Additional lemmas for Red-black trees

def Std.RBNode.depth {α : Type u_1} : Std.RBNode α → Nat

O(n). depth t is the maximum number of nodes on any path to a leaf. It is an upper bound on most tree operations.

Equations

• Std.RBNode.nil.depth = 0
• (Std.RBNode.node c a v b).depth = max a.depth b.depth + 1

theorem Std.RBNode.size_lt_depth {α : Type u_1} (t : Std.RBNode α) : t.size < 2 ^ t.depth

def Std.RBNode.depthLB : Std.RBColor → Nat → Nat

depthLB c n is the best upper bound on the depth of any balanced red-black tree with root colored c and black-height n.

Equations

• Std.RBNode.depthLB x✝ x = match x✝, x with | Std.RBColor.red, n => n + 1 | Std.RBColor.black, n => n

theorem Std.RBNode.depthLB_le (c : Std.RBColor) (n : Nat) : n ≤ Std.RBNode.depthLB c n

def Std.RBNode.depthUB : Std.RBColor → Nat → Nat

depthUB c n is the best upper bound on the depth of any balanced red-black tree with root colored c and black-height n.

Equations

• Std.RBNode.depthUB x✝ x = match x✝, x with | Std.RBColor.red, n => 2 * n + 1 | Std.RBColor.black, n => 2 * n

theorem Std.RBNode.depthUB_le (c : Std.RBColor) (n : Nat) : Std.RBNode.depthUB c n ≤ 2 * n + 1

theorem Std.RBNode.depthUB_le_two_depthLB (c : Std.RBColor) (n : Nat) : Std.RBNode.depthUB c n ≤ 2 * Std.RBNode.depthLB c n

theorem Std.RBNode.Balanced.depth_le {α : Type u_1} {t : Std.RBNode α} {c : Std.RBColor} {n : Nat} : t.Balanced c n → t.depth ≤ Std.RBNode.depthUB c n

theorem Std.RBNode.Balanced.le_size {α : Type u_1} {t : Std.RBNode α} {c : Std.RBColor} {n : Nat} : t.Balanced c n → 2 ^ Std.RBNode.depthLB c n ≤ t.size + 1

theorem Std.RBNode.Balanced.depth_bound {α : Type u_1} {t : Std.RBNode α} {c : Std.RBColor} {n : Nat} (h : t.Balanced c n) : t.depth ≤ 2 * (t.size + 1).log2

theorem Std.RBNode.WF.depth_bound {α : Type u_1} {cmp : α → α → Ordering} {t : Std.RBNode α} (h : Std.RBNode.WF cmp t) : t.depth ≤ 2 * (t.size + 1).log2

A well formed tree has t.depth ∈ O(log t.size), that is, it is well balanced. This justifies the O(log n) bounds on most searching operations of RBSet.

@[simp]

theorem Std.RBNode.min?_reverse {α : Type u_1} (t : Std.RBNode α) : t.reverse.min? = t.max?

@[simp]

theorem Std.RBNode.max?_reverse {α : Type u_1} (t : Std.RBNode α) : t.reverse.max? = t.min?

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theorem Std.RBNode.mem_nil {α : Type u_1} {x : α} : ¬x ∈ Std.RBNode.nil

@[simp]

theorem Std.RBNode.mem_node {α : Type u_1} {y : α} {c : Std.RBColor} {a : Std.RBNode α} {x : α} {b : Std.RBNode α} : y ∈ Std.RBNode.node c a x b ↔ y = x ∨ y ∈ a ∨ y ∈ b

theorem Std.RBNode.All_def {α : Type u_1} {p : α → Prop} {t : Std.RBNode α} : Std.RBNode.All p t ↔ ∀ (x : α), x ∈ t → p x

theorem Std.RBNode.Any_def {α : Type u_1} {p : α → Prop} {t : Std.RBNode α} : Std.RBNode.Any p t ↔ ∃ (x : α), x ∈ t ∧ p x

theorem Std.RBNode.memP_def : ∀ {α : Type u_1} {cut : α → Ordering} {t : Std.RBNode α}, Std.RBNode.MemP cut t ↔ ∃ (x : α), x ∈ t ∧ cut x = Ordering.eq

theorem Std.RBNode.mem_def : ∀ {α : Type u_1} {cmp : α → α → Ordering} {x : α} {t : Std.RBNode α}, Std.RBNode.Mem cmp x t ↔ ∃ (y : α), y ∈ t ∧ cmp x y = Ordering.eq

theorem Std.RBNode.mem_congr {α : Type u_1} {cmp : α → α → Ordering} {x : α} {y : α} [Std.TransCmp cmp] {t : Std.RBNode α} (h : cmp x y = Ordering.eq) : Std.RBNode.Mem cmp x t ↔ Std.RBNode.Mem cmp y t

theorem Std.RBNode.isOrdered_iff' {α : Type u_1} {cmp : α → α → Ordering} {L : Option α} {R : Option α} [Std.TransCmp cmp] {t : Std.RBNode α} : Std.RBNode.isOrdered cmp t L R = true ↔ (∀ (a : α), a ∈ L → Std.RBNode.All (fun (x : α) => Std.RBNode.cmpLT cmp a x) t) ∧ (∀ (a : α), a ∈ R → Std.RBNode.All (fun (x : α) => Std.RBNode.cmpLT cmp x a) t) ∧ (∀ (a : α), a ∈ L → ∀ (b : α), b ∈ R → Std.RBNode.cmpLT cmp a b) ∧ Std.RBNode.Ordered cmp t

theorem Std.RBNode.isOrdered_iff {α : Type u_1} {cmp : α → α → Ordering} [Std.TransCmp cmp] {t : Std.RBNode α} : Std.RBNode.isOrdered cmp t none = true ↔ Std.RBNode.Ordered cmp t

instance Std.RBNode.instDecidableOrderedOfTransCmp {α : Type u_1} (cmp : α → α → Ordering) [Std.TransCmp cmp] (t : Std.RBNode α) : Decidable (Std.RBNode.Ordered cmp t)

Equations

• Std.RBNode.instDecidableOrderedOfTransCmp cmp t = decidable_of_iff (Std.RBNode.isOrdered cmp t none = true) ⋯

class Std.RBNode.IsCut {α : Sort u_1} (cmp : α → α → Ordering) (cut : α → Ordering) : Prop

A cut is like a homomorphism of orderings: it is a monotonic predicate with respect to cmp, but it can make things that are distinguished by cmp equal. This is sufficient for find? to locate an element on which cut returns .eq, but there may be other elements, not returned by find?, on which cut also returns .eq.

• le_lt_trans : ∀ {x y : α} [inst : Std.TransCmp cmp], cmp x y ≠ Ordering.gt → cut x = Ordering.lt → cut y = Ordering.lt

  The set {x | cut x = .lt} is downward-closed.

• le_gt_trans : ∀ {x y : α} [inst : Std.TransCmp cmp], cmp x y ≠ Ordering.gt → cut y = Ordering.gt → cut x = Ordering.gt

  The set {x | cut x = .gt} is upward-closed.

theorem Std.RBNode.IsCut.le_lt_trans {α : Sort u_1} {cmp : α → α → Ordering} {cut : α → Ordering} [self : Std.RBNode.IsCut cmp cut] {x : α} {y : α} [Std.TransCmp cmp] : cmp x y ≠ Ordering.gt → cut x = Ordering.lt → cut y = Ordering.lt

The set {x | cut x = .lt} is downward-closed.

theorem Std.RBNode.IsCut.le_gt_trans {α : Sort u_1} {cmp : α → α → Ordering} {cut : α → Ordering} [self : Std.RBNode.IsCut cmp cut] {x : α} {y : α} [Std.TransCmp cmp] : cmp x y ≠ Ordering.gt → cut y = Ordering.gt → cut x = Ordering.gt

The set {x | cut x = .gt} is upward-closed.

theorem Std.RBNode.IsCut.lt_trans : ∀ {α : Sort u_1} {cmp : α → α → Ordering} {cut : α → Ordering} {x y : α} [inst : Std.RBNode.IsCut cmp cut] [inst : Std.TransCmp cmp], cmp x y = Ordering.lt → cut x = Ordering.lt → cut y = Ordering.lt

theorem Std.RBNode.IsCut.gt_trans : ∀ {α : Sort u_1} {cmp : α → α → Ordering} {cut : α → Ordering} {x y : α} [inst : Std.RBNode.IsCut cmp cut] [inst : Std.TransCmp cmp], cmp x y = Ordering.lt → cut y = Ordering.gt → cut x = Ordering.gt

theorem Std.RBNode.IsCut.congr : ∀ {α : Sort u_1} {cmp : α → α → Ordering} {cut : α → Ordering} {x y : α} [inst : Std.RBNode.IsCut cmp cut] [inst : Std.TransCmp cmp], cmp x y = Ordering.eq → cut x = cut y

class Std.RBNode.IsStrictCut {α : Sort u_1} (cmp : α → α → Ordering) (cut : α → Ordering) extends Std.RBNode.IsCut : Prop

IsStrictCut upgrades the IsCut property to ensure that at most one element of the tree can match the cut, and hence find? will return the unique such element if one exists.

• le_lt_trans : ∀ {x y : α} [inst : Std.TransCmp cmp], cmp x y ≠ Ordering.gt → cut x = Ordering.lt → cut y = Ordering.lt
• le_gt_trans : ∀ {x y : α} [inst : Std.TransCmp cmp], cmp x y ≠ Ordering.gt → cut y = Ordering.gt → cut x = Ordering.gt
• exact : ∀ {x y : α} [inst : Std.TransCmp cmp], cut x = Ordering.eq → cmp x y = cut y

  If cut = x, then cut and x have compare the same with respect to other elements.

theorem Std.RBNode.IsStrictCut.exact {α : Sort u_1} {cmp : α → α → Ordering} {cut : α → Ordering} [self : Std.RBNode.IsStrictCut cmp cut] {x : α} {y : α} [Std.TransCmp cmp] : cut x = Ordering.eq → cmp x y = cut y

If cut = x, then cut and x have compare the same with respect to other elements.

instance Std.RBNode.instIsStrictCut {α : Sort u_1} (cmp : α → α → Ordering) (a : α) : Std.RBNode.IsStrictCut cmp (cmp a)

A "representable cut" is one generated by cmp a for some a. This is always a valid cut.

Equations

• ⋯ = ⋯

theorem Std.RBNode.find?_some_eq_eq {α : Type u_1} {x : α} {cut : α → Ordering} {t : Std.RBNode α} : x ∈ Std.RBNode.find? cut t → cut x = Ordering.eq

theorem Std.RBNode.find?_some_mem {α : Type u_1} {x : α} {cut : α → Ordering} {t : Std.RBNode α} : x ∈ Std.RBNode.find? cut t → x ∈ t

theorem Std.RBNode.find?_some_memP {α : Type u_1} {x : α} {cut : α → Ordering} {t : Std.RBNode α} (h : x ∈ Std.RBNode.find? cut t) : Std.RBNode.MemP cut t

theorem Std.RBNode.Ordered.memP_iff_find? {α : Type u_1} {cmp : α → α → Ordering} {cut : α → Ordering} {t : Std.RBNode α} [Std.TransCmp cmp] [Std.RBNode.IsCut cmp cut] (ht : Std.RBNode.Ordered cmp t) : Std.RBNode.MemP cut t ↔ ∃ (x : α), Std.RBNode.find? cut t = some x

theorem Std.RBNode.Ordered.unique {α : Type u_1} {cmp : α → α → Ordering} {t : Std.RBNode α} {x : α} {y : α} [Std.TransCmp cmp] (ht : Std.RBNode.Ordered cmp t) (hx : x ∈ t) (hy : y ∈ t) (e : cmp x y = Ordering.eq) : x = y

theorem Std.RBNode.Ordered.find?_some {α : Type u_1} {cmp : α → α → Ordering} {cut : α → Ordering} {t : Std.RBNode α} {x : α} [Std.TransCmp cmp] [Std.RBNode.IsStrictCut cmp cut] (ht : Std.RBNode.Ordered cmp t) : Std.RBNode.find? cut t = some x ↔ x ∈ t ∧ cut x = Ordering.eq

theorem Std.RBNode.lowerBound?_le' {α : Type u_1} {lb : Option α} {cut : α → Ordering} {x : α} {t : Std.RBNode α} (H : ∀ {x : α}, x ∈ lb → cut x ≠ Ordering.lt) : Std.RBNode.lowerBound? cut t lb = some x → cut x ≠ Ordering.lt

The value x returned by lowerBound? is less or equal to the cut.

theorem Std.RBNode.lowerBound?_le {α : Type u_1} {cut : α → Ordering} {x : α} {t : Std.RBNode α} : Std.RBNode.lowerBound? cut t none = some x → cut x ≠ Ordering.lt

The value x returned by lowerBound? is less or equal to the cut.

