Require ZArith.
Require Sumbool.
Require WhyCoqCompat.
Definition annot_bool :
(b:bool) { b':bool | if b' then b=true else b=false }.
Proof.
Intro b.
Exists b. Case b; Trivial.
Save.
Definition if_then_else [A:Set; a:bool; b,c:A] := if a then b else c.
Implicits if_then_else.
Definition spec_and := [A,B,C,D:Prop][b:bool]if b then A /\ C else B \/ D.
Definition prog_bool_and :
(Q1,Q2:bool->Prop) (sig bool Q1) -> (sig bool Q2)
-> { b:bool | if b then (Q1 true) /\ (Q2 true)
else (Q1 false) \/ (Q2 false) }.
Proof.
Intros Q1 Q2 H1 H2.
Elim H1. Intro b1. Elim H2. Intro b2.
Case b1; Case b2; Intros.
Exists true; Auto.
Exists false; Auto. Exists false; Auto. Exists false; Auto.
Save.
Definition spec_or := [A,B,C,D:Prop][b:bool]if b then A \/ C else B /\ D.
Definition prog_bool_or :
(Q1,Q2:bool->Prop) (sig bool Q1) -> (sig bool Q2)
-> { b:bool | if b then (Q1 true) \/ (Q2 true)
else (Q1 false) /\ (Q2 false) }.
Proof.
Intros Q1 Q2 H1 H2.
Elim H1. Intro b1. Elim H2. Intro b2.
Case b1; Case b2; Intros.
Exists true; Auto. Exists true; Auto. Exists true; Auto.
Exists false; Auto.
Save.
Definition spec_not:= [A,B:Prop][b:bool]if b then B else A.
Definition prog_bool_not :
(Q:bool->Prop) (sig bool Q)
-> { b:bool | if b then (Q false) else (Q true) }.
Proof.
Intros Q H.
Elim H. Intro b.
Case b; Intro.
Exists false; Auto. Exists true; Auto.
Save.
Conversely, we build a sumbool out of a boolean, for the need of validations
|
Definition btest
[q:bool->Prop][b:bool][p:(q b)] : {(q true)}+{(q false)} :=
(<[b:bool](q b)->{(q true)}+{(q false)}>
Cases b of true => [p](left (q true) (q false) p)
| false => [p](right (q true) (q false) p)
end
p).
| Equality over booleans |
Definition B_eq_dec : (x,y:bool){x=y}+{~x=y}.
Proof. Decide Equality. Qed.
Definition B_eq_bool :=
[x,y:bool](bool_of_sumbool (B_eq_dec x y)).
Definition B_noteq_bool :=
[x,y:bool](bool_of_sumbool (sumbool_not ? ? (B_eq_dec x y))).
| Equality over type unit |
Definition U_eq_dec : (x,y:unit){x=y}+{~x=y}.
Proof. Decide Equality. Qed.
Definition U_eq_bool :=
[x,y:unit](bool_of_sumbool (U_eq_dec x y)).
Definition U_noteq_bool :=
[x,y:unit](bool_of_sumbool (sumbool_not ? ? (U_eq_dec x y))).