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.
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.
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.
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.
(* Andrea: perche' hint non lo trova ?*)
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_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.
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.
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.
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.
(* 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.
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).
+i < n \to f i \divides pi n f.
intros 3.elim n.apply False_ind.apply not_le_Sn_O i H.
simplify.
apply le_n_Sm_elim (S i) n1 H1.
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.
(* 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.
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.
(* 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.
| (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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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