author paulson Sat, 13 Jun 2015 12:31:23 +0100 changeset 60441 c483f8e1805a parent 60440 3c6acb281c38 (current diff) parent 60439 b765e08f8bc0 (diff) child 60442 310853646597
Merge
--- a/src/HOL/Number_Theory/Euclidean_Algorithm.thy	Sat Jun 13 12:30:12 2015 +0100
+++ b/src/HOL/Number_Theory/Euclidean_Algorithm.thy	Sat Jun 13 12:31:23 2015 +0100
@@ -6,9 +6,63 @@
imports Complex_Main
begin

-context semiring_div
+context semidom_divide
+begin
+
+lemma mult_cancel_right [simp]:
+  "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
+proof (cases "c = 0")
+  case True then show ?thesis by simp
+next
+  case False
+  { assume "a * c = b * c"
+    then have "a * c div c = b * c div c"
+      by simp
+    with False have "a = b"
+      by simp
+  } then show ?thesis by auto
+qed
+
+lemma mult_cancel_left [simp]:
+  "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
+  using mult_cancel_right [of a c b] by (simp add: ac_simps)
+
+end
+
+context semidom_divide
begin
+
+lemma div_self [simp]:
+  assumes "a \<noteq> 0"
+  shows "a div a = 1"
+  using assms nonzero_mult_divide_cancel_left [of a 1] by simp

+lemma dvd_div_mult_self [simp]:
+  "a dvd b \<Longrightarrow> b div a * a = b"
+  by (cases "a = 0") (auto elim: dvdE simp add: ac_simps)
+
+lemma dvd_mult_div_cancel [simp]:
+  "a dvd b \<Longrightarrow> a * (b div a) = b"
+  using dvd_div_mult_self [of a b] by (simp add: ac_simps)
+
+lemma div_mult_swap:
+  assumes "c dvd b"
+  shows "a * (b div c) = (a * b) div c"
+proof (cases "c = 0")
+  case True then show ?thesis by simp
+next
+  case False from assms obtain d where "b = c * d" ..
+  moreover from False have "a * divide (d * c) c = ((a * d) * c) div c"
+    by simp
+  ultimately show ?thesis by (simp add: ac_simps)
+qed
+
+lemma dvd_div_mult:
+  assumes "c dvd b"
+  shows "b div c * a = (b * a) div c"
+  using assms div_mult_swap [of c b a] by (simp add: ac_simps)
+
+
text \<open>Units: invertible elements in a ring\<close>

abbreviation is_unit :: "'a \<Rightarrow> bool"
@@ -144,13 +198,7 @@
lemma unit_mult_left_cancel:
assumes "is_unit a"
shows "a * b = a * c \<longleftrightarrow> b = c" (is "?P \<longleftrightarrow> ?Q")
-proof
-  assume ?Q then show ?P by simp
-next
-  assume ?P then have "a * b div a = a * c div a" by simp
-  moreover from assms have "a \<noteq> 0" by auto
-  ultimately show ?Q by simp
-qed
+  using assms mult_cancel_left [of a b c] by auto

lemma unit_mult_right_cancel:
"is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
@@ -217,34 +265,33 @@
"associated a b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)

-lemma associated_iff_div_unit:
-  "associated a b \<longleftrightarrow> (\<exists>c. is_unit c \<and> a = c * b)"
-proof
-  assume "associated a b"
-  show "\<exists>c. is_unit c \<and> a = c * b"
-  proof (cases "a = 0")
-    assume "a = 0"
-    then show "\<exists>c. is_unit c \<and> a = c * b" using associated a b
-        by (intro exI[of _ 1], simp add: associated_def)
-  next
-    assume [simp]: "a \<noteq> 0"
-    hence [simp]: "a dvd b" "b dvd a" using associated a b
-        unfolding associated_def by simp_all
-    hence "1 = a div b * (b div a)"
-      by (simp add: div_mult_swap)
-    hence "is_unit (a div b)" ..
-    moreover have "a = (a div b) * b" by simp
-    ultimately show ?thesis by blast
-  qed
-next
-  assume "\<exists>c. is_unit c \<and> a = c * b"
-  then obtain c where "is_unit c" and "a = c * b" by blast
-  hence "b = a * (1 div c)" by (simp add: algebra_simps)
-  hence "a dvd b" by simp
-  moreover from a = c * b have "b dvd a" by simp
-  ultimately show "associated a b" unfolding associated_def by simp
+lemma is_unit_associatedI:
+  assumes "is_unit c" and "a = c * b"
+  shows "associated a b"
+proof (rule associatedI)
+  from a = c * b show "b dvd a" by auto
+  from is_unit c obtain d where "c * d = 1" by (rule is_unitE)
+  moreover from a = c * b have "d * a = d * (c * b)" by simp
+  ultimately have "b = a * d" by (simp add: ac_simps)
+  then show "a dvd b" ..
qed

