X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Flibrary%2Fnat%2Ford.ma;h=340e33a865fb809d8beb35e632f4e129fa2bd0dc;hb=a957099550619f87a58be467b9b11f2ad6501378;hp=807d26733116f1ff5bab4d36e231c14444a77cdb;hpb=04e27500136c94e4f2ac072a5e4d330b75da35f0;p=helm.git

diff --git a/matita/library/nat/ord.ma b/matita/library/nat/ord.ma
index 807d26733..340e33a86 100644
--- a/matita/library/nat/ord.ma
+++ b/matita/library/nat/ord.ma
@@ -12,14 +12,12 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/nat/log".
+set "baseuri" "cic:/matita/nat/ord".
 
 include "datatypes/constructors.ma".
 include "nat/exp.ma".
-include "nat/lt_arith.ma".
-include "nat/primes.ma".
-
-(* this definition of log is based on pairs, with a remainder *)
+include "nat/nth_prime.ma".
+include "nat/relevant_equations.ma". (* required by autobatch paramod *)
 
 let rec p_ord_aux p n m \def
   match n \mod m with
@@ -68,6 +66,7 @@ rewrite > H1.
 apply p_ord_aux_to_Prop.
 assumption.
 qed.
+
 (* questo va spostato in primes1.ma *)
 theorem p_ord_exp: \forall n,m,i. O < m \to n \mod m \neq O \to
 \forall p. i \le p \to p_ord_aux p (m \sup i * n) m = pair nat nat i n.
@@ -142,4 +141,318 @@ rewrite > H3.
 apply p_ord_aux_to_Prop1.
 assumption.assumption.assumption.
 qed.
+
+theorem p_ord_exp1: \forall p,n,q,r. O < p \to \lnot p \divides r \to
+n = p \sup q * r \to p_ord n p = pair nat nat q r.
+intros.unfold p_ord.
+rewrite > H2.
+apply p_ord_exp
+ [assumption
+ |unfold.intro.apply H1.
+  apply mod_O_to_divides[assumption|assumption]
+ |apply (trans_le ? (p \sup q)).
+  cut ((S O) \lt p).
+  elim q.simplify.apply le_n_Sn.
+  simplify.
+  generalize in match H3.
+  apply (nat_case n1).simplify.
+  rewrite < times_n_SO.intro.assumption.
+  intros.
+  apply (trans_le ? (p*(S m))).
+  apply (trans_le ? ((S (S O))*(S m))).
+  simplify.rewrite > plus_n_Sm.
+  rewrite < plus_n_O.
+  apply le_plus_n.
+  apply le_times_l.
+  assumption.
+  apply le_times_r.assumption.
+  apply not_eq_to_le_to_lt.
+  unfold.intro.apply H1.
+  rewrite < H3.
+  apply (witness ? r r ?).simplify.apply plus_n_O.
+  assumption.
+  rewrite > times_n_SO in \vdash (? % ?).
+  apply le_times_r.
+  change with (O \lt r).
+  apply not_eq_to_le_to_lt.
+  unfold.intro.
+  apply H1.rewrite < H3.
+  apply (witness ? ? O ?).rewrite < times_n_O.reflexivity.
+  apply le_O_n.
+  ]
+qed.
+
+theorem p_ord_to_exp1: \forall p,n,q,r. (S O) \lt p \to O \lt n \to p_ord n p = pair nat nat q r\to 
+\lnot p \divides r \land n = p \sup q * r.
+intros.
+unfold p_ord in H2.
+split.unfold.intro.
+apply (p_ord_aux_to_not_mod_O n n p q r).assumption.assumption.
+apply le_n.symmetry.assumption.
+apply divides_to_mod_O.apply (trans_lt ? (S O)).
+unfold.apply le_n.assumption.assumption.
+apply (p_ord_aux_to_exp n).apply (trans_lt ? (S O)).
+unfold.apply le_n.assumption.symmetry.assumption.
+qed.
+
+theorem p_ord_times: \forall p,a,b,qa,ra,qb,rb. prime p 
+\to O \lt a \to O \lt b 
+\to p_ord a p = pair nat nat qa ra  
+\to p_ord b p = pair nat nat qb rb
+\to p_ord (a*b) p = pair nat nat (qa + qb) (ra*rb).
+intros.
+cut ((S O) \lt p).
+elim (p_ord_to_exp1 ? ? ? ? Hcut H1 H3).
+elim (p_ord_to_exp1 ? ? ? ? Hcut H2 H4).
+apply p_ord_exp1.
+apply (trans_lt ? (S O)).unfold.apply le_n.assumption.
+unfold.intro.
+elim (divides_times_to_divides ? ? ? H H9).
