Prototype of min_aux changed.

Now (min_aux off n f) find the smallest i such that
* f i = true or i = n+off
* \forall j,  n <= j <= n+off,  f j = false

The new function does no longer compute with off. Thus we obtain
for free a great computational speed-up in every function defined
in terms of it.

diff --git a/matita/library/nat/chinese_reminder.ma b/matita/library/nat/chinese_reminder.ma
index 31b976dcf4abd98169bd73a39068ba172871889c..766b85f715324398d2cde65bddf9c96f3c8da90c 100644 (file)
--- a/matita/library/nat/chinese_reminder.ma
+++ b/matita/library/nat/chinese_reminder.ma
@@ -223,14 +223,15 @@ simplify.
 intro.apply False_ind.
 apply not_eq_true_false.apply sym_eq.assumption.
 apply (f_min_aux_true 
-(\lambda x. andb (eqb (x \mod m) a) (eqb (x \mod n) b)) (pred (m*n)) (pred (m*n))).
+(\lambda x. andb (eqb (x \mod m) a) (eqb (x \mod n) b)) (pred (m*n)) O).
 elim (and_congruent_congruent_lt m n a b).
 apply (ex_intro ? ? a1).split.split.
-rewrite < minus_n_n.apply le_O_n.
+apply le_O_n.
 elim H3.apply le_S_S_to_le.apply (trans_le ? (m*n)).
 assumption.apply (nat_case (m*n)).apply le_O_n.
 intro.
-rewrite < pred_Sn.apply le_n.
+rewrite < pred_Sn.
+rewrite < plus_n_O.apply le_n.
 elim H3.elim H4.
 apply andb_elim.
 cut (a1 \mod m = a).

diff --git a/matita/library/nat/minimization.ma b/matita/library/nat/minimization.ma
index 2d273e1809e69595d9ba34ba7c8d95de6f36f698..8b4b83c2fbc7d910fc9c3d91dbd4dc130e713c30 100644 (file)
--- a/matita/library/nat/minimization.ma
+++ b/matita/library/nat/minimization.ma
@@ -161,19 +161,19 @@ elim n
 qed.
  
 let rec min_aux off n f \def
-  match f (n-off) with 
-  [ true \Rightarrow (n-off)
+  match f n with 
+  [ true \Rightarrow n
   | false \Rightarrow 
       match off with
       [ O \Rightarrow n
-      | (S p) \Rightarrow min_aux p n f]].
+      | (S p) \Rightarrow min_aux p (S n) f]].
 
 definition min : nat \to (nat \to bool) \to nat \def
-\lambda n.\lambda f. min_aux n n f.
+\lambda n.\lambda f. min_aux n O f.
 
 theorem min_aux_O_f: \forall f:nat \to bool. \forall i :nat.
 min_aux O i f = i.
-intros.simplify.rewrite < minus_n_O.
+intros.simplify.
 elim (f i).reflexivity.
 simplify.reflexivity.
 qed.
@@ -184,34 +184,35 @@ intro.apply (min_aux_O_f f O).
 qed.
 
 theorem min_aux_S : \forall f: nat \to bool. \forall i,n:nat.
-(f (n -(S i)) = true \land min_aux (S i) n f = (n - (S i))) \lor 
-(f (n -(S i)) = false \land min_aux (S i) n f = min_aux i n f).
-intros.simplify.elim (f (n - (S i))).
+((f n) = true \land min_aux (S i) n f = n) \lor 
+((f n) = false \land min_aux (S i) n f = min_aux i (S n) f).
+intros.simplify.elim (f n).
 simplify.left.split.reflexivity.reflexivity.
 simplify.right.split.reflexivity.reflexivity.
 qed.
 
 theorem f_min_aux_true: \forall f:nat \to bool. \forall off,m:nat.
-(\exists i. le (m-off) i \land le i m \land f i = true) \to
+(\exists i. le m i \land le i (off + m) \land f i = true) \to
 f (min_aux off m f) = true. 
 intros 2.
 elim off.elim H.elim H1.elim H2.
 cut (a = m).
 rewrite > (min_aux_O_f f).rewrite < Hcut.assumption.
-apply (antisym_le a m).assumption.rewrite > (minus_n_O m).assumption.
+apply (antisym_le a m).assumption.assumption.
 simplify.
 apply (bool_ind (\lambda b:bool.
-(f (m-(S n)) = b) \to (f (match b in bool with
-[ true \Rightarrow m-(S n)
-| false  \Rightarrow (min_aux n m f)])) = true)).
-simplify.intro.assumption.
-simplify.intro.apply H.
+(f m = b) \to (f (match b in bool with
+[ true \Rightarrow m
+| false  \Rightarrow (min_aux n (S m) f)])) = true)).
+intro; assumption.
+intro. apply H.
 elim H1.elim H3.elim H4.
-elim (le_to_or_lt_eq (m-(S n)) a H6). 
+elim (le_to_or_lt_eq ? a H6). 
 apply (ex_intro nat ? a).
 split.split.
-apply lt_minus_S_n_to_le_minus_n.assumption.
-assumption.assumption.
+assumption.
+rewrite < plus_n_Sm; assumption.
+assumption.
 absurd (f a = false).rewrite < H8.assumption.
 rewrite > H5.
 apply not_eq_true_false.
@@ -219,45 +220,67 @@ reflexivity.
 qed.
 
