Dependent tree map lemmas

This file contains lemmas about Std.DTreeMap. Most of the lemmas require TransCmp cmp for the comparison function cmp.

@[instance_reducible]

def Std.DTreeMap.instCoeTypeForall_3 {α : Type u} : Coe (Type v) (α → Type v)

Equations

• Std.DTreeMap.instCoeTypeForall_3 = { coe := fun (γ : Type ?u.1) (x : α) => γ }

@[simp]

theorem Std.DTreeMap.isEmpty_emptyc {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} : ∅.isEmpty = true

@[simp]

theorem Std.DTreeMap.isEmpty_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.insert k v).isEmpty = false

theorem Std.DTreeMap.mem_iff_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {k : α} : k ∈ t ↔ t.contains k = true

@[simp]

theorem Std.DTreeMap.contains_iff_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {k : α} : t.contains k = true ↔ k ∈ t

theorem Std.DTreeMap.contains_congr {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k k' : α} (hab : cmp k k' = Ordering.eq) : t.contains k = t.contains k'

theorem Std.DTreeMap.contains_eq_false_iff_not_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {k : α} : t.contains k = false ↔ ¬k ∈ t

theorem Std.DTreeMap.mem_congr {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k k' : α} (hab : cmp k k' = Ordering.eq) : k ∈ t ↔ k' ∈ t

@[simp]

theorem Std.DTreeMap.contains_emptyc {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {k : α} : ∅.contains k = false

@[simp]

theorem Std.DTreeMap.not_mem_emptyc {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {k : α} : ¬k ∈ ∅

theorem Std.DTreeMap.contains_of_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {a : α} : t.isEmpty = true → t.contains a = false

theorem Std.DTreeMap.not_mem_of_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {a : α} : t.isEmpty = true → ¬a ∈ t

theorem Std.DTreeMap.isEmpty_eq_false_iff_exists_contains_eq_true {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.isEmpty = false ↔ ∃ (a : α), t.contains a = true

theorem Std.DTreeMap.isEmpty_eq_false_iff_exists_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.isEmpty = false ↔ ∃ (a : α), a ∈ t

theorem Std.DTreeMap.isEmpty_eq_false_of_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {a : α} (hc : t.contains a = true) : t.isEmpty = false

theorem Std.DTreeMap.isEmpty_iff_forall_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.isEmpty = true ↔ ∀ (a : α), t.contains a = false

theorem Std.DTreeMap.isEmpty_iff_forall_not_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.isEmpty = true ↔ ∀ (a : α), ¬a ∈ t

@[simp]

theorem Std.DTreeMap.insert_eq_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {p : (a : α) × β a} : Insert.insert p t = t.insert p.fst p.snd

@[simp]

theorem Std.DTreeMap.singleton_eq_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {p : (a : α) × β a} : {p} = ∅.insert p.fst p.snd

@[simp]

theorem Std.DTreeMap.contains_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k a : α} {v : β k} : (t.insert k v).contains a = (cmp k a == Ordering.eq || t.contains a)

@[simp]

theorem Std.DTreeMap.mem_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k a : α} {v : β k} : a ∈ t.insert k v ↔ cmp k a = Ordering.eq ∨ a ∈ t

theorem Std.DTreeMap.contains_insert_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.insert k v).contains k = true

theorem Std.DTreeMap.mem_insert_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : k ∈ t.insert k v

theorem Std.DTreeMap.contains_of_contains_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k a : α} {v : β k} : (t.insert k v).contains a = true → cmp k a ≠ Ordering.eq → t.contains a = true

theorem Std.DTreeMap.mem_of_mem_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k a : α} {v : β k} : a ∈ t.insert k v → cmp k a ≠ Ordering.eq → a ∈ t

@[simp]

theorem Std.DTreeMap.size_emptyc {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} : ∅.size = 0

theorem Std.DTreeMap.isEmpty_eq_size_eq_zero {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} : t.isEmpty = (t.size == 0)

theorem Std.DTreeMap.size_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.insert k v).size = if t.contains k = true then t.size else t.size + 1

theorem Std.DTreeMap.size_le_size_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : t.size ≤ (t.insert k v).size

theorem Std.DTreeMap.size_insert_le {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.insert k v).size ≤ t.size + 1

@[simp]

theorem Std.DTreeMap.erase_emptyc {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {k : α} : ∅.erase k = ∅

@[simp]

theorem Std.DTreeMap.isEmpty_erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} : (t.erase k).isEmpty = (t.isEmpty || t.size == 1 && t.contains k)

theorem Std.DTreeMap.isEmpty_eq_isEmpty_erase_and_not_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (k : α) : t.isEmpty = ((t.erase k).isEmpty && !t.contains k)

theorem Std.DTreeMap.isEmpty_eq_false_of_isEmpty_erase_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} (he : (t.erase k).isEmpty = false) : t.isEmpty = false

@[simp]

theorem Std.DTreeMap.contains_erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k a : α} : (t.erase k).contains a = (cmp k a != Ordering.eq && t.contains a)

@[simp]

theorem Std.DTreeMap.mem_erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k a : α} : a ∈ t.erase k ↔ cmp k a ≠ Ordering.eq ∧ a ∈ t

theorem Std.DTreeMap.contains_of_contains_erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k a : α} : (t.erase k).contains a = true → t.contains a = true

theorem Std.DTreeMap.mem_of_mem_erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k a : α} : a ∈ t.erase k → a ∈ t

theorem Std.DTreeMap.size_erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} : (t.erase k).size = if t.contains k = true then t.size - 1 else t.size

theorem Std.DTreeMap.size_erase_le {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} : (t.erase k).size ≤ t.size

theorem Std.DTreeMap.size_le_size_erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} : t.size ≤ (t.erase k).size + 1

@[simp]

theorem Std.DTreeMap.containsThenInsert_fst {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.containsThenInsert k v).fst = t.contains k

@[simp]

theorem Std.DTreeMap.containsThenInsert_snd {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.containsThenInsert k v).snd = t.insert k v

@[simp]

theorem Std.DTreeMap.containsThenInsertIfNew_fst {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.containsThenInsertIfNew k v).fst = t.contains k

@[simp]

theorem Std.DTreeMap.containsThenInsertIfNew_snd {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.containsThenInsertIfNew k v).snd = t.insertIfNew k v

@[simp]

theorem Std.DTreeMap.get?_emptyc {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} : ∅.get? a = none

theorem Std.DTreeMap.get?_of_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} : t.isEmpty = true → t.get? a = none

theorem Std.DTreeMap.get?_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a k : α} {v : β k} : (t.insert k v).get? a = if h : cmp k a = Ordering.eq then some (cast ⋯ v) else t.get? a

@[simp]

theorem Std.DTreeMap.get?_insert_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {v : β k} : (t.insert k v).get? k = some v

theorem Std.DTreeMap.contains_eq_isSome_get? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} : t.contains a = (t.get? a).isSome

@[simp]

theorem Std.DTreeMap.isSome_get?_eq_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} : (t.get? a).isSome = t.contains a

theorem Std.DTreeMap.mem_iff_isSome_get? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} : a ∈ t ↔ (t.get? a).isSome = true

@[simp]

theorem Std.DTreeMap.isSome_get?_iff_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} : (t.get? a).isSome = true ↔ a ∈ t

theorem Std.DTreeMap.get?_eq_some_iff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {v : β k} : t.get? k = some v ↔ ∃ (h : k ∈ t), t.get k h = v

theorem Std.DTreeMap.get?_eq_none_of_contains_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} : t.contains a = false → t.get? a = none

theorem Std.DTreeMap.get?_eq_none {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} : ¬a ∈ t → t.get? a = none

theorem Std.DTreeMap.get?_erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k a : α} : (t.erase k).get? a = if cmp k a = Ordering.eq then none else t.get? a

@[simp]

theorem Std.DTreeMap.get?_erase_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} : (t.erase k).get? k = none

@[simp]

theorem Std.DTreeMap.Const.get?_emptyc {α : Type u} {cmp : α → α → Ordering} {β : Type v} [TransCmp cmp] {a : α} : get? ∅ a = none

theorem Std.DTreeMap.Const.get?_of_isEmpty {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {a : α} : t.isEmpty = true → get? t a = none

theorem Std.DTreeMap.Const.get?_insert {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {a k : α} {v : β} : get? (t.insert k v) a = if cmp k a = Ordering.eq then some v else get? t a

@[simp]

theorem Std.DTreeMap.Const.get?_insert_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {v : β} : get? (t.insert k v) k = some v

theorem Std.DTreeMap.Const.contains_eq_isSome_get? {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {a : α} : t.contains a = (get? t a).isSome

@[simp]

theorem Std.DTreeMap.Const.isSome_get?_eq_contains {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {a : α} : (get? t a).isSome = t.contains a

theorem Std.DTreeMap.Const.mem_iff_isSome_get? {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {a : α} : a ∈ t ↔ (get? t a).isSome = true

theorem Std.DTreeMap.Const.get?_eq_some_iff {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {v : β} : get? t k = some v ↔ ∃ (h : k ∈ t), get t k h = v

@[simp]

theorem Std.DTreeMap.Const.isSome_get?_iff_mem {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {a : α} : (get? t a).isSome = true ↔ a ∈ t

theorem Std.DTreeMap.Const.get?_eq_none_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {a : α} : t.contains a = false → get? t a = none

theorem Std.DTreeMap.Const.get?_eq_none {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {a : α} : ¬a ∈ t → get? t a = none

theorem Std.DTreeMap.Const.get?_erase {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k a : α} : get? (t.erase k) a = if cmp k a = Ordering.eq then none else get? t a

@[simp]

theorem Std.DTreeMap.Const.get?_erase_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} : get? (t.erase k) k = none

theorem Std.DTreeMap.Const.get?_eq_get? {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [LawfulEqCmp cmp] [TransCmp cmp] {a : α} : get? t a = t.get? a

theorem Std.DTreeMap.Const.get?_congr {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {a b : α} (hab : cmp a b = Ordering.eq) : get? t a = get? t b

theorem Std.DTreeMap.get_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k a : α} {v : β k} {h₁ : a ∈ t.insert k v} : (t.insert k v).get a h₁ = if h₂ : cmp k a = Ordering.eq then cast ⋯ v else t.get a ⋯

theorem Std.DTreeMap.toList_insert_perm {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [BEq α] [TransCmp cmp] [LawfulBEqCmp cmp] {k : α} {v : β k} : (t.insert k v).toList.Perm (⟨k, v⟩ :: List.filter (fun (x : (a : α) × β a) => decide ¬(k == x.fst) = true) t.toList)

theorem Std.DTreeMap.toArray_insert_perm {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [BEq α] [TransCmp cmp] [LawfulBEqCmp cmp] {k : α} {v : β k} : (t.insert k v).toArray.Perm ((Array.filter (fun (x : (a : α) × β a) => decide ¬(k == x.fst) = true) t.toArray).push ⟨k, v⟩)

theorem Std.DTreeMap.Const.toList_insert_perm {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [BEq α] [TransCmp cmp] [LawfulBEqCmp cmp] {k : α} {v : β} : (toList (t.insert k v)).Perm ((k, v) :: List.filter (fun (x : α × β) => decide ¬(k == x.fst) = true) (toList t))

theorem Std.DTreeMap.Const.toArray_insert_perm {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [BEq α] [TransCmp cmp] [LawfulBEqCmp cmp] {k : α} {v : β} : (toArray (t.insert k v)).Perm ((Array.filter (fun (x : α × β) => decide ¬(k == x.fst) = true) (toArray t)).push (k, v))

theorem Std.DTreeMap.keys_insertIfNew_perm {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [BEq α] [TransCmp cmp] [LawfulBEqCmp cmp] {k : α} {v : β k} : (t.insertIfNew k v).keys.Perm (if k ∈ t then t.keys else k :: t.keys)

theorem Std.DTreeMap.keysArray_insertIfNew_perm {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [BEq α] [TransCmp cmp] [LawfulBEqCmp cmp] {k : α} {v : β k} : (t.insertIfNew k v).keysArray.Perm (if k ∈ t then t.keysArray else t.keysArray.push k)

@[simp]

theorem Std.DTreeMap.get_insert_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {v : β k} : (t.insert k v).get k ⋯ = v

@[simp]

theorem Std.DTreeMap.get_erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k a : α} {h' : a ∈ t.erase k} : (t.erase k).get a h' = t.get a ⋯

theorem Std.DTreeMap.get?_eq_some_get {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} (h' : a ∈ t) : t.get? a = some (t.get a h')

theorem Std.DTreeMap.get_eq_get_get? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} {h : a ∈ t} : t.get a h = (t.get? a).get ⋯

theorem Std.DTreeMap.get_get? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} {h : (t.get? a).isSome = true} : (t.get? a).get h = t.get a ⋯

theorem Std.DTreeMap.Const.get_insert {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k a : α} {v : β} {h₁ : a ∈ t.insert k v} : get (t.insert k v) a h₁ = if h₂ : cmp k a = Ordering.eq then v else get t a ⋯

@[simp]

theorem Std.DTreeMap.Const.get_insert_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {v : β} : get (t.insert k v) k ⋯ = v

@[simp]

theorem Std.DTreeMap.Const.get_erase {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k a : α} {h' : a ∈ t.erase k} : get (t.erase k) a h' = get t a ⋯

theorem Std.DTreeMap.Const.get?_eq_some_get {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {a : α} (h : a ∈ t) : get? t a = some (get t a h)

theorem Std.DTreeMap.Const.get_eq_get_get? {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {a : α} {h : a ∈ t} : get t a h = (get? t a).get ⋯

theorem Std.DTreeMap.Const.get_get? {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {a : α} {h : (get? t a).isSome = true} : (get? t a).get h = get t a ⋯

theorem Std.DTreeMap.Const.get_eq_get {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} {h : a ∈ t} : get t a h = t.get a h

theorem Std.DTreeMap.Const.get_congr {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {a b : α} (hab : cmp a b = Ordering.eq) {h' : a ∈ t} : get t a h' = get t b ⋯

@[simp]

theorem Std.DTreeMap.get!_emptyc {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} [Inhabited (β a)] : ∅.get! a = default

theorem Std.DTreeMap.get!_of_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} [Inhabited (β a)] : t.isEmpty = true → t.get! a = default

theorem Std.DTreeMap.get!_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k a : α} [Inhabited (β a)] {v : β k} : (t.insert k v).get! a = if h : cmp k a = Ordering.eq then cast ⋯ v else t.get! a

@[simp]

theorem Std.DTreeMap.get!_insert_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} [Inhabited (β a)] {b : β a} : (t.insert a b).get! a = b

theorem Std.DTreeMap.get!_eq_default_of_contains_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} [Inhabited (β a)] : t.contains a = false → t.get! a = default

theorem Std.DTreeMap.get!_eq_default {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} [Inhabited (β a)] : ¬a ∈ t → t.get! a = default

theorem Std.DTreeMap.get!_erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k a : α} [Inhabited (β a)] : (t.erase k).get! a = if cmp k a = Ordering.eq then default else t.get! a

@[simp]

theorem Std.DTreeMap.get!_erase_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} [Inhabited (β k)] : (t.erase k).get! k = default

theorem Std.DTreeMap.get?_eq_some_get!_of_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} [Inhabited (β a)] : t.contains a = true → t.get? a = some (t.get! a)

theorem Std.DTreeMap.get?_eq_some_get! {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} [Inhabited (β a)] : a ∈ t → t.get? a = some (t.get! a)

theorem Std.DTreeMap.get!_eq_get!_get? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} [Inhabited (β a)] : t.get! a = (t.get? a).get!

theorem Std.DTreeMap.get_eq_get! {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} [Inhabited (β a)] {h : a ∈ t} : t.get a h = t.get! a

@[simp]

theorem Std.DTreeMap.Const.get!_emptyc {α : Type u} {cmp : α → α → Ordering} {β : Type v} [TransCmp cmp] [Inhabited β] {a : α} : get! ∅ a = default

theorem Std.DTreeMap.Const.get!_of_isEmpty {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {a : α} : t.isEmpty = true → get! t a = default

theorem Std.DTreeMap.Const.get!_insert {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {k a : α} {v : β} : get! (t.insert k v) a = if cmp k a = Ordering.eq then v else get! t a

@[simp]

theorem Std.DTreeMap.Const.get!_insert_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {k : α} {v : β} : get! (t.insert k v) k = v

theorem Std.DTreeMap.Const.get!_eq_default_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {a : α} : t.contains a = false → get! t a = default

theorem Std.DTreeMap.Const.get!_eq_default {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {a : α} : ¬a ∈ t → get! t a = default

theorem Std.DTreeMap.Const.get!_erase {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {k a : α} : get! (t.erase k) a = if cmp k a = Ordering.eq then default else get! t a

@[simp]

theorem Std.DTreeMap.Const.get!_erase_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {k : α} : get! (t.erase k) k = default

theorem Std.DTreeMap.Const.get?_eq_some_get!_of_contains {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {a : α} : t.contains a = true → get? t a = some (get! t a)

theorem Std.DTreeMap.Const.get?_eq_some_get! {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {a : α} : a ∈ t → get? t a = some (get! t a)

theorem Std.DTreeMap.Const.get!_eq_get!_get? {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {a : α} : get! t a = (get? t a).get!

theorem Std.DTreeMap.Const.get_eq_get! {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {a : α} {h : a ∈ t} : get t a h = get! t a

theorem Std.DTreeMap.Const.get!_eq_get! {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited β] {a : α} : get! t a = t.get! a

theorem Std.DTreeMap.Const.get!_congr {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {a b : α} (hab : cmp a b = Ordering.eq) : get! t a = get! t b

@[simp]

theorem Std.DTreeMap.getD_emptyc {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} {fallback : β a} : ∅.getD a fallback = fallback

theorem Std.DTreeMap.getD_of_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} {fallback : β a} : t.isEmpty = true → t.getD a fallback = fallback

theorem Std.DTreeMap.getD_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k a : α} {fallback : β a} {v : β k} : (t.insert k v).getD a fallback = if h : cmp k a = Ordering.eq then cast ⋯ v else t.getD a fallback

@[simp]

theorem Std.DTreeMap.getD_insert_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} {fallback b : β a} : (t.insert a b).getD a fallback = b

theorem Std.DTreeMap.getD_eq_fallback_of_contains_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} {fallback : β a} : t.contains a = false → t.getD a fallback = fallback

theorem Std.DTreeMap.getD_eq_fallback {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} {fallback : β a} : ¬a ∈ t → t.getD a fallback = fallback

theorem Std.DTreeMap.getD_erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k a : α} {fallback : β a} : (t.erase k).getD a fallback = if cmp k a = Ordering.eq then fallback else t.getD a fallback

@[simp]

theorem Std.DTreeMap.getD_erase_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {fallback : β k} : (t.erase k).getD k fallback = fallback

theorem Std.DTreeMap.get?_eq_some_getD_of_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} {fallback : β a} : t.contains a = true → t.get? a = some (t.getD a fallback)

theorem Std.DTreeMap.get?_eq_some_getD {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} {fallback : β a} : a ∈ t → t.get? a = some (t.getD a fallback)

theorem Std.DTreeMap.getD_eq_getD_get? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} {fallback : β a} : t.getD a fallback = (t.get? a).getD fallback

theorem Std.DTreeMap.get_eq_getD {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} {fallback : β a} {h : a ∈ t} : t.get a h = t.getD a fallback

theorem Std.DTreeMap.get!_eq_getD_default {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} [Inhabited (β a)] : t.get! a = t.getD a default

@[simp]

theorem Std.DTreeMap.Const.getD_emptyc {α : Type u} {cmp : α → α → Ordering} {β : Type v} [TransCmp cmp] {a : α} {fallback : β} : getD ∅ a fallback = fallback

theorem Std.DTreeMap.Const.getD_of_isEmpty {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {a : α} {fallback : β} : t.isEmpty = true → getD t a fallback = fallback

theorem Std.DTreeMap.Const.getD_insert {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k a : α} {fallback v : β} : getD (t.insert k v) a fallback = if cmp k a = Ordering.eq then v else getD t a fallback

@[simp]

theorem Std.DTreeMap.Const.getD_insert_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {fallback v : β} : getD (t.insert k v) k fallback = v

theorem Std.DTreeMap.Const.getD_eq_fallback_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {a : α} {fallback : β} : t.contains a = false → getD t a fallback = fallback

theorem Std.DTreeMap.Const.getD_eq_fallback {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {a : α} {fallback : β} : ¬a ∈ t → getD t a fallback = fallback

theorem Std.DTreeMap.Const.getD_erase {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k a : α} {fallback : β} : getD (t.erase k) a fallback = if cmp k a = Ordering.eq then fallback else getD t a fallback

@[simp]

theorem Std.DTreeMap.Const.getD_erase_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {fallback : β} : getD (t.erase k) k fallback = fallback

theorem Std.DTreeMap.Const.get?_eq_some_getD_of_contains {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {a : α} {fallback : β} : t.contains a = true → get? t a = some (getD t a fallback)

theorem Std.DTreeMap.Const.get?_eq_some_getD {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {a : α} {fallback : β} : a ∈ t → get? t a = some (getD t a fallback)

theorem Std.DTreeMap.Const.getD_eq_getD_get? {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {a : α} {fallback : β} : getD t a fallback = (get? t a).getD fallback

theorem Std.DTreeMap.Const.get_eq_getD {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {a : α} {fallback : β} {h : a ∈ t} : get t a h = getD t a fallback

theorem Std.DTreeMap.Const.get!_eq_getD_default {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {a : α} : get! t a = getD t a default

theorem Std.DTreeMap.Const.getD_eq_getD {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] {a : α} {fallback : β} : getD t a fallback = t.getD a fallback

theorem Std.DTreeMap.Const.getD_congr {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {a b : α} {fallback : β} (hab : cmp a b = Ordering.eq) : getD t a fallback = getD t b fallback

@[simp]

theorem Std.DTreeMap.getKey?_emptyc {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {a : α} : ∅.getKey? a = none

theorem Std.DTreeMap.getKey?_of_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {a : α} : t.isEmpty = true → t.getKey? a = none

theorem Std.DTreeMap.getKey?_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {a k : α} {v : β k} : (t.insert k v).getKey? a = if cmp k a = Ordering.eq then some k else t.getKey? a

@[simp]

theorem Std.DTreeMap.getKey?_insert_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.insert k v).getKey? k = some k

theorem Std.DTreeMap.contains_eq_isSome_getKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {a : α} : t.contains a = (t.getKey? a).isSome

@[simp]

theorem Std.DTreeMap.isSome_getKey?_eq_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {a : α} : (t.getKey? a).isSome = t.contains a

theorem Std.DTreeMap.mem_iff_isSome_getKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {a : α} : a ∈ t ↔ (t.getKey? a).isSome = true

@[simp]

theorem Std.DTreeMap.isSome_getKey?_iff_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {a : α} : (t.getKey? a).isSome = true ↔ a ∈ t

theorem Std.DTreeMap.mem_of_getKey?_eq_some {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k k' : α} : t.getKey? k = some k' → k' ∈ t

theorem Std.DTreeMap.getKey?_eq_some_iff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k k' : α} : t.getKey? k = some k' ↔ ∃ (h : k ∈ t), t.getKey k h = k'

theorem Std.DTreeMap.getKey?_eq_none_of_contains_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {a : α} : t.contains a = false → t.getKey? a = none

theorem Std.DTreeMap.getKey?_eq_none {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {a : α} : ¬a ∈ t → t.getKey? a = none

theorem Std.DTreeMap.getKey?_erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k a : α} : (t.erase k).getKey? a = if cmp k a = Ordering.eq then none else t.getKey? a

@[simp]

theorem Std.DTreeMap.getKey?_erase_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} : (t.erase k).getKey? k = none

theorem Std.DTreeMap.compare_getKey?_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} : Option.all (fun (x : α) => decide (cmp x k = Ordering.eq)) (t.getKey? k) = true

theorem Std.DTreeMap.getKey?_congr {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k k' : α} (h' : cmp k k' = Ordering.eq) : t.getKey? k = t.getKey? k'

theorem Std.DTreeMap.getKey?_eq_some_of_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} (h' : t.contains k = true) : t.getKey? k = some k

theorem Std.DTreeMap.getKey?_eq_some {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} (h' : k ∈ t) : t.getKey? k = some k

theorem Std.DTreeMap.getKey_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k a : α} {v : β k} {h₁ : a ∈ t.insert k v} : (t.insert k v).getKey a h₁ = if h₂ : cmp k a = Ordering.eq then k else t.getKey a ⋯

@[simp]

theorem Std.DTreeMap.getKey_insert_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.insert k v).getKey k ⋯ = k

@[simp]

theorem Std.DTreeMap.getKey_erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k a : α} {h' : a ∈ t.erase k} : (t.erase k).getKey a h' = t.getKey a ⋯

theorem Std.DTreeMap.getKey?_eq_some_getKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {a : α} (h' : a ∈ t) : t.getKey? a = some (t.getKey a h')

theorem Std.DTreeMap.getKey_eq_get_getKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {a : α} {h : a ∈ t} : t.getKey a h = (t.getKey? a).get ⋯

@[simp]

theorem Std.DTreeMap.get_getKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {a : α} {h : (t.getKey? a).isSome = true} : (t.getKey? a).get h = t.getKey a ⋯

theorem Std.DTreeMap.compare_getKey_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h' : k ∈ t) : cmp (t.getKey k h') k = Ordering.eq

theorem Std.DTreeMap.getKey_congr {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k₁ k₂ : α} (h' : cmp k₁ k₂ = Ordering.eq) (h₁ : k₁ ∈ t) : t.getKey k₁ h₁ = t.getKey k₂ ⋯

@[simp]

theorem Std.DTreeMap.getKey_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} (h' : k ∈ t) : t.getKey k h' = k

@[simp]

theorem Std.DTreeMap.getKey!_emptyc {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {a : α} [Inhabited α] : ∅.getKey! a = default

theorem Std.DTreeMap.getKey!_of_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {a : α} : t.isEmpty = true → t.getKey! a = default

theorem Std.DTreeMap.getKey!_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k a : α} {v : β k} : (t.insert k v).getKey! a = if cmp k a = Ordering.eq then k else t.getKey! a

@[simp]

theorem Std.DTreeMap.getKey!_insert_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {a : α} {b : β a} : (t.insert a b).getKey! a = a

theorem Std.DTreeMap.getKey!_eq_default_of_contains_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {a : α} : t.contains a = false → t.getKey! a = default

theorem Std.DTreeMap.getKey!_eq_default {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {a : α} : ¬a ∈ t → t.getKey! a = default

theorem Std.DTreeMap.getKey!_erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k a : α} : (t.erase k).getKey! a = if cmp k a = Ordering.eq then default else t.getKey! a

@[simp]

theorem Std.DTreeMap.getKey!_erase_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} : (t.erase k).getKey! k = default

theorem Std.DTreeMap.getKey?_eq_some_getKey!_of_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {a : α} : t.contains a = true → t.getKey? a = some (t.getKey! a)

theorem Std.DTreeMap.getKey?_eq_some_getKey! {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {a : α} : a ∈ t → t.getKey? a = some (t.getKey! a)

theorem Std.DTreeMap.getKey!_eq_get!_getKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {a : α} : t.getKey! a = (t.getKey? a).get!

theorem Std.DTreeMap.getKey_eq_getKey! {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {a : α} {h : a ∈ t} : t.getKey a h = t.getKey! a

theorem Std.DTreeMap.getKey!_congr {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k k' : α} (h' : cmp k k' = Ordering.eq) : t.getKey! k = t.getKey! k'

theorem Std.DTreeMap.getKey!_eq_of_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {k : α} (h' : t.contains k = true) : t.getKey! k = k

theorem Std.DTreeMap.getKey!_eq_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {k : α} (h' : k ∈ t) : t.getKey! k = k

@[simp]

theorem Std.DTreeMap.getKeyD_emptyc {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {a fallback : α} : ∅.getKeyD a fallback = fallback

theorem Std.DTreeMap.getKeyD_of_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {a fallback : α} : t.isEmpty = true → t.getKeyD a fallback = fallback

theorem Std.DTreeMap.getKeyD_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k a fallback : α} {v : β k} : (t.insert k v).getKeyD a fallback = if cmp k a = Ordering.eq then k else t.getKeyD a fallback

@[simp]

theorem Std.DTreeMap.getKeyD_insert_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {a fallback : α} {b : β a} : (t.insert a b).getKeyD a fallback = a

theorem Std.DTreeMap.getKeyD_eq_fallback_of_contains_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {a fallback : α} : t.contains a = false → t.getKeyD a fallback = fallback

theorem Std.DTreeMap.getKeyD_eq_fallback {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {a fallback : α} : ¬a ∈ t → t.getKeyD a fallback = fallback

theorem Std.DTreeMap.getKeyD_erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k a fallback : α} : (t.erase k).getKeyD a fallback = if cmp k a = Ordering.eq then fallback else t.getKeyD a fallback

@[simp]

theorem Std.DTreeMap.getKeyD_erase_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k fallback : α} : (t.erase k).getKeyD k fallback = fallback

theorem Std.DTreeMap.getKey?_eq_some_getKeyD_of_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {a fallback : α} : t.contains a = true → t.getKey? a = some (t.getKeyD a fallback)

theorem Std.DTreeMap.getKey?_eq_some_getKeyD {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {a fallback : α} : a ∈ t → t.getKey? a = some (t.getKeyD a fallback)

theorem Std.DTreeMap.getKeyD_eq_getD_getKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {a fallback : α} : t.getKeyD a fallback = (t.getKey? a).getD fallback

theorem Std.DTreeMap.getKey_eq_getKeyD {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {a fallback : α} {h : a ∈ t} : t.getKey a h = t.getKeyD a fallback

theorem Std.DTreeMap.getKey!_eq_getKeyD_default {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {a : α} : t.getKey! a = t.getKeyD a default

theorem Std.DTreeMap.getKeyD_congr {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k k' fallback : α} (h' : cmp k k' = Ordering.eq) : t.getKeyD k fallback = t.getKeyD k' fallback

theorem Std.DTreeMap.getKeyD_eq_of_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k fallback : α} (h' : t.contains k = true) : t.getKeyD k fallback = k

theorem Std.DTreeMap.getKeyD_eq_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k fallback : α} (h' : k ∈ t) : t.getKeyD k fallback = k

@[simp]

theorem Std.DTreeMap.isEmpty_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.insertIfNew k v).isEmpty = false

@[simp]

theorem Std.DTreeMap.contains_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k a : α} {v : β k} : (t.insertIfNew k v).contains a = (cmp k a == Ordering.eq || t.contains a)

@[simp]

theorem Std.DTreeMap.mem_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k a : α} {v : β k} : a ∈ t.insertIfNew k v ↔ cmp k a = Ordering.eq ∨ a ∈ t

theorem Std.DTreeMap.contains_insertIfNew_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.insertIfNew k v).contains k = true

theorem Std.DTreeMap.mem_insertIfNew_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : k ∈ t.insertIfNew k v

theorem Std.DTreeMap.contains_of_contains_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k a : α} {v : β k} : (t.insertIfNew k v).contains a = true → cmp k a ≠ Ordering.eq → t.contains a = true

theorem Std.DTreeMap.mem_of_mem_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k a : α} {v : β k} : a ∈ t.insertIfNew k v → cmp k a ≠ Ordering.eq → a ∈ t

theorem Std.DTreeMap.mem_of_mem_insertIfNew' {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k a : α} {v : β k} : a ∈ t.insertIfNew k v → ¬(cmp k a = Ordering.eq ∧ ¬k ∈ t) → a ∈ t

