\land
gcd_aux (S m1) m (S m1) \divides (S m1)).
apply divides_gcd_aux_mn.
-simplify.apply le_S_S.apply le_O_n.
+unfold lt.apply le_S_S.apply le_O_n.
assumption.apply le_n.
simplify.intro.
apply (nat_case1 m).
gcd_aux (S m1) n (S m1) \divides S m1).
elim Hcut.split.assumption.assumption.
apply divides_gcd_aux_mn.
-simplify.apply le_S_S.apply le_O_n.
-apply not_lt_to_le.simplify.intro.apply H.
+unfold lt.apply le_S_S.apply le_O_n.
+apply not_lt_to_le.unfold Not. unfold lt.intro.apply H.
rewrite > H1.apply (trans_le ? (S n)).
apply le_n_Sn.assumption.apply le_n.
qed.
intros.
change with (d \divides gcd_aux (S m1) m (S m1)).
apply divides_gcd_aux.
-simplify.apply le_S_S.apply le_O_n.assumption.apply le_n.assumption.
+unfold lt.apply le_S_S.apply le_O_n.assumption.apply le_n.assumption.
rewrite < H2.assumption.
apply (nat_case1 m).simplify.intros.assumption.
intros.
change with (d \divides gcd_aux (S m1) n (S m1)).
apply divides_gcd_aux.
-simplify.apply le_S_S.apply le_O_n.
+unfold lt.apply le_S_S.apply le_O_n.
apply lt_to_le.apply not_le_to_lt.assumption.apply le_n.assumption.
rewrite < H2.assumption.
qed.
a*(S m1) - b*m = (gcd_aux (S m1) m (S m1))
\lor b*m - a*(S m1) = (gcd_aux (S m1) m (S m1))).
apply eq_minus_gcd_aux.
-simplify. apply le_S_S.apply le_O_n.
+unfold lt. apply le_S_S.apply le_O_n.
assumption.apply le_n.
apply (nat_case1 m).
simplify.intros.
apply (ex_intro ? ? a).
left.assumption.
apply eq_minus_gcd_aux.
-simplify. apply le_S_S.apply le_O_n.
+unfold lt. apply le_S_S.apply le_O_n.
apply lt_to_le.apply not_le_to_lt.assumption.
apply le_n.
qed.
generalize in match (gcd_O_to_eq_O m n H1).
intros.elim H2.
rewrite < H4 in \vdash (? ? %).assumption.
-intros.simplify.apply le_S_S.apply le_O_n.
+intros.unfold lt.apply le_S_S.apply le_O_n.
qed.
theorem symmetric_gcd: symmetric nat gcd.
apply divides_to_le.
apply lt_O_gcd.
rewrite > (times_n_O O).
-apply lt_times.simplify.apply le_S_S.apply le_O_n.assumption.
+apply lt_times.unfold lt.apply le_S_S.apply le_O_n.assumption.
apply divides_d_gcd.
apply (transitive_divides ? (S m1)).
apply divides_gcd_m.
theorem gcd_SO_n: \forall n:nat. gcd (S O) n = (S O).
intro.
-apply antisym_le.apply divides_to_le.simplify.apply le_n.
+apply antisym_le.apply divides_to_le.unfold lt.apply le_n.
apply divides_gcd_n.
cut (O < gcd (S O) n \lor O = gcd (S O) n).
elim Hcut.assumption.
theorem prime_to_gcd_SO: \forall n,m:nat. prime n \to n \ndivides m \to
gcd n m = (S O).
-intros.simplify in H.change with (gcd n m = (S O)).
+intros.unfold prime in H.change with (gcd n m = (S O)).
elim H.
apply antisym_le.
apply not_lt_to_le.
apply eq_minus_gcd.
assumption.assumption.
apply (decidable_divides n p).
-apply (trans_lt ? (S O)).simplify.apply le_n.
-simplify in H.elim H. assumption.
+apply (trans_lt ? (S O)).unfold lt.apply le_n.
+unfold prime in H.elim H. assumption.
qed.
theorem eq_gcd_times_SO: \forall m,n,p:nat. O < n \to O < p \to
apply (lt_to_le_to_lt ? (smallest_factor (gcd m (n*p)))).
apply lt_SO_smallest_factor.assumption.
apply divides_to_le.
-rewrite > H2.simplify.apply le_n.
+rewrite > H2.unfold lt.apply le_n.
apply divides_d_gcd.assumption.
apply (transitive_divides ? (gcd m (n*p))).
apply divides_smallest_factor_n.
-apply (trans_lt ? (S O)). simplify. apply le_n. assumption.
+apply (trans_lt ? (S O)). unfold lt. apply le_n. assumption.
apply divides_gcd_n.
apply (not_le_Sn_n (S O)).
change with ((S O) < (S O)).
apply (lt_to_le_to_lt ? (smallest_factor (gcd m (n*p)))).
apply lt_SO_smallest_factor.assumption.
apply divides_to_le.
-rewrite > H3.simplify.apply le_n.
+rewrite > H3.unfold lt.apply le_n.
apply divides_d_gcd.assumption.
apply (transitive_divides ? (gcd m (n*p))).
apply divides_smallest_factor_n.
-apply (trans_lt ? (S O)). simplify. apply le_n. assumption.
+apply (trans_lt ? (S O)). unfold lt. apply le_n. assumption.
apply divides_gcd_n.
apply divides_times_to_divides.
apply prime_smallest_factor_n.
assumption.
apply (transitive_divides ? (gcd m (n*p))).
apply divides_smallest_factor_n.
-apply (trans_lt ? (S O)). simplify. apply le_n. assumption.
+apply (trans_lt ? (S O)).unfold lt. apply le_n. assumption.
apply divides_gcd_m.
change with (O < gcd m (n*p)).
apply lt_O_gcd.