src/HOL/GCD.thy
author wenzelm
Mon Mar 16 18:24:30 2009 +0100 (2009-03-16)
changeset 30549 d2d7874648bd
parent 30242 aea5d7fa7ef5
child 30738 0842e906300c
permissions -rw-r--r--
simplified method setup;
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(*  Title:      HOL/GCD.thy
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    Author:     Christophe Tabacznyj and Lawrence C Paulson
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    Copyright   1996  University of Cambridge
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*)
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header {* The Greatest Common Divisor *}
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theory GCD
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imports Plain Presburger Main
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begin
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text {*
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  See \cite{davenport92}. \bigskip
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*}
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subsection {* Specification of GCD on nats *}
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definition
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  is_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where -- {* @{term gcd} as a relation *}
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  [code del]: "is_gcd m n p \<longleftrightarrow> p dvd m \<and> p dvd n \<and>
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    (\<forall>d. d dvd m \<longrightarrow> d dvd n \<longrightarrow> d dvd p)"
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text {* Uniqueness *}
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lemma is_gcd_unique: "is_gcd a b m \<Longrightarrow> is_gcd a b n \<Longrightarrow> m = n"
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  by (simp add: is_gcd_def) (blast intro: dvd_anti_sym)
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text {* Connection to divides relation *}
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lemma is_gcd_dvd: "is_gcd a b m \<Longrightarrow> k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd m"
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  by (auto simp add: is_gcd_def)
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text {* Commutativity *}
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lemma is_gcd_commute: "is_gcd m n k = is_gcd n m k"
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  by (auto simp add: is_gcd_def)
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subsection {* GCD on nat by Euclid's algorithm *}
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fun
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  gcd  :: "nat => nat => nat"
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where
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  "gcd m n = (if n = 0 then m else gcd n (m mod n))"
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lemma gcd_induct [case_names "0" rec]:
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  fixes m n :: nat
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  assumes "\<And>m. P m 0"
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    and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n"
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  shows "P m n"
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proof (induct m n rule: gcd.induct)
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  case (1 m n) with assms show ?case by (cases "n = 0") simp_all
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qed
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lemma gcd_0 [simp, algebra]: "gcd m 0 = m"
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  by simp
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lemma gcd_0_left [simp,algebra]: "gcd 0 m = m"
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  by simp
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lemma gcd_non_0: "n > 0 \<Longrightarrow> gcd m n = gcd n (m mod n)"
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  by simp
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lemma gcd_1 [simp, algebra]: "gcd m (Suc 0) = Suc 0"
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  by simp
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lemma nat_gcd_1_right [simp, algebra]: "gcd m 1 = 1"
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  unfolding One_nat_def by (rule gcd_1)
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declare gcd.simps [simp del]
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text {*
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  \medskip @{term "gcd m n"} divides @{text m} and @{text n}.  The
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  conjunctions don't seem provable separately.
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*}
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lemma gcd_dvd1 [iff, algebra]: "gcd m n dvd m"
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  and gcd_dvd2 [iff, algebra]: "gcd m n dvd n"
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  apply (induct m n rule: gcd_induct)
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     apply (simp_all add: gcd_non_0)
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  apply (blast dest: dvd_mod_imp_dvd)
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  done
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text {*
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  \medskip Maximality: for all @{term m}, @{term n}, @{term k}
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  naturals, if @{term k} divides @{term m} and @{term k} divides
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  @{term n} then @{term k} divides @{term "gcd m n"}.
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*}
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lemma gcd_greatest: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd gcd m n"
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  by (induct m n rule: gcd_induct) (simp_all add: gcd_non_0 dvd_mod)
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text {*
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  \medskip Function gcd yields the Greatest Common Divisor.
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*}
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lemma is_gcd: "is_gcd m n (gcd m n) "
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  by (simp add: is_gcd_def gcd_greatest)
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subsection {* Derived laws for GCD *}
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lemma gcd_greatest_iff [iff, algebra]: "k dvd gcd m n \<longleftrightarrow> k dvd m \<and> k dvd n"
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  by (blast intro!: gcd_greatest intro: dvd_trans)
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lemma gcd_zero[algebra]: "gcd m n = 0 \<longleftrightarrow> m = 0 \<and> n = 0"
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  by (simp only: dvd_0_left_iff [symmetric] gcd_greatest_iff)
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lemma gcd_commute: "gcd m n = gcd n m"
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  apply (rule is_gcd_unique)
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   apply (rule is_gcd)
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  apply (subst is_gcd_commute)
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  apply (simp add: is_gcd)
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  done
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lemma gcd_assoc: "gcd (gcd k m) n = gcd k (gcd m n)"
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  apply (rule is_gcd_unique)
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   apply (rule is_gcd)
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  apply (simp add: is_gcd_def)
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  apply (blast intro: dvd_trans)
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  done
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lemma gcd_1_left [simp, algebra]: "gcd (Suc 0) m = Suc 0"
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  by (simp add: gcd_commute)
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lemma nat_gcd_1_left [simp, algebra]: "gcd 1 m = 1"
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  unfolding One_nat_def by (rule gcd_1_left)
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text {*
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  \medskip Multiplication laws
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*}
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lemma gcd_mult_distrib2: "k * gcd m n = gcd (k * m) (k * n)"
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    -- {* \cite[page 27]{davenport92} *}
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  apply (induct m n rule: gcd_induct)
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   apply simp
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  apply (case_tac "k = 0")
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   apply (simp_all add: mod_geq gcd_non_0 mod_mult_distrib2)
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  done
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lemma gcd_mult [simp, algebra]: "gcd k (k * n) = k"
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  apply (rule gcd_mult_distrib2 [of k 1 n, simplified, symmetric])
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  done
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lemma gcd_self [simp, algebra]: "gcd k k = k"
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  apply (rule gcd_mult [of k 1, simplified])
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  done
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lemma relprime_dvd_mult: "gcd k n = 1 ==> k dvd m * n ==> k dvd m"
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  apply (insert gcd_mult_distrib2 [of m k n])
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  apply simp
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  apply (erule_tac t = m in ssubst)
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  apply simp
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  done
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lemma relprime_dvd_mult_iff: "gcd k n = 1 ==> (k dvd m * n) = (k dvd m)"
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  by (auto intro: relprime_dvd_mult dvd_mult2)
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lemma gcd_mult_cancel: "gcd k n = 1 ==> gcd (k * m) n = gcd m n"
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  apply (rule dvd_anti_sym)
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   apply (rule gcd_greatest)
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    apply (rule_tac n = k in relprime_dvd_mult)
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     apply (simp add: gcd_assoc)
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     apply (simp add: gcd_commute)
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    apply (simp_all add: mult_commute)
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  done
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text {* \medskip Addition laws *}
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lemma gcd_add1 [simp, algebra]: "gcd (m + n) n = gcd m n"
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  by (cases "n = 0") (auto simp add: gcd_non_0)
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lemma gcd_add2 [simp, algebra]: "gcd m (m + n) = gcd m n"
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proof -
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  have "gcd m (m + n) = gcd (m + n) m" by (rule gcd_commute)
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  also have "... = gcd (n + m) m" by (simp add: add_commute)
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  also have "... = gcd n m" by simp
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  also have  "... = gcd m n" by (rule gcd_commute)
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  finally show ?thesis .
