src/HOL/NumberTheory/Chinese.thy
 changeset 11049 7eef34adb852 parent 9508 4d01dbf6ded7 child 11468 02cd5d5bc497
```--- a/src/HOL/NumberTheory/Chinese.thy	Sat Feb 03 17:43:34 2001 +0100
+++ b/src/HOL/NumberTheory/Chinese.thy	Sun Feb 04 19:31:13 2001 +0100
@@ -1,55 +1,260 @@
-(*  Title:	Chinese.thy
+(*  Title:      HOL/NumberTheory/Chinese.thy
ID:         \$Id\$
-    Author:	Thomas M. Rasmussen
-    Copyright	2000  University of Cambridge
+    Author:     Thomas M. Rasmussen
+    Copyright   2000  University of Cambridge
*)

-Chinese = IntPrimes +
+header {* The Chinese Remainder Theorem *}
+
+theory Chinese = IntPrimes:
+
+text {*
+  The Chinese Remainder Theorem for an arbitrary finite number of
+  equations.  (The one-equation case is included in theory @{text
+  IntPrimes}.  Uses functions for indexing.\footnote{Maybe @{term
+  funprod} and @{term funsum} should be based on general @{term fold}
+  on indices?}
+*}
+
+
+subsection {* Definitions *}

consts
-  funprod     :: (nat => int) => nat => nat => int
-  funsum      :: (nat => int) => nat => nat => int
+  funprod :: "(nat => int) => nat => nat => int"
+  funsum :: "(nat => int) => nat => nat => int"

primrec
-  "funprod f i 0        = f i"
-  "funprod f i (Suc n)  = (f (Suc (i+n)))*(funprod f i n)"
+  "funprod f i 0 = f i"
+  "funprod f i (Suc n) = f (Suc (i + n)) * funprod f i n"

primrec
-  "funsum f i 0         = f i"
-  "funsum f i (Suc n)   = (f (Suc (i+n)))+(funsum f i n)"
-
+  "funsum f i 0 = f i"
+  "funsum f i (Suc n) = f (Suc (i + n)) + funsum f i n"

consts
-  m_cond      :: [nat,nat => int] => bool
-  km_cond     :: [nat,nat => int,nat => int] => bool
-  lincong_sol :: [nat,nat => int,nat => int,nat => int,int] => bool
+  m_cond :: "nat => (nat => int) => bool"
+  km_cond :: "nat => (nat => int) => (nat => int) => bool"
+  lincong_sol ::
+    "nat => (nat => int) => (nat => int) => (nat => int) => int => bool"

-  mhf         :: (nat => int) => nat => nat => int
-  xilin_sol   :: [nat,nat,nat => int,nat => int,nat => int] => int
-  x_sol       :: [nat,nat => int,nat => int,nat => int] => int
+  mhf :: "(nat => int) => nat => nat => int"
+  xilin_sol ::
+    "nat => nat => (nat => int) => (nat => int) => (nat => int) => int"
+  x_sol :: "nat => (nat => int) => (nat => int) => (nat => int) => int"

defs
-  m_cond_def   "m_cond n mf ==
-                   (ALL i. i<=n --> #0 < mf i) &
-                   (ALL i j. i<=n & j<=n & i ~= j --> zgcd(mf i,mf j) = #1)"
+  m_cond_def:
+    "m_cond n mf ==
+      (\<forall>i. i \<le> n --> #0 < mf i) \<and>
+      (\<forall>i j. i \<le> n \<and> j \<le> n \<and> i \<noteq> j --> zgcd (mf i, mf j) = #1)"
+
+  km_cond_def:
+    "km_cond n kf mf == \<forall>i. i \<le> n --> zgcd (kf i, mf i) = #1"
+
+  lincong_sol_def:
+    "lincong_sol n kf bf mf x == \<forall>i. i \<le> n --> zcong (kf i * x) (bf i) (mf i)"
+
+  mhf_def:
+    "mhf mf n i ==
+      if i = 0 then funprod mf 1 (n - 1)
+      else if i = n then funprod mf 0 (n - 1)
+      else funprod mf 0 (i - 1) * funprod mf (i + 1) (n - 1 - i)"
+
+  xilin_sol_def:
+    "xilin_sol i n kf bf mf ==
+      if 0 < n \<and> i \<le> n \<and> m_cond n mf \<and> km_cond n kf mf then
