X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Fnat%2Fdiv_and_mod.ma;h=73344c7c46b0cf1466b0aea949755ced792fc13a;hb=ab44166935d77276c04fcce50aa8281292776e29;hp=2b00a97abcf4c867d7b24bb5704beaa6d2ee562e;hpb=5e1fd9ee5ced5737c7fd4f25fca47feda1fda8e9;p=helm.git

diff --git a/helm/matita/library/nat/div_and_mod.ma b/helm/matita/library/nat/div_and_mod.ma
index 2b00a97ab..73344c7c4 100644
--- a/helm/matita/library/nat/div_and_mod.ma
+++ b/helm/matita/library/nat/div_and_mod.ma
@@ -15,8 +15,6 @@
 set "baseuri" "cic:/matita/nat/div_and_mod".
 
 include "nat/minus.ma".
-include "nat/orders_op.ma".
-include "nat/compare.ma".
 
 let rec mod_aux p m n: nat \def
 match (leb m n) with
@@ -24,7 +22,7 @@ match (leb m n) with
 | false \Rightarrow
   match p with
   [O \Rightarrow m
-  |(S q) \Rightarrow mod_aux q (minus m (S n)) n]].
+  |(S q) \Rightarrow mod_aux q (m-(S n)) n]].
 
 definition mod : nat \to nat \to nat \def
 \lambda n,m.
@@ -38,7 +36,7 @@ match (leb m n) with
 | false \Rightarrow
   match p with
   [O \Rightarrow O
-  |(S q) \Rightarrow S (div_aux q (minus m (S n)) n)]].
+  |(S q) \Rightarrow S (div_aux q (m-(S n)) n)]].
 
 definition div : nat \to nat \to nat \def
 \lambda n,m.
@@ -47,15 +45,15 @@ match m with
 | (S p) \Rightarrow div_aux n n p]. 
 
 theorem le_mod_aux_m_m: 
-\forall p,n,m. (le n p) \to (le (mod_aux p n m) m).
+\forall p,n,m. n \leq p \to (mod_aux p n m) \leq m.
 intro.elim p.
-apply le_n_O_elim n H (\lambda n.(le (mod_aux O n m) m)).
+apply le_n_O_elim n H (\lambda n.(mod_aux O n m) \leq m).
 simplify.apply le_O_n.
 simplify.
 apply leb_elim n1 m.
 simplify.intro.assumption.
 simplify.intro.apply H.
-cut (le n1 (S n)) \to (le (minus n1 (S m)) n).
+cut n1 \leq (S n) \to n1-(S m) \leq n.
 apply Hcut.assumption.
 elim n1.
 simplify.apply le_O_n.
@@ -63,7 +61,7 @@ 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. (lt O m) \to (lt (mod n m) m).
+theorem lt_mod_m_m: \forall n,m. O < m \to (mod n 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.
@@ -71,7 +69,7 @@ apply le_n.
 qed.
 
 theorem div_aux_mod_aux: \forall p,n,m:nat. 
-(eq nat n (plus (times (div_aux p n m) (S m)) (mod_aux p n m) )).
+(n=(div_aux p n m)*(S m) + (mod_aux p n m)).
 intro.elim p.
 simplify.elim leb n m.
 simplify.apply refl_eq.
@@ -81,38 +79,36 @@ apply leb_elim n1 m.
 simplify.intro.apply refl_eq.
 simplify.intro.
 rewrite > assoc_plus. 
-elim (H (minus n1 (S m)) m).
-change with (eq nat n1 (plus (S m) (minus n1 (S m)))).
+elim (H (n1-(S m)) m).
+change with (n1=(S m)+(n1-(S m))).
 rewrite < sym_plus.
 apply plus_minus_m_m.
-change with lt m n1.
+change with m < n1.
 apply not_le_to_lt.exact H1.
 qed.
 
-theorem div_mod: \forall n,m:nat. 
-(lt O m) \to (eq nat n (plus (times (div n m) m) (mod n m))).
+theorem div_mod: \forall n,m:nat. O < m \to n=(div n m)*m+(mod n m).
 intros 2.elim m.elim (not_le_Sn_O O H).
 simplify.
 apply div_aux_mod_aux.
 qed.
 
 inductive div_mod_spec (n,m,q,r:nat) : Prop \def
-div_mod_spec_intro: 
-(lt r m) \to (eq nat n (plus (times q m) r)) \to (div_mod_spec n m q r).
+div_mod_spec_intro: r < m \to n=q*m+r \to (div_mod_spec n m q r).
 
 (* 
 definition div_mod_spec : nat \to nat \to nat \to nat \to Prop \def
-\lambda n,m,q,r:nat.(And (lt r m) (eq nat n (plus (times q m) r))).
+\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 Not (eq nat m O).
+theorem div_mod_spec_to_not_eq_O: \forall n,m,q,r.(div_mod_spec n m q r) \to \lnot m=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. (lt 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 (div n m) (mod n m)).
 intros.
 apply div_mod_spec_intro.
 apply lt_mod_m_m.assumption.
@@ -125,20 +121,19 @@ theorem div_mod_spec_to_eq :\forall a,b,q,r,q1,r1.
 intros.elim H.elim H1.
 apply nat_compare_elim q q1.intro.
 apply False_ind.
-cut eq nat (plus (times (minus q1 q) b) r1) r.
-cut le b (plus (times (minus q1 q) b) r1).
-cut le b r.
+cut eq nat ((q1-q)*b+r1) r.
+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 ? (times (minus 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.
@@ -153,19 +148,19 @@ intros.assumption.
 (* the following case is symmetric *)
 intro.
 apply False_ind.
-cut eq nat (plus (times (minus q q1) b) r) r1.
-cut le b (plus (times (minus q q1) b) r).
-cut le b r1.
+cut eq nat ((q-q1)*b+r) r1.
+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 ? (times (minus 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.
@@ -175,4 +170,108 @@ apply H5.
 apply le_times_r.
 apply lt_to_le.
 apply H6.
-qed.
\ No newline at end of file
+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.
+
+(* some properties of div and mod *)
+theorem div_times: \forall n,m:nat. div ((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 div n n = S O.
+intros.
+apply div_mod_spec_to_eq n n (div n n) (mod n 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_n_n: \forall n:nat. O < n \to mod n n = O.
+intros.
+apply div_mod_spec_to_eq2 n n (div n n) (mod n 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 (mod n m) < m \to 
+(mod (S n) m) = S (mod n m).
+intros.
+apply div_mod_spec_to_eq2 (S n) m (div (S n) m) (mod (S n) m) (div n m) (S (mod n 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.mod O 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.
+rewrite < div_times n.
+rewrite < div_times n q.
+apply eq_f2.assumption.
+reflexivity.
+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.
+apply inj_times_r n p q.
+rewrite < sym_times.
+rewrite < sym_times q.
+assumption.
+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.