made executable again

[helm.git] / helm / software / matita / library / nat / div_and_mod.ma

diff --git a/helm/software/matita/library/nat/div_and_mod.ma b/helm/software/matita/library/nat/div_and_mod.ma
index f7f2883d590a8118ae6a39a8c348a75dcb2ff565..cbab87206ee5457cf967e25505b3772ed9180b15 100644 (file)
--- a/helm/software/matita/library/nat/div_and_mod.ma
+++ b/helm/software/matita/library/nat/div_and_mod.ma
@@ -12,12 +12,9 @@
 (*                                                                        *)
 (**************************************************************************)
 
-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
 match (leb m n) with
 [ true \Rightarrow m
@@ -32,8 +29,7 @@ match m with
 [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
@@ -49,8 +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.
@@ -152,11 +147,14 @@ apply le_plus_n.
 rewrite < sym_times.
 rewrite > distr_times_minus.
 rewrite > plus_minus.
+lapply(plus_to_minus ??? H3); demodulate all.
+(*
 rewrite > sym_times.
 rewrite < H5.
 rewrite < sym_times. 
 apply plus_to_minus.
 apply H3.
+*)
 apply le_times_r.
 apply lt_to_le.
 apply H6.
@@ -200,8 +198,8 @@ 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. 
@@ -223,6 +221,7 @@ 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_intro[assumption|reflexivity]
   ]
 qed.
+
 (* some properties of div and mod *)
 theorem div_times: \forall n,m:nat. ((S n)*m) / (S n) = m.
 intros.
@@ -247,8 +246,8 @@ 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.
@@ -264,8 +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 
@@ -312,16 +311,28 @@ rewrite > (div_mod ? (S O)) in \vdash (? ? ? %)
   ]
 qed.
 
-theorem le_div: \forall n,m. O < n \to m/n \le m.
+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.
-rewrite > (div_mod m n) in \vdash (? ? %)
-  [apply (trans_le ? (m/n*n))
-    [rewrite > times_n_SO in \vdash (? % ?).
-     apply le_times
-      [apply le_n|assumption]
-    |apply le_plus_n_r
+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
     ]
-  |assumption
   ]
 qed.
 
@@ -370,6 +381,7 @@ 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 *)