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qed
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lemma gcd_add2' [simp, algebra]: "gcd m (n + m) = gcd m n"
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  apply (subst add_commute)
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  apply (rule gcd_add2)
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  done
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lemma gcd_add_mult[algebra]: "gcd m (k * m + n) = gcd m n"
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  by (induct k) (simp_all add: add_assoc)
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lemma gcd_dvd_prod: "gcd m n dvd m * n" 
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  using mult_dvd_mono [of 1] by auto
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text {*
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  \medskip Division by gcd yields rrelatively primes.
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*}
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lemma div_gcd_relprime:
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  assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
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  shows "gcd (a div gcd a b) (b div gcd a b) = 1"
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proof -
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  let ?g = "gcd a b"
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  let ?a' = "a div ?g"
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  let ?b' = "b div ?g"
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  let ?g' = "gcd ?a' ?b'"
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  have dvdg: "?g dvd a" "?g dvd b" by simp_all
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  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
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  from dvdg dvdg' obtain ka kb ka' kb' where
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      kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
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    unfolding dvd_def by blast
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  then have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'" by simp_all
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  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
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    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
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      dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
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  have "?g \<noteq> 0" using nz by (simp add: gcd_zero)
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  then have gp: "?g > 0" by simp
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  from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
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  with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
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qed
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lemma gcd_unique: "d dvd a\<and>d dvd b \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
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proof(auto)
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  assume H: "d dvd a" "d dvd b" "\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d"
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  from H(3)[rule_format] gcd_dvd1[of a b] gcd_dvd2[of a b] 
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  have th: "gcd a b dvd d" by blast
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  from dvd_anti_sym[OF th gcd_greatest[OF H(1,2)]]  show "d = gcd a b" by blast 
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qed
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lemma gcd_eq: assumes H: "\<forall>d. d dvd x \<and> d dvd y \<longleftrightarrow> d dvd u \<and> d dvd v"
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  shows "gcd x y = gcd u v"
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proof-
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  from H have "\<forall>d. d dvd x \<and> d dvd y \<longleftrightarrow> d dvd gcd u v" by simp
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  with gcd_unique[of "gcd u v" x y]  show ?thesis by auto
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qed
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lemma ind_euclid: 
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  assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0" 
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  and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)" 
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  shows "P a b"
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proof(induct n\<equiv>"a+b" arbitrary: a b rule: nat_less_induct)
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  fix n a b
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  assume H: "\<forall>m < n. \<forall>a b. m = a + b \<longrightarrow> P a b" "n = a + b"
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  have "a = b \<or> a < b \<or> b < a" by arith
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  moreover {assume eq: "a= b"