+        (SOME x. #0 \<le> x \<and> x < mf i \<and> zcong (kf i * mhf mf n i * x) (bf i) (mf i))
+      else #0"
+
+  x_sol_def:
+    "x_sol n kf bf mf == funsum (\<lambda>i. xilin_sol i n kf bf mf * mhf mf n i) 0 n"
+
+
+text {* \medskip @{term funprod} and @{term funsum} *}
+
+lemma funprod_pos: "(\<forall>i. i \<le> n --> #0 < mf i) ==> #0 < funprod mf 0 n"
+  apply (induct n)
+   apply auto
+  done
+
+lemma funprod_zgcd [rule_format (no_asm)]:
+  "(\<forall>i. k \<le> i \<and> i \<le> k + l --> zgcd (mf i, mf m) = #1) -->
+    zgcd (funprod mf k l, mf m) = #1"
+  apply (induct l)
+   apply simp_all
+  apply (rule impI)+
+  apply (subst zgcd_zmult_cancel)
+  apply auto
+  done

-  km_cond_def  "km_cond n kf mf == (ALL i. i<=n --> zgcd(kf i,mf i) = #1)"
+lemma funprod_zdvd [rule_format]:
+    "k \<le> i --> i \<le> k + l --> mf i dvd funprod mf k l"
+  apply (induct l)
+   apply auto
+    apply (rule_tac [2] zdvd_zmult2)
+    apply (rule_tac [3] zdvd_zmult)
+    apply (subgoal_tac "i = k")
+    apply (subgoal_tac [3] "i = Suc (k + n)")
+    apply (simp_all (no_asm_simp))
+  done
+
+lemma funsum_mod:
+    "funsum f k l mod m = funsum (\<lambda>i. (f i) mod m) k l mod m"
+  apply (induct l)
+   apply auto
+  apply (rule trans)
+  apply simp
+  done

-  lincong_sol_def "lincong_sol n kf bf mf x ==
-                   (ALL i. i<=n --> zcong ((kf i)*x) (bf i) (mf i))"
+lemma funsum_zero [rule_format (no_asm)]:
+    "(\<forall>i. k \<le> i \<and> i \<le> k + l --> f i = #0) --> (funsum f k l) = #0"
+  apply (induct l)
+   apply auto
+  done
+
+lemma funsum_oneelem [rule_format (no_asm)]:
+  "k \<le> j --> j \<le> k + l -->
+    (\<forall>i. k \<le> i \<and> i \<le> k + l \<and> i \<noteq> j --> f i = #0) -->
+    funsum f k l = f j"
+  apply (induct l)
+   prefer 2
+   apply clarify
+   defer
+   apply clarify
+   apply (subgoal_tac "k = j")
+    apply (simp_all (no_asm_simp))
+  apply (case_tac "Suc (k + n) = j")
+   apply (subgoal_tac "funsum f k n = #0")
+    apply (rule_tac [2] funsum_zero)
+    apply (subgoal_tac [3] "f (Suc (k + n)) = #0")
+     apply (subgoal_tac [3] "j \<le> k + n")
+      prefer 4
+      apply arith
+     apply auto
+  done
+
+
+subsection {* Chinese: uniqueness *}

-  mhf_def  "mhf mf n i == (if i=0 then (funprod mf 1 (n-1))
-                           else (if i=n then (funprod mf 0 (n-1))
-                           else ((funprod mf 0 (i-1)) *
-                                 (funprod mf (i+1) (n-1-i)))))"
+lemma aux:
+  "m_cond n mf ==> km_cond n kf mf
+    ==> lincong_sol n kf bf mf x ==> lincong_sol n kf bf mf y
+    ==> [x = y] (mod mf n)"
+  apply (unfold m_cond_def km_cond_def lincong_sol_def)
+  apply (rule iffD1)
+   apply (rule_tac k = "kf n" in zcong_cancel2)
+    apply (rule_tac [3] b = "bf n" in zcong_trans)
+     prefer 4
+     apply (subst zcong_sym)
+     defer
+     apply (rule order_less_imp_le)
+     apply simp_all
+  done
+
+lemma zcong_funprod [rule_format]:
+  "m_cond n mf --> km_cond n kf mf -->
+    lincong_sol n kf bf mf x --> lincong_sol n kf bf mf y -->
+    [x = y] (mod funprod mf 0 n)"
+  apply (induct n)
+   apply (simp_all (no_asm))
