src/HOL/Computational_Algebra/Euclidean_Algorithm.thy
 author Manuel Eberl Tue Dec 12 10:01:14 2017 +0100 (18 months ago) changeset 67167 88d1c9d86f48 parent 66817 0b12755ccbb2 child 67399 eab6ce8368fa permissions -rw-r--r--
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```     1 (*  Title:      HOL/Computational_Algebra/Euclidean_Algorithm.thy
```
```     2     Author:     Manuel Eberl, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 section \<open>Abstract euclidean algorithm in euclidean (semi)rings\<close>
```
```     6
```
```     7 theory Euclidean_Algorithm
```
```     8   imports Factorial_Ring
```
```     9 begin
```
```    10
```
```    11 subsection \<open>Generic construction of the (simple) euclidean algorithm\<close>
```
```    12
```
```    13 class normalization_euclidean_semiring = euclidean_semiring + normalization_semidom
```
```    14 begin
```
```    15
```
```    16 lemma euclidean_size_normalize [simp]:
```
```    17   "euclidean_size (normalize a) = euclidean_size a"
```
```    18 proof (cases "a = 0")
```
```    19   case True
```
```    20   then show ?thesis
```
```    21     by simp
```
```    22 next
```
```    23   case [simp]: False
```
```    24   have "euclidean_size (normalize a) \<le> euclidean_size (normalize a * unit_factor a)"
```
```    25     by (rule size_mult_mono) simp
```
```    26   moreover have "euclidean_size a \<le> euclidean_size (a * (1 div unit_factor a))"
```
```    27     by (rule size_mult_mono) simp
```
```    28   ultimately show ?thesis
```
```    29     by simp
```
```    30 qed
```
```    31
```
```    32 context
```
```    33 begin
```
```    34
```
```    35 qualified function gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```    36   where "gcd a b = (if b = 0 then normalize a else gcd b (a mod b))"
```
```    37   by pat_completeness simp
```
```    38 termination
```
```    39   by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)
```
```    40
```
```    41 declare gcd.simps [simp del]
```
```    42
```
```    43 lemma eucl_induct [case_names zero mod]:
```
```    44   assumes H1: "\<And>b. P b 0"
```
```    45   and H2: "\<And>a b. b \<noteq> 0 \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b"
```
```    46   shows "P a b"
```
```    47 proof (induct a b rule: gcd.induct)
```
```    48   case (1 a b)
```
```    49   show ?case
```
```    50   proof (cases "b = 0")
```
```    51     case True then show "P a b" by simp (rule H1)
```
```    52   next
```
```    53     case False
```
```    54     then have "P b (a mod b)"
```
```    55       by (rule "1.hyps")
```
```    56     with \<open>b \<noteq> 0\<close> show "P a b"
```
```    57       by (blast intro: H2)
```
```    58   qed
```
```    59 qed
```
```    60
```
```    61 qualified lemma gcd_0:
```
```    62   "gcd a 0 = normalize a"
```
```    63   by (simp add: gcd.simps [of a 0])
```
```    64
```
```    65 qualified lemma gcd_mod:
```
```    66   "a \<noteq> 0 \<Longrightarrow> gcd a (b mod a) = gcd b a"
```
```    67   by (simp add: gcd.simps [of b 0] gcd.simps [of b a])
```
```    68
```
```    69 qualified definition lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```    70   where "lcm a b = normalize (a * b) div gcd a b"
```
```    71
```
```    72 qualified definition Lcm :: "'a set \<Rightarrow> 'a" \<comment>
```
```    73     \<open>Somewhat complicated definition of Lcm that has the advantage of working
```
```    74     for infinite sets as well\<close>
```
```    75   where
```
```    76   [code del]: "Lcm A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then
```
```    77      let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =
```
```    78        (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)
```
```    79        in normalize l
```
```    80       else 0)"
```
```    81
```
```    82 qualified definition Gcd :: "'a set \<Rightarrow> 'a"
```
```    83   where [code del]: "Gcd A = Lcm {d. \<forall>a\<in>A. d dvd a}"
```
```    84
```
```    85 end
```
```    86
```
```    87 lemma semiring_gcd:
```
```    88   "class.semiring_gcd one zero times gcd lcm
```
```    89     divide plus minus unit_factor normalize"
```
```    90 proof
```
```    91   show "gcd a b dvd a"
```
```    92     and "gcd a b dvd b" for a b
```
```    93     by (induct a b rule: eucl_induct)
```
```    94       (simp_all add: local.gcd_0 local.gcd_mod dvd_mod_iff)
```
```    95 next
```
```    96   show "c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> c dvd gcd a b" for a b c
```
```    97   proof (induct a b rule: eucl_induct)
```
```    98     case (zero a) from \<open>c dvd a\<close> show ?case
```
