Commit f75bc28d authored by Samuel Ben Hamou's avatar Samuel Ben Hamou
Browse files

Elucubrations sur les singletons.

parent 3d63606d
......@@ -243,8 +243,42 @@ Proof.
apply R_Ax. apply in_eq.
Qed.
(* utiliser l'axiome de la paire et défaire puis refaire quantificateurs
OU
paire -> existence de {x,x}; skolem; séparation -> {x} = {x,x}
*)
Lemma singleton : IsTheorem Intuiti ZF (∀∃∀ (#0 #1 <-> #0 = #2)).
Admitted.
Proof.
thm.
exists [pairing].
split; auto.
- simpl. rewrite Forall_forall. intros. destruct H.
+ rewrite<- H. unfold IsAx. left. compute; intuition.
+ inversion H.
- apply R_All_i with (x := "x"); [ calc | cbn ].
apply R_Ex_i with (t := FVar "s"); cbn.
apply R_All_i with (x := "y"); [ calc | cbn ].
assert (Pr Intuiti ([pairing] pairing)).
{ apply R_Ax. apply in_eq. }
unfold pairing in H.
apply R_All_e with (t := FVar "x") in H; cbn in H.
apply R_All_e with (t := FVar "x") in H; cbn in H.
(* il se passe un truc ici *)
apply R_And_i.
+ apply R_Imp_i.
apply R_Imp_e with (A := FVar "y" FVar "s" \/ FVar "y" FVar "s"); [ | apply R_Or_i1; apply R_Ax; compute; intuition ].
(* set (L := _ _ -> _).
assert (Pr Intuiti ([pairing] pairing)).
{ apply R_Ax. apply in_eq. }
unfold pairing in H.
apply R_All_e with (t := FVar "x") in H; cbn in H.
apply R_All_e with (t := FVar "x") in H; cbn in H.
apply R_Ex_e with (x := "s") (B := (#0 #1 -> #0 = FVar "x" \/ #0 = FVar "x") /\
(#0 = FVar "x" \/ #0 = FVar "x" -> #0 #1)) in H; [ cbn | calc | cbn ].
* apply R_All_e with (t := FVar "y") in H; cbn in H.
admit.
* apply R_And_i.*)
Lemma union : IsTheorem Intuiti ZF (∀∀∃∀ (#0 #1 <-> #0 #2 \/ #0 #3)).
Admitted.
\ No newline at end of file
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment