include "datatypes/constructors.ma".
include "nat/exp.ma".
-include "nat/gcd.ma".
-include "nat/relevant_equations.ma". (* required by auto paramod *)
+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
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.
apply (absurd ? ? H10 H7).
(* rewrite > H6.
rewrite > H8. *)
-auto paramodulation.
+autobatch paramodulation.
unfold prime in H. elim H. assumption.
qed.
apply exp_n_SO.
qed.
-(* spostare *)
-theorem le_plus_to_le:
-\forall a,n,m. a + n \le a + m \to n \le m.
-intro.
-elim a
- [assumption
- |apply H.
- apply le_S_S_to_le.assumption
- ]
-qed.
-
-theorem le_times_to_le:
-\forall a,n,m. O < a \to a * n \le a * m \to n \le m.
-intro.
-apply nat_elim2;intros
- [apply le_O_n
- |apply False_ind.
- rewrite < times_n_O in H1.
- generalize in match H1.
- apply (lt_O_n_elim ? H).
- intros.
- simplify in H2.
- apply (le_to_not_lt ? ? H2).
- apply lt_O_S
- |apply le_S_S.
- apply H
- [assumption
- |rewrite < times_n_Sm in H2.
- rewrite < times_n_Sm in H2.
- apply (le_plus_to_le a).
- assumption
- ]
- ]
-qed.
-
-theorem le_exp_to_le:
-\forall a,n,m. S O < a \to exp a n \le exp a m \to n \le m.
-intro.
-apply nat_elim2;intros
- [apply le_O_n
- |apply False_ind.
- apply (le_to_not_lt ? ? H1).
- simplify.
- rewrite > times_n_SO.
- apply lt_to_le_to_lt_times
- [assumption
- |apply lt_O_exp.apply lt_to_le.assumption
- |apply lt_O_exp.apply lt_to_le.assumption
- ]
- |simplify in H2.
- apply le_S_S.
- apply H
- [assumption
- |apply (le_times_to_le a)
- [apply lt_to_le.assumption|assumption]
- ]
- ]
-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
apply (lt_to_not_eq O ? H).
rewrite > H7.
rewrite < H10.
- auto
+ autobatch
]
|elim c
[rewrite > sym_gcd.
|apply lt_O_exp.apply lt_to_le.assumption
|rewrite > sym_gcd.
(* hint non trova prime_to_gcd_SO e
- auto non chiude il goal *)
+ autobatch non chiude il goal *)
apply prime_to_gcd_SO
[assumption|assumption]
|assumption
|apply lt_O_exp.apply lt_to_le.assumption
|rewrite > sym_gcd.
(* hint non trova prime_to_gcd_SO e
- auto non chiude il goal *)
+ autobatch non chiude il goal *)
apply prime_to_gcd_SO
[assumption|assumption]
|rewrite > sym_gcd. assumption
]
|elim H2.assumption
]
-qed.
+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).
|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.