Binary tree #

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

References #

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

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@[protected, instance]

def tree.decidable_eq (α : Type u) [a : decidable_eq α] :

decidable_eq (tree α)

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@[protected, instance]

meta def tree.has_reflect (α : Type) [a : has_reflect α] [a_1 : reflected α] :

has_reflect (tree α)

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inductive tree (α : Type u) :

Type u

nil : Π {α : Type u}, tree α
node : Π {α : Type u}, α → tree α → tree α → tree α

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

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def tree.repr {α : Type u} [has_repr α] :

tree α → string

Construct a string representation of a tree. Provides a has_repr instance.

Equations

(tree.node a t1 t2).repr = "tree.node " ++ has_repr.repr a ++ " (" ++ t1.repr ++ ") (" ++ t2.repr ++ ")"
tree.nil.repr = "nil"

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@[protected, instance]

def tree.has_repr {α : Type u} [has_repr α] :

has_repr (tree α)

Equations

tree.has_repr = {repr := tree.repr _inst_1}

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@[protected, instance]

def tree.inhabited {α : Type u} :

inhabited (tree α)

Equations

tree.inhabited = {default := tree.nil α}

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def tree.of_rbnode {α : Type u} :

rbnode α → tree α

Makes a tree α out of a red-black tree.

Equations

tree.of_rbnode (l.black_node a r) = tree.node a (tree.of_rbnode l) (tree.of_rbnode r)
tree.of_rbnode (l.red_node a r) = tree.node a (tree.of_rbnode l) (tree.of_rbnode r)
tree.of_rbnode rbnode.leaf = tree.nil

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def tree.index_of {α : Type u} (lt : α → α → Prop) [decidable_rel lt] (x : α) :

tree α → option pos_num

Finds the index of an element in the tree assuming the tree has been constructed according to the provided decidable order on its elements. If it hasn't, the result will be incorrect. If it has, but the element is not in the tree, returns none.

Equations

tree.index_of lt x (tree.node a t₁ t₂) = tree.index_of._match_1 (tree.index_of lt x t₁) (tree.index_of lt x t₂) (cmp_using lt x a)
tree.index_of lt x tree.nil = none
tree.index_of._match_1 _f_1 _f_2 ordering.gt = pos_num.bit1 <$> _f_2
tree.index_of._match_1 _f_1 _f_2 ordering.eq = some pos_num.one
tree.index_of._match_1 _f_1 _f_2 ordering.lt = pos_num.bit0 <$> _f_1

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def tree.get {α : Type u} :

pos_num → tree α → option α

Retrieves an element uniquely determined by a pos_num from the tree, taking the following path to get to the element:

bit0 - go to left child
bit1 - go to right child
pos_num.one - retrieve from here

Equations

tree.get n.bit0 (tree.node a t₁ t₂) = tree.get n t₁
tree.get ᾰ.bit0 tree.nil = none
tree.get n.bit1 (tree.node a t₁ t₂) = tree.get n t₂
tree.get ᾰ.bit1 tree.nil = none
tree.get pos_num.one (tree.node a t₁ t₂) = some a
tree.get pos_num.one tree.nil = none

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def tree.get_or_else {α : Type u} (n : pos_num) (t : tree α) (v : α) :

Retrieves an element from the tree, or the provided default value if the index is invalid. See tree.get.

Equations

tree.get_or_else n t v = (tree.get n t).get_or_else v

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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. TODO: implement traversable tree.

Equations

tree.map f (tree.node a l r) = tree.node (f a) (tree.map f l) (tree.map f r)
tree.map f tree.nil = tree.nil