(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/nat/primes".
-
include "nat/div_and_mod.ma".
include "nat/minimization.ma".
include "nat/sigma_and_pi.ma".
inductive divides (n,m:nat) : Prop \def
witness : \forall p:nat.m = times n p \to divides n m.
-interpretation "divides" 'divides n m = (cic:/matita/nat/primes/divides.ind#xpointer(1/1) n m).
-interpretation "not divides" 'ndivides n m =
- (cic:/matita/logic/connectives/Not.con (cic:/matita/nat/primes/divides.ind#xpointer(1/1) n m)).
+interpretation "divides" 'divides n m = (divides n m).
+interpretation "not divides" 'ndivides n m = (Not (divides n m)).
theorem reflexive_divides : reflexive nat divides.
unfold reflexive.
theorem divides_plus: \forall n,p,q:nat.
n \divides p \to n \divides q \to n \divides p+q.
intros.
-elim H.elim H1. apply (witness n (p+q) (n2+n1)).
+elim H.elim H1. apply (witness n (p+q) (n1+n2)).
rewrite > H2.rewrite > H3.apply sym_eq.apply distr_times_plus.
qed.
theorem divides_minus: \forall n,p,q:nat.
divides n p \to divides n q \to divides n (p-q).
intros.
-elim H.elim H1. apply (witness n (p-q) (n2-n1)).
+elim H.elim H1. apply (witness n (p-q) (n1-n2)).
rewrite > H2.rewrite > H3.apply sym_eq.apply distr_times_minus.
qed.
theorem divides_times: \forall n,m,p,q:nat.
n \divides p \to m \divides q \to n*m \divides p*q.
intros.
-elim H.elim H1. apply (witness (n*m) (p*q) (n2*n1)).
+elim H.elim H1. apply (witness (n*m) (p*q) (n1*n2)).
rewrite > H2.rewrite > H3.
-apply (trans_eq nat ? (n*(m*(n2*n1)))).
-apply (trans_eq nat ? (n*(n2*(m*n1)))).
+apply (trans_eq nat ? (n*(m*(n1*n2)))).
+apply (trans_eq nat ? (n*(n1*(m*n2)))).
apply assoc_times.
apply eq_f.
-apply (trans_eq nat ? ((n2*m)*n1)).
+apply (trans_eq nat ? ((n1*m)*n2)).
apply sym_eq. apply assoc_times.
-rewrite > (sym_times n2 m).apply assoc_times.
+rewrite > (sym_times n1 m).apply assoc_times.
apply sym_eq. apply assoc_times.
qed.
theorem transitive_divides: transitive ? divides.
unfold.
intros.
-elim H.elim H1. apply (witness x z (n2*n)).
+elim H.elim H1. apply (witness x z (n1*n)).
rewrite > H3.rewrite > H2.
apply assoc_times.
qed.
theorem antisymmetric_divides: antisymmetric nat divides.
unfold antisymmetric.intros.elim H. elim H1.
-apply (nat_case1 n2).intro.
+apply (nat_case1 n1).intro.
rewrite > H3.rewrite > H2.rewrite > H4.
rewrite < times_n_O.reflexivity.
intros.
(* divides le *)
theorem divides_to_le : \forall n,m. O < m \to n \divides m \to n \le m.
-intros. elim H1.rewrite > H2.cut (O < n2).
-apply (lt_O_n_elim n2 Hcut).intro.rewrite < sym_times.
+intros. elim H1.rewrite > H2.cut (O < n1).
+apply (lt_O_n_elim n1 Hcut).intro.rewrite < sym_times.
simplify.rewrite < sym_plus.
apply le_plus_n.
-elim (le_to_or_lt_eq O n2).
+elim (le_to_or_lt_eq O n1).
assumption.
absurd (O<m).assumption.
rewrite > H2.rewrite < H3.rewrite < times_n_O.
simplify.exact (not_le_Sn_n O).
qed.