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    from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq by simp}
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  moreover
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  {assume lt: "a < b"
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    hence "a + b - a < n \<or> a = 0"  using H(2) by arith
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    moreover
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    {assume "a =0" with z c have "P a b" by blast }
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    moreover
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    {assume ab: "a + b - a < n"
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      have th0: "a + b - a = a + (b - a)" using lt by arith
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      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
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      have "P a b" by (simp add: th0[symmetric])}
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    ultimately have "P a b" by blast}
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  moreover
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  {assume lt: "a > b"
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    hence "b + a - b < n \<or> b = 0"  using H(2) by arith
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    moreover
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    {assume "b =0" with z c have "P a b" by blast }
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    moreover
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    {assume ab: "b + a - b < n"
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      have th0: "b + a - b = b + (a - b)" using lt by arith
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      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
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      have "P b a" by (simp add: th0[symmetric])
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      hence "P a b" using c by blast }
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    ultimately have "P a b" by blast}
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ultimately  show "P a b" by blast
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qed
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lemma bezout_lemma: 
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  assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x = b * y + d \<or> b * x = a * y + d)"
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  shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and> (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
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using ex
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apply clarsimp
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apply (rule_tac x="d" in exI, simp add: dvd_add)
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apply (case_tac "a * x = b * y + d" , simp_all)
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apply (rule_tac x="x + y" in exI)
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apply (rule_tac x="y" in exI)
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apply algebra
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apply (rule_tac x="x" in exI)
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apply (rule_tac x="x + y" in exI)
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apply algebra
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done
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lemma bezout_add: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x = b * y + d \<or> b * x = a * y + d)"
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apply(induct a b rule: ind_euclid)
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apply blast
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apply clarify
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apply (rule_tac x="a" in exI, simp add: dvd_add)
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apply clarsimp
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apply (rule_tac x="d" in exI)
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apply (case_tac "a * x = b * y + d", simp_all add: dvd_add)
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apply (rule_tac x="x+y" in exI)
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apply (rule_tac x="y" in exI)
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apply algebra
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apply (rule_tac x="x" in exI)
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apply (rule_tac x="x+y" in exI)
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apply algebra
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done
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lemma bezout: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x - b * y = d \<or> b * x - a * y = d)"
chaieb@27669
   304
using bezout_add[of a b]
chaieb@27669
   305
apply clarsimp
chaieb@27669
   306
apply (rule_tac x="d" in exI, simp)
chaieb@27669
   307
apply (rule_tac x="x" in exI)
chaieb@27669
   308
apply (rule_tac x="y" in exI)
chaieb@27669
   309
apply auto
chaieb@27669
   310
done
chaieb@27669
