wenzelm@21256: (*  Title:      HOL/GCD.thy
wenzelm@21256:     Author:     Christophe Tabacznyj and Lawrence C Paulson
wenzelm@21256:     Copyright   1996  University of Cambridge
wenzelm@21256: *)
wenzelm@21256: 
wenzelm@21256: header {* The Greatest Common Divisor *}
wenzelm@21256: 
wenzelm@21256: theory GCD
haftmann@29655: imports Plain Presburger Main
wenzelm@21256: begin
wenzelm@21256: 
wenzelm@21256: text {*
haftmann@23687:   See \cite{davenport92}. \bigskip
wenzelm@21256: *}
wenzelm@21256: 
haftmann@23687: subsection {* Specification of GCD on nats *}
wenzelm@21256: 
wenzelm@21263: definition
haftmann@23687:   is_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where -- {* @{term gcd} as a relation *}
haftmann@28562:   [code del]: "is_gcd m n p \<longleftrightarrow> p dvd m \<and> p dvd n \<and>
haftmann@23687:     (\<forall>d. d dvd m \<longrightarrow> d dvd n \<longrightarrow> d dvd p)"
haftmann@23687: 
haftmann@23687: text {* Uniqueness *}
haftmann@23687: 
haftmann@27556: lemma is_gcd_unique: "is_gcd a b m \<Longrightarrow> is_gcd a b n \<Longrightarrow> m = n"
haftmann@23687:   by (simp add: is_gcd_def) (blast intro: dvd_anti_sym)
haftmann@23687: 
haftmann@23687: text {* Connection to divides relation *}
wenzelm@21256: 
haftmann@27556: lemma is_gcd_dvd: "is_gcd a b m \<Longrightarrow> k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd m"
haftmann@23687:   by (auto simp add: is_gcd_def)
haftmann@23687: 
haftmann@23687: text {* Commutativity *}
haftmann@23687: 
haftmann@27556: lemma is_gcd_commute: "is_gcd m n k = is_gcd n m k"
haftmann@23687:   by (auto simp add: is_gcd_def)
haftmann@23687: 
haftmann@23687: 
haftmann@23687: subsection {* GCD on nat by Euclid's algorithm *}
haftmann@23687: 
chaieb@27568: fun
chaieb@27568:   gcd  :: "nat => nat => nat"
chaieb@27568: where
haftmann@27556:   "gcd m n = (if n = 0 then m else gcd n (m mod n))"
haftmann@27556: lemma gcd_induct [case_names "0" rec]:
haftmann@23687:   fixes m n :: nat
haftmann@23687:   assumes "\<And>m. P m 0"
haftmann@23687:     and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n"
haftmann@23687:   shows "P m n"
haftmann@27556: proof (induct m n rule: gcd.induct)
haftmann@27556:   case (1 m n) with assms show ?case by (cases "n = 0") simp_all
haftmann@27556: qed
wenzelm@21256: 
chaieb@27669: lemma gcd_0 [simp, algebra]: "gcd m 0 = m"
wenzelm@21263:   by simp
wenzelm@21256: 
chaieb@27669: lemma gcd_0_left [simp,algebra]: "gcd 0 m = m"
haftmann@23687:   by simp
haftmann@23687: 
haftmann@27556: lemma gcd_non_0: "n > 0 \<Longrightarrow> gcd m n = gcd n (m mod n)"
haftmann@23687:   by simp
haftmann@23687: 
huffman@30082: lemma gcd_1 [simp, algebra]: "gcd m (Suc 0) = Suc 0"
wenzelm@21263:   by simp
wenzelm@21256: 
huffman@30082: lemma nat_gcd_1_right [simp, algebra]: "gcd m 1 = 1"
huffman@30082:   unfolding One_nat_def by (rule gcd_1)
huffman@30082: 
wenzelm@21256: declare gcd.simps [simp del]
wenzelm@21256: 
wenzelm@21256: text {*
haftmann@27556:   \medskip @{term "gcd m n"} divides @{text m} and @{text n}.  The
wenzelm@21256:   conjunctions don't seem provable separately.
wenzelm@21256: *}
wenzelm@21256: 
chaieb@27669: lemma gcd_dvd1 [iff, algebra]: "gcd m n dvd m"
chaieb@27669:   and gcd_dvd2 [iff, algebra]: "gcd m n dvd n"
wenzelm@21256:   apply (induct m n rule: gcd_induct)
wenzelm@21263:      apply (simp_all add: gcd_non_0)
wenzelm@21256:   apply (blast dest: dvd_mod_imp_dvd)
wenzelm@21256:   done
wenzelm@21256: 
wenzelm@21256: text {*
wenzelm@21256:   \medskip Maximality: for all @{term m}, @{term n}, @{term k}
wenzelm@21256:   naturals, if @{term k} divides @{term m} and @{term k} divides
haftmann@27556:   @{term n} then @{term k} divides @{term "gcd m n"}.
wenzelm@21256: *}
wenzelm@21256: 
haftmann@27556: lemma gcd_greatest: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd gcd m n"
wenzelm@21263:   by (induct m n rule: gcd_induct) (simp_all add: gcd_non_0 dvd_mod)
wenzelm@21256: 
wenzelm@21256: text {*
wenzelm@21256:   \medskip Function gcd yields the Greatest Common Divisor.
