X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Farithmetics%2Fbigops.ma;h=97bda394da6d144854b122000456d94146ce0a22;hb=1f946a70bc439ced80c695f0fd0c210df0d3b767;hp=5e34160fe91a4998ac11014e1d75db94467c6ef4;hpb=ddc80515997a3f56085c6234d4db326141e189aa;p=helm.git

diff --git a/matita/matita/lib/arithmetics/bigops.ma b/matita/matita/lib/arithmetics/bigops.ma
index 5e34160fe..97bda394d 100644
--- a/matita/matita/lib/arithmetics/bigops.ma
+++ b/matita/matita/lib/arithmetics/bigops.ma
@@ -70,7 +70,7 @@ notation "\big  [ op , nil ]_{ ident j ∈ [a,b[ } f"
 for @{'bigop ($b-$a) $op $nil (λ${ident j}.((λ${ident j}.true) (${ident j}+$a)))
   (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.  
  
-(* notation "\big  [ op , nil ]_{( term 50) a ≤ ident j < b | p } f"
+(* notation "\big  [ op , nil ]_{( term 55) a ≤ ident j < b | p } f"
   with precedence 80
 for @{\big[$op,$nil]_{${ident j} < ($b-$a) | ((λ${ident j}.$p) (${ident j}+$a))}((λ${ident j}.$f)(${ident j}+$a))}.
 *)
@@ -99,7 +99,7 @@ qed.
 
 theorem pad_bigop: ∀k,n,p,B,nil,op.∀f:nat→B. n ≤ k → 
 \big[op,nil]_{i < n | p i}(f i)
-  = \big[op,nil]_{i < k | if_then_else ? (leb n i) false (p i)}(f i).
+  = \big[op,nil]_{i < k | if leb n i then false else p i}(f i).
 #k #n #p #B #nil #op #f #lenk (elim lenk) 
   [@same_bigop #i #lti // >(not_le_to_leb_false …) /2/
   |#j #leup #Hind >bigop_Sfalse >(le_to_leb_true … leup) // 
@@ -114,8 +114,8 @@ record Aop (A:Type[0]) (nil:A) : Type[0] ≝
 
 theorem bigop_sum: ∀k1,k2,p1,p2,B.∀nil.∀op:Aop B nil.∀f,g:nat→B.
 op (\big[op,nil]_{i<k1|p1 i}(f i)) \big[op,nil]_{i<k2|p2 i}(g i) =
-      \big[op,nil]_{i<k1+k2|if_then_else ? (leb k2 i) (p1 (i-k2)) (p2 i)}
-        (if_then_else ? (leb k2 i) (f (i-k2)) (g i)).
+      \big[op,nil]_{i<k1+k2|if leb k2 i then p1 (i-k2) else p2 i}
+        (if leb k2 i then f (i-k2) else g i).
 #k1 #k2 #p1 #p2 #B #nil #op #f #g (elim k1)
   [normalize >nill @same_bigop #i #lti 
    >(lt_to_leb_false … lti) normalize /2/
@@ -141,7 +141,7 @@ a ≤ b → b ≤ c →
   op (\big[op,nil]_{i ∈ [b,c[ |p i}(f i)) 
       \big[op,nil]_{i ∈ [a,b[ |p i}(f i).
 #a #b # c #p #B #nil #op #f #leab #lebc 
->(plus_minus_m_m (c-a) (b-a)) {⊢ (??%?)} /2/
+>(plus_minus_m_m (c-a) (b-a)) in ⊢ (??%?); /2/
 >minus_plus >(commutative_plus a) <plus_minus_m_m //
 >bigop_sum (cut (∀i. b -a ≤ i → i+a = i-(b-a)+b))
   [#i #lei >plus_minus // <plus_minus1 
@@ -173,11 +173,11 @@ theorem bigop_prod: ∀k1,k2,p1,p2,B.∀nil.∀op:Aop B nil.∀f: nat →nat →
 #n #Hind cases(true_or_false (p1 n)) #Hp1
   [>bigop_Strue // >Hind >bigop_sum @same_bigop
    #i #lti @leb_elim // #lei cut (i = n*k2+(i-n*k2)) /2/
-   #eqi [|#H] >eqi {⊢ (???%)}
-     >div_plus_times /2/ >Hp1 >(mod_plus_times …) /2/ normalize //
+   #eqi [|#H] >eqi in ⊢ (???%);
+     >div_plus_times /2/ >Hp1 >(mod_plus_times …) /2/
   |>bigop_Sfalse // >Hind >(pad_bigop (S n*k2)) // @same_bigop
    #i #lti @leb_elim // #lei cut (i = n*k2+(i-n*k2)) /2/
-   #eqi >eqi {⊢ (???%)} >div_plus_times /2/ 
+   #eqi >eqi in ⊢ (???%); >div_plus_times /2/ 
   ]
 qed.
 
@@ -192,7 +192,7 @@ op (\big[op,nil]_{i<k|p i}(f i)) (\big[op,nil]_{i<k|p i}(g i)) =
 #k #p #B #nil #op #f #g (elim k) [normalize @nill]
 -k #k #Hind (cases (true_or_false (p k))) #H
   [>bigop_Strue // >bigop_Strue // >bigop_Strue //
-   normalize <assoc <assoc {⊢ (???%)} @eq_f >assoc >comm {⊢ (??(????%?)?)}
+   normalize <assoc <assoc in ⊢ (???%); @eq_f >assoc >comm in ⊢ (??(????%?)?);
    <assoc @eq_f @Hind
   |>bigop_Sfalse // >bigop_Sfalse // >bigop_Sfalse //
   ]
@@ -263,7 +263,7 @@ theorem bigop_iso: ∀n1,n2,p1,p2,B.∀nil.∀op:ACop B nil.∀f1,f2.
   iso B (mk_range B f1 n1 p1) (mk_range B f2 n2 p2) →
   \big[op,nil]_{i<n1|p1 i}(f1 i) = \big[op,nil]_{i<n2|p2 i}(f2 i).
 #n1 #n2 #p1 #p2 #B #nil #op #f1 #f2 * #h * #k * * #same
-@(le_gen ? n1) #i generalize {match p2} (elim i) 
+@(le_gen ? n1) #i lapply p2 (elim i) 
   [(elim n2) // #m #Hind #p2 #_ #sub1 #sub2
    >bigop_Sfalse 
     [@(Hind ? (le_O_n ?)) [/2/ | @(transitive_sub … (sub_lt …) sub2) //]
@@ -280,7 +280,7 @@ theorem bigop_iso: ∀n1,n2,p1,p2,B.∀nil.∀op:ACop B nil.∀f1,f2.
       |#j #ltj #p2j (cases (sub2 j ltj (andb_true_r …p2j))) * 
        #ltkj #p1kj #eqj % // % // 
        (cases (le_to_or_lt_eq …(le_S_S_to_le …ltkj))) //
-       #eqkj @False_ind generalize {match p2j} @eqb_elim 
+       #eqkj @False_ind lapply p2j @eqb_elim 
        normalize /2/
       ]
     |>bigop_Sfalse // @(Hind ? len) 
@@ -298,7 +298,7 @@ qed.
 record Dop (A:Type[0]) (nil:A): Type[0] ≝
   {sum : ACop A nil; 
    prod: A → A →A;
-   null: ∀a. prod a nil = nil; 
+   null: \forall a. prod a nil = nil; 
    distr: ∀a,b,c:A. prod a (sum b c) = sum (prod a b) (prod a c)
   }.