(* the following factorization algorithm looks for the largest prime
factor. *)
definition max_prime_factor \def \lambda n:nat.
-(max n (\lambda p:nat.eqb (mod n (nth_prime p)) O)).
+(max n (\lambda p:nat.eqb (n \mod (nth_prime p)) O)).
(* max_prime_factor is indeed a factor *)
theorem divides_max_prime_factor_n: \forall n:nat. (S O) < n \to
nth_prime (max_prime_factor n) \divides n.
intros.apply divides_b_true_to_divides.
apply lt_O_nth_prime_n.
-apply f_max_true (\lambda p:nat.eqb (mod n (nth_prime p)) O) n.
+apply f_max_true (\lambda p:nat.eqb (n \mod (nth_prime p)) O) n.
cut \exists i. nth_prime i = smallest_factor n.
elim Hcut.
apply ex_intro nat ? a.
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 (mod n (nth_prime p)) O)) \le
-(max m (\lambda p:nat.eqb (mod m (nth_prime p)) O)).
+(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)).
apply f_m_to_le_max.
apply trans_le ? n.
apply le_max_n.apply divides_to_le.assumption.assumption.
assumption.
change with nth_prime (max_prime_factor n) \divides r \to False.
intro.
-cut mod r (nth_prime (max_prime_factor n)) \neq O.
+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.
match n1 with
[ O \Rightarrow nfa_one
| (S n2) \Rightarrow
- let p \def (max (S(S n2)) (\lambda p:nat.eqb (mod (S(S n2)) (nth_prime p)) O)) in
+ let p \def (max (S(S n2)) (\lambda p:nat.eqb ((S(S n2)) \mod (nth_prime p)) O)) in
match p_ord (S(S n2)) (nth_prime p) with
[ (pair q r) \Rightarrow
nfa_proper (factorize_aux p r (nf_last (pred q)))]]].
apply nat_case n.reflexivity.
intro.apply nat_case m.reflexivity.
intro.change with
-let p \def (max (S(S m1)) (\lambda p:nat.eqb (mod (S(S m1)) (nth_prime p)) O)) in
+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
nfa_proper (factorize_aux p r (nf_last (pred q)))])=(S(S m1)).
| false \Rightarrow
match p with
[O \Rightarrow n
- |(S q) \Rightarrow gcd_aux q n (mod m n)]].
+ |(S q) \Rightarrow gcd_aux q n (m \mod n)]].
definition gcd : nat \to nat \to nat \def
\lambda n,m:nat.
| (S p) \Rightarrow gcd_aux (S p) n (S p) ]].
theorem divides_mod: \forall p,m,n:nat. O < n \to p \divides m \to p \divides n \to
-p \divides mod m n.
+p \divides (m \mod n).
intros.elim H1.elim H2.
-apply witness ? ? (n2 - n1*(div m n)).
+apply witness ? ? (n2 - n1*(m / n)).
rewrite > distr_times_minus.
rewrite < H3.
rewrite < assoc_times.
apply plus_to_minus.
rewrite > div_mod m n in \vdash (? ? %).
rewrite > sym_times.
-apply eq_plus_to_le ? ? (mod m n).
+apply eq_plus_to_le ? ? (m \mod n).
reflexivity.
assumption.
rewrite > sym_times.
qed.
theorem divides_mod_to_divides: \forall p,m,n:nat. O < n \to
-p \divides mod m n \to p \divides n \to p \divides m.
+p \divides (m \mod n) \to p \divides n \to p \divides m.
intros.elim H1.elim H2.
-apply witness p m ((n1*(div m n))+n2).
+apply witness p m ((n1*(m / n))+n2).
rewrite > distr_times_plus.
rewrite < H3.
rewrite < assoc_times.
change with
(match divides_b n1 m with
[ true \Rightarrow n1
-| false \Rightarrow gcd_aux n n1 (mod m n1)]) \divides m \land
+| false \Rightarrow gcd_aux n n1 (m \mod n1)]) \divides m \land
(match divides_b n1 m with
[ true \Rightarrow n1
-| false \Rightarrow gcd_aux n n1 (mod m n1)]) \divides n1.
