Goal:
  
  A : Type
  P : list A → list A → A → A → Type
  Γ : list A
  Γ' : list A
  a : A
  Δ : list A
  a0 : A
  Δ' : list A
  H : S #|Δ| = S #|Δ'|
  IHΔ : All2_fold P (Δ ++ Γ) (Δ' ++ Γ')
        → All2_fold P Γ Γ'
          × All2_fold (λ Δ Δ' : list A, P (Δ ++ Γ) (Δ' ++ Γ')) Δ Δ'
  H'1' : list A
  H'1 : list A
  H' : All2_fold P (pr1 {| pr1 := H'1'; pr2 := H'1 |})
         (pr2 {| pr1 := H'1'; pr2 := H'1 |})
  ============================
   ({| pr1 := a :: Δ ++ Γ; pr2 := a0 :: Δ' ++ Γ' |} =
    {| pr1 := H'1'; pr2 := H'1 |}
    → let H := CoreTactics.block in
      let H0 := CoreTactics.block in
      All2_fold P Γ Γ'
      × All2_fold (λ Δ Δ' : list A, P (Δ ++ Γ) (Δ' ++ Γ')) 
          (a :: Δ) (a0 :: Δ'))


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