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

theorem Std.DTreeMap.size_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.insertIfNew k v).size = if k ∈ t then t.size else t.size + 1

theorem Std.DTreeMap.size_le_size_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : t.size ≤ (t.insertIfNew k v).size

theorem Std.DTreeMap.size_insertIfNew_le {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.insertIfNew k v).size ≤ t.size + 1

theorem Std.DTreeMap.get?_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k a : α} {v : β k} : (t.insertIfNew k v).get? a = if h : cmp k a = Ordering.eq ∧ ¬k ∈ t then some (cast ⋯ v) else t.get? a

theorem Std.DTreeMap.get_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k a : α} {v : β k} {h₁ : a ∈ t.insertIfNew k v} : (t.insertIfNew k v).get a h₁ = if h₂ : cmp k a = Ordering.eq ∧ ¬k ∈ t then cast ⋯ v else t.get a ⋯

theorem Std.DTreeMap.get!_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k a : α} [Inhabited (β a)] {v : β k} : (t.insertIfNew k v).get! a = if h : cmp k a = Ordering.eq ∧ ¬k ∈ t then cast ⋯ v else t.get! a

theorem Std.DTreeMap.getD_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k a : α} {fallback : β a} {v : β k} : (t.insertIfNew k v).getD a fallback = if h : cmp k a = Ordering.eq ∧ ¬k ∈ t then cast ⋯ v else t.getD a fallback

theorem Std.DTreeMap.Const.get?_insertIfNew {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k a : α} {v : β} : get? (t.insertIfNew k v) a = if cmp k a = Ordering.eq ∧ ¬k ∈ t then some v else get? t a

theorem Std.DTreeMap.Const.get_insertIfNew {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k a : α} {v : β} {h₁ : a ∈ t.insertIfNew k v} : get (t.insertIfNew k v) a h₁ = if h₂ : cmp k a = Ordering.eq ∧ ¬k ∈ t then v else get t a ⋯

theorem Std.DTreeMap.Const.get!_insertIfNew {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {k a : α} {v : β} : get! (t.insertIfNew k v) a = if cmp k a = Ordering.eq ∧ ¬k ∈ t then v else get! t a

theorem Std.DTreeMap.Const.getD_insertIfNew {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k a : α} {fallback v : β} : getD (t.insertIfNew k v) a fallback = if cmp k a = Ordering.eq ∧ ¬k ∈ t then v else getD t a fallback

theorem Std.DTreeMap.getKey?_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k a : α} {v : β k} : (t.insertIfNew k v).getKey? a = if cmp k a = Ordering.eq ∧ ¬k ∈ t then some k else t.getKey? a

theorem Std.DTreeMap.getKey_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k a : α} {v : β k} {h₁ : a ∈ t.insertIfNew k v} : (t.insertIfNew k v).getKey a h₁ = if h₂ : cmp k a = Ordering.eq ∧ ¬k ∈ t then k else t.getKey a ⋯

theorem Std.DTreeMap.getKey!_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k a : α} {v : β k} : (t.insertIfNew k v).getKey! a = if cmp k a = Ordering.eq ∧ ¬k ∈ t then k else t.getKey! a

theorem Std.DTreeMap.getKeyD_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k a fallback : α} {v : β k} : (t.insertIfNew k v).getKeyD a fallback = if cmp k a = Ordering.eq ∧ ¬k ∈ t then k else t.getKeyD a fallback

@[simp]

theorem Std.DTreeMap.getThenInsertIfNew?_fst {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {v : β k} : (t.getThenInsertIfNew? k v).fst = t.get? k

@[simp]

theorem Std.DTreeMap.getThenInsertIfNew?_snd {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {v : β k} : (t.getThenInsertIfNew? k v).snd = t.insertIfNew k v

theorem Std.DTreeMap.mem_of_get_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [LawfulEqCmp cmp] {k : α} {v : β k} {w : k ∈ t} : t.get k w = v → k ∈ t

@[simp]

theorem Std.DTreeMap.Const.getThenInsertIfNew?_fst {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {v : β} : (getThenInsertIfNew? t k v).fst = get? t k

@[simp]

theorem Std.DTreeMap.Const.getThenInsertIfNew?_snd {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {v : β} : (getThenInsertIfNew? t k v).snd = t.insertIfNew k v

@[simp]

theorem Std.DTreeMap.length_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.keys.length = t.size

@[simp]

theorem Std.DTreeMap.size_keysArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.keysArray.size = t.size

@[simp]

theorem Std.DTreeMap.isEmpty_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} : t.keys.isEmpty = t.isEmpty

@[simp]

theorem Std.DTreeMap.isEmpty_keysArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} : t.keysArray.isEmpty = t.isEmpty

@[simp]

theorem Std.DTreeMap.contains_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [BEq α] [LawfulBEqCmp cmp] [TransCmp cmp] {k : α} : t.keys.contains k = t.contains k

@[simp]

theorem Std.DTreeMap.contains_keysArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [BEq α] [LawfulBEqCmp cmp] [TransCmp cmp] {k : α} : t.keysArray.contains k = t.contains k

@[simp]

theorem Std.DTreeMap.mem_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [LawfulEqCmp cmp] [TransCmp cmp] {k : α} : k ∈ t.keys ↔ k ∈ t

@[simp]

theorem Std.DTreeMap.mem_keysArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [LawfulEqCmp cmp] [TransCmp cmp] {k : α} : k ∈ t.keysArray ↔ k ∈ t

theorem Std.DTreeMap.mem_of_mem_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : k ∈ t.keys) : k ∈ t

theorem Std.DTreeMap.mem_of_mem_keysArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : k ∈ t.keysArray) : k ∈ t

theorem Std.DTreeMap.distinct_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : List.Pairwise (fun (a b : α) => ¬cmp a b = Ordering.eq) t.keys

theorem Std.DTreeMap.nodup_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.keys.Nodup

theorem Std.DTreeMap.ordered_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : List.Pairwise (fun (a b : α) => cmp a b = Ordering.lt) t.keys

@[simp]

theorem Std.DTreeMap.map_fst_toList_eq_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} : List.map Sigma.fst t.toList = t.keys

@[simp]

theorem Std.DTreeMap.map_fst_toArray_eq_keysArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} : Array.map Sigma.fst t.toArray = t.keysArray

@[simp]

theorem Std.DTreeMap.length_toList {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.toList.length = t.size

@[simp]

theorem Std.DTreeMap.size_toArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.toArray.size = t.size

@[simp]

theorem Std.DTreeMap.isEmpty_toList {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} : t.toList.isEmpty = t.isEmpty

@[simp]

theorem Std.DTreeMap.isEmpty_toArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} : t.toArray.isEmpty = t.isEmpty

@[simp]

theorem Std.DTreeMap.mem_toList_iff_get?_eq_some {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {v : β k} : ⟨k, v⟩ ∈ t.toList ↔ t.get? k = some v

@[simp]

theorem Std.DTreeMap.mem_toArray_iff_get?_eq_some {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {v : β k} : ⟨k, v⟩ ∈ t.toArray ↔ t.get? k = some v

theorem Std.DTreeMap.find?_toList_eq_some_iff_get?_eq_some {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {v : β k} : List.find? (fun (x : (a : α) × β a) => cmp x.fst k == Ordering.eq) t.toList = some ⟨k, v⟩ ↔ t.get? k = some v

theorem Std.DTreeMap.find?_toArray_eq_some_iff_get?_eq_some {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {v : β k} : Array.find? (fun (x : (a : α) × β a) => cmp x.fst k == Ordering.eq) t.toArray = some ⟨k, v⟩ ↔ t.get? k = some v

theorem Std.DTreeMap.find?_toList_eq_none_iff_contains_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} : List.find? (fun (x : (a : α) × β a) => cmp x.fst k == Ordering.eq) t.toList = none ↔ t.contains k = false

theorem Std.DTreeMap.find?_toArray_eq_none_iff_contains_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} : Array.find? (fun (x : (a : α) × β a) => cmp x.fst k == Ordering.eq) t.toArray = none ↔ t.contains k = false

@[simp]

theorem Std.DTreeMap.find?_toList_eq_none_iff_not_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} : List.find? (fun (x : (a : α) × β a) => cmp x.fst k == Ordering.eq) t.toList = none ↔ ¬k ∈ t

@[simp]

theorem Std.DTreeMap.find?_toArray_eq_none_iff_not_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} : Array.find? (fun (x : (a : α) × β a) => cmp x.fst k == Ordering.eq) t.toArray = none ↔ ¬k ∈ t

theorem Std.DTreeMap.distinct_keys_toList {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : List.Pairwise (fun (a b : (a : α) × β a) => ¬cmp a.fst b.fst = Ordering.eq) t.toList

theorem Std.DTreeMap.ordered_keys_toList {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : List.Pairwise (fun (a b : (a : α) × β a) => cmp a.fst b.fst = Ordering.lt) t.toList

@[simp]

theorem Std.DTreeMap.Const.map_fst_toList_eq_keys {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} : List.map Prod.fst (toList t) = t.keys

@[simp]

theorem Std.DTreeMap.Const.map_fst_toArray_eq_keysArray {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} : Array.map Prod.fst (toArray t) = t.keysArray

@[simp]

theorem Std.DTreeMap.Const.length_toList {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} : (toList t).length = t.size

@[simp]

theorem Std.DTreeMap.Const.size_toArray {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} : (toArray t).size = t.size

@[simp]

theorem Std.DTreeMap.Const.isEmpty_toList {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} : (toList t).isEmpty = t.isEmpty

@[simp]

theorem Std.DTreeMap.Const.isEmpty_toArray {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} : (toArray t).isEmpty = t.isEmpty

@[simp]

theorem Std.DTreeMap.Const.mem_toList_iff_get?_eq_some {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {v : β} : (k, v) ∈ toList t ↔ get? t k = some v

@[simp]

theorem Std.DTreeMap.Const.mem_toArray_iff_get?_eq_some {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {v : β} : (k, v) ∈ toArray t ↔ get? t k = some v

@[simp]

theorem Std.DTreeMap.Const.mem_toList_iff_getKey?_eq_some_and_get?_eq_some {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {v : β} : (k, v) ∈ toList t ↔ t.getKey? k = some k ∧ get? t k = some v

@[simp]

theorem Std.DTreeMap.Const.mem_toArray_iff_getKey?_eq_some_and_get?_eq_some {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {v : β} : (k, v) ∈ toArray t ↔ t.getKey? k = some k ∧ get? t k = some v

theorem Std.DTreeMap.Const.get?_eq_some_iff_exists_compare_eq_eq_and_mem_toList {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {v : β} : get? t k = some v ↔ ∃ (k' : α), cmp k k' = Ordering.eq ∧ (k', v) ∈ toList t

theorem Std.DTreeMap.Const.get?_eq_some_iff_exists_compare_eq_eq_and_mem_toArray {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {v : β} : get? t k = some v ↔ ∃ (k' : α), cmp k k' = Ordering.eq ∧ (k', v) ∈ toArray t

theorem Std.DTreeMap.Const.find?_toList_eq_some_iff_getKey?_eq_some_and_get?_eq_some {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k k' : α} {v : β} : List.find? (fun (x : α × β) => cmp x.fst k == Ordering.eq) (toList t) = some (k', v) ↔ t.getKey? k = some k' ∧ get? t k = some v

theorem Std.DTreeMap.Const.find?_toArray_eq_some_iff_getKey?_eq_some_and_get?_eq_some {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k k' : α} {v : β} : Array.find? (fun (x : α × β) => cmp x.fst k == Ordering.eq) (toArray t) = some (k', v) ↔ t.getKey? k = some k' ∧ get? t k = some v

theorem Std.DTreeMap.Const.find?_toList_eq_none_iff_contains_eq_false {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} : List.find? (fun (x : α × β) => cmp x.fst k == Ordering.eq) (toList t) = none ↔ t.contains k = false

theorem Std.DTreeMap.Const.find?_toArray_eq_none_iff_contains_eq_false {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} : Array.find? (fun (x : α × β) => cmp x.fst k == Ordering.eq) (toArray t) = none ↔ t.contains k = false

@[simp]

theorem Std.DTreeMap.Const.find?_toList_eq_none_iff_not_mem {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} : List.find? (fun (x : α × β) => cmp x.fst k == Ordering.eq) (toList t) = none ↔ ¬k ∈ t

@[simp]

theorem Std.DTreeMap.Const.find?_toArray_eq_none_iff_not_mem {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} : Array.find? (fun (x : α × β) => cmp x.fst k == Ordering.eq) (toArray t) = none ↔ ¬k ∈ t

theorem Std.DTreeMap.Const.distinct_keys_toList {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] : List.Pairwise (fun (a b : α × β) => ¬cmp a.fst b.fst = Ordering.eq) (toList t)

theorem Std.DTreeMap.Const.ordered_keys_toList {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] : List.Pairwise (fun (a b : α × β) => cmp a.fst b.fst = Ordering.lt) (toList t)

theorem Std.DTreeMap.foldlM_eq_foldlM_toList {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {δ : Type w} {m : Type w → Type w'} [Monad m] [LawfulMonad m] {f : δ → (a : α) → β a → m δ} {init : δ} : foldlM f init t = List.foldlM (fun (a : δ) (b : (a : α) × β a) => f a b.fst b.snd) init t.toList

theorem Std.DTreeMap.foldlM_eq_foldlM_toArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {δ : Type w} {m : Type w → Type w'} [Monad m] [LawfulMonad m] {f : δ → (a : α) → β a → m δ} {init : δ} : foldlM f init t = Array.foldlM (fun (a : δ) (b : (a : α) × β a) => f a b.fst b.snd) init t.toArray

theorem Std.DTreeMap.foldl_eq_foldl_toList {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {δ : Type w} {f : δ → (a : α) → β a → δ} {init : δ} : foldl f init t = List.foldl (fun (a : δ) (b : (a : α) × β a) => f a b.fst b.snd) init t.toList

theorem Std.DTreeMap.foldl_eq_foldl_toArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {δ : Type w} {f : δ → (a : α) → β a → δ} {init : δ} : foldl f init t = Array.foldl (fun (a : δ) (b : (a : α) × β a) => f a b.fst b.snd) init t.toArray

theorem Std.DTreeMap.foldrM_eq_foldrM_toList {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {δ : Type w} {m : Type w → Type w'} [Monad m] [LawfulMonad m] {f : (a : α) → β a → δ → m δ} {init : δ} : foldrM f init t = List.foldrM (fun (a : (a : α) × β a) (b : δ) => f a.fst a.snd b) init t.toList

theorem Std.DTreeMap.foldrM_eq_foldrM_toArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {δ : Type w} {m : Type w → Type w'} [Monad m] [LawfulMonad m] {f : (a : α) → β a → δ → m δ} {init : δ} : foldrM f init t = Array.foldrM (fun (a : (a : α) × β a) (b : δ) => f a.fst a.snd b) init t.toArray

theorem Std.DTreeMap.foldr_eq_foldr_toList {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {δ : Type w} {f : (a : α) → β a → δ → δ} {init : δ} : foldr f init t = List.foldr (fun (a : (a : α) × β a) (b : δ) => f a.fst a.snd b) init t.toList

theorem Std.DTreeMap.foldr_eq_foldr_toArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {δ : Type w} {f : (a : α) → β a → δ → δ} {init : δ} : foldr f init t = Array.foldr (fun (a : (a : α) × β a) (b : δ) => f a.fst a.snd b) init t.toArray

@[simp]

theorem Std.DTreeMap.forM_eq_forM {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {m : Type w → Type w'} [Monad m] [LawfulMonad m] {f : (a : α) → β a → m PUnit} : forM f t = ForM.forM t fun (a : (a : α) × β a) => f a.fst a.snd

theorem Std.DTreeMap.forM_eq_forM_toList {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {m : Type w → Type w'} [Monad m] [LawfulMonad m] {f : (a : α) × β a → m PUnit} : ForM.forM t f = ForM.forM t.toList f

theorem Std.DTreeMap.forM_eq_forM_toArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {m : Type w → Type w'} [Monad m] [LawfulMonad m] {f : (a : α) × β a → m PUnit} : ForM.forM t f = ForM.forM t.toArray f

@[simp]

theorem Std.DTreeMap.forIn_eq_forIn {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {δ : Type w} {m : Type w → Type w'} [Monad m] [LawfulMonad m] {f : (a : α) → β a → δ → m (ForInStep δ)} {init : δ} : forIn f init t = ForIn.forIn t init fun (a : (a : α) × β a) (b : δ) => f a.fst a.snd b

theorem Std.DTreeMap.forIn_eq_forIn_toList {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {δ : Type w} {m : Type w → Type w'} [Monad m] [LawfulMonad m] {f : (a : α) × β a → δ → m (ForInStep δ)} {init : δ} : ForIn.forIn t init f = ForIn.forIn t.toList init f

@[simp]

theorem Std.DTreeMap.forIn_toList {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} (c : DTreeMap α β cmp) : ForIn.toList c = c.toList

instance Std.DTreeMap.instPureForInSigmaOfLawfulMonad {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {m : Type w → Type w'} [Monad m] [LawfulMonad m] : Internal.PureForIn m (DTreeMap α β cmp) ((a : α) × β a)

theorem Std.DTreeMap.forIn_eq_forIn_toArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {δ : Type w} {m : Type w → Type w'} [Monad m] [LawfulMonad m] {f : (a : α) × β a → δ → m (ForInStep δ)} {init : δ} : ForIn.forIn t init f = ForIn.forIn t.toArray init f

theorem Std.DTreeMap.foldlM_eq_foldlM_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {δ : Type w} {m : Type w → Type w'} [Monad m] [LawfulMonad m] {f : δ → α → m δ} {init : δ} : foldlM (fun (d : δ) (a : α) (x : β a) => f d a) init t = List.foldlM f init t.keys

theorem Std.DTreeMap.foldlM_eq_foldlM_keysArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {δ : Type w} {m : Type w → Type w'} [Monad m] [LawfulMonad m] {f : δ → α → m δ} {init : δ} : foldlM (fun (d : δ) (a : α) (x : β a) => f d a) init t = Array.foldlM f init t.keysArray

theorem Std.DTreeMap.foldl_eq_foldl_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {δ : Type w} {f : δ → α → δ} {init : δ} : foldl (fun (d : δ) (a : α) (x : β a) => f d a) init t = List.foldl f init t.keys

theorem Std.DTreeMap.foldl_eq_foldl_keysArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {δ : Type w} {f : δ → α → δ} {init : δ} : foldl (fun (d : δ) (a : α) (x : β a) => f d a) init t = Array.foldl f init t.keysArray

theorem Std.DTreeMap.foldrM_eq_foldrM_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {δ : Type w} {m : Type w → Type w'} [Monad m] [LawfulMonad m] {f : α → δ → m δ} {init : δ} : foldrM (fun (a : α) (x : β a) (d : δ) => f a d) init t = List.foldrM f init t.keys

theorem Std.DTreeMap.foldrM_eq_foldrM_keysArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {δ : Type w} {m : Type w → Type w'} [Monad m] [LawfulMonad m] {f : α → δ → m δ} {init : δ} : foldrM (fun (a : α) (x : β a) (d : δ) => f a d) init t = Array.foldrM f init t.keysArray

theorem Std.DTreeMap.foldr_eq_foldr_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {δ : Type w} {f : α → δ → δ} {init : δ} : foldr (fun (a : α) (x : β a) (d : δ) => f a d) init t = List.foldr f init t.keys

theorem Std.DTreeMap.foldr_eq_foldr_keysArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {δ : Type w} {f : α → δ → δ} {init : δ} : foldr (fun (a : α) (x : β a) (d : δ) => f a d) init t = Array.foldr f init t.keysArray

theorem Std.DTreeMap.forM_eq_forM_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {m : Type w → Type w'} [Monad m] [LawfulMonad m] {f : α → m PUnit} : (ForM.forM t fun (a : (a : α) × β a) => f a.fst) = t.keys.forM f

theorem Std.DTreeMap.forM_eq_forM_keysArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {m : Type w → Type w'} [Monad m] [LawfulMonad m] {f : α → m PUnit} : (ForM.forM t fun (a : (a : α) × β a) => f a.fst) = Array.forM f t.keysArray

theorem Std.DTreeMap.forIn_eq_forIn_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {δ : Type w} {m : Type w → Type w'} [Monad m] [LawfulMonad m] {f : α → δ → m (ForInStep δ)} {init : δ} : (ForIn.forIn t init fun (a : (a : α) × β a) (d : δ) => f a.fst d) = ForIn.forIn t.keys init f

theorem Std.DTreeMap.forIn_eq_forIn_keysArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {δ : Type w} {m : Type w → Type w'} [Monad m] [LawfulMonad m] {f : α → δ → m (ForInStep δ)} {init : δ} : (ForIn.forIn t init fun (a : (a : α) × β a) (d : δ) => f a.fst d) = ForIn.forIn t.keysArray init f

theorem Std.DTreeMap.Const.foldlM_eq_foldlM_toList {α : Type u} {cmp : α → α → Ordering} {δ : Type w} {m : Type w → Type w'} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [Monad m] [LawfulMonad m] {f : δ → α → β → m δ} {init : δ} : foldlM f init t = List.foldlM (fun (a : δ) (b : α × β) => f a b.fst b.snd) init (toList t)

theorem Std.DTreeMap.Const.foldlM_eq_foldlM_toArray {α : Type u} {cmp : α → α → Ordering} {δ : Type w} {m : Type w → Type w'} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [Monad m] [LawfulMonad m] {f : δ → α → β → m δ} {init : δ} : foldlM f init t = Array.foldlM (fun (a : δ) (b : α × β) => f a b.fst b.snd) init (toArray t)

theorem Std.DTreeMap.Const.foldl_eq_foldl_toList {α : Type u} {cmp : α → α → Ordering} {δ : Type w} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {f : δ → α → β → δ} {init : δ} : foldl f init t = List.foldl (fun (a : δ) (b : α × β) => f a b.fst b.snd) init (toList t)

theorem Std.DTreeMap.Const.foldl_eq_foldl_toArray {α : Type u} {cmp : α → α → Ordering} {δ : Type w} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {f : δ → α → β → δ} {init : δ} : foldl f init t = Array.foldl (fun (a : δ) (b : α × β) => f a b.fst b.snd) init (toArray t)

theorem Std.DTreeMap.Const.foldrM_eq_foldrM_toList {α : Type u} {cmp : α → α → Ordering} {δ : Type w} {m : Type w → Type w'} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [Monad m] [LawfulMonad m] {f : α → β → δ → m δ} {init : δ} : foldrM f init t = List.foldrM (fun (a : α × β) (b : δ) => f a.fst a.snd b) init (toList t)

theorem Std.DTreeMap.Const.foldrM_eq_foldrM_toArray {α : Type u} {cmp : α → α → Ordering} {δ : Type w} {m : Type w → Type w'} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [Monad m] [LawfulMonad m] {f : α → β → δ → m δ} {init : δ} : foldrM f init t = Array.foldrM (fun (a : α × β) (b : δ) => f a.fst a.snd b) init (toArray t)

theorem Std.DTreeMap.Const.foldr_eq_foldr_toList {α : Type u} {cmp : α → α → Ordering} {δ : Type w} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {f : α → β → δ → δ} {init : δ} : foldr f init t = List.foldr (fun (a : α × β) (b : δ) => f a.fst a.snd b) init (toList t)

theorem Std.DTreeMap.Const.foldr_eq_foldr_toArray {α : Type u} {cmp : α → α → Ordering} {δ : Type w} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {f : α → β → δ → δ} {init : δ} : foldr f init t = Array.foldr (fun (a : α × β) (b : δ) => f a.fst a.snd b) init (toArray t)

theorem Std.DTreeMap.Const.forM_eq_forMUncurried {α : Type u} {cmp : α → α → Ordering} {m : Type w → Type w'} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [Monad m] [LawfulMonad m] {f : α → β → m PUnit} : forM f t = forMUncurried (fun (a : α × β) => f a.fst a.snd) t

theorem Std.DTreeMap.Const.forMUncurried_eq_forM_toList {α : Type u} {cmp : α → α → Ordering} {m : Type w → Type w'} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [Monad m] [LawfulMonad m] {f : α × β → m PUnit} : forMUncurried f t = (toList t).forM f

theorem Std.DTreeMap.Const.forMUncurried_eq_forM_toArray {α : Type u} {cmp : α → α → Ordering} {m : Type w → Type w'} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [Monad m] [LawfulMonad m] {f : α × β → m PUnit} : forMUncurried f t = Array.forM f (toArray t)

theorem Std.DTreeMap.Const.forIn_eq_forInUncurried {α : Type u} {cmp : α → α → Ordering} {δ : Type w} {m : Type w → Type w'} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [Monad m] [LawfulMonad m] {f : α → β → δ → m (ForInStep δ)} {init : δ} : forIn f init t = forInUncurried (fun (a : α × β) (b : δ) => f a.fst a.snd b) init t

theorem Std.DTreeMap.Const.forInUncurried_eq_forIn_toList {α : Type u} {cmp : α → α → Ordering} {δ : Type w} {m : Type w → Type w'} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [Monad m] [LawfulMonad m] {f : α × β → δ → m (ForInStep δ)} {init : δ} : forInUncurried f init t = ForIn.forIn (toList t) init f

theorem Std.DTreeMap.Const.forInUncurried_eq_forIn_toArray {α : Type u} {cmp : α → α → Ordering} {δ : Type w} {m : Type w → Type w'} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [Monad m] [LawfulMonad m] {f : α × β → δ → m (ForInStep δ)} {init : δ} : forInUncurried f init t = ForIn.forIn (toArray t) init f

@[simp]

theorem Std.DTreeMap.insertMany_nil {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} : t.insertMany [] = t

@[simp]

theorem Std.DTreeMap.insertMany_list_singleton {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {k : α} {v : β k} : t.insertMany [⟨k, v⟩] = t.insert k v

theorem Std.DTreeMap.insertMany_cons {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {l : List ((a : α) × β a)} {k : α} {v : β k} : t.insertMany (⟨k, v⟩ :: l) = (t.insert k v).insertMany l

theorem Std.DTreeMap.insertMany_append {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {l₁ l₂ : List ((a : α) × β a)} : t.insertMany (l₁ ++ l₂) = (t.insertMany l₁).insertMany l₂

@[simp]

theorem Std.DTreeMap.contains_insertMany_list {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List ((a : α) × β a)} {k : α} : (t.insertMany l).contains k = (t.contains k || (List.map Sigma.fst l).contains k)

@[simp]

theorem Std.DTreeMap.mem_insertMany_list {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List ((a : α) × β a)} {k : α} : k ∈ t.insertMany l ↔ k ∈ t ∨ (List.map Sigma.fst l).contains k = true

theorem Std.DTreeMap.mem_of_mem_insertMany_list {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List ((a : α) × β a)} {k : α} : k ∈ t.insertMany l → (List.map Sigma.fst l).contains k = false → k ∈ t

theorem Std.DTreeMap.get?_insertMany_list_of_contains_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List ((a : α) × β a)} {k : α} (contains_eq_false : (List.map Sigma.fst l).contains k = false) : (t.insertMany l).get? k = t.get? k

theorem Std.DTreeMap.get?_insertMany_list_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List ((a : α) × β a)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) {v : β k} (distinct : List.Pairwise (fun (a b : (a : α) × β a) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : ⟨k, v⟩ ∈ l) : (t.insertMany l).get? k' = some (cast ⋯ v)

theorem Std.DTreeMap.get_insertMany_list_of_contains_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List ((a : α) × β a)} {k : α} (contains : (List.map Sigma.fst l).contains k = false) {h' : k ∈ t.insertMany l} : (t.insertMany l).get k h' = t.get k ⋯

theorem Std.DTreeMap.get_insertMany_list_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {l : List ((a : α) × β a)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) {v : β k} (distinct : List.Pairwise (fun (a b : (a : α) × β a) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : ⟨k, v⟩ ∈ l) {h' : k' ∈ t.insertMany l} : (t.insertMany l).get k' h' = cast ⋯ v

theorem Std.DTreeMap.get!_insertMany_list_of_contains_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List ((a : α) × β a)} {k : α} [Inhabited (β k)] (contains_eq_false : (List.map Sigma.fst l).contains k = false) : (t.insertMany l).get! k = t.get! k

theorem Std.DTreeMap.get!_insertMany_list_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {l : List ((a : α) × β a)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) {v : β k} [Inhabited (β k')] (distinct : List.Pairwise (fun (a b : (a : α) × β a) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : ⟨k, v⟩ ∈ l) : (t.insertMany l).get! k' = cast ⋯ v

theorem Std.DTreeMap.getD_insertMany_list_of_contains_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List ((a : α) × β a)} {k : α} {fallback : β k} (contains_eq_false : (List.map Sigma.fst l).contains k = false) : (t.insertMany l).getD k fallback = t.getD k fallback

theorem Std.DTreeMap.getD_insertMany_list_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {l : List ((a : α) × β a)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) {v : β k} {fallback : β k'} (distinct : List.Pairwise (fun (a b : (a : α) × β a) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : ⟨k, v⟩ ∈ l) : (t.insertMany l).getD k' fallback = cast ⋯ v

theorem Std.DTreeMap.getKey?_insertMany_list_of_contains_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List ((a : α) × β a)} {k : α} (contains_eq_false : (List.map Sigma.fst l).contains k = false) : (t.insertMany l).getKey? k = t.getKey? k

theorem Std.DTreeMap.getKey?_insertMany_list_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {l : List ((a : α) × β a)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) (distinct : List.Pairwise (fun (a b : (a : α) × β a) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : k ∈ List.map Sigma.fst l) : (t.insertMany l).getKey? k' = some k

theorem Std.DTreeMap.getKey_insertMany_list_of_contains_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List ((a : α) × β a)} {k : α} (contains_eq_false : (List.map Sigma.fst l).contains k = false) {h' : k ∈ t.insertMany l} : (t.insertMany l).getKey k h' = t.getKey k ⋯

theorem Std.DTreeMap.getKey_insertMany_list_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {l : List ((a : α) × β a)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) (distinct : List.Pairwise (fun (a b : (a : α) × β a) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : k ∈ List.map Sigma.fst l) {h' : k' ∈ t.insertMany l} : (t.insertMany l).getKey k' h' = k

theorem Std.DTreeMap.getKey!_insertMany_list_of_contains_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] [Inhabited α] {l : List ((a : α) × β a)} {k : α} (contains_eq_false : (List.map Sigma.fst l).contains k = false) : (t.insertMany l).getKey! k = t.getKey! k

theorem Std.DTreeMap.getKey!_insertMany_list_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {l : List ((a : α) × β a)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) (distinct : List.Pairwise (fun (a b : (a : α) × β a) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : k ∈ List.map Sigma.fst l) : (t.insertMany l).getKey! k' = k

theorem Std.DTreeMap.getKeyD_insertMany_list_of_contains_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List ((a : α) × β a)} {k fallback : α} (contains_eq_false : (List.map Sigma.fst l).contains k = false) : (t.insertMany l).getKeyD k fallback = t.getKeyD k fallback

theorem Std.DTreeMap.getKeyD_insertMany_list_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {l : List ((a : α) × β a)} {k k' fallback : α} (k_eq : cmp k k' = Ordering.eq) (distinct : List.Pairwise (fun (a b : (a : α) × β a) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : k ∈ List.map Sigma.fst l) : (t.insertMany l).getKeyD k' fallback = k

theorem Std.DTreeMap.size_insertMany_list {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List ((a : α) × β a)} (distinct : List.Pairwise (fun (a b : (a : α) × β a) => ¬cmp a.fst b.fst = Ordering.eq) l) : (∀ (a : α), a ∈ t → (List.map Sigma.fst l).contains a = false) → (t.insertMany l).size = t.size + l.length

theorem Std.DTreeMap.size_le_size_insertMany_list {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {l : List ((a : α) × β a)} : t.size ≤ (t.insertMany l).size

theorem Std.DTreeMap.size_insertMany_list_le {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {l : List ((a : α) × β a)} : (t.insertMany l).size ≤ t.size + l.length

@[simp]

theorem Std.DTreeMap.isEmpty_insertMany_list {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {l : List ((a : α) × β a)} : (t.insertMany l).isEmpty = (t.isEmpty && l.isEmpty)

@[simp]

theorem Std.DTreeMap.Const.insertMany_nil {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} : insertMany t [] = t

@[simp]

theorem Std.DTreeMap.Const.insertMany_list_singleton {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {k : α} {v : β} : insertMany t [(k, v)] = t.insert k v

theorem Std.DTreeMap.Const.insertMany_cons {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {l : List (α × β)} {k : α} {v : β} : insertMany t ((k, v) :: l) = insertMany (t.insert k v) l

theorem Std.DTreeMap.Const.insertMany_append {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {l₁ l₂ : List (α × β)} : insertMany t (l₁ ++ l₂) = insertMany (insertMany t l₁) l₂

@[simp]

theorem Std.DTreeMap.Const.contains_insertMany_list {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List (α × β)} {k : α} : (insertMany t l).contains k = (t.contains k || (List.map Prod.fst l).contains k)

@[simp]

theorem Std.DTreeMap.Const.mem_insertMany_list {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List (α × β)} {k : α} : k ∈ insertMany t l ↔ k ∈ t ∨ (List.map Prod.fst l).contains k = true

theorem Std.DTreeMap.Const.mem_of_mem_insertMany_list {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List (α × β)} {k : α} : k ∈ insertMany t l → (List.map Prod.fst l).contains k = false → k ∈ t

theorem Std.DTreeMap.Const.getKey?_insertMany_list_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List (α × β)} {k : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) : (insertMany t l).getKey? k = t.getKey? k

theorem Std.DTreeMap.Const.getKey?_insertMany_list_of_mem {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {l : List (α × β)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) (distinct : List.Pairwise (fun (a b : α × β) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : k ∈ List.map Prod.fst l) : (insertMany t l).getKey? k' = some k

theorem Std.DTreeMap.Const.getKey_insertMany_list_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List (α × β)} {k : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) {h' : k ∈ insertMany t l} : (insertMany t l).getKey k h' = t.getKey k ⋯

theorem Std.DTreeMap.Const.getKey_insertMany_list_of_mem {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {l : List (α × β)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) (distinct : List.Pairwise (fun (a b : α × β) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : k ∈ List.map Prod.fst l) {h' : k' ∈ insertMany t l} : (insertMany t l).getKey k' h' = k

theorem Std.DTreeMap.Const.getKey!_insertMany_list_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] [Inhabited α] {l : List (α × β)} {k : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) : (insertMany t l).getKey! k = t.getKey! k

theorem Std.DTreeMap.Const.getKey!_insertMany_list_of_mem {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited α] {l : List (α × β)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) (distinct : List.Pairwise (fun (a b : α × β) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : k ∈ List.map Prod.fst l) : (insertMany t l).getKey! k' = k

theorem Std.DTreeMap.Const.getKeyD_insertMany_list_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List (α × β)} {k fallback : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) : (insertMany t l).getKeyD k fallback = t.getKeyD k fallback

theorem Std.DTreeMap.Const.getKeyD_insertMany_list_of_mem {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {l : List (α × β)} {k k' fallback : α} (k_eq : cmp k k' = Ordering.eq) (distinct : List.Pairwise (fun (a b : α × β) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : k ∈ List.map Prod.fst l) : (insertMany t l).getKeyD k' fallback = k

theorem Std.DTreeMap.Const.size_insertMany_list {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List (α × β)} (distinct : List.Pairwise (fun (a b : α × β) => ¬cmp a.fst b.fst = Ordering.eq) l) : (∀ (a : α), a ∈ t → (List.map Prod.fst l).contains a = false) → (insertMany t l).size = t.size + l.length

theorem Std.DTreeMap.Const.size_le_size_insertMany_list {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {l : List (α × β)} : t.size ≤ (insertMany t l).size

theorem Std.DTreeMap.Const.size_insertMany_list_le {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {l : List (α × β)} : (insertMany t l).size ≤ t.size + l.length