+lemma associated_is_unitE:
+  assumes "associated a b"
+  obtains c where "is_unit c" and "a = c * b"
+proof (cases "b = 0")
+  case True with assms have "is_unit 1" and "a = 1 * b" by simp_all
+  with that show thesis .
+next
+  case False
+  from assms have "a dvd b" and "b dvd a" by (auto dest: associatedD1 associatedD2)
+  then obtain c d where "b = a * d" and "a = b * c" by (blast elim: dvdE)
+  then have "a = c * b" and "(c * d) * b = 1 * b" by (simp_all add: ac_simps)
+  with False have "c * d = 1" using mult_cancel_right [of "c * d" b 1] by simp
+  then have "is_unit c" by auto
+  with a = c * b that show thesis by blast
+qed
+
lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff
dvd_div_unit_iff unit_div_mult_swap unit_div_commute
unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel
@@ -264,78 +311,78 @@
\item division with remainder
\item a size function such that @{term "size (a mod b) < size b"}
for any @{term "b \<noteq> 0"}
-  \item a normalisation factor such that two associated numbers are equal iff
-        they are the same when divd by their normalisation factors.
+  \item a normalization factor such that two associated numbers are equal iff
+        they are the same when divd by their normalization factors.
\end{itemize}
The existence of these functions makes it possible to derive gcd and lcm functions
for any Euclidean semiring.
*}
class euclidean_semiring = semiring_div +
fixes euclidean_size :: "'a \<Rightarrow> nat"
-  fixes normalisation_factor :: "'a \<Rightarrow> 'a"
+  fixes normalization_factor :: "'a \<Rightarrow> 'a"
assumes mod_size_less [simp]:
"b \<noteq> 0 \<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 normalisation_factor_is_unit [intro,simp]:
-    "a \<noteq> 0 \<Longrightarrow> is_unit (normalisation_factor a)"
-  assumes normalisation_factor_mult: "normalisation_factor (a * b) =
-    normalisation_factor a * normalisation_factor b"
-  assumes normalisation_factor_unit: "is_unit a \<Longrightarrow> normalisation_factor a = a"
-  assumes normalisation_factor_0 [simp]: "normalisation_factor 0 = 0"
+  assumes normalization_factor_is_unit [intro,simp]:
+    "a \<noteq> 0 \<Longrightarrow> is_unit (normalization_factor a)"
+  assumes normalization_factor_mult: "normalization_factor (a * b) =
+    normalization_factor a * normalization_factor b"
+  assumes normalization_factor_unit: "is_unit a \<Longrightarrow> normalization_factor a = a"
+  assumes normalization_factor_0 [simp]: "normalization_factor 0 = 0"
begin

-lemma normalisation_factor_dvd [simp]:
-  "a \<noteq> 0 \<Longrightarrow> normalisation_factor a dvd b"
+lemma normalization_factor_dvd [simp]:
+  "a \<noteq> 0 \<Longrightarrow> normalization_factor a dvd b"
by (rule unit_imp_dvd, simp)

-lemma normalisation_factor_1 [simp]:
-  "normalisation_factor 1 = 1"
-  by (simp add: normalisation_factor_unit)
+lemma normalization_factor_1 [simp]:
+  "normalization_factor 1 = 1"
+  by (simp add: normalization_factor_unit)

-lemma normalisation_factor_0_iff [simp]:
-  "normalisation_factor a = 0 \<longleftrightarrow> a = 0"
+lemma normalization_factor_0_iff [simp]:
+  "normalization_factor a = 0 \<longleftrightarrow> a = 0"
proof
-  assume "normalisation_factor a = 0"
-  hence "\<not> is_unit (normalisation_factor a)"
+  assume "normalization_factor a = 0"
+  hence "\<not> is_unit (normalization_factor a)"
by simp
then show "a = 0" by auto
qed simp

-lemma normalisation_factor_pow:
-  "normalisation_factor (a ^ n) = normalisation_factor a ^ n"
-  by (induct n) (simp_all add: normalisation_factor_mult power_Suc2)
+lemma normalization_factor_pow:
+  "normalization_factor (a ^ n) = normalization_factor a ^ n"
+  by (induct n) (simp_all add: normalization_factor_mult power_Suc2)

-lemma normalisation_correct [simp]:
-  "normalisation_factor (a div normalisation_factor a) = (if a = 0 then 0 else 1)"
+lemma normalization_correct [simp]:
+  "normalization_factor (a div normalization_factor a) = (if a = 0 then 0 else 1)"
proof (cases "a = 0", simp)
assume "a \<noteq> 0"
-  let ?nf = "normalisation_factor"
-  from normalisation_factor_is_unit[OF a \<noteq> 0] have "?nf a \<noteq> 0"
+  let ?nf = "normalization_factor"
+  from normalization_factor_is_unit[OF a \<noteq> 0] have "?nf a \<noteq> 0"
by auto
have "?nf (a div ?nf a) * ?nf (?nf a) = ?nf (a div ?nf a * ?nf a)"
-    by (simp add: normalisation_factor_mult)
+    by (simp add: normalization_factor_mult)
also have "a div ?nf a * ?nf a = a" using a \<noteq> 0
by simp
also have "?nf (?nf a) = ?nf a" using a \<noteq> 0
-    normalisation_factor_is_unit normalisation_factor_unit by simp
-  finally have "normalisation_factor (a div normalisation_factor a) = 1"
+    normalization_factor_is_unit normalization_factor_unit by simp
+  finally have "normalization_factor (a div normalization_factor a) = 1"
using ?nf a \<noteq> 0 by (metis div_mult_self2_is_id div_self)
with a \<noteq> 0 show ?thesis by simp
qed