+(*a variant of or_div_mod *)
+theorem or_div_mod1: \forall n,q. O < q \to
+(divides q (S n)) \land S n = (S (div n q)) * q \lor
+(\lnot (divides q (S n)) \land S n= (div n q) * q + S (n\mod q)).
+intros.elim (or_div_mod n q H);elim H1
+ [left.split
+ [apply (witness ? ? (S (n/q))).
+ rewrite > sym_times.assumption
+ |assumption
+ ]
+ |right.split
+ [intro.
+ apply (not_eq_O_S (n \mod q)).
+ (* come faccio a fare unfold nelleipotesi ? *)
+ cut ((S n) \mod q = O)
+ [rewrite < Hcut.
+ apply (div_mod_spec_to_eq2 (S n) q (div (S n) q) (mod (S n) q) (div n q) (S (mod n q)))
+ [apply div_mod_spec_div_mod.
+ assumption
+ |apply div_mod_spec_intro;assumption
+ ]
+ |apply divides_to_mod_O;assumption
+ ]
+ |assumption
+ ]
+ ]
+qed.
+
+theorem divides_to_div: \forall n,m.divides n m \to m/n*n = m.
+intro.
+elim (le_to_or_lt_eq O n (le_O_n n))
+ [rewrite > plus_n_O.
+ rewrite < (divides_to_mod_O ? ? H H1).
+ apply sym_eq.
+ apply div_mod.
+ assumption
+ |elim H1.
+ generalize in match H2.
+ rewrite < H.
+ simplify.
+ intro.
+ rewrite > H3.
+ reflexivity
+ ]
+qed.
+
+theorem divides_div: \forall d,n. divides d n \to divides (n/d) n.
+intros.
+apply (witness ? ? d).
+apply sym_eq.
+apply divides_to_div.
+assumption.
+qed.
+
+theorem div_div: \forall n,d:nat. O < n \to divides d n \to
+n/(n/d) = d.
+intros.
+apply (inj_times_l1 (n/d))
+ [apply (lt_times_n_to_lt d)
+ [apply (divides_to_lt_O ? ? H H1).
+ |rewrite > divides_to_div;assumption
+ ]
+ |rewrite > divides_to_div
+ [rewrite > sym_times.
+ rewrite > divides_to_div
+ [reflexivity
+ |assumption
+ ]
+ |apply (witness ? ? d).
+ apply sym_eq.
+ apply divides_to_div.
+ assumption
+ ]
+ ]
+qed.
+
+theorem divides_to_eq_times_div_div_times: \forall a,b,c:nat.
+O \lt b \to c \divides b \to a * (b /c) = (a*b)/c.
+intros.
+elim H1.
+rewrite > H2.
+rewrite > (sym_times c n1).
+cut(O \lt c)
+[ rewrite > (lt_O_to_div_times n1 c)
+ [ rewrite < assoc_times.
+ rewrite > (lt_O_to_div_times (a *n1) c)
+ [ reflexivity
+ | assumption
+ ]
+ | assumption
+ ]
+| apply (divides_to_lt_O c b);
+ assumption.
+]
+qed.
+
+theorem eq_div_plus: \forall n,m,d. O < d \to
+divides d n \to divides d m \to
+(n + m ) / d = n/d + m/d.
+intros.
+elim H1.
+elim H2.
+rewrite > H3.rewrite > H4.
+rewrite < distr_times_plus.
+rewrite > sym_times.
+rewrite > sym_times in ⊢ (? ? ? (? (? % ?) ?)).
+rewrite > sym_times in ⊢ (? ? ? (? ? (? % ?))).
+rewrite > lt_O_to_div_times
+ [rewrite > lt_O_to_div_times
+ [rewrite > lt_O_to_div_times
+ [reflexivity
+ |assumption
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+qed.
+
(* boolean divides *)
definition divides_b : nat \to nat \to bool \def
\lambda n,m :nat. (eqb (m \mod n) O).
