Goal:
  
  A : Type
  B : Type
  f : nat → A → B
  l : list A
  a : A
  l' : list A
  IHl' : ∀ n : nat,
           mapi_rec f l' (#|l| + n) =
           mapi_rec (λ (n0 : nat) (x : A), f (#|l| + n0) x) l' n
  n : nat
  ============================
   (f (#|l| + n) a :: mapi_rec (λ (n : nat) (x : A), f (#|l| + n) x) l' (S n) =
    mapi_rec (λ (n : nat) (x : A), f (#|l| + n) x) (a :: l') n)


DEBUG:Going to execute:
let rec use_hyps :=
            match goal with
            | H:_ ∧ _ |- _ => exact H || (destruct_hyp H; use_hyps)
            | H:_ * _ |- _ => exact H || (destruct_hyp H; use_hyps)
            | H:_ |- _ => solve [ inversion H ]
            | _ => idtac
            end
           in
           let do_atom := solve
           [ trivial
           with eq_true
           | reflexivity
           | symmetry; trivial
           | contradiction
           | congruence ]
           in
           let rec do_ccl :=
            try do_atom; repeat (do_intro; try do_atom); try arith_congr;
             (solve [ split; do_ccl ])
           in
           (solve [ do_atom | use_hyps; do_ccl ]) ||
             fail "Cannot solve this goal"