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

Fin preuve emptyset.

parent 13ff6682
...@@ -4,7 +4,7 @@ ...@@ -4,7 +4,7 @@
This file is released under the CC0 License, see the LICENSE file *) This file is released under the CC0 License, see the LICENSE file *)
Require Import ROmega. Require Import ROmega.
Require Import Defs NameProofs Mix Meta Theories PreModels Models. Require Import Defs NameProofs Mix Meta Theories PreModels Models Peano.
Import ListNotations. Import ListNotations.
Local Open Scope bool_scope. Local Open Scope bool_scope.
Local Open Scope eqb_scope. Local Open Scope eqb_scope.
...@@ -215,8 +215,6 @@ Definition ZF := ...@@ -215,8 +215,6 @@ Definition ZF :=
Import ZFAx. Import ZFAx.
Ltac thm := unfold IsTheorem; split; [ unfold Wf; split; [ auto | split; auto ] | ].
Lemma emptyset : IsTheorem Intuiti ZF (∃∀ ~(#0 #1)). Lemma emptyset : IsTheorem Intuiti ZF (∃∀ ~(#0 #1)).
Proof. Proof.
thm. thm.
...@@ -225,8 +223,25 @@ Proof. ...@@ -225,8 +223,25 @@ Proof.
- simpl. rewrite Forall_forall. intros. destruct H. - simpl. rewrite Forall_forall. intros. destruct H.
+ rewrite<- H. unfold IsAx. left. compute; intuition. + rewrite<- H. unfold IsAx. left. compute; intuition.
+ inversion H. + inversion H.
- apply R_Ex_e with (A := infinity) (x := "a"). - apply R_Ex_e with (A := ( (#0 #1 /\ ~(#0 #1)) /\ (#0 #1 -> ( (#0 #2 /\ ( (#0 #1 <-> #0 = #2 \/ #0 #2))))))) (x := "a").
+ intro. cbn in H. + 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)). Lemma singleton : IsTheorem Intuiti ZF (∀∃∀ (#0 #1 <-> #0 = #2)).
Admitted. Admitted.
......
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