lean4-htt/src/Init/Coe.lean
2020-01-28 14:09:04 -08:00

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/-
Copyright (c) 2020 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
prelude
import Init.HasCoe -- import legacy HasCoe
import Init.Core
universes u v w
class Coe (α : Sort u) (β : Sort v) :=
(coe : α → β)
class CoeDep (α : Sort u) (a : α) (β : Sort v) :=
(coe : β)
/-- Auxiliary class that contains the transitive closure of `Coe` and `CoeDep`. -/
class CoeT (α : Sort u) (a : α) (β : Sort v) :=
(coe : β)
class CoeFun (α : Sort u) (a : α) (γ : outParam (Sort v)) :=
(coe : γ)
class CoeSort (α : Sort u) (a : α) (β : outParam (Sort v)) :=
(coe : β)
abbrev coeB {α : Sort u} {β : Sort v} (a : α) [Coe α β] : β :=
@Coe.coe α β _ a
abbrev coeD {α : Sort u} {β : Sort v} (a : α) [CoeDep α a β] : β :=
@CoeDep.coe α a β _
/-- Apply coercion manually. -/
abbrev coe {α : Sort u} {β : Sort v} (a : α) [CoeT α a β] : β :=
@CoeT.coe α a β _
abbrev coeFun {α : Sort u} {γ : Sort v} (a : α) [CoeFun α a γ] : γ :=
@CoeFun.coe α a γ _
abbrev coeSort {α : Sort u} {γ : Sort v} (a : α) [CoeSort α a γ] : γ :=
@CoeSort.coe α a γ _
instance coeDepTrans {α : Sort u} {β : Sort v} {δ : Sort w} (a : α) [CoeT α a β] [CoeDep β (coe a) δ] : CoeT α a δ :=
{ coe := coeD (coe a : β) }
instance coeTrans {α : Sort u} {β : Sort v} {δ : Sort w} (a : α) [CoeT α a β] [Coe β δ] : CoeT α a δ :=
{ coe := coeB (coe a : β) }
instance coeBase {α : Sort u} {β : Sort v} (a : α) [Coe α β] : CoeT α a β :=
{ coe := coeB a }
instance coeDepBase {α : Sort u} {β : Sort v} (a : α) [CoeDep α a β] : CoeT α a β :=
{ coe := coeD a }
instance coeFunTrans {α : Sort u} {β : Sort v} {γ : Sort w} (a : α) [CoeT α a β] [CoeFun β (coe a) γ] : CoeFun α a γ :=
{ coe := coeFun (coe a : β) }
instance coeSortTrans {α : Sort u} {β : Sort v} {δ : Sort w} (a : α) [CoeT α a β] [CoeSort β (coe a) δ] : CoeFun α a δ :=
{ coe := coeSort (coe a : β) }
/- Basic instances -/
instance boolToProp : Coe Bool Prop :=
{ coe := fun b => b = true }
instance coeDecidableEq (x : Bool) : Decidable (coe x) :=
inferInstanceAs (Decidable (x = true))
instance decPropToBool (p : Prop) [Decidable p] : CoeDep Prop p Bool :=
{ coe := decide p }
instance optionCoe {α : Type u} : Coe α (Option α) :=
{ coe := some }
instance subtypeCoe {α : Sort u} {p : α → Prop} (v : { x // p x }) : CoeT { x // p x } v α :=
{ coe := v.val }