src/HOL/NumberTheory/Chinese.ML
author paulson
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(*  Title:	Chinese.ML
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    ID:         $Id$
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    Author:	Thomas M. Rasmussen
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    Copyright	2000  University of Cambridge
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The Chinese Remainder Theorem for an arbitrary finite number of equations. 
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(The one-equation case is included in 'IntPrimes')
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Uses functions for indexing. Maybe 'funprod' and 'funsum'
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should be based on general 'fold' on indices?
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*)
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(*** funprod and funsum ***)
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Goal "(ALL i. i <= n --> #0 < mf i) --> #0 < funprod mf 0 n";
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by (induct_tac "n" 1);
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by Auto_tac;
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by (asm_full_simp_tac (simpset() addsimps [int_0_less_mult_iff]) 1);
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qed_spec_mp "funprod_pos";
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Goal "(ALL i. k<=i & i<=(k+l) --> zgcd (mf i, mf m) = #1) --> \
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\     zgcd (funprod mf k l, mf m) = #1";
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by (induct_tac "l" 1);
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by (ALLGOALS Simp_tac);
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by (REPEAT (rtac impI 1));
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by (stac zgcd_zmult_cancel 1);
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by Auto_tac;
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qed_spec_mp "funprod_zgcd";
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Goal "k<=i --> i<=(k+l) --> (mf i) dvd (funprod mf k l)";     
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by (induct_tac "l" 1);
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by Auto_tac;
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by (rtac zdvd_zmult2 2);
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by (rtac zdvd_zmult 3);
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by (subgoal_tac "i=k" 1);
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by (subgoal_tac "i=Suc (k + n)" 3);
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by (ALLGOALS Asm_simp_tac);
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qed_spec_mp "funprod_zdvd";
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Goal "(funsum f k l) mod m = (funsum (%i. (f i) mod m) k l) mod m";
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by (induct_tac "l" 1);
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by Auto_tac;
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by (rtac trans 1);
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by (rtac zmod_zadd1_eq 1);
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by (Asm_simp_tac 1);
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by (rtac (zmod_zadd_right_eq RS sym) 1);
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qed "funsum_mod";
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Goal "(ALL i. k<=i & i<=(k+l) --> (f i) = #0) --> (funsum f k l) = #0";
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by (induct_tac "l" 1);
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by Auto_tac;
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qed_spec_mp "funsum_zero";
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Goal "k<=j --> j<=(k+l) --> \
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\     (ALL i. k<=i & i<=(k+l) & i~=j --> (f i) = #0) --> \
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\     (funsum f k l) = (f j)";
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by (induct_tac "l" 1);
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by (ALLGOALS Clarify_tac);
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by (subgoal_tac "k=j" 1 THEN ALLGOALS Asm_simp_tac);
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by (case_tac "Suc (k+n) = j" 1);
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by (subgoal_tac "funsum f k n = #0" 1);
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by (rtac funsum_zero 2);
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by (subgoal_tac "f (Suc (k+n)) = #0" 3);
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by (subgoal_tac "j<=k+n" 3);
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by (arith_tac 4);
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by Auto_tac;
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qed_spec_mp "funsum_oneelem";
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4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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(*** Chinese: Uniqueness ***)
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Goalw [m_cond_def,km_cond_def,lincong_sol_def]
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      "[| m_cond n mf; km_cond n kf mf; \
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\         lincong_sol n kf bf mf x; lincong_sol n kf bf mf y |] \
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\     ==>  [x=y] (mod mf n)";
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by (rtac iffD1 1);
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by (res_inst_tac [("k","kf n")] zcong_cancel2 1);
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by (res_inst_tac [("b","bf n")] zcong_trans 3);
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by (stac zcong_sym 4);
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by (rtac order_less_imp_le 1);
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by (ALLGOALS Asm_simp_tac);
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val lemma = result();
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Goal "m_cond n mf --> km_cond n kf mf --> \
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\     lincong_sol n kf bf mf x --> lincong_sol n kf bf mf y --> \
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\     [x=y] (mod funprod mf 0 n)";
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by (induct_tac "n" 1);
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by (ALLGOALS Simp_tac);
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by (blast_tac (claset() addIs [lemma]) 1);
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by (REPEAT (rtac impI 1));
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by (rtac zcong_zgcd_zmult_zmod 1);
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by (blast_tac (claset() addIs [lemma]) 1);
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by (stac zgcd_commute 2);
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by (rtac funprod_zgcd 2);
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by (auto_tac (claset(), 
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              simpset() addsimps [m_cond_def,km_cond_def,lincong_sol_def]));  
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qed_spec_mp "zcong_funprod";
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(* Chinese: Existence *)
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Goal "[| 0<n; i<=n; m_cond n mf; km_cond n kf mf |] \
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\     ==> EX! x. #0<=x & x<(mf i) & \
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\                [(kf i)*(mhf mf n i)*x = bf i] (mod mf i)";
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by (rtac zcong_lineq_unique 1);
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by (stac zgcd_zmult_cancel 2);
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by (rewrite_goals_tac [m_cond_def,km_cond_def,mhf_def]);
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by (ALLGOALS Asm_simp_tac);
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by Safe_tac; 
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by (stac zgcd_zmult_cancel 3);
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by (ALLGOALS (rtac funprod_zgcd));
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by Safe_tac;
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by (ALLGOALS Asm_full_simp_tac);
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by (subgoal_tac "ia<=n" 3);
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by (arith_tac 4);
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by (subgoal_tac "i<n" 1);
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by (arith_tac 2);
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by (case_tac "i" 2);
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by (ALLGOALS Asm_full_simp_tac);
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qed "unique_xi_sol";
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Goalw [mhf_def] "[| 0<n; i<=n; j<=n; j~=i |] ==> (mf j) dvd (mhf mf n i)";
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by (case_tac "i=0" 1);
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by (case_tac "i=n" 2);
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by (ALLGOALS Asm_simp_tac);
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by (case_tac "j<i" 3);
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by (rtac zdvd_zmult2 3);
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by (rtac zdvd_zmult 4);
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by (ALLGOALS (rtac funprod_zdvd));
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by (ALLGOALS arith_tac);
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val lemma = result();
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74817560a8e8 tidied using arith_tac
paulson
parents: 10658
diff changeset
   134
Goalw [x_sol_def]
74817560a8e8 tidied using arith_tac
paulson
parents: 10658
diff changeset
   135
     "[| 0<n; i<=n |] \
9508
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   136
\     ==> (x_sol n kf bf mf) mod (mf i) = \
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   137
\         (xilin_sol i n kf bf mf)*(mhf mf n i) mod (mf i)";
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   138
by (stac funsum_mod 1);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   139
by (stac funsum_oneelem 1);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   140
by Auto_tac;
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   141
by (stac (zdvd_iff_zmod_eq_0 RS sym) 1);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   142
by (rtac zdvd_zmult 1);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   143
by (rtac lemma 1);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   144
by Auto_tac;
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   145
qed "x_sol_lin";
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   146
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   147
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   148
(* Chinese *)
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   149
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   150
Goal "[| 0<n; m_cond n mf; km_cond n kf mf |] \
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   151
\     ==> (EX! x. #0 <= x & x < (funprod mf 0 n) & \
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   152
\                 (lincong_sol n kf bf mf x))";
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   153
by Safe_tac;
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   154
by (res_inst_tac [("m","funprod mf 0 n")] zcong_zless_imp_eq 2);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   155
by (rtac zcong_funprod 6);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   156
by Auto_tac;
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   157
by (res_inst_tac [("x","(x_sol n kf bf mf) mod (funprod mf 0 n)")] exI 1);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   158
by (rewtac lincong_sol_def);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   159
by Safe_tac;
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   160
by (stac zcong_zmod 3);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   161
by (stac zmod_zmult_distrib 3);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   162
by (stac zmod_zdvd_zmod 3);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   163
by (stac x_sol_lin 5);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   164
by (stac (zmod_zmult_distrib RS sym) 7);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   165
by (stac (zcong_zmod RS sym) 7);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   166
by (subgoal_tac "#0<=(xilin_sol i n kf bf mf) & \
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   167
\                (xilin_sol i n kf bf mf)<(mf i) & \
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   168
\                [(kf i)*(mhf mf n i)*(xilin_sol i n kf bf mf) = bf i] \
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   169
\                  (mod mf i)" 7);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   170
by (asm_full_simp_tac (simpset() addsimps zmult_ac) 7);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   171
by (rewtac xilin_sol_def);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   172
by (Asm_simp_tac 7);
10175
76646fc8b1bf ex_someI -> someI_ex
nipkow
parents: 9943
diff changeset
   173
by (rtac (ex1_implies_ex RS someI_ex) 7);
9508
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   174
by (rtac unique_xi_sol 7);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   175
by (rtac funprod_zdvd 4);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   176
by (rewtac m_cond_def);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   177
by (rtac (funprod_pos RS pos_mod_sign) 1);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   178
by (rtac (funprod_pos RS pos_mod_bound) 2);
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   179
by Auto_tac;
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   180
qed "chinese_remainder";