Coe.lean

/-
Copyright (c) 2020 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
prelude
import Init.Prelude
set_option linter.missingDocs true -- keep it documented

/-!
# Coercion

Lean uses a somewhat elaborate system of typeclasses to drive the coercion system.
Here a *coercion* means an invisible function that is automatically inserted
to fix what would otherwise be a type error. For example, if we have:
```
def f (x : Nat) : Int := x
```
then this is clearly not type correct as is, because `x` has type `Nat` but
type `Int` is expected, and normally you will get an error message saying exactly that.
But before it shows that message, it will attempt to synthesize an instance of
`CoeT Nat x Int`, which will end up going through all the other typeclasses defined
below, to discover that there is an instance of `Coe Nat Int` defined.

This instance is defined as:
```
instance : Coe Nat Int := ⟨Int.ofNat⟩
```
so Lean will elaborate the original function `f` as if it said:
```
def f (x : Nat) : Int := Int.ofNat x
```
which is not a type error anymore.

You can also use the `↑` operator to explicitly indicate a coercion. Using `↑x`
instead of `x` in the example will result in the same output.

Because there are many polymorphic functions in Lean, it is often ambiguous where
the coercion can go. For example:
```
def f (x y : Nat) : Int := x + y
```
This could be either `↑x + ↑y` where `+` is the addition on `Int`, or `↑(x + y)`
where `+` is addition on `Nat`, or even `x + y` using a heterogeneous addition
with the type `Nat → Nat → Int`. You can use the `↑` operator to disambiguate
between these possibilities, but generally Lean will elaborate working from the
"outside in", meaning that it will first look at the expression `_ + _ : Int`
and assign the `+` to be the one for `Int`, and then need to insert coercions
for the subterms `↑x : Int` and `↑y : Int`, resulting in the `↑x + ↑y` version.

Note that unlike most operators like `+`, `↑` is always eagerly unfolded at
parse time into its definition. So if we look at the definition of `f` from
before, we see no trace of the `CoeT.coe` function:
```
def f (x : Nat) : Int := x
#print f
-- def f : Nat → Int :=
-- fun (x : Nat) => Int.ofNat x
```

## Important typeclasses

Lean resolves a coercion by either inserting a `CoeDep` instance
or chaining `CoeHead? CoeOut* Coe* CoeTail?` instances.
(That is, zero or one `CoeHead` instances, an arbitrary number of `CoeOut`
instances, etc.)

The `CoeHead? CoeOut*` instances are chained from the "left" side.
So if Lean looks for a coercion from `Nat` to `Int`, it starts by trying coerce
`Nat` using `CoeHead` by looking for a `CoeHead Nat ?α` instance, and then
continuing with `CoeOut`.  Similarly `Coe* CoeTail?` are chained from the "right".

These classes should be implemented for coercions:

* `Coe α β` is the most basic class, and the usual one you will want to use
  when implementing a coercion for your own types.
  The variables in the type `α` must be a subset of the variables in `β`
  (or out-params of type class parameters),
  because `Coe` is chained right-to-left.

* `CoeOut α β` is like `Coe α β` but chained left-to-right.
  Use this if the variables in the type `α` are a superset of the variables in `β`.

* `CoeTail α β` is like `Coe α β`, but only applied once.
  Use this for coercions that would cause loops, like `[Ring R] → CoeTail Nat R`.

* `CoeHead α β` is similar to `CoeOut α β`, but only applied once.
  Use this for coercions that would cause loops, like `[SetLike S α] → CoeHead S (Set α)`.

* `CoeDep α (x : α) β` allows `β` to depend not only on `α` but on the value
  `x : α` itself. This is useful when the coercion function is dependent.
  An example of a dependent coercion is the instance for `Prop → Bool`, because
  it only holds for `Decidable` propositions. It is defined as:
  ```
  instance (p : Prop) [Decidable p] : CoeDep Prop p Bool := ...
  ```

* `CoeFun α (γ : α → Sort v)` is a coercion to a function. `γ a` should be a
  (coercion-to-)function type, and this is triggered whenever an element
  `f : α` appears in an application like `f x` which would not make sense since
  `f` does not have a function type.
  `CoeFun` instances apply to `CoeOut` as well.

* `CoeSort α β` is a coercion to a sort. `β` must be a universe, and this is
  triggered when `a : α` appears in a place where a type is expected, like
  `(x : a)` or `a → a`.
  `CoeSort` instances apply to `CoeOut` as well.

