(**
 List notations for wider connectives; the Prop connectives have a fixed
 width so as to avoid iterated destruction (we go up to width 5 for /\, and
 width 4 for or). The bool connectives have arbitrary widths, but denote
 expressions that associate to the RIGHT. This is consistent with the right
 associativity of list expressions and thus more convenient in most proofs.  **)

Inductive and3 (P1 P2 P3 : Prop) : Prop := And3 of P1 & P2 & P3.(**
 List notations for wider connectives; the Prop connectives have a fixed
 width so as to avoid iterated destruction (we go up to width 5 for /\, and
 width 4 for or). The bool connectives have arbitrary widths, but denote
 expressions that associate to the RIGHT. This is consistent with the right
 associativity of list expressions and thus more convenient in most proofs.  **)

Inductive and3 (P1 P2 P3 : Prop) : Prop := And3 of P1 & P2 & P3.Inductive and4 (P1 P2 P3 P4 : Prop) : Prop := And4 of P1 & P2 & P3 & P4.Inductive and4 (P1 P2 P3 P4 : Prop) : Prop := And4 of P1 & P2 & P3 & P4.Inductive and5 (P1 P2 P3 P4 P5 : Prop) : Prop :=
  And5 of P1 & P2 & P3 & P4 & P5.Inductive and5 (P1 P2 P3 P4 P5 : Prop) : Prop :=
  And5 of P1 & P2 & P3 & P4 & P5.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.Notation "[ /\ P1 & P2 ]" := (and P1 P2) (only parsing) : type_scope.Notation "[ /\ P1 , P2 & P3 ]" := (and3 P1 P2 P3) : type_scope.Notation "[ /\ P1 , P2 , P3 & P4 ]" := (and4 P1 P2 P3 P4) : type_scope.Notation "[ /\ P1 , P2 , P3 , P4 & P5 ]" := (and5 P1 P2 P3 P4 P5) : type_scope.Notation "[ \/ P1 | P2 ]" := (or P1 P2) (only parsing) : type_scope.Notation "[ \/ P1 , P2 | P3 ]" := (or3 P1 P2 P3) : type_scope.Notation "[ \/ P1 , P2 , P3 | P4 ]" := (or4 P1 P2 P3 P4) : type_scope.Notation "[ && b1 & c ]" := (b1 && c) (only parsing) : bool_scope.Notation "[ && b1 , b2 , .. , bn & c ]" := (b1 && (b2 && .. (bn && c) .. ))
  : bool_scope.Notation "[ || b1 | c ]" := (b1 || c) (only parsing) : bool_scope.Notation "[ || b1 , b2 , .. , bn | c ]" := (b1 || (b2 || .. (bn || c) .. ))
  : bool_scope.Notation "[ ==> b1 , b2 , .. , bn => c ]" :=
   (b1 ==> (b2 ==> .. (bn ==> c) .. )) : bool_scope.Notation "[ ==> b1 => c ]" := (b1 ==> c) (only parsing) : bool_scope.Local Notation a P := (forall x, P x).

Lemma addbC : commutative addb.