Ltac 模式匹配：为什么 `forall x, ?P x` 不匹配 `forall x, x`？

Question

Ltac checkForall H :=
  let T := type of H in
  match T with
  | forall x, ?P x =>
    idtac
  | _ =>
    fail 1 "not a forall"
  end.

Example test : (forall x, x) -> True.
Proof.
  intros H.
  Fail checkForall H. (* not a forall *)
Abort.

我天真地期望 checkForall H 会成功，但事实并非如此。

在 Adam Chlipala Certified Programming with Dependent Types 一书中，discusses 模式匹配对依赖类型的限制：

The problem is that unification variables may not contain locally bound variables.

这是我在这里看到的行为的原因吗？

Answer 1

H的类型是forall x, x，不是forall x, P x。

Ltac checkForall H :=
  let T := type of H in
  match T with
  | forall x, ?P x =>
    idtac
  | forall x, x =>
    idtac "this matches"
  | _ =>
    fail 1 "not a forall"
  end.

Example test : (forall x, x) -> True.
Proof.
  intros H.
  checkForall H. (* not a forall *)

Abort.

或匹配您的 checkForall

Example test {T} (f:T->Prop)  : (forall x, f x) -> True.
Proof.
  intros H.
  checkForall H.

Answer 2

正如 larsr 所解释的，模式 ?P x 只能匹配在句法上属于应用程序的术语，它不涵盖您正在考虑的情况。但是，Ltac 确实提供了您正在寻找的匹配项的功能。正如 user manual 所说：

There is also a special notation for second-order pattern-matching problems: in an applicative pattern of the form @?id id1 …idn, the variable id matches any complex expression with (possible) dependencies in the variables id1 …idn and returns a functional term of the form fun id1 …idn => term.

因此，我们可以编写如下证明脚本：

Goal (forall x : Prop, x) -> False.
intros H.
match goal with
| H : forall x : Prop, @?P x |- _ => idtac P
end.

打印 (fun x : Prop => x).

Ltac 模式匹配：为什么 `forall x, ?P x` 不匹配 `forall x, x`？

Ltac pattern matching: why does `forall x, ?P x` not match `forall x, x`?

coq

coq-tactic

ltac