(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/nat/gcd".
-
include "nat/primes.ma".
+include "nat/lt_arith.ma".
let rec gcd_aux p m n: nat \def
match divides_b n m with
theorem divides_gcd_nm: \forall n,m.
gcd n m \divides m \land gcd n m \divides n.
intros.
-(*CSC: simplify simplifies too much because of a redex in gcd *)
change with
(match leb n m with
[ true \Rightarrow
exact (proj1 ? ? (divides_gcd_nm n m)).
qed.
+
+theorem divides_times_gcd_aux: \forall p,m,n,d,c.
+O \lt c \to O < n \to n \le m \to n \le p \to
+d \divides (c*m) \to d \divides (c*n) \to d \divides c*gcd_aux p m n.
+intro.
+elim p
+[ absurd (O < n)
+ [ assumption
+ | apply le_to_not_lt.
+ assumption
+ ]
+| simplify.
+ 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
+ [ simplify.
+ apply H
+ [ assumption
+ | 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 le_to_or_lt_eq.
+ apply le_O_n
+ ]
+ | apply lt_to_le.
+ apply lt_mod_m_m.
+ assumption
+ | apply le_S_S_to_le.
+ apply (trans_le ? n1)
+ [ change with (m \mod n1 < n1).
+ apply lt_mod_m_m.
+ assumption
+ | assumption
+ ]
+ | assumption
+ | rewrite < times_mod
+ [ rewrite < (sym_times c m).
+ rewrite < (sym_times c n1).
+ apply divides_mod
+ [ rewrite > (S_pred c)
+ [ rewrite > (S_pred n1)
+ [ apply (lt_O_times_S_S)
+ | assumption
+ ]
+ | assumption
+ ]
+ | assumption
+ | assumption
+ ]
+ | assumption
+ | assumption
+ ]
+ ]
+ | assumption
+ | assumption
+ ]
+ ]
+ | apply (decidable_divides n1 m).
+ assumption
+ ]
+]
+qed.
+
+(*a particular case of the previous theorem (setting c=1)*)
theorem divides_gcd_aux: \forall p,m,n,d. O < n \to n \le m \to n \le p \to
d \divides m \to d \divides n \to d \divides gcd_aux p m n.
-intro.elim p.
-absurd (O < n).assumption.apply le_to_not_lt.assumption.
-simplify.
-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.
-simplify.
-apply H.
-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 le_to_or_lt_eq.apply le_O_n.
-apply lt_to_le.
-apply lt_mod_m_m.assumption.
-apply le_S_S_to_le.
-apply (trans_le ? n1).
-change with (m \mod n1 < n1).
-apply lt_mod_m_m.assumption.assumption.
-assumption.
-apply divides_mod.assumption.assumption.assumption.
-assumption.assumption.
-apply (decidable_divides n1 m).assumption.
+intros.
+rewrite > (times_n_SO (gcd_aux p m n)).
+rewrite < (sym_times (S O)).
+apply (divides_times_gcd_aux)
+[ apply (lt_O_S O)
+| assumption
+| assumption
+| assumption
+| rewrite > (sym_times (S O)).
+ rewrite < (times_n_SO m).
+ assumption
+| rewrite > (sym_times (S O)).
+ rewrite < (times_n_SO n).
+ assumption
+]
qed.
-theorem divides_d_gcd: \forall m,n,d.
-d \divides m \to d \divides n \to d \divides gcd n m.
+theorem divides_d_times_gcd: \forall m,n,d,c.
+O \lt c \to d \divides (c*m) \to d \divides (c*n) \to d \divides c*gcd n m.
intros.
-(*CSC: here simplify simplifies too much because of a redex in gcd *)
change with
-(d \divides
+(d \divides c *
match leb n m with
[ true \Rightarrow
match n with
match m with
[ O \Rightarrow n
| (S p) \Rightarrow gcd_aux (S p) n (S p) ]]).
-apply (leb_elim n m).