+| false \Rightarrow gcd_aux n n1 (m \mod n1)]) \divides n1.
elim Hcut.rewrite > divides_to_divides_b_true.
simplify.
split.assumption.apply witness n1 n1 (S O).apply times_n_SO.
assumption.assumption.
rewrite > not_divides_to_divides_b_false.
change with
-gcd_aux n n1 (mod m n1) \divides m \land
-gcd_aux n n1 (mod m n1) \divides n1.
-cut gcd_aux n n1 (mod m n1) \divides n1 \land
-gcd_aux n n1 (mod m n1) \divides mod m n1.
+gcd_aux n n1 (m \mod n1) \divides m \land
+gcd_aux n n1 (m \mod n1) \divides n1.
+cut gcd_aux n n1 (m \mod n1) \divides n1 \land
+gcd_aux n n1 (m \mod n1) \divides mod m n1.
elim Hcut1.
split.apply divides_mod_to_divides ? ? n1.
assumption.assumption.assumption.assumption.
apply H.
-cut O \lt mod m n1 \lor O = mod m n1.
+cut O \lt m \mod n1 \lor O = mod m n1.
elim Hcut1.assumption.
apply False_ind.apply H4.apply mod_O_to_divides.
assumption.apply sym_eq.assumption.
apply lt_mod_m_m.assumption.
apply le_S_S_to_le.
apply trans_le ? n1.
-change with mod m n1 < n1.
+change with m \mod n1 < n1.
apply lt_mod_m_m.assumption.assumption.
assumption.assumption.
apply decidable_divides n1 m.assumption.
d \divides
(match divides_b n1 m with
[ true \Rightarrow n1
-| false \Rightarrow gcd_aux n n1 (mod m n1)]).
+| false \Rightarrow gcd_aux n n1 (m \mod n1)]).
cut n1 \divides m \lor n1 \ndivides m.
elim Hcut.
rewrite > divides_to_divides_b_true.
simplify.assumption.
assumption.assumption.
rewrite > not_divides_to_divides_b_false.
-change with d \divides gcd_aux n n1 (mod m n1).
+change with d \divides gcd_aux n n1 (m \mod n1).
apply H.
-cut O \lt mod m n1 \lor O = mod m n1.
+cut O \lt m \mod n1 \lor O = m \mod n1.
elim Hcut1.assumption.
absurd n1 \divides m.apply mod_O_to_divides.
assumption.apply sym_eq.assumption.assumption.
apply lt_mod_m_m.assumption.
apply le_S_S_to_le.
apply trans_le ? n1.
-change with mod m n1 < n1.
+change with m \mod n1 < n1.
apply lt_mod_m_m.assumption.assumption.
assumption.
apply divides_mod.assumption.assumption.assumption.
\exists a,b.
a*n1 - b*m = match divides_b n1 m with
[ true \Rightarrow n1
-| false \Rightarrow gcd_aux n n1 (mod m n1)]
+| false \Rightarrow gcd_aux n n1 (m \mod n1)]
\lor
b*m - a*n1 = match divides_b n1 m with
[ true \Rightarrow n1
-| false \Rightarrow gcd_aux n n1 (mod m n1)].
+| false \Rightarrow gcd_aux n n1 (m \mod n1)].
elim Hcut1.
rewrite > divides_to_divides_b_true.
simplify.
rewrite > not_divides_to_divides_b_false.
change with
\exists a,b.
-a*n1 - b*m = gcd_aux n n1 (mod m n1)
+a*n1 - b*m = gcd_aux n n1 (m \mod n1)
\lor
-b*m - a*n1 = gcd_aux n n1 (mod m n1).
+b*m - a*n1 = gcd_aux n n1 (m \mod n1).
cut
\exists a,b.
-a*(mod m n1) - b*n1= gcd_aux n n1 (mod m n1)
+a*(m \mod n1) - b*n1= gcd_aux n n1 (m \mod n1)
\lor
-b*n1 - a*(mod m n1) = gcd_aux n n1 (mod m n1).
+b*n1 - a*(m \mod n1) = gcd_aux n n1 (m \mod n1).
elim Hcut2.elim H5.elim H6.
