A few changes to factorization and gcd.

diff --git a/helm/matita/library/nat/factorization.ma b/helm/matita/library/nat/factorization.ma
index c9669e2ba0a8071e58cf3e07a407f8d9e77b9478..f9f0289a0bee7e2f7d9db983d021aec87e43d112 100644 (file)
--- a/helm/matita/library/nat/factorization.ma
+++ b/helm/matita/library/nat/factorization.ma
@@ -14,7 +14,7 @@
 
 set "baseuri" "cic:/matita/nat/factorization".
 
-include "nat/log.ma".
+include "nat/ord.ma".
 include "nat/gcd.ma".
 include "nat/nth_prime.ma".
 
@@ -72,9 +72,9 @@ apply divides_max_prime_factor_n.
 assumption.
 qed.
 
-theorem plog_to_lt_max_prime_factor: \forall n,p,q,r. O < n \to
+theorem p_ord_to_lt_max_prime_factor: \forall n,p,q,r. O < n \to
 p = max_prime_factor n \to 
-(pair nat nat q r) = plog n (nth_prime p) \to
+(pair nat nat q r) = p_ord n (nth_prime p) \to
 (S O) < r \to max_prime_factor r < p.
 intros.
 rewrite > H1.
@@ -90,7 +90,7 @@ intro.
 cut mod r (nth_prime (max_prime_factor n)) \neq O.
 apply Hcut1.apply divides_to_mod_O.
 apply lt_O_nth_prime_n.assumption.
-apply plog_aux_to_not_mod_O n n ? q r.
+apply p_ord_aux_to_not_mod_O n n ? q r.
 apply lt_SO_nth_prime_n.assumption.
 apply le_n.
 rewrite < H1.assumption.
@@ -99,13 +99,13 @@ apply divides_to_max_prime_factor.
 assumption.assumption.
 apply witness r n ((nth_prime p) \sup q).
 rewrite < sym_times.
-apply plog_aux_to_exp n n ? q r.
+apply p_ord_aux_to_exp n n ? q r.
 apply lt_O_nth_prime_n.assumption.
 qed.
 
-theorem plog_to_lt_max_prime_factor1: \forall n,p,q,r. O < n \to
+theorem p_ord_to_lt_max_prime_factor1: \forall n,p,q,r. O < n \to
 max_prime_factor n \le p \to 
-(pair nat nat q r) = plog n (nth_prime p) \to
+(pair nat nat q r) = p_ord n (nth_prime p) \to
 (S O) < r \to max_prime_factor r < p.
 intros.
 cut max_prime_factor n < p \lor max_prime_factor n = p.
@@ -113,10 +113,10 @@ elim Hcut.apply le_to_lt_to_lt ? (max_prime_factor n).
 apply divides_to_max_prime_factor.assumption.assumption.
 apply witness r n ((nth_prime p) \sup q).
 rewrite > sym_times.
-apply plog_aux_to_exp n n.
+apply p_ord_aux_to_exp n n.
 apply lt_O_nth_prime_n.
 assumption.assumption.
-apply plog_to_lt_max_prime_factor n ? q.
+apply p_ord_to_lt_max_prime_factor n ? q.
 assumption.apply sym_eq.assumption.assumption.assumption.
 apply le_to_or_lt_eq ? p H1.
 qed.
@@ -136,7 +136,7 @@ let rec factorize_aux p n acc \def
   match p with 
   [ O \Rightarrow acc
   | (S p1) \Rightarrow 
-    match plog n (nth_prime p1) with
+    match p_ord n (nth_prime p1) with
     [ (pair q r) \Rightarrow 
       factorize_aux p1 r (nf_cons q acc)]].
   
@@ -148,7 +148,7 @@ definition factorize : nat \to nat_fact_all \def \lambda n:nat.
       [ O \Rightarrow nfa_one
     | (S n2) \Rightarrow 
       let p \def (max (S(S n2)) (\lambda p:nat.eqb (mod (S(S n2)) (nth_prime p)) O)) in
-      match plog (S(S n2)) (nth_prime p) with
+      match p_ord (S(S n2)) (nth_prime p) with
       [ (pair q r) \Rightarrow 
            nfa_proper (factorize_aux p r (nf_last (pred q)))]]].
            