@[simp]

theorem Std.DTreeMap.Const.isEmpty_insertMany_list {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {l : List (α × β)} : (insertMany t l).isEmpty = (t.isEmpty && l.isEmpty)

theorem Std.DTreeMap.Const.get?_insertMany_list_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List (α × β)} {k : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) : get? (insertMany t l) k = get? t k

theorem Std.DTreeMap.Const.get?_insertMany_list_of_mem {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {l : List (α × β)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) {v : β} (distinct : List.Pairwise (fun (a b : α × β) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : (k, v) ∈ l) : get? (insertMany t l) k' = some v

theorem Std.DTreeMap.Const.get?_insertMany_list {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List (α × β)} {k : α} : get? (insertMany t l) k = (List.findSomeRev? (fun (x : α × β) => match x with | (a, b) => if cmp a k = Ordering.eq then some b else none) l).or (get? t k)

theorem Std.DTreeMap.Const.get_insertMany_list_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List (α × β)} {k : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) {h' : k ∈ insertMany t l} : get (insertMany t l) k h' = get t k ⋯

theorem Std.DTreeMap.Const.get_insertMany_list_of_mem {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {l : List (α × β)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) {v : β} (distinct : List.Pairwise (fun (a b : α × β) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : (k, v) ∈ l) {h' : k' ∈ insertMany t l} : get (insertMany t l) k' h' = v

theorem Std.DTreeMap.Const.get!_insertMany_list_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] [Inhabited β] {l : List (α × β)} {k : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) : get! (insertMany t l) k = get! t k

theorem Std.DTreeMap.Const.get!_insertMany_list_of_mem {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {l : List (α × β)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) {v : β} (distinct : List.Pairwise (fun (a b : α × β) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : (k, v) ∈ l) : get! (insertMany t l) k' = v

theorem Std.DTreeMap.Const.getD_insertMany_list_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List (α × β)} {k : α} {fallback : β} (contains_eq_false : (List.map Prod.fst l).contains k = false) : getD (insertMany t l) k fallback = getD t k fallback

theorem Std.DTreeMap.Const.getD_insertMany_list_of_mem {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {l : List (α × β)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) {v fallback : β} (distinct : List.Pairwise (fun (a b : α × β) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : (k, v) ∈ l) : getD (insertMany t l) k' fallback = v

@[simp]

theorem Std.DTreeMap.Const.insertManyIfNewUnit_nil {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} : insertManyIfNewUnit t [] = t

@[simp]

theorem Std.DTreeMap.Const.insertManyIfNewUnit_list_singleton {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} {k : α} : insertManyIfNewUnit t [k] = t.insertIfNew k ()

theorem Std.DTreeMap.Const.insertManyIfNewUnit_cons {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} {l : List α} {k : α} : insertManyIfNewUnit t (k :: l) = insertManyIfNewUnit (t.insertIfNew k ()) l

@[simp]

theorem Std.DTreeMap.Const.contains_insertManyIfNewUnit_list {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List α} {k : α} : (insertManyIfNewUnit t l).contains k = (t.contains k || l.contains k)

@[simp]

theorem Std.DTreeMap.Const.mem_insertManyIfNewUnit_list {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List α} {k : α} : k ∈ insertManyIfNewUnit t l ↔ k ∈ t ∨ l.contains k = true

theorem Std.DTreeMap.Const.mem_of_mem_insertManyIfNewUnit_list {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List α} {k : α} (contains_eq_false : l.contains k = false) : k ∈ insertManyIfNewUnit t l → k ∈ t

theorem Std.DTreeMap.Const.getKey?_insertManyIfNewUnit_list_of_not_mem_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List α} {k : α} : ¬k ∈ t → l.contains k = false → (insertManyIfNewUnit t l).getKey? k = none

theorem Std.DTreeMap.Const.getKey?_insertManyIfNewUnit_list_of_not_mem_of_mem {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} [TransCmp cmp] {l : List α} {k k' : α} (k_eq : cmp k k' = Ordering.eq) : ¬k ∈ t → List.Pairwise (fun (a b : α) => ¬cmp a b = Ordering.eq) l → k ∈ l → (insertManyIfNewUnit t l).getKey? k' = some k

theorem Std.DTreeMap.Const.getKey?_insertManyIfNewUnit_list_of_mem {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} [TransCmp cmp] {l : List α} {k : α} : k ∈ t → (insertManyIfNewUnit t l).getKey? k = t.getKey? k

theorem Std.DTreeMap.Const.getKey_insertManyIfNewUnit_list_of_mem {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} [TransCmp cmp] {l : List α} {k : α} {h' : k ∈ insertManyIfNewUnit t l} (contains : k ∈ t) : (insertManyIfNewUnit t l).getKey k h' = t.getKey k contains

theorem Std.DTreeMap.Const.getKey_insertManyIfNewUnit_list_of_not_mem_of_mem {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} [TransCmp cmp] {l : List α} {k k' : α} (k_eq : cmp k k' = Ordering.eq) {h' : k' ∈ insertManyIfNewUnit t l} : ¬k ∈ t → List.Pairwise (fun (a b : α) => ¬cmp a b = Ordering.eq) l → k ∈ l → (insertManyIfNewUnit t l).getKey k' h' = k

theorem Std.DTreeMap.Const.getKey!_insertManyIfNewUnit_list_of_not_mem_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] [Inhabited α] {l : List α} {k : α} : ¬k ∈ t → l.contains k = false → (insertManyIfNewUnit t l).getKey! k = default

theorem Std.DTreeMap.Const.getKey!_insertManyIfNewUnit_list_of_not_mem_of_mem {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} [TransCmp cmp] [Inhabited α] {l : List α} {k k' : α} (k_eq : cmp k k' = Ordering.eq) : ¬k ∈ t → List.Pairwise (fun (a b : α) => ¬cmp a b = Ordering.eq) l → k ∈ l → (insertManyIfNewUnit t l).getKey! k' = k

theorem Std.DTreeMap.Const.getKey!_insertManyIfNewUnit_list_of_mem {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} [TransCmp cmp] [Inhabited α] {l : List α} {k : α} : k ∈ t → (insertManyIfNewUnit t l).getKey! k = t.getKey! k

theorem Std.DTreeMap.Const.getKeyD_insertManyIfNewUnit_list_of_not_mem_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List α} {k fallback : α} : ¬k ∈ t → l.contains k = false → (insertManyIfNewUnit t l).getKeyD k fallback = fallback

theorem Std.DTreeMap.Const.getKeyD_insertManyIfNewUnit_list_of_not_mem_of_mem {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} [TransCmp cmp] {l : List α} {k k' fallback : α} (k_eq : cmp k k' = Ordering.eq) : ¬k ∈ t → List.Pairwise (fun (a b : α) => ¬cmp a b = Ordering.eq) l → k ∈ l → (insertManyIfNewUnit t l).getKeyD k' fallback = k

theorem Std.DTreeMap.Const.getKeyD_insertManyIfNewUnit_list_of_mem {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} [TransCmp cmp] {l : List α} {k fallback : α} : k ∈ t → (insertManyIfNewUnit t l).getKeyD k fallback = t.getKeyD k fallback

theorem Std.DTreeMap.Const.size_insertManyIfNewUnit_list {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List α} (distinct : List.Pairwise (fun (a b : α) => ¬cmp a b = Ordering.eq) l) : (∀ (a : α), a ∈ t → l.contains a = false) → (insertManyIfNewUnit t l).size = t.size + l.length

theorem Std.DTreeMap.Const.size_le_size_insertManyIfNewUnit_list {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} [TransCmp cmp] {l : List α} : t.size ≤ (insertManyIfNewUnit t l).size

theorem Std.DTreeMap.Const.size_insertManyIfNewUnit_list_le {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} [TransCmp cmp] {l : List α} : (insertManyIfNewUnit t l).size ≤ t.size + l.length

@[simp]

theorem Std.DTreeMap.Const.isEmpty_insertManyIfNewUnit_list {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} [TransCmp cmp] {l : List α} : (insertManyIfNewUnit t l).isEmpty = (t.isEmpty && l.isEmpty)

@[simp]

theorem Std.DTreeMap.Const.get?_insertManyIfNewUnit_list {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List α} {k : α} : get? (insertManyIfNewUnit t l) k = if k ∈ t ∨ l.contains k = true then some () else none

@[simp]

theorem Std.DTreeMap.Const.get_insertManyIfNewUnit_list {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} {l : List α} {k : α} {h' : k ∈ insertManyIfNewUnit t l} : get (insertManyIfNewUnit t l) k h' = ()

@[simp]

theorem Std.DTreeMap.Const.get!_insertManyIfNewUnit_list {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} {l : List α} {k : α} : get! (insertManyIfNewUnit t l) k = ()

@[simp]

theorem Std.DTreeMap.Const.getD_insertManyIfNewUnit_list {α : Type u} {cmp : α → α → Ordering} {t : DTreeMap α (fun (x : α) => Unit) cmp} {l : List α} {k : α} {fallback : Unit} : getD (insertManyIfNewUnit t l) k fallback = ()

@[simp]

theorem Std.DTreeMap.ofList_nil {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} : ofList [] cmp = ∅

@[simp]

theorem Std.DTreeMap.ofList_singleton {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {k : α} {v : β k} : ofList [⟨k, v⟩] cmp = ∅.insert k v

theorem Std.DTreeMap.ofList_cons {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {k : α} {v : β k} {tl : List ((a : α) × β a)} : ofList (⟨k, v⟩ :: tl) cmp = (∅.insert k v).insertMany tl

theorem Std.DTreeMap.ofList_eq_insertMany_empty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {l : List ((a : α) × β a)} : ofList l cmp = ∅.insertMany l

@[simp]

theorem Std.DTreeMap.contains_ofList {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List ((a : α) × β a)} {k : α} : (ofList l cmp).contains k = (List.map Sigma.fst l).contains k

@[simp]

theorem Std.DTreeMap.mem_ofList {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List ((a : α) × β a)} {k : α} : k ∈ ofList l cmp ↔ (List.map Sigma.fst l).contains k = true

theorem Std.DTreeMap.get?_ofList_of_contains_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [TransCmp cmp] [LawfulEqCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List ((a : α) × β a)} {k : α} (contains_eq_false : (List.map Sigma.fst l).contains k = false) : (ofList l cmp).get? k = none

theorem Std.DTreeMap.get?_ofList_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [TransCmp cmp] [LawfulEqCmp cmp] {l : List ((a : α) × β a)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) {v : β k} (distinct : List.Pairwise (fun (a b : (a : α) × β a) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : ⟨k, v⟩ ∈ l) : (ofList l cmp).get? k' = some (cast ⋯ v)

theorem Std.DTreeMap.get_ofList_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [TransCmp cmp] [LawfulEqCmp cmp] {l : List ((a : α) × β a)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) {v : β k} (distinct : List.Pairwise (fun (a b : (a : α) × β a) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : ⟨k, v⟩ ∈ l) {h : k' ∈ ofList l cmp} : (ofList l cmp).get k' h = cast ⋯ v

theorem Std.DTreeMap.get!_ofList_of_contains_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [TransCmp cmp] [LawfulEqCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List ((a : α) × β a)} {k : α} [Inhabited (β k)] (contains_eq_false : (List.map Sigma.fst l).contains k = false) : (ofList l cmp).get! k = default

theorem Std.DTreeMap.get!_ofList_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [TransCmp cmp] [LawfulEqCmp cmp] {l : List ((a : α) × β a)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) {v : β k} [Inhabited (β k')] (distinct : List.Pairwise (fun (a b : (a : α) × β a) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : ⟨k, v⟩ ∈ l) : (ofList l cmp).get! k' = cast ⋯ v

theorem Std.DTreeMap.getD_ofList_of_contains_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [TransCmp cmp] [LawfulEqCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List ((a : α) × β a)} {k : α} {fallback : β k} (contains_eq_false : (List.map Sigma.fst l).contains k = false) : (ofList l cmp).getD k fallback = fallback

theorem Std.DTreeMap.getD_ofList_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [TransCmp cmp] [LawfulEqCmp cmp] {l : List ((a : α) × β a)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) {v : β k} {fallback : β k'} (distinct : List.Pairwise (fun (a b : (a : α) × β a) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : ⟨k, v⟩ ∈ l) : (ofList l cmp).getD k' fallback = cast ⋯ v

theorem Std.DTreeMap.getKey?_ofList_of_contains_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List ((a : α) × β a)} {k : α} (contains_eq_false : (List.map Sigma.fst l).contains k = false) : (ofList l cmp).getKey? k = none

theorem Std.DTreeMap.getKey?_ofList_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [TransCmp cmp] {l : List ((a : α) × β a)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) (distinct : List.Pairwise (fun (a b : (a : α) × β a) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : k ∈ List.map Sigma.fst l) : (ofList l cmp).getKey? k' = some k

theorem Std.DTreeMap.getKey_ofList_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [TransCmp cmp] {l : List ((a : α) × β a)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) (distinct : List.Pairwise (fun (a b : (a : α) × β a) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : k ∈ List.map Sigma.fst l) {h' : k' ∈ ofList l cmp} : (ofList l cmp).getKey k' h' = k

theorem Std.DTreeMap.getKey!_ofList_of_contains_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [TransCmp cmp] [Inhabited α] [BEq α] [LawfulBEqCmp cmp] {l : List ((a : α) × β a)} {k : α} (contains_eq_false : (List.map Sigma.fst l).contains k = false) : (ofList l cmp).getKey! k = default

theorem Std.DTreeMap.getKey!_ofList_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [TransCmp cmp] [Inhabited α] {l : List ((a : α) × β a)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) (distinct : List.Pairwise (fun (a b : (a : α) × β a) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : k ∈ List.map Sigma.fst l) : (ofList l cmp).getKey! k' = k

theorem Std.DTreeMap.getKeyD_ofList_of_contains_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List ((a : α) × β a)} {k fallback : α} (contains_eq_false : (List.map Sigma.fst l).contains k = false) : (ofList l cmp).getKeyD k fallback = fallback

theorem Std.DTreeMap.getKeyD_ofList_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [TransCmp cmp] {l : List ((a : α) × β a)} {k k' fallback : α} (k_eq : cmp k k' = Ordering.eq) (distinct : List.Pairwise (fun (a b : (a : α) × β a) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : k ∈ List.map Sigma.fst l) : (ofList l cmp).getKeyD k' fallback = k

theorem Std.DTreeMap.size_ofList {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [TransCmp cmp] {l : List ((a : α) × β a)} (distinct : List.Pairwise (fun (a b : (a : α) × β a) => ¬cmp a.fst b.fst = Ordering.eq) l) : (ofList l cmp).size = l.length

theorem Std.DTreeMap.size_ofList_le {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [TransCmp cmp] {l : List ((a : α) × β a)} : (ofList l cmp).inner.size ≤ l.length

@[simp]

theorem Std.DTreeMap.isEmpty_ofList {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [TransCmp cmp] {l : List ((a : α) × β a)} : (ofList l cmp).inner.isEmpty = l.isEmpty

@[simp]

theorem Std.DTreeMap.Const.ofList_nil {α : Type u} {cmp : α → α → Ordering} {β : Type v} : ofList [] cmp = ∅

@[simp]

theorem Std.DTreeMap.Const.ofList_singleton {α : Type u} {cmp : α → α → Ordering} {β : Type v} {k : α} {v : β} : ofList [(k, v)] cmp = ∅.insert k v

theorem Std.DTreeMap.Const.ofList_cons {α : Type u} {cmp : α → α → Ordering} {β : Type v} {k : α} {v : β} {tl : List (α × β)} : ofList ((k, v) :: tl) cmp = insertMany (∅.insert k v) tl

theorem Std.DTreeMap.Const.ofList_eq_insertMany_empty {α : Type u} {cmp : α → α → Ordering} {β : Type v} {l : List (α × β)} : ofList l cmp = insertMany ∅ l

@[simp]

theorem Std.DTreeMap.Const.contains_ofList {α : Type u} {cmp : α → α → Ordering} {β : Type v} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List (α × β)} {k : α} : (ofList l cmp).contains k = (List.map Prod.fst l).contains k

@[simp]

theorem Std.DTreeMap.Const.mem_ofList {α : Type u} {cmp : α → α → Ordering} {β : Type v} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List (α × β)} {k : α} : k ∈ ofList l cmp ↔ (List.map Prod.fst l).contains k = true

theorem Std.DTreeMap.Const.get?_ofList_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} {β : Type v} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List (α × β)} {k : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) : get? (ofList l cmp) k = none

theorem Std.DTreeMap.Const.get?_ofList_of_mem {α : Type u} {cmp : α → α → Ordering} {β : Type v} [TransCmp cmp] {l : List (α × β)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) {v : β} (distinct : List.Pairwise (fun (a b : α × β) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : (k, v) ∈ l) : get? (ofList l cmp) k' = some v

theorem Std.DTreeMap.Const.get_ofList_of_mem {α : Type u} {cmp : α → α → Ordering} {β : Type v} [TransCmp cmp] {l : List (α × β)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) {v : β} (distinct : List.Pairwise (fun (a b : α × β) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : (k, v) ∈ l) {h : k' ∈ ofList l cmp} : get (ofList l cmp) k' h = v

theorem Std.DTreeMap.Const.get!_ofList_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} {β : Type v} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List (α × β)} {k : α} [Inhabited β] (contains_eq_false : (List.map Prod.fst l).contains k = false) : get! (ofList l cmp) k = default

theorem Std.DTreeMap.Const.get!_ofList_of_mem {α : Type u} {cmp : α → α → Ordering} {β : Type v} [TransCmp cmp] {l : List (α × β)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) {v : β} [Inhabited β] (distinct : List.Pairwise (fun (a b : α × β) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : (k, v) ∈ l) : get! (ofList l cmp) k' = v

theorem Std.DTreeMap.Const.getD_ofList_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} {β : Type v} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List (α × β)} {k : α} {fallback : β} (contains_eq_false : (List.map Prod.fst l).contains k = false) : getD (ofList l cmp) k fallback = fallback

theorem Std.DTreeMap.Const.getD_ofList_of_mem {α : Type u} {cmp : α → α → Ordering} {β : Type v} [TransCmp cmp] {l : List (α × β)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) {v fallback : β} (distinct : List.Pairwise (fun (a b : α × β) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : (k, v) ∈ l) : getD (ofList l cmp) k' fallback = v

theorem Std.DTreeMap.Const.getKey?_ofList_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} {β : Type v} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List (α × β)} {k : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) : (ofList l cmp).getKey? k = none

theorem Std.DTreeMap.Const.getKey?_ofList_of_mem {α : Type u} {cmp : α → α → Ordering} {β : Type v} [TransCmp cmp] {l : List (α × β)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) (distinct : List.Pairwise (fun (a b : α × β) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : k ∈ List.map Prod.fst l) : (ofList l cmp).getKey? k' = some k

theorem Std.DTreeMap.Const.getKey_ofList_of_mem {α : Type u} {cmp : α → α → Ordering} {β : Type v} [TransCmp cmp] {l : List (α × β)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) (distinct : List.Pairwise (fun (a b : α × β) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : k ∈ List.map Prod.fst l) {h' : k' ∈ ofList l cmp} : (ofList l cmp).getKey k' h' = k

theorem Std.DTreeMap.Const.getKey!_ofList_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} {β : Type v} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] [Inhabited α] {l : List (α × β)} {k : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) : (ofList l cmp).getKey! k = default

theorem Std.DTreeMap.Const.getKey!_ofList_of_mem {α : Type u} {cmp : α → α → Ordering} {β : Type v} [TransCmp cmp] [Inhabited α] {l : List (α × β)} {k k' : α} (k_eq : cmp k k' = Ordering.eq) (distinct : List.Pairwise (fun (a b : α × β) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : k ∈ List.map Prod.fst l) : (ofList l cmp).getKey! k' = k

theorem Std.DTreeMap.Const.getKeyD_ofList_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} {β : Type v} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List (α × β)} {k fallback : α} (contains_eq_false : (List.map Prod.fst l).contains k = false) : (ofList l cmp).getKeyD k fallback = fallback

theorem Std.DTreeMap.Const.getKeyD_ofList_of_mem {α : Type u} {cmp : α → α → Ordering} {β : Type v} [TransCmp cmp] {l : List (α × β)} {k k' fallback : α} (k_eq : cmp k k' = Ordering.eq) (distinct : List.Pairwise (fun (a b : α × β) => ¬cmp a.fst b.fst = Ordering.eq) l) (mem : k ∈ List.map Prod.fst l) : (ofList l cmp).getKeyD k' fallback = k

theorem Std.DTreeMap.Const.size_ofList {α : Type u} {cmp : α → α → Ordering} {β : Type v} [TransCmp cmp] {l : List (α × β)} (distinct : List.Pairwise (fun (a b : α × β) => ¬cmp a.fst b.fst = Ordering.eq) l) : (ofList l cmp).size = l.length

theorem Std.DTreeMap.Const.size_ofList_le {α : Type u} {cmp : α → α → Ordering} {β : Type v} [TransCmp cmp] {l : List (α × β)} : (ofList l cmp).size ≤ l.length

@[simp]

theorem Std.DTreeMap.Const.isEmpty_ofList {α : Type u} {cmp : α → α → Ordering} {β : Type v} [TransCmp cmp] {l : List (α × β)} : (ofList l cmp).isEmpty = l.isEmpty

@[simp]

theorem Std.DTreeMap.Const.unitOfList_nil {α : Type u} {cmp : α → α → Ordering} : unitOfList [] cmp = ∅

@[simp]

theorem Std.DTreeMap.Const.unitOfList_singleton {α : Type u} {cmp : α → α → Ordering} {k : α} : unitOfList [k] cmp = ∅.insertIfNew k ()

theorem Std.DTreeMap.Const.unitOfList_cons {α : Type u} {cmp : α → α → Ordering} {hd : α} {tl : List α} : unitOfList (hd :: tl) cmp = insertManyIfNewUnit (∅.insertIfNew hd ()) tl

theorem Std.DTreeMap.Const.unitOfList_eq_insertManyIfNewUnit_empty {α : Type u} {cmp : α → α → Ordering} {l : List α} : unitOfList l cmp = insertManyIfNewUnit ∅ l

@[simp]

theorem Std.DTreeMap.Const.contains_unitOfList {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List α} {k : α} : (unitOfList l cmp).contains k = l.contains k

@[simp]

theorem Std.DTreeMap.Const.mem_unitOfList {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List α} {k : α} : k ∈ unitOfList l cmp ↔ l.contains k = true

theorem Std.DTreeMap.Const.getKey?_unitOfList_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List α} {k : α} (contains_eq_false : l.contains k = false) : (unitOfList l cmp).getKey? k = none

theorem Std.DTreeMap.Const.getKey?_unitOfList_of_mem {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {l : List α} {k k' : α} (k_eq : cmp k k' = Ordering.eq) (distinct : List.Pairwise (fun (a b : α) => ¬cmp a b = Ordering.eq) l) (mem : k ∈ l) : (unitOfList l cmp).getKey? k' = some k

theorem Std.DTreeMap.Const.getKey_unitOfList_of_mem {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {l : List α} {k k' : α} (k_eq : cmp k k' = Ordering.eq) (distinct : List.Pairwise (fun (a b : α) => ¬cmp a b = Ordering.eq) l) (mem : k ∈ l) {h' : k' ∈ unitOfList l cmp} : (unitOfList l cmp).getKey k' h' = k

theorem Std.DTreeMap.Const.getKey!_unitOfList_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] [Inhabited α] {l : List α} {k : α} (contains_eq_false : l.contains k = false) : (unitOfList l cmp).getKey! k = default

theorem Std.DTreeMap.Const.getKey!_unitOfList_of_mem {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] [Inhabited α] {l : List α} {k k' : α} (k_eq : cmp k k' = Ordering.eq) (distinct : List.Pairwise (fun (a b : α) => ¬cmp a b = Ordering.eq) l) (mem : k ∈ l) : (unitOfList l cmp).getKey! k' = k

theorem Std.DTreeMap.Const.getKeyD_unitOfList_of_contains_eq_false {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List α} {k fallback : α} (contains_eq_false : l.contains k = false) : (unitOfList l cmp).getKeyD k fallback = fallback

theorem Std.DTreeMap.Const.getKeyD_unitOfList_of_mem {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {l : List α} {k k' fallback : α} (k_eq : cmp k k' = Ordering.eq) (distinct : List.Pairwise (fun (a b : α) => ¬cmp a b = Ordering.eq) l) (mem : k ∈ l) : (unitOfList l cmp).getKeyD k' fallback = k

theorem Std.DTreeMap.Const.size_unitOfList {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {l : List α} (distinct : List.Pairwise (fun (a b : α) => ¬cmp a b = Ordering.eq) l) : (unitOfList l cmp).size = l.length

theorem Std.DTreeMap.Const.size_unitOfList_le {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {l : List α} : (unitOfList l cmp).size ≤ l.length

@[simp]

theorem Std.DTreeMap.Const.isEmpty_unitOfList {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {l : List α} : (unitOfList l cmp).isEmpty = l.isEmpty

@[simp]

theorem Std.DTreeMap.Const.get?_unitOfList {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] [BEq α] [LawfulBEqCmp cmp] {l : List α} {k : α} : get? (unitOfList l cmp) k = if l.contains k = true then some () else none

@[simp]

theorem Std.DTreeMap.Const.get_unitOfList {α : Type u} {cmp : α → α → Ordering} {l : List α} {k : α} {h : k ∈ unitOfList l cmp} : get (unitOfList l cmp) k h = ()

@[simp]

theorem Std.DTreeMap.Const.get!_unitOfList {α : Type u} {cmp : α → α → Ordering} {l : List α} {k : α} : get! (unitOfList l cmp) k = ()

@[simp]

theorem Std.DTreeMap.Const.getD_unitOfList {α : Type u} {cmp : α → α → Ordering} {l : List α} {k : α} {fallback : Unit} : getD (unitOfList l cmp) k fallback = ()

@[simp]

theorem Std.DTreeMap.union_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} : t₁.union t₂ = t₁ ∪ t₂

@[simp]

theorem Std.DTreeMap.contains_union {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} : (t₁ ∪ t₂).contains k = (t₁.contains k || t₂.contains k)

theorem Std.DTreeMap.mem_union_of_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} : k ∈ t₁ → k ∈ t₁ ∪ t₂

theorem Std.DTreeMap.mem_union_of_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} : k ∈ t₂ → k ∈ t₁ ∪ t₂

@[simp]

theorem Std.DTreeMap.mem_union_iff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} : k ∈ t₁ ∪ t₂ ↔ k ∈ t₁ ∨ k ∈ t₂

theorem Std.DTreeMap.mem_of_mem_union_of_not_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} : k ∈ t₁ ∪ t₂ → ¬k ∈ t₂ → k ∈ t₁

theorem Std.DTreeMap.mem_of_mem_union_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} : k ∈ t₁ ∪ t₂ → ¬k ∈ t₁ → k ∈ t₂

theorem Std.DTreeMap.Equiv.union_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ t₃ : DTreeMap α β cmp} [TransCmp cmp] (equiv : t₁.Equiv t₂) : (t₁ ∪ t₃).Equiv (t₂ ∪ t₃)

theorem Std.DTreeMap.Equiv.union_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ t₃ : DTreeMap α β cmp} [TransCmp cmp] (equiv : t₂.Equiv t₃) : (t₁ ∪ t₂).Equiv (t₁ ∪ t₃)

theorem Std.DTreeMap.Equiv.union_congr {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ t₃ t₄ : DTreeMap α β cmp} [TransCmp cmp] (equiv₁ : t₁.Equiv t₃) (equiv₂ : t₂.Equiv t₄) : (t₁ ∪ t₂).Equiv (t₃ ∪ t₄)

theorem Std.DTreeMap.union_insert_right_equiv_insert_union {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {p : (a : α) × β a} : (t₁ ∪ t₂.insert p.fst p.snd).Equiv ((t₁ ∪ t₂).insert p.fst p.snd)

theorem Std.DTreeMap.get?_union {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} : (t₁ ∪ t₂).get? k = (t₂.get? k).or (t₁.get? k)

theorem Std.DTreeMap.get?_union_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} (not_mem : ¬k ∈ t₁) : (t₁ ∪ t₂).get? k = t₂.get? k

theorem Std.DTreeMap.get?_union_of_not_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} (not_mem : ¬k ∈ t₂) : (t₁ ∪ t₂).get? k = t₁.get? k

theorem Std.DTreeMap.get_union_of_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} (mem : k ∈ t₂) : (t₁ ∪ t₂).get k ⋯ = t₂.get k mem

theorem Std.DTreeMap.get_union_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} (not_mem : ¬k ∈ t₁) {h' : k ∈ t₁ ∪ t₂} : (t₁ ∪ t₂).get k h' = t₂.get k ⋯

theorem Std.DTreeMap.get_union_of_not_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} (not_mem : ¬k ∈ t₂) {h' : k ∈ t₁ ∪ t₂} : (t₁ ∪ t₂).get k h' = t₁.get k ⋯

theorem Std.DTreeMap.getD_union {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {fallback : β k} : (t₁ ∪ t₂).getD k fallback = t₂.getD k (t₁.getD k fallback)

theorem Std.DTreeMap.getD_union_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {fallback : β k} (not_mem : ¬k ∈ t₁) : (t₁ ∪ t₂).getD k fallback = t₂.getD k fallback

theorem Std.DTreeMap.getD_union_of_not_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {fallback : β k} (not_mem : ¬k ∈ t₂) : (t₁ ∪ t₂).getD k fallback = t₁.getD k fallback

theorem Std.DTreeMap.get!_union {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} [Inhabited (β k)] : (t₁ ∪ t₂).get! k = t₂.getD k (t₁.get! k)

theorem Std.DTreeMap.get!_union_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} [Inhabited (β k)] (not_mem : ¬k ∈ t₁) : (t₁ ∪ t₂).get! k = t₂.get! k

theorem Std.DTreeMap.get!_union_of_not_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} [Inhabited (β k)] (not_mem : ¬k ∈ t₂) : (t₁ ∪ t₂).get! k = t₁.get! k

theorem Std.DTreeMap.getKey?_union {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} : (t₁ ∪ t₂).getKey? k = (t₂.getKey? k).or (t₁.getKey? k)

theorem Std.DTreeMap.getKey?_union_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (not_mem : ¬k ∈ t₁) : (t₁ ∪ t₂).getKey? k = t₂.getKey? k

theorem Std.DTreeMap.getKey?_union_of_not_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (not_mem : ¬k ∈ t₂) : (t₁ ∪ t₂).getKey? k = t₁.getKey? k

theorem Std.DTreeMap.getKey_union_of_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (mem : k ∈ t₂) : (t₁ ∪ t₂).getKey k ⋯ = t₂.getKey k mem

theorem Std.DTreeMap.getKey_union_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (not_mem : ¬k ∈ t₁) {h' : k ∈ t₁ ∪ t₂} : (t₁ ∪ t₂).getKey k h' = t₂.getKey k ⋯

theorem Std.DTreeMap.getKey_union_of_not_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (not_mem : ¬k ∈ t₂) {h' : k ∈ t₁ ∪ t₂} : (t₁ ∪ t₂).getKey k h' = t₁.getKey k ⋯

theorem Std.DTreeMap.getKeyD_union {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k fallback : α} : (t₁ ∪ t₂).getKeyD k fallback = t₂.getKeyD k (t₁.getKeyD k fallback)

theorem Std.DTreeMap.getKeyD_union_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k fallback : α} (mem : ¬k ∈ t₁) : (t₁ ∪ t₂).getKeyD k fallback = t₂.getKeyD k fallback

theorem Std.DTreeMap.getKeyD_union_of_not_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k fallback : α} (mem : ¬k ∈ t₂) : (t₁ ∪ t₂).getKeyD k fallback = t₁.getKeyD k fallback

theorem Std.DTreeMap.getKey!_union {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} : (t₁ ∪ t₂).getKey! k = t₂.getKeyD k (t₁.getKey! k)

theorem Std.DTreeMap.getKey!_union_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [Inhabited α] [TransCmp cmp] {k : α} (not_mem : ¬k ∈ t₁) : (t₁ ∪ t₂).getKey! k = t₂.getKey! k

theorem Std.DTreeMap.getKey!_union_of_not_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [Inhabited α] [TransCmp cmp] {k : α} (not_mem : ¬k ∈ t₂) : (t₁ ∪ t₂).getKey! k = t₁.getKey! k

theorem Std.DTreeMap.size_union_of_not_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] : (∀ (a : α), a ∈ t₁ → ¬a ∈ t₂) → (t₁ ∪ t₂).size = t₁.size + t₂.size

theorem Std.DTreeMap.size_left_le_size_union {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] : t₁.size ≤ (t₁ ∪ t₂).size

theorem Std.DTreeMap.size_right_le_size_union {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] : t₂.size ≤ (t₁ ∪ t₂).size

theorem Std.DTreeMap.size_union_le_size_add_size {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] : (t₁ ∪ t₂).size ≤ t₁.size + t₂.size