-lemma normalisation_0_iff [simp]:
-  "a div normalisation_factor a = 0 \<longleftrightarrow> a = 0"
+lemma normalization_0_iff [simp]:
+  "a div normalization_factor a = 0 \<longleftrightarrow> a = 0"
by (cases "a = 0", simp, subst unit_eq_div1, blast, simp)

-lemma mult_div_normalisation [simp]:
-  "b * (1 div normalisation_factor a) = b div normalisation_factor a"
+lemma mult_div_normalization [simp]:
+  "b * (1 div normalization_factor a) = b div normalization_factor a"
by (cases "a = 0") simp_all

lemma associated_iff_normed_eq:
-  "associated a b \<longleftrightarrow> a div normalisation_factor a = b div normalisation_factor b"
-proof (cases "b = 0", simp, cases "a = 0", metis associated_0(1) normalisation_0_iff, rule iffI)
-  let ?nf = normalisation_factor
+  "associated a b \<longleftrightarrow> a div normalization_factor a = b div normalization_factor b"
+proof (cases "b = 0", simp, cases "a = 0", metis associated_0(1) normalization_0_iff, rule iffI)
+  let ?nf = normalization_factor
assume "a \<noteq> 0" "b \<noteq> 0" "a div ?nf a = b div ?nf b"
hence "a = b * (?nf a div ?nf b)"
apply (subst (asm) unit_eq_div1, blast, subst (asm) unit_div_commute, blast)
@@ -343,23 +390,24 @@
done
with a \<noteq> 0 b \<noteq> 0 have "\<exists>c. is_unit c \<and> a = c * b"
by (intro exI[of _ "?nf a div ?nf b"], force simp: mult_ac)
-  with associated_iff_div_unit show "associated a b" by simp
+  then obtain c where "is_unit c" and "a = c * b" by blast
+  then show "associated a b" by (rule is_unit_associatedI)
next
-  let ?nf = normalisation_factor
+  let ?nf = normalization_factor
assume "a \<noteq> 0" "b \<noteq> 0" "associated a b"
-  with associated_iff_div_unit obtain c where "is_unit c" and "a = c * b" by blast
+  then obtain c where "is_unit c" and "a = c * b" by (blast elim: associated_is_unitE)
then show "a div ?nf a = b div ?nf b"
-    apply (simp only: a = c * b normalisation_factor_mult normalisation_factor_unit)
+    apply (simp only: a = c * b normalization_factor_mult normalization_factor_unit)
apply (rule div_mult_mult1, force)
done
qed

lemma normed_associated_imp_eq:
-  "associated a b \<Longrightarrow> normalisation_factor a \<in> {0, 1} \<Longrightarrow> normalisation_factor b \<in> {0, 1} \<Longrightarrow> a = b"
+  "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)

-lemmas normalisation_factor_dvd_iff [simp] =
-  unit_dvd_iff [OF normalisation_factor_is_unit]
+lemmas normalization_factor_dvd_iff [simp] =
+  unit_dvd_iff [OF normalization_factor_is_unit]

lemma euclidean_division:
fixes a :: 'a and b :: 'a
@@ -389,7 +437,7 @@

function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
where
-  "gcd_eucl a b = (if b = 0 then a div normalisation_factor a else gcd_eucl b (a mod b))"
+  "gcd_eucl a b = (if b = 0 then a div normalization_factor a else gcd_eucl b (a mod b))"
by (pat_completeness, simp)
termination by (relation "measure (euclidean_size \<circ> snd)", simp_all)

@@ -403,7 +451,7 @@

definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
where
-  "lcm_eucl a b = a * b div (gcd_eucl a b * normalisation_factor (a * b))"
+  "lcm_eucl a b = a * b div (gcd_eucl a b * normalization_factor (a * b))"

(* Somewhat complicated definition of Lcm that has the advantage of working
for infinite sets as well *)
@@ -413,7 +461,7 @@
"Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then
let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l =
(LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)
-       in l div normalisation_factor l
+       in l div normalization_factor l
else 0)"

definition Gcd_eucl :: "'a set \<Rightarrow> 'a"
@@ -436,11 +484,11 @@
by (rule gcd_red)

lemma gcd_0_left:
-  "gcd 0 a = a div normalisation_factor a"
+  "gcd 0 a = a div normalization_factor a"
by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, subst gcd_eucl.simps, simp add: Let_def)

lemma gcd_0:
-  "gcd a 0 = a div normalisation_factor a"
+  "gcd a 0 = a div normalization_factor a"
by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, simp add: Let_def)

lemma gcd_dvd1 [iff]: "gcd a b dvd a"
@@ -492,8 +540,8 @@
"gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+

