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[helm.git] / matita / matita / lib / arithmetics / div_and_mod.ma

diff --git a/matita/matita/lib/arithmetics/div_and_mod.ma b/matita/matita/lib/arithmetics/div_and_mod.ma
index f2b0849afa7b6fe087ef26f039cb00073f9106af..966ea3c84bc05e34c4aa6f426e0a85369c9f0d8e 100644 (file)
--- a/matita/matita/lib/arithmetics/div_and_mod.ma
+++ b/matita/matita/lib/arithmetics/div_and_mod.ma
@@ -136,7 +136,6 @@ theorem div_times: ∀a,b:nat. O < b → a*b/b = a.
 @(div_mod_spec_to_eq (a*b) b … O (div_mod_spec_div_mod …))
 // @div_mod_spec_intro // qed.
 
-(*
 theorem div_n_n: ∀n:nat. O < n → n / n = 1.
 /2/ qed.
 
@@ -148,13 +147,13 @@ theorem eq_div_O: ∀n,m. n < m → n / m = O.
 theorem mod_n_n: ∀n:nat. O < n → n \mod n = O.
 #n #posn 
 @(div_mod_spec_to_eq2 n n … 1 0 (div_mod_spec_div_mod …))
-/2/ qed. *)
+/2/ qed. 
 
 theorem mod_S: ∀n,m:nat. O < m → S (n \mod m) < m → 
 ((S n) \mod m) = S (n \mod m).
 #n #m #posm #H 
 @(div_mod_spec_to_eq2 (S n) m … (n / m) ? (div_mod_spec_div_mod …))
-// @div_mod_spec_intro// (applyS eq_f) //
+// @div_mod_spec_intro// applyS eq_f // 
 qed.
 
 theorem mod_O_n: ∀n:nat.O \mod n = O.
@@ -178,7 +177,7 @@ theorem or_div_mod: ∀n,q. O < q →
   ((S (n \mod q)<q) ∧ S n = (div n q) * q + S (n\mod q))).
 #n #q #posq 
 (elim (le_to_or_lt_eq ?? (lt_mod_m_m n q posq))) #H
-  [%2 % // (applyS eq_f) //
+  [%2 % // applyS eq_f // 
   |%1 % // /demod/ <H in ⊢(? ? ? (? % ?)) @eq_f//
   ]
 qed.
@@ -368,30 +367,11 @@ split
 ]
 qed. *)
 
-theorem le_plus_to_minus_r: ∀a,b,c. a + b ≤ c → a ≤ c -b.
-#a #b #c #H @(le_plus_to_le_r … b) /2/
-qed.
-
-theorem le_minus_to_plus_r: ∀a,b,c. c ≤ b → a ≤ b - c → a + c ≤ b.
-#a #b #c #Hlecb #H >(plus_minus_m_m … Hlecb) /2/
-qed.
-
-theorem lt_minus_to_plus: ∀a,b,c. a - b < c → a < c + b.
-#a #b #c #H @not_le_to_lt 
-@(not_to_not … (lt_to_not_le …H)) /2/
-qed.
-
-theorem lt_minus_to_plus_r: ∀a,b,c. c ≤ a → 
-  a < b + c → a - c < b.
-#a #b #c #lea #H @not_le_to_lt 
-@(not_to_not … (lt_to_not_le …H)) /2/
-qed. 
-
 theorem lt_to_le_times_to_lt_S_to_div: ∀a,c,b:nat.
 O < b → (b*c) ≤ a → a < (b*(S c)) → a/b = c.
 #a #c #b #posb#lea #lta
 @(div_mod_spec_to_eq … (a-b*c) (div_mod_spec_div_mod … posb …))
-@div_mod_spec_intro [@lt_minus_to_plus_r // |/2/]
+@div_mod_spec_intro [@lt_plus_to_minus // |/2/]
 qed.
 
 theorem div_times_times: ∀a,b,c:nat. O < c → O < b → 
@@ -407,7 +387,7 @@ O < c → O < b → (a*c) \mod (b*c) = c*(a\mod b).
 @(div_mod_spec_to_eq2 (a*c) (b*c) (a/b) ((a*c) \mod (b*c)) (a/b) (c*(a \mod b)))
   [>(div_times_times … posc) // @div_mod_spec_div_mod /2/
   |@div_mod_spec_intro
-    [(applyS monotonic_lt_times_l) /2/
+    [applyS (monotonic_lt_times_r … c posc) /2/
     |(applyS (eq_f …(λx.x*c))) //
     ]
   ]