--- a/src/HOL/NumberTheory/Chinese.thy Tue Sep 29 22:15:54 2009 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,259 +0,0 @@
-(* Title: HOL/NumberTheory/Chinese.thy
- ID: $Id$
- Author: Thomas M. Rasmussen
- Copyright 2000 University of Cambridge
-*)
-
-header {* The Chinese Remainder Theorem *}
-
-theory Chinese
-imports IntPrimes
-begin
-
-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"
-
-primrec
- "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"
-
-definition
- m_cond :: "nat => (nat => int) => bool" where
- "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))"
-
-definition
- km_cond :: "nat => (nat => int) => (nat => int) => bool" where
- "km_cond n kf mf = (\<forall>i. i \<le> n --> zgcd (kf i) (mf i) = 1)"
-
-definition
- lincong_sol ::
- "nat => (nat => int) => (nat => int) => (nat => int) => int => bool" where
- "lincong_sol n kf bf mf x = (\<forall>i. i \<le> n --> zcong (kf i * x) (bf i) (mf i))"
-
-definition
- mhf :: "(nat => int) => nat => nat => int" where
- "mhf mf n i =
- (if i = 0 then funprod mf (Suc 0) (n - Suc 0)
- else if i = n then funprod mf 0 (n - Suc 0)
- else funprod mf 0 (i - Suc 0) * funprod mf (Suc i) (n - Suc 0 - i))"
-
-definition
- xilin_sol ::
- "nat => nat => (nat => int) => (nat => int) => (nat => int) => int" where
- "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)"
-
-definition
- x_sol :: "nat => (nat => int) => (nat => int) => (nat => int) => int" where
- "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
- apply (simp add: zero_less_mult_iff)
- 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
-
-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 (subgoal_tac "i = Suc (k + l)")
- 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 (rule mod_add_eq)
- apply simp
- apply (rule mod_add_right_eq [symmetric])
- done
-
-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 + l) = j")
- apply (subgoal_tac "funsum f k l = 0")
- apply (rule_tac [2] funsum_zero)
- apply (subgoal_tac [3] "f (Suc (k + l)) = 0")
- apply (subgoal_tac [3] "j \<le> k + l")
- prefer 4
- apply arith
- apply auto
- done
-
-
-subsection {* Chinese: uniqueness *}
-
-lemma zcong_funprod_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: zcong_funprod_aux)
- apply (rule impI)+
- apply (rule zcong_zgcd_zmult_zmod)
- apply (blast intro: zcong_funprod_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 "i<n")
- prefer 2
- apply arith
- apply (case_tac [2] i)
- apply simp_all
- done
-
-lemma x_sol_lin_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] dvd_mult2)
- apply (rule_tac [4] dvd_mult)
- apply (rule_tac [!] funprod_zdvd)
- apply arith
- apply arith
- apply arith
- apply arith
- apply arith
- apply arith
- apply arith
- 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 dvd_eq_mod_eq_0 [symmetric])
- apply (rule dvd_mult)
- apply (rule x_sol_lin_aux)
- apply auto
- done
-
-
-subsection {* Chinese *}
-
-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 "mod_mult_eq") 3 *})
- apply (tactic {* stac (thm "mod_mod_cancel") 3 *})
- apply (tactic {* stac (thm "x_sol_lin") 4 *})
- apply (tactic {* stac (thm "mod_mult_eq" RS sym) 6 *})
- apply (tactic {* stac (thm "zcong_zmod" RS sym) 6 *})
- apply (subgoal_tac [6]
- "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 6
- apply (simp add: zmult_ac)
- apply (unfold xilin_sol_def)
- apply (tactic {* asm_simp_tac @{simpset} 6 *})
- apply (rule_tac [6] ex1_implies_ex [THEN someI_ex])
- apply (rule_tac [6] unique_xi_sol)
- apply (rule_tac [3] funprod_zdvd)
- apply (unfold m_cond_def)
- apply (rule funprod_pos [THEN pos_mod_sign])
- apply (rule_tac [2] funprod_pos [THEN pos_mod_bound])
- apply auto
- done
-
-end