theorem pred_Sn : \forall n:nat.
(eq nat n (pred (S n))).
-intros.reflexivity.
+intros; reflexivity.
qed.
theorem injective_S : \forall n,m:nat.
(eq nat (S n) (S m)) \to (eq nat n m).
-intros.
-rewrite > pred_Sn n.
+intros;
+rewrite > pred_Sn;
rewrite > pred_Sn m.
-apply f_equal. assumption.
+apply f_equal; assumption.
qed.
theorem not_eq_S : \forall n,m:nat.
Not (eq nat n m) \to Not (eq nat (S n) (S m)).
-intros. simplify.intros.
-apply H.apply injective_S.assumption.
+intros; simplify; intros;
+apply H; apply injective_S; assumption.
qed.
definition not_zero : nat \to Prop \def
| (S p) \Rightarrow True ].
theorem O_S : \forall n:nat. Not (eq nat O (S n)).
-intros.simplify.intros.
-cut (not_zero O).exact Hcut.rewrite > H.
-exact I.
+intros; simplify; intros;
+cut (not_zero O); [ exact Hcut | rewrite > H; exact I ].
qed.
theorem n_Sn : \forall n:nat. Not (eq nat n (S n)).
| (S p) \Rightarrow S (plus p m) ].
theorem plus_n_O: \forall n:nat. eq nat n (plus n O).
-intros.elim n.simplify.reflexivity.
-simplify.apply f_equal.assumption.
+intros;elim n;
+ [ simplify;reflexivity
+ | simplify;apply f_equal;assumption ].
qed.
theorem plus_n_Sm : \forall n,m:nat. eq nat (S (plus n m)) (plus n (S m)).
\forall n,m:nat. eq nat (plus n (times n m)) (times n (S m)).
intros.elim n.simplify.reflexivity.
simplify.apply f_equal.rewrite < H.
-transitivity (plus (plus e m) (times e m)).symmetry.
-apply assoc_plus.transitivity (plus (plus m e) (times e m)).
+transitivity (plus (plus e1 m) (times e1 m)).symmetry.
+apply assoc_plus.transitivity (plus (plus m e1) (times e1 m)).
apply f_equal2.
apply sym_plus.reflexivity.apply assoc_plus.
qed.
theorem sym_times :
\forall n,m:nat. eq nat (times n m) (times m n).
intros.elim n.simplify.apply times_n_O.
-simplify.rewrite < sym_eq ? ? ? H.apply times_n_Sm.
+simplify.rewrite > H.apply times_n_Sm.
qed.
let rec minus n m \def
theorem le_Sn_n : \forall n:nat. Not (le (S n) n).
intros.elim n.apply le_Sn_O.simplify.intros.
-cut le (S e) e.apply H.assumption.apply le_S_n.assumption.
+cut le (S e1) e1.apply H.assumption.apply le_S_n.assumption.
qed.
theorem le_antisym : \forall n,m:nat. (le n m) \to (le m n) \to (eq nat n m).