| rewrite > H1;
apply le_smallest_factor_n; ]
| rewrite > H1;
+ (*CSC: simplify here does something nasty! *)
change with (divides_b (smallest_factor n) n = true);
apply divides_to_divides_b_true;
[ apply (trans_lt ? (S O));
[ unfold lt; apply le_n;
| apply lt_SO_smallest_factor; assumption; ]
- | apply divides_smallest_factor_n;
+ | letin x \def le.auto.
+ (*
+ apply divides_smallest_factor_n;
apply (trans_lt ? (S O));
[ unfold lt; apply le_n;
- | assumption; ] ] ]
- | apply prime_to_nth_prime;
+ | assumption; ] *) ] ]
+ | letin x \def prime. auto.
+ (*
+ apply prime_to_nth_prime;
apply prime_smallest_factor_n;
- assumption; ] ]
+ assumption; *) ] ]
qed.
theorem divides_to_max_prime_factor : \forall n,m. (S O) < n \to O < m \to n \divides m \to
max_prime_factor n \le max_prime_factor m.
-intros.change with
-((max n (\lambda p:nat.eqb (n \mod (nth_prime p)) O)) \le
-(max m (\lambda p:nat.eqb (m \mod (nth_prime p)) O))).
+intros.unfold max_prime_factor.
apply f_m_to_le_max.
apply (trans_le ? n).
apply le_max_n.apply divides_to_le.assumption.assumption.
cut (prime (nth_prime (max_prime_factor n))).
apply lt_O_nth_prime_n.apply prime_nth_prime.
cut (nth_prime (max_prime_factor n) \divides n).
-apply (transitive_divides ? n).
-apply divides_max_prime_factor_n.
-assumption.assumption.
-apply divides_b_true_to_divides.
-apply lt_O_nth_prime_n.
-apply divides_to_divides_b_true.
-apply lt_O_nth_prime_n.
-apply divides_max_prime_factor_n.
-assumption.
+auto.
+auto.
+(*
+ [ apply (transitive_divides ? n);
+ [ apply divides_max_prime_factor_n.
+ assumption.
+ | assumption.
+ ]
+ | apply divides_b_true_to_divides;
+ [ apply lt_O_nth_prime_n.
+ | apply divides_to_divides_b_true;
+ [ apply lt_O_nth_prime_n.
+ | apply divides_max_prime_factor_n.
+ assumption.
+ ]
+ ]
+ ]
+*)
qed.
theorem p_ord_to_lt_max_prime_factor: \forall n,p,q,r. O < n \to
absurd (nth_prime (max_prime_factor n) \divides r).
rewrite < H4.
apply divides_max_prime_factor_n.
-assumption.
-change with (nth_prime (max_prime_factor n) \divides r \to False).
+assumption.unfold Not.
intro.
-cut (r \mod (nth_prime (max_prime_factor n)) \neq O).
-apply Hcut1.apply divides_to_mod_O.
-apply lt_O_nth_prime_n.assumption.
-apply (p_ord_aux_to_not_mod_O n n ? q r).
-apply lt_SO_nth_prime_n.assumption.
-apply le_n.
-rewrite < H1.assumption.
+cut (r \mod (nth_prime (max_prime_factor n)) \neq O);
+ [unfold Not in Hcut1.auto.
+ (*
+ apply Hcut1.apply divides_to_mod_O;
+ [ apply lt_O_nth_prime_n.
+ | assumption.
+ ]
+ *)
+ |letin z \def le.
+ cut(pair nat nat q r=p_ord_aux n n (nth_prime (max_prime_factor n)));
+ [2: rewrite < H1.assumption.|letin x \def le.auto width = 4]
+ (* CERCA COME MAI le_n non lo applica se lo trova come Const e non Rel *)
+ ].
+(*
+ apply (p_ord_aux_to_not_mod_O n n ? q r);
+ [ apply lt_SO_nth_prime_n.
+ | assumption.
+ | apply le_n.
+ | rewrite < H1.assumption.
+ ]
+ ].
+*)
+cut (n=r*(nth_prime p)\sup(q));
+ [letin www \def le.letin www1 \def divides.
+ auto.
+(*
apply (le_to_or_lt_eq (max_prime_factor r) (max_prime_factor n)).
apply divides_to_max_prime_factor.
assumption.assumption.
apply (witness r n ((nth_prime p) \sup q)).
+*)
+ |
rewrite < sym_times.
apply (p_ord_aux_to_exp n n ? q r).
apply lt_O_nth_prime_n.assumption.
+]
qed.
theorem p_ord_to_lt_max_prime_factor1: \forall n,p,q,r. O < n \to
| nfa_one \Rightarrow (S O)
| (nfa_proper g) \Rightarrow defactorize_aux g O].
-theorem lt_O_defactorize_aux: \forall f:nat_fact.\forall i:nat.
-O < defactorize_aux f i.
-intro.elim f.simplify.unfold lt.
-rewrite > times_n_SO.
-apply le_times.
-change with (O < nth_prime i).
-apply lt_O_nth_prime_n.
-change with (O < exp (nth_prime i) n).
-apply lt_O_exp.
-apply lt_O_nth_prime_n.
-simplify.unfold lt.
-rewrite > times_n_SO.
-apply le_times.
-change with (O < exp (nth_prime i) n).
-apply lt_O_exp.
-apply lt_O_nth_prime_n.