wenzelm@21256: *}
wenzelm@21256: 
haftmann@27556: lemma is_gcd: "is_gcd m n (gcd m n) "
haftmann@23687:   by (simp add: is_gcd_def gcd_greatest)
wenzelm@21256: 
wenzelm@21256: 
haftmann@23687: subsection {* Derived laws for GCD *}
wenzelm@21256: 
chaieb@27669: lemma gcd_greatest_iff [iff, algebra]: "k dvd gcd m n \<longleftrightarrow> k dvd m \<and> k dvd n"
haftmann@23687:   by (blast intro!: gcd_greatest intro: dvd_trans)
haftmann@23687: 
chaieb@27669: lemma gcd_zero[algebra]: "gcd m n = 0 \<longleftrightarrow> m = 0 \<and> n = 0"
haftmann@23687:   by (simp only: dvd_0_left_iff [symmetric] gcd_greatest_iff)
wenzelm@21256: 
haftmann@27556: lemma gcd_commute: "gcd m n = gcd n m"
wenzelm@21256:   apply (rule is_gcd_unique)
wenzelm@21256:    apply (rule is_gcd)
wenzelm@21256:   apply (subst is_gcd_commute)
wenzelm@21256:   apply (simp add: is_gcd)
wenzelm@21256:   done
wenzelm@21256: 
haftmann@27556: lemma gcd_assoc: "gcd (gcd k m) n = gcd k (gcd m n)"
wenzelm@21256:   apply (rule is_gcd_unique)
wenzelm@21256:    apply (rule is_gcd)
wenzelm@21256:   apply (simp add: is_gcd_def)
wenzelm@21256:   apply (blast intro: dvd_trans)
wenzelm@21256:   done
wenzelm@21256: 
huffman@30082: lemma gcd_1_left [simp, algebra]: "gcd (Suc 0) m = Suc 0"
haftmann@23687:   by (simp add: gcd_commute)
wenzelm@21256: 
huffman@30082: lemma nat_gcd_1_left [simp, algebra]: "gcd 1 m = 1"
huffman@30082:   unfolding One_nat_def by (rule gcd_1_left)
huffman@30082: 
wenzelm@21256: text {*
wenzelm@21256:   \medskip Multiplication laws
wenzelm@21256: *}
wenzelm@21256: 
haftmann@27556: lemma gcd_mult_distrib2: "k * gcd m n = gcd (k * m) (k * n)"
wenzelm@21256:     -- {* \cite[page 27]{davenport92} *}
wenzelm@21256:   apply (induct m n rule: gcd_induct)
wenzelm@21256:    apply simp
wenzelm@21256:   apply (case_tac "k = 0")
wenzelm@21256:    apply (simp_all add: mod_geq gcd_non_0 mod_mult_distrib2)
wenzelm@21256:   done
wenzelm@21256: 
chaieb@27669: lemma gcd_mult [simp, algebra]: "gcd k (k * n) = k"
wenzelm@21256:   apply (rule gcd_mult_distrib2 [of k 1 n, simplified, symmetric])
wenzelm@21256:   done
wenzelm@21256: 
chaieb@27669: lemma gcd_self [simp, algebra]: "gcd k k = k"
wenzelm@21256:   apply (rule gcd_mult [of k 1, simplified])
wenzelm@21256:   done
wenzelm@21256: 
haftmann@27556: lemma relprime_dvd_mult: "gcd k n = 1 ==> k dvd m * n ==> k dvd m"
wenzelm@21256:   apply (insert gcd_mult_distrib2 [of m k n])
wenzelm@21256:   apply simp
wenzelm@21256:   apply (erule_tac t = m in ssubst)
wenzelm@21256:   apply simp
wenzelm@21256:   done
wenzelm@21256: 
haftmann@27556: lemma relprime_dvd_mult_iff: "gcd k n = 1 ==> (k dvd m * n) = (k dvd m)"
haftmann@27651:   by (auto intro: relprime_dvd_mult dvd_mult2)
wenzelm@21256: 
haftmann@27556: lemma gcd_mult_cancel: "gcd k n = 1 ==> gcd (k * m) n = gcd m n"
wenzelm@21256:   apply (rule dvd_anti_sym)
wenzelm@21256:    apply (rule gcd_greatest)
wenzelm@21256:     apply (rule_tac n = k in relprime_dvd_mult)
wenzelm@21256:      apply (simp add: gcd_assoc)
wenzelm@21256:      apply (simp add: gcd_commute)
wenzelm@21256:     apply (simp_all add: mult_commute)
wenzelm@21256:   done
wenzelm@21256: 
wenzelm@21256: 
wenzelm@21256: text {* \medskip Addition laws *}
wenzelm@21256: 
haftmann@27676: lemma gcd_add1 [simp, algebra]: "gcd (m + n) n = gcd m n"
haftmann@27676:   by (cases "n = 0") (auto simp add: gcd_non_0)
wenzelm@21256: 
chaieb@27669: lemma gcd_add2 [simp, algebra]: "gcd m (m + n) = gcd m n"
wenzelm@21256: proof -
haftmann@27556:   have "gcd m (m + n) = gcd (m + n) m" by (rule gcd_commute)
haftmann@27556:   also have "... = gcd (n + m) m" by (simp add: add_commute)
haftmann@27556:   also have "... = gcd n m" by simp
haftmann@27556:   also have  "... = gcd m n" by (rule gcd_commute)
wenzelm@21256:   finally show ?thesis .
wenzelm@21256: qed
wenzelm@21256: 
chaieb@27669: lemma gcd_add2' [simp, algebra]: "gcd m (n + m) = gcd m n"
wenzelm@21256:   apply (subst add_commute)
wenzelm@21256:   apply (rule gcd_add2)
wenzelm@21256:   done
wenzelm@21256: 
chaieb@27669: lemma gcd_add_mult[algebra]: "gcd m (k * m + n) = gcd m n"
wenzelm@21263:   by (induct k) (simp_all add: add_assoc)
wenzelm@21256: 
chaieb@27669: lemma gcd_dvd_prod: "gcd m n dvd m * n" 
haftmann@23687:   using mult_dvd_mono [of 1] by auto
chaieb@22027: 
wenzelm@22367: text {*
wenzelm@22367:   \medskip Division by gcd yields rrelatively primes.
