--- a/src/HOL/Number_Theory/Euclidean_Algorithm.thy Thu Jun 25 12:41:43 2015 +0200
+++ b/src/HOL/Number_Theory/Euclidean_Algorithm.thy Thu Jun 25 15:01:41 2015 +0200
@@ -22,8 +22,8 @@
class euclidean_semiring = semiring_div +
fixes euclidean_size :: "'a \<Rightarrow> nat"
fixes normalization_factor :: "'a \<Rightarrow> 'a"
- assumes mod_size_less [simp]:
- "b \<noteq> 0 \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b"
+ assumes mod_size_less:
+ "b \<noteq> 0 \<Longrightarrow> \<not> b dvd a \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b"
assumes size_mult_mono:
"b \<noteq> 0 \<Longrightarrow> euclidean_size (a * b) \<ge> euclidean_size a"
assumes normalization_factor_is_unit [intro,simp]:
@@ -107,48 +107,102 @@
lemma normed_associated_imp_eq:
"associated a b \<Longrightarrow> normalization_factor a \<in> {0, 1} \<Longrightarrow> normalization_factor b \<in> {0, 1} \<Longrightarrow> a = b"
by (simp add: associated_iff_normed_eq, elim disjE, simp_all)
-
+
+lemma normed_dvd [iff]:
+ "a div normalization_factor a dvd a"
+proof (cases "a = 0")
+ case True then show ?thesis by simp
+next
+ case False
+ then have "a = a div normalization_factor a * normalization_factor a"
+ by (auto intro: unit_div_mult_self)
+ then show ?thesis ..
+qed
+
+lemma dvd_normed [iff]:
+ "a dvd a div normalization_factor a"
+proof (cases "a = 0")
+ case True then show ?thesis by simp
+next
+ case False
+ then have "a div normalization_factor a = a * (1 div normalization_factor a)"
+ by (auto intro: unit_mult_div_div)
+ then show ?thesis ..
+qed
+
+lemma associated_normed:
+ "associated (a div normalization_factor a) a"
+ by (rule associatedI) simp_all
+
+lemma normalization_factor_dvd' [simp]:
+ "normalization_factor a dvd a"
+ by (cases "a = 0", simp_all)
+
lemmas normalization_factor_dvd_iff [simp] =
unit_dvd_iff [OF normalization_factor_is_unit]
lemma euclidean_division:
fixes a :: 'a and b :: 'a
- assumes "b \<noteq> 0"
+ assumes "b \<noteq> 0" and "\<not> b dvd a"
obtains s and t where "a = s * b + t"
and "euclidean_size t < euclidean_size b"
proof -
- from div_mod_equality[of a b 0]
+ from div_mod_equality [of a b 0]
have "a = a div b * b + a mod b" by simp
- with that and assms show ?thesis by force
+ with that and assms show ?thesis by (auto simp add: mod_size_less)
qed
lemma dvd_euclidean_size_eq_imp_dvd:
assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"
shows "a dvd b"
-proof (subst dvd_eq_mod_eq_0, rule ccontr)
- assume "b mod a \<noteq> 0"
+proof (rule ccontr)
+ assume "\<not> a dvd b"
+ then have "b mod a \<noteq> 0" by (simp add: mod_eq_0_iff_dvd)
from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)
from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast
with \<open>b mod a \<noteq> 0\<close> have "c \<noteq> 0" by auto
with \<open>b mod a = b * c\<close> have "euclidean_size (b mod a) \<ge> euclidean_size b"
using size_mult_mono by force
- moreover from \<open>a \<noteq> 0\<close> have "euclidean_size (b mod a) < euclidean_size a"
+ moreover from \<open>\<not> a dvd b\<close> and \<open>a \<noteq> 0\<close>
+ have "euclidean_size (b mod a) < euclidean_size a"
using mod_size_less by blast
ultimately show False using size_eq by simp
qed
function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
where
- "gcd_eucl a b = (if b = 0 then a div normalization_factor a else gcd_eucl b (a mod b))"
+ "gcd_eucl a b = (if b = 0 then a div normalization_factor a
+ else if b dvd a then b div normalization_factor b
+ else gcd_eucl b (a mod b))"
by (pat_completeness, simp)
-termination by (relation "measure (euclidean_size \<circ> snd)", simp_all)
+termination
+ by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less)
declare gcd_eucl.simps [simp del]
-lemma gcd_induct: "\<lbrakk>\<And>b. P b 0; \<And>a b. 0 \<noteq> b \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b\<rbrakk> \<Longrightarrow> P a b"
