The W construction as a multivariate polynomial functor.

W types are well-founded tree-like structures. They are defined as the least fixpoint of a polynomial functor.

Main definitions

• W_mk - constructor
• W_dest - destructor
• W_rec - recursor: basis for defining functions by structural recursion on P.W α
• W_rec_eq - defining equation for W_rec
• W_ind - induction principle for P.W α

Implementation notes

Three views of M-types:

• wp: polynomial functor
• W: data type inductively defined by a triple: shape of the root, data in the root and children of the root
• W: least fixed point of a polynomial functor

Specifically, we define the polynomial functor wp as:

• A := a tree-like structure without information in the nodes
• B := given the tree-like structure t, B t is a valid path (specified inductively by W_path) from the root of t to any given node.

As a result wp α is made of a dataless tree and a function from its valid paths to values of α

Reference

• Jeremy Avigad, Mario M. Carneiro and Simon Hudon. [Data Types as Quotients of Polynomial Functors][avigad-carneiro-hudon2019]

inductive MvPFunctor.WPath {n : ℕ} (P : MvPFunctor.{u} (n + 1)) : P.last.W → Fin2 n → Type u

A path from the root of a tree to one of its node

• root {n : ℕ} {P : MvPFunctor.{u} (n + 1)} (a : P.A) (f : P.last.B a → P.last.W) (i : Fin2 n) (c : P.drop.B a i) : P.WPath (WType.mk a f) i
• child {n : ℕ} {P : MvPFunctor.{u} (n + 1)} (a : P.A) (f : P.last.B a → P.last.W) (i : Fin2 n) (j : P.last.B a) (c : P.WPath (f j) i) : P.WPath (WType.mk a f) i

instance MvPFunctor.WPath.inhabited {n : ℕ} (P : MvPFunctor.{u} (n + 1)) (x : P.last.W) {i : Fin2 n} [I : Inhabited (P.drop.B x.head i)] : Inhabited (P.WPath x i)

def MvPFunctor.wPathCasesOn {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α : TypeVec.{u_1} n} {a : P.A} {f : P.last.B a → P.last.W} (g' : (P.drop.B a).Arrow α) (g : (j : P.last.B a) → TypeVec.Arrow (P.WPath (f j)) α) : TypeVec.Arrow (P.WPath (WType.mk a f)) α

Specialized destructor on WPath

def MvPFunctor.wPathDestLeft {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α : TypeVec.{u_1} n} {a : P.A} {f : P.last.B a → P.last.W} (h : TypeVec.Arrow (P.WPath (WType.mk a f)) α) : (P.drop.B a).Arrow α

Specialized destructor on WPath

def MvPFunctor.wPathDestRight {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α : TypeVec.{u_1} n} {a : P.A} {f : P.last.B a → P.last.W} (h : TypeVec.Arrow (P.WPath (WType.mk a f)) α) (j : P.last.B a) : TypeVec.Arrow (P.WPath (f j)) α

Specialized destructor on WPath

theorem MvPFunctor.wPathDestLeft_wPathCasesOn {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α : TypeVec.{u_1} n} {a : P.A} {f : P.last.B a → P.last.W} (g' : (P.drop.B a).Arrow α) (g : (j : P.last.B a) → TypeVec.Arrow (P.WPath (f j)) α) : P.wPathDestLeft (P.wPathCasesOn g' g) = g'

theorem MvPFunctor.wPathDestRight_wPathCasesOn {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α : TypeVec.{u_1} n} {a : P.A} {f : P.last.B a → P.last.W} (g' : (P.drop.B a).Arrow α) (g : (j : P.last.B a) → TypeVec.Arrow (P.WPath (f j)) α) : P.wPathDestRight (P.wPathCasesOn g' g) = g