@@ -165,6 +165,47 @@ match f with
 | nfa_one \Rightarrow (S O)
 | (nfa_proper g) \Rightarrow defactorize_aux g O]. 
 
+theorem lt_O_defactorize_aux: \forall f:nat_fact.\forall i:nat.
+O < defactorize_aux f i.
+intro.elim f.simplify.
+rewrite > times_n_SO.
+apply le_times.
+change with O < nth_prime i.
+apply lt_O_nth_prime_n.
+change with O < exp (nth_prime i) n.
+apply lt_O_exp.
+apply lt_O_nth_prime_n.
+simplify.
+rewrite > times_n_SO.
+apply le_times.
+change with O < exp (nth_prime i) n.
+apply lt_O_exp.
+apply lt_O_nth_prime_n.
+change with O < defactorize_aux n1 (S i).
+apply H.
+qed.
+
+theorem lt_SO_defactorize_aux: \forall f:nat_fact.\forall i:nat.
+S O < defactorize_aux f i.
+intro.elim f.simplify.
+rewrite > times_n_SO.
+apply le_times.
+change with S O < nth_prime i.
+apply lt_SO_nth_prime_n.
+change with O < exp (nth_prime i) n.
+apply lt_O_exp.
+apply lt_O_nth_prime_n.
+simplify.
+rewrite > times_n_SO.
+rewrite > sym_times.
+apply le_times.
+change with O < exp (nth_prime i) n.
+apply lt_O_exp.
+apply lt_O_nth_prime_n.
+change with S O < defactorize_aux n1 (S i).
+apply H.
+qed.
+
 theorem defactorize_aux_factorize_aux : 
 \forall p,n:nat.\forall acc:nat_fact.O < n \to
 ((n=(S O) \land p=O) \lor max_prime_factor n < p) \to
@@ -175,12 +216,12 @@ rewrite > sym_times. apply times_n_SO.
 apply False_ind.apply not_le_Sn_O (max_prime_factor n) H2.
 simplify.
 (* generalizing the goal: I guess there exists a better way *)
-cut \forall q,r.(pair nat nat q r) = (plog_aux n1 n1 (nth_prime n)) \to
-defactorize_aux match (plog_aux n1 n1 (nth_prime n)) with
+cut \forall q,r.(pair nat nat q r) = (p_ord_aux n1 n1 (nth_prime n)) \to
+defactorize_aux match (p_ord_aux n1 n1 (nth_prime n)) with
 [(pair q r)  \Rightarrow (factorize_aux n r (nf_cons q acc))] O =
 n1*defactorize_aux acc (S n).
-apply Hcut (fst ? ? (plog_aux n1 n1 (nth_prime n)))
-(snd ? ? (plog_aux n1 n1 (nth_prime n))).
+apply Hcut (fst ? ? (p_ord_aux n1 n1 (nth_prime n)))
+(snd ? ? (p_ord_aux n1 n1 (nth_prime n))).
 apply sym_eq.apply eq_pair_fst_snd.
 intros.
 rewrite < H3.
@@ -198,7 +239,7 @@ apply le_to_or_lt_eq.apply le_O_n.
 cut (S O) < r \lor (S O) \nlt r.
 elim Hcut1.
 right.
-apply plog_to_lt_max_prime_factor1 n1 ? q r.
+apply p_ord_to_lt_max_prime_factor1 n1 ? q r.
 assumption.elim H2.
 elim H5.
 apply False_ind.
@@ -225,7 +266,7 @@ apply le_to_or_lt_eq r (S O).
 apply not_lt_to_le.assumption.
 apply decidable_lt (S O) r.
 rewrite > sym_times.
-apply plog_aux_to_exp n1 n1.
+apply p_ord_aux_to_exp n1 n1.
 apply lt_O_nth_prime_n.assumption.
 qed.
 
