coq - 帮助子序列的 Coq 证明

Question

我有定义的归纳类型:

Inductive InL (A:Type) (y:A) : list A -> Prop := 
  | InHead : forall xs:list A, InL y (cons y xs) 
  | InTail : forall (x:A) (xs:list A), InL y xs -> InL y (cons x xs).

Inductive SubSeq (A:Type) : list A -> list A -> Prop :=
 | SubNil : forall l:list A, SubSeq nil l
 | SubCons1 : forall (x:A) (l1 l2:list A), SubSeq l1 l2 -> SubSeq l1 (x::l2)
 | SubCons2 : forall (x:A) (l1 l2:list A), SubSeq l1 l2 -> SubSeq (x::l1) (x::l2).

现在我必须证明该归纳类型的一系列属性，但我一直被卡住。

Lemma proof1: forall (A:Type) (x:A) (l1 l2:list A), SubSeq l1 l2 -> InL x l1 -> InL x l2.
Proof.
 intros.
 induction l1.
 induction l2.
 exact H0.

Qed.

有人可以帮助我前进吗？

Answer 1

其实直接对SubSet判断做归纳比较容易。
但是，您需要尽可能通用，所以这是我的建议:

Lemma proof1: forall (A:Type) (x:A) (l1 l2:list A), 
  SubSeq l1 l2 -> InL x l1 -> InL x l2.
(* first introduce your hypothesis, but put back x and In foo
   inside the goal, so that your induction hypothesis are correct*)
intros. 
revert x H0. induction H; intros.
(* x In [] is not possible, so inversion will kill the subgoal *)
inversion H0.

(* here it is straitforward: just combine the correct hypothesis *)
apply InTail; apply IHSubSeq; trivial.

(* x0 in x::l1 has to possible sources: x0 == x or x0 in l1 *)
inversion H0; subst; clear H0.
apply InHead.
apply InTail; apply IHSubSeq; trivial.
Qed.

“反转”是一种检查归纳项的策略，并为您提供构建此类项的所有可能方法!!没有任何归纳假设!!
它只给你 build 性的前提。

你可以直接通过对 l1 然后 l2 的归纳来完成它，但是你必须手工构建正确的反演实例，因为你的归纳假设真的很弱。

希望能帮助到你，
五、