Binary tree

Provides binary tree storage for values of any type, with O(lg n) retrieval. See also Lean.Data.RBTree for red-black trees - this version allows more operations to be defined and is better suited for in-kernel computation.

We also specialize for Tree Unit, which is a binary tree without any additional data. We provide the notation a △ b for making a Tree Unit with children a and b.

References

https://leanprover-community.github.io/archive/stream/113488-general/topic/tactic.20question.html

inductive Tree (α : Type u) : Type u

A binary tree with values stored in non-leaf nodes.

• nil {α : Type u} : Tree α
• node {α : Type u} : α → Tree α → Tree α → Tree α

instance instDecidableEqTree {α✝ : Type u_1} [DecidableEq α✝] : DecidableEq (Tree α✝)

instance instReprTree {α✝ : Type u_1} [Repr α✝] : Repr (Tree α✝)

instance Tree.instInhabited {α : Type u} : Inhabited (Tree α)

def Tree.traverse {m : Type u_1 → Type u_2} [Applicative m] {α : Type u_3} {β : Type u_1} (f : α → m β) : Tree α → m (Tree β)

Do an action for every node of the tree. Actions are taken in node -> left subtree -> right subtree recursive order. This function is the traverse function for the Traversable Tree instance.

def Tree.map {α : Type u} {β : Type u_1} (f : α → β) : Tree α → Tree β

Apply a function to each value in the tree. This is the map function for the Tree functor.

theorem Tree.id_map {α : Type u} (t : Tree α) : map id t = t

theorem Tree.comp_map {α : Type u} {β : Type u_1} {γ : Type u_2} (f : α → β) (g : β → γ) (t : Tree α) : map (g ∘ f) t = map g (map f t)

theorem Tree.traverse_pure {α : Type u} (t : Tree α) {m : Type u → Type u_1} [Applicative m] [LawfulApplicative m] : traverse pure t = pure t

def Tree.numNodes {α : Type u} : Tree α → ℕ

The number of internal nodes (i.e. not including leaves) of a binary tree

def Tree.numLeaves {α : Type u} : Tree α → ℕ

The number of leaves of a binary tree

def Tree.height {α : Type u} : Tree α → ℕ

The height - length of the longest path from the root - of a binary tree

theorem Tree.numLeaves_eq_numNodes_succ {α : Type u} (x : Tree α) : x.numLeaves = x.numNodes + 1

theorem Tree.numLeaves_pos {α : Type u} (x : Tree α) : 0 < x.numLeaves

theorem Tree.height_le_numNodes {α : Type u} (x : Tree α) : x.height ≤ x.numNodes

def Tree.left {α : Type u} : Tree α → Tree α

The left child of the tree, or nil if the tree is nil

def Tree.right {α : Type u} : Tree α → Tree α

The right child of the tree, or nil if the tree is nil

def Tree.«term_△_» : Lean.TrailingParserDescr

A node with Unit data

def Tree.unitRecOn {motive : Tree Unit → Sort u_1} (t : Tree Unit) (base : motive nil) (ind : (x y : Tree Unit) → motive x → motive y → motive (node () x y)) : motive t

Induction principle for Tree Units

theorem Tree.left_node_right_eq_self {x : Tree Unit} (_hx : x ≠ nil) : node () x.left x.right = x