@@ -235,17 +276,17 @@ apply nat_case n.reflexivity.
 intro.apply nat_case m.reflexivity.
 intro.change with  
 let p \def (max (S(S m1)) (\lambda p:nat.eqb (mod (S(S m1)) (nth_prime p)) O)) in
-defactorize (match plog (S(S m1)) (nth_prime p) with
+defactorize (match p_ord (S(S m1)) (nth_prime p) with
 [ (pair q r) \Rightarrow 
    nfa_proper (factorize_aux p r (nf_last (pred q)))])=(S(S m1)).
 intro.
 (* generalizing the goal; find a better way *)
-cut \forall q,r.(pair nat nat q r) = (plog (S(S m1)) (nth_prime p)) \to
-defactorize (match plog (S(S m1)) (nth_prime p) with
+cut \forall q,r.(pair nat nat q r) = (p_ord (S(S m1)) (nth_prime p)) \to
+defactorize (match p_ord (S(S m1)) (nth_prime p) with
 [ (pair q r) \Rightarrow 
    nfa_proper (factorize_aux p r (nf_last (pred q)))])=(S(S m1)).
-apply Hcut (fst ? ? (plog (S(S m1)) (nth_prime p)))
-(snd ? ? (plog (S(S m1)) (nth_prime p))).
+apply Hcut (fst ? ? (p_ord (S(S m1)) (nth_prime p)))
+(snd ? ? (p_ord (S(S m1)) (nth_prime p))).
 apply sym_eq.apply eq_pair_fst_snd.
 intros.
 rewrite < H.
@@ -259,7 +300,7 @@ cut (S (pred q)) = q.
 rewrite > Hcut2.
 rewrite > sym_times.
 apply sym_eq.
-apply plog_aux_to_exp (S(S m1)).
+apply p_ord_aux_to_exp (S(S m1)).
 apply lt_O_nth_prime_n.
 assumption.
 (* O < q *)
@@ -273,7 +314,7 @@ cut (S(S m1)) = r.
 rewrite > Hcut3 in \vdash (? (? ? %)).
 change with nth_prime p \divides r \to False.
 intro.
-apply plog_aux_to_not_mod_O (S(S m1)) (S(S m1)) (nth_prime p) q r.
+apply p_ord_aux_to_not_mod_O (S(S m1)) (S(S m1)) (nth_prime p) q r.
 apply lt_SO_nth_prime_n.
 simplify.apply le_S_S.apply le_O_n.apply le_n.
 assumption.
@@ -288,7 +329,7 @@ apply le_to_or_lt_eq.apply le_O_n.assumption.
 cut (S O) < r \lor S O \nlt r.
 elim Hcut2.
 right. 
-apply plog_to_lt_max_prime_factor1 (S(S m1)) ? q r.
+apply p_ord_to_lt_max_prime_factor1 (S(S m1)) ? q r.
 simplify.apply le_S_S. apply le_O_n.
 apply le_n.
 assumption.assumption.
@@ -315,7 +356,7 @@ rewrite > Hcut.rewrite < H1.rewrite < times_n_O.reflexivity.
 apply le_to_or_lt_eq.apply le_O_n.
 (* prova del cut *)
 goal 20.
-apply plog_aux_to_exp (S(S m1)).
+apply p_ord_aux_to_exp (S(S m1)).
 apply lt_O_nth_prime_n.
 assumption.
 (* fine prova cut *)
@@ -407,6 +448,41 @@ apply divides_times_to_divides.
 apply prime_nth_prime.assumption.
 qed.
 