 theorem lt_min_aux_to_false : \forall f:nat \to bool. 
-\forall n,off,m:nat. (n-off) \leq m \to m < (min_aux off n f) \to f m = false.
+\forall n,off,m:nat. n \leq m \to m < (min_aux off n f) \to f m = false.
 intros 3.
-elim off.absurd (le n m).rewrite > minus_n_O.assumption.
-apply lt_to_not_le.rewrite < (min_aux_O_f f n).assumption.
-generalize in match H1.
-elim (min_aux_S f n1 n).
-elim H3.
-absurd (n - S n1 \le m).assumption.
-apply lt_to_not_le.rewrite < H6.assumption.
-elim H3.
-elim (le_to_or_lt_eq (n -(S n1)) m).
-apply H.apply lt_minus_S_n_to_le_minus_n.assumption.
-rewrite < H6.assumption.
-rewrite < H7.assumption.
-assumption.
+generalize in match n; clear n.
+elim off.absurd (le n1 m).assumption.
+apply lt_to_not_le.rewrite < (min_aux_O_f f n1).assumption.
+elim (le_to_or_lt_eq ? ? H1);
+ [ unfold lt in H3;
+   apply (H (S n1));
+   [ assumption
+   | elim (min_aux_S f n n1);
+     [ elim H4;
+       elim (not_le_Sn_n n1);
+       rewrite > H6 in H2;
+       apply (lt_to_le (S n1) n1 ?).
+       apply (le_to_lt_to_lt (S n1) m n1 ? ?);
+        [apply (H3).
+        |apply (H2).
+        ]
+     | elim H4;
+       rewrite < H6;
+       assumption
+     ]
+   ]
+ | rewrite < H3 in H2 ⊢ %.
+   elim (min_aux_S f n n1);
+    [ elim H4;
+      rewrite > H6 in H2;
+      unfold lt in H2;
+      elim (not_le_Sn_n ? H2)
+    | elim H4;
+      assumption
+    ]
+ ]
 qed.
 
-theorem le_min_aux : \forall f:nat \to bool. 
-\forall n,off:nat. (n-off) \leq (min_aux off n f).
+
+lemma le_min_aux : \forall f:nat \to bool. 
+\forall n,off:nat. n \leq (min_aux off n f).
 intros 3.
-elim off.rewrite < minus_n_O.
-rewrite > (min_aux_O_f f n).apply le_n.
-elim (min_aux_S f n1 n).
+generalize in match n. clear n.
+elim off.
+rewrite > (min_aux_O_f f n1).apply le_n.
+elim (min_aux_S f n n1).
 elim H1.rewrite > H3.apply le_n.
 elim H1.rewrite > H3.
-apply (trans_le (n-(S n1)) (n-n1)).
-apply monotonic_le_minus_r.
-apply le_n_Sn.
-assumption.
+apply (transitive_le ? (S n1));
+ [ apply le_n_Sn
+ | apply (H (S n1))
+ ]
 qed.
 
 theorem le_min_aux_r : \forall f:nat \to bool. 
-\forall n,off:nat. (min_aux off n f) \le n.
+\forall n,off:nat. (min_aux off n f) \le n+off.
 intros.
-elim off.simplify.rewrite < minus_n_O.
-elim (f n).simplify.apply le_n.
-simplify.apply le_n.
-simplify.elim (f (n -(S n1))).
-simplify.apply le_plus_to_minus.
-rewrite < sym_plus.apply le_plus_n.
-simplify.assumption.
-qed.
+generalize in match n. clear n.
+elim off.simplify.
+elim (f n1).simplify.rewrite < plus_n_O.apply le_n.
+simplify.rewrite < plus_n_O.apply le_n.
+simplify.elim (f n1).
+simplify.
+apply (le_plus_n_r (S n) n1).
+simplify.rewrite < plus_n_Sm.
+apply (H (S n1)).
+qed.
\ No newline at end of file

diff --git a/matita/library/nat/nth_prime.ma b/matita/library/nat/nth_prime.ma
index 9d01e99f211e0b3a6e65cc7cb567c732aeff1c6e..f675e80ba45717620d95a25c5081ca93bcd956c7 100644 (file)
--- a/matita/library/nat/nth_prime.ma
+++ b/matita/library/nat/nth_prime.ma
@@ -74,11 +74,9 @@ match n with
   | (S p) \Rightarrow
     let previous_prime \def (nth_prime p) in
     let upper_bound \def S previous_prime! in
-    min_aux (upper_bound - (S previous_prime)) upper_bound primeb].
+    min_aux upper_bound (S previous_prime) primeb].
     