```    99       by (rule dvd_trans) (simp add: local.gcd_0)
```
```   100   next
```
```   101     case (mod a b)
```
```   102     then show ?case
```
```   103       by (simp add: local.gcd_mod dvd_mod_iff)
```
```   104   qed
```
```   105 next
```
```   106   show "normalize (gcd a b) = gcd a b" for a b
```
```   107     by (induct a b rule: eucl_induct)
```
```   108       (simp_all add: local.gcd_0 local.gcd_mod)
```
```   109 next
```
```   110   show "lcm a b = normalize (a * b) div gcd a b" for a b
```
```   111     by (fact local.lcm_def)
```
```   112 qed
```
```   113
```
```   114 interpretation semiring_gcd one zero times gcd lcm
```
```   115   divide plus minus unit_factor normalize
```
```   116   by (fact semiring_gcd)
```
```   117
```
```   118 lemma semiring_Gcd:
```
```   119   "class.semiring_Gcd one zero times gcd lcm Gcd Lcm
```
```   120     divide plus minus unit_factor normalize"
```
```   121 proof -
```
```   122   show ?thesis
```
```   123   proof
```
```   124     have "(\<forall>a\<in>A. a dvd Lcm A) \<and> (\<forall>b. (\<forall>a\<in>A. a dvd b) \<longrightarrow> Lcm A dvd b)" for A
```
```   125     proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")
```
```   126       case False
```
```   127       then have "Lcm A = 0"
```
```   128         by (auto simp add: local.Lcm_def)
```
```   129       with False show ?thesis
```
```   130         by auto
```
```   131     next
```
```   132       case True
```
```   133       then obtain l\<^sub>0 where l\<^sub>0_props: "l\<^sub>0 \<noteq> 0" "\<forall>a\<in>A. a dvd l\<^sub>0" by blast
```
```   134       define n where "n = (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)"
```
```   135       define l where "l = (SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)"
```
```   136       have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n"
```
```   137         apply (subst n_def)
```
```   138         apply (rule LeastI [of _ "euclidean_size l\<^sub>0"])
```
```   139         apply (rule exI [of _ l\<^sub>0])
```
```   140         apply (simp add: l\<^sub>0_props)
```
```   141         done
```
```   142       from someI_ex [OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l"
```
```   143         and "euclidean_size l = n"
```
```   144         unfolding l_def by simp_all
```
```   145       {
```
```   146         fix l' assume "\<forall>a\<in>A. a dvd l'"
```
```   147         with \<open>\<forall>a\<in>A. a dvd l\<close> have "\<forall>a\<in>A. a dvd gcd l l'"
```
```   148           by (auto intro: gcd_greatest)
```
```   149         moreover from \<open>l \<noteq> 0\<close> have "gcd l l' \<noteq> 0"
```
```   150           by simp
```
```   151         ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and>
```
```   152           euclidean_size b = euclidean_size (gcd l l')"
```
```   153           by (intro exI [of _ "gcd l l'"], auto)
```
```   154         then have "euclidean_size (gcd l l') \<ge> n"
```
```   155           by (subst n_def) (rule Least_le)
```
```   156         moreover have "euclidean_size (gcd l l') \<le> n"
```
```   157         proof -
```
```   158           have "gcd l l' dvd l"
```
```   159             by simp
```
```   160           then obtain a where "l = gcd l l' * a" ..
```
```   161           with \<open>l \<noteq> 0\<close> have "a \<noteq> 0"
```
```   162             by auto
```
```   163           hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)"
```
```   164             by (rule size_mult_mono)
```
```   165           also have "gcd l l' * a = l" using \<open>l = gcd l l' * a\<close> ..
```
```   166           also note \<open>euclidean_size l = n\<close>
```
```   167           finally show "euclidean_size (gcd l l') \<le> n" .
```
```   168         qed
```
```   169         ultimately have *: "euclidean_size l = euclidean_size (gcd l l')"
```
```   170           by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)
```
```   171         from \<open>l \<noteq> 0\<close> have "l dvd gcd l l'"
```
```   172           by (rule dvd_euclidean_size_eq_imp_dvd) (auto simp add: *)
```
```   173         hence "l dvd l'" by (rule dvd_trans [OF _ gcd_dvd2])
```
```   174       }
```
```   175       with \<open>\<forall>a\<in>A. a dvd l\<close> and \<open>l \<noteq> 0\<close>
```
```   176         have "(\<forall>a\<in>A. a dvd normalize l) \<and>
```
```   177           (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> normalize l dvd l')"
```
```   178         by auto
```
```   179       also from True have "normalize l = Lcm A"
```
```   180         by (simp add: local.Lcm_def Let_def n_def l_def)
```
```   181       finally show ?thesis .
