The M construction as a multivariate polynomial functor.
M types are potentially infinite tree-like structures. They are defined as the greatest fixpoint of a polynomial functor.
Main definitions
M.mk- constructorM.dest- destructorM.corec- corecursor: useful for formulating infinite, productive computationsM.bisim- bisimulation: proof technique to show the equality of infinite objects
Implementation notes
Dual view of M-types:
Mp: polynomial functorM: greatest fixed point of a polynomial functor
Specifically, we define the polynomial functor Mp as:
- A := a possibly infinite tree-like structure without information in the nodes
- B := given the tree-like structure
t,B tis a valid path from the root oftto any given node.
As a result Mp.obj α is made of a dataless tree and a function from
its valid paths to values of α
The difference with the polynomial functor of an initial algebra is
that A is a possibly infinite tree.
Reference
- [Jeremy Avigad, Mario M. Carneiro and Simon Hudon, Data Types as Quotients of Polynomial Functors][avigad-carneiro-hudon2019]
- root : Π {n : ℕ} (P : mvpfunctor (n + 1)) (x : P.last.M) (a : P.A) (f : P.last.B a → P.last.M), x.dest = ⟨a, f⟩ → Π (i : fin2 n), P.drop.B a i → mvpfunctor.M.path P x i
- child : Π {n : ℕ} (P : mvpfunctor (n + 1)) (x : P.last.M) (a : P.A) (f : P.last.B a → P.last.M), x.dest = ⟨a, f⟩ → Π (j : P.last.B a) (i : fin2 n), mvpfunctor.M.path P (f j) i → mvpfunctor.M.path P x i
A path from the root of a tree to one of its node
@[instance]
def
mvpfunctor.M.path.inhabited
{n : ℕ}
(P : mvpfunctor (n + 1))
(x : P.last.M)
{i : fin2 n}
[inhabited (P.drop.B x.head i)] :
inhabited (mvpfunctor.M.path P x i)
Equations
- mvpfunctor.M.path.inhabited P x = {default := mvpfunctor.M.path.root x x.head x.children _ i (inhabited.default (P.drop.B x.head i))}
Polynomial functor of the M-type of P. A is a data-less
possibly infinite 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
n-ary M-type for P
@[instance]
Equations
- P.mvfunctor_M = id P.Mp.mvfunctor
@[instance]
def
mvpfunctor.inhabited_M
{n : ℕ}
(P : mvpfunctor (n + 1))
{α : typevec n}
[I : inhabited P.A]
[Π (i : fin2 n), inhabited (α i)] :
Equations
construct through corecursion the shape of an M-type without its contents
Equations
- mvpfunctor.M.corec_shape P g₀ g₂ = pfunctor.M.corec (λ (b : β), ⟨g₀ b, g₂ b⟩)
Proof of type equality as a function
Equations
- P.cast_lastB h = λ (b : P.last.B a), h.rec_on b
def
mvpfunctor.M.corec_contents
{n : ℕ}
(P : mvpfunctor (n + 1))
{α : typevec n}
{β : Type u}
(g₀ : β → P.A)
(g₁ : Π (b : β), (P.drop.B (g₀ b)).arrow α)
(g₂ : Π (b : β), P.last.B (g₀ b) → β)
(x : P.last.M)
(b : β) :
x = mvpfunctor.M.corec_shape P g₀ g₂ b → typevec.arrow (mvpfunctor.M.path P x) α
Using corecursion, construct the contents of an M-type
Equations