+lemma not_eq_nf_last_nf_cons: \forall g:nat_fact.\forall n,m,i:nat.
+\lnot (defactorize_aux (nf_last n) i= defactorize_aux (nf_cons m g) i).
+intros.
+change with 
+exp (nth_prime i) (S n) = defactorize_aux (nf_cons m g) i \to False.
+intro.
+cut S(max_p g)+i= i.
+apply not_le_Sn_n i.
+rewrite < Hcut in \vdash (? ? %).
+simplify.apply le_S_S.
+apply le_plus_n.
+apply injective_nth_prime.
+(* uffa, perche' semplifica ? *)
+change with nth_prime (S(max_p g)+i)= nth_prime i.
+apply divides_exp_to_eq ? ? (S n).
+apply prime_nth_prime.apply prime_nth_prime.
+rewrite > H.
+change with divides (nth_prime ((max_p (nf_cons m g))+i)) 
+(defactorize_aux (nf_cons m g) i).
+apply divides_max_p_defactorize.
+qed.
+
+lemma not_eq_nf_cons_O_nf_cons: \forall f,g:nat_fact.\forall n,i:nat.
+\lnot (defactorize_aux (nf_cons O f) i= defactorize_aux (nf_cons (S n) g) i).
+intros.
+simplify.rewrite < plus_n_O.
+intro.
+apply not_divides_defactorize_aux f i (S i) ?.
+simplify.apply le_n.
+rewrite > H.
+rewrite > assoc_times.
+apply witness ? ? ((exp (nth_prime i) n)*(defactorize_aux g (S i))).
+reflexivity.
+qed.
+
 theorem eq_defactorize_aux_to_eq: \forall f,g:nat_fact.\forall i:nat.
 defactorize_aux f i = defactorize_aux g i \to f = g.
 intro.
@@ -417,46 +493,13 @@ apply eq_f.
 apply inj_S. apply inj_exp_r (nth_prime i).
 apply lt_SO_nth_prime_n.
 assumption.
-absurd defactorize_aux (nf_last n) i =
-defactorize_aux (nf_cons n1 n2) i.
-rewrite > H2.reflexivity.
-absurd nth_prime (S(max_p n2)+i) \divides defactorize_aux (nf_cons n1 n2) i.
-apply divides_max_p_defactorize.
-rewrite < H2.
-change with 
-(nth_prime (S(max_p n2)+i) \divides (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).
-apply prime_nth_prime.apply prime_nth_prime.assumption.
-change with nth_prime (S (max_p n2) + i) = nth_prime i \to False.
-intro.apply not_le_Sn_n i.
-cut S(max_p n2)+i= i.
-rewrite < Hcut in \vdash (? ? %).
-simplify.apply le_S_S.
-apply le_plus_n.
-apply injective_nth_prime ? ? H4.
+apply False_ind.
+apply not_eq_nf_last_nf_cons n2 n n1 i H2.
 generalize in match H1.
 elim g.
-absurd defactorize_aux (nf_last n2) i =
-defactorize_aux (nf_cons n n1) i.
+apply False_ind.
+apply not_eq_nf_last_nf_cons n1 n2 n i.
 apply sym_eq. assumption.
-absurd nth_prime (S(max_p n1)+i) \divides defactorize_aux (nf_cons n n1) i.
-apply divides_max_p_defactorize.
-rewrite > H2.
-change with 
-(nth_prime (S(max_p n1)+i) \divides (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).
-apply prime_nth_prime.apply prime_nth_prime.assumption.
-change with nth_prime (S (max_p n1) + i) = nth_prime i \to False.
-intro.apply not_le_Sn_n i.
-cut S(max_p n1)+i= i.
-rewrite < Hcut in \vdash (? ? %).
-simplify.apply le_S_S.
-apply le_plus_n.
-apply injective_nth_prime ? ? H4.
 simplify in H3.
 generalize in match H3.
 apply nat_elim2 (\lambda n,n2.
@@ -470,24 +513,11 @@ simplify in H4.
 rewrite > plus_n_O.
 rewrite > plus_n_O (defactorize_aux n3 (S i)).assumption.
 apply False_ind.
-apply not_divides_defactorize_aux n1 i (S i).
-simplify. apply le_n.
-simplify in H5.
-rewrite > plus_n_O (defactorize_aux n1 (S i)).
-rewrite > H5.
-rewrite > assoc_times.
-apply witness ? ? (((nth_prime i) \sup n5)*(defactorize_aux n3 (S i))).
-reflexivity.
+apply not_eq_nf_cons_O_nf_cons n1 n3 n5 i.assumption.
 intros.
 apply False_ind.
-apply not_divides_defactorize_aux n3 i (S i).
-simplify. apply le_n.
-simplify in H4.
-rewrite > plus_n_O (defactorize_aux n3 (S i)).
-rewrite < H4.
-rewrite > assoc_times.
-apply witness ? ? (((nth_prime i) \sup n4)*(defactorize_aux n1 (S i))).
-reflexivity.
+apply not_eq_nf_cons_O_nf_cons n3 n1 n4 i.
+apply sym_eq.assumption.
 intros.
 cut nf_cons n4 n1 = nf_cons m n3.
 cut n4=m.
@@ -517,13 +547,67 @@ intros.
 apply eq_defactorize_aux_to_eq f g i H.
 qed.
 
