| Classical Predicate Logic on Set |
Require Import Classical_Prop.
Section Generic.
Variable U : Set.
| de Morgan laws for quantifiers |
Lemma not_all_ex_not :
forall P:U -> Prop, ~ (forall n:U, P n) -> exists n : U, ~ P n.
Proof.
unfold not in |- *; intros P notall.
apply NNPP; unfold not in |- *.
intro abs.
cut (forall n:U, P n); auto.
intro n; apply NNPP.
unfold not in |- *; intros.
apply abs; exists n; trivial.
Qed.
Lemma not_all_not_ex :
forall P:U -> Prop, ~ (forall n:U, ~ P n) -> exists n : U, P n.
Proof.
intros P H.
elim (not_all_ex_not (fun n:U => ~ P n) H); intros n Pn; exists n.
apply NNPP; trivial.
Qed.
Lemma not_ex_all_not :
forall P:U -> Prop, ~ (exists n : U, P n) -> forall n:U, ~ P n.
Proof.
unfold not in |- *; intros P notex n abs.
apply notex.
exists n; trivial.
Qed.
Lemma not_ex_not_all :
forall P:U -> Prop, ~ (exists n : U, ~ P n) -> forall n:U, P n.
Proof.
intros P H n.
apply NNPP.
red in |- *; intro K; apply H; exists n; trivial.
Qed.
Lemma ex_not_not_all :
forall P:U -> Prop, (exists n : U, ~ P n) -> ~ (forall n:U, P n).
Proof.
unfold not in |- *; intros P exnot allP.
elim exnot; auto.
Qed.
Lemma all_not_not_ex :
forall P:U -> Prop, (forall n:U, ~ P n) -> ~ (exists n : U, P n).
Proof.
unfold not in |- *; intros P allnot exP; elim exP; intros n p.
apply allnot with n; auto.
Qed.
End Generic.