theorem Std.RBNode.All.lowerBound?_lb {α : Type u_1} {p : α → Prop} {lb : Option α} {cut : α → Ordering} {x : α} {t : Std.RBNode α} (hp : Std.RBNode.All p t) (H : ∀ {x : α}, x ∈ lb → p x) : Std.RBNode.lowerBound? cut t lb = some x → p x

theorem Std.RBNode.All.lowerBound? {α : Type u_1} {p : α → Prop} {cut : α → Ordering} {x : α} {t : Std.RBNode α} (hp : Std.RBNode.All p t) : Std.RBNode.lowerBound? cut t none = some x → p x

theorem Std.RBNode.lowerBound?_mem_lb {α : Type u_1} {cut : α → Ordering} {lb : Option α} {x : α} {t : Std.RBNode α} (h : Std.RBNode.lowerBound? cut t lb = some x) : x ∈ t ∨ x ∈ lb

theorem Std.RBNode.lowerBound?_mem {α : Type u_1} {cut : α → Ordering} {x : α} {t : Std.RBNode α} (h : Std.RBNode.lowerBound? cut t none = some x) : x ∈ t

theorem Std.RBNode.lowerBound?_of_some {α : Type u_1} {cut : α → Ordering} {y : α} {t : Std.RBNode α} : ∃ (x : α), Std.RBNode.lowerBound? cut t (some y) = some x

theorem Std.RBNode.Ordered.lowerBound?_exists {α : Type u_1} {cmp : α → α → Ordering} {cut : α → Ordering} {t : Std.RBNode α} [Std.TransCmp cmp] [Std.RBNode.IsCut cmp cut] (h : Std.RBNode.Ordered cmp t) : (∃ (x : α), Std.RBNode.lowerBound? cut t none = some x) ↔ ∃ (x : α), x ∈ t ∧ cut x ≠ Ordering.lt

theorem Std.RBNode.Ordered.lowerBound?_least_lb {α : Type u_1} {cmp : α → α → Ordering} {cut : α → Ordering} {t : Std.RBNode α} {lb : Option α} {x : α} {y : α} [Std.TransCmp cmp] [Std.RBNode.IsCut cmp cut] (h : Std.RBNode.Ordered cmp t) (hlb : ∀ {x : α}, lb = some x → Std.RBNode.All (fun (x_1 : α) => Std.RBNode.cmpLT cmp x x_1) t) : Std.RBNode.lowerBound? cut t lb = some x → y ∈ t → cut x = Ordering.gt → cmp x y = Ordering.lt → cut y = Ordering.lt

theorem Std.RBNode.Ordered.lowerBound?_least {α : Type u_1} {cmp : α → α → Ordering} {cut : α → Ordering} {t : Std.RBNode α} {x : α} {y : α} [Std.TransCmp cmp] [Std.RBNode.IsCut cmp cut] (ht : Std.RBNode.Ordered cmp t) (H : Std.RBNode.lowerBound? cut t none = some x) (hy : y ∈ t) (xy : cmp x y = Ordering.lt) (hx : cut x = Ordering.gt) : cut y = Ordering.lt

A statement of the least-ness of the result of lowerBound?. If x is the return value of lowerBound? and it is strictly less than the cut, then any other y > x in the tree is in fact strictly greater than the cut (so there is no exact match, and nothing closer to the cut).

theorem Std.RBNode.Ordered.memP_iff_lowerBound? {α : Type u_1} {cmp : α → α → Ordering} {cut : α → Ordering} {t : Std.RBNode α} [Std.TransCmp cmp] [Std.RBNode.IsCut cmp cut] (ht : Std.RBNode.Ordered cmp t) : Std.RBNode.MemP cut t ↔ ∃ (x : α), Std.RBNode.lowerBound? cut t none = some x ∧ cut x = Ordering.eq

theorem Std.RBNode.Ordered.lowerBound?_lt {α : Type u_1} {cmp : α → α → Ordering} {cut : α → Ordering} {t : Std.RBNode α} {x : α} {y : α} [Std.TransCmp cmp] [Std.RBNode.IsStrictCut cmp cut] (ht : Std.RBNode.Ordered cmp t) (H : Std.RBNode.lowerBound? cut t none = some x) (hy : y ∈ t) : cmp x y = Ordering.lt ↔ cut y = Ordering.lt

A stronger version of lowerBound?_least that holds when the cut is strict.

theorem Std.RBNode.foldr_cons {α : Type u_1} (t : Std.RBNode α) (l : List α) : Std.RBNode.foldr (fun (x : α) (x_1 : List α) => x :: x_1) t l = t.toList ++ l

@[simp]

theorem Std.RBNode.toList_nil {α : Type u_1} : Std.RBNode.nil.toList = []

@[simp]

theorem Std.RBNode.toList_node {α : Type u_1} {c : Std.RBColor} {a : Std.RBNode α} {x : α} {b : Std.RBNode α} : (Std.RBNode.node c a x b).toList = a.toList ++ x :: b.toList

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theorem Std.RBNode.toList_reverse {α : Type u_1} (t : Std.RBNode α) : t.reverse.toList = t.toList.reverse

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theorem Std.RBNode.mem_toList {α : Type u_1} {x : α} {t : Std.RBNode α} : x ∈ t.toList ↔ x ∈ t

@[simp]

theorem Std.RBNode.mem_reverse {α : Type u_1} {a : α} {t : Std.RBNode α} : a ∈ t.reverse ↔ a ∈ t

theorem Std.RBNode.min?_eq_toList_head? {α : Type u_1} {t : Std.RBNode α} : t.min? = t.toList.head?

theorem Std.RBNode.max?_eq_toList_getLast? {α : Type u_1} {t : Std.RBNode α} : t.max? = t.toList.getLast?

theorem Std.RBNode.foldr_eq_foldr_toList {α : Type u_1} : ∀ {α_1 : Type u_2} {f : α → α_1 → α_1} {init : α_1} {t : Std.RBNode α}, Std.RBNode.foldr f t init = List.foldr f init t.toList

theorem Std.RBNode.foldl_eq_foldl_toList {α : Type u_1} : ∀ {α_1 : Type u_2} {f : α_1 → α → α_1} {init : α_1} {t : Std.RBNode α}, Std.RBNode.foldl f init t = List.foldl f init t.toList

theorem Std.RBNode.foldl_reverse {α : Type u_1} {β : Type u_2} {t : Std.RBNode α} {f : β → α → β} {init : β} : Std.RBNode.foldl f init t.reverse = Std.RBNode.foldr (flip f) t init

theorem Std.RBNode.foldr_reverse {α : Type u_1} {β : Type u_2} {t : Std.RBNode α} {f : α → β → β} {init : β} : Std.RBNode.foldr f t.reverse init = Std.RBNode.foldl (flip f) init t

theorem Std.RBNode.forM_eq_forM_toList {m : Type u_1 → Type u_2} {α : Type u_3} {f : α → m PUnit} [Monad m] [LawfulMonad m] {t : Std.RBNode α} : Std.RBNode.forM f t = t.toList.forM f

theorem Std.RBNode.foldlM_eq_foldlM_toList {m : Type u_1 → Type u_2} {α : Type u_3} : ∀ {a : Type u_1} {f : a → α → m a} {init : a} [inst : Monad m] [inst_1 : LawfulMonad m] {t : Std.RBNode α}, Std.RBNode.foldlM f init t = List.foldlM f init t.toList

theorem Std.RBNode.forIn_visit_eq_bindList {m : Type u_1 → Type u_2} {α : Type u_3} : ∀ {α_1 : Type u_1} {f : α → α_1 → m (ForInStep α_1)} {init : α_1} [inst : Monad m] [inst_1 : LawfulMonad m] {t : Std.RBNode α}, Std.RBNode.forIn.visit f t init = ForInStep.bindList f t.toList (ForInStep.yield init)

theorem Std.RBNode.forIn_eq_forIn_toList {m : Type u_1 → Type u_2} {α : Type u_3} : ∀ {α_1 : Type u_1} {init : α_1} {f : α → α_1 → m (ForInStep α_1)} [inst : Monad m] [inst_1 : LawfulMonad m] {t : Std.RBNode α}, forIn t init f = forIn t.toList init f

theorem Std.RBNode.Stream.foldr_cons {α : Type u_1} (t : Std.RBNode.Stream α) (l : List α) : Std.RBNode.Stream.foldr (fun (x : α) (x_1 : List α) => x :: x_1) t l = t.toList ++ l

@[simp]

theorem Std.RBNode.Stream.toList_nil {α : Type u_1} : Std.RBNode.Stream.nil.toList = []

@[simp]

theorem Std.RBNode.Stream.toList_cons {α : Type u_1} {x : α} {r : Std.RBNode α} {s : Std.RBNode.Stream α} : (Std.RBNode.Stream.cons x r s).toList = x :: r.toList ++ s.toList

theorem Std.RBNode.Stream.foldr_eq_foldr_toList {α : Type u_1} : ∀ {α_1 : Type u_2} {f : α → α_1 → α_1} {init : α_1} {s : Std.RBNode.Stream α}, Std.RBNode.Stream.foldr f s init = List.foldr f init s.toList

theorem Std.RBNode.Stream.foldl_eq_foldl_toList {α : Type u_1} : ∀ {α_1 : Type u_2} {f : α_1 → α → α_1} {init : α_1} {t : Std.RBNode.Stream α}, Std.RBNode.Stream.foldl f init t = List.foldl f init t.toList

theorem Std.RBNode.Stream.forIn_eq_forIn_toList {m : Type u_1 → Type u_2} {α : Type u_3} : ∀ {α_1 : Type u_1} {init : α_1} {f : α → α_1 → m (ForInStep α_1)} [inst : Monad m] [inst_1 : LawfulMonad m] {t : Std.RBNode α}, forIn t init f = forIn t.toList init f

theorem Std.RBNode.toStream_toList' {α : Type u_1} {t : Std.RBNode α} {s : Std.RBNode.Stream α} : (t.toStream s).toList = t.toList ++ s.toList

@[simp]

theorem Std.RBNode.toStream_toList {α : Type u_1} {t : Std.RBNode α} : t.toStream.toList = t.toList

theorem Std.RBNode.Stream.next?_toList {α : Type u_1} {s : Std.RBNode.Stream α} : Option.map (fun (x : α × Std.RBNode.Stream α) => match x with | (a, b) => (a, b.toList)) s.next? = s.toList.next?

theorem Std.RBNode.ordered_iff {α : Type u_1} {cmp : α → α → Ordering} {t : Std.RBNode α} : Std.RBNode.Ordered cmp t ↔ List.Pairwise (Std.RBNode.cmpLT cmp) t.toList

theorem Std.RBNode.Ordered.toList_sorted {α : Type u_1} {cmp : α → α → Ordering} {t : Std.RBNode α} : Std.RBNode.Ordered cmp t → List.Pairwise (Std.RBNode.cmpLT cmp) t.toList

theorem Std.RBNode.min?_mem {α : Type u_1} {a : α} {t : Std.RBNode α} (h : t.min? = some a) : a ∈ t

theorem Std.RBNode.Ordered.min?_le {α : Type u_1} {cmp : α → α → Ordering} {a : α} {t : Std.RBNode α} [Std.TransCmp cmp] (ht : Std.RBNode.Ordered cmp t) (h : t.min? = some a) (x : α) (hx : x ∈ t) : cmp a x ≠ Ordering.gt

theorem Std.RBNode.max?_mem {α : Type u_1} {a : α} {t : Std.RBNode α} (h : t.max? = some a) : a ∈ t

theorem Std.RBNode.Ordered.le_max? {α : Type u_1} {cmp : α → α → Ordering} {a : α} {t : Std.RBNode α} [Std.TransCmp cmp] (ht : Std.RBNode.Ordered cmp t) (h : t.max? = some a) (x : α) (hx : x ∈ t) : cmp x a ≠ Ordering.gt

@[simp]

theorem Std.RBNode.setBlack_toList {α : Type u_1} {t : Std.RBNode α} : t.setBlack.toList = t.toList

@[simp]

theorem Std.RBNode.setRed_toList {α : Type u_1} {t : Std.RBNode α} : t.setRed.toList = t.toList

@[simp]

theorem Std.RBNode.balance1_toList {α : Type u_1} {l : Std.RBNode α} {v : α} {r : Std.RBNode α} : (l.balance1 v r).toList = l.toList ++ v :: r.toList

@[simp]

theorem Std.RBNode.balance2_toList {α : Type u_1} {l : Std.RBNode α} {v : α} {r : Std.RBNode α} : (l.balance2 v r).toList = l.toList ++ v :: r.toList

@[simp]

theorem Std.RBNode.balLeft_toList {α : Type u_1} {l : Std.RBNode α} {v : α} {r : Std.RBNode α} : (l.balLeft v r).toList = l.toList ++ v :: r.toList

@[simp]

theorem Std.RBNode.balRight_toList {α : Type u_1} {l : Std.RBNode α} {v : α} {r : Std.RBNode α} : (l.balRight v r).toList = l.toList ++ v :: r.toList

theorem Std.RBNode.size_eq {α : Type u_1} {t : Std.RBNode α} : t.size = t.toList.length

@[simp]

theorem Std.RBNode.reverse_size {α : Type u_1} (t : Std.RBNode α) : t.reverse.size = t.size

@[simp]

theorem Std.RBNode.find?_reverse {α : Type u_1} (t : Std.RBNode α) (cut : α → Ordering) : Std.RBNode.find? cut t.reverse = Std.RBNode.find? (fun (x : α) => (cut x).swap) t

def Std.RBNode.setRoot {α : Type u_1} (v : α) : Std.RBNode α → Std.RBNode α

Auxiliary definition for zoom_ins: set the root of the tree to v, creating a node if necessary.