wenzelm@22367: *}
chaieb@22027: 
chaieb@22027: lemma div_gcd_relprime:
wenzelm@22367:   assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
haftmann@27556:   shows "gcd (a div gcd a b) (b div gcd a b) = 1"
wenzelm@22367: proof -
haftmann@27556:   let ?g = "gcd a b"
chaieb@22027:   let ?a' = "a div ?g"
chaieb@22027:   let ?b' = "b div ?g"
haftmann@27556:   let ?g' = "gcd ?a' ?b'"
chaieb@22027:   have dvdg: "?g dvd a" "?g dvd b" by simp_all
chaieb@22027:   have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
wenzelm@22367:   from dvdg dvdg' obtain ka kb ka' kb' where
wenzelm@22367:       kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
chaieb@22027:     unfolding dvd_def by blast
wenzelm@22367:   then have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'" by simp_all
wenzelm@22367:   then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
wenzelm@22367:     by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
wenzelm@22367:       dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
chaieb@22027:   have "?g \<noteq> 0" using nz by (simp add: gcd_zero)
wenzelm@22367:   then have gp: "?g > 0" by simp
wenzelm@22367:   from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
wenzelm@22367:   with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
chaieb@22027: qed
chaieb@22027: 
chaieb@27669: 
chaieb@27669: 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"
chaieb@27669: proof(auto)
chaieb@27669:   assume H: "d dvd a" "d dvd b" "\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d"
chaieb@27669:   from H(3)[rule_format] gcd_dvd1[of a b] gcd_dvd2[of a b] 
chaieb@27669:   have th: "gcd a b dvd d" by blast
chaieb@27669:   from dvd_anti_sym[OF th gcd_greatest[OF H(1,2)]]  show "d = gcd a b" by blast 
chaieb@27669: qed
chaieb@27669: 
chaieb@27669: lemma gcd_eq: assumes H: "\<forall>d. d dvd x \<and> d dvd y \<longleftrightarrow> d dvd u \<and> d dvd v"
chaieb@27669:   shows "gcd x y = gcd u v"
chaieb@27669: proof-
chaieb@27669:   from H have "\<forall>d. d dvd x \<and> d dvd y \<longleftrightarrow> d dvd gcd u v" by simp
chaieb@27669:   with gcd_unique[of "gcd u v" x y]  show ?thesis by auto
chaieb@27669: qed
chaieb@27669: 
chaieb@27669: lemma ind_euclid: 
chaieb@27669:   assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0" 
chaieb@27669:   and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)" 
chaieb@27669:   shows "P a b"
chaieb@27669: proof(induct n\<equiv>"a+b" arbitrary: a b rule: nat_less_induct)
chaieb@27669:   fix n a b
chaieb@27669:   assume H: "\<forall>m < n. \<forall>a b. m = a + b \<longrightarrow> P a b" "n = a + b"
chaieb@27669:   have "a = b \<or> a < b \<or> b < a" by arith
chaieb@27669:   moreover {assume eq: "a= b"
chaieb@27669:     from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq by simp}
chaieb@27669:   moreover
chaieb@27669:   {assume lt: "a < b"
chaieb@27669:     hence "a + b - a < n \<or> a = 0"  using H(2) by arith
chaieb@27669:     moreover
chaieb@27669:     {assume "a =0" with z c have "P a b" by blast }
chaieb@27669:     moreover
chaieb@27669:     {assume ab: "a + b - a < n"
chaieb@27669:       have th0: "a + b - a = a + (b - a)" using lt by arith
chaieb@27669:       from add[rule_format, OF H(1)[rule_format, OF ab th0]]
chaieb@27669:       have "P a b" by (simp add: th0[symmetric])}
chaieb@27669:     ultimately have "P a b" by blast}
chaieb@27669:   moreover
chaieb@27669:   {assume lt: "a > b"
chaieb@27669:     hence "b + a - b < n \<or> b = 0"  using H(2) by arith
chaieb@27669:     moreover
chaieb@27669:     {assume "b =0" with z c have "P a b" by blast }
chaieb@27669:     moreover
chaieb@27669:     {assume ab: "b + a - b < n"
chaieb@27669:       have th0: "b + a - b = b + (a - b)" using lt by arith
chaieb@27669:       from add[rule_format, OF H(1)[rule_format, OF ab th0]]
chaieb@27669:       have "P b a" by (simp add: th0[symmetric])
chaieb@27669:       hence "P a b" using c by blast }
chaieb@27669:     ultimately have "P a b" by blast}
chaieb@27669: ultimately  show "P a b" by blast
chaieb@27669: qed
chaieb@27669: 
chaieb@27669: lemma bezout_lemma: 
chaieb@27669:   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)"
chaieb@27669:   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)"
chaieb@27669: using ex
chaieb@27669: apply clarsimp
chaieb@27669: apply (rule_tac x="d" in exI, simp add: dvd_add)
chaieb@27669: apply (case_tac "a * x = b * y + d" , simp_all)
chaieb@27669: apply (rule_tac x="x + y" in exI)
chaieb@27669: apply (rule_tac x="y" in exI)
chaieb@27669: apply algebra
chaieb@27669: apply (rule_tac x="x" in exI)
chaieb@27669: apply (rule_tac x="x + y" in exI)
chaieb@27669: apply algebra
chaieb@27669: done
chaieb@27669: 
chaieb@27669: 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)"
chaieb@27669: apply(induct a b rule: ind_euclid)
chaieb@27669: apply blast
chaieb@27669: apply clarify
chaieb@27669: apply (rule_tac x="a" in exI, simp add: dvd_add)