```
```   182     qed
```
```   183     then show dvd_Lcm: "a \<in> A \<Longrightarrow> a dvd Lcm A"
```
```   184       and Lcm_least: "(\<And>a. a \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> Lcm A dvd b" for A and a b
```
```   185       by auto
```
```   186     show "a \<in> A \<Longrightarrow> Gcd A dvd a" for A and a
```
```   187       by (auto simp add: local.Gcd_def intro: Lcm_least)
```
```   188     show "(\<And>a. a \<in> A \<Longrightarrow> b dvd a) \<Longrightarrow> b dvd Gcd A" for A and b
```
```   189       by (auto simp add: local.Gcd_def intro: dvd_Lcm)
```
```   190     show [simp]: "normalize (Lcm A) = Lcm A" for A
```
```   191       by (simp add: local.Lcm_def)
```
```   192     show "normalize (Gcd A) = Gcd A" for A
```
```   193       by (simp add: local.Gcd_def)
```
```   194   qed
```
```   195 qed
```
```   196
```
```   197 interpretation semiring_Gcd one zero times gcd lcm Gcd Lcm
```
```   198     divide plus minus unit_factor normalize
```
```   199   by (fact semiring_Gcd)
```
```   200
```
```   201 subclass factorial_semiring
```
```   202 proof -
```
```   203   show "class.factorial_semiring divide plus minus zero times one
```
```   204      unit_factor normalize"
```
```   205   proof (standard, rule factorial_semiring_altI_aux) \<comment> \<open>FIXME rule\<close>
```
```   206     fix x assume "x \<noteq> 0"
```
```   207     thus "finite {p. p dvd x \<and> normalize p = p}"
```
```   208     proof (induction "euclidean_size x" arbitrary: x rule: less_induct)
```
```   209       case (less x)
```
```   210       show ?case
```
```   211       proof (cases "\<exists>y. y dvd x \<and> \<not>x dvd y \<and> \<not>is_unit y")
```
```   212         case False
```
```   213         have "{p. p dvd x \<and> normalize p = p} \<subseteq> {1, normalize x}"
```
```   214         proof
```
```   215           fix p assume p: "p \<in> {p. p dvd x \<and> normalize p = p}"
```
```   216           with False have "is_unit p \<or> x dvd p" by blast
```
```   217           thus "p \<in> {1, normalize x}"
```
```   218           proof (elim disjE)
```
```   219             assume "is_unit p"
```
```   220             hence "normalize p = 1" by (simp add: is_unit_normalize)
```
```   221             with p show ?thesis by simp
```
```   222           next
```
```   223             assume "x dvd p"
```
```   224             with p have "normalize p = normalize x" by (intro associatedI) simp_all
```
```   225             with p show ?thesis by simp
```
```   226           qed
```
```   227         qed
```
```   228         moreover have "finite \<dots>" by simp
```
```   229         ultimately show ?thesis by (rule finite_subset)
```
```   230       next
```
```   231         case True
```
```   232         then obtain y where y: "y dvd x" "\<not>x dvd y" "\<not>is_unit y" by blast
```
```   233         define z where "z = x div y"
```
```   234         let ?fctrs = "\<lambda>x. {p. p dvd x \<and> normalize p = p}"
```
```   235         from y have x: "x = y * z" by (simp add: z_def)
```
```   236         with less.prems have "y \<noteq> 0" "z \<noteq> 0" by auto
```
```   237         have normalized_factors_product:
```
```   238           "{p. p dvd a * b \<and> normalize p = p} =
```
```   239              (\<lambda>(x,y). x * y) ` ({p. p dvd a \<and> normalize p = p} \<times> {p. p dvd b \<and> normalize p = p})" for a b
```
```   240         proof safe
```
```   241           fix p assume p: "p dvd a * b" "normalize p = p"
```
```   242           from dvd_productE[OF p(1)] guess x y . note xy = this
```
```   243           define x' y' where "x' = normalize x" and "y' = normalize y"
```
```   244           have "p = x' * y'"
```
```   245             by (subst p(2) [symmetric]) (simp add: xy x'_def y'_def normalize_mult)
```
```   246           moreover from xy have "normalize x' = x'" "normalize y' = y'" "x' dvd a" "y' dvd b"
```
```   247             by (simp_all add: x'_def y'_def)
```
```   248           ultimately show "p \<in> (\<lambda>(x, y). x * y) `
```
```   249             ({p. p dvd a \<and> normalize p = p} \<times> {p. p dvd b \<and> normalize p = p})"
```
```   250             by blast
```
```   251         qed (auto simp: normalize_mult mult_dvd_mono)
```
```   252         from x y have "\<not>is_unit z" by (auto simp: mult_unit_dvd_iff)
```
```   253         have "?fctrs x = (\<lambda>(p,p'). p * p') ` (?fctrs y \<times> ?fctrs z)"
```
```   254           by (subst x) (rule normalized_factors_product)
```
```   255         also have "\<not>y * z dvd y * 1" "\<not>y * z dvd 1 * z"
```
```   256           by (subst dvd_times_left_cancel_iff dvd_times_right_cancel_iff; fact)+
```
```   257         hence "finite ((\<lambda>(p,p'). p * p') ` (?fctrs y \<times> ?fctrs z))"
```
```   258           by (intro finite_imageI finite_cartesian_product less dvd_proper_imp_size_less)
```
```   259              (auto simp: x)
```
```   260         finally show ?thesis .
