--- a/src/HOL/Library/GCD.thy Fri Jul 11 23:17:25 2008 +0200
+++ b/src/HOL/Library/GCD.thy Mon Jul 14 11:04:42 2008 +0200
@@ -18,64 +18,62 @@
definition
is_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where -- {* @{term gcd} as a relation *}
- [code func del]: "is_gcd p m n \<longleftrightarrow> p dvd m \<and> p dvd n \<and>
+ [code func del]: "is_gcd m n p \<longleftrightarrow> p dvd m \<and> p dvd n \<and>
(\<forall>d. d dvd m \<longrightarrow> d dvd n \<longrightarrow> d dvd p)"
text {* Uniqueness *}
-lemma is_gcd_unique: "is_gcd m a b \<Longrightarrow> is_gcd n a b \<Longrightarrow> m = n"
+lemma is_gcd_unique: "is_gcd a b m \<Longrightarrow> is_gcd a b n \<Longrightarrow> m = n"
by (simp add: is_gcd_def) (blast intro: dvd_anti_sym)
text {* Connection to divides relation *}
-lemma is_gcd_dvd: "is_gcd m a b \<Longrightarrow> k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd m"
+lemma is_gcd_dvd: "is_gcd a b m \<Longrightarrow> k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd m"
by (auto simp add: is_gcd_def)
text {* Commutativity *}
-lemma is_gcd_commute: "is_gcd k m n = is_gcd k n m"
+lemma is_gcd_commute: "is_gcd m n k = is_gcd n m k"
by (auto simp add: is_gcd_def)
subsection {* GCD on nat by Euclid's algorithm *}
-fun
- gcd :: "nat \<times> nat => nat"
-where
- "gcd (m, n) = (if n = 0 then m else gcd (n, m mod n))"
+fun gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
+ "gcd m n = (if n = 0 then m else gcd n (m mod n))"
-lemma gcd_induct:
+thm gcd.induct
+
+lemma gcd_induct [case_names "0" rec]:
fixes m n :: nat
assumes "\<And>m. P m 0"
and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n"
shows "P m n"
-apply (rule gcd.induct [of "split P" "(m, n)", unfolded Product_Type.split])
-apply (case_tac "n = 0")
-apply simp_all
-using assms apply simp_all
-done
+proof (induct m n rule: gcd.induct)
+ case (1 m n) with assms show ?case by (cases "n = 0") simp_all
+qed
-lemma gcd_0 [simp]: "gcd (m, 0) = m"
+lemma gcd_0 [simp]: "gcd m 0 = m"
by simp
-lemma gcd_0_left [simp]: "gcd (0, m) = m"
+lemma gcd_0_left [simp]: "gcd 0 m = m"
by simp
-lemma gcd_non_0: "n > 0 \<Longrightarrow> gcd (m, n) = gcd (n, m mod n)"
+lemma gcd_non_0: "n > 0 \<Longrightarrow> gcd m n = gcd n (m mod n)"
by simp
-lemma gcd_1 [simp]: "gcd (m, Suc 0) = 1"
+lemma gcd_1 [simp]: "gcd m (Suc 0) = 1"
by simp
declare gcd.simps [simp del]
text {*
- \medskip @{term "gcd (m, n)"} divides @{text m} and @{text n}. The
+ \medskip @{term "gcd m n"} divides @{text m} and @{text n}. The
conjunctions don't seem provable separately.
*}
-lemma gcd_dvd1 [iff]: "gcd (m, n) dvd m"
- and gcd_dvd2 [iff]: "gcd (m, n) dvd n"
+lemma gcd_dvd1 [iff]: "gcd m n dvd m"
+ and gcd_dvd2 [iff]: "gcd m n dvd n"
apply (induct m n rule: gcd_induct)
apply (simp_all add: gcd_non_0)
apply (blast dest: dvd_mod_imp_dvd)
@@ -84,50 +82,50 @@
text {*
\medskip Maximality: for all @{term m}, @{term n}, @{term k}
naturals, if @{term k} divides @{term m} and @{term k} divides
- @{term n} then @{term k} divides @{term "gcd (m, n)"}.
