (hl₁ : DistinctKeys l₁) (hl₂ : DistinctKeys l₂) : beqModel l₁ l₂ = true ↔ l₁.Perm l₂

theorem Std.Internal.List.Const.beqModel_iff_perm {α : Type u} {β : Type v} [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β] {l₁ l₂ : List ((_ : α) × β)} (hl₁ : DistinctKeys l₁) (hl₂ : DistinctKeys l₂) : beqModel l₁ l₂ = true ↔ l₁.Perm l₂

theorem Std.Internal.List.Const.getKey_getValue_mem {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {l : List ((_ : α) × β)} {k : α} {h : containsKey k l = true} : ⟨getKey k l h, getValue k l h⟩ ∈ l

theorem Std.Internal.List.Const.forall_mem_iff_forall_contains_getKey_getValue {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {l : List ((_ : α) × β)} {p : α → β → Prop} (distinct : DistinctKeys l) : (∀ (x : (_ : α) × β), x ∈ l → p x.fst x.snd) ↔ ∀ (a : α) (h : containsKey a l = true), p (getKey a l h) (getValue a l h)

theorem Std.Internal.List.Const.getValue?_filterMap {α : Type u} {β : Type v} {γ : Type w} [BEq α] [EquivBEq α] {f : α → β → Option γ} {l : List ((_ : α) × β)} (distinct : DistinctKeys l) {k : α} : getValue? k (List.filterMap (fun (p : (_ : α) × β) => Option.map (fun (x : γ) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) = (getValue? k l).pbind fun (v : β) (h : getValue? k l = some v) => f (getKey k l ⋯) v

theorem Std.Internal.List.Const.getValue?_filterMap_of_getKey?_eq_some {α : Type u} {β : Type v} {γ : Type w} [BEq α] [EquivBEq α] {f : α → β → Option γ} {l : List ((_ : α) × β)} (distinct : DistinctKeys l) {k k' : α} : getKey? k l = some k' → getValue? k (List.filterMap (fun (p : (_ : α) × β) => Option.map (fun (x : γ) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) = (getValue? k l).bind fun (x : β) => f k' x

theorem Std.Internal.List.Const.getValue!_filterMap {α : Type u} {β : Type v} {γ : Type w} [BEq α] [EquivBEq α] [Inhabited γ] {f : α → β → Option γ} {l : List ((_ : α) × β)} (distinct : DistinctKeys l) {k : α} : getValue! k (List.filterMap (fun (p : (_ : α) × β) => Option.map (fun (x : γ) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) = ((getValue? k l).pbind fun (v : β) (h : getValue? k l = some v) => f (getKey k l ⋯) v).get!

theorem Std.Internal.List.Const.getValue!_filterMap_of_getKey?_eq_some {α : Type u} {β : Type v} {γ : Type w} [BEq α] [EquivBEq α] [Inhabited γ] {f : α → β → Option γ} {l : List ((_ : α) × β)} (distinct : DistinctKeys l) {k k' : α} : getKey? k l = some k' → getValue! k (List.filterMap (fun (p : (_ : α) × β) => Option.map (fun (x : γ) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) = ((getValue? k l).bind fun (x : β) => f k' x).get!

theorem Std.Internal.List.Const.getValueD_filterMap {α : Type u} {β : Type v} {γ : Type w} [BEq α] [EquivBEq α] {fallback : γ} {f : α → β → Option γ} {l : List ((_ : α) × β)} (distinct : DistinctKeys l) {k : α} : getValueD k (List.filterMap (fun (p : (_ : α) × β) => Option.map (fun (x : γ) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) fallback = ((getValue? k l).pbind fun (v : β) (h : getValue? k l = some v) => f (getKey k l ⋯) v).getD fallback

theorem Std.Internal.List.Const.getValueD_filterMap_of_getKey?_eq_some {α : Type u} {β : Type v} {γ : Type w} [BEq α] [EquivBEq α] {f : α → β → Option γ} {l : List ((_ : α) × β)} (distinct : DistinctKeys l) {k k' : α} {fallback : γ} : getKey? k l = some k' → getValueD k (List.filterMap (fun (p : (_ : α) × β) => Option.map (fun (x : γ) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) fallback = ((getValue? k l).bind fun (x : β) => f k' x).getD fallback

theorem Std.Internal.List.Const.getValue?_filter {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {f : α → β → Bool} {l : List ((_ : α) × β)} (distinct : DistinctKeys l) {k : α} : getValue? k (List.filter (fun (p : (_ : α) × β) => f p.fst p.snd) l) = (getValue? k l).pfilter fun (v : β) (h : getValue? k l = some v) => f (getKey k l ⋯) v

theorem Std.Internal.List.Const.getValue?_filter_of_getKey?_eq_some {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {f : α → β → Bool} {l : List ((_ : α) × β)} (distinct : DistinctKeys l) {k k' : α} : getKey? k l = some k' → getValue? k (List.filter (fun (p : (_ : α) × β) => f p.fst p.snd) l) = Option.filter (fun (x : β) => f k' x) (getValue? k l)

theorem Std.Internal.List.Const.getValue!_filter {α : Type u} {β : Type v} [BEq α] [EquivBEq α] [Inhabited β] {f : α → β → Bool} {l : List ((_ : α) × β)} (distinct : DistinctKeys l) {k : α} : getValue! k (List.filter (fun (p : (_ : α) × β) => f p.fst p.snd) l) = ((getValue? k l).pfilter fun (v : β) (h : getValue? k l = some v) => f (getKey k l ⋯) v).get!

theorem Std.Internal.List.Const.getValue!_filter_of_getKey?_eq_some {α : Type u} {β : Type v} [BEq α] [EquivBEq α] [Inhabited β] {f : α → β → Bool} {l : List ((_ : α) × β)} (distinct : DistinctKeys l) {k k' : α} : getKey? k l = some k' → getValue! k (List.filter (fun (p : (_ : α) × β) => f p.fst p.snd) l) = (Option.filter (fun (x : β) => f k' x) (getValue? k l)).get!

theorem Std.Internal.List.Const.getValueD_filter {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {fallback : β} {f : α → β → Bool} {l : List ((_ : α) × β)} (distinct : DistinctKeys l) {k : α} : getValueD k (List.filter (fun (p : (_ : α) × β) => f p.fst p.snd) l) fallback = ((getValue? k l).pfilter fun (v : β) (h : getValue? k l = some v) => f (getKey k l ⋯) v).getD fallback

theorem Std.Internal.List.Const.getValueD_filter_of_getKey?_eq_some {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {fallback : β} {f : α → β → Bool} {l : List ((_ : α) × β)} (distinct : DistinctKeys l) {k k' : α} (h : getKey? k l = some k') : getValueD k (List.filter (fun (p : (_ : α) × β) => f p.fst p.snd) l) fallback = (Option.filter (fun (x : β) => f k' x) (getValue? k l)).getD fallback

theorem Std.Internal.List.Const.getValue?_map {α : Type u} {β : Type v} {γ : Type w} [BEq α] [EquivBEq α] {f : α → β → γ} {l : List ((_ : α) × β)} (hl : DistinctKeys l) {k : α} : getValue? k (List.map (fun (p : (_ : α) × β) => ⟨p.fst, f p.fst p.snd⟩) l) = Option.pmap (fun (v : β) (h : containsKey k l = true) => f (getKey k l h) v) (getValue? k l) ⋯

theorem Std.Internal.List.Const.getValue?_map_of_getKey?_eq_some {α : Type u} {β : Type v} {γ : Type w} [BEq α] [EquivBEq α] {f : α → β → γ} {l : List ((_ : α) × β)} (hl : DistinctKeys l) {k k' : α} (h : getKey? k l = some k') : getValue? k (List.map (fun (p : (_ : α) × β) => ⟨p.fst, f p.fst p.snd⟩) l) = Option.map (f k') (getValue? k l)

theorem Std.Internal.List.Const.getValue!_map {α : Type u} {β : Type v} {γ : Type w} [BEq α] [EquivBEq α] [Inhabited γ] {f : α → β → γ} {l : List ((_ : α) × β)} (hl : DistinctKeys l) {k : α} : getValue! k (List.map (fun (p : (_ : α) × β) => ⟨p.fst, f p.fst p.snd⟩) l) = (Option.pmap (fun (v : β) (h : containsKey k l = true) => f (getKey k l h) v) (getValue? k l) ⋯).get!

theorem Std.Internal.List.Const.getValue!_map_of_getKey?_eq_some {α : Type u} {β : Type v} {γ : Type w} [BEq α] [EquivBEq α] [Inhabited γ] {f : α → β → γ} {l : List ((_ : α) × β)} (hl : DistinctKeys l) {k k' : α} (h : getKey? k l = some k') : getValue! k (List.map (fun (p : (_ : α) × β) => ⟨p.fst, f p.fst p.snd⟩) l) = (Option.map (f k') (getValue? k l)).get!

theorem Std.Internal.List.Const.getValueD_map {α : Type u} {β : Type v} {γ : Type w} [BEq α] [EquivBEq α] {fallback : γ} {f : α → β → γ} {l : List ((_ : α) × β)} (hl : DistinctKeys l) {k : α} : getValueD k (List.map (fun (p : (_ : α) × β) => ⟨p.fst, f p.fst p.snd⟩) l) fallback = (Option.pmap (fun (v : β) (h : containsKey k l = true) => f (getKey k l h) v) (getValue? k l) ⋯).getD fallback

theorem Std.Internal.List.Const.getValueD_map_of_getKey?_eq_some {α : Type u} {β : Type v} {γ : Type w} [BEq α] [EquivBEq α] {f : α → β → γ} {l : List ((_ : α) × β)} (hl : DistinctKeys l) {k k' : α} {fallback : γ} (h : getKey? k l = some k') : getValueD k (List.map (fun (p : (_ : α) × β) => ⟨p.fst, f p.fst p.snd⟩) l) fallback = (Option.map (f k') (getValue? k l)).getD fallback

theorem Std.Internal.List.Const.getKey?_filterMap {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {γ : Type w} {f : α → β → Option γ} {l : List ((_ : α) × β)} {k : α} (hl : DistinctKeys l) : getKey? k (List.filterMap (fun (p : (_ : α) × β) => Option.map (fun (x : γ) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) = (getKey? k l).pfilter fun (x : α) (h : getKey? k l = some x) => (f x (getValue x l ⋯)).isSome

theorem Std.Internal.List.Const.getKey!_filterMap {α : Type u} [BEq α] [EquivBEq α] [Inhabited α] {β : Type v} {γ : Type w} {f : α → β → Option γ} {l : List ((_ : α) × β)} {k : α} (hl : DistinctKeys l) : getKey! k (List.filterMap (fun (p : (_ : α) × β) => Option.map (fun (x : γ) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) = ((getKey? k l).pfilter fun (x : α) (h : getKey? k l = some x) => (f x (getValue x l ⋯)).isSome).get!

