X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Fnat%2Fprimes.ma;h=6913562439f6e0f45d195241fa7fcecd8d3bc353;hb=aeb7f0539398561dc84cadf38df14a051dd1ba75;hp=644cae978f58a419e9d2d53ca2188b7fbcbe1322;hpb=78044035b4419e569df0d7f6a7f96fa32d21a19d;p=helm.git

diff --git a/helm/matita/library/nat/primes.ma b/helm/matita/library/nat/primes.ma
index 644cae978..691356243 100644
--- a/helm/matita/library/nat/primes.ma
+++ b/helm/matita/library/nat/primes.ma
@@ -22,6 +22,10 @@ include "nat/factorial.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)).
+
 theorem reflexive_divides : reflexive nat divides.
 simplify.
 intros.
@@ -29,7 +33,7 @@ exact witness x x (S O) (times_n_SO x).
 qed.
 
 theorem divides_to_div_mod_spec :
-\forall n,m. O < n \to divides n 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.
@@ -37,8 +41,8 @@ rewrite < plus_n_O.
 rewrite > div_times.apply sym_times.
 qed.
 
-theorem div_mod_spec_to_div :
-\forall n,m,p. div_mod_spec m n p O \to divides n m.
+theorem div_mod_spec_to_divides :
+\forall n,m,p. div_mod_spec m n p O \to n \divides m.
 intros.elim H.
 apply witness n m p.
 rewrite < sym_times.
@@ -46,34 +50,38 @@ rewrite > plus_n_O (p*n).assumption.
 qed.
 
 theorem divides_to_mod_O:
-\forall n,m. O < n \to divides n 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  divides n 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.
-(* perche' hint non lo trova ?*)
+(* Andrea: perche' hint non lo trova ?*)
 apply div_mod.
 assumption.
 qed.
 
-theorem divides_n_O: \forall n:nat. divides n O.
+theorem divides_n_O: \forall n:nat. n \divides O.
 intro. apply witness n O O.apply times_n_O.
 qed.
 
-theorem divides_SO_n: \forall n:nat. divides (S O) n.
+theorem divides_n_n: \forall n:nat. n \divides n.
+intro. apply witness n n (S O).apply times_n_SO.
+qed.
+
+theorem divides_SO_n: \forall n:nat. (S O) \divides n.
 intro. apply witness (S O) n n. simplify.apply plus_n_O.
 qed.
 
 theorem divides_plus: \forall n,p,q:nat. 
-divides n p \to divides n q \to divides n (p+q).
+n \divides p \to n \divides q \to n \divides p+q.
 intros.
 elim H.elim H1. apply witness n (p+q) (n2+n1).
 rewrite > H2.rewrite > H3.apply sym_eq.apply distr_times_plus.
@@ -87,7 +95,7 @@ rewrite > H2.rewrite > H3.apply sym_eq.apply distr_times_minus.
 qed.
 
 theorem divides_times: \forall n,m,p,q:nat. 
-divides n p \to divides m q \to divides (n*m) (p*q).
+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).
 rewrite > H2.rewrite > H3.
@@ -101,16 +109,49 @@ rewrite > sym_times n2 m.apply assoc_times.
 apply sym_eq. apply assoc_times.
 qed.
 
-theorem transitive_divides: \forall n,m,p. 
-divides n m \to divides m p \to divides n p.
+theorem transitive_divides: transitive ? divides.
+unfold.
 intros.
-elim H.elim H1. apply witness n p (n2*n1).
+elim H.elim H1. apply witness x z (n2*n).
 rewrite > H3.rewrite > H2.
 apply assoc_times.
 qed.
 
+variant trans_divides: \forall n,m,p. 
+ n \divides m \to m \divides p \to n \divides p \def transitive_divides.
+
+theorem eq_mod_to_divides:\forall n,m,p. O< p \to
+mod n p = mod m p \to divides p (n-m).
+intros.
+cut n \le m \or \not n \le m.
+elim Hcut.
+cut n-m=O.
+rewrite > Hcut1.
+apply witness p O O.
+apply times_n_O.
+apply eq_minus_n_m_O.
+assumption.
+apply witness p (n-m) ((div n p)-(div m p)).
+rewrite > distr_times_minus.
+rewrite > sym_times.
+rewrite > sym_times p.
+cut (div n p)*p = n - (mod n p).
+rewrite > Hcut1.
+rewrite > eq_minus_minus_minus_plus.
+rewrite > sym_plus.
+rewrite > H1.
+rewrite < div_mod.reflexivity.
+assumption.
+apply sym_eq.
+apply plus_to_minus.
+rewrite > sym_plus.
+apply div_mod.
+assumption.
+apply decidable_le n m.
+qed.
+
 (* divides le *)
-theorem divides_to_le : \forall n,m. O < m \to divides n m \to n \le m.
+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.
 simplify.rewrite < sym_plus.
@@ -123,7 +164,7 @@ apply not_le_Sn_n O.
 apply le_O_n.
 qed.
 