- mvpfunctor.M.corec_contents P g₀ g₁ g₂ x b h i (mvpfunctor.M.path.child x a f h' j i c) = have h₀ : a = g₀ b, from _, have h₁ : f j = mvpfunctor.M.corec_shape P g₀ g₂ (g₂ b (P.cast_lastB h₀ j)), from _, mvpfunctor.M.corec_contents P g₀ g₁ g₂ (f j) (g₂ b (P.cast_lastB h₀ j)) h₁ i c
- mvpfunctor.M.corec_contents P g₀ g₁ g₂ x b h i (mvpfunctor.M.path.root x a f h' i c) = have this : a = g₀ b, from _, g₁ b i (P.cast_dropB this i c)
def
mvpfunctor.M.corec'
{n : ℕ}
(P : mvpfunctor (n + 1))
{α : typevec n}
{β : Type u}
(g₀ : β → P.A) :
Corecursor for M-type of P
Equations
- mvpfunctor.M.corec' P g₀ g₁ g₂ = λ (b : β), ⟨mvpfunctor.M.corec_shape P g₀ g₂ b, mvpfunctor.M.corec_contents P g₀ g₁ g₂ (mvpfunctor.M.corec_shape P g₀ g₂ b) b _⟩
Corecursor for M-type of P
Equations
- mvpfunctor.M.corec P g = mvpfunctor.M.corec' P (λ (b : β), (g b).fst) (λ (b : β), typevec.drop_fun (g b).snd) (λ (b : β), typevec.last_fun (g b).snd)
def
mvpfunctor.M.path_dest_left
{n : ℕ}
(P : mvpfunctor (n + 1))
{α : typevec n}
{x : P.last.M}
{a : P.A}
{f : P.last.B a → P.last.M} :
x.dest = ⟨a, f⟩ → typevec.arrow (mvpfunctor.M.path P x) α → (P.drop.B a).arrow α
Implementation of destructor for M-type of P
Equations
- mvpfunctor.M.path_dest_left P h f' = λ (i : fin2 n) (c : P.drop.B a i), f' i (mvpfunctor.M.path.root x a f h i c)
def
mvpfunctor.M.path_dest_right
{n : ℕ}
(P : mvpfunctor (n + 1))
{α : typevec n}
{x : P.last.M}
{a : P.A}
{f : P.last.B a → P.last.M}
(h : x.dest = ⟨a, f⟩)
(f' : typevec.arrow (mvpfunctor.M.path P x) α)
(j : P.last.B a) :
typevec.arrow (mvpfunctor.M.path P (f j)) α
Implementation of destructor for M-type of P
Equations
- mvpfunctor.M.path_dest_right P h f' = λ (j : P.last.B a) (i : fin2 n) (c : mvpfunctor.M.path P (f j) i), f' i (mvpfunctor.M.path.child x a f h j i c)
def
mvpfunctor.M.dest'
{n : ℕ}
(P : mvpfunctor (n + 1))
{α : typevec n}
{x : P.last.M}
{a : P.A}
{f : P.last.B a → P.last.M} :
x.dest = ⟨a, f⟩ → typevec.arrow (mvpfunctor.M.path P x) α → P.obj (α ::: P.M α)
Destructor for M-type of P
Equations
- mvpfunctor.M.dest' P h f' = ⟨a, typevec.split_fun (mvpfunctor.M.path_dest_left P h f') (λ (x_1 : (P.B a).last), ⟨f x_1, mvpfunctor.M.path_dest_right P h f' x_1⟩)⟩
Destructor for M-types
Equations
- mvpfunctor.M.dest P x = mvpfunctor.M.dest' P _ x.snd
Constructor for M-types
Equations
- mvpfunctor.M.mk P = mvpfunctor.M.corec P (λ (i : P.obj (α ::: P.M α)), mvfunctor.map (typevec.id ::: mvpfunctor.M.dest P) i)
theorem
mvpfunctor.M.dest'_eq_dest'
{n : ℕ}
(P : mvpfunctor (n + 1))
{α : typevec n}
{x : P.last.M}
{a₁ : P.A}
{f₁ : P.last.B a₁ → P.last.M}
(h₁ : x.dest = ⟨a₁, f₁⟩)
{a₂ : P.A}
{f₂ : P.last.B a₂ → P.last.M}
(h₂ : x.dest = ⟨a₂, f₂⟩)
(f' : typevec.arrow (mvpfunctor.M.path P x) α) :
mvpfunctor.M.dest' P h₁ f' = mvpfunctor.M.dest' P h₂ f'
theorem
mvpfunctor.M.dest_eq_dest'
{n : ℕ}