@[simp]

theorem Std.DTreeMap.isEmpty_union {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] : (t₁ ∪ t₂).isEmpty = (t₁.isEmpty && t₂.isEmpty)

theorem Std.DTreeMap.Const.get?_union {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} : get? (t₁ ∪ t₂) k = (get? t₂ k).or (get? t₁ k)

theorem Std.DTreeMap.Const.get?_union_of_not_mem_left {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} (not_mem : ¬k ∈ t₁) : get? (t₁ ∪ t₂) k = get? t₂ k

theorem Std.DTreeMap.Const.get?_union_of_not_mem_right {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} (not_mem : ¬k ∈ t₂) : get? (t₁ ∪ t₂) k = get? t₁ k

theorem Std.DTreeMap.Const.get_union_of_mem_right {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} (mem : k ∈ t₂) : get (t₁ ∪ t₂) k ⋯ = get t₂ k mem

theorem Std.DTreeMap.Const.get_union_of_not_mem_left {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} (not_mem : ¬k ∈ t₁) {h' : k ∈ t₁ ∪ t₂} : get (t₁ ∪ t₂) k h' = get t₂ k ⋯

theorem Std.DTreeMap.Const.get_union_of_not_mem_right {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} (not_mem : ¬k ∈ t₂) {h' : k ∈ t₁ ∪ t₂} : get (t₁ ∪ t₂) k h' = get t₁ k ⋯

theorem Std.DTreeMap.Const.getD_union {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {fallback : β} : getD (t₁ ∪ t₂) k fallback = getD t₂ k (getD t₁ k fallback)

theorem Std.DTreeMap.Const.getD_union_of_not_mem_left {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {fallback : β} (not_mem : ¬k ∈ t₁) : getD (t₁ ∪ t₂) k fallback = getD t₂ k fallback

theorem Std.DTreeMap.Const.getD_union_of_not_mem_right {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {fallback : β} (not_mem : ¬k ∈ t₂) : getD (t₁ ∪ t₂) k fallback = getD t₁ k fallback

theorem Std.DTreeMap.Const.get!_union {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {k : α} : get! (t₁ ∪ t₂) k = getD t₂ k (get! t₁ k)

theorem Std.DTreeMap.Const.get!_union_of_not_mem_left {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {k : α} (not_mem : ¬k ∈ t₁) : get! (t₁ ∪ t₂) k = get! t₂ k

theorem Std.DTreeMap.Const.get!_union_of_not_mem_right {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {k : α} (not_mem : ¬k ∈ t₂) : get! (t₁ ∪ t₂) k = get! t₁ k

@[simp]

theorem Std.DTreeMap.inter_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} : t₁.inter t₂ = t₁ ∩ t₂

@[simp]

theorem Std.DTreeMap.contains_inter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} : (t₁ ∩ t₂).contains k = (t₁.contains k && t₂.contains k)

@[simp]

theorem Std.DTreeMap.mem_inter_iff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} : k ∈ t₁ ∩ t₂ ↔ k ∈ t₁ ∧ k ∈ t₂

theorem Std.DTreeMap.not_mem_inter_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : ¬k ∈ t₁) : ¬k ∈ t₁ ∩ t₂

theorem Std.DTreeMap.not_mem_inter_of_not_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : ¬k ∈ t₂) : ¬k ∈ t₁ ∩ t₂

theorem Std.DTreeMap.Equiv.inter_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ t₃ : DTreeMap α β cmp} [TransCmp cmp] (equiv : t₁.Equiv t₂) : (t₁ ∩ t₃).Equiv (t₂ ∩ t₃)

theorem Std.DTreeMap.Equiv.inter_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ t₃ : DTreeMap α β cmp} [TransCmp cmp] (equiv : t₂.Equiv t₃) : (t₁ ∩ t₂).Equiv (t₁ ∩ t₃)

theorem Std.DTreeMap.Equiv.inter_congr {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ t₃ t₄ : DTreeMap α β cmp} [TransCmp cmp] (equiv₁ : t₁.Equiv t₃) (equiv₂ : t₂.Equiv t₄) : (t₁ ∩ t₂).Equiv (t₃ ∩ t₄)

theorem Std.DTreeMap.get?_inter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} : (t₁ ∩ t₂).get? k = if k ∈ t₂ then t₁.get? k else none

theorem Std.DTreeMap.get?_inter_of_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} (h : k ∈ t₂) : (t₁ ∩ t₂).get? k = t₁.get? k

theorem Std.DTreeMap.get?_inter_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} (h : ¬k ∈ t₁) : (t₁ ∩ t₂).get? k = none

theorem Std.DTreeMap.get?_inter_of_not_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} (h : ¬k ∈ t₂) : (t₁ ∩ t₂).get? k = none

@[simp]

theorem Std.DTreeMap.get_inter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {h_mem : k ∈ t₁ ∩ t₂} : (t₁ ∩ t₂).get k h_mem = t₁.get k ⋯

theorem Std.DTreeMap.getD_inter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {fallback : β k} : (t₁ ∩ t₂).getD k fallback = if k ∈ t₂ then t₁.getD k fallback else fallback

theorem Std.DTreeMap.getD_inter_of_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {fallback : β k} (h : k ∈ t₂) : (t₁ ∩ t₂).getD k fallback = t₁.getD k fallback

theorem Std.DTreeMap.getD_inter_of_not_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {fallback : β k} (h : ¬k ∈ t₂) : (t₁ ∩ t₂).getD k fallback = fallback

theorem Std.DTreeMap.getD_inter_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {fallback : β k} (h : ¬k ∈ t₁) : (t₁ ∩ t₂).getD k fallback = fallback

theorem Std.DTreeMap.get!_inter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} [Inhabited (β k)] : (t₁ ∩ t₂).get! k = if k ∈ t₂ then t₁.get! k else default

theorem Std.DTreeMap.get!_inter_of_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} [Inhabited (β k)] (h : k ∈ t₂) : (t₁ ∩ t₂).get! k = t₁.get! k

theorem Std.DTreeMap.get!_inter_of_not_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} [Inhabited (β k)] (h : ¬k ∈ t₂) : (t₁ ∩ t₂).get! k = default

theorem Std.DTreeMap.get!_inter_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} [Inhabited (β k)] (h : ¬k ∈ t₁) : (t₁ ∩ t₂).get! k = default

theorem Std.DTreeMap.getKey?_inter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} : (t₁ ∩ t₂).getKey? k = if k ∈ t₂ then t₁.getKey? k else none

theorem Std.DTreeMap.getKey?_inter_of_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : k ∈ t₂) : (t₁ ∩ t₂).getKey? k = t₁.getKey? k

theorem Std.DTreeMap.getKey?_inter_of_not_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : ¬k ∈ t₂) : (t₁ ∩ t₂).getKey? k = none

theorem Std.DTreeMap.getKey?_inter_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : ¬k ∈ t₁) : (t₁ ∩ t₂).getKey? k = none

@[simp]

theorem Std.DTreeMap.getKey_inter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} {h_mem : k ∈ t₁ ∩ t₂} : (t₁ ∩ t₂).getKey k h_mem = t₁.getKey k ⋯

theorem Std.DTreeMap.getKeyD_inter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k fallback : α} : (t₁ ∩ t₂).getKeyD k fallback = if k ∈ t₂ then t₁.getKeyD k fallback else fallback

theorem Std.DTreeMap.getKeyD_inter_of_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k fallback : α} (h : k ∈ t₂) : (t₁ ∩ t₂).getKeyD k fallback = t₁.getKeyD k fallback

theorem Std.DTreeMap.getKeyD_inter_of_not_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k fallback : α} (h : ¬k ∈ t₂) : (t₁ ∩ t₂).getKeyD k fallback = fallback

theorem Std.DTreeMap.getKeyD_inter_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k fallback : α} (h : ¬k ∈ t₁) : (t₁ ∩ t₂).getKeyD k fallback = fallback

theorem Std.DTreeMap.getKey!_inter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [Inhabited α] [TransCmp cmp] {k : α} : (t₁ ∩ t₂).getKey! k = if k ∈ t₂ then t₁.getKey! k else default

theorem Std.DTreeMap.getKey!_inter_of_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [Inhabited α] [TransCmp cmp] {k : α} (h : k ∈ t₂) : (t₁ ∩ t₂).getKey! k = t₁.getKey! k

theorem Std.DTreeMap.getKey!_inter_of_not_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [Inhabited α] [TransCmp cmp] {k : α} (h : ¬k ∈ t₂) : (t₁ ∩ t₂).getKey! k = default

theorem Std.DTreeMap.getKey!_inter_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [Inhabited α] [TransCmp cmp] {k : α} (h : ¬k ∈ t₁) : (t₁ ∩ t₂).getKey! k = default

theorem Std.DTreeMap.size_inter_le_size_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] : (t₁ ∩ t₂).size ≤ t₁.size

theorem Std.DTreeMap.size_inter_le_size_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] : (t₁ ∩ t₂).size ≤ t₂.size

theorem Std.DTreeMap.size_inter_eq_size_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] (h : ∀ (a : α), a ∈ t₁ → a ∈ t₂) : (t₁ ∩ t₂).size = t₁.size

theorem Std.DTreeMap.size_inter_eq_size_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] (h : ∀ (a : α), a ∈ t₂ → a ∈ t₁) : (t₁ ∩ t₂).size = t₂.size

theorem Std.DTreeMap.size_add_size_eq_size_union_add_size_inter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] : t₁.size + t₂.size = (t₁ ∪ t₂).size + (t₁ ∩ t₂).size

@[simp]

theorem Std.DTreeMap.isEmpty_inter_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] (h : t₁.isEmpty = true) : (t₁ ∩ t₂).isEmpty = true

@[simp]

theorem Std.DTreeMap.isEmpty_inter_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] (h : t₂.isEmpty = true) : (t₁ ∩ t₂).isEmpty = true

theorem Std.DTreeMap.isEmpty_inter_iff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] : (t₁ ∩ t₂).isEmpty = true ↔ ∀ (k : α), k ∈ t₁ → ¬k ∈ t₂

theorem Std.DTreeMap.isEmpty_inter_comm {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] : (t₁ ∩ t₂).isEmpty = (t₂ ∩ t₁).isEmpty

theorem Std.DTreeMap.Const.get?_inter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} : get? (t₁ ∩ t₂) k = if k ∈ t₂ then get? t₁ k else none

theorem Std.DTreeMap.Const.get?_inter_of_mem_right {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} (h : k ∈ t₂) : get? (t₁ ∩ t₂) k = get? t₁ k

theorem Std.DTreeMap.Const.get?_inter_of_not_mem_left {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} (h : ¬k ∈ t₁) : get? (t₁ ∩ t₂) k = none

theorem Std.DTreeMap.Const.get?_inter_of_not_mem_right {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} (h : ¬k ∈ t₂) : get? (t₁ ∩ t₂) k = none

@[simp]

theorem Std.DTreeMap.Const.get_inter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {h_mem : k ∈ t₁ ∩ t₂} : get (t₁ ∩ t₂) k h_mem = get t₁ k ⋯

theorem Std.DTreeMap.Const.getD_inter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {fallback : β} : getD (t₁ ∩ t₂) k fallback = if k ∈ t₂ then getD t₁ k fallback else fallback

theorem Std.DTreeMap.Const.getD_inter_of_mem_right {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {fallback : β} (h : k ∈ t₂) : getD (t₁ ∩ t₂) k fallback = getD t₁ k fallback

theorem Std.DTreeMap.Const.getD_inter_of_not_mem_right {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {fallback : β} (h : ¬k ∈ t₂) : getD (t₁ ∩ t₂) k fallback = fallback

theorem Std.DTreeMap.Const.getD_inter_of_not_mem_left {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {fallback : β} (h : ¬k ∈ t₁) : getD (t₁ ∩ t₂) k fallback = fallback

theorem Std.DTreeMap.Const.get!_inter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {k : α} : get! (t₁ ∩ t₂) k = if k ∈ t₂ then get! t₁ k else default

theorem Std.DTreeMap.Const.get!_inter_of_mem_right {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {k : α} (h : k ∈ t₂) : get! (t₁ ∩ t₂) k = get! t₁ k

theorem Std.DTreeMap.Const.get!_inter_of_not_mem_right {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {k : α} (h : ¬k ∈ t₂) : get! (t₁ ∩ t₂) k = default

theorem Std.DTreeMap.Const.get!_inter_of_not_mem_left {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {k : α} (h : ¬k ∈ t₁) : get! (t₁ ∩ t₂) k = default

theorem Std.DTreeMap.Equiv.beq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {m₁ m₂ : DTreeMap α β cmp} [(k : α) → BEq (β k)] [LawfulEqCmp cmp] [TransCmp cmp] [∀ (k : α), ReflBEq (β k)] (h : m₁.Equiv m₂) : (m₁ == m₂) = true

theorem Std.DTreeMap.equiv_of_beq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {m₁ m₂ : DTreeMap α β cmp} [(k : α) → BEq (β k)] [LawfulEqCmp cmp] [TransCmp cmp] [∀ (k : α), LawfulBEq (β k)] (h : (m₁ == m₂) = true) : m₁.Equiv m₂

theorem Std.DTreeMap.beq_iff_equiv {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {m₁ m₂ : DTreeMap α β cmp} [(k : α) → BEq (β k)] [LawfulEqCmp cmp] [TransCmp cmp] [∀ (k : α), LawfulBEq (β k)] : (m₁ == m₂) = true ↔ m₁.Equiv m₂

theorem Std.DTreeMap.Equiv.beq_congr {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {m₁ m₂ : DTreeMap α β cmp} [(k : α) → BEq (β k)] [LawfulEqCmp cmp] [TransCmp cmp] {m₃ m₄ : DTreeMap α β cmp} (w₁ : m₁.Equiv m₃) (w₂ : m₂.Equiv m₄) : (m₁ == m₂) = (m₃ == m₄)

theorem Std.DTreeMap.Const.Equiv.beq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {m₁ m₂ : DTreeMap α (fun (x : α) => β) cmp} [BEq β] [TransCmp cmp] [ReflBEq β] (h : m₁.Equiv m₂) : Const.beq m₁ m₂ = true

theorem Std.DTreeMap.Const.equiv_of_beq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {m₁ m₂ : DTreeMap α (fun (x : α) => β) cmp} [BEq β] [TransCmp cmp] [LawfulEqCmp cmp] [LawfulBEq β] (h : beq m₁ m₂ = true) : m₁.Equiv m₂

theorem Std.DTreeMap.Const.beq_iff_equiv {α : Type u} {cmp : α → α → Ordering} {β : Type v} {m₁ m₂ : DTreeMap α (fun (x : α) => β) cmp} [BEq β] [TransCmp cmp] [LawfulEqCmp cmp] [LawfulBEq β] : beq m₁ m₂ = true ↔ m₁.Equiv m₂

theorem Std.DTreeMap.Const.Equiv.beq_congr {α : Type u} {cmp : α → α → Ordering} {β : Type v} {m₁ m₂ : DTreeMap α (fun (x : α) => β) cmp} [BEq β] [TransCmp cmp] {m₃ m₄ : DTreeMap α (fun (x : α) => β) cmp} (w₁ : m₁.Equiv m₃) (w₂ : m₂.Equiv m₄) : Const.beq m₁ m₂ = Const.beq m₃ m₄

@[simp]

theorem Std.DTreeMap.diff_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} : t₁.diff t₂ = t₁ \ t₂

@[simp]

theorem Std.DTreeMap.contains_diff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} : (t₁ \ t₂).contains k = (t₁.contains k && !t₂.contains k)

@[simp]

theorem Std.DTreeMap.mem_diff_iff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} : k ∈ t₁ \ t₂ ↔ k ∈ t₁ ∧ ¬k ∈ t₂

theorem Std.DTreeMap.not_mem_diff_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : ¬k ∈ t₁) : ¬k ∈ t₁ \ t₂

theorem Std.DTreeMap.not_mem_diff_of_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : k ∈ t₂) : ¬k ∈ t₁ \ t₂

theorem Std.DTreeMap.Equiv.diff_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ t₃ : DTreeMap α β cmp} [TransCmp cmp] (equiv : t₁.Equiv t₂) : (t₁ \ t₃).Equiv (t₂ \ t₃)

theorem Std.DTreeMap.Equiv.diff_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ t₃ : DTreeMap α β cmp} [TransCmp cmp] (equiv : t₂.Equiv t₃) : (t₁ \ t₂).Equiv (t₁ \ t₃)

theorem Std.DTreeMap.Equiv.diff_congr {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ t₃ t₄ : DTreeMap α β cmp} [TransCmp cmp] (equiv₁ : t₁.Equiv t₃) (equiv₂ : t₂.Equiv t₄) : (t₁ \ t₂).Equiv (t₃ \ t₄)

theorem Std.DTreeMap.get?_diff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} : (t₁ \ t₂).get? k = if k ∈ t₂ then none else t₁.get? k

theorem Std.DTreeMap.get?_diff_of_not_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} (h : ¬k ∈ t₂) : (t₁ \ t₂).get? k = t₁.get? k

theorem Std.DTreeMap.get?_diff_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} (h : ¬k ∈ t₁) : (t₁ \ t₂).get? k = none

theorem Std.DTreeMap.get?_diff_of_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} (h : k ∈ t₂) : (t₁ \ t₂).get? k = none

theorem Std.DTreeMap.get_diff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {h_mem : k ∈ t₁ \ t₂} : (t₁ \ t₂).get k h_mem = t₁.get k ⋯

theorem Std.DTreeMap.getD_diff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {fallback : β k} : (t₁ \ t₂).getD k fallback = if k ∈ t₂ then fallback else t₁.getD k fallback

theorem Std.DTreeMap.getD_diff_of_not_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {fallback : β k} (h : ¬k ∈ t₂) : (t₁ \ t₂).getD k fallback = t₁.getD k fallback

theorem Std.DTreeMap.getD_diff_of_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {fallback : β k} (h : k ∈ t₂) : (t₁ \ t₂).getD k fallback = fallback

theorem Std.DTreeMap.getD_diff_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {fallback : β k} (h : ¬k ∈ t₁) : (t₁ \ t₂).getD k fallback = fallback

theorem Std.DTreeMap.get!_diff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} [Inhabited (β k)] : (t₁ \ t₂).get! k = if k ∈ t₂ then default else t₁.get! k

theorem Std.DTreeMap.get!_diff_of_not_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} [Inhabited (β k)] (h : ¬k ∈ t₂) : (t₁ \ t₂).get! k = t₁.get! k

theorem Std.DTreeMap.get!_diff_of_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} [Inhabited (β k)] (h : k ∈ t₂) : (t₁ \ t₂).get! k = default

theorem Std.DTreeMap.get!_diff_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} [Inhabited (β k)] (h : ¬k ∈ t₁) : (t₁ \ t₂).get! k = default

theorem Std.DTreeMap.getKey?_diff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} : (t₁ \ t₂).getKey? k = if k ∈ t₂ then none else t₁.getKey? k

theorem Std.DTreeMap.getKey?_diff_of_not_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : ¬k ∈ t₂) : (t₁ \ t₂).getKey? k = t₁.getKey? k

theorem Std.DTreeMap.getKey?_diff_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : ¬k ∈ t₁) : (t₁ \ t₂).getKey? k = none

theorem Std.DTreeMap.getKey?_diff_of_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : k ∈ t₂) : (t₁ \ t₂).getKey? k = none

theorem Std.DTreeMap.getKey_diff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} {h_mem : k ∈ t₁ \ t₂} : (t₁ \ t₂).getKey k h_mem = t₁.getKey k ⋯

theorem Std.DTreeMap.getKeyD_diff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k fallback : α} : (t₁ \ t₂).getKeyD k fallback = if k ∈ t₂ then fallback else t₁.getKeyD k fallback

theorem Std.DTreeMap.getKeyD_diff_of_not_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k fallback : α} (h : ¬k ∈ t₂) : (t₁ \ t₂).getKeyD k fallback = t₁.getKeyD k fallback

theorem Std.DTreeMap.getKeyD_diff_of_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k fallback : α} (h : k ∈ t₂) : (t₁ \ t₂).getKeyD k fallback = fallback

theorem Std.DTreeMap.getKeyD_diff_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k fallback : α} (h : ¬k ∈ t₁) : (t₁ \ t₂).getKeyD k fallback = fallback

theorem Std.DTreeMap.getKey!_diff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [Inhabited α] [TransCmp cmp] {k : α} : (t₁ \ t₂).getKey! k = if k ∈ t₂ then default else t₁.getKey! k

theorem Std.DTreeMap.getKey!_diff_of_not_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [Inhabited α] [TransCmp cmp] {k : α} (h : ¬k ∈ t₂) : (t₁ \ t₂).getKey! k = t₁.getKey! k

theorem Std.DTreeMap.getKey!_diff_of_mem_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [Inhabited α] [TransCmp cmp] {k : α} (h : k ∈ t₂) : (t₁ \ t₂).getKey! k = default

theorem Std.DTreeMap.getKey!_diff_of_not_mem_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [Inhabited α] [TransCmp cmp] {k : α} (h : ¬k ∈ t₁) : (t₁ \ t₂).getKey! k = default

theorem Std.DTreeMap.size_diff_le_size_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] : (t₁ \ t₂).size ≤ t₁.size

theorem Std.DTreeMap.size_diff_eq_size_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] (h : ∀ (a : α), a ∈ t₁ → ¬a ∈ t₂) : (t₁ \ t₂).size = t₁.size

theorem Std.DTreeMap.size_diff_add_size_inter_eq_size_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] : (t₁ \ t₂).size + (t₁ ∩ t₂).size = t₁.size

@[simp]

theorem Std.DTreeMap.isEmpty_diff_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] (h : t₁.isEmpty = true) : (t₁ \ t₂).isEmpty = true

theorem Std.DTreeMap.isEmpty_diff_iff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] : (t₁ \ t₂).isEmpty = true ↔ ∀ (k : α), k ∈ t₁ → k ∈ t₂

theorem Std.DTreeMap.Const.get?_diff {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} : get? (t₁ \ t₂) k = if k ∈ t₂ then none else get? t₁ k

theorem Std.DTreeMap.Const.get?_diff_of_not_mem_right {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} (h : ¬k ∈ t₂) : get? (t₁ \ t₂) k = get? t₁ k

theorem Std.DTreeMap.Const.get?_diff_of_not_mem_left {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} (h : ¬k ∈ t₁) : get? (t₁ \ t₂) k = none

theorem Std.DTreeMap.Const.get?_diff_of_mem_right {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} (h : k ∈ t₂) : get? (t₁ \ t₂) k = none

theorem Std.DTreeMap.Const.get_diff {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {h_mem : k ∈ t₁ \ t₂} : get (t₁ \ t₂) k h_mem = get t₁ k ⋯

theorem Std.DTreeMap.Const.getD_diff {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {fallback : β} : getD (t₁ \ t₂) k fallback = if k ∈ t₂ then fallback else getD t₁ k fallback

theorem Std.DTreeMap.Const.getD_diff_of_not_mem_right {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {fallback : β} (h : ¬k ∈ t₂) : getD (t₁ \ t₂) k fallback = getD t₁ k fallback

theorem Std.DTreeMap.Const.getD_diff_of_mem_right {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {fallback : β} (h : k ∈ t₂) : getD (t₁ \ t₂) k fallback = fallback

theorem Std.DTreeMap.Const.getD_diff_of_not_mem_left {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {fallback : β} (h : ¬k ∈ t₁) : getD (t₁ \ t₂) k fallback = fallback

theorem Std.DTreeMap.Const.get!_diff {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {k : α} : get! (t₁ \ t₂) k = if k ∈ t₂ then default else get! t₁ k

theorem Std.DTreeMap.Const.get!_diff_of_not_mem_right {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {k : α} (h : ¬k ∈ t₂) : get! (t₁ \ t₂) k = get! t₁ k

theorem Std.DTreeMap.Const.get!_diff_of_mem_right {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {k : α} (h : k ∈ t₂) : get! (t₁ \ t₂) k = default

theorem Std.DTreeMap.Const.get!_diff_of_not_mem_left {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {k : α} (h : ¬k ∈ t₁) : get! (t₁ \ t₂) k = default

theorem Std.DTreeMap.isEmpty_alter_eq_isEmpty_erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : Option (β k) → Option (β k)} : (t.alter k f).isEmpty = ((t.erase k).isEmpty && (f (t.get? k)).isNone)

@[simp]

theorem Std.DTreeMap.isEmpty_alter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : Option (β k) → Option (β k)} : (t.alter k f).isEmpty = ((t.isEmpty || t.size == 1 && t.contains k) && (f (t.get? k)).isNone)

theorem Std.DTreeMap.contains_alter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k k' : α} {f : Option (β k) → Option (β k)} : (t.alter k f).contains k' = if cmp k k' = Ordering.eq then (f (t.get? k)).isSome else t.contains k'

theorem Std.DTreeMap.mem_alter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k k' : α} {f : Option (β k) → Option (β k)} : k' ∈ t.alter k f ↔ if cmp k k' = Ordering.eq then (f (t.get? k)).isSome = true else k' ∈ t

theorem Std.DTreeMap.mem_alter_of_compare_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k k' : α} {f : Option (β k) → Option (β k)} (he : cmp k k' = Ordering.eq) : k' ∈ t.alter k f ↔ (f (t.get? k)).isSome = true

@[simp]

theorem Std.DTreeMap.contains_alter_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : Option (β k) → Option (β k)} : (t.alter k f).contains k = (f (t.get? k)).isSome

@[simp]

theorem Std.DTreeMap.mem_alter_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : Option (β k) → Option (β k)} : k ∈ t.alter k f ↔ (f (t.get? k)).isSome = true

theorem Std.DTreeMap.contains_alter_of_not_compare_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k k' : α} {f : Option (β k) → Option (β k)} (he : ¬cmp k k' = Ordering.eq) : (t.alter k f).contains k' = t.contains k'

theorem Std.DTreeMap.mem_alter_of_not_compare_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k k' : α} {f : Option (β k) → Option (β k)} (he : ¬cmp k k' = Ordering.eq) : k' ∈ t.alter k f ↔ k' ∈ t

theorem Std.DTreeMap.size_alter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : Option (β k) → Option (β k)} : (t.alter k f).size = if k ∈ t ∧ (f (t.get? k)).isNone = true then t.size - 1 else if ¬k ∈ t ∧ (f (t.get? k)).isSome = true then t.size + 1 else t.size

theorem Std.DTreeMap.size_alter_eq_add_one {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : Option (β k) → Option (β k)} (h₁ : ¬k ∈ t) (h₂ : (f (t.get? k)).isSome = true) : (t.alter k f).size = t.size + 1

theorem Std.DTreeMap.size_alter_eq_sub_one {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : Option (β k) → Option (β k)} (h₁ : k ∈ t) (h₂ : (f (t.get? k)).isNone = true) : (t.alter k f).size = t.size - 1

theorem Std.DTreeMap.size_alter_eq_self_of_not_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : Option (β k) → Option (β k)} (h₁ : ¬k ∈ t) (h₂ : (f (t.get? k)).isNone = true) : (t.alter k f).size = t.size

theorem Std.DTreeMap.size_alter_eq_self_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : Option (β k) → Option (β k)} (h₁ : k ∈ t) (h₂ : (f (t.get? k)).isSome = true) : (t.alter k f).size = t.size

theorem Std.DTreeMap.size_alter_le_size {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : Option (β k) → Option (β k)} : (t.alter k f).size ≤ t.size + 1

theorem Std.DTreeMap.size_le_size_alter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : Option (β k) → Option (β k)} : t.size - 1 ≤ (t.alter k f).size

theorem Std.DTreeMap.get?_alter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k k' : α} {f : Option (β k) → Option (β k)} : (t.alter k f).get? k' = if h : cmp k k' = Ordering.eq then cast ⋯ (f (t.get? k)) else t.get? k'

@[simp]

theorem Std.DTreeMap.get?_alter_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : Option (β k) → Option (β k)} : (t.alter k f).get? k = f (t.get? k)

theorem Std.DTreeMap.get_alter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k k' : α} {f : Option (β k) → Option (β k)} {hc : k' ∈ t.alter k f} : (t.alter k f).get k' hc = if heq : cmp k k' = Ordering.eq then cast ⋯ ((f (t.get? k)).get ⋯) else t.get k' ⋯

@[simp]

theorem Std.DTreeMap.get_alter_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : Option (β k) → Option (β k)} {hc : k ∈ t.alter k f} : (t.alter k f).get k hc = (f (t.get? k)).get ⋯

theorem Std.DTreeMap.get!_alter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k k' : α} [Inhabited (β k')] {f : Option (β k) → Option (β k)} : (t.alter k f).get! k' = if heq : cmp k k' = Ordering.eq then (Option.map (cast ⋯) (f (t.get? k))).get! else t.get! k'

@[simp]

theorem Std.DTreeMap.get!_alter_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} [Inhabited (β k)] {f : Option (β k) → Option (β k)} : (t.alter k f).get! k = (f (t.get? k)).get!

theorem Std.DTreeMap.getD_alter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k k' : α} {fallback : β k'} {f : Option (β k) → Option (β k)} : (t.alter k f).getD k' fallback = if heq : cmp k k' = Ordering.eq then (Option.map (cast ⋯) (f (t.get? k))).getD fallback else t.getD k' fallback

@[simp]

theorem Std.DTreeMap.getD_alter_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {fallback : β k} {f : Option (β k) → Option (β k)} : (t.alter k f).getD k fallback = (f (t.get? k)).getD fallback

theorem Std.DTreeMap.getKey?_alter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k k' : α} {f : Option (β k) → Option (β k)} : (t.alter k f).getKey? k' = if cmp k k' = Ordering.eq then if (f (t.get? k)).isSome = true then some k else none else t.getKey? k'

theorem Std.DTreeMap.getKey?_alter_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : Option (β k) → Option (β k)} : (t.alter k f).getKey? k = if (f (t.get? k)).isSome = true then some k else none

theorem Std.DTreeMap.getKey!_alter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {k k' : α} {f : Option (β k) → Option (β k)} : (t.alter k f).getKey! k' = if cmp k k' = Ordering.eq then if (f (t.get? k)).isSome = true then k else default else t.getKey! k'

theorem Std.DTreeMap.getKey!_alter_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {k : α} {f : Option (β k) → Option (β k)} : (t.alter k f).getKey! k = if (f (t.get? k)).isSome = true then k else default

@[simp]

theorem Std.DTreeMap.getKey_alter_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {k : α} {f : Option (β k) → Option (β k)} {hc : k ∈ t.alter k f} : (t.alter k f).getKey k hc = k

theorem Std.DTreeMap.getKeyD_alter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k k' fallback : α} {f : Option (β k) → Option (β k)} : (t.alter k f).getKeyD k' fallback = if cmp k k' = Ordering.eq then if (f (t.get? k)).isSome = true then k else fallback else t.getKeyD k' fallback

@[simp]

theorem Std.DTreeMap.getKeyD_alter_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {k fallback : α} {f : Option (β k) → Option (β k)} : (t.alter k f).getKeyD k fallback = if (f (t.get? k)).isSome = true then k else fallback

theorem Std.DTreeMap.Const.isEmpty_alter_eq_isEmpty_erase {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : Option β → Option β} : (alter t k f).isEmpty = ((t.erase k).isEmpty && (f (get? t k)).isNone)

@[simp]

theorem Std.DTreeMap.Const.isEmpty_alter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : Option β → Option β} : (alter t k f).isEmpty = ((t.isEmpty || t.size == 1 && t.contains k) && (f (get? t k)).isNone)

theorem Std.DTreeMap.Const.contains_alter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k k' : α} {f : Option β → Option β} : (alter t k f).contains k' = if cmp k k' = Ordering.eq then (f (get? t k)).isSome else t.contains k'

theorem Std.DTreeMap.Const.mem_alter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k k' : α} {f : Option β → Option β} : k' ∈ alter t k f ↔ if cmp k k' = Ordering.eq then (f (get? t k)).isSome = true else k' ∈ t

theorem Std.DTreeMap.Const.mem_alter_of_compare_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k k' : α} {f : Option β → Option β} (he : cmp k k' = Ordering.eq) : k' ∈ alter t k f ↔ (f (get? t k)).isSome = true

