More notation here and there.

[helm.git] / helm / matita / library / nat / factorization.ma

diff --git a/helm/matita/library/nat/factorization.ma b/helm/matita/library/nat/factorization.ma
index c8127a1fa8cde6467be67ca32b6549466b56a566..43b942a5b828ea0305d7d3442e9f1db824c3ca8e 100644 (file)
--- a/helm/matita/library/nat/factorization.ma
+++ b/helm/matita/library/nat/factorization.ma
@@ -97,7 +97,7 @@ rewrite < H1.assumption.
 apply le_to_or_lt_eq (max_prime_factor r)  (max_prime_factor n).
 apply divides_to_max_prime_factor.
 assumption.assumption.
-apply witness r n (exp (nth_prime p) q).
+apply witness r n ((nth_prime p) \sup q).
 rewrite < sym_times.
 apply plog_aux_to_exp n n ? q r.
 apply lt_O_nth_prime_n.assumption.
@@ -111,7 +111,7 @@ intros.
 cut max_prime_factor n < p \lor max_prime_factor n = p.
 elim Hcut.apply le_to_lt_to_lt ? (max_prime_factor n).
 apply divides_to_max_prime_factor.assumption.assumption.
-apply witness r n (exp (nth_prime p) q).
+apply witness r n ((nth_prime p) \sup q).
 rewrite > sym_times.
 apply plog_aux_to_exp n n.
 apply lt_O_nth_prime_n.
@@ -154,9 +154,9 @@ definition factorize : nat \to nat_fact_all \def \lambda n:nat.
            
 let rec defactorize_aux f i \def
   match f with
-  [ (nf_last n) \Rightarrow exp (nth_prime i) (S n)
+  [ (nf_last n) \Rightarrow (nth_prime i) \sup (S n)
   | (nf_cons n g) \Rightarrow 
-      (exp (nth_prime i) n)*(defactorize_aux g (S i))].
+      (nth_prime i) \sup n *(defactorize_aux g (S i))].
       
 definition defactorize : nat_fact_all \to nat \def
 \lambda f : nat_fact_all. 
@@ -185,7 +185,7 @@ apply sym_eq.apply eq_pair_fst_snd.
 intros.
 rewrite < H3.
 simplify.
-cut n1 = r*(exp (nth_prime n) q).
+cut n1 = r * (nth_prime n) \sup q.
 rewrite > H.
 simplify.rewrite < assoc_times.
 rewrite < Hcut.reflexivity.
@@ -251,10 +251,10 @@ intros.
 rewrite < H.
 change with 
 defactorize_aux (factorize_aux p r (nf_last (pred q))) O = (S(S m1)).
-cut (S(S m1)) = (exp (nth_prime p) q)*r.
+cut (S(S m1)) = (nth_prime p) \sup q *r.
 cut O <r.
 rewrite > defactorize_aux_factorize_aux.
-change with r*(exp (nth_prime p) (S (pred q))) = (S(S m1)).
+change with r*(nth_prime p) \sup (S (pred q)) = (S(S m1)).
 cut (S (pred q)) = q.
 rewrite > Hcut2.
 rewrite > sym_times.
@@ -334,27 +334,27 @@ match f with
 theorem divides_max_p_defactorize: \forall f:nat_fact.\forall i:nat. 
 divides (nth_prime ((max_p f)+i)) (defactorize_aux f i).
 intro.
-elim f.simplify.apply witness ? ? (exp (nth_prime i) n).
+elim f.simplify.apply witness ? ? ((nth_prime i) \sup n).
 reflexivity.
 change with 
 divides (nth_prime (S(max_p n1)+i)) 
-((exp (nth_prime i) n)*(defactorize_aux n1 (S i))).
+((nth_prime i) \sup n *(defactorize_aux n1 (S i))).
 elim H (S i).
 rewrite > H1.
 rewrite < sym_times.
 rewrite > assoc_times.
 rewrite < plus_n_Sm.
-apply witness ? ? (n2*(exp (nth_prime i) n)).
+apply witness ? ? (n2* (nth_prime i) \sup n).
 reflexivity.
 qed.
 
 theorem divides_exp_to_divides: 
 \forall p,n,m:nat. prime p \to 
-divides p (exp n m) \to divides p n.
+divides p (n \sup m) \to divides p n.
 intros 3.elim m.simplify in H1.
 apply transitive_divides p (S O).assumption.
 apply divides_SO_n.
-cut divides p n \lor divides p (exp n n1).
+cut divides p n \lor divides p (n \sup n1).
 elim Hcut.assumption.
 apply H.assumption.assumption.
 apply divides_times_to_divides.assumption.
@@ -363,7 +363,7 @@ qed.
 
