Goal:
  
  A : Type
  a : A
  l : list A
  n : nat
  P : A → bool
  d : A
  IHl : ∀ n : nat, forallb P l → n < #|l| → ∃ x : A, nth n l d = x ∧ P x
  ============================
   (P a && forallb P l → S n < S #|l| → ∃ x : A, nth n l d = x ∧ P x)


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