chaieb@27669: apply clarsimp
chaieb@27669: apply (rule_tac x="d" in exI)
chaieb@27669: apply (case_tac "a * x = b * y + d", simp_all add: dvd_add)
chaieb@27669: apply (rule_tac x="x+y" in exI)
chaieb@27669: apply (rule_tac x="y" in exI)
chaieb@27669: apply algebra
chaieb@27669: apply (rule_tac x="x" in exI)
chaieb@27669: apply (rule_tac x="x+y" in exI)
chaieb@27669: apply algebra
chaieb@27669: done
chaieb@27669: 
chaieb@27669: 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: using bezout_add[of a b]
chaieb@27669: apply clarsimp
chaieb@27669: apply (rule_tac x="d" in exI, simp)
chaieb@27669: apply (rule_tac x="x" in exI)
chaieb@27669: apply (rule_tac x="y" in exI)
chaieb@27669: apply auto
chaieb@27669: done
chaieb@27669: 
chaieb@27669: 
chaieb@27669: text {* We can get a stronger version with a nonzeroness assumption. *}
chaieb@27669: lemma divides_le: "m dvd n ==> m <= n \<or> n = (0::nat)" by (auto simp add: dvd_def)
chaieb@27669: 
chaieb@27669: lemma bezout_add_strong: assumes nz: "a \<noteq> (0::nat)"
chaieb@27669:   shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
chaieb@27669: proof-
chaieb@27669:   from nz have ap: "a > 0" by simp
chaieb@27669:  from bezout_add[of a b] 
chaieb@27669:  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:  moreover
chaieb@27669:  {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
chaieb@27669:    from H have ?thesis by blast }
chaieb@27669:  moreover
chaieb@27669:  {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
chaieb@27669:    {assume b0: "b = 0" with H  have ?thesis by simp}
chaieb@27669:    moreover 
chaieb@27669:    {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
chaieb@27669:      from divides_le[OF H(2)] b have "d < b \<or> d = b" using le_less by blast
chaieb@27669:      moreover
chaieb@27669:      {assume db: "d=b"
chaieb@27669:        from prems have ?thesis apply simp
chaieb@27669: 	 apply (rule exI[where x = b], simp)
chaieb@27669: 	 apply (rule exI[where x = b])
chaieb@27669: 	by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
chaieb@27669:     moreover
chaieb@27669:     {assume db: "d < b" 
chaieb@27669: 	{assume "x=0" hence ?thesis  using prems by simp }
chaieb@27669: 	moreover
chaieb@27669: 	{assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
chaieb@27669: 	  
chaieb@27669: 	  from db have "d \<le> b - 1" by simp
chaieb@27669: 	  hence "d*b \<le> b*(b - 1)" by simp
chaieb@27669: 	  with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
chaieb@27669: 	  have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
chaieb@27669: 	  from H (3) have "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)" by algebra
chaieb@27669: 	  hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
chaieb@27669: 	  hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)" 
chaieb@27669: 	    by (simp only: diff_add_assoc[OF dble, of d, symmetric])
chaieb@27669: 	  hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
chaieb@27669: 	    by (simp only: diff_mult_distrib2 add_commute mult_ac)
chaieb@27669: 	  hence ?thesis using H(1,2)
chaieb@27669: 	    apply -
chaieb@27669: 	    apply (rule exI[where x=d], simp)
chaieb@27669: 	    apply (rule exI[where x="(b - 1) * y"])
chaieb@27669: 	    by (rule exI[where x="x*(b - 1) - d"], simp)}
chaieb@27669: 	ultimately have ?thesis by blast}
chaieb@27669:     ultimately have ?thesis by blast}
chaieb@27669:   ultimately have ?thesis by blast}
chaieb@27669:  ultimately show ?thesis by blast
chaieb@27669: qed
chaieb@27669: 
chaieb@27669: 
chaieb@27669: lemma bezout_gcd: "\<exists>x y. a * x - b * y = gcd a b \<or> b * x - a * y = gcd a b"
chaieb@27669: proof-
chaieb@27669:   let ?g = "gcd a b"
chaieb@27669:   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:   from d(1,2) have "d dvd ?g" by simp
chaieb@27669:   then obtain k where k: "?g = d*k" unfolding dvd_def by blast
chaieb@27669:   from d(3) have "(a * x - b * y)*k = d*k \<or> (b * x - a * y)*k = d*k" by blast 
chaieb@27669:   hence "a * x * k - b * y*k = d*k \<or> b * x * k - a * y*k = d*k" 
chaieb@27669:     by (algebra add: diff_mult_distrib)
chaieb@27669:   hence "a * (x * k) - b * (y*k) = ?g \<or> b * (x * k) - a * (y*k) = ?g" 
chaieb@27669:     by (simp add: k mult_assoc)
chaieb@27669:   thus ?thesis by blast
chaieb@27669: qed
chaieb@27669: 
chaieb@27669: lemma bezout_gcd_strong: assumes a: "a \<noteq> 0" 
chaieb@27669:   shows "\<exists>x y. a * x = b * y + gcd a b"
chaieb@27669: proof-
chaieb@27669:   let ?g = "gcd a b"
chaieb@27669:   from bezout_add_strong[OF a, of b]
chaieb@27669:   obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
chaieb@27669:   from d(1,2) have "d dvd ?g" by simp
chaieb@27669:   then obtain k where k: "?g = d*k" unfolding dvd_def by blast
chaieb@27669:   from d(3) have "a * x * k = (b * y + d) *k " by algebra
chaieb@27669:   hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