Equations

• Std.RBNode.setRoot v x = match x with | Std.RBNode.nil => Std.RBNode.node Std.RBColor.red Std.RBNode.nil v Std.RBNode.nil | Std.RBNode.node c a v_1 b => Std.RBNode.node c a v b

def Std.RBNode.delRoot {α : Type u_1} : Std.RBNode α → Std.RBNode α

Auxiliary definition for zoom_ins: set the root of the tree to v, creating a node if necessary.

Equations

• x.delRoot = match x with | Std.RBNode.nil => Std.RBNode.nil | Std.RBNode.node c a v b => a.append b

def Std.RBNode.Path.listL {α : Type u_1} : Std.RBNode.Path α → List α

The list of elements to the left of the hole. (This function is intended for specification purposes only.)

Equations

• Std.RBNode.Path.root.listL = []
• (Std.RBNode.Path.left c parent v r).listL = parent.listL
• (Std.RBNode.Path.right c l v parent).listL = parent.listL ++ (l.toList ++ [v])

def Std.RBNode.Path.listR {α : Type u_1} : Std.RBNode.Path α → List α

The list of elements to the right of the hole. (This function is intended for specification purposes only.)

Equations

• Std.RBNode.Path.root.listR = []
• (Std.RBNode.Path.left c parent v r).listR = v :: r.toList ++ parent.listR
• (Std.RBNode.Path.right c l v parent).listR = parent.listR

@[reducible, inline]

abbrev Std.RBNode.Path.withList {α : Type u_1} (p : Std.RBNode.Path α) (l : List α) : List α

Wraps a list of elements with the left and right elements of the path.

Equations

• p.withList l = p.listL ++ l ++ p.listR

theorem Std.RBNode.Path.rootOrdered_iff {α : Type u_1} {cmp : α → α → Ordering} {v : α} {p : Std.RBNode.Path α} (hp : Std.RBNode.Path.Ordered cmp p) : Std.RBNode.Path.RootOrdered cmp p v ↔ (∀ (a : α), a ∈ p.listL → Std.RBNode.cmpLT cmp a v) ∧ ∀ (a : α), a ∈ p.listR → Std.RBNode.cmpLT cmp v a

theorem Std.RBNode.Path.ordered_iff {α : Type u_1} {cmp : α → α → Ordering} {p : Std.RBNode.Path α} : Std.RBNode.Path.Ordered cmp p ↔ List.Pairwise (Std.RBNode.cmpLT cmp) p.listL ∧ List.Pairwise (Std.RBNode.cmpLT cmp) p.listR ∧ ∀ (x : α), x ∈ p.listL → ∀ (y : α), y ∈ p.listR → Std.RBNode.cmpLT cmp x y

theorem Std.RBNode.Path.zoom_zoomed₁ : ∀ {α : Type u_1} {cut : α → Ordering} {t : Std.RBNode α} {path : Std.RBNode.Path α} {t' : Std.RBNode α} {path' : Std.RBNode.Path α}, Std.RBNode.zoom cut t path = (t', path') → Std.RBNode.OnRoot (fun (x : α) => cut x = Ordering.eq) t'

@[simp]

theorem Std.RBNode.Path.fill_toList {α : Type u_1} {t : Std.RBNode α} {p : Std.RBNode.Path α} : (p.fill t).toList = p.withList t.toList

theorem Std.RBNode.zoom_toList {α : Type u_1} {cut : α → Ordering} {t' : Std.RBNode α} {p' : Std.RBNode.Path α} {t : Std.RBNode α} (eq : Std.RBNode.zoom cut t = (t', p')) : p'.withList t'.toList = t.toList

@[simp]

theorem Std.RBNode.Path.ins_toList {α : Type u_1} {t : Std.RBNode α} {p : Std.RBNode.Path α} : (p.ins t).toList = p.withList t.toList

@[simp]

theorem Std.RBNode.Path.insertNew_toList {α : Type u_1} {v : α} {p : Std.RBNode.Path α} : (p.insertNew v).toList = p.withList [v]

theorem Std.RBNode.Path.insert_toList {α : Type u_1} {t : Std.RBNode α} {v : α} {p : Std.RBNode.Path α} : (p.insert t v).toList = p.withList (Std.RBNode.setRoot v t).toList

theorem Std.RBNode.Path.Balanced.insert {α : Type u_1} {c₀ : Std.RBColor} {n₀ : Nat} {c : Std.RBColor} {n : Nat} {t : Std.RBNode α} {v : α} {path : Std.RBNode.Path α} (hp : Std.RBNode.Path.Balanced c₀ n₀ path c n) : t.Balanced c n → ∃ (c : Std.RBColor), ∃ (n : Nat), (path.insert t v).Balanced c n

theorem Std.RBNode.Path.Ordered.insert {α : Type u_1} {cmp : α → α → Ordering} {v : α} {path : Std.RBNode.Path α} {t : Std.RBNode α} : Std.RBNode.Path.Ordered cmp path → Std.RBNode.Ordered cmp t → Std.RBNode.All (Std.RBNode.Path.RootOrdered cmp path) t → Std.RBNode.Path.RootOrdered cmp path v → Std.RBNode.OnRoot (Std.RBNode.cmpEq cmp v) t → Std.RBNode.Ordered cmp (path.insert t v)

theorem Std.RBNode.Path.Ordered.erase {α : Type u_1} {cmp : α → α → Ordering} {path : Std.RBNode.Path α} {t : Std.RBNode α} : Std.RBNode.Path.Ordered cmp path → Std.RBNode.Ordered cmp t → Std.RBNode.All (Std.RBNode.Path.RootOrdered cmp path) t → Std.RBNode.Ordered cmp (path.erase t)

theorem Std.RBNode.Path.zoom_ins {α : Type u_1} {v : α} {path : Std.RBNode.Path α} {t' : Std.RBNode α} {path' : Std.RBNode.Path α} {t : Std.RBNode α} {cmp : α → α → Ordering} : Std.RBNode.zoom (cmp v) t path = (t', path') → path.ins (Std.RBNode.ins cmp v t) = path'.ins (Std.RBNode.setRoot v t')

theorem Std.RBNode.Path.insertNew_eq_insert : ∀ {α : Type u_1} {cmp : α → α → Ordering} {t : Std.RBNode α} {path : Std.RBNode.Path α} {v : α}, Std.RBNode.zoom (cmp v) t = (Std.RBNode.nil, path) → path.insertNew v = (Std.RBNode.insert cmp t v).setBlack

