Classical Predicate Logic on Type |
Require
Classical_Prop.
Section
Generic.
Variable
U: Type.
de Morgan laws for quantifiers |
Lemma
not_all_ex_not : (P:U->Prop)(~(n:U)(P n)) -> (EXT n:U | ~(P n)).
Proof
.
Unfold not; Intros P notall.
Apply NNPP; Unfold not.
Intro abs.
Cut ((n:U)(P n)); Auto.
Intro n; Apply NNPP.
Unfold not; Intros.
Apply abs; Exists n; Trivial.
Qed
.
Lemma
not_all_not_ex : (P:U->Prop)(~(n:U)~(P n)) -> (EXT n:U | (P n)).
Proof
.
Intros P H.
Elim (not_all_ex_not [n:U]~(P n) H); Intros n Pn; Exists n.
Apply NNPP; Trivial.
Qed
.
Lemma
not_ex_all_not : (P:U->Prop)(~(EXT n:U | (P n))) -> (n:U)~(P n).
Proof
.
Unfold not; Intros P notex n abs.
Apply notex.
Exists n; Trivial.
Qed
.
Lemma
not_ex_not_all : (P:U->Prop)(~(EXT n:U | ~(P n))) -> (n:U)(P n).
Proof
.
Intros P H n.
Apply NNPP.
Red; Intro K; Apply H; Exists n; Trivial.
Qed
.
Lemma
ex_not_not_all : (P:U->Prop) (EXT n:U | ~(P n)) -> ~(n:U)(P n).
Proof
.
Unfold not; Intros P exnot allP.
Elim exnot; Auto.
Qed
.
Lemma
all_not_not_ex : (P:U->Prop) ((n:U)~(P n)) -> ~(EXT n:U | (P n)).
Proof
.
Unfold not; Intros P allnot exP; Elim exP; Intros n p.
Apply allnot with n; Auto.
Qed
.
End
Generic.