--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/library_autobatch/nat/chinese_reminder".
+
+include "auto/nat/exp.ma".
+include "auto/nat/gcd.ma".
+include "auto/nat/permutation.ma".
+include "auto/nat/congruence.ma".
+
+theorem and_congruent_congruent: \forall m,n,a,b:nat. O < n \to O < m \to
+gcd n m = (S O) \to ex nat (\lambda x. congruent x a m \land congruent x b n).
+intros.
+cut (\exists c,d.c*n - d*m = (S O) \lor d*m - c*n = (S O))
+[ elim Hcut.
+ elim H3.
+ elim H4
+ [
+ apply (ex_intro nat ? ((a+b*m)*a1*n-b*a2*m)).
+ split
+ [ (* congruent to a *)
+ cut (a1*n = a2*m + (S O))
+ [ rewrite > assoc_times.
+ rewrite > Hcut1.
+ rewrite < (sym_plus ? (a2*m)).
+ rewrite > distr_times_plus.
+ rewrite < times_n_SO.
+ rewrite > assoc_plus.
+ rewrite < assoc_times.
+ rewrite < times_plus_l.
+ rewrite > eq_minus_plus_plus_minus
+ [ autobatch
+ (*rewrite < times_minus_l.
+ rewrite > sym_plus.
+ apply (eq_times_plus_to_congruent ? ? ? ((b+(a+b*m)*a2)-b*a2))
+ [ assumption
+ | reflexivity
+ ]*)
+ | apply le_times_l.
+ apply (trans_le ? ((a+b*m)*a2))
+ [ apply le_times_l.
+ apply (trans_le ? (b*m));autobatch
+ (*[ rewrite > times_n_SO in \vdash (? % ?).
+ apply le_times_r.
+ assumption
+ | apply le_plus_n
+ ]*)
+ | apply le_plus_n
+ ]
+ ]
+ | apply minus_to_plus
+ [ apply lt_to_le.
+ change with (O + a2*m < a1*n).
+ apply lt_minus_to_plus.
+ autobatch
+ (*rewrite > H5.
+ unfold lt.
+ apply le_n*)
+ | assumption
+ ]
+ ]
+ | (* congruent to b *)
+ cut (a2*m = a1*n - (S O))
+ [ rewrite > (assoc_times b a2).
+ rewrite > Hcut1.
+ rewrite > distr_times_minus.
+ rewrite < assoc_times.
+ rewrite < eq_plus_minus_minus_minus
+ [ autobatch
+ (*rewrite < times_n_SO.
+ rewrite < times_minus_l.
+ rewrite < sym_plus.
+ apply (eq_times_plus_to_congruent ? ? ? ((a+b*m)*a1-b*a1))
+ [ assumption
+ | reflexivity
+ ]*)
+ | rewrite > assoc_times.
+ apply le_times_r.
+ (*autobatch genera un'esecuzione troppo lunga*)
+ apply (trans_le ? (a1*n - a2*m));autobatch
+ (*[ rewrite > H5.
+ apply le_n
+ | apply (le_minus_m ? (a2*m))
+ ]*)
+ | apply le_times_l.
+ apply le_times_l.
+ autobatch
+ (*apply (trans_le ? (b*m))
+ [ rewrite > times_n_SO in \vdash (? % ?).
+ apply le_times_r.
+ assumption
+ | apply le_plus_n
+ ]*)
+ ]
+ | apply sym_eq.
+ apply plus_to_minus.
+ rewrite > sym_plus.
+ apply minus_to_plus
+ [ apply lt_to_le.
+ change with (O + a2*m < a1*n).
+ apply lt_minus_to_plus.
+ autobatch
+ (*rewrite > H5.
+ unfold lt.
+ apply le_n*)
+ | assumption
+ ]
+ ]
+ ]
+ | (* and now the symmetric case; the price to pay for working
+ in nat instead than Z *)
+ apply (ex_intro nat ? ((b+a*n)*a2*m-a*a1*n)).
