[apply lt_O_nth_prime_n
|apply (lt_O_n_elim ? H).
intro.
- apply (witness ? ? (r*(nth_prime p \sup m))).
+ apply (witness ? ? ((r*(nth_prime p) \sup m))).
rewrite < assoc_times.
rewrite < sym_times in \vdash (? ? ? (? % ?)).
rewrite > exp_n_SO in \vdash (? ? ? (? (? ? %) ?)).
defactorize_aux f i = defactorize_aux g i \to f = g.
intro.
elim f.
-generalize in match H.
-elim g.
+elim g in H ⊢ %.
apply eq_f.
apply inj_S. apply (inj_exp_r (nth_prime i)).
apply lt_SO_nth_prime_n.
assumption.
apply False_ind.
-apply (not_eq_nf_last_nf_cons n2 n n1 i H2).
-generalize in match H1.
-elim g.
+apply (not_eq_nf_last_nf_cons n2 n n1 i H1).
+elim g in H1 ⊢ %.
apply False_ind.
apply (not_eq_nf_last_nf_cons n1 n2 n i).
apply sym_eq. assumption.
-simplify in H3.
-generalize in match H3.
+simplify in H2.
+generalize in match H2.
apply (nat_elim2 (\lambda n,n2.
((nth_prime i) \sup n)*(defactorize_aux n1 (S i)) =
((nth_prime i) \sup n2)*(defactorize_aux n3 (S i)) \to
[ (nf_last m) \Rightarrow m
| (nf_cons m g) \Rightarrow m ] = m).
rewrite > Hcut.simplify.reflexivity.
-apply H4.simplify in H5.
+apply H3.simplify in H4.
apply (inj_times_r1 (nth_prime i)).
apply lt_O_nth_prime_n.
rewrite < assoc_times.rewrite < assoc_times.assumption.
unfold injective.
change with (\forall f,g.defactorize f = defactorize g \to f=g).
intro.elim f.
-generalize in match H.elim g.
+elim g in H ⊢ %.
(* zero - zero *)
reflexivity.
(* zero - one *)
simplify in H1.
apply False_ind.
-apply (not_eq_O_S O H1).
+apply (not_eq_O_S O H).
(* zero - proper *)
simplify in H1.
apply False_ind.
apply (not_le_Sn_n O).
-rewrite > H1 in \vdash (? ? %).
+rewrite > H in \vdash (? ? %).
change with (O < defactorize_aux n O).
apply lt_O_defactorize_aux.
-generalize in match H.
-elim g.
+elim g in H ⊢ %.
(* one - zero *)
simplify in H1.
apply False_ind.
simplify in H1.
apply False_ind.
apply (not_le_Sn_n (S O)).
-rewrite > H1 in \vdash (? ? %).
+rewrite > H in \vdash (? ? %).
change with ((S O) < defactorize_aux n O).
apply lt_SO_defactorize_aux.
-generalize in match H.elim g.
+elim g in H ⊢ %.
(* proper - zero *)
simplify in H1.
apply False_ind.
apply (not_le_Sn_n O).
-rewrite < H1 in \vdash (? ? %).
+rewrite < H in \vdash (? ? %).
change with (O < defactorize_aux n O).
apply lt_O_defactorize_aux.
(* proper - one *)
simplify in H1.
apply False_ind.
apply (not_le_Sn_n (S O)).
-rewrite < H1 in \vdash (? ? %).
+rewrite < H in \vdash (? ? %).
change with ((S O) < defactorize_aux n O).
apply lt_SO_defactorize_aux.
(* proper - proper *)
apply eq_f.
apply (injective_defactorize_aux O).
-exact H1.
+exact H.
qed.
theorem factorize_defactorize: