Handling let in hypothesis

Question

As an exercise in Coq, I'm trying to prove that the following function returns a pair of lists of equal length.

Require Import List.

Fixpoint split (A B:Set)(x:list (A*B)) : (list A)*(list B) :=
match x with
|nil => (nil, nil)
|cons (a,b) x1 => let (ta, tb) := split A B x1 in (a::ta, b::tb)
end.

Theorem split_eq_len : forall (A B:Set)(x:list (A*B))(y:list A)(z:list B),(split A B x)=(y,z) -> length y = length z.
Proof.
intros A B x.
elim x.
simpl.
intros y z.
intros H.
injection H.
intros H1 H2.
rewrite <- H1.
rewrite <- H2.
reflexivity.
intros hx.
elim hx.
intros a b tx H y z.
simpl.
intro.

After the last step I get a hypothesis with a let statement inside, which I do not know how to handle:

1 subgoals
A : Set
B : Set
x : list (A * B)
hx : A * B
a : A
b : B
tx : list (A * B)
H : forall (y : list A) (z : list B),
    split A B tx = (y, z) -> length y = length z
y : list A
z : list B
H0 : (let (ta, tb) := split A B tx in (a :: ta, b :: tb)) = (y, z)
______________________________________(1/1)
length y = length z

Answer 1

You want to do destruct (split A B tx). This will break it up, binding the two pieces to ta and tb