set "baseuri" "cic:/matita/nat/plus".
-include "logic/equality.ma".
include "nat/nat.ma".
let rec plus n m \def
[ O \Rightarrow m
| (S p) \Rightarrow S (plus p m) ].
-theorem plus_n_O: \forall n:nat. eq nat n (plus n O).
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "natural plus" 'plus x y = (cic:/matita/nat/plus/plus.con x y).
+
+theorem plus_n_O: \forall n:nat. n = n+O.
intros.elim n.
simplify.reflexivity.
simplify.apply eq_f.assumption.
qed.
-theorem plus_n_Sm : \forall n,m:nat. eq nat (S (plus n m)) (plus n (S m)).
+theorem plus_n_Sm : \forall n,m:nat. S (n+m) = n+(S m).
intros.elim n.
simplify.reflexivity.
simplify.apply eq_f.assumption.
qed.
-(* some problem here: confusion between relations/symmetric
-and functions/symmetric; functions symmetric is not in
-functions.moo why?
-theorem symmetric_plus: symmetric nat plus. *)
-
-theorem sym_plus: \forall n,m:nat. eq nat (plus n m) (plus m n).
+theorem sym_plus: \forall n,m:nat. n+m = m+n.
intros.elim n.
simplify.apply plus_n_O.
simplify.rewrite > H.apply plus_n_Sm.
simplify.apply eq_f.assumption.
qed.
-theorem assoc_plus : \forall n,m,p:nat. eq nat (plus (plus n m) p) (plus n (plus m p))
+theorem assoc_plus : \forall n,m,p:nat. (n+m)+p = n+(m+p)
\def associative_plus.
-theorem injective_plus_r: \forall n:nat.injective nat nat (\lambda m.plus n m).
+theorem injective_plus_r: \forall n:nat.injective nat nat (\lambda m.n+m).
intro.simplify.intros 2.elim n.
exact H.
apply H.apply inj_S.apply H1.
qed.
-theorem inj_plus_r: \forall p,n,m:nat.eq nat (plus p n) (plus p m) \to (eq nat n m)
+theorem inj_plus_r: \forall p,n,m:nat. p+n = p+m \to n=m
\def injective_plus_r.
-theorem injective_plus_l: \forall m:nat.injective nat nat (\lambda n.plus n m).
+theorem injective_plus_l: \forall m:nat.injective nat nat (\lambda n.n+m).
intro.simplify.intros.
-(* qui vorrei applicare injective_plus_r *)
-apply inj_plus_r m.
+apply injective_plus_r m.
rewrite < sym_plus.
rewrite < sym_plus y.
assumption.
qed.
-theorem inj_plus_l: \forall p,n,m:nat.eq nat (plus n p) (plus m p) \to (eq nat n m)
+theorem inj_plus_l: \forall p,n,m:nat. n+p = m+p \to n=m
\def injective_plus_l.