X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Fnat%2Fdiv_and_mod.ma;h=26e3dcc829a604fa0b1453fe249928a2dac154dd;hb=5c6b8eec9db4119a87eb4fd4055f1ac31a713d90;hp=619ba68151a48183109400077fc9e1d034d6919a;hpb=7bbce6bc163892cfd99cfcda65db42001b86789f;p=helm.git

diff --git a/helm/matita/library/nat/div_and_mod.ma b/helm/matita/library/nat/div_and_mod.ma
index 619ba6815..26e3dcc82 100644
--- a/helm/matita/library/nat/div_and_mod.ma
+++ b/helm/matita/library/nat/div_and_mod.ma
@@ -1,5 +1,5 @@
 (**************************************************************************)
-(*       ___	                                                            *)
+(*       ___	                                                          *)
 (*      ||M||                                                             *)
 (*      ||A||       A project by Andrea Asperti                           *)
 (*      ||T||                                                             *)
@@ -15,8 +15,6 @@
 set "baseuri" "cic:/matita/nat/div_and_mod".
 
 include "nat/minus.ma".
-include "nat/le_arith.ma".
-include "nat/compare.ma".
 
 let rec mod_aux p m n: nat \def
 match (leb m n) with
@@ -32,6 +30,9 @@ match m with
 [O \Rightarrow m
 | (S p) \Rightarrow mod_aux n n p]. 
 
+interpretation "natural remainder" 'module x y =
+  (cic:/matita/nat/div_and_mod/mod.con x y).
+
 let rec div_aux p m n : nat \def
 match (leb m n) with
 [ true \Rightarrow O
@@ -46,6 +47,9 @@ match m with
 [O \Rightarrow S n
 | (S p) \Rightarrow div_aux n n p]. 
 
+interpretation "natural divide" 'divide x y =
+  (cic:/matita/nat/div_and_mod/div.con x y).
+
 theorem le_mod_aux_m_m: 
 \forall p,n,m. n \leq p \to (mod_aux p n m) \leq m.
 intro.elim p.
@@ -63,7 +67,7 @@ simplify.apply trans_le ? n2 n.
 apply le_minus_m.apply le_S_S_to_le.assumption.
 qed.
 
-theorem lt_mod_m_m: \forall n,m. O < m \to (mod n m) < m.
+theorem lt_mod_m_m: \forall n,m. O < m \to (n \mod m) < m.
 intros 2.elim m.apply False_ind.
 apply not_le_Sn_O O H.
 simplify.apply le_S_S.apply le_mod_aux_m_m.
@@ -89,7 +93,7 @@ change with m < n1.
 apply not_le_to_lt.exact H1.
 qed.
 
-theorem div_mod: \forall n,m:nat. O < m \to n=(div n m)*m+(mod n m).
+theorem div_mod: \forall n,m:nat. O < m \to n=(n / m)*m+(n \mod m).
 intros 2.elim m.elim (not_le_Sn_O O H).
 simplify.
 apply div_aux_mod_aux.
@@ -103,14 +107,14 @@ definition div_mod_spec : nat \to nat \to nat \to nat \to Prop \def
 \lambda n,m,q,r:nat.r < m \land n=q*m+r).
 *)
 
-theorem div_mod_spec_to_not_eq_O: \forall n,m,q,r.(div_mod_spec n m q r) \to \lnot m=O.
+theorem div_mod_spec_to_not_eq_O: \forall n,m,q,r.(div_mod_spec n m q r) \to m \neq O.
 intros 4.simplify.intros.elim H.absurd le (S r) O.
 rewrite < H1.assumption.
 exact not_le_Sn_O r.
 qed.
 
 theorem div_mod_spec_div_mod: 
-\forall n,m. O < m \to (div_mod_spec n m (div n m) (mod n m)).
+\forall n,m. O < m \to (div_mod_spec n m (n / m) (n \mod m)).
 intros.
 apply div_mod_spec_intro.
 apply lt_mod_m_m.assumption.
@@ -128,20 +132,18 @@ cut b \leq (q1-q)*b+r1.
 cut b \leq r.
 apply lt_to_not_le r b H2 Hcut2.
 elim Hcut.assumption.
-apply trans_le ? ((q1-q)*b) ?.
+apply trans_le ? ((q1-q)*b).
 apply le_times_n.
 apply le_SO_minus.exact H6.
 rewrite < sym_plus.
 apply le_plus_n.
 rewrite < sym_times.
 rewrite > distr_times_minus.
-(* ATTENZIONE ALL' ORDINAMENTO DEI GOALS *)
-rewrite > plus_minus ? ? ? ?.
+rewrite > plus_minus.
 rewrite > sym_times.
 rewrite < H5.
 rewrite < sym_times.
 apply plus_to_minus.
-apply eq_plus_to_le ? ? ? H3.
 apply H3.
 apply le_times_r.
 apply lt_to_le.
@@ -156,39 +158,93 @@ cut b \leq (q-q1)*b+r.
 cut b \leq r1.
 apply lt_to_not_le r1 b H4 Hcut2.
 elim Hcut.assumption.
-apply trans_le ? ((q-q1)*b) ?.
+apply trans_le ? ((q-q1)*b).
 apply le_times_n.
 apply le_SO_minus.exact H6.
 rewrite < sym_plus.
 apply le_plus_n.
 rewrite < sym_times.
 rewrite > distr_times_minus.
-rewrite > plus_minus ? ? ? ?.
+rewrite > plus_minus.
 rewrite > sym_times.
 rewrite < H3.
 rewrite < sym_times.
 apply plus_to_minus.
-apply eq_plus_to_le ? ? ? H5.
 apply H5.
 apply le_times_r.
 apply lt_to_le.
 apply H6.
 qed.
 