theorem Std.Internal.List.Const.getKeyD_filterMap {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {γ : Type w} {f : α → β → Option γ} {l : List ((_ : α) × β)} {k fallback : α} (hl : DistinctKeys l) : getKeyD k (List.filterMap (fun (p : (_ : α) × β) => Option.map (fun (x : γ) => ⟨p.fst, x⟩) (f p.fst p.snd)) l) fallback = ((getKey? k l).pfilter fun (x : α) (h : getKey? k l = some x) => (f x (getValue x l ⋯)).isSome).getD fallback

theorem Std.Internal.List.Const.getValue_eq_get_getValue? {α : Type u} [BEq α] {β : Type v} {a : α} {l : List ((_ : α) × β)} {h : containsKey a l = true} : getValue a l h = (getValue? a l).get ⋯

theorem Std.Internal.List.Const.getKey?_filter {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {f : α → β → Bool} {l : List ((_ : α) × β)} {k : α} (hl : DistinctKeys l) : getKey? k (List.filter (fun (p : (_ : α) × β) => f p.fst p.snd) l) = (getKey? k l).pfilter fun (x : α) (h : getKey? k l = some x) => f x (getValue x l ⋯)

theorem Std.Internal.List.Const.getKey!_filter {α : Type u} [BEq α] [EquivBEq α] [Inhabited α] {β : Type v} {f : α → β → Bool} {l : List ((_ : α) × β)} {k : α} (hl : DistinctKeys l) : getKey! k (List.filter (fun (p : (_ : α) × β) => f p.fst p.snd) l) = ((getKey? k l).pfilter fun (x : α) (h : getKey? k l = some x) => f x (getValue x l ⋯)).get!

theorem Std.Internal.List.Const.getKeyD_filter {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {f : α → β → Bool} {l : List ((_ : α) × β)} {k fallback : α} (hl : DistinctKeys l) : getKeyD k (List.filter (fun (p : (_ : α) × β) => f p.fst p.snd) l) fallback = ((getKey? k l).pfilter fun (x : α) (h : getKey? k l = some x) => f x (getValue x l ⋯)).getD fallback

theorem Std.Internal.List.Const.length_filterMap_eq_length_iff {α : Type u} {β : Type v} {γ : Type w} [BEq α] [EquivBEq α] {f : α → β → Option γ} {l : List ((_ : α) × β)} (distinct : DistinctKeys l) : (List.filterMap (fun (p : (_ : α) × β) => Option.map (fun (x : γ) => ⟨p.fst, x⟩) (f p.fst p.snd)) l).length = l.length ↔ ∀ (a : α) (h : containsKey a l = true), (f (getKey a l h) (getValue a l h)).isSome = true

theorem Std.Internal.List.Const.length_filter_eq_length_iff {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {f : α → β → Bool} {l : List ((_ : α) × β)} (distinct : DistinctKeys l) : (List.filter (fun (p : (_ : α) × β) => f p.fst p.snd) l).length = l.length ↔ ∀ (a : α) (h : containsKey a l = true), f (getKey a l h) (getValue a l h) = true

theorem Std.Internal.List.Const.length_filter_key_eq_length_iff {α : Type u} {β : Type v} [BEq α] [EquivBEq α] {f : α → Bool} {l : List ((_ : α) × β)} (distinct : DistinctKeys l) : (List.filter (fun (p : (_ : α) × β) => f p.fst) l).length = l.length ↔ ∀ (a : α) (h : containsKey a l = true), f (getKey a l h) = true

theorem Std.Internal.List.Const.isEmpty_filterMap_eq_true {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {γ : Type w} {f : α → β → Option γ} {l : List ((_ : α) × β)} (distinct : DistinctKeys l) : (List.filterMap (fun (p : (_ : α) × β) => Option.map (fun (x : γ) => ⟨p.fst, x⟩) (f p.fst p.snd)) l).isEmpty = true ↔ ∀ (k : α) (h : containsKey k l = true), f (getKey k l h) (getValue k l h) = none

theorem Std.Internal.List.Const.isEmpty_filterMap_eq_false {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {γ : Type w} {f : α → β → Option γ} {l : List ((_ : α) × β)} (distinct : DistinctKeys l) : (List.filterMap (fun (p : (_ : α) × β) => Option.map (fun (x : γ) => ⟨p.fst, x⟩) (f p.fst p.snd)) l).isEmpty = false ↔ ∃ (k : α), ∃ (h : containsKey k l = true), (f (getKey k l h) (getValue k l h)).isSome = true

theorem Std.Internal.List.Const.isEmpty_filter_eq_true {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {f : α → β → Bool} {l : List ((_ : α) × β)} (distinct : DistinctKeys l) : (List.filter (fun (p : (_ : α) × β) => f p.fst p.snd) l).isEmpty = true ↔ ∀ (k : α) (h : containsKey k l = true), f (getKey k l h) (getValue k l h) = false

theorem Std.Internal.List.Const.isEmpty_filter_eq_false {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {f : α → β → Bool} {l : List ((_ : α) × β)} (distinct : DistinctKeys l) : (List.filter (fun (p : (_ : α) × β) => f p.fst p.snd) l).isEmpty = false ↔ ∃ (k : α), ∃ (h : containsKey k l = true), f (getKey k l h) (getValue k l h) = true

theorem Std.Internal.List.Const.isEmpty_filter_key_eq_true {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {f : α → Bool} {l : List ((_ : α) × β)} (distinct : DistinctKeys l) : (List.filter (fun (p : (_ : α) × β) => f p.fst) l).isEmpty = true ↔ ∀ (k : α) (h : containsKey k l = true), f (getKey k l h) = false

theorem Std.Internal.List.Const.isEmpty_filter_key_eq_false {α : Type u} [BEq α] [EquivBEq α] {β : Type v} {f : α → Bool} {l : List ((_ : α) × β)} (distinct : DistinctKeys l) : (List.filter (fun (p : (_ : α) × β) => f p.fst) l).isEmpty = false ↔ ∃ (k : α), ∃ (h : containsKey k l = true), f (getKey k l h) = true

theorem Std.Internal.List.Const.toList_map' {α : Type u} {β : Type v} {γ : Type w} {f : α → β → γ} {l : List ((_ : α) × β)} : List.map (fun (p : (_ : α) × β) => (p.fst, f p.fst p.snd)) l = List.map (fun (p : (_ : α) × γ) => (p.fst, p.snd)) (List.map (fun (p : (_ : α) × β) => ⟨p.fst, f p.fst p.snd⟩) l)

theorem Std.Internal.List.Const.toList_map {α : Type u} {β : Type v} {γ : Type w} {f : α → β → γ} {l : List ((_ : α) × β)} : (List.map (fun (p : (_ : α) × β) => (p.fst, f p.fst p.snd)) l).Perm (List.map (fun (p : (_ : α) × γ) => (p.fst, p.snd)) (List.map (fun (p : (_ : α) × β) => ⟨p.fst, f p.fst p.snd⟩) l))

def Std.Internal.List.minEntry? {α : Type u} {β : α → Type v} [Ord α] (xs : List ((a : α) × β a)) : Option ((a : α) × β a)

Like List.min?, but using an Ord typeclass instead of a Min typeclass.

def Std.Internal.List.minKey? {α : Type u} {β : α → Type v} [Ord α] (xs : List ((a : α) × β a)) : Option α

Returns the smallest key in an associative list.

theorem Std.Internal.List.DistinctKeys.eq_of_mem_of_beq {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] {a b : (a : α) × β a} {l : List ((a : α) × β a)} (hma : a ∈ l) (hmb : b ∈ l) (he : (a.fst == b.fst) = true) (hd : DistinctKeys l) : a = b

theorem Std.Internal.List.minEntry?_eq_head? {α : Type u} {β : α → Type v} [Ord α] {l : List ((a : α) × β a)} (hl : List.Pairwise (fun (a b : (a : α) × β a) => compare a.fst b.fst = Ordering.lt) l) : minEntry? l = l.head?

@[simp]

theorem Std.Internal.List.minEntry?_nil {α : Type u} {β : α → Type v} [Ord α] : minEntry? [] = none

theorem Std.Internal.List.isSome_minEntry?_of_isEmpty_eq_false {α : Type u} {β : α → Type v} [Ord α] {l : List ((a : α) × β a)} (hl : l.isEmpty = false) : (minEntry? l).isSome = true

theorem Std.Internal.List.minEntry?_eq_some_iff {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] (a : (a : α) × β a) {l : List ((a : α) × β a)} (hd : DistinctKeys l) : minEntry? l = some a ↔ a ∈ l ∧ ∀ (b : α), containsKey b l = true → (compare a.fst b).isLE = true

theorem Std.Internal.List.minKey?_eq_some_iff_getKey?_eq_self_and_forall {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {l : List ((a : α) × β a)} (hd : DistinctKeys l) : minKey? l = some k ↔ getKey? k l = some k ∧ ∀ (k' : α), containsKey k' l = true → (compare k k').isLE = true

theorem Std.Internal.List.minKey?_eq_some_iff_mem_and_forall {α : Type u} {β : α → Type v} [Ord α] [LawfulEqOrd α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {l : List ((a : α) × β a)} (hd : DistinctKeys l) : minKey? l = some k ↔ containsKey k l = true ∧ ∀ (k' : α), containsKey k' l = true → (compare k k').isLE = true

theorem Std.Internal.List.minEntry?_of_perm {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l l' : List ((a : α) × β a)} (hl : DistinctKeys l) (hp : l.Perm l') : minEntry? l = minEntry? l'

theorem Std.Internal.List.minKey?_of_perm {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l l' : List ((a : α) × β a)} (hl : DistinctKeys l) (hp : l.Perm l') : minKey? l = minKey? l'

theorem Std.Internal.List.minEntry?_eq_none_iff_isEmpty {α : Type u} {β : α → Type v} [Ord α] {l : List ((a : α) × β a)} : minEntry? l = none ↔ l.isEmpty = true

theorem Std.Internal.List.minKey?_eq_none_iff_isEmpty {α : Type u} {β : α → Type v} [Ord α] {l : List ((a : α) × β a)} : minKey? l = none ↔ l.isEmpty = true

theorem Std.Internal.List.minKey?_of_isEmpty {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] {l : List ((a : α) × β a)} (he : l.isEmpty = true) : minKey? l = none

theorem Std.Internal.List.isNone_minEntry?_eq_isEmpty {α : Type u} {β : α → Type v} [Ord α] {l : List ((a : α) × β a)} : (minEntry? l).isNone = l.isEmpty

theorem Std.Internal.List.isNone_minKey?_eq_isEmpty {α : Type u} {β : α → Type v} [Ord α] {l : List ((a : α) × β a)} : (minKey? l).isNone = l.isEmpty

theorem Std.Internal.List.isSome_minEntry?_eq_not_isEmpty {α : Type u} {β : α → Type v} [Ord α] {l : List ((a : α) × β a)} : (minEntry? l).isSome = !l.isEmpty

theorem Std.Internal.List.isSome_minKey?_eq_not_isEmpty {α : Type u} {β : α → Type v} [Ord α] {l : List ((a : α) × β a)} : (minKey? l).isSome = !l.isEmpty

theorem Std.Internal.List.isSome_minKey?_iff_isEmpty_eq_false {α : Type u} {β : α → Type v} [Ord α] {l : List ((a : α) × β a)} : (minKey? l).isSome = true ↔ l.isEmpty = false

theorem Std.Internal.List.minEntry?_map {α : Type u} {β : α → Type v} [Ord α] (l : List ((a : α) × β a)) (f : (a : α) × β a → (a : α) × β a) (hf : ∀ (e₁ e₂ : (a : α) × β a), compare e₁.fst e₂.fst = compare (f e₁).fst (f e₂).fst) : minEntry? (List.map f l) = Option.map f (minEntry? l)

theorem Std.Internal.List.replaceEntry_eq_map {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hl : DistinctKeys l) : replaceEntry k v l = List.map (fun (e : (a : α) × β a) => if (e.fst == k) = true then ⟨k, v⟩ else e) l

theorem Std.Internal.List.minEntry?_replaceEntry {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hl : DistinctKeys l) : minEntry? (replaceEntry k v l) = Option.map (fun (e : (a : α) × β a) => if (e.fst == k) = true then ⟨k, v⟩ else e) (minEntry? l)

theorem Std.Internal.List.isSome_minEntry?_of_contains {α : Type u} {β : α → Type v} [Ord α] [BEq α] {l : List ((a : α) × β a)} {b : α} (hb : containsKey b l = true) : (minEntry? l).isSome = true

theorem Std.Internal.List.minEntry?_insertEntry {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hl : DistinctKeys l) : minEntry? (insertEntry k v l) = some (match minEntry? l with | none => ⟨k, v⟩ | some w => if (compare k w.fst).isLE = true then ⟨k, v⟩ else w)

theorem Std.Internal.List.minKey?_insertEntry {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hl : DistinctKeys l) : minKey? (insertEntry k v l) = some ((minKey? l).elim k fun (k' : α) => if (compare k k').isLE = true then k else k')

theorem Std.Internal.List.isSome_minEntry?_insert {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hl : DistinctKeys l) : (minEntry? (insertEntry k v l)).isSome = true

theorem Std.Internal.List.isSome_minKey?_insertEntry {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hl : DistinctKeys l) : (minKey? (insertEntry k v l)).isSome = true

theorem Std.Internal.List.isSome_minEntry?_of_containsKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {l : List ((a : α) × β a)} (hc : containsKey k l = true) : (minEntry? l).isSome = true

theorem Std.Internal.List.isSome_minKey?_of_containsKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {l : List ((a : α) × β a)} (hc : containsKey k l = true) : (minKey? l).isSome = true

theorem Std.Internal.List.getEntry?_minKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {em : (a : α) × β a} (hem : minEntry? l = some em) : getEntry? em.fst l = some em

theorem Std.Internal.List.minKey?_bind_getEntry? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) : ((minKey? l).bind fun (k : α) => getEntry? k l) = minEntry? l

theorem Std.Internal.List.getKey?_minKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {km : α} (hkm : minKey? l = some km) : getKey? km l = some km

theorem Std.Internal.List.getKey_minKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {km : α} {hc : containsKey km l = true} (hkm : (minKey? l).get ⋯ = km) : getKey km l hc = km

theorem Std.Internal.List.getKey!_minKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [Inhabited α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {km : α} (hkm : minKey? l = some km) : getKey! km l = km

theorem Std.Internal.List.getKeyD_minKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {km fallback : α} (hkm : minKey? l = some km) : getKeyD km l fallback = km

theorem Std.Internal.List.minKey?_bind_getKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) : ((minKey? l).bind fun (k : α) => getKey? k l) = minKey? l

theorem Std.Internal.List.containsKey_minKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {km : α} (hkm : minKey? l = some km) : containsKey km l = true

theorem Std.Internal.List.min?_keys {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [LawfulEqOrd α] [LE α] [LawfulOrderOrd α] [Min α] [LawfulOrderLeftLeaningMin α] (l : List ((a : α) × β a)) : (keys l).min? = minKey? l

theorem Std.Internal.List.head?_keys {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [LawfulEqOrd α] [LE α] [LawfulOrderOrd α] [Min α] [LawfulOrderLeftLeaningMin α] (l : List ((a : α) × β a)) (hs : List.Pairwise (fun (a b : (a : α) × β a) => compare a.fst b.fst = Ordering.lt) l) : (keys l).head? = minKey? l

theorem Std.Internal.List.minKey?_eraseKey_eq_iff_beq_minKey?_eq_false {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {l : List ((a : α) × β a)} (hd : DistinctKeys l) : minKey? (eraseKey k l) = minKey? l ↔ ∀ {km : α}, minKey? l = some km → (k == km) = false

theorem Std.Internal.List.minKey?_eraseKey_eq_of_beq_minKey?_eq_false {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {l : List ((a : α) × β a)} (hd : DistinctKeys l) (hc : ∀ {km : α}, minKey? l = some km → (k == km) = false) : minKey? (eraseKey k l) = minKey? l

theorem Std.Internal.List.minKey?_insertEntry_le_minKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hl : DistinctKeys l) {km kmi : α} (hkm : minKey? l = some km) (hkmi : (minKey? (insertEntry k v l)).get ⋯ = kmi) : (compare kmi km).isLE = true

theorem Std.Internal.List.minKey?_insertEntry_le_self {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hl : DistinctKeys l) {kmi : α} (hkmi : (minKey? (insertEntry k v l)).get ⋯ = kmi) : (compare kmi k).isLE = true

theorem Std.Internal.List.minKey?_le_of_containsKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k km : α} {l : List ((a : α) × β a)} (hd : DistinctKeys l) (hc : containsKey k l = true) (hkm : (minKey? l).get ⋯ = km) : (compare km k).isLE = true

theorem Std.Internal.List.le_minKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {l : List ((a : α) × β a)} (hd : DistinctKeys l) : (∀ (k' : α), minKey? l = some k' → (compare k k').isLE = true) ↔ ∀ (k' : α), containsKey k' l = true → (compare k k').isLE = true

theorem Std.Internal.List.isSome_minKey?_of_isSome_minKey?_eraseKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {l : List ((a : α) × β a)} (hs : (minKey? (eraseKey k l)).isSome = true) : (minKey? l).isSome = true

theorem Std.Internal.List.containsKey_minKey?_eraseKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {l : List ((a : α) × β a)} (hd : DistinctKeys l) {kme : α} (hkme : minKey? (eraseKey k l) = some kme) : containsKey kme l = true

theorem Std.Internal.List.minKey?_le_minKey?_eraseKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k km kme : α} {l : List ((a : α) × β a)} (hd : DistinctKeys l) (hkme : minKey? (eraseKey k l) = some kme) (hkm : (minKey? l).get ⋯ = km) : (compare km kme).isLE = true

theorem Std.Internal.List.minKey?_cons {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] (e : (a : α) × β a) (l : List ((a : α) × β a)) : minKey? (e :: l) = some (match minKey? l with | none => e.fst | some w => if (compare e.fst w).isLE = true then e.fst else w)

theorem Std.Internal.List.minEntry?_insertEntryIfNew {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hd : DistinctKeys l) : minEntry? (insertEntryIfNew k v l) = some (match minEntry? l with | none => ⟨k, v⟩ | some e => if compare k e.fst = Ordering.lt then ⟨k, v⟩ else e)

theorem Std.Internal.List.minKey?_insertEntryIfNew {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hd : DistinctKeys l) : minKey? (insertEntryIfNew k v l) = some ((minKey? l).elim k fun (k' : α) => if compare k k' = Ordering.lt then k else k')

theorem Std.Internal.List.isSome_minEntry?_insertIfNew {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hl : DistinctKeys l) : (minEntry? (insertEntryIfNew k v l)).isSome = true

theorem Std.Internal.List.isSome_minKey?_insertEntryIfNew {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hl : DistinctKeys l) : (minKey? (insertEntryIfNew k v l)).isSome = true

theorem Std.Internal.List.minKey?_insertEntryIfNew_le_minKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hl : DistinctKeys l) {km kmi : α} (hkm : minKey? l = some km) (hkmi : (minKey? (insertEntryIfNew k v l)).get ⋯ = kmi) : (compare kmi km).isLE = true

theorem Std.Internal.List.minKey?_insertEntryIfNew_le_self {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k kmi : α} {v : β k} {l : List ((a : α) × β a)} (hl : DistinctKeys l) (hkmi : (minKey? (insertEntryIfNew k v l)).get ⋯ = kmi) : (compare kmi k).isLE = true

theorem Std.Internal.List.minKey?_eq_head?_keys {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (ho : List.Pairwise (fun (a b : (a : α) × β a) => compare a.fst b.fst = Ordering.lt) l) : minKey? l = (keys l).head?

theorem Std.Internal.List.minKey?_modifyKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {k : α} {f : β k → β k} {l : List ((a : α) × β a)} (hd : DistinctKeys l) : minKey? (modifyKey k f l) = minKey? l

theorem Std.Internal.List.minKey?_alterKey_eq_self {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {k : α} {f : Option (β k) → Option (β k)} {l : List ((a : α) × β a)} (hd : DistinctKeys l) : minKey? (alterKey k f l) = some k ↔ (f (getValueCast? k l)).isSome = true ∧ ∀ (k' : α), containsKey k' l = true → (compare k k').isLE = true

theorem Std.Internal.List.Const.minKey?_modifyKey {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {f : β → β} {l : List ((_ : α) × β)} (hd : DistinctKeys l) : minKey? (modifyKey k f l) = Option.map (fun (km : α) => if (km == k) = true then k else km) (minKey? l)

theorem Std.Internal.List.Const.minKey?_modifyKey_eq_minKey? {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {k : α} {f : β → β} {l : List ((_ : α) × β)} (hd : DistinctKeys l) : minKey? (modifyKey k f l) = minKey? l

theorem Std.Internal.List.Const.isSome_minKey?_modifyKey {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {f : β → β} {l : List ((_ : α) × β)} : (minKey? (modifyKey k f l)).isSome = !l.isEmpty

theorem Std.Internal.List.Const.isSome_minKey?_modifyKey_eq_isSome {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {f : β → β} {l : List ((_ : α) × β)} : (minKey? (modifyKey k f l)).isSome = (minKey? l).isSome

theorem Std.Internal.List.Const.minKey?_modifyKey_beq {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {f : β → β} {km kmm : α} {l : List ((_ : α) × β)} (hd : DistinctKeys l) (hkm : minKey? l = some km) (hkmm : (minKey? (modifyKey k f l)).get ⋯ = kmm) : (kmm == km) = true

theorem Std.Internal.List.Const.minKey?_alterKey_eq_self {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {f : Option β → Option β} {l : List ((_ : α) × β)} (hd : DistinctKeys l) : minKey? (alterKey k f l) = some k ↔ (f (getValue? k l)).isSome = true ∧ ∀ (k' : α), containsKey k' l = true → (compare k k').isLE = true

def Std.Internal.List.minKey {α : Type u} {β : α → Type v} [Ord α] (xs : List ((a : α) × β a)) (h : xs.isEmpty = false) : α