@[simp]

theorem Std.DTreeMap.Const.contains_alter_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : Option β → Option β} : (alter t k f).contains k = (f (get? t k)).isSome

@[simp]

theorem Std.DTreeMap.Const.mem_alter_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : Option β → Option β} : k ∈ alter t k f ↔ (f (get? t k)).isSome = true

theorem Std.DTreeMap.Const.contains_alter_of_not_compare_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k k' : α} {f : Option β → Option β} (he : ¬cmp k k' = Ordering.eq) : (alter t k f).contains k' = t.contains k'

theorem Std.DTreeMap.Const.mem_alter_of_not_compare_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k k' : α} {f : Option β → Option β} (he : ¬cmp k k' = Ordering.eq) : k' ∈ alter t k f ↔ k' ∈ t

theorem Std.DTreeMap.Const.size_alter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : Option β → Option β} : (alter t k f).size = if k ∈ t ∧ (f (get? t k)).isNone = true then t.size - 1 else if ¬k ∈ t ∧ (f (get? t k)).isSome = true then t.size + 1 else t.size

theorem Std.DTreeMap.Const.size_alter_eq_add_one {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : Option β → Option β} (h₁ : ¬k ∈ t) (h₂ : (f (get? t k)).isSome = true) : (alter t k f).size = t.size + 1

theorem Std.DTreeMap.Const.size_alter_eq_sub_one {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : Option β → Option β} (h₁ : k ∈ t) (h₂ : (f (get? t k)).isNone = true) : (alter t k f).size = t.size - 1

theorem Std.DTreeMap.Const.size_alter_eq_self_of_not_mem {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : Option β → Option β} (h₁ : ¬k ∈ t) (h₂ : (f (get? t k)).isNone = true) : (alter t k f).size = t.size

theorem Std.DTreeMap.Const.size_alter_eq_self_of_mem {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : Option β → Option β} (h₁ : k ∈ t) (h₂ : (f (get? t k)).isSome = true) : (alter t k f).size = t.size

theorem Std.DTreeMap.Const.size_alter_le_size {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : Option β → Option β} : (alter t k f).size ≤ t.size + 1

theorem Std.DTreeMap.Const.size_le_size_alter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : Option β → Option β} : t.size - 1 ≤ (alter t k f).size

theorem Std.DTreeMap.Const.get?_alter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k k' : α} {f : Option β → Option β} : get? (alter t k f) k' = if cmp k k' = Ordering.eq then f (get? t k) else get? t k'

@[simp]

theorem Std.DTreeMap.Const.get?_alter_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : Option β → Option β} : get? (alter t k f) k = f (get? t k)

theorem Std.DTreeMap.Const.get_alter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k k' : α} {f : Option β → Option β} {hc : k' ∈ alter t k f} : get (alter t k f) k' hc = if heq : cmp k k' = Ordering.eq then (f (get? t k)).get ⋯ else get t k' ⋯

@[simp]

theorem Std.DTreeMap.Const.get_alter_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : Option β → Option β} {hc : k ∈ alter t k f} : get (alter t k f) k hc = (f (get? t k)).get ⋯

theorem Std.DTreeMap.Const.get!_alter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k k' : α} [Inhabited β] {f : Option β → Option β} : get! (alter t k f) k' = if cmp k k' = Ordering.eq then (f (get? t k)).get! else get! t k'

@[simp]

theorem Std.DTreeMap.Const.get!_alter_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} [Inhabited β] {f : Option β → Option β} : get! (alter t k f) k = (f (get? t k)).get!

theorem Std.DTreeMap.Const.getD_alter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k k' : α} {fallback : β} {f : Option β → Option β} : getD (alter t k f) k' fallback = if cmp k k' = Ordering.eq then (f (get? t k)).getD fallback else getD t k' fallback

@[simp]

theorem Std.DTreeMap.Const.getD_alter_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {fallback : β} {f : Option β → Option β} : getD (alter t k f) k fallback = (f (get? t k)).getD fallback

theorem Std.DTreeMap.Const.getKey?_alter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k k' : α} {f : Option β → Option β} : (alter t k f).getKey? k' = if cmp k k' = Ordering.eq then if (f (get? t k)).isSome = true then some k else none else t.getKey? k'

theorem Std.DTreeMap.Const.getKey?_alter_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : Option β → Option β} : (alter t k f).getKey? k = if (f (get? t k)).isSome = true then some k else none

theorem Std.DTreeMap.Const.getKey!_alter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited α] {k k' : α} {f : Option β → Option β} : (alter t k f).getKey! k' = if cmp k k' = Ordering.eq then if (f (get? t k)).isSome = true then k else default else t.getKey! k'

theorem Std.DTreeMap.Const.getKey!_alter_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited α] {k : α} {f : Option β → Option β} : (alter t k f).getKey! k = if (f (get? t k)).isSome = true then k else default

theorem Std.DTreeMap.Const.getKey_alter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited α] {k k' : α} {f : Option β → Option β} {hc : k' ∈ alter t k f} : (alter t k f).getKey k' hc = if heq : cmp k k' = Ordering.eq then k else t.getKey k' ⋯

@[simp]

theorem Std.DTreeMap.Const.getKey_alter_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited α] {k : α} {f : Option β → Option β} {hc : k ∈ alter t k f} : (alter t k f).getKey k hc = k

theorem Std.DTreeMap.Const.getKeyD_alter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k k' fallback : α} {f : Option β → Option β} : (alter t k f).getKeyD k' fallback = if cmp k k' = Ordering.eq then if (f (get? t k)).isSome = true then k else fallback else t.getKeyD k' fallback

@[simp]

theorem Std.DTreeMap.Const.getKeyD_alter_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited α] {k fallback : α} {f : Option β → Option β} : (alter t k f).getKeyD k fallback = if (f (get? t k)).isSome = true then k else fallback

@[simp]

theorem Std.DTreeMap.isEmpty_modify {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : β k → β k} : (t.modify k f).isEmpty = t.isEmpty

theorem Std.DTreeMap.contains_modify {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k k' : α} {f : β k → β k} : (t.modify k f).contains k' = t.contains k'

theorem Std.DTreeMap.mem_modify {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k k' : α} {f : β k → β k} : k' ∈ t.modify k f ↔ k' ∈ t

theorem Std.DTreeMap.size_modify {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : β k → β k} : (t.modify k f).size = t.size

theorem Std.DTreeMap.get?_modify {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k k' : α} {f : β k → β k} : (t.modify k f).get? k' = if h : cmp k k' = Ordering.eq then cast ⋯ (Option.map f (t.get? k)) else t.get? k'

@[simp]

theorem Std.DTreeMap.get?_modify_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : β k → β k} : (t.modify k f).get? k = Option.map f (t.get? k)

theorem Std.DTreeMap.get_modify {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k k' : α} {f : β k → β k} {hc : k' ∈ t.modify k f} : (t.modify k f).get k' hc = if heq : cmp k k' = Ordering.eq then cast ⋯ (f (t.get k ⋯)) else t.get k' ⋯

@[simp]

theorem Std.DTreeMap.get_modify_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : β k → β k} {hc : k ∈ t.modify k f} : (t.modify k f).get k hc = f (t.get k ⋯)

theorem Std.DTreeMap.get!_modify {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k k' : α} [hi : Inhabited (β k')] {f : β k → β k} : (t.modify k f).get! k' = if heq : cmp k k' = Ordering.eq then (Option.map (cast ⋯) (Option.map f (t.get? k))).get! else t.get! k'

@[simp]

theorem Std.DTreeMap.get!_modify_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} [Inhabited (β k)] {f : β k → β k} : (t.modify k f).get! k = (Option.map f (t.get? k)).get!

theorem Std.DTreeMap.getD_modify {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k k' : α} {fallback : β k'} {f : β k → β k} : (t.modify k f).getD k' fallback = if heq : cmp k k' = Ordering.eq then (Option.map (cast ⋯) (Option.map f (t.get? k))).getD fallback else t.getD k' fallback

@[simp]

theorem Std.DTreeMap.getD_modify_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {fallback : β k} {f : β k → β k} : (t.modify k f).getD k fallback = (Option.map f (t.get? k)).getD fallback

theorem Std.DTreeMap.getKey?_modify {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k k' : α} {f : β k → β k} : (t.modify k f).getKey? k' = if cmp k k' = Ordering.eq then if k ∈ t then some k else none else t.getKey? k'

theorem Std.DTreeMap.getKey?_modify_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : β k → β k} : (t.modify k f).getKey? k = if k ∈ t then some k else none

theorem Std.DTreeMap.getKey!_modify {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {k k' : α} {f : β k → β k} : (t.modify k f).getKey! k' = if cmp k k' = Ordering.eq then if k ∈ t then k else default else t.getKey! k'

theorem Std.DTreeMap.getKey!_modify_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {k : α} {f : β k → β k} : (t.modify k f).getKey! k = if k ∈ t then k else default

theorem Std.DTreeMap.getKey_modify {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {k k' : α} {f : β k → β k} {hc : k' ∈ t.modify k f} : (t.modify k f).getKey k' hc = if cmp k k' = Ordering.eq then k else t.getKey k' ⋯

@[simp]

theorem Std.DTreeMap.getKey_modify_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {k : α} {f : β k → β k} {hc : k ∈ t.modify k f} : (t.modify k f).getKey k hc = k

theorem Std.DTreeMap.getKeyD_modify {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k k' fallback : α} {f : β k → β k} : (t.modify k f).getKeyD k' fallback = if cmp k k' = Ordering.eq then if k ∈ t then k else fallback else t.getKeyD k' fallback

theorem Std.DTreeMap.getKeyD_modify_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {k fallback : α} {f : β k → β k} : (t.modify k f).getKeyD k fallback = if k ∈ t then k else fallback

@[simp]

theorem Std.DTreeMap.Const.isEmpty_modify {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {k : α} {f : β → β} : (modify t k f).isEmpty = t.isEmpty

theorem Std.DTreeMap.Const.contains_modify {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {k k' : α} {f : β → β} : (modify t k f).contains k' = t.contains k'

theorem Std.DTreeMap.Const.mem_modify {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {k k' : α} {f : β → β} : k' ∈ modify t k f ↔ k' ∈ t

theorem Std.DTreeMap.Const.size_modify {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {k : α} {f : β → β} : (modify t k f).size = t.size

theorem Std.DTreeMap.Const.get?_modify {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {k k' : α} {f : β → β} : get? (modify t k f) k' = if cmp k k' = Ordering.eq then Option.map f (get? t k) else get? t k'

@[simp]

theorem Std.DTreeMap.Const.get?_modify_self {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {k : α} {f : β → β} : get? (modify t k f) k = Option.map f (get? t k)

theorem Std.DTreeMap.Const.get_modify {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {k k' : α} {f : β → β} {hc : k' ∈ modify t k f} : get (modify t k f) k' hc = if heq : cmp k k' = Ordering.eq then f (get t k ⋯) else get t k' ⋯

@[simp]

theorem Std.DTreeMap.Const.get_modify_self {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {k : α} {f : β → β} {hc : k ∈ modify t k f} : get (modify t k f) k hc = f (get t k ⋯)

theorem Std.DTreeMap.Const.get!_modify {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {k k' : α} [hi : Inhabited β] {f : β → β} : get! (modify t k f) k' = if cmp k k' = Ordering.eq then (Option.map f (get? t k)).get! else get! t k'

@[simp]

theorem Std.DTreeMap.Const.get!_modify_self {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {k : α} [Inhabited β] {f : β → β} : get! (modify t k f) k = (Option.map f (get? t k)).get!

theorem Std.DTreeMap.Const.getD_modify {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {k k' : α} {fallback : β} {f : β → β} : getD (modify t k f) k' fallback = if cmp k k' = Ordering.eq then (Option.map f (get? t k)).getD fallback else getD t k' fallback

@[simp]

theorem Std.DTreeMap.Const.getD_modify_self {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {k : α} {fallback : β} {f : β → β} : getD (modify t k f) k fallback = (Option.map f (get? t k)).getD fallback

theorem Std.DTreeMap.Const.getKey?_modify {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {k k' : α} {f : β → β} : (modify t k f).getKey? k' = if cmp k k' = Ordering.eq then if k ∈ t then some k else none else t.getKey? k'

theorem Std.DTreeMap.Const.getKey?_modify_self {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {k : α} {f : β → β} : (modify t k f).getKey? k = if k ∈ t then some k else none

theorem Std.DTreeMap.Const.getKey!_modify {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [Inhabited α] {k k' : α} {f : β → β} : (modify t k f).getKey! k' = if cmp k k' = Ordering.eq then if k ∈ t then k else default else t.getKey! k'

theorem Std.DTreeMap.Const.getKey!_modify_self {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [Inhabited α] {k : α} {f : β → β} : (modify t k f).getKey! k = if k ∈ t then k else default

theorem Std.DTreeMap.Const.getKey_modify {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [Inhabited α] {k k' : α} {f : β → β} {hc : k' ∈ modify t k f} : (modify t k f).getKey k' hc = if cmp k k' = Ordering.eq then k else t.getKey k' ⋯

@[simp]

theorem Std.DTreeMap.Const.getKey_modify_self {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [Inhabited α] {k : α} {f : β → β} {hc : k ∈ modify t k f} : (modify t k f).getKey k hc = k

theorem Std.DTreeMap.Const.getKeyD_modify {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {k k' fallback : α} {f : β → β} : (modify t k f).getKeyD k' fallback = if cmp k k' = Ordering.eq then if k ∈ t then k else fallback else t.getKeyD k' fallback

theorem Std.DTreeMap.Const.getKeyD_modify_self {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [Inhabited α] {k fallback : α} {f : β → β} : (modify t k f).getKeyD k fallback = if k ∈ t then k else fallback

@[simp]

theorem Std.DTreeMap.minKey?_emptyc {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} : ∅.minKey? = none

theorem Std.DTreeMap.minKey?_of_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (he : t.isEmpty = true) : t.minKey? = none

@[simp]

theorem Std.DTreeMap.minKey?_eq_none_iff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.minKey? = none ↔ t.isEmpty = true

theorem Std.DTreeMap.minKey?_eq_some_iff_getKey?_eq_self_and_forall {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {km : α} : t.minKey? = some km ↔ t.getKey? km = some km ∧ ∀ (k : α), k ∈ t → (cmp km k).isLE = true

theorem Std.DTreeMap.minKey?_eq_some_iff_mem_and_forall {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {km : α} : t.minKey? = some km ↔ km ∈ t ∧ ∀ (k : α), k ∈ t → (cmp km k).isLE = true

@[simp]

theorem Std.DTreeMap.isNone_minKey?_eq_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.minKey?.isNone = t.isEmpty

@[simp]

theorem Std.DTreeMap.isSome_minKey?_eq_not_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.minKey?.isSome = !t.isEmpty

theorem Std.DTreeMap.isSome_minKey?_iff_isEmpty_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.minKey?.isSome = true ↔ t.isEmpty = false

theorem Std.DTreeMap.minKey?_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.insert k v).minKey? = some (t.minKey?.elim k fun (k' : α) => if (cmp k k').isLE = true then k else k')

theorem Std.DTreeMap.isSome_minKey?_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.insert k v).minKey?.isSome = true

theorem Std.DTreeMap.minKey_insert_of_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} (he : t.isEmpty = true) : (t.insert k v).minKey ⋯ = k

theorem Std.DTreeMap.minKey?_insert_of_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} (he : t.isEmpty = true) : (t.insert k v).minKey? = some k

theorem Std.DTreeMap.minKey!_insert_of_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} {v : β k} (he : t.isEmpty = true) : (t.insert k v).minKey! = k

theorem Std.DTreeMap.minKeyD_insert_of_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} (he : t.isEmpty = true) {fallback : α} : (t.insert k v).minKeyD fallback = k

theorem Std.DTreeMap.minKey_insertIfNew_of_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} (he : t.isEmpty = true) : (t.insertIfNew k v).minKey ⋯ = k

theorem Std.DTreeMap.minKey?_insertIfNew_of_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} (he : t.isEmpty = true) : (t.insertIfNew k v).minKey? = some k

theorem Std.DTreeMap.minKey!_insertIfNew_of_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} {v : β k} (he : t.isEmpty = true) : (t.insertIfNew k v).minKey! = k

theorem Std.DTreeMap.minKeyD_insertIfNew_of_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} (he : t.isEmpty = true) {fallback : α} : (t.insertIfNew k v).minKeyD fallback = k

theorem Std.DTreeMap.minKey?_insert_le_minKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} {km kmi : α} (hkm : t.minKey? = some km) (hkmi : (t.insert k v).minKey?.get ⋯ = kmi) : (cmp kmi km).isLE = true

theorem Std.DTreeMap.minKey?_insert_le_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} {kmi : α} (hkmi : (t.insert k v).minKey?.get ⋯ = kmi) : (cmp kmi k).isLE = true

theorem Std.DTreeMap.contains_minKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {km : α} (hkm : t.minKey? = some km) : t.contains km = true

@[simp]

theorem Std.DTreeMap.min?_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Min α] [LE α] [LawfulOrderCmp cmp] [LawfulOrderMin α] [LawfulOrderLeftLeaningMin α] [LawfulEqCmp cmp] : t.keys.min? = t.minKey?

@[simp]

theorem Std.DTreeMap.head?_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Min α] [LE α] [LawfulOrderCmp cmp] [LawfulOrderMin α] [LawfulOrderLeftLeaningMin α] [LawfulEqCmp cmp] : t.keys.head? = t.minKey?

theorem Std.DTreeMap.minKey?_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {km : α} (hkm : t.minKey? = some km) : km ∈ t

theorem Std.DTreeMap.isSome_minKey?_of_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} (hc : t.contains k = true) : t.minKey?.isSome = true

theorem Std.DTreeMap.isSome_minKey?_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} : k ∈ t → t.minKey?.isSome = true

theorem Std.DTreeMap.minKey?_le_of_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k km : α} (hc : t.contains k = true) (hkm : t.minKey?.get ⋯ = km) : (cmp km k).isLE = true

theorem Std.DTreeMap.minKey?_le_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k km : α} (hc : k ∈ t) (hkm : t.minKey?.get ⋯ = km) : (cmp km k).isLE = true

theorem Std.DTreeMap.le_minKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} : (∀ (k' : α), t.minKey? = some k' → (cmp k k').isLE = true) ↔ ∀ (k' : α), k' ∈ t → (cmp k k').isLE = true

theorem Std.DTreeMap.getKey?_minKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {km : α} (hkm : t.minKey? = some km) : t.getKey? km = some km

theorem Std.DTreeMap.getKey_minKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {km : α} {hc : t.contains km = true} (hkm : t.minKey?.get ⋯ = km) : t.getKey km hc = km

theorem Std.DTreeMap.getKey!_minKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {km : α} (hkm : t.minKey? = some km) : t.getKey! km = km

theorem Std.DTreeMap.getKeyD_minKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {km fallback : α} (hkm : t.minKey? = some km) : t.getKeyD km fallback = km

@[simp]

theorem Std.DTreeMap.minKey?_bind_getKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.minKey?.bind t.getKey? = t.minKey?

theorem Std.DTreeMap.minKey?_erase_eq_iff_not_compare_eq_minKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} : (t.erase k).minKey? = t.minKey? ↔ ∀ {km : α}, t.minKey? = some km → ¬cmp k km = Ordering.eq

theorem Std.DTreeMap.minKey?_erase_eq_of_not_compare_eq_minKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} (hc : ∀ {km : α}, t.minKey? = some km → ¬cmp k km = Ordering.eq) : (t.erase k).minKey? = t.minKey?

theorem Std.DTreeMap.isSome_minKey?_of_isSome_minKey?_erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} (hs : (t.erase k).minKey?.isSome = true) : t.minKey?.isSome = true

theorem Std.DTreeMap.minKey?_le_minKey?_erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k km kme : α} (hkme : (t.erase k).minKey? = some kme) (hkm : t.minKey?.get ⋯ = km) : (cmp km kme).isLE = true

theorem Std.DTreeMap.minKey?_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.insertIfNew k v).minKey? = some (t.minKey?.elim k fun (k' : α) => if cmp k k' = Ordering.lt then k else k')

theorem Std.DTreeMap.isSome_minKey?_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.insertIfNew k v).minKey?.isSome = true

theorem Std.DTreeMap.minKey?_insertIfNew_le_minKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} {km kmi : α} (hkm : t.minKey? = some km) (hkmi : (t.insertIfNew k v).minKey?.get ⋯ = kmi) : (cmp kmi km).isLE = true

theorem Std.DTreeMap.minKey?_insertIfNew_le_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} {kmi : α} (hkmi : (t.insertIfNew k v).minKey?.get ⋯ = kmi) : (cmp kmi k).isLE = true

theorem Std.DTreeMap.minKey?_eq_head?_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.minKey? = t.keys.head?

theorem Std.DTreeMap.minKey?_eq_getElem?_keysArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.minKey? = t.keysArray[0]?

@[simp]

theorem Std.DTreeMap.minKey?_modify {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : β k → β k} : (t.modify k f).minKey? = t.minKey?

theorem Std.DTreeMap.minKey?_alter_eq_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : Option (β k) → Option (β k)} : (t.alter k f).minKey? = some k ↔ (f (t.get? k)).isSome = true ∧ ∀ (k' : α), k' ∈ t → (cmp k k').isLE = true

theorem Std.DTreeMap.Const.minKey?_modify {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : β → β} : (modify t k f).minKey? = Option.map (fun (km : α) => if cmp km k = Ordering.eq then k else km) t.minKey?

@[simp]

theorem Std.DTreeMap.Const.minKey?_modify_eq_minKey? {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : β → β} : (modify t k f).minKey? = t.minKey?

theorem Std.DTreeMap.Const.isSome_minKey?_modify {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : β → β} : (modify t k f).minKey?.isSome = !t.isEmpty

theorem Std.DTreeMap.Const.isSome_minKey?_modify_eq_isSome {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : β → β} : (modify t k f).minKey?.isSome = t.minKey?.isSome

theorem Std.DTreeMap.Const.compare_minKey?_modify_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : β → β} {km kmm : α} (hkm : t.minKey? = some km) (hkmm : (modify t k f).minKey?.get ⋯ = kmm) : cmp kmm km = Ordering.eq

theorem Std.DTreeMap.Const.minKey?_alter_eq_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : Option β → Option β} : (alter t k f).minKey? = some k ↔ (f (get? t k)).isSome = true ∧ ∀ (k' : α), k' ∈ t → (cmp k k').isLE = true

theorem Std.DTreeMap.minKey_eq_get_minKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {he : t.isEmpty = false} : t.minKey he = t.minKey?.get ⋯

theorem Std.DTreeMap.minKey?_eq_some_minKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {he : t.isEmpty = false} : t.minKey? = some (t.minKey he)

theorem Std.DTreeMap.minKey_eq_iff_getKey?_eq_self_and_forall {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {he : t.isEmpty = false} {km : α} : t.minKey he = km ↔ t.getKey? km = some km ∧ ∀ (k : α), k ∈ t → (cmp km k).isLE = true

theorem Std.DTreeMap.minKey_eq_iff_mem_and_forall {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {he : t.isEmpty = false} {km : α} : t.minKey he = km ↔ km ∈ t ∧ ∀ (k : α), k ∈ t → (cmp km k).isLE = true

theorem Std.DTreeMap.minKey_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.insert k v).minKey ⋯ = t.minKey?.elim k fun (k' : α) => if (cmp k k').isLE = true then k else k'

theorem Std.DTreeMap.minKey_insert_le_minKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} {he : t.isEmpty = false} : (cmp ((t.insert k v).minKey ⋯) (t.minKey he)).isLE = true

theorem Std.DTreeMap.minKey_insert_le_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (cmp ((t.insert k v).minKey ⋯) k).isLE = true

theorem Std.DTreeMap.contains_minKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {he : t.isEmpty = false} : t.contains (t.minKey he) = true

theorem Std.DTreeMap.minKey_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {he : t.isEmpty = false} : t.minKey he ∈ t

theorem Std.DTreeMap.minKey_le_of_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} (hc : t.contains k = true) : (cmp (t.minKey ⋯) k).isLE = true

theorem Std.DTreeMap.minKey_le_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} (hc : k ∈ t) : (cmp (t.minKey ⋯) k).isLE = true

theorem Std.DTreeMap.le_minKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {he : t.isEmpty = false} : (cmp k (t.minKey he)).isLE = true ↔ ∀ (k' : α), k' ∈ t → (cmp k k').isLE = true

@[simp]

theorem Std.DTreeMap.getKey?_minKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {he : t.isEmpty = false} : t.getKey? (t.minKey he) = some (t.minKey he)

@[simp]

theorem Std.DTreeMap.getKey_minKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {he : t.isEmpty = false} {hc : t.minKey he ∈ t} : t.getKey (t.minKey he) hc = t.minKey he

@[simp]

theorem Std.DTreeMap.getKey!_minKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {he : t.isEmpty = false} : t.getKey! (t.minKey he) = t.minKey he

@[simp]

theorem Std.DTreeMap.getKeyD_minKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {he : t.isEmpty = false} {fallback : α} : t.getKeyD (t.minKey he) fallback = t.minKey he

@[simp]

theorem Std.DTreeMap.minKey_erase_eq_iff_not_compare_eq_minKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {he : (t.erase k).isEmpty = false} : (t.erase k).minKey he = t.minKey ⋯ ↔ ¬cmp k (t.minKey ⋯) = Ordering.eq

theorem Std.DTreeMap.minKey_erase_eq_of_not_compare_eq_minKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {he : (t.erase k).isEmpty = false} (hc : ¬cmp k (t.minKey ⋯) = Ordering.eq) : (t.erase k).minKey he = t.minKey ⋯

theorem Std.DTreeMap.minKey_le_minKey_erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {he : (t.erase k).isEmpty = false} : (cmp (t.minKey ⋯) ((t.erase k).minKey he)).isLE = true

theorem Std.DTreeMap.minKey_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.insertIfNew k v).minKey ⋯ = t.minKey?.elim k fun (k' : α) => if cmp k k' = Ordering.lt then k else k'

theorem Std.DTreeMap.minKey_insertIfNew_le_minKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} {he : t.isEmpty = false} : (cmp ((t.insertIfNew k v).minKey ⋯) (t.minKey he)).isLE = true

theorem Std.DTreeMap.minKey_insertIfNew_le_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (cmp ((t.insertIfNew k v).minKey ⋯) k).isLE = true

theorem Std.DTreeMap.minKey_eq_head_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {he : t.isEmpty = false} : t.minKey he = t.keys.head ⋯

theorem Std.DTreeMap.minKey_eq_getElem_keysArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {he : t.isEmpty = false} : t.minKey he = t.keysArray[0]

@[simp]

theorem Std.DTreeMap.minKey_modify {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : β k → β k} {he : (t.modify k f).isEmpty = false} : (t.modify k f).minKey he = t.minKey ⋯

theorem Std.DTreeMap.minKey_alter_eq_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : Option (β k) → Option (β k)} {he : (t.alter k f).isEmpty = false} : (t.alter k f).minKey he = k ↔ (f (t.get? k)).isSome = true ∧ ∀ (k' : α), k' ∈ t → (cmp k k').isLE = true

theorem Std.DTreeMap.Const.minKey_modify {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : β → β} {he : (modify t k f).isEmpty = false} : (modify t k f).minKey he = if cmp (t.minKey ⋯) k = Ordering.eq then k else t.minKey ⋯

@[simp]

theorem Std.DTreeMap.Const.minKey_modify_eq_minKey {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : β → β} {he : (modify t k f).isEmpty = false} : (modify t k f).minKey he = t.minKey ⋯

theorem Std.DTreeMap.Const.compare_minKey_modify_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : β → β} {he : (modify t k f).isEmpty = false} : cmp ((modify t k f).minKey he) (t.minKey ⋯) = Ordering.eq

@[simp]

theorem Std.DTreeMap.Const.ordCompare_minKey_modify_eq {α : Type u} {β : Type v} [Ord α] [TransOrd α] {t : DTreeMap α (fun (x : α) => β) compare} {k : α} {f : β → β} {he : (modify t k f).isEmpty = false} : compare ((modify t k f).minKey he) (t.minKey ⋯) = Ordering.eq

theorem Std.DTreeMap.Const.minKey_alter_eq_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : Option β → Option β} {he : (alter t k f).isEmpty = false} : (alter t k f).minKey he = k ↔ (f (get? t k)).isSome = true ∧ ∀ (k' : α), k' ∈ t → (cmp k k').isLE = true

theorem Std.DTreeMap.minKey?_eq_some_minKey! {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) : t.minKey? = some t.minKey!

theorem Std.DTreeMap.minKey_eq_minKey! {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {he : t.isEmpty = false} : t.minKey he = t.minKey!

theorem Std.DTreeMap.minKey!_eq_default {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = true) : t.minKey! = default

theorem Std.DTreeMap.minKey!_eq_iff_getKey?_eq_self_and_forall {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {km : α} : t.minKey! = km ↔ t.getKey? km = some km ∧ ∀ (k : α), k ∈ t → (cmp km k).isLE = true

theorem Std.DTreeMap.minKey!_eq_iff_mem_and_forall {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] (he : t.isEmpty = false) {km : α} : t.minKey! = km ↔ km ∈ t ∧ ∀ (k : α), k ∈ t → (cmp km k).isLE = true

theorem Std.DTreeMap.minKey!_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} {v : β k} : (t.insert k v).minKey! = t.minKey?.elim k fun (k' : α) => if (cmp k k').isLE = true then k else k'

theorem Std.DTreeMap.minKey!_insert_le_minKey! {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {k : α} {v : β k} : (cmp (t.insert k v).minKey! t.minKey!).isLE = true

theorem Std.DTreeMap.minKey!_insert_le_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} {v : β k} : (cmp (t.insert k v).minKey! k).isLE = true

theorem Std.DTreeMap.contains_minKey! {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) : t.contains t.minKey! = true

theorem Std.DTreeMap.minKey!_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) : t.minKey! ∈ t

theorem Std.DTreeMap.minKey!_le_of_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} (hc : t.contains k = true) : (cmp t.minKey! k).isLE = true

theorem Std.DTreeMap.minKey!_le_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} (hc : k ∈ t) : (cmp t.minKey! k).isLE = true

theorem Std.DTreeMap.le_minKey! {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {k : α} : (cmp k t.minKey!).isLE = true ↔ ∀ (k' : α), k' ∈ t → (cmp k k').isLE = true

theorem Std.DTreeMap.getKey?_minKey! {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) : t.getKey? t.minKey! = some t.minKey!

theorem Std.DTreeMap.getKey_minKey! {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {hc : t.minKey! ∈ t} : t.getKey t.minKey! hc = t.minKey!

@[simp]

theorem Std.DTreeMap.getKey_minKey!_eq_minKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {hc : t.minKey! ∈ t} : t.getKey t.minKey! hc = t.minKey ⋯

theorem Std.DTreeMap.getKey!_minKey! {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) : t.getKey! t.minKey! = t.minKey!

theorem Std.DTreeMap.getKeyD_minKey! {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {fallback : α} : t.getKeyD t.minKey! fallback = t.minKey!

theorem Std.DTreeMap.minKey!_erase_eq_of_not_compare_minKey!_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} (he : (t.erase k).isEmpty = false) (heq : ¬cmp k t.minKey! = Ordering.eq) : (t.erase k).minKey! = t.minKey!

theorem Std.DTreeMap.minKey!_le_minKey!_erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} (he : (t.erase k).isEmpty = false) : (cmp t.minKey! (t.erase k).minKey!).isLE = true

theorem Std.DTreeMap.minKey!_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} {v : β k} : (t.insertIfNew k v).minKey! = t.minKey?.elim k fun (k' : α) => if cmp k k' = Ordering.lt then k else k'

theorem Std.DTreeMap.minKey!_insertIfNew_le_minKey! {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {k : α} {v : β k} : (cmp (t.insertIfNew k v).minKey! t.minKey!).isLE = true

theorem Std.DTreeMap.minKey!_insertIfNew_le_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} {v : β k} : (cmp (t.insertIfNew k v).minKey! k).isLE = true

theorem Std.DTreeMap.minKey!_eq_head!_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] : t.minKey! = t.keys.head!

theorem Std.DTreeMap.minKey!_eq_getElem!_keysArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] : t.minKey! = t.keysArray[0]!

@[simp]

theorem Std.DTreeMap.minKey!_modify {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {k : α} {f : β k → β k} : (t.modify k f).minKey! = t.minKey!

theorem Std.DTreeMap.minKey!_alter_eq_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {k : α} {f : Option (β k) → Option (β k)} (he : (t.alter k f).isEmpty = false) : (t.alter k f).minKey! = k ↔ (f (t.get? k)).isSome = true ∧ ∀ (k' : α), k' ∈ t → (cmp k k').isLE = true

theorem Std.DTreeMap.Const.minKey!_modify {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited α] {k : α} {f : β → β} (he : (modify t k f).isEmpty = false) : (modify t k f).minKey! = if cmp t.minKey! k = Ordering.eq then k else t.minKey!

