(* Specifications: *)
Lemma or_false : forall P : Prop, P -> (P \/ False).
Lemma if_sum_bool_fun_comm : forall A B C D (b : {A}+{B}) (c1 c2 : C) (f : C -> D), f (if b then c1 else c2) = if b then f c1 else f c2.
Lemma if_decider_true : forall A B (P : A -> Prop) (dec : forall x, {P x} + {~ P x}) a (b1 b2 : B), P a -> (if dec a then b1 else b2) = b1.
Lemma if_decider_false : forall A B (P : A -> Prop) (dec : forall x, {P x} + {~ P x}) a (b1 b2 : B), ~ P a -> (if dec a then b1 else b2) = b2.
Lemma leb_false_lt : forall m n, leb m n = false -> n < m.
Lemma leb_true_le : forall m n, leb m n = true -> m <= n.
Lemma ltb_false_le : forall m n, m <? n = false -> n <= m.
Lemma ltb_true_lt : forall m n, m <? n = true -> m < n.