-lemma normalisation_factor_gcd [simp]:
-  "normalisation_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)" (is "?f a b = ?g a b")
+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)"
@@ -502,7 +550,7 @@

lemma gcdI:
"k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> (\<And>l. l dvd a \<Longrightarrow> l dvd b \<Longrightarrow> l dvd k)
-    \<Longrightarrow> normalisation_factor k = (if k = 0 then 0 else 1) \<Longrightarrow> k = gcd a b"
+    \<Longrightarrow> normalization_factor k = (if k = 0 then 0 else 1) \<Longrightarrow> k = gcd a b"
by (intro normed_associated_imp_eq) (auto simp: associated_def intro: gcd_greatest)

sublocale gcd!: abel_semigroup gcd
@@ -517,7 +565,7 @@
moreover have "gcd (gcd a b) c dvd c" by simp
ultimately show "gcd (gcd a b) c dvd gcd b c"
by (rule gcd_greatest)
-    show "normalisation_factor (gcd (gcd a b) c) =  (if gcd (gcd a b) c = 0 then 0 else 1)"
+    show "normalization_factor (gcd (gcd a b) c) =  (if gcd (gcd a b) c = 0 then 0 else 1)"
by auto
fix l assume "l dvd a" and "l dvd gcd b c"
with dvd_trans[OF _ gcd_dvd1] and dvd_trans[OF _ gcd_dvd2]
@@ -532,7 +580,7 @@
qed

lemma gcd_unique: "d dvd a \<and> d dvd b \<and>
-    normalisation_factor d = (if d = 0 then 0 else 1) \<and>
+    normalization_factor d = (if d = 0 then 0 else 1) \<and>
(\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
by (rule, auto intro: gcdI simp: gcd_greatest)

@@ -546,29 +594,29 @@
by (rule sym, rule gcdI, simp_all)

lemma gcd_proj2_if_dvd:
-  "b dvd a \<Longrightarrow> gcd a b = b div normalisation_factor b"
+  "b dvd a \<Longrightarrow> gcd a b = b div normalization_factor b"
by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0)

lemma gcd_proj1_if_dvd:
-  "a dvd b \<Longrightarrow> gcd a b = a div normalisation_factor a"
+  "a dvd b \<Longrightarrow> gcd a b = a div normalization_factor a"
by (subst gcd.commute, simp add: gcd_proj2_if_dvd)

-lemma gcd_proj1_iff: "gcd m n = m div normalisation_factor m \<longleftrightarrow> m dvd n"
+lemma gcd_proj1_iff: "gcd m n = m div normalization_factor m \<longleftrightarrow> m dvd n"
proof
-  assume A: "gcd m n = m div normalisation_factor m"
+  assume A: "gcd m n = m div normalization_factor m"
show "m dvd n"
proof (cases "m = 0")
assume [simp]: "m \<noteq> 0"
-    from A have B: "m = gcd m n * normalisation_factor m"
+    from A have B: "m = gcd m n * normalization_factor m"
by (simp add: unit_eq_div2)
show ?thesis by (subst B, simp add: mult_unit_dvd_iff)
qed (insert A, simp)
next
assume "m dvd n"
-  then show "gcd m n = m div normalisation_factor m" by (rule gcd_proj1_if_dvd)
+  then show "gcd m n = m div normalization_factor m" by (rule gcd_proj1_if_dvd)
qed

-lemma gcd_proj2_iff: "gcd m n = n div normalisation_factor n \<longleftrightarrow> n dvd m"
+lemma gcd_proj2_iff: "gcd m n = n div normalization_factor n \<longleftrightarrow> n dvd m"
by (subst gcd.commute, simp add: gcd_proj1_iff)

lemma gcd_mod1 [simp]:
@@ -579,21 +627,21 @@
"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 normalisation_factor_dvd' [simp]:
-  "normalisation_factor a dvd a"
+lemma normalization_factor_dvd' [simp]:
+  "normalization_factor a dvd a"
by (cases "a = 0", simp_all)

lemma gcd_mult_distrib':
-  "k div normalisation_factor k * gcd a b = gcd (k*a) (k*b)"
+  "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: normalisation_factor_mult gcd_0 algebra_simps div_mult_div_if_dvd)
+    then show ?thesis by (simp add: normalization_factor_mult gcd_0 algebra_simps div_mult_div_if_dvd)
next
case False
-    hence "k div normalisation_factor k * gcd a b =  gcd (k * b) (k * (a mod b))"
+    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)
@@ -602,13 +650,13 @@
qed

lemma gcd_mult_distrib:
-  "k * gcd a b = gcd (k*a) (k*b) * normalisation_factor k"
+  "k * gcd a b = gcd (k*a) (k*b) * normalization_factor k"
proof-
-  let ?nf = "normalisation_factor"
+  let ?nf = "normalization_factor"
from gcd_mult_distrib'
have "gcd (k*a) (k*b) = k div ?nf k * gcd a b" ..
also have "... = k * gcd a b div ?nf k"
-    by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalisation_factor_dvd)
+    by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalization_factor_dvd)
finally show ?thesis
by simp
qed
@@ -648,7 +696,7 @@
apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps)
apply (rule gcd_dvd2)
apply (rule gcd_greatest, simp add: unit_simps, assumption)
-  apply (subst normalisation_factor_gcd, simp add: gcd_0)
+  apply (subst normalization_factor_gcd, simp add: gcd_0)
done

lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c"
@@ -660,7 +708,7 @@
lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c"
by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2)