```
```   261       qed
```
```   262     qed
```
```   263   next
```
```   264     fix p
```
```   265     assume "irreducible p"
```
```   266     then show "prime_elem p"
```
```   267       by (rule irreducible_imp_prime_elem_gcd)
```
```   268   qed
```
```   269 qed
```
```   270
```
```   271 lemma Gcd_eucl_set [code]:
```
```   272   "Gcd (set xs) = fold gcd xs 0"
```
```   273   by (fact Gcd_set_eq_fold)
```
```   274
```
```   275 lemma Lcm_eucl_set [code]:
```
```   276   "Lcm (set xs) = fold lcm xs 1"
```
```   277   by (fact Lcm_set_eq_fold)
```
```   278
```
```   279 end
```
```   280
```
```   281 hide_const (open) gcd lcm Gcd Lcm
```
```   282
```
```   283 lemma prime_elem_int_abs_iff [simp]:
```
```   284   fixes p :: int
```
```   285   shows "prime_elem \<bar>p\<bar> \<longleftrightarrow> prime_elem p"
```
```   286   using prime_elem_normalize_iff [of p] by simp
```
```   287
```
```   288 lemma prime_elem_int_minus_iff [simp]:
```
```   289   fixes p :: int
```
```   290   shows "prime_elem (- p) \<longleftrightarrow> prime_elem p"
```
```   291   using prime_elem_normalize_iff [of "- p"] by simp
```
```   292
```
```   293 lemma prime_int_iff:
```
```   294   fixes p :: int
```
```   295   shows "prime p \<longleftrightarrow> p > 0 \<and> prime_elem p"
```
```   296   by (auto simp add: prime_def dest: prime_elem_not_zeroI)
```
```   297
```
```   298
```
```   299 subsection \<open>The (simple) euclidean algorithm as gcd computation\<close>
```
```   300
```
```   301 class euclidean_semiring_gcd = normalization_euclidean_semiring + gcd + Gcd +
```
```   302   assumes gcd_eucl: "Euclidean_Algorithm.gcd = GCD.gcd"
```
```   303     and lcm_eucl: "Euclidean_Algorithm.lcm = GCD.lcm"
```
```   304   assumes Gcd_eucl: "Euclidean_Algorithm.Gcd = GCD.Gcd"
```
```   305     and Lcm_eucl: "Euclidean_Algorithm.Lcm = GCD.Lcm"
```
```   306 begin
```
```   307
```
```   308 subclass semiring_gcd
```
```   309   unfolding gcd_eucl [symmetric] lcm_eucl [symmetric]
```
```   310   by (fact semiring_gcd)
```
```   311
```
```   312 subclass semiring_Gcd
```
```   313   unfolding  gcd_eucl [symmetric] lcm_eucl [symmetric]
```
```   314     Gcd_eucl [symmetric] Lcm_eucl [symmetric]
```
```   315   by (fact semiring_Gcd)
```
```   316
```
```   317 subclass factorial_semiring_gcd
```
```   318 proof
```
```   319   show "gcd a b = gcd_factorial a b" for a b
```
```   320     apply (rule sym)
```
```   321     apply (rule gcdI)
```
```   322        apply (fact gcd_lcm_factorial)+
```
```   323     done
```
```   324   then show "lcm a b = lcm_factorial a b" for a b
```
```   325     by (simp add: lcm_factorial_gcd_factorial lcm_gcd)
```
```   326   show "Gcd A = Gcd_factorial A" for A
```
```   327     apply (rule sym)
```
```   328     apply (rule GcdI)
```
```   329        apply (fact gcd_lcm_factorial)+
```
```   330     done
```
```   331   show "Lcm A = Lcm_factorial A" for A
```
```   332     apply (rule sym)
```
```   333     apply (rule LcmI)
```
```   334        apply (fact gcd_lcm_factorial)+
```
```   335     done
```
```   336 qed
```
```   337
```
```   338 lemma gcd_mod_right [simp]:
```
```   339   "a \<noteq> 0 \<Longrightarrow> gcd a (b mod a) = gcd a b"
```
```   340   unfolding gcd.commute [of a b]
```
```   341   by (simp add: gcd_eucl [symmetric] local.gcd_mod)
```
```   342
```
```   343 lemma gcd_mod_left [simp]:
```
```   344   "b \<noteq> 0 \<Longrightarrow> gcd (a mod b) b = gcd a b"
```
```   345   by (drule gcd_mod_right [of _ a]) (simp add: gcd.commute)
```
```   346
```
```   347 lemma euclidean_size_gcd_le1 [simp]:
```
```   348   assumes "a \<noteq> 0"
```
```   349   shows "euclidean_size (gcd a b) \<le> euclidean_size a"
```
```   350 proof -
```
```   351   from gcd_dvd1 obtain c where A: "a = gcd a b * c" ..