Given a proof that the list is nonempty, returns the smallest key in an associative list.

theorem Std.Internal.List.minKey_of_perm {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l l' : List ((a : α) × β a)} {hl : l.isEmpty = false} (hd : DistinctKeys l) (hp : l.Perm l') : minKey l hl = minKey l' ⋯

theorem Std.Internal.List.minKey_eq_get_minKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} {he : l.isEmpty = false} : minKey l he = (minKey? l).get ⋯

theorem Std.Internal.List.minKey?_eq_some_minKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} {he : l.isEmpty = false} : minKey? l = some (minKey l he)

theorem Std.Internal.List.minKey_eq_iff_getKey?_eq_self_and_forall {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he : l.isEmpty = false} {km : α} : minKey l he = km ↔ getKey? km l = some km ∧ ∀ (k : α), containsKey k l = true → (compare km k).isLE = true

theorem Std.Internal.List.minKey_eq_iff_mem_and_forall {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he : l.isEmpty = false} {km : α} : minKey l he = km ↔ containsKey km l = true ∧ ∀ (k : α), containsKey k l = true → (compare km k).isLE = true

theorem Std.Internal.List.minKey_insertEntry {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} : minKey (insertEntry k v l) ⋯ = (minKey? l).elim k fun (k' : α) => if (compare k k').isLE = true then k else k'

theorem Std.Internal.List.minKey_insertEntry_le_minKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} {he : l.isEmpty = false} : (compare (minKey (insertEntry k v l) ⋯) (minKey l he)).isLE = true

theorem Std.Internal.List.minKey_insertEntry_le_self {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} : (compare (minKey (insertEntry k v l) ⋯) k).isLE = true

theorem Std.Internal.List.containsKey_minKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he : l.isEmpty = false} : containsKey (minKey l he) l = true

theorem Std.Internal.List.minKey_le_of_containsKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} (hc : containsKey k l = true) : (compare (minKey l ⋯) k).isLE = true

theorem Std.Internal.List.le_minKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {he : l.isEmpty = false} : (compare k (minKey l he)).isLE = true ↔ ∀ (k' : α), containsKey k' l = true → (compare k k').isLE = true

theorem Std.Internal.List.getKey?_minKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he : l.isEmpty = false} : getKey? (minKey l he) l = some (minKey l he)

theorem Std.Internal.List.getKey_minKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he : l.isEmpty = false} : getKey (minKey l he) l ⋯ = minKey l he

theorem Std.Internal.List.getKey!_minKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he : l.isEmpty = false} : getKey! (minKey l he) l = minKey l he

theorem Std.Internal.List.getKeyD_minKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he : l.isEmpty = false} {fallback : α} : getKeyD (minKey l he) l fallback = minKey l he

theorem Std.Internal.List.minKey_eraseKey_eq_iff_beq_minKey_eq_false {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {he : (eraseKey k l).isEmpty = false} : minKey (eraseKey k l) he = minKey l ⋯ ↔ (k == minKey l ⋯) = false

theorem Std.Internal.List.minKey_eraseKey_eq_of_beq_minKey_eq_false {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {he : (eraseKey k l).isEmpty = false} (hc : (k == minKey l ⋯) = false) : minKey (eraseKey k l) he = minKey l ⋯

theorem Std.Internal.List.minKey_le_minKey_erase {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {he : (eraseKey k l).isEmpty = false} : (compare (minKey l ⋯) (minKey (eraseKey k l) he)).isLE = true

theorem Std.Internal.List.minKey_insertEntryIfNew {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} : minKey (insertEntryIfNew k v l) ⋯ = (minKey? l).elim k fun (k' : α) => if compare k k' = Ordering.lt then k else k'

theorem Std.Internal.List.minKey_insertEntryIfNew_le_minKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} {he : l.isEmpty = false} : (compare (minKey (insertEntryIfNew k v l) ⋯) (minKey l he)).isLE = true

theorem Std.Internal.List.minKey_insertEntryIfNew_le_self {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} : (compare (minKey (insertEntryIfNew k v l) ⋯) k).isLE = true

theorem Std.Internal.List.minKey_eq_head_keys {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (ho : List.Pairwise (fun (a b : (a : α) × β a) => compare a.fst b.fst = Ordering.lt) l) {he : l.isEmpty = false} : minKey l he = (keys l).head ⋯

theorem Std.Internal.List.minKey_modifyKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {k : α} {f : β k → β k} {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he : (modifyKey k f l).isEmpty = false} : minKey (modifyKey k f l) he = minKey l ⋯

theorem Std.Internal.List.minKey_alterKey_eq_self {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {f : Option (β k) → Option (β k)} {he : (alterKey k f l).isEmpty = false} : minKey (alterKey k f l) he = k ↔ (f (getValueCast? k l)).isSome = true ∧ ∀ (k' : α), containsKey k' l = true → (compare k k').isLE = true

theorem Std.Internal.List.Const.minKey_modifyKey {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : β → β} {he : (modifyKey k f l).isEmpty = false} : minKey (modifyKey k f l) he = if (minKey l ⋯ == k) = true then k else minKey l ⋯

theorem Std.Internal.List.Const.minKey_modifyKey_eq_minKey {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : β → β} {he : (modifyKey k f l).isEmpty = false} : minKey (modifyKey k f l) he = minKey l ⋯

theorem Std.Internal.List.Const.minKey_modifyKey_beq {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : β → β} {he : (modifyKey k f l).isEmpty = false} : (minKey (modifyKey k f l) he == minKey l ⋯) = true

theorem Std.Internal.List.Const.minKey_alterKey_eq_self {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : Option β → Option β} {he : (alterKey k f l).isEmpty = false} : minKey (alterKey k f l) he = k ↔ (f (getValue? k l)).isSome = true ∧ ∀ (k' : α), containsKey k' l = true → (compare k k').isLE = true

def Std.Internal.List.minKey! {α : Type u} {β : α → Type v} [Ord α] [Inhabited α] (xs : List ((a : α) × β a)) : α

Returns the smallest key in an associative list or panics if the list is empty.

theorem Std.Internal.List.minKey!_of_perm {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l l' : List ((a : α) × β a)} (hd : DistinctKeys l) (hp : l.Perm l') : minKey! l = minKey! l'

theorem Std.Internal.List.minKey!_eq_get!_minKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} : minKey! l = (minKey? l).get!

theorem Std.Internal.List.minKey_eq_minKey! {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} {he : l.isEmpty = false} : minKey l he = minKey! l

theorem Std.Internal.List.minKey?_eq_some_minKey! {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (he : l.isEmpty = false) : minKey? l = some (minKey! l)

theorem Std.Internal.List.minKey!_eq_default {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (h : l.isEmpty = true) : minKey! l = default

theorem Std.Internal.List.minKey!_eq_iff_getKey?_eq_self_and_forall {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {km : α} : minKey! l = km ↔ getKey? km l = some km ∧ ∀ (k : α), containsKey k l = true → (compare km k).isLE = true

theorem Std.Internal.List.minKey!_eq_iff_mem_and_forall {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {km : α} : minKey! l = km ↔ containsKey km l = true ∧ ∀ (k : α), containsKey k l = true → (compare km k).isLE = true

theorem Std.Internal.List.minKey!_insertEntry {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} : minKey! (insertEntry k v l) = (minKey? l).elim k fun (k' : α) => if (compare k k').isLE = true then k else k'

theorem Std.Internal.List.minKey!_insertEntry_le_minKey! {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {k : α} {v : β k} : (compare (minKey! (insertEntry k v l)) (minKey! l)).isLE = true

theorem Std.Internal.List.minKey!_insertEntry_le_self {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} : (compare (minKey! (insertEntry k v l)) k).isLE = true

theorem Std.Internal.List.containsKey_minKey! {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) : containsKey (minKey! l) l = true

theorem Std.Internal.List.minKey!_le_of_containsKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} (hc : containsKey k l = true) : (compare (minKey! l) k).isLE = true

theorem Std.Internal.List.le_minKey! {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {k : α} : (compare k (minKey! l)).isLE = true ↔ ∀ (k' : α), containsKey k' l = true → (compare k k').isLE = true

theorem Std.Internal.List.getKey?_minKey! {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) : getKey? (minKey! l) l = some (minKey! l)

theorem Std.Internal.List.getKey_minKey! {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he : containsKey (minKey! l) l = true} : getKey (minKey! l) l he = minKey! l

theorem Std.Internal.List.getKey_minKey!_eq_minKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he : containsKey (minKey! l) l = true} : getKey (minKey! l) l he = minKey l ⋯

theorem Std.Internal.List.getKey!_minKey! {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) : getKey! (minKey! l) l = minKey! l

theorem Std.Internal.List.getKeyD_minKey! {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {fallback : α} : getKeyD (minKey! l) l fallback = minKey! l

theorem Std.Internal.List.minKey!_eraseKey_eq_iff_beq_minKey!_eq_false {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} (he : (eraseKey k l).isEmpty = false) : minKey! (eraseKey k l) = minKey! l ↔ (k == minKey! l) = false

theorem Std.Internal.List.minKey!_eraseKey_eq_of_beq_minKey!_eq_false {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} (he : (eraseKey k l).isEmpty = false) (heq : (k == minKey! l) = false) : minKey! (eraseKey k l) = minKey! l

theorem Std.Internal.List.minKey!_le_minKey!_erase {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} (he : (eraseKey k l).isEmpty = false) : (compare (minKey! l) (minKey! (eraseKey k l))).isLE = true

theorem Std.Internal.List.minKey!_insertEntryIfNew {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} : minKey! (insertEntryIfNew k v l) = (minKey? l).elim k fun (k' : α) => if compare k k' = Ordering.lt then k else k'

theorem Std.Internal.List.minKey!_insertEntryIfNew_le_minKey! {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {k : α} {v : β k} : (compare (minKey! (insertEntryIfNew k v l)) (minKey! l)).isLE = true

theorem Std.Internal.List.minKey!_insertEntryIfNew_le_self {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} : (compare (minKey! (insertEntryIfNew k v l)) k).isLE = true

theorem Std.Internal.List.minKey!_eq_head!_keys {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (ho : List.Pairwise (fun (a b : (a : α) × β a) => compare a.fst b.fst = Ordering.lt) l) : minKey! l = (keys l).head!