+apply (absurd ? ? H10 H5).
+apply (absurd ? ? H10 H7).
+(* rewrite > H6.
+rewrite > H8. *)
+autobatch paramodulation.
+unfold prime in H. elim H. assumption.
+qed.
+
+theorem fst_p_ord_times: \forall p,a,b. prime p 
+\to O \lt a \to O \lt b 
+\to fst ? ? (p_ord (a*b) p) = (fst ? ? (p_ord a p)) + (fst ? ? (p_ord b p)).
+intros.
+rewrite > (p_ord_times p a b (fst ? ? (p_ord a p)) (snd ? ? (p_ord a p))
+(fst ? ? (p_ord b p)) (snd ? ? (p_ord b p)) H H1 H2).
+simplify.reflexivity.
+apply eq_pair_fst_snd.
+apply eq_pair_fst_snd.
+qed.
+
+theorem p_ord_p : \forall p:nat. (S O) \lt p \to p_ord p p = pair ? ? (S O) (S O).
+intros.
+apply p_ord_exp1.
+apply (trans_lt ? (S O)). unfold.apply le_n.assumption.
+unfold.intro.
+apply (absurd ? ? H).
+apply le_to_not_lt.
+apply divides_to_le.unfold.apply le_n.assumption.
+rewrite < times_n_SO.
+apply exp_n_SO.
+qed.
+
+(* p_ord and divides *)
+theorem divides_to_p_ord: \forall p,a,b,c,d,n,m:nat. 
+O < n \to O < m \to prime p 
+\to divides n m \to p_ord n p = pair ? ? a b \to
+p_ord m p = pair ? ? c d \to divides b d \land a \le c.
+intros.
+cut (S O < p)
+  [lapply (p_ord_to_exp1 ? ? ? ? Hcut H H4).
+   lapply (p_ord_to_exp1 ? ? ? ? Hcut H1 H5).
+   elim Hletin. clear Hletin.
+   elim Hletin1. clear Hletin1.
+   rewrite > H9 in H3.
+   split
+    [apply (gcd_SO_to_divides_times_to_divides (exp p c))
+      [elim (le_to_or_lt_eq ? ? (le_O_n b))
+        [assumption
+        |apply False_ind.
+         apply (lt_to_not_eq O ? H).
+         rewrite > H7.
+         rewrite < H10.
+         autobatch
+        ]
+      |elim c
+        [rewrite > sym_gcd.
+         apply gcd_SO_n
+        |simplify.
+         apply eq_gcd_times_SO
+          [apply lt_to_le.assumption
+          |apply lt_O_exp.apply lt_to_le.assumption
+          |rewrite > sym_gcd.
+           (* hint non trova prime_to_gcd_SO e
+              autobatch non chiude il goal *)
+           apply prime_to_gcd_SO
+            [assumption|assumption]
+          |assumption
+          ]
+        ]
+      |apply (trans_divides ? n)
+        [apply (witness ? ? (exp p a)).
+         rewrite > sym_times.
+         assumption
+        |assumption
+        ]
+      ]
+    |apply (le_exp_to_le p)
+      [assumption
+      |apply divides_to_le
+        [apply lt_O_exp.apply lt_to_le.assumption
+        |apply (gcd_SO_to_divides_times_to_divides d)
+          [apply lt_O_exp.apply lt_to_le.assumption
+          |elim a
+            [apply gcd_SO_n
+            |simplify.rewrite < sym_gcd.
+             apply eq_gcd_times_SO
+              [apply lt_to_le.assumption
+              |apply lt_O_exp.apply lt_to_le.assumption
+              |rewrite > sym_gcd.
+              (* hint non trova prime_to_gcd_SO e
+                 autobatch non chiude il goal *)
+               apply prime_to_gcd_SO
+                [assumption|assumption]
+              |rewrite > sym_gcd. assumption
+              ]
+            ]
+          |apply (trans_divides ? n)
+            [apply (witness ? ? b).assumption
+            |rewrite > sym_times.assumption
+            ]
+          ]
+        ]
+      ]
+    ]
+  |elim H2.assumption
+  ]
+qed.    
+
+(* p_ord and primes *)
+theorem not_divides_to_p_ord_O: \forall n,i.
+Not (divides (nth_prime i) n) \to p_ord n (nth_prime i) = 
+pair nat nat O n.
+intros.
+apply p_ord_exp1
+  [apply lt_O_nth_prime_n
+  |assumption
+  |autobatch
+  ]
+qed.
+
+theorem p_ord_O_to_not_divides: \forall n,i,r.
+O < n \to
+p_ord n (nth_prime i) = pair nat nat O r 
+\to Not (divides (nth_prime i) n).
