Commit 13ff6682 authored by Samuel Ben Hamou's avatar Samuel Ben Hamou
Browse files

Debut preuve emptyset.

parent 9e00370c
......@@ -226,7 +226,6 @@ Fixpoint lift t :=
end.
(* +1 sur les dB >= k *)
Fixpoint lift_above k t :=
match t with
| BVar n => if (k <=? n)%nat then BVar (S n) else t
......
......@@ -208,10 +208,28 @@ End ZFAx.
Local Open Scope string.
Local Open Scope formula_scope.
Definition ZFTheory :=
Definition ZF :=
{| sign := ZFSign;
IsAxiom := ZFAx.IsAx;
WfAxiom := ZFAx.WfAx |}.
Import ZFAx.
Import ZFAx.
\ No newline at end of file
Ltac thm := unfold IsTheorem; split; [ unfold Wf; split; [ auto | split; auto ] | ].
Lemma emptyset : IsTheorem Intuiti ZF (∃∀ ~(#0 #1)).
Proof.
thm.
exists [infinity].
split; auto.
- simpl. rewrite Forall_forall. intros. destruct H.
+ rewrite<- H. unfold IsAx. left. compute; intuition.
+ inversion H.
- apply R_Ex_e with (A := infinity) (x := "a").
+ intro. cbn in H.
Lemma singleton : IsTheorem Intuiti ZF (∀∃∀ (#0 #1 <-> #0 = #2)).
Admitted.
Lemma union : IsTheorem Intuiti ZF (∀∀∃∀ (#0 #1 <-> #0 #2 \/ #0 #3)).
Admitted.
\ No newline at end of file
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