theorem Std.Internal.List.minKey!_modifyKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {f : β k → β k} : minKey! (modifyKey k f l) = minKey! l

theorem Std.Internal.List.minKey!_alterKey_eq_self {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {f : Option (β k) → Option (β k)} (he : (alterKey k f l).isEmpty = false) : minKey! (alterKey k f l) = k ↔ (f (getValueCast? k l)).isSome = true ∧ ∀ (k' : α), containsKey k' l = true → (compare k k').isLE = true

theorem Std.Internal.List.Const.minKey!_modifyKey {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : β → β} (he : (modifyKey k f l).isEmpty = false) : minKey! (modifyKey k f l) = if (minKey! l == k) = true then k else minKey! l

theorem Std.Internal.List.Const.minKey!_modifyKey_eq_minKey! {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] [Inhabited α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : β → β} : minKey! (modifyKey k f l) = minKey! l

theorem Std.Internal.List.Const.minKey!_modifyKey_beq {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : β → β} : (minKey! (modifyKey k f l) == minKey! l) = true

theorem Std.Internal.List.Const.minKey!_alterKey_eq_self {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : Option β → Option β} (he : (alterKey k f l).isEmpty = false) : minKey! (alterKey k f l) = k ↔ (f (getValue? k l)).isSome = true ∧ ∀ (k' : α), containsKey k' l = true → (compare k k').isLE = true

def Std.Internal.List.minKeyD {α : Type u} {β : α → Type v} [Ord α] (xs : List ((a : α) × β a)) (fallback : α) : α

Returns the smallest key in an associative list or fallback if the list is empty.

theorem Std.Internal.List.minKeyD_of_perm {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l l' : List ((a : α) × β a)} {fallback : α} (hd : DistinctKeys l) (hp : l.Perm l') : minKeyD l fallback = minKeyD l' fallback

theorem Std.Internal.List.minKeyD_eq_getD_minKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} {fallback : α} : minKeyD l fallback = (minKey? l).getD fallback

theorem Std.Internal.List.minKey_insertEntry_of_isEmpty {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hl : DistinctKeys l) (he : l.isEmpty = true) : minKey (insertEntry k v l) ⋯ = k

theorem Std.Internal.List.minKey?_insertEntry_of_isEmpty {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hl : DistinctKeys l) (he : l.isEmpty = true) : minKey? (insertEntry k v l) = some k

theorem Std.Internal.List.minKeyD_insertEntry_of_isEmpty {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hl : DistinctKeys l) (he : l.isEmpty = true) {fallback : α} : minKeyD (insertEntry k v l) fallback = k

theorem Std.Internal.List.minKey_eq_minKeyD {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} {he : l.isEmpty = false} {fallback : α} : minKey l he = minKeyD l fallback

theorem Std.Internal.List.minKey?_eq_some_minKeyD {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} {fallback : α} (he : l.isEmpty = false) : minKey? l = some (minKeyD l fallback)

theorem Std.Internal.List.minKey!_eq_minKeyD_default {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} : minKey! l = minKeyD l default

theorem Std.Internal.List.minKey!_insertEntry_of_isEmpty {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hl : DistinctKeys l) (he : l.isEmpty = true) : minKey! (insertEntry k v l) = k

theorem Std.Internal.List.minKey_insertEntryIfNew_of_isEmpty {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hl : DistinctKeys l) (he : l.isEmpty = true) : minKey (insertEntryIfNew k v l) ⋯ = k

theorem Std.Internal.List.minKey?_insertEntryIfNew_of_isEmpty {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hl : DistinctKeys l) (he : l.isEmpty = true) : minKey? (insertEntryIfNew k v l) = some k

theorem Std.Internal.List.minKeyD_insertEntryIfNew_of_isEmpty {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hl : DistinctKeys l) (he : l.isEmpty = true) {fallback : α} : minKeyD (insertEntryIfNew k v l) fallback = k

theorem Std.Internal.List.minKey!_insertEntryIfNew_of_isEmpty {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hl : DistinctKeys l) (he : l.isEmpty = true) : minKey! (insertEntryIfNew k v l) = k

theorem Std.Internal.List.minKeyD_eq_fallback {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} {fallback : α} (h : l.isEmpty = true) : minKeyD l fallback = fallback

theorem Std.Internal.List.minKeyD_eq_iff_getKey?_eq_self_and_forall {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {km fallback : α} : minKeyD l fallback = km ↔ getKey? km l = some km ∧ ∀ (k : α), containsKey k l = true → (compare km k).isLE = true

theorem Std.Internal.List.minKeyD_eq_iff_mem_and_forall {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {km fallback : α} : minKeyD l fallback = km ↔ containsKey km l = true ∧ ∀ (k : α), containsKey k l = true → (compare km k).isLE = true

theorem Std.Internal.List.minKeyD_insertEntry {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} {fallback : α} : minKeyD (insertEntry k v l) fallback = (minKey? l).elim k fun (k' : α) => if (compare k k').isLE = true then k else k'

theorem Std.Internal.List.minKeyD_insertEntry_le_minKeyD {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {k : α} {v : β k} {fallback : α} : (compare (minKeyD (insertEntry k v l) fallback) (minKeyD l fallback)).isLE = true

theorem Std.Internal.List.minKeyD_insertEntry_le_self {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} {fallback : α} : (compare (minKeyD (insertEntry k v l) fallback) k).isLE = true

theorem Std.Internal.List.containsKey_minKeyD {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {fallback : α} : containsKey (minKeyD l fallback) l = true

theorem Std.Internal.List.minKeyD_le_of_containsKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} (hc : containsKey k l = true) {fallback : α} : (compare (minKeyD l fallback) k).isLE = true

theorem Std.Internal.List.le_minKeyD {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {k fallback : α} : (compare k (minKeyD l fallback)).isLE = true ↔ ∀ (k' : α), containsKey k' l = true → (compare k k').isLE = true

theorem Std.Internal.List.getKey?_minKeyD {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {fallback : α} : getKey? (minKeyD l fallback) l = some (minKeyD l fallback)

theorem Std.Internal.List.getKey_minKeyD {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {fallback : α} {he : containsKey (minKeyD l fallback) l = true} : getKey (minKeyD l fallback) l he = minKeyD l fallback

theorem Std.Internal.List.getKey_minKeyD_eq_minKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {fallback : α} {he : containsKey (minKeyD l fallback) l = true} : getKey (minKeyD l fallback) l he = minKey l ⋯

theorem Std.Internal.List.getKey!_minKeyD {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {fallback : α} : getKey! (minKeyD l fallback) l = minKeyD l fallback

theorem Std.Internal.List.getKeyD_minKeyD {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {fallback fallback' : α} : getKeyD (minKeyD l fallback) l fallback' = minKeyD l fallback

theorem Std.Internal.List.minKeyD_eraseKey_eq_iff_beq_minKeyD_eq_false {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k fallback : α} (he : (eraseKey k l).isEmpty = false) : minKeyD (eraseKey k l) fallback = minKeyD l fallback ↔ (k == minKeyD l fallback) = false

theorem Std.Internal.List.minKeyD_eraseKey_eq_of_beq_minKeyD_eq_false {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k fallback : α} (he : (eraseKey k l).isEmpty = false) (heq : (k == minKeyD l fallback) = false) : minKeyD (eraseKey k l) fallback = minKeyD l fallback

theorem Std.Internal.List.minKeyD_le_minKeyD_erase {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} (he : (eraseKey k l).isEmpty = false) {fallback : α} : (compare (minKeyD l fallback) (minKeyD (eraseKey k l) fallback)).isLE = true

theorem Std.Internal.List.minKeyD_insertEntryIfNew {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} {fallback : α} : minKeyD (insertEntryIfNew k v l) fallback = (minKey? l).elim k fun (k' : α) => if compare k k' = Ordering.lt then k else k'

theorem Std.Internal.List.minKeyD_insertEntryIfNew_le_minKeyD {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {k : α} {v : β k} {fallback : α} : (compare (minKeyD (insertEntryIfNew k v l) fallback) (minKeyD l fallback)).isLE = true

theorem Std.Internal.List.minKeyD_insertEntryIfNew_le_self {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} {fallback : α} : (compare (minKeyD (insertEntryIfNew k v l) fallback) k).isLE = true

theorem Std.Internal.List.minKeyD_eq_headD_keys {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (ho : List.Pairwise (fun (a b : (a : α) × β a) => compare a.fst b.fst = Ordering.lt) l) {fallback : α} : minKeyD l fallback = (keys l).headD fallback

theorem Std.Internal.List.minKeyD_modifyKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {f : β k → β k} {fallback : α} : minKeyD (modifyKey k f l) fallback = minKeyD l fallback

theorem Std.Internal.List.minKeyD_alterKey_eq_self {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {f : Option (β k) → Option (β k)} {fallback : α} (he : (alterKey k f l).isEmpty = false) : minKeyD (alterKey k f l) fallback = k ↔ (f (getValueCast? k l)).isSome = true ∧ ∀ (k' : α), containsKey k' l = true → (compare k k').isLE = true

theorem Std.Internal.List.Const.minKeyD_modifyKey {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : β → β} (he : (modifyKey k f l).isEmpty = false) {fallback : α} : minKeyD (modifyKey k f l) fallback = if (minKeyD l fallback == k) = true then k else minKeyD l fallback

theorem Std.Internal.List.Const.minKeyD_modifyKey_eq_minKeyD {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : β → β} {fallback : α} : minKeyD (modifyKey k f l) fallback = minKeyD l fallback

theorem Std.Internal.List.Const.minKeyD_modifyKey_beq {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : β → β} {fallback : α} : (minKeyD (modifyKey k f l) fallback == minKeyD l fallback) = true

theorem Std.Internal.List.Const.minKeyD_alterKey_eq_self {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : Option β → Option β} (he : (alterKey k f l).isEmpty = false) {fallback : α} : minKeyD (alterKey k f l) fallback = k ↔ (f (getValue? k l)).isSome = true ∧ ∀ (k' : α), containsKey k' l = true → (compare k k').isLE = true