   311
chaieb@27669
   312
chaieb@27669
   313
text {* We can get a stronger version with a nonzeroness assumption. *}
chaieb@27669
   314
lemma divides_le: "m dvd n ==> m <= n \<or> n = (0::nat)" by (auto simp add: dvd_def)
chaieb@27669
   315
chaieb@27669
   316
lemma bezout_add_strong: assumes nz: "a \<noteq> (0::nat)"
chaieb@27669
   317
  shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
chaieb@27669
   318
proof-
chaieb@27669
   319
  from nz have ap: "a > 0" by simp
chaieb@27669
   320
 from bezout_add[of a b] 
chaieb@27669
   321
 have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or> (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
chaieb@27669
   322
 moreover
chaieb@27669
   323
 {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
chaieb@27669
   324
   from H have ?thesis by blast }
chaieb@27669
   325
 moreover
chaieb@27669
   326
 {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
chaieb@27669
   327
   {assume b0: "b = 0" with H  have ?thesis by simp}
chaieb@27669
   328
   moreover 
chaieb@27669
   329
   {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
chaieb@27669
   330
     from divides_le[OF H(2)] b have "d < b \<or> d = b" using le_less by blast
chaieb@27669
   331
     moreover
chaieb@27669
   332
     {assume db: "d=b"
chaieb@27669
   333
       from prems have ?thesis apply simp
chaieb@27669
   334
	 apply (rule exI[where x = b], simp)
chaieb@27669
   335
	 apply (rule exI[where x = b])
chaieb@27669
   336
	by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
chaieb@27669
   337
    moreover
chaieb@27669
   338
    {assume db: "d < b" 
chaieb@27669
   339
	{assume "x=0" hence ?thesis  using prems by simp }
chaieb@27669
   340
	moreover
chaieb@27669
   341
	{assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
chaieb@27669
   342
	  
chaieb@27669
   343
	  from db have "d \<le> b - 1" by simp
chaieb@27669
   344
	  hence "d*b \<le> b*(b - 1)" by simp
chaieb@27669
   345
	  with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
chaieb@27669
   346
	  have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
chaieb@27669
   347
	  from H (3) have "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)" by algebra
chaieb@27669
   348
	  hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
chaieb@27669
   349
	  hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)" 
chaieb@27669
   350
	    by (simp only: diff_add_assoc[OF dble, of d, symmetric])
chaieb@27669
   351
	  hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
chaieb@27669
   352
	    by (simp only: diff_mult_distrib2 add_commute mult_ac)
chaieb@27669
   353
	  hence ?thesis using H(1,2)
chaieb@27669
   354
	    apply -
chaieb@27669
   355
	    apply (rule exI[where x=d], simp)
chaieb@27669
   356
	    apply (rule exI[where x="(b - 1) * y"])
chaieb@27669
   357
	    by (rule exI[where x="x*(b - 1) - d"], simp)}
chaieb@27669
   358
	ultimately have ?thesis by blast}
chaieb@27669
   359
    ultimately have ?thesis by blast}
chaieb@27669
   360
  ultimately have ?thesis by blast}
chaieb@27669
   361
 ultimately show ?thesis by blast
chaieb@27669
   362
qed
chaieb@27669
   363
chaieb@27669
   364
chaieb@27669
   365
lemma bezout_gcd: "\<exists>x y. a * x - b * y = gcd a b \<or> b * x - a * y = gcd a b"
chaieb@27669
   366
proof-
chaieb@27669
   367
  let ?g = "gcd a b"
chaieb@27669
   368
  from bezout[of a b] obtain d x y where d: "d dvd a" "d dvd b" "a * x - b * y = d \<or> b * x - a * y = d" by blast
chaieb@27669
   369
  from d(1,2) have "d dvd ?g" by simp
chaieb@27669
   370
  then obtain k where k: "?g = d*k" unfolding dvd_def by blast
chaieb@27669
   371
  from d(3) have "(a * x - b * y)*k = d*k \<or> (b * x - a * y)*k = d*k" by blast 
chaieb@27669
   372
  hence "a * x * k - b * y*k = d*k \<or> b * x * k - a * y*k = d*k" 
chaieb@27669
   373
    by (algebra add: diff_mult_distrib)
chaieb@27669
   374
  hence "a * (x * k) - b * (y*k) = ?g \<or> b * (x * k) - a * (y*k) = ?g" 
chaieb@27669
   375
    by (simp add: k mult_assoc)
chaieb@27669
   376
  thus ?thesis by blast
chaieb@27669
   377
qed
chaieb@27669
   378
chaieb@27669
   379
lemma bezout_gcd_strong: assumes a: "a \<noteq> 0" 
chaieb@27669
   380
  shows "\<exists>x y. a * x = b * y + gcd a b"
chaieb@27669
   381
proof-
chaieb@27669
   382
  let ?g = "gcd a b"
chaieb@27669
   383
  from bezout_add_strong[OF a, of b]
chaieb@27669
   384
  obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
chaieb@27669
   385
  from d(1,2) have "d dvd ?g" by simp
chaieb@27669
   386
  then obtain k where k: "?g = d*k" unfolding dvd_def by blast
chaieb@27669
   387
  from d(3) have "a * x * k = (b * y + d) *k " by algebra
chaieb@27669
   388
  hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
chaieb@27669
   389
  thus ?thesis by blast
chaieb@27669
   390
qed
chaieb@27669
   391
chaieb@27669
   392
lemma gcd_mult_distrib: "gcd(a * c) (b * c) = c * gcd a b"
chaieb@27669
   393
by(simp add: gcd_mult_distrib2 mult_commute)
chaieb@27669
   394
chaieb@27669
   395
lemma gcd_bezout: "(\<exists>x y. a * x - b * y = d \<or> b * x - a * y = d) \<longleftrightarrow> gcd a b dvd d"
chaieb@27669
   396
  (is "?lhs \<longleftrightarrow> ?rhs")
chaieb@27669
   397
proof-
chaieb@27669
   398
  let ?g = "gcd a b"
chaieb@27669
   399
  {assume H: ?rhs then obtain k where k: "d = ?g*k" unfolding dvd_def by blast
