X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fnat%2Ford.ma;h=7dd4fb362bce6a7803caf1837ec2430e6370ddf2;hb=25564c06c570e5ab9be455904c0b381842f8d4c4;hp=ba08a0dcf5ba21536cb4830f10c22e88b5eac00a;hpb=af329cf0e0db4521e5f3d333802928e1425e9f75;p=helm.git

diff --git a/helm/software/matita/library/nat/ord.ma b/helm/software/matita/library/nat/ord.ma
index ba08a0dcf..7dd4fb362 100644
--- a/helm/software/matita/library/nat/ord.ma
+++ b/helm/software/matita/library/nat/ord.ma
@@ -12,14 +12,11 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/nat/ord".
-
 include "datatypes/constructors.ma".
 include "nat/exp.ma".
-include "nat/gcd.ma".
-include "nat/relevant_equations.ma". (* required by auto paramod *)
-
-(* this definition of log is based on pairs, with a remainder *)
+include "nat/nth_prime.ma".
+include "nat/gcd.ma". (* for some properties of divides *)
+include "nat/relevant_equations.ma". (* required by autobatch paramod *)
 
 let rec p_ord_aux p n m \def
   match n \mod m with
@@ -168,8 +165,7 @@ apply p_ord_exp
   apply le_times_l.
   assumption.
   apply le_times_r.assumption.
-alias id "not_eq_to_le_to_lt" = "cic:/matita/algebra/finite_groups/not_eq_to_le_to_lt.con".
-apply not_eq_to_le_to_lt.
+  apply not_eq_to_le_to_lt.
   unfold.intro.apply H1.
   rewrite < H3.
   apply (witness ? r r ?).simplify.apply plus_n_O.
@@ -198,6 +194,46 @@ apply (p_ord_aux_to_exp n).apply (trans_lt ? (S O)).
 unfold.apply le_n.assumption.symmetry.assumption.
 qed.
 
+theorem eq_p_ord_q_O: ∀p,n,q. p_ord n p = pair ? ? q O → n=O ∧ q=O.
+ intros 2;
+ cases n;
+  [ cases p; simplify; intros; destruct H; split; reflexivity;
+  | cases p;
+     [ intros;
+       simplify in H;
+       destruct H
+     | cases n2;
+        [ intros;
+          simplify in H;
+          rewrite < minus_n_O in H;
+          simplify in H;
+          change in H:(? ? match % return ? with [_⇒?|_⇒?] ?) with (mod n1 (S O));
+          rewrite > mod_SO in H;
+          simplify in H;
+          change in H:(? ? match ? ? (? %) ? return ? with [_⇒?] ?) with (div n1 (S O));
+          rewrite > div_SO in H;
+          simplify in H;
+          
+          letin H1 ≝ (p_ord_aux_to_exp n1 (S n1) 1); clearbody H1;
+          elim (p_ord_aux n1 (S n1) 1) in H H1 (q' r'); simplify in H H1;
+          destruct H;
+          lapply (H1 q' O (le_n ?) (refl_eq ? ?));
+          rewrite < times_n_O in Hletin;
+          destruct Hletin
+        | intros;
+          lapply (p_ord_to_exp1 ? ? ? ? ? ? H);
+          [ apply le_S_S;
+            apply le_S_S;
+            apply le_O_n
+          | apply le_S_S;
+            apply le_O_n
+          | cases Hletin;
+            elim H1;
+            apply (witness ? O O);
+            rewrite < times_n_O;
+            reflexivity]]]]
+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  
@@ -213,9 +249,10 @@ unfold.intro.
 elim (divides_times_to_divides ? ? ? H H9).
 apply (absurd ? ? H10 H5).
 apply (absurd ? ? H10 H7).
-(* rewrite > H6.
-rewrite > H8. *)
-auto paramodulation library=yes.
+(* REGRESSION *)
+rewrite > H6. 
+rewrite > H8.
+autobatch paramodulation.
 unfold prime in H. elim H. assumption.
 qed.
 