```
```   352   with assms have "c \<noteq> 0"
```
```   353     by auto
```
```   354   moreover from this
```
```   355   have "euclidean_size (gcd a b) \<le> euclidean_size (gcd a b * c)"
```
```   356     by (rule size_mult_mono)
```
```   357   with A show ?thesis
```
```   358     by simp
```
```   359 qed
```
```   360
```
```   361 lemma euclidean_size_gcd_le2 [simp]:
```
```   362   "b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b"
```
```   363   by (subst gcd.commute, rule euclidean_size_gcd_le1)
```
```   364
```
```   365 lemma euclidean_size_gcd_less1:
```
```   366   assumes "a \<noteq> 0" and "\<not> a dvd b"
```
```   367   shows "euclidean_size (gcd a b) < euclidean_size a"
```
```   368 proof (rule ccontr)
```
```   369   assume "\<not>euclidean_size (gcd a b) < euclidean_size a"
```
```   370   with \<open>a \<noteq> 0\<close> have A: "euclidean_size (gcd a b) = euclidean_size a"
```
```   371     by (intro le_antisym, simp_all)
```
```   372   have "a dvd gcd a b"
```
```   373     by (rule dvd_euclidean_size_eq_imp_dvd) (simp_all add: assms A)
```
```   374   hence "a dvd b" using dvd_gcdD2 by blast
```
```   375   with \<open>\<not> a dvd b\<close> show False by contradiction
```
```   376 qed
```
```   377
```
```   378 lemma euclidean_size_gcd_less2:
```
```   379   assumes "b \<noteq> 0" and "\<not> b dvd a"
```
```   380   shows "euclidean_size (gcd a b) < euclidean_size b"
```
```   381   using assms by (subst gcd.commute, rule euclidean_size_gcd_less1)
```
```   382
```
```   383 lemma euclidean_size_lcm_le1:
```
```   384   assumes "a \<noteq> 0" and "b \<noteq> 0"
```
```   385   shows "euclidean_size a \<le> euclidean_size (lcm a b)"
```
```   386 proof -
```
```   387   have "a dvd lcm a b" by (rule dvd_lcm1)
```
```   388   then obtain c where A: "lcm a b = a * c" ..
```
```   389   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_eq_0_iff)
```
```   390   then show ?thesis by (subst A, intro size_mult_mono)
```
```   391 qed
```
```   392
```
```   393 lemma euclidean_size_lcm_le2:
```
```   394   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)"
```
```   395   using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps)
```
```   396
```
```   397 lemma euclidean_size_lcm_less1:
```
```   398   assumes "b \<noteq> 0" and "\<not> b dvd a"
```
```   399   shows "euclidean_size a < euclidean_size (lcm a b)"
```
```   400 proof (rule ccontr)
```
```   401   from assms have "a \<noteq> 0" by auto
```
```   402   assume "\<not>euclidean_size a < euclidean_size (lcm a b)"
```
```   403   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a"
```
```   404     by (intro le_antisym, simp, intro euclidean_size_lcm_le1)
```
```   405   with assms have "lcm a b dvd a"
```
```   406     by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_eq_0_iff)
```
```   407   hence "b dvd a" by (rule lcm_dvdD2)
```
```   408   with \<open>\<not>b dvd a\<close> show False by contradiction
```
```   409 qed
```
```   410
```
```   411 lemma euclidean_size_lcm_less2:
```
```   412   assumes "a \<noteq> 0" and "\<not> a dvd b"
```
```   413   shows "euclidean_size b < euclidean_size (lcm a b)"
```
```   414   using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps)
```
```   415
```
```   416 end
```
```   417
```
```   418 lemma factorial_euclidean_semiring_gcdI:
```
```   419   "OFCLASS('a::{factorial_semiring_gcd, normalization_euclidean_semiring}, euclidean_semiring_gcd_class)"
```
```   420 proof
```
```   421   interpret semiring_Gcd 1 0 times
```
```   422     Euclidean_Algorithm.gcd Euclidean_Algorithm.lcm
```
```   423     Euclidean_Algorithm.Gcd Euclidean_Algorithm.Lcm
```
```   424     divide plus minus unit_factor normalize
```
```   425     rewrites "dvd.dvd op * = Rings.dvd"
```
```   426     by (fact semiring_Gcd) (simp add: dvd.dvd_def dvd_def fun_eq_iff)
```
```   427   show [simp]: "Euclidean_Algorithm.gcd = (gcd :: 'a \<Rightarrow> _)"
```
