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

diff --git a/matita/matita/lib/arithmetics/minimization.ma b/matita/matita/lib/arithmetics/minimization.ma
index 33a75b67bad55c4ed5fe63693d1dd50c1c44c66a..143104c50a75b98371aa59b9225ad86834b40c60 100644 (file)
--- 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.
@@ -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 >Hind //
+#m #Hind #ext normalize >ext normalize {Hind} >Hind //
 #i #ltim @ext /2/
 qed.
 
@@ -190,7 +190,7 @@ 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) //
+    [lapply (true_min …fb) #H >H >H //
     |>false_min // >false_min // @Hind /2/
     ]
   ]
@@ -230,8 +230,8 @@ 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 //]
+    [#ltbm >false_min /2/ @Hind //
+      [#i #H #H1 @ismin /2/ | >eqt normalize //] 
     |#eqbm >true_min //
     ]
   ]
@@ -298,7 +298,7 @@ min n b g ≤ min n b f.
 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 /2/
 qed.