--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/nat/gcd_properties1".
+
+include "nat/gcd.ma".
+
+(* this file contains some important properites of gcd in N *)
+
+(* an alternative characterization for gcd *)
+theorem gcd1: \forall a,b,c:nat.
+c \divides a \to c \divides b \to
+(\forall d:nat. d \divides a \to d \divides b \to d \divides c) \to (gcd a b) = c.
+intros.
+elim (H2 ((gcd a b)))
+[ apply (antisymmetric_divides (gcd a b) c)
+ [ apply (witness (gcd a b) c n2).
+ assumption
+ | apply divides_d_gcd;
+ assumption
+ ]
+| apply divides_gcd_n
+| rewrite > sym_gcd.
+ apply divides_gcd_n
+]
+qed.
+
+
+theorem eq_gcd_times_times_times_gcd: \forall a,b,c:nat.
+(gcd (c*a) (c*b)) = c*(gcd a b).
+intros.
+apply (nat_case1 c)
+[ intros.
+ simplify.
+ reflexivity
+| intros.
+ rewrite < H.
+ apply gcd1
+ [ apply divides_times
+ [ apply divides_n_n
+ | apply divides_gcd_n.
+ ]
+ | apply divides_times
+ [ apply divides_n_n
+ | rewrite > sym_gcd.
+ apply divides_gcd_n
+ ]
+ | intros.
+ apply (divides_d_times_gcd)
+ [ rewrite > H.
+ apply lt_O_S
+ | assumption
+ | assumption
+ ]
+ ]
+]
+qed.
+
+theorem associative_nat_gcd: associative nat gcd.
+change with (\forall a,b,c:nat. (gcd (gcd a b) c) = (gcd a (gcd b c))).
+intros.
+apply gcd1
+[ apply divides_d_gcd
+ [ apply (trans_divides ? (gcd b c) ?)
+ [ apply divides_gcd_m
+ | apply divides_gcd_n
+ ]
+ | apply divides_gcd_n
+ ]
+| apply (trans_divides ? (gcd b c) ?)
+ [ apply divides_gcd_m
+ | apply divides_gcd_m
+ ]
+| intros.
+ cut (d \divides a \land d \divides b)
+ [ elim Hcut.
+ cut (d \divides (gcd b c))
+ [ apply (divides_d_gcd (gcd b c) a d Hcut1 H2)
+ | apply (divides_d_gcd c b d H1 H3)
+ ]
+ | split
+ [ apply (trans_divides d (gcd a b) a H).
+ apply divides_gcd_n
+ | apply (trans_divides d (gcd a b) b H).
+ apply divides_gcd_m
+ ]
+ ]
+]
+qed.
+
+
+theorem eq_gcd_div_div_div_gcd: \forall a,b,m:nat.
+O \lt m \to m \divides a \to m \divides b \to
+(gcd (a/m) (b/m)) = (gcd a b) / m.
+intros.
+apply (inj_times_r1 m H).
+rewrite > (sym_times m ((gcd a b)/m)).
+rewrite > (divides_to_div m (gcd a b))
+[ rewrite < eq_gcd_times_times_times_gcd.
+ rewrite > (sym_times m (a/m)).
+ rewrite > (sym_times m (b/m)).
+ rewrite > (divides_to_div m a H1).
+ rewrite > (divides_to_div m b H2).
+ reflexivity
+| apply divides_d_gcd;
+ assumption
+]
+qed.
+
+
+
+theorem divides_times_to_divides_div_gcd: \forall a,b,c:nat.
+a \divides (b*c) \to (a/(gcd a b)) \divides c.
+intros.
+apply (nat_case1 a)
+[ intros.
+ apply (nat_case1 b)
+ [ (*It's an impossible situation*)
+ intros.
+ simplify.
+ apply divides_SO_n
+ | intros.
+ cut (c = O)
+ [ rewrite > Hcut.
+ apply (divides_n_n O)
+ | apply (lt_times_eq_O b c)
+ [ rewrite > H2.
+ apply lt_O_S
+ | apply antisymmetric_divides
+ [ apply divides_n_O
+ | rewrite < H1.
+ assumption
+ ]
+ ]
+ ]
+ ]
+| intros.
+ rewrite < H1.
+ elim H.
+ cut (O \lt a)
+ [ cut (O \lt (gcd a b))
+ [ apply (gcd_SO_to_divides_times_to_divides (b/(gcd a b)) (a/(gcd a b)) c)
+ [ apply (O_lt_times_to_O_lt (a/(gcd a b)) (gcd a b)).
+ rewrite > (divides_to_div (gcd a b) a)
+ [ assumption
+ | apply divides_gcd_n
+ ]
+ | rewrite < (div_n_n (gcd a b)) in \vdash (? ? ? %)
+ [ apply eq_gcd_div_div_div_gcd
+ [ assumption
+ | apply divides_gcd_n
+ | apply divides_gcd_m
+ ]
+ | assumption
+ ]
+ | apply (witness ? ? n2).
+ apply (inj_times_r1 (gcd a b) Hcut1).
+ rewrite < assoc_times.
+ rewrite < sym_times in \vdash (? ? (? % ?) ?).
+ rewrite > (divides_to_div (gcd a b) b)
+ [ rewrite < assoc_times in \vdash (? ? ? %).
+ rewrite < sym_times in \vdash (? ? ? (? % ?)).
+ rewrite > (divides_to_div (gcd a b) a)
+ [ assumption
+ | apply divides_gcd_n
+ ]
+ | apply divides_gcd_m
+ ]
+ ]
+ | rewrite > sym_gcd.