+ split
+ [(* congruent to a *)
+ cut (a1*n = a2*m - (S O))
+ [ rewrite > (assoc_times a a1).
+ rewrite > Hcut1.
+ rewrite > distr_times_minus.
+ rewrite < assoc_times.
+ rewrite < eq_plus_minus_minus_minus
+ [ autobatch
+ (*rewrite < times_n_SO.
+ rewrite < times_minus_l.
+ rewrite < sym_plus.
+ apply (eq_times_plus_to_congruent ? ? ? ((b+a*n)*a2-a*a2))
+ [ assumption
+ | reflexivity
+ ]*)
+ | rewrite > assoc_times.
+ apply le_times_r.
+ apply (trans_le ? (a2*m - a1*n));autobatch
+ (*[ rewrite > H5.
+ apply le_n
+ | apply (le_minus_m ? (a1*n))
+ ]*)
+ | rewrite > assoc_times.
+ rewrite > assoc_times.
+ apply le_times_l.
+ autobatch
+ (*apply (trans_le ? (a*n))
+ [ rewrite > times_n_SO in \vdash (? % ?).
+ apply le_times_r.
+ assumption.
+ | apply le_plus_n.
+ ]*)
+ ]
+ | apply sym_eq.
+ apply plus_to_minus.
+ rewrite > sym_plus.
+ apply minus_to_plus
+ [ apply lt_to_le.
+ change with (O + a1*n < a2*m).
+ apply lt_minus_to_plus.
+ autobatch
+ (*rewrite > H5.
+ unfold lt.
+ apply le_n*)
+ | assumption
+ ]
+ ]
+ | (* congruent to b *)
+ cut (a2*m = a1*n + (S O))
+ [ rewrite > assoc_times.
+ rewrite > Hcut1.
+ rewrite > (sym_plus (a1*n)).
+ rewrite > distr_times_plus.
+ rewrite < times_n_SO.
+ rewrite < assoc_times.
+ rewrite > assoc_plus.
+ rewrite < times_plus_l.
+ rewrite > eq_minus_plus_plus_minus
+ [ autobatch
+ (*rewrite < times_minus_l.
+ rewrite > sym_plus.
+ apply (eq_times_plus_to_congruent ? ? ? ((a+(b+a*n)*a1)-a*a1))
+ [ assumption
+ | reflexivity
+ ]*)
+ | apply le_times_l.
+ apply (trans_le ? ((b+a*n)*a1))
+ [ apply le_times_l.
+ autobatch
+ (*apply (trans_le ? (a*n))
+ [ rewrite > times_n_SO in \vdash (? % ?).
+ apply le_times_r.
+ assumption
+ | apply le_plus_n
+ ]*)
+ | apply le_plus_n
+ ]
+ ]
+ | apply minus_to_plus
+ [ apply lt_to_le.
+ change with (O + a1*n < a2*m).
+ apply lt_minus_to_plus.
+ autobatch
+ (*rewrite > H5.
+ unfold lt.
+ apply le_n*)
+ | assumption
+ ]
+ ]
+ ]
+ ]
+(* proof of the cut *)
+| (* qui autobatch non conclude il goal *)
+ rewrite < H2.
+ apply eq_minus_gcd
+]
+qed.
+
+theorem and_congruent_congruent_lt: \forall m,n,a,b:nat. O < n \to O < m \to
+gcd n m = (S O) \to
+ex nat (\lambda x. (congruent x a m \land congruent x b n) \land
+ (x < m*n)).
+intros.
+elim (and_congruent_congruent m n a b)
+[ elim H3.
+ apply (ex_intro ? ? (a1 \mod (m*n))).
+ split
+ [ split
+ [ apply (transitive_congruent m ? a1)
+ [ unfold congruent.
+ apply sym_eq.
+ change with (congruent a1 (a1 \mod (m*n)) m).
+ rewrite < sym_times.
+ autobatch
+ (*apply congruent_n_mod_times;assumption*)
+ | assumption
+ ]
+ | apply (transitive_congruent n ? a1)
+ [ unfold congruent.