-
+
theorem divides_b_to_Prop :
\forall n,m:nat. O < n \to
match divides_b n m with
intro.simplify.unfold Not.intro.apply H1.apply divides_to_mod_O.assumption.assumption.
qed.
-theorem divides_b_true_to_divides :
+theorem divides_b_true_to_divides1:
\forall n,m:nat. O < n \to
(divides_b n m = true ) \to n \divides m.
intros.
assumption.
qed.
-theorem divides_b_false_to_not_divides :
+theorem divides_b_true_to_divides:
+\forall n,m:nat. divides_b n m = true \to n \divides m.
+intros 2.apply (nat_case n)
+ [apply (nat_case m)
+ [intro.apply divides_n_n
+ |simplify.intros.apply False_ind.
+ apply not_eq_true_false.apply sym_eq.
+ assumption
+ ]
+ |intros.
+ apply divides_b_true_to_divides1
+ [apply lt_O_S|assumption]
+ ]
+qed.
+
+theorem divides_b_false_to_not_divides1:
\forall n,m:nat. O < n \to
(divides_b n m = false ) \to n \ndivides m.
intros.
assumption.
qed.
+theorem divides_b_false_to_not_divides:
+\forall n,m:nat. divides_b n m = false \to n \ndivides m.
+intros 2.apply (nat_case n)
+ [apply (nat_case m)
+ [simplify.unfold Not.intros.
+ apply not_eq_true_false.assumption
+ |unfold Not.intros.elim H1.
+ apply (not_eq_O_S m1).apply sym_eq.
+ assumption
+ ]
+ |intros.
+ apply divides_b_false_to_not_divides1
+ [apply lt_O_S|assumption]
+ ]
+qed.
+
theorem decidable_divides: \forall n,m:nat.O < n \to
decidable (n \divides m).
intros.unfold decidable.
absurd (n \divides m).assumption.assumption.
qed.
+theorem divides_to_divides_b_true1 : \forall n,m:nat.
+O < m \to n \divides m \to divides_b n m = true.
+intro.
+elim (le_to_or_lt_eq O n (le_O_n n))
+ [apply divides_to_divides_b_true
+ [assumption|assumption]
+ |apply False_ind.
+ rewrite < H in H2.
+ elim H2.
+ simplify in H3.
+ apply (not_le_Sn_O O).
+ rewrite > H3 in H1.
+ assumption
+ ]
+qed.
+
theorem not_divides_to_divides_b_false: \forall n,m:nat. O < n \to
\lnot(n \divides m) \to (divides_b n m) = false.
intros.
reflexivity.
qed.
+theorem divides_b_div_true:
+\forall d,n. O < n \to
+ divides_b d n = true \to divides_b (n/d) n = true.
+intros.
+apply divides_to_divides_b_true1
+ [assumption
+ |apply divides_div.
+ apply divides_b_true_to_divides.
+ assumption
+ ]
+qed.
+
+theorem divides_b_true_to_lt_O: \forall n,m. O < n \to divides_b m n = true \to O < m.
+intros.
+elim (le_to_or_lt_eq ? ? (le_O_n m))
+ [assumption
+ |apply False_ind.
+ elim H1.
+ rewrite < H2 in H1.
+ simplify in H1.
+ apply (lt_to_not_eq O n H).
+ apply sym_eq.
+ apply eqb_true_to_eq.
+ assumption
+ ]
+qed.
+
(* divides and pi *)
theorem divides_f_pi_f : \forall f:nat \to nat.\forall n,m,i:nat.
m \le i \to i \le n+m \to f i \divides pi n f m.
(S O) < i \to i \le n \to i \ndivides S n!.
intros.
apply divides_b_false_to_not_divides.
-apply (trans_lt O (S O)).apply (le_n (S O)).assumption.
unfold divides_b.
-rewrite > mod_S_fact.simplify.reflexivity.
-assumption.assumption.
