--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / Matita is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/nat/log".
+
+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 *)
+
+let rec p_ord_aux p n m \def
+ match (mod n m) with
+ [ O \Rightarrow
+ match p with
+ [ O \Rightarrow pair nat nat O n
+ | (S p) \Rightarrow
+ match (p_ord_aux p (div n m) m) with
+ [ (pair q r) \Rightarrow pair nat nat (S q) r]]
+ | (S a) \Rightarrow pair nat nat O n].
+
+(* p_ord n m = <q,r> if m divides n q times, with remainder r *)
+definition p_ord \def \lambda n,m:nat.p_ord_aux n n m.
+
+theorem p_ord_aux_to_Prop: \forall p,n,m. O < m \to
+ match p_ord_aux p n m with
+ [ (pair q r) \Rightarrow n = m \sup q *r ].
+intro.
+elim p.
+change with
+match (
+match (mod n m) with
+ [ O \Rightarrow pair nat nat O n
+ | (S a) \Rightarrow pair nat nat O n] )
+with
+ [ (pair q r) \Rightarrow n = m \sup q * r ].
+apply nat_case (mod n m).
+simplify.apply plus_n_O.
+intros.
+simplify.apply plus_n_O.
+change with
+match (
+match (mod n1 m) with
+ [ O \Rightarrow
+ match (p_ord_aux n (div n1 m) m) with
+ [ (pair q r) \Rightarrow pair nat nat (S q) r]
+ | (S a) \Rightarrow pair nat nat O n1] )
+with
+ [ (pair q r) \Rightarrow n1 = m \sup q * r].
+apply nat_case1 (mod n1 m).intro.
+change with
+match (
+ match (p_ord_aux n (div n1 m) m) with
+ [ (pair q r) \Rightarrow pair nat nat (S q) r])
+with
+ [ (pair q r) \Rightarrow n1 = m \sup q * r].
+generalize in match (H (div n1 m) m).
+elim p_ord_aux n (div n1 m) m.
+simplify.
+rewrite > assoc_times.
+rewrite < H3.rewrite > plus_n_O (m*(div n1 m)).
+rewrite < H2.
+rewrite > sym_times.
+rewrite < div_mod.reflexivity.
+assumption.assumption.
+intros.simplify.apply plus_n_O.
+qed.
+
+theorem p_ord_aux_to_exp: \forall p,n,m,q,r. O < m \to
+ (pair nat nat q r) = p_ord_aux p n m \to n = m \sup q * r.
+intros.
+change with
+match (pair nat nat q r) with
+ [ (pair q r) \Rightarrow n = m \sup q * r ].
+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 mod n m \neq O \to
+\forall p. i \le p \to p_ord_aux p (m \sup i * n) m = pair nat nat i n.
+intros 5.
+elim i.
+simplify.
+rewrite < plus_n_O.
+apply nat_case p.
+change with
+ match (mod n m) with
+ [ O \Rightarrow pair nat nat O n
+ | (S a) \Rightarrow pair nat nat O n]
+ = pair nat nat O n.
+elim (mod n m).simplify.reflexivity.simplify.reflexivity.
+intro.
+change with
+ match (mod n m) with
+ [ O \Rightarrow
+ match (p_ord_aux m1 (div n m) m) with
+ [ (pair q r) \Rightarrow pair nat nat (S q) r]
+ | (S a) \Rightarrow pair nat nat O n]
+ = pair nat nat O n.
+cut O < mod n m \lor O = mod n m.
+elim Hcut.apply lt_O_n_elim (mod n m) H3.
+intros. simplify.reflexivity.
+apply False_ind.
+apply H1.apply sym_eq.assumption.
+apply le_to_or_lt_eq.apply le_O_n.
+generalize in match H3.
+apply nat_case p.intro.apply False_ind.apply not_le_Sn_O n1 H4.
+intros.
+change with
+ match (mod (m \sup (S n1) *n) m) with
+ [ O \Rightarrow
+ match (p_ord_aux m1 (div (m \sup (S n1) *n) m) m) with
+ [ (pair q r) \Rightarrow pair nat nat (S q) r]
+ | (S a) \Rightarrow pair nat nat O (m \sup (S n1) *n)]
+ = pair nat nat (S n1) n.
+cut (mod (m \sup (S n1)*n) m) = O.
+rewrite > Hcut.
+change with
+match (p_ord_aux m1 (div (m \sup (S n1)*n) m) m) with
+ [ (pair q r) \Rightarrow pair nat nat (S q) r]
+ = pair nat nat (S n1) n.
+cut div (m \sup (S n1) *n) m = m \sup n1 *n.
+rewrite > Hcut1.
+rewrite > H2 m1. simplify.reflexivity.
+apply le_S_S_to_le.assumption.
+(* div_exp *)
+change with div (m* m \sup n1 *n) m = m \sup n1 * n.
+rewrite > assoc_times.
+apply lt_O_n_elim m H.
+intro.apply div_times.
+(* mod_exp = O *)
+apply divides_to_mod_O.
+assumption.
+simplify.rewrite > assoc_times.
+apply witness ? ? (m \sup n1 *n).reflexivity.
+qed.
+
+theorem p_ord_aux_to_Prop1: \forall p,n,m. (S O) < m \to O < n \to n \le p \to
+ match p_ord_aux p n m with
+ [ (pair q r) \Rightarrow mod r m \neq O].
+intro.elim p.absurd O < n.assumption.
+apply le_to_not_lt.assumption.
+change with
+match
+ (match (mod n1 m) with
+ [ O \Rightarrow
+ match (p_ord_aux n(div n1 m) m) with
+ [ (pair q r) \Rightarrow pair nat nat (S q) r]
+ | (S a) \Rightarrow pair nat nat O n1])
+with
+ [ (pair q r) \Rightarrow mod r m \neq O].
+apply nat_case1 (mod n1 m).intro.
+generalize in match (H (div n1 m) m).
+elim (p_ord_aux n (div n1 m) m).
+apply H5.assumption.
+apply eq_mod_O_to_lt_O_div.
+apply trans_lt ? (S O).simplify.apply le_n.
+assumption.assumption.assumption.
+apply le_S_S_to_le.
+apply trans_le ? n1.change with div n1 m < n1.
+apply lt_div_n_m_n.assumption.assumption.assumption.
+intros.
+change with mod n1 m \neq O.
+rewrite > H4.
+(* Andrea: META NOT FOUND !!!
+rewrite > sym_eq. *)
+simplify.intro.
+apply not_eq_O_S m1.
+rewrite > H5.reflexivity.
+qed.
+
+theorem p_ord_aux_to_not_mod_O: \forall p,n,m,q,r. (S O) < m \to O < n \to n \le p \to
+ pair nat nat q r = p_ord_aux p n m \to mod r m \neq O.
+intros.
+change with
+ match (pair nat nat q r) with
+ [ (pair q r) \Rightarrow mod r m \neq O].
+rewrite > H3.
+apply p_ord_aux_to_Prop1.
+assumption.assumption.assumption.
+qed.
+
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