Set Implicit Arguments.
Set Strict Implicit.
Hint Rewrite negb_orb negb_andb negb_involutive if_negb : bool_rw.

Lemma negb_false : forall a, negb a = false -> a = true.
Ltac do_bool' runner :=
( autorewrite with bool_rw in * );
repeat match goal with
| [ H : negb _ = true |- _ ] => apply negb_true in H
| [ H : negb _ = false |- _ ] => apply negb_false in H
| [ H : andb _ _ = true |- _ ] =>
apply andb_true_iff in H; destruct H
| [ H : orb _ _ = false |- _ ] =>
apply orb_false_iff in H; destruct H
| [ H : true = andb _ _ |- _ ] =>
symmetry in H; apply andb_true_iff in H; destruct H
| [ H : false = orb _ _ |- _ ] =>
symmetry in H; apply orb_false_iff in H; destruct H
| [ H : andb _ _ = false |- _ ] =>
apply andb_false_iff in H; runner H
| [ H : orb _ _ = true |- _ ] =>
apply orb_true_iff in H; runner H
| [ H : false = andb _ _ |- _ ] =>
symmetry in H; apply andb_false_iff in H; runner H
| [ H : true = orb _ _ |- _ ] =>
symmetry in H; apply orb_true_iff in H; runner H
end.