(* first case *)
rewrite < H7.
-apply ex_intro ? ? (a1+a*(div m n1)).
+apply ex_intro ? ? (a1+a*(m / n1)).
apply ex_intro ? ? a.
right.
rewrite < sym_plus.
assumption.
(* second case *)
rewrite < H7.
-apply ex_intro ? ? (a1+a*(div m n1)).
+apply ex_intro ? ? (a1+a*(m / n1)).
apply ex_intro ? ? a.
left.
(* clear Hcut2.clear H5.clear H6.clear H. *)
rewrite < minus_n_n.reflexivity.
apply le_n.
assumption.
-apply H n1 (mod m n1).
-cut O \lt mod m n1 \lor O = mod m n1.
+apply H n1 (m \mod n1).
+cut O \lt m \mod n1 \lor O = m \mod n1.
elim Hcut2.assumption.
absurd n1 \divides m.apply mod_O_to_divides.
assumption.
apply lt_mod_m_m.assumption.
apply le_S_S_to_le.
apply trans_le ? n1.
-change with mod m n1 < n1.
+change with m \mod n1 < n1.
apply lt_mod_m_m.
assumption.assumption.assumption.assumption.
apply decidable_divides n1 m.assumption.
(* div *)
-theorem eq_mod_O_to_lt_O_div: \forall n,m:nat. O < m \to O < n\to mod n m = O \to O < div n m.
+theorem eq_mod_O_to_lt_O_div: \forall n,m:nat. O < m \to O < n\to n \mod m = O \to O < n / m.
intros 4.apply lt_O_n_elim m H.intros.
apply lt_times_to_lt_r m1.
rewrite < times_n_O.
-rewrite > plus_n_O ((S m1)*(div n (S m1))).
+rewrite > plus_n_O ((S m1)*(n / (S m1))).
rewrite < H2.
rewrite < sym_times.
rewrite < div_mod.
simplify.apply le_S_S.apply le_O_n.
qed.
-theorem lt_div_n_m_n: \forall n,m:nat. (S O) < m \to O < n \to div n m \lt n.
+theorem lt_div_n_m_n: \forall n,m:nat. (S O) < m \to O < n \to n / m \lt n.
intros.
-apply nat_case1 (div n m).intro.
+apply nat_case1 (n / m).intro.
assumption.intros.rewrite < H2.
rewrite > (div_mod n m) in \vdash (? ? %).
-apply lt_to_le_to_lt ? ((div n m)*m).
-apply lt_to_le_to_lt ? ((div n m)*(S (S O))).
+apply lt_to_le_to_lt ? ((n / m)*m).
+apply lt_to_le_to_lt ? ((n / m)*(S (S O))).
rewrite < sym_times.
rewrite > H2.
simplify.
(* this definition of log is based on pairs, with a remainder *)
let rec p_ord_aux p n m \def
- match (mod n m) with
+ match n \mod m with
[ O \Rightarrow
match p with
[ O \Rightarrow pair nat nat O n
| (S p) \Rightarrow
- match (p_ord_aux p (div n m) m) with
+ match (p_ord_aux p (n / m) m) with
[ (pair q r) \Rightarrow pair nat nat (S q) r]]
| (S a) \Rightarrow pair nat nat O n].
elim p.
change with
match (
-match (mod n m) with
+match n \mod m with
[ O \Rightarrow pair nat nat O n
| (S a) \Rightarrow pair nat nat O n] )
with
[ (pair q r) \Rightarrow n = m \sup q * r ].
-apply nat_case (mod n m).
+apply nat_case (n \mod m).
simplify.apply plus_n_O.
intros.
simplify.apply plus_n_O.
change with
match (
-match (mod n1 m) with
+match n1 \mod m with
[ O \Rightarrow
- match (p_ord_aux n (div n1 m) m) with
+ match (p_ord_aux n (n1 / m) m) with
[ (pair q r) \Rightarrow pair nat nat (S q) r]
| (S a) \Rightarrow pair nat nat O n1] )
with
[ (pair q r) \Rightarrow n1 = m \sup q * r].
-apply nat_case1 (mod n1 m).intro.
