Goal:
  
  A : Type
  P : list A → A → Type
  a : A
  Γ : list A
  Δ : list A
  IHΓ : All_fold P (Γ ++ Δ)
        → All_fold P Δ × All_fold (λ Γ : list A, P (Γ ++ Δ)) Γ
  X : All_fold P (Γ ++ Δ)
  p : P (Γ ++ Δ) a
  ============================
   (All_fold (λ Γ : list A, P (Γ ++ Δ)) Γ)


DEBUG:Going to execute:
<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>)