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