chaieb@27669:   thus ?thesis by blast
chaieb@27669: qed
chaieb@27669: 
chaieb@27669: lemma gcd_mult_distrib: "gcd(a * c) (b * c) = c * gcd a b"
chaieb@27669: by(simp add: gcd_mult_distrib2 mult_commute)
chaieb@27669: 
chaieb@27669: 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:   (is "?lhs \<longleftrightarrow> ?rhs")
chaieb@27669: proof-
chaieb@27669:   let ?g = "gcd a b"
chaieb@27669:   {assume H: ?rhs then obtain k where k: "d = ?g*k" unfolding dvd_def by blast
chaieb@27669:     from bezout_gcd[of a b] obtain x y where xy: "a * x - b * y = ?g \<or> b * x - a * y = ?g"
chaieb@27669:       by blast
chaieb@27669:     hence "(a * x - b * y)*k = ?g*k \<or> (b * x - a * y)*k = ?g*k" by auto
chaieb@27669:     hence "a * x*k - b * y*k = ?g*k \<or> b * x * k - a * y*k = ?g*k" 
chaieb@27669:       by (simp only: diff_mult_distrib)
chaieb@27669:     hence "a * (x*k) - b * (y*k) = d \<or> b * (x * k) - a * (y*k) = d"
chaieb@27669:       by (simp add: k[symmetric] mult_assoc)
chaieb@27669:     hence ?lhs by blast}
chaieb@27669:   moreover
chaieb@27669:   {fix x y assume H: "a * x - b * y = d \<or> b * x - a * y = d"
chaieb@27669:     have dv: "?g dvd a*x" "?g dvd b * y" "?g dvd b*x" "?g dvd a * y"
chaieb@27669:       using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all
nipkow@30042:     from nat_dvd_diff[OF dv(1,2)] nat_dvd_diff[OF dv(3,4)] H
chaieb@27669:     have ?rhs by auto}
chaieb@27669:   ultimately show ?thesis by blast
chaieb@27669: qed
chaieb@27669: 
chaieb@27669: lemma gcd_bezout_sum: assumes H:"a * x + b * y = d" shows "gcd a b dvd d"
chaieb@27669: proof-
chaieb@27669:   let ?g = "gcd a b"
chaieb@27669:     have dv: "?g dvd a*x" "?g dvd b * y" 
chaieb@27669:       using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all
chaieb@27669:     from dvd_add[OF dv] H
chaieb@27669:     show ?thesis by auto
chaieb@27669: qed
chaieb@27669: 
chaieb@27669: lemma gcd_mult': "gcd b (a * b) = b"
chaieb@27669: by (simp add: gcd_mult mult_commute[of a b]) 
chaieb@27669: 
chaieb@27669: lemma gcd_add: "gcd(a + b) b = gcd a b" 
chaieb@27669:   "gcd(b + a) b = gcd a b" "gcd a (a + b) = gcd a b" "gcd a (b + a) = gcd a b"
chaieb@27669: apply (simp_all add: gcd_add1)
chaieb@27669: by (simp add: gcd_commute gcd_add1)
chaieb@27669: 
chaieb@27669: lemma gcd_sub: "b <= a ==> gcd(a - b) b = gcd a b" "a <= b ==> gcd a (b - a) = gcd a b"
chaieb@27669: proof-
chaieb@27669:   {fix a b assume H: "b \<le> (a::nat)"
chaieb@27669:     hence th: "a - b + b = a" by arith
chaieb@27669:     from gcd_add(1)[of "a - b" b] th  have "gcd(a - b) b = gcd a b" by simp}
chaieb@27669:   note th = this
chaieb@27669: {
chaieb@27669:   assume ab: "b \<le> a"
chaieb@27669:   from th[OF ab] show "gcd (a - b)  b = gcd a b" by blast
chaieb@27669: next
chaieb@27669:   assume ab: "a \<le> b"
chaieb@27669:   from th[OF ab] show "gcd a (b - a) = gcd a b" 
chaieb@27669:     by (simp add: gcd_commute)}
chaieb@27669: qed
chaieb@27669: 
chaieb@27669: 
haftmann@23687: subsection {* LCM defined by GCD *}
wenzelm@22367: 
chaieb@27669: 
haftmann@23687: definition
haftmann@27556:   lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@23687: where
chaieb@27568:   lcm_def: "lcm m n = m * n div gcd m n"
haftmann@23687: 
haftmann@23687: lemma prod_gcd_lcm:
haftmann@27556:   "m * n = gcd m n * lcm m n"
haftmann@23687:   unfolding lcm_def by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod])
haftmann@23687: 
haftmann@27556: lemma lcm_0 [simp]: "lcm m 0 = 0"
haftmann@23687:   unfolding lcm_def by simp
haftmann@23687: 
haftmann@27556: lemma lcm_1 [simp]: "lcm m 1 = m"
haftmann@23687:   unfolding lcm_def by simp
haftmann@23687: 
haftmann@27556: lemma lcm_0_left [simp]: "lcm 0 n = 0"
haftmann@23687:   unfolding lcm_def by simp
haftmann@23687: 
haftmann@27556: lemma lcm_1_left [simp]: "lcm 1 m = m"
haftmann@23687:   unfolding lcm_def by simp
haftmann@23687: 
haftmann@23687: lemma dvd_pos:
haftmann@23687:   fixes n m :: nat
haftmann@23687:   assumes "n > 0" and "m dvd n"
haftmann@23687:   shows "m > 0"
haftmann@23687: using assms by (cases m) auto
haftmann@23687: 
haftmann@23951: lemma lcm_least:
haftmann@23687:   assumes "m dvd k" and "n dvd k"
haftmann@27556:   shows "lcm m n dvd k"
haftmann@23687: proof (cases k)
haftmann@23687:   case 0 then show ?thesis by auto
haftmann@23687: next
haftmann@23687:   case (Suc _) then have pos_k: "k > 0" by auto
haftmann@23687:   from assms dvd_pos [OF this] have pos_mn: "m > 0" "n > 0" by auto
haftmann@27556:   with gcd_zero [of m n] have pos_gcd: "gcd m n > 0" by simp
haftmann@23687:   from assms obtain p where k_m: "k = m * p" using dvd_def by blast
haftmann@23687:   from assms obtain q where k_n: "k = n * q" using dvd_def by blast
haftmann@23687:   from pos_k k_m have pos_p: "p > 0" by auto
haftmann@23687:   from pos_k k_n have pos_q: "q > 0" by auto
haftmann@27556:   have "k * k * gcd q p = k * gcd (k * q) (k * p)"