 theorem divides_exp_to_eq: 
 \forall p,q,m:nat. prime p \to prime q \to
-divides p (exp q m) \to p = q.
+divides p (q \sup m) \to p = q.
 intros.
 simplify in H1.
 elim H1.apply H4.
@@ -376,7 +376,7 @@ theorem  not_divides_defactorize_aux: \forall f:nat_fact. \forall i,j:nat.
 i < j \to \not divides (nth_prime i) (defactorize_aux f j).
 intro.elim f.
 change with
-divides (nth_prime i) (exp (nth_prime j) (S n)) \to False.
+divides (nth_prime i) ((nth_prime j) \sup (S n)) \to False.
 intro.absurd (nth_prime i) = (nth_prime j).
 apply divides_exp_to_eq ? ? (S n).
 apply prime_nth_prime.apply prime_nth_prime.
@@ -386,9 +386,9 @@ intro.cut i = j.
 apply not_le_Sn_n i.rewrite > Hcut in \vdash (? ? %).assumption.
 apply injective_nth_prime ? ? H2.
 change with 
-divides (nth_prime i) ((exp (nth_prime j) n)*(defactorize_aux n1 (S j))) \to False.
+divides (nth_prime i) ((nth_prime j) \sup n *(defactorize_aux n1 (S j))) \to False.
 intro.
-cut divides (nth_prime i) (exp (nth_prime j) n)
+cut divides (nth_prime i) ((nth_prime j) \sup n)
 \lor divides (nth_prime i) (defactorize_aux n1 (S j)).
 elim Hcut.
 absurd (nth_prime i) = (nth_prime j).
@@ -424,7 +424,7 @@ absurd divides (nth_prime (S(max_p n2)+i)) (defactorize_aux (nf_cons n1 n2) i).
 apply divides_max_p_defactorize.
 rewrite < H2.
 change with 
-(divides (nth_prime (S(max_p n2)+i)) (exp (nth_prime i) (S n))) \to False.
+(divides (nth_prime (S(max_p n2)+i)) ((nth_prime i) \sup (S n))) \to False.
 intro.
 absurd nth_prime (S (max_p n2) + i) = nth_prime i.
 apply divides_exp_to_eq ? ? (S n).
@@ -445,7 +445,7 @@ absurd divides (nth_prime (S(max_p n1)+i)) (defactorize_aux (nf_cons n n1) i).
 apply divides_max_p_defactorize.
 rewrite > H2.
 change with 
-(divides (nth_prime (S(max_p n1)+i)) (exp (nth_prime i) (S n2))) \to False.
+(divides (nth_prime (S(max_p n1)+i)) ((nth_prime i) \sup (S n2))) \to False.
 intro.
 absurd nth_prime (S (max_p n1) + i) = nth_prime i.
 apply divides_exp_to_eq ? ? (S n2).
@@ -460,8 +460,8 @@ apply injective_nth_prime ? ? H4.
 simplify in H3.
 generalize in match H3.
 apply nat_elim2 (\lambda n,n2.
-(exp (nth_prime i) n)*(defactorize_aux n1 (S i)) =
-(exp (nth_prime i) n2)*(defactorize_aux n3 (S i)) \to
+((nth_prime i) \sup n)*(defactorize_aux n1 (S i)) =
+((nth_prime i) \sup n2)*(defactorize_aux n3 (S i)) \to
 nf_cons n n1 = nf_cons n2 n3).
 intro.
 elim n4. apply eq_f.
@@ -476,7 +476,7 @@ simplify in H5.
 rewrite > plus_n_O (defactorize_aux n1 (S i)).
 rewrite > H5.
 rewrite > assoc_times.
-apply witness ? ? ((exp (nth_prime i) n5)*(defactorize_aux n3 (S i))).
+apply witness ? ? (((nth_prime i) \sup n5)*(defactorize_aux n3 (S i))).
 reflexivity.
 intros.
 apply False_ind.
@@ -486,7 +486,7 @@ simplify in H4.
 rewrite > plus_n_O (defactorize_aux n3 (S i)).
 rewrite < H4.
 rewrite > assoc_times.
-apply witness ? ? ((exp (nth_prime i) n4)*(defactorize_aux n1 (S i))).
+apply witness ? ? (((nth_prime i) \sup n4)*(defactorize_aux n1 (S i))).
 reflexivity.
 intros.
 cut nf_cons n4 n1 = nf_cons m n3.