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 P Δ)


DEBUG:Going to execute:
<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