+lemma gcd_eucl_induct [case_names zero mod]:
+ assumes H1: "\<And>b. P b 0"
+ and H2: "\<And>a b. b \<noteq> 0 \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b"
+ shows "P a b"
proof (induct a b rule: gcd_eucl.induct)
- case ("1" m n)
- then show ?case by (cases "n = 0") auto
+ case ("1" a b)
+ show ?case
+ proof (cases "b = 0")
+ case True then show "P a b" by simp (rule H1)
+ next
+ case False
+ have "P b (a mod b)"
+ proof (cases "b dvd a")
+ case False with \<open>b \<noteq> 0\<close> show "P b (a mod b)"
+ by (rule "1.hyps")
+ next
+ case True then have "a mod b = 0"
+ by (simp add: mod_eq_0_iff_dvd)
+ then show "P b (a mod b)" by simp (rule H1)
+ qed
+ with \<open>b \<noteq> 0\<close> show "P a b"
+ by (blast intro: H2)
+ qed
qed
definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
@@ -179,7 +233,8 @@
lemma gcd_red:
"gcd a b = gcd b (a mod b)"
- by (metis gcd_eucl.simps mod_0 mod_by_0 gcd_gcd_eucl)
+ by (cases "b dvd a")
+ (auto simp add: gcd_gcd_eucl gcd_eucl.simps [of a b] gcd_eucl.simps [of 0 a] gcd_eucl.simps [of b 0])
lemma gcd_non_0:
"b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)"
@@ -195,22 +250,9 @@
lemma gcd_dvd1 [iff]: "gcd a b dvd a"
and gcd_dvd2 [iff]: "gcd a b dvd b"
-proof (induct a b rule: gcd_eucl.induct)
- fix a b :: 'a
- assume IH1: "b \<noteq> 0 \<Longrightarrow> gcd b (a mod b) dvd b"
- assume IH2: "b \<noteq> 0 \<Longrightarrow> gcd b (a mod b) dvd (a mod b)"
-
- have "gcd a b dvd a \<and> gcd a b dvd b"
- proof (cases "b = 0")
- case True
- then show ?thesis by (cases "a = 0", simp_all add: gcd_0)
- next
- case False
- with IH1 and IH2 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff)
- qed
- then show "gcd a b dvd a" "gcd a b dvd b" by simp_all
-qed
-
+ by (induct a b rule: gcd_eucl_induct)
+ (simp_all add: gcd_0 gcd_non_0 dvd_mod_iff)
+
lemma dvd_gcd_D1: "k dvd gcd m n \<Longrightarrow> k dvd m"
by (rule dvd_trans, assumption, rule gcd_dvd1)
@@ -220,16 +262,12 @@
lemma gcd_greatest:
fixes k a b :: 'a
shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b"
-proof (induct a b rule: gcd_eucl.induct)
- case (1 a b)
- show ?case
- proof (cases "b = 0")
- assume "b = 0"
- with 1 show ?thesis by (cases "a = 0", simp_all add: gcd_0)
- next
- assume "b \<noteq> 0"
- with 1 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff)
- qed
+proof (induct a b rule: gcd_eucl_induct)
+ case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_0)
+next
+ case (mod a b)
+ then show ?case
+ by (simp add: gcd_non_0 dvd_mod_iff)
qed
lemma dvd_gcd_iff:
@@ -244,11 +282,8 @@
lemma normalization_factor_gcd [simp]:
"normalization_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)" (is "?f a b = ?g a b")
-proof (induct a b rule: gcd_eucl.induct)
- fix a b :: 'a
- assume IH: "b \<noteq> 0 \<Longrightarrow> ?f b (a mod b) = ?g b (a mod b)"
- then show "?f a b = ?g a b" by (cases "b = 0", auto simp: gcd_non_0 gcd_0)
-qed
+ by (induct a b rule: gcd_eucl_induct)
+ (auto simp add: gcd_0 gcd_non_0)
lemma gcdI:
"k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> (\<And>l. l dvd a \<Longrightarrow> l dvd b \<Longrightarrow> l dvd k)
@@ -329,25 +364,24 @@
"gcd a (b mod a) = gcd a b"
by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff)
-lemma normalization_factor_dvd' [simp]:
- "normalization_factor a dvd a"
- by (cases "a = 0", simp_all)
-
lemma gcd_mult_distrib':
- "k div normalization_factor k * gcd a b = gcd (k*a) (k*b)"
-proof (induct a b rule: gcd_eucl.induct)
- case (1 a b)
- show ?case
- proof (cases "b = 0")
- case True
- then show ?thesis by (simp add: normalization_factor_mult gcd_0 algebra_simps div_mult_div_if_dvd)
- next
- case False
- hence "k div normalization_factor k * gcd a b = gcd (k * b) (k * (a mod b))"
- using 1 by (subst gcd_red, simp)
- also have "... = gcd (k * a) (k * b)"
- by (simp add: mult_mod_right gcd.commute)
- finally show ?thesis .
