Polynomial functors

This file defines polynomial functors and the W-type construction as a polynomial functor. (For the M-type construction, see pfunctor/M.lean.)

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structure PFunctor : Type (u+1)

• The head type

  A : Type u

• The child family of types

  B : A → Type u

A polynomial functor P is given by a type A and a family B of types over A. P maps any type α to a new type P.obj α, which is defined as the sigma type Σ x, P.B x → α.

An element of P.obj α is a pair ⟨a, f⟩, where a is an element of a type A and f : B a → α. Think of a as the shape of the object and f as an index to the relevant elements of α.

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instance PFunctor.instInhabitedPFunctor : Inhabited PFunctor

Equations

• PFunctor.instInhabitedPFunctor = { default := { A := default, B := default } }

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def PFunctor.Obj (P : PFunctor) (α : Type u_2) : Type (maxu_1u_2u_1)

Applying P to an object of Type

Equations

• PFunctor.Obj P α = ((x : P.A) × (PFunctor.B P x → α))

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def PFunctor.map (P : PFunctor) {α : Type u_2} {β : Type u_3} (f : α → β) : PFunctor.Obj P α → PFunctor.Obj P β

Applying P to a morphism of Type

Equations

• PFunctor.map P f x = match x with | { fst := a, snd := g } => { fst := a, snd := f ∘ g }

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instance PFunctor.Obj.inhabited (P : PFunctor) {α : Type u} [inst : Inhabited P.A] [inst : Inhabited α] : Inhabited (PFunctor.Obj P α)

Equations

• PFunctor.Obj.inhabited P = { default := { fst := default, snd := default } }

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instance PFunctor.instFunctorObj (P : PFunctor) : Functor (PFunctor.Obj P)

Equations

• PFunctor.instFunctorObj P = { map := @PFunctor.map P, mapConst := fun {α β} => PFunctor.map P ∘ Function.const β }

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theorem PFunctor.map_eq (P : PFunctor) {α : Type u_1} {β : Type u_1} (f : α → β) (a : P.A) (g : PFunctor.B P a → α) : f <$> { fst := a, snd := g } = { fst := a, snd := f ∘ g }

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theorem PFunctor.id_map (P : PFunctor) {α : Type u_1} (x : PFunctor.Obj P α) : id <$> x = id x

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theorem PFunctor.comp_map (P : PFunctor) {α : Type u_1} {β : Type u_1} {γ : Type u_1} (f : α → β) (g : β → γ) (x : PFunctor.Obj P α) : (g ∘ f) <$> x = g <$> f <$> x

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instance PFunctor.instLawfulFunctorObjInstFunctorObj (P : PFunctor) : LawfulFunctor (PFunctor.Obj P)

Equations

• PFunctor.instLawfulFunctorObjInstFunctorObj = PFunctor.instLawfulFunctorObjInstFunctorObj.proof_1

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def PFunctor.W (P : PFunctor) : Type u_1

re-export existing definition of W-types and adapt it to a packaged definition of polynomial functor

Equations

• PFunctor.W P = WType P.B

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def PFunctor.W.head {P : PFunctor} : PFunctor.W P → P.A

root element of a W tree

Equations

• PFunctor.W.head x = match x with | WType.mk a _f => a

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def PFunctor.W.children {P : PFunctor} (x : PFunctor.W P) : PFunctor.B P (PFunctor.W.head x) → PFunctor.W P

children of the root of a W tree

Equations

• PFunctor.W.children x = match (motive := (x : PFunctor.W P) → PFunctor.B P (PFunctor.W.head x) → PFunctor.W P) x with | WType.mk _a f => f

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def PFunctor.W.dest {P : PFunctor} : PFunctor.W P → PFunctor.Obj P (PFunctor.W P)

destructor for W-types

Equations

• PFunctor.W.dest x = match x with | WType.mk a f => { fst := a, snd := f }

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def PFunctor.W.mk {P : PFunctor} : PFunctor.Obj P (PFunctor.W P) → PFunctor.W P

constructor for W-types

Equations

• PFunctor.W.mk x = match x with | { fst := a, snd := f } => WType.mk a f

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@[simp]

theorem PFunctor.W.dest_mk {P : PFunctor} (p : PFunctor.Obj P (PFunctor.W P)) : PFunctor.W.dest (PFunctor.W.mk p) = p

