Commit 2ce24ec1 authored by Samuel Ben Hamou's avatar Samuel Ben Hamou
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Fin de la preuve de la commutativité de + (après avoir enregistré les...

Fin de la preuve de la commutativité de + (après avoir enregistré les modifications ça marche mieux...)
parent d28c98b8
...@@ -294,10 +294,47 @@ Lemma Comm : ...@@ -294,10 +294,47 @@ Lemma Comm :
(∀∀ #0 + #1 = #1 + #0)). (∀∀ #0 + #1 = #1 + #0)).
Proof. Proof.
thm. thm.
rec. rec; set (Γ' := _ :: _ :: Γ).
set (Γ' := _ :: _ :: Γ). + app_R_All_i "x".
+ cbm.
app_R_All_i "x".
cbm.
trans (FVar "x").
- sym.
assert (Pr Intuiti (Γ' # 0 = # 0 + Zero)). { apply R_Ax. simpl in *; intuition. }
apply R_All_e with (t := FVar "x") in H; auto.
- sym.
axiom ax9 AX9.
apply R_All_e with (t := FVar "x") in AX9; auto.
+ app_R_All_i "y".
cbm.
apply R_Imp_i.
app_R_All_i "x".
cbm.
trans (Succ (FVar "x" + FVar "y")).
- sym.
assert (Pr Intuiti (( # 0 + FVar "y" = FVar "y" + # 0) :: Γ' Succ (#1 + #0) = #1 + Succ (#0))). { apply R_Ax. simpl in *; intuition. }
apply R_All_e with (t := FVar "x") in H; auto.
apply R_All_e with (t := FVar "y") in H; auto.
- trans (Succ (FVar "y" + FVar "x")).
* ahered.
assert (Pr Intuiti (( #0 + FVar "y" = FVar "y" + #0) :: Γ' #0 + FVar "y" = FVar "y" + #0)). { apply R_Ax. apply in_eq. }
apply R_All_e with (t := FVar "x") in H; auto.
* sym.
axiom ax10 AX10.
apply R_All_e with (t := FVar "y") in AX10; auto.
apply R_All_e with (t := FVar "x") in AX10; auto.
Qed.
Lemma Commutativity : IsTheorem Intuiti PeanoTheory (∀∀ #0 + #1 = #1 + #0).
Proof.
apply ModusPonens with (A := (∀∀ Succ(#1 + #0) = #1 + Succ(#0))).
+ apply ModusPonens with (A := #0 = #0 + Zero).
* apply Comm.
* apply ZeroRight.
+ apply SuccRight.
Qed.
(** A Coq model of this Peano theory, based on the [nat] type *) (** A Coq model of this Peano theory, based on the [nat] type *)
Definition PeanoFuns : modfuns nat := Definition PeanoFuns : modfuns nat :=
......
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