-lemma gcd_idem: "gcd a a = a div normalisation_factor a"
+lemma gcd_idem: "gcd a a = a div normalization_factor a"
by (cases "a = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all)

lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b"
@@ -692,7 +740,7 @@
assumes "gcd c b = 1" and "c dvd a * b"
shows "c dvd a"
proof -
-  let ?nf = "normalisation_factor"
+  let ?nf = "normalization_factor"
from assms gcd_mult_distrib [of a c b]
have A: "a = gcd (a * c) (a * b) * ?nf a" by simp
from c dvd a * b show ?thesis by (subst A, simp_all add: gcd_greatest)
@@ -710,7 +758,7 @@
with A show "gcd a b dvd c" by (rule dvd_trans)
have "gcd c d dvd d" by simp
with A show "gcd a b dvd d" by (rule dvd_trans)
-  show "normalisation_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)"
+  show "normalization_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)"
by simp
fix l assume "l dvd c" and "l dvd d"
hence "l dvd gcd c d" by (rule gcd_greatest)
@@ -871,8 +919,8 @@
using div_gcd_coprime by (subst sym, auto simp: div_gcd_coprime)
hence "(gcd a b) ^ n = (gcd a b) ^ n * ..." by simp
also note gcd_mult_distrib
-  also have "normalisation_factor ((gcd a b)^n) = 1"
-    by (simp add: normalisation_factor_pow A)
+  also have "normalization_factor ((gcd a b)^n) = 1"
+    by (simp add: normalization_factor_pow A)
also have "(gcd a b)^n * (a div gcd a b)^n = a^n"
by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp)
also have "(gcd a b)^n * (b div gcd a b)^n = b^n"
@@ -994,13 +1042,13 @@
qed

lemma lcm_gcd:
-  "lcm a b = a * b div (gcd a b * normalisation_factor (a*b))"
+  "lcm a b = a * b div (gcd a b * normalization_factor (a*b))"
by (simp only: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)

lemma lcm_gcd_prod:
-  "lcm a b * gcd a b = a * b div normalisation_factor (a*b)"
+  "lcm a b * gcd a b = a * b div normalization_factor (a*b)"
proof (cases "a * b = 0")
-  let ?nf = normalisation_factor
+  let ?nf = normalization_factor
assume "a * b \<noteq> 0"
hence "gcd a b \<noteq> 0" by simp
from lcm_gcd have "lcm a b * gcd a b = gcd a b * (a * b div (?nf (a*b) * gcd a b))"
@@ -1015,8 +1063,8 @@
proof (cases "a*b = 0")
assume "a * b \<noteq> 0"
hence "gcd a b \<noteq> 0" by simp
-  let ?c = "1 div normalisation_factor (a * b)"
-  from a * b \<noteq> 0 have [simp]: "is_unit (normalisation_factor (a * b))" by simp
+  let ?c = "1 div normalization_factor (a * b)"
+  from a * b \<noteq> 0 have [simp]: "is_unit (normalization_factor (a * b))" by simp
from lcm_gcd_prod[of a b] have "lcm a b * gcd a b = a * ?c * b"
by (simp add: div_mult_swap unit_div_commute)
hence "lcm a b * gcd a b div gcd a b = a * ?c * b div gcd a b" by simp
@@ -1030,7 +1078,7 @@
lemma lcm_least:
"\<lbrakk>a dvd k; b dvd k\<rbrakk> \<Longrightarrow> lcm a b dvd k"
proof (cases "k = 0")
-  let ?nf = normalisation_factor
+  let ?nf = normalization_factor
assume "k \<noteq> 0"
hence "is_unit (?nf k)" by simp
hence "?nf k \<noteq> 0" by (metis not_is_unit_0)
@@ -1072,7 +1120,7 @@
lemma lcm_zero:
"lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
proof -
-  let ?nf = normalisation_factor
+  let ?nf = normalization_factor
{
assume "a \<noteq> 0" "b \<noteq> 0"
hence "a * b div ?nf (a * b) \<noteq> 0" by (simp add: no_zero_divisors)
@@ -1089,30 +1137,30 @@