+ "c div normalization_factor c * gcd a b = gcd (c * a) (c * b)"
+proof (cases "c = 0")
+ case True then show ?thesis by (simp_all add: gcd_0)
+next
+ case False then have [simp]: "is_unit (normalization_factor c)" by simp
+ show ?thesis
+ proof (induct a b rule: gcd_eucl_induct)
+ case (zero a) show ?case
+ proof (cases "a = 0")
+ case True then show ?thesis by (simp add: gcd_0)
+ next
+ case False then have "is_unit (normalization_factor a)" by simp
+ then show ?thesis
+ by (simp add: gcd_0 unit_div_commute unit_div_mult_swap normalization_factor_mult is_unit_div_mult2_eq)
+ qed
+ case (mod a b)
+ then show ?case by (simp add: mult_mod_right gcd.commute)
qed
qed
@@ -1421,11 +1455,14 @@
"euclid_ext a b =
(if b = 0 then
let c = 1 div normalization_factor a in (c, 0, a * c)
- else
+ else if b dvd a then
+ let c = 1 div normalization_factor b in (0, c, b * c)
+ else
case euclid_ext b (a mod b) of
- (s,t,c) \<Rightarrow> (t, s - t * (a div b), c))"
+ (s, t, c) \<Rightarrow> (t, s - t * (a div b), c))"
by (pat_completeness, simp)
- termination by (relation "measure (euclidean_size \<circ> snd)", simp_all)
+ termination by (relation "measure (euclidean_size \<circ> snd)")
+ (simp_all add: mod_size_less)
declare euclid_ext.simps [simp del]
@@ -1435,51 +1472,41 @@
lemma euclid_ext_non_0:
"b \<noteq> 0 \<Longrightarrow> euclid_ext a b = (case euclid_ext b (a mod b) of
- (s,t,c) \<Rightarrow> (t, s - t * (a div b), c))"
- by (subst euclid_ext.simps) simp
+ (s, t, c) \<Rightarrow> (t, s - t * (a div b), c))"
+ apply (subst euclid_ext.simps)
+ apply (auto simp add: split: if_splits)
+ apply (subst euclid_ext.simps)
+ apply (auto simp add: split: if_splits)
+ done
definition euclid_ext' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"
where
"euclid_ext' a b = (case euclid_ext a b of (s, t, _) \<Rightarrow> (s, t))"
lemma euclid_ext_gcd [simp]:
- "(case euclid_ext a b of (_,_,t) \<Rightarrow> t) = gcd a b"
-proof (induct a b rule: euclid_ext.induct)
- case (1 a b)
- then show ?case
- proof (cases "b = 0")
- case True
- then show ?thesis by
- (simp add: euclid_ext_0 unit_div mult_ac unit_simps gcd_0)
- next
- case False with 1 show ?thesis
- by (simp add: euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)
- qed
-qed
+ "(case euclid_ext a b of (_, _ , t) \<Rightarrow> t) = gcd a b"
+ by (induct a b rule: gcd_eucl_induct)
+ (simp_all add: euclid_ext_0 gcd_0 euclid_ext_non_0 ac_simps split: prod.split prod.split_asm)
lemma euclid_ext_gcd' [simp]:
"euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b"
by (insert euclid_ext_gcd[of a b], drule (1) subst, simp)
lemma euclid_ext_correct:
- "case euclid_ext a b of (s,t,c) \<Rightarrow> s*a + t*b = c"
-proof (induct a b rule: euclid_ext.induct)
- case (1 a b)
- show ?case
- proof (cases "b = 0")
- case True
- then show ?thesis by (simp add: euclid_ext_0 mult_ac)
- next
- case False
- obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)"
- by (cases "euclid_ext b (a mod b)", blast)
- from 1 have "c = s * b + t * (a mod b)" by (simp add: stc False)
- also have "... = t*((a div b)*b + a mod b) + (s - t * (a div b))*b"
- by (simp add: algebra_simps)
- also have "(a div b)*b + a mod b = a" using mod_div_equality .
- finally show ?thesis
- by (subst euclid_ext.simps, simp add: False stc)
- qed
+ "case euclid_ext a b of (s, t, c) \<Rightarrow> s * a + t * b = c"
+proof (induct a b rule: gcd_eucl_induct)
+ case (zero a) then show ?case
+ by (simp add: euclid_ext_0 ac_simps)
+next
+ case (mod a b)
+ obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)"
+ by (cases "euclid_ext b (a mod b)") blast
+ with mod have "c = s * b + t * (a mod b)" by simp
+ also have "... = t * ((a div b) * b + a mod b) + (s - t * (a div b)) * b"
+ by (simp add: algebra_simps)
+ also have "(a div b) * b + a mod b = a" using mod_div_equality .
+ finally show ?case
+ by (subst euclid_ext.simps) (simp add: stc mod ac_simps)
qed
lemma euclid_ext'_correct: