Commit 01b6b4a2 authored by Samuel Ben Hamou's avatar Samuel Ben Hamou
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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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