Definition prod_eq_dec :
  forall A B
    (A_eq_dec : forall x y : A, {x = y} + {x <> y})
    (B_eq_dec : forall x y : B, {x = y} + {x <> y})
    (x y : A * B),
    {x = y} + {x <> y}.

Lemma equates_4 : forall y1 (P:A0->forall(x1:A1)(x2:A2 x1)(x3:A3 x2),Prop) x1 x2 x3 x4, P y1 x2 x3 x4 -> x1 = y1 -> P x1 x2 x3 x4.