帮助SubSequences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).

现在我必须证明这种归纳类型的一系列属性,但我一直陷入困境.

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.

有人能帮助我前进吗？

Answer 1

实际上,直接对SubSet判断进行归纳更容易.但是,你需要尽可能一般,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"是一种检查归纳术语的策略,并为您提供了建立这样一个术语的所有可能方法!!没有任何归纳假设!它只给你建设性的前提.

您可以通过l1然后l2的归纳直接完成它,但是您必须手动构建正确的反演实例,因为您的归纳假设本来就很弱.

希望它有所帮助,V.