src/HOL/NumberTheory/Residues.thy
 author haftmann Wed Sep 26 20:27:55 2007 +0200 (2007-09-26) changeset 24728 e2b3a1065676 parent 21404 eb85850d3eb7 child 29948 cdf12a1cb963 permissions -rw-r--r--
moved Finite_Set before Datatype
```     1 (*  Title:      HOL/Quadratic_Reciprocity/Residues.thy
```
```     2     ID:         \$Id\$
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```     3     Authors:    Jeremy Avigad, David Gray, and Adam Kramer
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```     4 *)
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```     5
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```     6 header {* Residue Sets *}
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```     7
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```     8 theory Residues imports Int2 begin
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```     9
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```    10 text {*
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```    11   \medskip Define the residue of a set, the standard residue,
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```    12   quadratic residues, and prove some basic properties. *}
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```    13
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```    14 definition
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```    15   ResSet      :: "int => int set => bool" where
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```    16   "ResSet m X = (\<forall>y1 y2. (y1 \<in> X & y2 \<in> X & [y1 = y2] (mod m) --> y1 = y2))"
```
```    17
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```    18 definition
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```    19   StandardRes :: "int => int => int" where
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```    20   "StandardRes m x = x mod m"
```
```    21
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```    22 definition
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```    23   QuadRes     :: "int => int => bool" where
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```    24   "QuadRes m x = (\<exists>y. ([(y ^ 2) = x] (mod m)))"
```
```    25
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```    26 definition
```
```    27   Legendre    :: "int => int => int" where
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```    28   "Legendre a p = (if ([a = 0] (mod p)) then 0
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```    29                      else if (QuadRes p a) then 1
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```    30                      else -1)"
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```    31
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```    32 definition
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```    33   SR          :: "int => int set" where
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```    34   "SR p = {x. (0 \<le> x) & (x < p)}"
```
```    35
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```    36 definition
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```    37   SRStar      :: "int => int set" where
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```    38   "SRStar p = {x. (0 < x) & (x < p)}"
```
```    39
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```    40
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```    41 subsection {* Some useful properties of StandardRes *}
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```    42
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```    43 lemma StandardRes_prop1: "[x = StandardRes m x] (mod m)"
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```    44   by (auto simp add: StandardRes_def zcong_zmod)
```
```    45
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```    46 lemma StandardRes_prop2: "0 < m ==> (StandardRes m x1 = StandardRes m x2)
```
```    47       = ([x1 = x2] (mod m))"
```
```    48   by (auto simp add: StandardRes_def zcong_zmod_eq)
```
```    49
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```    50 lemma StandardRes_prop3: "(~[x = 0] (mod p)) = (~(StandardRes p x = 0))"
```
```    51   by (auto simp add: StandardRes_def zcong_def zdvd_iff_zmod_eq_0)
```
```    52
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```    53 lemma StandardRes_prop4: "2 < m
```
```    54      ==> [StandardRes m x * StandardRes m y = (x * y)] (mod m)"
```
```    55   by (auto simp add: StandardRes_def zcong_zmod_eq
```
```    56                      zmod_zmult_distrib [of x y m])
```
```    57
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```    58 lemma StandardRes_lbound: "0 < p ==> 0 \<le> StandardRes p x"
```
```    59   by (auto simp add: StandardRes_def pos_mod_sign)
```
```    60
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```    61 lemma StandardRes_ubound: "0 < p ==> StandardRes p x < p"
```
```    62   by (auto simp add: StandardRes_def pos_mod_bound)
```
```    63
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```    64 lemma StandardRes_eq_zcong:
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```    65    "(StandardRes m x = 0) = ([x = 0](mod m))"
```
```    66   by (auto simp add: StandardRes_def zcong_eq_zdvd_prop dvd_def)
```
```    67
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```    68
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```    69 subsection {* Relations between StandardRes, SRStar, and SR *}
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```    70
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```    71 lemma SRStar_SR_prop: "x \<in> SRStar p ==> x \<in> SR p"
```
```    72   by (auto simp add: SRStar_def SR_def)
```
```    73
```
```    74 lemma StandardRes_SR_prop: "x \<in> SR p ==> StandardRes p x = x"
```
```    75   by (auto simp add: SR_def StandardRes_def mod_pos_pos_trivial)
```
```    76
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```    77 lemma StandardRes_SRStar_prop1: "2 < p ==> (StandardRes p x \<in> SRStar p)
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```    78      = (~[x = 0] (mod p))"
```
```    79   apply (auto simp add: StandardRes_prop3 StandardRes_def
```
```    80                         SRStar_def pos_mod_bound)