+ @{term n} then @{term k} divides @{term "gcd m n"}.
*}
-lemma gcd_greatest: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd gcd (m, n)"
+lemma gcd_greatest: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd gcd m n"
by (induct m n rule: gcd_induct) (simp_all add: gcd_non_0 dvd_mod)
text {*
\medskip Function gcd yields the Greatest Common Divisor.
*}
-lemma is_gcd: "is_gcd (gcd (m, n)) m n"
+lemma is_gcd: "is_gcd m n (gcd m n) "
by (simp add: is_gcd_def gcd_greatest)
subsection {* Derived laws for GCD *}
-lemma gcd_greatest_iff [iff]: "k dvd gcd (m, n) \<longleftrightarrow> k dvd m \<and> k dvd n"
+lemma gcd_greatest_iff [iff]: "k dvd gcd m n \<longleftrightarrow> k dvd m \<and> k dvd n"
by (blast intro!: gcd_greatest intro: dvd_trans)
-lemma gcd_zero: "gcd (m, n) = 0 \<longleftrightarrow> m = 0 \<and> n = 0"
+lemma gcd_zero: "gcd m n = 0 \<longleftrightarrow> m = 0 \<and> n = 0"
by (simp only: dvd_0_left_iff [symmetric] gcd_greatest_iff)
-lemma gcd_commute: "gcd (m, n) = gcd (n, m)"
+lemma gcd_commute: "gcd m n = gcd n m"
apply (rule is_gcd_unique)
apply (rule is_gcd)
apply (subst is_gcd_commute)
apply (simp add: is_gcd)
done
-lemma gcd_assoc: "gcd (gcd (k, m), n) = gcd (k, gcd (m, n))"
+lemma gcd_assoc: "gcd (gcd k m) n = gcd k (gcd m n)"
apply (rule is_gcd_unique)
apply (rule is_gcd)
apply (simp add: is_gcd_def)
apply (blast intro: dvd_trans)
done
-lemma gcd_1_left [simp]: "gcd (Suc 0, m) = 1"
+lemma gcd_1_left [simp]: "gcd (Suc 0) m = 1"
by (simp add: gcd_commute)
text {*
\medskip Multiplication laws
*}
-lemma gcd_mult_distrib2: "k * gcd (m, n) = gcd (k * m, k * n)"
+lemma gcd_mult_distrib2: "k * gcd m n = gcd (k * m) (k * n)"
-- {* \cite[page 27]{davenport92} *}
apply (induct m n rule: gcd_induct)
apply simp
@@ -135,26 +133,26 @@
apply (simp_all add: mod_geq gcd_non_0 mod_mult_distrib2)
done
-lemma gcd_mult [simp]: "gcd (k, k * n) = k"
+lemma gcd_mult [simp]: "gcd k (k * n) = k"
apply (rule gcd_mult_distrib2 [of k 1 n, simplified, symmetric])
done
-lemma gcd_self [simp]: "gcd (k, k) = k"
+lemma gcd_self [simp]: "gcd k k = k"
apply (rule gcd_mult [of k 1, simplified])
done
-lemma relprime_dvd_mult: "gcd (k, n) = 1 ==> k dvd m * n ==> k dvd m"
+lemma relprime_dvd_mult: "gcd k n = 1 ==> k dvd m * n ==> k dvd m"
apply (insert gcd_mult_distrib2 [of m k n])
apply simp
apply (erule_tac t = m in ssubst)
apply simp
done
-lemma relprime_dvd_mult_iff: "gcd (k, n) = 1 ==> (k dvd m * n) = (k dvd m)"
+lemma relprime_dvd_mult_iff: "gcd k n = 1 ==> (k dvd m * n) = (k dvd m)"
apply (blast intro: relprime_dvd_mult dvd_trans)
done
-lemma gcd_mult_cancel: "gcd (k, n) = 1 ==> gcd (k * m, n) = gcd (m, n)"
+lemma gcd_mult_cancel: "gcd k n = 1 ==> gcd (k * m) n = gcd m n"
apply (rule dvd_anti_sym)
