Goal:
  
  A : Type
  B : Type
  P : A → B → Type
  x : A
  y : B
  l : list A
  l' : list B
  r : P x y
  HF : All2 P l l'
  IHHF : Alli
           (λ (i : nat) (x : A),
              ∑ y : B,
                (nth_error l i = Some x ∧ nth_error l' i = Some y) × P x y) 0
           l
  ============================
   (∑ y0 : B,
      (nth_error (x :: l) 0 = Some x ∧ nth_error (y :: l') 0 = Some y0)
      × P x y0)


DEBUG:Going to execute:
match reverse goal with
           | id:?X1
             |- _ =>
                 <Coq.Init.Tauto.is_conj> flags X1; elim id; do 2 intro;
                  clear id
           | id:_ ↔ _ |- _ => elim id; do 2 intro; clear id
           | id:¬ _ |- _ => red in id
           | id:?X1
             |- _ =>
                 <Coq.Init.Tauto.is_disj> flags X1; elim id; intro; clear id
           | _ =>
               <Coq.Init.Tauto.find_cut>
                ltac:(fun id0 id1 X2 =>
                        assert X2; [ exact (id0 id1) | clear id0 ])
           | id:?X1 → ?X2
             |- _ =>
                 <Coq.Init.Tauto.is_unit_or_eq> flags X1; Ltac.cut X2;
                  [ intro; clear id
                  | Ltac.cut X1; [ exact id | constructor 1; fail ] ]
           | id:?X1 → ?X2
             |- _ =>
                 <Coq.Init.Tauto.flatten_contravariant_conj> flags X1 X2 id
           | id:?X1 ↔ ?X2 → ?X3
             |- _ =>
                 assert ((X1 → X2) → (X2 → X1) → X3) by
                  (do 2 intro; apply id; Ltac.split; Ltac.assumption); clear
                  id
           | id:?X1 → ?X2
             |- _ =>
                 <Coq.Init.Tauto.flatten_contravariant_disj> flags X1 X2 id
           | |- ?X1 => <Coq.Init.Tauto.is_conj> flags X1; Ltac.split
           | |- _ ↔ _ => Ltac.split
           | |- ¬ _ => red
           end; <Coq.Init.Tauto.not_dep_intros>