(P : mvpfunctor (n + 1))
{α : typevec n}
{x : P.last.M}
{a : P.A}
{f : P.last.B a → P.last.M}
(h : x.dest = ⟨a, f⟩)
(f' : typevec.arrow (mvpfunctor.M.path P x) α) :
mvpfunctor.M.dest P ⟨x, f'⟩ = mvpfunctor.M.dest' P h f'
theorem
mvpfunctor.M.dest_corec'
{n : ℕ}
(P : mvpfunctor (n + 1))
{α : typevec n}
{β : Type u}
(g₀ : β → P.A)
(g₁ : Π (b : β), (P.drop.B (g₀ b)).arrow α)
(g₂ : Π (b : β), P.last.B (g₀ b) → β)
(x : β) :
mvpfunctor.M.dest P (mvpfunctor.M.corec' P g₀ g₁ g₂ x) = ⟨g₀ x, typevec.split_fun (g₁ x) (mvpfunctor.M.corec' P g₀ g₁ g₂ ∘ g₂ x)⟩
theorem
mvpfunctor.M.dest_corec
{n : ℕ}
(P : mvpfunctor (n + 1))
{α : typevec n}
{β : Type u}
(g : β → P.obj (α ::: β))
(x : β) :
mvpfunctor.M.dest P (mvpfunctor.M.corec P g x) = mvfunctor.map (typevec.id ::: mvpfunctor.M.corec P g) (g x)
theorem
mvpfunctor.M.bisim_lemma
{n : ℕ}
(P : mvpfunctor (n + 1))
{α : typevec n}
{a₁ : P.Mp.A}
{f₁ : (P.Mp.B a₁).arrow α}
{a' : P.A}
{f' : (P.B a').drop.arrow α}
{f₁' : (P.B a').last → P.M α} :
mvpfunctor.M.dest P ⟨a₁, f₁⟩ = ⟨a', typevec.split_fun f' f₁'⟩ → (∃ (g₁' : P.last.B a' → P.last.M) (e₁' : pfunctor.M.dest a₁ = ⟨a', g₁'⟩), f' = mvpfunctor.M.path_dest_left P e₁' f₁ ∧ f₁' = λ (x : P.last.B a'), ⟨g₁' x, mvpfunctor.M.path_dest_right P e₁' f₁ x⟩)
theorem
mvpfunctor.M.bisim
{n : ℕ}
(P : mvpfunctor (n + 1))
{α : typevec n}
(R : P.M α → P.M α → Prop)
(h : ∀ (x y : P.M α), R x y → (∃ (a : P.A) (f : (P.B a).drop.arrow (α ::: P.M α).drop) (f₁ f₂ : (P.B a).last → (α ::: P.M α).last), mvpfunctor.M.dest P x = ⟨a, typevec.split_fun f f₁⟩ ∧ mvpfunctor.M.dest P y = ⟨a, typevec.split_fun f f₂⟩ ∧ ∀ (i : (P.B a).last), R (f₁ i) (f₂ i)))
(x y : P.M α) :
R x y → x = y
theorem
mvpfunctor.M.bisim₀
{n : ℕ}
(P : mvpfunctor (n + 1))
{α : typevec n}
(R : P.M α → P.M α → Prop)
(h₀ : equivalence R)
(h : ∀ (x y : P.M α), R x y → mvfunctor.map (typevec.id ::: quot.mk R) (mvpfunctor.M.dest P x) = mvfunctor.map (typevec.id ::: quot.mk R) (mvpfunctor.M.dest P y))
(x y : P.M α) :
R x y → x = y
theorem
mvpfunctor.M.bisim'
{n : ℕ}
(P : mvpfunctor (n + 1))
{α : typevec n}
(R : P.M α → P.M α → Prop)
(h : ∀ (x y : P.M α), R x y → mvfunctor.map (typevec.id ::: quot.mk R) (mvpfunctor.M.dest P x) = mvfunctor.map (typevec.id ::: quot.mk R) (mvpfunctor.M.dest P y))
(x y : P.M α) :
R x y → x = y
theorem
mvpfunctor.M.dest_map
{n : ℕ}
(P : mvpfunctor (n + 1))
{α β : typevec n}
(g : α.arrow β)
(x : P.M α) :
mvpfunctor.M.dest P (mvfunctor.map g x) = mvfunctor.map (g ::: λ (x : P.M α), mvfunctor.map g x) (mvpfunctor.M.dest P x)
theorem
mvpfunctor.M.map_dest
{n : ℕ}
(P : mvpfunctor (n + 1))
{α β : typevec n}
(g : (α ::: P.M α).arrow (β ::: P.M β))
(x : P.M α) :
(∀ (x : P.M α), typevec.last_fun g x = mvfunctor.map (typevec.drop_fun g) x) → mvfunctor.map g (mvpfunctor.M.dest P x) = mvpfunctor.M.dest P (mvfunctor.map (typevec.drop_fun g) x)