Ltac bool_eq_sound_case :=
match goal with
| [ H: false = true |- _ ] => exact (false_not_true _ H)
| [ H: _ && _ = true |- _ ] => apply andb_prop in H; destruct H; bool_eq_sound_case
| [ H: proj_sumbool ?a = true |- _ ] => apply proj_sumbool_true in H; bool_eq_sound_case
| [ |- ?C ?x1 ?x2 ?x3 ?x4 = ?C ?y1 ?y2 ?y3 ?y4 ] => apply f_equal4; auto
| [ |- ?C ?x1 ?x2 ?x3 = ?C ?y1 ?y2 ?y3 ] => apply f_equal3; auto
| [ |- ?C ?x1 ?x2 = ?C ?y1 ?y2 ] => apply f_equal2; auto
| [ |- ?C ?x1 = ?C ?y1 ] => apply f_equal; auto
| [ |- ?x = ?x ] => reflexivity
| _ => idtac
end.
Ltac bool_eq_sound :=
let Hrec := fresh "Hrec" in let x := fresh "x" in let y := fresh "y" in
fix Hrec 1; intros x y; destruct x, y; simpl; intro; bool_eq_sound_case.

(* Specifications: *)
Lemma dec_eq_from_bool_eq:
forall (A: Type) (f: A -> A -> bool)
(f_refl: forall x, f x x = true) (f_sound: forall x y, f x y = true -> x = y),
forall (x y: A), {x=y} + {x<>y}.