Commit 3d63606d by Samuel Ben Hamou

### 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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