src/HOL/Library/Primes.thy
 author wenzelm Sat Jun 09 14:22:08 2001 +0200 (2001-06-09) changeset 11369 2c4bb701546a parent 11368 9c1995c73383 child 11374 2badb9b2a8ec permissions -rw-r--r--
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```     1 (*  Title:      HOL/Library/Primes.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Christophe Tabacznyj and Lawrence C Paulson
```
```     4     Copyright   1996  University of Cambridge
```
```     5 *)
```
```     6
```
```     7 header {*
```
```     8   \title{The Greatest Common Divisor and Euclid's algorithm}
```
```     9   \author{Christophe Tabacznyj and Lawrence C Paulson}
```
```    10 *}
```
```    11
```
```    12 theory Primes = Main:
```
```    13
```
```    14 text {*
```
```    15   See \cite{davenport92}.
```
```    16   \bigskip
```
```    17 *}
```
```    18
```
```    19 consts
```
```    20   gcd  :: "nat \<times> nat => nat"  -- {* Euclid's algorithm *}
```
```    21
```
```    22 recdef gcd  "measure ((\<lambda>(m, n). n) :: nat \<times> nat => nat)"
```
```    23   "gcd (m, n) = (if n = 0 then m else gcd (n, m mod n))"
```
```    24
```
```    25 constdefs
```
```    26   is_gcd :: "nat => nat => nat => bool"  -- {* @{term gcd} as a relation *}
```
```    27   "is_gcd p m n == p dvd m \<and> p dvd n \<and>
```
```    28     (\<forall>d. d dvd m \<and> d dvd n --> d dvd p)"
```
```    29
```
```    30   coprime :: "nat => nat => bool"
```
```    31   "coprime m n == gcd (m, n) = 1"
```
```    32
```
```    33   prime :: "nat set"
```
```    34   "prime == {p. 1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p)}"
```
```    35
```
```    36
```
```    37 lemma gcd_induct:
```
```    38   "(!!m. P m 0) ==>
```
```    39     (!!m n. 0 < n ==> P n (m mod n) ==> P m n)
```
```    40   ==> P (m::nat) (n::nat)"
```
```    41   apply (induct m n rule: gcd.induct)
```
```    42   apply (case_tac "n = 0")
```
```    43    apply simp_all
```
```    44   done
```
```    45
```
```    46
```
```    47 lemma gcd_0 [simp]: "gcd (m, 0) = m"
```
```    48   apply simp
```
```    49   done
```
```    50
```
```    51 lemma gcd_non_0: "0 < n ==> gcd (m, n) = gcd (n, m mod n)"
```
```    52   apply simp
```
```    53   done
```
```    54
```
```    55 declare gcd.simps [simp del]
```
```    56
```
```    57 lemma gcd_1 [simp]: "gcd (m, 1) = 1"
```
```    58   apply (simp add: gcd_non_0)
```
```    59   done
```
```    60
```
```    61 text {*
```
```    62   \medskip @{term "gcd (m, n)"} divides @{text m} and @{text n}.  The
```
```    63   conjunctions don't seem provable separately.
```
```    64 *}
```
```    65
```
```    66 lemma gcd_dvd_both: "gcd (m, n) dvd m \<and> gcd (m, n) dvd n"
```
```    67   apply (induct m n rule: gcd_induct)
```
```    68    apply (simp_all add: gcd_non_0)
```
```    69   apply (blast dest: dvd_mod_imp_dvd)
```
```    70   done
```
```    71
```
```    72 lemmas gcd_dvd1 [iff] = gcd_dvd_both [THEN conjunct1, standard]
```
```    73 lemmas gcd_dvd2 [iff] = gcd_dvd_both [THEN conjunct2, standard]
```
```    74
```
```    75 lemma gcd_zero: "(gcd (m, n) = 0) = (m = 0 \<and> n = 0)"
```
```    76 proof
```
```    77   have "gcd (m, n) dvd m \<and> gcd (m, n) dvd n" by simp
```
```    78   also assume "gcd (m, n) = 0"
```
```    79   finally have "0 dvd m \<and> 0 dvd n" .
```
```    80   thus "m = 0 \<and> n = 0" by (simp add: dvd_0_left)
```
```    81 next
```
```    82   assume "m = 0 \<and> n = 0"
```
```    83   thus "gcd (m, n) = 0" by simp
```
```    84 qed
```
```    85
```
```    86
```
```    87 text {*
```
```    88   \medskip Maximality: for all @{term m}, @{term n}, @{term k}
```
```    89   naturals, if @{term k} divides @{term m} and @{term k} divides
```
```    90   @{term n} then @{term k} divides @{term "gcd (m, n)"}.
```
```    91 *}
```
```    92
```
```    93 lemma gcd_greatest: "k dvd m ==> k dvd n ==> k dvd gcd (m, n)"
```
```    94   apply (induct m n rule: gcd_induct)
```
```    95    apply (simp_all add: gcd_non_0 dvd_mod)
```
```    96   done
```
```    97
```
```    98 lemma gcd_greatest_iff [iff]: "(k dvd gcd (m, n)) = (k dvd m \<and> k dvd n)"
```
```    99   apply (blast intro!: gcd_greatest intro: dvd_trans)
```
```   100   done
```
```   101
```
```   102
```
```   103 text {*
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```   104   \medskip Function gcd yields the Greatest Common Divisor.
```
```   105 *}
```
```   106
```
```   107 lemma is_gcd: "is_gcd (gcd (m, n)) m n"
```
```   108   apply (simp add: is_gcd_def gcd_greatest)
```
```   109   done
```
```   110
```
```   111 text {*
```
```   112   \medskip Uniqueness of GCDs.
```
```   113 *}
```
```   114
```
```   115 lemma is_gcd_unique: "is_gcd m a b ==> is_gcd n a b ==> m = n"
```
```   116   apply (simp add: is_gcd_def)
```
```   117   apply (blast intro: dvd_anti_sym)
```
```   118   done
```
```   119
```
```   120 lemma is_gcd_dvd: "is_gcd m a b ==> k dvd a ==> k dvd b ==> k dvd m"
```
```   121   apply (auto simp add: is_gcd_def)
```
```   122   done
```
```   123
```
```   124
```
```   125 text {*
```
```   126   \medskip Commutativity
```
```   127 *}
```
```   128
```
```   129 lemma is_gcd_commute: "is_gcd k m n = is_gcd k n m"
```
```   130   apply (auto simp add: is_gcd_def)
```
```   131   done
```
```   132
```
```   133 lemma gcd_commute: "gcd (m, n) = gcd (n, m)"
```
```   134   apply (rule is_gcd_unique)
```
```   135    apply (rule is_gcd)
```
```   136   apply (subst is_gcd_commute)
```
```   137   apply (simp add: is_gcd)
```
```   138   done
```
```   139
```
```   140 lemma gcd_assoc: "gcd (gcd (k, m), n) = gcd (k, gcd (m, n))"
```
```   141   apply (rule is_gcd_unique)
```
```   142    apply (rule is_gcd)
```
```   143   apply (simp add: is_gcd_def)
```
```   144   apply (blast intro: dvd_trans)
```
```   145   done
```
```   146
```
```   147 lemma gcd_0_left [simp]: "gcd (0, m) = m"
```
```   148   apply (simp add: gcd_commute [of 0])
```
```   149   done
```
```   150
```
```   151 lemma gcd_1_left [simp]: "gcd (1, m) = 1"
```
```   152   apply (simp add: gcd_commute [of 1])
```
```   153   done
```
```   154
```
```   155
```
```   156 text {*
```
```   157   \medskip Multiplication laws
```
```   158 *}
```
```   159
```
```   160 lemma gcd_mult_distrib2: "k * gcd (m, n) = gcd (k * m, k * n)"
```
```   161     -- {* \cite[page 27]{davenport92} *}
```
```   162   apply (induct m n rule: gcd_induct)
```
```   163    apply simp
```
```   164   apply (case_tac "k = 0")
```
```   165    apply (simp_all add: mod_geq gcd_non_0 mod_mult_distrib2)
```
```   166   done
```
```   167
