X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Farithmetics%2Fminimization.ma;h=5d4fc9bce9a5aa03002dfdd6151cd0bc14324af7;hb=29c979302cf398410bf6f10c11b146ebe42bd54a;hp=19ea144403e0994531bca78005ea77dd2bfdc528;hpb=16a95f57b09ae92ea24ab2addd02c1d0be80f109;p=helm.git

diff --git a/matita/matita/lib/arithmetics/minimization.ma b/matita/matita/lib/arithmetics/minimization.ma
index 19ea14440..5d4fc9bce 100644
--- a/matita/matita/lib/arithmetics/minimization.ma
+++ b/matita/matita/lib/arithmetics/minimization.ma
@@ -40,7 +40,7 @@ qed.
 
 theorem lt_max_n : ∀f.∀n. O < n → max n f < n.
 #f #n #posn @(lt_O_n_elim ? posn) #m
-normalize (cases (f m)) normalize apply le_S_S //
+normalize (cases (f m)) normalize @le_S_S //
 @le_max_n qed.
 
 theorem le_to_le_max : ∀f.∀n,m.
@@ -85,7 +85,7 @@ qed.
 lemma max_not_exists: ∀f.∀n.
  (∀i. i < n → f i = false) → max n f = O.
 #f #n #ffalse @(le_gen ? n) #i (elim i) // #j #Hind #ltj
-normalize >(ffalse j …) // @Hind /2/
+normalize >ffalse // @Hind /2/
 qed.
 
 lemma fmax_false: ∀f.∀n,m.max n f = m → f m = false → m = O. 
@@ -122,7 +122,7 @@ qed.
 theorem max_f_g: ∀f,g,n.(∀i. i < n → f i = g i) → 
   max n f = max n g.
 #f #g #n (elim n) //
-#m #Hind #ext normalize >(ext … (le_n …)) >(Hind …) //
+#m #Hind #ext normalize >ext normalize in Hind; >Hind //
 #i #ltim @ext /2/
 qed.
 
@@ -130,14 +130,47 @@ theorem le_max_f_max_g: ∀f,g,n. (∀i. i < n → f i = true → g i =true) →
 max n f ≤ max n g.
 #f #g #n (elim n) //
 #m #Hind #ext normalize (cases (true_or_false (f m))) #Heq >Heq 
-  [>(ext m …) //
+  [>ext //
   |(cases (g m)) normalize [@le_max_n] @Hind #i #ltim @ext /2/
 qed.
 
 theorem f_max_true : ∀ f.∀n.
 (∃i:nat. i < n ∧ f i = true) → f (max n f) = true. 
 #f #n cases(max_to_max_spec f n (max n f) (refl …)) //
-#Hall * #x * #ltx #fx @False_ind @(absurd … fx) >(Hall x ?) /2/
+#Hall * #x * #ltx #fx @False_ind @(absurd … fx) >Hall /2/
+qed.
+
+theorem f_false_to_le_max: ∀f,n,p. (∃i:nat.i<n∧f i=true) →
+  (∀m. p < m → f m = false) → max n f ≤ p.
+#f #n #p #H1 #H2 @not_lt_to_le % #H3
+@(absurd ?? not_eq_true_false) <(H2 ? H3) @sym_eq
+@(f_max_true ? n H1)
+qed.
+
+theorem exists_forall_lt:∀f,n. 
+(∃i. i < n ∧ f i = true) ∨ (∀i. i < n → f i = false).
+#f #n elim n
+  [%2 #i #lti0 @False_ind @(absurd ? lti0) @le_to_not_lt // 
+  |#n1 *
+    [* #a * #Ha1 #Ha2 %1 %{a} % // @le_S //
+    |#H cases (true_or_false (f n1)) #HfS >HfS
+      [%1 %{n1} /2/
+      |%2 #i #lei 
+       cases (le_to_or_lt_eq ?? lei) #Hi 
+        [@H @le_S_S_to_le @Hi | destruct (Hi) //] 
+      ]
+    ]
+  ]
+qed.
+     
+theorem exists_max_forall_false:∀f,n. 
+((∃i. i < n ∧ f i = true) ∧ (f (max n f) = true))∨
+((∀i. i < n → f i = false) ∧ (max n f) = O).
+#f #n
+cases (exists_forall_lt f n)
+  [#H %1 % // @(f_max_true f n) @H
+  |#H %2 % [@H | @max_not_exists @H 
+  ]
 qed.
 