chaieb@27669
   400
    from bezout_gcd[of a b] obtain x y where xy: "a * x - b * y = ?g \<or> b * x - a * y = ?g"
chaieb@27669
   401
      by blast
chaieb@27669
   402
    hence "(a * x - b * y)*k = ?g*k \<or> (b * x - a * y)*k = ?g*k" by auto
chaieb@27669
   403
    hence "a * x*k - b * y*k = ?g*k \<or> b * x * k - a * y*k = ?g*k" 
chaieb@27669
   404
      by (simp only: diff_mult_distrib)
chaieb@27669
   405
    hence "a * (x*k) - b * (y*k) = d \<or> b * (x * k) - a * (y*k) = d"
chaieb@27669
   406
      by (simp add: k[symmetric] mult_assoc)
chaieb@27669
   407
    hence ?lhs by blast}
chaieb@27669
   408
  moreover
chaieb@27669
   409
  {fix x y assume H: "a * x - b * y = d \<or> b * x - a * y = d"
chaieb@27669
   410
    have dv: "?g dvd a*x" "?g dvd b * y" "?g dvd b*x" "?g dvd a * y"
chaieb@27669
   411
      using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all
nipkow@30042
   412
    from nat_dvd_diff[OF dv(1,2)] nat_dvd_diff[OF dv(3,4)] H
chaieb@27669
   413
    have ?rhs by auto}
chaieb@27669
   414
  ultimately show ?thesis by blast
chaieb@27669
   415
qed
chaieb@27669
   416
chaieb@27669
   417
lemma gcd_bezout_sum: assumes H:"a * x + b * y = d" shows "gcd a b dvd d"
chaieb@27669
   418
proof-
chaieb@27669
   419
  let ?g = "gcd a b"
chaieb@27669
   420
    have dv: "?g dvd a*x" "?g dvd b * y" 
chaieb@27669
   421
      using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all
chaieb@27669
   422
    from dvd_add[OF dv] H
chaieb@27669
   423
    show ?thesis by auto
chaieb@27669
   424
qed
chaieb@27669
   425
chaieb@27669
   426
lemma gcd_mult': "gcd b (a * b) = b"
chaieb@27669
   427
by (simp add: gcd_mult mult_commute[of a b]) 
chaieb@27669
   428
chaieb@27669
   429
lemma gcd_add: "gcd(a + b) b = gcd a b" 
chaieb@27669
   430
  "gcd(b + a) b = gcd a b" "gcd a (a + b) = gcd a b" "gcd a (b + a) = gcd a b"
chaieb@27669
   431
apply (simp_all add: gcd_add1)
chaieb@27669
   432
by (simp add: gcd_commute gcd_add1)
chaieb@27669
   433
chaieb@27669
   434
lemma gcd_sub: "b <= a ==> gcd(a - b) b = gcd a b" "a <= b ==> gcd a (b - a) = gcd a b"
chaieb@27669
   435
proof-
chaieb@27669
   436
  {fix a b assume H: "b \<le> (a::nat)"
chaieb@27669
   437
    hence th: "a - b + b = a" by arith
chaieb@27669
   438
    from gcd_add(1)[of "a - b" b] th  have "gcd(a - b) b = gcd a b" by simp}
chaieb@27669
   439
  note th = this
chaieb@27669
   440
{
chaieb@27669
   441
  assume ab: "b \<le> a"
chaieb@27669
   442
  from th[OF ab] show "gcd (a - b)  b = gcd a b" by blast
chaieb@27669
   443
next
chaieb@27669
   444
  assume ab: "a \<le> b"
chaieb@27669
   445
  from th[OF ab] show "gcd a (b - a) = gcd a b" 
chaieb@27669
   446
    by (simp add: gcd_commute)}
chaieb@27669
   447
qed
chaieb@27669
   448
chaieb@27669
   449
haftmann@23687
   450
subsection {* LCM defined by GCD *}
wenzelm@22367
   451
chaieb@27669
   452
haftmann@23687
   453
definition
haftmann@27556
   454
  lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@23687
   455
where
chaieb@27568
   456
  lcm_def: "lcm m n = m * n div gcd m n"
haftmann@23687
   457
haftmann@23687
   458
lemma prod_gcd_lcm:
haftmann@27556
   459
  "m * n = gcd m n * lcm m n"
haftmann@23687
   460
  unfolding lcm_def by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod])
haftmann@23687
   461
haftmann@27556
   462
lemma lcm_0 [simp]: "lcm m 0 = 0"
haftmann@23687
   463
  unfolding lcm_def by simp
haftmann@23687
   464
haftmann@27556
   465
lemma lcm_1 [simp]: "lcm m 1 = m"
haftmann@23687
   466
  unfolding lcm_def by simp
haftmann@23687
   467
haftmann@27556
   468
lemma lcm_0_left [simp]: "lcm 0 n = 0"
haftmann@23687
   469
  unfolding lcm_def by simp
haftmann@23687
   470
haftmann@27556
   471
lemma lcm_1_left [simp]: "lcm 1 m = m"
haftmann@23687
   472
  unfolding lcm_def by simp
haftmann@23687
   473
haftmann@23687
   474
lemma dvd_pos:
haftmann@23687
   475
  fixes n m :: nat
haftmann@23687
   476
  assumes "n > 0" and "m dvd n"
haftmann@23687
   477
  shows "m > 0"
haftmann@23687
   478
using assms by (cases m) auto
haftmann@23687
   479
haftmann@23951
   480
lemma lcm_least:
haftmann@23687
   481
  assumes "m dvd k" and "n dvd k"
haftmann@27556
   482
  shows "lcm m n dvd k"
haftmann@23687
   483
proof (cases k)
haftmann@23687
   484
  case 0 then show ?thesis by auto
haftmann@23687
   485
next
haftmann@23687
   486
  case (Suc _) then have pos_k: "k > 0" by auto
haftmann@23687
   487
  from assms dvd_pos [OF this] have pos_mn: "m > 0" "n > 0" by auto
haftmann@27556
   488
  with gcd_zero [of m n] have pos_gcd: "gcd m n > 0" by simp
haftmann@23687
   489
  from assms obtain p where k_m: "k = m * p" using dvd_def by blast
haftmann@23687
   490
  from assms obtain q where k_n: "k = n * q" using dvd_def by blast
haftmann@23687
   491
  from pos_k k_m have pos_p: "p > 0" by auto
haftmann@23687
   492
  from pos_k k_n have pos_q: "q > 0" by auto
haftmann@27556
   493
  have "k * k * gcd q p = k * gcd (k * q) (k * p)"
haftmann@23687
   494
    by (simp add: mult_ac gcd_mult_distrib2)
haftmann@27556
   495
  also have "\<dots> = k * gcd (m * p * q) (n * q * p)"
haftmann@23687
   496
    by (simp add: k_m [symmetric] k_n [symmetric])
haftmann@27556
   497
  also have "\<dots> = k * p * q * gcd m n"
haftmann@23687
   498
    by (simp add: mult_ac gcd_mult_distrib2)
haftmann@27556