apply (rule gcd_greatest)
apply (rule_tac n = k in relprime_dvd_mult)
@@ -167,29 +165,29 @@
text {* \medskip Addition laws *}
-lemma gcd_add1 [simp]: "gcd (m + n, n) = gcd (m, n)"
+lemma gcd_add1 [simp]: "gcd (m + n) n = gcd m n"
apply (case_tac "n = 0")
apply (simp_all add: gcd_non_0)
done
-lemma gcd_add2 [simp]: "gcd (m, m + n) = gcd (m, n)"
+lemma gcd_add2 [simp]: "gcd m (m + n) = gcd m n"
proof -
- have "gcd (m, m + n) = gcd (m + n, m)" by (rule gcd_commute)
- also have "... = gcd (n + m, m)" by (simp add: add_commute)
- also have "... = gcd (n, m)" by simp
- also have "... = gcd (m, n)" by (rule gcd_commute)
+ have "gcd m (m + n) = gcd (m + n) m" by (rule gcd_commute)
+ also have "... = gcd (n + m) m" by (simp add: add_commute)
+ also have "... = gcd n m" by simp
+ also have "... = gcd m n" by (rule gcd_commute)
finally show ?thesis .
qed
-lemma gcd_add2' [simp]: "gcd (m, n + m) = gcd (m, n)"
+lemma gcd_add2' [simp]: "gcd m (n + m) = gcd m n"
apply (subst add_commute)
apply (rule gcd_add2)
done
-lemma gcd_add_mult: "gcd (m, k * m + n) = gcd (m, n)"
+lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n"
by (induct k) (simp_all add: add_assoc)
-lemma gcd_dvd_prod: "gcd (m, n) dvd m * n"
+lemma gcd_dvd_prod: "gcd m n dvd m * n"
using mult_dvd_mono [of 1] by auto
text {*
@@ -198,12 +196,12 @@
lemma div_gcd_relprime:
assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
- shows "gcd (a div gcd(a,b), b div gcd(a,b)) = 1"
+ shows "gcd (a div gcd a b) (b div gcd a b) = 1"
proof -
- let ?g = "gcd (a, b)"
+ let ?g = "gcd a b"
let ?a' = "a div ?g"
let ?b' = "b div ?g"
- let ?g' = "gcd (?a', ?b')"
+ let ?g' = "gcd ?a' ?b'"
have dvdg: "?g dvd a" "?g dvd b" by simp_all
have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
from dvdg dvdg' obtain ka kb ka' kb' where
@@ -222,28 +220,28 @@
subsection {* LCM defined by GCD *}
definition
- lcm :: "nat \<times> nat \<Rightarrow> nat"
+ lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat"
where
- lcm_prim_def: "lcm = (\<lambda>(m, n). m * n div gcd (m, n))"
+ lcm_prim_def: "lcm m n = m * n div gcd m n"
lemma lcm_def:
- "lcm (m, n) = m * n div gcd (m, n)"
+ "lcm m n = m * n div gcd m n"
unfolding lcm_prim_def by simp
lemma prod_gcd_lcm:
- "m * n = gcd (m, n) * lcm (m, n)"
+ "m * n = gcd m n * lcm m n"
unfolding lcm_def by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod])
-lemma lcm_0 [simp]: "lcm (m, 0) = 0"
+lemma lcm_0 [simp]: "lcm m 0 = 0"
unfolding lcm_def by simp
-lemma lcm_1 [simp]: "lcm (m, 1) = m"
+lemma lcm_1 [simp]: "lcm m 1 = m"
unfolding lcm_def by simp
-lemma lcm_0_left [simp]: "lcm (0, n) = 0"
+lemma lcm_0_left [simp]: "lcm 0 n = 0"
unfolding lcm_def by simp
-lemma lcm_1_left [simp]: "lcm (1, m) = m"
