src/HOL/Old_Number_Theory/Chinese.thy
author blanchet
Thu Sep 11 18:54:36 2014 +0200 (2014-09-11)
changeset 58306 117ba6cbe414
parent 57514 bdc2c6b40bf2
child 58889 5b7a9633cfa8
permissions -rw-r--r--
renamed 'rep_datatype' to 'old_rep_datatype' (HOL)
     1 (*  Title:      HOL/Old_Number_Theory/Chinese.thy
     2     Author:     Thomas M. Rasmussen
     3     Copyright   2000  University of Cambridge
     4 *)
     5 
     6 header {* The Chinese Remainder Theorem *}
     7 
     8 theory Chinese 
     9 imports IntPrimes
    10 begin
    11 
    12 text {*
    13   The Chinese Remainder Theorem for an arbitrary finite number of
    14   equations.  (The one-equation case is included in theory @{text
    15   IntPrimes}.  Uses functions for indexing.\footnote{Maybe @{term
    16   funprod} and @{term funsum} should be based on general @{term fold}
    17   on indices?}
    18 *}
    19 
    20 
    21 subsection {* Definitions *}
    22 
    23 primrec funprod :: "(nat => int) => nat => nat => int"
    24 where
    25   "funprod f i 0 = f i"
    26 | "funprod f i (Suc n) = f (Suc (i + n)) * funprod f i n"
    27 
    28 primrec funsum :: "(nat => int) => nat => nat => int"
    29 where
    30   "funsum f i 0 = f i"
    31 | "funsum f i (Suc n) = f (Suc (i + n)) + funsum f i n"
    32 
    33 definition
    34   m_cond :: "nat => (nat => int) => bool" where
    35   "m_cond n mf =
    36     ((\<forall>i. i \<le> n --> 0 < mf i) \<and>
    37       (\<forall>i j. i \<le> n \<and> j \<le> n \<and> i \<noteq> j --> zgcd (mf i) (mf j) = 1))"
    38 
    39 definition
    40   km_cond :: "nat => (nat => int) => (nat => int) => bool" where
    41   "km_cond n kf mf = (\<forall>i. i \<le> n --> zgcd (kf i) (mf i) = 1)"
    42 
    43 definition
    44   lincong_sol ::
    45     "nat => (nat => int) => (nat => int) => (nat => int) => int => bool" where
    46   "lincong_sol n kf bf mf x = (\<forall>i. i \<le> n --> zcong (kf i * x) (bf i) (mf i))"
    47 
    48 definition
    49   mhf :: "(nat => int) => nat => nat => int" where
    50   "mhf mf n i =
    51     (if i = 0 then funprod mf (Suc 0) (n - Suc 0)
    52      else if i = n then funprod mf 0 (n - Suc 0)
    53      else funprod mf 0 (i - Suc 0) * funprod mf (Suc i) (n - Suc 0 - i))"
    54 
    55 definition
    56   xilin_sol ::
    57     "nat => nat => (nat => int) => (nat => int) => (nat => int) => int" where
    58   "xilin_sol i n kf bf mf =
    59     (if 0 < n \<and> i \<le> n \<and> m_cond n mf \<and> km_cond n kf mf then
    60         (SOME x. 0 \<le> x \<and> x < mf i \<and> zcong (kf i * mhf mf n i * x) (bf i) (mf i))
    61      else 0)"
    62 
    63 definition
    64   x_sol :: "nat => (nat => int) => (nat => int) => (nat => int) => int" where
    65   "x_sol n kf bf mf = funsum (\<lambda>i. xilin_sol i n kf bf mf * mhf mf n i) 0 n"
    66 
    67 
    68 text {* \medskip @{term funprod} and @{term funsum} *}
    69 
    70 lemma funprod_pos: "(\<forall>i. i \<le> n --> 0 < mf i) ==> 0 < funprod mf 0 n"
    71 by (induct n) auto
    72 
    73 lemma funprod_zgcd [rule_format (no_asm)]:
    74   "(\<forall>i. k \<le> i \<and> i \<le> k + l --> zgcd (mf i) (mf m) = 1) -->
    75     zgcd (funprod mf k l) (mf m) = 1"
