A type for VM-erased data

This file defines a type Erased α which is classically isomorphic to α, but erased in the VM. That is, at runtime every value of Erased α is represented as 0, just like types and proofs.

def Erased (α : Sort u) : Sort (max 1 u)

Erased α is the same as α, except that the elements of Erased α are erased in the VM in the same way as types and proofs. This can be used to track data without storing it literally.

Equations

• Erased α = ((s : α → Prop) ×' ∃ (a : α), (fun (b : α) => a = b) = s)

@[inline]

def Erased.mk {α : Sort u_1} (a : α) : Erased α

Erase a value.

Equations

• Erased.mk a = ⟨fun (b : α) => a = b, ⋯⟩

noncomputable def Erased.out {α : Sort u_1} : Erased α → α

Extracts the erased value, noncomputably.

Equations

• x.out = match x with | ⟨fst, h⟩ => Classical.choose h

@[reducible, inline]

abbrev Erased.OutType (a : Erased (Sort u)) : Sort u

Extracts the erased value, if it is a type.

Note: (mk a).OutType is not definitionally equal to a.

Equations

• a.OutType = a.out

theorem Erased.out_proof {p : Prop} (a : Erased p) : p

Extracts the erased value, if it is a proof.

@[simp]

theorem Erased.out_mk {α : Sort u_1} (a : α) : (Erased.mk a).out = a

@[simp]

theorem Erased.mk_out {α : Sort u_1} (a : Erased α) : Erased.mk a.out = a

theorem Erased.out_inj {α : Sort u_1} (a : Erased α) (b : Erased α) (h : a.out = b.out) : a = b

noncomputable def Erased.equiv (α : Sort u_1) : Erased α ≃ α

Equivalence between Erased α and α.

Equations

• Erased.equiv α = { toFun := Erased.out, invFun := Erased.mk, left_inv := ⋯, right_inv := ⋯ }

instance Erased.instRepr (α : Type u) : Repr (Erased α)

Equations

• Erased.instRepr α = { reprPrec := fun (x : Erased α) (x : ℕ) => Std.Format.text "Erased" }

instance Erased.instToString (α : Type u) : ToString (Erased α)

Equations

• Erased.instToString α = { toString := fun (x : Erased α) => "Erased" }

def Erased.choice {α : Sort u_1} (h : Nonempty α) : Erased α

Computably produce an erased value from a proof of nonemptiness.

Equations

• Erased.choice h = Erased.mk (Classical.choice h)

@[simp]

theorem Erased.nonempty_iff {α : Sort u_1} : Nonempty (Erased α) ↔ Nonempty α

instance Erased.instInhabitedOfNonempty {α : Sort u_1} [h : Nonempty α] : Inhabited (Erased α)

Equations

• Erased.instInhabitedOfNonempty = { default := Erased.choice h }

def Erased.bind {α : Sort u_1} {β : Sort u_2} (a : Erased α) (f : α → Erased β) : Erased β

(>>=) operation on Erased.

This is a separate definition because α and β can live in different universes (the universe is fixed in Monad).

Equations

• a.bind f = ⟨fun (b : β) => (f a.out).fst b, ⋯⟩

@[simp]

theorem Erased.bind_eq_out {α : Sort u_1} {β : Sort u_2} (a : Erased α) (f : α → Erased β) : a.bind f = f a.out

def Erased.join {α : Sort u_1} (a : Erased (Erased α)) : Erased α

Collapses two levels of erasure.

Equations

• a.join = a.bind id

@[simp]

theorem Erased.join_eq_out {α : Sort u_1} (a : Erased (Erased α)) : a.join = a.out

def Erased.map {α : Sort u_1} {β : Sort u_2} (f : α → β) (a : Erased α) : Erased β

(<$>) operation on Erased.

This is a separate definition because α and β can live in different universes (the universe is fixed in Functor).

Equations

• Erased.map f a = a.bind (Erased.mk ∘ f)

@[simp]

theorem Erased.map_out {α : Sort u_1} {β : Sort u_2} {f : α → β} (a : Erased α) : (Erased.map f a).out = f a.out

instance Erased.Monad : Monad Erased

Equations

• Erased.Monad = Monad.mk

@[simp]

theorem Erased.pure_def {α : Type u_1} : pure = Erased.mk

@[simp]

theorem Erased.bind_def {α : Type u_1} {β : Type u_1} : (fun (x : Erased α) (x_1 : α → Erased β) => x >>= x_1) = Erased.bind

@[simp]

theorem Erased.map_def {α : Type u_1} {β : Type u_1} : (fun (x : α → β) (x_1 : Erased α) => x <$> x_1) = Erased.map

instance Erased.instLawfulMonad : LawfulMonad Erased

Equations

• Erased.instLawfulMonad = ⋯