X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fnat%2Fdiv_and_mod.ma;h=e31f2d678cd739a075e5e33b5847998157f5d4fc;hb=0137a346eaaf9ae7a0b23c7a3b4c6628073b7dfb;hp=e9831f82ad1ec5cc01decf9e920f9e80518c3f64;hpb=55b82bd235d82ff7f0a40d980effe1efde1f5073;p=helm.git

diff --git a/helm/software/matita/library/nat/div_and_mod.ma b/helm/software/matita/library/nat/div_and_mod.ma
index e9831f82a..e31f2d678 100644
--- a/helm/software/matita/library/nat/div_and_mod.ma
+++ b/helm/software/matita/library/nat/div_and_mod.ma
@@ -12,8 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/nat/div_and_mod".
-
+include "datatypes/constructors.ma".
 include "nat/minus.ma".
 
 let rec mod_aux p m n: nat \def
@@ -27,11 +26,10 @@ match (leb m n) with
 definition mod : nat \to nat \to nat \def
 \lambda n,m.
 match m with 
-[O \Rightarrow m
+[O \Rightarrow n
 | (S p) \Rightarrow mod_aux n n p]. 
 
-interpretation "natural remainder" 'module x y =
-  (cic:/matita/nat/div_and_mod/mod.con x y).
+interpretation "natural remainder" 'module x y = (mod x y).
 
 let rec div_aux p m n : nat \def
 match (leb m n) with
@@ -47,8 +45,7 @@ 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).
+interpretation "natural divide" 'divide x y = (div x y).
 
 theorem le_mod_aux_m_m: 
 \forall p,n,m. n \leq p \to (mod_aux p n m) \leq m.
@@ -99,6 +96,16 @@ simplify.
 apply div_aux_mod_aux.
 qed.
 
+theorem eq_times_div_minus_mod:
+\forall a,b:nat. O \lt b \to
+(a /b)*b = a - (a \mod b).
+intros.
+rewrite > (div_mod a b) in \vdash (? ? ? (? % ?))
+[ apply (minus_plus_m_m (times (div a b) b) (mod a b))
+| assumption
+]
+qed.
+
 inductive div_mod_spec (n,m,q,r:nat) : Prop \def
 div_mod_spec_intro: r < m \to n=q*m+r \to (div_mod_spec n m q r).
 
@@ -140,11 +147,14 @@ apply le_plus_n.
 rewrite < sym_times.
 rewrite > distr_times_minus.
 rewrite > plus_minus.
+lapply(plus_to_minus ??? H3); demodulate. reflexivity.
+(*
 rewrite > sym_times.
 rewrite < H5.
-rewrite < sym_times.
+rewrite < sym_times. 
 apply plus_to_minus.
 apply H3.
+*)
 apply le_times_r.
 apply lt_to_le.
 apply H6.
@@ -188,26 +198,56 @@ qed.
 
 theorem div_mod_spec_times : \forall n,m:nat.div_mod_spec ((S n)*m) (S n) m O.
 intros.constructor 1.
-unfold lt.apply le_S_S.apply le_O_n.
-rewrite < plus_n_O.rewrite < sym_times.reflexivity.
+unfold lt.apply le_S_S.apply le_O_n. demodulate. reflexivity.
+(*rewrite < plus_n_O.rewrite < sym_times.reflexivity.*)
+qed.
+
+lemma div_plus_times: \forall m,q,r:nat. r < m \to  (q*m+r)/ m = q. 
+intros.
+apply (div_mod_spec_to_eq (q*m+r) m ? ((q*m+r) \mod m) ? r)
+  [apply div_mod_spec_div_mod.
+   apply (le_to_lt_to_lt ? r)
+    [apply le_O_n|assumption]
+  |apply div_mod_spec_intro[assumption|reflexivity]
+  ]
+qed.
+
+lemma mod_plus_times: \forall m,q,r:nat. r < m \to  (q*m+r) \mod m = r. 
+intros.
+apply (div_mod_spec_to_eq2 (q*m+r) m ((q*m+r)/ m) ((q*m+r) \mod m) q r)
+  [apply div_mod_spec_div_mod.
+   apply (le_to_lt_to_lt ? r)
+    [apply le_O_n|assumption]
+  |apply div_mod_spec_intro[assumption|reflexivity]
+  ]
 qed.
 