+lemma lcm_1_left [simp]: "lcm 1 m = m"
unfolding lcm_def by simp
lemma dvd_pos:
@@ -254,38 +252,38 @@
lemma lcm_least:
assumes "m dvd k" and "n dvd k"
- shows "lcm (m, n) dvd k"
+ shows "lcm m n dvd k"
proof (cases k)
case 0 then show ?thesis by auto
next
case (Suc _) then have pos_k: "k > 0" by auto
from assms dvd_pos [OF this] have pos_mn: "m > 0" "n > 0" by auto
- with gcd_zero [of m n] have pos_gcd: "gcd (m, n) > 0" by simp
+ with gcd_zero [of m n] have pos_gcd: "gcd m n > 0" by simp
from assms obtain p where k_m: "k = m * p" using dvd_def by blast
from assms obtain q where k_n: "k = n * q" using dvd_def by blast
from pos_k k_m have pos_p: "p > 0" by auto
from pos_k k_n have pos_q: "q > 0" by auto
- have "k * k * gcd (q, p) = k * gcd (k * q, k * p)"
+ have "k * k * gcd q p = k * gcd (k * q) (k * p)"
by (simp add: mult_ac gcd_mult_distrib2)
- also have "\<dots> = k * gcd (m * p * q, n * q * p)"
+ also have "\<dots> = k * gcd (m * p * q) (n * q * p)"
by (simp add: k_m [symmetric] k_n [symmetric])
- also have "\<dots> = k * p * q * gcd (m, n)"
+ also have "\<dots> = k * p * q * gcd m n"
by (simp add: mult_ac gcd_mult_distrib2)
- finally have "(m * p) * (n * q) * gcd (q, p) = k * p * q * gcd (m, n)"
+ finally have "(m * p) * (n * q) * gcd q p = k * p * q * gcd m n"
by (simp only: k_m [symmetric] k_n [symmetric])
- then have "p * q * m * n * gcd (q, p) = p * q * k * gcd (m, n)"
+ then have "p * q * m * n * gcd q p = p * q * k * gcd m n"
by (simp add: mult_ac)
- with pos_p pos_q have "m * n * gcd (q, p) = k * gcd (m, n)"
+ with pos_p pos_q have "m * n * gcd q p = k * gcd m n"
by simp
with prod_gcd_lcm [of m n]
- have "lcm (m, n) * gcd (q, p) * gcd (m, n) = k * gcd (m, n)"
+ have "lcm m n * gcd q p * gcd m n = k * gcd m n"
by (simp add: mult_ac)
- with pos_gcd have "lcm (m, n) * gcd (q, p) = k" by simp
+ with pos_gcd have "lcm m n * gcd q p = k" by simp
then show ?thesis using dvd_def by auto
qed
lemma lcm_dvd1 [iff]:
- "m dvd lcm (m, n)"
+ "m dvd lcm m n"
proof (cases m)
case 0 then show ?thesis by simp
next
@@ -297,16 +295,16 @@
next
case (Suc _)
then have npos: "n > 0" by simp
- have "gcd (m, n) dvd n" by simp
- then obtain k where "n = gcd (m, n) * k" using dvd_def by auto
- then have "m * n div gcd (m, n) = m * (gcd (m, n) * k) div gcd (m, n)" by (simp add: mult_ac)
+ have "gcd m n dvd n" by simp
+ then obtain k where "n = gcd m n * k" using dvd_def by auto
+ then have "m * n div gcd m n = m * (gcd m n * k) div gcd m n" by (simp add: mult_ac)
also have "\<dots> = m * k" using mpos npos gcd_zero by simp
finally show ?thesis by (simp add: lcm_def)
qed
qed
lemma lcm_dvd2 [iff]:
- "n dvd lcm (m, n)"
+ "n dvd lcm m n"
proof (cases n)
case 0 then show ?thesis by simp
next
@@ -318,9 +316,9 @@
next
case (Suc _)
