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
  ============================
   ((nth_error (x :: l) 0 = Some x ∧ nth_error (y :: l') 0 = Some y) × P x y)


DEBUG:Going to execute:
fun flags t_reduce t_solver =>
             let rec t_tauto_intuit :=
              <Coq.Init.Tauto.simplif> flags;
               <Coq.Init.Tauto.axioms> flags ||
                 match reverse goal with
                 | id:(?X1 → ?X2) → ?X3
                   |- _ =>
                       Ltac.cut X3;
                        [ intro; clear id; t_tauto_intuit
                        | Ltac.cut (X1 → X2);
                           [ exact
                           id
                           | generalize (λ y : X2, id (λ x : X1, y)); intro;
                              clear id; (solve [ t_tauto_intuit ]) ] ]
                 | id:¬ ?X1 → ?X3
                   |- _ =>
                       Ltac.cut X3;
                        [ intro; clear id; t_tauto_intuit
                        | Ltac.cut (¬ X1);
                           [ exact
                           id
                           | clear id; intro; (solve [ t_tauto_intuit ]) ] ]
                 | |- ?X1 =>
                       <Coq.Init.Tauto.is_disj> flags X1; (solve
                        [ Ltac.left; t_tauto_intuit
                        | Ltac.right; t_tauto_intuit ])
                 end ||
                   match goal with
                   | |- _ → _ => intro; t_tauto_intuit
                   | |- _ => t_reduce; t_solver
                   end || t_solver
             in
             t_tauto_intuit