mathlib documentation

data.pfunctor.multivariate.basic

Multivariate polynomial functors.

Multivariate polynomial functors are used for defining M-types and W-types. They map a type vector α to the type Σ a : A, B a ⟹ α, with A : Type and B : A → typevec n. They interact well with Lean's inductive definitions because they guarantee that occurrences of α are positive.

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structure mvpfunctor (n : ℕ) :
Type (u+1)

multivariate polynomial functors

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def mvpfunctor.obj {n : ℕ} (P : mvpfunctor n) (α : typevec n) :
Type u

Applying P to an object of Type

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def mvpfunctor.map {n : ℕ} (P : mvpfunctor n) {α β : typevec n} (f : α ⟹ β) (a : P.obj α) :
P.obj β

Applying P to a morphism of Type

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  • P.map f = λ (_x : P.obj α), mvpfunctor.map._match_1 P f _x
  • mvpfunctor.map._match_1 P f ⟨a, g⟩ = ⟨a, f ⊚ g⟩
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@[instance]
def mvpfunctor.inhabited {n : ℕ} :
inhabited (mvpfunctor n)

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@[instance]
def mvpfunctor.obj.inhabited {n : ℕ} (P : mvpfunctor n) {α : typevec n} [inhabited P.A] [Π (i : fin2 n), inhabited (α i)] :
inhabited (P.obj α)

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@[instance]
def mvpfunctor.mvfunctor {n : ℕ} (P : mvpfunctor n) :
mvfunctor P.obj

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theorem mvpfunctor.map_eq {n : ℕ} (P : mvpfunctor n) {α β : typevec n} (g : α ⟹ β) (a : P.A) (f : P.B a ⟹ α) :
g <$$> ⟨a, f⟩ = ⟨a, g ⊚ f⟩

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theorem mvpfunctor.id_map {n : ℕ} (P : mvpfunctor n) {α : typevec n} (x : P.obj α) :
typevec.id <$$> x = x

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theorem mvpfunctor.comp_map {n : ℕ} (P : mvpfunctor n) {α β γ : typevec n} (f : α ⟹ β) (g : β ⟹ γ) (x : P.obj α) :
(g ⊚ f) <$$> x = g <$$> f <$$> x

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@[instance]
def mvpfunctor.is_lawful_mvfunctor {n : ℕ} (P : mvpfunctor n) :
is_lawful_mvfunctor P.obj

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def mvpfunctor.const (n : ℕ) (A : Type u) :
mvpfunctor n

Constant functor where the input object does not affect the output

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def mvpfunctor.const.mk (n : ℕ) {A : Type u} (x : A) {α : typevec n} :
(mvpfunctor.const n A).obj α

Constructor for the constant functor

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def mvpfunctor.const.get {n : ℕ} {A : Type u} {α : typevec n} (x : (mvpfunctor.const n A).obj α) :
A

Destructor for the constant functor

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@[simp]
theorem mvpfunctor.const.get_map {n : ℕ} {A : Type u} {α β : typevec n} (f : α ⟹ β) (x : (mvpfunctor.const n A).obj α) :
mvpfunctor.const.get (f <$$> x) = mvpfunctor.const.get x

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@[simp]
theorem mvpfunctor.const.get_mk {n : ℕ} {A : Type u} {α : typevec n} (x : A) :
mvpfunctor.const.get (mvpfunctor.const.mk n x) = x

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@[simp]
theorem mvpfunctor.const.mk_get {n : ℕ} {A : Type u} {α : typevec n} (x : (mvpfunctor.const n A).obj α) :
mvpfunctor.const.mk n (mvpfunctor.const.get x) = x

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def mvpfunctor.comp {n m : ℕ} (P : mvpfunctor n) (Q : fin2 n → mvpfunctor m) :
mvpfunctor m