 (* minimization *)
@@ -190,8 +223,8 @@ n ≤ m  → ∀b.min n b f ≤ min m b f.
   [#n #leSO @False_ind /2/ 
   |#n #m #Hind #leSS #b
    (cases (true_or_false (f b))) #fb 
-    [lapply (true_min …fb) //
-    |>(false_min f n b fb) >(false_min f m b fb) @Hind /2/
+    [lapply (true_min …fb) #H >H >H //
+    |>false_min // >false_min // @Hind /2/
     ]
   ]
 qed.
@@ -217,8 +250,8 @@ theorem fmin_true: ∀f.∀n,m,b.
   [#m #b normalize #eqmb >eqmb #leSb @(False_ind) 
    @(absurd … leSb) //
   |#n #Hind #m #b (cases (true_or_false (f b))) #caseb
-    [>(true_min f b caseb (S n)) //
-    |>(false_min … caseb) #eqm #ltm @(Hind m (S b)) /2/
+    [>true_min //
+    |>false_min // #eqm #ltm @(Hind m (S b)) /2/
     ]
   ]
 qed.
@@ -230,9 +263,9 @@ lemma min_exists: ∀f.∀t,m. m < t → f m = true →
   [#b #lebm #ismin #eqtb @False_ind @(absurd … lebm) <eqtb
    @lt_to_not_le //
   |#d #Hind #b #lebm #ismin #eqt cases(le_to_or_lt_eq …lebm)
-    [#ltbm >(false_min …) /2/ @Hind // 
-      [#i #H #H1 @ismin /2/ | >eqt normalize //]
-    |#eqbm >(true_min …) //
+    [#ltbm >false_min /2 by le_n/ @Hind //
+      [#i #H #H1 @ismin /2/ | >eqt normalize //] 
+    |#eqbm >true_min //
     ]
   ]
 qed.
@@ -240,8 +273,8 @@ qed.
 lemma min_not_exists: ∀f.∀n,b.
  (∀i. b ≤ i → i < n + b → f i = false) → min n b f = n + b.
 #f #n (elim n) // 
-#p #Hind #b #ffalse >(false_min …)
-  [>(Hind …) // #i #H #H1 @ffalse /2/
+#p #Hind #b #ffalse >false_min
+  [>Hind // #i #H #H1 @ffalse /2/
   |@ffalse //
   ]
 qed.
@@ -252,8 +285,8 @@ lemma fmin_false: ∀f.∀n,b.let m ≝ min n b f in
 #i #Hind #b normalize cases(true_or_false … (f b)) #fb >fb
 normalize 
   [#eqm @False_ind @(absurd … fb) // 
-  |>(plus_n_Sm …) @Hind]
-qed. 
+  |>plus_n_Sm @Hind]
+qed.
 
 inductive min_spec (n,b:nat) (f:nat→bool) : nat→Prop ≝
  | found : ∀m:nat. b ≤ m → m < n + b → f m =true →
@@ -284,7 +317,7 @@ qed.
 theorem min_f_g: ∀f,g,n,b.(∀i. b ≤ i → i < n + b → f i = g i) → 
   min n b f = min n b g.
 #f #g #n (elim n) //
-#m #Hind #b #ext normalize >(ext b (le_n b) ?) // >(Hind …) //
+#m #Hind #b #ext normalize >(ext b (le_n b) ?) // >Hind //
 #i #ltib #ltim @ext /2/
 qed.
 
@@ -292,13 +325,14 @@ theorem le_min_f_min_g: ∀f,g,n,b. (∀i. b ≤ i → i < n +b → f i = true 
 min n b g ≤ min n b f.
 #f #g #n (elim n) //
 #m #Hind #b #ext normalize (cases (true_or_false (f b))) #Heq >Heq 
-  [>(ext b …) //
+  [>ext //
   |(cases (g b)) normalize /2/ @Hind #i #ltb #ltim #fi
     @ext /2/
+  ]
 qed.
 
 theorem f_min_true : ∀ f.∀n,b.
-(∃i:nat. b ≤ i ∧ i < n ∧ f i = true) → f (min n b f) = true. 
+(∃i:nat. b ≤ i ∧ i < n + b ∧ f i = true) → f (min n b f) = true. 
 #f #n #b cases(min_to_min_spec f n b (min n b f) (refl …)) //
-#Hall * #x * * #leb #ltx #fx @False_ind @(absurd … fx) >(Hall x ??) /2/
+#Hall * #x * * #leb #ltx #fx @False_ind @(absurd … fx) >Hall /2/
 qed.