   499
  finally have "(m * p) * (n * q) * gcd q p = k * p * q * gcd m n"
haftmann@23687
   500
    by (simp only: k_m [symmetric] k_n [symmetric])
haftmann@27556
   501
  then have "p * q * m * n * gcd q p = p * q * k * gcd m n"
haftmann@23687
   502
    by (simp add: mult_ac)
haftmann@27556
   503
  with pos_p pos_q have "m * n * gcd q p = k * gcd m n"
haftmann@23687
   504
    by simp
haftmann@23687
   505
  with prod_gcd_lcm [of m n]
haftmann@27556
   506
  have "lcm m n * gcd q p * gcd m n = k * gcd m n"
haftmann@23687
   507
    by (simp add: mult_ac)
haftmann@27556
   508
  with pos_gcd have "lcm m n * gcd q p = k" by simp
haftmann@23687
   509
  then show ?thesis using dvd_def by auto
haftmann@23687
   510
qed
haftmann@23687
   511
haftmann@23687
   512
lemma lcm_dvd1 [iff]:
haftmann@27556
   513
  "m dvd lcm m n"
haftmann@23687
   514
proof (cases m)
haftmann@23687
   515
  case 0 then show ?thesis by simp
haftmann@23687
   516
next
haftmann@23687
   517
  case (Suc _)
haftmann@23687
   518
  then have mpos: "m > 0" by simp
haftmann@23687
   519
  show ?thesis
haftmann@23687
   520
  proof (cases n)
haftmann@23687
   521
    case 0 then show ?thesis by simp
haftmann@23687
   522
  next
haftmann@23687
   523
    case (Suc _)
haftmann@23687
   524
    then have npos: "n > 0" by simp
haftmann@27556
   525
    have "gcd m n dvd n" by simp
haftmann@27556
   526
    then obtain k where "n = gcd m n * k" using dvd_def by auto
haftmann@27556
   527
    then have "m * n div gcd m n = m * (gcd m n * k) div gcd m n" by (simp add: mult_ac)
haftmann@23687
   528
    also have "\<dots> = m * k" using mpos npos gcd_zero by simp
haftmann@23687
   529
    finally show ?thesis by (simp add: lcm_def)
haftmann@23687
   530
  qed
haftmann@23687
   531
qed
haftmann@23687
   532
haftmann@23687
   533
lemma lcm_dvd2 [iff]: 
haftmann@27556
   534
  "n dvd lcm m n"
haftmann@23687
   535
proof (cases n)
haftmann@23687
   536
  case 0 then show ?thesis by simp
haftmann@23687
   537
next
haftmann@23687
   538
  case (Suc _)
haftmann@23687
   539
  then have npos: "n > 0" by simp
haftmann@23687
   540
  show ?thesis
haftmann@23687
   541
  proof (cases m)
haftmann@23687
   542
    case 0 then show ?thesis by simp
haftmann@23687
   543
  next
haftmann@23687
   544
    case (Suc _)
haftmann@23687
   545
    then have mpos: "m > 0" by simp
haftmann@27556
   546
    have "gcd m n dvd m" by simp
haftmann@27556
   547
    then obtain k where "m = gcd m n * k" using dvd_def by auto
haftmann@27556
   548
    then have "m * n div gcd m n = (gcd m n * k) * n div gcd m n" by (simp add: mult_ac)
haftmann@23687
   549
    also have "\<dots> = n * k" using mpos npos gcd_zero by simp
haftmann@23687
   550
    finally show ?thesis by (simp add: lcm_def)
haftmann@23687
   551
  qed
haftmann@23687
   552
qed
haftmann@23687
   553
chaieb@27568
   554
lemma gcd_add1_eq: "gcd (m + k) k = gcd (m + k) m"
chaieb@27568
   555
  by (simp add: gcd_commute)
chaieb@27568
   556
chaieb@27568
   557
lemma gcd_diff2: "m \<le> n ==> gcd n (n - m) = gcd n m"
chaieb@27568
   558
  apply (subgoal_tac "n = m + (n - m)")
chaieb@27669
   559
  apply (erule ssubst, rule gcd_add1_eq, simp)  
chaieb@27568
   560
  done
chaieb@27568
   561
haftmann@23687
   562
haftmann@23687
   563
subsection {* GCD and LCM on integers *}
wenzelm@22367
   564
wenzelm@22367
   565
definition
haftmann@27556
   566
  zgcd :: "int \<Rightarrow> int \<Rightarrow> int" where
haftmann@27556
   567
  "zgcd i j = int (gcd (nat (abs i)) (nat (abs j)))"
wenzelm@22367
   568
chaieb@27669
   569
lemma zgcd_zdvd1 [iff,simp, algebra]: "zgcd i j dvd i"
nipkow@29700
   570
by (simp add: zgcd_def int_dvd_iff)
chaieb@22027
   571
chaieb@27669
   572
lemma zgcd_zdvd2 [iff,simp, algebra]: "zgcd i j dvd j"
nipkow@29700
   573
by (simp add: zgcd_def int_dvd_iff)
chaieb@22027
   574
haftmann@27556
   575
lemma zgcd_pos: "zgcd i j \<ge> 0"
nipkow@29700
   576
by (simp add: zgcd_def)
wenzelm@22367
   577
chaieb@27669
   578
lemma zgcd0 [simp,algebra]: "(zgcd i j = 0) = (i = 0 \<and> j = 0)"
nipkow@29700
   579
by (simp add: zgcd_def gcd_zero)
chaieb@22027
   580
haftmann@27556
   581
lemma zgcd_commute: "zgcd i j = zgcd j i"
nipkow@29700
   582
unfolding zgcd_def by (simp add: gcd_commute)
wenzelm@22367
   583
chaieb@27669
   584
lemma zgcd_zminus [simp, algebra]: "zgcd (- i) j = zgcd i j"
nipkow@29700
   585
unfolding zgcd_def by simp
wenzelm@22367
   586
chaieb@27669
   587
lemma zgcd_zminus2 [simp, algebra]: "zgcd i (- j) = zgcd i j"
nipkow@29700
   588
unfolding zgcd_def by simp
wenzelm@22367
   589
chaieb@27669
   590
  (* should be solved by algebra*)
haftmann@27556
   591
lemma zrelprime_dvd_mult: "zgcd i j = 1 \<Longrightarrow> i dvd k * j \<Longrightarrow> i dvd k"
haftmann@27556
   592
  unfolding zgcd_def
wenzelm@22367
   593
proof -
haftmann@27556
   594
  assume "int (gcd (nat \<bar>i\<bar>) (nat \<bar>j\<bar>)) = 1" "i dvd k * j"
haftmann@27556
   595
  then have g: "gcd (nat \<bar>i\<bar>) (nat \<bar>j\<bar>) = 1" by simp
wenzelm@22367
   596
  from `i dvd k * j` obtain h where h: "k*j = i*h" unfolding dvd_def by blast
chaieb@22027
   597
  have th: "nat \<bar>i\<bar> dvd nat \<bar>k\<bar> * nat \<bar>j\<bar>"
wenzelm@22367
   598
    unfolding dvd_def
wenzelm@22367
   599
    by (rule_tac x= "nat \<bar>h\<bar>" in exI, simp add: h nat_abs_mult_distrib [symmetric])