-change with (O < defactorize_aux n1 (S i)).
-apply H.
+theorem lt_O_defactorize_aux:
+ \forall f:nat_fact.
+ \forall i:nat.
+ O < defactorize_aux f i.
+intro; elim f;
+[1,2:
+ simplify; unfold lt;
+ rewrite > times_n_SO;
+ apply le_times;
+ [ change with (O < nth_prime i);
+ apply lt_O_nth_prime_n;
+ |2,3:
+ change with (O < exp (nth_prime i) n);
+ apply lt_O_exp;
+ apply lt_O_nth_prime_n;
+ | change with (O < defactorize_aux n1 (S i));
+ apply H; ] ]
qed.
theorem lt_SO_defactorize_aux: \forall f:nat_fact.\forall i:nat.
left.split.assumption.reflexivity.
intro.right.rewrite > Hcut2.
simplify.unfold lt.apply le_S_S.apply le_O_n.
-cut (r \lt (S O) \or r=(S O)).
+cut (r < (S O) ∨ r=(S O)).
elim Hcut2.absurd (O=r).
apply le_n_O_to_eq.apply le_S_S_to_le.exact H5.
unfold Not.intro.
cut (O=n1).
apply (not_le_Sn_O O).
-rewrite > Hcut3 in \vdash (? ? %).
+rewrite > Hcut3 in ⊢ (? ? %).
assumption.rewrite > Hcut.
rewrite < H6.reflexivity.
assumption.
intro.
apply (nat_case n).reflexivity.
intro.apply (nat_case m).reflexivity.
-intro.change with
+intro.(*CSC: simplify here does something really nasty *)
+change with
(let p \def (max (S(S m1)) (\lambda p:nat.eqb ((S(S m1)) \mod (nth_prime p)) O)) in
defactorize (match p_ord (S(S m1)) (nth_prime p) with
[ (pair q r) \Rightarrow
apply sym_eq.apply eq_pair_fst_snd.
intros.
rewrite < H.
-change with
-(defactorize_aux (factorize_aux p r (nf_last (pred q))) O = (S(S m1))).
+simplify.
cut ((S(S m1)) = (nth_prime p) \sup q *r).
cut (O<r).
rewrite > defactorize_aux_factorize_aux.
+(*CSC: simplify here does something really nasty *)
change with (r*(nth_prime p) \sup (S (pred q)) = (S(S m1))).
cut ((S (pred q)) = q).
rewrite > Hcut2.
unfold lt.apply le_S_S.apply le_S_S. apply le_O_n.
cut ((S(S m1)) = r).
rewrite > Hcut3 in \vdash (? (? ? %)).
+(*CSC: simplify here does something really nasty *)
change with (nth_prime p \divides r \to False).
intro.
apply (p_ord_aux_to_not_mod_O (S(S m1)) (S(S m1)) (nth_prime p) q r).
intro.absurd ((nth_prime i) = (nth_prime j)).
apply (divides_exp_to_eq ? ? (S n)).
apply prime_nth_prime.apply prime_nth_prime.
-assumption.
-change with ((nth_prime i) = (nth_prime j) \to False).
+assumption.unfold Not.
intro.cut (i = j).
apply (not_le_Sn_n i).rewrite > Hcut in \vdash (? ? %).assumption.
apply (injective_nth_prime ? ? H2).
-change with
-(nth_prime i \divides (nth_prime j) \sup n *(defactorize_aux n1 (S j)) \to False).
+unfold Not.simplify.
intro.
cut (nth_prime i \divides (nth_prime j) \sup n
\lor nth_prime i \divides defactorize_aux n1 (S j)).
absurd ((nth_prime i) = (nth_prime j)).
apply (divides_exp_to_eq ? ? n).
apply prime_nth_prime.apply prime_nth_prime.
-assumption.
-change with ((nth_prime i) = (nth_prime j) \to False).
+assumption.unfold Not.
intro.
cut (i = j).
apply (not_le_Sn_n i).rewrite > Hcut1 in \vdash (? ? %).assumption.
simplify.apply le_S_S.
apply le_plus_n.
apply injective_nth_prime.
-(* uffa, perche' semplifica ? *)
-change with (nth_prime (S(max_p g)+i)= nth_prime i).
apply (divides_exp_to_eq ? ? (S n)).
apply prime_nth_prime.apply prime_nth_prime.
rewrite > H.
theorem injective_defactorize_aux: \forall i:nat.
injective nat_fact nat (\lambda f.defactorize_aux f i).
-change with (\forall i:nat.\forall f,g:nat_fact.
-defactorize_aux f i = defactorize_aux g i \to f = g).
+simplify.
intros.
-apply (eq_defactorize_aux_to_eq f g i H).
+apply (eq_defactorize_aux_to_eq x y i H).
qed.
theorem injective_defactorize:
injective nat_fact_all nat defactorize.
-change with (\forall f,g:nat_fact_all.
-defactorize f = defactorize g \to f = g).
+unfold injective.
+change with (\forall f,g.defactorize f = defactorize g \to f=g).
intro.elim f.
generalize in match H.elim g.
(* zero - zero *)
\forall f,g: nat_fact_all. factorize (defactorize f) = f.
intros.
apply injective_defactorize.
-(* uffa: perche' semplifica ??? *)
-change with (defactorize(factorize (defactorize f)) = (defactorize f)).
apply defactorize_factorize.
qed.
-
projects / helm.git / blobdiff