lemma gcd_lcm:
assumes "lcm a b \<noteq> 0"
-  shows "gcd a b = a * b div (lcm a b * normalisation_factor (a * b))"
+  shows "gcd a b = a * b div (lcm a b * normalization_factor (a * b))"
proof-
from assms have "gcd a b \<noteq> 0" by (simp add: lcm_zero)
-  let ?c = "normalisation_factor (a * b)"
+  let ?c = "normalization_factor (a * b)"
from lcm a b \<noteq> 0 have "?c \<noteq> 0" by (intro notI, simp add: lcm_zero no_zero_divisors)
hence "is_unit ?c" by simp
from lcm_gcd_prod [of a b] have "gcd a b = a * b div ?c div lcm a b"
by (subst (2) div_mult_self2_is_id[OF lcm a b \<noteq> 0, symmetric], simp add: mult_ac)
also from is_unit ?c have "... = a * b div (lcm a b * ?c)"
-    by (metis ?c \<noteq> 0 div_mult_mult1 dvd_mult_div_cancel mult_commute normalisation_factor_dvd')
+    by (metis ?c \<noteq> 0 div_mult_mult1 dvd_mult_div_cancel mult_commute normalization_factor_dvd')
finally show ?thesis .
qed

-lemma normalisation_factor_lcm [simp]:
-  "normalisation_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
+lemma normalization_factor_lcm [simp]:
+  "normalization_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
proof (cases "a = 0 \<or> b = 0")
case True then show ?thesis
by (auto simp add: lcm_gcd)
next
case False
-  let ?nf = normalisation_factor
+  let ?nf = normalization_factor
from lcm_gcd_prod[of a b]
have "?nf (lcm a b) * ?nf (gcd a b) = ?nf (a*b) div ?nf (a*b)"
-    by (metis div_by_0 div_self normalisation_correct normalisation_factor_0 normalisation_factor_mult)
+    by (metis div_by_0 div_self normalization_correct normalization_factor_0 normalization_factor_mult)
also have "... = (if a*b = 0 then 0 else 1)"
by simp
finally show ?thesis using False by simp
@@ -1123,7 +1171,7 @@

lemma lcmI:
"\<lbrakk>a dvd k; b dvd k; \<And>l. a dvd l \<Longrightarrow> b dvd l \<Longrightarrow> k dvd l;
-    normalisation_factor k = (if k = 0 then 0 else 1)\<rbrakk> \<Longrightarrow> k = lcm a b"
+    normalization_factor k = (if k = 0 then 0 else 1)\<rbrakk> \<Longrightarrow> k = lcm a b"
by (intro normed_associated_imp_eq) (auto simp: associated_def intro: lcm_least)

sublocale lcm!: abel_semigroup lcm
@@ -1174,8 +1222,8 @@
assume "is_unit a \<and> is_unit b"
hence "a dvd 1" and "b dvd 1" by simp_all
hence "is_unit (lcm a b)" by (rule lcm_least)
-  hence "lcm a b = normalisation_factor (lcm a b)"
-    by (subst normalisation_factor_unit, simp_all)
+  hence "lcm a b = normalization_factor (lcm a b)"
+    by (subst normalization_factor_unit, simp_all)
also have "\<dots> = 1" using is_unit a \<and> is_unit b
by auto
finally show "lcm a b = 1" .
@@ -1191,7 +1239,7 @@

lemma lcm_unique:
"a dvd d \<and> b dvd d \<and>
-  normalisation_factor d = (if d = 0 then 0 else 1) \<and>
+  normalization_factor d = (if d = 0 then 0 else 1) \<and>
(\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
by (rule, auto intro: lcmI simp: lcm_least lcm_zero)

@@ -1204,43 +1252,43 @@
by (metis lcm_dvd2 dvd_trans)

lemma lcm_1_left [simp]:
-  "lcm 1 a = a div normalisation_factor a"
+  "lcm 1 a = a div normalization_factor a"
by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)

lemma lcm_1_right [simp]:
-  "lcm a 1 = a div normalisation_factor a"
+  "lcm a 1 = a div normalization_factor a"
using lcm_1_left [of a] by (simp add: ac_simps)

lemma lcm_coprime:
-  "gcd a b = 1 \<Longrightarrow> lcm a b = a * b div normalisation_factor (a*b)"
+  "gcd a b = 1 \<Longrightarrow> lcm a b = a * b div normalization_factor (a*b)"
by (subst lcm_gcd) simp

lemma lcm_proj1_if_dvd:
-  "b dvd a \<Longrightarrow> lcm a b = a div normalisation_factor a"
+  "b dvd a \<Longrightarrow> lcm a b = a div normalization_factor a"
by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)

lemma lcm_proj2_if_dvd:
-  "a dvd b \<Longrightarrow> lcm a b = b div normalisation_factor b"
+  "a dvd b \<Longrightarrow> lcm a b = b div normalization_factor b"
using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps)

lemma lcm_proj1_iff:
-  "lcm m n = m div normalisation_factor m \<longleftrightarrow> n dvd m"
+  "lcm m n = m div normalization_factor m \<longleftrightarrow> n dvd m"
proof
-  assume A: "lcm m n = m div normalisation_factor m"
+  assume A: "lcm m n = m div normalization_factor m"
show "n dvd m"
proof (cases "m = 0")
assume [simp]: "m \<noteq> 0"
-    from A have B: "m = lcm m n * normalisation_factor m"
+    from A have B: "m = lcm m n * normalization_factor m"
by (simp add: unit_eq_div2)
show ?thesis by (subst B, simp)
qed simp
next
assume "n dvd m"
-  then show "lcm m n = m div normalisation_factor m" by (rule lcm_proj1_if_dvd)
+  then show "lcm m n = m div normalization_factor m" by (rule lcm_proj1_if_dvd)
qed