-apply (nat_case1 n).simplify.intros.assumption.
-intros.
-change with (d \divides gcd_aux (S m1) m (S m1)).
-apply divides_gcd_aux.
-unfold lt.apply le_S_S.apply le_O_n.assumption.apply le_n.assumption.
-rewrite < H2.assumption.
-apply (nat_case1 m).simplify.intros.assumption.
+apply (leb_elim n m)
+[ apply (nat_case1 n)
+ [ simplify.
+ intros.
+ assumption
+ | intros.
+ change with (d \divides c*gcd_aux (S m1) m (S m1)).
+ apply divides_times_gcd_aux
+ [ assumption
+ | unfold lt.
+ apply le_S_S.
+ apply le_O_n
+ | assumption
+ | apply (le_n (S m1))
+ | assumption
+ | rewrite < H3.
+ assumption
+ ]
+ ]
+| apply (nat_case1 m)
+ [ simplify.
+ intros.
+ assumption
+ | intros.
+ change with (d \divides c * gcd_aux (S m1) n (S m1)).
+ apply divides_times_gcd_aux
+ [ unfold lt.
+ change with (O \lt c).
+ assumption
+ | apply lt_O_S
+ | apply lt_to_le.
+ apply not_le_to_lt.
+ assumption
+ | apply (le_n (S m1)).
+ | assumption
+ | rewrite < H3.
+ assumption
+ ]
+ ]
+]
+qed.
+
+(*a particular case of the previous theorem (setting c=1)*)
+theorem divides_d_gcd: \forall m,n,d.
+d \divides m \to d \divides n \to d \divides gcd n m.
intros.
-change with (d \divides gcd_aux (S m1) n (S m1)).
-apply divides_gcd_aux.
-unfold lt.apply le_S_S.apply le_O_n.
-apply lt_to_le.apply not_le_to_lt.assumption.apply le_n.assumption.
-rewrite < H2.assumption.
+rewrite > (times_n_SO (gcd n m)).
+rewrite < (sym_times (S O)).
+apply (divides_d_times_gcd)
+[ apply (lt_O_S O)
+| rewrite > (sym_times (S O)).
+ rewrite < (times_n_SO m).
+ assumption
+| rewrite > (sym_times (S O)).
+ rewrite < (times_n_SO n).
+ assumption
+]
qed.
theorem eq_minus_gcd_aux: \forall p,m,n.O < n \to n \le m \to n \le p \to
\exists a,b. a*n - b*m = gcd_aux p m n \lor b*m - a*n = gcd_aux p m n.
intro.
-elim p.
-absurd (O < n).assumption.apply le_to_not_lt.assumption.
-cut (O < m).
-cut (n1 \divides m \lor n1 \ndivides m).
-simplify.
-elim Hcut1.
-rewrite > divides_to_divides_b_true.
-simplify.
-apply (ex_intro ? ? (S O)).
-apply (ex_intro ? ? O).
-left.simplify.rewrite < plus_n_O.
-apply sym_eq.apply minus_n_O.
-assumption.assumption.
-rewrite > not_divides_to_divides_b_false.
-change with
-(\exists a,b.
-a*n1 - b*m = gcd_aux n n1 (m \mod n1)
-\lor
-b*m - a*n1 = gcd_aux n n1 (m \mod n1)).
-cut
-(\exists a,b.
-a*(m \mod n1) - b*n1= gcd_aux n n1 (m \mod n1)
-\lor
-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*(m / n1))).
-apply (ex_intro ? ? a).
-right.
-rewrite < sym_plus.
-rewrite < (sym_times n1).
-rewrite > distr_times_plus.
-rewrite > (sym_times n1).
-rewrite > (sym_times n1).
-rewrite > (div_mod m n1) in \vdash (? ? (? % ?) ?).
-rewrite > assoc_times.
-rewrite < sym_plus.
-rewrite > distr_times_plus.