```
```    81   apply (subgoal_tac "0 < p")
```
```    82   apply (drule_tac a = x in pos_mod_sign, arith, simp)
```
```    83   done
```
```    84
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```    85 lemma StandardRes_SRStar_prop1a: "x \<in> SRStar p ==> ~([x = 0] (mod p))"
```
```    86   by (auto simp add: SRStar_def zcong_def zdvd_not_zless)
```
```    87
```
```    88 lemma StandardRes_SRStar_prop2: "[| 2 < p; zprime p; x \<in> SRStar p |]
```
```    89      ==> StandardRes p (MultInv p x) \<in> SRStar p"
```
```    90   apply (frule_tac x = "(MultInv p x)" in StandardRes_SRStar_prop1, simp)
```
```    91   apply (rule MultInv_prop3)
```
```    92   apply (auto simp add: SRStar_def zcong_def zdvd_not_zless)
```
```    93   done
```
```    94
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```    95 lemma StandardRes_SRStar_prop3: "x \<in> SRStar p ==> StandardRes p x = x"
```
```    96   by (auto simp add: SRStar_SR_prop StandardRes_SR_prop)
```
```    97
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```    98 lemma StandardRes_SRStar_prop4: "[| zprime p; 2 < p; x \<in> SRStar p |]
```
```    99      ==> StandardRes p x \<in> SRStar p"
```
```   100   by (frule StandardRes_SRStar_prop3, auto)
```
```   101
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```   102 lemma SRStar_mult_prop1: "[| zprime p; 2 < p; x \<in> SRStar p; y \<in> SRStar p|]
```
```   103      ==> (StandardRes p (x * y)):SRStar p"
```
```   104   apply (frule_tac x = x in StandardRes_SRStar_prop4, auto)
```
```   105   apply (frule_tac x = y in StandardRes_SRStar_prop4, auto)
```
```   106   apply (auto simp add: StandardRes_SRStar_prop1 zcong_zmult_prop3)
```
```   107   done
```
```   108
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```   109 lemma SRStar_mult_prop2: "[| zprime p; 2 < p; ~([a = 0](mod p));
```
```   110      x \<in> SRStar p |]
```
```   111      ==> StandardRes p (a * MultInv p x) \<in> SRStar p"
```
```   112   apply (frule_tac x = x in StandardRes_SRStar_prop2, auto)
```
```   113   apply (frule_tac x = "MultInv p x" in StandardRes_SRStar_prop1)
```
```   114   apply (auto simp add: StandardRes_SRStar_prop1 zcong_zmult_prop3)
```
```   115   done
```
```   116
```
```   117 lemma SRStar_card: "2 < p ==> int(card(SRStar p)) = p - 1"
```
```   118   by (auto simp add: SRStar_def int_card_bdd_int_set_l_l)
```
```   119
```
```   120 lemma SRStar_finite: "2 < p ==> finite( SRStar p)"
```
```   121   by (auto simp add: SRStar_def bdd_int_set_l_l_finite)
```
```   122
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```   123
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```   124 subsection {* Properties relating ResSets with StandardRes *}
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```   125
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```   126 lemma aux: "x mod m = y mod m ==> [x = y] (mod m)"
```
```   127   apply (subgoal_tac "x = y ==> [x = y](mod m)")
```
```   128   apply (subgoal_tac "[x mod m = y mod m] (mod m) ==> [x = y] (mod m)")
```
```   129   apply (auto simp add: zcong_zmod [of x y m])
```
```   130   done
```
```   131
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```   132 lemma StandardRes_inj_on_ResSet: "ResSet m X ==> (inj_on (StandardRes m) X)"
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```   133   apply (auto simp add: ResSet_def StandardRes_def inj_on_def)
```
```   134   apply (drule_tac m = m in aux, auto)
```
```   135   done
```
```   136
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```   137 lemma StandardRes_Sum: "[| finite X; 0 < m |]
```
```   138      ==> [setsum f X = setsum (StandardRes m o f) X](mod m)"
```
```   139   apply (rule_tac F = X in finite_induct)
```
```   140   apply (auto intro!: zcong_zadd simp add: StandardRes_prop1)
```
```   141   done
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```   142
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```   143 lemma SR_pos: "0 < m ==> (StandardRes m ` X) \<subseteq> {x. 0 \<le> x & x < m}"
```
```   144   by (auto simp add: StandardRes_ubound StandardRes_lbound)
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```   145
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```   146 lemma ResSet_finite: "0 < m ==> ResSet m X ==> finite X"
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```   147   apply (rule_tac f = "StandardRes m" in finite_imageD)
```
```   148   apply (rule_tac B = "{x. (0 :: int) \<le> x & x < m}" in finite_subset)
```
```   149   apply (auto simp add: StandardRes_inj_on_ResSet bdd_int_set_l_finite SR_pos)
```
```   150   done
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```   151
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```   152 lemma mod_mod_is_mod: "[x = x mod m](mod m)"
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```   153   by (auto simp add: zcong_zmod)
```
```   154
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```   155 lemma StandardRes_prod: "[| finite X; 0 < m |]
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```   156      ==> [setprod f X = setprod (StandardRes m o f) X] (mod m)"
```
```   157   apply (rule_tac F = X in finite_induct)
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```   158   apply (auto intro!: zcong_zmult simp add: StandardRes_prop1)
```
```   159   done
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```   160
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```   161 lemma ResSet_image:
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```   162   "[| 0 < m; ResSet m A; \<forall>x \<in> A. \<forall>y \<in> A. ([f x = f y](mod m) --> x = y) |] ==>
```
```   163     ResSet m (f ` A)"
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```   164   by (auto simp add: ResSet_def)
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```   165
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```   166
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```   167 subsection {* Property for SRStar *}
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```   168
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```   169 lemma ResSet_SRStar_prop: "ResSet p (SRStar p)"
```
```   170   by (auto simp add: SRStar_def ResSet_def zcong_zless_imp_eq)
```
```   171
```
```   172 end
```