-(* it works, but nth_prime 4 takes already a few minutes -
-it must compute factorial of 7 ...*)
-
+(* it works
 theorem example11 : nth_prime (S(S O)) = (S(S(S(S(S O))))).
 normalize.reflexivity.
 qed.
@@ -91,10 +89,11 @@ theorem example13 : nth_prime (S(S(S(S O)))) = (S(S(S(S(S(S(S(S(S(S(S O)))))))))
 normalize.reflexivity.
 qed.
 
-(*
-theorem example14 : nth_prime (S(S(S(S(S O))))) = (S(S(S(S(S(S(S(S(S(S(S O))))))))))).
+alias num (instance 0) = "natural number".
+theorem example14 : nth_prime 18 = 67.
 normalize.reflexivity.
-*) 
+qed.
+*)
 
 theorem prime_nth_prime : \forall n:nat.prime (nth_prime n).
 intro.
@@ -104,19 +103,17 @@ intro.
 change with
 (let previous_prime \def (nth_prime m) in
 let upper_bound \def S previous_prime! in
-prime (min_aux (upper_bound - (S previous_prime)) upper_bound primeb)).
+prime (min_aux upper_bound (S previous_prime) primeb)).
 apply primeb_true_to_prime.
 apply f_min_aux_true.
 apply (ex_intro nat ? (smallest_factor (S (nth_prime m)!))).
 split.split.
-cut (S (nth_prime m)!-(S (nth_prime m)! - (S (nth_prime m))) = (S (nth_prime m))).
-rewrite > Hcut.exact (smallest_factor_fact (nth_prime m)).
-(* maybe we could factorize this proof *)
-apply plus_to_minus.
-apply plus_minus_m_m.
-apply le_S_S.
-apply le_n_fact_n.
-apply le_smallest_factor_n.
+apply smallest_factor_fact.
+apply transitive_le;
+ [2: apply le_smallest_factor_n
+ | skip
+ | apply (le_plus_n_r (S (nth_prime m)) (S (fact (nth_prime m))))
+ ].
 apply prime_to_primeb_true.
 apply prime_smallest_factor_n.unfold lt.
 apply le_S_S.apply le_SO_fact.
@@ -129,15 +126,9 @@ intros.
 change with
 (let previous_prime \def (nth_prime n) in
 let upper_bound \def S previous_prime! in
-(S previous_prime) \le min_aux (upper_bound - (S previous_prime)) upper_bound primeb).
+(S previous_prime) \le min_aux upper_bound (S previous_prime) primeb).
 intros.
-cut (upper_bound - (upper_bound -(S previous_prime)) = (S previous_prime)).
-rewrite < Hcut in \vdash (? % ?).
 apply le_min_aux.
-apply plus_to_minus.
-apply plus_minus_m_m.
-apply le_S_S.
-apply le_n_fact_n.
 qed.
 
 variant lt_nth_prime_n_nth_prime_Sn :\forall n:nat. 
@@ -184,14 +175,13 @@ intros.
 apply primeb_false_to_not_prime.
 letin previous_prime \def (nth_prime n).
 letin upper_bound \def (S previous_prime!).
-apply (lt_min_aux_to_false primeb upper_bound (upper_bound - (S previous_prime)) m).
-cut (S (nth_prime n)!-(S (nth_prime n)! - (S (nth_prime n))) = (S (nth_prime n))).
-rewrite > Hcut.assumption.
-apply plus_to_minus.
-apply plus_minus_m_m.
-apply le_S_S.
-apply le_n_fact_n.
+apply (lt_min_aux_to_false primeb (S previous_prime) upper_bound m).
 assumption.
+unfold lt.
+apply (transitive_le (S m) (nth_prime (S n)) (min_aux (S (fact (nth_prime n))) (S (nth_prime n)) primeb) ? ?);
+  [apply (H1).
+  |apply (le_n (min_aux (S (fact (nth_prime n))) (S (nth_prime n)) primeb)).
+  ]
 qed.
 