@[reducible, inline]

abbrev Std.Internal.List.maxKey? {α : Type u} {β : α → Type v} [Ord α] (xs : List ((a : α) × β a)) : Option α

Returns the largest key in an associative list.

theorem Std.Internal.List.maxKey?_eq_some_iff_getKey?_eq_self_and_forall {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {l : List ((a : α) × β a)} (hd : DistinctKeys l) : maxKey? l = some k ↔ getKey? k l = some k ∧ ∀ (k' : α), containsKey k' l = true → (compare k' k).isLE = true

theorem Std.Internal.List.maxKey?_eq_some_iff_mem_and_forall {α : Type u} {β : α → Type v} [Ord α] [LawfulEqOrd α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {l : List ((a : α) × β a)} (hd : DistinctKeys l) : maxKey? l = some k ↔ containsKey k l = true ∧ ∀ (k' : α), containsKey k' l = true → (compare k' k).isLE = true

theorem Std.Internal.List.maxKey?_of_perm {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l l' : List ((a : α) × β a)} (hl : DistinctKeys l) (hp : l.Perm l') : maxKey? l = maxKey? l'

theorem Std.Internal.List.maxKey?_eq_none_iff_isEmpty {α : Type u} {β : α → Type v} [Ord α] {l : List ((a : α) × β a)} : maxKey? l = none ↔ l.isEmpty = true

theorem Std.Internal.List.maxKey?_of_isEmpty {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] {l : List ((a : α) × β a)} (he : l.isEmpty = true) : maxKey? l = none

theorem Std.Internal.List.isNone_maxKey?_eq_isEmpty {α : Type u} {β : α → Type v} [Ord α] {l : List ((a : α) × β a)} : (maxKey? l).isNone = l.isEmpty

theorem Std.Internal.List.isSome_maxKey?_eq_not_isEmpty {α : Type u} {β : α → Type v} [Ord α] {l : List ((a : α) × β a)} : (maxKey? l).isSome = !l.isEmpty

theorem Std.Internal.List.isSome_maxKey?_iff_isEmpty_eq_false {α : Type u} {β : α → Type v} [Ord α] {l : List ((a : α) × β a)} : (maxKey? l).isSome = true ↔ l.isEmpty = false

theorem Std.Internal.List.maxKey?_insertEntry {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hd : DistinctKeys l) : maxKey? (insertEntry k v l) = some ((maxKey? l).elim k fun (k' : α) => if (compare k' k).isLE = true then k else k')

theorem Std.Internal.List.isSome_maxKey?_insertEntry {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hd : DistinctKeys l) : (maxKey? (insertEntry k v l)).isSome = true

theorem Std.Internal.List.isSome_maxKey?_of_containsKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {l : List ((a : α) × β a)} (hc : containsKey k l = true) : (maxKey? l).isSome = true

theorem Std.Internal.List.getKey?_maxKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {km : α} (hkm : maxKey? l = some km) : getKey? km l = some km

theorem Std.Internal.List.getKey_maxKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {km : α} {hc : containsKey km l = true} (hkm : (maxKey? l).get ⋯ = km) : getKey km l hc = km

theorem Std.Internal.List.getKey!_maxKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [Inhabited α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {km : α} (hkm : maxKey? l = some km) : getKey! km l = km

theorem Std.Internal.List.getKeyD_maxKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {km fallback : α} (hkm : maxKey? l = some km) : getKeyD km l fallback = km

theorem Std.Internal.List.maxKey?_bind_getKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) : ((maxKey? l).bind fun (k : α) => getKey? k l) = maxKey? l

theorem Std.Internal.List.containsKey_maxKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {km : α} (hkm : maxKey? l = some km) : containsKey km l = true

theorem Std.Internal.List.maxKey?_eraseKey_eq_iff_beq_maxKey?_eq_false {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {l : List ((a : α) × β a)} (hd : DistinctKeys l) : maxKey? (eraseKey k l) = maxKey? l ↔ ∀ {km : α}, maxKey? l = some km → (k == km) = false

theorem Std.Internal.List.maxKey?_eraseKey_eq_of_beq_maxKey?_eq_false {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {l : List ((a : α) × β a)} (hd : DistinctKeys l) (hc : ∀ {km : α}, maxKey? l = some km → (k == km) = false) : maxKey? (eraseKey k l) = maxKey? l

theorem Std.Internal.List.maxKey?_le_maxKey?_insertEntry {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hd : DistinctKeys l) {km kmi : α} (hkm : maxKey? l = some km) (hkmi : (maxKey? (insertEntry k v l)).get ⋯ = kmi) : (compare km kmi).isLE = true

theorem Std.Internal.List.self_le_maxKey?_insertEntry {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hd : DistinctKeys l) {kmi : α} (hkmi : (maxKey? (insertEntry k v l)).get ⋯ = kmi) : (compare k kmi).isLE = true

theorem Std.Internal.List.maxKey?_le_of_containsKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k km : α} {l : List ((a : α) × β a)} (hd : DistinctKeys l) (hc : containsKey k l = true) (hkm : (maxKey? l).get ⋯ = km) : (compare k km).isLE = true

theorem Std.Internal.List.maxKey?_le {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {l : List ((a : α) × β a)} (hd : DistinctKeys l) : (∀ (k' : α), maxKey? l = some k' → (compare k' k).isLE = true) ↔ ∀ (k' : α), containsKey k' l = true → (compare k' k).isLE = true

theorem Std.Internal.List.isSome_maxKey?_of_isSome_maxKey?_eraseKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {l : List ((a : α) × β a)} (hs : (maxKey? (eraseKey k l)).isSome = true) : (maxKey? l).isSome = true

theorem Std.Internal.List.containsKey_maxKey?_eraseKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {l : List ((a : α) × β a)} (hd : DistinctKeys l) {kme : α} (hkme : maxKey? (eraseKey k l) = some kme) : containsKey kme l = true

theorem Std.Internal.List.maxKey?_eraseKey_le_maxKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k km kme : α} {l : List ((a : α) × β a)} (hd : DistinctKeys l) (hkme : maxKey? (eraseKey k l) = some kme) (hkm : (maxKey? l).get ⋯ = km) : (compare kme km).isLE = true

theorem Std.Internal.List.maxKey?_insertEntryIfNew {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hd : DistinctKeys l) : maxKey? (insertEntryIfNew k v l) = some ((maxKey? l).elim k fun (k' : α) => if compare k' k = Ordering.lt then k else k')

theorem Std.Internal.List.isSome_maxKey?_insertEntryIfNew {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hd : DistinctKeys l) : (maxKey? (insertEntryIfNew k v l)).isSome = true

theorem Std.Internal.List.maxKey?_le_maxKey?_insertEntryIfNew {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k} {l : List ((a : α) × β a)} (hd : DistinctKeys l) {km kmi : α} (hkm : maxKey? l = some km) (hkmi : (maxKey? (insertEntryIfNew k v l)).get ⋯ = kmi) : (compare km kmi).isLE = true

theorem Std.Internal.List.self_le_maxKey?_insertEntryIfNew {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k kmi : α} {v : β k} {l : List ((a : α) × β a)} (hd : DistinctKeys l) (hkmi : (maxKey? (insertEntryIfNew k v l)).get ⋯ = kmi) : (compare k kmi).isLE = true

theorem Std.Internal.List.reverse_keys {α : Type u} {β : α → Type v} {l : List ((a : α) × β a)} : (keys l).reverse = keys l.reverse

theorem Std.Internal.List.maxKey?_eq_getLast?_keys {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (ho : List.Pairwise (fun (a b : (a : α) × β a) => compare a.fst b.fst = Ordering.lt) l) : maxKey? l = (keys l).getLast?

theorem Std.Internal.List.maxKey?_modifyKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {k : α} {f : β k → β k} {l : List ((a : α) × β a)} (hd : DistinctKeys l) : maxKey? (modifyKey k f l) = maxKey? l

theorem Std.Internal.List.maxKey?_alterKey_eq_self {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {k : α} {f : Option (β k) → Option (β k)} {l : List ((a : α) × β a)} (hd : DistinctKeys l) : maxKey? (alterKey k f l) = some k ↔ (f (getValueCast? k l)).isSome = true ∧ ∀ (k' : α), containsKey k' l = true → (compare k' k).isLE = true

theorem Std.Internal.List.Const.maxKey?_modifyKey {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {f : β → β} {l : List ((_ : α) × β)} (hd : DistinctKeys l) : maxKey? (modifyKey k f l) = Option.map (fun (km : α) => if (km == k) = true then k else km) (maxKey? l)

theorem Std.Internal.List.Const.maxKey?_modifyKey_eq_maxKey? {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {k : α} {f : β → β} {l : List ((_ : α) × β)} (hd : DistinctKeys l) : maxKey? (modifyKey k f l) = maxKey? l

theorem Std.Internal.List.Const.isSome_maxKey?_modifyKey {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {f : β → β} {l : List ((_ : α) × β)} : (maxKey? (modifyKey k f l)).isSome = !l.isEmpty

theorem Std.Internal.List.Const.isSome_maxKey?_modifyKey_eq_isSome {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {f : β → β} {l : List ((_ : α) × β)} : (maxKey? (modifyKey k f l)).isSome = (maxKey? l).isSome

theorem Std.Internal.List.Const.maxKey?_modifyKey_beq {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {f : β → β} {km kmm : α} {l : List ((_ : α) × β)} (hd : DistinctKeys l) (hkm : maxKey? l = some km) (hkmm : (maxKey? (modifyKey k f l)).get ⋯ = kmm) : (kmm == km) = true

theorem Std.Internal.List.Const.maxKey?_alterKey_eq_self {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {f : Option β → Option β} {l : List ((_ : α) × β)} (hd : DistinctKeys l) : maxKey? (alterKey k f l) = some k ↔ (f (getValue? k l)).isSome = true ∧ ∀ (k' : α), containsKey k' l = true → (compare k' k).isLE = true

@[reducible, inline]

abbrev Std.Internal.List.maxKey {α : Type u} {β : α → Type v} [Ord α] (xs : List ((a : α) × β a)) (h : xs.isEmpty = false) : α

Given a proof that the list is nonempty, returns the largest key in an associative list.