    76   apply (induct l)
    77    apply simp_all
    78   apply (rule impI)+
    79   apply (subst zgcd_zmult_cancel)
    80   apply auto
    81   done
    82 
    83 lemma funprod_zdvd [rule_format]:
    84     "k \<le> i --> i \<le> k + l --> mf i dvd funprod mf k l"
    85   apply (induct l)
    86    apply auto
    87   apply (subgoal_tac "i = Suc (k + l)")
    88    apply (simp_all (no_asm_simp))
    89   done
    90 
    91 lemma funsum_mod:
    92     "funsum f k l mod m = funsum (\<lambda>i. (f i) mod m) k l mod m"
    93   apply (induct l)
    94    apply auto
    95   apply (rule trans)
    96    apply (rule mod_add_eq)
    97   apply simp
    98   apply (rule mod_add_right_eq [symmetric])
    99   done
   100 
   101 lemma funsum_zero [rule_format (no_asm)]:
   102     "(\<forall>i. k \<le> i \<and> i \<le> k + l --> f i = 0) --> (funsum f k l) = 0"
   103   apply (induct l)
   104    apply auto
   105   done
   106 
   107 lemma funsum_oneelem [rule_format (no_asm)]:
   108   "k \<le> j --> j \<le> k + l -->
   109     (\<forall>i. k \<le> i \<and> i \<le> k + l \<and> i \<noteq> j --> f i = 0) -->
   110     funsum f k l = f j"
   111   apply (induct l)
   112    prefer 2
   113    apply clarify
   114    defer
   115    apply clarify
   116    apply (subgoal_tac "k = j")
   117     apply (simp_all (no_asm_simp))
   118   apply (case_tac "Suc (k + l) = j")
   119    apply (subgoal_tac "funsum f k l = 0")
   120     apply (rule_tac [2] funsum_zero)
   121     apply (subgoal_tac [3] "f (Suc (k + l)) = 0")
   122      apply (subgoal_tac [3] "j \<le> k + l")
   123       prefer 4
   124       apply arith
   125      apply auto
   126   done
   127 
   128 
   129 subsection {* Chinese: uniqueness *}
   130 
   131 lemma zcong_funprod_aux:
   132   "m_cond n mf ==> km_cond n kf mf
   133     ==> lincong_sol n kf bf mf x ==> lincong_sol n kf bf mf y
   134     ==> [x = y] (mod mf n)"
   135   apply (unfold m_cond_def km_cond_def lincong_sol_def)
   136   apply (rule iffD1)
   137    apply (rule_tac k = "kf n" in zcong_cancel2)
   138     apply (rule_tac [3] b = "bf n" in zcong_trans)
   139      prefer 4
   140      apply (subst zcong_sym)
   141      defer
   142      apply (rule order_less_imp_le)
   143      apply simp_all
   144   done
   145 
   146 lemma zcong_funprod [rule_format]:
   147   "m_cond n mf --> km_cond n kf mf -->
   148     lincong_sol n kf bf mf x --> lincong_sol n kf bf mf y -->
   149     [x = y] (mod funprod mf 0 n)"
   150   apply (induct n)
   151    apply (simp_all (no_asm))
   152    apply (blast intro: zcong_funprod_aux)
   153   apply (rule impI)+
   154   apply (rule zcong_zgcd_zmult_zmod)
   155     apply (blast intro: zcong_funprod_aux)
   156     prefer 2
   157     apply (subst zgcd_commute)
   158     apply (rule funprod_zgcd)
   159    apply (auto simp add: m_cond_def km_cond_def lincong_sol_def)
   160   done
   161 
   162 
   163 subsection {* Chinese: existence *}
   164 
   165 lemma unique_xi_sol:
   166   "0 < n ==> i \<le> n ==> m_cond n mf ==> km_cond n kf mf
   167     ==> \<exists>!x. 0 \<le> x \<and> x < mf i \<and> [kf i * mhf mf n i * x = bf i] (mod mf i)"
   168   apply (rule zcong_lineq_unique)
   169    apply (tactic {* stac @{thm zgcd_zmult_cancel} 2 *})