+   apply (blast intro: aux)
+  apply (rule impI)+
+  apply (rule zcong_zgcd_zmult_zmod)
+    apply (blast intro: aux)
+    prefer 2
+    apply (subst zgcd_commute)
+    apply (rule funprod_zgcd)
+   apply (auto simp add: m_cond_def km_cond_def lincong_sol_def)
+  done
+
+
+subsection {* Chinese: existence *}
+
+lemma unique_xi_sol:
+  "0 < n ==> i \<le> n ==> m_cond n mf ==> km_cond n kf mf
+    ==> \<exists>!x. #0 \<le> x \<and> x < mf i \<and> [kf i * mhf mf n i * x = bf i] (mod mf i)"
+  apply (rule zcong_lineq_unique)
+   apply (tactic {* stac (thm "zgcd_zmult_cancel") 2 *})
+    apply (unfold m_cond_def km_cond_def mhf_def)
+    apply (simp_all (no_asm_simp))
+  apply safe
+    apply (tactic {* stac (thm "zgcd_zmult_cancel") 3 *})
+     apply (rule_tac [!] funprod_zgcd)
+     apply safe
+     apply simp_all
+    apply (subgoal_tac [3] "ia \<le> n")
+     prefer 4
+     apply arith
+     apply (subgoal_tac "i<n")
+     prefer 2
+     apply arith
+    apply (case_tac [2] i)
+     apply simp_all
+  done

-  xilin_sol_def "xilin_sol i n kf bf mf ==
-                  (if 0<n & i<=n & m_cond n mf & km_cond n kf mf then
-                    (@ x. #0<=x & x<(mf i) &
-                          zcong ((kf i)*(mhf mf n i)*x) (bf i) (mf i))
-                    else #0)"
+lemma aux:
+    "0 < n ==> i \<le> n ==> j \<le> n ==> j \<noteq> i ==> mf j dvd mhf mf n i"
+  apply (unfold mhf_def)
+  apply (case_tac "i = 0")
+   apply (case_tac [2] "i = n")
+    apply (simp_all (no_asm_simp))
+    apply (case_tac [3] "j < i")
+     apply (rule_tac [3] zdvd_zmult2)
+     apply (rule_tac [4] zdvd_zmult)
+     apply (rule_tac [!] funprod_zdvd)
+          apply arith+
+  done
+
+lemma x_sol_lin:
+  "0 < n ==> i \<le> n
+    ==> x_sol n kf bf mf mod mf i =
+      xilin_sol i n kf bf mf * mhf mf n i mod mf i"
+  apply (unfold x_sol_def)
+  apply (subst funsum_mod)
+  apply (subst funsum_oneelem)
+     apply auto
+  apply (subst zdvd_iff_zmod_eq_0 [symmetric])
+  apply (rule zdvd_zmult)
+  apply (rule aux)
+  apply auto
+  done
+
+
+subsection {* Chinese *}

-  x_sol_def "x_sol n kf bf mf ==
-              (funsum (%i. (xilin_sol i n kf bf mf)*(mhf mf n i)) 0 n)"
+lemma chinese_remainder:
+  "0 < n ==> m_cond n mf ==> km_cond n kf mf
+    ==> \<exists>!x. #0 \<le> x \<and> x < funprod mf 0 n \<and> lincong_sol n kf bf mf x"
+  apply safe
+   apply (rule_tac [2] m = "funprod mf 0 n" in zcong_zless_imp_eq)
+       apply (rule_tac [6] zcong_funprod)
+          apply auto
+  apply (rule_tac x = "x_sol n kf bf mf mod funprod mf 0 n" in exI)
+  apply (unfold lincong_sol_def)
+  apply safe
+    apply (tactic {* stac (thm "zcong_zmod") 3 *})
+    apply (tactic {* stac (thm "zmod_zmult_distrib") 3 *})
+    apply (tactic {* stac (thm "zmod_zdvd_zmod") 3 *})
+      apply (tactic {* stac (thm "x_sol_lin") 5 *})
+        apply (tactic {* stac (thm "zmod_zmult_distrib" RS sym) 7 *})
+        apply (tactic {* stac (thm "zcong_zmod" RS sym) 7 *})
+        apply (subgoal_tac [7]
+          "#0 \<le> xilin_sol i n kf bf mf \<and> xilin_sol i n kf bf mf < mf i
+          \<and> [kf i * mhf mf n i * xilin_sol i n kf bf mf = bf i] (mod mf i)")
+         prefer 7