```   428   proof (rule ext)+
```
```   429     fix a b :: 'a
```
```   430     show "Euclidean_Algorithm.gcd a b = gcd a b"
```
```   431     proof (induct a b rule: eucl_induct)
```
```   432       case zero
```
```   433       then show ?case
```
```   434         by simp
```
```   435     next
```
```   436       case (mod a b)
```
```   437       moreover have "gcd b (a mod b) = gcd b a"
```
```   438         using GCD.gcd_add_mult [of b "a div b" "a mod b", symmetric]
```
```   439           by (simp add: div_mult_mod_eq)
```
```   440       ultimately show ?case
```
```   441         by (simp add: Euclidean_Algorithm.gcd_mod ac_simps)
```
```   442     qed
```
```   443   qed
```
```   444   show [simp]: "Euclidean_Algorithm.Lcm = (Lcm :: 'a set \<Rightarrow> _)"
```
```   445     by (auto intro!: Lcm_eqI GCD.dvd_Lcm GCD.Lcm_least)
```
```   446   show "Euclidean_Algorithm.lcm = (lcm :: 'a \<Rightarrow> _)"
```
```   447     by (simp add: fun_eq_iff Euclidean_Algorithm.lcm_def semiring_gcd_class.lcm_gcd)
```
```   448   show "Euclidean_Algorithm.Gcd = (Gcd :: 'a set \<Rightarrow> _)"
```
```   449     by (simp add: fun_eq_iff Euclidean_Algorithm.Gcd_def semiring_Gcd_class.Gcd_Lcm)
```
```   450 qed
```
```   451
```
```   452
```
```   453 subsection \<open>The extended euclidean algorithm\<close>
```
```   454
```
```   455 class euclidean_ring_gcd = euclidean_semiring_gcd + idom
```
```   456 begin
```
```   457
```
```   458 subclass euclidean_ring ..
```
```   459 subclass ring_gcd ..
```
```   460 subclass factorial_ring_gcd ..
```
```   461
```
```   462 function euclid_ext_aux :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> ('a \<times> 'a) \<times> 'a"
```
```   463   where "euclid_ext_aux s' s t' t r' r = (
```
```   464      if r = 0 then let c = 1 div unit_factor r' in ((s' * c, t' * c), normalize r')
```
```   465      else let q = r' div r
```
```   466           in euclid_ext_aux s (s' - q * s) t (t' - q * t) r (r' mod r))"
```
```   467   by auto
```
```   468 termination
```
```   469   by (relation "measure (\<lambda>(_, _, _, _, _, b). euclidean_size b)")
```
```   470     (simp_all add: mod_size_less)
```
```   471
```
```   472 abbreviation (input) euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<times> 'a) \<times> 'a"
```
```   473   where "euclid_ext \<equiv> euclid_ext_aux 1 0 0 1"
```
```   474
```
```   475 lemma
```
```   476   assumes "gcd r' r = gcd a b"
```
```   477   assumes "s' * a + t' * b = r'"
```
```   478   assumes "s * a + t * b = r"
```
```   479   assumes "euclid_ext_aux s' s t' t r' r = ((x, y), c)"
```
```   480   shows euclid_ext_aux_eq_gcd: "c = gcd a b"
```
```   481     and euclid_ext_aux_bezout: "x * a + y * b = gcd a b"
```
```   482 proof -
```
```   483   have "case euclid_ext_aux s' s t' t r' r of ((x, y), c) \<Rightarrow>
```
```   484     x * a + y * b = c \<and> c = gcd a b" (is "?P (euclid_ext_aux s' s t' t r' r)")
```
```   485     using assms(1-3)
```
```   486   proof (induction s' s t' t r' r rule: euclid_ext_aux.induct)
```
```   487     case (1 s' s t' t r' r)
```
```   488     show ?case
```
```   489     proof (cases "r = 0")
```
```   490       case True
```
```   491       hence "euclid_ext_aux s' s t' t r' r =
```
```   492                ((s' div unit_factor r', t' div unit_factor r'), normalize r')"
```
```   493         by (subst euclid_ext_aux.simps) (simp add: Let_def)
```
```   494       also have "?P \<dots>"
```
```   495       proof safe
```
```   496         have "s' div unit_factor r' * a + t' div unit_factor r' * b =
```
```   497                 (s' * a + t' * b) div unit_factor r'"
```
```   498           by (cases "r' = 0") (simp_all add: unit_div_commute)
```
```   499         also have "s' * a + t' * b = r'" by fact
```
```   500         also have "\<dots> div unit_factor r' = normalize r'" by simp
```
```   501         finally show "s' div unit_factor r' * a + t' div unit_factor r' * b = normalize r'" .