haftmann@23687:     by (simp add: mult_ac gcd_mult_distrib2)
haftmann@27556:   also have "\<dots> = k * gcd (m * p * q) (n * q * p)"
haftmann@23687:     by (simp add: k_m [symmetric] k_n [symmetric])
haftmann@27556:   also have "\<dots> = k * p * q * gcd m n"
haftmann@23687:     by (simp add: mult_ac gcd_mult_distrib2)
haftmann@27556:   finally have "(m * p) * (n * q) * gcd q p = k * p * q * gcd m n"
haftmann@23687:     by (simp only: k_m [symmetric] k_n [symmetric])
haftmann@27556:   then have "p * q * m * n * gcd q p = p * q * k * gcd m n"
haftmann@23687:     by (simp add: mult_ac)
haftmann@27556:   with pos_p pos_q have "m * n * gcd q p = k * gcd m n"
haftmann@23687:     by simp
haftmann@23687:   with prod_gcd_lcm [of m n]
haftmann@27556:   have "lcm m n * gcd q p * gcd m n = k * gcd m n"
haftmann@23687:     by (simp add: mult_ac)
haftmann@27556:   with pos_gcd have "lcm m n * gcd q p = k" by simp
haftmann@23687:   then show ?thesis using dvd_def by auto
haftmann@23687: qed
haftmann@23687: 
haftmann@23687: lemma lcm_dvd1 [iff]:
haftmann@27556:   "m dvd lcm m n"
haftmann@23687: proof (cases m)
haftmann@23687:   case 0 then show ?thesis by simp
haftmann@23687: next
haftmann@23687:   case (Suc _)
haftmann@23687:   then have mpos: "m > 0" by simp
haftmann@23687:   show ?thesis
haftmann@23687:   proof (cases n)
haftmann@23687:     case 0 then show ?thesis by simp
haftmann@23687:   next
haftmann@23687:     case (Suc _)
haftmann@23687:     then have npos: "n > 0" by simp
haftmann@27556:     have "gcd m n dvd n" by simp
haftmann@27556:     then obtain k where "n = gcd m n * k" using dvd_def by auto
haftmann@27556:     then have "m * n div gcd m n = m * (gcd m n * k) div gcd m n" by (simp add: mult_ac)
haftmann@23687:     also have "\<dots> = m * k" using mpos npos gcd_zero by simp
haftmann@23687:     finally show ?thesis by (simp add: lcm_def)
haftmann@23687:   qed
haftmann@23687: qed
haftmann@23687: 
haftmann@23687: lemma lcm_dvd2 [iff]: 
haftmann@27556:   "n dvd lcm m n"
haftmann@23687: proof (cases n)
haftmann@23687:   case 0 then show ?thesis by simp
haftmann@23687: next
haftmann@23687:   case (Suc _)
haftmann@23687:   then have npos: "n > 0" by simp
haftmann@23687:   show ?thesis
haftmann@23687:   proof (cases m)
haftmann@23687:     case 0 then show ?thesis by simp
haftmann@23687:   next
haftmann@23687:     case (Suc _)
haftmann@23687:     then have mpos: "m > 0" by simp
haftmann@27556:     have "gcd m n dvd m" by simp
haftmann@27556:     then obtain k where "m = gcd m n * k" using dvd_def by auto
haftmann@27556:     then have "m * n div gcd m n = (gcd m n * k) * n div gcd m n" by (simp add: mult_ac)
haftmann@23687:     also have "\<dots> = n * k" using mpos npos gcd_zero by simp
haftmann@23687:     finally show ?thesis by (simp add: lcm_def)
haftmann@23687:   qed
haftmann@23687: qed
haftmann@23687: 
chaieb@27568: lemma gcd_add1_eq: "gcd (m + k) k = gcd (m + k) m"
chaieb@27568:   by (simp add: gcd_commute)
chaieb@27568: 
chaieb@27568: lemma gcd_diff2: "m \<le> n ==> gcd n (n - m) = gcd n m"
chaieb@27568:   apply (subgoal_tac "n = m + (n - m)")
chaieb@27669:   apply (erule ssubst, rule gcd_add1_eq, simp)  
chaieb@27568:   done
chaieb@27568: 
haftmann@23687: 
haftmann@23687: subsection {* GCD and LCM on integers *}
wenzelm@22367: 
wenzelm@22367: definition
haftmann@27556:   zgcd :: "int \<Rightarrow> int \<Rightarrow> int" where
haftmann@27556:   "zgcd i j = int (gcd (nat (abs i)) (nat (abs j)))"
wenzelm@22367: 
chaieb@27669: lemma zgcd_zdvd1 [iff,simp, algebra]: "zgcd i j dvd i"
nipkow@29700: by (simp add: zgcd_def int_dvd_iff)
chaieb@22027: 
chaieb@27669: lemma zgcd_zdvd2 [iff,simp, algebra]: "zgcd i j dvd j"
nipkow@29700: by (simp add: zgcd_def int_dvd_iff)
chaieb@22027: 
haftmann@27556: lemma zgcd_pos: "zgcd i j \<ge> 0"
nipkow@29700: by (simp add: zgcd_def)
wenzelm@22367: 
chaieb@27669: lemma zgcd0 [simp,algebra]: "(zgcd i j = 0) = (i = 0 \<and> j = 0)"
nipkow@29700: by (simp add: zgcd_def gcd_zero)
chaieb@22027: 
haftmann@27556: lemma zgcd_commute: "zgcd i j = zgcd j i"
nipkow@29700: unfolding zgcd_def by (simp add: gcd_commute)
wenzelm@22367: 
chaieb@27669: lemma zgcd_zminus [simp, algebra]: "zgcd (- i) j = zgcd i j"
nipkow@29700: unfolding zgcd_def by simp
wenzelm@22367: 
chaieb@27669: lemma zgcd_zminus2 [simp, algebra]: "zgcd i (- j) = zgcd i j"
nipkow@29700: unfolding zgcd_def by simp
wenzelm@22367: 
chaieb@27669:   (* should be solved by algebra*)
haftmann@27556: lemma zrelprime_dvd_mult: "zgcd i j = 1 \<Longrightarrow> i dvd k * j \<Longrightarrow> i dvd k"
haftmann@27556:   unfolding zgcd_def
wenzelm@22367: proof -
haftmann@27556:   assume "int (gcd (nat \<bar>i\<bar>) (nat \<bar>j\<bar>)) = 1" "i dvd k * j"
haftmann@27556:   then have g: "gcd (nat \<bar>i\<bar>) (nat \<bar>j\<bar>) = 1" by simp
wenzelm@22367:   from `i dvd k * j` obtain h where h: "k*j = i*h" unfolding dvd_def by blast
