Goal:
  
  A : Type
  B : Type
  C : Type
  P : nat → A → B → Type
  Q : B → C → Type
  R : nat → A → C → Type
  S : nat → A → C → Type
  n : nat
  c1' : nat
  c1 : sigma (λ _ : list A, list C)
  c : All2i R (pr1 {| pr1 := c1'; pr2 := c1 |})
        (pr1 (pr2 {| pr1 := c1'; pr2 := c1 |}))
        (pr2 (pr2 {| pr1 := c1'; pr2 := c1 |}))
  ============================
   ({| pr1 := n; pr2 := {| pr1 := []; pr2 := [] |} |} =
    {| pr1 := c1'; pr2 := c1 |}
    → let H := CoreTactics.block in
      (∀ (n : nat) (x : A) (y : B) (z : C),
         P n x y → Q y z → R n x z → S n x z)
      → let H0 := CoreTactics.block in All2i S n [] [])


DEBUG:Going to execute:
fun id id' =>
             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