-rewrite < eq_minus_minus_minus_plus.
-rewrite < sym_plus.
-rewrite < plus_minus.
-rewrite < minus_n_n.reflexivity.
-apply le_n.
-assumption.
-(* second case *)
-rewrite < H7.
-apply (ex_intro ? ? (a1+a*(m / n1))).
-apply (ex_intro ? ? a).
-left.
-(* clear Hcut2.clear H5.clear H6.clear H. *)
-rewrite > sym_times.
-rewrite > distr_times_plus.
-rewrite > sym_times.
-rewrite > (sym_times n1).
-rewrite > (div_mod m n1) in \vdash (? ? (? ? %) ?).
-rewrite > distr_times_plus.
-rewrite > assoc_times.
-rewrite < eq_minus_minus_minus_plus.
-rewrite < sym_plus.
-rewrite < plus_minus.
-rewrite < minus_n_n.reflexivity.
-apply le_n.
-assumption.
-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.
-symmetry.assumption.assumption.
-apply le_to_or_lt_eq.apply le_O_n.
-apply lt_to_le.
-apply lt_mod_m_m.assumption.
-apply le_S_S_to_le.
-apply (trans_le ? n1).
-change with (m \mod n1 < n1).
-apply lt_mod_m_m.
-assumption.assumption.assumption.assumption.
-apply (decidable_divides n1 m).assumption.
-apply (lt_to_le_to_lt ? n1).assumption.assumption.
+elim p
+ [absurd (O < n)
+ [assumption
+ |apply le_to_not_lt.assumption
+ ]
+ |cut (O < m)
+ [cut (n1 \divides m \lor n1 \ndivides m)
+ [simplify.
+ elim Hcut1
+ [rewrite > divides_to_divides_b_true
+ [simplify.
+ apply (ex_intro ? ? (S O)).
+ apply (ex_intro ? ? O).
+ left.
+ simplify.
+ rewrite < plus_n_O.
+ apply sym_eq.
+ apply minus_n_O
+ |assumption
+ |assumption
+ ]
+ |rewrite > not_divides_to_divides_b_false
+ [change with
+ (\exists a,b.a*n1 - b*m = gcd_aux n n1 (m \mod n1)
+ \lor b*m - a*n1 = gcd_aux n n1 (m \mod n1)).
+ cut
+ (\exists a,b.a*(m \mod n1) - b*n1= gcd_aux n n1 (m \mod n1)
+ \lor 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*(m / n1))).
+ apply (ex_intro ? ? a).
+ right.
+ rewrite < sym_plus.
+ rewrite < (sym_times n1).
+ rewrite > distr_times_plus.
+ rewrite > (sym_times n1).
+ rewrite > (sym_times n1).
+ rewrite > (div_mod m n1) in \vdash (? ? (? % ?) ?)
+ [rewrite > assoc_times.
+ rewrite < sym_plus.
+ rewrite > distr_times_plus.
+ rewrite < eq_minus_minus_minus_plus.
+ rewrite < sym_plus.
+ rewrite < plus_minus
+ [rewrite < minus_n_n.reflexivity
+ |apply le_n
+ ]
+ |assumption
+ ]
+ |(* second case *)
+ rewrite < H7.
+ apply (ex_intro ? ? (a1+a*(m / n1))).
+ apply (ex_intro ? ? a).
+ left.
+ (* clear Hcut2.clear H5.clear H6.clear H. *)
+ rewrite > sym_times.
+ rewrite > distr_times_plus.
+ rewrite > sym_times.
+ rewrite > (sym_times n1).
+ rewrite > (div_mod m n1) in \vdash (? ? (? ? %) ?)
+ [rewrite > distr_times_plus.
+ rewrite > assoc_times.
+ rewrite < eq_minus_minus_minus_plus.
+ rewrite < sym_plus.