theorem Std.RBNode.Path.ins_eq_fill {α : Type u_1} {c₀ : Std.RBColor} {n₀ : Nat} {c : Std.RBColor} {n : Nat} {path : Std.RBNode.Path α} {t : Std.RBNode α} : Std.RBNode.Path.Balanced c₀ n₀ path c n → t.Balanced c n → path.ins t = (path.fill t).setBlack

theorem Std.RBNode.Path.zoom_insert {α : Type u_1} {c : Std.RBColor} {n : Nat} {cmp : α → α → Ordering} {t' : Std.RBNode α} {v : α} {path : Std.RBNode.Path α} {t : Std.RBNode α} (ht : t.Balanced c n) (H : Std.RBNode.zoom (cmp v) t = (t', path)) : (path.insert t' v).setBlack = (Std.RBNode.insert cmp t v).setBlack

theorem Std.RBNode.Path.zoom_del {α : Type u_1} {cut : α → Ordering} {path : Std.RBNode.Path α} {t' : Std.RBNode α} {path' : Std.RBNode.Path α} {t : Std.RBNode α} : Std.RBNode.zoom cut t path = (t', path') → path.del (Std.RBNode.del cut t) (match t with | Std.RBNode.node c l v r => c | x => Std.RBColor.red) = path'.del t'.delRoot (match t' with | Std.RBNode.node c l v r => c | x => Std.RBColor.red)

def Std.RBNode.Path.AllL {α : Type u_1} (p : α → Prop) : Std.RBNode.Path α → Prop

Asserts that p holds on all elements to the left of the hole.

Equations

• Std.RBNode.Path.AllL p Std.RBNode.Path.root = True
• Std.RBNode.Path.AllL p (Std.RBNode.Path.left c parent v r) = Std.RBNode.Path.AllL p parent
• Std.RBNode.Path.AllL p (Std.RBNode.Path.right c l v parent) = (Std.RBNode.All p l ∧ p v ∧ Std.RBNode.Path.AllL p parent)

def Std.RBNode.Path.AllR {α : Type u_1} (p : α → Prop) : Std.RBNode.Path α → Prop

Asserts that p holds on all elements to the right of the hole.

Equations

• Std.RBNode.Path.AllR p Std.RBNode.Path.root = True
• Std.RBNode.Path.AllR p (Std.RBNode.Path.left c parent v r) = (Std.RBNode.Path.AllR p parent ∧ p v ∧ Std.RBNode.All p r)
• Std.RBNode.Path.AllR p (Std.RBNode.Path.right c l v parent) = Std.RBNode.Path.AllR p parent

theorem Std.RBNode.insert_toList_zoom {α : Type u_1} {c : Std.RBColor} {n : Nat} {cmp : α → α → Ordering} {t' : Std.RBNode α} {p : Std.RBNode.Path α} {v : α} {t : Std.RBNode α} (ht : t.Balanced c n) (e : Std.RBNode.zoom (cmp v) t = (t', p)) : (Std.RBNode.insert cmp t v).toList = p.withList (Std.RBNode.setRoot v t').toList

theorem Std.RBNode.insert_toList_zoom_nil {α : Type u_1} {c : Std.RBColor} {n : Nat} {cmp : α → α → Ordering} {p : Std.RBNode.Path α} {v : α} {t : Std.RBNode α} (ht : t.Balanced c n) (e : Std.RBNode.zoom (cmp v) t = (Std.RBNode.nil, p)) : (Std.RBNode.insert cmp t v).toList = p.withList [v]

theorem Std.RBNode.exists_insert_toList_zoom_nil {α : Type u_1} {c : Std.RBColor} {n : Nat} {cmp : α → α → Ordering} {p : Std.RBNode.Path α} {v : α} {t : Std.RBNode α} (ht : t.Balanced c n) (e : Std.RBNode.zoom (cmp v) t = (Std.RBNode.nil, p)) : ∃ (L : List α), ∃ (R : List α), t.toList = L ++ R ∧ (Std.RBNode.insert cmp t v).toList = L ++ v :: R

theorem Std.RBNode.insert_toList_zoom_node {α : Type u_1} {c : Std.RBColor} {n : Nat} {cmp : α → α → Ordering} {c' : Std.RBColor} {l : Std.RBNode α} {v' : α} {r : Std.RBNode α} {p : Std.RBNode.Path α} {v : α} {t : Std.RBNode α} (ht : t.Balanced c n) (e : Std.RBNode.zoom (cmp v) t = (Std.RBNode.node c' l v' r, p)) : (Std.RBNode.insert cmp t v).toList = p.withList (Std.RBNode.node c l v r).toList

theorem Std.RBNode.exists_insert_toList_zoom_node {α : Type u_1} {c : Std.RBColor} {n : Nat} {cmp : α → α → Ordering} {c' : Std.RBColor} {l : Std.RBNode α} {v' : α} {r : Std.RBNode α} {p : Std.RBNode.Path α} {v : α} {t : Std.RBNode α} (ht : t.Balanced c n) (e : Std.RBNode.zoom (cmp v) t = (Std.RBNode.node c' l v' r, p)) : ∃ (L : List α), ∃ (R : List α), t.toList = L ++ v' :: R ∧ (Std.RBNode.insert cmp t v).toList = L ++ v :: R

theorem Std.RBNode.mem_insert_self {α : Type u_1} {c : Std.RBColor} {n : Nat} {v : α} {cmp : α → α → Ordering} {t : Std.RBNode α} (ht : t.Balanced c n) : v ∈ Std.RBNode.insert cmp t v

theorem Std.RBNode.mem_insert_of_mem {α : Type u_1} {c : Std.RBColor} {n : Nat} {v' : α} {cmp : α → α → Ordering} {v : α} {t : Std.RBNode α} (ht : t.Balanced c n) (h : v' ∈ t) : v' ∈ Std.RBNode.insert cmp t v ∨ cmp v v' = Ordering.eq

theorem Std.RBNode.exists_find?_insert_self {α : Type u_1} {cmp : α → α → Ordering} {cut : α → Ordering} {c : Std.RBColor} {n : Nat} {v : α} [Std.TransCmp cmp] [Std.RBNode.IsCut cmp cut] {t : Std.RBNode α} (ht : t.Balanced c n) (ht₂ : Std.RBNode.Ordered cmp t) (hv : cut v = Ordering.eq) : ∃ (x : α), Std.RBNode.find? cut (Std.RBNode.insert cmp t v) = some x

theorem Std.RBNode.find?_insert_self {α : Type u_1} {cmp : α → α → Ordering} {cut : α → Ordering} {c : Std.RBColor} {n : Nat} {v : α} [Std.TransCmp cmp] [Std.RBNode.IsStrictCut cmp cut] {t : Std.RBNode α} (ht : t.Balanced c n) (ht₂ : Std.RBNode.Ordered cmp t) (hv : cut v = Ordering.eq) : Std.RBNode.find? cut (Std.RBNode.insert cmp t v) = some v

theorem Std.RBNode.mem_insert {α : Type u_1} {cmp : α → α → Ordering} {c : Std.RBColor} {n : Nat} {v' : α} {v : α} [Std.TransCmp cmp] {t : Std.RBNode α} (ht : t.Balanced c n) (ht₂ : Std.RBNode.Ordered cmp t) : v' ∈ Std.RBNode.insert cmp t v ↔ v' ∈ t ∧ Std.RBNode.find? (cmp v) t ≠ some v' ∨ v' = v