lemma lcm_proj2_iff:
-  "lcm m n = n div normalisation_factor n \<longleftrightarrow> m dvd n"
+  "lcm m n = n div normalization_factor n \<longleftrightarrow> m dvd n"
using lcm_proj1_iff [of n m] by (simp add: ac_simps)

lemma euclidean_size_lcm_le1:
@@ -1282,7 +1330,7 @@
apply (rule dvd_trans[of _ "b * a"], simp, rule lcm_dvd1)
apply (rule lcm_dvd2)
apply (rule lcm_least, simp add: unit_simps, assumption)
-  apply (subst normalisation_factor_lcm, simp add: lcm_zero)
+  apply (subst normalization_factor_lcm, simp add: lcm_zero)
done

lemma lcm_mult_unit2:
@@ -1327,11 +1375,11 @@

lemma dvd_Lcm [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm A"
and Lcm_dvd [simp]: "(\<forall>a\<in>A. a dvd l') \<Longrightarrow> Lcm A dvd l'"
-  and normalisation_factor_Lcm [simp]:
-          "normalisation_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
+  and normalization_factor_Lcm [simp]:
+          "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)"
proof -
have "(\<forall>a\<in>A. a dvd Lcm A) \<and> (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> Lcm A dvd l') \<and>
-    normalisation_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)
+    normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis)
proof (cases "\<exists>l. l \<noteq>  0 \<and> (\<forall>a\<in>A. a dvd l)")
case False
hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def)
@@ -1373,13 +1421,13 @@
hence "l dvd l'" by (blast dest: dvd_gcd_D2)
}

-    with (\<forall>a\<in>A. a dvd l) and normalisation_factor_is_unit[OF l \<noteq> 0] and l \<noteq> 0
-      have "(\<forall>a\<in>A. a dvd l div normalisation_factor l) \<and>
-        (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> l div normalisation_factor l dvd l') \<and>
-        normalisation_factor (l div normalisation_factor l) =
-        (if l div normalisation_factor l = 0 then 0 else 1)"
+    with (\<forall>a\<in>A. a dvd l) and normalization_factor_is_unit[OF l \<noteq> 0] and l \<noteq> 0
+      have "(\<forall>a\<in>A. a dvd l div normalization_factor l) \<and>
+        (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> l div normalization_factor l dvd l') \<and>
+        normalization_factor (l div normalization_factor l) =
+        (if l div normalization_factor l = 0 then 0 else 1)"
by (auto simp: unit_simps)
-    also from True have "l div normalisation_factor l = Lcm A"
+    also from True have "l div normalization_factor l = Lcm A"
by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def)
finally show ?thesis .
qed
@@ -1387,12 +1435,12 @@

{fix a assume "a \<in> A" then show "a dvd Lcm A" using A by blast}
{fix l' assume "\<forall>a\<in>A. a dvd l'" then show "Lcm A dvd l'" using A by blast}
-  from A show "normalisation_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast
+  from A show "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast
qed

lemma LcmI:
"(\<And>a. a\<in>A \<Longrightarrow> a dvd l) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. a dvd l') \<Longrightarrow> l dvd l') \<Longrightarrow>
-      normalisation_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Lcm A"
+      normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Lcm A"
by (intro normed_associated_imp_eq)
(auto intro: Lcm_dvd dvd_Lcm simp: associated_def)

@@ -1453,8 +1501,8 @@
apply (simp add: l\<^sub>0_props)
done
from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all
-    hence "l div normalisation_factor l \<noteq> 0" by simp
-    also from ex have "l div normalisation_factor l = Lcm A"
+    hence "l div normalization_factor l \<noteq> 0" by simp
+    also from ex have "l div normalization_factor l = Lcm A"
by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def)
finally show False using Lcm A = 0 by contradiction
qed
@@ -1508,7 +1556,7 @@
using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps)

lemma Lcm_singleton [simp]:
-  "Lcm {a} = a div normalisation_factor a"
+  "Lcm {a} = a div normalization_factor a"
by simp

lemma Lcm_2 [simp]:
@@ -1519,21 +1567,21 @@
lemma Lcm_coprime:
assumes "finite A" and "A \<noteq> {}"
assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1"
-  shows "Lcm A = \<Prod>A div normalisation_factor (\<Prod>A)"
+  shows "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"
using assms proof (induct rule: finite_ne_induct)
case (insert a A)
have "Lcm (insert a A) = lcm a (Lcm A)" by simp
-  also from insert have "Lcm A = \<Prod>A div normalisation_factor (\<Prod>A)" by blast
+  also from insert have "Lcm A = \<Prod>A div normalization_factor (\<Prod>A)" by blast
also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2)
also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto
-  with insert have "lcm a (\<Prod>A) = \<Prod>(insert a A) div normalisation_factor (\<Prod>(insert a A))"
+  with insert have "lcm a (\<Prod>A) = \<Prod>(insert a A) div normalization_factor (\<Prod>(insert a A))"
by (simp add: lcm_coprime)
finally show ?case .
qed simp

