Question

作为一个粗略而无条件的背景，在HoTT中，可以推断出归纳定义的类型

Inductive paths {X : Type } : X -> X -> Type :=
 | idpath : forall x: X, paths x x.

允许非常一般的构造

Lemma transport {X : Type } (P : X -> Type ){ x y : X} (γ : paths x y):
  P x -> P y.
Proof.
 induction γ.
 exact (fun a => a).
Defined.

Lemma transport 将成为HoTT“替换”或“重写”策略的核心;据我所知，这个伎俩就是假设你或我可以抽象地认出的目标

...
H : paths x y
[ Q : (G x) ]
_____________
(G y)

找出必要的依赖类型G，以便我们可以apply (transport G H)。到目前为止，我所知道的只是那个

Ltac transport_along γ :=
match (type of γ) with 
| ?a ~~> ?b =>
 match goal with
 |- ?F b => apply (transport F γ)
 | _ => idtac "apparently couldn't abstract" b "from the goal."  end 
| _ => idtac "Are you sure" γ "is a path?" end.

不够通用。也就是说，第一个idtac经常被使用。

问题是

[有没有|什么是正确的事情？

Answer 1

对于类型中的关系使用重写有bug，这样您就可以只说rewrite <- y.

同时，

Ltac transport_along γ :=
  match (type of γ) with 
    | ?a ~~> ?b => pattern b; apply (transport _ y)
    | _ => idtac "Are you sure" γ "is a path?"
  end.

可能会做你想要的。

Answer 2

Tom Prince在his answer中提到的功能请求已被授予：

Require Import Coq.Setoids.Setoid Coq.Classes.CMorphisms.
Inductive paths {X : Type } : X -> X -> Type :=
| idpath : forall x: X, paths x x.

Lemma transport {X : Type } (P : X -> Type ){ x y : X} (γ : paths x y):
  P x -> P y.
Proof.
  induction γ.
  exact (fun a => a).
Defined.

Global Instance paths_Reflexive {A} : Reflexive (@paths A) := idpath.
Global Instance paths_Symmetric {A} : Symmetric (@paths A).
Proof. intros ?? []; constructor. Defined.

Global Instance proper_paths {A} (x : A) : Proper paths x := idpath x.
Global Instance paths_subrelation
       (A : Type) (R : crelation A)
       {RR : Reflexive R}
  : subrelation paths R.
Proof.
  intros ?? p.
  apply (transport _ p), RR.
Defined.
Global Instance reflexive_paths_dom_reflexive
       {B} {R' : crelation B} {RR' : Reflexive R'}
       {A : Type}
  : Reflexive (@paths A ==> R')%signature.
Proof. intros ??? []; apply RR'. Defined.

Goal forall (x y : nat) G, paths x y -> G x -> G y.
  intros x y G H Q.
  rewrite <- H.
  exact Q.
Qed.

我通过Set Typeclasses Debug setoid_rewrite <- H和H : paths x y时H : eq x y与C++的日志进行比较，找到了所需的实例。

在目标类型的子项上进行抽象抽象

2 个答案: