I don't know how to prove a simple theorem used with fixpoint in Coq
You are making your life difficult by defining things in convoluted ways. Here's how a better definition of the same thing makes the proof easy.

Fixpoint f_better n0 len : Prop :=
  match len with
  | 0 =&gt; True
  | S k =&gt; x n0 /\ f_better (S n0) k
end.

Theorem t1_better: forall n0 len,
    x n0 /\ f_better (S n0) len &lt;-&gt; f_better n0 (S len).
Proof.
  easy.
Qed.


If you really insist on using your version, we can prove that as well, but it'll be painful.