Goal:
  
  A : Type
  P : list A → list A → A → A → Type
  n : nat
  a : A
  Γ : list A
  Γ' : list A
  d : A
  IHn : ∀ (Γ Γ' : list A) (d : A),
          All2_fold P Γ Γ'
          → nth_error Γ n = Some d
            → ∑ d' : A,
                nth_error Γ' n = Some d'
                × (let Γs := skipn (S n) Γ in
                   let Γs' := skipn (S n) Γ' in
                   All2_fold P Γs Γs' × P Γs Γs' d d')
  Hrel1' : list A
  Hrel1 : list A
  Hrel : All2_fold P (pr1 {| pr1 := Hrel1'; pr2 := Hrel1 |})
           (pr2 {| pr1 := Hrel1'; pr2 := Hrel1 |})
  ============================
   ({| pr1 := a :: Γ; pr2 := Γ' |} = {| pr1 := Hrel1'; pr2 := Hrel1 |}
    → let H := CoreTactics.block in
      nth_error (a :: Γ) (S n) = Some d
      → let H0 := CoreTactics.block in
        ∑ d' : A,
          nth_error Γ' (S n) = Some d'
          × (let Γs := skipn (S (S n)) (a :: Γ) in
             let Γs' := skipn (S (S n)) Γ' in
             All2_fold P Γs Γs' × P Γs Γs' d d'))


DEBUG:Going to execute:
hnf in id;
            match type of id with
            | sigma _ =>
                let xn := fresh id "'" in
                destruct id as [xn id]; DepElim.decompose_exists xn id;
                 cbv beta iota delta [TODO projection TODO projection]
                  in id, id'; DepElim.decompose_exists id id'
            | _ =>
                cbv beta iota delta [TODO projection TODO projection]
                 in id, id'
            end