-theorem divides_to_lt_O : \forall n,m. O < m \to divides n m \to O < n.
+theorem divides_to_lt_O : \forall n,m. O < m \to n \divides m \to O < n.
 intros.elim H1.
 elim le_to_or_lt_eq O n (le_O_n n).
 assumption.
@@ -134,18 +175,18 @@ qed.
 
 (* 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
 match divides_b n m with
-[ true \Rightarrow divides n m
-| false \Rightarrow \lnot (divides n m)].
+[ true \Rightarrow n \divides m
+| false \Rightarrow n \ndivides m].
 intros.
 change with 
-match eqb (mod m n) O with
-[ true \Rightarrow divides n m
-| false \Rightarrow \lnot (divides n m)].
+match eqb (m \mod n) O with
+[ true \Rightarrow n \divides m
+| false \Rightarrow n \ndivides m].
 apply eqb_elim.
 intro.simplify.apply mod_O_to_divides.assumption.assumption.
 intro.simplify.intro.apply H1.apply divides_to_mod_O.assumption.assumption.
@@ -153,107 +194,110 @@ qed.
 
 theorem divides_b_true_to_divides :
 \forall n,m:nat. O < n \to
-(divides_b n m = true ) \to divides n m.
+(divides_b n m = true ) \to n \divides m.
 intros.
 change with 
 match true with
-[ true \Rightarrow divides n m
-| false \Rightarrow \lnot (divides n m)].
+[ true \Rightarrow n \divides m
+| false \Rightarrow n \ndivides m].
 rewrite < H1.apply divides_b_to_Prop.
 assumption.
 qed.
 
 theorem divides_b_false_to_not_divides :
 \forall n,m:nat. O < n \to
-(divides_b n m = false ) \to \lnot (divides n m).
+(divides_b n m = false ) \to n \ndivides m.
 intros.
 change with 
 match false with
-[ true \Rightarrow divides n m
-| false \Rightarrow \lnot (divides n m)].
+[ true \Rightarrow n \divides m
+| false \Rightarrow n \ndivides m].
 rewrite < H1.apply divides_b_to_Prop.
 assumption.
 qed.
 
 theorem decidable_divides: \forall n,m:nat.O < n \to
-decidable (divides n m).
-intros.change with (divides n m) \lor \not (divides n m).
+decidable (n \divides m).
+intros.change with (n \divides m) \lor n \ndivides m.
 cut 
 match divides_b n m with
-[ true \Rightarrow divides n m
-| false \Rightarrow \not (divides n m)] \to (divides n m) \lor \not (divides n m).
+[ true \Rightarrow n \divides m
+| false \Rightarrow n \ndivides m] \to n \divides m \lor n \ndivides m.
 apply Hcut.apply divides_b_to_Prop.assumption.
 elim (divides_b n m).left.apply H1.right.apply H1.
 qed.
 
 theorem divides_to_divides_b_true : \forall n,m:nat. O < n \to
-divides n m \to divides_b n m = true.
+n \divides m \to divides_b n m = true.
 intros.
 cut match (divides_b n m) with
-[ true \Rightarrow (divides n m)
-| false \Rightarrow \not (divides n m)] \to ((divides_b n m) = true).
+[ true \Rightarrow n \divides m
+| false \Rightarrow n \ndivides m] \to ((divides_b n m) = true).
 apply Hcut.apply divides_b_to_Prop.assumption.
 elim divides_b n m.reflexivity.
-absurd (divides n m).assumption.assumption.
+absurd (n \divides m).assumption.assumption.
 qed.
 
 theorem not_divides_to_divides_b_false: \forall n,m:nat. O < n \to
-\not(divides n m) \to (divides_b n m) = false.
+\lnot(n \divides m) \to (divides_b n m) = false.
 intros.
 cut match (divides_b n m) with
-[ true \Rightarrow (divides n m)
-| false \Rightarrow \not (divides n m)] \to ((divides_b n m) = false).
+[ true \Rightarrow n \divides m
+| false \Rightarrow n \ndivides m] \to ((divides_b n m) = false).
 apply Hcut.apply divides_b_to_Prop.assumption.
 elim divides_b n m.
-absurd (divides n m).assumption.assumption.
+absurd (n \divides m).assumption.assumption.
 reflexivity.
 qed.
 