+theorem div_mod_spec_to_eq2 :\forall a,b,q,r,q1,r1.
+(div_mod_spec a b q r) \to (div_mod_spec a b q1 r1) \to 
+(eq nat r r1).
+intros.elim H.elim H1.
+apply inj_plus_r (q*b).
+rewrite < H3.
+rewrite > div_mod_spec_to_eq a b q r q1 r1 H H1.
+assumption.
+qed.
+
 theorem div_mod_spec_times : \forall n,m:nat.div_mod_spec ((S n)*m) (S n) m O.
 intros.constructor 1.
 simplify.apply le_S_S.apply le_O_n.
 rewrite < plus_n_O.rewrite < sym_times.reflexivity.
 qed.
 
-theorem div_times: \forall n,m:nat. div ((S n)*m) (S n) = m.
+(* some properties of div and mod *)
+theorem div_times: \forall n,m:nat. ((S n)*m) / (S n) = m.
 intros.
 apply div_mod_spec_to_eq ((S n)*m) (S n) ? ? ? O.
+goal 15. (* ?11 is closed with the following tactics *)
 apply div_mod_spec_div_mod.
 simplify.apply le_S_S.apply le_O_n.
 apply div_mod_spec_times.
 qed.
 
+theorem div_n_n: \forall n:nat. O < n \to n / n = S O.
+intros.
+apply div_mod_spec_to_eq n n (n / n) (n \mod n) (S O) O.
+apply div_mod_spec_div_mod.assumption.
+constructor 1.assumption.
+rewrite < plus_n_O.simplify.rewrite < plus_n_O.reflexivity.
+qed.
+
+theorem eq_div_O: \forall n,m. n < m \to n / m = O.
+intros.
+apply div_mod_spec_to_eq n m (n/m) (n \mod m) O n.
+apply div_mod_spec_div_mod.
+apply le_to_lt_to_lt O n m.
+apply le_O_n.assumption.
+constructor 1.assumption.reflexivity.
+qed.
+
+theorem mod_n_n: \forall n:nat. O < n \to n \mod n = O.
+intros.
+apply div_mod_spec_to_eq2 n n (n / n) (n \mod n) (S O) O.
+apply div_mod_spec_div_mod.assumption.
+constructor 1.assumption.
+rewrite < plus_n_O.simplify.rewrite < plus_n_O.reflexivity.
+qed.
+
+theorem mod_S: \forall n,m:nat. O < m \to S (n \mod m) < m \to 
+((S n) \mod m) = S (n \mod m).
+intros.
+apply div_mod_spec_to_eq2 (S n) m ((S n) / m) ((S n) \mod m) (n / m) (S (n \mod m)).
+apply div_mod_spec_div_mod.assumption.
+constructor 1.assumption.rewrite < plus_n_Sm.
+apply eq_f.
+apply div_mod.
+assumption.
+qed.
+
+theorem mod_O_n: \forall n:nat.O \mod n = O.
+intro.elim n.simplify.reflexivity.
+simplify.reflexivity.
+qed.
+
+
+(* injectivity *)
 theorem injective_times_r: \forall n:nat.injective nat nat (\lambda m:nat.(S n)*m).
 change with \forall n,p,q:nat.(S n)*p = (S n)*q \to p=q.
 intros.
@@ -201,6 +257,16 @@ qed.
 variant inj_times_r : \forall n,p,q:nat.(S n)*p = (S n)*q \to p=q \def
 injective_times_r.
 
+theorem lt_O_to_injective_times_r: \forall n:nat. O < n \to injective nat nat (\lambda m:nat.n*m).
+change with \forall n. O < n \to \forall p,q:nat.n*p = n*q \to p=q.
+intros 4.
+apply lt_O_n_elim n H.intros.
+apply inj_times_r m.assumption.
+qed.
+
+variant inj_times_r1:\forall n. O < n \to \forall p,q:nat.n*p = n*q \to p=q
+\def lt_O_to_injective_times_r.
+
 theorem injective_times_l: \forall n:nat.injective nat nat (\lambda m:nat.m*(S n)).
 change with \forall n,p,q:nat.p*(S n) = q*(S n) \to p=q.
 intros.
@@ -212,3 +278,13 @@ qed.
 
 variant inj_times_l : \forall n,p,q:nat. p*(S n) = q*(S n) \to p=q \def
 injective_times_l.
+
+theorem lt_O_to_injective_times_l: \forall n:nat. O < n \to injective nat nat (\lambda m:nat.m*n).
+change with \forall n. O < n \to \forall p,q:nat.p*n = q*n \to p=q.
+intros 4.
+apply lt_O_n_elim n H.intros.
+apply inj_times_l m.assumption.
+qed.
+
+variant inj_times_l1:\forall n. O < n \to \forall p,q:nat.p*n = q*n \to p=q
+\def lt_O_to_injective_times_l.