Global Instance type_sigT : type (sigT F) :=
  { equal := fun x y =>
    equal (projT1 x) (projT1 y) /\
    exists pf : projT1 y = projT1 x,
      equal (projT2 x) (match pf in _ = t return F t with
                          | eq_refl => projT2 y
                        end)
  ; proper := fun x => proper (projT1 x) /\ proper (projT2 x)
  }.Global Instance typeOk_sigT : typeOk type_sigT.Global Instance proper_existT a b : proper a -> proper b -> proper (existT F a b).Global Instance proper_projT1 a : proper a -> proper (projT1 a).Global Instance proper_projT2 a : proper a -> proper (projT2 a).Global Instance Injective_existT a b d
    : Injective (existT F a b = existT F a d) | 1.Global Instance Injective_existT_dif a b c d
  : Injective (existT F a b = existT F c d) | 2.

Lemma eq_sigT_rw : forall T U F (a b : T) (pf : a = b) s, match pf in _ = x return @sigT U (F x) with | eq_refl => s end = @existT U (F b) (projT1 s) match pf in _ = x return F x (projT1 s) with | eq_refl => (projT2 s) end.