then have mpos: "m > 0" by simp
- have "gcd (m, n) dvd m" by simp
- then obtain k where "m = gcd (m, n) * k" using dvd_def by auto
- then have "m * n div gcd (m, n) = (gcd (m, n) * k) * n div gcd (m, n)" by (simp add: mult_ac)
+ have "gcd m n dvd m" by simp
+ then obtain k where "m = gcd m n * k" using dvd_def by auto
+ then have "m * n div gcd m n = (gcd m n * k) * n div gcd m n" by (simp add: mult_ac)
also have "\<dots> = n * k" using mpos npos gcd_zero by simp
finally show ?thesis by (simp add: lcm_def)
qed
@@ -330,35 +328,35 @@
subsection {* GCD and LCM on integers *}
definition
- igcd :: "int \<Rightarrow> int \<Rightarrow> int" where
- "igcd i j = int (gcd (nat (abs i), nat (abs j)))"
+ zgcd :: "int \<Rightarrow> int \<Rightarrow> int" where
+ "zgcd i j = int (gcd (nat (abs i)) (nat (abs j)))"
-lemma igcd_dvd1 [simp]: "igcd i j dvd i"
- by (simp add: igcd_def int_dvd_iff)
+lemma zgcd_dvd1 [simp]: "zgcd i j dvd i"
+ by (simp add: zgcd_def int_dvd_iff)
-lemma igcd_dvd2 [simp]: "igcd i j dvd j"
- by (simp add: igcd_def int_dvd_iff)
+lemma zgcd_dvd2 [simp]: "zgcd i j dvd j"
+ by (simp add: zgcd_def int_dvd_iff)
-lemma igcd_pos: "igcd i j \<ge> 0"
- by (simp add: igcd_def)
+lemma zgcd_pos: "zgcd i j \<ge> 0"
+ by (simp add: zgcd_def)
-lemma igcd0 [simp]: "(igcd i j = 0) = (i = 0 \<and> j = 0)"
- by (simp add: igcd_def gcd_zero) arith
+lemma zgcd0 [simp]: "(zgcd i j = 0) = (i = 0 \<and> j = 0)"
+ by (simp add: zgcd_def gcd_zero) arith
-lemma igcd_commute: "igcd i j = igcd j i"
- unfolding igcd_def by (simp add: gcd_commute)
+lemma zgcd_commute: "zgcd i j = zgcd j i"
+ unfolding zgcd_def by (simp add: gcd_commute)
-lemma igcd_neg1 [simp]: "igcd (- i) j = igcd i j"
- unfolding igcd_def by simp
+lemma zgcd_neg1 [simp]: "zgcd (- i) j = zgcd i j"
+ unfolding zgcd_def by simp
-lemma igcd_neg2 [simp]: "igcd i (- j) = igcd i j"
- unfolding igcd_def by simp
+lemma zgcd_neg2 [simp]: "zgcd i (- j) = zgcd i j"
+ unfolding zgcd_def by simp
-lemma zrelprime_dvd_mult: "igcd i j = 1 \<Longrightarrow> i dvd k * j \<Longrightarrow> i dvd k"
- unfolding igcd_def
+lemma zrelprime_dvd_mult: "zgcd i j = 1 \<Longrightarrow> i dvd k * j \<Longrightarrow> i dvd k"
+ unfolding zgcd_def
proof -
- assume "int (gcd (nat \<bar>i\<bar>, nat \<bar>j\<bar>)) = 1" "i dvd k * j"
- then have g: "gcd (nat \<bar>i\<bar>, nat \<bar>j\<bar>) = 1" by simp
+ assume "int (gcd (nat \<bar>i\<bar>) (nat \<bar>j\<bar>)) = 1" "i dvd k * j"
+ then have g: "gcd (nat \<bar>i\<bar>) (nat \<bar>j\<bar>) = 1" by simp
from `i dvd k * j` obtain h where h: "k*j = i*h" unfolding dvd_def by blast
have th: "nat \<bar>i\<bar> dvd nat \<bar>k\<bar> * nat \<bar>j\<bar>"
unfolding dvd_def
@@ -375,34 +373,34 @@
done
qed