+intros.
+lapply (p_ord_to_exp1 ? ? ? ? ? ? H1)
+  [apply lt_SO_nth_prime_n
+  |assumption
+  |elim Hletin.
+   simplify in H3.
+   rewrite > H3.
+   rewrite < plus_n_O.
+   assumption
+  ]
+qed.
+
+theorem p_ord_to_not_eq_O : \forall n,p,q,r.
+  (S O) < n \to
+  p_ord n (nth_prime p) = pair nat nat q r \to
+  Not (r=O).
+intros.
+unfold.intro.
+cut (O < n)
+  [lapply (p_ord_to_exp1 ? ? ? ? ? ? H1)
+    [apply lt_SO_nth_prime_n.
+    |assumption
+    |elim Hletin.
+     apply (lt_to_not_eq ? ? Hcut).
+     rewrite > H4.
+     rewrite > H2.
+     apply times_n_O
+    ]
+  |apply (trans_lt ? (S O))[apply lt_O_S|assumption]
+  ]
+qed.
+
+definition ord :nat \to nat \to nat \def
+\lambda n,p. fst ? ? (p_ord n p).
+
+definition ord_rem :nat \to nat \to nat \def
+\lambda n,p. snd ? ? (p_ord n p).
+         
+theorem divides_to_ord: \forall p,n,m:nat. 
+O < n \to O < m \to prime p 
+\to divides n m 
+\to divides (ord_rem n p) (ord_rem m p) \land (ord n p) \le (ord m p).  
+intros.
+apply (divides_to_p_ord p ? ? ? ? n m H H1 H2 H3)
+  [unfold ord.unfold ord_rem.apply eq_pair_fst_snd
+  |unfold ord.unfold ord_rem.apply eq_pair_fst_snd
+  ]
+qed.
+
+theorem divides_to_divides_ord_rem: \forall p,n,m:nat. 
+O < n \to O < m \to prime p \to divides n m \to
+divides (ord_rem n p) (ord_rem m p).
+intros.
+elim (divides_to_ord p n m H H1 H2 H3).assumption.
+qed.
+
+theorem divides_to_le_ord: \forall p,n,m:nat. 
+O < n \to O < m \to prime p \to divides n m \to
+(ord n p) \le (ord m p).  
+intros.
+elim (divides_to_ord p n m H H1 H2 H3).assumption.
+qed.
+
+theorem exp_ord: \forall p,n. (S O) \lt p 
+\to O \lt n \to n = p \sup (ord n p) * (ord_rem n p).
+intros.
+elim (p_ord_to_exp1 p n (ord n p) (ord_rem n p))
+  [assumption
+  |assumption
+  |assumption
+  |unfold ord.unfold ord_rem.
+   apply eq_pair_fst_snd
+  ]
+qed.
+
+theorem divides_ord_rem: \forall p,n. (S O) < p \to O < n 
+\to divides (ord_rem n p) n. 
+intros.
+apply (witness ? ? (p \sup (ord n p))).
+rewrite > sym_times. 
+apply exp_ord[assumption|assumption]
+qed.
+
+theorem lt_O_ord_rem: \forall p,n. (S O) < p \to O < n \to O < ord_rem n p. 
+intros.
+elim (le_to_or_lt_eq O (ord_rem n p))
+  [assumption
+  |apply False_ind.
+   apply (lt_to_not_eq ? ? H1).
+   lapply (divides_ord_rem ? ? H H1).
+   rewrite < H2 in Hletin.
+   elim Hletin.
+   rewrite > H3.
+   reflexivity
+  |apply le_O_n
+  ]
+qed.
+
+(* p_ord_inv is the inverse of ord *)
+definition p_ord_inv \def
+\lambda p,m,x.
+  match p_ord x p with
+  [pair q r \Rightarrow r*m+q].
   
+theorem  eq_p_ord_inv: \forall p,m,x.
+p_ord_inv p m x = (ord_rem x p)*m+(ord x p).
+intros.unfold p_ord_inv. unfold ord_rem.
+unfold ord.
+elim (p_ord x p).
+reflexivity.
+qed.
+
+theorem div_p_ord_inv: 
+\forall p,m,x. ord x p < m \to p_ord_inv p m x / m = ord_rem x p.
+intros.rewrite > eq_p_ord_inv.
+apply div_plus_times.
+assumption.
+qed.
+
+theorem mod_p_ord_inv: 
+\forall p,m,x. ord x p < m \to p_ord_inv p m x \mod m = ord x p.
+intros.rewrite > eq_p_ord_inv.
+apply mod_plus_times.
+assumption.
+qed.