+
(**************************************************************************)
(* __ *)
(* ||M|| *)
\forall n,m:nat. n+(n*m) = n*(S m).
intros.elim n.
simplify.reflexivity.
-simplify.apply eq_f.rewrite < H.
+simplify.
+demodulate all.
+(*
+apply eq_f.rewrite < H.
transitivity ((n1+m)+n1*m).symmetry.apply assoc_plus.
transitivity ((m+n1)+n1*m).
apply eq_f2.
apply sym_plus.
reflexivity.
apply assoc_plus.
+*)
qed.
theorem times_O_to_O: \forall n,m:nat.n*m = O \to n = O \lor m= O.
qed.
theorem times_n_SO : \forall n:nat. n = n * S O.
-intros.
+intros. demodulate. reflexivity. (*
rewrite < times_n_Sm.
rewrite < times_n_O.
rewrite < plus_n_O.
-reflexivity.
+reflexivity.*)
qed.
theorem times_SSO_n : \forall n:nat. n + n = S (S O) * n.
theorem symmetric_times : symmetric nat times.
unfold symmetric.
-intros.elim x.
-simplify.apply times_n_O.
+intros.elim x;
+ [ simplify. apply times_n_O.
+ | demodulate. reflexivity. (*
(* applyS times_n_Sm. *)
-simplify.rewrite > H.apply times_n_Sm.
+simplify.rewrite > H.apply times_n_Sm.*)]
qed.
variant sym_times : \forall n,m:nat. n*m = m*n \def
unfold distributive.
intros.elim x.
simplify.reflexivity.
-simplify.rewrite > H. rewrite > assoc_plus.rewrite > assoc_plus.
+simplify.
+demodulate all.
+(*
+rewrite > H. rewrite > assoc_plus.rewrite > assoc_plus.
apply eq_f.rewrite < assoc_plus. rewrite < (sym_plus ? z).
-rewrite > assoc_plus.reflexivity.
+rewrite > assoc_plus.reflexivity. *)
qed.
variant distr_times_plus: \forall n,m,p:nat. n*(m+p) = n*m + n*p
unfold associative.
intros.
elim x. simplify.apply refl_eq.
-simplify.rewrite < sym_times.
+simplify.
+demodulate all.
+(*
+rewrite < sym_times.
rewrite > distr_times_plus.
rewrite < sym_times.
rewrite < (sym_times (times n y) z).
rewrite < H.apply refl_eq.
+*)
qed.
variant assoc_times: \forall n,m,p:nat. (n*m)*p = n*(m*p) \def
lemma times_times: ∀x,y,z. x*(y*z) = y*(x*z).
intros.
-demodulate. reflexivity;
+demodulate. reflexivity.
(* autobatch paramodulation. *)
qed.