X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Fnat%2Fdiv_and_mod.ma;h=f79891b48a39b7394493d8cf7cc2cfa547450c81;hb=8b55faddb06e3c4b0a13839210bb49170939b33e;hp=73344c7c46b0cf1466b0aea949755ced792fc13a;hpb=ab44166935d77276c04fcce50aa8281292776e29;p=helm.git

diff --git a/helm/matita/library/nat/div_and_mod.ma b/helm/matita/library/nat/div_and_mod.ma
index 73344c7c4..f79891b48 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||                                                             *)
@@ -30,6 +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
@@ -44,6 +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.
@@ -61,7 +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.
@@ -87,7 +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.
@@ -101,14 +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.
@@ -138,7 +144,6 @@ 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.
@@ -165,7 +170,6 @@ 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.
@@ -189,7 +193,7 @@ 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.
+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 *)
@@ -198,26 +202,35 @@ 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.
+theorem div_n_n: \forall n:nat. O < n \to 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_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 mod_n_n: \forall n:nat. O < n \to mod n n = O.
+theorem eq_div_O: \forall n,m. n < m \to n / m = O.
 intros.
-apply div_mod_spec_to_eq2 n n (div n n) (mod n n) (S O) O.
+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 (mod n m) < m \to 
-(mod (S n) m) = S (mod n m).
+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 (div (S n) m) (mod (S n) m) (div n m) (S (mod n m)).
+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.
@@ -225,11 +238,19 @@ apply div_mod.
 assumption.
 qed.
 
-theorem mod_O_n: \forall n:nat.mod O n = O.
+theorem mod_O_n: \forall n:nat.O \mod n = O.
 intro.elim n.simplify.reflexivity.
 simplify.reflexivity.
 qed.
 
+theorem lt_to_eq_mod:\forall n,m:nat. n < m \to n \mod m = n.
+intros.
+apply div_mod_spec_to_eq2 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.
 
 (* injectivity *)
 theorem injective_times_r: \forall n:nat.injective nat nat (\lambda m:nat.(S n)*m).