@[simp]

theorem Std.DTreeMap.Const.minKey!_modify_eq_minKey! {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {k : α} {f : β → β} : (modify t k f).minKey! = t.minKey!

theorem Std.DTreeMap.Const.compare_minKey!_modify_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited α] {k : α} {f : β → β} : cmp (modify t k f).minKey! t.minKey! = Ordering.eq

@[simp]

theorem Std.DTreeMap.Const.ordCompare_minKey!_modify_eq {α : Type u} {β : Type v} [Ord α] [TransOrd α] {t : DTreeMap α (fun (x : α) => β) compare} [Inhabited α] {k : α} {f : β → β} : compare (modify t k f).minKey! t.minKey! = Ordering.eq

theorem Std.DTreeMap.Const.minKey!_alter_eq_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited α] {k : α} {f : Option β → Option β} (he : (alter t k f).isEmpty = false) : (alter t k f).minKey! = k ↔ (f (get? t k)).isSome = true ∧ ∀ (k' : α), k' ∈ t → (cmp k k').isLE = true

theorem Std.DTreeMap.minKey?_eq_some_minKeyD {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (he : t.isEmpty = false) {fallback : α} : t.minKey? = some (t.minKeyD fallback)

theorem Std.DTreeMap.minKeyD_eq_fallback {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (he : t.isEmpty = true) {fallback : α} : t.minKeyD fallback = fallback

theorem Std.DTreeMap.minKey!_eq_minKeyD_default {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] : t.minKey! = t.minKeyD default

theorem Std.DTreeMap.minKeyD_eq_iff_getKey?_eq_self_and_forall {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (he : t.isEmpty = false) {km fallback : α} : t.minKeyD fallback = km ↔ t.getKey? km = some km ∧ ∀ (k : α), k ∈ t → (cmp km k).isLE = true

theorem Std.DTreeMap.minKeyD_eq_iff_mem_and_forall {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] (he : t.isEmpty = false) {km fallback : α} : t.minKeyD fallback = km ↔ km ∈ t ∧ ∀ (k : α), k ∈ t → (cmp km k).isLE = true

theorem Std.DTreeMap.minKeyD_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} {fallback : α} : (t.insert k v).minKeyD fallback = t.minKey?.elim k fun (k' : α) => if (cmp k k').isLE = true then k else k'

theorem Std.DTreeMap.minKeyD_insert_le_minKeyD {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (he : t.isEmpty = false) {k : α} {v : β k} {fallback : α} : (cmp ((t.insert k v).minKeyD fallback) (t.minKeyD fallback)).isLE = true

theorem Std.DTreeMap.minKeyD_insert_le_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} {fallback : α} : (cmp ((t.insert k v).minKeyD fallback) k).isLE = true

theorem Std.DTreeMap.contains_minKeyD {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (he : t.isEmpty = false) {fallback : α} : t.contains (t.minKeyD fallback) = true

theorem Std.DTreeMap.minKeyD_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (he : t.isEmpty = false) {fallback : α} : t.minKeyD fallback ∈ t

theorem Std.DTreeMap.minKeyD_le_of_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} (hc : t.contains k = true) {fallback : α} : (cmp (t.minKeyD fallback) k).isLE = true

theorem Std.DTreeMap.minKeyD_le_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} (hc : k ∈ t) {fallback : α} : (cmp (t.minKeyD fallback) k).isLE = true

theorem Std.DTreeMap.le_minKeyD {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (he : t.isEmpty = false) {k fallback : α} : (cmp k (t.minKeyD fallback)).isLE = true ↔ ∀ (k' : α), k' ∈ t → (cmp k k').isLE = true

theorem Std.DTreeMap.getKey?_minKeyD {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (he : t.isEmpty = false) {fallback : α} : t.getKey? (t.minKeyD fallback) = some (t.minKeyD fallback)

theorem Std.DTreeMap.getKey_minKeyD {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {fallback : α} {hc : t.minKeyD fallback ∈ t} : t.getKey (t.minKeyD fallback) hc = t.minKeyD fallback

theorem Std.DTreeMap.getKey!_minKeyD {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {fallback : α} : t.getKey! (t.minKeyD fallback) = t.minKeyD fallback

theorem Std.DTreeMap.getKeyD_minKeyD {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (he : t.isEmpty = false) {fallback fallback' : α} : t.getKeyD (t.minKeyD fallback) fallback' = t.minKeyD fallback

theorem Std.DTreeMap.minKeyD_erase_eq_of_not_compare_minKeyD_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k fallback : α} (he : (t.erase k).isEmpty = false) (heq : ¬cmp k (t.minKeyD fallback) = Ordering.eq) : (t.erase k).minKeyD fallback = t.minKeyD fallback

theorem Std.DTreeMap.minKeyD_le_minKeyD_erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} (he : (t.erase k).isEmpty = false) {fallback : α} : (cmp (t.minKeyD fallback) ((t.erase k).minKeyD fallback)).isLE = true

theorem Std.DTreeMap.minKeyD_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} {fallback : α} : (t.insertIfNew k v).minKeyD fallback = t.minKey?.elim k fun (k' : α) => if cmp k k' = Ordering.lt then k else k'

theorem Std.DTreeMap.minKeyD_insertIfNew_le_minKeyD {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (he : t.isEmpty = false) {k : α} {v : β k} {fallback : α} : (cmp ((t.insertIfNew k v).minKeyD fallback) (t.minKeyD fallback)).isLE = true

theorem Std.DTreeMap.minKeyD_insertIfNew_le_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} {fallback : α} : (cmp ((t.insertIfNew k v).minKeyD fallback) k).isLE = true

theorem Std.DTreeMap.minKeyD_eq_headD_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {fallback : α} : t.minKeyD fallback = t.keys.headD fallback

theorem Std.DTreeMap.minKeyD_eq_getD_keysArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {fallback : α} : t.minKeyD fallback = t.keysArray.getD 0 fallback

@[simp]

theorem Std.DTreeMap.minKeyD_modify {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : β k → β k} {fallback : α} : (t.modify k f).minKeyD fallback = t.minKeyD fallback

theorem Std.DTreeMap.minKeyD_alter_eq_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : Option (β k) → Option (β k)} (he : (t.alter k f).isEmpty = false) {fallback : α} : (t.alter k f).minKeyD fallback = k ↔ (f (t.get? k)).isSome = true ∧ ∀ (k' : α), k' ∈ t → (cmp k k').isLE = true

theorem Std.DTreeMap.Const.minKeyD_modify {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : β → β} (he : (modify t k f).isEmpty = false) {fallback : α} : (modify t k f).minKeyD fallback = if cmp (t.minKeyD fallback) k = Ordering.eq then k else t.minKeyD fallback

@[simp]

theorem Std.DTreeMap.Const.minKeyD_modify_eq_minKeyD {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : β → β} {fallback : α} : (modify t k f).minKeyD fallback = t.minKeyD fallback

theorem Std.DTreeMap.Const.compare_minKeyD_modify_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : β → β} {fallback : α} : cmp ((modify t k f).minKeyD fallback) (t.minKeyD fallback) = Ordering.eq

@[simp]

theorem Std.DTreeMap.Const.ordCompare_minKeyD_modify_eq {α : Type u} {β : Type v} [Ord α] [TransOrd α] {t : DTreeMap α (fun (x : α) => β) compare} {k : α} {f : β → β} {fallback : α} : compare ((modify t k f).minKeyD fallback) (t.minKeyD fallback) = Ordering.eq

theorem Std.DTreeMap.Const.minKeyD_alter_eq_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : Option β → Option β} (he : (alter t k f).isEmpty = false) {fallback : α} : (alter t k f).minKeyD fallback = k ↔ (f (get? t k)).isSome = true ∧ ∀ (k' : α), k' ∈ t → (cmp k k').isLE = true

@[simp]

theorem Std.DTreeMap.maxKey?_emptyc {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} : ∅.maxKey? = none

theorem Std.DTreeMap.maxKey?_of_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (he : t.isEmpty = true) : t.maxKey? = none

@[simp]

theorem Std.DTreeMap.maxKey?_eq_none_iff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.maxKey? = none ↔ t.isEmpty = true

theorem Std.DTreeMap.maxKey?_eq_some_iff_getKey?_eq_self_and_forall {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {km : α} : t.maxKey? = some km ↔ t.getKey? km = some km ∧ ∀ (k : α), k ∈ t → (cmp k km).isLE = true

theorem Std.DTreeMap.maxKey?_eq_some_iff_mem_and_forall {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {km : α} : t.maxKey? = some km ↔ km ∈ t ∧ ∀ (k : α), k ∈ t → (cmp k km).isLE = true

@[simp]

theorem Std.DTreeMap.isNone_maxKey?_eq_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.maxKey?.isNone = t.isEmpty

@[simp]

theorem Std.DTreeMap.isSome_maxKey?_eq_not_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.maxKey?.isSome = !t.isEmpty

theorem Std.DTreeMap.isSome_maxKey?_iff_isEmpty_eq_false {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.maxKey?.isSome = true ↔ t.isEmpty = false

theorem Std.DTreeMap.maxKey?_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.insert k v).maxKey? = some (t.maxKey?.elim k fun (k' : α) => if (cmp k' k).isLE = true then k else k')

theorem Std.DTreeMap.isSome_maxKey?_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.insert k v).maxKey?.isSome = true

theorem Std.DTreeMap.maxKey?_le_maxKey?_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} {km kmi : α} (hkm : t.maxKey? = some km) (hkmi : (t.insert k v).maxKey?.get ⋯ = kmi) : (cmp km kmi).isLE = true

theorem Std.DTreeMap.self_le_maxKey?_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} {kmi : α} (hkmi : (t.insert k v).maxKey?.get ⋯ = kmi) : (cmp k kmi).isLE = true

theorem Std.DTreeMap.contains_maxKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {km : α} (hkm : t.maxKey? = some km) : t.contains km = true

theorem Std.DTreeMap.maxKey?_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {km : α} (hkm : t.maxKey? = some km) : km ∈ t

theorem Std.DTreeMap.isSome_maxKey?_of_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} (hc : t.contains k = true) : t.maxKey?.isSome = true

theorem Std.DTreeMap.isSome_maxKey?_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} : k ∈ t → t.maxKey?.isSome = true

theorem Std.DTreeMap.le_maxKey?_of_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k km : α} (hc : t.contains k = true) (hkm : t.maxKey?.get ⋯ = km) : (cmp k km).isLE = true

theorem Std.DTreeMap.le_maxKey?_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k km : α} (hc : k ∈ t) (hkm : t.maxKey?.get ⋯ = km) : (cmp k km).isLE = true

theorem Std.DTreeMap.maxKey?_le {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} : (∀ (k' : α), t.maxKey? = some k' → (cmp k' k).isLE = true) ↔ ∀ (k' : α), k' ∈ t → (cmp k' k).isLE = true

theorem Std.DTreeMap.getKey?_maxKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {km : α} (hkm : t.maxKey? = some km) : t.getKey? km = some km

theorem Std.DTreeMap.getKey_maxKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {km : α} {hc : t.contains km = true} (hkm : t.maxKey?.get ⋯ = km) : t.getKey km hc = km

theorem Std.DTreeMap.getKey!_maxKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {km : α} (hkm : t.maxKey? = some km) : t.getKey! km = km

theorem Std.DTreeMap.getKeyD_maxKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {km fallback : α} (hkm : t.maxKey? = some km) : t.getKeyD km fallback = km

@[simp]

theorem Std.DTreeMap.maxKey?_bind_getKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.maxKey?.bind t.getKey? = t.maxKey?

theorem Std.DTreeMap.maxKey?_erase_eq_iff_not_compare_eq_maxKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} : (t.erase k).maxKey? = t.maxKey? ↔ ∀ {km : α}, t.maxKey? = some km → ¬cmp k km = Ordering.eq

theorem Std.DTreeMap.maxKey?_erase_eq_of_not_compare_eq_maxKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} (hc : ∀ {km : α}, t.maxKey? = some km → ¬cmp k km = Ordering.eq) : (t.erase k).maxKey? = t.maxKey?

theorem Std.DTreeMap.isSome_maxKey?_of_isSome_maxKey?_erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} (hs : (t.erase k).maxKey?.isSome = true) : t.maxKey?.isSome = true

theorem Std.DTreeMap.maxKey?_erase_le_maxKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k km kme : α} (hkme : (t.erase k).maxKey? = some kme) (hkm : t.maxKey?.get ⋯ = km) : (cmp kme km).isLE = true

theorem Std.DTreeMap.maxKey?_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.insertIfNew k v).maxKey? = some (t.maxKey?.elim k fun (k' : α) => if cmp k' k = Ordering.lt then k else k')

theorem Std.DTreeMap.isSome_maxKey?_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.insertIfNew k v).maxKey?.isSome = true

theorem Std.DTreeMap.maxKey?_le_maxKey?_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} {km kmi : α} (hkm : t.maxKey? = some km) (hkmi : (t.insertIfNew k v).maxKey?.get ⋯ = kmi) : (cmp km kmi).isLE = true

theorem Std.DTreeMap.self_le_maxKey?_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} {kmi : α} (hkmi : (t.insertIfNew k v).maxKey?.get ⋯ = kmi) : (cmp k kmi).isLE = true

theorem Std.DTreeMap.maxKey?_eq_getLast?_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.maxKey? = t.keys.getLast?

theorem Std.DTreeMap.maxKey?_eq_back?_keysArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] : t.maxKey? = t.keysArray.back?

@[simp]

theorem Std.DTreeMap.maxKey?_modify {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : β k → β k} : (t.modify k f).maxKey? = t.maxKey?

theorem Std.DTreeMap.maxKey?_alter_eq_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : Option (β k) → Option (β k)} : (t.alter k f).maxKey? = some k ↔ (f (t.get? k)).isSome = true ∧ ∀ (k' : α), k' ∈ t → (cmp k' k).isLE = true

theorem Std.DTreeMap.Const.maxKey?_modify {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : β → β} : (modify t k f).maxKey? = Option.map (fun (km : α) => if cmp km k = Ordering.eq then k else km) t.maxKey?

@[simp]

theorem Std.DTreeMap.Const.maxKey?_modify_eq_maxKey? {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : β → β} : (modify t k f).maxKey? = t.maxKey?

theorem Std.DTreeMap.Const.isSome_maxKey?_modify {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : β → β} : (modify t k f).maxKey?.isSome = !t.isEmpty

theorem Std.DTreeMap.Const.isSome_maxKey?_modify_eq_isSome {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : β → β} : (modify t k f).maxKey?.isSome = t.maxKey?.isSome

theorem Std.DTreeMap.Const.compare_maxKey?_modify_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : β → β} {km kmm : α} (hkm : t.maxKey? = some km) (hkmm : (modify t k f).maxKey?.get ⋯ = kmm) : cmp kmm km = Ordering.eq

theorem Std.DTreeMap.Const.maxKey?_alter_eq_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : Option β → Option β} : (alter t k f).maxKey? = some k ↔ (f (get? t k)).isSome = true ∧ ∀ (k' : α), k' ∈ t → (cmp k' k).isLE = true

theorem Std.DTreeMap.maxKey_eq_get_maxKey? {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {he : t.isEmpty = false} : t.maxKey he = t.maxKey?.get ⋯

theorem Std.DTreeMap.maxKey?_eq_some_maxKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {he : t.isEmpty = false} : t.maxKey? = some (t.maxKey he)

theorem Std.DTreeMap.maxKey_eq_iff_getKey?_eq_self_and_forall {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {he : t.isEmpty = false} {km : α} : t.maxKey he = km ↔ t.getKey? km = some km ∧ ∀ (k : α), k ∈ t → (cmp k km).isLE = true

theorem Std.DTreeMap.maxKey_eq_iff_mem_and_forall {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {he : t.isEmpty = false} {km : α} : t.maxKey he = km ↔ km ∈ t ∧ ∀ (k : α), k ∈ t → (cmp k km).isLE = true

theorem Std.DTreeMap.maxKey_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.insert k v).maxKey ⋯ = t.maxKey?.elim k fun (k' : α) => if (cmp k' k).isLE = true then k else k'

theorem Std.DTreeMap.maxKey_le_maxKey_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} {he : t.isEmpty = false} : (cmp (t.maxKey he) ((t.insert k v).maxKey ⋯)).isLE = true

theorem Std.DTreeMap.self_le_maxKey_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (cmp k ((t.insert k v).maxKey ⋯)).isLE = true

theorem Std.DTreeMap.contains_maxKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {he : t.isEmpty = false} : t.contains (t.maxKey he) = true

theorem Std.DTreeMap.maxKey_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {he : t.isEmpty = false} : t.maxKey he ∈ t

theorem Std.DTreeMap.le_maxKey_of_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} (hc : t.contains k = true) : (cmp k (t.maxKey ⋯)).isLE = true

theorem Std.DTreeMap.le_maxKey_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} (hc : k ∈ t) : (cmp k (t.maxKey ⋯)).isLE = true

theorem Std.DTreeMap.maxKey_le {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {he : t.isEmpty = false} : (cmp (t.maxKey he) k).isLE = true ↔ ∀ (k' : α), k' ∈ t → (cmp k' k).isLE = true

@[simp]

theorem Std.DTreeMap.getKey?_maxKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {he : t.isEmpty = false} : t.getKey? (t.maxKey he) = some (t.maxKey he)

@[simp]

theorem Std.DTreeMap.getKey_maxKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {he : t.isEmpty = false} {hc : t.maxKey he ∈ t} : t.getKey (t.maxKey he) hc = t.maxKey he

@[simp]

theorem Std.DTreeMap.getKey!_maxKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {he : t.isEmpty = false} : t.getKey! (t.maxKey he) = t.maxKey he

@[simp]

theorem Std.DTreeMap.getKeyD_maxKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {he : t.isEmpty = false} {fallback : α} : t.getKeyD (t.maxKey he) fallback = t.maxKey he

@[simp]

theorem Std.DTreeMap.maxKey_erase_eq_iff_not_compare_eq_maxKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {he : (t.erase k).isEmpty = false} : (t.erase k).maxKey he = t.maxKey ⋯ ↔ ¬cmp k (t.maxKey ⋯) = Ordering.eq

theorem Std.DTreeMap.maxKey_erase_eq_of_not_compare_eq_maxKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {he : (t.erase k).isEmpty = false} (hc : ¬cmp k (t.maxKey ⋯) = Ordering.eq) : (t.erase k).maxKey he = t.maxKey ⋯

theorem Std.DTreeMap.maxKey_erase_le_maxKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {he : (t.erase k).isEmpty = false} : (cmp ((t.erase k).maxKey he) (t.maxKey ⋯)).isLE = true

theorem Std.DTreeMap.maxKey_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (t.insertIfNew k v).maxKey ⋯ = t.maxKey?.elim k fun (k' : α) => if cmp k' k = Ordering.lt then k else k'

theorem Std.DTreeMap.maxKey_le_maxKey_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} {he : t.isEmpty = false} : (cmp (t.maxKey he) ((t.insertIfNew k v).maxKey ⋯)).isLE = true

theorem Std.DTreeMap.self_le_maxKey_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} : (cmp k ((t.insertIfNew k v).maxKey ⋯)).isLE = true

theorem Std.DTreeMap.maxKey_eq_getLast_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {he : t.isEmpty = false} : t.maxKey he = t.keys.getLast ⋯

theorem Std.DTreeMap.maxKey_eq_back_keysArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {he : t.isEmpty = false} : t.maxKey he = t.keysArray.back ⋯

@[simp]

theorem Std.DTreeMap.maxKey_modify {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : β k → β k} {he : (t.modify k f).isEmpty = false} : (t.modify k f).maxKey he = t.maxKey ⋯

theorem Std.DTreeMap.maxKey_alter_eq_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : Option (β k) → Option (β k)} {he : (t.alter k f).isEmpty = false} : (t.alter k f).maxKey he = k ↔ (f (t.get? k)).isSome = true ∧ ∀ (k' : α), k' ∈ t → (cmp k' k).isLE = true

theorem Std.DTreeMap.Const.maxKey_modify {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : β → β} {he : (modify t k f).isEmpty = false} : (modify t k f).maxKey he = if cmp (t.maxKey ⋯) k = Ordering.eq then k else t.maxKey ⋯

@[simp]

theorem Std.DTreeMap.Const.maxKey_modify_eq_maxKey {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : β → β} {he : (modify t k f).isEmpty = false} : (modify t k f).maxKey he = t.maxKey ⋯

theorem Std.DTreeMap.Const.compare_maxKey_modify_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : β → β} {he : (modify t k f).isEmpty = false} : cmp ((modify t k f).maxKey he) (t.maxKey ⋯) = Ordering.eq

@[simp]

theorem Std.DTreeMap.Const.ordCompare_maxKey_modify_eq {α : Type u} {β : Type v} [Ord α] [TransOrd α] {t : DTreeMap α (fun (x : α) => β) compare} {k : α} {f : β → β} {he : (modify t k f).isEmpty = false} : compare ((modify t k f).maxKey he) (t.maxKey ⋯) = Ordering.eq

theorem Std.DTreeMap.Const.maxKey_alter_eq_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : Option β → Option β} {he : (alter t k f).isEmpty = false} : (alter t k f).maxKey he = k ↔ (f (get? t k)).isSome = true ∧ ∀ (k' : α), k' ∈ t → (cmp k' k).isLE = true

theorem Std.DTreeMap.maxKey?_eq_some_maxKey! {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) : t.maxKey? = some t.maxKey!

theorem Std.DTreeMap.maxKey_eq_maxKey! {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {he : t.isEmpty = false} : t.maxKey he = t.maxKey!

theorem Std.DTreeMap.maxKey!_eq_default {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = true) : t.maxKey! = default

theorem Std.DTreeMap.maxKey!_eq_iff_getKey?_eq_self_and_forall {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {km : α} : t.maxKey! = km ↔ t.getKey? km = some km ∧ ∀ (k : α), k ∈ t → (cmp k km).isLE = true

theorem Std.DTreeMap.maxKey!_eq_iff_mem_and_forall {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] (he : t.isEmpty = false) {km : α} : t.maxKey! = km ↔ km ∈ t ∧ ∀ (k : α), k ∈ t → (cmp k km).isLE = true

theorem Std.DTreeMap.maxKey!_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} {v : β k} : (t.insert k v).maxKey! = t.maxKey?.elim k fun (k' : α) => if (cmp k' k).isLE = true then k else k'

theorem Std.DTreeMap.maxKey!_le_maxKey!_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {k : α} {v : β k} : (cmp t.maxKey! (t.insert k v).maxKey!).isLE = true

theorem Std.DTreeMap.self_le_maxKey!_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} {v : β k} : (cmp k (t.insert k v).maxKey!).isLE = true

theorem Std.DTreeMap.contains_maxKey! {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) : t.contains t.maxKey! = true

theorem Std.DTreeMap.maxKey!_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) : t.maxKey! ∈ t

theorem Std.DTreeMap.le_maxKey!_of_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} (hc : t.contains k = true) : (cmp k t.maxKey!).isLE = true

theorem Std.DTreeMap.le_maxKey!_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} (hc : k ∈ t) : (cmp k t.maxKey!).isLE = true

theorem Std.DTreeMap.maxKey!_le {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {k : α} : (cmp t.maxKey! k).isLE = true ↔ ∀ (k' : α), k' ∈ t → (cmp k' k).isLE = true

theorem Std.DTreeMap.getKey?_maxKey! {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) : t.getKey? t.maxKey! = some t.maxKey!

theorem Std.DTreeMap.getKey_maxKey! {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {hc : t.maxKey! ∈ t} : t.getKey t.maxKey! hc = t.maxKey!

@[simp]

theorem Std.DTreeMap.getKey_maxKey!_eq_maxKey {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {hc : t.maxKey! ∈ t} : t.getKey t.maxKey! hc = t.maxKey ⋯

theorem Std.DTreeMap.getKey!_maxKey! {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) : t.getKey! t.maxKey! = t.maxKey!

theorem Std.DTreeMap.getKeyD_maxKey! {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {fallback : α} : t.getKeyD t.maxKey! fallback = t.maxKey!

theorem Std.DTreeMap.maxKey!_erase_eq_of_not_compare_maxKey!_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} (he : (t.erase k).isEmpty = false) (heq : ¬cmp k t.maxKey! = Ordering.eq) : (t.erase k).maxKey! = t.maxKey!

theorem Std.DTreeMap.maxKey!_erase_le_maxKey! {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} (he : (t.erase k).isEmpty = false) : (cmp (t.erase k).maxKey! t.maxKey!).isLE = true

theorem Std.DTreeMap.maxKey!_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} {v : β k} : (t.insertIfNew k v).maxKey! = t.maxKey?.elim k fun (k' : α) => if cmp k' k = Ordering.lt then k else k'

theorem Std.DTreeMap.maxKey!_le_maxKey!_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {k : α} {v : β k} : (cmp t.maxKey! (t.insertIfNew k v).maxKey!).isLE = true

theorem Std.DTreeMap.self_le_maxKey!_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} {v : β k} : (cmp k (t.insertIfNew k v).maxKey!).isLE = true

theorem Std.DTreeMap.maxKey!_eq_getLast!_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] : t.maxKey! = t.keys.getLast!

theorem Std.DTreeMap.maxKey!_eq_back!_keysArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] : t.maxKey! = t.keysArray.back!

@[simp]

theorem Std.DTreeMap.maxKey!_modify {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {k : α} {f : β k → β k} : (t.modify k f).maxKey! = t.maxKey!

theorem Std.DTreeMap.maxKey!_alter_eq_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {k : α} {f : Option (β k) → Option (β k)} (he : (t.alter k f).isEmpty = false) : (t.alter k f).maxKey! = k ↔ (f (t.get? k)).isSome = true ∧ ∀ (k' : α), k' ∈ t → (cmp k' k).isLE = true

theorem Std.DTreeMap.Const.maxKey!_modify {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited α] {k : α} {f : β → β} (he : (modify t k f).isEmpty = false) : (modify t k f).maxKey! = if cmp t.maxKey! k = Ordering.eq then k else t.maxKey!

@[simp]

theorem Std.DTreeMap.Const.maxKey!_modify_eq_maxKey! {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {k : α} {f : β → β} : (modify t k f).maxKey! = t.maxKey!

theorem Std.DTreeMap.Const.compare_maxKey!_modify_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited α] {k : α} {f : β → β} : cmp (modify t k f).maxKey! t.maxKey! = Ordering.eq

@[simp]

theorem Std.DTreeMap.Const.ordCompare_maxKey!_modify_eq {α : Type u} {β : Type v} [Ord α] [TransOrd α] {t : DTreeMap α (fun (x : α) => β) compare} [Inhabited α] {k : α} {f : β → β} : compare (modify t k f).maxKey! t.maxKey! = Ordering.eq

theorem Std.DTreeMap.Const.maxKey!_alter_eq_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited α] {k : α} {f : Option β → Option β} (he : (alter t k f).isEmpty = false) : (alter t k f).maxKey! = k ↔ (f (get? t k)).isSome = true ∧ ∀ (k' : α), k' ∈ t → (cmp k' k).isLE = true

theorem Std.DTreeMap.maxKey?_eq_some_maxKeyD {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (he : t.isEmpty = false) {fallback : α} : t.maxKey? = some (t.maxKeyD fallback)

theorem Std.DTreeMap.maxKeyD_eq_fallback {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (he : t.isEmpty = true) {fallback : α} : t.maxKeyD fallback = fallback

theorem Std.DTreeMap.maxKey!_eq_maxKeyD_default {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] : t.maxKey! = t.maxKeyD default

theorem Std.DTreeMap.maxKeyD_eq_iff_getKey?_eq_self_and_forall {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (he : t.isEmpty = false) {km fallback : α} : t.maxKeyD fallback = km ↔ t.getKey? km = some km ∧ ∀ (k : α), k ∈ t → (cmp k km).isLE = true

theorem Std.DTreeMap.maxKeyD_eq_iff_mem_and_forall {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] (he : t.isEmpty = false) {km fallback : α} : t.maxKeyD fallback = km ↔ km ∈ t ∧ ∀ (k : α), k ∈ t → (cmp k km).isLE = true

theorem Std.DTreeMap.maxKeyD_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} {fallback : α} : (t.insert k v).maxKeyD fallback = t.maxKey?.elim k fun (k' : α) => if (cmp k' k).isLE = true then k else k'

theorem Std.DTreeMap.maxKeyD_le_maxKeyD_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (he : t.isEmpty = false) {k : α} {v : β k} {fallback : α} : (cmp (t.maxKeyD fallback) ((t.insert k v).maxKeyD fallback)).isLE = true

theorem Std.DTreeMap.self_le_maxKeyD_insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} {fallback : α} : (cmp k ((t.insert k v).maxKeyD fallback)).isLE = true

theorem Std.DTreeMap.contains_maxKeyD {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (he : t.isEmpty = false) {fallback : α} : t.contains (t.maxKeyD fallback) = true

theorem Std.DTreeMap.maxKeyD_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (he : t.isEmpty = false) {fallback : α} : t.maxKeyD fallback ∈ t

theorem Std.DTreeMap.le_maxKeyD_of_contains {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} (hc : t.contains k = true) {fallback : α} : (cmp k (t.maxKeyD fallback)).isLE = true

theorem Std.DTreeMap.le_maxKeyD_of_mem {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} (hc : k ∈ t) {fallback : α} : (cmp k (t.maxKeyD fallback)).isLE = true

theorem Std.DTreeMap.maxKeyD_le {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (he : t.isEmpty = false) {k fallback : α} : (cmp (t.maxKeyD fallback) k).isLE = true ↔ ∀ (k' : α), k' ∈ t → (cmp k' k).isLE = true

theorem Std.DTreeMap.getKey?_maxKeyD {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (he : t.isEmpty = false) {fallback : α} : t.getKey? (t.maxKeyD fallback) = some (t.maxKeyD fallback)

theorem Std.DTreeMap.getKey_maxKeyD {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {fallback : α} {hc : t.maxKeyD fallback ∈ t} : t.getKey (t.maxKeyD fallback) hc = t.maxKeyD fallback

theorem Std.DTreeMap.getKey!_maxKeyD {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {fallback : α} : t.getKey! (t.maxKeyD fallback) = t.maxKeyD fallback

theorem Std.DTreeMap.getKeyD_maxKeyD {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (he : t.isEmpty = false) {fallback fallback' : α} : t.getKeyD (t.maxKeyD fallback) fallback' = t.maxKeyD fallback

theorem Std.DTreeMap.maxKeyD_erase_eq_of_not_compare_maxKeyD_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k fallback : α} (he : (t.erase k).isEmpty = false) (heq : ¬cmp k (t.maxKeyD fallback) = Ordering.eq) : (t.erase k).maxKeyD fallback = t.maxKeyD fallback

theorem Std.DTreeMap.maxKeyD_erase_le_maxKeyD {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} (he : (t.erase k).isEmpty = false) {fallback : α} : (cmp ((t.erase k).maxKeyD fallback) (t.maxKeyD fallback)).isLE = true

theorem Std.DTreeMap.maxKeyD_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} {fallback : α} : (t.insertIfNew k v).maxKeyD fallback = t.maxKey?.elim k fun (k' : α) => if cmp k' k = Ordering.lt then k else k'

theorem Std.DTreeMap.maxKeyD_le_maxKeyD_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] (he : t.isEmpty = false) {k : α} {v : β k} {fallback : α} : (cmp (t.maxKeyD fallback) ((t.insertIfNew k v).maxKeyD fallback)).isLE = true

theorem Std.DTreeMap.self_le_maxKeyD_insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {k : α} {v : β k} {fallback : α} : (cmp k ((t.insertIfNew k v).maxKeyD fallback)).isLE = true

theorem Std.DTreeMap.maxKeyD_eq_getLastD_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {fallback : α} : t.maxKeyD fallback = t.keys.getLastD fallback

theorem Std.DTreeMap.maxKeyD_eq_getD_back?_keysArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {fallback : α} : t.maxKeyD fallback = t.keysArray.back?.getD fallback

@[simp]

theorem Std.DTreeMap.maxKeyD_modify {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : β k → β k} {fallback : α} : (t.modify k f).maxKeyD fallback = t.maxKeyD fallback

theorem Std.DTreeMap.maxKeyD_alter_eq_self {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : Option (β k) → Option (β k)} (he : (t.alter k f).isEmpty = false) {fallback : α} : (t.alter k f).maxKeyD fallback = k ↔ (f (t.get? k)).isSome = true ∧ ∀ (k' : α), k' ∈ t → (cmp k' k).isLE = true

theorem Std.DTreeMap.Const.maxKeyD_modify {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : β → β} (he : (modify t k f).isEmpty = false) {fallback : α} : (modify t k f).maxKeyD fallback = if cmp (t.maxKeyD fallback) k = Ordering.eq then k else t.maxKeyD fallback