 (* divides and pi *)
-theorem divides_f_pi_f : \forall f:nat \to nat.\forall n,i:nat. 
-i < n \to divides (f i) (pi n f).
-intros 3.elim n.apply False_ind.apply not_le_Sn_O i H.
+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.
+intros 5.elim n.simplify.
+cut i = m.rewrite < Hcut.apply divides_n_n.
+apply antisymmetric_le.assumption.assumption.
 simplify.
-apply le_n_Sm_elim (S i) n1 H1.
-intro.
-apply transitive_divides ? (pi n1 f).
-apply H.simplify.apply le_S_S_to_le. assumption.
-apply witness ? ? (f n1).apply sym_times.
-intro.cut i = n1.
-rewrite > Hcut.
-apply witness ? ? (pi n1 f).reflexivity.
-apply inj_S.assumption.
+cut i < S n1+m \lor i = S n1 + m.
+elim Hcut.
+apply transitive_divides ? (pi n1 f m).
+apply H1.apply le_S_S_to_le. assumption.
+apply witness ? ? (f (S n1+m)).apply sym_times.
+rewrite > H3.
+apply witness ? ? (pi n1 f m).reflexivity.
+apply le_to_or_lt_eq.assumption.
 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.
 apply divides_to_mod_O.apply trans_lt O (S O).apply le_n (S O).assumption.
 apply divides_f_pi_f.assumption.
 qed.
+*)
 
 (* divides and fact *)
 theorem divides_fact : \forall n,i:nat. 
-O < i \to i \le n \to divides i (fact n).
+O < i \to i \le n \to i \divides n!.
 intros 3.elim n.absurd O<i.assumption.apply le_n_O_elim i H1.
 apply not_le_Sn_O O.
-change with divides i ((S n1)*(fact n1)).
+change with i \divides (S n1)*n1!.
 apply le_n_Sm_elim i n1 H2.
 intro.
-apply transitive_divides ? (fact n1).
+apply transitive_divides ? n1!.
 apply H1.apply le_S_S_to_le. assumption.
 apply witness ? ? (S n1).apply sym_times.
 intro.
 rewrite > H3.
-apply witness ? ? (fact n1).reflexivity.
+apply witness ? ? n1!.reflexivity.
 qed.
 
 theorem mod_S_fact: \forall n,i:nat. 
-(S O) < i \to i \le n \to mod (S (fact n)) i = (S O).
-intros.cut mod (fact 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.
@@ -263,11 +307,11 @@ assumption.
 qed.
 
 theorem not_divides_S_fact: \forall n,i:nat. 
-(S O) < i \to i \le n \to \not (divides i (S (fact n))).
+(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.
-change with (eqb (mod (S (fact n)) i) O) = false.
+change with (eqb ((S n!) \mod i) O) = false.
 rewrite > mod_S_fact.simplify.reflexivity.
 assumption.assumption.
 qed.
@@ -275,7 +319,7 @@ qed.
 (* prime *)
 definition prime : nat \to  Prop \def
 \lambda n:nat. (S O) < n \land 
-(\forall m:nat. divides m n \to (S O) < m \to  m = n).
+(\forall m:nat. m \divides n \to (S O) < m \to  m = n).
 
 theorem not_prime_O: \lnot (prime O).
 simplify.intro.elim H.apply not_le_Sn_O (S O) H1.
@@ -293,7 +337,7 @@ match n with
 | (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))).
@@ -315,13 +359,13 @@ apply nat_case n.intro.apply False_ind.apply not_le_Sn_O (S O) H.
 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.
 rewrite > Hcut.
 apply le_min_aux.
-apply sym_eq.apply plus_to_minus.apply le_S.apply le_n_Sn.
+apply sym_eq.apply plus_to_minus.
 rewrite < sym_plus.simplify.reflexivity.
 qed.
 
@@ -337,7 +381,7 @@ apply le_S_S.apply le_O_n.
 qed.
 
 theorem divides_smallest_factor_n : 
-\forall n:nat. O < n \to divides (smallest_factor n) n.
+\forall n:nat. O < n \to smallest_factor n \divides n.
 intro.
 apply nat_case n.intro.apply False_ind.apply not_le_Sn_O O H.
 intro.apply nat_case m.intro. simplify.
@@ -346,8 +390,8 @@ intros.
 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.
@@ -368,20 +412,19 @@ simplify.apply le_S_S.apply le_O_n.
 qed.
 
