Inductive Types

We see more precisely what are valid (and not valid) definitions for inductive types. We will also see the weak and strong generalizations: parametric inductive types and indexed inductive types.

We have constructed a few examples. We first see some more from Lean core.

Simplest ones are enumerations

inductive Bool : Type where
  | false : Bool
  | true : Bool

In general, the construtors let us construct terms of the inductive type being introduced using terms of that type.

inductive Nat where
  | zero : Nat
  | succ (n : Nat) : Nat

#check Nat.zero -- ℕ #check Nat.succ -- ℕ → ℕ

Two more inductive types:

abbrev Unit : Type := PUnit

inductive PUnit : Sort u where
  | unit : PUnit

inductive Empty : Type

Disallowed types

• constructors should have resulting type the inductive type being introduced.

The above is disallowed because with a construction of the above form, we would have an injection (in this case a biection) from the power set on the Cantor set to itself.

If we had such an inductive type, we could define

diag (x :Cantor): Bool :=  match x with
| Cantor.mk f => ¬ f x

Apply this to x:= Cantor.mk diag to get

diag (Cantor.mk diag) = ¬ diag (Cantor.mk diag)

A non-trivial positive occurence, which is hence allowed, is in the following binary tree.

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inductive NatBinTree : Type

• leaf: ℕ → NatBinTree
• node: (Bool → NatBinTree) → NatBinTree

Parametrized inductive types

Here we define a family of inductive types. However each member of the family is "separately defined", i.e., all constructors only involve that parameter.

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inductive InfiniteTree (α : Type u) : Type u

• leaf: {α : Type u} → α → InfiniteTree α
• node: {α : Type u} → (ℕ → InfiniteTree α) → InfiniteTree α

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inductive FiniteTree (α : Type u) : Type u

• leaf: {α : Type u} → α → FiniteTree α
• node: {α : Type u} → List (FiniteTree α) → FiniteTree α

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partial def FiniteTree.flatten {α : Type u} : FiniteTree α → List α

Indexed inductive type

We define types Vec α n for α : Type u and n: ℕ with terms of Vec α n n-tuples in α.

• α will be a parameter.
• n will be an index.

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inductive Vec (α : Type u) : ℕ → Type (u+1)

• nil: {α : Type u} → Vec α 0
• cons: {α : Type u} → {n : ℕ} → α → Vec α n → Vec α (n + 1)

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def Vec.to_list {α : Type u} {n : ℕ} : Vec α n → List α

Equations

• Vec.to_list Vec.nil = []
• Vec.to_list (Vec.cons head tail) = head :: Vec.to_list tail

List is a parametrized inductive type

inductive List (α : Type u) where
  /-- `[]` is the empty list. -/
  | nil : List α
  /-- If `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
  list whose first element is `a` and with `l` as the rest of the list. -/
  | cons (head : α) (tail : List α) : List α

Sometimes types include conditions

structure Fin (n : Nat) where
  /-- If `i : Fin n`, then `i.val : ℕ` is the described number. It can also be
  written as `i.1` or just `i` when the target type is known. -/
  val  : Nat
  /-- If `i : Fin n`, then `i.2` is a proof that `i.1 < n`. -/
  isLt : LT.lt val n

Conditions can be given by a subtype of the type.

def Vector (α : Type u) (n : ℕ) :=
  { l : List α // l.length = n }

When an (indexed) inductive type is introduced,

• the type (or family of types) is defined.
• the constructors are defined.
• a recursor is defined.
• a rule for simplification of applications of the recursor is introduced.

The recursor can be conveniently used by pattern matching.

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def Bool.disagree (b : Bool) : Bool

Equations

• Bool.disagree b = match b with | false => true | true => false

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def egFamily (b : Bool) : Type

Equations

• egFamily b = match b with | false => Unit | true => ℕ

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def egDepFunction (b : Bool) : egFamily b

Equations

• egDepFunction b = match b with | false => () | true => id 3