prove that a list returned from a recursively defined function is fixed length in Coq

Question

How do I prove a lemma like the following:

Require Import Coq.Lists.List.

Lemma len_seq_n : forall start n, length (seq start n)=n.

I tried

Proof.
induction n.
simpl. auto. simpl.

and at this point Coq gives me

1 subgoal
start, n : nat
IHn : length (seq start n) = n
______________________________________(1/1)
S (length (seq (S start) n)) = S n

I'm not sure how to proceed from there.

score 3 · Accepted Answer · answered Sep 29 '17 at 03:07

3

The problem is that your induction hypothesis is not general enough. You need the following statement instead:

IHn : forall start', length (seq start' n) = n

To obtain this hypothesis, simply generalize over start before doing induction on n with the revert tactic.

Proof.
  intros start n.
  revert start.
  induction n.
  (* Continue as previously *)

(Next time, please provide a complete example so that we can help you better. Your question was missing the definition of seq.)

answered Sep 29 '17 at 03:07

1

IMVHO a slightly better style would be to reverse the declaration order of the parameters, this way you save the revert. – ejgallego Sep 29 '17 at 10:17
And there is the `Lemma seq_length : forall len start, length (seq start len) = len.` there. So, indeed, they stated it as ejgallego suggested. – Anton Trunov Sep 30 '17 at 12:06

1 Answers1