theorem Std.Internal.List.maxKey_of_perm {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l l' : List ((a : α) × β a)} {hl : l.isEmpty = false} (hd : DistinctKeys l) (hp : l.Perm l') : maxKey l hl = maxKey l' ⋯

theorem Std.Internal.List.maxKey_eq_get_maxKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} {he : l.isEmpty = false} : maxKey l he = (maxKey? l).get ⋯

theorem Std.Internal.List.maxKey?_eq_some_maxKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} {he : l.isEmpty = false} : maxKey? l = some (maxKey l he)

theorem Std.Internal.List.maxKey_eq_iff_getKey?_eq_self_and_forall {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he : l.isEmpty = false} {km : α} : maxKey l he = km ↔ getKey? km l = some km ∧ ∀ (k : α), containsKey k l = true → (compare k km).isLE = true

theorem Std.Internal.List.maxKey_eq_iff_mem_and_forall {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he : l.isEmpty = false} {km : α} : maxKey l he = km ↔ containsKey km l = true ∧ ∀ (k : α), containsKey k l = true → (compare k km).isLE = true

theorem Std.Internal.List.maxKey_insertEntry {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} : maxKey (insertEntry k v l) ⋯ = (maxKey? l).elim k fun (k' : α) => if (compare k' k).isLE = true then k else k'

theorem Std.Internal.List.maxKey_le_maxKey_insertEntry {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} {he : l.isEmpty = false} : (compare (maxKey l he) (maxKey (insertEntry k v l) ⋯)).isLE = true

theorem Std.Internal.List.self_le_maxKey_insertEntry {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} : (compare k (maxKey (insertEntry k v l) ⋯)).isLE = true

theorem Std.Internal.List.containsKey_maxKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he : l.isEmpty = false} : containsKey (maxKey l he) l = true

theorem Std.Internal.List.le_maxKey_of_containsKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} (hc : containsKey k l = true) : (compare k (maxKey l ⋯)).isLE = true

theorem Std.Internal.List.maxKey_le {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {he : l.isEmpty = false} : (compare (maxKey l he) k).isLE = true ↔ ∀ (k' : α), containsKey k' l = true → (compare k' k).isLE = true

theorem Std.Internal.List.getKey?_maxKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he : l.isEmpty = false} : getKey? (maxKey l he) l = some (maxKey l he)

theorem Std.Internal.List.getKey_maxKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he : l.isEmpty = false} : getKey (maxKey l he) l ⋯ = maxKey l he

theorem Std.Internal.List.getKey!_maxKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he : l.isEmpty = false} : getKey! (maxKey l he) l = maxKey l he

theorem Std.Internal.List.getKeyD_maxKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he : l.isEmpty = false} {fallback : α} : getKeyD (maxKey l he) l fallback = maxKey l he

theorem Std.Internal.List.maxKey_eraseKey_eq_iff_beq_maxKey_eq_false {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {he : (eraseKey k l).isEmpty = false} : maxKey (eraseKey k l) he = maxKey l ⋯ ↔ (k == maxKey l ⋯) = false

theorem Std.Internal.List.maxKey_eraseKey_eq_of_beq_maxKey_eq_false {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {he : (eraseKey k l).isEmpty = false} (hc : (k == maxKey l ⋯) = false) : maxKey (eraseKey k l) he = maxKey l ⋯

theorem Std.Internal.List.maxKey_eraseKey_le_maxKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {he : (eraseKey k l).isEmpty = false} : (compare (maxKey (eraseKey k l) he) (maxKey l ⋯)).isLE = true

theorem Std.Internal.List.maxKey_insertEntryIfNew {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} : maxKey (insertEntryIfNew k v l) ⋯ = (maxKey? l).elim k fun (k' : α) => if compare k' k = Ordering.lt then k else k'

theorem Std.Internal.List.maxKey_le_maxKey_insertEntryIfNew {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} {he : l.isEmpty = false} : (compare (maxKey l he) (maxKey (insertEntryIfNew k v l) ⋯)).isLE = true

theorem Std.Internal.List.self_le_maxKey_insertEntryIfNew {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} : (compare k (maxKey (insertEntryIfNew k v l) ⋯)).isLE = true

theorem Std.Internal.List.maxKey_eq_getLast_keys {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (ho : List.Pairwise (fun (a b : (a : α) × β a) => compare a.fst b.fst = Ordering.lt) l) {he : l.isEmpty = false} : maxKey l he = (keys l).getLast ⋯

theorem Std.Internal.List.maxKey_modifyKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {k : α} {f : β k → β k} {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he : (modifyKey k f l).isEmpty = false} : maxKey (modifyKey k f l) he = maxKey l ⋯

theorem Std.Internal.List.maxKey_alterKey_eq_self {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {f : Option (β k) → Option (β k)} {he : (alterKey k f l).isEmpty = false} : maxKey (alterKey k f l) he = k ↔ (f (getValueCast? k l)).isSome = true ∧ ∀ (k' : α), containsKey k' l = true → (compare k' k).isLE = true

theorem Std.Internal.List.Const.maxKey_modifyKey {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : β → β} {he : (modifyKey k f l).isEmpty = false} : maxKey (modifyKey k f l) he = if (maxKey l ⋯ == k) = true then k else maxKey l ⋯

theorem Std.Internal.List.Const.maxKey_modifyKey_eq_maxKey {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : β → β} {he : (modifyKey k f l).isEmpty = false} : maxKey (modifyKey k f l) he = maxKey l ⋯

theorem Std.Internal.List.Const.maxKey_modifyKey_beq {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : β → β} {he : (modifyKey k f l).isEmpty = false} : (maxKey (modifyKey k f l) he == maxKey l ⋯) = true

theorem Std.Internal.List.Const.maxKey_alterKey_eq_self {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : Option β → Option β} {he : (alterKey k f l).isEmpty = false} : maxKey (alterKey k f l) he = k ↔ (f (getValue? k l)).isSome = true ∧ ∀ (k' : α), containsKey k' l = true → (compare k' k).isLE = true

def Std.Internal.List.maxKey! {α : Type u} {β : α → Type v} [Ord α] [Inhabited α] (xs : List ((a : α) × β a)) : α

Given a proof that the list is nonempty, returns the smallest key in an associative list.

theorem Std.Internal.List.maxKey!_of_perm {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l l' : List ((a : α) × β a)} (hd : DistinctKeys l) (hp : l.Perm l') : maxKey! l = maxKey! l'

theorem Std.Internal.List.maxKey!_eq_get!_maxKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} : maxKey! l = (maxKey? l).get!

theorem Std.Internal.List.maxKey_eq_maxKey! {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} {he : l.isEmpty = false} : maxKey l he = maxKey! l

theorem Std.Internal.List.maxKey?_eq_some_maxKey! {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (he : l.isEmpty = false) : maxKey? l = some (maxKey! l)

theorem Std.Internal.List.maxKey!_eq_default {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (h : l.isEmpty = true) : maxKey! l = default

theorem Std.Internal.List.maxKey!_eq_iff_getKey?_eq_self_and_forall {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {km : α} : maxKey! l = km ↔ getKey? km l = some km ∧ ∀ (k : α), containsKey k l = true → (compare k km).isLE = true

theorem Std.Internal.List.maxKey!_eq_iff_mem_and_forall {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {km : α} : maxKey! l = km ↔ containsKey km l = true ∧ ∀ (k : α), containsKey k l = true → (compare k km).isLE = true

theorem Std.Internal.List.maxKey!_insertEntry {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} : maxKey! (insertEntry k v l) = (maxKey? l).elim k fun (k' : α) => if (compare k' k).isLE = true then k else k'

theorem Std.Internal.List.maxKey!_le_maxKey!_insertEntry {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {k : α} {v : β k} : (compare (maxKey! l) (maxKey! (insertEntry k v l))).isLE = true

theorem Std.Internal.List.self_le_maxKey!_insertEntry {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} : (compare k (maxKey! (insertEntry k v l))).isLE = true

theorem Std.Internal.List.containsKey_maxKey! {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) : containsKey (maxKey! l) l = true

theorem Std.Internal.List.le_maxKey!_of_containsKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} (hc : containsKey k l = true) : (compare k (maxKey! l)).isLE = true

theorem Std.Internal.List.maxKey!_le {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {k : α} : (compare (maxKey! l) k).isLE = true ↔ ∀ (k' : α), containsKey k' l = true → (compare k' k).isLE = true

theorem Std.Internal.List.getKey?_maxKey! {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) : getKey? (maxKey! l) l = some (maxKey! l)

theorem Std.Internal.List.getKey_maxKey! {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he : containsKey (maxKey! l) l = true} : getKey (maxKey! l) l he = maxKey! l

theorem Std.Internal.List.getKey_maxKey!_eq_maxKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he : containsKey (maxKey! l) l = true} : getKey (maxKey! l) l he = maxKey l ⋯

theorem Std.Internal.List.getKey!_maxKey! {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) : getKey! (maxKey! l) l = maxKey! l

theorem Std.Internal.List.getKeyD_maxKey! {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {fallback : α} : getKeyD (maxKey! l) l fallback = maxKey! l

theorem Std.Internal.List.maxKey!_eraseKey_eq_iff_beq_maxKey!_eq_false {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} (he : (eraseKey k l).isEmpty = false) : maxKey! (eraseKey k l) = maxKey! l ↔ (k == maxKey! l) = false

theorem Std.Internal.List.maxKey!_eraseKey_eq_of_beq_maxKey!_eq_false {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} (he : (eraseKey k l).isEmpty = false) (heq : (k == maxKey! l) = false) : maxKey! (eraseKey k l) = maxKey! l

theorem Std.Internal.List.maxKey!_erase_le_maxKey! {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} (he : (eraseKey k l).isEmpty = false) : (compare (maxKey! (eraseKey k l)) (maxKey! l)).isLE = true

theorem Std.Internal.List.maxKey!_insertEntryIfNew {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} : maxKey! (insertEntryIfNew k v l) = (maxKey? l).elim k fun (k' : α) => if compare k' k = Ordering.lt then k else k'

theorem Std.Internal.List.maxKey!_le_maxKey!_insertEntryIfNew {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {k : α} {v : β k} : (compare (maxKey! l) (maxKey! (insertEntryIfNew k v l))).isLE = true

theorem Std.Internal.List.self_le_maxKey!_insertEntryIfNew {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} : (compare k (maxKey! (insertEntryIfNew k v l))).isLE = true

theorem Std.Internal.List.maxKey!_eq_getLast!_keys {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (ho : List.Pairwise (fun (a b : (a : α) × β a) => compare a.fst b.fst = Ordering.lt) l) : maxKey! l = (keys l).getLast!

theorem Std.Internal.List.maxKey!_modifyKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {f : β k → β k} : maxKey! (modifyKey k f l) = maxKey! l

theorem Std.Internal.List.maxKey!_alterKey_eq_self {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {f : Option (β k) → Option (β k)} (he : (alterKey k f l).isEmpty = false) : maxKey! (alterKey k f l) = k ↔ (f (getValueCast? k l)).isSome = true ∧ ∀ (k' : α), containsKey k' l = true → (compare k' k).isLE = true

theorem Std.Internal.List.Const.maxKey!_modifyKey {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : β → β} (he : (modifyKey k f l).isEmpty = false) : maxKey! (modifyKey k f l) = if (maxKey! l == k) = true then k else maxKey! l

theorem Std.Internal.List.Const.maxKey!_modifyKey_eq_maxKey! {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] [Inhabited α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : β → β} : maxKey! (modifyKey k f l) = maxKey! l

theorem Std.Internal.List.Const.maxKey!_modifyKey_beq {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : β → β} : (maxKey! (modifyKey k f l) == maxKey! l) = true

theorem Std.Internal.List.Const.maxKey!_alterKey_eq_self {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : Option β → Option β} (he : (alterKey k f l).isEmpty = false) : maxKey! (alterKey k f l) = k ↔ (f (getValue? k l)).isSome = true ∧ ∀ (k' : α), containsKey k' l = true → (compare k' k).isLE = true

@[reducible, inline]

abbrev Std.Internal.List.maxKeyD {α : Type u} {β : α → Type v} [Ord α] (xs : List ((a : α) × β a)) (fallback : α) : α

Returns the smallest key in an associative list or fallback if the list is empty.

theorem Std.Internal.List.maxKeyD_of_perm {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l l' : List ((a : α) × β a)} {fallback : α} (hd : DistinctKeys l) (hp : l.Perm l') : maxKeyD l fallback = maxKeyD l' fallback

theorem Std.Internal.List.maxKeyD_eq_getD_maxKey? {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} {fallback : α} : maxKeyD l fallback = (maxKey? l).getD fallback

theorem Std.Internal.List.maxKey_eq_maxKeyD {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} {he : l.isEmpty = false} {fallback : α} : maxKey l he = maxKeyD l fallback

theorem Std.Internal.List.maxKey?_eq_some_maxKeyD {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} {fallback : α} (he : l.isEmpty = false) : maxKey? l = some (maxKeyD l fallback)

theorem Std.Internal.List.maxKey!_eq_maxKeyD_default {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} : maxKey! l = maxKeyD l default

theorem Std.Internal.List.maxKeyD_eq_fallback {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} {fallback : α} (h : l.isEmpty = true) : maxKeyD l fallback = fallback

theorem Std.Internal.List.maxKeyD_eq_iff_getKey?_eq_self_and_forall {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {km fallback : α} : maxKeyD l fallback = km ↔ getKey? km l = some km ∧ ∀ (k : α), containsKey k l = true → (compare k km).isLE = true

theorem Std.Internal.List.maxKeyD_eq_iff_mem_and_forall {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {km fallback : α} : maxKeyD l fallback = km ↔ containsKey km l = true ∧ ∀ (k : α), containsKey k l = true → (compare k km).isLE = true

theorem Std.Internal.List.maxKeyD_insertEntry {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} {fallback : α} : maxKeyD (insertEntry k v l) fallback = (maxKey? l).elim k fun (k' : α) => if (compare k' k).isLE = true then k else k'

theorem Std.Internal.List.maxKeyD_le_maxKeyD_insertEntry {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {k : α} {v : β k} {fallback : α} : (compare (maxKeyD l fallback) (maxKeyD (insertEntry k v l) fallback)).isLE = true

theorem Std.Internal.List.self_le_maxKeyD_insertEntry {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} {fallback : α} : (compare k (maxKeyD (insertEntry k v l) fallback)).isLE = true

theorem Std.Internal.List.containsKey_maxKeyD {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {fallback : α} : containsKey (maxKeyD l fallback) l = true

theorem Std.Internal.List.le_maxKeyD_of_containsKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} (hc : containsKey k l = true) {fallback : α} : (compare k (maxKeyD l fallback)).isLE = true

theorem Std.Internal.List.maxKeyD_le {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {k fallback : α} : (compare (maxKeyD l fallback) k).isLE = true ↔ ∀ (k' : α), containsKey k' l = true → (compare k' k).isLE = true

theorem Std.Internal.List.getKey?_maxKeyD {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {fallback : α} : getKey? (maxKeyD l fallback) l = some (maxKeyD l fallback)

theorem Std.Internal.List.getKey_maxKeyD {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {fallback : α} {he : containsKey (maxKeyD l fallback) l = true} : getKey (maxKeyD l fallback) l he = maxKeyD l fallback

theorem Std.Internal.List.getKey_maxKeyD_eq_maxKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {fallback : α} {he : containsKey (maxKeyD l fallback) l = true} : getKey (maxKeyD l fallback) l he = maxKey l ⋯

theorem Std.Internal.List.getKey!_maxKeyD {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {fallback : α} : getKey! (maxKeyD l fallback) l = maxKeyD l fallback

theorem Std.Internal.List.getKeyD_maxKeyD {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {fallback fallback' : α} : getKeyD (maxKeyD l fallback) l fallback' = maxKeyD l fallback

theorem Std.Internal.List.maxKeyD_eraseKey_eq_iff_beq_maxKeyD_eq_false {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k fallback : α} (he : (eraseKey k l).isEmpty = false) : maxKeyD (eraseKey k l) fallback = maxKeyD l fallback ↔ (k == maxKeyD l fallback) = false

theorem Std.Internal.List.maxKeyD_eraseKey_eq_of_beq_maxKeyD_eq_false {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k fallback : α} (he : (eraseKey k l).isEmpty = false) (heq : (k == maxKeyD l fallback) = false) : maxKeyD (eraseKey k l) fallback = maxKeyD l fallback

theorem Std.Internal.List.maxKeyD_eraseKey_le_maxKeyD {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} (he : (eraseKey k l).isEmpty = false) {fallback : α} : (compare (maxKeyD (eraseKey k l) fallback) (maxKeyD l fallback)).isLE = true

theorem Std.Internal.List.maxKeyD_insertEntryIfNew {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} {fallback : α} : maxKeyD (insertEntryIfNew k v l) fallback = (maxKey? l).elim k fun (k' : α) => if compare k' k = Ordering.lt then k else k'

theorem Std.Internal.List.maxKeyD_le_maxKeyD_insertEntryIfNew {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {k : α} {v : β k} {fallback : α} : (compare (maxKeyD l fallback) (maxKeyD (insertEntryIfNew k v l) fallback)).isLE = true

theorem Std.Internal.List.self_le_maxKeyD_insertEntryIfNew {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {v : β k} {fallback : α} : (compare k (maxKeyD (insertEntryIfNew k v l) fallback)).isLE = true

theorem Std.Internal.List.maxKeyD_eq_getLastD_keys {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (ho : List.Pairwise (fun (a b : (a : α) × β a) => compare a.fst b.fst = Ordering.lt) l) {fallback : α} : maxKeyD l fallback = (keys l).getLastD fallback

theorem Std.Internal.List.maxKeyD_modifyKey {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {f : β k → β k} {fallback : α} : maxKeyD (modifyKey k f l) fallback = maxKeyD l fallback

theorem Std.Internal.List.maxKeyD_alterKey_eq_self {α : Type u} {β : α → Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α} {f : Option (β k) → Option (β k)} {fallback : α} (he : (alterKey k f l).isEmpty = false) : maxKeyD (alterKey k f l) fallback = k ↔ (f (getValueCast? k l)).isSome = true ∧ ∀ (k' : α), containsKey k' l = true → (compare k' k).isLE = true

theorem Std.Internal.List.Const.maxKeyD_modifyKey {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : β → β} (he : (modifyKey k f l).isEmpty = false) {fallback : α} : maxKeyD (modifyKey k f l) fallback = if (maxKeyD l fallback == k) = true then k else maxKeyD l fallback

theorem Std.Internal.List.Const.maxKeyD_modifyKey_eq_maxKeyD {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : β → β} {fallback : α} : maxKeyD (modifyKey k f l) fallback = maxKeyD l fallback

theorem Std.Internal.List.Const.maxKeyD_modifyKey_beq {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : β → β} {fallback : α} : (maxKeyD (modifyKey k f l) fallback == maxKeyD l fallback) = true

theorem Std.Internal.List.Const.maxKeyD_alterKey_eq_self {α : Type u} {β : Type v} [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k : α} {f : Option β → Option β} (he : (alterKey k f l).isEmpty = false) {fallback : α} : maxKeyD (alterKey k f l) fallback = k ↔ (f (getValue? k l)).isSome = true ∧ ∀ (k' : α), containsKey k' l = true → (compare k' k).isLE = true

def Std.Internal.List.interSmallerFn {α : Type u} {β : α → Type v} [BEq α] (l sofar : List ((a : α) × β a)) (k : α) : List ((a : α) × β a)

Internal implementation detail of the hash map

theorem Std.Internal.List.distinctKeys_interSmallerFn {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] (l sofar : List ((a : α) × β a)) (k : α) (hs : DistinctKeys sofar) : DistinctKeys (interSmallerFn l sofar k)

theorem Std.Internal.List.getEntry?_interSmallerFn {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] (l sofar : List ((a : α) × β a)) (k k' : α) : getEntry? k' (interSmallerFn l sofar k) = (Option.filter (fun (kv : (a : α) × β a) => k == kv.fst) (getEntry? k' l)).or (getEntry? k' sofar)

def Std.Internal.List.interSmaller {α : Type u} {β : α → Type v} [BEq α] (l₁ l₂ : List ((a : α) × β a)) : List ((a : α) × β a)

Internal implementation detail of the hash map

@[simp]

theorem Std.Internal.List.Option.filter_false {α : Type u} {o : Option α} : Option.filter (fun (x : α) => false) o = none

theorem Std.Internal.List.Option.filter_or {α : Type u} {o : Option α} {p q : α → Bool} : Option.filter (fun (a : α) => p a || q a) o = (Option.filter p o).or (Option.filter q o)

theorem Std.Internal.List.getEntry?_interSmaller {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] (l₁ l₂ : List ((a : α) × β a)) (k : α) : getEntry? k (interSmaller l₁ l₂) = Option.filter (fun (kv : (a : α) × β a) => containsKey kv.fst l₂) (getEntry? k l₁)

theorem Std.Internal.List.distinctKeys_interSmaller {α : Type u} {β : α → Type v} [BEq α] [PartialEquivBEq α] {l₁ l₂ : List ((a : α) × β a)} : DistinctKeys (interSmaller l₁ l₂)

theorem Std.Internal.List.interSmaller_perm_filter {α : Type u} {β : α → Type v} [BEq α] [EquivBEq α] (l₁ l₂ : List ((a : α) × β a)) (h₁ : DistinctKeys l₁) : (interSmaller l₁ l₂).Perm (List.filter (fun (kv : (a : α) × β a) => containsKey kv.fst l₂) l₁)