 (* 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.
-unfold lt.apply le_S_S.apply le_O_n.
-apply div_mod_spec_times.
+apply (div_mod_spec_to_eq ((S n)*m) (S n) ? ? ? O);
+[2: apply div_mod_spec_div_mod.
+    unfold lt.apply le_S_S.apply le_O_n.
+|   skip
+|   apply div_mod_spec_times
+]
+qed.
+
+(*a simple variant of div_times theorem *)
+theorem lt_O_to_div_times: \forall a,b:nat. O \lt b \to
+a*b/b = a.
+intros.
+rewrite > sym_times.
+rewrite > (S_pred b H).
+apply div_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.
+constructor 1.assumption. demodulate. reflexivity. (*
+rewrite < plus_n_O.simplify.rewrite < plus_n_O.reflexivity.*)
 qed.
 
 theorem eq_div_O: \forall n,m. n < m \to n / m = O.
@@ -223,8 +263,8 @@ 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.
+constructor 1.assumption. demodulate. reflexivity.(*
+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 
@@ -252,6 +292,50 @@ constructor 1.
 assumption.reflexivity.
 qed.
 
+theorem mod_SO: \forall n:nat. mod n (S O) = O.
+intro.
+apply sym_eq.
+apply le_n_O_to_eq.
+apply le_S_S_to_le.
+apply lt_mod_m_m.
+apply le_n.
+qed.
+
+theorem div_SO: \forall n:nat. div n (S O) = n.
+intro.
+rewrite > (div_mod ? (S O)) in \vdash (? ? ? %)
+  [rewrite > mod_SO.
+   rewrite < plus_n_O.
+   apply times_n_SO
+  |apply le_n
+  ]
+qed.
+
+theorem or_div_mod: \forall n,q. O < q \to
+((S (n \mod q)=q) \land S n = (S (div n q)) * q \lor
+((S (n \mod q)<q) \land S n= (div n q) * q + S (n\mod q))).
+intros.
+elim (le_to_or_lt_eq ? ? (lt_mod_m_m n q H))
+  [right.split
+    [assumption
+    |rewrite < plus_n_Sm.
+     apply eq_f.
+     apply div_mod.
+     assumption
+    ]
+  |left.split
+    [assumption
+    |simplify.
+     rewrite > sym_plus.
+     rewrite < H1 in ⊢ (? ? ? (? ? %)).
+     rewrite < plus_n_Sm.
+     apply eq_f.
+     apply div_mod.
+     assumption
+    ]
+  ]
+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).
@@ -266,7 +350,7 @@ 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).
+simplify.
 intros 4.
 apply (lt_O_n_elim n H).intros.
 apply (inj_times_r m).assumption.
@@ -276,11 +360,11 @@ 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).
+simplify.
 intros.
-apply (inj_times_r n p q).
+apply (inj_times_r n x y).
 rewrite < sym_times.
-rewrite < (sym_times q).
+rewrite < (sym_times y).
 assumption.
 qed.
 
@@ -288,7 +372,7 @@ 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).
+simplify.
 intros 4.
 apply (lt_O_n_elim n H).intros.
 apply (inj_times_l m).assumption.
@@ -296,3 +380,19 @@ 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.
+
+      
+(* n_divides computes the pair (div,mod) *)
+
+(* p is just an upper bound, acc is an accumulator *)
+let rec n_divides_aux p n m acc \def
+  match n \mod m with
+  [ O \Rightarrow 
+    match p with
+      [ O \Rightarrow pair nat nat acc n
+      | (S p) \Rightarrow n_divides_aux p (n / m) m (S acc)]
+  | (S a) \Rightarrow pair nat nat acc n].
+
+(* n_divides n m = <q,r> if m divides n q times, with remainder r *)
+definition n_divides \def \lambda n,m:nat.n_divides_aux n n m O.
+