Commit 8ceb9473 authored by Samuel Ben Hamou's avatar Samuel Ben Hamou
Browse files

Fin des preuves de singleton et WfAx (améliorée).

parent 8ebb63b7
......@@ -163,8 +163,7 @@ Proof.
easy.
+ red. rewrite nForall_level.
cbn -[Nat.max].
rewrite !pred_max.
simpl.
rewrite !pred_max. simpl.
assert (H := level_lift_form_above B 1).
apply Nat.sub_0_le.
repeat apply Nat.max_lub; omega with *.
......@@ -178,19 +177,21 @@ Proof.
+ rewrite nForall_check. cbn.
rewrite !check_lift_form_above, HC.
easy.
(* todo *)
+ red. rewrite nForall_level. repeat rewrite !aux2, !aux'.
+ red. rewrite nForall_level.
cbn -[Nat.max].
simpl (Nat.max 1 _).
rewrite !pred_max. simpl.
assert (H := level_lift_form_above C 1).
assert (H' := level_lift_form_above C 2).
romega with *.
+ apply nForall_fclosed. red. unfold Names.Empty. intro a. intro.
rewrite !aux2' in H. cbn -[Names.union] in H.
rewrite !fvars_lift_form_above in H.
unfold FClosed in HC'. unfold Names.Empty in HC'. destruct HC' with (a := a).
namedec.
Qed.
apply Nat.sub_0_le.
repeat apply Nat.max_lub; omega with *.
+ apply nForall_fclosed. red.
apply-> Names.is_empty_spec. cbn -[Names.union].
rewrite fvars_lift_form_above.
apply empty_alt in HC'.
assert (H := fvars_lift_form_above C 2).
rewrite H, !HC'.
reflexivity.
Qed.
End ZFAx.
......@@ -241,6 +242,17 @@ Proof.
+ cbn. fold x.
set (y := FVar "y"). reIff.
apply R_Ex_i with (t := y); cbn; reIff. fold x; fold y.
apply R_All_i with (x := "z"); [ calc | ].
set (z := FVar "z"). apply R'_All_e with (t := z); auto.
cbn. fold x. fold y. fold z.
apply R_And_i; apply R'_And_e.
* apply R_Imp_i.
apply R_Or_e with (A := z = x) (B := z = x); [ | apply R_Ax; calc | apply R_Ax; calc ].
apply R_Imp_e with (A := z y); apply R_Ax; calc.
* apply R_Imp_i.
apply R_Imp_e with (A := z = x \/ z = x); [ apply R_Ax; calc | ].
apply R_Or_i1; apply R_Ax; calc.
Qed.
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