-lemma int_nat_abs: "int (nat (abs x)) = abs x" by arith
+lemma int_nat_abs: "int (nat (abs x)) = abs x" by arith
-lemma igcd_greatest:
+lemma zgcd_greatest:
assumes "k dvd m" and "k dvd n"
- shows "k dvd igcd m n"
+ shows "k dvd zgcd m n"
proof -
let ?k' = "nat \<bar>k\<bar>"
let ?m' = "nat \<bar>m\<bar>"
let ?n' = "nat \<bar>n\<bar>"
from `k dvd m` and `k dvd n` have dvd': "?k' dvd ?m'" "?k' dvd ?n'"
unfolding zdvd_int by (simp_all only: int_nat_abs zdvd_abs1 zdvd_abs2)
- from gcd_greatest [OF dvd'] have "int (nat \<bar>k\<bar>) dvd igcd m n"
- unfolding igcd_def by (simp only: zdvd_int)
- then have "\<bar>k\<bar> dvd igcd m n" by (simp only: int_nat_abs)
- then show "k dvd igcd m n" by (simp add: zdvd_abs1)
+ from gcd_greatest [OF dvd'] have "int (nat \<bar>k\<bar>) dvd zgcd m n"
+ unfolding zgcd_def by (simp only: zdvd_int)
+ then have "\<bar>k\<bar> dvd zgcd m n" by (simp only: int_nat_abs)
+ then show "k dvd zgcd m n" by (simp add: zdvd_abs1)
qed
-lemma div_igcd_relprime:
+lemma div_zgcd_relprime:
assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
- shows "igcd (a div (igcd a b)) (b div (igcd a b)) = 1"
+ shows "zgcd (a div (zgcd a b)) (b div (zgcd a b)) = 1"
proof -
from nz have nz': "nat \<bar>a\<bar> \<noteq> 0 \<or> nat \<bar>b\<bar> \<noteq> 0" by arith
- let ?g = "igcd a b"
+ let ?g = "zgcd a b"
let ?a' = "a div ?g"
let ?b' = "b div ?g"
- let ?g' = "igcd ?a' ?b'"
- have dvdg: "?g dvd a" "?g dvd b" by (simp_all add: igcd_dvd1 igcd_dvd2)
- have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by (simp_all add: igcd_dvd1 igcd_dvd2)
+ let ?g' = "zgcd ?a' ?b'"
+ have dvdg: "?g dvd a" "?g dvd b" by (simp_all add: zgcd_dvd1 zgcd_dvd2)
+ have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by (simp_all add: zgcd_dvd1 zgcd_dvd2)
from dvdg dvdg' obtain ka kb ka' kb' where
kab: "a = ?g*ka" "b = ?g*kb" "?a' = ?g'*ka'" "?b' = ?g' * kb'"
unfolding dvd_def by blast
@@ -411,35 +409,36 @@
by (auto simp add: zdvd_mult_div_cancel [OF dvdg(1)]
zdvd_mult_div_cancel [OF dvdg(2)] dvd_def)
have "?g \<noteq> 0" using nz by simp
- then have gp: "?g \<noteq> 0" using igcd_pos[where i="a" and j="b"] by arith
- from igcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
+ then have gp: "?g \<noteq> 0" using zgcd_pos[where i="a" and j="b"] by arith
+ from zgcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
with zdvd_mult_cancel1 [OF gp] have "\<bar>?g'\<bar> = 1" by simp
- with igcd_pos show "?g' = 1" by simp
+ with zgcd_pos show "?g' = 1" by simp
qed
-definition "ilcm = (\<lambda>i j. int (lcm(nat(abs i),nat(abs j))))"
+definition zlcm :: "int \<Rightarrow> int \<Rightarrow> int" where
+ "zlcm i j = int (lcm (nat (abs i)) (nat (abs j)))"
-lemma dvd_ilcm_self1[simp]: "i dvd ilcm i j"
-by(simp add:ilcm_def dvd_int_iff)