On top of these instances this file defines several auxiliary type classes:
  * `CoeTC := Coe*`
  * `CoeOTC := CoeOut* Coe*`
  * `CoeHTC := CoeHead? CoeOut* Coe*`
  * `CoeHTCT := CoeHead? CoeOut* Coe* CoeTail?`
  * `CoeDep := CoeHead? CoeOut* Coe* CoeTail? | CoeDep`

-/

universe u v w w'

/--
`Coe α β` is the typeclass for coercions from `α` to `β`. It can be transitively
chained with other `Coe` instances, and coercion is automatically used when
`x` has type `α` but it is used in a context where `β` is expected.
You can use the `↑x` operator to explicitly trigger coercion.
-/
class Coe (α : semiOutParam (Sort u)) (β : Sort v) where
  /-- Coerces a value of type `α` to type `β`. Accessible by the notation `↑x`,
  or by double type ascription `((x : α) : β)`. -/
  coe : α → β
attribute [coe_decl] Coe.coe

/--
Auxiliary class implementing `Coe*`.
Users should generally not implement this directly.
-/
class CoeTC (α : Sort u) (β : Sort v) where
  /-- Coerces a value of type `α` to type `β`. Accessible by the notation `↑x`,
  or by double type ascription `((x : α) : β)`. -/
  coe : α → β
attribute [coe_decl] CoeTC.coe

instance [Coe β γ] [CoeTC α β] : CoeTC α γ where coe a := Coe.coe (CoeTC.coe a : β)
instance [Coe α β] : CoeTC α β where coe a := Coe.coe a
instance : CoeTC α α where coe a := a

/--
`CoeOut α β` is for coercions that are applied from left-to-right.
-/
class CoeOut (α : Sort u) (β : semiOutParam (Sort v)) where
  /-- Coerces a value of type `α` to type `β`. Accessible by the notation `↑x`,
  or by double type ascription `((x : α) : β)`. -/
  coe : α → β
attribute [coe_decl] CoeOut.coe

/--
Auxiliary class implementing `CoeOut* Coe*`.
Users should generally not implement this directly.
-/
class CoeOTC (α : Sort u) (β : Sort v) where
  /-- Coerces a value of type `α` to type `β`. Accessible by the notation `↑x`,
  or by double type ascription `((x : α) : β)`. -/
  coe : α → β
attribute [coe_decl] CoeOTC.coe

instance [CoeOut α β] [CoeOTC β γ] : CoeOTC α γ where coe a := CoeOTC.coe (CoeOut.coe a : β)
instance [CoeTC α β] : CoeOTC α β where coe a := CoeTC.coe a
instance : CoeOTC α α where coe a := a

-- Note: ^^ We add reflexivity instances for CoeOTC/etc. so that we avoid going
-- through a user-defined CoeTC/etc. instance.  (Instances like
-- `CoeTC F (A →+ B)` apply even when the two sides are defeq.)

/--
`CoeHead α β` is for coercions that are applied from left-to-right at most once
at beginning of the coercion chain.
-/
class CoeHead (α : Sort u) (β : semiOutParam (Sort v)) where
  /-- Coerces a value of type `α` to type `β`. Accessible by the notation `↑x`,
  or by double type ascription `((x : α) : β)`. -/
  coe : α → β
attribute [coe_decl] CoeHead.coe

/--
Auxiliary class implementing `CoeHead CoeOut* Coe*`.
Users should generally not implement this directly.
-/
class CoeHTC (α : Sort u) (β : Sort v) where
  /-- Coerces a value of type `α` to type `β`. Accessible by the notation `↑x`,
  or by double type ascription `((x : α) : β)`. -/
  coe : α → β
attribute [coe_decl] CoeHTC.coe

instance [CoeHead α β] [CoeOTC β γ] : CoeHTC α γ where coe a := CoeOTC.coe (CoeHead.coe a : β)
instance [CoeOTC α β] : CoeHTC α β where coe a := CoeOTC.coe a
instance : CoeHTC α α where coe a := a

/--
`CoeTail α β` is for coercions that can only appear at the end of a
sequence of coercions. That is, `α` can be further coerced via `Coe σ α` and
`CoeHead τ σ` instances but `β` will only be the expected type of the expression.
-/
class CoeTail (α : semiOutParam (Sort u)) (β : Sort v) where
  /-- Coerces a value of type `α` to type `β`. Accessible by the notation `↑x`,
  or by double type ascription `((x : α) : β)`. -/
  coe : α → β
attribute [coe_decl] CoeTail.coe

/--
Auxiliary class implementing `CoeHead* Coe* CoeTail?`.
Users should generally not implement this directly.
-/
class CoeHTCT (α : Sort u) (β : Sort v) where
  /-- Coerces a value of type `α` to type `β`. Accessible by the notation `↑x`,
  or by double type ascription `((x : α) : β)`. -/
  coe : α → β
attribute [coe_decl] CoeHTCT.coe

instance [CoeTail β γ] [CoeHTC α β] : CoeHTCT α γ where coe a := CoeTail.coe (CoeHTC.coe a : β)
instance [CoeHTC α β] : CoeHTCT α β where coe a := CoeHTC.coe a
instance : CoeHTCT α α where coe a := a

/--
`CoeDep α (x : α) β` is a typeclass for dependent coercions, that is, the type `β`
can depend on `x` (or rather, the value of `x` is available to typeclass search
so an instance that relates `β` to `x` is allowed).