wenzelm@22367
   600
  from relprime_dvd_mult [OF g th] obtain h' where h': "nat \<bar>k\<bar> = nat \<bar>i\<bar> * h'"
chaieb@22027
   601
    unfolding dvd_def by blast
chaieb@22027
   602
  from h' have "int (nat \<bar>k\<bar>) = int (nat \<bar>i\<bar> * h')" by simp
huffman@23431
   603
  then have "\<bar>k\<bar> = \<bar>i\<bar> * int h'" by (simp add: int_mult)
chaieb@22027
   604
  then show ?thesis
nipkow@30042
   605
    apply (subst abs_dvd_iff [symmetric])
nipkow@30042
   606
    apply (subst dvd_abs_iff [symmetric])
chaieb@22027
   607
    apply (unfold dvd_def)
wenzelm@22367
   608
    apply (rule_tac x = "int h'" in exI, simp)
chaieb@22027
   609
    done
chaieb@22027
   610
qed
chaieb@22027
   611
haftmann@27556
   612
lemma int_nat_abs: "int (nat (abs x)) = abs x" by arith
wenzelm@22367
   613
haftmann@27556
   614
lemma zgcd_greatest:
wenzelm@22367
   615
  assumes "k dvd m" and "k dvd n"
haftmann@27556
   616
  shows "k dvd zgcd m n"
wenzelm@22367
   617
proof -
chaieb@22027
   618
  let ?k' = "nat \<bar>k\<bar>"
chaieb@22027
   619
  let ?m' = "nat \<bar>m\<bar>"
chaieb@22027
   620
  let ?n' = "nat \<bar>n\<bar>"
wenzelm@22367
   621
  from `k dvd m` and `k dvd n` have dvd': "?k' dvd ?m'" "?k' dvd ?n'"
nipkow@30042
   622
    unfolding zdvd_int by (simp_all only: int_nat_abs abs_dvd_iff dvd_abs_iff)
haftmann@27556
   623
  from gcd_greatest [OF dvd'] have "int (nat \<bar>k\<bar>) dvd zgcd m n"
haftmann@27556
   624
    unfolding zgcd_def by (simp only: zdvd_int)
haftmann@27556
   625
  then have "\<bar>k\<bar> dvd zgcd m n" by (simp only: int_nat_abs)
nipkow@30042
   626
  then show "k dvd zgcd m n" by simp
chaieb@22027
   627
qed
chaieb@22027
   628
haftmann@27556
   629
lemma div_zgcd_relprime:
wenzelm@22367
   630
  assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
haftmann@27556
   631
  shows "zgcd (a div (zgcd a b)) (b div (zgcd a b)) = 1"
wenzelm@22367
   632
proof -
chaieb@25112
   633
  from nz have nz': "nat \<bar>a\<bar> \<noteq> 0 \<or> nat \<bar>b\<bar> \<noteq> 0" by arith 
haftmann@27556
   634
  let ?g = "zgcd a b"
chaieb@22027
   635
  let ?a' = "a div ?g"
chaieb@22027
   636
  let ?b' = "b div ?g"
haftmann@27556
   637
  let ?g' = "zgcd ?a' ?b'"
chaieb@27568
   638
  have dvdg: "?g dvd a" "?g dvd b" by (simp_all add: zgcd_zdvd1 zgcd_zdvd2)
chaieb@27568
   639
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by (simp_all add: zgcd_zdvd1 zgcd_zdvd2)
wenzelm@22367
   640
  from dvdg dvdg' obtain ka kb ka' kb' where
wenzelm@22367
   641
   kab: "a = ?g*ka" "b = ?g*kb" "?a' = ?g'*ka'" "?b' = ?g' * kb'"
chaieb@22027
   642
    unfolding dvd_def by blast
wenzelm@22367
   643
  then have "?g* ?a' = (?g * ?g') * ka'" "?g* ?b' = (?g * ?g') * kb'" by simp_all
wenzelm@22367
   644
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
wenzelm@22367
   645
    by (auto simp add: zdvd_mult_div_cancel [OF dvdg(1)]
wenzelm@22367
   646
      zdvd_mult_div_cancel [OF dvdg(2)] dvd_def)
chaieb@22027
   647
  have "?g \<noteq> 0" using nz by simp
haftmann@27556
   648
  then have gp: "?g \<noteq> 0" using zgcd_pos[where i="a" and j="b"] by arith
haftmann@27556
   649
  from zgcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
wenzelm@22367
   650
  with zdvd_mult_cancel1 [OF gp] have "\<bar>?g'\<bar> = 1" by simp
haftmann@27556
   651
  with zgcd_pos show "?g' = 1" by simp
chaieb@22027
   652
qed
chaieb@22027
   653
chaieb@27669
   654
lemma zgcd_0 [simp, algebra]: "zgcd m 0 = abs m"
chaieb@27568
   655
  by (simp add: zgcd_def abs_if)
chaieb@27568
   656
chaieb@27669
   657
lemma zgcd_0_left [simp, algebra]: "zgcd 0 m = abs m"
chaieb@27568
   658
  by (simp add: zgcd_def abs_if)
chaieb@27568
   659
chaieb@27568
   660
lemma zgcd_non_0: "0 < n ==> zgcd m n = zgcd n (m mod n)"
chaieb@27568
   661
  apply (frule_tac b = n and a = m in pos_mod_sign)
chaieb@27568
   662
  apply (simp del: pos_mod_sign add: zgcd_def abs_if nat_mod_distrib)
chaieb@27568
   663
  apply (auto simp add: gcd_non_0 nat_mod_distrib [symmetric] zmod_zminus1_eq_if)
chaieb@27568
   664
  apply (frule_tac a = m in pos_mod_bound)
chaieb@27568
   665
  apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2 nat_le_eq_zle)
chaieb@27568
   666
  done
chaieb@27568
   667
chaieb@27568
   668
lemma zgcd_eq: "zgcd m n = zgcd n (m mod n)"
chaieb@27568
   669
  apply (case_tac "n = 0", simp add: DIVISION_BY_ZERO)
chaieb@27568
   670
  apply (auto simp add: linorder_neq_iff zgcd_non_0)
chaieb@27568
   671
  apply (cut_tac m = "-m" and n = "-n" in zgcd_non_0, auto)
chaieb@27568
   672
  done
chaieb@27568
   673
chaieb@27669
   674
lemma zgcd_1 [simp, algebra]: "zgcd m 1 = 1"
chaieb@27568
   675
  by (simp add: zgcd_def abs_if)
chaieb@27568
   676
chaieb@27669
   677
lemma zgcd_0_1_iff [simp, algebra]: "zgcd 0 m = 1 \<longleftrightarrow> \<bar>m\<bar> = 1"
chaieb@27568
   678
  by (simp add: zgcd_def abs_if)
chaieb@27568
   679
chaieb@27669
   680
lemma zgcd_greatest_iff[algebra]: "k dvd zgcd m n = (k dvd m \<and> k dvd n)"
chaieb@27568
   681
  by (simp add: zgcd_def abs_if int_dvd_iff dvd_int_iff nat_dvd_iff)
chaieb@27568
   682
chaieb@27669
   683
lemma zgcd_1_left [simp, algebra]: "zgcd 1 m = 1"
chaieb@27568
   684
  by (simp add: zgcd_def gcd_1_left)
chaieb@27568
   685
chaieb@27568
   686
lemma zgcd_assoc: "zgcd (zgcd k m) n = zgcd k (zgcd m n)"
