Création de ZF.v et ajout d'axiomes.

ZF.v

+(** * The Zermelo Fraenkel set theory and its Coq model *)
+
+(** The NatDed development, Pierre Letouzey, 2019.
+    This file is released under the CC0 License, see the LICENSE file *)
+
+Require Import Defs NameProofs Mix Meta Theories PreModels Models.
+Import ListNotations.
+Local Open Scope bool_scope.
+Local Open Scope eqb_scope.
+
+(** The Peano axioms *)
+
+Definition ZFSign := Finite.to_infinite zf_sign.
+
+Notation "x = y" := (Pred "=" [x;y]) : formula_scope.
+Notation "x ∈ y" := (Pred "∈" [x;y]) : formula_scope.
+
+Module ZFAx.
+Local Open Scope formula_scope.
+
+Definition eq_refl := ∀ (#0 = #0).
+Definition eq_sym := ∀∀ (#1 = #0 -> #0 = #1).
+Definition eq_trans := ∀∀∀ (#2 = #1 /\ #1 = #0 -> #2 = #0).
+Definition compat_left := ∀∀∀ (#0 = #1 /\ #0 ∈ #2 -> #1 ∈ #2).
+Definition compat_right := ∀∀∀ (#0 ∈ #1 /\ #1 = #2 -> #0 ∈ #2).
+
+Definition ext := ∀∀ (∀ (#2 ∈ #0 <-> #2 ∈ #1) -> #0 = #1).
+Definition pairing := ∀∀∃∀ (#3 ∈ #2 <-> #3 = #0 \/ #3 = #1).
+Definition union := ∀∃∀ (#2 ∈ #1 <-> ∃ (#3 ∈ #0 /\ #2 ∈ #3).
+Definition powerset := ∀∃∀ (#2 ∈ #1 <-> ∀ (#3 ∈ #2 -> #3 ∈ #0)).
+Definition infinity := ∃ (∃ (#1 ∈ #0 /\ ∀ not #2 ∈ #1) /\ ∀ (#3 ∈ #0 -> (∃ (#4 ∈ #0 /\ (∀ (#5 ∈ #4 <-> #5 = #3 \/ #5 ∈ #3)))))).
+
+Definition axioms_list :=
+ [ eq_refl; eq_symp; eq_trans; compat_left; compat_right; ext; pairing; union; powerset; infinity ].
+
+(** Beware, [bsubst] is ok below for turning [#0] into [Succ #0], but
+    only since it contains now a [lift] of substituted terms inside
+    quantifiers.
+    And the unconventional [∀] before [A[0]] is to get the right
+    bounded vars (Hack !). *)
+
+Definition comprehension_schema A :=
+  nForall
+    (Nat.pred (level A))
+    (
+
+Definition induction_schema A_x :=
+  let A_0 := bsubst 0 Zero A_x in
+  let A_Sx := bsubst 0 (Succ(#0)) A_x in
+  nForall
+    (Nat.pred (level A_x))
+    (((∀ A_0) /\ (∀ (A_x -> A_Sx))) -> ∀ A_x).
+
+Local Close Scope formula_scope.
+
+Definition IsAx A :=
+  List.In A axioms_list \/
+  exists B, A = induction_schema B /\
+            check PeanoSign B = true /\
+            FClosed B.
+
+Lemma WfAx A : IsAx A -> Wf PeanoSign A.
+Proof.
+ intros [ IN | (B & -> & HB & HB')].
+ - apply Wf_iff.
+   unfold axioms_list in IN.
+   simpl in IN. intuition; subst; reflexivity.
+ - repeat split; unfold induction_schema; cbn.
+   + rewrite nForall_check. cbn.
+     rewrite !check_bsubst, HB; auto.
+   + red. rewrite nForall_level. cbn.
+     assert (level (bsubst 0 Zero B) <= level B).
+     { apply level_bsubst'. auto. }
+     assert (level (bsubst 0 (Succ(BVar 0)) B) <= level B).
+     { apply level_bsubst'. auto. }
+     omega with *.
+   + apply nForall_fclosed. red. cbn.
+     assert (FClosed (bsubst 0 Zero B)).
+     { red. rewrite bsubst_fvars.
+       intro x. rewrite Names.union_spec. cbn. red in HB'. intuition. }
+     assert (FClosed (bsubst 0 (Succ(BVar 0)) B)).
+     { red. rewrite bsubst_fvars.
+       intro x. rewrite Names.union_spec. cbn - [Names.union].
+       rewrite Names.union_spec.
+       generalize (HB' x) (@Names.empty_spec x). intuition. }
+     unfold FClosed in *. intuition.
+Qed.
+
+End PeanoAx.
+
+Local Open Scope string.
+Local Open Scope formula_scope.
+
+Definition PeanoTheory :=
+ {| sign := PeanoSign;
+    IsAxiom := PeanoAx.IsAx;
+    WfAxiom := PeanoAx.WfAx |}.
+
+
+Import PeanoAx.