Commit 3d63606d authored by Samuel Ben Hamou's avatar Samuel Ben Hamou
Browse files

Fin preuve emptyset.

parent 13ff6682
......@@ -4,7 +4,7 @@
This file is released under the CC0 License, see the LICENSE file *)
Require Import ROmega.
Require Import Defs NameProofs Mix Meta Theories PreModels Models.
Require Import Defs NameProofs Mix Meta Theories PreModels Models Peano.
Import ListNotations.
Local Open Scope bool_scope.
Local Open Scope eqb_scope.
......@@ -215,8 +215,6 @@ Definition ZF :=
Import ZFAx.
Ltac thm := unfold IsTheorem; split; [ unfold Wf; split; [ auto | split; auto ] | ].
Lemma emptyset : IsTheorem Intuiti ZF (∃∀ ~(#0 #1)).
Proof.
thm.
......@@ -225,8 +223,25 @@ Proof.
- 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.
- apply R_Ex_e with (A := ( (#0 #1 /\ ~(#0 #1)) /\ (#0 #1 -> ( (#0 #2 /\ ( (#0 #1 <-> #0 = #2 \/ #0 #2))))))) (x := "a").
+ calc.
+ apply R_Ax; auto. unfold infinity. intuition.
+ cbn.
apply R_Ex_e with (A := #0 FVar "a" /\ ( ~ #0 #1)) (x := "x").
* calc.
* apply R'_Ex_e with (x := "y"); [ calc | ].
cbn.
set (rem := _ -> _).
apply R_Ex_i with (t := FVar "y").
cbn.
apply R_And_e1 with (B := rem).
apply R_Ax. apply in_eq.
* cbn.
apply R_Ex_i with (t := FVar "x").
cbn.
apply R_And_e2 with (A := FVar "x" FVar "a").
apply R_Ax. apply in_eq.
Qed.
Lemma singleton : IsTheorem Intuiti ZF (∀∃∀ (#0 #1 <-> #0 = #2)).
Admitted.
......
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