src/HOL/Library/Primes.thy
 author paulson Sat Jun 09 08:41:25 2001 +0200 (2001-06-09) changeset 11363 a548865b1b6a child 11368 9c1995c73383 permissions -rw-r--r--
moved Primes.thy from NumberTheory to Library
```     1 (*  Title:      HOL/NumberTheory/Primes.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Christophe Tabacznyj and Lawrence C Paulson
```
```     4     Copyright   1996  University of Cambridge
```
```     5 *)
```
```     6
```
```     7 header {* The Greatest Common Divisor and Euclid's algorithm *}
```
```     8
```
```     9 theory Primes = Main:
```
```    10
```
```    11 text {*
```
```    12   (See H. Davenport, "The Higher Arithmetic".  6th edition.  (CUP, 1992))
```
```    13
```
```    14   \bigskip
```
```    15 *}
```
```    16
```
```    17 consts
```
```    18   gcd  :: "nat * nat => nat"  -- {* Euclid's algorithm *}
```
```    19
```
```    20 recdef gcd  "measure ((\<lambda>(m, n). n) :: nat * nat => nat)"
```
```    21   "gcd (m, n) = (if n = 0 then m else gcd (n, m mod n))"
```
```    22
```
```    23 constdefs
```
```    24   is_gcd :: "nat => nat => nat => bool"  -- {* @{term gcd} as a relation *}
```
```    25   "is_gcd p m n == p dvd m \<and> p dvd n \<and>
```
```    26     (\<forall>d. d dvd m \<and> d dvd n --> d dvd p)"
```
```    27
```
```    28   coprime :: "nat => nat => bool"
```
```    29   "coprime m n == gcd (m, n) = 1"
```
```    30
```
```    31   prime :: "nat set"
```
```    32   "prime == {p. 1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p)}"
```
```    33
```
```    34
```
```    35 lemma gcd_induct:
```
```    36   "(!!m. P m 0) ==>
```
```    37     (!!m n. 0 < n ==> P n (m mod n) ==> P m n)
```
```    38   ==> P (m::nat) (n::nat)"
```
```    39   apply (induct m n rule: gcd.induct)
```
```    40   apply (case_tac "n = 0")
```
```    41    apply simp_all
```
```    42   done
```
```    43
```
```    44
```
```    45 lemma gcd_0 [simp]: "gcd (m, 0) = m"
```
```    46   apply simp
```
```    47   done
```
```    48
```
```    49 lemma gcd_non_0: "0 < n ==> gcd (m, n) = gcd (n, m mod n)"
```
```    50   apply simp
```
```    51   done
```
```    52
```
```    53 declare gcd.simps [simp del]
```
```    54
```
```    55 lemma gcd_1 [simp]: "gcd (m, 1) = 1"
```
```    56   apply (simp add: gcd_non_0)
```
```    57   done
```
```    58
```
```    59 text {*
```
```    60   \medskip @{term "gcd (m, n)"} divides @{text m} and @{text n}.  The
```
```    61   conjunctions don't seem provable separately.
```
```    62 *}
```
```    63
```
```    64 lemma gcd_dvd_both: "gcd (m, n) dvd m \<and> gcd (m, n) dvd n"
```
```    65   apply (induct m n rule: gcd_induct)
```
```    66    apply (simp_all add: gcd_non_0)
```
```    67   apply (blast dest: dvd_mod_imp_dvd)
```
```    68   done
```
```    69
```
```    70 lemmas gcd_dvd1 [iff] = gcd_dvd_both [THEN conjunct1, standard]
```
```    71 lemmas gcd_dvd2 [iff] = gcd_dvd_both [THEN conjunct2, standard]
```
```    72
```
```    73
```
```    74 text {*
```
```    75   \medskip Maximality: for all @{term m}, @{term n}, @{term k}
```
```    76   naturals, if @{term k} divides @{term m} and @{term k} divides
```
```    77   @{term n} then @{term k} divides @{term "gcd (m, n)"}.
```
```    78 *}
```
```    79
```
```    80 lemma gcd_greatest: "k dvd m ==> k dvd n ==> k dvd gcd (m, n)"
```
```    81   apply (induct m n rule: gcd_induct)
```
```    82    apply (simp_all add: gcd_non_0 dvd_mod)
```
```    83   done
```
```    84
```
```    85 lemma gcd_greatest_iff [iff]: "(k dvd gcd (m, n)) = (k dvd m \<and> k dvd n)"
```
```    86   apply (blast intro!: gcd_greatest intro: dvd_trans)
```
```    87   done
```
```    88
```
```    89
```
```    90 text {*
```
```    91   \medskip Function gcd yields the Greatest Common Divisor.
```
```    92 *}
```
```    93
```
```    94 lemma is_gcd: "is_gcd (gcd (m, n)) m n"
```
```    95   apply (simp add: is_gcd_def gcd_greatest)
```
```    96   done
```
```    97
```
```    98 text {*
```
```    99   \medskip Uniqueness of GCDs.