@[simp]

theorem Std.RBSet.val_toList {α : Type u_1} {cmp : α → α → Ordering} {t : Std.RBSet α cmp} : t.val.toList = t.toList

@[simp]

theorem Std.RBSet.mkRBSet_eq {α : Type u_1} {cmp : α → α → Ordering} : Std.mkRBSet α cmp = ∅

@[simp]

theorem Std.RBSet.empty_eq {α : Type u_1} {cmp : α → α → Ordering} : Std.RBSet.empty = ∅

@[simp]

theorem Std.RBSet.default_eq {α : Type u_1} {cmp : α → α → Ordering} : default = ∅

@[simp]

theorem Std.RBSet.empty_toList {α : Type u_1} {cmp : α → α → Ordering} : ∅.toList = []

@[simp]

theorem Std.RBSet.single_toList {α : Type u_1} {cmp : α → α → Ordering} {a : α} : (Std.RBSet.single a).toList = [a]

theorem Std.RBSet.mem_toList {α : Type u_1} {cmp : α → α → Ordering} {x : α} {t : Std.RBSet α cmp} : x ∈ t.toList ↔ x ∈ t.val

theorem Std.RBSet.mem_congr {α : Type u_1} {cmp : α → α → Ordering} {x : α} {y : α} [Std.TransCmp cmp] {t : Std.RBSet α cmp} (h : cmp x y = Ordering.eq) : x ∈ t ↔ y ∈ t

theorem Std.RBSet.mem_iff_mem_toList {α : Type u_1} {cmp : α → α → Ordering} {x : α} {t : Std.RBSet α cmp} : x ∈ t ↔ ∃ (y : α), y ∈ t.toList ∧ cmp x y = Ordering.eq

theorem Std.RBSet.mem_of_mem_toList {α : Type u_1} {cmp : α → α → Ordering} {x : α} [Std.OrientedCmp cmp] {t : Std.RBSet α cmp} (h : x ∈ t.toList) : x ∈ t

theorem Std.RBSet.foldl_eq_foldl_toList {α : Type u_1} {cmp : α → α → Ordering} : ∀ {α_1 : Type u_2} {f : α_1 → α → α_1} {init : α_1} {t : Std.RBSet α cmp}, Std.RBSet.foldl f init t = List.foldl f init t.toList

theorem Std.RBSet.foldr_eq_foldr_toList {α : Type u_1} {cmp : α → α → Ordering} : ∀ {α_1 : Type u_2} {f : α → α_1 → α_1} {init : α_1} {t : Std.RBSet α cmp}, Std.RBSet.foldr f init t = List.foldr f init t.toList

theorem Std.RBSet.foldlM_eq_foldlM_toList {m : Type u_1 → Type u_2} {α : Type u_3} {cmp : α → α → Ordering} : ∀ {a : Type u_1} {f : a → α → m a} {init : a} [inst : Monad m] [inst_1 : LawfulMonad m] {t : Std.RBSet α cmp}, Std.RBSet.foldlM f init t = List.foldlM f init t.toList

theorem Std.RBSet.forM_eq_forM_toList {m : Type u_1 → Type u_2} {α : Type u_3} {cmp : α → α → Ordering} {f : α → m PUnit} [Monad m] [LawfulMonad m] {t : Std.RBSet α cmp} : Std.RBSet.forM f t = t.toList.forM f

theorem Std.RBSet.forIn_eq_forIn_toList {m : Type u_1 → Type u_2} {α : Type u_3} {cmp : α → α → Ordering} : ∀ {α_1 : Type u_1} {init : α_1} {f : α → α_1 → m (ForInStep α_1)} [inst : Monad m] [inst_1 : LawfulMonad m] {t : Std.RBSet α cmp}, forIn t init f = forIn t.toList init f

theorem Std.RBSet.toStream_eq {α : Type u_1} {cmp : α → α → Ordering} {t : Std.RBSet α cmp} : toStream t = t.val.toStream

@[simp]

theorem Std.RBSet.toStream_toList {α : Type u_1} {cmp : α → α → Ordering} {t : Std.RBSet α cmp} : (toStream t).toList = t.toList

theorem Std.RBSet.isEmpty_iff_toList_eq_nil {α : Type u_1} {cmp : α → α → Ordering} {t : Std.RBSet α cmp} : t.isEmpty = true ↔ t.toList = []

theorem Std.RBSet.toList_sorted {α : Type u_1} {cmp : α → α → Ordering} {t : Std.RBSet α cmp} : List.Pairwise (Std.RBNode.cmpLT cmp) t.toList

theorem Std.RBSet.findP?_some_eq_eq {α : Type u_1} {cmp : α → α → Ordering} {cut : α → Ordering} {y : α} {t : Std.RBSet α cmp} : t.findP? cut = some y → cut y = Ordering.eq

theorem Std.RBSet.find?_some_eq_eq {α : Type u_1} {cmp : α → α → Ordering} {x : α} {y : α} {t : Std.RBSet α cmp} : t.find? x = some y → cmp x y = Ordering.eq

theorem Std.RBSet.findP?_some_mem_toList {α : Type u_1} {cmp : α → α → Ordering} {cut : α → Ordering} {y : α} {t : Std.RBSet α cmp} (h : t.findP? cut = some y) : y ∈ t.toList

theorem Std.RBSet.find?_some_mem_toList {α : Type u_1} {cmp : α → α → Ordering} {x : α} {y : α} {t : Std.RBSet α cmp} (h : t.find? x = some y) : y ∈ t.toList

theorem Std.RBSet.findP?_some_memP {α : Type u_1} {cmp : α → α → Ordering} {cut : α → Ordering} {y : α} {t : Std.RBSet α cmp} (h : t.findP? cut = some y) : Std.RBSet.MemP cut t

theorem Std.RBSet.find?_some_mem {α : Type u_1} {cmp : α → α → Ordering} {x : α} {y : α} {t : Std.RBSet α cmp} (h : t.find? x = some y) : x ∈ t

