Goal:
  
  A : Type
  B : Type
  P : A → nat → B → Type
  a : B
  ls1 : list B
  IH : ∀ (ls2 : list A) (n : nat),
         All (λ x : A, Alli (P x) n ls1) ls2 <~>
         Alli (λ (n0 : nat) (y : B), All (λ x : A, P x n0 y) ls2) n ls1
  a0 : A
  ls2 : list A
  n : nat
  X : All (λ x : A, Alli (P x) n (a :: ls1)) (a0 :: ls2)
  X0 : Alli (P a0) n (a :: ls1)
  X1 : All (λ x : A, Alli (P x) n (a :: ls1)) ls2
  ============================
   (Alli (λ (n : nat) (y : B), All (λ x : A, P x n y) (a0 :: ls2)) (S n) ls1)


DEBUG:Going to execute:
match goal with
           | H:¬ _ |- _ => solve [ case H; trivial ]
           end
Level 1: Backtrace:

No such assumption.
Level 0: Backtrace:

No such assumption.
Level 0: Backtrace:

No such assumption.
Level 0: Backtrace:

No such assumption.