 (* nth_prime enumerates all primes *)

diff --git a/matita/library/nat/primes.ma b/matita/library/nat/primes.ma
index d2e89b8f1b56de4e2de72fc7673f008d909e01cf..a95b2e88fddda2344f0fafc448593de57404d470 100644 (file)
--- a/matita/library/nat/primes.ma
+++ b/matita/library/nat/primes.ma
@@ -397,18 +397,18 @@ match n with
 | (S p) \Rightarrow 
   match p with
   [ O \Rightarrow (S O)
-  | (S q) \Rightarrow min_aux q (S(S q)) (\lambda m.(eqb ((S(S q)) \mod m) O))]].
+  | (S q) \Rightarrow min_aux q (S (S O)) (\lambda m.(eqb ((S(S q)) \mod m) O))]].
 
-(* it works ! 
-theorem example1 : smallest_prime_factor (S(S(S O))) = (S(S(S O))).
+(* it works !
+theorem example1 : smallest_factor (S(S(S O))) = (S(S(S O))).
 normalize.reflexivity.
 qed.
 
-theorem example2: smallest_prime_factor (S(S(S(S O)))) = (S(S O)).
+theorem example2: smallest_factor (S(S(S(S O)))) = (S(S O)).
 normalize.reflexivity.
 qed.
 
-theorem example3 : smallest_prime_factor (S(S(S(S(S(S(S O))))))) = (S(S(S(S(S(S(S O))))))).
+theorem example3 : smallest_factor (S(S(S(S(S(S(S O))))))) = (S(S(S(S(S(S(S O))))))).
 simplify.reflexivity.
 qed. *)
 
@@ -419,7 +419,7 @@ apply (nat_case n).intro.apply False_ind.apply (not_le_Sn_O (S O) H).
 intro.apply (nat_case m).intro. apply False_ind.apply (not_le_Sn_n (S O) H).
 intros.
 change with 
-(S O < min_aux m1 (S(S m1)) (\lambda m.(eqb ((S(S m1)) \mod m) O))).
+(S O < min_aux m1 (S (S O)) (\lambda m.(eqb ((S(S m1)) \mod m) O))).
 apply (lt_to_le_to_lt ? (S (S O))).
 apply (le_n (S(S O))).
 cut ((S(S O)) = (S(S m1)) - m1).
@@ -449,15 +449,18 @@ apply (witness ? ? (S O)). simplify.reflexivity.
 intros.
 apply divides_b_true_to_divides.
 change with 
-(eqb ((S(S m1)) \mod (min_aux m1 (S(S m1)) 
+(eqb ((S(S m1)) \mod (min_aux m1 (S (S O)) 
   (\lambda m.(eqb ((S(S m1)) \mod m) O)))) O = true).
 apply f_min_aux_true.
 apply (ex_intro nat ? (S(S m1))).
 split.split.
-apply le_minus_m.apply le_n.
-rewrite > mod_n_n.reflexivity.
-apply (trans_lt ? (S O)).apply (le_n (S O)).unfold lt.
-apply le_S_S.apply le_S_S.apply le_O_n.
+apply (le_S_S_to_le (S (S O)) (S (S m1)) ?).
+apply (minus_le_O_to_le (S (S (S O))) (S (S (S m1))) ?).
+apply (le_n O).
+rewrite < sym_plus. simplify. apply le_n.
+apply (eq_to_eqb_true (mod (S (S m1)) (S (S m1))) O ?).
+apply (mod_n_n (S (S m1)) ?).
+apply (H).
 qed.
   
 theorem le_smallest_factor_n : 
@@ -478,12 +481,9 @@ intro.apply (nat_case m).intro. apply False_ind.apply (not_le_Sn_n (S O) H).
 intros.
 apply divides_b_false_to_not_divides.
 apply (lt_min_aux_to_false 
-(\lambda i:nat.eqb ((S(S m1)) \mod i) O) (S(S m1)) m1 i).
-cut ((S(S O)) = (S(S m1)-m1)).
-rewrite < Hcut.exact H1.
-apply sym_eq. apply plus_to_minus.
-rewrite < sym_plus.simplify.reflexivity.
-exact H2.
+(\lambda i:nat.eqb ((S(S m1)) \mod i) O) (S (S O)) m1 i).
+assumption.
+assumption.
 qed.
 
 theorem prime_smallest_factor_n : 

diff --git a/matita/library/nat/totient.ma b/matita/library/nat/totient.ma
index 730ec8b56cfc05bad765600ccad3418e5d71dbac..03e2587a8f02b1bc17eb3f3e8d0a8f932ad5417c 100644 (file)
--- a/matita/library/nat/totient.ma
+++ b/matita/library/nat/totient.ma
@@ -63,7 +63,8 @@ apply (nat_case1 n)
    [intros.unfold cr_pair.
         apply (le_to_lt_to_lt ? (pred ((S m2)*(S m1))))
           [unfold min.
-           apply le_min_aux_r
+           apply transitive_le;
+            [2: apply le_min_aux_r | skip | apply le_n]
           |unfold lt.
            apply (nat_case ((S m2)*(S m1)))
             [apply le_n|intro.apply le_n]