chaieb@22027:   have th: "nat \<bar>i\<bar> dvd nat \<bar>k\<bar> * nat \<bar>j\<bar>"
wenzelm@22367:     unfolding dvd_def
wenzelm@22367:     by (rule_tac x= "nat \<bar>h\<bar>" in exI, simp add: h nat_abs_mult_distrib [symmetric])
wenzelm@22367:   from relprime_dvd_mult [OF g th] obtain h' where h': "nat \<bar>k\<bar> = nat \<bar>i\<bar> * h'"
chaieb@22027:     unfolding dvd_def by blast
chaieb@22027:   from h' have "int (nat \<bar>k\<bar>) = int (nat \<bar>i\<bar> * h')" by simp
huffman@23431:   then have "\<bar>k\<bar> = \<bar>i\<bar> * int h'" by (simp add: int_mult)
chaieb@22027:   then show ?thesis
nipkow@30042:     apply (subst abs_dvd_iff [symmetric])
nipkow@30042:     apply (subst dvd_abs_iff [symmetric])
chaieb@22027:     apply (unfold dvd_def)
wenzelm@22367:     apply (rule_tac x = "int h'" in exI, simp)
chaieb@22027:     done
chaieb@22027: qed
chaieb@22027: 
haftmann@27556: lemma int_nat_abs: "int (nat (abs x)) = abs x" by arith
wenzelm@22367: 
haftmann@27556: lemma zgcd_greatest:
wenzelm@22367:   assumes "k dvd m" and "k dvd n"
haftmann@27556:   shows "k dvd zgcd m n"
wenzelm@22367: proof -
chaieb@22027:   let ?k' = "nat \<bar>k\<bar>"
chaieb@22027:   let ?m' = "nat \<bar>m\<bar>"
chaieb@22027:   let ?n' = "nat \<bar>n\<bar>"
wenzelm@22367:   from `k dvd m` and `k dvd n` have dvd': "?k' dvd ?m'" "?k' dvd ?n'"
nipkow@30042:     unfolding zdvd_int by (simp_all only: int_nat_abs abs_dvd_iff dvd_abs_iff)
haftmann@27556:   from gcd_greatest [OF dvd'] have "int (nat \<bar>k\<bar>) dvd zgcd m n"
haftmann@27556:     unfolding zgcd_def by (simp only: zdvd_int)
haftmann@27556:   then have "\<bar>k\<bar> dvd zgcd m n" by (simp only: int_nat_abs)
nipkow@30042:   then show "k dvd zgcd m n" by simp
chaieb@22027: qed
chaieb@22027: 
haftmann@27556: lemma div_zgcd_relprime:
wenzelm@22367:   assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
haftmann@27556:   shows "zgcd (a div (zgcd a b)) (b div (zgcd a b)) = 1"
wenzelm@22367: proof -
chaieb@25112:   from nz have nz': "nat \<bar>a\<bar> \<noteq> 0 \<or> nat \<bar>b\<bar> \<noteq> 0" by arith 
haftmann@27556:   let ?g = "zgcd a b"
chaieb@22027:   let ?a' = "a div ?g"
chaieb@22027:   let ?b' = "b div ?g"
haftmann@27556:   let ?g' = "zgcd ?a' ?b'"
chaieb@27568:   have dvdg: "?g dvd a" "?g dvd b" by (simp_all add: zgcd_zdvd1 zgcd_zdvd2)
chaieb@27568:   have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by (simp_all add: zgcd_zdvd1 zgcd_zdvd2)
wenzelm@22367:   from dvdg dvdg' obtain ka kb ka' kb' where
wenzelm@22367:    kab: "a = ?g*ka" "b = ?g*kb" "?a' = ?g'*ka'" "?b' = ?g' * kb'"
chaieb@22027:     unfolding dvd_def by blast
wenzelm@22367:   then have "?g* ?a' = (?g * ?g') * ka'" "?g* ?b' = (?g * ?g') * kb'" by simp_all
wenzelm@22367:   then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
wenzelm@22367:     by (auto simp add: zdvd_mult_div_cancel [OF dvdg(1)]
wenzelm@22367:       zdvd_mult_div_cancel [OF dvdg(2)] dvd_def)
chaieb@22027:   have "?g \<noteq> 0" using nz by simp
haftmann@27556:   then have gp: "?g \<noteq> 0" using zgcd_pos[where i="a" and j="b"] by arith
haftmann@27556:   from zgcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
wenzelm@22367:   with zdvd_mult_cancel1 [OF gp] have "\<bar>?g'\<bar> = 1" by simp
haftmann@27556:   with zgcd_pos show "?g' = 1" by simp
chaieb@22027: qed
chaieb@22027: 
chaieb@27669: lemma zgcd_0 [simp, algebra]: "zgcd m 0 = abs m"
chaieb@27568:   by (simp add: zgcd_def abs_if)
chaieb@27568: 
chaieb@27669: lemma zgcd_0_left [simp, algebra]: "zgcd 0 m = abs m"
chaieb@27568:   by (simp add: zgcd_def abs_if)
chaieb@27568: 
chaieb@27568: lemma zgcd_non_0: "0 < n ==> zgcd m n = zgcd n (m mod n)"
chaieb@27568:   apply (frule_tac b = n and a = m in pos_mod_sign)
chaieb@27568:   apply (simp del: pos_mod_sign add: zgcd_def abs_if nat_mod_distrib)
chaieb@27568:   apply (auto simp add: gcd_non_0 nat_mod_distrib [symmetric] zmod_zminus1_eq_if)
chaieb@27568:   apply (frule_tac a = m in pos_mod_bound)
chaieb@27568:   apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2 nat_le_eq_zle)
chaieb@27568:   done
chaieb@27568: 
chaieb@27568: lemma zgcd_eq: "zgcd m n = zgcd n (m mod n)"
chaieb@27568:   apply (case_tac "n = 0", simp add: DIVISION_BY_ZERO)
chaieb@27568:   apply (auto simp add: linorder_neq_iff zgcd_non_0)
chaieb@27568:   apply (cut_tac m = "-m" and n = "-n" in zgcd_non_0, auto)
chaieb@27568:   done
chaieb@27568: 
chaieb@27669: lemma zgcd_1 [simp, algebra]: "zgcd m 1 = 1"
chaieb@27568:   by (simp add: zgcd_def abs_if)
chaieb@27568: 
chaieb@27669: lemma zgcd_0_1_iff [simp, algebra]: "zgcd 0 m = 1 \<longleftrightarrow> \<bar>m\<bar> = 1"
chaieb@27568:   by (simp add: zgcd_def abs_if)
chaieb@27568: 
chaieb@27669: lemma zgcd_greatest_iff[algebra]: "k dvd zgcd m n = (k dvd m \<and> k dvd n)"
chaieb@27568:   by (simp add: zgcd_def abs_if int_dvd_iff dvd_int_iff nat_dvd_iff)
chaieb@27568: 
chaieb@27669: lemma zgcd_1_left [simp, algebra]: "zgcd 1 m = 1"