```
```   502       next
```
```   503         from "1.prems" True show "normalize r' = gcd a b"
```
```   504           by simp
```
```   505       qed
```
```   506       finally show ?thesis .
```
```   507     next
```
```   508       case False
```
```   509       hence "euclid_ext_aux s' s t' t r' r =
```
```   510              euclid_ext_aux s (s' - r' div r * s) t (t' - r' div r * t) r (r' mod r)"
```
```   511         by (subst euclid_ext_aux.simps) (simp add: Let_def)
```
```   512       also from "1.prems" False have "?P \<dots>"
```
```   513       proof (intro "1.IH")
```
```   514         have "(s' - r' div r * s) * a + (t' - r' div r * t) * b =
```
```   515               (s' * a + t' * b) - r' div r * (s * a + t * b)" by (simp add: algebra_simps)
```
```   516         also have "s' * a + t' * b = r'" by fact
```
```   517         also have "s * a + t * b = r" by fact
```
```   518         also have "r' - r' div r * r = r' mod r" using div_mult_mod_eq [of r' r]
```
```   519           by (simp add: algebra_simps)
```
```   520         finally show "(s' - r' div r * s) * a + (t' - r' div r * t) * b = r' mod r" .
```
```   521       qed (auto simp: algebra_simps minus_mod_eq_div_mult [symmetric] gcd.commute)
```
```   522       finally show ?thesis .
```
```   523     qed
```
```   524   qed
```
```   525   with assms(4) show "c = gcd a b" "x * a + y * b = gcd a b"
```
```   526     by simp_all
```
```   527 qed
```
```   528
```
```   529 declare euclid_ext_aux.simps [simp del]
```
```   530
```
```   531 definition bezout_coefficients :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"
```
```   532   where [code]: "bezout_coefficients a b = fst (euclid_ext a b)"
```
```   533
```
```   534 lemma bezout_coefficients_0:
```
```   535   "bezout_coefficients a 0 = (1 div unit_factor a, 0)"
```
```   536   by (simp add: bezout_coefficients_def euclid_ext_aux.simps)
```
```   537
```
```   538 lemma bezout_coefficients_left_0:
```
```   539   "bezout_coefficients 0 a = (0, 1 div unit_factor a)"
```
```   540   by (simp add: bezout_coefficients_def euclid_ext_aux.simps)
```
```   541
```
```   542 lemma bezout_coefficients:
```
```   543   assumes "bezout_coefficients a b = (x, y)"
```
```   544   shows "x * a + y * b = gcd a b"
```
```   545   using assms by (simp add: bezout_coefficients_def
```
```   546     euclid_ext_aux_bezout [of a b a b 1 0 0 1 x y] prod_eq_iff)
```
```   547
```
```   548 lemma bezout_coefficients_fst_snd:
```
```   549   "fst (bezout_coefficients a b) * a + snd (bezout_coefficients a b) * b = gcd a b"
```
```   550   by (rule bezout_coefficients) simp
```
```   551
```
```   552 lemma euclid_ext_eq [simp]:
```
```   553   "euclid_ext a b = (bezout_coefficients a b, gcd a b)" (is "?p = ?q")
```
```   554 proof
```
```   555   show "fst ?p = fst ?q"
```
```   556     by (simp add: bezout_coefficients_def)
```
```   557   have "snd (euclid_ext_aux 1 0 0 1 a b) = gcd a b"
```
```   558     by (rule euclid_ext_aux_eq_gcd [of a b a b 1 0 0 1])
```
```   559       (simp_all add: prod_eq_iff)
```
```   560   then show "snd ?p = snd ?q"
```
```   561     by simp
```
```   562 qed
```
```   563
```
```   564 declare euclid_ext_eq [symmetric, code_unfold]
```
```   565
```
```   566 end
```
```   567
```
```   568
```
```   569 subsection \<open>Typical instances\<close>
```
```   570
```
```   571 instance nat :: normalization_euclidean_semiring ..