+ apply lt_O_gcd.
+ assumption
+ ]
+ | rewrite > H1.
+ apply lt_O_S
+ ]
+]
+qed.
+
+theorem gcd_plus_times_gcd: \forall a,b,d,m:nat.
+(gcd (a+m*b) b) = (gcd a b).
+intros.
+apply gcd1
+[ apply divides_plus
+ [ apply divides_gcd_n
+ | apply (trans_divides ? b ?)
+ [ apply divides_gcd_m
+ | rewrite > sym_times.
+ apply (witness b (b*m) m).
+ reflexivity
+ ]
+ ]
+| apply divides_gcd_m
+| intros.
+ apply divides_d_gcd
+ [ assumption
+ | rewrite > (minus_plus_m_m a (m*b)).
+ apply divides_minus
+ [ assumption
+ | apply (trans_divides ? b ?)
+ [ assumption
+ | rewrite > sym_times.
+ apply (witness b (b*m) m).
+ reflexivity
+ ]
+ ]
+ ]
+]
+qed.
+
+
+
+theorem gcd_SO_to_divides_to_divides_to_divides_times: \forall c,e,f:nat.
+(gcd e f) = (S O) \to e \divides c \to f \divides c \to
+(e*f) \divides c.
+intros.
+apply (nat_case1 c); intros
+[ apply divides_n_O
+| rewrite < H3.
+ elim H1.
+ elim H2.
+ rewrite > H5.
+ rewrite > (sym_times e f).
+ apply (divides_times)
+ [ apply (divides_n_n)
+ | rewrite > H5 in H1.
+ apply (gcd_SO_to_divides_times_to_divides f e n)
+ [ rewrite < H5 in H1.
+ rewrite > H3 in H1.
+ apply (divides_to_lt_O e (S m))
+ [ apply lt_O_S
+ | assumption
+ ]
+ | assumption
+ | assumption
+ ]
+ ]
+]
+qed.
+(* the following theorem shows that gcd is a multiplicative function in
+ the following sense: if a1 and a2 are relatively prime, then
+ gcd(a1·a2, b) = gcd(a1, b)·gcd(a2, b).
+ *)
+theorem gcd_SO_to_eq_gcd_times_times_gcd_gcd: \forall a,b,c:nat.
+(gcd a b) = (S O) \to (gcd (a*b) c) = (gcd a c) * (gcd b c).
+intros.
+apply gcd1
+[ apply divides_times;
+ apply divides_gcd_n
+| apply (gcd_SO_to_divides_to_divides_to_divides_times c (gcd a c) (gcd b c))
+ [ apply gcd1
+ [ apply divides_SO_n
+ | apply divides_SO_n
+ | intros.
+ cut (d \divides a)
+ [ cut (d \divides b)
+ [ rewrite < H.
+ apply (divides_d_gcd b a d Hcut1 Hcut)
+ | apply (trans_divides d (gcd b c) b)
+ [ assumption
+ | apply (divides_gcd_n)
+ ]
+ ]
+ | apply (trans_divides d (gcd a c) a)
+ [ assumption
+ | apply (divides_gcd_n)
+ ]
+ ]
+ ]
+ | apply (divides_gcd_m)
+ | apply (divides_gcd_m)
+ ]
+| intros.
+ rewrite < (eq_gcd_times_times_times_gcd b c (gcd a c)).
+ rewrite > (sym_times (gcd a c) b).
+ rewrite > (sym_times (gcd a c) c).
+ rewrite < (eq_gcd_times_times_times_gcd a c b).
+ rewrite < (eq_gcd_times_times_times_gcd a c c).
+ apply (divides_d_gcd)
+ [ apply (divides_d_gcd)
+ [ rewrite > (times_n_SO d).
+ apply (divides_times)
+ [ assumption
+ | apply divides_SO_n
+ ]
+ | rewrite > (times_n_SO d).
+ apply (divides_times)
+ [ assumption
+ | apply divides_SO_n
+ ]
+ ]
+ | apply (divides_d_gcd)
+ [ rewrite > (times_n_SO d).
+ rewrite > (sym_times d (S O)).
+ apply (divides_times)
+ [ apply (divides_SO_n)
+ | assumption
+ ]
+ | rewrite < (sym_times a b).
+ assumption
+ ]
+ ]
+]
+qed.
+
+
+theorem eq_gcd_gcd_minus: \forall a,b:nat.
+a \lt b \to (gcd a b) = (gcd (b - a) b).
+intros.
+apply sym_eq.
+apply gcd1
+[ apply (divides_minus (gcd a b) b a)
+ [ apply divides_gcd_m
+ | apply divides_gcd_n
+ ]
+| apply divides_gcd_m
+| intros.
+ elim H1.
+ elim H2.
+ cut(b = (d*n2) + a)
+ [ cut (b - (d*n2) = a)
+ [ rewrite > H4 in Hcut1.
+ rewrite < (distr_times_minus d n n2) in Hcut1.
+ apply divides_d_gcd
+ [ assumption
+ | apply (witness d a (n - n2)).
+ apply sym_eq.
+ assumption
+ ]
+ | apply (plus_to_minus ? ? ? Hcut)
+ ]
+ | rewrite > sym_plus.
+ apply (minus_to_plus)
+ [ apply lt_to_le.
+ assumption
+ | assumption
+ ]
+ ]
+]
+qed.
+
projects / helm.git / blobdiff