-(*
 theorem injective_defactorize: 
 injective nat_fact_all nat defactorize.
 change with \forall f,g:nat_fact_all.
 defactorize f = defactorize g \to f = g.
 intro.elim f.
 generalize in match H.elim g.
-reflexivity.simplify in H1.
-*)
+(* zero - zero *)
+reflexivity.
+(* zero - one *)
+simplify in H1.
+apply False_ind.
+apply not_eq_O_S O H1.
+(* zero - proper *)
+simplify in H1.
+apply False_ind.
+apply not_le_Sn_n O.
+rewrite > H1 in \vdash (? ? %).
+change with O < defactorize_aux n O.
+apply lt_O_defactorize_aux.
+generalize in match H.
+elim g.
+(* one - zero *)
+simplify in H1.
+apply False_ind.
+apply not_eq_O_S O.apply sym_eq. assumption.
+(* one - one *)
+reflexivity.
+(* one - proper *)
+simplify in H1.
+apply False_ind.
+apply not_le_Sn_n (S O).
+rewrite > H1 in \vdash (? ? %).
+change with (S O) < defactorize_aux n O.
+apply lt_SO_defactorize_aux.
+generalize in match H.elim g.
+(* proper - zero *)
+simplify in H1.
+apply False_ind.
+apply not_le_Sn_n O.
+rewrite < H1 in \vdash (? ? %).
+change with O < defactorize_aux n O.
+apply lt_O_defactorize_aux.
+(* proper - one *)
+simplify in H1.
+apply False_ind.
+apply not_le_Sn_n (S O).
+rewrite < H1 in \vdash (? ? %).
+change with (S O) < defactorize_aux n O.
+apply lt_SO_defactorize_aux.
+(* proper - proper *)
+apply eq_f.
+apply injective_defactorize_aux O.
+exact H1.
+qed.
+
+theorem factorize_defactorize: 
+\forall f,g: nat_fact_all. factorize (defactorize f) = f.
+intros.
+apply injective_defactorize.
+(* uffa: perche' semplifica ??? *)
+change with defactorize(factorize (defactorize f)) = (defactorize f).
+apply defactorize_factorize.
+qed.
 

diff --git a/helm/matita/library/nat/gcd.ma b/helm/matita/library/nat/gcd.ma
index 8475fc06aa026aacff1fd29c46c264ff25575947..99c97b09add8d506bc1566c5250da6fd3b76969f 100644 (file)
--- a/helm/matita/library/nat/gcd.ma
+++ b/helm/matita/library/nat/gcd.ma
@@ -297,6 +297,7 @@ rewrite < H7.
 apply ex_intro ? ? (a1+a*(div m n1)).
 apply ex_intro ? ? a.
 left.
+(* clear Hcut2.clear H5.clear H6.clear H. *)
 rewrite > sym_times.
 rewrite > distr_times_plus.
 rewrite > sym_times.
@@ -331,28 +332,7 @@ qed.
 theorem eq_minus_gcd:
  \forall m,n.\exists a,b.a*n - b*m = (gcd n m) \lor b*m - a*n = (gcd n m).
 intros.
-change with
-\exists a,b.
-a*n - b*m = 
-match leb n m with
-  [ true \Rightarrow 
-    match n with 
-    [ O \Rightarrow m
-    | (S p) \Rightarrow gcd_aux (S p) m (S p) ]
-  | false \Rightarrow 
-    match m with 
-    [ O \Rightarrow n
-    | (S p) \Rightarrow gcd_aux (S p) n (S p) ]]
-\lor b*m - a*n = 
-match leb n m with
-  [ true \Rightarrow 
-    match n with 
-    [ O \Rightarrow m
-    | (S p) \Rightarrow gcd_aux (S p) m (S p) ]
-  | false \Rightarrow 
-    match m with 
-    [ O \Rightarrow n
-    | (S p) \Rightarrow gcd_aux (S p) n (S p) ]].
+unfold gcd.
 apply leb_elim n m.
 apply nat_case1 n.
 simplify.intros.

diff --git a/helm/matita/library/nat/nat.ma b/helm/matita/library/nat/nat.ma
index e2bf56542a96c2a9e9b18b392988b60c17e24da2..e36c1beaa95303f52e693f11e4ac7f3d913b6928 100644 (file)
--- a/helm/matita/library/nat/nat.ma
+++ b/helm/matita/library/nat/nat.ma
@@ -26,7 +26,7 @@ definition pred: nat \to nat \def
 | (S p) \Rightarrow p ].
 
 theorem pred_Sn : \forall n:nat.n=(pred (S n)).
-intros; reflexivity.
+intros. reflexivity.
 qed.
 
 theorem injective_S : injective nat nat S.