```
```   572
```
```   573 instance nat :: euclidean_semiring_gcd
```
```   574 proof
```
```   575   interpret semiring_Gcd 1 0 times
```
```   576     "Euclidean_Algorithm.gcd" "Euclidean_Algorithm.lcm"
```
```   577     "Euclidean_Algorithm.Gcd" "Euclidean_Algorithm.Lcm"
```
```   578     divide plus minus unit_factor normalize
```
```   579     rewrites "dvd.dvd op * = Rings.dvd"
```
```   580     by (fact semiring_Gcd) (simp add: dvd.dvd_def dvd_def fun_eq_iff)
```
```   581   show [simp]: "(Euclidean_Algorithm.gcd :: nat \<Rightarrow> _) = gcd"
```
```   582   proof (rule ext)+
```
```   583     fix m n :: nat
```
```   584     show "Euclidean_Algorithm.gcd m n = gcd m n"
```
```   585     proof (induct m n rule: eucl_induct)
```
```   586       case zero
```
```   587       then show ?case
```
```   588         by simp
```
```   589     next
```
```   590       case (mod m n)
```
```   591       then have "gcd n (m mod n) = gcd n m"
```
```   592         using gcd_nat.simps [of m n] by (simp add: ac_simps)
```
```   593       with mod show ?case
```
```   594         by (simp add: Euclidean_Algorithm.gcd_mod ac_simps)
```
```   595     qed
```
```   596   qed
```
```   597   show [simp]: "(Euclidean_Algorithm.Lcm :: nat set \<Rightarrow> _) = Lcm"
```
```   598     by (auto intro!: ext Lcm_eqI)
```
```   599   show "(Euclidean_Algorithm.lcm :: nat \<Rightarrow> _) = lcm"
```
```   600     by (simp add: fun_eq_iff Euclidean_Algorithm.lcm_def semiring_gcd_class.lcm_gcd)
```
```   601   show "(Euclidean_Algorithm.Gcd :: nat set \<Rightarrow> _) = Gcd"
```
```   602     by (simp add: fun_eq_iff Euclidean_Algorithm.Gcd_def semiring_Gcd_class.Gcd_Lcm)
```
```   603 qed
```
```   604
```
```   605 lemma prime_factorization_Suc_0 [simp]: "prime_factorization (Suc 0) = {#}"
```
```   606   unfolding One_nat_def [symmetric] using prime_factorization_1 .
```
```   607
```
```   608 instance int :: normalization_euclidean_semiring ..
```
```   609
```
```   610 instance int :: euclidean_ring_gcd
```
```   611 proof
```
```   612   interpret semiring_Gcd 1 0 times
```
```   613     "Euclidean_Algorithm.gcd" "Euclidean_Algorithm.lcm"
```
```   614     "Euclidean_Algorithm.Gcd" "Euclidean_Algorithm.Lcm"
```
```   615     divide plus minus unit_factor normalize
```
```   616     rewrites "dvd.dvd op * = Rings.dvd"
```
```   617     by (fact semiring_Gcd) (simp add: dvd.dvd_def dvd_def fun_eq_iff)
```
```   618   show [simp]: "(Euclidean_Algorithm.gcd :: int \<Rightarrow> _) = gcd"
```
```   619   proof (rule ext)+
```
```   620     fix k l :: int
```
```   621     show "Euclidean_Algorithm.gcd k l = gcd k l"
```
```   622     proof (induct k l rule: eucl_induct)
```
```   623       case zero
```
```   624       then show ?case
```
```   625         by simp
```
```   626     next
```
```   627       case (mod k l)
```
```   628       have "gcd l (k mod l) = gcd l k"
```
```   629       proof (cases l "0::int" rule: linorder_cases)
```
```   630         case less
```
```   631         then show ?thesis
```
```   632           using gcd_non_0_int [of "- l" "- k"] by (simp add: ac_simps)
```
```   633       next
```
```   634         case equal
```
```   635         with mod show ?thesis
```
```   636           by simp
```
```   637       next
```
```   638         case greater
```
```   639         then show ?thesis
```
```   640           using gcd_non_0_int [of l k] by (simp add: ac_simps)
```
```   641       qed
```
```   642       with mod show ?case
```
```   643         by (simp add: Euclidean_Algorithm.gcd_mod ac_simps)
```
```   644     qed
```
```   645   qed
```
```   646   show [simp]: "(Euclidean_Algorithm.Lcm :: int set \<Rightarrow> _) = Lcm"
```
```   647     by (auto intro!: ext Lcm_eqI)
```
```   648   show "(Euclidean_Algorithm.lcm :: int \<Rightarrow> _) = lcm"
```
```   649     by (simp add: fun_eq_iff Euclidean_Algorithm.lcm_def semiring_gcd_class.lcm_gcd)
```
```   650   show "(Euclidean_Algorithm.Gcd :: int set \<Rightarrow> _) = Gcd"
```
```   651     by (simp add: fun_eq_iff Euclidean_Algorithm.Gcd_def semiring_Gcd_class.Gcd_Lcm)
```
```   652 qed
```
```   653
```
```   654 end
```