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@[simp]

theorem PFunctor.W.mk_dest {P : PFunctor} (p : PFunctor.W P) : PFunctor.W.mk (PFunctor.W.dest p) = p

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def PFunctor.IdxCat (P : PFunctor) : Type u_1

Idx identifies a location inside the application of a pfunctor. For F : pfunctor, x : F.obj α and i : F.Idx, i can designate one part of x or is invalid, if i.1 ≠ x.1

Equations

• PFunctor.IdxCat P = ((x : P.A) × PFunctor.B P x)

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instance PFunctor.IdxCat.inhabited (P : PFunctor) [inst : Inhabited P.A] [inst : Inhabited (PFunctor.B P default)] : Inhabited (PFunctor.IdxCat P)

Equations

• PFunctor.IdxCat.inhabited P = { default := { fst := default, snd := default } }

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def PFunctor.Obj.iget {P : PFunctor} [inst : DecidableEq P.A] {α : Type u_2} [inst : Inhabited α] (x : PFunctor.Obj P α) (i : PFunctor.IdxCat P) : α

x.iget i takes the component of x designated by i if any is or returns a default value

Equations

• PFunctor.Obj.iget x i = if h : i.fst = x.fst then Sigma.snd P.A (fun x => PFunctor.B P x → α) x (cast (_ : PFunctor.B P i.fst = PFunctor.B P x.fst) i.snd) else default

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@[simp]

theorem PFunctor.fst_map {P : PFunctor} {α : Type u} {β : Type u} (x : PFunctor.Obj P α) (f : α → β) : (f <$> x).fst = x.fst

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@[simp]

theorem PFunctor.iget_map {P : PFunctor} [inst : DecidableEq P.A] {α : Type u} {β : Type u} [inst : Inhabited α] [inst : Inhabited β] (x : PFunctor.Obj P α) (f : α → β) (i : PFunctor.IdxCat P) (h : i.fst = x.fst) : PFunctor.Obj.iget (f <$> x) i = f (PFunctor.Obj.iget x i)

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def PFunctor.comp (P₂ : PFunctor) (P₁ : PFunctor) : PFunctor

functor composition for polynomial functors

Equations

• One or more equations did not get rendered due to their size.

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def PFunctor.comp.mk (P₂ : PFunctor) (P₁ : PFunctor) {α : Type} (x : PFunctor.Obj P₂ (PFunctor.Obj P₁ α)) : PFunctor.Obj (PFunctor.comp P₂ P₁) α

constructor for composition

Equations

• One or more equations did not get rendered due to their size.

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def PFunctor.comp.get (P₂ : PFunctor) (P₁ : PFunctor) {α : Type} (x : PFunctor.Obj (PFunctor.comp P₂ P₁) α) : PFunctor.Obj P₂ (PFunctor.Obj P₁ α)

destructor for composition

Equations

• One or more equations did not get rendered due to their size.

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theorem PFunctor.liftp_iff {P : PFunctor} {α : Type u} (p : α → Prop) (x : PFunctor.Obj P α) : Functor.Liftp p x ↔ ∃ a f, x = { fst := a, snd := f } ∧ ((i : PFunctor.B P a) → p (f i))

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theorem PFunctor.liftp_iff' {P : PFunctor} {α : Type u} (p : α → Prop) (a : P.A) (f : PFunctor.B P a → α) : Functor.Liftp p { fst := a, snd := f } ↔ (i : PFunctor.B P a) → p (f i)

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theorem PFunctor.liftr_iff {P : PFunctor} {α : Type u} (r : α → α → Prop) (x : PFunctor.Obj P α) (y : PFunctor.Obj P α) : Functor.Liftr r x y ↔ ∃ a f₀ f₁, x = { fst := a, snd := f₀ } ∧ y = { fst := a, snd := f₁ } ∧ ((i : PFunctor.B P a) → r (f₀ i) (f₁ i))

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theorem PFunctor.supp_eq {P : PFunctor} {α : Type u} (a : P.A) (f : PFunctor.B P a → α) : Functor.supp { fst := a, snd := f } = f '' Set.univ