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Inductive seq : nat -> Set :=
| niln : seq 0
| consn : forall n : nat, nat -> seq n -> seq (S n).
Fixpoint length (n : nat) (s : seq n) {struct s} : nat :=
match s with
| niln => 0
| consn i _ s' => S (length i s')
end.
Theorem length_corr : forall (n : nat) (s : seq n), length n s = n.
Proof.
intros n s.
(* reasoning by induction over s. Then, we have two new goals
corresponding on the case analysis about s (either it is
niln or some consn *)
induction s.
(* We are in the case where s is void. We can reduce the
term: length 0 niln *)
simpl.
(* We obtain the goal 0 = 0. *)
trivial.
(* now, we treat the case s = consn n e s with induction
hypothesis IHs *)
simpl.
(* The induction hypothesis has type length n s = n.
So we can use it to perform some rewriting in the goal: *)
rewrite IHs.
(* Now the goal is the trivial equality: S n = S n *)
trivial.
(* Now all sub cases are closed, we perform the ultimate
step: typing the term built using tactics and save it as
a witness of the theorem. *)
Qed.
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