theorem MvPFunctor.wPathCasesOn_eta {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α : TypeVec.{u_1} n} {a : P.A} {f : P.last.B a → P.last.W} (h : TypeVec.Arrow (P.WPath (WType.mk a f)) α) : P.wPathCasesOn (P.wPathDestLeft h) (P.wPathDestRight h) = h

theorem MvPFunctor.comp_wPathCasesOn {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α : TypeVec.{u_1} n} {β : TypeVec.{u_2} n} (h : α.Arrow β) {a : P.A} {f : P.last.B a → P.last.W} (g' : (P.drop.B a).Arrow α) (g : (j : P.last.B a) → TypeVec.Arrow (P.WPath (f j)) α) : TypeVec.comp h (P.wPathCasesOn g' g) = P.wPathCasesOn (TypeVec.comp h g') fun (i : P.last.B a) => TypeVec.comp h (g i)

def MvPFunctor.wp {n : ℕ} (P : MvPFunctor.{u} (n + 1)) : MvPFunctor.{u} n

Polynomial functor for the W-type of P. A is a data-less well-founded tree whereas, for a given a : A, B a is a valid path in tree a so that Wp.obj α is made of a tree and a function from its valid paths to the values it contains

def MvPFunctor.W {n : ℕ} (P : MvPFunctor.{u} (n + 1)) (α : TypeVec.{u} n) : Type u

W-type of P

instance MvPFunctor.mvfunctorW {n : ℕ} (P : MvPFunctor.{u} (n + 1)) : MvFunctor P.W

First, describe operations on W as a polynomial functor.

def MvPFunctor.wpMk {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α : TypeVec.{u} n} (a : P.A) (f : P.last.B a → P.last.W) (f' : TypeVec.Arrow (P.WPath (WType.mk a f)) α) : P.W α

Constructor for wp

def MvPFunctor.wpRec {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α : TypeVec.{u_2} n} {C : Type u_1} (g : (a : P.A) → (f : P.last.B a → P.last.W) → TypeVec.Arrow (P.WPath (WType.mk a f)) α → (P.last.B a → C) → C) (x : P.last.W) : TypeVec.Arrow (P.WPath x) α → C

theorem MvPFunctor.wpRec_eq {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α : TypeVec.{u_2} n} {C : Type u_1} (g : (a : P.A) → (f : P.last.B a → P.last.W) → TypeVec.Arrow (P.WPath (WType.mk a f)) α → (P.last.B a → C) → C) (a : P.A) (f : P.last.B a → P.last.W) (f' : TypeVec.Arrow (P.WPath (WType.mk a f)) α) : P.wpRec g (WType.mk a f) f' = g a f f' fun (i : P.last.B a) => P.wpRec g (f i) (P.wPathDestRight f' i)

theorem MvPFunctor.wp_ind {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α : TypeVec.{u_1} n} {C : (x : P.last.W) → TypeVec.Arrow (P.WPath x) α → Prop} (ih : ∀ (a : P.A) (f : P.last.B a → P.last.W) (f' : TypeVec.Arrow (P.WPath (WType.mk a f)) α), (∀ (i : P.last.B a), C (f i) (P.wPathDestRight f' i)) → C (WType.mk a f) f') (x : P.last.W) (f' : TypeVec.Arrow (P.WPath x) α) : C x f'

Now think of W as defined inductively by the data ⟨a, f', f⟩ where

• a : P.A is the shape of the top node
• f' : P.drop.B a ⟹ α is the contents of the top node
• f : P.last.B a → P.last.W are the subtrees

def MvPFunctor.wMk {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α : TypeVec.{u} n} (a : P.A) (f' : (P.drop.B a).Arrow α) (f : P.last.B a → P.W α) : P.W α

Constructor for W

def MvPFunctor.wRec {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α : TypeVec.{u} n} {C : Type u_1} (g : (a : P.A) → (P.drop.B a).Arrow α → (P.last.B a → P.W α) → (P.last.B a → C) → C) : P.W α → C

