(*  The built-in "remember" tactic is weaker than this one!
  The built-in one can lead to "Error: The correctness of the conclusion relies on the body of a"
  where this one will succeed.
  [this comment may be obsolete, perhaps from Coq 8.2 or before
Tactic Notation "remember" constr(a) "as" ident(x) :=
   let x := fresh x in
  let H := fresh "Heq" x in
  (set (x:=a) in *; assert (H: x=a) by reflexivity; clearbody x).
*)

Ltac simple_if_tac := 
  match goal with |- context [if ?A then _ else _] => 
    lazymatch type of A with
    | bool => destruct A 
    | sumbool _ _ => fail "Use if_tac instead of simple_if_tac, since your expression "A" has type sumbool"
    | ?t => fail "Use simple_if_tac only for bool; your expression"A" has type" t
  end end.

Lemma predicate_max: forall (F: nat -> Prop) (Fdec: forall n, {F n}+{~ F n}) n, F 0%nat -> ~ F n -> exists i, F i /\ (i<n)%nat /\ ~ F (S i).