 theorem lt_smallest_factor_to_not_divides: \forall n,i:nat. 
-(S O) < n \to (S O) < i \to i < (smallest_factor n) \to \lnot (divides i n).
+(S O) < n \to (S O) < i \to i < (smallest_factor n) \to i \ndivides n.
 intros 2.
 apply nat_case n.intro.apply False_ind.apply not_le_Sn_O (S O) H.
 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.
-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.
-apply le_S.apply le_n_Sn.
 rewrite < sym_plus.simplify.reflexivity.
 exact H2.
 qed.
@@ -389,13 +432,13 @@ qed.
 theorem prime_smallest_factor_n : 
 \forall n:nat. (S O) < n \to prime (smallest_factor n).
 intro. change with (S(S O)) \le n \to (S O) < (smallest_factor n) \land 
-(\forall m:nat. divides m (smallest_factor n) \to (S O) < m \to  m = (smallest_factor n)).
+(\forall m:nat. m \divides smallest_factor n \to (S O) < m \to  m = (smallest_factor n)).
 intro.split.
 apply lt_SO_smallest_factor.assumption.
 intros.
 cut le m (smallest_factor n).
 elim le_to_or_lt_eq m (smallest_factor n) Hcut.
-absurd divides m n.
+absurd m \divides n.
 apply transitive_divides m (smallest_factor n).
 assumption.
 apply divides_smallest_factor_n.
@@ -417,7 +460,7 @@ intro.apply nat_case m.intro.apply False_ind.apply not_prime_SO H.
 intro.
 change with 
 (S O) < (S(S m1)) \land 
-(\forall m:nat. divides m (S(S m1)) \to (S O) < m \to  m = (S(S m1))) \to 
+(\forall m:nat. m \divides S(S m1) \to (S O) < m \to  m = (S(S m1))) \to 
 smallest_factor (S(S m1)) = (S(S m1)).
 intro.elim H.apply H2.
 apply divides_smallest_factor_n.
@@ -455,7 +498,7 @@ qed. *)
 theorem primeb_to_Prop: \forall n.
 match primeb n with
 [ true \Rightarrow prime n
-| false \Rightarrow \not (prime n)].
+| false \Rightarrow \lnot (prime n)].
 intro.
 apply nat_case n.simplify.intro.elim H.apply not_le_Sn_O (S O) H1.
 intro.apply nat_case m.simplify.intro.elim H.apply not_le_Sn_n (S O) H1.
@@ -463,13 +506,13 @@ intro.
 change with 
 match eqb (smallest_factor (S(S m1))) (S(S m1)) with
 [ true \Rightarrow prime (S(S m1))
-| false \Rightarrow \not (prime (S(S m1)))].
+| false \Rightarrow \lnot (prime (S(S m1)))].
 apply eqb_elim (smallest_factor (S(S m1))) (S(S m1)).
 intro.change with prime (S(S m1)).
 rewrite < H.
 apply prime_smallest_factor_n.
 simplify.apply le_S_S.apply le_S_S.apply le_O_n.
-intro.change with \not (prime (S(S m1))).
+intro.change with \lnot (prime (S(S m1))).
 change with prime (S(S m1)) \to False.
 intro.apply H.
 apply prime_to_smallest_factor.
@@ -481,27 +524,27 @@ primeb n = true \to prime n.
 intros.change with
 match true with 
 [ true \Rightarrow prime n
-| false \Rightarrow \not (prime n)].
+| false \Rightarrow \lnot (prime n)].
 rewrite < H.
 apply primeb_to_Prop.
 qed.
 
 theorem primeb_false_to_not_prime : \forall n:nat.
-primeb n = false \to \not (prime n).
+primeb n = false \to \lnot (prime n).
 intros.change with
 match false with 
 [ true \Rightarrow prime n
-| false \Rightarrow \not (prime n)].
+| false \Rightarrow \lnot (prime n)].
 rewrite < H.
 apply primeb_to_Prop.
 qed.
 
 theorem decidable_prime : \forall n:nat.decidable (prime n).
-intro.change with (prime n) \lor \not (prime n).
+intro.change with (prime n) \lor \lnot (prime n).
 cut 
 match primeb n with
 [ true \Rightarrow prime n
-| false \Rightarrow \not (prime n)] \to (prime n) \lor \not (prime n).
+| false \Rightarrow \lnot (prime n)] \to (prime n) \lor \lnot (prime n).
 apply Hcut.apply primeb_to_Prop.
 elim (primeb n).left.apply H.right.apply H.
 qed.
@@ -511,18 +554,18 @@ prime n \to primeb n = true.
 intros.
 cut match (primeb n) with
 [ true \Rightarrow prime n
-| false \Rightarrow \not (prime n)] \to ((primeb n) = true).
+| false \Rightarrow \lnot (prime n)] \to ((primeb n) = true).
 apply Hcut.apply primeb_to_Prop.
 elim primeb n.reflexivity.
 absurd (prime n).assumption.assumption.
 qed.
 
 theorem not_prime_to_primeb_false: \forall n:nat. 
-\not(prime n) \to primeb n = false.
+\lnot(prime n) \to primeb n = false.
 intros.
 cut match (primeb n) with
 [ true \Rightarrow prime n
-| false \Rightarrow \not (prime n)] \to ((primeb n) = false).
+| false \Rightarrow \lnot (prime n)] \to ((primeb n) = false).
 apply Hcut.apply primeb_to_Prop.
 elim primeb n.
 absurd (prime n).assumption.assumption.