Functor composition on polynomial functors

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  • P.comp Q = {A := Σ (a₂ : P.A), Π (i : fin2 n), P.B a₂ i → (Q i).A, B := λ (a : Σ (a₂ : P.A), Π (i : fin2 n), P.B a₂ i → (Q i).A) (i : fin2 m), Σ (j : fin2 n) (b : P.B a.fst j), (Q j).B (a.snd j b) i}
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def mvpfunctor.comp.mk {n m : ℕ} {P : mvpfunctor n} {Q : fin2 n → mvpfunctor m} {α : typevec m} (x : P.obj (λ (i : fin2 n), (Q i).obj α)) :
(P.comp Q).obj α

Constructor for functor composition

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def mvpfunctor.comp.get {n m : ℕ} {P : mvpfunctor n} {Q : fin2 n → mvpfunctor m} {α : typevec m} (x : (P.comp Q).obj α) :
P.obj (λ (i : fin2 n), (Q i).obj α)

Destructor for functor composition

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theorem mvpfunctor.comp.get_map {n m : ℕ} {P : mvpfunctor n} {Q : fin2 n → mvpfunctor m} {α β : typevec m} (f : α ⟹ β) (x : (P.comp Q).obj α) :
mvpfunctor.comp.get (f <$$> x) = (λ (i : fin2 n) (x : (Q i).obj α), f <$$> x) <$$> mvpfunctor.comp.get x

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@[simp]
theorem mvpfunctor.comp.get_mk {n m : ℕ} {P : mvpfunctor n} {Q : fin2 n → mvpfunctor m} {α : typevec m} (x : P.obj (λ (i : fin2 n), (Q i).obj α)) :
mvpfunctor.comp.get (mvpfunctor.comp.mk x) = x

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@[simp]
theorem mvpfunctor.comp.mk_get {n m : ℕ} {P : mvpfunctor n} {Q : fin2 n → mvpfunctor m} {α : typevec m} (x : (P.comp Q).obj α) :
mvpfunctor.comp.mk (mvpfunctor.comp.get x) = x

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theorem mvpfunctor.liftp_iff {n : ℕ} {P : mvpfunctor n} {α : typevec n} (p : Π ⦃i : fin2 n⦄, α i → Prop) (x : P.obj α) :
mvfunctor.liftp p x ↔ ∃ (a : P.A) (f : P.B a ⟹ α), x = ⟨a, f⟩ ∧ ∀ (i : fin2 n) (j : P.B a i), p (f i j)

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theorem mvpfunctor.liftp_iff' {n : ℕ} {P : mvpfunctor n} {α : typevec n} (p : Π ⦃i : fin2 n⦄, α i → Prop) (a : P.A) (f : P.B a ⟹ α) :
mvfunctor.liftp p ⟨a, f⟩ ↔ ∀ (i : fin2 n) (x : P.B a i), p (f i x)

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theorem mvpfunctor.liftr_iff {n : ℕ} {P : mvpfunctor n} {α : typevec n} (r : Π ⦃i : fin2 n⦄, α i → α i → Prop) (x y : P.obj α) :
mvfunctor.liftr r x y ↔ ∃ (a : P.A) (f₀ f₁ : P.B a ⟹ α), x = ⟨a, f₀⟩ ∧ y = ⟨a, f₁⟩ ∧ ∀ (i : fin2 n) (j : P.B a i), r (f₀ i j) (f₁ i j)

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theorem mvpfunctor.supp_eq {n : ℕ} {P : mvpfunctor n} {α : typevec n} (a : P.A) (f : P.B a ⟹ α) (i : fin2 n) :
mvfunctor.supp ⟨a, f⟩ i = f i '' set.univ

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def mvpfunctor.drop {n : ℕ} (P : mvpfunctor (n + 1)) :
mvpfunctor n

Split polynomial functor, get a n-ary functor from a n+1-ary functor

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def mvpfunctor.last {n : ℕ} (P : mvpfunctor (n + 1)) :
pfunctor

Split polynomial functor, get a univariate functor from a n+1-ary functor

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def mvpfunctor.append_contents {n : ℕ} (P : mvpfunctor (n + 1)) {α : typevec n} {β : Type u_1} {a : P.A} (f' : P.drop.B a ⟹ α) (f : P.last.B a → β) :
P.B a ⟹ α ::: β

append arrows of a polynomial functor application

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