+ rewrite < plus_minus
+ [rewrite < minus_n_n.reflexivity
+ |apply le_n
+ ]
+ |assumption
+ ]
+ ]
+ |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
+ |symmetry.assumption
+ ]
+ |assumption
+ ]
+ ]
+ |apply le_to_or_lt_eq.
+ apply le_O_n
+ ]
+ |apply lt_to_le.
+ apply lt_mod_m_m.
+ assumption
+ |apply le_S_S_to_le.
+ apply (trans_le ? n1)
+ [change with (m \mod n1 < n1).
+ apply lt_mod_m_m.
+ assumption
+ |assumption
+ ]
+ ]
+ ]
+ |assumption
+ |assumption
+ ]
+ ]
+ |apply (decidable_divides n1 m).
+ assumption
+ ]
+ |apply (lt_to_le_to_lt ? n1);assumption
+ ]
+ ]
qed.
theorem eq_minus_gcd:
qed.
theorem symmetric_gcd: symmetric nat gcd.
-(*CSC: bug here: unfold symmetric does not work *)
change with
(\forall n,m:nat. gcd n m = gcd m n).
intros.
|apply (trans_lt ? (S O))[apply le_n|assumption]
|assumption
]
- |elim (le_to_or_lt_eq O n2 (le_O_n n2))
+ |elim (le_to_or_lt_eq O n2 (le_O_n n2));
[assumption
|apply False_ind.
apply (le_to_not_lt n (S O))
apply le_to_or_lt_eq.apply le_O_n.
qed.
+(* primes and divides *)
theorem divides_times_to_divides: \forall n,p,q:nat.prime n \to n \divides p*q \to
n \divides p \lor n \divides q.
intros.
]
qed.
+theorem divides_exp_to_divides:
+\forall p,n,m:nat. prime p \to
+p \divides n \sup m \to p \divides n.
+intros 3.elim m.simplify in H1.
+apply (transitive_divides p (S O)).assumption.
+apply divides_SO_n.
+cut (p \divides n \lor p \divides n \sup n1).
+elim Hcut.assumption.
+apply H.assumption.assumption.
+apply divides_times_to_divides.assumption.
+exact H2.
+qed.
+
+theorem divides_exp_to_eq:
+\forall p,q,m:nat. prime p \to prime q \to
+p \divides q \sup m \to p = q.
+intros.
+unfold prime in H1.
+elim H1.apply H4.
+apply (divides_exp_to_divides p q m).
+assumption.assumption.
+unfold prime in H.elim H.assumption.
+qed.
+
theorem eq_gcd_times_SO: \forall m,n,p:nat. O < n \to O < p \to
gcd m n = (S O) \to gcd m p = (S O) \to gcd m (n*p) = (S O).
intros.
|apply (decidable_divides n p).
assumption.
]
+qed.
+
+(*
+theorem divides_to_divides_times1: \forall p,q,n. prime p \to prime q \to p \neq q \to
+divides p n \to divides q n \to divides (p*q) n.
+intros.elim H3.
+rewrite > H5 in H4.
+elim (divides_times_to_divides ? ? ? H1 H4)
+ [elim H.apply False_ind.
+ apply H2.apply sym_eq.apply H8
+ [assumption
+ |apply prime_to_lt_SO.assumption
+ ]
+ |elim H6.
+ apply (witness ? ? n1).
+ rewrite > assoc_times.
+ rewrite < H7.assumption
+ ]
+qed.
+*)
+
+theorem divides_to_divides_times: \forall p,q,n. prime p \to p \ndivides q \to
+divides p n \to divides q n \to divides (p*q) n.
+intros.elim H3.
+rewrite > H4 in H2.
+elim (divides_times_to_divides ? ? ? H H2)
+ [apply False_ind.apply H1.assumption
+ |elim H5.
+ apply (witness ? ? n1).
+ rewrite > sym_times in ⊢ (? ? ? (? % ?)).
+ rewrite > assoc_times.
+ rewrite < H6.assumption
+ ]
qed.
\ No newline at end of file
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