Inductive IfProp (A B:Prop) : bool -> Prop :=
  | Iftrue : A -> IfProp A B true
  | Iffalse : B -> IfProp A B false.Inductive IfProp (A B:Prop) : bool -> Prop :=
  | Iftrue : A -> IfProp A B true
  | Iffalse : B -> IfProp A B false.Inductive IfProp (A B:Prop) : bool -> Prop :=
  | Iftrue : A -> IfProp A B true
  | Iffalse : B -> IfProp A B false.

(* Specifications: *)
Lemma Iftrue_inv : forall (A B:Prop) (b:bool), IfProp A B b -> b = true -> A.
Lemma Iffalse_inv : forall (A B:Prop) (b:bool), IfProp A B b -> b = false -> B.
Lemma IfProp_true : forall A B:Prop, IfProp A B true -> A.
Lemma IfProp_false : forall A B:Prop, IfProp A B false -> B.
Lemma IfProp_or : forall (A B:Prop) (b:bool), IfProp A B b -> A \/ B.
Lemma IfProp_sum : forall (A B:Prop) (b:bool), IfProp A B b -> {A} + {B}.