Recursor for W

theorem MvPFunctor.wRec_eq {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α : TypeVec.{u} n} {C : Type u_1} (g : (a : P.A) → (P.drop.B a).Arrow α → (P.last.B a → P.W α) → (P.last.B a → C) → C) (a : P.A) (f' : (P.drop.B a).Arrow α) (f : P.last.B a → P.W α) : P.wRec g (P.wMk a f' f) = g a f' f fun (i : P.last.B a) => P.wRec g (f i)

Defining equation for the recursor of W

theorem MvPFunctor.w_ind {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α : TypeVec.{u} n} {C : P.W α → Prop} (ih : ∀ (a : P.A) (f' : (P.drop.B a).Arrow α) (f : P.last.B a → P.W α), (∀ (i : P.last.B a), C (f i)) → C (P.wMk a f' f)) (x : P.W α) : C x

Induction principle for W

theorem MvPFunctor.w_cases {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α : TypeVec.{u} n} {C : P.W α → Prop} (ih : ∀ (a : P.A) (f' : (P.drop.B a).Arrow α) (f : P.last.B a → P.W α), C (P.wMk a f' f)) (x : P.W α) : C x

def MvPFunctor.wMap {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α β : TypeVec.{u} n} (g : α.Arrow β) : P.W α → P.W β

W-types are functorial

theorem MvPFunctor.wMk_eq {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α : TypeVec.{u} n} (a : P.A) (f : P.last.B a → P.last.W) (g' : (P.drop.B a).Arrow α) (g : (j : P.last.B a) → TypeVec.Arrow (P.WPath (f j)) α) : (P.wMk a g' fun (i : P.last.B a) => ⟨f i, g i⟩) = ⟨WType.mk a f, P.wPathCasesOn g' g⟩

theorem MvPFunctor.w_map_wMk {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α β : TypeVec.{u} n} (g : α.Arrow β) (a : P.A) (f' : (P.drop.B a).Arrow α) (f : P.last.B a → P.W α) : MvFunctor.map g (P.wMk a f' f) = P.wMk a (TypeVec.comp g f') fun (i : P.last.B a) => MvFunctor.map g (f i)

@[reducible, inline]

abbrev MvPFunctor.objAppend1 {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α : TypeVec.{u} n} {β : Type u} (a : P.A) (f' : (P.drop.B a).Arrow α) (f : P.last.B a → β) : ↑P (α ::: β)

Constructor of a value of P.obj (α ::: β) from components. Useful to avoid complicated type annotation

theorem MvPFunctor.map_objAppend1 {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α γ : TypeVec.{u} n} (g : α.Arrow γ) (a : P.A) (f' : (P.drop.B a).Arrow α) (f : P.last.B a → P.W α) : MvFunctor.map (g ::: P.wMap g) (P.objAppend1 a f' f) = P.objAppend1 a (TypeVec.comp g f') fun (x : P.last.B a) => P.wMap g (f x)

Yet another view of the W type: as a fixed point for a multivariate polynomial functor. These are needed to use the W-construction to construct a fixed point of a qpf, since the qpf axioms are expressed in terms of map on P.

def MvPFunctor.wMk' {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α : TypeVec.{u} n} : ↑P (α ::: P.W α) → P.W α

Constructor for the W-type of P

def MvPFunctor.wDest' {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α : TypeVec.{u} n} : P.W α → ↑P (α ::: P.W α)

Destructor for the W-type of P

theorem MvPFunctor.wDest'_wMk {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α : TypeVec.{u} n} (a : P.A) (f' : (P.drop.B a).Arrow α) (f : P.last.B a → P.W α) : P.wDest' (P.wMk a f' f) = ⟨a, TypeVec.splitFun f' f⟩

theorem MvPFunctor.wDest'_wMk' {n : ℕ} (P : MvPFunctor.{u} (n + 1)) {α : TypeVec.{u} n} (x : ↑P (α ::: P.W α)) : P.wDest' (P.wMk' x) = x