+ apply sym_eq.
+ change with (congruent a1 (a1 \mod (m*n)) n).
+ autobatch
+ (*apply congruent_n_mod_times;assumption*)
+ | assumption
+ ]
+ ]
+ | apply lt_mod_m_m.
+ autobatch
+ (*rewrite > (times_n_O O).
+ apply lt_times;assumption*)
+ ]
+| assumption
+| assumption
+| assumption
+]
+qed.
+
+definition cr_pair : nat \to nat \to nat \to nat \to nat \def
+\lambda n,m,a,b.
+min (pred (n*m)) (\lambda x. andb (eqb (x \mod n) a) (eqb (x \mod m) b)).
+
+theorem cr_pair1: cr_pair (S (S O)) (S (S (S O))) O O = O.
+reflexivity.
+qed.
+
+theorem cr_pair2: cr_pair (S(S O)) (S(S(S O))) (S O) O = (S(S(S O))).
+autobatch.
+(*simplify.
+reflexivity.*)
+qed.
+
+theorem cr_pair3: cr_pair (S(S O)) (S(S(S O))) (S O) (S(S O)) = (S(S(S(S(S O))))).
+reflexivity.
+qed.
+
+theorem cr_pair4: cr_pair (S(S(S(S(S O))))) (S(S(S(S(S(S(S O))))))) (S(S(S O))) (S(S O)) =
+(S(S(S(S(S(S(S(S(S(S(S(S(S(S(S(S(S(S(S(S(S(S(S O))))))))))))))))))))))).
+reflexivity.
+qed.
+
+theorem mod_cr_pair : \forall m,n,a,b. a \lt m \to b \lt n \to
+gcd n m = (S O) \to
+(cr_pair m n a b) \mod m = a \land (cr_pair m n a b) \mod n = b.
+intros.
+cut (andb (eqb ((cr_pair m n a b) \mod m) a)
+ (eqb ((cr_pair m n a b) \mod n) b) = true)
+[ generalize in match Hcut.
+ apply andb_elim.
+ apply eqb_elim;intro
+ [ rewrite > H3.
+ simplify.
+ intro.
+ autobatch
+ (*split
+ [ reflexivity
+ | apply eqb_true_to_eq.
+ assumption
+ ]*)
+ | simplify.
+ intro.
+ (* l'invocazione di autobatch qui genera segmentation fault *)
+ apply False_ind.
+ (* l'invocazione di autobatch qui genera segmentation fault *)
+ apply not_eq_true_false.
+ (* l'invocazione di autobatch qui genera segmentation fault *)
+ 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))).
+ elim (and_congruent_congruent_lt m n a b)
+ [ apply (ex_intro ? ? a1).
+ split
+ [ split
+ [ autobatch
+ (*rewrite < minus_n_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.
+ autobatch
+ (*rewrite < pred_Sn.
+ apply le_n*)
+ ]
+ ]
+ ]
+ | elim H3.
+ elim H4.
+ apply andb_elim.
+ cut (a1 \mod m = a)
+ [ cut (a1 \mod n = b)
+ [ rewrite > (eq_to_eqb_true ? ? Hcut).
+ rewrite > (eq_to_eqb_true ? ? Hcut1).
+ (* l'invocazione di autobatch qui non chiude il goal *)
+ simplify.
+ reflexivity
+ | rewrite < (lt_to_eq_mod b n);
+ assumption
+ ]
+ | rewrite < (lt_to_eq_mod a m);assumption
+ ]
+ ]
+ | autobatch
+ (*apply (le_to_lt_to_lt ? b)
+ [ apply le_O_n
+ | assumption
+ ]*)
+ | autobatch
+ (*apply (le_to_lt_to_lt ? a)
+ [ apply le_O_n
+ | assumption
+ ]*)
+ | assumption
+ ]
+]
+qed.
\ No newline at end of file
projects / helm.git / blobdiff