@[simp]

theorem Std.DTreeMap.Const.maxKeyD_modify_eq_maxKeyD {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {f : β → β} {fallback : α} : (modify t k f).maxKeyD fallback = t.maxKeyD fallback

theorem Std.DTreeMap.Const.compare_maxKeyD_modify_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : β → β} {fallback : α} : cmp ((modify t k f).maxKeyD fallback) (t.maxKeyD fallback) = Ordering.eq

@[simp]

theorem Std.DTreeMap.Const.ordCompare_maxKeyD_modify_eq {α : Type u} {β : Type v} [Ord α] [TransOrd α] {t : DTreeMap α (fun (x : α) => β) compare} {k : α} {f : β → β} {fallback : α} : compare ((modify t k f).maxKeyD fallback) (t.maxKeyD fallback) = Ordering.eq

theorem Std.DTreeMap.Const.maxKeyD_alter_eq_self {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {f : Option β → Option β} (he : (alter t k f).isEmpty = false) {fallback : α} : (alter t k f).maxKeyD fallback = k ↔ (f (get? t k)).isSome = true ∧ ∀ (k' : α), k' ∈ t → (cmp k' k).isLE = true

@[simp]

theorem Std.DTreeMap.Equiv.rfl {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} : t.Equiv t

theorem Std.DTreeMap.Equiv.symm {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} : t₁.Equiv t₂ → t₂.Equiv t₁

theorem Std.DTreeMap.Equiv.trans {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ t₃ : DTreeMap α β cmp} : t₁.Equiv t₂ → t₂.Equiv t₃ → t₁.Equiv t₃

@[instance_reducible]

instance Std.DTreeMap.Equiv.instTrans {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} : Trans Equiv Equiv Equiv

Equations

• Std.DTreeMap.Equiv.instTrans = { trans := ⋯ }

theorem Std.DTreeMap.Equiv.comm {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} : t₁.Equiv t₂ ↔ t₂.Equiv t₁

theorem Std.DTreeMap.Equiv.congr_left {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ t₃ : DTreeMap α β cmp} (h : t₁.Equiv t₂) : t₁.Equiv t₃ ↔ t₂.Equiv t₃

theorem Std.DTreeMap.Equiv.congr_right {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ t₃ : DTreeMap α β cmp} (h : t₁.Equiv t₂) : t₃.Equiv t₁ ↔ t₃.Equiv t₂

theorem Std.DTreeMap.Equiv.isEmpty_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} (h : t₁.Equiv t₂) : t₁.isEmpty = t₂.isEmpty

theorem Std.DTreeMap.Equiv.contains_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : t₁.Equiv t₂) : t₁.contains k = t₂.contains k

theorem Std.DTreeMap.Equiv.mem_iff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : t₁.Equiv t₂) : k ∈ t₁ ↔ k ∈ t₂

theorem Std.DTreeMap.Equiv.size_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} (h : t₁.Equiv t₂) : t₁.size = t₂.size

theorem Std.DTreeMap.Equiv.get?_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} (h : t₁.Equiv t₂) : t₁.get? k = t₂.get? k

theorem Std.DTreeMap.Equiv.get_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {hk : k ∈ t₁} (h : t₁.Equiv t₂) : t₁.get k hk = t₂.get k ⋯

theorem Std.DTreeMap.Equiv.get!_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} [Inhabited (β k)] (h : t₁.Equiv t₂) : t₁.get! k = t₂.get! k

theorem Std.DTreeMap.Equiv.getD_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {k : α} {fallback : β k} (h : t₁.Equiv t₂) : t₁.getD k fallback = t₂.getD k fallback

theorem Std.DTreeMap.Equiv.getKey?_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : t₁.Equiv t₂) : t₁.getKey? k = t₂.getKey? k

theorem Std.DTreeMap.Equiv.getKey_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} {hk : k ∈ t₁} (h : t₁.Equiv t₂) : t₁.getKey k hk = t₂.getKey k ⋯

theorem Std.DTreeMap.Equiv.getKey!_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} (h : t₁.Equiv t₂) : t₁.getKey! k = t₂.getKey! k

theorem Std.DTreeMap.Equiv.getKeyD_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k fallback : α} (h : t₁.Equiv t₂) : t₁.getKeyD k fallback = t₂.getKeyD k fallback

theorem Std.DTreeMap.Equiv.toList_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] (h : t₁.Equiv t₂) : t₁.toList = t₂.toList

theorem Std.DTreeMap.Equiv.toArray_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] (h : t₁.Equiv t₂) : t₁.toArray = t₂.toArray

theorem Std.DTreeMap.Equiv.keys_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] (h : t₁.Equiv t₂) : t₁.keys = t₂.keys

theorem Std.DTreeMap.Equiv.keysArray_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] (h : t₁.Equiv t₂) : t₁.keysArray = t₂.keysArray

theorem Std.DTreeMap.Equiv.foldlM_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} {δ : Type w} {m : Type w → Type w'} [TransCmp cmp] [Monad m] [LawfulMonad m] {f : δ → (a : α) → β a → m δ} {init : δ} (h : t₁.Equiv t₂) : foldlM f init t₁ = foldlM f init t₂

theorem Std.DTreeMap.Equiv.foldl_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} {δ : Type w} [TransCmp cmp] {f : δ → (a : α) → β a → δ} {init : δ} (h : t₁.Equiv t₂) : foldl f init t₁ = foldl f init t₂

theorem Std.DTreeMap.Equiv.foldrM_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} {δ : Type w} {m : Type w → Type w'} [TransCmp cmp] [Monad m] [LawfulMonad m] {f : (a : α) → β a → δ → m δ} {init : δ} (h : t₁.Equiv t₂) : foldrM f init t₁ = foldrM f init t₂

theorem Std.DTreeMap.Equiv.foldr_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} {δ : Type w} [TransCmp cmp] {f : (a : α) → β a → δ → δ} {init : δ} (h : t₁.Equiv t₂) : foldr f init t₁ = foldr f init t₂

theorem Std.DTreeMap.Equiv.forIn_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} {δ : Type w} {m : Type w → Type w'} [TransCmp cmp] [Monad m] [LawfulMonad m] {b : δ} {f : (a : α) × β a → δ → m (ForInStep δ)} (h : t₁.Equiv t₂) : ForIn.forIn t₁ b f = ForIn.forIn t₂ b f

theorem Std.DTreeMap.Equiv.forM_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} {m : Type w → Type w'} [TransCmp cmp] [Monad m] [LawfulMonad m] {f : (a : α) × β a → m PUnit} (h : t₁.Equiv t₂) : ForM.forM t₁ f = ForM.forM t₂ f

theorem Std.DTreeMap.Equiv.any_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {p : (a : α) → β a → Bool} (h : t₁.Equiv t₂) : t₁.any p = t₂.any p

theorem Std.DTreeMap.Equiv.all_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {p : (a : α) → β a → Bool} (h : t₁.Equiv t₂) : t₁.all p = t₂.all p

theorem Std.DTreeMap.Equiv.minKey?_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] (h : t₁.Equiv t₂) : t₁.minKey? = t₂.minKey?

theorem Std.DTreeMap.Equiv.minKey_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {h' : t₁.isEmpty = false} (h : t₁.Equiv t₂) : t₁.minKey h' = t₂.minKey ⋯

theorem Std.DTreeMap.Equiv.minKey!_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (h : t₁.Equiv t₂) : t₁.minKey! = t₂.minKey!

theorem Std.DTreeMap.Equiv.minKeyD_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {fallback : α} (h : t₁.Equiv t₂) : t₁.minKeyD fallback = t₂.minKeyD fallback

theorem Std.DTreeMap.Equiv.maxKey?_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] (h : t₁.Equiv t₂) : t₁.maxKey? = t₂.maxKey?

theorem Std.DTreeMap.Equiv.maxKey_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {h' : t₁.isEmpty = false} (h : t₁.Equiv t₂) : t₁.maxKey h' = t₂.maxKey ⋯

theorem Std.DTreeMap.Equiv.maxKey!_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] (h : t₁.Equiv t₂) : t₁.maxKey! = t₂.maxKey!

theorem Std.DTreeMap.Equiv.maxKeyD_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {fallback : α} (h : t₁.Equiv t₂) : t₁.maxKeyD fallback = t₂.maxKeyD fallback

theorem Std.DTreeMap.Equiv.minEntry?_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] (h : t₁.Equiv t₂) : t₁.minEntry? = t₂.minEntry?

theorem Std.DTreeMap.Equiv.minEntry_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {he : t₁.isEmpty = false} (h : t₁.Equiv t₂) : t₁.minEntry he = t₂.minEntry ⋯

theorem Std.DTreeMap.Equiv.minEntry!_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [Inhabited ((a : α) × β a)] (h : t₁.Equiv t₂) : t₁.minEntry! = t₂.minEntry!

theorem Std.DTreeMap.Equiv.minEntryD_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {fallback : (a : α) × β a} (h : t₁.Equiv t₂) : t₁.minEntryD fallback = t₂.minEntryD fallback

theorem Std.DTreeMap.Equiv.maxEntry?_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] (h : t₁.Equiv t₂) : t₁.maxEntry? = t₂.maxEntry?

theorem Std.DTreeMap.Equiv.maxEntry_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {he : t₁.isEmpty = false} (h : t₁.Equiv t₂) : t₁.maxEntry he = t₂.maxEntry ⋯

theorem Std.DTreeMap.Equiv.maxEntry!_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [Inhabited ((a : α) × β a)] (h : t₁.Equiv t₂) : t₁.maxEntry! = t₂.maxEntry!

theorem Std.DTreeMap.Equiv.maxEntryD_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {fallback : (a : α) × β a} (h : t₁.Equiv t₂) : t₁.maxEntryD fallback = t₂.maxEntryD fallback

theorem Std.DTreeMap.Equiv.entryAtIdx?_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {i : Nat} (h : t₁.Equiv t₂) : t₁.entryAtIdx? i = t₂.entryAtIdx? i

theorem Std.DTreeMap.Equiv.entryAtIdx_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {i : Nat} {h' : i < t₁.size} (h : t₁.Equiv t₂) : t₁.entryAtIdx i h' = t₂.entryAtIdx i ⋯

theorem Std.DTreeMap.Equiv.entryAtIdx!_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [Inhabited ((a : α) × β a)] {i : Nat} (h : t₁.Equiv t₂) : t₁.entryAtIdx! i = t₂.entryAtIdx! i

theorem Std.DTreeMap.Equiv.entryAtIdxD_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {i : Nat} {fallback : (a : α) × β a} (h : t₁.Equiv t₂) : t₁.entryAtIdxD i fallback = t₂.entryAtIdxD i fallback

theorem Std.DTreeMap.Equiv.keyAtIdx?_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {i : Nat} (h : t₁.Equiv t₂) : t₁.keyAtIdx? i = t₂.keyAtIdx? i

theorem Std.DTreeMap.Equiv.keyAtIdx_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {i : Nat} {h' : i < t₁.size} (h : t₁.Equiv t₂) : t₁.keyAtIdx i h' = t₂.keyAtIdx i ⋯

theorem Std.DTreeMap.Equiv.keyAtIdx!_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {i : Nat} (h : t₁.Equiv t₂) : t₁.keyAtIdx! i = t₂.keyAtIdx! i

theorem Std.DTreeMap.Equiv.keyAtIdxD_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {i : Nat} {fallback : α} (h : t₁.Equiv t₂) : t₁.keyAtIdxD i fallback = t₂.keyAtIdxD i fallback

theorem Std.DTreeMap.Equiv.getEntryGE?_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : t₁.Equiv t₂) : t₁.getEntryGE? k = t₂.getEntryGE? k

theorem Std.DTreeMap.Equiv.getEntryGE_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} {h' : ∃ (a : α), a ∈ t₁ ∧ (cmp a k).isGE = true} (h : t₁.Equiv t₂) : t₁.getEntryGE k h' = t₂.getEntryGE k ⋯

theorem Std.DTreeMap.Equiv.getEntryGE!_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [Inhabited ((a : α) × β a)] {k : α} (h : t₁.Equiv t₂) : t₁.getEntryGE! k = t₂.getEntryGE! k

theorem Std.DTreeMap.Equiv.getEntryGED_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} {fallback : (a : α) × β a} (h : t₁.Equiv t₂) : t₁.getEntryGED k fallback = t₂.getEntryGED k fallback

theorem Std.DTreeMap.Equiv.getEntryGT?_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : t₁.Equiv t₂) : t₁.getEntryGT? k = t₂.getEntryGT? k

theorem Std.DTreeMap.Equiv.getEntryGT_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} {h' : ∃ (a : α), a ∈ t₁ ∧ cmp a k = Ordering.gt} (h : t₁.Equiv t₂) : t₁.getEntryGT k h' = t₂.getEntryGT k ⋯

theorem Std.DTreeMap.Equiv.getEntryGT!_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [Inhabited ((a : α) × β a)] {k : α} (h : t₁.Equiv t₂) : t₁.getEntryGT! k = t₂.getEntryGT! k

theorem Std.DTreeMap.Equiv.getEntryGTD_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} {fallback : (a : α) × β a} (h : t₁.Equiv t₂) : t₁.getEntryGTD k fallback = t₂.getEntryGTD k fallback

theorem Std.DTreeMap.Equiv.getEntryLE?_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : t₁.Equiv t₂) : t₁.getEntryLE? k = t₂.getEntryLE? k

theorem Std.DTreeMap.Equiv.getEntryLE_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} {h' : ∃ (a : α), a ∈ t₁ ∧ (cmp a k).isLE = true} (h : t₁.Equiv t₂) : t₁.getEntryLE k h' = t₂.getEntryLE k ⋯

theorem Std.DTreeMap.Equiv.getEntryLE!_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [Inhabited ((a : α) × β a)] {k : α} (h : t₁.Equiv t₂) : t₁.getEntryLE! k = t₂.getEntryLE! k

theorem Std.DTreeMap.Equiv.getEntryLED_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} {fallback : (a : α) × β a} (h : t₁.Equiv t₂) : t₁.getEntryLED k fallback = t₂.getEntryLED k fallback

theorem Std.DTreeMap.Equiv.getEntryLT?_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : t₁.Equiv t₂) : t₁.getEntryLT? k = t₂.getEntryLT? k

theorem Std.DTreeMap.Equiv.getEntryLT_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} {h' : ∃ (a : α), a ∈ t₁ ∧ cmp a k = Ordering.lt} (h : t₁.Equiv t₂) : t₁.getEntryLT k h' = t₂.getEntryLT k ⋯

theorem Std.DTreeMap.Equiv.getEntryLT!_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [Inhabited ((a : α) × β a)] {k : α} (h : t₁.Equiv t₂) : t₁.getEntryLT! k = t₂.getEntryLT! k

theorem Std.DTreeMap.Equiv.getEntryLTD_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} {fallback : (a : α) × β a} (h : t₁.Equiv t₂) : t₁.getEntryLTD k fallback = t₂.getEntryLTD k fallback

theorem Std.DTreeMap.Equiv.getKeyGE?_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : t₁.Equiv t₂) : t₁.getKeyGE? k = t₂.getKeyGE? k

theorem Std.DTreeMap.Equiv.getKeyGE_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} {h' : ∃ (a : α), a ∈ t₁ ∧ (cmp a k).isGE = true} (h : t₁.Equiv t₂) : t₁.getKeyGE k h' = t₂.getKeyGE k ⋯

theorem Std.DTreeMap.Equiv.getKeyGE!_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} (h : t₁.Equiv t₂) : t₁.getKeyGE! k = t₂.getKeyGE! k

theorem Std.DTreeMap.Equiv.getKeyGED_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k fallback : α} (h : t₁.Equiv t₂) : t₁.getKeyGED k fallback = t₂.getKeyGED k fallback

theorem Std.DTreeMap.Equiv.getKeyGT?_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : t₁.Equiv t₂) : t₁.getKeyGT? k = t₂.getKeyGT? k

theorem Std.DTreeMap.Equiv.getKeyGT_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} {h' : ∃ (a : α), a ∈ t₁ ∧ cmp a k = Ordering.gt} (h : t₁.Equiv t₂) : t₁.getKeyGT k h' = t₂.getKeyGT k ⋯

theorem Std.DTreeMap.Equiv.getKeyGT!_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} (h : t₁.Equiv t₂) : t₁.getKeyGT! k = t₂.getKeyGT! k

theorem Std.DTreeMap.Equiv.getKeyGTD_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k fallback : α} (h : t₁.Equiv t₂) : t₁.getKeyGTD k fallback = t₂.getKeyGTD k fallback

theorem Std.DTreeMap.Equiv.getKeyLE?_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : t₁.Equiv t₂) : t₁.getKeyLE? k = t₂.getKeyLE? k

theorem Std.DTreeMap.Equiv.getKeyLE_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} {h' : ∃ (a : α), a ∈ t₁ ∧ (cmp a k).isLE = true} (h : t₁.Equiv t₂) : t₁.getKeyLE k h' = t₂.getKeyLE k ⋯

theorem Std.DTreeMap.Equiv.getKeyLE!_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} (h : t₁.Equiv t₂) : t₁.getKeyLE! k = t₂.getKeyLE! k

theorem Std.DTreeMap.Equiv.getKeyLED_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k fallback : α} (h : t₁.Equiv t₂) : t₁.getKeyLED k fallback = t₂.getKeyLED k fallback

theorem Std.DTreeMap.Equiv.getKeyLT?_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} (h : t₁.Equiv t₂) : t₁.getKeyLT? k = t₂.getKeyLT? k

theorem Std.DTreeMap.Equiv.getKeyLT_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k : α} {h' : ∃ (a : α), a ∈ t₁ ∧ cmp a k = Ordering.lt} (h : t₁.Equiv t₂) : t₁.getKeyLT k h' = t₂.getKeyLT k ⋯

theorem Std.DTreeMap.Equiv.getKeyLT!_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {k : α} (h : t₁.Equiv t₂) : t₁.getKeyLT! k = t₂.getKeyLT! k

theorem Std.DTreeMap.Equiv.getKeyLTD_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] {k fallback : α} (h : t₁.Equiv t₂) : t₁.getKeyLTD k fallback = t₂.getKeyLTD k fallback

theorem Std.DTreeMap.Equiv.insert {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] (h : t₁.Equiv t₂) (k : α) (v : β k) : (t₁.insert k v).Equiv (t₂.insert k v)

theorem Std.DTreeMap.Equiv.erase {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] (h : t₁.Equiv t₂) (k : α) : (t₁.erase k).Equiv (t₂.erase k)

theorem Std.DTreeMap.Equiv.insertIfNew {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] (h : t₁.Equiv t₂) (k : α) (v : β k) : (t₁.insertIfNew k v).Equiv (t₂.insertIfNew k v)

theorem Std.DTreeMap.Equiv.alter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] (h : t₁.Equiv t₂) (k : α) (f : Option (β k) → Option (β k)) : (t₁.alter k f).Equiv (t₂.alter k f)

theorem Std.DTreeMap.Equiv.modify {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] (h : t₁.Equiv t₂) (k : α) (f : β k → β k) : (t₁.modify k f).Equiv (t₂.modify k f)

theorem Std.DTreeMap.Equiv.filter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} (h : t₁.Equiv t₂) (f : (a : α) → β a → Bool) : (DTreeMap.filter f t₁).Equiv (DTreeMap.filter f t₂)

theorem Std.DTreeMap.Equiv.map {α : Type u} {β : α → Type v} {γ : α → Type w} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} (h : t₁.Equiv t₂) (f : (a : α) → β a → γ a) : (DTreeMap.map f t₁).Equiv (DTreeMap.map f t₂)

theorem Std.DTreeMap.Equiv.filterMap {α : Type u} {β : α → Type v} {γ : α → Type w} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} (h : t₁.Equiv t₂) (f : (a : α) → β a → Option (γ a)) : (DTreeMap.filterMap f t₁).Equiv (DTreeMap.filterMap f t₂)

theorem Std.DTreeMap.Equiv.insertMany_list {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] (h : t₁.Equiv t₂) (l : List ((a : α) × β a)) : (t₁.insertMany l).Equiv (t₂.insertMany l)

theorem Std.DTreeMap.Equiv.eraseMany_list {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] (h : t₁.Equiv t₂) (l : List α) : (t₁.eraseMany l).Equiv (t₂.eraseMany l)

theorem Std.DTreeMap.Equiv.mergeWith {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ t₃ t₄ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] (f : (a : α) → β a → β a → β a) (h : t₁.Equiv t₂) (h' : t₃.Equiv t₄) : (DTreeMap.mergeWith f t₁ t₃).Equiv (DTreeMap.mergeWith f t₂ t₄)

theorem Std.DTreeMap.Equiv.constGet?_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} (h : t₁.Equiv t₂) : Const.get? t₁ k = Const.get? t₂ k

theorem Std.DTreeMap.Equiv.constGet_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {hk : k ∈ t₁} (h : t₁.Equiv t₂) : Const.get t₁ k hk = Const.get t₂ k ⋯

theorem Std.DTreeMap.Equiv.constGet!_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {k : α} (h : t₁.Equiv t₂) : Const.get! t₁ k = Const.get! t₂ k

theorem Std.DTreeMap.Equiv.constGetD_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {fallback : β} (h : t₁.Equiv t₂) : Const.getD t₁ k fallback = Const.getD t₂ k fallback

theorem Std.DTreeMap.Equiv.constToList_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] (h : t₁.Equiv t₂) : Const.toList t₁ = Const.toList t₂

theorem Std.DTreeMap.Equiv.constToArray_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] (h : t₁.Equiv t₂) : Const.toArray t₁ = Const.toArray t₂

theorem Std.DTreeMap.Equiv.values_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] (h : t₁.Equiv t₂) : t₁.values = t₂.values

theorem Std.DTreeMap.Equiv.valuesArray_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] (h : t₁.Equiv t₂) : t₁.valuesArray = t₂.valuesArray

theorem Std.DTreeMap.Equiv.constMinEntry?_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] (h : t₁.Equiv t₂) : Const.minEntry? t₁ = Const.minEntry? t₂

theorem Std.DTreeMap.Equiv.constMinEntry_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {he : t₁.isEmpty = false} (h : t₁.Equiv t₂) : Const.minEntry t₁ he = Const.minEntry t₂ ⋯

theorem Std.DTreeMap.Equiv.constMinEntry!_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited (α × β)] (h : t₁.Equiv t₂) : Const.minEntry! t₁ = Const.minEntry! t₂

theorem Std.DTreeMap.Equiv.constMinEntryD_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {fallback : α × β} (h : t₁.Equiv t₂) : Const.minEntryD t₁ fallback = Const.minEntryD t₂ fallback

theorem Std.DTreeMap.Equiv.constMaxEntry?_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] (h : t₁.Equiv t₂) : Const.maxEntry? t₁ = Const.maxEntry? t₂

theorem Std.DTreeMap.Equiv.constMaxEntry_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {he : t₁.isEmpty = false} (h : t₁.Equiv t₂) : Const.maxEntry t₁ he = Const.maxEntry t₂ ⋯

theorem Std.DTreeMap.Equiv.constMaxEntry!_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited (α × β)] (h : t₁.Equiv t₂) : Const.maxEntry! t₁ = Const.maxEntry! t₂

theorem Std.DTreeMap.Equiv.constMaxEntryD_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {fallback : α × β} (h : t₁.Equiv t₂) : Const.maxEntryD t₁ fallback = Const.maxEntryD t₂ fallback

theorem Std.DTreeMap.Equiv.constEntryAtIdx?_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {i : Nat} (h : t₁.Equiv t₂) : Const.entryAtIdx? t₁ i = Const.entryAtIdx? t₂ i

theorem Std.DTreeMap.Equiv.constEntryAtIdx_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {i : Nat} {h' : i < t₁.size} (h : t₁.Equiv t₂) : Const.entryAtIdx t₁ i h' = Const.entryAtIdx t₂ i ⋯

theorem Std.DTreeMap.Equiv.constEntryAtIdx!_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited (α × β)] {i : Nat} (h : t₁.Equiv t₂) : Const.entryAtIdx! t₁ i = Const.entryAtIdx! t₂ i

theorem Std.DTreeMap.Equiv.constEntryAtIdxD_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {i : Nat} {fallback : α × β} (h : t₁.Equiv t₂) : Const.entryAtIdxD t₁ i fallback = Const.entryAtIdxD t₂ i fallback

theorem Std.DTreeMap.Equiv.constGetEntryGE?_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} (h : t₁.Equiv t₂) : Const.getEntryGE? t₁ k = Const.getEntryGE? t₂ k

theorem Std.DTreeMap.Equiv.constGetEntryGE_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {h' : ∃ (a : α), a ∈ t₁ ∧ (cmp a k).isGE = true} (h : t₁.Equiv t₂) : Const.getEntryGE t₁ k h' = Const.getEntryGE t₂ k ⋯

theorem Std.DTreeMap.Equiv.constGetEntryGE!_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited (α × β)] {k : α} (h : t₁.Equiv t₂) : Const.getEntryGE! t₁ k = Const.getEntryGE! t₂ k

theorem Std.DTreeMap.Equiv.constGetEntryGED_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {fallback : α × β} (h : t₁.Equiv t₂) : Const.getEntryGED t₁ k fallback = Const.getEntryGED t₂ k fallback

theorem Std.DTreeMap.Equiv.constGetEntryGT?_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} (h : t₁.Equiv t₂) : Const.getEntryGT? t₁ k = Const.getEntryGT? t₂ k

theorem Std.DTreeMap.Equiv.constGetEntryGT_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {h' : ∃ (a : α), a ∈ t₁ ∧ cmp a k = Ordering.gt} (h : t₁.Equiv t₂) : Const.getEntryGT t₁ k h' = Const.getEntryGT t₂ k ⋯

theorem Std.DTreeMap.Equiv.constGetEntryGT!_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited (α × β)] {k : α} (h : t₁.Equiv t₂) : Const.getEntryGT! t₁ k = Const.getEntryGT! t₂ k

theorem Std.DTreeMap.Equiv.constGetEntryGTD_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {fallback : α × β} (h : t₁.Equiv t₂) : Const.getEntryGTD t₁ k fallback = Const.getEntryGTD t₂ k fallback

theorem Std.DTreeMap.Equiv.constGetEntryLE?_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} (h : t₁.Equiv t₂) : Const.getEntryLE? t₁ k = Const.getEntryLE? t₂ k

theorem Std.DTreeMap.Equiv.constGetEntryLE_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {h' : ∃ (a : α), a ∈ t₁ ∧ (cmp a k).isLE = true} (h : t₁.Equiv t₂) : Const.getEntryLE t₁ k h' = Const.getEntryLE t₂ k ⋯

theorem Std.DTreeMap.Equiv.constGetEntryLE!_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited (α × β)] {k : α} (h : t₁.Equiv t₂) : Const.getEntryLE! t₁ k = Const.getEntryLE! t₂ k

theorem Std.DTreeMap.Equiv.constGetEntryLED_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {fallback : α × β} (h : t₁.Equiv t₂) : Const.getEntryLED t₁ k fallback = Const.getEntryLED t₂ k fallback

theorem Std.DTreeMap.Equiv.constGetEntryLT?_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} (h : t₁.Equiv t₂) : Const.getEntryLT? t₁ k = Const.getEntryLT? t₂ k

theorem Std.DTreeMap.Equiv.constGetEntryLT_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {h' : ∃ (a : α), a ∈ t₁ ∧ cmp a k = Ordering.lt} (h : t₁.Equiv t₂) : Const.getEntryLT t₁ k h' = Const.getEntryLT t₂ k ⋯

theorem Std.DTreeMap.Equiv.constGetEntryLT!_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited (α × β)] {k : α} (h : t₁.Equiv t₂) : Const.getEntryLT! t₁ k = Const.getEntryLT! t₂ k

theorem Std.DTreeMap.Equiv.constGetEntryLTD_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {k : α} {fallback : α × β} (h : t₁.Equiv t₂) : Const.getEntryLTD t₁ k fallback = Const.getEntryLTD t₂ k fallback

theorem Std.DTreeMap.Equiv.constAlter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] (h : t₁.Equiv t₂) (k : α) (f : Option β → Option β) : (Const.alter t₁ k f).Equiv (Const.alter t₂ k f)

theorem Std.DTreeMap.Equiv.constModify {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] (h : t₁.Equiv t₂) (k : α) (f : β → β) : (Const.modify t₁ k f).Equiv (Const.modify t₂ k f)

theorem Std.DTreeMap.Equiv.constInsertMany_list {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] (h : t₁.Equiv t₂) (l : List (α × β)) : (Const.insertMany t₁ l).Equiv (Const.insertMany t₂ l)

theorem Std.DTreeMap.Equiv.constInsertManyIfNewUnit_list {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {t₁ t₂ : DTreeMap α (fun (x : α) => Unit) cmp} (h : t₁.Equiv t₂) (l : List α) : (Const.insertManyIfNewUnit t₁ l).Equiv (Const.insertManyIfNewUnit t₂ l)

theorem Std.DTreeMap.Equiv.constMergeWith {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ t₃ t₄ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] (f : α → β → β → β) (h : t₁.Equiv t₂) (h' : t₃.Equiv t₄) : (Const.mergeWith f t₁ t₃).Equiv (Const.mergeWith f t₂ t₄)

theorem Std.DTreeMap.Equiv.of_forall_get?_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] (h : ∀ (k : α), t₁.get? k = t₂.get? k) : t₁.Equiv t₂

theorem Std.DTreeMap.Equiv.of_forall_getKey_eq_of_forall_constGet?_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] (hk : ∀ (k : α) (hk : k ∈ t₁) (hk' : k ∈ t₂), t₁.getKey k hk = t₂.getKey k hk') (hv : ∀ (k : α), Const.get? t₁ k = Const.get? t₂ k) : t₁.Equiv t₂

theorem Std.DTreeMap.Equiv.of_forall_constGet?_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] (h : ∀ (k : α), Const.get? t₁ k = Const.get? t₂ k) : t₁.Equiv t₂

theorem Std.DTreeMap.Equiv.of_forall_getKey?_unit_eq {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {t₁ t₂ : DTreeMap α (fun (x : α) => Unit) cmp} (h : ∀ (k : α), t₁.getKey? k = t₂.getKey? k) : t₁.Equiv t₂

theorem Std.DTreeMap.Equiv.of_forall_contains_unit_eq {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] [LawfulEqCmp cmp] {t₁ t₂ : DTreeMap α (fun (x : α) => Unit) cmp} (h : ∀ (k : α), t₁.contains k = t₂.contains k) : t₁.Equiv t₂

theorem Std.DTreeMap.Equiv.of_forall_mem_unit_iff {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] [LawfulEqCmp cmp] {t₁ t₂ : DTreeMap α (fun (x : α) => Unit) cmp} (h : ∀ (k : α), k ∈ t₁ ↔ k ∈ t₂) : t₁.Equiv t₂

@[instance_reducible]

def Std.DTreeMap.isSetoid (α : Type u) (β : α → Type v) (cmp : α → α → Ordering := by exact compare) : Setoid (DTreeMap α β cmp)