lemma Lcm_coprime':
"card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1)
-    \<Longrightarrow> Lcm A = \<Prod>A div normalisation_factor (\<Prod>A)"
+    \<Longrightarrow> Lcm A = \<Prod>A div normalization_factor (\<Prod>A)"
by (rule Lcm_coprime) (simp_all add: card_eq_0_iff)

lemma Gcd_Lcm:
@@ -1542,8 +1590,8 @@

lemma Gcd_dvd [simp]: "a \<in> A \<Longrightarrow> Gcd A dvd a"
and dvd_Gcd [simp]: "(\<forall>a\<in>A. g' dvd a) \<Longrightarrow> g' dvd Gcd A"
-  and normalisation_factor_Gcd [simp]:
-    "normalisation_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
+  and normalization_factor_Gcd [simp]:
+    "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
proof -
fix a assume "a \<in> A"
hence "Lcm {d. \<forall>a\<in>A. d dvd a} dvd a" by (intro Lcm_dvd) blast
@@ -1553,13 +1601,13 @@
hence "g' dvd Lcm {d. \<forall>a\<in>A. d dvd a}" by (intro dvd_Lcm) blast
then show "g' dvd Gcd A" by (simp add: Gcd_Lcm)
next
-  show "normalisation_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
+  show "normalization_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)"
by (simp add: Gcd_Lcm)
qed

lemma GcdI:
"(\<And>a. a\<in>A \<Longrightarrow> l dvd a) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. l' dvd a) \<Longrightarrow> l' dvd l) \<Longrightarrow>
-    normalisation_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Gcd A"
+    normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Gcd A"
by (intro normed_associated_imp_eq)
(auto intro: Gcd_dvd dvd_Gcd simp: associated_def)

@@ -1600,12 +1648,15 @@
"Gcd (set xs) = fold gcd xs 0"
using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)

-lemma Gcd_singleton [simp]: "Gcd {a} = a div normalisation_factor a"
+lemma Gcd_singleton [simp]: "Gcd {a} = a div normalization_factor a"
by (simp add: gcd_0)

lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"
by (simp only: Gcd_insert Gcd_empty gcd_0) (cases "b = 0", simp, rule gcd_div_unit2, simp)

+subclass semiring_gcd
+  by unfold_locales (simp_all add: gcd_greatest_iff)
+
end

text {*
@@ -1620,6 +1671,8 @@

subclass euclidean_ring ..

+subclass ring_gcd ..
+
lemma gcd_neg1 [simp]:
"gcd (-a) b = gcd a b"
by (rule sym, rule gcdI, simp_all add: gcd_greatest)
@@ -1665,7 +1718,7 @@
function euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" where
"euclid_ext a b =
(if b = 0 then
-        let c = 1 div normalisation_factor a in (c, 0, a * c)
+        let c = 1 div normalization_factor a in (c, 0, a * c)
else
case euclid_ext b (a mod b) of
(s,t,c) \<Rightarrow> (t, s - t * (a div b), c))"
@@ -1675,7 +1728,7 @@
declare euclid_ext.simps [simp del]

lemma euclid_ext_0:
-  "euclid_ext a 0 = (1 div normalisation_factor a, 0, a div normalisation_factor a)"
+  "euclid_ext a 0 = (1 div normalization_factor a, 0, a div normalization_factor a)"
by (subst euclid_ext.simps) (simp add: Let_def)

lemma euclid_ext_non_0:
@@ -1739,7 +1792,7 @@
lemma bezout: "\<exists>s t. s * a + t * b = gcd a b"
using euclid_ext'_correct by blast

-lemma euclid_ext'_0 [simp]: "euclid_ext' a 0 = (1 div normalisation_factor a, 0)"
+lemma euclid_ext'_0 [simp]: "euclid_ext' a 0 = (1 div normalization_factor a, 0)"
by (simp add: bezw_def euclid_ext'_def euclid_ext_0)

lemma euclid_ext'_non_0: "b \<noteq> 0 \<Longrightarrow> euclid_ext' a b = (snd (euclid_ext' b (a mod b)),
@@ -1756,7 +1809,7 @@
"euclidean_size_nat = (id :: nat \<Rightarrow> nat)"

definition [simp]:
-  "normalisation_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)"
+  "normalization_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)"

instance proof
qed simp_all
@@ -1770,7 +1823,7 @@
"euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"

definition [simp]:
-  "normalisation_factor_int = (sgn :: int \<Rightarrow> int)"
+  "normalization_factor_int = (sgn :: int \<Rightarrow> int)"

instance proof
case goal2 then show ?case by (auto simp add: abs_mult nat_mult_distrib)