```
```   100 *}
```
```   101
```
```   102 lemma is_gcd_unique: "is_gcd m a b ==> is_gcd n a b ==> m = n"
```
```   103   apply (simp add: is_gcd_def)
```
```   104   apply (blast intro: dvd_anti_sym)
```
```   105   done
```
```   106
```
```   107 lemma is_gcd_dvd: "is_gcd m a b ==> k dvd a ==> k dvd b ==> k dvd m"
```
```   108   apply (auto simp add: is_gcd_def)
```
```   109   done
```
```   110
```
```   111
```
```   112 text {*
```
```   113   \medskip Commutativity
```
```   114 *}
```
```   115
```
```   116 lemma is_gcd_commute: "is_gcd k m n = is_gcd k n m"
```
```   117   apply (auto simp add: is_gcd_def)
```
```   118   done
```
```   119
```
```   120 lemma gcd_commute: "gcd (m, n) = gcd (n, m)"
```
```   121   apply (rule is_gcd_unique)
```
```   122    apply (rule is_gcd)
```
```   123   apply (subst is_gcd_commute)
```
```   124   apply (simp add: is_gcd)
```
```   125   done
```
```   126
```
```   127 lemma gcd_assoc: "gcd (gcd (k, m), n) = gcd (k, gcd (m, n))"
```
```   128   apply (rule is_gcd_unique)
```
```   129    apply (rule is_gcd)
```
```   130   apply (simp add: is_gcd_def)
```
```   131   apply (blast intro: dvd_trans)
```
```   132   done
```
```   133
```
```   134 lemma gcd_0_left [simp]: "gcd (0, m) = m"
```
```   135   apply (simp add: gcd_commute [of 0])
```
```   136   done
```
```   137
```
```   138 lemma gcd_1_left [simp]: "gcd (1, m) = 1"
```
```   139   apply (simp add: gcd_commute [of 1])
```
```   140   done
```
```   141
```
```   142
```
```   143 text {*
```
```   144   \medskip Multiplication laws
```
```   145 *}
```
```   146
```
```   147 lemma gcd_mult_distrib2: "k * gcd (m, n) = gcd (k * m, k * n)"
```
```   148     -- {* Davenport, page 27 *}
```
```   149   apply (induct m n rule: gcd_induct)
```
```   150    apply simp
```
```   151   apply (case_tac "k = 0")
```
```   152    apply (simp_all add: mod_geq gcd_non_0 mod_mult_distrib2)
```
```   153   done
```
```   154
```
```   155 lemma gcd_mult [simp]: "gcd (k, k * n) = k"
```
```   156   apply (rule gcd_mult_distrib2 [of k 1 n, simplified, symmetric])
```
```   157   done
```
```   158
```
```   159 lemma gcd_self [simp]: "gcd (k, k) = k"
```
```   160   apply (rule gcd_mult [of k 1, simplified])
```
```   161   done
```
```   162
```
```   163 lemma relprime_dvd_mult: "gcd (k, n) = 1 ==> k dvd m * n ==> k dvd m"
```
```   164   apply (insert gcd_mult_distrib2 [of m k n])
```
```   165   apply simp
```
```   166   apply (erule_tac t = m in ssubst)
```
```   167   apply simp
```
```   168   done
```
```   169
```
```   170 lemma relprime_dvd_mult_iff: "gcd (k, n) = 1 ==> (k dvd m * n) = (k dvd m)"
```
```   171   apply (blast intro: relprime_dvd_mult dvd_trans)
```
```   172   done
```
```   173
```
```   174 lemma prime_imp_relprime: "p \<in> prime ==> \<not> p dvd n ==> gcd (p, n) = 1"
```
```   175   apply (auto simp add: prime_def)
```
```   176   apply (drule_tac x = "gcd (p, n)" in spec)
```
```   177   apply auto
```
```   178   apply (insert gcd_dvd2 [of p n])
```
```   179   apply simp
```
```   180   done
```
```   181
```
```   182 text {*
```
```   183   This theorem leads immediately to a proof of the uniqueness of
```
```   184   factorization.  If @{term p} divides a product of primes then it is
```
```   185   one of those primes.
```
```   186 *}
```
```   187
```
```   188 lemma prime_dvd_mult: "p \<in> prime ==> p dvd m * n ==> p dvd m \<or> p dvd n"
```
```   189   apply (blast intro: relprime_dvd_mult prime_imp_relprime)
```
```   190   done
```
```   191
```
```   192
```
```   193 text {* \medskip Addition laws *}
```
```   194
```
```   195 lemma gcd_add1 [simp]: "gcd (m + n, n) = gcd (m, n)"
```
```   196   apply (case_tac "n = 0")
```
```   197    apply (simp_all add: gcd_non_0)
```
```   198   done
```
```   199
```
```   200 lemma gcd_add2 [simp]: "gcd (m, m + n) = gcd (m, n)"
```
```   201   apply (rule gcd_commute [THEN trans])
```
```   202   apply (subst add_commute)
```
```   203   apply (simp add: gcd_add1)
```
```   204   apply (rule gcd_commute)
```
```   205   done
```
```   206
```
```   207 lemma gcd_add2' [simp]: "gcd (m, n + m) = gcd (m, n)"
```
```   208   apply (subst add_commute)
```
```   209   apply (rule gcd_add2)
```
```   210   done
```
```   211
```
```   212 lemma gcd_add_mult: "gcd (m, k * m + n) = gcd (m, n)"
```
```   213   apply (induct k)
```
```   214    apply (simp_all add: gcd_add2 add_assoc)
```
```   215   done
```
```   216
```
```   217
```
```   218 text {* \medskip More multiplication laws *}
```
```   219
```
```   220 lemma gcd_mult_cancel: "gcd (k, n) = 1 ==> gcd (k * m, n) = gcd (m, n)"
```
```   221   apply (rule dvd_anti_sym)
```
```   222    apply (rule gcd_greatest)
```
```   223     apply (rule_tac n = k in relprime_dvd_mult)
```
```   224      apply (simp add: gcd_assoc)
```
```   225      apply (simp add: gcd_commute)
```
```   226     apply (simp_all add: mult_commute gcd_dvd1 gcd_dvd2)
```
```   227   apply (blast intro: gcd_dvd1 dvd_trans)
```
```   228   done
```
```   229
```
```   230 end
```