+apply nat_case1 (n1 \mod m).intro.
change with
match (
- match (p_ord_aux n (div n1 m) m) with
+ match (p_ord_aux n (n1 / m) m) with
[ (pair q r) \Rightarrow pair nat nat (S q) r])
with
[ (pair q r) \Rightarrow n1 = m \sup q * r].
-generalize in match (H (div n1 m) m).
-elim p_ord_aux n (div n1 m) m.
+generalize in match (H (n1 / m) m).
+elim p_ord_aux n (n1 / m) m.
simplify.
rewrite > assoc_times.
-rewrite < H3.rewrite > plus_n_O (m*(div n1 m)).
+rewrite < H3.rewrite > plus_n_O (m*(n1 / m)).
rewrite < H2.
rewrite > sym_times.
rewrite < div_mod.reflexivity.
assumption.
qed.
(* questo va spostato in primes1.ma *)
-theorem p_ord_exp: \forall n,m,i. O < m \to mod n m \neq O \to
+theorem p_ord_exp: \forall n,m,i. O < m \to n \mod m \neq O \to
\forall p. i \le p \to p_ord_aux p (m \sup i * n) m = pair nat nat i n.
intros 5.
elim i.
rewrite < plus_n_O.
apply nat_case p.
change with
- match (mod n m) with
+ match n \mod m with
[ O \Rightarrow pair nat nat O n
| (S a) \Rightarrow pair nat nat O n]
= pair nat nat O n.
-elim (mod n m).simplify.reflexivity.simplify.reflexivity.
+elim (n \mod m).simplify.reflexivity.simplify.reflexivity.
intro.
change with
- match (mod n m) with
+ match n \mod m with
[ O \Rightarrow
- match (p_ord_aux m1 (div n m) m) with
+ match (p_ord_aux m1 (n / m) m) with
[ (pair q r) \Rightarrow pair nat nat (S q) r]
| (S a) \Rightarrow pair nat nat O n]
= pair nat nat O n.
-cut O < mod n m \lor O = mod n m.
-elim Hcut.apply lt_O_n_elim (mod n m) H3.
+cut O < n \mod m \lor O = n \mod m.
+elim Hcut.apply lt_O_n_elim (n \mod m) H3.
intros. simplify.reflexivity.
apply False_ind.
apply H1.apply sym_eq.assumption.
apply nat_case p.intro.apply False_ind.apply not_le_Sn_O n1 H4.
intros.
change with
- match (mod (m \sup (S n1) *n) m) with
+ match ((m \sup (S n1) *n) \mod m) with
[ O \Rightarrow
- match (p_ord_aux m1 (div (m \sup (S n1) *n) m) m) with
+ match (p_ord_aux m1 ((m \sup (S n1) *n) / m) m) with
[ (pair q r) \Rightarrow pair nat nat (S q) r]
| (S a) \Rightarrow pair nat nat O (m \sup (S n1) *n)]
= pair nat nat (S n1) n.
-cut (mod (m \sup (S n1)*n) m) = O.
+cut ((m \sup (S n1)*n) \mod m) = O.
rewrite > Hcut.
change with
-match (p_ord_aux m1 (div (m \sup (S n1)*n) m) m) with
+match (p_ord_aux m1 ((m \sup (S n1)*n) / m) m) with
[ (pair q r) \Rightarrow pair nat nat (S q) r]
= pair nat nat (S n1) n.
-cut div (m \sup (S n1) *n) m = m \sup n1 *n.
+cut (m \sup (S n1) *n) / m = m \sup n1 *n.
rewrite > Hcut1.
rewrite > H2 m1. simplify.reflexivity.
apply le_S_S_to_le.assumption.
(* div_exp *)
-change with div (m* m \sup n1 *n) m = m \sup n1 * n.
+change with (m* m \sup n1 *n) / m = m \sup n1 * n.
rewrite > assoc_times.
apply lt_O_n_elim m H.
intro.apply div_times.
theorem p_ord_aux_to_Prop1: \forall p,n,m. (S O) < m \to O < n \to n \le p \to
match p_ord_aux p n m with
- [ (pair q r) \Rightarrow mod r m \neq O].
