Commit 01b6b4a2 by Samuel Ben Hamou

### Preuve de Succ dans ZF.v

parent c9f6e71c
 ... ... @@ -241,7 +241,7 @@ Proof. apply R_Or_i1; apply R_Ax; calc. Qed. Lemma union : IsTheorem Intuiti ZF (∀∀∃∀ (#0 ∈ #1 <-> #0 ∈ #3 \/ #0 ∈ #2)). Lemma unionset : IsTheorem Intuiti ZF (∀∀∃∀ (#0 ∈ #1 <-> #0 ∈ #3 \/ #0 ∈ #2)). Proof. thm. exists [ pairing; union; compat_right; eq_refl ]. ... ... @@ -256,15 +256,12 @@ Proof. app_R_All_i "B" B. inst_axiom pairing [A; B]; simp. eapply R_Ex_e with (x := "C"); [ | exact H | ]; calc. set (C := FVar "C"). simp. clear H. set (C := FVar "C"). simp. clear H. inst_axiom union [C]; simp. eapply R_Ex_e with (x := "U"); [ | exact H | ]; calc. set (U := FVar "U"). simp. clear H. apply R_Ex_i with (t := U). simp. app_R_All_i "x" x. simp. apply R_Ex_i with (t := U). simp. app_R_All_i "x" x. simp. apply R_And_i. + apply R_Imp_i. apply R_Ex_e with (A := #0 ∈ C /\ x ∈ #0) (x := "y"); set (y := FVar "y"). ... ... @@ -320,10 +317,54 @@ Proof. Qed. Lemma Succ : IsTheorem Intuiti ZF (∀∃∀ (#0 ∈ #1 <-> #0 = #2) -> ∀∀∃∀ (#0 ∈ #1 <-> #0 ∈ #3 \/ #0 ∈ #2) ((∀∃∀ (#0 ∈ #1 <-> #0 = #2)) -> (∀∀∃∀ (#0 ∈ #1 <-> #0 ∈ #3 \/ #0 ∈ #2)) -> ∀∃ succ (#1) (#0)). Admitted. Proof. thm. exists [ ]. split; auto. repeat apply R_Imp_i. set (Un := ∀ ∀ _). set (Sing := ∀ ∃ _). set (Γ := Un :: Sing :: []). app_R_All_i "x" x. simp. inst_axiom Sing [x]; simp. eapply R_Ex_e with (x := "A"); [ | exact H | ]; calc. set (A := FVar "A"). simp. clear H. inst_axiom Un [ A; x ]. simp. eapply R_Ex_e with (x := "U"); [ | exact H | ]; calc. set (U := FVar "U"). simp. clear H. apply R_Ex_i with (t := U). simp. app_R_All_i "y" y. simp. apply R_And_i. - apply R_Imp_i. apply R_Or_e with (A := y ∈ A) (B := y ∈ x). + set (Ax := ∀ _ ∈ U <-> _). inst_axiom Ax [ y ]. simp. apply R_And_e1 in H. apply R_Imp_e with (A := y ∈ U) in H; [ intuition | apply R'_Ax ]. + apply R_Or_i1. set (Ax := ∀ _ ∈ A <-> _). inst_axiom Ax [ y ]. simp. apply R_And_e1 in H. apply R_Imp_e with (A := y ∈ A) in H; [ intuition | apply R'_Ax ]. + apply R_Or_i2. apply R'_Ax. - apply R_Imp_i. apply R_Imp_e with (A := y ∈ A \/ y ∈ x). + set (Ax := ∀ _ ∈ U <-> _). inst_axiom Ax [ y ]. simp. apply R_And_e2 in H. assumption. + apply R'_Or_e. * apply R_Or_i1. set (Ax := ∀ _ ∈ A <-> _). inst_axiom Ax [ y ]. simp. apply R_And_e2 in H. apply R_Imp_e with (A := y = x) in H; [ intuition | apply R'_Ax ]. * apply R_Or_i2. apply R'_Ax. Qed. Lemma Successor : IsTheorem Intuiti ZF (∀∃ succ (#1) (#0)). Admitted. \ No newline at end of file Proof. apply ModusPonens with (A := ∀∀∃∀ #0 ∈ #1 <-> #0 ∈ #3 \/ #0 ∈ #2); [ | apply unionset ]. apply ModusPonens with (A := ∀∃∀ #0 ∈ #1 <-> #0 = #2); [ apply Succ | apply singleton ]. Qed. \ No newline at end of file
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