   170     apply (unfold m_cond_def km_cond_def mhf_def)
   171     apply (simp_all (no_asm_simp))
   172   apply safe
   173     apply (tactic {* stac @{thm zgcd_zmult_cancel} 3 *})
   174      apply (rule_tac [!] funprod_zgcd)
   175      apply safe
   176      apply simp_all
   177    apply (subgoal_tac "ia<n")
   178     prefer 2
   179     apply arith
   180    apply (case_tac [2] i)
   181     apply simp_all
   182   done
   183 
   184 lemma x_sol_lin_aux:
   185     "0 < n ==> i \<le> n ==> j \<le> n ==> j \<noteq> i ==> mf j dvd mhf mf n i"
   186   apply (unfold mhf_def)
   187   apply (case_tac "i = 0")
   188    apply (case_tac [2] "i = n")
   189     apply (simp_all (no_asm_simp))
   190     apply (case_tac [3] "j < i")
   191      apply (rule_tac [3] dvd_mult2)
   192      apply (rule_tac [4] dvd_mult)
   193      apply (rule_tac [!] funprod_zdvd)
   194      apply arith
   195      apply arith
   196      apply arith
   197      apply arith
   198      apply arith
   199      apply arith
   200      apply arith
   201      apply arith
   202   done
   203 
   204 lemma x_sol_lin:
   205   "0 < n ==> i \<le> n
   206     ==> x_sol n kf bf mf mod mf i =
   207       xilin_sol i n kf bf mf * mhf mf n i mod mf i"
   208   apply (unfold x_sol_def)
   209   apply (subst funsum_mod)
   210   apply (subst funsum_oneelem)
   211      apply auto
   212   apply (subst dvd_eq_mod_eq_0 [symmetric])
   213   apply (rule dvd_mult)
   214   apply (rule x_sol_lin_aux)
   215   apply auto
   216   done
   217 
   218 
   219 subsection {* Chinese *}
   220 
   221 lemma chinese_remainder:
   222   "0 < n ==> m_cond n mf ==> km_cond n kf mf
   223     ==> \<exists>!x. 0 \<le> x \<and> x < funprod mf 0 n \<and> lincong_sol n kf bf mf x"
   224   apply safe
   225    apply (rule_tac [2] m = "funprod mf 0 n" in zcong_zless_imp_eq)
   226        apply (rule_tac [6] zcong_funprod)
   227           apply auto
   228   apply (rule_tac x = "x_sol n kf bf mf mod funprod mf 0 n" in exI)
   229   apply (unfold lincong_sol_def)
   230   apply safe
   231     apply (tactic {* stac @{thm zcong_zmod} 3 *})
   232     apply (tactic {* stac @{thm mod_mult_eq} 3 *})
   233     apply (tactic {* stac @{thm mod_mod_cancel} 3 *})
   234       apply (tactic {* stac @{thm x_sol_lin} 4 *})
   235         apply (tactic {* stac (@{thm mod_mult_eq} RS sym) 6 *})
   236         apply (tactic {* stac (@{thm zcong_zmod} RS sym) 6 *})
   237         apply (subgoal_tac [6]
   238           "0 \<le> xilin_sol i n kf bf mf \<and> xilin_sol i n kf bf mf < mf i
   239           \<and> [kf i * mhf mf n i * xilin_sol i n kf bf mf = bf i] (mod mf i)")
   240          prefer 6
   241          apply (simp add: ac_simps)
   242         apply (unfold xilin_sol_def)
   243         apply (tactic {* asm_simp_tac @{context} 6 *})
   244         apply (rule_tac [6] ex1_implies_ex [THEN someI_ex])
   245         apply (rule_tac [6] unique_xi_sol)
   246            apply (rule_tac [3] funprod_zdvd)
   247             apply (unfold m_cond_def)
   248             apply (rule funprod_pos [THEN pos_mod_sign])
   249             apply (rule_tac [2] funprod_pos [THEN pos_mod_bound])
   250             apply auto
   251   done
   252 
   253 end