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

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

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} : Std.RBNode.Balanced t c n → Std.RBNode.depth t ≤ Std.RBNode.depthUB c n

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

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

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

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

Equations

• Std.RBNode.instDecidableOrdered 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.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.

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 = Std.RBNode.toList t ++ l

@[simp]

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

@[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.toList (Std.RBNode.node c a x b) = Std.RBNode.toList a ++ x :: Std.RBNode.toList b

@[simp]

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

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 (Std.RBNode.toList t)

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 (Std.RBNode.toList 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 = List.forM (Std.RBNode.toList t) 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 (Std.RBNode.toList t)

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 (Std.RBNode.toList t) (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 (Std.RBNode.toList t) 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 = Std.RBNode.Stream.toList t ++ l

@[simp]

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

@[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.toList (Std.RBNode.Stream.cons x r s) = x :: Std.RBNode.toList r ++ Std.RBNode.Stream.toList s

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 (Std.RBNode.Stream.toList s)

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 (Std.RBNode.Stream.toList t)

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 (Std.RBNode.toList t) init f

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

@[simp]

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

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, Std.RBNode.Stream.toList b)) (Std.RBNode.Stream.next? s) = List.next? (Std.RBNode.Stream.toList s)

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) (Std.RBNode.toList t)

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) (Std.RBNode.toList t)

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

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

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

@[inline, reducible]

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

• Std.RBNode.Path.withList p l = Std.RBNode.Path.listL p ++ l ++ Std.RBNode.Path.listR p

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 ∈ Std.RBNode.Path.listL p → Std.RBNode.cmpLT cmp a v) ∧ ∀ (a : α), a ∈ Std.RBNode.Path.listR p → 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) (Std.RBNode.Path.listL p) ∧ List.Pairwise (Std.RBNode.cmpLT cmp) (Std.RBNode.Path.listR p) ∧ ∀ (x : α), x ∈ Std.RBNode.Path.listL p → ∀ (y : α), y ∈ Std.RBNode.Path.listR p → Std.RBNode.cmpLT cmp x y

@[simp]

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

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')) : Std.RBNode.Path.withList p' (Std.RBNode.toList t') = Std.RBNode.toList t

@[simp]

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

@[simp]

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

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

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 : Std.RBNode.Balanced t c n) (e : Std.RBNode.zoom (cmp v) t = (t', p)) : Std.RBNode.toList (Std.RBNode.insert cmp t v) = Std.RBNode.Path.withList p (Std.RBNode.toList (Std.RBNode.setRoot v t'))

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 : Std.RBNode.Balanced t c n) (e : Std.RBNode.zoom (cmp v) t = (Std.RBNode.nil, p)) : Std.RBNode.toList (Std.RBNode.insert cmp t v) = Std.RBNode.Path.withList p [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 : Std.RBNode.Balanced t c n) (e : Std.RBNode.zoom (cmp v) t = (Std.RBNode.nil, p)) : ∃ (L : List α), ∃ (R : List α), Std.RBNode.toList t = L ++ R ∧ Std.RBNode.toList (Std.RBNode.insert cmp t v) = 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 : Std.RBNode.Balanced t c n) (e : Std.RBNode.zoom (cmp v) t = (Std.RBNode.node c' l v' r, p)) : Std.RBNode.toList (Std.RBNode.insert cmp t v) = Std.RBNode.Path.withList p (Std.RBNode.toList (Std.RBNode.node c l v r))

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 : Std.RBNode.Balanced t c n) (e : Std.RBNode.zoom (cmp v) t = (Std.RBNode.node c' l v' r, p)) : ∃ (L : List α), ∃ (R : List α), Std.RBNode.toList t = L ++ v' :: R ∧ Std.RBNode.toList (Std.RBNode.insert cmp t v) = 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 : Std.RBNode.Balanced t 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 : Std.RBNode.Balanced t 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 : Std.RBNode.Balanced t 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 : Std.RBNode.Balanced t 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 : Std.RBNode.Balanced t 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} : Std.RBNode.toList t.val = Std.RBSet.toList t

@[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} : Std.RBSet.toList ∅ = []

@[simp]

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

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

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 (Std.RBSet.toList t)

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 (Std.RBSet.toList t)

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 (Std.RBSet.toList t) init f

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

@[simp]

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

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

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

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 : Std.RBSet.find? t x = some y) : y ∈ Std.RBSet.toList 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 : Std.RBSet.find? t 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 ∈ Std.RBSet.toList t) (hy : y ∈ Std.RBSet.toList t) (e : cmp x y = Ordering.eq) : x = y

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} : Std.RBSet.find? t x = some y ↔ y ∈ Std.RBSet.toList t ∧ cmp x y = Ordering.eq

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), Std.RBSet.find? t 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} : Std.RBSet.contains t x = true ↔ x ∈ t

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

Equations

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

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

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

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 ∈ Std.RBSet.insert t 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' ∈ Std.RBSet.insert t 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' ∈ Std.RBSet.toList t) : v' ∈ Std.RBSet.toList (Std.RBSet.insert t v) ∨ 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' ∈ Std.RBSet.toList t) : v' ∈ Std.RBSet.insert t 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' ∈ Std.RBSet.insert t 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' ∈ Std.RBSet.toList (Std.RBSet.insert t v) ↔ v' ∈ Std.RBSet.toList t ∧ Std.RBSet.find? t 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' ∈ Std.RBSet.insert t 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) : Std.RBSet.find? t v₁ = Std.RBSet.find? t v₂

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) : Std.RBSet.find? (Std.RBSet.insert t v) v' = some v

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) : Std.RBSet.find? (Std.RBSet.insert t v) v' = Std.RBSet.find? t v'

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) : Std.RBSet.find? (Std.RBSet.insert t v) v' = if cmp v' v = Ordering.eq then some v else Std.RBSet.find? t v'