--- 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)