+lemma dvd_zlcm_self1[simp]: "i dvd zlcm i j"
+by(simp add:zlcm_def dvd_int_iff)
-lemma dvd_ilcm_self2[simp]: "j dvd ilcm i j"
-by(simp add:ilcm_def dvd_int_iff)
+lemma dvd_zlcm_self2[simp]: "j dvd zlcm i j"
+by(simp add:zlcm_def dvd_int_iff)
-lemma dvd_imp_dvd_ilcm1:
- assumes "k dvd i" shows "k dvd (ilcm i j)"
+lemma dvd_imp_dvd_zlcm1:
+ assumes "k dvd i" shows "k dvd (zlcm i j)"
proof -
have "nat(abs k) dvd nat(abs i)" using `k dvd i`
by(simp add:int_dvd_iff[symmetric] dvd_int_iff[symmetric] zdvd_abs1)
- thus ?thesis by(simp add:ilcm_def dvd_int_iff)(blast intro: dvd_trans)
+ thus ?thesis by(simp add:zlcm_def dvd_int_iff)(blast intro: dvd_trans)
qed
-lemma dvd_imp_dvd_ilcm2:
- assumes "k dvd j" shows "k dvd (ilcm i j)"
+lemma dvd_imp_dvd_zlcm2:
+ assumes "k dvd j" shows "k dvd (zlcm i j)"
proof -
have "nat(abs k) dvd nat(abs j)" using `k dvd j`
by(simp add:int_dvd_iff[symmetric] dvd_int_iff[symmetric] zdvd_abs1)
- thus ?thesis by(simp add:ilcm_def dvd_int_iff)(blast intro: dvd_trans)
+ thus ?thesis by(simp add:zlcm_def dvd_int_iff)(blast intro: dvd_trans)
qed
@@ -453,35 +452,35 @@
lemma lcm_pos:
assumes mpos: "m > 0"
- and npos: "n>0"
- shows "lcm (m,n) > 0"
+ and npos: "n > 0"
+ shows "lcm m n > 0"
proof(rule ccontr, simp add: lcm_def gcd_zero)
-assume h:"m*n div gcd(m,n) = 0"
-from mpos npos have "gcd (m,n) \<noteq> 0" using gcd_zero by simp
-hence gcdp: "gcd(m,n) > 0" by simp
+assume h:"m * n div gcd m n = 0"
+from mpos npos have "gcd m n \<noteq> 0" using gcd_zero by simp
+hence gcdp: "gcd m n > 0" by simp
with h
-have "m*n < gcd(m,n)"
- by (cases "m * n < gcd (m, n)") (auto simp add: div_if[OF gcdp, where m="m*n"])
+have "m*n < gcd m n"
+ by (cases "m * n < gcd m n") (auto simp add: div_if[OF gcdp, where m="m*n"])
moreover
-have "gcd(m,n) dvd m" by simp
- with mpos dvd_imp_le have t1:"gcd(m,n) \<le> m" by simp
- with npos have t1:"gcd(m,n)*n \<le> m*n" by simp
- have "gcd(m,n) \<le> gcd(m,n)*n" using npos by simp
- with t1 have "gcd(m,n) \<le> m*n" by arith
+have "gcd m n dvd m" by simp
+ with mpos dvd_imp_le have t1:"gcd m n \<le> m" by simp
+ with npos have t1:"gcd m n*n \<le> m*n" by simp
+ have "gcd m n \<le> gcd m n*n" using npos by simp
+ with t1 have "gcd m n \<le> m*n" by arith
ultimately show "False" by simp
qed
-lemma ilcm_pos:
+lemma zlcm_pos:
assumes anz: "a \<noteq> 0"
and bnz: "b \<noteq> 0"
- shows "0 < ilcm a b"
+ shows "0 < zlcm a b"
proof-
let ?na = "nat (abs a)"
let ?nb = "nat (abs b)"
have nap: "?na >0" using anz by simp
have nbp: "?nb >0" using bnz by simp
- have "0 < lcm (?na,?nb)" by (rule lcm_pos[OF nap nbp])
- thus ?thesis by (simp add: ilcm_def)
+ have "0 < lcm ?na ?nb" by (rule lcm_pos[OF nap nbp])
+ thus ?thesis by (simp add: zlcm_def)
qed
end