Implementation detail of the tree map

Equations

• Std.DTreeMap.isSetoid α β cmp = { r := Std.DTreeMap.Equiv, iseqv := ⋯ }

theorem Std.DTreeMap.equiv_empty_iff_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} : t.Equiv empty ↔ t.isEmpty = true

theorem Std.DTreeMap.inter_equiv_empty_comm {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] : (t₁ ∩ t₂).Equiv ∅ ↔ (t₂ ∩ t₁).Equiv ∅

theorem Std.DTreeMap.empty_equiv_iff_isEmpty {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} : empty.Equiv t ↔ t.isEmpty = true

theorem Std.DTreeMap.equiv_iff_toList_perm {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} : t₁.Equiv t₂ ↔ t₁.toList.Perm t₂.toList

theorem Std.DTreeMap.equiv_iff_toArray_perm {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} : t₁.Equiv t₂ ↔ t₁.toArray.Perm t₂.toArray

theorem Std.DTreeMap.Equiv.of_toList_perm {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} (h : t₁.toList.Perm t₂.toList) : t₁.Equiv t₂

theorem Std.DTreeMap.Equiv.of_toArray_perm {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} (h : t₁.toArray.Perm t₂.toArray) : t₁.Equiv t₂

theorem Std.DTreeMap.equiv_iff_toList_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] : t₁.Equiv t₂ ↔ t₁.toList = t₂.toList

theorem Std.DTreeMap.equiv_iff_toArray_eq {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α β cmp} [TransCmp cmp] : t₁.Equiv t₂ ↔ t₁.toArray = t₂.toArray

theorem Std.DTreeMap.insertMany_list_equiv_foldl {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ : DTreeMap α β cmp} {l : List ((a : α) × β a)} : (t₁.insertMany l).Equiv (List.foldl (fun (acc : DTreeMap α β cmp) (p : (a : α) × β a) => acc.insert p.fst p.snd) t₁ l)

theorem Std.DTreeMap.ofList_equiv_foldl {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {l : List ((a : α) × β a)} : (ofList l cmp).Equiv (List.foldl (fun (acc : DTreeMap α β cmp) (p : (a : α) × β a) => acc.insert p.fst p.snd) ∅ l)

theorem Std.DTreeMap.insertManyIfNew_list_equiv_foldl {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ : DTreeMap α β cmp} {l : List ((a : α) × β a)} : (t₁.insertManyIfNew l).Equiv (List.foldl (fun (acc : DTreeMap α β cmp) (p : (a : α) × β a) => acc.insertIfNew p.fst p.snd) t₁ l)

theorem Std.DTreeMap.Const.equiv_iff_toList_perm {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} : t₁.Equiv t₂ ↔ (toList t₁).Perm (toList t₂)

theorem Std.DTreeMap.Const.equiv_iff_toArray_perm {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} : t₁.Equiv t₂ ↔ (toArray t₁).Perm (toArray t₂)

theorem Std.DTreeMap.Const.equiv_iff_toList_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] : t₁.Equiv t₂ ↔ toList t₁ = toList t₂

theorem Std.DTreeMap.Const.equiv_iff_toArray_eq {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] : t₁.Equiv t₂ ↔ toArray t₁ = toArray t₂

theorem Std.DTreeMap.Const.equiv_iff_keys_unit_perm {α : Type u} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α (fun (x : α) => Unit) cmp} : t₁.Equiv t₂ ↔ t₁.keys.Perm t₂.keys

theorem Std.DTreeMap.Const.equiv_iff_keysArray_unit_perm {α : Type u} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α (fun (x : α) => Unit) cmp} : t₁.Equiv t₂ ↔ t₁.keysArray.Perm t₂.keysArray

theorem Std.DTreeMap.Const.equiv_iff_keys_unit_eq {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {t₁ t₂ : DTreeMap α (fun (x : α) => Unit) cmp} : t₁.Equiv t₂ ↔ t₁.keys = t₂.keys

theorem Std.DTreeMap.Const.equiv_iff_keysArray_unit_eq {α : Type u} {cmp : α → α → Ordering} [TransCmp cmp] {t₁ t₂ : DTreeMap α (fun (x : α) => Unit) cmp} : t₁.Equiv t₂ ↔ t₁.keysArray = t₂.keysArray

theorem Std.DTreeMap.Equiv.of_constToList_perm {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} : (Const.toList t₁).Perm (Const.toList t₂) → t₁.Equiv t₂

theorem Std.DTreeMap.Equiv.of_constToArray_perm {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ t₂ : DTreeMap α (fun (x : α) => β) cmp} : (Const.toArray t₁).Perm (Const.toArray t₂) → t₁.Equiv t₂

theorem Std.DTreeMap.Equiv.of_keys_unit_perm {α : Type u} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α (fun (x : α) => Unit) cmp} : t₁.keys.Perm t₂.keys → t₁.Equiv t₂

theorem Std.DTreeMap.Equiv.of_keysArray_unit_perm {α : Type u} {cmp : α → α → Ordering} {t₁ t₂ : DTreeMap α (fun (x : α) => Unit) cmp} : t₁.keysArray.Perm t₂.keysArray → t₁.Equiv t₂

theorem Std.DTreeMap.Const.insertMany_list_equiv_foldl {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t₁ : DTreeMap α (fun (x : α) => β) cmp} {l : List (α × β)} : (insertMany t₁ l).Equiv (List.foldl (fun (acc : DTreeMap α (fun (x : α) => β) cmp) (p : α × β) => acc.insert p.fst p.snd) t₁ l)

theorem Std.DTreeMap.Const.ofList_equiv_foldl {α : Type u} {cmp : α → α → Ordering} {β : Type v} {l : List (α × β)} : (ofList l cmp).Equiv (List.foldl (fun (acc : DTreeMap α (fun (x : α) => β) cmp) (p : α × β) => acc.insert p.fst p.snd) ∅ l)

theorem Std.DTreeMap.Const.insertManyIfNewUnit_list_equiv_foldl {α : Type u} {cmp : α → α → Ordering} {t₁ : DTreeMap α (fun (x : α) => Unit) cmp} {l : List α} : (insertManyIfNewUnit t₁ l).Equiv (List.foldl (fun (acc : DTreeMap α (fun (x : α) => Unit) cmp) (a : α) => acc.insertIfNew a ()) t₁ l)

theorem Std.DTreeMap.Const.unitOfList_equiv_foldl {α : Type u} {cmp : α → α → Ordering} {l : List α} : (unitOfList l cmp).Equiv (List.foldl (fun (acc : DTreeMap α (fun (x : α) => Unit) cmp) (a : α) => acc.insertIfNew a ()) ∅ l)

@[simp]

theorem Std.DTreeMap.toList_filterMap {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} {f : (a : α) → β a → Option (γ a)} : (filterMap f t).toList = List.filterMap (fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)) t.toList

@[simp]

theorem Std.DTreeMap.toArray_filterMap {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} {f : (a : α) → β a → Option (γ a)} : (filterMap f t).toArray = Array.filterMap (fun (p : (a : α) × β a) => Option.map (fun (x : γ p.fst) => ⟨p.fst, x⟩) (f p.fst p.snd)) t.toArray

theorem Std.DTreeMap.isEmpty_filterMap_iff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Option (γ a)} : (filterMap f t).isEmpty = true ↔ ∀ (k : α) (h : k ∈ t), f k (t.get k h) = none

theorem Std.DTreeMap.isEmpty_filterMap_eq_false_iff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Option (γ a)} : (filterMap f t).isEmpty = false ↔ ∃ (k : α), ∃ (h : k ∈ t), (f k (t.get k h)).isSome = true

theorem Std.DTreeMap.contains_filterMap {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Option (γ a)} {k : α} : (filterMap f t).contains k = Option.any (fun (x : β k) => (f k x).isSome) (t.get? k)

theorem Std.DTreeMap.mem_filterMap {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Option (γ a)} {k : α} : k ∈ filterMap f t ↔ ∃ (h : k ∈ t), (f k (t.get k h)).isSome = true

theorem Std.DTreeMap.contains_of_contains_filterMap {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] {f : (a : α) → β a → Option (γ a)} {k : α} : (filterMap f t).contains k = true → t.contains k = true

theorem Std.DTreeMap.mem_of_mem_filterMap {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] {f : (a : α) → β a → Option (γ a)} {k : α} : k ∈ filterMap f t → k ∈ t

theorem Std.DTreeMap.size_filterMap_le_size {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] {f : (a : α) → β a → Option (γ a)} : (filterMap f t).size ≤ t.size

theorem Std.DTreeMap.size_filterMap_eq_size_iff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Option (γ a)} : (filterMap f t).size = t.size ↔ ∀ (a : α) (h : a ∈ t), (f a (t.get a h)).isSome = true

@[simp]

theorem Std.DTreeMap.get?_filterMap {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Option (γ a)} {k : α} : (filterMap f t).get? k = (t.get? k).bind (f k)

theorem Std.DTreeMap.isSome_apply_of_mem_filterMap {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Option (γ a)} {k : α} (h' : k ∈ filterMap f t) : (f k (t.get k ⋯)).isSome = true

@[simp]

theorem Std.DTreeMap.get_filterMap {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Option (γ a)} {k : α} {h' : k ∈ filterMap f t} : (filterMap f t).get k h' = (f k (t.get k ⋯)).get ⋯

@[simp]

theorem Std.DTreeMap.get!_filterMap {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Option (γ a)} {k : α} [Inhabited (γ k)] : (filterMap f t).get! k = ((t.get? k).bind (f k)).get!

@[simp]

theorem Std.DTreeMap.getD_filterMap {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Option (γ a)} {k : α} {fallback : γ k} : (filterMap f t).getD k fallback = ((t.get? k).bind (f k)).getD fallback

theorem Std.DTreeMap.getKey?_filterMap {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Option (γ a)} {k : α} : (filterMap f t).getKey? k = (t.getKey? k).pfilter fun (x : α) (h' : t.getKey? k = some x) => (f x (t.get x ⋯)).isSome

@[simp]

theorem Std.DTreeMap.getKey_filterMap {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] {f : (a : α) → β a → Option (γ a)} {k : α} {h' : k ∈ filterMap f t} : (filterMap f t).getKey k h' = t.getKey k ⋯

theorem Std.DTreeMap.getKey!_filterMap {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {f : (a : α) → β a → Option (γ a)} {k : α} : (filterMap f t).getKey! k = ((t.getKey? k).pfilter fun (x : α) (h' : t.getKey? k = some x) => (f x (t.get x ⋯)).isSome).get!

theorem Std.DTreeMap.getKeyD_filterMap {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Option (γ a)} {k fallback : α} : (filterMap f t).getKeyD k fallback = ((t.getKey? k).pfilter fun (x : α) (h' : t.getKey? k = some x) => (f x (t.get x ⋯)).isSome).getD fallback

theorem Std.DTreeMap.Const.isEmpty_filterMap_iff {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Option γ} : (filterMap f t).isEmpty = true ↔ ∀ (k : α) (h : k ∈ t), f (t.getKey k h) (get t k h) = none

theorem Std.DTreeMap.Const.isEmpty_filterMap_eq_false_iff {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Option γ} : (filterMap f t).isEmpty = false ↔ ∃ (k : α), ∃ (h : k ∈ t), (f (t.getKey k h) (get t k h)).isSome = true

theorem Std.DTreeMap.Const.mem_filterMap {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Option γ} {k : α} : k ∈ filterMap f t ↔ ∃ (h : k ∈ t), (f (t.getKey k h) (get t k h)).isSome = true

theorem Std.DTreeMap.Const.size_filterMap_eq_size_iff {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Option γ} : (filterMap f t).size = t.size ↔ ∀ (k : α) (h : k ∈ t), (f (t.getKey k h) (get t k h)).isSome = true

@[simp]

theorem Std.DTreeMap.Const.get?_filterMap {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Option γ} {k : α} : get? (filterMap f t) k = (get? t k).pbind fun (x : β) (h' : get? t k = some x) => f (t.getKey k ⋯) x

theorem Std.DTreeMap.Const.get?_filterMap' {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] {f : α → β → Option γ} {k : α} : get? (filterMap f t) k = (get? t k).bind fun (x : β) => f k x

Simpler variant of get?_filterMap when LawfulEqCmp is available.

theorem Std.DTreeMap.Const.get?_filterMap_of_getKey?_eq_some {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Option γ} {k k' : α} (h : t.getKey? k = some k') : get? (filterMap f t) k = (get? t k).bind (f k')

theorem Std.DTreeMap.Const.isSome_apply_of_mem_filterMap {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Option γ} {k : α} (h : k ∈ filterMap f t) : (f (t.getKey k ⋯) (get t k ⋯)).isSome = true

@[simp]

theorem Std.DTreeMap.Const.get_filterMap {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Option γ} {k : α} {h : k ∈ filterMap f t} : get (filterMap f t) k h = (f (t.getKey k ⋯) (get t k ⋯)).get ⋯

theorem Std.DTreeMap.Const.get_filterMap' {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] {f : α → β → Option γ} {k : α} {h : k ∈ filterMap f t} : get (filterMap f t) k h = (f k (get t k ⋯)).get ⋯

Simpler variant of get_filterMap when LawfulEqCmp is available.

theorem Std.DTreeMap.Const.get!_filterMap {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited γ] {f : α → β → Option γ} {k : α} : get! (filterMap f t) k = ((get? t k).pbind fun (x : β) (h' : get? t k = some x) => f (t.getKey k ⋯) x).get!

theorem Std.DTreeMap.Const.get!_filterMap' {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited γ] {f : α → β → Option γ} {k : α} : get! (filterMap f t) k = ((get? t k).bind (f k)).get!

Simpler variant of get!_filterMap when LawfulEqCmp is available.

theorem Std.DTreeMap.Const.get!_filterMap_of_getKey?_eq_some {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited γ] {f : α → β → Option γ} {k k' : α} (h : t.getKey? k = some k') : get! (filterMap f t) k = ((get? t k).bind (f k')).get!

theorem Std.DTreeMap.Const.getD_filterMap {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Option γ} {k : α} {fallback : γ} : getD (filterMap f t) k fallback = ((get? t k).pbind fun (x : β) (h' : get? t k = some x) => f (t.getKey k ⋯) x).getD fallback

theorem Std.DTreeMap.Const.getD_filterMap' {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] {f : α → β → Option γ} {k : α} {fallback : γ} : getD (filterMap f t) k fallback = ((get? t k).bind (f k)).getD fallback

Simpler variant of getD_filterMap when LawfulEqCmp is available.

theorem Std.DTreeMap.Const.getD_filterMap_of_getKey?_eq_some {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Option γ} {k k' : α} {fallback : γ} (h : t.getKey? k = some k') : getD (filterMap f t) k fallback = ((get? t k).bind (f k')).getD fallback

theorem Std.DTreeMap.Const.toList_filterMap {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} {f : α → β → Option γ} : toList (filterMap (fun (k : α) (v : β) => f k v) t) = List.filterMap (fun (p : α × β) => Option.map (fun (x : γ) => (p.fst, x)) (f p.fst p.snd)) (toList t)

theorem Std.DTreeMap.Const.toArray_filterMap {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} {f : α → β → Option γ} : toArray (filterMap (fun (k : α) (v : β) => f k v) t) = Array.filterMap (fun (p : α × β) => Option.map (fun (x : γ) => (p.fst, x)) (f p.fst p.snd)) (toArray t)

theorem Std.DTreeMap.Const.getKey?_filterMap {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Option γ} {k : α} : (filterMap f t).getKey? k = (t.getKey? k).pfilter fun (x : α) (h' : t.getKey? k = some x) => (f x (get t x ⋯)).isSome

theorem Std.DTreeMap.Const.getKey!_filterMap {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited α] {f : α → β → Option γ} {k : α} : (filterMap f t).getKey! k = ((t.getKey? k).pfilter fun (x : α) (h' : t.getKey? k = some x) => (f x (get t x ⋯)).isSome).get!

theorem Std.DTreeMap.Const.getKeyD_filterMap {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Option γ} {k fallback : α} : (filterMap f t).getKeyD k fallback = ((t.getKey? k).pfilter fun (x : α) (h' : t.getKey? k = some x) => (f x (get t x ⋯)).isSome).getD fallback

theorem Std.DTreeMap.filterMap_equiv_filter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {f : (a : α) → β a → Bool} : (filterMap (fun (k : α) => Option.guard fun (v : β k) => f k v) t).Equiv (filter f t)

@[simp]

theorem Std.DTreeMap.toList_filter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {f : (a : α) → β a → Bool} : (filter f t).toList = List.filter (fun (p : (a : α) × β a) => f p.fst p.snd) t.toList

@[simp]

theorem Std.DTreeMap.toArray_filter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {f : (a : α) → β a → Bool} : (filter f t).toArray = Array.filter (fun (p : (a : α) × β a) => f p.fst p.snd) t.toArray

theorem Std.DTreeMap.keys_filter_key {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {f : α → Bool} : (filter (fun (k : α) (x : β k) => f k) t).keys = List.filter f t.keys

theorem Std.DTreeMap.keysArray_filter_key {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {f : α → Bool} : (filter (fun (k : α) (x : β k) => f k) t).keysArray = Array.filter f t.keysArray

theorem Std.DTreeMap.isEmpty_filter_iff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Bool} : (filter f t).isEmpty = true ↔ ∀ (k : α) (h : k ∈ t), f k (t.get k h) = false

theorem Std.DTreeMap.isEmpty_filter_eq_false_iff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Bool} : (filter f t).isEmpty = false ↔ ∃ (k : α), ∃ (h : k ∈ t), f k (t.get k h) = true

theorem Std.DTreeMap.isEmpty_filter_key_iff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {f : α → Bool} : (filter (fun (a : α) (x : β a) => f a) t).isEmpty = true ↔ ∀ (k : α) (h : k ∈ t), f (t.getKey k h) = false

theorem Std.DTreeMap.isEmpty_filter_key_eq_false_iff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {f : α → Bool} : (filter (fun (a : α) (x : β a) => f a) t).isEmpty = false ↔ ∃ (k : α), ∃ (h : k ∈ t), f (t.getKey k h) = true

theorem Std.DTreeMap.contains_filter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Bool} {k : α} : (filter f t).contains k = Option.any (f k) (t.get? k)

theorem Std.DTreeMap.mem_filter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Bool} {k : α} : k ∈ filter f t ↔ ∃ (h : k ∈ t), f k (t.get k h) = true

theorem Std.DTreeMap.mem_filter_key {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {f : α → Bool} {k : α} : k ∈ filter (fun (a : α) (x : β a) => f a) t ↔ ∃ (h : k ∈ t), f (t.getKey k h) = true

theorem Std.DTreeMap.contains_of_contains_filter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {f : (a : α) → β a → Bool} {k : α} : (filter f t).contains k = true → t.contains k = true

theorem Std.DTreeMap.mem_of_mem_filter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {f : (a : α) → β a → Bool} {k : α} : k ∈ filter f t → k ∈ t

theorem Std.DTreeMap.size_filter_le_size {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {f : (a : α) → β a → Bool} : (filter f t).size ≤ t.size

theorem Std.DTreeMap.size_filter_eq_size_iff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Bool} : (filter f t).size = t.size ↔ ∀ (k : α) (h : k ∈ t), f k (t.get k h) = true

theorem Std.DTreeMap.filter_equiv_self_iff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Bool} : (filter f t).Equiv t ↔ ∀ (k : α) (h : k ∈ t), f k (t.get k h) = true

theorem Std.DTreeMap.filter_key_equiv_self_iff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {f : α → Bool} : (filter (fun (k : α) (x : β k) => f k) t).Equiv t ↔ ∀ (k : α) (h : k ∈ t), f (t.getKey k h) = true

theorem Std.DTreeMap.size_filter_key_eq_size_iff {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {f : α → Bool} : (filter (fun (k : α) (x : β k) => f k) t).size = t.size ↔ ∀ (k : α) (h : k ∈ t), f (t.getKey k h) = true

@[simp]

theorem Std.DTreeMap.get?_filter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Bool} {k : α} : (filter f t).get? k = Option.filter (f k) (t.get? k)

@[simp]

theorem Std.DTreeMap.get_filter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Bool} {k : α} {h' : k ∈ filter f t} : (filter f t).get k h' = t.get k ⋯

theorem Std.DTreeMap.get!_filter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Bool} {k : α} [Inhabited (β k)] : (filter f t).get! k = (Option.filter (f k) (t.get? k)).get!

theorem Std.DTreeMap.getD_filter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Bool} {k : α} {fallback : β k} : (filter f t).getD k fallback = (Option.filter (f k) (t.get? k)).getD fallback

theorem Std.DTreeMap.keys_filter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Bool} : (filter f t).keys = (List.filter (fun (x : { x : α // x ∈ t.keys }) => match x with | ⟨x, h'⟩ => f x (t.get x ⋯)) t.keys.attach).unattach

theorem Std.DTreeMap.getKey?_filter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Bool} {k : α} : (filter f t).getKey? k = (t.getKey? k).pfilter fun (x : α) (h' : t.getKey? k = some x) => f x (t.get x ⋯)

theorem Std.DTreeMap.getKey?_filter_key {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {f : α → Bool} {k : α} : (filter (fun (k : α) (x : β k) => f k) t).getKey? k = Option.filter f (t.getKey? k)

@[simp]

theorem Std.DTreeMap.getKey_filter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {f : (a : α) → β a → Bool} {k : α} {h' : k ∈ filter f t} : (filter f t).getKey k h' = t.getKey k ⋯

theorem Std.DTreeMap.getKey!_filter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {f : (a : α) → β a → Bool} {k : α} : (filter f t).getKey! k = ((t.getKey? k).pfilter fun (x : α) (h' : t.getKey? k = some x) => f x (t.get x ⋯)).get!

theorem Std.DTreeMap.getKey!_filter_key {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [Inhabited α] {f : α → Bool} {k : α} : (filter (fun (k : α) (x : β k) => f k) t).getKey! k = (Option.filter f (t.getKey? k)).get!

theorem Std.DTreeMap.getKeyD_filter {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → Bool} {k fallback : α} : (filter f t).getKeyD k fallback = ((t.getKey? k).pfilter fun (x : α) (h' : t.getKey? k = some x) => f x (t.get x ⋯)).getD fallback

theorem Std.DTreeMap.getKeyD_filter_key {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} [TransCmp cmp] {f : α → Bool} {k fallback : α} : (filter (fun (k : α) (x : β k) => f k) t).getKeyD k fallback = (Option.filter f (t.getKey? k)).getD fallback

theorem Std.DTreeMap.Const.isEmpty_filter_iff {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Bool} : (filter f t).isEmpty = true ↔ ∀ (k : α) (h : k ∈ t), f (t.getKey k h) (get t k h) = false

theorem Std.DTreeMap.Const.isEmpty_filter_eq_false_iff {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Bool} : (filter f t).isEmpty = false ↔ ∃ (k : α), ∃ (h : k ∈ t), f (t.getKey k h) (get t k h) = true

theorem Std.DTreeMap.Const.mem_filter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Bool} {k : α} : k ∈ filter f t ↔ ∃ (h' : k ∈ t), f (t.getKey k h') (get t k h') = true

theorem Std.DTreeMap.Const.size_filter_le_size {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Bool} : (filter f t).size ≤ t.size

theorem Std.DTreeMap.Const.size_filter_eq_size_iff {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Bool} : (filter f t).size = t.size ↔ ∀ (a : α) (h : a ∈ t), f (t.getKey a h) (get t a h) = true

theorem Std.DTreeMap.Const.filter_equiv_self_iff {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Bool} : (filter f t).Equiv t ↔ ∀ (k : α) (h : k ∈ t), f (t.getKey k h) (get t k h) = true

theorem Std.DTreeMap.Const.get?_filter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Bool} {k : α} : get? (filter f t) k = (get? t k).pfilter fun (x : β) (h' : get? t k = some x) => f (t.getKey k ⋯) x

@[simp]

theorem Std.DTreeMap.Const.get?_filter' {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] {f : α → β → Bool} {k : α} : get? (filter f t) k = Option.filter (f k) (get? t k)

Simpler variant of get?_filter when LawfulEqCmp is available.

theorem Std.DTreeMap.Const.get?_filter_of_getKey?_eq_some {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Bool} {k k' : α} : t.getKey? k = some k' → get? (filter f t) k = Option.filter (fun (x : β) => f k' x) (get? t k)

@[simp]

theorem Std.DTreeMap.Const.get_filter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Bool} {k : α} {h' : k ∈ filter f t} : get (filter f t) k h' = get t k ⋯

theorem Std.DTreeMap.Const.get!_filter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {f : α → β → Bool} {k : α} : get! (filter f t) k = ((get? t k).pfilter fun (x : β) (h' : get? t k = some x) => f (t.getKey k ⋯) x).get!

theorem Std.DTreeMap.Const.get!_filter' {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited β] {f : α → β → Bool} {k : α} : get! (filter f t) k = (Option.filter (f k) (get? t k)).get!

Simpler variant of get!_filter when LawfulEqCmp is available.

theorem Std.DTreeMap.Const.get!_filter_of_getKey?_eq_some {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited β] {f : α → β → Bool} {k k' : α} : t.getKey? k = some k' → get! (filter f t) k = (Option.filter (fun (x : β) => f k' x) (get? t k)).get!

theorem Std.DTreeMap.Const.getD_filter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Bool} {k : α} {fallback : β} : getD (filter f t) k fallback = ((get? t k).pfilter fun (x : β) (h' : get? t k = some x) => f (t.getKey k ⋯) x).getD fallback

theorem Std.DTreeMap.Const.getD_filter' {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] {f : α → β → Bool} {k : α} {fallback : β} : getD (filter f t) k fallback = (Option.filter (f k) (get? t k)).getD fallback

Simpler variant of getD_filter when LawfulEqCmp is available.

theorem Std.DTreeMap.Const.getD_filter_of_getKey?_eq_some {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Bool} {k k' : α} {fallback : β} : t.getKey? k = some k' → getD (filter f t) k fallback = (Option.filter (fun (x : β) => f k' x) (get? t k)).getD fallback

@[simp]

theorem Std.DTreeMap.Const.toList_filter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {f : α → β → Bool} : toList (filter f t) = List.filter (fun (p : α × β) => f p.fst p.snd) (toList t)

@[simp]

theorem Std.DTreeMap.Const.toArray_filter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} {f : α → β → Bool} : toArray (filter f t) = Array.filter (fun (p : α × β) => f p.fst p.snd) (toArray t)

theorem Std.DTreeMap.Const.keys_filter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Bool} : (filter f t).keys = (List.filter (fun (x : { x : α // x ∈ t.keys }) => match x with | ⟨x, h'⟩ => f x (get t x ⋯)) t.keys.attach).unattach

theorem Std.DTreeMap.Const.getKey?_filter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Bool} {k : α} : (filter f t).getKey? k = (t.getKey? k).pfilter fun (x : α) (h' : t.getKey? k = some x) => f x (get t x ⋯)

theorem Std.DTreeMap.Const.getKey!_filter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited α] {f : α → β → Bool} {k : α} : (filter f t).getKey! k = ((t.getKey? k).pfilter fun (x : α) (h' : t.getKey? k = some x) => f x (get t x ⋯)).get!

theorem Std.DTreeMap.Const.getKeyD_filter {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → Bool} {k fallback : α} : (filter f t).getKeyD k fallback = ((t.getKey? k).pfilter fun (x : α) (h' : t.getKey? k = some x) => f x (get t x ⋯)).getD fallback

theorem Std.DTreeMap.map_id_equiv {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} : (map (fun (x : α) (v : β x) => v) t).Equiv t

theorem Std.DTreeMap.map_map_equiv {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} {δ : α → Type w'} {f : (a : α) → β a → γ a} {g : (a : α) → γ a → δ a} : (map g (map f t)).Equiv (map (fun (k : α) (v : β k) => g k (f k v)) t)

@[simp]

theorem Std.DTreeMap.toList_map {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} {f : (a : α) → β a → γ a} : (map f t).toList = List.map (fun (p : (a : α) × β a) => ⟨p.fst, f p.fst p.snd⟩) t.toList

@[simp]

theorem Std.DTreeMap.toArray_map {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} {f : (a : α) → β a → γ a} : (map f t).toArray = Array.map (fun (p : (a : α) × β a) => ⟨p.fst, f p.fst p.snd⟩) t.toArray

@[simp]

theorem Std.DTreeMap.keys_map {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} {f : (a : α) → β a → γ a} : (map f t).keys = t.keys

@[simp]

theorem Std.DTreeMap.keysArray_map {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} {f : (a : α) → β a → γ a} : (map f t).keysArray = t.keysArray

theorem Std.DTreeMap.filterMap_equiv_map {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] {f : (a : α) → β a → γ a} : (filterMap (fun (k : α) (v : β k) => some (f k v)) t).Equiv (map f t)

@[simp]

theorem Std.DTreeMap.isEmpty_map {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] {f : (a : α) → β a → γ a} : (map f t).isEmpty = t.isEmpty

theorem Std.DTreeMap.contains_map {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] {f : (a : α) → β a → γ a} {k : α} : (map f t).contains k = t.contains k

theorem Std.DTreeMap.contains_of_contains_map {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] {f : (a : α) → β a → γ a} {k : α} : (map f t).contains k = true → t.contains k = true

@[simp]

theorem Std.DTreeMap.mem_map {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] {f : (a : α) → β a → γ a} {k : α} : k ∈ map f t ↔ k ∈ t

theorem Std.DTreeMap.mem_of_mem_map {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] {f : (a : α) → β a → γ a} {k : α} : k ∈ map f t → k ∈ t

@[simp]

theorem Std.DTreeMap.size_map {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] {f : (a : α) → β a → γ a} : (map f t).size = t.size

@[simp]

theorem Std.DTreeMap.get?_map {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → γ a} {k : α} : (map f t).get? k = Option.map (f k) (t.get? k)

@[simp]

theorem Std.DTreeMap.get_map {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → γ a} {k : α} {h' : k ∈ map f t} : (map f t).get k h' = f k (t.get k ⋯)

theorem Std.DTreeMap.get!_map {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → γ a} {k : α} [Inhabited (γ k)] : (map f t).get! k = (Option.map (f k) (t.get? k)).get!

theorem Std.DTreeMap.getD_map {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] [LawfulEqCmp cmp] {f : (a : α) → β a → γ a} {k : α} {fallback : γ k} : (map f t).getD k fallback = (Option.map (f k) (t.get? k)).getD fallback

@[simp]

theorem Std.DTreeMap.getKey?_map {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] {f : (a : α) → β a → γ a} {k : α} : (map f t).getKey? k = t.getKey? k

@[simp]

theorem Std.DTreeMap.getKey_map {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] {f : (a : α) → β a → γ a} {k : α} {h' : k ∈ map f t} : (map f t).getKey k h' = t.getKey k ⋯

@[simp]

theorem Std.DTreeMap.getKey!_map {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] [Inhabited α] {f : (a : α) → β a → γ a} {k : α} : (map f t).getKey! k = t.getKey! k

@[simp]

theorem Std.DTreeMap.getKeyD_map {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} {γ : α → Type w} [TransCmp cmp] {f : (a : α) → β a → γ a} {k fallback : α} : (map f t).getKeyD k fallback = t.getKeyD k fallback

@[simp]

theorem Std.DTreeMap.Const.get?_map {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] {f : α → β → γ} {k : α} : get? (map f t) k = Option.map (f k) (get? t k)

@[simp]

theorem Std.DTreeMap.Const.get?_map' {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → γ} {k : α} : get? (map f t) k = Option.pmap (fun (v : β) (h' : k ∈ t) => f (t.getKey k h') v) (get? t k) ⋯

Variant of get?_map that holds without LawfulEqCmp.

theorem Std.DTreeMap.Const.get?_map_of_getKey?_eq_some {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → γ} {k k' : α} (h : t.getKey? k = some k') : get? (map f t) k = Option.map (f k') (get? t k)

@[simp]

theorem Std.DTreeMap.Const.get_map {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] {f : α → β → γ} {k : α} {h' : k ∈ map f t} : get (map f t) k h' = f k (get t k ⋯)

@[simp]

theorem Std.DTreeMap.Const.get_map' {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → γ} {k : α} {h' : k ∈ map f t} : get (map f t) k h' = f (t.getKey k ⋯) (get t k ⋯)

Variant of get_map that holds without LawfulEqCmp.

theorem Std.DTreeMap.Const.get!_map {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited γ] {f : α → β → γ} {k : α} : get! (map f t) k = (Option.map (f k) (get? t k)).get!

theorem Std.DTreeMap.Const.get!_map' {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited γ] {f : α → β → γ} {k : α} : get! (map f t) k = (Option.pmap (fun (v : β) (h : k ∈ t) => f (t.getKey k h) v) (get? t k) ⋯).get!

Variant of get!_map that holds without LawfulEqCmp.

theorem Std.DTreeMap.Const.get!_map_of_getKey?_eq_some {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited γ] {f : α → β → γ} {k k' : α} (h : t.getKey? k = some k') : get! (map f t) k = (Option.map (f k') (get? t k)).get!

theorem Std.DTreeMap.Const.getD_map {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [LawfulEqCmp cmp] {f : α → β → γ} {k : α} {fallback : γ} : getD (map f t) k fallback = (Option.map (f k) (get? t k)).getD fallback

theorem Std.DTreeMap.Const.getD_map' {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] {f : α → β → γ} {k : α} {fallback : γ} : getD (map f t) k fallback = (Option.pmap (fun (v : β) (h : k ∈ t) => f (t.getKey k h) v) (get? t k) ⋯).getD fallback

Variant of getD_map that holds without LawfulEqCmp.

theorem Std.DTreeMap.Const.getD_map_of_getKey?_eq_some {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} [TransCmp cmp] [Inhabited γ] {f : α → β → γ} {k k' : α} {fallback : γ} (h : t.getKey? k = some k') : getD (map f t) k fallback = (Option.map (f k') (get? t k)).getD fallback

@[simp]

theorem Std.DTreeMap.Const.toList_map {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} {f : α → β → γ} : toList (map f t) = List.map (fun (p : α × β) => (p.fst, f p.fst p.snd)) (toList t)

@[simp]

theorem Std.DTreeMap.Const.toArray_map {α : Type u} {cmp : α → α → Ordering} {β : Type v} {γ : Type w} {t : DTreeMap α (fun (x : α) => β) cmp} {f : α → β → γ} : toArray (map f t) = Array.map (fun (p : α × β) => (p.fst, f p.fst p.snd)) (toArray t)

theorem Std.DTreeMap.Const.toArray_toList {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} : (toList t).toArray = toArray t

theorem Std.DTreeMap.Const.toList_toArray {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : DTreeMap α (fun (x : α) => β) cmp} : (toArray t).toList = toList t

theorem Std.DTreeMap.toArray_toList {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} : t.toList.toArray = t.toArray

theorem Std.DTreeMap.toList_toArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} : t.toArray.toList = t.toList

theorem Std.DTreeMap.toArray_keys {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} : t.keys.toArray = t.keysArray

theorem Std.DTreeMap.toList_keysArray {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : DTreeMap α β cmp} : t.keysArray.toList = t.keys