+rewrite > mod_S_fact[simplify.reflexivity|assumption|assumption].
qed.
(* prime *)
unfold Not.unfold prime.intro.elim H.apply (not_le_Sn_n (S O) H1).
qed.
+theorem prime_to_lt_O: \forall p. prime p \to O < p.
+intros.elim H.apply lt_to_le.assumption.
+qed.
+
+theorem prime_to_lt_SO: \forall p. prime p \to S O < p.
+intros.elim H.
+assumption.
+qed.
+
(* smallest factor *)
definition smallest_factor : nat \to nat \def
\lambda n:nat.
| (S p) \Rightarrow
match p with
[ O \Rightarrow (S O)
- | (S q) \Rightarrow min_aux q (S(S q)) (\lambda m.(eqb ((S(S q)) \mod m) O))]].
+ | (S q) \Rightarrow min_aux q (S (S O)) (\lambda m.(eqb ((S(S q)) \mod m) O))]].
-(* it works !
-theorem example1 : smallest_prime_factor (S(S(S O))) = (S(S(S O))).
+(* it works !
+theorem example1 : smallest_factor (S(S(S O))) = (S(S(S O))).
normalize.reflexivity.
qed.
-theorem example2: smallest_prime_factor (S(S(S(S O)))) = (S(S O)).
+theorem example2: smallest_factor (S(S(S(S O)))) = (S(S O)).
normalize.reflexivity.
qed.
-theorem example3 : smallest_prime_factor (S(S(S(S(S(S(S O))))))) = (S(S(S(S(S(S(S O))))))).
+theorem example3 : smallest_factor (S(S(S(S(S(S(S O))))))) = (S(S(S(S(S(S(S O))))))).
simplify.reflexivity.
qed. *)
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 ((S(S m1)) \mod m) O))).
+(S O < min_aux m1 (S (S O)) (\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 (witness ? ? (S O)). simplify.reflexivity.
intros.
apply divides_b_true_to_divides.
-apply (lt_O_smallest_factor ? H).
change with
-(eqb ((S(S m1)) \mod (min_aux m1 (S(S m1))
+(eqb ((S(S m1)) \mod (min_aux m1 (S (S O))
(\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.
-apply le_minus_m.apply le_n.
-rewrite > mod_n_n.reflexivity.
-apply (trans_lt ? (S O)).apply (le_n (S O)).unfold lt.
-apply le_S_S.apply le_S_S.apply le_O_n.
+apply (le_S_S_to_le (S (S O)) (S (S m1)) ?).
+apply (minus_le_O_to_le (S (S (S O))) (S (S (S m1))) ?).
+apply (le_n O).
+rewrite < sym_plus. simplify. apply le_n.
+apply (eq_to_eqb_true (mod (S (S m1)) (S (S m1))) O ?).
+apply (mod_n_n (S (S m1)) ?).
+apply (H).
qed.
theorem le_smallest_factor_n :
\forall n:nat. smallest_factor n \le n.
-intro.apply (nat_case n).simplify.reflexivity.
-intro.apply (nat_case m).simplify.reflexivity.
+intro.apply (nat_case n).simplify.apply le_n.
+intro.apply (nat_case m).simplify.apply le_n.
intro.apply divides_to_le.
unfold lt.apply le_S_S.apply le_O_n.
apply divides_smallest_factor_n.
intro.apply (nat_case m).intro. apply False_ind.apply (not_le_Sn_n (S O) H).
intros.
apply divides_b_false_to_not_divides.
-apply (trans_lt O (S O)).apply (le_n (S O)).assumption.unfold divides_b.
apply (lt_min_aux_to_false
-(\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.
-rewrite < sym_plus.simplify.reflexivity.
-exact H2.
+(\lambda i:nat.eqb ((S(S m1)) \mod i) O) (S (S O)) m1 i).
+assumption.
+assumption.
qed.
theorem prime_smallest_factor_n :
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