chaieb@27568
   687
  by (simp add: zgcd_def gcd_assoc)
chaieb@27568
   688
chaieb@27568
   689
lemma zgcd_left_commute: "zgcd k (zgcd m n) = zgcd m (zgcd k n)"
chaieb@27568
   690
  apply (rule zgcd_commute [THEN trans])
chaieb@27568
   691
  apply (rule zgcd_assoc [THEN trans])
chaieb@27568
   692
  apply (rule zgcd_commute [THEN arg_cong])
chaieb@27568
   693
  done
chaieb@27568
   694
chaieb@27568
   695
lemmas zgcd_ac = zgcd_assoc zgcd_commute zgcd_left_commute
chaieb@27568
   696
  -- {* addition is an AC-operator *}
chaieb@27568
   697
chaieb@27568
   698
lemma zgcd_zmult_distrib2: "0 \<le> k ==> k * zgcd m n = zgcd (k * m) (k * n)"
chaieb@27568
   699
  by (simp del: minus_mult_right [symmetric]
chaieb@27568
   700
      add: minus_mult_right nat_mult_distrib zgcd_def abs_if
chaieb@27568
   701
          mult_less_0_iff gcd_mult_distrib2 [symmetric] zmult_int [symmetric])
chaieb@27568
   702
chaieb@27568
   703
lemma zgcd_zmult_distrib2_abs: "zgcd (k * m) (k * n) = abs k * zgcd m n"
chaieb@27568
   704
  by (simp add: abs_if zgcd_zmult_distrib2)
chaieb@27568
   705
chaieb@27568
   706
lemma zgcd_self [simp]: "0 \<le> m ==> zgcd m m = m"
chaieb@27568
   707
  by (cut_tac k = m and m = 1 and n = 1 in zgcd_zmult_distrib2, simp_all)
chaieb@27568
   708
chaieb@27568
   709
lemma zgcd_zmult_eq_self [simp]: "0 \<le> k ==> zgcd k (k * n) = k"
chaieb@27568
   710
  by (cut_tac k = k and m = 1 and n = n in zgcd_zmult_distrib2, simp_all)
chaieb@27568
   711
chaieb@27568
   712
lemma zgcd_zmult_eq_self2 [simp]: "0 \<le> k ==> zgcd (k * n) k = k"
chaieb@27568
   713
  by (cut_tac k = k and m = n and n = 1 in zgcd_zmult_distrib2, simp_all)
chaieb@27568
   714
chaieb@27568
   715
chaieb@27568
   716
definition "zlcm i j = int (lcm(nat(abs i)) (nat(abs j)))"
chaieb@23244
   717
chaieb@27669
   718
lemma dvd_zlcm_self1[simp, algebra]: "i dvd zlcm i j"
haftmann@27556
   719
by(simp add:zlcm_def dvd_int_iff)
nipkow@23983
   720
chaieb@27669
   721
lemma dvd_zlcm_self2[simp, algebra]: "j dvd zlcm i j"
haftmann@27556
   722
by(simp add:zlcm_def dvd_int_iff)
nipkow@23983
   723
chaieb@23244
   724
haftmann@27556
   725
lemma dvd_imp_dvd_zlcm1:
haftmann@27556
   726
  assumes "k dvd i" shows "k dvd (zlcm i j)"
nipkow@23983
   727
proof -
nipkow@23983
   728
  have "nat(abs k) dvd nat(abs i)" using `k dvd i`
nipkow@30042
   729
    by(simp add:int_dvd_iff[symmetric] dvd_int_iff[symmetric])
haftmann@27556
   730
  thus ?thesis by(simp add:zlcm_def dvd_int_iff)(blast intro: dvd_trans)
nipkow@23983
   731
qed
nipkow@23983
   732
haftmann@27556
   733
lemma dvd_imp_dvd_zlcm2:
haftmann@27556
   734
  assumes "k dvd j" shows "k dvd (zlcm i j)"
nipkow@23983
   735
proof -
nipkow@23983
   736
  have "nat(abs k) dvd nat(abs j)" using `k dvd j`
nipkow@30042
   737
    by(simp add:int_dvd_iff[symmetric] dvd_int_iff[symmetric])
haftmann@27556
   738
  thus ?thesis by(simp add:zlcm_def dvd_int_iff)(blast intro: dvd_trans)
nipkow@23983
   739
qed
nipkow@23983
   740
chaieb@23994
   741
chaieb@23244
   742
lemma zdvd_self_abs1: "(d::int) dvd (abs d)"
chaieb@23244
   743
by (case_tac "d <0", simp_all)
chaieb@23244
   744
chaieb@23244
   745
lemma zdvd_self_abs2: "(abs (d::int)) dvd d"
chaieb@23244
   746
by (case_tac "d<0", simp_all)
chaieb@23244
   747
chaieb@23244
   748
(* lcm a b is positive for positive a and b *)
chaieb@23244
   749
chaieb@23244
   750
lemma lcm_pos: 
chaieb@23244
   751
  assumes mpos: "m > 0"
chaieb@27568
   752
  and npos: "n>0"
haftmann@27556
   753
  shows "lcm m n > 0"
chaieb@23244
   754
proof(rule ccontr, simp add: lcm_def gcd_zero)
chaieb@27568
   755
assume h:"m*n div gcd m n = 0"
haftmann@27556
   756
from mpos npos have "gcd m n \<noteq> 0" using gcd_zero by simp
haftmann@27556
   757
hence gcdp: "gcd m n > 0" by simp
chaieb@23244
   758
with h
haftmann@27556
   759
have "m*n < gcd m n"
haftmann@27556
   760
  by (cases "m * n < gcd m n") (auto simp add: div_if[OF gcdp, where m="m*n"])
chaieb@23244
   761
moreover 
haftmann@27556
   762
have "gcd m n dvd m" by simp
haftmann@27556
   763
 with mpos dvd_imp_le have t1:"gcd m n \<le> m" by simp
chaieb@27568
   764
 with npos have t1:"gcd m n *n \<le> m*n" by simp
haftmann@27556
   765
 have "gcd m n \<le> gcd m n*n" using npos by simp
haftmann@27556
   766
 with t1 have "gcd m n \<le> m*n" by arith
chaieb@23244
   767
ultimately show "False" by simp
chaieb@23244
   768
qed
chaieb@23244
   769
haftmann@27556
   770
lemma zlcm_pos: 
nipkow@23983
   771
  assumes anz: "a \<noteq> 0"
nipkow@23983
   772
  and bnz: "b \<noteq> 0" 
haftmann@27556
   773
  shows "0 < zlcm a b"
chaieb@23244
   774
proof-
chaieb@23244
   775
  let ?na = "nat (abs a)"
chaieb@23244
   776
  let ?nb = "nat (abs b)"
nipkow@23983
   777
  have nap: "?na >0" using anz by simp
nipkow@23983
   778
  have nbp: "?nb >0" using bnz by simp
haftmann@27556
   779
  have "0 < lcm ?na ?nb" by (rule lcm_pos[OF nap nbp])
haftmann@27556
   780
  thus ?thesis by (simp add: zlcm_def)
chaieb@23244
   781
qed
chaieb@23244
   782
haftmann@28562
   783
lemma zgcd_code [code]:
haftmann@27651
   784
  "zgcd k l = \<bar>if l = 0 then k else zgcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
haftmann@27651
   785
  by (simp add: zgcd_def gcd.simps [of "nat \<bar>k\<bar>"] nat_mod_distrib)
haftmann@27651
   786
wenzelm@21256
   787
end