+ [ (pair q r) \Rightarrow r \mod m \neq O].
intro.elim p.absurd O < n.assumption.
apply le_to_not_lt.assumption.
change with
match
- (match (mod n1 m) with
+ (match n1 \mod m with
[ O \Rightarrow
- match (p_ord_aux n(div n1 m) m) with
+ match (p_ord_aux n(n1 / m) m) with
[ (pair q r) \Rightarrow pair nat nat (S q) r]
| (S a) \Rightarrow pair nat nat O n1])
with
- [ (pair q r) \Rightarrow mod r m \neq O].
-apply nat_case1 (mod n1 m).intro.
-generalize in match (H (div n1 m) m).
-elim (p_ord_aux n (div n1 m) m).
+ [ (pair q r) \Rightarrow r \mod m \neq O].
+apply nat_case1 (n1 \mod m).intro.
+generalize in match (H (n1 / m) m).
+elim (p_ord_aux n (n1 / m) m).
apply H5.assumption.
apply eq_mod_O_to_lt_O_div.
apply trans_lt ? (S O).simplify.apply le_n.
assumption.assumption.assumption.
apply le_S_S_to_le.
-apply trans_le ? n1.change with div n1 m < n1.
+apply trans_le ? n1.change with n1 / m < n1.
apply lt_div_n_m_n.assumption.assumption.assumption.
intros.
-change with mod n1 m \neq O.
+change with n1 \mod m \neq O.
rewrite > H4.
(* Andrea: META NOT FOUND !!!
rewrite > sym_eq. *)
qed.
theorem p_ord_aux_to_not_mod_O: \forall p,n,m,q,r. (S O) < m \to O < n \to n \le p \to
- pair nat nat q r = p_ord_aux p n m \to mod r m \neq O.
+ pair nat nat q r = p_ord_aux p n m \to r \mod m \neq O.
intros.
change with
match (pair nat nat q r) with
- [ (pair q r) \Rightarrow mod r m \neq O].
+ [ (pair q r) \Rightarrow r \mod m \neq O].
rewrite > H3.
apply p_ord_aux_to_Prop1.
assumption.assumption.assumption.
qed.
theorem divides_to_div_mod_spec :
-\forall n,m. O < n \to n \divides m \to div_mod_spec m n (div m n) O.
+\forall n,m. O < n \to n \divides m \to div_mod_spec m n (m / n) O.
intros.elim H1.rewrite > H2.
constructor 1.assumption.
apply lt_O_n_elim n H.intros.
qed.
theorem divides_to_mod_O:
-\forall n,m. O < n \to n \divides m \to (mod m n) = O.
-intros.apply div_mod_spec_to_eq2 m n (div m n) (mod m n) (div m n) O.
+\forall n,m. O < n \to n \divides m \to (m \mod n) = O.
+intros.apply div_mod_spec_to_eq2 m n (m / n) (m \mod n) (m / n) O.
apply div_mod_spec_div_mod.assumption.
apply divides_to_div_mod_spec.assumption.assumption.
qed.
theorem mod_O_to_divides:
-\forall n,m. O< n \to (mod m n) = O \to n \divides m.
+\forall n,m. O< n \to (m \mod n) = O \to n \divides m.
intros.
-apply witness n m (div m n).
-rewrite > plus_n_O (n*div m n).
+apply witness n m (m / n).
+rewrite > plus_n_O (n * (m / n)).
rewrite < H1.
rewrite < sym_times.
(* Andrea: perche' hint non lo trova ?*)
(* boolean divides *)
definition divides_b : nat \to nat \to bool \def
-\lambda n,m :nat. (eqb (mod m n) O).
+\lambda n,m :nat. (eqb (m \mod n) O).
theorem divides_b_to_Prop :
\forall n,m:nat. O < n \to
| false \Rightarrow n \ndivides m].
intros.
change with
-match eqb (mod m n) O with
+match eqb (m \mod n) O with
[ true \Rightarrow n \divides m
| false \Rightarrow n \ndivides m].
apply eqb_elim.
qed.
theorem mod_S_pi: \forall f:nat \to nat.\forall n,i:nat.