```
```   168 lemma gcd_mult [simp]: "gcd (k, k * n) = k"
```
```   169   apply (rule gcd_mult_distrib2 [of k 1 n, simplified, symmetric])
```
```   170   done
```
```   171
```
```   172 lemma gcd_self [simp]: "gcd (k, k) = k"
```
```   173   apply (rule gcd_mult [of k 1, simplified])
```
```   174   done
```
```   175
```
```   176 lemma relprime_dvd_mult: "gcd (k, n) = 1 ==> k dvd m * n ==> k dvd m"
```
```   177   apply (insert gcd_mult_distrib2 [of m k n])
```
```   178   apply simp
```
```   179   apply (erule_tac t = m in ssubst)
```
```   180   apply simp
```
```   181   done
```
```   182
```
```   183 lemma relprime_dvd_mult_iff: "gcd (k, n) = 1 ==> (k dvd m * n) = (k dvd m)"
```
```   184   apply (blast intro: relprime_dvd_mult dvd_trans)
```
```   185   done
```
```   186
```
```   187 lemma prime_imp_relprime: "p \<in> prime ==> \<not> p dvd n ==> gcd (p, n) = 1"
```
```   188   apply (auto simp add: prime_def)
```
```   189   apply (drule_tac x = "gcd (p, n)" in spec)
```
```   190   apply auto
```
```   191   apply (insert gcd_dvd2 [of p n])
```
```   192   apply simp
```
```   193   done
```
```   194
```
```   195 text {*
```
```   196   This theorem leads immediately to a proof of the uniqueness of
```
```   197   factorization.  If @{term p} divides a product of primes then it is
```
```   198   one of those primes.
```
```   199 *}
```
```   200
```
```   201 lemma prime_dvd_mult: "p \<in> prime ==> p dvd m * n ==> p dvd m \<or> p dvd n"
```
```   202   apply (blast intro: relprime_dvd_mult prime_imp_relprime)
```
```   203   done
```
```   204
```
```   205 lemma prime_dvd_square: "p \<in> prime ==> p dvd m^2 ==> p dvd m"
```
```   206   apply (auto dest: prime_dvd_mult)
```
```   207   done
```
```   208
```
```   209
```
```   210 text {* \medskip Addition laws *}
```
```   211
```
```   212 lemma gcd_add1 [simp]: "gcd (m + n, n) = gcd (m, n)"
```
```   213   apply (case_tac "n = 0")
```
```   214    apply (simp_all add: gcd_non_0)
```
```   215   done
```
```   216
```
```   217 lemma gcd_add2 [simp]: "gcd (m, m + n) = gcd (m, n)"
```
```   218   apply (rule gcd_commute [THEN trans])
```
```   219   apply (subst add_commute)
```
```   220   apply (simp add: gcd_add1)
```
```   221   apply (rule gcd_commute)
```
```   222   done
```
```   223
```
```   224 lemma gcd_add2' [simp]: "gcd (m, n + m) = gcd (m, n)"
```
```   225   apply (subst add_commute)
```
```   226   apply (rule gcd_add2)
```
```   227   done
```
```   228
```
```   229 lemma gcd_add_mult: "gcd (m, k * m + n) = gcd (m, n)"
```
```   230   apply (induct k)
```
```   231    apply (simp_all add: gcd_add2 add_assoc)
```
```   232   done
```
```   233
```
```   234
```
```   235 text {* \medskip More multiplication laws *}
```
```   236
```
```   237 lemma gcd_mult_cancel: "gcd (k, n) = 1 ==> gcd (k * m, n) = gcd (m, n)"
```
```   238   apply (rule dvd_anti_sym)
```
```   239    apply (rule gcd_greatest)
```
```   240     apply (rule_tac n = k in relprime_dvd_mult)
```
```   241      apply (simp add: gcd_assoc)
```
```   242      apply (simp add: gcd_commute)
```
```   243     apply (simp_all add: mult_commute gcd_dvd1 gcd_dvd2)
```
```   244   apply (blast intro: gcd_dvd1 dvd_trans)
```
```   245   done
```
```   246
```
```   247 end
```