Dependent coercions do not participate in the transitive chaining process of
regular coercions: they must exactly match the type mismatch on both sides.
-/
class CoeDep (α : Sort u) (_ : α) (β : Sort v) where
  /-- The resulting value of type `β`. The input `x : α` is a parameter to
  the type class, so the value of type `β` may possibly depend on additional
  typeclasses on `x`. -/
  coe : β
attribute [coe_decl] CoeDep.coe

/--
`CoeT` is the core typeclass which is invoked by Lean to resolve a type error.
It can also be triggered explicitly with the notation `↑x` or by double type
ascription `((x : α) : β)`.

A `CoeT` chain has the grammar `CoeHead? CoeOut* Coe* CoeTail? | CoeDep`.
-/
class CoeT (α : Sort u) (_ : α) (β : Sort v) where
  /-- The resulting value of type `β`. The input `x : α` is a parameter to
  the type class, so the value of type `β` may possibly depend on additional
  typeclasses on `x`. -/
  coe : β
attribute [coe_decl] CoeT.coe

instance [CoeHTCT α β] : CoeT α a β where coe := CoeHTCT.coe a
instance [CoeDep α a β] : CoeT α a β where coe := CoeDep.coe a
instance : CoeT α a α where coe := a

/--
`CoeFun α (γ : α → Sort v)` is a coercion to a function. `γ a` should be a
(coercion-to-)function type, and this is triggered whenever an element
`f : α` appears in an application like `f x`, which would not make sense since
`f` does not have a function type.
`CoeFun` instances apply to `CoeOut` as well.
-/
class CoeFun (α : Sort u) (γ : outParam (α → Sort v)) where
  /-- Coerces a value `f : α` to type `γ f`, which should be either be a
  function type or another `CoeFun` type, in order to resolve a mistyped
  application `f x`. -/
  coe : (f : α) → γ f
attribute [coe_decl] CoeFun.coe

instance [CoeFun α fun _ => β] : CoeOut α β where coe a := CoeFun.coe a

/--
`CoeSort α β` is a coercion to a sort. `β` must be a universe, and this is
triggered when `a : α` appears in a place where a type is expected, like
`(x : a)` or `a → a`.
`CoeSort` instances apply to `CoeOut` as well.
-/
class CoeSort (α : Sort u) (β : outParam (Sort v)) where
  /-- Coerces a value of type `α` to `β`, which must be a universe. -/
  coe : α → β
attribute [coe_decl] CoeSort.coe

instance [CoeSort α β] : CoeOut α β where coe a := CoeSort.coe a

/--
`↑x` represents a coercion, which converts `x` of type `α` to type `β`, using
typeclasses to resolve a suitable conversion function. You can often leave the
`↑` off entirely, since coercion is triggered implicitly whenever there is a
type error, but in ambiguous cases it can be useful to use `↑` to disambiguate
between e.g. `↑x + ↑y` and `↑(x + y)`.
-/
syntax:1024 (name := coeNotation) "↑" term:1024 : term

/-- `⇑ t` coerces `t` to a function. -/
syntax:1024 (name := coeFunNotation) "⇑" term:1024 : term

/-- `↥ t` coerces `t` to a type. -/
syntax:1024 (name := coeSortNotation) "↥" term:1024 : term

/-! # Basic instances -/

instance boolToProp : Coe Bool Prop where
  coe b := Eq b true

instance boolToSort : CoeSort Bool Prop where
  coe b := b

instance decPropToBool (p : Prop) [Decidable p] : CoeDep Prop p Bool where
  coe := decide p

instance optionCoe {α : Type u} : Coe α (Option α) where
  coe := some

instance subtypeCoe {α : Sort u} {p : α → Prop} : CoeOut (Subtype p) α where
  coe v := v.val

/-! # Coe bridge -/

/--
Helper definition used by the elaborator. It is not meant to be used directly by users.

This is used for coercions between monads, in the case where we want to apply
a monad lift and a coercion on the result type at the same time.
-/
@[coe_decl] abbrev Lean.Internal.liftCoeM {m : Type u → Type v} {n : Type u → Type w} {α β : Type u}
    [MonadLiftT m n] [∀ a, CoeT α a β] [Monad n] (x : m α) : n β := do
  let a ← liftM x
  pure (CoeT.coe a)

/--
Helper definition used by the elaborator. It is not meant to be used directly by users.

This is used for coercing the result type under a monad.
-/
@[coe_decl] abbrev Lean.Internal.coeM {m : Type u → Type v} {α β : Type u}
    [∀ a, CoeT α a β] [Monad m] (x : m α) : m β := do
  let a ← x
  pure (CoeT.coe a)