Goal:
  
  A : Type
  B : Type
  P : A → B → Type
  t : A
  y : B
  l : list A
  l' : list B
  r : P t y
  Hall : All2 P l l'
  IHHall : ∀ n : nat,
             nth_error l n = Some t
             → ∑ t' : B, nth_error l' n = Some t' × P t t'
  ============================
   (Some y = Some y × P t y)


DEBUG:Going to execute:
fun flags =>
             <Coq.Init.Tauto.not_dep_intros>;
              repeat
               (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>)