-i < n \to (S O) < (f i) \to mod (S (pi n f)) (f i) = (S O).
-intros.cut mod (pi n f) (f i) = O.
+i < n \to (S O) < (f i) \to (S (pi n f)) \mod (f i) = (S O).
+intros.cut (pi n f) \mod (f i) = O.
rewrite < Hcut.
apply mod_S.apply trans_lt O (S O).apply le_n (S O).assumption.
rewrite > Hcut.assumption.
qed.
theorem mod_S_fact: \forall n,i:nat.
-(S O) < i \to i \le n \to mod (S n!) i = (S O).
-intros.cut mod n! i = O.
+(S O) < i \to i \le n \to (S n!) \mod i = (S O).
+intros.cut n! \mod i = O.
rewrite < Hcut.
apply mod_S.apply trans_lt O (S O).apply le_n (S O).assumption.
rewrite > Hcut.assumption.
intros.
apply divides_b_false_to_not_divides.
apply trans_lt O (S O).apply le_n (S O).assumption.
-change with (eqb (mod (S n!) i) O) = false.
+change with (eqb ((S n!) \mod i) O) = false.
rewrite > mod_S_fact.simplify.reflexivity.
assumption.assumption.
qed.
| (S p) \Rightarrow
match p with
[ O \Rightarrow (S O)
- | (S q) \Rightarrow min_aux q (S(S q)) (\lambda m.(eqb (mod (S(S q)) m) O))]].
+ | (S q) \Rightarrow min_aux q (S(S q)) (\lambda m.(eqb ((S(S q)) \mod m) O))]].
(* it works !
theorem example1 : smallest_prime_factor (S(S(S O))) = (S(S(S O))).
intro.apply nat_case m.intro. apply False_ind.apply not_le_Sn_n (S O) H.
intros.
change with
-S O < min_aux m1 (S(S m1)) (\lambda m.(eqb (mod (S(S m1)) m) O)).
+S O < min_aux m1 (S(S m1)) (\lambda m.(eqb ((S(S m1)) \mod m) O)).
apply lt_to_le_to_lt ? (S (S O)).
apply le_n (S(S O)).
cut (S(S O)) = (S(S m1)) - m1.
apply divides_b_true_to_divides.
apply lt_O_smallest_factor ? H.
change with
-eqb (mod (S(S m1)) (min_aux m1 (S(S m1))
- (\lambda m.(eqb (mod (S(S m1)) m) O)))) O = true.
+eqb ((S(S m1)) \mod (min_aux m1 (S(S m1))
+ (\lambda m.(eqb ((S(S m1)) \mod m) O)))) O = true.
apply f_min_aux_true.
apply ex_intro nat ? (S(S m1)).
split.split.
intros.
apply divides_b_false_to_not_divides.
apply trans_lt O (S O).apply le_n (S O).assumption.
-change with (eqb (mod (S(S m1)) i) O) = false.
+change with (eqb ((S(S m1)) \mod i) O) = false.
apply lt_min_aux_to_false
-(\lambda i:nat.eqb (mod (S(S m1)) i) O) (S(S m1)) m1 i.
+(\lambda i:nat.eqb ((S(S m1)) \mod i) O) (S(S m1)) m1 i.
cut (S(S O)) = (S(S m1)-m1).
rewrite < Hcut.exact H1.
apply sym_eq. apply plus_to_minus.
(* p is just an upper bound, acc is an accumulator *)
let rec n_divides_aux p n m acc \def
- match (mod n m) with
+ match n \mod m with
[ O \Rightarrow
match p with
[ O \Rightarrow pair nat nat acc n
- | (S p) \Rightarrow n_divides_aux p (div n m) m (S acc)]
+ | (S p) \Rightarrow n_divides_aux p (n / m) m (S acc)]
| (S a) \Rightarrow pair nat nat acc n].
(* n_divides n m = <q,r> if m divides n q times, with remainder r *)
match n_divides_aux p n m a with
[ (pair q r) \Rightarrow n = m \sup a *r].
intros.
-apply nat_case (mod n m). *)
+apply nat_case (n \mod m). *)
projects / helm.git / commitdiff