@@ -240,4 +277,380 @@ apply le_to_not_lt.
 apply divides_to_le.unfold.apply le_n.assumption.
 rewrite < times_n_SO.
 apply exp_n_SO.
-qed.
\ No newline at end of file
+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;assumption;
+        ]
+      |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.
+
+(* properties of ord e ord_rem *)
+
+theorem ord_times: \forall p,m,n.O<m \to O<n \to prime p \to
+ord (m*n) p = ord m p+ord n p.
+intros.unfold ord.
+rewrite > (p_ord_times ? ? ? (ord m p) (ord_rem m p) (ord n p) (ord_rem n p))
+  [reflexivity
+  |assumption
+  |assumption
+  |assumption
+  |unfold ord.unfold ord_rem.apply eq_pair_fst_snd
+  |unfold ord.unfold ord_rem.apply eq_pair_fst_snd
+  ]
+qed.
+
+theorem ord_exp: \forall p,m.S O < p \to
+ord (exp p m) p = m.
+intros.
+unfold ord.
+rewrite > (p_ord_exp1 p (exp p m) m (S O))
+  [reflexivity
+  |apply lt_to_le.assumption
+  |intro.apply (lt_to_not_le ? ? H).
+   apply divides_to_le
+    [apply lt_O_S
+    |assumption
+    ]
+  |apply times_n_SO
+  ]
+qed.
+
+theorem not_divides_to_ord_O: 
+\forall p,m. prime p \to \lnot (divides p m) \to
+ord m p = O.
+intros.unfold ord.
+rewrite > (p_ord_exp1 p m O m)
+  [reflexivity
+  |apply prime_to_lt_O.assumption
+  |assumption
+  |simplify.apply plus_n_O
+  ]
+qed.
+
+theorem ord_O_to_not_divides: 
+\forall p,m. O < m \to prime p \to ord m p = O \to 
+\lnot (divides p m).
+intros.
+lapply (p_ord_to_exp1 p m (ord m p) (ord_rem m p))
+  [elim Hletin.
+   rewrite > H4.
+   rewrite > H2.
+   rewrite > sym_times.
+   rewrite < times_n_SO.
+   assumption
+  |apply prime_to_lt_SO.assumption
+  |assumption
+  |unfold ord.unfold ord_rem.apply eq_pair_fst_snd
+  ]
+qed.
+
+theorem divides_to_not_ord_O: 
+\forall p,m. O < m \to prime p \to divides p m \to
+\lnot(ord m p = O).
+intros.intro.
+apply (ord_O_to_not_divides ? ? H H1 H3).
+assumption.
+qed.
+
+theorem not_ord_O_to_divides: 
+\forall p,m. O < m \to prime p \to \lnot (ord m p = O) \to 
+divides p m.
+intros.
+rewrite > (exp_ord p) in ⊢ (? ? %)
+  [apply (trans_divides ? (exp p (ord m p)))
+    [generalize in match H2.
+     cases (ord m p);intro
+      [apply False_ind.apply H3.reflexivity
+      |simplify.autobatch
+      ]
+    |autobatch
+    ]
+  |apply prime_to_lt_SO.
+   assumption
+  |assumption
+  ]
+qed.
+
+theorem not_divides_ord_rem: \forall m,p.O< m \to S O < p \to
+\lnot (divides p (ord_rem m p)).
+intros.
+elim (p_ord_to_exp1 p m (ord m p) (ord_rem m p))
+  [assumption
+  |assumption
+  |assumption
+  |unfold ord.unfold ord_rem.apply eq_pair_fst_snd
+  ]
+qed.
+
+theorem ord_ord_rem: \forall p,q,m. O < m \to 
+prime p \to prime q \to
+q < p \to ord (ord_rem m p) q = ord m q.
+intros.
+rewrite > (exp_ord p) in ⊢ (? ? ? (? % ?))
+  [rewrite > ord_times
+    [rewrite > not_divides_to_ord_O in ⊢ (? ? ? (? % ?))
+      [reflexivity
+      |assumption
+      |intro.
+       apply (lt_to_not_eq ? ? H3).
+       apply (divides_exp_to_eq ? ? ? H2 H1 H4)
+      ]
+    |apply lt_O_exp.
+     apply (ltn_to_ltO ? ? H3)
+    |apply lt_O_ord_rem
+      [elim H1.assumption
+      |assumption
+      
+      ]
+    |assumption
+    ]
+  |elim H1.assumption
+  |assumption
+  ]
+qed.
+
+theorem lt_ord_rem: \forall n,m. prime n \to O < m \to
+divides n m \to ord_rem m n < m.
+intros.
+elim (le_to_or_lt_eq (ord_rem m n) m)
+  [assumption
+  |apply False_ind.
+   apply (ord_O_to_not_divides ? ? H1 H ? H2).
+   apply (inj_exp_r n)
+    [apply prime_to_lt_SO.assumption
+    |apply (inj_times_l1 m)
+      [assumption
+      |rewrite > sym_times in ⊢ (? ? ? %).
+       rewrite < times_n_SO.
+       rewrite < H3 in ⊢ (? ? (? ? %) ?).
+       apply sym_eq.
+       apply exp_ord
+        [apply prime_to_lt_SO.assumption
+        |assumption
+        ]
+      ]
+    ]
+  |apply divides_to_le
+    [assumption
+    |apply divides_ord_rem
+      [apply prime_to_lt_SO.assumption
+      |assumption
+      ]
+    ]
+  ]
+qed.
+
+(* p_ord_inv is used to encode the pair ord and rem into 
+   a single natural number. *)
+ 
+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.