src/HOL/NumberTheory/Chinese.thy
author paulson
Tue Jun 28 15:27:45 2005 +0200 (2005-06-28)
changeset 16587 b34c8aa657a5
parent 16417 9bc16273c2d4
child 19670 2e4a143c73c5
permissions -rw-r--r--
Constant "If" is now local
     1 (*  Title:      HOL/NumberTheory/Chinese.thy
     2     ID:         $Id$
     3     Author:     Thomas M. Rasmussen
     4     Copyright   2000  University of Cambridge
     5 *)
     6 
     7 header {* The Chinese Remainder Theorem *}
     8 
     9 theory Chinese imports IntPrimes begin
    10 
    11 text {*
    12   The Chinese Remainder Theorem for an arbitrary finite number of
    13   equations.  (The one-equation case is included in theory @{text
    14   IntPrimes}.  Uses functions for indexing.\footnote{Maybe @{term
    15   funprod} and @{term funsum} should be based on general @{term fold}
    16   on indices?}
    17 *}
    18 
    19 
    20 subsection {* Definitions *}
    21 
    22 consts
    23   funprod :: "(nat => int) => nat => nat => int"
    24   funsum :: "(nat => int) => nat => nat => int"
    25 
    26 primrec
    27   "funprod f i 0 = f i"
    28   "funprod f i (Suc n) = f (Suc (i + n)) * funprod f i n"
    29 
    30 primrec
    31   "funsum f i 0 = f i"
    32   "funsum f i (Suc n) = f (Suc (i + n)) + funsum f i n"
    33 
    34 consts
    35   m_cond :: "nat => (nat => int) => bool"
    36   km_cond :: "nat => (nat => int) => (nat => int) => bool"
    37   lincong_sol ::
    38     "nat => (nat => int) => (nat => int) => (nat => int) => int => bool"
    39 
    40   mhf :: "(nat => int) => nat => nat => int"
    41   xilin_sol ::
    42     "nat => nat => (nat => int) => (nat => int) => (nat => int) => int"
    43   x_sol :: "nat => (nat => int) => (nat => int) => (nat => int) => int"
    44 
    45 defs
    46   m_cond_def:
    47     "m_cond n mf ==
    48       (\<forall>i. i \<le> n --> 0 < mf i) \<and>
    49       (\<forall>i j. i \<le> n \<and> j \<le> n \<and> i \<noteq> j --> zgcd (mf i, mf j) = 1)"
    50 
    51   km_cond_def:
    52     "km_cond n kf mf == \<forall>i. i \<le> n --> zgcd (kf i, mf i) = 1"
    53 
    54   lincong_sol_def:
    55     "lincong_sol n kf bf mf x == \<forall>i. i \<le> n --> zcong (kf i * x) (bf i) (mf i)"
    56 
    57   mhf_def:
    58     "mhf mf n i ==
    59       if i = 0 then funprod mf (Suc 0) (n - Suc 0)
    60       else if i = n then funprod mf 0 (n - Suc 0)
    61       else funprod mf 0 (i - Suc 0) * funprod mf (Suc i) (n - Suc 0 - i)"
    62 
    63   xilin_sol_def:
    64     "xilin_sol i n kf bf mf ==
    65       if 0 < n \<and> i \<le> n \<and> m_cond n mf \<and> km_cond n kf mf then
    66         (SOME x. 0 \<le> x \<and> x < mf i \<and> zcong (kf i * mhf mf n i * x) (bf i) (mf i))
    67       else 0"
    68 
    69   x_sol_def:
    70     "x_sol n kf bf mf == funsum (\<lambda>i. xilin_sol i n kf bf mf * mhf mf n i) 0 n"
    71 
    72 
    73 text {* \medskip @{term funprod} and @{term funsum} *}
    74 
    75 lemma funprod_pos: "(\<forall>i. i \<le> n --> 0 < mf i) ==> 0 < funprod mf 0 n"
    76   apply (induct n)
    77    apply auto
    78   apply (simp add: zero_less_mult_iff)
    79   done
    80 
    81 lemma funprod_zgcd [rule_format (no_asm)]:
    82   "(\<forall>i. k \<le> i \<and> i \<le> k + l --> zgcd (mf i, mf m) = 1) -->
    83     zgcd (funprod mf k l, mf m) = 1"
    84   apply (induct l)
    85    apply simp_all
    86   apply (rule impI)+
    87   apply (subst zgcd_zmult_cancel)
    88   apply auto
    89   done
    90 
    91 lemma funprod_zdvd [rule_format]:
    92     "k \<le> i --> i \<le> k + l --> mf i dvd funprod mf k l"
    93   apply (induct l)
    94    apply auto
    95     apply (rule_tac [1] zdvd_zmult2)
    96     apply (rule_tac [2] zdvd_zmult)
    97     apply (subgoal_tac "i = Suc (k + l)")
    98     apply (simp_all (no_asm_simp))
    99   done
   100 
   101 lemma funsum_mod:
   102     "funsum f k l mod m = funsum (\<lambda>i. (f i) mod m) k l mod m"
   103   apply (induct l)
   104    apply auto
   105   apply (rule trans)
   106    apply (rule zmod_zadd1_eq)
   107   apply simp
   108   apply (rule zmod_zadd_right_eq [symmetric])
   109   done
   110 
   111 lemma funsum_zero [rule_format (no_asm)]:
   112     "(\<forall>i. k \<le> i \<and> i \<le> k + l --> f i = 0) --> (funsum f k l) = 0"
   113   apply (induct l)
   114    apply auto
   115   done
   116 
   117 lemma funsum_oneelem [rule_format (no_asm)]:
   118   "k \<le> j --> j \<le> k + l -->
   119     (\<forall>i. k \<le> i \<and> i \<le> k + l \<and> i \<noteq> j --> f i = 0) -->
   120     funsum f k l = f j"
   121   apply (induct l)
   122    prefer 2
   123    apply clarify
   124    defer
   125    apply clarify
   126    apply (subgoal_tac "k = j")
   127     apply (simp_all (no_asm_simp))
   128   apply (case_tac "Suc (k + l) = j")
   129    apply (subgoal_tac "funsum f k l = 0")
   130     apply (rule_tac [2] funsum_zero)
   131     apply (subgoal_tac [3] "f (Suc (k + l)) = 0")
   132      apply (subgoal_tac [3] "j \<le> k + l")
   133       prefer 4
   134       apply arith
   135      apply auto
   136   done
   137 
   138 
   139 subsection {* Chinese: uniqueness *}
   140 
   141 lemma zcong_funprod_aux:
   142   "m_cond n mf ==> km_cond n kf mf
   143     ==> lincong_sol n kf bf mf x ==> lincong_sol n kf bf mf y
   144     ==> [x = y] (mod mf n)"
   145   apply (unfold m_cond_def km_cond_def lincong_sol_def)
   146   apply (rule iffD1)
   147    apply (rule_tac k = "kf n" in zcong_cancel2)
   148     apply (rule_tac [3] b = "bf n" in zcong_trans)
   149      prefer 4
   150      apply (subst zcong_sym)
   151      defer
   152      apply (rule order_less_imp_le)
   153      apply simp_all
   154   done
   155 
   156 lemma zcong_funprod [rule_format]:
   157   "m_cond n mf --> km_cond n kf mf -->
   158     lincong_sol n kf bf mf x --> lincong_sol n kf bf mf y -->
   159     [x = y] (mod funprod mf 0 n)"
   160   apply (induct n)
   161    apply (simp_all (no_asm))
   162    apply (blast intro: zcong_funprod_aux)
   163   apply (rule impI)+
   164   apply (rule zcong_zgcd_zmult_zmod)
   165     apply (blast intro: zcong_funprod_aux)
   166     prefer 2
   167     apply (subst zgcd_commute)
   168     apply (rule funprod_zgcd)
   169    apply (auto simp add: m_cond_def km_cond_def lincong_sol_def)
   170   done
   171 
   172 
   173 subsection {* Chinese: existence *}
   174 
   175 lemma unique_xi_sol:
   176   "0 < n ==> i \<le> n ==> m_cond n mf ==> km_cond n kf mf
   177     ==> \<exists>!x. 0 \<le> x \<and> x < mf i \<and> [kf i * mhf mf n i * x = bf i] (mod mf i)"
   178   apply (rule zcong_lineq_unique)
   179    apply (tactic {* stac (thm "zgcd_zmult_cancel") 2 *})
   180     apply (unfold m_cond_def km_cond_def mhf_def)
   181     apply (simp_all (no_asm_simp))
   182   apply safe
   183     apply (tactic {* stac (thm "zgcd_zmult_cancel") 3 *})
   184      apply (rule_tac [!] funprod_zgcd)
   185      apply safe
   186      apply simp_all
   187    apply (subgoal_tac "i<n")
   188     prefer 2
   189     apply arith
   190    apply (case_tac [2] i)
   191     apply simp_all
   192   done
   193 
   194 lemma x_sol_lin_aux:
   195     "0 < n ==> i \<le> n ==> j \<le> n ==> j \<noteq> i ==> mf j dvd mhf mf n i"
   196   apply (unfold mhf_def)
   197   apply (case_tac "i = 0")
   198    apply (case_tac [2] "i = n")
   199     apply (simp_all (no_asm_simp))
   200     apply (case_tac [3] "j < i")
   201      apply (rule_tac [3] zdvd_zmult2)
   202      apply (rule_tac [4] zdvd_zmult)
   203      apply (rule_tac [!] funprod_zdvd)
   204           apply arith+
   205   done
   206 
   207 lemma x_sol_lin:
   208   "0 < n ==> i \<le> n
   209     ==> x_sol n kf bf mf mod mf i =
   210       xilin_sol i n kf bf mf * mhf mf n i mod mf i"
   211   apply (unfold x_sol_def)
   212   apply (subst funsum_mod)
   213   apply (subst funsum_oneelem)
   214      apply auto
   215   apply (subst zdvd_iff_zmod_eq_0 [symmetric])
   216   apply (rule zdvd_zmult)
   217   apply (rule x_sol_lin_aux)
   218   apply auto
   219   done
   220 
   221 
   222 subsection {* Chinese *}
   223 
   224 lemma chinese_remainder:
   225   "0 < n ==> m_cond n mf ==> km_cond n kf mf
   226     ==> \<exists>!x. 0 \<le> x \<and> x < funprod mf 0 n \<and> lincong_sol n kf bf mf x"
   227   apply safe
   228    apply (rule_tac [2] m = "funprod mf 0 n" in zcong_zless_imp_eq)
   229        apply (rule_tac [6] zcong_funprod)
   230           apply auto
   231   apply (rule_tac x = "x_sol n kf bf mf mod funprod mf 0 n" in exI)
   232   apply (unfold lincong_sol_def)
   233   apply safe
   234     apply (tactic {* stac (thm "zcong_zmod") 3 *})
   235     apply (tactic {* stac (thm "zmod_zmult_distrib") 3 *})
   236     apply (tactic {* stac (thm "zmod_zdvd_zmod") 3 *})
   237       apply (tactic {* stac (thm "x_sol_lin") 5 *})
   238         apply (tactic {* stac (thm "zmod_zmult_distrib" RS sym) 7 *})
   239         apply (tactic {* stac (thm "zcong_zmod" RS sym) 7 *})
   240         apply (subgoal_tac [7]
   241           "0 \<le> xilin_sol i n kf bf mf \<and> xilin_sol i n kf bf mf < mf i
   242           \<and> [kf i * mhf mf n i * xilin_sol i n kf bf mf = bf i] (mod mf i)")
   243          prefer 7
   244          apply (simp add: zmult_ac)
   245         apply (unfold xilin_sol_def)
   246         apply (tactic {* Asm_simp_tac 7 *})
   247         apply (rule_tac [7] ex1_implies_ex [THEN someI_ex])
   248         apply (rule_tac [7] unique_xi_sol)
   249            apply (rule_tac [4] funprod_zdvd)
   250             apply (unfold m_cond_def)
   251             apply (rule funprod_pos [THEN pos_mod_sign])
   252             apply (rule_tac [2] funprod_pos [THEN pos_mod_bound])
   253             apply auto
   254   done
   255 
   256 end