Inductive or3 (P1 P2 P3 : Prop) : Prop := Or31 of P1 | Or32 of P2 | Or33 of P3.Inductive or3 (P1 P2 P3 : Prop) : Prop := Or31 of P1 | Or32 of P2 | Or33 of P3.Inductive or3 (P1 P2 P3 : Prop) : Prop := Or31 of P1 | Or32 of P2 | Or33 of P3.Inductive or3 (P1 P2 P3 : Prop) : Prop := Or31 of P1 | Or32 of P2 | Or33 of P3.Inductive or4 (P1 P2 P3 P4 : Prop) : Prop :=
  Or41 of P1 | Or42 of P2 | Or43 of P3 | Or44 of P4.Inductive or4 (P1 P2 P3 P4 : Prop) : Prop :=
  Or41 of P1 | Or42 of P2 | Or43 of P3 | Or44 of P4.Inductive or4 (P1 P2 P3 P4 : Prop) : Prop :=
  Or41 of P1 | Or42 of P2 | Or43 of P3 | Or44 of P4.Inductive or4 (P1 P2 P3 P4 : Prop) : Prop :=
  Or41 of P1 | Or42 of P2 | Or43 of P3 | Or44 of P4.Inductive or4 (P1 P2 P3 P4 : Prop) : Prop :=
  Or41 of P1 | Or42 of P2 | Or43 of P3 | Or44 of P4.Local Notation a P := (forall x, P x).

Lemma andb_id2r (a b c : bool) : (b -> a = c) -> a && b = c && b.