45 lines
1.1 KiB
Coq
45 lines
1.1 KiB
Coq
(* Ouverture d’une section *)
|
||
Section CalculPredicats.
|
||
(* Définition de 2 domaines pour les prédicats *)
|
||
Variable A B : Type.
|
||
|
||
(* Formule du second ordre : Quantification des prédicats phi et psi *)
|
||
Theorem Thm_8 : forall (P Q : A -> Prop),
|
||
(forall x1 : A, (P x1) /\ (Q x1))
|
||
-> (forall x2 : A, (P x2)) /\ (forall x3 : A, (Q x3)).
|
||
intros.
|
||
split.
|
||
intro x2.
|
||
destruct (H x2).
|
||
exact H0.
|
||
intro x3.
|
||
destruct (H x3).
|
||
exact H1.
|
||
Qed.
|
||
|
||
(* Formule du second ordre : Quantification des prédicats phi et psi *)
|
||
Theorem Thm_9 : forall (P : A -> B -> Prop),
|
||
(exists x1 : A, forall y1 : B, (P x1 y1))
|
||
-> forall y2 : B, exists x2 : A, (P x2 y2).
|
||
intros.
|
||
destruct H.
|
||
exists x.
|
||
generalize y2.
|
||
exact H.
|
||
Qed.
|
||
|
||
|
||
(* Formule du second ordre : Quantification des prédicats phi et psi *)
|
||
Theorem Thm_10 : forall (P Q : A -> Prop),
|
||
(exists x1 : A, (P x1) -> (Q x1))
|
||
-> (forall x2 : A, (P x2)) -> exists x3 : A, (Q x3).
|
||
intros.
|
||
destruct H.
|
||
exists x.
|
||
cut (P x).
|
||
exact H.
|
||
generalize x.
|
||
exact H0.
|
||
Qed.
|
||
|
||
End CalculPredicats. |