chaieb@27568:   by (simp add: zgcd_def gcd_1_left)
chaieb@27568: 
chaieb@27568: lemma zgcd_assoc: "zgcd (zgcd k m) n = zgcd k (zgcd m n)"
chaieb@27568:   by (simp add: zgcd_def gcd_assoc)
chaieb@27568: 
chaieb@27568: lemma zgcd_left_commute: "zgcd k (zgcd m n) = zgcd m (zgcd k n)"
chaieb@27568:   apply (rule zgcd_commute [THEN trans])
chaieb@27568:   apply (rule zgcd_assoc [THEN trans])
chaieb@27568:   apply (rule zgcd_commute [THEN arg_cong])
chaieb@27568:   done
chaieb@27568: 
chaieb@27568: lemmas zgcd_ac = zgcd_assoc zgcd_commute zgcd_left_commute
chaieb@27568:   -- {* addition is an AC-operator *}
chaieb@27568: 
chaieb@27568: lemma zgcd_zmult_distrib2: "0 \<le> k ==> k * zgcd m n = zgcd (k * m) (k * n)"
chaieb@27568:   by (simp del: minus_mult_right [symmetric]
chaieb@27568:       add: minus_mult_right nat_mult_distrib zgcd_def abs_if
chaieb@27568:           mult_less_0_iff gcd_mult_distrib2 [symmetric] zmult_int [symmetric])
chaieb@27568: 
chaieb@27568: lemma zgcd_zmult_distrib2_abs: "zgcd (k * m) (k * n) = abs k * zgcd m n"
chaieb@27568:   by (simp add: abs_if zgcd_zmult_distrib2)
chaieb@27568: 
chaieb@27568: lemma zgcd_self [simp]: "0 \<le> m ==> zgcd m m = m"
chaieb@27568:   by (cut_tac k = m and m = 1 and n = 1 in zgcd_zmult_distrib2, simp_all)
chaieb@27568: 
chaieb@27568: lemma zgcd_zmult_eq_self [simp]: "0 \<le> k ==> zgcd k (k * n) = k"
chaieb@27568:   by (cut_tac k = k and m = 1 and n = n in zgcd_zmult_distrib2, simp_all)
chaieb@27568: 
chaieb@27568: lemma zgcd_zmult_eq_self2 [simp]: "0 \<le> k ==> zgcd (k * n) k = k"
chaieb@27568:   by (cut_tac k = k and m = n and n = 1 in zgcd_zmult_distrib2, simp_all)
chaieb@27568: 
chaieb@27568: 
chaieb@27568: definition "zlcm i j = int (lcm(nat(abs i)) (nat(abs j)))"
chaieb@23244: 
chaieb@27669: lemma dvd_zlcm_self1[simp, algebra]: "i dvd zlcm i j"
haftmann@27556: by(simp add:zlcm_def dvd_int_iff)
nipkow@23983: 
chaieb@27669: lemma dvd_zlcm_self2[simp, algebra]: "j dvd zlcm i j"
haftmann@27556: by(simp add:zlcm_def dvd_int_iff)
nipkow@23983: 
chaieb@23244: 
haftmann@27556: lemma dvd_imp_dvd_zlcm1:
haftmann@27556:   assumes "k dvd i" shows "k dvd (zlcm i j)"
nipkow@23983: proof -
nipkow@23983:   have "nat(abs k) dvd nat(abs i)" using `k dvd i`
nipkow@30042:     by(simp add:int_dvd_iff[symmetric] dvd_int_iff[symmetric])
haftmann@27556:   thus ?thesis by(simp add:zlcm_def dvd_int_iff)(blast intro: dvd_trans)
nipkow@23983: qed
nipkow@23983: 
haftmann@27556: lemma dvd_imp_dvd_zlcm2:
haftmann@27556:   assumes "k dvd j" shows "k dvd (zlcm i j)"
nipkow@23983: proof -
nipkow@23983:   have "nat(abs k) dvd nat(abs j)" using `k dvd j`
nipkow@30042:     by(simp add:int_dvd_iff[symmetric] dvd_int_iff[symmetric])
haftmann@27556:   thus ?thesis by(simp add:zlcm_def dvd_int_iff)(blast intro: dvd_trans)
nipkow@23983: qed
nipkow@23983: 
chaieb@23994: 
chaieb@23244: lemma zdvd_self_abs1: "(d::int) dvd (abs d)"
chaieb@23244: by (case_tac "d <0", simp_all)
chaieb@23244: 
chaieb@23244: lemma zdvd_self_abs2: "(abs (d::int)) dvd d"
chaieb@23244: by (case_tac "d<0", simp_all)
chaieb@23244: 
chaieb@23244: (* lcm a b is positive for positive a and b *)
chaieb@23244: 
chaieb@23244: lemma lcm_pos: 
chaieb@23244:   assumes mpos: "m > 0"
chaieb@27568:   and npos: "n>0"
haftmann@27556:   shows "lcm m n > 0"
chaieb@23244: proof(rule ccontr, simp add: lcm_def gcd_zero)
chaieb@27568: assume h:"m*n div gcd m n = 0"
haftmann@27556: from mpos npos have "gcd m n \<noteq> 0" using gcd_zero by simp
haftmann@27556: hence gcdp: "gcd m n > 0" by simp
chaieb@23244: with h
haftmann@27556: have "m*n < gcd m n"
haftmann@27556:   by (cases "m * n < gcd m n") (auto simp add: div_if[OF gcdp, where m="m*n"])
chaieb@23244: moreover 
haftmann@27556: have "gcd m n dvd m" by simp
haftmann@27556:  with mpos dvd_imp_le have t1:"gcd m n \<le> m" by simp
chaieb@27568:  with npos have t1:"gcd m n *n \<le> m*n" by simp
haftmann@27556:  have "gcd m n \<le> gcd m n*n" using npos by simp
haftmann@27556:  with t1 have "gcd m n \<le> m*n" by arith
chaieb@23244: ultimately show "False" by simp
chaieb@23244: qed
chaieb@23244: 
haftmann@27556: lemma zlcm_pos: 
nipkow@23983:   assumes anz: "a \<noteq> 0"
nipkow@23983:   and bnz: "b \<noteq> 0" 
haftmann@27556:   shows "0 < zlcm a b"
chaieb@23244: proof-
chaieb@23244:   let ?na = "nat (abs a)"
chaieb@23244:   let ?nb = "nat (abs b)"
nipkow@23983:   have nap: "?na >0" using anz by simp
nipkow@23983:   have nbp: "?nb >0" using bnz by simp
haftmann@27556:   have "0 < lcm ?na ?nb" by (rule lcm_pos[OF nap nbp])
haftmann@27556:   thus ?thesis by (simp add: zlcm_def)
chaieb@23244: qed
chaieb@23244: 
haftmann@28562: lemma zgcd_code [code]:
haftmann@27651:   "zgcd k l = \<bar>if l = 0 then k else zgcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
haftmann@27651:   by (simp add: zgcd_def gcd.simps [of "nat \<bar>k\<bar>"] nat_mod_distrib)
haftmann@27651: 
wenzelm@21256: end