theorem Std.RBSet.mem_toList_unique {α : Type u_1} {cmp : α → α → Ordering} {x : α} {y : α} [Std.TransCmp cmp] {t : Std.RBSet α cmp} (hx : x ∈ t.toList) (hy : y ∈ t.toList) (e : cmp x y = Ordering.eq) : x = y

theorem Std.RBSet.findP?_some {α : Type u_1} {cmp : α → α → Ordering} {cut : α → Ordering} {y : α} [Std.TransCmp cmp] [Std.RBNode.IsStrictCut cmp cut] {t : Std.RBSet α cmp} : t.findP? cut = some y ↔ y ∈ t.toList ∧ cut y = Ordering.eq

theorem Std.RBSet.find?_some {α : Type u_1} {cmp : α → α → Ordering} {x : α} {y : α} [Std.TransCmp cmp] {t : Std.RBSet α cmp} : t.find? x = some y ↔ y ∈ t.toList ∧ cmp x y = Ordering.eq

theorem Std.RBSet.memP_iff_findP? {α : Type u_1} {cmp : α → α → Ordering} {cut : α → Ordering} [Std.TransCmp cmp] [Std.RBNode.IsCut cmp cut] {t : Std.RBSet α cmp} : Std.RBSet.MemP cut t ↔ ∃ (y : α), t.findP? cut = some y

theorem Std.RBSet.mem_iff_find? {α : Type u_1} {cmp : α → α → Ordering} {x : α} [Std.TransCmp cmp] {t : Std.RBSet α cmp} : x ∈ t ↔ ∃ (y : α), t.find? x = some y

@[simp]

theorem Std.RBSet.contains_iff {α : Type u_1} {cmp : α → α → Ordering} {x : α} [Std.TransCmp cmp] {t : Std.RBSet α cmp} : t.contains x = true ↔ x ∈ t

instance Std.RBSet.instDecidableMemOfTransCmp {α : Type u_1} {cmp : α → α → Ordering} {x : α} [Std.TransCmp cmp] {t : Std.RBSet α cmp} : Decidable (x ∈ t)

Equations

• Std.RBSet.instDecidableMemOfTransCmp = decidable_of_iff (t.contains x = true) ⋯

theorem Std.RBSet.size_eq {α : Type u_1} {cmp : α → α → Ordering} (t : Std.RBSet α cmp) : t.size = t.toList.length

theorem Std.RBSet.mem_toList_insert_self {α : Type u_1} {cmp : α → α → Ordering} (v : α) (t : Std.RBSet α cmp) : v ∈ (t.insert v).toList

theorem Std.RBSet.mem_insert_self {α : Type u_1} {cmp : α → α → Ordering} [Std.OrientedCmp cmp] (v : α) (t : Std.RBSet α cmp) : v ∈ t.insert v

theorem Std.RBSet.mem_insert_of_eq {α : Type u_1} {cmp : α → α → Ordering} {v : α} {v' : α} [Std.TransCmp cmp] (t : Std.RBSet α cmp) (h : cmp v v' = Ordering.eq) : v' ∈ t.insert v

theorem Std.RBSet.mem_toList_insert_of_mem {α : Type u_1} {cmp : α → α → Ordering} {v' : α} (v : α) {t : Std.RBSet α cmp} (h : v' ∈ t.toList) : v' ∈ (t.insert v).toList ∨ cmp v v' = Ordering.eq

theorem Std.RBSet.mem_insert_of_mem_toList {α : Type u_1} {cmp : α → α → Ordering} {v' : α} [Std.OrientedCmp cmp] (v : α) {t : Std.RBSet α cmp} (h : v' ∈ t.toList) : v' ∈ t.insert v

theorem Std.RBSet.mem_insert_of_mem {α : Type u_1} {cmp : α → α → Ordering} {v' : α} [Std.TransCmp cmp] (v : α) {t : Std.RBSet α cmp} (h : v' ∈ t) : v' ∈ t.insert v

theorem Std.RBSet.mem_toList_insert {α : Type u_1} {cmp : α → α → Ordering} {v' : α} {v : α} [Std.TransCmp cmp] {t : Std.RBSet α cmp} : v' ∈ (t.insert v).toList ↔ v' ∈ t.toList ∧ t.find? v ≠ some v' ∨ v' = v

theorem Std.RBSet.mem_insert {α : Type u_1} {cmp : α → α → Ordering} {v' : α} {v : α} [Std.TransCmp cmp] {t : Std.RBSet α cmp} : v' ∈ t.insert v ↔ v' ∈ t ∨ cmp v v' = Ordering.eq

theorem Std.RBSet.find?_congr {α : Type u_1} {cmp : α → α → Ordering} {v₁ : α} {v₂ : α} [Std.TransCmp cmp] (t : Std.RBSet α cmp) (h : cmp v₁ v₂ = Ordering.eq) : t.find? v₁ = t.find? v₂

theorem Std.RBSet.findP?_insert_of_eq {α : Type u_1} {cmp : α → α → Ordering} {cut : α → Ordering} {v : α} [Std.TransCmp cmp] [Std.RBNode.IsStrictCut cmp cut] (t : Std.RBSet α cmp) (h : cut v = Ordering.eq) : (t.insert v).findP? cut = some v

theorem Std.RBSet.find?_insert_of_eq {α : Type u_1} {cmp : α → α → Ordering} {v' : α} {v : α} [Std.TransCmp cmp] (t : Std.RBSet α cmp) (h : cmp v' v = Ordering.eq) : (t.insert v).find? v' = some v

theorem Std.RBSet.findP?_insert_of_ne {α : Type u_1} {cmp : α → α → Ordering} {cut : α → Ordering} {v : α} [Std.TransCmp cmp] [Std.RBNode.IsStrictCut cmp cut] (t : Std.RBSet α cmp) (h : cut v ≠ Ordering.eq) : (t.insert v).findP? cut = t.findP? cut

theorem Std.RBSet.find?_insert_of_ne {α : Type u_1} {cmp : α → α → Ordering} {v' : α} {v : α} [Std.TransCmp cmp] (t : Std.RBSet α cmp) (h : cmp v' v ≠ Ordering.eq) : (t.insert v).find? v' = t.find? v'

theorem Std.RBSet.findP?_insert {α : Type u_1} {cmp : α → α → Ordering} [Std.TransCmp cmp] (t : Std.RBSet α cmp) (v : α) (cut : α → Ordering) [Std.RBNode.IsStrictCut cmp cut] : (t.insert v).findP? cut = if cut v = Ordering.eq then some v else t.findP? cut

theorem Std.RBSet.find?_insert {α : Type u_1} {cmp : α → α → Ordering} [Std.TransCmp cmp] (t : Std.RBSet α cmp) (v : α) (v' : α) : (t.insert v).find? v' = if cmp v' v = Ordering.eq then some v else t.find? v'